Joint location–inventory problem with economies and diseconomies of scale: A general model and an efficient column generation algorithm

Abstract

This paper studies the location–inventory problem, which jointly optimizes the location, inventory, and allocation of distribution centers (DCs) to serve uncertain demand of multiple retailers. Problems of this kind have been extensively studied to carefully tradeoff various costs, including the fixed facility location cost and the variable facility operating cost Most existing works assumed the facility operating cost to be either a linear or a concave function of the volume of demand allocated. However, practical evidence suggests that when the volume of allocated demand exceeds a certain level, the marginal cost may increase due to overtime charge or facility congestion, leading to a complicated cost function. In this paper, we capture the facility operating cost as a general inverse S-shaped function of the volume of allocated demand, which is first concave due to economies of scale and then convex due to diseconomies of scale. We formulate this problem as a set-covering model and propose a column generation (CG) algorithm to solve its linear relaxation. The corresponding pricing problem has nice structural properties and can be solved efficiently by a branch-and-bound (B&B) method. The efficiency of the proposed algorithm is validated through extensive numerical studies, where two types of inverse S-shaped functions are considered to capture facility operating cost under various situations. Sensitivity analysis is also carried out to reveal managerial insights and analyze the delicate tradeoffs between several cost components.