ELECTRE TRI 19 IRIS ELECTRE TRI ELECTRE TRI A = a 1 a 2 a m g = g 1 g 2 g n g j a i a i g j C = C 1 C 2 C q C 1 C

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1 JOURNAL OF MANAGEMENT SCIENCES IN CHINA Vol 16 No 2 Feb 蔡付龄, 廖貅武, 杨娜 针对多准则分类问题中决策者难以提供分类模型参数以及多人参与决策的复杂性, 给 出了基于案例信息的多准则群决策分类方法 该方法首先根据效用函数分类模型的基本性质 定义了案例信息的一致性, 根据决策者的不同决策权重定义了案例信息的可信度 然后, 建立整数规划模型从各决策者的案例信息中挑选出可信度最大的一致案例信息, 并利用该信息和 一致性条件对所有待评价方案分类 在分类过程中, 直接利用案例信息分类, 不需要设置分类边界参数 最后, 通过一个 MBA 项目分类的实例说明了方法的可行性与合理性 多准则分类 ; 群决策 ; 案例信息 ; 决策权重 ; 一致性 C934 A UTA- 5 DIS 12 MHDIS N - TOMIC 14 ELECTRE TRI Flow- Sort 17 ELECTRE TRI - C caifuling@ stu xjtu edu cn

2 ELECTRE TRI 19 IRIS ELECTRE TRI ELECTRE TRI A = a 1 a 2 a m g = g 1 g 2 g n g j a i a i g j C = C 1 C 2 C q C 1 C k k k 21 E = e 1 e s e L L λ = λ 1 λ s λ L L λ s = 1 λ s > 0 s = 1 A * 22 s A e s ABC e s a i A * s I si C I si g b 2 A B j * = min g j a i g * j = max g j a i a i A a i A b g j U a i = n w j u j g j a i j = 1

3 g j U a < g j g j min U a i U b U a < U b a a u j g j a i = g* j - g j a i i C k+1 A 1 b A C a < C b C a < U b A g * j - g j * C a 2 g j g j a i C p A a j C p +1 A p = 1 q g j u j g j a i = g j a i - g j * g * j - g j * w = w 1 w 2 w n w j 0 1 n w j = 1 w j = C a i a i C a < C b U a < U b A * s l i = min I si = max I si s s 1 A a a i a i A * s C a a A U a i C l i a i A a b AC a < C b U a < a i C U b A a i 1 A a 2 bel k i a i C k C a a A C k A = a 1 a i A bel k i = max λ s min λ s λ s s I si = k s I si = l i s I si = C a i = C k A a i C p A a j C p +1 A p = 1 q - 1 U a i < U a j k 2 bel k i 1 a b A C a < C b C b - C a = t 1 C a = C k C b = C k +l a i C p A a j C p+1 A p = 1 q U a i < U a j C a i = C p U a i < U a j min U a i < < a i C k+1 A 1 < C a j = C p bel k 1 i > 0 bel k 2 i > 0 k 1 < > 0 k k 1 k k bel k i 1 k 1 2 q e s1 e s2 a C k 1 C k 2 a C k 1 C k 2 C k 0

4 2 25 a i C l i C C k l i k q = 5 A a C 2 B a C 4 19 a C 3 0 a bel 0 C 3 C 1 C 5 1 C 1 C 5 a C 3 bel k i y k i 2 i a i A * k a C 确定群体的一致案例集 θ θ θ C a < C b U a < U b 1 d r a r a r C l r MLP 0 max bel 0 = ρ i bel k i y k i - i d i θ y k i k = 1 q 1 1 a r C l r i l i k 1 4 a U a i -U a r +ε M 1-y k i +M 1 - y k+1 r C a r > C h r r a i d r a r C h r a r A * s l r -1 k h r - 1 l i k a r U a i -U a r -d r M 1 -y l r i +My l r r a i a r A * s l i l r 1 3 C l r C h r 1 5 s t U a r -U a i -d r M 1-y h r i +My h r r a i 0 < U a 1 a A a r A * s l i h r < U a i - U a r < 1 M M > 1 ε 1 y k i 1 a i A * s 1 5 ε 1 = 0 01 n ρ A * s /ε 1 1 w j = 1 d i 0 y k i 0 1 i = 1 m j = 1 2 ε 1 k = l i l i y k i 0-1 y k i = 1 2 ρ A * s /ε 1 MLP 0 a i a i C k y * bel 0 y k i = 0 a i C k h i bel k y l i i = y l i +1 i = = y y k i i i = 0 a i i l i y * MLP 0 a i y k i bel k i y k i k l i + 1 l i ρ C h r a r C l r C h r 1 3 a r MLP 0 C l r C a r < C l r d r i

5 y' bel k i y k i i bel k i y k i y < * bel k i y k i y ' bel i y k i y * y = y * bel k i y k i y - ' i ρ d i bel 0 y' - bel 0 y * bel k i y k i y = * bel k i y k i y - y k ' i > y * A * s 1 - U a i < A * s > 0 y * MLP 0 MLP 1 2 MLP 0 y k i 0-1 y k i = 1 a i C k y k i = 0 a i C k MLP cen k -1 C k -1 C k a i C k 2 5 a i C k -1 C k d - ik a i cen k -1 1 a i cen k -1 d - ik = 0 A 2 6 a i C k C k +1 d + ik a i cen k 计算待分类方案的分类 a A * i cen k d + ik = MLP a C k A * i k = 1 2 q C k C a i a i a i C 1 C 2 MLP 1 min d = q-1 d + ik i a i A-A * k =1 + d - i k+1 U a i -U a r +ε 0 a i a r A * C a i = C k C a r = C k+1 k = 1 q U a i - U a r + ε M 1 - y k r a i A * C a i = C k-1 a r A - A * k =2 q 22 cen k s t U a r -cen k U a r -U a t +ε M 1 - y k r a t A * C a t = C k+1 a r A - A * k =1 q-1 23 cen k-1 U a = 1 i 2 a i C k A * + C k A * k = 1 q - 1 k = 1 q - 1 U a i a i C k+1 A * C k+1 A * d + rk +ε M 1-y k r a r A - A * U a r - d - rk M 1 - y k r a r A - A * k = 2 q n w j = 1 j =1 d ik + dik 0 y - k i 0 1 y k i =1 a i A - A a i 2 6 C 1 C 2 MLP 1 d - ik d + ik a i *

6 U * a MLP 1 1 Fig 1 The demonstration of the sorting results W * W * = w U a i -U a r +ε 0 a i C k A * a r C k+1 A * k = 1 q - 1 n w j = 1 3 a r a i w W * U a i U a r a i A * a r 3 a i A * a r MLP 1 C a i C a r MLP 1 w * d * a r C a r = C k MLP 1a t C k -1 A * U * a r > U * a t U * a i > U * a t a i C k -1 d ' rk = cen * k -1 - U * a r d ' ik -1 = U * a i - cen * k -1 + ε d ' rk j =1 > 0 w * d * C k U * a v U * a r cen * k -1 a v - d ' rk U * a v U * a i a i C k d ' rk 0 d ' r k -1 - d ' rk + ε > 0 a i C k -1 Fig 2 Sorting process for group decision-making C k -1 d + * ik -1 > 0 a i C k 0 w * d * A g = g 1 g 2 g n a i C k C a i C a r g j a i λ s 1 MLP 1 C = C 1 C 2 C q 1 e s 3 A * s A * 3 A * MLP 2 MLP 1 2 A * s MLP

7 A - A * 2 I 11 = 5 I 12 = 4 I 1 12 = 3 I 1 14 = 5 4 A * s = a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 12 a 14 1 a MBA = 0 4 MBA 1 30 MBA 5 A = a 1 a 2 a 30 C 1 = C 2 = C 3 = C 4 = C 5 MBA 3 g 1 = g 2 = g 3 = 3 e 1 e 2 e MLP 0 2 bel 5 1 = bel 4 2 = bel 5 14 = 0 2 bel 4 3 = bel 3 4 = bel 3 5 = 0 35 bel 2 6 = bel 2 7 = bel 1 8 = 0 4 bel 3 12 = 0 2 bel ε 1 =0 01 u 3 g 3 a i = g 3 a i MBA bel 1 29 = 0 4 bel 2 29 = bel 3 29 = bel 4 29 = 0 35 g 1 g 2 g 3 = u 1 g 1 a i = g 1 a i u 2 g 2 a i = g 2 a i MBA Table 1 The evaluation scores of the MBA programs No g 1 g 2 g 3 No g 1 g 2 g 3 1 LBS Insead MIT Sloan NYU Stern YSM NUK UNC KFlagler RSM UT Rotman SU GSB UCLA Anderson UCB Haas UO Said UC GSB UM Smith UI Tippie CM Tepper UM Ross CBS UP Wharton CU Johnson UR Simon DU Fuqua UV Darden EBS UWO Ivey HBS VU Owen IBS YU Schulich Table 2 The assignment examples given by the DMs e 1 e 2 e 3 a 1 a 2 a 12 a 14 a 3 a 4 a 5 a 29 a 6 a 7 a 8 a 12 a 29 C 5 C 4 C 3 C 5 C 4 C 3 C 3 C 4 C 2 C 2 C 1 C 4 C 1

8 2 29 A * k θ = 1 ε = 0 01 M =2ρ = = ε MLP 0 Lingo 9 0 y 5 1 = y 4 2 = y 4 3 = y 3 5 = y 2 6 = y 2 7 = y 1 8 = y 4 12 = y 5 14 = y 1 29 = 1 y 3 4 = y 3 12 = y 2 29 = y 3 29 = y 4 29 = 0 a 4 a 12 a 29 a 4 a 12 a 29 e 3 e 3 e 3 3 A * A * = a 1 a 2 a 3 a 5 a 6 a 7 MLP 1 a 8 a 12 a 14 a 29 C a 1 =C a 14 = 3 a 8 a 9 a 13 a 19 a 23 a 28 a 29 C 1 C 5 C a 2 = C a 3 = C a 12 = C 4 C a 5 = C 3 a 6 a 7 a 11 a 26 C 2 a 5 a 27 a 30 C 3 C a 6 = C a 7 = C 2 C a 8 = C a 29 = C 1 a 2 a 3 a 4 a 12 a 15 a 16 a 17 a 18 a 21 a 24 C 4 a 1 a 10 a 14 a 20 a 22 a 25 C 5 C 1 C 2 C 3 C 4 C 5 C 1 C 2 C 3 C 4 C 5 a 1 a 16 a 2 a 17 a 3 a 18 a 4 a 19 a 5 a 20 a 6 a 21 a 7 a 22 a 8 a 23 a 9 a 24 a 10 a 25 a 11 a 26 a 12 a 27 a 13 a 28 a 14 a 29 a 15 a 30 3 Fig 3 The group assignment results of the alternatives e 1 10 Table 3 The assignment examples given by the DM a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 C 5 C 4 C 4 C 3 C 3 C 2 C 2 C 1 10

9 a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10 a 11 C 1 C 2 C 3 C 4 C 5 a 12 a 13 a 14 a 15 a 16 a 17 a 18 a 19 a 20 a 21 a 22 a 23 a 24 a 25 a 26 a 27 a 28 a 29 a Fig 4 Comparison of sorting results Table 4 Comparisons between multi-criteria group sorting methods

10 UTADIS 1 Zopounidis C Doumpos M Multicriteria classification and sorting methods A literature review J European Journal of Operational Research Zopounidis C Doumpos M A multi-criteria decision aid methodology for the assessment of country risk J European Research on Management and Business Economics Zopounidis C Doumpos M A preference disaggregation decision support system for financial classification problems J European Journal of Operational research J Yang Haijun Tai Lei Predicting financial distress of listed corporations based on fuzzy support vector machine J Journal of Management Sciences in China in Chinese 5 Chen Y Li K W Kilgour D M et al A case-based distance model for multiple criteria ABC analysis J Computers and Operations Research Chen Y Hipel K W Kilgour D M Multiple criteria sorting using case-based distance models with an application in water resources management J IEEE Transactions on Systems Man and Cybernetics Vetschera R Chen Y Keith W et al Robustness and information levels in case-based multiple criteria sorting J European journal of Operational research J Cai Fuling Liao Xiuwu Multi-criteria sorting method with uncertainty in assignment examples J Systems Engineering- Theory & Practice in Chinese 9 Almeida-Dias J Figueira J R Roy B ELECTRE TRI-C A multiple criteria sorting method based on characteristic reference actions J European Journal of Operational Research K ksalan M Mousseau V zpeynirci et al A new outranking-based approach for assigning alternatives to ordered classes J Naval Research Logistics J Liao Xiuwu Li Yuan Lei Hongzhen Model and approach for determining the compromise weight of multi-attribute group decision-making problem J Journal of Management Sciences in China in Chinese 12 Zopounidis C Doumpos M A multicriteria decision aid methodology for sorting decision problems The case of financial

11 distress J Computational Economics Doumpos M Zanakis S Zopounidis C Multicriteria preference disaggregation for classification problems with an application to global investing risk J Decision Sciences Massaglia R Ostanello A N-tomic A support system for multicriteria segmentation problems M / /Multiple Criteria Decision Support Berlin Springer Verlag Mousseau V Slowinski R Inferring an ELECTRE TRI model from assignment examples J Journal of Global Optimization Dias L Mousseau V Figueira J et al An aggregation /disaggregation approach to obtain robust conclusion with ELECTRE TRI J European Journal of Operational Research Nemery P Lamboray C FlowSort A flow-based sorting method with limiting or central profiles J TOP Dias L C Climaco J N ELECTRE TRI for groups witmprecise information on parameter values J Group Decision and Negotiation Damart S Dias L C Mousseau V Supporting groups in sorting decisions Methodology and use of a multi-criteria aggregation /disaggregation DSS J Decision Support Systems Jabeur K Martel J M An ordinal sorting method for group decision-making J European Journal of Operational Research Greco S Mousseau V Slowinski R The possible and the necessary for multiple criteria group decision J Lecture Notes in Computer Science Rocha C Dias L An algorithm for ordinal sorting based on ELECTRE with categories defined by examples J Journal of Global Optimization Bous G Fortemps P Glineur F et al ACUTA A novel method for eliciting additive value functions on the basis of holistic preference statements J European Journal of Operational Research Angilella S Greco S Matarazzo B Non-additive robust ordinal regression A multiple criteria decision model based on the Choquet integral J European Journal of Operational Research k- J Zhang Ling Zhou Dequn Multiple attributes decision making based on k-additive fuzzy measures J Journal of Management Sciences in China in Chinese Multi-criteria sorting method based on assignment examples in the group decision context CAI Fu-ling LIAO Xiu-wu YANG Na School of Management The Key Lab of Education for Process Control & Efficiency Xi an Jiaotong University Xi an China Abstract In the multi-criteria group sorting decision-making problems the decision makers are often reluctant to provide parameter values of the group sorting model and the decision complexity increases as multiple decision makers are involved In this paper a group sorting method based on assignment examples is proposed Firstly the definition of consistency among assignment examples is given according to the property of the utility-function-based sorting methods and the definition of belief degree on assignment examples is given to reflect the different decision priorities of the decision makers Then a mixed integer linear programming model is built to find the consistent assignment example set with the biggest belief degree Based on the chosen example set and the definition of consistency all alternatives are assorted through solving another mixed integer linear programming model During the sorting process the assignment examples are directly utilized for sorting sorting boundaries are no longer needed At last a sorting example on MBA program is provided to demonstrate the feasibility and rationality of the proposed method Key words multi-criteria sorting group decision-making assignment examples decision priority consistency