Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Linear programming based decomposition approach in evaluating priorities from pairwise comparisons and error analysis

Abstract

One of the most difficult issues in many real-life decisionmaking problems is how to estimate the pertinent data. An approach which uses pairwise comparisons was proposed by Saaty and is widely accepted as an effective way of determining these data. Suppose that two matrices with pairwise comparisons are available. Furthermore, suppose that there is an overlapping of the elements compared in these two matrices. The problem examined in this paper is how to combine the comparisons of the two matrices in order to derive the priorities of the elements considered in both matrices. A simple approach and a linear programming approach are formulated and analyzed in solving this problem. Computational results suggest that the LP approach, under certain conditions, is an effective way for dealing with this problem. The proposed approach is of critical importance because it can also result in a reduction of the total required number of comparisons.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Zadeh, L. A.,Fuzzy Sets, Information and Control, Vol. 8, pp. 338–353, 1965.

  2. 2.

    Zadeh, L. A.,Fuzzy Algorithms, Information and Control, Vol. 12, pp. 94–102, 1968.

  3. 3.

    Bellman, R. E., andZadeh, L. A.,Decision-Making in a Fuzzy Environment, Management Science, Vol. 17, pp. 140–164, 1970.

  4. 4.

    Chang, S. K.,Fuzzy Programs, Proceedings of the Brooklyn Polytechnical Institute Symposium on Computers and Automata, Brooklyn, New York, New York, Vol. 21, pp. 124–135, 1971.

  5. 5.

    Dubois, D., andPrade, H.,Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, New York, 1980.

  6. 6.

    Gupta, M. M., Ragade, R. K., andYager, R. Y., Editors,Fuzzy Set Theory and Applications, North-Holland, New York, New York, 1979.

  7. 7.

    Xie, L. A., andBerdosian, S. D.,The Information in a Fuzzy Set and the Relation between Shannon and Fuzzy Information, Proceedings of the 7th Annual Conference on Information Sciences and Systems, Johns Hopkins University, Baltimore, Maryland, pp. 212–233, 1983.

  8. 8.

    Zadeh, L. A., Fu, K. S., Tanaka, K., andShimura, M., Editors,Fuzzy Sets and Their Applications to Cognitive and Decision Processes, Academic Press, New York, New York, 1975.

  9. 9.

    Zadeh, L. A.,Fuzzy Sets as a Basis for a Theory of Possibility, Fuzzy Sets and Systems, Vol. 1, pp. 3–28, 1978.

  10. 10.

    Zadeh, L. A.,A Theory of Approximate Reasoning, Machine Intelligence, Edited by J. Hayes, D. Michie, and L. I. Mikulich, Wiley, New York, New York, Vol. 9, pp. 149–194, 1979.

  11. 11.

    Zadeh, L. A.,The Concept of a Linguistic Variable and Its Applications to Approximate Reasoning, Parts 1–3, Information Sciences, Vol. 8, pp. 199–249, 1975; Vol. 8, pp. 301–357, 1975; and Vol. 9, pp. 43–80, 1976.

  12. 12.

    Zimmermann, H. J.,Fuzzy Set Theory and Its Applications, Kluwer Academic Publishers, Amsterdam, Holland, 1985.

  13. 13.

    Lee, C. T. R.,Fuzzy Logic and the Resolution Principle, 2nd International Joint Conference on Artificial Intelligence, London, England, pp. 560–567, 1971.

  14. 14.

    Lee, S. N., Grize, Y. L., andDehnad, K.,Quantitative Models for Reasoning under Uncertainty in Knowledge-Based Expert Systems, International Journal of Intelligent Systems, Vol. 2, pp. 15–38, 1987.

  15. 15.

    Prade, H., andNegoita, C. V., Editors,Fuzzy Logic in Knowledge Engineering, Verlag TUV Rheinland, Berlin, Germany, 1986.

  16. 16.

    Ramsay, A.,Formal Methods in Artificial Intelligence, Cambridge University Press, Cambridge, England, 1988.

  17. 17.

    Zadeh, L. A.,Can Expert Systems Be Designed Without Using Fuzzy Logic?, Proceedings of the 17th Annual Conference on Information Sciences and Systems, Johns Hopkins University, Baltimore, Maryland, pp. 23–31, 1983.

  18. 18.

    Lootsma, F. A., Pardalos, P. M., andTriantaphyllou, E., Editors,Fuzzy Decision Making, Kluwer Academic Publishers, Amsterdam, Holland, 1995 (to appear).

  19. 19.

    Saaty, T. L.,A Scaling Method for Priorities in Hierarchical Structures, Journal of Mathematical Psychology, Vol. 15, pp. 234–281, 1977.

  20. 20.

    Saaty, T. L.,Exploring the Interface between Hierarchies, Multiple Objects, and Fuzzy Sets, Journal of Fuzzy Sets and Systems, Vol. 1, pp. 57–68, 1978.

  21. 21.

    Saaty, T. L.,The Analytic Hierarchy Process, McGraw-Hill International, New York, New York, 1980.

  22. 22.

    Chu, A. T. W., Kalaba, R. E., andSpingarn, K.,A Comparison of Two Methods for Determining the Weights of Belonging to Fuzzy Sets, Journal of Optimization Theory and Applications, Vol. 27, pp. 531–538, 1979.

  23. 23.

    Federov, V. V., Kuzmin, V. B., andVereskov, A. I.,Membership Degrees Determination from Saaty Matrix Totalities, Approximate Reasoning in Decision Analysis, Edited by M. M. Gupta and E. Sanchez, North-Holland Publishing Company, Amsterdam, Holland, pp. 23–30, 1982.

  24. 24.

    Khurgin, J. I., andPolyakov, V. V.,Fuzzy Analysis of the Group Concordance of Expert Preferences Defined by Saaty Matrices, Mathematical Research/Mathematische Forschung, Vol. 30, pp. 111–115, 1986.

  25. 25.

    Khurgin, J. I., andPolyakov, V. V.,Fuzzy Approach to the Analysis of Expert Data, Mathematical Research/Mathematische Forschung, Vol. 30, pp. 116–124, 1986.

  26. 26.

    Lootsma, F. A.,Numerical Scaling of Human Judgment in Pairwise-Comparison Methods for Fuzzy Multi-Criteria Decision Analysis, Mathematical Models for Decision Support, Springer, Berlin, Germany, Vol. 48, pp. 57–88, 1988.

  27. 27.

    Lootsma, F. A., Mensch, T. C. A., andVos, F. A.,Multi-Criteria Analysis and Budget Reallocation in Long-Term Research Planning, European Journal of Operational Research, Vol. 47, pp. 293–305, 1990.

  28. 28.

    Triantaphyllou, E., Lootsma, F. A., Pardalos, P. M., andMann, S. H.,On the Evaluation and Application of Different Scales for Quantifying Pairwise Comparisons in Fuzzy Sets, Journal of Multi-Criteria Decision Analysis, Vol. 3, 1994.

  29. 29.

    Vargas, L. G.,Reciprocal Matrices with Random Coefficients, Mathematical Modeling, Vol. 3, pp. 69–81, 1982.

  30. 30.

    Triantaphyllou, E., andMann, S. H.,An Examination of the Effectiveness of Multi-Dimensional Decision-Making Methods: A Decision-Making Paradox, International Journal of Decision Support Systems, Vol. 5, pp. 303–312, 1989.

  31. 31.

    Lootsma, F. A.,The French and the American School in Multi-Criteria Decision Analysis, Recherche Operationnelle/Operations Research, Vol. 24, pp. 263–285, 1990.

  32. 32.

    Lootsma, F. A.,Scale Sensitivity and Rank Preservation in a Multiplicative Variant of the AHP and SMART, Report 91-67, Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, Holland, 1991.

  33. 33.

    Triantaphyllou, E., Pardalos, P. M., andMann, S. H.,A Minimization Approach to Membership Evaluation in Fuzzy Sets and Error Analysis, Journal of Optimization Theory and Applications, Vol. 66, pp. 275–287, 1990.

  34. 34.

    Harker, P. T.,Incomplete Pairwise Comparisons in the Analytic Hierarchy Process, Mathematical Modeling, Vol. 9, pp. 837–848, 1987.

  35. 35.

    Triantaphyllou, E., andMann, S. H.,An Evaluation of the Eigenvalue Approach for Determining the Membership Values in Fuzzy Sets, Fuzzy Sets and Systems, Elsevier Science Publishers, Vol. 35, pp. 295–301, 1990.

  36. 36.

    Triantaphyllou, E., andMann, S. H.,A Computational Evaluation of the AHP and the Revised AHP When the Eigenvalue Method Is Used under a Continuity Assumption, Computers and Industrial Engineering, Vol. 26, pp. 609–618, 1994.

  37. 37.

    Triantaphyllou, E., Pardalos, P. M., andMann, S. H.,The Problem of Determining Membership Values in Fuzzy Sets in Real World Situations, Operations Research and Artificial Intelligence: The Integration of Problem Solving Strategies, Edited by D. E. Brown and C. C. White III, Kluwer Academic Publishers, Amsterdam, Holland, pp. 197–214, 1990.

  38. 38.

    Sanchez, A., andTriantaphyllou, E.,Identification of the Critical Criteria When Using the Analytic Hierarchy Process, Working Paper, Department of Industrial and Manufacturing Systems Engineering, Louisiana State University, Baton Rouge, Louisiana, 1994.

  39. 39.

    Sanchez, A., andTriantaphyllou, E.,Identification of the Critical Criteria in Deterministic Multi-Criteria Decision Making, Working Paper, Department of Industrial and Manufacturing Systems Engineering, Louisiana State University, Baton Rouge, Louisiana, 1994.

Download references

Author information

Additional information

The author would like to thank Professors Stuart H. Mann, Pennsylvania State University, and Panos M. Pardalos, University of Florida, for their support and valuable comments during the early stages of this research.

Communicated by R. E. Kalaba

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Triantaphyllou, E. Linear programming based decomposition approach in evaluating priorities from pairwise comparisons and error analysis. J Optim Theory Appl 84, 207–234 (1995). https://doi.org/10.1007/BF02191743

Download citation

Key Words

  • Pairwise comparisons
  • eigenvectors
  • analytic hierarchy process
  • linear programming
  • fuzzy sets
  • membership values
  • artificial intelligence