Models and column generation approach for the resource-constrained minimum cost path problem with relays

In this paper, we introduce the resource-constrained minimum cost path problem with relays (RMCPR), which deals with multiple-resource constraint on the path with relays. The RMCPR consists of finding a path from a source to a destination, and locating relays on some nodes of the path, such that the total cost of setting arcs and relays is minimized, and for each resource, the total consumption between the source and the first relay, any two consecutive relays, and the last relay and the destination do not exceed a predefined upper bound of resource consumption. The RMCPR is a single-commodity resource-constrained network design problem with relays.
We present a pattern-chain formulation and develop a column generation based exact approach for the RMCPR. We design a Lagrangian relaxation based method to efficiently price out columns to enter the basis. We present computational results on three sets of 560 randomly generated instances with different properties. Computational results demonstrate that our proposed algorithm is an efficient exact method for solving the RMCPR.