Discrete-time analysis of levelled order release and staffing in order picking systems

Mohring, Uta; Baumann, Marion; Furmans, Kai Article Discrete-time analysis of levelled order release and staffing in order picking systems Logistics Research Provided in Cooperation with: Bundesvereinigung Logistik (BVL) e.V., Bremen Suggested Citation: Mohring, Uta; Baumann, Marion; Furmans, Kai (2020) : Discrete-time analysis of levelled order release and staffing in order picking systems, Logistics Research, ISSN 1865-0368, Bundesvereinigung Logistik (BVL), Bremen, Vol. 13, Iss. 1, pp. 1-20, https://doi.org/10.23773/2020_9 This Version is available at: https://hdl.handle.net/10419/297185 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Received:07 December 2019 /Accepted:05 August 2020 /Publishedonline:03 September2020 ©TheAuthor(s)2020 This articleis publishedwith Open Access at www.bvl.de/lore Discrete-TimeAnalysis of Levelled OrderReleaseandStaffing in OrderPickingSystems UtaMohring, Marion Baumann, KaiFurmans ABSTRACT Orderpickingsystemsareconfronted with avolatile demand andshortdelivery timerequirements. Manufacturing companies face theincreasing variabilityrequirements with Heijunka-levelling, one method of theToyota Production System.Theobjectives of this publicationareto develop a levelling conceptfor order pickingsystems, to analyseitsperformancebased on a discrete-timeanalytical modelandto develop a staffingalgorithmdetermining therequired workforce levelin an orderpicking system with levelled order release. Thelevelling conceptfor orderpicking systems resultsfrom theexisting models of Heijunka-levelling in theliterature, whichareadoptedandexpanded regarding thespecificrequirements of order picking systems. The orderpickingsystem with levelled order releaseis depicted as adiscrete-timeMarkov chain. To analyseitsperformance,we derive severalperformance measures,such as servicelevel, backlog duration and system utilisation, from thesteady-state distribution of theMarkov chain. The staffing algorithmis a binary search algorithmbasedon theMarkov chain. Themodels developed in this publicationenable aquantitative evaluation of theimpact of several system parameters, such as variabilityof customer demand, workforce level andtraffic intensity, on the performance measures of the orderpicking system. Furthermore, thestaffingalgorithmdetermines the workforce levelwhichis required to guarantee acertain system performance, such as aservicelevelof 99%,in an orderpickingsystem with levelled orderrelease. By comparinglevelled orderreleaseto FCFS-based order releasestrategies in anumericalexample, we show the benefitsof levelled order release. KEYWORDS:discrete-timemodelling·Markovchain· orderpicking · levelling · staffing 1. INTRODUCTION Orderpickingsystemsareconfronted with constantly increasing order volumes, volatilecustomer demand andpermanentcost pressure.Sincethe orderpicking process directlydepends on thecustomer orders, thevariationof customer demand is assigned to the workload of orderpickingsystems. Furthermore, customers requirehigh flexibilityandshort delivery times (cf. [1]).Orderpicking systemsoftendifferentiate thecustomer ordersregardingtheir lead timein express andstandard orders. A typical examplearethedifferent delivery conditions of online retailersin B2Csectors. However, this differentiationis also common in B2B sectors: Forexamplein theautomotive aftermarket sector, express orders correspond to unexpected breakdowns of vehicles,whereasstandard orders referto planned andregularmaintenance measures of vehicles. Dueto theseflexibility requirements,many warehousesstillprefer manual orderpickingsystems with workerspicking therequired itemsfrom shelves or palletswhiledriving or walking throughthewarehouse insteadof innovative,partly or fullyautomated order picking systems. Automated systemsensure higher picking ratesthan manual systems, butthey require ahigh homogeneity of items. Furthermore, especially LogisticsResearch (2020) 13:9 DOI_10.23773/2020_9 UtaMohring, M.Sc. eMail:[email protected] Dr.-Ing. Marion Baumann eMail:[email protected] Prof.Dr.-Ing. Kai Furmans eMail:[email protected] InstituteforMaterialHandling andLogistics (IFL) KarlsruheInstituteof Technology (KIT) Karlsruhe, Germany 2 workers, are not necessary.It is also ensuredthat the customer orders areprocessedwithin therequiredlead times. In this publication, we focus on theproposedapproach of levelled orderrelease. Theapproach of flexible workforce planning will notbe considered further. The objectives of this publicationareto develop a levelling conceptfor orderpicking systems, to analyse its performancebasedon a discrete-timeanalytical model and andto develop a staffing algorithm determining the required workforce level in an order picking system with levelled order release. Theremainderof this publicationis structured as follows: Section2gives a short literaturereview on related research topics. In Section3, we derive a levelling conceptfor orderpicking systems. To analyse theperformanceof this concept, we depictthe order pickingsystem with levelled orderreleaseas adiscretetime Markov chainandwe derive severalperformance measures from itssteady-state distributionin Section 4. In Section5, we derive astaffingalgorithmfor orderpickingsystemswith levelled orderrelease. The numericalstudiesin Section6give insights into system behaviourandshow thebenefits of levelled orderrelease compared to FCFS-based order releasestrategies in a numericalexample. Section7summarisestheinsights andprovides directions forfuture research. 2. LITERATURE REVIEW Review papers on warehouseoperation(cf. [4], [5], [6]) and orderpickingsystems(cf. [2]) state that past research focusedonspecificwarehouse configurations or specificdecision problems,e.g. routing, storageand batching policies.Thereis alack of global models in contract logistics, automated orderpickingsystems arenot profitable,since thecontract periods aretoo short andthecustomers’ productranges aretoodiverse. However, manual orderpickingis cost-intensive, especiallyin high-wagecountries. Accordingto [2] and[3], on average45-55% of totalwarehousing costs canbe assigned to the orderpicking process.Therefore, workforce efficiencyis an importantleverto reduce costsandto face theupcoming challenge of skills shortage. Flexible workforce planning andlevelled orderrelease aretwodifferentappropriate solution approaches to face thesepartly opposite requirementsin manual orderpickingsystems(cf. Figure 1).Thekeyidea of flexibleworkforce planning is to coverthevolatile workload of the order pickingsystem as preciselyas possible with theavailableworkforce capacities of thesystem by combining methodsof flexibleshift scheduling andflexiblework time models.In this way, customer orders areprocessedin atimely manner and the orderpickingsystem canguaranteeboth short lead times with ahigh servicelevelandhigh workforce efficiency. In contrast, thekeyidea of levelled order releaseis to convertthevolatile workload of the order picking system into asmooth andregularworkload per time interval. Forthispurpose, thedeployed workforce capacities pertime interval areconstant andlevelled orderreleasetakesadvantageof thedifferentlead times of thecustomer orders whichallow a certaintime flexibility to determine the timeof processingof the orders.By consideringtheduedatesof thecustomer orders,levelled orderreleasecompensates peaks in the workload of the order pickingsystem resultingfrom peaks of thecustomer demand.Thus,therequired workforce levelis constant andexpensiveadditional workforce capacities,such as overtime andtemporary Main requirements of an order picking system Fulfill avolatile customer demand Efficientstaffing to minimize operational costs Guarantee shortleadtimes with ahigh servicelevel Flexible workforceplanning Cover thevolatile workload of the order picking system as preciselyaspossiblewiththe availableworkforcecapacitiesofthe system Combineflexibleshift schedulingand flexible work time models Levelledorderrelease Convert thevolatile workload of the order pickingsysteminto asmoothand regular workload pertimeinterval Use fixedworkforcecapacitiesper time interval and exploittimeflexibilityoforder processing duetothe different lead timesofthe orders Fig. 1: Problemstatementand possible solution approaches 3 Discrete-Time Analysis of Levelled OrderReleaseandStaffingin OrderPickingSystems [23],[24] and[25] providecomprehensiveliterature reviews on personnel scheduling categorizing the publications accordingto thesolution method,the applicationarea andseveralsystem characteristics. [25] focuseson staffingandscheduling approaches for systemswith non-stationary demand. Thereareonly few publications on staffingin warehouses, although researchersagreeon theimportance of workforce planningin warehouses(cf. [6], [26]). [27] develops a time series forecasting method to predict theworkload in azone orderpickingsystem.Basedon thepredicted workload andtheproductivity of one worker,the requiredworkforce levelis calculated. To thebest of ourknowledge, thisis thefirst publicationdealingwith levelled orderreleasein order picking systems, analysingitsperformancebased on an analytical,stochastic modelanddeterminingthe workforce levelof this system. 3. LEVELLING CONCEPT FOR ORDER PICKINGSYSTEMS In this section, we derive alevellingconceptfor order picking systemsbasedon the keyideasof Heijunkalevellingin production systems. We initially describe theideasandtheprocedureof Heijunka-levelling in productionsystems. Subsequently,we identify the differencesbetween production systemsand order picking systemsregardinggeneralconditions and decision problem.Finally, we present thelevelling conceptfor orderpickingsystems. 3.1. Heijunka-levelling in Production Systems Heijunka-levelling is asimple andwidespread concept for order releasein production systemsto manage theproduction of severaldifferentproducts on one common production line. Thekeyidea of Heijunkalevelling is to convertthevolatile customer demand into aregular, recurringandstandardized production schedule to guaranteean even load of thegiven production capacity.Heijunka-levelling smoothsboth thevolume andtheproductmix of theproduction system (cf. [11]). Theplanning procedureof Heijunka-levelling refers to one planningperiod of theproduction system (e.g.one month), whichis subdivided into smaller scheduling intervals (e.g.one week,one day, one shift). It consists of thefollowingplanning steps: System parametrisationandoperationalplanning(cf. Figure 2).System parametrisationtakesplace at thebeginning of each planning period anddeals with smoothingboth theproduction volume andtheproductmix.Volume smoothingdetermines theproduction capacity per product perscheduling intervalwhichis reserved fortheproduction of this productin each scheduling interval. Forthis purpose, thetotalcustomer demand of theplanning period of each product is evenly distributed on thescheduling intervals. Thereserved andgeneralprocedures for orderpickingsystems. Furthermore, past research predominantlyfocused on deterministic warehouse configurationsassuming that alldata is givenin advance(cf. [2]).However, severalquestionsin practice in warehouse operation include stochastics, e.g. stochastic customer demand andstochastic processing times, whichis notexplored in literatureyet(cf. [7]). An order pickingpolicy describestheprinciple accordingto which orders areprocessedin the order pickingsystem.Literaturedifferentiates between strict orderpicking, batchpicking,wave pickingandzone picking(cf. [5], [8]). In contrast, an orderreleasestrategy describestheprincipleaccordingto which orders are released forprocessing.Orderreleasestrategies in warehouseshave hardly been investigated in literature so far. [9]differentiates betweenwave-based and waveless releasepolicies: A wave-based releasepolicy groups orders into batches by some criteria andthese batches arereleased in asequential manner.In case of waveless orderrelease, individual orders arereleased continuously.[9]focuses on waveless orderreleasefor warehouses with an automatedsorter and[10] deals with waved-based orderreleasein orderfulfillment systemswith deadlines. In contrastto warehouseliterature, levelling customer demand andsystem workload in production systemsis abundantlydiscussed in literature. The most knownlevellingconceptin production systemsis the Heijunka-levelling approach of theToyota Production System.Past research on Heijunka-levelling canbe classifiedinto thefollowing areas: •Procedure models, •Stochastic models and •Models forlevelscheduling. Procedure models of Heijunka-levelling, such as [11], [12] and[13],qualitativelydescribe theconceptof Hejunka-levelling. Stochastic models,such as [14], [15], [16] and[17],focus on buffer sizing of aHeijunkalevelled Kanban system.[14]–[17]usediscrete-time analytical models includingstochastic parameters, such as productioncapacity andcustomer demand, to depict theHeijunka-levelledKanbansystem and to computeseveralperformance measures,such as servicelevelandbuffer size.Theresearch area of levelscheduling covers static optimizationproblems forproduction sequence planning in Heijunka-levelled production systems(cf. [18],[19]). Thephilosophyandthemethodsof theToyota Production System aretransferredto other fields such as supply anddistributionlogistics. Theso called “LeanLogistics” and“LeanWarehousing”are studiedin academic literatureto alimitedextent: [20] describesthebasicconcepts of lean warehousing,[21] develops alean assessment tool forwarehousesand [22] investigates theimpact of lean warehousing on thewarehouse performance. None of thesepublications describesthedifferent methodsof lean warehousing, such as levelling, in detail. 4 when thecustomer demand of this productis belowits reserved production capacity (cf. Figure 3, [16], [17]). 3.2. Delimitation from Heijunka-levelling in Production Systems Theconceptof Heijunka-levelling in production systemscannot be directlyappliedin orderpicking systemsbecausethegeneralconditions andthedecision problemof orderpickingsystemsdiffer from thoseof productionsystemsto some extent:First, orderpicking systemsandproduction systemsdiffer in termsof lot size.Since orders arecustomer-specific concerning producttype andproductvolume, the orderlotsize in orderpicking systemsis usually one, whereasfor reasonsof setup times, lotsizesin production systems areoften higher than one (cf. [22]). Second,setup times between differentproducts arerelevant in production systems, whereassetup times betweenthe order picking processes of different customer orders are negligible small(cf. [22]). Third, Heijunka-levelling in production systemspredominantlyfocuseson Make- To-Stock processes,whichdecouple workload and customer demand by abuffer,whereastheworkload of orderpicking systemsdirectly depends on thecustomer demand.Thus, orderpicking canbe considered as a production capacity of aproductcorresponds to the averagecustomer demand perscheduling interval of this product. Productmix smoothingdetermines the production sequence of thedifferentproducts within a scheduling interval.Common objectives forproduction sequence planning areminimizing setup times or maximizing theregularity of theproductmix.These decision problems arecoveredin detail by theresearch area of levelscheduling.Thereserved production capacities andtheproduction sequence perscheduling interval arevisualised in thelevellingpatternon the Heijunka-board (cf. Figure 3).Basedonthelevelling pattern,theoperationalplanning takesplace at the beginningof each scheduling interval: Theincoming customer ordersof thecurrentscheduling interval arefulfilled by taking therequired products from the finished-goods-supermarket. Theassociated kanbans arereturned to theHeijunka-board.Thesekanbans areassigned to thereserved productioncapacity of thecorrespondingproduct on theHeijunka-board accordingto First-Come-First-Served(FCFS).If the customer demand of aproduct exceedsitsreserved production capacity in thecurrentscheduling interval, theassociated kanbansarekept in an overflow box. They areassigned to future scheduling intervals, System Parametrisation Operational Planning Smoothing of Production Volume Smoothing of Product Mix Time slots Overflow 12345678 Products A22 21 B223 C3 D3 Finished-goods supermarket Heijunka-board Levellingpattern: A-A-B-C-A-A-B-D Machine / productionresource Customer Levelled production sequence Fig. 2: Planning procedure of Heijunka-levelling Fig. 3: Model of a Heijunka-levelled kanban system 5 Discrete-Time Analysis of Levelled OrderReleaseandStaffingin OrderPickingSystems is reserved for order processing of this ordertype in each scheduling interval(smoothing of volume) and theprocessingsequence of thedifferentordertypes within a scheduling interval (smoothing of product mix). Theprocedure of both stepsequals theone of Heijunka-levelling. Theresultinglevellingpattern is visualisedon theHeijunka-board (cf. Figure 5), whichis thestarting pointof theoperationalplanning. Operationalplanningtakesplace at thebeginningof each scheduling intervalandallocates thepicking orders of thedifferent ordertypesto thecorresponding reserved pickingcapacities in thelevellingpattern. The pool of assignable picking orders covers theincoming picking orders of thecurrentscheduling intervaland theremainingunprocessedpicking orders of previous scheduling intervals stored in theoverflowbox. To determine theprocessingsequence of picking orders within one ordertype,their duedates areconsidered as follows: •Picking orders areprocessedaccordingto ascendingduedates. •Picking orders with identicalduedatesare processedin accordance of FCFS. •Pickingorders become lost sales,when theirdue date exceedsthemaximumaccepted backlog duration. If thenumber of assignable picking orders of an order type exceedsitsreserved pickingcapacity in thecurrent scheduling interval, the orderbacklog in the overflow box increases by the corresponding number of orders. Otherwise, the remaining capacity is used for training, maintenance and continuous improvement measures. To sumup,themain characteristics of thelevelling conceptfor orderpickingsystemsarethefollowing: •Thereis afixed pickingcapacity per ordertype perscheduling intervalwhichis reserved for orderprocessing of picking orders of this order type in each scheduling interval. •Size andsequence of thereserved picking capacities within one scheduling interval arevisualisedin thelevelling patternonthe Heijunka-board. Make-To-Orderprocess. Finally, productioncapacity in productionsystemsis fixed in theshort term, dueto aconstant number of machines with agiven performance. In manual orderpickingsystems, capacity mainly depends on thenumber of assigned workers, whichis rather flexiblein short term. Thus, capacity canbe easily adjusted to thecurrentworkload of the orderpickingsystem. Thedecision problemof Heijunka-levelling in productionsystemsfocuses on choosing an appropriate buffer size to guaranteetherequired servicelevel, wherebyproductioncapacity is constant.On the contrary,thedecision problemof levellingin order pickingsystemsdetermines theappropriate workforce capacity to guarantee therequiredservicelevel. Dueto thesedifferencesbetween orderpicking systemsandproduction systems, some adjustmentsand extensionsare necessaryto derive alevellingconcept for orderpickingbased on thekeyideasof Heijunkalevelling.Furthermore, theabove mentioned discretetime stochastic models of Heijunka-levelled production systemsfocus on a differentdecision problemand cannot be used fortheperformance analysis of order pickingsystemswith levelled orderrelease. 3.3. Levelling Conceptfor OrderPicking Systems Thelevelling conceptfor orderpicking systems determines thereleaseprinciples forpicking orders. Thepicking orders result from theincomingcustomer orders dependingontheorderpickingpolicy of the considered system.In thesimplest case of strict order picking,each customer ordercorresponds to one picking order. Forsome orderpickingsystems, it canbe reasonable to classify thepicking orders into different ordertypes,e.g. regardinglead time (express vs. standardpicking orders)orregardingused picking technology(picker-to-parts vs. parts-to-picker).Due to theMake-To-Ordercharacterof orderpicking, the levellingconcepthasto consider theindividualdue dates of thepicking ordersresultingfrom theduedates of thecorrespondingcustomer orders.Consequently, picking orders aredifferentiated regardingtheirdue date into orders withoutfailed duedates andthose with failed duedates(cf. Figure4).Picking orders withfailed duedates arefurthermore differentiated into backordersandlost sales: Backordersrepresent picking orders with failedduedates whichstillhave to be fulfilled,whereaslost sales correspond to picking orders with failedduedates whichareremovedfrom thesystem withoutbeingprocessed, since their due date exceeds a certainmaximumbacklog duration. Followingtheprinciples of Heijunka-levelling, the planning procedure of levelled order releasein order pickingsystemsconsists of theplanning stepssystem parametrisationandoperationalplanning(cf. Figure 2).System parametrisationtakesplaceat thebeginning of each planning period anddetermines thepicking capacity per ordertype perscheduling intervalwhich Picking orders without failed duedates Picking Orders Backorders Lost Sales Picking orders with failed duedates Fig. 4: Classification of picking orders regarding duedate 6 allow a more detailedanalysis of thesystem:The performanceanalysis is not limitedto expected values, butcomplete probabilitydistributions arecomputed (cf. [28]). 4.2. System Description Theanalytical modeldepicts orderprocessing of one ordertype.An isolatedconsiderationof each order type is possible,sincetheanalytical modelfocuseson operationalplanning.When operationalplanning takes place, thelevellingpatternof the orderpickingsystem hasalreadybeen determined. Thus,thereserved picking capacity per ordertype is fixed and orderprocessing of thedifferent ordertypes is independentof each other. We assume that thereis one orderincome of picking orders perscheduling interval whichis alreadyknown at thebeginningof thescheduling interval. Thegeneralconditions of the orderpickingsystem aredescribedby thefollowing parameters(cf. Table 1):Customer demand is specified by itsvolume anditslead time: Random variable Adescribesthe number of incoming picking orders perscheduling intervalandrandom variable Especifies thelead time associated to one picking orderwhen arriving at the orderpicking system.We specifythe order picking process by theparametersLandc:Theindividual pickingperformance Ldepictsthenumber of picking orders one worker is able to completely fulfill within onescheduling interval, whereasccorresponds to thenumber of workersassigned to the order picking system.Furthermore, parameter Nspecifies the maximumbacklogduration. •During each scheduling interval, thereserved pickingcapacity per ordertype is used to process picking orders of this ordertypeaccordingto ascendingduedates. 4. DISCRETE-TIMEANALYSIS OF LEVELLED ORDERRELEASE In this section, we depict an orderpickingsystem with levelled orderreleaseas adiscrete-time, analytical modelto analysetheperformanceof thedeveloped levellingconcept. We firstly explainthereasons forchoosing adiscrete-timeMarkov chain. After describing thegeneralconditions of thestudied orderpickingsystemswith levelled orderrelease, we introducethecorrespondingMarkov chain. Finally, severalperformance measures of interest arederived from thesteady-state distributionof theMarkov chain. 4.1. Model Choice We choose adiscrete-timeMarkov chainto analyse theperformanceof thelevellingconceptfor order pickingsystemsdueto thefollowing aspects: On the contrary to static models, Markov chains areable to depict thestochastic characterof severalparameters. Furthermore, performance measures derivedfrom the steady-state distributionof theMarkov chainareexact in contrastto theapproximate resultsof simulation models. A discrete-time modelis preferredto continuous-time models,since therelevant parameters have discrete-timecharacter anddiscrete-time models Fig. 5: Modelofan orderpicking systemwith levelled order release Furtherprocessingsteps in the warehouse Heijunka-board Levelling pattern:A-A-B-C-A-A-B-D Order picking Customer Levelled order release of picking orders … Stochasticincoming customerorders Processed customer orders Processed picking orders Time slots Overflow 12345678 Ordertypes A22 21 B223 C3 D3 7 Discrete-Time Analysis of Levelled OrderReleaseandStaffingin OrderPickingSystems foreach lead timek∈ E (second component in equation (2)) and •theprobabilityof having gkpicking orders with a lead time of kscheduling intervalsforeach lead time k∈E(third componentin equation (2)): (2) To aggregate allrelevant informationregarding the orderpickingprocess in asingle parameter,we introducerandom variable B.This variable specifies thetotalpickingperformance perscheduling interval anddepends on thenumber of workerscandtheir individual picking performanceL.Assuming an identicalandindependentdistribution of theindividual performanceof allworkers, theprobabilitydistribution of Bis computed as c-foldconvolutionof theprobability distributionof L. We assume that theconsidered orderpickingsystem is stable. A system is stable, if its traffic intensity Uis smallerthan one: (3) 4.3. Discrete-timeMarkov Chain Thediscrete-timeMarkov chaindepictsthetemporal developmentof the orderbacklog in an orderpicking system withlevelled order release. We observethe number of picking orders at discrete-timepoints in time ,whichcorrespond to thestarting points of thescheduling intervalsof thelevellingconcept. To aggregate allrelevant informationregardingthe customer demand in asingle parameter, we introduce random vector G= (G − N ... Gemax ). It characterises theincoming picking orders perscheduling interval, wherebyrandom variable Gkcorresponds to the number of incomingpicking orders with alead time of kscheduling intervals,k∈{−N, ..., emax }.The valuerangeGof Gis defined based on thefollowing conditions: •No incoming picking order has a negative lead time (firstcomponent in equation (1)). •Thevaluerangeof each vector componentgk, k∈ {0,. .., emax }, is defined by zero and the maximum number of incoming picking orders amax per scheduling interval (second component in equation (1)). •Thetotalnumber of incoming picking orders corresponds to arealisationof thenumber of incomingpicking orders Aper scheduling interval(third componentin equation (1)). (1) TheprobabilityP(G=g)of realisation g= (g − N ... gemax )depends on •theprobabilityof having incoming picking orders perscheduling interval(first component in equation (2)), •thenumber of possibilitiesof having gkpicking orders with alead time of kscheduling intervals Tab. 1: Overview of parameters of the discrete-timeMarkov chain Parameter Variable Value range Parameters specifying the customer demand Number of incoming picking orders AA={amin,...,a max} per scheduling interval Lead time of apicking order EE={emin,...,e max} Incoming picking orders per scheduling interval G=G−N... G e max G Parameters specifying the order picking process Individual picking performance LL={lmin,...,l max} per scheduling interval Number of workers cN Total picking performance per scheduling interval BB={(c·lmin),...,(c·l max)} Parameters specifying the levelling concept Maximum backlog duration NN 8 date of emaxscheduling intervals at thebeginning of scheduling interval (t+1) equals thenumber of incomingpicking orders gemax with alead time of emax scheduling intervals at thebeginningof scheduling interval (t+1). Usingthespecificstructureof thestate transition, we derive an upperboundforthestate spaceX:The number of unprocessedpicking orders of aparticular duedate is at itsmaximum, •if thenumber of incoming picking orders per scheduling interval of this duedate is at its maximum, •if theresidualpickingperformance perscheduling intervalassigned to thisduedate is at itsminimum and •if thenumberof remainingunprocessedpicking orders of thepreviousscheduling intervalassigned to this duedate is at itsmaximum. TheupperboundOof system state Xis defined by (9) Consequently,theMarkovchainis finiteanditsstate spaceXis defined by (10) Several performance measures of interestcanbe derivedfrom thesteady-state distribution π with πx=P(X=x)of theMarkov chain. Foraperiodic, finite, irreducibleandtherefore ergodic discrete-time Markov chains,thesteady-state distribution is obtained by solving a setof linear equations(cf. [29]). TheMarkov chain modelled in thispublicationis aperiodicforallconsidered practicalapplications. Accordingto equation (10),its state spaceis finite. Furthermore, it is possible to reacheverystate of theMarkovchainfrom everyother state either by a direct state transition or by an indirect transition via afinitenumber of other states.In case of unreachable states,thesestates areexcluded from thecomputations. Consequently,it is possible to findan irreducible subset of thestate spacewhichis used as starting pointfor subsequentcomputations.Thesteady-state distribution πis therefore computed by solvingthefollowing setof linear equations (11) Sincethisset of linear equations is overdetermined by one equation, one equation of (11) is omitted when solvingthesystem of equations. To obtaintheexact steady-state distribution, this setof linear equations is solved by usingtheGaussian Elimination (cf. [29]). System state Xof theMarkov chaindepicts the number of unprocessed picking orders of the order pickingsystem: (4) wherebyXkcorresponds to thenumber of unprocessed picking orderswith aduedate of kscheduling intervals. Thestate transition from an arbitrary state Xt=xat thebeginningof scheduling intervaltto astate Xt+1 =y at thebeginningof scheduling interval (t+1) depends on •thetotalpicking performancebof scheduling intervalt, •theincoming picking orders gat thebeginningof scheduling interval (t+1) and •theprinciples of thelevelling concept. Assuming independenceof incoming picking orders Gandtotalpicking performance Bperscheduling interval, thetransition probabilityis computed as follows (5) with (6) Thenumber of unprocessedpicking orders ykwith a duedate of kscheduling intervals at thebeginningof scheduling interval (t+1) is thesumof thenumber of incomingpicking orders gkwith alead time of k scheduling intervals at thebeginningof scheduling interval (t+1) andnumber of unprocessed picking orders (7) with aduedate of (k+1) scheduling intervals at theend of scheduling intervalt.Thenumber of unprocessed picking orderswith aduedate of (k+1) scheduling intervals at theendof scheduling intervaltis either zero or it corresponds to thedifference of thenumber of unprocessedpicking orders xk+1 with aduedate of (k+1) scheduling intervals at thebeginningof scheduling intervaltandtheresidualpicking performance (8) of scheduling intervaltremainingafter allpicking orders with a duedate of l < (k+1) scheduling intervals have alreadybeen processed.Thenumber of unprocessed picking ordersyemax with adue 15 Discrete-Time Analysis of Levelled OrderReleaseandStaffingin OrderPickingSystems 6.2. NumericalPerformanceAnalysisfor DifferentLeadTimes To investigate theimpactof thelead time of apicking order on theperformance measuresof the orderpicking system with levelled orderrelease, we examineseveral expected values of thelead time of apicking orderin therangebetween E(E) = 0 andE(E) = 3 fordifferent traffic intensities of thesystem U∈ {0.4762,0.7143, 0.9524}(cf. Table4). Express orders have to be processedwithin thesame day (E(E) = 0),whereasfor replenishment orders,thelead time is typicallygreater than one day. Figure 9shows an increase in β-service levelSLβand γ-service levelSLγwith increasing expected valueof thelead time E(E)of apicking order. Thelead time of apicking orderaffects thetime flexibility of the order pickingsystem to determine thetime of processing of this picking order: Thetime flexibility to determinethe time of processing of apicking orderwith along lead time is higher than theone of apicking order with a short lead time. In an orderpicking system with agiven pickingperformanceand a givennumber of incoming picking orders,an increasing time flexibility increases thenumber of on timeprocessed picking orders and thus resultsin an increase of theperformance of the orderpickingsystem (cf. Figure 9). Furthermore, Figure 9shows a disproportionate increase in β-service levelSLβandγ-service levelSLγ with increasing traffic intensityU. In orderpicking systemswith alowtraffic intensity, the average available pickingperformance E(B)exceedstheaverage needed picking performanceE(A)to aremarkable extent, so that thereis enough pickingperformanceto process almost everypicking orderon time independentof its lead time. In orderpickingsystemswith ahigh traffic intensity, theaverageavailablepickingperformance E(B)is only slightly greater than theaverage needed picking performanceE(A), so that a higher time flexibility has a remarkable positive effect on the theworkload has a negative impact on theperformance of the orderpickingsystem (cf. Figure 7). Furthermore, Figure 7shows a disproportionate decrease in β-service levelSLβandγ-service level SLγwith increasing traffic intensityU. In order pickingsystemswith alowtraffic intensity,the averageavailablepicking performanceE(B)exceeds theaverage needed pickingperformance E(A)to a remarkable extent. This performance excess is used to compensate occurringpeaks of theworkload.On the contrary,in orderpickingsystemswith ahigh traffic intensity, theaverageavailablepickingperformance E(B)is only slightly higher than theaverage needed pickingperformance E(A), so that thereis less picking performanceto compensate occurringpeaks of the workload.Consequently,the negative impact of an increasing volatilityof theworkload on theperformance of the orderpickingsystem increaseswith increasing traffic intensity of the system (cf. Figure 7). Figure 8shows a positive correlationbetween the variabilityof processedpicking orders c2(F)and thevariabilityof incoming picking orders c2(A). For c2(A)>0, thevalueof thevariabilityof processed picking ordersis smallerthan thecorresponding valueof thevariabilityof incoming picking orders, especiallyfor medium andhigh traffic intensities. Thus,thelevelled orderreleasesucceedsin reducing thevolatilityof theworkload of the order picking system.Thesmoothingeffect of levelled order release increaseswith increasing traffic intensity. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.9 1.01.1 Service level SLβ,SLγ Variability of numberofincomingpicking orders c2(A) SLβ, U=0.4762 SLγ, U=0.4762 SLβ, U=0.7143 SLγ, U=0.7143 SLβ, U=0.9524 SLγ, U=0.9524 Fig.7: Impact of variability of incoming picking orders c2(A) andtraffic intensity U on β-servicelevel SLβandγ-servicelevel SLγ(Note: Thereis no value of SLβ,SLγforscenario (E(A) = 1.5, c2(A) = 0.0), sincethevaluerange of A is integer.) -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 -0.1 0.00.1 0.2 0.3 0.4 0.5 0.60.7 0.8 0.9 1.0 1.1 Variability of numberofprocessed picking ordersc 2 (F) Variability of number of incoming picking ordersc 2 (A) U=0.4762 U=0.7143 U=0.9524 Fig.8: Impact of variability of incoming picking orders c2(A)and traffic intensity Uon variability of processedpickingorders c2(F )(Note:Thereis no value of c2(F )forscenario (E(A) = 1.5, c2(A) = 0.0), sincethevaluerange of A is integer.) AE(A)∈{1.0,1.5,2.0}c 2 (A)=0.5 EE(E)∈{0.0,1.0,2.0,3.0}c 2 (E)=0 LE(L)=1.05 c2(L)=0.315 c2 N2 Tab. 4: Parameter settingfornumerical performanceanalysisfordifferentlead times 16 Forthis order pickingsystem, weaimto quantify the benefitsoflevelled orderrelease compared to FCFS- basedreference orderreleasestrategies.Therefore, we comparelevelled orderreleasewith the orderrelease strategies FCFS-DD andFCFS-RAND: •FCFS-DD: Orders arriving at differentpoints in timearereleased accordingto FCFS and orders arriving at thesame pointin time arereleased accordingto ascendingduedates. •FCFS-RAND: Orders arriving at differentpoints in time arereleased accordingto FCFS,whereas thereleasesequence of orders arriving at thesame pointin time is determined randomly. We initiallyhave to derive theparametersof the Markovchain(cf. Table1) from thegivendata.There aretwopossibilitiesto modelthedifferent ordertypes of thenumericalexample: On the one hand, we can consider each order type separately modellingone Markovchainforeach ordertype.On theother hand, the ordertypes only differ regardingtheir lead time in this example. We candepictthesedifferences in the probability distribution of thelead timeandthus the differentiationof two ordertypes is not necessary any further. Forreasonsof simplicity,we choose thelatter number of on time processed picking orders.Thus, thepositive impact of an increasing lead time of the picking orders on theperformanceof the orderpicking system increaseswith increasing traffic intensity of the system (cf. Figure 9). Theseinteraction effectsbetween thelead time of a picking order, thetraffic intensity of thesystem andits performance couldbe relevant forpractitionerswhen negotiatingtheir supply agreements with customers: To guarantee ahigh servicelevel, either thecustomers have to transmit their orderswith asufficientlylong lead time, or thesystem hasto be run at a sufficiently smalltraffic intensity. Themodels developed in this publicationenable aquantitative evaluation of theseinteraction effects: Forinstance,for a given probability distributionof thelead timeof apicking order, we areable to determine themaximumpossible traffic intensity of the orderpickingsystem while guaranteeingtherequired servicelevelof thecustomer at thesame time. 6.3. Comparison of OrderRelease Strategies in aNumericalExample We investigate a manual order pickingsystem of a German companyof theautomotive aftermarket sector. Theobservationperiod lastsfrom 2 January2015 to 31 December 2015.Thetime series of the orderdata within this period consists of 260data sets(cf. Figure 10). Each data set represents the orderdata of one workingdayandincludesthedaily order volume as well as thelead times of thecustomer orders.Thedaily ordervolume fluctuates between itsminimum of 1144 ordersperdayanditsmaximumof 14193 orders per day. Theaverage ordervolume is 7842 orders perday. Thecustomer orders aresubdivided into thefollowing ordertypes:express orders with an averageleadtime of 0.84 days andstandard orders with an averagelead time of 4.04 days.Regardingthepickingperformance, it is knownthat one worker picks on average 14 orders perhourandhas a dailyworkingtime of 8hours. The required performance of this orderpickingsystem is measured by β-servicelevelanditstarget value is 98%. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Service level SLβ,SLγ Expected value of lead timeE(E) SLβ, U=0.4762 SLγ, U=0.4762 SLβ, U=0.7143 SLγ, U=0.7143 SLβ, U=0.9524 SLγ, U=0.9524 0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000 Number of orders Fig. 9: Impact of expected value of lead time E(E) andtraffic intensity U on β-service level SLβand γ-servicelevel SLγ Fig. 10:Time series of order volume in 2015 17 Discrete-Time Analysis of Levelled OrderReleaseandStaffingin OrderPickingSystems when usinglevelled orderreleaseinsteadof FCFS-DD order release or by sixworkers(6.29%) when using levelled orderreleaseinsteadof FCFS-RAND order releaserespectively. Furthermore, we comparethedifferent order releasestrategies regardingβ-service levelfor a given workforce level. For a workforce levelof 84 workers, theβ-service levelof the orderpicking system is 99.09%in case of levelled order release, 96.30%in case of FCFS-DD orderreleaseand93.87% in caseof FCFSRAND orderrelease. This corresponds to an increase in β-service levelby 2.79%when usinglevelled order and modelthe orderpickingsystem of thenumerical examplebasedon a single Markov chain. The scheduling intervalof thelevellingconceptis one day, in particular one workingday, dueto thetimeintervalof the orderdata.Furthermore, thedata of thedaily order volume is classifiedinto 14 equally-sizedclassesin the rangefrom 1000 to 15000 orders perday, wherebythe mean of each classis chosen as classrepresentative. Theresulting probability distribution of thenumber of incoming picking ordersperscheduling interval is shown in Figure 11.Theprobabilitydistributionof thelead time of apicking orderis derivedfrom the absolute frequencydistribution of the orderlead time. Thepeaks of theprobability distribution for a lead time of one dayand a lead time of four days indicate the twodifferent ordertypes (cf. Figure 12). Regarding theindividual picking performanceperscheduling interval, the mean valueis givenby 112picking orders perscheduling interval.Basedonourexperienceson common probabilitydistributions of processing times in orderpicking systems, we assume a discretelognormal distribution with avariabilityof c2(L) = 0.4as appropriate probabilitydistributionfortheindividual picking performanceperscheduling interval(cf. Figure 13). Since thereis no informationregardingthe maximumbacklogduration, we assume that it equals themaximumlead time of an order, whichis 8days. Based on theseparameters, thestate spaceof the correspondingMarkov chainconsistsof 8.65 ·1083 states.Forreasonsof computingtimeand memory, it is not possible to analyse levelled orderreleasein the orderpicking system of thenumericalexample by meansof theMarkovchain. Instead, we exploit a simulation modelwhichhasbeen validatedbased on acomparisonwith theMarkovchainfornumerous examplescenarios. Theresultsof aChi-Square Goodness-of-Fit Test with a5% levelof significance show that theempiricalsteady-state distribution resultingfrom thesimulation modeldeviates from the exactone to a negligible smallextent. Both FCFS-based orderreleasestrategies FCFS-DD andFCFS-RAND arealso each implemented in asimulation model. To obtain robust results, we perform ten replications each andcomputetherequired workforce leveland theperformance measures as averagevalues of these replicates. Initially, we comparethedifferent orderrelease strategies regardingtheworkforce levelwhichis required to guarantee aβ-service levelof 98%in the orderpickingsystem of thenumericalexample. When usinglevelled orderrelease, at least84 workershave to be assigned to the orderpickingsystem.In contrast, in case of FCFS-DD orderrelease, at least87 workers andin case of FCFS-RAND orderrelease, at least 90 workershave to be assigned to the orderpicking system.Therespective values of theperformance measures aresummarized in Table5. Thus,in the order pickingsystem of thenumericalexample, therequired workforce leveldecreasesby threeworkers(3.13%) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 1500 2500 3500 4500 5500 6500 7500 8500 9500 10500 11500 12500 13500 14500 Probability Number of incomingpicking ordersper scheduling interval 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 35 45 55 65 75 85 95 105115 125135 145155 165175 185195 205215225 235 Probability Individual picking performance per scheduling interval Fig. 11: Number of incoming picking orders per scheduling interval Fig. 12:Lead time of a picking order Fig. 13:Individualpicking performance perscheduling interval 18 release, system utilisationdecreaseswithincreasing number of assigned workers. Thus,by assigning more than therequiredminimum number of workersto the orderpicking system,system utilisationof the order picking system decreases. However, includingthis additional requirementregarding a maximumpossible system utilisationraises a conflictof interests:Since system utilisationis independentof theselected order releasestrategy,assigningadditional workersto the orderpicking system reducesthebenefitsof levelled orderreleasecompared to FCFS-based orderrelease strategies. 7. CONCLUSION ANDFUTURE DIRECTIONS In thispublication, we investigate theapproach of levelled orderreleasein orderpickingsystems. Based on theconceptof Heijunka-levelling in production systems, we derive alevelling conceptfor orderpicking systems: Thereis a fixed picking capacity per order type perscheduling intervalwhichis reserved for orderprocessingof picking orders of this ordertype in each scheduling interval. Size andsequence of the reserved picking capacities within one scheduling intervalarevisualisedin thelevelling pattern. During each scheduling interval, thereserved picking capacity per ordertype is used to processpicking orders of this order type accordingto ascendingduedates. To analysetheperformance of thelevellingconcept, we depict the orderpickingsystem with levelled order releaseas adiscrete-timeMarkov chainandwe derive severalperformance measures from itssteady-state releaseinsteadof FCFS-DD orderrelease or by 5.22% when usinglevelled orderreleaseinsteadof FCFSRAND order releaserespectively. In conclusion,forthenumericalexample, we find that the orderpickingsystem benefitsfrom levelled orderrelease: Compared to FCFS-based orderrelease strategies,levelled orderreleaseenables either a decrease in theworkforce levelrequired to guarantee acertainsystem performance or it enablesan increase in system performance for a givenworkforcelevel. Foroperationalplanning andcontrolof order pickingsystems, further performance measures besides servicelevels arerelevant.Forinstance,the system utilisationof the orderpickingsystem may not exceed 90%. Asshownin Figure 14 forlevelled order LEVELLING FCFS-DD FCFS-RAND Workforce level Numberofworkers c83.5000 86.2000 89.1000 Performancemeasures based on unprocessed orders Number of unprocessed picking orders Q12206.8671 10338.0943 9606.7460 Number of unprocessed backorders M127.5067 113.7588 121.2315 Number of lost sales S000 System utilisation ˜ U0.9426 0.9130 0.8845 Performancemeasures based on processed orders Number of processed picking orders F7879.7851 7892.9886 7893.6698 Number of processed backorders Fbacklog 102.9403 103.0287 113.7604 Number of processed picking orders without failed due date Fbuffer 7776.8447 7789.9599 7779.9094 Time difference to order deadline of aprocessed picking order D1.7239 1.9634 2.0560 Backlog duration of aprocessed backorder Dbacklog 1.2045 1.0989 1.0486 Time buffer of aprocessed pickingorder Dbuffer 2.0430 2.0039 2.1012 Service level β-service level SLβ98.77% 98.81% 98.70% γ-service level SLγ99.81% 99.84% 99.83% 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 70 75 80 85 90 95 100 System performance SLβ,SLγ,U Workforcelevel c SLβ SLγ U Tab. 5: Workforcelevel andperformancemeasures of the orderpickingsystem of thenumericalexample fordifferentorderreleasestrategies:levelled orderreleasestrategy LEVELLING and FCFS-based referenceorderreleasestrategies FCFS-DDandFCFS-RAND(average values of tenreplications) Fig. 14:Impact of workforcelevel c on several performancemeasures SLβ,SLγ, Ũ of theorder picking system with levelled order release in thenumericalexample 19 Discrete-Time Analysis of Levelled OrderReleaseandStaffingin OrderPickingSystems study. Other future directions concerntheextension of thelevelling conceptto thewholewarehouse. Levellingtheglobal workload of awarehouse can be a meaningful approach,if workerscan be flexibly assigned to differentprocessesof thewarehouse and switches between differentprocesses within one shiftarepossible. We mentioned flexibleworkforce planningandlevelled orderreleaseas twoappropriate solution approachesto face thecurrentrequirements in manual orderpickingsystems(cf. Figure 1) andonly focusedonlevelled orderreleasein this publication. Thecombinationof thesetwoapproaches wouldbe a meaningful further potential future field of research. ACKNOWLEDGMENTS Theauthors wish to thanktwoanonymousreferees fortheir many helpfulcomments whichledto amuch improved form of thepaper. This research is supportedby theresearch project “Smoothing andLevelling in OrderPickingSystems” (originaltitle: “GlättenundNivellierenin der Kommissionierung”) by theBundesministeriumfür Wirtschaft undEnergie(BMWi) (reference number 20509N). CONFLICT OF INTEREST Theauthors declarethat they have no conflictof interest. REFERENCES [1]N. Boysen, R. de Koster,andF. Weidinger. “Warehousing in thee-commerce era: A survey”. In:European Journal of OperationalResearch (2018).DOI:10.1016/ j.ejor.2018.08.023. [2]R. de Koster,T. Le-Duc,andK. J. Roodbergen. “Designandcontrolof warehouse orderpicking: Aliteraturereview”. In:European Journal of OperationalResearch 182.2 (2007),pp.481–501. DOI:10.1016/j.ejor.2006.07.009. [3]C. Huber. “Throughput Analysis of Manual Order PickingSystemswith Congestion Consideration”. PhDthesis.Karlsruhe: KarlsruheInstituteof Technology (KIT), 2011. [4]B. Rouwenhorstet al.“Warehouse design and control: Framework andliteraturereview”. In:European journalofoperationalresearch 122.3(2000), pp.515–533.DOI:10.1016/S0377- 2217(99)00020-X. [5]J. Gu, M. Goetschalckx, andL. F. McGinnis. “Researchonwarehouse operation: A comprehensivereview”. In:European Journal of OperationalResearch 177.1(2007),pp.1–21. DOI:0.1016/j.ejor.2006.02.025. distribution, such as thenumber of unprocessed and processedpicking orders,thenumber of lost sales,the time difference to order deadline, thesystem utilisation andtheservicelevels.Subsequently, we developa staffingalgorithmfor orderpickingsystemswith levelled orderreleaseas binary search based on the Markov chain. In contrastto themajority of staffing models in practice,ourstaffingalgorithmis notonly a rough estimation basedon averagedata andempirical knowledge, but it considersthestochastic variations of customer demand andpickingperformance. Furthermore, ourapproachestablishes a direct link between theperformanceof the orderpickingsystem andtherequired workforce level. Themodels developed in this publicationcanbe used fortwodifferent purposes:On theone hand, thediscretetime Markov chain exactlydetermines thesystem performance of a given order pickingsystem with levelled orderrelease. On the other hand, thestaffingalgorithmdetermines theworkforce levelwhichis required to guarantee a certainsystem performancein an order picking system with levelled orderrelease. Numericalstudiesshow that thevariabilityof incomingpicking orders has a negative impact on the system performance, whereastheexpected valueof the lead time of apicking orderhas a positive impact on thesystem performance. Theseeffectsincrease with increasing traffic intensity of thesystem.Furthermore, thenumericalstudiesshowthesmoothingeffect of the levelling concept: Thevariabilityof processedpicking orders is smallerthan thevariabilityof incoming picking orders,especiallyforsystemswith ahigh traffic intensity.Thestructureof theseeffectsis as expected. However, the models developedin this publication enable an exactandquantitative evaluation of these effects. Thus,we areable to quantify theimpact of variations in thesystem parameters, such as variability of incoming picking orders,workforce level or traffic intensity,ontheperformance measuresof the order pickingsystem. Thecomparison of levelled order releasewith FCFS- based orderreleasestrategies in anumericalexample showsthebenefitsof levelled orderrelease: Compared to FCFS-based orderreleasestrategies,levelled order releaseenables either adecrease in theworkforce level required to guarantee a certainsystem performance or it enablesan increase in system performance for a givenworkforce level. Further research canbe conductedongeneralising the modelsdescribed here. Different types of workers regardingperformance andwork time models canbe includedin thestaffingalgorithm, since thealgorithm actually assumes an identicalpicking performanceand full-timejobs foreach worker.Additionally,the models abstractfrom seasonal fluctuationsandauto correlation of thecustomer demand. Furthermore, to investigate thebenefitsof levelled orderrelease, theconceptof levelled orderreleaseshould be compared to alternative orderreleasestrategies in acomprehensivenumerical 20 [18] T. N. DhamalaandW. Kubiak.“A brief survey of just-in-time sequencing formixed-model systems”.In:InternationalJournal of Operations Research 2.2(2005), pp.38–47. [19] N. Boysen, M. Fliedner,andA. Scholl.“Level Scheduling forbatchedJITsupply”. In:Flexible Servicesand ManufacturingJournal21.1-2 (2009),pp.31–50. DOI:10.1007/s10696-009- 9058-z. [20] D.T. Jones,P. Hines,andN.Rich.“LeanLogistics”. In:InternationalJournalofPhysical Distribution &LogisticsManagement 27.3/4 (1997),pp.153– 173. DOI:10.1108/09600039710170557. [21] E. B. Sobanski.“Assessing Lean Warehousing: DevelopmentandValidation of AleanAssessment Tool”. PhDthesis. Oklahoma: Oklahoma State University,2009. [22] P. Dehdari. “Measuring theImpact of Lean Techniques on PerformanceIndicators in LogisticsOperations”. PhDthesis.Karlsruhe: KarlsruheInstituteof Technology(KIT), 2014. [23] A. T. Ernstet al.“Staff scheduling androstering: Areview of applications, methodsand models”. In:European Journal of OperationalResearch 153.1(2004),pp.3–27. [24] J. VandenBerghet al.“Personnel scheduling: Aliteraturereview”. In:European Journal of OperationalResearch 226.3(2013),pp.367–385. [25] M. Defraeye andI. VanNieuwenhuyse. “Staffing andscheduling undernonstationarydemand forservice: Aliteraturereview”. In:Omega58 (2016),pp.4–25. [26] S. S. Chakravorty.“Improving distribution operations:Implementationof material handling systems”.In:Internationaljournalofproduction economics122.1(2009),pp.89–106.DOI: 10.1016/j.ijpe.2008.12.026. [27] T. vanGilset al.“The useof timeseries forecasting in zone orderpickingsystemsto predict order pickers’ workload”. In:InternationalJournal of Production Research 55.21(2017),pp.6380– 6393.DOI:10.1080/00207543.2016.1216659. [28] M. Schleyer andK. Furmans. “Ananalytical method forthecalculationof thewaitingtime distributionof adiscrete time G/G/1-queueing system with batcharrivals”. In:OR Spectrum 29.4 (2007),pp.745–763. [29] G. Bolch. Queueing networksand Markov chains: Modelingand performanceevaluation with computer scienceapplications.2. ed.Hoboken, NJ:Wiley-Interscience,2006. [30] T. H. Cormen et al.Introduction to algorithms. 3. ed.Cambridge, Mass.: MITPress, 2009. [6] H. DavarzaniandA. Norrman. “Toward a relevant agenda forwarehousing research: literature review andpractitioners’input”. In: LogisticsResearch 8.1(2015), p. 78.DOI:10.1007/ s12159-014-0120-1. [7]Y. Gong andR. B. M. de Koster.“A review on stochastic models andanalysis of warehouse operations”. In:LogisticsResearch 3.4(2011),pp. 191–205. DOI:10.1007/s12159-011-0057-6. [8] C. G. Petersen. “An evaluation of orderpicking policies formail ordercompanies”.In:Production andOperations Management 9.4(2000), pp.319– 335. [9]J. GallienandT. Weber. “ToWave or Notto Wave?OrderReleasePolicies forWarehouses with an Automated Sorter”. In:Manufacturing & ServiceOperations Management 12.4 (2010), pp. 642–662. DOI:10.1287/ msom.1100.0291. [10] E. C¸ even andK. R. Gue. “Optimal Wave Release Times forOrderFulfillment Systemswith Deadlines”. In:Transportation Science51.1 (2017),pp.52–66. DOI:10.1287/trsc.2015.0642. [11] J. K. Liker. TheToyotaway: 14 management principlesfromtheworld’sgreatestmanufacturer. NewYork,NY:McGraw-Hill, 2004. [12] A. Smalleyet al.Creating level pull: A lean production-system improvementguide for production-control,operations,andengineering professionals ; a lean toolkit method and workbook.Version1.0, March2009.Cambridge, MA:Lean Enterprise Institute, 2004. [13] Y. Monden.Toyotaproduction system:An integrated approach to just-in-time.4. ed.Boca Raton, Fla. andNorcross, Ga.: CRCPressandIIE Inst.of Industrial Engineers, 2012. [14] K. Furmans. “Modelsof heijunka-levelled kanbansystems”.In:5thInternational Conferenceon Analysis of Manufacturing Systems-Production and Management.2005,pp. 243–248. [15]C. R. LippoltandK. Furmans. “Sizingof Heijunka-controlledProduction Systemswith Unreliable Production Processes”.In:Lean Business Systems and Beyond.Ed.by T. Koch. Boston, MA:International Federation for InformationProcessing, 2008,pp.11–20. [16] J. Matzka, M. Di Mascolo, andK. Furmans. “Buffersizing of aHeijunka Kanban system”. In: Journal of IntelligentManufacturing23.1 (2012), pp.49–60. DOI:10.1007/s10845-009-0317-3. [17] K. Furmansand M. Veit.“Modelsof Leveling for Lean ManufacturingSystems”.In:Handbook of stochastic modelsandanalysis of manufacturing system operations.Ed.by J. M. SmithandB. Tan. NewYork,NY:Springer,2013,pp.115–138.