List Of Pareto Optimal Solutions of a Biobjective Shortest Path Problem

Many applications in practice involve the search for a shortest path in a network by optimizing two conflicting objective functions. Such problems often are referred to as biobjective optimization problems. Their goal is to find special optimal paths that are nondominated and are also known in the specialized literature as to as Pareto optimal. While most of the existing methods aim to find the minimum complete set of Pareto optimal paths, we propose an approach that is able to generate a list of all Pareto optimal solutions in a given network. The described method solves the biobjective optimization problem in the case in which the first objective function is a linear (MINSUM), while the second objective function is from the “bottleneck” type (MAXMIN). The presented approach is based on two modifications of the Dijkstra's shortest path algorithm that solve the MINSUM and the MAXMIN problems respectively. We prove the correctness and the computational complexity of the presented algorithms. Also, we provide detailed numerical examples that illustrate their execution.