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Article Excerpt
In this paper, we address the optimal joint control of inventory and transshipment for a firm that produces in two locations and faces capacity uncertainty. Capacity uncertainty (e.g., due to downtime, quality problems, yield, etc.) is a common feature of many production systems, but its effects have not been explored in the context of a firm that has multiple production facilities. We first characterize the optimal production and transshipment policies and show that uncertain capacity leads the firm to ration the inventory that is available for transshipment to the other location and characterize the structure of this rationing policy. Then, we characterize the optimal production policies at both locations, which are defined by state-dependent produce-up-to thresholds. We also describe sensitivity of the optimal production and transshipment policies to problem parameters and, in particular, explain how uncertain capacity can lead to counterintuitive behavior, such as produce-up-to limits decreasing for locations that face stochastically higher demand. We finally explore, through a numerical study, when the optimal policy is most likely to yield significant benefits compared to simple policies.
Subject classifications: dynamic programming: application; inventory/production: uncertainty/stochastic; reliability; capacity: uncertainty.
Area of review: Manufacturing, Service, and Supply Chain Operations.
History: Received July 2004; revisions received July 2006, December 2006; accepted December 2006.
1. Introduction
Consider a firm that produces the same product in multiple locations, but faces demand and capacity uncertainty. The capacity uncertainty is caused by factors such as downtime, quality problems, yield, etc. The firm faces two related decisions: (1) How much should it produce at each location? and (2) How much should it transship from one location to another? Even though the literature on transshipment is rich, it usually ignores the effects of capacity uncertainty, and our aim is to gain insight into how capacity uncertainty affects both these decisions.
The problem we describe is very common in industry. For example, we recently worked with a diesel engine manufacturer that has multiple locations where castings are made. The capacity of the plants making the castings in any week was random due to quality problems and, therefore, the company was exploring transshipment from one location to another to satisfy engine plants' demands for castings. We observed similar issues in the case of a major paper manufacturer that produces paper cups in multiple locations in the United States, as well as a major newspaper ink manufacturer with over 20 plants in the United States. In all cases, products would be transshipped from one plant to another plant's markets when capacity in a plant was low in a given period. However, we observed that the actual production policies of the plants did not take into account the fact that such transshipments may occur. In some situations, we also observed that plant management was reluctant to transship beyond a certain amount, due to fear that they may face a shortage next period if their inventory levels are down significantly. All of these observations motivated us to explore how optimal transshipment and production decisions should be made jointly and how the level of demand and capacity uncertainty affects the behavior of optimal policies.
We consider a centralized system with two facilities that operate in two markets, produce the same product, and sell it at constant prices. Both facilities, in addition to demand uncertainty, face uncertain production capacities. Inventory can be held from period to period, but unsatisfied demand is lost. All decisions are made by a central planner who has full access to the stock status at the two facilities. Her objective is to maximize the expected discounted joint profits over a finite horizon. At the beginning of any period, she determines the production quantities for both facilities. After the production and demand uncertainties are revealed, she decides how much inventory should be transshipped from one location to another. Demands are satisfied after the transshipment. (1)
We examine the structure of the optimal production and transshipment policies for both facilities and find it different from the previous research. Under various assumptions, Robinson (1990), Tagaras (1989), and others verify optimality of the "complete pooling" policy for transshipment: transshipment occurs when one location has excess stock and the other is short and the transshipped quantity is equal to the minimum of the surplus and the shortage. Due to uncertain capacity in our setting, even when all the complete pooling assumptions (listed at the end of [section]4.1) are satisfied, the whole system may be better off with one facility keeping some safety stock and not satisfying the shortage of the other facility. Also, it may be beneficial to ship some inventory from the facility with higher holding cost to the other one even when the latter does not need it to decrease holding cost across multiple periods.
Unlike the base-stock policy established in the literature, our optimal production policies for two facilities are based on switching curves--each facility's production quantity is a nonincreasing function of the other facility's starting inventory--and may also depend on its own starting inventory. In the special case when one of the facilities has infinite capacity, the optimal policy for that facility is an up-to level. The up-to level is decreasing in the inventory level of the facility facing uncertain capacity. In addition to showing that the uncertainty in capacity changes the structure of the optimal production policy, we also show that optimal policies behave differently (sometimes counterintuitively) in the presence of uncertain capacity--e.g., stochastically larger demand, lower holding costs, or higher revenue may result in strictly lower production targets. This is because when a site needs more inventory, either because its demand or revenue increased, it may choose to transship it from the more reliable site to ensure that the material will be there on time. In our numerical study, we examine both the direction and size of impact that different parameters have on the policy and on the total multiperiod profit. Because the optimal policy is fairly complicated, we consider two simple straw policies (often used in practice) and compare them to the optimal policy, which allows us to describe when the optimal policy is most beneficial and when simple policies perform well.
We review the literature in [section]2 and state the assumptions and formulate the model in [section]3. In [section]4, we establish the structure of the optimal transshipment and production policies. Then, we derive analytical results for the sensitivity of the optimal policy in [section]5. In [section]6, we use a numerical study to describe additional properties of the optimal policy. Finally, in [section]7 we discuss possible extensions.
2. Literature Review
In multilocation stochastic inventory systems, lateral transshipment across locations allows one to better match supply and demand. Typically, transshipment helps a firm to deal with potential shortage of products and takes place after demand is realized but before it is satisfied. The commonly considered costs include linear production, holding, shortage, and transshipment costs. Krishnan and Rao (1965) study a single-period two-location problem and include an extension to the N-location scenario. The costs at all locations are equal. Robinson (1990) extends their work to the multiperiod multilocation case, with varying costs across the outlets. Tagaras (1989) considers a similar model and focuses on the pooling effects created by allowing transshipment on service levels in a two-location system. He also establishes a set of assumptions to guarantee the complete pooling (i.e., in our vocabulary, complete transshipment without rationing).
Gross (1963) may be the first to consider a two-location problem in which transshipment occurs before the demand realization. The corresponding multiperiod multilocation problem is studied by Karmarkar and Patel (1977), Showers (1979), and Karmarkar (1987). Das (1975) allows one-time transshipment in the middle of the period when demand is partially disclosed. Lee (1987) and Axsater (1990) examine a continuous-review system with transshipment triggered by ts-review system with transshipment triggered by stockouts. Archibald et al (1997) combine Das (1975) work with Lee's (1987) and Axsater's (1990) to a multiperiod two-location periodic-review model in which demand is disclosed continuously during the review period and the transshipment or emergency-order decision is made whenever stockout occurs.
A number of extensions have been studied in the literature. Fixed joint replenishment costs are explicitly considered by Herer and Rashit (1999) in a two-location single-period problem. Tagaras and Cohen (1992) study the effect of replenishment lead times. Axsater (2003) considers a centralized system with more than two locations and develops an effective heuristic decision rule for lateral transshipment. The first paper we are aware of that consider decentralized decision makers is Rudi et al. (2001), where two locations maximize their own single-period profits. The authors identify transshipment prices that induce both locations to choose inventory levels consistent with joint profit maximization. Hu et al. (2007) addresses a similar problem, allowing for both certain and uncertain capacity. More general multiple-location decentralized distribution systems are studied by Anupindi et al. (2001). Granot and Sosic (2003), and Zhao et al. (2005).
All of the above papers consider stochastic demand and assume that replenishment capacity is infinite and certain. Capacitated inventory systems and systems with uncertain capacity are considered in a separate group of papers, but these papers do not consider transshipment among locations. We first list papers that consider deterministic but limited capacity. Federgruen and Zipkin (1986a, b) study a system with the stationary stochastic demand and capacity restrictions and show that order-up-to policies are optimal for the infinite-horizon case. Glasserman and Tayur (1994, 1995, 1996) assume order-up-to policies for a multiechelon system and describe how to find optimal up-to levels. Parker and Kapuscinski (2003) show that the up-to policy is optimal for a 2-echelon capacitated system. Kapuscinski and Tayur (1998) consider a capacitated production-inventory system with nonstationary demand. A multiproduct version is analyzed by DeCroix and Arreola-Risa (1998). None of these papers considers multiple locations or uncertain capacity.
Capacity/production uncertainty has been modeled in two different ways. One approach has been heavily influenced by yield issues in electronics manufacturing and uses the concept of stochastically proportional yield, or random yield, as defined in Henig and Gerchak (1990). Random-yield models assume that a random fraction of a quantity ordered (or attempted to produce) is actually good. This is an appropriate model when the uncertainty is due to uncertain quality of individual items produced in a batch.
The other approach regards the capacity in a given time interval as a random variable. Ciarallo et al. (1994) and Duenyas et al. (1997) consider that maximum production in any given time interval is uncertain. These papers, however, do not consider lateral transshipment. In a chapter of a recent dissertation, Zhao (2003) allows for transshipment between two M/M/1 queues and characterizes optimal policies.
In this paper, we focus on an inventory/production model with general demand and capacity distributions. We follow the approach in Ciarallo et al. (1994) and Duenyas et al. (1997) and model the firm's capacity as a random variable. Unlike in those two papers, however, we focus on a firm that produces in two locations instead of one. Our aim is to explore how the level of capacity uncertainty affects optimal transshipment decisions and optimal production when goods can be transshipped.
3. The Model
We consider two manufacturing facilities, each serving its individual market, through multiple time periods. The facilities face uncertain capacity--they do not know with certainty how much they will be able to produce in any given period (e.g., due to machine downtime or quality and yield issues). The uncertain capacities in each facility are characterized by capacity distributions that are independent in time and of each other. The facilities also face demand uncertainty. The stochastic demand distributions are independent in time but can be correlated for any given period across the two facilities. This assumption is relaxed in [section]7 by allowing dependence across the time periods using a Markov-modulated process.
In any period, production decisions are made first: the firm decides how much it will attempt to produce in each of the facilities that period. Then, the capacities and demands are realized for both facilities. The actual production is the minimum of planned production and the realized capacity. Finally, decisions are made regarding transshipment of inventory between facilities. We assume that demand that is unsatisfied after transshipment is lost. The firm earns linear revenues on satisfied demand and incurs linear production, holding, and transshipment costs. The objective is to maximize the joint discounted profit for both facilities. Let i, j = 1, 2 denote the facilities and
[c.sub.i] = variable production cost for facility i
[h.sub.i] = variable holding cost for facility i
[s.sub.ij] = variable transshipment cost from facility i to j, i [not equal to] j
[r.sub.i] = unit revenue for facility i.
We assume that marginal profit is always higher when...
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