Optimal production and rationing policies of a make-to-stock production system with batch demand and backordering

Abstract

In this paper, we consider the stock rationing problem of a single-item make-to-stock production/inventory system with multiple demand classes. Demand arrives as a Poisson process with a randomly distributed batch size. It is assumed that the batch demand can be partially satisfied. The facility can produce a batch up to a certain capacity at the same time. Production time follows an exponential distribution. We show that the optimal policy is characterized by multiple rationing levels.

Introduction

This paper considers a dynamic production system that maintains the inventory of a single product to satisfy the demands of different classes of customers. Customers are classified into classes according to their priorities which can be expressed in a variety of ways e.g., economic value to the manufacturer, terms of supply contracts etc. In periods of high demand a commonly used practice is to ration the limited stock among different demand classes. Stock rationing strategies have been widely used in many industries, including manufacturing and retail industries for allocating goods to the contracted versus walk-in customers, airlines for rationing seat inventories, as well as hotels for renting rooms. Since customers arrive sequentially, an important decision problem for managers is to decide whether or not they should fill a current demand with a lower priority in favor of reserving the stock for later demand with a higher priority. Determining the optimal policy for stock rationing, and furthermore, analyzing simple structures for the optimal policy is therefore a key decision problem with real life significance.

Veinott [18] first investigated the inventory problem of several demand classes and introduced the concept of critical level policy. In a system with multiple classes and unit demand, the critical level policy operates as follows. Demand from a particular class is satisfied from stock on-hand if the inventory level exceeds the critical level associated with this class [13]. The critical level policy is simple to implement and has been widely adopted in practice and is also well studied in the academic literature.

The critical level policy has been shown to be an optimal stock rationing policy under certain conditions. For the periodic review models, [17] demonstrated the optimality of the critical level policy for both the backordering case and the lost-sales case. Evans [6] also obtained the same results for problems with two demand classes. In a continuous review setting, several papers use the Markov decision process to formulate the problem from the stand point of make-to-stock queue, and establish the optimality of the critical level policy. For example, [9] examined the lost-sales case with multiple demand classes and showed that the lot-for-lot production policy and the critical level rationing policy are optimal. Ha [10] further extended the previous results to the case of Erlang distributed processing time. Ha [11] studied a make-to-stock system with two demand classes and backordering. Assuming Poisson demand and exponentially distributed production time, he showed that the critical level policy is optimal and the critical level is non-increasing in the number of backorders of the low-priority class. de Vericourt et al. [5] extended the two-class case in [11] to the case with multiple (more than two) demand classes. They showed that the optimal policy is characterized by multiple critical levels (they use the term Multilevel Rationing (ML) policy). Huang & Iravani [12] extended the models in [9], [11] and demonstrated the optimality of the critical level policy for the case of fixed size demand.

In general, the optimal rationing policy may have a complicated structure and cannot be characterized simply by the critical level policy (see [2], [21], [7]). Some papers assume a critical level policy, though the optimal policy is unknown, and focus on optimizing policy parameters. Nahmias & Demmy [16] examined the case of multiple demand classes in a continuous review inventory model. They assumed Poisson demand with backordering and two demand classes. Moon & Kang [15] generalized a similar model with compound Poisson demand and used a ($R,Q$) model to derive the approximate expressions for the fill rates of the demand classes. Deshpande et al. [4] analyzed the same ($R,Q$) inventory rationing model with two demand classes without any restriction on the number of outstanding orders. Recently, Arslan et al. [1] considered a single-product inventory system without any restriction on the number of demand classes.

The current paper is most closely related to [11], [5], [12]. We seek to investigate the optimality of the critical level policy under more general scenarios. Our paper is distinct from the previous papers in that we consider the stock rationing production/inventory systems with batch production, multiple demand classes and batch arrival. The relationship between this paper and the relevant literature is illustrated in Table 1.

First, we extend Ha [11] (two-class, unit demand), de Vericourt et al. [5] (multiple-class, unit demand) and Huang & Iravani [12] (two-class, batch demand) to the multiple-class, batch demand problem. This is a significant problem context which is commonly encountered in real life especially in the semiconductor industry. Second, we assume that the facility can produce all units in a batch up to its capacity concurrently, while all the above papers consider unit production and therefore are not applicable to the batch production case. To our knowledge, batch production has not been previously reported in the make-to-stock queue literature.

Batch demand is very common in the wholesale market and the manufacturing industries such as electronics. In the basic model, we assume that batch arrivals can be partially accepted, i.e., given a demand for $a$$\ge 1$ units, the manufacturer can sell any quantity $k$ in the range $0\le k\le a$, and the customer is willing to accept any quantity in this range. We believe that this is a reasonable assumption. For example, an electronic equipment manufacturer produces spare parts that are used to replace defective components for some specified equipment. The production lead time is stochastic due to uncertain production environment. Since some components are vital while others are auxiliary, different spare parts should be treated differently resulting in their being accorded different priorities. When an order for spare parts comes in, the manager needs to determine the extent to which it should be satisfied or whether it should be totally rejected and inventory reserved for later demand from more critical equipment. For the partially met demand, the customer will typically choose to accept it to repair the defective equipment as far as possible. This behavior can also be observed among retailers of companies like Apple Computers that market trendy products. When Apple releases its new line of iPhones or iPods, typically a huge demand builds up in the initial launch period which for reasons described above cannot be satisfied immediately. Independent retailers are accorded a lower priority as compared to Apple operated company stores. Since Apple can only satisfy partial orders from independent retailers, the balance demand is backordered and is fulfilled in future periods [19].

Similar extension from the unit demand case to the batch demand case has also been found in the airline yield management literature where [3] considered a batch size demand for the airline booking problem. Our work differs significantly from theirs in several ways. First, we consider both stock rationing and production for an infinite-horizon problem while they consider a finite-period problem in context of seat capacity allocation. Their model is a one-dimension semi-Markov decision process and ours is a multi-dimension Markov decision process. Second, structures of the optimal policies are essentially different. In their paper, the switching curve, so-called booking curve therein, is non-stationary and monotone with respect to time. In our paper, the optimal policy is characterized by multiple rationing levels.