NS  Vol.4 No.5 , May 2012
On consistency and ranking of alternatives in uncertain AHP
Author(s) Liang Lin, Chao Wang
ABSTRACT
This paper introduces uncertainty theory to deal with non-deterministic factors in ranking alternatives. The uncertain variable method (UVM) and the definition of consistency for uncertainty comparison matrices are proposed. A simple yet pragmatic approach for testing whether or not an uncertainty comparison matrix is consistent is put forward. In cases where an uncertainty comparison matrix is inconsistent, an algorithm is used to generate consistent matrix. And then the consistent uncertainty comparison matrix can derive the uncertainty weights. The final ranking is given by uncertainty weighs if they are acceptable; otherwise we rely on the ranks of expected values of uncertainty weights instead. Three numerical examples including a hierarchical (AHP) decision problem are examined to illustrate the validity and practicality of the proposed methods.

Cite this paper
Lin, L. and Wang, C. (2012) On consistency and ranking of alternatives in uncertain AHP. Natural Science, 4, 340-348. doi: 10.4236/ns.2012.45047.
References
[1]   Saaty, T.L. (1977) A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology, 15, 234-281. doi:10.1016/0022-2496(77)90033-5

[2]   Van Laarhoven, P.J.M. and Pedrycz, W. (1983) A fuzzy extension of Saaty’s priority theory. Fuzzy Sets and Systems, 11, 229-241.

[3]   Buckley, J.J. (1985) Fuzzy hierarchical analysis. Fuzzy Sets and Systems, 17, 233-247. doi:10.1016/0165-0114(85)90090-9

[4]   Boender, C.G.E., De Graan, J.G. and Lootsma, F.A. (1989) Multicretia decision analysis with fuzzy pairwise comparisons. Fuzzy Sets and Systems, 29, 133-143. doi:10.1016/0165-0114(89)90187-5

[5]   Kwiesielewicz, M. (1998) A note on the fuzzy extension of Saaty’s priority theory. Fuzzy Sets and Systems, 95, 161-172. doi:10.1016/S0165-0114(96)00329-6

[6]   Leung, L.C. and Cao, D. (2000) On consistency and ranking of alternatives in fuzzy AHP. European Journal of Operational Research, 124, 102-113. doi:10.1016/S0377-2217(99)00118-6

[7]   Xu, Z.S. and Da, Q.L. (2005) A least deviation method to obtain a priority vector of a fuzzy preference relation. European Journal of Operational Research, 164, 206- 216. doi:10.1016/j.ejor.2003.11.013

[8]   Saaty, T.L. and Vargas, L.G. (1987) Uncertainty and rank order in the analytic hierarchy process. European Journal of Operational Research, 32, 107-117. doi:10.1016/0377-2217(87)90275-X

[9]   Arbel, A. (1989) Approximate articulation of preference and priority derivation. European Journal of Operational Research, 43, 317-326. doi:10.1016/0377-2217(89)90231-2

[10]   Kress, M. (1991) Approximate articulation of preference and priority derivation—A comment. European Journal of Operational Research, 52, 382-383. doi:10.1016/0377-2217(91)90174-T

[11]   Islam, R., Biswal, M.P. and Alam, S.S. (1997) Preference programming and inconsistent interval judgments. European Journal of Operational Research, 97, 53-62. doi:10.1016/S0377-2217(95)00377-0

[12]   Haines, L.M. (1998) A statistical approach to the analytic hierarchy process with interval judgments. (I). Distributions on feasible regions. European Journal of Operational Research, 110, 112-125. doi:10.1016/S0377-2217(97)00245-2

[13]   Wang, Y.M., Yang, J.B. and Xu, D.L. (2005) A two-stage logarithmic goal programming method for generating weights from interval comparison matrices. Fuzzy Sets and Systems, 152, 475-498. doi:10.1016/j.fss.2004.10.020

[14]   Juan, A. and José María, M.-J. (2003) The geometric consistency index: Approximated thresholds. European Journal of Operational Research, 147, 137-145. doi:10.1016/S0377-2217(02)00255-2

[15]   Wang, Y.M., Yang, J.B. and Xu, D.L. (2005) Interval weight generation approaches based on consistency test and interval comparison matrices. Applied Mathematics and Computation, 167, 252-273. doi:10.1016/j.amc.2004.06.080

[16]   Liu, B.D. (2007) Uncertainty theory. 2nd Edition, Springer-Verlag, Berlin.

[17]   Liu, B.D. (2010) Uncertainty theory: A branch of mathematics for modeling human uncertainty. Springer-Verlag, Berlin.

[18]   Liu, B.D. (2010) Uncertainty theory. 4th Edition, Springer-Verlag, Berlin. doi:10.1007/978-3-642-13959-8

[19]   Wang, C., Lin, L. and Liu, J.J. (2012) Uncertainty weight generation approach based on uncertainty comparison matrices. Applied Mathematics, 3.

[20]   Crawford, G. and Williams, C. (1985) A note on the analysis of subjective judgment matrices. European Journal of Operational Research, 29, 387-405.

[21]   Liu, Y.H. and Ha, M.H. (2010) Expected value of function of uncertain variables. Journal of Uncertain Systems, 4, 181-186.

[22]   Arbel, A. and Vargas, L.G. (1993) Preference simulation and preference programming: Robustness issues in priority deviation. European Journal of Operational Research, 69, 200-209. doi:10.1016/0377-2217(93)90164-I

[23]   Kwiesielewicz, M. and Van Uden, E. (2004) Inconsistent and contradictory judgments in pairwise comparison method in the AHP. Computers and Operations Research, 31, 713-719. doi:10.1016/S0305-0548(03)00022-4

 
 
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