Cost analysis of an (S−1,S) inventory system with two demand classes and rationingby K. P. Sapna Isotupa

Annals of Operations Research


Decision Sciences (all) / Management Science and Operations Research


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Inventory rationing for a system with heterogeneous customer classes

Paul Enders, Ivo Adan, Alan Scheller-Wolf, Geert-Jan van Houtum


Ann Oper Res

DOI 10.1007/s10479-013-1407-3

Cost analysis of an (S − 1,S) inventory system with two demand classes and rationing

K.P. Sapna Isotupa © Springer Science+Business Media New York 2013

Abstract In this paper we analyze a continuous review, lost sales (S − 1, S) inventory system with two demand classes—high priority and low priority. We compare two different policies—one where the two classes of customers are treated exactly alike and the other where a threshold rationing policy is used. We prove that under certain conditions there is a sub-optimal rationing policy which yields a lower cost for the supplier and higher service levels for both the high priority and low priority customers than the optimal policy where the two customers are treated alike.

Keywords Markov chain · Stochastic processes · Inventory · Multiple classes of customers · Rationing · Cost comparison 1 Introduction

Suppliers often need to provide different service levels to different customers based on their contracts. For example, a supplier may have 25 different customers who have fill rate requirements between 85 % and 98 %. Another example is a case where different customers pay different prices for the same product and so the supplier has an incentive to meet more of the demand of the customer paying a higher price than a customer who pays a lower price. There are many other situations where it would be financially beneficial for a supplier to provide different levels of service to different customers. An often used method to deal with these types of situations is to group customers into a finite number of categories. The service levels for all customers within a category are the same and are different from the service levels for customers external to that category. The supplier’s objective is to meet the service level requirements of the different categories while keeping costs low.

The literature on inventory systems with multiple classes of customers was very sparse before 1990s. However, assigning different amount of product to different classes had been

K.P. Sapna Isotupa ()

School of Business & Economics, Wilfrid Laurier University, Waterloo, Ontario N2L 3C5, Canada e-mail:

Ann Oper Res well studied in the yield management literature and Kimes (1989) provides a literature review of yield management practices.

In the field of inventory systems an early paper was by Vienott (1965) which dealt with a periodic review inventory model with multiple demand classes and zero lead time. After this there were a few papers which dealt with periodic review inventory systems with multiple classes of demand which are summarized in the paper by Kleijn and Dekker (1999).

In the continuous review framework, the first paper which dealt with stock rationing was by Nahmias and Demmy (1981) who analyzed a system with two classes of demand, fixed lead times and backlogging. Their model was generalized by Moon and Kang (1998) who modeled demand as having a compound Poisson distribution and by Arsalan et al. (2007) who extended the model to more than two demand classes. Sivakumar and Arivarignan (2008) extended the model of Nahmias and Demmy (1981) to the case of perishable items where demands occur according to a Markovian arrival process. Deshpande et al. (2003) compared four different policies—priority clearing of backlog, threshold clearing of backlog, hybrid policy and optimal rationing policy for systems with multiple demand classes.

They showed that in most cases, the hybrid policy performs better. None of these papers proved the optimality of the (s,Q) policy.

Sapna Isotupa (2006) compared two different policies for a Markovian (s,Q) inventory system with two classes of customers and determined conditions under which a threshold rationing policy where the two classes of customers are treated differently is cost optimal for the supplier when compared to a policy where the two customers are treated alike. Isotupa (2011) analyzes the same system as the one in Sapna Isotupa (2006) but with a different objective. In this paper the objective was to determine conditions under which the supplier and both types of customers experience benefits when the threshold rationing policy is adopted when compared to the policy where the two classes of customers are treated alike.

The first paper that deals with an (S − 1, S) inventory system with multiple classes of customers was by Ha (1997a) who analyzed a make-to-stock production system with multiple classes of demand and lost sales and showed that for a certain class of problems, the (S − 1, S) policy with rationing is optimal. Ha (1997b) analyzed a system similar to that of

Ha (1997a) but for the backordering case and proved the optimality of the (S − 1, S) policy in this case as well. Ha (2000) extends the work of Ha (1997a) to include the case of

Erlang lead times. Dekker et al. (2002) analyzed an (S − 1, S) lost sales inventory system with multiple demand classes, Poisson demands and fixed lead times. They present efficient solution methods for obtaining optimal policies with and without service level constraints.

Numerical examples where significant cost reductions were achieved by distinguishing between demand classes are presented. We analyze the same model as this with the difference that the lead time is exponentially distributed. Kranenburg and van Houtum (2007) analyzed an (S − 1, S) lost sales inventory system with multiple demands classes and provide three heuristic algorithms for finding optimal values of the critical levels for a given S that minimizes the holding and shortage costs.

The model assumptions of this paper are very similar to the paper by Kranenburg and van Houtum (2007) with the difference that we consider only two classes of demand. The objective of this paper is exactly the same as the objective of Isotupa (2011) and the difference between the two papers is that Isotupa (2011) deals with an (s,Q) system whereas in this paper, we model the inventory process using an (S − 1, S) system.