Research on similarity values relevant for fuzzy matrix consistency check.

Lavic, Zedina ; Dukic, Nedzad ; Pasic, Mugdim 等

1. Introduction

Numerous different methods are developed to solve multicriteria decision making problems, mostly related to the strict numerical valuation of selected criteria--examples of papers using such methods written by the authors relate to multicriteria evaluation of supervisory boards[1], selection of buildings' insulation [2] or selection of best alternative for highway tunnel doors [3].

One of frequently used multicriteria decision making aid methods is Analytic Hierarchy Process (classical, but also fuzzy). Limited research is done so far regarding evaluation of consistency in forming the pairwise comparison matrices, with few papers available dealing with the issue of consistency of fuzzy pairwise comparison matrices (examples are [4, 5, 6, 7]). According to the work of Buckley [8], fuzzy pairwise comparison matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is consistent if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where i,j,k = 1,2, ..., n, and where "[cross product]" represents multiplication of fuzzy numbers, while "[approximately equal to]" means "fuzzy equal" [9].

The concept of fuzzy equality has the same meaning as the notion of similarity, since the similarity in essence is a generalization of the concept of equivalence [10]. Similarity is a relation which is an extension to the concept of functions belonging to the set of elements, where, abandoning the idea of exact equality, the elements of the domain are considered to have different degrees of similarity [11].

The application of fuzzy numbers in resolving problems requires the application of different similarity measures, calculating the degree of similarity between two fuzzy number; similarity measures appropriate for solving one type of problem often are not appropriate for solving the other problem or type of problems. Therefore, various similarity measures of fuzzy numbers exist [11].. The similarity measure of triangular fuzzy numbers proposed by Chen and Lin ([7]) is based on distances between the midpoints and the appropriate boundary points of the support interval: if A = ([a.sup.l], [a.sup.m], [a.sup.u]) and B = ([b.sup.l], [b.sup.m], [b.sup.u]) are triangular fuzzy numbers, similarity can be calculated with

[S.sub.MB] (A, B) = 1 - [absolute value of ([a.sup.l] - [b.sup.l])] + [absolute value of ([a.sup.m] - [b.sup.m])] + [absolute value of ([a.sup.u] - [b.sup.u])]/3 (1)

Hsieh and Chen have presented the utility value concept of triangular fuzzy numbers, and based on these values have also defined the similarity [9]:

[S.sub.UV](A,B) = [U.sub.UV] (A)x [U.sub.UV] (B)/max([([U.sub.UV](A)).sup.2], [([U.sub.UV](B)).sup.2]). (2)

The similarity measure is function fulfilling the following two conditions [9]:

* Reflexivity, i. e. = 1 and

* Symmetry, i.e. S(A, B) = S(B, A).

Similarity measure based on the distance is used in this paper (as defined in [9]):

S(A,B)= 1/1 + d(A, B), (3)

where d (A, B) is a measure of the distance defined with:

d(A, B) = [absolute value of ([a.sup.l] - [b.sup.l])] + [absolute value of ([a.sup.m] - [b.sup.m])] + [absolute value of ([a.sup.u] - [b.sup.u])], (4)

so the similarity measure used is

S(A,B) = 1/1 + [absolute value of ([a.sup.l.] - [b.sup.l])] + [absolute value of ([a.sup.m] - [b.sup.m])] + [absolute value of ([a.sup.u] - [b.sup.u])]]. (5)

This similarity measure fulfils the conditions of reflexivity and symmetry.

2. Research

Pairwise comparison matrices of the fuzzy Analytic Hierarchy Process (fAHP) based decision making models are positive and reciprocal. The smallest matrix in fAHP is 3x3 pairwise comparison matrix, square, positive and reciprocal:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

The elements [a.sub.ij] (i,j=1,2,3) are obtained using appropriate fuzzification of scale given by Saaty [13]. Original Saaty's scale and fuzzified Saaty's scale are presented with Tables 1 & 2 respectively.

In order to explore relationship between the similarity measures value and consistency of fuzzy matrices, Java application is created and used. The application creates [17.sup.n(n-1)/] 2 matrices, where n is dimension of the particular matrix explored. For each created matrix application calculates similarity of fuzzy numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for every i,j,k = 1,2, ..., n. After that, for each matrix the application finds minimum, calculates the average similarity and forms MS Excel file where all the fuzzy matrices with the corresponding minimum and average similarities are given. The application also calculates and presents CR value (consistency ratio) [13] for classical matrices derived from the fuzzy matrices. Matrices are formed using classical matrices' elements [a.sub.ij] representing the midpoint of the fuzzy numbers interval [[??].sub.ij] = ([a.sup.l.sub.ij], [a.sup.m.sub.ij], [a.sup.u.sub.ij]). Thus, the elements of the classical matrix are [a.sub.ij] = [a.sup.m.sub.ij] for all i,j = 1,2,...,n. Similarity values (minimal and average) and CR values for fuzzy and classical matrices 3x3 respectively are explored. It should be noted that these matrices, as the corresponding values of displayed indicators, would be different if fuzzy scales would be different than those presented in the Table 2.

3. Results and Discussion

For the case of selected fuzzy scale, all the values of minimal similarity of matrices 3x3 are rounded to 4 decimal places and they belong to the interval [0.0041,1]. There are total of 162 minimal similarity values, with the smallest popup value of 0.0041, for the selected fuzzy scale. Analysis of the output Excel file, containing all matrices so as corresponding minimal and average similarities and CR values of correspondent classic matrices, is performed. It was found that for the fuzzy matrices whose minimal similarities belong to the interval [0.0041,0.0294] all correspondent classical matrices have CR values greater than 10%, what means that they are not consistent. The total number of these matrices is 3576 and indicates the level of the problem of inconsistency check with matrices of dimension three. The total number of different minimal similarities within the range [0.0041,0.0294] is 122. For these matrices, the lowest average similarity is 0.6577, the largest average similarity is 0.8297, and the smallest CR is 10.1587%. The focus of further analysis of the minimal similarities values is at the values belonging to the interval [0.0303,1]. The values for these similarities (a total of 40 values) are presented with Table 3.

Analysis of the entire range of minimal values of similarities [0.0041. 1] has shown that:

* For the minimal similarities lower than 0.0303 there is no consistent classical matrix.

* For some minimal similarities within the range (0.0303, 0.1429) there are corresponding average similarities for which both consistent and inconsistent classical matrices exist.

* For some minimal similarities within the range (0.0303, 0.1] there are corresponding average similarities for which classical matrices are always consistent (Table 4).

* When the minimal similarities are higher than 0.1 (except if the minimal similarity is 0.1429 and corresponding average similarity is 0.7873, when CR of classical matrix is 12.7714%). classical matrices are consistent.,

* For minimal similarities higher than 0.1429 all the classic matrices are consistent.

4. Conclusion

Both for classical and for fuzzy Analytic Hierarchy Process, a problem of inconsistency of decision maker appears when forming the pairwise comparison matrices and this paper focuses on resolving it. Appropriate fuzzy scale, obtained by fuzzification of Saaty's scale, is used in this paper. Research on similarities of fuzzy numbers, relevant in terms of consistency check, resulted with the conclusion that the inconsistency issues appear even with matrices of smaller dimensions. For the case of selected fuzzy scale, all the values of minimal similarity of matrices 3x3, rounded to 4 decimal places, belong to the interval [0.0041, 1]. It is found that for the fuzzy matrices whose minimal similarities are within the interval [0.0041, 0.0294] all correspondent classical matrices have CR values greater than 10%, what means they are not consistent. Total number of these matrices is 3576 (total number of fuzzy matrices is 4913). For some minimal similarities within the range (0.0303, 0.1] there are corresponding average similarities for which classical matrices are always consistent. For minimal similarities higher than 0.1 (except if the minimal similarity is 0.1429 and corresponding average similarity is 0.7873, when CR of classical matrix is 12.7714%). classical matrices are consistent, and for minimal similarities higher than 0.1429 all the related classic matrices are consistent.

Since the matrix 3x3 is the basic element in fuzzy (and classical) Analytic Hierarchy Process, consistency check of fuzzy matrices of dimensions larger than 3 is feasible using decomposition of larger matrices into the matrices 3x3, what will be the subject of the further research.

DOI: 10.2507/27th.daaam.proceedings.093

5. References

[1] Karic E., Vucijak B., Pasic M., Bijelonja I. (2011): "Multi-Criteria Evaluation of Supervisory Board Effectiveness", Proceedings of the 22nd DAAAM International Symposium on Intelligent Manufacturing and Automation, Volume 22, Vienna, 23-26 November 2011, 251-252

[2] Civic A. and Vucijak B. (2013): "Multi-Criteria Optimization of Insulation Options for Warmth of Buildings to Increase Energy Efficiency", 24th DAAAM International Symposium on Intelligent Manufacturing and Automation, Zadar, 23-26 October 2013, Procedia Engineering 69 (2014) 911-920

[3] Vucijak B., Pasic. M., Zorlak A. (2015): "Use of Multi-Criteria Decision Aid Methods for Selection of the Best Alternative for the Highway Tunnel Doors ", 25th DAAAM International Symposium on Intelligent Manufacturing and Automation, Zadar, 23-26 October 2013, Procedia Engineering 100 (2015) 656-665

[4] Ramik, J., Vlach, M. (2013). Measuring consistency and inconsistency of pair comparison systems, Kybernetika 49(3): 465-486 (2013)

[5] Ramik, J. (2014). Incomplete fuzzy preference matrix and its application to ranking of alternatives, Intelligent Systems 29, 8 (2014), 787-806

[6] Ramik, 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 (2010)

[7] Ramik, J. (2009). Consistency of Pair-wise Comparison Matrix with Fuzzy Elements. IFSA/EUSFLAT Conf. 2009: 98-101

[8] Buckley, J. J. (1985), Fuzzy hierarchical analysis. Fuzzy Sets and Systems, 17(2), 233-247.

[9] Javanbarg, M. B., Scawthorn, C., Kiyono, J., Shahbodaghkhan, B. (2012), Fuzzy AHP-based multicriteria decision making systems using particle swarm optimization, Expert Systems vith Applications, 39 (2012), 960-966

[10] Zadeh, L. A. (1971). Similarity Relations and Fuzzy Orderings. Information Sciences 3, p. 177-200.

[11] Dukic, N. (2006), The Semantic Distance, Fuzzy Dependency and Fuzzy Formulas, Sarajevo Journal of Mathematics, Vol. 2 (15), 137-146.

[12] Zhang, X., Ma, W. and Chen, L. (2014), New Similarity of Triangular Fuzzy Number and Its Application, Hindawi Publishing Corporation, The Scientific World Journal, Volume 2014

[13] Saaty, T. L. (1980), Analytic hierarchy process, New York, McGraw-Hill. Table 1. Scale given by Saaty Definition Importance Reciprocals intensity Equal importance 1 1 Moderate importance 3 1/3 Strong importance 5 1/5 Very strong importance 7 1/7 Extreme importance 9 1/9 Intermediate values 2, 4, 6 and 8 1/2, 1/4, 1/6 and 1/8 Table 2. Fuzzified Saaty's scale Definition Importance intensity Equal importance (1, 1, 1) Moderate importance (2, 3, 4) Strong importance (4, 5, 6) Very strong importance (6, 7, 8) Extreme importance (9, 9, 9) Intermediate values (1, 2, 3), (3, 4, 5), (5, 6, 7) and (7, 8, 9) Definition Reciprocals Equal importance (1, 1, 1) Moderate importance (1/4, 1/3, 1/2) Strong importance (1/6, 1/5, 1/4) Very strong importance (1/8, 1/7, 1/6) Extreme importance (1/9, 1/9, 1/9) Intermediate values (1/3, 1/2, 1), (1/5, 1/4, 1/3), (1/7, 1/6, 1/5) and (1/9, 1/8, 1/7) Table 3. Values of the minimal similarities from the interval [0.0303 to 1] Ordinal Minimal Number Similarity 1. 0.0303 2. 0.0321 3. 0.0323 4. 0.0333 5. 0.0346 6. 0.0355 7. 0.0357 8. 0.0370 9. 0.0397 10. 0.0400 11. 0.0417 12. 0.0435 13. 0.0455 14. 0.0476 15. 0.0500 16. 0.0522 17. 0.0526 18. 0.0556 19. 0.0588 20. 0.0625 21. 0.0667 22. 0.0714 23. 0.0759 24. 0.0769 25. 0.0833 26. 0.0909 27. 0.1000 28. 0.1111 29. 0.125 30. 0.1429 31. 0.2069 32. 0.2500 33. 0.2727 34. 0.4000 35. 0.4839 36. 0.5455 37. 0.5932 38. 0.6316 39. 0.6632 40. 1 Table 4. Minimal and average similarities in the range (0.0303 to 0.1]. CR <10% Minimal Similarity Average Similarity 0.0333 0.7919 0.0357 0.8154 0.796 0.037 0.7545 0.7627 0.7641 0.0417 0.7583 0.7652 0.7962 0.7969 0.0435 0.7673 0.7985 0.0476 0.7494 0.7513 0.05 0.7621 0.7984 0.0526 0.7731 0.0556 0.7245 0.7425 0.7637 0.8135 0.0588 0.7559 0.7766 0.0625 0.7699 0.8017 0.0667 0.7312 0.7459 0.7762 0.0714 0.7316 0.7485 0.7647 0.7658 0.7999 0.0769 0.7677 0.7888 0.7966 0.8338 0.0833 0.7159 0.7767 0.7992 0.0909 0.7105 0.7403 0.7492 0.7872 0.1 0.7126 0.7586 0.7734 0.7808 0.8015 0.8166 0.8509