Generalized Point Operators for Aggregating Intuitionistic Fuzzy Information

Generalized Point Operators for Aggregating Intuitionistic Fuzzy Information Meimei Xia, Zeshui Xu School of Economics and Management, Southeast University, Naning 211189, People s Republic of China We first develop a series of intuitionistic fuzzy point operators, and then based on the idea of generalized aggregation Yager RR. Generalized OWA aggregation operators. Fuzzy Optim Decis Making 2004;3:93 107 and Zhao H, Xu ZS, Ni MF, Liu SS. Generalized aggregation operators for intuitionistic fuzzy sets. Int J Intell Syst 2010;25:1 30, we develop various generalized intuitionistic fuzzy point aggregation operators, such as the generalized intuitionistic fuzzy point weighted averaging GIFPWA operators, generalized intuitionistic fuzzy point ordered weighted averaging GIFPOWA operators, and generalized intuitionistic fuzzy point hybrid averaging GIFPHA operators, which can control the certainty degrees of the aggregated arguments with some parameters. Furthermore, we study the properties and special cases of our operators. C 2010 Wiley Periodicals, Inc. 1. INTRODUCTION Intuitionistic fuzzy set IFS, which is characterized by a membership function and a non-membership function, was introduced by Atanassov. 1 Since it is powerful in dealing with uncertainty, imprecision and vagueness, IFS has attracted much attention. 2,3 For example, Chen and Tan 4 gave the score function of IFS. Hong and Choi 5 gave the accuracy function of IFS. Both the functions were applied to deal with the multi-criteria decision making problem. Atanassov 6 defined some operators for controlling the certainty degree of IFS by parameters. Liu and Wang 7 gave some point operators to translate an IFS into another one so as to reduce its uncertainty degree. In the last decades, a lot of research has been done about the aggregation methods. 8 11 Yager 12 gave the ordered weighted averaging OWA operator and then extended it to propose a class of generalized ordered weighted averaging aggregation GOWA operators. For the intuitionistic fuzzy values IFVs, Xu 13 developed some aggregation operators, such as the intuitionistic fuzzy weighted averaging IFWA operator, intuitionistic fuzzy ordered weighted averaging IFOWA Author to whom all correspondence should be addressed: e-mail: xu zeshui@263.net. e-mail: meimxia@163.com. INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 25, 1061 1080 2010 C 2010 Wiley Periodicals, Inc. View this article online at wileyonlinelibrary.com..20439

1062 XIA AND XU operator, intuitionistic fuzzy hybrid aggregation IFHA operator, and established various properties of these operators. Then Xu and Yager 14 proposed some geometric aggregation operators for IFVs, such as the intuitionistic fuzzy weighted geometric operator, the intuitionistic fuzzy weighted geometric IFWG operator, intuitionistic fuzzy ordered weighted geometric IFOGA operator, and intuitionistic fuzzy hybrid geometric IFHG operator. Zhao et al. 15 combined Xu and Yager s operators to develop some generalized aggregation operators, such as the generalized intuitionistic fuzzy weighted averaging GIFWA operator, generalized intuitionistic fuzzy ordered weighted averaging GIFOWA operator, and H κ α,λ α α. It should be noted that all the above aggregation operators are only based on the original information, and thus cannot reduce the uncertainty of the aggregated arguments. However, in some situations with intuitionistic fuzzy information, it is necessary to reduce the uncertainty of IFSs. 6,7 In this article, we first give some point aggregation operators for IFVs, which can reduce the uncertainty degree of IFVs, and then based on the idea of generalized aggregation, 15,16 we develop various generalized intuitionistic fuzzy point aggregation operators, and study the properties and some special cases of these developed operators. 2. BASIC CONCEPTS 2.1. Intuitionistic Fuzzy Set Let a set X be fixed, Atanassov 1 gave the concept of IFS A on X as follows: A {<x, μ A x,ν A x > x X} 1 where the functions μ A x and v A x denote the degrees of membership and nonmembership of the element x X to the set A respectively, with the condition: 0 μ A x 1, 0 ν A x 1, 0 μ A x + ν A x 1 2 and π A x 1 μ A x ν A x is usually called the degree of indeterminacy of x to A. For convenience, Xu and Yager 13 named α μ α,ν α as an IFV. In this article, let V be the set of all IFVs. For α, α 1,α 2 V, Xu and Yager 13,14 gave some operational laws by which we can get other IFVs: α 1 α 2 μ α1 + μ α2 μ α1 μ α2,ν α1 ν α2 ; α 1 α 2 μ α1 μ α2,ν α1 + ν α2 ν α1 ν α2 ; λα 1 1 μ α λ,να λ,λ>0; αλ μ λ α, 1 1 ν α λ,λ>0. Chen and Tan 4 introduced the score function sα μ α ν α to get the score of α. Then Hong and Choi 5 defined the accuracy function hα μ α + ν α to evaluate the accuracy degree of α. Based on the score function s and the accuracy function h, Xu and Yager 14 gave an order relation between two IFVs α and β: 1 If sα <sβ, then α<β;

GENERALIZED POINT OPERATORS AND FUZZY INFORMATION 1063 2 If sα sβ, then i If hα hβ, then α β; ii If hα <hβ, then α<β. 2.2. The GOWA and GIFOWA Operators Since its appearance, the OWA operator introduced by Yager 12 has received more and more attention. 8 14 In Ref. 16, Yager extended it and defined a GOWA operator as follows: DEFINITION 1. 16 The GOWA operator of dimension m is a mapping GOWA: I m I, which has the following form: 1/η m GOWA a 1,a 2,...,a m w b η 3 where η [, + ], w w 1,w 2,...,w m T is the associate weighting vector with w 0, 1, 2,...,m, m 1 w 1, b is the th largest of a i i 1, 2,...,m, and I [0, 1]. Zhao et al. 15 extended the GOWA operator to accommodate the situations where the input arguments are IFVs. A GIFOWA operator of dimension m is a mapping GIFOWA: V m V, which has the following form: DEFINITION 2. 15 GIFOWA w α 1,α 2,...,α m w 1 α η σ 1 w 2α η σ 2 w mα η σ m 1/η 4 where η>0, w w 1,w 2,...,w m T is the weight vector of α 1,α 2,...,α m,w 0, 1, 2,...,m, m 1 w 1, and α σ is the th largest of α i i 1, 2,...,m. However, the above aggregation operators only use the original information. In some situations, we should get more information from the original one, 6,7 so it is necessary to deal with the original information first. In the next sections, we shall develop some point aggregation operators for IFVs to control the membership degree or nonmembership degree of the IFVs using different parameters. 1 3. POINT OPERATORS FOR AGGREGATING IFVS For an IFS A {<x, μ A x,ν A x > x X}, let κ, λ [0, 1], Atanassov 6 gave the following operators: 1 D κ A {x, μ A x + κπ A x,ν A x + 1 κπ A x x X}. 2 F κ,λ A {x, μ A x + κπ A x,ν A x + λπ A x x X}, where κ + λ 1. 3 G κ,λ A {x, κμ A x,λν A x x X}. 4 H κ,λ A {x, κμ A x,ν A x + λπ A x x X}. 5 H κ,λ A {x, κμ Ax,ν A x + λ1 κμ A x ν A x x X}.

1064 XIA AND XU 6 J κ,λ A {x, μ A x + κπ A x,λν A x x X}. 7 J κ,λ A {x, μ Ax + κ1 μ A x λν A x,λν A x x X}. 8 P κ,λ A {x, maxκ, μ A x, minλ, ν A x x X}, where κ + λ 1. 9 Q κ,λ A {x, minκ, μ A x, maxλ, ν A x x X}, where κ + λ 1. Let IFSX be the set of all IFSs on X.ForA IFSX, Burillo and Bustince 17 defined an operator D κx A for each point x X: D κx A {x, μ A x + κ x π A x,ν A x + 1 κ x π A x x X} 5 where κ x [0, 1]. Then, Liu and Wang 7 defined an IF point operator for aggregating IFSs: DEFINITION 3. 7 Let A IFSX, for each point x X, taking κ x,λ x [0, 1] and κ x + λ x 1, then an IF point operator F κx,λ x A: IFSX IFSX is as follows: F κx,λ x A {x, μ A x + κ x π A x,ν A x + λ x π A x x X} 6 and if let F 0 κ x,λ x A A, then F n κ x,λ x A { x, μ A x + κ x π A x 1 1 κ x λ x n, κ x + λ x ν A x + λ x π A x 1 1 κ x λ x n κ x + λ x x X} 7 In the following, we define some new point operators for aggregating IFVs: DEFINITION 4. ForanIFVα μ α,ν α, let κ α,λ α [0, 1], we define some point operators: IFV IFV as follows: 1 D κα α μ α + κ α π α,ν α + 1 κ α π α. 2 F κα,λ α α μ α + κ α π α,ν α + λ α π α,whereκ α + λ α 1. 3 G κα,λ α α κ α μ α,λ α ν α. 4 H κα,λ α α κ α μ α,ν α + λ α π α. 5 H κ α,λ α α κ α μ α,ν α + λ α 1 κ α μ α ν α. 6 J κα,λ α α μ α + κ α π α,λ α ν α. 7 J κ α,λ α α μ α + κ α 1 μ α λ α ν α,λ α ν α. 8 P κα,λ α α maxκ α,μ α, minλ α,ν α,whereκ α + λ α 1. 9 Q κα,λ α α minκ α,μ α, maxλ α,ν α,whereκ α + λ α 1. Based on Definition 4, let F 0 κ α,λ α A D 0 κ α A G 0 κ α,λ α A H 0 κ α,λ α A H,0 κ α,λ α A then we have the following theorem: J 0 κ α,λ α A J,0 κ α,λ α A P 0 κ α,λ α A Q 0 κ α,λ α A A 8

GENERALIZED POINT OPERATORS AND FUZZY INFORMATION 1065 THEOREM 1. Let α μ α,ν α be an IFV, and n be a positive integer, taking κ α,λ α [0, 1], then 1 Dκ n α α μ α + κ α π α,ν α + 1 κ α π α. 2 Fκ n 1 1 κ α,λ α α μ α + κ α π α λ α n α κ α+λ α,ν α + λ α π α 1 1 κ α λ α n κ α+λ α,whereκ α + λ α 1. 3 G n κ α,λ α α κ n α μ α,λ n α ν α. 4 H n κ α,λ α α κ n α μ α,ν α + 1 ν α 1 1 λ α n μ α λ α n 1 t0 κn 1 t α 1 λ α t. 5 H κ α,λ α α κα nμ α,ν α + 1 ν α 1 1 λ α n μ α κ α λ α n 1 t0 κn 1 t 6 Jκ n α,λ α α μ α + 1 μ α 1 1 κ α n ν α κ α n 1 t0 λn 1 t α t0 λn 1 t α 7 J κ α,λ α α μ α + 1 μ α 1 1 κ α n ν α κ α λ α n 1 8 Pκ n α,λ α α maxκ α,μ α, minλ α,ν α,whereκ α + λ α 1. 9 Q n κ α,λ α α minκ α,μ α, maxλ α,ν α,whereκ α + λ α 1. α 1 λ α t. 1 κ α t,λ n α ν α. 1 κ α t,λ n α ν α. By Definition 4, we can easily get that the point operators translate an IFV to another one. In the following section, we combine the developed point operators with Zhao et al. s operators 15 to develop a new class of aggregation operators, which we denote as generalized intuitionistic fuzzy point averaging GIFPA operators. 4. THE GIFPWA, GIFPOWA, AND GIFPHA OPERATORS 4.1. The GIFPWA Operators DEFINITION 5. Let α μ α,ν α 1, 2,...,m be a collection of IFVs, and n be a positive integer, taking κ α,λ α [0, 1], 1, 2,...,m. Let the generalized intuitionistic fuzzy point weighted averaging GIFPWA: V m V,if 1 GIFPWAD n w α 1,α 2,...,α m η η η 1/η w 1 D n κ α1 α 1 w2 D n κ α2 α 2 wm D n κ αm α m 2 GIFPWAF n w α 1,α 2,...,α m η η η 1/η w 1 F n κ α1,λ α1 α 1 w2 F n κ α2,λ α2 α 2 wm F n κ αm,λ αm α m where κ α + λ α 1, 1, 2,...,m. 3 GIFPWAG n w α 1,α 2,...,α m η η η 1/η w 1 G n κ α1,λ α1 α 1 w2 G n κ α2,λ α2 α 2 wm G n κ αm,λ αm α m 4 GIFPWAH n w α 1,α 2,...,α m η η η 1/η w 1 H n κ α1,λ α1 α 1 w2 H n κ α2,λ α2 α 2 wm H n κ αm,λ αm α m

1066 XIA AND XU 5 GIFPWAH w α 1,α 2,...,α m η η η 1/η w 1 H κ α1,λ α1 α 1 w2 H κ α2,λ α2 α 2 wm H κ αm,λ αm α m 6 GIFPWAJ n w α 1,α 2,...,α n η η η 1/η w 1 J n κ α1,λ α1 α 1 w2 J n κ α2,λ α2 α 2 wm J n κ αm,λ αm α m 7 GIFPWAJ w α 1,α 2,...,α m η η η 1/η w 1 J κ α1,λ α1 α 1 w2 J κ α2,λ α2 α 2 wm J κ αm,λ αm α m 8 GIFPWAP n w α 1,α 2,...,α m η η η 1/η w 1 P n κ α1,λ α1 α 1 w2 P n κ α2,λ α2 α 2 wm P n κ αm,λ αm α m where κ α + λ α 1, 1, 2,...,m. 9 GIFPWAQ n w α 1,α 2,...,α m η η η 1/η w 1 Q n κ α1,λ α1 α 1 w2 Q n κ α2,λ α2 α 2 wm Q n κ αm,λ αm α m where κ α + λ α 1, 1, 2,...,m. Then, the functions GIFPWAD n w, GIFPWAFn w, GIFPWAGn w, GIFPWAHn w, GIFPWAH w, GIFPWAJn w, GIFPWAJ w, GIFPWAPn w, and GIFPWAQn w are called the GIFPWA operators, where η>0, w w 1,w 2,...,w m T is a weight vec- associated with the GIFPWA operators, with w 0, 1, 2,...,n, and tor n 1 w 1. THEOREM 2. Let α μ α,ν α 1, 2,...,m be a collection of IFVs, and n be a positive integer, κ α,λ α [0, 1], 1, 2,...,m, η>0, w w 1,w 2,...,w m T is a weight vector associated with the GIFPWA operators, with w 0 and m 1 w 1, then the aggregated values by using the GIFPWA operators are also IFVs, and 1 GIFPWAD n w α 1,α 2,...,α m 1/η 1 m 1 μ α + κ α π α η w, 1 1 1/η 1 m 1 1 ν α 1 κ α π α η w 1

GENERALIZED POINT OPERATORS AND FUZZY INFORMATION 1067 2 GIFPWAF n w α 1,α 2,...,α m 1 m w 1/η η 1 μ η F 1 κα n α,λα, 1 1 m w 1/η 1 1 ν F n 1 κα α,λα where κ α + λ α 1, 1, 2,...,m, and 3 GIFPWAG n w α 1,α 2,...,α m μ F n κα,λα α μ α + κ α π α 1 1 κ α λ α n ν F n κα,λα α ν α + λ α π α 1 1 κ α λ α n κ α + λ α 9 κ α + λ α 10 1 m η w 1/η 1 κ n α 1 μ α, 1 1 m η w 1/η 1 1 λ n α 1 ν α 4 GIFPWAH n w α 1,α 2,...,α m 1 m η w 1/η η 1 κ n α 1 μ α, 1 1 m w 1/η 1 1 ν H n 1 κα α,λα where n 1 ν H n κα α,λα ν α + 1 ν α 1 1 λ α n μ α λ α κ n 1 t α 1 λ α t t0 11 5 GIFPWAH w α 1,α 2,...,α m 1 m η w 1/η η 1 κ n α 1 μ α, 1 1 m w 1/η 1 1 ν H α 1 κα,λα where ν H κα,λα α ν α + 1 ν α 1 1 λ α n μ α κ α λ α n 1 t0 κ n 1 t α 1 λ α t 12 6 GIFPWAJ n w α 1,α 2,...,α m 1 m w 1/η 1 μ η J 1 κα n α,λα, 1 1 m η w 1/η 1 1 λ n α 1 ν α

1068 XIA AND XU where μ J n κα α,λα μ α + 1 μ α 1 n n 1 1 κ α να κ α λ n 1 t α 1 μ α t t0 13 7 GIFPWAJ w α 1,α 2,...,α m 1 m w 1/η 1 μ η 1 J, 1 1 m η w 1/η α κα,λα 1 1 λ n α 1 ν α where μ J κα,λα α μ α + 1 μ α 1 1 κ α n να κ α λ α n 1 t0 λ n 1 t α 1 μ α t 14 8 GIFPWAP n w α 1,α 2,...,α m 1 m 1 maxκα,μ α 1/η 1/η η w, 1 1 m 1 1 minλ α,ν α η w 1 1 where κ α + λ α 1, 1, 2,...,m. 9 GIFPWAQ n w α 1,α 2,...,α m 1/η 1/η 1 m 1 minκ α,μ α η w, 1 1 m 1 1 maxλ α,ν α η w 1 1 where κ α + λ α 1, 1, 2,...,m. Moreover, from Definition 5 and the operational laws given in Section 2.1, we can easily prove that the aggregated values by using the GIFPWA operators are also IFVs. THEOREM 3. Let α μ α,ν α 1, 2,...,m be a collection of IFVs, n be a positive integer, κ α,λ α [0, 1], 1, 2,...,m, η>0, w w 1,w 2,...,w m T be the weight vector associated with the GIFPWA operators, with w 0, 1, 2,...,m, m 1 w 1. If all α 1, 2,...,m are equal, that is, α α, for all, then 1 GIFPWAD n w α 1,α 2,...,α m Dκ n α α. 2 GIFPWAF n w α 1,α 2,...,α m Fκ n α,λ α α,whereκ α + λ α 1, 1, 2,...,m. 3 GIFPWAG n w α 1,α 2,...,α m G n κ α,λ α α. 4 GIFPWAH n w α 1,α 2,...,α m Hκ n α,λ α α. 5 GIFPWAH w α 1,α 2,...,α m H κ α,λ α α. 6 GIFPWAJ n w α 1,α 2,...,α m Jκ n α,λ α α. 7 GIFPWAJ w α 1,α 2,...,α m J κ α,λ α α. 8 GIFPWAP n w α 1,α 2,...,α m Pκ n α,λ α α,whereκ α + λ α 1, 1, 2,...,m.

GENERALIZED POINT OPERATORS AND FUZZY INFORMATION 1069 9 GIFPWAQ n w α 1,α 2,...,α m Q n κ α,λ α α,whereκ α + λ α 1, 1, 2,...,m. THEOREM 4. Let α μ α,ν α 1, 2,...,m be a collection of IFVs, n be a positive integer, κ α,λ α [0, 1], 1, 2,...,m, η>0, w w 1,w 2,...,w m T be the weight vector related to the GIFPWA operators, with w 0, 1, 2,...,m, and m 1 w 1, then 1 α D n GIFPWAD n w α 1,α 2,...,α m α + D n. 2 α F n GIFPWAF n w α 1,α 2,...,α m α + F n,whereκ α + λ α 1, 1, 2,...,m. 3 α G n GIFPWAG n w α 1,α 2,...,α m α + G n. 4 α H n GIFPWAH n w α 1,α 2,...,α m α + H n. 5 α Hn GIFPWAH w α 1,α 2,...,α m α + Hn. 6 α J n GIFPWAJ n w α 1,α 2,...,α m α + J n. 7 α Jn GIFPWAJ w α 1,α 2,...,α m α + Jn. 8 α P n GIFPWAP n w α 1,α 2,...,α m α + P n,whereκ α + λ α 1, 1, 2,...,m. 9 α Q n GIFPWAQ n w α 1,α 2,...,α m α + Q n,whereκ α + λ α 1, 1, 2,...,m, and α D n min α F n min α G n min α H n min μ D n κα,λα α, max μ F n κα,λα α, max μ G n κα,λα α, max μ H n κα,λα α, max α H min n μ H α κα,λα, max α J n min μ J n κα,λα α, max α J min n μ J α κα,λα, max α P n min α Q n min μ P n κα,λα α, max μ Q n κα,λα α, max ν D n κα α,λα ν F n κα α,λα ν G n κα α,λα ν H n κα α,λα ν H,α + D n,α + F n,α + G n,α + H n maxμ D n κα α,λα, minν D n κα α,λα maxμ F n κα α,λα, minν F n κα α,λα maxμ G n κα α,λα, minν G n κα α,λα maxμ H n κα α,λα, minν H n κα α,λα α κα,λα,α + H max n μ H α κα,λα, min ν J n κα α,λα ν J,α + J n ν H α κα,λα maxμ J n κα α,λα, minν J n κα α,λα α κα,λα,α + J max n μ J α κα,λα, min ν P n κα α,λα ν Q n κα α,λα,α + P n,α + Q n ν J α κα,λα maxμ P n κα α,λα, minν P n κα α,λα maxμ Q n κα α,λα, minν Q n κα α,λα THEOREM 5. Let α μ α,ν α 1, 2,...,m and α μ α,ν α

1070 XIA AND XU 1, 2,...,m be two collections of IFVs, n be a positive integer, κ α,λ α [0, 1], 1, 2,...,m, η>0, w w 1,w 2,...,w m T be the weight vector related to the GIFPWA operators, where w 0, 1, 2,...,m, and m 1 w 1, then 1 If μ D n κα α μ D n α and ν D n κα α ν D n α, for all,then GIFP W AD n w α 1,α 2,...,α m GIFP W AD n w α 1,α 2,...,α m. 2 If μ F n κα,λα α μ F n α and ν F n κα,λα α ν F n α, for all,then GIFP W AF n w α 1,α 2,...,α m GIFP W AF n w α 1,α 2,...,α n where κ α + λ α 1, 1, 2,...,m. 3 If μ G n κα α,λα μ G n,λ α α and ν G n κα α,λα ν G n,λ α α, for all,then GIFP W AG n w α 1,α 2,...,α m GIFP W AG n w α 1,α 2,...,α m. 4 If μ H n κα,λα α μ H n α and ν H n κα,λα α ν H n α, for all,then GIFP W AH n w α 1,α 2,...,α m GIFP W AH n w α 1,α 2,...,α m. 5 If μ H κα,λα α μ H κ α α and ν H κα,λα α ν H κ α α, for all,then GIFP W AH w α 1,α 2,...,α m GIFP W AH w α 1,α 2,...,α m. 6 If μ J n κα,λα α μ J n α and ν J n κα,λα α ν J n α, for all,then GIFP W AJ n w α 1,α 2,...,α m GIFP W AJ n w α 1,α 2,...,α m. 7 If μ J κα,λα α μ J κ α α and ν J κα,λα α ν J κ α α, for all,then GIFP W AJ w α 1,α 2,...,α m GIFP W AJ w α 1,α 2,...,α m. 8 If μ P n κα,λα α μ P n α and ν P n κα,λα α ν P n α, for all,then GIFP W AP n w α 1,α 2,...,α m GIFP W AP n w α 1,α 2,...,α m where κ α + λ α 1, 1, 2,...,m. 9 If μ Q n κα α,λα μ Q n,λ α α and ν Q n κα α,λα ν Q n,λ α α, for all,then GIFP W AQ n w α 1,α 2,...,α m GIFP W AQ n w α 1,α 2,...,α m where κ α + λ α 1, 1, 2,...,m.

GENERALIZED POINT OPERATORS AND FUZZY INFORMATION 1071 We now look at some special cases obtained by using different choices of the parameters n, w, and η. THEOREM 6. Let α μ α,ν α 1, 2,...,m be a collection of IFVs, n be a positive integer, κ α,λ α [0, 1], 1, 2,...,m, η>0, and w w 1,w 2,...,w m T be the weight vector related to the GIFPWA operators with w 0, 1, 2,...,m, and m 1 w 1, then 1 If n 0, then the GIFPWA operators reduce to the following: GIF W A w α 1,α 2,...,α m w 1 α η 1 w 2α η 2 w mα η m which is called a GIFWA operator. 15 2 If η 1, n 0, then the GIFPWA operators reduce to the following: 1/η IFWA w α 1,α 2,...,α m w 1 α 1 w 2 α 2 w m α m which is called an IFWA operator. 13 3 If w 1/m,1/m,...,1/m T, n 0, and η 1, then the GIFPWA operators reduce to the following: IFA w α 1,α 2,...,α m 1 m α 1 α 2 α m which is called an intuitionistic fuzzy averaging IFA operator. 13 4.2. The GIFPOWA Operators DEFINITION 6. Let α μ α,ν α, 1, 2,...,m be a collection of IFVs, n be a positive integer, κ α,λ α [0, 1], 1, 2,...,m, η>0,w w 1,w 2,...,w m T is an associated weighting vector such that w 0, 1, 2,...,m, and m 1 w 1, let the generalized intuitionistic fuzzy point ordered weighted averaging GIFPOWA: V m V,if 1 GIFPOWAD n w α 1,α 2,...,α m η η η 1/η w 1 D n κ ασ α σ 1 w2 D n 1 κ ασ α σ 2 wm D n 2 κ ασ α σ m m where D n κ ασ α σ is the th largest of D n κ αi α i i 1, 2,...,m 2 GIFPOWAF n w α 1,α 2,...,α m η η η 1/η w 1 F n κ ασ 1 α σ 1 w2 F n 1 κ ασ 2 α σ 2 wm F n 2 κ ασ m α σ m m where κ ασ + λ ασ 1, 1, 2,...,m, F n κ ασ α σ is the thlargest of F n κ αi,λ αi α i i 1, 2,...,m

1072 XIA AND XU 3 GIFPOWAG n w α 1,α 2,...,α m η η η 1/η w 1 G n κ ασ 1 α σ 1 w2 G n 1 κ σ 2,λ σ 2 α σ 2 wm G n κ ασ m α σ m m where G n κ ασ α σ is the th largest of G n κ αi,λ αi α i i 1, 2,...,m. 4 GIFPOWAH n w α 1,α 2,...,α m η η η 1/η w 1 H n κ σ 1,λ σ 1 α σ 1 w2 H n κ σ 2,λ σ 2 α σ 2 wm H n κ ασ m α m m where H n κ ασ α σ is the th largest of H n κ αi,λ αi α i i 1, 2,...,m. 5 GIFPOWAH w α 1,α 2,...,α m η η η 1/η w 1 H κ ασ 1 α σ 1 w2 H 1 κ ασ 2 α σ 2 wm H 2 κ ασ m α σ m m where H κ ασ α σ is the th largest of H κ αi,λ αi α i i 1, 2,...,m. 6 GIFPOWAJ n w α 1,α 2,...,α m η η η 1/η w 1 J n κ ασ 1 α σ 1 w2 J n 1 κ ασ 2 α σ 2 wm J n 2 κ ασ m α σ m m where J n κ ασ α σ is the th largest of J n κ αi,λ αi α i i 1, 2,...,m. 7 GIFPOWAJ w α 1,α 2,...,α m η η η 1/η w 1 J κ ασ 1 α σ 1 w2 J 1 κ ασ 2 α σ 2 wm J 2 κ ασ m α σ m m where J κ ασ α σ is the th largest of J κ αi,λ αi α i i 1, 2,...,m. 8 GIFPOWAP n w α 1,α 2,...,α m η η η 1/η w 1 P n κ ασ 1 α σ 1 w2 P n 1 κ ασ 2 α σ 2 wm P n 2 κ ασ m α σ m m where κ ασ + λ ασ 1, 1, 2,...,m, P n κ ασ α σ is the thlargest of P n κ αi,λ αi α i i 1, 2,...,m. 9 GIFPOWAQ n w α 1,α 2,...,α m η η η 1/η w 1 Q n κ ασ 1 α σ 1 w2 Q n 1 κ ασ 2 α σ 2 wm Q n 2 κ ασ m α σ m m where κ ασ + λ ασ 1, 1, 2,...,m, Q n κ ασ α σ is the thlargest of Q n κ αi,λ αi α i i 1, 2,...,m. Then the functions GIFPOWAD n w, GIFPOWAFn w, GIFPOWAGn w, GIFPOWAHn w, GIFPOWAH w, GIFPOWAJn w, GIFPOWAJ w, GIFPOWAPn w, and GIFPOWAQn w are called the GIFPOWA operators.

GENERALIZED POINT OPERATORS AND FUZZY INFORMATION 1073 The GIFPOWA operators have some properties similar to those of the GIFPWA operators. THEOREM 7. Let α μ α,ν α 1, 2,...,m be a collection of IFVs, n be a positive integer, κ α,λ α [0, 1], 1, 2,...,m, η>0, w w 1,w 2,...,w m T be an associated weight vector such that w 0, 1, 2,...,m, and m 1 w 1, then the aggregated value by using the GIFPOWA operators are also IFVs, and 1 GIFPOWAD n w α 1,α 2,...,α m 1 m η 1/η w 1 μασ + κ ασ π ασ, 1 1 1 m η 1/η w 1 1 νασ 1 κ ασ π ασ 1 2 GIFPOWAF n w α 1,α 2,...,α m 1 m w 1/η 1 μ η F 1 κα n α σ,λα σ σ, η 1 1 m w 1/η 1 1 ν F n 1 κασ α,λα σ σ where κ ασ + λ ασ 1, 1, 2,...,m, and μ F n κασ,λα σ α σ μ ασ + κ ασ π ασ 1 1 κ ασ λ ασ n κ ασ + λ ασ 15 ν F n κασ,λα σ α σ ν ασ + λ ασ π ασ 1 1 κ ασ λ ασ n 3 GIFPOWAG n w α 1,α 2,...,α m κ ασ + λ ασ 16 1 m η w 1/η 1 κ n α 1 σ μ ασ, 1 1 m η w 1/η 1 1 λ n α 1 σ ν ασ 4 GIFPOWAH n w α 1,α 2,...,α m 1 m η w 1/η 1 κ n α 1 σ μ ασ, η 1 1 m w 1/η 1 1 ν H n 1 κασ α,λα σ σ

1074 XIA AND XU where ν H n κασ,λα σ α σ ν ασ + 1 ν ασ 1 1 λασ n n 1 μ ασ λ ασ κ n 1 t n α σ 1 λασ t0 17 5 GIFPOWAH w α 1,α 2,...,α m 1 m η w 1/η 1 κ n α 1 σ μ ασ, 1 η 1 m w 1/η 1 1 ν H α 1 κα σ,λα σ σ where ν H κα σ,λα σ α σ ν ασ + 1 ν ασ 1 1 λασ n n 1 μ ασ κ ασ λ ασ κ n 1 t n α σ 1 λασ t0 18 6 GIFPOWAJ n w α 1,α 2,...,α m 1 m w 1/η 1 μ η J 1 κα n α σ,λα σ σ, 1 1 m η w 1/η 1 1 λ n α 1 σ ν ασ where μ J n κασ,λα σ α σ μ ασ + 1 μ ασ 1 1 κ ασ n n 1 ν ασ κ ασ λ n 1 t α σ 1 κ ασ n t0 19 7 GIFPOWAJ w α 1,α 2,...,α m 1 m w 1/η 1 μ η 1 J, 1 1 m η w 1/η α κα σ,λα σ 1 1 λ n α σ 1 σ ν ασ where μ J κα σ,λα σ α σ μ ασ + 1 μ ασ 1 1 κ ασ n n 1 ν ασ κ ασ λ ασ λ n 1 t α σ 1 κ ασ n t0 20

GENERALIZED POINT OPERATORS AND FUZZY INFORMATION 1075 8 GIFPOWAP n w α 1,α 2,...,α m 1 m 1 maxκασ,μ ασ 1/η η w, 1 1 1/η 1 m 1 1 minλ ασ,ν ασ η w 1 where κ ασ + λ ασ 1, 1, 2,...,m. 9 GIFPOWAQ n w α 1,α 2,...,α m 1/η 1 m 1 minκ ασ,μ ασ η w, 1 1 1/η 1 m 1 1 maxλ ασ,ν ασ η w 1 where κ ασ + λ ασ 1, 1, 2,...,m. THEOREM 8. Let α μ α,ν α 1, 2,...,m be a collection of IFVs, n be a positive integer, κ α,λ α [0, 1], 1, 2,...,m, η>0, w w 1,w 2,...,w m T be the weight vector related to the GIFPOWA operators, with w 0, 1, 2,...,m, and m 1 w 1. If all α 1, 2,...,m are equal, that is, α α, for all, then 1 GIFPOWAD n w α 1,α 2,...,α m Dκ n α α. 2 GIFPOWAF n w α 1,α 2,...,α m Fκ n α,λ α α,whereκ α + λ α 1, 1, 2,...,m. 3 GIFPOWAG n w α 1,α 2,...,α m G n κ α,λ α α. 4 GIFPOWAH n w α 1,α 2,...,α m Hκ n α,λ α α. 5 GIFPOWAH w α 1,α 2,...,α m H κ α,λ α α. 6 GIFPOWAJ n w α 1,α 2,...,α m Jκ n α,λ α α. 7 GIFPOWAJ w α 1,α 2,...,α m J κ α,λ α α. 8 GIFPOWAP n w α 1,α 2,...,α m Pκ n α,λ α α,whereκ α + λ α 1, 1, 2,...,m. 9 GIFPOWAQ n w α 1,α 2,...,α m Q n κ α,λ α α,whereκ α + λ α 1, 1, 2,...,m. THEOREM 9. Let α μ α,ν α 1, 2,...,m be a collection of IFVs, n be a positive integer, κ α,λ α [0, 1], 1, 2,...,m, η>0, w w 1,w 2,...,w m T be the weight vector related to the GIFPOWA operators, with w 0, 1, 2,...,m, and m 1 w 1, then 1 α D n GIFPOWAD n w α 1,α 2,...,α m α + D n. 2 α F n GIFPOWAF n w α 1,α 2,...,α m α + F n,whereκ α + λ α 1, 1, 2,...,m. 3 α G n GIFPOWAG n w α 1,α 2,...,α m α + G n. 4 α H n GIFPOWAH n w α 1,α 2,...,α m α + H n. 5 α H n GIFPOWAH w α 1,α 2,...,α m α + H n.

1076 XIA AND XU 6 α J n GIFPOWAJ n w α 1,α 2,...,α m α + J n. 7 α Jn GIFPOWAJ w α 1,α 2,...,α m α + Jn. 8 α P n GIFPOWAP n w α 1,α 2,...,α m α + P n,whereκ α + λ α 1, 1, 2,...,m. 9 α Q n GIFPOWAQ n w α 1,α 2,...,α m α + Q n,whereκ α + λ α 1, 1, 2,...,m. THEOREM 10. Let α μ α,ν α 1, 2,...,m, α μ α,ν α 1, 2,...,m be two collections of IFVs, n be a positive integer, κ α,λ α [0, 1], 1, 2,...,m, η>0, w w 1,w 2,...,w m T be the weight vector related to the GIFPOWA operators, with w 0, 1, 2,...,m, and m 1 w 1, then 1 If μ D n κα α μ D n α and ν D n κα α ν D n α, for all,then GIFPOWAD n w α 1,α 2,...,α m GIFPOWAD n w α 1,α 2,...,α m. 2 If μ F n κα,λα α μ F n α and ν F n κα,λα α ν F n α, for all,then GIFPOWAF n w α 1,α 2,...,α m GIFPOWAF n w α 1,α 2,...,α m where κ α + λ α 1, 1, 2,...,m. 3 If μ G n κα α,λα μ G n,λ α α and ν G n κα α,λα ν G n,λ α α, for all,then GIFPOWAG n w α 1,α 2,...,α m GIFPOWAG n w α 1,α 2,...,α m. 4 If μ H n κα,λα α μ H n α and ν H n κα,λα α ν H n α, for all,then GIFPOWAH n w α 1,α 2,...,α m GIFPOWAH n w α 1,α 2,...,α m. 5 If μ H κα,λα α μ H κ α α and ν H κα,λα α ν H κ α α, for all,then GIFPOWAH w α 1,α 2,...,α m GIFPOWAH w α 1,α 2,...,α m. 6 If μ J n κα,λα α μ J n α and ν J n κα,λα α ν J n α, for all,then GIFPOWAJ n w α 1,α 2,...,α m GIFPOWAJ n w α 1,α 2,...,α m. 7 If μ J κα,λα α μ J κ α α and ν J κα,λα α ν J κ α α, for all,then GIFPOWAJ w α 1,α 2,...,α m GIFPOWAJ w α 1,α 2,...,α m. 8 If μ P n κα,λα α μ P n α and ν P n κα,λα α ν P n α, for all,then GIFPOWAP n w α 1,α 2,...,α m GIFPOWAP n w α 1,α 2,...,α m

GENERALIZED POINT OPERATORS AND FUZZY INFORMATION 1077 where κ α + λ α 1, 1, 2,...,m. 9 If μ Q n κα α,λα μ Q n,λ α α and ν Q n κα α,λα ν Q n,λ α α, for all,then GIFPOWAQ n w α 1,α 2,...,α m GIFPOWAQ n w α 1,α 2,...,α m where κ α + λ α 1, 1, 2,...,m. THEOREM 11. Let α μ α,ν α 1, 2,...,m and α μ α,ν α 1, 2,...,m be two collections of IFVs, n be a positive integer, κ α,λ α [0, 1], 1, 2,...,m, η>0, w w 1,w 2,...,w m T be the weight vector related to the GIF- POWA operators, with w 0, 1, 2,...,m, m 1 w 1, and α 1,α 2...,α m be any permutation of α 1,α 2...,α m, then 1 GIFPOWAD n w α 1,α 2...,α m GIFPOWAD n w α 1,α 2...,α m. 2 GIFPOWAF n w α 1,α 2...,α m GIFPOWAF n w α 1,α 2...,α m,whereκ α + λ α 1, 1, 2,...,m. 3 GIFPOWAG n w α 1,α 2...,α m GIFPOWAG n w α 1,α 2...,α m. 4 GIFPOWAH n w α 1,α 2...,α m GIFPOWAH n w α 1,α 2...,α m. 5 GIFPOWAH w α 1,α 2...,α m GIFPOWAD w α 1,α 2...,α m. 6 GIFPOWAJ n w α 1,α 2...,α m GIFPOWAJ n w α 1,α 2...,α m. 7 GIFPOWAJ w α 1,α 2...,α m GIFPOWAJ w α 1,α 2...,α m. 8 GIFPOWAP n w α 1,α 2...,α m GIFPOWAP n w α 1,α 2...,α m,whereκ α + λ α 1, 1, 2,...,m. 9 GIFPOWAQ n w α 1,α 2...,α m GIFPOWAQ n w α 1,α 2...,α m,whereκ α + λ α 1, 1, 2,...,m We now look at some special cases obtained by using different choices of the parameters n, w, and η: THEOREM 12. Let α μ α,ν α 1, 2,...,m be a collection of IFVs, n be a positive integer, κ α,λ α [0, 1], 1, 2,...,m, η>0, and w w 1,w 2,...,w m T be the weight vector related to the GIFPOWA operators, with w 0, 1, 2,...,m, and m 1 w 1, then 1 If n 0, then the GIFPOWA operators reduce to the GIFOWA operator. 15 2 If η 1,n 0, then the GIFPOWA operators reduce to the IFOWA operator. 13 3 If w 1/m,1/m,...,1/m T, n 0,η 1, then the GIFPOWA operators reduce to the IFA operator. 13 4 If w 1, 0,...,0 T,n 0, then the GIFPOWA operator reduce to the intuitionistic fuzzy maximum operator. 4 5 If w 0, 0,...,1 T,n 0, then the GIFPOWA operators reduce to the intuitionistic fuzzy minimum operator. 4 4.3. The GIFPHA Operators Consider that the GIFPWA operators weight only the IFVs, while the GIF- POWA operators weight only the ordered positions of the IFVs instead of weighting

1078 XIA AND XU the IFVs themselves. To overcome this limitation, motivated by the idea of combining the weighted averaging operator and the OWA operators, 10,18 in what follows, we develop the generalized intuitionistic fuzzy point hybrid aggregation GIFPHA operators, which weight each given IFV and its ordered position. DEFINITION 7. Let α μ α,ν α 1, 2,...,m be a collection of IFVs, and n be a positive integer, taking κ α,λ α [0, 1], 1, 2,...,m, η>0. Letω ω 1,ω 2,...,ω m T be the weight vector of α with ω 0, m 1 ω 1, and m be the balancing coefficient, which plays a role of balance, then the GIFPHA operators of dimension m is a mapping GIFPHA: V m V, which has an associated vector w w 1,w 2,...,w m T, with w 0, 1, 2,...,m, and m 1 w 1, such that 1 GIFPHAD n w,ω α 1,α 2,...,α m η η η 1/η w 1 D n κ ασ α σ 1 w2 D n 1 κ ασ α σ 2 wm D n 2 κ ασ α σ m m where D n κ ασ α σ is the th largest of mω i D n κ αi α i i 1, 2,...,m. 2 GIFPHAF n w,ω α 1,α 2,...,α m η η η 1/η w 1 F n κ ασ 1 α σ 1 w2 F n 1 κ ασ 2 α σ 2 wm F n 2 κ ασ m α σ m m where κ ασ + λ ασ 1, 1, 2,...,m, F n κ ασ α σ is the th largest of mω i F n κ αi,λ αi α i i 1, 2,...,m. 3 GIFPHAG n w,ω α 1,α 2,...,α m η η 1/η η w 1 G n κ ασ 1 α σ 1 w2 G n 1 κ ασ α σ 2 w m G n 2 2 κ ασ m α σ m m where G n κ ασ α σ is the th largest of mω i G n κ αi,λ αi α i i 1, 2,...,m. 4 GIFPHAH n w,ω α 1,α 2,...,α m η η η 1/η w 1 H n κ σ 1,λ σ 1 α σ 1 w2 H n κ σ 2,λ σ 2 α σ 2 wm H n κ ασ m α m m where H n κ ασ α σ is the th largest of mω i H n κ αi,λ αi α i i 1, 2,...,m. 5 GIFPHAH w,ω α 1,α 2,...,α m η η η 1/η w 1 H κ ασ 1 α σ 1 w2 H 1 κ ασ 2 α σ 2 wm H 2 κ ασ m α σ m m where H κ ασ α σ is the th largest of mω i H κ αi,λ αi α i i 1, 2,...,m. 6 GIFPHAJ n w,ω α 1,α 2,...,α m η η η 1/η w 1 J n κ ασ 1 α σ 1 w2 J n 1 κ ασ 2 α σ 2 wm J n 2 κ ασ m α σ m m

GENERALIZED POINT OPERATORS AND FUZZY INFORMATION 1079 where J n κ ασ α σ is the th largest of mω i J n κ αi,λ αi α i i 1, 2,...,m. 7 GIFPHAJ w,ω α 1,α 2,...,α m η η η 1/η w 1 J κ ασ 1 α σ 1 w2 J 1 κ ασ 2 α σ 2 wm J 2 κ ασ m α σ m m where J κ ασ α σ is the th largest of mω i J κ αi,λ αi α i i 1, 2,...,m. 8 GIFPHAP n w,ω α 1,α 2,...,α m η η η 1/η w 1 P n κ ασ 1 α σ 1 w2 P n 1 κ ασ 2 α σ 2 wm P n 2 κ ασ m α σ m m where κ ασ + λ ασ 1, 1, 2,...,m, P n κ ασ α σ is the th largest of mω i P n κ αi,λ αi α i i 1, 2,...,m. 9 GIFPHAQ n w,ω α 1,α 2,...,α m η η η 1/η w 1 Q n κ ασ 1 α σ 1 w2 Q n 1 κ ασ 2 α σ 2 wm Q n 2 κ ασ m α σ m m where κ ασ + λ ασ 1, 1, 2,...,m, Q n κ ασ α σ is the th largest of mω i Q n κ αi,λ αi α i i 1, 2,...,m. THEOREM 13. Let α μ α,ν α 1, 2,...,m be a collection of IFVs, ω ω 1,ω 2,...,ω m T be the weight vector of α with ω 0, m 1 ω 1, m be the balancing coefficient, which plays a role of balance, and m be a positive integer. Let κ α,λ α [0, 1], 1, 2,...,m, η>0, and w w 1,w 2,...,w m T be the weight vector related to the GIFPHA operators, with w 0, 1, 2,...,m, and m 1 w 1, then 1 If w 1/m,1/m,...,1/m T, n 0, then the GIFPHA operators reduce to the GIFWA operators. 15 2 If ω 1/m,1/m,...,1/m T, n 0, then the GIFPHA operators reduce to the GIFOWA operator. 15 3 If η 1, n 0, then GIFPHA operators reduce to the following: IFHA w,ω α 1,α 2,...,α m 1 m 1 μ ασ w, m ν w α σ 1 1 which is called an IFHA operator. 13 5. CONCLUDING REMARKS In this article, we have developed some new point aggregating operators for IFVs, which can raise the certainty degree of the IFVs. Based on the point operators, we have developed a new class of aggregation operators including the GIFPWA, GIFPOWA, and GIFPHA operators which can control the membership degree and

1080 XIA AND XU the nonmembership degree with some parameters. Then we have studied the properties of these developed operators and discussed their specific forms. With different choices of the parameters n, η, and w, we can get some special cases of the developed aggregation operators. It is worth noting that all the GOWA operators defined by Yager 16 and the generalized intuitionistic fuzzy aggregation operators introduced by Zhao et al. 15 are also the special cases of our operators. Moreover, the results in this article can be further extended to accommodate interval-valued intuitionsitic fuzzy environments. Acknowledgments The work was supported by the National Science Fund for Distinguished Young Scholars of China No.70625005. References 1. Atanassov K. Intuitionistic fuzzy sets. Fuzzy Sets Syst 1986;20:87 96. 2. Gau WL, Buehrer DL. Vague sets. IEEE Trans Syst Man Cybern 1993;23:610 614. 3. Bustince H, Burillo P. Vague sets are intuitionistic fuzzy sets. Fuzzy Sets Syst 1996;79:403 405. 4. Chen SM, Tan JM. Handling multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst 1994;67:163 172. 5. Hong DH, Choi CH. Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst 2000;114:103 113. 6. Atanassov K. Remark on the intuitionistic fuzzy sets-iii. Fuzzy Sets Syst 1995;75:401 402. 7. Liu HW, Wang GJ. Multi-criteria decision making methods based on intuitionistic fuzzy sets. Eur J Oper Res 2007;197:220 233. 8. Calvo T, Mayor G, Mesiar R. Aggregation operators: New trends and applications. Heidelberg, Germany: Kluwer, 2002. 9. Yager RR, Kacprzyk J. The ordered weighted averaging operator: Theory and application. Norwell, MA: Kluwer, 1997. 10. Xu ZS, Da QL. An overview of operators for aggregating information. Int J Intell Syst 2003;18:953 969. 11. Xu ZS. An overview of methods for determining OWA weights. Int J Intell Syst 2005;20:843 865. 12. Yager RR. On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Trans Syst Man Cybern 1988;18:183 190. 13. Xu ZS. Intuitionistic fuzzy aggregation operators. IEEE Trans Fuzzy Syst 2007;15:1179 1187. 14. Xu ZS, Yager RR. Some geometric aggregation operators based on intuitionistic fuzzy sets. Int J Gen Syst 2006;35:417 433. 15. Zhao H, Xu ZS, Ni MF, Liu SS. Generalized aggregation operators for intuitionistic fuzzy sets. Int J Intell Syst 2010;25:1 30. 16. Yager RR. Generalized OWA aggregation operators. Fuzzy Optim Decis Making 2004;3:93 107. 17. Burillo P, Bustince H. Construction theorems of intuitionistic fuzzy sets. Fuzzy Sets Syst 1996;84:271 281. 18. Torra V. The weighted OWA operator. Int J Intell Syst 1997;12:153 166.