Intuitionistic fuzzy Choquet aggregation operator based on Einstein operation laws

Transcription

1 Scientia Iranica E (3 (6, 9{ Sharif University of Technology Scientia Iranica Transactions E: Industrial Engineering Intuitionistic fuzzy Choquet aggregation operator based on Einstein operation laws Deian Yu School of Information, Zheiang University of Finance and Economics, Hangzhou, 38, China. Received 9 October ; received in revised form 8 ugust ; accepted 4 May 3 KEYWORDS Multi-criteria decision making; Intuitionistic fuzzy set; ggregation operator; Choquet integral; Einstein operation laws. bstract. In this paper, we study the intuitionistic fuzzy information aggregation operators based on Einstein operation laws under the condition in which the aggregated arguments are independent. The Einstein-based Intuitionistic Fuzzy Choquet veraging (EIFC operator is proposed. Furthermore, the relationship between the EIFC operator and the IFC operator is investigated. The desirable properties of the EIFC operator, such as boundeness, monotonicity, shift-invariance and homogeneity are discussed. multi-criteria decision making approach, based on the EIFC operator is proposed under intuitionistic fuzzy environment. comparative example is given for demonstrating the applicability of the proposed decision procedure and for nding links with other operatorsbased decision approach. c 3 Sharif University of Technology. ll rights reserved.. Introduction Intuitionistic Fuzzy Sets (IFS [-], introduced by tanassov, is a generalization of the concept of fuzzy set. IFS is characterized by three functions, expressing degrees of membership, non-membership and indeterminacy, so it is more powerful to deal with uncertainty and vagueness in real applications than type- fuzzy set [3-4], type-fuzzy set [3], fuzzy multiset [5-6] and hesitant fuzzy set [7-8] which only consider the membership degree. For example, if a boy wants to nd a girlfriend and evaluates the girl from ten aspects, there are six aspects which satises the boy, three aspects do not satisfy, and he is uncertain with one aspect of the girl. In such a case, other types of fuzzy set can only reect the satised aspects which lose some uncertain information, while intuitionistic fuzzy set can describe all the satised, unsatised and uncertain information. Because of its appearance, IFS has attracted much attention [9-34]. Research on the intuitionistic fuzzy information *. address: yudeian6@6.com aggregation method is one of the hot topics of the IFS theory. Based on the famous OW operator [35-36] and the GOW operator [37] proposed by merican scholar Yager, many extended operators have been appeared. For example, Xu [5] proposed the Intuitionistic Fuzzy Weighted veraging (IFW operator, ordered IFW (IFOW operator and the intuitionistic hybrid aggregation (IFH operator. From the geometric point of view, Xu and Yager [38] introduced the Intuitionistic Fuzzy Weighted Geometric (IFWG operator, ordered IFWG (IFOWG operator and the Intuitionistic Fuzzy Hybrid Geometric (IFHG operator. Zhao et al. [39] proposed the generalized forms of the IFW, IFOW and IFH operators, and they proofed that the operators proposed by Xu [5] are special cases of the operators. The above aggregation operators for intuitionistic fuzzy information are under the condition in which the aggregated arguments are independent. However, it cannot meet the requirement of real situations. Choquet integral [4,4] is a powerful tool to deal with this situation, based on which Xu [4], Tan and Chen [] and Tan [43] developed some intuitionistic fuzzy Choquet operators such as

2 Deian Yu/Scientia Iranica, Transactions E: Industrial Engineering (3 9{ the Intuitionistic Fuzzy Choquet verage (IFC operator and the Intuitionistic Fuzzy Choquet Geometric (IFCG operator. Tan and Chen [] proposed the induced intuitionistic fuzzy Choquet integral operator and applied it to decision making. The prominent characteristic of those operators is that they cannot only adapt to the intuitionistic fuzzy environment, but also reect the interrelationship of the individual criteria. It needs to be pointed out that the basic operational laws of IFS for the above aggregation operators are the algebraic operational laws. Einstein product and Einstein sum are good alternatives for the algebraic product and algebraic sum, respectively [44]. The purpose of this paper is to investigate intuitionistic fuzzy information aggregation methods based on the Einstein product and Einstein sum under the assumption that the aggregated arguments are correlative. To do this, the remainder of this paper is constructed as follows: Section briey reviews some basic concepts. In Section 3, we propose the Einstein-based Intuitionistic Fuzzy Choquet veraging (EIFC operator; the desirable properties of the EIFC are also studied in this section. n approach to multi-criteria decision making based on the proposed operator is proposed in Section 4; a comparative example is also illustrated in this section. S ection 5 gives some conclusion remarks.. Some basic concepts s a generalization of fuzzy set, Intuitionistic Fuzzy Set (IFS assigns to each element a membership degree and a non-membership degree. tanassov [] gave the denition of Intuitionistic Fuzzy Set (IFS as follows. Denition. If a set X be xed, the concept of Intuitionistic Fuzzy Set (IFS on X is dened as follows: f< x; (x; v (x > x X g ; ( where the functions (x and v (x denote the degrees of membership and non-membership of the element x X to the set, respectively, with the condition that (x, v (x and (x+v (x. For convenience, Xu [5] named ( ; v an Intuitionistic Fuzzy Value (IFV. In this paper, we let V be the set of all IFVs. For three IFVs,, V, some Einstein operational laws were given as follows [44]:. " + + ; v v +( v, ( v,. " +( ( ; v +v +v v 3. ( + ; (+v ( v (+v +( v, 4. (+ ( v (+ +( ; ( v +v. Chen and Tan [45] introduced the score function s( v to get the score of, then Hong and Choi [46] dened the accuracy function h( + v to evaluate the accuracy degree of. Based on the score function s and the accuracy function h, Xu and Yager [38] gave an order relation between two IFVs and :. If s( < s(, then < ;. If s( s(, then: i If h( h(, then ; ii If h( < h(, then <. Let (fx i g (i ; ; ; n be the weights of the elements x i X (i ; ; ; n where is a fuzzy measure; Sugeno [47], Wang and Klir [48] and Denneberg [49] dened a fuzzy measure as follows. Denition. fuzzy measure on the set X is a function : (X [; ] satisfying the following axioms:. (;, (X ;. B C implies (B (C, for all B; C X; 3. (B[C (B+(C+ (B(C, for all B; C X and B \ C ;, where >. Based on Denition, Xu [4], Tan and Chen [] and Tan [43] dened Intuitionistic Fuzzy Choquet veraging (IFC and Intuitionistic Fuzzy Choquet Geometric (IFCG operators as follows. Denition 3. Let be a fuzzy measure on X, and (x ( (x ; v (x ( ; ; ; n be a collection of intuitionistic fuzzy sets, then: IFC((x ; (x ; ; (x n n (B ( B ( (x ( ( ( (B ( B ( ; (v ( (B ( B ( ; ( IFCG((x ; (x ; ; (x n n (x ( (B ( B ( ( ( (B ( B ( ; ( v ( (B ( B ( ; (3

3 Deian Yu/Scientia Iranica, Transactions E: Industrial Engineering (3 9{ are called the Intuitionistic Fuzzy Choquet veraging (IFC and Intuitionistic Fuzzy Choquet Geometric (IFCG operators, respectively, where ((; (; ; (n is a permutation of (; ; ; n, such that (x ( (x ( (x (n, B (k fx ( kg, for k, and B ( ;. 3. Intuitionistic fuzzy Choquet operator based on Einstein operation laws In this section, we shall investigate the intuitionistic fuzzy information aggregation operator combined with Choquet integral. Based on the Einstein operational laws described by Denition, we give the denition of the Einstein-based Intuitionistic Fuzzy Choquet veraging (EIFC operator as follows: Denition 4. Let be a fuzzy measure on X and ( ; ; ; n be a collection of IFVs, an Einstein-based Intuitionistic Fuzzy Choquet veraging (EIFC operator is a mapping V n V, and: E(C Z d EIFC((x ; (x ; ; (x n n (B ( B ( (x ( ; (4 where (C R d denotes the Choquet integral and ((; (; ; (n is a permutation of (; ; ; n, such that (x ( (x ( (x (n, B (k fx ( kg, for k, and B ( ;. Based on the operational laws of the IFVs described in Section, we can derive Theorem easily. Theorem. Let be a fuzzy measure on X, ( ; v ( ; ; ; n be a collection of IFVs and ( be the th largest of them, then their aggregated value by using the EIFC operator is also an IFVs, and is dened in Eq. (5 which is shown in Box I. Proof. The proof including Eqs. (6 to (3 is shown in Box II. It should be noted that the proof of Theorem was done by many references such as [5,39-44]. In order to analyze the relationship between the EIFC and IFC operators proposed by Tan and Chen [] and Xu [4], we introduce the following lemma [5]. Lemma P. Let x >, >, ; ; ; n and n, then: x x ; (4 with equality if and only if x x x n. Theorem. Let be a fuzzy measure on X, ( ; v ( ; ; ; n be a collection of IFVs and ( be the th largest of them, then: EIFC( ; ; ; n IFC( ; ; ; n : (5 Proof. On P one hand, since (B ( B ( for n all and (B ( B (, then based on Lemma, we have: + ( (B( B ( + (B( B ( ( + + (B ( B ( + ( (B ( B ( ( (B ( B ( + ( (B ( B ( ( (B ( B ( EIFC ( ; ; ; n Q n (+ ( (B ( B ( Q n ( ( (B ( B ( Q n (+ ( (B ( B ( + Q n ( ( (B ( B ( ; Q n v (B ( B ( ( Q n ( v ( (B ( B ( + Q n v (B ( B ( : (5 ( Box I.

4 Deian Yu/Scientia Iranica, Transactions E: Industrial Engineering (3 9{ EIFC ( ; ; ; n by using mathematical induction on n: For n : since: (B ( B ( (x ( Q n (+ ( (B ( B ( Q n ( ( (B ( B ( Q n (+ ( (B ( B ( + Q n ( ( (B ( B ( ; Q n v (B ( B ( ( Q n ( v ( (B ( B ( + Q n v (B ( B ( (+ ( (B ( B ( ( ( (B ( B ( (+ ( (B ( B ( +( ( (B ( B ( ; (B ( B ( (x ( Then: (+ ( (B ( B ( ( ( (B ( B ( (+ ( (B ( B ( +( ( (B ( B ( ; ( ; (6 v (B ( B ( ( ( v ( (B ( B ( (B +v ( B ( ; (7 ( v (B ( B ( ( ( v ( (B ( B ( (B +v ( B ( : (8 ( (B ( B ( (x ( (B ( B ( (x ( (+ ( (B ( B ( ( ( (B ( B ( (+ ( (B ( B ( +( ( (B ( B ( + (+ ( (B ( B ( ( ( (B ( B ( (+ ( (B ( B ( +( ( (B ( B ( + (+ ( (B ( B ( ( ( (B ( B ( (+ ( (B ( B ( ( ( (B ( B ( (+ ( (B ( B ( +( ( (B ( B ( (+ ( (B ( B ( +( ( (B ( B ( + v (B ( B ( ( ( v ( (B ( B ( (B +v ( B ( ( (B ( B ( ( ( v ( (B ( B ( (B +v ( B ( ( v (B ( B ( ( ( v ( (B ( B ( (B +v ( B ( ( v ( (B ( B ( (B +v ( ( v (B ( ( B ( B ( C (9 Q (+ ( (B ( B ( Q ( ( (B ( B ( Q (+ ( (B ( B ( + Q ( ( (B ( B ( ; Q v (B ( B ( ( Q ( v ( (B ( B ( + Q v (B ( B (. ( ( + + (B ( B ( ( (B ( B ( ( (B ( B ( : (6 Therefore, Eq. (7 is given in Box III. On the other hand, since: Box II. Continued on the next page. ( v ( (B ( B ( + + v(b ( B ( ( (B ( B( ( v ( (B ( B ( v ( ; (8 then Eq. (9 is given in Box IV. Let EIFC( ; ;

5 Deian Yu/Scientia Iranica, Transactions E: Industrial Engineering (3 9{ 3 If Eq. (6 holds for n k, that is: k Q k (B (+ ( (B ( B ( Q k ( ( (B ( B ( ( B ( (x ( Q k (+ ( (B ( B ( + Q k ( ( (B ( B ( ; then, when n k +, by the operational laws of IFVs, we have: k+ (B ( B ( (x ( k Q k v (B ( B ( ( Q k ( v ( (B ( B ( + Q k v (B ( B ( ; ( ( (B ( B ( (x ( (B (k+ B (k (x (k+ Q k+ (+ ( (B ( B ( Q k+ ( ( (B ( B ( Q k+ (+ ( (B ( B ( + Q k+ ( ( (B ( B ( ; Q k+ v(b ( B ( ( ( Q k+ ( v ( (B ( B ( + Q k+ v(b ( B ( ; ( i.e. Eq. (6 holds for n k +. Thus, Eq. (6 holds for all n. Then: Q n (+ ( EIFC ( ; ; (B ( B ( Q n ( ( (B ( B ( ; n Q n (+ ( (B ( B ( + Q n ( ( (B ( B ( ; Q n v (B ( B ( ( Q n ( v ( (B ( B ( + Q n v (B ( B ( ; (3 ( which completes the proof of Theorem. Box II. Continued. Q n +( (B( B ( Q n ( (B( B ( Q n +( (B( B ( + Q n ( (B( B ( Q n ( (B( B ( Q n +( (B( B ( + Q n ( (B( B ( Q n ( (B( B ( ; (7 where the equality holds if and only if ( ( ; ; ; n are equal. Box III. ; n ( ; v and IFC( ; ; ; n ( ; v, then Eq. (9 can be transformed to: ; and v v : ( Based on Eq. (, we have: s( v v s( : ( If s( < s( +, then: EIFC( ; ; ; n <IFC( ; ; ; n : ( If s( v v s(, i.e. Eqs. (3 to (5 are given in Box V. Therefore: h( + v h ( + v : (6 Thus, is follows that: EIFC( ; ; ; n IFC( ; ; ; n : (7 From Eqs. ( and (7, we know that Eq. (5 always holds.

6 4 Deian Yu/Scientia Iranica, Transactions E: Industrial Engineering (3 9{ Q n v (B ( B ( ( Q n ( v ( (B ( B ( + Q n v (B ( B ( Q n v(b ( B ( ( : (9 ( Box IV. Q n (+ ( (B ( B ( Q n ( ( (B ( B ( Q n (+ ( (B ( B ( + Q n ( ( (B ( B ( Q n v (B ( B ( ( Q n ( v ( (B ( B ( + Q n v (B ( B ( ( Q n ( (B( B ( Q n v(b ( B ( ( ; ; ; n (3 Then we have: Q n (+ ( (B ( B ( Q n ( ( (B ( B ( Q n (+ ( (B ( B ( + Q n ( ( (B ( B ( Q n ( (B( B ( (4 Q n v (B ( B ( Q ( n Q n ( v ( (B ( B ( + Q n v (B ( B ( v(b ( B ( ( (5 ( Box V. Theorem 3 (Idempotency. Let be a fuzzy measure on X, ( ; v ( ; ; ; n be a collection of IFVs and ( be the th largest of them. If all ( ; ; ; n are equal, i.e., for all, then: EIFC( ; ; ; n : (8 Proof. By Denition 4, we have: EIFC( ; ; ; n n (B ( B ( (x ( (B ( B ( (x ( (B ( B ( (x ( (B (n B (n (x (n (B ( B ( (B ( B ( (B (n B n X (B ( B ( (B (n B ( : (9 Theorem 4 (Boundeness. Let be a fuzzy measure on X, ( ; v ( ; ; ; n be a collection of IFVs and ( be the th largest of them, and also let: + Then: min( ( ; max(v ( max( ( ; min(v ( EIFC ( ; ; ; n + : (3 Proof. Let f(x +x x, x [; ], then f (x (+x <, i.e. f(x is a decreasing function. Since, min( ( max( ( for all, then: i.e.: f(max( ( f( ( f(min( ( ; (3 max( ( + max( ( ( ( min( + ( ( ( + min( ( ; ; ; ; n: (3 ; :

7 Deian Yu/Scientia Iranica, Transactions E: Industrial Engineering (3 9{ 5 Since (B ( B ( for all, then we Thus: max + max ( ( ( ( ( ( + ( min ( ( + min ( ( (B ( B ( (B( B ( (B ( B ( ; ; ; max ( ( + max( ( max + ( ( + ( min ( ( + min( ( ( ( ( ( np (B ( B ( (B( B ( (B ( B ( (B ( B ( (B( ( ( B ( + ( ( min ( ( + min max( ( ( ( + max ( ( np (B ( B ( (B( ( ( B ( + ( ( ; ; (34 ; (35 min ( ( + min ( ( ; (36 + max( ( Y (B( B ( n ( ( + + ( ( + min ( ( ; (37 + min( ( + Q n (( +( ( (B( B ( + max( ( ; (38 + min ( ( + Q n (( +( ( (B( B ( + max( ( ; (39 min ( ( + Q n (( +( ( (B( B ( max( ( : (4 nd Eq. (4 is shown in Box VI. Let g(y y y, y (; ], then g (y y <, i.e. g(y is a decreasing function. Since min(v ( v max(v (, for all, then: f i.e.: max (v ( max(v ( max(v ( f v ( f v ( v ( min min (v ( (v ( min (v ( ; (4 ; ; ; n: (43 ;

8 6 Deian Yu/Scientia Iranica, Transactions E: Industrial Engineering (3 9{ min ( ( Q n (+ ( (B ( B ( Q n ( ( (B ( B ( Q n (+ ( (B ( B ( + Q n ( ( (B ( B ( max ( ( : (4 Box VI. Since (B ( B ( for all, then we max max (v ( (v ( (B ( B ( v ( v ( min (v ( min (v max max max (v ( max(v (B ( B ( (B ( B ( v ( v ( (B( B( min (v ( min(v ( (v ( (v ( v ( v min (v ( min (v ( max(v ( max(v ( v ( v ( np (B ( B ( (B ( B ( np (B ( B ( np (B ( B ( np (B ( B ( ; (44 ; (45 ; (46 max(v ( v ( v ( np (B ( B ( + min(v ( ; (48 min(v ( v( v ( max min (v ( P n (B ( B ( + (v ( ; (49 v( v ( P n (B ( B ( + max(v ( ; (5 and Eq. (5 is shown in Box VII. Let: EIFC( ; ; ; n ( ; v ; (5 we have: min( ( max( ( ; (53 min(v ( v max(v ( ; (54 s( v max( ( min(v ( s( + ; (55 min (v ( min (v ( ; (47 s( v min( ( max(v ( s( : (56

9 Deian Yu/Scientia Iranica, Transactions E: Industrial Engineering (3 9{ 7 min(v ( Q n v (B ( B ( ( Q n ( v ( (B ( B ( + Q n v (B ( B ( ( Box VII. max(v ( : (5 If s( < s( + and s( > s(, then by order relation between two IFVs which was described in Section, we have: < EIFC( ; ; ; n < + : (57 If s( s( +, i.e. v max( ( min(v (, we have: max( ( ; v min(v ( : (58 Therefore: h( +v max( ( +min(v ( h( + ; (59 then: EIFC( ; ; ; n + : (6 If s(s(, i.e. then we have: v min( ( max(v (, min( ( ; v max(v ( : (6 Therefore: h( + min Thus, it follows that: ( ( +max( ( h( : (6 EIFC( ; ; ; n : (63 From Eqs. (57, (6 and (63, we know that Eq. (3 always holds. It should be noted that the proof of Theorems and 4 are referred to the proved methods provided by Wang and Liu [44]. The Monotonicity of the EIFC operator can be obtained by a similar proving method. Theorem 5 (Monotonicity. Let be a fuzzy measure on X, ( ; v ( ; ; ; n and ( ; v ( ; ; ; n be two collections of IFVs, ( be the th largest of ( ; ; ; n and ( be the th largest of ( ; ; ; n, >. If ( ( and v ( v (, for all, then: EIFC( ; ; ; n EIFC( ; ; ; n: (64 Theorem 6. Let be a fuzzy measure on X, ( ; v ( ; ; ; n be a collection of IFVs and ( be the th largest of them. If( ; ; ; n is any permutation of ( ; ; ; n, then: EIFC( ; ; ; n EIFC( ; ; ; n: (65 Proof. ccording to Denition 4, we have: EIFC((x ; (x ; ; (x n and: n (B ( B ( (x ( ; (66 EIFC( (x ; (x ; ; (x n n (B ( B ( (x ( : (67 Since ( ; ; ; n is any permutation of, we have: ( (i ( (i i ; ; ; n: (68 Therefore, EIFC( ; ; ; n EIFC( ; ; ; n, which completes the proof of Theorem 6. From Denition 4, the following property of the EIFC operator can be obtained easily. Property (Shift-invariance. Let be a fuzzy measure on X and ( ; v ( ; ; ; n be a collection of IFVs. If ( ; is an intuitionistic fuzzy value on X, then: EIFC( ; ; ; n EIFC( ; ; ; n : (69 Property (Homogeneity. Let be a fuzzy measure on X and ( ; v ( ; ; ; n be a collection of IFVs. If >, then: EIFC( ; ; ; n EIFC( ; ; ; n : (7

10 8 Deian Yu/Scientia Iranica, Transactions E: Industrial Engineering (3 9{ Property 3. Let be a fuzzy measure on X and ( ; v ( ; ; ; n be a collection of IFVs. If ( ; is an intuitionistic fuzzy value on X and if >, then: EIFC( ; ; ; n EIFC( ; ; ; n : (7 Property 4. Let be a fuzzy measure on X, ( ; v and ( ; v ( ; ; ; n be two collections of IFVs, then: EIFC( ; ; ; n n EIFC( ; ; ; n EIFC( ; ; ; n : (7 4. n approach to multi-criteria decision making under intuitionistic fuzzy environment The multi-criteria decision making is a very practical method in the real world, and have very signicant eect on both theory and practical. It aims to nd the best alternatives from a nite number of options. Multi-criteria decision making, using IFVs, is an important variety of decision making theory [5-8]. In the following, we utilize the EIFC operator to develop an approach for multi-criteria decision making using IFVs, which includes the following steps: Step. The intuitionistic fuzzy decision making matric B (b i mn always needs to be standardized, since there are cost attributes (the smaller the attribute values the better. Xu and Hu [5] have proposed a standardization method. ( b i ; for benet attribute C r i ( i ; v i bi ; for cos t attribute C i ; ; ; m ; ; ; n; (73 where b i is the complement of b i such that: bi (f i ; t i ; i ; ; ; m; ; ; ; n: (74 Based on Eqs. (94 and (95, the standardized decision making matric R (r i mn can be obtained. Step. By the order relation between IFVs, r i is reordered such that r i( r i( r i(n. Step 3. Based on Choquet integral, calculate the correlations between the criteria, using the method given in Section [,]. Step 4. Utilize the EIFC operator to aggregate all the performance values r i ( ; ; ; n of the ith line, and get the overall performance value r i corresponding to the alternative i. Step 5. Calculate the score function and accuracy function of the overall values r i (i ; ; ; m, then utilize order relation between IFVs described in Section to rank the overall performance values r i and select the best one. Example. Suppose a multinational corporation in China is planning its nancial strategy for the next year, according to the group strategy obective. fter preliminary screening, the four alternatives are produced. : To invest in the Southeast sian markets; : To invest in the Eastern European market; 3 : To invest in the North merican market; 4 : To invest in the local market. This evaluation proceeds from the following four aspects, such as the growth analysis (c, the risk analysis (c, the sociopolitical impact analysis (c 3 and the environmental impact analysis (c 4. The four alternatives i (i ; ; ; 4 are to be evaluated by corresponding experts, using the IFVs, as shown in Table. In order to choose the most appropriate investment program, the main steps are as follows (based on the EIFC operator: Step : Since the criteria c and c 4 are the cost criteria, the decision matrix need normalization. Normalized decision matrix is shown in Table. Step : ccording to Table, by the order relation between IFVs, the evaluation r i of the candidate i such that r i( r i( r i(n (i ; ; ; 4, is Table. The evaluation information on the proects. c c c3 c4 (.6,. (.,.8 (.8,. (.5,.3 (.4,. (.,.6 (.5,. (.,.8 3 (.7,.3 (.,.8 (.6,.3 (.,.4 4 (.5,.5 (.,.9 (.8,. (.,.7

11 Deian Yu/Scientia Iranica, Transactions E: Industrial Engineering (3 9{ 9 Table. The normalized evaluation information on the proects. c c c3 c4 (.6,. (.8,. (.8,. (.3,.5 (.4,. (.6,. (.5,. (.8,. 3 (.7,.3 (.8,. (.6,.3 (.4,. 4 (.5,.5 (.9,. (.8,. (.7,. reordered as follows: r ( (:8; :; r ( (:8; :; r (3 (:6; :; r (4 (:3; :5; r ( (:8; :; r ( (:6; :; r (3 (:5; :; r (4 (:4; :; r 3( (:8; :; r 3( (:7; :3; r 3(3 (:6; :3; r 3(4 (:4; :; r 4( (:9; :; r 4( (:8; :; r 4(3 (:7; :; r 4(4 (:5; :5: Step 3. Suppose the fuzzy measures of criteria of C and criteria sets of C are as follows: (; ; (c :38; (c :7; (c 3 :36; (c 4 :; (c ; c :77; (c ; c 3 :64; (c ; c 4 :5; (c ; c 3 :44; (c ; c 4 :3; (c 3 ; c 4 :45; (c ; c ; c 3 :83; (c ; c ; c 4 :69; (c ; c 3 ; c 4 :78; (c ; c 3 ; c 4 :55; (c ; c ; c 3 ; c 4 : Step 4. Utilize the EIFC operator to aggregate all the performance values r i ( ; ; ; 4 of the ith line, and get the overall performance value r i corresponding to the alternative i (i ; ; 3; 4.r is calculated and shown in Box VIII. Similarly: r (:584; :37; r 3 (:6879; :5; r 4 (:79; :38; Step 5. Calculate the scores of r i (i ; ; 3; 4, respectively: S :489; S :443; S 3 :468; S 4 :4973: Since: S 4 > S > S 3 > S ; we have: 4 3 : Hence, the best nancial strategy is 4, i.e. to invest in the local market. If we use the IFC operator proposed by Xu [4] and Tan and Chen [] (i.e., Eq. ( described in Section to get the overall values ri of the options i (i ; ; 3; 4 then: r (:6757; :8; r (:595; :366; r 3 (:69; :3; r 4 (:738; :7: ccording to the overall values ri (i ; ; 3; 4, by the order relations of IFVs, we can obtain that: i.e.: r 4 > r > r 3 > r ; 4 3 : Hence, the best nancial strategy is 4, i.e. to invest in the local market. The optimal nancial strategy and the ranking of the nancial strategies are the same as the ones obtained by EIFC Operator. Furthermore, we nd that: r (:6676; :857 < (:6757; :8 r ; r (:584; :37 < (:595; :366 r ; r 3 (:6879; :5 < (:69; :3 r 3 ; r 4 (:79; :38 < (:738; :7 r 4 : These results satisfy Theorem. 5. Concluding remarks In this paper, we have proposed the Einstein-based Intuitionistic Fuzzy Choquet veraging (EIFC operator. Various properties of EIFC operator have been studied in this paper. Furthermore, the relationships between EIFC operator and some existing operator such as IFC have been discussed. The EIFC operator were distinguished from the existing operators

12 Deian Yu/Scientia Iranica, Transactions E: Industrial Engineering (3 9{ r (+:8 :36 (+:8 :44 :36 (+:6 :83 :44 (+:3 :83 ( :8 :36 ( :8 :44 :36 ( :6 :83 :44 ( :3 :83 (+:8 :36 (+:8 :44 :36 (+:6 :83 :44 (+:3 :83 +( :8 :36 ( :8 :44 :36 ( :6 :83 :44 ( :3 :83, : :36 : (:44 :36 : (:83 :44 :5 ( :83 ( : :36 ( : (:44 :36 ( : (:83 :44 ( :5 ( :83 +: :36 : (:44 :36 : (:83 :44 :5 ( :83 (:6676; :857 Box VIII. not only due to the EIFC operator, using the Einstein operations, but also due to the consideration of the inter-dependent phenomena among the evaluated criteria, which makes the method proposed in this paper to have more wide practical application potentials. The results of this paper can be extended to the dual hesitant fuzzy environment and applied to supplier selection, personnel evaluation and so on. cknowledgments This work has been supported by China National Natural Science Foundation (No.734 and Zheiang Natural Science Foundation of China (No. LQ3G4. References. tanassov, K. \Intuitionistic fuzzy sets", Fuzzy Sets and Systems, (, pp (986.. tanassov, K., Intuitionistic Fuzzy Sets: Theory and pplications, Physica-Verlag, Heidelberg ( Dubois, D. and Prade, H., Fuzzy Sets and Systems: Theory and pplications, New York, cademic Press ( Miyamoto, S. \Remarks on basics of fuzzy sets and fuzzy multisets", Fuzzy Sets and Systems, 56(3, pp (5. 5. Yager, R.R. \On the theory of bags", International Journal of General Systems, 3(, pp ( Miyamoto, S. \Multisets and fuzzy multisets", In Soft Computing and Human-Centered Machines, Z.-Q. Liu, S. Miyamoto, Eds., Berlin, Springer (. 7. Torra, V. and Narukawa, Y. \On hesitant fuzzy sets and decision", The 8th IEEE International Conference on Fuzzy Systems, Jeu Island, Korea, pp (9. 8. Torra, V. \Hesitant fuzzy sets", International Journal of Intelligent Systems, 5(6, pp ( 9. Xu, Z.S. and Cai, X.Q. \Nonlinear optimization models for multiple attribute group decision making with intuitionistic fuzzy information", International Journal of Intelligent Systems, 5(6, pp (. Tan, C.Q. and Chen, X.H. \Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making", Expert Systems with pplications, 37(, pp (.. Tan, C.Q. and Chen, X.H. \Induced intuitionistic fuzzy Choquet integral operator for multicriteria decision making", International Journal of Intelligent Systems, 6(7, pp (.. Zhang, Q.S. and Jiang, S.Y. \Relationships between entropy and similarity measure of interval-valued intuitionistic fuzzy sets", International Journal of Intelligent Systems, 5(, pp. -4 (. 3. Chen, T.Y. \ comparative analysis of score functions for multiple criteria decision making in intuitionistic fuzzy settings", Information Sciences, 8(7, pp (. 4. Beliakov, G., Bustince, H., Goswami, D.P., Mukheree, U.K. and Pal, N.R. \On averaging operators for tanassov's intuitionistic fuzzy sets", Information Sciences, 8(6, pp. 6-4 (. 5. Xu, Z.S. \Intuitionistic fuzzy aggregation operators", IEEE Transactions on Fuzzy Systems, 5(6, pp (7. 6. Xu, Z.S. \ deviation-based approach to intuitionistic fuzzy multiple attribute group decision making", Group Decision and Negotiation, 9(, pp (a. 7. Yager, R.R. \OW aggregation of intuitionistic fuzzy set", International Journal of General Systems, 38(6, pp (9. 8. Liu, P.D. \Multi-attribute decision-making method research based on interval vague set and TOPSIS method", Technological and Economic Development of Economy, 5(3, pp (9. 9. Bustince, H., Barrenechea, E. and Pagola, M. \Generation of interval-valued fuzzy and atanassov's intuitionistic fuzzy connectives from fuzzy connectives and from K operators: Laws for conunctions and disunctions, amplitude", International Journal of Intelligent Systems, 3(6, pp (8.. Chen, T.Y. and Li, C.H. \Determining obective weights with intuitionistic fuzzy entropy measures: a comparative analysis", Information Sciences, 8(, pp (.. Li, D.F. \Multi-attribute decision making models and methods using intuitionistic fuzzy sets", Journal of Computer and System Sciences, 7(, pp (5.. Li, D.F. \ ratio ranking method of triangular intuitionistic fuzzy numbers and its application to MDM

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14 Deian Yu/Scientia Iranica, Transactions E: Industrial Engineering (3 9{ 5. Xu, Z.S. and Hu, H. \Proection models for intuitionistic fuzzy multiple attribute decision making", International Journal of Information Technology and Decision Making, 9(, pp (. Biography Deian Yu received the BS degree in Information Management and Information System from Heilongiang institute of Science and Technology, Harbin, China, in 6. The MS degree in System Engineering from Naning University of eronautics and stronautics, Naning, China, in, and the PhD degree in Management Science and Engineering from Southeast University, Naning, China, in. He is currently a Lecturer with the School of Information, Zheiang University of Finance and Economics, Hangzhou, China. His current research interests include aggregation operators, information fusion, and multi-criteria decision making.