Applying simheuristics to minimize overall costs of an MRP planned production system

  Ci a ion: Sei inge , W.; Cas aneda, J.; Al endo e , K.; Panade o J.; Juan, A.A. Applying Simheu is ics o Minimize O e all Cos s o an MRP Planned P oduc ion Sys em. Algo i hms 2022,15, 40. h ps:// doi.o g/10.3390/a15020040 Academic Edi o : Meng Liu Recei ed: 13 Decembe 2021 Accep ed: 25 Janua y 2022 Published: 27 Janua y 2022 Publishe ’s No e: MDPI s ays neu al wi h ega d o ju isdic ional claims in published maps and ins i u ional a il- ia ions. Copy igh : © 2022 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h ps:// c ea i ecommons.o g/licenses/by/ 4.0/). algo i hms A icle Applying Simheu is ics o Minimize O e all Cos s o an MRP Planned P oduc ion Sys em Wol gang Sei inge 1,* , Juliana Cas aneda 2, Klaus Al endo e 1, Ja ie Panade o 2and Angel A. Juan 3 1School o Business and Managemen , Uni e si y o Applied Sciences Uppe Aus ia, Weh g abengasse 1-3, 4400 S ey , Aus ia; klaus.al endo e @ h-s ey .a 2IN3—Compu e Science Depa men , Uni e si a Obe a de Ca alunya, Rambla Poblenou 156, 08018 Ba celona, Spain; [email p o ec ed] (J.C.); jpanade [email p o ec ed] (J.P.) 3Depa men o S a is ics and OR, Uni e si a Poli ècnica de València, Plaza Fe andiz y Ca bonell, 03801 Alcoy, Spain; [email p o ec ed] *Co espondence: wol gang.sei inge @ h-s ey .a Abs ac : Looking a cu en en e p ise esou ce planning sys ems shows ha ma e ial equi emen s planning (MRP) is one o he main p oduc ion planning app oaches implemen ed he e. The MRP planning pa ame e s lo size, sa e y s ock, and planned lead ime, ha e o be iden i ied o each MRP planned ma e ial. Wi h inc easing p oduc ion sys em complexi y, mo e planning pa ame e s ha e o be de ined. Simula ion-based op imiza ion is known as a aluable ool o op imizing hese MRP planning pa ame e s o he unde lying p oduc ion sys em. In his a icle, a as and easy- o-apply simheu is ic was de eloped wi h he objec i e o minimize o e all cos s. The simheu is ic se s he planning pa ame e s lo size, sa e y s ock, and planned lead ime o he simula ed s ochas ic p oduc ion sys ems. The de eloped simheu is ic applies aspec s o simula ion annealing (SA) o an e icien me aheu is ic-based solu ion pa ame e sampling. Addi ionally, an in elligen simula ion budge managemen (SBM) concep is in oduced, which skips eplica ions o no p omising i e a ions. A comp ehensi e simula ion s udy o a mul i-i em and mul i-s aged p oduc ion sys em s uc u e is conduc ed o e alua e i s pe o mance. Di e en simheu is ic combina ions and pa ame e s a e es ed, wi h he esul ha he combina ion o SA and SBM led o he lowes o e all cos s. The con ibu ions o his a icle a e an easy implemen able simheu is ic o MRP pa ame e op imiza ion and a p omising concep o in elligen ly manage simula ion budge . Keywo ds: MRP; planning pa ame e ; op imiza ion; simula ion budge ; heu is ic 1. In oduc ion Fo manu ac u ing companies, an en e p ise esou ce planning sys em (ERP) is he cen al sys em o plan and con ol p oduc ion- ela ed esou ces. Compa ed o he ech- nological de elopmen o ERP sys ems, he used planning algo i hms ha e no changed ha much du ing he las decade. Mos comme cial ERP sys ems s ill use he hie a chical p oduc ion planning app oach o ma e ial equi emen s planning (MRP) o gene a e p o- duc ion o de s [ 1 ]. The applicabili y in di e en indus ies and he s aigh o wa d and scalable logic—independen om he p oduc complexi y— os e ed MRP’s impo ance in indus y and also science. The h ee planning pa ame e s o con ol MRP a e lo size, planned lead ime, and sa e y s ock. Many o he pa ame e s mus be se up and de ined o ge a p oduc ion sys em on which an MRP can be applied, such as a bill o ma e ial (BOM), p ocessing ime, se up ime, planning pe iod, machine a ailabili y, and many mo e. The mas e p oduc ion schedule (MPS), which is based on he cus ome demands, de ines he quan i ies and ime pe iods o he o de ed ma e ials and speci ies he g oss equi e- men s o he MRP algo i hm. E en hough he MRP logic is no complica ed, keeping an MRP sys em up o da e is a challenging ask. Suppose aw ma e ial is a ailable in he ERP sys em, bu canno be handed o p oduc ion. This ype o misin o ma ion can Algo i hms 2022,15, 40. h ps://doi.o g/10.3390/a15020040 h ps://www.mdpi.com/jou nal/algo i hms Algo i hms 2022,15, 40 2 o 18 equi e upda ing he comple e MRP sys em wi h e-planning, including scheduled eceip s and planned o de eleases [ 1 ]. Howe e , hese a e no sophis ica ed asks compa ed o selec ing op imal MRP planning pa ame e s. To o e come a wo sening se ice le el, i is, o example, possible o inc ease planned lead ime wi h he consequence o inc easing s ocking le el [ 2 ]. Conside ing he e ec s o changing he alue o one MRP planning pa ame e is manageable, bu doing his in a sys ema ic way o lo size, planned lead ime, and sa e y s ock need o be suppo ed by me hods om he ield o p oduc ion sys em simula ion and heu is ics. Using p oduc ion sys em simula ion o e alua e he pe o - mance o a p oduc ion sys em elying on an MRP p o ides in e es ing insigh s in o he beha io o he p oduc ion sys em pe o mance. In pa icula , he sys ema ic explo a ion o sui able MRP pa ame e combina ions will help ind op imal alues wi h espec o gi en pe o mance indica o s, such as in en o y and a diness cos s. P oduc ion sys ems a e cha ac e ized by a high unce ain y le el, which is associa ed wi h hei planning. This may be due o in e nal ac o s, such as s ochas ic p ocessing imes, machine ailu es, limi ed a ailabili y o esou ces—ei he wo ke s o aw ma e ials—e c. I can also be due o ex e nal ac o s associa ed wi h cus ome s’ demand, such as equi ed lead imes, changes in o de s eques ed, o cancella ions [ 3 ]. Gi en his complexi y, he planning o he MRP pa ame e s in a p oduc ion sys em equi es he de elopmen o me hodologies capable o dealing wi h sys ems unde unce ain y. The implemen a ion o op imiza ion echniques based on hyb id me hods be ween me aheu is ic and simula ion echniques has p o en o be a me hodology wi h he po en ial o es ablish he s ochas ic MRP pa- ame e s a nea op imal le els [ 4 ]. The e o e, his pape p oposes a simula ion heu is ic (simheu is ic) o es ablish he alues o he MRP pa ame e s in a mul i-s age, mul i-i em p oduc ion sys em. A disc e e e en simula ion model simula ing a s ochas ic MRP sys em is combined wi h a me aheu is ic using a daemon-like p ocedu al c i e ion and simula ion budge managemen (SBM). The s ochas ic MRP mimics a p oduc ion plan exposed o he unce ain ies associa ed wi h s ochas ic o de amoun s, he cus ome - equi ed lead imes, and machine se up imes. He eby, he ocus o his a icle is o in es iga e how o apply di e en simheu is ic app oaches o sys ema ically ind he bes MRP pa ame e se ings in o de o minimize o e all cos s. F om an algo i hmic pe spec i e, he challenge is o in eg a e he simheu is ics in o he used simula ion amewo k. Da a om he ac ual i e a ion mus be s o ed and p o ide he ounda ion o subsequen i e a ions. Fo SBM, i is necessa y o keep ack o p e ious solu ion quali y. This in o ma ion is used o decide i he emaining eplica ions o an i e a ion a e consumed. P oduc ion sys em modeling associa ed challenges a e he selec ion o app op ia e s a ing pa ame e s including he anges o he MRP pa ame e s. The ackled p oblem is o iden i y he combina ion o MRP pa ame e s p o iding he lowes o e all cos s by applying he de eloped simheu is ics. Hence, he pe o med simula ion s udy is used o answe he ollowing h ee esea ch ques ions. To answe hese ques ions he de eloped simheu is ics a e e alua ed using an MRP-based p oduc ion sys em: Q1: Does he applica ion o an in elligen simula ion budge managemen (SBM) p o ide lowe o e all cos s compa ed o a ixed numbe o eplica ions? Q2: In he con ex o MRP pa ame e s se ing, can he use o simula ed annealing (SA) help a oid ge ing apped in a local minimum ela ed o he o e all cos s? Q3: Is he combina ion o SBM and SA leading o lowe minimum o e all cos s and, i so, wha is he e ec o inc easing simula ion budge ? The emainde o he a icle is s uc u ed as ollows: Sec ion 2p esen s a li e a u e e iew on he use o simula ion and simula ion-based op imiza ion me hods o deal wi h s ochas ic MRP sys ems. Sec ion 3in oduces he simula ion-based op imiza ion heu is ic used o sol e he p oblem. Sec ion 4explains he cha ac e is ics o he s ochas ic MRP sys em used in his wo k o es ou me hodology. Sec ion 5desc ibes he simula ion- op imiza ion expe imen s pe o med. Sec ion 6discusses he esul s ob ained. Finally, Sec ion 7concludes on he indings o his wo k and discusses u u e lines o esea ch. Algo i hms 2022,15, 40 3 o 18 2. Rela ed Wo k P oduc ion sys ems a e cha ac e ized by a high complexi y gi en by he la ge numbe o pa ame e s in ol ed in hei wo king p ocesses. In gene al, MRP sys ems calcula e he lo size o be p oduced o o de ed o each componen in each pe iod based on he i ems’ demand. The e is a BOM in which hey a e o ganized hie a chically as componen s, and i also includes he numbe o each pa pe inal p oduc [ 3 ]. This sec ion p esen s a e iew o some o he wo ks ca ied ou abou he op imiza ion o he pa ame e s s udied in his wo k, which a e associa ed wi h MRP. The objec i e is o iden i y he mos s udied pa ame e s, hei combina ion, and he analysis o solu ion me hods. Please no e, ha he objec i e unc ion o he op imiza ion p oblem discussed in his pape includes in en o y and backo de cos s. This includes se ice and ime pe o mance e ec s, i.e., low se ice le el leads o high backo de cos s and high p oduc ion lead imes lead o high in en o y cos s (and p obably o highe backo de cos s). Consequen ly, he objec i e unc ion in his a icle is in line wi h a lo o MRP pa ame e op imiza ion p oblems om he p esen ed li e a u e. The MRP in a p oduc ion line is modeled o simpli y he sys em wi h a se o as- sump ions. They include all hose pa ame e s ha allow us o model he unce ain y and complexi y o he sys em i sel . The op imiza ion o hese models—ei he wi h exac , heu is ic, me aheu is ic, o e en hyb id me hods—enables us o ind nea -op imal and e icien solu ions o he sys em. Among he mos s udied pa ame e s unde unce ain y in he MRP li e a u e a e he sa e y s ock and he planned lead ime [ 5 ]. The app oach employed o manage an MRP unde unce ain y depends on he ype o unce ain y, he signi icance o i s e ec , and he manage s’ p e e ences [ 6 ]. Table 1p esen s some o he op imiza ion wo ks on one o mo e o he MRP pa ame e s s udied in his pape , wi h hei espec i e solu ion echniques. F om he li e a u e e iew, i can be concluded ha his a icle ackles he MRP pa ame e op imiza ion p oblem, which is al eady ea ed by o he au ho s. Table 1. Rela ed wo k e iewed. Au ho App oach Pa ame e s Me hod o Analysis Lead Time Sa e y S ock Lo Size Demand Whyba k and Williams [7], Buzaco and Shan hikuma [8], Enns [9]X X X Simula ion Molinde [10] X X X X Hyb id: simula ed annealing & simula ion Al endo e [2] X X Heu is ic Al endo e e al. [11] X X Simula ion Teo e al. [12] X X Non-linea op imiza ion Libe opoulos and Koukoumialos [13] X X Simula ion Al endo e and Minne [14] X X Gene al op imiza ion model Al endo e [15] X X X Heu is ic Ba ios e al. [4] X Hyb id: heu is ic & simula ion Gans e e e al. [5] X X X Hyb id: Va iable neighbo sea ch & simula ion Ka de e al. [16] X X X Hyb id: Gene ic Algo i hm & simula ion Among he ela ed wo k analyzed, i is ound ha simula ion echniques ha e been widely applied in MRP models since hey allow us o obse e he beha io o he supply Algo i hms 2022,15, 40 4 o 18 chain h ough he c ea ion o expe imen s based on he model o he sys em unde s udy. A he same ime, i allows us o obse e he e ec s o changes in MRP pa ame e s. I o e s he possibili y o planning and con olling he p oduc ion sys ems and imp o ing s a egies a di e en o ganiza ional le els. Likewise, disc e e-e en simula ion mimics he dynamics o sys ems in he eal wo ld, which has made i a e y popula me hod o MRP modeling [ 17 ]. Whyba k and Williams [7] ha e been among he i s o p opose a amewo k o cha ac e ize and s udy he unce ain y ha can a ec in en o y in es men and se ice le el pe o mance in an MRP sys em. They used simula ion o compa e wo pa ame e s, sa e y s ock and sa e y lead ime. They also es ed whe he he e is be e in en o y con ol ha mee s ma ke unce ain y. In he same di ec ion, Buzaco and Shan hikuma [8] , Enns [9] s udied he in luence o sa e y s ock size and planned lead ime. Likewise, hese au ho s de eloped models ha simul aneously decide on he lo size and planned lead ime in an en i onmen wi h cons an cus ome demand and s ochas ic lead imes [ 18 ]. O he models a e de eloped o add ess he isk gene a ed by unce ain y in lead imes and demand, while ocusing on deciding which o hese wo pa ame e s should be used o make decisions [ 10 ]. The design o hese simula ion models o explo e he e ec o demand unce ain y on he sys em pe o mance also a ies depending on whe he we a e conside ing a single-s age manu ac u ing sys em [ 2 ] o a mul i-i em, mul i-s age MRP p oduc ion sys em [11]. Typically, MRP simula ion models a e s udied o know he esponses (o ou pu s o he sys em) o he ini ially de ined pa ame e s [ 19 , 20 ]. Howe e , his echnique does no allow de e mining he op imal alues o he pa ame e s o achie e a gi en objec i e [ 21 ]. Pa ame e op imiza ion o con igu e he sys ems and ope a e as e icien ly as possible is ecommended [ 10 ]. In he li e a u e, some models es ablish planned lead imes based on capaci y equi emen s planning, using nonlinea op imiza ion o ind he op imal alues o he planning pa ame e s minimizing p oduc ion- ela ed cos s [ 12 ]. O he models ely on simula ion-based op imiza ion o pe o m nume ical explo a ions. The in eg a ion o op imiza ion me hods and simula ion allows us o calib a e he pa ame e s o he s ochas ic model. In his op imiza ion p ocess, he objec i e unc ion has an associa ed measu e o he expe imen al simula ion ha is op imized [ 22 ]. Fo ins ance, i includes he op imiza ion o planned s ock le els and lead ime in p oduc ion, as well as he op imiza ion o in en o y sys ems wi h ad anced demand in o ma ion and some se ice le el equi emen s [ 13 ]. The e a e also analy ical models o he simul aneous op imiza ion o capaci y and planned deli e y ime in a wo-s age p oduc ion sys em wi h di e en cus ome due da es [ 14 ]. Ano he wo k in es iga ing he applica ion o analy ical me hods and heu is ics was p esen ed in [ 15 ]. The in en o y and he a diness cos s we e analyzed. In addi ion, a heu is ic-based app oach o sol e he associa ed op imiza ion p oblem was used. All h ee MRP pa ame e s wi h he limi a ion o only one p oduc ion s age and capaci y, we e in es iga ed. One o he me hods o simula ion-based op imiza ion is simheu is ics. I is a solu ion me hod based on algo i hms combining simula ion me hods wi hin a heu is ic/me aheu is ic op imiza ion amewo k. I allows o deal wi h s ochas ic combina o ial op imiza ion p ob- lems [ 23 , 24 ]. Ba ios e al. [4] demons a ed ha simheu is ics a e a p omising app oach o sol e a wo-s age s ochas ic MRP. They iden i ied he sa e y s ock le el o each i em o minimize he o al expec ed manu ac u ing cos . Thei esul s demons a e ha he use o simheu is ics p o ides ad an ages o e he use o simula ion app oaches alone. Some app oaches a y be ween hem because o he p oposed op imiza ion heu is ics and me aheu is ics. Fo example, Molinde [10] p oposed o de e mine he sa e y lead ime wi h a hyb id app oach combining simula ion wi h op imiza ion based on he simula ed an- nealing algo i hm. Gans e e e al. [5] p esen ed a simula ion-based op imiza ion app oach wi h a sea ch p ocedu e based on a iable neighbo hood sea ch wi h a obus p oduc ion planning as a esul . They ocused on he simul aneous op imiza ion o h ee pa ame e s, planned lead ime, sa e y s ock, and lo sizes. In addi ion, Ka de e al. [16] op imized he MRP pa ame e s, applied, and compa ed wo di e en e sions o e icien global Algo i hms 2022,15, 40 5 o 18 op imiza ion o single-objec i e and mul i-objec i e unc ions. Thei esul s demons a e how bo h app oaches a e compe i i e wi h each o he . Thus, some o he wo ks ela ed o he op imiza ion o he MRP sys em ha e ocused mainly on pa ame e s such as lead ime, ollowed by sa e y s ock and lo size. The li e - a u e ha has analyzed he h ee pa ame e s has always used hyb id simula ion-based op imiza ion app oaches o analyze and sol e he model. I is because inc easing he numbe o pa ame e s inc eases he complexi y. The simples analysis me hods usually in ol e only one o a mos wo pa ame e s. Simula ion is he mos widely used analysis me hod, ollowed by hyb id simula ion-based op imiza ion me hods. Mo eo e , he la e is also one o he mos ecen solu ion me hods in he li e a u e. Conside ing he end o ela ed wo ks, his pape p esen s a simula ion-based op imiza ion me hodology o ind he alues o he s udied pa ame e s o MRP ha gua an ee an e icien pe o mance o he p oduc ion sys em, wi h he op imiza ion a ge o minimum o e all cos s. 3. Simula ion Heu is ic In he con ex o MRP pa ame e se ing, simheu is ic algo i hms [ 25 ] a e a meaning ul app oach o e alua e a b oad ange o pa ame e combina ions wi h a de ined objec i e (e.g., minimizing o e all cos ). A i s simple simheu is ic wi h wo ex ensions o minimize o e all cos s was p esen ed in [ 26 ]. Howe e , hei app oach is limi ed in he sense ha i excludes lo sizing. Thei simheu is ic is signi ican ly ex ended in his pape and applied also o he lo sizing decision which has a majo impac on he o e all cos . In addi ion, he new simheu is ic algo i hm is u he in eg a ed in o a disc e e-e en simula ion model, which implemen s a p oduc ion sys em elying on MRP. Du ing a simula ion expe imen , he simheu is ic is applied o compu e new planning pa ame e s o each planned ma e ial. The nex simula ion i e a ion is hen pe o med wi h he changed planning pa ame e s. In Sec ion 3.1, h ee e sions o he p oposed simheu is ic a e desc ibed. They all ely on Algo i hm 1. 3.1. Ini ial Simheu is ic Algo i hm The ini ial simheu is ic depic ed in Algo i hm 1allows us o de elop di e en a ian s, named: s a ic ange (STR), exponen ial ange educ ion (ERR), and bes solu ion-se base ange (BSBR). The ini ial simheu is ic p o ides a desc ip ion o he algo i hmic low, and his gene ic s uc u e acili a es o subsequen ly in eg a e he a o emen ioned a ian s. The algo i hm s a s by selec ing he bes solu ion o all p e iously inished i e a ions. The bes solu ion ep esen s he minimum o e all cos o all inished i e a ions and p o ides a lis wi h he used mode, lowe bound (LB), and uppe bound (UB). Fo he STR e sion, a iangula dis ibu ion is used o compu e new alues using he LB, he UB, and he mode gi en by he midpoin o he ange. Fo all h ee a ian s, he possible alues o he MRP pa ame e s (planned lead ime and sa e y s ock) lie wi hin he LB and he UB. While in [ 26 ] he lo size is no included, in his pape , we conside i s pa ame e sampling o op imiza ion in he same way as o he pa ame e s planned lead ime and sa e y s ock. The second simheu is ic e sion, named ERR, upda es LB and UB using he bes solu ion. This makes i possible o each a sea ch space ou side he o iginal bounds. The solu ion ange is de ined as 0= mode − UB o bo h MRP pa ame e s. The new bounds o he MRP pa ame e s and o each i em a e compu ed as i= ( 1 −α)∗ i−1 , whe e, iis he simula ion i e a ion numbe and α is he ange educ ion le el. The goal o he ERR a ian is o educe he LB and UB when he numbe o i e a ions inc ease, so he possible solu ion ange is i e a i ely sh inking. In he hi d e sion, named BSBR, he LB and he UB a e compu ed using he minimum and maximum o he bes nsolu ions. Algo i hms 2022,15, 40 6 o 18 Algo i hm 1 Base Simheu is ic Algo i hm. 1: i Ini ializa ion hen 2: α←e.g., 0.025 3: opN ←e.g., 7 4: ini I e a ions ←e.g., 50 5: ini Pa ame e Range(lb,ub,mode) 6: n←0 7: endIni Phase ← alse 8: cu en Solu ion ←T iangula Dis ibu ion(pa ame e Range) 9: baseSolu ion ←cu en Solu ion 10: end i 11: while n≤maxReplica ions do 12: cos (cu en Solu ion)←Simula eMRP(cu en Solu ion) 13: i n>ini I e a ions hen 14: endIni Phase ← ue 15: end i 16: i cos (baseSolu ion)<cos (cu en Solu ion) hen 17: baseSolu ion ←cu en Solu ion 18: end i 19: baseSolu ion ←Ge Bes Solu ion() 20: i simheu is ic =STR ∨endIni Phase = alse hen 21: pa ame e Range.mode ←baseSolu ion 22: end i 23: i simheu is ic =ERR ∧endIni Phase = ue hen 24: oldRange ←(pa ame e Range.ub −pa ame e Range.lb)/2 25: newRange ←(1−α)∗oldRange 26: pa ame e Range.mode ←baseSolu ion 27: pa ame e Range.lb ←pa ame e Range.mode −newRange 28: pa ame e Range.ub ←pa ame e Range.mode +newRange 29: end i 30: i simheu is ic =BSBR ∧endIni Phase = ue hen 31: bes NSolu ions ←Ge Bes NSolu ions( opN) 32: pa ame e Range.mode ←baseSolu ion 33: pa ame e Range.lb ←min(bes NSolu ions) 34: pa ame e Range.ub ←max(bes NSolu ions) 35: end i 36: baseSolu ion ←T iangula Dis ibu ion(pa ame e Range) 37: n←n+ eplica ionsPe I e a ion 38: end while The cons an α is se o he ERR ange educ ion and opN is equi ed only o he BSBR e sion. An ini ializa ion phase can be applied o bo h he ERR and BSBR e sions. The numbe o i e a ions is ep esen ed by he cons an ini I e a ions. The a iable endIni Phase is used o con ol he end o he ini ializa ion phase. No ice ha , by design, ERR and BSBR a e ex ensions o STR. The e o e, STR is applied du ing he ini I e a ions. Thus, o example, 50 i e a ions ou o a maximum o 300 i e a ions can be used o ind s a ing alues o ange educ ion and he bes n solu ions. A key componen in he algo i hmic low is he a ay pa ame e Range, as i ep esen s he s a ing LB and UB alues, as well as he mode o each MRP pa ame e and planned i em. In addi ion, his a ay is con inuously upda ed du ing he simula ion. The inal s ep o each i e a ion in each e sion is o pass he adap ed pa ame e Range o he iangula dis ibu ion. This allows us o gene a e new MRP pa ame e alues and apply hem du ing he MRP simula ion. A he beginning o each i e a ion, he base solu ion is upda ed o he cu en Solu ion in case his is he new bes - ound solu ion. The bes solu ion is ep esen ed by he minimum o e all cos o he p e ious i e a ions, compu ed using he a e age o e all cos o he simula ion uns. The objec i e is o minimize o e all cos s. A limi a ion is ha logis ic objec i es such as se ice Algo i hms 2022,15, 40 7 o 18 le el and lead ime a e no conside ed. E en hough he o e all cos c i e ia a e common in a lo o s udies (see Sec ion 2, e.g., [ 5 , 13 , 15 ]), his limi a ion implies ha he e ec o he op imized planning pa ame e s on o he impo an key pe o mance indica o s is no analyzed which es ic s he discussion o one dimension. The algo i hm is epea ed un il a maximum numbe o uns (maxReplica ions) is eached. Hence, o ins ance, o 100 i e a ions wi h 20 uns, he alue o maxReplica- ions is 2000. The s a emen Simula eMRP is hen pe o med 20 imes pe i e a ion. As shown in [ 26 ], e en basic e sions o hese concep s can be use ul o iden i y pa ame e se ings ha minimize he o e all cos . In his pape , we ex end he ini ial concep s o de elop a holis ic app oach o MRP pa ame e op imiza ion. Thus, he ollowing no el aspec s ha e been conside ed in ou s udy: (i) he MRP pa ame e lo size has been added; (ii) he o iginal heu is ic has been ex ended in o a ull simheu is ic; and (iii) an in elligen simula ion budge managemen is included. The a ge o he simula ion budge managemen is o es solu ion quali y a e each un wi hin an i e a ion. In o he wo ds, o each pa ame e se , he o e all cos is compa ed o he pas esul s, and only i he solu ion is su icien ly good, u he uns o he cu en i e a ion a e conduc ed. The a ge is o pe o m mo e i e a ions wi h he same o e all numbe o uns—i.e., he same simula ion budge —by a oiding unnecessa y uns. 3.2. Simula ed Annealing Escaping a local minimum and explo ing a new solu ion ange equi es a conc e e s a egy. A biased- andomiza ion algo i hm [ 27 ] o iangula dis ibu ion mode is de- sc ibed in Algo i hm 2. This algo i hm applies aspec s o he SA amewo k [ 28 ]. In addi ion, he eedom aspec o a demon algo i hm is used. Such a demon-based beha io is explained in [ 29 ]. A e each i e a ion, he cos s o he base solu ion a e compa ed o he cos s o he bes solu ion. Each ime he cos s o he base solu ion a e g ea e o equal o he bes solu ion, a ese coun e is dec eased by one. When he ese coun e eaches 0, he base solu ion is ese o he bes solu ion. The ese o he bes solu ion a oids was ing simula ion budge in a no p omising solu ion space. Algo i hm 2 Reac i e biased- andomiza ion o iangula dis ibu ion mode. 1: i Ini ializa ion hen 2: ese Coun e ←ini ialValue e.g., 5 3: bes Solu ion ←baseSolu ion 4: end i 5: δ=cos (baseSolu ion)−cos (cu en Solu ion) 6: i δ>0 hen 7: c edi ←δ 8: baseSolu ion ←cu en Solu ion 9: i cos (baseSolu ion)<cos (bes Solu ion) hen 10: bes Solu ion ←baseSolu ion 11: end i 12: else i −δ≤c edi hen 13: baseSolu ion ←cu en Solu ion 14: c edi ←0 15: end i 16: i cos (cu en Solu ion)≥cos (bes Solu ion) hen 17: ese Coun e −− 18: end i 19: i ese Coun e =0 hen 20: baseSolu ion ←bes Solu ion 21: ese Coun e ←ini alValue 22: end i Algo i hms 2022,15, 40 8 o 18 3.3. Simula ion Budge Managemen (Sbm) Limi ing he simula ion expe imen by a maximum numbe o i e a ions and uns pe i e a ion inc eases he p obabili y o exclude po en ial good solu ions, as hey a e excluded due o he un ou o simula ion i e a ions. Always pe o ming a cons an numbe o uns pe i e a ion will no necessa ily inc ease he po en ial o ind a new bes solu ion. Consequen ly, i would be mo e meaning ul o in oduce a cons ain o skipping he ac ual i e a ion and in es he emaining simula ion budge o explo e new solu ions wi h addi ional i e a ions and a changed se o pa ame e alues. A simple simula ion budge managemen pseudocode, which acili a es o un he simula ion expe imen , is desc ibed in Algo i hm 3. This algo i hm allows as many i e a ions as budge is a ailable, and will s op he las i e a ion when he maximum numbe o uns is eached. The bes solu ion always has he maximum numbe o he gi en eplica ions due o he s opping c i e ia. The a e age cos o he p e ious i e a ions (a gI e a ionO e allCos s) is equal o 1 n∑n io e allCos si , wi h n= maximum uns pe i e a ion (maxReplica ionsPe I e a ion). The a gI e a ionO e allCos s is compa ed o he pe cen ile alue o all pas i e a ions, which depends on he eplica ion numbe . The pe cen ile alue is compu ed using Ge Pe cen ilValue(se O AllSolu ions, β) , which equi es he p e ious o e all cos o be equal o se O AllSolu ions, and a alue educed a e each un om a UB, e u ning he alue associa ed wi h he posi ion in he passed solu ions. These alues a e ep esen ed by β and pe cen ileS ep. A e each eplica ion, he o e all cos is added o he alues o he cu en i e a ion and he new pe cen ile alue is compu ed. Compa ed o he ini ial Algo i hm 1, we do no always pe o m a ixed numbe o uns. The SBM s ops a e he simula ionBudge is exhaus ed. Algo i hm 3 Simula ion budge managemen . 1: i Ini ializa ion hen 2: o alReplica ionCoun ←0 3: simula ionBudge ←maxI e a ions ∗maxReplica ionsPe I e a ion 4: s opCu en I e a ion ← alse 5: cu en Replica ionCoun ←1 6: pe cen ileS ep ←0.0175 7: end i 8: β←0.4 9: while cu en Replica ionCoun ≤maxReplica ionsPe I e a ion ∧ 10: s opCu en I e a ion = alse do 11: o alReplica ionCoun ++ 12: cu en Replica ionCoun ++ 13: i a gI e a ionO e allCos s >Ge Pe cen ilValue(se O AllSolu ions,β)∧ cu en Replica ionCoun >3 hen 14: s opCu en I e a ion ← ue 15: end i 16: β=β−pe cen ilS ep 17: end while 18: s opCu en I e a ion ← alse 19: eplica ionsPe I e a ion ←cu en Replica ionCoun 4. Modeling a S ochas ic M p-Based P oduc ion Sys em To model a s ochas ic MRP sys em, disc e e e en simula ion can be used [ 30 ]. F om a echnical pe spec i e, his equi es agen s o simula e he cus ome s’ o de s beha io and a simula ion amewo k capable o eading he gi en expe imen planning pa ame e s and change hem a e each i e a ion. The expe imen planning pa ame e s include s a alues and bounda ies o he MRP- ela ed planning pa ame e s o lo size, planned lead ime, and sa e y s ock. In addi ion, MRP ela ed se ings, such as planning ho izon and addi ionally echnical pa ame e s a e equi ed o he simula ion expe imen s, such as he numbe o simula ion uns and algo i hm i e a ions [ 31 ]. Cus ome ’s o de agen s Algo i hms 2022,15, 40 9 o 18 a e passed o a queue. Whene e a cus ome ’s o de canno be comple ed be o e he deadline, i is classi ied as ‘delayed’. The same logic is applied o he p oduc ion o de s. The p oduc ion o de s a e gene a ed based on cus ome ’s o de s, which a e he g oss equi emen s o he applied MRP un. The ou pu o each MRP un is he p oduc ion o de s, which p o ide he in o ma ion on quan i y, i em, s a and end da e. A e each i e a ion he solu ion quali y is e alua ed, i.e., he a e age o all eplica ions’ o e all cos s is calcula ed, and he simheu is ic is applied o se he new MRP planning pa ame e alues o he nex i e a ion. The simula ion esul s o each i e a ion (MRP pa ame e s, backo de cos s, in en o y cos s, i e a ion coun , eplica ion coun , . . . ) a e s o ed a e each i e a ion in he linked in-memo y da abase. These alues a e hen a ailable o he subsequen i e a ions. The esul s o he cu en i e a ion including each eplica ion a e hold in he wo king memo y o he simula ion compu e , un il he las eplica ion is pe o med. The esul s o he in-memo y da abase a e s o ed as da abase ile and a e he basis o he pe o mance analysis o he unde aken simula ion s udy. 4.1. S ochas ic M p Se ing Ou MRP simula ion model can handle s ochas ic demands as well as andom p o- cessing imes, and p o ides he desc ibed s anda d MRP logic o ea cus ome s’ o de s. S ochas ic beha io is in oduced and con olled using log-no mal p obabili y dis ibu ions wi h expec ed alues ( µi ) and a iances ( σ2 i ) o machine se up ime, cus ome equi ed lead ime and cus ome ’s expec ed o de amoun . The andom a iables can be ound in Table 2, which also includes he espec i e coe icien s o a ia ion (CVi). Table 2. Random a iables wi h log-no mal beha io . Pa ame e I em µiσ2 iCVi o de amoun 10 10 2 0.1414 o de amoun 11 15 6 0.1633 cus ome equi ed lead ime 10 6 9 0.5000 cus ome equi ed lead ime 11 6 10 0.5270 machine se up ime all 12 36 0.5000 The simula ed p oduc ion sys em is illus a ed in Figu e 1. The ole o he de eloped simheu is ic is o se lo size, planned lead ime and sa e y s ock pa ame e s in o de o minimize he o e all cos , which is he sum o in en o y and a diness cos s. Despi e being a simple BOM ha conside s only h ee le els, he example allows o demons a e how simheu is ics can be meaning ully in eg a ed in o an MRP sys em o op imize o e all cos s o all h ee planning pa ame e s. Two inal p oduc s, p oduc s 10 and 11, a low-le el-code (LLC) 0 a e p oduced on machine M 2. The wo semi- inished p oduc s, ma e ials 20 and 21, a e p oduced on machine M 1. Fo one uni o inal p oduc 10, 1 semi- inished p oduc 20 is needed. One piece o semi- inished p oduc 21 is needed o inal p oduc 11. The aw ma e ial 100, which is a pu chased p oduc , is needed o he semi- inished p oduc s 20 and 21. This aw ma e ial is assumed o be always a ailable. Fo his simple p oduc ion sys em, 12 di e en pa ame e s ha e o be op imized because du ing simula ion o each o he ma e ials (10, 11, 20 and 21), all h ee MRP planning pa ame e s ha e o be es ed o op imali y. Table 3shows he s a -up alues and espec i e anges o op imiza ion pa ame e s. Please no e ha he MRP planning pa ame e alues a e ea ed and e alua ed as in ege uni s. This means ha in he simula ion esul s, he inc ease o dec ease o o e all cos s ela ed o each indi idual MRP pa ame e can be acked. Fu he mo e, he in e ela ion be ween he single planning pa ame e s is in eg a ed in he simula ion model, e.g., bo h a highe planned lead ime and a highe sa e y s ock lead o highe in en o y cos s and lowe a diness cos s, also compa e o Buzaco and Shan hikuma [8] . A dec ease in lo size, o example, leads o highe se up e o s and, he e o e, also a ec s he p oduc ion Algo i hms 2022,15, 40 16 o 18 mo e MRP planning pa ame e s, i.e., mo e decision a iables. This equi es o upscale he sys em o pa ame e handling and in e media e esul compu a ion. Fu he mo e, an e icien MRP pa ame e upda e s a egy has o be de eloped. Resul s o such an implemen- a ion wi h eal da a can p o ide aluable esul s abou op imal pa ame e combina ions and he pe o mance issues o he espec i e p oduc ion sys em. The au oma ic compu- a ion o op imal MRP planning pa ame e based on op imized in en o y and backo de cos s p o ides use ul in o ma ion o p oduc ion planning ela ed decision making. F om a scien i ic poin o iew, especially he gene al app oach o SBM is eusable o o he simula ion s udies and can help consume he a ailable simula ion budge o es la ge solu ion spaces. Figu e 3. Compa ison be ween he bes esul ob ained in each simheu is ic. The p esen ed simheu is ic does no explici ly conside he in e ela ion be ween op imized planning pa ame e s. Igno ing he pa ame e dependency can ha e nega i e e ec s on he op imized pa ame e s and he inal esul . Howe e , pa ame e in e ela ion is also igno ed by o he heu is ics, compa e wi h [ 16 ]. In es iga ing MRP pa ame e in e ela ion may p o ide addi ional insigh in o op imiza ion-based p oduc ion sys em simula ions, bu is ou o con ex o his a icle. 7. Conclusions and Fu u e Wo k This a icle p o ides insigh s in o he applica ion o simheu is ics in iden i ying op i- mal MRP planning pa ame e alues o lo size, planned lead ime and sa e y s ock. The aim is o ind he minimal o e all cos (in en o y cos s plus backo de cos s) compu ed du ing he applica ion o he MRP planning algo i hm in a simula ion en i onmen . This a icle is a u he de elopmen o he esul s based on h ee di e en simheu is ic e sions o iden i y he minimal o e all cos only o he MRP planning pa ame e sa e y s ock and planned lead ime. The i s ex ension is ha op imal alues o all h ee MRP planning pa ame e s a e sea ched. A second ex ension is he de elopmen o a SA-based simheu is ic ex ension. The hi d ex ension ocuses on inding op imal pa ame e s wi hou was ing simula ion budge o non-p omising i e a ions. Hence, an in elligen simula ion budge managemen (SBM) is in oduced o consume he a ailable eplica ions only o po en ial bes solu ions, hus skipping solu ions ou side he de ined pe cen ile ange. SBM was combined wi h he SA ex ension o addi ionally explo e new a eas o he solu ion space and o escape local minima. The applica ion o he de eloped SBM and SA ex ensions, combined wi h all h ee planning pa ame e s, allows o a sys ema ic iden i ica ion o op imal planning pa ame e s wi h a clea op imiza ion a ge . The nume ical esul s show ha he bes pe o ming e sion o his applica ion is ERR. F om a p oduc ion planning Algo i hms 2022,15, 40 17 o 18 pe spec i e, he de eloped simheu is ic e sions, speci ically ERR, p o ides a as and well pe o ming simula ion-based op imiza ion app oach which is simple o apply in p ac ical en i onmen s. The pe o med simula ion s udy shows he imp o emen o he exis ing simheu is ic e sions ega ding he iden i ica ion o he minimal o e all cos wi h he join applica ion o SBM and SA. This inding is suppo ed by he lowes o e all cos ob ained wi h he highes simula ion budge and he join applica ion o SBM and SA o all h ee simheu is ic e sions. Limi a ions o his s udy a e ela ed o he applica ion ield and he p oduc ion sys em wi hin which he simheu is ics a e es ed, i.e., only one applica ion ield is e alua ed in his s udy and he espec i e p oduc ion sys em is s ill a he simple. How- e e , his a icle p o ides he ounda ion o u u e in es iga ions, o sys ema ically apply he simheu is ics concep o mo e complex p oduc ion sys em s uc u es and o o he ap- plica ion ields. The de eloped simheu is ics ocus on a single objec i e unc ion, howe e , o eal p oduc ion sys ems, i is no always possible o ans o m all objec i e dimensions in o cos s. The e o e, u u e wo k could in es iga e how o ex end he simheu is ic o a mul i-objec i e app oach. A b oad sensi i i y analysis o p o e he p omising esul s o SBM is also planned in u u e wo k. Au ho Con ibu ions: Concep ualiza ion, K.A. and A.A.J.; me hodology, K.A. and J.P.; so wa e, W.S.; alida ion, K.A., W.S.; o mal analysis, K.A., W.S. and J.C.; in es iga ion, W.S. and J.C.; esou ces, W.S.; da a cu a ion, W.S.; w i ing—o iginal d a p epa a ion, W.S. and J.C.; w i ing— e iew and edi ing, K.A., W.S., J.C. and A.A.J.; isualiza ion, W.S. and J.C.; supe ision, K.A. and J.P.; p ojec adminis a ion, K.A. and J.P. All au ho s ha e ead and ag eed o he published e sion o he manusc ip . Funding: This wo k has been pa ially unded by he Spanish Minis y o Science (PID2019-111100RB- C21/AEI/10.13039/501100011033 and RED2018-102642-T), he E asmus+ P og am (2019-I-ES01- KA103-062602), and unded by he Aus ian Science Fund (FWF): P32954-G. Ins i u ional Re iew Boa d S a emen : No applicable. In o med Consen S a emen : No applicable. Da a A ailabili y S a emen : No applicable. Con lic s o In e es : The au ho s decla e no con lic o in e es . The unde s had no ole in he design o he s udy; in he collec ion, analyses, o in e p e a ion o da a; in he w i ing o he manusc ip , o in he decision o publish he esul s. Abb e ia ions The ollowing abb e ia ions a e used in his manusc ip : BOM Bill o Ma e ial CU Cos Uni CV Coe icien o a ia ion ERP En e p ise Resou ce Planning FOP Fixed O de Pe iod FOQ Fixed O de Quan i y LB Lowe Bound MRP Ma e ial Requi emen s Planning UB Uppe Bound SA Simula ed Annealing SBM Simula ion Budge Managemen Re e ences 1. Hopp, W.J.; Spea man, M.L. Fac o y Physics, 3 d ed.; Wa eland P ess: Long G o e, IL, USA, 2011. 2. Al endo e , K. In luence o lo size and planned lead ime on se ice le el and in en o y o a single-s age p oduc ion sys em wi h ad ance demand in o ma ion and andom equi ed lead imes. In . J. P od. Econ. 2015,170, 478–488. [C ossRe ] 3. 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