Valid inequalities and facets of the capacitated plant location problem

Recently, several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods, and polyhedral characterizations, of integer programming problems. In this paper, we investigate the polyhedral structure of the capacitated plant location problem. Our purpose is to identify facets and valid inequalities for a wide range of capacitated fixed charge problems that contain this prototype problem as a substructure.The first part of the paper introduces a family of facets for a version of the capacitated plant location problem with a constant capacity for all plants. These facet inequalities depend on the capacity and thus differ fundamentally from the valid inequalities for the uncapacited version of the problem.We also introduce a second formulation for a model with indivisible customer demand and show that it is equivalent to a vertex packing problem on a derived graph. We identify facets and valid inequalities for this version of the problem by applying known results for the vertex packing polytope.

[1]  R. Soland,et al.  A Branch-and-Bound Algorithm for Multi-Level Fixed-Charge Problems , 1969 .

[2]  A. Manne.  Plant Location Under Economies-of-Scale---Decentralization and Computation , 1964 .

[3]  George L. Nemhauser,et al.  The uncapacitated facility location problem , 1990 .

[4]  Bezalel Gavish,et al.  Topological design of centralized computer networks - formulations and algorithms , 1982, Networks.

[5]  Dorit S. Hochbaum,et al.  Database Location in Computer Networks , 1980, JACM.

[6]  Martin Grötschel,et al.  Facets of the linear ordering polytope , 1985, Math. Program..

[7]  Ellis L. Johnson,et al.  Solving Large-Scale Zero-One Linear Programming Problems , 1983, Oper. Res..

[8]  Laurence A. Wolsey,et al.  Valid Linear Inequalities for Fixed Charge Problems , 1985, Oper. Res..

[9]  W. R. Pulleyblank,et al.  Polyhedral Combinatorics , 1989, ISMP.

[10]  Nicos Christofides,et al.  Extensions to a Lagrangean relaxation approach for the capacitated warehouse location problem , 1983 .

[11]  P. S. Davis,et al.  A branch‐bound algorithm for the capacitated facilities location problem , 1969 .

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

[13]  J. G. Riggs,et al.  An Application of Facilities Location Theory to the Design of Forest Harvesting Areas , 1977 .

[14]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[15]  David L. Kelly,et al.  Capacitated Warehouse Location with Concave Costs , 1982 .

[16]  Gerhard Reinelt,et al.  A Cutting Plane Algorithm for the Linear Ordering Problem , 1984, Oper. Res..

[17]  Eitan Zemel,et al.  Lifting the facets of zero–one polytopes , 1978, Math. Program..

[18]  Manfred W. Padberg,et al.  On the facial structure of set packing polyhedra , 1973, Math. Program..

[19]  Søren Kruse Jacobsen,et al.  Heuristics for the capacitated plant location model , 1983 .

[20]  Leon F. McGinnis.  A Survey of Recent Results for a Class of Facilities Location Problems , 1977 .

[21]  Robert M. Nauss,et al.  An Improved Algorithm for the Capacitated Facility Location Problem , 1978 .

[22]  R. Boorstyn,et al.  Large-Scale Network Topological Optimization , 1977, IEEE Trans. Commun..

[23]  Linus Schrage,et al.  Subset Coefficient Reduction Cuts for 0/1 Mixed-Integer Programming , 1985, Oper. Res..

[24]  D. Chinhyung Cho,et al.  On the Uncapacitated Plant Location Problem. II: Facets and Lifting Theorems , 1983, Math. Oper. Res..

[25]  B. M. Khumawala.  An Efficient Branch and Bound Algorithm for the Warehouse Location Problem , 1972 .

[26]  B. M. Khumawala,et al.  An Efficient Branch and Bound Algorithm for the Capacitated Warehouse Location Problem , 1977 .

[27]  Graciano Sá,et al.  Branch-and-Bound and Approximate Solutions to the Capacitated Plant-Location Problem , 1969, Oper. Res..

[28]  Aaron Kershenbaum,et al.  Centralized teleprocessing network design , 1976, Networks.

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

[30]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

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

[32]  E. Balas,et al.  Set Partitioning: A survey , 1976 .

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

[34]  Laurence A. Wolsey,et al.  Valid inequalities for mixed 0-1 programs , 1986, Discret. Appl. Math..

[35]  A. M. Geoffrion,et al.  Lagrangean Relaxation Applied to Capacitated Facility Location Problems , 1978 .

[36]  Laurence A. Wolsey,et al.  Technical Note - Facets and Strong Valid Inequalities for Integer Programs , 1976, Oper. Res..

[37]  Monique Guignard-Spielberg,et al.  A direct dual method for the mixed plant location problem with some side constraints , 1979, Math. Program..

[38]  Michael M. Kostreva,et al.  Solving 0-1 Integer Programming Problems Arising from Large Scale Planning Models , 1985, Oper. Res..

[39]  Barrie M. Baker,et al.  Linear Relaxations of the Capacitated Warehouse Location Problem , 1982 .

[40]  J. K. Lenstra,et al.  Combinatorial optimization : annotated bibliographies , 1985 .

[41]  Martin Grötschel,et al.  On the symmetric travelling salesman problem I: Inequalities , 1979, Math. Program..

[42]  P. Gray,et al.  Solving Fixed Charge Location-Allocation Problems with Capacity and Configuration Constraints , 1971 .

[43]  Manfred W. Padberg,et al.  On the symmetric travelling salesman problem: A computational study , 1980 .

[44]  Martin Grötschel,et al.  On the symmetric travelling salesman problem II: Lifting theorems and facets , 1979, Math. Program..

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

[46]  Kurt Spielberg,et al.  Algorithms for the Simple Plant-Location Problem with Some Side Conditions , 1969, Oper. Res..

[47]  L. Wolsey,et al.  Valid inequalities and separation for uncapacitated fixed charge networks , 1985 .

[48]  George B. Dantzig,et al.  Solution of a Large-Scale Traveling-Salesman Problem , 1954, Oper. Res..

[49]  L. A. Wolsey,et al.  Further facet generating procedures for vertex packing polytopes , 1976, Math. Program..

[50]  M. Padberg.  Covering, Packing and Knapsack Problems , 1979 .

[51]  Andrew S. Tanenbaum,et al.  Computer Networks , 1981 .

[52]  Leslie E. Trotter,et al.  A class of facet producing graphs for vertex packing polyhedra , 1975, Discret. Math..

[53]  Thomas L. Magnanti,et al.  Decomposition methods for facility location problems , 1986 .

[54]  Michel Balinski,et al.  Integer Programming: Methods, Uses, Computations , 1965 .

[55]  Gérard Cornuéjols,et al.  Some facets of the simple plant location polytope , 1982, Math. Program..

[56]  J. Stollsteimer.  A Working Model for Plant Numbers and Locations , 1963 .

[57]  V. Chandru,et al.  Inverse Optimization: An Application to the Capacitated Plant Location Problem , 1981 .

[58]  T. L. Ray,et al.  A Branch-Bound Algorithm for Plant Location , 1966, Oper. Res..

[59]  Leslie E. Trotter,et al.  Properties of vertex packing and independence system polyhedra , 1974, Math. Program..

[60]  H. Crowder,et al.  Solving Large-Scale Symmetric Travelling Salesman Problems to Optimality , 1980 .

[61]  Tony J. Van Roy,et al.  A Cross Decomposition Algorithm for Capacitated Facility Location , 1986, Oper. Res..

[62]  W. M. Hirsch,et al.  The fixed charge problem , 1968 .

[63]  Egon Balas,et al.  Facets of the knapsack polytope , 1975, Math. Program..

[64]  Richard M. Karp,et al.  On Linear Characterizations of Combinatorial Optimization Problems , 1982, SIAM J. Comput..

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

[66]  J. Krarup,et al.  The simple plant location problem: Survey and synthesis , 1983 .

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

[68]  Alfred A. Kuehn,et al.  A Heuristic Program for Locating Warehouses , 1963 .

[69]  Leslie E. Trotter,et al.  Vertex packings: Structural properties and algorithms , 1975, Math. Program..

[70]  M. Y. Leung Janny.  Polyhedral structure of capacitated fixed change problems and problem in delivery , 1986 .

[71]  G. Ross,et al.  Modeling Facility Location Problems as Generalized Assignment Problems , 1977 .

[72]  C. J. McCallum,et al.  Facility location models for planning a transatlantic communications network , 1981 .

[73]  G. Nemhauser,et al.  On the Uncapacitated Location Problem , 1977 .

[74]  M. R. Rao,et al.  An Algorithm for the Fixed-Charge Assigning Users to Sources Problem , 1983 .

[75]  M. Guignard.  Fractional vertices, cuts and facets of the simple plant location problem , 1980 .

[76]  G. D. Eppen,et al.  Solving Multi-Item Capacitated Lot-Sizing Problems Using Variable Redefinition , 1987, Oper. Res..

[77]  Tony J. Van Roy,et al.  Cross decomposition for mixed integer programming , 1983, Math. Program..

[78]  Polly Bart,et al.  Heuristic Methods for Estimating the Generalized Vertex Median of a Weighted Graph , 1968, Oper. Res..

[79]  Manfred Padberg,et al.  On the Uncapacitated Plant Location Problem. I: Valid Inequalities and Facets , 1983, Math. Oper. Res..

[80]  Vasek Chvátal,et al.  Edmonds polytopes and a hierarchy of combinatorial problems , 1973, Discret. Math..

[81]  T. L. Ray,et al.  Warehouse Location Under Continuous Economies of Scale , 1966 .

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