Multicriteria spatial price equilibrium network design: Theory and computational results

In this paper we consider the problem of determining the optimal design of a transportation network using a vector valued criterion function when the flow pattern is assumed to correspond to a spatial price equilibrium. This problem arises in rail and freight network design, where the spatial price equilibrium is a better behavioral description than the Wardropian user equilibrium characteristic of urban transportation applications. We describe two alternative heuristic solution techniques for the multicriteria spatial price equilibrium network design problem. The first is based on iteration between a pure spatial price equilibrium model and a vector optimization model with only nonnegativity constraints. The second solution technique employs the Hooke and Jeeves algorithm for nonlinear programming to solve a vector optimization model with implicit constraints guaranteeing a spatial price equilibrium flow pattern. In these solution procedures, rather than represent the equilibrium problem as a mathematical program, as is normally done for the Wardropian traffic assignment problems used in urban applications, we employ an original nonlinear complementarity formulation of the spatial price equilibrium problem written entirely in terms of nodal and arc variables and solved extremely efficiently through the iterative application of a linear complementarity algorithm. The nonlinear complementarity formulation allows us to address problems with asymmetric transportation cost, commodity demand and commodity supply functions without the specialized diagonalization/relaxation algorithms required by other approaches.