A Fast Calculation Method for the Partial Group of All Pareto Solutions at a Three-Objective Network

Abstract

The multi-objective network model can be applied to common problems with many conditions, for example, optimal routing of a network connection to the Internet services, optimally scheduling of a production and distribution systems, and multistage-structured modeling for supply chain management systems, etc. The shortest path problem is an optimization problem for finding the shortest path connecting two specific nodes of a directed or undirected graph. When considering not only the distances between the nodes but also other information, for example, toll, fuel cost, or gradient, the problem is formulated as a multi-objective shortest path problem that involves multiple conflicting objective functions. To solve the optimization problem of multi-objective network is difficult because the multiple objectives have to be simultaneously optimized. Thus, few algorithms for this problem have been proposed. In this study, we use a three-objective shortest path problem to find the shortest path between two terminal nodes on a network, and we propose three efficient algorithms for obtaining the Pareto solutions based on the extended Dijkstra’s algorithm. We use our algorithms to find the partial group of all Pareto solutions. The results of the numerical experiments suggest that the proposed algorithms reduce the computing time and the memory size for obtaining the Pareto solutions. In addition, we show the algorithms could find almost set of solution in the Pareto solutions.