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Network design problem with congestion effects: A case of bilevel programming.(English)Zbl 0604.90053

The problem under consideration is the minimum cost design of a transportation network where arc capacities and flow variables have to be optimized simultaneously and a user optimal routing has to be produced. The emphasis is on bi-level programming techniques, where the two levels correspond to flow optimization and capacity optimization respectively.
A number of theoretical properties of the optimal solution are derived. Assuming a separable capacity cost function g, and congestion functions on the links S, then the optimization of capacities for fixed flows has a linear objective function. If g is linear and proportional to length and capacity and if the traversal times are homogeneous, nonnegative and proportional to arc length then the system-optimal solution is also user- optimal. Furthermore, the optimal flow is an extremal flow. This last result is extended to the case of concave g.
A number of heuristics are then proposed. (1) Use the capacities of the system-optimal network, and find a user-optimal flow. (2) Alternately solve the capacity and flow problem until convergence. (3) Find a capacity vector for which the system-optimal flow is an equilibrium flow. (4) Generalize 2 and 3 to the case of a convex network optimization problem whose solution is constrained to be user-optimal.
In each case, a worst-case bound is computed for the accuracy of the heuristic with respect to the optimal solution. Computational results are presented for a 9-node 36-arc network. It is found that heuristics 1 and 2 are nearly optimal, and that the others, more complex ones, are not needed.
Reviewer: A.Girard

MSC:

90B10 Deterministic network models in operations research
90C35 Programming involving graphs or networks
65K10 Numerical optimization and variational techniques
90C30 Nonlinear programming
90C08 Special problems of linear programming (transportation, multi-index, data envelopment analysis, etc.)

Cite

References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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