European Journal of Operational Research

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1 European Journal of Operatonal Research 221 (2012) Contents lsts avalable at ScVerse ScenceDrect European Journal of Operatonal Research journal homepage: Producton, Manufacturng and Logstcs Enhanced lateral transshpments n a mult-locaton nventory system Coln Paterson a,, Ruud Teunter b,1, Kevn Glazebrook a,2 a Lancaster Unversty Management School, Lancaster, LA1 4YX, UK b Unversty of Gronngen, PO Box 800, 9700 AV, Gronngen, The Netherlands artcle nfo abstract Artcle hstory: Receved 25 November 2010 Accepted 4 March 2012 Avalable onlne 12 March 2012 Keywords: Inventory control Lateral transshpments Dynamc programmng In managng an nventory network, two approaches to the poolng of stock have been proposed. Reactve transshpments respond to shortages at a locaton by movng nventory from elsewhere wthn the network, whle proactve stock redstrbuton seeks to mnmze the chance of future stockouts. Ths paper s the frst to propose an enhanced reactve approach n whch ndvdual transshpments are vewed as an opportunty for proactve stock redstrbuton. We adopt a quas-myopc approach to the development of a strongly performng enhanced reactve transshpment polcy. In comparson to a purely reactve approach to transshpment, servce levels are mproved whle a reducton n safety stock levels s acheved. The aggregate costs ncurred n managng the system are sgnfcantly reduced, especally so for large networks. Moreover, an optmal polcy s determned for small networks and t s shown that the enhanced reactve polcy substantally closes the gap to optmalty. Ó 2012 Elsever B.V. All rghts reserved. 1. Introducton Lateral transshpments (LTs) are stock movements between locatons n the same echelon of an nventory system. They provde a valuable tool to supply chan managers who are lookng to reduce the penaltes assocated wth a lack of stock at one or more nventory ponts. By strategcally reallocatng excess stock t can be possble to mprove the system wde servce levels and/or lower the cost of operatng the system. These goals have tradtonally been sought wthn spare part networks, where there s a hgh penalty attached to a shortage. However the benefts of LTs have also been realzed n sectors rangng from retal to energy generaton. The challenge that LTs brng s n managng when and where t s benefcal to nstgate a stock movement. An LT may reduce the short term shortage rsk at the recevng locaton but t nevtably ncreases the longer term rsk at the sendng locaton. A transshpment polcy must therefore balance these contrastng rsks and decde when the cost of transshpment s outweghed by the beneft t s expected to delver. The sutablty of a gven LT polcy wll often depend on the attrbutes of the nventory system n whch t s employed. However, a key dstncton wthn the lterature on LTs s that between reactve and proactve polces. Reactve LTs are performed when a shortage or potental shortage occurs, by shppng ether the whole Correspondng author. Tel.: E-mal addresses: c.paterson@lancaster.ac.uk (C. Paterson), r.h.teunter@rug.nl (R. Teunter), k.glazebrook@lancaster.ac.uk (K. Glazebrook). 1 Tel.: Tel.: customer demand or the number of unts short from a dfferent locaton. Proactve transshpments are performed perodcally to rebalance the whole system s stock levels. Ths paper s prncple motvaton s n consderng an enhanced reactve polcy whch falls between these two dstnct sets so as to maxmze the beneft each transshpment can delver. Rather than merely lookng to meet the excess demand, the proposed polcy vews each trggered transshpment as an opportunty to proactvely rebalance the two nteractng locatons nventory. Often when a transshpment occurs the cost assocated wth the stock movement wll prmarly be a fxed cost, ndependent of the sze of the transshpment. The reason for ths s that regardless of whether one tem s transported or several, the costs such as usng a vehcle and the assocated fuel cost of nstgatng the journey wll be hghest for the frst tem. The margnal cost for subsequent tems wll typcally be much lower. When such cost structures exst t s mportant to know how best to carry out transshpments. Economes of scale are consdered throughout nventory management, from orderng n batches to centralzng warehouses. It s therefore natural to want to know how best to operate a transshpment polcy when the opportunty to extract smlar benefts exsts. Wthn the exstng lterature Reactve LTs have been studed under both a perodc and contnuous nventory revew settng. For perodc revew models, Krshnan and Rao (1965) develop optmal transshpments n a sngle perod for a system wth two locatons. Ths s expanded to a mult-locaton, mult-perod settng by Robnson (1990), although here the optmal reactve soluton can only be determned for ether two locatons or multple dentcal locatons and when the transshpment cost structure does not /$ - see front matter Ó 2012 Elsever B.V. All rghts reserved.

2 318 C. Paterson et al. / European Journal of Operatonal Research 221 (2012) nclude a fxed element. Ths hghlghts the complexty of determnng optmal transshpment polces. These papers perform LTs once all demand s known but before t has to be satsfed. In contrast, Archbald (2007) and Archbald et al. (2009, 2010) develop approxmately optmal polces whch can respond to contnuous demand wthn each perod. The former proposes heurstcs to deal wth the transshpment decson process, whle the latter papers look to mprove upon ths and relax some of the restrctons usng dynamc programmng polcy mprovement technques. The results obtaned from these polces show them to be reasonably close to optmal when used n small networks. Ths method of valdaton s one whch ths paper looks to emulate. The above models focus on sngle echelon centralzed models. However, addtonal research n the perodc settng consders the beneft of LTs wthn two echelon models (e.g. Dong and Rud, 2004), decentralzed models (e.g. Rud et al., 2001) and producton based models (e.g. Zhao et al., 2008). The latter of these s closely related to ths paper as t also consders an enhanced transshpment polcy. However, t consders a producton model where stock can be reallocated upon producton whlst also allowng reactve transshpments. It therefore falls between a producton allocaton model and a reactve transshpment model. Much of the lterature on reactve LTs n a contnuous order revew settng s motvated by applcatons n the spare parts ndustry. Here, practcal settngs nclude electronc component manufacturng and electrcty generaton companes. Buldng on the METRIC repars model of Sherbrooke (1968), Lee (1987) proposes a model whch uses complete poolng wthn preset groups of dentcal locatons. Ths shows the beneft of LTs wthn the area and the model s expanded by Axsäter (1990) to allow non-dentcal locatons. Several papers have been wrtten whch further expand these deas by relaxng or tghtenng some constrants such as makng repar capacty fnte (Jung et al., 2003), usng lost sales rather than backorderng (Dada, 1992) or consderng a model where backorders have to be mnmzed rather than costs (Sherbrooke, 1986). In addton to ths, nventory systems that supply more than one type of tem are nvestgated by papers such as Wong et al. (2006b) and Kranenburg and van Houtum (2009). The latter examnes the benefts of partal poolng, where only certan transshpments are performed. All of these papers assume an order-up-to replenshment polcy for each locaton. Wjk et al. (2009) consder a sngle tem system where parts are repared at each locaton and use dynamc programmng to determne an optmal transshpment polcy. Kukreja and Schmdt (2005) consder a more general (s,s) polcy, but have to resort to a smulaton based approach to determne the optmal order polcy. Away from spare parts, Archbald et al. (1997) shows that n a perodc revew model wthout fxed order costs, an order-up-to polcy s optmal. However, postve order costs or mnmum order quanttes often suggest that an (R,Q) polcy s more approprate n practce. Several papers take ths approach. Evers (2001) and Mnner et al. (2003) develop heurstcs that can be used to determne when and how much to transshp for systems wth lost sales. Axsäter (2003) does the same, but for a model wth backorders. He proposes a decson rule whch s constructed to make optmal decsons under an assumpton that no further transshpments wll be made. Ths assumpton enables the exact myopc beneft of transshppng to be calculated and optmzed. A related model by Axsäter et al. (2010) consders an (R, Q) nventory system n practce. They determne approxmately optmal replenshment polcy parameters when transshpments are sourced from a support warehouse. Research on proactve LTs explores ther use to rebalance an entre system s stock on hand. Ths rebalancng s done at a set pont durng a revew perod and before all demand has been realzed. Allen (1958) and Agrawal et al. (2004) consder ths problem ndependently of replenshment decsons. Allen (1958) looks to perform the transshpments at the start of the demand perod, whlst Agrawal et al. (2004) devse a method to calculate the best tme to redstrbute stock durng the perod. Other authors study proactve transshpment and replenshment decsons together. However, due to the perodc nature of proactve redstrbuton all known studes only consder ther use alongsde a perodc revew replenshment polcy. Gross (1963) provdes optmalty results for a two-locaton system, where both orderng and redstrbuton decsons take place at the begnnng of the revew perod. Ths dea s further developed by Das (1975), who allows the redstrbuton pont to occur at an arbtrary tme durng the revew perod. Gross and Das both assume neglgble transshpment tmes. Jönsson and Slver (1987) and Bertrand and Bookbnder (1998) allow postve transshpment tmes. The man dfference between these two studes s that Jönsson and Slver (1987) consder how best to meet servce levels whlst Bertrand and Bookbnder (1998) examne the goal of cost reducton. For a detaled overvew of the lterature we refer to Paterson et al. (2011). However, the hghlghted lterature shows that both reactve and proactve LTs provde cost benefts, but the cost benefts of proactve LTs have only been exploted n a perodc revew settng. In ths study, we analyze the frst hybrd transshpment polcy whch tres to secure the benefts of both under a contnuous revew replenshment polcy by enhancng a tradtonal reactve approach. Our polcy can quckly react to shortages by allowng transshpments at any tme when they occur, as for prevously proposed reactve LT polces. However, the polcy also seeks to proactvely redstrbute stock between the sendng and recevng locatons whenever such an LT s trggered. Ths wll allow maxmum beneft to be extracted from each transshpment nstance and wll be especally benefcal n systems where there s a sgnfcant fxed cost nvolved n carryng out a transshpment. The specfc settng that we consder s as n Axsäter (2003), wth backorderng and an arbtrary number of stockng locatons whch all apply (R, Q) orderng polces. Axsäter (2003) derves an algorthm that determnes near-optmal reactve transshpment decsons. These are shown n a smulaton study on small networks (wth two and three locatons) to provde a sgnfcant cost beneft compared to not transshppng at all and to applyng a smpler transshpment polcy. In ths paper, we generalze ths algorthm wth the goal of determnng an approxmately optmal enhanced reactve transshpment polcy that allows addtonal stock redstrbuton when reactng to a stock out. The results of a comparatve numercal study show that, for small networks, the enhanced polcy sgnfcantly outperforms the orgnal Axsäter reactve proposal, achevng an average 1.6% cost savng over 600 experments. Such a recurrent savng s of major practcal mportance, consderng that nventory costs typcally account for a substantal proporton of a busness s total turnover. To analyze the closeness to optmalty of our enhanced polces, we also develop a dynamc programmng (DP) approach to fndng an -optmal transshpment polcy whch also allows for a proactve element n each transshpment. More sgnfcantly we show through numercal results that the optmalty gap s closed by over 95% on average compared to a polcy of not transshppng and by 88% compared to the orgnal reactve polcy. Ths s strong evdence that our development of an enhanced reactve approach makes an mportant contrbuton to the applcaton of transshpments and enables benefts whch are close to optmal. In a further numercal study, we compare the tradtonal and enhanced algorthms for larger networks wth 5 20 locatons. The exact DP algorthm s too numercally ntensve to be appled n these experments. The results of a comparson of the polces show that the mprovement of the enhanced reactve polcy over the tradtonal reactve polcy s even larger than for small

3 C. Paterson et al. / European Journal of Operatonal Research 221 (2012) networks, wth a cost reducton of over 6.4% on average. It also provdes an average savng of 14.4% over not transshppng at all. A senstvty study provdes further nsghts nto when the cost reducton s most sgnfcant. The study also hghlghts the addtonal benefts that an mproved transshpment polcy can delver. The average servce level wthn the large network study s mproved by 1.5% ponts by the enhanced polcy over the purely reactve polcy and the amount of safety stock requred s reduced by over half when compared to a polcy of no transshpments. The remander of the paper s organzed as follows. In the next secton, we descrbe the model and the algorthm for computng our approxmately optmal enhanced reactve transshpment polcy. The cost beneft of the enhanced reactve polcy s tested numercally n Secton 3 for networks wth two locatons. Secton 4 descrbes the exact DP algorthm, and the optmalty gap for small networks s nvestgated n Secton 5. Larger networks are explored n Secton 6. We end wth conclusons and drectons for further research n Secton An enhanced transshpment polcy An approxmately optmal polcy for determnng purely reactve transshpment decsons n an nventory system, n whch all locatons follow a contnuous revew (R, Q) orderng polcy s derved by Axsäter (2003). The algorthm determnng the polcy s constructed by makng use of an assumpton that the consdered transshpment wll be the last one ever made. Whenever a locaton experences a shortage, the algorthm calculates the most cost effcent amount and locaton to transshp from under ths assumpton. However, the derved polcy only looks to react to a shortage, not to be proactve n future shortage preventon. The polcy proposed n ths paper allows more stock to be transshpped than s needed to meet the mmedate shortage. Ths permts the two locatons whch are partes to a transshpment to redstrbute ther stock and balance future rsk. For the sake of clarty n the remander of the paper we refer to the proposed polcy as the enhanced polcy, wth the exstng polcy referred to as the reactve polcy. Table 1 provdes a lst of notaton needed to defne the nventory system. Table 1 Lst of Notaton. N R Q t s, IL IP L D (L ) A h b c,j (y) c f ;j c u ;j k l f,j f n ;j P n ;j C C (k) X X 0 (d) V (X,t) a (X ) b (IP ) c (X ) Number of demand locatons: locaton 2 {1,...,N} Reorder pont at Batch sze of orders placed at Tme untl sth unt of stock becomes avalable at Inventory level at Inventory poston at Lead tme of orders placed at Stochastc lead tme demand at Order cost at (per order) Holdng cost at (per tem per unt tme) Backorder cost at (per tem per unt tme) Cost of transshppng y unts from to j Fxed cost of transshppng from to j Cost per unt of transshppng from to j Arrval rate of demand nstances at Average sze of each demand at Probablty that a demand at wll be of sze j Probablty that j unts are demanded by n customers at Probablty that nth customer demands the jth unt at Steady state cost of locaton Expected cost rate at tme L gven current IP = k State varable at locaton State varable at locaton less d unts of stock Expected total cost at durng the nterval [0,t] Expected lead tme bas at gven startng state X Expected bas after lead tme at gven current IP Total expected bas at gven startng state X The system has N stockng locatons whch all experence ndependent compound Posson demand wth locaton havng arrval ntensty k. Modelng demand as a compound Posson process enables the polcy to be appled n a wde range of nventory systems that allow customers to demand more than one tem at a tme. The probablty that a demand at locaton s of sze j s denoted by f,j and we do not prescrbe any dstrbutonal form. It should also be noted that P n ;j, the probablty that the nth customer demands at locaton the jth unt of stock, can be calculated usng the recurson P n ;j ¼ Pn ;j 1 f n ;j 1 þ f n 1 ;j 1. To replensh ts stock locaton 2 {1,...,N} places an order of sze Q at the central suppler whenever ts nventory poston drops to or below reorder level R. Ths suppler s assumed to have suffcent nventory capacty so that ths order can always be met and takes a fxed lead tme L to arrve. If a locaton does not have suffcent stock on hand to satsfy a demand, then tems may be transshpped to that locaton from a dfferent locaton wth neglgble lead tme. Ths setup s vald as stockng ponts are often geographcally closer than that of the suppler and do not have the external logstcal delay of placng an order wth a suppler. Any demand that cannot be met mmedately (after transshppng) s backordered. Costs are ncurred for orderng (A ), holdng stock on hand (h per tem and per tme unt), backorderng (b per tem and per tme unt), and for transshppng. The cost for transshppng y unts from locaton to j s gven by c ;j ðyþ ¼c f þ ;j ycu ;j, where cf ;j s the fxed cost per transshpment and c u ;j s the cost per unt transshpped. Under the assumpton that no transshpments take place, Axsäter (2003) derves an expresson for the bas assocated wth each system state. Ths measures the transent effect on costs of startng the system n that state. Performng a transshpment wll nstantly move the system to a new state, so the beneft of a gven transshpment can be dentfed by comparng the bas of the current state wth the aggregate of the bas f the transshpment s enacted and the cost to enact t. By maxmzng the dfference between these quanttes over all possble locatons and transshpment quanttes, the best myopc decson can be dentfed. The lmtaton of the formulaton gven by Axsäter (2003) s that t does not allow the sze of the transshpment to be larger than the shortage. Ths restrcton s mathematcally convenent n that t ensures that the nventory poston of any locaton never exceeds R + Q and thus provdes a closed range on whch to consder the future bas. However, when there s a sgnfcant fxed cost per transshpment t seems ntutve that allowng larger transshpments may well delver greater cost benefts. Not every such enhanced transshpment wll take the nventory poston above R + Q, but to provde a completely general account and to calculate the beneft assocated wth every possble decson t s necessary to extend the methodology to account for such scenaros. Please note that, whle ths extenson s certanly needed n general, we shall encounter n Secton 4 two locaton set-ups where almost all of the (very consderable) benefts from enhanced transshpments can actually be acheved whle keepng the nventory below R + Q. We now descrbe how the above deas can be developed to yeld near-optmal decsons n our enhanced approach. In steady state under an (R,Q ) replenshment polcy wthout transshpments t s known that locaton s nventory poston s unformly dstrbuted over the range [R +1,...,R + Q ]. Wth current (tme zero) nventory poston k and stochastc lead tme demand D (L ) then the mean nventory cost rate at L s gven by C ðkþ ¼h Eðk D ðl ÞÞ þ þ b Eðk D ðl ÞÞ ¼ðh þ b ÞEðk D ðl ÞÞ þ þ b EðD ðl Þ kþ X k 1 ¼ðh þ b Þe k L ðk jþ Xj ðk L Þ n f n ;j n! þ b ðk l L kþ: j¼0 n¼0 ð2:1þ

4 320 C. Paterson et al. / European Journal of Operatonal Research 221 (2012) We nfer that the steady state cost rate for locaton s gven by C ¼ 1 Q R PþQ k¼r þ1 C ðkþ: ð2:2þ The current state X of locaton ncorporates nformaton on the current nventory poston (IP ) together wth the tmes at whch each nventory tem becomes avalable. We wrte V (X,t) for the total expected cost ncurred under an (R,Q ) orderng polcy wth no transshpments durng the tme nterval [0, t] when the system s n state X at tme 0. The bas assocated wth X s defned as c ðx Þ, lm t!1 fv ðx ; tþ tc g: ð2:3þ Suppose that t > 0 and the random state of locaton at tme L s denoted by X L ; we then have V ðx ; L þ tþ ¼V ðx ; L ÞþEV ðx L ; tþ: We now easly see that ð2:4þ c ðx Þ¼lm t!1 fv ðx ; L þ tþ ðl þ tþc g; ð2:5þ has a decomposton gven by c ðx Þ¼a ðx Þþb ðip Þ; where a ðx Þ, V ðx ; L Þ L C ; ð2:6þ ð2:7þ and n o b ðip Þ, lm t!1 EV ðx L ; tþ tc : ð2:8þ The calculaton of a (X ) s gven n the Appendx and s unchanged by the fact that here we may have IP > R + Q. However the earler method for calculatng b s dependent on the nventory poston at tme L beng n the range [R +1,...,R + Q ]. A partcularly large proactve transshpment may nvaldate ths assumpton. Therefore n our formulaton we must take account of the addtonal bas that wll be arse f the nventory poston s taken above R + Q. We now descrbe the calculaton of b. We frst observe that, snce X L s stochastcally ndependent of all nformaton n X save only IP, then b depends upon X only through IP. More specfcally, IL L, the nventory level at tme L, can be calculated through the relaton IL L ¼ IP D ðl Þ: ð2:9þ We also note from Axsäter (2003) the followng notaton to show the relatonshp between IL and IP at any pont n tme. For a gven IL = k the correspondng nventory poston can be gven by hk ¼k þ nq ; ð2:10þ where n s the smallest nteger such that hk P R +1. As locaton evolves under an (R,Q ) orderng polcy wth no transshpments, the assocated nventory poston process regenerates upon every entry nto state R + 1 (or ndeed any other state R + 1 s chosen for convenence). In (2.8) we now deem X L to be the locaton state at tme zero and wrte T for the frst subsequent tme at whch the correspondng nventory poston, gven by rearrangng (2.9), s n the regeneraton state R + 1. Standard theory, for example n Tjms (1986), parttons the nfnte horzon consdered n (2.8) as [0,1) = [0,T) [ [T,1) and demonstrates that the contrbuton to b (IP ) from the latter nterval s zero. Ths reasonng yelds both b ðip Þ¼EV ðx L ; TÞ EðTÞC ; and ð2:11þ b ðr þ 1Þ ¼0: ð2:12þ We now fx IP = k > R + 1 and develop an expresson for b (k) by condtonng upon the sze of the frst demand to occur after tme zero. The expected tme at whch ths frst demand occurs s k 1 and from (2.11) the cost rate contrbuton to b (k) pror to then s C (k) C. Followng demand d the nventory poston at wll be changed to hk d and hence we obtan b ðkþ ¼ fc ðkþ C g k þ P1 d¼1 f ;d b ½< k d >Š: ð2:13þ We now use (2.13) to recover b as the lmt of an teratve scheme whch s akn to DP value teraton. It proceeds as follows: b 0 ðkþ ¼0; k 2½R þ 1;...; R þ Q ;...; SŠ; ð2:14þ b n ðr þ 1Þ ¼0; n P 0; ð2:15þ b nþ1 ðkþ ¼ fc ðkþ C g k þ P1 d¼1 f ;d b n ðhk dþ; k 2½R þ 2;...; R þ Q ;...; SŠ; ð2:16þ where S s a large nventory poston state such that no hgher state s reached. The scheme must converge geometrcally fast, wth b the lmt. We can now use the complete bas functons c for each locaton to dentfy the transshpment that delvers the most cost beneft. We use the notatonal shorthand X 0 ðdþ for the resultng state of a locaton once d unts of nventory have been wthdrawn, ether through customer demand or for transshpment. Suppose that some demand d occurs at locaton when n state X and that ths causes a shortage. The long term cost beneft of transshppng y unts from locaton j (n current state X j ) to locaton n comparson wth performng no transshpment and allowng all demand to be absorbed at s dentfed as the quantty n Dðj; yþ ¼lm V ðx 0 t!1 ðdþ; tþþv o jðx j ; tþ V X 0 ðd yþ; t V j X 0 j ðyþ; t c j; ðyþ yða j =Q j A =Q Þ: From (2.3) we have Dðj; yþ ¼c X 0 ðdþ þ cj ðx j Þ c X 0 ðd yþ cj X 0 j ðyþ c j; ðyþ yða j =Q j A =Q Þ: ð2:17þ ð2:18þ We note that, pror to (2.17), we have taken no account of the order costs n our calculatons. A transshpment of y unts from j to has the effect of adjustng the long-term cost burden from orders by y(a j /Q j A /Q ). We wrte D ¼ max max Dðj; yþ: 16y6IL j j ð2:19þ Our enhanced polcy s as follows: If D 6 0, do not transshp. If D > 0 transshp y unts from j to, where y and j are the maxmzers n (2.19). Ths decson rule s nvoked every tme a demand occurs at a locaton whch would result n a shortage. Ths polcy s quas-myopc n that t s cost mnmzng f no further transshpment s permtted after the current shortage s dealt wth. As the reactve polcy of Axsäter (2003) s a more constraned case of our general enhanced polcy, under the myopc assumpton any enhanced decson wll be at least as good as that of the purely reactve polcy. 3. Two locaton smulaton study: reactve vs. enhanced polcy To analyze the performance of the enhanced transshpment polcy, an ntal smulaton study s conducted. Ths study s restrcted to networks wth two dentcal locatons. In a second study, of

5 C. Paterson et al. / European Journal of Operatonal Research 221 (2012) whch the detals and results wll be presented n later sectons, larger and more vared networks wll be consdered. The two reasons for the more restrcted ntal exploraton are as follows. Frst, there are many model parameters and varyng all of them for larger networks s tme consumng. Ths ntal exploraton allows us to observe how the parameters mpact on costs so that the larger network can focus on key ssues. Secondly, the restrcton wll allow us to determne the optmal transshpment polcy usng DP n Secton 5 for the same set of experments, and therefore to study the relatve reducton of the optmalty gap from the orgnal reactve polcy to the new enhanced transshpment polcy. Snce the two locatons are assumed dentcal, we wll drop the locaton dentfyng subscrpt for cost parameters n the remander of ths secton. We wll do the same for the polcy parameters, and assume that both locatons use the same replenshment and transshpment polcy. Although the transshpment polcy mght have some effect on the optmal orderng quantty, n practce there are often fxed or mnmal order szes. For ths reason and n lne wth prevous studes, pffffffffffffffffffffffffffffffffffffffffffffffff we fx the order cost ($100) and use the EOQ formula ð2aklþ=ðhþ to determne the order sze. The valdty of ths decson s dscussed below. For the szes of successve demands wthn the study we use a geometrc dstrbuton such that f j = p(1 p) j 1, j P 1. The holdng cost rate ($1) s used as the unt cost and all other model parameters are vared n a full factoral study. The full range of parameters examned s shown n Table 2. There are 600 parameter combnatons and hence 600 experments n total. For both the orgnal reactve and enhanced transshpment polces, the transshpment decsons are determned by the algorthms developed n Secton 2. To obtan further nsghts we also consder the no transshpment polcy as a benchmark. For all three (no, reactve and enhanced) transshpment polces, the optmal value of R s found post hoc by conductng smulaton studes on the full range of possble values. Over the 600 experments an average savng of 1.59% s observed for the enhanced polcy when compared to the reactve polcy. Ths s broken down by each parameter n Table 3. For example, the 200 experments whch have an arrval rate of 2.4 customers per unt tme yeld an average savng of 1.77%. The results n Table 3 confrm ntuton n several ways. Once a transshpment s nstgated, a polcy whch looks to transshp more wll see a greater beneft f the margnal cost of addng an extra unt s low. Ths s supported by the c u results. Further, f a polcy s performng each transshpment more effcently then t s natural that the savng wll ncrease when there are more transshpment opportuntes. By ncreasng parameters k or L, or decreasng parameter p we rase the lead tme demand varablty and thus the chance of a shortage. Unsurprsngly, ncreasng the shortage rsk through varyng these parameters n the manner descrbed leads to an observed gan n savngs. Consderng the backorder costs, we see that the savngs are greater as the penalty for not mmedately meetng a demand ncreases. However, as the fxed transshpment cost ncreases the dynamcs are less straghtforward. As the cost per transshpment Table 2 Parameter values. Arrval rate (k) 0.8, 2.4, 4.0 Geometrc dstrbuton parameter for demand sze (p) 0.6, 0.8 Lead tme (L) 2, 3 Backorder cost (b) 10, 20, 30, 40, 50($) Transshpment cost [per tem] (c u ) 1, 2($) Transshpment cost [per transshpment] (c f ) 10, 20, 30, 40, 50($) Table 3 Enhanced rule vs reactve rule. k p L b c f c u % % % % % % % % % % % % % % % % % % % ncreases then naturally so does the reward for carryng out more effectve transshpments but at the same tme the beneft assocated wth carryng out any transshpment decreases. Ths can be observed n Table 3 where the beneft conferred by the enhanced approach ncreases up to a pont before decreasng. Ths s supported by the average number of transshpments per 100 unts of tme, whch falls from 11.2 when c f = 10 to only 3.3 when c f = 50. More detaled results are provded n the Appendx for a sample of parameter values (27 experments). All statstcal comparsons have used pared t-tests at a 95% confdence level, wth common random numbers used for all polces. Overall, the average savng of the new enhanced polcy compared to one whch uses no transshpments s 4.21%, renforcng the vew that transshppng s worthwhle. The standard errors of the costs for the reactve and the new enhanced polces are also gven n Table A.1 (n brackets). There are a few low demand cases where there s no statstcally sgnfcant dfference between the reactve and enhanced polces but no cases were found where the enhanced polcy was outperformed by the exstng polcy. We are not able to prove that the enhanced polcy always outperforms the reactve polcy but we have studed a very large number of problem nstances and have as yet found no counterexamples. A comparson of the safety stock levels R provdes nsght nto how the cost reducton s acheved. The average reorder level s 9.8 unts wthout transshpments, 8.9 unts for the reactve transshpment polcy and 8.2 unts for the new enhanced transshpment polcy. Better transshpment decsons reduce the negatve cost effects of shortages, thereby allowng the system to functon wth lower safety stocks. Further exploraton also showed that n general the enhanced transshpment polcy s more cost effcent, wth transshpments happenng less frequently but wth larger quanttes (4.7 unts for the enhanced polcy compared to 2.1 unts for the reactve polcy on average). To examne the effect of the batch sze on the relatve performance of the polcy, the reorder pont R, whch was optmsed usng a value of Q set by the EOQ, was fxed and used to determne the correspondng optmal batch sze n a smlar fashon. It should be noted that ths par may stll not be the global optmum. The results from ths showed that the dfference between the local optmal value of Q and the EOQ value s relatvely consstent between the polces, wth a dfference of 2 unts for the reactve polcy and 2.2 for the enhanced polcy. In fact the relatve performance of the two polces remaned constant, wth the performance gan remanng at 1.6%. Ths ndcates that the observed gans are robust when the batch sze s vared and that usng the EOQ value gves a sutable overvew of the polces performance. The results n ths secton have clearly shown that the new enhanced transshpment polcy sgnfcantly outperforms the orgnal reactve transshpment polcy. However, t remans of real nterest to dscover how close (n cost terms) t s to an optmal transshpment polcy. To explore ths, we wll develop a dynamc programmng (DP) formulaton for fndng the optmal transshpment polcy n the next secton, and then compare ts cost to that of the new enhanced transshpment polcy n Secton 5, usng the same set of experments that were nvestgated n ths secton.

6 322 C. Paterson et al. / European Journal of Operatonal Research 221 (2012) Table 4 Addtonal notaton. d Tme quantum between each state transton f f;j Truncated probablty of demand j at locaton such that R Q < IL s always true Y Number of quanta untl the outstandng order arrves at locaton Z Locaton of most recent demand where Z = 0 ndcates no demand occurred n the precedng perod I H Indcator functon where H s a logcal statement: f true I H = 1 else I H =0 4. Dynamc programmng formulaton In ths secton, we provde a DP formulaton that can be used to fnd the optmal transshpment polcy usng value teraton. In our system, the resultng polcy determnes how much to transshp, gven the locatons nventory levels and remanng lead tmes of outstandng orders, n order to mnmze the overall cost rate ncurred. We remark that our DP formulaton utlzes a dscrete tme approxmaton of the actual contnuous revew system. However, for a suffcently small tme quantum the approxmaton s very good. Ths wll be verfed n the numercal nvestgaton of Secton 5. For presentatonal ease, we wll refer to the transshpment polcy that mnmzes the cost under the DP formulaton as the optmal transshpment polcy. Our formulaton s for a two locaton system setup. Whle t s possble to generalze the approach to larger systems, value teraton becomes computatonally ntractable very quckly due to the rapd growth n the number of states. In order to lmt the state space, only one order s allowed to be outstandng at each locaton at any tme. Ths assumpton s supported by real world practce and by the smulaton results obtaned n Secton 3. These showed that condtonal upon a locaton havng orders outstandng the probablty of multple orders outstandng was less than 1%. Our smulatons also showed that lmtng the enhanced polcy n a two locaton network so that a locaton could only ever have a maxmum stock of R + Q resulted n no statstcally sgnfcant cost dfference. The reader s referred to the related dscusson n Secton 2. Ths result s due to the fact that t s unlkely that one locaton would have enough excess stock to be able to completely refll the other s safety stock, and thus make IP > R + Q. Therefore t s feasble to lmt the nventory state space to a range of R ± Q.It should be noted that n a mult-locaton network, and n partcular wth non-dentcal locatons, ths assumpton wll not hold. Recall from Secton 3 that we wll apply the DP algorthm for the same set of experments as descrbed there. To develop the DP model we ntroduce addtonal notaton and formulae n Table 4. For suffcently small tme quantum d, we may assume that n a sngle tme slot the system experences an nstance of demand at ether one locaton or nether locaton. The probablty of a demand of sze j at locaton s dk f f;j durng each perod whle the probablty of no demand n the system s 1 d(k 1 + k 2 ). In the lmt d? 0 these probabltes converge to the exact Posson probabltes State defnton A fve dmensonal system state ncorporates the nventory level and the tme untl the outstandng order arrves (f there s one) at both locatons. The ffth dmenson ndcates the locaton where any current demand has occurred. We wrte state s as follows s ¼hIL 1 ; IL 2 ; Y 1 ; Y 2 ; Z ð4:1þ Recall that we do not allow the nventory poston to go above R + Q. Further, the decson to allow a maxmum of one outstandng order bounds the nventory level below. Hence we have that R Q < IL 6 R + Q. Moreover, we clearly have 0 6 Y j < L j d Acton space defnton The acton a ( a) s the amount to transshp from locatons 1(2) 2(1). For a gven state s, the set of actons Act(s) s bounded by zero, and by the mnmum of the amount avalable to transshp and the amount that can be stored at the recevng locaton (.e. that does not take the nventory poston above ts maxmum). We summarze these constrants by. If IL 2 >0,IL 1 < 0 and Z =1: a 2f mnðil 2 ; R 1 þ I ðy1 ¼0Þ Q 1 IL 1 Þ;...; 0g If IL 1 >0,IL 2 < 0 and Z =2: a 2f0;...; mnðil 1 ; R 2 þ I ðy2 ¼0Þ Q 2 IL 2 Þg Else: a 2 {0}. Hence, for example, f IL 1 >0,IL 2 <0,Z = 2 and Y 2 > 0 then a s an nteger n the range from 0 to mn(il 1,R 2 IL 2 ) Cost functon The cost (f s (a)) assocated wth beng n state s and choosng acton a s obtaned by aggregatng the cost of holdng (or backorderng) the current level of nventory after the transshppng acton has taken place wth the cost of the acton tself. There s also the addtonal cost of placng any replenshment order Table 5 Optmalty gap analyss. k p L Parameter value Reactve Enhanced Improvement Parameter value Reactve Enhanced Improvement Parameter value Reactve Enhanced b c f c u Improvement

7 C. Paterson et al. / European Journal of Operatonal Research 221 (2012) whch s ncluded n the perod drectly after the order has been nstgated. Hence we have f s ðaþ ¼h 1 ðil 1 aþ þ þ h 2 ðil 2 þ aþ þ þ b 1 ðil 1 aþ þ b 2 ðil 2 þ aþ þ A 1 I ðy1 ¼ L 1 d 1Þ þ A 2I ðy2 ¼ L 2 d 1Þ þ cu jajþc f I jaj> State transtons ð4:2þ If the current state s s = hil 1, IL 2, Y 1, Y 2, Z, acton a s undertaken and demand d occurs at locaton 1 then the new state wll be s 0 ¼hIL 0 1 ; IL0 2 ; Y0 1 ; Y 0 2, where IL 0 1 ¼ IL 1 a d þ Q 1 I ðy1 ¼1Þ; IL 0 2 ¼ IL 2 þ a þ Q 2 I ðy2 ¼1Þ; Y 0 1 ¼ðY 1 1ÞI þ L 1 ð16y1 6 L 1 d 1Þ d 1 Y 0 2 ¼ðY 2 1ÞI þ L 2 ð16y2 6 L 2 d 1Þ d 1 I ðil1 a d6r 1 &Y 1 ¼0Þ; I ðil2 þa6r 2 &Y 2 ¼0Þ: ð4:3þ Smlar transtons can be dentfed for the cases when a demand occurs at locaton 2 and when no demand occurs anywhere n the system Value teraton The above are deployed n a value teraton n whch the value functon (V n (s)) s the mnmal cost ncurred over an n-perod horzon from ntal state s. If we wrte w(s,s 0 ) for the probablty of movng from state s to s 0 then the optmalty prncple gves! V n ðsþ ¼ mn a2actðsþ f s ðaþþ X s 0 2Swðs; s 0 ÞV n 1 ðs 0 Þ : ð4:4þ We develop the (V n ) np1 usng backwards nducton and utlze the stoppng crteron recommended by Tjms (1986). The mnmzng actons n the fnal teraton yeld the -optmal polcy. 5. Optmalty gap For each of the 600 experments, the dynamc programmng model was used to determne the optmal transshpment polcy. Ths was then used wthn the smulaton model of Secton 3 along wth the two approxmate polces and the polcy of not transshppng. Ths ensured that lke for lke comparsons could be made usng the same set of randomly generated events. It was necessary to gather results for a range of values of the reorder pont R so that the optmal value could be used for each parameter set Results The beneft of usng the enhanced polcy can now be seen n the large step t takes towards the performance of the -optmal polcy. Table 5 shows that over a range of parameter values there s a consstent level of mprovement n cost performance acheved by ncludng a proactve stock rebalancng element to a reactve polcy. Compared to a polcy of no transshpment the enhanced polcy closes the suboptmalty gap by over 95% over the entre data set. Compared to the reactve polcy t s closed by 89%; t s ths relatve mprovement over the optmalty gap whch s broken down n Table 5. The enhanced polcy appears to perform strongly n comparson to the optmal polcy. In fact n 90 of the 600 experments there s no statstcal dfference between the two polces. It also dsplays very smlar characterstcs to the dynamc programmng based optmal polcy. Both ts average reorder pont (8.2 vs. 8.1) and average transshpment sze (both 4.7) are smlar. The one dfference s the number of transshpment events. For every 100 tme unts the -optmal polcy carres out one extra transshpment on average (6.4 vs. 7.5). Ths mples that the lack of future nformaton results n a slghtly more conservatve approach n the enhanced polcy. A subsecton of the full results can be found n Table A.2. Fg. 1 clearly demonstrates the superorty n performance of the enhanced polcy over the reactve polcy, more so for larger values of the fxed transshpment cost. It also gves an ndcaton of the greater consstency the enhanced polcy acheves n performance Accuracy of Dscrete Tme Assumpton & Computaton Tme The above results arsng from the DP mplementaton all use dscrete tme quantum d ¼ 1. We confrm the acceptablty of ths 8 choce by resolvng wth d ¼ 1 and observng that the resultng 16 changes n cost rates are mnmal and nowhere statstcally sgnfcant. Whle usng the optmal polcy may reduce nventory costs ts development s computatonally expensve when compared to the heurstc polces, whch can be obtaned very rapdly n real tme. The tme taken to compute each experments optmal polcy was recorded. Table 6 gves a breakdown of the tme needed to develop an optmal polcy by arrval rate. It dsplays how halvng the sze of d from d ¼ 1 ncreases the computatonal tme by a factor of 5 or Opt Gap 8 6 Opt Gap No Trans Reactve Enhanced Polcy No Trans Reactve Enhanced Polcy Fg. 1. Senstvty Analyss for c f = 10 (left) and c f = 50 (rght).

8 324 C. Paterson et al. / European Journal of Operatonal Research 221 (2012) Table 6 Computatonal tme (mns). Arrval rate d ¼ 1 8 d ¼ 1 16 Multple Overall more on average. Moreover, the computaton tme ncreases rapdly wth the arrval rate. Table 8 Percentage cost savngs for the enhanced polcy over the reactve polcy: large networks. Average k Identcal Two ter Large network study Table 7 Parameter values. Number of locatons 5, 10, 20 Arrval rate [Identcal Locatons](k) 1.4, 2.2, 3.0, 3.8, 4.6 Low arrval rate [Dfferent Locatons](k l ) 1.0, 1.8, 2.6, 3.4, 4.2 Hgh arrval rate [Dfferent Locatons](k h ) 2.0, 2.8, 3.6, 4.4, 5.2 Dstrbuton of order sze (p) 0.8 Lead tme (L) 3 Backorder cost (b) 30($) Transshpment cost [per tem] (c u ) 1($) Transshpment cost [per transshpment] (c f ) 10($) Fg. 2. Savngs acheved by the enhanced polcy over no transshpment. Havng shown that the enhanced polcy mproves the orgnal reactve polcy and makes a large step towards closng the optmalty gap whle reducng some of the varablty n performance, the next step s to consder ts performance n larger networks. In a smulaton study, desgned n a smlar way to the above small network study, nventory systems wth 5, 10 and 20 locaton are consdered. Rather than a full factoral study the focus s now put on how the sze of the network and the arrval rate (and hence the lead tme demand varablty) nfluences polcy performance. Eght dfferent arrval rates are consdered n networks wth dentcal locatons. Addtonally, networks whch had two dfferent levels of arrvals are consdered. These networks have 40% of the locatons at a hgher demand rate than the other 60%. In the latter case, the overall system arrval rate s set equal to a correspondng dentcal locaton confguraton so that comparsons could be farly drawn (see Table 7). A full set of results s gven n Tables A.3 and A.4 wthn the Appendx. The average percentage savngs, broken down by assgned values of the arrval rate, are gven n Table 8. The overall average results show a savng over the reactve polcy of 6.38% for dentcal locatons and 6.42% for the networks wth a two ter arrval rate structure. As for smaller networks, the greatest savngs occur when the arrval rate s large. The dfference n results between the dentcal locaton setup and the two ter setup s small. Ideally a system wth many dfferent arrval rates could be consdered but ths s challengng to mplement due to the necessty of determnng sutable values of R for each locaton va a post hoc optmzaton. The enhanced polcy offers a consstent level of cost mprovement. Average costs are reduced by between 11% and 17% n comparson wth no transshpment. In the case of the reactve polcy, the cost savng can be as lttle as 6%. The greater consstency n performance of the enhanced polcy aganst ncreasng arrval rate s llustrated by Fg. 2. Whle costs are an mportant part of nventory systems t s not the only performance measure of nterest. Servce levels wthn a system are also a key consderaton and the fll rates (the percentage of demand flled mmedately from stock on hand or va transshpment) were also recorded for the large network study. For the reactve polcy a servce level of 96.8% was acheved, but the enhanced polcy ncreased ths to 98.3%. The large network study also renforces other fndngs from the smaller network results. For dentcal locatons, safety stock s reduced from 12.6 unts on average wth no transshpments to 9.3 unts under the reactve polcy and to 5.6 unts under the new enhanced polcy. Smlar results are obtaned for the two ter networks. The average sze of transshpment agan ncreases, from 1.3 unts to 6.0 unts. These results llustrate the greater effcency possble from antcpatng shortages rather than merely respondng to them. 7. Conclusons We have shown that the benefts of reactve transshpments can be enhanced by the development of a new type of polcy, whch ncorporate a proactve element. System costs can be reduced and the effcency of the transshpment process mproved. Ths has been observed through an extensve study of both small and large (R, Q) replenshment polcy nventory networks, wth the benefts growng wth the number of stock holdng locatons. Moreover the mprovements that ths enhanced transshpment polcy can brng have been shown to sgnfcantly reduce the optmalty gap. The comparson to optmalty has been acheved through a dynamc programmng model that enables the calculaton of an - optmal transshpment polcy and the resultng costs. Whlst ths formulaton s restrcted to small systems t s an mportant step n understandng the transshpment process and n evaluatng the performance of the more easly developed enhanced polcy. More mportantly, the numercal results show that the enhanced polcy performs close to optmal over the examned parameters, wth some scenaros showng no sgnfcant dfference between the enhanced polcy and the -optmal polcy. One possble way to further mprove the enhanced polcy s to relax the myopc assumpton that underpns t. Another avenue would be to develop the redstrbuton element by consderng transshpments at tmes other than at those when shortages occur.

9 C. Paterson et al. / European Journal of Operatonal Research 221 (2012) One clear lmtaton wthn batch orderng systems wth transshpments s the challenge to fnd approprate replenshment polcy parameter values, even more so n systems wth non-dentcal locatons. Our results have shown that the enhanced transshpment polcy can sgnfcantly alter the optmal reorder pont when compared to no-transshpments. Indeed part of the savngs acheved s a consequence of beng able to lower the amount of safety stock requred throughout the system. Future work could develop analytcal approaches to the determnaton of reorder ponts. Ths would enable the full benefts of the mprovements n the transshpment polcy to be realzed n more complex nventory systems wth a larger number of stockng locatons and nondentcal demand rates. Appendx A. Detaled results Tables A.1 A.4. Table A.1 Two locaton results: p = 0.8, L =3,c u =1. k b c f Q R No tran. R Reactve (Err) Savng R Enhanced (Err) Savng Improve (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.03) (0.02) (0.02) (0.03) (0.02) (0.02) (0.03) (0.02) (0.02) (0.03) (0.02) (0.02) (0.03) (0.03) (0.02) (0.03) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.03) (0.02) (0.02) (0.03) (0.02) (0.02) (0.03) (0.02) (0.02) (0.03) (0.03) (0.02) (0.03) (0.02) (0.02) (0.03) (0.03) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.03) (0.02) (0.02) (0.03) (0.03) (0.02) (0.03) (0.03) (0.02) (0.04) (0.03) (0.03) (0.04) (0.03) (0.03) (0.04) (0.03) (0.02) Table A.2 Two locaton results: optmalty comparson p = 0.8, L =3,c u =1. k b c f Q Opt. No tran. Gap Reactve Gap Enhanced Gap

10 326 C. Paterson et al. / European Journal of Operatonal Research 221 (2012) Table A.3 Large network results: dentcal locatons. No. k Q R No tran. R Reactve Savng R Enhanced Savng Improve Table A.4 Large network results: two ter locatons. No. k Q R No tran. R Reactve Savng R Enhanced Savng Improve Appendx B. Calculatng a (X ) If t s s the tme when the sth unt becomes avalable at locaton then at each pont n tme a locaton s state can be descrbed by a varable X, where X ¼ðIP ; t IP ; t IP 1 ;...; t 1 ðil Þ Þ: ðb:1þ For locaton the pdf and cdf of the dstrbuton of the tme when the nth customer arrval nstant occurs can be respectvely gven as: For s 6 0, t s 6 L, x ðs; t s Þ¼b t s : For s 6 0, t s > L, x ðs; t s Þ¼b L : We now defne quantty U j ðtþ as ðb:6þ ðb:7þ g n ðtþ ¼kn tn 1 e k t ; ðn 1Þ! ðb:2þ G n Xn 1 ðk tþ k ðtþ ¼1 e kt : k! ðb:3þ k¼1 Usng these dstrbutons t s possble to obtan the pdf and cdf of the tme when the jth unt s demanded at locaton : r j ðtþ ¼Xj P n ;j gn ðtþ; n¼1 R j ðtþ ¼Xj P n ;j Gn ðtþ; n¼1 ðb:4þ ðb:5þ It s now possble to calculate a (X ). We let x (s,t s ) be the expected holdng and backorder costs, assocated wth the sth tem of stock demanded durng the lead tme L.Ifs 6 0 then the tem has already been demanded, wth s = 0 the most recently demanded tem. Ths gves four specfc cases. The frst two beng: U j ðtþ ¼ Z t 0 r j Xj ðuþudu ¼ P n ;j Gnþ1 ðtþ n : ðb:8þ k n¼1 Ths allows us to express the other two cases as: For s >0,t s 6 L, x ðs; t s Þ¼h For s >0,t s > L, Z L t s r s ðuþðu t sþdu þ 1 R s ðl Þ ðl t s Þ Z ts þ b r s ðuþðt s uþdu; 0 ¼ h U s ðl Þ U s ðt sþ t s ð1 R s ðt sþþ þl ð1 R s ðl ÞÞ þ b R s ðt sþt s U s ðt sþ : ðb:9þ

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