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An analytic approach for obtaining maximal entropy OWA operator weights ∗ Robert Fullér [email protected] Péter Majlender [email protected] Abstract One important issue in the theory of Ordered Weighted Averaging (OWA) operators is the determination of the associated weights. One of the first approaches, suggested by O’Hagan, determines a special class of OWA operators having maximal entropy of the OWA weights for a given level of orness; algorithmically it is based on the solution of a constrained optimization problem. In this paper, using the method of Lagrange multipliers, we shall solve this constrained optimization problem analytically and derive a polinomial equation which is then solved to determine the optimal weighting vector. Keywords: OWA operator, dispersion, degree of orness 1 Introduction An OWA operator of dimension n is a mapping F : Rn → R that has an associated weighting vector W = (w1 , . . . , wn )T of having the properties w1 + · · · + wn = 1, 0 ≤ wi ≤ 1, i = 1, . . . , n, and such that F (a1 , . . . , an ) = n X wi bi , i=1 where bj is the jth largest element of the collection of the aggregated objects {a1 , . . . , an }. ∗ The final version of this paper appeared in: Fuzzy Sets and Systems, 124(2001) 53-57. 1 In [3], Yager introduced two characterizing measures associated with the weighting vector W of an OWA operator. The first one, the measure of orness of the aggregation, is defined as n 1 X orness(W ) = (n − i)wi . n−1 i=1 and it characterizes the degree to which the aggregation is like an or operation. It is clear that orness(W ) ∈ [0, 1] holds for any weighting vector. The second one, the measure of dispersion of the aggregation, is defined as disp(W ) = − n X wi ln wi i=1 and it measures the degree to which W takes into account all information in the aggregation. It is clear that the actual type of aggregation performed by an OWA operator depends upon the form of the weighting vector. A number of approaches have been suggested for obtaining the associated weights, i.e., quantifier guided aggregation [3, 4], exponential smoothing [6] and learning [5]. Another approach, suggested by O’Hagan [2], determines a special class of OWA operators having maximal entropy of the OWA weights for a given level of orness. This approach is based on the solution of he following mathematical programming problem: n X − wi ln wi maximize i=1 subject to 1 n−1 n X n X (n − i)wi = α, 0 ≤ α ≤ 1 (1) i=1 wi = 1, 0 ≤ wi ≤ 1, i = 1, . . . , n. i=1 Using the method of Lagrange multipliers we shall transfer problem (1) to a polinomial equation which is then solved to determine the optimal weighting vector. 2 2 Obtaining maximal entropy weights First we note that disp(W ) is meaningful if wi > 0 and by letting wi ln wi to zero if wi = 0, problem (1) turns into disp(W ) → max; subject to {orness(W ) = α, w1 + · · · + wn = 1, 0 ≤ α ≤ 1}. If n = 2 then from orness(w1 , w2 ) = α we get w1 = α and w2 = 1 − α. Furthemore, if α = 0 or α = 1 then the associated weighting vectors are uniquely defined as (0, 0, . . . , 0, 1)T and (1, 0, . . . , 0, 0)T , respectively, with value of dispersion zero. Suppose now that n ≥ 3 and 0 < α < 1. Let us L(W, λ1 , λ2 ) = − n X wi ln wi + λ1 i=1 X n i=1 X n n−i wi − α + λ2 wi − 1 . n−1 i=1 denote the Lagrange function of constrained optimization problem (1), where λ1 and λ2 are real numbers. Then the partial derivatives of L are computed as ∂L n−j = − ln wj − 1 + λ1 + λ2 = 0, ∀j ∂wj n−1 n X ∂L = wi − 1 = 0 ∂λ1 (2) i=1 n X ∂L n−i = wi − α = 0. ∂λ2 n−1 i=1 For j = n equation (2) turns into − ln wn − 1 + λ1 = 0 ⇐⇒ λ1 = ln wn + 1, and for j = 1 we get − ln w1 − 1 + λ1 + λ2 = 0, and, therefore, λ2 = ln w1 + 1 − λ1 = ln w1 + 1 − ln wn − 1 = ln w1 − ln wn . For 1 ≤ j ≤ n we find j−1 n−j ln wj = ln wn + ln w1 ⇒ wj = n−1 n−1 3 q w1n−j wnj−1 . n−1 (3) If w1 = wn then (3) gives w1 = w2 = · · · = wn = 1 ⇒ disp(W ) = ln n, n which is the optimal solution to (1) for α = 0.5 (actually, this is the global optimal value for the dispersion of all OWA operators of dimension n). Suppose now that w1 6= wn . Let us introduce the notations 1 1 u1 = w1n−1 , un = wnn−1 . Then we may rewrite (3) as wj = u1n−j uj−1 n , for 1 ≤ j ≤ n. From the first condition, orness(W ) = α, we find n n X X n−i wi = α ⇐⇒ (n − i)u1n−i uni−1 = (n − 1)α, n−1 i=1 i=1 and from n−1 X 1 n i n−i (n − 1)u1 − u1 un u1 − un i=1 1 − unn−1 un−1 1 n = (n − 1)u1 − u1 un u1 − un u1 − un 1 n n n (n − 1)u1 (u1 − un ) − u1 un + u1 un = (u1 − un )2 1 n+1 n n = (n − 1)u1 − nu1 un + u1 un , (u1 − un )2 n X i−1 = (n − i)un−i 1 un i=1 we get (n − 1)un+1 − nun1 un + u1 unn = (n − 1)α(u1 − un )2 1 nun1 − u1 = (n − 1)α(u1 − un ) 1 n un = ((n − 1)α + 1)u1 − nu1 (n − 1)α un (n − 1)α + 1 − nw1 = . (4) u1 (n − 1)α 4 From the second condition, w1 + · · · + wn = 1, we get n X j−1 un−j = 1 ⇐⇒ 1 un j=1 un1 − unn =1 u1 − un ⇐⇒ un1 − unn = u1 − un ⇐⇒ u1n−1 − un un × un−1 =1− n u1 u1 (5) (6) Comparing equations (4) and (6) we find w1 − (n − 1)α + 1 − nw1 nw1 − 1 × wn = (n − 1)α (n − 1)α and, therefore, wn = ((n − 1)α − n)w1 + 1 . (n − 1)α + 1 − nw1 (7) Let us rewrite equation (5) as un1 − unn = u1 − un u1 (w1 − 1) = un (wn − 1) w1 (w1 − 1)n−1 = wn (wn − 1)n−1 n−1 w1 (w1 − 1) (n − 1)α(w1 − 1 n−1 ((n − 1)α − n)w1 + 1 = × (n − 1)α + 1 − nw1 (n − 1)α + 1 − nw1 w1 [(n − 1)α + 1 − nw1 ]n = ((n − 1)α)n−1 [((n − 1)α − n)w1 + 1]. (8) So the optimal value of w1 should satisfy equation (8). Once w1 is computed then wn can be determined from equation (7) and the other weights are obtained from equation (3). Remark 2.1 If n = 3 then from (3) we get w2 = √ w1 w3 independently of the value of α, which means that the optimal value of w2 is always the geometric mean of w1 and w3 . 5 3 Computing the optimal weights Let us introduce the notations f (w1 ) = w1 [(n − 1)α + 1 − nw1 ]n , g(w1 ) = ((n − 1)α)n−1 [((n − 1)α − n)w1 + 1]. Then to find the optimal value for the first weight we have to solve the following equation f (w1 ) = g(w1 ), where g is a line and f is a polinom of w1 of dimension n + 1. Without loss of generality we can assume that α < 0.5, because if a weighting vector W is optimal for problem (1) under some given degree of orness, α < 0.5, then its reverse, denoted by W R , and defined as wiR = wn−i+1 is also optimal for problem (1) under degree of orness (1 − α). Really, as was shown by Yager [4], we find that disp(W R ) = disp(W ) and orness(W R ) = 1 − orness(W ). Therefore, for any α > 0.5, we can solve problem (1) by solving it with level of orness (1 − α) and then taking the reverse of that solution. From the equations 1 1 0 1 0 1 f =g and f =g n n n n we get that g is always a tangency line to f at the point w1 = 1/n. But if w1 = 1/n then w1 = · · · = wn = 1/n also holds, and that is the optimal solution for α = 0.5. Consider the the graph of f . It is clear that f (0) = 0 and by solving the equation f 0 (w1 ) = [(n − 1)α + 1 − nw1 ]n − n2 w1 [(n − 1)α + 1 − nw1 ]n−1 = 0 we find that its unique solution is ŵ1 = (n − 1)α + 1 1 < , n(n + 1) n and its second derivative, f 00 (ŵ1 ) is negative, which means that ŵ1 is the only maximizing point of f on the segment [0, 1/n]. 6 Figure 1: Graph of f withr n = 4 and α = 0.2. We prove now that g can intersect f only once in the open interval (0, 1/n). It will guarantee the uniqueness of the optimal solution of problem (1). Really, from the equation f 00 (w1 ) = −2n2 [(n−1)α+1−nw1 ]n−1 +n3 (n−1)w1 [(n−1)α+1−nw1 ]n−2 = 0 we find that its unique solution is w̄1 = 2 (n − 1)α + 1 1 = 2ŵ1 < , (since α < 0.5). n(n + 1) n with the meaning that f is strictly concave on (0, w̄1 ), has an inflextion point at w̄1 , and f is strictly convex on (w̄1 , 1/n). Therefore, the graph of g should lie below the graph of g if ŵ1 < w1 < 1/n and g can cross f only once in the interval (0, ŵ1 ). 4 Illustrations Let us suppose that n = 5 and α = 0.6. Then from the equation w1 [4 × 0.6 + 1 − 5w1 ]5 = (4 × 0.6)4 [1 − (5 − 4 × 0.6)w1 ]. we find w1∗ = 0.2884 (4 × 0.6) − 5)w1∗ + 1 = 0.1278 4 × 0.6 + 1 − 5w1∗ q w2∗ = 4 (w1∗ )3 w5∗ = 0.2353, q w3∗ = 4 (w1∗ )2 (w5∗ )2 = 0.1920, q w4∗ = 4 (w1∗ )(w5∗ )3 = 0.1566. w5∗ = and, disp(W ∗ ) = 1.5692. Using exponential smoothing [1], Filev and Yager [6] obtained the following weighting vector W 0 = (0.41, 0.10, 0.13, 0.16, 0.20), 7 with disp(W 0 ) = 1.48 and orness(W 0 ) = 0.5904. We first note that the weights computed from the constrained optimization problem have better dispersion than those ones obtained by Filev and Yager in [6], however the (heuristic) technology suggested in [6] needs less computational efforts. Other interesting property here is that small changes in the required level of orness, α, can cause a big variation in weighting vectors of near optimal dispersity, (for example, compare the weighting vectors W ∗ and W 0 ). References [1] R.G. Brown, Smoothing, Forecasting and Prediction of Discrete Time Series (Prentice-Hall, Englewood Cliffs, 1963). [2] M. O’Hagan, Aggregating template or rule antecedents in real-time expert systems with fuzzy set logic, in: Proc. 22nd Annual IEEE Asilomar Conf. Signals, Systems, Computers, Pacific Grove, CA, 1988 81-689. [3] R.R.Yager, Ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Trans. on Systems, Man and Cybernetics, 18(1988) 183-190. [4] R.R.Yager, Families of OWA operators, Fuzzy Sets and Systems, 59(1993) 125-148. [5] R. R. Yager and D. Filev, Induced ordered weighted averaging operators, IEEE Trans. on Systems, Man and Cybernetics – Part B: Cybernetics, 29(1999) 141-150. [6] D. Filev and R. R. Yager, On the issue of obtaining OWA operator weights, Fuzzy Sets and Systems, 94(1998) 157-169 8 5 Citations [A13] Robert Fullér and Péter Majlender, An analytic approach for obtaining maximal entropy OWA operator weights, FUZZY SETS AND SYSTEMS, 124(2001) 53-57. [Zbl.0989.03057] in journals A13-c134 J.-W. Liu; C.-H. Cheng; Y.H. Chen; S.-F. Huang, OWA based PCA information fusion method for classification problem, INTERNATIONAL JOURNAL OF INFORMATION AND MANAGEMENT SCIENCES, 21(2010), Issue 2, pp. 209-225. 2010 A13-c133 Chang, Liang; Shi, Zhong-Zhi; Chen, Li-Min; Niu, Wen-Jia, Family of extended dynamic description logics, Ruan Jian Xue Bao (Journal of Software). Vol. 21, no. 1, pp. 1-13. 2010 A13-c132 R. A. Nasibova; E. N. Nasibov, Linear Aggregation with Weighted Ranking, AUTOMATIC CONTROL AND COMPUTER SCIENCES, 44(2010), Number 2, pp. 96-102. 2010 http://dx.doi.org/10.3103/S0146411610020057 A13-c131 Li-Gang Zhou and Hua-you Chen, Generalized ordered weighted logarithm aggregation operators and their applications to group decision making, INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, 25(2010), issue 7, pp. 683-707. 2010 http://dx.doi.org/10.1002/int.20419 A13-c130 Victor M Vergara, Shan Xia, Minimization of uncertainty for ordered weighted average, INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS 25(2010), pp. 581-595. 2010 http://dx.doi.org/10.1002/int.20422 There are several methods that find the OWA weights. [3,912] Yager [3] proposed finding weights that maximizes their entropy (also known as weights dispersion). Later, Fuller and Majlender [A13] found an analytical solution to Yager’s theory based on Lagrange multipliers. In addition, these two authors proposed minimizing the variance of weights [A10] instead of maximizing their entropy. The goal of these methods is to maximize similarity. Maximizing the entropy or minimizing the variance of the weights {w1 , w2 , . . . , wn } makes each weight wi more similar in magnitude tot he others. (page 582) 9 A13-c129 Yao-Hsien Chen, Ching-Hsue Cheng, Jing-Wei Liu, Intelligent preference selection model based on NRE for evaluating student learning achievement, COMPUTERS & EDUCATION, 54(2010), pp. 916926. 2010 http://dx.doi.org/10.1016/j.compedu.2009.09.020 This step computes the aggregate values of the test-score examples and typical score examples, for different orders of evaluation items by using the OWA operator. First, according to the Fullér and Majlenders’ equation introduced in Section 3, we can obtain a setP of OWA weights Wα {w1 , w2 , . . . , wn }, where 0 ≤ w1 ≤ 1, ni=1 wi = 1 and α ∈ [0, 1]. Second, to compute the aggregate values, we multiply the values of the evaluation items, which are permuted by all possible orders, by the corresponding OWA weights, and then sum up these multiplication values. (page 920) A13-c128 A. Emrouznejad, G.R. Amin, Improving minimax disparity model to determine the OWA operator weights, INFORMATION SCIENCES, 180(2010) 1477-1485. 2010 http://dx.doi.org/10.1016/j.ins.2009.11.043 A13-c127 Xiao-Yong Li, Xiao-Lin Gui, Cognitive Model of Dynamic Trust Forecasting, JOURNAL OF SOFTWARE, 21(2010), number 1, pp. 163-176 (in Chinese). 2010 http://dx.doi.org/10.3724/SP.J.1001.2010.03558 A13-c126 Kuei-Hu Chang, Ta-Chun Wen, A novel efficient approach for DFMEA combining 2-tuple and the OWA operator, EXPERT SYSTEMS WITH APPLICATIONS, Volume 37, Issue 3, 15 March 2010, pp. 2362-2370. 2010 http://dx.doi.org/10.1016/j.eswa.2009.07.026 A13-c125 Jing-Wei Liu, Ching-Hsue Cheng, Yao-Hsien Chen, Tai-Liang Chen, OWA rough set model for forecasting the revenues growth rate of the electronic industry, EXPERT SYSTEMS WITH APPLICATIONS, 37(2010), pp. 610-617. 2010 http://dx.doi.org/10.1016/j.eswa.2009.06.020 Fullér and Majlender (2001) proposed a new method by using Lagrange multipliers to improve the problem in Eq. (5) and got the following: (page 611) 10 A13-c124 Pan Yuhou; Yao Shuang; Guo Yajun, Comprehensive Multiindex Group Evaluation Method for Scheme of Oilfield Development Adjustment and Its Application, TECHNOLOGY ECONOMICS, 29(2010), number 3, pp. 31-34 (in Chinese). 2010 http://d.wanfangdata.com.cn/Periodical_jsjj201003007.aspx A13-c123 Yung-Chia Chang, Kuie-Hu Chang, Cheng-Shih Liaw, Innovative reliability allocation using the maximal entropy ordered weighted averaging method, COMPUTERS & INDUSTRIAL ENGINEERING, 57(2009), Issue 4, pp. 1274-1281. 2009 http://dx.doi.org/10.1016/j.cie.2009.06.007 Additionally, Fuller and Majlender (2001) used Lagrange multipliers on Yager’s OWA equation to derive a polynomial equation, which determines the optimal weighting vector under maximal entropy (ME-OWA operator). The proposed approach thus determines the optimal weighting vector under maximal entropy, and the OWA operator ascertains the optimal reliability allocation rating after an aggregation process. This method is both a simple and effective approach that can efficiently resolve the shortcomings of the FOO technique and average weighting allocation. (page 1275) A13-c122 S. Zadrozny, J. Kacprzyk, Issues in the practical use of the OWA operators in fuzzy querying, JOURNAL OF INTELLIGENT INFORMATION SYSTEMS, 33(2009), No. 3, pp. 307-325. 2009 http://dx.doi.org/10.1007/s10844-008-0068-1 A13-c121 Yu Yi, Thomas Fober, Eyke Hüllermeier, Fuzzy Operator Trees for Modeling Rating Functions, INTERNATIONAL JOURNAL OF COMPUTATIONAL INTELLIGENCE AND APPLICATIONS, 8(2009), pp. 423-428. 2009 http://dx.doi.org/10.1142/S1469026809002679 A13-c120 E. Cables Pérez, M.Teresa Lamata, OWA weights determination by means of linear functions, MATHWARE & SOFT COMPUTING, 16(2009), 107-122. 2009 http://ic.ugr.es/Mathware/index.php/Mathware/article/view/398/pdf-162-art1-final A13-c119 Kuei-Hu Chang, Evaluate the orderings of risk for failure problems using a more general RPN methodology, MICROELECTRONICS RELIABILITY, Volume 49, Issue 12, pp. 1586-1596. 2009 11 http://dx.doi.org/10.1016/j.microrel.2009.07.057 A13-c118 Jian Wu, Bo-Liang Sun, Chang-Yong Liang, Shan-Lin Yang, A linear programming model for determining ordered weighted averaging operator weights with maximal Yager’s entropy, COMPUTERS & INDUSTRIAL ENGINEERING, Volume 57, Issue 3, pp. 742-747. 2009 http://dx.doi.org/10.1016/j.cie.2009.02.001 A13-c117 Byeong Seok Ahn, Some remarks on the LSOWA approach for obtaining OWA operator weights, INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Volume 24 Issue 12, Pages 1265-1279. 2009 http://dx.doi.org/10.1002/int.20384 A13-c116 R.R. Yager, On the dispersion measure of OWA operators, INFORMATION SCIENCES, 179(2009), pp. 3908-3919. 2009 http://dx.doi.org/10.1016/j.ins.2009.07.015 A13-c115 Badredine Arfi, Probing the Democratic Peace Argument Using Linguistic Fuzzy Logic, INTERNATIONAL INTERACTIONS: EMPIRICAL AND THEORETICAL RESEARCH IN INTERNATIONAL RELATIONS, 35(2009), pp. 30-57. 2009 http://dx.doi.org/10.1080/03050620902743838 A13-c114 S. Yao, Y.-J. Guo, P.-T. Yi, Multi-variable induced ordered weighted averaging operator and its application, Dongbei Daxue Xuebao/Journal of Northeastern University, 30(2009), pp. 298-301. 2009 A13-c113 F. Szidarovszky, M. Zarghami, Combining fuzzy quantifiers and neat operators for soft computing, IRANIAN JOURNAL OF FUZZY SYSTEMS, 6(2009), pp. 15-25. 2009 A13-c112 B.S. Ahn, H. Park, An efficient pruning method for decision alternatives of OWA operators, IEEE Transactions on Fuzzy Systems, 16 (2009), pp. 1542-1549. 2009 http://dx.doi.org/10.1109/TFUZZ.2008.2005012 A13-c111 Ching-Hsue Cheng, Jia-Wen Wang, Ming-Chang Wua, OWAweighted based clustering method for classification problem, EXPERT SYSTEMS WITH APPLICATIONS, 36(2009), pp. 4988-4995. 2009 http://dx.doi.org/10.1016/j.eswa.2008.06.013 2.1.2. Fullér and Majlender’s OWA Fullér and Majlender (2001) transform Yager’s OWA equation to a polynomial equation by using Lagrange multipliers. 12 According to their approach, the associated weighting vector can be obtained by (5)-(7). A13-c110 YAO Shuang; GUO Ya-jun; YI Ping-tao, Multi-variable Induced Ordered Weighted Averaging Operator and Its Application, JOURNAL OF NORTHEASTERN UNIVERSITY(NATURAL SCIENCE), 30(2009), number 2, pp. (in Chinese). 2009 http://d.wanfangdata.com.cn/Periodical_dbdxxb200902037.aspx A13-c109 LU Zhen-bang; ZHOU Li-hua, Probabilistic Fuzzy Cognitive Maps Based on Ordered Weighted Averaging Operators COMPUTER SCIENCE, 35(2008), number 12, pp. 187-189 (in Chinese). 2008 http://d.wanfangdata.com.cn/Periodical_jsjkx200812049.aspx A13-c108 ZHOU Rong-xi, LIU Shan-cun, QIU Wan-hua, Survey of applications of entropy in decision analysis, CONTROL AND DECISION, 23(2008), number 4, pp. 361-366 (in Chinese), 2008 http://d.wanfangdata.com.cn/Periodical_kzyjc200804001.aspx A13-c107 XU Jian-Rong; ZOU Rong-Xi, A model of weight variables for obtaining an MEOWA operator based on the constraint of interval orness measure, JOURNAL OF BEIJING UNIVERSITY OF CHEMICAL TECHNOLOGY (NATURAL SCIENCE EDITION), 35(2008), number 3, pp. 100-103 (in Chinese). 2008 http://d.wanfangdata.com.cn/Periodical_bjhgdxxb200803023.aspx A13-c106 Robert I. John, Shang-Ming Zhou, Jonathan M. Garibaldi and Francisco Chiclana, Automated Group Decision Support Systems Under Uncertainty: Trends and Future Research, INTERNATIONAL JOURNAL OF COMPUTATIONAL INTELLIGENCE RESEARCH, 4(2008), pp. 357-371. 2008 http://www.cci.dmu.ac.uk/preprintPDF/Franciscov4i4p5.pdf A13-c105 K. -H. Chang; C. -H. Cheng; Y. -C. Chang, Reliability assessment of an aircraft propulsion system using IFS and OWA tree, ENGINEERING OPTIMIZATION, 40(2008) 907-921. 2008 http://dx.doi.org/10.1080/03052150802132914 A13-c104 Konstantinos Anagnostopoulos, Haris Doukas, John Psarras, A linguistic multicriteria analysis system combining fuzzy sets theory, 13 ideal and anti-ideal points for location site selection, EXPERT SYSTEMS WITH APPLICATIONS, 35 (2008) pp. 2041-2048. 2008 http://dx.doi.org/10.1016/j.eswa.2007.08.074 In this paper we use a special class of OWA operators which have maximum entropy for a given level of orness (O’Hagan, 1988). The weighing vector of an OWA operator with maximum entropy is calculated applying the results of Fullér and Majlender (2001). (page 2044) A13-c103 Ali Emrouznejad, MP-OWA: The most preferred OWA operator, KNOWLEDGE-BASED SYSTEMS, 21(2008), Issue 8, pp. 847-851. 2008 http://dx.doi.org/10.1016/j.knosys.2008.03.057 However to apply the OWA operator for decision making, a very crucial issue is to determine its weights. O’Hagan [16] suggested a maximum entropy method as the rst approach to determine OWA operator weights in which he formulated the OWA operator weight problem to a constrained nonlinear optimization model with a predened degree of orness. Fullér and Majlender [A13] transformed the maximum entropy method into a polynomial equation that can be solved analytically. 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(page 599) 14 A13-c98 Xinwang Liu and Da Qingli, On the properties of regular increasing monotone (RIM) quantifiers with maximum entropy, INTERNATIONAL JOURNAL OF GENERAL SYSTEMS, Volume 37, Issue 2, pp. 167-179. 2008 http://dx.doi.org/10.1080/03081070701192675 A13-c97 Byeong Seok Ahn, Some Quantier Functions From Weighting Functions With Constant Value of Orness, IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICSPART B: CYBERNETICS, 38: (2) 540-546. 2008 http://dx.doi.org/10.1109/TSMCB.2007.912743 A13-c96 Xinwang Liu and Shilian Han, Orness and parameterized RIM quantifier aggregation with OWA operators: A summary, INTERNATIONAL JOURNAL OF APPROXIMATE REASONING, Volume 48, Issue 1, Pages 77-97. 2008 http://dx.doi.org/10.1016/j.ijar.2007.05.006 A13-c95 B.S. Ahn, Preference relation approach for obtaining OWA operators weights, INTERNATIONAL JOURNAL OF APPROXIMATE REASONING, 47 (2), pp. 166-178. 2008 http://dx.doi.org/10.1016/j.ijar.2007.04.001 The resulting weights are called maximum entropy OWA (MEOWA) weights for a given degree of orness and analytic forms and property for these weights are further investigated by several researchers [25,A13]. (page 167) A13-c94 B.S. Ahn, H. Park, Least-squared ordered weighted averaging operator weights, INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, 23(1), pp. 33-49. 2008 http://dx.doi.org/10.1002/int.20257 A13-c93 D.H. Hong and K.T. Kim, A note on the maximum entropy weighting function problem, JOURNAL OF APPLIED MATHEMATICS AND COMPUTING, 23(2007), No. 1-2, pp. 547-552. 2007 http://www.mathnet.or.kr/mathnet/thesis_file/DHHong0613F.pdf A13-c90 Zhenbang Lv and Lihua Zhou, Advanced Fuzzy Cognitive Maps Based on OWA Aggregation, INTERNATIONAL JOURNAL OF COMPUTATIONAL COGNITION, 5(2007) pp. 31-34. 2007 http://www.yangsky.com/ijcc/pdf/ijcc524.pdf 15 A13-c89 Xinwang Liu, Hongwei Lou, On the equivalence of some approaches to the OWA operator and RIM quantier determination, FUZZY SETS AND SYSTEMS, vol. 159, pp. 1673-1688. 2007 http://dx.doi.org/10.1016/j.fss.2007.12.024 Filev and Yager [2] further analyzed the properties of MEOWA operators, and proposed a method for obtaining weights as a function of one parameter. Recently, Fullér and Majlender [A13] proposed an analytical solution by transforming the problem into a polynomial equation. (page 1677) A13-c88 X. Liu, Some OWA operator weights determination methods with RIM quantifier, JOURNAL OF SOUTHEAST UNIVERSITY (ENGLISH EDITION), vol. 23 (SUPPL.), pp. 76-82. 2007 A13-c61 Z.-B. Lu, L.-H. Zhou, Hybrid fuzzy cognitive maps, JOURNAL OF XIDIAN UNIVERSITY (NATURAL SCIENCE), 34 (5), pp. 779783 (in Chinese). 2007 A13-c60 B.S. Ahn, The OWA aggregation with uncertain descriptions on weights and input arguments, IEEE TRANSACTIONS ON FUZZY SYSTEMS, 15 (6), pp. 1130-1134. 2007 http://dx.doi.org/10.1109/TFUZZ.2007.895945 A13-c87 L. Zhenbang, Z. Lihua, A hybrid fuzzy cognitive model based on weighted OWA operators and single-antecedent rules, INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, 22 (11), pp. 11891196. 2007 http://dx.doi.org/10.1002/int.20243 A13-c86 Sadiq, R., Tesfamariam, S., Probability density functions based weights for ordered weighted averaging (OWA) operators: An example of water quality indices, EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, 182 (3), pp. 1350-1368. 2007 http://dx.doi.org/10.1016/j.ejor.2006.09.041 A13-c85 Llamazares, B., Choosing OWA operator weights in the field of Social Choice, INFORMATION SCIENCES, 177 (21), pp. 4745-4756. 2007 http://dx.doi.org/10.1016/j.ins.2007.05.015 In the field of OWA operators, one of the first approaches, suggested by O’Hagan [18], lies in selecting the vector that maximizes the entropy of the OWA weights for a given level of orness. This methodology has also been used by Fullér and Majlender [A13]. (page 4745) 16 A13-c84 Wang YM, Luo Y, Liu XW, Two new models for determining OWA operator weights COMPUTERS & INDUSTRIAL ENGINEERING 52 (2): 203-209 MAR 2007 http://dx.doi.org/10.1016/j.cie.2006.12.002 Fullér and Majlender (2001) showed that the maximum entropy model could be transformed into a polynomial equation that can be solved analytically. (page 203) A13-c83 Yeh DY, Cheng CH, Yio HW, Empirical research of the principal component analysis and ordered weighted averaging integrated evaluation model on software projects CYBERNETICS AND SYSTEMS, 38 (3): 289-303 2007 http://dx.doi.org/10.1080/01969720601187347 A13-c82 Wang YM, Parkan C, A preemptive goal programming method for aggregating OWA operator weights in group decision making INFORMATION SCIENCES, 177 (8): 1867-1877 APR 15 2007 http://dx.doi.org/10.1016/j.ins.2006.07.023 Fullér and Majlender [A13] showed that the maximum entropy model could be converted into a polynomial equation that can be solved analytically. (page 1867) A13-c81 Wu J, Liang CY, Huang YQ, An argument-dependent approach to determining OWA operator weights based on the rule of maximum entropy INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, 22 (2): 209-221 FEB 2007 http://dx.doi.org/10.1002/int.20201 Fullér and Majlender [A13] transformed the maximum entropy model into a polynomial equation that can be solved analytically. (page 209) A13-c80 Xu ZS, Chen J, An interactive method for fuzzy multiple attribute group decision making INFORMATION SCIENCES, 177 (1): 248263 JAN 1 2007 http://dx.doi.org/10.1016/j.ins.2006.03.001 Fullér and Majlender [A13] used the method of Lagrange multipliers to solve O’Hagan’s procedure analytically. (page 251) A13-c79 Sadiq, R., Tesfamariam, S. Probability density functions based weights for ordered weighted averaging (OWA) operators: An example 17 of water quality indices EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, 182 (3), pp. 1350-1368. 2007 http://dx.doi.org/10.1016/j.ejor.2006.09.041 Yager and Filev (1999) suggested an algorithm to obtain the OWA weights from a collection of samples with the relevant aggregated data. Fullér and Majlender (2001) used the method of Lagrange multipliers to solve O’Hagan’s procedure analytically. (page 1356) A13-c78 Liu, X., The solution equivalence of minimax disparity and minimum variance problems for OWA operators, INTERNATIONAL JOURNAL OF APPROXIMATE REASONING, 45 (1), pp. 68-81. 2007 http://dx.doi.org/10.1016/j.ijar.2006.06.004 Recently, Fullér [A13] transformed the maximum entropy model into a polynomial equation, which can be solved in an analytical way. (page 69) A13-c77 Cheng CH, Chang JR, Ho TH Dynamic fuzzy OWA model for evaluating the risks of software development CYBERNETICS AND SYSTEMS, 37 (8): 791-813 DEC 2006 http://dx.doi.org/10.1080/01969720600939797 A13-c76 Xu ZS, A note on linguistic hybrid arithmetic averaging operator in multiple attribute group decision making with linguistic information GROUP DECISION AND NEGOTIATION, 15 (6): 593-604 NOV 2006 http://dx.doi.org/10.1007/s10726-005-9008-4 Fullér and Majlender (2001) used the method of Lagrange multipliers to solve O’Hagan’s procedure analytically. (page 595) A13-c75 Liu XW, Lou HW Parameterized additive neat OWA operators with different orness levels INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, 21(10): 1045-1072 OCT 2006 http://dx.doi.org/10.1002/int.20176 The maximum entropy OWA operator was first suggested by O’Hagan [40] and later was discussed by Filev and Yager [21] and Fullér and Majlender [A13]. (page 1055) A13-c74 Liu XW, On the properties of equidifferent OWA operator INTERNATIONAL JOURNAL OF APPROXIMATE REASONING, 43 (1): 90-107 SEP 2006 18 http://dx.doi.org/10.1016/j.ijar.2005.11.003 The consistent condition of geometric (maximum entropy) OWA operator was proved, some properties associated with the orness level are discussed, which extended the results of O’Hagan [14], Filev and Yager [6,7], Fullér and Majlender [A13]. (page 91) A13-c73 Cheng CH, Chang JR, MCDM aggregation model using situational ME-OWA and ME-OWGA operators, INTERNATIONAL JOURNAL OF UNCERTAINTY FUZZINESS AND KNOWLEDGE-BASED SYSTEMS, 14 (4): 421-443 AUG 2006 http://dx.doi.org/ 10.1142/S0218488506004102 A13-c72 Ahn BS, On the properties of OWA operator weights functions with constant level of orness, IEEE TRANSACTIONS ON FUZZY SYSTEMS, 14(4): 511-515 AUG 2006 http://dx.doi.org/10.1109/TFUZZ.2006.876741 A13-c71 Xu ZH, Induced uncertain linguistic OWA operators applied to group decision making, INFORMATION FUSION, 7(2): 231-238 JUN 2006 http://dx.doi.org/10.1016/j.inffus.2004.06.005 A13-c70 Marchant T, Maximal orness weights with a fixed variability for owa operators, INTERNATIONAL JOURNAL OF UNCERTAINTY FUZZINESS AND KNOWLEDGE-BASED SYSTEMS, 14(3): 271276 JUN 2006 http://dx.doi.org/10.1142/S021848850600400X A13-c69 Wang JW, Chang JR, Cheng CH, Flexible fuzzy OWA querying method for hemodialysis database, SOFT COMPUTING, 10 (11): 1031-1042 SEP 2006 http://dx.doi.org/10.1007/s00500-005-0030-x Fullér and Majlender [A13] use the method of Lagrange multipliers to transfer Eq. (12) to a polynomial equation, which can determine the optimal weighting vector. By their method, the associated weighting vector is easily obtained by Eqs. (13)-(18). (page 1033) A13-c68 Xinwang Liu, On the maximum entropy parameterized interval approximation of fuzzy numbers, FUZZY SETS AND SYSTEMS, 157, pp. 869-878. 2006 http://dx.doi.org/10.1016/j.fss.2005.09.010 19 A13-c67 J.-R. Chang, T.-H. Ho, C.-H. Cheng, A.-P. Chen, Dynamic fuzzy OWA model for group multiple criteria decision making, SOFT COMPUTING, 10 543-554. 2006 http://dx.doi.org/10.1007/s00500-005-0484-x To resolve this problem, this study proposes a dynamic OWA aggregation model based on the faster OWA operator, which has been introduced by Fullér and Majlender [A13] and can work like a magnifying lens and adjust its focus based on the sparest information to change the dynamic attribute weights to revise the weight of each attribute based on aggregation situation, and then to provide suggestions to decision maker (DM). (page 544) A13-c66 Liu XW, Some properties of the weighted OWA operator, IEEE TRANSACTIONS ON SYSTEMS MAN AND CYBERNETICS PART B-CYBERNETICS, 36(1): 118-127 FEB 2006 http://dx.doi.org/10.1109/TSMCA.2005.854496 Comparing the researches on the weights obtaining methods in OWA operator, such as the quantifier guided aggregation [2], [37], exponential smoothing [14], learning [25], especially the maximum entropy method [16], [28], [38], [A13], the WOWA aggregation methods are relatively rare [30], [40]. (pages 118-119) A13-c65 Nasibov, E.N., Aggregation of fuzzy information on the basis of decompositional representation, CYBERNETICS AND SYSTEMS ANALYSIS, 41 (2), pp. 309-318. 2005 http://dx.doi.org/10.1007/s10559-005-0065-0 A13-c64 Xu ZS, An overview of methods for determining OWA weights, INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, 20 (8): 843-865 AUG 2005 http://dx.doi.org/10.1002/int.20097 Fullér and Majlender [A13] used the method of Lagrange multipliers to solve problem 12 analytically and got the following: 1. If n = 2 then w1 = α and w2 = 1 − α. 2. If α = 0 or α = 1 then the associated weighting vectors are uniquely defined as w = (0, 0, . . . , , 1)T and w = (1, 0, . . . , 0)T respectively, with value of dispersion zero. 20 3. If n ≥ 3 and 0 < α < 1 then q n−1 wj = w1n−j wnj−1 ((n − 1)α − n)w1 + 1 (n − 1)α + 1 − nw1 w1 [(n − 1)α + 1 − nw1 ]n = ((n − 1)α)n−1 [((n − 1)α − n)w1 + 1] wn = (15) (16) (17) Solving Equations 15-17, the optimal OWA weights can be determined. (page 847) A13-c63 Lan H, Ding Y, Hong J, Decision support system for rapid prototyping process selection through integration of fuzzy synthetic evaluation and an expert system INTERNATIONAL JOURNAL OF PRODUCTION RESEARCH, 43 (1): 169-194 JAN 1 2005 http://dx.doi.org/10.1080/00207540410001733922 A13-c60 Arfi B, Fuzzy decision making in politics: A linguistic fuzzy-set approach (LFSA), POLITICAL ANALYSIS, 13 (1): 23-56 WIN 2005 http://dx.doi.org/10.1093/pan/mpi002 A13-c59 Ying-Ming Wang, Celik Parkan, A minimax disparity approach for obtaining OWA operator weights, INFORMATION SCIENCES, 175(2005) 20-29. 2005 http://dx.doi.org/10.1016/j.ins.2004.09.003 Fullér and Majlender [A13] showed that the maximum entropy model could be transformed into a polynomial equation that can be solved analytically. (page 21) A13-c58 Liu Xinwang, Preference Representation with Geometric OWA Operator, SYSTEMS ENGINEERING, 22(2004), number 9, pp. 8286 (in Chinese). 2004 http://d.wanfangdata.com.cn/Periodical_xtgc200409019.aspx A13-c57 Liu Xinwang, Three methods for generating monotonic OWA operator weights with given orness level, JOURNAL OF SOUTHEAST UNIVERSITY (ENGLISH EDITION), Vol. 20 No. 3, pp. 369-373. 2004 http://www.wanfangdata.com.cn/qikan/periodical.articles/dndxxb-e/dndx2004/0403/040321.htm A13-c56 Liu XW, On the methods of decision making under uncertainty with probability information, INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, 19(12): 1217-1238 DEC 2004 http://dx.doi.org/10.1002/int.20045 21 The maximum entropy OWA operator was first suggested by O’Hagan [11] and later was discussed by Filev and Yager [10] and Fullér and Majlender [A13]. (page 1225) A13-c55 Xinwang Liu and Lianghua Chen, On the properties of parametric geometric OWA operator, INTERNATIONAL JOURNAL OF APPROXIMATE REASONING, 35 pp. 163-178. 2004 http://dx.doi.org/10.1016/j.ijar.2003.09.001 Recently, Fullér and Majlender [A13] proposed another method to generate MEOWA weights, the method get the weights by solving a polynomial equation. (page 164) A13-c54 Chiclana F, Herrera-Viedma E, Herrera F, et al. Induced ordered weighted geometric operators and their use in the aggregation of multiplicative preference relations INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, 19(3): 233-255 MAR 2004 http://dx.doi.org/10.1002/int.10172 A13-c53 Beliakov G., How to build aggregation operators from data INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, 18(8): 903-923 AUG 2003 http://dx.doi.org/10.1002/int.10120 A13-c52 Xu, Z., Da, Q. Approaches to obtaining the weights of the ordered weighted aggregation operators Dongnan Daxue Xuebao (Ziran Kexue Ban)/Journal of Southeast University (Natural Science Edition), 33 (1), pp. 94-96. 2003 in proceedings and in edited volumes A13-c32 Gleb Beliakov, Optimization and Aggregation Functions, in: Weldon A Lodwick, Janusz Kacprzyk eds., Fuzzy Optimization: Recent Advances and Applications, Studies in Fuzziness and Soft Computing vol. 254/2010, Springer [ISBN 978-3-642-13934-5], pp. 77-108. 2010 http://dx.doi.org/10.1007/978-3-642-13935-2_4 A13-c31 Jia-Wen Wang and Jing-Wen Chang, A Fusion Approach for Multicriteria Evaluation, in: Ngoc Thanh Nguyen, Radosław Katarzyniak, and Shyi-Ming Chen eds., Advances in Intelligent Information and Database Systems, Studies in Computational Intelligence, vol. 283/2010, Springer Berlin / Heidelberg, [ISBN 978-3-642-12089-3], pp. 349358. 2010 22 http://dx.doi.org/10.1007/978-3-642-12090-9_30 A13-c30 X. Liu, On the methods of OWA operator determination with different dimensional instantiations, 6th International Conference on Fuzzy Systems and Knowledge Discovery, FSKD 2009, 14-16 August 2009, Tianjin, China, Volume 7, [ISBN 978-076953735-1], Article number 5359982, pp. 200-204. 2009 http://dx.doi.org/10.1109/FSKD.2009.312 A13-c29 Mostafa Keikha and Fabio Crestani, Effectiveness of Aggregation Methods in Blog Distillation, in: Troels Andreasen, Ronald R.Yager, Henrik Bulskov, Henning Christiansen, Henrik Legind Larsen eds., Flexible Query Answering Systems, Lecture Notes in Computer Science, vol. 5822/2009, Springer, [ISBN 978-3-642-04956-9], pp. 157167. 2009 http://dx.doi.org/10.1007/978-3-642-04957-6_14 A13-c28 Matteo Brunelli, Michele Fedrizzi, A Fuzzy Approach to Social Network Analysis, Social Network Analysis and Mining, International Conference on Advances in Social Network Analysis and Mining, Athens, Greece, July 20-July 22, [ISBN 978-0-7695-3689-7] pp. 225-230. 2009 http://doi.ieeecomputersociety.org/10.1109/ASONAM.2009.72 A13-c27 B. Fonooni; S. Moghadam, Applying induced aggregation operator in designing intelligent monitoring system for financial market, IEEE Symposium on Computational Intelligence for Financial Engineering, March 30, 2009 - April 2, 2009, Nashville, TN, [ISBN 978-14244-2774-1], pp. 80-84. 2009 http://dx.doi.org/10.1109/CIFER.2009.4937506 A13-c26 Benjamin Fonooni, Seied Javad Mousavi Moghadam, Automated trading based on uncertain OWA in financial markets, in: Proceedings of the 10th WSEAS international conference on Mathematics and computers in business and economics, Prague, Czech Republic, pp. 21-25. 2009 A13-c25 Victor M. Vergara, Shan Xia, and Thomas P. Caudell, Information fusion across expert groups with dependent and independent components, Multisensor, Multisource Information Fusion: Architectures, Algorithms, and Applications 2009, Proceedings of SPIE - The International Society for Optical Engineering, 7345, art. no. 73450C. http://dx.doi.org/10.1117/12.818669 23 A13-c24 Ming Li, Yan-Tao Zheng, Shou-Xun Lin, Yong-Dong Zhang and Tat-Seng Chua, Multimedia evidence fusion for video concept detection via OWA operator, Lecture Notes in Computer Science, vol. 5371/2008, pp. 208-216. 2008 http://dx.doi.org/10.1007/978-3-540-92892-8_21 We therefore formulate the multi-modal fusion as an information aggregation task in the framework of group decision making (GMD) problem. Specifically, we employ the Ordered Weighted Average (OWA) operator to aggregate the group of decisions by uni-modal detectors, as it has been reported to be an effective solution for GMD problem [A13]. (page 209) In Dispersion Maxspace represented by Eqn (8), the weights of different Orness are given with maximal dispersion which means most individual criteria are being used in the aggregation that gives more robustness [A13]. (page 212) A13-c23 Yao-Hsien Chen, Jing-Wei Liu, Ching-Huse Cheng, Intelligent Preference Selection for Evaluating Studentsapos; Learning Achievement Third International Conference on Convergence and Hybrid Information Technology, Volume 2, Article number 4682412, 11-13 November, 2008, pp. 1214 -1219. 2008 http://dx.doi.org/10.1109/ICCIT.2008.410 A13-c22 K.-M. Björk, Obtaining minimum variability OWA operators under a fuzzy level of orness, ICINCO 2008 - Proceedings of the 5th International Conference on Informatics in Control, Automation and Robotics ICSO, Volume ICSO, pp. 114-119. 2008 A13-c21 Xinwang Liu; Xiaoguang Yang; Yong Fang, The relationships between two kinds of OWA operator determination methods, IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2008), pp. 264270, 1-6 June 2008 http://dx.doi.org/10.1109/FUZZY.2008.4630375 A13-c20 B. Fonooni; S. J. Moghadam, Designing financial market intelligent monitoring system based on OWA, in: Proceedings of the WSEAS international Conference on Applied Computing, (Istanbul, Turkey, May 27 - 30, 2008). M. Demiralp, W. B. Mikhael, A. A. Caballero, N. Abatzoglou, M. N. Tabrizi, R. Leandre, M. I. Garcia-Planas, and R. 24 S. Choras, Eds. Mathematics And Computers In Science And Engineering. World Scientific and Engineering Academy and Society (WSEAS), Stevens Point, Wisconsin, pp. 35-39. 2008 A13-c19 Calvo, T., Beliakov, G., Identification of weights in aggregation operators, in: Bustince, Humberto; Herrera, Francisco; Montero, Javier (Eds.) Fuzzy Sets and Their Extensions: Representation, Aggregation and Models Intelligent Systems from Decision Making to Data Mining, Web Intelligence and Computer Vision Series: Studies in Fuzziness and Soft Computing , Vol. 220 Springer, [ISBN: 978-3-540-73722-3] 2008, pp. 145-162. 2008 http://dx.doi.org/10.1007/978-3-540-73723-0_8 A13-c18 B. Llamazares, J.L. Garcia-Lapresta, Extension of some voting systems to the field of gradual preferences, in: Bustince, Humberto; Herrera, Francisco; Montero, Javier (Eds.) Fuzzy Sets and Their Extensions: Representation, Aggregation and Models Intelligent Systems from Decision Making to Data Mining, Web Intelligence and Computer Vision Series: Studies in Fuzziness and Soft Computing , Vol. 220 Springer, [ISBN: 978-3-540-73722-3] 2008, pp. 297-315. 2008 http://dx.doi.org/10.1007/978-3-540-73723-0_15 A13-c17 Benjamin Fonooni, Rational-Emotional Agent Decision Making Algorithm Design with OWA, 19th IEEE International Conference on Tools with Artificial Intelligence, October 29-31, 2007, Paris, France, pp. 63-66. 2007 http://doi.ieeecomputersociety.org/10.1109/ICTAI.2007.123 A13-c16 Cheng, Ching-Huse; Liu, Jing-Wei; Wu, Ming-Chang, OWA Based Information Fusion Techniques for Classification Problem, 2007 International Conference on Machine Learning and Cybernetics, 19-22 Aug. 2007, vol.3, [ISBN 978-1-4244-0973-0 ], pp.1383-1388. 2007 http://dx.doi.org/10.1109/ICMLC.2007.4370360 Fullér and Majlender [A13] transform Yager’s OWA equation to a polynomial equation by using Lagrange multipliers. According to their approach, the associated weighting vector can be obtained by (2) - (4). (page 1384) A13-c15 Wang, Jia-Wen; Cheng, Ching-Hsue, Information Fusion Technique for Weighted Time Series Model, International Conference on 25 Machine Learning and Cybernetics, 19-22 Aug. 2007, [ISBN 978-14244-0973-0 ], vol.4, pp.1860-1865. 2007 http://dx.doi.org/10.1109/ICMLC.2007.4370451 Fullér and Majlender use the method of Lagrange multipliers to transfer equation (7) to a polynomial equation, which can determine the optimal weighting vector. By their method, the associated weighting vector is easily obtained by (8)-(9) [A13]. (page 1861) A13-c14 Na Cai; Ming Li; Shouxun Lin; Yongdong Zhang; Sheng Tang; AP-Based Adaboost in High Level Feature Extraction at TRECVID, in: Proceedings of the 2nd International Conference on Pervasive Computing and Applications, (ICPCA 2007), 26-27 July 2007, pp. 194-198. 2007 http://dx.doi.org/10.1109/ICPCA.2007.4365438 A13-c13 Chang, Jing-Rong; Liao, Shu-Ying; Cheng, Ching-Hsue, Situational ME-LOWA Aggregation Model for Evaluating the Best Main Battle Tank, 2007 International Conference on Machine Learning and Cybernetics, 19-22 Aug. 2007, vol.4, pp.1866-1870. 2007 http://dx.doi.org/10.1109/ICMLC.2007.4370452 A13-c12 Ching-Huse Cheng, Jing-Wei Liu, OWA Rough Set to Forecast the Industrial Growth Rate, International Conference on Convergence Information Technology, 21-23 Nov. 2007, pp. 1862-1867. 2007 http://doi.ieeecomputersociety.org/10.1109/ICCIT.2007.233 A13-c11 Benjamin Fonooni, Behzad Moshiri and Caro Lucas, Applying Data Fusion in a Rational Decision Making with Emotional Regulation, in: 50 Years of Artificial Intelligence, Essays Dedicated to the 50th Anniversary of Artificial Intelligence, Lecture Notes in Computer Science, Volume 4850/2007, Springer, 2007, pp. 320-331. 2007 http://dx.doi.org/10.1007/978-3-540-77296-5_29 A13-c10 Beliakov, G., Pradera, A., Calvo, T., Other types of aggregation and additional properties, in: Aggregation Functions: A Guide for Practitioners, Studies in Fuzziness and Soft Computing, Vol. 221, [ISBN 978-3-540-73720-9], Springer, pp. 297-304. 2007 http://dx.doi.org/10.1007/978-3-540-73721-6_7 26 A13-c9 Eric Levrat, Jean Renaud, Christian Fonteix, Decision compromise modelling based on OWA operators, Ninth IFAC Symposium on Automated Systems Based on Human Skill and Knowledge, Automated Systems Based on Human Skill and Knowledge, May 22-24, 2006, [ISBN 978-3-902661-05-0], volume 9, part I. 2006 http://www.ifac-papersonline.net/Detailed/38831.html A13-c8 Yeh, Duen-Yian Cheng, Ching-Hsue Yio, Hwei-Wun, OWA and PCA integrated assessment model in software project, in: 2006 World Automation Congress, WAC’06, 24-26 July 2006, Budapest, Hungary, art. no. 4259935, pp. 1-6. 2006 http://dx.doi.org/10.1109/WAC.2006.376019 A13-c7 Zadrozny S, Kacprzyk J, On tuning OWA operators in a flexible querying interface, in: Flexible Query Answering Systems, 7th International Conference, FQAS 2006, LECTURE NOTES IN COMPUTER SCIENCE 4027: 97-108 2006 http://dx.doi.org/10.1007/11766254_9 Filev and Yager [11] simplified this optimization problem using the Lagrange multipliers method. Then the problem boils down to finding the root of a polynomial of degree m − 1. Fullér and Majlender [A13], assuming the same approach, proposed a simpler formulae for the weight vector W . (page 101) A13-c6 Xu ZS, Dependent OWA operators, in: Modeling Decisions for Artificial Intelligence, Third International Conference, MDAI 2006, LECTURE NOTES IN ARTIFICIAL INTELLIGENCE 3885: 172178 2006 http://dx.doi.org/10.1007/11681960_18 A13-c5 Troiano L, Yager RR On the relationship between the quantifier threshold and OWA operators, LECTURE NOTES IN ARTIFICIAL INTELLIGENCE 3885: 215-226 2006 http://dx.doi.org/10.1007/11681960_22 A13-c4 Ching-Hsue Cheng, Jing-Rong Chang, Tien-Hwa Ho, and An-Pin Chen, Evaluating the Airline Service Quality by Fuzzy OWA Operators in: Vincent Torra, Yasuo Narukawa, Sadaaki Miyamoto (Eds.): Proceedings of the Modeling Decisions for Artificial Intelligence: Second 27 International Conference, MDAI 2005, Tsukuba, Japan, July 25-27, 2005, LNAI 3558, Springer, pp. 77-88. 2005 http://dx.doi.org/10.1007/11526018_9 Fullér and Majlender [A13] used the method of Lagrange multipliers to transfer Yager’s OWA equation to a polynomial equation, which can determine the optimal weighting vector. By their method, the associated weighting vector is easily obtained by (5)-(7). q j−1 n−j ln wj = ln wn + ln w1 ⇒ wj = n−1 w1n−j wnj−1 n−1 n−1 ((n − 1)α − n)w1 + 1 and wn = (n − 1)α + 1 − nw1 then w1 [(n − 1)α + 1 − nw1 ]n = ((n − 1)α)n−1 [((n − 1)α − n)w1 + 1] (5) (6) (7) (pages 79-80) A13-c3 L. Troiano and R.R. Yager, A meaure of dispersion for OWA operators, in: Y. Liu, G. Chen and M. Ying eds., Proceedings of the Eleventh International Fuzzy systems Association World Congress, July 28-31, 2005, Beijing, China, 2005 Tsinghua University Press and Springer, [ISBN 7-302-11377-7] pp. 82-87. 2005 Entropy has been generally adopted as a measure of weight dispersion of the OWA operators. O’Hagan [2], in his ground braking work, suggests to select the vector that maximizes the entropy of OWA weights (ME-OWA). Analytical solutions to this problem have been proposed by Filev and Yager [3], and Fullér and Majlender [A13]. (page 82) A13-c2 Liu, X.-W., Chen, L.-H. The equivalence of maximum entropy OWA operator and geometric OWA operator, International Conference on Machine Learning and Cybernetics, 5, pp. 2673-2676. 2003 http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1259989 A13-c1 H.B. Mitchell, Data Mining Using a Probabilistic Weighted Ordered Weighted Average (PWOWA) Operator, in: Vicenc Torra ed., Information Fusion in Data Mining Series: Studies in Fuzziness and Soft Computing , Vol. 123, Springer, [ISBN: 978-3-540-00676-3] pp. 41-58. 2003 28