Stochastic Perishable Inventory Systems: Dual-Balancing and Look-Ahead Approaches

Sochasic Perishable Invenory Sysems: Dual-Balancing and Look-Ahead Approaches by Yuhe Diao A hesis presened o he Universiy Of Waerloo in fulfilmen of he hesis requiremen for he degree of Maser of Applied Science in Managemen Sciences Waerloo, Onario, Canada, 2016 Yuhe Diao 2016

Auhor's Declaraion I hereby declare ha I am he sole auhor of his hesis. This is a rue copy of he hesis, including any required final revisions, as acceped by my examiners. I undersand ha my hesis may be made elecronically available o he public. ii

Absrac We sudy a single-iem, muli-period, sochasic perishable invenory problem under boh backlogging and los-sales circumsances, wih and wihou an order capaciy consrain in each period. We firs model he problem as a dynamic program and hen develop wo heurisics namely, Dual-Balancing (DB) and Look-Ahead (LA) policies, o approximae he opimal invenory level a he beginning of each period. To characerize he holding and backlog cos funcions under he proposed polices, we inroduce a runcaed marginal holding cos for he marginal cos accouning scheme. Our numerical examples demonsrae ha boh DB and LA policies have a possible wors-case performance guaranee of wo in perishable invenory sysems under differen assumpions, and he LA policy significanly ouperforms he DB policy in mos siuaions. We also analyze he arge invenory level in each period (he invenory level a he beginning of each period) under differen policies. We observe ha he arge invenory level under he LA policy is no larger han he opimal one in each period in sysems wihou an order capaciy consrain. iii

Acknowledgemens Firs and foremos, I would like o express my sincere graiude o my supervisor Dr. Hossein Abouee Mehrizi for his consan encouragemen, professional insrucions and paience. Wihou his help, he compleion of his hesis would no have been possible. I feel very lucky o have been a suden of such an insighful professor. I also hank Dr. Bissan Ghaddar and Dr. Fama Gzara for heir ime and paience in reviewing my hesis. Secondly, I wan o hank all my friends a Waerloo for he grea ime. Yangzi and Haoran have also helped me a lo. Finally, my hanks would go o my beloved Mum, Dad and Xiaohan, who have always been helping and supporing me ou of difficulies. iv

Table of Conens Lis of Figures... vii Lis of Tables... viii 1 Inroducion... 1 2 Lieraure Review... 5 3 Models and Mehodologies... 8 3.1 Perishable Invenory Sysems... 8 3.1.1 Sysem Evoluion in Each Period... 10 3.1.2 Objecive... 16 3.1.3 Cos Transformaion... 17 3.2 Dynamic Programming Approach... 18 3.3 Heurisic Policies... 19 3.3.1 Marginal Cos Accouning... 19 3.3.2 Dual-Balancing and Look-Ahead Approaches... 31 3.4 Algorihms... 33 3.4.1 Dynamic Programming Algorihm... 33 3.4.2 Dual-Balancing Algorihm... 34 3.4.3 Look-Ahead Algorihm... 35 4 Numerical Sudy... 36 4.1 Sysems wih Backlogs and No Ordering Capaciy Consrain... 37 4.1.1 Performance of DB and LA Policies... 37 4.1.2 Opimal and Heurisic Policies... 41 v

4.2 Sysems wih Los-Sales and No Ordering Capaciy Consrain... 46 4.2.1 Performance of DB and LA Policies... 46 4.2.2 Opimal and Heurisic Policies... 48 4.3 Sysems wih Backlogs and an Ordering Capaciy Consrain... 51 4.3.1 Performance of DB and LA Policies... 51 4.3.2 Opimal and heurisic Policies... 52 4.4 Sysems wih Los-Sales and an Ordering Capaciy Consrain... 55 4.4.1 Performance of DB and LA Policies... 55 4.4.2 Opimal and Heurisic Policies... 56 5 Conclusion... 58 References... 59 vi

Lis of Figures Figure 1. ρ DB for Uniform Demand... 40 Figure 2. ρ DB for Binomial Demand... 40 Figure 3. ρ DB for Uniform Demand... 41 Figure 4. ρ DB for Binomial Demand... 41 Figure 5. Cos for h=10, o=5... 45 Figure 6. Cos for h=0.1, o=1... 45 Figure 7. ρ DB for Uniform Demand... 47 Figure 8. ρ DB for Binomial Demand... 47 Figure 9. ρ DB for Uniform Demand... 48 Figure 10. ρ DB for Binomial Demand... 48 Figure 11. Cos for h=10, o=5 wih an Order Capaciy Consrain... 55 vii

Lis of Tables Table 1. Per-Uni Cos Combinaions... 37 Table 2. ρ LA and ρ DB wih Backlogs and No Capaciy Consrain (%, T=6)... 38 Table 3. ρ LA and ρ DB wih Backlogs and No Capaciy Consrain (%, T=8)... 39 Table 4. Average Opimal Targe Invenory Level (T=6)... 42 Table 5. Average Opimal Targe Invenory Level (T=8)... 43 Table 6. Targe Invenory Level under LA and Opimal Policies (T=6)... 43 Table 7. Targe Invenory Level under LA and Opimal Policies (T=8)... 44 Table 8. ρ LA and ρ DB wih Los-Sales and No Capaciy Consrain (%, T=6)... 46 Table 9. ρ LA and ρ DB wih Los-Sales and No Capaciy Consrain (%, T=8)... 47 Table 10. Average Opimal Targe Invenory Level (T=6)... 49 Table 11. Average Opimal Targe Invenory Level (T=8)... 49 Table 12. Targe Invenory Level under LA and Opimal Policies (T=6)... 50 Table 13. Targe Invenory Level under LA and Opimal Policies (T=8)... 50 Table 14. ρ LA and ρ DB wih Backlogs and a Capaciy Consrain (%)... 51 Table 15. Targe Invenory Level under he Opimal Policy... 52 Table 16. Comparison of Opimal Expeced Toal Cos... 53 Table 17. Targe Invenory Level under he LA Policy... 53 Table 18. Targe Invenory Level under LA and Opimal Policies... 54 Table 19. ρ LA and ρ DB wih Los-Sales and a Capaciy Consrain (%)... 55 Table 20. Targe Invenory Level under he Opimal Policy... 56 Table 21. Comparison of Opimal Expeced Toal Cos... 56 Table 22. Targe Invenory Level under LA and Opimal Policies... 57 viii

1 Inroducion Perishable producs are very common in he real life. The main difference beween perishable and non-perishable producs is ha he former mus be consumed wihin a relaively shor period of ime before hey become oudaed, while he laer one can be kep in invenory for a long ime, unil hey are used o saisfied demand. For example, food producs, such as fresh frui and dairy produc, usually have a shor qualiy guaranee period. Besides, blood producs are also perishable and need o be used in a shor ime. For example, blood plaeles are very precious and mus be used wihin six-days [1], and blood plasma also deerioraes wih ime [2]. If perishable producs are no consumed in ime, oudaing can lead o a significan financial loss. For example, abou 321,000 unis of apheresis plaeles oudaed in US in 2011, accouning for 12.8% of he oal processed/produced amoun [3]. Anoher repor indicaes ha he average oudaing rae of perishable producs for reailers and disribuors was around 1.21% in 2007 [4]. On he oher hand, approximaely 50% of he causes of oudaing across a value chain can be miigaed effecively hrough improved planning pracices [4]. These facs indicae he imporance of developing efficien invenory managemen policies for perishable producs. In his sudy, we sudy a sochasic periodic-review invenory sysem over a finie planning horizon wih a perishable produc, which has a fixed lifeime. Inuiively, he lifeime ha we consider needs o be no longer han he lengh of he planning horizon. Also, if he lifeime is 1 period, here will be no excess invenory lef a he end of each period, no maer wheher all demand has been saisfied or no, and we can solve he problem for one period a a ime, separaely. Therefore, in our sudy, we assume ha he lifeime of he producs is longer han 1 period. We assume ha producs in he invenory are used based on he firs-ordered firsconsumed (FOFC) policy o simplify he modeling of he problem. This is a common 1

assumpion in he vas majoriy of he lieraure ha producs in he invenory are used based on he FOFC policy [5], which is quie reasonable in pracice. For example, grocery managers usually pu he oldes producs in he mos convenien place on shelves for cusomers, in order o reduce he oudaing rae. The FOFC policy is also he foundaion of many algorihms and policies for perishable invenory problems. For example, i is a criical assumpion for he marginal cos accouning scheme ha maches he available producs and demand in perishable sysems [6], [7]. In general, here are wo differen ways o rea unme demand a he end of each period, eiher backlog hem (sysems wih backlogs) or lose hem (sysems wih los-sales). In backlogging problems, unme demand is backlogged and should be saisfied in he nex period while in los-sales problems, unme demand is los. This difference impacs he period sae ransiion and he amoun of demand a he beginning of each period, which is zero in los-sales problems and he number of backlogs in backlogging problems. Order capaciy is anoher feaure in he invenory problems, which can limi he order quaniy in each period and, hus, affec he ordering policy and he oal cos of he sysem. In his research, we consider boh backlogging and los-sales problems, wih and wihou an order capaciy consrain in each period. The opimal invenory policy for he muli-period invenory sysems wih a perishable produc can be characerized using a dynamic program. However, he dimension of he dynamic program depends on he lengh of he lifeime, and he opimal order quaniy depends on boh he age disribuion of he on-hand invenory and he ime of he curren period. Therefore, he dynamic problem usually has a very large number of sae variables, acion spaces and oucome spaces [1], and he compuaion of he program is usually inracable due o he curse-ofdimensionaliy. Thus, many effors were pu ino developing and examining effecive and efficien heurisics o approximae he opimal policy. For example, he Look-Ahead (LA) 2

policy has been proposed for sysems wihou perishable producs [8] and he Dual-Balancing (DB) policy has been applied o boh problems wih and wihou perishable producs [6], [7]. In his sudy, we firs propose algorihms based on boh DB and LA policies for he invenory conrol of he perishable producs wih limied order capaciy. Then, we invesigae he performance of he proposed algorihms using numerical examples. The main conribuions of his hesis are summarized as follows. Algorihms. We use he marginal cos accouning scheme and nesed srucure o exend DB and LA policies for perishable invenory problems wih differen modeling feaures. When here is no order capaciy consrain, he marginal cos accouning was presened in he lieraure [7], [9], and a nesed srucure was used o model problems wih perishable producs [7]. We exend he marginal cos accouning scheme and nesed srucure o perishable problems wih an order capaciy consrain in each period. Wors-case Performance Based on Numerical Examples. For a perishable invenory sysem wihou an order capaciy consrain, he DB policy was proved o have a wors-case performance guaranee of wo in he lieraure [10]. In his sudy, based on our numerical examples, we observe ha he expeced oal coss under he LA policy in perishable sysems wihou an order capaciy consrain, as well as boh DB and LA policies in sysems wih an order capaciy consrain, are less han wice he opimal expeced oal cos. Performance of LA and DB Policies. We compare performance of boh DB and LA policies in perishable invenory sysems under differen assumpions. We observe ha in perishable invenory sysems wihou an order capaciy consrain, he LA policy has much beer performance, on average, han he DB policy, under boh backlogging and los-sales assumpions. In problems wih an order capaciy consrain, while he LA policy ouperforms he DB policy under he backlogging assumpion, i does no always ouperform he DB policy under he los-sales assumpion. 3

Truncaed marginal holding cos. We inroduce a new runcaed marginal holding cos for he marginal cos accouning scheme. We show ha wih he runcaed marginal holding cos, we do no need o consider he impac of one decision on he holding cos in all following periods. We also examine he effeciveness of his runcaed marginal holding cos. The res of his hesis is organized as follows. In Chaper 2, we review he lieraure relaed o he sochasic periodic-review perishable invenory sysems and he DB and LA policies. In Chaper 3, we model a perishable invenory problem wih boh backlogs and los-sales, wih and wihou an order capaciy consrain. In Chaper 4, we provide some insighs ino he perishable invenory sysems. In Chaper 5, we conclude our resuls and presen some possible direcions for he fuure sudy. 4

2 Lieraure Review Sochasic periodic-review perishable invenory sysems have araced many researchers since 1960 s. A early sages, he focus of he sudies was on developing opimal policies using dynamic programs. The fundamenal characerizaion of he opimal ordering policy wih a woperiod lifeime is provided by Nahmias and Pierskalla [11]. Then, Nahmias [12] and Fries [13] independenly sudy he opimal policy for he general lifeime problem, focusing on backlogging and los-sales problems, respecively. Nahmias [12] uses a per-uni oudaing cos, ordering cos, holding cos, and per-uni per-period shorage cos, which is a sandard cos srucure [14]. They show ha, due o perishabiliy, he srucure of he opimal policy is quie complex. Also, he compuaion of he opimal policy using a dynamic problem is racable only if he lifeime of he produc is shor. This complexiy of he opimal policy is reinforced by Cohen [15], who derives an explici closed-form soluion for a wo-period problem, and discusses procedures o obain he soluion for he m-period case. Many researchers pu effor ino developing heurisic policies for boh backlogging and los-sales models, which would be easier o define and implemen while remaining close o he opimal policy [14]. Brodheim e al. [16] evaluae a class of invenory policies and obain some measures for he invenory sysem, such as he probabiliy of shorage, as well as he easily compuable bounds for he policies. Nahmias [17] proposes hree heurisics, including he criical number and linear approximaions, and compares heir performance using a simulaion model. Nahmias [18] consrucs a bound on he oudaing cos which is only a funcion of he oal on-hand invenory, and develops a myopic policy by subsiuing he oudaing cos wih he bound. Then, Nahmias [19] compares he wo models developed by Nahmias [12] and Fries [13] wih he myopic policy, and shows ha when he remaining periods in he horizon is more han 5

he lifeime, he opimal policy in Fries [13] is he same as a discouned version of he one characerized in Nahmias [12]. Nahmias [20] uses a bounded expeced oudaing funcion and a refined ransfer funcion o approximae he problem, focusing on a wo-period problem, which yields o a one-dimensional sae variable. For problems wih los-sales, Nandakumar and Moron [21] derive upper and lower bounds for he order quaniy in each period, and examine Nahmias approximaions wih a weighed average of he upper and lower bounds. The resuls show ha all heurisics have good performance. Cooper [22] considers a fixedcriical number ordering policy and derives new families of bounds on he expeced number of oudaes per period. The performance of he criical-number policies is close o he opimal policy based on his numerical sudy. Recenly, Chen e al. [23] propose wo heurisic policies for join invenory and pricing conrol problems wih boh coninuous and discree demand, and boh backlogging and los-sales cases. They indicae some monooniciy properies of he opimal policy and idenify bounds for he opimal order-up-o invenory level. Li e al. [24] analyze he opimal soluion-srucure of a wo-period lifeime problem and develop a basesock/lis-price heurisic policy. In addiion o he above lieraure focusing on periodic-review perishable invenory sysems, many oher aspecs of perishable invenory problems were considered as well. We refer readers o Kempf e al. [14] for a comprehensive lieraure review. Recenly, Levi e al. [6] uilize a marginal cos accouning scheme and cos balancing idea o develop an approximae policy for sochasic periodic-review invenory problems, and propose a Dual-Balancing as well as a randomized Dual-Balancing algorihms. They show ha boh he Dual-Balancing and randomized Dual-Balancing policies have a wors-case performance guaranee of wo. Then, Levi e al. [25] apply he marginal cos accouning scheme and cos balancing idea o a sochasic invenory problem wih an order capaciy consrain. For los-sales problems, Levi e al. [10] prove ha boh he Dual-Balancing policy and randomized 6

Dual-Balancing policy have a wors-case performance guaranee of wo. They also consider a capaciaed model using forced los-sales cos and prove ha he Dual-Balancing policy has a wors-case performance guaranee of wo, even when here is a capaciy consrain on ordering in each period. The marginal cos accouning scheme and Dual-Balancing policy have been also used in perishable invenory sysems. Chao e al. [7] develop he firs approximaion algorihm for a periodic-review perishable invenory sysem wih a perishable produc for boh backlogging and los-sales models. They propose a proporional-balancing policy and a dualbalancing policy wih discouned marginal holding cos, and indicae ha boh policies have a compuaional complexiy of O(mT), which is very efficien compared o he opimal policy. Zhang e al. [9] develop a marginal-cos Dual-Balancing policy for a periodic-review, fixedlifeime perishable invenory conrol problem, and prove ha i has a wors-case performance guaranee of wo. This resul works for boh backlogging and los-sales problems due o a zero lead ime. Then, hey prove ha a myopic policy under he marginal cos accouning scheme provides a lower bound on he opimal ordering quaniy, based on a given on-hand invenory. Anoher approach relaed o our sudy is he Look-Ahead opimizaion approach. Truong [8] sudies a single-iem, muli-period sochasic invenory problem and shows ha he Look- Ahead policy has a wors-case performance guaranee of wo. Also, he Look-Ahead policy, on average, performs well wihin 3.9% of he opimal policy, compared o 19.7% for he Dual- Balancing policy, based on he numerical sudy. 7

3 Models and Mehodologies 3.1 Perishable Invenory Sysems We consider a periodic-review sochasic perishable invenory problem, in a finie planning horizon wih T periods, where each period is denoed by, =1,, T. We disinguish a random variable and is realizaion by using capial leers and lower case leers, respecively. The demand in period is denoed by D, =1,, T, which is a random variable. A he beginning of each period, an informaion se f can be observed, conaining he informaion we can have a ha ime. The informaion se has wo pars: realized demands d 1,, d -1, and all oher informaion w 1,, w relaed o he problem in each period. Thus, he informaion se f is one specific realizaion of he random vecor F =(D 1,, D -1 ; W 1,, W ). Furhermore, when f is observed, he fuure demands will have a known condiional join disribuion, including he upcoming curren demand D s, which means E[ D f ] is welldefined and finie for each period s, s=,, T. When we consider differen policies, he informaion se f and he condiional join disribuion is independen of a specific policy [6]. Le lead ime be 0, so he unis ordered in one period can arrive and become available o mee demand in he same period before demand occurs. We apply he firs-ordered firsconsumed (FOFC) policy o calculae he on-hand invenory a he end of each period. This is a common assumpion in perishable problems, because unis in perishable problem have a lifeime, and hose having a shorer remaining lifeime are usually used o saisfy cusomer demand prior o hose arriving laer. The lifeime of he produc is denoed by m, m=2,, T. This means ha afer one uni eners he sysem, i can say for a mos m periods if i has no been used o mee cusomer demand by he end of he m-h periods. For example, suppose here is no on-hand invenory a he beginning of period, and q unis are ordered. These q unis 8

arrive in he curren period and are used o mee he demand d. Le m equal o 3 and assume ha he invenory level a he beginning of period is zero. Then, if q >d +d +1 +d +2, here will be sill some excess unis of q a he end of period +2. Bu, hey canno be reained in he sysem because hey have been saying for 3 periods,, +1 and +2. As a resul, he remaining q -(d +d +1 +d +2 ) unis will perish and leave he sysem a he end of period +2. On he oher hand, if q <d +d +1 +d +2, all q unis have been used before hey reach heir lifeime and perish. Some new unis may arrive in period +1 and +2, bu hey need o be kep in he sysem before all q unis arriving in period have been consumed, due o he FOFC policy. A he beginning of period, he on-hand invenory consiss of m-1 ypes of unis, whose remaining lifeime is from m-1 periods o 1 period. Le x,i represen he number of unis having a remaining lifeime of i periods, i=1,, m-1, in he invenory a he beginning of period. Then, he oal invenory a he beginning of period is X m1, i i1 x. Le non-negaive ineger q denoe he number of unis ordered in period. Because lead ime is 0, he q unis will arrive and become available in curren period before demand occurs, wih a lifeime of m periods. So in period, he oal invenory used o mee demand is Y =X +q, which is he basesock level in period. As we discussed in Secion 1, wo general feaures of invenory problems in he lieraure are wheher excess demand is backlogged or los, and wheher here is a limi on he ordering level in each period. In problems wihou an ordering capaciy consrain, he number of ordered unis can be any non-negaive ineger, bu in problems wih an ordering capaciy consrain u, he order quaniy canno surpass he capaciy limi. In a periodic-review sochasic perishable invenory conrol problem, several ypes of cos are incurred. A per-uni ordering cos c ˆ occurs when one uni is ordered in period. A per- ˆ uni holding cos is denoed by h, which occurs a he end of period for each excess uni ha 9

is sill in invenory, represening a cos of keeping his uni from period o +1. These excess unis can be held in invenory for a long ime if he demands in he following periods are no high enough o consume all available unis. Also, because one uni can say in he sysem for a mos m periods, he holding cos incurred by one order occurs in a mos m-1 following periods. If he demand in period canno be saisfied immediaely, here will be a penaly, which is ˆ b per uni of unsaisfied demand. This cos has differen meanings in he backlogging and lossales problems. In a backlogging problem, he unsaisfied demand is backlogged in he sysem and can be eliminaed by ordering more in he following period. Therefore, his penaly means he compensaion o he unsaisfied demand and will exis in only one period, if here is no ordering capaciy consrain. Sysems wih an ordering capaciy consrain in each period will be discussed in more deail laer. In a los-sales problem, he unfulfilled demand will leave he sysem a he end of he period. Thus, he penaly is he loss of value caused by he unsaisfied demand and will occur once. For simpliciy, he per-uni penaly in boh ypes of problems are denoed by ˆ b. The fourh cos in a perishable problem is a per-uni oudaing cos o ˆ. I occurs when a uni has been in he sysem for m periods and perishes, before been used o mee demand. 3.1.1 Sysem Evoluion in Each Period 4.1.2.1 Sysems wih Backlogs and No Ordering Capaciy Consrain In his secion, we discuss a complee process of a perishable invenory sysem wih backlogs, ineger demand and order quaniy, and no ordering capaciy consrain. 1. A he beginning of period, =1,, T, informaion se f and on-hand invenory x,i, i=1,, m-1, are observed, where i, x i 1,, m 1 0, 1, m1, i x 0, x 0, i1,, m 2 {2,, T}, means backlogs in period 1 i, {2,, }, means x 0, i1,, m1 T no backlogs in period 1 Therefore, he oal invenory level is X 10 m1, i i1 x. The condiional join disribuion of.

upcoming demand D is known based on he informaion se f. 2. The sysem orders q unis, and hey arrive and become available, having a lifeime of m periods. In sysems wihou an ordering capaciy consrain, he feasible range of q is {0,, }. The oal cos of ordering is cq ˆ. Also, he on-hand invenory level increases from X o Y =X +q. 3. Demand d occurs, and he on-hand invenory is used o mee his demand. The x,1 unis having only 1-period lifeime, are consumed firs. If x,1 >d, he remaining x,1 -d unis will perish and an oudaing cos of,1 oˆ ( x d ) occurs. Oherwise, he x,2 unis having 2- period lifeime are used o mee he excess d -x,1 unis of demand, and so on, unil all invenory is consumed. If d >Y, he demand canno be saisfied compleely by he curren invenory. The remaining d -Y unis of demand becomes backlogs ha need o be me in period +1, wih a backlogging penaly of bˆ ( d Y ). If d <Y, he remaining Y -d unis resul in a holding cos of hˆ ( Y d ). 4. A he end of period, he remaining lifeime of he excess unis of he on-hand invenory 1 decreases by 1 period. Noe ha if, i i, j x d x j1, where x max{ x,0}, all x,i 1 unis will be used o mee demand. Oherwise,, i i, j x d x j1 unis remain for period +1, becoming x +1,i-1, i=2,, m-1. In period, he q unis have a lifeime of m periods, and hey will mee he excess demand afer all X have been used. Therefore, in a m1, j backlogging problem, if q d x j1 m1, j backlogs is q d x j1, backlogs occur and he number of. These backlogs will reain in he sysem for period +1 as a negaive invenory. So he new invenory level a he beginning of period +1 can be calculaed using (similar o Eq. (1) in [7]) 11

i 1, i, i1, j x x d x, i {1,, m 2}, j1 x q d x j1 m1 1, m1, j. (1) Then, d will be added ino f +1, which will be he given informaion for period +1, as well as x +1,i, i=1,, m-1. 4.1.2.2 Sysems wih Los-Sales and No Ordering Capaciy Consrain The process of a los-sales problem wihou an order capaciy consrain can be considered as follows. 1. A he beginning of period, =1,, T, informaion se f and on-hand invenory x,i are observed, where i, x i1,, m1 x i, 0, 1. 0, i 1,, m 1 {2,, T} The oal invenory level is X m1, i i1 x. The condiional join disribuion of he upcoming demand D is known based on he informaion se f. 2. The sysem orders q unis, and hey arrive and become available, having a lifeime of m periods. In sysems wihou an order capaciy consrain, he feasible range of q is {0,, }. The oal cos of ordering is cq ˆ. Also, he on-hand invenory level increases from X o Y =X +q. 3. Demand d occurs, and he on-hand invenory is used o mee his demand. The x,1 unis having only 1-period lifeime are consumed firs. If x,1 >d, he remaining x,1 -d unis will perish and an oudaing cos of,1 oˆ ( x d ) occurs. Oherwise, he x,2 unis having 2- period lifeime are used o mee he excess d -x,1 unis of demand, and so on, unil all invenory is consumed. If d >Y, he demand canno be saisfied compleely by he curren invenory. The remaining d -Y unis of demand will be los and leave he sysem, wih a 12

los-sales penaly of bˆ ( d Y ). If d <Y, he remaining Y -d unis resul in a holding cos of hˆ ( Y d ). 4. A he end of period, he remaining lifeime of he excess unis of he on-hand invenory 1 decreases by 1 period. Noe ha if, i i, j x d x j1 1 demand. Oherwise,, i i, j x d x j1, all x,i unis will be used o mee unis remain for period +1, becoming x +1,i-1, i=2,, m-1. In period, he q unis have a lifeime of m periods, and hey will mee he excess demand afer all X have been used. Therefore, in a los-sales problem, if 1, m j1, he excess demand will leave he sysem. If 1 m, j q d x j1 q d x j m1, j he q d x j1, remaining unis will become x +1,m-1. So he new invenory level a he beginning of period +1 can be calculaed using (similar o Eq. (18) in [7]) i 1, i, i1, j x x d x, i {1,, m 2}, j1 x q d x j1 m1 1, m1, j. (2) 4.1.2.3 Sysems wih Backlogs and an Ordering Capaciy Consrain In a backlogging problem wih an order capaciy consrain, he order quaniy is limied and he complee process in period can be lised as follows. 1. A he beginning of period, =1,, T, informaion se f and on-hand invenory x,i, i=1,, m-1, are observed, where i, x i 1,, m 1 0, 1, m1, i x 0, x 0, i1,, m 2 {2,, T}, means backlogs in period 1 i, {2,, }, means x 0, i1,, m1 T no backlogs in period 1 The oal invenory level is X m1, i i1 x and he condiional join disribuion of. upcoming demand D is known based on he informaion se f. 2. The sysem orders q unis, and hey arrive and become available, having a lifeime of m 13

periods. In a sysem wih an order capaciy consrain u, he feasible range of q is {0,, u }. The oal cos of ordering is cq ˆ. Also, he on-hand invenory level increases from X o Y =X +q. 3. Demand d occurs, and he on-hand invenory is used o mee his demand. The x,1 unis having only 1-period lifeime are consumed firs. If x,1 >d, he remaining x,1 -d unis will perish and an oudaing cos of,1 oˆ ( x d ) occurs. Oherwise, he x,2 unis having 2- period lifeime are used o mee he excess d -x,1 unis of demand, and so on, unil all invenory is consumed. If d >Y, he demand canno be saisfied compleely using he curren invenory. Therefore, he remaining d -Y unis of demand becomes backlogs ha need o be me in period +1, and a backlogging penaly of bˆ ( d Y ) occurs. If d <Y, he remaining Y -d unis resul in a holding cos of hˆ ( Y d ). 4. A he end of period, he remaining lifeime of he excess unis of he on-hand invenory 1 decreases by 1 period. Noe ha if, i i, j x d x j1 1 demand. Oherwise,, i i, j x d x j1, all x,i unis will be used o mee unis remain for period +1, becoming x +1,i-1, i=2,, m-1. In period, he q unis have a lifeime of m periods, and hey will mee he excess demand afer all X have been used. Therefore, in a backlogging problem, if j1 m1, j q d x, backlogs occur and he number of backlogs is m1, j q d x j1. These backlogs will reain in he sysem for period +1 as a negaive invenory. So he new invenory level a he beginning of period +1 can be calculaed using (similar o Eq. (1)) i 1, i, i1, j x x d x, i {1,, m 2}, j1 x q d x j1 m1 1, m1, j 14.

4.1.2.4 Sysems wih Los-Sales and an Ordering Capaciy Consrain The process of a los-sales problem wih an order capaciy consrain can be considered as follows. 1. A he beginning of period, =1,, T, informaion se f and on-hand invenory x,i are observed, where i, x i1,, m1 x i, 0, 1. 0, i 1,, m 1 {2,, T} Then oal invenory level is X m1, i x and he condiional join disribuion of he i1 upcoming demand D is known based on he informaion se f. 2. The sysem orders q unis, and hey arrive and become available, having a lifeime of m periods. In a sysem wih an order capaciy consrain u, he feasible range of q is {0,, u }. The oal cos of ordering is cq ˆ. Also, he on-hand invenory level increases from X o Y =X +q. 3. Demand d occurs, and he on-hand invenory is used o mee his demand. The x,1 unis having only 1-period lifeime are consumed firs. If x,1 >d, he remaining x,1 -d unis will perish and an oudaing cos of,1 oˆ ( x d ) occurs. Oherwise, he x,2 unis having 2- period lifeime are used o mee he excess d -x,1 unis of demand, and so on, unil all invenory is consumed. If d >Y, he demand canno be saisfied compleely using he curren invenory. Therefore, he remaining d -Y unis of demand will be los and leave he sysem, wih a los-sales penaly of bˆ ( d Y ). If d <Y, he remaining Y -d unis resul in a holding cos of hˆ ( Y d ). 4. A he end of period, he remaining lifeime of he excess unis of he on-hand invenory 1 decreases by 1 period. Noe ha if, i i, j x d x j1 1 demand. Oherwise,, i i, j x d x j1, all x,i unis will be used o mee unis remain for period +1, becoming x +1,i-1, 15

i=2,, m-1. In period, he q unis have a lifeime of m periods, and hey will mee he excess demand afer all X have been used. Therefore, in a los-sales problem, if 1, m j1, he excess demand will leave he sysem. If 1 m, j q d x j1 q d x j m1, j he q d x j1, remaining unis of q will become x +1,m-1. Therefore, he new invenory level a he beginning of period +1 can be calculaed using (similar o Eq. (2)) i 1, i, i1, j x x d x, i {1,, m 2}, j1 x q d x j1 m1 1, m1, j. 3.1.2 Objecive We consider a finie horizon problem wih T periods. The oal cos occurring in each period includes holding, backlogging/los-sales penaly, and oudaing coss. A he beginning of period, he informaion se f and on-hand invenory x,i, i=1,, m-1 are observed, and order quaniy q is deermined, which causes an ordering cos of cq ˆ. The holding cos occurs when here are some unis lef afer fulfilling he demand, which means ha Y -d >0. Therefore, he holding cos in period is ˆ ( h Y ) D. If Y -d <0, no all demands are saisfied, and a backlogging/los-sales penaly of ˆ ( b D Y ) occurs. Also, in each period, perishable unis cause an oudaing cos of,1 oˆ ( x D ). Le α, 0 1, denoe he discoun facor. Then, because demand in each period is uncerain, he expeced oal cos in period under policy P can be obained using ˆ ˆ ˆ ˆ, 1,1 S (P) E c q h ( Y D ) b ( D Y ) o ( x D ) f where m1, i. i1 Y x q A he end of T periods, some unis may remain in he sysem, which can be sold in a salvage marke. Le ˆv denoe he per-uni salvage value. Then, he oal salvage value a he 16

end of T periods is T m1 T 1, i vˆ i1 x. The objecive is o find he opimal policy based on he informaion se f and on-hand invenory x,i, i=1,, m-1, ha minimizes he expeced oal cos over T periods (similar o Eq. (2) in [7]): C(P) T m1 T T 1, i = S (P) E vˆ x 1 i1 T m1 1 ˆ ˆ,1 T T 1, i E cˆ q h ( Y D ) b ( D Y ) oˆ ( x D ) vˆ x. 1 i1 (3) 3.1.3 Cos Transformaion Suppose ha all per-uni coss are consan over he T periods and denoe ĉ, ĥ, ˆb and ô as he consan per-uni ordering cos, holding cos, backlogging/los-sales penaly and oudaing cos. Then, ˆ ˆ ˆ ˆ,, ˆ ˆ c c h h b b, oˆ oˆ, {1,, T}. A cos ransformaion can be carried ou o make ordering cos c=0. Le salvage value ˆv be equal o ĉ, and se non-negaive cos parameers o c=0, h hˆ (1 ) cˆ, b bˆ (1 ) cˆ and o oˆ cˆ. Then, i has been proved ha he expeced oal cos described in (3) can be rewrien as (see Proposiion 1 in [7]) C(P) E h( Y D ) b( D Y ) o( x D ) ce D 1 1 T T 1,1 1 ˆ. Noe ha 1 T cˆe D 1 is a consan and does no depend on he invenory policy. To simplify he noaion, we le α=1 and only consider he expeced cos per period and expeced oal cos as: S h Y D b D Y o x D f,1 (P) E ( ) ( ) ( ), T C S h Y D b D Y o x D f T,1 (P) (P)=E ( ) ( ) ( ). 1 1 17

3.2 Dynamic Programming Approach Dynamic programming approach is usually used o describe a muli-period sochasic invenory conrol problem, which is defined recursively and deermines he order quaniy q in period o minimize he expeced oal cos over he ime horizon. Noe ha he sae a he beginning of period consiss of he informaion se f and he on-hand invenory x,i, i=1,, m-1, and he expeced cos in period is.,1 S (P) E h( Y D ) b( D Y ) o( x D ) f Le V x x f,1, m1 (,,, ) denoe he opimal expeced cos over periods o T, and suppose ha V x x f j,1 j, m1 j(,,, j), j=+1,, T, have been calculaed. The Bellman formulaion for calculaing he opimal expeced cos in period for he perishable invenory problem wihou an order capaciy consrain is V x x f,1, m1 (,,, ) 1,1 1, m1 min S E V 1(,,, 1) x x F f q {0,, },1 E h( Y D ) b( D Y ) o( x D ) f min, q {0,, } 1,1 1, m1 E V 1( x,, x, F 1) f and for he perishable invenory problem wih an order capaciy consrain is V x x f,1, m1 (,,, ) 1,1 1, m1 min S E V 1(,,, 1) x x F f q {0,, u },1 E h( Y D ) b( D Y ) o( x D ) f min. q {0,, u } 1,1 1, m1 E V 1( x,, x, F 1) f Considering Eq. (1) and (2), he sae ransiion equaions for a problem wih backlogs are i 1, i, i1, j x x D x, i {1,, m 2}, j1 x q D x j1 m1 1, m1, j and for a problem wih los-sales are 18,

i 1, i, i1, j x x d x, i {1,, m 2}, j1 x q d x j1 m1 1, m1, j. Due o he curse-of-dimensionaliy, solving he dynamic program discussed above o obain he opimal order quaniy in each period is no sraighforward [6]. Therefore, in he nex secion, we propose wo simple efficien policies based on he marginal cos accouning scheme o approximae he opimal invenory level a he beginning of each period. 3.3 Heurisic Policies 3.3.1 Marginal Cos Accouning In he marginal cos accouning scheme, he cos incurred by a decision is no considered periodically. Insead, he oal cos incurred by one decision over all he periods from when he decision is made o he end of he planning horizon is calculaed. There are hree differen ypes of coss caused by ordering q unis in period, which are he holding, backlogging/los-sales penaly, and oudaing coss. When hey occur, he number of periods hey will las for depends on he specific model assumpions. Also, noe ha he cos occurring in one period may conain he marginal coss caused by decisions made in previous periods. In his secion, we firs characerize hese marginal coss. Recall ha he FOFC policy is applied here, meaning ha unis ordered earlier should be used o saisfy demand prior o hose ordered laer. One imporan feaure abou using he marginal cos accouning is ha once one decision is made, he coss caused by i from he curren period o he end of he planning horizon are only affeced by he demand, bu no he decisions in he subsequen periods. This feaure ensures ha he expeced oal coss associaed o differen decisions can be obained before he decision is made, so we can compare he 19

differen decisions and heir corresponding expeced oal coss, and choose he bes one. 3.3.1.1 Sysems wih Backlogs and No Ordering Capaciy Consrain a) Marginal holding cos accouning Holding cos occurs when some unis have no been consumed a he end of one period. Le he order quaniy in period be q. Because hese q unis have a lifeime of m periods in a perishable problem when hey become available, hey can say in he sysem for a mos m periods, from period o +m-1, if hey are no used o mee demand. Therefore, in he marginal holding cos accouning, he holding cos incurred by hese q unis may occurs a mos from period o period +m-1. In order o calculae he marginal holding cos incurred by hese q unis, we need o know how many unis of hese q unis can reain in he sysem a he end of each period. Recall ha a he beginning of period, he on-hand available invenory is X m1, i x. i1 These X unis need o be consumed prior o q due o he FOFC policy. Therefore, wheher q unis will be used in period depends on he sign of D -X. If D -X >0, he on-hand invenory is no enough o mee demand in period, hus, some unis of q are needed. If D -X 0, all q unis will be kep in he sysem for he nex period. Therefore, he remaining unis of q a he end of period is m 1, i q D x i1. In period +1, afer unused unis of x,1, if here are any, have perished, he number of unme D is,1 ( D x ), which will be me by m1 i, i2 x. Then, D +1 occurs and his will also be me by m1 i, x before q is used. Therefore, he remaining unis of q a he end of period +1 is i2 m 1 1,1, i q D D x x i2. The number of remaining unis of q a he end of period +2,, +m-1 can be calculaed 20

similarly. For he simpliciy of expression, a nesed srucure is used o describe hese resuls (see e.g., [7]). Le B ( x, i) denoe he ousanding demand in period +i-1 afer he unis having a lifeime of i periods or less have been depleed. Then, (similar o Eq. (5) in [7]) B ( x, i) : 0 i 0. (4) (, 1) {1,, 1} i 1, i D B x i x i m Therefore, he number of remaining unis of q a he end of period j, j=,, +m-1, can be calculaed, using m 1 j, i q D B x, j x. i j1 Considering ha here are T periods in he horizon, he oal marginal holding cos over period o +m-1 incurred by ordering q unis in period can be defined as (similar o Eq. (4) in [7]) where x y min{ x, y}. ( m1) T m 1 j, i H ( q ) : h q D B ( x, j ) x j, i j1 In pracical siuaions, a runcaed marginal holding cos can be calculaed. Insead of adding all holding cos incurred by q from period o ( m 1) T, he runcaed marginal holding cos considers he holding cos from period o +l-1, l=1,, m. Le, l H ( q ), l=1,, m, denoe he runcaed marginal holding cos, which represens he marginal holding cos incurred by q over periods o ( m 1) T. When l=1, only he marginal holding cos in period is considered. Thus, he runcaed marginal holding cos incurred by q in period wih l=1 is m 1,1, i H ( q ) h q D x i1. When l=2 and <T,,2 H ( q ) can be obained by adding he marginal holding cos in period 21

and +1, which is H,2 ( q ) h q D B ( x, j ) x j i2 ( 1) T m1 j, i m1 m1, i 1,1, i h q D x h q D D x x i1 i2 If =T, we only need o consider he marginal holding cos in period T, and i is easy o see ha in period T, no maer wha l is, he runcaed marginal holding cos will be. m 1 T, l T T T T, i H ( q ) h q D x i1. For l=3,, m-1, he runcaed marginal holding cos can be obained using a similar mehod, and when l=m, he runcaed marginal holding cos is exacly equal o he marginal holding cos. In general, he runcaed marginal holding cos is ( l1) T m 1, l j, i H ( q ) : h q D B ( x, j ) x, l {1,, m} j. i j1 An inuiive explanaion for he runcaed marginal holding cos is ha because he ordering cos is consan, ordering more in one period for he fuure demand will cos more han ordering in fuure periods. Therefore, here is no moivaion o order more in he curren period for he fuure demand unless he variabiliy of demand is high. Hereafer, we will use, l H ( q ), l=1,, m, insead of H ( q ). b) Marginal backlogging penaly accouning The marginal backlogging penaly caused by ordering q unis only occurs in period in a backlogging problem wihou an order capaciy consrain. Before he demand is observed, he available on-hand invenory ha can be used o mee he demand is m1, i i1 x q. Therefore, he unsaisfied demand is 22

m1, i D x q, i1 and he marginal backlogging penaly is (similar o Eq. (7) in [7]) m1, i ( q ) : bd x q. (5) i1 c) Marginal oudaing cos accouning The marginal oudaing cos incurred by ordering q in period occurs in period +m-1 if hese q unis have no been consumed compleely by he end of period +m-1. This oudaing cos can only be incurred by orders placed from period o T-m+1 due o he m-period lifeime. The unis arriving in period T-m+2 o T will have no reached heir lifeime of m periods by he end of period T, so he oudaing cos for hese orders is 0. For he q unis ordered in period, =1,, T-m+1, hey will be used o mee demand only when all m1 i, x have been used up. Noe ha x,i unis have a lifeime of i periods, so hey i1 will perish and leave he sysem a he end of period +i-1 if hey have no been used by hen. Therefore, he nesed srucure in he marginal holding cos is also needed o calculae he oudaing cos. Wih he same definiion of B ( x, i) given in he marginal holding cos, he marginal oudaing cos can be obained using (similar o eq. (6) in [7]) m1 ( ): (, 1), {1,, 1} O q o q D B x m T m. (6) 3.3.1.2 Sysems wih Los-Sales and No Ordering Capaciy Consrain a) Marginal holding cos accouning The difference beween sysems wih backlog and los-sales is he way he excess demand is reaed. However, holding cos only occurs when excess invenory exiss. In period, if he on-hand invenory m1, i x q is larger han d, here will be no excess demand a he end i1 of he period. Therefore, he runcaed marginal holding cos in los-sales problems wihou an 23

order capaciy consrain is ( l1) T m 1, l j, i H ( q ) : h q D B ( x, j ) x, l {1,, m} j. i j1 b) Marginal los-sales penaly accouning When he demand in period is no saisfied compleely, he excess par is los a he end of period and leads o a los-sales penaly. This los-sales penaly only occurs once a he end of period. Alhough in a backlogging problem his excess demand reain in he sysem, he calculaion of he amoun of he excess demand is he same for boh sysems wih backlogs and los-sales, which is m1, i D x q. i1 Therefore, he marginal los-sales penaly for ordering q unis in period is (see Eq. (5)) m1, i ( q ) : bd x q. i1 c) Marginal oudaing cos accouning The marginal oudaing cos incurred by ordering q in period occurs in period +m-1 if hese q unis have no been consumed compleely by he end of period +m-1. If an oudaing cos occurs, he calculaion is he same as he one given in Secion 3.3.1.1, which is (see Eq. (6)) m1 ( ): (, 1), {1,, 1} O q o q D B x m T m. 3.3.1.3 Sysems wih Backlogs and an Ordering Capaciy Consrain a) Marginal holding cos accouning For any given order quaniy a he beginning of period, q, he available invenory a he beginning of period is m1, i i1 x q. Therefore, he runcaed marginal holding cos, which is based on he available invenory and observed demand, follows he discussion in Secion 24

3.3.1.1 and can be obained using ( l1) T m 1, l j, i H ( q ) : h q D B ( x, j ) x, l {1,, m} j. i j1 b) Marginal backlogging penaly accouning To obain he marginal backlogging penaly in sysems wih an ordering capaciy consrain, a forced shorage is inroduced in [25]. Noe ha if he order capaciy in period, u, is less han he order quaniy q P under policy P in a problem wihou an ordering capaciy consrain, he order capaciy makes q o be less han q P. Therefore, he number of backlogs, m1, i ha should be considered o calculae he backlogging penaly is no D x q i1 because he order quaniy q canno reach q P due o he capaciy consrain. Also, he penaly of he unsaisfied demand beyond u canno be calculaed as he backlogging penaly caused by he decision of ordering q unis. Le q denoe he slack capaciy in period, where q u q, 0 q u. In period, we use a forced shorage, which is he number of unis ha could have been ordered more o avoid he backlogging penaly, o calculae he cos of he backlogs, insead of comparing he available invenory and demand direcly. Le W,s denoe he forced shorage, which occurs in period s, s=,, T, because of he ordering decision in period. Then, when s=, we ge m 1,, i W min q, D x q i1. (7) If all demand in period is saisfied, or we order u unis in period, W, will be 0. If backlog occurs in period, he capaciy u +1 should be considered o see wheher he sum of he backlogs and new demand d +1 in period +1 can be saisfied by ordering q +1 unis under he order capaciy u +1. If so, no backlog will exis a he end of period +1. If no, backlogs from period should be saisfied wih a higher prioriy due o he FOFC policy. Afer x,1 unis have 25

perished a he end of period, he remaining demand is m1, i i2 x increases o D x,1, and he invenory is q. Suppose ha u +1 unis are ordered in period +1. Then, he available invenory m1, i 1 i2 x q u +1 increases o D 1 D x,1 1,1 m1, i 1 D D x x q u. Afer demand D +1 is observed, he oal demand in period. Comparing he available invenory and demand, if i2, ordering q unis in period causes no backlogging penaly in period +1. Oherwise, ordering q unis in period is no enough o mee he cumulaive demand D [, 1], where D [, s] s i i D, s=,, T, even hough he unis of he maximum feasible order quaniy u +1 are ordered in period +1. Noe ha m1 i2 1,1, i 1 D D x x q u is he shorage of invenory in period +1. Similar o period, we compare his value wih he slack capaciy caused by ordering q unis, which is q o deermine he forced shorage m1, 1 1,1, i 1 W min q, D D x x q u i2. Because u +1 is he maximum feasible order quaniy in period +1, he forced shorage W,+1 using u +1 is no larger han ha using oher order quaniies under he ordering capaciy consrain in period +1. The analysis for he subsequen periods o period +1 is similar. We define a new vecor x, which consiss of unis ha are already in he invenory in period and ha will come ino he invenory, as,1, m 1 1 T : x,, x x ; q ; u,, u, where =1,, T-1. If =T, no u unis will be ordered by he end of T periods, so we have T T,1 T, m 1 T x : x,, x ; q. Noe ha x consiss of invenory x,i, q and order capaciy u j, for j=+1,...,t. Therefore, here are T-+m componens in oal. The orders of q and u j, j=+1,...,t, unis are placed from period 26

o T, from when hey will have a lifeime of m periods. We can consider he periods from o he period when he q and u j, j=+1,...,t, unis are ordered as an exra lifeime for hese unis. For example, consider period and assume ha u +1 unis will be ordered in period +1. The one period from o +1 can be considered as an exra lifeime for he u +1 unis. Thus, in period, we can hink ha he u +1 unis have a lifeime of m+1 periods. Therefore, generally, in period, he I, x unis, I=1,,T-+m, have a lifeime from 1 period o T-+m periods. Bu, when hey come ino he invenory and become available, hey all have a lifeime of only m periods. Then, we modify he definiion of B ( x, i) in Eq. (4) o B ( x, I), using x and I, as B ( x, I) : 0 I 0 1, I I D B ( x, I 1) x I 1,, T m. Noe ha B ( x, I) denoes he ousanding demand in period +I-1 afer he unis having a lifeime of I periods or less have been depleed, similar o B ( x, i). The difference beween B ( x, I) and B ( x, i) is ha he former considers order quaniies placed in periods +1,, T. Therefore, a general expression for he forced shorage in period s, s=,, T, which is caused by ordering q unis in period is s m, s s, I W min q, D B ( x, s ) x, s {,, T}, I s1 and he corresponding backlogging penaly in he problem wih an order capaciy consrain is T, s ( q ) : b W. s c) Marginal oudaing cos accouning In a backlogging problem wih an order capaciy consrain, he order capaciy does no impac he calculaion of he marginal oudaing cos. If he order quaniy is smaller han ha in a problem wihou an order capaciy consrain, he probabiliy of oudaing decreases. Therefore, he oudaing cos can be calculaed using (see Eq. (6)) 27

m1 ( ): (, 1), {1,, 1} O q o q D B x m T m. 3.3.1.4 Sysems wih Los-Sales and an Ordering Capaciy Consrain a) Marginal holding cos accouning The derivaion of he runcaed marginal holding cos is based on he available invenory a he beginning of a period and observed demand during ha period. Therefore, following he discussion in Secion 3.3.1.1, we ge ( l1) T m 1, l j, i H ( q ) : h q D B ( x, j ) x, l {1,, m} j. i j1 b) Marginal los-sales penaly accouning In a los-sales problem wih an order capaciy consrain, similar o Secion 3.3.1.3, he excess demand in period canno be used o calculae he los-sales penaly incurred by ordering q unis in period direcly. This is because his excess demand may surpass he difference beween q and he order capaciy u. Therefore, he forced shorage is needed o obain he marginal los-sales penaly. Similar o he backlogging problem wih an ordering capaciy consrain, if we always order a he capaciy level in period s, s=+1,, T, and unsaisfied demand occurs in a period, his los-sales penaly is associaed o he shorage in period. 1, In period, he excess demand is m D x i q. We compare his excess demand i1 wih he slack capaciy q u q, 0 q u, o deermine he exac los-sales quaniy ha is caused by ordering q unis in period, which is (see Eq. (7)) m 1,, i W min q, D x q i1. Because he excess demand will be los in a los-sales problem, so in period +1 he only demand ha needs o be me is D +1. Noe ha he unused unis of x,1 will perish a he end of period, and should be deduced from he invenory. Therefore, a he end of period, afer x,1 unis have perished, he excess invenory is 28