LATERAL TRANSSHIPMENTS IN INVENTORY MODELS

Transcription

1 LATERAL TRANSSHIPMENTS IN INVENTORY MODELS MIN CHEN

2 BMI THESIS Lateral Transshpments n Inventory Models MIN CHEN II

3 PREFACE The BMI Thess s the fnal report to acqure the Master's degree n Busness Mathematcs and Informatcs. Busness Mathematcs and Informatcs (BMI) s a mult-dscplnary master program that s supplyng the students wth the ntegrated knowledge of Busness Economcs, Mathematcs and Computer Scence prncples. It teaches students to solve the management and operatonal problems n the ndustry wth a quanttatve background. Ths BMI thess starts wth the bref ntroducton of lateral transshpments. Dfferent knds of lateral transshpment polces are studed by the researchers. Based on the lateral transshpment models, we are nterested n recognzng how the dfferent models mprove the servce level and cut the total cost by applyng the lateral transshpment polcy. After readng ths thess, one s able to know the detals about lateral transshpment polces, and how these lateral transshpment polces maxmze customer satsfacton and reduce management costs n nventory systems. I would lke to express my sncere grattude to my supervsor Marco Bjvank for hs gudance, encouragement, constructve crtques, enthusasm and patence throughout ths research. He has been a great mentor and gude to me. Ths thess would not have been possble wthout hm. A few words of thanks seem nadequate to express my apprecaton for those who have supported and helped me over the last two years durng my study at VU Unversty Amsterdam, The Netherlands. Mn Chen Amstelveen, 2008 III

4 SUMMARY Ths paper focuses on the lterature study on lateral transshpment polces n nventory models. Emergency lateral transshpments combned wth dfferent types of replenshment polces can help to mprove the servce level of nventory systems. The problem of how to determne the optmal stockng levels under certan servce level constrants s also presented n ths paper. Furthermore, preventve lateral transshpments appear to be more effectve n reducng the rsk of a stock out before the demand s arrvng. The new lateral transshpment polcy (also referred to as servce level adjustment) combnes emergency and preventve lateral transshpments to effcently respond to customer demands. The algorthm how a servce level adjustment polcy can be appled n an nventory system s llustrated and the relevant system cost s derved. The objectve of ths paper s to gve a complete overvew of the dfferent types of lateral transshpment polces n an nventory system, and how they reduce the cost and mprove the servce level. We consder the use of a perodc revew system for ths study, but addtonal studes of lateral transshpments are needed to nclude other characterstcs of the supply chan as well. For example, we have assumed that the shpment tmes from the depot to the bases are the same for all bases. Future work should relax ths assumpton, and ncorporate more realstc and accurate factors. IV

5 CONTENTS Chapter 1 Introducton Research objectve...2 Chapter 2 Emergency Lateral Transshpment (ELT) Introducton Emergency lateral transshpment wth ( S 1, S) replenshment polcy Assumptons and notatons Model formulaton and specfcaton Determnaton of optmal stockng levels Emergency lateral transshpment wth ( rq), replenshment polcy Assumptons and notatons The analytcal model Optmzng reorder pont...12 Chapter 3 Preventve Lateral Transshpment (PLT) Introducton Assumptons and notatons Lateral transshpments based on avalablty Lateral transshpments based on nventory equalzaton System performance analyss...16 Chapter 4 Servce Level Adjustment Polcy (SLA) Introducton Assumptons and notatons Servce level adjustment analyss System performance analyss...22 Chapter 5 Conclusons and Future Work...24 References...25 V

6 Chapter 1 Introducton In an nventory system, spare parts supply chans often consst of multple local warehouses and one or a few central warehouses. Local warehouses, located close to customers, are used to ensure prompt replenshments of spare parts to customers. The central warehouse(s) replenshes the local warehouse, but they can also perform an emergency delvery to the customers when a local warehouse runs out of stock. Snce emergency delveres from one central warehouse to a customer cost a lot of tme and money, alternatve solutons are common n practce. In ths paper, we consder the alternatve of lateral transshpments. Lateral transshpment s defned as a local warehouse whch provdes stocked tems to another local warehouse whch s out of stock or to prevent out-of-stock occurrences. In other words, these local warehouses exchange ther nventory on the same echelon level. Dfferent local warehouses can operate ndvdually and rely on regular or emergency replenshments from the central warehouse when they are out of stock. It may however be more useful to have a qucker backup from other local warehouses as well. Lateral transshpments can be seen a form of poolng. Physcally, there are multple stock ponts, but they have access to each other s nventory when needed. Cost savngs can be expected because of ths poolng. In case a local warehouse s out of stock at the moment of a customer's request, t frst tres to obtan the requred parts from a neghborng local warehouse. Ths costs less tme than an emergency delvery from the central warehouse. In many ndustres and servce organzatons, the relance on two-echelon nventory systems for reparng and supplyng recoverable tems s becomng more and more prevalent. Ths paper focuses on dscussng the two-echelon nventory system nvolvng a central warehouse (or suppler) and multple local warehouses (or retalers) wth lateral transshpments as an opton under the varous nventory replenshment polces. The model s depcted n Fgure 1. Fgure 1: Lateral transshpment n nventory system. 1

7 1.1 Research objectve Lateral transshpments can be dvded nto emergency lateral transshpments (ELTs), preventve lateral transshpments (PLTs), and servce level adjustments (SLAs). ELT mandates emergency redstrbuton from a retaler wth ample stock to a retaler that has reached a stock out. PLT reduces ths rsk by redstrbutng stock to prevent a stock out at a retaler before demand exceeds the nventory level. In other words, ELT responds to actual stock outs whle PLT reduces the rsk of possble future stock outs. The SLA polcy combnes the ELT and PLT polces together to reduce the rsk of stock outs n advance and effcently respond to actual stock outs. In ths paper, we gve a complete overvew on how PLT, ELT and SLA polces play an mportant role n mprovng servce levels and reducng nventory cost. The followng research objectves are studed: To mprove servce levels and reduce nventory cost wth emergency lateral transshpments n a system wth one-for-one ( S 1, S) and (, rq) replenshment polces. To acheve equal margnal cost over all retalers just before the replenshment perod wth preventve lateral transshpments. To effcently respond to customer demands wth a servce level adjustment polcy by ntegratng emergency and preventve lateral transshpments. The remander of ths paper s organzed as follows. In Chapter 2, emergency lateral transshpments wth two dfferent replenshment polces are dscussed. The mathematcal formulatons to calculate the expected number of backorders and the total system cost are developed for each replenshment polcy. Next, the optmal stockng levels are determned based on servce level constrants. In Chapter 3, we gve an overvew and examnaton of the effectveness for preventve lateral transshpments. We propose servce level adjustments usng the servce level n the remanng perod (SLRP) as the servce level n Chapter 4. Fnally, we compare the three lateral transshpment polces n Chapter 5, and gve our conclusons and the drectons for further research. 2

8 Chapter 2 Emergency Lateral Transshpment (ELT) 2.1 Introducton Mult-echelon nventory systems are usually used to provde servce support for products where customers are dstrbuted over an extensve geographcal regon. These systems can be characterzed by lower echelons (bases) that serve as a frst level of product support, and a hgher echelon, consstng of a depot that serves as a second level of support n case the bases are not able to meet the customers demand. The depot can also serve as a dstrbuton center for the replenshment of stock to the bases. When stock s shpped from one base to another when t faces an out of stock occurrence t s called an emergency lateral transshpment. There s a lot of lterature that descrbes the emergency lateral transshpment polcy n nventory models. See for example, Lee [1], Axsäter [2], Sherbroke [3], Alfrodsson and Verrjdt [4], Grahovac and Chakravarty [5], Kukreja et al. [6] and Wong et al. [7]. These papers focus on a one-for-one ( S 1, S) replenshment polcy. Such polces are defned as polces, n whch a replenshment order s placed as soon as the customer wthdraws an tem. S s called the base stock level. It s usually appled n supply chan systems n whch the manufacturer supples expensve, low-demand tems to vertcally ntegrated or autonomous retalers va a central depot. Another type of replenshment polcy s the (, rq) replenshment polcy whch s consdered as a batch orderng polcy, where r s the reorder level and q the replenshment quantty. It means that an order of a fxed sze q s placed whenever the nventory poston reaches the reorder pont r. Ths polcy s a very common and relatvely straghtforward polcy for an nventory system. For contnuous revew (, rq) polcy, Needham and Evers [8], Evers [9], Xu, et al.[10], Axsäter [11] and Huo [12] study how to make transshpment decsons when replenshment parameters (such as the order pont and order sze) are gven. Table 1 presents a bref revew of the prevous studes on mult-echelon nventory systems wth an emergency lateral transshpment polcy. The objectve of ths chapter s to determne the optmal stockng levels at each echelon based on servce performance measures (e.g. expected shortages) for a mult-echelon system wth emergency lateral transshpments. 3

9 Table 1: A bref revew of the lterature on the mult-echelon nventory systems 2.2 Emergency lateral transshpment wth ( S 1, S) replenshment polcy Emergency lateral transshpments from the same echelon level are one of the popular tools to handle stock outs at local warehouses. Snce the cost of a lateral transshpment s generally lower than the shortage cost and the cost of an emergency delvery from the depot. Therefore emergency lateral transshpments reduce the total system cost. Smultaneously, the transshpment tme s shorter than the regular replenshment lead tme and therefore they ncrease the fll rate at the local warehouses. It s used by lots of companes to form a poolng group between retalers to reduce the rsk of shortages and provde better servce at lower cost. For reparable tems wth low demand volume and hgh value, a one-for-one ( S 1, S) replenshment polcy s commonly used n modelng the mult-echelon nventory 4

10 system Assumptons and notatons Consder a mult-echelon nventory system for reparable tems wth one depot (or central warehouse) and M bases (or local warehouses) whch are grouped nto n dsjont groups. Each of these groups s dentcal and shares the same poolng stocks. Let m be the number of dentcal bases n poolng group. A one-for-one ( S 1, S) replenshment polcy s used and emergency lateral transshpments between dentcal bases are allowed. If one of the bases can not satsfy a customer demand wth ts stock on hand, the emergency lateral transshpment polcy wll be appled to fll the demand from another base n the same group whch has enough stock on hand. If the emergency lateral transshpment s mpossble due to group-wde stock out, the demand s backordered. For the stuaton of more than one base wth stock on hand, there are three prortzed sourcng rules [16] to determne the suppler for the emergency lateral transshpments. The three prortzed rules are random choce, choce based on maxmum stock on hand and choce based on the smallest number of outstandng orders. In ths model, we assume the random choce as the man prortzed rule for emergency lateral transshpments. The followng notatons and assumptons are used n ths chapter: the demand rates at the bases n the same poolng group are dentcal, all faled tems can be repared n the depot whch s assumed to have an nfnte number of servers. It s accordng to an exponental dstrbuton wth mean repar tme (1/ μ ) for a faled tem, orders are comng as frst-n frst-out (FIFO) at both depot and bases, the transportaton tme for a lateral transshpment to poolng group s determnstc and denoted as T for a poolng group, demand at each base of poolng group s accordng to a Posson dstrbuton wth rate λ. So, n = 1 λ = m λ represents the total arrval rate to all bases and the depot, the emergency lateral transshpment tmes are substantally lower than the normal resupply tme. Otherwse, the emergency transshpment wll not be chosen, a s the unt cost for a lateral transshpment n poolng group, 5

11 s the average number of emergency lateral transshpments per unt tme at N poolng group. The model can be seen n Fgure 2. Fgure 2: Emergency lateral transshpment wth one-for-one replenshment polcy Model formulaton and specfcaton In ths model, we are nterested n calculatng the expected number of backorders and the quantty of emergency lateral transshpments n order to determne the optmal nventory stockng levels S n the mult-echelon nventory systems. The measurement of the models performance wll be accordng to the expected total number of backorders, the total system cost and the mprovement of the customer servce level. Accordng to Lee s theory [1], the expected number of emergency lateral transshpments N per unt tme at poolng group s equal to the proporton ( α ) of arrvng demands met by emergency lateral transshpments multpled by the total arrval rate ( m λ ) at poolng group : N m = λα. (2.1) The total number of backorders n the system conssts of backorders that are to be met by emergency lateral transshpments at poolng group and to be met by the depot. Followng the well-known result of Palm (1938), the number of tems on order at the 6

12 depot s Posson dstrbuted wth the parameter ( λ / μ ) as long as successve resupply tmes are ndependent. Hence, the expected number of backorders at the depot can be determned as (accordng to the theorem provded by Graves [17] and Axsäter [2]): E( B ) = E( Q S ) λ ( λ/ μ) = k ( k S0)exp( ), (2.2) k= S μ k! 0 where Q s the number of orders outstandng at poolng group at tme t, S0 s the base stock level at the depot, and [ x ] + denotes max[0, x ]. Combnng Equaton (2.2) wth Lttle s formula, whch provdes the expected number of tems n transt for emergency lateral transshpments as NT, the total expected number of backorders ( ) EB at poolng group s derved by Lee [1] as: + E( B ) = E( Q ms ) + NT k mλ ( mλ / μ) = ( k ms )exp( ) + NT, (2.3) μ k! k= ms where S s the base-stock level at a base of poolng group,( = 1,..., n). The frst part of Equaton (2.3) expresses the expected number of backorders at poolng group that are to be met by the depot. The second part s the expected number of tems n transt for emergency lateral transshpments. So, by addng Equaton (2.2) for all poolng groups and Equaton (2.3), we can get the total expected number of backorders nvolved n ths system. In ths secton, we evaluate the performance of nventory system n terms of the overall transportaton, nventory holdng, and backorder costs. Equaton (2.4) expresses the total system cost per unt tme (derved by Lee [1]): n n n C = h( S + ms ) + v E( B) + a N, (2.4) total 0 = 1 = 1 = 1 where h s holdng cost per unt tme, v s the backorder cost per unt backordered. The frst summaton represents all nventory holdng costs. The second summaton 7

13 represents all watng costs or backorder costs that customers transfer to the supply chan. The thrd summaton s the total addtonal cost comng from the emergency shpments and transshpments n the nventory system Determnaton of optmal stockng levels The key component n the desgn of mult-echelon nventory systems for recoverable tems s the determnaton of the base-stock levels at each echelon. There s a two phase method to determne these optmal base-stock levels at the bases n each poolng group and the depot (Lee [1]). The frst phase s to fnd the value of S at each poolng group to mnmze the cost C n Equaton (2.4) for each gven S0 S0 0. The second phase s to optmze the value of at the depot. For a gven S, total the optmal base-stock level S at each poolng group, s the one that mnmzes Equaton (2.5). C ( S S ) = hms + ve( B) + α N 0 + = hm S + ve( Q m S ) + ( vt + a ) m λα. (2.5) * The optmal cost C for poolng group s denoted as C ( S ) for a gven value 0 of S 0, where C ( S ) = mn C ( S S0 ). (2.6) * 0 S By solvng Equaton (2.6), we always fnd the best value of optmal cost. S whch result n the The second phase n the method s to look for the optmal value of wth the followng proposton as stoppng crteron (from Lee [1]): Suppose S 0 s such that S 0 at the depot, n n * * 0 0 = 1 = 1. (2.7) h C ( S ) C ( S ) Then all S0 > S 0 can not be optmal. 8

14 2.3 Emergency lateral transshpment wth (, rq) replenshment polcy The (, rq) nventory replenshment polcy s a generalzaton of the one-for-one or base-stock polcy, and represents a more realstc model of a common manufacturng problem faced by management. Comparng to the one-for-one stock polcy, the (, rq) replenshment polcy s optmal when there are costs nvolved for every orders. The one-for-one polcy s also costly when there s a shppng cost assocated wth each replenshment shpment to the nventory. Emergency lateral transshpments wth an (, rq) replenshment polcy can result n much lower costs and hgh servce levels. In ths secton, we focus on a mult-echelon nventory system wthout poolng groups Assumptons and notatons Consder a mult-echelon model wth a depot (central warehouse) and M bases (local warehouses). Spare parts are stored n dfferent locatons n the system. The demand at each base s accordng to a Posson dstrbuton wth demand rate λ at each base. The (, rq) replenshment polcy s used. Emergency lateral transshpments between each base at a lower echelon level are allowed. When there s more than one base avalable for the emergency lateral transshpments, the base performng the transshpment s assumed to be chosen randomly. In other words, f there are no parts avalable at one base, the part s ordered wth an emergency lateral transshpment from another randomly chosen base whch has stock on hand. The followng notatons and assumptons wll be used: the average lead tme for a replenshment from the depot to base s denoted as L, D( λ ) denotes as lead tme demand, whch s a random varable followng as a Posson dstrbuton wth mean λ L, the order quantty s denoted by q for base, the reorder pont at base s represented by r for the (, rq) replenshment polcy, 9

15 the number of bases s m, the requred hgh servce level s ξ, all the nventory bases are the same. The model can be seen n Fgure 3. Fgure 3: Emergency lateral transshpment wth ( rq, ) replenshment polcy The analytcal model The decson rule n ths model s to fnd the reorder ponts at all locatons that mnmze the sum of nventory holdng and transshpment costs. Wth emergency lateral transshpments, the demand at base can be satsfed under three condtons. The demand at base s met by an emergency lateral transshpment wth probablty α, demand at base s met mmedately by stock on hand wth probablty β, or the demand s met by a backorder wth probablty θ. Evdently, α + β + θ = 1. (2.8) Each base serves the customers of ts desgnated terrtory and controls ts nventory level ndependently and contnuously wth a reorder-pont, fxed order-quantty (, rq) polcy. The base places a replenshment order of sze q at the depot or at another base whch has ample capacty to fulfll all replenshment requests wthn a determnstc replenshment lead tme. 10

16 As defned, base has a postve on hand nventory level durng the proporton β of the tme, and no nventory on hand durng the remanng proporton 1 of the tme. In other words, when the nventory s postve, the base s facng the Posson β demand λ plus a demand from other bases due to emergency transshpments wth an average rate α λ / β. When the base has no nventory on hand, the demand that has to be backordered comes n wth rate αλ / (1 β) whch s the rate λ mnus the demand satsfed by the other bases due to emergency transshpments. Thus, the demand rate at base when t has postve nventory on hand s denoted as: g = λ (1 + α / β ) = λ (1 θ ) / β. (2.9) Smlarly, the demand rate at base when t has negatve nventory on hand s denoted as: η = λ (1 α / (1 β )) = λθ / (1 β ), (2.10) So, the expected number of backorders EB ( ) at a base s expressed as Equaton (2.11) from [14] when there s no postve on-hand nventory level: ( k x)( η L) 1! EB ( ) = q k r+ q k= x k k x= r + 1 ηl k= x ( ) k!. (2.11) Accordng to Axsäter s theory [13] and Huo s theory [14], the reasonable approxmaton for the expected number of backorders n the whole system consstng of the expected number of backorders n depot EB ( 0) and n base, EB ( ), s determned by Equaton (2.12): EB ( ) = EB ( ) + EB ( ) = ( M 1) EB ( ). (2.12) 0 Axsäter s and Huo s methods to approxmate the number of backorders n the whole system E( B ) ndcates a stuaton n whch the system s just about to leave or has just reached a state where no ndvdual base has postve stock on hand. Ths means that one of the bases has no stock on hand and no backorders. Some of the other M 1 bases may have backorders, though, and a reasonable approxmaton for the 11

17 total expected number of backorders n the whole system s accordng to Equaton (2.12) Optmzng reorder pont The objectve of ths secton s to fnd the optmal reorder ponts for all locatons such that the total system cost s mnmzed subject to a certan servce level constrant ξ. The total system cost conssts of nventory holdng, and backorder costs. The average on-hand nventory level at base s denoted as G ( x) when the current nventory poston s x. It s determned accordng to Equaton (2.13) from [14] G( x) = E ( D( λ) x) x 1 k ( λl ) = ( x k)exp( λl ), (2.13) k! k = 0 where D ( λ ) denotes the lead tme demand, whch s a random varable that follows a Posson dstrbuton wth mean λ L at base. Equaton (2.14) expresses the total system cost per unt of tme (from [14]). r+ q 1 C = Mh G ( x) + Mα λ a, (2.15) total q x= r + 1 where h s the holdng cost, a s the lateral transshpments cost per unt. Accordng to Axsäter s theory [13], we assume the order quantty s determned n advance for each base n order to solve Equaton (2.15). After that, we can always fnd the best reorderng ponts r that mnmze the total system cost subject q to a certan servce level constrant ξ n ths system. 12

18 Chapter 3 Preventve Lateral Transshpment (PLT) 3.1 Introducton In ths chapter we consder a poolng group consstng of multple stockng locatons at the same echelon whch places regular orders at a depot accordng to perodc revews wth a base stock replenshment polcy. The local warehouses of the poolng group may resort to lateral transshpments when there s a stock out, or when there appears to be sgnfcant rsk of an mmnent stock out n the near future. The decson whether to move nventores between these stock ponts n a poolng group to antcpate to a possble stock out s consdered to be a preventve lateral transshpment (PLT). There are two major redstrbuton polces towards makng a preventve lateral transshpment decson at the same echelon level: transshpment based on avalablty (called a TBA polcy) and transshpment based on nventory equalzaton (called a TIE polcy). In ths secton, we wll gve an overvew on how these two polces can be appled n nventory systems ncludng ther relevant system cost. 3.2 Assumptons and notatons Consder a two-echelon model wth a depot (or suppler) at the hgher echelon and M bases (or retalers) at the lower echelon. We assume that the suppler has unlmted nventory at the depot, and no other supply sources exst for the retalers at the bases. Although, several products are stocked at each base, we focus on a sngle tem. Demand for the product at each base s stochastc and statonary. The delvery lead tme from the depot to any bases s the same and determnstc. Lateral transshpments between any of the retal locatons are possble and lead tmes are neglgbly small. The followng assumptons and notatons are used: expected daly demand durng a revew perod at base s λ, the length of the revew perod s R, supply lead tme from depot to any base s L, the avalable nventory at base at tme t s I ( t ), the unt holdng cost per perod at any base s h, shortage (or backorder) cost per unt and perod s v, unt lateral transshpment cost between two bases s a, 13

19 the lateral transshpments lead tmes are neglgble. At the begnnng, we defne the reorder pont for a lateral transshpment to base ( = 1, 2,..., M) n ths system as: S = λ 1, = 1,2,..., M. (3.1) Snce demand at each base s statonary, a transshpment order pont can be easly mplemented at each retal locaton. When the avalable nventory at one or more bases drops down to order level S, t ndcates the lkelhood of an mpendng shortage. So, the lateral transshpments can be appled for ths stuaton to reduce the rsk of a shortage. At a tme t, we defne a set of bases J, whch requre a lateral transshpment: J = { I ( t) S}, = 1,2,..., M. (3.2) The shortage at base j for j J equals the expected needed transshpment quantty: SH () t = λ I (), t j J. (3.3) j j j When a base s not ncluded n J, t does not mean that tems can be transshpped to those bases. Therefore, the safety stock level has to be determned for each base, whch should be suffcent to fulfll the demand untl the recept of the next delvery. Ths s gven by Equaton (3.4) from [15]: E[ I ( t)] = λ ( τ t+ R), = 1, 2,..., M. (3.4) t, R where τ tr, s the determnstcally known scheduled tme of recept of the next shpment from the depot at hgher echelon. Thus, the bases whch have excess nventory on hand can be used for makng lateral transshpments to the bases whch have a shortage. Therefore, we defne a set of bases K, whch have stock levels above the average stock level to fulfll demand: The excess nventory for the base transshpments at tme t : K = { I ( t) > E[ I ( t)]}, = 1,2,..., M. (3.5) k for k K can be used for lateral A ( t) = I ( t) E[ I ( t)], k K. (3.6) k k k Thus, the transshpments can be performed from any bases k K to any bases j J. 14

20 3.3 Lateral transshpments based on avalablty The lateral transshpment based on avalablty (TBA) polcy s one type of preventve lateral transshpment n order to reduce the rsk of a stock out. The objectve of ths polcy s to redstrbute stock to retalers wth ample stock levels to fulfll customer demands. Ths polcy can also be consdered as a reactve polcy. The dffculty s to determne the desred stock levels correctly such that future demand s satsfed. We denote the tme nstance at whch the avalable stock falls down to the transshpment order pont at a base for the frst tme as t. At ths tme nstance t the shortage the other bases. SH j () t for base j s calculated, as well as the excess stock Ak ( t) for The followng procedure s used to perform the lateral transshpment: 1) Determne the set J and K of the bases wth shortages and excess stock, respectvely. Next, set the excess quanttes A ( t) for base k K, and k shortage quanttes SH j () t for base j J. 2) Rank the bases wth excess stock accordng to the avalable quanttes n descendng order, and rank the bases wth shortages n descendng order. Consequently, [1] [2] [ K ] Ak ( t) A () t A () t... A () t, SH () t SH () t... SH () t. [1] [2] [ J ] 3) Next, the total shpment quantty at tme t from the base wth the largest excess stock to the base whch needs the most replenshments s determned by Equaton (3.7) from [15]: Q = mn{ A ( t), SH ( t)}. (3.7) [1][1] [1] [1] 4) The quantty of Equaton (3.5) s reallocated accordng to step (3). Repeat step (1) to step (3) untl Q [1][1] = 0. As a result, ether all transshpments are performed or the total amount of stock avalable for the transshpments s exhausted. 3.4 Lateral transshpments based on nventory equalzaton The lateral transshpment based on nventory equalzaton (TIE) polcy redstrbutes stocks to match the rato of average demand of each retaler whenever there are retalers wth less than desrable stock levels. Ths polcy can be also consdered as the proactve polcy where transshpment decsons are based on the concept of 15

21 nventory balancng or equalzaton through stock redstrbuton. The ratonale behnd ths polcy s to correctly determne the transshpment quanttes n every base. The transshpped quanttes must be those that equalze the probablty of a stock out for every base durng one revew perod. Durng one revew perod, nventores are redstrbuted among the bases through lateral transshpments, such that all bases have an equal number of days supply. Accordng to the TIE polcy, the expected equalzed nventory level EI [ ( t)] Equaton (3.4) at base at tme t can be further expressed as Equaton (3.8) from [15]. λ E[ I( t)] = I( t), (3.8) M M λ 1 1 = = where I ( t ) s the avalable nventory at locaton at tme t. n For achevng nventory equalzaton at base for M ; f I () t E[ I ()] t < 0, the j j shortage at base j for j J equals the expected needed transshpment quantty s calculated from Equaton (3.3); f I ( t) E[ I ( t)] > 0, the excess nventory for the k k bases k for k K can be used for lateral transshpments at tme t as determned by Equaton (3.6). 3.5 System performance analyss The expected total system cost per perod s adopted here as the crteron for evaluatng the performance of the system. In general, the total cost conssts of the transportaton cost from the central depot, nventory holdng cost h, backorder cost v and lateral transshpments cost a. As the transportaton cost for regular supply from the central depot to all bases s the same, the expected total transportaton cost s constant, ndependent of the base stocks and the transshpment polcy, and can be dsregarded. Thus, assume the system has reached steady state, the expected cost per perod can be wrtten as Equaton (3.9) from [16]. M M C = h OH + v BO + a X, (3.9) total, j = 1 j= 1 where OH, BO denote as the expected on-hand nventory and expected backorder level at base, X s the expected quantty transshpped from base to, j j at the end of a perod after redstrbuton, respectvely. 16

22 The economcally optmal soluton to the problem of operatng the nventory system under consderaton s the combnaton of orderng and transshpment polces that mnmzes the total expected cost per perod C total. The decson varable that completely determnes the orderng polcy s the order-up-to levels. S At the begnnng, n Equaton (3.1), we assume the base stock level S equals to expected daly demand rate mnus 1. But, ths s not optmal base stock level n the nventory system. Suppose the base faces random demand wth mean λ and standard devaton ndependent at other bases. σ per perod. The demand s statonary, normally dstrbuted and The cost Ctotal of an ndependent retaler followng an order-up-to S perodc revew polcy conssts only of the holdng, backorder and lateral transshpments costs can be expressed as Equaton (3.9). Based on the assumpton of normally dstrbuted demand, the on-hand nventory OH and backorder level by Equaton (3.10) and (3.11) from [16] respectvely. S BO can be determned OH ( ) ( ) = S y 0 y f y dy, (3.10) = BO = ( y S ) f ( y ) dy, (3.11) y= S where y and f ( y ) are the demand durng L + 1 perods and ts normal densty functon, respectvely. The expected quantty X, j transshpped from base to j at the end of a perod after redstrbuton s expressed as Equaton (3.12) from [15]. X, j= mn{ max( A( t)), max( SH j( t))}. (3.12) In Equaton (3.12), the quantty X, j at tme t s transshpped from the base wth the largest excess stock to the base j whch needs the most replenshment. Equaton (3.13) from [16] wth the effectve safety factors ω mples the mnmum expected cost per perod C total. 17

23 S = λ ( L + 1) + ωσ L + 1. (3.13) The optmal S value s computed for all bases per perod, and the mnmum Ctotal s determned. The total fll rate ϑ s defned as Equaton (3.14) from [16]. M M ϑ = 1 ( BO λ ). (3.14) = 1 = 1 So, the redstrbuton of nventory between bases before the realzaton of demand s to reduce the rsk of future stock outs. From Equaton (3.13), we can determne the optmal order-up-to levels S to reduce the total system cost by Equaton (3.9) and ncrease the fll rate and better servce at the bases by Equaton (3.14). 18

24 Chapter 4 Servce Level Adjustment Polcy (SLA) 4.1 Introducton A new lateral transshpment polcy s called the servce level adjustment or servce level agreement (SLA), and combnes emergency lateral transshpments wth preventve lateral transshpments. Ths polcy can reduce the rsk by forecastng stock outs n advance and effcently respond to actual stock outs. In ths secton, we gve a descrpton on how ths polcy can be appled n an nventory system and what ts relevant total system costs are. 4.2 Assumptons and notatons Consder a sngle tem two-echelon nventory system wth one depot and M bases. In ths system, the tem has hgh stock out cost. Demand for the tem at base s normally dstrbuted and ndependent wth mean λ and standard devaton σ. The average lead tme L requred to replensh each base from the central depot s assumed to be equal and determnstc. We assume that the lead tme of emergency lateral transshpments between each base s extremely shorter than the lead tme of replenshments from the depot to the bases. In ths chapter, we focus on the servce level adjustment polcy operates based on several replenshment scenaros durng one revew replenshment perod. The followng notatons and assumptons wll be used n ths chapter: expected normal demand rate durng a revew perod at base s λ, standard devaton s σ, the length of the revew perod s R, t ( t R) s the ndex for tme wthn a revew perod R, supply lead tme from depot to any base s L, the reorder pont for a lateral transshpment to base ( = 1, 2,..., M) s, the current stock or nventory level at base at perod t s I ( t ), on-hand stock of base at perod t s OH ( t), quantty for backorder of base at perod t s BO( t ), 19 S

25 X k, j s the quantty of lateral transshpments from base k to base j, the unt holdng cost per perod at any base s h, shortage (or backorder) cost per unt and perod s v, unt lateral transshpment cost between two bases s a, orderng cost per order s c, nt(.) s a functon to return the nteger value, Φ(.) s the cumulatve dstrbuton functon of standard normal dstrbuton wth mean 0 and varance 1, z α s z-value wth proporton α of the area under the normal curve, where α =Φ(( λ) / σ), z α β s defned as the target level of SLRP, γ s defned as the lower level of SLRP, the lateral transshpments lead tmes are neglgble. In ths paper, we use another defnton of servce level whch s dfferent from prevous studes. We defne the new servce level to concde wth the servce level adjustment polcy to determne the desred stock level. For base durng one replenshment perod R, we defne RP() t as the remanng tme untl next replenshment perod starts. It can be seen from Fgure 4 and s expressed as Equaton (4.1). RP() t = R t. (4.1) If a base does not have enough stock to react to future demands before recevng ordered products, they satsfy emergency demands by havng products transshpped from other bases wth ample stock. Accordng to Lee s theory [18], the new servce level s defned as SLRP whch s the abbrevaton of servce level remanng perod. The SLRP ndcates the probablty that no stock outs occur at base durng the remanng perod RP() t. It can be seen from Fgure 4. The mathematcal formulaton for the SLRP at base durng the remanng perod RP() t s accordng to Equaton (4.2) from [18]. SLRP[ RP( t)] = P( D < I ( t)), RP( t) I() t λ( RP()) t =Φ, σ ( RP( t)) (4.2) 20

26 where D s the demand of base durng the perod RP() t. RPt, ( ) Ths equaton expresses the probablty that the demand durng the remanng perod RP() t at base s less than the current stock level. In other words, t corresponds to the probablty that base RP() t. has not reached a stock out durng the remanng perod Fgure 4: SLRP (fgure from Lee [18]) 4.3 Servce level adjustment analyss The servce level adjustment polcy s performed n the mult-echelon nventory system based on several replenshment scenaros. Based on the assumptons and notatons above, the servce level adjustment polcy determnes the quanttes for lateral transshpments at a revewng perod accordng to the followng steps. In case, there s replenshment from the central depot n the current perod t : 1. f OH () t > λ () t, the base can mmedately meet the customer demand wth on-hand stock. 2. f OH () t < λ () t, the base can meet the customer demand by replenshment from the central depot and also by lateral transshpments from other bases wth on-hand stock. In case, there s no replenshment n the current perod t : Frst, we use the upper and lower levels of the SLRP to search for base k (k K) wth a hgh stock level that satsfes Equaton (4.3) and for base j (j J ) wth a low stock level that satsfes Equaton (4.4) (Lee [18]). 1. { λ σ } nt ( RP()) t + z ( RP()) t < OH () t BO () t. (4.3) α 2. { λ γσ } nt ( RP()) t + z ( RPt ()) OH() t BO() t. (4.4) 21

27 Secondly, calculate the quantty for lateral transshpments between bases accordng to Equaton (4.5) and (4.6) from Lee [18]: 1. The excess nventory for the bases k K can be used for lateral transshpments at perod t. The possble amounts to transshp are { { λ ασ } } A ( t) = OH ( t) max nt ( RP( t)) + z ( RP( t)) + 0.5,0, k K. (4.5) k k k k 2. The shortage at base j J equals the expected needed transshpment quantty at perod t. The amounts shortage are { { λ βσ } } SH ( t) = max nt ( RPt ( )) + z ( RPt ( )) + 0.5,0 ( OH ( t) BO( t)), j J. (4.6) j j j j j Thrdly, we adjust the quantty for stock level at each base when the avalable stock s smaller than the quantty of requred stock or stock outs. It s followed Equaton (4.7) to adjust requred stock quantty. SH () t nt( BO () t ( A ())/ t BO ()) t =. (4.7) j j k k j j Fourthly, redstrbute the hghest avalable stock from base k to the hghest requred stock at base j. The total quantty of lateral transshpments X, () t from base k j k to base j at the current perod t s expressed n Equaton (4.8) (from Lee [18]). X k, j( t) = mn[max( SH j( t)), max( Ak( t ))]. (4.8) Ffthly, update the quantty of requred stock and avalable stock accordng to Expressons (4.9) and (4.10). SH () t X () t SH () t. (4.9) j k, j j A () t X () t A () t. (4.10) k k, j k Fnally, f the total quantty of shortage stock at base j satsfes Equaton (4.11), the servce level adjustment polcy n the current perod s completed. SH j () t = 0. (4.11) j 4.4 System performance analyss The fundamental ratonale for the lateral transshpments s the potental for achevng hgher customer servce levels. In ths model, the SLRP servce level s based on the 22

28 concept of the safety stock. So the holdng cost should be consdered nsde the model. Furthermore, the backorder, transportaton and orderng costs should also be consdered n the system. The total cost durng whole perod can be wrtten as Equaton (4.12) (from Lee [18]): C total = ( h OH ( t) + v BO ( t)) + ( a Xk, j ( t) + c ek, j ( )) t k j t, (4.12) where ek, j() t s a constant value of 1 when lateral transshpment s performed between base k and base j at perod t. 23

29 Chapter 5 Conclusons and Future Work We have presented models for two-echelon nventory systems n whch lateral transshpments are allowed. A very common problem n nventory theory s to determne the optmal reoder pont and order quantty n order to mnmze total costs. In ths paper, we llustrate three types of lateral transshpment polces: emergency lateral transshpments, preventve lateral transshpments and servce level adjustments. For the model whch apples emergency lateral transshpments together wth a ( S 1, S) or (, rq) replenshment polcy, we present the calculatons to characterze the servce performance (e.g. expected shortages). For the model whch uses preventve lateral transshpments, we have examned the relatve effectveness of two rather smple lateral transshpment methodologes: lateral transshpments based on avalablty and lateral transshpments based on nventory equalzaton. Fnally, we propose a new lateral transshpment polcy, called servce level adjustment, whch reduces the rsk of stock outs n advance and effcently responds to customer demands. These three lateral transshpments polces can maxmze customer satsfacton and reduce management costs n certan nventory systems. We consdered the usage of a perodc revew system for ths study, but addtonal studes of lateral transshpments are needed to nclude the characterstcs that ncorporate nventory polcy systems accordng to the characterstcs of products n the supply chan. Furthermore, our fndngs are obvously vald only under the specfc operatng assumptons of the model studed n ths chapter. There are many varatons of that model whch present practcal nterests and consttute potental topcs of future research. Some of the most extensons are the followng: (a) non-neglgble transshpment tmes; (b) dfferent costs at each base; (c) dependent demand; (d) more than one depot n the nventory system; (e) alternatve transshpment polces; (f) lost sales. Better understandng of the propertes and performance of more complex systems wth some of these characterstcs may have sgnfcant mplcatons for the desgn and operaton of supply chans. 24

30 References 1. H Lee, A mult-echelon nventory model for reparable tems wth emergency lateral transshpments, Management Scence 33 (1987), pp S Axsäter, Modelng emergency lateral transshpments n nventory systems, Management Scence 36 (1990), pp C Sherbrooke, Mult-echelon nventory systems wth lateral supply, Naval Research Logstcs 39 (1992), pp P Alfredsson and J Verrjdt, Modelng emergency supply flexblty n a two-echelon nventory system, Management Scence 45 (1999), pp J Grahovac and A Chakravarty, Sharng and lateral transshpment of nventory n a supply chan wth expensve lowdemand tems, Management Scence 47 (2001), pp A Kukreja, CP Schmdt and DM Mller, Stockng decsons for low-usage tems n a multlocaton nventory system, Management Scence 47 (2001), pp H Wong, GJ Van Houtum, D Cattrysse and D Van Oudheusden, Mult-tem spare parts systems wth lateral transshpments and watng tme constrants, European Journal of Operatonal Research 171 (2006), pp PM Needham and PT Evers, The nfluence of ndvdual cost factors on the use of emergency transshpments, Transportaton Research E34 (1998), pp PT Evers, Heurstcs for assessng emergency transshpments, European Journal of Operatonal Research 129 (2001), pp

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