Sequential versus Integrated Optimization: Production, Location, Inventory Control and Distribution

Transcription

1 Sequenial versus Inegraed Opimizaion: Producion, Locaion, Invenory Conrol and Disribuion Maryam Darvish Leandro C. Coelho July 2017 Documen de ravail égalemen publié par la Faculé des sciences de l adminisraion de l Universié Laval, sous le numéro FSA

2 Sequenial versus Inegraed Opimizaion: Lo Sizing, Invenory Conrol and Disribuion Ɨ Maryam Darvish *, Leandro C. Coelho Ineruniversiy Research Cenre on Enerprise Neworks, Logisics and Transporaion (CIRRELT) and Deparmen of Operaions and Decision Sysems, 2325 de la Terrasse, Universié Laval, Québec, Canada G1V 0A6 Absrac. Tradiionally, a ypical approach owards supply chain planning has been he sequenial one. Ignoring he links beween decisions, his approach leads o each deparmen of a company making is own decisions, regardless of wha ohers are doing, and overlooking he synergy of a global sraegy. However, companies are realizing ha significan improvemens can occur by exploiing an inegraed approach, where various decisions are simulaneously aken ino consideraion and joinly opimized. Moivaed by a real case, in his paper, we consider a producion-disribuion sysem ha deals wih locaion, producion, invenory, and disribuion decisions. Muliple producs are produced in a number of plans, ransferred o disribuion ceners, and finally shipped o cusomers. The objecive is o minimize oal coss while saisfying demands wihin a delivery ime window. We firs describe and model he problem and hen solve i, using boh sequenial and inegraed approaches. To solve he problem sequenially, we exploi hree commonly used procedures based on separaely solving each par of he problem. The inegraed problem is solved by boh an exac mehod and a maheurisic approach. Our exensive compuaional experimens and analysis compare soluion coss obained from he wo approaches, highligh he value of an inegraed approach, and provide ineresing managerial insighs. Keywords. Logisics, inegraed opimizaion, sequenial decision making, delivery ime window, locaion analysis. Acknowledgemens. This projec was parly funded by he Naural Sciences and Engineering Research Council of Canada (NSERC) under gran This suppor is grealy acknowledged. The suppor rendered by our indusrial parner is also highly appreciaed. Ɨ Revised version of CIRRELT Resuls and views expressed in his publicaion are he sole responsibiliy of he auhors and do no necessarily reflec hose of CIRRELT. Les résulas e opinions conenus dans cee publicaion ne reflèen pas nécessairemen la posiion du CIRRELT e n'engagen pas sa responsabilié. * Corresponding auhor: Maryam.Darvish@cirrel.ca Dépô légal Bibliohèque e Archives naionales du Québec Bibliohèque e Archives Canada, 2017 Darvish, Coelho and CIRRELT, 2017

3 1. Inroducion The ulimae goal of any producion sysem is o fulfill he demand of is cusomers quickly and e cienly. This goal is achieved hrough e ecive and e cien supply chain planning. Hisorically, supply chain planning has been conduced in a sequenial or hierarchical fashion. This approach reas each supply chain decision separaely from he ohers. Therefore, in such a disinegraed planning sysem, even despie he high cos associaed wih holding socks, invenory plays an imporan role in saisfying he demand in a imely manner and linking di eren funcions of he supply chain. In recen years, he increasing compeiion among supply chains has forced companies o seek soluions ha resul in saving cos and improving he e ciency on he one hand, and o ering even faser and more flexible service o he cusomers on he oher. Invenory opimizaion has become he main arge for cos reducion iniiaives. Recen emphasis on invenory cos reducion coupled wih he growing ransporaion cos and compeiive delivery daes accenuae he imporance of coordinaion and inegraion of supply chain funcions and decisions (Fumero and Vercellis, 1999). Under an inegraed approach, various funcions and decisions wihin a supply chain are simulaneously reaed and joinly opimized. In he sequenial approach, ypically known as managemen in silos, he soluion obained from one level is imposed o he nex one in he hierarchy of decisions (Vogel e al., 2016). Ignoring he links beween decisions, his approach resuls in sub-opimal soluions. On he conrary, mos research and case sudies on supply chain inegraion confirm he posiive e ec of inegraion on business performance (Adulyasak e al., 2015; Coelho e al., 2014). Hence, supply chain inegraion is recognized as he linchpin of success for oday s companies (Archei e al., 2011). In his paper, we describe, model, and solve a muli-produc, muli-plan, muli-period, muliechelon inegraed producion, invenory, and disribuion problem. This inegraed problem has hree disinc feaures of direc shipmen, delivery ime windows, and dynamic locaion decisions for disribuion ceners (DCs). The ransporaion decision in inegraed producion and disribuion lieraure is considered as eiher direc shipmen (full-ruck loads) or vehicle roues (milk runs). Wih he large number of firms ousourcing he ransporaion funcion o hird pary logisics service providers (Amorim e al., 2012), direc shipmen is considered in his paper. 2

4 Owing o is significan research and pracical poenial, much aenion is devoed o ime windows. We consider a delivery ime window, meaning ha he demand mus be saisfied wihin a specific ime frame. Faciliy locaion planning has always been a criical sraegic decision. Once he locaions are deermined, all oher decisions such as producion quaniies, invenories, and ransporaion can be made. In modern days, cusomers always impose igher delivery ime windows, herefore, keeping a high service level and managing invenory require simulaneous producion and dynamic faciliy locaion planning. The inegraed producion-locaion problem has become so prevalen ha flexible nework inegraion is idenified as one of he imporan recen rends in logisics (Speranza, 2016). Hence, following his rend, in his paper, we sudy a flexible supply nework by considering geographically dispersed DCs available o be rened for a specific period of ime. The objecive is o operae a producion-disribuion sysem ha minimizes producion, locaion, invenory, and disribuion coss while saisfying demands wihin a predeermined delivery ime window. To he bes of our knowledge, his rich problem has no ye been sudied in he lieraure. Th problem is inspired by a real-world case. Our indusrial parner is facing a seady bu gradual increase in demand, which requires expanding he operaions. To dae, he company has invesed abundan capial on is producion and sorage faciliies and herefore, producion capaciies exceed he demand of he company for he momen. However, wih he increasing demand growh rae, capaciy consrains seem o be faed. Currenly, he producion manager makes decisions on he producion scheduling and quaniies, which are laer used by he ransporaion manager o plan he disribuion. A his poin, he company is ineresed in how o conduc producion planning o save on coss, bu a he same ime o mainain a high service level. To solve he problem in a sequenial manner, we exploi hree commonly used procedures. These procedures decompose he main problem ino easier subproblems and hen solve each of hem separaely. Two of hese procedures mimic he curren siuaion in companies while one is a lower bound procedure used as a benchmark. Taking an inegraive approach, we solve he problem by boh an exac mehod and a maheurisic. Our maheurisic combines an adapive large neighborhood search (ALNS) heurisic wih an exac mehod. In summary, he main conribuions of his paper are as follows. Firs, we describe and model 3

5 a real-life problem in which producion, invenory, disribuion, and faciliy locaion decisions are simulaneously aken ino consideraion. Second, sequenial and inegraed opimizaion approaches are applied. We exploi an exac and a heurisic mehod o solve he inegraed problem. Finally, we demonsrae he value of he inegraed approach by comparing is coss wih hose obained from he sequenial approach. Moreover, we evaluae he qualiy and performance of all hese mehods by comparing hem wih he soluions obained from he exac mehods. The remainder of his paper is organized as follows. Secion 2 provides an overview of he relevan inegraed producion-disribuion lieraure. In Secion 3, we formally describe and model he problem a hand. This is followed by a descripion of he procedures used o solve he problem sequenially in Secion 4. Our proposed inegraed maheurisic is explained in Secion 5. We presen he resuls of exensive compuaional experimens in Secion 6, followed by he conclusions in Secion Lieraure review Despie he abundance of concepual and empirical sudies on supply chain inegraion and coordinaion, e.g., Power (2005); Musafa Kamal and Irani (2014), unil recenly, inegraed models of supply chains have been sparse in he operaions research lieraure. Simulaneous opimizaion of criical supply chain decisions, by inegraing hem ino a single problem, has been such a complex and di cul ask ha he common approach o solving any inegraed problem was o rea each decision separaely. Mainly due o heir naure, operaional level decisions are he arges for inegraion, among which producion and disribuion decisions are he mos imporan ones. Independenly, boh producion and disribuion problems have several well-sudied varians, and so does heir inegraion. As of now, few reviews on various inegraed producion-disribuion models exis, e.g., Sarmieno and Nagi (1999); Mula e al. (2010); Chen (2010); Fahimnia e al. (2013), and Adulyasak e al. (2015). Focusing on he sudies ha inegrae producion wih direc shipmen, in wha follows, we briefly review he relevan lieraure. A lis of hese papers wih heir feaures is presened in Table 1. Ekşioğlu e al. (2006) formulae he producion and ransporaion planning problem as a nework flow and propose a primal-dual based heurisic o solve i. In heir model, plans are muli-funcional, producion and seup coss vary from one plan o anoher as well as from 4

6 Number of Reference Producs Echelons Periods Plans Invenory Seup Locaion Ekşioğlu e al. (2006) S S M M P X Akbalik and Penz (2011) S S M M P,C X Sharkey e al. (2011) S S M M P X Jolayemi and Olorunniwo (2004) S S M M C X Darvish e al. (2016) S S M M P X X Park (2005) M S M M P,C X Ekşioğlu e al. (2007) M S M M P X Melo and Wolsey (2012) M S M S C X Nezhad e al. (2013) M S S M X X De Maa e al. (2015) M S/M M M DC X Liang e al. (2015) M S M M C X X Barbarosoğlu and Özgür (1999) M M M S P X Jayaraman and Pirkul (2001) M M S M X X This paper M M M M P,DC X X Number of producs, echelons, periods and plans: S: Single - M: Muliple Invenory a: P: Plan - DC: Disribuion cener - C: Cusomer Table 1: Inegraed producion-disribuion problems one period o he nex, and ransporaion coss are concave. Aiming o compare jus-in-ime and ime window policies, Akbalik and Penz (2011) consider delivery ime windows. Wih he jus-in-ime policy, cusomers receive a fixed amoun whereas, wih he ime window policy he deliveries are consrained by he ime windows. In heir model, coss change over ime and a fixed ransporaion cos per vehicle is assumed. A dynamic programming (DP) algorihm is used o solve he problem. The resuls show ha he ime window policy has lower cos han he jus-in-ime one, furhermore, by comparing he mixed ineger linear programming (MILP) and DP mehods, he auhors show ha even for large size insances he DP ouperforms he MILP. Sharkey e al. (2011) apply a branch-and-price mehod for an inegraion of locaion and producion planning in a single sourcing model. The findings show he poenial benefis of inegraing faciliy locaion decisions wih he producion planning. The proposed branch-andprice algorihm works beer when he raio of he number of cusomers o he number of plans is low. Darvish e al. (2016) invesigae a rich inegraed capaciaed lo sizing problem (LSP) wih a single-produc, muli-plan, and muli-period seing. They incorporae direc shipmen, delivery ime windows, and faciliy locaion decisions. They use a branch-and-bound approach 5

7 o solve he problem. Assessing he rade-o s beween coss and fas deliveries, hey show he compeiive advanage of he inegraed approach, boh in erms of oal coss and service level. In he profi maximizaion model presened by Jolayemi and Olorunniwo (2004), any shorfall in demand can be overcome by eiher increasing capaciy or subconracing. They inroduce a procedure o reduce he size of he zero-one MILP and, using a numerical example, hey show ha he reduced and full-size models generae exacly he same resuls. Anoher paper wih a profi maximizaion objecive funcion is ha of Park (2005). The model allows sockou and uses homogeneous vehicles for direc shipmens. They develop a wo-phase heurisic; in he firs phase, he producion and disribuion plans are idenified while in he second, he plans are improved by consolidaing he deliveries ino full ruckloads. Only for he small insances, he heurisic generaes good resuls. The paper also invesigaes he benefis of he inegraed approach compared o he decoupled planning procedure, concluding ha wih he inegraed approach boh he profi and he demand fill rae increase. Ekşioğlu e al. (2007) exend he problem sudied in Ekşioğlu e al. (2006) by considering muliple producs. They apply a Lagrangian decomposiion heurisic o solve he problem. The problem invesigaed by Melo and Wolsey (2012) is similar o ha of Park (2005). They develop formulaions and heurisics ha yield soluions wih 10% gap for insances wih limied ransporaion capaciy bu up o 40% for insances wih join producion/sorage capaciy resricions. Nezhad e al. (2013) ackle an inegraion of locaion, producion wih seup coss, and disribuion decisions. In heir problem plans are single-source and no capaciaed. They propose Lagrangian-based heurisics o solve he problem. The inegraed producion-disribuion problem addressed in De Maa e al. (2015) assumes ha each plan uses eiher direc shipmen or a consolidaed delivery mode provided by a hird pary logisics firm. They use Benders decomposiion o selec he delivery mode and o simulaneously schedule he producion. Liang e al. (2015) allow backlogging in he model and propose a hybrid column generaion and relax-and-fix mehod, he exac approach provides he lower bounds while he decomposiion yields he upper bounds. In Barbarosoğlu and Özgür (1999), a Lagrangian-based heurisic is applied o solve an inegraed producion-disribuion problem. They propose a decomposiion echnique o divide he problem ino wo subproblems and o opimize each of hem separaely. Jayaraman and Pirkul (2001) incorporae procuremen of he raw maerial and supply side decisions ino he model. Generaing several insances, firs, hey compare he bounds from he Lagrangian approach wih 6

8 he opimal soluion obained by a commercial solver. Then, hey apply he proposed mehod o he daa obained from a real case. 3. Problem descripion and mahemaical formulaion We now formally describe he inegraed producion, faciliy locaion, invenory managemen, and disribuion wih delivery ime windows problem. We consider a se of plans, available over a finie ime horizon, producing muliple producs. Saring a new lo incur a seup cos a each plan where a variable cos proporional o he quaniy produced is also considered. Each plan owns a warehouse where he producs are sored. An invenory holding cos is due for he producs kep a hese warehouses. The producs are hen sen o DCs, o be shipped o he final cusomers. There is a se of poenial DCs from which some are seleced o be rened. A fixed cos is due and he DC remains rened for a given duraion of ime. DCs charge an invenory holding cos per uni per period. The producs are finally shipped o he geographically scaered cusomers o saisfy heir demand. There is a maximum allowed laeness for he delivery of hese producs o cusomers, meaning ha he demand mus be me wihin he predeermined delivery ime window. A service provider is in charge of all shipmens, from plans o DCs and from DCs o final cusomers. The ransporaion cos is proporional o he disance, he load, and he ype of produc being shipped. Formally, he problem is defined on a graph G =(N, A) wheren = {1,...,n} is he node se and A = {(i, j) :i, j 2N,i 6= j} is he arc se. The node se N is pariioned ino a plan se N p,adcsen d, and a cusomer se N c, such ha N = N p [N d [N c. Le P be he se of P producs, and T be he se of discree periods of he planning horizon of lengh T. The invenory holding cos of produc p a node i 2N p [N d is denoed as h pi, he uni shipping cos of produc p from he plan i o he DC j is c pij, and he uni shipping cos of produc p from he DC j o he cusomer k is c 0 pjk. Le also f i be he fixed renal fee for DC i; once seleced, he DC will remain rened for he nex g periods. Le s pi be he fixed seup cos per period for produc p in plan i, v pi be he variable producion cos of produc p a plan i, and d pi be he demand of cusomer i for produc p in period. The demand occurring in period mus be fulfilled unil period + r, as r represens he delivery ime window. For ease of represenaion le D be he oal demand for all producs from all cusomers in all periods, i.e., D = P P P d pi. 2T p2p i2n c 7

9 To solve his rich inegraed problem, in each period of he planning horizon, one needs o deermine: he produc(s) and quaniies o be produced in each plan, he DCs o be seleced, he invenory levels in plans and DCs, he quaniy of producs sen from plans o DCs, if he demand of cusomer is saisfied or delayed, and he quaniy of producs sen from DCs o cusomers. We formulae he problem wih he following binary variables. Le pi be equal o one if produc p is produced a plan i in period, and zero oherwise; i be equal o one if and only if DC i is chosen o be rened in period, o be used for g consecuive periods, and! i be equal o one o indicae wheher DC i in period is in is leasing period. Ineger variables o represen quaniies produced and shipped are defined as follows. Le 0 pij be he quaniy of produc p delivered from DC i o cusomer j in period o saisfy he demand of period 0,wih represen he quaniy of produc p produced a plan i in period, 0, pi pij represen he quaniy of produc p delivered from plan i o DC j in period, apple pi as he amoun of produc p held in invenory a DC i a he end of period, and µ pi, he amoun of produc p held in invenory a plan i a he end of period. Table 2 summarizes he noaion used in our model. 8

10 Table 2: Noaion used in he model Parameers h pi c pij c 0 pjk f i s pi v pi d pi Ses N c N p N d T P Variables pi i invenory holding cos of produc p a node i 2N p [N d uni shipping cos of produc p from plan i o DC j uni shipping cos of produc p from DC j o cusomer k fixed rening cos for DC i fixed seup cos per period for produc p in plan i variable producion cos of produc p in plan i demand of cusomer i for produc p in period Se of cusomers Se of plans Se of DCs Se of periods Se of producs equals o one if produc p is produced a plan i in period equals o one if DC i is chosen in period o be used for g consecuive periods! i equals o one o indicae wheher DC i in period is in is leasing period 0 pij quaniy of produc p delivered from DC i o cusomer j in period, o saisfy he demand of period 0 pi pij apple pi quaniy of produc p produced a plan i in period quaniy of produc p delivered from plan i o DC j in period amoun of produc p held in invenory a DC i a he end of period µ pi amoun of produc p held in invenory a plan i a he end of period Indices p, Produc index 0, Period index i, j Node index The problem is hen formulaed as follows: min X X X v pi pi + X X X s pi pi + X X X h pi apple pi + X X X h pi µ pi+ p2p i2n p 2T p2p i2n p 2T p2p i2n d 2T p2p i2n p 2T X X f i i + X X X X c pij pij + X X X X X c 0 pij pij 0 2T p2p 2T p2p 2T 0 2T i2n d i2n p j2n d i2n d j2n c (1) subjec o: pi apple pid i 2N p,2t,p2p (2) 9

11 X 0 =+g 0 =+g X µ pi = µ 1 pi + pi µ 1 pi = 1 pi apple pi = apple 1 pi apple 1 pi = X j2n p X j2n d + X j2n p 1 pji X j2n d pij p 2P,i2N p,2t \{1} (3) 1 pij p 2P,i2N p (4) pji X X X j2n c 0 = r 0 pij p 2P,i2N d,2t \{1} (5) j2n c 11 pij p 2P i 2N d (6) X apple pi apple! id i 2N d 2T (7) p2p X p2p X 0 = g X 0 = X 0 = 1, 0 applet sx 1, 0 applet X apple pi apple! +1 i D i 2N d 2T \{T } (8) 0 i! i i 2N d 2T (9)! 0 i pij 0 i2n d =1 0 =1 Xs+r 0 i2n d =1,<T 0 =1 X X p2p j2n c 0 =1 imin(g, T ) i 2N d 2T (10) 0 i apple 1 i 2N d 2T (11) apple sx d pj p 2P j 2N c s 2T (12) =1 pij =0 p 2P i 2N d j 2N c 2T 0 2{0,..., r}[{,..., T } (13) sx sx d pj p 2P j 2N c s 2T (14) X X X i2n d 2T! i, pi, 0 pij,apple pi, 0 ijp, 0 pij =1 0 pij apple D! i i 2N d 2T (15) 0 pij = d 0 pj p 2P j 2N c 0 2T (16) i 2 {0, 1} (17) ijp 2 Z. (18) The objecive funcion (1) minimizes he oal cos consising of he producion seup and variable coss, invenory holding coss, renal fees, and ransporaion coss, from plans o DCs and also from DCs o final cusomers. Consrains (2) guaranee ha only producs se up for producion are produced. Consrains (3) and (4) ensure he invenory conservaion a each plan. Similarly, consrains (5) and (6) are applied o DCs. Consrains (7) and (8) guaranee ha he remaining invenory a he DC is ransferred o he nex period only if he DC is rened 10

12 in he nex period. Consrains (9) (11) ensure ha once a DC is seleced, i will remain rened for he nex g consecuive periods. Consrains (11) make sure ha he renal fee for each g period is paid only once. Consrains (12) and (13) guaranee ha no demand is saisfied in advance, while consrains (14) impose r periods as he maximum allowed laeness for fulfilling he demand. Thus, he oal demand up o period mus be delivered by period + r. No delivery o cusomers can ake place from a DC if i is no rened, as ensured by consrains (15). Consrains (16) make sure ha every single demand is delivered o he cusomers. Finally, consrains (17) and (18) define he domain and naure of he variables. 4. Sequenial and lower bound procedures In his secion we propose hree sequenial procedures o solve he problem. Their moivaion is wofold. Firs, we wan o mimic producion sysems managed in silos, as inspired and currenly conduced by our indusrial parner. Second, we wan o assess how a sequenial algorihm performs compared o he inegraed one proposed in his paper. These comparisons are presened in Secion 6. In wha follows, in Secion 4.1 we presen a Top-down procedure, for he cases in which producion is he mos imporan par of he process and has prioriy in deermining how he sysem works. This decision is hen followed by invenory allocaion o DCs and finally by disribuion decisions. In Secion 4.2 we describe a Boom-up procedure, simulaing he alernaive scenario in which disribuion has prioriy. The disribuion decisions are followed by DC allocaion, and lasly by producion decisions. Finally, in Secion 4.3 we describe an Equal power procedure, in which all hree deparmens would have similar posiions in he hierarchy of power; we explain how his procedure yields a lower bound on he opimal cos Top-down procedure In he Top-down procedure, producion managers have he mos power and herefore, hey can deermine how he res of he sysem works. This mehod, which observed as he curren pracice of our indusrial parner, works as follows. Firs, minimize only producion coss P P v pi pi + P P p2p i2n p 2T P P p2p i2n p 2T s pi pi subjec o (2) (18). An opimal soluion o his problem deermines he bes producion plan wihou any ineracion wih downsream decisions. Le hese opimal decision values be pi and pi. Noe 11

13 ha because hese producion decisions are made considering he whole feasible region, deermined by (2) (18), feasibiliy is ensured. The second phase works by considering a minimizaion objecive funcion consising of only DC-relaed erms, namely P i, subjec o (2) (18), and o pi = pi and pi = pi. In P f i i2n d 2T his problem, invenory allocaion decisions are made subjec o he feasible region of he overall problem and he producion decisions ha had prioriy over he invenory ones. Le he value of hese decision variables be i and! i. The final phase consiss of deermining he bes way o disribue he invenory o he cusomers, given fixed producion and allocaion plans. This is accomplished by minimizing P P P P c 0 pij 0 pij, and subjec o he feasible region of he original problem (2) (18), P p2p i2n d j2n c 2T 0 2T and o pi = pi, pi = pi, i = i and! i =! i. By puing ogeher all hree levels of decisions, one can obain he overall soluion and easily compue he cos of he soluion yielded by he Top-down procedure. A pseudocode of his procedure is presened in Algorihm 1. Algorihm 1 Top-down procedure 1: Consider all consrains of he problem formulaion from Secion 3, (2) (18). 2: Build an objecive funcion wih producion variables pi and pi : P P P v pi pi + P P P s pi pi. p2p i2n p 2T p2p i2n p 2T 3: Opimize he problem, obain opimal values pi and pi. 4: Fix pi and pi o heir obained values. 5: Add DC-relaed variables i o he objecive funcion: P P f i i. i2n d 2T 6: Opimize he problem, obain opimal values i and! i. 7: Fix variables i and! i o heir obained values. 8: Add all variables o he objecive funcion, as i is defined in (1). 9: Opimize he problem, obain opimal values for all variables. 10: Reurn he objecive funcion value Boom-up procedure In he Boom-up procedure, we suppose ha he disribuion managers have he mos power and can, herefore, deermine how he res of he sysem works. This is done by aking all consrains (2) (18) ino accoun, bu opimizing he objecive funcion only for he disribuion 12

14 variables. Once disribuion decisions are made and fixed, invenory allocaion decisions, namely when and which DCs o ren, are opimized. As menioned earlier, feasibiliy is guaraneed. We now solve he same problem wih a new se of fixed decisions (relaed o disribuion), and opimize only DC-relaed coss. When his par is deermined, all he decisions are fixed and no longer change. Finally, once DCs have been seleced, and all disribuion and DC variables are known, we can opimize he remaining variables of he problem. By puing ogeher all hree levels of decisions, one can obain he overall soluion and easily compue he cos of he soluion yielded by he Boom-up procedure. A pseudocode of his procedure is presened in Algorihm 2. Algorihm 2 Boom-up procedure 1: Consider all consrains of he problem formulaion from Secion 3, (2) (18). 2: Build an objecive funcion wih disribuion variables 0 P P P P P c 0 pij 0 pij. p2p i2n d j2n c 2T 0 2T 3: Opimize he problem, obain opimal values 0 pij. 4: Fix pij 0 o heir obained values. 5: Add DC-relaed variables i o he objecive funcion: P P f i i. i2n d 2T 6: Opimize he problem, obain opimal values i and! i. 7: Fix variables i and! i o heir obained values. 8: Add all variables o he objecive funcion, as i is defined in (1). 9: Opimize he problem, obain opimal values for all variables. 10: Reurn he objecive funcion value. pij Equal power procedure In he Equal power procedure, we assume ha all hree decision levels have equal power. Therefore, informaion is shared wih all deparmens a he same ime bu decisions are made in parallel and each deparmen opimizes is own decisions. This procedure will likely yield an infeasible soluion since each par of he problem is opimized individually. However, he sum of he coss of all hree levels indicaes he opimal decision for each level, when he coss of he oher levels are no considered. Having all hree decision levels pu ogeher, if hese yield a feasible soluion, i is obviously opimal, oherwise, heir sum consiues a valid lower bound on he coss of he problem. Algorihm 3 describes he pseudocode for his procedure. 13

15 Algorihm 3 Equal power procedure 1: Consider all consrains of he problem formulaion from Secion 3, (2) (18). 2: Build an objecive funcion wih disribuion variables 0 pij. 3: Opimize he problem, obain opimal values 0 pij, and opimal disribuion soluion z c. 4: Build an objecive funcion wih DC-relaed variables i, pij, and apple pi. 5: Opimize he problem, obain opimal values for i,! i, pij, and apple pi, and opimal DC soluion z d. 6: Build an objecive funcion wih plan-relaed variables pi, pi, and µ pi. 7: Opimize he problem, obain opimal values for pi, pi, and µ pi, and opimal producion soluion z p. 8: if he combinaion of all hree decisions is feasible hen 9: Reurn opimal soluion and is cos z = z p + z d + z c. 10: else 11: Reurn lower bound value z = z p + z d + z c. 12: end if 5. Inegraed soluion algorihm The problem a hand is reducible o he muli-plan uncapaciaed LSP and also he joinreplenishmen problem, an exension of he uncapaciaed fixed charge nework flow. The joinreplenishmen problem is known o be NP-hard (Cunha and Melo, 2016), as are mos varians of he LSP. Alhough he uncapciaed LSP is easier o solve, he muli-plan version is sill NP-complee (Sambasivan and Schmid, 2002). As is he case of many oher NP-hard problems, exac mehods can solve small-size insances o opimaliy in a reasonable ime bu o obain good soluions for larger insances, one mus develop ad hoc heurisic algorihms. To solve he problem a hand, we propose a maheurisic based on a hybrid of ALNS and exac mehods. The ALNS inroduced by Ropke and Pisinger (2006) has shown ousanding resuls in solving various supply chain problems. ALNS, as a very e cien and flexible algorihm, explores large complex neighborhoods and avoids local opima. Hence, because of is generaliy and flexibiliy, i is highly suiable for he problem a hand. Our conribuion, however, lies in cusomizing and applying his mehod o our problem. We propose a hree-level maheurisic approach in which he problem is divided ino wo subproblems ha are hen solved in an ieraive manner. In he firs level, we apply he ALNS heurisic in order o decide which plans and DCs should be seleced, and o deermine which 14

16 producs have o be produced in any of he seleced plans. Once hese decisions are fixed, he problem becomes a minimum cos nework flow (MCNF) problem. The MCNF finds a feasible flow wih minimum cos on a graph in which a cos is associaed o each arc (Goldberg, 1997). Therefore, in he second level, all he oher remaining decisions on deliveries from seleced plans o rened DCs, and from rened DCs o he cusomers, as well as he invenory level held a plans and rened DCs are obained exacly by solving an ineger linear programming subproblem. This is done e cienly by exploiing he branch-and-bound algorihm and applying i o he MCNF problem. Finally, if needed and o avoid local opimum soluions, we improve he obained soluion and move i oward he global opimal by solving he model presened in Secion 3 wih exac mehods for a very shor period of ime. framework is as follows. The deailed algorihmic Iniial soluion: we sar wih generaing a feasible iniial soluion by making all plans and DCs seleced in all periods. This feasible iniial soluion is quickly improved by deselecing as many faciliies as possible while mainaining feasibiliy. A his sep, coss are no ye of concern and in order o improve he soluion, we ake all he consrains of (2) (18) and solve he problem wih he following objecive funcion: min P P pi + P P p2p i2n p 2T We obain he iniial soluion s and is corresponding cos z(s) o be improved. P i2n d 2T Large neighborhood: a each ieraion, one operaor from he lis described in Secion 5.1 is seleced. Operaors work for any ype of faciliy; herefore, plans and DCs have he same chance of being seleced. To diversify he search, each operaor is repeaed n imes, n being drawn from a semi-riangular disribuion and bounded beween [1,a]. We compue n as in (19) where b is a random number in he [0,1] inerval and a is an ineger number, saring wih a value of one and increasing hroughou he ieraions. i. n = j a p (1 b)(a 1) k (19) Adapive search engine: he operaors are seleced according o a roulee-wheel mechanism. A weigh is associaed o previous performances of each of he operaors, modulaing heir chances of being seleced. Accepance crieria: To diversify he soluions, a simulaed annealing-based accepance 15

17 rule is applied. 0The curren soluion 1 s is acceped over he incumben soluion s 0 wih z(s0 ) z(s) A H probabiliy of e,whereh is he curren emperaure. The emperaure is decreased a every ieraion by, where 0 < <1. Once he emperaure reaches he final emperaure, H final, i is rese o he iniial emperaure, H sar. Adapive weigh adjusmen: A score and a weigh are assigned o each of he operaors. The weigh marix, which has an iniial value of one, is updaed a every ' ieraions. I is updaed using he scores each operaor has accumulaed. The score marix is iniially se o zero, and he beer he operaor performs, he higher score i accumulaes. We define 1 > 2 > 3 > 0. If an operaor finds a soluion beer han he bes soluion obained so far, a score of 1 will be assigned o i. If he obained soluion by he operaor is no he bes bu i is beer han he incumben soluion, he score will be updaed by 2. Finally, if he soluion is no beer han he incumben soluion bu i is sill wihin he accepable range, he operaor will be given a 3 score. Periodic pos-opimizaion: if no improvemen is achieved for more han 2' ieraions, we use he bes soluion as an inpu o he model of Secion 3 and solve i for 20 seconds wih he exac mehod. If his pos-opimizaion aemp yields an opimal soluion, he algorihm sops, as he global opimum has been found; oherwise, if i improves he soluion, he improved soluion is passed o he ALNS framework and he procedure coninues. Sopping crieria: he algorihm will sop, if eiher he maximum number of ieraions ier max or he maximum alloed ime is reached, we limi he running ime o one hour. I will also sop, if he soluion does no change in more han iermax 2 ieraions. Moreover, i will sop when he opimal soluion is obained in he periodic pos-opimizaion sep Lis of operaors The operaors we have designed o explore he search space wih he ALNS framework are as follows. 1. Random: his operaor selecs a plan, a produc and a period (or a DC and a period), and flips is curren saus; if he faciliy is no in use i becomes in use, and vice-versa. 16

18 2. Based on shipping coss: firs, for each produc we compare he shipping coss from plans ha are no producing any produc o all currenly rened DCs, and hen we idenify he combinaion of produc, plan, and period wih he lowes cos. The corresponding produc is hen assigned o be produced a ha plan in ha period. Similarly, he highes shipping cos induces a produc o have is producion sopped a he given plan and period. 3. Based on uni coss: among all plans, we idenify he plan and he produc wih he highes uni producion cos; we sop producion of he idenified produc in he seleced plan; for DCs, we sop rening he one wih he highes uni invenory cos. 4. Based on demand: firs, we idenify he produc and period wih he highes demand, hen we make all he plans produce ha produc in ha period. Similarly, we idenify he produc and period wih he minimum demand, and sop is producion in he idenified period. 5. Based on delivery quaniy o DCs: we idenify he plan delivering he leas (mos) and he DC receiving he leas (mos) per period. Faciliies wih he leas usage will no be in use; for hose wih he larges usage, a random DC is rened in he same period, and producion for he same plan is se up for all producs in he following period. 6. Based on invenory level: we idenify he plan and period wih he maximum invenory, and ensure i says in use in ha period. If he plan is already in use, we keep i in use also in he nex period. For DCs, we sop rening he one wih he lowes invenory level during is g leasing periods. 7. Based on producion quaniy: we idenify he produc/plan/period combinaion in which he maximum (minimum) producion occurs; we sop producion of ha produc in he plan wih he smalles producion in he idenified period bu assign he plan and he produc wih he maximum producion o is nex period. We also idenify he period wih he highes producion, and ren an exra DC. 8. Based on delivery quaniy o cusomers: we idenify all DC/period combinaions wih deliveries lower han a percenage of he oal demand and among hem, we selec a DC and end is lease for ha period (and consequenly he nex g periods). Similarly, for plans, we selec a random one and sop producion of all producs in he previously idenified period. 17

19 5.2. Parameer seings and he pseudocode We have esed di eren combinaions of parameers and uned hem mainly by rial and error. The iniial emperaure H sar is se o (r + 1) 100, 000. This iniial emperaure is cooled down unil i reaches he final emperaure H final =0.01. The cooling rae,, isuned o In our implemenaion, ieraion coun is one of he sopping crieria, and i is saisfied once 3,000,000 ieraions are performed. We se ' o 1,000 ieraions and updae he scores wih 1 = 10, 2 = 4, and 3 = 3. The pseudocode for he proposed maheurisic is provided in Algorihm Compuaional Experimens We now describe he deails relaed o he compuaional experimens used o evaluae our algorihms. All compuaions are conduced on Inel Core i7 processor running a 3.4 GHz wih 64 GB of RAM insalled, wih he Ubunu Linux operaing sysem. A single hread was used for up o one hour, i.e., a ime limi of 3600 seconds was imposed on all algorihms. The algorihms are coded in C++ and we use IBM Concer Technology and CPLEX as he MIP solver. Secion 6.1 describes how he insances are generaed, deailed compuaional resuls are provided in Secion 6.2, and sensiiviy analysis and he managerial insighs are provided in Secion Generaion of he insances By consulaion wih our indusrial parner, we have generaed a large daa se by varying he number of producs, periods, plans, DCs, and cusomers. Our es bed is generaed as shown in Table 3. The number of plans and DCs are deermined by he number of periods: if T = 5, hen N d = 8 and N p = 5, if T = 10, hen N d = 15 and N p = 10, and finally if T = 50, hen N d = 25 and N p = 15. For each of 11 combinaions, we generae five random insances. For each insance we consider a delivery ime window r = 0, 1, 2, or 5 periods. Thus, we solve 220 insances in oal Resuls of he compuaional experimens We now presen he resuls of exensive compuaional experimens carried ou o evaluae he performance of all algorihms, and o draw meaningful conclusions for he problem a hand. 18

20 Algorihm 4 Proposed maheurisic 1: Iniialize weighs o 1, scores o 0, H H sar. 2: s s bes iniial soluion. 3: while sopping crieria are no me do 4: s 0 s 5: Selec an operaor and apply i o s 0 6: Solve he remaining flow problem, obain soluion z(s 0 ) 7: if z(s 0 ) <z(s) hen 8: if z(s 0 ) <z(s bes ) hen 9: s bes s 0 10: updae he score for he operaor used wih 1 11: else 12: updae he score for he operaor used wih 2 13: end if 14: else 15: if s 0 is acceped by he simulaed annealing crierion hen 16: updae he scores for he operaor used wih 3 17: s s 0 18: end if 19: end if 20: H H 21: if ieraions is a muliple of ' hen 22: updae weighs and rese scores of all operaors 23: if no improvemen found in las 2' ieraions hen 24: if H<H final hen 25: H H sar 26: if no improvemen found for z(s 0 ) hen 27: Inpu s bes ino he MIP in Secion 3 and solve i for 20 seconds 28: end if 29: end if 30: else 31: s s bes 32: end if 33: end if 34: end while 35: Reurn s bes 19

21 Table 3: Inpu parameer values Name Parameer Values Producs P 1, 5, 10 Periods T 5, 10, 50 Plans N p 5, 10, 15 DCs N d 8, 15, 25 Cusomers N c 20, 50, 100 Delivery ime window r 0, 1, 2, 5 DC acive period g T 5 Demand d pk [0, 2] Plan seup cos s pi [10, 15] Plan variable cos v pi [1, 10] Fixed DC rening cos f j [100, 150] Invenory holding cos h pj [1, 4] Shipping cos (plans-dc) c pij [10, 100] Shipping cos (DC-cusomers) c 0 pjk [10, 1000] We firs describe he resuls of he experimens wih he mahemaical model proposed in Secion 3. This is followed by he comparison of he performance of he sequenial procedures proposed in Secion 4, and our inegraed hybrid maheurisic from Secion 5 wih ha of he exac algorihms. Average compuaional resuls using he CPLEX branch-and-bound algorihm are presened in Table 4. For each insance, we repor he average of he gaps (G) wihrespecohe Upper Bound Lower Bound lower bound, calculaed as 100, he number of cases solved Lower Bound o opimaliy (O), and he average running ime (T ) in seconds for each predeermined ime window r. As presened in Table 4, only he small insances, mosly hose wih fewer han five producs or periods, could be solved o opimaliy. The parameer conrolling he number of periods seems o have a srong e ec on he performance of he exac mehod. Indeed, i has a huge e ec on he size of he problem as measured by he number of variables and consrains. Moreover, he lengh of he delivery ime window a ecs he number of insances solved o opimaliy, he average gap, and he running ime. To evaluae he performance of he sequenial procedures versus our proposed maheurisics 20

22 Table 4: Resuls from he branch-and-bound algorihm Insance r =0 r =1 r =2 r =5 P -T -N c -N d -N p G(%) (O) T (s) G(%) (O) T (s) G(%) (O) T (s) G(%) (O) T (s) (5) (5) (5) (5) (1) 3, (0) 3, (0) 3, (0) 3, (5) (0) 3, (0) 3, (0) 3, (0) 3, (0) 3, (0) 3, (0) 3, (5) (4) 1, (3) 1, (5) (0) 3, (0) 3, (0) 3, (0) 3, (0) 3, (0) 3, (0) 3, (0) 3, (0) 3, (0) 3, (0) 3, (5) 1, (0) 3, (0) 3, (0) 3, (0) 3, (0) 3, (0) 3, (0) 3, (0) 3,611 Average 8.30 (0.32) 2, (0.18) 3, (0.16) 3, (0.30) 2,675 and o gain insigh ino managerial decisions relaed o he problem a hand, we presen heir resuls in Tables 5 8, one able per value of he delivery ime window r. The improvemens wih respec o he soluion obained from he exac algorihm by he Top-down, Boom-up, and maheurisic algorihms are presened along wih heir running imes. For each mehod, his improvemen is obained as 100 Upper Bound CPLEX Cos mehod. Upper Bound CPLEX Table 5 presens he resuls obained wih no delivery ime window, i.e., r = 0. On average, he proposed mehod ges slighly beer soluions han CPLEX. When only one produc is involved, no maer of he number of cusomers or periods, our proposed mehod always ouperforms CPLEX. Boh he Top-down and Boom-up procedures are very fas, bu he coss obained by hese mehods are much higher han he ones from CPLEX. As indicaed in Table 5, he Boom-up procedure ouperforms he Top-down on almos all large insances wih muliple producs, more han five periods and 50 cusomers. Alhough on average he Boom-up procedure akes less running ime, he resuls obained by his procedure are abou 1.5 imes worse han he ones from he Top-down. 21

23 Table 5: Heurisics resuls for r = 0 Insance Top-down Boom-up proposed mehod P -T -N c -N d -N p 1 I(%) T(s) I(%) T(s) I(%) T(s) , , , , , , , , ,610 Average ,829 When r = 1, as indicaed in Table 6, our approach always ouperforms CPLEX, wih an average improvemen of 4.17%. For a large insance wih one produc, 50 periods, and 100 cusomers, his di erence is up o 27.03%. Alhough he soluions obained by boh sequenial mehods have slighly worsened, he exra delivery period has dramaically increased he running ime for he Top-down procedure, wih almos no significan e ec on he Boom-up. Table 6: Heurisics resuls for r = 1 Insance Top-down Boom-up proposed mehod P -T -N c -N d -N p I(%) T(s) I(%) T(s) I(%) T(s) , , , , , , , , , , , , ,604 Average ,115 22

24 Table 7 shows he resuls obained by considering wo-day delivery ime window, i.e., r = 2. On average our algorihm improves he soluion by 5.29%. As before, he bigges improvemen is observed for he large insance wih one produc, 50 periods and 100 cusomers, bu small insances are eiher solved o opimaliy as CPLEX or has been slighly improved. As he ime window grows, he performance of boh Top-down and Boom-up procedures declines bu compared o he r = 1 case, he running ime slighly increases. Table 7: Heurisics resuls for r = 2 Insance Top-down Boom-up proposed mehod P -T -N c -N d -N p I(%) T(s) I(%) T(s) I(%) T(s) , , , , , , , , , , , , ,601 Average ,174 The bes resuls of our proposed mehod are obained for r = 5. As presened in Table 8, our mehod improves he resuls by 7.62%. The di erence in performance of he wo mehods becomes even more eviden for he big insances wih 50 periods and 100 cusomers, in which our proposed mehod improves he soluion obained by CPLEX up o 49.66%. As before, he wo sequenial procedures can quickly provide feasible soluions, bu of very poor qualiy. 23

25 Table 8: Heurisics resuls for r = 5 Insance Top-down Boom-up proposed mehod P -T -N c -N d -N p I(%) T(s) I(%) T(s) I(%) T(s) , , , , , , , , , , ,601 Average , Sensiiviy analysis and managerial insighs We now perform sensiiviy analysis o derive imporan managerial insighs. From Table 4, we observe ha he more flexible he delivery ime windows ges, he harder o solve he problem becomes. Also, as he number of producs, periods, and cusomers increases, he problem becomes harder o be solved o opimaliy. Small insances wih P = 1, T = 5, and N c = 20 are easily solved o opimaliy, however, insances wih only one produc bu T > 5 canno be solved o opimaliy under he presence of any delivery ime window. This di culy in solving he problem when delivery ime windows exis shows wo ineresing aspecs of he business problem. The firs one is relaed o he poenial cos saving if one is o properly exploi he added flexibiliy of ime windows. This is eviden since all soluions wihou ime windows are sill valid o he cases in which hey are considered. However, o ake advanage of such flexibiliy, using a ailored mehod seems necessary. As shown already, modeling he problem ino a commercial solver or using a sequenial mehod does no yield any good soluions. In fac, he qualiy of soluions degrades as he size of he problem and he added flexibiliy increase. Figure 1 provides an overview on he comparison of our maheurisic and CPLEX for di eren delivery ime windows. We compare he performance of boh mehods over he lower bound obained by CPLEX. As observed in his figure, on average over all insances, our proposed 24

26 algorihm works beer when he delivery ime window enlarges. The resuls reveal ha for large insances our maheurisic ouperforms he exac algorihm. The highes average improvemen is obained for r = 5. For all insances ha could be solved o opimaliy by CPLEX, our algorihm also obains he opimal soluion. Time (s) Gap (%) 3,500 CPLEX Maheurisic 40 CPLEX Maheurisic ,000 3,174 3,115 3,018 3,056 2,829 2,6752,685 2, , ,000 r0 r1 r2 r5 r0 r1 r2 r5 Figure 1: Comparison beween ime (s) and gap (%) of CPLEX and he proposed maheurisic Considering he processing ime, CPLEX performs slighly beer, mainly because he ieraive heurisic reaches he ime limi o search he soluion area, aiming o improve he soluion obained. However, as presened in Table 9, our algorihm akes on average less han 20 minues o find is bes soluion, which is ofen beer han he ones from he exac algorihm. Table 9: Average ime for he proposed mehod o obain is bes soluion Time window r =0 r =1 r =2 r =5 Average Average ime (s) 1,160 1,159 1, ,063 Regarding our proposed maheurisic, Tables 5 8 also reveal ha aking an inegraive approach owards producion, locaion, invenory, and disribuion decisions can lead o enormous cos reducions. For all ime windows, he average resuls obained from he proposed mehod are always beer han he sequenial ones. I is ineresing o noe ha on average he soluions obained by Top-down procedure are lower han he ones from he Boom-up approach; however, he Boom-up procedure is much faser. As presened in Tables 5 8, he Boom-up 25

27 procedure generaes beer resuls, in less ime, han he Top-down when P>1and N c > 20. As expeced, applying he Equal power procedure, where each deparmen of he company is focused only on is own decisions, resuls in no even one insance wih a feasible soluion. Comparing he soluions obained by his procedure o he lower bounds of he exac algorihm, on average his infeasible soluion from he Equal power procedure is 48.11% worse han he lower bound, which forgoes any hopes ha his approach would yield any good soluion. For his reason we do no provide deailed resuls from his mehod. 7. Conclusions This paper invesigaes a challenging and pracical problem of inegraed producion, locaion, invenory, and disribuion, in which muliple producs are produced over a discree ime horizon, sored a he DCs before being shipped o final cusomers. The paper conribues o he inegraed opimizaion lieraure as i combines disinc feaures of delivery ime windows, disribuion wih direc shipmen, and dynamic locaion decisions. A sae of he ar commercial solver is able o find opimum soluions for very small insances of our problem, however, i does no prove opimaliy in a reasonable ime for larger insances. To achieve beer soluions in an accepable compuaion ime, we have proposed a mahuerisric algorihm. Several insances are generaed and he soluions are compared o he opimal ones (if any) obained by he exac mehod. On average he soluions obained wih our algorihm improve he ones from of he exac mehod by up o 49.66%, generally in only a hird of he running ime. In his paper, we have also evaluaed how a ypical managemen in silos would perform, by deriving and implemening sequenial soluion mehods. Our resuls confirm he cos benefis of he inegraed approach owards decision making. Boh Top-down and Boom-up procedures perform worse han he exac mehods as well as our proposed mehod. However, beween hese wo procedures, he Boom-up works beer for insances wih larger planning horizons and more producs and cusomers, while Top-down is preferred when here is only one produc and fewer han 20 cusomers. Using our randomly generaed insances validaed by an indusrial parner, we have shown he benefis of an inegraed managemen, as opposed o he sequenial one. Moreover, we have shown ha for complex and rich inegraed problems inspired by real-world cases, such as he one sudied here, neiher a hierarchical soluion approach nor modeling and solving he problem 26