Analysis of Consistency Indices of Pairwise Comparison Methods

Öz

İkili karşılaştırma yöntemi, karar verme sürecinde faktörlerin karşılıklı, kolay ve etkili bir şekilde değerlendirilmesinde önemli bir araçtır. Tutarlılık indeksi ve tutarlılık oran değerleri karar vericiler tarafından yapılan ikili karşılaştırmaların geçişlilik ve karşılıklılık özelliklerine göre yapılıp yapılmadığını belirler. Literatürdeki tutarlılık ölçüm yöntemleri, ikili karşılaştırmaların geçerliliğini değerlendirmek için farklı hesaplama yöntemleri kullanır. Literatürden seçilen 14 farklı tutarlılık yöntemi, ikili karşılaştırmaların geçerliliğini kabul etmek için farklı tutarlılık indeksi ve eşik değerleri tanımlar. Bu çalışma aynı ve farklı ikili karşılaştırma matrisi boyutlarında 14 farklı tutarlılık yönteminin tutarlılık indekslerinin davranışını ve ilişkilerini gözlemlemeyi amaçlar. Yöntemlerin tutarlılık indeksleri tüm boyutlarda karşılaştırılır ve yöntemlerin farklı boyutlardaki rassal indeksleri hesaplanır. Saaty'nin tutarlılık oranı eşik değerine (≤0.1) göre 8 farklı boyutta 14 farklı yöntemin tutarlılık indeksleri için eşik değerler tanımlanır. Böylece karar vericilerin farklı yöntemlerde ve farklı boyutlarda ikili karşılaştırmaların tutarlılığını daha kolay belirlemelerine yardımcı olunur.

Anahtar Kelimeler

• pairwise comparison matrix
• consistency index
• random index
• consistency ratio
• consistency threshold value

Kaynakça

1. [1] Kahraman C., Cebeci U., Ulukan Z., (2003) Multi‐criteria supplier selection using fuzzy AHP, Logistics Information Management, 16(6) 382-394.
2. [2] Saaty T. L., (2008) Decision making with the analytic hierarchy process, International Journal of Services Sciences, (1) 83–98.
3. [3] Kahraman C., Onar S.C., Oztaysi B., (2015) Fuzzy multicriteria decision-making: a literature review, International Journal of Computational Intelligence Systems, (8) 637–66.
4. [4] Brunelli M., (2018) A survey of inconsistency indices for pairwise comparisons, International Journal of General Systems, (47) 751–71.
5. [5] Brunelli M., Fedrizzi M., (2015) Boundary properties of the inconsistency of pairwise comparisons in group decisions, European Journal of Operational Research, (240) 765–73.
6. [6] Seker S., Kahraman C., (2021) Socio-economic evaluation model for sustainable solar PV panels using a novel integrated MCDM methodology: A case in Turkey, Socio-Economic Planning Sciences, (77) 100998. [7] Brunelli M., (2014) Introduction to the analytic hierarchy process, Springer.
7. [8] Csató L., (2018) Characterization of an inconsistency ranking for pairwise comparison matrices, Annals of Operations Research, (261) 155–65.
8. [9] Cavallo B., D’Apuzzo L., (2009) A general unified framework for pairwise comparison matrices in multicriterial methods, International Journal of Intelligent Systems, (24) 377–98.

9. [10] Stein W. E., Mizzi P.J., (2007) The harmonic consistency index for the analytic hierarchy process, European journal of Operational Research. (177) 488–97.
10. [11] Saaty T.L., (1977) A scaling method for priorities in hierarchical structures, Journal of Mathematical Psychology, (15) 234–81.
11. [12] Saaty R.W., (1987) The analytic hierarchy process—what it is and how it is used, Mathematical Modelling, (9) 161–76.
12. [13] Crawford G.B., (1987) The geometric mean procedure for estimating the scale of a judgement matrix, Mathematical Modelling, (9) 327–34.
13. [14] Koczkodaj W.W., (1993) A new definition of consistency of pairwise comparisons, Mathematical and Computer Modelling, (18) 79–84.
14. [15] Harker P.T., (1987) Derivatives of the Perron root of a positive reciprocal matrix: with application to the analytic hierarchy process, Applied Mathematics and Computation, (22) 217–32.
15. [16] Golden B.L., Wang Q., (1989) An alternate measure of consistency The analytic hierarchy process, Springer, 68–81.
16. [17] Shiraishi S., Obata T., Daigo M., (1998) Properties of a positive reciprocal matrix and their application to AHP, Journal of the Operations Research Society of Japan, (41) 404–14.
17. [18] Wedley W.C., (1993) Consistency prediction for incomplete AHP matrices, Mathematical and Computer Modelling, (17) 151–61.
18. [19] Takeda E., (1993) A note on consistent adjustments of pairwise comparison judgments, Mathematical and Computer Modelling, (17) 29–35.
19. [20] Salo A.A., Hämäläinen R.P., (1995) Preference programming through approximate ratio comparisons, European Journal of Operational Research, (82) 458–75.
20. [21] Fedrizzi M., Ferrari F., (2018) A chi-square-based inconsistency index for pairwise comparison matrices, Journal of the Operational Research Society, (69) 1125–34.
21. [22] Kou G., Lin C., (2014) A cosine maximization method for the priority vector derivation in AHP, European Journal of Operational Research, (235) 225–32.
22. [23] Gass S.I., Rapcsák T., (2004) Singular value decomposition in AHP, European Journal of Operational Research, (154) 573–84.
23. [24] Kułakowski K., (2015) Notes on order preservation and consistency in AHP, European Journal of Operational Research, (245) 333–7.
24. [25] Barzilai J., (1998) Consistency measures for pairwise comparison matrices, Journal of Multi‐Criteria Decision Analysis, (7) 123–32.
25. [26] Ozdemir M.S., (2005) Validity and inconsistency in the analytic hierarchy process, Applied Mathematics and Computation, (161) 707–20.
26. [27] Aguarón J., Escobar M.T., Moreno-Jiménez J.M., (2021) Reducing inconsistency measured by the geometric consistency index in the analytic hierarchy process, European Journal of Operational Research, (288) 576–83.
27. [28] Alonso J.A., Lamata M.T., (2006) Consistency in the analytic hierarchy process: a new approach, International Journal of Uncertainty, Fuzziness and Knowledge-based Systems, (14) 445–59. [29] Apostolou B., Hassell J.M., (2002) Note on consistency ratio: a reply, Mathematical and Computer Modelling, (35) 1081–1083.
28. [30] Lane E.F., Verdini W.A., (1989) A consistency test for AHP decision makers, Decision Sciences, (20) 575–90.
29. [31] Şahin B., Yazır D., (2019) An analysis for the effects of different approaches used to determine expertise coefficients on improved fuzzy analytical hierarchy process method, Journal of the Faculty of Engineering and Architecture of Gazi University, (34) 89–102.
30. [32] Aguarón J., Escobar M.T., Moreno-Jiménez J.M., Turón A., (2020) The Triads Geometric Consistency Index in AHP-Pairwise Comparison Matrices, Mathematics, (8) 926.
31. [33] Liu Y., Eckert C.M., Earl C., (2020) A review of fuzzy AHP methods for decision-making with subjective judgements, Expert Systems with Applications, (161) 113738.
32. [34] Bozóki S., Fülöp J., Poesz A., (2011) On pairwise comparison matrices that can be made consistent by the modification of a few elements, Central European Journal of Operations Research, (19) 157–75.
33. [35] Zhang J., Kou G., Peng Y., Zhang Y., (2021) Estimating priorities from relative deviations in pairwise comparison matrices, Information Sciences, (552) 310–27.
34. [36] Dijkstra T.K., (2013) On the extraction of weights from pairwise comparison matrices, Central European Journal of Operations Research, (21) 103–23.
35. [37] Li K.W., Wang Z.J., Tong X., (2016) Acceptability analysis and priority weight elicitation for interval multiplicative comparison matrices, European Journal of Operational Research, (250) 628–38.
36. [38] Crawford G., Williams C., (1985) A note on the analysis of subjective judgment matrices, Journal of Mathematical Psychology, (29) 387–405. [39] Ramík J., Korviny P., (2010) Inconsistency of pair-w ise comparison matrix with fuzzy elements based on geometric mean, Fuzzy Sets and Systems, (161) 1604–1613.
37. [40] Basile L, D’Apuzzo L., (2006) Transitive matrices, strict preference order and ordinal evaluation operators, Soft Computing, (10) 933.
38. [41] Ji P., Jiang R., (2003) Scale transitivity in the AHP, Journal of the Operational Research Society, (54) 896–905.
39. [42] Franek J., Kresta A., (2014) Judgment scales and consistency measure in AHP, Procedia Economics and Finance, (12) 164–73.
40. [43] Kwiesielewicz M., Van Uden E., (2004) Inconsistent and contradictory judgements in pairwise comparison method in the AHP, Computers and Operations Research, (31) 713–9.
41. [44] Rao T.V. M., Wan Y., (1994) On the mean random inconsistency index of analytic hierarchy process (AHP), Computers and Industrial Engineering, (27) 401–404.
42. [45] Saaty T.L., (1994) How to make a decision: the analytic hierarchy process, Interfaces, (24) 19–43.
43. [46] Aguarón J., Moreno-Jiménez J.M., (2003) The geometric consistency index: Approximated thresholds, European Journal of Operational Research, (147) 137–45.
44. [47] Amenta P., Lucadamo A., Marcarelli G., (2020) On the transitivity and consistency approximated thresholds of some consistency indices for pairwise comparison matrices, Information Sciences, (507) 274–87.
45. [48] Duszak Z., Koczkodaj W.W., (1994) Generalization of a new definition of consistency for pairwise comparisons, Information Processing Letters, (52) 273–6.
46. [49] Grzybowski A.Z., (2016) New results on inconsistency indices and their relationship with the quality of priority vector estimation, Expert Systems with Applications, (43) 197–212.
47. [50] Saaty T., (1980) The analytic hierarchy process (AHP) for decision making Kobe, Japan 1–69.
48. [51] Noble E.E., Sanchez P.P., (1993) A note on the information content of a consistent pairwise comparison judgment matrix of an AHP decision maker, Theory and Decision, (34) 99–108.
49. [52] Forman E.H., (1990) Random indices for incomplete pairwise comparison matrices, European Journal of Operational Research, (48) 153–5.