Stochastic Transportation-Inventory Network Design Problem

In this paper, we study the stochastic transportation-inventory network design problem involving one supplier and multiple retailers. Each retailer faces some uncertain demand. Due to this uncertainty, some amount of safety stock must be maintained to achieve suitable service levels. However, risk-pooling benefits may be achieved by allowing some retailers to serve as distribution centers (and therefore inventory storage locations) for other retailers. The problem is to determine which retailers should serve as distribution centers and how to allocate the other retailers to the distribution centers. Shen et al. (2000) and Daskin et al. (2001) formulated this problem as a setcovering integer-programming model. The pricing subproblem that arises from the column generation algorithm gives rise to a new class of submodular function minimization problem. They only provided efficient algorithms for two special cases, and assort to ellipsoid method to solve the general pricing problem, which run in O(n7log(n)) time, where n is the number of retailers. In this paper, we show that by exploiting the special structures of the pricing problem, we can solve it in O(n2 log n) time. Our approach implicitly utilizes the fact that the set of all lines in 2-D plane has low VC-dimension. Computational results show that moderate size transportation-inventory network design problem can be solved efficiently via this approach.