Bi-level programming model and solution method for mixed transportation network design problem

All authors:
Haozhi, Z., Gao, Z.
Host organisation: CoE Beijing, P. R. China
Country: China
Publication year: 2009
Published in:
Journal of Systems Science and Complexity, 2009, Vol. 22, pp446‐459.
Research theme: Management and Control, Policy and Planning
Keywords:
bilevel programming, network design, optimal-value function, penalty function method
Type:
Research article
Summary:
By handling the travel cost function artfully, the authors formulate the transportation mixed network design problem (MNDP) as a mixed-integer, nonlinear bilevel programming problem, in which the lower-level problem, comparing with that of conventional bilevel DNDP models, is not a side constrained user equilibrium assignment problem, but a standard user equilibrium assignment problem. Then, the bilevel programming model for MNDP is reformulated as a continuous version of bilevel programming problem by the continuation method. By virtue of the optimal-value function, the lower-level assignment problem can be expressed as a nonlinear equality constraint. Therefore, the bilevel programming model for MNDP can be transformed into an equivalent single-level optimization problem. By exploring the inherent nature of the MNDP, the optimal-value function for the lower-level equilibrium assignment problem is proved to be continuously di®erentiable and its functional value and gradient can be obtained e±ciently. Thus, a continuously di®erentiable but still nonconvex optimization formulation of the MNDP is created, and then a locally convergent algorithm is proposed by applying penalty function method. The inner loop of solving the subproblem is mainly to implement an all-or-nothing assignment. Finally, a small-scale transportation network and a large-scale network are presented to verify the proposed model and algorithm.
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