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

Abstract

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.

NSGA-II for Solving Multiobjective Integer Minimum Cost Flow Problem with Probabilistic Tree-Based Representation

“…However, the PbR method has serious drawbacks. To counteract the limitations of PbR, the PTbR (Probabilistic Tree based Representation) is introduced in [8]. The PTbR chromosome has n-1 sub-chromosomes (Sub.Ch) and the value of each gene is a random number between 0 and 1 which is then accumulated to 1 in each sub-chromosome.…”

“…Then, two parents swap the selected sub-chromosomes to generate new offspring. To perform mutation, a random parent is selected and the randomly chosen sub-chromosome is regenerated to create a new offspring [8].…”

“…Note that, for each node size (n), five different networks are randomly generated. The number of supply/demand for nodes 1/n are set to q=20/-20 in the test instances up to 20 nodes and for all other test problems supply/demand are set to q=30/-30 [8]. The parameter settings for NSGA-II are as follows: maximum number of iterations (It max =200), population size (pop size = min{n × 10, 300}), crossover rate (P c =0.95), mutation rate (P m =0.3), maximum number of function evaluations (N F Es=100,000) and the termination criterion β = 30 [8].…”

“…The number of supply/demand for nodes 1/n are set to q=20/-20 in the test instances up to 20 nodes and for all other test problems supply/demand are set to q=30/-30 [8]. The parameter settings for NSGA-II are as follows: maximum number of iterations (It max =200), population size (pop size = min{n × 10, 300}), crossover rate (P c =0.95), mutation rate (P m =0.3), maximum number of function evaluations (N F Es=100,000) and the termination criterion β = 30 [8]. For NSGA-II(H) only α = 10% of the initial individuals are generated using the heuristic method explained in Section 3.2 and the rest are generated randomly.…”

Probabilistic tree-based representation for solving minimum cost integer flow problems with nonlinear non-convex cost functions

“…Building on our preliminary work [32], where only single-source and singlesink network instances were considered, this paper provides a more detailed account of the probabilistic tree-based representation (PTbR) scheme, which provides the key ingredients allowing us to search the entire feasible search space more effectively, as compared with the conventional priority-based representation (PbR) scheme. Furthermore, we introduce an improved decoding procedure that allows more thorough sampling of the search space.…”