|
Article Excerpt
Motivated by logistics practices, we consider a retailer that replenishes its inventory by making a delivery request without specifying a quantity, then deciding the quantity when the delivery vehicle arrives after one period. A fixed cost is incurred whenever a delivery request is made, regardless of the quantity ordered later. The new feature of this research relative to previous work is the separation of the delivery request and the quantity decision, or the postponement of ordering until one-period demand information is observed. Due to such separation, both the state space and the action space must be augmented in the model. We show that the optimal policy for delivery requests is of a threshold type: A delivery request is made if and only if the inventory on hand is below a threshold. The optimal decision on ordering is more complex, and there might be multiple order-up-to levels. Our numerical studies show, nonetheless, that the cost of an ordering policy that considers (at most) two order-up-to levels is close to the minimal when the planning horizon is not too short. We also identify conditions under which a base-stock policy is optimal for ordering. To understand the effects of ordering postponement, we compare our model with the traditional model in which the two decisions must be made at the same time. We show that postponement leads not only to a lower cost, but also a higher threshold for making delivery requests.
Subject classifications: periodic-review inventory systems; optimal policy; postponement; quasi-K-convex; single crossing; dynamic programming; Markov decision programming.
Area of review: Manufacturing, Service, and Supply Chain Operations.
History: Received January 2007; revisions received October 2007, July 2008, October 2008; accepted December 2008.
1. Introduction
In many logistics systems, inventory replenishment by retailers consists of two distinct stages. In the first stage, retailers may make a delivery request in advance without making a firm commitment on order quantity. In the second stage, the retailers are allowed to make an adjustment to the quantity requested right before the shipping schedule is finalized, or are even allowed not to decide the quantity until a delivery vehicle has arrived at their receiving docks. In contrast to more-traditional approaches by which delivery request and quantity decisions must be made at the same time, this approach allows retailers to postpone their quantity decision until better demand information is available, and, as a result, the mismatch between supply and demand is reduced. In this paper, we consider the inventory problem faced by a retailer who replenishes its stock by using this approach.
The practice described above has been observed in various settings. Menlo Logistics, a U.S.-based contract logistics provider, delivers products to a number of destinations along a fixed route. Instead of loading trucks with a well-defined shipment quantity for each destination, Menlo defers the quantity decisions until its trucks arrive at each destination. The trucks are described as rolling warehouses. Orient Overseas Container Line (OOCL), a major shipping company in the Asia-Pacific region, adopts a similar approach for their sea freight transport. Their ships are called "floating warehouses." Similar practices have been documented at Wal-Mart, the U.S. Defense Department (Kumar et al. 1995), Waste Management Inc. in subscription service areas (Sahoo et al. 2005), and the American Red Cross (Ballou 2004). Whether implemented on road or sea, all practices involve the following two stages. A delivery route must first be designed based on delivery requests from retailers. Second, vehicles will travel along the delivery route, and the amount of inventory that needs to be unloaded is determined when the vehicles arrive at each destination. Therefore, the latest demand information can be utilized in the quantity decision making.
This important practice, sometimes known as milk run or fixed-route delivery, has been studied analytically in the literature, but from a very different angle. The existing work considers multiple retailers and focuses on their coordination. For example, Kumar et al. (1995) studied the benefits of allocating stock sequentially to the retailers over allocating it at the moment when the vehicle leaves the warehouse. Cheung and Lee (2002) examined the benefits of using information on retailers' inventory positions to coordinate shipments from the supplier to enjoy economies of scale in shipments, as well as to rebalance the retailers' stocking positions. To implement the sequential allocations in Kumar et al. (1995), or shipment coordination and stock rebalancing in Cheung and Lee (2002), all retailers and the supplier must belong to the same firm so that they can all be managed by a central planner whose objective is maximizing the total profit of the entire supply chain. The hospitals replenished by the American Red Cross along fixed routes, however, are independently owned and managed. In fact, in many cases of fixed-route delivery, retailers and suppliers are often different firms with different objectives. Independent retailers would often prefer to maintain control over inventory replenishment. In a decentralized environment, to coordinate the replenishment of the supply chain, we must understand how independent retailers replenish their stock. This is the focus of our study.
Specifically, we consider an inventory system operating in a periodic mode. In each period, the retailer must decide whether a delivery request should be made. (Alternatively, the retailer must decide whether it wants to be included in a delivery route, or place an order that can later be changed without any restriction.) A delivery request incurs a fixed cost. The delivery vehicle will arrive one period after the request is made, and the retailer can place an order without a capacity constraint from the vehicle. The inventory-ordering cost is linear in quantity. We further assume that inventory ordering is instantaneous and unmet demand is backlogged. We consider both finite and infinite planning horizons. The objective of this study is to find an optimal policy for delivery requests and inventory ordering such that the total discounted cost is minimized.
The concept of postponement has been applied in various contexts of supply chain management. Perhaps because of its applicability, there has been an ongoing stream of research on postponement strategies (e.g., Lee et al. 1993; Lee and Tang 1997; Aviv and Federgruen 2001a, b, and the literature review therein). Our work is also related to the literature on dynamic inventory control with fixed costs. Research in this area has a long history and dates back to more than 40 years ago. Scarf (1960) proved the optimally of an (s, S) policy with the concept of K-convexity: An order should be placed if and only if the initial inventory is below s, and, when an order is placed, the inventory level should be brought up to S. The basic model considered by Scarf (1960) has been extended along various directions, including an infinite horizon (Iglehart 1963), computational issues (Veinott and Wagner 1965), a more general concave increasing ordering cost (Porteus 1971), and stochastic lead times (Ehrhardt 1984). Porteus (1990) provided a comprehensive review of the early literature. Recent interest in this area is quite diverse, including, for example, capacity constraints (Chen and Lambreeht 1996, Gallego and Scheller-Wolf 2000), multiple classes of demand (Sobel and Zhang 2001, Frank et al. 2003), dynamic pricing (Chen and Simchi-Levi 2004, Feng and Chen 2007), and dual sourcing (Yang et al. 2005, Fox et al. 2006). Note that none of the work in this literature considers the separation of delivery requests and ordering, which is the theme of our model.
To our best knowledge, this is the only study of inventory systems that allows the separation of delivery requests and inventory ordering. Because the model includes fixed costs and both action and state spaces are two dimensional, the problem is technically challenging. We show that if the demand has a [PF.sub.2] distribution, then the optimal policy for delivery requests is of a threshold type: A delivery request is made if and only if the inventory on hand is below a threshold. Inventory ordering, however, is more complex. The optimal policy, in general, may lead to multiple order-up-to levels, and extra conditions are needed for a base-stock policy to be optimal. Because the optimal ordering policy is complex, we devise a heuristic that considers (at most) two order-up-to levels. The heuristic is tested numerically. The results show that the cost of implementing the heuristic is very close to the minimal when the planning horizon is not too short. Finally, we examine the effects of ordering postponement. We show that postponement leads to not only a lower cost, but also a higher threshold for making delivery requests.
Our model reveals three types of substitutability the retailer must consider in managing its replenishment. First, the threshold structure for...
|
