Assessing the Cost of Assortment Complexity in
Consumer Goods Supply Chains by Reconfiguration of
Inventory and Production Planning Parameters in
Response to Assortment Changes
Dissertation
zur Erlangung der Würde eines
Doktors der Wirtschaftswissenschaften
(Dr. rer. pol.)
der Universität Paderborn
vorgelegt von
Dipl.-Wirt.-Inf. Christoph Danne
33102 Paderborn
Paderborn, September 2009
Dekan: Prof. Dr. Peter F. E. Sloane
Referent: Prof. Dr.-Ing. habil. Wilhelm Dangelmaier
Korreferentin: Prof. Dr. Leena Suhl
i
Vorwort
Die vorliegende Arbeit entstand während meiner Zeit als Stipendiat der Interna-
tional Graduate School Paderborn und im Rahmen meiner Tätigkeit am Lehrstuhl
Wirtschaftsinformatik, insb. CIM und bei der Freudenberg Haushaltsprodukte KG.
Das Vorhaben hatte viele Wegbegleiter, denen ich an dieser Stelle danken möchte.
Mein erster Dank gilt meiner Frau Nathalie. Du warst stets mein größter Rückhalt,
hast mit mir in der Zeit “so manchen Streifen mitgemacht” und viel Verständnis für
verlorene gemeinsame Stunden und meine Launen aufgebracht. Die größten Heraus-
forderungen während dieser Zeit waren nicht wissenschaftlicher Natur. Durch deine
Unterstützung hast du zur Entstehung dieses Buches sehr viel beigetragen.
Ich danke allen fachlichen Wegbegleitern, insbesondere meinem Doktorvater Prof. Dr.-
Ing. habil. Wilhelm Dangelmaier für die wissenschaftliche Betreuung, die stets offen
stehende Tür und die zahlreichen Diskussionen. Frau Prof. Dr. Leena Suhl möchte ich
für die konstruktiven Anmerkungen bei Vorträgen während der Promotionszeit und für
die Übernahme des Zweitgutachtens danken. Herrn Prof. Dr. Eckhard Steffen danke
ich für das Mitwirken in der Prüfungskommission. Den Kollegen in der Arbeitsgruppe
Wirtschaftsinformatik, insb. CIM sei für alle fachlichen Diskussionen und für den
selten guten Humor am Lehrstuhl gedankt. Wie beruhigend zu wissen, dass man auch
“am Rande des Wahnsinns” noch in guter Gesellschaft ist.
Bei der Freudenberg Haushaltsprodukte KG danke ich Herrn Dr. Alexander Moker,
der mir mit einem großen Vertrauensvorschuss die Möglichkeit für diese Praxiskoop-
eration gab und als Ansprechpartner und Mitglied der Prüfungskommission stets mit
wertvollem Feedback zur Seite stand. Mein besonderer Dank gilt Herrn Oliver Neu-
bert, der die Arbeit auf praktischer Seite mit großem Einsatz federführend betreute
und sich immer wieder Zeit für Diskussionen nahm. Frau Dr. Petra Häusler möchte
ich für viele Diskussionen und für die Durchsicht der Dissertation danken. Herrn Frank
Mönikes von FHP Augsburg danke ich für die Einblicke in die Tücherproduktion. Allen
Kollegen bei der FHP danke ich für die angenehme und offene Arbeitsatmosphäre.
Ich danke meinen Eltern, die mich während meiner gesamten Ausbildung gefördert und
in meinem Weg bestärkt haben. Schließlich möchte ich mich bei meiner ganzen Familie
und meinen Freunden bedanken, die mir stets Rückhalt gegeben haben. Bewusst oder
unbewusst habt ihr mir gerade während der “Tiefs” immer wieder gezeigt, dass sich
die Welt aus gutem Grund nicht nur um diese Arbeit dreht.
Paderborn, im November 2009
iii
Contents
List of Figures vii
List of Tables ix
Acronyms xi
List of symbols xiii
1 Introduction 1
2 Problem Statement 3
2.1 Production and Distribution in Consumer Goods Supply Chains . . . . 3
2.1.1 Organisation and Material Coordination Processes . . . . . . . . 5
2.1.2 Production Processes . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Assortment Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Notions and Concepts . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.2 Benefits of Assortment Complexity . . . . . . . . . . . . . . . . 17
2.2.3 Costs of Assortment Complexity . . . . . . . . . . . . . . . . . . 18
2.2.3.1 Effects on Uncertainty . . . . . . . . . . . . . . . . . . 20
2.2.3.2 Cost Effects in Inventory Management . . . . . . . . . 22
2.2.3.3 Cost Effects in Production Execution . . . . . . . . . . 23
2.3 Assessing the Cost of Assortment Complexity . . . . . . . . . . . . . . 24
2.3.1 A Model to Assess the Cost Effects of Assortment Complexity
Changes............................... 30
2.3.2 Inventory Allocation in Production and Distribution Networks . 32
2.3.3 Determination of Planning Buffers and Planned Production
Quantities.............................. 34
2.4 StructureofthisWork ........................... 37
3 State of the Art 39
3.1 Assessing the Cost Effects of Assortment Complexity . . . . . . . . . . 39
3.1.1 Models of Assortment Complexity . . . . . . . . . . . . . . . . . 39
3.1.2 Quantitative Assessment of Complexity-Related Cost . . . . . . 42
3.1.2.1 Descriptive and Empirical Works . . . . . . . . . . . . 42
3.1.2.2 Cost Estimation with Simplifying Assumptions . . . . 44
3.1.2.3 Activity Based Costing and Related Approaches . . . . 46
iv Contents
3.2 Inventory Allocation in Production and Distribution Networks . . . . . 49
3.2.1 Single Installation Systems . . . . . . . . . . . . . . . . . . . . . 50
3.2.2 Multi-Echelon Systems with Various Products . . . . . . . . . . 52
3.2.2.1 Stochastic Service Models . . . . . . . . . . . . . . . . 53
3.2.2.2 Guaranteed Service Models . . . . . . . . . . . . . . . 54
3.2.3 Inventory Allocation as a Combinatorial Optimisation Problem 57
3.2.4 Risk Pooling Effects in Inventory Management . . . . . . . . . . 61
3.3 Determination of Planning Buffers and Planned Production Quantities 65
3.3.1 Determination of MRP Production Parameters . . . . . . . . . 65
3.3.2 Suitability of Standard Lot-Sizing Models . . . . . . . . . . . . 66
4 Required Work 69
4.1 A Model to Assess the Cost Effects of Assortment Complexity Changes 69
4.2 Inventory Allocation in Production and Distribution Networks . . . . . 70
4.3 Determination of Planning Buffers and Planned Production Quantities 71
5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains 73
5.1 A Model to Assess the Effects of Assortment Complexity Changes . . . 73
5.1.1 Production and Distribution Networks as a Model for Assort-
mentComplexity .......................... 73
5.1.1.1 Model Elements . . . . . . . . . . . . . . . . . . . . . 74
5.1.1.2 Model Generation . . . . . . . . . . . . . . . . . . . . 84
5.1.2 Alternative Assortment Scenarios . . . . . . . . . . . . . . . . . 90
5.1.2.1 Definition of Alternative Assortment Scenarios . . . . 90
5.1.2.2 Application to a Baseline Model . . . . . . . . . . . . 94
5.1.3 Cost Assessment for a Given Assortment . . . . . . . . . . . . . 99
5.2 Inventory Allocation in Production and Distribution Networks . . . . . 103
5.2.1 Inventory Model and Optimisation Objective . . . . . . . . . . . 103
5.2.2 Domain Knowledge for Heuristic Inventory Allocation . . . . . . 109
5.2.2.1 Item Characteristics . . . . . . . . . . . . . . . . . . . 109
5.2.2.2 Network Structure . . . . . . . . . . . . . . . . . . . . 113
5.2.2.3 Aggregation to a Single Indicator of Stockpoint Eligibility114
5.2.3 A Tabu Search Heuristic for Inventory Allocation . . . . . . . . 115
5.2.3.1 Definition of an Initial Solution . . . . . . . . . . . . . 116
5.2.3.2 Definition of Moves and Neighbourhoods . . . . . . . . 117
5.2.3.3 Evaluation of Inventory Cost for a Given Solution . . . 119
5.3 Determination of Planning Buffers and Planned Production Quantities 123
5.3.1 Optimisation Model . . . . . . . . . . . . . . . . . . . . . . . . 123
5.3.2 Estimating Average Sequence-Dependent Setup Cost . . . . . . 128
5.3.3 Calculating Penalty Cost for Inventories Incurred by Planning
Buffers................................130
5.3.4 Integration with the Inventory Allocation Problem . . . . . . . 135
6 Application and Validation via Examples 139
6.1 Implementation...............................139
Contents v
6.2 Specification of Example Models . . . . . . . . . . . . . . . . . . . . . . 141
6.2.1 Product Assortments . . . . . . . . . . . . . . . . . . . . . . . . 141
6.2.2 Alternative Assortment Scenarios . . . . . . . . . . . . . . . . . 144
6.2.3 Production Processes . . . . . . . . . . . . . . . . . . . . . . . . 146
6.2.4 Parameter Settings . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.3 Application and Results . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.3.1 Cost Effects in Inventory Allocation . . . . . . . . . . . . . . . . 150
6.3.2 Cost Effects in Production Execution . . . . . . . . . . . . . . . 154
6.3.3 Conclusions: Cost Effects of Assortment Changes . . . . . . . . 157
6.4 Performance of the Optimisation Methods . . . . . . . . . . . . . . . . 159
6.4.1 Inventory Allocation Heuristic . . . . . . . . . . . . . . . . . . . 159
6.4.1.1 Configurations . . . . . . . . . . . . . . . . . . . . . . 159
6.4.1.2 Computational Performance . . . . . . . . . . . . . . . 161
6.4.2 Production Parameter Optimisation . . . . . . . . . . . . . . . . 163
6.4.2.1 Configurations . . . . . . . . . . . . . . . . . . . . . . 163
6.4.2.2 Computational Performance . . . . . . . . . . . . . . . 164
7 Conclusions and Future Research 167
7.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.2 Limitations and Outlook on Potential Extensions . . . . . . . . . . . . 169
Bibliography 171
A Implementation 187
A.1 Model Building and Management . . . . . . . . . . . . . . . . . . . . . 190
A.2 Scenario Building and Management . . . . . . . . . . . . . . . . . . . . 193
A.3 Optimisation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
A.4 CostComparison ..............................195
B Example Networks 197
vii
List of Figures
2.1 Overview of supply chain planning processes . . . . . . . . . . . . . . . 8
2.2 The planning buffer concept . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Planning buffer, replenishment lead times and average setup times . . . 13
2.4 Cost and benefit trade-off of assortment complexity . . . . . . . . . . . 16
2.5 Assortment complexity influences supply chain cost via its effects on
uncertainty, inventory management and production planning . . . . . . 19
2.6 Types of uncertainty at demand, supply and process points . . . . . . . 21
2.7 Interdependency of lot sizes, replenishment lead times and production
cycles .................................... 24
2.8 Cost model: components and parameters . . . . . . . . . . . . . . . . . 26
2.9 Input, output and steps of the analysis process . . . . . . . . . . . . . . 28
2.10 Inventory allocation problem . . . . . . . . . . . . . . . . . . . . . . . . 33
2.11 Rationale for the determination of planning buffers and planned pro-
ductionquantities.............................. 34
2.12 Relations between subproblems . . . . . . . . . . . . . . . . . . . . . . 36
2.13 Structure of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1 Views and dimensions of assortment complexity . . . . . . . . . . . . . 40
3.2 Basic model of guaranteed service approaches . . . . . . . . . . . . . . 55
5.1 Production and distribution network example . . . . . . . . . . . . . . 83
5.2 Simple examples of scenario definition operations . . . . . . . . . . . . 93
5.3 Reorder point, safety stock level and average inventory level . . . . . . 101
5.4 Inventory allocation and times between adjacent items . . . . . . . . . 105
5.5 Example of incremental solution evaluation . . . . . . . . . . . . . . . . 121
5.6 Additional inventory cost due to increasing planning buffers . . . . . . 131
5.7 Integration of the optimisation methods . . . . . . . . . . . . . . . . . 136
6.1 Distribution structure of cloth assortment (model M2) .........143
6.2 Production process steps . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.3 Number of materials processed on each production process step . . . . 148
6.4 Inventory cost comparisons . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.5 Introducing additional stockpoints for finished products . . . . . . . . . 153
6.6 Inventory cost changes on location level for M1/S6............ 154
6.7 Comparison of setup, scrap and production cycle stock costs . . . . . . 156
6.8 Development of production costs for production process step ‘Refine 3’
in model M2.................................157
viii List of Figures
6.9 Total cost changes for all scenarios . . . . . . . . . . . . . . . . . . . . 158
6.10 Performance of the inventory allocation heuristic . . . . . . . . . . . . . 161
A.1 Overview of the Complana software tool . . . . . . . . . . . . . . . . . 187
A.2 Network generation dialog . . . . . . . . . . . . . . . . . . . . . . . . . 190
A.3 Modelmanagement............................. 191
A.4 Editing possibilities for existing models . . . . . . . . . . . . . . . . . . 192
A.5 Network visualisation dialog . . . . . . . . . . . . . . . . . . . . . . . . 193
A.6 Scenario definition in a spreadsheet file . . . . . . . . . . . . . . . . . . 194
A.7 Optimisation dialog wizard . . . . . . . . . . . . . . . . . . . . . . . . . 195
A.8 Selection of models for cost comparison . . . . . . . . . . . . . . . . . . 196
B.1 Network visualisation for M1........................198
B.2 Network visualisation for M1/S1......................199
B.3 Network visualisation for M1/S2......................200
B.4 Network visualisation for M1/S3......................201
B.5 Network visualisation for M1/S4......................202
B.6 Network visualisation for M1/S5......................203
B.7 Network visualisation for M1/S6......................204
ix
List of Tables
3.1 Classification of studies regarding MRP parameter settings . . . . . . . 67
5.1 Product and distribution structure model elements . . . . . . . . . . . 74
5.2 Timemodelelements............................ 77
5.3 Material transformation, transport and coordination model elements . . 78
5.4 Customer demand model elements . . . . . . . . . . . . . . . . . . . . . 81
5.5 Costparameters...............................100
6.1 Overview of example assortment models . . . . . . . . . . . . . . . . . 142
6.2 Scenariodefinitions.............................145
6.3 Number of materials processed on each production process step . . . . 148
6.4 Model generation parameter data sources . . . . . . . . . . . . . . . . . 149
6.5 Comparison of inventory cost . . . . . . . . . . . . . . . . . . . . . . . 150
6.6 Comparison of production execution costs . . . . . . . . . . . . . . . . 155
6.7 Neighbourhood definitions . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.8 Runtime of optimisation, averaged over all planning buffer candidates
(in[ms])...................................164
A.1 Complana software packages . . . . . . . . . . . . . . . . . . . . . . . . 189
xi
Acronyms
ABC activity based costing
BCU basic consumer unit
BOM bill of material
CLSP capacitated lot-sizing problem
ERP enterprise resource planning
FMCG fast moving consumer goods
GSA guaranteed service approach
GUI graphical user interface
MAD mean absolute deviation
MIP mixed integer programming
MPS master production schedule
MRP material requirements planning
PDN production and distribution network
PPS production process step
RCP rich client platform
RLT replenishment lead time
SA simulated annealing
SCOR supply chain operations reference model
SKU stock keeping unit
SSA stochastic service approach
TS tabu search
TSU trade sales unit
xiii
List of symbols
Production and distribution network structure
Lset of locations, indexed l∈ L
Mset of materials, indexed m∈ M
Nset of items, indexed i, j ∈ N
NP ROC set of procurement items
NP ROD set of production items
NDIST set of distribution items
mat(i)material of item i
loc(i)location of item i
Vset of links vconnecting two items, V ⊂ N ×N
wi,j quantity relationship for material flow from ito j
PR(i)set of direct predecessor items of i
SC(i)set of direct successor items of i
UP(i)set of direct and indirect predecessor items of i
DN(i)set of direct and indirect successor items of i
LS(i)set of end product items at sales locations that igoes into
Sset of process steps
SP ROD set of production process steps, indexed s∈ SP ROD
ST RANS set of transport process steps, indexed s∈ ST RANS
pi,s Binary indicator showing if iis processed on s
Nsset of items processed on s
Siset of processes related to i
Bibill of material of i
PDN(N,V,S)production and distribution network
xiv List of symbols
Time model
Tset of discrete mid-term time periods t∈ T ={1, . . . , T}
TSnumber of short-term periods that constitute one mid-term period
ttrans
itransport time of i
TTithroughput time of i
RLTireplenishment lead time of i
STiservice time that ipresents to its successors
∆ticoverage time that has to be covered with inventory at i
External processes - demands and service levels
dext
i,t expected primary demand for iin period t
di,t expected total demand for iin period t
Di,t random variable describing the actual demand for iin period t
FDi,t random variable describing the forecast deviation for iin period t
σd
i,t standard deviation of forecast deviations at iin period t
σdext
i,t standard deviation of forecast deviations for primary demands at i
in period t
FDmad
irelative mean absolute deviation of forecasts for item i
αSL
iα-service level for i
βSL
iβ-service level for i
STmax
imaximum delivery time for iif irepresents an end product sold to
customers
Internal processes
Kscapacity provided by sin one short-term period
kscapacity coefficient for production of ion production process step s
Qrnd
ilot-size rounding value for production quantities of i∈ NP ROD
PBpen
s,pbspenalty costs for choosing planning buffer pbson s
xv
Scenario definition
Aset of new item additions
Rfin set of material replacement definitions for end products
Rrs set of material replacement definitions for raw and semi-finished
materials
Mnew set of all materials that are added via the elements of A
dratio
kdemand ratio for an item replacement
cvkquantity conversion factor for an item replacement
Cost assessment
Ciinternal accounting value for one basic unit of i
Cinv inventory holding cost rate for one period t∈ T
Cwhsg
lwarehousing cost for one storage unit at location lover one period
t∈ T
Cinv
itotal inventory cost for iin one period t∈ T
Cstp
i,s,pbsaverage setup cost incurred by production of ion production process
step swith a planning buffer of pbs
scrpiquantity of scrap produced during a production run of ion s
qppinumber of basic units of ithat can be stored on one storage unit at
loc(i)
Decision variables
SP set of stockpoints
zi,t safety factor to determine safety stock levels as multiples of lead
time demand variation for iin period t
RPi,t reorder point for iin period t
Iss
i,t safety stock level for iin period t
Ics
i,t cycle inventory for iin period t
Qi,s,t planned production quantity of iin time period t
pbsplanning buffer for production scheduling on s∈ SP ROD
Xi,s,t binary variable indicating planned production of ion sin period t
1
CHAPTER 1
Introduction
What’s it going to be then, eh?
Anthony Burgess
Assortment complexity has a major impact on the complexity of the entire supply
chain, as the number of products and product variants affects the complexity of pro-
duction and distribution systems in several ways. Especially for repetitive manufac-
turing companies like consumer goods manufacturers, it greatly affects planning of
production processes and material management.
Managerial decisions on changes of assortment are driven by the trade-off between
additional benefits in terms of sales and additional costs incurred by the increased
complexity. This has led to much research effort in the area of complexity manage-
ment and assortment variety in particular. However, the assessment of the effects of
assortment-related decisions on the underlying production and distribution network
has not yet been systematically undertaken.1As the assortment of a company evolves
continuously by introducing new or discontinuing existing products, the most impor-
tant question is what effects on the configuration of the production and distribution
network and related costs can be expected if the assortment is changed in a particular
way. The answer to this question would give valuable decision support for assortment-
related decisions.
Methods from costing, especially activity-based costing, seek to assign costs fairly to
single product variants according to the input involved. This approach is not suitable
for the intended what-if analysis, even if the cost assignment were perfectly fair, be-
cause of interdependencies between the single product variants that cannot be mapped
into a single cost value per product. For example, if a certain degree of standardisation
of packaging options across several market regions is reached, it becomes favourable to
1 See Ramdas (2003, p. 49).
21 Introduction
keep inventories at a central warehouse instead of local stocks. Thus, statements about
potential cost effects derived from assortment changes can only be made for complete
assortment change scenarios rather than on a per product basis.
In order to assess these effects monetarily, the question what changes in the configu-
ration of the production and distribution network can be expected in response to the
assortment changes must be answered first. Only on the basis of a production and
distribution network adapted to the new assortment statements about cost effects of
these changes can be made. Hence the requirement for methods to adapt the plan-
ning and control parameters to the new situation in order to assess the optimisation
potential offered by assortment reduction exists. Accordingly, the method presented
in this work first seeks a cost-optimal configuration of the most relevant assortment-
dependent parameters in the production and distribution network, in order to assess
the cost effects of assortment changes on the basis of this adapted network configu-
ration. Thereby the above-mentioned what-if analyses can be performed to support
assortment-related decisions.
Assortment complexity incurs costs in almost all areas of a company’s operation and
has therefore been analysed extensively for distinct functional areas and by different
methods.2This work seeks to evaluate the most relevant assortment-dependent cost
positions for the entire production and distribution network, focusing on the areas
of inventory management and production execution. This is due to the fact that
cost effects in response to assortment changes are particularly expected as a result of
changes in inventory requirements and related inventory holding costs as well as setup
costs and scrap in the production area. As a consequence, the parameter optimisation
for the alternative assortment scenarios adapts the inventory allocation as well as
the material requirements planning (MRP) production planning parameters for the
production process steps of the network.
In accordance with the statement above that any what-if analysis can only be con-
ducted for concrete assortment change scenarios and on the basis of an optimally
adapted production and distribution network, this work presents a method to assess
the cost effects of assortment complexity in inventory management and production
operation in consumer goods supply chains by adjustment of inventory allocations and
production parameters in response to assortment changes.
2 For example, Kestel (1995) analyses the effects on logistics operations like transport and picking,
Rathnow (1993) describes the cost effects along the entire supply chain and Bräutigam (2004),
Heina (1999), Köhler (1988), Schuh (2005) all focus on production operations and analyse the
effects of product variety for different types or production systems with different methods.
3
CHAPTER 2
Problem Statement
Had no objections, sir, my only
questions were ‘Where do I go and
will I know when I’m there?’
The Thermals
The subject-matter of this work is the quantitative relationship between assortment
complexity and related costs in consumer goods supply chains. This chapter defines
the problem to be solved in three parts: Firstly, consumer goods supply chains as
the system under consideration are described with all relevant notions and concepts
(Section 2.1). Secondly, assortment complexity as the phenomenon to be analysed
is described and its effects on different aspects of consumer goods supply chains are
presented (Section 2.2). Thirdly, the problem is defined and decomposed into smaller
subproblems that have to be solved. For each subproblem, the aim and requirements
for its solution are defined and a possible solution approach is outlined (Section 2.3).
The problem decomposition and corresponding solution approaches together constitute
the concept for the required cost assessment.
2.1 Production and Distribution in Consumer Goods Supply
Chains
Analysis of assortment-related cost can be seen in the context of supply chain man-
agement research.1This huge research field has emerged due to the fact that most
1 For a condensed introduction to the subject matters of supply chain management, see the funda-
mental works of Davis (1993) and Lambert and Cooper (2000).
42 Problem Statement
companies nowadays form part of a larger network consisting of suppliers, manufac-
turers, logistic service providers and retailers, among others. Such a network is called
a supply chain2.
Definition 2.1 (Supply Chain): The network of all parties involved in satisfying cus-
tomer demand for a certain product assortment. These parties comprise component
and raw material suppliers, manufacturers, logistic service providers, sales organisa-
tions and retailers. They are involved in a variety of partly value-adding processes and
are interconnected by upstream and downstream flows of material and information. All
supply chain members pursue the aim of producing and delivering end products that
satisfy customer demand.
With respect to the subject-matter of this work, Definition 2.1 captures the most
relevant elements from the multitude of definitions presented in the literature and
accords with the common understanding of a supply chain3.
In this work we restrict our view to supply chains that produce physical final products
and exclude the special case of supply chains for non-material services. In particular,
this work deals with consumer goods supply chains, which implies several characteris-
tics for the organisation of the supply chain, the products it produces and the produc-
tion processes involved.4Consumer goods are products purchased by individuals for
final personal use. In contrast to durable goods and major appliances, consumer goods
are comparably inexpensive and consumed shortly after purchase. This is why they
are usually purchased repeatedly at smaller intervals and therefore often referred to as
fast moving consumer goods (FMCG). They are often classified according to the three
major categories of packaged foods, cosmetics and toiletries, and household products.
The corresponding supply chains share most characteristics and the greater part of this
work can be applied to all three types of supply chains. This work focuses however on
non-food products and household products in particular. It does not take into account
the additional peculiarities of food supply chains, like the increased perishability and
very limited shelf life of products.
2 Note that although the term supply chain suggests a set of sequentially interacting organisations,
supply chains usually have a network structure of interconnected organisations (Lambert and
Cooper, 2000, p. 65). Although alternative terms like supply networks can be found in literature,
none of them prevailed. We shall therefore use the commonly accepted notion of a supply chain
in this work, despite its slight imprecision.
3 Elements of this definition can also be found in the definitions given by Lee and Billington (1995),
Chopra and Meindl (2004), Christopher (2005), Sahin and Robinson (2002), Stevens (1989), Hopp
(2006) and the Supply Chain Council (2007). The reader is referred to Van der Vorst (2000, p.
22) for a more comprehensive list of supply chain definitions in the literature.
4 For general descriptions of the characteristics of consumer goods supply chains, the reader is
referred to Soman et al. (2004, pp. 227-229) and Fleischmann and Meyr (2003, pp. 463-465).
2.1 Production and Distribution in Consumer Goods Supply Chains 5
The remainder of this section follows the definition of a supply chain as the system
under consideration. Section 2.1.1 describes the participants in consumer goods supply
chains as well as the organisation of material and information flows between them.
Section 2.1.2 describes the characteristics of the value-adding production processes
required to produce the end products.
2.1.1 Organisation and Material Coordination Processes
This work considers the supply chain from the manufacturer’s point of view. We
distinguish five types of supply chain participants (in the order found upstream in the
supply chain):
Customers For consumer goods manufacturers, customers are retailers and whole-
salers. Note that the term customer is not a synonym for consumer here. While
the customers are retailers, consumers are those who purchase the products from
a retail store and eventually consume them.
Sales companies Customers order at sales companies that ship the products from
distribution warehouses to the customers’ stores or central warehouses.
Production facilities A number of specialised production facilities each produce a
certain product range and supply the sales companies.
Suppliers External suppliers supply the production facilities or sales companies with
raw and packaging material as well as selected semi-finished components or fin-
ished products.
Logistic service providers The supply chain actors usually make use of the transport
and warehousing services offered by third-party logistic providers. These service
providers may carry out transports, provide physical storage space and perform
order picking and dispatching tasks.
For consumer goods manufacturers, the retailers or retail chains are considered the
customers. All customers order the required finished products at sales companies and
manage the delivery of these products to the actual consumers.
Sales companies usually supply customers in a defined region, e.g. a single country.
These additional storage and transshipment points in terms of sales companies are
required for two main reasons: customers like retail stores usually expect very short
order fulfilment within 1-2 days from the manufacturers. As shipments directly from
the production sites would usually take too long, finished goods are shipped from local
62 Problem Statement
distribution warehouses, which may e.g. be located at the sales company sites. Trans-
port costs are, of course, an important consideration in consumer goods distribution.
Given the comparatively low value and large volume of the products, transport costs
are particularly high and constitute a considerable fraction of total costs. As customer
orders contain products that are likely to be produced at several production sites, an
additional point for transshipment and consolidation of goods is required to achieve
efficient transport.
Considering that consumer goods are fairly inexpensive, production facilities are com-
paratively capital intensive. They are thus often specialised in production of a certain
product range and supply many, if not all, sales companies with these types of product.
Consequently, sales companies procure finished goods for their sales activities from a
number of production facilities at various locations to supply all customers in their
sales region.
Many consumer good supply chains comprise a focal company, typically the main
manufacturer and owner of the product’s consumer brands.5This focal company has
to oversee the entire supply chain and manage all relations with raw and packaging
material suppliers and customers. The sales companies typically belong to the focal
company as well and manage all relations with the customers, i.e. the retail stores.
Thus we consider a decentralised supply chain in which central coordination and control
can be assumed to exist between the production sites and sales companies.
The central control is implemented as a common material coordination system that
integrates the production and distribution processes. If there is the possibility of em-
ploying a central material control system, it is reasonable and common to use program
orientated material coordination, like the well-known MRP6system. The starting
point of MRP-based planning is a master production schedule (MPS) that defines the
required production quantities.
The finished products have to be available from stock in local warehouses at the sales
companies to meet the short delivery times and high service levels. Alternatively, the
replenishment lead times at the sales companies must be extremely short to allow
procurement from production sites only on customer orders. As the total lead time
for procurement and production operations is generally much longer than the delivery
times accepted by customers, the material coordination is mainly based in forecasts
issued by the sales companies. These forecasts contain the expected sales quantities
5 For more detail on supply chains with focal companies, see Lambert and Cooper (2000).
6 See Orlicky (1975). Modern material requirements planning systems usually employ the extended
MRP II concept (Vollmann et al., 2004).
2.1 Production and Distribution in Consumer Goods Supply Chains 7
for a certain period so that procurement and production activities can be planned with
the required lead time based on these forecasts. The MPS required for MRP based
demand planning is then derived from the forecasts issued by the sales companies.
It must always be expected that forecasts are not totally correct and actual require-
ments differ from the forecasted values.7The fact that most production and pro-
curement activities are based on uncertain forecasts requires some means of buffering
against the uncertainties that are inherent in forecasts in order to meet the required
service levels and delivery times to customers. A variety of strategies can be employed
to hedge against these uncertainties.8The most widely used means of achieving this
is the installation of safety stocks at determined points in the supply chain.
Definition 2.2 (Safety stock): Safety stock is stock used to protect against uncer-
tainty that arises from internal processes like production lead time, from unknown
customer demand and from uncertain supplier lead times.9
From Definition 2.2 it follows that the main drivers for safety stock requirements are
production and transport disruptions, forecasting errors, and lead time variations.
Safety stock is held to improve customer service and to avoid lost sales and loss of
goodwill10. The decision where to place which amount of safety stock in order to meet
all customer requirements at minimal cost is an important tactical decision problem
in supply chain management.
Figure 2.1 depicts the material coordination and planning processes described so far,
together with the effects of safety stock placement decisions. At the very final stage,
sales companies are faced with uncertain customer demand for all products of the
assortment offered. As these demands may change daily, forecasts are made for a
mid-term period to allow the calculation of planned requirements for the upstream
stages11. As the replenishment lead times (RLTs) agreed with preceding stages have
7 As Nahmias states: “the main characteristic of forecasts is that they are usually wrong” (Nahmias,
1997, p. 61). For a more detailed analysis of problems in forecasting consumer demand in the
fast-moving consumer goods industry, the reader is referred to the empirical study of Adebanjo
(2000), who investigates the demand-forecasting systems of some leading UK food companies.
8 Extensive discussion of competing techniques and literature reviews on uncertainty handling in
MRP-driven manufacturing systems are provided by Koh et al. (2002), Whybark and Williams
(1976), Guide and Srivastava (2000), Buzacott and Shanthikumar (1994) and Enns (2002).
9 See Stadtler and Kilger (2005, p. 61).
10 See ibid., p. 61.
11 Since common in supply chain literature, we shall use the terms upstream and downstream
throughout this work to refer to directions of material flow in the supply chain. Downstream
refers to the flow from external procurement to the customer stages, while upstream refers to the
opposite direction.
82 Problem Statement
Fixed forecasts
(replenishment lead time)
Throughput
time
0
10
20
Period
Demand
0
10
20
Period
Demand
Period
0
10
20
Demand
0
10
20
Period
Demand
0
10
20
Period
Demand
Stage 1 Stage 2 Stage 3
Distribution
Distribution
Procure-
ment Production Distribution
Fixed forecasts over RLT
covered with safety stock
Deterministic demands
resulting from fixed forecasts
Uncertain forecasts
Figure 2.1: Overview of supply chain planning processes
to be respected, these forecasts have to be transferred to fixed orders according to
these replenishment lead times. If the required delivery time is shorter than the re-
plenishment lead times, the stage has to keep safety stocks to buffer against demand
fluctuations over this time interval.
This principle is especially clear for the interaction of sales companies with production
facilities, as delivery times to customers are short and replenishment lead times are
longer due to the lead time required by production and transport. However, this
principle can be applied to any stage of the supply chain, including production stages.
Some stages can operate without inventories since they operate on orders that successor
stages have issued by fixing uncertain forecast values over their replenishment lead
times. These successor stages then need to keep safety stock to buffer against the
variations of actual demands from these forecasted values.
If a stage has fixed its forecasts over a certain replenishment lead time, this fixed time
period is reduced at each preceding stage by that stage’s throughput time. As long as
a positive difference remains, preceding stages can operate deterministically. The fixed
time period is consumed successively and eventually upstream stages are required to
keep inventories again as the demands they receive are forecasts that are still subject
to change. The trade-off in such a system is as follows: the longer the replenishment
lead times of the inventory carrying stages, the more preceding stages can operate on
fixed orders, as their throughput times are covered by these fixed planning intervals.
2.1 Production and Distribution in Consumer Goods Supply Chains 9
However, longer replenishment lead times also increase the safety stock required at the
inventory carrying stages.
This high-level description of the planning and material coordination processes reveals
that there are many interrelations between decisions on the required inventories and
agreed throughput and replenishment lead times of all stages in a supply chain, which
leads to complex decision tasks, especially if complex assortments are considered.
2.1.2 Production Processes
As production capacities are comparably expensive in relation to the product value,
production systems for consumer goods are designed to efficiently produce large quan-
tities of certain goods. Production processes for consumer goods accordingly follow a
serial production oriented layout12. Typical characteristics of such production systems
are that13
•product development and engineering activities are independent of customer or-
ders
•end products are produced to stock based on demand forecasts, independently
of concrete customer orders
•customer orders are filled immediately from stocks of finished goods
•stocks of finished goods cause high levels of working capital and a high risk of
obsolescence due to imprecise forecasts of future demands
•precise forecasts are of crucial importance
In such production systems large batches of certain goods are produced with lot sizes
1as each changeover from the production of given goods to another requires some
expensive setup procedures. The reduction of these unproductive setup times and
thereby the increase of the production system’s utilisation rate is one of the primary
goals of production planning in serial production systems.
The production process of any finished goods typically comprises several distinguish-
able production process steps. A production process step (PPS) takes a set of input
components and transforms them into a certain quantity of an output material. Both
12 See Hopp and Spearman (2000, pp. 8-10) for an overview of production layout types. Sometimes,
the term batch production is used synonymously.
13 See Dangelmaier and Warnecke (1997, pp. 10-12).
10 2 Problem Statement
the input and output materials may be physically stored as inventories. A produc-
tion process step typically represents the production process on one major production
line.
Production of consumer goods can be subdivided into the two major parts of base
production and converting. Base production comprises all activities required to pro-
duce the base products in high volumes. Typically, base production stages use volume,
weight or length as the unit of measure in contrast to the quantities in discrete man-
ufacturing. The output of this part of the production process is large units of the
actual final goods that have not yet been split up into smaller sellable units and that
therefore can be stored efficiently.
Consequently the main activity of the converting stages is packaging, possibly after
some smaller assembly and converting procedures. Single production steps of the
converting stage comprise the division of the batch quantities received from base pro-
duction into smaller units as sold to the consumer and the subsequent packaging pro-
cedures. Components for converting processes comprise the output of base production
and mostly externally procured packaging materials like boxes, paper and foil for end
consumer units and larger boxes to form batches for shipment to retail stores.
Example 2.1 In cloth production, fibres and chemicals are first mixed and
compacted at the base production stage. The resulting raw cloth is further
refined by applying additional coatings, prints and sometimes imprinting
structures. The converting steps then use the wide cloth rolls that form the
output of the base production part as input components and the rolls are
cut into smaller lanes and single small cloth. In the final step, cloths are
folded, packed into foil and then into boxes in larger batches for distribution
to the sales companies.
Base production stages are capital intensive and require little labour. Converting
stages are typically more labour intensive, as assembly and packaging activities may
require some manual work, depending on the degree of automation of the production
process. Often there are highly automated processes for some products and manual
operations for others, depending on their value and volume. There may also be various
ways of producing a single product via alternative routings that vary in their degree
of automation.
All production stages face long and often sequence-dependent setup and cleaning times.
The degree to which setups are critical and sequence-dependent varies between the
single process steps and with the machines available. Generally, both the absolute
2.1 Production and Distribution in Consumer Goods Supply Chains 11
importance and also the sequence dependence of setups is more critical in base pro-
duction than in converting stages since the setups required for each changeover are
much longer and comprise cleaning activities that depend on the predecessor product
in the production sequence. However, the principle of sequence-dependent setups also
holds in the converting stages and causes a similar decrease in the machine’s utilisation
rates, although their absolute value may not be as high due to the less capital-intensive
production resources.
Production stages also produce scrap, either as a fixed amount during the start and
end of production runs or as scrap that occurs as a fraction of the volumes produced.
Unlike the former scrap type, the latter cannot be avoided or reduced by production
planning and sequencing decisions and therefore is not of interest in this work.
The cost implications of these setup and scrap related characteristics result in the
requirement for big production lots, as each changeover and start of a new production
run incurs setup and scrap cost. Apart from this economical minimum lot size, there
are technical restrictions to lot sizes on different production lines. Depending on the
type of production output, there exist rounding values for the lot sizes, so that each
production run must produce a multiple of this rounding value. Such restrictions result
from the physical storage characteristics of the output, which e.g. require full pallets.
The sequence-dependent setup costs are the main reason for installing a planning
buffer, i.e. a time interval in which the actual production order can be shifted. This
gives the planner the flexibility to create sequence-optimised production plans and
thereby reduce setup costs.
Definition 2.3 (Production planning buffer): The planning buffer of a production
process step determines the calculation of requirement dates for all materials processed
at that stage. It is the time between the provision of all components (earliest produc-
tion start date) and the latest production start date of a production order. Within this
time interval the actual production start can be shifted in order to create cost-optimal
production sequences.
Figure 2.2 illustrates the calculation of requirement dates for a simple example with 2
production stages and one procurement stage. With a given requirement date for the
finished product, the requirement dates for all components are determined by calculat-
ing the latest possible production start date with the actual processing time for that
order. In addition, the production planning buffer of the corresponding production
process step is added to determine the requirement dates for all required components.
12 2 Problem Statement
processing
time
requirement
date
latest
production
start date
Finished
product time
processing
time
requirement
date
latest
production
start date
Components time
requirement
date
latest
order date
supplier service
+ transportation
time
Raw
materials
Total replenishment lead
time for finished product
time
Figure 2.2: The planning buffer concept
Likewise, the requirement dates for all externally procured raw materials are deter-
mined at the second production stage. With the delivery time of the supplier and the
required transport time, the latest order date for the raw materials can be determined.
The difference between this latest order date and the requirement date of the finished
product represents the total replenishment lead time for that product.
This simple example demonstrates the trade-off related to the determination of the
planning buffers: the shorter the buffer interval, the fewer possibilities to create
sequence-optimised production plans. Therefore longer planning buffers can reduce
the average setup costs incurred by the changeovers required to process a certain set of
production orders. However, this relationship is non-linear, as the additional benefit of
longer planning buffer decreases. On the other hand, increasing planning buffers also
increase the throughput times for that production stage. This also affects subsequent
stages, as their replenishment lead times increase linearly with that throughput time.
For those stages that hold safety stock, these increased replenishment lead times may
result in longer coverage times and increasing safety stock levels. Figure 2.3 illustrates
these relationships.
2.2 Assortment Complexity 13
Planning buffer
length [days or shifts]
Replenishment
lead times [days]
Average setup
times [minutes]
Figure 2.3: Planning buffer, replenishment lead times and average setup times
2.2 Assortment Complexity
2.2.1 Notions and Concepts
The aim of this work is to assess the effects of assortment complexity in consumer
goods supply chains. The characteristics of the latter having been described in the
preceding section, this section discusses the concept of assortment complexity in more
detail. First of all it must be noted that many works on this topic use the term
product variety rather than assortment complexity. While these terms are sometimes
used synonymously, we favour the latter throughout this work14 since both the terms
product and variety may be misleading.
Firstly, the term ‘product’ is generally used to refer to finished goods requested by
customers to serve a certain purpose. However, identical products may still vary in
their packaging, which is an important complexity driver for consumer goods and
therefore also considered in this work. We thus prefer the term ‘assortment’ as defined
below.
Definition 2.4 (Assortment): The set of packed end products distributed to a set of
sales locations to satisfy customer demand.
Secondly, the term ‘variety’ suggests that only the mere number of different product
variants is relevant. It can be stated however that relations between materials, both
where-used relations in production and transport links in distribution, affect supply
14 As there is no commonly accepted definition for either of these terms, references made to other
works also include works that use a different notion.
14 2 Problem Statement
chain cost as well. These relations thus have to be considered since they directly result
from the design of the assortment. This leads to the concept of complexity.
The concept of complexity lacks a commonly adapted definition in literature. It is
discussed in many disciplines, including system theory, computer science, philosophy
and economics and thus different definitions are given.15 Approaches to a definition
of complexity in general agree that it is best defined via the notions of elements and
relations, as described in the comprehensive discussion of complexity given by Luh-
mann16. One general definition that is applicable to all types of systems and consistent
with most of the specialised definitions from different disciplines is given by Buhr and
Klaus17:
Definition 2.5 (Complexity): The complexity of a system is determined by the number
of elements in the system and the relations between these elements.
Definitions 2.4 and 2.5 underline that assortment complexity refers both to the size
of the assortment in terms of all end products with their packaging variants as well
as the relations between materials in terms of where-used relations in production and
transport relations at the distribution stages.
The relevance of assortment complexity is supported by the fact that is has a pre-
dominant effect on the total supply chain complexity. This supply chain complexity
may be defined in terms of the dimensions structure, products and processes.18 Due
to its generality, Definition 2.5 may be used to describe complexity in the structure
and process dimensions as well. The structural complexity is captured e.g. via the
number of suppliers, production stages, warehouses and customers and the relations
between them. Process complexity may be considered as the number and design of
material flow or information flow processes. Supply chain complexity management has
attracted increasing attention of both scientists and practitioners, as this complexity is
assumed to diminish total supply chain performance.19 With an increasing complexity
15 An overview of literature on complexity theory is given by Anderson (1999).
16 Luhmann (1980).
17 Buhr and Klaus (1975). A similar definition is also given by Ulrich and Probst (1988).
18 See the internal complexity in the framework presented by Blecker et al. (2005) and the framework
given by Danne et al. (2008). Adam (2004) also claims that complexity can hardly be formalised
since it comprises many dimensions. He argues that e.g. the number of products, parts, customers,
suppliers and value adding processes are all interrelated factors that make up the complexity of
a supply chain (Adam, 2004, p. 20). Further classifications of supply chain complexity can be
found in Wilding (1998), Meier and Hanenkamp (2004) and Perona and Miragliotta (2004).
19 In a comprehensive empirical study in the household appliances industry Perona and Miragliotta
(2004) provide evidence for the claim that a direct relation between logistical complexity and
supply chain performance holds.
2.2 Assortment Complexity 15
of the supply chain, its size and the number of interactions grow and planning20 and
controlling21 becomes more and more difficult.
Definition 2.1 states that the ultimate goal of all supply chain activities is to satisfy
customer demand for a certain product assortment. Decision on size and structure of
the assortment directly affects the supply chain in its products, structures and pro-
cesses and thereby the complexity in the product dimension mainly determines the
complexity in all other dimensions as well. This observation is especially true for con-
sumer goods supply chains, where assortment complexity has already been identified
as a chief determinant of complexity.22 In contrast to other industries that suffer from
high product variety, each variant of consumer goods is not only a theoretical vari-
ant that the production system must be able to produce in case of a given customer
order, but a real variant that has to be produced, distributed and kept available for
customers at possibly different locations at all times. Therefore assortment-related
decisions directly affect inventory management23 and the production process.
Depending on the industry and the types of product considered, different drivers of
assortment complexity can be identified.24 For consumer goods, the most important
drivers at the design level are variations in colour, size and materials used. Another
important driver especially found in the case of consumer goods is variety in packaging
options. Packaging takes place in several steps and has to be considered on various
levels: The logistical units shipped between production and sales locations are not
identical to those purchased by consumers. The latter, called basic consumer units
(BCUs), are packed units of single or few products that are purchased by the con-
sumers. The so-called trade sales units (TSUs) constitute the logistical unit shipped
from production locations to sales companies and to customers. These TSUs are larger
20 Planning is defined as the process of solving the problem that a current or anticipated state
deviates from the desired goal state by anticipating future actions that create the desired goal
state (Klein and Scholl, 2004, p. 1). For an overview of the multitude of planning tasks in a
supply chain and their classification according to the time horizon affected (short, mid and long-
term planning) and the related supply chain process (procurement, production, distribution and
sales) see the supply chain planning matrix (Rohde et al. (2000), Stadtler and Kilger (2005, p.
87)).
21 Controlling is a process in a system in which one or more input values influence output values
according to the system’s principles (DIN19226, 1968).
22 Hoole reports that “Product proliferation is the leading driver of supply chain complexity”, ac-
cording to the management executives responding to a supply chain complexity survey conducted
by the consulting firm PRTM (Hoole, 2006, p. 3).
23 Inventory management comprises the tasks of “Deciding which inventories are to be held at which
stage, in what quantity and for what purpose [...]. It is basically a resource allocation decision -
establishing the overall inventory ’budget’ that firms can afford and setting targets for how much
will be allocated to each inventory type” (Warner, 2001, p. 2395).
24 For an elaborate list of product variation features, see Ramdas (2003, p. 5).
16 2 Problem Statement
batches of several BCUs. Thus variety in packaging can first be introduced in the in-
dividual packaging of the BCUs, e.g. via localised packaging with country-specific
imprints. Secondly, it can be introduced in the batch sizes and case dimensions of the
TSUs.
There is an undoubted trend to a continuously increasing number of stock keeping
units (SKUs), which is often referred to as product proliferation. Baker notes that this
“trend is most noticeable in the consumer packaged goods industry”25. This observa-
tion is supported by various empirical studies on different types of consumer goods
that have indicated a strong product proliferation and accompanying increase of as-
sortment complexity in recent years.26 The number of products available in large
supermarkets has increased from the order of 1000 in the 1950s to 30,000 in a modern
supermarket27.
Costs
Revenue
Profit
Degree of
assortment
complexity
Costs
Revenue
Maximum
profit
Optimal
assortment
complexity
Profit
Figure 2.4: Cost and benefit trade-off of assortment complexity28
Assortment complexity creates both potential benefits as well as increasing costs via
various levers. The trade-off results from the fact that increasing assortment complex-
ity allows the exploitation of economies of scope, while reduced assortment complexities
allows the exploitation of economies of scale, especially in the areas of production and
inventory management.29 This leads to the question of an optimal assortment, i.e. the
one that yields the highest profit, as depicted in Figure 2.4. As assortment complexity
affects both costs as well as revenue, there is at least one optimal assortment complex-
25 See the entry product proliferation in Baker (2002)
26 See Quelch and Kenny (1994) and the recent study conducted by The Economist Intelligence
Unit (2008).
27 See Thonemann and Bradley (2002, p. 594).
28 Adapted from Herrmann and Peine (2007, p. 654) and Rathnow (1993, p. 44).
29 See Lancaster (1998, p. 8), Lancaster (1990, p. 191).
2.2 Assortment Complexity 17
ity where the difference of revenue and costs is at maximum. The following sections
briefly outline the factors involved in this trade-off.
2.2.2 Benefits of Assortment Complexity
Benefits of assortment complexity are reflected in gains in market share and increasing
revenue. They can be summarised as follows:
Meeting heterogeneous consumer preferences If we describe each product via the
combination of characteristics or attributes it provides, then individual con-
sumers have different preferences with regard to these combinations. A large
product portfolio better meets the preferences of different consumers and can in-
crease turnovers, both due to larger sales quantities as well as better acceptance
of higher prices for the preferred products.30
Price discrimination Different consumers are willing to pay different prices for a cer-
tain type of products. Offering a wide range of product variants at different
prices thus allows the seller to address a larger consumer target group.31
Variety seeking Consumers seek diversity in their choice of goods from time to time.
A wide product portfolio can therefore help to prevent consumers from changing
to competitor products.32
Scatter shot approach Offering a great variety of products may be used as a way of
gaining information about the consumers’ preferences. As consumer preferences
are difficult to foresee, companies may use the so-called scatter shot approach
and launch more products that they expect to sustain in the long-term, just to
see how consumers accept the different variants.33
Demarcation from competitors Product variants are used as means of demarcating
products of a certain brand from their competitors. This reduces competitive
pressure and avoid direct price competition.34 At the same time, the active occu-
pation of market segments raises market entry barriers to prevent the emergence
of new competitors in that area.
30 See Kahn and Morales (2001, p. 64).
31 See Ulrich (2006, p. 7).
32 An extensive review of the different type of variety seeking with explanations from both psychol-
ogy and marketing literature is given by Kahn (1998).
33 See Lancaster (1998, pp. 15-16).
34 See Lancaster (1990).
18 2 Problem Statement
Globalisation Given that most consumer goods manufacturers conduct sales activities
in different countries, product differentiation due to different languages and legal
regulations becomes necessary.
Shelf-space effects Consumer goods manufacturers or brand owners compete for
shelf-space in retail stores, as the shelf-space occupied by products of one brand
is assumed to correlate with the corresponding turnover. It is thus alluring to
widen the product portfolio to occupy more shelf-space.
Trade pressure Direct customers in terms of retail stores and trade may demand a
high variety in end products. Such requirements may comprise different packag-
ing sizes or styles to fit their logistical or marketing needs, as well as customised
product variants that are unique to their store and also impede a direct compar-
ison with competitor offerings by the consumer.35
All the above-mentioned potential benefits are hardly measurable. Eventually, all
approaches that try to assess the benefits of assortment complexity have to answer the
question how consumers change their behaviour and purchasing habits when confronted
with more or fewer product variants. Research approaches mainly comprise qualitative
methods and empirical studies mostly discussed in marketing literature.36
2.2.3 Costs of Assortment Complexity
There is consensus that assortment complexity is a major influencing factor for supply
chain wide costs and supply chain performance.37 Kluge et al.38 show empirical
evidence that successful companies are characterised by a stronger focus within their
assortment, compared with less successful companies. Quelch and Kenny39 believe
that many manufacturing companies have allowed their assortments to proliferate too
much. Indeed, some major companies including large consumer goods manufacturers
have already taken measures in order drastically to reduce assortment complexity
35 See Quelch and Kenny (1994, pp. 154-155).
36 See e.g. Riemenschneider (2006) and Kahn (1998) and the references given there.
37 Homburg and Daum provide a systematic overview of complexity-induced cost and corresponding
levers to influence them (Homburg and Daum, 1997). More evidence for this hypotheses is given
by Klaus (2005) and Hoole (2006).
38 See Kluge et al. (1994, p. 41).
39 See Quelch and Kenny (1994).
2.2 Assortment Complexity 19
and obtain considerable cost savings40. The assessment of cost incurred by assortment
complexity interrelates with almost all aspects of a company’s logistics and production
activities.41 Figure 2.5 illustrates how assortment complexity affects the areas of supply
chain uncertainty, inventory management and production planning and related costs.
increases
Assortment
complexity
influences
Structure
Products
Processes
requires buffering
Forecast accuracy
Supplier
performance
Production yield and
lead time variability
Production and converting
Transportation
Procurement
Planning & administration
Inventory and warehousing
Obsoletes
Volumes
Lead times
Lot sizes
Production cycles
Delivery capability and
reliability
Positioning task
Locations
Dimensioning task
Levels
Figure 2.5: Assortment complexity influences supply chain cost via its effects on
uncertainty, inventory management and production planning
Assortment complexity directly affects costs in the areas of procurement, planning and
administration. External suppliers usually offer quantity-dependent price discounts.
The more variants of an externally procured material exist, the smaller the procure-
ment quantities and the less these price discounts can be exploited. Furthermore, all
materials in an assortment have to be planned and managed, i.e. there is considerable
effort in master data maintenance, administration and the periodic planning tasks.
Planning tasks include demand planning, revision of inventory levels and planning of
replenishment to ensure material availability.
40 For example, Schiller et al. (1996) report concrete measures taken by Procter & Gamble to re-
duce the existing assortment complexity and limit further product proliferation. Hoole (2005)
propose several complexity reduction measures along the processes of the Supply chain opera-
tions reference model (SCOR) (Supply Chain Council, 2006) and reports on their application in
practice.
41 Rathnow provides a graphical overview of complexity-incurred costs along the value processes
‘research and development’, ‘procurement’, ‘production’, ‘sales’ and ‘customer service’ (Rathnow,
1993, p. 24).
20 2 Problem Statement
Besides these direct cost impacts, there are several levers that indirectly incur addi-
tional cost. Figure 2.5 shows how these costs are created via effects in the area of supply
chain uncertainty and via changes in the area of production planning and operation
as well as inventory management, which we discuss in the following sections.42
2.2.3.1 Effects on Uncertainty
The more complex an assortment and the corresponding supply chain become, the more
possibilities for unforeseen events are introduced and the harder it gets to forecast the
effects of the actions taken. Generally, this is called uncertainty: Van der Vorst defines
supply chain uncertainty as “decision-making situations in the supply chain in which
the decision-maker lacks effective control actions or is unable to accurately predict the
impact of possible control actions on system behaviour”43. It is an inherent property
of complex systems that the multitude of interrelations between the elements makes it
hard to foresee the ultimate effects of certain actions.
In the context of production and logistic processes, different types of uncertainty can
be distinguished: Uncertainty in
Demand Future customer requirements are generally unknown.
Supply Suppliers or supplying production and distribution stages do not always fulfil
their deliveries as expected.
Processes No production or distribution process is deterministic in its outcome.
All these types of uncertainty become apparent as fluctuations in quantity and time.
In order to alleviate the problems caused by uncertainty, companies have to install
costly buffers in terms of material, capacity and time. An increasing complexity thus
increases uncertainty and thereby costs via various levers.44 Davis45 states that inher-
ent uncertainty is one of the major problems in planning and controlling supply chains
and greatly affects their performance.
42 Similar categorisations of costs incurred by assortment complexity can be found in literature:
Randall and Ulrich distinguish production costs and market mediation costs, where the former
comprise all costs of “materials, labour, manufacturing overhead, and process technology invest-
ments”. The latter consist of “inventory holding costs and product mark-down costs occurring
when supply exceeds demand, and the costs of lost sales when demand exceeds supply”, which
occur due to “presence of demand uncertainty” (Randall and Ulrich, 2001, p. 1588).
43 Van der Vorst (2000, p. 74).
44 For a more detailed discussion of this causal relationships, see Wilding (1998).
45 Davis (1993).
2.2 Assortment Complexity 21
•
Production lead time
•Supplier availability
•Deliver
y
date
Production
lead
time
•Production output
•Capacity availability (machine
breakdowns, downtimes, labour) •Supplier
availability
•Deliver
y
date
•Inventory data accuracy
•Perishability of products
y
•Delivery quantity
•Quality
•Demand quantities
•Demand dates
y
•Delivery quantity
•Quality
•Demand quantities
•Demand dates
Process step Buffer /
Stockpoint
Figure 2.6: Types of uncertainty at demand, supply and process points
A more complex assortment and related supply chain complexity increase uncertainty
in the above-mentioned dimensions of demand (forecast accuracy), supply (supplier
performance) and internal processes (production yield and lead time variability). These
impacts are either direct or via the increasing total supply chain complexity.
Sales forecasts for single variants and sales regions are generally less accurate than
aggregate forecasts from different regions and for products with commonality. The
average sales quantity for each product decreases and natural fluctuations in demand
or deviations in forecasts cannot be compensated for by corresponding fluctuations in
other market regions. A high assortment complexity impedes the exploitation of risk
pooling effects46 and thereby increases demand uncertainty, as all different variants are
held on stock at different locations and forecasts are made on a per material and per
location basis.
Process uncertainties become apparent in larger variations of production yield, trans-
port times, information availability and accuracy. With an increasing number of mate-
rials produced, the learning effects and experience for the production of single materials
decrease and create a larger variety in production yield and processing times. Consid-
ering the entire production process, structural features like the number of production
steps also have an impact, as these effects accumulate over the various steps. In dis-
tribution stages, the uncertainty in transport times are linked to structural features
like transport links and facility locations. With respect to information availability and
accuracy, it may be expected that the error-proneness of a business information system
increases with the number of materials and locations involved.
46 See Nahmias (1997, p. 61) and Zipkin (2000). A more detailed discussion of risk pooling is
presented in Section 3.2.4.
22 2 Problem Statement
Supply risks regarding supply availability and reliability are introduced via structural
complexity with each additional supplier. Increasing assortment complexity in terms
of materials procured externally also increases supply uncertainty since suppliers face
greater challenges to comply with the required service levels and delivery times if the
number of materials ordered increases.
2.2.3.2 Cost Effects in Inventory Management
Increasing assortment complexity affects inventory management by creating increasing
inventory requirements. The determination of the total inventory requirements com-
prises a positioning and a dimensioning task, which are both affected by a changing
assortment.
The positioning of stockpoints47 in the supply chain changes due to the creation of
new potential stockpoints with each additional material. Each amplification of the
assortment with additional end products may require additional stockpoints to assure
the timely delivery of these products to customers at the required service level. For each
additional variant, stock of end products and possibly of components and packaging
materials on upstream stages has to be built.
With respect to the dimensioning task, it must be noted that inventory requirements
are mainly driven by replenishment lead times, demand quantities and demand un-
certainty. As inventories in terms of safety stocks are one of the major means of
buffering against the uncertainties mentioned above48, we can expect an increase in
the total amount of inventory distributed along the supply chain to provide the same
service to customers, despite the increased levels of uncertainty in demand, supply and
processes.
These changes in the required inventory levels directly affect costs. Firstly, costs are
incurred in terms of opportunity costs for working capital bound in inventories as well
as warehousing costs for the provision of physical storage space. Secondly, inventories
always risk obsolescence due to seasonality or limited product lifecycles which then
incurs costs of scrapping and write-offs.
47 We again refer to stockpoints at the individual material level, as defined in Section 2.1.1.
48 See Section 2.1.
2.2 Assortment Complexity 23
2.2.3.3 Cost Effects in Production Execution
In the production area, we consider flow shop oriented batch production systems that
do not produce customer individual variants with lot size 1, but have to plan produc-
tion orders of reasonable lot sizes for each variant.49 In this setting, changes in the
assortment and thereby the set of materials produced on one production process step
as well as the demand volumes of these materials heavily affect the costs incurred for
their production.
A decrease in required production volumes per material is the logical consequence of
increasing assortment complexity. A large number of materials with small demand
volumes conflicts with the requirements for large production lots and avoidance of
setup and scrap cost. Due to a larger number of materials processed, lead times on
the corresponding production process step become larger and / or production cycles50
increase. Smaller production volumes tend to result in smaller average lot sizes. This
leads to more frequent changeovers in the production sequence, each of which incurs
cost for setup operations in terms of labour and opportunity costs for the unused
production capacity during the changeover. The decrease in lot sizes may be alleviated
by an increase in production cycles, which means that demands of longer time periods
have to be aggregated to form larger production lots. However, this results in increasing
replenishment lead times of successor stages, as the agreed service time for a certain
material depends on the maximum time it may take to schedule a production order
of that material. Figure 2.7 illustrates the interdependency of lot sizes, replenishment
lead times and production cycles.
The set of materials processed on a certain production process step influences the plan-
ning buffer required to enable production planners to generate production sequences
with reasonable setup costs. The larger the number of production orders to be sched-
uled, the more difficult it becomes to create optimal production sequences and longer
planning buffers are required to allow the generation of cost-efficient production se-
quences.
These production planning and operation decisions are also interrelated with the area of
inventory management in a variety of ways, as the decision which materials to produce
to stock influences production parameters such as production cycles. Furthermore, the
lead times that result from the selection of planning buffers influence the amount of
49 See Section 2.1.2
50 For this chain of reasoning, we use the concept of fixed production cycles per material. Even if
such production cycles may not be used in practice, the rationale remains valid for the average
time between two production orders of the same material.
24 2 Problem Statement
Assortment complexity /
number of materials processed
Production cycles /
replenishment lead times
of successor stages
[days]
Average lot sizes
[pieces, meters, kg, ...]
Figure 2.7: Interdependency of lot sizes, replenishment lead times and produc-
tion cycles
inventory required at different locations and can thereby change the optimal inventory
allocation.
2.3 Assessing the Cost of Assortment Complexity in Production
and Distribution Networks
This work seeks to evaluate the most relevant assortment-dependent cost positions
and therefore focuses on the areas of inventory management and production execu-
tion. Accordingly, we also use the term production and distribution network (PDN)
instead of supply chain, as this implies that only production and distribution pro-
cesses and their related costs are the subject-matter of this analysis. This focus is
selected for the following reasons: firstly, cost effects particularly are expected as a
result of changing inventory requirements and related inventory holding costs as well
as changing setup times and scrap quantities in the production area. Secondly, the
effects of assortment complexity on these costs are particularly difficult to assess by
conventional cost-accounting methods and therefore often remain disregarded when it
comes to assortment-related decisions. Quelch and Kenny51 find that the cost factors
not considered correctly and / or sufficiently when taking assortment-related decisions
comprise:
•increased production complexity resulting from shorter production runs and more
frequent line changeovers
51 Quelch and Kenny (1994, p. 156).
2.3 Assessing the Cost of Assortment Complexity 25
•more errors in forecasting demand
•increased logistics complexity, resulting in increased remnants and larger buffer
inventories to avoid stockouts
The problem of assessing and foreseeing these costs correctly results from the inter-
dependencies between single variants. These interdependencies make it impossible to
have a single cost value per material reflecting the exact cost for its production and
distribution and thus the savings that can be expected if this single material is dis-
continued.52 To underline this point, the following list gives examples where a simple
summation of the costs assigned to each individual discontinued material does not
represent the real savings potential of e.g. an assortment reduction.
•If end product packaging is standardised, e.g. by introducing multilingual pack-
aging, a central warehouse with finished products, e.g. at the production site,
may become preferable to decentralised stocks at local sales warehouses.
•If the set of semi-finished materials for one category of end products can be
reduced, it may become preferable to keep inventories of semi-finished products
instead of raw materials. Furthermore, it may be possible to reduce the planning
buffer of the corresponding production process step without significant increases
in the average setup costs due to less complex production sequencing.
•If the set of raw and / or packaging materials is standardised to a certain degree,
it may become preferable to keep inventories of these materials to reduce the
total replenishment lead times for the end products.
As the cost effects depend on such concrete assortment changes, they cannot be re-
flected by traditional cost-accounting systems, even if they allocate the costs fairly
according to the inputs involved. In order to provide a reliable decision support for
assortment-related decisions, this work also answers the question
How do costs in inventory management and production execution change in response
to assortment changes?
With this guiding research question, the methodic approach of this work is mainly
quantitative. In order to allow the type of what-if analysis implied by this question, a
method of assessment of costs incurred in a production and distribution network by a
52 Quelch and Kenny also point out that “the costs of line-extension proliferation remain hidden”,
as “traditional cost-accounting systems allocate overheads to items in proportion to their sales.
These systems [...] overburden the high sellers and undercharge the slow movers” (ibid., p. 156).
26 2 Problem Statement
certain assortment is required. Given such a method, arbitrary possible assortments
can be assessed and compared. Figure 2.8 shows how the task of carrying out such a
cost assessment is broken down according to the three cost areas to be assessed.
Aim
Cost
components
Parameters of
cost function
Given parameter
Parameter to be
determined for a
given assortment
Stockpoints
Safety stock levels
per period
Inventory holding
and warehousing
cost rates
Inventory costs
Reorder points per
period
Planned production
quantities per
period
Demand rates per
period
Inventory holding
and warehousing
cost rates
Production cycle stock
costs
Scrap cost per
product and
production order
Planned production
quantities per
period
Planning buffers
Average setup cost
per product and
production order
Setup and scrap costs at
production process steps
Replenishment
lead times
Assess the costs incurred in a production and
distribution network by a certain assortment
Figure 2.8: Cost model: components and parameters
In accordance with the focus on inventory management and production execution, the
cost areas can be summarised as inventory cost, setup and scrap cost, and cycle stock
costs. For each such cost component, the figure lists the factors that determine the
individual cost positions.
For the assessment of inventory costs, we can distinguish between cost incurred by
general inventories and costs incurred by production cycle stocks. For the former, both
the positioning of stockpoints in the network and the required inventory levels at each
such stockpoint have to be known. For the latter, the position of the reorder points and
the corresponding replenishment lead times have to be known to estimate the average
cycle stock levels. These factors are partly interdependent, as the replenishment lead
times influence the values of both the reorder point as well as the required inventory
levels for the considered material. At the same time the chosen safety stock level
influences the replenishment lead times of successor materials, which is why this is a
2.3 Assessing the Cost of Assortment Complexity 27
bidirectional dependency. Given that these factors are known, inventory costs can be
easily calculated with the known inventory holding and warehousing cost rates.
For the assessment of setup and scrap costs, planned production quantities per period
have to be known for each material that is produced internally in order to derive the
number of production runs per period. With the number of production runs, the
corresponding setup and scrap costs can be estimated via the known cost parameters
for scrap and average setup costs. As described in Section 2.1.2, the average setup cost
parameter also depends on the choice of the planning buffer, which makes the decision
variables interdependent.
For the assessment of production cycle stock costs, the difference of planned production
quantities and actual demand quantities has to be calculated for each period. Thus
relevant factors comprise planned production quantities, demand rates and the inven-
tory holding and warehousing cost rates already used for the assessment of general
inventory costs.
These factors can be subdivided into two groups: firstly, there are fixed parameters
in terms of cost rates or externally given demands, represented by white boxes in
Figure 2.8. These parameters either remain constant across assortments or change in
a known way. While cost parameters generally remain constant, changes of demand
quantities may be defined on the basis of expected customer behaviour.
Secondly, there are several variable factors that have to be adapted and determined
for each individual assortment, represented by grey shaded boxes in Figure 2.8. These
variables are the means of taking into account the various ways in which the complexity
of the given assortment influences the costs mentioned above. As the aim is to provide
a method to assess these costs for an arbitrary assortment, we cannot rely on actual
values from the existing production and distribution system: an optimisation method
that determines these variables for any given production and distribution network is
required.
By providing such optimisation method that is able to determine these variables for
arbitrary assortments, this work not only provides means of analysing the cost effects,
but also identifies the effects on the production and distribution processes by answering
the question
What changes in the configuration of production and inventory management
parameters may be expected in response to the assortment changes?
28 2 Problem Statement
The fact that there are several variable factors in the cost function requires a multi-
stage analysis process. Figure 2.9 summarises the input, steps of the analysis process
and output of the the method developed.
I
Mdlli d
1
2
3
4
Sl
Sl
I
nventory
allocation
Setting production
parameters
Cost analysis
and comparison
M
o
d
e
lli
ng an
d
scenario building
1
2
3
4
Bli
S
e
l
ect
•Stockpoint
locations
Ridi t
S
e
l
ect
•Planning buffers
•Planned production
titi
Scenarios
B
ase
li
ne
model Scenarios
•
R
equ
i
re
d
i
nven
t
ory
levels for given
customer service
and demand
Baseline
model
quan
titi
es on
aggregate level
Recalculate in entor
forecast deviations
•Material master data
•
Production and sales
Recalculate
in
v
entor
y
levels with new
replenishment lead
times •Inventories
Pd l tk
Production
and
sales
locations
•Sales volumes and
forecasts
•
Distribution
relations
•
P
ro
d
. cyc
l
e s
t
oc
k
s
•Setups and scrap
Distribution
relations
•Bills of material
•Production process steps
•Cost rates
Figure 2.9: Input, output and steps of the analysis process
In the first step, a model53 of the system54 under consideration has to be built. This
model has to represent the production and distribution network of a particular as-
sortment with the entire product structure. From this baseline model, which is most
likely to represent the current as-is situation, scenarios representing theoretical alter-
native scenarios are derived. They differ from the baseline models in terms of the set
of materials in the assortment, the product structures, the distribution relations and
/ or assumptions about the demand distribution and demand uncertainty. Both the
baseline models and all scenarios can be represented with the same formal model.
53 “A model is an abstract description of the real world giving an approximate representation of
more complex functions of physical systems” (Papalambros and Wilde, 2000, p. 4). A similar
definition is also given by Buhr and Klaus (1975, p. 805).
54 A system are “groups of interacting, interrelated, or interdependent elements forming a complex
whole” (Pickett, 2000). Common characteristics of a system are to
1. consists of interacting components
2. be hierarchically structured
3. be associated with a function it is intended to perform.
From the fact that there are numerous types of systems that each require a specific definition,
Cassandras and Lafortune (2007, p. 2) conclude that “‘system’ is one of those primitive concepts
[...] whose understanding might be left to intuition rather than an exact definition”.
2.3 Assessing the Cost of Assortment Complexity 29
In the second step, the inventory allocation in both the baseline model and all relevant
scenarios has to be determined. We define an inventory allocation as the positioning of
stockpoints and the determination of inventory levels and corresponding replenishment
lead times for each material.
In the third step, planning buffers and planned production quantities have to be set. As
planning buffers are also part of the throughput times of each production process step,
the second and third step are interrelated: after the throughput times have changed
due to the adapted planning buffers, replenishment lead times may change for a set
of affected materials and the required inventory levels and reorder points have to be
adapted. Accordingly, steps 3 and 4 may either have to be solved in an integrated way
or may have to be solved repeatedly.
In the fourth and final step the required cost analysis can be performed on the basis of
the adapted configurations of the production and distribution networks represented by
the baseline model and the scenarios. This step enables the assessment of the effects of
assortment complexity by a pairwise comparison of the assortments and cost changes
of the baseline model and the various scenarios.
Complex problems are solved by decomposing them into a set of smaller subproblems
that have a clear relationship to each other.55 The structuring of the cost assessment
and analysis process allows us to identify three major subproblems to be solved in the
context of this work:
1. Modelling of assortments, assortment scenarios and production and distribution
related cost
2. Inventory allocation in production and distribution networks
3. Determination of planning buffers and planned production quantities
The following sections discuss these subproblems by describing the related tasks to be
addressed and solution requirements for each of them.
55 See Pärli (1980).
30 2 Problem Statement
2.3.1 A Model to Assess the Cost Effects of Assortment Complexity
Changes
This subproblem may be summarised by the question
How can the production and distribution structure of a certain assortment be
represented by a formal model that serves as a basis for what-if analyses?
The specific tasks that result from this question may be summarised as follows:
1. to provide a model that serves as a basis for the analysis of assortment complexity-
related cost
2. to provide a formalism to define assortment scenarios on that model
3. to enable the assessment of setup, scrap, inventory and cycle stock costs on the
basis of any given model instance with all parameters set
The solution developed to address these tasks must fulfil the following requirements:
Model completeness for the aspired analysis The model has to serve as a basis for
the entire analysis process. Its level of abstraction must be defined to explicitly
represent all the information required for the analysis and optimisation steps. In
particular, this information comprises
•the assortment structure with all materials, their where-used relations and
distribution relations between various locations,
•the production process steps required to produce the assortment under con-
sideration, i.e. a mapping of single materials to production resources to-
gether with their resource requirements,
•the characteristics of internal and external processes: external supply, ex-
ternal customer demand, production and transport processes,
•demand and demand uncertainty data over a certain period of time and at
an aggregate level. As it should be possible to define scenarios that also
change the distribution of demands over the end products and materials,
this information should be defined as a demand distribution over aggregate
time periods. A stochastic model for the uncertainty inherent in the demand
forecasts is also required and must be available at all elements of the PDN.
2.3 Assessing the Cost of Assortment Complexity 31
At the same time, the model must remain manageable in its complexity.56
Manageable effort for model building Models of real-world assortments easily be-
come complex themselves. In order to be able to create such models for practical
use, this process must be automated as fully as possible. This seems especially
reasonable as almost all companies can be expected to use some IT based enter-
prise resource planning (ERP) system that already contains most of the required
data. It must be possible automatically to create network models from this
information based on the definition of a set of end products.
Configurable definition of consistent scenarios The definition of alternative assort-
ment scenarios must allow addition of new materials as well as discontinuation
and / or replacement of existing materials both at end product level and on the
level of raw and semi-finished materials. It must be possible to also include the
expected effects of the assortment changes on demand volumes and distributions.
The application of a scenario definition to an existing baseline model must ensure
that the resulting scenario is consistent in terms of structure as well as demand
information.
Assessment of relevant cost areas The cost assessment must consider all relevant
areas defined in Section 2.3 and allow their assessment for a concrete assortment
model or scenario.
An approach to attaining the aims stated above may be based on a mathematical
model57. Existing approaches to representations of assortment complexity are analysed
in Section 3.1. Once such a model is defined, algorithms can be developed to derive
such a model from standard data available in ERP systems such as bills of materials
(BOMs), routings and procurement information. Based on such a model, a formalism
for definition of assortment scenarios and a cost function that includes all relevant cost
areas can be defined.
56 This trade-off between the level of detail and model complexity holds for all types of model and
is discussed e.g. by Dangelmaier (2003, p. 41).
57 A mathematical model is “a model that represents a system by mathematical relations.” (Pa-
palambros and Wilde, 2000).
32 2 Problem Statement
2.3.2 Inventory Allocation in Production and Distribution Networks
This subproblem may be summarised by the question
What is an optimal inventory allocation in a production and distribution network of a
certain assortment?
The term inventory allocation here refers to both the inventory positions in terms of
stockpoints in a production and distribution network and the corresponding inventory
levels. The answer to the question above thus provides the values for the variables
required to assess inventory costs as defined in Section 2.3.
The solution developed to find optimal inventory allocations must fulfil the following
requirements:
Make decisions on the detailed level of individual materials Inventory allocation prob-
lems can be considered at various levels of abstraction. Nodes and potential
stockpoints can represent physical locations, products at different levels of ag-
gregation, or combinations of these.58 Given that the inventory allocation is used
to compare assortments that differ in the set of materials they contain, it has to
operate on the very detailed level of individual materials at different locations.
Work on variable demand data on aggregate level Demand for consumer goods can-
not be assumed to follow a certain known probability distribution, as factors like
seasonality, promotions and generally short product lifecycles result in extremely
volatile demand. The method therefore must be able to make inventory alloca-
tion decisions on the basis of expected demand quantities that vary over time
and are known for a set of aggregate discrete time periods.59
Usage and integration of domain knowledge Given that the model on which the
inventory allocation decisions are made cannot contain every detail about the
underlying production and distribution system, it must be possible to integrate
further domain knowledge into the allocation decisions to obtain solutions that
are feasible in practice. This might be necessary to adequately represent situ-
ations in which certain materials cannot be put on stock due to their physical
characteristics or the characteristics of the underlying production process.
58 See Zipkin (2000, p. 107).
59 See Section 2.3.1
2.3 Assessing the Cost of Assortment Complexity 33
Computational efficiency for realistic network sizes The combinatorial complexity
of the optimisation problem considered requires the application of computational
optimisation methods. As this problem has to be solved for each alternative
assortment scenario that is to be analysed, the optimisation procedure must be
able to find solutions even for large assortments in acceptable time. Considering
the combinatorial complexity for large networks and the fact that the result of
the optimisation serves as the basis for a cost analysis in the first place, the
guaranteed optimality of the solution is not a primary requirement.
Figure 2.10 illustrates how the inventory allocation problem can be seen as the selection
of stockpoint nodes in a network together with the determination of the corresponding
inventory levels and delivery times between adjacent stages.
Delivery time to
successors / customers
Safety stock level
Stockpoint
1
4
2
Delivery time
to successors
5
36
end customers
end customers
end customers
supplier
supplier
supplier
7
8
9
Delivery time to
end customers
Delivery time to
end customers
Delivery time to
end customers
Figure 2.10: Inventory allocation problem
On this basis, an optimisation model is defined to determine the cost-minimal inventory
allocation. Available approaches to similar problems are evaluated and analysed in
Section 3.2 with respect to the questions how their models and assumptions match
this application scenario. A solution method is developed based on existing techniques
for computational optimisation. Heuristic techniques may have to be considered in
this context due to the complexity of the optimisation problems.
34 2 Problem Statement
2.3.3 Determination of Planning Buffers and Planned Production
Quantities
This subproblem may be summarised by the question
What are optimal planning buffers and planned production quantities such that total
setup, scrap and inventory costs are minimal?
The trade-offs to be solved in this context are illustrated in Figure 2.11.
Materials
Safety stock costs
€
Inventory costs
Materials
processed
Production
Cycle stock costs
ÆSubproblem 1: Estimation of
additional inventory costs on
bttdt
€
PPS 1
Production
process step
Dd
t
y
Available
capacity
su
b
sequen
t
s
t
ages
d
ue
t
o
increased planning buffers
D
eman
d
s per
mid-term period
Quanti
t
Mid-term
Planning buffers
Planned production quantities
Decision variables
capacity
?
Quantity
planning
period
o
duction
order
S
Production costs
Scheduling
Pr
o
S
etup costs
Scrap costs
ÆSubproblem 2: Estimation of
average setup costs for given
€
Short-term
period
Scheduling
Short-term
period
average
setup
costs
for
given
planning buffer
Figure 2.11: Rationale for the determination of planning buffers and planned
production quantities
For each production process step, the set of materials processed and their corresponding
demands are known, giving a set of demands for each period that have to be covered
by corresponding production quantities. In a real-world setting, concrete production
orders are the result of a quantity planning and scheduling process that takes into
account the requirement dates of the orders from which the demands originate. The
planning buffers define the set of potential production start dates for the orders and
thereby the optimisation potential for the sequencing task. The actual production cost
that should be assessed is incurred when such a detailed production plan is carried out
2.3 Assessing the Cost of Assortment Complexity 35
and the setup and scrap cost that depend on the lot sizes and the sequence of the
production orders are actually known.
One problem is that we cannot expect to have sufficiently detailed demand information
to derive concrete requirement dates. As outlined in Section 2.3.1, demand information
for the assortment models and especially the theoretical alternative scenarios is only
available at an aggregate level. The assessment of operational cost factors would actu-
ally require detailed production planning, which is not possible with the information
available. An appropriate handling of this situation will be recorded as a requirement
for the solution to this subproblem.
Both the planning buffers and planned production quantities that should be deter-
mined have cost effects which conflict. This creates two trade-offs to be solved: firstly,
increasing the length of the planning buffer reduces the average setup costs incurred in
the production execution. At the same time, safety stock requirements for materials on
the same or subsequent production step increase due to increased replenishment lead
times created by the planning buffers. Secondly, large planned production quantities
lead to bigger lot sizes and consequently fewer changeovers that incur setup and scrap
cost. At the same time, the large planned production quantities increase the average
cycle stocks as the produced quantities are only consumed successively.
A solution to the problem of finding the optimal trade-offs in this setting must:
Assess operational production cost from demand data on aggregate level As de-
scribed above, the solution has to assess operational production cost without any
exact knowledge of the detailed production plans.
Consider interrelation of setup, scrap and cycle stock costs The determination of
planned production quantities has to consider the cost trade-off between setup,
scrap and cycle stock cost and find solutions where these total costs are minimal.
Consider effects of planning buffers on inventory requirements The determination
of suitable planning buffers has to consider that larger planning buffers also in-
crease inventory requirements. The planning buffers must be chosen such that
the total costs are minimal.
Computational efficiency for realistic model sizes Although number of potential con-
figuration increases rapidly with the number of materials, time periods considered
and potential planning buffers, the solution method must be able to determine
suitable variable values in acceptable time even for the large model instances
expected in practical applications.
36 2 Problem Statement
Consider the interrelation to the inventory allocation subproblem Figure 2.12 shows
how the optimisations of the different variables relate to each other. As indicated
Inventory positioning
Where-used relations
Demand volumes
Demand uncertainties
Stockpoints
Stockpoints Stockpoints
Throughput
times
Setting production planning parameters
Setup costs
Scrap costs
Inventory costs
Planning buffers
Planned production
quantities
Inventory dimensioning
Demand uncertainty
Demand volumes
Replenishment lead times
Required customer
service levels
Safety stock
levels
Reorder points
Figure 2.12: Relations between subproblems
by the solid arrows, the results of the inventory placement are input parameters
for both the optimisation of planning buffers and for the dimensioning of inven-
tory levels. In the former case, the positions of the stockpoints influence the
effect of planning buffers on inventory costs. In the latter case, stockpoints have
to be determined before the required inventory levels can be calculated. As the
determination of planning buffers also increases the throughput times of produc-
tion stages and thereby the replenishment lead times of successor stages, these
results are also a required input for the dimensioning of inventory levels. In
addition, the results of the dimensioning of inventories are a required input to
evaluate the solutions of the other subproblems. A solution to this dependency
problem is another requirement for the overall optimisation procedure.
To find an optimal configuration for the variables described, optimisation methods from
mixed integer programming can be used and thus a mathematical optimisation model
is formulated. In particular, the determination of planned production quantities shows
parallels to big-bucket lot-sizing models. In order to comply with the requirement of
efficient solvability for large model instances, the model is defined so that it can be
solved with standard optimisation software. This implies that the model should be
formulated as a linear optimisation problem and that linearisation is applied where
necessary and applicable. In order to handle the aggregate demand data, the relations
between some of the cost factors may have to be estimated where applicable.
2.4 Structure of this Work 37
2.4 Structure of this Work
Figure 2.13 summarises how the structure of the remainder of this work corresponds to
the subproblems described above. Existing works from the relevant fields mentioned
5 Concept
Inventory allocation in
production and
distribution networks
Determination of
planning buffers and
planned production
quantities
A model to assess the
cost effects of
assortment complexity
changes
Inventory allocation in
production and
distribution networks
Determination of
planning buffers and
planned production
quantities
Assessing the cost
effects of assortment
complexity
Inventory allocation in
production and
distribution networks
Determination of
planning buffers and
planned production
quantities
3 State of the art
4 Required Work
A model to assess the cost effects of assortment complexity changes
Production and
distribution networks as
a model for assortment
complexity
Cost assessment for
given assortments
Defining alternative
assortment scenarios
6 Validation via Examples
Figure 2.13: Structure of this work
in Sections 2.3.1 to 2.3.3 are evaluated with respect to their applicability in the cor-
responding sections of Chapter 3. Upon these findings, Chapter 4 summarises the
remaining tasks to be addressed. Chapter 5 then develops the individual solutions in
the structure depicted in Figure 2.13. The validation in Chapter 6 shows how the soft-
ware implementation of the methods developed can be used to apply these methods to
real-world problems and shows both the applicability and validity of the solution.
39
CHAPTER 3
State of the Art
All in all you’re just another
brick in the wall.
Pink Floyd
This chapter summarises the state of the art in all areas relevant to this work. Sec-
tion 3.1 first analyses existing work on the assessment of the cost effects of assortment
complexity in order to see if the intended what-if analysis is a viable approach. Sec-
tion 3.2 then focuses on work on optimisation methods for the inventory allocation
problem. Finally, Section 3.3 provides a literature review on the optimisation of se-
lected production planning parameters that are similar to the planning buffers and
planned production quantities considered in this work.
3.1 Assessing the Cost Effects of Assortment Complexity
Since assortment complexity is omnipresent in virtually all manufacturing companies
and due to its far reaching implications, its management has gained considerable atten-
tion in both literature and management practice. Against the background of this work,
this section first reviews formal models of assortment complexity and then evaluates
existing work on the assessment of complexity-related costs.
3.1.1 Models of Assortment Complexity
Different models of assortment complexity can be grouped according to their view
on the assortment. Figure 3.1 illustrates these different views and dimensions of as-
sortment complexity. The first view on assortment complexity is to consider the end
40 3 State of the Art
Product
porogram
Product
structure
Product group
Product line A Product line B
Model 4600 Model 4700
V. 4701 V. 4702 V. 4710 V. 4711
Dyeing
Cutting
Customer view: which
variants are offered?
Variety in product
program and structure.
Internal view: which
features create what
number of variants?
Process view: where in the
production process is
variety introduced?
red
Model
4700 yellow
blue
135x12
110x10
90x8
135x12
110x10
Colour
Size 90
120
90
120
90
120
Weight
80
100
120
Figure 3.1: Views and dimensions of assortment complexity
product level with all product groups, product lines, the product program for an in-
dividual product line and finally the individual variants. This can be seen as the
‘customer view’, as the individual variants of finished products represent the variety
as it is visible to the customer. The design level describes the individual product fea-
tures and possible combinations of these features. The feature tree helps to determine
which features create what number of finished product variants and is the most com-
mon way to represent assortment complexity.1The production process view combines
these features with the actual production processes required to manufacture a given
set of product variants. Depending on the organisation of these production processes,
variants may emerge at different production stages. For a given product structure,
there are many possibilities to organise the production process so that variety may be
introduced at different steps.
One particular class of formal models of assortment complexity are market oriented
models that describe the variety of products offered by a company within a certain
market. The aim of these models is to analyse the suitability of an assortment in a
given market setting. The two basic models that can be distinguished in this context
are those given by Chamberlin2with its formalisation by Dixit and Stiglitz3and the
1 See e.g. Moore et al. (1986).
2 Chamberlin (1933).
3 Dixit and Stiglitz (1977).
3.1 Assessing the Cost Effects of Assortment Complexity 41
model by Hotelling4with its adaptations by Lancaster5. The Chamberlin-Dixit model
analyses how product differentiation may lead to markets with monopolistic competi-
tion, as customers do not perceive the products of different brands as substitutes due
do their differentiating characteristics. The Hotelling-Lancaster model describes each
product via its characteristics in a virtual space of product characteristics. This way
similarities and substitution effects between products can be represented. Customer
preferences in terms of desired product characteristics can be modelled analogously by
defining points in that space and measuring their distance to existing products. Chen
et al.6build on this model to address product line design and product pricing problems.
For a more detailed description of these models, the reader is referred to the work of
Lancaster7. Although these models have been used and enhanced extensively, they do
not provide a suitable approach in the context of this work since they do not contain
any information about the cost of the product variety, but take a market oriented view
to describe suitable assortments given assumptions about customer preferences.
In their analysis of the effects of brand width on brand market share, Chong et al.8
use a simple product tree to represent the assortment of a company. This tree roughly
corresponds to the feature tree depicted in Figure 3.1. On the basis of this tree,
they derive different measures for brand width, comprising the number of SKUs, the
number of feature levels and the number of distinct product variants. While this model
is intuitive and sufficient for their analysis, it does not contain sufficient information
about the involved production and distribution processes to be used for the purpose
of this work.
Schuh9proposes the variant tree as an extension of the product tree to include infor-
mation about the assembly sequences of car variants. This variant tree also serves as
the basis for an accounting method developed by the same author to assess complexity-
related cost, as described in Section 3.1.2.3. While this is a first approach to include
information about the production processes into a model of assortment complexity,
it lacks a representation of the required production facilities, subsequent distribution
structures and is especially tailored to the automotive industry.
4 Hotelling (1929).
5 Lancaster (1979).
6 Chen et al. (1998).
7 Lancaster (1998).
8 Chong et al. (1998).
9 Schuh (2005, pp. 158-159).
42 3 State of the Art
3.1.2 Quantitative Assessment of Complexity-Related Cost
Several reasons have led to the large number of works that deal with the effects of
assortment complexity. Firstly, the topic has already been discussed since the early
19th century.10 Secondly, research in this area has gained increasing attention during
the last decades due to an increasing need for complexity management methods.11 As
this chapter cannot give an exhaustive literature review, we classify relevant works into
three categories and describe some selected examples for each of these categories in
the following sections. These exemplary works have been chosen to match the problem
addressed in this work as closely as possible.
There exist several more comprehensive reviews on this topic. For example, Ramdas12
provides a “framework for managerial decisions about variety” to structure the large
amount of literature available. This framework distinguishes between four decision
themes in variety creation (dimensions of variety, product architecture, degree of cus-
tomisation and timing) and three decision themes in variety implementation (process
and organisational capabilities, points of variation and day-to-day decisions). Further
comprehensive literature surveys are provided by Lancaster13, Herrmann and Peine14,
Bräutigam15 and Heina16. The current state of the art is also summed up in the edited
volumes by Adam17 and by Ho and Tang18, who provide collections of relevant papers
with a focus on controlling and operations research methods, respectively.
3.1.2.1 Descriptive and Empirical Works
The majority of relevant publications do not provide quantitative models to directly as-
sess the costs of assortment complexity. Moreover, the effects of an increasing number
of product variants are described on an abstract level and / or analysed in empir-
ical studies. This section briefly describes some selected works from this category.
10 See e.g. Wolter (1937).
11 See the explanations in Section 2.2.
12 Ramdas (2003).
13 Lancaster (1990).
14 Herrmann and Peine (2007).
15 Bräutigam (2004).
16 Heina (1999).
17 Adam (1998).
18 Ho and Tang (1998).
3.1 Assessing the Cost Effects of Assortment Complexity 43
For further examples, the reader is referred to the works of Marti19, Rathnow20 and
Kirchhof21.
Kestel22 describes the effects of increasing assortment complexity in logistic systems in
a detailed analysis of its implications on warehousing, picking, transport and informa-
tion flow processes. Furthermore, he considers the impact on several functional areas
like production and procurement. While all considerations are made on a detailed
level, no quantitative measures for the relationship between the number of variants in
the logistic system and its performance are provided.
Davila and Wouters23 provide a model to assess the effects of product standardis-
ation and show its application in a case study. The performance model comprises
key indicators for the areas of inventory reduction, service improvement and quality
improvements. This model is used in a case study with a major hard disk drive manu-
facturer. Despite the relevance of the performance indicators used, it has to be noted
that the entire model is designed to assess the effects of standardisation based on in-
formation from the post-standardisation phase. It is thus unsuitable to predict the
effects of potential standardisation as the basis for assortment-related decisions.
Prillmann24 discusses assortment complexity as a strategic managerial task and de-
scribes the levers that determine both assortment and total supply chain complexity.
He formulates several hypotheses to describe the success factors for effective complex-
ity management. These hypotheses are validated with several empirical studies from
the electronics industry to show that companies which consider these success factors
appropriately are generally more successful.
Randall and Ulrich25 describe the effects of assortment complexity on supply chain
performance via a case study in the U.S. bicycle industry. They distinguish produc-
tion costs and market mediation costs as the main performance metrics. Production
costs also include the fixed investments associated with providing additional prod-
uct variants. Market mediation costs are incurred by all measures required to match
supply and demand despite the presence of demand uncertainty. These costs include
the variety-related inventory holding cost, obsolescence cost as well as lost sales. The
19 Marti (2007).
20 Rathnow (1993).
21 Kirchhof (2002).
22 Kestel (1995).
23 Davila and Wouters (2003).
24 Prillmann (1996).
25 Randall and Ulrich (2001), Ulrich et al. (1998).
44 3 State of the Art
particular interest of the authors is to investigate how companies can match their as-
sortment complexity with the structure of their supply chain to reduce these costs.
Many authors conjecture that complexity-induced costs increase progressively with
the degree of assortment complexity. This progressive cost behaviour leads into the
well-known complexity trap described by Adam and Johannwille26. Likewise, Klaus27
claims that there is a certain degree of complexity beyond which an over-complexity
phenomenon arises and complexity-related costs even increase exponentially. However,
the analytical considerations of Bräutigam28 suggest that this progressive cost increase
does not always hold and that degressive cost functions are possible in certain settings
as well. This disagreement about the general nature of the cost function shows that
it is almost impossible to define a generic, monovariable cost function for the effects
of assortment complexity, an argument also supported by the review of quantitative
approaches in the following section.
3.1.2.2 Cost Estimation with Simplifying Assumptions
Adam29 points out that complexity-related costs have some unpleasant characteristics
that impede a precise assessment. Firstly, they are overhead costs and thus cannot be
assigned precisely to the causing product variant. Within certain intervals of numbers
of variants, complexity-induced costs can be fixed and increase only when the provided
management and coordination capacities do not suffice any more.30 This way, slow-
moving items are generally subsidised by the standard products with high volumes.
Secondly, complexity consists of many interdependent factors and drivers31, whose
interrelations have to be known in order to measure the effects of a change. To solve
this cost assignment problem, some authors propose a monovariable approach and only
consider the number of product variants, assuming the ceteris paribus assumption for
all other factors. This may lead to problems, as this assumption generally does not
hold in practice.32
Bräutigam33 analyses the cost behaviour in production systems with a high degree of
product variety. He argues that the number of product variants offered is a strate-
26 Adam and Johannwille (1998).
27 Klaus (2005).
28 Bräutigam (2004).
29 Adam (2004).
30 See ibid., pp. 22-23.
31 See Section 2.2.
32 Adam (2004, p. 21).
33 Bräutigam (2004).
3.1 Assessing the Cost Effects of Assortment Complexity 45
gic decision parameter of uttermost importance for all manufacturing companies and
therefore pursues the aim to derive a generic production cost function that depends
only on this single parameter. The main determinants of this cost function are setup
costs caused by changeovers between production runs of different variants. In order
to estimate the number of changeovers required for a certain set of product variants,
the author requires that the occurrences of demand for the variants are classified in
the dimensions time (uniform, random, accumulated) and quantity (uniform, accu-
mulated).34 Given a certain number of variants and classifications of their demand
according to this scheme, he then derives expected production sequences and corre-
sponding estimator functions for the number of changeovers.
Although such a generic estimator function would provide a simple way to support
assortment-related decisions, its simplicity can only be obtained at the expense of far
reaching assumptions about all other influencing factors. Since the estimator function
is monovariable, i.e. only depends on the number of variants, it requires that all other
influencing factors have to be assumed constant and known a priori.35 In the work of
Bräutigam, these assumptions include
Identical value of all variants If only the number of variants to add or discontinue is
considered instead of a concrete set of variants, statements about cost changes
can only be interpreted if all variants are assumed to have the same value.
Fixed production sequences In order to estimate the number of changeovers required
to produce a certain number of variants, fixed and cyclic production sequences
have to be assumed, based on the demand characteristics of the variants.
Identical setup costs The total setup costs incurred are only estimated via the ex-
pected number of changeovers caused by a given number of variants, which in
turn implies that all changeovers are assumed to incur identical setup costs.
These assumptions can be questioned and they probably bias the results of applications
of this method in practice.
Thonemann and Bradley36 provide a model to assess the effects of product variety on
supply chain performance in a setting with a single manufacturer and multiple retailers.
Supply chain performance is measured in terms of setup costs and inventory costs at
the retail stores. The authors consider a setting where setup times are important and
focus on the effects of increased lead times due to more frequent changeovers caused by
34 See ibid., pp. 101-104.
35 See ibid., pp. 96-99.
36 Thonemann and Bradley (2002).
46 3 State of the Art
increasing variety. They conclude that the lead time effect of product variety on cost
is “substantially greater than that suggested by the risk-pooling literature”.37 While the
aim of their work coincides in many aspects with the aim of this work, they obtain
their general results on the expense of assumptions about the production system and
inventory policies. Demands are all assumed to be identically Poisson distributed.
Production planning is done according to a simple batching strategy where production
of a certain good is only initiated when orders for a certain minimum quantity are
present. Inventory management is done according to a simple (S−1, S)strategy at the
retail stores. They only consider one-stage production systems with direct distribution
relations to retail stores. The complex closed-form analysis makes it hard to adapt
this approach to other settings.
In the context of assortment complexity, the concept of postponement becomes relevant.
Lee and Tang38 provide a quantitative model to assess the costs and benefits obtained
by shifting the point of differentiation downstream in the production process by making
more options common. The costs comprise required investments, additional material
and processing costs. A more qualitative analysis of the effects of postponement on
supply chains is provided by Yang and Burns39. Venkatesh and Swaminathan40 discuss
postponement as a key strategy to manage product variety and provide a case study
that shows a successful implementation at Benneton. However, no model for such an
assessment in the general case is provided.
3.1.2.3 Activity Based Costing and Related Approaches
One approach to assess the cost of assortment complexity is to precisely calculate
the actual costs incurred by the production of each single product variant. On that
basis, the cost incurred by a given variant can be compared to the revenue it generates
in order to assess its profitability. Adam and Johannwille41 argue that the results
of traditional accounting approaches based on overhead costing may be misleading
when they are used for variety-related decisions. New variants and slow movers are
usually cross-subsidised by the standard products since the fixed overhead cost rates
are applied to all products. There two main reasons for this are:
37 Ibid., p. 563. We discuss risk pooling in Section 3.2.4.
38 Lee and Tang (1997).
39 Yang and Burns (2003) and partly their more recent works (Yang et al., 2004a,b).
40 Venkatesh and Swaminathan (2003).
41 Adam and Johannwille (1998, pp. 16-19).
3.1 Assessing the Cost Effects of Assortment Complexity 47
•Additional overhead costs created by the introduction of new variants are equally
split up over all products, including the existing variants. Such additional over-
head costs should only be assigned to the products that cause them in order to
make them only appear profitable if they actually cover these expenses.
•Overhead costs are not calculated according to the actual resource consump-
tion of each individual product, but proportionally to the single unit costs. A
cause-fair distribution of overhead costs must consider the time and quantity
requirements for resources of each product to avoid cross-subsidisation.
The inability of traditional accounting approaches to correctly assess assortment
complexity-induced costs has led to much research in the development of alternative
accounting systems that circumvent these drawbacks. Most of these approaches come
from the area of activity based costing (ABC), an accounting principle mainly devel-
oped by Johnson and Kaplan42, Cooper and Kaplan43 and Horváth and Mayer44.
Activity based costing tries to improve the assignment of overhead costs by focussing
on the resource utilisation of each cost unit. Within each cost centre, the corresponding
processes are analysed, documented and later aggregated over cost centres to several
main processes. For each main process, the costs incurred by the required resources
are derived. Process costs can be quantity-invariant or quantity-dependent. For the
latter type a process cost rate is derived that reflects the resource costs for the output
quantity of a single process execution.45
The usage of process cost rates in place of fixed overhead cost rates make it possible
to consider the fact that the materials produced have different characteristics and
different resource requirements. In the context of a correct assessment of assortment
complexity-induced costs, three different effects can be observed:46
Allocation effect In contrast to traditional accounting systems, overhead costs are
not fixed, but assigned according to the utilisation of operational resources.
Complexity effect More complex products are drivers of overhead costs as they in-
crease the effort for indirect activities related to their production. The applica-
42 Johnson and Kaplan (1991).
43 Cooper and Kaplan (1988).
44 Horváth and Mayer (1989), who established activity based costing especially in the German
speaking countries.
45 A more detailed description is beyond the scope of this summary and can be found e.g. in
Coenberg and Fischer (1991).
46 See Heina (1999, pp. 66-67).
48 3 State of the Art
tion of activity based costing thus leads to lower cost rates for standard products
and higher rates for slow movers.
Degression effect Unit costs decrease with higher production volume due to the fact
that constant process cost are split up. This leads to a degressive cost function
depending on the order volumes, whereas traditional accounting systems imply
constant overhead costs per unit.
Due to these advantages, activity based costing has been widely used in the context
of variety management. Horváth and Mayer47 provide several examples of how variant
costs can be calculated with the help of activity based costing. Kaiser48 first uses it
to develop a more comprehensive variety management method based on a profitability
analysis for each variant. Such applications have shown that activity based costing
tends to result in much better estimates for variant costs, i.e. slow movers are assigned
a higher share of the overhead costs, while standard products get cheaper, compared
to the cost rates that result from traditional overhead calculation.
The basic idea of activity based costing has been extended in many ways. Schuh49
presents an extension called resource oriented activity based costing. The main dif-
ferences are that there is no aggregation of single processes to main processes, but a
complete process hierarchy is built such that each sub-process is assigned exactly one
resource. Furthermore, he distinguishes between technical and value-related resource
utilisation. The core element of this approach is the functional description of the re-
source utilisation by a certain sub-process. These relationships are defined in so-called
nomograms, which combine functions to describe the resource utilisation and cost de-
velopments. The resource oriented activity based costing has also been combined with
target costing approaches50, especially by Horváth and Seidenschwarz51 and Schuh52.
Heina53 presents an integrated method to determine the optimal product variety. In
the context of the required cost assessment, he mainly builds on activity based costing
methods and adapts them in several ways to make them decision-oriented. Firstly, he
proposes to reduce the required effort by assigning costs to individual product char-
acteristics rather than product variants. Secondly, he splits the entire cost calculation
47 Horváth and Mayer (1989, p. 217).
48 Kaiser (1995).
49 Schuh (2005).
50 For a description of target costing, see Seidenschwarz (1991).
51 Horváth and Seidenschwarz (1992).
52 Schuh (2005).
53 Heina (1999).
3.2 Inventory Allocation in Production and Distribution Networks 49
into a basic and variant calculation such that only the relevant characteristics that
differ between products are considered in the detailed analysis.
Adam and Johannwille54 argue that both traditional accounting approaches as well as
activity based costing approaches and its extensions cannot fulfil the requirements of
an analysis of complexity-related costs. Traditional overhead calculations suffer from
the fixed and equal overhead cost rates assigned to each variant. The activity based
costing approaches discussed here suffer from various drawbacks as well:
•The definition of the processes requires standardised and repetitive activities.
Especially the production of exotic variants often requires non-standard activities
for which no general processes can be defined in the first place.
•It is difficult to correctly define the relation between the materials and required
resources, especially for those activities that are only indirectly related to the
value adding processes. Direct relationships are usually only defined for produc-
tion processes in terms of material routings.
•Process complexity and costs increase with the number of variants. Therefore,
the problem that standard products are assigned overhead costs that they do not
cause still remains, as these products now face increased unit cost rates.
•There is no way to consider fixed step costs. Additional complexity may re-
quire to create additional coordination resources which cannot be included in
the calculations since only currently existing resources are considered.
A more detailed discussion of these existing drawbacks is provided by Adam and Jo-
hannwille55 and Glaser56.
3.2 Inventory Allocation in Production and Distribution
Networks
Inventory management is an important pillar of supply chain management research.
Neale et al.57 analyse the impact of inventory management on supply chain perfor-
mance and stress its importance. Lee and Billington58 describe common pitfalls and
54 Adam and Johannwille (1998, pp. 14-22).
55 Ibid.
56 Glaser (1993).
57 Neale et al. (2003).
58 Lee and Billington (1992).
50 3 State of the Art
opportunities in inventory management and give practical suggestions how logistics
managers can identify optimisation potential in this area. Simchi-Levi et al.59 present
a case study showing how the optimisation of the “location of inventory across the
various stages of the manufacturing and assembly process” and the “quantity of safety
stock for each component at each stage” can significantly reduce total cost.
The question what savings can be achieved in inventory management by altering the
product assortment also leads to the question of optimal stock positions and dimensions
in the production and distribution network under consideration. Chung et al.60 define
this problem as “the task [. . . ] to deploy, possibly subject to constraints, inventory at
a number of stages in a supply network through the exploitation of economies of scale
and flexibility so as to serve customer demand optimally”. An important distinction
for inventory models is between single installation (i.e. single product and location)
and multi-echelon systems.61 As any multi-echelon model builds on some model for
the single installations, we will briefly summarise these models before discussing mutli-
echelon systems relevant in the context of this work.
3.2.1 Single Installation Systems
The ultimate cause of inventory are imbalances between supply and demand processes.
These imbalances may be desired or not. Desired or intentional inventory results from
the exploitation of economies of scale, where batching production or order quantities
yields savings in setup and ordering cost as well as supplier discounts. Other reasons
comprise technological restrictions like minimum lot sizes. These inventories are called
cycle stocks. Another class of inventory is pipeline or work in progress stock caused
by finite production and transport rates. Further anticipation stocks may be caused
by capacity limits, production smoothing or expected price fluctuations. Besides these
intentionally planned inventories installed to decrease costs of production and distri-
bution, safety stocks62 are required as means of buffering against uncertainties present
in demand, supply and transformation processes.63
For the context of this work, models for inventory positioning and dimensioning are
relevant. The base stock model is one of the most important inventory models for single
59 Simchi-Levi et al. (2005, pp. 305–313).
60 See Chung et al. (2006, p. 178).
61 See the discussions of different system types in Zipkin (2000) and Axsäter (2006).
62 See the explanations on safety stocks in Section 2.2.
63 This is a generally accepted classification of inventory types in literature, e.g. in Chopra and
Meindl (2004, pp. 57-59).
3.2 Inventory Allocation in Production and Distribution Networks 51
installation systems and serves as the basis for many multi-echelon models. Given a
single installation that faces stationary stochastic demand, the stock is increased to
a base stock level yin each discrete time period. Demand is assumed to follow a
probability distribution with mean µand standard deviation σ. The base stock level
can then be written as
y∗=µ+z·σ(3.1)
The base stock is set to the expected demand µplus a safety stock, measured in
multiples of the demand standard deviation. This multiple is called the safety factor,
denoted z. If a positive lead time Texists, the base stock level has to cover the entire
lead time demand over T, which results in the base stock formula
y∗=µ·T+z·σ·√T(3.2)
Given that µ,σand Tare known, the safety factor needs to be specified. As the
safety factor determines the degree to which the inventory buffers against demand
fluctuations, it has to be set in accordance with some objective. Such objectives
comprise required service levels that must be met or the minimisation of inventory
holding and backordering costs.
If demand is assumed to follow a certain probability distribution and the safety stock
should be dimensioned to guarantee a given stockout probability (α-service level)64
or fill rate (β-service level)65, standard procedures to determine the required safety
factor exist. If no predefined service level exists and if both inventory holding and
backordering costs can be quantified, the safety factor can be determined to minimise
the sum of inventory and backordering costs incurred. If excess inventory incurs a
holding cost of hmonetary units per period and unmet demand is backordered at cost
of pmonetary units per period, then the optimal α-service level can be calculated as
αSL =p
p+h(3.3)
Equation 3.3 is also called the newsboy ratio because the single period, single instal-
lation base stock model is often referred to as the newsboy problem in which a news
vendor has to determine the optimal stock level for newspapers, given that unmet
demand can be quantified as lost sales and excess newspapers incur penalty costs.66
64 The α-service level is defined as the probability of having sufficient inventory to meet all demand
in a given period. In other words, it is the proportion of periods where no stockout occurs.
65 The β-service level is defined as the proportion of demand that is met from stock and can be
delivered in time
66 See Axsäter (2006, pp. 114-116).
52 3 State of the Art
With this ratio, the cost minimisation problem can be solved with the methods for
α-service levels.
For the common cases where demand is assumed to follow a normal distributed N(µ, σ),
the required safety factor can be determined straightforward. If the required service
level is given as an α-service level, the density function F(·)of this probability distri-
bution can be used to calculate the corresponding safety factor67, while the standard
loss function L(·)is used for the case of the β-service level criterion68.
Extensions to these models include the integration of other sources of uncertainty, like
processing delays or imperfect supplier service. If these additional uncertainties can
also be described via probability distributions, the distribution of the aggregate uncer-
tainties is determined as the convolution of these individual distributions. Generally,
this distribution has to be known to derive an appropriate safety factor. The convolu-
tion of several distributions easily becomes statistically and computationally complex.
Even if an aggregate distribution can be derived, the determination of appropriate
safety factors remains a difficult task, which is why the majority of works on safety
stock models either considers one type of uncertainty only or assumes all uncertainties
to be normally distributed. For the latter case, the aggregate uncertainties are again
normally distributed and the above-mentioned models can be used.
3.2.2 Multi-Echelon Systems with Various Products
For the extension of single installation systems to multi-echelon systems with various
products and stochastic demand, two main branches of research can be identified. The
most active research on inventory allocation is built on the work of Simpson Jr.69.
The second branch of research is based on the work of Clark and Scarf70. Both works
propose a modelling approach for serial production and distribution systems to analyse
possible inventory allocations and have been extended successively.
The approaches differ in terms of modelling the replenishment mechanism and the
resulting service time characteristics. In the stochastic service approach (SSA) pro-
posed by Clark and Scarf, the service (=delivery) times at one stage are stochastic
and vary based on the material availability at supplying stages. In the guaranteed
67 See Axsäter (2006, p. 96), (Nahmias, 1997, pp. 272 et seqq.) and Zipkin (2000).
68 See van Ryzin (2001).
69 Simpson Jr. (1958).
70 Clark and Scarf (1960).
3.2 Inventory Allocation in Production and Distribution Networks 53
service approach (GSA) proposed by Simpson Jr., each stage quotes a service time to
its successor which it can always satisfy.71
Both approaches work on network models in which each node irepresents an item in a
supply chain that performs some operation like a production or transport process. Each
such item has a known and deterministic throughput time and is a potential stockpoint
that can hold safety stock after the processing is finished. The only real source of
uncertainty is stochastic customer demand, which is represented as some probability
distribution with known demand and standard deviation. None of the original models
considers any capacity constraints, i.e. production stages are assumed to produce
arbitrary quantities within one period and external suppliers can always deliver any
quantity ordered. The inventory cost is assumed to be linear in the quantities held on
stock.
3.2.2.1 Stochastic Service Models
The stochastic service approach makes the following additional assumptions:
•serial network structure
•linear backordering cost rates known for each item
•internal shortages immediately affect service time to succeeding location
The third assumption is the main model property: The replenishment lead time of
an item becomes stochastic due to the stochastic service time of its predecessor. This
makes the determination of the required safety stock levels much harder since the
safety stock has to buffer against demand variability over a stochastic replenishment
lead time. While the external service level is given as a parameter, the internal service
levels are an additional decision variable in the optimisation problem.
Clark and Scarf72 introduce the concept of echelon stock, defined as “the stock at any
given installation plus stock in transit to or on hand at a lower installation”. They
show that under the above-mentioned assumptions, the optimal stock policy is an
order-up-to policy with order-up-to levels (Y∗
1, . . . , Y ∗
n). This means that each item
takes into account the amount of stock available at all items further downstream the
network and fills orders such that the echelon stock at that location ireaches its order-
up-to-level Y∗
i. The calculation of the optimal order-up-to-levels is computationally
71 The notions stochastic service approach and stochastic service approach were not coined in the
original works but by other authors, e.g. Klosterhalfen and Minner (2006).
72 Clark and Scarf (1960, p. 1784).
54 3 State of the Art
very complex, partly since it requires numerical integration. Clark and Scarf present
a dynamic programming algorithm to determine these levels sequentially, beginning
with the most downstream stockpoint.
The stochastic service approach has some practical drawbacks. Firstly, the analytical
model is mathematically difficult due to the stochastic modelling of service times.
This makes it very hard to extend this model to more complex, especially non-serial
networks. Secondly, the model does not distinguish explicit stockpoints, but assumes
that each stage holds stock and periodically fills this stock up to its order-up-to level.
This may become a problem depending on the level of abstraction of the network
model. It is reasonable if the network items represent physical production or storage
locations, but it is unrealistic if the network is a detailed model at the single product
level. In this case, this approach prescribes an order-up-to-level for all products at all
stages, including raw and semi-finished materials. Thirdly, the echelon stock concept
assumes central control, as planners at each installation have to consider stock at
downstream locations.
There are some works that extend the approach of Clark and Scarf in different ways.
As the suitability of this model for the problem considered in this work is limited,
the reader is referred to van Houtum et al.73 and Minner74 for an overview of such
extensions. Among other extensions, the approach has been adapted to some more
realistic network structures like distribution and assembly structures. However, there
is currently no extension to general networks in which each item can have arbitrary
numbers of predecessors and successors. Other works present adaptations to different
service measures, e.g. the fulfilment of a certain α-service-level instead of backordering
costs.75
3.2.2.2 Guaranteed Service Models
The area of most active research on inventory placement is built on the work of Simp-
son Jr.76, who proposes a model for serial production and distribution systems to anal-
yse possible stockpoint allocations. In contrast to the approach of Clark and Scarf,
Simpson makes several simplifying assumptions that facilitate the analysis:
73 van Houtum et al. (1996).
74 Minner (2000, pp. 123-125).
75 See Lagodimos and Anderson (1993).
76 Simpson Jr. (1958).
3.2 Inventory Allocation in Production and Distribution Networks 55
•Each item quotes a constant and deterministic service time to its successor that
it can always satisfy.
•Demand covered by safety stock is bound by a maximum reasonable demand
determined via the α-service level required for that item. The α-service level
defines the maximum reasonable demand via the probability that actual demand
exceeds this value. With assumptions about the demand distribution, the maxi-
mum reasonable demand can be computed.77
•All demand that exceeds this maximum reasonable demand is assumed to be
handled by operating flexibility, e.g. emergency orders, accelerated or overtime
production and rescheduling of the existing production sequence, so that stock-
outs do not affect the agreed delivery time at successor stages.
•The total replenishment lead time for each stage is deterministic and equals the
service time of its successor plus its own processing time.
The assumption of operational flexibility and the resulting deterministic service times
are discussed controversially. While replenishment lead times are surely not always
deterministic in reality, there probably is some operational flexibility that can be used
to react to unexpected demand fluctuations. In this sense, the guaranteed service
approach is more realistic as it does not consider safety stocks as the only mean to
react to demand uncertainty. This has led to a bias towards GSA models, as their
assumptions are justifiable and tremendously facilitate the analysis.
Raw
material
Finished
product
ST0ST1ST2
STn-1
STn=0
1
λ
2
λ
1−n
λ
n
λ
Customer
In process
inventories
Figure 3.2: Basic model of guaranteed service approaches
Figure 3.2 depicts the basic model and the main parameters for a serial network. Each
stage ihas a processing time λiand quotes a service time STito its successor. The
replenishment lead time of ican thus be computed as STi−1+λi. The time interval that
77 Simpson Jr. uses this approach in his work for normally distributed demand.
56 3 State of the Art
has to be covered with safety stocks equals this replenishment lead time, diminished
by i’s service time78:STi−1+λi−STi.
The aim to find combinations of service times and corresponding safety stock levels
that minimise the total safety stock cost79 can now be written as
min
n
X
i=1
ci·σi·zi·qSTi−1+λi−STi
s.t. STi≤STi−1+λi∀i∈ {1, . . . , n}
STi≥0∀i∈ {1, . . . , n}
ST0= 0
STn= 0
The objective function seeks to minimise the safety stock cost required to guarantee the
required service level, which determines the safety factors zi. The constraints assure
that the service times are none-negative and that a stockpoint’s service time is not
longer than its replenishment lead time. The central result of Simpson’s work is the
extreme point property, which states that in all optimal solutions
STi∈ {0, STi−1+λi}(3.4)
holds for all i= 1, . . . , n. Each item either covers its entire replenishment lead time
with safety stock (and has a service time of 0) or does not hold any safety stock and
passes its entire replenishment lead time plus the throughput time to its successor
(and has a service time of STi−1+λi). This is also referred to as the all or nothing
policy since the service times are set to cover either the maximum or minimum feasible
values.
While the original work of Simpson only addresses serial systems and does not propose
any solution technique for the optimisation problem apart from testing all possible
combinations, many extensions have been proposed in past years. If some restrictive
assumptions regarding the structure of the network are made, dynamic programming
can be used to solve the optimisation problem efficiently. The above-mentioned ex-
treme point property is the basis for all such approaches that make use of the fact
that the optimal service time of each item can be expressed via the service times of
78 The same modelling approach can be found in Simchi-Levi et al. (2005, pp. 194-196), who calls
this time interval the net lead time.
79 See Simpson Jr. (1958), Minner (2000, p. 93).
3.2 Inventory Allocation in Production and Distribution Networks 57
the downstream and upstream items.80 The safety stock levels can be calculated via
the replenishment lead times that are not yet covered by safety stock at upstream
items.81
Inderfurth82 proposes such approaches for convergent and divergent network structures
and later extends these models to different service criteria83. Graves and Willems84
propose a special algorithm for the case that the network is a spanning tree. They
also extend the problem formulation to include strategic supply chain configuration
decisions.85 These decisions comprise the selection of suppliers, parts, processes and
modes of transport, given that for each stage there are different options that can
be distinguished by their lead times and costs added. The optimisation model then
chooses a sourcing option for each stage as to minimise the total cost, comprising
safety stock costs.86 Minner87 provides an overview of how the original model can be
extended to different special network types and how dynamic programming algorithms
are used to solve the safety stock allocation problem optimally.
Despite the progress in this area of research, no optimal algorithm for general networks
exists so far. General networks are those networks where each node can have an
arbitrary number of predecessors and successors, which is generally the case if the
network is used to model an inventory system on the single product level.
3.2.3 Inventory Allocation as a Combinatorial Optimisation Problem
The extreme point property from Equation 3.4 makes the inventory allocation problem
acombinatorial optimisation problem. In this subclass of general optimisation problems
the decision variables are discrete, i.e. “the solution is a set, or a sequence of integers or
other discrete objects” 88. According to Blum and Roli89, a combinatorial optimisation
problem is defined by
•a set of variables X={x1, . . . , xn}
•variable domains D1, . . . , Dn
80 See Minner (1997).
81 See Inderfurth (1992, p. 23).
82 Inderfurth (1991).
83 Inderfurth and Minner (1998).
84 Graves and Willems (1996, 2000).
85 Graves and Willems (2003).
86 Ibid., p. 121.
87 Minner (2000).
88 Reeves (1993, p. 2).
89 Blum and Roli (2003, p. 269).
58 3 State of the Art
•constraints among variables
•an objective function f:D1×D1···×Dn7→ R+to be minimised.
The set of feasible solutions is called the solution space and is given by
S={s={(x1, v1),...,(xn, vn)}| vi∈Di, s satisfies all the constraints}(3.5)
Among this set of candidate solutions, there is a subset of optimal solutions S∗⊆ S
with minimum objective function values: S∗={s∗|f(s∗)≤f(s)∀s∈ S}.
Given that each item in the network either covers its entire replenishment lead time
with safety stock or does not hold any safety stock and passes its entire replenishment
lead time plus the processing time to its successor90, the remaining decision is a binary
stockpoint or no stockpoint decision. The safety stock allocation problem described in
Section 2.3.2 can thus be mapped to a combinatorial optimisation problem. The set of
decision variables is X={spi, . . . , spn}with binary stockpoint indicators spi∀i∈ N
as the decision variables. All domains are Di={0,1} ∀i∈ N and the solution space
is S={s={(spi, v1),...,(spn, vn)}| vi∈Di, s satisfies all the constraints}. The set
of stockpoint nodes SP then is SP ={i∈ N| spi= 1}.
For each item, the only decision that has to be taken is whether or not it holds
safety stock. Simpson states that due to this property, there are 2n−1possible
combinations for a serial network with nitems and a predefined service time of STn= 0.
More generally, the problem complexity is O(2n)for general networks regardless of the
assumptions made with respect to the service times of items without successors. The
exponential increase of the problem size makes the computation of optimal solutions for
realistic problem sizes with complete optimisation methods91 impossible in reasonable
time.
Combinatorial optimisation problems are particularly suited to be addressed with ap-
proximate or heuristic92 solution techniques. While each heuristic has to be prob-
90 See Section 3.2.2.2.
91 “Complete algorithms are guaranteed to find for every finite size instance of a combinatorial
optimisation problem an optimal solution in bounded time” (Blum and Roli, 2003, p. 269).
92 “A heuristic is a technique which seeks good (i.e. near-optimal) solutions at reasonable computa-
tional cost without being able to guarantee either feasibility or optimality, or even in many cases
to state how close to optimality a particular feasible solution is” (Reeves, 1993, p. 6). The term
heuristic is derived from the Greece heuriskein (υρισκιν), which means to find.
3.2 Inventory Allocation in Production and Distribution Networks 59
lem specific to some extend, several high-level strategies called meta-heuristics93 have
emerged that are aimed at efficiently and effectively exploring the solution space of
a combinatorial optimisation problem. Among these, the most relevant are evolu-
tionary computation including genetic algorithms, ant colony systems, iterated local
search, scatter search, greedy randomized adaptive search, simulated annealing and
tabu search.94
The only application of meta-heuristics to the inventory allocation problem is presented
by Minner95, who uses the local search heuristics tabu search and simulated annealing
to determine stockpoints in a general network.
Local search algorithms start from some initial solution and iteratively try to improve
the current solution by replacing it with a solution from an appropriately defined
neighbourhood. Given a current feasible solution s, a local search heuristic examines a
neighbourhood U(s)and may select one of its members to be the new current solution.
The neighbourhood is defined as the set of solutions that can be reached by applying
a single move or operation to the current solution. These moves usually comprise
changes to the values of certain elements in the current solution. The main problem
of local search heuristics is the risk of getting stuck in so-called local optima, i.e. in
a solution s /∈ S∗whose neighbourhood does not contain any solution with a better
objective value: f(s)≤f(ˆs)∀ˆs∈U(s).
Meta-heuristics employing local search provide different strategies to prevent the op-
timisation from getting stuck in such local optima. The simplest approach is the one
of iterated local search, which restarts the search with a modified initial solution af-
ter a local optimum has been found. The above-mentioned meta-heuristics simulated
annealing (SA) and tabu search (TS) pursue more sophisticated strategies.
The basic idea of SA is to accept inferior neighbourhood solutions ˆswith a certain
probability. The difference f(ˆs)−f(s)together with the current temperature value
determine the probability at which an inferior neighbourhood solution ˆsis accepted.
The temperature value decreases during the search process according to a temperature
93 “A meta-heuristic is an iterative master process that guides and modifies the operations of subor-
dinate heuristics to efficiently produce high-quality solutions. It may manipulate a complete (or
incomplete)single solution or a collection of solutions at each iteration. The subordinate heuris-
tics may be high (or low) level procedures, or a simple local search, or just a construction method”
(Voß, 2001, p. 5). For a description of common characteristics of meta-heuristics, see Blum and
Roli (2003, pp. 270-271).
94 For more detailed descriptions of the above-mentioned meta-heuristics and their applications, see
Dréo et al. (2006), Reeves (1993), Blum and Roli (2003) and Voß (2001). Specifically, we refer to
Dorigo et al. (1999) for ant colony systems and Holland (1975) for genetic algorithms.
95 Minner (2000, pp. 154-169).
60 3 State of the Art
function, so that large deteriorations are accepted during the first iterations, while the
probability that a non-improving solution is accepted is successively reduced.96
Like SA, tabu search is not a pure improvement procedure, but always accepts the
best solution in the current neighbourhood, even if it is worse than the current solu-
tion. The best solution found so far is stored during the entire process and returned
when a stopping criterion is met. Tabu search is a memory-based approach that uses
information about the search history to determine future moves. This is done via a
tabu list, which defines a set of forbidden moves that must not be executed for a cer-
tain number of iterations, called tabu tenure, in order to prevent cyclic computations.
In many implementations, this constraint may be violated if an aspiration criterion
is met, allowing the move to be performed despite its tabu status. In order to avoid
getting stuck in local optima, TS may also make use of diversification strategies that
encourage the search process to examine regions of the solution space that have not
been visited so far by generating new initial solutions that differ significantly from the
solutions visited before. On the other hand, intensification strategies make the search
examine the neighbourhood of the best solutions found so far or generate new solutions
by combining the best solutions visited.97
Minner uses both meta-heuristics for the inventory allocation problem, as they only
differ in the strategy to avoid local optima and can use the same solution representation
and neighbourhood definition. He defines the solution representation for the safety
stock allocation problem as a binary vector whose entries indicates whether or not
safety stock is held at an item i∈ N. The neighbourhood comprises all solutions that
can be reached by any zero/one switch for each single bit within the binary vector.98
U(s) = {ˆs|ˆsi= 1 −sifor one i∈ N,ˆsj=sj∀j∈ N\{i}} (3.6)
A move thus corresponds to changing the stockpoint status of one node from non-
stockpoint to stockpoint or vice versa. With this neighbourhood definition, the author
uses both the simulated annealing and the tabu search meta-heuristic to escape local
optima in the search process. Minner tests both approaches and shows that near
optimal results can be obtained for some problem instances. No significant differences
between the performance of TS and TS can be observed.
While this use of meta-heuristics for the inventory allocation problem appears success-
ful, the size of the networks used for testing is not clearly stated. Since the author
96 For a more elaborate tutorial on SA, see Eglese (1990).
97 See Glover (1989, 1990) for a comprehensive tutorial on TS.
98 See Minner (2000, p. 154).
3.2 Inventory Allocation in Production and Distribution Networks 61
claims that the comparison is done against optimal solutions that “have been deter-
mined by enumeration”99, the network size nmust have been very limited to allow the
full enumeration of all 2npossible configurations. Therefore, this can only be consid-
ered a first proof of the general suitability of local search based meta-heuristics to the
safety stock allocation problem. In order to be able to perform this optimisation in
large networks, the search process should employ more sophisticated moves than all
possible binary switches on the stockpoint statuses.
3.2.4 Risk Pooling Effects in Inventory Management
Some of the expected positive effects of assortment reduction and standardisation are
based on the concept known as risk pooling. The statistical phenomenon underlying
this concept has been summarised by Nahmias100 as follows: “the variance of the
average of a collection of independent identically distributed random variables is lower
than the variance of each of the random variables; that is, the variance of the sample
mean is smaller than the population variance”.
This concept helps to understand many economic phenomena, e.g. in banking and
insurance, as well as in supply chain management. Particularly, the effects obtained
by consolidating multiple random demands have been observed and analysed in the
context of inventory management.101 This consolidation may take many forms, either
a geographical consolidation of multiple inventories at one physical location, or the
consolidation at the product level by rationalising product lines.102 Hopp103 charac-
terises risk pooling as follows: “the combination of sources of variability [. . .] reduces
the total amount of buffering required to achieve a given level of performance”.
In the context of this work, risk pooling occurs as the reduction in demand variability
and forecast deviations. Sobel104 states that “the standard deviation of a sum of
interdependent random demands can be lower than the sum of the standard deviations
of the component demands”. With respect to forecast accuracy, Nahmias105 finds that
“aggregate forecasts are more accurate. [...] On the percentage basis, the error made
99 Minner (2000, p. 166).
100 Nahmias (1997, p. 61).
101 A detailed overview of risk pooling application areas and especially risk pooling in inventory
management for different stock policies is provided by Sobel (2008).
102 Hopp describes the four applications areas warehouse centralisation, product standardisation,
postponement and worksharing (Hopp, 2006, pp. 121-126).
103 Hopp (2006, p. 120).
104 Sobel (2008, p. 155).
105 Nahmias (1997, p. 61).
62 3 State of the Art
in forecasting sales for an entire product line is generally less than the error made in
forecasting sales for an individual item”. Simchi-Levi et al.106 make an important point
by underlining that risk pooling is also obtained by commonality in product structure
and therefore “demand for a component used by a number of finished products has
smaller variability and uncertainty than that of the finished goods”. These insights
have led to a great research interest quantitative models of the costs and benefits of
risk pooling.
Research on the effects of risk pooling in inventory management was initiated by Ep-
pen107, who analyses the effects of consolidating normally distributed demands from
several facilities in a multi-location newsboy model, assuming linear holding and back-
ordering costs108. He provides a model to derive expected inventory holding and back-
ordering penalty costs for different demand parameters. Eppen’s results have proven
to be valid in several works and remain the key insights with respect to risk pooling
in inventory management. He calls these findings the statistical economies of scale,
which can be summarised as follows:
1. Total cost of centralised systems are lower compared to decentralised systems.
2. The magnitude of these savings depends on the correlation between demands.
3. For identical and uncorrelated demands, cost decreases by the factor 1
√nwhen
consolidating ndemands.
Finding 1 can be traced back to effects of pooling multiple random demands. Such
pooling can be achieved via various means. Apart from consolidation of physical ware-
houses, a reduction of assortment complexity can yield the same effects, as either end
customer demand is concentrated on fewer end products or individual end product
demands can be pooled as aggregated demand for semi-finished products and raw
materials. As this concept applies both to the end product level as well as to all inter-
mediate components, demand for a component used by a number of finished products
is less volatile and uncertain than that for finished goods. Accordingly, postponement
strategies109 are also seen as one way to achieve the desired risk pooling effects.110
106 Simchi-Levi et al. (2005, p. 312).
107 Eppen (1979).
108 See Section 3.2.1.
109 See Venkatesh and Swaminathan (2003) for a general introduction to the concept of postpone-
ment.
110 Yang et al. (2004a,b) analyse the implications of postponement on various types of uncertainty
in supply chains and also provide an extensive literature review on this topic.
3.2 Inventory Allocation in Production and Distribution Networks 63
Finding 2 captures the fact that the magnitude of the risk pooling effect depends on
the correlation between the stochastic factors. For example, if two demands are highly
correlated, their aggregation hardly affects the total variability. If we consider normal
demands, this can be shown easily, as the folding of two normal random variables
N1(µ1, σ1)and N2(µ2, σ2)with a correlation coefficient of ρ∈[−1,1] is defined as
N1(µ1, σ1)N2(µ2, σ2)=Nµ1+µ1,qσ2
1+σ2
2+ 2ρσ1σ2(3.7)
For fully correlated variables (ρ= 1), there clearly is no risk pooling effect since the
fluctuations of the two stochastic elements cannot compensate each other. On the other
extreme, risk pooling effects are biggest if the variables are fully negatively correlated
(ρ=−1). Sobel111 concludes that “the advantage grows as the correlation shifts from
strongly positive to strongly negative”. For examples and graphical illustrations of this
concept, the reader is referred to Alicke112.
Finding 3 is closely linked to the above-mentioned conclusions about correlation and
has become a well-known rule for both researches as well as practitioners. The so-
called square root formula has become a standard technique in inventory management
and can be found in any textbook on this topic113. When aggregating demands or
consolidating warehouses while maintaining the same service level and service times,
total system costs decrease as a strictly monotonically decreasing convex function of the
number of demands or warehouses. So the relative benefit of consolidation decreases
with the number of items or warehouses consolidated.
In order to quantitatively evaluate risk pooling effects for a wider range of realistic
application scenarios, the model used by Eppen has been extended successively. Eppen
and Schrage114 introduce positive lead times. Chang and Lin115 extend the model with
the consideration of transport costs. This is one of the most important extensions, as
the implementation of physical inventory pooling in central warehouses goes along with
an increase in transport costs since the distances from the centralised warehouses to
the customers are increased.116 These additional costs have to be offset by the savings
achieved via inventory reduction.
111 Sobel (2008, p. 159).
112 Alicke (2005, pp. 160-162).
113 See Chopra and Meindl (2004, pp. 313–317), Tempelmeier (2005, pp. 156 et seq.).
114 Eppen and Schrage (1981).
115 Chang and Lin (1991).
116 See Simchi-Levi et al. (2007, p. 232).
64 3 State of the Art
Recently, Corbett and Rajaram117 generalised the findings to almost arbitrary multivariate-
dependent demand distributions. They show how pooling of inventories can be anal-
ysed without the need to resort to assumptions of independence and normal distribu-
tions. For these generalised assumptions, they prove that centralisation or pooling of
inventories is more valuable when demands are less positively correlated.
Alfaro and Corbett118 discuss how risk pooling effects change if the system under
consideration does not operate under an optimal inventory policy. They argue that
in practice, an optimal policy is normally impossible to find and also consider non-
normal demand distributions in this context. They provide a model to compare the
relative value of implementing some form of inventory pooling against the value of
improving a suboptimal inventory policy. The authors conclude that there is always
a uniquely defined interval within which pooling leads to greater cost reductions than
optimizing inventory policy, while the reverse is true outside that interval. Apart from
this concrete model, the work also presents an elaborate literature review on inventory
pooling.
Gerchak and He119 show that the benefits of risk pooling increase with the variability of
the original demands. This goes in accordance with intuition, as the positive effects of
risk pooling increase with the risks that are aggregated. They prove this observation
for certain mean-preserving variations of demand variability. Simchi-Levi et al.120
draw similar conclusions and use the coefficient of variation of the demand process as a
metric to evaluate the effects of risk pooling. The higher the coefficient of variation, the
greater the effects of risk pooling and the inventory reduction. As a consequence, they
suggest to evaluate the opportunities to foster consolidation for slow-moving items.
Kulkarni et al.121 investigate risk pooling effects on a strategic level for the question of
network configurations and evaluate the trade-off between logistic costs and risk pool-
ing benefits in production networks. The alternatives considered are product networks
with component manufacturing being spread over all plant and process networks with
component manufacturing being consolidated in a single plant. They show that the
process plant networks offer significant risk pooling advantages under a wide range
of conditions, even without accounting for the benefits of economies of scale. They
conclude that companies should consider this network configuration due to the risk
pooling benefits offered.
117 Corbett and Rajaram (2006).
118 Alfaro and Corbett (2003).
119 Gerchak and He (2003).
120 Simchi-Levi et al. (2007, pp. 318–319).
121 Kulkarni et al. (2004, 2005)
3.3 Determination of Planning Buffers and Planned Production Quantities 65
Benjaafar et al.122 extend the analysis of risk pooling in pure inventory systems to
production-inventory systems, in which lead time demands are not exogenous, but
influenced by the lead times that result from the sharing of the production supply
process. Due to the consideration of the production stages with their characteristics
like utilisation, there can be significant correlation in the lead-time demands of the
different items, even if the individual demand processes are independent. This is
due to the correlation between the lead times, which is particularly significant if the
production system’s utilisation is high. As correlation influences the potential benefits
of risk pooling, they use the model to analyse how factors like utilisation, demand and
service time variability and supply structure affect the benefits of risk pooling. Finally,
they use the model to compare the benefits of inventory pooling to those of capacity
pooling at the production stages.
As supply chains often do not have any central control, Hartman and Dror123 analyse
the fair allocation of benefits gained from inventory centralisation among various par-
ticipants. They consider a system of retail stores that should be supplied from a central
inventory location and use a game theoretic approach to share the savings among all
participants such that no participant or subset of participants (called coalitions) has
any incentive to order separately, even if the holding and backordering penalty costs
are not the same at all stores and for all coalitions.
3.3 Determination of Planning Buffers and Planned Production
Quantities
3.3.1 Determination of MRP Production Parameters
The planning buffer as defined in Section 2.1.2 can be seen as one particular param-
eter for production planning in MRP systems. The corresponding literature review
therefore focuses on approaches that analyse the optimal determination of such plan-
ning parameters and their impact on production costs. The literature review shows
that there is a certain set of parameters that have frequently been analysed. These
parameters can be summarised as
•MPS frozen interval,
122 Benjaafar et al. (2005).
123 Hartman and Dror (2005).
66 3 State of the Art
•MPS replanning frequency,
•MPS planning horizon,
•product structure,
•forecast error,
•safety stock and
•lot-sizing rules.
Table 3.1 provides an overview of works that analyse the performance of MRP produc-
tion systems under variation of different parameters. This comparative composition
of relevant works is based on the extensive literature review conducted by Yeung and
Wong124. Table 3.1 has been updated and extended by more recent works reviewed
for this purpose.
Although the planning buffer is an important planning parameter that is also imple-
mented in commonly used ERP systems125, no work exists to the best of our knowledge
that explicitly addresses its optimal determination. In particular, the trade-offs de-
scribed in Section 2.1.2 have not yet been addressed in the literature.
3.3.2 Suitability of Standard Lot-Sizing Models
The determination of planned production quantities for all materials and several time
periods shows many similarities to standard lot-sizing problems. Lot-sizing models
determine production lots based on primary demand and under consideration of avail-
able resources, setup times and several cost components. The application context of
this work does not require the determination of production lots and sequences on the
detailed planning level. Moreover, we consider planned production quantities for each
production stage and the respective materials over a larger time horizon. These con-
ditions coincide with the characteristics of existing big-bucket lot-sizing models that
allow production of different products within one time period without making any
statements about their production sequence. Suerie126 summarises the basic assump-
tions of the Capacitated lot-sizing problem (CLSP) as the most common multi-item
basic big-bucket model as:
124 Yeung and Wong (1998).
125 For example, SAP R/3 allows the definition of this parameter as the scheduling margin key, which
determines float periods before and after production orders (Dickersbach et al., 2005, p. 259).
126 Suerie (2005, p. 14).
3.3 Determination of Planning Buffers and Planned Production Quantities 67
Frozen interval
Replanning frequency
Planning horizon
Product structure
Forecast error
Safety stock
Lot-sizing rules
Kunreuther and Morton (1973, 1974) # # # # # #
Whybark and Williams (1976) # # # # # #
Baker (1977) # # # # # #
Carlson et al. (1979) # # # # # #
Kropp et al. (1979) # # # # # #
Baker and Peterson (1979) # # # # # #
Blackburn and Millen (1980) # # # # # #
Carlson et al. (1982) # # # # # #
Biggs and Campion (1982) # # # # # #
Blackburn and Millen (1982a,b) # # # # # #
DeBodt and Wassenhove (1983) # # # #
Wemmerlov and Whybark (1984) # # # # # #
Chung and Krajewski (1984) # # # # # #
Benton and Srivastava (1985) # # # # # #
Wemmerlov (1985) # # # # # #
Yano and Carlson (1985) # # # # #
Lee and Adam (1986) # # # # #
Chung and Krajewski (1986) # # # # # #
Carlson and Yano (1986) # # # # # #
Wemmerlov (1986) # # # # # #
Sridharan et al. (1987) # # # # # #
Yano and Carlson (1987) # # # # # #
Sridharan and Berry (1990a,b) # # # # #
Ristroph (1990) # # # # # #
Barrett and LaForge (1991) # # # # # #
Benton (1991) # # # # # #
Bregman (1991a,b) # # # # # #
Lin and Krajewski (1992) # # # #
Lee et al. (1993) # # # # # #
Zhao and Lee (1993) # # # #
Russell and Urban (1993) # # # # # #
Lin et al. (1994) # # # #
Sridharan and LaForge (1994a,b) # # # # # #
Kadipasaoglu (1995) # # # # #
Zhao et al. (1995) # # # # #
Zhao and Lee (1996) # # # #
Molinder (1997) # # # #
Enns (2001) # # # # #
Table 3.1: Classification of studies regarding MRP parameter settings
68 3 State of the Art
•Several products are produced on one shared resource with limited capacity.
•The planning horizon is finite and divided into discrete periods.
•All products face a deterministic dynamic demand.
•If a product is produced in a certain period, the resource has to be set up for
this product in this period.
•Setups consume resource capacity and incur a setup cost.
•The aim is to minimise the sum of holding costs and setup costs.
All these assumptions coincide with the characteristics of the corresponding problem
considered in this work. For each production process step with a limited capacity,
planned production quantities have to be determined for a sequence of time periods,
where multiple materials are produced within one period. As setup costs should be
estimated via the number of products with strictly positive production volumes in that
periods, the assumptions regarding setup costs also holds. Based on these assumptions,
the CLSP can be formulated as a mathematical optimisation problem:
min X
j∈JX
t∈T
hi,t ·Ij,t +X
j∈JX
t∈T
scj·Yj,t
s.t. Ij,t−1+Xj,t =dj,t +Ij,t ∀j∈J, t ∈T
X
j∈J
aj·Xj,t +X
j∈J
stj·Yj,t ≤ct∀t∈T
Xj,t ≤bj,t ·Yj,t
Xj,t ≥0, Ij,t ≥0, Ij,0= 0, Yj,t ∈ {0,1} ∀j∈J, t ∈T
Following the notation used by Suerie, the model determines optimal lot sizes Xj,t for
each product j∈Jand period t∈T. The objective function seeks to minimise total
setup and holding costs by incurring holding costs hj,t for the inventories Ij,t present in
each period and incurring setup costs scjif product jis produced in period t, as indi-
cated by Yj,t. In the order of occurrence, the restrictions ensure the inventory balance,
consideration of available capacities and enforce the binary production indicators Yj,t
to be equal to 1 where required. For a more in-depth discussion of this basic model
and its extensions, the reader is referred to the works of Suerie127 and Tempelmeier128.
Due to the similarities in the underlying assumptions, the CLSP provides a good basis
to build the required optimisation model for planned production quantities upon.
127 Ibid.
128 Tempelmeier (2005).
69
CHAPTER 4
Required Work
Promise less or do more.
The Whitest Boy Alive
In this chapter we deduce the conceptual tasks to be addressed in Chapter 5 via a
comparative analysis of the findings from Chapter 3 and the requirements identified
for each subproblem in Section 2.3. We also show how the solutions to these tasks can
contribute to and extend the current state of the art.
4.1 A Model to Assess the Cost Effects of Assortment
Complexity Changes
Having reviewed existing approaches to assessing the costs of assortment complexity,
we may conclude that these approaches fall into two categories. Firstly, some use
accounting methods to fairly assign the costs to the product variants according to
the input involved. Some drawbacks of these approaches have already been discussed
in Section 3.1.2.3. But even if it were possible to assign all overhead costs perfectly
accurately to each variant, the interrelations between the single variants are always
neglected. Consequently, these approaches cannot evaluate cost effects that occur if
certain combinations of assortment change decisions are evaluated. None of these
methods tries to analyse the changes to the optimal configuration of the production
and distribution network that an assortment change might bring. Secondly, some ap-
proaches try to derive general cost functions with the number of variants as the only
independent variable. While this approach is in principle suited for what-if analyses,
using the number of product variant as the only input variable requires many simplify-
ing assumptions that make this type of analysis very imprecise. If concrete assortment
changes form the input of the analysis, potential changes to the configuration of the
70 4 Required Work
production and distribution network are not considered. There is currently no ap-
proach that allows what-if analyses in response to assortment changes to evaluate the
cost effects of concrete changes in detail.
Against this background, this work defines a new model for production and distribution
networks to represent assortments and assortment scenarios. For its practical usability,
algorithms to generate these models from existing ERP system data are developed to
facilitate model generation. On the basis of this model, operations to derive scenarios
are defined and algorithms to apply such scenario definitions to a model are developed.
A cost function to assess the relevant costs for an arbitrary assortment model or
scenario is defined. These elements form the basis for the novel approach to use what-
if analyses quantitatively to assess assortment complexity. Reviewing the analysis
process in Figure 2.9, they support the steps 1 and 4. The optimisation problems
posed by steps 2 and 3 are treated as separate subproblems.
4.2 Inventory Allocation in Production and Distribution
Networks
Considering existing works in the area of inventory allocation, the basic model of
Simpson Jr.1is identified as a suitable approach that can be adapted and extended
for use in this work. The assumptions made coincide well with the actual practice
of MRP-based material planning in the production and distribution networks under
consideration. These commonalities comprise a similar network structure and service
times as the main scheduling parameter between adjacent stages. The assumption
of operating flexibility to guarantee constant replenishment lead times can also be
justified.
On this basis, the existing model has to be adapted to accurately fulfil all differing
requirements. These comprise a different demand representation, where individual
demand volumes and consequently safety stock levels are considered for each planning
period. This eliminates the need to assume strictly normally distributed demands,
which cannot be expected to exist in practical applications. Further extensions are
required to handle different service level measures like the β-service level as well as
more complex network structures. In particular, networks cannot only be assumed to
be linear, divergent or convergent, but have to allow arbitrary non-cyclic structures to
represent arbitrary assortments.
1 See Section 3.2.2.2.
4.3 Determination of Planning Buffers and Planned Production Quantities 71
It has to be pointed out that there is no known approach that is computationally
feasible in large networks like those we have to expect when considering the inventory
allocation problem at the product level. Given the results of Chapter 3.2, the use of
heuristic approaches to determine optimal inventory allocations in real-world scenarios
is necessary and has proven to be promising. This is especially true if strict optimality
is not the primary goal, as is the case here, where the resulting parameters rather
serve as a basis for a cost analysis. The task therefore is to define a heuristic solution
method that is able to make use of domain knowledge to find reasonable and near-
optimal solutions in acceptable time. The considerations in Section 3.2.3 indicate
that tabu search is a proven meta-heuristic where domain knowledge can easily be
incorporated into the definition and selection of moves.
4.3 Determination of Planning Buffers and Planned Production
Quantities
The subproblem of setting planning buffers in MRP manufacturing environments has
not yet been addressed to the best of our knowledge. While several works treat the
optimal determination of different MRP parameters, none addresses any parameter
similar to the planning buffers described in Section 2.1.2. By contrast, the determina-
tion of planned production quantities shows certain similarities to known big-bucket
lot-sizing problems. Against the background of the current state of the art, the task
of determining optimal planning buffers and planned production quantities remains as
described in Section 2.3.3. The trade-offs described in that section may be modelled as
an optimisation problem, which should use elements of existing big-bucket lot-sizing
models where possible. These models must be required to include all relevant cost
components and especially integrate the planning buffers and their inventory cost ef-
fects as additional elements. Despite these profound extensions, the overall aim is to
guarantee the solvability of the model with standard optimisation software.
73
CHAPTER 5
Configuration of Inventory Management and
Production Planning Parameters to Assess
the Effects of Assortment Complexity in
Consumer Goods Supply Chains
I keep up with the racing rats
and do my best to win.
Editors
5.1 A Model to Assess the Effects of Assortment Complexity
Changes
To assess the effects of assortment complexity changes, Section 5.1.1 first defines an
assortment model on which to base the cost assessment. On the basis of this model, a
formalism to define assortment scenarios is derived in Section 5.1.2. A cost model for
the actual assessment is then defined in Section 5.1.3.
5.1.1 Production and Distribution Networks as a Model for Assortment
Complexity
This section defines the common basic model of a production and distribution network
that serves as the basis for all subordinate optimisation and cost models. Accordingly,
only the common elements are introduced here, while additional elements are intro-
duced in later sections along with the optimisation or evaluation problems where they
74 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
are used. As a convention, we use calligraphic letters for sets, while the corresponding
capital letters denote the cardinality of the set. General variables and parameters are
denoted both by upper or lowercase letters. Superscripts are used to further define the
variable or parameter as required.
5.1.1.1 Model Elements
The definition of model elements is structured according to the categories product and
distribution structure, time, internal processes (material transformation and transport)
and external processes (customer demand).
Product and distribution structure An assortment defines a certain product and dis-
tribution structure which we represent in a network model. Table 5.1 summarises the
notations used for the structural elements of that network model.
Table 5.1: Product and distribution structure model elements
Symbol Description
Lset of physical locations of production sites and sales locations
Mset of materials
Nset of items each representing a material at a certain location
Vset of links connecting two items, V ⊂ N ×N
w(i,j)quantity relationship for material flow from ito j
NP ROC set of first-stage items procured externally
NP ROD set of intermediate-stage production items
NDIST set of intermediate or last-stage distribution items
SP set of stockpoint items SP ⊆ N where inventory is held
The set of locations is denoted L. A location l∈ L refers to the physical location
and organisational unit of a production or distribution site of one of the supply chain
actors. According to the function of the organisational unit, we distinguish production
and sales locations. We only distinguish physical locations at the level of different
production or sales sites, while different storage locations within the same plant or
warehouse are considered the same physical location.
The set of materials is denoted M. A material m∈ M is goods at any production or
distribution stage and can be procured, produced, consumed in the production process
5.1 A Model to Assess the Effects of Assortment Complexity Changes 75
or be distributed. Thus we can distinguish between finished, semi-finished, raw and
packaging materials. Finished materials are also called end products and describe the
subset of materials that are not processed any further and sold to customers. Semi-
finished materials are all intermediate goods produced in the production process of the
end products. Raw and packaging materials are used to produce the semi-finished and
sometimes finished materials and are procured externally.
The central element of model are the items N, which are valid combinations of materi-
als at their respective physical locations, and thus N ⊂ M×L. Valid here means that
the material is considered for planning at the respective location. An item i= (p, l)
thus indicates that product p∈ M is procured or produced at or distributed to lo-
cation l∈ L. The distinction between materials Mand items Nis necessary as the
resulting model should represent both the assortment and distribution structure and
so each material may be relevant in more than one location.
The set V ⊂ N×N represents the links between pairs of items i, j ∈ N to represent the
product and distribution structure. Such a link (i, j)∈ V may either represent a where-
used relation as defined in an entry of j’s bill of material (BOM), or a distribution
relation between two locations. In the first case, the locations of iand jare identical
and their products differ. Item ithen is a component in the bill of material of j. In
the latter case, demand for jis filled by ordering the required quantities from i, thus
the items’ locations differ while their products are identical and there is a transport
relation between the two locations. The weight w(i,j)represents the quantity relation
for material flows from ito j. Depending on whether the link v= (i, j)represents a
where-used or distribution relation, wvis the production coefficient that describes the
quantity of irequired to produce one basic unit of j, or it is 1, respectively.
In order to achieve the change of either material characteristics or physical locations
between adjacent items, each item i∈ N may have some type of process related to it.
These processes may either be a production process step that transforms a set of input
components into a resulting product, or a transport process that transfers a material
from one location to another. More details on the characteristics of these processes
are given in Section 5.1.1.1.
With this definition of a network structure and the processes attached to single items,
the set of items Ncan be further classified into disjunctive subsets. Firstly, set NP ROC
contains all items that are procured externally and thereby mark the system boundary
at one side of the network. All products and related production and transport processes
further upstream in the supply chain are not considered in the model. Consequently,
nodes i∈ NP ROC have no predecessors in the network representation, which we write
76 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
as PR(i) = ∅. Any practical analysis will focus on a limited part of the production and
distribution network and consider some materials externally procured. Although it is
theoretically possible to extend the model to all suppliers and sub-suppliers, such a
limitation may be required to limit model complexity or due to information availability
about the structures and processes at the suppliers.
Secondly, the set NP ROD contains all intermediate stage items that have a production
process related to them. Such a node can have an arbitrary number greater or equal
to 1 of predecessors and successors.
Thirdly, the set NDIST contains all items with distribution processes attached. With
respect to their position in the network, it can only be assured that all last-stage
items iwith no successors SC(i) = ∅are elements of NDIST , while not all elements
of NDIST are necessarily last-stage nodes. Those items with no successors generally
represent the end products at the sales locations that are requested by and shipped to
customers from there. But there may be distribution structures that span more than
one sales location, as not all sales locations necessarily procure their end products
directly from the production locations. Moreover, they can procure products from
other sales locations if the quantities procured by a sales location do not suffice to
organise full truck load transports from the production location. In this case, it may
be reasonable to use a nearby sales location as a transshipment point. In any case, we
assume that each distribution node i∈ NDIST has exactly one predecessor and thus
|PR(i)|= 1. This means that the material at a distribution item is either procured
from the corresponding production location, or from another sales location. It is
obvious that these three sets form the set of all items N=NP ROC ∪NP ROD ∪NDIST .
A given set of materials and corresponding links suffices to represent a certain assort-
ment with its product and distribution structure. This network forms the basis for
the PDN. Each item i∈ N is a potential stockpoint, which can be interpreted as the
decision to keep inventory of the respective item at the respective physical location
to uncouple the supply and demand processes.1The set of all stockpoints is denoted
SP ⊂ N.
Time In order to describe any sequence of events (i.e. state changes) in a system, a
time model is required.2The analyses carried out in this work must always refer to a
1 Hopp defines stockpoints as “locations in the supply chain where inventories are held” (Hopp,
2006, p. 2). This definition describes stockpoints on the higher aggregation level of physical
location that does not allow to distinguish which materials and products are held in stock at
these locations.
2 See Dangelmaier (2003, p. 224).
5.1 A Model to Assess the Effects of Assortment Complexity Changes 77
certain period of time, as they cannot use information from an infinite past nor can
they extend endlessly to the future. All assumptions made and all conclusions derived
refer to a certain period of time that has a sound interpretation in the real world.
Time can be modelled as continuous or discrete. As we do not consider states of the
system at a single moment, but only make statements about events in certain time
intervals3, we consider discrete time periods of equal length. The set of all discrete
time periods is denoted T={1, . . . , T}. The analysis then always refers to the time
interval of lengths T.
Table 5.2: Time model elements
Symbol Description
Tset of mid-term time periods under consideration T={1, . . . , T}
TSnumber of short-term periods that constitute one mid-term period in their
interpretation in the real world
In order to discretise a time interval, one has to determine the lengths of the single time
periods t∈ T. For the application considered here we find that some operations like
demand forecasting and tracing are made on a timely aggregated level, while planning
parameters like planning buffers and replenishment lead times are defined at a more
detailed level, i.e. in units of smaller time periods. The discrete time periods t∈ T refer
to the aggregated level and represent the mid-term periods used to forecast and trace
demand. In order to be able to relate the short-term periods used to define planning
parameters to the mid-term periods, the parameter TSdefines the number of short-
term planning periods within one mid-term period. That is, TSshort-term periods
describe the same time span as one mid-term period t∈ T in the real world.4
Internal processes: Material transformation and transport In order to describe the pro-
cesses that transform adjacent items in a network, we define the set Sof production
and transport processes. Each production process step s∈ SP ROD ⊆ S describes an
arbitrary set of consecutive operations required to produce any related material from
its input components. In flow production systems such a process step might describe
3 For example demand in a certain month, production quantities in single shifts etc.
4 In practice, e.g. months or weeks may be chosen as the mid-term period, while all operational
parameters are defined in days as the short-term periods. The set of all time periods T=
{1,...,12}may then describe a year and the number of short-term periods per mid-term period
would be TS= 7 or TS= 30 respectively. If there are no operations on weekend days, TSmust
be adjusted accordingly.
78 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
Table 5.3: Material transformation, transport and coordination model elements
Symbol Description
Sset of process steps
SP ROD set of production process steps SP ROD ⊆ S to produce items in NP ROD
ST RANS set of transport process steps ST RANS ⊆ S to distribute items in NDIST
Nsset of items processed on s∈ S
Kscapacity available for production process step s∈ SP ROD in one short-term
period
ki,s fraction of Ksrequired to produce one basic unit of mat(i),i∈ Nson s
Qi,t planned production quantity of iin time period t
Qrnd
i,s lot-size rounding value for production quantities of i
pbsplanning buffer for production scheduling on s∈ SP ROD
processing on a particular production line. We assume that the set of resources re-
quired by different production process steps are disjunctive, such that each production
process step can be assigned an independent capacity. In flow production systems the
bottleneck resources are the production lines. Separate lines with identical processing
capabilities are modelled as separate production process steps. Additional machines
required for pre or postprocessing on more than one line should not be any bottle-
neck resources, so that the assumption of disjunctive resource requirements can be
justified.
The processing of an item i∈ NP ROD on a production process step s∈ SP ROD is
further defined in terms of capacities. The capacity available on a production process
step s∈ SP ROD in one short-term period is denoted Ks, defined in the common
basic unit of measure for all items assigned to s. While the units of measure used
to quantify production quantities usually differ between production process steps, it
can be assumed that the different items processed on one production process step can
be measured with the same unit, e.g. pieces, kilogrammes, metres etc. The total
capacity in one midterm period is therefore Ks·TS. The capacity coefficient ki,s of
the mapping from ito sprovides information about the capacities required to process
iand is defined as the fraction of Ksrequired to process one unit of i.
In order to fulfil demands for production items, these quantities have to be produced
at a given time. Accordingly, planned production quantities Qi,s,t are defined for each
item i∈ NP ROD and related production process step s. These planned production
quantities are an assignment of demand quantities to a production process step in a
5.1 A Model to Assess the Effects of Assortment Complexity Changes 79
certain period, which is either the period in which the demand occurs or any earlier
period. These quantities are defined at the aggregate level of mid-term periods, as this
is the aggregation level on which demand information is available. Keeping in mind
that this model is used to asses cost for real or theoretical assortments, we cannot
assume that demand information in terms of concrete order dates and quantities is
available, but only at such an aggregate level. The definition of planned production
quantities for all demand quantities describes a theoretical production plan at the
aggregate level of the time periods t∈ T. The problem of assessing operational
production cost without any knowledge of the operational production plan is addressed
in Section 5.1.3.
Each production process step s∈ SP ROD has a planning buffer5pbs, defined as the
time buffer between the provision of components and the requirement date of an order,
measured in short-term periods.
A transport process step s∈ ST RANS ⊆ S is characterised by its start and end lo-
cations as well as a transport time ttrans
s, measured in short-term periods. The re-
quired resources of means of transport are not modelled explicitly and are assumed
to have unlimited capacity, as they are usually provided by third-party logistic service
providers. Additional capacity can therefore be ordered from these service providers
at the aggregate planning level considered here.
Each item iis that is not a procurement item is assigned to at least one process step. If
the item is a production item i∈ NP ROD, this process is a production process step s∈
SP ROD, while it is a transport process step s∈ ST RANS if i∈ NDIST . Production items
can be assigned more than one production process step as there may be alternative
routings that allow processing an item on various machines.6Distribution items i∈
NDIST are assigned exactly one transport process step s∈ ST RANS .
The relations between the items and the process steps are modelled via binary indicator
variables pi,s, which is 1 if item iis assigned to process step sand 0 otherwise. We
can then define the set of all processes assigned to one item as
Si={s∈ S | pi,s = 1},(5.1)
5 See Section 2.1.2 for a definition and description of planning buffers.
6 Alternative routings sometimes require alternative BOMs, as different machines require slightly
different input components. However, these differences are negligible in this context and therefore
only one BOM per item is considered.
80 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
and analogously the set of all items assigned to one process step as
Ns={i∈ N | pi,s = 1}.(5.2)
In this way, each node can be assigned to a subset of available processes. For model
consistency, we require that
•Si⊆ SP ROD and |Si| ≥ 1if i∈ NP ROD: Each production item is mapped to one
or more production process steps
•Si∈ ST RANS and |Si|= 1 if i∈ NDIST : Each distribution item is mapped to
exactly one transport process step.
There are no restrictions in the sets Ns, as an arbitrary number of items can be assigned
to all process steps.
In order to describe material flows in the PDN, we need some notion of the duration of
activities carried out at the items and their related processes. We define a throughput
time TTifor each item i∈ N, which is the time required to carry out all operations
of the related process steps, plus supplier lead times for procurement items and any
in-plant transports from the physical storage location of all predecessor items to the
physical location for production items. Transport times, both of in-plant and inter-
location transport, are assumed to include any time required at the destination for
goods receipt processing. In total, TTiis the time required to make the respective
materials available for planning at i, given that all predecessors have the respective
materials available for planning at their respective locations. Depending on the type
of item i, it is defined as
TTi=
supplier delivery time (order lead time) +transport time i∈ NP ROC
max. in-plant transport of components +|Si|−1·Ps∈Sipbsi∈ NP ROD
ttrans
swith s∈ Sii∈ NDIST
For procurement nodes, the throughput time comprises the supplier order lead time
as well as the transport time from the supplier to the destination location. As dis-
tribution nodes form the system boundary, the suppliers and the respective transport
processes are not modelled explicitly and are only included in these throughput times
of procurement nodes.
5.1 A Model to Assess the Effects of Assortment Complexity Changes 81
For production nodes the throughput time comprises the maximum in-plant transport
time of all components and the average planning buffer of all related production pro-
cess steps. Taking the average planning buffer introduces some imprecision, as actual
throughput times may be both longer and shorter in particular cases. There are some
possible alternatives: Firstly, one could use the maximum planning buffer of all related
processes maxs∈Si(pbs)instead of the average. This would result in a throughput time
that represents the maximum time required to make the materials available for plan-
ning, but would overestimate that time in general. Secondly, the planning buffers can
be weighted with the probabilities wi,s that process step swill be used to produce a
concrete production order of i. While this approach is the most exact, it will generally
be difficult to define these probabilities. If the probabilities are unknown, they may be
approximated with the ratio of the capacity provided by sover the capacity provided
by all related process steps:
wi,s =Ks
Pp∈SiKp
(5.3)
The precision of this approximation increases as capacity utilisation approaches 100%,
as other factors like different machine cost rates then become negligible and the dis-
tribution of the production volumes over the production process steps follows the
distribution of capacities.
For distribution nodes, the throughput time only comprises the transport time of the
related transport process step.
Table 5.4: Customer demand model elements
Symbol Description
dext
i,t expected external market demand (primary demand) for iin period t
di,t expected total demand for iin period t
FDmad
irelative mean absolute deviation of forecasts for i
Di,t random variable describing the actual demand for iin period t
σd
i,t standard deviation of forecast deviation distribution of total demand for iin
period t
σdext
i,t standard deviation of forecast deviation distribution of primary demand for
iin period t
αSL
iα-service level required by customers for i
βSL
iβ-service level required by customers for i
STmax
imaximum delivery time for iif irepresents an end product sold to customers
82 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
External factors: Material supply and customer demand Customer demand is the main
external factor considered in the model. Both the cost assessment as well as optimi-
sation problems described in Section 2.3 require information on the demand volumes
and distribution at each item. We assume that information about the demands Di,t
for item iis available at the aggregation level of the mid-term periods t∈ T. This
demand information may be derived both from historical data or assumptions about
theoretical demand developments. Practically, historical data can serve as a starting
point to define realistic demand quantities for each item.
For end products there is primary demand dext
i,t that originates from the marked demand
in terms of customer orders, for which no explicit model element exists. Furthermore,
demand at an item may be dependent demand originating from successor items within
the network. In most cases, we can expect that an item is either an end product at
a sales location that faces primary demand or represents an arbitrary material at a
production location, which faces dependent demand only from its successors, be it from
the adjacent distribution items for end products or from the next production step in
case of semi-finished and raw materials. However, we cannot explicitly assume that, as
there may be cases where both types of demand are present, e.g. when a sales location
supplies another sales location that does not procure its materials directly from the
production location, of if semi-finished products are also sold on the marked.
As the expected demand di,t for an item iis based on forecasts, a probabilistic model
of the uncertainty related to the forecast is required. Demand is forecasted for the
mid-term periods, i.e. for each period t∈ T , demand is forecasted in period t−1.
If we consider the forecast error FDi,t, i.e. the deviation of forecasted from actual
demand for item iin period tas a random variable, we can describe the uncertainty
via the probability distribution of this variable. While there are no assumptions about
the distribution of the actual demand over the mid-term periods, we do assume that
the forecast error is normally distributed FDi,t ∼N(0, σd
i,t)with mean 0 and standard
deviation σd
i,t. The zero mean is reasonable as the long-term error should not be biased
to any side. This assumptions holds for most statistical forecasting methods and is
also reasonable if the forecasts are made or at least corrected by human planners, as
they should neither over nor underestimate the demand on the long-term average. The
actual measure of demand uncertainty is the standard deviation σd
i,t, which indicates
how the forecast errors are distributed around their zero mean. If an item faces primary
demand, we denote the corresponding forecast error standard deviation with σdext
i,t .
With the expected demands di,t and a model of their variation, the actual demand
in a single period t∈ T can be considered a random variable Di,t, whose realisations
5.1 A Model to Assess the Effects of Assortment Complexity Changes 83
consist of the deterministic expected demand di,t and an error distributed according
to FDi,t. Consequently, Di,t also follows a normal distribution Di,t ∼N(di,t, σd
i,t)with
mean di,t and standard deviation σd
i,t. It is important to note that this model does
not assume that all Di,t are identically distributed for different t. The actual demand
is normally distributed within in each period, which is why the corresponding random
variable Di,t is indexed over the periods t∈ T. The sequence of demands for one item
iover all periods is therefore a stochastic process Dtwith parameter t=(1, . . . , T),
rather than a single random variable that follows one particular distribution.
Production and distribution network model With all the model elements defined in the
preceding sections, we can define the production and distribution network model as
PDN(N,V,S)7. Figure 5.1 shows a small example of the structure of a PDN.
s1s3
s2
Period
Demand
External market
demand
(m1,l1)
Period
Quantity
Required production
quantities
Production DistributionProcurement
NPROC NPROD NDIST
...
...
...
...
Production process steps
(m2,l1)
(m3,l1)
(m4,l1)
(m5,l1)
(m6,l1)
(m7,l1)
(m8,l1)
(m9,l1)
(m10,l1)
(m8,l2)
(m8,l3)
(m9,l4)
(m9,l5)
(m9,l3)
(m10,l2)
Transportation p. steps
s4s5
...
Figure 5.1: Production and distribution network example
This simplified example shows the production and distribution structure for 3 end prod-
ucts {m8, m9, m10}which are all produced at the production site l1and distributed to
7 We shall use PDN(·)as a shorthand for this full notation.
84 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
the sales locations l2, l3, l4and l5. The two last mentioned locations are not supplied
directly from the production site but procure m9via l3as a transhipment point. All
items are either procurement, production or distribution items. All demands originate
from the external market demands at the sales locations and are propagated to all
items of the network. Each production item is assigned to one production process step
(e.g. S(m6,l1)={s1}), while each distribution items is assigned to one transport process
step (e.g. S(m8,l3)=S(m9,l3)={s4}). For one particular production process step like s1,
the required production quantities for each period can be calculated by aggregating the
demands of all related items Ns1={(m5, l1),(m6, l1)}. In this example the set of stock-
point is selected to be SP ={(m1, l1),(m2, l1),(m6, l1),(m9, l4),(m9, l5),(m10, l2)}.
5.1.1.2 Model Generation
Considering the intended usage of the model, it is obvious that the representation of
realistic assortments results in large networks that easily contain hundreds or thousands
of nodes. For its practical application it is desirable that model generation is at least
semi-automatic. The widespread use of ERP systems makes it reasonable to use the
available data to generate the network structures with less effort. This section shows
how parts of the PDN models can be built from data available in standard ERP
systems. We firstly show how the network structure can be generated and secondly how
demand and forecast deviation data can be calculated consistently over the network.
Network structure The following list defines the required input data that can be ob-
tained from most ERP systems.
Bills of material Each item i∈ NP ROD that is produced at location l∈ L has a
BOM Bithat defines a set of a of input components and the respective quantities
required of each component jto produce one basic unit of i. A BOM entry b∈ Bi
thus is a tuple (c1, q1)defining that c1∈ M is a required input component for i
and that q1basic units of c1are required to produce one basic unit of i.
Distribution relations Each material i∈ NDIST defines a location src(i) = l∈ L as
its procurement source. This may be interpreted as the production or distribution
site that supplies i’s location loc(i)with the respective material mat(i). Such
distribution relations are usually part of the masterdata maintained for each
material to allow the automatic generation of purchase proposals by the ERP
system.
5.1 A Model to Assess the Effects of Assortment Complexity Changes 85
Routings A routing for i∈ NP ROD defines a set of operations carried out on different
work centres to produce the respective material.
As described in Algorithm 5.1, the structure of the production and distribution net-
work can be derived from the documents mentioned above. As input parameters, the
algorithm requires a set of starting materials M∗⊂ M that contains only end prod-
ucts and thus finished materials. The selection of this set determines for which part of
the assortment the PDN is built. For practical applications this allows invocation of
the generation of an assortment model e.g. for a certain product category by simply
providing a list of the corresponding end product material numbers as an input.
Furthermore, it requires that all processes, i.e. production and transport process steps,
are defined a priori. A production process step is not necessarily a single physical
production resource and thus may comprise several work centres. If the routings
from an ERP system are used to map the items to the production process steps, the
production process steps have to be defined such that each s∈ SP ROD comprises all
work centres required by the operations in each routing for an item iif s∈ Si. This
implies that for each routing of i, there is at most one s∈ SP ROD assigned to i. This
correspondence of routings to production process steps guarantees that if multiple
production process steps are assigned to an item, they always represent alternative
routings and thus alternative production possibilities.
The set Ncontains all items that still have to be processed, i.e. added to the network
and possibly expanded if there are any supplying items or component predecessors to
be defined. Initially, this set is filled with all items that represent any material from
M∗at all the locations (lines 3 to 6) where it is produced, distributed to and / or
sold: N={i∈ N | mat(i)∈ M∗}. All items in this set are then iteratively processed
(line 7). Each item is added to the network and then handled differently depending
on the type of item encountered.
For distribution items (lines 10 to 18), a new item with the same material at the sup-
plying plant is added to the set of nodes and to the set of items to process if necessary.
The link between the two items is added with a weighting of one, as the materials
are not altered during the transport process. The process mapping is extended by the
mapping of ito the respective transport process.
For production items (lines 19 to 30), i’s BOM is exploded and for each of its compo-
nents, a new item with the material at the same production location is added to the
set of nodes and to the set of items to process if necessary. The corresponding links
86 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
Algorithm 5.1: Generate a production and distribution network
Input:M∗,S
Result:PDN(N,V,S)
N ← ∅1
N ← ∅2
foreach m∈ M∗do3
foreach location lwhere mis defined do4
N ← N ∪(m, l)5
6
while N 6=∅do7
select i∈ N8
N ← N ∪i9
if i is procured from a production location then10
j←(mat(i), src(i))11
if j /∈ N then12
N ← N ∪j13
N ← N ∪j14
V ← V ∪(j, i)15
w(j,i)←116
get s∈ ST RANS that represents transport from loc(j)to loc(i)17
pi,s ←118
if i is produced at l then19
foreach BOM entry (c, q)∈ Bido20
j←(c, l)21
if j /∈ N then22
N ← N ∪j23
N ← N ∪j24
V ← V ∪(j, i)25
w(j,i)←q26
foreach routing r defined for i do27
get s∈ SP ROD that comprises all work centres referenced in r28
pi,s ←129
30
if i is procured from a external supplier then31
// nothing to do
remove ifrom N32
33
5.1 A Model to Assess the Effects of Assortment Complexity Changes 87
between the component items and iare added and weighted with the production coef-
ficients from the corresponding BOM entry. Finally, the routings defined for iare used
to determine the set of production process steps to be assigned to ivia the indicators
pi,s.8
For procurement items (lines 31) no changes have to be made to the network since the
network is constructed upstream, i.e. for each item predecessors are determined and
the corresponding links are created. Procurement items do not have any predecessors
and are connected to their successors during the processing of these successors, which
by definition are either production or distribution items.
After an item has been processed, it is removed from the set of items to be processed
(line 32). In this way the set of items to be processed will eventually be empty, as each
successive explosion of BOMs will result in procurement items that are not further
exploded and will only be removed from N. This guarantees the termination of the
algorithm due to the loop’s exit condition N 6=∅.
Demands and forecast deviations In addition to the network structure, the demand and
forecast data of each item usually can be obtained from an ERP system. Clearly, all this
data can also be defined “manually” based on assumptions about the development of
customer demand and forecast accuracy for each period. However, as this requires huge
effort9, we propose an alternative method that uses historical data to automatically
derive these assumptions. It takes into account two practical considerations:
1. If demand information is to be based on historical data, it can only be based on
the primary demands. Dependent demands, which mainly occur for end products
at production locations and semi- as well as raw materials within the production
stages are influenced by historical planning decisions. Orders from sales locations
are aggregated to obtain full truck load transports, and the material flow in the
production area is influenced by lot-sizing decisions.
2. The same applies to information about forecast deviations. Forecast deviations
are only traced for those items that face primary demands, as the forecasts are
only made for these items and then propagated to all predecessors in the material
requirements planning process.
8 By definition there is only one production process step required to perform the operations defined
in one routing. See Section 5.1.1.1 for the definition of production process steps.
9 For example, consider a network with 500 items, for which an analysis should be made over a
period of one year. If we select months as the mid-term periods, a total of 500 ·12 = 6000 values
as demand estimations have to be defined.
88 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
Due to these observations, we propose a method to calculate demands and forecast
deviations only from information about primary demands dext
i,t and the related forecast
deviations. Both demand and forecast deviations are propagated upstream the net-
work, aggregating at each item the potential primary demand, where applicable, and
the dependent demands of its successors. With this logic, the di,t are calculated as
di,t =dext
i,t +X
j∈SC(i)
w(i,j)·dj,t (5.4)
While it may be reasonable to use historical demand data in terms of the expected
demands in each period, this is unreasonable for stochastic values like the forecast
deviations. As we consider the demand deviations FDi,t in each period to be separate
random variables, we cannot assume that these single distributions are known. More-
over, we want to derive them from an average forecast error observed over the entire
time horizon T. We therefore assume that a long-term measure FDmad
ifor the relative
mean absolute forecast deviation is available for each item that faces primary demand.
This measure is defined as the long-term average of
|actual primary demand in period t−primary demand forecasted in period (t−1)|
actual primary demand in period t(5.5)
This is a much more intuitive and commonly-used long-term measure of the forecast
deviations, as it expresses the long-term mean deviation as a percentage of demand.10
As defined in Section 5.1.1.1, the demand for an item iis represented as a series of
random variables Di,t ∼N(di,t, σd
i,t)over the periods t∈ T . As expected demand
can be calculated as described above, the remaining question is to derive the standard
deviations σd
i,t for a normally distributed forecast error from the relative mean absolute
error FDmad
i. For a normally distributed random variable X∼N(µ, σ), there is a
constant relation between the mean absolute deviation δ=Pi|xi−µ|of realisations
xiof Xfrom its mean and the standard deviation σ. Burrows and Talbot11 show
that
δ=σs2
π(5.6)
and consequently the standard deviation can be obtained from the mean absolute
deviation as
σ=δrπ
2(5.7)
10 For example, “forecasts deviate from the actual demand by FDmad
i= 30% on average”.
11 Burrows and Talbot (1985, p. 89).
5.1 A Model to Assess the Effects of Assortment Complexity Changes 89
Transferred to our model, we can now calculate the standard deviations of the forecast
errors on the basis of the relative mean absolute deviations. As the latter refer to
primary demands, we obtain the standard deviations of forecast errors for the external
market demands as
σdext
i,t =rπ
2·FDmad
i·dext
i,t (5.8)
similar to the demand calculation in Equation 5.4, we have to aggregate forecast devi-
ations from primary demands and dependent demands at an item. This aggregation is
conducted by forming the convolution of the relevant forecast deviation distributions.
For normally distributed random variables, the convolution equals the sum of these
variables.12 As the means of forecast deviation distributions are always 0, we only
have to consider the summation of the standard deviations related to the primary and
successors’ demands:
σd
i,t =v
u
u
tσdext
i,t 2+X
j∈SC(i)σd
j,t2(5.9)
As expected, Equation 5.9 yields σd
i,t =σdext
i,t if SC(i) = ∅, i.e. demand uncertainty
only originates from external market demands if an item has no successors.
Algorithm 5.2: Propagating demands and forecast deviations over a PDN
Input: a directed graph G(N,V)as part of a PDN
Result: updated graph Gwith demand and forecast information at each i∈ N
L←TopologicalSorting(G)1
Invert(L)2
foreach item iin Ldo3
with successors SC(i)4
begin5
foreach period t∈ T do6
calculate di,t according to Equation 5.47
calculate σd
i,t according to Equation 5.9
8
Di,t ←N(di,t, σd
i,t)
9
10
end11
12
Algorithm 5.2 shows the propagation of demands and forecast deviations in a PDN
via the iteration over the items of the graph G(N,V). The most important operations
12 The sum Y=X1+X2of two normally distributed random variables X1and X2with means µ1
and µ2and standard deviations σ1and σ2is Y∼N(µ1+µ2,pσ2
1+σ2
2).
90 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
are the topological sorting13 and the inversion of the result list in lines 1 and 2, which
guarantees that no item is processed before the calculations are finished for all its suc-
cessors. This is important as the calculation of the σd
i,t for an item irequires that the
σd
j,t have already been calculated for all j∈SC(i)(compare Equation 5.9). Note that
the results of these calculations are only determined by the external primary demands
dext
i,t and the relative mean absolute forecast deviations FDmad
iobserved for these de-
mands. This implies that Algorithm 5.2 can also be used to update the demands and
forecast deviations at all items if primary demands change, a feature we can also use
for the scenario generation, as described in the Section 5.1.2.2.
5.1.2 Alternative Assortment Scenarios
An assortment scenario (or scenario) is an assortment model that is derived from
an existing assortment model (called baseline model) by changing the assortment is
represents. All changes are made via one of the operations
1. Adding new materials and items
2. Replacing and / or discontinuing existing materials
3. Changing demand and forecast deviation information.
Operation 3 is trivial and not described in more detail, as it only changes some pa-
rameters in the model. This section describes how scenarios based on changes with
operations 1 and 2 are defined and applied to a baseline model.
5.1.2.1 Definition of Alternative Assortment Scenarios
The changes made to the assortment of the baseline model are encoded in a scenario
definition, which consists of
•A: a set of new item additions each describing a possibly new material at a
respective production or sales location
•Rfin: a set of material replacement definitions for end products
•Rrs: a set of material replacement definitions for raw and semi-finished materials
13 A topological sorting of a directed acyclic graph is a linear ordering of its nodes in which each
node comes before all its successors in the graph. Algorithms to compute such orderings are well
known and described e.g. by Cormen et al. (2001, pp. 549–551).
5.1 A Model to Assess the Effects of Assortment Complexity Changes 91
The set Acontains item additions, each of which describes an item that is to be
added to the network. If the item refers to a material that does not yet exist in the
assortment, it first has to be added at a production location and be integrated into the
product structure and production process steps. If the corresponding material already
exists, the addition defines that it is to be distributed to a new location and the item
has to be integrated into the distribution structure. The set of all materials that
are added via the elements of Ais denoted Mnew. In general, each material addition
a=m, lm,B(m,lm), rm, src((m, lm)), dext
(m,lm), FDmad
(m,lm)∈ Acontains the definition of
•m: the material to be added to the assortment
•lm: the production or distribution location where the material is to be added
•B(m,lm): a bill of material if lmis a production location, otherwise not specified
•rm: a routing if lmis a production location, otherwise not specified
•src((m, lm)): a procurement source if lmis a sales location, otherwise not specified
•dext
(m,lm),t: primary demands for mat location lmfor each t∈ T
•FDmad
(m,lm): relative mean absolute forecast deviation for mat lm
For the definition we have to distinguish between new materials at production and
sales locations. For production locations, a BOM has to be specified and the primary
demands and related forecast deviations are likely all to be 0. New materials at sales
locations require the definition of a procurement source and will probably have positive
demands and forecast deviations. The definition of material additions at sales locations
is only required if the material really is an additional material in the assortment and
does not serve as a replacement for any other material. In the latter case, it is sufficient
to define one material addition for the new material at the production location and
define appropriate material replacements that affect the products at the sales locations,
as described below.
The set Rfin ⊂ M×(M∪Mnew)n∪◦×Rn∪◦×Rn∪◦ contains material replacement
definitions for end products, each of which specifies how a finished material should
be partly replaced and possibly discontinued in the scenario. Such a replacement
definition r=mr, Mrep
r, Dratio
r, CVris given via
•mr∈ M: finished material to be replaced and possibly discontinued
•Mrep
r: tuple of replacement materials
•Dratio
r: tuple of replacement material demand ratios
92 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
•CVr: tuple of replacement materials conversion factors
An end product can be discontinued without replacement, indicated as Mrep
r=◦, or
it can be replaced by n≥1replacement materials Mrep
r=mrepl
1, . . . , mrepl
n. The re-
placement materials can either be existing products or new products defined in Mnew,
which leads to Mrep
r∈(M∪Mnew)n∪ ◦. The placeholder element ◦indicates that
there is no replacement and the corresponding product is just removed from the as-
sortment. If n≥1, the tuple Dratio
r=dratio
1, . . . , dratio
nspecifies for each replacement
material, what percentage dratio
iof primary demand of mris transferred to each mrepl
i
and added to its primary demand. These percentages need not sum up to 1 (or 100%)
over all replacement materials. If Pidratio
i<1, the remaining demand ratio is inter-
preted as lost sales, which means that customers are expected to buy less of the new
material. Analogously, if Pidratio
i>1, the surplus is interpreted as additional demand
expected as a result of the replacement. The conversion factors CVr=(cv1, . . . , cvn)
specify how demand quantities of mrare converted to demand quantities of each mrep
i.
Such a conversion factor is 6= 1 if the finished products represented by these materials
mrand mrep
iare packed end products with different packaging sizes. This is relevant
both for products on the BCU and TSU packaging level.14
For end products we do not have to pose any restrictions on the number of replacement
definitions per material due to the structure of the subgraphs spanned by all direct
and indirect successors DN(i)of any end product item i. By definition, iis not
processed any further on any production process step and is only distributed to different
locations. Therefore the structure of the sub-network of all items in DN(i)is always
strictly divergent15 for any and end product item i, so that one item can be replaced
by multiple items without creating any ambiguities with respect to the material flows
represented by that subgraph. Nodes and links are duplicated according to number of
replacement definitions.
For all materials that are not end products, a substitution of one material by several
replacement materials causes ambiguities with respect to their use as input components
on a production process step. If such a material m1
rwere to be replaced with multiple
materials mrepl
r1, mrepl
r2, there would be no sound interpretation of how these materials
are used as input components for the production of other materials. For this reason, the
set of material replacement definitions Rrs ⊂ M×M∪Mnew ×Ris more restrictive
with respect to the definition of replacements for one material, as it requires that
14 For example, if the materials are both packed at TSU level and the material that is to be replaced is
a batch of 20 single BCUs, while the replacement material only contains 10 BCUs, the conversion
factor would be 2. See Section 2.2 for a description of different packaging levels.
15 A directed graph is called divergent if each node has at most one predecessor.
5.1 A Model to Assess the Effects of Assortment Complexity Changes 93
there is at most one replacement definition r=(mr, mrep
r, cvr)for one particular raw
or semi-finished material:
mr16=mr2∀r1, r2∈ Rrs (5.10)
In contrast to the case of end products, an indicator for discontinued materials (◦) is
not allowed and each replacement definition requires a specific replacement material
mrep
r∈ M∪Mnew. As demand for raw and semi-finished products is allways dependent
demand, no demand has to be transferred from mrto mrep
rand the dratio
rcan be
omitted. The conversion factor cvrdefines how production coefficients of the BOM
entries that specify mras the input component have to be adapted.
(m1,l1) (m1,l2)
(m2,l1) (m2,l3)
(m1,l1) (m1,l2)
(m2,l1)
Add new materials Replace finished materials Replace raw / semi-finished
materials
Initial situationScenario result Scenario
definition
(m6,l1)(m4,l1)(m2,l1)
(m1,l1) (m3,l1) (m5,l1)
(m6,l1)(m4,l1)(m2,l1)
(m1,l1) (m3,l1)(m5,l1)
(m2,l1)
(m1,l1)
(m1,l1)
(m2,l3)
(m2,l2)
Replace material
m1with m2
Replace material
m3with m4
Add new material
m2at location l1
Added item and link Removed item and link
Figure 5.2: Simple examples of scenario definition operations
Figure 5.2 illustrates a simple example of each of the operations that can be used to
define a scenario. For the sake of simplicity of illustration the depicted examples are
the simplest possible cases, with single replacements and simple distribution structures
only.
94 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
5.1.2.2 Application to a Baseline Model
To obtain a scenario, a scenario definition is applied to a baseline model. In order to
maintain the consistency of the model the sequence in which changes are applied to
the baseline model is important and must follow the steps:
1. Add the new materials as defined in A. Assure that those additions with new
materials at production locations are made first.
2. Replace end products as defined in Rfin
3. Replace raw and semi-finished products as defined in Rrs
4. Update all demands and forecast deviations by propagating external (primary)
demands and forecast deviations over the graph G(N,V)according to Algo-
rithm 5.2.
We define the algorithms for each of these steps in the next sections. Step 4 is an
exception, as it is already fully defined via Algorithm 5.2 in Section 5.1.1.2.
Adding new materials The addition of new items described in Algorithm 5.3 processes
each item addition and first creates the new item iand adds it to the network (lines 1
to 3). For production items, the corresponding BOM Biis exploded and the entries
are used to determine the predecessor items in the production network and to add
the required links (lines 5 to 8). For each routing defined for i, the corresponding
production process steps are determined and the mapping of ito these process steps
is created (lines 9 to 11). If iis a distribution item, the supplying predecessor is
determined and linked to the new item. The items in Aare ordered to process those
item additions at production locations first to ensure that the predecessors already exist
even if mat(i)∈ Mnew. On the basis of this distribution relation, the corresponding
transport process step is assigned to the new item (lines 13 to 16).
Algorithm 5.3 assumes that for new production items, both a BOM as well as at least
one routing are defined for each new material. For practical applications, the tedious
task of explicitly specifying all this information can be alleviated by deriving each new
material from a template material. All information required for the assignments of
predecessors and production process steps is copied from that material and adapted
as required. This approach is especially suitable if the assortment changes represent a
standardisation of several items to one generic variant. In this case, the new generic
5.1 A Model to Assess the Effects of Assortment Complexity Changes 95
Algorithm 5.3: Scenario generation: adding new materials
Input: a baseline model PDN(N,V,S), a set A
Result: the scenario model PDN∗(N,V,S)including the new items defined in A
foreach a∈ A do1
create item i= (m, lm)2
N ← N ∪i3
if lmis a production location then4
foreach BOM entry (c, q)∈ Bido5
j←(c, lr)6
V=V ∪(j, i)7
w(j,i)←q8
foreach routing rmdefined for ido9
select s∈ SP ROD that comprises all work centres referenced in rm
10
pi,s ←111
12
else13
j←(m, src(i))14
V ← V ∪(j, i)15
select s∈ ST RANS that represents transport from loc(j)to lm
16
pi,s ←117
18
19
variant can be defined as a new material with all information copied from that existing
variant that is most similar to it.
Discontinuing or replacing end products Each end product replacement defined in Rfin
specifies the replacement for one material and therefore affects all items that represent
that material at any location. For each replacement Algorithm 5.4 first processes all
items that represent the corresponding material at a sales location. For each such item
several replacements can be specified, for each of which the following operations are
performed (lines 6 to 19):
1. Add the replacement item and predecessor link to the network as required.
2. Update the demand information at the replacement item.
3. Update the forecast deviation information at the replacement item.
4. Remove the item and related links from the network.
96 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
Algorithm 5.4: Scenario generation: replacing end products
Input: a baseline model PDN(N,V,S), a set Rfin
Result: a scenario model PDN∗(N,V,S)
foreach r∈ Rfin do1
foreach Sales location lwhere mris defined do2
i←(mr, l)3
if Mrepl
r6=◦then4
for idx = 1,...n do5
j←(mrepl
idx , l)6
if j /∈ N then7
N ← N ∪j8
v←(PR(i), j)9
wv←110
V ← V ∪v11
foreach t∈ T do12
dext
j,t ←dext
i,t ·dratio
idx ·cvidx
13
FDmad
j←FDmad
i
14
else15
foreach t∈ T do16
dext
i,t ←dext
i,t ·dratio
r·cvr
17
dext
j,t ←dext
i,t +dext
j,t
18
FDmad
j←q(F Dmad
i·Pt∈T dext
i,t )2+(F Dmad
j·Pt∈T dext
j,t )2
Pt∈T (dext
i,t +dext
j,t )·
19
20
21
N ← N\{i}22
V ← V\{(k, i)∈ V|k∈PR(i)}23
foreach Production location lwhere mris defined do24
k←(mr, l)25
if |SC(k)|= 0 then26
RemoveObsoleteItems(k)27
28
29
30
Demands are calculated separately for each period t∈ T. The relative mean absolute
forecast deviations FDmad
iare used to aggregate forecast deviation information for
all periods. As there is a defined relationship between the single primary demand
distribution standard deviations σdext
tand the relative mean absolute forecast deviation
FDmad
i, it is sufficient to update the latter value. The single demand distribution
5.1 A Model to Assess the Effects of Assortment Complexity Changes 97
standard deviations σd
i,t can then be calculated according to Equations 5.8 and 5.9.
The algorithm distinguishes two cases, depending on whether the replacement item j
already exists in the network or not. If it does not exist, the replacement material
was not sold at that sales location before and the corresponding item jis added to
the network. A distribution link to this item is added, where the source of the link
is determined by the predecessor of iand thus the distribution structure of the old
product is maintained. If mrwas procured directly from the corresponding production
location, mrepl
ris procured from the same production location. The case where products
are procured via another sales location is handled analogously.16 The external demands
at the new item jdirectly result from the demands at i, adapted with the demand
rate and conversion factor. The mean relative forecast deviation in this case can be
taken from iwithout modifications as it is a percentage value that does not require
any quantity adaptations.
If the material was already sold at the sales location, no structural changes have to
be made to the network. However, the adaptation of demands and forecast deviations
now has to consolidate the information from both iand the existing replacement
item j. In the notation of the algorithm, dext
i,t holds the demand quantity that is
transferred from ito jin period t, considering demand ratios and conversion rates.
It is added to j’s previous demand to yield the new demand quantity dext
j,t . In order
to aggregate relative the relative mean absolute forecast deviations one has to take
into account that different demand quantities determine the degree to which a certain
forecast deviation influences the aggregate deviation. Thus FDmad
iand FDmad
jare
aggregated by weighting them with the relevant total demands Pt∈T dext
i,t and Pt∈T dext
j,t ,
respectively (line 19).
Procedure RemoveObsoleteItems(k)
foreach j∈PR(k)do1
if |SC(j) = 1|then2
RemoveObsoleteItems(j)3
4
N ← N\i5
V ← V\(j, i)6
After all replacements for an item ihave been finished, the item is removed from the
network, together with all links that have ias their target (lines 22 to 22). After
16 Note that as we consider distribution items here, they have only one predecessor by definition.
See Section 5.1.1.1 on page 76.
98 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
all items that represent the material of a given material replacement definition at a
sales location have been processed, the material can only be found at a production
location. As the material is no longer sold at any sales location, there is no dependent
demand at the production location and the item can be removed from the network.
If an end product is not produced any more, there may be some semi-finished, raw
and packaging materials that were only used for that product and consequently can
be removed from the assortment now. Therefore all direct and indirect predecessors
of the item in terms of production and procurement items can be removed as long as
they do not form input components of any other materials. To achieve this each such
item kis removed via the function RemoveObsoleteItems(k)if it does not have any
successors any more. This function then removes kfrom the network and is called
recursively for all predecessors j∈PR(k)if kis their only successor and thus they
are not input components to any other materials. The recursive invocation guarantees
that all unnecessary items are removed from the assortment.
Algorithm 5.6: Scenario generation: replacing raw and semi-finished products
Input: a baseline model PDN(N,V,S), a set Rrs
Result: a scenario model PDN∗(N,V,S)
foreach mr∈ Rrs do1
foreach production location lwhere mris defined do2
i←(mr, l)3
j←(mrep
r)4
foreach k∈SC(i)do5
V ← V ∪(j, k)6
w(j,k)←w(i,k)·cvr
7
V ← V\(i, k)8
RemoveObsoleteItems(i)9
10
11
Replacing materials in procurement and production stages The replacement of raw and
semi-finished products defined in Algorithm 5.6 is similar to the replacement of end
products described above. The main differences are simplifications that result from
the more restrictive material replacement definitions17 and the fact that no primary
demands exist at the replaced items. In particular, these differences are:
17 See Section 5.1.2.1.
5.1 A Model to Assess the Effects of Assortment Complexity Changes 99
1. No items have to be added to the network, as they either already exist or have
been added to the network during the addition of new materials.18
2. There is no processing of multiple replacement items per replacement definition.
3. Due to the absence of primary demands, no demands and forecast information
has to be transferred between the discontinued item and its replacement. This
information will result from the propagation of demands and forecast deviations
made when all structural changes have been applied.
For a production item ithat is to be replaced with an item j, all links that have ias
their source have to be replaced by links with jas their source. Thus any material that
had mras an input component now has mrep
rin that place. Changes in the quantities
required are considered via the conversion factor cvrand included by adapting the
corresponding link weights. Finally, the item iis removed from the network via the
RemoveObsoleteItems(i)function already introduced in the previous section, which
guarantees that all direct and indirect predecessors are also removed recursively.
5.1.3 Cost Assessment for a Given Assortment
Based on the model from Section 5.1.1.1, we can define the cost model used to assess
the cost incurred by a certain assortment, represented as a production and distribution
network. As described in Section 2.3, we distinguish inventory, scrap and setup cost
as well as cycle stock costs, such that the total cost Ccan be written as
C=CINV +CST SCRP +CCY ST (5.11)
Table 5.5 summarises the notation for the cost parameters used in this section.
Each item has a value Cithat represents the internal accounting value of the mate-
rial mat(i)at location loc(i). This value generally increases with each production or
distribution stage, due to the value added over these stages. For a production item
i∈ NP ROD,Ciis at least as large as the sum of values of its input components
Ci≥Pj∈P R(i)Cj·wj,i and generally greater than this, as the production process itself
adds more value to the material and the cost incurred by the production is reflected in
Ci. For distribution items i∈ NDIST the cost incurred by the transport is generally re-
flected in a value increase for the material at the target location Ci≥Cj∀j∈PR(i).
18 See Section 5.1.2.2: new items are added to the network at their corresponding production location
and connected to their predecessors via explosion of their BOM.
100 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
Table 5.5: Cost parameters
Symbol Description
Ciinternal accounting value for one basic unit of i
Cinv inventory holding cost rate to account for capital binding over one period t∈ T
Cwhsg
lwarehousing cost for one storage unit at location l∈ L over one period t∈ T
qppinumber of basic units of ithat can be stored on one storage unit at loc(i)
Cstp
i,s,pbsaverage setup cost incurred by production of ion production process step s
with a planning buffer of pbs
scrpi,s quantity of scrap produced during start and end of a production run of ion s
Inventory costs comprise inventory holding and warehousing costs. The inventory
holding cost rate Cinv is an interest rate used to express the opportunity cost incurred
by the capital tied up in inventory over one single period t∈ T . Warehousing costs are
incurred for the provision of physical space used to store the inventory. They are given
for the space required for one storage unit, typically a pallet in the case of consumer
goods. In order to break these costs down to one basic unit of a certain item, we denote
the number of basic units of ithat can be stored on one storage unit qppi. This value
is maintained on the item level rather than per material, as different locations may
use different types and sizes of storage units and thus these values may differ between
locations. These costs can be aggregated into a single total inventory cost rate Cinv
i
for one basic unit of iover one period t:
Cinv
i=Ci·Cinv +Cwhsg
loc(i)
qppi
(5.12)
Within a single period inventory levels are not constant. For our cost assessment, it is
sufficient to derive average inventory levels per period. According to the requirements
from Section 2.3, inventory costs are assessed on the basis of fixed safety stock levels and
reorder points per periods, which determine the average inventory level at an item iin
a given period t. The reorder point RPi,t is set so that at the moment a replenishment
order is placed, the remaining quantity on stock covers the lead time demand, i.e. the
demand over the replenishment lead time RLTi.19 Given that demands are given on
19 This implies that there are no outstanding orders at the time the net inventory is compared
with the reorder point. We thus assume that an item does not place new orders while there
are outstanding orders. If this assumption does not hold, the reorder point is compared to the
inventory position rather than the net inventory on hand. For the concept of inventory position,
see Zipkin (2000, pp. 30-32) or van Ryzin (2001).
5.1 A Model to Assess the Effects of Assortment Complexity Changes 101
the level mid-term periods, while replenishment lead times are measured in short-term
periods, it is given as
lead time demand of iin t=RLTi
TS·di,t (5.13)
If this lead time demand is uncertain, iis a stockpoint i∈ SP and holds an additional
quantity Iss
i,t of safety stock. The reorder point is given by the lead time demand plus
the defined safety stock.
RPi,t =RLTi
TS·di,t
| {z }
demand over RLTi
+Iss
i,t (5.14)
Figure 5.3 shows how the reorder point and safety stock level relate and determine the
average inventory level.
Inventory
level
Short-term periods
Mid-term periods
RPi,t
tt+1
RLTi
Ii,tSS
ti
Si
d
T
RLT
,
⋅
Figure 5.3: Reorder point, safety stock level and average inventory level
The cycle stock build to cover the expected demand over the replenishment lead time
is assumed to be half the demand quantity on average. The total inventory cost can
then be calculated on the basis of safety and average cycle stock as
CINV =X
i∈N X
t∈T RLTi
2·TS·di,t ·Cinv
i+X
i∈SP X
t∈T
Iss
i,t ·Cinv
i(5.15)
Setup and scrap cost are operational cost incurred during the execution of a concrete
production plan. Given that such an operational production plan is not known20,
estimations of average cost rates to be used in the assessment are required. Let scrpi
20 See the discussion of planned production quantities as a theoretical production plan on an aggre-
gate level in Section 5.1.1.1.
102 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
denote the average quantity of scrap produced during the start and end phase of each
production run of i. We do not consider additional scrap produced during the entire
production execution as it only represents a loss caused by technical characteristics
of the production process which cannot be influenced by any decisions related to the
distribution of planned production quantities or the assortment and are thus irrelevant
to this analysis.
As setup costs are generally sequence-dependent, a precise assessment is only possible
on the basis of a concrete production plan with sequence information. As we do
not assume any knowledge of production sequences within the periods, we base the
assessment of setup costs on values Cstp
i,s,pbsthat express the average setup cost for one
production run of ion production process step swith a planning buffer of pbs. Given
that planned production quantities Qi,s,t are defined for each item, production process
step and period, these average scrap quantities and setup costs per production run can
be used to calculate setup and scrap costs as
CST SCRP =X
s∈SP ROD X
i∈NsX
t∈T Cstp
i,s,pbs+scrpi·Ci·Xi,s,t (5.16)
where Xi,s,t is a binary indicator to control whether or not a production run is required
for the indicated combination of item, production process step and period, defined as
Xi,s,t =
1if Qi,s,t >0
0if Qi,s,t = 0
With this definition of Xi,s,t, we assume that if a positive planned production quantity
is determined for a certain period, the entire quantity is produced in a single batch.
While this is reasonable to keep setup and scrap costs within the periods low, there may
be reasons to split up these quantities into more than one batch within a period, which
would result in higher setup and scrap costs than those calculated in Equation 5.16.
One such reason are capacity bottlenecks, which lead to the inability or unwillingness
to block a certain production resource for the time required to produce the entire
planned production quantity. The cost function used here thus is conservative and
only incurs setup and scrap costs once per period. Alternatives are possible but are
highly application-dependent and therefore must be defined individually. Ideally, rules
can be defined to obtain more precise estimation of the number of production runs
required to produce a certain planned production quantity.
5.2 Inventory Allocation in Production and Distribution Networks 103
Following the traditional lot-sizing logic, it may be reasonable to produce some ma-
terials in advance, i.e. before the period in which the demand originates in order to
reduce setup and scrap costs. If planned production quantities do not match the de-
mand quantities exactly but exceed them in any period, cycle stock Ics
i,t is built up.
The costs for these cycle stocks are assessed analogously to the general inventory cost
in Equation 5.15, which yields
CCY ST =X
s∈SP ROD X
i∈NsX
t∈T
Ics
i,t−1+Ics
i,t
2·Cinv
i(5.17)
The evaluation of Equations 5.11 to 5.17 for an arbitrary baseline model or scenario
represents the ultimate goal of this work. To make this evaluation possible, the follow-
ing Sections 5.2 and 5.3 describe optimisation methods to set those parameters that
are not known in advance and have to be adapted for each model and scenario.
5.2 Inventory Allocation in Production and Distribution
Networks
This section presents an optimisation method that yields a near optimal inventory
allocation in terms of a set of stockpoints SP, replenishment lead times RLTi, corre-
sponding safety stock levels Iss
i,t and reorder points RPi,t on the basis of a given model
PDN(N,V,S). The method addresses all requirements described in Section 2.3.3.
5.2.1 Inventory Model and Optimisation Objective
The basic prerequisite to finding an optimal inventory allocation in a production and
distribution network is an inventory model that determines the required amount of
safety stocks and average inventory levels at a stockpoint depending on the inventory
positions and levels at the adjacent items. The key concept to connect the inventory
requirements at a stockpoint with the decisions about inventory placements on adjacent
items are the replenishment lead times.
As all items i∈ N have positive throughput times21 TTi, demand for these items
cannot in general be met immediately. Accordingly, each item iquotes a service time
STito its successors. This order lead time between adjacent items is the time required
21 See Section 5.1.1.1 for the definition of throughput times for different item types.
104 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
for ito fill orders from successor items (reaction time). We assume that there are no
fixed order cycles such that orders can be placed in any short-term period. For each
item the sum of the maximum service time of any predecessor plus the throughput
time of that item corresponds to its total replenishment lead time RLTi:
RLTi= max
j∈P R(i)STj+TTi(5.18)
At a given item iit takes replenishment lead time short-term periods from the place-
ment of a replenishment order until the ordered quantity is available for planning at
the respective location.22 This means that an item ihas to transfer its forecasted
demand volumes into fixed orders to all its predecessors at least RLTishort-term pe-
riods before the requirements date. This replenishment lead time thus corresponds to
a frozen forecast interval in which the available quantities at the end of this interval
cannot be influenced any more.
Given the short delivery times requested by customers, the service times of the final
distribution stages are generally not long enough to cover the total throughput times
of all upstream stages. It follows that there must be some items in the network that
cannot pass their replenishment lead time to their successor as the service time, which
results in a positive coverage time ∆ti=RLTi−STiat these items. Over this coverage
time, demand information is uncertain. Any production or distribution operations for
an item ican only be planned with fixed order quantities over time STi. In order to be
able to deliver in time STiif STi< RLTi, this item has to replenish its inventory upon
forecasts and buffer against demand uncertainty over the time interval ∆tiwith safety
stock. Figure 5.4 illustrates with a small example how replenishment lead, service and
coverage times interrelate.
Example 5.1 In the depicted example, n1is selected as a stockpoint and
covers its entire replenishment lead time with safety stock to ensure a
service time of STn1= 0. The replenishment lead time of the successor
item n2therefore consists only of its own throughput time, which is passed
as the service time STn2=RLTn2to the successors n3and n4. Item n3adds
its throughput time to that service time and again passes the sum further
22 Analogously to the approach of Simpson Jr. (1958) we assume the existence of operating flexibility
to ensure deterministic and constant replenishment lead times. This is a justifiable assumption
as it is practically irrelevant to what extent this flexibility actually exists. Without any demand
bound stocks would grow infinitely. Demand thus has to be bounded by some service level and
any excessive demand can either be handled by operating flexibility or leads to stockouts. If such
excessive demand originates from a single customer, other measures like agreements on longer
delivery times are also possible.
5.2 Inventory Allocation in Production and Distribution Networks 105
Inventory
coverage time
Inventory
coverage time
n1n2
n3
n4
Throughput time TTi
Service time STi
Maximum predecessor
service time
Coverage time ∆ti
Figure 5.4: Inventory allocation and times between adjacent items
downstream as its own service time STn3=STn2+TTn3. By contrast,
n4is selected as a stockpoint and covers n2’s service time plus its own
throughput time with safety stock to present a service time STn4= 0 to
its successors.
Now that the interrelations between the nodes can be expressed depending on the
selected inventory allocation, a model to determine the required safety stock levels
at a single item has to be defined. As demands for an item i∈ N may fluctuate
considerably over the periods T, we consider safety stock levels Iss
i,t for each single
mid-term period t∈ T. This is reasonable as we assume that these periods have
been chosen to represent the demand planning and forecasting periods23 and thus
the information required to determine safety stock levels is available at this level of
aggregation.
The determination of safety stock levels Iss
i,t has to consider the replenishment lead
time as well as the service time at an item. Depending on the length of the resulting
coverage time ∆ti, the safety stock levels have to be dimensioned for each period. The
reorder points RPi,t are incremented by this amount of safety stock.24 The safety stock
levels are measured as multiples of the demand standard deviation over the coverage
time in which demand is uncertain:25
Iss
i,t =zi,t ·σd
i,t ·s∆ti
TS(5.19)
23 See Section 5.1.1.1.
24 See Equation 5.14 on page 101.
25 Compare the models for single installation systems discussed in the state of the art in Section 3.2.1.
106 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
The task is to determine a value for zi,t that yields a safety stock level that guarantees
an average β-service level26 of βSL
i. In order to determine the proportion of demand
that cannot be met with a certain safety stock level, we note that actual demand in each
period Di,t is a random variable and use the partial expectation of the corresponding
probability distribution. The partial expectation E(X−x)+of a random variable X
with respect to a threshold value xis defined as the long run average of the positive
part of the difference X−x. It tells us how much, on average, the realisations of X
exceed the threshold value x.
Transferred to our problem, this enables us to evaluate what proportion of demand
cannot be met on average and what β-service level can be achieved with a given
safety stock level Iss
i,t and a given demand distribution over the replenishment lead
time RLTi. For this evaluation, we consider the partial expectation of the lead time
demand random variable with respect to the reorder point as the threshold value.
The reorder point RPi,t results from a certain safety stock level, replenishment lead
time and expected demand.27 Knowing the demand distribution Di,t for one mid-term
periodt28, the lead time demand for an arbitrary lead time interval within that period
can be expressed as a random variable Drlt
i,t , defined as
Drlt
i,t ∼N
di,t
RLTi
TS, σd
i,t ·s∆ti
TS
(5.20)
We use the partial expectation to calculate how much, on average, the lead time
demand exceeds the reorder point. As this is the amount of unmet demand, we can
set it in relation to the expected lead time demand to obtain the percentage of unmet
demand, which then relates to the β-service level as our target criterion as:
βSL
i= 1 −EDrlt
i,t −RPi,t
di,t ·RLTi
TS
(5.21)
The partial expectation is hard to compute for normally-distributed random variables
with arbitrary means and standard deviations. However, there are approximations as
well as tabular values for the case of a standard normal random variable Z∼N(0,1)
with zero mean and a standard deviation of one. The partial expectation E(Z−z)+
of such a standard normal variable is denoted L(z)and Lis called the standard loss
26 For the proportion of demand that can be met within a defined time interval, see Section 3.2.1.
27 See Equation 5.14.
28 See the definition of demand distributions Di,t in Section 5.1.1.1.
5.2 Inventory Allocation in Production and Distribution Networks 107
function. The partial expectation of any normally distributed variable X∼N(µ, σ)
can be expressed via this standard loss function as29
E(X−x)+=σL(z)with z=x−µ
σ(5.22)
If we express the safety stock level Iss
i,t via the reorder point and the lead time demand
(see Equation 5.14), we can rearrange Equation 5.19 to
zi,t =RPi,t −di,t ·RLTi
TS
σd
i,tq∆ti
TS
(5.23)
This shows that zi,t fits into the normalisation scheme shown in Equation 5.22 and
thus Equation 5.21 can be rewritten as
βSL
i= 1 −σd
i,t ·q∆ti
TS·L(zi,t)
di,t ·RLTi
TS
,(5.24)
With the partial expectation, the numerator represents the expected amount of unmet
demand, while the denominator represents the expected demand volume over the con-
sidered time period. Subtracting this ratio from 1 equals the desired β-service level.
Rearranging this equation yields
L(zi,t) = (1 −βSL
i)di,t RLTi
TS
σd
i,tq∆ti
TS
(5.25)
This equation gives a relation between the parameters service level, replenishment
lead time, expected demand, demand variation and the standard loss function value
of the corresponding safety factor zi,t. To obtain the value for zi,t, we have to inverse
the standard loss function. However, as tabular values for L(·)are available, the
corresponding zi,t values can easily be looked up and we do not go into detail of the
computation of this function and its inverse. With a method to calculate the zi,t values,
Equation 5.19 can be used to calculate the required safety stock levels Iss
i,t. We can
now write the total inventory cost equation used in the cost assessment as a function
of the coverage times at each stockpoint:
CINV
i(∆ti) = X
t∈T
RLTi
2·TS·di,t +zi,t ·σd
i,t ·s∆ti
TS
·Cinv
i(5.26)
29 For a proof of this equality, see van Ryzin (2001, p. 12).
108 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
The complete inventory allocation problem can now be formulated as
min X
i∈N X
t∈T
RLTi
2·TS·di,t +zi,t ·σd
i,t ·s∆ti
TS
·Cinv
i(5.27a)
s.t. RLTi≥max
j∈P R(i)STj+TTi∀i∈ N (5.27b)
STi≤STmax
i∀i∈ N (5.27c)
STi≤RLTi∀i∈ N (5.27d)
STi≥0∀i∈ N (5.27e)
The objective function seeks to minimise the total cost incurred by the average inven-
tory held by all nodes over all time periods. It corresponds to the sum of the inventory
costs according to Equation 5.26 over all items. As Equation 5.26 is a function of the
coverage times ∆ti, the actual decision variables in the model are the service times
STi, which determine the values of the ∆ti. It is notable that the required customer
service levels do not appear in any restriction, as they are already used to determine
the safety factors zi,t in the objective function according to Equation 5.25.
Restriction 5.27b assures that the replenishment lead times are defined according to
Equation 5.18. For all items that represent end products and thus are shipped to
customers, this service time cannot be decided on independently but is bounded by
service agreements with these customers. Therefore the service time of each last-stage
distribution item imust not be greater than STmax
iand thus is bounded accordingly
by restriction 5.27c. For items where there is no STmax
idefined, this restriction may be
omitted or STmax
imay be set to a sufficiently large value. Restrictions 5.27d and 5.27e
require the values of STito be positive and at most as long as i’s replenishment lead
time.
A closer analysis of this optimisation problem shows that it is non-linear. Both the
safety stock factors zi,t and the coverage times ∆tidepend on the service times STi.
Since the STiare decision variables and the safety stock factors and the square-root
term with the coverage times are multiplied in the objective function, the problem
is non-linear. The revision of the state of the art in Section 3.2.3 showed that for
this type of inventory model, the extreme point property holds and can be used to
transform this problem into a combinatorial optimisation problem. The solution space
grows exponentially with the problem size, i.e. the number of items. Therefore we
cannot expect to find an efficient exact solution algorithm and must consider a heuris-
tic approach. In the next section we first discuss the use of domain knowledge to
5.2 Inventory Allocation in Production and Distribution Networks 109
find good inventory allocations and then present a concrete heuristic based on these
considerations in Section 5.2.3.
5.2.2 Domain Knowledge for Heuristic Inventory Allocation
An important consideration for the development of any heuristic is the integration of as
much problem-specific domain knowledge as possible in order to avoid the evaluation
of practically irrelevant or infeasible solutions during the optimisation process. For
the given problem, there are several such considerations. From the inventory model
presented in the previous section, it results that some items are better suited as stock-
points than others. Here, suited means that they contribute to covering throughput
times with inventory levels that incur comparably little cost. Comparably means com-
pared with other items that could be used to cover part of or the same throughput
time.
The aim of incorporating domain knowledge is to identify these items in advance, i.e.
without testing all possibilities to determine those favourable. The domain knowledge
is used via a set of quantitative criteria that can be divided into the two categories
•item characteristics and
•network structure.
For each particular criterion, a key indicator for the eligibility of the considered item as
a stockpoint with respect to this criterion is derived. A key indicator is a measurable
value that contains aggregated information and is used to put different values into
relation with each other for purposes of comparison. Finally, all these key indicators
are aggregated to obtain a single measure of an item’s stockpoint eligibility, which can
be used in a heuristic for inventory allocation. With the aim to have one standardised
measure of stockpoint eligibility, all key indicators have to be defined on the common
interval [0,1] and are dimensionless.
5.2.2.1 Item Characteristics
The characteristics of the demand for a certain item mainly determine the total inven-
tory requirements and especially safety stock requirements in case the item is chosen
to be a stockpoint. Accordingly, we base the assessment of an item’s eligibility as a
stockpoint on the following characteristics of the demand process:
110 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
•demand volume
•demand variation over time
•forecast deviation
For each of these criteria, the corresponding information already exists in our model.
In order to derive a normalised key indicator on the interval [0,1], we have to put these
values of each item into relation with the values of other items. Here it is not always
reasonable to compare the values of one particular item with the values of all other
items in the network, as the following problems may occur:
•For demand volumes, the units of measure may differ and thus quantities are not
directly comparable. Comparing valuated quantities, i.e. demand volumes mea-
sured in a monetary unit, does not help either, as materials on later production
stages naturally have higher volumes and thus bias the comparison.
•A single model may contain products and related materials from different product
categories and ranges, for which demand quantities and values are not compara-
ble.
To circumvent these problems, we define a partition30 Π = {M1, . . . , Mn}over the set
of items Nand assess each item i∈Mkonly relative to all other items in Mk. The
definition of such a partition must follow a reasonable logic to ensure that items within
each subset can be compared unbiasedly. Possible approaches to define the subsets Mk
are:
•Group production items that are processed on the same production process step:
Π = nNs|s∈ SP RODo.
•Group procurement items according to the type of material, e.g. raw materials
and packaging materials.
•Group distribution items according to product range or category of the end
products they represent.
•Generally group items according to their basic unit of measure, such that all
items in one Mk∈Πare comparable with respect to their quantities.
30 A partition P={S1, . . . , Sn}of a set Sis a family of subsets of Sthat are mutually exclusive and
jointly exhaustive. That is, no element of Sis present in more than one of the subsets, and all the
subsets together contain all the members of the original set: Si=1...n Si=S∧Si∩Sj=∅for i6=j.
5.2 Inventory Allocation in Production and Distribution Networks 111
With these subsets, we can derive normalised key indicators on the interval [0,1] for
each criterion by calculating the ratio of the value of the considered item i∈Mkand a
reference value from all items in Mk. These reference values are maximum or minimum
values, depending on the definition of the key indicator. However, this approach has
to consider potential outliers in the comparison. As the model data may be derived
practically from historical data, there may be items with exceptionally high or low key
indicator values, e.g. high demand quantities due to promotions or forecast deviations
of 100% for items that have not been forecasted at all. If such an item is present in
one subset Mk, all other items receive particular low ratings. Thus reference values
are adjusted by taking a high percentile ≤100, e.g. the 95th percentile of the relevant
value.31 This approach is general enough to allow the use of the maximum (or minimum
value) by taking the 100th percentile. At the same time it bears the risk that values
>1become possible for those items that do not fall below the percentile boundary.
For these items, stockpoint eligibility for the corresponding criterion is bounded by
definition to 1.
The first criterion is the total expected demand quantity over the periods T, denoted
dΣ
i=Pt∈T di,t. Items with high demands are considered more eligible as stockpoints
than items with lower demands. This rationale is also commonly used in approaches
to deciding about inventory strategies based on ABC-analyses.32 With Pdem
Mkdenoting
the chosen percentile of the values {dΣ
j|j∈Mk}we measure the stockpoint eligibility
of item iwith respect to the demand quantity as
SEdem
i= min
dP
i
Pdem
Mk
,1
i∈Mk(5.28)
The second criterion is the variation of the demand distribution over the time period.
As the corresponding key indicator, we use the sample coefficient of variation as the
common measure of variation of a time series.33 The sample coefficient of variation for
31 The Nth percentile of a set of values is defined as the value below which Npercent of observa-
tions fall. So here e.g. the 95th percentile of the demand volumes would be the smallest value
below which 95% of all observed demand quantities may be found. Percentiles are often used in
descriptive statistics as a means of avoiding the inclusion of outliers in an analysis.
32 See Tempelmeier (2003, pp. 31-33).
33 The sample coefficient of variation is a measure of the volatility of a time series, defined as the
ratio of the sample standard deviation over the sample mean:
CV (x1, . . . , xn) = qn−1Pi(xi−x)2
x
with x=n−1Pixi(Brockwell and Davis, 1991, p. 212).
112 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
the expected demand over the time periods is defined as
CV dem
i=rT−1Pt∈T di,t −di2
di
(5.29)
where di=T−1Pt∈T di,t. Analogous to the definitions above, Pdemvol
Mkdenotes the
chosen percentile of the values {CV dem
j|j∈Mk}. As items with smaller demand
volatility are better suited as stockpoints than those with high demand volatility, the
corresponding key indicator is defined as the distance to 1 and therefore has to be
bound to 0 for those items that do not fall below the percentile boundary:
SEdemvar
i= max 1−CV dem
i
Pdemvol
Mk
,0!i∈Mk(5.30)
The third criterion is forecast uncertainty as measured by the relative mean absolute
forecast deviations FDmad
i. The better demand of an item can be forecasted, the
smaller the safety stock requirements of that item in case it is chosen as a stockpoint,
which indicates that it is comparatively suitable as a stockpoint in the sense defined
above. With Pfd
Mkdefining the chosen percentile of the values {FDmad
j|j∈Mk}, we
can define the corresponding key indicator as
SEfd
i= min
FDmad
i
Pfd
Mk
,1
i∈Mk(5.31)
The criteria presented above are interdependent. It can generally be expected that
there is correlation between their values, as
1. items with high demand volumes generally have less volatile demand distributions
and
2. items with stable demand can be forecast with greater precision and have smaller
relative mean absolute forecast deviations.
In summary, the first two criteria are expected to determine the value of the third
criterion, which raises the question why the consideration of forecast deviations is not
sufficient. We argue that all three criteria have to be evaluated and considered, as
there are reasonable exceptions to the relations described above. Sporadic demand
that occurs e.g. for promotion items is interpreted as volatile, but may be forecast
very precisely due to fixed promotion quantities. Furthermore, demand quantities and
volatility are purely external factors, while the forecast deviations result from past
5.2 Inventory Allocation in Production and Distribution Networks 113
planning decisions if working with historical data. Thus relying only on the forecast
deviations may lead to wrong assumptions about the eligibility of that item due to
particular good or bad forecasts in the past. Accordingly, only the combination of all
three factors provides a strong indicator of stockpoint eligibility, as it identifies items
with high and steady demands that can be forecast with high precision.
5.2.2.2 Network Structure
The second class of criteria is derived from the structure of the assortment and the
distribution relations. Considering a single item, we analyse
•the number of direct successors and end products that the corresponding material
goes into,
•the number of direct predecessors and
•whether there is only a single option for further use of that item.
Items of high commonality are more eligible as stockpoints than items of low com-
monality. As described in Section 3.2.4, item commonality plays an important role in
inventory allocation, as external demand uncertainties can be pooled and thus reduced
by distributing finished products to multiple locations or using raw and semi-finished
materials as input components for production of different materials. These effects have
already been quantified in Equation 5.9, which shows how the standard deviations of
the demand distributions at an item result from the aggregation of the deviations of
all successor items. Another reason is that multiple successor items are dependent on
the service time of the considered item, which means that holding inventory at this
single central location can reduce the service time to a large number of successor items
at comparatively low cost. In the key indicator for this concept, we consider two types
of item commonality:
•Item commonality in the strict sense of the number of successors |SC(i)|of an
item i.
•Item commonality in the wider sense of the number of distribution items rep-
resenting end products that an item is related to. Each item in the network is
connected, directly of indirectly, to a set of ≥1items that represent end products
at sales locations.34 We denote the set of these end items to which a particular
item iis related LS(i).
34 Otherwise, the item could be removed from the network as it is not required for any end product
and thus cannot face any dependent demands.
114 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
When considering network structures, there is no comparison bias due to outliers
as in the case of the demand characteristics. Thus we can safely use the minimum
and maximum values from the reference set to calculate the key indicators, which
for item commonality is calculated from the indicators for the two different types of
commonality as
SEcomm
i=1
2 |SC(i)|
maxj∈Mk|SC(j)|+|LS(i)|
maxj∈Mk|LS(j)|!i∈Mk(5.32)
Analogous considerations can be made for the number of predecessors of an item.
Placing inventory at an item with a high number of predecessors makes the item
less dependent on the material availability of all the supplying items. Accordingly,
stockpoint eligibility is also evaluated based on the number of predecessors of an item
ias
SEpred
i=|PR(i)|
maxj∈Mk|PR(j)|i∈Mk(5.33)
Apart from the advantages of inventory placements at item with many predecessors
or successors, we also consider single usage structures, where an item ihas only one
successor j.
In this case, the downstream item jshould be considered a preferred stockpoint, as
the absolute inventory requirements are not higher for that item. This can be assured
because demand quantities and forecast deviations are directly propagated from jto
i.35 The only causes that might make inventory at imore economical are increased
inventory costs at j. These increased costs may result either from higher inventory
holding costs due to the value added during a corresponding production stage, or from
higher warehousing cost rates at loc(j)if the link between the two items represents a
distribution relation. In either case, the increased inventory costs may outweigh the
savings generated by reduced replenishment lead times of items further downstream.
In contrast to all the criteria described above, this consideration is not a soft criterion
to be expressed in an eligibility rating, but gives a direct hint to try to move inventory
positions downstream if there is only a single usage relation.
5.2.2.3 Aggregation to a Single Indicator of Stockpoint Eligibility
The eligibility of an item as a possible stockpoint depends on all the criteria expressed
via the corresponding key indicators described in the preceding sections that have
35 Compare Section 5.1.1.2.
5.2 Inventory Allocation in Production and Distribution Networks 115
to be combined into a single metric. As all key indicators are already dimensionless
and normalised to [0,1], we can use the straightforward approach of weighting each
individual key indicator and using the weighted sum as the overall key indicator SEi
for the stockpoint eligibility of an item i:
SEi=wdemSEdem
i+wdemvarSEdemvar
i+wfdSEfd
i+wcommSEcomm
i+wpredSEpred
i(5.34)
where the weights must sum up to one wdem +wdemvar +wfd +wcomm +wpred = 1 to
ensure SEi∈[0,1].
5.2.3 A Tabu Search Heuristic for Inventory Allocation
Tabu search is a proven neighbourhood search meta-heuristic36 that gained popularity
in recent years. Neighbourhood search heuristics define a neighbourhood as the set of
solutions that can be reached by applying a single move or operation to the current
solution. These moves usually consist of replacing a certain number of elements in the
current solution with new ones. Tabu search seems especially suited to the problem
at hand as it is well proven in general and has already yielded promising results in
its application to similar problems.37 Moreover, the possibility of defining moves on
a given solution to improve it makes it easy to incorporate the domain knowledge
described above into the optimisation process.
Algorithm 5.7 shows the tabu search38 implementation proposed for this problem.
A solution is represented by a set of stockpoints SP. The current best solution is
denoted SP∗and is set to be the initial solution when the optimisation starts. The
most important step of each iteration is the generation of the moves based on the
current solution SP (lines 4 to 6). The definition of the three neighbourhoods used
here is discussed in detail in Section 5.2.3.2. From the set M(SP)of all generated
moves that are not tabu at that moment, the algorithm selects the move mthat yields
the best objective value. The objective values are evaluated by a function f(SP, m),
which gives the objective value of the solution SP after move mhas been applied to
it. If the solution in the current iteration is better than the best solution found so far,
the latter is updated. At the end of an iteration, the selected solution is set as the new
current solution for the next iteration and the tabu list is updated. This update adds
36 For the definition and description of meta-heuristics, see Section 3.2.3.
37 See Section 3.2.3.
38 For a more detailed description of the tabu search approach, see e.g. Glover (1989, 1990).
116 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
Algorithm 5.7: Inventory allocation tabu search heuristic
Input: a model PDN(N,V,S), an initial solution SP
Result: a near optimal set of stockpoints SP∗
TL ← ∅1
SP∗← SP2
while stopping criterion is not met do3
generate moves M1(SP)based on stockpoint eligibility ratings4
generate moves M2(SP)moving inventory downstream where appropriate5
generate moves M3(SP)by randomly switching the stockpoint status6
M(SP)← {M1(SP)∪M2(SP)∪M3(SP)}\TL7
select m∈M(SP)as m= arg minm∈M(SP)nf(SP, m)o
8
SP ← apply move mto SP9
if f(SP)< f(SP∗)then10
SP∗← SP11
SP ← SP12
update TL13
return SP∗
14
the selected move mto the tabu list and removes all moves that have been in the tabu
list for the length of the tabu tenure.
Further implementation details which are not of conceptual interest are omitted here
and described in the validation Chapter 6. Such details include the selection of an
appropriate stopping criterion, the length of the tabu list and the usage of an aspiration
criterion.
5.2.3.1 Definition of an Initial Solution
Algorithm 5.7 requires an initial solution SP, i.e. an initial set of stockpoints as an
input parameter. There are different ways to define such an initial solution:
Random selection For each item, decide randomly if it is a stockpoint in the initial
solution or not. The ratio of stockpoints to the total number of items can be
controlled via the probability that an item is decided to be a stockpoint. This is
an advisable strategy if no assumptions about favourable inventory distributions
can be made. The configuration via the probabilities also allows us to obtain
extreme cases where there are no stockpoints at all or all items are stockpoints
initially.
5.2 Inventory Allocation in Production and Distribution Networks 117
Last-stage nodes only Make all finished products at sales locations stockpoints. This
guarantees service times of 0 at all sales location items and immediate fulfilment
of customer orders. The optimisation procedure then searches for possibilities
to reduce total inventory cost by introducing additional stockpoints at upstream
stages. This is an advisable strategy if material commonality in the assortment
represented by the model is very low and therefore inventories are primarily held
at the end product level.
Based on stockpoint eligibilities Use the calculated stockpoint eligibilities to deter-
mine the stockpoints in the initial solution. This can either be done by definition
of a threshold value above which items are set to be stockpoints, or by defini-
tion of a ratio of items that should be stockpoints in the initial solution. In the
latter case, the required number of stockpoints with highest eligibility ratings
are selected. This strategy incorporates most of the domain knowledge and pro-
vides good starting solutions. However, it also risks quickly getting stuck in local
optima if the whole optimisation process also relies on the eligibility ratings.
While many more alternatives are possible, only these three strategies are used in this
work, as they already provide the possibility to define many different starting solutions
via their parameterisation.
5.2.3.2 Definition of Moves and Neighbourhoods
In each iteration, the neighbourhood comprises all solutions that can be reached by
applying a move m∈M(SP)to the current solution. For our application, a move
should be able to represent the following operations:
•add inventories at a stockpoint
•remove inventories from existing stockpoints
•move inventories from one stockpoint to another
In the formal representation, each move m= (SP+, SP−)consists of a set of items
SP+that are made stockpoints in the new solution, and a set of items SP−that are
made non-stockpoints in the new solution. In this way, a move can express an arbitrary
change to the solution, including the above-mentioned operations. A move is applied
to a solution by updating the set of stockpoints as
SP ← nSP ∪SP+o\SP−(5.35)
118 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
One effective means of reducing the search space is to fix the stockpoint status of
items where possible. For example, there may be items for which inventories are
always required, as service times of zero to customers must be guaranteed. On the
other hand, there may be items where keeping inventory is impossible due to technical
characteristics of the production processes that impede the installation of physical
buffers for these items. This may be the case in highly-automated production systems
where the output of one production process step is directly passed to the next step.
For the sake of simplicity in the description of the neighbourhoods, we do not explicitly
consider these fixed stockpoint status, as they do not present any conceptual problem.
If such status is maintained for each item, the optimisation procedure has only to
ensure that these items are excluded during the generation of moves so that their
stockpoint status is never changed.
The set of available moves consists of distinct subsets that pursue different strate-
gies. The moves in M1are based on the domain knowledge encoded in the stockpoint
eligibility ratings described in Section 5.2.2.3. It contains moves that either
•make an item a stockpoint that is a non-stockpoint in the current solution and
has a high stockpoint eligibility rating, or
•make an item a non-stockpoint that is a stockpoint in the current solution and
has a low stockpoint eligibility rating.
To define the set of moves, we consider a list of all items non-stockpoint items N\SP
in descending order of their stockpoint eligibility ratings. We denote the position of
an item iin that list as r+(i). Analogously, we denote the position of an item in the
list of all stockpoints SP in ascending order of their stockpoint eligibility ratings as
r−(i). We can then define the first set of moves as M1=M+
1∪M−
1with
M+
1=nm= (SP+, SP−)|SP −=∅, SP+={i} ⇔ i /∈ SP ∧r+(i)> r+(j)−m1∀j∈ N\SPo
M−
1=nm= (SP+, SP−)|SP −=∅, SP+={i} ⇔ i∈ SP ∧r−(i)> r−(j)−m1∀j∈ SPo
This definition yields moves that select the most eligible stockpoints and deselect the
most ineligible stockpoints under consideration of their current stockpoint status. The
size of the neighbourhood created by M1is determined by the parameter m1and equal
to 2·m1, as both M+
1and M−
1include the m1changeable items with the highest and
lowest ratings, respectively.
The set of moves M2is defined as
M2=nm= (SP+, SP−)|i∈SP+⇔ |SC(j)|= 1∀j∈PR(i), SP−=PR(i)o
5.2 Inventory Allocation in Production and Distribution Networks 119
The moves m∈M2pursue the strategy to move the inventory positions downstream
as described in Section 5.2.2.2. If all predecessors of an item ihave ias their only
successor, inventory is placed at iand removed from all of i’s predecessors.
The neighbourhoods that result from the two above-mentioned sets of moves are fully
dependent on the appropriateness of the stockpoint eligibility ratings or limited in
their scope as they only consider certain network structures. Thus the moves in M3
add a stochastic element and randomly switch the stockpoint status of single items.
Each move only changes one item, as there is no reason to assume that moves with
multiple, randomly selected items create any beneficial combinations. The set of all
possible single item switches is
M3=nm= (SP+, SP−)|i /∈ SP ⇒ SP+={i}∧SP−=∅, i ∈ SP ⇒ SP −={i}∧SP+=∅o
The resulting neighbourhood contains all solutions in which the stockpoint status of
one item has been changed.39 It is our aim to avoid this full evaluation of all possible
moves as it requires considerable effort to evaluate such a big neighbourhood in each
iteration. Accordingly, M3only contains a subset of m3randomly selected elements
from M3.
M3=nm3moves randomly selected from M3o(5.36)
The full neighbourhood used in the tabu search is defined by the union of the single
neighbourhoods. The sizes of neighbourhoods defined via M1and M3are controlled
via the parameters m1and m3. The definition of appropriate neighbourhood combi-
nations and sizes significantly affects the performance of the optimisation and will be
discussed in the validation Chapter 6. It has to be noted that the neighbourhoods
cannot be expected to be disjunctive, i.e. a certain move can appear twice in different
neighbourhoods during a single iteration of the tabu search. In order to avoid dupli-
cate evaluations of the same neighbourhood solution, all evaluated moves are stored
during one iteration and then checked before any move is evaluated. The next section
discusses how relevant neighbourhood solutions are actually evaluated.
5.2.3.3 Evaluation of Inventory Cost for a Given Solution
In line 8, the tabu search algorithm has to select the move that yields the solution
with the best (i.e. minimal) objective value, which requires that all solutions of a
neighbourhood are evaluated. Each solution is evaluated with an objective function
39 Note that this also comprises the moves from M1.
120 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
f:P(N)7→ Rthat assesses the total inventory cost f(SP)caused with a certain
configuration SP. This assessment is made according to Equation 5.15, which requires
that each stockpoint i∈ SP has a replenishment lead time RLTiand safety stock levels
Iss
i,t defined.
Algorithm 5.8:f(SP): full evaluation of a solution
Input: a model PDN(N,V,S), a current solution SP
Result: total inventory cost for PDN(N,V,S)with solution SP
L←TopologicalSorting(G=(N,V))1
foreach item iin Ldo2
calculate RLTiaccording to Equation 5.183
if i∈ SP then4
STi←05
calculate safety stock levels Iss
i,t for each t∈ T according to Equation 5.19
6
else7
STi←RLTi
8
set safety stock levels Iss
i,t = 0 for each t∈ T
9
10
return total inventory cost calculated according to Equation 5.1511
Algorithm 5.8 shows how replenishment lead times, service times and safety stock levels
are calculated successively for each item to prepare the evaluation of Equation 5.15. For
each item the replenishment lead time is first updated. Depending on the stockpoint
status of the current item, the algorithm sets its service time to zero and calculates the
required safety stock levels, or it sets the service times at equal to its replenishment
lead time and the safety stock levels to zero.
This full evaluation computes the replenishment lead times and service times for all
items in the network. The topological sorting is used to ensure that the replenishment
lead times and service times of an item are not computed before they have been
computed for all predecessors of that item.40 However, moves generally only change
the stockpoint status of a small subset of the items. These changes often do not affect
the entire network, but only a small number of items compared to the total number of
items in the network. Thus, the evaluation of a neighbourhood solution can be sped
up if we only calculate the changes in the inventory levels and related cost of these
particular items.
Example 5.2 Consider the part of a network depicted in Figure 5.5. The
table shows the throughput, replenishment lead and service times of all
40 See Section 5.1.1.2 for a definition and a similar application of a topological sorting.
5.2 Inventory Allocation in Production and Distribution Networks 121
Item TT RLT ST
n12 5 5
n22 7 7
n33 5 0
n41 8 8
n5210 10
n6212 0
n71 9 0
n82 2 2
n91 1 1
n2
n3
n4
n5
n1
n7
n6
n8
n9
Item TT RLT ST
n12 5 0
n22 2 2
n33 5 5
n41 6 6
n52 8 8
n6210 0
n71 7 0
n82 2 2
n91 1 1
n2
n3
n4
n5
n1
n7
n6
n8
n9
m = ({n1},{n3})
Figure 5.5: Example of incremental solution evaluation
items. The applied move m=({n1},{n3})adds a stockpoint at item
n1and removes the stockpoint at n3. The replenishment lead times and
service times of all direct and indirect predecessors in the network remain
unchanged, and so neither are their inventory requirements affected. Only
direct or indirect successor items that can be reached from n1or n3via a
directed path might be affected in terms of changing replenishment lead
times. From this set, we can further exclude all items that can only be
reached via a path that contains a stockpoint on an intermediate item in
the path. In the example, items n8and n9are not affected, as the increased
replenishment lead time is fully covered by increased inventory levels at n7.
The stockpoints n6and n7are reached on paths that start in n1or n3, and
reflect the changes in their replenishment lead times in changing inventory
levels, while all their successors are no longer affected due to the defined
service time of 0 for all stockpoints. The table on the right side shows
the adapted throughput, service and replenishment lead times after the
application of the move, where only the highlighted values have actually
changed.
122 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
Algorithm 5.9:f(SP, m): incremental evaluation of a move on a solution
Input: a model PDN(N,V,S), a current solution SP and a move m= (SP+, SP−)
Result: Total inventory cost for network PDN(N,V,S)after applying move mto
SP
apply move mto SP1
foreach item i∈SP+∪SP−changed by mdo2
recalculate RLTiaccording to Equation 5.183
if i∈ SP then4
STi←05
calculate safety stock levels Iss
i,t for each t∈ T according to Equation 5.19
6
else7
STi←RLTi
8
set safety stock levels Iss
i,t = 0 for each t∈ T
9
foreach j∈SC(i)do10
RecalculateInventory (j)11
12
return total inventory cost calculated according to Equation 5.1513
Procedure RecalculateInventory(i)
recalculate RLTiaccording to Equation 5.181
if i∈ SP then2
STi←03
calculate safety stock levels Iss
i,t for each t∈ T according to Equation 5.19
4
else5
STi←RLTi
6
set safety stock levels Iss
i,t = 0 for each t∈ T
7
if i /∈ SP then8
foreach j∈SC(i)do9
RecalculateInventory (j)10
11
12
Algorithm 5.9 together with procedure RecalculateInventory(i)shows how the in-
sights from the example can be used to define an incremental evaluation of a neigh-
bourhood solution. The objective function f(SP, m)represented by this algorithm
evaluates the application of a given move mto a current solution SP. In contrast to
the full evaluation of a given solution in Algorithm 5.8, this approach evaluates the
inventory cost in the PDN after a given move has been applied, which is just what
the tabu search algorithm has to do for each neighbourhood solution (line 8). Starting
from all items that are directly changed by the move, their service times and inventory
5.3 Determination of Planning Buffers and Planned Production Quantities 123
levels are updated and RecalculateInventory(i)is invoked on all their successors.
This procedure updates the replenishment lead time, service time and inventory levels
of the processed item and invokes the procedure recursively on all successors until a
stockpoint is encountered on the path or no more successors are found. With this pro-
cedure, it is perfectly possible that one item is re-evaluated several times, like items
n4to n7in the example given above. This does not cause any inconsistency, as the re-
plenishment lead time of each item always is determined by the maximum predecessor
service time after the entire evaluation41, independently of the sequence in which they
are re-evaluated. At the end of the recursion, the relevant values have been updated for
all affected items and the costs can be evaluated with Equation 5.15. This limitation
in terms of the number of items considered in the evaluation significantly reduces the
number of recalculations required to evaluate a single move.
5.3 Determination of Planning Buffers and Planned Production
Quantities
This section presents an optimisation method that yields optimal production parame-
ters in terms of planning buffers for all s∈ SP ROD and planned production quantities
for all i∈ NP ROD on the basis of a given model PDN(N,V,S).
5.3.1 Optimisation Model
The problem is formulated as a mixed-integer mathematical programming model. This
model is designed to solve to optimality the trade-offs described in Section 2.3.3 and
can be written as:
41 See Equation 5.18.
124 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
min X
s∈SP ROD X
i∈NsX
t∈T Cstp
i,s,pbs+scrpi·Ci·Xi,s,t (5.37a)
+X
s∈SP ROD X
i∈NsX
t∈T
Ics
i,t−1+Ics
i,t
2·Cinv
i(5.37b)
+X
s∈SP ROD X
t∈T
PBpen
s,pbs(5.37c)
s.t.
t=t∗
X
t=1 X
s∈Si
Qi,s,t ≥
t=t∗
X
t=1
di,t ∀t∗∈ T, i ∈ NP ROD
(5.38a)
M·Xi,s,t ≥Qi,s,t ∀i∈ NP ROD, s ∈ SP ROD, t ∈ T
(5.38b)
Ics
i,t−1+X
s∈SP ROD
Qi,s,t −di,t =Ics
i,t ∀i∈ NP ROD, t ∈ T
(5.38c)
X
i∈Ns
ki,s ·Qi,s,t ≤Ks∀s∈ SP ROD, t ∈ T
(5.38d)
Qi,s,t ·Qrnd
i,s =k·Qrnd
i,s k∈N0,∀i∈ NP ROD, s ∈ SP ROD, t ∈ T
(5.38e)
Qrnd
i,s ·M≥Qrnd
i,s ∀i∈ NP ROD, s ∈ SP ROD
(5.38f)
Qrnd
i,s ∈ {0,1} ∀i∈ NP ROD, s ∈ SP ROD
(5.38g)
pbs∈npbmin
s, . . . , pbmax
so∀s∈ SP ROD
(5.38h)
Xi,t ∈ {0,1}, Qi,s,t ∈R, Ics
i,t ∈R∀i∈ NP ROD, s ∈ SP ROD, t ∈ T
(5.38i)
The model is intended to determine the parameters that serve as the basis of the cost
assessment. The objective function comprises the corresponding setup, scrap and cycle
stock costs (5.37a and 5.37b), analogously to the calculations of the cost components
5.3 Determination of Planning Buffers and Planned Production Quantities 125
CST SCRP and CCY ST in Equations 5.16 and 5.17, respectively. In addition, penalty
costs for inventories incurred via the length of the chosen planning buffer are also
added to the objective function (5.37c). In this way the relevant trade-off between
the length of the planning buffer and the amount of setup cost incurred is encoded in
the objective function. With increasing planning buffers pbs, the average setup costs
Cstp
i,s,pbsand therefore the value of 5.37a decrease. At the same time the penalty costs
in 5.37c increase with pbs.
Restriction 5.38a assures that the planned production quantities cover the actual de-
mand quantities. The planned production quantities have to be scheduled at latest in
the periods where the corresponding demand occurs. Demand quantities for an item i
can be covered by allocating planned production quantities on any of the production
process steps assigned to i. This is implicitly assured by restriction 5.38a, as for any
production item i, only planned production quantities on related production process
steps s∈ Siare summed. Thus any planned production quantities allocated to other
infeasible process steps are automatically set to 0. The binary production indicator
variable Xi,s,t is used in the objective function to incur setup and scrap cost where
appropriate. It is therefore forced to a value of 1 by restriction 5.38b in case there are
positive production quantities defined. The parameter Mhere refers to an arbitrary
but sufficiently large number MQi,s,t for any planned production quantity.
The inventory balance restriction 5.38c assures that the variables Ics
i,t hold the correct
level of cycle stock for item iat the end of period t. For the cost assessment it is
sufficient to determine these cycle stocks for each item and period, which is why the
planned production quantities are summed up over all relevant production process
steps.
The planned production quantities describe a theoretical production plan at an aggre-
gate level. This plan has to be defined in a way that makes it possible to transfer it into
an operational production plan by scheduling production orders for all planned produc-
tion quantities assigned to one period and sequencing their execution. Thus planned
production quantities must already consider all capacity and lot-sizing constraints.
Restriction 5.38d assures that planned production quantities are distributed over the
available production process steps so that the capacity available on each production
process step is not exceeded. We assume that the total capacity available suffices to
produce the required quantities, which is particularly reasonable if the model data is
derived from historical data where observed demands did not exceed the capacity of
the production resources.
126 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
Further lot-sizing restrictions may result from technical characteristics of the produc-
tion process that only allows economical production of multiples of certain quantities.42
Therefore, Qrnd
i,s defines the rounding value for actual lot sizes and consequently also
the planned production quantities. Restriction 5.38e assures that all production quan-
tities for an item ion a production process step smust be multiples of this rounding
value Qrnd
i,s . This restriction can be made ineffective via the additional binary indicator
Qrnd
i,s (5.38g), which is set to 0 if there are no specific rounding values and Qrnd
i,s is 0.
Otherwise, it is forced to 1 by restriction 5.38f, where Magain denotes an arbitrary
but sufficiently large value. The additional indicator Qrnd
i,s is necessary to avoid making
the model infeasible by impeding positive planned production quantities on production
process steps without such rounding constraints in restriction 5.38e.
Restrictions 5.38g to 5.38i define the domains of all variables. Potential planning
buffers are restricted to a maximum and minimum value. This is required as the
average setup cost for each material must be available for each potential planning
buffer, so that a finite set of potential planning buffers has to be defined a priori.
This is also reasonable from a practical point of view, as the additional benefit of
increasing planning buffers decreases. A production planner should be able define
what minimum planning buffer is possible and beyond which planning buffer length
no additional benefit is to be expected.
One requirement for the solution of the subprobem addressed in this section is the
possibility to solve the optimisation problem with standard software. The presented
model contains several binary and integer variables that make it a mixed-integer op-
timisation problem, which does not impede the use of standard solvers. However, the
objective function part 5.37a is non-linear due to the relation between average setup
costs Cstp
i,s,pbsand production indicators Xi,s,t. These are all decision variables since the
former depend on the choice of a planning buffer, while the latter are determined by
the corresponding planned production quantities.
As this is a problem for the use of standard solvers, we propose two approaches to
deal with this particular non-linearity. Both are based on the idea of changing the
status of the planning buffers pbsfrom decision variables to fixed parameters in order
to make the objective function part 5.37a linear. The resulting problem instances can
then be solved separately. The remaining questions are what combinations of planning
buffers to choose to obtain this set of optimisation problem instances and how the best
solution from this set relates to the optimal solution of the original model.
42 This is especially the case in base production stages where the output cannot be measured in
pieces.
5.3 Determination of Planning Buffers and Planned Production Quantities 127
The first possibility is to try all combinations of planning buffer candidates for all pro-
duction process steps. This is possible as both the number of production process steps
SP RODand the number of planning buffer candidates npbs=|{pbmin
s, . . . , pbmax
s}| are
finite and yields the same exact results as the original problem formulation. However,
the number of resulting subproblems then is npb|SP ROD|
sand grows exponentially with
the number of production process steps. This approach is thus only applicable if ei-
ther the number of planning buffer candidates or the number of considered production
process steps is very small. Considering practical applications, this cannot be taken
for granted.
One way to alleviate this complexity problem is to constrain the set of combinations of
planning buffers tested by explicitly imposing restrictions on these combinations. For
instance, a production process step s1always works with the same planning buffer as
another production process step s2, or if pbs1is larger than a certain value, pbs2must
not be greater than a certain value. The biggest simplification that can be made using
this approach is to require that all production process steps operate with the same
planning buffer. This results in only npbs·SP RODseparate problems to be solved,
but at the same time this equality of planning buffers is a strong assumption that may
lead to suboptimal solutions. Whether or not such rules can be formulated to reduce
the number of tested combinations sufficiently is highly application-dependent.
An alternative approach to reduce the number of problem instances is to formulate the
problem only for a single production process step. Considering the critical objective
function part 5.37a, there is no reason that requires all production process steps being
optimised in an integrated model. With this approach, a separate problem instance
is solved for each planning buffer candidate of each production process step. This
decoupling of the production process steps allows to test each planning buffer candidate
on each production process step with reasonable effort, since the total number of
problem instances to be solved again is npbs·SP ROD. However, there are two points
to consider.
Firstly, it may lead to suboptimal results if there are alternative production possibilities
and some items are assigned to more than one production process step |Si|>1for
some i∈ NP ROD. In this case, demand quantities of these items have to be split
up and distributed over the corresponding production process steps a priori. This
may make the overall result suboptimal, as the optimisation model cannot distribute
required production quantities arbitrarily over the available production process steps
to find assignments at minimum setup and scrap costs.
128 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
Secondly, the precise determination of the penalty cost parameters PBpen
s,pbsmay become
impossible since the required penalties for a given production process step and planning
buffer may depend on the choice of planning buffers for other production process
steps. Objective function part 5.37c thus requires the integrated optimisation of all
production process steps. We discuss this problem in Section 5.3.3 and present two
approaches to solve this interdependency by approximation of the planning buffer
penalties.
In summary, we conclude that the optimisation model can be transformed into a linear
model by solving it either for a subset of planning buffer combinations or separately for
each production process step. If the number of production process steps and planning
buffers is too big to try all combinations and if no reasonable subset of these combina-
tions can be defined, the second approach described above represents the best trade-off
between the approximations made and the practical solvability of the problem. Meth-
ods for the approximation of planing buffer penalties required for this approach are
presented in Section 5.3.3.
5.3.2 Estimating Average Sequence-Dependent Setup Cost
The relation between the length of the planning buffer and the average setup costs
on a certain production process step is represented by the parameters Cstp
i,s,pbs. As
these values are generally not known, this section presents a method of deriving an
estimation based on the data available from ERP systems. The observation that
average setup costs can be reduced with larger planning buffers, i.e. the series
(Cstp
i,s,pbmin
s, . . . , Cstp
i,s,pbmax
s)is monotonically decreasing, is due to the fact that the se-
quencing procedure has more possibilities to bring forward or postpone production
orders so they can be combined sequenced after orders for similar materials. The
method of generating estimations for the Cstp
i,s,pbsuses this logic.
We assume that setup cost information is available in terms of setup matrices for each
production process step. In such a setup matrix, the entries cstp
s,i,j represent the setup
costs incurred by a changeover from i∈ Nsto j∈ Nson production process step s.
We propose a simulation approach to generate the estimated values from these setup
matrices by evaluating a large number of setup sequences. Algorithm 5.11 shows the
steps of the calculation and is invoked for each s∈ SP ROD and each potential planning
buffer candidate pbs∈ {pbmin
s, . . . , pbmax
s}.
5.3 Determination of Planning Buffers and Planned Production Quantities 129
Algorithm 5.11: Simulation approach to estimate average setup costs
Input:s∈ SP ROD,pbs∈ {pbmin
s, . . . , pbmax
s},zpbs
Result: average setup costs Cstp
i,s,pbs∀i∈ Ns
while not nsim
i≥nsim ∀i∈ Nsdo1
N ← zpbsitems randomly selected from Ns
2
Create cost-optimal production sequence for items in N3
foreach i∈ N that is not the first item in the optimised sequence do4
cstp
i←5
cstp
i+cstp
s,j,i where jis the predecessor of iin the optimised production sequence
nsim
i←nsim
i+ 16
7
foreach i∈ Nsdo8
Cstp
i,s,pbs←cstp
i/nsim
i
9
10
The algorithm makes use of the fact that the average setup costs incurred by a set
of production orders depends in the size of this set. At any given time, there is a
certain number of production orders available for which all required components have
been provided by the predecessor stages and which are therefore available for release
to production. This number depends on the length of the planning buffer: The longer
the planned time span between the provision of all components and the requirement
date of a production order, the more production orders are available for release to
production at any given time. This expected number of production orders within the
planning buffer interval pbsis denoted zpbs.43
The basic idea of this approach is to imitate this situation that a production planner
faces when generating production sequences for a certain production process step with
a given planning buffer. The algorithm randomly selects zpbsitems from the set of items
processed on the considered production process step. This subset represents the set
of production orders that have to be executed within the planning buffer interval. For
this set an optimal production sequence with respect to the total setup cost incurred
is created. Due to the very limited number of orders to be scheduled, this optimisation
problem does not pose a major problem. For realistic values of zpbs, the problem can
still be solved optimally, either by full enumeration of the possibilities or with standard
solvers. If the number of orders within a single planning buffer interval should really
43 If this number is not known, an historical average can be used or it can be estimated. For example,
if the planning buffer interval comprises 6 shifts, and in each shift 1,5 different production orders
are processed on average, we can use zpbs= 9.
130 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
get too large in a practical application, there are plenty of well-established heuristics
for the sequencing problem44 and potentially suboptimal solutions of such heuristics
do not cause major problems in this context of parameter estimation.
For each item in that sequence, the observed setup costs are taken as a single sample
for the estimation of the average setup costs. Over the iterations of the simulation, the
total setup cost incurred for a certain item is recorded as cstp
i(line 5). The cost incurred
for each item is the actual setup cost taken from the setup matrix, given the predecessor
in the optimised production sequence. For the first item in the optimised sequence no
sample is recorded since we only consider a limited section of an actual production
plan. The number of samples observed for each item is recorded as nsim
i. This number
serves as the stopping criterion and the simulation stops when nsim samples have been
recorded for each item. It furthermore is used to calculate the estimated average setup
costs from the total setup costs observed over the iterations of the simulation (line 9).
5.3.3 Calculating Penalty Cost for Inventories Incurred by Planning Buffers
One integral part of determining appropriate planning buffers is the integration of
penalty costs for the length of the planning buffer. The task addressed in this section
is the determination of appropriate values for the planning buffer penalty costs PBpen
s,pbs.
These penalty costs for planning buffers are included in the objective function (5.37c)
to ensure that not all planning buffers are set to their maximum values pbmax
s, which
would ensure the minimal setup costs (5.37a), but not necessarily minimum overall
cost if we also consider inventory costs in the network. The planning buffer penalty
PBpen
s,pbsmust therefore represent the additional inventory cost incurred in the entire
PDN if soperates with a planning buffer of pbs, compared to the configuration where
the planning buffer is 0.
Additional inventory costs are incurred at items which are stockpoints and which are
affected by a change of the planning buffer of production process step s. An item i
is affected if changes of the considered planning buffer pbschange its replenishment
lead time RLTi. The logic to find these affected items is similar to that logic used to
update the replenishment lead times and inventory levels in the incremental evaluation
presented in Section 5.2.3.3 and we can reuse part of that logic. All affected items can
be found on a critical path downstream in the network. A critical path is a maximum
sequence (n1, . . . , nk)of items with an item n1∈ Nsthat is processed on sat its
beginning and which contains at most one stockpoint at the end of the path, i.e.
44 See e.g. Fleischmann and Meyr (1997), Haase (1996) and Gupta and Magnusson (2005).
5.3 Determination of Planning Buffers and Planned Production Quantities 131
n1, . . . , nk−1/∈ SP. The replenishment lead times of all items processed on sare
changed directly by the increase of the planning buffer. This increased lead time is
passed downstream as an increased service time until a stockpoint is reached that
covers the additional replenishment lead time with an increased inventory level.
n1n2n5
n6
n4
n3
pbss
I
STn3
STn2
TT
n1In5
n4
ss/cs
In6
ss/cs
ss/cs const.
Figure 5.6: Additional inventory cost due to increasing planning buffers
Example 5.3 Figure 5.6 shows which items are affected in a network if
the planning buffer of production process step sis increased. Firstly, the
throughput times and thereby the replenishment lead times of all items
processed on sare increased, which in this case applies only to n1. As n1
is not a stockpoint itself, it propagates this increased throughput time as
an increased service time to its successors. This logic is applied recursively,
until a stockpoint item is reached, in this case n4,n5and n6. These items
may be affected by the changing service time of one of their predecessors,
but only if that service time is the maximum predecessor service time that
directly affects their replenishment lead time.45 In this case, n4and n5are
affected as they only have one predecessor that determines their replenish-
ment lead time. Item n4, however, though reachable on a path from n1,
is not affected as its replenishment lead time is determined by the service
time of n3due to STn3> STn4and therefore maxj∈P R(n4)STj=STn3.
For the calculation of PBpen
s,pbsfor each production process step sand planning buffer
candidate pbs, two characteristics have to be considered:
Nonlinearity As the inventory cost function is non-linear, inventory costs do not in-
crease linearly with the planning buffers and replenishment lead times.
45 See Equation 5.18.
132 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
Interdependence The degree to which the replenishment lead time of a particular
item iis increased depends on the planning buffer decisions on all production
process steps related to ivia any direct or indirect predecessor.
The combination of these two characteristics impedes an exact evaluation of the inven-
tory penalty costs. Firstly, the nonlinearity is a problem due to the interdependencies.
It suggests evaluation of the PBpen
s,pbsexplicitly for each planning buffer candidate. This
would however be necessary for all possible combinations of planning buffer configura-
tions for the remaining production process steps and thus lead to the same complexity
already discussed in the context of the nonlinearity of the optimisation model in Sec-
tion 5.3.1. We already proposed to circumvent this problem by solving the optimisation
model for each production process step separately, which renders the exact assessment
with consideration of all combinations impossible. Secondly, the interdependencies
are a problem due to the nonlinearity of the inventory cost function. With a linear
cost function, the configuration for the remaining planning buffers would be irrelevant
when considering one particular production process step, as costs would increase lin-
early anyway. With the combination of both characteristics, none of these approaches
suffices.
This analysis suggests two feasible approaches to approximately assess penalty costs.
Firstly, the interdependencies may be neglected and penalty costs are evaluated for
each planning buffer candidate of each individual production process step separately.
Secondly, the inventory cost function can be linearly approximated, which renders the
interdependencies irrelevant and allows assessment of the penalties for a single pro-
duction process step without consideration of the configurations of remaining planning
buffers.
The first approach is defined in Algorithm 5.12. For each production process step and
planning buffer candidate the cumulative penalty costs for all critical paths that start
in one of the related production items Nsare calculated. During the iteration over all
related production items, two cases have to be distinguished: If the considered item
iis a stockpoint itself, the penalty costs can be calculated directly as the difference
of inventory costs (Equation 5.26) with and without the considered planning buffer
of length pbs. If it is not a stockpoint, the penalty cost for all paths that start in i
are calculated by summing the results of procedure CalcPBPenalty(i, j, pbs)over all
successor items j. This procedure first checks if an increase of i’s service time would
affect the replenishment lead time of the considered successor item j. If this is not the
case, the penalty costs amount to 0 and no further successor has to be considered from
here. Otherwise the same logic as already described above applies. If jis a stockpoint,
5.3 Determination of Planning Buffers and Planned Production Quantities 133
Algorithm 5.12: Calculation of planning buffer penalties disregarding interdepen-
dencies
Input: a model PDN(N,V,S)with an inventory allocation SP
Result: the planning buffer penalty costs PBpen
s,pbs
foreach s∈ SP ROD do1
foreach pbs∈ {pbmin
s, . . . , pbmax
s}do2
foreach i∈ Nsdo3
if i∈ SP then4
PBpen
s,pbs←PBpen
s,pbs+CINV
i(∆ti+pbs)−CINV
i(∆ti)
5
else6
PBpen
s,pbs←PBpen
s,pbs+Pj∈SC(i)CalcPBPenalty(i, j, pbs)
7
8
9
10
11
Procedure CalcPBPenalty(i, j, pbs)
if arg maxp∈P R(j)STp=ithen1
if j∈ SP then2
return CINV
j(∆tj+pbs)−CINV
j(∆tj)
3
else4
return Pk∈SC(j)CalcPBPenalty(j, k, pbs)
5
6
else7
return 08
9
the penalty costs are calculated and returned immediately, while the calculation is
invoked recursively over the successors of jif jis not a stockpoint.
The second feasible approach is linearly to approximate the inventory cost function.
If inventory costs are assumed to increase linearly with replenishment lead times, a
single penalty factor that is independent of the configuration of planning buffers at the
remaining production process steps can be calculated. For fixed safety stock factors
zi,t the inventory cost function is concave due to the square root of the coverage time
interval.46 Concave functions can be linearly approximated by introducing 2 reference
46 See Equation 5.27a on page 108.
134 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
points that mark the relevant interval of that function.47. For the coverage times ∆ti,
we can limit this interval to
0≤∆ti≤∆tmax
i,with ∆tmax
i= max
w∈W(NP ROC ,i)X
j∈w
TTj(5.39)
The set W(NP ROC , i)denotes all paths from a procurement node to i. The upper
bound ∆tmax
iis the maximum replenishment lead time that can be expected, based on
the maximum sum of throughput times of the nodes on any path w∈W(NP ROC , i).
The convex part of the inventory cost formula is given by the square root term, which
can now be approximated over the interval [0,∆tmax
i]as48
s∆ti
TS≈1
q∆tmax
i·∆ti
√TS(5.40)
As expected, this approximation yields exact results only for the two references points
0 and ∆tmax
iat the extremes of the interval. Applying this approximation to the
inventory cost Equation 5.26 yields
˜
CINV
i(∆ti) = X
t∈T
RLTi
2·TS·di,t +zi,t ·σd
i,t
q∆tmax
i·∆ti
√TS
·Cinv
i(5.41)
which in contrast to the original equation is linear in ∆ti. This allows us to derive
exactly one planning buffer penalty value per item that approximately shows how
inventory costs increase if the replenishment lead time and thereby the coverage time
∆tiincrease by 1 short-term period. For any ∆ti∈[0,∆tmax
i], this value is constant
and equal to
˜
CINV
i(∆ti+ 1) −˜
CINV
i(∆ti)(5.42)
Algorithm 5.14 defines the calculations of the second approach and shows how to
determine the planning buffer penalties based on the presented linearisation.
As this algorithm is similar to Algorithm 5.12, we limit explanation to the differences.
As only one linear penalty cost factor is derived, the entire procedure is only carried
out once per production process step. For each affected item at which additional
47 Minner also proposes such a linearisation approach to transform a inventory allocation problem
to a linear optimisation problem (Minner, 2000, pp. 152-154).
48 This is the simplest form of such a linearisation using only two reference points. More exact
approximations can be obtained by increasing the number of reference points and thereby mak-
ing the approximation a piecewise linear function. For a description of the required additional
variables and constraints, see Domschke and Drexl (2005, pp. 206-208).
5.3 Determination of Planning Buffers and Planned Production Quantities 135
Algorithm 5.14: Calculation of planning buffer penalties using linearisation
Input: a model PDN(N,V,S)with an inventory allocation SP
Result: the planning buffer penalty costs PBpen
s,pbs
foreach s∈ SP ROD do1
foreach i∈ Nsdo2
if i∈ SP then3
PBpen
s←PBpen
s+˜
CINV
i(∆ti+ 1) −˜
CINV
i(∆ti)
4
else5
PBpen
s←PBpen
s+Pj∈SC(i)LinearPBPenalty(i, j)
6
7
foreach pbs∈ {pbmin
s, . . . , pbmax
s}do8
PBpen
s,pbs←PBpen
s·pbs
9
10
11
Procedure LinearPBPenalty(i, j)
if arg maxp∈P R(j)STp=ithen1
if j∈ SP then2
return ˜
CINV
i(∆ti+ 1) −˜
CINV
i(∆ti)3
else4
return Pk∈SC(j)LinearPBPenalty(j, k)
5
6
else7
return 08
9
costs are incurred, the linear approximation for the penalty cost is used according to
Equation 5.42 (compare line 4 in the algorithm and line 3 in the procedure). After all
items of the considered production process step have been processed, the value PBpen
s,pbs
contains the penalty cost for the increase of the corresponding planning buffer by 1.
As we assume linear cost increases in this approach, the corresponding penalty costs
for each individual planning buffer PBpen
s,pbscan be determined by multiplication with
each planning buffer candidate pbs∈ {pbmin
s, . . . , pbmax
s}(lines 8 to 9).
5.3.4 Integration with the Inventory Allocation Problem
This section addresses the remaining open task of integrating the two optimisation
models presented. According to the requirements formulated, this integration also has
136 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
to solve the problem of interdependencies between the variables of both models as
described in Section 2.3.3.
The only viable solution is a repeated sequential invocation of both optimisation meth-
ods, where each execution uses the results of the previous solution of the other opti-
misation problem. Figure 5.7 shows what the intended sequence of invocations looks
like.
Build model or scenario
(Algorithms 5.1, 5.3, 5.4)
A PDN model with throughput times TTi
Optimise inventory allocation
based on current throughput times
(Algorithm 5.7)
PDN model with updated parameters
SP, RLTi, STi
Optimise planning buffers and production process steps
(Optimisation model from Section 5.3.1)
Model with planning buffers
and planned production quantities set.
Updated throughput times TTi
Recalculate RLTi, STi, Iiss
stopping criterion met
stopping criterion
not met
End: PDN model with all
parameters for the cost analysis set
Start: Input data for
model generation
available
Figure 5.7: Integration of the optimisation methods
Once a model or scenario has been built, the inventory allocation method is used to
determine the stockpoints SP, replenishment lead times RLTiand service times STi
for each item. The set of stockpoints and the determined replenishment lead times
5.3 Determination of Planning Buffers and Planned Production Quantities 137
are required as input parameters for the parameter estimation (Algorithms 5.12 and
5.14) in the context of the optimisation of planning buffers and planned production
quantities. After this second optimisation step has been performed, the network model
contains defined planning buffers and planned production quantities. The changes
of planning buffers also affect the throughput times TTi. The stockpoint allocation
determined previously may therefore no longer be optimal, as it was based on different
input parameters.
On the basis of the updated network, the sequence provides a loop back to the in-
ventory allocation optimisation method, which is then invoked again to consider the
updated throughput times. In any repeated invocation, the previous stockpoint allo-
cation can be used as a starting solution. The fact that only a small part of the input
parameters in terms of planning buffers has actually changed suggests that the new
optimal solution is closely similar to the previous one. These repeated evaluations are
thus much faster than the initial optimisation. The same applies to the subsequent re-
peated solution of the second optimisation model where the determination of planned
production quantities is not affected by the parameters changed during the repeated
inventory allocation optimisation. Therefore the planned production quantities deter-
mined in the initial optimisation can be fixed and only planning buffers are optimised
in subsequent optimisation runs, which reduces the problem complexity enormously.
As indicated in the flow chart, some stopping criterion is required to limit the number
of repeated invocations and guarantee an eventual termination. There are several
stopping criteria possible, e.g.
No more changes in decision variables No repeated invocation is required if no more
changes in the optimal configuration of stockpoints and planning buffers are
observed. This stopping criterion is desirable, as it guarantees that an overall
optimal configuration is found despite the sequential solution of the optimisation
problems. Practically, the fulfilment of this criterion is even likely since the
stockpoint allocation is only relevant for the second optimisation model precisely
to estimate the penalty costs for increasing planning buffers. If these cost rates
slightly change, it does not have too big an influence on the overall objective value
and optimal planning buffers might well remain constant. Without changing
planning buffers, the throughput times are also constant and there is no need to
invoke the inventory allocation again.
Convergence of total cost As a relaxed version of the first stopping criterion, we do
no longer require that no changes are made to the decision variables any more,
but that the sum of both objective function values converges. Conversion can
138 5 Assessing the Effects of Assortment Complexity in Consumer Goods Supply Chains
be defined in different ways, e.g. if the value remains constant over a certain
number of iterations or does not improve beyond a certain percentage threshold.
Maximum number of iterations In the unlikely case that none of the stopping crite-
ria mentioned is ever met, a maximum number of iterations should be defined to
guarantee the termination after a finite period of time.
This sequence shows how the two separate optimisation models and solution ap-
proaches presented in this work can be connected to obtain a basis for the assess-
ment of assortment-related costs. Following the sequence described in the flowchart, a
production and distribution network model or scenario can be built and optimised so
that all parameters required for the cost assessment are determined nearly optimally.
The interdependencies between the single steps are resolved by a repeated invocation,
which has been shown to lead to a globally near-optimal configuration after a limited
number of iterations.
139
CHAPTER 6
Application and Validation via Examples
We have the facts and
we’re voting ‘yes’.
Death Cab for Cutie
The next Section 6.1 gives a short overview of the software developed to test the
concepts presented in Chapter 5. A more detailed description of the technical im-
plementation can be found in appendix A. The remainder of this chapter provides a
validation and exemplary application of these concepts in three steps: Firstly, Sec-
tion 6.2 defines the example assortments and scenarios used. Secondly, Section 6.3
then describes and discusses results of the analyses performed. Finally, Section 6.4
provides some details on the performance of the optimisation methods as observed
during the example application.
6.1 Implementation
For this exemplary application and validation, the entire concept presented in Chap-
ter 5 was implemented in a the software tool Complana (COMPLexityANAlyser). It
provides a graphical user interface that supports the entire analysis process with the
following steps:
Data import and management The data required to build the production and dis-
tribution network models can be imported into the Complana database from an
SAP ERP system. Import functionality is provided for material master and ac-
counting data, production and sales locations with relevant customer service and
cost parameters, demand and forecast accuracy data, bills of material, routings
and production process steps with the corresponding data on setup and scrap
140 6 Application and Validation via Examples
times and quantities. Existing data can be viewed and edited to allow quick
adaptations of the relevant parameters.
Model building and management The automatic generation of production and dis-
tribution networks can be invoked with a set of end products as the main in-
put. Generated models are stored in the database along with descriptive meta-
information. Existing models can be edited in terms of adapting demands and
forecast deviations, either individually for single items or collectively for sets of
selected items.
Scenario building and management For a selected baseline model, scenarios can be
generated from a scenario definition. The software tool supports the user by
automatically calculating the conversion factors from material master data and
taking care of peculiarities like currency conversions in case the replacement
material is valued in a currency other than the original material. To facilitate
the selection of sets of materials to be replaced, standard ABC/XYZ analyses
can be carried out on a selected model. Scenarios can also be edited just like the
corresponding models.
Visualisation Models and scenarios can be visualised as networks. This provides the
means visually to inspect the production and distribution relations and the data
at each individual item. Furthermore, it helps to compare the complexity of
different assortments by visualising their items and relations between them.1
Optimisation The optimisation methods can be parameterized and invoked for all
models and scenarios in the database. The optimised models and scenarios are
stored back into the database afterwards and a detailed result report is written
into a spreadsheet file.
Comparative analysis Optimised models and scenarios can be used for comparative
analyses. For pairs of an optimised model and corresponding scenario, a cost
comparison is carried out and the results are visualised in different types of
charts. Information about the changes in the number of materials at different
locations, on different production process steps and of different material types is
calculated.
1 See Definition 2.5.
6.2 Specification of Example Models 141
6.2 Specification of Example Models
In order to show the practical applicability for real-world business problems, this vali-
dation uses example cases based on the assortment of an international household prod-
uct manufacturer. The considered company produces, distributes and sells mechanical
household products in different parts of the world. Several specialised production sites
each produce a certain product range and supply a number of sales locations, which in
turn manage sales to customers in their market region. Such a market region usually
comprises a certain country or set of adjacent countries. Customer orders are ful-
filled by shipment of goods from dedicated sales warehouses that are typically situated
close to the sales locations. Customers include retail stores and wholesalers as well as
cleaning companies and hotels, thus separate products for consumer and professional
business divisions are available. The supply chain employs decentralised control, i.e.
sales and production locations each manage stock-keeping, purchasing and distribution
autonomously, though integrated via a central material coordination system.
6.2.1 Product Assortments
We consider two baseline assortment models M1and M2, which are both based on the
household cloth products of an international household product manufacturer.
M1- window cloths The first model was built from all window cloths end products
offered at any sales location.
M2- complete cloths assortment The second model was built from all cloths end
products offered at any sales location. The first model is therefore a subset of
this model.
This particular product category was chosen as it has grown for many years and there-
fore suffers from particularly high complexity. The fact that both the entire cloth
assortment as well as a smaller subset are considered separately allows validation of
different aspects of the methods developed. Due to its manageable size, the first model
M1allows a detailed inspection of the results to detect any unexplainable output as
well as a detailed sensitivity analysis to ensure the model reacts as expected. The sec-
ond example model M2is in turn used to show the feasibility of the methods for large
problem instances. Table 6.1 shows an overview of the assortments considered. Clearly
model M2comprises many more elements in all respects; in particular it also comprises
142 6 Application and Validation via Examples
some locations and production process steps that are not used for the production and
distribution of window cloths in M1.
Table 6.1: Overview of example assortment models
M1M2
No. of materials 293 2006
thereof end products 78 588
No. of items 419 3330
finished material items thereof 195 1810
semi-finished material items thereof 66 649
packaging material items thereof 115 764
raw material items thereof 43 107
No. of locations 23 27
No. of PPS 10 15
Almost all cloth end products are produced at a single production location that exter-
nally procures raw and packaging materials and performs all required steps to produce
the end products. The end products are then distributed to the sales locations from
where they are sold to customers. In some cases semi-finished products, bulk goods or
finished products on a non-TSU packaging level are shipped to sales companies and
simple converting activities are carried out at these sales locations. The production
costs of these converting activities are not considered in our model since the convert-
ing processes mainly comprise manual assembly and packaging activities that incur
only negligible setup and scrap costs. Moreover, their consideration would even fur-
ther increase the data requirements with information about the production processes
at each of the converting sites. Technically this means that there are no production
process steps defined at any location except the considered production site. We do,
however, consider the related BOMs at these locations and use them to determine
product structures just as for the actual production location.
Figure 6.1 illustrates the distribution structure of the cloths assortment as derived
from model M2. At each location, the number of cloth materials handled at that
location is shown in a small bar graph, separated according to material type.2From
the production location in Germany, mainly finished products are distributed to a
number of sales locations abroad. In some cases further transshipment is carried out
at the sales location to consolidate transport for countries with small sales volumes.
2 Note that these bars have a different scaling for the production and sales locations.
6.2 Specification of Example Models 143
Distribution
Production
Raw / Pack 662
Semi 510
Finished 263
Location DE (Prod.)
Raw / Pack 2
Semi 12
Finished 142
Location DE (Sales)
Finished 104
Location Other (Export)
Raw / Pack 56
Semi 28
Finished 146
Location BE
Raw / Pack 14
Semi 1
Finished 16
Location CA
Raw / Pack 60
Semi 11
Finished 90
Location ES
Finished 72
Location FR
Raw / Pack 7
Semi 0
Finished 79
Location GB
Raw / Pack 4
Semi 3
Finished 64
Location IT
Raw / Pack 9
Semi 13
Finished 13
Location NL
Raw / Pack 36
Semi 11
Finished 42
Location TR
Finished 50
Location PT
Finished 49
Location SE
Finished 39
Location FI
Finished 5
Location US
Figure 6.1: Distribution structure of cloth assortment (model M2)
144 6 Application and Validation via Examples
6.2.2 Alternative Assortment Scenarios
For each assortment model, several alternative assortment scenarios are defined. Ta-
ble 6.2 summarises the scenario definitions and shows the changes made to the network
by the scenario application in terms of materials discontinued. For each scenario, the
number of discontinued end products and the resulting numbers of discontinued items
of different types are shown both as absolute and relative figures, the latter compared
with the corresponding baseline model.
The scenarios for M1represent different assortment reduction strategies on the end
product level, based on the demand characteristics of the products. As a basis for
scenario definitions, the demand series of end products at the sales locations are anal-
ysed with standard ABC and XYZ analyses. The parameters for the ABC analysis are
chosen such that the A, B and C products contribute 80%, 15% and 5% respectively to
the total cumulative demand. For the XYZ analysis, the classification of an item ito
one of the classes is made on the basis of the coefficient of variation cviof its demand
distribution. The intervals are chosen as cvi≤0.5,0.5< cvi≤1.0and 1< cvi, for
classes X, Y and Z respectively. The scenarios for M1are defined with the goal of
finding out what items cause the highest complexity-related costs. Scenario S1dis-
continues all products from class C. Scenario S2additionally discontinues the class B
products with medium demand volumes. S3analyses the effects of standardisation of
A products with high demands. Analogously to the scenarios on the basis of the ABC
analysis, S4and S5are based on the XYZ analysis and discontinue those products with
volatile and thus hardly predictable demands. S6describes the largest standardisation
possible in that model with of a reduction to a single end product which is then sold
at all locations.
For the full cloth assortment model M2it is to be noted that it comprises cloths from
different categories like window cloths, floor cloths and micro fibre cloths, among oth-
ers. Since cloths of different types may not substitute each other, standardisation to
one end product is made only within these categories in the first scenario S1, leaving
one standard product per category. While this may still be a reasonable example from
practice, scenario S2finally considers the hypothetical case of standardisation to a sin-
gle multi-purpose cloth. Although it is practically unrealistic to reduce such a complex
assortment with different types of product to one single product, it provides insights
into what fraction of total costs is caused by the existing assortment complexity.
The term replace in the scenario definitions means that all affected products are dis-
continued and replaced by one newly-added replacement product. For this newly-
6.2 Specification of Example Models 145
Table 6.2: Scenario definitions
Input Scenario application result
Name Scenario description
discontinued
end products
total no.
discontinued
items
discontinued
finished
material items
discontinued
semi-finished
material items
discontinued
raw and pack.
material items
M1/S1Replace all C end products
with the C end product with
the highest total demand
53
67.95%
173
41.29%
103
52.82%
17
25.70%
53
33.54%
M1/S2Replace all C and B end prod-
ucts with the B end product
with the highest total demand
66
84.62%
256
61.10%
138
70.77%
33
50.00%
85
53.80%
M1/S3Replace all A end products
with the A end product with
the highest total demand
9
11.54%
46
10.98%
25
12.82%
4
6.06%
17
11.04%
M1/S4Replace all Z end products
with the Z end product with
the highest total demand
44
56.41%
110
26.25%
77
39.49%
6
9.09%
27
17.09%
M1/S5Replace all Z and Y end prod-
ucts with the Y end product
with the highest total demand
62
79.49%
218
52.03%
134
68.72%
21
31.82%
63
39.87%
M1/S6Replace all end products with
the end product with highest
total demand
77
98.72%
346
82.58%
176
90.26%
46
69.70%
124
78.48%
M2/S1Replace end products within
each product category with
the end product with highest
total demand in that category
570
96.94%
2788
83.72%
1533
84.70%
511
78.74%
744
85.42%
M2/S2Replace all end products with
the end product with highest
total demand
587
99.83%
3287
98.71%
1788
98.78%
645
99.38%
854
98.05%
introduced product, all master data and the production BOM are taken from the
material with highest total demand in the set of replaced products. In the scenarios
all demands of discontinued products are assumed to be transferred to the respective
replacement products, i.e. we do not consider any lost sales. This is reasonable to
assure a direct comparability of the absolute costs indicated by the cost assessment.
146 6 Application and Validation via Examples
During the calculation of demand for the replacement products we also consider dif-
ferent packaging sizes for discontinued and replacement products. The software tool
automatically calculates the conversion factor3for each replacement relation to trans-
fer the correct demand quantities based on the number of individual products, i.e. on
BCU level4.
The possibility of visualising all models and scenarios provides the means for a mere
visual comparison of the network complexity. Appendix B contains various visual-
isations of the example models used. For example, Figure B.1 shows the graphical
representation of the smaller model M1, where one can distinguish procurement, base
production, converting and distribution stages in the network structure. As a com-
parison, Figure B.7 shows the visualisation of the related scenario M1/S6, where the
complexity reduction becomes clearly visible. See the appendix for more such visuali-
sation examples.
6.2.3 Production Processes
The products of the cloth product category are mainly produced at a specialised pro-
duction facility in Germany. For both models, the same production system at that
location is considered. It comprises a number of production process steps that can
roughly be grouped into the categories base production and converting.5Production
costs in terms of setup, scrap and production cycle stock costs are assessed for all of
these production process steps. Figure 6.2 shows the considered production process
steps, their categorisation as well as their logical sequence in the production process.
The base production starts with two production process steps that create the paint
and fibre mixtures from the raw materials. These production steps are of less interest
when considering assortment complexity, as they only do the preliminary work for the
subsequent production lines in terms of composing the raw materials according to the
corresponding recipes and thus operate fully in line with the demands of these sub-
sequent process steps. At the actual base production stage these fibres and chemical
mixtures are compacted to a basic raw cloth produced on large rolls. For these oper-
ations three production lines (Line 1, 2, 3) are used, each mainly for one particular
cloth type. Some of the resulting raw cloths are further refined by applying additional
3 See Section 5.1.2.1.
4 See Section 2.2 for a description of packaging levels.
5 See Section 2.1.2.
6.2 Specification of Example Models 147
Cloth lanes
at certain
width
Line 1
Line 2
Line 3
Refine 1
Refine 2
Refine 3
Lane
Cutting
Pack 1
Pack 2
Pack 3
Cut/Fold/Pack
Fold 1
Fold 2
Paint
Mix
Fibre
Mix
Raw cloth
on rolls
Refined cloth
on rolls
Single
folded
cloth
Finished product
packed for
shipment
Base production Coverting
Figure 6.2: Production process steps
coatings, prints and imprinting structures into them. Depending on the type of refine-
ment, one of the three alternative machines (Refine 1, 2, 3) is used. The result of this
step then is the finished cloth material, again produced in large rolls.
In the converting stage, the wide rolls are first cut into smaller lanes on a separate
production process step (Lane Cutting). These lanes then form the input components
to two cutting and folding machines (Fold 1, 2), which output the finished unpacked
cloth. In the final packaging step, these cloths are then packed into foil and then
into cases in certain batches for shipment to the sales locations, which is done on
three packaging machines (Pack 1, 2, 3). Apart from the described separate cutting,
folding and packaging steps, one additional fully automated production process step
(Cut/Fold/Pack) is considered, which is capable of performing all converting operations
in-line, i.e. from the refined cloth on rolls to the packaged end products on TSU level.
For each of these production process steps, setup, scrap and cycle stock costs are as-
sessed on the basis of optimised planning buffers and planned production quantities.
The observed cost developments are based on the changing number of materials pro-
cessed in each production process step in each scenario. Table 6.3 shows the number of
materials processed on each individual production process step in the baseline models
and all related scenarios. Figures 6.3(a) and 6.3(b) visualise these numbers for models
M1and M2and their scenarios, respectively. It can be seen that for the window cloth
assortment, only 10 out of the 15 production process steps are required.
148 6 Application and Validation via Examples
Table 6.3: Number of materials processed on each production process step
PPS M1M1/S1M1/S2M1/S3M1/S4M1/S5M1/S6M2M2/S1M2/S2
Fibre Mix 2 2 2 2 2 2 1 23 11 1
Paint Mix 21 19 17 21 21 19 15 104 56 2
Line 1 4 3 2 4 4 3 1 30 9 -
Line 2 - - - - - - - 24 3 1
Line 3 3 2 1 3 3 2 1 66 13 -
Refine 1 - - - - - - - 18 8 -
Refine 2 - - - - - - - 2 - -
Refine 3 6 3 2 6 5 3 1 37 13 -
Lane Cutting 5 3 - 5 5 3 - 38 3 -
Fold 1 7 3 3 6 6 4 1 126 14 -
Fold 2 2 2 2 2 2 2 1 93 16 -
Pack 1 19 15 9 12 13 7 1 71 4 -
Pack 2 - - - - - - - 68 6 -
Pack 3 1 1 1 - 1 - - 54 5 -
Cut/Fold/Pack - - - - - - - 47 6 1
Idx PPS M1 M1/S1 M1/S2 M1/S3 M1/S4 M1/S5 M1/S6
14 Pack 3 111 1
12 Pack 1 19 15 9 12 13 7 1
11 Fold 2 2222221
10 Fold 1 7336641
9Lane Cutting 53 553
8Refine 3 6326531
5Line 3 3213321
3Line 1 4324431
2Paint Mix 21 19 17 21 21 19 15
1Fibre Mix 2222221
Pack3
Pack1
Fold2
Fold1
LaneCutting
Refine3
Line3
Line1
PaintMix
FibreMix
Cut/Fold/Pack
Pack3
Pack2
Pack1
Fold2
Fold1
LaneCutting
Refine3
Refine2
Refine1
Line3
Line2
Line1
PaintMix
FibreMix
0 5 10 15 20
Pack3
Pack1
Fold2
Fold1
LaneCutting
Refine3
Line3
Line1
PaintMix
FibreMix
M1 M1/S1 M1/S2 M1/S3
M1/S4 M1/S5 M1/S6
0 25 50 75 100 125
Cut/Fold/Pack
Pack3
Pack2
Pack1
Fold2
Fold1
LaneCutting
Refine3
Refine2
Refine1
Line3
Line2
Line1
PaintMix
FibreMix
M2 M2/S1 M2/S2
(a) M1and related scenarios
Idx PPS M1 M1/S1 M1/S2 M1/S3 M1/S4 M1/S5 M1/S6
14 Pack 3 111 1
12 Pack 1 19 15 9 12 13 7 1
11 Fold 2 2222221
10 Fold 1 7336641
9Lane Cutting 53 553
8Refine 3 6326531
5Line 3 3213321
3Line 1 4324431
2Paint Mix 21 19 17 21 21 19 15
1Fibre Mix 2222221
Pack3
Pack1
Fold2
Fold1
LaneCutting
Refine3
Line3
Line1
PaintMix
FibreMix
Cut/Fold/Pack
Pack3
Pack2
Pack1
Fold2
Fold1
LaneCutting
Refine3
Refine2
Refine1
Line3
Line2
Line1
PaintMix
FibreMix
0 5 10 15 20
Pack3
Pack1
Fold2
Fold1
LaneCutting
Refine3
Line3
Line1
PaintMix
FibreMix
M1 M1/S1 M1/S2 M1/S3
M1/S4 M1/S5 M1/S6
0 25 50 75 100 125
Cut/Fold/Pack
Pack3
Pack2
Pack1
Fold2
Fold1
LaneCutting
Refine3
Refine2
Refine1
Line3
Line2
Line1
PaintMix
FibreMix
M2 M2/S1 M2/S2
(b) M2and related scenarios
Figure 6.3: Number of materials processed on each production process step
6.2 Specification of Example Models 149
6.2.4 Parameter Settings
Table 6.4 summarises the most important parameters used in the model and scenario
generation and explains from where the corresponding values are derived. In order to
base the analysis on real-world data and realistic assumptions, most data were taken
from the practice of the household product manufacturer considered. The parameters
are set according to company policy or are derived from historical data as described
below.
Table 6.4: Model generation parameter data sources
Parameter Data source
Demands Historical demands observed at the sales locations for each
month from January to December 2008.
Forecast
deviations
Historical forecast deviations observed at the sales locations
for each month from January to December 2008. The fore-
cast deviations are available as mean absolute deviations, from
which forecast error distributions are derived as described in
Section 5.1.1.2.
Service levels The service levels for all items are set to the internal target
service level aspired for customer orders at the sales locations.
Inventory cost
rates
Inventory costs are derived for each material at each location
on the basis of the internal average cost of capital and the
warehousing cost rates for one pallet at each location.
Setup cost
rates
Setup cost rates are calculated based on setup times, labour
and machine cost at the production site. On some occasions,
average costs had to be used due to insufficient data availabil-
ity. This does not affect the validity of the presented results
or the general correctness of the presented methods.
Scrap cost
rates
Scrap cost rates are derived from average scrap quantities per
production run on the different production process step and
the corresponding material values per basic unit.
150 6 Application and Validation via Examples
6.3 Application and Results
6.3.1 Cost Effects in Inventory Allocation
This section presents the results of the inventory allocation for the two baseline models
and their corresponding scenarios and discusses the cost changes per scenario on an
aggregate level. For one selected scenario a more detailed analysis is provided. The
complete numerical results are shown in Table 6.5. For each baseline model or scenario,
it shows the inventory costs in the optimised production and distribution network,
additionally broken down according to material type. For each scenario, it also provides
the relative change compared with the cost of the corresponding baseline model, again
for each material type and scenario.
Table 6.5: Comparison of inventory cost
Inventory cost [e] Relative change [%]
Model Finished Semi R./P. Total Finished Semi R./P. Total
M1 30,715.14 2,388.37 4,935.87 38,039.38
M1/S1 28,657.01 1,734.58 4,378.80 34,770.40 -6.70 -27.37 -11.29 -8.59
M1/S2 22.085,66 2.475,29 4,472.50 29.033,44 -28,10 3.64 -9.39 -23,68
M1/S3 22,616.20 1,635.71 2,852.00 27,103.91 -26.37 -31.51 -42.22 -28.75
M1/S4 27,817.16 2,189.80 4,784.21 34,791.17 -9.44 -8.31 -3.07 -8.54
M1/S5 22,840.53 504.11 4,363.92 27,708.56 -25.64 -78.89 -11.59 -27.16
M1/S6 12.797,18 0,00 1.666,36 14.463,54 -58,34 -100.00 -66.24 -61,98
M2 367.329,11 35.770,81 28.867,32 431.967,25
M2/S1 119.212,57 5.199,66 15.701,32 140.113,55 -67,55 -85,46 -45,61 -67,56
M2/S2 25.601,97 84,77 1.661,19 27.347,93 -93,03 -99,76 -94,25 -93,67
Figure 6.4(a) visualises the inventory cost data for M1and all corresponding scenarios,
sorted by their total cost value. It first can be noted that all forms of standardisation
lead to cost savings in the area of inventory management. The scenarios S4(Z materi-
als) and S1(C materials) lead to similar changes in total inventory costs. This can be
explained by closer comparison of these product sets, which reveals that they coincide
at 74.074%6. Analogously, scenarios S2and S5with the combined replacements of all
6 That is, 40 out of the 45 Z products are also part of the set of C products with 54 elements.
6.3 Application and Results 151
Model Fin.atsales Fin.atproduction Fin Semi
M1 Baseline 30,699.47€ 15.67€ 30,715.14€ 2,388.37€
M1/S4 Z27,112.32€ 704.85€ 27,817.16€ 2,189.80€
M1/S1 C27,810.56€ 846.45€ 28,657.01€ 1,734.58€
M1/S2 C+B 20,293.13€ 1,792.53€ 22,085.66€ 2,475.29€
M1/S5 Z+Y 19,911.96€ 2,928.58€ 22,840.53€ 504.11€
M1/S3 A19,281.20€ 3,335.00€ 22,616.20€ 1,635.71€
M1/S6 All 8,242.05€ 4,555.13€ 12,797.18€ ‐ €
M2 347,966.50€ 19,362.61€ 367,329.11€ 35,770.81€
M2/S1 102,505.37€ 16,707.20€ 119,212.57€ 5,199.66€
M2/S2 19,850.15€ 5,751.82€ 25,601.97€ 84.77€
Invent
o
‐€
5.000€
10.000€
15.000€
20.000€
25.000€
30.000€
35.000€
40.000€
M1 M1/S4 M1/S1 M1/S2 M1/S5 M1/S3 M1/S6
Raw /Pack
Semi
Fin at production
Fin at sales
‐€
5.000€
10.000€
15.000€
20.000€
25.000€
30.000€
35.000€
40.000€
M1 M1/S4 M1/S1 M1/S2 M1/S5 M1/S3 M1/S6
Raw/Pack Semi Fin.atproduction Fin.atsales
‐€
50.000€
100.000€
150.000€
200.000€
250.000€
300.000€
350.000€
400.000€
450.000€
M2 M2/S1 M2/S2
Raw/Pack Semi Fin.atproduction Fin.atsales
(a) Comparison for M1
Model Fin.atsales Fin.atproduction Fin Semi
M1 Baseline 30,699.47€ 15.67€ 30,715.14€ 2,388.37€
M1/S4 Z27,112.32€ 704.85€ 27,817.16€ 2,189.80€
M1/S1 C27,810.56€ 846.45€ 28,657.01€ 1,734.58€
M1/S2 C+B 20,293.13€ 1,792.53€ 22,085.66€ 2,475.29€
M1/S5 Z+Y 19,911.96€ 2,928.58€ 22,840.53€ 504.11€
M1/S3 A19,281.20€ 3,335.00€ 22,616.20€ 1,635.71€
M1/S6 All 8,242.05€ 4,555.13€ 12,797.18€ ‐ €
M2 347,966.50€ 19,362.61€ 367,329.11€ 35,770.81€
M2/S1 102,505.37€ 16,707.20€ 119,212.57€ 5,199.66€
M2/S2 19,850.15€ 5,751.82€ 25,601.97€ 84.77€
Invent
o
‐€
5.000€
10.000€
15.000€
20.000€
25.000€
30.000€
35.000€
40.000€
M1 M1/S4 M1/S1 M1/S2 M1/S5 M1/S3 M1/S6
Raw /Pack
Semi
Fin at production
Fin at sales
‐€
5.000€
10.000€
15.000€
20.000€
25.000€
30.000€
35.000€
40.000€
M1 M1/S4 M1/S1 M1/S2 M1/S5 M1/S3 M1/S6
Raw/Pack Semi Fin.atproduction Fin.atsales
‐€
50.000€
100.000€
150.000€
200.000€
250.000€
300.000€
350.000€
400.000€
450.000€
M2 M2/S1 M2/S2
Raw/Pack Semi Fin.atproduction Fin.atsales
(b) Comparison for M2
Figure 6.4: Inventory cost comparisons
C+B and Z+Y materials show similar total savings, where in this case the material
sets coincide at 85.07%7.
Despite the fact that there are considerably fewer A products than C and B products
combined, assortment reduction by all materials in that class yields comparable and
even slightly higher savings. The same applies in the comparison with the combined
replacement of all Z and Y products. This is mainly due to the higher demand quanti-
ties for the discontinued A products, which result in much higher theoretically required
safety stocks and thus higher savings potential, even for a small number of discontinued
A products. However, it has to be noted that in practice, average inventories for C and
7 That is, 57 out of 63 Z and Y products are also part of the set of C and B products with 67
elements.
152 6 Application and Validation via Examples
sometimes B products often turn out to be higher than the theoretically calculated
levels from the model. In this case the savings potential in a real-world application may
be even greater than indicated here. Reasons for higher inventory levels for C products
include additional restrictions like minimum order sizes or psychological factors that
cause planners keep a higher level of base stock. Furthermore, our model assumes that
safety stock levels can be adapted on the level of mid-term periods, which can be a
practical problem for products with minimum order sizes or production batch sizes
and sporadic demand.
Analogously to the analysis for M1, figure 6.4(b) depicts the results of the inventory
allocation for M2and the corresponding scenarios. Notably, the cost change of scenario
M2/S1coincides very well with the changes calculated for the corresponding scenario
M1/S6of the window cloth model (67.56% versus 61.98%), which supports the validity
of this figure. It indicates that the window cloths scenario is representative also of the
other cloth product categories, as the joint standardisation within each such category
yields approximately the same savings ratio. In the scenario with only one multi-
purpose cloth for all application types (M2/S2), almost all inventory costs can be
eliminated. While this change is admittedly unlikely to be implemented in practice, it
shows that the model correctly represents the fact that a production and distribution
system with only one standard, fast-moving product can operate almost without any
inventory.
It is also notable that the allocation and type of inventory vary with different degrees
of standardisation. Generally, increasing standardisation also increases the ratio of
inventory held at production sites, which manifests itself in increasing inventory costs
for raw and packaging materials, semi-finished materials and especially finished ma-
terials held at production locations. In the baseline models there are practically no
or very little finished material inventories at the production site, as there is almost
no item commonality in the diversified assortment and therefore inventories are better
kept locally at the sales locations to reduce service time to the customers. This ratio
changes with increasing standardisation. In scenario M1/S6,35.59% of finished mate-
rial inventories are held at the production location and inventories at the production
location account for 50.01% of total inventory costs.
This effect can be traced back to the risk pooling effects of assortment standardisa-
tions.8With increasing item commonality, forecast deviations decrease upstream in
the network and inventories at early stages become preferable. A detailed analysis
8 See Section 3.2.4 for a description of risk pooling and Section 5.1.1.2 for a description of how
these effects are quantified in the underlying model.
6.3 Application and Results 153
Stockpoint
Non-Stockpoint
No inventory of
finished products at
production location
Inventory of finished
products at
production location
M1M1/S6
Figure 6.5: Introducing additional stockpoints for finished products
shows that this is especially the case for end products. Replacing various variants of
a certain product with one standard product makes it favourable to keep inventories
of the latter at the production location to decrease replenishment lead times to sales
locations. The additional costs for inventory at the production location are outweighed
by the savings generated at sales locations. Figure 6.5 shows a detail of the network
visualisations of M1and M1/S6. On the left, inventory for each specialised product
variant is only held at the sales locations, whereas on the right, an additional stock-
point is introduced for the newly-added finished product at the sales location, which
reduces replenishment lead times at the sales locations and allows them to lower their
safety stock levels.
Further analysis of the cost changes in the network supports this conclusion. Figure 6.6
illustrates the inventory cost changes for each location. All sales locations experience a
cost decrease, partly due to standardisations in their local assortments, but mainly due
to shorter replenishment lead times for the new replacement product. The only location
that faces an increase in inventory cost is the main production location. However, the
figure shows that this is by far outweighed by the savings generated, such that a total
cost decrease of 61.98% remains. Furthermore, this cost increase at the production
location is practically negligible (4,45%). This is due to the fact that the increased
154 6 Application and Validation via Examples
30.000
€
35.000 €
40.000 €
15 000
€
20.000 €
25.000 €
- 61.98 %
5.000 €
10.000 €
15
.
000
€
M
1
M
1/S6
0 €
Location
M
M
Baseline model Savings Additional costs Scenario
Figure 6.6: Inventory cost changes on location level for M1/S6
cost for additional finished material inventory are almost completely outweighed by the
inventory reductions for semi-finished, raw and packaging materials in the production
area, which result from the fact that many of these materials are no longer present in
the scenario.
6.3.2 Cost Effects in Production Execution
For all baseline models and scenarios, the optimisation of planned production quantities
and planning buffers provides insight into the development of production costs as
a response to assortment changes. Table 6.6 shows the resulting setup, scrap and
cycle stock costs for all models and scenarios in their optimal configuration. For each
scenario, the relative changes in comparison with the corresponding baseline model are
also indicated.
Figures 6.7 summarise the cost changes in a bar chart, separated by cost types and
ordered by their total cost value. Comparing the order in figure 6.7(a) with the corre-
sponding figure 6.4(a) obtained for the cost assessment in the area of inventory man-
agement, we note that it is identical except that M1/S3has moved and now appears
6.3 Application and Results 155
Table 6.6: Comparison of production execution costs
Production cost [e] Relative change [%]
Model Setup Scrap Cycle Total Setup Scrap Cycle Total
M1 28,608.94 9,546.90 12,735.13 50,890.97
M1/S1 26,100.67 8,144.05 9,946.22 44,190.94 -8.77 -14.69 -21.90 -13.17
M1/S2 21,684.11 7,917.24 6,286.56 35,887.91 -24.21 -17.07 -50.64 -29.48
M1/S3 20,071.41 7,399.14 11,591.51 39,062.06 -29.84 -22.50 -8.98 -23.24
M1/S4 27,377.88 9,338.92 11,212.30 47,929.10 -4.30 -2.18 -11.96 -5.82
M1/S5 21,011.14 8,580.18 5,716.77 35,308.09 -26.56 -10.13 -55.11 -30.62
M1/S6 10,315.17 3,760.30 1,161.29 15,236.77 -63.94 -60.61 -90.88 -70.06
M2 285,906.71 110,178.83 226,426.34 622,511.88
M2/S1 96,415.46 37,971.70 29,457.20 163,844.36 -66.28 -65.54 -86.99 -73.68
M2/S2 2,259.51 804.92 231.76 3,296.19 -99.21 -99.27 -99.90 -99.47
before M1/S2and M1/S5, i.e. now yields lower savings than these two scenarios. This
is one indicator for the observation that in the area of production costs, the relative
importance of the two factors number of materials discontinued and demand value of
the discontinued materials shifts towards the factor number of materials discontinued.
While inventory costs are heavily influenced by the demand rates of the materials,
demand-invariant fixed costs for setups and scrap have a higher importance here and
result in higher savings obtained by a standardisation of both C and B products than
those obtained by a standardisation of the high-volume A products. Furthermore, fig-
ure 6.7(a) shows that the commonalities in the scenario inputs again lead to a similar
cost behaviour for scenarios S1/S4and S2/S5.
All costs shown here are based on the optimal choices for planning buffers for the
production process steps considered in each respective model and scenario.9For
each production process step s, the set of candidate planning buffers was defined as
pbs∈ {0,...,6}. From this set, the planning buffer that yields minimal total cost was
chosen and consequently also used in the calculation of the throughput times TTiof
all materials processed on s.
In order to illustrate the cost behaviour when changing the planning buffer at a single
production process step, Figure 6.8 shows the production costs and planning buffer
penalties for production process step ‘Refine 3’ in the base production area for the large
baseline model M2. Each bar indicates the total cost as well as the proportion of each
9 Compare Table 6.3 for an overview of the usage of the production process steps in the models
and scenarios.
156 6 Application and Validation via Examples
Model Setup Scrap Cyclestock Total
M1 Baseline 28,608.94€ 9,546.90€ 12,735.13€ 50,890.97€
M1/S4 Z27,377.88€ 9,338.92€ 11,212.30€ 47,929.10€
M1/S1 C26,100.67€ 8,144.05€ 9,946.22€ 44,190.94€
M1/S3 A20,071.41€ 7,399.14€ 11,591.51€ 39,062.06€
M1/S2 C+B 21,684.11€ 7,917.24€ 6,286.56€ 35,887.91€
M1/S5
Z+Y
21,011.14
€
8,580.18
€
5,716.77
€
35,308.09
€
M1/S5
Z+Y
21
,
011
.
14
€
8
,
580
.
18
€
5
,
716
.
77
€
35
,
308
.
09
€
M1/S6 All 10,315.17€ 3,760.30€ 1,161.29€ 15,236.77€
M2 285,906.71€ 110,178.83€ 226,426.34€ 622,511.88€
M2/S1 96,415.46€ 37,971.70€ 29,457.20€ 163,844.36€
M2/S2 2,259.51€ 804.92€ 231.76€ 3,296.19€
30.000€
40.000€
50.000€
‐€
10.000€
20.000€
30.000€
40.000€
50.000€
M1 M1/S4 M1/S1 M1/S3 M1/S2 M1/S5 M1/S6
Setup Scrap Cyclestock
‐€
10.000€
20.000€
30.000€
40.000€
50.000€
M1 M1/S4 M1/S1 M1/S3 M1/S2 M1/S5 M1/S6
Setup Scrap Cyclestock
300.000€
400.000€
500.000€
600.000€
‐€
10.000€
20.000€
30.000€
40.000€
50.000€
M1 M1/S4 M1/S1 M1/S3 M1/S2 M1/S5 M1/S6
Setup Scrap Cyclestock
‐€
100.000€
200.000€
300.000€
400.000€
500.000€
600.000€
M2 M2/S1 M2/S2
Setup Scrap Cyclestock
‐€
10.000€
20.000€
30.000€
40.000€
50.000€
M1 M1/S4 M1/S1 M1/S3 M1/S2 M1/S5 M1/S6
Setup Scrap Cyclestock
‐€
100.000€
200.000€
300.000€
400.000€
500.000€
600.000€
M2 M2/S1 M2/S2
Setup Scrap Cyclestock
(a) Comparison for M1
Model Setup Scrap Cyclestock Total
M1 Baseline 28,608.94€ 9,546.90€ 12,735.13€ 50,890.97€
M1/S4 Z27,377.88€ 9,338.92€ 11,212.30€ 47,929.10€
M1/S1 C26,100.67€ 8,144.05€ 9,946.22€ 44,190.94€
M1/S2 C+B 21,684.11€ 7,917.24€ 6,286.56€ 35,887.91€
M1/S3 A20,071.41€ 7,399.14€ 11,591.51€ 39,062.06€
M1/S5 Z+Y 21,011.14€ 8,580.18€ 5,716.77€ 35,308.09€
M1/S6 All 10,315.17€ 3,760.30€ 1,161.29€ 15,236.77€
M2 285,906.71€ 110,178.83€ 226,426.34€ 622,511.88€
M2/S1 96,415.46€ 37,971.70€ 29,457.20€ 163,844.36€
M2/S2 2,259.51€ 804.92€ 231.76€ 3,296.19€
‐€
10.000€
20.000€
30.000€
40.000€
50.000€
M1 M1/S4 M1/S1 M1/S2 M1/S3 M1/S5 M1/S6
Setup Scrap Cyclestock
‐€
10.000€
20.000€
30.000€
40.000€
50.000€
M1 M1/S4 M1/S1 M1/S2 M1/S3 M1/S5 M1/S6
Setup Scrap Cyclestock
‐€
100.000€
200.000€
300.000€
400.000€
500.000€
600.000€
M2 M2/S1 M2/S2
Setup Scrap Cyclestock
(b) Comparison for M2
Figure 6.7: Comparison of setup, scrap and production cycle stock costs
constituent. One can clearly see the anti-proportional developing of setup costs and
planning buffer penalties as planning buffers increase. This developing quantitatively
represents the trade-off between long planning buffers to reduce setup costs by sequence
optimisation versus the additional inventories made necessary by the longer throughput
times. In this case, the total cost is minimal for pbs= 5, which is therefore selected
as the optimal planning buffer for this production process step in baseline model M2.
The same consideration was made in the optimisation for each production process step
in each model and scenario.
6.3 Application and Results 157
38.697
34.781
32.210
30.607
29.796
28.562
27.991
9.114
9.145
9.199
9.158
9.326
9.397
9.393
18.620
19.025
18.196
18.423
17.549
17.352
17.714
1.686
3.212
4.649
6.014
7.272
8.481
0 €
10.000 €
20.000 €
30.000 €
40.000 €
50.000 €
60.000 €
70.000 €
0
1
2
3
4
5
6
Planning Buffer [days]
Setup
Scrap
Cycle stock
Penalty
min. total cost
Figure 6.8: Development of production costs for production process step ‘Re-
fine 3’ in model M2
6.3.3 Conclusions: Cost Effects of Assortment Changes
From this experimental evaluation, we can derive some general conclusions about the
cost effects of assortment changes. In the area of inventory management, we can
identify considerable differences in the ratio of the number of products discontinued
and the obtained savings when considering products with different demand volumes.
This ratio is, according to the model, much higher for large volume products. However,
in practical applications it might well be that the savings in inventory costs turn out
to be higher due to comparably higher safety stock levels before standardisation. The
advantage of the replacement of products with highly volatile demand distributions
may be reduced by the likelihood that these products will also be low in demand
and thus offer little potential for inventory reduction. Considering the analysis of
execution production cost, the requirement to address the high volume products to
obtain considerable savings is not as big as in the area of inventory management. This
is mainly because setup costs are fixed costs per production order, which are also
incurred for the production of low volumes.
Figure 6.9 combines the results obtained from the two analysis areas, showing the
total relative cost savings per scenario and the composition of these savings. For each
scenario the pie chart indicates what ratio of the total savings is due to what cost
area. It can be seen that these distributions vary among the scenarios. However,
158 6 Application and Validation via Examples
Total Savings
11,21%
M1/S1
Total Savings
27,00%
M1/S2
Total Savings
25,60%
M1/S3
Total Savings
6,98%
M1/S4
Total Savings
29,14%
M1/S5
Total Savings
66,60%
M1/S6
Total Savings
71,17%
M2/S1
Total Savings
97,09%
M2/S2
Invent.;
52,31%
Setup;
19,82%
Scrap;
3,35%
Cycle
stock;
24,52%
Invent.;
32,79%
Setup;
25,16%
Scrap;
14,07%
Cycle
stock;
27,98%
Invent.;
37,51%
Setup;
28,84%
Scrap;
6,79%
Cycle
stock;
26,86%
Invent.;
39,80%
Setup;
30,89%
Scrap;
9,77%
Cycle
stock;
19,54%
Invent.;
38,89%
Setup;
25,25%
Scrap;
9,62%
Cycle
stock;
26,24%
Invent.;
39,52%
Setup;
27,70%
Scrap;
10,68%
Cycle
stock;
22,09%
Invent.;
39,87%
Setup;
29,32%
Scrap;
3,73%
Cycle
stock;
27,08%
Invent.;
48,04%
Setup;
37,50%
Scrap;
9,43%
Cycle
stock;
5,02%
Figure 6.9: Total cost changes for all scenarios
6.4 Performance of the Optimisation Methods 159
these variations can be explained via the set of products replaced in each scenario.
For example, the particularly small fraction of savings in cycle stock costs for M1/S3
may be explained by the fact that A products do not cause high cycle stock cost in
the baseline model and therefore there is only little potential for any savings in this
area.
In summary, it has to be noted that the multitude of influencing factors make it
impossible to draw universal conclusions about the actual cost effects of assortment
changes. Consequently, there exist no general statements about recommended assort-
ment changes either. However, the results of the exemplary applications show how the
methods developed in this work help to derive recommendations for concrete change
scenarios. This goes in accordance with the initial motive for this approach, stating
that such individual assessments provide much better decision support than general
statements about cost developments with increasing numbers of product variants.
6.4 Performance of the Optimisation Methods
6.4.1 Inventory Allocation Heuristic
6.4.1.1 Configurations
The analyses and experiments were carried out on a laptop computer equipped with a
Dual Core processor at 1.66 GHz and 2 GB RAM. The two processor kernels were never
used concurrently, as the underlying software does not support parallel computations.
Statements about the configurations and performance of the optimisation methods all
refer to the optimisation runs carried out with the baseline models M1and M2. This
is sufficient, as all scenarios generally comprise smaller networks and therefore cannot
represent more complex optimisation problems as the corresponding baseline models.
The described configurations were used identically for the corresponding scenarios of
each baseline model.
One of the most crucial decisions for the tabu search presented in Section 5.2.3 is
the definition of neighbourhood sizes and compositions. Extensive tests with various
configurations have shown that the best results are obtained with a neighbourhood
composed of a limited number of solutions based on all the presented strategies. Ta-
ble 6.7 shows the parameters for neighbourhood generation chosen for the two baseline
models and all related scenarios.
160 6 Application and Validation via Examples
Table 6.7: Neighbourhood definitions
Model Random switches S.E. switches move downstream?
M150 50 Yes
M2100 100 Yes
The neighbourhoods for both models use the stockpoint eligibility ratings to construct
part of the neighbourhood solutions. While this helps quickly to find good first moves,
it risks getting stuck in local optima, as the generated moves are all based on the
static stockpoint eligibility ratings. Accordingly, randomly selected moves are also
added to keep the search process exploring new areas of the solution space. The third
set of moves for the neighbourhoods is based on the strategy of moving inventories
downstream if there is no item commodity in the network structures. As the network
structure remains unchanged during the optimisation, these items are identified only
once before the optimisation starts and then stored in memory to avoid a repeated
search in each iteration.
As a meta-heuristic, tabu search leaves many additional degrees of freedom to fine-
tune the search process. For all optimisations described in this section, the following
configuration was used.
Initial solution From the set of possible starting solutions described in Section 5.2.3.1,
the one that worked best for the considered model was to preselect all end
products as stockpoints. This can be reasonably explained from the applica-
tion context since the sales locations provide immediate order fulfilment to their
customers, which requires inventories of the finished products in most cases.
Tabu list The tabu list is defined to forbid moves that have already been performed
during the tabu tenure. To assure this, the changed items and the direction of the
stockpoint switches are stored in the tabu list for each selected move. The length
of the tabu tenure were set to 100 and 200 for models M1and M2, respectively.
Aspiration criterion To assure that the tabu list does not impede the discovery of
improved solutions, the search employs the best ever aspiration criterion. When
a move that is tabu would lead to a solution which is better than the current
best solution, the move is permitted despite its tabu status.
Stopping criterion The search is bounded by a time limit, which was set to 300 sec-
onds for all optimisation runs. However, in many applications the search con-
verged long before this limit was reached. Accordingly, the convergence of the
6.4 Performance of the Optimisation Methods 161
search process was used as an alternative stopping criterion and the search ter-
minated if no new best solution was found during the latest 500 tabu search
iterations.
6.4.1.2 Computational Performance
Figure 6.10 shows the runtimes and obtained objective values for the inventory allo-
cation optimisation of the baseline models M1and M2. For presentation purposes we
restrict the description to the performance on the baseline models here, as all sce-
narios can be seen as subsets of these models10 and thus do not pose any additional
challenges for the optimisation methods. The figures compare the performance of the
heuristic developed in Section 5.2 with a comparable tabu-search heuristic based on a
simple one-switch neighbourhood. This alternative heuristic was developed as an im-
plementation of the state of the art with respect to the optimisation methods used so
far for similar inventory problems as described in Sections 3.2.2.2 and 3.2.3. Thereby,
the effects of the enhanced neighbourhood definition is made visible. In addition, the
effect of the partial solution recalculation as the second major enhancements of the
new heuristic is analysed as we test the benchmark heuristic both with and without
this feature.
Convergence200
CustomT.S. Benchmark1 Benchmark2
M1 Runtime 11.04 14.14 140.9
Objective 38,039.38 38,039.38 38,039.38
M2 Runtime 80.15 245.82 300
300sec Objective 431967.25 431967.25 457008.89
300conv
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CustomT.S. Benchmark1Benchmark2
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455.000€
460.000€
0 100 200 300
CustomT.S. Benchmark1Benchmark2
time
limit
(a) Performance for M1
Convergence200
CustomT.S. Benchmark1 Benchmark2
M1 Runtime 11.04 14.14 140.9
Objective 38,039.38 38,039.38 38,039.38
M2 Runtime 80.15 245.82 300
300sec Objective 431967.25 431967.25 457008.89
300conv
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0 50 100 150
CustomT.S. Benchmark1Benchmark2
0€
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0 50 100 150
CustomT.S. Benchmark1Benchmark2
430.000€
435.000€
440.000€
445.000€
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0 100 200 300
CustomT.S. Benchmark1Benchmark2
time
limit
(b) Performance for M2
Figure 6.10: Performance of the inventory allocation heuristic
10 Except for the newly-added replacement products.
162 6 Application and Validation via Examples
The results show the effects of the network size on the performance of the different op-
timisation methods. For baseline model M1, figure 6.10(a) shows that all methods find
an equally good solution within the time limit. The advances of the new tabu search
are reflected in the fact that it achieves faster convergence. This time advance is quite
small for the benchmark heuristic that also uses the solution recalculation, which can
be explained by the small network where both neighbourhoods always almost comprise
all eligible items and thus the intelligent selection of stockpoint items cannot make a
big difference. The consideration of the benchmark without the solution recalculation
shows the runtime advantage obtained by this feature, as its runtime is more than 12
times longer than the runtime of our heuristic.
Figure 6.10(b) shows the analog results for the large model M2. The larger network
clearly increases runtimes, as the neighbourhood generation as well as recalculation of
the objective value has to consider far more items. Our heuristic converges after 80.15
seconds with the best known objective value. The benchmark heuristic that employs
the partial solution recalculation manages to find a solution of the same quality, but
requires 245.82 seconds to converge, which is more than 3 times as long. This speedup
is fully attributable to the enhanced neighbourhood definition. Considering the bench-
mark heuristic without the partial solution recalculation, we see that no solution of
equal quality can be found within the time limit of 300 seconds and the best objective
value deteriorates to 457,008.89.
In summary, the heuristic developed in Section 5.2 has proven to fulfil its require-
ments11 during its use for the various optimisation instances. It provides acceptable
solutions and runtimes for all tested practical instances. In particular, it generates
good solutions even for extremely big networks like M2within few seconds or at most
minutes. With respect to the absolute solution quality, we cannot guarantee its opti-
mality, as a total enumeration of all solutions is impossible within reasonable time12
and standard solvers cannot easily be used due to the problem’s nonlinearity. How-
ever, the fact that the benchmark heuristic with an entirely different neighbourhood
structure yields the same objective value may be interpreted as an indicator of the
good quality, if not optimality of these solutions. This is evidence that the solutions
are of sufficient quality to serve as the basis for a cost analysis.
The predominance of the new heuristic becomes more apparent with increasing net-
work sizes. While the benchmark with the simple one switch neighbourhood can still
11 See Section 2.3.2.
12 As noted earlier, the solution space is of dimension 2nwhere nis the number of items in the
network. For the example models under consideration M1and M2, this results in 2419 ≈1.3538×
10126 and 23330 ≈2.6908 ×101002 possible combinations, respectively.
6.4 Performance of the Optimisation Methods 163
compete for M1, its inferiority becomes visible in instances with larger networks. As
intended, the heuristic is particularly suited to large networks that describe complex
assortments. The configurations described in the previous section have proven efficient
for the considered example models. It has to be noted however that the configuration of
such heuristics is always problem-dependent and may have to be adapted for different
problem instances to obtain competitive results.
6.4.2 Production Parameter Optimisation
6.4.2.1 Configurations
For the optimisation of planning buffers and planned production quantities, standard
solvers for linear optimisation problems are used on a computer equipped with an Intel
Core 2 Duo processor at 3.33 GHz and 8 GB of RAM. In particular, the commercial
solvers CPLEX13 version 11.0.1 and the free alternative GLPK14 version 4.34 have
been tested. The model is implemented using the MathProg modelling language15,
which provides a subset of the commercial AMPL language16. This subset is sufficient
to represent the optimisation model defined in Section 5.3.1. This modelling language
is used to maintain the flexibility easily to use different solvers. The GLPK solver
natively supports MathProg models as its input. For the usage of CPLEX, concrete
instances of the optimisation problem are generated in the CPLEX LP format with the
help of GLPG and written to disk. The CPLEX solver is then invoked from within the
Complana tool and instructed to read these files and solve the corresponding problem
instances. In both cases the optimisation results are written to a file and parsed
again by the Complana software. As one would expect, the commercial CPLEX solver
showed significantly better performance and therefore only these numbers are presented
in this validation. The solver was invoked with its default settings for mixed integer
optimisation problems. In addition, a time limit of 15 minutes was set for each single
optimisation problem instance, i.e. for each production process step and planning
buffer candidate.
13 CPLEX is a state of the art solver for linear problems, developed and distributed by ILOG, an
IBM company. See http://www.ilog.com/cplex/ for further details.
14 The GLPK solver is part of the Gnu Linear Programming Kit, which is an open-source project
licensed under the Lesser Gnu Public Licence (LGPL).
15 Like the GLPK solver, MathProg is part of the Gnu Linear Programming Kit.
16 AMPL is a formal modelling language used to describe mathematical optimisation problems. See
Fourer et al. (1993).
164 6 Application and Validation via Examples
6.4.2.2 Computational Performance
Table 6.8 shows the average solution times for each model or scenario and the contained
production process steps. The runtimes are averaged over the solution times for each
planning buffer candidate and shown in milliseconds.
Table 6.8: Runtime of optimisation, averaged over all planning buffer candidates
(in [ms])
PPS M1M1/S1M1/S2M1/S3M1/S4M1/S5M1/S6M2M2/S1M2/S2
Fibre Mix 2 3 2 2 2 2 2 12 6 1
Paint Mix 1 1 1 1 1 1 0 2 1 1
Line 1 25 80 10 92 33 6 2 3.44% 2,332 -
Line 2 - - - - - - - 0.46% 17 2
Line 3 13 22 2 10 23 2 2 9.42% 6,453 -
Refine 1 - - - - - - - 294 9 -
Refine 2 - - - - - - - 10 - -
Refine 3 46 30 3 50 34 6 2 3.42% 4,734 -
Lane Cutting 28 13 - 35 31 16 - 1.04% 3 -
Fold 1 103 19 23 15 109 8 2 14.07% 3,564 -
Fold 2 3 3 3 3 3 3 2 10.08% 368 -
Pack 1 306 214 33 185 133 19 2 0.01% 12 -
Pack 2 - - - - - - - 1.18 7 -
Pack 3 2 2 2 - 2 - - 1558 7 -
Cut/Fold/Pack - - - - - - - 1.20% 13 2
Sum 529 387 79 393 371 63 14 17,526 6
The table shows that the majority of the optimisation problems are solved to optimality
almost instantaneously. All problems related to the smaller model M1are solved
on average in less that half a second. The total of the sums in the last row can
be seen as the average solution time for the entire model (optimising all contained
production process steps). Even for a large number of planning buffer candidates,
total optimisation time would still be in the range of a few seconds for these problem
instances. It however has to be noted that this solution time is required for each
planning buffer candidate and thus has to be multiplied with the number of planning
buffer candidates to obtain an estimate for the total computation time.
6.4 Performance of the Optimisation Methods 165
Considering the problem instances related to the larger model M2, one can clearly see
the effect of an increasing number of materials on the production process step. The
extreme scenario M2/S2with only one end product is solved instantaneously due to
the very small number of materials. The scenario M2/S2shows runtimes of up to 6.45
seconds for some production process steps, which is still a very quick response time.
Only for the full model M2with the entire cloths assortment the solver does not find a
proven optimal solution within the short time limit for some of the production process
steps. In these cases, the table cells are marked grey and show the maximum gap of the
solution to the optimal solution. For example, it shows that for production process
step ‘Line 1’ the objective value of the best solution found within the time limit is
at most 3.44% worse than that of the optimal solution. Considering these cases, we
can see that in most cases the solutions found are very close (<5%) to the optimal
solution. Only for 3 production process steps we observe gaps of 9.42%,10.07% and
14.07%. For this exemplary analysis, we state that these solutions are sufficient to
serve as the basis for the cost analysis. If more exact solutions are required, the time
limit can be further extended.
It can be concluded that for practically relevant problem sizes, solutions can be ob-
tained with the use of standard solvers in acceptable times. Even for the largest
problem instances, the solutions show a sufficiently high quality within comparably
short time limits. Given that this type of analysis is performed for non time-critical,
strategic or tactical rather than operational decisions, there is also the possibility of
accepting longer runtimes to obtain proven optimal solutions.
167
CHAPTER 7
Conclusions and Future Research
So keep moving, onward,
run through that open door.
Dredg
7.1 Summary and Conclusions
Assortment complexity is an ubiquitous topic in any industrial enterprise. Product
proliferation and assortment complexity have increased impressively in the past as
markets become more global and customer-oriented. This raises the importance of the
managerial task of deciding the breadth and width of assortment.
Approaches to address this task in its entirety all fall short, as a true optimal assort-
ment depends on far too many factors to be assessed precisely. Restricting the focus
to the cost of assortment complexity in the areas of inventory management and pro-
duction execution, we identify the lack of an integrated method to assess the cost of
assortment complexity based on concrete assortment change scenarios. This approach
promises to deliver more precise assessments than any monovariable model, defined
e.g. on the number of products. At the same time it has more practical relevance since
any assortment-related decision starts with some status quo and decisions are to be
taken on concrete changes rather than on an entirely new assortment.
Such a method was presented in this work. The conceptual solution starts with the de-
velopment of a formal model of a production and distribution network which captures
all relevant characteristics of product and distribution structures as well as production
and distribution processes. To facilitate practical usage, algorithms were formulated
automatically to build such networks from data typically available in modern ERP
systems. On the basis of this model, assortment change scenarios can be defined to de-
scribe theoretical assortment changes in terms of material additions and replacements
at all production and distribution stages.
168 7 Conclusions and Future Research
An inherent advantage of the cost assessment on the basis of concrete assortment
change scenarios is that important parameters in the areas of inventory management
and production operations can be adjusted to the new situation. This permits a more
precise cost assessment since it can be based on a production and distribution system
adapted to the new situation. This approach requires to address the determination
of stockpoints with corresponding inventory levels as well as the determination of
production planning buffers and planned production quantities as related optimisation
problems.
For comparison of inventory costs, an inventory allocation model was formulated as
a combinatorial optimisation problem on the basis of the production and distribution
network model. This optimisation model captures the interrelation between the posi-
tions of stockpoints in the network and the replenishment lead times observed at each
individual item. For the solution of this optimisation problem, a tabu search heuristic
was developed that makes use of special domain knowledge to select promising subsets
of items as stockpoint candidates to guide the search process.
For the comparison of production execution costs, a mathematical optimisation model
was formulated to decide the cost-optimal planning buffers and to determine planned
production quantities to minimise setup, scrap and production cycle stock costs. Since
the resulting model is non-linear, approaches to transfer it into a linear model were
discussed to allow the use of standard software. Aware of possible obstacles to the
practical applicability of this method, we also addressed the question of cost parameter
definitions in this context and provided methods to estimate setup cost developments
for given planning buffers.
In order to validate both the individual methods as well as the overall approach,
the entire concept was implemented in a software tool and validated in cooperation
with an international household product manufacturer. This practical background of
the validation assures that the input parameters are near to real-world application
scenarios. On the basis of this data two assortments were analysed considering their
current as-is status as well as various alternative scenarios. The developed optimisation
models and solution methods proved to be feasible for real-world problem instances.
The results have also proven to be valid, as far as this can be assured by sensitivity
analyses. In this context, the experience of logistics and production managers could
be used to assure the correspondence to reality of the numerical results and ensure
that the model responds as expected to parameter changes. The validations showed
how the developed method can be used to support managerial decisions on concrete
assortment changes.
7.2 Limitations and Outlook on Potential Extensions 169
7.2 Limitations and Outlook on Potential Extensions
Any work that addresses a far-reaching problem like the effects of assortment com-
plexity naturally has its limitations. First of all, this work does not provide a method
to calculate an optimal assortment, as this is not believed to be a very promising ap-
proach at the current state of research. The question of finding an optimal assortment
complexity is addressed indirectly by providing decision support via cost estimations
for theoretical assortment changes.
As stated in Chapter 2, assortment complexity incurs cost in virtually every functional
unit and every process of a company. Accordingly, the view on assortment-related costs
had to be limited for the scope of this work, which was imposed by focusing on inven-
tory management and production execution as functional areas with generalisable and
quantifiable cost components. Further extension may include other general areas like
product development (R&D), quality assurance or master data maintenance. However,
the corresponding processes in these areas are probably harder to generalise to obtain
a generic cost assessment, making it a rather case-specific task to describe the corre-
sponding processes, assess the related costs and describe their relation to assortment
complexity.
All results in terms of cost estimations can only be approximate. This is inevitable
given that much data that the estimations rely on are assumptions about future devel-
opments, e.g. expected demand volumes and their distribution over time. Furthermore,
the assessment of effects in production planning makes some simplifying assumptions
about the production planning and scheduling processes. This opens up opportunities
for further research by making the underlying models more detailed, i.e. either include
additional information or decrease the aggregation level. For example, the assessment
of production-related costs could be refined if it were based on more detailed demand
data and also included production sequence planning to avoid the usage of average cost
rates. Furthermore, there is additional loss of accuracy due to the usage of heuristic
optimisation methods. However, the practical applications have shown that they can
be assumed to be negligible in this application context.
This work focuses on consumer goods supply chains. The transferability to other in-
dustries is only partly given. Of the methods developed, the assessment of inventory
costs may easily be transferred to other industries as the underlying assumptions are
rather weak and apply to almost any production and distribution network with in-
ventories in it. However, the assessment of production-related cost assumes certain
170 7 Conclusions and Future Research
characteristics of the production layout as found in consumer goods supply chains,
making it less transferable to other industries. In summary, the method and results
are limited to a certain type of supply chain or industry as outlined in Section 2.1,
which definitely covers all consumer goods supply chains, both in food and non-food
industries. Its extension to other industries forms another field of further research.
171
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187
APPENDIX A
Implementation
Figure A.1 provides an overview of the Complana functional modules and their inter-
actions. The software support for the single steps of the analysis process are described
in more detail in the following sections. Table A.1 shows the separate packages that
make up the Complana software and summarises the functionality provided by each
of them.
Figure A.1: Overview of the Complana software tool
The Complana tool was developed using the Java programming language and the
Eclipse Rich client platform (RCP) as the underlying development framework1. Fur-
ther third-party technologies used in the implementation are
1 The Eclipse RCP is an open tools platform for building and deploying rich client applications.
See http://www.eclipse.org/rcp/ for details.
188 A Implementation
•Microsoft SQL Server for data persistence2
•Hibernate framework for object-relational mapping3
•yEd for network visualisation4
•jFreeChart for results visualisation5
•OpenTS as a framework for the development of the tabu search heuristic6
•Standard solvers and modelling languages for the mixed-integer optimisation
problems, as described in Section 6.4.2.1.
2 See http://www.microsoft.com/sql/ for details.
3 See http://www.hibernate.org for details.
4 See http://www.yworks.com/yed/ for details.
5 See http://www.jfree.org/jfreechart/ for details.
6 See http://www.coin-or.org/ots/ for details.
189
Table A.1: Complana software packages
Package Functional description
complana.app This package contains all elements of the rich client
application. In particular, this includes all GUI
elements like perspectives, view and user dialogs.
complana.imports This package provides the functionality to import
the major part of the required input data from ex-
isting ERP systems. In particular, the import from
the SAP ERP system was implemented and tested.
complana.model This package contains all classes that represent
data elements in the application domain, like ma-
terials, plants, production process steps etc. Fur-
thermore, these classes use special annotations to
specify all the information needed to configure their
object relational mapping to and from the database
via the Hibernate framework.
complana.network This package comprises the network data model,
i.e. all classes used to represent production and
distribution networks with all their elements. Fur-
thermore, it provides several features to work with
these network models, like their transformation
into the GraphML format for visualisation and
their serialisation for permanent storage on the
filesystem or in the underlying database.
complana.prodeval.optimisation This package contains all classes required to gener-
ate MIP optimisation problems from existing pro-
duction and distribution networks to optimise plan-
ning buffers and planned production quantities.
After generation of these problems, the standard
solvers CPLEX and GLPK can be invoked from
the classes provided.
complana.ssplacement This package contains all classes required to imple-
ment the inventory allocation tabu search heuris-
tic. As it uses the OpenTS framework, the classes
are implemented according the interfaces defined
by that framework and thus provide custom im-
plementation for the objective function, solutions,
moves and neighbourhood managers.
complana.util This package contains general utility classes used
at different points in the entire software. This com-
prises classes that provide logging, time measuring,
and reading and writing Microsoft Excel spread-
sheet files.
190 A Implementation
A.1 Model Building and Management
The automatic generation of new assortment models from imported master data can
be invoked via the GUI. Figure A.2 shows the corresponding dialog with its input
parameters. The main inputs comprise the specification of the time interval for which
demands should be considered, the definition of a set of finished products to start with
as well as an optional list of locations to be considered in the model. If not specified,
materials at all locations are included.
Figure A.2: Network generation dialog
Once a model has been generated, it is stored in the underlying database along with
optimal descriptive meta data. Via the models management perspective shown in
Figure A.3, all models stored in the database can be viewed, edited, visualised and
deleted. As all models are stored in a central database, this also allows to easily share
models and thus facilitates collaboration.
A.1 Model Building and Management 191
Figure A.3: Model management
192 A Implementation
Models can be edited in terms of the external demands and forecast deviations stored
at each item as well as in terms of their structure. In order to allow easy editing of
demands and forecast deviations for a large number of items, a list of items together
with their demands and forecast deviations is written to a spreadsheet file. In this
file, the demand parameters can be adjusted, before the file is parsed again by the
Complana tool and the changes defined in the spreadsheet are applied to the model.
Figure A.4(a) shows a detail of such a model editing spreadsheet file, while figure A.4(b)
shows how the network structure can be changed by adapting the predecessors of single
items.
Plant
Material
No.
Material
Type NodeType
SumExternal
Demands
NewTotalexternal
Demand
Adjustdemand
percentage
Externalforecast
deviation
Newexternalforecast
deviation
ForecastDeviation
Multiplier
ES10 11200
6
FIN DIS
T
25 25 100% 100% 100% 1
ES11 11200
6
FIN PROD 0 0 100% 0% 0% 1
DE40 12625
0
FIN DIS
T
728 728 100% 100% 100% 1
DE30 12625
0
FIN DIS
T
0 0 100% 0% 0% 1
DE31 12625
0
FIN PROD 0 0 100% 0% 0% 1
BE10 112441 FIN DIS
T
2394 2394 100% 33% 33% 1
BE11 112441 FIN PROD 0 0 100% 0% 0% 1
BE10 11244
4
FIN DIS
T
115 115 100% 74% 74% 1
BE11 11244
4
FIN PROD 0 0 100% 0% 0% 1
BE10 120829 FIN DIS
T
1379 1379 100% 28% 28% 1
BE11 120829 FIN PROD 0 0 100% 0% 0% 1
BE10 121311 FIN DIS
T
2312 2312 100% 58% 58% 1
BE11 121311 FIN PROD 0 0 100% 0% 0% 1
TR10 12190
4
FIN PROD 153 153 100% 307% 307% 1
TR10 121902 FIN PROD 1269 1269 100% 151% 151% 1
TR10 121903 FIN PROD 467 467 100% 186% 186% 1
TR10 121905 FIN PROD 1248 1248 100% 72% 72% 1
TR10 12364
4
FIN PROD 4970 4970 100% 44% 44% 1
TR10 12364
6
FIN PROD 1979 1979 100% 85% 85% 1
TR10 108408 FIN PROD 0 0 100% 0% 0% 1
ES10 10155
4
FIN DIS
T
38 38 100% 97% 97% 1
ES11 10155
4
FIN PROD 0 0 100% 0% 0% 1
PT10 125847 FIN DIS
T
518 518 100% 148% 148% 1
PT10 100073 FIN DIS
T
55 55 100% 85% 85% 1
ES10 100073 FIN DIS
T
2563 2563 100% 64% 64% 1
ES11 100073 FIN PROD 0 0 100% 0% 0% 1
DE20 10180
6
FIN DIS
T
2322 2322 100% 26% 26% 1
DE40 10180
6
FIN DIS
T
253 253 100% 67% 67% 1
DE44 10180
6
FIN DIS
T
44 44 100% 45% 45% 1
IT20 10180
6
FIN DIS
T
140 140 100% 53% 53% 1
FI10 10180
6
FIN DIS
T
42 42 100% 29% 29% 1
SE10 10180
6
FIN DIS
T
3 3 100% 100% 100% 1
DE30 10180
6
FIN DIS
T
0 0 100% 0% 0% 1
DE31 10180
6
FIN PROD 0 0 100% 0% 0% 1
PT10 100072 FIN DIS
T
10 10 100% 100% 100% 1
ES10 100072 FIN DIS
T
449 449 100% 59% 59% 1
ES11 100072 FIN PROD 0 0 100% 0% 0% 1
ES10 10154
4
FIN DIS
T
12 12 100% 100% 100% 1
ES11 10154
4
FIN PROD 0 0 100% 0% 0% 1
ES10 101545 FIN DIS
T
8 8 100% 100% 100% 1
ES11 101545 FIN PROD 0 0 100% 0% 0% 1
ES10 10154
6
FIN DIS
T
11 11 100% 100% 100% 1
ES11 10154
6
FIN PROD 0 0 100% 0% 0% 1
ES10 101547 FIN DIS
T
15 15 100% 100% 100% 1
ES11 101547 FIN PROD 0 0 100% 0% 0% 1
ES10 101549 FIN DIS
T
5 5 100% 100% 100% 1
ES11 101549 FIN PROD 0 0 100% 0% 0% 1
ES10 10155
0
FIN DIS
T
55 55 100% 100% 100% 1
ES11 10155
0
FIN PROD 0 0 100% 0% 0% 1
ES10 101551 FIN DIS
T
12 12 100% 100% 100% 1
ES11 101551 FIN PROD 0 0 100% 0% 0% 1
ES10 101558 FIN DIS
T
5 5 100% 100% 100% 1
ES11 101558 FIN PROD 0 0 100% 0% 0% 1
ES10 101559 FIN DIS
T
10 10 100% 100% 100% 1
ES11 101559 FIN PROD 0 0 100% 0% 0% 1
ES10 101563 FIN DIS
T
8 8 100% 100% 100% 1
ES11 101563 FIN PROD 0 0 100% 0% 0% 1
ES10 101565 FIN DIS
T
4 4 100% 100% 100% 1
ES11 101565 FIN PROD 0 0 100% 0% 0% 1
ES10 101568 FIN DIS
T
12 12 100% 100% 100% 1
ES11 101568 FIN PROD 0 0 100% 0% 0% 1
ES10 10157
0
FIN DIS
T
13 13 100% 100% 100% 1
ES11 10157
0
FIN PROD 0 0 100% 0% 0% 1
ES10 101571 FIN DIS
T
24 24 100% 100% 100% 1
ES11 101571 FIN PROD 0 0 100% 0% 0% 1
ES10 112008 FIN DIS
T
4 4 100% 100% 100% 1
ES11 112008 FIN PROD 0 0 100% 0% 0% 1
ES10 11201
0
FIN DIS
T
4 4 100% 100% 100% 1
ES11 11201
0
FIN PROD 0 0 100% 0% 0% 1
ES10 112012 FIN DIS
T
4 4 100% 100% 100% 1
ES11 112012 FIN PROD 0 0 100% 0% 0% 1
ES10 117063 FIN DIS
T
4 4 100% 100% 100% 1
(a) Model editing spreadsheet file
(b) Changing predecessor
relations of an item
Figure A.4: Editing possibilities for existing models
Another useful feature for working with production and distribution networks is the
visualisation component. Existing models can be visualised by automatically trans-
A.2 Scenario Building and Management 193
forming their network structure into the GraphML file format and opening that file
with the third-party tool yEd. Figure A.5 shows the corresponding dialog in the Com-
plana tool. The visualisation can be further parameterised, e.g. by omitting raw and
packaging materials to reduce the network complexity. Furthermore, an arbitrary set
of material numbers can be specified such that only items related to these materials
over some connection are included in the visualisation. This opens the possibility to
use the visualisation as a graphical where-used list, which can be used e.g. to deter-
mine the set of end products and locations that would be affected by a change in a
base material or some packaging.
Figure A.5: Network visualisation dialog
A.2 Scenario Building and Management
Scenarios are defined in a spreadsheet file based on a certain template. In three
distinct sections new material additions, finished material replacements as well as
replacements of raw/packaging/semi-finished materials are defined. While especially
194 A Implementation
the replacement of finished products is highly configurable, it is always possible to
only define the material numbers of the discontinued and of the replacement materials,
leaving all other options to their defaults. Conversion factors for replacements can be
calculated automatically from within the software tool. Figure A.6 shows a detail of a
scenario file for replacement of finished materials.
Plants
[optional]
Material
No. Description[autofill]
#Products
[autofill]
Replacement
MaterialNo. Description[autofill]
#Products
[autofill]
ProcurementSource
SPT[optional]
Demand
Percentage
[optional]
ConversionFactor
[optional/autofill]
121311 CLTWND/_/T/_12Bx__2P/390x360/200x180/YW 24 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 D3 1.67
126250 CLTWND/P/T/_30Bx__2P/395x365/200x185/YW 60 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.67
121902 CLTWND/_/T/_4Tx_80Bx__1P/390x340/195x170 80 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.50
115021 CLTWND/_/T/_20Bx__5P/550x430/270x190/BN 100 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.40
125847 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 1.00
120829 CLTWND/P/T/_12Bx__2P/390x360/200x180/YW 24 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 1.67
100072 CLTWND/_/T/_6Sx60Bx1P/550x395/235x200/YW 60 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.67
112441 CLTWND/_/T/_15Bx__1P/360x390/180x200/YW 15 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 2.67
121905 CLTWND/_/T/_20Bx__2P/390x340/195x170/YW 40 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 1.00
112439 CLTWND/_/T/_10Bx__1P/360x490/YW 10 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 4.00
112431 CLTWND/_/T/10Bx_30Bx__1P/395x365/200x185 300 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.13
113571 CLTWND/_/T/100P/395x365/YW/C 100 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.40
115332 CLTWND/_/T/_30Bx__1P/550x395/270x190/YW 30 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 1.33
101550 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
121904 CLTWND/_/T/_4Tx_80Bx__2P/390x340/195x170 160 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.25
115331 CLTWND/_/T/_40Bx__1P/395x365/270x190/YW 40 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 1.00
115950 CLTWND/P/T/_10Bx__1P/395x365/200x185/YW 10 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 4.00
112636 CLTWND/_/T/_10Bx__5P/490x360/YW 50 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.80
101554 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
121625 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
117516 CLTWND/_/T/_48Bx__1P/365x395/185x200/YW 48 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 D3 0.83
119719 CLTWND/_/T/_12Bx__1P/395x365/200x185/YW 12 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 3.33
121903 CLTWND/_/T/_20Bx__1P/390x340/195x170/YW 20 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 2.00
113945 CLTWND/_/T/1000P/440x370/220x185/CS 1000 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.04
112006 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
101571 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
112444 CLTWND/_/T/10Bx_30Bx__1P/390x360/200x180 300 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.13
112635 CLTWND/_/T/_10Bx__3P/390x360/200x180/YW 30 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 1.33
101547 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
112442 CLTWND/_/T/_15Bx__2P/360x390/180x200/YW 30 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 1.33
101570 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
121651 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
101551 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
101544 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
101546 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
101568 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
101559 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
114292 CLTWND/_/T/300P/490x360/YW/C 300 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.13
101545 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
119575 #CLTWND/_/T/_12Bx__1P/400x360/195x175/BN 12 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 D3 3.33
101563 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
101549 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
112012 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
101558 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
121650 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
121627 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
117063 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
112008 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
112010 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
101565 CLTWND/_/T/250Bx__1P/395x330/200x170/YW 250 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.16
111282 CLTWND/_/T/_10Bx__1P/365x395/200x185/YW 10 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 4.00
102027 CLTWND/_/T/_3Sx30Bx1P/360x400/180x200/YW 30 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 1.33
101187 CLTWND/_/T/_15Bx__3P/400x350/200x175/EXP 45 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 0.89
117561 CLTWND/_/T/_12Bx__1P/400X400/140X140/RS 12 119999 CLTWND/P/T/_2Sx_20Bx__2P/395x365/200x185 40 3.33
Figure A.6: Scenario definition in a spreadsheet file
Once a scenario definition is applied to a baseline model, the same viewing, editing
and visualisation features can be used as already described for the baseline models in
the preceding section.
A.3 Optimisation Methods
For all generated models and scenarios, the optimisation of inventory allocation, plan-
ning buffer and planned production quantities can be invoked. Figure A.7 shows the
user dialogs in the GUI with the possible parameter settings for each optimisation run.
When the optimisation is finished, the resulting optimised networks can be stored in
the database as new models or the existing models can be updated with the optimised
values. Furthermore, the results of the optimisation are written to a spreadsheet file
to be used for more in-depth analysis.
A.4 Cost Comparison 195
(a) Inventory allocation parameters (b) Production optimisation parameters
Figure A.7: Optimisation dialog wizard
A.4 Cost Comparison
For all generated models and scenarios, a comparative cost analysis can be invoked.
The Complana tool compares the selected model and scenario with respect to the
number of items for the different material types, inventory and production cost. Fig-
ures A.8 shows the selection of models and scenarios to be compared, as well as an
example of a chart that compares their inventory costs.
196 A Implementation
Figure A.8: Selection of models for cost comparison
197
APPENDIX B
Example Networks
This section provides some exemplary visualisations of the assortment models and
scenarios used for the validation in Chapter 6. The visualisations of model M2and its
related scenarios have been skipped due to their mere size, as their visual representation
is far too large to be scaled on a single page. These visualisation examples should
serve the purpose to allow a visual comparison of the complexity of the models and
the corresponding scenarios.
198 B Example Networks
Figure B.1: Network visualisation for M1
199
Figure B.2: Network visualisation for M1/S1
200 B Example Networks
Figure B.3: Network visualisation for M1/S2
201
Figure B.4: Network visualisation for M1/S3
202 B Example Networks
Figure B.5: Network visualisation for M1/S4
203
Figure B.6: Network visualisation for M1/S5
204 B Example Networks
Figure B.7: Network visualisation for M1/S6
