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Global Optimization in Practice: An Application to Interactive Multiple Objective Linear Programming

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Abstract

A multiple objective linear programming problem (P) involves the simultaneous maximization of two or more conflicting linear objective functions over a nonempty polyhedron X. Many of the most popular methods for solving this type of problem, including many well-known interactive methods, involve searching the efficient set X E of the problem. Generally, however, X E is a complicated, nonconvex set. As a result, concepts and methods from global optimization may be useful in searching X E. In this paper, we will explain in theory, and show via an actual application to citrus rootstock selection in Florida, how the potential usefulness of the well-known interactive method STEM for solving problem (P) in this way, can depend crucially upon how accurately certain global optimization problems involving minimizations over X E are solved. In particular, we will show both in theory and in practice that the choice of whether to use the popular but unreliable ‘payoff table’ approach or to use one of the lesser known, more accurate global optimization methods to solve these problems can determine whether STEM succeeds or fails as a decision aid. Several lessons and conclusions of transferable value derived from this research are also given.

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References

  1. Aksoy, Y. (1990), Interactive Multiple Objective Decision Making: A Bibliography (1965–1988), Management Research News 2, 1–8.

    Google Scholar 

  2. Anderson, A. M. and Earle, M. D. (1983), Diet Planning in the Third World by Linear and Goal Programming, Journal of the Operational Research Society 34, 9–16.

    Google Scholar 

  3. Armand, P. and Malivert, C. (1991), Determination of the Efficient Set in Multiobjective Linear Programming, Journal of Optimization Theory and Applications 70, 467–489.

    Google Scholar 

  4. Armann, R. (1989), Solving Multiobjective Programming Problems by Discrete Representation, Optimization 20, 483–492.

    Google Scholar 

  5. Bazaraa, M. S. and Bouzaher, A. (1981), A Linear Goal Programming Model for Developing Economies with an Illustration from the Agricultural Sector in Egypt, Management Science 27, 396–413.

    Google Scholar 

  6. Belenson, S. and Kapur, K. C. (1973), An Algorithm for Solving Multicriterion Linear Programming Problems with Examples, Operational Research Quarterly 24, 65–77.

    Google Scholar 

  7. Benayoun, R., De Montgolfier, J., Tergny, J., and Laritchev, O. (1971), Linear Programming with Multiple Objective Functions: Step Method (STEM), Mathematical Programming 1, 366–375.

    Google Scholar 

  8. Benjamin, C. O. (1985), A Linear Goal-Programming Model for Public-Sector Project Selection, Journal of the Operational Research Society 36, 13–23.

    Google Scholar 

  9. Benson, H. P. (1978), Existence of Efficient Solutions for Vector Maximization Problems, Journal of Optimization Theory and Applications 26, 569–580.

    Google Scholar 

  10. Benson, H. P. (1986), An Algorithm for Optimizing over the Weakly-Efficient Set, European Journal of Operational Research 25, 192–199.

    Google Scholar 

  11. Benson, H. P. (1991), An All-Linear Programming Relaxation Algorithm for Optimization over the Efficient Set, Journal of Global Optimization 1, 83–104.

    Google Scholar 

  12. Benson, H. P. (1992), A Finite, Nonadjacent Extreme Point Search Algorithm for Optimization over the Efficient Set, Journal of Optimization Theory and Applications 73, 47–64.

    Google Scholar 

  13. Benson, H. P. (1993), A Bisection-Extreme Point Search Algorithm for Optimizing over the Efficient Set in the Linear Dependence Case, Journal of Global Optimization 3, 95–111.

    Google Scholar 

  14. Benson, H. P. (1995), Concave Minimization: Theory, Applications and Algorithms, in R. Horst and P. M. Pardalos (eds.), Handbook of Global Optimization, Kluwer Academic Publishers, Dordrecht, pp. 43–148.

    Google Scholar 

  15. Benson, H. P. (1995), A Geometrical Analysis of the Efficient Outcome Set in Multiple Objective Convex Programs with Linear Criterion Functions, Journal of Global Optimization 6, 231–251.

    Google Scholar 

  16. Benson, H. P. and Aksoy, Y. (1991), Using Efficient Feasible Directions in Interactive Multiple Objective Linear Programming, Operations Research Letters 10, 203–209.

    Google Scholar 

  17. Benson, H. P. and Lee, D. (1996), Outcome-Based Algorithm for Optimizing over the Efficient Set of a Bicriteria Linear Programming Problem, Journal of Optimization Theory and Applications 88, 77–105.

    Google Scholar 

  18. Benson, H. P., Lee, D. and McClure, J. P. (1992), Applying Multiple Criteria Decision Making in Practice: The Citrus Rootstock Selection Problem in Florida, Discussion Paper, University of Florida, Department of Decision and Information Sciences, Gainesville, Florida.

    Google Scholar 

  19. Benson, H. P. and Morin, T. L. (1987), A Bicriteria Mathematical Programming Model for Nutrition Planning in Developing Nations, Management Science 33, 1593–1601.

    Google Scholar 

  20. Benson, H. P. and Sayin, S. (1993), A Face Search Heuristic Algorithm for Optimizing over the Efficient Set, Naval Research Logistics 40, 103–116.

    Google Scholar 

  21. Benson, H. P. and Sayin, S. (1994), Optimization over the Efficient Set: Four Special Cases, Journal of Optimization Theory and Applications 80, 3–18.

    Google Scholar 

  22. Benson, H. P. and Sayin, S. (1997), Towards Finding Global Representations of the Efficient Set in Multiple Objective Mathematical Programming, Naval Research Logistics 44, 47–67.

    Google Scholar 

  23. Bitran, G. R. and Magnanti, T. L. (1979), The Structure of Admissible Points with Respect to Cone Dominance, Journal of Optimization Theory and Applications 29, 573–614.

    Google Scholar 

  24. Bolintineanu, S. (1993), Minimization of a Quasi-Concave Function over an Efficient Set, Mathematical Programming 61, 89–110.

    Google Scholar 

  25. Brlansky, R. H., Pelosi, R. R., Garnsey, S. H., Youtsey, C. O., Lee, R. F., Yokomi, R. K. and Sonoda, R. M. (1986), Tristeza Quick Decline Epidemic in South Florida, Proceedings of the Florida State Horticultural Society 99, 66–69.

    Google Scholar 

  26. Candler, W. and Boehije, M. (1971), Use of Linear Programming in Capital Budgeting with Multiple Goals, American Journal of Agricultural Economics 53, 325–330.

    Google Scholar 

  27. Castle, W. S., Tucker, D. P. H., Krezdorn, A. H. and Youtsey, C. O. (1989), Rootstocks for Florida Citrus, University of Florida, Institute of Food and Agricultural Sciences, Gainesville, Florida.

    Google Scholar 

  28. Changkong, V. and Haimes, Y. Y. (1983), Multiobjective Decision Making, North-Holland Publishing Company, Amsterdam.

    Google Scholar 

  29. Cohon, J. L. (1978), Multiobjective Programming and Planning, Academic Press, New York.

    Google Scholar 

  30. Dauer, J. P. (1991), Optimization over the Efficient Set Using an Active Constraint Approach, Zeitschrift für Operations Research 35, 185–195.

    Google Scholar 

  31. Dauer, J. P. and Fosnaugh, T. A. (1995), Optimization over the Efficient Set, Journal of Global Optimization 7, 261–277.

    Google Scholar 

  32. Dessouky, M. I., Ghiassi, M. and Davis, W. J. (1986), Estimates of the Minimum Nondominated Criterion Values in Multiple-Criteria Decision-Making, Engineering Costs and Production Economics 10, 95–104.

    Google Scholar 

  33. Eatman, J. L. and Sealey, C. W. (1979), A Multiobjective Linear Programming Model for Commercial Bank Balance Sheet Management, Journal of Banking Research 9, 227–236.

    Google Scholar 

  34. Ecker, J. G., Hegner, N. S. and Kouada, I. A. (1980), Generating All Maximal Efficient Faces for Multiple Objective Linear Programs, Journal of Optimization Theory and Applications 30, 353–381.

    Google Scholar 

  35. Evans, G. W. (1984), An Overview of Techniques for Solving Multiobjective Mathematical Programs, Management Science 30, 1268–1282.

    Google Scholar 

  36. Evans, J. P. and Steuer, R. E. (1973), A Revised Simplex Method for Linear Multiple Objective Programs, Mathematical Programming 5, 54–72.

    Google Scholar 

  37. Fulop, J. (1994), A Cutting Plane Method for Linear Optimization over the Efficient Set, in S. Komlosi, T. Rapcsak and S. Schaible (eds.), Generalized Convexity, Springer Verlag, Berlin, pp. 374–385.

    Google Scholar 

  38. Geoffrion, A. M. (1968), Proper Efficiency and the Theory of Vector Maximization, Journal of Mathematical Analysis and Applications 22, 618–630.

    Google Scholar 

  39. Geoffrion, A. M., Dyer, J. S. and Feinberg, A. (1972), An Interactive Approach for Multi-Criterion Optimization with an Application to the Operation of an Academic Department, Management Science 19, 357–368.

    Google Scholar 

  40. Ghiassi, M., De Vor, R. E., Dessouky, M. I. and Kijowski, B. A. (1984), An Application of Multiple Criteria Decision Making Principles for Planning Machining Operations, IIE Transactions 16, 106–114.

    Google Scholar 

  41. Goicoechea, A., Hansen, D. R. and Duckstein, L. (1982), Multiobjective Decision Analysis with Engineering and Business Applications, John Wiley and Sons, New York.

    Google Scholar 

  42. Henig, M. I. (1990), Value Functions, Domination Cones and Proper Efficiency in Multicriteria Optimization, Mathematical Programming 46, 205–217.

    Google Scholar 

  43. Horst, R. and Tuy, H. (1993), Global Optimization: Deterministic Approaches (2nd edition), Springer Verlag, Berlin.

    Google Scholar 

  44. Isermann, H. (1977), The Enumeration of the Set of All Efficient Solutions for a Linear Multiple Objective Program, Operational Research Quarterly 28, 711–725.

    Google Scholar 

  45. Isermann, H. and Steuer, R. E. (1987), Computational Experience Concerning Payoff Tables and Minimum Criteria Values over the Efficient Set, European Journal of Operational Research 33, 91–97.

    Google Scholar 

  46. Joiner, J. (1955), Extension Circular No. 132, University of Florida, Institute of Food and Agricultural Sciences, Gainesville, Florida.

    Google Scholar 

  47. Kok, M. and Lootsma, F. A. (1985), Pairwise-Comparison Methods in Multiple Objective Programming, with Application in a Long-Term Energy-Planning Model, European Journal of Operational Research 22, 44–55.

    Google Scholar 

  48. Korhonen, P., Salo, S. and Steuer, R. E. (1996), A Heuristic for Estimating Nadir Criterion Values in Multiple Objective Linear Programming, Working Paper, Helsinki School of Economics, Helsinki, Finland.

    Google Scholar 

  49. Lawrence, F. and Bridges, D. (1974), Extension Circular No. 394, University of Florida, Institute of Food and Agricultural Sciences, Gainesville, Florida.

    Google Scholar 

  50. Lawrence, K. D. and Burbridge, J. J. (1976), A Multiple Goal Linear Programming Model for Coordinated Production and Logistics Planning, International Journal of Production Research 14, 215–222.

    Google Scholar 

  51. Loucks, D. P. (1977), An Application of Interactive Water Resources Planning, Interfaces 8, 70–85.

    Google Scholar 

  52. Masud, A. S. and Hwang, C. L. (1981), Interactive Sequential Goal Programming, Journal of the Operational Research Society 32, 391–400.

    Google Scholar 

  53. Philip, J. (1972), Algorithms for the Vector Maximization Problem, Mathematical Programming 2, 207–229.

    Google Scholar 

  54. Reeves, G. and Reid, R. (1988), Minimum Values Over the Efficient Set in Multiple Objective Decision Making, European Journal of Operational Research 36, 334–338.

    Google Scholar 

  55. Ringuest, J. L. (1992), Multiobjective Optimization: Behavioral and Computational Considerations, Kluwer Academic Publishers, Dordrecht.

    Google Scholar 

  56. Rosenthal, R. E. (1985), Principles of Multiobjective Optimization, Decision Sciences 16, 133–152.

    Google Scholar 

  57. Rouse, R. E., Holcomb, E. D., Tucker, D. P. H. and Youtsey, C. O. (1990), Freeze Damage Sustained by 27 Citrus Cultivars on 21 Rootstocks in the Budwood Foundation Grove, Immokalee, Proceedings of the Florida State Horticultural Society 103, 63–67.

    Google Scholar 

  58. Sawaragi, Y., Nakayama, H. and Tanino, T. (1985), Theory of Multiobjective Optimization, Academic Press, Orlando, Florida.

    Google Scholar 

  59. Schrage, L. (1991), LINDO User's Manual, Release 5.0, Scientific Press, San Francisco.

    Google Scholar 

  60. Shin, W. S. and Ravindran, A. (1991), Interactive Multiple Objective Optimization: Survey I-Continuous Case, Computers and Operations Research 18, 97–114.

    Google Scholar 

  61. Smith, G. S., Hutchison, D. J. and Henderson, C. T. (1987), Screening Sweet Orange Citrus Cultivars for Relative Susceptibility to Phytophthora Foot Rot, Proceedings of the Florida State Horticultural Society 100, 64–66.

    Google Scholar 

  62. Soland, R. M. (1979), Multicriteria Optimization: A General Characterization of Efficient Solutions, Decision Sciences 10, 26–38.

    Google Scholar 

  63. Steuer, R. E. (1986), Multiple Criteria Optimization: Theory, Computation, and Application, John Wiley and Sons, New York.

    Google Scholar 

  64. Steuer, R. E. and Schuler, A. T. (1978), An Interactive Multiple Objective Linear Programming Approach to a Problem in Forest Management, Operations Research 25, 254–269.

    Google Scholar 

  65. Wallenius, J. (1975), Comparative Evaluation for Some Interactive Approaches to Multicriterion Optimization, Management Science 21, 1387–1396.

    Google Scholar 

  66. Weistroffer, H. R. (1985), Careful Usage of Pessimistic Values is Needed in Multiple Objectives Optimization, Operations Research Letters 4, 23–25.

    Google Scholar 

  67. Young, R. H., Albrigo, L. G., Tucker, D. P. H. and Williams, G. (1980), Incidence of Citrus Blight on Carrizo Citrange and Some Other Rootstocks, Proceedings of the Florida State Horticultural Society 93, 14–17.

    Google Scholar 

  68. Young, R. H., Albrigo, L. G., Cohen, M. and Castle, W. S. (1982), Rates of Blight Incidence in Trees on Carrizo Citrange and Other Rootstocks, Proceedings of the Florida State Horticultural Society 95, 76–78.

    Google Scholar 

  69. Youtsey, C. O. (1986), Incidence of Citrus Blight in Florida's Citrus Budwood Foundation Grove, Proceedings of the Florida State Horticultural Society 99, 71–73.

    Google Scholar 

  70. Yu, P. L. (1985), Multiple Criteria Decision Making, Plenum Press, New York.

    Google Scholar 

  71. Yu, P. L. (1989), Multiple Criteria Decision Making: Five Basic Concepts, in G. L. Nemhauser, A. H. G. Rinooy Kan and M. J. Tod (eds.), Optimization, North-Holland Publishing Company, Amsterdam, pp. 663–699.

    Google Scholar 

  72. Yu, P. L. and Zeleny, M. (1975), The Set of All Nondominated Solutions in Linear Cases and a Multicriteria Simplex Method, Journal of Mathematical Analysis and Applications 49, 430–468.

    Google Scholar 

  73. Zeleny, M. (1982), Multiple Criteria Decision Making, McGraw Hill, New York.

    Google Scholar 

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Authors and Affiliations

  1. Department of Decision and Information Sciences, University of Florida, 351 Business Building, Gainesville, FL, 32611, USA

    Harold P. Benson

  2. Korea Industrial Technology Association, KSTC Building, 635-4, Yeogsam-dong, Kangnam-gu, Seoul, 135-703, Korea

    Dongyeup Lee

  3. Becker Groves Inc., Box 1240, Fort Pierce, FL, 34954, USA

    J. Peter McClure

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  1. Harold P. Benson
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  2. Dongyeup Lee
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  3. J. Peter McClure
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Benson, H.P., Lee, D. & McClure, J.P. Global Optimization in Practice: An Application to Interactive Multiple Objective Linear Programming. Journal of Global Optimization 12, 353–372 (1998). https://doi.org/10.1023/A:1008285515867

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