This paper addresses the implications of considering power demand pattern and backorders in the one-warehouse NN-retailer problem. Specifically, we assume that items at the retailers are withdrawn from the inventory following a power pattern. The objective function consists of the sum of ordering, holding and backordering costs at all installations. This objective function depends on the class of inventory policies that we choose for the formulation of the problem. In general, obtaining the cost function is an arduous task since the average inventory at the warehouse could be difficult to determine. However, if we apply single-cycle policies the computation of the total average cost becomes possible. Under these assumptions we show that the one-warehouse NN-retailer system with power demand pattern and backorders can be formulated as a mixed non linear programming problem.
In practice, inventory systems usually involve a number of installations which are coupled to each other. For instance, one central warehouse can supply goods to a chain of stores. This class of inventory systems arises in both distribution and production context. In case of a distribution system, products are delivered to a set of installations located over large geographical areas. Accordingly, local stock points should be established close to the customers to guarantee a satisfactory service level. These local sites replenish their stocks from a central warehouse close to the production facility. In the production framework, stocks of raw materials, components and finished products are coupled to each other in a similar way. It is worth noting that in these situations decisions made by a member of a chain can affect to the rest of locations. Hence, it is necessary that all members of the supply chain collaborate and integrate their order policies to achieve a more efficient inventory control.
This paper specifically focuses on an inventory/distribution system consisting of one central warehouse, which supplies products to a set of N retailers that in turn cover customer demand. The one-warehouse N-retailer problem has been extensively analyzed in the literature by authors such as Schwarz (1973), Graves and Schwarz (1977), Williams, 1981 and Williams, 1983, Roundy (1985), Muckstadt and Roundy (1993) and Abdul-Jalbar et al., 2003, Abdul-Jalbar et al., 2005 and Abdul-Jalbar et al., 2006, among others. For an introduction to this problem the reader is referred to Silver et al. (1998), Zipkin (2000) and Axsäter (2000). In most cases, the problem assumes that customer demand at the retailers occurs at constant rate and is uniformly distributed through the planning horizon. Additionally, it is assumed that demand should be satisfied without backorders. Nevertheless, Mitchell (1987), Atkins and Sun (1995), Sun and Atkins (1997) and Chen, 1998, Chen, 1999 and Chen, 2000 extended the model to allow shortages.
However, assuming that customer demand at each retailer is uniformly distributed through the planning horizon is an idealistic assumption which is not always appropriate or applicable. In real life situations the demand of an item toward the beginning of a period, e.g., a week, can be smaller or greater than the demand at the end of the period. The goal of this paper is to analyze the one-warehouse NN-retailer problem under this situation, that is, assuming that customer demand at each retailer follows a power pattern. Moreover, we also admit that unmet demand in one cycle can be satisfied by orders in subsequent periods, i.e., we allow backorders. Thus, we should compute a joint cost function integrating the ordering, holding and backordering costs at each location. Unfortunately, obtaining this expression is, in general, a quite complex task since the computation of the average inventory at the warehouse is not straightforward. Hence we focus our attention on a special class of ordering policies, namely single-cycle policies, which simplifies the formulation of the problem. In particular, we show that after some rearranging and calculations the formulation of the one-warehouse NN-retailer problem with power demand pattern and backorders yields the formulation of the basic one-warehouse NN-retailer problem. In addition, it is also shown that the expression of the total average cost generalizes the one obtained for the single-level case.
In the next section we introduce the notation required to state the problem. In Section 3 the problem is formulated as a mixed non linear programming problem by using single-cycle policies. In addition, we show that all methods that have been proposed in the literature to solve the one-warehouse NN-retailer problem with constant demand and without backorders can be easily modified to solve the problem addressed in this paper. Finally, in Section 4, concluding remarks are reported and future research is proposed.
