Multiobjective Optimization in Transportation: The Case of Equilibrium Network Design

Abstract

In this paper we offer a statement of the continuous multiobjective equilibrium network design problem and discuss how it may be solved by an adaptation of Geoffrion’s interactive method. The formulation is fully multimodal and employs no symmetry restrictions.

The equilibrium network design problem is concerned with selecting those link additions or improvements which minimize system-wide costs subject to flow conservation constraints and a budget constraint, together with the requirement that the flow pattern be in accord with Wardrop’s first principle or user optimization rule. This problem may be formulated with either discrete or continuous decision variables, depending on one’s assumptions regarding the divisibility of link improvements. Leblanc (1975) is the classic reference for the discrete case; Abdulaal and Leblanc (1979) give a discussion of the continuous case. In this paper we will be concerned solely with the continuous formulation.

The model presented here differs from other equilibrium network design models in that it explicitly treats multiple objectives. A thorough study of the multiobjective network design problem without user equilibrium restrictions was made be Agarwal (1973). Agarwal’s work may be characterized as a “vector systems optimization model” and is the natural extension of Wardrop’s second principal or systems optimization rule to a multiobjective setting. No other significant literature on multiobjective network design problems exists. It is well known that user optimization and systems optimization yield distinct results for networks with flow dependent costs. Moreover, it is generally agreed that user optimization is the better framework for network planning and design; consequently the model reported in this paper fills a significant gap in the literature.