Multi-criteria group decision making method based

115

Journal of Intelligent & Fuzzy Systems 26 (2014) 115–125 DOI:10.3233/IFS-120719 IOS Press

Multi-criteria group decision making method based on intuitionistic linguistic aggregation operators Xin-Fan Wanga,b , Jian-Qiang Wanga,∗ and Wu-E Yanga a School b School

of Business, Central South University, Changsha, Hunan, China of Science, Hunan University of Technology, Zhuzhou, Hunan, China

Abstract. For multi-criteria group decision making problems with intuitionistic linguistic information, we define a new score function and a new accuracy function of intuitionistic linguistic numbers, and propose a simple approach for the comparison between two intuitionistic linguistic numbers. Based on the intuitionistic linguistic weighted arithmetic averaging (ILWAA) operator, we define two new intuitionistic linguistic aggregation operators, such as the intuitionistic linguistic ordered weighted averaging (ILOWA) operator and the intuitionistic linguistic hybrid aggregation (ILHA) operator, and establish various properties of these operators. The ILOWA operator weights the ordered positions of the intuitionistic linguistic numbers instead of weighting the arguments themselves. The ILHA operator generalizes both the ILWAA operator and the ILOWA operator at the same time, and reflects the importance degrees of both the given intuitionistic linguistic numbers and the ordered positions of these arguments. Furthermore, based on the ILHA operator and the ILWAA operator, we develop a multi-criteria group decision making approach, in which the criteria values are intuitionistic linguistic numbers and the criteria weight information is known completely. Finally, an example is given to illustrate the feasibility and effectiveness of the developed method. Keywords: Multi-criteria group decision making, intuitionistic linguistic number, intuitionistic linguistic ordered weighted averaging (ILOWA) operator, intuitionistic linguistic hybrid aggregation (ILHA) operator

1. Introduction In the socio-economic activities, there are a lot of multi-criteria decision making problems. A multicriteria decision making problem is to find a desirable solution from a finite number of feasible alternatives assessed on multiple criteria, both quantitative and qualitative. Depending on quantitative aspects presented by each decision making problem we can handle different types of precise numerical values, but in other cases, the ∗ Corresponding

author. Jian-Qiang Wang, School of Business, Central South University, Changsha, Hunan 410083, China. Tel.: +86 731 88830594; E-mail: [email protected].

problems present qualitative aspects that are complex to assess by means of exact values. In the latter case, the use of the fuzzy linguistic approaches [1-14] has provided very good results. It handles qualitative aspects that are represented in qualitative terms by means of linguistic variables, that is, variables whose values are not numbers but linguistic terms, such as “poor”, “slightly poor”, “fair”, “slightly good”, “good”, etc. In the last decades, a number of linguistic multicriteria decision making problems were studied and many linguistic aggregation operators were presented [2, 5, 15–24]. Herrera et al. [2] proposed a consensus model for group decision making under linguistic assessments information. Herrera et al. [15] developed

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X.-F. Wang et al. / GMCDM method based on intuitionistic linguistic aggregation operators

several group decision making processes using linguistic ordered weighted averaging (LOWA) operator. Herrera and Herrera-Viedma [5] presented three aggregation operators for weighted linguistic information, such as the linguistic weighted disjunction (LWD) operator, linguistic weighted conjunction (LWC) operator and linguistic weighted averaging (LWA) operator. Herrera and Herrera-Viedma [16] proposed three steps to follow in the linguistic decision analysis of group decision making problems with linguistic information. Xu [17] presented the linguistic hybrid aggregation (LHA) operator and applied it to group decision making. Tang and Zheng [18] developed a new linguistic modeling technique based on semantic similarity relation among linguistic labels. Wei [19] proposed some 2-tuple linguistic aggregation operators, such as the 2tuple linguistic weighted harmonic averaging (TWHA) operator, 2-tuple linguistic ordered weighted harmonic averaging (TOWHA) operator and 2-tuple linguistic combined weighted harmonic averaging (TCWHA) operator, and developed a multi-criteria group decision making method based on the TWHA and TCWHA operators. Wei [20] proposed a grey relational analysis method for 2-tuple linguistic multi-criteria group decision making with incomplete weight information. Atanassov [25] extended Zadeh’ fuzzy set [26] and introduced the notion of intuitionistic fuzzy set (IFS), which is characterized by a membership function and a non-membership function. Gau and Buehrer [27] presented the concept of vague set, but Bustine and Burillo [28] pointed out that vague sets are intuitionistic fuzzy sets. The IFS is more useful and effective in dealing with vagueness and uncertainty than the traditional fuzzy set [29–32], and has already been applied in many fields, such as decision analysis [33, 34], cluster analysis [35–37], pattern recognition [38, 39], forecasting [40], manufacturing grid [41] and so on. Aggregation operators of intuitionistic fuzzy information and their applications to multi-criteria decision making is a new branch of IFS theory, which has attracted significant interest from researchers in recent years. Xu [34] proposed some intiutionistic fuzzy arithmetic aggregation operators, such as the intuitionistic fuzzy weighted averaging (IFWA) operator, intuitionistic fuzzy ordered weighted averaging (IFOWA) operator and intuitionistic fuzzy hybrid aggregation (IFHA) operator. Zhao et al. [42] extended the IFWA, IFOWA and IFHA operators to provide a new class of operators referred to as the generalized IFWA (GIFWA) operator, generalized IFOWA (GIFOWA) operator and generalized IFHA (GIFHA) operator. Xia and Xu [43]

developed 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. Xu and Xia [44] proposed some induced generalized intuitionistic fuzzy aggregation operators, including the induced generalized intuitionistic fuzzy Choquet integral operators and induced generalized intuitionistic fuzzy Dempster-Shafer operators. In addition, they established various properties of these operators and applied them in multi-criteria decision making fields. Shu, Wang, et al. extended the IFS and defined the concepts of intuitionistic triangular fuzzy number [45], intuitionistic trapezoidal fuzzy number [46] and intuitionistic linguistic number [47], respectively. Shu et al. [45] defined four operations of the intuitionistic triangular fuzzy numbers and used them in fault tree analysis. Wang and Li [47] proposed the intuitionistic linguistic weighted arithmetic averaging (ILWAA) operator and developed a multi-criteria decision making approach in which the criteria values are intuitionistic linguistic numbers. Wang and Zhang [48], Wan and Dong [49] introduced some intuitionistic trapezoidal arithmetic aggregation operators, such as the intuitionistic trapezoidal weighted averaging (ITWA) operator, intuitionistic trapezoidal ordered weighted averaging (ITOWA) operator and intuitionistic trapezoidal hybrid aggregation (ITHA) operator, and developed several multi-criteria decision making approaches in which the criteria values are intuitionistic trapezoidal fuzzy numbers. From the [47], we know that an intuitionistic linguistic number, characterized by a linguistic term, a membership function and a non-membership function, is a generalization of linguistic term and intuitionistic fuzzy number. In processes of cognition of things, people may not possess a precise or sufficient level of knowledge of the problem domain, due to the increasing complexity of the socio-economic environments. In such a case, they usually have some hesitation and indeterminacy in providing their linguistic evaluation values over the objects considered, which makes the results of cognitive performance reveal the characteristics of affirmation, negation and hesitation. As the linguistic term and intuitionistic fuzzy number cannot be used to completely express all the information in a situation as such, their applications are limited. The intuitionistic linguistic number can describe the fuzzy


characters of things more detailedly and comprehensively, therefore, it is more suitable and reasonable to express decision making information taking the form of intuitionistic linguistic numbers rather than linguistic terms and intuitionistic fuzzy numbers. In addition, the ILWAA operator presented in this reference weights only the intuitionistic linguistic numbers. To solve the drawback, we shall propose an intuitionistic linguistic ordered weighted averaging (ILOWA) operator. The ILOWA operator first reorders all the given intuitionistic linguistic numbers in descending order and then weights these ordered arguments, and finally aggregates all these ordered weighted arguments into a collective one. The fundamental characteristic of the ILOWA operator is to weight the ordered positions of the intuitionistic linguistic numbers instead of weighting the arguments themselves. Furthermore, weights represent different aspects in both the ILWAA operator and the ILOWA operator, and both the operators consider only one of them. Thus, in the following we shall propose another new aggregation operator called the intuitionistic linguistic hybrid aggregation (ILHA) operator. The ILHA operator first weights the given intuitionistic linguistic numbers, and then reorders the weighted arguments in descending order and weights these ordered arguments by the ILHA weights, and finally aggregates all the weighted arguments into a collective one. Obviously, the ILHA operator generalizes both the ILWAA operator and the ILOWA operator at the same time, and reflects the importance degrees of both the given intuitionistic linguistic numbers and the ordered positions of these arguments. In order to do so, we organize the paper as follows. In Section 2, we define a new score function and a new accuracy function of intuitionistic linguistic numbers, and based on these two functions, propose a simple approach for the comparison between two intuitionistic linguistic numbers. In Section 3, we propose two new aggregation operators called the intuitionistic linguistic ordered weighted averaging (ILOWA) operator and the intuitionistic linguistic hybrid aggregation (ILHA) operator, and study some desirable properties of these two operators. In Section 4, based on the ILHA operator and the ILWAA operator, we develop a multi-criteria group decision making approach, in which the criteria values are intuitionistic linguistic numbers and the criteria weight information is known completely. In Section 5, an illustrative example is given to verify the developed approach and to demonstrate its feasibility and effectiveness. Finally, conclusions of this paper are presented in Section 6.

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2. Preliminaries Definition 1 [2, 14]. Let S = {sθ |θ = 0, 1, · · · , 2l}, in which l is a positive integer, sθ represents a possible value for a linguistic variable, and it should satisfy the following characteristics: 1) The set is ordered: sa > sb if a > b; 2) There is the negation operator: neg(sa ) = sb such that a + b = 2l. then we call S a discrete linguistic term set. For example, let l = 4, then S can be defined as: S = { s0 = extremely poor, s1 = very poor, s2 = poor, s3 = slightly poor, s4 = fair, s5 = slightly good, s6 = good, s7 = very good, s8 = extremely good}. To preserve all the given information, the discrete linguistic term set S should be extended to a continuous linguistic term set S˜ = {sθ |θ ∈ [0, q]}, in which sa > sb if a > b, and q (q > 2l) is a sufficiently large positive integer. If sθ ∈ S, then we call sθ the original linguistic term, otherwise, we call sθ the virtual linguistic term. ˜ then the following operational laws Let sa , sb , sθ ∈ S, are valid [14]: 1) sa + sb = sa+b ; 2) λsθ = sλθ , λ ∈ [0, 1]. Definition 2 [25]. Let X be a universe of discourse, then an IFS V in X is given by: V = {(x, µV (x), νV (x)) |x ∈ X} where the functions µV : X → [0, 1] and νV : X → [0, 1] determine the degree of membership and the degree of non-membership of the element x to V , respectively, and for every x ∈ X: 0 ≤ µV (x) + νV (x) ≤ 1 Let πV (x) = 1 − µV (x) − νV (x) for all x ∈ X, then πV (x) is called the degree of hesitancy or indeterminacy of x to V . Generally, for convenience, α = (µα , να ) is called an intuitionistic fuzzy number, where µα ∈ [0, 1], να ∈ [0, 1] and µα + να ≤ 1. Based on the linguistic term set and the IFS, Wang and Li [47] defined the intuitionistic linguistic number set as follows. Definition 3 [47]. Let X be a universe of discourse, ˜ then an intuitionistic linguistic number set A sθ(x) ∈ S, in X is an object having the following form: A = x, sθ(x) , µA (x), νA (x) |x ∈ X

118


which is characterized by a linguistic term sθ(x) , a membership function µA and a non-membership function νA of the element x to sθ(x) , where µA : X → S˜ → [0, 1], x → sθ(x) → µA (x) νA : X → S˜ → [0, 1], x → sθ(x) → νA (x) with the condition 0 ≤ µA (x) + νA (x) ≤ 1, x ∈ X Let πA (x) = 1 − µA (x) − νA (x) for all x ∈ X, then πA (x) is called the degree of hesitancy of x to sθ(x) . When µA (x) = 1, νA (x) = 0, the intuitionistic linguistic number set is reduced to the linguistic term set. For convenience, β = sθ(β) , µ(β), ν(β) is called an intuitionistic linguistic number, where sθ(β) is a linguistic term, µ(β) ∈ [0, 1], ν(β) ∈ [0, 1], µ(β) + ν(β) ≤ 1, and let be the set of all intuitionistic linguistic numbers. For example, β = s5 , 0.6, 0.3 is an intuitionistic linguistic number, and from it, we know that the degree that the evaluation object belongs to s5 is 0.6, the degree that the evaluation object doesn’t belong to s5 is 0.3 and the degree of decision maker’s hesitancy is 0.1. Obviously, the intuitionistic linguistic number generalizes both the linguistic term and the intuitionistic fuzzy number at the same time, and it is more practical and reasonable to express decision making information taking the form of intuitionistic linguistic numbers rather than linguistic terms and intuitionistic fuzzy numbers. In the following, we introduce some operational laws of the intuitionistic linguistic numbers. Definition 4 [47]. Let β1 = sθ(β1 ) , µ(β1 ), ν(β1 ) and β2 = sθ(β2 ) , µ(β2 ), ν(β2 ) be two intuitionistic linguistic numbers, then 1) β1 ⊕ β2 = sθ(β1 )+θ(β2 ) , θ(β1 )µ(β1 )+θ(β2 )µ(β2 ) θ(β1 )ν(β1 )+θ(β2 )ν(β2 ) ; , θ(β1 )+θ(β2 ) θ(β1 )+θ(β2 ) 2) λβ1 = sλθ(β1 ) , µ(β1 ), ν(β1 ), λ ∈ [0, 1]. It can be easily proven that all the above results are also intuitionistic linguistic numbers. Based on Definition 4, we can further obtain the following relation. 1) 2) 3) 4)

β1 ⊕ β2 = β2 ⊕ β1 ; (β1 ⊕ β2 ) ⊕ β3 = β1 ⊕ (β2 ⊕ β3 ); λ(β1 ⊕ β2 ) = λβ1 ⊕ λβ2 , λ ∈ [0, 1]; λ1 β1 ⊕ λ2 β1 = (λ1 + λ2 )β1 , λ1 , λ2 ∈ [0, 1].

The basis of the concepts of maximum expected values, minimum expected values and compromise expected

values, Wang and Li [47] defined a score function and an accuracy function for the comparison between two intuitionistic linguistic numbers, but these two functions are cumbersome. In this paper, we define a new score function and a new accuracy function of the intuitionistic linguistic numbers. Definition 5. Let β = sθ(β) , µ(β), ν(β) be an intuitionistic linguistic number, the score of β can be evaluated by a new score function h(β) shown as h(β) = θ(β) (µ(β) − ν(β))

(1)

where h(β) ∈ [−q, q]. Definition 6. Let β = sθ(β) , µ(β), ν(β) be an intuitionistic linguistic number, the degree of accuracy of β can be evaluated by a new accuracy function H(β) shown as H(β) = θ(β) (µ(β) + ν(β))

(2)

where H(β) ∈ [0, q]. Hong and Choi [33] showed that the relation between the score function and the accuracy function is similar to the relation between mean and variance in statistics. So, by using expectation-variance principle, we propose a simple method for the comparison between two intuitionistic linguistic numbers, which is based on the score function h(β) and the accuracy function H(β) and defined as follows. Definition 7. Let β1 = sθ(β1 ) , µ(β1 ), ν(β1 ) and β2 = sθ(β2 ) , µ(β2 ), ν(β2 ) be two intuitionistic linguistic numbers, then 1) if h(β1 ) < h(β2 ), then β1 is smaller than β2 , denoted by β1 < β2 ; 2) if h(β1 ) = h(β2 ), then (a) if H(β1 ) = H(β2 ), then β1 is equal to β2 , denoted by β1 = β2 ; (b) if H(β1 ) < H(β2 ), then β1 is smaller than β2 , denoted by β1 < β2 ; (c) if H(β1 ) > H(β2 ), then β1 is bigger than β2 , denoted by β1 > β2 .

3. Intuitionistic linguistic aggregation operators Based on Definition 4, Wang and Li [47] presented the ILWAA operator, but the ILWAA operator weights only the intuitionistic linguistic numbers. In the following, we shall propose two new intuitionistic linguistic aggregation operators, such as the ILOWA operator and


the ILHA operator, and investigate various properties of these operators. Definition 8 [47]. Let βj = sθ(βj ) , µ(βj ), ν(βj ) (j = 1, 2, · · · , n) be a collection of intuitionistic linguistic numbers and let ILWAA: n → , if

then ILWAAw (β1 , β2 ) = w1 β1 ⊕ w2 β2 = sw1 θ(β1 )+w2 θ(β2 ) , w1 θ(β1 )µ(β1 ) + w2 θ(β2 )µ(β2 ) , w1 θ(β1 ) + w2 θ(β2 ) w1 θ(β1 )ν(β1 ) + w2 θ(β2 )ν(β2 ) w1 θ(β1 ) + w2 θ(β2 )

ILWAAw (β1 , β2 , · · · , βn ) = w1 β1 ⊕ w2 β2 ⊕ · · · ⊕ wn βn

(3)

then ILWAA is called an intuitionistic linguistic weighted arithmetic averaging operator of dimension n, where w = (w1 , w2 , · · · , wn )T is the weight vector n of βj (j = 1, 2, · · · , n), with wj ∈ [0, 1] and wj = j=1 1. Especially, if w = (1 n, 1 n, · · · , 1 n)T , then the ILWAA operator is reduced to the intuitionistic linguistic arithmetic averaging (ILAA) operator of dimension n, which is defined as follows:

=

s 2

2 j=1

(4)

,

j=1

wj θ(βj )

wj θ(βj )ν(βj )

2

. wj θ(βj )

2) If equation (5) holds for n = k, that is ILWAAw (β1 , β2 , · · · , βk ) =

s k

, wj θ(βj )

j=1

k

ILWAA operator is also an intuitionistic linguistic number, and

j=1

k

wj θ(βj )µ(βj )

k

ILWAAw (β1 , β2 , · · · , βn )

j=1

,

wj θ(βj )µ(βj )

2

j=1

j=1

n s

,

j=1

wj θ(βj )

j=1

Theorem 1. Let βj = sθ(βj ) , µ(βj ), ν(βj ) (j = 1, 2, · · · , n) be a collection of intuitionistic linguistic numbers, and w = (w1 , w2 , · · · , wn )T be the weight vector of βj (j = 1, 2, · · · , n), with wj ∈ [0, 1] and n wj = 1, then their aggregated value by using the

=

2

ILAAw (β1 , β2 , · · · , βn ) 1 = (β1 ⊕ β2 ⊕ · · · ⊕ βn ) n

119

,

j=1

wj θ(βj )ν(βj )

k

wj θ(βj )

j=1

. wj θ(βj )

Then, when n = k + 1, by Definition 4, we have

wj θ(βj )

j=1

n j=1

n

wj θ(βj )µ(βj )

n

j=1

, wj θ(βj )

j=1

ILWAAw (β1 , β2 , · · · , βk , βk+1 )

wj θ(βj )ν(βj )

n

j=1

(5) wj θ(βj )

Proof. Obviously, from Definition 4, the aggregated value by using the ILWAA operator is also an intuitionistic linguistic number. In the following, we prove Equation (5) by using mathematical induction on n. 1) For n = 2: since w1 β1 = sw1 θ(β1 ) , µ(β1 ), ν(β1 ) w2 β2 = sw2 θ(β2 ) , µ(β2 ), ν(β2 )

k

=

s k

,

j=1

wj θ(βj )

j=1

k j=1

wj θ(βj )µ(βj )

k j=1

, wj θ(βj )

wj θ(βj )ν(βj )

k j=1

⊕ wj θ(βj )

wk+1 sθ(βk+1 ) , µ(βk+1 ), ν(βk+1 )

120


=

s k

ILOWAω (β1 , β2 , · · · , βn )

,

j=1

k j=1

=

j=1 k j=1

j=1

j=1

sk+1

,

j=1

wj θ(βj )

j=1

j=1

wj θ(βj )µ(βj )

k+1 j=1

j=1

n j=1

, ωj θ(βτ(j) )

ωj θ(βτ(j) )ν(βτ(j) )

n j=1

(7) ωj θ(βτ(j) )

j=1

OWA weights (for example, we can use the normal distribution based method [50]). The ILOWA operator has the following properties.

, wj θ(βj )

wj θ(βj )ν(βj )

k+1

ωj θ(βτ(j) )

ωj θ(βτ(j) )µ(βτ(j) )

where ω = (ω1 , ω2 , · · · , ωn )T is the weight vector related to the ILOWA operator, with ωj ∈ [0, 1] and n ωj = 1, which can be determined similar to the

wj θ(βj ) + wk+1 θ(βk+1 ) k+1

k+1

n

wj θ(βj ) + wk+1 θ(βk+1 )

,

j=1

j=1

,

wj θ(βj )ν(βj ) + wk+1 θ(βk+1 )ν(βk+1 )

k

=

n s

wj θ(βj )µ(βj ) + wk+1 θ(βk+1 )µ(βk+1 ) k

n

wj θ(βj )+wk+1 θ(βk+1 )

. wj θ(βj )

i.e. (5) holds for n = k + 1. Thus, based on 1) and 2), (5) holds for all n ∈ N, which completes the proof of Theorem 1. Definition 9. Let βj = sθ(βj ) , µ(βj ), ν(βj ) (j = 1, 2, · · · , n) be a collection of intuitionistic linguistic numbers. An intuitionistic linguistic ordered weighted averaging (ILOWA) operator of dimension n is a mapping ILOWA: n → , that has an associated weight vector ω = (ω1 , ω2 , · · · , ωn )T such that ωj ∈ [0, 1] n and ωj = 1. Furthermore, j=1

Theorem 3. Let βj = sθ(βj ) , µ(βj ), ν(βj ) (j = 1, 2, · · · , n) be a collection of intuitionistic linguistic numbers and ω = (ω1 , ω2 , · · · , ωn )T is the weight vector related to the ILOWA operator, with ωj ∈ [0, 1] n and ωj = 1, then j=1

1) (Idempotency) If all βj (j = 1, 2, · · · , n) are equal, i.e. βj = β for all j, then ILOWAω (β1 , β2 , · · · , βn ) = β. 2) (Boundary) β− ≤ ILOWAω (β1 , β2 , · · · , βn ) ≤ β+ . where β− = min{sθ(βj ) }, min{µ(βj )}, max{ν(βj )}, j

ILOWAω (β1 , β2 , · · · , βn )

j

j

+

(6)

β = max{sθ(βj ) }, max{µ(βj )}, min{ν(βj )}.

where (τ(1), τ(2), · · · , τ(n)) is a permutation of (1, 2, · · · , n) such that βτ(j−1) ≥ βτ(j) for all j. Especially, if ω = (1 n, 1 n, · · · , 1 n)T , then the ILOWA operator is reduced to the ILAA operator. Similar to Theorem 1, we have the following.

3) (Monotonicity) Let βj∗ = sθ(βj∗ ) , µ(βj∗ ), ν(βj∗ ) (j = 1, 2, · · · , n) be a collection of intuitionistic linguistic numbers, if sθ(βj ) ≤ sθ(β∗ ) , µ(βj ) ≤ j µ(βj∗ ), ν(βj ) ≥ ν(βj∗ ), for all j, then

= ω1 βτ(1) ⊕ ω2 βτ(2) ⊕ · · · ⊕ ωn βτ(n)

Theorem 2. Let βj = sθ(βj ) , µ(βj ), ν(βj ) (j = 1, 2, · · · , n) be a collection of intuitionistic linguistic numbers, then their aggregated value by using the ILOWA operator is also an intuitionistic linguistic number, and

j

j

j

ILOWAω (β1 , β2 , · · · , βn ) ≤ ILOWAω (β1∗ , β2∗ , · · · , βn∗ ). 4) (Commutativity) Let β˜ j = sθ(β˜ j ) , µ(β˜ j ), ν(β˜ j ) (j = 1, 2, · · · , n) be a collection of intuitionistic


linguistic numbers, then

wj ∈ [0, 1] and

ILOWAω (β1 , β2 , · · · , βn )

Theorem 4. Let βj = sθ(βj ) , µ(βj ), ν(βj ) (j = 1, 2, · · · , n) be a collection of intuitionistic linguistic numbers and ω = (ω1 , ω2 , · · · , ωn )T is the weight vector related to the ILOWA operator, with ωj ∈ [0, 1] and n ωj = 1, then

j=1

1) If ω = (1, 0, · · · , 0)T , then

ILHAw,ω (β1 , β2 , · · · , βn ) n

=

n s

j=1

2) If ω = (0, 0, · · · , 1)T , then ILOWAω (β1 , β2 , · · · , βn ) = min{βj }.

) ωj θ(βτ(j)

,

j=1

3) If ωj = 1, ωi = 0, and i = / j, then

n j=1

) ωj θ(βτ(j)

,

n

) ωj θ(βτ(j)

(9)

and the aggregated value derived by using the ILHA operator is also an intuitionistic linguistic number.

ILOWAω (β1 , β2 , · · · , βn ) = βτ(j) where βτ(j) is the largest of βj (j = 1, 2, · · · , n). From Definition 8 and 9, we know that the ILWAA operator weights only the intuitionistic linguistic numbers, whereas the ILOWA operator weights only the ordered positions of the intuitionistic linguistic numbers instead of weighting the arguments themselves. To overcome this limitation, in what follows, we develop an ILHA operator, which weights both the given intuitionistic linguistic numbers and their ordered positions. Definition 10. Let βj = sθ(βj ) , µ(βj ), ν(βj ) (j = 1, 2, · · · , n) be a collection of intuitionistic linguistic numbers. An intuitionistic linguistic hybrid aggregation (ILHA) operator of dimension n is a mapping ILHA: n → , which has an associated vector ω = (ω1 , ω2 , · · · , ωn )T with ωj ∈ [0, 1] and n ωj = 1, such that j=1

Theorem 6. If ω = (1 n, 1 n, · · · , 1 n)T , the ILHA operator is reduced to the ILWAA operator. Theorem 7. If w = (1 n, 1 n, · · · , 1 n)T , the ILHA operator is reduced to the ILOWA operator. Obviously, the ILHA operator generalizes both the ILWAA operator and the ILOWA operator at the same time, and reflects the importance degrees of both the given intuitionistic linguistic numbers and the ordered positions of these arguments. Example. Suppose β1 = s4 , 0.8, 0.1, β2 = s6 , 0.7, 0.2, β3 = s5 , 0.9, 0.1, β4 = s3 , 0.7, 0.1 and β5 = s3 , 0.8, 0.2 are five intuitionistic linguistic numbers, and w = (0.22, 0.26, 0.15, 0.17, 0.20)T is the weight vector of βj (j = 1, 2, · · · , 5), then by the operational law in Definition 4, we get the weighted intuitionistic linguistic numbers as β1 = s4.40 , 0.8, 0.1, β2 = s7.80 , 0.7, 0.2,

ILHAw,ω (β1 , β2 , · · · , βn ) ⊕ · · · ⊕ ωn βτ(n)

)µ(β ) ωj θ(βτ(j) τ(j)

)ν(β )

ωj θ(βτ(j) τ(j)

j=1

j

βτ(j)

Theorem 5. Let βj = sθ(βj ) , µ(βj ), ν(βj ) (j = 1, 2, · · · , n) be a collection of intuitionistic linguistic ), ν(β ) (j = numbers and βτ(j) = sθ(β ) , µ(βτ(j) τ(j) τ(j) 1, 2, · · · , n), then

n

j

=

wj = 1, and n is the balancing

j=1

ILOWAω (β1 , β2 , · · · , βn ) = max{βj }.

⊕ ω2 βτ(2)

j=1

coefficient, which plays a role of balance.

= ILOWAω (β˜ 1 , β˜ 2 , · · · , β˜ n ). where β˜ 1 , β˜ 2 , · · · , β˜ n is any permutation of (β1 , β2 , · · · , βn ).

ω1 βτ(1)

n

121

β3 = s3.75 , 0.9, 0.1, β4 = s2.55 , 0.7, 0.1, (8)

is the jth largest of weighted intuitionistic where linguistic numbers (nw1 β1 , nw2 β2 , · · · , nwn βn ), w = (w1 , w2 , · · · , wn )T is the weight vector of βj , with

β5 = s3.00 , 0.8, 0.2. By Equation (1) in Definition 5, we calculate the scores of βj (j = 1, 2, · · · , 5):

122


h(β1 ) = 3.08, h(β2 ) = 3.90, h(β3 ) = 3.00, h(β4 )

= 1.53,

h(β5 )

To get the best alternative(s), the group decision making procedure is given as follows.

= 1.80.

Since h(β2 ) > h(β1 ) > h(β3 ) > h(β5 ) > h(β4 ) then βτ(1) = s7.80 , 0.7, 0.2, βτ(2) = s4.40 , 0.8, 0.1, βτ(3) = s3.75 , 0.9, 0.1, βτ(4) = s3.00 , 0.8, 0.2, = s2.55 , 0.7, 0.1. βτ(5)

Suppose that ω = (0.1117, 0.2365, 0.3036, 0.2365, 0.1117)T is the weight vector related to the ILHA operator (derived by the normal distribution based method [50]), then, by (9), it follows that ILHAw,ω (β1 , β2 , β3 , β4 , β5 )

Step 1. Normalize the decision making information in the matrix Rk (k = 1, 2, · · · , t). For the benefittype criteria, we do nothing; for the cost-type criteria, we utilize the linguistic negation operator s˜ θ(β(k) ) = ij

neg sθ(β(k) ) = s2l−θ(β(k) ) to make linguistic evaluaij

ij

tion values be normalized. For convenience, the normalized criteria values of the alternatives Ai (i = 1, 2, · · · , m) with respect to the (k) criteria Bj (j = 1, 2, · · · , n) are also denoted by βij = (k)

(k)

sθ(β(k) ) , µ(βij ), ν(βij ). ij

Step 2. Utilize the ILWAA operator (k)

(k)

(k)

(k)

βi = ILWAAw (βi1 , βi2 , · · · , βin )

(10)

to aggregate the criteria values of the ith row of the decision matrix Rk and derive the individual overall (k) values βi of the alternatives Ai , given by the decision makers dk (i = 1, 2, · · · , m; k = 1, 2, · · · , t).

= s4.0447 , 0.7996, 0.1391 . 4. A group decision making method based on the ILHA and ILWAA operator

Step 3. Utilize the ILHA operator (1)

(2)

(t)

βi = ILHAe,v (βi , βi , · · · , βi ) =

In the following, based on the ILHA and ILWAA operator, we develop a multi-criteria group decision making method, in which the criteria values are intuitionistic linguistic numbers and the criteria weight information is known completely. Let A = {A1 , A2 , · · · , Am } be a set of alternatives, and B = {B1 , B2 , · · · , Bn } be a set of criteria, w = (w1 , w2 , · · · , wn )T is the weight vector of Bj (j = n 1, 2, · · · , n), where wj ∈ [0, 1], wj = 1. Let D =

to aggregate the individual overall values βi , (2) (t) βi , · · · , βi and derive the collective overall values βi of the alternatives Ai (i = 1, 2, · · · , m), where v = (v1 , v2 , · · · , vt )T is the weighting vector of the ILHA t (τ(k)) vk = 1, and βi operator with vk ∈ [0, 1] and

{d1 , d2 , · · · , dt } be a set of decision makers, and e = (e1 , e2 , · · · , et )T be the weight vector of dk (k = t ek = 1. Assume 1, 2, · · · , t) with ek ∈ [0, 1] and

is the kth largest of the weighted intuitionistic lin(1) (2) (t) guistic numbers (te1 βi , te2 βi , · · · , tet βi ), where (τ(1), τ(2), · · · , τ(n)) is a permutation of (1, 2, · · · , n) and t is the balancing coefficient.

j=1

k=1

that decision makers D = {d1 , d2 , · · · , dt } represent the characteristics of the alternatives Ai (i = 1, 2, · · · , m) by the intuitionistic linguistic numbers (k) (k) (k) βij = sθ(β(k) ) , µ(βij ), ν(βij ) with respect to the criij

teria Bj (j = 1, 2, · · · , n), and derive the decision (k) matrix Rk = (βij )m×n (k = 1, 2, · · · , t), where sθ(β(k) ) ij

are the linguistic evaluation values of Ai with respect to (k) (k) Bj , µ(βij ) and ν(βij ) indicate, respectively, the degree of membership and the degree of non-membership that Ai attach to sθ(β(k) ) with respect to Bj (i = ij

1, 2, · · · , m; j = 1, 2, · · · , n; k = 1, 2, · · · , t).

(τ(1))

v1 βi

(τ(2))

⊕ v2 βi

(τ(t))

⊕ · · · ⊕ vt β i

(11) (1)

k=1

Step 4. Utilize equation (1) to calculate the scores h(βi ) of the collective overall values βi of the alternatives Ai (i = 1, 2, · · · , m). Step 5. By Definition 7, utilize the scores h(βi ) (i = 1, 2, · · · , m) to rank the alternatives Ai (i = 1, 2, · · · , m), and then select the best one(s) (if there is no difference between two scores h(βi ) and h(βp ), then we need to calculate the accuracy degrees H(βi ) and H(βp ) of the collective overall values βi and βp by equation (2), respectively, and then rank the alternatives Ai and Ap , in accordance with the accuracy degrees H(βi ) and H(βp )).

X.-F. Wang et al. / GMCDM method based on intuitionistic linguistic aggregation operators Table 1 Decision matrix R1

5. Illustrative example In this section, a multi-criteria group decision making problem involving a company’s making a project investment is used to illustrate the developed method using the ILHA and ILWAA operator. Consider that a risk investment company wants to make a project investment, which will be chosen from four alternative enterprises Ai (i = 1, 2, 3, 4) by three decision makers whose weight vector is e = (0.3, 0.4, 0.3)T . In assessing the potential contribution of each enterprise, three factors are considered, B1 −profitability, B2 −competitiveness, and B3 −risk affordability. Suppose that the weight vector of Bj (j = 1, 2, 3) is w =(0.3727, 0.3500, 0.2773)T , and the characteristic information of the alternatives Ai (i = 1, 2, 3, 4) with respect to the criteria Bj (j = 1, 2, 3) given by decision makers dk (k = 1, 2, 3) are repre(k) sented by the intuitionistic linguistic numbers βij = (k) (k) sθ(β(k) ) , µ(βij ), ν(βij )

listed in Tables 1, 2 and 3.

ij

Because all criteria Bj (j = 1, 2, 3) are benefit-type, their values need not to be normalized. In the following, we utilize the developed approach to get the best investment enterprise.

Step 1. Utilize Equation (10) to aggregate the criteria values of the ith row of the decision matrix Rk and (k) derive the individual overall values βi of the alternatives Ai by the decision makers dk (i = 1, 2, 3, 4; k = 1, 2, 3).

A1 A2 A3 A4

B1

B2

B3

s4 , 0.8, 0.1 s5 , 0.8, 0.2 s5 , 0.7, 0.1 s4 , 0.8, 0.1

s4 , 0.7, 0.2 s4 , 0.8, 0.1 s5 , 0.7, 0.3 s4 , 0.9, 0.1

s5 , 0.7, 0.3 s4 , 0.8, 0.2 s6 , 0.9, 0.1 s5 , 0.8, 0.1

Table 2 Decision matrix R2 A1 A2 A3 A4

B1

B2

B3

s4 , 0.9, 0.1 s3 , 0.8, 0.2 s4 , 0.7, 0.1 s5 , 0.8, 0.2

s6 , 0.7, 0.2 s5 , 0.7, 0.1 s7 , 0.8, 0.2 s5 , 0.9, 0.1

s4 , 0.8, 0.2 s5 , 0.9, 0.1 s5 , 0.7, 0.2 s4 , 0.8, 0.1

Table 3 Decision matrix R3 A1 A2 A3 A4

B1

B2

B3

s4 , 0.7, 0.3 s3 , 0.7, 0.2 s5 , 0.8, 0.2 s3 , 0.9, 0.1

s6 , 0.7, 0.3 s5 , 0.8, 0.1 s7 , 0.9, 0.1 s5 , 0.7, 0.2

s5 , 0.9, 0.1 s6 , 0.8, 0.2 s5 , 0.7, 0.2 s6 , 0.9, 0.1

Step 2. Utilize Equation (11) to aggregate all the (1) (2) (3) individual overall values βi , βi , βi and derive the collective overall values βi of the alternatives Ai (i = 1, 2, 3, 4), where the weight vector of ILHA operator is ω = (0.2429, 0.5142, 0.2429)T which is determined by the weighting method based on normal distribution [50].

(1)

β1 = s4.2773 , 0.7349, 0.1976,

β1 = s4.6084 , 0.7608, 0.2122,

(2) β1

= s4.7000 , 0.7870, 0.1683,

β2 = s4.2933 , 0.7855, 0.1527,

β1 = s4.9773 , 0.7557, 0.2443,

(3)

β3 = s5.3924 , 0.7854, 0.1634,

(1)

β4 = s4.3467 , 0.8318, 0.1213.

β2 = s4.3727 , 0.8000, 0.1680, (2)

β2 = s4.2546 , 0.7915, 0.1263, (3)

β2 = s4.5319 , 0.7753, 0.1614, (1) β3

= s5.2773 , 0.7631, 0.1663,

(2) β3

= s5.3273 , 0.7460, 0.1720,

(3) β3

= s5.7000 , 0.8187, 0.1570,

(1) β4

= s4.2773 , 0.8327, 0.1000,

(2) β4

= s4.7227 , 0.8371, 0.1395,

(3)

123

β4 = s4.5319 , 0.8228, 0.1386.

Step 3. By Equation (1), we calculate the scores h(βi ) of alternatives Ai (i = 1, 2, 3, 4). h(β1 ) = 2.5280, h(β2 ) = 2.7166, h(β3 ) = 3.3540, h(β4 ) = 3.0885. Step 4. By Definition 7, we obtain that h(β3 ) > h(β4 ) > h(β2 ) > h(β1 ) then A3 A4 A2 A1 . So the best investment enterprise is A3 .

124


6. Conclusions In this paper, we have presented a simple approach for the comparison between two intuitionistic linguistic numbers depending on a new score function and a new accuracy function. Based on the ILWAA operator, we have proposed two new intuitionistic linguistic aggregation operators, such as the ILOWA operator and the ILHA operator, and established various properties of these operators. The ILWAA operator weights only the intuitionistic linguistic numbers, whereas the ILOWA operator weights only the ordered positions of the intuitionistic linguistic numbers instead of weighting the arguments themselves, the ILHA operator weights both the given intuitionistic linguistic numbers and their ordered positions, and generalizes both the ILWAA operator and the ILOWA operator at the same time. In addition, either of the ILOWA weights or the ILHA weights can be derived from the normal distribution based method, which can relieve the influence of unfair arguments on the final results by assigning low weights to those unduly high or unduly low ones. Furthermore, based on the ILHA operator and the ILWAA operator, we have developed a multi-criteria group decision making method under intuitionistic linguistic environment. The work in this paper develops the theories of the aggregation operators and fuzzy decision making theory, and it can be widely applied in the supply chain selection, investment decisions, project evaluation and economic benefits comprehensive evaluation and other related decision making.

Acknowledgment This work was supported by the National Natural Science Foundation of China (No. 71271218, 71221061 and 61174075), the Humanities and Social Science Foundation of the Ministry of Education of China (No. 12YJA630114 and 10YJC630338), and the Natural Science Foundation of Hunan Province of China (No. 11JJ6068). The authors are very grateful to the Editor-in-Chief and the anonymous reviewers for their constructive comments and suggestions that have led to an improved version of this paper. References [1] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning. Part 1, Information Sciences 8(3) (1975), 199–249.

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