A fuzzy extension of Analytic Hierarchy Process based on the constrained fuzzy arithmetic

Fuzzy Optimization and Decision Making volume 16, pages89–110(2017)Cite this article

Abstract

The aim of the paper is to highlight the necessity of applying the concept of constrained fuzzy arithmetic instead of the concept of standard fuzzy arithmetic in a fuzzy extension of Analytic Hierarchy Process (AHP). Emphasis is put on preserving the reciprocity of pairwise comparisons during the computations. For deriving fuzzy weights from a fuzzy pairwise comparison matrix, we consider a fuzzy extension of the geometric mean method and simplify the formulas proposed by Enea and Piazza (Fuzzy Optim Decis Mak 3:39–62, 2004). As for the computation of the overall fuzzy weights of alternatives, we reveal the inappropriateness of applying the concept of standard fuzzy arithmetic and propose the proper formulas where the interactions among the fuzzy weights are taken into account. The advantage of our approach is elimination of the false increase of uncertainty of the overall fuzzy weights. Finally, we advocate the validity of the proposed fuzzy extension of AHP; we show by an illustrative example that by neglecting the information about uncertainty of intensity of preferences we lose an important part of knowledge about the decision making problem which can cause the change in ordering of alternatives.

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References

  1. Buckley, J. J. (1985). Fuzzy hierarchical analysis. Fuzzy Sets and Systems, 17(3), 233–247.

    MathSciNet  Article  MATH  Google Scholar 

  2. Chang, D. Y. (1996). Applications of the extent analysis method on fuzzy AHP. European Journal of Operational Research, 95(3), 649–655.

    Article  MATH  Google Scholar 

  3. Crawford, G., & Williams, C. A. (1985). A note on the analysis of subjective judgment matrices. Journal of Mathematical Psychology, 29(4), 387–405.

    Article  MATH  Google Scholar 

  4. Dubois, D. (2011). The role of fuzzy sets in decision sciences: Old techniques and new directions. Fuzzy Sets and Systems, 184(1), 3–28.

    MathSciNet  Article  MATH  Google Scholar 

  5. Enea, M., & Piazza, T. (2004). Project selection by constrained fuzzy AHP. Fuzzy Optimization and Decision Making, 3(1), 39–62.

    Article  MATH  Google Scholar 

  6. Güngör, Z., Serhadlioglu, G., & Kesen, S. E. (2009). A fuzzy AHP approach to personnel selection problem. Applied Soft Computing, 9(2), 641–646.

    Article  Google Scholar 

  7. Ishizaka, A., & Nguyen, N. H. (2013). Calibrated fuzzy AHP for current bank account selection. Expert Systems with Applications, 40, 3775–3783.

    Article  Google Scholar 

  8. Klir, G. J., & Yuan, B. (1998). Constrained fuzzy arithmetic: Basic questions and some answers. Soft Computing, 2(2), 100–108.

    Article  Google Scholar 

  9. Krejčí, J., & Stoklasa, J. (2015). Fuzzified AHP in the evaluation of scientific monographs. Central European Journal of Operations Research. doi:10.1007/s10100-015-0399-2.

  10. Pan, N.-F. (2008). Fuzzy AHP approach for selecting the suitable bridge construction method. Automation in Construction, 17(8), 958–965.

    Article  Google Scholar 

  11. Pavlačka, O. (2011). Modeling uncertain variables of the weighted average operation by fuzzy vectors. Information Sciences, 181(22), 4969–4992.

    MathSciNet  Article  MATH  Google Scholar 

  12. Pavlačka, O. (2014). On various approaches to normalization of interval and fuzzy weights. Fuzzy Sets and Systems, 243, 110–130.

    MathSciNet  Article  MATH  Google Scholar 

  13. Ramík, J., & Korviny, P. (2010). Inconsistency of pair-wise comparison matrix with fuzzy elements based on geometric mean. Fuzzy Sets and Systems, 161(11), 1604–1613.

    MathSciNet  Article  MATH  Google Scholar 

  14. Ramík, J. (2015). Isomorphisms between fuzzy pairwise comparison matrices. Fuzzy Optimization and Decision Making, 14(2), 199–209.

    MathSciNet  Article  Google Scholar 

  15. Saaty, T. L. (1980). The analytic hierarchy process. New York: McGraw Hill.

    Google Scholar 

  16. Saaty, T. L. (2006). There is no mathematical validity for using fuzzy number crunching in the analytic hierarchy process. Journal of Systems Science and Systems Engineering, 15(4), 457–464.

    Article  Google Scholar 

  17. Saaty, T. L., & Tran, L. T. (2007). On the invalidity of fuzzifying numerical judgments in the analytic hierarchy process. Mathematical and Computer Modelling, 46, 962–975.

    MathSciNet  Article  MATH  Google Scholar 

  18. Subramanian, N., & Ramanathan, R. (2012). A review of applications of analytic hierarchy process in operations management. International Journal of Production Economics, 138, 215–241.

    Article  Google Scholar 

  19. Tesfamariam, S., & Sadiq, R. (2006). Risk-based environmental decision-making fuzzy analytic hierarchy process (F-AHP). Stochastic Environmental Research and Risk Assessment, 21, 35–50.

    MathSciNet  Article  Google Scholar 

  20. Tiryaki, F., & Ahlatcioglu, A. (2009). Fuzzy portfolio selection using fuzzy analytic hierarchy process. Information Sciences, 179(1–2), 53–69.

    Article  MATH  Google Scholar 

  21. Vahidnia, M. H., Alesheikh, A. A., & Alimohammadi, A. (2009). Hospital site selection using fuzzy AHP and its derivatives. Journal of Environmental Management, 90(10), 3048–3056.

    Article  Google Scholar 

  22. Van Laarhoven, P. J. M., & Pedrycz, W. (1983). A fuzzy extension of Saaty’s priority theory. Fuzzy Sets and Systems, 11(1–3), 199–227.

    MathSciNet  MATH  Google Scholar 

  23. Wang, Y. M., & Elhag, T. M. S. (2006). On the normalization of interval and fuzzy weights. Fuzzy Sets and Systems, 157(18), 2456–2471.

    MathSciNet  Article  MATH  Google Scholar 

  24. Wang, Y. M., Luo, Y., & Hua, Z. (2008). On the extent analysis method for fuzzy AHP and its applications. European Journal of Operational Research, 186(2), 735–747.

    Article  MATH  Google Scholar 

  25. Yuen, K. K. F. (2014). Combining compound linguistic ordinal scale and cognitive pairwise comparison in the rectified fuzzy TOPSIS method for group decision making. Fuzzy Optimization and Decision Making, 13(1), 105–130.

    MathSciNet  Article  Google Scholar 

  26. Zadeh, L. A. (1975). Concept of a linguistic variable and its application to approximate reasoning I. Information Sciences, 8(3), 199–249.

    MathSciNet  Article  MATH  Google Scholar 

  27. Zadeh, L. A. (1975). Concept of a linguistic variable and its application to approximate reasoning II. Information Sciences, 8(4), 301–357.

    MathSciNet  Article  MATH  Google Scholar 

  28. Zadeh, L. A. (1975). Concept of a linguistic variable and its application to approximate reasoning III. Information Sciences, 9(1), 43–80.

    MathSciNet  Article  MATH  Google Scholar 

  29. Zheng, G., Zhu, N., Tian, Z., Chen, Y., & Sun, B. (2012). Application of a trapezoidal fuzzy AHP method for work safety evaluation and early warning rating of hot and humid environments. Safety Science, 50(2), 228–239.

    Article  Google Scholar 

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Author information

Affiliations

  1. Department of Industrial Engineering, University of Trento, Via Sommarive 9, 38122, Trento, Italy

    Jana Krejčí

  2. Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University Olomouc, 17. listopadu 12, 771 46, Olomouc, Czech Republic

    Ondřej Pavlačka & Jana Talašová

Authors
  1. Jana Krejčí
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  2. Ondřej Pavlačka
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  3. Jana Talašová
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Corresponding author

Correspondence to Ondřej Pavlačka.

Additional information

Research has been supported by the project No. GA 14-02424S Methods of operations research for decision support under uncertainty of the Grant Agency of the Czech Republic.

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Krejčí, J., Pavlačka, O. & Talašová, J. A fuzzy extension of Analytic Hierarchy Process based on the constrained fuzzy arithmetic. Fuzzy Optim Decis Making 16, 89–110 (2017). https://doi.org/10.1007/s10700-016-9241-0

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Keywords

  • Fuzzy Analytic Hierarchy Process
  • Fuzzy pairwise comparison matrices
  • Triangular fuzzy numbers
  • Constrained fuzzy arithmetic
  • Fuzzy weighted average

Mathematics Subject Classification

  • 90B50