A Probabilistic Tree-Based Representation for Non-convex Minimum Cost Flow Problems

Network flow optimisation has many real-world applications. The minimum cost flow problem (MCFP) is one of the most common network flow problems. Mathematical programming methods often assume the linearity and convexity of the underlying cost function, which is not realistic in many real-world situations. Solving large-sized MCFPs with nonlinear non-convex cost functions poses a much harder problem. In this paper, we propose a new representation scheme for solving non-convex MCFPs using genetic algorithms (GAs). The most common representation scheme for solving the MCFP in the literature using a GA is priority-based encoding, but it has some serious limitations including restricting the search space to a small part of the feasible set. We introduce a probabilistic tree-based representation scheme (PTbR) that is far superior compared to the priority-based encoding. Our extensive experimental investigations show the advantage of our encoding compared to previous methods for a variety of cost functions.