Dynamic supply chain design with inventory

Abstract

In this paper, we deal with a facility location problem where we build new facilities or close down already existing facilities at two different distribution levels over a given time horizon. In addition, we allow to carry over stock in warehouses between consecutive periods. Our model intends to minimize the total costs, including transportation and inventory holding costs for products as well as fixed and operating costs for facilities.

After formulating the problem, we propose a Lagrangian approach which relaxes the constraints connecting the distribution levels. A procedure is developed to solve the resulting, independent subproblems and, based on this solution, to construct a feasible solution for the original problem.

Introduction

Discrete location problems are an important group of problems within operational research. Especially in the context of strategic supply chain management, location problems experience more and more attention (see [1], [2], [3]).

A supply chain network comprises a number of facilities (e.g., manufacturing plants, distribution centers, warehouses, etc.) that perform a set of operations ranging from the acquisition of raw materials, the transformation of these materials into intermediate and finished products, to the distribution of the finished goods to the customers. Fig. 1 shows an example of a network with suppliers, plants, warehouses, and customers. Arrows indicate that products are shipped between two facilities in a particular time period. Note that operating facilities can change from one time period to another. In the following, we will concentrate on the distribution part (without procurement); therefore, we deal with a two-echelon network structure. The optimization of the complete logistics network is accomplished through efficient planning decisions. Strategic decisions, on the one hand, include facility location, among others, and have a long-lasting effect on a company. On the other hand, the transportation pattern to be followed in each time period is considered as a tactical decision.

Most location models deal with the redesign of supply chain networks by deciding which existing facilities should be closed and where new facilities should be established. However, during this redesign process, one is often faced with the problem of how to transform an initial supply chain structure into a new one. For example, a company wishes to adapt the locations of its warehouses throughout Europe to meet changes in the customer behavior. This process has to be completed in five years, where the initial and the desired final state are known, and a time-dependent dynamic solution which transforms one into the other is sought. Hence, multi-period models have to be employed in order to address these questions about the redesign process. To cope with these types of problems, several approaches have already been proposed in the literature (see [4], [5], [6], [7]). However, some typical features of supply chain networks, such as a multi-echelon structure or capacity and inventory aspects, have been considered only partially in the literature until now (see [5], [6], [8], [9], [10], [11], [12], [13]). A recent modeling paper that covers most of the above-mentioned issues but that does not go into algorithmic considerations is Melo et al. [14].

In this paper, we investigate a dynamic two-echelon multi-commodity location model where potential new facilities can be opened and existing facilities can be closed. Assuming a high fixed cost for establishing and closing a facility, we do not allow a facility which has been closed once to be reopened and consequently, a facility opened during the planning horizon cannot be closed again. This model is an extension of the problem considered in Hinojosa et al. [15], where a multi-period distribution systems of perishable goods has been modeled and no outsourcing has been allowed to cover demand. Under these hypotheses, items do not carry over to consecutive time periods. However, when modeling distribution systems for non-perishable goods, inventories are an essential decision aspect. Moreover, it is unrealistic not to allow outsourcing, because in any distribution system it is nearly always possible to buy products from outside suppliers if capacities are insufficient. Note that this feature also avoids the infeasibility of some patterns of demand that may occur in the previous model. The integration of these two characteristics, inventories and outsourcing, makes the model more general, respectively more realistic, although the mathematical treatment is more difficult. To the best of our knowledge, this model has not been addressed in the literature yet.

The goal of this problem is to minimize the total cost of designing the supply chain network and of the distribution activities in order to fulfill the customer demands. The problem is modeled as a mixed-integer linear program. However, since approaches dealing directly with such formulations lead to extensive computation times, we propose the following alternative solution approach. First, we employ a Lagrangian Relaxation scheme incorporating a dual ascent method to obtain a lower bound on the optimal objective value (see [16], [17], [18], [19], [20], [21] for applications of this method in different contexts). Afterward, based on the solution of the relaxed problem, we construct a heuristic solution, and hence an upper bound, for the problem. At last, this upper bound is improved using an interchange heuristic. Although we address a more general problem than in [15], we obtain solutions of similar quality.

The remaining paper is organized as follows. In Section 2, a mathematical formulation of the problem is presented. In Section 3, we introduce a Lagrangian Relaxation for this model. Section 4 contains the procedure to construct heuristic solutions based on the relaxed ones; moreover, the Interchange method is detailed. Computational results are presented in Section 5. The paper ends with some conclusions, an outlook to future research, and an Appendix where some technical results are included.