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Article Excerpt
1. Introduction
A stock and capacity allocation problem occurs when a common stock and the production capacity of a supplier must be shared among different markets/customers. Such problems arise in a number of supply chain settings. For instance, delayed product differentiation often results in maintaining a stock of generic components for multiple end-products (De Vericourt et al., 2002). Spare parts inventory management, as in Deshpande et al. (2003) for instance, is another situation where inventory allocation is critical. The design of supply contracts in the settings with different retailers can also entail a stock allocation problem at the supplier (Cachon and Lariviere, 1999).
Stock and capacity allocation problems are very challenging and are sometimes considered intractable, as explained by Tsay et al. (1999), especially when customer demands can be backordered. Even when optimal allocation strategies can be characterized, they are typically hard to implement. Indeed, the supplier needs to take many dimensions into account when deciding to allocate stock: The inventory level, the number of waiting demands in the system, the current status of the production process, etc. The complexity of such problems depends on the number of customers sharing the common stock (Ha, 1997a), and on the nature of the production cycle time (Ha, 2000).
In this paper, we consider the model of a supplier that produces a standard item in a make-to-stock environment for several classes of customers. Demands for each class are Poisson processes and item processing times have an Erlang distribution. The supplier has a finite production capacity and has some information on the status of the current production. Different customer classes generate different backorder penalties for the supplier. The objective is to find the stock and capacity allocation policies to minimize the expected discounted (or average) holding and backorder costs over an infinite horizon. At each time instant, the optimal decision depends on the inventory level, the number of waiting demands of each class and the current production stage.
We provide a partial characterization of the optimal stock and production policy for the above described system which is an M/[E.sub.r]/1 make-to-stock queue with backorders. While this characterization yields some basic properties of the optimal policy, the model turns out to be too challenging for a finer characterization. In order to further enhance the understanding of this model, we then focus on a related problem where production cannot be interrupted but excess inventory can be diverted to an ample market at no cost. For this auxiliary problem, it is shown that the optimal stock and capacity allocation policy can be completely characterized: there exist work-storage thresholds for each class that determine how production and inventory should be allocated in a simple way. In addition, these threshold parameters are easily computable. To our knowledge, this is the first such characterization for a multi-dimensional make-to-stock queue problem with non-exponential production times. Finally, it turns out that similar threshold policies lead to extremely effective heuristics for the standard problem.
The ample salvage market that allows absorption of production excess can be considered an approximation for the original model where production is stopped whenever required. The approximate model is more amenable to analysis than the original model. Furthermore, the approximate model may be of interest in itself if the salvage market assumption is justified. One example of this may be the situation where the supplier can divert inventory to a speculative (spot) market. In recent years, speculative markets for non-commodity items have developed rapidly. For instance, Milner and Kouvelis (2007) mention that 80% of electronic component parts (e.g., memory chips) are sold through contract purchasing while the rest are diverted to a spot market. In particular, suppliers may still conduct their main business through long-term contracts with established customers but can also easily get rid of excess inventories in the speculative market (for which no backorder cost exists). The assumption that the system never stops working is also relevant when the production setup cost is very high. Gupta and Wang (2007) present a model motivated by an integrated steel mill where primary processes remain continuously operational but production has to be allocated between contract and spot (i.e., transactional) customers. Of course, such spot markets may manifest other complications such as different lead time requirements or fluctuating prices that are not taken into account in our approximate model.
Stock and capacity allocation problems were first introduced in the context of inventory control. Topkis (1968) provides one of the earliest formulations of an optimal stock rationing problem for an uncapacitated system in discrete time. He analyzes a system with two classes of customers and shortage costs. Since then, there has been considerable research on similar systems under the assumption of exogenous lead times (uncapacitated replenishment) problems. Deshpande et al. (2003) present a brief review of this literature.
In the case of endogenous replenishment lead times or limited production capacity, queuing-based models provide a powerful framework which allows explicit modeling of the production capacity and the randomness of the supply process (see, Buzacott and Shanthikumar (1993)). We follow this approach and model our system as a single server, single-product, make-to-stock queue with multiple demand classes as introduced by Ha (1997a. 1997b) in the stock rationing context.
Rationing strategies also appear in inventory transshipment problems, which have attracted a lot of attention from researchers and practitioners recently. Zhao et al. (2008) characterize the structure of the optimal stock allocation and production policies for a problem with two make-to-stock queues. In this system, each processor primarily serves its own class of customers but is also allowed to serve the other class at an additional cost. Hu et al. (2008) study a similar problem in discrete time where production capacity in each period is uncertain. They characterize optimal transhipment and rationing policies.
Ha (1997b) characterizes the optimal rationing and production policy of a multi-class M/M/1 make-to-stock queue with lost sales. He shows that there are thresholds for each customer class such that it is optimal to reject an arriving demand from a customer if the on-hand inventory is below the threshold for that customer (and to satisfy the demand with the stock otherwise). Carr and Duenyas (2000) analyze the structure of the optimal admission/sequencing policy for a related problem where demands from one class can be rejected. Lee and Hong (2003) numerically study the performance of a lost-sales system with Coxian processing times operating under critical level rationing policies. Huang and Iravani (2007) investigate rationing decisions for a two-echelon supply chain with batch ordering and characterize the optimal policy. Gayon et al. (2009) investigate a rationing problem with imperfect advance demand information. Cil et al. (2009) present some structural results for batch demand arrivals in the lost-sales case with exponential processing times.
When backorders are allowed, the problem of characterizing the optimal policy becomes significantly more difficult because the number of waiting demands has to be tracked for each customer class. For the backorder case, Ha (1997a) shows that the optimal stock and capacity allocation for two customer classes has a monotone structure. De Vericourt et al. (2002) generalize this result and provide a full characterization of the optimal stock and capacity allocation for n customer classes. The optimal policy specifies threshold levels such that it is optimal to satisfy an arriving demand from a customer if the on-hand inventory is above the threshold for that customer and to backorder the demand otherwise. These threshold levels also determine production priority for waiting demands in a simple way.
The models in Ha (1997a, 1997b) and De Vericourt et al. (2002) assume exponential processing times. Because of the memoryless property of the exponential distribution, the supplier does not need the current production status (i.e., elapsed processing time) in that case. Information technologies in real production systems, however, can provide constant access to information on the status of the production process which would be needed with non-exponential processing times. In this paper, we consider a multi-class M/[E.sub.r]/1 make-to-stock queue (with an Erlang-r processing time). We assume the supplier knows the current stage (phase) of the Erlang distribution exactly. This allows us to model the information on the production status. In addition, Erlang distributions provide some flexibility in modeling the production process variability. De Vericourt et al. (2001) provide insights about the benefit of stock allocation policies when the utilization rate and the relative importance of the customer classes vary. Because of the exponential assumption therein, the impact of production time variability in this comparison is not addressed. In this paper, we evaluate the performance of optimal stock rationing policies when the production time variability increases and the mean stays constant. These two features of the Erlang distribution (information on the production status and production time variability) yield insights that cannot be obtained under the exponential distribution assumption.
To our knowledge Ha (2000) is the only paper that addresses optimality issues in a stock allocation problem for the make-to-stock queue where the processing time has an Erlang distribution. He assumes lost sales and shows that a single state variable, the work storage level, can fully capture the inventory level and the status of the current production of the system. This reduces the problem to a single-dimensional Markov Decision Process model. The optimal stock allocation policy is then fully characterized: for each customer class there exists a work-storage threshold level at which it is optimal to reject a demand of this class. More recently, Abouee-Mehrizi et al. (2008) investigate the rationing problem for an M/G/l make-to-stock queue from a performance evaluation perspective and compare different policies.
Our model differs from that of Ha (2000) in the assumption that demands are backordered. The backordering assumption is fundamental from an inventory management perspective and merits attention but it makes the analysis much more challenging for two reasons. First, as mentioned earlier, we deal with an (n + 1)-dimensional state space since we need to keep track of the waiting demands of each class. Second, backorders require addressing a new type of decision which corresponds to the production allocation in the presence of waiting demands from different classes. This issue does not exist when demands are lost.
When the production surplus cannot be sold in a salvage market, we obtain some partial structural results for the optimal stock allocation policy. Although these results uncover certain useful properties of the optimal policy,...
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