An exact and polynomial approach for a bi-objective integer programming problem regarding network flow routing

•Load balancing strategies are needed to improve performance, and make effective use of the network capillarity.•The unfairness in the path length caused by the load balancing has little effect in general.•Our exact approach is a valuable tool for evaluating heuristic solutions. This paper presents a bi-objective network flow routing problem regarding network load balancing and flow path length. The problem is formulated within the integer programming framework and an exact and polynomial approach based on the ϵ-constraint technique is presented. In each iteration, our algorithm solves a single-objective linear integer programming subproblem. The goal of our approach is to obtain a minimal complete set of Pareto-optimal solutions. Our proposal was evaluated through several computational experiments, which included grid and random networks. The random topology was generated by the Barabási–Albert model and the settings of the network flows defined here are those commonly used in wireless sensor networks and wireless mesh networks. The analysis of the computational results provide a decision maker with valuable information about which factors most affect the solutions, for example, the smallest and largest bottleneck, the size of increment in the shortest path of the last Pareto-optimal solution and the difference between the path lengths of the first and last Pareto-optimal solutions generated.