On The Primal And Dual Formulations For Elastic-demand Stochastic Traffic Assignment Problems

In an elastic-demand stochastic traffic assignment problem,the equilibrium conditions are achieved on two levels,namely,the stochastic user equilibrium or system optimum on the path level and the supply-demand equilibrium on the origin-destination level.By recognizing both equilibrium levels imply random individual perceptions and decision makings,this paper reinvestigates the mathematical formulations of this kind of problems in both the system optimum and user equilibrium principles.Different from previous research that was devoted to the development of solution methods for some specific versions of traffic assignment problems,our focus is given to a pair of new generalized formulations that have a duality relationship to each other.We found that the equilibrium or optimality conditions of an elastic-demand stochastic traffic assignment problem can be redefined as a combination of three sets of equations and an arbitrary feasible solution of either the primal or dual formulation satisfies only two of them.We also rigorously proved the solution equivalency and uniqueness of both the primal and dual formulations,by using derivative-based techniques.While the two formulations pose their respective modeling advantages and drawbacks,our preliminary algorithmic analysis and numerical test results indicate that in terms of solution efficiency,the dual formulation-based algorithm,i.e.,the Cauchy algorithm,can be more readily implemented for large-scale problems and converge evidently faster than the primal formulation-based one,i.e.,the Frank-Wolfe algorithm.