An integer linear programming formulation and heuristics for the minmax relative regret robust shortest path problem

The well-known Shortest Path problem (SP) consists in finding a shortest path from a source to a destination such that the total cost is minimized. The SP models practical and theoretical problems. However, several shortest path applications rely on uncertain data. The Robust Shortest Path problem (RSP) is a generalization of SP. In the former, the cost of each arc is defined by an interval of possible values for the arc cost. The objective is to minimize the maximum relative regret of the path from the source to the destination. This problem is known as the minmax relative regret RSP and it is NP-Hard. We propose a mixed integer linear programming formulation for this problem. The CPLEX branch-and-bound algorithm based on this formulation is able to find optimal solutions for all instances with 100 nodes, and has an average gap of 17 % on the instances with up to 1,500 nodes. We also develop heuristics with emphasis on providing efficient and scalable methods for solving large instances for the minmax relative regret RSP, based on Pilot method and random-key genetic algorithms. To the best of our knowledge, this is the first work to propose a linear formulation, an exact algorithm and metaheuristics for the minmax relative regret RSP.