Consistency issues of interval pairwise comparison matrices

Soft Computing

Volumn 19, Issue 8, 2015, Pages 2321-2335

  Dong, Yucheng ^a   Chen, Xia ^a   Li, Cong Cong ^a   Hong, Wei Chiang ^b   Xu, Yinfeng ^a  

^a Sichuan University   (China)

^b ORIENTAL INSTITUTE OF TECHNOLOGY   (Taiwan)

Author keywords

Consistency; Interval pairwise comparison matrix; Linear programming; Multiple attribute decision making

Indexed keywords

DECISION MAKING; LINEAR PROGRAMMING;

CONSISTENCY; CONSISTENCY INDEX; LINEAR PROGRAMMING MODELS; MANHATTAN DISTANCE; MULTIPLE ATTRIBUTE DECISION MAKING; PAIR-WISE COMPARISON; PAIRWISE COMPARISON MATRICES; STRONG CONSISTENCY;

MATRIX ALGEBRA;

EID: 84937526631     PISSN: 14327643     EISSN: 14337479     Source Type: Journal    
DOI: 10.1007/s00500-014-1426-2     Document Type: Article

References (77)

1. Aguarón J, Moreno-Jiménez JM (2003) The geometric consistency index: approximated thresholds. Eur J Oper Res 147:137–145
2. Alonso JA, Lamata MT (2006) Consistency in the analytic hierarchy process: a new approach. Int J Uncertain Fuzziness Knowl Based Syst 14(4):445–459
3. Alonso S, Chiclana F, Herrera F, Herrera-Viedma E, Alcal$$\acute{a}$$a´-Fdez J, Porcel C (2008) A consistency-based procedure to estimate missing pairwise preference values. Int J Intell Syst 23:155–175
4. Alonso S, Cabrerizo FJ, Chiclana F, Herrera F, Herrera-Viedma E (2009) Group decision-making with incomplete fuzzy linguistic preference relations. Int J Intell Syst 24:201–222
5. Alonso S, Pérez IJ, Cabrerizo FJ, Herrera-Viedma E (2013) A linguistic consensus model for Web 2.0 communities. Appl Soft Comput 13:149–157
6. Apostolou B, Hassell JM (2002) Note on consistency ratio: a reply. Math Comput Model 35:1081–1083
7. Arbel A (1989) Approximate articulation of preference and priority derivation. Eur J Oper Res 43:317–326
8. Arbel A (1991) A linear programming approach for processing approximate articulation of preference. In: Korhonen P, Lewandowski A, Wallenius J (eds) Multiple criteria decision support. Lecture Notes in Economics and Mathematical Systems, vol 356. Springer, Berlin, pp 79–86
9. Arbel A, Vargas LG (1993) Preference simulation and preference programming: robustness issues in priority deviation. Eur J Oper Res 69:200–209
10. Barrenechea E, Fernandez J, Pagola M, Chiclana F, Bustince H (2014) Construction of interval-valued fuzzy preference relations from ignorance functions and fuzzy preference relations. Knowl Based Syst 58:33–44
11. Bozóki S, F$$\ddot{u}$$u¨l$$\ddot{o}$$o¨p J, Koczkodaj WW (2011) An LP-based inconsistency monitoring of pairwise comparison matrices. Math Comput Model 54:789–793
12. Brunelli M, Canal L, Fedrizzi M (2013) Inconsistency indices for pairwise comparison matrices: a numerical study. Ann Oper Res 211(1):493–509
13. Cabrerizo FJ, Herrera-Viedma E, Pedrycz W (2013) A method based on PSO and granular computing of linguistic information to solve group decision making problems defined in heterogeneous contexts. Eur J Oper Res 230:624–633
14. Cabrerizo FJ, Moreno JM, Pérez IJ, Herrera-Viedma E (2010a) Analyzing consensus approaches in fuzzy group decision making: advantages and drawbacks. Soft Comput 14:451–463
15. Cabrerizo FJ, Pérez IJ, Herrera-Viedma E (2010b) Managing the consensus in group decision making in an unbalanced fuzzy linguistic context with incomplete information. Knowl Based Syst 23:169–181
16. Chiclana F, Herrera F, Herrera-Viedma E (1998) Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations. Fuzzy Sets Syst 97:33–48
17. Chiclana F, Herrera F, Herrera-Viedma E (2001) Integrating multiplicative preference relations in a multipurpose decision-making model based on fuzzy preference relations. Fuzzy Sets Syst 122:277–291
18. Chiclana F, Herrera-Viedma E, Alonso S, Herrera F (2009) Cardinal consistency of reciprocal preference relations: a characterization of multiplicative transitivity. IEEE Trans Fuzzy Syst 17:14–23
19. Chiclana F, Tapia Garcia JM (2013) A statistical comparative study of different similarity measures of consensus in group decision making. Inf Sci 221:110–123
20. Cao D, Leung LC, Law JS (2008) Modifying inconsistent comparison matrix in analytic hierarchy process: a heuristic approach. Decis Support Syst 44:944–953
21. Chandran B, Golden B, Wasil E (2005) Linear programming models for estimating weights in the analytic hierarchy process. Comput Oper Res 32:2235–2254
22. Choo EU, Wedley WC (2004) A common framework for deriving preference values from pairwise comparison matrices. Comput Oper Res 31:893–908
23. Conde E, Pérez M (2010) A linear optimization problem to derive relative weights using an interval judgement matrix. Eur J Oper Res 201:537–544
24. Cox MAA (2007) Examining alternatives in the interval analytic hierarchy process using complete enumeration. Eur J Oper Res 180:957–962
25. Crawford G, Williams C (1985) A note on the analysis of subjective judgement matrices. J Math Psychol 29:387–405
26. Chu P, Liu JKH (2002) Note on consistency ratio. Math Comput Model 35:1077–1080
27. Dantzig GB (1963) Linear programming and extensions. Princeton University Press, New York
28. Dong YC, Fan Z, Yu S (2014) Consensus building in a local context for the AHP-GDM with the individual numerical scale and prioritization method. IEEE Trans Fuzzy Syst. doi:10.1109/TFUZZ.2014.2312974
29. Dong YC, Hong WC, Xu YF (2013) Measuring consistency of linguistic preference relations: a 2-tuple linguistic approach. Soft Comput 17(11):2117–2130
30. Dong YC, Hong WC, Xu YF, Yu S (2011) Selecting the individual numerical scale and prioritization method in the analytic hierarchy process: a 2-tuple fuzzy linguistic approach. IEEE Trans Fuzzy Syst 19:13–26
31. Dong YC, Hong WC, Xu YF, Yu S (2013) Numerical scales generated individually for analytic hierarchy process. Eur J Oper Res 229:654–662
32. Dong YC, Xu YF, Li HY (2008) On consistency measures of linguistic preference relations. Eur J Oper Res 189:430–444
33. Dong YC, Xu YF, Li HY, Dai M (2008) A comparative study of the numerical scales and the prioritization methods in AHP. Eur J Oper Res 186:229–242
34. Dong YC, Xu YF, Yu S (2009) Linguistic multiperson decision making based on the use of multiple preference relations. Fuzzy Sets Syst 160:603–623
35. Dong YC, Zhang GQ, Hong WC, Xu YF (2010) Consensus models for AHP group decision making under row geometric mean prioritization method. Decis Support Syst 49:281–289
36. Escobar MT, Aguarón J, Moreno-Jiménez JM (2004) A note on AHP group consistency for the row geometric mean priorization procedure. Eur J Oper Res 153:318–322
37. Fedrizzi M, Giove S (2007) Incomplete pairwise comparison and consistency optimization. Eur J Oper Res 183:303–313
38. Finan JS, Hurley WJ (1997) The analytic hierarchy process: does adjusting a pairwise comparison matrix to improve the consistency ratio help. Comput Oper Res 24:749–755
39. Golden BL, Wang Q (1989) An alternate measure of consistency. In: Golden BL, Wasil EA, Harker PT (eds) The analytic hierarchy process: applications and studies. Springer, Berlin
40. Haines LM (1998) A statistical approach to the analytic hierarchy process with interval judgments. (I) distributions on feasible regions. Eur J Oper Res 110:112–125
41. Herrera-Viedma E, Cabrerizo FJ, Kacprzyk J, Pedrycz W (2014) A review of soft consensus models in a fuzzy environment. Inf Fusion 17:4–13
42. Herrera-Viedma E, Chiclana F, Herrera F, Alonso S (2007) Group decision-making model with incomplete fuzzy preference relations based on additive consistency. IEEE Trans Syst Man Cybern Part B Cybern 31(2):227–234
43. Herrera-Viedma E, Herrera F, Chiclana F, Luque M (2004) Some issues on consistency of fuzzy preference relations. Eur J Oper Res 154:98–109
44. Islam R, Biswal MP, Alam SS (1997) Preference programming and inconsistent interval judgments. Eur J Oper Res 97:53–62
45. Koczkodaj WW (1993) A new definition of consistency of pairwise comparisons. Math Comput Model 18:79–84
46. Kress M (1991) Approximate articulation of preference and priority derivation: a comment. Eur J Oper Res 52:382–383
47. Lan JB, Lin J, Cao LJ (2009) An information mining method for deriving weights from an interval comparison matrix. Math Comput Model 50:393–400
48. Lane EF, Verdini WA (1989) A consistency test for AHP decision makers. Decis Sci 20:575–590
49. Liu F (2009) Acceptable consistency analysis of interval reciprocal comparison matrices. Fuzzy Sets Syst 160:2686–2700
50. Mikhailov L (2002) Fuzzy analytical approach to partnership selection in formation of virtual enterprises. Omega 30:393–401
51. Mikhailov L (2004) A fuzzy approach to deriving priorities from interval pairwise comparison judgments. Eur J Oper Res 159:687–704
52. Moreno-Jiménez JM, Aguarón J, Escobar MT (2008) The core of consistency in AHP-group decision making. Group Decis Negotiat 17:249–265
53. Peláez JI, Lamata MT (2003) A new measure of consistency for positive reciprocal matrices. Comput Math Appl 46:1839–1845
54. Pérez IJ, Cabrerizo FJ, Alonso S, Herrera-Viedma E (2014) A new consensus model for group decision making problems with non homogeneous experts. IEEE Trans Syst Man Cybern Syst 44:494–498
55. Pérez IJ, Cabrerizo FJ, Herrera-Viedma E (2010) A mobile decision support system for dynamic group decision making problems. IEEE Trans Syst Man Cybern Part A Syst Hum 40:1244–1256
56. Pérez IJ, Wikstr$$\ddot{o}$$o¨m R, Mezei J, Carlsson C, Herrera-Viedma E (2013) A new consensus model for group decision making using fuzzy ontology. Soft Comput 17:1617–1627
57. Podinovski VV (2007) Interval articulation of superiority and precise elicitation of priorities. Eur J Oper Res 180:406–417
58. Saaty TL (1977) A scaling method for priorities in hierarchical structures. J Math Psychol 15:234–281
59. Saaty TL (1980) The analytic hierarchy process. McGraw-Hill, New York
60. Saaty TL (2003) Decision-making with the AHP: why is the principal eigenvector necessary. Eur J Oper Res 145:85–91
61. Saaty TL, Vargas LG (1987) Uncertainty and rank order in the analytic hierarchy process. Eur J Oper Res 32:107–117
62. Srdjevic B (2005) Combining different prioritization methods in the analytic hierarchy process synthesis. Comput Oper Res 32:1897–1919
63. Srdjevic B, Srdjevic Z (2013) Synthesis of individual best local priority vectors in AHP-group decision making. Appl Soft Comput 13:2045–2056
64. Salo A, Hämäläinen RP (1992) Processing interval judgments in the analytic hierarchy process. In: Goicoechea A, Duckstein L, Zoints S (eds) Multiple criteria decision making. Proceedings of the ninth international conference held in Fairfax. Springer, New York, pp 359–372
65. Sugihara K, Ishii H, Tanaka H (2004) Interval priorities in AHP by interval regression analysis. Eur J Oper Res 158:745–754
66. Stein WE, Mizzi PJ (2007) The harmonic consistency index for the analytic hierarchy process. Eur J Oper Res 177(1):488–497
67. Tapia García JM, Del Moral MJ, Martínez MA, Herrera-Viedma E (2012) A consensus model for group decision-making problems with interval fuzzy preference relations. Int J Inf Technol Decis Mak 11(4):709–725
68. Wang YM, Yang JB, Xu DL (2005a) A two-stage logarithmic goal programming method for generating weights from interval comparison matrices. Fuzzy Sets Syst 152:475–498
69. Wang YM, Yang JB, Xu DL (2005b) Interval weight generation approaches based on consistency test and interval comparison matrices. Appl Math Comput 167:252–273
70. Wang YM (2006) On lexicographic goal programming method for generating weights from inconsistent interval comparison matrices. Appl Math Comput 173:985–991
71. Wang YM, Elhag TMS (2007) A goal programming method for obtaining interval weights from an interval comparison matrix. Eur J Oper Res 177:458–471
72. Wu J, Chiclana F (2014) A social network analysis trust-consensus based approach to group decision-making problems with interval-valued fuzzy reciprocal preference relations. Knowl Based Syst 59:97–107
73. Xu Y, Li KW, Wang H (2013) Consistency test and weight generation for additive interval fuzzy preference relations. Soft Comput. doi:10.1007/s00500-013-1156-x
74. Xu Y, Wang H (2014) A comment on “Incomplete fuzzy linguistic preference relations under uncertain environments”. Inf Fusion 20:2–5
75. Xu ZS (2000) On consistency of the weighted geometric mean complex judgement matrix in AHP. Eur J Oper Res 126:683–687
76. Xu ZS, Wei CP (1999) A consistency improving method in analytic hierarchy process. Eur J Oper Res 116:443–449
77. Zhang GQ, Dong YC, Xu YF (2014) Consistency and consensus measures for linguistic preference relations based on distribution assessments. Inf Fusion 17:46–55