L-shaped Decomposition Method for Multi-stage Stochastic Concentrator Location Problem

A stochastic version of a concentrator location problem is dealt with in which traffic demand at each terminal location is uncertain. The concentrator location problem is defined as to determine the following: (i) the numbers and locations of concentators that are to be open, and (ii) the allocation of terminals to concentrator sites. The problem is formulated as a stochastic multi-stage integer linear program, with first stage binary variables concerning network design and continuous recourse variables concerning expansion of capacity. Given a first stage decision, the series of realization of traffic demand may possibly imply a violation of the capacity constraint of the concentrator. Therefore from the second stage to the last stage, recourse action is taken to correct the violation. The objective function minimizes the cost of connecting terminals and the cost of opening concentrators and the expected recourse cost of capacity expansion. We propose a new algorithm which combines an L-shaped method and a branch-and-bound method. Under some assumptions it decomposes the problem into a set of problems as many as the number of stages in parallel. Finally we demonstrate the computational efficiency of our algorithm for the multi-stage model.