Dynamic Capacity Management with General Upgrading

Yueshan Yu, Xin Chen, Fuqiang Zhang
2015 Operations Research  
This paper studies a capacity management problem with upgrading. A firm needs to procure multiple classes of capacities and then allocate the capacities to satisfy multiple classes of customers that arrive over time. A general upgrading rule is considered, i.e., unmet demand can be satisfied using multi-step upgrade. No replenishment is allowed and the firm has to make the allocation decisions without observing future demand. We first characterize the structure of the optimal allocation policy,
more » ... allocation policy, which consists of parallel allocation and then sequential rationing. Specifically, the firm first uses capacity to satisfy the same-class demand as much as possible, then considers possible upgrading decisions in a sequential manner. We also propose a heuristic based on certainty equivalence control to solve the problem. Numerical analysis shows that the heuristic is fast and delivers close-to-optimal profit for the firm. Finally, we conduct extensive numerical studies to derive insights into the problem. It is found that under the proposed heuristic, the value of using sophisticated multi-step upgrading can be quite significant; however, using simple approximations for the initial capacity leads to negligible profit loss, which suggests that the firm's profit is not sensitive to the initial capacity decision if the optimal upgrading policy is used. Operations Research; manuscript no. (Please, provide the mansucript number!) demand. This paper studies the influential practice of upgrading, where higher-quality products can be used to satisfy demand for a lower-quality product that is sold out. Such a practice takes advantage of risk pooling (product substitution essentially allows product/demand pooling), which results in several immediate benefits: First, it increases revenue by serving more demand; second, it enhances customer service by reducing lost sales; third, it may lead to lower inventory investment by hedging against demand uncertainty. The practice of upgrading or substitution has been widely adopted in the business world. In the automobile industry, firms may shift demand for a dedicated capacity to a flexible capacity when the dedicated capacity is constrained (Wall 2003). In the semiconductor industry, faster memory chips can substitute for slower chips when the latter are no longer available (Leachman 1987). More examples in production/inventory control settings can be found in Bassok et al. (1999) and Shumsky and Zhang (2009). Similar practice is ubiquitous in the service industries as well. For instance, airlines may assign business-class seats to economy-class passengers, car rental companies may upgrade customers to more expensive cars, and hotels may use luxury rooms to satisfy demand for standard rooms. Both practitioners and academics surely understand the importance of the upgrading practice. Substantial research has been conducted on how to manage upgrading in a variety of problem settings. Here we contribute to this large body of literature by studying a dynamic capacity management problem under general upgrading structure. For convenience, we use the terms "product" and "capacity" exchangeably throughout the paper, and similarly for "upgrading" and "substitution" (strictly speaking, upgrading is one-way substitution). A brief description of our problem is as follows. Consider a firm selling N products with differentiated qualities in a fixed horizon consisting of T periods. There are N classes of customers who arrive randomly in each period. Each customer requests one unit of the product; in the case of a stock-out, the customer can be satisfied with a higher-quality product at no extra charge. Unsatisfied demand is backlogged and the firm incurs a goodwill cost. The firm needs to first determine the procurement quantity of each product at the beginning of the horizon, and then decide how to distribute the products among incoming customers. Due to long ordering lead time, the firm cannot replenish inventory before the end of the horizon; as a result, the firm must dynamically allocate the products over time, before observing future demand. This paper represents an extension of the recent work by Shumsky and Zhang (2009, referred to as SZ hereafter). As one of the first studies that incorporate dynamic allocation into substitution models, SZ make a simplifying assumption to maintain tractability. Specifically, they consider single-step upgrading, i.e., a demand can only be upgraded by the adjacent product. Clearly, this is a restrictive assumption because in many practical situations firms may have incentives to use Yu, Chen, and Zhang: Dynamic Capacity Management with General Upgrading Article submitted to Operations Research; manuscript no. (Please, provide the mansucript number!) 3 multi-step upgrading to satisfy demand. Thus there is a need for a theoretical model that captures the realistic upgrading structure. The purpose of this paper is to fill this gap in the literature. While relaxing the single-step upgrading assumption, we attempt to address the following questions as in SZ: What is the optimal initial capacity? How should the products be allocated among customers over time? Are there any effective and efficient heuristics for solving the capacity management problem? The main findings from this paper are summarized as follows. We start with the dynamic capacity allocation problem. In each period, the firm needs to use the available products to satisfy the realized demand. When a product is depleted while there is still demand for that product, the firm may use upgrading to satisfy customers. How to make such upgrading decisions is a key in substitution models. With the general upgrading structure, the optimal allocation policy is complicated by the fact that the upgrading decisions within a period are interdependent. Under the backlog assumption, we show that a Parallel and Sequential Rationing (PSR) policy is optimal among all possible policies. The PSR policy consists of two stages: In stage 1, the firm uses parallel allocation to satisfy demand as much as possible (i.e., demand is satisfied by the same-class capacity). Then in stage 2, the firm sequentially upgrades leftover demand, starting from the highest demand class; when upgrading a given demand class, the firm starts with the lowest capacity class. The optimality of such a sequential rationing scheme depends on an important property. That is, when using a particular class of capacity to upgrade demand, the upgrading decision does not depend on the status of the portion of the system below that class. The PSR can greatly reduce the computational complexity because the upgrading decisions do not have to be solved jointly. As an extension, we also consider the multi-horizon model with capacity replenishment and show that the PSR policy remains optimal in each horizon. Our theoretical results, though intuitive, turn out to be very challenging to prove. Indeed, our proofs rely on intricate arguments and fully exploit the special structure of the upgrading problem. Despite the simplified policy structure given by the PSR, solving the problem is still quite involved due to the curse of dimensionality. So there is a need to search for fast heuristics that perform well for the firm. We present a heuristic that adapts certainty equivalence control (CEC) to exploit the sequential rationing property in the PSR policy. Such a heuristic is more appealing than the commonly used CEC heuristic, and we call it refined certainty equivalence control (RCEC) heuristic. Through extensive numerical experiments, we find that the RCEC heuristic delivers close-to-optimal profit for the firm. The RCEC heuristic enables us to solve large problems effectively. Thus we can use numerical studies based on such a heuristic to derive more insights into the dynamic capacity management problem. First, compared to single-step upgrading, general upgrading (multi-step upgrading) can be highly valuable, especially when the capacities are severely unbalanced. Second, our numerical Yu, Chen, and Zhang: Dynamic Capacity Management with General Upgrading 4 Article submitted to Operations Research; manuscript no. (Please, provide the mansucript number!) studies indicate that the firm's profit is not sensitive to the initial capacity decision, given that the optimal upgrading policy is used. For instance, either the newsvendor capacities (calculated assuming no upgrading) or the static capacities (calculated assuming complete demand information) provide nearly optimal profit for the firm. However, the negative impact of using suboptimal allocation policies could be quite significant. These findings suggest that from the practical perspective, deriving the optimal allocation policy should receive a higher priority than calculating the optimal initial capacity. The remainder of the paper is organized as follows. Section 2 reviews the related literature. Section 3 describes the model setting. The optimal allocation policy is characterized by Sections 4 and 5. Section 6 extends the base model to multiple horizons with capacity replenishment. Section 7 proposes the RCEC heuristic and Section 8 presents the findings from numerical studies. The paper concludes with Section 9. All proofs are given in the appendix. Literature Review This paper falls in the vast literature on how to match supply with demand when there are multiple classes of uncertain demand. To facilitate the review, we may divide this literature into four major categories using the following criteria: (1) whether there are multiple capacity types or a single capacity type; and (2) whether the nature of capacity allocation is static or dynamic. A problem is called static if capacity allocation can be made after observing full demand information. The category that involves the single capacity and static allocation essentially reduces to the newsvendor model that is less relevant. Thus, our review below focuses on the representative studies from the other three categories. The first category of studies involves multiple capacity types and static capacity allocation. In these studies, firms invest in capacities before demand is realized and then allocate capacities to customers after observing all demand. Due to the existence of multiple capacity types, the issue of substitution naturally arises. Van Mieghem (2003) and Yao and Zheng (2003) provide comprehensive surveys of this category of studies, which can be further divided into two groups. One group of papers study the optimal capacity investment and/or allocation decisions under substitution. Parlar and Goyal (1984) and Pasternack and Drezner (1991) are among the first to consider the simplest substitution structure with two products. Bassok et al. (1999) extend the problem to the general multi-product case. Hsu and Bassok (1999) introduce random yield into the substitution problem. By assuming single-level substitution, Netessine et al. (2002) study the impact of demand correlation on the optimal capacity levels. Van Mieghem and Rudi (2002) propose the notion of newsvendor networks that consist of multiple newsvendors and multiple periods of demand. Similar Yu, Chen, and Zhang: Dynamic Capacity Management with General Upgrading Article submitted to Operations Research; manuscript no. (Please, provide the mansucript number!) 5 settings can be found in the studies on multi-period inventory models with transshipment, including Robinson (1990), Archibald et al. (1997), and Axsäter (2003) . Although these studies involve multiple periods, replenishment is allowed and capacity allocation in each period is made with full demand information. The other group of studies focuses on the value of capacity flexibility. Fine and Freund (1990) and Van Mieghem (1998) consider two types of capacities (dedicated and flexible) and study the optimal investment in flexibility. Bish and Wang (2004) and Chod and Rudi (2005)
doi:10.1287/opre.2015.1446 fatcat:5kugtqqmtvgcrgyyzoqxf2ot7a