Two-Level Capacitated Lot Sizing in
Production Control to Guarantee Availability,
Considering Multidimensional Restrictions
Dissertation
zur Erlangung der Würde eines
Doktors der Wirtschaftswissenschaften
(Dr. rer. pol.)
der Universität Paderborn
vorgelegt von
Dipl.-Wirt.-Inf. Daniel Brodkorb
33102 Paderborn
Paderborn, März 2011
Dekan: Prof. Dr. Peter F.E. Sloane
Referent: Prof. Dr.-Ing. habil. Wilhelm Dangelmaier
Korreferentin: Prof. Dr. Leena Suhl
Acknowledgements
This doctoral thesis resulted from the research I conducted in cooperation with the de-
partment of Business Computing, especially CIM at the Heinz Nixdorf Institute, the
International Graduate School Paderborn and the plant in of Keiper GmbH & Co.KG
Rockenhausen, Germany.
First, I would like to thank my advisor and academic teacher Prof. Dr.-Ing. habil. Wil-
helm Dangelmaier for scientific support and valuable advice. He provided the possibil-
ity to realize the thesis project in cooperation with an industrial partner. That made it
possible to combine research and industrial practice and to give insight into the clash
between scientific methods and their application and acceptance in practice. I would
like to thank my supervisor for his support in many constructive discussions with practi-
tioners at Keiper. I also would like to thank my second supervisor Prof. Dr. Suhl for her
advice and support.
Then, I would like to thank the Keiper Rockenhausen plant manager Martin Queck. His
ability to understand methodologies quickly, his patience and his openness for new
methods helped me to complete the project. I would also like to thank Martin Westrich,
formerly Manager of Logistics, for ideas and advice during the project. Many thanks go
to other Keiper employees: Stefan Boller, Hans-Peter Buchmann, Marcel Frazao,
Volker Klein, Margit Krauss, Joachim Lanzer, Manuel Nickel, Thomas Schwind, and
Ute Zache. They helped not only with their high technical skills and expertise but also
with their openness and kindness and motivated me to sustain constant progress on my
work.
I wish to express my appreciation for my colleagues at the working group Business
Computing for the good atmosphere and the possibility to discuss problems that ap-
peared in detail. I would like to thank all the students who supported my scientific work
and helped me to realize the research project.
My deepest gratitude goes to my family – my parents, my sister and my wife. Their
love, patience, and permanent encouragement gave me support, hope, and motivation to
finish my education. I would like to give special thanks to my wife Janet-Lucia for the
understanding she showed for the time I had to spend to complete the work and for her
encouragement during difficult times.
For Janet-Lucia
Table of Contents
Illustrations ................................................................................................................ v
Abbreviations and Symbols ....................................................................................... vii
1
Introduction ........................................................................................................... 1
2
Problem Statement / Problem Decomposition ...................................................... 3
2.1
The Necessity of Guaranteeing Availability to Customers ......................... 4
2.1.1
Characterization of Customer Orders ................................................... 5
2.1.2
Flexibility vs. Costs .............................................................................. 6
2.2
Restrictions .................................................................................................. 10
2.2.1
Workforce ............................................................................................. 11
2.2.2
Machines ............................................................................................... 14
2.2.3
Dies ....................................................................................................... 17
2.2.4
Raw Material ........................................................................................ 19
2.2.5
Loading Equipment .............................................................................. 21
2.2.6
Batches.................................................................................................. 22
2.2.7
Lots ....................................................................................................... 23
2.3
Two-Level Capacitated Lot Sizing in Production Control .......................... 24
2.3.1
Mid-Range Level .................................................................................. 24
2.3.2
Short-Range Level ................................................................................ 26
3
State of Art ............................................................................................................ 27
3.1
Improvement of Delivery Service Availability ........................................... 27
3.1.1
Evaluation of Delivery Service Level .................................................. 28
3.1.2
Methods for Improving Supply Availability ........................................ 29
3.2
Flexibility vs. Costs ..................................................................................... 31
3.2.1
Total or Complete Planning .................................................................. 31
3.2.2
Cyclic Planning .................................................................................... 32
3.2.3
Rolling Planning ................................................................................... 32
3.3
Methods for Planning Requirements ........................................................... 33
3.3.1
Workforce Planning .............................................................................. 33
3.3.2
Machine Planning ................................................................................. 36
3.3.3
Maintenance Planning of Dies .............................................................. 41
ii
3.3.4
Raw Material Procurement Planning ................................................... 42
3.4
Two-Level Capacitated Lot Sizing in Production Planning ....................... 44
3.4.1
Decomposition Approaches and Hierarchical Production Planning .... 44
3.4.2
Mid-Term Lot Sizing ........................................................................... 46
3.4.3
Short-Term Lot Sizing ......................................................................... 49
3.4.4
Integrated Mid- and Short-Term Lot Sizing ........................................ 51
4
Action Points ........................................................................................................ 55
4.1
Mid-Term Lot Sizing .................................................................................. 56
4.2
Short-Term Lot Sizing and Scheduling ...................................................... 57
4.3
Coupling of Mid-Term and Short-Term Planning ...................................... 57
5
Concept ................................................................................................................. 58
5.1
Goals, Requirement Prioritization and Decomposition of the Problem ...... 58
5.2
Mid-Term Lot Sizing Considering Multidimensional Restrictions ............ 60
5.2.1
Input ..................................................................................................... 60
5.2.2
Output .................................................................................................. 65
5.2.3
Model ................................................................................................... 66
5.3
Short-Term Scheduling Considering Multidimensional Restrictions ......... 73
5.3.1
Input ..................................................................................................... 73
5.3.2
Output .................................................................................................. 76
5.3.3
Model ................................................................................................... 76
5.4
Coupling of Partial Models and Integration into Real Production .............. 88
5.4.1
Coupling of Partial Models .................................................................. 88
5.4.2
Integration into Real Production .......................................................... 90
5.4.3
Determination of Relevant Short-Term Planning Subsets ................... 92
5.5
Techniques to Improve Solution Time ........................................................ 93
5.5.1
Mid-Term Lot Size Planning ............................................................... 94
5.5.2
Short-Term Schedule Planning ............................................................ 94
6
Realization ............................................................................................................ 97
6.1
System Description ..................................................................................... 97
6.1.1
Overall Architecture ............................................................................. 97
6.1.2
Database and Data Structure ................................................................ 99
6.1.3
Software Structure................................................................................ 101
6.1.4
Application Flow.................................................................................. 102
iii
6.2
SAP Integration and User Interface ............................................................. 106
6.3
Evaluation .................................................................................................... 110
6.3.1
Planning Results ................................................................................... 110
6.3.2
Manual vs. Automatically Generated Plans ......................................... 117
7
Summary and Outlook .......................................................................................... 119
7.1
Summary ...................................................................................................... 119
7.2
Future Outlook ............................................................................................. 121
8
Bibliography .......................................................................................................... 122
v
Illustrations
Figure 1: Product Stages .................................................................................................... 3
Figure 2: Customer Demand Characterization ................................................................. 6
Figure 3: Flexibility Types According to REFA ................................................................ 7
Figure 4: Flexibility Types According to Sethi and Sethi ................................................ 8
Figure 5: Flexibility Types According to Wildemann ...................................................... 8
Figure 6: Classification of Production Factors according to Gutenberg ...................... 11
Figure 7: Cost Factors of Different Day and Shift Types ................................................ 14
Figure 8: Classification of Cost Factors at Machines ..................................................... 16
Figure 9: Loading Equipment Mapping according to Part Properties ........................... 22
Figure 10: Buying Criteria according to Pawellek ........................................................... 28
Figure 11: Demand Planning Framework by Kilger and Wagner ................................... 29
Figure 12: Differentiation of Personnel Planning Approaches according to Rossi ..... 34
Figure 13: Classification of Order Release Methods according to Lödding ................. 36
Figure 14: Comparison of Order Release Methods ......................................................... 39
Figure 15: Classification of Maintenance Strategies ....................................................... 41
Figure 16: Split Planning Horizon ...................................................................................... 55
Figure 17: Visualization of Ending Inventory ................................................................... 63
Figure 18: Erroneous Modeling of Set-up State Changes at Coupled Products .......... 80
Figure 19: Correct Modeling of Set-up State Changes for Coupled Products .............. 82
Figure 20: Illustration of Maintenance Interconnection of Partial Models .................... 89
Figure 21: Summary of Partial Model Interconnection .................................................... 91
Figure 22: Overall System Architecture ............................................................................ 98
Figure 23: Database Diagram ............................................................................................. 100
Figure 24: Software Structure ............................................................................................ 102
Figure 25: Overall Application Flow Diagram .................................................................. 103
Figure 26: Sub-process: Database Update Application Flow Diagram ......................... 104
Figure 27: Sub-process: Instantiation and Import Flow Diagram .................................. 105
Figure 28: Sub-process: Lot Sizing and Scheduling Flow Diagrams ............................ 105
Figure 29: Set-up Parameter Screen ................................................................................. 107
Figure 30: Set-up State Parameterization ......................................................................... 107
Figure 31: Visualization of Optimization Process ............................................................ 108
Figure 32: Mid-Term Planning Results in SAP ................................................................. 109
Figure 33: Mid-Term Lots in SAP ....................................................................................... 109
Figure 34: Short-Term Schedule Visualization in SAP .................................................... 110
vi
Figure 35: Production Amounts: Weingarten I Stamping Machine ................................ 111
Figure 36: Visualization of Production Amounts Weingarten I ...................................... 111
Figure 37: Production Amounts: Weingarten II Stamping Machine ............................... 112
Figure 38: Visualization of Production Amounts Weingarten II ..................................... 112
Figure 39: Combined Utilization Chart (1) ......................................................................... 113
Figure 40: Combined Utilization Chart (2) ......................................................................... 113
Figure 41: Personnel Mid-Term Plan ................................................................................. 114
Figure 42: Raw Material Units Procurement Plan ............................................................ 116
Figure 43: Short-Term Planning Result Visualization ..................................................... 117
Figure 44: Comparison of Manual and Automatic Planning ........................................... 118
Figure 45: Appraisal of Presented Method ....................................................................... 120
vii
Abbreviations and Symbols
Sets:
: ( , )
TM TM
=
⊑
Time-Periods: Mid-Term Planning
: ( , )
TS TS
=
⊑
Time-Periods: Short-Term Planning
p
CP
Set of products which are produced in coupled production with
p P
∈
C
Coils
D
Dies
LE
Loading Equipment
M
Machines
P
Products
ST
P P
⊆
Subset of products, relevant for short-term planning
R
Raw Materials
S
Shifts
Parameters and Variables:
0
, ,
m p tm
bcM ∈
N
Counting variable for completed batches at machine
m M
∈
, product
p P
∈
finished during mid-term planning time-period
tm TM
∈
{
}
, ,
0,1
mtnc
m p ts
binS ∈
=1, if maintenance of product
p P
∈
is activated in short-term period
ts TS
∈
{
}
,
0,1
cw
m ts
binS ∈
=1, if a coil is being changed at machine
m M
∈
during short-term period
ts TS
∈
(0, otherwise)
{
}
,
0,1
prod
m ts
binS ∈
=1, if machine
m M
∈
is producing during short-term period
ts TS
∈
(0,
otherwise)
{
}
,
0,1
r
m ts
binS ∈
=1, if machine
m M
∈
is being set up during short-term period
ts TS
∈
(0,
otherwise)
{
}
, ,
0,1
m p tm
binmM ∈
=1, if die for machine
m M
∈
, product
p P
∈
is maintained during mid-
term planning time-period
tm TM
∈
(0, otherwise)
{
}
, ,
0,1
m p tm
binsM ∈
=1, if set-up is executed to produce product
p P
∈
at machine
m M
∈
dur-
ing mid-term planning time-period
tm TM
∈
(0, otherwise)
{
}
, ,
0,1
m p tm
binsrM ∈
=1, if a set-up with low effort is executed to produce product
p P
∈
at ma-
chine
m M
∈
during mid-term planning time-period
tm TM
∈
(0, other-
wise)
{
}
, ,
0,1
m p tm
binxM ∈
=1, if machine
m M
∈
produces product
p P
∈
during mid-term planning
time-period
tm TM
∈
(0, otherwise)
p
bs
Batch size of product
p P
∈
tm
btM
Break time in mid-term planning time-period
tm TM
∈
viii
p
chw
Charge weight of product
p P
∈
,
mtnc
d tm
cM
Maintenance cost of die
d D
∈
in mid-term planning time-period
tm TM
∈
ir
cM
Interest rate for capital commitment for a mid-term time-period
,
prod
p tm
cM
Production costs of product
p P
∈
in mid-term planning time-period
tm TM
∈
,
setup
p tm
cM
Set-up cost average of product
p P
∈
in mid-term planning time-period
tm TM
∈
inv
p
cM
Inventory holding costs and capital commitment of a product stored
p P
∈
in
one mid-term planning time-period
so
p
cM
Imputed stock-out costs for product
p P
∈
w
p
cM
Warehousing costs of a product for a mid-term time-period
yw
p
cM
Yearly warehousing costs of product
p P
∈
yir
cM
Yearly interest rate for capital commitment
lt
capaLT
Capacity of a loading equipment type
lt LE
∈
, , ,
setup
m p q ts
c
Sequence-dependent set-up costs at machine
m M
∈
from product
p P
∈
to
product
q P
∈
in short-term period
ts TS
∈
, ,
cc
m p ts
c
Costs for coil changes at machine
m M
∈
, product
p P
∈
in short-term
period
ts TS
∈
, ,
mtnc
m p ts
c
Maintenance costs for one short-term period
ts TS
∈
related with a machine
and product
, ,
prod
m p ts
c
Variable machine production costs of product
p P
∈
at machine
m M
∈
during short-term period
ts TS
∈
[
]
, ,
0,1
m p tm
cmM ∈
Percentage of die maintenance ready for machine
m M
∈
, product
p P
∈
is
maintained at the end of mid-term planning time-period
tm TM
∈
[
]
, ,
0,1
mtnc
m p ts
cmS ∈
Cumulated progress of maintenance in per cent of product
p P
∈
in short-
term period
ts TS
∈
of the rolling horizon
inv
p
c
Inventory holding costs and capital commitment of
p P
∈
in one short-term
period
[
]
, , ,
0,1
m p q ts
csS ∈
Set-up progress of a set-up from product
p P
∈
to product
q P
∈
at machine
m M
∈
in short-term period
ts TS
∈
in per cent
team
ts
c
Cost factor of a set-up team for one short-term period
ts TS
∈
{
}
, ,
0,1
m p ts
cwS ∈
=1, if a coil is currently changed at machine
m M
∈
for
p P
∈
in short-term
period
ts TS
∈
(0, otherwise)
,
p tm
dM
Announced demand of product
p P
∈
in mid-term planning time-period
tm TM
∈
ix
,
p ts
dS
Short-term planning demand of product
p P
∈
in short-term period
ts TS
∈
p
dfM
Monthly demand forecast of product
p P
∈
p
eiM
Ending inventory of product
p P
∈
at the end of mid-term planning time-
period
max
n TM
=
of the rolling horizon
p
eiS
Ending inventory of product
p P
∈
at the end of short-term planning time-
period
max
TM TS
∈
of the rolling horizon
{
}
, ,
0,1
m p tm
fmM ∈
=1, if die maintenance for machine
m M
∈
, product
p P
∈
is finished dur-
ing mid-term planning time-period
tm TM
∈
(0, otherwise)
{
}
, ,
0,1
mtnc
m p ts
fmS ∈
Binary state for a completed maintenance of product
p P
∈
in short-term
period
ts TS
∈
0
,
p tm
iM
∈
N
Inventory of product
p P
∈
at the end of mid-term planning time-period
tm TM
∈
0
,
p ts
iS
∈
N
Inventory of product
p P
∈
in short-term period
ts TS
∈
0
, ,
m p tm
lotM
∈
N
Lot variable to memorize produced amount of product
p P
∈
until the end of
mid-term planning period
tm TM
∈
on machine
m M
∈
0
, ,
m p ts
lotS
∈
N
Lot that is the cumulative production of product
p P
∈
in short-term period
ts TS
∈
of product
p P
∈
at machine
m M
∈
p
maxlot
Maximum lot size of product
p P
∈
0
,
p n
miM
∈
N
Missing ending inventory of product
p P
∈
at the end of mid-term planning
time-period
max
n TM
=
of the rolling horizon
p
minlot
Minimum lot size of product
p P
∈
{
}
, ,
0,1
m p ts
mlS ∈
Variable for minimal lot size achievement.
=1, if minimal lot size currently achieved at machine
m M
∈
is producing
product
p P
∈
in the first short-term period
ts TS
∈
(0, otherwise)
,
m p
mpM Maintenance progress in per cent of die for machine
m M
∈
in one mid-term
planning time-period
t TM
∈
{
}
, , ,
0,1
m p q ts
mstS ∈
Minimal set-up time variable.
=1, if a set-up from product
p P
∈
to product
q P
∈
at machine
m M
∈
was finished in short-term period
ts TS
∈
(0, otherwise)
p
mtM
Maintenance time in mid-term periods
p
mtnc
Maintenance time in hours
,
p m
pptS
.
Products per short-term period for product
p P
∈
at machine
m M
∈
p
price
Selling price of product
p P
∈
x
{
}
, ,
0,1
m p ts
prodS ∈
=1, if machine
m M
∈
is producing product
p P
∈
in short-term period
ts TS
∈
(0, otherwise)
p
pt
Production time in minutes of product
p P
∈
{
}
, , ,
0,1
m p q ts
rS ∈
=1, if machine
m M
∈
is currently being set up from product
p P
∈
to
product
q P
∈
in short-term period
ts TS
∈
, whereas
p q
≠
(0,otherwise)
0
, ,
m p ts
reS ∈
N
Number of completely used steel coils of the actual lot relevant for machine
m M
∈
, product
p P
∈
in short-term period
ts TS
∈
0
lt
reqLT
∈
N
Number of required loading equipment entities of type
lt LT
∈
{
}
, ,
0,1
m p ts
sS ∈
=1, if machine
m M
∈
is set up for product
p P
∈
in short-term period
ts TS
∈
(0, otherwise)
0
, ,
m p ts
slS
∈
N
Slack variable, representing the cumulative quantity of uncompleted batches
relevant for machine
m M
∈
, product
p P
∈
in the first short-term period
ts TS
∈
,
p q
stMin
Set-up time of product
p P
∈
to
q P
∈
in minutes
p
stM
Average set-up time in hours for product
p P
∈
,
p q
st
Number of short-term planning periods to represent set-up time of product
p P
∈
to
q P
∈
tm
tM
Available time-based capacity of mid-term planning time-period
tm TM
∈
ts
teamLimS
Limit of available set-up teams during short-term period
ts TS
∈
0
ts
teamsS
∈
N
Number of required set-up teams during short-term period
ts TS
∈
m
udM
Maximum degree of utilization of machine
m M
∈
,
p lt
verbPLT
Usage of loading equipment for a product
p P
∈
in loading equipment type
lt LE
∈
0
, ,
m p tm
xM
∈
N
Production amount of
p P
∈
produced on machine
m M
∈
during in mid-
term planning time-period
tm TM
∈
0
, ,
m p ts
xS
∈
N
Production output of product
p P
∈
at machine
m M
∈
in short-term period
ts TS
∈
,
m p
ReS
ϖ
Initialization of the number of completely used steel coils of the actual lot
relevant for machine
m M
∈
, product
p P
∈
in the first short-term period
min
TS TS
∈
,
m p
SlS
ϖ
Initialization of the slack variable, representing the cumulative quantity of
uncompleted batches relevant for machine
m M
∈
, product
p P
∈
in the
first short-term period
min
TS TS
∈
,
mtnc
m p
binS
ϖ
Initial maintenance binary state of product
p P
∈
in the first short-term peri-
od
min
TS TS
∈
of the rolling horizon
xi
,
m p
binxM
ϖ
Initial production binary state of product
p P
∈
in the first mid-term planning
time-period of the rolling horizon
,
mtnc
m p
cmS
ϖ
Initial cumulated progress of maintenance in per cent of product
p P
∈
in the
first short-term period
min
TS TS
∈
of the rolling horizon
, ,
m p q
csS
ϖ
Initialization of set-up progress of a set-up from product
p P
∈
to product
q P
∈
at machine
m M
∈
in the first short-term period
min
TS TS
∈
in per
cent
,
m p
cwS
ϖ
Initialization of coil change status.
=1, if a coil is currently changed at machine
m M
∈
for
p P
∈
in the first
short-term period
min
TS TS
∈
(0, otherwise)
,
mtnc
m p
fmS
ϖ
Initial binary state for a completed maintenance of product
p P
∈
in the first
short-term period
min
TS TS
∈
of the rolling horizon
p
iM
ϖ
Initial inventory of product
p P
∈
in the first mid-term planning time-period
of the rolling horizon
p
iS
ϖ
Initial inventory of product
p P
∈
in the first short-term period
min
TS TS
∈
of the rolling horizon
,
m p
lotM
ϖ
Initial lot of product
p P
∈
in the first mid-term planning time-period of the
rolling horizon
,
m p
lotS
ϖ
Initial lot of product
p P
∈
in the first short-term period
min
TS TS
∈
of the
rolling horizon
p
mbinM
ϖ
Initial maintenance binary state of product
p P
∈
in the first mid-term plan-
ning time-period of the rolling horizon
,
m p
mlS
ϖ
Initialization of minimal lot size achievement.
=1, if minimal lot size was currently achieved at machine
m M
∈
is produc-
ing product
p P
∈
in the first short-term period
min
TS TS
∈
(0, otherwise)
p
mpM
ϖ
Initial maintenance percentage of product
p P
∈
in the first mid-term plan-
ning time-period of the rolling horizon
p
mpS
ϖ
Initial maintenance percentage of product
p P
∈
in the first short-term period
min
TS TS
∈
of the rolling horizon
, ,
m p q
mstS
ϖ
Initialization of the minimal set-up time variable.
=1, if a set-up from product
p P
∈
to product
q P
∈
at machine
m M
∈
was finished in the first short-term period
min
TS TS
∈
(0, otherwise)
,
m p
prodS
ϖ
Initialization of production.
=1, if machine
m M
∈
is producing product
p P
∈
in the first short-term
period
min
TS TS
∈
(0, otherwise)
, ,
m p q
rS
ϖ
Initialization of machine set-up.
=1, if machine
m M
∈
is currently being set up from product
p P
∈
to
product
q P
∈
in the first short-term period
min
TS TS
∈
(0, otherwise)
xii
,
m p
sS
ϖ
Initialization of machine status.
=1, if machine
m M
∈
is set up for product
p P
∈
in the first short-term
period
min
TS TS
∈
(0, otherwise)
ts
teamsS
ϖ
Initialization of the number of set-up teams in the first short-term period
min
TS TS
∈
tsM
Mid-term period length in hours
tsS
Short-term period length in hours
1
1 Introduction
Customer satisfaction is of substantial interest for companies, which want to sustain
their success.
1
This prioritization determines the targets of production planning and con-
trol, as it is a part of corporate planning.
2
Lot sizing and scheduling are related to pro-
duction planning,
3
especially in multi-variant serial shop fabrication.
4
Due to the influ-
ence on lead times, on flexibility and on the adherence to promised delivery dates, lot
sizing and scheduling has an impact on delivery serviceability. Despite its importance
for productivity, lots and schedules are often planned without using mathematical meth-
ods that will guarantee the optimality of the plans.
The basic problem has already been formulated as a mixed-integer linear program
known as the Discrete Lotsizing and Scheduling Problem (DLSP)
5
. This formulation
cannot, however, be applied in practice
6
as important aspects are disregarded. Dynami-
cally changing customer demands and unexpected events
7
complicate the basic prob-
lem. Inventory costs, sequence-dependent set-up costs, time-dependent production costs
and so on are further examples of complicating factors. Personnel planning has to be
focused as it influences overall costs significantly, especially selecting cheaper shifts for
personnel-intensive tasks. Several technical and organizational restrictions in produc-
tion, like the consideration of sequence-dependent set-up times, batched production,
maximum lot sizes and maintenance of dies, make the calculation of feasible solutions
more difficult. In this work, a lot sizing approach is presented, which considers all the
mentioned as well as further aspects.
Due to the constantly changing environment, it is not useful to spend too much effort
calculating detailed lot sizes and schedules for long planning horizons. Accordingly, the
relevant operative and rolling planning horizon is split into two consecutive levels: On
the first level, rough mid-term production plans are calculated, taking into consideration
all the relevant costs and constraints using an extension of the basic Capacitated Lot-
1 See e.g. [LG09].
2 See [Kur11], p.29, or [Paw07].
3 See e.g. [KS01], pp.40–91.
4 See [Tem06], p.1 and [AIKTF08], p.110.
5 See e.g. [Fle90].
6 The research project was executed in cooperation with a supplier to the automotive industry. Extensions
are based on practical circumstances.
7 In production practice, unexpected events can be machine or die malfunctions resulting in smaller pro-
duction outputs and capacity reductions.
2
Sizing Problem (CLSP).
8
For the short-term, the resulting lots are detailed within the
next planning level. An extension of the DLSP determines maintenance of the dies, per-
sonnel schedules, raw material and loading equipment, as well as procurement and de-
tailed production scheduling, all of which minimize the overall costs.
8 See e.g. [BY82].
3
2 Problem Statement / Problem Decomposition
The subject matter of this work is to give appropriate operative production plans which
define lot sizes and schedules in capacitated production environments. The considered
plant consists of different product stages:
Figure 1: Product Stages
First, supplied raw material is stocked. The raw material is then processed in the mold-
ing presses stage. After that, half-finished parts are stocked before they are washed, pol-
ished and/or hardened. The need for passing parts through sub-processes as well as the
material flow through sub-processes depends on the part. These steps, pooled in the
black box (see figure 1), are neglected in this work, as the processing lead times are
similar and because of high capacities. After that, finished parts are stocked and later
assembled as final products, which are dispatched to the final customer. In this work,
the focus is laid on the molding presses stage. The following sections describe the prob-
lem in further detail.
As competitiveness can only be sustained by satisfying customer needs, availability of
supply is of great importance. The first section is dedicated to describing the obligatory
guarantee of availability and characterized customer demands. In order to satisfy de-
mands, several manufacturing resources are required. Being one of the major cost driv-
ers, human resources have to be considered in production plans. Available machines as
well as raw material and molding tools, from now on referred to as “dies”, must be used
as efficiently as possible in order to produce at minimum cost. Lots and batches underlie
constraints induced by production requirements which are described in the restrictions
section. Lastly, the problem is broken down into smaller sub-problems that must be
solved, and which are detailed in the last section. The goal and the necessities for each
sub-problem are outlined. The broken-down problem and solution approaches for the
sub-problems constitute the lot sizing concept.
Half-Finished
Part Stock
Molding Blackbox
Assembling
Dispatching
Raw Material
Stock
Finished Part
Stock
Finished Product
Stock
4
2.1 The Necessity of Guaranteeing Availability to Customers
The long-term goal of manufacturing producing companies is to be successful. In par-
ticular, six strategic factors of success or competitive advantages
9
are named in the liter-
ature: costs, quality, flexibility, time, product variety and service.
10
An aspect of logisti-
cal service quality is the company’s ability to deliver the correct amount of ordered
products at the agreed time. This becomes more important as customers have higher
exigencies towards supply availability due to the request of higher flexibility at low
costs in a volatile and competitive environment.
The change from stock-oriented to more flexible just-in-time or even just-in-sequence
production,
11
which is induced by the shift from a sellers’ to a buyers’ market, forces
the necessity of coupling production systems
12
along the supply chain in order to fulfill
changing customer demands as quickly as possible. An established communication be-
tween partners is a precondition for that. The basic interconnection between customers
and their suppliers is the transfer of orders. Therefore, orders placed and the way in
which they are placed have to be examined.
In particular, this work deals with the production and delivery of parts for the automo-
tive industry. The sample company is a first and second tier supplier which produces
seat parts and seat components for cars. The problem properties, which are described in
the next sub-sections, can be found at other automotive suppliers and even in different
industries. First, the customers’ orders are characterized, and then the flexibility is ex-
plained and put into contrast with induced costs.
9
According to[Sim88], a competitive advantage is a performance which is better than the performance of
a competitor if the following criteria are met:
1. The performance has to be an important feature for the customer
2. The performance has to be recognized and realized by the customer
3. It should not be possible for competitors to copy the performance quickly and the performance
should be sustainable
10 See [KB05] (p.6 et seqq.), [KG83] (p.27 et seqq.), [Eid91] or [BGG89].
11 The main concept of just-in-time production is the initiation of goods and services by a customer order
[Dan09] (p.1300). Just-in-sequence production is often considered as an evolution of just-in-time pro-
duction for a production environment with a high number of variants [TDS9].
12 The composition of a production system is described in [Dan09](translation): “A production system
consists of (elementary) working systems, which represent the smallest unit of a combination of poten-
tial factors operating resources and workforce and which can execute one or more classes of transfor-
mations.”
5
2.1.1 Characterization of Customer Orders
As the fulfillment of customer demands is a competitive advantage from the viewpoint
of an automotive supplier, it is considered in this work and analyzed in the case in ques-
tion in further detail in this sub-section. Customers of the selected company are Original
Equipment Manufacturers (OEM) as well as first tier suppliers. The plant which is be-
ing examined delivers its products not only to external customers but also to other
plants within the company, called internal customers.
Customer orders
13
are transmitted and updated electronically via the installed Enterprise
Resource Planning (ERP) system. This enables the customer to be rather flexible and to
change orders quickly. Although there exist long-term forecasts of sales for each prod-
uct, which are necessary to dimension required capacities correctly, demands vary de-
pending on the final consumer demand. Especially on the mid- and short-term horizon,
seasonal reasons, marketing campaigns, stock increments or reductions along the supply
chain or other accounts effect fluctuation of demands. This work concentrates on the
operative level. A major problem to consider is therefore the satisfaction of altering cus-
tomer orders with available but limited and fixed capacities at minimal costs on a short-
term horizon.
An analysis of customer orders and order changes made some characteristics observa-
ble. Demand planning is carried out in a hierarchical way. As stated before, forecasts
exist for about two years which serve as a basis for calculating required manufacturing
capacities as well as required human resources. The results are input for shorter time-
periods. Yearly demand forecasts are apportioned to each month. These calculations are
adapted and corrected, applying the expert knowledge of production planners and using
statistical methods, so that suitable monthly demand forecasts are available. More prob-
lematic are demand forecasts in shorter time-periods. As most customers apply just-in-
time or even just-in-sequence principles in their production systems, orders are often
modified regarding the amount or/and the exact delivery time due to changes in the cus-
tomer environment. Relevant information about customer disruption concerning produc-
tion or supply is not transmitted instantly. The differences between real and forecast
orders depend on the time to the planned delivery.
13 In this work, dependent demands as well as independent demands (see [OLL93] for a description of
both terms) are available in the considered production. As the considered stage is at the beginning of
the production process, the types do not have to be differentiated.
6
Figure 2: Customer Demand Characterization
The farther away the planned order date, the greater the variance of the exact delivery
time and ordered amount. Experience has shown that starting from today orders from
day one to three are not changed by the customer and can be considered as fixed. From
day four to fourteen, orders change slightly. After that, order time and amount are no
longer guaranteed and the production is confronted with high order variation.
14
As the customer usually does not reveal information about process problems to the sup-
plier but instead asks for just-in-time supply, the supplier has to adapt to this situation.
The resulting questions are how the supplier can adapt to such volatile environments
and how much the expected flexibility costs.
2.1.2 Flexibility vs. Costs
Due to the development from a sellers’ market to a buyers’ market, production princi-
ples have changed from push to pull. Today, material and products are no longer pushed
into production (push principle) but available orders are realized (pull principle). This
requires flexibility in production as the equalization of the order inflow with the produc-
tion plan needs rapid reactions.
15
Automotive OEMs in particular exact accurate and on-
14 See [Tha97] for a basic description of delivery request systems. See [VDa96], [VDA91] and [VDA96]
for further details.
15 See [Wan05].
80
100
120
140
160
180
Today Day 2 Day 4 Day 6 Day 8 Day 10 Day 12 Day 14 Day 16 Day 18 Day 20
Contractually Guaranteed Maximum Demand
Expected Demand
Contractually Guaranteed Minimum Demand
No Changes of
Demand
High Demand
Variation
7
time delivery of ordered goods at low costs. This sub-section is dedicated to the prob-
lem each supplier has to deal with: the balance between flexibility and costs.
The great number of different definitions
16
is a result of heterogeneous terms and di-
verse definitions about the dimensions of flexibility and the varied understandings of
the delimitations of flexibility stretching to other terms like agility or adaptability.
17
Horvárth and Mayer give a definition in the context of manufacturing. They consider
flexibility as the ability to advance production in the short term and to keep freedom of
action in the long term. As bordering areas like personnel management, finance or pur-
chasing have a great impact on flexibility, they have to be reconciled with production.
18
Schmigalla defines flexibility as the capability of a production system, which is consid-
ered to be fixed during a defined time horizon, to adapt to changing requirements in-
duced by the range of products and the technological process without changing the
numbers of elements and without changing the structure.
19
Handrich combines two def-
initions and describes flexibility as the ability to adapt to changed environmental condi-
tions which can occur in the future. Flexibility can generally be described as the ability
to change within defined dimensions and scenarios.
20
A standardized classification of flexibility types in entrepreneurial practice does not
exist. Some authors make a classification on a time basis and others classify flexibility
types according to system-dependent dimensions. A classification of flexibility types,
which also groups the types according their time frame, is given by REFA.
21
Flexibility Type Quantitative Description Time Frame
Flexibility of extension Effort to make extensions Long term
Flexibility of adaptations Effort to make modifications
Flexibility of products Number of different component parts,
degree of freedom at machine scheduling
Short term Redundancy of production Number of alternative means of production
Flexibility of amount Restrictions of additional shifts or reduced
hours
Figure 3: Flexibility Types According to REFA
16 See [SM98].
17 See [KK05],[KB05].
18 See [HM86].
19 See [Sch95].
20 See [Han02].
21 See [Rog09].
8
A classification of flexibility by Sethi and Sethi
22
is made according to system depend-
ent dimensions. Eleven flexibility types are differentiated and scopes of flexibility are
identified within group flexibility types.
Scope of Flexi-
bility Type of Flexibility Description
Flexibility of
compo-
nent/basis
Flexibility of machine Variety of operations at one machine without
set-up
Flexibility of material flow Ability to produce various parts efficiently us-
ing different flow paths
Flexibility of workflow Possibility of different workflows
Flexibilities of
the system
Flexibility of process Ability to produce various parts without recon-
figuration or rebuilding within the system
Flexibility of process se-
quence
Possibility of producing a part in different se-
quences
Flexibility of product
range Ease of introducing new products
Flexibility of production
quantities
Ability to work economically at different work-
loads
Flexibility of extensions Effort to adapt the flexibility and the ability to
work
Aggregated
Flexibilities
Flexibility of production
program
Stability of the system to produce different
variants without changing resources
Flexibility of production Variety of production of the system to produce
parts without rebuilding but with set-ups
Flexibility towards market Ability of the system to react to market changes
Figure 4: Flexibility Types According to Sethi and Sethi
This way of classifying flexibility by means of system-dependent dimensions is also
used in a similar classification carried out by Tempelmeier.
23
Another classification is made by Wildemann,
24
which is based on a differentiation be-
tween quantitative, qualitative and time flexibility.
Group Quantitative Flexibility Qualitative Flexibility Time Flexibility
Differentiation Adaptation to varied
quantities and structures
Adaptations to new manu-
facturing tasks
Time necessary to change
production tasks
Characteristics
Ability to
- Expand
- Compensate
- Store
- Versatility, ability to set
up
- Manufacturing redundan-
cy
- Ability to rebuild
- Permissive throughputs
- Automated changes
Figure 5: Flexibility Types According to Wildemann
Essentially, most definitions refer to the availability of freedom of action, the availabil-
ity of freedom for decisions, or the possibility of varying something in conjunction with
22 See [SS90].
23 See [TK93].
24 See [Wil87].
9
changes.
25
Since uncertainty as well as unpredictable environmental changes overbur-
den the technical and organizational adaptability, it is important to consider the change-
ability of production systems regarding structure and available resources. All in all, it
can be said that flexibility of a production system, which is characterized by its adapta-
bility and changeability to counteract environmental changes, creates and extends the
technical and organizational scope for action.
26
The initial generation of flexibility and
sometimes the sustainment of flexibility are related to time
27
and consequently to
costs.
28
By considering only some aspects of Sethi and Sethi’s classification, it is easy
to find examples:
1. Flexibility of machine: A machine which can execute a variety of operations
without set-up is more expensive than a simple machine designed to do only one
task.
2. Flexibility of production quantities: In order to cope with different workloads,
capacities of production requirements have to be adapted. Capacity extensions
are often related to investments (i.e. machines) and take time. Capacity reduc-
tions are also limited as former invested capital is bound up in, amongst other
things, buildings, machines or the specialized know-how of the personnel.
3. Flexibility of production program: Changes in the production program influence
many entities of the production system. Overall flexibility costs are induced by
the sum of the flexibility costs for all influenced entities. If considering, for ex-
ample, only influenced machines, the sum of adaptation costs has to be calculat-
ed.
Flexibility enables adaption to market dynamics, in other words, changing customer
demands. Investments in capacities to gain flexibility have to generate an adequate ad-
vantage. Instead of obtaining further entities to increase production capacity, the usage
of the available ones should be analyzed and improved. A way to improve productivity
quickly and with less financial investments is to automate and optimize planning. One
goal of this work is to free capacity as a consequence of optimized planning resulting in
larger flexibility to satisfy varying customer demands. Another goal is to make deci-
25 See [Dor86].
26 See [Rog09].
27 [Hop89] identifies several types of period for an activity for improving flexibility to take effect. The
time to perceive a change is taken into account, as well as the time to decide the activity, to realize the
activity and finally the time the activity needs to take effect.
28 See [Hal99].
10
sions faster and to reduce the delay time of activities executed as a reaction to chang-
es.
29
In the next section, requirements which are necessary for production in the considered
production stage are described in detail. The flexibility of each requirement is analyzed
in order to describe planning decisions’ degrees of freedom.
2.2 Restrictions
In [Ame06], a restriction is defined as “Something that restricts; a regulation or limita-
tion.” A more precise and suitable definition is given in [Agn02]: “A condition that im-
poses a constraint on the possible values of a variable or in
the domain of arguments of
a function.” This definition in the m
athematical sense is useful for the purpose of this
work.
In production there are many technological and organizational aspects which restrict
decisions in many dimensions. Restrictions complicate decision making significantly.
Without restrictions, it would be easy to satisfy all customer demands in the considered
practical case. The required workforce, machines, dies, raw materials and loading
equipment mentioned and described are necessary to produce and, consequently, they
are also necessary to satisfy customer demands. These resources are not ubiquitous;
they are only available in limited amounts. Different flexibility degrees pose a further
challenge during decision making, as every requirement has to be considered individual-
ly and the interrelation between the requirements complicates the problem. Hence, the
availability of resources has to be considered over time. As production does not run
without production factors, the consideration of existing limitations of production fac-
tors
30
is essential during the preparation or planning
31
of production. Gutenberg
32
char-
acterizes and groups production factors as follows:
29 See [Hop89] for further details.
30 Production factors are the inputs of a production [Dan09].
31 According to [WVW00], “planning is a notional anticipation of future events” (translated). In [Lut],
planning is defined as “basic management function involving formulation of one or more detailed
plans to achieve optimum balance of needs or demands with the available resources.” In [Hah96]
(translated), production planning is defined as notional anticipation of future events through a system-
atic preparation of decisions and a systematic decision taking. It contains the decision process to
search, evaluate and choose between solution alternatives to solve a problem in a target oriented way”.
He further states that planning as well as control are the most important leading and management tasks.
32 See [Gut83].
11
Human Work Manufacturing re-
sources Material
Management Planning Organization
Object-
related
work
Machines,
tools, build-
ings,
grounds
Supplies
Raw material, half
finished products,
finished products
Figure 6: Classification of Production Factors according to Gutenberg
First, production factors can be subdivided into dispositive factors and elementary fac-
tors, which have direct influence on the production process. Object-related work direct-
ly influences the production process and manufacturing resources as well as raw materi-
al. Human work, which is dedicated to management and control of the companies’ busi-
ness processes, is further subdivided by Gutenberg into original factors and derivative
factors. The following sub-sections describe those production factors which are relevant
for the considered production stage. Starting with a detailed description of the human
workforce, the most important manufacturing resources as well as needed materials are
described.
2.2.1 Workforce
This section is dedicated to defining and describing relevant problem details about the
workforce. Regarding Gutenberg’s classification of production factors, planning and
object related work is relevant. Beginning with a definition, human work and related
processes in the actual practical case are described. Lastly, available flexibility and costs
are mentioned and described.
A workforce is “the total number of workers employed by a company on a specific job,
project, etc.”
33
or “all the people working or available to work, as in a nation, company,
industry, or on a project.”
34
There exist several other words expressing the workforce in
a company. Definitions for personnel are similar: “The body of persons employed by or
active in an organization, business, or service”
35
or “persons employed in any work,
33 See [Agn02].
34 See [Ame06].
35 See [Ame06].
Dispositive factors Elementary factors
Original
factors
Derivative factors
Potential factors Consumable factors
12
enterprise, service, establishment, etc.”
36
Another term often used to describe the pro-
duction factor “Human work” is “Human Resource”: the “scarcest and most crucial re-
source that creates the largest and longest lasting advantage for an organization. It re-
sides in the knowledge, skills, and motivation of people, is the least mobile of the four
factors of production, and (under right conditions) learns and grows better with age and
experience which no other resource can.”
37
For the purpose of this work, there can be
found a suitable definition in [Beu96] (translation):”Human work is a potential factor
with the inborn and trained ability to do corporal and mental work”.
In the case study, different types of workers are necessary in order to keep production
running. In this document, only those workforce types are mentioned and described
which are relevant to keeping production running on an operative timescale. First, pro-
duction planners will be described. Production planners are responsible for planning and
scheduling production on a specified subset of machines.
38
As production planners pos-
sess detailed expert knowledge about products and processes, it is difficult to replace
them. Machines are not necessarily compatible with each other. Commutability of plan-
ner–machine assignments is therefore impeded. Because of the emerging risk, the re-
sulting dependency of the company on specialized workers is not desirable. In the case
study, workers with specialized skills are required to change the dies of the machines.
These workers, called machine operators, are assigned to a single machine. Despite
comparably expensive working hours, machine operators are not explicitly considered
during production planning and scheduling. If the assigned machine is already set up,
the machine operator controls the production process, books the number of produced
parts and replaces used and empty raw material units. Machine operators also help other
machine operators during the die change at their machines. Although the time needed to
change dies can be significantly reduced by having machine operators as set-up helpers,
it does allow for the planning of parallel changes of dies at different machines. A set-up
includes all tasks involved in the changing of one part for another. The used die has to
be released and transported to the maintenance department. The new die has to be car-
ried to and mounted on the machine. As a precondition before serial production can
start, first a number of produced parts have to be quality checked. Dimensions are
measured and compared with specifications. Depending on the part, this is sometimes
done by machine operators and sometimes by specialized personnel. In both cases a
limited number of measurement instruments are required to do this. Stackers, who are
36 See [Agn02].
37 See [Lut].
38 In the case study, every production planner has to create production plan schedules for between two
and six machines.
13
only needed for bigger or more delicate parts, are another type of worker. Stackers do
not have to be highly skilled and their working hours are comparably cheap. They pick
up formed parts and put them into boxes. The transport of finished parts to their next
process destination is done by forklift operators, who are not studied in this work.
As in many processes where automation using machines instead of humans is not prof-
itable, production often relies on the availability of a workforce. Human work is, com-
pared to the machines and other intangible assets, flexible and tasks for workers can be
changed to a certain extent. Nevertheless, human creativity cannot be replaced by ma-
chines, and human work is one of the major cost drivers in production. Therefore, in-
duced costs have to be considered and minimized during production planning. Costs of
workers basically depend on the workers’ experience, responsibilities, and on the work
that is carried out as well as on the time and day a worker is deployed. Hourly wages
depend on the type of worker. The working hours of stackers, for example, are less cost-
ly than the hourly wages of machine operators. Another difference between working
types is how costs are treated. Stackers have to be available at the machine for the
whole production time. Some parts do not need stackers, as they simply fall into boxes.
In the case study, costs for stackers are part of the manufacturing costs of the parts.
39
In
contrast, planners, machine operators and measurement personnel are not calculated as
direct manufacturing costs at the part level but rather as indirect manufacturing costs. In
the long term, it is possible to change labor capacity by dismissing or employing people
or by qualifying already available employees. Flexibility regarding labor capacity in the
short term can be achieved by using more or less production shifts
40
within given con-
straints.
41
The day is divided into three shifts
42
and there exist three day types
43
with
different cost factors for working hours.
39 In the case study there exist four cost types for parts: direct material costs, indirect material costs, direct
manufacturing costs per piece and indirect manufacturing costs.
40 Short-term manpower planning on an individual level has to consider legal, organizational and personal
aspects which are disregarded in this work.
41 Labor capacity can be adapted only within defined limits. Long-term employment contracts limit reduc-
tion in labor capacity, and required technical schooling and limited availability of appropriate workers
restricts an increase in labor capacity.
42 A shift can be defined as follows: 1. A group of workers who work for a specific period 2. the period of
time worked by such a group.[But03]
43 Shifts in case study: morning shift: 06:00–14:00; late shift: 14:00–22:00; night shift: 22:00–06:00.
14
Workday
Cost Factor: 1
Sunday
Cost Factor: 1.7
Bank Holiday
Cost Factor: 2.5
Morning Shift
Cost Factor: 1 1 1.7 2.5
Late Shift
Cost Factor: 1.1 1.1 1.87 2.75
Night Shift
Cost Factor: 1.2 1.2 2.04 3
Figure 7: Cost Factors of Different Day and Shift Types
Besides object-related work, other production factors are relevant. The next sub-sections
describe consumable production factors.
2.2.2 Machines
According to Gutenberg,
44
machines are elementary production factors. This sub-
section is dedicated to defining and describing relevant problem details about machines.
First, a definition is formulated. Then, the elementary production factor itself and relat-
ed processes in the case study are described. Lastly, the available flexibility and costs
are mentioned and described.
A device that applies force, changes the direction of a force, or changes the
strength of a force, in order to perform a task, generally involving work done
on a load. Machines are often designed to yield a high mechanical advantage
to reduce the effort needed to do that work. A simple machine is a wheel, a
lever, or an inclined plane. All other machines can be built using combina-
tions of these simple machines; for example, a drill uses a combination of
gears (wheels) to drive helical inclined planes (the drill-bit) to split a material
and carve a hole in it.45
In this work, the focus is laid on the molding presses production stage. This first stage
influences the rest of the production and can be seen as a bottleneck as all products have
to be processed at this stage and the available machines are limited in production ca-
pacity. The analyzed production depends on the machines as only machines can apply
appropriate pressures
46
on the molds to cut and form steel parts. Before production
starts, steel coils
47
have to be fixed in the coiler. The machine pressure, production
speed and diverse other adjustments have to be carried out by machine operators. Serial
44 See [Gut83].
45 See [The05].
46 Depending on the machine, pressures between 500 tons and 1,500 tons can be applied.
47 See sub-section 2.2.4 for further details on raw materials.
15
production can start after checking the quality and dimensions of the part. Every pass,
48
the machine pulls raw material by a defined infeed and applies pressure on the installed
die,
49
which cuts and forms the parts. The produced half-finished parts fall onto a short
conveyor belt. Bigger or more delicate parts can be picked from there by stacking per-
sonnel. Other parts fall directly into boxes. The pass counter is used to determine the
number of produced parts. Filled boxes, provided with a control card,
50
are then placed
by machine operators in a dedicated space, where they are collected and transported by
forklift operators. On each machine, several dies,
51
which have to be compatible with
the machine, can be installed. The initial installation of a die on a machine is very time
intensive
52
as precision adjustments have to be made by machine operators and mainte-
nance personnel. As the used stamping machines are relatively huge,
53
high investments
leading to high capital commitment have to be made. Monetary aspects and limited
space impede fast adaptations of available machine production capacity. In contrast to
machine-related fixed costs, which are important for making strategic investment deci-
sions but less relevant during operative production planning, variable costs, including
amongst other things operating supplies, have to be considered in order to calculate
time-dependent production costs.
The sample machine’s cost center positions are grouped into four cost categories: pri-
mary or secondary variable costs, and primary or secondary fixed costs. The following
table shows how these costs are defined in terms of the case study.
48 Depending on the machine, on the installed die and on the part which has to be produced, 15–30 passes
per minute are possible.
49 See sub-section 2.2.3 for detailed information about dies.
50 The control card contains information on the content of a box as well as the next production steps.
51 In the case study, there are about 10–20 dies assigned to each machine.
52 As it is not very easy for the initial installation process to be standardized and it is difficult to estimate
time for precision adjustments, between two and four shifts have to be reserved.
53 Depending on the type, machines are about 14m x 4m x 5m in size.
16
Variable Costs Fixed Costs
Primary
Costs
- Variable Wages
- Variable Personal Extra Expenses
- Tools and Dies
- Basic and Maintenance Material
- Consumable Material
- Maintenance
- External Labor
- Fixed Wages
- Fixed Personal Extra Expenses
- Depreciation
- Debt service
Secondary
Costs
- Energy
- Maintenance
- Overheads and Management
- Maintenance
- Production Planning
- Occupancy
- Security
- Cleaning
- Overheads and Management
- Further Internal Services
- Maintenance
Figure 8: Classification of Cost Factors at Machines
The investments which have to be put into machines are high. It follows that deprecia-
tion and debt services are of high relevance in the sample cost center.
54
Consequently,
adaptations to available machine capacity are only possible in the long term. Flexibility
can only be gained by other means.
Because of the number of produced variants and because of changes to the products and
the product portfolio, it is not possible to obtain one specialized machine for each prod-
uct as this would generate high investment costs and small capacity utilizations. A way
of obtaining flexibility in production, at the same time keeping investment costs at a low
level, is to assign multiple products to a single machine
55
using different dies. Although
there are machines which are constructed in the same way, they are not identical. Thus,
time-intensive initial precision adjustments of dies are machine-specific. Consequently,
machine-die assignments are set on a mid-term time horizon and considered as fixed for
operative planning. The production speed, in this case expressed by passes per minute,
is also set during the initial installation of a die on a machine. The production speed has
an upper limit. Higher speeds result in lower quality of parts.
To sum it up, the capacity of a single machine can be flexibly shared so that multiple
parts can be produced on one machine in a limited way. Considering flexibility types by
Sethi and Sethi,
56
the described machines match with different ones. There is a certain
54 In the case study, primary and secondary costs are nearly equal. More than 50 % of the primary costs
consist of depreciation and debt service of the machine. The next highest primary costs are maintenance
costs and costs for basic and maintenance material. The sum of variable and fixed personal expenses is
about 10 % of the total primary costs. The sum of overheads and management, maintenance and occu-
pancy costs set 75 % of secondary costs. The sum of energy, production planning and further internal
service costs are about 20 % of total secondary costs.
55 Twenty to forty products are assigned to one machine in the case study.
56 See [SS90].
17
flexibility of product range, as several products can be produced on one machine. Quan-
tities can be adjusted to a limited extent (flexibility of production quantities). Extensions
can be installed using new dies that are compatible with the machine (flexibility of ex-
tensions) and new parts can then be produced (flexibility of program / flexibility of pro-
duction). Capacity adaptations are possible to a limited extent. These enable machines
to adapt towards market changes (flexibility towards market). Other flexibility types,
like machine, material flow and workflow, process or process sequence flexibility are
not available from the considered machines. In order to be able to produce different
products on a single machine, exchangeable dies are required. In the next sub-section,
dies are defined and described in further detail.
2.2.3 Dies
Other elementary production factors are the dies. In this section, it is explained what
dies are. Further, it is described which production processes require a die and it is clari-
fied which restrictions exist.
In the [Ame06], a die is defined as “a device used for cutting out, forming, or stamping
material.” The definition is specified by further explanations describing what a die is. A
die is “an engraved metal piece used for impressing a design onto a softer metal, as in
coining money”; “one of several component pieces that are fitted into a diestock to cut
threads on screws or bolts”; “a part on a machine that punches shaped holes in, cuts, or
forms sheet metal, cardboard, or other stock”; or “a metal block containing small coni-
cal holes through which plastic, metal, or other ductile material is extruded or drawn.”
In [Agn02] a die is defined as “a shaped block of metal or other hard material used to
cut or form metal in a drop forge, press, or similar device” or “a tool of metal, silicon
carbide, or other hard material with a conical hole through which wires, rods, or tubes
are drawn to reduce their diameter.” From the perspective of the case study, the first
definition fits in particular. In this case, the die is a device for cutting out, forming and
stamping metal.
For forming and shaping parts out of steel, exchangeable dies are used. As dies
57
allow
the production of several parts by a single machine, and as they are as cost-intensive as
a whole machine, usage provides the ability to cope with product variety. Nevertheless,
investments have to be done to construct a new die with a mold to form and shape parts.
57 The dies used in the case study are about 8.5m x 2.5m x 2m in size and can weigh up to 10 tons.
18
One die can produce either two or four identical parts, or two or four different parts.
58
The dies used are multi-stage dies which complete several process steps or stages
59
in
succession without any buffers between each stage. The raw material is transformed
into half-finished parts, which only need to be washed, polished, and/or hardened. Each
sub-stage of a die is responsible for a part-specific transformation. Raw material is cut
and formed according to part specifications. Oil is used during the cutting process in
order to cool the material and the die, to improve the quality of the cuts and to reduce
abrasion of the dies’ molds and tools. Each stage has to be considered during initial pre-
cision adjustments when a die is firstly installed on a machine as well as during adjust-
ments after setting up the die on its standard machine. That is the reason for the long
set-up and adjustment times.
60
Initial precision adjustments have to be finished by peo-
ple in the maintenance department together with machine operators. Regular set-ups of
dies can be completed by machine operators. Two different types of set-ups can be dif-
ferentiated: internal set-ups and external set-ups. Internal set-ups are completed in the
machine. That means that both halves of the die remain installed in the machine during
the changing of molds and/or tools of the die. Internal set-ups are only possible if the
set-up’s starting part and the target part use the same base and the set-up can be com-
pleted by simply changing some molds and/or tools. Alternatively, a set-up is completed
externally. An external set-up is carried out by uninstalling the whole die from the ma-
chine. Molds and/or tools are changed outside the machine, which requires use of a set-
up table whose availability is limited.
61
Whereas internal set-ups block the machine dur-
ing the whole set-up time, production can continue during external set-ups. Neverthe-
less, internal set-ups applied to similar parts can reduce adjustment times. Besides the
set-up table, further resources are needed during set-up like the crane for the dies and
the measuring room as well as measuring personnel for first part checks. Set-up times
are sequence-dependent, obeying the triangle inequality.
62
The sequence of production
therefore has an influence on the loss of production capacity, personnel costs for set-ups
and the usage of previously named shared set-up resources. The dies’ cutting compo-
58 In the case study, if different parts are produced, they always have a certain relation to each other dur-
ing the next production steps. Usually the left-hand part and the right-hand part are produced simulta-
neously with a single die, in the knowledge that both parts will later be needed simultaneously.
59 According to the classification of manufacturing methods presented in [Dan09] (p.300), the process
steps of the multi-stage dies in the case study are different types of metal forming and cutting. Surfac-
ing, modification of material properties and assembling of parts is carried out in separate machines in
further process steps as described at the beginning of chapter 2.
60 Regular set-ups on the standard machine take 1.5–8 hours.
61 A set-up table is a special piece of equipment which is used to prepare dies outside the machine.
62 The triangle inequality for set-ups states that a set-up from a to b to c always takes more time than a
direct set-up from a to c. For detailed geometrical explanations see [KK01] chapter 1.3.
19
nents and tools for foraminating steel become frayed,
63
so they have to be maintained.
During maintenance, the components of the dies are replaced, polished or sharpened.
Maintenance is carried out every time the die is dismounted from the machine.
The use of exchangeable dies enables the sequenced production of multiple products on
a single machine and facilitates savings in machine investments. Nevertheless, dies also
have to be designed, built and the initial sample inspection has to be done, which is re-
lated to costs.
64
The investment expenses impede the stocking up of a number of alter-
native dies for a product. Hence, operational flexibility is reduced due to the required
maintenance of dies, which takes several days.
65
In some cases, dies can be adapted by
changing only some tools or molds in the die in order to produce similar products. This
is done to reduce initial investments for dies but reduces flexibility as one base die is
used for more than one product and maintenance intervals have to be considered for all
produced products. Although in this case only some components of the die have to be
replaced, the machine is blocked for several hours if set-up is done internally.
2.2.4 Raw Material
Raw material is a consumable production factor. In this section, raw material is defined
and described. Only relevant processes which are related to raw material are described.
After that, costs and flexibility aspects of raw material are described.
There exist several definitions for raw material. “Basic substance in its natural, modi-
fied, or semi-processed state, used as an input to a production process for subsequent
modification or transformation into a finished good.”
66
In [Agn02], raw material is de-
fined by two alternatives: “material still in its natural or original state, before processing
or manufacture” or “anything that is capable of being processed, converted, changed,
etc. to produce something else.”
67
Another two different definitions are as follows: “an
unprocessed natural product used in manufacture” or “unprocessed material of any
kind.”
68
As the raw material in this case is already processed and the steel is not in its
natural or original state, none but the first definition can be applied. In this case, raw
material is a basic substance used as an input to a production process for subsequent
transformation.
63 In the case study, dies have a durability of approximately 50,000 parts.
64 In the case study, dies cost between €500,000 and €2,000,000.
65 Maintenance of one die takes three days in the case study.
66 See [Lut].
67 See [Agn02].
68 See [Ame06].
20
The most important raw material for the analyzed production is steel. In this case, it is
delivered in coils of band steel.
69
Other operating supplies, like oil, energy or cleaning
supplies are not considered in this work. Depending on the part, different steel types of
different compositions and dimensions
70
are required for production. Raw material is
ordered one year in advance, depending on demand forecasts. The supply of material is
guaranteed within contractually defined increases or decreases of demands within a def-
inite time horizon. The time between order and delivery on an operative timescale is one
day. In the case study, a local inventory covering the next three days of production is
sufficient to guarantee supply availability in the short term. The required steel has to be
transported on a crane driven by a machine operator to the machine which is running
out of raw material. Some machines have a dedicated space where the next steel coil can
be placed some time before it is needed. If this is the case, the change can be executed
fluently without the disruption of other machines running out of steel at the same time.
The raw material unit is then put into the decoiler, and fixed and adjusted to the ma-
chine. During the fixing stage,
71
the production at the machine has to be stopped. Be-
cause of relatively small tolerances of the steel, production can usually continue as be-
fore. In exceptional cases, a die cannot be adapted to the used steel coil. Then, the coil
has to be replaced, if possible. Removal of a steel coil from the decoiler is very danger-
ous as the high tension force of the furled band steel is difficult to control and can seri-
ously injure workers. Another reason to avoid coil removal is that there is a possibility
that removed raw material can no longer be used. Problems during the fixing stage of a
previously used steel coil occur especially if the size of the coil falls below 50 % of the
maximum diameter. The unusable raw material has to be scrapped. Very important for
the stamping process is that the composition, thickness and width of the used band steel
are always within defined tolerances. Variations in the length of the coiled band steel
and variations of the coil weight are not important for product quality. But these varia-
tions influence the output of one coil regarding the amount of produced parts without
changing the coil.
As the half-finished parts after the stamping presses are at the beginning of the value
chain, raw material costs make up a major percentage of the value of the parts. In the
present case study, between 60 % and 85 % and an average of 77 % of the half-finished
part value consists of raw material costs. The cost structure of the parts cannot be
changed due to lot size planning. In the considered case, only two of the 16 used raw
material types are shared among six parts. Hence, the usage of alternative raw material
69 About 10 steel suppliers deliver requested, specialized steel.
70 In the case study, steel is between 0.8 cm and 1.5 cm thick and between 31.3 cm and 65.7 cm wide.
71 The changing of the coil including required tasks takes about 15 minutes.
21
for the production of a part is constrained because of this technological restriction.
Without flexibility in raw material sharing among different products, lot size planning
has only a very small influence over improving the availability of half-finished parts
whenever raw material is missing. The supply of raw material is a precondition for pro-
duction. Although the costs for raw material and the proportion of raw material costs to
the half-finished part costs cannot be reduced by lot size planning, scrap can be reduced
if production batches and lots consider the coil size. The availability of raw material can
also be improved by giving production plans in advance.
2.2.5 Loading Equipment
Loading equipment, that is, the boxes or cases used in production, is part of the sup-
plies. As loading equipment is also relevant, this section is dedicated to describing the
loading equipment used and to explain relevant related processes. Costs and flexibility
of loading equipment are described as well.
There exist different types
72
of loading equipment. Depending on the stamped part and
subsequent processes, a specific loading equipment type is chosen. Box types with dif-
ferent capacities can be classified into four major groups: small boxes, medium boxes,
large boxes
73
and non-returnable cardboard boxes, whose size will not be distinguished
any further. First, the selection of the box type depends on the size of the produced part.
Stability of the parts restricts the amount of parts put into one box as well as the way
boxes have to be filled. Small, stable parts can fall into boxes, whereas others have to be
picked from the belt by stackers and put into cushioned boxes in separated layers. Other
boxes are filled with parts by robots. The subsequent process steps of the parts also in-
fluence the type of loading equipment used. The most important example is parts which
are transported to internal or external customers overseas. As the return of empty boxes
takes too much time and is costly, only non-returnable cardboard boxes are used. An-
other example is that some parts have to be cleaned of the oil used to improve the
stamping process. These parts often pass through the washing system inside the boxes,
with the consequence that the varnish of the boxes is damaged. It is therefore preferred
to use boxes without varnish for these parts.
The following table summarizes the main part-loading equipment type relations.
72 About six different types of relevant loading equipment are used.
73 Depending on the used loading equipment, between 130 and 3,000 parts are bundled into one loading
equipment unit.
22
Part Properties
Small
Box Medium Box Large
Box
Non-
returnable
cardboard
box
Size Stability
Small Damageable X (only with inlays)
All (Overseas
Destinations)
Stable X X
Large Damageable X
Stable X
Figure 9: Loading Equipment Mapping according to Part Properties
Boxes become oily and dirty over time. Dirty loading equipment deteriorates the quality
of the contained parts. Consequently, boxes have to be cleaned. As the cleaning process
is outsourced in the case study, lead and transport times have to be considered when
guaranteeing availability of the correct boxes at the desired time.
Because of part properties, it is not possible to use every loading equipment type for
every part. Flexible substitution of loading equipment types is impeded. The flexible
use of different loading equipment types is also reduced by successive processes. As
described before, parts with overseas destinations have to be packed into non-returnable
loading equipment and parts to be washed should be placed into boxes without varnish.
Investments needed in loading equipment are much less than for dies or machines. Nev-
ertheless, fixed capital has to be minimized. The limited available space required for
loading equipment is also a problem, reducing the possibility of reserving large amounts
of loading equipment of every type. Besides investment costs for loading equipment,
other costs are relevant for loading equipment including loading equipment manage-
ment costs, cleaning and loading equipment maintenance costs. These costs are not de-
pendent on production planning. Therefore, they can be disregarded in this work.
2.2.6 Batches
According to [Ame06], a batch is “an amount produced at one baking” or “a quantity
required for or produced as the result of one operation.” The most suitable definition, in
[Agn02], states that a batch is a “group or set of usually similar objects or people, espe-
cially if sent off, handled, or arriving at the same time.”
The molding presses production stage considered has restrictions regarding batch sizes
produced. As the changing of steel coils reduces the time available for production the
changing of coils should be avoided when they are not completely used. Another reason
is the danger posed to workers, due to the steel coils’ tension force, if they have to
change a coil which has not been completely used. Consequently, the batch size is de-
fined by the size of the coil currently used. The exact amount of parts which can be pro-
23
duced with one coil can only be estimated by dividing the coil weight by the charge
weight of the part which has to be produced. In coupled production, the charge weight
of all simultaneously produced parts has to be taken into account. Although inevitable
small variations of the material fall within part production tolerance margins, they ac-
cumulate and influence the output amount of one coil. Consequently the exact produc-
tion output and also the exact production time for one batch can only be estimated. In
practice, the average coil size is calculated for planning production output and time and
this is precise enough to estimate batch production ends. On the basis of the estimated
production output and time, a batch-wise production can be planned, in which produc-
tivity reductions due to coil changes, the estimated time of a coil change, and required
raw material units, can be planned.
2.2.7 Lots
Among other definitions, the [Ame06] defines a lot as “Miscellaneous articles sold as
one unit.” This definition is not appropriate for the purpose of this work. A precise defi-
nition for a production lot, which is suitable for this work, can be found in [Dep01].
There, a lot is defined as “Specifically, a quantity of material all of which was manufac-
tured under identical conditions and assigned an identifying lot number.”
Lot sizes at the considered molding presses production stage have to obey restrictions as
well. The smallest lot size is defined by the smallest possible batch size which is in turn
estimated using average coil sizes. As the production is executed batch-wise, lot sizes
can only be integer multiples of raw material units, that is, the coils. The maximum lot
size depends on the die’s lifespan. In order to keep the quality of the parts high and to
prevent broken dies, the number of produced parts is limited. This number defines the
maximum lot size. Since in some cases different parts use the same die, the sum of the
cumulated production quantity for all these parts has to obey the maximum lot size. In
other cases, different parts are produced simultaneously in coupled production. In this
case, the cumulated production has to be considered separately, although both parts are
using the same die. This is because the parts produced in coupled production use differ-
ent cavities of the die. These cavities are frayed equally during production and not addi-
tionally. As the maintenance of the dies has great influence on the lot restrictions, dif-
ferent applicable maintenance trigger methods have to be taken into account. The first
alternative would be to start maintenance just after dismounting a die from the machine
after production. Another possibility is to carry out maintenance on the die before the
defined maximum lot size is about to exceed. In this case, the cumulative production
24
quantity has to be memorized between productions. The cumulative production quantity
is in both cases reset during maintenance.
Apart from restrictions induced by raw material and die maintenance, set-ups have to be
considered. Required times for set-ups are mainly influenced by the changing of the
dies. A set-up consists of two main tasks: the changing of the die and the adjustment of
the die. The changing of the die comprises the provision of required set-up material,
including the new die, the dismounting of the installed die, the mounting of the new die
and the removal of the old die and set-up material. The set-up and adjustment effort
depend on the sequence of mounting the dies on the machine. As described in section
2.2.3, times depend on whether set-ups are executed internally or externally, too. As
follows, set-up times are sequence dependent. In 2.2.1, it was described that skilled,
specialized personnel are needed to carry out the set-ups. The limited availability of
these personnel has to be taken into account during the definition of lots.
In summary, the lots’ starting and ending times depend on the workforce and machine
capacities, the sequence-dependent set-ups of dies, the dies’ lifespans, and die mainte-
nance, as well as the size of raw material units.
2.3 Two-Level Capacitated Lot Sizing in Production Control
Changes in the production environment on an operative timescale, especially changing
customer demands,
74
make it senseless to define detailed production schedules for the
long term. In order to cope with decision complexity and speed up planning, the calcu-
lation effort is reduced by splitting the planning horizon into time-based levels. Being a
flexible but also costly resource, the production factor of human work is considered
within both planning levels on a different level of detail. Depending on the planning
level, requirements and related restrictions are considered in different ways. The follow-
ing two sub-sections describe how the two planning levels are defined and separated in
practice. It is also described which decisions on the basis of which data have to be taken
at each level and which of the formerly described restrictions have to be considered.
2.3.1 Mid-Range Level
On an operative time-horizon, there are still many decisions to be taken which have a
great impact on the success or failure of satisfying customer demands at minimal costs.
Because of changing customer demands and other changes in the production system,
74 See 2.1.1 for a detailed description of how customers make their orders.
25
like inventory differences due to rejections or refinishing operations, a planning horizon
of two weeks is enough to guarantee the availability of required production factors.
Today, planners consider different input data to define production lots and schedules.
First, monthly demand estimates, which are generated by program planners with yearly
demand forecasts and customer contracts, are considered. The use of monthly demand
estimates to plan production lots for the next two weeks guarantees that demand chang-
es on a tactical time horizon are regarded during lot sizing. Product start-ups or run-offs
or seasonal demand fluctuations can easily be managed. Although demand estimates do
not meet short-term customer demands, they enable production planners to create plans
which can satisfy customer demands with higher success. Secondly, production plan-
ners consider the declared customer orders of the next two weeks, which are fixed with
small tolerance margins.
75
At the mid-range planning level it is decided which amount
of which part is produced on which day during the next two weeks in order to fulfill
customer demands. Production lots of parts which are personnel-intensive are ideally
positioned in those days
76
which are cheaper in terms of workforce costs. The capacity
of machines limits the production amount per day whereas different production speeds
of different parts are taken into account. The availability of the dies, which is first and
foremost determined by maintenance, is also planned. Maintenance intervals and maxi-
mum lot sizes restrict planners’ decisions. Planned lots already have to be dimensioned
in a way that enables complete coils to be used. Otherwise, the capacity utilization as
well as the produced amounts would not be calculated correctly and the plans would not
be suitable for practice. With determined production amounts for the whole mid-range
planning horizon, it is possible to order the required amounts of raw material coils. The
disposition of loading equipment depends on the information about production amounts
and times defined by production planners, as loading equipment has to be cleaned of
residual oil and dirt before usage, which takes time.
77
The mid-range planning horizon slides forward every day by one day. This rolling plan-
ning horizon scheme guarantees that changes in the production environment are proper-
ly taken into account.
78
On the basis of the calculated mid-range planning results, a de-
tailed short-range scheduling is carried out. Since the results of the mid-range planning,
that is, the production lots, take capacity restrictions into account, it can be guaranteed
75 See 2.1.1 for a detailed description of customer demands and contracted change tolerances.
76 See 2.2.1 for a chart of shifts’ cost factors.
77 As in the case study, cleaning is carried out by a specialized company, and loading equipment has to be
transported, both of which take time. See chapter 2.2.5 for details.
78 One alternative to a continuous rolling planning horizon scheme is a connected planning scheme. A
description of both concepts can be found in [Ste07].
26
that short-range planning tasks are feasible. The next sub-section describes the decisions
made for the short-range planning horizon and which data are used to determine produc-
tion schedules.
2.3.2 Short-Range Level
As detailed lot sizing and scheduling is complex and takes time, it is not practicable to
create plans far in advance, which then have to be recalculated every time something
affecting the production changes. Hence, detailed planning is only used for a time hori-
zon where as many parameters as possible are fixed. In this case, the contractual fixing
of customer demands for the next three days is a suitable limitation for a detailed plan-
ning horizon.
A plan generated for this short range of three days has to take into account all the re-
strictions that the mid-range planning considers, plus those restrictions which are neces-
sary to calculate feasible detailed schedules. First, there are the workforce restrictions
and costs. In contrast to mid-range planning, where workforce distribution is done on a
daily basis, short-range plans are able to allocate the workforce to smaller time units.
Cost differences for shifts have to be taken into account. The usage of the limited ma-
chine production capacity is calculated for the short-range level to a higher level of de-
tail, taking the same parameters into account as in mid-range planning. Sequence-
dependent set-up times have to be regarded. The time used for set-ups of dies reduces
production capacity at the machine which is currently set-up. Additionally, maintenance
times and intervals are important in short-range planning. The calculation is made as in
mid-range planning but to a shorter, more detailed timescale. The timing and the point
of time for coil changes are planned in the short-range timescale. The reduction of pro-
duction capacity is therefore automatically taken into account. Last but not least, load-
ing equipment is planned depending on the planned production.
27
3 State of Art
After having described the problem in the previous chapter, the current approaches
available in the literature are reviewed. First, available concepts and methods designed
to improve service availability are presented. After that, available approaches to im-
prove flexibility are presented. Then, several methods for planning the requirements are
listed. Available decomposition approaches as well as lot sizing methods are described
in the last sub-section.
3.1 Improvement of Delivery Service Availability
As the customers’ purchasing decisions are influenced by the suppliers’ delivery service
availability, the importance of logistical service quality has increased during recent
years.
79
According to Zibell,
80
the logistical service level can be evaluated by the fol-
lowing components:
- Delivery time: Time between the order and the delivery
- Willingness to supply: Proportion of orders which can be promised to be deliv-
ered
- Delivery reliability: Proportion of deliveries delivered on or before the promised
date
- Delivery flexibility: Time-based scope for the customer to change orders
- Delivery quality: Quality and state (e.g. damage) of the delivered goods
- Willingness and readiness to provide information on the status of the customer
order
79 Compare [Paw07].
80 See [Zib40].
28
Figure 10: Buying Criteria according to Pawellek
In order to be able to improve service availability, evaluation methods for service avail-
ability, which are presented in the next sub-section, are needed. Depending on the situa-
tion, different methods for improving supply service availability can be applied. These
are presented afterwards.
3.1.1 Evaluation of Delivery Service Level
In order to improve the service availability, the service level has to be evaluated.
Pawellek
81
defines a basic performance indicator for the service level:
*100
Number of Deliveries within Agreed Time
Service Level Number of Orders
=
The number of deliveries/orders can be replaced by the monetary value.
A more differentiated evaluation is presented in the VDA recommendation 5001.
82
With
the presented method, it is possible to differentiate quantity variance as well as delivery
schedule variance. A method for measuring flexibility and comparing it with completed
deliveries is also presented.
81 See [Paw07].
82 See [VDA94].
3,58
3,24
3,10
3,02 3,00
2,7
2,8
2,9
3
3,1
3,2
3,3
3,4
3,5
3,6
3,7
End Product
Properties
Logistical Service
Level
Image Consulting Price
• Products fulfill customers’ expectations with marginal difference
• “Time” is gaining higher importance because of short innovation cy-
cles and due to the necessity to adapt to new market requirements
Buying decisions are getting influenced more and more by logistical
service quality, like:
- delivery time
- delivery reliability
- delivery flexibility
- willingness to provide information
Importance of most important buying criteria from the customers' point of view
29
The next sub-section is dedicated to methods which can be used to improve supply
availability.
3.1.2 Methods for Improving Supply Availability
There are several ways to improve supply availability. One way to improve supply
availability is to plan demands using statistical methods. The first sub-section describes
the methods used for demand planning. In the second sub-section, the methods used for
fulfilling demands are described.
3.1.2.1 Demand Planning
On the one hand, there exists demand uncertainty, induced by the variation in planned
or estimated demand and realized sales. On the other hand, the goal is to fulfill customer
demand. Many decisions, including, for example, those on the procurement of raw ma-
terial or components with long lead times, have to be made before the customer submits
his order.
83
Therefore, demand planning is necessary in order to “improve decisions
affecting demand accuracy and the calculation of buffer or safety stocks to reach a pre-
defined service level.”
84
Depending on the planning horizon, different methods can be
applied to obtain results for demand planning tasks, which can be structured in the same
way as in the demand planning framework presented by Kilger and Wagner.
85
Demand
Planning
Structures
- Structuring products, customers and time
- Structuring input and output of demand plan-
ning
- Aggregation and disaggregation
Demand
Planning
Process
- Phases of demand planning process
- Participants in demand planning process
- Statistical forecasting
- Judgemental and consensus forecasting
Demand
Planning
Controlling
- Definition of basic metrics
- Aggregation rules for forecast accuracy metrics
- Dealing with exceptions
- Technical implementation of KPIs
- Incentives and responsibility
Figure 11: Demand Planning Framework by Kilger and Wagner
83 See [SK08].
84 See [SK05], p.139.
85 See [SK08], p. 133.
30
In order to improve accuracy, demand planning data is structured, often on the basis of
products or product families, customers or regions, and time. Demand planning is sub-
divided into a long-term aggregated demand prognosis level, in which demands for sev-
eral periods are forecast and subdivided into a short-term prognosis level.
86
In order to
plan demands, statistical forecasting techniques are used.
87
A problematic aspect of
forecasting techniques, however, is that they are usually wrong.
88
Uncertainty about real
demands has to be considered. Demand planning has to be controlled using defined
basic metrics and key performance indicators.
89
The next sub-section is dedicated to determining how the actual customer demand can
be satisfied.
3.1.2.2 Demand Fulfillment
The planning process dedicated to determining how actual customer demands are satis-
fied is called demand fulfillment. “The demand fulfillment process determines the first
promise date for customer orders.”
90
Traditionally, the inventory is checked and orders
are quoted against it. If there is not enough inventory available, production lead times
are taken into account in order to provide achievable order promises. As constraints e.g.
capacity limitations are not taken into account, infeasible quotes may be calculated.
Nowadays, demand fulfillment solutions contain more sophisticated methods, which
improve the generation of reliable quotes, the searching for feasible quotes and the in-
crease of profitability. These methods
91
generate plans for future supplies from the sup-
pliers on the basis of demand forecasts, even beyond the already existing scheduled or-
ders.
92
Depending on the product and the production environment, demands are satisfied from
stock (make-to-stock) or produced after the receipt of the order (make-to-order). In
make-to-stock environments, production is forecast driven. Customer orders can then be
served with short lead times as only transport and order processing times arise. The
86 See [GT09], p.148. Data warehouses and online analytical processing (OLAP) tools can be used for this
purpose. See [SK05], p.142.
87 Compare e.g. [GLM04], [SK05], [GT09].
88 See [Nah97].
89 In section 3.1.1 some metrics for logistical service quality were introduced.
90 See [SK05], p.179.
91 The newest approaches can be found under the available-to-promise concept. Examples of improved
available-to-promise approaches can be found in [CZB02], [JSJK02], [XTKC03]; an overview is pre-
sented in [Pib05].
92 See [SK05].
31
main restriction to fulfilling an order is the availability of stock. In make-to-order envi-
ronments, procurement is driven by forecast; production is driven by customer orders.
Consequently, order fulfillment depends on procurement time and capacities. Produc-
tion time and capacities have to be considered as well.
93
Planned demands and feasible order promises are preconditions to planning the pro-
curement of resources as well as production.
3.2 Flexibility vs. Costs
Planning can be considered as the notional anticipation of future actions in order to
achieve set objectives in an economically advantageous way
94
. Consequently, plans can
reduce costs if actions are executed in compliance with the planned specifications. Pro-
duction control, which is one of the most important leading and management tasks
95
, is
defined as the reaction on the actual events and the resulting plan deviations on a short-
term
96
. But otherwise, adaptations, which may be necessary due to environmental
changes, are limited and therefore decision flexibility is reduced. Demands are planned,
and feasible order promises are given, on the basis of uncertain parameters. The main
causes of uncertainty are:
97
- Exact demand is not assured
- Actual times (e.g. replenishment) differ from planned times
- Real amounts (e.g. production or delivery quantities) differ from planned times
- Documentation is erroneous (e.g. available stock)
All uncertainties can be reduced by investments or contracts but they are never com-
pletely eliminated and are related to costs. Consequently, the flexibility required to be
able to adapt to upcoming situations has to be obtained by other means. There exist dif-
ferent possibilities for improving flexibility through different planning approaches.
3.2.1 Total or Complete Planning
Ideally, complete, unchangeable information is used to plan cost-optimal activities for a
long horizon. During total planning, it is assumed that the whole problem can be solved
93 See [SK05].
94 A definition of planning can be found in chapter 2.2.
95 See [Hah96].
96 Translated from [Krü96].
97 Following [GT09].
32
in one planning step and this means that the planning horizon equates to the length of
the total horizon T
total
.
98
Therefore, all interdependencies have to be known in ad-
vance,
99
which is usually not the case in practice.
3.2.2 Cyclic Planning
If the cyclic planning approach is applied, the total horizon T
total
is subdivided into
smaller, consecutive, non-overlapping planning horizons T
c
consisting of several peri-
ods. Planning for the next planning horizon T
c’
is carried out after |T
c
| periods. Actual-
ized data as well as system state information gained from the previous planning hori-
zons are used. Decisions made are fixed for all periods of the planning horizon T
c’.100
3.2.3 Rolling Planning
The rolling planning approach minimizes the problems of information dynamics and
time-based interdependencies related to the previously described approaches. At each
planning step, decisions for π periods are fixed. Decisions related to the other |T
c
|-π pe-
riods are revised and corrected depending on actualized data. Decisions for the π peri-
ods are implemented. Consequently, |T
c
|/π planning steps are executed and |T
c
|/π-1 are
fixed once. Comparable to the cyclic planning approach, several plans are generated
considering the end state of the planned system. In contrast to cyclic planning, the roll-
ing planning approach enables flexible reaction to environmental changes.
A result of applying the rolling horizon approach, when considering changes in infor-
mation, is that less planning errors are made. Due to frequent changes in plans, high
flexibility is expected from the planned resources. These adaptations, also known as
planning nervousness, lead to organizational difficulties in fulfilling the changed plans
and a consequence of this may be fewer acceptances of the planning procedure.
101
An additional problem related to the rolling planning approach, whenever a planning
horizon smaller than the relevant planning horizon is taken into account,
102
is that in-
ventories at the end of the horizon are minimized in order to reduce inventory holding
costs for the actual plan. This negatively influences adherence to delivery dates after the
98 See [SKH03].
99 See [Bre04].
100 See [SKH03], [KS01].
101 See [SKH03].
102 See [Heu03].
33
planning horizon and possibly leads to higher set-up and production costs. In order to
improve planning quality and to reduce nervousness, ending inventories have to be set
for each planning step.
There already exist methods for reducing the negative side effects of rolling planning
approaches. Fisher et al. present a concept which calculates an ending inventory on the
basis of the economic order quantity (EOQ).
103
Information about future average de-
mands after the planning horizon has to be available. Heuvel extends the planning hori-
zon so that amounts can be calculated using the Wagner–Whitin algorithm.
104
The Peri-
odic Order Quantity (POQ) is the quotient of the average demand and the EOQ and
determines the extension of the horizon. Another approach, which is also based on in-
formation available after the defined planning horizon, is presented by Stadtler.
105
Using
the heuristic by Groff,
106
the Time Between Orders (TBO) value is calculated, in order
to determine for how many periods the amount produced within one period can meet the
demands. With an adapted Wagner–Whitin algorithm and the usage of the calculated
TBO, an inventory level can be determined to reduce set-up costs which otherwise
would occur.
So far, planning approaches with different flexibility and cost-optimality characteristics
as well as methods to reduce negative side-effects of the rolling planning approach have
been presented. Still missing are the methods for how workforce, production, set-ups,
maintenance and coil changes are planned within the planning horizon. Alternatives for
these factors will be described in the next sections.
3.3 Methods for Planning Requirements
3.3.1 Workforce Planning
Workforce or personnel planning can be defined as an ordered, information-processing
process, whereas during its progress, the values of personnel variables are set anticipato-
rily, so that entrepreneurial targets are met.
107
Personnel variables can represent all as-
pects of availability and specificity problems on the individual or categorical level.
108
103 See [FRZ01], for details about their presented Ending Inventory Valuation concept.
104 See [WW58].
105 See [Sta00].
106 See [Gro79].
107 Translated from [Kos93].
108 See [Spe].
34
Depending on the focal point of personnel planning variables, nine categories of per-
sonnel planning can be differentiated, which are determined by combinations of variable
characteristics. In this work, personnel planning characterized by variables determining
the availability of personnel on a categorical level is relevant.
109
As follows, methods
for the so-called collective personnel planning
110
are presented and evaluated. Catego-
ries can be differentiated into categories of activities and categories of qualifications.
111
Moreover, a goal is to integrate personnel planning into corporate planning including
the calibration of all planning areas. For that reason the simultaneous planning approach
has been introduced, in order to guarantee optimality. Depending on the case in ques-
tion, theoretical simultaneous planning approaches may be able to be used in practice
because of difficulties in obtaining relevant data and the high calculation effort required.
Consequently, the traditional approaches of using successive planning still dominate
planning procedures.
In the literature there exist several approaches and methods for workforce planning. The
approaches can be distinguished by their area of application:
112
Figure 12: Differentiation of Personnel Planning Approaches according to Rossi
The three types of workforce planning approaches with general application areas can be
distinguished by their time-based relationships. If the operating time is longer than the
daily working time of employees, shift scheduling is necessary, and the working time is
organized in shifts. In shift scheduling, it is decided which shifts are required to satisfy
the necessary workforce. Decisions about working time and time points as well as
breaks are made. If the operating time lasts longer than the average period of working
109 Specificity problems, like skill enhancement planning or the design of incentives, are not considered
here as they are not relevant for lot sizing. Availability planning on an individual level, that is, indi-
vidual worker disposition, is omitted as well. Although it is relevant to dispose each worker in accord-
ance to company agreements (e.g. maximum working hours, holidays etc.), this problem is disregard-
ed, too.
110 See [Kos75] and [Dru75].
111 See [Kos75], [Str76] (S.28 ff), [Vie99] (S.18 ff).
112 See [Ros07].
Personnel Planning Approaches
Specific Application Approaches
Tour Scheduling
Bus Driver Scheduling
Crew Scheduling
Nurse Scheduling
Days Off Scheduling
Course Scheduling
Timetabling
Audit Staff Scheduling
Generali
zed Approaches
Shift Scheduling
35
days of employees, days off have to be respected. Days off scheduling is dedicated to
matching the off or working days of workers, taking into consideration days off during
the week or at weekends over a period of several weeks. The combination of shift and
days off scheduling is known as tour scheduling. In tour scheduling, shifts as well as
days off are planned for each worker. Furthermore, there are models with specific appli-
cation areas. Because of special characteristics and further restrictions, it is difficult to
classify them into general approaches. Crew scheduling, bus driver scheduling, nurse
scheduling, course scheduling, timetabling or audit-staff scheduling can be differentiat-
ed. In [EJKOS04], there is an overview of workforce planning approaches.
Crew scheduling examples can be found in [BMR04], where a new solution is presented
to calculate multiple depot crew schedules which takes into consideration the time it
takes for a crew to return to the starting depot, and limits of elapsed time and working
time. Another crew scheduling approach is presented by [SFD98], in which the opera-
tional airline crew scheduling problem is described. The described problem consists of
modifying personalized monthly assignments planned for airline crew members on an
operative timescale in response to a given flight plan. Crew scheduling is a problem
which is often analyzed from an airline perspective. Among other scheduling problems,
especially for airlines, crew scheduling approaches are described in [Suh95].
Examples of methods for bus driver scheduling are [VH02], [BGL01], and [WW95],
where schedules are calculated for bus drivers on an operational timescale. Different
approaches to improving the performance of the solving of presented problems like heu-
ristics or column generation methods are also presented.
An overview of nurse scheduling problems is given in [BCBv04]. The authors discuss
the role of nurse scheduling in hospitals’ personnel planning and review several nurse
scheduling approaches in the literature.
Course scheduling, timetabling and audit staff methods and reviews are presented in
[Bor00], [Sch99], [PVH03], [Hib01], [Wer97], [DE97], [Sal95] and [Fun02]. They will
not be explained here as their restrictions and the practical background does not match
the purpose of this work.
The available approaches concentrate on workforce scheduling. These generalized ap-
proaches do not consider any production restrictions. These methods have to be adapted
in order to be usable for the presented problem. The methods with defined application
backgrounds do not precisely match the problem described.
36
3.3.2 Machine Planning
In this section, machine planning approaches are subdivided into order release methods
and scheduling methods. Relevant approaches are presented and discussed.
3.3.2.1 Order Release Methods
The order release determines the point in time at which the production can handle an
order. An order release starts with material procurement and after this has happened the
material usually cannot be used for other orders. The order release influences inventory
and machine utilization. Order release methods can be classified as follows:
113
Without leveling of work-
ing system-specific utiliza-
tion
With leveling of working
system-specific utilization
Centralized
- Conwip
- Bottleneck control
- Workload control
- Load-dependent order
release
- Order release with linear
programming
Decentralized
- Polca
- Decentralized invento-
ry-
based manufacturing
control
a) Order Release Categories b) Subcategories of Inventory Controlling Order Release Methods
Figure 13: Classification of Order Release Methods according to Lödding
As the immediate order release ignores utilization, lead times and inventory, they will
not be analyzed in this work. The appointment-based order release is the basis of most
production planning and control systems. A precondition is that superordinate planning,
which determines a list of orders and starting appointments, is provided in advance. It is
possible to describe the appointment-based order release by the following rule:114
In the appointment-based order release, an order is released when its planned start
time has been reached or exceeded and the required material is available.
113 See [LW05].
114 Translated from [LW05] p. 313.
Categories of Order
Release Methods
Immediate Order Re-
lease
Appointment-Based
Order Release
Order Release Me-
thods Controlling In-
37
A centralized order release method controlling the inventory is the constant work in
process approach (CONWIP).
115
This procedure is controlled by the following rule:
An order is released whenever the inventory of the considered production line falls be-
low a defined threshold. The order with the highest priority is then selected from the
order list. The order list contains unreleased orders with a planned start time, which is
situated within a defined planning horizon.
Another centralized order release method is the bottleneck control. The basic rule of this
approach is as follows:
Whenever an order has been finished by the bottleneck working system, a new order is
released.
The bottleneck control approach subdivides the manufacturing into an inventory-
controlled part which incorporates the bottleneck working system instead of being be-
hind the bottleneck working system. A centralized order release approach supporting the
leveling of the working system-specific utilization is called workload control.
116
The
main parameters for this procedure are inventory limits of the working systems. Its
basic idea can be described thus:
Detain orders which pass overloaded manufacturing entities. The load of the manufac-
turing entities is based on the analysis of inventory and already released orders.
The load-dependent order release
117
is centralized and considers system-specific utiliza-
tion. Its basic rule can be summarized as follows:
An order is released whenever the utilization threshold or inventory threshold is not
exceeded adding another order.
In its basic approach a periodic order release was proposed. An event-based order re-
lease is possible as well. A centralized order release considering the utilization of the
working system is the order release using linear programming.
118
The basic rules are:
A list containing all unreleased orders is available. Release orders if the inventory dif-
fers from a previously planned level.
The workload is balanced using optimization software and requires a lot of parameters.
The number of parameters complicates the method but an adaptation to a specific pro-
duction system is possible. An example of a decentralized order release approach with-
115 See [SWH03], [SZ03], [HS96].
116 [Jen78], [BW81], [KTH89].
117 [Bec80], [Wie92].
118 [ID74].
38
out leveling working system-specific utilization is the POLCA
119
control (Paired-Cell
Overlapping Loops of Cards with Authorization). The production is subdivided into
closed loops, where cards are used to control the inventory. A precondition of the POL-
CA control is the availability of an order list which has been generated in advance. A
POLCA card, which provides the authorization of production, is assigned to a pair of
manufacturing sections. A superordinate production planning and control system de-
fines earliest order release dates using backward scheduling. The following rules are
used in the POLCA concept:
1. A manufacturing entity is allowed to execute an order, when the order release
date is exceeded and a card is available. Otherwise, the order is blocked
2. The manufacturing entity checks whether other orders can be executed, if an or-
der is blocked.
3. One card is added to the executed order at the first manufacturing stage and
stays until the order reaches the last manufacturing stage. Then the card is freed
and can be used for the next order.
The decentralized inventory-based manufacturing control is another decentralized order
release method without leveling of working system-specific utilization. On the basis of
customer orders, a list of orders has to be generated in advance. The orders are stored in
a list and released by decentralized inventory control cycles on the basis of the invento-
ry from the next manufacturing stage. The exact rules used in this approach are availa-
ble in [Löd01].
The following table summarizes the evaluation of the discussed approaches on the basis
of the description presented in [Löd01]:
119 [Sur98].
39
Order Release Method
Appointment based
Conwip
Bottleneck control
Polca
Decentralized
inv.
based man. cont.
Workload control
Load dependent
Linear progra
m-
ming
Criteria of Manufacturing co
n-
trol methods
Inventory level should be controlled - o
o
+
+
+
+
+
Inventory deviations should be minimized - o
o
+
+
o
o
o
Blocked inventories should be minimized +
+
+
- - - - -
Balancing of utilization and capacity o
+
+
+
+
+
- +
New sequencing should be minimized +
+
+
- o
- o
-
Backlogs should be minimized and controlled - - - - o
+
- -
Bottleneck principle should be considered - - +
o
o
+
o
o
Simplicity +
+
+
o
- o
- -
Figure 14: Comparison of Order Release Methods
The presented approaches are suitable for planning production at machine level. The
presented approaches are suitable for planning production at machine level, although
many practical restrictions are not taken into consideration as they are outside the scope
of this study.
3.3.2.2 Sequencing Methods
Sequencing methods determine which of the orders in the queue is processed next. Se-
quencing has a great influence on the logistical service quality, especially in situations
where the order queue is long or inventory is high.
120
The first sequencing approach is the First-In-First-Out (FIFO) rule. In this case, it is not
possible to re-sequence the orders. Disadvantages are the interdiction of adaptations to
planned schedules or the enforcement of standard lead times in every case. Although
improvements responding to changes to orders cannot be achieved and flexibility is not
available, several advantages can be obtained by using this method, like simplicity and
calculability of lead times.
Further, there exist the earliest planned start date and the earliest planned end date rules.
These rules change the order according to the planned execution date of an order and
120 See [Löd01] p. 443–457 for more information on the described methods.
40
can improve e.g. serviceability towards the customer. It has to be assumed that the start
and end dates of orders have been calculated in advance.
Another sequencing approach is the selection of an order by the minimal slack. The
term “slack” is defined as the time until the planned end date of the order, which is not
used for processing or minimal transition times. It is calculated as follows:
0 ,
plan i min i
max
i I i I
i I
Slack time ED T PT TT
∈∈
≠
= − − −
∑ ∑
plan
ED
=
Planned end date of an order
0
T
=
Planning date
i
PT
=
Processing time of process
i I
∈
,
min i
TT
=
Minimal processing time of process
i I
∈
I
=
Set of processes
The basic idea is that delays are more probable for orders with a smaller slack time val-
ue than orders with a higher slack time value. On this basis, it is possible to consider
future disturbances during order sequencing. One disadvantage is that the order se-
quence can be changed although there are no variations to the planned schedule.
In order to improve the performance, sequences can be improved using various simple
methods. If set-up times are sequence dependent, the order with the lowest set-up costs
is selected. The application of this approach risks orders related with high set-up times
being delayed for a relatively long time.
The Extended Work in Next Queue (XWINQ)
121
is another approach. The basic order
prioritization criterion is the inventory of the precedent and the subsequent working
system. The lower the inventory, the higher the priority of an order. This method aims
to reduce material flow breaks at consecutive manufacturing stages. Disadvantages are
that the inventory is not a suitable criterion for reducing material flow breaks in an envi-
ronment where numerous machines have to be controlled. The method does not differ-
entiate between bottleneck systems and non-bottleneck systems. Moreover, planned
order dates are ignored.
With the shortest operation time rule, the orders are sequenced according to their pro-
cessing time. Orders with less processing time have a higher priority. Advantages are
low inventories, short to medium lead times, a low medium order delay, and high ser-
121 This concept is presented in [CMM03].
41
viceability. Disadvantages are that positive effects depend on inventory levels and that
unimportant and important orders are equally prioritized.
3.3.3 Maintenance Planning of Dies
Maintenance takes time and restricts productive time but is necessary in order to guar-
antee error-free production. In this work, maintenance of the dies required for produc-
tion has to be considered, as production is not possible during maintenance. Conse-
quently, maintenance influences availability and productivity and the selection of an
appropriate maintenance strategy is important.
122
According to [RF10] and [Mat02],
maintenance strategies can be differentiated as follows:
Figure 15: Classification of Maintenance Strategies
In [War09], maintenance strategies are differentiated as follows:
- Condition-Based Maintenance:
It is possible for sensors or trained personnel to control and monitor the status of
a component and to change the component in good time.
- Time-Based Maintenance:
Inspection of component is done after prescribed time-periods, which is deter-
mined by experience.
- Damage-Based Maintenance:
The maintenance of a component is executed after the component is damaged.
Reduced availability is the consequence.
As it is not possible to monitor the status of all the components of a die during produc-
tion in the analyzed case, condition-based maintenance planning is not applicable.
Availability is the most important. This is the reason why damage-based maintenance is
excluded. In this work, maintenance does not depend on time but on production lots and
the number of produced parts, as the components of the dies are frayed during the
122 See [Mat02] for practical relevance of the selection of the best maintenance strategy.
Maintenance Strategies
Reactive Maintenance Preventive Maintenance
Periodic Condition
Dependent
Anticipatory
42
stamping process. As maintenance can be regarded as a logistical process,
123
planning
improves the serviceability of maintained production factors.
There are already several reviews on maintenance planning approaches available. One
example of such a review is [CP91], which deals with maintenance and replacement
models for multi-unit systems. Approaches are partitioned into topical categories like
machine interference/repair models or inspection/maintenance models. Other, newer
reviews on maintenance planning approaches are presented by Dekker et al. who differ-
entiate approaches by their stationary or dynamic character or by the type of their appli-
ance in case studies or in decision support systems.
124
[SK09] presents, in a generalized way, how modern information technologies can be
applied to improve maintenance processes, especially planning. It is said that real bene-
fits arise when maintenance planning tools become integrated communicatively with
other planning systems. Therefore, concepts in which production planning as well as
maintenance planning are executed simultaneously are relevant for the purpose of this
work. In [AJA07], an integrated lot sizing and preventive maintenance strategy satisfy-
ing demands without the allowance of backlogging minimizing production and mainte-
nance costs is presented. The authors make use of a mixed-integer linear program to
solve experiments in order to obtain an optimal integrated production and maintenance
strategy. Another approach is presented in [ST10]. The authors propose a method to
determine simultaneously the period of preventive maintenance and the job sequence
for two parallel machines in order to minimize the makespan with the result that a shop
improves coordination between maintenance planning and production scheduling and
improves shop efficiency.
Nevertheless, the consideration of relevant restrictions is not sufficiently integrated and
therefore the presented approaches cannot be adapted to solve the problem described in
this work.
3.3.4 Raw Material Procurement Planning
The purpose of raw material procurement planning is to satisfy the demands of produc-
tion factors resulting from previously planned lots and generated schedules in a cost-
effective way taking into consideration already existing suppliers.
125
Corsten identifies the main goals of procurement planning in general:
126
123 See [RF10].
124 Examples of reviews on maintenance planning approaches are [Dek96], [DWv97], [DS98].
125 See [Ste05].
43
- Guarantee of supply (assurance of material quality, flexibility and quantity;
spreading risks of procurement, maintaining independence, etc.)
- Cost effectiveness (low capital commitment, reduction of costs, etc.)
- Safe disposal (ecologically acceptable materials)
There exist several approaches designed for cost-effective procurement planning. One
of the first such approaches in systematic procurement planning was introduced by
Andler,
127
in which the economic order quantity was defined. Many other research pa-
pers were published on the topic of purchase order sizing, but assumptions were made
which do not reflect practice, like constant demand, unlimited capacities, constant prices
and quantity discounts as well as multiple suppliers, none of which can be considered in
the scope of this study.
Therefore, existing approaches were extended and can be found in the literature. Ap-
proaches to order sizing under quantity discounts are classified in [BP96] or [MR98]. A
review on lot sizing models considering dynamic demands
128
is given in [BGv84].
In the Uncapacitated Multi-Supplier Order Quantity Problem with Time-Varying All-
units Discounts
129
the sum of inventory costs and order costs, which consists of fixed
and variable costs, is minimized. Besides other constraints, it is guaranteed that no de-
lays can occur. Supplier-dependent discount levels are introduced. Last but not least, a
heuristic is presented to solve the model. Besides constraints, which were already inte-
grated in [T02], further aspects like supplier capacity limits, limited customer inventory
capacities, limited period-dependent supplier capacities and supplier-dependent mini-
mum purchase quantities are modeled in [Rei02]. With the approach presented in
[Sta07], multiple products as well as different discount types are supported.
Although integrated procurement and production planning concepts are available,
130
it is
not desirable in the analyzed case to influence the production plans since procurement
as the raw material replenishment method can be disregarded for production in this case.
Another more important argument against the available integrated approaches is that,
according to the author’s reviews, the literature is missing approaches which consider
all relevant aspects of procurement and production planning simultaneously.
131
126 See [CC95], p.573–586.
127 See [And29].
128 Examples are [Laf85], [Ben86], [HM02], [BS93], [CHK96], [Sil79], [CHK00], [KFW03a], [HS03].
129 See [T02].
130 Overview examples are [GD92].
131 Examples of further integrated procurement production approaches are [Lee05], [Bal99], [San11],
[BAS]. The latest developments of advanced planning systems (e.g. [Sta05]) do not consider practical
restrictions in the required detail.
44
The reviewed approaches might be suitable for the cases described and provide concepts
which are suitable for developing further procurement planning methods, but the inte-
gration into a planning method designed to solve the problem as was described in chap-
ter 2 is not available. The consideration of immanent technical and organizational re-
strictions available is not supported by the presented approaches.
3.4 Two-Level Capacitated Lot Sizing in Production Planning
Minimizing the sum of set-up and inventory holding costs was already picked out as a
central theme in [And29]. Since the assumptions made, such as endless production ca-
pacities and inventory capacities as well as static demands, are not practicable in most
cases, further approaches have been developed. An extension of the economic order
quantity considering dynamic demands was presented in [WW58]. According to the
knowledge of the author, the first approach to solving the capacitated lot sizing problem
with dynamic demands, which is considered as one of the most important and at the
same time most difficult problems in production planning,
132
was presented in [Eis75].
Several adaptations added further aspects to the basic capacitated lot sizing problem in
order to model further aspects, and planning results have become more practicable. But
the consideration of further practical constraints is often related to higher model com-
plexity. In order to reduce complexity, problem decomposition is often used as an ac-
cepted approach in practice. The first sub-section below is dedicated to decomposition
and hierarchical production planning approaches. After that, mid-range lot sizing meth-
ods are depicted and their suitability for the previously described problem is analyzed.
Short-term lot sizing methods, which have to define more detailed schedules, are de-
scribed in the sub-section after that. Last but not least, available integrated short- and
mid-term approaches are briefly explained and their usability for solving the problem is
discussed.
3.4.1 Decomposition Approaches and Hierarchical Production
Planning
A basic approach to solving complex planning problems is their division into partial
models. Optimization problems can be obtained which are solvable with less effort.
133
132 See for example [KFW03b]. See [BY82], [FLR80] for complexity analyses of the capacitated lot siz-
ing problem in the single-item case, and [CT90] for a multi-item complexity analysis of the capacitat-
ed lot sizing problem. In [MMv91], it is shown that finding a feasible solution is NP-complete for
problems with set-up times.
133 See [Sta88].
45
According to [KS89], discrepancies between theoretical recommendations of operations
research and practical requirements as well as practical limitations of production plan-
ning can be solved using hierarchical production planning by employing three devices:
1. Separation of distinct planning areas defined by organizational units and coordi-
nation by a few, controlled interfaces
2. Use of the natural time-structure of the planning process
3. Reduction of data by aggregation
The first approaches of hierarchical production planning and scheduling were presented
in [HM73] and [Gab76]. In [HM73], the authors describe a hierarchical planning and
scheduling system for a multiple plant and multiple products with a seasonal demand
situation. Optimal decisions at an aggregate level, which are termed “planning”, provide
constraints for the detailed decision-making level at which schedules are defined. Alt-
hough the described restrictions do not match the problem previously described, the
presented concept of decomposing the problem in planning and scheduling decisions
seems to be useful. Based on this work, a similar approach is presented in [Gab76].
Both references form the basis of later works on hierarchical production planning and
scheduling.
134
First, developments on the provided basis were reviewed [HO85], includ-
ing another proposition for a method for manufacturing control which subdivides medi-
um-term and short-term decisions. Examples of newer approaches in hierarchical pro-
duction planning are usually specific and designed to solve a particular problem. Exam-
ples are [Sta88], in which a method of hierarchical lot sizing is proposed, [KS89], which
provides a review on problems and methods to solve production planning in hierarchies,
[HG01], which deals with a hierarchical and product-based decomposition to plan pro-
duction of a steel plant, and [WI07], which presents a hierarchical production planning
method looking at uncertainty in demands. Another hierarchical production planning
approach using Karmarkar’s algorithm
135
is available in [YZJ04]. The hierarchical pro-
duction planning approach in [ASv11] only considers production capacities on a weekly
basis and does not define schedules. The most promising approach is presented in
[OT07]. Many practical circumstances, like the multi-product environment or the batch
processes, are similar to those available in the previously described problem. But most
aspects are still missing. Examples are the consideration of lots, and maintenance- or
sequence-dependent set-up times during the scheduling process. According to the
knowledge of the author, there are no approaches available which simultaneously con-
134 See [BT93] for further information on hierarchical production planning.
135 See [ARVK89] for a detailed description of how Karmarkar’s algorithm can be used to solve linear
programs.
46
sider production and scheduling restrictions as well as personnel planning aspects in an
integrated hierarchical method.
3.4.2 Mid-Term Lot Sizing
In the last sub-section, available hierarchical production planning and decomposition
approaches were briefly described and their suitability to the problem in question ana-
lyzed. As the available approaches do not cover every requirement, it is necessary to
analyze concepts which are used for lot sizing at a mid-term level only, without integra-
tion into a hierarchical planning approach.
In mid-range lot sizing, detailed schedules are not necessary, as dynamic input parame-
ters in reality change very often.
136
But in the analyzed case, it is necessary, when plan-
ning production, to consider the available capacities, which are mainly reduced by set-
ups and coil changes. Maintenance has to be considered as well as different cost factors.
Capacitated lot sizing can therefore be modeled with big buckets.
137
The basic capaci-
tated lot sizing model can be described thus:
138
Assumptions:
- Several products J are produced on a single shared resource.
- The resource is limited in capacity.
- The planning horizon is finite and divided into T periods.
- The demand is dynamic but deterministic.
- Production depends on machine state, which can be changed by set-up.
- Resource capacity is reduced by set-ups. A set-up incurs set-up costs.
- The target is the minimization of the sum of holding and set-up costs
Sets:
J
Set of products
T
Set of periods
136 Besides dynamic demands, production is related to uncertainties. The initial system state, that is, for
example, initial inventories or initial machine states, can change over time and the longer the planning
horizon the greater the difference from planned system state to real system state.
137 The differentiation into small- and big-bucket models concerns the relative length of the time-periods
with respect to the expected length of a production lot [Sue05]. “‘Big’ and ‘small’ bucket indicates
how long a period of a calendar, which is used in a production system, a node or a point in a model, is
in relation to the density of events set to the original production.” (translated from [Dan99] p. 255).
138 Compare the formulation and explanation available in [Sue05].
47
Data:
j
a
Consumption of capacity to produce one unit of item
j J
∈
(=production
coefficient)
jt
b
Large number, not limiting feasible production quantities of product
j J
∈
in period
t T
∈
t
c
Available capacity in period
t T
∈
jt
d
Demand for
j J
∈
in period
t T
∈
(with
jT
d
including final inventory, if
given for the planning horizon T)
jt
h
Inventory holding cost for one unit of
j J
∈
in period
t T
∈
j
sc
Set-up cost for product
j J
∈
j
st
Set-up time for product
j J
∈
Variables:
jt
I
Inventory of
j J
∈
at the end of
t T
∈
jt
X
Production quantity of item
j J
∈
in period
t T
∈
(lot size)
jt
Y
Set-up variable (=1, if a set-up operation for item
j J
∈
is performed in
period
t
=0 otherwise)
* *
jt jt j jt
j J t T j J t T
Min h I sc Y
∈ ∈ ∈ ∈
+
∑∑ ∑∑
(1)
1
jt jt jt jt
I X d I
−
+ = +
,
j J t T
∀ ∈ ∈
(2)
* *
j jt j jt t
j J j J
a X st Y c
∈ ∈
+ ≤
∑ ∑
t T
∀ ∈
(3)
*
jt jt jt
X b Y
≤
,
j J t T
∀ ∈ ∈
(4)
0
0, 0, 0
jt jt j
X I I
≥ ≥ =
,
j J t T
∀ ∈ ∈
(5)
{
}
0;1
jt
Y∈
,
j J t T
∀ ∈ ∈
(6)
The sum of holding and set-up costs are minimized in the objective function (1). The
inventory balance constraints (2) guarantee that all demands are met in time. Available
capacity is shared by production (
jt
X
) and set-up operations (
jt
Y
) due to constraints
(3). Production variables
jt
X
are coupled with set-up operations
jt
Y
, by constraint (4).
Equations (5) and (6) define non-negativity as well as binary conditions.
Reviews of the capacitated lot sizing problem are presented.
139
In [BRG87], the authors
present a classification of production planning problems differentiating between single-
level and multiple-level problems which are then subdivided into problem groups with
139 The author does not claim that the list of reviews is complete. The list is a small subset of available
reviews.
48
unconstrained or constrained resources. They evaluate research work using computa-
tional effort, generalization, optimality, simplicity and testing as evaluation criteria. In
[Mv88], the authors compare available heuristic approaches for solving the multi-item
single-level capacitated lot sizing problem. They differentiate between single resource
heuristics and methods based on mathematical programming. On the basis of computa-
tional results, suggestions are given for the appliance of the diverse heuristics in indus-
try. In many review papers, extensions to the basic problem formulations were dis-
cussed in order to model practical aspects. In [TTM89], a review of capacitated lot siz-
ing models is presented including set-up times. [KSv94] provide a structure for batching
research and models on the basis of a distinction of batching issues and related decision
levels. The authors define process design, activity planning and activity control to clus-
ter research results. In [YL95] lot sizing models with random yields in production were
reviewed and procurement costs were considered as well. [KFW03a] concentrate on
single-level lot sizing problems and variations. Moreover, heuristic and exact solution
approaches are discussed. Extensions to basic lot sizing models for industrial applica-
tions are collected and summarized in [JDZ05]. Examples of actual reviews formed on
the basis of the latest research results are [QK08] and [UP10]. In [QK08], a literature
review suitable for practitioners as well as scientists is presented, including formula-
tions of capacitated lot sizing problems with back-orders, set-up carry-over, sequencing,
parallel machines, multi-level product structures and overtime. A classification contain-
ing various approaches available in the literature and based on the characteristics of the
planning horizon, the number of items, the order quantity, the frequency of review, lead
times, capacities, demand properties, and stocking points is presented in [UP10].
According to the knowledge of the author, formulations which precisely match the
problem are not available. Nevertheless, parts of other formulations can be used. There
is only some literature available regarding simultaneous lot sizing and personnel plan-
ning. One example is [JMN05]. The target of the authors was to minimize the costs re-
lated to human resources needed in the process, linked with a lot sizing production plan.
Another example of model extensions is the use of linked lot sizes in order to correctly
represent capacity consumption due to set-ups. Basic models using linked lot sizes can
be found in [DEWZ93], [Haa94] or [Tv85], in which a heuristic approach is presented
to solve the previously presented problem. An example of a modeling framework which
includes set-up carry-over is available in [GMS95]. In [SG99], multiple products are
supported to be produced in one period. In the literature, capacitated lot sizing models
and solution approaches considering batch-wise production are available. Examples are
[AES93], [SWS06] and [van07]. Sample approaches, which are dedicated to mainte-
nance and planning production lots simultaneously, are [CK05], [CRR08], [NFM10]
and [BBH10].
49
All the presented approaches are well suited for the specifically analyzed and solved
problems. But, according to the knowledge of the author, there is no approach available
which integrates all required points into a single concept.
3.4.3 Short-Term Lot Sizing
In this section, available reviews and approaches for short-term lot sizing are described.
Selected lot sizing approaches have to support scheduling and other aspects which are
described in chapter 2. Therefore, small-bucket
140
problems in particular will be de-
scribed in this section. Although it was stated
141
that the discrete lot sizing and schedul-
ing problem has an edge over the continuous set-up lot sizing problem
142
regarding per-
formance and practical relevance, other small-bucket problems and their extensions will
be analyzed in order to find a solution to the problem.
One of the first contributions to the research on the discrete lot sizing and scheduling
problem was presented in [LT71].
143
In [Sch82], the first extensions were formulated to
model sequence-dependent set-up costs and a product-based decomposition approach
was presented. In [vKKSv90], the complexity of the discrete lot sizing and scheduling
problem was analyzed in further detail. A general formulation of the discrete lot sizing
problem is available in [Fle90]:
144
Assumptions:
- Products
∈
j J
are produced on a single shared resource.
- The resource is limited in its capacity.
- The finite planning horizon is divided into T periods.
- The demand is dynamic but deterministic.
- Only one product can be produced in a period.
- Full capacity is used if a product is produced within a given period (all-or-
nothing assumption).
- The change of a set-up state of a resource incurs set-up costs.
- The minimization of the sum of holding costs and set-up costs is the target.
140 A differentiating definition is provided in [Dan99] p. 255.
141 See [SKKW91].
142 See [KS85].
143 The application of the discrete lot sizing problem in practice has been important ever since the first
contributions to research. The model formulated in [LT71], for example, was used in an automated
production-scheduling system for a tire company. In another very early example, presented in [vV83],
sequence-dependent set-up times are modeled.
144 The model and its explanations are adaptations of the formulations presented in [Sue05] and [Fle90].
50
Sets:
J
Set of products
T
Set of periods
Data:
j
p
Production speed for
j J
∈
jt
d
Demand of
j J
∈
in period
t T
∈
j
h
Inventory holding cost for one unit of
j J
∈
per period
j
sc
Set-up cost for product
j J
∈
jt
ss
Safety stock of product
j J
∈
at the end of period
t T
∈
Variables:
jt
I
Inventory of
j J
∈
at the end of
t T
∈
(with
0
j
I
for the initial inventory)
jt
Z
Set-up state variable (=1, if item j is set-up at the end of period t, = 0 oth-
erwise) (with
0
j
Z
representing the initial set-up state)
(
)
, 1
0,
j jt j t j jt
j J t T
Min sc max Z Z h I
=
∈ ∈
− +
∑∑
(1)
, 1
jt j t j jt jt
I I p Z d
−
= + −
,
j J t T
∀ ∈ ∈
(2)
1
jt
j J
Z
∈
≤
∑
t T
∀ ∈
(3)
{
}
,
0,1
jt jt jt
I ss Z≥ ∈
,
j J t T
∀ ∈ ∈
(4)
The target of the model is to minimize the sum of holding and set-up costs (1). As pro-
duction has to be at full capacity or not at all in each period, the model formulation does
not rely on production variables
,
jt
X
which are replaced by
*
j jt
p Z
in the inventory
balance constraints (1). Constraint (3) limits the number of simultaneous set-up states
jt
Z
in one period. Constraints (3) and (4) define non-negativity and binary conditions
on the decision variables.
Reviews of the discrete lot sizing and scheduling problem are available in the literature.
Examples are [DK97], which also contains reviews of big-bucket models, and [JD08],
which contains a review of relevant extensions to lot sizing and scheduling models, es-
pecially for industrial applications.
51
Several extensions to the discrete lot sizing and scheduling model were formulated to
cover practical constraints.
145
In [JD98], the discrete lot sizing and scheduling problem
is solved with sequence-dependent set-up costs and times on a single machine. [JD04]
provide an adaptation of the basic model supporting start-up times, which can be frac-
tions of the time bucket, multiple alternative machines with different efficiencies, mul-
tiple capacitated resources and backlogging. Another approach, which supports set-up
times as well as earliness and tardiness penalties, is presented in [SLM10]. One of the
first approaches of batch-oriented scheduling can be found in [AADT92], in which the
problem is stated, a complexity analysis given, and a heuristic solution approach is pro-
vided. Batch production and consequent complexity is considered in [BJH00], too. The
paper [JD98] is another example of batch-oriented scheduling.
The papers [AGH99], [AGH98] and [LC00] are examples of scheduling problems
which consider the maintenance of machines but not the maintenance of the required
resources, which is the case with the dies in the current problem.
The most suitable approach implementing several relevant aspects of the discussed
scheduling problem is presented by Suerie.
146
In his work, basic concepts for modeling
period-overlapping actions are introduced. With the presented fundamental model,
building blocks, batch production, and maximum lot sizes, as well as period-
overlapping set-up times and maintenance, can be introduced in any formulation by
adapting them to the specific situation.
Although all aspects required to solve the presented problem have already been dis-
cussed in the literature, there is no contribution available which combines all the re-
quired aspects into a single approach. Also, the most promising work by Suerie
147
has to
be adapted in many directions. Sequence-dependent set-up times, and further activities
like coil changes or the use of set-up personnel, are only a small excerpt of the charac-
teristics it is necessary to add.
3.4.4 Integrated Mid- and Short-Term Lot Sizing
Another promising approach is the combination of mid-term and short-term planning,
termed a general lot sizing and scheduling problem. A fundamental research contribu-
tion is formulated in [FM97]. In this approach, the schedules are independent of prede-
145 The list of approaches is not complete. The depicted approaches are only a small subset of the ap-
proaches available in the literature.
146 See [Sue05].
147 See [Sue05].
52
fined time-periods and hence a generalization of known models using restricted time
structures is provided.
148
Assumptions:
- Products
∈
j J
are produced on a single shared resource.
- The resource is limited in its capacity in each big-bucket period t.
- Each big-bucket period consists of a set of small-bucket periods s.
- The finite planning horizon is divided into T big-bucket periods.
- The demand is dynamic but deterministic. Demand data is based on the big-
bucket periods.
- In each small-bucket period s at most one product has to be produced.
- Set-up states are maintained across periods.
- The change of a set-up state of a resource incurs set-up costs and consumes re-
source capacity.
- The number of set-up operations per big-bucket periods is not limited by the
number of products as the triangle inequality
149
does not have to hold.
Sets:
J
Set of products
t
S
Set of (small-bucket) periods forming a (big-bucket) period t
T
Set of periods
Data:
j
a
Consumption of capacity to produce one unit of item
j J
∈
(=production
coefficient)
t
c
Available capacity in period
t T
∈
jt
d
Demand of
j J
∈
in period
t T
∈
j
h
Inventory holding cost for one unit of
j J
∈
per period
j
minlot
Minimal lot size for product
j J
∈
sd
ij
sc
Sequence-dependent set-up cost, if a set-up operation from product
i J
∈
to product
j J
∈
is performed
sd
ij
st
Sequence-dependent set-up time, if a set-up operation from product
i J
∈
to product
j J
∈
is performed
148 See [Sue05].
149 For detailed geometrical explanations of the triangle inequality see [KK01] chapter 1.3.
53
Variables:
jt
I
Inventory of
j J
∈
at the end of
t T
∈
(with
0
j
I
for the initial inventory)
jt
Z
Set-up state variable (=1, if item
j J
∈
is set-up at the end of period
t T
∈
, = 0 otherwise) (with
0
j
Z
representing the initial set-up state)
sd
ijt
Y
Sequence-dependent set-up variable (=1, if a set-up operation from
i J
∈
to
j J
∈
is performed in period
t T
∈
, =0 otherwise)
* *
sd sd
jt jt ij ijt
j J
j J t T i J t T
j i
Min h I sc Y
∈
∈ ∈ ∈ ∈
≠
+
∑∑ ∑∑∑
(1)
1
t
jt js jt jt
s S
I X d I
−
∈
+ = +
∑
,
j J t T
∀ ∈ ∈
(2)
* * *
t t
sd
j js j ijt t
j J
s S j J s S i J
j i
a X st Y c
∈
∈ ∈ ∈ ∈
≠
≤
∑∑ ∑∑∑
t T
∀ ∈
(3)
*
t
js js
j
c
X Z
a
≤
, ,
t
j J t T s S
∀ ∈ ∈ ∈
(4)
1
js
j J
Z
∈
=
∑
,
j J t T
∀ ∈ ∈
(5)
1
1
sd
ijt is js
Y Z Z
−
≥ + −
,
j J t T
∀ ∈ ∈
(6)
1
*( )
jt j is js
X minlot Z Z
−
≥ +
, ,
t
j J t T s S
∀ ∈ ∈ ∈
(7)
0
0, 0, 0
jt jt j
X I I
≥ ≥ =
,
j J t T
∀ ∈ ∈
(8)
{
}
0
0, 0;1 , 0
sd
ijt is i
Y Z Z
≥ ∈ =
, , ,
t
i j J t T s S
∀ ∈ ∈ ∈
(9)
Minimizing the sum of the holding costs and sequence-dependent set-up costs is the
objective, represented in (1). In (2) inventory balance constraints are formulated. The
capacity limitation in (3) is based on big-bucket periods and guarantees that production
as well as set-up activities do not exceed available limits. Production variables
jt
X
and
set-up state variables
is
Z
are coupled in (4). Restriction (5) is introduced to guarantee
that the set-up state at the end of each small-bucket period is well defined. Set-up opera-
tion
sd
ijt
Y
variables and set-up state variables
is
Z
are coupled in (6). Constraint (7) have
to be introduced because of a missing triangle inequality.
150
A minimal production is
enforced in order to avoid direct set-up changes (
i j k
→ →
, instead of
i k
→
) without
production and without consuming capacity. Non-negativity and binary conditions are
stated in (8) and (9).
150 The triangle inequality does not have to hold in every situation. Especially in the chemical industry,
where cleaning processes are modeled by an additional “cleaning” product, the triangle inequality no
longer holds [FM97].
54
In [KS05], it is shown that the basic general lot sizing and scheduling approach is lim-
ited to the case where the production state between two consecutive periods is con-
served if the available capacity of the proceeding period exceeds the minimum batch
quantity. Minimum batch sizes are modeled in [JZ08] but batch-oriented production is
not supported.
In [Mey00], sequence-dependent set-up times were added to the basic general lot sizing
and scheduling problem. The method was only tested with 18 products and other as-
pects like batch-oriented production are missing.
According to the author’s research, further relevant model enhancements are not yet
available, although the general lot sizing and scheduling approach seems to be a promis-
ing methodology for modeling production planning problems, especially regarding per-
formance issues. Nevertheless, modeling of cross-machine constraints and aspects
which require time continuity like set-up times, maintenance times or coil change times
are supported in a better way by the fundaments of discrete lot sizing and scheduling.
55
4 Action Points
Although some ideas of the concepts and methods presented in the state of the art are
useful and can be transferred, they are not satisfactory for solving the described prob-
lem. Even the combination of the approaches does not suffice to solve the problem in
every detail. This section describes the action points—those things that still have to be
done—in order to be able to plan lot sizes in production control, considering relevant
restrictions and taking required resources sufficiently into account. As it is not useful to
calculate detailed schedules for long-term situations in volatile environments with
changing information, the planning horizon is split into two.
Figure 16: Split Planning Horizon
In the case study, detailed schedules are necessary for the first three days and infor-
mation about demands can be assured for the next 14 days. Therefore, the short-term
planning horizon starts today and ends with day three. The mid-term lot sizing starts at
the end of day 3 and ends with day 14.
This partition can be adapted. It is important that both planning horizons are intercon-
nected so that the information can be transferred. As shown in the state of the art, there
exist many research contributions based on linear programs which aim to solve produc-
tion planning tasks. The representation of practical problem instances can be realized in
a relatively short time. Available commercial solver software, developed over a number
of years, works efficiently with modern hardware, and implemented algorithms are test-
ed. Therefore, the implementation of individual algorithms for the described problem is
m = Index of mid-term planning peri-
ods
n = Index of short-term planning peri-
ods
l = Last short-term period
k = Index of first mid-
term period after
short-term planning horizon
n=l-1 n=l
n=1 n=2
m=0 m=k-2 m=k-
1
m=0 m=k-2 m=k-1 m=k m=k+1 … m=mmax
56
rejected and linear programming techniques are used.
151
The following chapters de-
scribe the aspects which have to be taken into account in each of the sub-problems, dis-
tinguished by their planning horizons. Last but not least, both approaches have to be
coupled.
4.1 Mid-Term Lot Sizing
The mid-term lot sizing method has to generate valid, cost-effective production plans so
that customer demands are satisfied. Cost factors influencing the results are all time-
dependent. Production costs, set-up costs, maintenance costs and inventory holding
costs have to be considered. Dynamics in demands and changes in the production sys-
tem have to be factored into the planning procedure. Therefore, a rolling planning hori-
zon is necessary, which takes updated information into account. In order to reduce nega-
tive side-effects of the rolling horizon scheme, like planning nervousness, simultaneous
out-of-stock situations for more than one product, or the resulting risk of losing supply
availability, a method is required which calculates expected ending inventories for each
product. During planning, several practical conditions have to be taken into account.
First, differences in production costs depending on the required personnel workload on
the day have to be taken into account. Machines only have limited capacity, which has
to be considered. Capacity is consumed by set-up or production activities and depends
on the part produced or set-up. For production of the parts, dies are necessary, which
have to be maintained after producing certain parts. Since during maintenance the dies
are not available for production, maintenance has to be considered in mid-term lot siz-
ing. Some dies provide two or more cavities for the same or different products which
are then produced in coupled production. The provision of raw material has to be im-
proved by calculating required steel coils. Production lots have to be integer multiples
of raw material units. As the declared customer demands can be much smaller than the
production output of one steel coil, production capacity consumption has to be calculat-
ed on the basis of the output of the steel coils.
151 In [Kur11] (p. 49) it is said that the difficulties of solving optimization problems for practical instances
have been reduced due to the development of efficient algorithms and improved hardware perfor-
mance. In [Kal02] (p.36) it is stated that practical problem instances can usually be solved to optimali-
ty using linear programs as the resulting problem matrix is sparse.
57
4.2 Short-Term Lot Sizing and Scheduling
The short-term lot sizing and scheduling method has to generate valid, cost-effective
operative production schedules. Set-up costs, inventory holding costs, production costs,
and personnel costs for set-up teams, as well as maintenance costs, have to be consid-
ered. The results of the mid-term planning horizon, represented by expected ending in-
ventories and declared customer demands, have to be used to calculate production lots
and schedules which satisfy customer demands. The use of costly set-up teams during
expensive shifts has to be minimized. As the set-up teams are available shift-wise, set-
ups should be concentrated into a few shifts if total costs are to be reduced. Production
of workforce-intensive parts should be carried out during cheaper shifts, but only if cus-
tomer demands permit this and the sum of the costs does not increase. Generated short-
term production plans have to consider the limited capacity of machines, which is re-
duced by production, set-ups or coil changes. Different production speeds depending on
the product, sequence-dependent set-up times, and times for coil changes, need to be
taken into account. In short-term lot sizing and scheduling, maintenance of the dies has
to be considered. During maintenance, production of the related parts is not possible.
Some dies have several cavities which produce different parts simultaneously. Hence,
coupled production has to be taken into account as well. Coil changes have to be
planned according to the coil size and the calculated production output.
4.3 Coupling of Mid-Term and Short-Term Planning
Another action point is to couple mid-term and short-term planning approaches. As the
mid-term planning method has broader information about upcoming demands, planned
production quantities have to be transferred to the short-term planning method. As the
short-term planning method possesses information about the actual system state, it is
necessary to communicate planning results from the short-term planning method to the
mid-term planning method in order to guarantee that both procedures are based on up-
to-date information and to be able to generate feasible and practicable planning results.
In summary, interfaces between both planning approaches and between the actual state
of the production system and the planning procedure have to be defined by using the
means available in linear programming.
58
5 Concept
After describing relevant action points, this chapter is dedicated to defining the concept
for solving the previously described problem. The first sub-chapter describes and ex-
plains a prioritization of goals and requirements. Another topic is the decomposition of
the problem. Then, mid-term lot size planning is explained. All required inputs and de-
termined outputs are defined and clarified. After that, short-term schedule planning is
explained using the same structure. Last but not least, coupling of both partial models as
well as the integration into the real production is clarified in the following sub-section.
5.1 Goals, Requirement Prioritization and Decomposition of
the Problem
As was said in section 2.1, it is essential for competitiveness that customer demands are
satisfied as much as possible. The guarantee of supply availability is therefore the high-
est goal which has to be considered in the planning procedure. The requirements for
achieving this goal are capacitated. The importance of each requirement depends on the
flexibility to adapt capacity and its costs. The next priorities in the planning procedure
are machine and workforce utilization. As the specialized machines entail high invest-
ments, the capacity of the machines, which cannot be adapted flexibly on a day-to-day
basis, has to be used in the best possible way. The influence on the operative variable
costs leads to the necessity of improving workforce capacity utilization, especially set-
up time utilization. Both priorities are closely related to each other as the available
workforce capacity influences the productivity of the machines and vice versa. Parallel
set-ups at different machines have to be avoided, if only one single set-up team is avail-
able. As the production depends on dies which are individually designed and construct-
ed for each product and are often only available in limited amounts, lot sizes are limited
due to maximum die life. In combination with the coil sizes, which are the raw material
units used, lot and batch sizes are restricted. These are the last two priorities which have
to be regarded in the planning procedure. Due to the quick response to demands of raw
material and loading equipment suppliers, availability does not affect planning and is
therefore not regarded during lot sizing. As described by Domschke, Scholl and Voß in
their textbook, the target is to minimize the overall costs. Other targets like the maximi-
59
zation of the service level or uniform capacity utilization are modeled using con-
straints.
152
As stated before, the production system states as well as inputs like demand data can
vary in reality and it is therefore not useful to spend a lot of effort in calculating detailed
plans for a long horizon. Hence, it makes sense to subdivide the planning problem into
sub-problems where the decision spectrum and decision detail of the sub-problems de-
pend on the time and relevance for the actual production. Considering the mentioned
priorities, two planning horizons are identified for operative production lot sizing. Dur-
ing mid-range planning, which starts on day 4 and ends with day 14 in the case study,
production amounts to satisfy customer demands are defined on a daily basis consider-
ing given restrictions for lots and batches. Machine capacity limits and capacity utiliza-
tion due to set-ups, coil changes and part production are calculated. As maintenance of
dies takes time and can impede the production of parts, it is also considered during mid-
range planning. Mid-range planning outputs serve as a planning basis for neighbored
processes like raw material supply, the personnel planning department, loading equip-
ment logistics, the die maintenance department etc. which then are responsible for
providing requirements. The production cost, maintenance cost and personnel cost dif-
ferences between days are considered, too. A more detailed differentiation of costs on
the basis of shifts, for instance, is not possible in mid-term planning, but it is in short-
term planning because of a higher planning granularity. In short-term planning, the
same restrictions are taken into account but in a more detailed way. Other restrictions
are added as well to generate a detailed production schedule for, in this case, the first
three days. Besides customer demands, machine utilization and worker utilization - es-
pecially utilization of set-up teams - are considered. The avoidance of parallel set-ups at
different machines reduces the loss of machine capacity due to missing set-up personnel
and keeps set-up teams continuously working. Therefore, set-up-dependent set-up times
are explicitly planned, as set-ups reduce production capacity. Maximum die life is never
exceeded. Coil changes are explicitly planned as they reduce production capacity, too.
The next sections describe how the sub-problem is modeled. Which inputs are required
to calculate the values of the desired output variables representing planning decisions is
also discussed.
152 In [DSV97], 11 criteria for classifying lot sizing problems were identified: the level of information,
time-based development of model parameters, the selection of the planning horizon, the number of
products, the number of production stages, the consideration of capacities, characterization of relevant
costs, the consideration of backlogs, production speeds, the transfer-type of the product and the targets
are differentiated to classify lot sizing problems.
60
5.2 Mid-Term Lot Sizing Considering Multidimensional Re-
strictions
Within this section, the model for mid-term lot size planning is described and explained
in detail. First, relevant inputs are defined and calculations to obtain specific parameters
are explained. The following sub-chapter is dedicated to the outputs expected. After
that, the model is elucidated in detail.
5.2.1 Input
This sub-section is dedicated to the inputs of the mid-term lot size planning. First, fi-
nancial parameters are explained and some are calculated by given parameters. After
that, parameters for variable initialization are determined. Then, production parameters
are explained.
5.2.1.1 Financial Parameters
Calculation of lots and the distribution of calculated lots along the mid-term planning
horizon require several input data. In order to generate optimal plans, relevant costs
have to be considered during lot size planning. Inventory holding costs
inv
p
cM
are prod-
uct dependent. They include capital commitment, which is calculated on the basis of the
selling price
p
price
of a product multiplied by a given interest rate
ir
cM
, as well as
warehousing costs
w
p
cM
applying to a part.
*
w ir inv
p p p
cM price cM cM
+ =
Next, set-up costs have to be taken into account. Sequence dependency of set-ups is not
considered in mid-term planning. Consequently, it is enough to consider average esti-
mates for set-up costs
,
setup
p tM
cM
. Maintenance costs
,
mtnc
d tm
cM
have to be taken into account
because otherwise, plans would be generated which provoke more maintenance, induc-
ing too-high maintenance costs. In order to be able to guarantee availability, an ending
inventory is set. Achieving this ending inventory has less priority than fulfilling an-
nounced, fixed customer demands. Consequently, it is desirable to use production ca-
pacity for announced demands instead of using it to fill the inventory. That means that it
is possible to fall below the desired ending inventory and to lose the guarantee of avail-
ability. This risk is taken into account with the cost factor for imputed stock-outs
so
p
cM
,
61
which consist of the selling price of a product
p
price
multiplied by a defined factor
representing the consequences
cc
of the inability to supply to customers:
*
so
p p
price cc cM
=
Lastly, production costs
,
prod
p tm
cM
have to be taken into account. The costs are period and
part dependent. They are based on given manufacturing costs and on the day-type fac-
tor.
, ,
*
prod
p t p tm
pco dtc cM=
5.2.1.2 Parameters for Variable Initialization
Besides costs, inventory, lot and maintenance parameters have to be considered. The
available inventory level has to be transmitted to the model. This is done by initializing
the parameter
p
iM
ϖ
, which defines the initial inventory level for each product. Planned
lots have to obey several restrictions. Lots can overlap periods and, among other varia-
bles, die maintenance is controlled and triggered by the lot variable. Therefore, it is nec-
essary to initialize the lot variable, too. The parameter
,
m p
lotM
ϖ
is used to transmit the
actual cumulative quantity of the lot. The initial production state is defined by
,
m p
binxM
ϖ
. Maintenance states and maintenance progress are considered in the model.
That is the reason why the parameters
p
mbinM
ϖ
and
p
mpM
ϖ
are necessary to initial-
ize whether maintenance is actually going on or not with respect to the maintenance
progress.
Another parameter which has to be defined is the desired ending inventory used to guar-
antee availability. The parameter
p
eiM
has to be set for each product
p P
∈
. Fixed val-
ues for ending inventories are not suitable because of missing adaptability and flexibil-
ity. Changes in customer demands or product run-outs are difficult to consider. Moreo-
ver, fixed-ending inventory levels are always either too small, resulting in supply short-
falls, or too large, resulting in high inventory holding costs and inventory risks. An al-
gorithm for calculating flexible, self-adapting ending inventory levels has been de-
signed. On the basis of the last production days and monthly demand forecasts
p
dfM
,
the ending inventories are calculated easily, although information about capacities, ex-
act demands and production times after the planning horizon are missing. As this situa-
tion fits the preconditions of the economic order quantity, the economic order quantity
can be used as a basic component:
62
,
*
2* *12*
( ) *
setup
p tm
t TM
p
yw yir
p p
cM
dfM TM
i p cM price cM
∈
=+
∑
Then, a function has to be defined, which determines the last production day of a prod-
uct:
Let
ϕ
:undefined mid-term period
{
}
:
Φ
ϕ
Let
(
)
(
)
(
)
: , Φ, , , :
P TM TM p tm p tm tm
λ λ
→ =
֏
∪
A function that determines the last production mid-term period
Φ
tm TM
∈
∪
of product
p P
∈
:
(
)
,p tm
λ τ
=
(F1)
Ending inventory of
p P
∈
is calculated depending on the value of
(
)
,p tm
λ τ
=
.
If
τ ϕ
≡
and
Φ
ϕ
∈
the last production mid-term period could not be defined.
153
Then,
the ending inventory is set to the economic order quantity
(
)
*
:
p
eiM i p
=
.
If
TM
τ
∈
, the ending inventory is interpolated assuming that the last production lot of
p P
∈
, ending in mid-term period
τ
, equates the economic order quantity
(
)
*
i p
and
assuming that the exit speed of a product
154
is constant with
30
p
dfM
:
( )
*
max
: *( )
30
p
p
dfM
eiM i p TM
τ
= − −
This method has two main advantages: First, the ending inventory is dynamically calcu-
lated. Monthly demand variations are taken into account so that the planned inventory is
always on an adequate level. The second advantage is the generation of stabilized pro-
duction sequences by considering the last production period. The planning method for
the mid-term horizon - the model on which the planning method is based is described in
the next sub-section - guarantees that the production capacity of one mid-term period is
never exceeded. The interpolation of every product’s stock depending on the last day of
production results in the avoidance of simultaneous stock-outs of too many products
which could not be produced due to resulting missing daily production capacity. The
consequence of avoiding simultaneous stock-outs of products is that the production se-
153 This is especially the case at product start-ups.
154 The exit speed of a product is calculated by dividing the monthly demand forecast by the length of a
month. To ease calculation an average of 30 days is used.
quence calculated
count,
after the planning horizon are absorbed. Cha
zon, which result in lesser acceptance in practice, are minimized.
The following illustration visualizes the method. For simplification
product is
I
n this example, the
information available about the days after day 15. Consequently, the relatively high d
mand on d
utilization for covering demands on day
ing inventory at the end of day 15, that is
tion is
quantity. It is assumed that the economic order quantity is available on stock at the end
of the last production day. The minimum inventory is interpolated for every day, assu
in
tory is then able to satisfy the demand on day 16.
5.2.1.3
Besides financial parameters and parameters for variable initialization, production p
rameters ar
consumption of production, the
because the capacity limitation is defined by
quence calculated
count,
is
virtually
after the planning horizon are absorbed. Cha
zon, which result in lesser acceptance in practice, are minimized.
The following illustration visualizes the method. For simplification
product is
displayed and used to explain
n this example, the
information available about the days after day 15. Consequently, the relatively high d
mand on d
ay 16 would not be considered
utilization for covering demands on day
ing inventory at the end of day 15, that is
tion is
avoided as monthly demand forecasts are used to calculate t
quantity. It is assumed that the economic order quantity is available on stock at the end
of the last production day. The minimum inventory is interpolated for every day, assu
in
g that the inventory is reduced every day by the average daily demand. The real inve
tory is then able to satisfy the demand on day 16.
5.2.1.3
Production Parameters
Besides financial parameters and parameters for variable initialization, production p
rameters ar
e used
consumption of production, the
because the capacity limitation is defined by
0
10000
20000
30000
40000
50000
60000
70000
3 (FP)
Inventory
quence calculated
in
the planning method
virtually
contin
after the planning horizon are absorbed. Cha
zon, which result in lesser acceptance in practice, are minimized.
The following illustration visualizes the method. For simplification
displayed and used to explain
Figure
n this example, the
mid
information available about the days after day 15. Consequently, the relatively high d
ay 16 would not be considered
utilization for covering demands on day
ing inventory at the end of day 15, that is
avoided as monthly demand forecasts are used to calculate t
quantity. It is assumed that the economic order quantity is available on stock at the end
of the last production day. The minimum inventory is interpolated for every day, assu
g that the inventory is reduced every day by the average daily demand. The real inve
tory is then able to satisfy the demand on day 16.
Production Parameters
Besides financial parameters and parameters for variable initialization, production p
e used
as inputs
consumption of production, the
because the capacity limitation is defined by
3 (FP)
5
Inventory
the planning method
contin
ued after the end of the planning horizon. Demand fluctuations
after the planning horizon are absorbed. Cha
zon, which result in lesser acceptance in practice, are minimized.
The following illustration visualizes the method. For simplification
displayed and used to explain
Figure
17
: Visuali
mid
-
term planning horizon ends on day 15. There is no detailed
information available about the days after day 15. Consequently, the relatively high d
ay 16 would not be considered
utilization for covering demands on day
ing inventory at the end of day 15, that is
avoided as monthly demand forecasts are used to calculate t
quantity. It is assumed that the economic order quantity is available on stock at the end
of the last production day. The minimum inventory is interpolated for every day, assu
g that the inventory is reduced every day by the average daily demand. The real inve
tory is then able to satisfy the demand on day 16.
Production Parameters
Besides financial parameters and parameters for variable initialization, production p
as inputs
to model the production.
consumption of production, the
production time for one part
because the capacity limitation is defined by
7
Demand
the planning method
, which takes practical constraints into a
ued after the end of the planning horizon. Demand fluctuations
after the planning horizon are absorbed. Cha
nges of the plans within the planning hor
zon, which result in lesser acceptance in practice, are minimized.
The following illustration visualizes the method. For simplification
displayed and used to explain
how
the method wor
: Visuali
z
ation of Ending Inventory
term planning horizon ends on day 15. There is no detailed
information available about the days after day 15. Consequently, the relatively high d
ay 16 would not be considered
in
the planning procedure and only capacity
utilization for covering demands on day
s
6 and 9 would be planned. By setting an en
ing inventory at the end of day 15, that is
,
at the end of the planning horizon, this situ
avoided as monthly demand forecasts are used to calculate t
quantity. It is assumed that the economic order quantity is available on stock at the end
of the last production day. The minimum inventory is interpolated for every day, assu
g that the inventory is reduced every day by the average daily demand. The real inve
tory is then able to satisfy the demand on day 16.
Production Parameters
Besides financial parameters and parameters for variable initialization, production p
to model the production.
production time for one part
because the capacity limitation is defined by
the
9 11
Production
, which takes practical constraints into a
ued after the end of the planning horizon. Demand fluctuations
nges of the plans within the planning hor
zon, which result in lesser acceptance in practice, are minimized.
The following illustration visualizes the method. For simplification
the method wor
ation of Ending Inventory
term planning horizon ends on day 15. There is no detailed
information available about the days after day 15. Consequently, the relatively high d
the planning procedure and only capacity
6 and 9 would be planned. By setting an en
at the end of the planning horizon, this situ
avoided as monthly demand forecasts are used to calculate t
quantity. It is assumed that the economic order quantity is available on stock at the end
of the last production day. The minimum inventory is interpolated for every day, assu
g that the inventory is reduced every day by the average daily demand. The real inve
tory is then able to satisfy the demand on day 16.
Besides financial parameters and parameters for variable initialization, production p
to model the production.
In order to calculate
production time for one part
the
available production
13
Production
Inventory Economic Order Quantity
, which takes practical constraints into a
ued after the end of the planning horizon. Demand fluctuations
nges of the plans within the planning hor
zon, which result in lesser acceptance in practice, are minimized.
The following illustration visualizes the method. For simplification
purpose
the method wor
ks:
ation of Ending Inventory
term planning horizon ends on day 15. There is no detailed
information available about the days after day 15. Consequently, the relatively high d
the planning procedure and only capacity
6 and 9 would be planned. By setting an en
at the end of the planning horizon, this situ
avoided as monthly demand forecasts are used to calculate t
he economic order
quantity. It is assumed that the economic order quantity is available on stock at the end
of the last production day. The minimum inventory is interpolated for every day, assu
g that the inventory is reduced every day by the average daily demand. The real inve
Besides financial parameters and parameters for variable initialization, production p
In order to calculate
production time for one part
,p m
pt
has to be determined
available production
15 (End)
Inventory Economic Order Quantity
Unknown
, which takes practical constraints into a
ued after the end of the planning horizon. Demand fluctuations
nges of the plans within the planning hor
purpose
s
, only one
term planning horizon ends on day 15. There is no detailed
information available about the days after day 15. Consequently, the relatively high d
the planning procedure and only capacity
6 and 9 would be planned. By setting an en
at the end of the planning horizon, this situ
he economic order
quantity. It is assumed that the economic order quantity is available on stock at the end
of the last production day. The minimum inventory is interpolated for every day, assu
g that the inventory is reduced every day by the average daily demand. The real inve
Besides financial parameters and parameters for variable initialization, production p
In order to calculate
the
capacity
has to be determined
available production
time in one per
17 19
Inventory Economic Order Quantity
Unknown
Horizon
63
, which takes practical constraints into a
c-
ued after the end of the planning horizon. Demand fluctuations
nges of the plans within the planning hor
i-
, only one
term planning horizon ends on day 15. There is no detailed
information available about the days after day 15. Consequently, the relatively high d
e-
the planning procedure and only capacity
6 and 9 would be planned. By setting an en
d-
at the end of the planning horizon, this situ
a-
he economic order
quantity. It is assumed that the economic order quantity is available on stock at the end
of the last production day. The minimum inventory is interpolated for every day, assu
m-
g that the inventory is reduced every day by the average daily demand. The real inve
n-
Besides financial parameters and parameters for variable initialization, production p
a-
capacity
has to be determined
time in one per
i-
Inventory Economic Order Quantity
64
od. In this case, one mid-term period is one day. One period
155
consists of the time
tm
tM
less break time
tm
btM
. The remaining time cannot be used completely for production.
There exist some tasks like cleaning or small maintenance tasks which cannot be
planned in detail but estimated on the basis of experience. These tasks reduce the max-
imum utilization. Therefore, it has to be reduced by the maximum degree of utilization
of the machine
m M
∈
factor
m
udM
. Monthly demand forecasts are available and rep-
resented by
p
dfM
. Exact demands announced by customers, which are contractually
fixed with small tolerances, are stored in
,
p tm
dM
. The average set-up time for a product
is expressed in
p
stM
. Maintenance of dies takes more than one period. The average
maintenance progress per mid-term time-period is defined by the parameter
,
m p
mpM
.
Production lots have to be designed as integer multiples of raw material units. The size
of the steel coils is stored in the batch size parameter
p
bs
for each product
p P
∈
. The
size of steel coils varies. As the variance is not crucial, and a direct consideration of the
individual coil size would increase complexity, average values are taken into account.
First, a set
p
C
is defined. This set groups the coils
c C
∈
which can be used as raw ma-
terial units for the production of a product
p P
∈
. With this set, it is possible to calcu-
late the average size, that is, the average weight of the coils
p
avgcs
, depending on their
matching with a part
p P
∈
:
1
: ( )*
p
pp
c C
avgcs size c p P
C
∈
= ∀ ∈
∑
The average weight of a coil is not suitable for defining lots directly. With the charge
weight of a product, the number of parts which can be produced from one coil is calcu-
lated (
p
cout
). In this calculation, the number of parts produced simultaneously has to be
taken into account. Any potential remainder has to be ignored.
1
: *
p
p
pp
avgcs
cout chw
CP
=
The interrelation between the mid-term planning method and the short-term planning
method prohibits the simple use of the average output of the coils
p
cout
as batch-size.
As explained in section 5.3, batches and lots have to fit to short-term periods. In order to
avoid discrepancies, this has to be considered during mid-term planning. The batch-size
155 In the case study, one mid-term period is defined by a day containing 24 hours of time-based capacity,
which is then further reduced by break time and machine utilization.
65
has to be defined in the same way as in short-term planning, where the size of short-
term periods, that is, the time-based length, is considered:
: * * *60
* *60
p
p p
p
cout
bs tsS pt
tsS pt
=
Last but not least, maximum and minimum lot sizes
p
maxlot
and
p
minlot
have to be
parameterized. The absolute minimum for a lot is the batch size
p
bs
. A higher mini-
mum lot size can also be set, if necessary. The maintenance of a die overlaps several
mid-term time-periods.
156
Therefore, the maintenance time has to be converted into a
number of maintenance periods:
:
p
p
mtnc
mtM tsM
=
Having defined and explained the calculation of all relevant inputs, the expected outputs
have to be declared and explained in further detail. This is done in the next sub-section.
5.2.2 Output
The result of the planning model, which is described in the next sub-section, is repre-
sented by values of defined variables. First, there is the production variable
0
, ,
m p tm
xM
∈
N
, which is used to store the amount of parts produced on machine
m M
∈
in a mid-term period
tm TM
∈
. If production is running on, the binary variable
, ,
m p tm
binxM
is activated. Then, the inventory variable
0
,
p tm
iM
∈
N
is used to get the
amount of parts available in the inventory. The variable
0
,
p n
miM
∈
N
is used to evaluate
the gap between the achievable inventory and the desired ending inventory level
p
eiM
at the end of the planning horizon. Another variable named
0
, ,
m p tm
lotM
∈
N
stores the
cumulative quantity of the amount actually produced since the last die maintenance.
Set-ups are managed by binary variables
, ,
m p tm
binsM
and
, ,
m p tm
binsrM
. The former is
activated if a product change is necessary. On this basis, capacity reductions and set-up
costs are planned. The latter represents the case where a set-up requiring only low effort
is executed. This is the case whenever products with different identifications but which
156 In the case study, maintenance takes about 70 hours. That corresponds to three mid-term periods in the
model.
66
are very similar or even identical are produced consecutively.
157
The variables
0
, ,
m p tm
bcM
∈
N
serve as counting variables for complete batches. Die maintenance is
managed by three other variables: the binary variable
, ,
m p tm
binmM
, indicating whether
maintenance is actually going on; a progress variable
, ,
m p tm
cmM
, storing the percentage
of the maintenance progress; and, last but not least, the binary variable
, ,
m p tm
fmM
, rep-
resenting the completion of maintenance.
5.2.3 Model
The mid-term horizon, which is in this case a planning period between day 4 and day
14, is modeled using a big-bucket linear programming model. In one time-period, it is
possible that multiple actions like production of several parts, set-ups or coil changes
are planned.
In order to define whether two products are produced in coupled production, whether
two products have to use the same die, or whether a product can be produced with a
machine, the following functions are used.
Let
{0,1}
=
B
and
(
)
(
)
: , , ( , ) , :
P P p q p q b
γ γ
→ =
֏
B
A function that determines whether two products
,
p q P
∈
are produced in
coupled production:
(
)
, 1
p q
γ
=
, if
,
p q
are produced in coupled production.
(F2)
Let
{0,1}
=
B
and
(
)
(
)
: , , ( , ) , :
P P p q p q b
δ δ
→ =
֏
B
A function that determines whether two products
,
p q P
∈
are produced with
the same die:
(
)
, 1
p q
δ
=
, if
,
p q
are produced with the same die.
(F3)
157 In the case study, there are several products which are identical but which have different identification
numbers. The reason for that is to ease the differentiation between different subsequent processes.
67
Let
{0,1}
=
B
and
(
)
(
)
: , , ( , ) , :
P M p m p m b
ρ ρ
→ =
֏
B
A function that determines whether a product
p P
∈
can be produced with
machine
m M
∈
:
(
)
, 1
p m
ρ
=
, if
p
can be produced with
m
.
(F4)
As products which are produced in coupled production use the same die, it is clear that
the following expression applies:
(
)
(
)
, ,
p q p q
γ δ
→
Target function:
min Z =
(
)
, , , , ,
*
setup
m p tm m p tm p tm
m M p Ptm TM
binsM binsrM cM
∈ ∈ ∈
−
∑ ∑ ∑
, , ,
*
prod
m p tm p tm
m M p Ptm TM
xM cM
∈ ∈ ∈
+
∑ ∑ ∑
, , ,
*
mtnc
m p tm d tm
m M p Ptm TM
binmM cM
∈ ∈ ∈
+
∑ ∑ ∑
,
*
inv
p tm p
p Ptm TM
iM cM
∈ ∈
+
∑ ∑
,
*
∈
=
+
∑
so
p n p
max
p P
n TM
miM cM
The target function of the mid-term lot sizing consists of several main components:
First, estimated set-up costs are calculated. The sequence dependency of set-up costs is
ignored, except in the case where no set-up is needed for two parts.
158
In addition, pro-
duction costs, which vary depending on the day,
159
are another component of the target
function. Being only an aggregated model, shift-based costs are ignored. Then, mainte-
nance costs are also considered and calculated. Then, inventory holding costs are con-
sidered. Last but not least, imputed stock-out costs for each product for the last period
of the rolling planning horizon are calculated and inserted into the target function.
Whenever
,
p n
miM
is greater than 0, the availability of product p cannot be guaranteed.
158 In the case study, there are parts which are identical but have different part numbers in order to differ-
entiate them from ones needed in subsequent production processes. These parts are produced sequen-
tially but do not induce set-ups at the machines.
159 Differentiation of production costs depending on the days can be read in 2.2.1.3.
68
The estimated sum of the costs for the consequences
160
is calculated by multiplying
so
p
cM
for every product.
The linear programming model consists of several restrictions describing practical con-
ditions. A basic condition for all lot sizing models is a constraint which sets all inputs
equal to outputs. In this case, this is done by the inventory balance equation.
, 1 , , , ,
p tm m p tm p tm p tm
m M
iM xM dM iM
−
∈
+ = +
∑
;
p P tm TM
∀ ∈ ∀ ∈
(1)
The inventory balance equation (1) guarantees that announced demands are covered
either by the inventory available until
t TM
∈
and/or by manufactured products during
the mid-term period
t TM
∈
. Hence, production is necessary to cover demands. But,
production capacity is limited and has to be restricted.
, ,
1
* *
m p tm p
p P
p
xM pt
CP
∈
∑
( )
, , , ,
1
* *
m p tm m p tm p
p P
p
binsM binsM stM
CP
∈
+ +
∑
(
)
*
tm tm m
tM btM udM
≤ −
;
m M tm TM
∀ ∈ ∀ ∈
(2)
Available production capacity, which is calculated on the basis of times, cannot be ex-
ceeded. The upper limit is calculated by the available day production time multiplied by
the maximum degree of utilization experienced which represents capacity losses due to
coil changes, cleaning, and other minor works on the machine which are not explicitly
planned. The rest of the available capacity is then shared by production and set-up
times, in both cases considering coupled production (2).
, , , ,
* 0
m p tm p m p tm
xM maxlot binxM
− ≤
; ;
m M p P tm TM
∀ ∈ ∀ ∈ ∀ ∈
(3)
, , , ,
0
m p tm m p tm
xM binxM
− ≥
; ;
m M p P tm TM
∀ ∈ ∀ ∈ ∀ ∈
(4)
Binary variables
, ,
m p tm
binxM
, which indicate the production of a product in a mid-term
period, are activated by restrictions (3) and (4). The maximum lot size parameter for
160 High contract penalties are the consequence in the short term. In the long term, competitive advantage
is endangered.
69
product p is taken as Big-M. The indication variables for production are necessary to
model further practical aspects and are used in other restrictions described later.
, , , ,
0
m p tm m q tm
lotM lotM
− =
; ;
m M tm TM
∀ ∈ ∀ ∈
(
)
, , 1
p q P p q p q
γ
∀ ∈ ∧ = ∧ ≠
(5)
, , , ,
0
m p tm m q tm
binmM binmM
− =
; ;
m M tm TM
∀ ∈ ∀ ∈
(
)
, , 1
p q P p q p q
γ
∀ ∈ ∧ = ∧ ≠
(6)
, , , ,
0
m p tm m q tm
fmM fmM
− =
; ;
m M tm TM
∀ ∈ ∀ ∈
(
)
, , 1
p q P p q p q
γ
∀ ∈ ∧ = ∧ ≠
(7)
, , , ,
0
m p tm m q tm
cmM cmM
− =
; ;
m M tm TM
∀ ∈ ∀ ∈
(
)
, , 1
p q P p q p q
γ
∀ ∈ ∧ = ∧ ≠
(8)
, , , ,
0
m p tm m q tm
xM xM
− =
; ;
m M tm TM
∀ ∈ ∀ ∈
(
)
, , 1
p q P p q p q
γ
∀ ∈ ∧ = ∧ ≠
(9)
Constraint (9) models coupled production. The solution space is reduced by valid ine-
qualities (5) to (8), which are based on the same function
(
)
,
p q
γ
calculating whether
two products are manufactured in coupled production.
, , , , , , 1
m p tm m p tm m p tm
lotM xM lotM
−
≤ +
; ;
m M p P tm TM
∀ ∈ ∀ ∈ ∀ ∈
(10)
, , , , , , 1 , ,
*(1 )
m p tm m p tm m p tm p m p tm
lotM xM lotM maxlot binmM
−
≥ + − −
; ;
m M p P tm TM
∀ ∈ ∀ ∈ ∀ ∈
(11)
( )
( )
, , , ,
, 1
, 0
1
m p tm m q tm p
q
q P
p q
p q
p q
lotM lotM maxlot
CP
δ
γ
∈
≠
=
=
+ ≤
∑
; ;
m M tm TM p P
∀ ∈ ∀ ∈ ∀ ∈
(12)
Constraints (10) and (11) define lot variables
, ,
m p t
lotM
. If
, ,
m p tm
binmM
is not active, that
means that no maintenance is planned in period
tm
; the combination of (10) and (11)
constrain
, , , , , , 1
m p tm m p tm m p tm
lotM xM lotM
−
= +
. Restriction (12) guarantees that the abra-
70
sion of a die is correctly taken into account. As a die can be used for multiple products,
the lot size, that is, the cumulative production amount, has to be calculated for all relat-
ed parts.
, , , ,
1
m p tm m p tm
binmM binxM
+ ≤
; ;
m M p P tm TM
∀ ∈ ∀ ∈ ∀ ∈
(13)
In (13), it is guaranteed that no production is planned simultaneously with die mainte-
nance.
, , , ,
0
m p tm m p tm
binmM cmM
− ≤
; ;
m M p P tm TM
∀ ∈ ∀ ∈ ∀ ∈
(14)
, , , , , 1 , , , ,
*
m p m p tm m p tm m p tm m p tm
mpM binmM cmM cmM fmM
−
+ = +
; ; : 0
m M p P tm TM t
∀ ∈ ∀ ∈ ∀ ∈ >
(15)
Constraints (14) and (15) manage the die maintenance progress, whereas (14) guaran-
tees that the progress variable
, ,
m p tm
cmM
is greater than 0 whenever die maintenance
, ,
m p tm
binmM
is active; (15) guarantees that the defined daily maintenance progress fac-
tor
,
m p
mpM
is correctly added to
, ,
m p tm
cmM
until die maintenance is finished, indicated
by
, ,
m p tm
fmM
.
, , , ,m p tm m q tm
binmM binmM
− =
0
; ;
m M tm TM
∀ ∈ ∀ ∈
(
)
, , 1
p q P p q p q
δ
∀ ∈ ∧ = ∧ ≠
(16)
Maintenance variables for all other parts produced with the same die are connected and
activated or deactivated by (16).
Maintenance can be triggered in two different ways: The first way is to start die mainte-
nance directly after a die change. Another possibility is to start die maintenance after a
defined maximum cumulative quantity. The first two restrictions describe the first op-
tion. The latter option is modeled by the following restriction.
( )
, , , , , , , 1
,
1*
m p tm mp tm m q tm m p tm
q P q
p q
binmM binxM binxM binxM
CP
δ
−
∈
∧
+ + ≥
∑
; , ; : 0
m M p q P tm TM tm
∀ ∈ ∀ ∈ ∀ ∈ >
(17)
71
Depending on the maintenance rule used, restriction (17) is either inserted into the mod-
el or not. Restriction (17) is necessary to represent the case where maintenance is started
after changing production to another product which is not being produced with the same
die. If maintenance in a previous period
, , 1
m p tm
binxM
−
is active, production is either con-
tinued or finished, indicated by
, ,
m p tm
binmM
, in the actual mid-term period
tm TM
∈
(17). If finished, maintenance is activated in the following mid-term period
tm TM
∈
,
indicated by
, ,
m p tm
binmM
.
, , 1 , , , , 1 , ,
2
m p tm m p tm m p tm m q tm
binmM binxM binxM binxM
+ −
− − − ≥ −
: 0;
tm TM tm m M
∀ ∈ > ∀ ∈
(
)
, , 1
p q P p q p q
δ
∀ ∈ ∧ = ∧ ≠
(18)
, , 1 , , , , 1 , , 1
2
m p tm m p tm m p tm m q tm
binmM binxM binxM binxM
+ − +
− − − ≥ −
: 0;
tm TM tm m M
∀ ∈ > ∀ ∈
(
)
, , 1
p q P p q p q
δ
∀ ∈ ∧ = ∧ ≠
(19)
Maintenance of the die used to produce
p P
∈
is started in a mid-term period
( 1)
tm TM
+ ∈
, when production of product
p P
∈
, which has been started in at least
two mid-term periods before, is changed to product
q P
∈
in a mid-term period
tm TM
∈
. In propositional logic, this can be represented like this:
(
)
, , , , 1 , , , , 1 , , 1
m p tm m p tm m q tm m q tm m p tm
binxM binxM binxM binxM binmM
− + +
∧ ∧ ∨ →
: 0;
tm TM tm m M
∀ ∈ > ∀ ∈
(
)
, , 1
p q P p q p q
δ
∀ ∈ ∧ = ∧ ≠
The algebraic formulation can be found in (18) and (19).
, , 1 , ,
0
m p tm m p tm
binxM binsrM
−
− ≥
; ; : 0
m M p P tm TM tm
∀ ∈ ∀ ∈ ∀ ∈ >
(20)
, , , ,
0
m p tm m p tm
binxM binsrM
− ≥
; ; : 0
m M p P tm TM tm
∀ ∈ ∀ ∈ ∀ ∈ >
(21)
, , 1 , ,
1
m p tm m q tm
binxM binsrM
−
− − ≥ −
m M
∀ ∈
(
)
, , 0
p q P p q p q
δ
∀ ∈ ∧ = ∧ ≠
: 0
tm TM tm
∀ ∈ >
(22)
72
, , , ,
1
m p tm m q tm
binxM binsrM
− − ≥ −
m M
∀ ∈
(
)
, , 0
p q P p q p q
δ
∀ ∈ ∧ = ∧ ≠
: 0
tm TM tm
∀ ∈ >
(23)
Period-overlapping lots are represented by restrictions (20) to (23). As no set-up is nec-
essary, no capacity reduction has to be calculated for the actual mid-term period if pro-
duction is continued from previous periods (20), (21). Restrictions (22) and (23) are the
algebraic formulation of the expression
, , , , 1 , ,
( )
m p tm m p tm m p tm
binsrM binxM binxM
−
→ ∧
: 0;
tm TM tm m M
∀ ∈ > ∀ ∈
(
)
, , 0
p q P p q p q
δ
∀ ∈ ∧ = ∧ ≠
This expression activates the set-up reduction whenever a product is produced in two
consecutive mid-term time-periods.
, , , ,
*
m p tm p m p t
lotM bs bcM m
=
; ;
m M p P tm TM
∀ ∈ ∀ ∈ ∀ ∈
(24)
As real capacity usage for production has to be calculated correctly,
161
batch-wise pro-
duction dependent on steel coil sizes has to be considered in mid-term lot sizing already,
constraint (24) is inserted. It constrains planned lots to integer multiples of raw material
units.
,
p TM p
min
iM iM
ϖ
=
;
min
p P TM TM
∀ ∈ ∈
(25)
, ,
p TM p TM p
max max
miM iM eiM
+ ≥
;
max
p P TM TM
∀ ∈ ∈
(26)
Restrictions (25) and (26) define starting and ending inventories. Missing amounts at
the end of the planning horizon are saved in
,
p n
miM
and inserted into the target func-
tion in order to consider the loss of guarantee of availability towards the customer.
161 Customer demands vary within the product portfolio. High runners’ demands are usually higher than
the output of a steel coil. In contrast, there exist some low runners, whose demands within the consid-
ered time horizon are smaller than the output of a steel coil. As it is a practical constraint to produce in
batches of raw material units, that is, in integer multiples of steel coil outputs, capacity utilization has
to be calculated on the basis of the production time for a whole coil instead of the production time for
a relatively small customer demand.
73
, , ,
m p TM m p
min
lotM lotM
ϖ
=
; ;
min
p P m M TM TM
∀ ∈ ∈ ∈
(27)
, , ,
m p TM m p
min
binxM binxM
ϖ
=
; ;
min
p P m M TM TM
∀ ∈ ∈ ∈
(28)
Equations (27) and (28) set the initial values for the lot variable or the state of the binary
production variable.
,
min
p TM p
mbinM mbinM
ϖ
=
;
min
p P TM TM
∀ ∈ ∈
(29)
,
min
p TM p
mbinM mbinM
ϖ
=
;
min
p P TM TM
∀ ∈ ∈
(30)
,
min
p TM p
mpM mpM
ϖ
=
;
min
p P TM TM
∀ ∈ ∈
(31)
Last but not least, restrictions (29) to (31) initialize the variables which are relevant for
maintenance regarding the first mid-term period of the rolling horizon.
5.3 Short-Term Scheduling Considering Multidimensional Re-
strictions
This section contains the description and explanations necessary for understanding the
model for short-term schedule planning. Parameters are elucidated in the first sub-
section. Then, the outputs are explained. Lastly, the model is explained in detail con-
taining all the restrictions required to consider practical conditions.
5.3.1 Input
In this sub-section, inputs of the short-term lot size planning are determined. First, fi-
nancial parameters are described and some are calculated by other parameters. After
that, parameters for variable initialization are determined. Lastly, production parameters
are explained.
74
5.3.1.1 Financial Parameters
During short-term schedule planning, diverse costs have to be taken into account in or-
der to achieve cost-optimal plans. First,
inv
p ST
c p P
∈
, inventory holding and capital com-
mitment costs of a product for a single short-term period are parameterized. Sequence-
dependent set-up costs
, , ,
setup
m p q ts
c
ST
p P
∈
have to be regarded as well. Set-up costs depend
on the time-period in which they are executed. This is similar with production costs
, ,
prod
m p ts ST
c p P
∈
and maintenance costs
, ,
mtnc
m p ts
c
ST
p P
∈
, which are machine-, product- and
period-dependent. In order to calculate the costs of the plans correctly, coil change costs
, ,
cc
m p ts
c
ST
p P
∈
are parameterized. Set-up team costs are calculated and stored separately
in
team
ts
c
ST
p P
∈
for each short-term period.
5.3.1.2 Parameters for Variable Initialization
In order to be able to link the system state in reality with planning, the relevant data has
to be transmitted to the planning method. This is done by parameter settings for variable
initialization. The following parameters have to be set with updated values determined
in real production.
First, there is the initial inventory
p
iS
ϖ
representing the actual inventory of products
ST
p P
∈
. In order to couple short-term and mid-term lot size planning,
p
eiS
is set. The
coupling of the partial models is explained in further detail in section 5.4. The current
real world production is initialized using
, ,
m p t ST
prodS p P
ϖ
∈
. Parameter
,
m p
lotS
ϖ
ST
p P
∈
is used to define the initial cumulative quantity of a production lot. If the mini-
mal lot size is exceeded in the initialization short-term period, the parameter
,
m p
mlS
ϖ
ST
p P
∈
is set to 1. Maintenance has to be considered in mid-term planning as well as in
short-term planning. Maintenance state and maintenance progress are initialized with
p
mbinS
ϖ
ST
p P
∈
and
p
mpS
ϖ
ST
p P
∈
. The actual set-up state of a machine is im-
portant for the planning method. The machine status is initialized with binary variable
,
m p
sS
ϖ
ST
p P
∈
. It is possible that a machine is in the process of being set up when
planning starts. This is done by setting the value of the binary variable
, ,
m p q
rS
ϖ
,
ST
p q P
∈
. In combination with the binary parameter
, , ,
m p q ts
mstS
ϖ
,
ST
p q P
∈
, which is
set to 1 if the set-up was finished in the short-term period
ts TS
∈
, and the parameter
, ,
m p q
csS
ϖ
,
ST
p q P
∈
, representing the set-up progress, the set-up of a machine
m M
∈
75
from product
ST
p P
∈
to product
ST
q P
∈
during the first planning period for short-term
planning can be transmitted completely from the real production world to the model.
The number of available set-up teams is initialized with parameter
teamsS
ϖ
and limited
by
ts
teamLimS
. Last but not least, real production statuses of coil changes have to be
transmitted to the model. The parameter
,
m p
ReS
ϖ
ST
p P
∈
initializes the number of
completely used steel coils of the actual lot. The initialization of the slack variable
,
m p
SlS
ϖ
ST
p P
∈
guarantees that the coil usage is modeled correctly, and then the pa-
rameter
,
m p
cwS
ϖ
ST
p P
∈
sets the value of the corresponding variable for the coil
change.
5.3.1.3 Production Parameters
Production parameters are necessary for describing the relevant production. Basically,
all parameters define capacity limits and capacity usages for different actions or entities.
The production speed parameter
,
p m
pptS
ST
p P
∈
defines how many products of
ST
p P
∈
can be produced on a machine
m M
∈
in one short-term period. The demand of
one product
ST
p P
∈
in the short-term period
ts TS
∈
is set. As in the actual practical
case, demands are available on a daily basis. As the periods of the short-term planning
are shorter, available demand data has to be transformed beforehand. In this case, de-
mands announced for the mid-term period
tm TM
∈
are transformed into demands of
the final short-term period of the mid-term period
( 1)
tm TM
− ∈
. Consequently, it is
guaranteed that ordered products are available on time. Maximum and minimum lot
sizes are parameterized, defining values for
p
maxlot
and
p
minlot
ST
p P
∈
, as in mid-
term planning. The calculation of the batch size parameter was already explained in
5.2.1.3. As the model, which is described in further detail in section 5.3.3, underlies the
all-or-nothing assumption, set-up times, which are given in minutes, have to be convert-
ed to a number of corresponding short-term planning periods using the parameter
tsS
,
which represents the length of a short-term period:
,
,
,
*60
p q
p q ST
stMin
st p q P
tsS
= ∈
As the parts’ usage of loading equipment differ from each other,
,
p lt
verbPLT
ST
p P
∈
is
determined. The parameter
lt
kapaLT
limits the capacity of a loading equipment type.
76
5.3.2 Output
The result of the planning method is represented by values of variables used in the mod-
el. In this sub-section, variables for short-term planning are described.
First, there are the variables
, ,
m p ts
prodS
ST
p P
∈
and
, ,
m p ts
xS
ST
p P
∈
. The values of the-
se variables determine whether production of a product
ST
p P
∈
is taking place at ma-
chine
m M
∈
in the short-term planning period
ts TS
∈
accordingly the production
amount. The set-up status is stored in variable
, ,
m p ts
sS
ST
p P
∈
. The calculated inventory
is stored for every product in each short-term period in
,
p ts
iS
ST
p P
∈
. The cumulative
quantity of produced products of the current lot can be obtained by reading variable
, ,
m p ts
lotS
.
ST
p P
∈
. The auxiliary variable
, ,
m p ts
mlS
ST
p P
∈
determines whether the min-
imal lot size was already exceeded. A product change is then possible. The number of
completed batches, that is, completely used coils, is stored in
, ,
m p ts
reS
ST
p P
∈
. In order
to be able to model coil usage on the basis of a cumulative production quantity, a slack
variable
, ,
m p ts
rlS
ST
p P
∈
is introduced. The binary variable
, ,
m p ts
cwS
ST
p P
∈
is set true
when a coil is changed. Sequence-dependent set-ups are represented by
, , ,
m p q ts
rS
,
ST
p q P
∈
. The corresponding binary variable is set true when a machine is being set up
from product
ST
p P
∈
to product
ST
q P
∈
during the short-term period
ts TS
∈
whereas
p q
≠
and
(
)
, 0
p q
γ
=
. With binary variables
, , ,
m p q t
mstS
and real variables
, , ,
m p q ts
cs
, the
end or the progress of a set-up are managed. The variable
t
teams
defines how many set-
up teams are required within a short-term period
ts TS
∈
. Binary indicator variables
,
r
m ts
binS
,
,
prod
m ts
binS
, and
,
cw
m ts
binS
enable the identification and separation of activities on
machines during a short-term period. Another binary variable
, ,
mtnc
m p ts
binS
ST
p P
∈
repre-
sents the maintenance activity.
5.3.3 Model
The short-term horizon, which is in this case a planning period between day 1 and day
3, is modeled using a small-bucket linear programming model. The all-or-nothing as-
sumption applies. Consequently, it is not possible that different actions like production,
set-up or coil changes are planned in one single time-period.
77
The model’s target function consists of several components. The first component is the
sum of set-up team costs. Then, the sum of sequence-dependent set-up costs is added.
The costs are activated depending on the value of the binary variables
, , ,
m p q ts
rS
,
ST
p q P
∈
. The sum of the coil changing costs is added as well as the sum of the pro-
duction costs and maintenance costs. These costs depend on the time-period in which
they are created. The sum of inventory holding costs is finally added.
Let
{0,1}
=
B
and
(
)
(
)
σ: TS,TS , ( , ) , :
u v u v b
σ
→ =
֏
B
The above is a function which determines whether two short-term periods
,
u v TS
∈
belong to the same shift
s S
∈
.
(
)
, 1
u v
σ
=
, if
,
u v
are part of the same shift.
(F5)
, 1 , , , ,
p t m p t p ts p ts
m M
iS xS dS iS
−
∈
+ = +
∑
; :
ST min
p P ts TS ts TS
∀ ∈ ∀ ∈ >
(1)
An inventory balance equation (1) guarantees that the input equals all outputs and that
the inventory is always correctly filled.
, , , , ,
*
p m m p ts m p ts
pptS prodS xS
=
; ;
ST
m M ts TS p P
∀ ∈ ∀ ∈ ∀ ∈
(2)
The production output
, ,
m p ts
x
is calculated by constraint (2), multiplying the production
amount per period with the corresponding binary production variable.
Target function:
min
Z
=
*
team
ts ts
ts TS
teams c
∈
∑
( )
, , , , , ,
, 0
*
ST
ruest
m p q ts m p q ts
q P
m M p P ts TS
p q
p q
rS c
γ
∈
∈ ∈ ∈
=
≠
+
∑ ∑ ∑ ∑
, , , ,
*
cw
m p ts m p ts
ST
m M p P ts TS
cwS c
∈ ∈ ∈
+
∑∑∑
, , , ,
*
prod
m p ts m p ts
ST
m M p P ts TS
xS c
∈ ∈ ∈
+
∑∑∑
, , , ,
*
mtnc mtnc
m p ts m p ts
ST
m M p P ts TS
binS c
∈ ∈ ∈
+
∑∑∑
,
*
p ts p
ST
p P ts TS
iS kl
∈ ∈
+
∑ ∑
78
, ,
1
1
| |
ST
m p ts
p P p
sS CP
∈
=
∑
;
m M ts TS
∀ ∈ ∀ ∈
(3)
Restriction (3) guarantees that during one short-term period, a machine is always set to
produce only one product, or one product with all other coupled products. Therefore,
set-up states are always well defined and it is not possible that a machine has no set-up
state.
, , , ,
m p ts m q ts
sS sS
−
=0
; ;
m M ts TS
∀ ∈ ∀ ∈
(
)
, , 1
ST
p q P p q p q
γ
∀ ∈ ∧ = ∧ ≠
(4)
, , , ,m p ts m q ts
prodS prodS
− =
0
; ;
m M ts TS
∀ ∈ ∀ ∈
(
)
, , 1
ST
p q P p q p q
γ
∀ ∈ ∧ = ∧ ≠
(5)
, , , ,
0
m p ts m p ts
prodS sS
− ≤
; ;
ST
m M p P ts TS
∀ ∈ ∀ ∈ ∀ ∈
(6)
The interconnection of set-up state variables and the interconnection of production vari-
ables in the case of coupled production are modeled with restrictions (4) and (5). As
production of a certain part requires the machine to be in the corresponding set-up state,
it is necessary to introduce (6), which ensures this.
, , 1 , , , , ,
1
m p ts m q ts m p q ts
sS sS rS
−
+ − ≤
(
)
, , 0 ;
ST
p q P p q p q
γ
∀ ∈ ∧ = ∧ ≠
; :
min
m M ts TS ts TS
∀ ∈ ∀ ∈ >
(7)
A set-up state change requires a set-up which is modeled with the variables
, , ,
m p q ts
rS
. In
order to represent the coherence of set-up state variables and set-up variables, (7) is es-
sential.
, , , ,
r
m p q ts m ts
rS binS
≤
(
)
, , 0 ;
ST
p q P p q p q
γ
∀ ∈ ∧ = ∧ ≠
;
m M ts TS
∀ ∈ ∀ ∈
(8)
79
( ) ( )
, , , ,
, 0 , 0
* max *
r
m p q ts m ts p q
ST ST
ST ST
q P q P
p P p P
q p p q q p p q
rS binS CP CP
γ γ
∈ ∈
∈ ∈
≠ ∧ = ≠ ∧ =
≤
∑ ∑
∑ ∑
;
m M ts TS
∀ ∈ ∈
(9)
( )
, , , ,
, 0
r
m p q ts m ts
ST
ST
q P
p P
q p p q
rS binS
γ
∈
∈
≠ ∧ =
≥
∑
∑
;
m M ts TS
∀ ∈ ∈
(10)
Restrictions (8) to (10) control the activation of the binary indication variable for set-
ups
,
r
m ts
binS
. In restriction (9), the maximum number of all possible
(
)
, : , 0
ST
p q P q p and p q
γ
∈ ≠ =
combinations is taken as Big-M.
, , ,
cw
m p ts m ts
cwS binS
≤
, ; ;
ST
p q P m M ts TS
∀ ∈ ∀ ∈ ∀ ∈
(11)
, , ,
*
cw
m p ts m ts ST
ST
p P
cwS binS P
∈
≤
∑
;
m M ts TS
∀ ∈ ∈
(12)
, , ,
cw
m p ts m ts
ST
p P
cwS binS
∈
≥
∑
;
m M ts TS
∀ ∈ ∈
(13)
Restrictions (11) to (13) are necessary to control the activation/deactivation of the bina-
ry indication variable for coil changes
,
cw
m ts
binS
. As Big-M, the cardinality of the product
set
ST
P
is considered.
, , ,
prod
m p ts m ts
prodS binS≤
; ;
ST
p P m M ts TS
∀ ∈ ∀ ∈ ∀ ∈
(14)
, , ,
*
prod
m p ts m ts ST
ST
p P
prodS binS P
∈
≤
∑
;
m M ts TS
∀ ∈ ∈
(15)
, , ,
prod
m p ts m ts
ST
p P
prodS binS
∈
≥
∑
;
m M ts TS
∀ ∈ ∈
(16)
80
Similar to the previously defined control of the binary indication variables,
,
prod
m ts
binS
is
activated/deactivated. Restrictions (10), (13) and (16) are redundant but improve model
performance by reducing solution space.
, , ,
1
prod r cw
m ts m ts m ts
binS binS binS
+ + ≤
;
m M ts TS
∀ ∈ ∈
(17)
Restriction (17) guarantees that the exclusive activities of production, set-up and coil
changes are never done simultaneously at one single machine. Modeling of sequence-
dependent set-up times in combination with coupled production has to be correctly
achieved. Erroneous set-up state changes, shown in the following illustration, in which
machine states are no longer well defined, have to be eliminated.
Part D
Part E
Part A
Part B
Part C
Die 0815
Die 4711 Die 4712
rm,A,D,t=1
rm,A,E,t=1
sm,A,t-setup time =1
sm,B,t-setup time=1
sm,C,t-setup time=1
sm,D,t+1=1
sm,E,t+1=1
sm,F,t+1=1
sm,G,t+1=1
Part GPart F
rm,C,F,t=1
rm,B,F,t=1
rm,C,G,t=1
rm,B,G,t=1
Figure 18: Erroneous Modeling of Set-up State Changes at Coupled Products
,
, , , , ,
p q
m p q ts m p ts st
rS sS
−
≤
(
)
; , : , 0;
ST
m M p q P p q
δ
∀ ∈ ∀ ∈ =
,
:
min p q
ts TS ts TS st
∀ ∈ > +
(18)
,
, , , , ,
p q
m p q ts m q ts st
rS sS
+
≤
(
)
; , : , 0;
ST
m M p q P p q
δ
∀ ∈ ∀ ∈ =
,
:
max p q
ts TS ts TS st
∀ ∈ < −
(19)
81
( )
( )
''
, ', , , '',
''
''' '
'' , 0
, '
' '
' 0
'
1
1| |*| |
m q ts m p q ts
ST ST
p P q P pq
p q q q
p q q q
p q
sS mstS
CP CP
γ
γ
∈∈
≠≠
≠=
=
≤ −
∑ ∑
; ; '
ST
m M ts TS q P
∀ ∈ ∀ ∈ ∀ ∈
(20)
Restrictions (18) to (20) prevent the set-up state variable from taking incorrect values
after set-up. With (18), a set-up from product
p
to
q
( ,
ST
p q P
∈
) is avoided whenever
the set-up state that exists before set-up
,
, ,
p q
m p t st
sS
−
is not set correctly to
p
. Restriction
(19) works in a similar way: a set-up from product
p
to
q
( ,
ST
p q P
∈
) is avoided
whenever the set-up state that exists after set-up
,
, ,
p q
m q t st
sS
+
is not correctly set to
q
.
These restrictions are only valid for a subset of
TS
as
,
, ,
max p q
m q TS st
sS
+
is not defined. Ad-
ditionally, (20) guarantees that the set-up state
, ',
m q ts
sS
is never set true when another
set-up is completed, indicated by , ,
'
,
'
m p q ts
mstS
.
, , , , , , 1
m p q ts m p q ts
csS rS
+
≤
; : ;
max
m M ts TS ts TS
∀ ∈ ∀ ∈ <
(
)
, , 0
ST
p q P p q p q
γ
∀ ∈ ∧ = ∧ ≠
(21)
, , , 1 , , , , , , , , ,
,
1
m p q ts m p q ts m p q ts m p q ts
p q
csS rS mstS csS
st
−
+ = +
; ;
m M ts TS
∀ ∈ ∀ ∈
(
)
, , 0
ST
p q P p q p q
γ
∀ ∈ ∧ = ∧ ≠
(22)
Inequality (21) sets the cumulative set-up time
, , ,
m p q ts
csS
, which represents the progress
of a set-up in per cent, to 0 whenever the set-up, managed by binary variable
, , ,
m p q ts
rS
, is
deactivated. Equality (22) saves the cumulative set-up time and sets the variable
, , ,
m p q ts
mstS
true, when a set-up was finished. For every period in which a set-up is taking
place, the progress percentage per set-up period
, ,
1 *60
p q p q
tsS
st stMin
=
is added to
, , ,
m p q ts
csS
.
, ', , , '', ,
m p q ts m p q ts
rS rS
=
; ;
m M ts TS
∀ ∈ ∀ ∈
(
)
(
)
(
)
', '', ', '' 1 ', 0 '', 0
ST
p p q P p p p q p q
γ γ γ
∀ ∈ ∧ = ∧ = ∧ = ∧
' '' ' ''
p p p q p q
≠ ∧ ≠ ∧ ≠
(23)
82
, , ', , , '',
m p q ts m p q ts
rS rS
=
; ;
m M ts TS
∀ ∈ ∀ ∈
(
)
(
)
, , : , '' 1 , 0
' '' ' '
ST
q q p P q q p q
γ γ
∀ ∈ = ∧ = ∧
(
)
, '' 0 ' '' ' ''
p q q q q p q p
γ
= ∧ ≠ ∧ ≠ ∧ ≠
(24)
The set-up of coupled products is managed by (23) and (24). These restrictions are nec-
essary to activate all set-up variables correctly in order to be able to model the practical
situation in which products are produced simultaneously with one single die. The fol-
lowing figure illustrates how the set-up and set-up state variables are set, so that product
changes of coupled products are correctly modeled.
Figure 19: Correct Modeling of Set-up State Changes for Coupled Products
'
, , , , '', ,
m p q ts m p q ts
mstS mstS
=
; ;
m M ts TS
∀ ∈ ∀ ∈
(
)
(
)
(
)
', '', ', '' 1 ', 0 '', 0
ST
p p q P p p p q p q
γ γ γ
∀ ∈ ∧ = ∧ = ∧ = ∧
'' ' ''
'
p p p q p q
≠ ∧ ≠ ∧ ≠
(25)
, , ', , , '',
m p q ts m p q ts
mstS mstS
=
; ;
m M ts TS
∀ ∈ ∀ ∈
(
)
(
)
', '', : ', '' 1 , ' 0
q q p P q q p q
γ γ
∀ ∈ = ∧ = ∧
(
)
, '' 0 ' '' ' ''
p q q q q p q p
γ
= ∧ ≠ ∧ ≠ ∧ ≠
(26)
, ', , , '', ,
m p q ts m p q ts
csS csS
=
; ;
m M ts TS
∀ ∈ ∀ ∈
(
)
(
)
(
)
', '', ', '' 1 ', 0 '', 0
ST
p p q p P p p p q p q
γ γ γ
∀ ∈ ∈ ∧ = ∧ = ∧ = ∧
'' ' ''
'
p p p q p q
≠ ∧ ≠ ∧ ≠
(27)
83
, , ', , , '',
m p q ts m p q ts
csS csS
=
; ;
m M ts TS
∀ ∈ ∀ ∈
(
)
(
)
', '', : ', 1 , ' 0
''
ST
q q p p P q q p q
γ γ
∀ ∈ ∈ = ∧ = ∧
(
)
, '' 0 ' '' ' ''
p q q q q p q p
γ
= ∧ ≠ ∧ ≠ ∧ ≠
(28)
Although restrictions (25) and (26) are sufficient to model the described case, further
redundant equalities (27) to (28) are introduced in order to make the solution space
smaller.
, , , ,
mtnc mtnc
m p ts m q ts
binS binS=
(
)
; ; , : , , 1
ST
m M ts TS p q p P p q p q
δ
∀ ∈ ∀ ∈ ∀ ∈ ∈ ≠ =
(29)
, , 1 , , ,
mtnc
m p ts m p q ts
binS mstS
+
≥
(
)
(
)
; ; , : , , 0 , 0
ST
m M ts TS p q p P p q p q p q
δ γ
∀ ∈ ∀ ∈ ∀ ∈ ∈ ≠ = ∧ =
(30)
Restrictions (29) and (30) activate maintenance. Equation (29) activates maintenance of
all products whose production is based on the same die. Inequality (30) activates the
maintenance variable after having terminated a set-up. This is the first way that mainte-
nance is triggered. This inequality is not introduced into the model, if the maintenance
is triggered by cumulative production.
, , 1 , , 1 , , , , ,
* *
mtnc mtnc mtnc mtnc
m p ts m p ts m p m p ts m p ts
tsS
cmS binS mpM fmS cmS
tsM
− +
+ = +
; ;
ST
m M ts TS p p P
∀ ∈ ∀ ∈ ∈ ∈
(31)
, , , ,
mtnc mtnc
m p ts m p ts
fmS binS≤
; ;
ST
m M ts TS p p P
∀ ∈ ∀ ∈ ∈ ∈
(32)
Maintenance progress is modeled with restrictions (31) and (32). Every maintenance
period,
,
*
m p
tsS mpM
tsM
is added to the cumulated maintenance progress variables
, ,
mtnc
m p ts
cmS
. When maintenance is finished, indicated by
, ,
mtnc
m p ts
fmS
set true, the progress
variable is reset to 0.
, , , , 1 , ,
m p ts m p ts m p ts
lotS lotS xS
−
≤ +
; : ;
min ST
m M ts TS ts TS p p P
∀ ∈ ∀ ∈ > ∀ ∈ ∈
(33)
84
, , , , 1 , , , ,
*
mtnc
m p ts m p ts m p ts p m p ts
lotS lotS xS maxlot binS
−
≥ + −
; ;
ST
m M ts TS p p P
∀ ∈ ∀ ∈ ∀ ∈ ∈
(34)
Restrictions (33) and (34) allow
, ,
m p ts
lotS
to be the cumulative production quantity until
the next maintenance takes place, which resets the lot variable.
( )
, , 1 , , , , ,
, 0
| |
p
m p ts m p t m p q ts
q
ST
q P
p q
p q
minlot
lotS xS rS
CP
δ
−
∈
≠
=
+ ≥ ∑
if
p
minlot
>
p
bs
, otherwise
( )
, , 1 , , , , ,
, 0
| |
p
m p ts m p ts m p q ts
q
ST
q P
p q
p q
bs
lotS xS rS
CP
δ
−
∈
≠
=
+ ≥ ∑
; : ;
min
m M ts TS ts TS
∀ ∈ ∀ ∈ >
ST
p P
∀ ∈
(35)
( )
( )
, , , ,
, 1
, 0
1
m p ts m q ts p
q
ST
q P
p q
p q
p q
lotS lotS maxlot
CP
δ
γ
∈
≠
=
=
+ ≤
∑
; ;
ST
m M ts TS p P
∀ ∈ ∀ ∈ ∀ ∈
(36)
Inequalities (35) and (36) set the lot variables correctly and guarantee that available
practical constraints regarding lots are considered. Depending on the relation between
the set minimal lot size and the batch size, a different restriction for the minimal lot size
is relevant for model (35). In (36) the lot size is constrained to the maximum lot size
defined by the die. As other products are produced by using and fretting the same die,
the cumulative production quantity of all products has to be considered.
, , , , , ,
*
m p ts p m p ts m p ts
lotS bs reS slS
= +
; ;
ST
m M ts TS p P
∀ ∈ ∀ ∈ ∀ ∈
(37)
, , , ,
*
m p ts p m p ts
rlS bs prodS
≤
; ;
ST
m M ts TS p P
∀ ∈ ∈ ∈
(38)
, , , , 1 , ,
*
p m p ts m p ts m p ts
bs cwS slS slS
−
≥ −
; : ;
min ST
m M ts TS ts TS p P
∀ ∈ ∀ ∈ > ∀ ∈
(39)
85
(
)
, , , , 1 , ,
*
p m p ts m p ts m p ts
bs prodS prodS slS
+
− ≤
; : ;
max ST
m M ts TS ts TS p P
∀ ∈ ∀ ∈ < ∀ ∈
(40)
, , 1 , , , , 1 , ,
*
p
mntc
m p ts m p ts m p ts m p ts
p
maxlot
binS cwS reS reS
bs
+ −
+ + =
; : ;
min max ST
m M ts TS ts TS ts TS p P
∀ ∈ ∀ ∈ > ∧ < ∀ ∈
(41)
Restrictions (37) to (38) model the coil-oriented production depending on the deter-
mined batch size
p
bs
. Equality (37) is used to model the relation between lots and
batches. The slack is introduced to be able to model period-overlapping batches. It can
be seen as a cumulative quantity of the actual batch. The variable , ,
0
m p ts
slS
>
can only
apply when production is going on, which is also modeled by valid inequality (39). Ine-
qualities (40) and (41) activate the coil change variable. For the activation of
, , 1
mntc
m p ts
binS
+
,
p
p
maxlot
bs
is calculated as Big-M. The reason for introducing the binary maintenance
variable into this equation is that the restriction has to be deactivated in the case of
maintenance, as otherwise, the model would be infeasible. This is because of the rela-
tion between the cumulative quantity of used coils and the variable for the cumulative
quantity of the lot
, ,
m p ts
lotS
, which is reset at the beginning of the die maintenance pro-
cess.
,
r
m ts ts
m M
binS teams
∈
≤
∑
ts TS
∀ ∈
(42)
ts ts
teams teamLimS
≤
ts TS
∀ ∈
(43)
u v
teams teams
=
(
)
, , 1
u v TS u v
σ
∈ ∧ =
(44)
The number of required set-up teams is determined by (42). A limitation of available
teams is modeled with inequality (43). Actually, the number of parallel set-ups at differ-
ent machines is limited by the number of available set-up teams, because set-up teams
are the most cost-intensive requirement. The upper limit could also be determined by
other requirements. Equality (44) is dedicated to activating the set-up team variables for
a whole shift. This is because in practice set-up personnel are only available shift-wise.
86
Depending on the capacity situation and on the cost benefit, set-ups are bundled into
cheaper shifts as a consequence of restriction (44).
,
, , , ,
0
p lt
m p lt m p ts
ts TS
lt
verbPLT
reqLT xS
kapaLT
∈
− + ≤
∑
; ;
ST
lt LT p P m M
∀ ∈ ∀ ∈ ∀ ∈
(45)
The department responsible for the disposition of loading equipment has to take care
that the correct type of loading equipment is available on time and in the correct
amounts. Required loading equipment is calculated in (45), taking into consideration the
capacity of the boxes. Consequently, management of loading equipment is simplified
and can be improved.
,
min
p TS p
iS iS
ϖ
=
ST
p P
∀ ∈
(46)
,
max
p TS p
iS eiS
≥
ST
p P
∀ ∈
(47)
Variables have to be initialized with practical values in order to link the real production
with the model for planned production. First, initialization equations (46) and (47) de-
termine the inventory at the beginning and the ending inventory of the short-term plan-
ning horizon. The latter one is important for the linkage of the mid-term and short-term
planning methods. Details about the interconnection can be read in the next sub-section.
, , ,
min
m p TS m p
lotS lotS
ϖ
=
;
ST
m M p P
∀ ∈ ∈
(48)
, , ,
min
m p TS m p
mlS mlS
ϖ
=
;
ST
m M p P
∀ ∈ ∈
(49)
, , ,
min
m p TS m p
ReS ReS
ϖ
=
;
ST
m M p P
∀ ∈ ∈
(50)
, , ,
min
m p TS m p
slS SlS
ϖ
=
;
ST
m M p P
∀ ∈ ∈
(51)
, , ,
min
m p TS m p
cwS cwS
ϖ
=
;
ST
m M p P
∀ ∈ ∈
(52)
87
Equations (48) to (52) initialize the values for lots and batches. Numerical as well as
binary variables are set to the values corresponding to the system state in reality.
, , ,
min
m p TS m p
sS sS
ϖ
=
;
ST
m M p P
∀ ∈ ∈
(53)
, , ,
min
m p TS m p
prodS prodS
ϖ
=
;
ST
m M p P
∀ ∈ ∈
(54)
Binary set-up state and production variables are set in (53) and (54).
, , , , ,
min
m p q TS m p q
rS rS
ϖ ϖ
=
(
)
; , : , 0
ST
m M p q p P p q
δ
∀ ∈ ∈ ∈ =
(55)
, , , , ,
min
m p q TS m p q
mstS mstS
ϖ
=
(
)
; , : , 0
ST
m M p q p P p q
δ
∀ ∈ ∈ ∈ =
(56)
, , , , ,
min
m p q TS m p q
csS csS
ϖ
=
(
)
; , : , 0
ST
m M p q p P p q
δ
∀ ∈ ∈ ∈ =
(57)
min
TS
teamsS teamsS
ϖ
=
(58)
Variables representing set-up state and the progress and finish of set-up as well as team
variables are initialized in equations (55) to (58).
, , ,
min
mtnc mtnc
m p TS m p
binS binS
ϖ
=
;
ST
m M p p P
∀ ∈ ∈ ∈
(59)
, , ,
min
mtnc mtnc
m p TS m p
cmS cmS
ϖ
=
;
ST
m M p p P
∀ ∈ ∈ ∈
(60)
, , ,
min
mtnc mtnc
m p TS m p
fmS fmS
ϖ
=
;
ST
m M p p P
∀ ∈ ∈ ∈
(61)
The maintenance state and progress is transferred to the model initializing the corre-
sponding variables (59) to (61).
In this section, the short-term schedule planning model was described in detail. First,
required inputs and calculated outputs were explained. As this model has several inter-
relations with the mid-term lot size planning model as well as with the actual production
88
system state, the coupling of the partial models as well as the integration into real-world
production will be explained in the next section.
5.4 Coupling of Partial Models and Integration into Real Pro-
duction
The last part of the concept contains two topics: First, there is the coupling of the previ-
ously described and explained planning models. The coupling is important as both mod-
els are interdependent. The interdependencies are clarified in order to give an under-
standing of how both models work together. The second topic is the integration or, in
technical terms, the interface of the planning models with production in the real world.
This is an important aspect in order to be able to transfer the developed theoretical mod-
els into practical production planning.
5.4.1 Coupling of Partial Models
The previously described partial models influence each other reciprocally by using their
output to constrain or even define the variable values of the other model. A data inter-
change is provided by the models’ input parameters for variables, which were described
in the input sections 5.2.1.2 and 5.3.1.2 for mid-term lot size planning and short-term
schedule planning respectively. Beginning with the mid-term lot size model, there are
the parameters
p
iM
ϖ
,
,
m p
lotM
ϖ
, which define the initial inventory or the initial lot,
,
m p
binxM
ϖ
, which sets the production status of a product, and
p
mbinM
ϖ
and
p
mpM
ϖ
which define relevant maintenance variable values. The values for these parameters are
obtained by calculating the results of the short-term schedule planning. The other inter-
face direction from the mid-term planning results to short-term planning method is done
by setting a single parameter value
p
eiS
. The precondition is that the end of the short-
term planning horizon equals the beginning of the mid-term horizon:
max min
TS TM
=
First, the starting inventory for the mid-term planning is set:
,
p ts p
iS iM
ϖ
→
; :
max
p P ts TS ts TS
∀ ∈ ∈ =
Second, the production activity is transmitted, in order to be able to consider the set-ups
in the mid-term planning horizon correctly:
89
, , ,
m p ts m p
prodS binxM
ϖ
→
; ; :
max
m M p P ts TS ts TS
∀ ∈ ∈ ∈ =
As maintenance can be controlled by the cumulative production quantity stored in the
, ,
m p ts
lotS
variables, it is important to transfer the values to the corresponding mid-term
parameters:
, , ,
m p ts m p
lotS lotM
ϖ
→
; ; :
max
m M p P ts TS ts TS
∀ ∈ ∈ ∈ =
After that, maintenance has to be transferred correctly, otherwise it may be overlooked
in mid-term planning that dies might not be available, and part production might be
planned in an infeasible way. The binary indicator variables as well as the maintenance
progress variables have to be transferred from the short-term to the mid-term planning
parameters. The transfer of the progress is more complicated than the other transfers as
the maintenance progress in the short-term planning depends on short-term planning
periods and the maintenance progress in the mid-term planning depends on larger mid-
term periods; a recalculation therefore has to be made before the values are transferred.
, ,
,
1
*
mtnc
m p ts
mtnc
m p
p p
cmS
cmS
mtM mtM
ϖ
→
, ,
mtnc
m p m p
cmS mbinM
ϖ ϖ
→
; ; :
max
m M p P ts TS ts TS
∀ ∈ ∈ ∈ =
This calculation is rather pessimistic. The maintenance progress during the first mainte-
nance mid-term planning period is ignored. The following figure illustrates the values
and the linkage with an example:
M-T Period M1 M2
M-T Mntnc 0 1
M-T Prog. 0 >0
S-T Period S01
S02
S03
S04
S05 S06
S07
S08
S09
S10
S-T Mntnc 0 0 1 1 1 1 1 1 1 1
S-T Prog. 0 0 >0 >0 >0 >0 >0 >0 >0 >0
M-T Period M3 M4 M5
M-T Mntnc 1 1 0
M-T Prog. >0 >0 1
S-T Period S11
S12
S13
S14
S15 S16
S17
S18
S19
S20 S21
S22
S23
S24
S25
S-T Mntnc 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
S-T Prog. >0 >0 >0 >0 >0 >0 >0 1 1 1 1 1 1 1 1
Figure 20: Illustration of Maintenance Interconnection of Partial Models
90
Although maintenance was started within mid-term period M1 in short-term period S03
and the progress value at the end of M1 stored in S05 is higher than 0, the mid-term
maintenance progress of M1 is still set to 0. This is because it is not possible to guaran-
tee in practice that maintenance starts exactly when it is purported to by the results of
the short-term planning. Consequently, maintenance time is reserved until the end of the
mid-term planning period M4 and production can restart with the maintained die in M5
instead of in the middle of M4 in short-term period S18.
The parameter settings discussed so far are all dedicated to transferring information
from the short-term planning results to the mid-term planning method. The setting for
an ending inventory of the short-term planning horizon is dedicated to covering the oth-
er direction. The value is obtained from previously calculated mid-term planning re-
sults. As the planning horizon moves on, the inventory levels calculated in the mid-term
planning can later be used in short-term planning.
This restriction, which was described in section 5.3.3, guarantees that production lots,
brought forward by mid-term planning, are correctly considered during short-term plan-
ning. After describing the interconnection of both partial models, the next sub-section is
dedicated to describing the integration of both models into real production.
5.4.2 Integration into Real Production
In the last section, it was described how the models are interconnected. In order to be
able to turn planning results into reality, it is necessary that changes in the production
reality are transmitted to the planning methods. Examples of changes can be inventory
changes due to scrap or retouching work, or demand changes generated by customers.
This section describes which parameters are changed in order to adapt the planning re-
sults to the production in reality.
The actual situation in production can be modeled in a summarized way by obtaining
and transferring only some relevant values. These parameters were described in 5.3.1.2.
Besides the initialization of the variables, the demand has to be taken into account.
Slight demand changes can be considered in the short-term planning. Therefore, the
demand
,
p tm
dM
is transferred to a short-term demand
,
p ts
dS
by mapping small and mid-
term time-periods.
,
max
p TS p
iS eiS
≥
p P
∀ ∈
θ θ
A function that determines
period
A
This function is then used to map the time
No mapping is
horizon are changed.
consider that production does not end with the planning horizon. Therefore, the param
ter
Forward
Interconnection
Backward
Interconnection
This illustration shows how forward and backward interconnection of parameters and
variables work. The integration of the planning models into the real world as well as the
coupling of the models is visualized.
162
(
)
: , ( ) :
TM TS tm tm ts
θ θ
→ =
A function that determines
period
tm TM
∈
A
function is defined which determines the last short
This function is then used to map the time
No mapping is
horizon are changed.
consider that production does not end with the planning horizon. Therefore, the param
ter
p
eiM
was defined
Interconnection
Real
Inv.
Real
Lot
Real
…
Interconnection
This illustration shows how forward and backward interconnection of parameters and
variables work. The integration of the planning models into the real world as well as the
coupling of the models is visualized.
162
The calculation of the ending inventory is explained in
Short
: , ( ) :
TM TS tm tm ts
θ θ
→ =
֏
A function that determines
tm TM
function is defined which determines the last short
This function is then used to map the time
No mapping is
necessary when demands which are situated
horizon are changed.
Besides the adaptation of the plans
consider that production does not end with the planning horizon. Therefore, the param
was defined
162
iS
ϖ
lotS
ϖ
…
Figure
This illustration shows how forward and backward interconnection of parameters and
variables work. The integration of the planning models into the real world as well as the
coupling of the models is visualized.
The calculation of the ending inventory is explained in
Short
-
Term Scheduling
(
)
: , ( ) :
TM TS tm tm ts
θ θ
→ =
֏
A function that determines
the last short
function is defined which determines the last short
This function is then used to map the time
ts tm ts TS tm TM
θ
= ∈ ∈
necessary when demands which are situated
Besides the adaptation of the plans
consider that production does not end with the planning horizon. Therefore, the param
162
and set.
p
iS
,
m p
lotS
lotS
Figure
21:
Summary of Partial Model Interconnection
This illustration shows how forward and backward interconnection of parameters and
variables work. The integration of the planning models into the real world as well as the
coupling of the models is visualized.
The calculation of the ending inventory is explained in
Term Scheduling
Mapping
TM TS tm tm ts
the last short
-
term period
function is defined which determines the last short
This function is then used to map the time
-
periods of the demands.
, ,
p tm p ts
dM dS
=
( ), ,
ts tm ts TS tm TM
θ
= ∈ ∈
necessary when demands which are situated
Besides the adaptation of the plans
consider that production does not end with the planning horizon. Therefore, the param
,p TS
max
iS
, ,m p TS
max
lotS
…
,p TS
max
iS
Summary of Partial Model Interconnection
This illustration shows how forward and backward interconnection of parameters and
variables work. The integration of the planning models into the real world as well as the
coupling of the models is visualized.
The calculation of the ending inventory is explained in
Mid
-
Mapping
term period
ts TS
∈
function is defined which determines the last short
-
term period of a mid
periods of the demands.
, ,
p tm p ts
dM dS
=
( ), ,
ts tm ts TS tm TM
= ∈ ∈
necessary when demands which are situated
Besides the adaptation of the plans
in the case study
consider that production does not end with the planning horizon. Therefore, the param
ϖ
max
lotM
ϖ
…
Summary of Partial Model Interconnection
This illustration shows how forward and backward interconnection of parameters and
variables work. The integration of the planning models into the real world as well as the
The calculation of the ending inventory is explained in
5.2.1.2.
-
Term Lot Sizing
ts TS
∈
of a mid
term period of a mid
periods of the demands.
necessary when demands which are situated
in
the mid
in the case study
consider that production does not end with the planning horizon. Therefore, the param
p
iM
ϖ
,
m p
lotM
…
p
eiS
iM
Summary of Partial Model Interconnection
This illustration shows how forward and backward interconnection of parameters and
variables work. The integration of the planning models into the real world as well as the
Term Lot Sizing
of a mid
-term
(F
term period of a mid
-
term period.
the mid
-
term planning
in the case study
, plans have to
consider that production does not end with the planning horizon. Therefore, the param
,p TM
max
iM
Summary of Partial Model Interconnection
This illustration shows how forward and backward interconnection of parameters and
variables work. The integration of the planning models into the real world as well as the
91
(F
6)
term period.
term planning
, plans have to
consider that production does not end with the planning horizon. Therefore, the param
e-
p
eiM
This illustration shows how forward and backward interconnection of parameters and
variables work. The integration of the planning models into the real world as well as the
92
5.4.3 Determination of Relevant Short-Term Planning Subsets
The lots calculated in the mid-term planning define which products are produced in
which mid-term period
tm TM
∈
. The short-term planning horizon only ranges over a
subset of TM. Because of practical restrictions regarding lots and batches and because of
the limitation of production capacity, only a subset of products
ST
P P
⊆
can be pro-
duced within the entire short-term planning horizon. Consequently, the short-term mod-
el size is significantly reduced in practical problem instances. This first sub-section de-
scribes how product subsets, which are relevant for short-term schedule planning, are
determined. The following sub-section explains how data excluded during short-term
schedule planning is extrapolated.
5.4.3.1 Determination of Short-Term Relevant Product Subset
Because of practical restrictions and limited production capacities, it is not possible in
practice to produce the whole product portfolio during the limited short-term schedule
planning horizon. The relevant subset of products has to be determined.
{
}
,
: | :
min
ST p TS p
P p p P iS eiS= ∈ <
∪
{
}
,
| : 1,
m p
p p P sS m M
ϖ
∈ = ∀ ∈
∪
(
)
{
}
| , : , 1
ST
p p q P p q
γ
∈ =
The set is defined by all products which have to be produced by the end of the short-
term planning horizon, determined by calculating the difference of the existing invento-
ry at the initialization period
min
TS
and the desired production amount at the end of the
short-term planning horizon
p
eiS
, and then adding to all coupled products all the prod-
ucts which are actually set up in the initialization period of the short-term planning
horizon
min
TS
, represented by
,
m p
sS
ϖ
.
5.4.3.2 Extrapolation of Short-Term Irrelevant Data Sets
Inventories, lots, batches, maintenance data, and so on are only calculated and updated
within the short-term model for those products
ST
p P
∈
which are relevant for short-
term scheduling, in order to reduce model size and consequently improve performance.
To guarantee data consistency and to be able to start planning at every point in time,
data for other products,
ST
p P
∉
, has to be extrapolated. The extrapolation of most data
is simple, as it is a simple copy process for all short-term periods:
93
,
p ts p
iS iS
ϖ
=
, ;
ST
p P p P ts TS
∀ ∈ ∉ ∈
(1)
, , ,
m p ts m p
lotS lotS
ϖ
=
; , ;
ST
m M p P p P ts TS
∀ ∈ ∈ ∉ ∈
(2)
, , ,
m p ts m p
mlS mlS
ϖ
=
; , ;
ST
m M p P p P ts TS
∀ ∈ ∈ ∉ ∈
(3)
, , ,
m p ts m p
reS ReS
ϖ
=
; , ;
ST
m M p P p P ts TS
∀ ∈ ∈ ∉ ∈
(4)
, , ,
m p ts m p
slS SlS
ϖ
=
; , ;
ST
m M p P p P ts TS
∀ ∈ ∈ ∉ ∈
(5)
, , ,
m p ts m p
cwS cwS
ϖ
=
; , ;
ST
m M p P p P ts TS
∀ ∈ ∈ ∉ ∈
(6)
The extrapolation of the maintenance variables is more sophisticated. Maintenance is
still going on in the background and the maintenance progress values have to be adapted
correspondingly. Therefore, the following algorithm is necessary:
1 Do ; ; ,
ST
m M ts TS p P p P
∀ ∈ ∀ ∈ ∈ ∉
2 Do
3
, , 1 , , ,
*
mtnc mtnc
m p ts m p m p ts
tsS
cmS mpM cmS
tsM
−
+ →
, ,
1
mtnc
m p ts
binS→
4 While
, , 1 ,
* 1
mtnc
m p ts m p
tsS
cmS mpM
tsM
−
+ <
5 If
, , 1 ,
* 1
mtnc
m p ts m p
tsS
cmS mpM
tsM
−
+ ==
then
6
, ,
1
mtnc
m p ts
fmS→
5.5 Techniques to Improve Solution Time
The linear programming models described in sub-sections 5.2.2 and 5.3.3 consider lots
of sets and many elements. Many relations between elements of the sets complicate the
model further. Moreover, lots of constraints are taken into account. Changes of the
94
model structure, decomposition techniques and relaxations are possible ways to improve
solution time. The next two sub-sections are dedicated to describing the methods used
for that purpose for mid-term lot size planning and short-term schedule planning.
5.5.1 Mid-Term Lot Size Planning
One way to improve the solution time of mid-term lot size planning models is to first
solve a precedent model of relaxation and use the found solutions as possible starting
solutions for the original modeled problem. The mid-term lot size planning can easily
be relaxed by ignoring the constraint for the guarantee of availability. The guarantee of
availability is modeled using the following constraint:
, ,
max max
p TM p TM p
miM iM eiM
+ ≥
;
max
p P TM TM
∀ ∈ ∈
Ignoring this inequality, valid solutions, which consider real announced demands and
inventory within the mid-term planning horizon, can be generated quickly. Although the
generated solution is a bad solution with no availability guarantee, it is useful for reduc-
ing the search space of the original problem.
Another way to improve solution performance is by introducing further inequalities,
known a priori after analyzing the problem. An inequality can be calculated by carrying
out a backward scheduling of the inventory.
, 1 ,
1
* * *
p tm p tm m
m M
p p
iM dM tsM udM
pt CP
−
∈
≥ − ∑
;
p P tm TM
∀ ∈ ∈
The restriction states that the inventory
, 1
p tm
iM
−
has to be greater than the difference of
the demand
,
p tm
dM
and the maximum production amount in period
tm TM
∈
. This ine-
quality is also applicable to improve short-term schedule planning.
5.5.2 Short-Term Schedule Planning
The short-term schedule planning model is very complex. Therefore, several approaches
are necessary to guarantee processing times suitable for practice. First, two decomposi-
tion approaches will be described. After that, valid inequalities, as well as the modeling
techniques used, are explained.
95
5.5.2.1 Using Past Solutions
One way to improve solution performance is to use past planning solutions. Not all pa-
rameters and system states are changed from one planning run to the next. The starting
solution has to be adapted by considering the new planning horizon. Although the val-
ues for variables relevant to the new part of the planning horizon are not set, heuristics
implemented in optimization software are able to find feasible solutions. Especially
when the time between two planning runs is short, this is a suitable method for generat-
ing a starting solution from past planning runs.
5.5.2.2 Decomposition by Time Axis
The first decomposition approach described is the decomposition by time axis. The
short-term planning horizon represented by the set of short-term periods
TS
can be sub-
divided according to their belonging to mid-term periods
TM
, which is determined by
the following function.
(
)
(
)
Θ: , ( ) Θ:
TS TM ts ts tm
→ =
֏
Is a function that determines the mid-term period
tm TM
∈
of a short-term
period
ts TS
∈
(F6)
With this function, the set of short-term periods can be partitioned into subsets
(
)
{
}
: |
Θ:
tm
TS tm TM tm ts ts TS
= ∀ ∈ = ∀ ∈
The short-term model can then be processed for each partition separately, as relevant
parameters regarding demands and minimum inventories at the end of each period are
obtained from the mid-term lot sizing solutions. After that, all solutions are merged, so
that a solution is obtained which corresponds to the whole set of short-term periods. The
merged solution is not optimal for the entire short-term planning horizon but suits as a
good starting solution.
5.5.2.3 Decomposition by Machines
Another decomposition approach is the decomposition of the problem by machines. The
optimization procedure is sequentially started considering only one element of the ma-
chine set M. The result is a schedule, valid for each machine. If this decomposition is
applied, restrictions of shared resources upon machines are ignored. In the case study,
this is the limitation of set-up teams. The fusion of the decomposed sub-solutions con-
96
siders this, and calculates the amount of required set-up teams. Although a very costly
starting solution is generated, it enables the reduction of the search space of the original
problem.
5.5.2.4 Valid Inequalities and Modeling Techniques
Besides decomposition approaches, further inequalities and modeling techniques can be
applied to improve the performance to solve the model. The Constantino
163
inequality,
for example, can be adapted to the current model. It models the logical conditions that
production can be active in period t-1, or that a set-up is going on in period t, or that
either production or set-up of another product is going on in period ts.
( )
, , 1 , , , , ', , ', ,
'
'
, 1
( ) 1
m p ts m p q ts m q ts m q q ts
q P q q q P
q p q p
p q
prodS rS prodS rS
γ
−
∈≠ ∈
≠ ≠
≠
+ + − ≤
∑ ∑
∑
, , ' ,
p q q P ts TS
∀ ∈ ∈
Another way to improve the solution is to use double variables instead of integer or
Boolean variables. This depends on the variable selection and has to be tested as no
general rule is applicable. In the case of the actual short-term model, the change of set-
up and production variables from Boolean to fractional variables with 0 and 1 as lower
and upper limits improved performance.
{
}
, , , , ,
, 0,1
m p q ts m p ts
rS prodS ∈
[
]
, , , , ,
, 0,1
m p q ts m p ts
rS prodS ∈
Another improvement method is the disaggregation of restrictions. This was already
mentioned in the model description. In this case only the restriction which guarantees
that excluding actions cannot be executed simultaneously is disaggregated.
164
163 See [Con00].
164 See section 5.3.3 for more details.
97
6 Realization
In order to transfer the previously described theoretical concept into production reality,
it has to be implemented and tested in realistic scenarios. In this chapter, the system
design is described and explained. The integration into SAP is described in the follow-
ing section; and then, calculated results are evaluated and compared with manual plan-
ning results.
6.1 System Description
In order to improve acceptance of the realized planning method, integration into the
existing ERP System is helpful. Users do not have to switch from one software tool to
another and this way redundant data management is avoided. The first sub-section de-
scribes the system architecture and explains the overall structure of the implemented
system. Although data redundancy is minimized, some data has to be stored in a system
database in order to improve data connection speed. Another argument for separate data
management is that data can easily be added, merged and obtained in a beneficial way
and calculated results can be stored quickly. The used database and its data structure are
described in the following sub-section. Finally, the software structure is described.
6.1.1 Overall Architecture
In order to understand how the system is used and how it is integrated into the business
environment, the overall system architecture is explained in this sub-section. The fol-
lowing figure illustrates the principle of operation of the system:
98
Presentation layer
Logic layerData layer
Figure 22: Overall System Architecture
The system can be subdivided into three main layers, as each layer works rather inde-
pendently of the others. The interconnection is achieved by interfaces which have to be
adapted to the systems used within the affected layers. The data layer is responsible for
supplying the system with up-to-date data. In order to facilitate the actualization, the
active ERP system should be used as a data source, as the actuality of the data stored
there has to be guaranteed due to other processes within the company. Only a small se-
lection of data is necessary for the lot sizing and scheduling optimization. First, the de-
fined sets of the models have to be filled with entities. A list of the parts, demands, ma-
chines, dies and molds, loading equipment, and raw material, as well as data about
available coils, have to be transmitted. Relations between the elements of the sets are
also important in order to be able to consider them in the planning. Besides sets, param-
eters have to be obtained from the ERP system used. Production times, part prices, and
machine costs, as well as inventory holding costs, have to be communicated. Parts’ uti-
lization of raw material and loading equipment are also saved in the ERP and can there-
fore be used. More details about the obtained input data transmitted from the used ERP
system can be read in the concept chapter.
The logical layer consists of the optimization method, which was implemented in Java
using IBM® ILOG CPLEX 12.1 optimization software. Both the mid-term lot sizing
method and the short-term scheduling method are part of this layer. The sets and param-
eters of both models are set by the interface connecting with the subjacent layer. Some
obtained data sets require calculations and set operations in order to transform them into
99
a suitable form. Some parameters, especially those which control the optimization pro-
cess, cannot be obtained directly or via calculations from the existent ERP data. These
have to be input at an individually designed user interface.
Individually designed user interfaces are part of the presentation layer. Although it
would be possible to present these as a web interface or in an individual application, the
acceptance of the lot sizing and scheduling tool is higher when it is directly integrated
into the ERP software, which is used daily. Parameters can be set by production plan-
ners who possess a great deal of process knowledge. Set-up times, machine availability
or the selection of shifts is done in specialized graphical user interfaces which are inte-
grated into the ERP system. The set parameters have to be communicated to the logical
layer. After executing the lot sizing and scheduling methods, the results have to be pre-
sented to the end users. Depending on the department, a different presentation of the
results is necessary. The production department gets schedules and lot plans. The
maintenance department receives maintenance plans for the dies and plans for required
raw material are transmitted to the purchasing department. There is also a view for re-
quired loading equipment in order to be able to better plan cleaning and transport.
6.1.2 Database and Data Structure
In this sub-section, the structure of the database is described, which works as fast back-
ground data storage. Although the database does not influence planning methods’ prin-
ciples of operation directly, it is useful for telling us how specific data from the ERP
system is abstracted, stored and used. The presented database structure is an example of
a way to manage sets, parameters and variables of a mathematical model in a relational
database, as a match to the models’ components is given. The following illustration
shows the database structure in detail. Explanations of contained entities are provided
afterwards.
100
SBSetup
PK ID
FK2,I4 Microperiod_ID
FK1,I1 Mach_ID
FK3,I2 PartA_ID
FK4,I3 PartB_ID
Coil_Raw
PK Coil_Raw_ID
FK1,I1 Coil_ID
FK2,I2 Raw_ID
Part_Die
PK,I2 Part_Die_ID
FK1,I1 Part_ID
FK2,I3 Die_ID
SBCumSetupTime
PK ID
FK2,I4 Mikroperiod_ID
FK1,I1 Mach_ID
FK3,I2 PartA_ID
FK4,I3 PartB_ID
Percentage
BBBinDieMtnc
PK ID
FK1,I3 Macroperiod_ID
FK3,I2 Part_ID
FK2,I1 Mach_ID
SBBatchcount
PK ID
FK2,I3 Microperiod_ID
FK3,I2 Part_ID
FK1,I1 Mach_ID
Amount
Die_Mach
PK Die_Mach_ID
FK2,I2 Die_ID
FK1,I1 Mach_ID
Die
PK,I1 Die_ID
I2 Die_SAPID
Die_name
Die_MaxLot
SBStatus
PK ID
FK2,I3 Microperiod_ID
FK3,I2 Part_ID
FK1,I1 Mach_ID
Macroperiod
PK,FK2 Macroperiod_ID
Start
End
FK2 Microperiod_ID
BBCoil
PK BBCoil_ID
FK1,I3 Macroperiod_ID
FK3,I2 Part_ID
FK2,I1 Mach_ID
Amount
SBInv
PK ID
FK1,I2 Mcroperiod_ID
FK2,I1 Part_ID
Amount
Inv
SBBatchslack
PK ID
FK2,I3 Microperiod_ID
FK3,I2 Part_ID
FK1,I1 Mach_ID
Amount
SBTeam
PK ID
FK1,I1 Microperiod_ID
Amount
Max
Machine
PK Macch_ID
I1 Mach_SAPID
Mach_Name
Fixed_costs
Variable_costs
Setupcosts
Degree_utilization
TimePerDay
BBInv
PK ID
FK1,I2 Macroperiod_ID
FK2,I1 Part_ID
Amount
Die_Mntc
PK,I1 ID
FK2,I3 Mach_ID
FK4,I5 Part_ID
FK5,I6 Die_ID
FK3,I4 Microperiod_ID
FK1,I2 Macroperiod_ID
Part_LE
PK,I3 Part_LE_ID
FK2,I2 Part_ID
FK1,I1 LE_ID
Part_LE_Capa
Raw_material
PK,I1 Raw_ID
I2 Raw_SAPID
Raw_name
Shift_type
PK Shift_type_ID
I1 SAP_ID
Shift_type
Additional_charge
Start
End
SBLot
PK ID
FK2,I3 Microperiod_ID
FK3,I2 Part_ID
FK1,I1 Macch_ID
Amount
Part_Mach
PK,I3 Part_MA_ID
FK2,I2 Part_ID
FK1,I1 Mach_ID
Productionspeed
BBLot
PK ID
FK1,I3 Macroperiod_ID
FK3,I2 Part_ID
FK2,I1 Mach_ID
Amount
Setup_time
PK Setup_ID
FK1,I1 Part_ID_A
FK2,I2 Part_ID_B
Time
coupled_production
Die_mntc_C
PK,FK2 Microperiod_ID
PK,I2 ID
FK1,I4 Mach_ID
I6 Part_ID
I3 Macroperiod_ID
SB_Progress
BB_Progress
I1 MaintenanceStart
FK3 Die_ID
SBMinSetupTime
PK ID
FK2,I2 Microperiod_ID
FK1,I1 Mach_ID
FK3,I3 PartA_ID
FK4,I4 PartB_ID
Microperiod
PK Microperiod_ID
Start
End
FK1 FK_Macroperiod
FK2 FK_Shift_type
Day_type
PK Day_type_ID
I1 SAP_ID
Day_type
Additional_charge
SBProduction
PK ID
FK2,I3 Microperiod_ID
FK3,I2 Part_ID
FK1,I1 Mach_ID
Amount
Part_Raw
PK,I3 Part_Raw_ID
FK2,I2 Part_ID
FK1,I1 Raw_ID
Consumption
Part
PK,FK1,I1 Part_ID
Part_SAPID
Part_name
Part_price
Teil_Manufacturing_costs
Teil_Material_costs
Minimal_Lot
Maximum_Lot
monthlyDemand
FK1 Microperiod_ID
SBCoilchange
PK ID
FK2,I3 Microperiod_ID
FK3,I2 Part_ID
FK1,I1 Mach_ID
Coil
PK,I1 Coil_ID
I2 Coil_SAPID
Coilweight
Die_mntc_M
PK,I1 SBDie_Mntnc_ID
FK2,I3 Mach_ID
FK4,I5 Part_ID
FK5,I6 Die_ID
FK3,I4 Microperiod_ID
FK1,I2 Macroperiod_ID
SAP_Demand
PK,I1 ID
FK1,I3 Macroperiod_ID
FK2,I2 Part_ID
Amount
LE
PK,I1 LE_ID
LE_SAPID
LE_name
Figure 23: Database Diagram
101
First, the database can be subdivided into two main components: input and output tables
for data storage. Visually, output tables can be distinguished from input tables by their
grey background. Looking at input tables first, different types of input tables can be
distinguished. There exist tables representing the sets, like parts, machines, dies, raw
materials, coils, mid-term periods and short-term periods; and then there are tables rep-
resenting relations between these sets. The machine and the parts tables occupy a central
position. Element-dependent properties like material costs or manufacturing costs, or
the degree of utilization and the time that a machine can be used for per day, are saved
in the parts table. The production speed is saved in the table representing the relation
between parts and machine, named Part_Machine. The relations of dies with machines
or dies with parts are defined in the similarly named tables. Parts cannot be directly re-
lated with coils because some parts consist of the same raw material. Therefore, a raw
material table has to be introduced. The Part_Raw relation, represented by another ta-
ble, contains data about the consumption of raw material of one part that is the charge
weight of a part. The relation to loading equipment, stored in table LE, is structured in a
similar way. Sequence-dependent set-up times are saved in the Setup_Time table. Tech-
nically realizable part-part set-up sequences, including set-up times, are stored in this
table. A Boolean value determines whether two parts are produced in coupled produc-
tion. Mid-term periods and short-term periods are saved in the tables titled Macroperiod
and Microperiod respectively. Both tables contain entries about the starting and ending
times of the periods. Additionally, every macro-period is linked to a day-type and every
micro-period is linked to a shift-type. In these tables, additional charges are saved. De-
mands transferred from the ERP system are stored in the SAP_Demand table. The out-
put tables are highlighted with a grey background. Only a selection of the models’ vari-
ables is stored as the rest can be calculated automatically. The variable tables, which are
relevant for mid-term planning, start with a BB in their name; tables relevant for short-
term planning are named starting with an SB. The die maintenance tables are hybrid
tables used for both models. Basically, all tables store the values in a similar way: If a
binary output variable is active in the method results, an entry is made in the corre-
sponding table. In the case of integer or fractional variables, entries are made in the cor-
responding table including the value in specified columns.
6.1.3 Software Structure
In this section, the overall structure of the software is briefly described. The following
figure illustrates the overall software structure.
102
Figure 24: Software Structure
The data package is dedicated to making data available for other packages. The sub-
package SystemEnvironment contains the classes representing production-relevant ob-
jects and relations. Properties and methods are saved in these objects. The packages
MidTerm_variables and ShortTerm_variables contain all classes representing the model
variables. This enables the transfer of the variables, calculations of calculated variable
values as well as an appropriate output of the results. In the Data package, classes are
defined, some of which control the data access to the ERP system, and some of which
manage the data storage. The models for short-term and mid-term planning are defined
in classes which are part of the control package. Interaction between the models and
control of communication between the top-level packages is managed by another class.
The output package contains the classes which manage the visualization of the calculat-
ed results. In this package, variable values are interpreted and transformed into appro-
priate, understandable charts. These charts can be presented in an integrated ERP, in
typical office suite-compatible spreadsheet formats or in an individually programmed
graphical user interface. The first two alternatives are already implemented in subordi-
nated classes.
6.1.4 Application Flow
Several steps have to be passed before planning results can be obtained. The following
activity chart represents the application flow:
103
Configure planning
Update database on basis of SAP
Import Data
Database
[updated]
Sets,
Parameters
[read, imported]
Short-term lot sizing / scheduling
Results in DB
[written]
Save results
Results in GUI
[written]
Mid-term lot sizing
Results in DB
[written]
Save results
Results in GUI
[written]
Configuration
[updated]
Unload application
Figure 25: Overall Application Flow Diagram
After starting the application, the planning has to be configured. The selection of the
planning horizon, the selection of relevant product groups, and the configuration of the
times for short-term and mid-term optimization, as well as the activation/deactivation of
preceding heuristics, are set. After that, the internal database has to be updated on the
basis of ERP data.
The following activity chart illustrates the steps which have to be completed:
104
Figure 26: Sub-process: Database Update Application Flow Diagram
After receiving the update request from the main application, the database update pro-
cess starts to update, write or remove data from the used ERP system into the applica-
tion database. First, the sets of relevant system objects are imported, which can be pro-
cessed parallel to one another as no interdependencies have to be considered when im-
porting these basic sets. The relations between these objects have to be updated in a
subsequent step, as the relations depend on the previously imported data. Last but not
least, inventories and demands are updated.
165
After updating the data of the application database, only relevant datasets have to be
read from the application database in order to present objects which are then much fast-
er to access. As there are interdependencies, not all import tasks can be executed in par-
allel. In particular those data sets representing relations need the linked objects in ad-
vance. The following flow diagram roughly visualizes the import and instantiation pro-
cess:
165 The interface required for obtaining the data from the SAP system was provided as a dynamic link
library by application developers of the case study partner.
105
Figure 27: Sub-process: Instantiation and Import Flow Diagram
The obtained data sets and parameters are stored in an object which encapsulates and
manages the data access. On the basis of imported data sets, the lot sizing and schedul-
ing procedures are started. The procedures of both the short-term lot sizing and schedul-
ing and the mid-term lot sizing are basically the same and differ merely in details of the
data and parameters required, variables built and restrictions modeled.
Figure 28: Sub-process: Lot Sizing and Scheduling Flow Diagrams
Results of both procedures are saved in the database and output. The mid-term lot sizing
method makes use of the short-term lot sizing procedure’s results in order to correctly
initialize, for example, inventory, lot or maintenance variables in the model.
106
6.2
SAP
The acceptance of a planning system can be improved
which is already being used
double
data inputs are minimized
data sets can be obtained from SAP.
The configuration screen
horizons,
like this:
As
production data acquisition is not available, many parameters have to be set in order
to transfer the up
which required parameters
the number of set
two examples
166 See
[HGH03].
167
The presented SAP screens were created in student projects by T
Blomen based on the
SAP
Integration
The acceptance of a planning system can be improved
which is already being used
data inputs are minimized
data sets can be obtained from SAP.
The configuration screen
horizons,
to
select relevant product groups
like this:
production data acquisition is not available, many parameters have to be set in order
to transfer the up
-
which required parameters
the number of set
-
two examples
of
the i
[HGH03].
The presented SAP screens were created in student projects by T
Blomen based on the
Integration
and User Interface
The acceptance of a planning system can be improved
which is already being used
for manual planning
data inputs are minimized
data sets can be obtained from SAP.
The configuration screen
in
SAP
select relevant product groups
production data acquisition is not available, many parameters have to be set in order
-
to-
date system state.
which required parameters
like set
-
up teams
can be
the i
ntegrated configuration screens.
The presented SAP screens were created in student projects by T
Blomen based on the
SAP Graphics library BC_FES_GRA
and User Interface
The acceptance of a planning system can be improved
for manual planning
data inputs are minimized
.
In the considered practical case, most parameters and
data sets can be obtained from SAP.
SAP
,
167
designed to define short
select relevant product groups
production data acquisition is not available, many parameters have to be set in order
date system state.
like set
-
up states,
can be
set and configured
ntegrated configuration screens.
The presented SAP screens were created in student projects by T
SAP Graphics library BC_FES_GRA
and User Interface
The acceptance of a planning system can be improved
for manual planning
on a daily basis
In the considered practical case, most parameters and
designed to define short
select relevant product groups
and to set optimization parameters
production data acquisition is not available, many parameters have to be set in order
date system state.
Accordingly, further screens are required in
up states,
die-
relevant maintenance parameters and
set and configured
ntegrated configuration screens.
The presented SAP screens were created in student projects by T
SAP Graphics library BC_FES_GRA
and User Interface
The acceptance of a planning system can be improved
by
integrating it
on a daily basis
In the considered practical case, most parameters and
designed to define short
-
and to set optimization parameters
production data acquisition is not available, many parameters have to be set in order
Accordingly, further screens are required in
relevant maintenance parameters and
set and configured
.
The following illustrations are
ntegrated configuration screens.
The presented SAP screens were created in student projects by T
homas Seebothe and
and the SAP Java Connector (JCo).
integrating it
into the system
on a daily basis
.
166
R
edundanc
In the considered practical case, most parameters and
-
and mid-
term planning
and to set optimization parameters
production data acquisition is not available, many parameters have to be set in order
Accordingly, further screens are required in
relevant maintenance parameters and
The following illustrations are
homas Seebothe and
and the SAP Java Connector (JCo).
into the system
edundanc
y
and
In the considered practical case, most parameters and
term planning
and to set optimization parameters
,
looks
production data acquisition is not available, many parameters have to be set in order
Accordingly, further screens are required in
relevant maintenance parameters and
The following illustrations are
homas Seebothe and
Benedict
and the SAP Java Connector (JCo).
into the system
and
In the considered practical case, most parameters and
term planning
looks
production data acquisition is not available, many parameters have to be set in order
Accordingly, further screens are required in
relevant maintenance parameters and
The following illustrations are
Benedict
The set
other are not
obtained and entered into this screen. Apart from set
be
The second screen defines the set
are used
the
The optimization process is visualized in a further screen:
168
The set
-
up parameter screen is necessary because the set
other are not
obtained and entered into this screen. Apart from set
be
input.
168
The second screen defines the set
are used
to
initialize
the
ir
background
The optimization process is visualized in a further screen:
168
The set-
up time table is
duction flag has to be defined for th
storage space and configuration effort for production planners, parameters, set
production are bundled in one screen.
up parameter screen is necessary because the set
other are not
yet
stored. Consequently, the sequence
obtained and entered into this screen. Apart from set
Figure
The second screen defines the set
initialize
the short
background
logic are programmed using SAP ABAP.
The optimization process is visualized in a further screen:
up time table is
large
duction flag has to be defined for th
storage space and configuration effort for production planners, parameters, set
production are bundled in one screen.
Figure
29
up parameter screen is necessary because the set
stored. Consequently, the sequence
obtained and entered into this screen. Apart from set
Figure
30
: Set
The second screen defines the set
-
up
the short
-
term planning procedure.
logic are programmed using SAP ABAP.
The optimization process is visualized in a further screen:
large
as the power set
duction flag has to be defined for th
e same product combination set. In order to reduce the required
storage space and configuration effort for production planners, parameters, set
production are bundled in one screen.
29
: Set-
up Parameter S
up parameter screen is necessary because the set
stored. Consequently, the sequence
obtained and entered into this screen. Apart from set
: Set
-
up State Parameterization
up
and activity
term planning procedure.
logic are programmed using SAP ABAP.
The optimization process is visualized in a further screen:
as the power set
of the
e same product combination set. In order to reduce the required
storage space and configuration effort for production planners, parameters, set
up Parameter S
creen
up parameter screen is necessary because the set
stored. Consequently, the sequence
-
dependent set
obtained and entered into this screen. Apart from set
-
up time, coupled product
up State Parameterization
and activity
state
of
term planning procedure.
logic are programmed using SAP ABAP.
The optimization process is visualized in a further screen:
of the
product set has to be displayed.
e same product combination set. In order to reduce the required
storage space and configuration effort for production planners, parameters, set
creen
-
up times from one part to a
dependent set
-
up times have to be
up time, coupled product
up State Parameterization
of
the
machine
All the presented s
logic are programmed using SAP ABAP.
product set has to be displayed.
e same product combination set. In order to reduce the required
storage space and configuration effort for production planners, parameters, set
up times from one part to a
up times have to be
up time, coupled product
ion has to
machine
s.
These settings
All the presented s
creens and
product set has to be displayed.
The coupled pr
e same product combination set. In order to reduce the required
storage space and configuration effort for production planners, parameters, set
-
up time and coupled
107
up times from one part to a
n-
up times have to be
ion has to
These settings
creens and
The coupled pr
o-
e same product combination set. In order to reduce the required
up time and coupled
108
The optimization process is visualized in a screen pre
mum and found integer solutions.
Intermediate Document objects are used to transfer the
to be able to communicate asynchronously. The mid
like this:
The optimization process is visualized in a screen pre
mum and found integer solutions.
Intermediate Document objects are used to transfer the
to be able to communicate asynchronously. The mid
like this:
Figure 31
:
The optimization process is visualized in a screen pre
mum and found integer solutions.
Intermediate Document objects are used to transfer the
to be able to communicate asynchronously. The mid
:
Vis
ualization
The optimization process is visualized in a screen pre
mum and found integer solutions.
Intermediate Document objects are used to transfer the
to be able to communicate asynchronously. The mid
ualization
of Optimization Process
The optimization process is visualized in a screen pre
Intermediate Document objects are used to transfer the
to be able to communicate asynchronously. The mid
of Optimization Process
The optimization process is visualized in a screen pre
senting the theoretical LP opt
Intermediate Document objects are used to transfer the
planning results to SAP
to be able to communicate asynchronously. The mid
-
term planning
of Optimization Process
senting the theoretical LP opt
planning results to SAP
term planning
visualization
senting the theoretical LP opt
planning results to SAP
in order
visualization
looks
senting the theoretical LP opt
i-
in order
looks
Production amounts of each part on every machine are visualized for each day in sep
rate bar charts. Exact production quantities can be
ing a more detailed lis
Production amounts of each part on every machine are visualized for each day in sep
rate bar charts. Exact production quantities can be
ing a more detailed lis
Figure
Production amounts of each part on every machine are visualized for each day in sep
rate bar charts. Exact production quantities can be
ing a more detailed lis
t:
Figure
32
: Mid
Production amounts of each part on every machine are visualized for each day in sep
rate bar charts. Exact production quantities can be
Figure
33
: Mid
-
Term Planning Results in SAP
Production amounts of each part on every machine are visualized for each day in sep
rate bar charts. Exact production quantities can be
33
: Mid-
Term Lots in SAP
Term Planning Results in SAP
Production amounts of each part on every machine are visualized for each day in sep
rate bar charts. Exact production quantities can be
checked in another overview contai
Term Lots in SAP
Term Planning Results in SAP
Production amounts of each part on every machine are visualized for each day in sep
checked in another overview contai
Term Lots in SAP
Production amounts of each part on every machine are visualized for each day in sep
checked in another overview contai
109
Production amounts of each part on every machine are visualized for each day in sep
a-
checked in another overview contai
n-
110
Short-
term schedules are presented in Gantt
The displayed user interfaces
material requirements planners, the loading equipment management department and the
personnel planning department. The management of rights and roles
views
can then be
6.3
Evalu
6.3.1
Planning Results
The following sub
tice. Planning results were transfe
sentative subset of planning results is depicted
show how the practical constraints are represented in the planning results.
Plans were generated for
Weingarten II 7800KN
mented Java method using IBM
cus
tomary
169
According to
170
The visualization of the results
term schedules are presented in Gantt
Figure
The displayed user interfaces
material requirements planners, the loading equipment management department and the
personnel planning department. The management of rights and roles
can then be
done by SAP managers.
Evalu
ation
Planning Results
The following sub
tice. Planning results were transfe
sentative subset of planning results is depicted
show how the practical constraints are represented in the planning results.
Plans were generated for
Weingarten II 7800KN
mented Java method using IBM
tomary
computer
According to
[ZB05]
The visualization of the results
term schedules are presented in Gantt
Figure
34
: Short
The displayed user interfaces
170
material requirements planners, the loading equipment management department and the
personnel planning department. The management of rights and roles
done by SAP managers.
ation
Planning Results
The following sub
-sections
present planning results which were transferred
tice. Planning results were transfe
sentative subset of planning results is depicted
show how the practical constraints are represented in the planning results.
Plans were generated for
the
Weingarten II 7800KN
.
All the tests were executed using the previousl
mented Java method using IBM
computer
with an 2,4 GHz Intel
[ZB05]
, Gantt charts are suitable
The visualization of the results
is
term schedules are presented in Gantt
: Short
-
Term Schedule Visualization in SAP
170
can
then b
material requirements planners, the loading equipment management department and the
personnel planning department. The management of rights and roles
done by SAP managers.
Planning Results
present planning results which were transferred
tice. Planning results were transfe
rred in
to practice for
sentative subset of planning results is depicted
show how the practical constraints are represented in the planning results.
the
two molding presses
All the tests were executed using the previousl
mented Java method using IBM
®
ILOG
with an 2,4 GHz Intel
, Gantt charts are suitable
is
integrated using the SAP Graphics library BC_FES_GRA
term schedules are presented in Gantt
charts
169
Term Schedule Visualization in SAP
then b
e
directly
material requirements planners, the loading equipment management department and the
personnel planning department. The management of rights and roles
done by SAP managers.
present planning results which were transferred
to practice for
sentative subset of planning results is depicted
in order to
show how the practical constraints are represented in the planning results.
two molding presses
All the tests were executed using the previousl
ILOG
CPLEX
with an 2,4 GHz Intel
®
I5 CPU with 2
, Gantt charts are suitable
for visualiz
ing
integrated using the SAP Graphics library BC_FES_GRA
169
integrated into SAP.
Term Schedule Visualization in SAP
directly
used by production planners, raw
material requirements planners, the loading equipment management department and the
personnel planning department. The management of rights and roles
present planning results which were transferred
to practice for
a period of
in order to
explain the results and to
show how the practical constraints are represented in the planning results.
two molding presses
Weingarten I 7800KN and
All the tests were executed using the previousl
12.1 as optimization software on a
I5 CPU with 2
GB
ing
machine schedules.
integrated using the SAP Graphics library BC_FES_GRA
integrated into SAP.
Term Schedule Visualization in SAP
used by production planners, raw
material requirements planners, the loading equipment management department and the
personnel planning department. The management of rights and roles
for the specific
present planning results which were transferred
a period of
one month. A repr
explain the results and to
show how the practical constraints are represented in the planning results.
Weingarten I 7800KN and
All the tests were executed using the previousl
y named impl
12.1 as optimization software on a
GB
RAM.
machine schedules.
integrated using the SAP Graphics library BC_FES_GRA
used by production planners, raw
material requirements planners, the loading equipment management department and the
for the specific
present planning results which were transferred
in
to pra
one month. A repr
explain the results and to
show how the practical constraints are represented in the planning results.
Lot Sizing
Weingarten I 7800KN and
y named impl
12.1 as optimization software on a
integrated using the SAP Graphics library BC_FES_GRA
.
used by production planners, raw
material requirements planners, the loading equipment management department and the
for the specific
to pra
c-
one month. A repr
e-
explain the results and to
Lot Sizing
Weingarten I 7800KN and
y named impl
e-
12.1 as optimization software on a
111
6.3.1.1 Sample Mid-Term Planning Results
First, mid-term planning results will be explained. Planned production amounts are out-
put in table form for each machine. In order to get a better overview, production
amounts are visualized using bar charts:
Date
29.01.2011
30.01.2011
31.01.2011
01.02.2011
02.02.2011
03.02.2011
04.02.2011
05.02.2011
06.02.2011
07.02.2011
08.02.2011
09.02.2011
Part ID
82411909-5,86 &
82352228-5,86 17070
3414
6828
82007975-5 &
82007975-4 5032
423615 &
523615 7546
82650191-4 &
82650191-5 10782
16173
16176
16173
16173
16173
10782
311357,85
8632
17264
82352228-4,86 &
82411909-4,86 17070
411359,85 &
411360,85 4857
Figure 35: Production Amounts: Weingarten I Stamping Machine
Figure 36: Visualization of Production Amounts Weingarten I
82411909-5.86
&
82352228-5.86
17070
82411909-5.86
&
82352228-5.86
3414
82411909-5.86
&
82352228-5.86
6828
82007975-5
&
82007975-4
5032
423615
&
523615
7546
82650191-4
&
82650191-5
10782
82650191-4
&
82650191-5
16173
82650191-4
&
82650191-5
16173
82650191-4
&
82650191-5
16173
82650191-4
&
82650191-5
16173
82650191-4
&
82650191-5
16173
82650191-4
&
82650191-5
10782
311357.85
8632
311357.85
17264
82352228-4.86
&
82411909-4.86
17070
411359.85
&
411360.85
4851
0
5000
10000
15000
20000
25000
29.1.11 30.1.11 31.1.11 1.2.11 2.2.11 3.2.11 4.2.11 5.2.11 6.2.11 7.2.11 8.2.11 9.2.11
Date
112
Date
29.01.2011
30.01.2011
31.01.2011
01.02.2011
02.02.2011
03.02.2011
04.02.2011
05.02.2011
06.02.2011
07.02.2011
08.02.2011
09.02.2011
Part ID
82028724-4.65 &
82028724-5.65 3295
3295 3295
430923 & 3234
520890 & 6468
82028724-4.64 &
82028724-5.64 6590
6590
82016865-4 5390
82650192-5 &
82650192-4 24150
9660
19320
24150
14490
24150
420889 & 420890 3234
82016885-4 7672
511359.85 &
511360.85 4851
82028724-4 &
82028724-5 6590
13180
19770
6590
9885
6590
Figure 37: Production Amounts: Weingarten II Stamping Machine
Figure 38: Visualization of Production Amounts Weingarten II
82028724-4.65
&
82028724-5.65
3295
82028724-4.65
&
82028724-5.65
3295
82028724-4.65
&
82028724-5.65
3295
430923
&
530923
3234
520890
&
520889
6468
82028724-4.64
&
82028724-5.64
6590 82028724-4.64
&
82028724-5.64
6590
82016865-4
5390
82650192-5
&
82650192-4
24150
82650192-5
&
82650192-4
9660
82650192-5
&
82650192-4
19320
82650192-5
&
82650192-4
24150
82650192-5
&
82650192-4
14490
82650192-5
&
82650192-4
24150
420889
&
420890
3234
82016885-4
7672
511359.85
&
511360.85
4851
82028724-4
&
82028724-5
6590
82028724-4
&
82028724-5
13180
82028724-4
&
82028724-5
19770
82028724-4
&
82028724-5
6590
82028724-4
&
82028724-5
9885
82028724-4
&
82028724-5
6590
0
5000
10000
15000
20000
25000
30000
29.01.2011
30.01.2011
31.01.2011
01.02.2011
02.02.2011
03.02.2011
04.02.2011
05.02.2011
06.02.2011
07.02.2011
08.02.2011
09.02.2011
Date
113
Production speeds differ from part to part. Therefore, the utilization of a machine cannot
be directly obtained by analyzing production quantities. For this purpose, a capacity
utilization bar chart is provided as well:
Figure 39: Combined Utilization Chart (1)
Figure 40: Combined Utilization Chart (2)
82411909-5.86
&
82352228-5.86
67%
82411909-5.86
&
82352228-5.86
13%
82411909-5.86
&
82352228-5.86
27%
423615
&
523615
31%
82650191-4
&
82650191-5
40%
82650191-4
&
82650191-5
61%
82650191-4
&
82650191-5
61%
311357.85
18%
82028724-4.65
&
82028724-5.65
11%
82028724-4.65
&
82028724-5.65
11%
82028724-4.65
&
82028724-5.65
11%
520889
&
520890
27%
82650192-4
&
82650192-5
79% 82650192-4
&
82650192-5
31%
82650192-4
&
82650192-5
63%
511360.85
&
511359.85
20%
82028724-4
&
82028724-5
22%
82028724-4
&
82028724-5
45%
82028724-4
&
82028724-5
67%
0%
10%
20%
30%
40%
50%
60%
70%
80%
Weingarten I
Weingarten II
Weingarten I
Weingarten II
Weingarten I
Weingarten II
Weingarten I
Weingarten II
Weingarten I
Weingarten II
Weingarten I
Weingarten II
29.01.2011 30.01.2011 31.01.2011 01.02.2011 02.02.2011 03.02.2011
82007975-4
&
82007975-5
18%
82650191-4
&
82650191-5
61%
82650191-4
&
82650191-5
61%
82650191-4
&
82650191-5
61% 82650191-4
&
82650191-5
40%
311357.85
36%
82352228-4.86
&
82411909-4.86
67%
411359.85
&
411360.85
20%
530923
&
430923
13%
82028724-4.64
&
82028724-5.64
22%
82028724-4.64
&
82028724-5.64
22%
82016865-4
22%
82650192-4
&
82650192-5
79%
82650192-4
&
82650192-5
47%
82650192-4
&
82650192-5
79%
420889
&
420890
13%
82016885-4
18% 82028724-4
&
82028724-5
22%
82028724-4
&
82028724-5
34%
82028724-4
&
82028724-5
22%
0%
10%
20%
30%
40%
50%
60%
70%
80%
Weingarten I
Weingarten II
Weingarten I
Weingarten II
Weingarten I
Weingarten II
Weingarten I
Weingarten II
Weingarten I
Weingarten II
Weingarten I
Weingarten II
04.02.2011 05.02.2011 06.02.2011 07.02.2011 08.02.2011 09.02.2011
114
The utilization charts show that the utilization capacity limit of 80 % is never exceeded.
In actual practice, the maximum utilization is set to 65 %. Thirty-five per cent of the
daily capacity is subtracted as a fixed rate for set-ups, coil changes and other machine-
related activities or smaller disruptions. In this method, a fixed set-up time is taken into
account so that capacity is only reduced by 20 % to cover activities like coil changes
and other machine-related activities or disruptions.
Taking customer demands, maintenance, and so on into account, the following cost-
optimal personnel plan was calculated:
Day type
Maximum Simulta-
neously Deployed
Staplers
Estimated Maximum of
Simultaneously Deployed
Set-up Teams
29.01.11
Working Day
2 0
30.01.11
Sunday 2 1
31.01.11
Working Day
4 1
01.02.11
Working Day
4 0
02.02.11
Working Day
4 0
03.02.11
Working Day
4 1
04.02.11
Working Day
4 0
05.02.11
Working Day
4 1
06.02.11
Sunday 2 0
07.02.11
Working Day
2 1
08.02.11
Working Day
2 1
09.02.11
Working Day
4 1
Figure 41: Personnel Mid-Term Plan
It can be seen that this method tries to reduce personnel deployment on more expensive
days (Sundays) in order to reduce personnel costs. As other constraints have to be con-
sidered, it is not possible to reduce the personnel needed to zero every Sunday. The ex-
act number of set-up teams can be determined during short-term planning as set-up ac-
tivities are precisely scheduled.
The presented production plans consider maintenance of the dies. In these plans,
maintenance is triggered after set down due to product change. There are some parts,
like 82028724-4, produced in coupled production with 82028724-5, which are produced
on several consecutive days in combination with other parts, that is 82028714-4.64 and
82028724-5.64, without activating maintenance. The IDs differ although the parts are
equal. Different IDs are used to distinguish successive processes. Accordingly, the parts
are produced with the same dies and no product change and no maintenance is neces-
sary. Three days are required for maintenance and during this time the production of the
related part is blocked.
115
Production quantities are based on raw material units. The number of required steel
coils can be calculated using stored part-raw material relations and the raw material
usage of the parts. The following table shows the planned maintenance for each die,
calculated on the basis of the stored die-part relations.
Furthermore, it provides a raw material procurement plan. As a result of the production
plans, required loading equipment can be calculated. The following table shows the
planned quantity for every loading equipment type for each day. Consequently, loading
equipment procurement is improved due to the reliable calculations provided.
116
Date
Resource Type Number
29.1.11
30.1.11
31.1.11
1.2.11
2.2.11
3.2.11
4.2.11
5.2.11
6.2.11
7.2.11
8.2.11
9.2.11
Raw Material
(number of required coils)
BD 0,80 X 425 GK`HC340LA
2
BD 0,85 X 574 GK`HC420LA
3 5 7
2 7 6
BD 0,90 X 420 GK`HC380LA
1
BD 1,00 X 574 GK`HC420LA
2
BD 1,00 X 648 GK`HC380LA
1
BD 1,50 X 418 GK`S420MC
2
1
BD 1,50 X 597 GK`HC500LA 4 1
2 3
BD 1,75 X 410 GK`S420MC
1
BD 2,00 X 657 GK`HC340LA
2
4
BD 2,10 X 335 GK`DC04-C290
1
1
BD 3,00 X 264 GK`HSM700HD 4 2
4 5 3
5
BD 3,00 X 313 GK`S500MC
2 3 3
3 3 3 2
Required
Loading Equip-
ment
790333 GLT VWK-BO 10 18 7
790334 GLT VWK-BO 70 40 70 140 210 146 70 46 106 84
790444 GLT TK3 76 30 60 76 46 76
790445 GLT TK3 34 50 50 50 50 34
E-303480 VCI-FOLIE 26 26 26 52 52
Die Maintenance Schedule
W001208000001 M M M
W000220000001 M M M
W080036000- M M M M M M
W020100200001 M M M
W020100300001 M M M M M M
W020072500001 M M M M M M
W020544700001 M M M M M
W000163100001 M M M
W080038000- M M M
W000158900001 M M M
W001200700001 M M M
Figure 42: Raw Material Units Procurement Plan
6.3.1.2 Sample Short-Term Planning Results
The standard form for presenting short-term planning results within the short-term lot
sizing and scheduling method are Gantt charts. The level of detail is determined by the
granularity of the short-term planning method, which is the size of small buckets. In this
case, each small bucket is 30 minutes long.
117
Figure 43: Short-Term Planning Result Visualization
The presented parts show how the number of set-up teams is minimized as simultaneous
set-ups at different machines are avoided. The number of stacking personal is mini-
mized and concentrated into shifts if possible. The different day and shift types are con-
sidered within the planning procedure and indicated by workday, Sunday, and bank hol-
iday, and further by morning shift (MS), late shift (LS) and night shift (NS). In the first
example, stackers are necessary for both machines. Sequence-dependent set-ups are
planned and coil changes integrated. The maintenance of the dies is activated after a
product change.
6.3.2 Manual vs. Automatically Generated Plans
Up until now, the plans have been created by experts without any mathematical tech-
niques. As the presented method is designed to give decision support, it has to be com-
petitive in comparison to the abilities of the planners, at least in regular cases in which
Start:
31.1.2011
6:0:0
31.1.2011
6:30:0
31.1.2011
7:0:0
31.1.2011
7:30:0
31.1.2011
8:0:0
31.1.2011
8:30:0
31.1.2011
9:0:0
31.1.2011
9:30:0
31.1.2011
10:0:0
31.1.2011
10:30:0
End:
31.1.2011
6:29:59
31.1.2011
6:59:59
31.1.2011
7:29:59
31.1.2011
7:59:59
31.1.2011
8:29:59
31.1.2011
8:59:59
31.1.2011
9:29:59
31.1.2011
9:59:59
31.1.2011
10:29:59
31.1.2011
10:59:59
Press1
Press2
Staplers
2
2
2
2
2
2
2
2
2
2
Setup
Teams
1111111111
Day Type
Workday
Workday
Workday
Workday
Workday
Workday
Workday
Workday
Workday
Workday
Shift Type
MS
MS
MS
MS
MS
MS
MS
MS
MS
MS
SETUP:
311357.85 & 311357.85 --> 82650191-
4 & 82650191-5
82650191-4 & 82650191-5
SETUP:
520889 & 520890 --> 511360.85 &
511359.85
511359.85 & 511360.85
31.1.2011
11:0:0
31.1.2011
11:30:0
31.1.2011
12:0:0
31.1.2011
12:30:0
31.1.2011
13:0:0
31.1.2011
13:30:0
31.1.2011
14:0:0
31.1.2011
14:30:0
31.1.2011
15:0:0
31.1.2011
15:30:0
31.1.2011
16:0:0
31.1.2011
11:29:59
31.1.2011
11:59:59
31.1.2011
12:29:59
31.1.2011
12:59:59
31.1.2011
13:29:59
31.1.2011
13:59:59
31.1.2011
14:29:59
31.1.2011
14:59:59
31.1.2011
15:29:59
31.1.2011
15:59:59
31.1.2011
16:29:59
COIL
COIL
2
2
2
2
2
2
2
2
2
2
2
111111
Workday
Workday
Workday
Workday
Workday
Workday
Workday
Workday
Workday
Workday
Workday
MS
MS
MS
MS
MS
MS
LS
LS
LS
LS
LS
82650191-4 & 82650191-5
82650191-4 & 82650191-5
SETUP:
511359.85 & 511360.85 --> 82028724-4
& 82028724-5
82028724-4 & 82028724-5
31.1.2011
17:0:0
31.1.2011
17:30:0
31.1.2011
18:0:0
31.1.2011
18:30:0
31.1.2011
19:0:0
31.1.2011
19:30:0
31.1.2011
20:0:0
31.1.2011
20:30:0
31.1.2011
21:0:0
31.1.2011
21:30:0
31.1.2011
22:0:0
31.1.2011
17:29:59
31.1.2011
17:59:59
31.1.2011
18:29:59
31.1.2011
18:59:59
31.1.2011
19:29:59
31.1.2011
19:59:59
31.1.2011
20:29:59
31.1.2011
20:59:59
31.1.2011
21:29:59
31.1.2011
21:59:59
31.1.2011
22:29:59
COIL COIL
COIL
2
2
2
2
2
2
2
2
2
2
Workday
Workday
Workday
Workday
Workday
Workday
Workday
Workday
Workday
Workday
Workday
LS
LS
LS
LS
LS
LS
LS
LS
LS
LS
NS
82650191-4 & 82650191-5
82028724-4 & 82028724-5
118
creative decisions and improvisation are not necessary. In this section, a comparison on
the basis of the most important costs between manual planning results and automatically
generated plans is provided.
The evaluation is based on inventory holding and capital commitment costs
ihcc
c
,
171
set-
up costs
setup
c
and manufacturing costs
man
c
:
ihcc
c
=
*
ir
p
t TM p P
cM price
∈ ∈
∑ ∑
setup
c
=
(
)
, , , , ,
*
setup
m p tm m p tm p tm
m M p P tm TM
binsM binsrM cM
∈ ∈ ∈
−
∑ ∑ ∑
man
c
=
, , ,
*
prod
m p tm p tm
m M p P tm TM
xM cM
∈ ∈ ∈
∑ ∑ ∑
Within an evaluation period of one month, the following results were obtained:
Figure 44: Comparison of Manual and Automatic Planning
171 The capital commitment and inventory holding cost rate is set to 40 %.
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
Manufacturing Costs Set-up Costs Inventory and
Capital Commitment
Costs
Total
Manual Planning Generated Plans
119
7 Summary and Outlook
7.1 Summary
In this work, a method is developed for solving capacitated lot sizing problems in pro-
duction control. In order to improve competitiveness, the guarantee of availability to-
wards the customer is the focus. Lot sizing in practice is restricted due to several organ-
izational and technological constraints. Accordingly, all the restrictions which lots and
batches are faced with are considered in the developed planning method. The considera-
tion of all available restrictions is necessary to generate feasible plans and schedules
which can be applied in practice.
As the production environment is subject to perpetual changes, and disruptions are
probable, it is not useful to calculate detailed plans for a long-term horizon. Conse-
quently, a decomposition approach was presented in this work which splits up the time
horizon according to the dynamics in demands. Rough planning of lots is carried out for
a longer mid-term horizon. Cost-optimal production lots are calculated, taking into con-
sideration restrictions for maximum lot sizes, maintenance of the dies and batched pro-
duction. With the applied rolling horizon approach, it is not possible to guarantee avail-
ability for demands set after the planning horizon. Flexibility of the rolling horizon ap-
proach is advantageous, as the plans are constantly updated. Nevertheless, these updates
cause high plan nervousness resulting in less user acceptance in practice. In this work, a
method is developed which calculates useful ending inventories on the basis of monthly
demand data to reduce problems related to the rolling horizon approach.
The planning results of the mid-term lot sizing approach are then used to determine de-
tailed schedules within the short-term lot sizing and scheduling procedure. Under con-
sideration of all constraints, detailed schedules are calculated on the basis of the actual
system state with the developed method.
The presented planning method simplifies lot sizing and scheduling. The competitive-
ness is improved, as relevant products are pre-produced. The negative aspects of the
rolling planning approach are avoided by the presented method. In an approval period of
one month, manual plans were replaced by the generated plans and overall costs, includ-
ing set-up team costs, production costs, inventory holding costs and sequence-
dependent set-up costs, were reduced significantly. The following table summarizes the
qualities of the method presented in this work. A ‘+’ indicates that the method supports
the mentioned aspect already. A ‘o’ indicates that the method can easily be adapted to
120
support this aspect. A ‘-‘ indicates that more effort and further research has to be done
to support this aspect.
Workforce
aspects
Consideration of limited resources (e.g. setup personnel) for special activi-
ties over time +
Consideration of shift and day dependent production costs (due to e.g. per-
sonnel costs) +
Categorical shift planning of setup personnel +
Personnel planning considering complete shifts o
Individual personnel planning -
Machine
aspects
Consideration of machine and part dependent production speeds and capac-
ities +
Manual deallocation of machines +
Capacity based production levelling +
Shift based machine planning -
Flexibly changing part-machine relations o
Dies &
Maintenance
aspects
Integrated preventive maintenance planning of resources (e.g. dies) +
Consideration of maintenance times +
Consideration of multiple dies to produce one product o
Randomly varying maintenance times -
Set-up
aspects
Coupled production +
Sequence dependent set-up costs and times +
Consideration of randomly varying set-up times -
Material
procurement
aspects
Input or output oriented lots (batched production) +
Consideration of capacity reductions due to batched production and input
unit changes +
Consideration of material inventory o
Consideration of input factors varying randomly in size -
Simultaneous orientation on inputs and outputs -
Demand
aspects
Consideration of dynamic demands +
Improvement of delivery service availability +
Consideration of product run-outs +
Customer based prioritization of demands o
Ease of integration into self-controlled productions +
Other
aspects
Consideration of capital commitment limitations o
Consideration of inventory limits o
Multi-Level consideration -
Consideration of the actual production system state +
Applicability coupling of the big bucket and small bucket lot sizing models
and decomposition approach in other concepts +
Figure 45: Appraisal of Presented Method
121
7.2 Future Outlook
The method was tested at a production plant of an automotive supplier. The object of
investigation was a subset of the machines within the molding presses production stage.
First, the method should be applied to plan further machines within the molding presses
stage. As the same conditions apply to similar machines, only parameters like produc-
tion speed or set-up times have to be adapted. After that, the method should be extended
to further production stages. In order to find an optimum for the whole production, sub-
sequent stages should be considered in a multi-stage lot sizing method. The mid-term
planning model could be replaced by an adapted MLCLSP
172
. As complexity grows it
will cause performance problems, and so heuristics and other decomposition approaches
as well as model improvements will be essential. Other technologies like constraint pro-
gramming could also be suitable for generating feasible starting solutions.
In summary, it is possible to improve production processes with intelligent planning
methods. The development and transfer of methods from operations research for real-
life scenarios is still at the beginning. Nevertheless, through the improvement of hard-
ware and software solutions, combined with the scientific progress of recent years, the
vision to optimize corporate planning in order to produce at maximum effectiveness
comes into reach.
172 See [Tem06].
122
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