An Inventory Rationing Method in a M-Store Regional Supply Chain Operating under the Order-up-to Level System

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1 IEMS Vol. 8, No. 2, pp , June An Invenory Raioning Mehod in a M-Sore Regional Supply Chain Operaing under he Order-up-o Level Sysem Chumpol Monhaipkul Graduae School of Managemen and Innovaion, King Mongku s Universiy of Technology Thonburi 126 Prachauid Road Bangmod Thrukru Bangkok 10140, THAILAND Tel: +66 (0) , Fax: +66 (0) , chumpol.mon@kmu.ac.h Received Dae, April 5, 2008; Acceped Dae, March 18, 2009 Absrac. This paper addresses he invenory raioning issue embedded in he regional supply chain invenory replenishmen problem (RSIRP). The concerned supply chain, which was fed by he naional supply chain, consised of a single warehouse disribuing a single produc o muliple sores (M-sores) wih independen and normally disribued cusomer demand. I was assumed ha he supply chain operaed under he order-up-o level invenory replenishmen sysem and had only one ruck a he regional warehouse. The ruck could make one replenishmen rip o one sore per period (a round rip per period). Based on curren invenories and he vehicle consrain, he warehouse mus make wo decisions in each period: which sore in he region o replenish and wha was he replenishmen quaniy? The obecive was o posiion invenories so as o minimize los sales in he region. The warehouse invenory was replenished in every fixed-inerval from a source ouside he region, bu he sore invenory could be replenished daily. The ruck desinaion (sore) in each period was seleced based on is maximum expeced shorage. The replenishmen quaniy was hen deermined based on he predeermined order-up-o level sysem. In case of insufficien warehouse invenories o fulfill all proeced sore demands, an invenory raioning rule mus be applied. In his paper, a new invenory raioning rule named Expeced Cos Minimizaion (ECM) was proposed based on he pracical purpose. The numerical resuls based on real daa from a selecive indusry show ha is performance was beer and more robus han he curren pracice and oher sharing rules in he exising lieraure. Keywords: Invenory Raioning Rule, Supply Chain, Order-up-o Level Sysem, Minimizaion, Heurisics 1. INTRODUCTION 1.1 Overview Nowadays, supply chains uilize a collecion of differen invenory replenishmen sysems. One such sysem configures regional warehouses ha are supplied from a higher-level (naional) supply chain. The regional warehouses hen operae as an invenory disribuion cener for several sores or cusomer poins wihin regions. In each region, he warehouse has an auhoriy o conrol is downsream members. The moivaion for his arrangemen is ha he unique demand behavior of each region is bes undersood and responded o a he regional level. This configuraion is popular wih several large and medium reailing chains and induces several imporan decision-making problems in supply chain managemen. As noed by Dooli e al. (2005) he configuraion of he supply chain arrangemen is essenial for business o pursue a compeiive advanage and o mee he marke demand. In his paper we sudy he invenory replenishmen problem as i relaes o regional supply chains. Specifically we solve he invenory raioning problem of a single ype of producs o M-sores from a single regional warehouse wih a ruck consrain. Figure 1 illusraes he concerned supply chain, which is modeled from a small frozen-food supply chain in Bangkok, Thailand. A regional warehouse receives periodical invenory replenishmen of frozen-food from facories, which are conrolled by he naional uni (Head Office). The naional uni uilizes a naionwide planning sysem (e.g. ERP) ha deermines he shipping quaniy and ime. They are based on invenories in he naional chain, he sales hisory and sales forecass for each re- : Corresponding Auhor

2 An Invenory Raioning Mehod in a M-Sore Regional Supply Chain Operaing under he Order-up-o Level Sysem 81 gion. Selecively, fixed-cycle (e.g. weekly, monhly) invenory replenishmens, known as he R, S sysem wih predeermined order-up-o level is adoped. The regional warehouse hen operaes a limied number of rucks ha ransfer producs o M regional sores. The socking poins are herefore M+1 and he produc ransfer among sores or back o he regional warehouse is no allowed. Each sore experiences an uncerain cusomer demand, and a los sale occurs when cusomer demand is no saisfied. Since he oal invenory in he regional supply chain is predeermined by he naional planning policy, he primary obecive of he regional warehouse is o minimize he oal number of los sales wihin he region. The replenishmen policy mus herefore posiion he invenory o mee cusomer demand, while a he same ime holding back invenories so as o be able o respond quickly when he demand a a sore rises. Obviously, when he available invenory a he warehouse far exceeds he regional demand hen he decision is rivial because each sore will be oversocked. Neverheless, when i is no sufficien he invenory raioning decision is vial because i affecs sysem performance. To simplify he analysis wihou loss of generaliy, he following addiional condiions are made: (i) a single ype of producs from he ineresing indusry is seleced (ii) he cusomer demand is independen, saionary and normally disribued (iii) he regional warehouse operaes only one ruck, (which can make only one replenishmen rip o only one sore per period ) (iv) sorage capaciies a any socking poins and on he ruck are sufficien (v) invenory replenishmens occur a he sar of each period and are available in ha period (vi) shipmen quaniy from he sources o he regional warehouse is known in advance and (vii) The sore o replenish is seleced based on is maximum expeced shorage The main concern in our research is he deerminaion of he replenishmen quaniy afer selecing he ruck desinaion so as o minimize he oal los sales in long run. In secion 2 we mahemaically describe he regional supply chain replenishmen problem. Secion 3 proposes a new invenory raioning rule and also presens exising invenory raioning rules. Secion 4 presens he soluion procedure. In secion 5 numerical resuls o evaluae he relaive performance of he proposed mehod agains he exising mehods are presened. Secion 6 concludes he resuls. 1.2 Relaed Lieraure So far, he invenory raioning problem has received much aenion from researchers. Earlier work on his issue belongs o Veino (1965), he focused on periodic review sysems. Veino analyzed an invenory raioning wih he concep of criical levels. His model was concerned wih muliple demand classes, zero lead ime, and backordering policy. Topkis (1968) sudied a similar model, bu he considered boh backorder and los sale siuaions. His dynamic programming showed ha, under cerain condiions, a base sock policy was opimal and he opimal raioning policy could be specified by a se of conrol limis. Kaplan (1969) made similar analysis bu for wo classes of cusomers. Nahmias and Demmy (1981) also specified criical levels for several demand classes. They firs sudied a single period invenory model, where demand was accouned a he end of a period hen exended o a muliple-period model. Demand was saisfied only if he invenory level was above he criical level, his way i was possible o reserve sock for possible fuure higher-prioriy demand. A woechelon divergen supply chain was firs considered by Eppen and Schrage (1981). Their model conained many Periodic replenishmens from naional supply chain Single Truck can make one sore replenishmen per period REGIONAL WAREHOUSE STORE-1 STORE-2 Produc sales for each sore is independen and normally disribued NATIONAL SUPPLY CHAIN REGIONAL SUPPLY CHAIN STORE-3 DECISION: WHICH STORE TO REPLENISH WITH WHAT QUANTITY? Figure 1. The single ruck M-sore regional supply chain replenishmen model.

3 82 Chumpol Monhaipkul assumpions: such as (i) he depo did no mainain invenories (ii) he holding-penaly cos raios a cusomers were idenical (iii) he cusomer demand was normally disribued (iv) lead imes of he cusomers were idenical and (v) for each allocaion, he depo received enough maerial from he supplier o be able o allocae he maerial o each cusomer such ha an equal fracile poin was achieved (presenly known as balanced invenories assumpion). Under hese assumpions, Eppen and Schrage (1981) derived an opimal order-up-o-policy a he depo, assuming no se-up coss. In case of fixed seup coss, an approximae opimal policy was derived based on an equal-sockou probabiliies allocaion rule. Federgruen and Zipkin (1984) relaxed some assumpions of Eppen and Schrage (1981). They saed he holding-penaly cos raios could be varied. The cusomer demand could be oher classes of demand disribuions (e.g. Erlang and gamma disribuions). The maerial allocaion was obained by solving a myopic allocaion rule, which aimed o minimize he expeced coss in he period ha he allocaion acually ook effec, ignoring coss in all subsequen periods. Zipkin (1984) proposed a dynamic program o allocae invenories from a single depo o many demand poins. His model was concerned wih normally disribued demands wih backlogging of unfulfilled demand. The oal cos included invenory holding coss and backordering coss. Jackson (1988) also sudied a sock allocaion in a woechelon disribuion sysem. He showed significan effecs of risk pooling beween replenishmens in a singlecycle model where he iniial sock level was given. In his model, he replenishmen cycle was divided ino many equal inervals. A he sar of each inerval, he invenory posiions of he local sockpoins were raised o heir order-up-o levels if sufficien cenral sock was available. McGavin e al. (1993) also considered allocaion policies in a single replenishmen cycle wih given iniial sock. Various ways were examined o exploi he effec of risk pooling beween replenishmens. They concluded ha benefis could be obained using a simple 50/25 heurisic he replenishmen cycle conained wo shipmen opporuniies, 25% of he mean replenishmen cycle demand was kep a he cenral depo for a second shipmen afer 50% of he replenishmen cycle had passed. Boh Jackson (1988), and McGavin e al. (1993) did no consider he issue of conrolling he oal sysem sock by adequae replenishmens o he cenral depo. A similar model wih a differen cos srucure, including he coss of shipmens beween he cenral depo and local warehouses, was analyzed by GÜllÜ and Erkip (1996). Wih a long hisory in his area an exensive review on divergen muli-echelon sysems can be seen in Diks e al. (1996). The auhors reviewed invenory raioning mehods, namely, Fair Share (FS) raioning, Appropriae Share (AS) raioning, Consisen Appropriae Share (CAS) raioning, Prioriy raioning, and oher raioning rules. The FS raioning policy raions he available maerial so as o mainain all end-sockpoins a a balanced posiion: all end-sockpoins have he same non-sockou probabiliy. The AS is o ensure ha a prespecified arge service level can be aained a a reailer. In he AS, he allocaion fracions can be chosen freely. In a resriced version of AS, which is called he CAS mehod, he allocaion fracion is chosen such ha he raio of he proeced ne invenory a any reailers over he sysem-wide proeced ne invenory consan a any ime. The Prioriy raioning uses a lis of reailers and raions from he available invenory so as o saisfy hem in he sequence hey were lised. The CAS rule is also inroduced in De Kok (1990). The auhor analyzed a woechelon divergen model wih sockless depo and ook ino accoun he service crierion (fill raes), which was more cusomer-oriened and herefore pracically more imporan. The similar models which allow he depo o mainain sock were analyzed by Seidel and De Kok (1990) and Verrid and De Kok (1996). The adapaion of he CAS policy o a one-depo/muli-reailer invenory sysem is presened in Diks and De Kok (1996). They used he Mone-Carlo simulaion o validae some heurisics which were used o deermine inegral orderup-o-levels, parameers of allocaion policy a he depo and of he rebalancing policy a he reailers so as he desired service levels were aained a he reailers for minimal expeced oal coss. Lagodimos (1992) proposed models o measure sysem service performances of woechelon divergen producion neworks consising of a cenral socking poin feeding a number of end socking poins. The sysem operaed under periodic review ordering policies and used fair sharing and prioriy rules when he cenral socking poin had insufficien invenories o cover is demands. Diks and De Kok (1999) exended he sudy o cover a divergen muli-echelon invenory sysem. They proposed a decomposiion-based algorihm o deermine he conrol parameers of a nearopimal replenishmen policy: he order-up-o-levels and he allocaion fracions. Diks and De Kok (1999) considered boh invenory holding cos and penaly cos for backorders. The exension version o cover muli-echelon sysems wih only end-sockpoin were allowed o hold sock was also sudied by Verrid and De Kok (1995). A divergen muli-echelon invenory sysem in which every member was allowed o hold sock was analyzed in Diks and De Kok (1998). Each node was responsible for allocaing maerials among is successor periodicalfixed lead ime. The model concerned fixed lead ime, periodical orders, and backlogging. The obecive funcion was o minimize he average coss per period in he long erm, which included penaly and invenory holding coss. A more complex sysem named Supply Chain Operaions Planning Problem (SCOP) was sudied by De kok and Fransoo (2003). SCOP aimed o coordinae he release of maerials and resources such ha supply chain coss were minimized under he required cusomer service. The auhor proposed he so-called Synchronized Base Sock (SBS) policy and compared i agains he

4 An Invenory Raioning Mehod in a M-Sore Regional Supply Chain Operaing under he Order-up-o Level Sysem 83 LP-based planning sysem. The SBS ouperformed he LP-based sysem because i used a more sophisicaed linear-allocaion invenory raioning rule. This rule coincides wih he one derived in Van der Heiden (1997), which is named Balanced Socking (BS) raioning. Van der Heiden (1997) proved ha he BS was accurae and more robus han he CAS raioning rule. The BS consiss of wo parameer ses: a se of raioning facors a he depo wih he purpose o minimize invenory imbalance and a se of order-up-o levels a he local sockpoin wih he purpose o achieve he arge fill-rae. If sufficien cenral sock is available, he invenory posiion of each local sockpoin is raised o he arge level and he remaining par is kep a he cenral depo for subsequen shipmens. If he cenral sock is less han required, hen he local invenory posiions are only raised o he arge level minus a raioning facor muliplied by he proeced shorage. An exensive numerical experimen o prove he effeciveness of he BS mehod on general muli-echelon disribuion sysems was performed by Van der Heiden e al. (1997). Van der Heiden (1999) considered shipmen frequencies on a woechelon divergen supply chain consising of a cenral depo and muliple non-idenical local warehouses. He showed ha a decomposiion approach used in Van der Heiden (1997) could be used o solve his model. Van der Heiden (2000) deal wih he opimizaion of sock levels in general divergen neworks operaing under a periodic-review/order-up-o level policy. The obecive was o reach arge fill raes wih minimum invenory holding cos. A procedure o deermine he conrol parameers was also proposed and hen esed by exensive numerical experimens. Recenly, a model wih several demand classes, as firs sudied by Veino (1965) was sill being invesigaed. Melchiors e al. (2000), and Melchiors (2003), analyzed an (s, Q) invenory model wih uni Poisson demand, several demand classes and los sales. He proposed he so-called Resriced Time-Remembering (RTR) policy, where he criical levels were allowed o be varied depending on ime. Deshpande e al. (2003) used a hreshold invenory raioning policy for service differeniaed demand classes. The cusomer service level was also furher analyzed by Zhang (2003). He considered muli-cusomer service level requiremens when he Prioriy Raioning rule was applied. He demonsraed ha he opimal purchasing quaniy for he wo-cusomer problem migh be found hrough a line search. For muli-cusomer problems, a formula o find a near-opimal purchasing quaniy was derived. Ayanso e al. (2006) also used a hreshold level invenory raioning policy. They considered non-perishable make-o-sock producs in he conex of e-commerce. The exension of he invenory raioning issue o cover he producion sage can be seen in Lee and Hong (2003), Huang and Iravani (2007), Ha (1997a, b, 2000), De Véricour e al. (2002), and Axsäer e al. (2004). Ha (1997b) considered he problem of sock raioning in a capaciaed make-o-sock producion sysem wih muliclass demand and los sales. He showed ha he hreshold-ype policies were opimal for boh producion and raioning decisions. Similar resuls were provided in Ha (1997a) for sysems wih wo demand classes and backlog. De Véricour e al. (2002) exended Ha s mo-del o muliple demand-class models. They demonsraed ha he benefi of invenory pooling could be realized only if he sock was efficienly allocaed. Axsäer e al. (2004) sudied a similar model in a coninuous review wo echelon invenory model. Lee and Hong (2003) sudied a producion model wih a (s, S) conrol policy. The producion ime o process an iem was assumed o follow a 2-phase Coxian disribuion. A sock raioning was se for each demand class and he demands were los if he invenory level was lower han he predeermined raioning level. The producion sysem was modeled as a coninuous-ime Markov chain, and an efficien algorihm o calculae he seady-sae probabiliy disribuion was hen proposed. The average operaing cos under he sock raioning policy was compared wih he one wihou sock raioning. Huang and Iravani (2007) considered a wo-echelon capaciaed supply chain wih wo non-idenical reailers and informaion sharing. They discussed he benefis of he opimal sock raioning policy over he firs come firs served (FCFS) and he modified echelon-sock raioning (MESR) policies. The lieraure review reveals ha while several papers consider he divergen supply chain, especially he wo-sage supply chain wih one upsream and muliple downsream members, hey are primarily focused on he invenory and disribuion coss, which ypically include invenory holding, invenory ordering, shipmen, and sockou coss. The problem sudied in his aricle differs in ha he invenory holding and disribuion coss are no a main concern. I is because he auhoriy faces a facing problem and his or her main concern is o provide he region wih as many producs as reasonably possible wih regard o he cusomer loyaly. Therefore, he main focus is on he los sale cos, which herefore affecs he replenishmen problem. More usificaion of his seing can be seen in Monhaipkul (2006). Furher, all previous sudies on he invenory raioning problem never considered he vehicle consrain, while his issue is of concern in his paper. This vehicle consrain is significan because i affecs he invenory replenishmen sysem in erm of some sores canno be replenished simulaneously due o lack of rucks. Thus, benefis of raioning some socks o hese sores need invesigaion. Finally, he exising lieraure normally solves he invenory raioning problem by using a criical level or a prioriy lis, equalizing he sockou probabiliy or arge service levels among sockpoins, solving a dynamic program, muliplying by a raioning fracion, uilizing he second shipmen opporuniy. In his paper, an invenory raioning rule based on expeced cos minimizaion is proposed.

5 84 Chumpol Monhaipkul 2. PROBLEM DESCRIPTION 2.1 The regional supply chain invenory replenishmen problem (RSIRP) The regional supply chain model under consideraion is illusraed in figure 1. I consiss of a cenralized regional warehouse supplying a single ype of produc o muliple regional sores (M-sore) wih a single ruck, which can make a round rip per period. The regional warehouse gahers he invenory saus of he enire regional supply chain and hen decides which sore in he region o replenish and also deermines he replenishmen quaniy. The following noaions are used o mahemaically describe he main decision problem a he regional warehouse. Noaion: sore index ( = 1, 2, 3,, M) ime period index ( = 1, 2, 3,, T) I invenories a he regional warehouse a he end of period V invenories a sore a he end of period A shipmen quaniy arriving a he regional warehouse a he sar of period, i is noed ha A is known in advance because he Head Office mus place orders a he facories in advance due o he order processing and ransporaion lead-ime. μ D mean cusomer per period demand for he produc a sore σ D sandard deviaion of cusomer demand per period for he produc a sore d acual demand or sales a sore in period L Los sales (sock ous) a sore in period C Cos of a los sale (sock ou) per uni a sore Main decision variables a he regional warehouse: X a binary variable represening which sore he ruck replenishes in period (if sore is supplied hen X = 1 else X = 0 ) Q replenishmen quaniy shipped o sore in period A he end of each period, he planner learns he invenory saus as [I, V = 1 o M]. A any poin in ime d is known for he hisory periods and unknown for he fuure periods, bu i can be described by he mean μ D and sandard deviaion σ D. Since parameers of he periodic review order-up-o level sysem are predeermined in a higher-level sysem configuraion (see secion 2.2), hence minimizing he invenory levels is no a primary obecive. Also, he supply chain operaes wih only one ruck, which can make only one round rip per period, he invenory posiioning o he righ place wih he righ quaniy o minimize he los sale cos ψ becomes a key performance meric. Hence, he obecive funcion of he regional warehouse can be expressed as an equaion 1. The ransporaion cos is excluded from he model because i is dominaed by he los sale cos. Thus, frequen replenishmens o sores so as o posiion invenories for a minimum ψ is reasonable. Minimize he los sale cos (ψ) = Min Σ C L (1) I is clear from he ruck consrain ha in any period a mos one X can be non-zero. Thus, he RSIRP a he regional warehouse can hen be described as selecing X and Q so as o minimize ψ given he curren sae [I, V ]. 2.2 The pre-designed order-up-o level sysem I is assumed ha he supply chain operaes under he periodic review order-up-o level sysem (R, S sysem). The warehouse invenory is replenished in every fixed-inerval FI (e.g. FI = week) from a source ouside he region, bu he sore invenory can be replenished daily by he single ruck. The invenory replenishmen a each sore occurs a he sar of each period and is available in ha period. Le S be he pre-designed order-upo level for a sore. I can be inuiively derived by he equaion 2. S = Mμ D + λ σ D M where λ 1 (2) Noe ha in his paper he simple parameer seing of S is adoped (for more advanced mehod, see Schneider and Rinks, 1991 and Schneider e al., 1995). The above equaion is modified from he so-called Modified Base Sock (MBS) policy which is clearly explained in De Kok and Fransoo (2003). I proecs ha he nex replenishmen o he sore will be M periods laer. S is herefore se equal o he cumulaive expeced demand plus a safey facor over he nex M periods. The safey facor a sore is represened by λ, and can be seleced o mee a specific sockou probabiliy. Similarly, he pre-designed order-up-o level a he warehouse S w mus be equal o (FI+TL) μ W + λ W σ W FI + TL, where μ W and σ W are average aggregae demand and sandard deviaion across all downsream members, respecively, TL is he sum of order processing ime and ransporaion lead-ime from he exernal source o he warehouse, and λ W is a safey sock facor a he warehouse. 2.3 Sore selecion and he concerned invenory raioning problem As menioned earlier, only one sore can be seleced a each period (Σ X = 1 for each period ). A mehod based on he basis of maximum expeced shorage is adoped here o deermine he variable X. Deails of such a mehod can be found in Monhaipkul (2006). I is briefly explained as follow. Deerminaion of X Le ES be he expeced shorage cos for sore in period. X is herefore deermined as follows:

6 An Invenory Raioning Mehod in a M-Sore Regional Supply Chain Operaing under he Order-up-o Level Sysem 85 ES = V, 1 P(y){(y -V -1 ) C }dy (3) X *, = 1 ES *, = Max {ES } else X = 0 (4) where P(y) is he normal probabiliy densiy funcion for he disribuion N(μ D, σ D ). Equaion 3 represens he expeced shorage cos a sore which is deermined by an inegraion funcion over he lower limi V -1 o infiniy. Equaion 4 shows ha he seleced sore mus have maximum ES. Once a sore is seleced hen he replenishmen quaniy Q mus be derived. I is obvious if he invenory a he warehouse is sufficien, he invenory raioning is no a mus. Neverheless, if he warehouse has limied invenories, an invenory raioning rule mus be imposed. I is because posiioning invenories o a wrong locaion wih wrong quaniy reurns poor sysem performance. This paper aims o propose a new invenory raioning rule, which will be presened in he nex secion, and o compare i agains he exising invenory raioning rules. 3. INVENTORY RATIONING RULES This research proposes a new invenory raioning rule. I s performance has also been compared o he exising invenory raioning rules. Even hough here are a lo of invenory raioning rules repored in he lieraure (see secion 1.2 for deails), none of hem is developed based on he concerned siuaion. Thus, his paper selecs some ineresing invenory raioning rules from he lieraure and applies only he main ideas behind hose rules for he curren circumsance. 3.1 Exising Invenory Raioning Rules The Curren Pracice rule (CP) The Curren Pracice rule is used in he frozen-food supply chain in Bangkok, Thailand. Basically, when his rule is adoped he produc will be ransferred o he seleced sore as much as possible unil is order-up-o level is reached. This rule represens he neglec of he invenory replenishmen opporuniy of he oher sores in he subsequen periods. The Fill-Rae Based Fair Share raioning rule (FRBFS) The FRBFS is seleced because i is a modified version of he Fair Share rule, which is widely sudied by many researchers (see Dik e al., 1996). The FRBFS conains wo main seps as follows (for deails see Monhaipkul, 2006); a) Calculaing he invenory sufficiency indicaor The invenory sufficiency indicaor is calculaed as a funcion of (i) S, (ii) PIA -he oal proeced available o promise invenory a he regional warehouse for he nex M-1 periods, and (iii) PSD -he oal proeced sore replenishmen demand for he curren period. I is noed ha PSD is calculaed based on he presen ime oher han he nex M-1 periods, because one canno deermine he demand a he fuure periods due o unavailabiliy of he on-hand invenories a he fuure periods. The second and hird facors are derived as follows: PIA = I -1 + Σ τ =, +M-1 A τ (5) PSD = Σ M {S - V -1 } (6) For he curren insance, he supply-demand raio given by PIA /PSD is indicaive of how consrained he replenishmen problem is. If he raio is greaer han or equal o one hen he regional warehouse is considered as having sufficien invenories, and he shipmen quaniy Q is seleced o mee he S arge. If he raio is less han one hen he regional warehouse is considered as having limied invenories, and he shipmen quaniy Q mus be re-calculaed by sep b). b) Deermining he replenishmen quaniy Q Since he raioning rule has effecs when he raio PIA /PSD < 1. The FRBFS suggess he deducion of he original shipmen quaniy Q by he raio PIA /PSD. Mahemaically, we have Q = (PIA /PSD )(S -V -1 ), where PIA /PSD < 1 (7) The Balanced Sock raioning rule (BS) The BS rule is seleced because i ouperforms many exising rules in he lieraure (see Van der Heiden, 1997 and De kok and Fransoo, 2003). This rule is applied only if PIA /PSD < 1. The general concep of he Balanced Sock raioning rule can be seen in Van der Heiden (2000). Briefly, when he shorage a he warehouse equals x-for his paper, x = PSD -PIA, when PIA / PSD < 1-he invenory level of is successors are raised o he levels S -p x, where p are raioning fracions such ha Σ p = 1, calculaed as seen in equaion 8. p = {1/2M } + { (σ D ) 2 / 2 Σ (σ D ) 2 } (8) 3.2 A Proposed Invenory Raioning Rule The Expeced Cos Minimizaion raioning rule (ECM) The ECM is a myopic allocaion rule, i aims o minimize he expeced cos in he period ha he allocaion akes effec and ignore all coss in all subsequen periods. Similar o he BS, his rule is applied only if PIA /PSD < 1. The ECM is applied by solving he problem P for each period. The problem P: Minimize Σ ES* (9) s.. ES* = P(y){(y-V V + Q -1 +Q )C }dy (10), 1,

7 86 Chumpol Monhaipkul Σ Q = I -1 + Σ τ =, +M-1 A τ (11) Q 0 for all (12) The obecive of funcion 9 is o minimize he overall expeced shorage of he enire supply chain. Equaion 10 shows a cumulaive cos funcion o esimae he shorage cos. The sign of Q is a plus sign because i represens a shipping quaniy of producs which are ransferred o he sores in order o increase he invenory levels of he sores. Therefore, he equaion 10 represens he expeced cumulaive cos afer ransferring he producs o each sore. Equaion 11 shows ha he oal proeced shipping quaniy mus be equal o he oal proeced available o promise invenory a he regional warehouse for he nex M-1 periods. Consrain 12 is he non-negaive condiion. A procedure o solve he problem P is presened in he nex secion. I is noed ha he Q is finally equal o all producs available in period (I -1 +A ) if i exceeds he sum of curren invenories I -1 plus shipmen quaniy A. 4. THE SOLUTION PROCEDURE Figure 2 presens an ieraive approach o solve he problem P. The drawback behind he flow char is he selecion of a giver and a aker from all sores. The giver will dedicae an amoun of is Q (represened by SCALE α) o he aker such ha he overall expeced shorage cos of he enire supply chain is decreased. The giver will be seleced if he increase in is expeced shorage cos ES* is a leas. In conras, he aker will be seleced if he decrease in is expeced cos is a mos. The giver-aker selecion process is sopped when here is no improvemen in he obecive funcion 9. The sepby-sep explanaion is as follow. Sep 1 Deermine he iniial soluion Q for all as a weigh raio [{S -V -1 } / Σ M {S -V -1 }] muliplied by he oal proeced available o promise PIA Sep 2 Deermine ES* for all using equaion 10 by a numerical inegraion Sep 3 Deermine he SCALE α. Iniially, α is se o 50% of Min {Q }. I is hen changed o Max {0.5α old, UM}, where UM is he smalles uni of measure of he produc, when new value of Σ new* ES* (Σ ES ) is greaer hen he old value of old * Σ ES* ( Σ ES ) Sep 4 Change he old α o he new α according o sep 3 Sep 5 Sop if he sopping condiion is saisfied. The new* procedure sops when α = UM and Σ ES > old * Σ ES Sep 6 Selec a GIVER and a TAKER. The GIVER will dedicae is fracion of Q (by he amoun of α) new* o he TAKER such ha Σ ES will be less han old * Σ ES. The GIVER is seleced only if is Q > 0 and he associaed erm ΔG is a leas among all sores. In conras, he TAKER is seleced when he relaed erm ΔT is a mos among all candidaes, where ΔG = ( V, 1 + Q, P(y){(y-V -1 +Q - )C }dy) - ( V, 1 + Q, P(y){(y-V -1 +Q )C }dy) (13) ΔT = ( V, 1 + Q, P(y){(y-V -1 +Q )C }dy) - ( P(y){(y-V -1 +Q + )C }dy) (14) V, 1 + Q, + In case of GIVER = TAKER, he GIVER will be mainained and a new TAKER which has he second maximum ΔT will be seleced. Sep 7 Decrease he GIVER by α and increase he TAKER by α, go back o sep 2 Sep 8 Keep Q for all which can give minimum Σ ES* as he final soluions 5. NUMERICAL EXPERIMENTS AND RESULT Performance of he proposed raioning rule was evaluaed agains he selecive raioning rules. The numerical experimens were conduced based on a real siuaion of a seleced supply chain in Bangkok, Thailand, namely he frozen-food supply chain. 5.1 Experimenal Base Case The concerned supply chain consiss of one regional warehouse and muliple sores supplying over 50 produc ypes o is cusomers. To simplify he analysis, he experimens were concerned wih only hree selecive sores (M = 3) and one ruck which was dedicaed o one selecive produc ype. The produc was seleced in which is cusomers demand a each sore could be assumed o follow normal disribuions (in he long erm). From he real daa se, he mean demands, sandard deviaions, and sock ou coss a he hree sores were as follows, μ D = {428, 418, 423), σ D /μ D = 0.1, and C = {6.00, 8.50, and 7.00}. The facor λ was se o 1.5 for all sore. The iniial invenories a he hree sores were V 0 = {450, 500, 475). The invenory replenishmen cycle a he regional warehouse was se o five (FI = 5). This was equivalen o a five-day delivery schedule. The order processing plus ransporaion leadime from he exernal source o he warehouse was equal o wo days (TL = 2). Thus, he Head Office mus place an order o he facories wo days in advance. While placing an order, he Head Office used he curren invenory saus of he regional supply chain. Our base problem used he safey facor λ W = 1. The iniial inven-

8 An Invenory Raioning Mehod in a M-Sore Regional Supply Chain Operaing under he Order-up-o Level Sysem 87 Q, 1. Deermine iniial soluion of for all -For each S V 1 Le Q = ( PIA ) ( ) ( S V ) M 1 -For each * 2. Deermine ES for all Le * ES = V, 1 + Q, P( y){( y V 1 + Q ) C } dy 4. Change YES 3. Change SCALE ( )? -Deerminaion of Define UM= Uni of measure of produc Deermine a) when saring he flow char min = ( )(50%) Q NO b ) when changing = max{ (50%)( new old ), UM } 8. Keep Q, which give minimum as final soluions ES * YES 5. STOP? NO new* Noe: he is changed when ES > -The sop condiion, The flow char is sopped when = UM new* old * and ES >, ES, ES old*, 6. Selec Q, 6.1 Selec a GIVER 6.2 Selec a TAKER 7. Find new soluions of by Q, 7.1 Decrease of 6.1 by If < 0 hen = /2 Q, 7.2 Increase of 6.2 by Q, Q, Figure 2. A flow char o solve he problem P. ory a he warehouse was 4500 (I 0 = 4500), and he experimen was run o cover 20 periods (T = 20). 5.2 Experimens In his paper, here were wo main experimenal obecives which would be proved by hree main ess. The deails are described as follows. 1 s Experimenal Obecive: Raioning Rule Performance Measuremen The firs numerical es was o compare efficiencies among he raioning rules under he base case. Afer he 20-day run, he following numbers were recorded for each rule; he oal los sale cos, he oal producs remaining in he sysem (REMAIN), and he difference beween he oal cusomer demand and he oal prod-

9 88 Chumpol Monhaipkul ucs enered ino he sysem (DIFF). The experimen was repeaed unil he 95%-confidence inervals of he observed average numbers did no exceed heir 5% errors. For he general purpose, he second es was performed o evaluae he efficiencies of he raioning rules under various parameer seings. The es was conduced wih 16 differen ses of mean cusomer demand μ D, sandard deviaion σ D, and cos of a los sale C. They were re-generaed wihin he 50% gap of hose used in he base case. For each parameer seing, he experimen was also repeaed unil he 95%-confidence inervals of he observed average numbers did no exceed heir 5% errors. 2 nd Experimenal Obecive: Effecs of Ineresing Parameers The hird numerical es was o sudy he effec of ineresing parameers, namely, cos of a los sale C and sandard deviaion σ D. The base case was used and hen he los sale cos C was varied ±40% (sep by 20%), while he oher parameers were kep consan. The effec of he sandard deviaion σ D was hen sudied. The raio σ D /μ D was increased from 0.1 o 0.5 (sep by 0.1). 5.3 Resuls Table 1 concluded he average numbers of he oal los sale coss from he 1 s and 2 nd numerical ess. The base case was shown on he firs row. The remaining 16 cases were he resuls of he 2 nd ess wih 16 differen ses of parameer seings. One-way ANOVA was applied and i could be concluded ha he raioning rule significanly affeced he oal los sale coss a a significan level of The muliple comparison es, known as he Duncan Mehod, was hen preformed o classify he raioning rules ino homogeneous subses. Their ranks were shown by he numbers in he parenheses. The review of able 1 showed ha he ECM ouperformed oher rules. I was always in he firs rank. The FRBFS and BS were almos in he second rank. Ineresingly, he CP rule, which was currenly used in he frozen-food supply chain in Bangkok, Thailand, provided he wors resuls. In fac, he CP rule represened ha he manager did no realize benefis of raioning some socks o some oher sores o ake advanages of he ransshipmen in he subsequen periods. I was proved by his research ha raioning some appropriae socks for fuure ransferring was beneficial and he curren pracice of he supply chain should be reviewed. The FRBFS and BS were poorer han he ECM because hey someimes allocaed oo less or much sock for he seleced sore. This was a resul of he calculaion which ook ino accoun of he demand disribuion funcions. The remaining sock was also kep a he regional warehouse due o lack of ruck. This remark coincides wih he paper of Van der Heiden (1997). Table 2 presened he average numbers of DIFF and REMAIN. DIFF numbers were obained by he oal Table 1. Toal los sale coss (average numbers) from he 1 s and 2 nd ess. Toal los sale cos (average numbers) ECM-Rule FRBFS-Rule BS-Rule CP-Rule Base Case 7462(1) * 7681(2) 7680(2) 7914(3) Case (1) 9006(1) 9147(2) 9405(3) (1) 7222(2) 7217(2) 7416(3) (1) 8704(2) 8661(2) 8930(3) (1) 6504(2) 6552(2) 6698(3) (1) 5243(1) 5316(2) 5488(3) (1) 5878(2) 5879(2) 5999(3) (1) 4713(1) 4887(2) 4995(3) (1) 6447(2) 6433(2) 6566(3) (1) 4095(2) 4120(2) 4124(2) (1) 6368(2) 6372(2) 6507(3) (1) 6543(2) 6535(2) 6746(3) (1) 5004(2) 5014(2) 5033(2) (1) 5520(2) 5545(2) 5699(3) (1) 4121(2) 4126(2) 4248(3) (1) 7405(2) 7382(2) 7698(3) (1) 4492(2) 4468(2) 4599(3) Noe) * represens he rank number of he rule classified by he muliple comparison es-duncan Mehod.

10 An Invenory Raioning Mehod in a M-Sore Regional Supply Chain Operaing under he Order-up-o Level Sysem 89 cusomer demand minus he oal supplied producs. The posiive number represened he insufficiency of he oal supply oward he oal demand. Noe ha DIFFs of he FRBFS, BS, and CP rules (columns 4, 6, and 8) were presened in erms of percenage gaps which were deermined using DIFF of he ECM as a base. For example, he -5% gap of he CP rule in he base case mean ha is DIFF was 5% less han ha of he ECM. REMAIN numbers showed he remaining producs in he sysem afer he 20-day running. The posiive numbers indicaed ha some producs were lef in he sysem and no sold o he cusomer. Table 2 provided some observaions as follows. (a) DIFF values could indicae he insufficiency of he sysem because hey were all posiive. Thus, raioning rules were a mus because he sysem was no oversocked. (b) CP-DIFFs were all less han ECM-DIFFs. These resuls implied ha more producs were always supplied o he sysem when he CP rule was used insead of he ECM rule. I was because he oal demand was kep consan for each replicaion across all he ess. A lower number of DIFF represened a smaller gap beween he oal demand and supply and consequenly referred o a higher supply level in he sysem. This observaion amplified he poor performance of he CP rule. I acquired more producs, bu reurned higher oal los sale coss. (c) DIFFs of FRBFS and BS rules could be more or less han hose of he ECM. I mean ha i was possible ha he sysem acquired more or less producs when he FRBFS or BS was in place. (d) REMAINs were all posiive for all raioning rules. I signified ha here were always producs remaining in he sysem afer he 20-day running even hough he sysem acquired insufficien producs. The remaining producs could be anywhere in he sysem and no sold o he cusomers. This condiion implied here were always shorages in he sysem because producs could be ransferred o wrong places, or a he wrong imes wih wrong quaniies. This phenomenon was a resul of demand uncerainy and could no be eliminaed by any raioning rules. Figure 3 and Figure 4 exhibied he average-number graphs of he oal los sale coss from he 3 rd numerical es when he parameer C and he raio σ D /μ D was varied, respecively. In Figure 3, he horizonal axis represened facors o muliply he parameer C. For example, he number 1 referred o he base case. The number 0.8 represens he sensiiviy analysis of he parameer C when 20% downside was concerned. In Figure 4, he horizonal axis represens he change of he raio σ D /μ D. For example, he number 0.1 referred o he base case. The number 0.2 represens he sensiiviy analysis of he raio σ D /μ D when 100% upside was concerned. Figure 3 and Figure 4 confirmed ha he ECM performance was beer and more robus han he oher rules hroughou he wide ranges of he concerned pa- Table 2. DIFF and REMAIN (average numbers) from he 1s and 2nd ess. DIFF/REMAIN (Average numbers) ECM-Rule FRBFS-Rule BS-Rule CP-Rule DIFF REMAIN DIFF REMAIN DIFF REMAIN DIFF REMAIN Base Case % * % % Case % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % Noe) * represens he DIFF value in percenage erm which is relaive o ha of he ECM rule.

11 90 Chumpol Monhaipkul 15,000 Toal Cos 12,000 9,000 6,000 3,000 ECM-Rule BS-Rule FRBFS-Rule CP-Rule A facor o muliply wih Cos of a Los-sale C Figure 3. Average oal cos graphs from he sensiiviy analysis of C Toal Cos ECM-Rule BS-Rule FRBFS-Rule CP-Rule The raio σd/ μd Figure 4. Average oal cos graphs from he sensiiviy analysis of he raio σ D /μ D. rameers. The CP rule, which was he curren pracice of he selecive frozen-food supply chain in Bangkok, Thailand, reurned poor performances. From Figure 3, i could be observed ha he oal cos had a posiive relaion wih he value of C for all concerned raioning rules because i was increased when C was increased. However, i did no have a relaionship wih he raio σ D /μ D (see Figure 4). 6. SUMMARY We sudied an invenory raioning problem embedded in he regional supply chain invenory replenishmen problem (RSIRP). Firs, he concerned regional supply chain, which operaed under he order-up-o level sysem, was inroduced. I was modified from a small frozen-food supply chain in Bangkok, Thailand. I was supplied by a naional uni hrough a naionwide planning sysem. To simplify he analysis, some condiions were made; he regional warehouse used single-ruck round rips o ransfer a single ype of produc o he sores in is region. When he warehouse had insufficien invenories, a raioning rule mus be imposed. In his paper, a new invenory raioning rule developed based on he logical purpose was compared agains he curren pracice and oher well-known raioning rules in he exising lieraure. Based on he experimens, i could be concluded ha he proposed raioning rule was beer and more robus han he oher rules. The curren research can be exended for fuure sudy in many direcions. The firs exension would be o consider he muliple produc case, which is cerainly a more common scenario in pracice. In such a case he replenishmen conflic will become wo dimensional since need of sores and producs may be in conflic. A second exension would be o increase he number of rucks, indicaing ha more han one sore can be replenished in a period. Typically, as he number of sores increase hen he number of rucks would also increase. Possibly, such a problem can be porioned ino several one ruck M-sore problems. A final exension could be

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