Abstract
We present an integer programming model for plant location with inventory costs. The linear programming relaxation has been solved by Dantzig-Wolfe decomposition. In this case the subproblems reduce to the minimum cut problem. We have used subgradient optimization to accelerate the convergence of the D-W algorithm. We present our experience with problems arising in the design of a distribution network for computer spare parts. In most cases, from a fractional solution we were able to derive integer solutions within 4% of optimality.
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R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows, Prentice-Hall, Englewood Cliffs, NJ, 1990.
R. Anbil, Personal communication, 1995.
M.L. Balinski, On finding integer solutions to linear programs, Mathematica, Princeton, NJ, 1964.
C. Barnhart, E. Johnson, G. Nemhauser, M. Savelsbergh, P. Vance, Branch-and-price: Column generation for huge integer programs, Oper. Res., to appear.
B.V. Cherkassky, A.V. Goldberg, On implementing push-relabel method for the maximum flow problem, in: E. Balas, J. Clausen (Eds.), Integer Programming and Combinatorial Optimization, Springer, Berlin, 1995, pp. 157–171.
D.C. Cho, E.L. Johnson, M.W. Padberg, M.R. Rao, On the uncapacitated plant location problem. I. Valid inequalities and facets, Math. Oper. Res. 8 (1983) 579–589.
D.C. Cho, M.W. Padberg, M.R. Rao, On the uncapacitated plant location problem. II. Facets and lifting theorems, Math. Oper. Res. 8 (1983) 590–612.
V. Chvátal, Linear Programming, Freeman, New York, 1993.
G. Cornuejols, M.L. Fisher, G.L., Nemhauser, Location of bank accounts to optimize float: An analytic study of exact and approximate algorithms, Management Sci. 23 (1977) 789–810.
G. Cornuejols, G.L. Nemhauser, L.A. Wolsey, The uncapacitated facility location problem, in: P.B. Mirchandani, R.L. Francis (Eds.), Discrete Location Theory, Wiley, New York, 1990.
G. Cornuejols, J.M. Thizy, Some facets of the simple plant location polytope, SIAM J. Algebraic Discrete Methods 3 (1982) 504–510.
IBM Corp., Optimization Subroutine Library: Guide and Reference, 1995.
G.B. Dantzig, P. Wolfe, Decomposition principle for linear programs, Oper. Res. 8 (1960) 101–111.
D. Erlenkotter, A dual-based procedure for uncapacitated facility location. Oper. Res. 26 (1978) 992–1009.
L.R. Ford, D.R. Fulkerson, Maximal flow through a network, Canad. J. Math. 8 (1956) 399–404.
R.S. Garfinkel, A.W. Neebe, M.R. Rao, An algorithm for them-median plant location problem, Transport. Sci. 8 (1974) 217–236.
A.M. Geoffrion, Lagrangean relaxation for integer programming, Math. Programming Study 2 (1974) 82–114.
M. Guignard, Fractional vertices, cuts and facets of the simple plant location problem, Math. Programming Study 12 (1980) 150–162.
M. Guignard, S. Zhu, A two-phase dual algorithm for solving lagrangean duals, Technical Report, The Wharton school, University of Pennsylvania, 1995.
M. Held, R.M. Karp, The travelling salesman problem and minimum spanning trees, Oper. Res. 18 (1970) 1138–1162.
M. Held, R.M. Karp, The travelling salesman problem and minimum spanning trees: Part ii, Math. Programming 1 (1971) 6–25.
M. Held, P. Wolfe, H.P. Crowder, Validation of subgradient optimization, Math. Programming 6 (1974) 62–88.
M. Jünger, S. Thienel, Introduction to ABACUS—a branch-and-cut system, Technical Report 97.263, Universität zu Köln, 1997.
P.B. Mirchandani, R.L. Francis, Discrete Location Theory, Wiley, New York, 1990.
B.T. Polyak, Minimization of unsmooth functionals, U.S.S.R. Comput. Math. and Math. Phys. 9 (1969) 509–521.
M.W.P. Savelsbergh, G.L. Nemhauser, Functional description of MINTO, a mixed integer optimizer, Technical Report, Georgia Institute of Technology, School of Industrial and Systems Engineering, 1993.
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Barahona, F., Jensen, D. Plant location with minimum inventory. Mathematical Programming 83, 101–111 (1998). https://doi.org/10.1007/BF02680552
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DOI: https://doi.org/10.1007/BF02680552