Marcotte, P. Network design problem with congestion effects: A case of bilevel programming. (English) Zbl 0604.90053 Math. Program. 34, 142-162 (1986). 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 Cited in 3 ReviewsCited in 53 Documents 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.) Keywords:Stackelberg games; decomposition; minimum cost design; transportation network; user optimal routing; bi-level programming; flow optimization; capacity optimization; worst-case bound; heuristic × Cite Format Result Cite Review PDF Full Text: DOI References: [1] H.A. Aashtiani and T.L. Magnanti, ”Equilibria on a congested transporation network”,SIAM Journal on Algebraic and Discrete Methods 2 (1981) 213–226. · Zbl 0501.90033 · doi:10.1137/0602024 [2] M. Abdulaal and L.J. LeBlanc, ”Continuous equilibrium network design models”,Transportation Research B 13B (1979) 19–32. · Zbl 0398.90042 · doi:10.1016/0191-2615(79)90004-3 [3] J.F. Bard, ”An Algorithm for solving the general bilevel programming problem”,Mathematics of Operations Research 8 (1983) 260–272. · Zbl 0516.90061 · doi:10.1287/moor.8.2.260 [4] J.F. Bard and J.E. Falk, ”An explicit solution to the multilevel programming problem”,Computers and Operations Research 9 (1982) 77–100. · doi:10.1016/0305-0548(82)90007-7 [5] D. Bertsekas and E.M. Gafni, ”Projection methods for variational inequalities with application to the traffic assignment problem”,Mathematical Programming Study 17 (1982) 139–159. · Zbl 0478.90071 [6] W.F. Bialas and M.H. Karwan, ”On two-level optimization”,IEEE Transactions on Automatic Control AC-27 (1982) 211–214. · Zbl 0487.90005 [7] J.W. Blankenship and J.E. Falk, ”Infinitely constrained optimization problems”,Journal of Optimization Theory and Applications 19 (1976) 261–281. · Zbl 0307.90071 · doi:10.1007/BF00934096 [8] W. Candler and R.J. Townsley, ”A linear two-level programming problem”,Computers and Operations Research 9 (1982) 59–76. · doi:10.1016/0305-0548(82)90006-5 [9] S.C. Dafermos, ”Traffic assignment and resource allocation in transportation networks”, Ph.D. Dissertation, Johns Hopkins University (Baltimore, Maryland, 1968). · Zbl 0183.37701 [10] S.C. Dafermos, ”Traffic equilibrium and variational inequalities”,Transportation Science 14 (1980) 42–54. · doi:10.1287/trsc.14.1.42 [11] C. Daganzo, ”Stochastic network equilibrium with multiple vehicle types and asymmetric, indefinite link cost Jacobians”,Transportation Science 17 (1983) 282–300. · doi:10.1287/trsc.17.3.282 [12] G.B. Dantzig et al., ”Formulating and solving the network design problem by decomposition”,Transportation Research B 13B (1979) 5–18. · doi:10.1016/0191-2615(79)90003-1 [13] R. Dionne and M. Florian, ”Exact and approximate algorithms for optimal network design”,Networks 9 (1979) 37–50. · Zbl 0397.94024 · doi:10.1002/net.3230090104 [14] M. Florian, ”An improved linear approximation algorithm for the network equilibrium (Packet Switching) problem”,Proceedings of the IEEE Conference on Decision and Control (1977), 812–818. [15] A. Haurie and P. Marcotte, ”On the relationship between Nash and Wardrop equilibrium”,Networks 15 (1985) 295–308. · Zbl 0579.90030 · doi:10.1002/net.3230150303 [16] H.H. Hoang, ”A computational approach to the selection of an optimal network”,Management Science 19 (1973) 488–498. · Zbl 0249.90024 · doi:10.1287/mnsc.19.5.488 [17] L.J. LeBlanc, ”Mathematical programming algorithms for large scale network equilibrium and network design problems”, Ph.D. Dissertation, Northwestern University (Evanston, IL, 1973). [18] M. Los, ”A discrete-convex programming approach to the simultaneous optimization of land-use and transportation”,Transportation Research B 13B (1979) 33–48. · doi:10.1016/0191-2615(79)90005-5 [19] P. Marcotte, ”Network optimization with continuous control parameters”,Transportation Science 17 (1983) 181–197. · doi:10.1287/trsc.17.2.181 [20] P. Marcotte, ”Design optimal d’un réseau de transport en présence d’effets de congestion”, Ph.D. Thesis, Université de Montréal (Montréal, Canada, 1981). [21] S. Nguyen, ”An algorithm for the traffic assignment problem”,Transportation Science 8 (1974) 203–216. · doi:10.1287/trsc.8.3.203 [22] G. Papavassilopoulos, ”Algorithms for leader-follower games”,Proceedings of the 18th Annual Allerton Conference on Communication Control and Computing (1980) 851–859. [23] G. Papavassilopoulos, ”Algorithms for static Stackelberg games with linear costs and polyhedral constraints”,Proceedings of the 21st IEEE Conference on Decisions and Control (1982) 647–652. [24] M.J. Smith, ”The existence, uniqueness and stability of traffic equilibrium”,Transportation Research B 13B (1979) 295–304. · doi:10.1016/0191-2615(79)90022-5 [25] P.A. Steenbrink,Optimization of transportation networks (Wiley, New York, 1974). · Zbl 0329.14007 [26] H.N. Tan, S.B. Gershwin and M. Athans, ”Hybrid optimization in urban traffic networks”, MIT Report DOT-TSC-RSPA-79–7 (1979). [27] J.G. Wardrop, ”Some theoretical aspects of road traffic research”,Proceedings of the Institute of Civil Engineers, Part II 1 (1952) 325–378. [28] T.L. Magnanti and R.T. Wong, ”Network design and transporation planning–Models and algorithms”,Transportation Science 18 (1984) 1–55. · doi:10.1287/trsc.18.1.1 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.