Dynamic inventory rationing for systems with multiple demand classes and general demand processes

Dynamic inventory rationing for systems with multiple demand classes and

general demand processes

Hui-Chih Hung

a,n

, Ek Peng Chew

b

, Loo Hay Lee

b

, Shudong Liu

b

a

Department of Industrial Engineering and Management, National Chiao Tung University, 1001 University Road, Hsinchu 30010, Taiwan

b_{Department of Industrial and Systems Engineering, University of Singapore, E1A-06-25, 1 Engineering Drive 2, Singapore 117576, Singapore}

a r t i c l e

i n f o

Article history: Received 11 June 2010 Accepted 16 May 2012 Available online 28 May 2012 Keywords:

Inventory rationing Dynamic critical level Backordering Multiperiod system

a b s t r a c t

We consider the dynamic rationing problem for inventory systems with multiple demand classes and general demand processes. We assume that backorders are allowed. Our aim is to find the threshold values for this dynamic rationing policy. For single period systems, dynamic critical level policy is developed and the detailed cost approximation subject to this policy is derived. For multiperiod systems, a dynamic rationing policy with periodic review is proposed. The numerical study shows that our dynamic critical level policies are close to being optimal for various parameter settings.

&_{2012 Elsevier B.V. All rights reserved.}

1. Introduction

Inventory is an important driver in modern supply chains and has traditionally been used to provide a buffer against demand uncertainty or increased service levels. However, there are costs associated with holding inventory, such as opportunity costs, storage costs, obsolescence costs, insurance costs, and damage costs. Hence, organizations face a trade-off between incurring inventory and servicing their customers. However, inventory can serve pur-poses beyond its traditional role because heterogeneous customers have different service needs and priorities. This means that firms can make tactical decisions with regard to the rationing of inventory and can set different pricing and service levels according to their customer service needs. By providing a differentiated service according to customer needs, firms can benefit, because this helps to increase market size, and thereby revenue. For example, firms can charge higher prices to customers who need immediate service and can charge less for customers who only need a normal service. This practice is common in many industries, such as the airline industry, online retailing, and the services parts industry. The airline industry usually charges different prices for the same seat, and online retailers, such as Amazon.com, provide expedited and normal shipping services. The services parts industry also charges customers according to services delivery contracts.

For a firm to successfully adopt a different pricing or service level strategy for the same inventory, the main assumption is that customers can be segmented according to their different service

needs and priorities. The key challenge is how to allocate the inventory to different segments of customers. For motivation, this paper uses the example of a firm that has an extensive network providing spare parts, which are used to maintain or replace failed equipment parts at the customer’s site. It has a major regional distribution center, which serves its customers. Requests for spare parts are prompted by parts failure and by scheduled mainte-nance. Requests prompted by parts failure must be rectified immediately, whereas those prompted by scheduled maintenance can wait. Hence, in any period, the distribution center may face these two types of demand from its customers. In this situation, a firm may adopt the rationing policy that when inventory is low, only urgent demand for parts is satisfied. The inventory level at which low-priority requests are rejected is sometimes known as the critical, or threshold level. The policy of reserving stock is termed the. Many researchers have explored practical examples of inventory rationing, such asKleijn and Dekker (1998),Deshpande et al. (2003), andCardo´s and Babiloni (2011).

There are two kinds of critical-level policies: stationary and dynamic. For stationary policies, the critical levels are constant. Much research has been carried out on stationary critical level policy. For make-to-stock production systems, the stationary critical level rationing policy is optimal for specific cases (Ha, 1997a, 1997b, 2000; Gayon et al., 2004). For exogenous inventory supply problems, researchers such as Melchiors et al. (2000),Deshpande et al. (2003), andArslan et al. (2007)propose stationary policies and then determine the optimal parameters for the critical levels that minimize inventory costs. Others such as

Nahmias and Demmy (1981),Moon and Kang (1998),Cohen et al. (1988), Dekker et al. (1998), and M ¨ollering and Thonemann (2009) determine the stationary critical levels for inventory systems operating under different service levels.

Contents lists available atSciVerse ScienceDirect

journal homepage:www.elsevier.com/locate/ijpe

Int. J. Production Economics

0925-5273/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2012.05.026

n

Corresponding author. Tel.: þ886 3 5712121x57305; fax: þ 886 3 5729101. E-mail addresses: hhc@nctu.edu.tw (H.-C. Hung),

isecep@nus.edu.sg (E.P. Chew), iseleelh@nus.edu.sg (L.H. Lee), g0300918@nus.edu.sg (S. Liu).

For dynamic policies, the critical levels may change over time.

Topkis (1968) considers dynamic inventory rationing policy for single period and multiple period systems with zero lead times. A dynamic programming model is proposed in which one period is divided into many small intervals. Topkis also shows that the optimal rationing policy is dynamic. However, he fails to show that the critical level is nonincreasing over time.Evans (1968)and

Kaplan (1969)extend results fromTopkis (1968)and explore two demand classes. Melchiors (2003) considers dynamic rationing policy under an inventory system with a Poisson demand process and an (s, Q) ordering policy in which backordering is not allowed.

Lee and Hersh (1993)consider dynamic rationing policy for an airline seating problem. Teunter and Klein Haneveld (2008)

develop a continuous time approach to determining the dynamic rationing policy for two Poisson demand classes under the assumption that there is no more than one outstanding order. However, its computational results are tractable only for limited settings.Fadiloglu and Bulut (2010)propose a heuristic rationing policy called ‘‘rationing with exponential replenishment flow’’ for continuous-review inventory systems. All except Topkis (1968)

consider only two demand classes. However, the limitation of his approach is that the state spaces grow exponentially large when the number of demand classes increases. Even for two demand classes, the state space can be very large. Moreover, many researchers assume a Poisson distribution.

In this paper, we develop an approximation approach to deriving the dynamic threshold level for inventory systems with multiple demand classes and general demand processes. This approximation approach is based on comparing the marginal costs of accepting and rejecting a demand class when it arrives. It is also assumed that when this demand class is rejected, all future demands from this class will be rejected until the next replenishment arrives. Unlike existing work, this method can deal with general demand processes, and is efficient in solving pro-blems with more than two demand classes. To illustrate the effectiveness of the proposed policy, we conduct numerical analysis. The results show that the proposed policies are close to being optimal under various parameter settings when demand follows a Poisson process. The Poisson process is used because we want to compare our solutions with the optimal solution.

Our paper is organized as follows. InSection 2, we consider a single period system with multiple demand classes. We derive the dynamic critical levels based on the concept of marginal cost. In

Section 3, we consider multiperiod systems with periodic review policy. The rational policy inSection 2is extended for multiperiod systems. InSection 4, numerical studies are conducted to inves-tigate the performance of the proposed approaches. InSection 5, we summarize the results and discuss some possible extensions.

2. Inventory rationing for single period systems

In this section, the inventory rationing problem for a single period system is considered. The goal is to find a good dynamic rationing policy. We first examine the dynamic critical levels for only two demand classes and then extend the results for a general number of demand classes. Then, we develop an approximation method for computing the expected total costs associated with our rationing policy.

2.1. Model formulation

Consider a single period inventory system with a period length of u. There is a single product with demands from K different customer classes. We assume that the demand processes of all customer classes are independent and stationary and that these

demands can be partially satisfied. We let t¼u denote the beginning of the period, we let t¼0 denote the end of the period, and we let X(t) denote the on-hand inventory at the remaining time t A ½0,u. For each unit stored, we assume a holding charge of h per unit of time. At the beginning of the period, we assume that the initial on-hand inventory is given, and is equal to x (i.e., XðuÞ ¼ x). During the period, each customer demand may either be satisfied or rejected according to our rationing policy. The rejected demand is backordered and a backorder cost is applied. We define the backorder cost for class i as

p

iþ ^

p

it, where t A ½0,u is the remaining time to the end of the period. Note that

p

iis a fixed penalty cost to reject a demand from class i, which may represent the loss of customers’ loyalty. Moreover, ^

p

iis the per-unit-of-time cost to hold a demand from class i to the end of the period without loss of goodwill or the order. Without loss of generality, we arrange demand classes in the order of nonincreasing backorder cost. That is,

p

iZ

p

_jand ^

p

_iZ

p

^_jfor demand classes ioj.

At the end of the period (t ¼ 0), we assume that all backorders must be fulfilled. We propose using the remaining inventory to fulfill these backorders first and if this is insufficient, we propose the purchase of additional units to fulfill the remaining back-orders from the open market. If the remaining inventory exceeds the backorders, the surplus is sold at salvage value on the same market. We assume that both the salvage value and the additional purchasing cost on the open market equal c0per unit of product at the end of the period.

To determine the dynamic rationing policy, we need to compute critical levels over time. Define siðtÞ to be the dynamic critical level of class i for the remaining time t A ½0,u. When XðtÞ 4siðtÞ, we satisfy the demand from class i. Otherwise, the demand from class i is rejected. As we have arranged demand classes in the order of nonincreasing backorder cost, we have siðtÞrsjðtÞ for classes ioj. We also define Hðt,XðtÞÞ as the expected cost for the remaining time t. Thus, our objective is to find the rationing policy that minimizes the expected total cost, which can be written as follows:

min siðtÞ i ¼ 1, . . . ,K

Hðu,xÞ: ð1Þ

Without loss of generality, let sn

iðtÞ denote the optimal critical level of class i and let Hn

ðu,xÞ denote the optimal total expected cost. We know that sn

1ðtÞ must be zero because class 1 has the highest backorder cost and there is no advantage in rejecting demand from class 1.

2.2. Dynamic critical levels for systems with two demand classes Consider a single period system with two demand classes ðK ¼ 2Þ and suppose that a customer demand has just arrived when the time remaining is t. If the demand is from class 1, it must be satisfied. If the demand is from class 2, it can either be satisfied or rejected. When this demand is satisfied, the total expected cost at the remaining time t is Hn

ðt,XðtÞ1Þ. If this demand is rejected, the total expected cost at the remaining time t is Hn

ðt,XðtÞÞ þe2ðtÞ, where e2ðtÞ ¼ c0þ

p

2þ ^

p

2t is the backorder cost. If Hn

ðt,XðtÞ1Þ 4Hn

ðt,XðtÞÞ þe2ðtÞ, then this demand should be rejected. Hence, the optimal dynamic critical level of class 2 is sn 2ðtÞ ¼ max XðtÞ9H n ðt,XðtÞÞ þ e2ðtÞHnðt,XðtÞ1Þo0   : ð2Þ

It is difficult to solve Eq. (2) because the closed form for Hn

ðt,XðtÞÞHn

ðt,XðtÞ1Þ cannot be found easily.

Thus, we adopt two important ideas to approximate sn 2ðtÞ. First, if a demand class is rejected, this demand class will be rejected for the remaining time until the end of the period. This is a reason-able approximation because when a demand class is rejected, it

intuitively implies that there is unlikely to be sufficient stock to cater for the less important classes for the remainder of the period.

Second, we use the marginal cost of rejecting the demand to determine the critical level. The marginal cost is computed by assuming that either this and all future demand classes are rejected or that the current demand is accepted but future demands are rejected. When a demand from class 2 arrives, we use this marginal cost to decide whether this demand is accepted or rejected.

Define

D

J2ðt,XðtÞÞ as the marginal cost of rejecting demand class 2 at the remaining time t when on-hand inventory is XðtÞ.

Without loss of generality, we consider two cases: XðtÞ ¼ s (when the demand is rejected at the remaining time t) and XðtÞ ¼ s1 (when the demand is accepted at the remaining time t, but all future demands of this class are rejected).

Fig. 1illustrates the on-hand inventory for both cases, where the solid line represents XðtÞ ¼ s and the dashed line represents XðtÞ ¼ s1. Note that, because we assume that all future demands from class 2 are rejected, the inventory decreases only when the demand of class 1 arrives. The line below the horizontal axis represents the backorder quantities of class 1.

Given on-hand inventory of s at the remaining time t, let

t

sbe the time it takes to run out of inventory. Define D1ðtÞ as the total demand of class 1 from the remaining time t to the end of the period.

The XðtÞ ¼ s case has higher holding costs than does the XðtÞ ¼ s1 case. If D1ðtÞos, the difference in holding costs between the two cases is hUt. If D1ðtÞ Z s, the difference in holding costs between the two cases is

t

sUh.

The XðtÞ ¼ s case has lower backorder cost than does the XðtÞ ¼ s1 case.

If D1ðtÞos, then D1ðtÞrs1. Thus, there is no backorder for both cases and there is a difference of one unit of product. This means that there is an extra salvage value, c0, for the XðtÞ ¼ s case. If D1ðtÞ Zs, the sth demand of class 1 can be satisfied in the XðtÞ ¼ s case. However, the sth demand of class 1 cannot be satisfied in the XðtÞ ¼ s1 case. Thus, this demand class has an extra backorder cost of c0þ

p

1þ ^

p

1Uðt

t

sÞin the XðtÞ ¼ s1 case.

In fact, these cost differences between the XðtÞ ¼ s and XðtÞ ¼ s1 cases are the marginal costs of rejecting demand class 2 at the remaining time t. Thus

D

J2ðt,sÞ ¼ tUhUPðD1ðtÞosÞþhUEð

t

s9D1ðtÞ Z sÞUPðD1ðtÞ Z sÞ  c0þ

p

1þ ^

p

1UE½ðt

t

sÞ9D1ðtÞ Z s   UPðD₁ðtÞ Z sÞ c0UPðD₁ðtÞosÞ ¼hUt½tUð ^

p

1þhÞ þ

p

1UPðD1ðtÞ Z sÞ þ ð ^

p

1þhÞUE½

t

s9D1ðtÞ Z sUPðD1ðtÞ Z sÞc0, ð3Þ

where PðD1ðtÞosÞ is the probability of D1ðtÞos and PðD1ðtÞ Z sÞ is the probability of D1ðtÞ Zs.

Now, we demonstrate the monotonicity property of

D

J2ðt,sÞ. Lemma 1. For the remaining time t and s Z0,

D

J2ðt,sÞ is nonde-creasing in s.

Proof. See Appendix.&

After deriving the marginal cost of rejecting demand class 2, we use the above two ideas to approximate sn

2ðtÞ. Define sa

2ðtÞ ¼ max s9

D

J2ðt,sÞ þ e2ðtÞo0

 

: ð4Þ

We show the relationship between sa

2ðtÞ and

D

J2ðt,sÞ.

Theorem 1. For demand class 2, there exists a unique critical level, sa

2ðtÞ, at the remaining time t such that

D

J2ðt,sÞ þ e2ðtÞ Z0 f or s 4 sa2ðtÞ, and

D

J2ðt,sÞ þ e2ðtÞo0 f or srsa2ðtÞ:

Proof. Theorem 1holds fromLemma 1. &

Theorem 1 shows that once the on-hand inventory drops below the unique critical level, sa

2ðtÞ, all future demands from demand class 2 will be rejected until the next replenishment arrives. This result is consistent with our assumptions. We now establish the monotonicity property of sa

2ðtÞ fromTheorem 1and Eqs. (3) and (4).

Theorem 2. For

p

1¼

p

2¼0 , sa2ðtÞ is nondecreasing in the remain-ing time t.

Proof. See Appendix. &

Theorem 2 shows that sa

2ðt1Þ Zsa2ðt2Þ for t1Zt2 when

p

1¼

p

2¼0. This means that when the remaining time becomes smaller, we only need to reserve fewer inventories for class 1. 2.3. Dynamic critical levels for systems with more than two demand classes

We now consider a single period system with more than two demand classes. We assume that a demand arrives at the remaining time t. If the demand is from class 1, then it must be satisfied. Otherwise, it can either be satisfied or rejected. Similarly to the case of two demand classes in the previous subsection, we generalize sa

2ðtÞ and approximate the dynamic critical level of class m by sa

mðtÞ. Hence, define sa

mðtÞ ¼ max XðtÞ9

D

Jmðt,XðtÞÞþ emðtÞo0, ð5Þ where

D

Jmðt,XðtÞÞ is the marginal cost of rejecting demand class m at the remaining time t.

To compute the critical levels for all demand classes, we adopt a sequential approach; i.e., we use the dynamic critical levels for demand classes 2 to m 1 to compute the dynamic critical level for demand class m. Note that from Subsection 2.1, we know that the critical level of class 1 is zero.

Suppose we want to compute the critical level for demand class m, assuming that the dynamic critical levels for demand classes 2, 3,y,m  1 are known. Similarly, we assume that if a demand class is rejected, that demand class will be rejected for the remainder of the period. Without loss of generality, we consider two cases: XðtÞ ¼ s (when the demand of class m is rejected at the remaining time t) and XðtÞ ¼ s1 (when the demand of class m is accepted at the remaining time t, but all future demands of this class will be rejected).Fig. 2illustrates a realization of on-hand inventory for both cases, where the solid line represents XðtÞ ¼ s and the dashed line represents XðtÞ ¼ s1.

s t ₀ Remaining time On-hand inventory s −1s End of period 0

Fig. 1. On-hand inventory after rejecting some order from class 2 at remaining time t.

This figure shows that, to compute the marginal cost difference between the two cases, we should consider two different situa-tions. In the first situation, all demands from class m 1 are accepted in both cases, and in the second situation, demands from class m  1 are rejected for the XðtÞ ¼ s1 case. In the first situation, the only cost difference is that in inventory holding costs. In the second situation, the cost difference is a weighted sum of the differences in inventory holding costs and backorder costs.

To compute the critical level for demand class m at the remaining time t, we assume that demand class m is rejected at the remaining time t. This implies that any demand class i, where i 4m, will be rejected. A demand class i, where iom, may be rejected in the future if the inventory level is below the respective dynamic critical level. To determine the conditions under which situation 1 applies, we must identify the time at which demand class m  1 is first rejected given XðtÞ ¼ s1; we denote this time as tm1

s1. This means that tm1s1 is the maximum amount of time remaining when Xðtm1

s1Þis equal to the dynamic critical level of demand class m1. When the remaining time is between t and tm1

s1, situation 1 applies. Hence, situation 2 applies when the remaining time is between tm1

s1 and 0.

For subsequent derivations, we must define tm1

s , which is the maximum amount of time remaining when Xðtm1

s Þis equal to the dynamic critical level of demand class m1. Note that tm1

s rtm1s1. Now, we examine the cost differences between the XðtÞ ¼ s case and the XðtÞ ¼ s1 case under two different situations; i.e., the remaining time intervals ½tm1

s1,t and ½0,t m1 s1Þ. (i) Situation 1: Remaining time interval ðtm1

s1,t

The difference in costs between the two cases in this interval is one unit of inventory cost. Thus, the cost difference is hUðttm1

s1Þ.

(ii) Situation 2: Remaining time interval ½0,tm1 s1

To derive the cost difference between the two cases, we assume that the approximated dynamic critical level of demand class m1 remains unchanged in the remaining time interval ½tm1 s ,tm1s1. That is, sam1ðt m1 s1Þ ¼sam1ðt m1 s Þ. Thus, for the XðtÞ ¼ s1 case, we have Xðtm1

s1Þ ¼sam1ðtm1s1 Þ ¼sam1ðtm1s Þ. For the XðtÞ ¼ s case, we have Xðtm1

s1Þ ¼sam1ðtm1s1Þ þ1 ¼ sam1 ðtm1

s Þ þ1.

Consider the XðtÞ ¼ s case. Demand class i, where i Zm1, will be rejected when the remaining time is less than tm1

s . This is because Xðtm1

s Þis less than or equal to the dynamic critical level of demand class m1.

Similarly, for the XðtÞ ¼ s1 case, when the remaining time is less than tm1

s1, demand class i, where i Z m1, will be rejected because Xðtm1

s1Þis less than or equal to the dynamic critical level of demand class m1.

Hence, in the interval ½tm1

s ,tm1s1, there is one extra unit of inventory in the XðtÞ ¼ s case. The marginal cost over the ½0,tm1

s  interval depends on the realization of the demand class at the remaining time tm1

s . (Note that by definition, this demand class cannot come from classes m and above because all these will be rejected and, thus, there will be no corresponding fall in inventory for the XðtÞ ¼ s case.) If the demand comes from classes 1 to m 2, that demand will be accepted in both cases, and so the marginal cost will be

D

Jm1ðtm1s ,sam1ðtm1s1ÞÞ. This is because the starting inventory at the remaining time tm1

s for the XðtÞ ¼ s case is always one more than the starting inventory for the XðtÞ ¼ s1 case, and it is equal to the critical level for demand class m 1; i.e., sa

m1ðtm1s1Þ. However, if the demand comes from class m 1, this demand will be accepted for the XðtÞ ¼ s case but will be rejected for the XðtÞ ¼ s1 case. Hence, the cost difference is em1 tm1s

  .

Thus, the difference in average costs between the two cases in situation 2 is hUðtm1 s1tm1s Þ þ

D

Jm1ðtm1s ,sam1ðtm1s1ÞÞU Pm2 i ¼ 1

l

i Pm1 i ¼ 1

l

i ! em1ðtm1s ÞU

l

m1 Pm1 i ¼ 1

l

i ! , ð6Þ

where ðPm2i ¼ 1

l

iÞ=ðPm1i ¼ 1

l

iÞ is the probability that the next demand is from class 1 to m  2, and

l

m1=ðPm1i ¼ 1

l

iÞ is the probability that the next demand is from class m1.

By combining situations 1 and 2, given the realization of tm1 s , the total marginal cost of rejecting demand class m at the remaining time t when the inventory level is s is

D

Jmðt,s9tm1s Þ ¼hUðttm1s Þ þ

D

Jmðtm1s ,sam1ðtm1s ÞÞU Pm2 i ¼ 1

l

i Pm1 i ¼ 1

l

i ! em1ðtm1s ÞU

l

m1 Pm1 i ¼ 1

l

i ! : ð7Þ Note that tm1

s is a random variable that depends on the on-hand inventory XðtÞ, the dynamic critical level sa

m1ðtÞ, and the demand arrival process. Let pðUÞ denote the probability density function of this random variable. It is possible that Xð0Þ 4sa

m1ð0Þ. If Xð0Þ 4 sa

m1ð0Þ, then inventory always exceeds the dynamic critical level of demand class m1. In this case, the marginal cost of rejecting demand from class m1 is ht c0. By defining P ¼ 1R0tpðtm1s Þdtm1s , which is the probability of Xð0Þ 4 sam1ð0Þ, we have

D

Jmðtc,sÞ  Z t

0

½

D

Jmðt,s9tm1s ÞUpðtm1s Þdtm1s þPUðhtc0Þ: ð8Þ Eq. (8) can be reduced to Eq. (4) when m¼2.

Because sa

m1ðtm1s Þ ¼max s9DJm1ðtsm1,Xðtm1s ÞÞ þem1ðtm1s Þo0

 

, it follows that

D

Jm1

X ðt,sÞ þ em1ðtÞ is approximately zero. More-over, because of our assumption that the dynamic critical level for class m 1 remains unchanged in the remaining time interval ½tm1

s ,tm1s1 (i.e., Xðtm1s Þ sam1ðtÞ), we can approximate

D

Jm1 ðtm1

s ,sam1ðt m1

s ÞÞ by em1ðtm1s Þ. Hence, Eq. (8) can be approxi-mated by

D

Jmðt,sÞ  Z t 0 ½hUðttm1 s Þem1ðtm1s ÞUpðtm1s Þdtm1s þPðhtc0Þ: ð9Þ From Eqs. (5) and (9), we can generate the approximate optimal dynamic critical level of class m from sa

2ðtÞ,sa3ðtÞ,y,sam1ðtÞ: Therefore, all dynamic critical levels can be generated sequen-tially starting from the lowest demand class index.

0 Remaining time Inventory s s −1 t

Critical level of class m−1 End of period

m−1 s−1 t

We propose the following algorithm to generate the dynamic critical levels for all demand classes.

Step 1: Set the optimal dynamic critical level of class 1 to be zero: sn

1ðtÞ ¼ 0.

Step 2: Considering only classes 1 and 2, useTheorem 1in

Section 2.2to estimate sa

2ðtÞ, t A ½0,u. Set m¼3. Step 3: Given sa

iðtcÞ, iA 2,:::,m1f g, use Eqs. (5) and (9) to find sa

mðtÞ, t A ½0,u. Set m¼mþ 1.

Step 4: Stop if mZK þ1. Otherwise, go to Step 3.

2.4. Expected total cost

In this subsection, we estimate the expected total cost by using an iterative method based on

Hðt,0Þ ¼X K i ¼ 1

p

iþ ^

p

iðtÞ, ð10aÞ Hðt,xÞ  Hðt,0Þ þX x j ¼ 1

D

J2ðt,jÞ for x A ð0,sa2ðtÞ, ð10bÞ Hðt,xÞ  Hðt,sa mðtÞÞ þ Xx j ¼ sa mðtÞ þ 1

D

Jm þ 1ðt,jÞ for x A ðsa mðtcÞ,sam þ 1ðtcÞ and 2rmoK, ð10cÞ Hðt,xÞ  Hðt,sa KðtÞÞ þ Xx j ¼ sa KðtÞ þ 1

D

JK þ 1ðt,jÞ for x A ðsaKðtcÞ,1Þ: ð10dÞ

Eq. (10a) represents the case in which there is no initial on-hand inventory, i.e., XðtÞ ¼ x ¼ 0. In this case, all demands should be rejected. The total cost will be equal to the sum of the total expected backorder costs of all the classes.

Eq. (10b) represents the case in which the initial on-hand inventory is below the critical level of class 2. In this case, only class 1 demand is accepted, and so the expected total cost can be approximated byPxj ¼ 1

D

J2ðt,jÞ, where

D

J2ðt,jÞ is given by Eq. (4). Eqs. (10c) and (10d) represent cases in which the initial on-hand inventory is between the critical levels of class m and class mþ1, where m is between two and K 1 and the initial on-hand inventory is above the critical level of class K. Similarly, the expected total cost can be approximated by the sum of the marginal costs given in Eq. (9).

3. Inventory rationing in multiperiod systems

In this section, we consider dynamic inventory rationing models for multiperiod systems with general demand process in which backordering is allowed.

3.1. Notation and model formulation

Consider an inventory system with a single type of product and K demand classes for an infinite time horizon. We assume that the demand process, the backorder cost, and the holding cost are the same as those of the single period model inSection 2. We adopt a policy of dynamic critical level rationing with periodic review ordering (R,S).

Under this ordering policy, the ordering opportunities, m¼0, 1, 2,y, occur at fixed intervals of time and the amount of time between two successive order opportunities is u, where R ¼ u. In addition, the deterministic lead time taken to replenish orders is L. The order placed at the mth order opportunity will arrive at time

lm¼mu þ L. The time interval (lm,lm þ 1) is termed the mth replen-ishment period. When the mth replenreplen-ishment order arrives at time lm, backorders are instantaneously fulfilled according to the follow-ing mechanism. Backorders from the most important (lowest index) demand class are always fulfilled first. Second, if replenishment is sufficient, less important demand classes are fulfilled.

With backorders fulfilled, we define ymas the net inventory and Bm as the remaining backorders at time lm. Note that Bm¼ bm1,. . .,b

m K

 

is the vector of backorders for all demand classes, where bm_i Z0 is the amount of backorders for demand class i. Under the above backorder fulfillment mechanism, ymmay be negative if some backorders remain unfulfilled. If ymZ0, then bmi ¼0 for all i. Otherwise, there is no on-hand inventory and PK

i ¼ 1b m

i ¼ ym. Thus, ðym,BmÞis the state variable at the begin-ning of the mth replenishment period and its distribution depends on the rationing policy and the base stock level S. Let PS,nðym,BmÞ denote the probability distribution of the state vari-able (ym,Bm) at the beginning of the mth replenishment period subject to the rationing policy

n

and the base stock level S.

Similarly, let Cnðym,BmÞ be the expected inventory cost incurred in the mth replenishment period subject to the state variable ðym,BmÞand the rationing policy

n

. For ymo0, Cnðym,BmÞ consists of backorder costs from Bm and from new backorders arising in this period.

For the multiperiod system, define ACðS,

n

Þ as the expected average cost incurred during a period in which the base stock level is S and the prevailing dynamic rationing policy is

n

. We can now model the optimization problem for the dynamic critical level rationing policy as follows:

min S,n ACðS,

n

Þ ¼Mlim-1 1 M X M1 m ¼ 0 X ym,Bm

Cnðym,BmÞUPS,nðym,BmÞ: ð11Þ The goal is to minimize the expected average cost in Eq. (11), which consists of inventory holding costs and backorder costs. Note that all rejected or unfulfilled orders are backlogged. Hence, we can ignore the variable ordering cost and set the salvage value to zero: c0¼0.

3.2. A solution to the optimization problem

In solving the rationing problem given by Eq. (11), the main issue is that the probability distribution for backorders, Bm, depends on the rationing policy

n

and the base stock level S. It is difficult to derive a closed form solution for the probability mass function for Bm. Hence, we propose a heuristic method to solve the rationing problem given by Eq. (11).

InSection 2, we proposed an approximate dynamic rationing policy for a single period problem. We now propose a rationing policy for a multiperiod problem. Let

n

abe the rationing policy for the multiperiod system, where each stock ration in each period is determined according to our corresponding approximate dynamic critical level. The goal is to locally minimize inventory costs in each period, ignoring subsequent periods.

We now develop an approximate expression for ACðS,

n

aÞ. Given the base stock level S and the rationing policy

n

a, the probability mass function of the state variable (ym,Bm) is PS,naðym,BmÞ ¼PðDL¼SymÞ, for ymZ0:

X Bm j j ¼ y

PS,naðym,BmÞ ¼PðDL¼SymÞ for ymo0, ð12Þ where Bj mj ¼PKi ¼ 1b

m

i and DLis the demand that arrives during the lead time L. Note that the probability mass functions in Eq. (12) are independent of m and

n

a.

Recall that Cnaðy,BÞ is the expected inventory cost incurred in the mth replenishment period subject to the state variable ðym,BmÞ

and the rationing policy

n

a. We now approximate Cnaðym,BmÞfor two different cases: ymZ0 and ymo0.

For ymZ0:

Cnaðym,BmÞ Hðu,ymÞ, ð13Þ where Hðu,ymÞ is the expected total cost in a single period of length u, given zero salvage cost (c0¼0) and initial on-hand inventory of ym. For ymo0: Cnaðym,BmÞ Hðu,0Þ þ XK i ¼ 1 bmi U^

p

iUu, ð14Þ where Cnaðym,BmÞconsists of backorder costs from Bm and new backorders arising during the period.

In a multiperiod system, one can preserve inventory for future use for more important demand classes by rejecting other demand classes during each period. It is assumed that backorders from more important demand classes are fulfilled using the mechanism described in the previous section. Therefore, for the mth replenishment period, given remaining backorders of ym, on average, there should be more backorders from less important demand classes. Moreover, in practice, ymo0 is unlikely. Thus, for ymo0, we can rewrite Eq. (14) as

Cnaðym,BmÞ Hðu,0ÞymU

p

^_KUu, ð15Þ where we assume that all backorders in ym are from the least important demand class K.

Based on Eqs. (12)–(15), we can approximate the average cost as ACðS,

n

aÞ  1 M X M1 m ¼ 0 XS ym¼0

Hðu,ymÞUPðDL¼SymÞ 0

@

þ X

1 ym¼ 1

ðHðu,0ÞymU^

p

KUuÞUPðDL¼SymÞ ! ¼ X S y ¼ 0 Hðu,yÞUPðDL¼SyÞ þ X 1 y ¼ 1

ðHðu,0ÞyU ^

p

KUuÞUPðD_L¼SyÞ:

For a given rationing policy

n

a, all replenishment periods are the same. By replacing ymwith y, we can rewrite Eq. (11) as min S ACðS,

n

aÞ minS XS y ¼ 0 Hðu,yÞUPðDL¼SyÞ þ X 1 y ¼ 1

ðHdyðu,0ÞyU ^

p

KUuÞUPðDL¼SyÞ: ð16Þ

4. Numerical study

In this section, we conduct numerical studies to investigate the effectiveness of our proposed methods. We develop bounds to evaluate the quality of our proposed solutions under different scenarios when the demands follow a Poisson process. These bounds are valid for both single and multiperiod problems.

We first consider the single period problem. Let

n

adenote the dynamic critical level implied by the proposed method and let

n

n denote the optimal dynamic critical level, which can be derived when the demand follows a Poisson process (Chew et al., 2011). Also, let Hnðu,xÞ denote the expected total cost incurred during

a single period of length u under rationing policy

n

starting with

on-hand inventory of x. We define the relative error for the expected total cost as

D

Hð

n

a,xÞ ¼

Hnaðu,xÞHnnðu,xÞ Hnnðu,xÞ

100%,

where

D

Hð

n

a,xÞ is determined by

n

aand x.

In this numerical study, we consider a base case in which there are three demand classes. In this base case, for the length of the period, we choose u ¼ 0:1, we choose the holding cost h ¼ 1, and we choose the salvage value c0¼0. For the backorder costs, we assume

p

1¼

p

2¼

p

3¼0 and ^

p

3¼1:5, where ^

p

1: ^

p

2: ^

p

3¼20 : 5 : 1:5. For the demand arrival rates, we let

l

1¼

l

2¼

l

3¼300. We also varied some parameters; different parameter settings are listed inTable 1.

For each new parameter setting, we compute the optimal dynamic critical level and the approximate dynamic critical level of the proposed method. Then, we use simulation to estimate their costs. To ensure the accuracy of the estimated costs, we repeat the simulation 20,000 times for each parameter setting. The results are illustrated inFigs. 3 and 4and reported inTable 1.

Table 1

Worst relative error under different operational conditions.

Factors Parameters DHðna,xnÞ (%) xn u l1:l2:l3 p^1: ^p2: ^p3 Base case 0.1 1:1:1 20:5:1.5 0.16 60 Changing ratios of backorder costs 0.1 1:1:1 5:2:1.5 0.03 63 10:3:1.5 0.08 45 40:8:1.5 0.50 53 100:10:1.5 1.31 55 Changing ratios of arrival

rates 0.1 1:2:3 20:5:1.5 0.24 45 1:3:6 0.24 38 1:10:100 0.29 10 3:2:1 0.10 58 6:3:1 0.08 58 100:10:1 0.01 70 3:6:1 0.23 58 10:100:1 0.37 43 1:6:3 0.32 30 1:100:10 0.44 20

Changing time horizon 0.05 1:1:1 20:5:1.5 0.85 32

0.15 0.13 80

0.2 0.10 118

0.3 0.06 160

Fig. 3shows both optimal and proposed dynamic critical levels for classes 2 and 3 in the base case. Note that both the optimal and proposed dynamic critical level for class 1 is zero. Curves

n

n_{_2} and

n

n_3 represent the optimal dynamic critical levels for classes 2 and 3, respectively. Curves

n

a_2 and

n

a_3 represent the proposed dynamic critical levels for classes 2 and 3, respectively. The time horizon u ¼ 0:1 is divided into 900 intervals.Fig. 3shows that the proposed critical levels are very close to the optimal critical levels.

Fig. 4illustrates the relative error,

D

Hð

n

a,xÞ, for different initial levels of on-hand inventory of x in the base case. Note that

D

Hð

n

a,xÞ is no higher than 0.16% and may be monotonic in x. Moreover,

D

Hð

n

a,xÞ is close to zero when x is either very small or very large. When x is large, there is always enough initial inventory to satisfy demands of all classes. Thus, the relative error induced by the proposed policy is negligible. Similarly, when x is small, most demand classes cannot be satisfied for any policy in a single period problem. Thus, the best strategy is to reserve all inventory for demand class 1. Hence, the relative error induced by the proposed policy is again negligible.

Define

D

Hð

n

a,xnÞ ¼maxx Z 0

D

Hð

n

a,xÞ as the worst relative error.

Table 1shows how several factors affect the worst relative error. In all cases reported inTable 1,

D

Hð

n

a,xnÞis small. Few factors significantly affect the worst relative error.

The factor that most significantly affects

D

Hð

n

a,xnÞ is the backorder cost ratio. As backorder costs for the more important classes increase,

D

Hð

n

a,xnÞ increases. This follows from our assumption that if some demand classes are rejected, these demand classes will be rejected for the remainder of the time period.

Note that when changing the ratios of arrival rates, we fix the total arrival rate at 900. Thus, the arrival rate ratio of 1:2:3 is implemented as 150:300:450 and the ratio of 1:3:6 is implemen-ted as 100:300:600. We approximate the arrival rate ratio of 1:10:100 by 9:81:810.

The numerical study shows that the proposed dynamic critical levels are very close to the optimal critical levels and that the relative errors for expected total cost are also small. However, when the number of demand classes increases, we may expect the relative errors to increase. This is because we determine the dynamic critical levels sequentially.

Note that the derived bounds reported inTable 1are also valid for the multiperiod problem. This is because the results for the simulated cases reported in Table 1 are based on a starting inventory that represents the worst-case scenario and because the probability of having negative inventory at the beginning of each period is negligible (around 10–4_{). Given that we have} considered extreme cases, in which the penalty cost is as much as 100 times higher than the inventory cost, these results are reasonable.

5. Conclusions and extensions

In this paper, we developed a heuristic approach to computing the dynamic critical levels for systems with general demand arrival processes. We first considered a single period problem and then extended this to a multiperiod system. The heuristic approach is based on two ideas. The first idea is that any demand class that is rejected in one period will be rejected for the remainder of the period. The second idea is that dynamic critical levels can be derived based on the difference in the marginal costs of accepting and rejecting a demand class. These two ideas have enabled us to deal with more general demand processes. Our numerical study shows that the outcomes generated by our proposed approach compare favorably with the optimal solutions under most parameter settings.

For future work, we will consider relaxing some of our model assumptions. For example, we could allow a demand from a class that is initially rejected to be accepted in the future. We could also consider backorders being satisfied before replenishment orders arrive.

Appendix

Proof of Lemma 1. From Eq. (3), we have

D

J2ðt,s þ 1Þ ¼ hUtc0½tUð ^

p

1þhÞ þ

p

1UPðD1ðtÞ Z s þ 1Þ

þ ð ^

p

1þhÞUE½

t

s þ 19D1ðtÞ Zs þ 1UPðD1ðtÞ Z s þ1Þ, ðA:1Þ and

D

J2ðt,sÞ ¼ hUtc0½tUð ^

p

1þhÞ þ

p

1UPðD1ðtÞ Z sÞ

þ ð ^

p

1þhÞUE½

t

s9D1ðtÞ Z sUPðD1ðtÞ Z sÞ: ðA:2Þ

From (A.1) and (A.2) we have

DJ2ðt,s þ 1ÞDJ2ðt,sÞ ¼ ½tUð ^

p

1þhÞ þ

p

1UPðD1ðtÞ ¼ sÞ

þ ð ^

p

1þhÞU E½

t

s þ 19D1ðtÞ Z s þ 1UPðD1ðtÞ Z s þ 1Þ E½

t

s9D1ðtÞ Z sUPðD1ðtÞ Z sÞ  : ðA:3Þ Note that E½

t

s9D1ðtÞ ZsUPðD1ðtÞ ZsÞ ¼X 1 i ¼ s

E½

t

s9D1ðtÞ ¼ iUPðD1ðtÞ ¼ iÞ

¼E½

t

s9D1ðtÞ Z s þ1UPðD1ðtÞ Zs þ 1Þ þ E½

t

s9D1ðtÞ ¼ sUPðD1ðtÞ ¼ sÞ: ðA:4Þ By substituting (A.4) into (A.3), we obtain

D

J2ðt,s þ 1Þ

D

J2ðt,sÞ ¼ ½tUð ^

p

1þhÞ þ

p

1UPðD1ðtÞ ¼ sÞ þ ð ^

p

1þhÞU E½

t

s þ 19D1ðtÞ Zs þ 1 UPðD₁ðtÞ Zs þ 1ÞE½

t

_s9D₁ðtÞ Zs þ 1 UPðD₁ðtÞ Zs þ 1Þð ^

p

1þhÞUE½

t

s9D1ðtÞ ¼ s UPðD₁ðtÞ ¼ sÞ ¼ ½tUð ^

p

₁þhÞ þ

p

₁ð ^

p

1þhÞ  UE½

t

s9D1ðtÞ ¼ sUPðD1ðtÞ ¼ sÞ þð ^

p

1þhÞ UE½

t

_{s þ 1}9D₁ðtÞ Z s þ1UPðD₁ðtÞ Zs þ 1Þ E½

t

s9D1ðtÞ Z s þ1UPðD1ðtÞ Zs þ 1Þ: ðA:5Þ

For E½

t

s9D1ðtÞ ¼ srt, we have ½tUð ^

p

1þhÞ þ

p

1ð ^

p

1þhÞUE½

t

s9D1ðtÞ ¼ s

¼ ð ^

p

1þhÞU tE½

t

s9D1ðtÞ ¼ sþ

p

1 Z0:

Thus, E½

t

s þ 19D1ðtÞ Z s þ 1UPðD1ðtÞ Z s þ 1Þ 4E½

t

s9D1ðtÞ Z s þ 1UPðD1 ðtÞ Z s þ1Þ. 0.00% 0.05% 0.10% 0.15% 0.20% 0 20 40 60 80 100 120 140 Initial inventory x Relative error ΔH (νa ,x )

From (A.5), we have

D

J2ðt,s þ1Þ

D

J2ðt,sÞ 4 0: &

Proof of Theorem 2. Lemma 1 implies that

D

J2ðt,sÞ is nonde-creasing in s. Thus,

D

J2ðt,sÞ þ e2ðtÞ is nondecreasing in s.

Next, we show the continuity of

D

J2ðt,sÞ þe2ðtÞ. Given s 40, J2ðt,sÞ is a continuous function of remaining time t. Moreover, e2ðtÞ ¼ c0þ

p

2þ ^

p

2t is a continuous function of remaining time t. Thus,

D

J2ðt,sÞ þe2ðtÞ is a continuous function of remaining time t.

Given a remaining time of t ¼ 0þ

, no more orders will arrive in this arbitrarily small remaining time interval. Thus, all orders from demand class 2 can be accepted at a remaining time of t ¼ 0þ

. This implies:

D

J2ð0 þ

,sÞ þ e2ð0þÞ40 ðA:6Þ

For given so1, if the remaining time t ¼ 1, then we must reject demands from class 2 because of limited on-hand inventory and infinite remaining time. Thus:

D

J2ð1,sÞ þ e2ð1Þo0 ðA:7Þ

From (A.6) and (A.7), and the continuity of

D

J2ðt,sÞ þe2ðtÞ, there exists some remaining time t0such that

D

J2ðt0,sÞ þ e2ðt0Þ ¼0 for given on-hand inventory s. Hence, define ts

2¼min t09

D

J2ðt0,sÞ þ 

e2ðt0Þ ¼0g.

From the continuity property and the definition of ts

2, we have

D

J2ðt,sÞ þ e2ðtÞo0 for t A½ts2,1Þ ðA:8Þ and

@½

D

J2ðts2,sÞ þe2ðts2Þ

@t r0: ðA:9Þ

Next, we show that inequality (A.9) holds for t A ½ts

2,1Þ. We know that

t

s is a continuous random variable. Let pð

t

sÞ be its probability density function. For the cumulative distribution function, we have FðtÞ ¼ Pð

t

srtÞ ¼ PðD1ðtÞ Z sÞ: ðA:10Þ Thus dPðD1ðtÞ Z sÞ dt ¼pð

t

s¼tÞ Z0 ðA:11Þ and dE½

t

s9D1ðtÞ ZsUPðD1ðtÞ ZsÞ dt ¼ d dt Z t 0

t

sUpð

t

sÞUd

t

s¼tUpð

t

s¼tÞ: ðA:12Þ From Eqs. (A.2) and (A.12), we have

@½

D

J2ðt,sÞ þ e2ðtÞ

@t ¼hð ^

p

1þhÞUPðD1ðtÞ Z sÞtUð ^

p

1þhÞUpð

t

s¼tÞ 

p

1Upð

t

_s¼tÞ þ ð ^

p

1þhÞUtUpð

t

s¼tÞ þ ^

p

2 ¼hð ^

p

1þhÞUPðD1ðtÞ Z sÞ

p

1Upð

t

s¼tÞ þ ^

p

2 ¼hð ^

p

1þhÞUPðD1ðtÞ Z sÞ þ ^

p

2,

which holds because

p

1¼

p

2¼0.

From (A.10), PðD1ðtÞ ZsÞ ¼ FðtÞ is a nondecreasing function of t. Thus, for any remaining time t A ½ts

2,1Þ @½

D

J2ðt,sÞ þ e2ðtÞ

@t r

@½

D

J2ðts2,sÞ þe2ðts2Þ

@t r0: ðA:13Þ

From (A.8) and (A.13), it follows that

D

J2ðt,sÞ þe2ðtÞo0 and that

D

J2ðt,sÞ þe2ðtÞ is a nonincreasing function of remaining time t A ðts

2,1Þ. Given that sa2ðtÞ ¼ max s9

D

J2ðt,sÞ þe2ðtÞo0 from Eq. (4), sa

2ðtÞ is nondecreasing in remaining time t. &

References

Arslan, H., Graves, S.C., Roemer, T., 2007. A single-product inventory model for multiple demand classes. Management Science 53 (9), 1486–1500. Cardo´s, M., Babiloni, E., 2011. Exact and approximate calculation of the cycle

service level in periodic review inventory policies. International Journal of Production Economics 131 (1), 63–68.

Chew, E.P., Lee, L.H., Liu, S., Dynamic rationing and ordering policies for multiple demand classes. OR Spectrum, http://dx.doi.org/10.1007/s00291-011-0239-2. Cohen, M.A., Kleindorfer, P., Lee, H.L., 1988. Service constrained (s, S) inventory systems with priority demand classes and lost sales. Management Science 34 (4), 482–499.

Dekker, R., Kleijn, M.J., Rooij, P.J.D., 1998. A spare parts stocking system based on equipment criticality. International Journal of Production Economics 56–57, 69–77.

Deshpande, V., Cohen, M.A., Donohue, K., 2003. A threshold inventory rationing policy for service-differentiated demand classes. Management Science 49 (6), 683–703.

Evans, R.V., 1968. Sales and restocking policies in a single item inventory system. Management Science 14 (7), 463–472.

Fadiloglu, M.M., Bulut, O¨ ., 2010. A dynamic rationing policy for continuous-review inventory systems. European Journal of Operational Research 202 (3), 675–685.

Gayon, J.P., Benjaafar, S., de Ve´ricourt, F., 2004. Stock rationing in a multi-class make-to-stock production system. In: Proceedings of the MSOM. Eindhoven. Ha, A.Y., 1997a. Inventory rationing in a make-to-stock production system with

several demand classes and lost sales. Management Science 43 (8), 1093–1103.

Ha, A.Y., 1997b. Stock rationing policy for a make-to-stock production system with two priority classes and backordering. Naval Research Logistics 44 (5), 457–472.

Ha, A.Y., 2000. Stock rationing in an M/Ek/1 make-to-stock queue. Management

Science 46 (1), 77–87.

Kaplan, A., 1969. Stock rationing. Management Science 15 (5), 260–267. Kleijn, M.J., Dekker, R., 1998. An Overview of Inventory Systems with Several

Demand Classes. Technical Report 9838/A. Econometric Institute, Erasmus University, Rotterdam, The Netherlands.

Lee, T.C., Hersh, M., 1993. A model for dynamic airline seat inventory control with multiple seat bookings. Transportation Science 27 (3), 252–265.

Melchiors, P.M., Dekker, R., Kleijn, M.J., 2000. Inventory rationing in an (s,Q) inventory model with lost sales and two demand classes. Journal of the Operational Research Society 51 (1), 111–122.

Melchiors, P.M., 2003. Restricted time-remembering policies for the inventory rationing problem. International Journal of Production Economics 81–82, 461–468.

M ¨ollering, T.K., Thonemann, U.W., 2009. An optimal constant level rationing policy under service level constraints. OR Spectrum 32 (2), 319–341.

Moon, I., Kang, S., 1998. Rationing policies for some inventory systems. Journal of the Operational Research Society 49 (5), 509–518.

Nahmias, S., Demmy, W., 1981. Operating characteristics of an inventory system with rationing. Management Science 27 (11), 1236–1245.

Teunter, R.H., Klein Haneveld, W.K., 2008. Dynamic inventory rationing strategies for inventory systems with two demand classes, Poisson demand and back-ordering. European Journal of Operational Research 190 (1), 156–178. Topkis, D.M., 1968. Optimal ordering and rationing policies in a non-stationary

dynamic inventory model with n demand classes. Management Science 15 (3), 160–176.