Most existing facility location and network design models assume the facility cost as either a fixed cost term or a linear/concave function of the volume of demand assigned to a facility. In particular, a concave cost function implies decreasing marginal cost and exhibits economies of scale. However, when the assigned demand volume exceeds a certain level, diseconomies of scale may occur due to facility congestions, overuse of resources, increasing operational complexity, and so on, leading to a convex cost function with increasing marginal cost. This paper studies a warehouse–retailer network design problem, which simultaneously optimizes the warehouse locations, the warehouse–retailer assignment, and the multi-echelon inventory replenishment policy to minimize the total cost, including the warehouse location and operating costs, the inventory replenishment cost, and the transportation cost. In this paper, the warehouse operating cost is defined as a general function of the volume of demand assigned to a warehouse, which could be concave first 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 algorithm to solve its linear relaxation. The pricing problem of the column generation process, although formulated as a nonlinear mixed integer program, has nice structural properties and can be solved efficiently by a branch-and-bound method. To validate the effectiveness and efficiency of the proposed column generation algorithm, we conduct numerical studies based on randomly generated instances. Numerical results also demonstrate the impact of economies and diseconomies of scale on network design decisions and warehouse operating cost.

现有的大多数设施位置和网络设计模型都将设施成本假定为固定成本项或分配给设施的需求量的线性/凹函数。特别地，凹成本函数意味着边际成本的降低，并表现出规模经济。但是，当分配的需求量超过一定水平时，由于设施拥挤，资源过度使用，操作复杂性增加等原因，可能会导致规模不经济，导致边际成本增加的凸成本函数。本文研究了一个仓库-零售商网络设计问题，该问题同时优化了仓库位置，仓库-零售商分配和多级库存补货策略，以使总成本（包括仓库位置和运营成本）最小化，库存补货成本​​和运输成本。在本文中，将仓库运营成本定义为分配给仓库的需求量的一般函数，由于规模经济，它可能首先是凹入的，而由于规模不经济性，它可能会凸出。我们将此问题公式化为集覆盖模型，并提出了一种解决其线性松弛问题的列生成算法。列生成过程的定价问题尽管被表述为非线性混合整数程序，但具有良好的结构特性，可以通过分支定界方法有效地解决。为了验证所提出的列生成算法的有效性和效率，我们基于随机生成的实例进行了数值研究。