Single Stage Heuristics for Perishable Inventory Control in Two-Echelon Supply Chains

Transcription

1 Sngle Stage Heurstcs for Pershable Inventory Control n Two-Echelon Supply Chans Erk Lystad* Mark Ferguson** The College of Management Georga Insttute of Technology 800 West Peachtree Street Atlanta, GA *Tel: (404) Fax: (404) Emal: Erk.Lystad@mgt.gatech.edu **Tel: (404) Fax: (404) Emal: Mark.Ferguson@mgt.gatech.edu Chrstos Alexopoulos H. Mlton Stewart School of Industral and Systems Engneerng Georga Insttute of Technology 765 Ferst Drve, NW Atlanta, GA Tel: (404) Fax: (404) Emal: Chrstos.Alexopoulos@sye.gatech.edu August, 006

2 Sngle Stage Heurstcs for Pershable Inventory Control n Two-Echelon Supply Chans We study the problem of determnng stockng levels for fxed-lfe pershable products n a two-echelon supply chan. We consder both seral chans and dstrbuton networks consstng of a warehouse and n non-dentcal retal locatons. Inventory retans constant utlty throughout ts lfetme, lead-tmes are determnstc, there are no fxed orderng costs, and unmet demand s backlogged. Although an extensve lterature exsts for the nonpershable product case, the consderaton of pershablty sgnfcantly complcates the problem. For nstance, a major complcaton s the need to track the age of nventory as well as ts poston n the supply chan, addng a dmenson to the already burdensome state space of dynamc programmng formulatons. We provde accurate sngle-stage heurstcs for determnng the stockng levels for two-echelon supply chans. We use these heurstcs to develop nsght and ntuton nto the proper management of pershable nventory. Our heurstcs are robust, easy-to-use, and smple enough to be mplemented usng spreadsheet applcatons. 1

3 1. Introducton The control of pershable products s ncreasngly mportant n supply chan management. In the past few decades, the need to properly manage such products has ncreased n many ndustres. For nstance, n the technology and fashon ndustres, contnually decreasng product lfecycles have ncreased the need for agle manufacturng practces. Shrnkng margns place ever ncreasng stress on managng the $00 bllon pershable product sales n the US grocery ndustry, whch loses up to 15% of product due to spolage. Numerous nstances of pershable products exst n practce, such as photographc flms, pharmaceutcals, blood, botechnology products, foodstuffs, radoactve materals, electronc wafer fabrcators, and many chemcals. Fashon and technology goods may also be vewed as deteroratng or pershable products over suffcently long tme horzons. Recognzng the economc and socal mportance of pershable products, researchers have conducted substantal work n pershable nventory theory. However, wth the excepton of a handful of stuatonally specfc papers, the multechelon pershable nventory problem has yet to be addressed. The followng example motvates our study. Example 1: A frm consstng of a two-echelon seral supply chan (a dstrbuton and retal stage) sells a pershable product from the retal stage. The frm purchases a product from an upstream suppler for $0 per unt, the lead-tme between the stages s a sngle perod, and unsold product pershes 3 perods after arrvng at the dstrbuton stage. The frm faces uncertan demand (per perod) that follows the negatve bnomal dstrbuton wth a mean of 10 unts per perod and a coeffcent of varaton of If demand exceeds the on-hand nventory at the retal stage, the frm ncurs a backorderng cost of $0 per unt. Holdng costs at the dstrbuton and retal stages are $0.5 and $1 per perod, respectvely. The frm seeks nventory stockng levels that mnmze ther expected long-term operatng costs per perod. The current state of the lterature gves scant gudance under ths scenaro. One approach s to treat the nventory as nonpershable. Usng the technque for determnng optmal base-stock levels n a seral supply chan descrbed n Federgruen and Zpkn (1984), the frm wll stock 1 unts at the warehouse and 3 unts at the retaler. Alternately, the frm may choose to use Nahmas (1979) sngle locaton heurstc for each stage. For the retal stage, ths procedure s straghtforward, whle for the warehouse, one

4 can utlze backorder costs analogously to the method descrbed n Shang and Song (003). Usng ths heurstc, the frm stocks 10 unts at the warehouse but only 1 unts at the retaler. Ether of these ad-hoc polces creates sgnfcant errors. For ths scenaro, the base-stock levels that result n the mnmal total costs are 4 unts at the warehouse and 4 at the retaler. The ncrease n costs assocated wth the two ad-hoc polces, for product lfetmes of, 3, and 5 perods, are presented n Fgure 1. Cost Increase for Alternate Polces % Addtonal Cost Lfetme Pershable Sngle-Stage Polces Seral Nonpershable Polces Fgure 1: Addtonal Costs Incurred by Exstng Polces Motvated by the large cost penaltes of the currently avalable polces, we provde managers wth a smple and robust heurstc for solvng problems such as the one descrbed above. Specfcally, our heurstc derves base-stock levels for two-level seral and dstrbuton systems when the product s pershable wth fxed lfetmes. To accomplsh ths, we lnk the calculaton of stockng polces for supply chans of nonpershable products wth the pershable nventory theory. We construct computatonally smple and ndependent sngle-stage stockng level heurstcs for each supply chan nstallaton, and test the performance of our heurstcs va smulaton. Our approach yelds costs that exceed the best found polcy (va the smulaton study presented 3

5 n Secton 5) by.18% for seral systems and.99% for dstrbuton systems. The smple structure of our heurstc allows us to develop nsghts nto the proper management of pershable nventory n a two-level supply chan. For nstance, our experments suggest that for seral supply chans, downstream nstallatons tend to behave as f they held nonpershable nventory; however, n dstrbuton systems, the stockng level of nventory at retal nstallatons must account for addtonal costs arsng from the allocaton decson. Thus, the nventory management of pershable products for a seral system s qute dfferent than for a dstrbuton system. We argue that the man drver of the dfferences n managng seral versus dstrbuton systems s, n the latter case, a sgnfcant opportunty cost s assessed for nventory exprng at one retaler rather than beng used to satsfy demand at a second retaler. For expensve products wth short lfetmes, ths opportunty cost s suffcently large to prevent explotaton of the common rsk-poolng effect at the warehouse stage. Thus, extendng the product s lfetme s most valuable when there are fewer downstream customers. The remander of the paper s organzed as follows. In Secton we revew the related lterature, and n Secton 3 we descrbe the settng and foundaton for our model. Secton 4 presents our heurstcs, and Secton 5 descrbes the smulaton methodology for testng them. In Secton 6 we present our numercal results and observatons based on a wde range of test cases over varous parameter values and demand dstrbutons. We conclude wth manageral nsghts n Secton 7.. Lterature Revew Our approach n fndng well performng crtcal number polces for the pershable twoechelon dstrbuton system draws from both pershable and nonpershable nventory theory. From the pershable nventory perspectve, we utlze the concept of echelon nventory, as ntroduced by Clark and Scarf (1960), and characterze our stockng polces as echelon base-stock levels. To set the warehouse echelon base-stock levels, we apply a regresson-based heurstc smlar to the one n Ehrhardt (1979) for nonpershable, sngle stage ( ss, ) polces. Our allocaton polcy and stockng heurstcs are smlar to those n Lystad and Ferguson (006), whch are based on newsvendor calculatons for calculatng 4

6 base-stock levels for dstrbuton network of nonpershable products. Ther approach s to construct a set of seral supply chans whose costs bound the cost of a nonpershable dstrbuton network from above and below, and average the resultng seral chan basestock polcy levels for each echelon. We use a smlar approach for determnng the warehouse stockng levels of pershable nventory, but mpose a further modfcaton when determnng the retaler base-stock levels. In comparson to the amount of work on managng nonpershable nventory, the work on how to manage pershable nventory s generally lmted to a small set of specfc applcatons. The semnal works n ths area are Van Zyl (1964), Nahmas (1975a) and Fres (1975). Van Zyl (1964) derves the optmal stockng polcy for products wth lfetmes of two perods. Nahmas (1975a) and Fres (1975) ndependently consder the expected sngle-perod cost of a sngle nstallaton controllng a sngle product wth a lfetme of r perods. In both of these studes, t s assumed that all tems arrve fresh from the suppler. Thus, n addton to the tradtonal per unt holdng and shortage costs, the frm ncurs an outdatng cost for unts of nventory held at the end of the ( r + 1) st perod after they were ordered. Ths fresh from the suppler assumpton may hold when the retaler s replenshed drectly from a manufacturer but s often volated n the more common case where a retaler s replenshed from a warehouse or dstrbuton center. An mportant fndng of Nahmas (1975a) s the exstence of a bounded orderng regon. Specfcally, he shows that once the system enters a partcular regon, t never leaves. The generaton of an optmal orderng polcy however, s complcated by the need to retan a mult-dmensonal state varable (the quantty of nventory held of ages ( 1,, r). Nahmas (198) notes that, due to ths complcaton, the computaton of optmal polces for large r s prohbtvely complex. Luckly, n lght of well performng myopc heurstcs, determnng an optmal polcy for the sngle locaton fxed-lfe pershable nventory problem s seldom necessary (Nahmas 1976, Lan and Lu 1999). Several approxmate methods to solve the snglestage pershable nventory problem have been ntroduced. Nahmas (1975b) consders three polces that only utlze nformaton on the total quantty of system nventory as 5

7 opposed to the quanttes of nventory at each possble remanng lfetme. Of these, a crtcal number polcy s found to be both accurate and smple to mplement. Nahmas (1976) approxmates the problem wth a myopc crtcal number polcy through the use of an upper bound on the expected quantty of product that wll persh before beng sold or consumed. The upper bound s determned through a modfed newsvendor polcy that we also utlze n our model. Cohen (1976) provdes an optmal crtcal number polcy for two perod lfetmes by constructng the statonary dstrbuton of nventory. Nandakumar and Morton (1993) create myopc upper and lower bounds and take a rato of the tghtest ones to select the order quantty. Tekn, Gurler, and Berk (001) show that, n a contnuous revew system, ncorporatng the age of nventory nto the orderng polcy mproves performance. Cooper (001) provdes addtonal bounds on the outdated quantty, and provdes numercal evdence that the crtcal number polces are nearly as good as the optmal polces. In ths work, we also utlze fxed crtcal number polces. Our work dffers from the aforementoned lterature prmarly by expandng the analyss to two-echelon systems. The consderaton of extended supply chans rases a number of addtonal complextes. The multdmensonal state vector now requres a second dmenson to account for the poston of each unt of nventory n the supply chan. Addtonally, the exstng lterature assumes that all ncomng nventory s fresh. Unfortunately, n mult-echelon problems where safety stock s held at the upper levels, downstream levels receve products wth varous nventory ages. Due n part to these dffcultes, papers consderng mult-echelon nventory theory for pershable products are sparse. Ferguson and Ketzenberg (006) and Ketzenberg and Ferguson (006) explctly consder the effects of uncertan remanng lfetme of nventory upon recept at the downstream stage of a two-echelon supply chan, and the value that nformaton sharng mparts to the system. Goh, Greensberg, and Matsuo (1993) consder a two-stage system where both supply and demand are stochastc, and nventory may fll two separate types of age segregated demand. Fujwara et al. (1997) also analyze a two-echelon seral system where the upstream stage holds a product that s decomposed nto multple subproducts. Ther model allows for emergency expedton of orders n the event of a stockout and for the lfetme of the product to vary by the nstallaton at whch t s held. 6

8 However, ther work s restrcted by the assumptons that the demands for all products are dentcally dstrbuted, and the subproducts are produced from a unt of the master product at a constant rato. Contrbutons consderng tradtonal dstrbuton systems are even less frequent. In addton to the two-dmensonal state space vector, arborescent systems also requre the specfcaton of an allocaton polcy. Prastacos (1981) extends the work of Yen (1965) by consderng the myopc allocaton polces of pershable nventory n dstrbuton networks. Prastacos shows that both stockouts and outages are mnmzed when nventory s allocated to equalze the probablty of demand exceedng nventory for each age at each locaton. Leberman (1958) and Perskalla and Roach (197) show that wth constant product utlty, ssung the oldest nventory frst (FIFO) s optmal. Prastacos (1981) allocaton, as well as ours, assumes constant product utlty; thus we also both employ FIFO polces. Unlke our work, Prastacos does not develop stockng polces, and assumes random supply. To the best of our knowledge, ths work s the frst to develop stockng polces for tradtonal two-echelon dstrbuton systems wth pershable nventory. 3. Model We consder a two-echelon supply chan wth n retal stes, labeled wth ndex { 1,,, n}, and a sngle warehouse denoted by W. Inventory s fresh when t arrves to the warehouse, but at the end of r perods after arrval, t must be dsposed of at a cost of p per unt. Before the age of the nventory exceeds r perods, t mantans a constant utlty to the customer over ts lfetme,.e., the customer values a two-day old unt the same as a three-day old unt. Let t D denote the stochastc demand over t perods at retaler (we omt the superscrpt when t = 1), wth respectve probablty and cumulatve ( ) t dstrbuton functons f and ( t) F. We assume that the demand process for each retaler s statonary over tme, wth the demand processes beng ndependent across retalers. In each perod, the followng sequence of events occurs: prevously shpped replenshments arrve at each level, demand occurs at each retaler, excess demand s fully backordered, nventory s aged (and dsposed of f necessary), replenshment orders are placed, costs 7

9 are assessed, and replenshment orders are shpped. The nventory postons are revewed every perod and a centralzed decson maker places replenshment orders based on knowledge of the entre supply chan s nventory postons. We assume lnear per unt local nventory holdng costs ( h ) and backorderng costs (b ), and zero orderng costs throughout the system. We also assume that nventory n transt from the warehouse to any of the retalers ncurs a holdng cost of h W per perod. Therefore we utlze echelon base-stock polces at each nstallaton wth reorder ponts s. Before costs are assessed n each perod, the followng varables are measured: B O J = number of unts backordered at nstallaton = nventory dsposed of n a over all nstallatons = on-hand nventory at nstallaton T = nventory n transt to nstallaton I = echelon nventory at nstallaton, I = J for = 1,, n n I W = JW + ( T + I ) = 1 IP = echelon nventory-transt poston at nstallaton = I B + T IO = nventory orders outstandng for nstallaton = max( s IP,0) The objectve s to mnmze the long-run expected total cost per perod, n mn E hwiw + bb + hi + po = 1 ( ) (1) Replenshments for each level n the supply chan arrve L perods after beng shpped. Product s shpped from the warehouse n a frst n, frst out (FIFO) order, mnmzng the outstandng orders (IO ) n successve allotments of ncreasng remanng lfe. By mnmzng the outstandng orders, the allocaton polcy allocates scarce nventory to nstallatons on the bass of ther relatve need. In other words, the FIFO polcy shps nventory that s more lkely to expre to retalers who are more lkely to use the nventory to satsfy demand before that unt s lfetme s exceeded. We do not clam that ths allocaton polcy s optmal wth regard to the total cost, although we note that such an allocaton mnmzes expected backorders and stockouts (Prastacos, 1981). The 8

10 polcy s nonpershable analog has also been used prevously by McGavn, Schwarz, and Ward (1993) for dentcal retalers and Lystad and Ferguson (006) for non-dentcal retalers. 4. Echelon Base-Stock Heurstc for Pershable Dstrbuton Networks In ths secton, we present a heurstc for determnng echelon base-stock levels for a twoechelon dstrbuton network. Based on the approach ntroduced by Lystad and Ferguson (006) for nonpershable nventory systems, we begn by constructng two seral supply chan systems whose mean costs bound the mean cost of the dstrbuton system from above and below. We then use a power approxmaton regresson model, constructed and tested usng smulaton, to dentfy robust echelon base-stock levels for these seral chans. Unfortunately, the seral chan stockng polces are not close approxmatons for the stockng levels n a dstrbuton networks. Thus, we ntroduce a second adjustment for determnng the retaler base-stock levels. The heurstc resultng from the power regressons and the retaler stockng level adjustment provdes near-optmal echelon basestock levels for the dstrbuton system. 4.1 Boundng Seral Systems To determne the upper bound, we remove rsk poolng opportuntes from the warehouse by constranng the centralzed decson maker such that he must specfy whch retaler each unt of nventory wll eventually be shpped to as that unt of nventory s ordered from the suppler. Ths approach was ntroduced by Graves (1996), who noted that snce t may be desrable to un-commt stock, ths assgnment rule wll not perform as well as a dynamc allocaton polcy. The restrcton decomposes the dstrbuton network nto a set of n ndependent seral systems, one system for each retaler. We ntroduce the labels W to denote a warehouse nstallaton that exclusvely serves retaler. We refer to these seral chans as decomposed. The constructon of a lower bound on the mean system cost s based on an approach used by Federgruen and Zpkn (1984), who assume nstantaneous and costless transshpment opportuntes wthn an echelon. Under ths assumpton, the dstnctons between nstallatons wthn an echelon are artfcal and the retalers may be treated 9

11 collectvely as a sngle vrtual nstallaton that flls all system demands, as shown on the left-hand sde of Fgure. We refer to ths system as collapsed. We use a combnaton of the stockng levels of these two seral systems as an approxmaton for the stockng levels of the dstrbuton system. W 1 n D α α = 1 W 1 D 1 W 1 1 W D 1 D N D n W n N D n Fgure : Collapsed and Decomposed Systems It stll remans, however, to determne stockng levels for the seral supply chan relaxatons when nventory s pershable. In our numercal studes, we found that a systematc bas was nduced by tradtonal seral chan stockng polces developed for nonpershable products as an approxmaton for the stockng levels of a seral chan carryng pershable products. To adjust for ths bas, we use a modfcaton of the power approxmaton method developed by Ehrhardt (1979). Our method s descrbed below. 4. Power Approxmaton for Pershable Inventory n Seral Systems We formulate a regresson model to approxmate the echelon base-stock levels for the seral chans. We ntally create two regressors to utlze n our model. Frst, we calculate a dstrbuton stage echelon base stock level by treatng the supply chan as a sngle nstallaton that receves fresh nventory from ts upstream suppler. Nahmas (1976) shows that a good heurstc for stockng levels n ths problem s the soluton to the equaton dg () s ( r ) ( r+ 1 + pf ( ) ) s pf ( s) = 0 () ds where 10

12 s ( ) = ( ) ( ) + ( ) ( ) G s h s x f x b x s f x x= 0 x= s+ 1 We next calculate retaler base stock levels by usng () assumng that the nventory ages by L perods (.e. the dstrbuton center stocks no nventory). Ths approach s motvated by notng that for nonpershable products, keepng no safety stock at the upstream stage of a two-echelon system s frequently a good and smple heurstc (Graves, 1996). Ths observaton suggests that the nventory arrvng to the downstream stage of a two-echelon network has a remanng lfetme of approxmately r L. Hence N we calculate Nahmas heurstc twce, frst for the warehouse base-stock level ( W ) N then for the retaler base-stock level ( s R ) wth the reduced remanng lfetme. s and Next we treat the nventory n the seral chans as nonpershable. In ths case, optmal solutons are gven by Clark and Scarf (1960) over a fnte horzon and Federgruen and Zpkn (1984) over an nfnte horzon. Approxmately optmal solutons may be calculated va the smple newsvendor heurstcs of Shang and Song (003). These heurstcs have the attractve property that they can be expressed n closed-form. Thus, we use the Shang and Song approxmaton for our second regressor by settng s S R b + h 1 W = FR b+ hw + h R (3) and 1 S 1 b 1 b sw = FW FW + b hw hr b h W for the retal and warehouse nstallatons, respectvely. An ntal plot study also suggested that demand varance ( σ ), product lfetme () r, backorderng cost (b), outdatng cost (p), and coeffcent of varaton ( σ µ ) may also be sgnfcant. These regressors were entered nto the two regresson models below ( ) ( ) ( σ ) ( ) ( ) ( ) ( σ µ ) E 1 N v1 v1 3 v S 1 4 v1 5 v1 6 v1 7 v1 8 R 1 R R s = v s s r b p (5) ( ) ( ) ( σ ) ( ) ( ) ( ) ( σ µ ) E 1 N V V 3 V S 4 V 5 V 6 V 7 V 8 W W W s = V s s r b p (6) (4) 11

13 where x v are varables to be ftted. We make (5) and (6) lnear by takng ther logarthms, and use least-squares regresson to ft the models over a grd of 144 values for E E s R and s W. These values were determned by the smulaton model descrbed n Secton 5, and were taken over the followng parameter values: demand followng the Posson and negatve bnomal dstrbutons wth mean µ {10, 0} and varance σ { µ, µ,4µ}, r {,3}, b {5, 10, 0}, and p {0.5, 5, 10, 0}. The Posson and negatve bnomal dstrbutons were selected because of ther versatlty n representng a spectrum of low to hgh coeffcents of varaton. Our cost parameters are smlar to those used n Nahmas (1976), and Nandakumar and Morton (1993) for pershable systems, and Cachon (001), Axsater et. al (00), Shang and Song (003) and Lystad and Ferguson (006) for nonpershable systems. The models (5) and (6) were refned by dscardng nsgnfcant factors ( p -value< 0.01) and were reftted, yeldng the approxmatons S ( R) ( σ ) ( ) R s = 1.40 s b (7) R ( ) ( ) ( σ ) ( ) ( ) R N S W W W s = s s b r (8) The proportons of explaned varaton for these regressons are and 0.997, respectvely (although we note that because the regressors are not ndependent, these values solely represent the ft of the resultng equatons). The model n (7) and (8) was tested aganst a second grd of 16 values for R s R and R s W, over the followng parameter values: demand followng the Posson and negatve bnomal dstrbutons wth mean µ {5, 15, 30} and varance σ { µ, µ,3µ}, r {,3}, b {, 8, 3}, and p {, 8, 3}. The average error n the resultng seral system costs s 3.43% compared to the best found results va the smulaton study n Secton 5 (full data are presented n Tables A.1 and A. n the Appendx). Investgaton of Equaton (7) gves rse to the followng observaton: Observaton 1: The retaler base-stock levels of a two-nstallaton seral chan may be calculated ndependently of any consderaton of pershablty. 1

14 Our ntutve explanaton s as follows. When the central decson maker sets the stockng level at the warehouse, s/he n effect determnes the total system stock and thus the lkelhood that demand over the lfetme of the product falls below ths level. Thus the full responsblty of outdatng s realzed at the warehouse. Wth the total system stock decson made, the retaler, unable to affect the outdatng process, operates as f t held nonpershable nventory. 4.3 Retaler Base-Stock Approxmatons for a Dstrbuton System Observaton 1 does not apply to dstrbuton systems. In the absence of a rebalancng relaxaton (where nventory s collected from the retalers and redstrbuted each perod), carryng an addtonal unt of nventory at one retaler may cause that unt to expre rather than beng used to satsfy demand at an alternate retaler. Ths opportunty cost s not present n seral systems. Thus, we nclude an adjustment to our seral system polcy for settng the retaler base-stock levels n a dstrbuton system. To develop ths adjustment, we relax our problem by makng the followng smplfyng assumptons: 1) All nventory unts at each retaler are as fresh as s possble (e.g., wth r 1 perods of lfe remanng). ) The warehouse carres suffcent stock to ensure that all orders are flled n the followng perod. These assumptons allow an approxmaton for the cost of allocatng a unt of nventory to a retal stage. Ths cost s comprsed of three elements: addtonal expected holdng costs, a reducton n potental backorderng costs, and an ncrease n expected outdatng costs. The frst element s smply the probablty that the demand n one perod s less than the amount of nventory held at the retaler tmes the echelon holdng cost at that retaler, hf ( s ). By movng a unt of nventory from the warehouse to the retaler, savngs n expected backorder costs are acheved. To capture ths, our second element s equal to the probablty a stockout occurs tmes the backorder cost rate: b( 1 F( s) ). The prevous two terms are analogous to the costs n the tradtonal newsvendor problem. To capture the mpact of pershable products, we add the ncrease n the cost assocated wth pershablty. The probablty that the unt expres at the retaler s F ( r L ) ( s ). 13

15 However, the unt pershes regardless of whether t was at the retaler or at the warehouse f the total cumulatve demand at each other locaton s less than the nventory held at that locaton. In ths case, we do not penalze the allocaton decson. Thus the fnal cost element n our adjustment s the outdate cost tmes the jont probablty that a unt expres at the retaler and at least one other retaler has suffcent demand so the unt could have avoded pershng: ( r L ) ( r L ( ) 1 ) j pf s Fj ( sj) j The combned cost of allocatng a unt of nventory to a retal stage s ( r L ( ) ( ( )) ) ( r L 1 ( ) 1 ) j hf s b F s + pf s Fj ( sj) (9) j We select as our stockng level adjustment, the nventory quantty that mnmzes the cost n (9): 4.4 Polcy A ( r L ( ) ( ( )) ) ( r L arg mn 1 ( ) 1 ) j s = hf s b F s + pf s Fj ( sj) j (10) Our heurstc calculates the echelon base-stock levels as smple averages of the precedng calculatons. For the warehouse, we follow the technque n Lystad and Ferguson (006) R and average the warehouse echelon base-stock level under the collapsed seral chan ( s W ) wth the sum of the warehouse echelon base-stock levels over the decomposed seral R chans ( s W ). That s, n H 1 R R sw = sw sw +. (11) = 1 For the retal stages, we take a weghted sum of the retaler base stock level under the decomposed seral chans ( s R W ) and the stockng level found by our approxmaton ( s A ). The weghts were calculated by regresson, fttng the model s = V s + V s (1) H 9 R 10 A wth data from experments taken over the followng parameter values: demand followng the Posson and negatve bnomal dstrbutons wth mean µ {10, 0} and 14

16 varance σ { µ, µ,4µ} resultng model,, r {,3}, b {5, 10, 0}, and p {0.5, 5, 10, 0}. The H R A s = 0.81s s 13) was tested aganst a second grd of values for H s over the followng parameter set: demand followng the Posson and negatve bnomal dstrbutons wth mean µ {5, 15} and varance σ { µ, µ,3µ}, r {,3}, b {, 8, 3}, and p {, 8, 3}. The average error n the resultng dstrbuton system costs s 3.51% (the entre set of results from ths experment s presented n Tables A.3 and A.4 n the Appendx). 5. Smulaton Methodology Our smulaton methodology s based on an unequal varance, two-stage screenng-subset selecton procedure presented n Nelson et al. (001). We frst create a set of base-stock level canddates coverng a range of the expected mnmzng base-stock levels (as predcted by our heurstc), ± at least 5 nventory unts for each nstallaton. For the parameter settngs n these examples, ths range covers at least 50% of the cumulatve dstrbuton of the lead-tme demand at each nstallaton. For each stockng level, we ntally make a long smulaton run of our model over 0,00 perods. We use the method of batch means (Law and Kelton, 000) by batchng perods nto groups of 0 to obtan sample averages (costs) that are approxmately..d. and normally dstrbuted. We remove the frst batch (0 perods) to mtgate ntalzaton effects. The remanng data ponts are used n the ntal screenng phase whch dentfes a temporary wnnng system wth the lowest estmated cost and elmnates poorlyperformng systems wth hgh costs. Potental sets of stockng levels that survve the ntal screenng are subjected to a second round of smulaton experments, where we retan our batch szes and generate a suffcent number of data ponts to elmnate all but one of the systems. After ths experment, the set of stockng levels that has the lowest per perod cost s selected. Wth a 1.5 GHz processor, the jont processes completes n an average of 7.3 seconds for each experment. Ths procedure ensures that the cost of the selected system s wthn δ of the cost of the (unknown) best system wth probablty 1 α. The ndfference 15

17 parameter δ was set at 0.% of the estmated cost assocated wth the temporary wnner from the ntal screenng stage. Snce δ s a realzaton of a random varable, ths method s an approxmate mplementaton of the method of Nelson et al. (001). 6. Problem Desgn and Expermental Results 6.1 Symmetrc Two-Echelon Networks Our frst expermental desgn consders two network topologes, wth ether two or four symmetrc retalers. We test the heurstcs usng a full factoral desgn over a range of backorder costs, outdate costs, lfetmes, and demand varances. We assume the total system demand s dstrbuted accordng to the Posson or the negatve bnomal dstrbuton wth a mean of 0 unts per perod across topologes, and that the demand processes at the retalers are..d. Our remanng parameters are { 5,10, 0} p { 5,10, 0}, r {,3}, and σ { µ,4µ} b, for negatve bnomal demand. The demand, backorder, and holdng cost parameter values are smlar to those used n nonpershable works by Jackson (1988), Cachon (001), Axsater et al. (00), Shang and Song (003), and Lystad and Ferguson (006). The demand, holdng cost, outdate cost, and backorder cost parameter values are smlar to those used by Nahmas (1976) and Nandakumar and Morton (1993) for sngle-stage pershable systems. In the latter works, the authors note that a pershable system quckly resembles a nonpershable system as lfetmes exceed two perods. We also observe ths convergence as noted n the followng observaton. Observaton : In both the seral supply chan and dstrbuton network cases, ncreasng lfetmes cause a system to quckly converge to ts nonpershable analog. Ths trend s apparent when comparng the nventory stockng levels n Tables A.1 and A. n the Appendx. In Table A.1, the best found stockng levels decrease quckly wth outdate cost (whle keepng other parameters constant). In the analogous results n Table A., the stockng levels decrease at a sgnfcantly slower rate, suggestng that the costs mposed by expred nventory are somewhat small. To llustrate the effects of 16

18 ncreasng product lfetmes on total system costs, we plot the average total cost per perod as lfetmes ncrease for systems wth b = 0, p = 0, h W = h = 0.5 and r {, 3, 5}. Seral systems wth µ = 10, σ { 10,0,40} dstrbuton systems wth µ = 10, { 10,0,40} are presented n Fgure 3. Two-retaler σ are presented n Fgure 4. Fgure 3: Effects of Pershablty for Seral Systems Fgure 4: Effects of Pershablty for Two-retaler Systems 17

19 Clearly, the decrease n costs assocated wth ncreasng lfetmes occurs due to a decreasng frequency of outdatng. The expected perodc cost converges to that of a nonpershable system as the lfetmes ncrease, but sgnfcant addtonal costs assocated wth pershablty exst when ether lfetmes are short or demand varance s hgh. Because the majorty of pershable assocated costs are elmnated by the thrd perod of lfetme, we set our lfetmes to ether or 3 perods. These assgnments are equvalent to the ones n Nahmas (1976) and Nandakumar and Morton (1993), adjusted to account for the lfetme lost n transt to the retal stages. The results of these tests are sumarzed n Tables 1 and, for the two and four retaler systems, respectvely, wth the entre set of results dsplayed n Tables A.5 and A.6 n the Appendx. % Error for Two-retaler Systems Lfetme p Table 1: Two-retaler Summary % Error for Four-retaler Systems Lfetme p Table : Four-retaler Summary 18

20 The results above lead to the followng observaton: Observaton 3: The heurstc performs well, wth small error rates. These errors are ncreasng as the number of retalers ncreases. The most sgnfcant errors correspond to low outdatng costs. Whle the heurstc s error can exceed an average of 5%, t does so only for low outdatng costs. Snce the local holdng cost rate at the retalers n these tests s set to one unt whle the outdatng cost s only 0.5 unts, the frm prefers to dspose of exprng nventory rather than carry t an addtonal perod. In practce such scenaros are rare, although we note the backorder to holdng cost ratos n our study are greater than 5. Neglectng the cases where the holdng cost exceeds the outdatng cost, the heurstc s error s approxmately.6%. The ncrease n the error when the number of retalers ncreases s partly due to dscretzaton effects, where roundng errors mposed to ensure nteger valued base-stock levels become more problematc because of the smaller nventory quanttes. However, these errors are only slghtly greater than those arsng from leadng heurstcs for nonpershable dstrbuton systems (see for comparson Lystad and Ferguson, 006). Observaton 4: All else held constant, ncreasng the outdate cost, backorder cost, or demand varance also ncreases the total system cost. Further, decreasng the lfetme tends to ncrease costs, except when the outdatng cost s less than the local holdng cost. Ths observaton s n lne wth ntuton and prevous work n both nonpershable multechelon and pershable sngle locaton systems. Increasng the varance of demand ncreases both the expected number of backorders and expected number of outdates. Increasng the costs of ether of these ncreases mean system costs drectly. Observaton 5: As product lfetme ncreases, the mean system cost drops faster for tworetaler systems than for four-retaler systems. 19

21 Prevous work n nonpershable mult-echelon systems suggests that ncreasng the number of retalers nhbts the explotaton of rsk-poolng benefts, leadng to ncreased costs. Ths effect s apparent when lfetmes are long and our stockng levels approach those of a nonpershable system. When lfetmes are short, however, less nventory s held at the retalers because of the opportunty cost effect. The greater drop n system costs assocated wth the two-retaler systems results from the ablty to explot rskpoolng opportuntes at the retalers. Observaton 6: As product lfetme decreases, the system-wde nventory savngs assocated wth rsk-poolng for systems wth fewer, large-volume retalers decrease. The ncrease n opportunty cost effects begns to domnate the rsk poolng advantages the two-retaler network enjoys compared to the four-retaler network. The downstream rsk-poolng savngs can no longer be captured n lght of the ncreased opportunty costs, thus the ncentves for developng transshpment opportuntes ncrease as lfetmes decrease. However, t should be mentoned that n practce transshpment opportuntes may consume valuable lead-tme. As noted above, unless lfetmes are very short, the system may be treated as f the nventory were nonpershable. Thus, transshpments that route nventory back through a dstrbuton pont, such as the tradtonal rebalancng relaxaton n nonpershable work (e.g., see Clark and Scarf, 1960; Federgruen and Zpkn, 1984; and Axsater et al., 00), are partcularly napproprate n a pershable context. Rather, t s the ablty to drectly satsfy customer demand from multple sources that becomes ncreasngly attractve rather than the ablty to rebalance nventores. 6. Asymmetrc Backorder Rates In ths secton, we consder a warehouse that serves the pershable product to two retalers wth dfferng backorder rates. We contnue to utlze Posson and negatve bnomal demands wth r {,3}, and σ { µ,4µ} µ = 10. We set b 1 = 5, b { }, { 0.5,5,10, 0} 10,0 p, for negatve bnomal demand dstrbutons. These assgnments generate a total of 48 cases. The results of the tests are summarzed n 0

22 Tables 3 and 4 for problems wth lfetmes of or 3 perods, respectvely. The complete set of results s gven n Tables A.7 and A.8 n the Appendx. % Errors for Two-perod Lfetmes b σ Table 3: Asymmetrc Backorder Cost Summary for Two-perod Lfetmes % Errors for Three-perod Lfetmes b σ Table 4: Asymmetrc Backorder Cost Summary for Three-perod Lfetmes Observaton 7: The proposed heurstc approach s robust to asymmetry n backorderng costs. The errors of the heurstc under asymmetrc backorder cost profles are smlar to those of under symmetrc backorder cost profles. Thus the heurstc s robust to asymmetry n backorderng costs of even up to 400%, atypcal for a vast majorty of practcal examples. Observaton 8: Although system-wde echelon nventores for asymmetrc backorder cases are approxmately the same as for symmetrc backorder cases, more nventory s held at the warehouse n the asymmetrc cases. Also, backorder asymmetry decreases total system costs. 1

23 Lystad and Ferguson (006) show for the management of nonpershable products, backorder cost asymmetry decreases echelon stockng levels and nventory costs. We fnd smlar results for pershable products. The system controller may explot vrtual poolng effects as nventores ncrease at the retaler wth hgh backorder cost. However, these vrtual poolng savngs are nhbted by the danger of nventory exprng at the hgh backorder retaler; hence the decson maker holds a porton of nventory at the warehouse rather than allocatng t to the retal stages. Ths lmts the cost savngs that may be acheved through the rsk-poolng effects. 6.3 Asymmetrc Demand Rates We next consder a warehouse that serves the pershable product to two retalers wth varyng demand rates. As before, we assume Posson or negatve bnomal demands, wth means µ 1 = 5 and p { 0.5,5,10, 0}, r {,3} µ = 15. The remanng parameters are { 5,10, 0} b,, and σ = µ for negatve bnomal demands. The assgnments result n a total of 48 problems. The results of these tests are summarzed n Tables 5 and 6 for problems wth lfetmes of or 3, respectvely. The complete set of results s contaned n Tables A.9 and A.10 n the Appendx. % Error for Two-perod Lfetmes b σ Table 5: Asymmetrc Demand Summary for Two-perod Lfetmes % Error for Three-perod Lfetmes σ b Table 6: Asymmetrc Demand Summary for Three-perod Lfetmes

24 Observaton 9: The proposed heurstc methodology s robust to asymmetry n demand. The errors n the heurstc under asymmetry n demand are approxmately the same as under symmetrc demand, and contnue to be on par wth those found for heurstcs for nonpershable nventory control. Observaton 10: We fnd no sgnfcant dfferences n total system costs between the symmetrc and asymmetrc demand cases. Slghtly greater nventory s held n the asymmetrc demand problems. Lystad and Ferguson (006) show that n nonpershable cases, demand rate asymmetry decreases echelon stockng levels and nventory costs. In ths case, we fnd a slght ncrease n the warehouse echelon base-stock level wth demand asymmetry. The presence of the opportunty cost effect prevents the explotaton of vrtual rsk-poolng as one retaler captures most of the system demand. 6.4 Importance of Pershable Inventory Polces In the prevous sectons, we presented heurstcs to set nventory base-stock levels n supply chans when nventory s pershable. We also argued that, beyond very short lfetmes, the systems behave essentally as f they held nonpershable products. A natural queston s: when does a frm need to consder the pershable aspect of ts nventory and utlze the more complex nventory control polces? To llustrate the mportance of nventory control polces that account for pershablty, we compare our heurstc to the nonpershable newsvendor heurstcs of Shang and Song (003) and Lystad and Ferguson (006). We consder scenaros wth b = 0, p = 0, h W = h = 0.5, and r {, 3, 5}. The results for seral systems (havng sngle retalers) wth µ = 10 and σ {10, 0, 40} are presented n Fgure 5. The results for two-retaler dstrbuton systems wth µ = 10 and σ {10, 0, 40} are presented n Fgure 6. 3

25 Cost Increase for Nonpershable Polces n Seral Systems 50 % Addtonal Cost Lfetme CV = CV =0.447 CV = 0.63 Fgure 5: Nonpershable Heurstcs n Pershable Seral Systems Cost Increase for Nonpershable Polces n Two-Retaler Dstrbuton Systems % Addtonal Cost Lfetme CV = CV = CV = 0.63 Fgure 6: Nonpershable Heurstcs n Pershable Two-retaler Dstrbuton Systems Fgures 5 and 6 renforce our clam that supply chans resemble nonpershable systems as the lkelhood of outdatng decreases. They addtonally show that when the opportunty of outdatng s sgnfcant, the use of nonpershable nventory control polces 4

26 yelds sgnfcantly hgher costs. Thus, falure to account for the pershable aspect of nventory may lead to costly mstakes. We found that the use of nonpershable polces ncreases costs by about 3% for products wth the shortest lfetmes and lowest demand varances; ths mpact decreases wth lfetme but ncreases wth the demand varance. 7. Concludng Remarks Managers faced wth pershable nventory may make costly mstakes when relyng on nonpershable polces and ntuton. In ths paper, we present a smple heurstc for twoechelon supply chans of fxed-lfe pershable nventory. We consder both seral chans and dstrbuton networks wth n non-dentcal retalers, and show the qualtatve behavors of the two types of topologes are dstnct. Our heurstc treats nstallatons wthn the supply chan ndependently, allowng for smple, closed-form solutons that may be appled usng spreadsheet applcatons. Relatve to the best found (va smulaton) base-stock levels, our heurstc yelds a.18% and.99% average errors for the mean total cost per perod n seral and dstrbuton systems, respectvely. Under seral systems, the retal stage of a two-echelon network behaves as f t holds nonpershable nventory because the warehouse determnes the expected number of outdates per perod through the settng of ts echelon base-stock level. For the parameter settngs under consderaton, the heurstc echelon base-stock levels were close to ther nonpershable nventory analogs. As the lfetme ncreases, the lkelhood of outdatng decreases quckly snce the probablty of stock exceedng demand over the lfetme of the nventory unt approaches zero. For dstrbuton systems, however, the allocaton of nventory to the retalers may ncrease the probablty of outdatng; a unt of nventory held n stock at one retaler may expre whle a younger unt s used to satsfy demand at another retaler. We refer to ths as an opportunty cost of carryng nventory at the retal stages, and fnd that ths opportunty cost has mportant mplcatons for the control of dstrbuton networks. In partcular, t nhbts the explotaton of rsk-poolng benefts at the warehouse. As the product lfetme ncreases, ths opportunty cost effect dmnshes, creatng greater savngs for systems wth fewer retalers. Compared to the nonpershable product case, the rsk-poolng effect (ncreasng the number of retalers whle keepng the total system 5

27 demand constant) has a smaller beneft on the stockng levels and the total amount of nventory n the system. Ths suggests that nvestments made to extend product lfetmes (make the product less pershable) are more valuable when sgnfcant rsk-poolng opportuntes exst. Alternately, strateges allowng customer demands to be flled from multple retal stes remove ths opportunty cost effect entrely, n addton to enablng the tradtonal cost savngs. Ths work opens a promsng stream of future research. For nstance, we have assumed throughout ths paper that nventory lfetmes are determnstc, whle random lfetmes are common n practce for some products. We have lkewse assumed that nventory expraton costs are constant, regardless of the locaton of the unts when they persh. In a producton process, the ncrease n nventory value suggests that outdates late n the process are more costly than outdates early n the process. Extensons beyond the prevous model varatons are also numerous. For nstance, by comparng the costs of varous network topologes, the value of expedton strateges, delayed dfferentaton, demand varance reducton, and forecastng accuracy n supply chans managng pershable goods may be explored. 6

28 Appendx Table A.1: Seral System to Create Regressons r µ σ b p * s W * s 1 C *

29 Table A.1, Contnued

30 Table A.1, Contnued

31 Table A.1, Contnued

32 Table A.: Seral System Expermental Data Used to Test Regresson Models r µ σ b p * s W * s 1 C * H W s H 1 s C H

33 Table A., Contnued