Optimizing Storage Capacity of Retailers in Stochastic Periodic Inventory Routing Problem

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1 Opimizing Sorage Capaciy of Reailers in Sochasic Periodic Invenory Rouing Problem Ehsan Yadollahi 1,2, El-Houssaine Aghezzaf 1, Joris Walraevens 2 and Birger Raa 1 1 Deparmen of Indusrial Sysems Engineering and Produc Design, Faculy of Engineering and Archiecure, Ghen Universiy, Gen, Belgium 2 Deparmen of Telecommunicaions and Informaion Processing (TELIN), Faculy of Engineering and Archiecure, Ghen Universiy, Gen, Belgium Keywords: Absrac: Safey Sock, Sorage Capaciy Limiaion, Sochasic Demands, Invenory Rouing Problem. A challenging quesion in Sochasic Periodic Invenory Rouing Problem (SPIRP) is how o deal wih sochasic demand raes, while minimizing he coss (ransporaion, invenory, and sorage) and finding he bes rouing sysem. In his paper, we reformulae he SPIRP model o a safey sock-based SPIRP where he invenory sorage capaciy a he reailers are considered as variables and reailer s demand rae is sochasic. The supply chain planner needs o find he bes rouing sysem o replenish he reailers wih he mos opimum level of invenory, while he service level is saisfied in a long erm planning horizon. Four differen policies for sorage capaciy opimizaion are presened, evaluaed, and compared in an illusraive example. The impac of sorage capaciy limiaion is considered based on he defined policies o measure heir compaibiliy for differen siuaions. 1 INTRODUCTION Invenory-Rouing Problem (IRP) inegraes invenory managemen and vehicle rouing decisions over several periods and has received increased aenion in recen years (Aghezzaf, 2007, Berazzi e al., 2013, Yadollahi e al., 2017, Federgruen and Zipkin, 1984, Bell e al., 1983). Bell e al., (1983) are one of he firs researchers who used VRP and invenory managemen ogeher o deal wih he case where only ransporaion coss are included, demand is sochasic, and cusomer invenory levels mus be me. Demand sochasiciy means ha shorages may occur since he supplier only knows a probabilisic disribuion of demand for he reailer. To avoid having sock-ous, a penaly is imposed whenever a reailer runs ou of sock, and his penaly is usually paid wih he unsaisfied demand (negaive invenory). Unsaisfied demand is eiher considered as los-sale or backlogged. More explanaion abou IRP and SPIRP can be found in (Coelho e al., 2014a, Coelho e al., 2014b). Variabiliy of service, uncerainy in demand, and delay are he well-known characerisics of SPIRP. The rade-off beween coss (ransporaion and invenory) and producs availabiliy makes SPIRP a hard problem o solve. Even hough here is a noiceable body of lieraure abou IRP and SPIRP, only few sudies have involved capaciy limiaion as consrains. Sacey e al., (2007) are one of he pioneers in specifying he significance of sorage capaciy on boh he rouing and invenory decisions in he conex of inbound ransporaion. They have evaluaed he benefis of applying sorage consrains a differen levels by developing wo new heurisics ha sequenially ake ino accoun he invenory level and rouing decisions. Pujawan e al., (2015) have proposed a new mehod o inegrae operaional and sraegic decision parameers, namely shipmen planning and sorage capaciy decision under uncerainy. Their objecive is o provide a close o opimal soluion o find he bes balance for logisics cos and produc availabiliy. The auhors develop a simulaion model o invesigae he effecs of various indicaors on coss and service levels in a disribuion sysem. The model mimics he ransporaion and disribuion problems of bulk cemen, consising of a silo a he por of origin, wo silos a wo pors of desinaion, and a number of ships ha ranspor he bulk cemen. The oucome of heir model clarifies he significan effec of he number of ships deployed, silo capaciy, 217 Yadollahi, E., Aghezzaf, E-H., Walraevens, J. and Raa, B. Opimizing Sorage Capaciy of Reailers in Sochasic Periodic Invenory Rouing Problem. DOI: / In Proceedings of he 7h Inernaional Conference on Operaions Research and Enerprise Sysems (ICORES 2018), pages ISBN: Copyrigh 2018 by SCITEPRESS Science and Technology Publicaions, Lda. All righs reserved

2 ICORES h Inernaional Conference on Operaions Research and Enerprise Sysems working hours of pors, and he dispaching rules of ships on boh oal logisics coss and service level. Finding he appropriae sorage capaciy is one of he main objecives of SPIRP ha desires more invesigaion. The comparison beween small and big sorage capaciy can be assessed from several differen aspecs such as coss, service level, silo availabiliy, produc s perishabiliy, ec. In addiion, differen opions for sorage capaciy a he reailers wih differen coss makes i more challenging for he supply chain decision maker o find he mos opimum soluion. The idea of having he capaciy opimized is a new concep in SPIRP and has no been reaed compleely in he lieraure. In his paper by involving sorage capaciy consrains ino SPIRP, we develop he soluions o deal wih sochasiciy in demand raes and coss minimizaions while service level is saisfied. Four differen policies for sorage capaciy allocaion are considered in his paper. The sraegies are evaluaed and compared by implemening hem on an illusraive example based on wo indicaors namely coss and compuaion ime. The oucome of hese soluions are discussed in deails for he shor and long erm planning horizon in order o have a beer insigh of heir influence on he whole sysem. The res of he paper is organized as follows; secion 2 presens he Safey Sock-based SPIRP ogeher wih he differen approaches for he capaciy opimizaion. In secion 3, we explain all he approaches and discuss he advanages and disadvanages. 2 SAFETY STOCK-BASED SPIRP MODEL WITH STORAGE CAPACITY LIMITATION POLICIES The invenory rouing sysem sudied in his paper consiss of a single depo and a se of geographically scaered reailers. The reailers are indexed by j, j {1, 2,.., m} (where m is he oal number of reailers) and he depo is indexed by r. Each reailer j has a sochasic independen demand rae of d j per uni of ime, ha is assumed o be approximaely based on Gamma disribuion Γ(α, β). Le S be he se of reailers indexed by i and j; and S + = S {r }. Le H = {1, 2,.., T } be he planning horizon indexed by, and H + = H {0} be he planning horizon ha includes period = 0. Le τ be he size in ime unis of each period, for example eigh working hours per day. For he deliveries, a flee of vehicles V, v {1,2,.., k} each wih a capaciy of κ is available. The supplier and each reailer j agree o a service level (SL j ) based on a predeermined invenory violaion rae of θ j during each period and reailer, and SL j = (1 θ j ). Le SG = {1,2,, G} be he number of available silos for each reailer j. Addiional parameers of he model are as follows: φ j : he fixed handling cos (in euros) per delivery a locaion j S + (reailers and depo) in period H. h j : he per uni holding cos of he produc a locaion j S (in euros per on) in period H; ψ v : he fixed operaing cos of vehicle v V (in euros per vehicle per use); δ v : ravel cos of vehicle v V (in euros per km); η v : average speed of vehicle v V (in km per hour); ij : duraion of a direc rip from reailer i S + o reailer j S + (in hours); I j0 : he iniial invenory levels a each reailer j S in period zero; CI j : The cos of using a silo for each reailer j S, in period H KI jg : maximum capaciy of each silo g SG, for reailer j S The variables of he model are defined as follows: Q vij : he quaniy of produc remaining in vehicle v V when i ravels direcly o locaion j S + from locaion i S + in period H. This quaniy equals zero when he rip (i, j) is no on any our of he roue ravelled by vehicle v V in period ; q j : he quaniy delivered o locaion j S in period H; I j : he invenory level a locaion j S by he end of period H; x vij : a binary variable se o 1 if locaion j S + is visied immediaely afer locaion i S + by vehicle v V in period H, and 0 oherwise; y v : a binary variable se o 1 if vehicle v V is being used in period, and 0 oherwise; IS jg : a binary variable se o 1 if silo g SG is being used for reailer j S in period, and 0 oherwise; The minimizaion objecive funcion is: 218

3 Opimizing Sorage Capaciy of Reailers in Sochasic Periodic Invenory Rouing Problem Subjec o: CV = [ψ v y v + H v V i S + j S + (δ v η v ij + φ j )x vij ] + h j I j + (KI jg CI j )IS jg T j S g SG x vij 1, v V i S + H j S (1) j S, H (2) x vij x vjk = 0, i S + k S + j S +, H, v V (3) ij x vij τ, H, v V (4) i S + j S + Q vij Q vjk = q j, j S, H (5) v V i S + v V k S + Q vij κ x vij, i S +, j S +, H, v V (6) I j0 + q jl = E(d jl ) + SS j + I j, j S, H + (7) l=1 l=1 I j0 I jt, j S, H (8) I j IS jg KI jg, j S, H (9) g SG x vrj y v, j S +, H, v V (10) x vij, y v, IS jg {0,1}, I j 0, Q vij 0, q j 0, The objecive funcion (1) shows he variables o minimize he level of coss in his replenishmen sysem. I includes five cos componens, namely, oal fixed operaing cos of using he vehicle(s), oal ransporaion cos, oal delivery handling cos, oal invenory holding cos a he end of each period, and oal cos of rening silos a he reailers. Consrains (2) assure ha each reailer is visied a mos once during each period. Consrains (3) guaranee ha a vehicle moves o he nex reailer/depo afer serving he curren one. Consrains (4) preven ha he ime required o complee each our does no exceed he duraion of he period. The quaniies o be delivered o each reailer are deermined by consrains (5). These consrains also avoid sub-our from occurring. Consrains (6) are capaciy consrains induced by he vehicles capaciies. Consrains (7) deermine he delivered number of producs from period 1 o ogeher wih he iniial invenory o be equal o he expeced demand s values from period 1 o, safey sock, and remaining invenory a he end of period for each reailer j. Consrains (8) insure ha he level j S +, H, v V of invenory a he end of las period is equal or larger han iniial invenory. Consrains (9) deermine he opimum number of required silos for each reailer during each period. Finally, consrains (11) specify a vehicle canno be assigned o serve reailers unless he relaed fixed cos is payed. Eq. (11) presens he safey sock calculaion model o be used in consrains (7). As is specified by equaion (11), safey sock is a funcion of service level parameer (z θj ), number of ime periods (), and sandard deviaion of demand (σ j ) for each reailer (j). The parameer z θj is he service facor deermined by reailer s requesed service level (SL j %) gained by he level of θ j. I is used as a muliplier wih he sandard deviaion and number of ime periods o calculae a specific quaniy (as safey sock) o mee he pre-se service level. SS j = z θj σ jl 2 l=1 (11) 219

4 ICORES h Inernaional Conference on Operaions Research and Enerprise Sysems 3 DIFFERENT APPROACHES FOR STORAGE CAPACITY ALLOCATION We propose 4 differen policies in his sudy. These policies are suggesed based on he requiremens in shor/long erm planning horizons and high variabiliy in demand raes o evaluae heir applicabiliy in disribuion sysems. Differen indusries have differen preferences in rening a silo. Therefore, presening various sraegies for silo allocaion could help he decision maker o decide wisely. In he reminder four proposed policies for silo allocaion are modelled and described. 3.1 Fixed Number of Silos This is he basic policy ha allocaes a cerain number of silos o he reailers during he whole planning horizon. Equaions (1)-(10) formulae he Safey Sock-based SPIRP for his policy. Number of silos are fixed from period 1 o he las period. I means he maximum required silos need o be rened in he beginning of he planning horizon based on he expeced level of invenory from he opimizaion model. In some disribuion cenres where he variabiliy of demand raes is high, and high level of service is promised o he cusomers, i is beer o ren a cerain number of silos for he whole planning horizon. Therefore, here is less risk of having limied space for he invenory during he planning horizon. The calculaed number of silos is based on he maximum expeced level of invenory, meaning here are some periods ha some silos are no full, bu he ren mus be paid. The allocaion of he silos o he reailers are based on he renal fee, and he rade-off beween invenory/silo coss and ransporaion coss. 3.2 Fixed Cumulaive Fixed-cumulaive approach opimizes he silo allocaion mechanism, in order o use he maximum capaciy of rened silos during he periods wih low invenory level a he reailers. In oher words, he cumulaive level of invenory from he beginning o period is aken ino accoun insead of he level of invenory for period. To have his sraegy applied in he Safey Sock-based SPIRP, consrains (9) needs o be replaced by consrains (13). In consrains (13) he invenory level for all he periods from 1 o need o be smaller or equal o sorage capaciy in one period muliplied by. Reailers wih higher variabiliy in demand raes and/or long erm planning horizon are more convenien o have his sraegy for rening he silos, since for hose reailer he risk of having excess invenory/demand in long erm is compensaed by oher periods wih lower demand rae. 3.3 Flexible Number of Silos In his policy he reailers are allowed o have differen number of silos for each period. I means he number of silos are differen during he planning horizon, bu 1he decision for each period is made only based on he invenory for ha period. I makes he invenory coss as low as possible since here is no need o pay he ren when he silo is no used. Equaion (12) involves his flexibiliy in he objecive funcion by summing up he silo fee coss for each period. Therefore, he model selecs he number of silos for each period differenly based on he maximum invenory level on ha period. Equaions (2-10) and (12), presen he Safey Sock-based SPIRP model wih flexible sorage capaciy. All hese decisions are made before he planning horizon, herefore his mechanism may cause risks for he reailers in erms of sock-ou occurrence. Generally, he reailers wih lower coefficien of variaion wih shor erm planning horizon are more preferred o apply his policy. 3.4 Flexible Cumulaive This mechanism is similar o Fixed-cumulaive, wih his difference ha in his policy he reailer does no CV = [ψ v y v + (δ v η v ij + φ j )x vij ] + h j I j H v V i S + j S + H j S (12) + (KI jg CI j )IS jg H j S g SG I js IS jg s=1 g SG KI jg, j S, H (13) 220

5 Reailers Opimizing Sorage Capaciy of Reailers in Sochasic Periodic Invenory Rouing Problem need o keep a cerain number of silos for he whole planning horizon. The idea is o be flexible in rening he silos as well as involving he variabiliy in invenory level among he periods o minimize he coss. Equaions (2-8), (10), (12), and (13) presen he Safey Sock-based SPIRP model wih flexiblecumulaive approach for silo allocaion. 4 ILLUSTRATIVE EXAMPLE We consider a disribuion cenre wih 8 reailers. There is a flee of vehicles wih 2 available vehicles, each one wih he capaciy of 40 ons. The vehicles work 8 hours per day wih an average speed of 50 km/h. Fix and variable coss of he vehicles are presened in able 1. The reailers are scaered randomly around he warehouse. Disances beween reailers hemselves and warehouse are shown in able 3. Table 1: Some elemens. Noaion Parameer Cos φ j Delivery coss 25 η j Invenory holding coss per uni per period 0.5 δ v Travel coss for 1 vehicle in Euro per KM ψ v Fix operaing cos 30 of vehicle ν v Average speed of vehicle 50 The demand rae for each reailer is considered sochasic and follows Gamma disribuion and all he sock-ous are fully backlogged. Table 2 presens he demand raes for 8 hours (1 period) and sandard deviaions as well as heir coefficien of variaions. The res of he parameers of his example are provided in able 1. We use CPLEX for solving all models. All he compuaions are performed on a 3.60 GHz Inel Xeon CPU. Table 2: Demand rae parameers per period. Average demand E(d j ) (on/day) Sandard deviaion σ j (on/day) CV (α) (β) RESULTS AND DISCUSSION The wo indicaors considered in his sudy are cos level and compuaion ime. Boh indicaors have been measured and evaluaed for he defined policies in his example o clarify he differences. Figure 1 shows he expeced coss for each policy during he whole planning horizon. As menioned in equaions (1) and (2), hese coss are fixed and variable coss of ransporaions, silos, and invenory. Figure 11 clearly indicaes he low level of cos for flexible cumulaive sraegy while fixed sraegy is he highes. Flexible cumulaive sraegy has saved 40% of he expeced coss in his model, while flexible sraegy reduces he coss by almos 30%. The cumulaive approach shows a big improvemen compared o periodic approach, by allocaing he silos and rucks properly as well as minimizing he invenory level a he reailers among he periods. Table 3: Duraion of a rip from reailer i S + o reailer j S + (in hour). warehouse c1 c2 c3 c4 c5 c6 c7 c8 warehouse c c c c c c c c

6 ICORES h Inernaional Conference on Operaions Research and Enerprise Sysems We also consider compuaion ime for each policy in order o verify he applicabiliy of he sraegy, paricularly for larger models. Table 4 presens he compuaion ime per policy for he whole disribuion sysem. Fixed and flexible silo allocaion models need he minimum ime among he oher sraegies, while when he model is cumulaive in sorage capaciy allocaion, he required ime becomes larger. Fixed cumulaive approach needs 87 seconds o achieve he opimized soluion, while i is even more wih Flexible Cumulaive approach wih 106 seconds. Higher compuaional ime specifies he model complexiy level and compuaion difficuly ha resuls in lower ineres o apply he complex soluions for large sysems. limiaion. The proposed safey sock-based SPIRP model involved sorage capaciy as a consrain in he model o opimize i wih regard o cos minimizaion. Four differen policies are proposed o deal wih sorage capaciy limiaion a reailers. The advanages and disadvanages of hese approaches have been discussed in his paper. Finding he balance beween ransporaion and invenory coss ogeher wih he sorage coss (silo ren) is he mos imporan facor in SPIRP model. Definiely i depends on he value of produc iself, silo fee, promised service level, demand variabiliy rae a he reailers, lengh of he planning horizon, ec., o allocae silos o he reailers. The illusraive example presened in his paper has revealed he advanages of flexible model among oher policies. In addiion for smaller disribuion cenres, fixed cumulaive approach seems o be an appropriae sraegy o opimize he sorage capaciy. As for fuure research, he applicabiliy of hese approaches will be evaluaed in some experimenal cases wih design of various experimens based on he variables. In addiion, heir impac on service level, invenory and ransporaion coss, and compuaional ime will be measured and discussed. REFERENCES Figure 1: Overall coss for each policy. Table 4: Compuaion ime per policy. Policies Time (seconds) Fixed 20 Fixed cumulaive 87 Flexible 23 Flexible Cumulaive 106 According o he resuls of he illusraive example, flexible approach has go he mos reasonable resuls for boh compuaion ime and cos reducion. Bu if he model is small in size, he fix cumulaive approach seems more reasonable, since i is more logical o ren a silo for he whole planning horizon. 6 CONCLUSIONS In his paper we considered Sochasic Periodic Invenory Rouing Problem wih sorage capaciy Aghezzaf, E. H Robus disribuion planning for supplier-managed invenory agreemens when demand raes and ravel imes are saionary. J Oper Res Soc, 59, Bell, W. J., Dalbero, L. M., Fisher, M. L., Greenfield, A. J., Jaikumar, R., Kedia, P., Mack, R. G. & Pruzman, P. J Improving he Disribuion of Indusrial Gases wih an On-Line Compuerized Rouing and Scheduling Opimizer. Inerfaces, 13, Berazzi, L., Bosco, A., Guerriero, F. & Laganà, D A sochasic invenory rouing problem wih sock-ou. Transporaion Research Par C: Emerging Technologies, 27, Coelho, L. C., Cordeau, J.-F. & Lapore, G. 2014a. Heurisics for dynamic and sochasic invenoryrouing. Compuers & Operaions Research, 52, Par A, Coelho, L. C., Cordeau, J.-F. & Lapore, G. 2014b. Thiry Years of Invenory Rouing. Transporaion Science, 48, Federgruen, A. & Zipkin, P A Combined Vehicle Rouing and Invenory Allocaion Problem. Operaions Research, 32, Pujawan, N., Arief, M. M., Tjahjono, B. & Krichanchai, D An inegraed shipmen planning and sorage capaciy decision under uncerainy A simulaion sudy. Inernaional Journal of Physical Disribuion & Logisics Managemen, 45,

7 Opimizing Sorage Capaciy of Reailers in Sochasic Periodic Invenory Rouing Problem Sacey, J., Malini, N. & Charles, S The sorage consrained, inbound invenory rouing problem. Inernaional Journal of Physical Disribuion & Logisics Managemen, 37, Yadollahi, E., Aghezzaf, E. H. & Raa, B Managing invenory and service levels in a safey sock based invenory rouing sysem wih sochasic reailer demands. Applied Sochasic Models in Business and Indusry. 223