Advertisement

Model Building in Linear and Integer Programming

  • H. P. Williams
Part of the NATO ASI Series book series (volume 15)

Abstract

This paper surveys the topic of model building in mathematical programming discussing, (i) the systematisation of model building, including the use of Matrix Generating Languages, (ii) the use of Boolean Algebra for formulating 0–1 integer programming models and the efficient formulation of integer programming models considering both their facial structure and the desirability of creating meaningful dichotemies for the branch-and-bound tree search, (iii) the desirability and possibility of converting models to network flow models, (iv) the building of stable models.

Keywords

Model Building Mathematical Programming 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ayles, P.S., E.M.L. Beale, R.C. Blues and S.J. Wild (1978). Mathematical Models for the Location of Government, Mathematical Programming Study 9, 59–74.Google Scholar
  2. 2.
    Balas, E. (1975). Facets of the knapsack polytope, Math. Prog., 8, 146–164.CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Beale, E.M.L. and J.A. Tomlin (1969). Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables, in J. Lawrence (Ed.), Proc. 5th Int. Conf. of Operations Research, Tavistock, London.Google Scholar
  4. 4.
    Beale, E.M.L. and J.A. Tomlin (1972). An integer programming approach to a class of combinatorial problems, Maths. Prog. 1, 339–344.CrossRefMathSciNetGoogle Scholar
  5. 5.
    Beale, E.M.L., G.C. Beare and P.B. Tatham (1974). The DOAE reinforcement and redeployment study: a case study in mathematical programming, in P.L. Hammer and G. Zoutendijk (Eds), Mathematical Programming in Theory and Practice, North Holland, Amsterdam.Google Scholar
  6. 6.
    Bixby, R.E. and W.H. Cunningham (1980). Converting Linear Programs to Network Problems, Mathematics of Operations Research, 5, 321–357.CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bradley, G.H., P.L. Hammer and L. Wolsey (1974). Coefficient reduction for inequalities in 0-1 variables, Math. Prog., 7, 263–282.CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Brearley, A.L., G. Mitra and H.P. Williams (1975). Analysis of mathematical programming problems prior to applying the simplex algorithm, Math. Prog., 8, 54–83.CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Brown, G.S., R.D. McBride and R.K. Wood (1984). Extracting Embedding Generalised Networks from Linear Programming Problems, Technical Report, Naval Postgraduate School, Monterey, California.Google Scholar
  10. 10.
    Cheshire, M., K.I.M. McKinnon and H.P. Williams (1984). The efficient allocation of private contractors to public works, Journal of the Operational Research Society, 35, 705–709.Google Scholar
  11. 11.
    Crowder, H., E.L. Johnson and M.W. Padberg (1981). Solving Large Scale Zero-One Linear Programming Problems, IBM Research Report RC8888, Yorktown Heights.Google Scholar
  12. 12.
    Daniel, R.C. (1978). Reducing computational effort in solving a hard integer programme, SIGMAP Association for Computing Machinery, Special Interest Group on Mathematical Programming, No. 25, December, 39–44.Google Scholar
  13. 13.
    Daniel, R.C. (1973). Phasing out capital equipment, Opl. Res. Q., 24, 113–116.CrossRefGoogle Scholar
  14. 14.
    Dantzig, G.B. (1969), A Hospital Admission Problem. Technical Report N. 69-15, Stanford University, California.Google Scholar
  15. 15.
    Day, R.E. and H.P. Williams (1982). Magic: The Design and Use of an Interactive Modelling Language for Mathematical Programming, Dept. Business Studies, Working Paper 2/82, University of Edinburgh.Google Scholar
  16. 16.
    Day, R.E. (1984). MAGIC: User’s Guide, Dept. Business Studies, University of EdinburgGoogle Scholar
  17. 17.
    Fourer, R. (1983). Modelling languages versus matrix generators for linear programming, ACM Transactions on Mathematical Software, 9, 143–183.CrossRefGoogle Scholar
  18. 18.
    Glover, F. and D. Klingman (1974). Real world applications of network problems and breakthroughs in solving them efficiently, Research Report CC5159, Center for Cybernetic Studies, University of Texas at Austin.Google Scholar
  19. 19.
    Glover, F. (1975). Improved Linear Integer Programming Formulations of Nonlinear Integer Problems, Management Science, 22, 455–459.CrossRefMathSciNetGoogle Scholar
  20. 20.
    Glover, F., J. Hultz, D. Klingman and J. Stutz (1978). Generalised Networks: A Fundamental Computer-Based Planning Tool, Management Science, 24, 1209–1220.CrossRefGoogle Scholar
  21. 21.
    Goldman, A.J. (1983). Linearisation on 0-1 variables. A clarification, Operations Research 31, 946–947.CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Guignard, M. and K. Spielberg (1981). Logical reduction methods in zero-one programming. Minimal preferred inequalities. Operations Research 29, 49–74.CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Hammer, P.L. and S. Nguyen (1972). “APOSS: A Partial Order in the Solution Space of Bivalent Programs”, C.R.M.-163, Centre de Recherches Mathematiques, Universite de Montreal.Google Scholar
  24. 24.
    Hammer, P.L., E.L. Johnson and U.N. Peled (1975). Facets of regular 0-1 polytopes, Math. Prog., 8, 179–206.CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Hoffman, K. and M. Padberg (1985), LP Based Combinatorial Problem Solving, This volume.Google Scholar
  26. 26.
    Jeffreys, M. (1976). The Next Generation of Branch and Bound Codes, Paper presented at IX International Symposium on Mathematical Programming, Budapest, Hungary, August.Google Scholar
  27. 27.
    Jeroslow, R.G. and J.K. Lowe (1983). Modelling with Integer Variables, Report No. 83270-OR, Institute für Okonometre und Operations Research, University of Bonn.Google Scholar
  28. 28.
    Jeroslow, R.G. and J.K. Lowe (1985). Experimental Results with the New Techniques for Integer Programming Formulations, Journal of the Operational Research Society, to appear.Google Scholar
  29. 29.
    Karwan, M.H., V. Lotfi, J. Teigen and S. Zionts (1983). Redundancy in Mathematical Programming; A State of the Art Survey, Springer-Verlag, New York.zbMATHGoogle Scholar
  30. 30.
    Meyer, R.R. (1981). A Theoretical and Computational Comparison of ‘Equivalent’ Mixed Integer Formulations, Naval Research Logistics Quarterly, 28, 115–131.CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Müller-Merbach, H. (1978). Entwurf von input-output-Modellen, in K. Brockoff et al. (Eds), Proz. Ops. Res., Physica-Verlag, Wurzberg, Wien, 7, 521–531.Google Scholar
  32. 32.
    Oley, L.A. and R.J. Sjoquist (1982). Automatic Reformulation of Mixed and Pure Integer Models to Reduce Solution Time in APEX IV, Paper presented at ORSA/TIMS Meeting, San Diego, California, October.Google Scholar
  33. 33.
    Padberg, M.W. (1979). Covering, Packing and Knapsack Problems, Annals of Discrete Mathematics, 4, 265–287.CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Padberg, M.W., T.J. Van Roy and L.A. Wolsey (1982). Valid Linear Inequalities for Fixed Charge Problems, Core Discussion Paper No. 8232.Google Scholar
  35. 35.
    Stecke, K.E. (1983). Formulation and Solution of Nonlinear Integer Production Planning Problems for Flexible Manufacturing Systems, Management Science, 29, 273–287.CrossRefzbMATHGoogle Scholar
  36. 36.
    Tomlin, J.A. and J.S. Welch (1982). A Pathological Case in the Reduction of Linear Programs, Operations Research Letters, to appear.Google Scholar
  37. 37.
    Veinott, A.F. and H.M. Wagner (1962). Optimal capacity scheduling — I, Ops. Res., 10, 518–532.CrossRefGoogle Scholar
  38. 38.
    Wagner, H.M. (1969). Principles of Operations Research, Prentice Hall, Englewood Cliffs.zbMATHGoogle Scholar
  39. 39.
    Williams, A.C. (1973). Some Modelling Principles for MIP’s, 8th International Mathematical Programming Symposium, Stanford, California.Google Scholar
  40. 40.
    Williams, H.P. (1978). The reformulation of two mixed integer programming problems, Math. Prog., 14, 325–331.CrossRefzbMATHGoogle Scholar
  41. 41.
    Williams, H.P. (1982). Models with Network Duals, J. Opl. Res. Soc, 33, 161–169.zbMATHGoogle Scholar
  42. 42.
    Williams, H.P. (1984). Restricted Vertex Generation Applied as a Crashing Procedure for Linear Programming, Computers and Operations Research Vol. 11, 401–407.CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    Williams, H.P. (1985). Model Building in Mathematical Programming, 2nd Edition, Wiley, New York.zbMATHGoogle Scholar
  44. 44.
    Wolsey, L.A. (1975). Faces for a linear inequality in 0-1 variables, Math. Prog., 8, 165–178.CrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    Bradley, G.H., Brown, C.G. and Graves (1977). Design and Implementation of Large Scale Primal Transhipment Algorithms, Management Science, 24, 1–34.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • H. P. Williams
    • 1
  1. 1.Faculty of Mathematical StudiesUniversity of SouthamptonUK

Personalised recommendations