Dynamic Drum-Buffer-Rope approach for production planningand control in capacitated ow-shop manufacturing systemsqPatroklos Georgiadis, Alexandra PolitouIndustrial Management Division, Department of Mechanical Engineering, Aristotle University of Thessaloniki, P.O. Box 461, 541 24 Thessaloniki, Greecearti cle i nfoArticle history:Received 28 March 2012Received in revised form 22 February 2013Accepted 23 April 2013Available online 3 May 2013Keywords:Drum-Buffer-RopeTime-bufferFlow-shopProduction planning and controlSimulationSystem dynamicsabstractDrum-Buffer-Rope-based production planning and control (PPC) approaches provide productionmanagers with effective tools to manage production disruptions and improve operational performance.Thecornerstoneoftheseapproachesistheproperselectionoftime-bufferswhichareconsideredasexogenously dened constant. However, the majority of real-world manufacturing systems arecharacterized by the dynamic change of demand and by stochastic production times. This fact calls fora dynamic approach in supporting the decision making on time-buffer policies. To this end, we study acapacitated, single-product, three-operation, ow-shopmanufacturingsystem. Weproposeadynamictime-buffer control mechanism for short/medium-term PPC with adaptive response to demand changesand robustness to sudden disturbances in both internal and external shop environment. By integratingthecontrol mechanismintotheow-shopsystem, wedevelopasystemdynamicsmodel tosupportthedecision-makingontime-bufferpolicies. Usingthemodel, westudytheeffectof policiesonshopperformance by means of analysis of variance. Extensive numerical investigation reveals the insensitivityof time-buffer policies to key factors related to demand, demand due date and operational characteristicssuch as protective capacity and production times. 2013 Elsevier Ltd. All rights reserved.1. IntroductionInsufcient productionplanning in manufacturing systems oftenturnsanon-bottleneckresourcetocapacityconstraint resource(CCR), whichoperates as a bottleneckwithonaverage excess capacity(Goldratt, 1988). Drum-Buffer-Rope (DBR)-based productionplanning andcontrol (PPC) approaches focus onthe synchronizationof resources andmaterial utilizationinCCRs of manufacturingsystems (Goldratt &Fox, 1986; Sivasubramanian, Selladurai, &Rajamramasamy, 2000). This synchronization calls for time-buffersthat protect the production plans of CCR fromthe effects ofdisruptionsat theprecedingproductionresources. Bymeansoftime-buffers (i.e. constraint, assembly, shipping time-buffers),buffermanagementmonitorstheinventoryinfrontofprotectedresources to effectively manage and improve systems performance(Schragenheim & Ronen, 1990; Schragenheim & Ronen, 1991).The research agenda on the efciency of DBR approach in PPC ofmanufacturingsystemshasreceivedincreasedattentionduringthe last decade. The basic assumption in all relative studies is theexogenous determination of time-buffers as a constant throughoutthe planning horizon. However, the majority of real-worldmanufacturing systems are characterized by the dynamic changeof demandandbystochastic productiontimes. Therefore, thedecision making on time-buffer policies calls for a dynamicmechanism. This is exactly the purpose of this paper. Morespecically, weconsideradynamic, capacitated, single-product,three-operation, ow-shop production system. We dene asproduction time-buffer (PTB), the total of constraint and shippingtime-buffers. We propose a dynamic, goal-seeking, feedbackmechanism to dene PTB for short/medium-term PPC. By integratingthe proposed mechanism into the ow-shop system, we develop asystemdynamics(SD)modeltosupportthedecisionmakingonPTBpolicies. Westudytheshopresponse(dynamicsof productows, inventories, performancemeasures)toPTBpoliciesunderstochastic demand and production times. Since the dynamicbehavior may be used to evaluate the efciency of a specic PTBpolicy, the SD model can be viewed as a decision support system(DSS) for PTB-related decisions. In particular, by continuousmonitoring, the actual level of PTB is adjusted to demand-drivendesired values. The innovative element of the control mechanismistheendogenousdenitionof desiredPTBvalues. Inaddition,the mechanism provides robustness to sudden disturbance occur-rences in demand and shop operations. This is a positive propertyto cope with uncertainty issues in both external and internal shopenvironment. UsingtheSDmodel, wedeterminePTBincrease/decreasepolicies throughout agivenplanninghorizonandwestudytheireffectonshopperformancebymeansof analysisof0360-8352/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.cie.2013.04.013qThis manuscript was processed by Area Editor Manoj Tiwari.Corresponding author. Tel.: +30 2310 996046; fax: +30 2310 996018.E-mail addresses: geopat@eng.auth.gr, geopat@auth.gr (P. Georgiadis), apolitou@auth.gr (A. Politou).Computers & Industrial Engineering 65 (2013) 689703ContentslistsavailableatSciVerseScienceDirectComputers & Industrial Engineeringj our nal homepage: www. el sevi er. com/ l ocat e/ cai evariance (ANOVA). The examination of results obtained byextensive numerical investigation reveals the insensitivity of PTBpoliciestokeyfactorsrelatedtodemand, demandduedateandoperational characteristics such as protective capacity and produc-tion times. This is an additional appealing feature of the proposedPTB control mechanism which provides production managers withexibility on PTB-related decisions.The rest of the paper is organized as follows. Section 2presents the literature review on DBR studies and applications inmanufacturing systems and justies the suitability of SDmethodology in developing dynamic DBR-based PPC systems.Section 3 contains the ow-shop systemunder study and its perfor-mance measures, the description of the SDmodel, the mathematicalformulation and the models validation. Section 4 presents the con-trol parameters under study along with their sets of values, whileSection5presentstheadaptabilityandrobustnesspropertiesofthe dynamic PTB control mechanism. The effect of PTB policies onthe shops performance obtained by numerical investigation is givenin Section 6. Finally, in Section 7 we wrap-up with a summary, thelimitations of our work and directions for model extensions.2. Literature reviewReview papers present a variety of PPC problems dealing withow-shop scheduling in manufacturing systems; either withsequence-independent set-up times or sequence-dependentset-up times (Hejazi and Saghaan, 2005; Zhu & Wilhelm, 2006).The proposed scheduling methods include: (i) exact methods suchas dynamic programming (Held & Karp, 1962), branch-and-bound(Grabowski, Skubalska, & Smutnicki, 1983), integer programming(Frieze & Yadegar, 1989) and complete enumeration; (ii) heuristicmethods such as Palmer algorithm (Palmer, 1965), Guptaalgorithm(Gupta, 1971), CDS algorithm(Campbell, Dudek, &Smith, 1970) and NEH algorithm (Nawaz, Enscore, & Ham, 1983);and(iii)metaheuristicmethodssuchassimulatedannealing-SA(Liu, 1999), genetic algorithms-GA (Reeves, 1995), tabu search-TA(Widmer & Hertz, 1989), greedy approaches (Carpov, Carlier, Nace,& Sirdey,2012),variable-depth search approach (Jin,Yang,& Ito,2006), pilot methods (Vob, Fink, &Duin, 2005), hill climbingprocedures(Nearchou, 2004), antcolonysystem-ACS(Rajendran&Ziegler, 2004), articial neural network-ANN(Lee &Shaw,2000) and hybrid algorithms (Wang & Zheng, 2003).For the specic make-to-order ow-shop environment, Stevenson,Hendry, andKingsman(2005) provideadetailedreviewontheemployed PPC approaches. The commonly used approaches includeConstant Work In Process-CONWIP (Framinan, Gonzlez, &Ruiz-Usano, 2003), Workload Control-WLC (Th} urer, Stevenson,&Silva, 2011), Material RequirementPlanning-MRP(Bertrand&Muntslag, 1993), Just-in-Time-JIT (Singh & Brar, 1992), Theory ofConstraints-TOC (Atwater, Stephens, & Chakravorty, 2004; Goldratt& Cox, 1984; Mabin & Balderstone, 2003), Paired cell OverlappingLoopsof Cards withAuthorization-POLCA(Riezebos, 2010) andweb- or e-based Supply Chain Management-SCM (Cagliano, Caniato,& Spina, 2003; Kehoe & Boughton, 2001). The comparison of MRP,TOC and JIT approaches justies the TOC to be more effective for apure ow shop or general ow shop system, when the bottleneckresources are stationary positionedinthe productionprocess.The effectiveness of TOC approaches is further discussed for highlycustomized industries facing difculties in estimating in advancethe processing times (Stevenson et al., 2005). This is due to the factthat TOC requires dataaccuracy only in CCR tocontrol theplantthroughput (Gupta & Snyder, 2009).TOCwas rst developedinthemid-1980s (Goldratt &Cox,1984; Gupta, 2003). It uses the DBR production schedulingapproach;productionprocessisscheduledtoruninaccordancewith the needs of the CCR, as CCR determines the performance ofthe whole production system. The advantages of TOC are discussedinvariousindustrialimplementationsreportingthereductionofinventories by 49% and the improvement of due date and nancialperformance by 60% (Gupta, 2003; Mabin & Balderstone,2003). Thestatisticalanalysisofasurveywithquestionnairestomanufacturing managers performing TOC, JIT and traditionalmethods provides further insights regardingthe superiorityofTOCapproachesonother approachesintermsof nancial andoperational performance measures (Sale & Inman, 2003). TOC hasalso been used for the determination of optimal, or near optimal,product mix decisions (Aryanezhad &Komijan, 2004; Souren,Ahn, & Schmitz, 2005).In certain studies, the usefulness of DBR logic in PPC ofmanufacturing systems is revealed by conducted simulation exper-imentsunder different manufacturingsettings. Inthesestudiestimebuffersarenotoptimizedandremainconstantthroughoutthe simulation process. In a make-to-stock environment,DBR-based PPC is combined to manufacturing expediting ofproducts (Schragenheim, Cox, & Ronen, 1994). In a make-to orderenvironment, DBR approach is combined with different orderreview/releasepolicies(Russell &Fry, 1997)anditiscomparedto the previously used approach in furniture manufacturing rms(Wu, Morris, &Gordon, 1994). Inaserial productionlinewithexponentially distributed processing times setting, DBR-basedPPC is compared to CONWIP approach (Gilland, 2002). The differ-encebetweenthetwoapproachesisthat inCONWIPapproachmaterial unitsarereleasedintothelineat arateequal tolinethroughput, while in DBR approach at the rate they are producedat CCR. The outperformance of DBR approach is proved to increaseas CCR moves closer to the rst operation of production as well aswhentherequiredthroughput or service level is closetothesystemscapacity(Framinanetal., 2003). Finally, inaow-shopsetting, Sirikri andYenradee(2006)employDBR-basedPPCandinvestigate how buffer sizes related to lead time up to CCR affectspecic performance measures.The analytical approaches to determine time-buffer sizes basedon queuing theory are limited in simple PPC manufacturingproblems. In these approaches the constraint resource is modeledasaM/M/1/Ksystem(Radovilsky, 1998), whiletheproductionsystem is modeled either as a determination model, where a treestructure represents the relationship between the constraintmachine and its feeder machines (Tu & Li, 1998; Ye & Han, 2008)or as multiproduct open queuing network in which the productionoperations are modeled as GI/G/m (Louw & Page, 2004).The applicability of DBR in real world case studies is denoted bya lot of DBRimplementations inmanufacturing rms; e.g. inOrko-PakinNetherlands that manufactures packagingmaterialfromcorrugatedcardboard(Riezebos, Korte, &Land, 2003), inOregon Freeze Dry processing products by removing water at lowtemperature and pressure (Umble, Umble, & von Deylen, 2001), inAlamedaNaval AviationDepot that remanufacturesaircraft, jetturbine engines, engine components and avionics equipment(Guide & Ghishelli, 1995), in a light assembly rm for heavy dutytrucks andtrailers (Pegels &Watrous, 2005) andinabearingmanufacturing company (Steele, Philipoom, Malhotra, & Fry,2005). DBRisevenusedinaghtersquadronof theIsraeli AirForceforbetterschedulingof itsmissionsandallocatingcrewsto aircraft (Ronen, Gur, & Pass, 1994).DBR literature suggests several performance measures inevaluating the efciency of the proposed approaches. Thesemeasures include: theaveragesystemthroughput, theaveragenished product inventory, the average number of stockouts(Duclos & Spencer, 1995); the throughput, the utilization ofmachines, the average wait time of items, the percentage ofmachineblocking(Mahapatra&Sahu, 2006); themeanpercent690 P. Georgiadis, A. Politou/ Computers & Industrial Engineering 65 (2013) 689703completionto schedule, the meanwork-in-process (WIP), themeanthroughput rate, theproductmeantimeinsystem (Guide,1996); the total system output, the average and standard deviationof ow time (Cook, 1994); and the average wait time of items, theaverage WIP in queue and the system throughput (Betterton & Cox,2009).In the above-mentioned DBR studies, time-buffer is consideredas an exogenously dened constant throughout the planninghorizon. Further limitation is the inability to handle the non-linear,non-stationaryanduncertainnatureof productionprocessthatcharacterizesthemajorityof ow-shopmanufacturingsystems.Theselimitationscall foradynamictime-buffercontrol mecha-nism, suitable for monitoring and adjusting time-buffer values todesired levels. Such a control mechanism can be provided by theusage of SD methodology. Therefore, SD is the primary modelingandanalysistoolusedinthispaper. Forrester(1961)introducedSDintheearly1960sasamodelingandcontinuoussimulationmethodologyfordecision-makingincomplexdynamicindustrialmanagement problems (GrBler, Thun, & Milling, 2008). Incontrast to the traditional discrete event simulation-basedDSS, the methodology provides an understanding of changesoccurringwithinamanufacturingenvironment, byfocusingonthe interaction between physical ows, information ows, delaysandpoliciesthat createthedynamics ofthevariables ofinterestandthereafter searches for policies toimprovesystemperfor-mance (Georgiadis &Michaloudis, 2012; Sterman, 2000). Thestructureof aSDmodel isdescribedbystocksandows. Thisstructure provides the PPC with a capability to capture thedynamics of material, product and information ows under causaleffectsoriginatedfromtheinternal andexternal shopenviron-ment. The SD discipline acknowledges at the outset that realisticrepresentations may include non-linear elements,so closed-formsolutions are bypassed in favor of a simulation methodology.Although the rich body of SD studies in PPC issues, DBR-basedapproaches are very limited. In particular, these studies introducethe potential use of SD theory along with the expected advantages.WixsonandMills (2003) present anumerical exampleof DBRproduction process and show how SD may help in understandingthesystemconstraints. Theyconsiderthetime-bufferofDBRasanexternal parameter and assume constant productiontimesand innite raw materials. The potential use of SD in developingDBR-based PPC systems for a make-to-order, ow-shop system ispresented by Politou and Georgiadis (2008). They assumeexponentiallydistributedproductiontimes, niterawmaterialsandconstant PTBthroughout theplanninghorizon. This papertakes the last research further by developing an endogenouscontrol mechanismforPTBandintegratingintoaSD-basedPPCsystemforow-shopoperations. Thenewpossibilitiesprovidedby the proposed SD model can be summarized in its ability to copewiththechallengesfordynamicDBRapproachesinastochasticow-shop environment. This ability contributes to the PPCfunctionof ow-shopmanufacturingsystems providingstable,controllableandadaptiveproductionplanswhichdeal withthenon-linear, non-stationary and uncertain nature of productionprocesses.3. The SD model3.1. The ow-shop system under studyWe consider a three-operation, capacitated ow-shop thatproduces a single product and purchases one type of raw material(referredasmaterialattheremainderofpaper). TheCCRoftheow-shopliesinitssecondoperationandproductionratesaredened by DBR approach. The demand follows a normaldistribution. Eachoperationof theow-shopisconsideredasaqueueing model M/M/1, in which product arrival is described by aPoisson process with the parameter k being equal to mean demand.The capacities of the three operations are described by a PoissonprocessandtheirmeanvaluesaresetequaltoCapi_M(i = 1, 2, 3).The mean value of the production time at each operation (1/li) isdened in Eq. (1) by means of the respective mean capacity:1li1Capi M1Therefore, production times followexponential distributionwithparameters liequaltoorgreater thank. Consequently, theshop operation is considered as a series of three queuing modelsM/M/1(Hillier &Lieberman, 1995). Thus, themeanproductiontime (MPT) which is the mean value of the total production timeof shops operations is given in Eq. (2):MPT X3i11liX3i11Capi M2Thenotationandtherespectiveunits of measure are giveninTable 1.Table 1Notation list and units of measure.Flow-shop variablesCapiCapacity of the i operation (i = 1, 2, 3), items/dayCapi_MMean value of Capi (i = 1, 2, 3), items/dayCapCCRCapacity of the CCR operation, which is equal to Cap2, items/dayCapCCR,mMean of CapCCR, items/dayD Demand, items/dayDmMean value of D, items/dayDSDStandard Deviation of D, items/dayDB Demand Backlog, itemsDBDR Demand Backlog Decrease Rate, items/dayDBIR Demand Backlog Increase Rate, items/dayDDD Demand Due Date, daysED Expected Demand, items/dayFPI Finished Product Inventory, itemsMCR Material Consumption Rate, kg/dayMF Material Factor (i.e. for production of one item, MF kg of rawmaterial are required), kg/itemMFO Material For Order, kgMFODR Material For Order Decrease Rate, kg/dayMFOIR Material For Order Increase Rate, kg/dayMI Material Inventory, kgMLT Material Lead Time, daysMO Material Order, kgMOR Material Order Rope, items/dayMP Material Procurement, kgMPR Material Procurement Rate, kg/dayMPT Mean Production Time, daysMRR Material Release Rate, kg/dayMRT Material Release Time, daysMUR Material Usage Rate, kg/dayOB Orders Backlog, itemsOR Order Release backlog, itemsORR Order Release Rate, items/dayORT Orders Rate, items/dayPDF Planned Demand Fulllment, itemsPORR Planned Order Release Rate, items/dayPRi Production Rate of the i operation (i = 1, 2, 3), items/dayPTB Production Time-Buffer, daysr Delay time used in computation of MOR, daysSR Shipments Rate, items/dayWIP0 MI that has been released in production process and waits to beprocessed at rst operation, kgWIPi Work-In-Process inventory at the i operation (i = 1, 2), items1/c Smoothing factor used in computation of ED, 1/days1/liMean value of production time of the i operation (i = 1, 2, 3),days/itemP. Georgiadis, A. Politou/ Computers & Industrial Engineering 65 (2013) 689703 6913.2. Performance measuresThe efciency of PTB policies in association with themanufacturing process is obtained at the end of a given planninghorizon using performance measures suggested by the DBRliterature. Besides, the efciency of PTB policies in association withtheevaluationprocess is obtainedat theendof theplanninghorizon using two performance measures related to PTB. All thesemeasures are shown in Table 2.3.3. Conceptual modelingIn SD discipline, causal-loop diagrams are maps of the systemsunder study showing the causal links among the incorporated vari-ables (Sterman, 2000). Thegeneric causal-loopdiagramof thedeveloped SD model is depicted in Fig. 1.ThecontrolmechanismsshowninFig. 1arepresentedinthefollowing subsections. To improve appearance and distinctionamong the variables in the causal-loop diagrams, we changed theletter style according to the variable style; stock (state) variablesare written in capital letters,ow variables in small plain lettersand auxiliary variables in small italic letters. Stocks integrate theirows, characterizethestateofthesystem, givesystemsinertiaand provide it with memory. The arrows (inuence lines) representthe relations among variables. The direction of the inuence linesdisplaysthedirectionoftheeffect. Thesign+ or oneachinuencelineexhibits thesignof theeffect. A+ () signsignies that the variables change in the same (opposite) direction.3.3.1. Material release rate control mechanismFig. 2 depicts the material release rate control mechanism (forthenotationthereadermustrefertoTable1). Sincethecornerstoneof DBRsystems is thesynchronizationof resources andmaterial utilization, thematerial releasescheduleisthedrivingforce for the production planning. Material release (MaterialReleaseRate, MRR)isbasedonorderrelease(OrderReleaseRate,ORR). Morespecically, theplanof orderrelease(PlannedOrderRelease Rate, PORR) is set by means of a pipeline delay of Demand(D) withadurationequal toMaterial ReleaseTime(MRT). PORRincreases ORDER RELEASE (OR) backlog, which is depleted by ORR.ORRislimitedbyMATERIALINVENTORY(MI). MRTisdenedbymeansof DemandDueDate(DDD), MeanProductionTime(MPT),DandPTB. The denitionof MRT is basedonthe schedulinglogic of backward innite loading (Park & Bobrowski, 1989;Sabuncuoglu&Karapinar, 1999); theMRTvalueispredictedbyback scheduling fromDDD by means ofMPT andPTB, in orderDto be fullled on time. The related equations are given in AppendixA (Eqs. (A.1)(A.6)).3.3.2. Material procurement control mechanismInFig. 3, theLoop1describestheDBRlogicforthematerialprocurement control mechanism. More specically, Material OrderRope (MOR) is theropeof theDBRlogic andit is denedbyassumingthat thematerial inventoryismonitoredfororderattherateatwhichmaterial isreleasedintheproductionprocessof theshop. Material For Order Increase Rate(MFOIR) increasesMATERIALFORORDER(MFO), whichisdepletedbyMaterial ForOrderDecreaseRate(MFODR). Material Order(MO) isdenedbymeans of MFO. Material Procurement (MP) is the pipeline delay ofMaterial Order(MO) withdurationequalstoMaterial LeadTime(MLT). Material Procurement Rate (MPR) increases MATERIALINVENTORY (MI), which is depleted by Material Usage Rate (MUR).TheMaterialprocurementcontrolmechanismreferstoareviewsystem applying the DBR logic. In particular, this system is a peri-odic order quantity review system with probabilistic demand andvariableorder quantitythat equals MFO. Theorder quantityisbasedonalot-for-lot approach(Silver, Pyke, &Peterson, 1998;Steele et al., 2005); it is equal to the material consumption of theTable 2Performance measures.Performance measures of manufacturing processARM Average value of material inventory, kgAWIP1 Average value of WIP1, itemsAWIP2 Average value of WIP2, itemsAFP Average value of nished product inventory, itemsADB Average value of demand backlog, itemsDBD Demand Backlog Delay; i.e. total time duration of demand backlogoccurrence, daysALT Average lead time, daysAPR Average value of CCR production rate, items/dayPI Production Index measuring the efciency of DBR-based PPCapproach; i.e. the average value of the ratio of actual CCR productionrate values over the magnitude assuming innite inventory WIP1,dimensionlessPerformance measures of PTB evaluation processAPTB Average value of PTB, daysPTB PTB at the end of the simulation process, daysFig. 1. Generic causal-loop diagram of the SD model.692 P. Georgiadis, A. Politou/ Computers & Industrial Engineering 65 (2013) 689703previousperiod. TherelatedequationsaregiveninAppendixA(Eqs. (A.7)(A.15)).3.3.3. PTB control mechanismThe PTB control mechanismis illustrated in Fig. 4. Morespecically, bymonitoringPTBvaluesadecisioniscontinuouslymade whether or not to increase or decrease its level and to whatextent. The values of PTB increase (PTB Increase) and decrease (PTBDecrease)dependonthediscrepancy(PTBDiscrepancy)betweentheDesiredPTBandtheactuallevelofPTB. DesiredPTBisbasedon DEMAND BACKLOG (DB), DDD, MPT and Expected Demand (ED),whichisaforecastedvaluebasedonthetimeseriesof D. Themagnitude of each increase or decrease is proportional to the PTBDiscrepancyat thespecictime. Specically, PTBDiscrepancyismultipliedbyparameters K1for increaseandK2for decrease,which characterize alternative PTB planning policies. Values of K1or K2 equal to1, inparticular, refer to PTBplanningpoliciescharacterizedbymatching-timeresponsiveness. Insuchpolicies,practically, PTB reaches Desired PTB in one time unit. Values of K1orK2greaterthan1refertoPTBplanningpoliciescharacterizedby high-time responsiveness. In such policies, PTB reaches DesiredPTB in less than one time unit. Values of K1 or K2 smallerthan1refertoPTBplanningpoliciescharacterizedbylow-timeresponsiveness(i.e. PTBreachesDesiredPTBinmoretimeunits).PTB control mechanismis based on the stock managementstructure suggested by Sterman (1989). Because of its central rolein the model, the related equations are given in Section 3.4.3.3.4. Flow-shop production control mechanismTheow-shopproductioncontrol mechanismis depictedinFig. 5. ProductionRate1(PR1)iscontrolledbyCapacity1(Cap1),WIP0 and Material Factor (MF), whereas Production Rate 2 (PR2) islimitedbyCapacity2(Cap2)andWIP1. ProductionRate3(PR3)islimitedbyCapacity3(Cap3) andWIP2. PR3increases FINISHEDPRODUCTINVENTORY(FPI). Thecontrol mechanismisbasedonlimitation functions considering the capacity and inventoryFig. 2. Causal-loop diagram of the material release rate control mechanism.Fig. 3. Causal-loop diagram of the material procurement control mechanism.Fig. 4. Causal-loop diagram of the PTB control mechanism.P. Georgiadis, A. Politou/ Computers & Industrial Engineering 65 (2013) 689703 693constraints (Georgiadis & Michaloudis, 2012). The relatedequations are given in Appendix A (Eqs. (A.16)(A.22)).3.3.5. Shipments rate control mechanismTheshipmentsratecontrol mechanismisdepictedinFig. 6.Loop 2 (DB Increase Rate (DBIR), DB, Shipments Rate (SR), DB IncreaseRate(DBIR)) controlstheSRof thedemandfulllment process.PlannedDemandFulllment(PDF)isthepipelinedelayofDwithduration equals toDDD. DBIR increasesDB, which isdepleted byDBDecreaseRate(DBDR). IncaseofproductSRislessthanPDF,DBIR gets a positive value. However, in case of delayed fulllmentof D, SR is greater than PDF and thus DBDR gets a positive value. SRdecreases ORDERBACKLOG(OB), whichis increasedbyD. Thecontrol mechanism is based on the assumption that all the demandissatised, evenwithdelay. TherelatedequationsaregiveninAppendix A (Eqs. (A.23)(A.29)).3.4. Mathematical formulationIn SD discipline the development of the mathematical model isusually presented as a stock-ow diagram that captures the modelstructure and the interrelationships among the variables (Sterman,2000). The stock-ow diagram is translated to a system of differen-tial equations, whichis thensolvedvia simulation. High-levelgraphical simulation programs support such an analysis. Theembeddedmathematical equations are dividedinto two maincategories: the stock (state) equations, relating the accumulationswithinthesystemofthenetowrates, andtherateequations,deningtheowsamongthestocksasfunctionsoftime. InSDmodels, the stock and ow perspective represents time asunfoldingcontinuously;eventscanhappenatanytime;changecan occur continuously. The general mathematical representationof stocks and ows is given by the following equations:Stockt Ztt0Inflowt Outflowtdt Stockt0 3Inflowt f Stockt; Et; P; Outflowt gStockt; Et; P 4where E(t) any exogenous variable and P system parameters.Thegenericstock-ow diagramofthedevelopedSDmodelisgiven in Fig. 7. The SD model consists of two modules: ow-shopmain module; and performance evaluation module which providesthe performance dynamics.Belowweprovidethemathematical formulationforthePTBcontrol mechanism:PTBt Ztt0PTB Increaset PTB Decreasetdt PTBt0;PTBt 0 DDD25EDt EDt dt 1c Dt EDt dt 6Desired PTBt minDDD; MPT EDtdt DBt 7PTB Increaset maxK1 PTB Discrepancyt; 0=dt 8PTB Decreaset maxK2 PTB Discrepancyt; 0=dt 9PTB Discrepancyt Desired PTBt PTBt 10By Eq. (6), ED is a rst-order exponential smoothing of D at time t,with smoothing factor 1/c. Desired PTB is dened in Eq. (7) by meansofDDD, MPT, EDandDB. PTBincrease rate (PTBIncrease)andPTBdecrease rate (PTB Decrease) are the increase and decrease decisionsper day. ByEq. (8) (byEq. (9)), PTBIncrease (PTBDecrease) isFig. 5. Causal-loop diagram of the ow-shop production control mechanism.Fig. 6. Causal-loop diagram of the shipments rate control mechanism.694 P. Georgiadis, A. Politou/ Computers & Industrial Engineering 65 (2013) 689703proportional to the positive (negative) part of PTB Discrepancybetweenthedesiredandactual PTB, multipliedbyK1(K2). Themagnitude of PTB Discrepancy is given by Eq. (10).Theequationsfor therest of control mechanisms showninFig. 1 are given in Appendix A. The appendix provides also the per-formancemeasures. PerformancemeasuresgiveninEqs. (A.30)(A.35) are based on common measures suggested by the literature(Betterton&Cox, 2009; Cook, 1994; Duclos &Spencer, 1995;Guide, 1996). DBD(DemandBacklog Delay) measures the timeduration of DB occurrence (Eq. (A.36)). ALT (Average Lead Time) isdeneddividingOB(Orders Backlog) byASR(AverageShipmentsRate) (Eq. (A.37)). PI (Production Index) is the average value of theratioof theCCRproductionrateoveritsrespectiveupperlimit.The latter is obtained by assuming the inventory WIP1 as innitive(Eq. (A.38)). Therefore, PI is a measure of how efciently the WIP1is managed in order to maximize the CCR production rate. Finally,APTB (Average PTB) is the average value of PTB (Eq. (A.39)) and PTBrefers to the value of PTB at the end of the simulation process (Eq.(A.40)).The entire mathematical model is a non-linear model of 13 statevariables, 21 ow variables and a considerable number of auxiliaryvariables and constants (2 array auxiliary variables, 44 scalarvariables and 13parameters). The model is developed inthesimulation software Powersim2.5c.3.5. Model validationTo build condence in themodel and to checkits quality, weusedtests suggestedby the SDliterature (Sterman, 2000). Inparticular, we tested that every equation of the model isdimensional consistent. Besides, we conducted extreme-conditiontests checking whether the model behaves realistically even underextremepolicies. For instance, we checkedthat if there is nodemandforproducts(D = 0), PR1, PR2, PR3, ED, DBandDesiredPTB equal zero; if there is no available capacity for production inthe rst operation of the shop (Cap1_M = 0), PR1, PR2, PR3equal zero, DBequalsthetotal amount of demandbackloggedthroughout the planning horizon and Desired PTB equals itsmaximum possible value (i.e. equals DDD). Integration error testswere subsequently conducted. In our model we used the Euler nu-mericmethodsincetheintegrationmethodRungeKuttashouldbe avoided in models with random disturbances such as this one(demand is not constant). We choose a simulation horizon of 300working days (the rst 50 days are considered as transient period)to be able to analyze short/medium-term decisions. Moreover, weset the integrating time step (dt) initially at 0.25 days, signicantlyshorter than the shortest value of the models time constants andranthemodel. Thenwecut thedt inhalf andranthemodelagain. The results did not signicantly change, so we chosedt = 0.125 days (=1 working hour).4. Control parameters and sets of valuesA complete numerical investigation of the SD models responsetoPTBpolicieswouldrequirethesystematicstudyof problemswithvarious levels of thesystemparameters. Sucha detailedexperimental design is practically impossible because of the largenumber of model parameters (in total 13). Consequently, weconcentrate on two parameters controlling the demand, oneparameter controlling the demand due date, two operationalparameterscontrollingtheshopandtwoparameterscontrollingthe PTB. These control parameters are presented in the followingsubsections. The rest of system parameters are shown in Table 3.Fig. 7. Generic stock-ow diagram of the SD model.P. Georgiadis, A. Politou/ Computers & Industrial Engineering 65 (2013) 689703 6954.1. Demand control parametersUsingtheparametersaandb, weconnectthemeanvalueofdemand(Dm) tothemeanvalueof CCRcapacity(CapCCR,m) andthe standard deviation of demand (DSD), as follows:Dm a CapCCR;m11DSD b Dm12The parameter b stands for the coefcient variance of demand. Theparameters a and b are examined in two levels; 0.9, 0.98 for a and 0,0.2 for b.4.2. Demand due date control parameterThe demand due date control parameter is the DDD. It isexamined intwolevels;2and4. ThesevaluesofDDDarebasedonmodel runningwithCap2_Mequals 10items/day, ProtectiveCapacityequals0.1andCapacitySwitchequals1. Forthiscase,bymeans of (2), themeanvalueof thetotal productiontime(MPT) equals 0.28 days/item. Additionally, if a = 0.9 (a = 0.98)meaning that the mean value of daily demand is 9 (9.8)items/order, MPT equals 2.52 (2.72) days/order.4.3. Flow-shop control parametersTheow-shopcontrol parametersaretheprotectivecapacityand the capacity switch. Protective Capacity (Atwater et al.,2004; Betterton & Cox, 2009) connects the mean values ofproductiontimeinCCRoperationandrest shopoperations, asfollows:Protective CapacityMean of production time in CCR Mean of production time in no CCR Mean of production time in CCRThe Protective Capacity is examined in two levels; 0.1 and 0.3.Capacity Switch may take the values of 0 and 1; 0 denotes thatall theproductiontimes of theshoparekept constant duringsimulationrun, whereas1denotesthatproductiontimesfollowexponential distributions.4.4. Production time-buffer control parametersThedecisionparametersthatfullydescribethePTBplanningpoliciesarethecontrol parametersK1andK2(seeEqs. (8)and(9)). Recall fromSection 3.3.3 that low-time responsive PTBpoliciesrefertovaluesofK1 < 1. Thesepoliciesleadtodelayingthe starting of production process (see Eq. (A.4) of MRT denitioninAppendixA) andconsequentlytolowerinventories. Forthisreason, the parameter K1 is examined in a range from 0 to 1. Withregard to K2, the parameter is examined in a range from 0 to 0.25.The shorter range of K2 compared to that of K1 is explained by thefact that we wish to be more conservative in reducing the values ofPTB in order to keep low the possibility of DB occurrence.5. Dynamics of PTB control mechanism and propertiesBy assigning specic values to the control parameters andrunningthemodel, weobtainthedynamicsofstocksandowsthroughouttheplanninghorizon. Weconsiderasthebasecasethe following set of control parameters: a = 0.9, b = 0.2, protectivecapacity = 0.1, capacity switch = 1. For the base case, the dynamicsof PTB control mechanism for PTB control parameters K1 = 0.2 andK2 = 0.15andDDD = 2 days are showninFig. 8a. The case ofDDD = 4 daysisshowninFig. 9a. Figs. 8aand9aillustratethedecisionstoincreaseordecreasePTBvaluesonadailybasis, forthe given set of PTB control parameters. Figs. 8b and 9b illustratethe dynamics of PTB under different K1 and K2 values.5.1. Transient response and dynamic equilibriumFig. 10 (Fig. 11a) illustrates the response of actual level of PTBfor three different sets of PTB control parameters K1 and K2and DDD = 2 days (DDD = 4 days). We observe that in case ofDDD = 2 days, the actual level of PTB does not reach anyequilibriumevenwhenthesimulationhorizonis doubled(i.e.600 days). This is explained by the fact that for a = 0.9, MPT equals2.52 days/order (see Section 4.2). Given that DDD = 2 days, there isnot enough time for the production to be completed on time.However, in case of DDD = 4 days, there is enough time for theproductiontobecompletedontimeandtheactuallevel ofPTBreachesadynamicequilibrium; i.e. thetotal increaseof PTBisbalanced by its total decrease throughout the simulation horizon.More specically, PTB reaches a dynamic equilibrium towards thevalueof0.34 dayswithdifferenttransientperiods. Thedifferenttransient periods are shown more clearly in Fig. 11b.Table 3System parameters remaining constant throughout the simulation process.Parameter Value Unitc 3 daysCapCCR,m10 items/dayInitial value of MI 120 kgMF 2 kg/itemMLT 3 daysr timestep daysFig. 8. Dynamic behavior of PTB (base case, DDD = 2 days).696 P. Georgiadis, A. Politou/ Computers & Industrial Engineering 65 (2013) 6897035.2. Adaptability to demand changesFig. 12illustratesthe response of the actual levelofPTB for astepincreaseindemand. Inparticular, theFigureillustratestheresults for a stepincrease withmagnitude 1item/day(about11%of themeanvalueof thedailydemand)onthe100thday.Fig. 12a(Fig. 12b) provesthat thestepincreaseindemandforDDD = 2 days(DDD = 4 days)resultsinagradualincreaseofPTB,whichisbalancedinahigher valuecomparedtothat withoutstepincrease. This increaseof PTBis explainedbyanincreaseofDesiredPTBduetotheincreaseofED, DBandD(seeEqs. (6)and(7)).5.3. Robustness to sudden disturbancesThe robustness of the PTB control mechanismto suddendisturbances in the internal and external environment is given inFigs. 13and 14. Fig. 13illustrates theresponse ofactuallevelofPTB when a CCR breakdown occurs from the 100th day up to the110th day. It is shown that breakdown results in increase of PTBandDB, whichare counterbalancedlater on, inbothcases ofDDD; counterbalance is completed earlier in the case ofDDD = 4 days.Fig. 14 illustrates the dynamics of PTB actual level in case of apulse increase in demand with magnitude 90 items/day (equal toFig. 10. Dynamic behavior of PTB under different K1 and K2 values (base case, DDD = 2 days).Fig. 9. Dynamic behavior of PTB (base case, DDD = 4 days).Fig. 11. Transient response and dynamic equilibrium of PTB under different K1 and K2 values (base case, DDD = 4 days).P. Georgiadis, A. Politou/ Computers & Industrial Engineering 65 (2013) 689703 697ten times the Dm) on the 100th day. The pulse increase in demandresults in a sharp increase of PTB and DB, that are counterbalancedlater on, in both cases of DDD; counterbalance is completed earlierin the case of DDD = 4 days .6. The effect of PTB policies on shops performance: Numericalinvestigation and concluding discussionInordertoevaluatethesystemsperformanceinadynamicequilibrium condition, data is collected after the transient period(50 days)toavoidirregularitiesduringthatperiod. Thesystemsperformance, in terms of performance measures given in Table 2,isexaminedunder32combinationsof demand, DDDandow-shop control parameters, generating by the sets of levels given inSections 4.1, 4.2 and 4.3. In every experiment, two critical decisionsare made: how much material to order (MO) on a daily basis andthe adjustment of PTB on hourly basis.At rst, we examine each of the above 32 combinations undertwosetsof levelsof PTBcontrol parameters/factorsK1andK2;K1inlevels0.5and1, andK2inlevels0.125and0.25. Foreachcombination of K1 and K2, three repeat simulation runs allow theuseofANOVAtodeterminewhetherthePTBcontrolparametersaffect signicantly the performance measures. Therefore, the totalnumber of simulation runs is 32 4 3 (=384).The ANOVA results (P-values and Partial Eta Squared) for thesesimulationruns(initial ANOVA)arepresentedinTable4. SinceP-values are the lowest signicance levels to reject the nullhypothesisthat theindependent parameter doesnot affect theindicated performance measures, P-values less than the 0.05 levelof signicanceshowstatistical signicance. Besides, Partial EtaSquared (PES) reects the signicance of the independentFig. 12. Adaptability of PTB in case of a step increase in demand (base case, K1 = 0.2 and K2 = 0.15).Fig. 13. Robustness of PTB in case of CCR breakdown (base case, K1 = 0.2 and K2 = 0.15).Fig. 14. Robustness of PTB in case of a pulse increase in demand (base case, K1 = 0.2 and K2 = 0.15).698 P. Georgiadis, A. Politou/ Computers & Industrial Engineering 65 (2013) 689703Table 4P-values of initial ANOVA for the effects of all control variables on performance measures.Indicates minor signicant effect (P-value 60.05 and PES 6 0.5).Indicates major signicant effect (P-value 60.05 and PES > 0.5).P. Georgiadis, A. Politou/ Computers & Industrial Engineering 65 (2013) 689703 699parameter comparedtotheerrors signicance; thehigher thevalues of PES, the higher is the effect of the independent parameterto the dependent factor. Table 4 presents all the rst, second andthird-orderresults(P-values)of theinitial ANOVAandonlythesignicant higher-order results. For the signicant effects (P-value60.05), the corresponding results are classied regarding the PESvalueintotwocategories: (i) PESvaluesequal orlessthan0.5denoting minor effect of the independent parameter to thedependent factor and(ii) PESvalues higher than0.5denotingmajor effect of the independent parameter to the dependent factor.The results presented in Table 4 indicate that the demand, DDDand ow-shop control parameters have signicant rst-ordereffects on the majority of performance measures of manufacturingand PTB evaluation processes. These rst-order effects (by meansof estimated marginal means of performance measures) arepresented in Table 5. For example, increase of the values of a andb results in increase of AWIP1, AWIP2, ADB, DBD and ALT; increaseof protective capacity value results in decrease of DBD; increase ofDDD value results in decrease of ADB and DBD.In addition, the results shown in Table 5 indicate the effects of K1and K2 onthe performance measures of manufacturing and PTB evalu-ation processes. It is noticeable that there is no signicant rst-ordereffect of K2. For signicant effects, Table 5 illustrates these resultsfor rst-order effects of K1 and K2. Table 5 indicates that: Parameter K1 has signicant effect on the majority ofperformancemeasuresof manufacturingprocess(i.e. onthemeasures ARM, AWIP1, AWIP2, DBD, APR and PI) Parameters K1 and K2 do not have signicant effect on the per-formance measures of PTB evaluation process. However, thesemeasures are inuenced by the demand, DDD andow-shop control parameters.The above observations lead to the necessity of morethoroughly examination of PTB control parameters/factors K1 andK2. Therefore, inasecondANOVA, weexamineeachof the32combinations of demand, DDD and ow-shop control parametersunder 100 sets of levels of PTB control parameters/factors K1 andK2 generating by the combinations of their levels: K1 from 0.1 to1 with step of 0.1;K2 from 0.025 to 0.25 with step of 0.025. Foreachcombinationof K1 andK2, three repeat simulationrunsallowtheuseof ANOVAtodeterminewhetherthePTBcontrolparameters affect signicantly the performance measuresand select their optimumvalues among the considered ones.Therefore, the total number of simulationruns is 32 100 3(=9,200).The ANOVA results for each of 32 combinations of demand, DDDandow-shopcontrolparameters(secondANOVA)indicatethatthe effectK1 K2 onall performancemeasures ofmanufacturingprocessis insignicant (for level of signicanceequal to0.05).Consequently, there is no meaning to track a specic combinationof K1andK2inorder tooptimizetheperformancemeasures.Althoughit is inprinciple riskytogeneralize onthe basis ofnumerical examples, the embedded PTB control mechanism leadsto the conjecture that the performance measures are indeed robustto moderate changes of the control parameters K1 and K2 for thestudied cases of demand, DDD and ow-shop control parameters ofthe shop.It is very interesting that the insensitivity of performancemeasures to changes in PTB control parameters that was identiedthrough the above ANOVA analysis is not coincidental, nor peculiartoaparticularcombinationof thedemand, DDDandow-shopcontrol parameters. Extensivesimulationresults, not shownforbrevity, revealthattheperformancemeasuresareindeedrobustto changes in PTB control parameters for a wide range of demand,Table 5First-order Estimated Marginal Means (EMMs) of performance measures connected to control variables (initial ANOVA).Indicates no signicant effect.700 P. Georgiadis, A. Politou/ Computers & Industrial Engineering 65 (2013) 689703DDDandow-shopcontrol parameters. Therobustness of theperformance measures to PTB control parameters and to demandcharacteristicsisanextremelypositivepropertyoftheproposedmethodology, sincefromonehandaccurateforecastsofdemandare in many real-world applications difcult to obtain and on theother hand this property provides production managers with ex-ibility in decision making with regard to parameters K1 and K2.Besides, as it is shown in Table 6, for all the 32 combinations,the effect K1 K2 is insignicant on the performance measures ofPTBevaluationprocess(for level of signicanceequal to0.05).The insignicance of K1 K2 effect on APTB and PTB reinforce theevidence that the dynamic PTB control mechanism is indeed robustto changes in PTB control parameters for a wide range of demand,DDD and ow-shop control parameters.7. Summary, limitations and directions for model extensionsThis paper was aimedto introduce a dynamic PTB controlmechanism for DBR-based PPC of ow-shop manufacturingsystems. We developed a SD model for a three-operation, single-product, capacitatedow-shopsystemthat purchasesonetypeof rawmaterial. Weconsideredanormallydistributeddemandandexponentiallydistributedproductiontimes. IntegratingthePTB control mechanism into the SD model, we examined the shopperformance under different demand, DDDand shop settings.We proved the insensitivity of performance measures ofmanufacturingandPTBevaluationprocesses tochanges inPTBcontrolparameters. TheresultsobtainedbyextensivenumericalinvestigationandANOVAanalysisrevealedthatthePTBcontrolparameters are indeed robust to changes in the values of demand,DDD and operational parameters.TheproposedPTBcontrol mechanismprovidesthefollowingpossibilities: (i) decision making on the magnitude of PTB withouttrackingaspeciccombinationofcontrolparametersK1andK2(byassuminginitial values for PTB, K1andK2, managers maydecide on PTB values which are based on the evaluation of DesiredPTB values), (ii) ability to integrate real-time disturbances(machine failures, demandincrease) intoPTB-relateddecisionsand(iii) dynamicadaptationof PTBtochangesininternal andexternal shopenvironment. Inaddition, the employedcontrolmechanism provides the ability to consider the non-linear,non-stationaryanduncertainnatureof productionprocessthatcharacterizesthemajorityof ow-shopmanufacturingsystems.Withoutthiscontrol mechanism, it wouldbeimpossibletogetthe adaptive behavior of the manufacturing systemtowardsall the possible changes. However, the learning period is aprecondition for successful real-world applications. This is alignedwiththetransientperiodoccurrencethroughoutthesimulationprocess presented in this paper. After all,the control mechanismis self-correcting; its feedback structure ensures that forecasterrors, changes in the structure of the shop environment and evenself-generated overreactions can eventually be corrected.The proposed real-time PTB control mechanism faceslimitations. The continuous monitoringandadjustment of PTBrequires the use of real-time controllers, whichinautomatedproduction systems are integrated in their IT infrastructure.However, ina more traditional owshop, the controllers arehuman-drivenandconsequentlythemonitoringandadjustmentof PTB practically takes place in longer periods, resulting insuboptimal shop performance. For example, for the base case, forDDD = 2 days and forK1 = 0.2 andK2 = 0.15, the average value ofdemand backlog (ADB) equals 0.51 items, when dt equals0.125 days. However, for dt = 0.5 days, ADB equals 19.37 items.Therefore, in automated ow-shop manufacturing systems, theoor manager in order to exploit the advantages of the developedmodel inapplyingDBR-basedPPCmust estimatethevaluesofDemandDueDateandPTBcontrol parametersandascertaintheinitial values of stock variables. In case of real-time disturbances,themanagerhastoensuretheupdatingofthevaluesofmodelparameters.In case of traditional ow-shop systems that apply DBR-basedPPCwithhuman-drivencontrollers, the oor manager, at thebeginning of each day, must decide upon the value of PTB. In thisdecision, it is suggestedtofollowasimplerule. Inparticular,basedonthePTBvalueofthepreviousdayandtheDesiredPTBvalue(seeEq. (7)), themanager has tocomputethevalueofPTBDiscrepancy(seeEq. (10)). Then, thecurrentvalueof PTBiscalculated (see Eqs. (5), (8), and (9)) by using the assumed valuesof control parameters K1 and K2. Finally, the current PTB value isusedtocalculatethevaluesof Material ReleaseTime(MRT)(seeEq. (A.4)), Planned Order Release Rate (PORR) (see Eq. (A.2)), OrderRelease Rate (ORR) (see Eq. (A.3)) andMaterial Release Rate (MRR)(see Eq. (A.6)), and to schedule the rawmaterial release inproduction line.The results presented in this paper certainly do not exhaust thepossibilitiesofinvestigating allthefactorsaffecting thedynamicDBR-based PPC of ow-shop manufacturing systems. For example,it is worthwhile to study the proposed PPC system assuming multiproducts under different shop settings and to integrate costelements into the proposed control mechanism. Finally, thedevelopment of self-adaptive mechanisms for PTB controlparameters (K1 and K2) may have added-value in the developmentof more comprehensive DBR-based PPC systems.Table 6Resultsof secondANOVAfortheeffectsof PTBcontrol variablesonperformancemeasures of PTB evaluation process.a b Prot.cap.Cap.switchDDD APTB PTBP-value PES P-value PES0.9 0 0.1 0 2* * * *0.9 0 0.1 0 4* * * *0.9 0 0.1 1 2 0.91 0.24 0.75 0.260.9 0 0.1 1 4* * * *0.9 0 0.3 0 2* * * *0.9 0 0.3 0 4* * * *0.9 0 0.3 1 2 0.77 0.26 0.48 0.290.9 0 0.3 1 4* * * *0.9 0.2 0.1 0 2 0.52 0.29 0.37 0.300.9 0.2 0.1 0 4 0.25 0.31 0.42 0.300.9 0.2 0.1 1 2 0.27 0.31 0.34 0.300.9 0.2 0.1 1 4 0.66 0.27 0.80 0.260.9 0.2 0.3 0 2 0.95 0.23 0.46 0.290.9 0.2 0.3 0 4 0.79 0.26 0.82 0.250.9 0.2 0.3 1 2 0.49 0.29 0.33 0.300.9 0.2 0.3 1 4 0.88 0.24 0.85 0.250.98 0 0.1 0 2* * * *0.98 0 0.1 0 4* * * *0.98 0 0.1 1 2 0.94 0.23 0.37 0.300.98 0 0.1 1 4 0.49 0.29* *0.98 0 0.3 0 2* * * *0.98 0 0.3 0 4* * * *0.98 0 0.3 1 2 0.44 0.29 0.64 0.270.98 0 0.3 1 4 0.46 0.29* *0.98 0.2 0.1 0 2 0.55 0.28 0.96 0.220.98 0.2 0.1 0 4 0.71 0.27 0.82 0.250.98 0.2 0.1 1 2 0.51 0.29 0.66 0.270.98 0.2 0.1 1 4 0.39 0.30 0.16 0.330.98 0.2 0.3 0 2 0.98 0.21 0.40 0.300.98 0.2 0.3 0 4 0.76 0.26 0.72 0.270.98 0.2 0.3 1 2 0.79 0.26 0.92 0.240.98 0.2 0.3 1 4 0.73 0.26 0.38 0.30*Indicates that the value of performance measure obtained by all simulation runsis the same.P. Georgiadis, A. Politou/ Computers & Industrial Engineering 65 (2013) 689703 701Appendix A. Equations of control mechanisms and performancemeasuresA.1. Material release rate control mechanismBased on backward innite loading scheduling (Park andBobrowski, 1989; Sabuncuoglu and Karapinar, 1999).ORt Ztt0PORRt ORRtdt ORt0; ORt 0 0 A:1PORRt Dt MRT A:2ORRt minORt=dt; MIt=MF=dt A:3MRTt max0; DDD MPT Dt dt PTBt A:4MPT X3i11Capi MA:5MRRt ORRt MF A:6A.2. Material procurement control mechanismBased on a periodic order quantity review systemwithprobabilistic demand and variable order quantity that equalsMFO (Silver et al., 1998; Steele et al., 2005).MORt ORRt r MF A:7MFOt Ztt0MFOIRt MFODRtdt MFOt0; MFOt 0 0 A:8MFOIRt MORt A:9MFODRt MFOtdt; ifan order is given at time t0; otherwise(A:10MOt MFOt; ifan order is given at time t0; otherwise

A:11MPt MOt MLT A:12MIt Ztt0MPRt MURtdt MIt0; MIt 0 0 A:13MPRt MPt=dt A:14MURt minMRRt; MIt=dt A:15A.3. Flow-shop production control mechanismBasedonlimitationfunctions considering the capacity andinventory constraints (Georgiadis and Michaloudis, 2012).WIP0t Ztt0MRRt MCRtdt WIP0t0; WIP0t 0 0 A:16MCRt PR1t MF A:17WIP1t Ztt0PR1t PR2tdt WIP1t0; WIP1t 0 0 A:18PR1t minWIP0tMF dt; Cap1t ; PR2t minWIP1tdt; Cap2t A:19WIP2t Ztt0PR2t PR3tdt WIP2t0; WIP2t 0 0 A:20PR3t minWIP2tdt; Cap3t A:21FPIt Ztt0PR3t SRtdt FPIt0; FPIt 0 0 A:22A.4. Shipments rate control mechanismBased on the assumption that all the demand is satised, evenwith delay.PDFt Dt DDD A:23DBt Ztt0DBIRt DBDRtdt DBt0; DBt 0 0 A:24DBIRt maxPDFt=dt SRt; 0 A:25DBDRt minmaxSRt PDFt=dt; 0; DBt=dt A:26SRt minPDFt DBt; FPIt=dt A:27OBt Ztt0ORTt SRtdt OBt0; OBt 0 0 A:28ORTt Dt=dt A:29A.5. Performance measures of manufacturing processARM PTt1MItTA:30AWIP1 PTt1WIP1;tTA:31AWIP2 PTt1WIP2;tTA:32AFP PTt1FPItTA:33ADB PTt1DBtTA:34APR PTt1PR2;tTA:35DBD XTt1tt ; where tt t; ifDBt> 00; otherwise

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