Paper:

# Improving Interval Weight Estimations in Interval AHP by Relaxations

## Masahiro Inuiguchi and Shigeaki Innan

Osaka University

1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan

From the viewpoint that the vagueness of a decision maker’s evaluation causes inconsistencies in a pairwise comparison matrix, interval weights have been estimated using the interval AHP. However, the estimated interval weights are often insufficient to express the vagueness of the decision maker’s evaluation. We propose three modified estimation methods for interval weights. The first is based on a relaxation of the optimality of estimated interval weights in the conventional method. The second employs a modified objective function and the third is based on a relaxation of the optimality with respect to the modified objective function. Two of the proposed methods include parameters with degrees of relaxation. Through numerical experiments with 100,000 pairwise comparison matrices generated from 100 true interval weight vectors, we demonstrate the advantages of the proposed methods over the conventional method, and determine the best method and the suitable degree of relaxation.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.21, No.7, pp. 1135-1143, 2017.

- [1] T. L. Saaty, “The Analytic Hierarchy Process,” McGraw-Hill, New York, 1980.
- [2] T. L. Saaty and C. G. Vargas, “Comparison of Eigenvalue, Logarithmic Least Squares and Least Squares Methods in Estimating Ratios,” Mathematical Modelling, Vol.5, pp. 309-324, 1984.
- [3] A. Arbel, “Approximate Articulation of Preference and Priority Derivation,” European J. of Operational Research, Vol.43, pp. 317-326, 1989.
- [4] A. Arbel and L. G. Vargas, “Preference Simulation and Preference Programming: Robustness Issues in Priority Derivation,” European J. of Operational Research, Vol.69, pp. 200-209, 1993.
- [5] A. Arbel and L. G. Vargas, “Interval Judgements and Euclidean Centers,” Mathematical and Computer Modelling, Vol.46, pp. 976-984, 2007.
- [6] M. Kress, “Approximate Articulation of Preference and Priority Deviation: A Comment,” European J. of Operational Research, Vol.52, pp. 382-383, 1991.
- [7] R. Islam, M. P. Biswal, and S. S. Alam, “Preference Programming and Inconsistent Interval Judgements,” European J. of Operational Research, Vol.97, pp. 53-62, 1997.
- [8] J. R. Yu, Y.-W. Hsiao, and H.-J. Sheu, “A Multiplicative Approach to Derive Weights in the Interval Analytic Hierarchy Process,” Int. J. of Fuzzy Systems, Vol.13, No.3, pp. 225-231, 2011.
- [9] K. Sugihara, H. Ishii and H. Tanaka, “Interval Priorities in AHP by Interval Regression Analysis,” European J. of Operational Research, Vol.158, No.3, pp. 745-754, 2004.
- [10] Y.-M. Wang and T. M. S. Elhag, “A Goal Programming Method for Obtaining Interval Weights from an Interval Comparison Matrix,” European J. of Operational Research, Vol.177, pp. 458-471, 2007.
- [11] P. J. M. van Laarhoven, and W. Pedrycz, “A Fuzzy Extension of Saaty’s Priority Theory,” Fuzzy Sets and Systems, Vol.11, pp. 199-227, 1983.
- [12] Y.-M. Wang, T. M. S. Elhag, Z. Hua, “A Modified Fuzzy Logarithmic Least Squares Method for Fuzzy Analytic Hierarchy Process,” Fuzzy Sets and Systems, Vol.157, pp. 3055-3071, 2006.
- [13] J. Ramík and R. Perzina, “A Method for Solving Fuzzy Multicriteria Decision Problems with Dependent Criteria,” Fuzzy Optimization and Decision Making, Vol.9, No.2, pp. 123-141, 2010.
- [14] J. J. Buckley, “Fuzzy Hierarchical Analysis,” Fuzzy Sets and Systems, Vol.17, pp. 233-247, 1985.
- [15] J. J. Buckley, T. Feuring, and Y. Hayashi, “Fuzzy Hierarchical Analysis Revisited,” European J. of Operational Research, Vol.129, pp. 48-64, 2001.
- [16] R. Csutora and J. J. Buckley, “Fuzzy Hierarchical Analysis: The Lambda-max Method,” Fuzzy Sets and Systems, Vol.120, pp. 181-195, 2001.
- [17] L. Mikhailov, “Fuzzy Analytical Approach to Partnership Selection in Formation of Virtual Enterprises,” Omega, Vol.30, pp. 393-401, 2002.
- [18] L. Mikhailov and P. Tsvetinov, “Evaluation of Services Using a Fuzzy Analytic Hierarchy Process,” Applied Soft Computing, Vol.5, pp. 23-33, 2004.
- [19] L. Mikhailov, “A Fuzzy Approach to Deriving Priorities from Interval Pairwise Comparison Judgments,” European J. of Operational Research, Vol.159, pp. 687-704, 2004.
- [20] E. Dopazo, S. C. K. Lui, and J. Guisse, “A Parametric Model for Determining Consensus Priority Vectors from Fuzzy Comparison Matrices,” Fuzzy Sets and Systems, Vol.246, pp. 49-61, 2014.
- [21] T. L. Saaty and C. G. Vargas, “Uncertainty and Rank Order in the Analytic Hierarchy Process,” European J. of Operational Research, Vol.32, pp. 107-117, 1987.
- [22] J. M. Moreno-Jimenez and L. G. Vargas, “A Probabilistic Study of Preference Structures in the Analytic Hierarchy Process with Interval Judgments,” Mathematical and Computer Modelling, Vol.17, No.4/5, pp. 73-81, 1993.
- [23] K. Sugihara and H. Tanaka, “Interval Evaluations in the Analytic Hierarchy Process by Possibilistic Analysis,” Computational Intelligence, Vol.17, pp. 567-579, 2001.
- [24] T. Entani and M. Inuiguchi, “Pairwise Comparison Based Interval Analysis for Group Decision Aiding with Multiple Criteria,” Fuzzy Sets and Systems, Vol.274, pp. 79-96, 2015.
- [25] H. Tanaka, K. Sugihara, and Y. Maeda, “Non-additive Measures by Interval Probability Functions,” Information Sciences, Vol.164, pp. 209-227, 2004.
- [26] T. Entani and K. Sugihara, “Uncertainty index based interval assignment by Interval AHP,” European J. of Operational Research, Vol.219, pp. 379-385, 2012.

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