A GP-AHP method for solving group decision-making fuzzy AHP ...

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1998 C.-S. Yu / Computers & Operations Research 29 (2002) 1969–2001 Table 6 The comparative results of solving Examples 4 and 5 Example 4 Example 5 Laarhoven et al. method [8] Buckley method [3] Boender et al. method [2] Chang method [4] Ruoning et al. [21] The proposed method The nal scores are fuzzy values (0:227; 0:398; 0:705), (0:168; 0:313; 0:579), (0:188; 0:289; 0:504) for Candidates A, B, and C, respectively The nal scores are fuzzy values (0:233; 0:401; 0:715), (0:174; 0:332; 0:498), (0:187; 0:293; 0:476) for Candidates A, B, and C, respectively The nal scores are fuzzy values (0:30; 0:40; 0:54), (0:22; 0:31; 0:42), (0:23; 0:29; 0:52) for Candidates A, B, and C, respectively The nal scores are crisp values 0:41; 0:28; 0:25 for Candidates A, B, and C, respectively The nal scores are fuzzy values (0:23; 0:42; 0:56), (0:20; 0:29; 0:43), (0:21; 0:32; 0:51) for Candidates A, B, and C, respectively In the above methods the average grades of the total membership functions are unavailable The derived scores are crisp values 0:467567; 0:258803; 0:2736 for Candidates A, B, and C, respectively. The average grade of total membership functions is 0.72461 ⎡ ⎤ 0:34170 0:22789 0:43041 ⎢ ⎥ ⎢ 0:59974 0:20013 0:20013 ⎥ =[0:24714; 0:37223; 0:14363; 0:23700] ⎢ ⎥ ⎢ ⎥ ⎣ 0:59974 0:20013 0:20113 ⎦ 0:23749 0:41252 0:25999 =[0:45722; 0:26444; 0:27834] = [w1;w2;w3] Their methods cannot treat Example 5 The derived scores are crisp values 0:45722; 0:26444; 0:27834 for Candidates A, B, and C, respectively. The average grade of total membership functions is 0.74416 which are the weighting scores for the candidates A; B; and C; respectively: The comparison analysis in solving Examples 4 and 5 between traditional fuzzy AHP methods and the proposed method are summarized in Table 6. As was already shown by many researchers [2–4,8,21,37–44], one suitable method for dealing with fuzzy AHP problems involving fuzzy, point, and interval ratings as well as some ratings without available information is the fuzzy LLSM. The comparative analysis of Table 6
C.-S. Yu / Computers & Operations Research 29 (2002) 1969–2001 1999 therefore focuses on this method while the other traditional fuzzy AHP methods cannot tackle this type of problems. Compared with conventional fuzzy AHP methods, numerical examples illustrate that the proposed GP-AHP method can concurrently treat the pair-to-pair comparison involving triangular, general concave and concave–convex mixed fuzzy estimates. By encompassing the trade-o consideration of optimizing decision-makers’ group inconsistency as well as individual opinion, the proposed method directly treats the interactions among the fuzzy variables and the decision-makers. The extracted corresponding priority vector therefore best re ects an overall preference and is progressively less sensitive and vulnerable to con icting judgments. 7. Conclusions Due to the di culty that a decision-maker faces in precisely assessing the relative importance of two objectives, this study extends AHP practical applications to tackle a broader range of fuzzy problems by developing a separable linear expression to represent general triangular, concave, convex, and concave–convex mixed vague ratings. Since the developed solution algorithm is simple and the problem formulation means is distinct, a user-friendly computer program can be easily developed to handle simple yet time-consuming linearization calculations. Moreover, many prevailing software packages like LINDO [35] or EXCEL [36] can conveniently compute the formulated models. As a result, the generated priority vector of the proposed method allows the decision-maker to perform sensitivity analysis. Hence, the proposed method is a promising and attractive alternative to the current fuzzy AHP methods. Acknowledgements This research is supported by the National Science Council of the Republic of China under contract NSC 89-2416-H-158-008. References [1] Satty TL, Vargas LG. Comparison of eigenvalue, logarithmic least squares and least squares methods in estimating ratios. Mathematical Modeling 1984;5:309–24. [2] Boender CGE, de Graan JG, Lootsma FA. Multi-criteria decision analysis with fuzzy pairwise comparisons. Fuzzy Sets and Systems 1989;29:133–43. [3] Buckly JJ. Fuzzy hierarchical analysis. Fuzzy Sets and Systems 1985;17:233–47. [4] Chang DY. Applications of the extent analysis method on fuzzy AHP. European Journal of Operational Research 1996;95:649–55. [5] Dong WM, Wong FS. Interactive fuzzy variables and fuzzy decisions. Fuzzy Sets and Systems 1989;29:1–19. [6] Haines LM. Astatistical approach to the analytic hierarchy process with interval judgements. European Journal of Operational Research 1998;110:112–25. [7] Kumar NV, Ganesh LS. An empirical analysis of the use of the analytic hierarchy process for estimating membership values in a fuzzy set. Fuzzy Sets and Systems 1996;82:1–16. [8] van Laarhoven PJM, Pedrycz W. Afuzzy extension of Satty’s priority theory. Fuzzy Sets and Systems 1983;11:229–41.
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