Lagrangian relaxation and ADMM based decomposition approaches for the road network design problem considering travel time variability

Abstract

Abstract We study an uncapacitated, multi-commodity network design problem with a construction budget constraint and a concave objective function. Instead of minimizing the expected travel time across all edges, the objective minimizes jointly the travel times that are standard deviation above the expected travel time of each commodity. The idea is that the decision-maker wants to minimize the travel times not only on average, but also to keep their variability as small as possible. Thus, another way to view the objective is as a linear combination of the mean and standard deviation of travel times. The proposed mean-standard deviation network design model is actually a nonlinear and concave integer program. At any rate, this problem is significantly harder to solve and cannot be tackled with off-the-shelf mixed-integer linear programming solvers. This study proposes two novel methods which are Lagrangian relaxation (LR) and augmented Lagrangian relaxation (ALR) to tackle this problem. The constraints that link the design with the flow variables are dualized in the objective function, resulting in a series of single-commodity reliable shortest path problems and a knapsack problem. The quadratic penalty terms are extended to the LR, and the alternating direction method of multipliers (ADMM) is introduced to decompose the ALR into routing and design optimizations. The routing optimization can be naturally decomposed into many single-commodity reliable shortest path subproblems solved by the Lagrangian substitution method. The primal heuristic uses the solution of knapsack problems that are solved in LR or ALR to generate upper bounds. The dual problems of LR and ALR are solved with the subgradient optimization method. Some computational results on three networks are present, showing that these methods achieve good integrality gaps.