Distribution network design with service mode selection

Abstract

Classical distribution network design models assume exogenous demand and gross profit rate of serving demand. In practice, however, both demand and profitability depend on various factors such as delivery method, selling price, and after-sales terms. To address this deficiency, this study adopts the service mode options to model these factors that have an impact on demand and profitability. In particular, the selection of a service mode for each served customer is considered in the strategic design of single- and two-echelon distributions networks, respectively. This leads to three profit maximization location-inventory models.

The basic single-echelon model considers the case that each customer demands a single product at a constant rate depending on the service mode used to serve this customer. An integer program with a nonlinear objective function is formulated to maximize the network-wide profit by jointly determining the location of distribution centers (DCs), the assignment of customers to DCs, the inventory decision of the DCs, and the service mode to serve each customer. To cope with the computational challenge brought by the nonlinear objective function, this study investigates the properties of the extended polymatroid inequalities based on submodular functions. It is found that the nonlinear terms in the objective function of the proposed model can be linearized by introducing an exponential number of the polymatroid inequalities. A cutting plane approach is then constructed to solve the linearized model with exponentially many constraints. By exploiting the structural properties of the model, a greedy algorithm is developed to solve the separation problem for the cutting plane approach in O(|I|^2|G|), where I and G represent the sets of the customers and the service modes, respectively.

Based on the basic single-echelon model, this study then investigates the generalised problem in which the customers have stochastic demands of multiple products. Through decomposing the objective function into components with submodularity, this study linearizes the objective function using polymatroid inequalities and applies the cutting plane approach to solve the resulting model. Although the master problem in each cutting plane iteration has more binary decision variables than the basic model, the separation problem is simplified and can be solved by a greedy algorithm in O(|I||G|).

The basic single-echelon model is further extended to the case in which inventories are held at both the DCs and the customer sites, resulting in the two-echelon model. As the total profit of the network can only be maximized when the DCs and the customers coordinate their ordering decisions, the one-warehouse multi-retailer model proposed by Roundy (1985) is employed to optimize the inventory decisions of an open DC and the set of customers served by this DC under given service modes. The submodular property is preserved for the resulting two-echelon inventory and distribution network design model with service mode selection. A cutting plane procedure is proposed to solve the network design model based on polymatroid inequalities. The corresponding separation problem can be solved in O(|I|^2|G|^2) by a tailor-made algorithm utilizing the specific structure of the optimal solution to the separation problem.