Int. J. Production Economics

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1 Int. J. Production Economic 139 (212) Content lit available at SciVere ScienceDirect Int. J. Production Economic journal homepage: Dynamic inventory rationing for ytem with multiple demand clae and general demand procee Hui-Chih Hung a,n, Ek Peng Chew b, Loo Hay Lee b, Shudong Liu b a Department of Indutrial Engineering and Management, National Chiao Tung Univerity, 11 Univerity Road, Hinchu 31, Taiwan b Department of Indutrial and Sytem Engineering, Univerity of Singapore, E1A-6-25, 1 Engineering Drive 2, Singapore , Singapore article info Article hitory: Received 11 June 21 Accepted 16 May 212 Available online 28 May 212 Keyword: Inventory rationing Dynamic critical level Backordering Multiperiod ytem abtract We conider the dynamic rationing problem for inventory ytem with multiple demand clae and general demand procee. We aume that backorder are allowed. Our aim i to find the threhold value for thi dynamic rationing policy. For ingle period ytem, dynamic critical level policy i developed and the detailed cot approximation ubject to thi policy i derived. For multiperiod ytem, a dynamic rationing policy with periodic review i propoed. The numerical tudy how that our dynamic critical level policie are cloe to being optimal for variou parameter etting. & 212 Elevier B.V. All right reerved. 1. Introduction Inventory i an important driver in modern upply chain and ha traditionally been ued to provide a buffer againt demand uncertainty or increaed ervice level. However, there are cot aociated with holding inventory, uch a opportunity cot, torage cot, obolecence cot, inurance cot, and damage cot. Hence, organization face a trade-off between incurring inventory and ervicing their cutomer. However, inventory can erve purpoe beyond it traditional role becaue heterogeneou cutomer have different ervice need and prioritie. Thi mean that firm can make tactical deciion with regard to the rationing of inventory and can et different pricing and ervice level according to their cutomer ervice need. By providing a differentiated ervice according to cutomer need, firm can benefit, becaue thi help to increae market ize, and thereby revenue. For example, firm can charge higher price to cutomer who need immediate ervice and can charge le for cutomer who only need a normal ervice. Thi practice i common in many indutrie, uch a the airline indutry, online retailing, and the ervice part indutry. The airline indutry uually charge different price for the ame eat, and online retailer, uch a Amazon.com, provide expedited and normal hipping ervice. The ervice part indutry alo charge cutomer according to ervice delivery contract. For a firm to uccefully adopt a different pricing or ervice level trategy for the ame inventory, the main aumption i that cutomer can be egmented according to their different ervice n Correponding author. Tel.: þ x5735; fax: þ addree: hhc@nctu.edu.tw (H.-C. Hung), iecep@nu.edu.g (E.P. Chew), ieleelh@nu.edu.g (L.H. Lee), g3918@nu.edu.g (S. Liu). need and prioritie. The key challenge i how to allocate the inventory to different egment of cutomer. For motivation, thi paper ue the example of a firm that ha an extenive network providing pare part, which are ued to maintain or replace failed equipment part at the cutomer ite. It ha a major regional ditribution center, which erve it cutomer. Requet for pare part are prompted by part failure and by cheduled maintenance. Requet prompted by part failure mut be rectified immediately, wherea thoe prompted by cheduled maintenance can wait. Hence, in any period, the ditribution center may face thee two type of demand from it cutomer. In thi ituation, a firm may adopt the rationing policy that when inventory i low, only urgent demand for part i atified. The inventory level at which low-priority requet are rejected i ometime known a the critical, or threhold level. The policy of reerving tock i termed the. Many reearcher have explored practical example of inventory rationing, uch a Kleijn and Dekker (1998), Dehpande et al. (23), and Cardó and Babiloni (211). There are two kind of critical-level policie: tationary and dynamic. For tationary policie, the critical level are contant. Much reearch ha been carried out on tationary critical level policy. For make-to-tock production ytem, the tationary critical level rationing policy i optimal for pecific cae (Ha, 1997a, 1997b, 2; Gayon et al., 24). For exogenou inventory upply problem, reearcher uch a Melchior et al. (2), Dehpande et al. (23), and Arlan et al. (27) propoe tationary policie and then determine the optimal parameter for the critical level that minimize inventory cot. Other uch a Nahmia and Demmy (1981), Moon and Kang (1998), Cohen et al. (1988), Dekker et al. (1998), and Möllering and Thonemann (29) determine the tationary critical level for inventory ytem operating under different ervice level /$ - ee front matter & 212 Elevier B.V. All right reerved.

2 352 H.-C. Hung et al. / Int. J. Production Economic 139 (212) For dynamic policie, the critical level may change over time. Topki (1968) conider dynamic inventory rationing policy for ingle period and multiple period ytem with zero lead time. A dynamic programming model i propoed in which one period i divided into many mall interval. Topki alo how that the optimal rationing policy i dynamic. However, he fail to how that the critical level i nonincreaing over time. Evan (1968) and Kaplan (1969) extend reult from Topki (1968) and explore two demand clae. Melchior (23) conider dynamic rationing policy under an inventory ytem with a Poion demand proce and an (, Q) ordering policy in which backordering i not allowed. Lee and Herh (1993) conider dynamic rationing policy for an airline eating problem. Teunter and Klein Haneveld (28) develop a continuou time approach to determining the dynamic rationing policy for two Poion demand clae under the aumption that there i no more than one outtanding order. However, it computational reult are tractable only for limited etting. Fadiloglu and Bulut (21) propoe a heuritic rationing policy called rationing with exponential replenihment flow for continuou-review inventory ytem. All except Topki (1968) conider only two demand clae. However, the limitation of hi approach i that the tate pace grow exponentially large when the number of demand clae increae. Even for two demand clae, the tate pace can be very large. Moreover, many reearcher aume a Poion ditribution. In thi paper, we develop an approximation approach to deriving the dynamic threhold level for inventory ytem with multiple demand clae and general demand procee. Thi approximation approach i baed on comparing the marginal cot of accepting and rejecting a demand cla when it arrive. It i alo aumed that when thi demand cla i rejected, all future demand from thi cla will be rejected until the next replenihment arrive. Unlike exiting work, thi method can deal with general demand procee, and i efficient in olving problem with more than two demand clae. To illutrate the effectivene of the propoed policy, we conduct numerical analyi. The reult how that the propoed policie are cloe to being optimal under variou parameter etting when demand follow a Poion proce. The Poion proce i ued becaue we want to compare our olution with the optimal olution. Our paper i organized a follow. In Section 2, we conider a ingle period ytem with multiple demand clae. We derive the dynamic critical level baed on the concept of marginal cot. In Section 3, we conider multiperiod ytem with periodic review policy. The rational policy in Section 2 i extended for multiperiod ytem. In Section 4, numerical tudie are conducted to invetigate the performance of the propoed approache. In Section 5, we ummarize the reult and dicu ome poible extenion. 2. Inventory rationing for ingle period ytem In thi ection, the inventory rationing problem for a ingle period ytem i conidered. The goal i to find a good dynamic rationing policy. We firt examine the dynamic critical level for only two demand clae and then extend the reult for a general number of demand clae. Then, we develop an approximation method for computing the expected total cot aociated with our rationing policy Model formulation Conider a ingle period inventory ytem with a period length of u. There i a ingle product with demand from K different cutomer clae. We aume that the demand procee of all cutomer clae are independent and tationary and that thee demand can be partially atified. We let t¼u denote the beginning of the period, we let t¼ denote the end of the period, and we let X(t) denote the on-hand inventory at the remaining time t A½,uŠ. For each unit tored, we aume a holding charge of h per unit of time. At the beginning of the period, we aume that the initial on-hand inventory i given, and i equal to x (i.e., XðuÞ¼x). During the period, each cutomer demand may either be atified or rejected according to our rationing policy. The rejected demand i backordered and a backorder cot i applied. We define the backorder cot for cla i a p i þ ^p i t, where t A½,uŠ i the remaining time to the end of the period. Note that p i i a fixed penalty cot to reject a demand from cla i, which may repreent the lo of cutomer loyalty. Moreover, ^p i i the per-unit-of-time cot to hold a demand from cla i to the end of the period without lo of goodwill or the order. Without lo of generality, we arrange demand clae in the order of nonincreaing backorder cot. That i, p i Zp j and ^p i Z ^p j for demand clae ioj. At the end of the period (t ¼ ), we aume that all backorder mut be fulfilled. We propoe uing the remaining inventory to fulfill thee backorder firt and if thi i inufficient, we propoe the purchae of additional unit to fulfill the remaining backorder from the open market. If the remaining inventory exceed the backorder, the urplu i old at alvage value on the ame market. We aume that both the alvage value and the additional purchaing cot on the open market equal c per unit of product at the end of the period. To determine the dynamic rationing policy, we need to compute critical level over time. Define i ðtþ to be the dynamic critical level of cla i for the remaining time t A½,uŠ. When XðtÞ4 i ðtþ, we atify the demand from cla i. Otherwie, the demand from cla i i rejected. A we have arranged demand clae in the order of nonincreaing backorder cot, we have i ðtþr j ðtþ for clae ioj. We alo define Hðt,XðtÞÞ a the expected cot for the remaining time t. Thu, our objective i to find the rationing policy that minimize the expected total cot, which can be written a follow: min i ðtþ i ¼ 1,...,K Hðu,xÞ: Without lo of generality, let n i ðtþ denote the optimal critical level of cla i and let H n ðu,xþ denote the optimal total expected cot. We know that n 1ðtÞ mut be zero becaue cla 1 ha the highet backorder cot and there i no advantage in rejecting demand from cla Dynamic critical level for ytem with two demand clae Conider a ingle period ytem with two demand clae ðk ¼ 2Þ and uppoe that a cutomer demand ha jut arrived when the time remaining i t. If the demand i from cla 1, it mut be atified. If the demand i from cla 2, it can either be atified or rejected. When thi demand i atified, the total expected cot at the remaining time t i H n ðt,xðtþ 1Þ. If thi demand i rejected, the total expected cot at the remaining time t i H n ðt,xðtþþþe 2 ðtþ, where e 2 ðtþ¼c þp 2 þ ^p 2 t i the backorder cot. If H n ðt,xðtþ 1Þ4H n ðt,xðtþþþe 2 ðtþ, then thi demand hould be rejected. Hence, the optimal dynamic critical level of cla 2 i n 2 ðtþ¼max XðtÞ9Hn ðt,xðtþþþe 2 ðtþ H n ðt,xðtþ 1Þo : ð2þ It i difficult to olve Eq. (2) becaue the cloed form for H n ðt,xðtþþ H n ðt,xðtþ 1Þ cannot be found eaily. Thu, we adopt two important idea to approximate n 2ðtÞ. Firt, if a demand cla i rejected, thi demand cla will be rejected for the remaining time until the end of the period. Thi i a reaonable approximation becaue when a demand cla i rejected, it ð1þ

3 H.-C. Hung et al. / Int. J. Production Economic 139 (212) intuitively implie that there i unlikely to be ufficient tock to cater for the le important clae for the remainder of the period. Second, we ue the marginal cot of rejecting the demand to determine the critical level. The marginal cot i computed by auming that either thi and all future demand clae are rejected or that the current demand i accepted but future demand are rejected. When a demand from cla 2 arrive, we ue thi marginal cot to decide whether thi demand i accepted or rejected. Define DJ 2 ðt,xðtþþ a the marginal cot of rejecting demand cla 2 at the remaining time t when on-hand inventory i XðtÞ. Without lo of generality, we conider two cae: XðtÞ¼ (when the demand i rejected at the remaining time t) and XðtÞ¼ 1 (when the demand i accepted at the remaining time t, but all future demand of thi cla are rejected). Fig. 1 illutrate the on-hand inventory for both cae, where the olid line repreent XðtÞ¼ and the dahed line repreent XðtÞ¼ 1. Note that, becaue we aume that all future demand from cla 2 are rejected, the inventory decreae only when the demand of cla 1 arrive. The line below the horizontal axi repreent the backorder quantitie of cla 1. Given on-hand inventory of at the remaining time t, let t be the time it take to run out of inventory. Define D 1 ðtþ a the total demand of cla 1 from the remaining time t to the end of the period. The XðtÞ¼ cae ha higher holding cot than doe the XðtÞ¼ 1 cae. If D 1 ðtþo, the difference in holding cot between the two cae i hut. IfD 1 ðtþz, the difference in holding cot between the two cae i t Uh. The XðtÞ¼ cae ha lower backorder cot than doe the XðtÞ¼ 1 cae. If D 1 ðtþo, then D 1 ðtþr 1. Thu, there i no backorder for both cae and there i a difference of one unit of product. Thi mean that there i an extra alvage value, c, for the XðtÞ¼ cae. If D 1 ðtþz, the th demand of cla 1 can be atified in the XðtÞ¼ cae. However, the th demand of cla 1 cannot be atified in the XðtÞ¼ 1 cae. Thu, thi demand cla ha an extra backorder cot of c þp 1 þ ^p 1 Uðt t Þ in the XðtÞ¼ 1 cae. In fact, thee cot difference between the XðtÞ¼ and XðtÞ¼ 1 cae are the marginal cot of rejecting demand cla 2 at the remaining time t. Thu DJ 2 ðt,þ¼tuhupðd 1 ðtþoþþhueðt 9D 1 ðtþzþupðd 1 ðtþzþ c þp 1 þ ^p 1 UE½ðt t Þ9D 1 ðtþzš UPðD1 ðtþzþ c UPðD 1 ðtþoþ ¼ hut ½tUð ^p 1 þhþþp 1 ŠUPðD 1 ðtþzþ þð^p 1 þhþue½t 9D 1 ðtþzšupðd 1 ðtþzþ c, 1 t τ End of period ð3þ On-hand inventory Remaining time Fig. 1. On-hand inventory after rejecting ome order from cla 2 at remaining time t. where PðD 1 ðtþoþ i the probability of D 1 ðtþo and PðD 1 ðtþzþ i the probability of D 1 ðtþz. Now, we demontrate the monotonicity property of DJ 2 ðt,þ. Lemma 1. For the remaining time t and Z, DJ 2 ðt,þ i nondecreaing in. Proof. See Appendix.& After deriving the marginal cot of rejecting demand cla 2, we ue the above two idea to approximate n 2ðtÞ. Define a 2 ðtþ¼max 9DJ 2ðt,Þþe 2 ðtþo : ð4þ We how the relationhip between a 2 ðtþ and DJ 2ðt,Þ. Theorem 1. For demand cla 2, there exit a unique critical level, a 2ðtÞ, at the remaining time t uch that DJ 2 ðt,þþe 2 ðtþz for 4 a 2ðtÞ, and DJ 2 ðt,þþe 2 ðtþo for r a 2 ðtþ: Proof. Theorem 1 hold from Lemma 1. Theorem 1 how that once the on-hand inventory drop below the unique critical level, a 2ðtÞ, all future demand from demand cla 2 will be rejected until the next replenihment arrive. Thi reult i conitent with our aumption. We now etablih the monotonicity property of a 2ðtÞ from Theorem 1 and Eq. (3) and (4). Theorem 2. For p 1 ¼ p 2 ¼, a 2ðtÞ i nondecreaing in the remaining time t. Proof. See Appendix. & Theorem 2 how that a 2 ðt 1ÞZ a 2 ðt 2Þ for t 1 Zt 2 when p 1 ¼ p 2 ¼. Thi mean that when the remaining time become maller, we only need to reerve fewer inventorie for cla Dynamic critical level for ytem with more than two demand clae We now conider a ingle period ytem with more than two demand clae. We aume that a demand arrive at the remaining time t. If the demand i from cla 1, then it mut be atified. Otherwie, it can either be atified or rejected. Similarly to the cae of two demand clae in the previou ubection, we generalize a 2ðtÞ and approximate the dynamic critical level of cla m by a mðtþ. Hence, define a m ðtþ¼max XðtÞ9DJ mðt,xðtþþþe m ðtþo, ð5þ where DJ m ðt,xðtþþ i the marginal cot of rejecting demand cla m at the remaining time t. To compute the critical level for all demand clae, we adopt a equential approach; i.e., we ue the dynamic critical level for demand clae 2 to m 1 to compute the dynamic critical level for demand cla m. Note that from Subection 2.1, we know that the critical level of cla 1 i zero. Suppoe we want to compute the critical level for demand cla m, auming that the dynamic critical level for demand clae 2, 3,y,m 1 are known. Similarly, we aume that if a demand cla i rejected, that demand cla will be rejected for the remainder of the period. Without lo of generality, we conider two cae: XðtÞ¼ (when the demand of cla m i rejected at the remaining time t) and XðtÞ¼ 1 (when the demand of cla m i accepted at the remaining time t, but all future demand of thi cla will be rejected). Fig. 2 illutrate a realization of on-hand inventory for both cae, where the olid line repreent XðtÞ¼ and the dahed line repreent XðtÞ¼ 1. &

4 354 H.-C. Hung et al. / Int. J. Production Economic 139 (212) Thi figure how that, to compute the marginal cot difference between the two cae, we hould conider two different ituation. In the firt ituation, all demand from cla m 1 are accepted in both cae, and in the econd ituation, demand from cla m 1 are rejected for the XðtÞ¼ 1 cae. In the firt ituation, the only cot difference i that in inventory holding cot. In the econd ituation, the cot difference i a weighted um of the difference in inventory holding cot and backorder cot. To compute the critical level for demand cla m at the remaining time t, we aume that demand cla m i rejected at the remaining time t. Thi implie that any demand cla i, where i4m, will be rejected. A demand cla i, where iom, may be rejected in the future if the inventory level i below the repective dynamic critical level. To determine the condition under which ituation 1 applie, we mut identify the time at which demand cla m 1 i firt rejected given XðtÞ¼ 1; we denote thi time a t m 1 1. Thi mean that tm 1 1 i the maximum amount of time remaining when Xðt 1 m 1 Þ i equal to the dynamic critical level of demand cla m 1. When the remaining time i between t and t m 1 1 t, ituation 1 applie. Hence, ituation 2 applie when the remaining time i between t m 1 1 and. For ubequent derivation, we mut define t m 1, which i the maximum amount of time remaining when Xðt m 1 Þ i equal to the dynamic critical level of demand cla m 1. Note that t m 1 rt m 1 1. Now, we examine the cot difference between the XðtÞ¼ cae and the XðtÞ¼ 1 cae under two different ituation; i.e., the remaining time interval ½t 1 m 1,tŠ and ½,tm 1 1 Þ. (i) Situation 1: Remaining time interval ðt m 1 1,tŠ The difference in cot between the two cae in thi interval i one unit of inventory cot. Thu, the cot difference i huðt t m 1 1 Þ. (ii) Situation 2: Remaining time interval ½,t 1 m 1Š To derive the cot difference between the two cae, we aume that the approximated dynamic critical level of demand cla m 1 remain unchanged in the remaining time interval ½t m 1,t m 1 1 Š. That i, a m 1 ðtm 1 1 Þ¼a m 1 ðtm 1 Þ. Thu, for the XðtÞ¼ 1 cae, we have Xðt m 1 1 Þ¼a m 1 ðtm 1 1 Þ¼a m 1 ðtm 1 Þ. For the XðtÞ¼ cae, we have Xðt 1 m 1Þ¼a m 1 ðtm 1 1 Þþ1 ¼ a m 1 Þþ1. ðt m 1 Critical level of cla m 1 m 1 t 1 End of period Fig. 2. On-hand inventory veru remaining time. Inventory Remaining time Conider the XðtÞ¼ cae. Demand cla i, where izm 1, will be rejected when the remaining time i le than t m 1. Thi i becaue Xðt m 1 Þ i le than or equal to the dynamic critical level of demand cla m 1. Similarly, for the XðtÞ¼ 1 cae, when the remaining time i le than t m 1 1, demand cla i, where izm 1, will be rejected becaue Xðt m 1 1 Þ i le than or equal to the dynamic critical level of demand cla m 1. Hence, in the interval ½t m 1,t m 1 1 Š, there i one extra unit of inventory in the XðtÞ¼ cae. The marginal cot over the ½,t m 1 Š interval depend on the realization of the demand cla at the remaining time t m 1. (Note that by definition, thi demand cla cannot come from clae m and above becaue all thee will be rejected and, thu, there will be no correponding fall in inventory for the XðtÞ¼ cae.) If the demand come from clae 1 to m 2, that demand will be accepted in both cae, and o the marginal cot will be DJ m 1 ðt m 1, a m 1 ðtm 1 1 ÞÞ. Thi i becaue the tarting inventory at the remaining time t m 1 for the XðtÞ¼ cae i alway one more than the tarting inventory for the XðtÞ¼ 1 cae, and it i equal to the critical level for demand cla m 1; i.e., a m 1 ðtm 1 1 Þ. However, if the demand come from cla m 1, thi demand will be accepted for the XðtÞ¼ cae but will be rejected for the XðtÞ¼ 1 cae. Hence, the cot difference i e m 1 t m 1. Thu, the difference in average cot between the two cae in ituation 2 i huðt m 1 1 tm 1 e m 1 ðt m 1 ÞU ÞþDJ m 1 ðt m 1 l m 1 P m 1 i ¼ 1 l i, a m 1 ðtm 1 1 ÞÞU P m 2 i ¼ 1 l! i P m 1 i ¼ 1 l i!, ð6þ where ð P m 2 i ¼ 1 l iþ=ð P m 1 i ¼ 1 l iþ i the probability that the next demand i from cla 1 to m 2, and l m 1 =ð P m 1 i ¼ 1 l iþ i the probability that the next demand i from cla m 1. By combining ituation 1 and 2, given the realization of t m 1, the total marginal cot of rejecting demand cla m at the remaining time t when the inventory level i i P m 2 DJ m ðt,9t m 1 Þ¼hUðt t m 1 ÞþDJ m ðt m 1, a m 1 ðtm 1 i ¼ 1 ÞÞU l! i P m 1 i ¼ 1 l i e m 1 ðt m 1 ÞU l m 1 P m 1 i ¼ 1 l i! : ð7þ Note that t m 1 i a random variable that depend on the onhand inventory XðtÞ, the dynamic critical level a m 1ðtÞ, and the demand arrival proce. Let pðuþ denote the probability denity function of thi random variable. It i poible that XðÞ4 a m 1 ðþ. If XðÞ4 a m 1ðÞ, then inventory alway exceed the dynamic critical level of demand cla m 1. In thi cae, the marginal cot of rejecting demand from cla m 1 i ht c. By defining P ¼ 1 R t we have DJ m ðt c,þ pðtm 1 Z t Þdt m 1 ½DJ m ðt,9t m 1, which i the probability of XðÞ4 a m 1 ðþ, ÞŠUpðt m 1 Þdt m 1 þpuðht c Þ: ð8þ Eq. (8) can be reduced to Eq. (4) when m¼2. Becaue Þ¼max 9DJ m 1 ðt m 1,Xðt m 1 ÞÞþe m 1 ðt m 1 Þo, a m 1 ðtm 1 it follow that DJ m 1 X ðt,þþe m 1 ðtþ i approximately zero. Moreover, becaue of our aumption that the dynamic critical level for cla m 1 remain unchanged in the remaining time interval ½t m 1,t m 1 1 ðt m 1, a m 1 ðtm 1 Š (i.e., Xðtm 1 Þ a m 1 ðtþ), we can approximate DJ m 1 ÞÞ by e m 1 ðt m 1 Þ. Hence, Eq. (8) can be approximated by DJ m ðt,þ Z t ½hUðt t m 1 Þ e m 1 ðt m 1 ÞŠUpðt m 1 Þdt m 1 þpðht c Þ: From Eq. (5) and (9), we can generate the approximate optimal dynamic critical level of cla m from a 2 ðtþ,a 3 ðtþ,y,a m 1 ðtþ: Therefore, all dynamic critical level can be generated equentially tarting from the lowet demand cla index. ð9þ

5 H.-C. Hung et al. / Int. J. Production Economic 139 (212) We propoe the following algorithm to generate the dynamic critical level for all demand clae. Step 1: Set the optimal dynamic critical level of cla 1 to be zero: n 1 ðtþ¼. Step 2: Conidering only clae 1 and 2, ue Theorem 1 in Section 2.2 to etimate a 2ðtÞ, ta½,uš. Set m¼3. Step 3: Given a i ðt cþ, iaf2,:::,m 1g, ue Eq. (5) and (9) to find a mðtþ, ta½,uš. Set m¼mþ1. Step 4: Stop if mzkþ1. Otherwie, go to Step Expected total cot In thi ubection, we etimate the expected total cot by uing an iterative method baed on Hðt,Þ¼ XK i ¼ 1 p i þ ^p i ðtþ, Hðt,xÞHðt,Þþ Xx j ¼ 1 ð1aþ DJ 2 ðt,jþ for xað, a 2ðtÞŠ, ð1bþ Xx Hðt,xÞHðt, a m ðtþþþ DJ m þ 1 ðt,jþ for j ¼ a m ðtþþ1 xað a m ðt cþ, a m þ 1 ðt cþš and 2rmoK, ð1cþ Hðt,xÞHðt, a K ðtþþþ Xx j ¼ a K ðtþþ1 DJ K þ 1 ðt,jþ for xað a K ðt cþ,1þ: ð1dþ Eq. (1a) repreent the cae in which there i no initial on-hand inventory, i.e., XðtÞ¼x¼. In thi cae, all demand hould be rejected. The total cot will be equal to the um of the total expected backorder cot of all the clae. Eq. (1b) repreent the cae in which the initial on-hand inventory i below the critical level of cla 2. In thi cae, only cla 1 demand i accepted, and o the expected total cot can be approximated by P x j ¼ 1 DJ 2ðt,jÞ, where DJ 2 ðt,jþ i given by Eq. (4). Eq. (1c) and (1d) repreent cae in which the initial onhand inventory i between the critical level of cla m and cla mþ1, where m i between two and K 1 and the initial on-hand inventory i above the critical level of cla K. Similarly, the expected total cot can be approximated by the um of the marginal cot given in Eq. (9). 3. Inventory rationing in multiperiod ytem In thi ection, we conider dynamic inventory rationing model for multiperiod ytem with general demand proce in which backordering i allowed Notation and model formulation Conider an inventory ytem with a ingle type of product and K demand clae for an infinite time horizon. We aume that the demand proce, the backorder cot, and the holding cot are the ame a thoe of the ingle period model in Section 2. We adopt a policy of dynamic critical level rationing with periodic review ordering (R,S). Under thi ordering policy, the ordering opportunitie, m¼, 1, 2,y, occur at fixed interval of time and the amount of time between two ucceive order opportunitie i u, where R ¼ u. In addition, the determinitic lead time taken to replenih order i L. The order placed at the mth order opportunity will arrive at time l m ¼ muþl. The time interval (l m,l m þ 1 )itermedthemth replenihment period. When the mth replenihment order arrive at time l m, backorder are intantaneouly fulfilled according to the following mechanim. Backorder from the mot important (lowet index) demand cla are alway fulfilled firt. Second, if replenihment i ufficient, le important demand clae are fulfilled. With backorder fulfilled, we define y m a the net inventory and B m a the remaining backorder at time l m. Note that B m ¼ b m 1,...,bm K i the vector of backorder for all demand clae, where b m i Z i the amount of backorder for demand cla i. Under the above backorder fulfillment mechanim, y m may be negative if ome backorder remain unfulfilled. If y m Z, then b m i ¼ for all i. Otherwie, there i no on-hand inventory and P K i ¼ 1 bm i ¼ y m. Thu, ðy m,b m Þ i the tate variable at the beginning of the mth replenihment period and it ditribution depend on the rationing policy and the bae tock level S. Let P S,n ðy m,b m Þ denote the probability ditribution of the tate variable (y m,b m ) at the beginning of the mth replenihment period ubject to the rationing policy n and the bae tock level S. Similarly, let C n ðy m,b m Þ be the expected inventory cot incurred in the mth replenihment period ubject to the tate variable ðy m,b m Þ and the rationing policy n. For y m o, C n ðy m,b m Þ conit of backorder cot from B m and from new backorder ariing in thi period. For the multiperiod ytem, define ACðS,nÞ a the expected average cot incurred during a period in which the bae tock level i S and the prevailing dynamic rationing policy i n. We can now model the optimization problem for the dynamic critical level rationing policy a follow: min S,n ACðS,nÞ¼ lim M-1 1 M XM 1 m ¼ y m,bm X C n ðy m,b m ÞUP S,n ðy m,b m Þ: ð11þ The goal i to minimize the expected average cot in Eq. (11), which conit of inventory holding cot and backorder cot. Note that all rejected or unfulfilled order are backlogged. Hence, we can ignore the variable ordering cot and et the alvage value to zero: c ¼ A olution to the optimization problem In olving the rationing problem given by Eq. (11), the main iue i that the probability ditribution for backorder, B m, depend on the rationing policy n and the bae tock level S. It i difficult to derive a cloed form olution for the probability ma function for B m. Hence, we propoe a heuritic method to olve the rationing problem given by Eq. (11). In Section 2, we propoed an approximate dynamic rationing policy for a ingle period problem. We now propoe a rationing policy for a multiperiod problem. Let n a be the rationing policy for the multiperiod ytem, where each tock ration in each period i determined according to our correponding approximate dynamic critical level. The goal i to locally minimize inventory cot in each period, ignoring ubequent period. We now develop an approximate expreion for ACðS,n a Þ. Given the bae tock level S and the rationing policy n a, the probability ma function of the tate variable (y m,b m ) i P S,na ðy m,b m Þ¼PðD L ¼ S y m Þ, for y m Z: X P S,na ðy m,b m Þ¼PðD L ¼ S y m Þ for y m o, ð12þ jbmj ¼ y where jb m j¼ P K i ¼ 1 bm i and D L i the demand that arrive during the lead time L. Note that the probability ma function in Eq. (12) are independent of m and n a. Recall that C na ðy,bþ i the expected inventory cot incurred in the mth replenihment period ubject to the tate variable ðy m,b m Þ

6 356 H.-C. Hung et al. / Int. J. Production Economic 139 (212) and the rationing policy n a. We now approximate C na ðy m,b m Þ for two different cae: y m Z and y m o. For y m Z: C na ðy m,b m ÞHðu,y m Þ, ð13þ where Hðu,y m Þ i the expected total cot in a ingle period of length u, given zero alvage cot (c ¼ ) and initial on-hand inventory of y m. For y m o: C na ðy m,b m ÞHðu,Þþ XK i ¼ 1 b m i U ^p i Uu, ð14þ where C na ðy m,b m Þ conit of backorder cot from B m and new backorder ariing during the period. In a multiperiod ytem, one can preerve inventory for future ue for more important demand clae by rejecting other demand clae during each period. It i aumed that backorder from more important demand clae are fulfilled uing the mechanim decribed in the previou ection. Therefore, for the mth replenihment period, given remaining backorder of y m,on average, there hould be more backorder from le important demand clae. Moreover, in practice, y m o i unlikely. Thu, for y m o, we can rewrite Eq. (14) a C na ðy m,b m ÞHðu,Þ y m U ^p K Uu, ð15þ where we aume that all backorder in y m are from the leat important demand cla K. Baed on Eq. (12) (15), we can approximate the average cot a ACðS,n a Þ 1 XM 1 S Hðu,y M m ÞUPðD L ¼ S y m Þ þ ¼ XS m ¼ X 1 y m ¼ 1 y ¼ þ y m ¼ ðhðu,þ y m U ^p K UuÞUPðD L ¼ S y m Þ Hðu,yÞUPðD L ¼ S yþ X 1 y ¼ 1 ðhðu,þ yu ^p K UuÞUPðD L ¼ S yþ: For a given rationing policy n a, all replenihment period are the ame. By replacing y m with y, we can rewrite Eq. (11) a min S X ACðS,n S a Þmin Hðu,yÞUPðD L ¼ S yþ S y ¼! on-hand inventory of x. We define the relative error for the expected total cot a DHðn a,xþ¼ H ðu,xþ H na n nðu,xþ 1%, H n nðu,xþ where DHðn a,xþ i determined by n a and x. In thi numerical tudy, we conider a bae cae in which there are three demand clae. In thi bae cae, for the length of the period, we chooe u ¼ :1, we chooe the holding cot h ¼ 1, and we chooe the alvage value c ¼. For the backorder cot, we aume p 1 ¼ p 2 ¼ p 3 ¼ and ^p 3 ¼ 1:5, where ^p 1 : ^p 2 : ^p 3 ¼ 2 : 5 : 1:5. For the demand arrival rate, we let l 1 ¼ l 2 ¼ l 3 ¼ 3. We alo varied ome parameter; different parameter etting are lited in Table 1. For each new parameter etting, we compute the optimal dynamic critical level and the approximate dynamic critical level of the propoed method. Then, we ue imulation to etimate their cot. To enure the accuracy of the etimated cot, we repeat the imulation 2, time for each parameter etting. The reult are illutrated in Fig. 3 and 4 and reported in Table 1. Table 1 Wort relative error under different operational condition. Factor Parameter DHðn a,x n Þ (%) u l 1 : l 2 : l 3 ^p 1 : ^p 2 : ^p 3 Bae cae.1 1:1:1 2:5: Changing ratio of backorder cot Changing ratio of arrival rate.1 1:1:1 5:2: :3: :8: :1: :2:3 2:5: :3: :1: :2: :3: :1: :6: :1: :6: :1: Changing time horizon.5 1:1:1 2:5: x n þ X 1 y ¼ 1 ðh dy ðu,þ yu ^p K UuÞUPðD L ¼ S yþ: ð16þ 4. Numerical tudy In thi ection, we conduct numerical tudie to invetigate the effectivene of our propoed method. We develop bound to evaluate the quality of our propoed olution under different cenario when the demand follow a Poion proce. Thee bound are valid for both ingle and multiperiod problem. We firt conider the ingle period problem. Let n a denote the dynamic critical level implied by the propoed method and let nn denote the optimal dynamic critical level, which can be derived when the demand follow a Poion proce (Chew et al., 211). Alo, let H n ðu,xþ denote the expected total cot incurred during a ingle period of length u under rationing policy n tarting with Fig. 3. Optimal and propoed critical level in the bae cae.

7 H.-C. Hung et al. / Int. J. Production Economic 139 (212) Relative error ΔH (ν a,x).2%.15%.1%.5%.% Initial inventory x Fig. 4. Relative error, DHðn a,xþ, in the bae cae. Fig. 3 how both optimal and propoed dynamic critical level for clae 2 and 3 in the bae cae. Note that both the optimal and propoed dynamic critical level for cla 1 i zero. Curve nn_2 and nn_3 repreent the optimal dynamic critical level for clae 2 and 3, repectively. Curve n a _2 and n a _3 repreent the propoed dynamic critical level for clae 2 and 3, repectively. The time horizon u ¼ :1 i divided into 9 interval. Fig. 3 how that the propoed critical level are very cloe to the optimal critical level. Fig. 4 illutrate the relative error, DHðn a,xþ, for different initial level of on-hand inventory of x in the bae cae. Note that DHðn a,xþ i no higher than.16% and may be monotonic in x. Moreover, DHðn a,xþ i cloe to zero when x i either very mall or very large. When x i large, there i alway enough initial inventory to atify demand of all clae. Thu, the relative error induced by the propoed policy i negligible. Similarly, when x i mall, mot demand clae cannot be atified for any policy in a ingle period problem. Thu, the bet trategy i to reerve all inventory for demand cla 1. Hence, the relative error induced by the propoed policy i again negligible. Define DHðn a,x n Þ¼max x Z DHðn a,xþ a the wort relative error. Table 1 how how everal factor affect the wort relative error. In all cae reported in Table 1, DHðn a,x n Þ i mall. Few factor ignificantly affect the wort relative error. The factor that mot ignificantly affect DHðn a,x n Þ i the backorder cot ratio. A backorder cot for the more important clae increae, DHðn a,x n Þ increae. Thi follow from our aumption that if ome demand clae are rejected, thee demand clae will be rejected for the remainder of the time period. Note that when changing the ratio of arrival rate, we fix the total arrival rate at 9. Thu, the arrival rate ratio of 1:2:3 i implemented a 15:3:45 and the ratio of 1:3:6 i implemented a 1:3:6. We approximate the arrival rate ratio of 1:1:1 by 9:81:81. The numerical tudy how that the propoed dynamic critical level are very cloe to the optimal critical level and that the relative error for expected total cot are alo mall. However, when the number of demand clae increae, we may expect the relative error to increae. Thi i becaue we determine the dynamic critical level equentially. Note that the derived bound reported in Table 1 are alo valid for the multiperiod problem. Thi i becaue the reult for the imulated cae reported in Table 1 are baed on a tarting inventory that repreent the wort-cae cenario and becaue the probability of having negative inventory at the beginning of each period i negligible (around 1 4 ). Given that we have conidered extreme cae, in which the penalty cot i a much a 1 time higher than the inventory cot, thee reult are reaonable. 5. Concluion and extenion In thi paper, we developed a heuritic approach to computing the dynamic critical level for ytem with general demand arrival procee. We firt conidered a ingle period problem and then extended thi to a multiperiod ytem. The heuritic approach i baed on two idea. The firt idea i that any demand cla that i rejected in one period will be rejected for the remainder of the period. The econd idea i that dynamic critical level can be derived baed on the difference in the marginal cot of accepting and rejecting a demand cla. Thee two idea have enabled u to deal with more general demand procee. Our numerical tudy how that the outcome generated by our propoed approach compare favorably with the optimal olution under mot parameter etting. For future work, we will conider relaxing ome of our model aumption. For example, we could allow a demand from a cla that i initially rejected to be accepted in the future. We could alo conider backorder being atified before replenihment order arrive. Appendix Proof of Lemma 1. From Eq. (3), we have DJ 2 ðt,þ1þ¼hut c ½tUð ^p 1 þhþþp 1 ŠUPðD 1 ðtþzþ1þ þð^p 1 þhþue½t þ 1 9D 1 ðtþzþ1šupðd 1 ðtþzþ1þ, and DJ 2 ðt,þ¼hut c ½tUð ^p 1 þhþþp 1 ŠUPðD 1 ðtþzþ þð^p 1 þhþue½t 9D 1 ðtþzšupðd 1 ðtþzþ: From (A.1) and (A.2) we have ða:1þ ða:2þ DJ 2 ðt,þ1þ DJ 2 ðt,þ¼½tuð^p 1 þhþþp 1 ŠUPðD 1 ðtþ¼þ þð^p 1 þhþu E½t þ 1 9D 1 ðtþzþ1šupðd 1 ðtþzþ1þ E½t 9D 1 ðtþzšupðd 1 ðtþzþ : ða:3þ Note that E½t 9D 1 ðtþzšupðd 1 ðtþzþ ¼ X1 i ¼ E½t 9D 1 ðtþ¼išupðd 1 ðtþ¼iþ ¼ E½t 9D 1 ðtþzþ1šupðd 1 ðtþzþ1þþe½t 9D 1 ðtþ¼šupðd 1 ðtþ¼þ: ða:4þ By ubtituting (A.4) into (A.3), we obtain DJ 2 ðt,þ1þ DJ 2 ðt,þ¼½tuð^p 1 þhþþp 1 ŠUPðD 1 ðtþ¼þ þð^p 1 þhþu E½t þ 1 9D 1 ðtþzþ1š UPðD 1 ðtþzþ1þ E½t 9D 1 ðtþzþ1š UPðD 1 ðtþzþ1þ ð ^p 1 þhþue½t 9D 1 ðtþ¼š UPðD 1 ðtþ¼þ¼ ½tUð ^p 1 þhþþp 1 Š ð ^p 1 þhþ UE½t 9D 1 ðtþ¼š UPðD 1 ðtþ¼þþð^p 1 þhþ U E½t þ 1 9D 1 ðtþzþ1šupðd 1 ðtþzþ1þ E½t 9D 1 ðtþzþ1šupðd 1 ðtþzþ1þ : ða:5þ For E½t 9D 1 ðtþ¼šrt, we have ½tUð ^p 1 þhþþp 1 Š ð ^p 1 þhþue½t 9D 1 ðtþ¼š ¼ð^p 1 þhþu t E½t 9D 1 ðtþ¼š þp1 Z: Thu, E½t þ 1 9D 1 ðtþzþ1šupðd 1 ðtþzþ1þ4e½t 9D 1 ðtþzþ1šupðd 1 ðtþz þ1þ.

8 358 H.-C. Hung et al. / Int. J. Production Economic 139 (212) From (A.5), we have DJ 2 ðt,þ1þ DJ 2 ðt,þ4: Proof of Theorem 2. Lemma 1 implie that DJ 2 ðt,þ i nondecreaing in. Thu, DJ 2 ðt,þþe 2 ðtþ i nondecreaing in. Next, we how the continuity of DJ 2 ðt,þþe 2 ðtþ. Given 4, J 2 ðt,þ i a continuou function of remaining time t. Moreover, e 2 ðtþ¼c þp 2 þ ^p 2 t i a continuou function of remaining time t. Thu, DJ 2 ðt,þþe 2 ðtþ i a continuou function of remaining time t. Given a remaining time of t ¼ þ, no more order will arrive in thi arbitrarily mall remaining time interval. Thu, all order from demand cla 2 can be accepted at a remaining time of t ¼ þ. Thi implie: DJ 2 ð þ,þþe 2 ð þ Þ4 & ða:6þ For given o1, if the remaining time t ¼1, then we mut reject demand from cla 2 becaue of limited on-hand inventory and infinite remaining time. Thu: DJ 2 ð1,þþe 2 ð1þo ða:7þ From (A.6) and (A.7), and the continuity of DJ 2 ðt,þþe 2 ðtþ, there exit ome remaining time t uch that DJ 2 ðt,þþe 2 ðt Þ¼ for given on-hand inventory. Hence, define t 2 ¼ min t 9DJ 2 ðt,þþ e 2 ðt Þ¼g. From the continuity property and the definition of t 2, we have DJ 2 ðt,þþe 2 ðtþo for t A½t 2,1Þ ða:8þ 2 ðt 2,Þþe 2ðt 2 ÞŠ ða:9þ Next, we how that inequality (A.9) hold for t A½t 2,1Þ. We know that t i a continuou random variable. Let pðt Þ be it probability denity function. For the cumulative ditribution function, we have FðtÞ¼Pðt rtþ¼pðd 1 ðtþzþ: ða:1þ Thu dpðd 1 ðtþzþ ¼ pðt ¼ tþz dt and de½t 9D 1 ðtþzšupðd 1 ðtþzþ ¼ d dt dt Z t From Eq. (A.2) and (A.12), we 2 ðt,þþe 2 t Upðt ÞUdt ¼ tupðt ¼ tþ: ¼ h ð ^p 1 þhþupðd 1 ðtþzþ tuð ^p 1 þhþupðt ¼ tþ p 1 Upðt ¼ tþþð^p 1 þhþutupðt ¼ tþþ ^p 2 ¼ h ð ^p 1 þhþupðd 1 ðtþzþ p 1 Upðt ¼ tþþ ^p 2 ¼ h ð ^p 1 þhþupðd 1 ðtþzþþ ^p 2, which hold becaue p 1 ¼ p 2 ¼. ða:11þ ða:12þ From (A.1), PðD 1 ðtþzþ¼fðtþ i a nondecreaing function of t. Thu, for any remaining time t A½t 2 ðt,þþe 2 2ðt 2,Þþe 2ðt 2 ÞŠ r: From (A.8) and (A.13), it follow that DJ 2 ðt,þþe 2 ðtþo and that DJ 2 ðt,þþe 2 ðtþ i a nonincreaing function of remaining time taðt 2,1Þ. Given that a 2 ðtþ¼max 9DJ 2ðt,Þþe 2 ðtþo from Eq. (4), a 2ðtÞ i nondecreaing in remaining time t. & Reference Arlan, H., Grave, S.C., Roemer, T., 27. A ingle-product inventory model for multiple demand clae. Management Science 53 (9), Cardó, M., Babiloni, E., 211. Exact and approximate calculation of the cycle ervice level in periodic review inventory policie. 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