Fifty Years of Integer Programming: A Review of the Solution Approaches

Many scientific, industrial engineering and economic problems may be cast into an integer-programming model. This paper attempts to review various approaches that have been developed for solving this model, emphasizing on the methodological view. For the sake of space, we restrict ourselves mainly to the pure integer-programming problem. The paper is expository in nature and it includes many references to give credit to essential results in the field and help readers obtain more detailed information on issues of interest.

[1]  Manfred W. Padberg,et al.  Improving LP-Representations of Zero-One Linear Programs for Branch-and-Cut , 1991, INFORMS J. Comput..

[2]  Dimitris Bertsimas,et al.  Optimization over integers , 2005 .

[3]  Ralph E. Gomory,et al.  Outline of an Algorithm for Integer Solutions to Linear Programs and An Algorithm for the Mixed Integer Problem , 2010, 50 Years of Integer Programming.

[4]  R. Gomory Some polyhedra related to combinatorial problems , 1969 .

[5]  Martin W. P. Savelsbergh,et al.  Branch-and-Price: Column Generation for Solving Huge Integer Programs , 1998, Oper. Res..

[6]  Fred W. Glover,et al.  General Purpose Heuristics for Integer Programming—Part II , 1997, J. Heuristics.

[7]  W. Wilhelm A Technical Review of Column Generation in Integer Programming , 2001 .

[8]  Michael Jünger,et al.  Branch-and-Cut Algorithms for Combinatorial Optimization and Their Implementation in ABACUS , 2000, Computational Combinatorial Optimization.

[9]  Pham Nguyen Anh Huy,et al.  Solving Integer Programming Problems Using Genetic Algorithms , 2004 .

[10]  C. Ribeiro,et al.  Essays and Surveys in Metaheuristics , 2002, Operations Research/Computer Science Interfaces Series.

[11]  A. Land,et al.  An Automatic Method for Solving Discrete Programming Problems , 1960, 50 Years of Integer Programming.

[12]  Laurence A. Wolsey,et al.  Solving Mixed Integer Programming Problems Using Automatic Reformulation , 1987, Oper. Res..

[13]  Richard D. Wollmer,et al.  Two stage linear programming under uncertainty with 0–1 integer first stage variables , 1980, Math. Program..

[14]  R E Gomory,et al.  ON THE RELATION BETWEEN INTEGER AND NONINTEGER SOLUTIONS TO LINEAR PROGRAMS. , 1965, Proceedings of the National Academy of Sciences of the United States of America.

[15]  A. M. Geoffrion Lagrangean Relaxation and Its Uses in Integer Programming , 1972 .

[16]  Cynthia Barnhart,et al.  Using Branch-and-Price-and-Cut to Solve Origin-Destination Integer Multicommodity Flow Problems , 2000, Oper. Res..

[17]  Maarten H. van der Vlerk,et al.  Stochastic integer programming:General models and algorithms , 1999, Ann. Oper. Res..

[18]  Mitsuo Gen,et al.  Hybridized Neural Network and Genetic Algorithms for Solving Nonlinear Integer Programming Problem , 1998, SEAL.

[19]  Giovanni Rinaldi,et al.  A branch-and-cut algorithm for the equicut problem , 1997, Math. Program..

[20]  Giovanni Rinaldi,et al.  A Branch-and-Cut Algorithm for the Resolution of Large-Scale Symmetric Traveling Salesman Problems , 1991, SIAM Rev..

[21]  David K. Smith Theory of Linear and Integer Programming , 1987 .

[22]  Martin W. P. Savelsbergh,et al.  A Branch-and-Price Algorithm for the Generalized Assignment Problem , 1997, Oper. Res..

[23]  Jacques Desrosiers,et al.  Selected Topics in Column Generation , 2002, Oper. Res..

[24]  Kourosh Eshghi,et al.  SPECIAL CLASSES OF FUZZY INTEGER PROGRAMMING MODELS WITH ALL-DIFFERENT CONSTRAINTS , 2009 .

[25]  Claude Lemaréchal,et al.  Lagrangian Relaxation , 2000, Computational Combinatorial Optimization.

[26]  Gregory Dobson,et al.  Worst-Case Analysis of Greedy Heuristics for Integer Programming with Nonnegative Data , 1982, Math. Oper. Res..

[27]  Renato F. Werneck,et al.  Robust Branch-and-Cut-and-Price for the Capacitated Vehicle Routing Problem , 2006, Math. Program..

[28]  Christos H. Papadimitriou,et al.  On the complexity of integer programming , 1981, JACM.

[29]  Manfred W. Padberg Technical Note - A Note on Zero-One Programming , 1975, Oper. Res..

[30]  Moncef Abbas,et al.  Cutting plane method for multiple objective stochastic integer linear programming , 2006, Eur. J. Oper. Res..

[31]  S. Vajda,et al.  Integer Programming and Network Flows , 1970 .

[32]  Martin W. P. Savelsbergh,et al.  Preprocessing and Probing Techniques for Mixed Integer Programming Problems , 1994, INFORMS J. Comput..

[33]  Donald L. Kreher,et al.  Combinatorial algorithms: generation, enumeration, and search , 1998, SIGA.

[34]  P. Aravindan,et al.  A Tabu Search Algorithm for Job Shop Scheduling , 2000 .

[35]  Leslie E. Trotter,et al.  Branch, Cut, and Price: Sequential and Parallel , 2000, Computational Combinatorial Optimization.

[36]  L. Stougie,et al.  Approximation in Stochastic integer programming , 2003 .

[37]  Renato F. Werneck,et al.  Robust Branch-and-Cut-and-Price for the Capacitated Vehicle Routing Problem , 2004, Math. Program..

[38]  Vasek Chvátal Cutting Planes in Combinatorics , 1985, Eur. J. Comb..

[39]  Michel Gendreau,et al.  Metaheuristics in Combinatorial Optimization , 2022 .

[40]  Egon Balas,et al.  Intersection Cuts - A New Type of Cutting Planes for Integer Programming , 1971, Oper. Res..

[41]  Ailsa H. Land,et al.  An Automatic Method of Solving Discrete Programming Problems , 1960 .

[42]  Jeremy F. Shapiro,et al.  Group Theoretic Algorithms for the Integer Programming Problem II: Extension to a General Algorithm , 1968, Oper. Res..

[43]  Didier Dubois,et al.  Fuzzy sets and systems ' . Theory and applications , 2007 .

[44]  John A. Tomlin,et al.  Technical Note - An Improved Branch-and-Bound Method for Integer Programming , 1971, Oper. Res..

[45]  James J. Buckley,et al.  Fuzzy Integer Programming , 2007 .

[46]  F. Herrera,et al.  EUROPEAN JOURNAL OF OPERATIONAL , 1992 .

[47]  Harvey M. Salkin Technical Note - A Note on Gomory Fractional Cuts , 1971, Oper. Res..

[48]  Dr. Zbigniew Michalewicz,et al.  How to Solve It: Modern Heuristics , 2004 .

[49]  Jeroen Beliën Exact and heuristic methodologies for scheduling in hospitals: problems, formulations and algorithms , 2007, 4OR.

[50]  Monique Guignard-Spielberg,et al.  Lagrangean decomposition: A model yielding stronger lagrangean bounds , 1987, Math. Program..

[51]  Egon Balas,et al.  Discrete Programming by the Filter Method , 1967, Oper. Res..

[52]  Kate Smith-Miles,et al.  Neural Networks for Combinatorial Optimization: A Review of More Than a Decade of Research , 1999, INFORMS J. Comput..

[53]  Maarten H. van der Vlerk,et al.  Stochastic Integer Programming Bibliography , 2007 .

[54]  Kate A. Smith,et al.  Neural Networks for Combinatorial Optimization: a Review of More Than a Decade of Research , 1999 .

[55]  John K. Karlof,et al.  Integer programming : theory and practice , 2005 .

[56]  John M. Wilson,et al.  Fuzzy extensions to Integer Programming models of cell-formation problems in machine scheduling , 2009, Ann. Oper. Res..

[57]  E.M.L. Beale,et al.  Branch and Bound Methods for Mathematical Programming Systems , 1977 .

[58]  Laurent Hérault,et al.  Solving a Real World Assignment Problem with a Metaheuristic , 1998, J. Heuristics.

[59]  Ellis L. Johnson,et al.  Some continuous functions related to corner polyhedra , 1972, Math. Program..

[60]  E. Andrew Boyd,et al.  Fenchel Cutting Planes for Integer Programs , 1994, Oper. Res..

[61]  Ellis L. Johnson,et al.  Corner Polyhedra and their connection with cutting planes , 2003, Math. Program..

[62]  Egon Balas,et al.  Gomory cuts revisited , 1996, Oper. Res. Lett..

[63]  Günther R. Raidl,et al.  Combining Metaheuristics and Exact Algorithms in Combinatorial Optimization: A Survey and Classification , 2005, IWINAC.

[64]  G. Dantzig Note on solving linear programs in integers , 1959 .

[65]  Santosh Kumar INFORMATION RECYCLING MATHEMATICAL METHODS FOR PROTEAN SYSTEMS: A PATH-WAY APPROACH , 2012 .

[66]  R. J. Dakin,et al.  A tree-search algorithm for mixed integer programming problems , 1965, Comput. J..

[67]  Santosh Kumar,et al.  INFORMATION RECYCLING MATHEMATICAL METHODS FOR PROTEAN SYSTEMS: A PATH-WAY APPROACH TO A GEOMETRIC PROGRAM , 2012 .

[68]  Guohe Huang,et al.  Grey fuzzy integer programming: An application to regional waste management planning under uncertainty , 1995 .