This paper for the first time presents a novel model to simultaneously optimize location, allocation, capacity, inventory, and routing decisions in a stochastic supply chain system. Each customer’s demand is uncertain and follows a normal distribution, and each distribution center maintains a certain amount of safety stock. To solve the model, first we present an exact solution method by casting the problem as a mixed integer convex program, and then we establish a heuristic method based on a hybridization of Tabu Search and Simulated Annealing. The results show that the proposed heuristic is considerably efficient and effective for a broad range of problem sizes.
A key driver of the overall productivity and profitability of a supply chain is its distribution network which can be used to achieve a variety of the supply chain objectives ranging from low cost to high responsiveness. Designing a distribution network consists of three subproblems: location–allocation problem, vehicle routing problem, and inventory control problem. Because of high dependency among these problems, in the literature there are several papers integrating two of the above problems: location–routing problems, inventory–routing problems, and location–inventory problems. Location–routing problems are surveyed and classified by Min et al., 1998 and Nagy and Salhi, 2007. Inventory–routing problems are studied in several papers, e.g. Baita et al., 1998, Jaillet et al., 2002, Kleywegt et al., 2002, Adelman, 2004, Gaur and Fisher, 2004, Zhao et al., 2008, Yu et al., 2008, Oppen and Loketangen, 2008 and Day et al., 2009. Also, several papers considered location–inventory problems, e.g. Erlebacher and Meller, 2000 and Daskin et al., 2002 and Shen (2005).
Recently, Shen and Qi (2007) modified the inventory–location model given in Daskin et al. (2002). The objective function of their model is the sum of the inventory–location cost and an approximate routing cost which depends only on the locations of the opened distribution centers. They showed that significant cost saving can be obtained by their model in comparison with the sequential approach. However, their model optimizes only the inventory and location decisions and does not determine transportation decisions. Furthermore, there is no guarantee that their model can be used for real-world cases since their approximation method is applicable only under some restrictive assumptions.
In this paper, for the first time we present a model which simultaneously optimizes location, allocation, capacity, inventory and routing decisions without any approximation. To solve the problem, first we present an optimal solution method by expressing the problem as a mixed integer convex program. Since location–routing problems have been shown to be NP-hard (Perl and Daskin, 1985), our problem belongs to the class of NP-hard problems too. Hence, in the following to solve the large-sized instances, a heuristic method is developed. The heuristic method is decomposed into two stages: constructive stage and improvement stage. In the constructive stage an initial solution is built at random. In the improvement stage we have two phases: location phase and routing phase, and a hybrid algorithm based on Tabu Search and Simulated Annealing is used to improve the initial solution in each phase.
The remainder of the paper is organized as follows. In Section 2, the mathematical formulation of the problem is given. Section 3 presents the solution methods for solving the problem. Section 4 studies the model under extra constraints. The computational results are presented in Section 5. We conclude the paper in Section 6.
Design of a supply chain distribution network consists of three major problems: distribution center location problem, vehicle routing problem and inventory control problem. In the literature, there are three problems considering the integration of two of the above problems: inventory–routing problem, location–routing problem and location–inventory problem. Recently, Shen and Qi (2007) presented an inventory–location model with the objective function including the location-inventory cost and an approximate routing cost which depend only on the locations of the opened distribution centers. They showed that significant cost saving can be obtained by their model instead of sequential approach. However, their model cannot determine the routing decisions; additionally, there is no guarantee that their model can be exploited for real-world cases since the approximation of routing cost is appropriate only under some assumptions, and there is no error analysis for the approximation.
The main contribution of this paper to the literature is to simultaneously optimize location, allocation, capacity, inventory and routing decisions without any approximation. We show that the expected saving achieved by our model is big compared to the model in Shen and Qi (2007). The range of improvement is 9.78–27.90% for different problem sizes.
We show that our model can be reformulated as a mixed integer convex program. Also, we present an effective heuristic method. The heuristic method is decomposed into two stages: constructive stage, where an initial solution is built at random, and improvement stage, where the solution is iteratively improved in two phases: location phase and routing phase. In the improvement stage, a hybrid algorithm based on Tabu Search and Simulated Annealing is used to improve the current solution in each phase. The computational results indicate that the heuristic method is effective for a wide variety of problem sizes and structures. Also, we study the model under extra constraints such as stochastic capacity constraints.
Afterward, we investigate the impacts of the objective weight factors associated with the inventory and routing costs on the number of the opened distribution centers. We observe that, when we increase the weight factor of the routing cost, the number of the opened distribution centers increases, and when we increase the weight factor of the inventory cost, the number of the opened distribution centers decreases.
For future work, it is interesting to develop more effective and elegant heuristic methods to solve the model. Moreover, the model can be extended in several realistic and practical directions.
