Smooth Priorities for Multi-Product Inventory Control

Transcription

1 Smooh rioriies for Muli-roduc Invenory Conrol Francisco José.A.V. Mendonça*. Carlos F. Bispo** *Insiuo Superior Técnico - Universidade Técnica de Lisboa ( favm@mega.is.ul.p) ** Insiuo de Sisemas e Robóica - Insiuo Superior Técnico ( cfb@isr.is.ul.p) Absrac: Whenever dealing wih periodic review muli-producs invenory conrol for capaciaed machines, one of he main issues ha has o be addressed concerns he dynamic capaciy allocaion. Tha is, how o assign capaciy o he several compeing producs ha require more han i is available. One ypical approach is o assign prioriies o he producs, according o some degree of relaive imporance. Whereas prioriy based capaciy allocaion is aracive, due o is simpliciy and for he fac ha i makes sense in a significan number of seings, we conend his o be oo unfair o lower prioriy producs, given heir access o producion ends o be highly variable and uncerain. Deparing from wha we call sric prioriies, we propose a prioriy based mechanism ha improves on his drawback, ermed smooh prioriies. This new policy for muli-produc, limied capaciy producion sysems wih sochasic demand is sudied. Theoreical comparisons are made o he common policies of sric prioriies and linear scaling. An opimizer based on Infiniesimal erurbaion Analysis, IA, simulaion is devised and resuls of pracical comparison beween smooh and sric prioriies are presened. The srucure of he cos funcion wih smooh prioriies is sudied hrough funcion plos obained from simulaion and numerical resuls show consisen beer performances han hose achieved under sric prioriies. Keywords: eriodic Review Invenory Conrol, Muliple producs and machines, Capaciaed Sysems, Base-sock Echelon olicies, Infiniesimal erurbaion Analysis. 1. INTRODUCTION In he conex of muli-produc, limied capaciy producion sysems, wih sochasic demand and periodic review, he opimal policy has defied closed form formulaions and shown o be excessively complex for numerical calculaion. For his reason, i is frequen o use sub-opimal policies characerized by a small se of parameers which may hen allow for simulaion based opimizaion. Wihin he subopimal policies normally used, he mos common are basesock policies. Whereas defining he base-socks is enough for single produc sysems, being hem composed of capaciaed or uncapaciaed machines, such is no he case for muliple produc sysems wih capaciaed machines. In hose cases one has sill o address he issue of dynamic capaciy allocaion when, for a given machine, he requess from all producs exceed is capaciy. Many are he schemes o allocae capaciy dynamically, ou of which decisions based on a prioriy lis is one of he mos common. However, a sric prioriy policy, while ineresing in simpliciy and for he high prioriy producs, is no as ineresing when i comes o saisfying he demand for lower prioriy producs. There may be periods when a subse of he higher prioriy producs exhauss all he available capaciy for hose periods, and he lower prioriy producs will have o wai for a subsequen period o see heir needs fully or parially saisfied. A consequence of his is he fac ha heir levels of safey sock end o increase o hedge agains a less reliable, wih higher variance, access o he resources. Furhermore, deermining he adequae lis of prioriies is a combinaorial problem of difficul resoluion, when he number of differen producs is significan. The objecive of his paper is o inroduce and sudy he smooh prioriies conce as an alernaive o sric prioriies. Smooh prioriies ry o overcome he unfairness of sric prioriies by allowing a slice of he available capaciy o all producs in all periods, irrespecive of being higher or lower prioriy. Being a prioriy inspired conce he slice each produc ges on a given period is also a funcion of a se of weighs, which express he relaive prioriies among he producs. Given he fac ha hose weighs can be defined as real numbers beween 0 and 1, ransforms he problem of deermining he relaive imporance of he producs ino a coninuous variable, non-linear programming problem. A grea uorial ha describes he curren sae of he ar in producion sysems is [Hou06]. Relevan resuls for he work presened are: he opimaliy of base-sock policies for simple cases proved in [ClarkScarf60], [FergZip86a] [FergZip86b], and [RodKap04]; he pracical approach used in [GlasTay94] o deermine sabiliy condiions, and in [GlasTay95] o validae sensiiviy analysis for a single produc muliechelon capaciaed producion sysem using a base-sock policy; he sudy of several common base-sock policies applied o re-enran muli-produc capaciaed sysems in [Bispo97]; he firs pracical sudy of smooh prioriies in [NunSouSou99]; and he opimaliy of base-sock policies in he muliple producs, single machine case approached in [JanNagVeer07].

2 To obain pracical resuls, a simulaion based opimizaion wih gradien esimaes provided by Infiniesimal erurbaion Analysis, (IA), ([Glass91]) is used. IA is based on he possibiliy of exchanging he expecaion and he derivaive operaors allowing for a gradien o be esimaed by he average of he period gradiens calculaed during simulaion. The opimizer was developed afer a review of several algorihms from [Luen03] and uses he Flecher-Reeves mehod o deermine he search direcion. Consrains are enforced by projecion and he golden secion algorihm is used for line searches. The simulaor was developed by adapaion of he algorihm and echniques from [CasLaf99]. More informaion on he implemenaion deails is available in [Men08]. In wha follows we will use simple subsiuion o prove ha our implemenaion of smooh prioriies is a generalizaion of boh he Linear Scaling Rule, [Bispo97], (LSR) and sric prioriies. The general model used along wih he LSR, sric prioriies, and smooh prioriies producion equaions, and relevan IA equaions are presened in Secion 2. The fac ha LSR and sric prioriies for wo producs consiue a subclass of smooh prioriies is proven in Secion 3. racical cases sudied, resuls obained, and drawn conclusions are presened in Secion 4. Conclusions are presened in Secion MODEL AND BASE-STOCK OLICIES The general model used in his paper has M machines wih finie capaciy, final producs ha are consiued by F p phase producs. Each phase produc can be assigned o be produced in any machine wihou resricion (i.e., no serial or oher srucure is imposed) and can use up differen amouns of capaciy o be produced (i.e., differen loads), raw maerials are always available (i.e., perfec delivery), which are used by he firs phase produc of a final produc, demand is coninuous and occurs on he las phase produc of a final produc afer producion. roduc quaniies are considered coninuous and linear holding and backlog coss are used. 1 Mach.1 Mach. 2 2 Mach Figure 1 - Illusraion of he modeling flexibiliy. Noaion: M - number of machines; - number of final producs; F p - number of phase producs of final produc p ; M m - se of phase producs produce in machine m ; - capaciy of machine m ; I - invenory of phase produc f of final produc p in ime period ; E - echelon invenory of phase produc f of final produc p in ime period ; - producion of phase produc f of final produc p in ime period ; p d - demand for final produc f a he end of ime period ; h - holding cos of phase produc f of final produc p ; b p - backlog cos of final produc p ; - load of phase produc f of final produc p ; p C - cos for final produc p in ime period ; C - oal cos in ime period ; z f - echelon base-sock level of phase produc f of final produc p ; - alernaive (o z ) se of variables ha relae invenory beween phases; y - shorfall of phase produc f of final produc p in ime period ; f - producion needs of phase produc f of final produc p in ime period ; f - smooh prioriies' parameer of phase produc f of final produc p. Equaions ha show how he sysem evolves from period o period: I f 1 =I f f f 1, F 1 p p d (1) F p E f 1 =E f f p d,e f = I i (2) i= f Equaions o deermine he cos in each period: F p C p F = min I p,0 b p max I f,0 h (3) C = p=1 f =1 C p (4) Equaions ha define, y and f f : =z z f 1, z F 1 p 0 (5) y f =max z f E f,0 (6) f f =min y f, I 1 (7) 2.1 Descripion of base-sock policies and heir producion equaions Linear Scaling Rule (LSR) - when producion needs canno be saisfied for all producs of a given machine, producion is linearly ={ scaled o fi he machine capaciy. f f, f j j f i, j M m K f m f j, (8) f j j i, j j M m i, j M m Sric rioriy - each produc has a prioriy assigned and fulfills producion necessiies by order of prioriy unil machine capaciy is fully used or here are no more producs o produce. In he following equaion we consider producs ordered by prioriy and j=1 means he produc has he highes prioriy.

3 j =min f j, j 1 i i i=0 j, 0 0 (9) Whereas we have addressed he drawbacks of sric prioriies in he Inroducion, we need now o refer ha he LSR may perform well under cerain circumsances, see [Bispo97], bu when firs phase producs ener he scaling erm of (8), hey end o crush he oher producs jus because here is no bound in raw maerials. Tha is, while heir ne needs, f, are exacly equal o he shorfall, he ne needs of oher produc may be bounded by he feeding invenory. This is specially relevan for re-enran sysems. Therefore, he LSR ends up working as if giving higher prioriy o he firs phase producs, which is known no o be a good producion sraegy. The idea behind he smooh prioriies concep is ha a any decision poin all producs should have a higher chance of geing par of heir needs saisfied. When deciding wih sric prioriies he smaller prioriies producs quie ofen do no ge any of heir needs saisfied. This happens when he higher prioriy producs exhaus all available capaciy. Therefore, o circumven his drawback, we propose ha he producion needs for all producs should be considered in he decision, proporionally o heir relaive prioriies. This way we reduce he likelihood of he lower prioriy producs no geing anyhing done a any given period. To inroduce an implemenaion of he smooh prioriies conce we resor o he LSR. There, each produc needs ener he scaling erm of (8) wih equal weigh. This means all producs needs are equally imporan. If insead we assign a weigh, beween 0 and 1, o each produc we may be able o regulae he relaive imporance of each produc in he scaling erm. Wih his idea in mind we inroduce now an implemenaion of he smooh prioriies concep. Smooh rioriies Each producion decision a any given period has wo decision seps where each is produced wih LSR. Each produc has a parameer ha indicaes he relaive quaniy of producion needs ha ener he firs phase LSR. If here is remaining capaciy afer he firs phase, 1 is he proporion ha eners he second phase LSR. f, I = f f f min j f j, 1 j (10) i, j M m i, j, I j f, II = 1 f f j M m min 1 j i, f j j, 1 (11) j M m f = f, I, II (12) I should be easy o see now ha, in any period when he sum of he ne needs for all producs a a given machine exceed is capaciy, if all he relaive weighs are differen from zero, all producs will ge a non zero share of he available capaciy, conrary o wha happens wih he sric prioriies policy. Also, given he fac ha in general i is no a good sraegy o give higher prioriy o firs phase producs, one should expec ha he opimal weighs should reflec his fac by reducing adequaely heir values for firs phase producs. 2.2 IA equaions The IA equaions are omied here due o space consrains, bu can be seen in [Men08]. They are easily obained applying he formal procedure described in [GlasTay95]. IA equaions and heir validaion for several policies in he muli-produc case is done in [Bispo97]. Since, our implemenaion of he smooh prioriies is essenially an LSR scheme in wo phases, he validaion procedure for ha policy carries hrough for he presen case. 3. THEORETICAL COMARISON OF OLICIES To show ha he LSR policy is conained in he smooh prioriies policy we show he equaliy of boh when smooh prioriies has he parameers equal for all producs. The smooh prioriies producion equaions (10), (11), and (12) can be expressed differenly as a disjuncion of all he possible cases which ges rid of he min operaors. f Smooh f ={f f 1 f if f if j j f K m j, f j j if f j j j f j j 1 j f j j j f j j f j j j f j j (13) By doing M n =, simplifying and joining he equally valued disjuncion we finally arrive a (14). Smooh, p f = LSR ={ f f, f f j j. (14) j, j j K j m f To show ha for wo producs sric prioriies are conained in smooh prioriies we show he equaliy of boh when he highes sric prioriy produc has =1 and he lowes sric prioriy produc has =0. For produc 1 wih =1 we ge a null second phase of producion and for produc 2 wih =0 we ge a null firs phase giving he following equaions

4 Smooh 1 = f 1 min f 1,1 1 (15) Smooh = f 2 min, I,1 (16) f 2 2 The equaions for he sric prioriy case wih only wo producs are he following Sric 1 =min f 1, (17) 1 Sric 2 =min f 2, 1 1 (18) 2 I can easily be seen ha hey are equivalen. I is also imporan o noe ha by exending smooh prioriies o have as many LSR phases as here are producs in he machine his proof could easily be generalized for any amoun of producs. 4. NUMERICAL RESULTS Firs, he previous resuls of [NunSouSou99] are reproduced as closely as possible. This is done wih he purpose of confirming he validiy of previous resuls and validaing he approach here proposed. Then a new simple case is sudied for several coss. This serves he purpose of showing ha sric prioriies consiue upper bounds in erms of cos relaive o he smooh prioriies, as was esablished in Secion 3, while showing ha here is a choice of smoohing parameers ha equals he behaviour of sric prioriies for only wo producs. A more complex case is sudied and a parallel o he resuls of [LuKu91] is made. To finish, he srucure of he cos funcion is sudied and he exrapolaion for he general case is aemped. 4.1 Comparison wih revious Resuls The case is his: - one machine, wih capaciy 25, produces wo producs - produc one has: average demand = 8; inverse variance coefficien = 3; backlog cos = 50; and holding cos = produc wo has: average demand = 12; inverse variance coefficien = 1; backlog cos = 20; and holding cos = 10. This is a case in which, according o [Bispo97], a sric prioriy policy would have beer resuls han he LSR and anoher rule proposed here bu omied here, he ESR (Equalized Shorfall Rule), since one of he producs has he lowes average demand, he highes backlog coss, and he leas variance. The resuls obained by he opimizer for sric prioriy for 1 s produc, smooh prioriy and sric prioriy for 2 nd produc are presened in Table 1. The firs and hird line of he able refer o he resuls obained for sric prioriies and he middle line refers o smooh prioriies. In he former case we only opimize he base-socks for boh producs, and in he laer we opimize he base-socks and he weighing parameers. The behaviour is as expeced in erms of he relaive prioriy bu he bes performance is no obained wih sric prioriies since we do no have 1 =1 nor 2 =0 for he smooh prioriies. Acually, he fac ha he opimal weighing facor for produc 2 is non zero, means ha his produc will have access o capaciy in every period when is needs are non zero, conrary o wha would happen under sric prioriies. Noe also ha, as expec from he discussion made in he Inroducion, he performance improvemen is done a he expense of increasing slighly he base-sock for produc 1 and reducing i for produc 2, when compared wih he firs line. Tha is, produc 2 does no have o hedge as much given i has a more reliable access o he producion resource. Table 1. Comparison beween sric and smooh prioriies Opimizaion Resuls Avg. Cos Differen Coss The case is his: - one machine, wih capaciy 100, produces wo producs - produc one has: average demand = 40; and exponenial demand disribuion. - produc wo has: average demand = 40; and exponenial demand disribuion. - he holding and backlog coss are varied. We presen a summary of he findings. Resuls show ha, for he cases where holding and backlog coss are similar for boh producs, smooh prioriies clearly achieves beer resuls and is equivalen o he use of he LSR. This was o be expeced since here is no differeniaing facor beween he producs. In general, smooh prioriies achieves beer resuls han sric prioriies bu do converge o sric prioriies in several of he cases where holding coss differ. For every case he ordering of he sric prioriies wih he bes resuls correspond o ordering of he alphas for smooh prioriies. As would be expeced, when holding or backlog coss rise for a produc, smooh prioriies ge closer o sric prioriies for ha produc. This pracically confirms ha he performance of sric prioriies consiues an upper bound on he performance of smooh prioriies as shown in Secion 3. Given he exen of he daa, for he sake of space, we omi presening he able wih all he resuls here. These can be seen in [Men08]. 4.3 Complex Sysem To explore he use of smooh prioriies in a more complex srucure, he sysem illusraed in Figure 1 is used.

5 The deails are he following: - 3 machines - machine capaciies: K 1 =60, K 2 =100, K 3 =65-1s final produc: average demand = 30; exponenial demand disribuion; 4 phase producs wih holding coss: h 1,1 =10, h 1,2 =15, h 1,3 =20, h 1,4 =25 ; and backlog cos = nd final produc: average demand = 20; exponenial demand disribuion; 3 phase producs wih holding coss: h 2,1 =10, h 2,2 =10, h 2,3 =10 ; and backlog cos = he firs machine produces: phase produc 1 of final produc 1; and phase produc 3 of final produc 2. - he second machine produces: phase produc 2 of final produc 1; phase produc 3 of final produc 1; and phase produc 1 of final produc 2. - he hird machine produces: phase produc 4 of final produc 1; and phase produc 2 of final produc 2. The sysem was opimized wih all he machines wih smooh prioriies and hen wih machine one wih sric prioriy for produc 2 and he remaining machines wih smooh prioriies. The resuls obained (rounded) are shown in Table 2. We show hose ha presen he mos ineresing behaviour for he single machine and wo producs in Fig. 2 and for he sysem of Fig. 1 in Fig. 3. Figure 2 - Cos funcion for varying alphas (single machine). Table 2. Smooh vs sric prioriy (produc 2, machine 1) Opimizaion Resuls prod.1 prod.2 prod.1 prod.2 f1 f2 f3 f4 f1 f2 f3 f1 f2 f3 f4 f1 f2 f3 Cos From he resuls we can see ha he alpha parameers give prioriy o he phase producs closer o he exernal demand. This brings o mind he "Las Buffer Firs Served" sric prioriies policy ha in [LuKu91] is he one ha achieves he bes performance (in erms of mean cycle ime), from hose sudied here, in a sysem wih a re-enran srucure. We should also ake noice ha applying sric prioriies in machine 1 when he alpha parameers of smooh prioriies give clear order of prioriy, bu no sric prioriy, leads o an increase in he cos. 4.4 Srucure of he cos funcion Several graphs where produced, for boh one of he simple and he more complex sysems, o empirically sudy he srucure of he cos funcion. Given he number of parameers involved, we can only look a projecions of he cos funcion. Therefore, we have o selec a pair of parameers o produce a cos surface as a funcion of he pair, while he remaining parameers are kep consan. Figure 3 - Cos funcion for varying alphas (Same machine for he sysem of Fig.1). Besides he wo here displayed, all graphs produced under he above menioned seings presen smooh surfaces. Alhough his does no consiue proof, hese observaions ogeher wih he condiions which allow he validaion of he IA approach reinforce he belief ha he same happens in general. Alhough he graphs for a single machine do no presen local minima, which permis us o speculae ha he funcion is quasi-convex for ha case, he graph in Fig. 3 exhibis local minima. Since from all he graphs we have produced his is he only one for a machine wih more han wo phase producs bu also in a re-enran sysem, his raises

6 he quesion if such behaviour is due o he re-enrance or if i does happen even for hree producs wih no re-enrance. When we sared his line of research one of he goals was o invesigae if he cos funcion has a single minimum in general. We seleced he above pair of graphs o show wo hings: as a funcion of he alpha parameers, he cos is no convex and may presen local minima. The numerical resuls are herefore coheren wih he heoreical expecaions ha he cos funcion is coninuous bu no wih he expecaions ha i is quasi-convex, a leas generally. This, unforunaely, means ha gradien based opimizaion is no suiable o obain he opimal parameers for sufficienly general sysems. Therefore, oher opimizaion echniques will have o be used in order o saisfacorily ensure opimaliy of he parameers obained. This fac also bears consequences ino he purpose of using smooh prioriies o obain he opimal se of prioriies if one sill inends o resor o sric prioriies. Alhough global opimaliy canno be ensured wih gradien based opimizaion, we sill believe ha he error commied in he precise value of he alpha parameers, under gradien based opimizaion, does no carry hrough o heir relaive values. Thus, for ha purpose we coninue o claim ha a combinaorial problem has been convered ino a non linear problem. 5. CONCLUSIONS A general sysem model ha can encompass boh serial and re-enran sysems was presened. Sofware was developed ha implemens he funcioning of he sysems, a general simulaor and a general opimizer allowing for heir opimizaion based on simulaion. Equaions ha express he funcioning of he sysem were presened and were used o heoreically show ha smooh prioriies are a generalizaion of he Linear Scaling Rule and of he sric prioriies for sysems where each machine processes only wo differen producs. Wih he use of he developed sofware, previous numerical resuls where confirmed. The fac ha smooh prioriies consisenly achieve beer resuls han sric prioriies was verified. The hypohesis ha he ordering of he alphas under smooh prioriies ranslae o an opimal ordering of he producs for sric prioriies was reinforced and he srucure of he cos funcion was sudied, leading o he conclusion ha i is mos likely coninuous bu, unforunaely, i is no quasi-convex in general. REFERENCES Bispo, Carlos Filipe Gomes. (1997). Re-Enran Flow Lines. hd Disseraion. Carnegie Mellon Universiy. Cassandras, Chrisos G. and Laforune, Séphane. (1999). Inroducion o Discree Even-Sysems. Springer Science+Business Media, Inc. Clark, A.J. and Scarf, H. (1960). Opimal policies for a muli-echelon invenory problem, Managemen Science, 6, Federgruen, A. and Zipkin,.H. (1986 a). An invenory model wih limied producion capaciy and uncerain demands I: The average cos crierion. Mahemaics of Operaions Research, 11, Federgruen, A. and Zipkin,.H. (1986 b) An invenory model wih limied producion capaciy and uncerain demands II: The discouned cos crierion. Mahemaics of Operaions Research, 11, Glasserman, aul. (1991) Gradien Esimaion Via erurbaion Analysis. Kluwer Academic ublishers. Glasserman, aul and Tayur, Sridhar. (1994). The sabiliy of a capaciaed, muli-echelon producion-invenory sysem under a base-sock policy. Operaions Research, 42, Glasserman, aul and Tayur, Sridhar. (1995). Sensiiviy Analysis for Base-sock Levels in Muli-echelon roducion-invenory Sysems, Managemen Science, 41, Houum, Geer-Jan van. (2006). Muli-echelon roducion/invenory Sysems: Opimal olicies, Heurisics, and Algorihms. INFORMS Janakiraman, Ganesh. Nagarajan, Mahesh and Veeraraghavan, Senhil. (March 9, 2007). Invenory olicies for Capaciaed Sysems wih Muliple roducs. Working paper. Lu, S.H. and Kumar,.R. (1991). Disribued Scheduling Based on Due-Daes and Buffer rioriies. IEEE Trans. on Auomaic Conrol, 36 (12), Luenberger, David G. (2003). Linear and Nonlinear rogramming, Second Ediion. Springer. Mendonça, Francisco José.A.V. (2008). rioridades Suavizadas. Maser Thesis. Insiuo Superior Técnico - Universidade Técnica de Lisboa, orugal. Nunes, Célia. Sousa, Isabel and Sousa, Sofia. (1999). Conrolo de Invenário para Múliplos roduos. Thesis. Insiuo Superior Técnico - Universidade Técnica de Lisboa, orugal. arker, Rodney. and Kapuscinski, Roman. (2004). Opimal olicies for a Capaciaed Two-Echelon Invenory Sysem. Operaions Research, 52, Fuure work will have o address oher implemenaions of he smooh prioriies concep ha may be more amenable o opimizaion, given he fac ha he numerical resuls show beyond doub ha here are gains o expec when deparing from sric prioriies.