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Parametric methods in integer linear programming

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Abstract

In contrast to methods of parametric linear programming which were developed soon after the invention of the simplex algorithm and are easily included as an extension of that method, techniques for parametric analysis on integer programs are not well known and require considerable effort to append them to an integer programming solution algorithm.

The paper reviews some of the theory employed in parametric integer programming, then discusses algorithmic work in this area over the last 15 years when integer programs are solved by different methods. A summary of applications is included and the article concludes that parametric integer programming is a valuable tool of analysis awaiting further popularization.

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References

  1. D.W. Ashley and R.L. Brunner, Capital equipment selection with mixed integer programming, presented at CORS/TIMS/ORSA Joint Meeting, Toronto (1981).

  2. M.G. Bailey and B.E. Gillett, Parametric integer programming analysis: a contraction approach, J. Oper. Res. Soc. 31 (1980) 257–262.

    Google Scholar 

  3. E. Balas, An additive algorithm for solving linear programs with zero-one variables, Oper. Res. 13 (1965) 517–546.

    Google Scholar 

  4. B. Bank and R. Mandel, Parametric integer optimization, Math. Res. (Berlin) 39 (1988).

  5. C.E. Blair and R.G. Jeroslow, The value function of a mixed integer program: I, Discr. Math. 19 (1977) 121–138.

    Google Scholar 

  6. C.E. Blair and R.G. Jeroslow, The value function of a mixed integer program: II, Discr. Math. 25 (1979) 7–19.

    Google Scholar 

  7. C.E. Blair and R.G. Jeroslow, The value function of an integer program, Math. Programming 23 (1982) 237–273.

    Google Scholar 

  8. C.E. Blair and R.G. Jeroslow, Constructive characterizations of the value-function of a mixed-integer program I, Discr. Appl. Math. 9 (1984) 217–233.

    Google Scholar 

  9. C.E. Blair and R.G. Jeroslow, Constructive characterizations of the value function of a mixed-integer program II, Discr. Appl. Math. 10 (1985) 227–240.

    Google Scholar 

  10. V.J. Bowman, Jr., Sensitivity analysis in linear integer programming, AIIE Technical Papers (1972).

  11. P.J. Carstensen, Complexity of some parametric integer and network programming problems, Math. Programming 26 (1983) 64–75.

    Google Scholar 

  12. Control Data Corporation,APEX-III Reference Manual, CDC Data Services Publications, Minneapolis, MN (1975).

    Google Scholar 

  13. W. Cook, A.M.H. Gerards, A. Schrijver and E. Tardos, Sensitivity theorems in integer linear programming, Math. Programming 34 (1986) 251–264.

    Google Scholar 

  14. M.W. Cooper, Postoptimality analysis in nonlinear integer programming: the right-hand side case, Naval Res. Logist. Quarterly 28 (1981) 301–307.

    Google Scholar 

  15. M.J. Eisner and D.J. Severance, Mathematical techniques for efficient record segmentation in large shared databases, J. ACM 23 (1976) 619–635.

    Google Scholar 

  16. P.G. Falk, Experiments in mixed integer linear programming concurrent to a manufacturing hierarchy, presented at TIMS/ORSA Joint Meeting, New York (1978).

  17. C.R. Frank, Jr., Parametric programming in integers, Oper. Res. Verfahren 3 (1967) 167–180.

    Google Scholar 

  18. A.M. Geoffrion and G.W. Graves, Multicommodity distribution system design by Benders decomposition, Management Sci. 20 (1974) 822–844.

    Google Scholar 

  19. A.M. Geoffrion and R. Nauss, Parametric and postoptimality analysis in integer linear programming, Management Sci. 23 (1977) 453–466.

    Google Scholar 

  20. R.E. Gomory and W.J. Baumol, Integer programming and pricing, Econometrica 28 (1960) 521–550.

    Google Scholar 

  21. D. Gusfield, Parametric combinatorial computing and a problem of program module distribution, J. ACM 30 (1983) 551–563.

    Google Scholar 

  22. R.S. Hiller, C.A. Holmes, T.M. Magee and J.F. Shapiro, Constructive duality for mixed integer programming: Part I — Theory, Working Paper No. OR 147–86, Massachusetts Institutes of Technology (1987).

  23. F.S. Hillier and G.J. Lieberman,Introduction to Operations Research, 3rd ed. (Holden-Day, San Francisco, CA, 1980).

    Google Scholar 

  24. S. Holm and D. Klein, Discrete right hand side parameterization for linear integer programs, Europ. J. Oper. Res. 2 (1978) 50–53.

    Google Scholar 

  25. S. Holm and D. Klein, Three methods for postoptimal analysis in integer linear programming, Math. Programming Study 21 (1984) 97–109.

    Google Scholar 

  26. International Business Machines Corporation,Mathematical Program Extended/370 (MPSX/370), Basic Reference Manual, IBM Technical Publications Department, White Plains, NY (1976).

    Google Scholar 

  27. L. Jenkins, Parametric mixed integer programming: an application to solid waste management, Management Sci. 28 (1982) 1270–1284.

    Google Scholar 

  28. L. Jenkins, Using parametric integer programming to plan the mix of an air transport fleet, INFOR 25 (1987) 117–135.

    Google Scholar 

  29. L. Jenkins and D. Peters, A computational comparison of Gomory and knapsack cuts, Comput. Oper. Res. 14 (1987) 449–456.

    Google Scholar 

  30. L. Jenkins, Parametric-objective integer programming using knapsack facets and Gomory cutting planes, Europ. J. Oper. Res. 31 (1987) 102–109.

    Google Scholar 

  31. S. Kim and S. Cho, A shadow price in integer programming for management decision, Europ. J. Oper. Res. 37 (1988) 328–335.

    Google Scholar 

  32. D. Klein and S. Holm, Integer programming post-optimal analysis with cutting planes, Management Sci. 25 (1979) 64–72.

    Google Scholar 

  33. E. Loukakis and A.P. Muhlemann, Parameterisation algorithms for the integer linear programs in binary variables, Europ. J. Oper. Res. 17, 104–115, (1984).

    Google Scholar 

  34. A.S. Manne, Two producing areas — constant cycle time policies, in:Investments for Capacity Expansion, ed. A.S. Manne (George Allen and Unwin Ltd., London, 1967).

    Google Scholar 

  35. R.E. Marsten and T.L. Morin, Parametric integer programming: the right-hand side case, Ann. Discr. Math. 1 (1977) 375–390.

    Google Scholar 

  36. R.D. McBride and J.S. Yormark, Finding all solutions for a class of parametric quadratic integer programming problems, Management Sci. 26 (1980) 784–795.

    Google Scholar 

  37. R.M. Nauss, Parametric integer programming, Ph.D. Dissertation, University of California at Los Angeles (1974); subsequently published by University of Missouri Press, Columbia, Miss. (1979).

    Google Scholar 

  38. H. Noltemeier, Sensitivitätsanalyse bei diskreten linearen Optimierungsproblemen,Lecture Notes in Operations Research and Mathematical Systems, vol. 30 (Springer, Berlin and New York, 1970).

    Google Scholar 

  39. Y. Ohtake and N. Nishida, A branch- and-bound algorithm for 0–1 parametric mixed integer programming, Oper. Res. Lett. 4 (1985) 41–45.

    Google Scholar 

  40. C.J. Piper and A.A. Zoltners, Implicit enumeration based algorithms for postoptimizing zero-one programs, Naval Res. Logist. Quarterly 22 (1975) 791–809.

    Google Scholar 

  41. C.J. Piper and A.A. Zoltners, Some easy postoptimality analysis for zero-one programming, Management Sci. 22 (1976) 759–765.

    Google Scholar 

  42. M.A. Radke, Sensitivity analysis in discrete optimization, Ph.D. Dissertation, available as Working Paper No. 240, Western Management Science Institute, UCLA (1975).

  43. G.M. Roodman, Postoptimality analysis in zero-one programming by implicit enumeration, Naval Res. Logist. Quarterly 19 (1972) 435–447.

    Google Scholar 

  44. G.M. Roodman, Postoptimality analysis in integer programming by implicit enumeration: the mixed integer case, Naval Res. Logist. Quarterly 21 (1974) 595–607.

    Google Scholar 

  45. S.L.K. Rountree and B.E. Gillet, Parametric integer linear programming: a synthesis of branch and bound with cutting planes, Europ. J. Oper. Res. 10 (1982) 183–189.

    Google Scholar 

  46. L. Schrage,User's Manual for LINDO (Scientific Press, Palo Alto, CA, 1981).

    Google Scholar 

  47. L. Schrage and L. Wolsey, Sensitivity analysis for branch and bound integer programming, Oper. Res. 33 (1985) 1008–1023.

    Google Scholar 

  48. A. Schrijver,Theory of Linear and Integer Programming (Wiley, New York, 1986).

    Google Scholar 

  49. G.K. Tayi, Sensitivity analysis of mixed integer programs: an application to environmental policy making, Europ. J. Oper. Res. 22 (1985) 224–233.

    Google Scholar 

  50. J. Teghem Jr., A survey of techniques for finding efficient solutions to multi-objective integer linear programming, Asia-Pacific J. Oper. Res. 3 (1986) 95–108.

    Google Scholar 

  51. A. Turgeon, An application of parametric mixed-integer linear programming to hydropower development, Water Resources Res. 23 (1987) 399–407.

    Google Scholar 

  52. C.P.M. Van Hoesel, A.W.J. Kolen, A.H.G. Rinnooy Kan and A.P.M. Wagelmans, Sensitivity analysis in combinatorial optimization: a bibliography, Report 8944/A, Erasmus University, Rotterdam (1989).

    Google Scholar 

  53. G.R. Wilson and H.K. Jain, An approach to postoptimality and sensitivity analysis of zero-one goal programs, Naval Res. Logist. Quarterly 35 (1988) 73–84.

    Google Scholar 

  54. L.A. Wolsey, Integer programming duality: price functions and sensitivity analysis, Math. Programming 20 (1981) 173–195.

    Google Scholar 

  55. L.A. Wolsey, The b-hull of an integer program, Discr. Appl. Math. 3 (1981) 193–201.

    Google Scholar 

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Authors and Affiliations

  1. Department of Engineering Management, Royal Military College of Canada, K7K 5L0, Kingston, Ontario, Canada

    Larry Jenkins

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Jenkins, L. Parametric methods in integer linear programming. Ann Oper Res 27, 77–96 (1990). https://doi.org/10.1007/BF02055191

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