A Lagrangian relaxation approach for expansion of a highway network

This paper deals with the problem of improving an existing road network in the context of strategic planning through the creation of new highway corridors. To address this problem we analyse three mixed integer programming models. The so‐called [P1] is the classical capacitated multicommodity network design model. The model named [P2] imposes on [P1] the location of a single (main path) highway corridor in the road network and [P3] adds to [P2] a set of sub‐tour breaking constraints. The stated goal is to minimize the total travel time for a known origin‐destination demand matrix with a given budget. In this paper we propose an efficient method for [P3], based on a Lagrangian relaxation, to obtain easily‐solved sub‐problems. A cutting‐plane method for solving the Lagrangian sub‐problems is proposed. This method generates valid cuts until an optimal solution is found. The Lagrangian dual problem is solved using the sub‐gradient optimization method. A case study has been carried out for the region of Castilla‐La Mancha (Spain). Computational comparisons between the proposed method and a state‐of‐the‐art mixed‐integer code are presented. The Lagrangian relaxation approach is found to be capable of generating good feasible solutions to the case study within a reasonable computational time.