The safety stock and inventory cost paradox in a stochastic lead time setting

Transcription

1 Dsney, S.M., Malz, A., Wang, X. and Warburon, R., (05), The safey soc and nvenory cos paradox n a sochasc lead me seng, 6 h Producon and Operaons Managemen Socey Annual Conference, Washngon, USA, May 8 h May h, 0 pages. The safey soc and nvenory cos paradox n a sochasc lead me seng Sephen Dsney, Arnold Malz, Xun Wang and Roger Warburon 3 ) Logscs Sysems Dynamcs Group, Cardff Busness School, Cardff Unversy, UK ) W.P. Carey School of Busness, Arzona Sae Unversy USA 3) Dep of Admnsrave Scences, Meropolan College, Boson Unversy, USA DsneySM@cardff.ac.u Absrac We sudy a sochasc lead-me problem movaed by real world global shppng daa. Replenshmen quanes are generaed by he Order-Up-To polcy whch ams o acheve a sraegc avalably arge. We show ha unle he consan lead-me case, mnmum safey socs do no always lead o mnmum coss under sochasc lead-mes. Keywords: order up o polcy, sochasc lead me, order crossover Inroducon Global sourcng ofen allows access o low-cos supply bu s ofen assocaed wh long and varable lead mes (Blacburn 0). These longer and more varable lead mes brng wh hem a number of complcaons and poenal pfalls, from boh cos and servce perspecves (Sal 006). We add o he leraure on plannng wh sochasc lead mes by formulang and esng a calculaon of safey soc ha reflecs hese real-world complcaons. Our mehod allows for order crossover and correlaon beween ppelne nvenory and replenshmen orders, facors ha are ofen gnored. Usng a lnear modfcaon of he famlar Order-Up-To (OUT) orderng polcy we fnd a soluon ha always resuls n lower nvenory holdng and baclog coss. Praccally, we have raced and analyzed logscs daa for global supply chans for boh maor forwarders and realers and were sruc by he volaons of he lead me normaly assumpon see Fgure. When lead mes s hghly varable we may also have ssues wh order crossover, where shpmens are receved n a dfferen sequence from whch hey were dspached. Mos nvenory models do no allow for hs order crossover, ye varable shpmen delays, clercal errors, and random cusom nspecons can easly delay a shpmen long enough for ohers o pass. From he analycal perspecve, wo prescrpons for nvenory managemen are wdely dssemnaed. These approaches eher use an average (or maxmum) lead me n he consan lead me reorder pon soluon or assume ha he demand s normally dsrbued and hen use he mean and varance of a random sum of random varables o deermne he reorder pon (Feller, 966). We show ha neher approach s well-sued o global supply chans wh long rans mes and mulple hand-offs due o he mul-modal nvenory dsrbuon. Ths paper develops an exac heorecal reamen of he mpac of he sochasc lead mes wh order crossover on he probably densy funcon of he ne soc levels. As we progressed n our nvesgaons, we also began quesonng he well-nown asseron (Kaplan 970) ha he OUT model s always a good f for global supply chans. We fnd ha, when here s order crossover, beer economc performance s possble when he orderng sraegy

2 Dsney, S.M., Malz, A., Wang, X. and Warburon, R., (05), The safey soc and nvenory cos paradox n a sochasc lead me seng, 6 h Producon and Operaons Managemen Socey Annual Conference, Washngon, USA, May 8 h May h, 0 pages. Fgure Emprcal por-o-por lead me dsrbuons from Chan o he USA. follows he lnear proporonal order-up-o (POUT) polcy (Dsney and Lambrech, 008). The srucure of our paper s as follows. We frs defne he POUT polcy and hen show how o capure he effec of he sochasc lead mes analycally. Fnally, n order o valdae our heorecal resuls, we numercally analyze a sochasc lead me problem wh lnear nvenory holdng and baclog coss. The proporonal order-up-o polcy We frs brefly revew he POUT polcy before movng on o he case of sochasc lead mes wh order crossover. We assume hroughou ha a lnear sysem exss and ha demand, D, a me, s an ndependenly and dencally dsrbued (..d.) random varable drawn from a normal dsrbuon wh a mean of and a sandard devaon of. The POUT polcy, O, generaes orders a me, wh he followng dfference equaon (Dsney and Towll 003), O T I W. ()

3 Dsney, S.M., Malz, A., Wang, X. and Warburon, R., (05), The safey soc and nvenory cos paradox n a sochasc lead me seng, 6 h Producon and Operaons Managemen Socey Annual Conference, Washngon, USA, May 8 h May h, 0 pages. Here, he varable T, s a safey soc he mean nvenory; s he average lead-me; I s he on-hand nvenory a me ; and W s he on-order nvenory or wor n progress, WIP. The varable 0 s a proporonal feedbac conroller ha regulaes he speed a whch devaons n he nvenory poson are recovered. When, he POUT polcy degeneraes no he OUT polcy. The nvenory balance equaon defnes he nvenory I I R D, () where R O s he receps receved, he ncomng orders afer he sochasc lead me and he sequence of even delay (he ). The sysem s compleed when we defne he WIP, W O. (3) Modellng sochasc lead mes wh order crossover As he OUT/POUT polcy operaes on dscree me he sochasc lead mes mus also be dscreely dsrbued, whch s why we presened he daa n Fgure as a dscree dsrbuon. Le p be he probably ha he lead me of a parcular order s perods long, s an neger greaer han or equal o zero. The maxmum lead me s, he smalles lead me s 0, and he average lead me s p. h 0 Le M be a bnary marx wh = o columns and o rows. Assgn he, elemen of M a value accordng o m,, (4) where. Here x s he celng funcon. Each row of he M marx represens a -uple of bnary dgs ha descrbes he sae of he WIP ppelne. A zero n elemen m, of marx M ndcaes ha for sae, he order placed perods ago has been receved (he order s closed), uny ndcaes ha he order placed perods ago has no ye been receved ( s open). Noe, he order placed perods ago s always closed, hus ndexes from o o denoe he lead mes = 0 o. There are rows o M, one for each possble sae of he order ppelne. The probably ha he WIP ppelne s n sae s gven by q, (5) p One can derve (5) by observng ha n he h ppelne sae, he probably ha an order placed perods ago s open/closed can be expressed unversally as 3

4 Dsney, S.M., Malz, A., Wang, X. and Warburon, R., (05), The safey soc and nvenory cos paradox n a sochasc lead me seng, 6 h Producon and Operaons Managemen Socey Annual Conference, Washngon, USA, May 8 h May h, 0 pages. q, m, pm, p ( ) p, (6) and ha q s he produc of q, over. We now requre he varance of he nvenory levels n each of he sub-processes. We oban hs by frs deermnng he varance of he WIP n each sub-process and hen each subprocess s combned wh a scaled order o oban somehng we call he scaled shorfall dsrbuon sub-process, from whch we can oban he nvenory dsrbuon. We can rearrange () o oban I T W O. (7) For OUT polcy (ha s, when ) we can see ha he nvenory dsrbuon s a refleced shorfall dsrbuon, W O, T (Zalnd, 978; ranslaed by Robnson e al., 00). When he O componen has become scaled by O, n whch case we call he dsrbuon of W O he scaled shorfall dsrbuon. We now requre he mean and he varance of he scaled shorfall dsrbuon of each sub-process. The complcang facors are ha O s auo-correlaed and ha he dsrbuons of W and O are correlaed wh each oher. As he sysem s lnear he smples way o proceed s o explo he z-ransform, whch s defned by F z Z f f z. (8) 0 To deermne he varance of he WIP n sub-process,, we frs noe ha he varance of he orders mananed by he POUT polcy s ndependen of he lead-me, as O z 0 z 0 Here, z s he z-ransform operaor, Z. (9) d s he nverse z- Z F z F z z z f C Fz. ransform of ransfer funcon, z z s he ransfer funcon of he orders mananed by he POUT polcy under..d. demand and mnmum mean squared error forecasng (Dsney and Towll 003). The relaonshp beween he varance rao and he sum of he squared mpulse response s nown as Tsypn s (964) relaonshp. The probably densy funcon of he normal dsrbuon wh an argumen of x, a mean of, and a sandard devaon of, s defned by x e x,. (0) Usng hs noaon, (9) leads o an order process descrbed by he pdf, 4

5 Dsney, S.M., Malz, A., Wang, X. and Warburon, R., (05), The safey soc and nvenory cos paradox n a sochasc lead me seng, 6 h Producon and Operaons Managemen Socey Annual Conference, Washngon, USA, May 8 h May h, 0 pages. Fgure. The bullwhp rao for dfferen O x,. () The bullwhp rao, O, s ploed n Fgure. The rao s uny a, zero a 0, a, srcly ncreasng, and convex n. Noe ha he bullwhp rao and order varance are no affeced by he sochasc lead me. The varance of WIP sub-process,, s gven by he varance of he sum of he mpulse responses of he open orders, W, z m, Z 0 z. () where m, s an elemen of he bnary marx M ha capures wheher an order s open or closed. The dsrbuon of he scaled orders, O, for all sub-processes, can be obaned usng O O Z z z 0 0, (3) whch leads o he followng expressons for s pdf,, O x. (4) The covarance beween he WIP sub-process and he scaled orders sub-process s hen z z cov,, c 0 z z. (5) W O Z m Z ov W, O NS,, he varance of sub-process n he nvenory dsrbuon, s equal o he varance of he shorfall dsrbuon, whch s gven by NS S W O,,, cov W, O. (6) 5

6 Dsney, S.M., Malz, A., Wang, X. and Warburon, R., (05), The safey soc and nvenory cos paradox n a sochasc lead me seng, 6 h Producon and Operaons Managemen Socey Annual Conference, Washngon, USA, May 8 h May h, 0 pages. The mean of he each of he sub-processes of he nvenory dsrbuon can be shown o be NS,, The complee pdf nvenory dsrbuon s gven by T m. (7) NS q x NS,, NS,. (8) The varance of he complee, mul-modal, nvenory s gven by NS NS T x dxq m, NS,, (9) Equaon (9) shows ha he mean demand has an nfluence on he varance of he nvenory levels, somehng ha does no happen wh consan lead mes. Furhermore, we can see ha he nvenory varance also conans a weghed sum of ndvdual sub-process s varances. A numercal example when 4 Consder he suaon when 4. Table deals he ppelne saes M, he varance of he ne soc, and he mean of each of he 6 ndvdual sub-processes o he nvenory dsrbuon. I can be easly shown ha each of he expressons for he varance (and he sandard devaons) of he nvenory sub-processes s nfne a 0,. Furhermore, each sub-process has a sngle unque mnmum,, whch s also dealed n Table. We can see ha exss only n he sub-processes ha do no conan order cross-overs. All of he sub-processes ha conan order-crossover have. Ths leads us o speculae ha n cases where order crossover exss ha he POUT wll produce nvenory pdfs wh less varance han he OUT polcy and ha he opmal les n he regon, 0. To mae he resuls n Table specfc, we frs need o specfy he lead me probables assume p0, p p p3 0, p4. The maxmum lead me s 4 and he average lead me s. Usng (5) we are hen able o deermne he probably ha he ppelne s n sae, s, q Noe ha n general, he probably ha he ppelne s n a parcular sae need no be, and almos never s, he same as he probably ha he ppelne s anoher sae. Consder now ha he followng nvenory cos funcon J, exss J Eh ns b ns, (0) where h s he un nvenory holdng cos, and b s he un baclog cos. I s possble o show ha 6

7 Dsney, S.M., Malz, A., Wang, X. and Warburon, R., (05), The safey soc and nvenory cos paradox n a sochasc lead me seng, 6 h Producon and Operaons Managemen Socey Annual Conference, Washngon, USA, May 8 h May h, 0 pages. Fgure 3 Dsrbuon of he nvenory levels mananed by he OUT polcy for 90% avalably when p, p p p 0, p,, Table. Generc characerscs of he sub-processes when 4 M NS, NS, T T T T T T T T T 3 T T 3 T 3 T 4 T T T

8 Dsney, S.M., Malz, A., Wang, X. and Warburon, R., (05), The safey soc and nvenory cos paradox n a sochasc lead me seng, 6 h Producon and Operaons Managemen Socey Annual Conference, Washngon, USA, May 8 h May h, 0 pages. Fgure 4. The percenage economc gan from usng he POUT polcy T 4 T T 8 T T b h e e e 3 dj 0, () d 64 mplyng ha for all cos combnaons he OUT polcy s never opmal as here always exss a whch s more economcal. Togeher wh he varances of he ndvdual sub-processes dealed n Table, (4) and (8) we are now able o oban an expresson of he pdf of he nvenory levels, whch we plo n Fgure 3. Fgure 3 plos wo cases of he OUT polcy, and n boh we have se he safey soc, T, o mnmze J when h =, b = 9. In he frs case 00, and we can clearly see ha here are fve modes n he nvenory pdf. In he oher case, 40 and he fve modes overlap somewha. Furhermore, he complee pdf of he 40 case, has less varance, and requres less safey soc, han he 00 case. When 00, he nvenory levels have a varance of 0,300 for he OUT polcy. Numercal expermens show us ha here s a sngle mnmum nvenory varance (or sandard devaon) a 0.73 and he ne soc varance s 0,80 0.% less han he OUT varance. Usng numercal echnques we can fnd he opmal proporonal feedbac conroller, and safey soc T, ha mnmzes he nvenory cos. When we have se,t opmally, Fgure 4 descrbes he percenage economc gan JOUT JPOUT J OUT 00%, from usng he POUT polcy. Whle he mprovemen s raher small (noe ha 0.8 means 0.8% no 80%), he POUT s always more economcal han he OUT polcy. The economc benef ncreases as decreases. When he cos rao s such ha near 00% avalably s desrable,. Fgure 5 plos for dfferen cos raos and dfferen mean demands. We see ha s near uny when he avalably arge s (very) near 0% or 00%, bu for mos avalably arges Ineresngly, almos always, mplyng ha he ghes nvenory conrol does no always lead o he mnmal cos. Fgure 6 shows he safey soc requremens when s used for dfferen cos raos. The mul-model naure of he 00 case resuls n rapd ncreases n he safey soc requremen a predcable pons on he avalably scales. These are relaed o he mul-model pdf of he nvenory levels. Furhermore, beween 40-60% avalably, he wo demand sengs requre very smlar amouns of safey soc. As s no possble o vsually dsngush 8

9 Dsney, S.M., Malz, A., Wang, X. and Warburon, R., (05), The safey soc and nvenory cos paradox n a sochasc lead me seng, 6 h Producon and Operaons Managemen Socey Annual Conference, Washngon, USA, May 8 h May h, 0 pages. beween he opmal POUT safey soc, T POUT, and he opmal safey soc for he OUT polcy, T OUT, we have ploed he dfference, TPOUT TOUT, n Fgure 7. Here we can see ha, alhough he POUT polcy s economcally superor, mnmum safey soc do no always concde wh leas coss. Ths s an nsgh ha s conrary o he consan lead me case where he leas cos soluon always has he smalles safey socs. Fgure 8 hghlghs he bullwhp rao acheved when s used n he POUT polcy. We can see ha a 40% reducon n bullwhp s possble beween 8% o 9% avalably. Ths s neresng as he obecve funcon consss only of nvenory relaed coss. If coss are assocaed wh bullwhp are also presen, hese wll also be reduced. Ths s an mporan resul as bullwhp coss are somewha harder o quanfy, even hough bullwhp s wdely recognzed as havng a negave effec on supply chan performance (Lee e al. 000). Fgure 5. The nvenory cos opmal n he POUT polcy Fgure 6. Safey soc wh he nvenory cos opmal POUT polcy Fgure 7. Safey soc dfference beween he OUT polcy and he nvenory cos opmal POUT polcy T T POUT OUT Fgure 8. Bullwhp wh he nvenory cos opmal Conclusons We have suded he mpac of a sochasc lead me on a lnear perodc revew OUT polcy. Our novel conrbuon s a new mehod o oban he dsrbuon of he nvenory levels n he presence of correlaon beween he WIP and orders, va he so-called scaled shorfall dsrbuon. Ths bulds upon anoher unque conrbuon he M-marx and he assocaed mehod o deermne he probably of he ppelne beng n each of s possble saes. In he consan lead me case, wll mnmze he varance (or equvalenly he sandard devaon) of he nvenory levels and resul n he mnmum nvenory coss when he 9

10 Dsney, S.M., Malz, A., Wang, X. and Warburon, R., (05), The safey soc and nvenory cos paradox n a sochasc lead me seng, 6 h Producon and Operaons Managemen Socey Annual Conference, Washngon, USA, May 8 h May h, 0 pages. safey soc s se o he crcal fracle (Brown 963). However, n he sochasc lead me case, mnmzng he varance of he nvenory levels, by unng, wll no always resul n mnmal coss. Whle he opmal,, s never uny, may be near uny, and changes sgnfcanly wh he avalably arge, see Fgure 5. The sochasc lead me case wh order crossover resuls n a surprsng paradox. Mnmzng nvenory coss does no always lead o mnmum safey socs. However, he relaonshp beween holdng and baclog coss and he avalably acheved a he mos economcal soluon does sll hold. Ths leads o an mporan nsgh: Coss should be used o desgn he sysem because focusng on mnmzng nvenory varance, or safey socs, can lead o an ncorrecly specfed sysem. We have demonsraed ha he OUT polcy s no he opmal polcy when order crossover exss, as he lnear POUT economcally ouperforms. We have no proven he opmaly of he POUT polcy self because we do no now wheher here exss a beer performng polcy, lnear or non-lnear. Indeed, s nown ha he opmal polcy s non-lnear (Srnvasan e al., 0). However, he POUT polcy has a long hsory and has been successfully mplemened n pracce. See Poer and Dsney (00) for deals of an mplemenaon a he UK grocery realer, Tesco and Dsney e al., (03) for an mplemenaon n he global prner manufacurer, Lexmar. References Blacburn, J. 0. Valung me n supply chans: Esablshng lms of me-based compeon. Journal of Operaons Managemen 30(5): Brown, R. G Smoohng Forecasng and Predcon of Dscree Tme Seres. Prence-Hall, Mchgan. Dsney, S. M., M. R. Lambrech On replenshmen rules, forecasng and he Bullwhp Effec n supply chans. Foundaons and Trends n Technology, Informaon and Operaons Managemen (): 80. Dsney, S. M., D. R. Towll On he Bullwhp and Invenory varance produced by an orderng polcy. OMEGA: The Inernaonal Journal of Managemen Scence 3(3): Dsney, S. M., L. Hosho, L. Polley, C. Wegel. 03. Removng bullwhp from Lexmar s oner operaons. Producon and Operaons Managemen Socey Annual Conference, May 3 rd 6 h, Denver, USA. Feller, W An Inroducon o Probably Theory and s Applcaons, Vol. I. John Wley & Sons, Inc., New Yor. Kaplan, R. S A dynamc nvenory model wh sochasc lead mes. Managemen Scence 6(7): Lee, H. L., K. C. So, C. S. Tang The value of nformaon sharng n a wo-level supply chan. Managemen Scence 46(5): Poer, A. and S. M. Dsney. 00. Removng bullwhp from he Tesco supply chan. Producon and Operaons Managemen Socey Annual Conference, May 7 h 0 h, Vancouver, Canada. Robnson, L. W., J. R. Bradley, L. J. Thomas. 00. Consequences of order crossover under order-up-o nvenory polces. Manufacurng and Servce Operaons Managemen 3(3): Srnvasan, M., R. Novac, D. Thomas. 0. Opmal and approxmae polces for nvenory sysems wh order crossover. Journal of Busness Logscs 3(): Sal, G Survvng he Chna rpde. Supply Chan Managemen Revew 0(4): 8 6. Tsypn, Y. Z Samplng sysems heory and s applcaon. Pergamon Press, Oxford. Zalnd, D Order-level nvenory sysems wh ndependen sochasc lead mes. Managemen Scence 4(3):