Some interval-valued intuitionistic uncertain linguistic ...

Journal of Systems Engineering and Electronics Vol. 25, No. 3, June 2014, pp.452–463

Some interval-valued intuitionistic uncertain linguistic hybrid Shapley operators Fanyong Meng1,* , Chunqiao Tan2 , and Qiang Zhang3 1. School of Management, Qingdao Technological University, Qingdao 266520, China; 2. School of Business, Central South University, Changsha 410083, China; 3. School of Management and Economics, Beijing Institute of Technology, Beijing 100081, China

Abstract: Two interval-valued intuitionistic uncertain linguistic hybrid operators called the induced interval-valued intuitionistic uncertain linguistic hybrid Shapley averaging (I-IIULHSA) operator and the induced interval-valued intuitionistic uncertain linguistic hybrid Shapley geometric (I-IIULHSG) operator are defined. These operators not only reflect the importance of elements and their ordered positions, but also consider the correlations among elements and their ordered positions. Since the fuzzy measures are defined on the power set, it makes the problem exponentially complex. In order to simplify the complexity of solving a fuzzy measure, we further define the induced interval-valued intuitionistic uncertain linguistic hybrid λ–Shapley averaging (I-IIULHλSA) operator and the induced interval-valued intuitionistic uncertain linguistic hybrid λ–Shapley geometric (I-IIULHλSG) operator. Moreover, an approach for multi-attribute group decision making under the interval-valued intuitionistic uncertain linguistic environment is developed. Finally, a numerical example is provided to verify the developed procedure and demonstrate its practicality and feasibility.

Keywords: multi-attribute group decision making, interval-valued intuitionistic uncertain linguistic set, hybrid operator, Shapley function.

DOI: 10.1109/JSEE.2014.00052

1. Introduction Aggregation operators as an important research topic in the decision making theory have been received considerable attention. One of the most important aggregation operators is the ordered weighted averaging (OWA) operator [1], whose fundamental aspect is a reordering step in which the input arguments are rearranged in descending order and the weight vector is merely associated with its ordered Manuscript received December 7, 2012. *Corresponding author. This work was supported by the National Natural Science Foundation of China (71201089) and the Natural Science Foundation Youth Project of Shandong Province (ZR2012GQ005).

position. Since it was proposed in 1988, many extending forms are developed [2–7]. Based on the geometric mean operator, [8] defined the ordered weighted geometric (OWG) operator to aggregate the arguments in a similar way as the OWA operator. Reference [9] developed the continuous OWG (COWG) operator. Reference [10] defined the induced COWG (ICOWG) operator and the induced linguistic OWG (ILOWG) operator. Furthermore, [11–13] proposed some geometric aggregation operators and some arithmetic aggregation operators on intervalvalued intuitionistic fuzzy sets (IVIFS). In a similar way as the quantitative aggregation operators, many linguistic aggregation operators are presented [14–24]. Some authors found the weighted operators only consider the importance of elements, while the ordered weighted operators only consider the importance of the elements’ ordered positions [24]. Reference [24] proposed the hybrid weighted averaging (HWA) operator, which not only considers the ordered positions of the arguments, but also gives the importance of them. After the pioneering work of [24], many developed forms are proposed, such as the hybrid weighted arithmetical averaging (HWAA) operator [25], the continuous hybrid weighted quasiarithmetical averaging (C-HWQA) operator [25], the generalized intuitionistic fuzzy hybrid aggregation (GIFHA) operator [26], the linguistic hybrid geometric averaging (LHGA) operator [27], the 2-tuple hybrid weighted averaging (T-HWA) operator [28], the uncertain linguistic hybrid geometric mean (ULHGM) operator [29], the intuitionistic uncertain linguistic hybrid geometric (IULHG) operator [30], the intuitionistic linguistic generalized dependent hybrid weighted aggregation (ILGDHWA) operator [31] and the interval-valued intuitionistic uncertain linguistic hybrid geometric (IIULHG) operator [32]. All above aggregation operators are based on the assumption that the elements in a set are independent. In order to deal with the situations where the elements in a set

Fanyong Meng et al.: Some interval-valued intuitionistic uncertain linguistic hybrid Shapley operators

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are correlative, many Choquet aggregation operators are proposed [33-42], which consider the interactive characteristics among elements. As far as we know, however, the research about aggregation operators based on fuzzy measures is mainly restricted on (intuitionistic) fuzzy sets. In this paper, based on the Shapley function, we will research some hybrid Shapley operators on interval- valued intuitionistic uncertain linguistic sets (IVIULS) [32]. Meantime, some special cases are discussed, and some desirable properties are studied. In order to simplify the complexity of solving a fuzzy measure and reflect interaction among elements, we further define the induced interval-valued intuitionistic uncertain linguistic hybrid λ–Shapley averaging (I-IIULHλSA) operator and the induced intervalvalued intuitionistic uncertain linguistic hybrid λ–Shapley geometric (I-IIULHλSG) operator. As a series of development, an approach to multi-attribute group decision making under interval-valued intuitionistic uncertain linguistic environment is developed. This paper is organized as follows: in Section 2, some basic concepts about IVIULSs are reviewed. In Section 3, the I-IIULHSA and I-IIULHSG operators are defined. Meantime, some desirable properties are briefly studied. In order to simplify the complexity of solving a fuzzy measure, the I-IIULHλSA and I-IIULHλSG operators are defined. In Section 4, an approach to multiattribute group decision making under the interval-valued intuitionistic uncertain linguistic environment is developed. In Section 5, an illustrative example is provided to illustrate the developed procedure. The conclusion is made in the last section.

term set S = {sα |s1  sα  st , α ∈ [1, t]}, whose elements also meet all the characteristics above. If s α ∈ S, then it is called the original linguistic term, otherwise, it is called the virtual linguistic term [21]. Definition 1 [32] An interval-valued intuitionistic uncertain linguistic set (IVIULS) A ∈ X = {x 1 , x2 , . . . , xn } is given as

2. Preliminaries The linguistic approach is an approximate technique, which represents qualitative aspects as linguistic values by means of linguistic variables. Let S = {s i |i = 1, 2, . . . , t} be a linguistic term set with odd cardinality. s i represents a possible value for a linguistic variable, and it should satisfy the following characteristics [43]: (i) The set is ordered: s i > sj , if i > j; (ii) Max operator: max (s i , sj ) = si , if si  sj ; (iii) Min operator: min (s i , sj ) = si , if si  sj . For example, S can be defined as

where the interval numbers [u l (α), uu (α)] and [vl (α), vu (α)] respectively represent the interval membership degree and the interval non-membership degree to the uncertain linguistic variable [s θ(α) , sτ (α) ] with [ul (α), uu (α)] ⊆ [0, 1], [vl (α), vu (α)] ⊆ [0, 1] and uu (α) + vu (α)  1. By IVIULN (X), we denote the set of all IVIULNs in X. Let α  = ([sθ(α) , sτ (α) ], [ul (α), uu (α)], [vl (α), vu (α)]) and β = ([sθ(β) , sτ (β) ], [ul (β), uu (β)], [vl (β), vu (β)]) be two IVIULNs in IVIULN(X), then some operations of α   and β are defined [32] by (i) α  ⊕ β = ([sθ(α)+θ(β) , sτ (α)+τ (β)], [1 − (1 −

S = {s1 : extremely poor, s2 : very poor,

ul (α))(1 − ul (β)), 1 − (1 − uu (α))(1 − uu (β))],

s3 : poor, s4 : slightly poor, s5 : fair,

[vl (α)vl (β), vu (α)vu (β)]) (ii) α  ⊗ β = ([sθ(α)θ(β) , sτ (α)τ (β) ], [ul (α)ul (β),

s6 : slightly good, s7 : good, s8 : very good,

uu (α)uu (β)], [1 − (1 − vl (α))(1 − vl (β)), 1 − (1 −

s9 : extremely good}. In order to preserve all the given information, Xu [21] extended the discrete term set S to a continuous linguistic

A = {xi |([sθ(xi ) , sτ (xi ) ], [ul (xi ), uu (xi )], [vl (xi ), vu (xi )])|xi ∈ X} where sθ(xi ) , sτ (xi ) ∈ S, the numbers [u l (xi ), uu (xi )] and [vl (xi ), vu (xi )] respectively represent the interval membership degree and the interval non-membership degree of the element x ∈ X to the uncertain linguistic variable [sθ(xi ) , sτ (xi ) ] with [ul (xi ), uu (xi )] ⊆ [0, 1], [vl (xi ), vu (xi )] ⊆ [0, 1] and uu (xi ) + vu (xi )  1 for each xi ∈ X. When ul (xi ) = uu (xi ) and vl (xi ) = vu (xi ) for each xi ∈ X, the IVIULS A degenerates to be the intuitionistic uncertain linguistic set [28]. A = {xi |([sθ(xi ) , sτ (xi ) ], uA (xi ), vA (xi ))|xi ∈ X} Furthermore, if s θ(xi ) = sτ (xi ) , then it reduces to be the intuitionistic linguistic set (ILS) [44]. A = {x|(sθ(xi ) , uA (xi ), vA (xi ))|xi ∈ X} Definition 2 [32] An interval-valued intuitionistic uncertain linguistic number (IVIULN) α  is defined by α  = ([sθ(α) , sτ (α) ], [ul (α), uu (α)], [vl (α), vu (α)])

vu (α))(1 − vu (β))]) (iii) λ α = ([sλθ(α) , sλτ (α) ], [1 − (1 − ul (α))λ , 1 − (1 − uu (α))λ ], [vl (α)λ , vu (α)λ ]), λ ∈ [0, 1]

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Journal of Systems Engineering and Electronics Vol. 25, No. 3, June 2014

(iv) α λ = ([sθ(α)λ , sτ (α)λ ], [ul (α)λ , uu (α)λ ], [1 − (1 − λ

λ

vl (α)) , 1 − (1 − vu (α)) ]), λ ∈ [0, 1] Proposition 1 [32] Let α  and β be two IVIULNs in IVIULN (X), then (i) α  ⊕ β = β ⊕ α  (ii) α  ⊗ β = β ⊗ α   (iii) λ( α ⊕ β) = λβ ⊕ λ α, λ ∈ [0, 1]

α = λ1 α  ⊕ λ2 α , λ1 , λ2 ∈ [0, 1] (iv) (λ1 + λ2 ) λ λ λ   (v) ( α ⊗ β) = β ⊗ α  , λ ∈ [0, 1] (vi) α λ1 +λ2 = α λ1 ⊗ α λ2 , λ1 , λ2 ∈ [0, 1] For any IVIULN α  = ([s θ(α) , sτ (α) ], [ul (α), uu (α)], [vl (α), vu (α)]), [32] defined the expected function E( α) of α  by E( α) = s (θ(α)+τ (α))(ul (α)+uu (α)+2−vl (α)−vu (α)) 8

and presented the accuracy function

3. New interval-valued intuitionistic uncertain linguistic hybrid operators In order to deal with the situations where the elements in a set are correlative, this section defines some new intervalvalued intuitionistic uncertain linguistic Shapley hybrid operators, which consider the interaction among elements. 3.1 Fuzzy measures and Shapley function

H( α) = s (θ(α)+τ (α))(ul (α)+uu (α)+vl (α)+vu (α)) 4

to evaluate the accuracy degree of α . Furthermore, [32] gave the following order relationship between IVIULNs α   and β.  then α If E( α) < E(β),  ≺ β 

interval-valued intuitionistic uncertain linguistic weighted geometric (IIULWG) operator [32]. Further, if ω i = 1/n for each i = 1, 2, . . . , n, then the IIULHG operator reduces to be the interval-valued intuitionistic uncertain linguistic ordered weighted geometric (IIULOWG) operator [32]. Although the IIULHG operator can be seen as an extension of the IIULWG and IIULOWG operators, it does not satisfy with boundary and idempotent, which are desirable properties for aggregating a finite collection of arguments. Further, it is based on the assumption that the elements in a set are independent.

 then If E( α) = E(β),

Fuzzy measures [45] as an effective tool for measuring the importance of the elements and the correlations among them have been deeply studied by many researchers and successfully used in many different fields. Definition 4 [45] A fuzzy measure on finite set N is a set function μ : P (N ) → [0, 1] satisfying (i) μ(∅) = 0, μ(N ) = 1

 ⇒α H( α) = H(β)  = β  H( α) < H(β) ⇒ α  ≺ β

(ii) If A, B ∈ P (N ) and A ⊆ B, then μ(A)  μ(B)

Based on the given operational laws on IVIULNs, [29] defined the following interval-valued intuitionistic uncertain linguistic hybrid geometric (IIULHG) operator. Definition 3 [32] Let α  i = ([sθ(αi ) , sτ (αi ) ], [ul (αi ), uu (αi )], [vl (αi ), vu (αi )]) be a collection of IVIULNs in IVIULN (X), an IIULHG operator of dimension n is a mapping IIULHG: IVIULN (X) n → IVIULN(X) which has an associated weight vector w = (w 1 , w2 , . . . , wn )T n  such that wj ∈ [0, 1] and wj = 1, denoted by

where P (N ) is the power set of N . In the multi-attribute decision making, μ(A) can be viewed as the importance of the attribute set A. Thus, in addition to the usual weights on the attribute set taken separately, weights on any combination of attribute set are also defined. In order to overall reflect the marginal contributions of a player in the game theory, [46] defined the following Shapley function: ϕi (μ, N ) =

j=1

IIULHG w, ω( α1 , α 2 , . . . , α n ) =

 (n−s−1)!s! (μ(S ∪ i) − μ(S)), ∀i ∈ N n!

S⊆N \i

n 

w rj j

j=1

where rj is the jth largest of the weighted arguments nω α j j (j = 1, 2, . . . , n), ω = (ω1 , ω2 , . . . , ωn )T is the weight vector of α  i (i = 1, 2, . . . , n) with ωi > 0 and n  ωi = 1, and n is the balancing coefficient. i=1

From Definition 3, we know when w i = 1/n for each i = 1, 2, . . . , n, the IIULHG operator degenerates to be the

(1) where μ is a fuzzy measure on a finite set N, s and n denote the cardinalities of S and N , respectively. When the Shapley function is applied in decision making, from (1) we know that it is an expectation value of the overall marginal contributions between the element i and any combination in N \ i. Further, it is not difficult to know that ϕ i (μ, N )  0 for any element i, and n  ϕi (μ, N ) = 1. Thus, {ϕi (μ, N )}i∈N is a weight i=1

vector.

Fanyong Meng et al.: Some interval-valued intuitionistic uncertain linguistic hybrid Shapley operators

3.2 I-IIULHSA and I-IIULHSG operators

1 , . . . , un , α n ) = I-IIULOSAμ (u1 , α

Definition 5 An I-IIULHSA operator is a mapping I-IIULHSA: IVIULN(X) n → IVIULN(X) defined on the set of second arguments of two tuples 1 , u2 , α 2 , . . . , un , α n  with a set of orderu1 , α inducing variables u i (i = 1, 2, . . . , n), denoted by 1 , . . . , un , α n ) = I-IIULHSAμ,v (u1 , α n 

ϕj (μ, N ) ε(j)

j=1 n 

455

n 

ϕj (μ, N ) α(j)

j=1

where u(j) is the jth largest of u i (i = 1, 2, . . . , n). 1 for each i = 1, Remark 4 When ϕi (v, N ) = n 2, . . . , n, the I-IIULHSA operator reduces to be the interval-valued intuitionistic uncertain linguistic Shapley averaging (IIULSA) operator.

(2)

 ϕj (μ, N )ϕαe(j) (v, A)

IIULSAv ( α1 , α 2 , . . . , α n ) =

Remark 5 If each IVIULN α  i (i = 1, 2, . . . , n) degenerates to be an intuitionistic uncertain linguistic number, namely, u l (αi ) = uu (αi ) and vl (αi ) = vu (αi ), then we get the induced intuitionistic uncertain linguistic hybrid Shapley averaging (I-IULHSA) operator. Theorem 1 Let α i = ([sθ(αi ) , sτ (αi ) ], [ul (αi ), uu (αi )], [vl (αi ), vu (αi )]) (i = 1, 2, . . . , n) be a collection of IVIULNs in IVIULN(X), and μ and v be a fuzzy  = { αi }i∈N , respecmeasure on N = {1, 2, . . . , n} and A tively. Then, their collective value by using the I-IIULHSA operator is also an IVIULN in IVIULN(X), denoted by

1 , . . . , un , α n ) = I-IIULHWA w, ω(u1 , α n 

wj ε(j)

j=1 n 

wj ωαe(j)

j=1

where wj = μ(j) and ωαej = v( αj ) for each j = 1, 2, . . . , n. Remark 2 When ui = uj for each pair (i, j) with i = j (i, j = 1, 2, . . . , n), the I-IIULHSA operator reduces to be the interval-valued intuitionistic uncertain linguistic hybrid ordered Shapley averaging (IIULHOSA) operator. IIULHOSAμ,v ( α1 , α 2 , . . . , α n ) = n 

ϕj (μ, N ) εj

j=1 n 

 ϕj (μ, N )ϕαej (v, A)

j=1

where εj is the jth largest of the Shapley weighted argu αi (i = 1, 2, . . . , n). ment ϕαei (v, A)  = 1 for each i = 1, Remark 3 When ϕαei (v, A) n 2, . . . , n, the I-IIULHSA operator reduces to be the induced interval-valued intuitionistic uncertain linguistic ordered Shapley averaging (I-IIULOSA) operator.

 αj ϕαej (v, A)

j=1

j=1

where u(j) is the jth largest of u i (i = 1, 2, . . . , n);  αi with εi is the Shapley weighted argument ϕ αei (v, A)  being the Shapley value with respect to (w.r.t.) ϕαei (v, A)  = { αi }i=1,2,...,n for α the fuzzy measure v on A i (i = 1, 2, . . . , n), and ϕj (μ, N ) is the Shapley value w.r.t. the fuzzy measure μ on ordered set N = {1, 2, . . . , n} for the jth position. Remark 1 If μ and v are both additive, then the IIIULHSA operator reduces to be the induced intervalvalued intuitionistic uncertain linguistic hybrid weighted averaging (I-IIULHWA) operator.

n 

I-IIULHSAμ,v (u1 , α 1 , u2 , α 2 , . . . , un , α n ) = ⎛ ⎜ ⎜[s n ⎝ P j=1

e ϕj (μ,N )ϕα e (j) (v,A)θ(α(j) ) n P

j=1

[1 −

e ϕj (μ,N )ϕα e (j) (v,A)

n 

,sP n

e ϕj (μ,N )ϕα e (j) (v,A)τ (α(j) )

j=1

n P

j=1

],

e ϕj (μ,N )ϕα e (j) (v,A)

e ϕj (μ,N )ϕα e (j) (v,A)

n P

(1 − ul (α(j) )) j=1

e ϕj (μ,N )ϕα e (j) (v,A)

,

j=1

1−

n 

e ϕj (μ,N )ϕα e (j) (v,A)

n P

(1 − uu (α(j) )) j=1

e ϕj (μ,N )ϕα e (j) (v,A)

],

j=1

[

n 

e ϕj (μ,N )ϕα e (j) (v,A)

n P

vl (α(j) ) j=1

e ϕj (μ,N )ϕα e (j) (v,A)

j=1 n  j=1

e ϕj (μ,N )ϕα e (j) (v,A)

n P

vu (α(j) ) j=1

e ϕj (μ,N )ϕα e (j) (v,A)

,

⎞ ⎟ ]⎟ ⎠.

(3)

Proof From Proposition 1, it is not difficult to get the conclusion.  Similar to the definition of the I-IIULHSA operator, we define the following I-IIULHSG operator.

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Definition 6 An I-IIULHSG operator is a mapping I-IIULHSG: IVIULN(X) n → IVIULN(X) defined on the set of second arguments of two tu1 , u2 , α 2 , . . . , un , α n  with a set of orderples u1 , α inducing variables u i (i = 1, 2, . . . , n), denoted by

α1 , α 2 , . . . , α n ) = IIULSGv (

I-IIULHSAμ,v (u1 , α 1 , . . . , un , α n ) = n  j=1

n P

ε(j)

e ϕj (μ,N )ϕα e (j) (v,A)

(4)

where u(j) is the jth largest of u i (i = 1, 2, . . . , n); εi is e

ϕα e (v,A)  with ϕαei (v, A) the Shapley weighted argument α i i being the Shapley value w.r.t. the fuzzy measure v on  = { A αi }i=1,2,...,n for α i (i = 1, 2, . . . , n), and ϕj (μ, N ) is the Shapley value w.r.t. the fuzzy measure μ on ordered set N = {1, 2, . . . , n} for the jth position. Remark 6 If μ and v are both additive, then the I-IIULHSG operator reduces to be the induced intervalvalued intuitionistic uncertain linguistic hybrid weighted geometric mean (I-IIULHWGM) operator.

j=1

n P j=1

ε(j)

1 , u2 , α 2 , . . . , un , α n ) = I-IIULHSGμ,v (u1 , α  [s n Q

s

(j)

n P j=1

εj

n 

n 

ϕj (μ,N )

[1 −

e ϕj (μ,N )ϕα e j (v,A)

ϕj (μ,N )ϕα e (j)

e ϕj (μ,N )ϕα e (j) (v,A)

n P

ul (α(j) ) j=1

φj (μ,N )φα e

(j)

e (v,A)

,

e ϕj (μ,N )ϕα e (j) (v,A)

n P

uu (α(j) ) j=1

e ϕj (μ,N )ϕα e (j) (v,A)

],

n 

e ϕj (μ,N )ϕα e (j) (v,A)

n P

(1 − vl (α(j) )) j=1

e ϕj (μ,N )ϕα e (j) (v,A)

j=1

where εj is the jth largest of the Shapley weighted argue ϕα e (v,A) α i i (i

= 1, 2, . . . , n).  = 1 (i = 1, 2, . . . , n), Remark 8 When ϕαei (v, A) n the I-IIULHSG operator reduces to be the induced intervalvalued intuitionistic uncertain linguistic ordered Shapley geometric (I-IIULOSG) operator. I-IIULOSGμ (u1 , α 1 , . . . , un , α n ) =

τ (α(j) ) j=1

],

e (v,A)

j=1

j=1

ment

ϕj (μ,N )ϕα e (j)

j=1

α1 , α 2 , . . . , α n ) = IIULHOSGμ,v ( n

θ(α(j) ) j=1

e (v,A)

e ϕj (μ,N )ϕα e (j) (v,A)

n Q

[

n P

n P

j=1

where wj = μ(j) and ωαej = v( αj ) for each j = 1, 2, . . . , n. Remark 7 When ui = uj (j = 1, 2, . . . , n), the I-IIULHSG operator reduces to be the interval-valued intuitionistic uncertain linguistic hybrid ordered Shapley geometric (IIULHOSG) operator.

,

e ϕj (μ,N )ϕα e (j) (v,A)

j=1

wj wj ω α e

e φα e j (v,A)

α j

Remark 10 If each IVIULN α  i (i = 1, 2, . . . , n) degenerates to be an intuitionistic uncertain linguistic number, namely, u l (αi ) = uu (αi ) and vl (αi ) = vu (αi ), then we get the induced intuitionistic uncertain linguistic hybrid Shapley geometric (I-IULHSG) operator. Theorem 2 Let α i = ([sθ(αi ) , sτ (αi ) ], [ul (αi ), uu (αi )], [vl (αi ), vu (αi )]) (i = 1, 2, . . . , n) be the collection of IVIULNs in IVIULN(X), and μ and v be the fuzzy mea = { αi }i∈N , respecsure on N = {1, 2, . . . , n} and A tively. Then their collective value using the I-IIULHSG operator is also an IVIULN in IVIULN(X) denoted by

I-IIULHWGMw, ω(u1 , α 1 , . . . , un , α n ) = n

n

j=1

ϕj (μ,N )

j=1

1 Remark 9 When ϕi (v, N ) = (i = 1, 2, . . . , n), the n I-IIULHSG operator reduces to be the interval-valued intuitionistic uncertain linguistic Shapley geometric (IIULSG) operator.

n

j=1

φ (μ,N )

j α (j)

where u(j) is the jth largest of u i (i = 1, 2, . . . , n).

1−

n  j=1

e ϕj (μ,N )ϕα e (j) (v,A)

n P

(1 − vu (α(j) )) j=1

e ϕj (μ,N )ϕα e (j) (v,A)

,

⎞ ⎟ ]⎟ ⎠.

Proof The proof of Theorem 2 is similar to that of Theorem 1.  Definition 7 Let α i = ([sθ(αi ) , sτ (αi ) ], [ul (αi ), uu (αi )], [vl (αi ), vu (αi )]) and βi = ([sθ(βi ) , sτ (βi ) ], [ul (βi ), uu (βi )], [vl (βi ), vu (βi )]) (i = 1, 2, . . . , n) be two collections of IVIULNs in IVIULN(X). α  i and βi are said to be comonotonic if α (1)  α (2)  . . .  α (n) iff β(1)  β(2)  . . .  β(n)

Fanyong Meng et al.: Some interval-valued intuitionistic uncertain linguistic hybrid Shapley operators

where (·) denotes a permutation on N . Theorem 3 Let α i = ([sθ(αi ) , sτ (αi ) ], [ul (αi ), uu (αi )], [vl (αi ), vu (αi )]) and βi = ([sθ(βi ) , sτ (βi ) ], [ul (βi ), uu (βi )], [vl (βi ), vu (βi )]) (i = 1, 2, . . . , n) be two collections of IVIULNs in IVIULN(X), and μ and v be the fuzzy mea = { αi }i∈N , respecsure on N = {1, 2, . . . , n} and A tively. (i) Commutativity  i , then If α i (i = 1, 2, . . . , n) is a permutation of α I-IIULHSAμ,v (u1 , α 1 , . . . , un , α n ) = 1 , . . . , un , α n ), I-IIULHSAμ,v (u1 , α I-IIULHSGμ,v (u1 , α 1 , . . . , un , α n ) =

457

for any A, B ⊆ N with A ∩ B = ∅, where λ > −1. For a finite set N , the λ-fuzzy measure g λ can be equivalently expressed by  ⎧  ⎪ 1  ⎪ ⎪ [1 + λgλ (i)] − 1 , λ = 0 ⎨ λ i∈A gλ (A) = . (5)  ⎪ ⎪ g (i), λ = 0 ⎪ λ ⎩ i∈A

From μ(N ) = 1, we know λ is determined by  [1 + λgλ (i)] = 1 + λ.

(6)

i∈N

(iii) Comonotonicity If α i and βi are comonotonic and (·) is a permutation on N , such that α (i)  β(i) for all i, then

Thus, when each g λ (i) is given, we can get the value of λ. From (5), for a set with n elements we only need n values to get a λ-fuzzy measure. Similar to Definitions 5 and Definitions 6, we give the following definitions for the I-IIULHλSA and IIIULHλSG operators. Definition 9 An I-IIULHλSA operator is a mapping I-IIULHλSA: IVIULN(X) n → IVIULN(X) defined on the set of second arguments of two tuples u1 , α 1 , u2 , α 2 , . . . , un , α n  with a set of orderinducing variables u i (i = 1, 2, . . . , n), denoted by

1 , . . . , un , α n )  I-IIULHSAμ,v (u1 , α

I-IIULHλSAλ1 ,λ2 (u1 , α 1 , . . . , un , α n ) =

1 , . . . , un , α n ). I-IIULHSGμ,v (u1 , α (ii) Idempotency i = α  for all If all α i (i = 1, 2, . . . , n) are equal, i.e., α i, then 1 , . . . , un , α n ) = α  I-IIULHSAμ,v (u1 , α I-IIULHSGμ,v (u1 , α 1 , . . . , un , α n ) = α .

I-IIULHSAμ,v (u1 , β1 , . . . , un , βn )

n 

1 , . . . , un , α n )  I-IIULHSGμ,v (u1 , α  I-IIULHSGμ,v (u1 , β1 , . . . , un , βn ).

j=1

(iv) Boundary For IVIULNs α i (i = 1, 2, . . . , n), it has min{ α1 , α 2 , . . . , α n }  1 , . . . , un , α n )  I-IIULHSAμ,v (u1 , α 2 , . . . , α n } max{ α1 , α min{ α1 , α 2 , . . . , α n }  1 , . . . , un , α n )  I-IIULHSGμ,v (u1 , α max{ α1 , α 2 , . . . , α n }. Since the properties given in Theorem 3 are easily proved, we here no longer give their proof in detail. 3.3 Special case From Definition 4, we know that the fuzzy measures are defined on the power set, which makes the problem exponentially complex. The λ-fuzzy measure [45] seems to well deal with this issue. Definition 8 [45] Let g λ : P (N ) → [0, 1] be a fuzzy measure. gλ is called a λ-fuzzy measure if gλ (A ∪ B) = gλ (A) + gλ (B) + λgλ (A)gλ (B)

n 

ϕj (gλ1 , N ) ε(j) (7)

 ϕj (gλ1 , N )ϕαe(j) (gλ2 , A)

j=1

where u(j) is the jth largest of u i (i = 1, 2, . . . , n);  αi with εi is the Shapley weighted argument ϕ αei (gλ2 , A)  being the Shapley value w.r.t. the λ-fuzzy ϕαei (gλ2 , A)  = { αi }i=1,...,n for α i (i = 1, 2, . . . , n); measure gλ2 on A and ϕj (gλ1 , N ) is the Shapley value w.r.t. the λ-fuzzy measure gλ1 on ordered set N = {1, 2, . . . , n} for the jth position. Definition 10 An I-IIULHλSG operator is a mapping I-IIULHλSG: IVIULN(X) n → IVIULN(X) defined on the set of second arguments of two tuples u1 , α 1 , u2 , α 2 , . . . , un , α n  with a set of orderinducing variables u i (i = 1, 2, . . . , n), denoted by 1 , . . . , un , α n ) = I-IIULHλSGλ1 ,λ2 (u1 , α n

j=1

ϕj (gλ ,N ) 1 Pn e ϕ (g ,N )ϕα e (j) (gλ2 ,A) j=1 j λ1

ε(j)

(8)

where u(j) is the jth largest of u i (i = 1, 2, . . . , n); e ϕα e i (gλ2 ,A)

i εi is the Shapley weighted argument α

with

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Journal of Systems Engineering and Electronics Vol. 25, No. 3, June 2014

 being the Shapley value w.r.t. the λ-fuzzy ϕαei (gλ2 , A)  = { measure gλ2 on A αi }i=1,...,n for α i (i = 1, 2, . . . , n); and ϕj (gλ1 , N ) is the Shapley value w.r.t. the λ-fuzzy measure gλ1 on ordered set N = {1, 2, . . . , n} for the jth position. Similar to the discussion about the I-IIULHSA and I-IIULHSG operators, the I-IIULHλSA and I-IIULHλSG operators also satisfy the properties studied in Section 3.2.

4. Approach to multi-attribute group decision making under interval-valued intuitionistic uncertain linguistic environment Multi-attribute group decision making (MAGDM) problem is the process of finding the best alternative(s) from all of the feasible alternatives where all the alternatives can be evaluated according to a number of attributes. In general, multi-attribute group decision making problem includes uncertain and imprecise data and information. Consider a multi-attribute group decision making problem, in which the experts and attributes are correlative, respectively. Let A = {a1 , a2 , . . . , am } be the set of alternatives, C = {c1 , c2 , . . . , cn } be the set of attributes, and E = {e1 , e2 , . . . , eq } be the set of experts. Assume that akij )m×n is the IVIULN matrix, where  a kij = Ak = ( k k k k ([sθ(akij ) , sτ (akij ) ], [ul (aij ), uu (aij )], [vl (aij ), vu (aij )]) is the IVIULN given by the expert e k , for the alternative ai ∈ A w.r.t. the attribute c j ∈ C. Based on the introduced hybrid operators, this section gives a new decision-making method to interval-valued intuitionistic uncertain linguistic multi-attribute group decision making. The main decision procedure can be described as follows: Step 1 Assume that the evaluation of the alternative ai w.r.t. the criterion c j is an IVIULN  akij = k k k k ([sθ(akij ) , sτ (akij ) ], [ul (aij ), uu (aij )], [vl (aij ), vu (aij )]) (i = 1, 2, . . . , m; j = 1, 2, . . . , n) given by the expert e k (k = 1, 2, . . . , q). Then, we obtain the following IVIULN matrix ⎛ ⎜ ⎜ A =⎜ ⎝ k

 ak11  ak21 .. .

 ak12  ak22 .. .

... ... .. .

 ak1n  ak2n .. .

 akm1

 akm2

...  akmn

⎞ ⎟ ⎟ ⎟. ⎠

Step 2 Confirm the fuzzy measure g λE (ek ) of each expert ek (k = 1, 2, . . . , q). According to (6), calculate the parameter λE of the expert set E, and compute their Shapley values by (1) and (5). Step 3 Confirm the fuzzy measure g λK (k) of each ordered position k. According to (6), calculate the parameter λK of the ordered set K = {1, 2, . . . , q}, and compute their Shapley values by (1) and (5).

Step 4 For each pair (i, j)(i = 1, 2, . . . , m; j = 1, 2, . . . , n), calculate aij ) = d( akij , 

1  |θ(akij ) − θ(aij )| + |τ (akij ) − τ (aij )| + 6 t

|ul (akij ) − ul (aij )| + |uu (akij ) − uu (aij )|+  |vl (akij ) − vl (aij )| + |vu (akij ) − vu (aij )|   q ul (akij ) , aij = [s P , s ], where  k k q θ(a ) q τ (a ) P ij ij q q q k=1 k=1 k=1    q q q  uu (akij ) vl (akij )  vu (akij ) , , . q q q k=1

k=1

k=1

akij ,  aij ) (k = 1, 2, . . . , q), use the Let uk = d( I-IIULHλSA operator or the I-IIULHλSG operator to get the comprehensive IVIULN  a ij = ([sθ(aij ) , sτ (aij ) ], [ul (aij ), uu (aij )], [vl (aij ), vu (aij )]) and the comprehensive matrix ⎞ ⎛ a12 . . .  a1n  a11  ⎜  a22 . . .  a2n ⎟ ⎟ ⎜ a21  A=⎜ . .. ⎟ . .. .. ⎝ .. . . ⎠  am1

 am2

...  amn

where ϕek (gλE , E) and ϕk (gλK , K) are the Shapley values of the expert e k and the ordered position k, respectively. ϕe (g E ,E) k λ

εij akij and εij akij k = ϕek (gλE , E) k =

Step 5 Confirm the fuzzy measure g λC (ci ) of each attribute cj (j = 1, 2, . . . , n). According to (6), calculate the parameter λC of the attribute set C, and compute their Shapley values by (1) and (5). Step 6 Confirm the fuzzy measure g λN (j) of each ordered position j. According to (6), calculate the parameter λN of the ordered position set N = {1, 2, . . . , n}, and compute their Shapley values by (1) and (5). Step 7 For each j = 1, 2, . . . , n, let   m  m  a+ , max ul (aij ), smax ,s m j = θ(a ) max τ (a ) i=1

ij

i=1

i=1

ij

  m  m m max uu (aij ) , max vl (aij ), max vu (aij ) . i=1

i=1

Calculate

i=1

a+ d( aij ,  j )= m

m

θ(aij )| + |τ (aij ) − max τ (aij )| 1  |θ(aij ) − max i=1 i=1 + 6 t m

m

|ul (aij ) − max ul (aij )| + |uu (aij ) − max uu (aij )|+ i=1 i=1  m m |vl (aij ) − max vl (aij )| + |vu (aij ) − max vu (aij )| i=1

i=1

for each pair (i, j) (i = 1, 2, . . . , m; j = 1, 2, . . . , n).

Fanyong Meng et al.: Some interval-valued intuitionistic uncertain linguistic hybrid Shapley operators

For each i = 1, 2, . . . , m, let uj = dij (j = 1, 2, . . . , n), again use the I-IIULHλSA operator or the IIIULHλSG operator to get the comprehensive IVIULN  ai = ([sθ(ai ) , sτ (ai ) ], [ul (ai ), uu (ai )], [vl (ai ), vu (ai )]) of the alternative ai (i = 1, 2, . . . , m), where ϕcj (gλC , C) and ϕj (gλN , N ) are the Shapley values of the attribute c j and the ordered position j, respectively. ϕc (gλC ,C)

aij and εij =  aij j εij = ϕcj (gλC , C) m

m ,s m ], [max ul (ai ),  a = ([smax θ(a ) max τ (a ) i

i=1

i=1

i=1

i

m

m

m

i=1

i=1

i=1

max uu (ai )], [min vl (ai ), max vu (ai )])  a− = ([s

m

min θ(ai )

,s

i=1

m

m

min τ (ai )

], [min ul (ai ), i=1

i=1

m

m

m

i=1

i=1

i=1

min uu (ai )], [min vl (ai ), max vu (ai )]). Calculate di =

a− ) d( ai ,  d( ai ,  a+ ) + d( ai ,  a− )

for each i = 1, 2, . . . , m, where a+ ) = d( ai ,  m

m

θ(ai )| + |τ (ai ) − max τ (ai )| 1  |θ(ai ) − max i=1 i=1 + 6 t m

m

|ul (ai ) − max ul (ai )| + |uu (ai ) − max uu (ai )|+ i=1 i=1  m m |vl (ai ) − min vl (ai )| + |vu (ai ) − min vu (ai )| i=1

and  1 6

i=1

a− ) = d( ai , 

 m m |θ(ai ) − min θ(ai )| + |τ (ai ) − min τ (ai )| i=1

i=1

t

m

m

i=1

i=1

|ul (ai ) − min ul (ai )| + |uu (ai ) − min uu (ai )|+ m

m

i=1

i=1

|vl (ai ) − max vl (ai )| + |vu (ai ) − max vu (ai )|. Rank the alternatives according to d i (i = 1, 2, . . . , m), and then select the best one(s).

5. Application example

Step 8 Let +

459

+

In this section, we consider the evaluation problem of the new rural developing level [32,47]. Developing new rural is an important measurement to develop rural economy and to improve rural living standards. Its goal can be described as follows: development of production, affluent living, rural civilization, clean and tidy village, and democratic management. Because the evaluation indexes of new rural developing level are mostly qualitative indicators, it is very suitable for proposed methods to evaluate the new rural developing level. Firstly, we adopt the evaluation index systems, including: (c1 ) development of production, (c 2 ) affluent living, (c3 ) rural civilization, (c 4 ) clean and tidy village, and (c 5 ) democratic management. In order to evaluate the new rural developing level of Shandong province, we invite three experts to have investigated 24 countryside townships and have visited 132 villagers, the village party branch secretaries and the deputy township mayors in Weifang (a 1 ), Yantai (a2 ), Binzhou (a 3 ) and Liaocheng (a 4 ) which are secondary cities belong to Shandong province, and obtain the first-hand information about the status of the new rural developing in Shandong province. Then, we have compiled data, and formed the evaluation information for four cities w.r.t. three experts (e 1 , e2 , e3 ) using the linguistic term set S = {s0 , s1 , s2 , s3 , s4 , s5 , s6 } under the above four attributes. The IVIULN matrices are listed in Table 1 – Table 3.

Table 1 IVIULN matrix given by the expert e1 c1 ([s4 , s5 ],[0.7,0.8],[0.1,0.2]) ([s5 , s5 ],[0.6,0.6],[0.1,0.2]) ([s4 , s4 ],[0.7,0.7],[0.2,0.2]) ([s3 , s4 ],[0.6,0.7],[0.2,0.3])

c2 ([s5 , s5 ],[0.6,0.6],[0.1,0.3]) ([s5 , s6 ],[0.7,0.7],[0.2,0.2]) ([s4 , s4 ],[0.7,0.8],[0.1,0.2]) ([s3 , s3 ],[0.5,0.6],[0.2,0.3])

c3 ([s5 , s6 ],[0.8,0.8],[0.1,0.1]) ([s4 , s5 ],[0.5,0.6],[0.2,0.3]) ([s5 , s5 ],[0.7,0.7],[0.1,0.2]) ([s4 , s4 ],[0.6,0.7],[0.2,0.3])

c4 ([s4 , s4 ],[0.8,0.8],[0.1,0.1]) ([s4 , s5 ],[0.5,0.6],[0.3,0.3]) ([s5 , s5 ],[0.7,0.8],[0.1,0.2]) ([s3 , s4 ],[0.7,0.7],[0.2,0.2])

c5 ([s5 , s5 ],[0.7,0.8],[0.1,0.2]) ([s4 ,s5 ],[0.9,0.9],[0.0,0.1]) ([s3 ,s4 ],[0.8,0.8],[0.1,0.1]) ([s5 ,s6 ],[0.6,0.8],[0.2,0.2])

Table 2 IVIULN matrix given by the expert e2 c1 ([s5 ,s6 ],[0.6,0.7],[0.1,0.1]) ([s5 ,s5 ],[0.5,0.7],[0.2,0.2]) ([s5 ,s5 ],[0.6,0.7],[0.0,0.2]) ([s5 ,s5 ],[0.7,0.8],[0.1,0.2])

c2 ([s5 ,s5 ],[0.7,0.7],[0.1,0.1]) ([s4 ,s5 ],[0.6,0.7],[0.2,0.2]) ([s4 ,s5 ],[0.8,0.9],[0.1,0.1]) ([s4 ,s4 ],[0.5,0.6],[0.2,0.3])

c3 ([s4 ,s5 ],[0.9,0.9],[0.0,0.1]) ([s4 ,s5 ],[0.7,0.7],[0.1,0.2]) ([s4 ,s4 ],[0.6,0.6],[0.2,0.2]) ([s3 ,s3 ],[0.9,0.9],[0.0,0.1])

c4 ([s5 ,s5 ],[0.7,0.8],[0.1,0.2]) ([s6 ,s6 ],[0.6,0.7],[0.1,0.1]) ([s4 ,s4 ],[0.7,0.7],[0.2,0.2]) ([s3 ,s4 ],[0.8,0.8],[0.1,0.2])

c5 ([s4 ,s6 ],[0.6,0.6],[0.1,0.1]) ([s5 ,s5 ],[0.8,0.9],[0.1,0.1]) ([s5 ,s5 ],[0.7,0.8],[0.1,0.2]) ([s4 ,s4 ],[0.8,0.8],[0.1,0.1])

460

Journal of Systems Engineering and Electronics Vol. 25, No. 3, June 2014 Table 3 IVIULN matrix given by the expert e3

c1 [s5 ,s5 ],[0.7,0.8],[0.1,0.1]) ([s5 ,s6 ],[0.6,0.7],[0.1,0.2]) ([s5 ,s5 ],[0.8,0.8],[0.0,0.1]) ([s4 ,s5 ],[0.8,0.9],[0.1,0.1])

c2 ([s5 ,s5 ],[0.8,0.9],[0.1,0.1]) ([s5 ,s6 ],[0.7,0.7],[0.1,0.2]) ([s5 ,s5 ],[0.7,0.8],[0.1,0.2]) ([s4 ,s4 ],[0.8,0.8],[0.0,0.2])

c3 ([s5 ,s5 ],[0.8,0.9],[0.1,0.1]) ([s5 ,s5 ],[0.8,0.8],[0.1,0.1]) ([s4 ,s4 ],[0.7,0.8],[0.1,0.2]) ([s4 ,s5 ],[0.8,0.8],[0.0,0.1])

c4 ([s5 ,s6 ],[0.7,0.8],[0.2,0.2]) ([s5 ,s5 ],[0.9,0.9],[0.1,0.1]) ([s4 ,s4 ],[0.7,0.8],[0.1,0.1]) ([s5 ,s5 ],[0.7,0.7],[0.1,0.2])

c5 ([s4 ,s4 ],[0.8,0.8],[0.1,0.1]) ([s5 ,s5 ],[0.8,0.8],[0.2,0.2]) ([s4 ,s4 ],[0.8,0.9],[0.0,0.1]) ([s4 ,s5 ],[0.7,0.8],[0.1,0.1])

Next, we can utilize the proposed procedure to get the most desirable alternative(s). Step 1 Assume that the fuzzy measure g λE (ek ) (k = 1, 2, 3) of each expert is given by g λE (e1 ) = 0.2, gλE (e2 ) = 0.35 and gλE (e3 ) = 0.15. From (6), we get λE = 1.76. By (5), we obtain the following fuzzy measures of combinations in expert set E.

Step 2 Assume that the fuzzy measure g λK (k) (k = 1, 2, 3) of each ordered position is given by g λK (1) = 0.2, gλK (2) = 0.3 and gλK (3) = 0.4. From (6), we can have λK = 0.37. By (5), we get the following fuzzy measures of combinations in ordered set K = {1, 2, 3}.

gλE (e1 , e2 ) = 0.67, gλE (e1 , e3 ) = 0.4

gλK (2, 3) = 0.74, gλK (1, 2, 3) = 1.

gλK (1, 2) = 0.52, gλK (1, 3) = 0.63

From (1), we get the ordered position Shapley values gλE (e2 , e3 ) = 0.59, gλE (e1 , e2 , e3 ) = 1. ϕ1 (gλK , K) = 0.24, ϕ2 (gλK , K) = 0.33

From (1), we get the expert Shapley values ϕe1 (gλE , E) = 0.30, ϕe2 (gλE , E) = 0.45 ϕe3 (gλE , E) = 0.23.  a11 =

ϕ3 (gλK , K) = 0.44. Step 3 Use the I-IIULHλSA operator, we get the comprehensive IVIULN as

a211 ⊕ ϕ2 (gλK , K)ϕe1 (gλE , E) a111 ⊕ ϕ3 (gλK , K)ϕe3 (gλE , E) a311 ϕ1 (gλK , K)ϕe2 (gλE , E) = ϕ1 (gλK , K)ϕe2 (gλE , E) + ϕ2 (gλK , K)ϕe1 (gλE , E) + ϕ3 (gλK , K)ϕe3 (gλE , E) ([s4.68 , s5.35 ], [0.67, 0.77], [0.1, 0.12]).

Similar to the calculation of  a 11 , we get the following

comprehensive IVIULN matrix in Table 4.

Table 4 Comprehensive IVIULN matrix c1 ([s4 ,s5 ],[0.7,0.8],[0.1,0.1]) ([s5 ,s5 ],[0.6,0.7],[0.1,0.2]) ([s5 ,s5 ],[0.7,0.7],[0.1,0.2]) ([s4 ,s5 ],[0.7,0.8],[0.1,0.2])

c2 ([s5 ,s5 ],[0.7,0.8],[0.1,0.1]) ([s5 ,s6 ],[0.7,0.7],[0.2,0.2]) ([s4 ,s5 ],[0.7,0.8],[0.1,0.2]) ([s4 ,s4 ],[0.6,0.7],[0.2,0.3])

c3 ([s5 ,s6 ],[0.8,0.9],[0.1,0.1]) ([s4 ,s5 ],[0.7,0.7],[0.1,0.2]) ([s4 ,s4 ],[0.7,0.7],[0.1,0.2]) ([s4 ,s4 ],[0.8,0.8],[0.1,0.1])

Step 4 Assume that the fuzzy measure g λC (cj ) (j = 1, 2, 3, 4, 5) of each attribute is given by

c4 ([s5 ,s5 ],[0.7,0.8],[0.1,0.2]) ([s5 ,s6 ],[0.7,0.8],[0.1,0.1]) ([s4 ,s4 ],[0.7,0.8],[0.1,0.2]) ([s3 ,s4 ],[0.7,0.8],[0.1,0.2])

c5 ([s4 ,s5 ],[0.7,0.8],[0.1,0.1]) ([s5 ,s5 ],[0.8,0.9],[0.1,0.1]) ([s4 ,s4 ],[0.8,0.8],[0.1,0.1]) ([s4 ,s5 ],[0.7,0.8],[0.1,0.1])

gλC (c2 , c3 ) = 0.57, gλC (c4 , c5 ) = 0.39 gλC (c1 , c2 , c3 ) = 0.73, gλC (c1 , c2 , c4 ) =

gλC (c1 ) = 0.2, gλC (c2 ) = 0.3, gλC (c3 ) = 0.3

gλC (c1 , c3 , c4 ) = gλC (c2 , c3 , c5 ) = 0.69

gλC (c4 ) = 0.25, gλC (c5 ) = 0.15.

gλC (c1 , c2 , c5 ) = gλC (c1 , c3 , c5 ) = 0.61

From (6), we get λ C = −0.34. By (5), we obtain the following fuzzy measures of combinations in attribute set C.

gλC (c1 , c4 , c5 ) = 0.56, gλC (c2 , c3 , c4 ) = 0.77 gλC (c2 , c4 , c5 ) = gλC (c3 , c4 , c5 ) = 0.65 gλC (c1 , c2 , c3 , c4 ) = 0.92, gλC (c1 , c2 , c3 , c5 ) = 0.84

gλC (c1 , c2 ) = gλC (c1 , c3 ) = 0.48, gλC (c1 , c4 ) =

gλC (c1 , c2 , c4 , c5 ) = gλC (c1 , c3 , c4 , c5 ) = 0.8

gλC (c2 , c5 ) = gλC (c3 , c5 ) = 0.43, gλC (c1 , c5 ) = 0.34

gλC (c2 , c3 , c4 , c5 ) = 0.88, gλC (C) = 1.

Fanyong Meng et al.: Some interval-valued intuitionistic uncertain linguistic hybrid Shapley operators

From (1), we get the attribute Shapley values ϕc1 (gλC , C) = 0.17, ϕc2 (gλC , C) = ϕc3 (gλC , C) = 0.25 ϕc4 (gλC , C) = 0.21, ϕc5 (gλC , C) = 0.12. Step 5 Assume that the fuzzy measure g λN (j) (j = 1, 2, 3, 4, 5) of each ordered position is given by gλN (1) = 0.15, gλN (2) = 0.25, gλN (3) = 0.3 gλN (4) = 0.35, gλN (5) = 0.4. From (6), we have λ N = −0.5. By (5), we get the following fuzzy measures of combinations in ordered set N = {1, 2, 3, 4, 5}

461

0.45, d3 = 0.31, d4 = 0.04, namely, d1 > d2 > d3 > d4 . Thus, Weifang (a 1 ) is the best choice. The ranking results obtained here are the same as that Liu’s. The method given in [29] does not consider the interaction among experts and among attributes. Furthermore, the IIULHG operator [29] does not satisfy boundary and idempotent. In Step 7, if we adopt the ranking method given in [29], from the comprehensive IVIULNs  a i (i = 1, 2, 3, 4), we get the following expectation values a2 ) = 4.05 E( a1 ) = 4.62, E( a4 ) = 3.09 E( a3 ) = 3.60, E(

gλN (1, 4) = 0.47, gλN (1, 5) = 0.52

which also show that Weifang (a 1 ) is the best choice. In the above example, if we apply the I-IIULHλSG operator, then the comprehensive IVIULNs of alternatives are obtained by

gλN (2, 3) = 0.51, gλN (2, 4) = 0.56

 a1 = ([s4.31 , s4.68 ], [0.76, 0.81], [0.09, 0.13])

gλN (2, 5) = gλN (3, 4) = 0.60

 a2 = ([s4.82 , s5.34 ], [0.67, 0.73], [0.14, 0.17])

gλN (3, 5) = 0.64, gλN (4, 5) = 0.68

 a3 = ([s4.28 , s4.47 ], [0.71, 0.78], [0.1, 0.17])

gλN (1, 2, 3) = 0.62, gλN (1, 2, 4) = 0.66

 a4 = ([s3.55 , s3.98 ], [0.73, 0.79], [0.11, 0.18]).

gλN (1, 2, 5) = 0.71, gλN (1, 3, 4) = 0.70

From the comprehensive IVIULNs  a i (i = 1, 2, 3, 4) of alternatives, it gets

gλN (1, 2) = 0.38, gλN (1, 3) = 0.43

gλN (1, 3, 5) = 0.74, gλN (1, 4, 5) = 0.78 gλN (2, 3, 4) = 0.77, gλN (2, 3, 5) = 0.81 gλN (2, 4, 5) = 0.85, gλN (3, 4, 5) = 0.88 gλN (1, 2, 3, 4) = 0.86, gλN (1, 2, 3, 5) = 0.90 gλN (1, 2, 4, 5) = 0.93, gλN (1, 3, 4, 5) = 0.96 gλN (2, 3, 4, 5) = gλN (1, 2, 3, 4, 5) = 1. From (1), we get the ordered position Shapley values ϕ1 (gλN , N ) = 0.1, ϕ2 (gλN , N ) = 0.15

d1 = 0.72, d2 = 0.63, d3 = 0.49, d4 = 0.21 namely, d1 > d2 > d3 > d4 , which also shows that Weifang (a1 ) is the best choice. However, if we use the ranking method given in [29], from the comprehensive IVIULNs  a i (i = 1, 2, 3, 4) we get the following expectation values: a2 ) = 3.92 E(˜ a1 ) = 3.76, E(˜ a4 )y = 3.03 E(˜ a3 ) = 3.52, E(

 a1 = ([s4.75 , s5.19 ], [0.88, 0.91], [0.03, 0.04])

which shows that Yantai (a 2 ) is the best choice. From the given example, we get the same ranking results by using TOPSIS method w.r.t the I-IIULHλSA and I-IIULHλSG operators. Howerer the different ranking results are obtained by applying the Liu’s ranking method. Thus, the decision maker should properly select the ranking methods according to his interest and the actual needs.

 a2 = ([s4.86 , s5.36 ], [0.7, 0.76], [0.14, 0.16])

6. Conclusions

 a3 = ([s4.34 , s4.51 ], [0.72, 0.79], [0.09, 0.16])

Based on the Shapley function, we have researched some hybrid Shapley operators on interval-valued intuitionistic uncertain linguistic sets, in which the interaction among elements is considered. Meantime, some special cases are discussed, and some desirable properties are studied,

ϕ3 (gλN , N ) = 0.2, ϕ4 (gλN , N ) = 0.24 ϕ5 (gλN , N ) = 0.3. Step 6 Again using the I-IIULHλSA operator, we get the following comprehensive IVIULNs of alternatives

 a4 = ([s3.69 , s4.1 ], [0.72, 0.77], [0.13, 0.19]). Step 7 From the comprehensive IVIULNs  a i (i = 1, 2, 3, 4) of alternatives, we get d 1 = 0.94, d2 =

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such as commutativity, idempotency, comonotonicity and boundary. Since fuzzy measures are defined on the power set, it makes the problem exponentially complex. In order to simplify the complexity of solving a fuzzy measure and reflect the interactive characteristics among elements, we further introduce two interval-valued intuitionistic uncertain linguistic hybrid λ-Shapley operators. Based on the given hybrid Shapley operators and TOPSIS method, an approach to multi-attribute group decision making under interval-valued intuitionistic uncertain linguistic environment is developed. It is worth pointing out that when there are no interactive characteristics among elements, we get the corresponding decision making method based on additive measures.

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Biographies Fanyong Meng was born in 1981. He received his Ph.D. degree in management science and engineering from Beijing Institute of Technology in 2011. Currently, he is an associate prrofessor in Qingdao Technological University. His current research interests include fuzzy mathematics, decision making and game theory. His current research interests include fuzzy mathematics, games theory and decision making. E-mail: [email protected] Chunqiao Tan was born in 1975. He received his Ph.D. degree in management science and engineering from Beijing Institute of Technology in 2006. Currently, he is an associate professor at School of Business, Central South University, Changsha, China. His current research interests include game theory, decision making under uncertainty. E-mail: [email protected] Qiang Zhang was born in 1955. He received his Ph.D. degree in School of Traffic and Transportation from Southwest Jiaotong University in 1999. Currently, he is a professor in School of Management and Economics, Beijing Institute of Technology. His current research interests include management decisions in quantitative theory and method, the modern logistics and supply chain management, uncertain system theory and application. E-mail: [email protected]