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.
Similar content being viewed by others
References
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.
Buckley, J. J. (1985). Fuzzy hierarchical analysis. Fuzzy Sets and Systems, 17, 233–247.
Chang, D. Y. (1996). Applications of the extent analysis method on fuzzy AHP. European Journal of Operation Research, 95, 649–655.
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.
Fedrizzi, M., & Giove, S. (2007). Incomplete pairwise comparison and consistency optimization. European Journal of Operation Research, 183(1), 303–313.
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.
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.
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.
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.
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.
Kwiesielewicz, M. (1996). The logarithmic least squares and the generalized pseudoinverse in estimating ratios. European Journal of Operation Research, 93(3), 611–619.
Melanie, M. (1996). An introduction to genetic algorithms. Massachusetts: MIT Press.
Mikhailov, L. (2000). A fuzzy programming method for deriving priorities in the analytic hierarchy process. Journal of the Operational Research Society, 51, 341–349.
Mikhailov, L. (2003). Deriving priorities from fuzzy pairwise comparison judgments. Fuzzy Sets and Systems, 134, 365–385.
Mohtashami, A. (2014). A novel-heuristic based method for deriving priorities from fuzzy pairwise comparison judgements. Applied Soft Computing, 23, 530–545.
Rao, R. V. (2007). Decision making in the manufacturing environment using graph theory and fuzzy multiple attribute decision making methods. London: Springer.
Saaty, T. L. (1980). The analytic hierarchy process. New York: McGraw-Hill.
Triantaphyllou, E., & Lin, C. T. (1996). Development and evaluation of five fuzzy multiattribute decision-making methods. International Journal of Approximate Reasoning, 14, 281–310.
Van Laarhoven, P. J. M., & Pedrycz, W. (1983). A fuzzy extension of Saaty’s priority theory. Fuzzy Sets and Systems, 11, 229–241.
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.
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.
Zadeh, L. A. (1965). Fuzzy sets. Information Control, 8, 338–353.
Zimmermann, H. J. (1994). Fuzzy set theory and its applications. Boston: Kluwer Academic Publishers.
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.
Zhang, H. (2016). A goal programming model of obtaining the priority weights from an interval preference relation. Information Sciences, 354, 197–210.
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Goyal, R.K., Kaushal, S. Deriving crisp and consistent priorities for fuzzy AHP-based multicriteria systems using non-linear constrained optimization. Fuzzy Optim Decis Making 17, 195–209 (2018). https://doi.org/10.1007/s10700-017-9267-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10700-017-9267-y