Approximation Algorithms for Perishable Inventory Systems

Transcription

1 Approximaion Algorihms for Perishable Invenory Sysems Xiuli Chao Deparmen of Indusrial and Operaions Engineering, Universiy of Michigan, Ann Arbor, MI 48109, Xiing Gong Deparmen of Sysems Engineering and Engineering Managemen, The Chinese Universiy of Hong Kong, Shain, N.T., Hong Kong, Cong Shi, Huanan Zhang Deparmen of Indusrial and Operaions Engineering, Universiy of Michigan, Ann Arbor, MI 48109, {shicong, We develop he firs approximaion algorihms wih wors-case performance guaranees for periodic-review perishable invenory sysems wih general produc lifeime, for boh backlogging and los-sales models. The demand process can be non-saionary and correlaed over ime, capuring such feaures as demand seasonaliy and forecas updaes. The opimal conrol policy for such sysems is nooriously complicaed, hus finding effecive heurisic policies is of pracical imporance. In his paper, we consruc a compuaionally efficien invenory conrol policy, called he proporional-balancing policy, for sysems wih an arbirarily correlaed demand process and show ha i has a wors-case performance guaranee less han 3. In addiion, when he demands are independen and sochasically non-decreasing over ime, we propose anoher policy, called he dual-balancing policy, which admis a wors-case performance guaranee of 2. We demonsrae hrough an exensive numerical sudy ha boh policies perform consisenly close o opimal. Key words : invenory, perishable producs, correlaed demands, approximaion algorihms, cos balancing, wors-case performance guaranee Received Ocober 2013; revisions received July 2014, December 2014; acceped March Inroducion and Summary In his paper, we sudy he classic periodic-review sochasic invenory sysems wih perishable producs. The produc lifeime is known and fixed. Iniial ineres in hese sysems was sparked by blood bank applicaions (see, e.g., Prasacos (1984), Pierskalla (2004), Karaesmen e al. (2011)), bu he scope of applicaions is far greaer. For example, perishable producs such as food iems and pharmaceuicals consiue he majoriy of sales revenue of grocery reailing indusry. Food Marke Insiue (2012) repored ha perishables accouned for 52.63% of he 2011 oal supermarke sales 1

2 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems 2 of abou $459 billion 1, and mismanagemen of perishable producs (such as spoilage and shrinkage) represens a major hrea o he profiabiliy of companies in grocery reailing indusry. A survey by he Naional Supermarke Research Group repored an average loss of $34 million a year due o spoilage in one major 300-sore grocery chain. 2 Thus, finding effecive invenory managemen policies for perishable producs has always been of grea ineres o boh praciioners and academic researchers. We resric our aenion o he firs-in-firs-ou (FIFO) issuing policy which is commonly adoped in he lieraure (see, e.g., Nahmias (2011) and Karaesmen e al. (2011)), i.e., he oldes invenory is consumed firs when demand arrives. This assumpion is reasonable for blood invenory sysems and online grocery nework (e.g., AmazonFresh). I also applies o he reailers who display only he oldes iems on he shelves. The demands in differen periods in our model can be non-saionary (or ime-dependen) and correlaed over ime, capuring such demand feaures as seasonaliy and forecas updaes as well as many oher demand processes of pracical ineres. Boh backlog model and los-sales model will be sudied. These sysems are fundamenal bu nooriously hard o analyze in boh heory and compuaion. As seen from our lieraure review below, he opimal policy is very complex even when he demands are independen and idenically disribued. The opimal order quaniy depends on boh he age disribuion of he on-hand invenory and he lengh ill he end of he planning horizon. The compuaion of he opimal policy using dynamic programming is in general inracable due o he curse-of-dimensionaliy. Thus, many researchers urned o seek effecive heurisic policies for hese problems. To he bes of our knowledge, almos all heurisics developed hus far have been focused on he case wih independen and idenically disribued demands. Moreover, none of he heurisic policies in he lieraure admis provably wors-case performance guaranees. In his paper, we propose he firs approximaion algorihm wih a wors-case performance guaranee of 2 for hese imporan sysems when he demands in differen periods are independen and sochasically non-decreasing over ime. The demand processes in pracical seings are ofen seasonal, forecas-based, or driven by he sae of he economy (e.g., Markov modulaed demand processes). The demands of hese processes are correlaed over ime. For example, many firms employ forecasing mehods o learn abou he fuure demands and periodically updae heir forecass; and such forecas-based demand processes can ofen be modeled by he maringale model of forecas evoluion (MMFE for shor, see, e.g., Graves e al. (1986), and Heah and Jackson (1994)), in which he updaed forecas is he original forecas plus an adjusmen (or random error) wih mean 0. In addiion, in pracice firms ofen

3 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems 3 receive advance demand informaion (ADI) from some cusomers for he fuure periods, so managers have o periodically incorporae such ADI in he fuure demands (see, e.g., Gallego and Özer (2001)). These models are pracically imporan, bu finding he opimal policies using brue-force dynamic programming is compuaionally inracable, since he sae space of he corresponding dynamic programs is usually large (see, e.g., Lu e al. (2006)). The compuaional burden is even more severe for perishable invenory sysems, given he fac ha he age disribuion of on-hand invenory has o be racked oo. To overcome his prohibiive compuaional challenge, we propose anoher approximaion algorihm for perishable invenory sysems wih arbirarily correlaed demand processes, and show ha i admis a wors-case performance guaranee less han 3. To he bes of our knowledge, no effecive heurisic policy has ever been developed in he lieraure for perishable invenory sysems wih correlaed demand processes Main Resuls and Conribuions The main resuls and conribuions of his paper are summarized as follows. Algorihms. We develop wo approximaion algorihms which admi provably wors-case performance guaranees for perishable invenory sysems. Firsly, we develop a proporional-balancing (PB) policy for he perishable invenory sysems wih produc lifeime of m ( 2) periods under an arbirarily non-saionary and correlaed demand ( ) process. We show ha he PB policy has a wors-case performance guaranee of 2 + (m 2)h, mh+θ where h = ĥ + (1 α)ĉ, θ = ˆθ + αĉ, and ĉ, ĥ, ˆθ, and α are he per-uni ordering, holding, oudaing coss, and one-period discoun facor, respecively. Tha is, for any insance of he problem, he ( ) expeced cos of he PB policy is a mos 2 + (m 2)h imes he expeced cos of an opimal policy. mh+θ Therefore, he heoreical wors-case performance guaranee is beween 2 and 3 and i equals 2 when he produc lifeime m = 2. Secondly, when he demand process is independen and sochasically non-decreasing over ime, we develop a dual-balancing (DB) policy which has a wors-case performance guaranee of 2, i.e., for any insance of he problem, he expeced cos of he DB policy is a mos wice he expeced cos of an opimal policy. To he bes of our knowledge, our proposed policies are he firs se of compuaionally efficien policies wih wors-case performance guaranees in sochasic periodic-review perishable invenory

4 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems 4 sysems. In conras, compuing he exac opimal policy using dynamic programming suffers from he well-known curse of dimensionaliy and is inracable even wih shor produc lifeimes (e.g., m = 4) and under independen and idenically disribued demand processes. Wors-case analysis. In our algorihmic design, we develop a nesed marginal cos accouning scheme for perishable invenory sysems. This scheme is similar in spiri o ha developed in Levi e al. (2007), bu has a much more complex and nesed srucure due o he muli-dimensional invenory sae represening he age disribuion of on-hand invenory. The main idea of his approach is o associae he coss wih each ordering decision insead of each period. However, he echniques developed for our wors-case performance analysis depar significanly from hose in he previous sudies (e.g., Levi e al. (2007), Levi e al. (2008a)), which heavily rely on he exisence of a one-o-one maching beween he supply and demand unis when he invenory unis are consumed in an firs-in-firs-ou manner. Tha is, when analyzing he performances of he approximaion algorihms, all he previous sudies geomerically mach produc unis in a one-o-one manner for he sysems operaing under wo differen policies; and he coss for each pair of mached unis can be readily compared. However, he perishabiliy of producs desroys his maching mechanism, hus he exising echniques developed for non-perishable invenory sysems are no longer applicable. To overcome his difficuly, we inroduce a key new concep, called he rimmed on-hand invenory level, defined as he par of on-hand invenory unis ordered before a paricular ime. This key concep enables us o compare he coss of wo perishable invenory sysems operaing under wo differen policies. Compared wih he previous geomeric approach, our new approach is purely algebraic and we expec i o be useful in sudying oher perishable invenory sysems. Empirical performances. Our exensive compuaional sudies show ha our proposed policies perform consisenly near-opimal for all he esed insances, which are significanly beer han he heoreical wors-case performance guaranees. More specifically, for independen and idenically disribued demand processes and shor produc lifeimes (for which he opimal policies can be numerically compued), our numerical resuls are comparable o hose repored in Nahmias (1976, 1977b) and are very close o he opimal (around 0.3% above he opimal cos); for long produc lifeimes for which compuing opimal policies is inracable, we compare he performance of our mehods wih Nahmias (1976, 1977b); he overall performance of our policies is also comparable o hose of Nahmias (1976, 1977b) and i improves as he frequency of oudaing increases. For non-saionary and correlaed demand processes including Markov modulaed demand process, Maringale models of forecas evoluion (MMFE), auoregressive (AR) models, and models wih

5 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems 5 advance demand informaion (ADI), he proposed algorihms also perform close o opimal for all he problem insances esed wih maximum performance error below 3%, and average error below 1%, of he opimal coss Lieraure Review The problem of managing sochasic periodic-review invenory sysems wih perishable producs has araced he aenion of many researchers over he years. The dominan paradigm in he exising lieraure has been o formulae hese models using a dynamic programming framework. Nahmias and Pierskalla (1973) characerized he srucure of he opimal ordering policy for a woperiod produc lifeime problem. Nahmias (1975) and Fries (1975) hen, independenly, sudied he opimal policy for he general lifeime problem wih independen and idenically disribued (i.i.d.) demands, in a backlogging model and a los-sales model, respecively. They showed ha he opimal ordering quaniy depends on boh he age disribuion of he curren invenory and he remaining ime unil he end of planning horizon. Thus, compuing he opimal policy using brueforce dynamic programming is inracable due o he muli-dimensional sae space. The complexiy of his problem is laer reinforced by Cohen (1976) who characerized he saionary disribuion of invenory for he wo-period lifeime problem, and showed ha he opimal policy is a saedependen policy. Recenly, Li and Yu (2014) revisied he srucural properies of he opimal invenory policy in perishable invenory sysems by employing he mulimodulariy concep; and Chen e al. (2014) sudied he join invenory conrol and pricing problem for perishable invenory sysems and characerized he srucural properies of he opimal invenory and pricing policies. Due o he complexiy of he opimal policies for perishable invenory sysems, a lo of effors have been dedicaed o he design of efficien heurisic policies for boh backlogging and los-sales models. When he demands are i.i.d., Nahmias (1976) consruced a bound on he oudaing cos which is a funcion of only he oal on-hand invenory, and developed a base-sock myopic policy using his bound. Nahmias (1977a) employed he same myopic policies o compare wo dynamic perishable invenory models developed by Nahmias (1975) and Fries (1975). Subsequenly, Nahmias (1977b) used a more refined approximae sae ransiion funcion which reas he produc lifeime as if i were only wo periods, hereby resuling in a one-dimensional sae variable. The numerical resuls for problems wih produc lifeimes of 2 and 3 periods are near-opimal under i.i.d. demands. Nandakumar and Moron (1993) derived upper and lower bounds for he dynamic programming formulaion of he los-sales model, and used a weighed average of he lower and upper bounds o consruc an efficien heurisic. The numerical resuls showed ha he heurisic again performs

6 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems 6 close o opimal for shor produc lifeimes and i.i.d. demands. Cooper (2001) derived bounds on he saionary disribuion of he number of oudaed unis in each period, under a fixed criical number order policy. Recenly, following he approximaion scheme of he oudaing coss developed by Nahmias (1976), Chen e al. (2014) proposed wo heurisic policies for he join invenory conrol and pricing models. Since he fuure demands depend on he fuure prices, hey approximaed he expeced demands and prices in he fuure periods by solving he corresponding deerminisic models. Their numerical sudy showed ha he wo heurisic policies perform very well when he demands are i.i.d., bu he performance error could go up o 15% for he independen bu imevarying demands. Li e al. (2009) also designed an effecive heurisic following he approximaion mehod of Nahmias (1976) for he join invenory conrol and pricing model. Li e al. (2013) proposed wo myopic heurisics for perishable invenory sysems wih las-in-firs-ou issuing policy and clearance sales. To reduce he sae space, he firs heurisic reas all on-hand invenories as if hey would expire in one period whereas he second heurisic keeps rack of he oal invenory level and he invenory level of iems whose remaining lifeime is one period. Their numerical resuls showed ha hese heurisics perform very close o opimal under i.i.d. demands. As seen from he lieraure above, almos all he exising heurisic policies have been focused on i.i.d. demands, and none of hem admis provably wors-case performance guaranees. There is also a large body of lieraure discussing oher aspecs of perishable invenory sysems. We pariion hese sudies ino he following caegories (our lis below is by no means exhausive): (a) coninuous-review perishable invenory sysems (see, e.g., Weiss (1980), Goh e al. (1993), Liu and Lian (1999), Perry (1999)); (b) perishable invenory sysems wih muliple producs or demands (see, e.g., Nahmias and Pierskalla (1976), Deuermeyer (1979), Ferguson and Koenigsberg (2007), Deniz e al. (2010), Cai and Zhou (2014)); (c) join invenory and pricing conrol of perishables (see, e.g., Li e al. (2009), Chen and Sapra (2013), Chen e al. (2014)); (d) perishable invenory sysems wih depleion or clearance sales (see, e.g., Cai e al. (2009), Xue e al. (2011), Li and Yu (2014), Li e al. (2013)); and (f) blood banks and healh-care applicaions (see, e.g., Prasacos (1984), Haijema e al. (2005, 2007), Zhou e al. (2011)). We also refer ineresed readers o Nahmias (1982, 2011), Goyal and Giri (2001), and Karaesmen e al. (2011) for comprehensive lieraure reviews. Our work is also closely relaed o he recen sream of lieraure on approximaion algorihms for sochasic periodic-review invenory sysems pioneered by Levi e al. (2007). The sysems allow for correlaed sochasic demand processes, including all of he known approaches o model dynamic demand forecas updaes (e.g., Gallego and Özer (2001), Iida and Zipkin (2006), and Lu e al. (2006)). Levi e al. (2007) firs inroduced he concep of marginal cos accouning which associaes

7 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems 7 he coss wih each decision a paricular policy makes. They proposed a 2-approximaion policy which admis a wors-case performance guaranee of 2 for he backlogging model wih generally correlaed demands. Subsequenly, Levi e al. (2008a) proposed a 2-approximaion algorihm for he los-sales models under independen demand processes; and Levi e al. (2008b) inroduced he concep of forced marginal backlogging cos accouning and proposed a 2-approximaion algorihm for he capaciaed sysems wih backlogging. More recenly, Levi and Shi (2013) and Shi e al. (2014) proposed wo approximaion algorihms for he lo-sizing backlogging models wihou and wih capaciy consrains, respecively; and Tao and Zhou (2014) proposed a 2-approximaion algorihm for invenory sysems wih remanufacuring. All hese previous sudies assume ha he invenory is non-perishable, and herefore here exiss a one-o-one maching beween he supply and demand when he invenory issuing policy is FIFO. Perishabiliy, however, desroys his maching mechanism and he exising echniques for he wors-case analysis canno be applied o perishable invenory sysems Srucure The remainder of his paper is organized as follows. In 2, we presen he mahemaical formulaion for he backlogging model. In 3, we design a nesed marginal cos accouning scheme for perishable invenory sysems. In 4, we consruc he proporional-balancing policy and he dual-balancing policy, and provide heir wors-case performance guaranees. In 5, we provide he main proofs, while leaving some of he more involved echnical deails in he supplemenal maerial. In 6, we conduc an exensive numerical sudy on our proposed policies. Finally, we conclude he paper in 7 wih some discussions on exensions and possible fuure research. Throughou his paper, for any real numbers x and y, we denoe x + = max{x, 0}, x y = max{x, y}, and x y = min{x, y}. In addiion, for a sequence x 1, x 2,... and any inegers and s wih s, we denoe x [,s] = s j= x j and x [,s) = s 1 j= x j. Also, we use expiraion, oudaing, and perishing, inerchangeably. 2. Sochasic Periodic-Review Perishable Invenory Conrol Problem In his secion, we provide he mahemaical formulaion of he sochasic periodic-review perishable invenory sysem over a planning horizon of T (possibly infinie) periods, indexed by = 1,..., T. The lifeime of he produc is m periods, i.e., a produc perishes afer saying m periods in sock. Our model allows for a non-saionary and generally correlaed demand process. We assume ha he order lead ime is zero, i.e., an order placed a he beginning of a period can be used in

8 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems 8 he same period. This is a common assumpion in he perishable invenory lieraure (see Karaesmen e al. (2011)). We shall focus our presenaion on he backlogging model bu will exend he analysis and resuls o he los-sales model in Secion 7. Demand srucure. The demands over he planning horizon are random, denoed by D 1,..., D T. The demands in differen periods can be non-saionary and correlaed over ime. A he beginning of each period, here is an observed informaion se denoed by f, which conains all of he informaion accumulaed up o period. More specifically, he informaion se f consiss of he realized demands d 1,..., d 1 in he firs 1 periods, and possibly some exogenous informaion denoed by (w 1,..., w ). The informaion se f in period is one specific realizaion in he se of all possible realizaions of he random vecor F = (D 1,..., D 1, W 1,..., W ). The se of all possible realizaions is denoed by F. Wih he informaion se f, he condiional join disribuion of he fuure demands (D,..., D T ) is known. We assume ha he condiional expecaions, given f, are well defined. Noe ha his demand srucure is very general, ha includes a wide range of demand processes such as Markov modulaed demand process (see, e.g. Song and Zipkin (1993) and Sehi and Cheng (1997)), MMFE (see, e.g., Heah and Jackson (1994)), AR(p), ARMA(p, q), ARIMA(p, r, q), (see, e.g., Mills (1990)), and models wih advance demand informaion (ADI) (see, e.g., Gallego and Özer (2001)), among ohers. Cos srucure. In each period, four ypes of coss may occur: a uni ordering cos ĉ, a uni holding cos ĥ for lefover invenory, a uni backlogging cos ˆb for unsaisfied demand, and a uni oudaing cos ˆθ for expired producs. There is also a one-period discoun facor α, wih 0 < α 1 when T < and 0 < α < 1 when T =. We assume ha ˆb > (1 α)ĉ and ˆθ + αĉ 0. Thus, ˆθ can be negaive, and in his case i can be inerpreed as uni salvage value. Following Nahmias (1975) we assume ha any remaining invenory a he end of he planning horizon can be salvaged wih a reurn of ĉ per uni and unsaisfied demand can be saisfied by an emergency order a a cos of ĉ per uni. We noe ha our analysis can be exended o he case wih a uni salvage value ˆv for any on-hand invenory and a uni penaly cos ˆp for any unsaisfied demand a he end of he planning horizon, as long as ˆv ĉ and ˆb + αˆp > ĉ. However, our analysis canno be direcly exended o he case wih age-dependen salvage values. Sysem dynamics. For each period, he sequence of evens is as follows. Firs, a he beginning of period, he informaion se f F and he invenory vecor x = (x,1,..., x,m 1 ) (1) are observed, where x,i is he quaniy of on-hand producs whose remaining lifeime is i periods, i = 1,..., m 2, and x,m 1 is he quaniy of on-hand producs whose remaining lifeime is m

9 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems 9 1 periods minus he quaniy of backlogged demands (if any). Thus, x,1,..., x,m 2 are always nonnegaive; while x,m 1 can be posiive or negaive. For simpliciy, we assume ha he invenory sysem is iniially empy a he beginning of period 1, i.e., x 1,i = 0, for all i = 1,..., m 1; bu our analysis and resuls can be exended o he case wih an arbirary iniial sae. Second, an order wih quaniy q is placed, incurring an ordering cos ĉq. Following he discussions in Levi e al. (2007), we assume ha q is a coninuous decision variable, bu i can be exended o he case of ineger values. Denoe y as he oal invenory level afer receiving he order in period. Then, y = m 1 i=1 x,i + q. Third, he demand in period is realized and saisfied as much as possible by he on-hand invenory using he FIFO issuing policy, i.e., he oldes invenory is consumed firs when demand arrives. A he end of period, if y D 0, hen he excess invenory incurs a holding cos ĥ(y D ). Following Nahmias (1975), we assume ha all excess invenory (including he invenory which perishes a he end of his period) incurs a holding cos. On he oher hand, if y D < 0, hen he sysem incurs a backlogging cos ˆb(D y ). Furhermore, if he invenory wih one period remaining life x,1 > D, hen e := (x,1 D ) + unis perish and incur an oudaing cos ˆθe. Finally, he sysem proceeds o he subsequen period + 1. By he definiion of he invenory vecor x and he FIFO issuing policy, we obain he following sae ransiion from x o x +1 : x +1,j = x,j+1 ( D ) + j x,i i=1 +, for 1 j m 2, ( ) m 1 + x +1,m 1 = q D x,i. (2) i=1 We remark ha in defining he invenory sae x in (1), i is convenien and naural o combine he invenory having m 1 periods of remaining life wih he number of backlogs in x,m 1. This is because when demand arrives, by he FIFO issuing policy, i is firs me by x,1, and when x,1 is consumed hen he remaining demand is me by x,2. This process coninues and when (and if) x,m 2 also deplees o 0, he remaining demand will be saisfied by x,m 1. Clearly, when he demand is large, his las number will coninue o go down afer reaching 0, represening he backlog level. We also noe ha invenory only oudaes hrough he firs dimension, x,1, of vecor x, while backlogs always say in he las dimension, x,m 1 (hence backlogs will no disappear afer m periods). Moreover, if in period here are backlogs (hus x,m 1 is negaive and x,j = 0 for j = 1,..., m 2), hen by (2), in he nex period x +1,j will be equal o 0 for all j = 1,..., m 2, bu x +1,m 1 can be posiive or negaive, depending on wheher q is greaer or less han D x,m 1.

10 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems 10 Objecive. For clariy, we ofen disinguish beween a random variable and is realizaion using a capial leer and a lowercase leer, respecively. Then he expeced oal discouned cos incurred under a given policy P ha orders q in period can be wrien as [ T ] ( C (P ) = E α 1 ĉq + ĥ(y D ) + + ˆb(D Y ) + + ˆθe ) m 1 α T ĉ X T +1,i. (3) =1 Noe ha, he quaniies q, Y, e, and X all depend on he policy P ; and whenever necessary, we shall make he dependency explici, i.e., wrie hem as q P, Y P, e P, and X P, respecively. The objecive is o coordinae he sequence of orders o minimize he expeced oal discouned cos. As discussed in Secion 1, i is known ha finding he exac opimal policy using dynamic programming is compuaionally inracable. Thus, our focus in his paper is o design easy-ocompue and near-opimal approximaion algorihms. Approximaion policy assessmen. To measure he effeciveness of an approximaion algorihm P, we define is performance measure by he raio C (P )/C (OP T ), where C (OP T ) is he cos under an opimal policy. Clearly, he value of his raio depends on he problem insance, and is a leas 1. If under algorihm P his raio is always equal o 1 for all problem insances, hen P is an opimal policy. Oherwise, if here exiss some number r (> 1) such ha his raio is less han or equal o r for any problem insance, hen we say ha he algorihm admis a wors-case performance guaranee of r, or simply call i an r-approximaion algorihm. As menioned earlier, we will presen efficien approximaion algorihms for he perishable invenory sysems wih wors-case performance guaranees of 2 and 3, respecively. Cos ransformaion. Nex we carry ou a cos ransformaion o obain an equivalen model wih he uni ordering cos equal o 0. This will enable us o assume, wihou loss of generaliy, ha he uni ordering cos is 0 in he subsequen analysis. The proof of he following proposiion is given in he online Appendix. i=1 Proposiion 1. For every perishable invenory sysem wih cos parameers ĉ, ĥ, ˆb and ˆθ, here is an equivalen sysem wih non-negaive cos parameers c = 0, h = ĥ + (1 α)ĉ, b = ˆb (1 α)ĉ and θ = ˆθ + αĉ. And he expeced oal discouned cos can be rewrien as [ T C (P ) = E α ( ) ] T 1 h(y D ) + + b(d Y ) + + θe + α 1 ĉe [D ]. (4) =1 =1

11 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems Nesed Marginal Cos Accouning Scheme The radiional cos accouning scheme given in (3) decomposes he oal cos by periods. Levi e al. (2007) presened a marginal cos accouning scheme for he classical non-perishable invenory sysems. In his secion, we develop a marginal cos accouning scheme for perishable invenory sysems, similar in spiri o ha in Levi e al. (2007). Our marginal cos accouning scheme exhibis a nesed srucure due o he muli-dimensionaliy of sysem sae. The main idea underlying his approach is o decompose he oal cos in erms of he marginal coss of individual decisions. Tha is, we associae he decision in period wih is affiliaed cos conribuions o he sysem. These marginal coss may include coss (associaed wih he decision) incurred in boh he curren and subsequen periods. Given he invenory vecor x = (x,1,..., x,m 1 ) a he beginning of period, and ha a policy P orders q, we aim o compue he marginal cos conribuions o he sysem by hese q unis on he holding, oudaing, and backlogging coss. To his end, for i = 1,..., m 1, we le B (x, i) denoe he number of oudaed unis in periods [, + i 1] given ha he invenory vecor a he beginning of period is x, wih he convenion ha B (x, 0) 0. Then, for 1 i m 1, we have { i } B (x, i) = max x,j D [,+i 1], B (x, i 1). (5) j=1 To see why his is rue, noe ha i j=1 x,j B (x, i 1) is he number of non-expired unis in x,1,..., x,i ha would mee demands in periods [, + i 1]. These unis, if no consumed, will expire a he end of period + i 1. Thus ( i j=1 x,j B (x, i 1) D [,+i 1] ) +, if posiive, would be he number of unis ha will expire a he end of period + i 1. Adding B (x, i 1) o i gives he oal number of expired unis in [, + i 1], which is (5). The nesed srucure in he auxiliary funcion B (, ) follows from he fac ha some invenory unis reach heir expiraion dae before meeing he demand, and have o be discarded from he on-hand invenory. Using his auxiliary funcion, he number of oudaed unis in period + i 1, for 1 i m 1, is given as ( i ) +, e +i 1 = x,j B (x, i 1) D [,+i 1] j=1 and he number of oudaed unis in period + m 1 is ( m 1 ) +. e +m 1 = q + x,j B (x, m 1) D [,+m 1] j=1

12 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems Nesed Marginal Holding Cos Accouning We firs focus on he marginal holding cos accouning of a given policy P. The holding cos for he q unis ordered in period may be incurred in any period from o + m 1 (afer which he remaining ones will perish), or T, whichever is smaller. Le H P (q ) be he discouned marginal holding cos (o period 1) incurred by hese q unis. Then i follows from he FIFO issuing policy ha (+m 1) T H P (q ) := h i= α (q i 1 ( m 1 D [,i] + B (x, i ) j=1 x,j ) + ) +, (6) where he auxiliary funcion B (x, i) is given recursively via (5). To see why (6) is valid, noe ha he oal number of unis in x ha do no expire unil + i is m 1 j=1 x,j B (x, i), hus he ne demand afer consuming he unis in x is ( D [,+i] ( m 1 x j=1,j B (x, i)) ) +. Hence, he number of unconsumed unis from q a he end of period + i is ( q ( D [,+i] + B (x, i) m 1 x + ) +. j=1,j) Because he marginal holding cos is compued based on he nesed srucure of he auxiliary funcion B (, ), we call i he nesed marginal holding cos accouning. I is imporan o noe ha he marginal holding cos associaed wih he q unis ordered in period is only affeced by he fuure demands bu no by he fuure decisions Nesed Marginal Oudaing Cos Accouning Similarly, we can compue he marginal oudaing cos associaed wih he q unis ordered by policy P in period using he following nesed scheme. For = 1,..., T m + 1, ( m 1 ) +, Θ P (q ) := α +m 2 θ e +m 1 = α +m 2 θ q + x,j B (x, m 1) D [,+m 1] (7) j=1 where B (, ) is defined in (5); and for = T m + 2,..., T, we have Θ P do no expire wihin he planning horizon. 0 since he ordered unis 3.3. Marginal Backlogging Cos Accouning For each period = 1,..., T, he discouned (o period 1) marginal backlogging cos of he q unis ordered in period by policy P can be expressed as ( m 1 ) +, Π P (q ) := α 1 b D x,i q (8) i=1

13 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems 13 which is exacly he same as he radiional backlogging cos using he period-by-period accouning scheme. The inuiion is ha any negaive consequence of under-ordering can be correced by placing an order in he nex period; hus i suffices o only consider he backlogging cos incurred in he curren period Toal Cos of a Given Policy Noe ha he marginal coss defined above, H P (q ), Θ P (q ), and Π P (q ), are random as hey depend on he fuure demands. Since he sysem is iniially empy, he expeced oal sysem cos C (P ) of policy P can be obained by summing (6), (7) and (8) over from 1 o T, and hen aking expecaions. Thus, by (4) we have [ T ( )] T C (P ) = E H P (q ) + Π P (q ) + Θ P (q ) + α 1 ĉe[d ]. (9) =1 =1 If we ignore he consan erms ha are independen of he policy, hen we can wrie he effecive cos of a policy P as [ T ( )] C(P ) = E H P (q ) + Π P (q ) + Θ P (q ). (10) =1 Clearly, o compare differen policies, we only need o compare heir effecive coss. 4. Balancing Policies and Wors-Case Performance Guaranees In his secion, we propose wo efficien cos-balancing algorihms for perishable invenory sysems wih general produc lifeime using he nesed cos accouning scheme defined in he previous secion. The firs one is for arbirary non-saionary and correlaed demand processes; while he second is for independen and sochasically non-decreasing demand processes. These policies will be shown o admi wors-case performance guaranees of 3 and 2, respecively Proporional-Balancing (PB) Policy For each period = 1,..., T, wih an observed informaion se f F, he proporional-balancing (PB) policy orders q P B = q ha balances a proporion of he expeced marginal holding and oudaing coss wih he expeced backlogging cos as follows: mh+θ E[H P B 2(m 1)h+θ (q ) + Θ P B (q ) f ] = E[Π P B (q ) f ]. (11)

14 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems 14 I can be verified ha he lef hand side (LHS) of (11) is an increasing convex funcion of he order quaniy q, which equals 0 when q = 0 and approaches infiniy when q ends o infiniy. On he oher hand, he righ hand side (RHS) of (11) is a decreasing convex funcion of he order quaniy q, which equals a non-negaive number when q = 0 and ends o 0 when q goes o infiniy. Since q can ake any non-negaive real value and boh funcions are coninuous, q in (11) is well defined. Furhermore, since LHS minus RHS of (11) is increasing in q, q P B compued using bisecion mehods. I should be noed ha q P B simpliciy we make his dependency implici. For he special case where he demands in differen periods are independen, q P B can be very efficienly is a funcion of f and x, bu for does no depend on he informaion se f, and i becomes a funcion of only he invenory vecor x a he beginning of period. Several sudies in he lieraure have analyzed he qualiaive properies of he opimal order quaniy q OP T on he saring invenory vecor of period for he case of independen and idenically disribued demands (see, e.g., Fries (1975) and Nahmias (1982)). Suppose he invenory vecor a he beginning of period is x = (x 1,..., x m 1 ). I has been shown ha q OP T decreases a a rae less han one when he produc invenory of any age group increases, and ha i decreases more rapidly in he invenory level of newer produc han ha of older produc. The following resul shows ha, he order quaniy q P B proof is provided in he online Appendix. under he PB policy saisfies hese properies as well. Is Proposiion 2. For each period, he order quaniy q P B under he PB policy saisfies 1 qp B qp B qp B 0. x m 1 x m 2 x 1 The more imporan quesion is how well he PB policy performs. In wha follows, we firs provide a heoreical wors-case performance guaranee; hen in Secion 6, we will provide a comprehensive numerical sudy o demonsrae is empirical performance. Theorem 1. For an arbirary non-saionary and correlaed demand process, he proporionalbalancing policy for he perishable invenory sysem wih m 2 periods of produc lifeime has ( ) a wors-case performance guaranee of 2 + (m 2)h, i.e., for any insance of he problem, he mh+θ ( ) expeced cos of he proporional-balancing policy is a mos 2 + (m 2)h imes he expeced cos mh+θ of an opimal policy. Theorem 1 shows ha, when he produc lifeime m = 2, he PB policy has a wors-case performance guaranee of 2; while for a general lifeime m, he PB policy has a wors-case performance guaranee beween 2 and 3.

15 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems 15 We remark ha he balancing coefficien on he LHS of (11) is chosen so ha he resuling PB policy admis our bes provable wors-case performance guaranee. If we selec a general posiive balancing coefficien β o consruc he PB policy, hen we can prove ha i admis a worscase performance guaranee of (β + 1)/ min{β, β 0 }, where β 0 = mh+θ. Since he wors-case 2(m 1)h+θ performance guaranee is minimized when β = β 0, we consruc he PB policy wih his opimized parameer Dual-Balancing (DB) Policy In his subsecion, we propose anoher approximaion policy, referred o as he dual-balancing policy, which has a wors-case performance of 2 for an arbirary fixed lifeime m when he demands D 1,..., D T are independen and sochasically non-decreasing over ime. The random variables D 1,..., D T are said o be sochasically non-decreasing if for any 1 s T, D is less han D s in he usual sochasic order, or equivalenly, Pr(D > d) Pr(D s > d) for all d. For more deailed discussions on sochasic orders, we refer ineresed readers o Shaked and Shanhikumar (2007). In he remainder of his secion, we assume ha he demands D 1,..., D T sochasically non-decreasing. are independen and To inroduce he dual-balancing policy, we firs define he discouned (o period 1) marginal holding cos for period for an arbirary policy P by Ĥ P (q ) := α 1 h ( m 1 ) + x,i + q D. i=1 Then, we define he discouned marginal oudaing and discouned marginal backlogging coss for a policy P in exacly he same way as hose in (7) and (8). In addiion, for each period, le S be he soluion of y o he equaion he[(y D ) + ] = be[(d y) + ], which depends only on he disribuion of D in period. Since he demands D 1,..., D T follows ha S is non-decreasing in. are sochasically non-decreasing, i The dual-balancing (DB) policy is described as follows: Suppose a he beginning of period he sae is x = (x,1,..., x,m 1 ), he DB policy orders q DB soluion of = q if m 1 i=1 x,i S, where q is he E[Ĥ DB (q ) + Θ DB (q ) x ] = E[Π DB (q ) x ], (12) and q DB = 0 oherwise. Noe ha, since he demands are independen random variables, he informaion se f can be removed, and q DB is only a funcion of he invenory vecor x.

16 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems 16 The q in (12) balances he expeced discouned marginal holding and oudaing coss wih he expeced marginal backlogging cos. I can be verified ha he LHS of (12) is an increasing convex funcion of he order quaniy q, which equals α 1 he[( m 1 i=1 x,i D ) + ] when q = 0 and approaches infiniy when q goes o infiniy. On he oher hand, he RHS of (12) is a decreasing convex funcion of q, which equals α 1 be[(d m 1 i=1 x,i) + ] when q = 0 and approaches 0 when q goes o infiniy. When m 1 i=1 x,i S, he quaniy q in (12) is well defined wih 0 q S m 1 i=1 x,i. When m 1 i=1 x,i > S, Equaion (12) does no have a nonnegaive soluion and in his case he DB policy orders q DB = 0. I is imporan o noe ha since S is non-decreasing in, if for some period we have m 1 i=1 x,i S, hen following he DB policy we have m 1 i=1 x,i S for all regardless of he demand realizaions of D,..., D T. This is because when m 1 i=1 x,i S, we have m 1 m 1 x +1,i = x,i + q D e S D e S S +1. i=1 i=1 This implies ha when he demands are independen and sochasically non-decreasing, he DB policy can perfecly balance he expeced marginal holding and oudaing coss wih he expeced marginal backlogging cos afer placing is firs order. If he demands are no independen or sochasically non-decreasing, hen S will no necessarily be monoonically non-decreasing in and as a resul, he DB policy will no have he above propery. This is he reason why we need o assume ha he demands are independen and sochasically non-decreasing over ime. The following resul shows ha, again, he desired properies exhibied by he opimal conrol policy for he perishable invenory sysem are inheried by he DB policy. Is proof is very similar o ha of Proposiion 2 and hence i is omied. Proposiion 3. For each period, he order quaniy q DB under he DB policy saisfies 1 qdb qdb qdb 0. x m 1 x m 2 x 1 The following heorem shows ha, he DB policy has a wors-case performance guaranee of 2 when he demands are independen and sochasically non-decreasing over ime. Theorem 2. For an arbirary independen and sochasically non-decreasing demand process, he dual-balancing policy for he perishable invenory sysem wih an arbirary fixed produc lifeime has a wors-case performance guaranee of 2, i.e., for any insance of he problem, he expeced cos of he dual-balancing policy is a mos wice he expeced cos of an opimal policy.

17 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems Wors-Case Analysis The argumens used in he lieraure on proving wors-case performance guaranees for approximaion algorihms uilize a uni-maching approach (see, e.g., Levi e al. (2007, 2008a,b, 2012), Levi and Shi (2013)). In a sense, he approach is geomeric, and i relies on he correspondence of unis in he sysems operaing under differen policies hroughou he planning horizon, and hen i compares he coss incurred by he mached unis in differen sysems. However, he uni-maching approach fails o work for perishable invenory sysems because he invenory unis can perish and he number of oudaing unis differs in sysems operaing under differen policies. To overcome his difficuly, we develop an algebraic approach for comparing differen sysems. A key concep in our approach is he rimmed on-hand invenory level, which is defined as he par of on-hand invenory unis ordered before any given paricular ime. These rimmed invenory levels serve as a generalizaion of he radiional invenory level, as hey provide criical (parial) informaion on he ages of he producs on-hand. Due o he naure of perishable sysems, i is impossible o quanify he effec of he decision made in he curren period on fuure coss only hrough he radiional oal invenory level Y. The rimmed invenory levels provide a racable way o analyze his effec, and also provide he righ framework for coupling he marginal holding and oudaing coss in differen sysems. More echnically, he difference beween he rimmed invenory levels of our policy and he opimal policy OP T can be bounded by he difference beween he oudaing unis of he wo policies. An essenial par of his wors-case analysis presened below is based on his new concep. The main ideas and argumens for he proofs of our key resuls are given below. We leave some of he more involved echnical analysis in he online Appendix. For simpliciy, whenever possible we will abbreviae he marginal coss H P (q ), Θ P (q ) and Π P (q ) by H P, Θ P and Π P, respecively, i.e., we make he ordering quaniy q in hese funcions implici. In he following, we firs sudy he PB policy and is wors-case performance, and hen sudy he DB policy Analysis of PB Policy We now compare he PB policy wih he opimal policy OPT. To his end, we make he dependency of he relevan quaniies on he policy, PB or OPT, explici. For each realizaion of demands D 1,..., D T and he exogenous informaion W 1,..., W T, we compare and analyze he invenory processes of he sysems operaing under hese wo policies.

18 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems 18 Given a realizaion f T F T, le T H be he se of periods in which he opimal policy has more oal invenory level han he PB policy does. In oher words, we denoe T H = { [1, T ] : Y OP T In addiion, we le is complemen se be denoed by } Y P B. T Π = { [1, T ] : Y OP T Lemma 1. For each realizaion f T F T, we have } < Y P B. H P B T H T =1 H OP T + (m 2) h θ T =1 Θ OP T. Lemma 1 is one of he key echnical resuls of his paper. Given is complee proof is complicaed and lenghy, we leave i in he online Appendix for ineresed readers. To illusrae he main ideas of our argumen, we provide below he proof for he following, weaker, resul: H P B T H T =1 H OP T + (m 1) h θ T =1 Θ OP T. (13) For ease of illusraion, we also assume ha α = 1. As said earlier, an imporan concep in proving our main echnical resul is wha we refer o as he rimmed on-hand invenory level, denoed by Y,s for any s 1, which is defined as he par of on-hand invenory a he beginning of period s which is ordered in period or earlier. From he definiion of Y,s, i holds ha Y,s = 0 when s + m, and Y,s = ( Y D [,s) e [,s) ) +, s =, + 1,..., + m 1. (14) For any period = 1,..., T, we define he noaion R() as follows: if he se {s T H : s > } is nonempy, hen R() := min{s T H : s > }; oherwise, R() := T + 1. In addiion, for any s 1, denoe H s as he par of he holding cos incurred in period s associaed wih he producs ordered in periods { : T H, s}. Since he lifeime of he producs is m, all producs ordered in period or earlier will leave he sysem by he end of period +m 1. Then, i follows ha for any T H, H s = 0 when + m s R() 1. Consequenly, by he definiions of H, Hs, and R(), we have T H H = T H R() 1 s= H s = T H (+m) R() 1 s= H s.

19 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems 19 For each period s [, R() 1], from is definiion, Hs is clearly no greaer han he par of he holding cos incurred in period s associaed wih he producs ordered in periods 1,...,, which can be expressed as h (Y,s D s ) +. Since T =1 H = h T s=1 (Y s,s D s ) + and Y,s Y s,s for any s, he following inequaliies hold for any policy: In paricular, we have H h T H T H (+m) R() 1 s= (Y,s D s ) + T H. (15) =1 H P B T H T =1 H OP T h T H h T H (+m) R() 1 s= (+m) R() 1 s= (Y P B,s D s ) + ; (16) (Y OP T,s D s ) +. (17) Subracing (17) from (16), we obain H P B T H T =1 H OP T h T H h T H h T H (+m) R() 1 s= (+m) R() 1 s= (+m) R() 1 s= ( (Y P B,s ( Y P B ( e OP T [,s) D s ) + (Y OP T,s D s ) +) Y OP T ) e P B [,s) + e OP T + [,s) e P B [,s)) +, (18) where he second inequaliy follows from (14) and a + b + (a b) + for any real numbers a and b, and he las one holds because Y P B Thus, i follows from (e OP T H P B T H T =1 H OP T e P B ) + e OP T h Y OP T when T H. T H and e OP T [,) (+m) R() 1 s=+1 (m 1)h = (m 1)h where he las inequaliy follows from e OP T Θ OP T T H T =m R() 2 l= e OP T e OP T [,s) = 0 ha h T H e OP T l (m 1)h = (m 1) h θ (+m) R() 1 T =1 s=+1 T =1 Θ OP T, e OP T e OP T [,R() 1) = 0 for m 1 as he sysem is iniially empy, = θe OP +m 1 T for 1 T m+1, and Θ OP T = 0 when T m+1 < T. This proves a weaker form of Lemma 2, i.e., he resul (13).

20 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems 20 Lemma 2. For each realizaion f T F T, we have T H Θ P B Proof. T =1 ΘOP T. For breviy, we only prove below he resul for he special case when α = 1. The complee proof for a general α [0, 1] is provided in he online Appendix. Since any producs ordered afer period T m + 1 do no perish wihin he planning horizon, we only need o consider periods = 1,..., T m + 1. For each realizaion f T and he resuling T H, we pariion he periods {1,..., T m + 1} as follows: Firs, sar in period T m + 1 and search backward for he laes period T H such ha Θ P B > Θ OP T. If no such period exiss, hen we erminae he pariion process. Oherwise, le be ha period and mark he periods, 1,..., ( m) Nex, repea he above procedure over periods 1,..., ( m) + unil he remaining se of periods is empy. As a resul, his procedure pariions he periods {1,..., T m + 1} ino marked and unmarked periods. Le T M denoe he se of marked periods. Θ P B We firs consider any period T H \ T M. Then, i follows from he definiion of T M Θ OP T. Consequenly Since he se T M T H \T M Θ P B T H \T M Θ OP T ha. (19) is made up of disjoin inervals, we consider a represenaive inerval wih is larges period being, i.e., his inerval consiss of periods ( m) + + 1,..., 1,. Then, by he consrucion of T M, T H and Θ P B > Θ OP T. Since Θ = θe +m 1, we have e P +m 1 B > e OP +m 1 T 0. Noe ha e is he number of perished producs in period and i saisfies he following ideniy for any feasible policy: e +m 1 = ( Y D [,+m 1] e [,+m 1) ) +. (20) Thus i follows from (20) and e P B +m 1 > 0 ha e P B [,+m 1] = ( ) Y P B D [,+m 1] e P B + [,+m 1) + e P B [,+m 1) On he oher hand, for he OPT policy we have = Y P B D [,+m 1]. (21) Subracing (22) from (21) yields e OP T [,+m 1] = ( ) Y OP T D [,+m 1] e OP T + [,+m 1) + e OP T [,+m 1) Y OP T D [,+m 1]. (22) e P B [ m,+m 1] e OP T [ m,+m 1] = e P B [,+m 1] e OP T [,+m 1] Y P B Y OP T 0,

21 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems 21 where he equaliy holds since e P B = e OP T T H. This proves, by Θ s = θe s+m 1 for any period s, ha for all, for 1 m 1 and he las inequaliy follows from Θ P B [( m+1) 1,] Θ OP T [( m+1),]. As he above resul holds for any of he disjoin inervals of T M, adding hem up yields Θ P B T M T M Θ OP T. (23) Finally, since T H (T H \ T M ) T M {1, 2,..., T }, we obain, using (19) and (23), ha T H Θ P B Θ P B T H \T M + T M Θ P B Θ OP T T H \T M + Θ OP T T M T =1 Θ OP T. This complees he proof of Lemma 2 when α = 1. Q.E.D. Noe ha for each perished uni ordered in periods 1,..., T, i mus say in he sysem for exacly m periods. Thus, for any policy, we have he following inequaliy mh T Θ θ =1 T H. (24) Combining his inequaliy wih Lemmas 1 and 2 leads o he following resul. Corollary 1. For each realizaion f T F T, we have ( H P B T H + Θ P B ) (1 + (m 2)h mh+θ =1 ) T =1 ( H OP T ) + Θ OP T. (25) Proof. We apply Lemmas 1 and 2, and (24) o obain ( H P B T H + Θ P B ) T = =1 T =1 ( 1 + H OP T (m 2)h + θ ( H OP T (m 2)h mh + θ T =1 Θ OP T + ) + Θ OP T (m 2)h + mh + θ ) T =1 ( H OP T T =1 Θ OP T ( 1 + mh θ ) + Θ OP T, ) T =1 Θ OP T hereby proving he corollary. Q.E.D.

22 Chao e al.: Approximaion Algorihms for Perishable Invenory Sysems 22 Lemma 3. For each realizaion f T F T, we have T Π Π P B T =1 ΠOP T. Proof. From he definiion of Π and T Π, we have Π P B T Π = b α 1 (D Y P B ) + b α 1 (D Y OP T ) + T Π T Π where he firs inequaliy holds since Y OP T T =1 Π OP T, < Y P B when T Π. Q.E.D. Wih he preparaions above, we are now ready o prove our firs main resul, i.e., Theorem 1. Proof of Theorem 1. For each period = 1,..., T, denoe Z P B as he condiional expeced balanced cos by he PB policy in period. Tha is, Noe ha Z P B Z P B = mh+θ E[H P B 2(m 1)h+θ + Θ P B F ] = E[Π P B F ]. is a random variable before period ; and in period, F = f is realized and is value is he expeced balanced cos condiional on he observed informaion se f. Using he marginal cos accouning scheme and a sandard argumen of condiional expecaions, we have C(P B) = T =1 E[H P B ( = 2 + (m 2)h mh+θ + Θ P B ) T =1 + Π P B ] = T =1 E[E[H P B + Θ P B + Π P B F ]] E[Z P B ]. (26) Applying Corollary 1, Lemma 3, and he fac ha { T H } and { T Π } are compleely deermined by F, we obain [ T C(OP T ) = E =1 (H OP T + Θ OP T ) + E (m 2)h mh+θ [ T 1 = E =1 1 + (m 2)h mh+θ [ T [ 1 = E E =1 1 + (m 2)h mh+θ = = [ T E =1 T = (m 2)h mh+θ (H P B T H T =1 ] Π OP T + Θ P B ) + Π P B T Π 1( T H )(H P B 1( T H )(H P B 1( T H )E [ H P B E [ (1( T H ) + 1( T Π ))Z P B ] = + Θ P B ) + 1( T Π )Π P B ] ] + Θ P B ) + 1( T Π )Π P B F + Θ P B T =1 ] F + 1( TΠ )E [ ] ] Π P B F E[Z P B ]. ]