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Fuzzy Optimization and Decision Making

, Volume 17, Issue 2, pp 195–209 | Cite as

Deriving crisp and consistent priorities for fuzzy AHP-based multicriteria systems using non-linear constrained optimization

  • Raman Kumar GoyalEmail author
  • Sakshi Kaushal
Article

Abstract

Fuzzy optimization models are used to derive crisp weights (priority vectors) for the fuzzy analytic hierarchy process (AHP) based multicriteria decision making systems. These optimization models deal with the imprecise judgements of decision makers by formulating the optimization problem as the system of constrained non linear equations. Firstly, a Genetic Algorithm based heuristic solution for this optimization problem is implemented in this paper. It has been found that the crisp weights derived from this solution for fuzzy-AHP system, sometimes lead to less consistent or inconsistent solutions. To deal with this problem, we have proposed a consistency based constraint for the optimization models. A decision maker can set the consistency threshold value and if the solution exists for that threshold value then crisp weights can be derived, otherwise it can be concluded that the fuzzy comparison matrix for AHP is not consistent for the given threshold. Three examples are considered to demonstrate the effectiveness of the proposed method. Results with the proposed constraint based fuzzy optimization model are more consistent than the existing optimization models.

Keywords

MCDM Fuzzy AHP Genetic Algorithms Pairwise comparison judgements Priority vector 

Notes

Acknowledgements

The authors would like to thank University Institute of Engineering and Technology, Panjab University, Chandigarh, India for providing the research facilities for carrying out this research. The author Raman Kumar Goyal would also like to thank Technical Education Quality Improvement Programme (TEQIP)-II for providing the financial assistance during the course of study.

References

  1. Boender, C. G. E., de Grann, J. G., & Lootsma, F. A. (1989). Multi-criteria decision analysis with fuzzy pairwise comparison. Fuzzy Sets and Systems, 29, 133–143.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Buckley, J. J. (1985). Fuzzy hierarchical analysis. Fuzzy Sets and Systems, 17, 233–247.MathSciNetCrossRefzbMATHGoogle Scholar
  3. Chang, D. Y. (1996). Applications of the extent analysis method on fuzzy AHP. European Journal of Operation Research, 95, 649–655.CrossRefzbMATHGoogle Scholar
  4. Deng, X., Hu, Y., Deng, Y., & Mahadevan, S. (2014). Supplier selection using AHP methodology extended by D numbers. Expert Systems with Applications, 41, 156–167.CrossRefGoogle Scholar
  5. Fedrizzi, M., & Giove, S. (2007). Incomplete pairwise comparison and consistency optimization. European Journal of Operation Research, 183(1), 303–313.CrossRefzbMATHGoogle Scholar
  6. Goyal, R. K., & Kaushal, S. (2016). A constrained non-linear optimization model for fuzzy pairwise comparison matrices using teaching learning based optimization. Applied Intelligence, 45(3), 652–661.CrossRefGoogle Scholar
  7. Herrera-Viedma, E., Herrera, F., Chiclana, F., & Luque, M. (2004). Some issues on consistency of fuzzy preference relations. European Journal of Operation Research, 154(1), 98–109.MathSciNetCrossRefzbMATHGoogle Scholar
  8. Jaganathan, S., Jinson, J. E., & Ker, J. (2007). Fuzzy analytic hierarchy process based group decision support system to select and evaluate new manufacturing technologies. International Journal of Advanced Manufacturing Technology, 32(11–12), 1253–1262.CrossRefGoogle Scholar
  9. Javanbarg, M. B., Scawthorn, C., Kiyono, J., & Shahbodaghkhan, B. (2012). Fuzzy AHP-based multicriteria decision making systems using particle swarm optimization. Expert System with Applications, 39(1), 960–966.CrossRefGoogle Scholar
  10. Krejčí, J., Pavlacka, O., & Talasová, J. (2016). A fuzzy extension of Analytic Hierarchy Process based on the constrained fuzzy arithmetic. Fuzzy Optimisation and Decision Making,. doi: 10.1007/s10700-016-9241-0.Google Scholar
  11. Kwiesielewicz, M. (1996). The logarithmic least squares and the generalized pseudoinverse in estimating ratios. European Journal of Operation Research, 93(3), 611–619.CrossRefzbMATHGoogle Scholar
  12. Melanie, M. (1996). An introduction to genetic algorithms. Massachusetts: MIT Press.zbMATHGoogle Scholar
  13. Mikhailov, L. (2000). A fuzzy programming method for deriving priorities in the analytic hierarchy process. Journal of the Operational Research Society, 51, 341–349.CrossRefzbMATHGoogle Scholar
  14. Mikhailov, L. (2003). Deriving priorities from fuzzy pairwise comparison judgments. Fuzzy Sets and Systems, 134, 365–385.MathSciNetCrossRefzbMATHGoogle Scholar
  15. Mohtashami, A. (2014). A novel-heuristic based method for deriving priorities from fuzzy pairwise comparison judgements. Applied Soft Computing, 23, 530–545.CrossRefGoogle Scholar
  16. Rao, R. V. (2007). Decision making in the manufacturing environment using graph theory and fuzzy multiple attribute decision making methods. London: Springer.zbMATHGoogle Scholar
  17. Saaty, T. L. (1980). The analytic hierarchy process. New York: McGraw-Hill.zbMATHGoogle Scholar
  18. Triantaphyllou, E., & Lin, C. T. (1996). Development and evaluation of five fuzzy multiattribute decision-making methods. International Journal of Approximate Reasoning, 14, 281–310.CrossRefzbMATHGoogle Scholar
  19. Van Laarhoven, P. J. M., & Pedrycz, W. (1983). A fuzzy extension of Saaty’s priority theory. Fuzzy Sets and Systems, 11, 229–241.MathSciNetCrossRefzbMATHGoogle Scholar
  20. Wang, L., Chu, J., & Wu, J. (2007). Selection of optimum maintenance strategies based on a fuzzy analytic hierarchy process. International Journal of Production Economics, 107, 151–163.CrossRefGoogle Scholar
  21. Wang, Y. M., Luo, Y., & Hua, Z. (2008). On the extent analysis method for fuzzy AHP and its applications. European Journal of Operation Research, 186, 735–747.CrossRefzbMATHGoogle Scholar
  22. Zadeh, L. A. (1965). Fuzzy sets. Information Control, 8, 338–353.MathSciNetCrossRefzbMATHGoogle Scholar
  23. Zimmermann, H. J. (1994). Fuzzy set theory and its applications. Boston: Kluwer Academic Publishers.Google Scholar
  24. Zhang, F., Ignatius, J., Lim, C. P., & Zhao, Y. (2014). A new method for deriving priority weights by extracting consistent numerical-valued matrices from interval-valued fuzzy judgement matrix. Information Sciences, 290, 280–300.MathSciNetCrossRefzbMATHGoogle Scholar
  25. Zhang, H. (2016). A goal programming model of obtaining the priority weights from an interval preference relation. Information Sciences, 354, 197–210.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Computer Science and Engineering, University Institute of Engineering and TechnologyPanjab UniversityChandigarhIndia

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