Inventory Planning with Forecast Updates: Approximate Solutions and Cost Error Bounds


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1 OPERATIONS RESEARCH Vol. 54, No. 6, November December 2006, pp issn X eissn informs doi /opre INFORMS Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds Xiangwen Lu Cisco Sysems, 210 Wes Tasman Drive, San Jose, California 95134, JingSheng Song Fuqua School of Business, Duke Universiy, Durham, Norh Carolina 27708, Amelia Regan Compuer ScienceSysems, School of Informaion and Compuer Science, Universiy of California, Irvine, California 92697, We consider a finiehorizon, periodicreview invenory model wih demand forecasing updaes following he maringale model of forecas evoluion (MMFE). The opimal policy is a saedependen basesock policy, which, however, is compuaionally inracable o obain. We develop racable bounds on he opimal basesock levels and use hem o devise a general class of heurisic soluions. Through his analysis, we idenify a necessary and sufficien condiion for he myopic policy o be opimal. Finally, o assess he effeciveness of he heurisic policies, we develop upper bounds on heir value loss relaive o opimal cos. These soluion bounds and cos error bounds also work for general dynamic invenory models wih nonsaionary and auocorrelaed demands. Numerical resuls are presened o illusrae he resuls. Subjec classificaions: invenory, forecasing, MMFE, approximaion, error bounds. Area of review: Manufacuring, Service, and Supply Chain Operaions. Hisory: Received June 2003; revisions received Augus 2004, May 2005; acceped November Inroducion Demand forecasing is essenial for invenory planning, especially when he demand environmen is highly dynamic and he procuremen lead imes are long. How o adjus invenory planning decisions according o demand forecasing updaes is of grea ineres o managers and for decades has araced many researchers. I is well known ha he opimal invenory policy for any dynamic forecasing invenory model is very complex, so boh in pracice and in he research lieraure considerable aenion has been given o much simpler myopic policies. For example, several auhors have eiher proposed using myopic policy as invenory policy (e.g., Graves 1999 and Aviv 2003) or esablished sufficien condiions under which a myopic policy can be opimal in specific demandforecasing models (e.g., Johnson and Thompson 1975, Miller 1986). Also, due o racabiliy, some recen works in he supply chain managemen lieraure employ he myopic policy in a demandforecasing environmen o gain various insighs such as he value of informaion sharing (e.g., Lee e al. 2000), collaboraive planning, forecasing and replenishmen (e.g., Aviv 2001, 2002), and quanifying bullwhip effec (e.g., Chen e al. 2000). However, some basic quesions remain. In paricular, are he insighs gained from he myopic policy sill valid in a sysem under an opimal policy? This is equivalen o asking how good he myopic policy is in general dynamic demandforecasing invenory models. Also, if he myopic policy is no good enough, are here any simple adjusmens ha can improve he performance significanly? More generally, how can we evaluae he performance of a myopic policy or any oher heurisic policy in erms of heir value loss relaive o opimal policy? These are he quesions we aim o address in his paper. We consider a singleiem, periodicreview invenory sysem wih demand forecasing updaes. The demand can be imecorrelaed and nonsaionary over ime or follow any demandforecasing model. For simpliciy, we assume ha he forecas evoluion follows he maringale model of forecas evoluion (MMFE), developed independenly by Graves e al. (1986) and Heah and Jackson (1994). The MMFE is quie sraighforward, general, and flexible. I can represen nonsaionary and imecorrelaed demands. I can also accommodae judgmenal forecass as well as commonly used ime series models such as he auoregressive moving average (ARMA) model. Oher feaures of he invenory model are sandard, such as full backlogging; a consan replenishmen lead ime; and linear ordering, invenoryholding, and backorderpenaly coss. Several auhors have adoped he MMFE o sudy producioninvenory planning issues. For insance, Güllü (1996) uses a woperiod MMFE o assess he value gained from using a dynamic demandforecasing model. Graves e al. (1998) address how o adjus he maerial requiremen schedule when he safeysock plans are modified from 1079
2 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds 1080 Operaions Research 54(6), pp , 2006 INFORMS period o period due o he modificaion of he demand forecass. Tokay and Wein (2001) focus on one ype of forecascorreced invenory policy in a sysem wih finie capaciy and obain closedform approximaions. Gallego and Özer (2001) consider a model of advance demand informaion and characerize he form of opimal policy. Their demand model can be viewed as a special case of he MMFE. Among he sudies using MMFE, Iida and Zipkin (2006) is mos closely relaed o ours. They show ha a demandforecas dependen basesock policy is opimal and develop bounds on he opimal basesock levels. They furher develop a piecewiselinear approximaion of he cos funcions and a simulaionbased echnique o solve he problem approximaely. Finally, hey esablish condiions under which he myopic policy is opimal. In his paper, we make wo major conribuions o he lieraure. The firs is he developmen of easierocompue bounds on opimal basesock levels, using a differen approach from ha of Iida and Zipkin (2006). Our approach also allows us o provide a necessary and sufficien condiion for he myopic policy o be opimal and gain deeper insighs. The second conribuion is he developmen of error bounds on he value loss of any heurisic policy relaive o he opimal cos, a subjec no sudied by Iida and Zipkin. These coserror bounds can also be used o evaluae heurisic policies in any dynamic demandforecasing invenory sysems. Our main idea for developing bounds on he opimal basesock levels is hrough a samplepah approach o approximae he firsorder condiion funcion in he dynamicprogram formulaion. We develop an explici expression of he firsorder condiion o see clearly he radeoff beween he marginal cos in he curren period and he marginal cos for he fuure periods. We use boh he probabiliy of oversock for a number of periods (i.e., no orders would be placed for a number of periods because he preorder invenory level is higher han he opimal basesock level in hese periods) and he magniudes of hese oversocks o esimae he marginal fuure cos. The noion of obaining bounds on he opimal basesock levels by esimaing he marginal fuure cos in dynamic invenory models is no new. See Moron (1978) and Moron and Penico (1995) for models wih nonsaionary and independen demands. These works use he probabiliy of oversock for a number of periods o develop lower bounds on he opimal basesock levels. By employing more informaion han he probabiliy of oversock, we obain significanly igher lower bounds; see, e.g., Example 7. Anoher approach o obaining bounds on he opimal basesock levels in dynamic invenory models is o allow disposal of sock earlier han he end of he horizon. This approach ransforms he original problem ino a shorer planning horizon problem. Solving he shorer planning horizon problem wih he upper or lower bound (on he marginal fuure cos) being erminal cos leads o he upper or lower bound on he opimal basesock levels. Using lower bound zero and upper bound he maximum salvage cos in he erminal period, Moron (1978) and Iida and Zipkin (2006) develop bounds on he opimal basesock levels for he independen demand model and MMFE, respecively. In fac, our approach is general enough o rea hese mehods as special cases; see Examples 5 and 6. However, solving a shorer planning horizon problem opimally is sill compuaionally challenging even for small problem sizes under MMFE. In he lieraure of dynamic invenory models, a commonly used mehod o evaluae he performance of a heurisic policy is o esimae he gap beween he lower and upper bounds on he opimal basesock levels. Moron (1978) and Iida and Zipkin (2006) show ha, under cerain condiions, he gap beween he upper and lower bounds by solving a shorer planning horizon (say k + 1 period) problem (wih zero or maximum salvage value as he erminal cos) goes o zero as k goes o infiniy. However, for he MMFE examined in his research, i is no pracical o solve a k + 1 period problem opimally even for small k. This moivaes us o explore an alernaive approach o evaluaing he effeciveness of heurisic policies. Our approach is a samplepah worscase approach. We develop upper bounds on he cos difference beween any heurisic policy and he opimal policy. We also develop lower bounds on he opimal cos. This leads o upper bounds on he cos error of any heurisic relaive o he opimal policy. To our knowledge, Lovejoy s (1990, 1992) are he only previous aemps in he dynamic invenory lieraure o esablish cos error bounds on subopimal policies. While Lovejoy focused on myopic and sopped myopic policies, we derive cos error bounds for any heurisic policy. Our echniques are also differen from his. When applied o myopic policies, our bounds are significanly igher. (Since he compleion of our sudy, Levi e al. (2004) have examined a similar invenory model o ours and show ha he cos of a dualbalancing policy is wihin 200% of he cos of opimal policy.) Several oher ypes of forecasing models have been sudied in he lieraure. One is a Bayesian model of updaing demand disribuion from pas hisory; see, e.g., Scarf (1959, 1960), Azoury (1985), and Lovejoy (1990). The second is a ime series approach he demand process is he ARMA process or he ARIMA (inegraed ARMA) process; see, e.g., Johnson and Thompson (1975), Miller (1986), Reyman (1989), and Graves (1999). The hird approach is o model he demand as a Markovmodulaed sochasic process; see, e.g., Lovejoy (1992), Song and Zipkin (1993), and Treharne and Sox (2002). For furher discussion of he lieraure, see Iida and Zipkin (2006). The res of his paper is organized as follows. Secion 2 inroduces he basic noaion and he model formulaion. Secion 3 discusses he firsorder condiion and he myopic policy. Secion 4 presens he soluion bounds, while 5
3 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds Operaions Research 54(6), pp , 2006 INFORMS 1081 develops he cos error bounds of any heurisic policy relaive o opimal policy. Finally, 6 presens numerical examples, and 7 concludes he paper. 2. Model and Formulaion We consider a T period periodicreview invenory sysem wih sochasic demand and zero replenishmen lead ime. (The exension o sysems wih a fixed consan lead ime can be done by following he sandard argumen.) Le D be he acual demand in period. The demand process D = 1 2 T can be nonsaionary and correlaed over ime. A any period, we generae forecass of he demand for all fuure periods in he horizon. A he beginning of each period, an ordering decision is made based on he invenory saus and he demand forecas. Then, he placed orders arrive. During he period, demand is realized and fulfilled as much as possible. Unsaisfied demand is fully backlogged. A he end of he period, invenoryholding and backorderpenaly coss are charged, and demand forecass are updaed. There are linear coss for ordering, invenory holding, and backlogging, respecively. We use ĉ, ĥ, and ˆb o represen he uni ordering, invenory holding, and backorderpenaly coss in period, 1 T, respecively. We also assume ha here is a salvage value ĉ T +1 a he end of period T. The objecive is o minimize he oal expeced cos. To updae he demand forecass, we can eiher follow a sandard forecasing ool such as a imeseries model, or use oher echniques such as exper judgmen, or do boh. Le D +i be he forecas made a he end of period for he demand in period + i, i = 0 T. Because forecass are made afer he curren demand informaion is revealed, D = D. Le D = demand forecas vecor made a he end of period = D +1 D T, where D 0 is he iniial forecas vecor. We consider wo ypes of forecas updaes: addiive and muliplicaive. For addiive updaes, define e +i = D +i D 1 +i as he forecas updae made a he end of period for demand in period + i. Denoe Var e = 2 and le e = demand forecas updae vecor made a he end of period = e e +1 e T. We assume ha he forecass are unbiased, i.e., E e s = 0 s. We also assume ha he forecas updaes e, = 1 2 T are independen over ime. The forecas updaes wihin a period, however, are no necessarily independen because hey migh rely on he same or relaed informaion. For muliplicaive updaes, similarly define e +i = D +i /D 1 +i. Here, E e +i = 1. Again, le e be he forecas updae vecor made a he end of period. Weassume ha boh D +i and e +i are posiive and he forecas updaes are independen over ime. As before, he forecas updaes wihin a period are no necessarily independen. For exposiion simpliciy, we mainly focus on he addiive model hroughou he paper (wih he excepion of 6.2). However, all he resuls hold rue for he muliplicaive model. We furher assume ha he forecas updaes have a coninuous disribuion, and hus he oneperiod cos has a unique minimum poin. The forecas updaes in differen periods can have differen disribuions. This model is broader han he original MMFE (e.g., Heah and Jackson 1994), which assumes he mulivariae normal disribuion. We now formulae he problem as a dynamic program. Le I (respecively, I ) be he invenory level a he beginning of period afer (respecively, before) ordering. The sae of he sysem a he beginning of period is I D 1. Because he lead ime is zero, he sysem dynamics are I +1 = I D = I D 1 e D +i = D 1 +i + e +i for 0 i T Le C y D 1 be he expeced holding and backorder coss charged o period, given ha I = y and he laes forecas for demand in period is D 1. Then, C y D 1 = ĥ E y D 1 + e + + ˆb E D 1 + e y + where x + = max x 0 and D 1 + e = D. Le V x D 1 be he opimal oal expeced coss from period hrough T, given ha I = x and he laes forecas for he fuure demands is D 1. We have he following recursive funcional equaions: V T +1 x D T = ĉ T +1 x V x D 1 = min ĉ y x + C y D 1 y x + E V +1 y D 1 + e D Nex, we make a ransformaion o simplify he exposiion. Se h = ĥ + ĉ ĉ +1 and b = ˆb ĉ ĉ +1. Le C y D 1 =h E y D 1 +e + +b E D 1 +e y + = ĉ ĉ +1 y + C y D 1 V x D 1 =ĉ x + V x D 1 We obain V T +1 x D T = 0 (1) V x D 1 = min y D 1 y x 1 T (2) where G y D 1 C y D 1 + E V +1 y D 1 + e D (3)
4 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds 1082 Operaions Research 54(6), pp , 2006 INFORMS From now on, we call he ransformed funcions C y D 1 and V x D 1 he oneperiod expeced cos and he opimal oal expeced cos from period o period T, respecively. I can be shown ha G y D 1 is convex in y. Le s D 1 be is minimizer. Then, he basesock policy wih ime and saedependen basesock levels s D 1 is opimal (e.g., Iida and Zipkin 2006). However, he mulidimensional Equaions (1) and (2) are exremely difficul o compue, and so is s D 1. In he res of his paper, we develop racable approximaions of s D 1 and provide error bounds on he cos of using he approximae policies. For simpliciy, we someimes suppress he argumen in he basesock levels. For example, we wrie s +i wih he undersanding of s +i D +i 1. Also, le D +i represen he cumulaive demand in periods 1 hrough + i 1, i.e., i 1 i 1 D + i = D +j = D +j +j j=0 j=0 Using his noaion, (3) can be rewrien as G y D 1 C y D 1 + E V +1 y D + 1 D Throughou his paper, for any funcion y D, wihou any confusion, we use y D o denoe he parial derivaive / y y D. In addiion, for any real numbers u and v, we denoe u v = max u v and u v = min u v. 3. The FirsOrder Condiion and Myopic Policies 3.1. The FirsOrder Condiion: A SamplePah View Because s D 1 is he soluion of he firsorder condiion G D 1 = 0, o develop approximaions of s D 1, we begin wih examining he consiuions of G D 1. Assume ha he invenory level afer ordering in period is y, i.e., I = y. By he dominaed convergence heorem, i is sraighforward o show ha E V +1 y D + 1 D = E V +1 y D + 1 D. Therefore, G y D 1 = C y D 1 + E V +1 y D + 1 D (4) Noe ha V +1 x D = G +1 x s+1 D D, which implies 0 x s V +1 x D +1 D = (5) G +1 x D >0 x>s+1 D So, V +1 = 0 if he opimal basesock level in period + 1, s+1 D, is reachable (Veino 1965), i.e., he preorder invenory level, I+1 = y D + 1, is no greaer han he opimal basesock level s+1 D. Therefore, he decision y a ime has no effec on he fuure cos if s+1 D is reachable. In oher words, y affecs (increases) he cos of periods + 1 and beyond only if he opimal basesock level in period + 1isno reachable, in which case V +1 is posiive. Similarly, for any fuure period + i, V +i is nonzero (posiive) only if none of s+1 s +2 s +i is reachable, i.e., I+j = y D +j >s +j, j = 1 2 i. This means ha he evens A +j y D 1 happen for all j = 1 2 i, where A +j y D 1 = y D + j >s +j D +j 1 (6) Wihou confusion, we suppress y D 1, he iniial condiion a ime, in his noaion mos of he ime. Le I A be he indicaor funcion of A and I A c = 1 I A. Noe ha he decision y a ime affecs period + k and beyond only if A +i happens for all i = 1 k. This implies E I A +1 A +i V +i y D + i D +i 1 > 0 and E I A +1 A +i 1 A c +i V +i y D +i D +i 1 =0 (7) The following proposiion provides a decomposiion of G. Proposiion 1. For any ime, G y D 1 = C y D 1 + y D 1 (8) where y D 1 = E V +1 y D + 1 D E I A +1 A +i T = i=1 Moreover, y D 1 0. C +i y D + i D +i 1 +i (9) Proof. According o (4), we need o work only on E V +1 y D + 1 D : E V +1 y D + 1 D = E I A +1 V +1 y D + 1 D + E I A c +1 V +1 y D + 1 D = E I A +1 V +1 y D + 1 D = E I A +1 C +1 y D + 1 D +1 + E I A +1 V +2 y D + 2 D +1
5 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds Operaions Research 54(6), pp , 2006 INFORMS 1083 = E I A +1 C +1 y D + 1 D +1 + E I A +1 A +2 V +2 y D + 2 D +1 + E I A +1 A +2 c V +2 y D + 2 D +1 = E I A +1 C +1 y D + 1 D +1 + E I A +1 A +2 V +2 y D + 2 D +1 T = = E I A +1 A +i C +i y D +i D +i 1 +i i=1 = y D 1 The second and he fifh equaliies are due o (7). Finally, y D 1 0 follows from (5). From Proposiion 1, he opimal invenory decision is a radeoff beween he marginal cos in he curren period C y D 1 and he marginal fuure cos y D 1. While he marginal curren cos can be posiive or negaive, from (5) he marginal fuure cos is always nonnegaive. Applying (8), he firsorder condiion for opimaliy is G s D 1 D 1 = C s D 1 D 1 + s D 1 D 1 = 0 (10) Noe ha C s D 1 D 1 = b + h P D 1 + e s D 1 b = b + h F s D 1 D 1 b where F is he cumulaive disribuion funcion of forecas error erm e. Treaing s D 1 D 1 as a known consan, from (10) we can express he opimal basesock level as ( s D 1 = D 1 + F 1 b s D ) 1 D 1 (11) b + h Because s D 1 D 1 depends on he enire forecas evoluion, i is very difficul o obain Myopic Policies One common approach o deal wih he difficuly of obaining s D 1 D 1 is o ignore i and use only he firs erm, C y D 1, o approximae G y D 1. This resuls in he socalled myopic policy a basesock policy wih he basesock level solving C y D 1 = 0. Le s m D 1 be he myopic basesock level a ime given ha he forecas vecor a he beginning of is D 1. Then, ( ) s m D 1 = D 1 + F 1 b (12) b + h If F is a normal disribuion and we le be he sandard normal disribuion funcion (remember ha is he sandard deviaion of he forecasing error erm e ), hen ( ) s m D 1 = D b (13) b + h Here, 1 b / b + h is ermed he safey facor. The myopic policy simply uses a lower bound zero o approximae he erm s D 1 D 1 0 and hus is an upper bound on s D 1, i.e., s D 1 s m D 1 Remark. In he case of a consan lead ime L, (12) becomes ( ) L s m D 1 = D 1 +i + F L+1 1 b (14) b + h i=0 where F L+1 is he cumulaive disribuion funcion of L j=i e +i +j. L i=0 The simple expressions of (12) and (13) furher explain why he myopic policy is popular and when i migh cause subopimaliy. A any ime, he policy parameer is he sum of wo erms: he laes forecas of he curren period demand D 1 and a safey sock ha depends only on he disribuion of forecas error e. Noe ha in mos forecasing models, i is reasonable o have =. If, furher, he cos parameers are saionary, hen he safey sock is a consan. Under hese seings, using he myopic policy, we would sock in he firs period he consan safey sock plus he forecased demand for he firs period, D 0 1. Then, in he subsequen periods, we need only o adjus he order quaniy according o he realized demand in he previous period and he laes demand forecas for he curren period. More specifically, if we have ordered in period 1, so he posorder invenory posiion in ha period I 1 = s 1 m and he preorder invenory posiion I = s 1 m D 1 1, hen q = s m I = D 1 D D 1 1 = D 1 + e 1 1 is he order quaniy in period if his quaniy is nonnegaive. If q is nonnegaive for all sample pahs for all, which means ha he myopic basesock level in each period is reachable, hen he cos in each period is minimized, so he myopic policy is opimal. I urns ou ha q 0is boh necessary and sufficien for he myopic policy o be opimal. In general, we have he following. Proposiion 2. The myopic policy is opimal if and only if P s m D 1 D + 1 >s m +1 D = 0 for all (15) If boh he cos parameers and he forecas updae process are saionary, (15) is equivalen o q = D 1 + e for all (16) Proof. The sufficien condiion can be shown easily by inducion; we omi he deails here. To show he necessary condiion, suppose ha P s m D >s+1 m >0. This implies ha E I s m D >s+1 m C +1 sm D D +1 > 0.
6 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds 1084 Operaions Research 54(6), pp , 2006 INFORMS Because V +2 0, we have G sm D 1 >0, which means ha he myopic policy canno be opimal a conradicion. Now assume saionary coss and forecas updaes. Condiion (16) requires ha he demand forecas for he curren period (period ) be large enough o offse he negaive deviaion of he forecas error in he previous period. I is conceivable ha his condiion can be me easily if he demand process has a nondecreasing rend. This is consisen wih he general undersanding of when he myopic policy is expeced o be opimal. I is ineresing o see ha even when demand has a decreasing rend, he myopic policy can sill be opimal. Indeed, Iida and Zipkin (2006) offers such an example (see heir independen, nonsaionary demand example), provided he following condiion holds: For every sample pah of forecas updaes, demands are nonnegaive, i.e., D = D 1 + e 0 for all (17) Because e 1 1 and e have he same disribuion and e and D 1 are uncorrelaed, (17) implies (16). Alhough he demand may have a decreasing rend, (17) ensures ha he forecas errors will be bounded by he lowes possible demand. In oher words, if he demand in one period is low, he forecas errors in all periods have o be sufficienly small. Noe ha because D 1 can be correlaed wih e 1 1, (16) does no necessarily imply (17). Therefore, Iida and Zipkin s condiion is sufficien bu no necessary. We now illusrae his by an example adaped from Güllü (1997). In his example, demands in wo consecuive periods are negaively correlaed. If he demand in he curren period is high (respecively, low), hen he demand in he nex period is expeced o be lower (respecively, higher). This ype of demand paern is common when cerain markeing effors are in place. For example, he demand righ afer a promoion period is expeced o be lower because of forward buying during he promoion period. Example 3. Le be he mean demand, and se D 0 = e = N 0 2 e +1 = e s = 0 for all s + 2 D = D 0 + e 1 + e for all Here, N 0 2 is he normal random variable N 0 2 runcaed a 25 and 25. (Obviously, given he large runcaion limis, N 0 2 is basically N 0 2.) Assume ha = 0 9, = 100, and = 40. Because e 1 = 0 9e 1 1 and , we have q = D 1 +e 1 1 = +e 1 +e 1 1 = Hence, he myopic policy is opimal. However, D 1 + e can be negaive, so (17) is no saisfied. When would myopic policies behave poorly? From he necessary and sufficien condiion (16) we see ha, for saionary coss and forecas updaes, he myopic policy is subopimal if here exis sample pahs in which q < 0 for some. This could happen if here is a sudden drop in demand or here is a decreasing demand rend so ha D 1 is oo small o offse he negaive deviaion of e 1 1. A similar observaion is made by Song and Zipkin (1996) using a Markov modulaed sochasic process o model demand facing obsolescence. 4. Bounds on Opimal BaseSock Levels Recall ha he myopic policy simply approximaes he nonnegaive erm s D 1 D 1 by zero and hus is an upper bound on he opimal basesock level s D 1. In his secion, we develop a general class of upper (lower) bounds on s D 1 D 1 (hese imply bounds on G ), which, according o (11), yield lower (upper) bounds on he opimal basesock level s D 1. Observe from he expression of (9) ha wha makes difficul o compue is is dependence on he opimal basesock levels from period + 1 on hrough he evens A +j in (6), j = 1 i. Our idea for approximaion is o replace he opimal basesock level s+j D +j 1 in A +j wih simplerocompue bounds. We presen a procedure o consruc he bounds recursively. We laer show hrough examples ha our procedure includes many exising bounds in he lieraure as special cases Consrucion of he Bounds Se T u D T 1 = T l D T 1 = 0 and st u D T 1 = st l D T 1 = st m D T 1 (he myopic basesock level in period T ). For any ime < T, suppose ha we have obained upper and lower bounds on he opimal basesock level in period + i, s+i u D +i 1 and s+i l D +i 1, respecively, for all i = 1 T. (For example, we may have s+i u D +i 1 = s+i m D +i 1 and s+i l D +i 1 = 0.) Assume ha I = y, and define A u +i y D 1 = y D + i > s u +i D +i 1 A l +i y D 1 = y D + i > s l +i D +i 1 A m +i y D 1 = y D + i > s m +i D +i 1 T l y D 1 = E I A u +1 Au +i u y D 1 i=1 C +i y D + i D +i 1 +i = E I A u +1 Am +1 C +1 y D + 1 D +1 E I A l +1 Al +i 1 Am +i T + i=2 C +j y D + i D +i 1 +i
7 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds Operaions Research 54(6), pp , 2006 INFORMS 1085 S l D 1 = zero poin of C y D 1 + u y D 1 S u D 1 = zero poin of C y D 1 + l y D 1 Noe ha l y D 1, u y D 1, S l D 1, and S u D 1 all depend on he known bounds s+i u D +i 1 and s+i l D +i 1, i = 1 T, implicily. If all he evens A m +1 y D 1 A m +i y D 1 happen, for example, he invenory level before ordering is greaer han he myopic basesock level in all periods +1 hrough +i given ha he invenory level afer ordering in period is y. Therefore, no order will be placed in hese periods under he myopic policy. Similar discussion applies o A u +i y D 1 and A l +i y D 1. As we shall show below, using he righ combinaion of A u +i y D 1, A l +i y D 1, and A m +i y D 1 o replace A +i y D 1 in y D 1 can resul in bounds on y D 1. We have Theorem 4. Theorem 4 (Bounds on Opimal BaseSock Levels). For any given, = 1 T, under he above assumpion and consrucion, we have l y D 1 y D 1 u y D 1 (18) This implies ha G y D 1 is bounded below and above by C y D 1 + l y D 1 and C y D 1 + u y D 1, respecively. Moreover, hese boundingfuncions are increasingfuncions of y, so heir zero poins S u D 1 and S l D 1 are unique and have he exac expression as in (11) wih replaced by l and u, respecively. Finally, S l D 1 s D 1 S u D 1 Proof. I is sufficien o show ha (18) holds; oher pars are sraighforward. Noe ha when = T, from he erminal condiion (1), all he inequaliies in (18) become equaliies. Now, consider <T. Firs, noe ha from he consrucion, A u +i A +i A l +i and Am +i A +i, implying I A u +i I A +i I A l +i and I A m +i I A +i for all i = 1 T. For every sample pah of D + 1, wehave C +1 y D + 1 D +1 <0 for s +1 D <y D + 1 <s m +1 D So, E I A +1 A u +1 Am +1 C +1 y D + 1 D Therefore, E I A +1 C +1 y D + 1 D +1 = E I A +1 A u +1 Am +1 C +1 y D + 1 D +1 + E I A u +1 Am +1 C +1 y D + 1 D +1 E I A u +1 Am +1 C +1 y D + 1 D +1 (19) Similarly, E I A +1 A +i C +i y D + i D +i 1 +i = E I A +1 A +i 1 A m +i C +i y D + i D +i 1 +i + E I A +1 A +i 1 A +i A m +i C +i y D + i D +i 1 +i E I A +1 A +i 1 A m +i C +i y D + i D +i 1 +i E I A l +1 Al +i 1 Am +i C +i y D + i D +i 1 +i i = 2 T (20) The firs inequaliy is due o he fac ha I A +i A m +i C +i y D + i D +i 1 +i 0. The second inequaliy is from I A m +i C +i y D + i D +i 1 +i 0. Combining (9), (19), and (20), we obain y D 1 u y D 1. On he oher hand, recall ha V +1 y D + 1 D is nonnegaive, so I A +i A u +i V +i y D + i D +i 1 0 (21) Then, E I A +1 V +1 y D + 1 D 1 = E I A u +1 V +1 y D + 1 D + E I A +1 A u +1 V +1 y D + 1 D E I A u +1 V +1 y D + 1 D = E I A u +1 G +1 y D + 1 D = E I A u +1 C +1 y D + 1 D +1 + E I A u +1 A +2 V +2 y D + 2 D +1 = E I A u +1 C +1 y D + 1 D +1 + E I A u +1 Au +2 V +2 y D + 2 D +1 + E I A u +1 A +2 A u +2 V +2 y D + 2 D +1 E I A u +1 C +1 y D + 1 D +1 + E I A u +1 Au +2 V +2 y D + 2 D +1 = E I A u +1 C +1 y D + 1 D +1 + E I A u +1 Au +2 G +2 y D + 2 D +1 = E I A u +1 C +1 y D + 1 D +1 + E I A u +1 Au +2 C +2 y D + 2 D +1 + E I A u +1 Au +2 A +3 V +3 y D + 3 D +2 T E I A u +1 Au +i C +i y D +i D +i 1 +i i=1 = l y D 1 (22) All he inequaliies are due o (21). Noe ha he second and fifh equaliies are due o he fac ha when A u +i happens, we have y D + i >s+i, so no order will be
8 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds 1086 Operaions Research 54(6), pp , 2006 INFORMS placed in period + i under he opimal policy. The hird and sixh equaliies are due o (7). From (9) and (22), we obain y D 1 l y D 1. To summarize, from Theorem 4, for any, based on any upper and lower bounds on he opimal basesock levels from period +1 unil T, we can consruc lower and upper bounds on he opimal basesock level in period. In his way, we develop a general class of bounds on he opimal basesock levels Connecions wih Exising Bounds Clearly, he only difference beween hese new approximaions and he myopic policy is he adjusmens in he fracile ha deermines he magniude of he safey sock. The myopic is a special case of our resul in which we se l y D 1 = 0. Below we provide several oher examples o illusrae he connecion of he bounds in Theorem 4 wih he exising bounds in he lieraure. The soluion o he shorerhorizon, k + 1 period problem wih zero erminal cos is referred o as he kperiod ahead soluion. I has been shown in he lieraure ha his soluion is an upper bound on opimal basesock level for he nonsaionary independen demand model (Moron 1978) and for he MMFE (Iida and Zipkin 2006). In Example 5, we show ha his resul can be viewed as a special case of our procedure. For simpliciy, we illusrae only he case of k = 1. The idea for general k is similar. Example 5 (OnePeriod Ahead Policy). Seing s+1 u = s+1 m and s+2 u = yields Au +2 = y D + 2 > s+2 u D +1 = y D + 2 > =. Therefore, E I A u +1 Au +i C +j y D + i D +i 1 +i = 0 for all i 2 and l y D = E I Am +1 C +1 y D + 1 D +1. According o Theorem 4, he soluion o C y D 1 + E I A m +1 C +1 y D + 1 D +1 = 0 is an upper bound on s D 1. In he following, we show ha our procedure generalizes and ighens some lower bounds in he lieraure. For insance, Example 6(b) has been shown by Moron (1978) and Iida and Zipkin (2006) for independen and MMFE demand models, respecively. Example 6(c) provides igher lower bounds on he opimal basesock level han ha of Example 6(b). Example 6. Le s+i D k +i 1 be he opimal soluion o a k + 1 period problem (from period unil period + k) for period + i wih erminal cos of T i=k+1 h +i. Then, (a) s+i D k +i 1 solves he following equaions recursively: C +i y D +i 1 +i + k +i y D +i 1 = 0 (23) where k y D 1 = E [ I ( y D + 1 >s k ) +1 C +1 y D + 1 D +1 ] [ + +E I ( y D + 1 >s k ) +1 I ( y D +k+1 >s k ) +k+1 ( T h +i )] (24) i=k+1 (b) s+i D k +i 1 s+i D +i 1, i = 0 k. (c) Replacing T i=k+1 h +i in (24) wih T i=k+1 h +ip y D + i >s+i m D +i 1, solving (23) leads o a igher lower bound for s D 1. Proof. By he same logic in developing Proposiion 1, we obain (a). Because he proof of (c) is similar o ha of (b), below we show only (b). We use inducion. Because C +k y D +k 1 +k + k +k y D +k 1 C +k y D +k 1 +k + +k y D +k 1 we know ha s k +k D +k 1 s+k D +k 1. Suppose ha s+j D k +j 1 s+j D +j 1 holds for i j k. We only need o show s+i 1 D k +i 2 s+i 1 D +i 2. For simpliciy, we show he case of i =1. Applying a similar proof of (22) o he k + 1 period problem under consideraion, we know ha replacing s+j D k +j 1 1 j k wih is upper bound s+j D +j 1 leads o a lower bound on k y D 1. Tha is, C y D 1 + k y D 1 k 1 C y D 1 + E I A +1 A +i i=1 C +i y D + i D +i 1 +i + E I ( y D + k + 1 >s k ) ( T )] +k+1 h +i i=k+1 C y D 1 + y D 1 [ I A +1 A +k Therefore s k D 1 s D 1 holds because s k D 1 solves C y D 1 + k y D 1 = 0 and s D 1 solves C y D 1 + y D 1 = 0. Nex, we illusrae ha our procedure generalizes and ighens oher lower bounds presened in he lieraure ha are obained hrough avenues differen from solving shorerhorizon problems.
9 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds Operaions Research 54(6), pp , 2006 INFORMS 1087 Example 7. Se s+1 u D = s+1 m D. According o Theorem 4, we have y D 1 u 1 y D 1, where u 1 y D 1 = E I A m +1 C +1 y D + 1 D +1 Because E I A l +1 Al +j 1 I Am +j T + j=2 C +j y D + j D +j 1 +j C +j y D + j D +j 1 +j = b +j + h +j P D +j 1 +j + e +j +j y D + j b +j h +j (25) we furher have u 1 y D 1 u 2 y D 1 u 3 y D 1, where u 2 u 3 T y D 1 =h +1 P A m +1 + h +j P A l +1 Al +j 1 Am +j j=2 T y D 1 =h +1 P A l +1 + h +j P A l +1 Al +j 1 Al +j According o Theorem 4, he soluion o each of he following hree equaions: j=2 C y D 1 + u i y D 1 = 0 i= (26) is a lower bound on he opimal basesock level. Furhermore, because u i y D 1 increases in y, wehave C y D 1 + u i y D 1 C y D 1 + u i s m D 1 D 1 y s m D 1 Suppose ha S l i D 1 solves C y D 1 + u i s m D 1 D 1 = 0. Then, S l i D 1 is newsvendor soluion, which is easier o compue bu is a looser lower bound (han he soluion o (26)): s D 1 S l 1 D 1 S l 2 D 1 S l 3 D 1 where S l i D 1 = D 1 + ( b u i s m D 1 D 1 b + h ) i = (27) The lower bound S l 3 D 1 is essenially he lower bound developed in Moron (1978) and Moron and Penico (1995). They developed he lower bound on he opimal basesock level for he case of independen demand invenory models and heir resul can be exended o he demandforecasing invenory models. In addiion o he probabiliy of oversock hey employ, we use more informaion such as he magniude of oversock, y D +j s+j D +j 1 given I = y, o esimae he marginal fuure cos. Observe from (25) ha h +j P A l +1 Al +j 1 Al +j can be much greaer han E I A l +1 Al +j 1 Am +j C +j, j = 1 2 T. So, u 1 y D 1 can be much smaller han u 3 y D 1, which could lead o a significanly igher lower bound on opimal basesock levels. Indeed, for a woperiod problem, one of our lower bounds, he soluion o (26) for he case of i = 1, is he exac opimal soluion, while heir lower bound S l 3 D 1 canno be opimal. In fac, none of he bounds in (27) can be opimal. Now we presen some easieroimplemen bounds, which are used in he numerical sudies in 6. Example 8. Se s u +1 D = s m +1 D, s u +2 D +1 = s m +2 D +1, and s u +3 D +2 =. We obain A u +3 = y D + 3 >su +3 D +1 = y D + 3 > = which implies E I A u +1 Au +j C +j y D + j D +j 1 +j = 0 j 3 According o Theorem 4, S u 1 D 1, he soluion o C y D 1 +E I A m +1 C +1 y D +1 D +1 +E I A m +1 Am +2 C +2 y D +2 D =0 (28) is an upper bound for s D 1. Example 9. Seing s u +1 D = s m +1 D and s l +i D +i 1 =, 2 i T, wehave A l +i = y D + i > sl +i D +i 1 = y D + i > = Therefore, E I A l +1 Al +i 1 Am +i C +i y D + i D +i 1 +i =E I A l +1 Am +i C +i y D +i D +i 1 +i j 3 Thus, S l 4 D 1, he soluion o C y D 1 + E I A m +1 C +1 y D + 1 D +1 E I A l +1 Am +j C +j y D +j D +j 1 +j =0 T + j=2 (29) is a lower bound on s D 1 according o Theorem 4.
10 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds 1088 Operaions Research 54(6), pp , 2006 INFORMS 5. CosError Bounds In his secion, we consider how o esimae he value loss of a heurisic relaive o opimal cos. Le V H x D 1 be he oal expeced cos of a given heurisic policy H in periods hrough T, assuming ha he preorder invenory level in period is x and he forecas vecor made a he end of period 1isD 1. The cos error of H relaive o he opimal cos is defined by err = V 1 H x D 0 V 1 x D 0 100% V 1 x D 0 Our approach is o develop an upper bound for V H x D 1 V x D 1 and a lower bound for V x D 1 because V x D 1 is usually compuaionally impossible o obain. Recall ha s m is an upper bound for he opimal basesock level; herefore, we assume ha s H D 1 s m D Upper Bound on V H x D 1 V x D 1 Take any period, and assume ha I = x. There are wo possible siuaions ha require differen reamens: s H D 1 s D 1 and s H D 1 s D 1. We sudy hem separaely and presen he resuls in Lemmas 10 and 11. Based on hese resuls, we develop upper bounds on V H x D 1 V x D 1, which is presened in Theorem 12. Lemma 10. If s H D 1 s D 1, we have x D 1 V x D 1 1 D 1 + E +1 x sh D 1 D D V +1 x s H D 1 D D (30) where 1 D 1 = C s H D 1 D 1 C s u D 1 s m D 1 D 1 (31) Proof. If s H D 1 s D 1, hen C s H D 1 D 1 C s D 1 D 1. Furhermore, because s m D 1 minimizes C D 1,wehave C x s H D 1 D 1 C x s D 1 D 1 1 D 1 (32) Noe ha x D 1 = C s H D 1 x D 1 + E +1 sh D 1 x D D and V x D 1 = C s D 1 x D 1 + E V +1 s D 1 x D D We obain x D 1 V x D 1 = C x s H D 1 D 1 C x s D 1 D 1 + E +1 x sh D 1 D D E V +1 x s D 1 D D 1 D 1 + E +1 x sh D 1 D D E V +1 x s D 1 D D 1 D 1 + E +1 x sh D 1 D D V +1 x s H D 1 D D Here, he firs inequaliy is due o (32). The second inequaliy is because x s H D 1 D x s D 1 D and V +1 z D increases in z. Lemma 11. If s H D 1 >s D 1, hen x D 1 V x D 1 2 D 1 + E +1 x sh D 1 D D V +1 x s H D 1 D D (33) and x D 1 V x D 1 3 D 1 + max E +1 s D 1 D D V +1 s D 1 D D E +1 x D D V +1 x D D (34) Here, 2 D 1 = 1 C s H D 1 D 1 + u s H D 1 D 1 1 = s H D 1 s l D 1 + T 3 D 1 = E I A h +1 Ah +k 1 Am +k 2C +k sh D 1 k=1 D + k D +k 1 +k A h +i = sh D 1 D + i > s H +i D +i 1 2 = s H D 1 s l D 1 s H D 1 D s H +1 D + (35) Proof. We firs show ha (33) holds. We have he following key observaion: x D 1 V x D 1 = s H D 1 x D 1 G s D 1 x D 1 = G x s H D 1 D 1 G x s D 1 D 1 + x s H D 1 D 1 G x s H D 1 D 1
11 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds Operaions Research 54(6), pp , 2006 INFORMS 1089 = G x s H D 1 D 1 G x s D 1 D 1 + C x s H D 1 D 1 C x s H D 1 D 1 + E +1 x sh D 1 D D E V +1 x s H D 1 D D = G x s H D 1 D 1 G x s D 1 D 1 + E +1 x sh D 1 D D E V +1 x s H D 1 D D (36) Noe ha G y D 1 s D 1 y s H D 1 0. We also have G x s H D 1 D 1 G x s D 1 D 1 = x s H D 1 x s D 1 G y D 1 y= x s D 1 x s H D 1 s H D 1 s D 1 + G y D 1 y=s H = s H D 1 s D 1 + C sh D 1 D 1 + s H D 1 D 1 1 C sh D 1 D 1 + u sh D 1 D 1 = 2 D 1 (37) The second inequaliy is due o (18). Combining (36) and (37) yields (33). We nex show ha (34) holds. If x s H D 1 s D 1, wehave x D 1 V x D 1 = C x D 1 C x D 1 + E +1 x D D E V +1 x D D = E +1 x D D E V +1 x D D (38) If x<s H D 1,wehave x D 1 = 0 D 1 (39) Because s m D 1 s H D 1, we also have C s H D 1 D 1 C s D 1 D 1 0 (40) Therefore, x D 1 V x D 1 = 0 D 1 V x D 1 0 D 1 V 0 D 1 = C s H D 1 D 1 C s D 1 D 1 + E +1 sh D 1 D D E V +1 s D 1 D D (41) E +1 sh D 1 D D E V +1 s D 1 D D (42) =E +1 sh D 1 D D E +1 s D 1 D D + E +1 s D 1 D D E V +1 s D 1 D D (43) The firs equaliy is due o (39). The firs inequaliy is due o V x D 1 increases in x, and he second inequaliy is due o (40). We furher show he following in he appendix: E +1 sh D 1 D D E +1 s D 1 D D 3 D 1 (44) From (43) and (44), if x<s H D 1, we have ha (34) holds. On he oher hand, if x s H D 1, from (38), (34) also holds. This complees he proof. Based on Lemmas 10 and 11 and noing ha n D 1, n = 1 2 3, do no depend on he iniial invenory x, we develop upper bounds on V H x D 1 V x D 1 as follows. Theorem 12 (Maximum Gap beween he Coss of Any Heurisic Policy and he Opimal Policy). For any given heurisic policy H and any, x D 1 V x D 1 1 D 1 n D 1 [ T ] + E 1 k D k 1 n k D k 1 n= 2 3 (45) k=+1 Proof. Suppose ha n = 2. For = T, because we assume ha st H sm T = s T, we do no need o consider sh T >s T. For he case st H s T, from Lemma 10, we have ha (45) holds. Suppose ha (45) holds for V+i H x D V +i x D, i = 1 T, for some. Also, recall ha he upper bound in (45) is independen of x. In he following, we show ha (45) holds for period, he proof will hen be compleed by inducion. No maer wheher s H D 1 is less han s D 1 or no, from (30) and (33), we have x D 1 V x D 1 1 D 1 2 D 1 + E +1 x sh D D V +1 x s H D D By inducion, (45) holds for n = 2. Similarly, based on (30) and (34), (45) also holds for n = 3. Remarks. (a) The maximum value loss of using he heurisic policy H is given by he righhand side of (45) wih = 1. (b) The righhand side of (45) does no depend on he preorder invenory level x, so wha we have is a worscase analysis. In oher words, he righhand side of (45) is an absolue upper bound on he cos difference beween he heurisic and he opimal policy. We can modify 1 D 1 n D 1 o develop a recursive upper bound for V1 H x D 1 V 1 x D 1, which depends on x. We illusrae his by modifying 1 D 1 2 D 1 ino
12 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds 1090 Operaions Research 54(6), pp , 2006 INFORMS 1 D 1 and 2 D 1 only. The oher is much more complex. 1 D 1 = C s H D 1 x D 1 C s u D 1 s m D 1 x D 1 2 D 1 = 2 D 1 = 1 C sh D 1 D 1 + u sh D 1 D 1 1 = s H D 1 x s l D 1 x + (c) In he numerical sudy, we use he following fac: T 3 D 1 E I A h +1 Am +k 2 k=1 C +k sh D 1 D + k D +k 1 +k (46) Taking any upper bound on he opimal basesock level ha is no greaer han he myopic basesock level (such as he myopic soluion) as a heurisic policy, we do no need o consider he case of s H D 1 s D 1. This yields he following corollary. Corollary 13. The maximum gap beween he coss of he opimal policy and any upperbound policy s u D 1, = 1 2 T, ha saisfies s u D 1 s m D 1 for all is 1 x D 0 V 1 x D 0 T 3 1 D 0 + E 3 D 1 where 3 1 D D 1 T = =2 T E 3 D 1 (47) =2 E s u D 1 D s u +1 D + I A m +k k=1 C +k su D 1 D + k D +k 1 +k (48) ( T h +k )E s u D 1 D s u +1 D + (49) k=1 In paricular, he coserror bounds for he myopic policy is given by replacing s u D 1 wih s m D 1 in (47). Noe ha, compared wih (46), he bound (48) is much easier o evaluae bu can be much looser (see he examples in 6). Using a differen approach, Lovejoy (1992) developed an upper bound for he cos difference beween he opimal policy and he myopic policy. Because s m D 1 s D 1, insead of developing 3, Lovejoy esimaes he cos of disposing of he oversock s m D s+1 m +, which yields an upper bound on he exra cos due o he decision made in period. For example, according o Lovejoy (1992), he disposal cos can be se a T j=+1 h j. Thus, Lovejoy s bound is o replace 3 D 1 wih T k=1 h +k E s u D 1 D s+1 u D +. From (49), he error bound developed in his paper can be much igher, as illusraed in he numerical examples in Lower Bound on V x D 1 The derivaion of a lower bound on V x D 1 is quie sraighforward. We assume ha an upper bound on opimal basesock level s u D 1 is known (recall ha s m D 1 is a special case of s u D 1 ). We have he following. Proposiion 14. V x D 1 C x s u D 1 s m D 1 D 1 E C +j s u +j D +j 1 T + j=1 s m +j D +j 1 D +j 1 +j (50) Proof. Because s+j D +j 1 s+j u D +j 1 s+j m D +j 1 s+j m D +j 1 for all j, wehave V x D 1 C x s D 1 D 1 E C +j s +j D +j 1 D +j 1 +j T + j=1 C x s u D 1 s m D 1 D 1 E C +j s u +j D +j 1 T + j=1 s m +j D +j 1 D +j 1 +j This lower bound is similar o hose in Lovejoy (1990, 1992) when s u D 1 is chosen o be s m D Numerical Sudy 6.1. Resul Illusraion In his subsecion, we presen a numerical sudy o illusrae he resuls developed in he previous secions. We also compare our coserror bounds wih hose developed in Lovejoy (1990, 1992). We use an AR(1) demand forecas model. Tha is, D = + D for all < 1 where E D =, he coefficien of correlaion of demands in wo successive periods is, and are i.i.d. N 0 2 random variables. A any ime period, afer D is revealed, we generae a new forecas for he demand in period + 1as D +1 = +1 + D
13 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds Operaions Research 54(6), pp , 2006 INFORMS 1091 Using he new forecas for period + 1, we obain a new demand forecas for period + 2as D +2 = +2 + D = D Similarly, we obain he new forecas for period + i as D +i = +i + i D Therefore, we have 0 i T D = D +1 D T e = e e T = T Noe ha = e = D D 1 is he onesep forecasing error. The AR(1) model has been adoped by several auhors in he recen supply chain managemen lieraure o sudy he value of informaion sharing and collaboraive forecasing (see, e.g., Aviv 2001 and Lee e al. 2000). Due o racabiliy, hese auhors focus on myopic policies. A naural quesion is: Can he findings of hese sudies also apply o a sysem under an opimal policy? This is equivalen o asking wheher he myopic policy is sufficienly good for sysems wih he AR(1) demand model. Previous research has esablished cerain sufficien condiions under which he myopic policy is opimal when he demand follows AR(1) (see, e.g., Johnson and Thompson 1975 and Iida and Zipkin 2006). To shed ligh on he above issues, our numerical sudy focuses on parameers which do no saisfy hese sufficien condiions. More specifically, he ime horizon T = 10 and here is a replenishmen lead ime L = 3. Two ses of cos parameers were chosen. One se is nonsaionary: h 2j+1 = 1, b 2j+1 = 19, h 2j = 9, b 2j = 11 j = 0 4. The oher se is saionary: h i = 1, b i = 20, i 1. Iniial Demand. We choose several values of D 0 : D 0 0 = 0 + p 0, where 0 = / 1 2 1/2, p For any fixed p, D 0 = + p 0 and D 0 = D 0 1 D 0 2 D 0 T. We se 0 / 1 = 0 3 or Wih hese parameers, he demand can demonsrae a wide range of variabiliy. For example, suppose ha 0 / 1 = 0 3, +1 = 5, 1 = 100, = 0 9, and p = 3. We have D 1 N and D 2 N D 13 N Demand Trend. We firs consider consan over ime: = 100 for all 1. We hen consider wih a decreasing rend: +i = +i 1 5, 1 = 100. We es differen values of Heurisics Evaluaed. We evaluae he performances of hree policies: he myopic policy s m, he upper bound policy S u 1 given in (28), and he heurisic policy S H defined by S H D 1 = S u 1 D S l 4 D (51) Here, S l 4 D 1 is given by (29). The parameer is chosen o minimize max 1 D 1 3 D 1 in period (i.e., o minimize he upper bound on he relaive cos error). Also, we use (31) o compue 1 D 1 and we use (35) and (46) o compue 3 D 1. To illusrae he heurisic S H D 1 under consideraion, le us consider he firsperiod problem for he case of decreasing demand rend, nonsaionary cos parameers, and 0 / 1 = 0 3. Suppose ha D 0 = = Wehaves1 m D 0 = , S u 1 1 D 0 = , and S l 4 1 D 0 = We furher have S1 H D 0 = wih he parameer = 0 68 (which means ha he heurisic is closer o S u 1 1 D 0 ). In evaluaing he heurisic S1 H D 0, we have 2 = 1 32, 1 1 D 0 = 2 43, 3 1 D 0 = 2 44, and max 1 1 D D 0 = Upper Bounds on Relaive Cos Error. Tables 1 4 show err, he upper bound on relaive cos error of he approximae policies, where err is defined by err = V 1 H x D 0 V 1 x D 0 100% V 1 x D 0 err = RHS of 45 RHS of % where RHS sands for righhand side. MLovejoy, Measy, and Myopic are upper bounds on relaive cos error for myopic policy when we apply Lovejoy s mehod defined in (49), he easierocompue upper bound defined in (48), and a general mehod defined in (47) o evaluae he myopic policy, respecively. Because Lovejoy s mehod is no for evaluaing a general heurisic policy, our mehod is used o evaluae S H D 1 given in (51). Specifically, we use (31), (35), and (46) o compue he upper bound on V1 H x D 0 V 1 x D 0 as defined in 45 for n = 3 o evaluae he heurisic which is referred o as S H D 1. Tables 1 and 3 repor some special cases, while Tables 2 and 4 provide he averages. As menioned before, we esed all cases of and p For he cases we do no repor, he resuls are usually beer, i.e., he upper bounds on he relaive cos errors are smaller for he same value of. Sofware Used. We used Malab version 6.5 for he numerical sudy. We employed wo rouines for inegraion: he single inegraion funcion quad( ) and double inegraion funcion dblquad( ). We also used he sandard normal probabiliy densiy and cumulaive disribuion funcions: normpdf( ) and normcdf( ). We se he maximum error o be for precision. We observe ha for nonsaionary cos parameers, he myopic policy can be far from opimal; in one case he value loss is as high as 45%. On he oher hand, wih saionary cos parameers, he myopic policy is very close o opimal. The maximum value loss is around 2%.
14 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds 1092 Operaions Research 54(6), pp , 2006 INFORMS Table 1. Upper bound on relaive cos error for nonsaionary cos parameers. MEasy MLovejoy Myopic S H MEasy MLovejoy Myopic S H p 0 / 1 = 0 3 Trend = 0 0 / 1 = 0 3 Trend = / 1 = 0 25 Trend = 0 0 / 1 = 0 25 Trend = The heurisic policy S H is very close o opimal wheher or no he cos parameers are saionary. The maximum value loss is % for nonsaionary coss and % for saionary coss. This also indicaes ha our upper bound on he cos difference beween he heurisic and opimal policy is very igh. This observaion can be furher verified by comparing he upper bound on he relaive cos error on he myopic policy beween our mehod and Lovejoy s (1990, 1992) mehod. The examples show ha our cos error bounds are usually 20% o 1% of Lovejoy s (1992) bounds. The reason is, when he oversock (by ordering up o he myopic basesock level) happens, he exra cos due o he oversock is usually much smaller han he oal cos for holding he iems from he curren period unil he end of he planning horizon he basis for Lovejoy s esimaion. The performance of S u 1 is also very close o opimal; is maximum value loss is 2.26% in all he cases examined. Thus, his policy is recommended o be used if he myopic policy fails perform well. Noe ha he woperiodahead policy is a slighly igher upper bound on he opimal basesock level han S u 1.So he good performance of S u 1 implies good performance of he woperiodahead policy. This is consisen wih he finding by Treharne and Sox (2002), who show ha he woperiodahead policy is very close o opimal for he Markov modulaed demand model. I is ineresing o observe ha, as decreases, he performances of all hree policies end o be beer (closer o opimal). We find his difficul o explain. For example, on one hand, if <0 and he demand in period is lower han, he expeced demand in period +1 will be higher han +1, so boh he probabiliy and he magniude of oversock decrease, which favors he myopic policy. On he oher hand, if he demand in period is higher han, he expeced demand in period + 1 will be lower han +1, which seems o be agains he myopic policy. Table 2. Upper bound on relaive cos error for nonsaionary cos parameers. MEasy MLovejoy Myopic S H MEasy MLovejoy Myopic S H 0 / 1 = 0 3 Trend = 0 0 / 1 = 0 3 Trend = / 1 = 0 25 Trend = 0 0 / 1 = 0 25 Trend =
15 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds Operaions Research 54(6), pp , 2006 INFORMS 1093 Table 3. Upper bound on relaive cos error for saionary cos parameers. MEasy MLovejoy Myopic S H MEasy MLovejoy Myopic S H p 0 / 1 = 0 3 Trend = 0 0 / 1 = 0 3 Trend = / 1 = 0 25 Trend = 0 0 / 1 = 0 25 Trend = The following example, however, sheds some ligh on why he myopic policy may be near opimal for near 1. Assume ha cos parameers are saionary and = 100 for all 1. Then, he safey sock, ss, for he myopic policy is he same across differen periods due o (14). More specifically, he forecas error of he lead ime demand is I has a normal disribuion wih mean zero and sandard deviaion where is he sandard deviaion of. Thus, he myopic policy has safey sock ss = ( ) b 1 b + h In paricular, s m = D ss s m +1 = D ss Now, consider he case = 0 9 and 0 / 1 = 0 3. Because D 1 1 has a normal disribuion, by direc compuaion, he probabiliy of oversock is max 3 p 3 P s m D s+1 m D 1 1 = p Var D Wih such a low (close o zero) probabiliy of oversock under he myopic policy, he myopic level is likely o be always reachable and hus likely o be opimal Comparison wih Iida and Zipkin (2006) In his subsecion, we compare our approximaions wih hose in Iida and Zipkin (2006). For his purpose, we follow heir choice of demand paern and cos parameers. More specifically, demand follows a muliplicaive model. The experimens include wo paerns of iniial forecass: rends and cycles. A firs, he iniial forecas for period 1 is 250. We hen adjus ha value and se he iniial forecass for periods 2 o 16 as follows: Trends: Linear rends wih hree slopes: +10, 0, and 25, numbered 1 hrough 3 (represened by rend in he able). Cycles: Four ypes of cycle: none, 50 sin, 50 cos, and 50 cos ; numbered 1 hrough 4 (represened by C in Tables 5 and 6). Table 4. Upper bound on relaive cos error: Average case for saionary cos parameers. MEasy MLovejoy Myopic S H MEasy MLovejoy Myopic S H 0 / 1 = 0 3 Trend = 0 0 / 1 = 0 3 Trend = / 1 = 0 25 Trend = 0 0 / 1 = 0 25 Trend =
16 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds 1094 Operaions Research 54(6), pp , 2006 INFORMS These paerns can produce nonposiive iniial forecass (he rend erm plus he cycle erm); such values are rese o one. The muliplicaive forecas updaes have mulidimensional lognormal disribuions. Le s = log e s. The s hen are joinnormally disribued wih covariance marix (52) and mean vecor Oher parameers are h = 2 and b = 10. The planning horizon T = We also examine he case of demand uncerainy being revealed early. To represen early resoluion of uncerainy, a new se of problems is consruced by reversing he order of he diagonal elemens in he covariance marix (52) as done in Iida and Zipkin. Wih hese daa, we compare our heurisic soluion S H D 1 defined in (51) wih Iida and Zipkin s (2006, 4.3) soluions obained by using heir wo approximaion echniques o solve he dynamic program. The resuls are presened in Tables 5 and 6, respecively. Here, 1 and 2 are drawn from Tables 1 and 2 of an earlier version of heir paper, daed December 20, 2004, and err is he relaive cos error of our heurisic policy given by (51). Noe ha 1 measures he error of funcional approximaions and 2 measures he sampling error in Iida and Zipkin. In oher words, 1 is he upper bound on he relaive cos error while 2 is he compuaional error. See ha earlier version of Iida and Zipkin for more deail. For he problems esed, our heurisic policy is near opimal; is maximum value loss is less han 1% in all he cases examined. When he demand uncerainy is revealed early, our policy appears even beer. The reason is ha when he demand uncerainy is revealed earlier, he safey sock is reduced and hus he possibiliy of oversock is reduced. As a consequence, he gap beween he upper and lower bounds on he opimal basesock level can be very small (or even zero), rendering our heurisic o be near opimal or even opimal. Ou of he oal 120 cases repored in Tables 5 and 6, in 109 cases (more han 90%) our mehod is beer han Iida and Zipkin s (judging from he magniudes of he errors). Our mehod is usually beer when (1) he planning horizon is shorer, or (2) he demand paern increases or says he same, or (3) he demand uncerainy reveals early. Even in oher cases, such as when he demand paern decreases, our mehod can sill be beer. 7. Conclusions We have examined a singleiem, periodicreview invenory sysem wih demandforecas updaes following he Maringale model of forecas evoluion (MMFE). The opimal policy is a saedependen basesock policy ha is compuaionally inracable o obain. Using a samplepah approach, we developed a general class of racable bounds on he opimal basesock levels, which generalized and improved he exising bounds in he lieraure. We hen used hese bounds o consruc nearopimal policies. Our numerical examples showed ha our heurisics ouperform he myopic policy significanly. The samplepah approach also allowed us o idenify a necessary and sufficien condiion for he myopic policy o be opimal, which sharpens our inuiion on his policy. Furhermore, our samplepah Table 5. Comparison wih Iida and Zipkin s (2006) mehod: Base case. Trend C T = 2 T = 4 T = 8 T = 12 T = 16 C T = 2 T = 4 T = 8 T = 12 T = err err err err err err
17 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds Operaions Research 54(6), pp , 2006 INFORMS 1095 Table 6. Comparison wih Iida and Zipkin s (2006) mehod: Early resoluion of demand uncerainy. Trend C T = 2 T = 4 T = 8 T = 12 T = 16 C T = 2 T = 4 T = 8 T = 12 T = err err err err err err approach enabled us o perform worscase analysis and derive upper bounds on he value loss of any heurisic policy (including he myopic policy). These appear o be he firs se of coserror bounds on he performance of any heurisic policies in dynamic invenory models. Numerical examples demonsraed ha our error bounds improve he exising error bounds in he lieraure for evaluaing he performance of myopic policies. Finally, boh he soluion bounds and he coserror bounds developed in his paper can be easily adaped o general dynamic invenory models wih nonsaionary or auocorrelaed demands. Appendix Proof of (44). Define Ĩ +i = invenory level afer ordering in period + i following he heurisic policy under consideraion given I +1 = s H D 1 D, Î +i =invenory level afer ordering in period + i following he heurisic policy under consideraion given I +1 = s D 1 D. We have Ĩ +i Î +i 0 i= 1 2 T (53) Ĩ +1 Î +1 s H D 1 D s H +1 D s D 1 D s H +1 D s H D 1 s D 1 s H D 1 D s H +1 D + = 2 (54) The reason is, considering period + 1, if he heurisic orders for boh I +1 = sh D 1 D and I +1 = s D 1 D, hen Ĩ +1 Î +1 = 0; if he heurisic places an order for only one siuaion, because s H D 1 D s D 1 D, we have Ĩ +1 Î +1 = s H D 1 D s+1 H D s H D 1 s D 1 ; if he heurisic does no place an order for boh siuaions, we have Ĩ +1 Î +1 = s H D 1 s D 1. We focus on developing an upper bound for E V+1 H s H D 1 D D E V+1 H s D 1 D D by developing an upper bound for C +i Ĩ +i C +i Î +i, i = 1 T. Le i 0 = min j s H D 1 D + j s H +j D +j 1 Then, period +i 0 is he firs period afer when he heurisic will place an order given I +1 = sh D 1 D. Thus, we have Ĩ +i0 = s H +i 0 D +i0 1. Because Î +i0 s H +i 0 D +i0 1 = Ĩ +i0, from (53), we have Ĩ +i0 = Î +i0. Therefore, C +j Ĩ +j C +j Î +j = 0 j = i 0 T (55) The following evens happen for all k<i 0 : A h +k = sh D 1 D + k > s H +k D +k 1 In any of hese periods, say period + k k < i 0,ifĨ +k s m +k D +k 1, because Î +k Ĩ +k s m +k D +k 1, and one period expeced cos C +k z decreases in z when z< s m +k D +k 1, wehave C +k Ĩ +k C +k Î +k 0 (56) If Ĩ +k >s m +k D +k 1 + 2, because Ĩ +k Î +k 2 (from (54)) and C +k z increases in z when z>s m +k D +k 1, we have C +k Ĩ +k C +k Î +k C +k Ĩ +k C +k Ĩ +k 2 (57)
18 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds 1096 Operaions Research 54(6), pp , 2006 INFORMS If s m +k D +k Ĩ +k > s m +k D +k 1, because C +k Î +k C +k s m +k D +k 1, wehave C +k Ĩ +k C +k Î +k C +k Ĩ +k C +k s m +k D +k 1 (58) In summary, if Ĩ +j >s H +j D +j 1, j = 1 k, and Ĩ +k > s m +k D +k 1, from (57) and (58), we have C +k Ĩ +k C +k Î +k C +k Ĩ +k C +k Ĩ +k 2 s m +k D +k 1 (59) Considering all hese possibiliies, we have E C +k Ĩ +k C +k Î +k = E I A h +1 Ah +k 1 Ah +k c C +k Ĩ +k C +k Î +k + E I A h +1 Ah +k 1 Ah +k C +k Ĩ +k C +k Î +k = E I A h +1 Ah +k 1 Ah +k C +k Ĩ +k C +k Î +k = E I A h +1 Ah +k 1 Am +k C +k Ĩ +k C +k Î +k + E I A h +1 Ah +k 1 Ah +k Am +k C +k Ĩ +k C +k Î +k E I A h +1 Ah +k 1 Am +k C +k Ĩ +k C +k Î +k E I A h +1 Ah +k 1 Am +k C +k Ĩ +k C +k Ĩ +k 2 s m +k D +k 1 = E I A h +1 Ah +k 1 Am +k C +k s H D 1 D + k D +k 1 +k C +k s H D 1 D + k 2 s m +k D +k 1 D +k 1 +k = E I A h +1 Ah +k 1 Am +k C +k D +k 1 +k 2 s H D 1 D + k s m +k D +k 1 s H D 1 D +k 2 s m +k D +k 1 s H D 1 D +k E I A h +1 Ah +k 1 Am +k 2 C +k sh D 1 D + k D +k 1 +k The second equaliy is due o (55), i.e., when A h +1 A h +k 1 Ah +k c happens, Ĩ +k = Î +k. The firs inequaliy is due o (56), and he second inequaliy is due o (59). The las equaliy is due o he medium heory, and he las inequaliy is due o C +k z D +k 1 +k increases in z and C +k z D +k 1 +k 0 when z s+k m D +k 1. Finally, we have E +1 sh D 1 D D E = E [ T k=1 { C +k Ĩ +k C +k Î +k +1 s D 1 D D }] T E I A h +1 Ah +k 1 Am +k 2C +k sh D 1 k=1 D + k D +k 1 +k = 3 D 1 Acknowledgmens The auhors hank he associae edior and wo anonymous referees for many helpful suggesions ha improved he paper s exposiion. This research was suppored in par by NSF grans DMI and DMI and he Naional Naural Science Foundaion of China award no References Aviv, Y The effec of collaboraive forecasing on supply chain performance. Managemen Sci. 47(10) Aviv, Y Gaining benefis from join forecasing and replenishmen processes: The case of auocorrelaed demand. Manufacuring Service Oper. Managemen 4(1) Aviv, Y A imeseries framework for supplychain invenory managemen. Oper. Res. 51(2) Azoury, K. S Bayes soluion o dynamic invenory models under unknown demand disribuion. Managemen Sci. 31(9) Chen, F., Z. Drezner, J. K. Ryan, D. SimchiLevi Quanifying he bullwhip effec in a simple supply chain: The impac of forecasing, lead imes, and informaion. Managemen Sci. 46(3) Gallego, G., O. Özer Inegraing replenishmen decision wih advance demand informaion. Managemen Sci. 47(10) Graves, S. C A singleiem invenory model for a nonsaionary demand process. Manufacuring Service Oper. Managemen 1(1) Graves, S. C., D. B. Kleer, W. B. Hezel A dynamic model for requiremens planning wih applicaion o supply chain opimizaion. Oper. Res. 46(3) S35 S49. Graves, S. C., S. Meal, Y. Dasu, Y. Qing Twosage producion planning in a dynamic environmen. S. Axsaer, C. Schneeweiss, E. Silver, eds. MuliSage Producion Planning and Conrol. Lecure Noes in Economics and Mahemaical Sysems. SpringerVerlag, Berlin, Güllü, R On he value of informaion in dynamic producion/invenory problems. Naval Res. Logis. 43(2) Güllü, R A woechelon allocaion model and he value of informaion under correlaed forecass and demands. Eur. J. Oper. Res. 99(2) Heah, D. C., P. L. Jackson Modeling he evoluion of demand forecass wih applicaion o safey sock analysis in producion/disribuion sysems. IIE Trans. 26(3) Iida, T., P. Zipkin Approximae soluions of a dynamic forecasinginvenory model. Manufacuring Service Oper. Managemen 8(4) Johnson, G., H. Thompson Opimaliy of myopic invenory policies for cerain dependen demand process. Managemen Sci Lee, H. L., K. C. So, C. S. Tang The value of informaion sharing in a wo level supply chain. Managemen Sci. 46(5) Levi, R., M. Pal, R. Roundy, D. Shmoys Approximaion algorihms for sochasic invenory conrol. Working paper, Cornell Universiy, Ihaca, NY. Lovejoy, W. S Sopped myopic policies in some invenory models wih uncerain demand disribuions. Managemen Sci. 36(6) Lovejoy, W. S Sopped myopic policies in some invenory models wih generalized demand processes. Managemen Sci. 38(5) Miller, B. L Scarf s sae reducion mehod, flexibiliy, and a dependen demand invenory model. Managemen Sci. 34(1) Moron, T. E The nonsaionary infinie horizon invenory problem. Managemen Sci. 24(14)
19 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds Operaions Research 54(6), pp , 2006 INFORMS 1097 Moron, T. E., D. W. Penico The finie horizon nonsaionary sochasic invenory problem: Nearmyopic bounds, heurisics, esing. Managemen Sci. 41(2) Reyman, G Sae reducion in a dependen demand invenory model given by a ime series. Eur. J. Oper. Res. 41(2) Scarf, H Bayes soluions of he saisical invenory problem. Ann. Mah. Sais. 30(2) Scarf, H Some remarks on Bayes soluion o he invenory problem. Naval Res. Logis. Quar. 7(4) Song, J. S., P. Zipkin Invenory conrol in a flucuaing demand environmen. Oper. Res. 41(2) Song, J. S., P. Zipkin Managing invenory wih he prospec of obsolescence. Oper. Res. 44(1) Tokay, L. B., L. M. Wein Analysis of a forecasingproducioninvenory sysem wih saionary demand. Managemen Sci. 47(9) Treharne, J. T., C. R. Sox Adapive invenory conrol for nonsaionary demand and parial informaion. Managemen Sci. 48(5) Veino, A. F., Jr Opimal policy for a muliproduc, dynamic, nonsaionary invenory problem. Managemen Sci. 12(3)
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