سهمیه بندی موجودی پویا برای سیستم های با کلاس تقاضای چندگانه و فرآیندهای تقاضای عمومی

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.

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 purposes 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 maintenance. 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 as Kleijn and Dekker (1998), Deshpande et al. (2003), and Cardós 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, Ha, 1997b, Ha, 2000 and Gayon et al., 2004). For exogenous inventory supply problems, researchers such as Melchiors et al. (2000), Deshpande et al. (2003), and Arslan 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 Möllering and Thonemann (2009) determine the stationary critical levels for inventory systems operating under different service levels. 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 from Topkis (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 problems 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. In Section 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 in Section 2 is extended for multiperiod systems. In Section 4, numerical studies are conducted to investigate the performance of the proposed approaches. In Section 5, we summarize the results and discuss some possible 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.