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Maximizing a Linear Fractional Function on a Pareto Efficient Frontier

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Abstract

We consider the problem of maximizing a linear fractional function on the Pareto efficient frontier of two other linear fractional functions. We present a finite pivoting-type algorithm that solves the maximization problem while computing simultaneously the efficient frontier. Application to multistage efficiency analysis is discussed. An example demonstrating the computational procedure is included.

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References

  1. BAZARAA, M. S., SHERALI, H. D., and SHETTY, C. M., Nonlinear Programming; Theory and Algorithms, 2nd Edition, John Wiley and Sons, New York, NY, 1993.

    Google Scholar 

  2. KONNO, H., YASUTOSHI, Y., and TOMOMI, M., Parametric Simplex Algorithms for Solving a Special Class of Nonconvex Minimization Problems, Journal of Global Optimization, Vol. 1, pp. 65-81, 1991.

    Google Scholar 

  3. SCHAIBLE, S., Fractional Programming, Handbook of Global Optimization, Edited by R. Horst and P. M. Pardalos, Kluwer Academic Publishers, Boston, Massachusetts, pp. 495-608, 1995.

    Google Scholar 

  4. CHARNES, A., COOPER, W. W., and RHODES, E., Measuring the Efficiency of Decision Making Units, European Journal of Operations Research, Vol. 2, pp. 429-444, 1978.

    Google Scholar 

  5. FRIED, H. O., LOVELL, C. A. K., and SCHMIDT, S. S., The Measurement of Productive Efficiency: Techniques and Applications, Oxford University Press, New York, NY, 1993.

    Google Scholar 

  6. CHARNES, A., COOPER, W. W., LEWIN, A. Y., and SEIFORD, L.M., Data Envelopment Analysis: Theory, Methodology, and Application, Kluwer Academic Publishers, Boston, Massachusetts, 1994.

    Google Scholar 

  7. HACKMAN, S. T., PASSY, U., and PLATZMAN, L.K., Explicit Representation of the Two-Dimensional Section of a Production Possibility Set, Journal of Productivity Analysis, Vol. 5, pp. 161-170, 1994.

    Google Scholar 

  8. ROSEN, D., SCHAFFNIT, C., and PARADI, J. C., Marginal Rates and Two-Dimensional Level Curves in DEA, Journal of Productivity Analysis, Vol. 8, pp. 205-232, 1998.

    Google Scholar 

  9. CHOO, E. U., and ATKINS, D. R., Bicriteria Linear Fractional Programming, Journal of Optimization Theory and Applications, Vol. 36, pp. 203-220, 1982.

    Google Scholar 

  10. WARBURTON, A. R., Parametric Solution of Bicriterion Linear Fractional Programs, Operations Research, Vol. 33, pp. 74-84, 1985.

    Google Scholar 

  11. CAMBINI, A., and MARTEIN, L., Linear Fractional and Bicriteria Linear Fractional Programs, Lecture Notes in Economics and Mathematical Systems, Edited by A. Cambini, E. Castagnoli, L. Martein, P. Mazzoleni and S. Schaible, Springer Verlag, Berlin, Germany, Vol. 345, pp. 155-166, 1988.

    Google Scholar 

  12. CAMBINI, A., MARTEIN, L., and STANCU-MINASIAN, I. M., A Survey of Bicriteria Fractional Problems, AMO-Advanced Modeling and Optimization, Vol. 1, pp. 9-23, 1999.

    Google Scholar 

  13. ARMAND, P., Finding All Maximal Efficient Faces in Multiobjective Linear Programing, Mathematical Programming, Vol. 61A, pp. 357-375, 1993.

    Google Scholar 

  14. KARLIN, S., Mathematical Methods and Theory in Games, Programming, and Economics, Addison-Wesley Publishing Company, New York, NY, 1959.

    Google Scholar 

  15. MARCHI, A., On the Relationships between Bicriteria Problems and Nonlinear Programming, Generalized Convexity, Lecture Notes in Economics and Mathematical Systems, Edited by S. Komlosi, L. Rapcsak, and S. Schaible, Springer Verlag, Berlin, Germany, Vol. 405, pp. 392-400, 1994.

    Google Scholar 

  16. GAL, T., Post-Optimal Analyses, Parametric Programming, and Related Topics: Degeneracy, Multicriteria Decision Making, Redundancy, 2nd Edition, Walter de Gruyter, Berlin, Germany, 1995.

    Google Scholar 

  17. BENSON, H. P., An All-Linear Programming Relaxation Algorithm for Optimizing over the Efficient Set, Journal of Global Optimization, Vol. 1, pp. 83-104, 1991.

    Google Scholar 

  18. BOLINTINEAU, S., Minimization of Quasiconcave Functions over an Efficient Set, Mathematical Programming, Vol. 61A, pp. 89-110, 1993.

    Google Scholar 

  19. DAUER, J. P., and FOSNAUGH, T. A., Optimization over the Efficient Set, Journal of Global Optimization, Vol. 3, pp. 261-277, 1995.

    Google Scholar 

  20. BENSON, H. P., and LEE, D., Outcome-Based Algorithm for Optimization over the Efficient Set of a Bicriteria Linear Programming Problem, Journal of Optimization Theory and Applications, Vol. 88, pp. 77-105, 1996.

    Google Scholar 

  21. HORST, R., and THOAI, N. V., Utility Function Maximization over the Efficient Set in Multiple-Objective Decision Making, Journal of Optimization Theory and Applications, Vol. 92, pp. 605-631, 1997.

    Google Scholar 

  22. BENSON, H. P., An Outer Approximation Algorithm for Generating All Efficient Extreme Points in the Outcome Set of a Multiple-Objective Linear Programming Problem, Journal of Global Optimization, Vol. 13, pp. 1-24, 1998.

    Google Scholar 

  23. FüLöP, J., and MUU, L. D., Branch-and-Bound Variant of an Outcome-Based Algorithm for Optimizing over the Efficient Set of a Bicriteria Linear Programming Problem, Journal of Optimization Theory and Applications, Vol. 105, pp. 37-54, 2000.

    Google Scholar 

  24. SAYIN, S., Optimizing over the Efficient Set Using a Top-Down Search of Faces, Operations Research, Vol. 48, pp. 65-72, 2000.

    Google Scholar 

  25. CAMBINI, A., MARTEIN, L., and SCHAIBLE, S., On Maximizing a Sum of Ratios, Journal of Information and Optimization Sciences, Vol. 10, pp. 65-79, 1989.

    Google Scholar 

  26. KONNO, H., and KUNO, T., Generalized Linear Multiplicative and Fractional Programming, Report IHSS 89-14, Institute of Human and Social Sciences, Tokyo Institute of Technology, 1989.

  27. BORISOVA, E. P., GOL'SHTEIN, E. G., and DUBSON, M. S., Dialogue System of Multiobjective Optimization for an Economic Objective, Central Economical and Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia, pp. 60-61, 1984 (in Russian).

    Google Scholar 

  28. KONNO, H., and YAJIMA, Y., Minimizing and Maximizing the Product of Linear Fractional Functions, Recent Advances in Global Optimization, Edited by C. Floudas and P. M. Pardalos, Princeton University Press, Princeton, New Jersey, pp. 259-273, 1992.

    Google Scholar 

  29. FALK, J. E., and PALOCSAY, S. W., Image Space Analysis of Generalized Fractional Programs, Journal of Global Optimization, Vol. 4, pp. 63-88, 1994.

    Google Scholar 

  30. CAMBINI, A., MARCHI, A., MARTEIN, L., and SCHAIBLE, S., An Analysis of the Falk-Palocsay Algorithm, Scalar and Vector Optimization in Economic and Financial Problems, Edited by E. Castagnoli and G. Giorgi, Report 95, Department of Statistics and Applied Mathematics, University of Pisa, Pisa, Italy, 1995.

    Google Scholar 

  31. GEOFFRION, A., Solving Bicriterion Mathematical Programs, Operations Research, Vol. 15, pp. 39-54, 1967.

    Google Scholar 

  32. PASTERNAK, H., and PASSY, U., Finding the Global Optimum of Bicriterion Mathematical Programs, Cahiers du Centre d'Etudes de Recherche Operationnelle, Vol. 16, pp. 67-80, 1974.

    Google Scholar 

  33. CHARNES, A., and COOPER, W. W., Programming with Linear Fractional Functions, Naval Research Logistics Quarterly, Vol. 9, pp. 181-186, 1962.

    Google Scholar 

  34. SCHAIBLE, S., Bicriteria Quasiconcave Programs, Cahiers du Centre d'Etudes de Recherche Operationnelle, Vol. 25, pp. 93-101, 1983.

    Google Scholar 

  35. HU, Y. D., and SUN, E. J., Connectedness of the Efficient Set in Strictly Quasiconcave Vector Maximization, Journal of Optimization Theory and Applications, Vol. 78, pp. 613-622, 1993.

    Google Scholar 

  36. SUN, E. J., On the Connectedness of the Efficient Set for Strictly Quasiconvex Vector Minimization Problems, Journal of Optimization Theory and Applications, Vol. 89, pp. 475-481, 1996.

    Google Scholar 

  37. DANIILIDIS, A., HADJISAVVAS, N., and SCHAIBLE, S., Connectedness of the Efficient Set for Three-Objective Quasiconcave Maximization Problems, Journal of Optimization Theory and Applications, Vol. 93, pp. 517-524, 1997.

    Google Scholar 

  38. BENOIST, J, Connectedness of the Efficient Set for Quasiconcave Sets, Journal of Optimization Theory and Applications, Vol. 96, pp. 627-654, 1998.

    Google Scholar 

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Authors and Affiliations

  1. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia

    S.T. Hackman (Associate Professor)

  2. Faculty of Industrial Engineering and Management, Technion, Israel Institute of Technology, Haifa, Israel

    U. Passy (Professor)

Authors
  1. S.T. Hackman
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  2. U. Passy
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Hackman, S., Passy, U. Maximizing a Linear Fractional Function on a Pareto Efficient Frontier. Journal of Optimization Theory and Applications 113, 83–103 (2002). https://doi.org/10.1023/A:1014857230393

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  • DOI: https://doi.org/10.1023/A:1014857230393

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