A Dual-Ascent Procedure for Large-Scale Uncapacitated Network Design

The fixed-charge network design problem arises in a variety of problem contexts including transportation, communication, and production scheduling. We develop a family of dual-ascent algorithms for this problem. This approach generalizes known ascent procedures for solving shortest path, plant location, Steiner network and directed spanning tree problems. Our computational results for several classes of test problems with up to 500 integer and 1.98 million continuous variables and constraints show that the dual-ascent procedure and an associated drop-add heuristic generate solutions that, in almost all cases, are guaranteed to be within 1 to 4% of optimality. Moreover, the procedure requires no more than 150 seconds on an IBM 3083 computer. The test problems correspond to dense and sparse networks, including some models that arise in freight transport.

[1]  Thomas L. Magnanti,et al.  Network Design and Transportation Planning: Models and Algorithms , 1984, Transp. Sci..

[2]  J. Krarup,et al.  Sharp Lower Bounds and Efficient Algorithms for the Simple Plant Location Problem , 1977 .

[3]  Thomas L. Magnanti,et al.  Valid inequalities and facets of the capacitated plant location problem , 1989, Math. Program..

[4]  Yves Pochet.  Valid inequalities and separation for capacitated economic lot sizing , 1988 .

[5]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[6]  Michael Florian,et al.  Exact and approximate algorithms for optimal network design , 1979, Networks.

[7]  H. P. Williams.  Experiments in the formulation of integer programming problems , 1974 .

[8]  Tony J. Van Roy,et al.  A Dual-Based Procedure for Dynamic Facility Location , 1982 .

[9]  George L. Nemhauser,et al.  Note--On "Location of Bank Accounts to Optimize Float: An Analytic Study of Exact and Approximate Algorithms" , 1979 .

[10]  A. M. Geoffrion,et al.  Multicommodity Distribution System Design by Benders Decomposition , 1974 .

[11]  Anantaram Balakrishnan,et al.  LP extreme points and cuts for the fixed-charge network design problem , 1987, Math. Program..

[12]  K. Spielberg,et al.  On a Production Allocation and Distribution Problem , 1978 .

[13]  G. Nemhauser,et al.  Exceptional Paper—Location of Bank Accounts to Optimize Float: An Analytic Study of Exact and Approximate Algorithms , 1977 .

[14]  John A. Tomlin,et al.  An integer programming approach to a class of combinatorial problems , 1972, Math. Program..

[15]  Donald Erlenkotter,et al.  A Dual-Based Procedure for Uncapacitated Facility Location , 1978, Oper. Res..

[16]  Laurence A. Wolsey,et al.  Uncapacitated lot-sizing: The convex hull of solutions , 1984 .

[17]  Laurence A. Wolsey,et al.  Lot-size models with backlogging: Strong reformulations and cutting planes , 1988, Math. Program..

[18]  Thomas L. Magnanti,et al.  Tailoring Benders decomposition for uncapacitated network design , 1986 .

[19]  J. Billheimer,et al.  Network Design with Fixed and Variable Cost Elements , 1973 .

[20]  T. B. Boffey,et al.  Solving the optimal network problem , 1979 .

[21]  Marc Los,et al.  Combinatorial Programming, Statistical Optimization and the Optimal Transportation Network Problem , 1980 .

[22]  Hoang Hai Hoc.  A Computational Approach to the Selection of an Optimal Network , 1973 .

[23]  Warren B. Powell,et al.  A LOWER BOUND FOR UNCAPACITATED, MULTICOMMODITY FIXED CHARGE NETWORK DESIGN PROBLEMS , 1986 .

[24]  Thomas L. Magnanti,et al.  Facets and algorithms for capacitated lot sizing , 1989, Math. Program..

[25]  Anantaram Balakrishnan,et al.  Valid inequalities and algorithms for the network design problem with an application to LTL consolidation , 1984 .

[26]  R. Weischedel,et al.  Optimal Network Problem: A Branch-and-Bound Algorithm , 1973 .

[27]  Richard T. Wong,et al.  Worst-Case Analysis of Network Design Problem Heuristics , 1980, SIAM J. Algebraic Discret. Methods.

[28]  M. Fisher,et al.  A multiplier adjustment method for the generalized assignment problem , 1986 .

[29]  Thomas L. Magnanti,et al.  TRANSPORTATION PLANNING: NETWORK MODELS AND THEIR IMPLEMENTATION , 1978 .

[30]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[31]  Donald Ervin Knuth,et al.  The Art of Computer Programming, Volume II: Seminumerical Algorithms , 1970 .

[32]  Jan Karel Lenstra,et al.  The complexity of the network design problem , 1978, Networks.

[33]  Richard T. Wong,et al.  A dual ascent approach for steiner tree problems on a directed graph , 1984, Math. Program..

[34]  Thomas L. Magnanti,et al.  Accelerating Benders Decomposition: Algorithmic Enhancement and Model Selection Criteria , 1981, Oper. Res..

[35]  Laurence A. Wolsey,et al.  Strong Formulations for Multi-Item Capacitated Lot Sizing , 1984 .