2-Tuple linguistic hybrid arithmetic aggregation operators and application to multi-attribute group decision making

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Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys

2-Tuple linguistic hybrid arithmetic aggregation operators and application to multi-attribute group decision making

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College of Information Technology, Jiangxi University of Finance and Economics, Nanchang, Jiangxi 330013, China

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a r t i c l e

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Shu-Ping Wan ⇑, Jiu-Ying Dong

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Article history: Received 22 April 2012 Received in revised form 28 January 2013 Accepted 1 February 2013 Available online xxxx

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Keywords: Multi-attribute group decision making Linguistic preference 2-Tuple linguistic information Hybrid aggregation operator

a b s t r a c t The focus of this paper is on multi-attribute group decision making (MAGDM) problems in which the attribute values, attribute weights, and expert weights are all in the form of 2-tuple linguistic information, which are solved by developing a new decision method based on 2-tuple linguistic hybrid arithmetic aggregation operator. First, the operation laws for 2-tuple linguistic information are defined and the related properties of the operation laws are studied. Hereby some hybrid arithmetic aggregation operators with 2-tuple linguistic information are developed, involving the 2-tuple hybrid weighted arithmetic average (THWA) operator, the 2-tuple hybrid linguistic weighted arithmetic average (T-HLWA) operator, and the extended 2-tuple hybrid linguistic weighted arithmetic average (ET-HLWA) operator. In the proposed decision method, the individual overall preference values of alternatives are derived by using the extend 2-tuple weighted arithmetic average operator (ET-WA). Using the ET-HLWA operator, all the individual overall preference values of alternatives are further integrated into the collective ones of alternatives, which are used to rank the alternatives. A real example of personnel selection is given to illustrate the developed method and the comparison analysis demonstrates the universality and flexibility of the method proposed in this paper. Ó 2013 Published by Elsevier B.V.

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1. Introduction Multi-attribute group decision making (MAGDM) problems with linguistic information arise from a wide range of real-world situations [1–33]. There are several kinds of researches on linguistic MAGDM, such as linguistic preference MAGDM [4–7], uncertain linguistic MAGDM [8–10], unbalanced linguistic MAGDM [11–14], and 2-tuple linguistic MAGDM [15–30]. Herrera et al. proposed 2-tuple linguistic representation model, which is composed of a linguistic term and a real number [15,17]. The 2-tuple linguistic model has exact characteristic in linguistic information processing. It avoided information distortion and losing which occur formerly in the linguistic information processing. In recent years, 2-tuple linguistic model has been extensively used in group decision making problems [15–30]. These researches can be roughly classified into three types. The first type is on information aggregation operators. Herrera and Martı´nez [17] developed 2-tuple arithmetic averaging (TAA) operator, 2-tuple weighted averaging (TWA) operator, 2-tuple ordered weighted averaging (TOWA) operator and extended 2-tuple weighted averaging (ET-WA) operator. Chang and Wen [18] proposed a novel technique combining 2-tuple and the ordered

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⇑ Corresponding author. Tel.: +86 13870620534.

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E-mail address: [email protected] (S.-P. Wan).

weighted averaging (OWA) operator for prioritization of failures in a product design failure mode and effect analysis. Wei [19] developed three new aggregation operators: generalized 2-tuple weighted average (G-2TWA) operator, generalized 2-tuple ordered weighted average (G-2TOWA) operator and induced generalized 2-tuple ordered weighted average (IG-2TOWA) operator. Zhang and Fan [20] proposed the extended 2-tuple ordered weighted averaging (ET-OWA) operator. Pei et al. [21] analyzed three kinds of weight information, i.e., belief degrees of linguistic evaluation values, weights of IAEA experts about indicators and strengths of indicators and proposed a weighted linguistic aggregation operator. Wei [22] proposed some new geometric aggregation operators: the extended 2-tuple weighted geometric (ET-WG) and the extended 2-tuple ordered weighted geometric (ET-OWG) operator and analyzed the properties of these operators. Then, a MAGDM method is presented based on the ET-WG and ET-OWG operators. Wei and Zhao [23] developed some dependent aggregation operators with 2-tuple linguistic information and applied to MAGDM. Dong et al. [24] suggested that the virtual linguistic variable and the 2-tuple linguistic variable can be mutually retranslated and thus proposed the OWA-based consensus operator under linguistic representation models. Xu et al. [25] adopted the virtual linguistic label to replace 2-tuple linguistic variable and proposed the linguistic power average operators including LPA, LPWA and LPOWA. They further developed the uncertain linguistic power average operators, such as ULPA, ULPWA and ULPOWA.

0950-7051/$ - see front matter Ó 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.knosys.2013.02.002

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Table 1 The collective overall preference values of alternatives obtained by the methods of [22] and this paper.

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Alternatives

A1

A2

A3

A4

A5

Ranking result

Wei [22] This paper

(s4, 0.25) (s4, 0.1923)

(s3, 0.43) (s4, 0.2500)

(s4, 0.15) (s5, 0.3269)

(s4, 0.33) (s5, 0.4116)

(s4, 0.32) (s4, 0.1308)

A4  A3  A1  A5  A2 A3  A4  A1  A5  A2

The second type is on multi-granularity linguistic information. Herrera and Martı´nez [26] proposed another method to solve the group decision making problem with multi-granularity linguistic information. They constructed linguistic hierarchy term sets and generalized transformation functions to unify the multi-granularity linguistic information into the linguistic 2-tuples. Herrera et al. [27] investigated a fusion method based on the linguistic 2-tuple representation model to handle the multi-granularity linguistic information. Gramajo and Martı´nez [28] proposed a linguistic decision support model for traffic prioritization in networking. The third type is on incomplete weight information. Wei et al. [29] investigated the MAGDM problems with 2-tuple linguistic assessment information, in which the information about attribute weights is incompletely known, and the attribute values take the form of linguistic assessment information. Wei [30] proposed the gray relational analysis method for 2-tuple linguistic MAGDM with incomplete weight information. In most of the proposals for solving MAGDM problems with 2tuple linguistic information found in literature, the importance degrees of experts (or decision makers) are usually represented by a numerical weighting vector or absolutely unknown (i.e., do not consider the importance degrees of different experts). In group decision making problems, if the weighting vector is known, weighted aggregating strategy is usually used to associate with the vector; if the weighting vector is absolutely unknown, the OWA strategy is often used. Different experts generally act as different roles in the decision making process since the experts have their different cultural, educational backgrounds, experiences and knowledge, and expertise related with the problem domain. In addition, it is more reasonable and natural to use linguistic variables to represent the importance degrees of experts, such as ‘‘very important’’, ‘‘important’’. However, most of existing aggregation operators for 2-tuples did not consider the weighted vector in the form of linguistic variables or 2-tuples. To overcome this drawback, this paper develops some new hybrid arithmetic aggregation operators for 2-tuples and then proposes a new method for MAGDM problems with 2-tuple linguistic assessments. The motivation of this paper is based on the following facts: (i) The existing aggregation operators with 2-tuple linguistic information are mainly focused on the weighted arithmetic (geometric) average and the ordered weighted arithmetic (geometric) average operators. There has less investigation about the hybrid aggregation operators with 2-tuple linguistic information. (ii) The hybrid aggregation operators can reflect the important degrees of both the given 2-tuples and the ordered positions of the 2-tuples, and thus are more generalized operators. They are usually used to integrate the individual overall preference values of alternatives into the collective ones of alternatives. To do so, each individual overall preference value should first be weighted by using the corresponding expert weight, which can sufficiently reflect the importance degrees of different experts. (iii) Wei [22] only considered the weight information of attributes in the form of the linguistic variables and didn’t consider the weight information of experts. The MAGDM

method [22] can not deal with the case that the weight information of attributes and experts takes the form of the 2-tuples. However, this case may appear in some real-life decision problems (see Section 5). These new hybrid arithmetic aggregation operators for 2-tuples proposed in this paper can be used to effectively dispose this case. (iv) The proposed method in this paper is more reasonable and flexible than the existing ones and can be applicable to real-life decision problems in many areas such as risk investment, performance evaluation of military system, engineering management, and supply chain.

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The rest of the paper is arranged as follows. Section 2 introduces the notions for 2-tuple linguistic information, and gives the operation laws and analyzes the properties of the operation laws. Section 3 presents the existing arithmetic aggregation operators for 2-tuple linguistic information and further develops some new 2-tuple linguistic hybrid arithmetic aggregation operators. Section 4 proposes the MAGDM method with 2-tuple linguistic assessments. A real personnel selection example is illustrated in Section 5. The comparison analyses with other methods are conducted in Section 6. Concluding remark is made in Section 7.

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2. 2-Tuple linguistic information

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2.1. Notions for 2-tuple linguistic information

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Definition 1 ([15,17]). Let S = {s0, s1, s2, . . . , st} be a finite and totally ordered discrete linguistic term set with odd cardinality, where si represents a possible value for a linguistic variable. b 2 [0, t] is a number value representing the aggregation result of linguistic symbolic. Then the function D used to obtain the 2-tuple linguistic information equivalent to b is defined as:

D : ½0; t ! S  ½0:5; 0:5Þ b ! DðbÞ ¼ ðsi ; aÞ

ð1Þ

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where i = round(b), a = b  i, a 2 [0.5, 0.5), round() is the usual round operation. si has the closest index label to b and a is the value of the symbolic translation.

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Definition 2 ([15,17]). Let S = {s0, s1, s2, . . . , st} be a linguistic term set and (si, a) be a linguistic 2-tuple. There is always a function D1, such that, from a 2-tuple it returns its equivalent numerical value b 2 [0, t]  R, which is

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D

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: S  ½0:5; 0:5Þ # ½0; t

D1 ðsi ; aÞ ¼ i þ a ¼ b:

ð2Þ

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From Definitions 1 and 2, we can conclude that the conversion of a linguistic term into a linguistic 2-tuple consists of adding a value 0 as symbolic translation:

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Dðsi Þ ¼ ðsi ; 0Þ:

ð3Þ

Definition 3 ([15,17]). Let (sk, ak) and (sl, al) be two 2-tuples, they should have the following properties

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(1) If k < l then (sk, ak) is smaller than (sl, al), denoted by (sk, ak) < (sl, al); (2) If k > l then (sk, ak) is bigger than (sl, al), denoted by (sk, ak) > (sl, al); (3) If k = l then (a) If ak = al, then (sk, ak) and (sl, al) represent the same information, denoted by (sk, ak) = (sl, al); (b) If ak < al, then (sk, ak) < (sl, al); (c) If ak > al, then (sk, ak) > (sl, al).

Example 2. (s1,0.1) (s3,0.2) = D(D1(s1,0.1)  D1(s3,0.2)) = D(3.52) = (s4 0.48).

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Example 3. 2(s 2 , 0.3) = D (2 D 1 (s 2 , 0.3)) = D (2  2.3) = D (4.6) = (s5 , 0.4).

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To preserve all the given information, we extend the discrete term set S to a continuous term set S ¼ fsl js0 6 sl 6 sq ; l 2 ½0; qg, where q P t and q is a sufficiently large positive integer, whose elements also meet all the characteristics above. If sl 2 S, then we call sl the original term, otherwise, we call sl the virtual term. In general, the decision maker uses the original linguistic term to evaluate attributes and alternatives, and the virtual linguistic terms can only appear in calculation [4,10].

Example 5. ðs1 ; 0:1Þðs3 ;0:2Þ ¼ DððD1 ðs1 ; 0:1ÞÞD ðs3 ;0:2Þ Þ ¼ Dð1:13:2 Þ ¼ Dð1:3566Þ ¼ ðs1 ; 0:3566Þ: From Definition 4, the following Theorem 1 can be easily proven:

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Theorem 1. Let (sk, ak), (sl, al) and (si, ai) be three 2-tuples and k, k P 0. The following equalities hold:

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(sk, ak)  (sl, al) = (sl, al)  (sk, ak); (sk, ak) (sl, al) = (sl, al) (sk, ak); k((sk, ak)  (sl, al)) = k (sk, ak)  k(sl, al); ((sk, ak)k)k = (sk, ak)kk, (sk, ak)k (sk, ak)k = (sk, ak)k+k; [(sk, ak)  (sl, al)] (si,ai) = [(sk, ak) (si, ai)]  [(sl,al) (si, ai)]; [(sk, ak) (sl, al)] (si, ai) = (sk, ak) [(sl, al) (si, ai)].

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Example (s5, 0.29).

4. (s2, 0.3)2 =

D((D1(s2, 0.3))2) = D(2.32) = D(5.29) =

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2.2. Operation laws and properties for 2-tuple linguistic information

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Definition 4. Let (sk, ak) and (sl, al) be two 2-tuples and k P 0. Then the operation laws for 2-tuples are defined as follows:

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(1) (2) (3) (4) (5)

(sk, ak)  (sl, al) = D(D1(sk, ak) + D1(sl, al)); (sk, ak) (sl, al) = D(D1(sk, ak)  D1(sl, al)); k(sk, ak) = D(kD1(sk, ak)); (sk, ak)k = D((D1(sk, ak))k); 1 ðsk ; ak Þðsl ;al Þ ¼ DððD1 ðsk ; ak ÞÞD ðsl ;al Þ Þ.

1

(1) (2) (3) (4) (5) (6)

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3. Some arithmetic aggregation operators with 2-tuple linguistic information

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3.1. The existing 2-tuple linguistic arithmetic aggregation operators

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Based on Definitions 2 and 3, the existing arithmetic aggregation operators with 2-tuple linguistic information are presented in this subsection. For convenience, let T be the set of all 2-tuples.

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Definition 5 [17]. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2tuples, the 2-tuple arithmetic averaging TAA is defined as

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Remark 1. It should be noted that if the 2-tuple linguistic information comes from different linguistic term sets (i.e. multi-granularity linguistic information), they have to be converted into the fuzzy sets defined in the basic linguistic term set by means of a transformation function [27], then they can be operated using the above operation laws. To avoid the operation results of Definition 4 being out of the scope [s0, sq], we can make the cardinality q + 1 of the extended continuous term set S large enough. If all 2-tuples (sk, ak) and (sl, al) in Definition 4 are reduced to linguistic labels sk and sl, i.e., ak = 0 and al = 0, then the operation laws in Definition 4 are reduced to the following operation laws: (1) (2) (3) (4) (5)

sk  sl = sk+l; sk sl ¼ skl ; ksk = skk; ðsk Þk ¼ skk ; ðsk Þsl ¼ skl .

The above are just the operation laws for linguistic labels defined in [4–6,32], which shows the justification of Definition 4 to some degree. As far as we know, however, there is less investigation on the operation laws of 2-tuples. Definition 4 gives the operation laws of 2-tuples, which can be used to directly compute for 2-tuple linguistic information. We insist that this is an interesting and valuable work for 2-tuples though it is a generalization of operation algorithm in [4–6,32], even if it is a formal transformation. In the following, suppose that a given linguistic term set is S = {s0, s1, s2, . . . , s8}, we give some examples to illustrate the above Definition 4.

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Example 1. (s1,0.1)  (s3,0.2) = D(D (s1,0.1) + D (s3,0.2)) = D(4.3) = (s4,0.3).

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ð4Þ 282

Definition 6 [17]. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2tuples, and w = (w1, w2, . . . , wn)T be the weight vector of 2-tuples (rj, aj)(j = 1,2, . . . , n), satisfying that 0 6 wj 6 1(j = 1, 2, . . . , n) and Pn j¼1 wj ¼ 1. The 2-tuple weighted average TWA is defined as follows:

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! n X TWAw ððr1 ; a1 Þ; ðr 2 ; a2 Þ; . . . ; ðrn ; an ÞÞ ¼ D wj D1 ðr j ; aj Þ :

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ð5Þ 290

j¼1

Especially, if wj = 1/n(j = 1, 2, . . . , n), then the TWA operator is reduced to the TAA operator.

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Definition 7 [17]. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2-tuples. The 2-tuple ordered weighted average (TOWA) operator of dimension n is a mapping TOWA:Tn ? T so that

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TOWAw ððr 1 ; a1 Þ; ðr 2 ; a2 Þ; . . . ; ðr n ; an ÞÞ ! n X ¼D wj D1 ðr rðjÞ ; arðjÞ Þ ;

ð6Þ 298

j¼1 1

277

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!

n 1X TAAððr 1 ; a1 Þ; ðr 2 ; a2 Þ; . . . ; ðr n ; an ÞÞ ¼ D D1 ðr j ; aj Þ : n j¼1

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T

where w = (w1, w2, . . . , wn) is the weighted vector correlating with Pn TOWA, satisfying that 0 6 wj 6 1(j = 1,2, . . . , n) and j¼1 wj ¼ 1.

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(r(1), r(2), . . . , r(n)) is a permutation of (1,2, . . . , n) such that (rr(j1), rr(j1), ar(j1)) P (rr(j), ar(j)) for any j.

THWAw;x ððr 1 ; a1 Þ; ðr 2 ; a2 Þ; ðr3 ; a3 Þ; ðr4 ; a4 ÞÞ ¼ D

4  X

!   wj D1 r 0rðjÞ ; a0rðjÞ

j¼1

354

¼ Dð2:968Þ ¼ ðs3 ; 0:032Þ:

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Definition 8 [[17]. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2tuples, and C = ((c1, b1), (c2, b2), . . . , (cn, bn))T be the linguistic weighting vector of 2-tuples (rj, aj)(j = 1, 2, . . . , n). The extended 2-tuple weighted average (ET  WA) operator is defined as follows:

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ET  WAC ððr 1 ; a1 Þ; ðr2 ; a2 Þ; . . . ; ðr n ; an ÞÞ ! n X D1 ðcj ; bj ÞD1 ðr j ; aj Þ ¼D : Pn 1 j¼1 j¼1 D ðc j ; bj Þ

303 304 305

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ð7Þ

Definition 9 [[20]. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2tuples. The extended 2-tuple ordered weighted average (ET  OWA) operator of dimension n is a mapping ET  OWA:Tn ? T so that

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ET  OWAL ððr 1 ; a1 Þ; ðr 2 ; a2 Þ; . . . ; ðr n ; an ÞÞ ! n X D1 ðlj ; gj ÞD1 ðr rðjÞ ; arðjÞ Þ : ¼D Pn 1 j¼1 j¼1 D ðlj ; gj Þ

ð8Þ

where L = ((l1, g1), (l2, g2), . . . , (ln, gn))T is the linguistic weighted vector correlating with ET  OWA, (r(1), r(2), . . . , r(n)) is a permutation of (1, 2, . . . , n) such that (rr(j1), ar(j1)) P (rr(j), ar(j)) for any j.

Theorem 2. The TOWA operator is a special case of the THWA operator.

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Proof. xj = 1/n(j = 1,2, . . . , n), then  0 0  Let r i ; ai ¼ nxi ðri ; ai Þ ¼ ðr i ; ai Þð1; 2; . . . ; nÞ. This completes the proof of Theorem 2. h

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Theorem 3. The TWA operator is a special case of the THWA operator.

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Proof. Let wj = 1/n(j = 1, 2, . . . , n), then

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! n  X 1 1 0 THWAw;x ððr 1 ; a1 Þ; ðr 2 ; a2 Þ; . . . ; ðr n ; an ÞÞ ¼ D D ðr rðjÞ ; a0rðjÞ Þ n j¼1 !    ! n n X 1 1 X 1 1 D ðnxj ðr j ; aj ÞÞ ¼D nxj D ððr j ; aj ÞÞ ¼D n n j¼1 j¼1 ! n X ðxj D1 ðrj ; aj ÞÞ ¼ TWAx ððr 1 ; a1 Þ; ðr2 ; a2 Þ; . . . ; ðr n ; an ÞÞ; ¼D j¼1

321 322 323 324 325 326 327 328 329 330 331 332

333

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3.2. The proposed hybrid arithmetic aggregation operators with 2tuple linguistic information It can be seen from Definitions 8 and 9 that the ET  WA operator weights the 2-tuple linguistic arguments while the ET  OWA operator weights the ordered positions of the 2-tuple linguistic arguments instead of weighting the arguments themselves. Therefore, weights represent different aspects in both the ET  WA and ET  OWA operators. However, both the operators consider only one of them. To solve this drawback, based on Definitions 2–4, some hybrid arithmetic aggregation operators with 2-tuple linguistic information are developed in the following. Definition 10. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2tuples. If THWA:Tn ? T so that

THWAw; x ððr 1 ; a1 Þ; ðr 2 ; a2 Þ; . . . ; ðr n ; an ÞÞ ! n X 1 0 0 ¼D ðwj D ðr rðjÞ ; arðjÞ ÞÞ ; j¼1

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where w = (w1, w2, . . . , wn)T is the weighted vector correlating with P THWA, satisfying that 0 6 wj 6 1(j = 1, 2, . . . , n) and  nj¼1 wj ¼ 1. ðr0rðjÞ ; a0rðjÞ Þ is the jth largest 2-tuple of 2-tuples r 0i ; a0i ð1; 2; . . . ; nÞ   with r 0i ; a0i ¼ nxi ðr i ; ai Þ. x = (x1, x2, . . . , xn)T is the weighting vector of 2-tuples (rj, aj)(j = 1, 2, . . . , n), satisfying that 0 6 xj Pn 6 1(j = 1, 2, . . . , n) and j¼1 xj ¼ 1. n is the balancing coefficient (in this case, if x = (x1, x2, . . . , xn)T goes to ((1/n,1/n, . . . , 1/n)),   then r0i ; a0i goes to (ri, ai)(1, 2, . . . , n). Then the function THWA is called the 2-tuple hybrid weighted arithmetic average operator of dimension n.

346

Example

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347 348 349 350 351

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6. Assume

h

that, (r1,a1) = (s1, 0.1), (r2,a2) = (s3,0.3), (r3,a3) = (s2,0.2), (r4,a4) = (s4,0.3),w = (0.2,0.3,0.3,0.2)T and x = (0.1,0.4,     0.3,0.2)T, then, r01 ; a01 ¼ 4  0:1ðs1 ; 0:1Þ ¼ ðs0 ; 0:44Þ; r02 ; a02 ¼ 1:6ðs3 ; 0:3Þ  0 0  0 0 ¼ ðs5 ; 0:28Þ; r3 ; a3 ¼ 1:2ðs  2 ; 0:2Þ  ¼ ðs3 ; 0:36Þ; r 4 ; a4 ¼  0:8ðs4 ; 0:3Þ ¼ 0 ðs3 ; 0:44Þ, therefore, r0rð1Þ ; a 0:28Þ; r 0rð2Þ ; a0rð2Þ ¼ ðs3 ; 0:44Þ; rð1Þ ¼ ðs5 ;   r 0rð3Þ ; a0rð3Þ ¼ ðs3 ; 0:36Þ and r 0rð4Þ ; a0rð4Þ ¼ ðs0 ; 0:44Þ. Thus,

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From Theorems 2 and 3, we know that, the THWA operator first weights the given arguments, then reorders the weighted arguments in descending order and weights these ordered arguments, and finally aggregates all the weighted arguments into a collective one. The THWA operator generalizes both the TWA and TOWA operators. The THWA operator reflects the important degrees of both the given 2-tuples and the ordered positions of the 2-tuples.

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Definition 11. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2-tuples. If T  HLWA:Tn ? T so that

374

T  HLWAL;x ððr 1 ; a1 Þ; ðr 2 ; a2 Þ; . . . ; ðr n ; an ÞÞ ¼D

n X j¼1

D1 ðlj ; gj Þ

Pn

j¼1 D

1

ðlj ; gj Þ

1

0

0

D ðr rðjÞ ; arðjÞ Þ

368 369 370 371 372 373

375

376

!! ;

ð10Þ

T

ð9Þ

358

364

which completes the proof of Theorem 3. 320

356

378

where L = ((l1, g1), (l2, g2), . . . , (ln, gn)) is the 2-tuple linguistic weighted vector correlating with T  HLWA. ðr 0rðjÞ ; a0rðjÞ Þ is the jth largest 2-tuple of 2-tuples ðr 0i ; a0i Þði ¼ 1; 2; . . . ; nÞ with ðr 0i ; a0i Þ ¼ nxi ðr i ; ai Þ; x ¼ ðx1 ; x2 ; . . . ; xn ÞT is the weighting vector of 2-tuples (rj, aj)(j = 1, 2, . . . , n), satisfying that 0 6 xj Pn 6 1(j = 1, 2, . . . , n) and j¼1 xj ¼ 1. n is the balancing coefficient T (in  0 this  case, if x = (x1, x2, . . . , xn) goes to (1/n,1/n, . . . , 1/n)), then 0 r i ; ai goes to (ri, ai)(i = 1, 2, . . . , n). Then the function T  HLWA is called the 2-tuple hybrid linguistic weighted arithmetic average operator of dimension n.

379

7. Assume that (l1, g1) = (s3, 0.4), (l2, g2) = (s2, 0.2), (l3, g3) = (s1, 0.1), (l4, g4) = (s5, 0.2), (r1, a1) = (s1, 0.1), (r2, a2) = (s3, 0.3), (r3, a3) = (s2, 0.2), (r4, a4) = (s4, 0.3) and x = (0.1, 0.4, 0.3, 0.2)T, then,  0 0   r ; a ¼ 4  0:1ðs1 ; 0:1Þ ¼ ðs0 ; 0:44Þ; r 02 ; a02 ¼ 1:6ðs3 ; 0:3Þ ¼ ðs5 ; 0:28Þ;  10 10   0 0 r 3 ; a3 ¼ 1:2ðs  2 ; 0:2Þ ¼ ðs3 ; 0:36Þ and  r4 ; a4 ¼ 0:8ðs4 ; 0:3Þ ¼ ðs3 ; 0:44Þ.  0 0 Therefore, r 0rð1Þ ; a0rð1Þ Þ ¼ ðs ¼ ðs3 ; 0:44Þ; r0rð3Þ ; a0rð3Þ 5 ; 0:28Þ; r rð2Þ ; arð2Þ ¼ ðs3 ; 0:36Þ and r 0rð4Þ ; a0rð4Þ ¼ ðs0 ; 0:44Þ. Thus,

389

Example

Q1 Please cite this article in press as: S.-P. Wan, J.-Y. Dong, 2-Tuple linguistic hybrid arithmetic aggregation operators and application to multi-attribute group decision making, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.02.002

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T  HLWAL;x ððr1 ; a1 Þ; ðr2 ; a2 Þ; ðr 3 ; a3 Þ; ðr 4 ; a4 ÞÞ !! 4   X D1 ðlj ; gj Þ 1 0 0 D r ; a ¼ Dð5:5808Þ ¼D P4 1 rðjÞ rðjÞ j¼1 j¼1 D ðlj ; gj Þ ¼ ðs6 ; 0:4192Þ:

398

399 400

401 402 403

404 405

Theorem 4. The ET  OWA operator is a special case of the T  HLWA operator. Proof. Let xj = 1/n(j = 1,2, . . . , n), then  0 0 ri ; ai ¼ nxi ðr i ; ai Þ ¼ ðri ; ai Þ; ði ¼ 1; 2; . . . ; nÞ. This completes the proof of Theorem 4. h Definition 12. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2tuples. If ET  HLWA:Tn ? T so that

406

ET  HLWAL;C ððr1 ; a1 Þ; ðr 2 ; a2 Þ; . . . ; ðrn ; an ÞÞ 0 0 11 C BX B  C B n B D1 ðlj ; gj Þ CC 1 0 0 CC; B ¼ DB D r ; a rðjÞ rðjÞ CC n B BX @ j¼1 @ AA D1 ðlj ; gj Þ

408

ð11Þ

j¼1

410

where L = ((l1, g1), (l2, g2), . . . , (ln, gn))T is the 2-tuple linguistic weighted   vector correlating with ET  HLWA. r 0rðjÞ ; a0rðjÞ is the jth largest 2-tuple

411

of

412

ðri ; ai Þ; C ¼ ððc1 ; b1 Þ; ðc2 ; b2 Þ; . . . ; ðcn ; bn ÞÞT is the 2-tuple linguistic weighting vector of 2-tuples (rj, aj), n is the balancing coefficient. Then the function ET  HLWA is called the extended 2-tuple hybrid linguistic weighted arithmetic average operator of dimension n.

409

413 414 415

416 417 418 419 420 421 422 423 424

2-tuples

ðr 0i ; a0i Þði ¼ 1; 2; . . . ; nÞ

with

ðr 0i ; a0i Þ ¼ nðci ; bi Þ

Example 8. Assume that (l1, g1) = (s3, 0.4), (l2, g2) = (s2, 0.2), (l3, g3) = (s1, 0.1), (l4, g4) = (s5, 0.2), (r1, a1) = (s3, 0.1), (r2, a2) = (s3, 0.3), (r3, a3) = (s1, 0.2), (r4, a4) = (s2, 0.3), (c1, b1) = (s1, 0.4), (c2, b2) = (s0,0.1), (c3, b3) = (s1, 0.2) and (c4, b4) = (s1,  0.3), then, ðr01 ; a01 Þ ¼ 4ðs1 ; 0:4Þ ðs3 ; 0:1Þ ¼ ðs3 ; 0:36Þ; ðr 02 ; a02 Þ ¼ 4ðs0 ; 0:1Þ ðs3 ; 0:3Þ ¼ ðs1 ; 0:32Þ; ðr 03 ; a03 Þ ¼ 4ðs1 ; 0:2Þ ðs1 ; 0:2Þ ¼ ðs6 ; 0:24Þ and ðr04 ; a04 Þ ¼ 4ðs1 ; 0:3Þ ðs2 ; 0:3Þ ¼ ðs6 ; 0:44Þ. Therefore, ðr 0rð1Þ ; a0rð1Þ Þ 0 0 0 0 ¼ ðs6 ; 0:44Þ; ðr rð2Þ ; arð2Þ Þ ¼ ðs6 ; 0:24Þ; ðr rð3Þ ; arð3Þ Þ ¼ ðs3 ; 0:36Þ and ðr0rð4Þ ; a0rð4Þ Þ ¼ ðs1 ; 0:32Þ. Thus,

425 427

428 429 430 431 432 433

434 435 436 437 438 439 440 441

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ET  HLWAL;C ððr 1 ; a1 Þ; ðr 2 ; a2 Þ; ðr 3 ; a3 Þ;ðr 4 ; a4 ÞÞ ¼ D ¼ Dð14:4712Þ ¼ ðs4 ; 0:2743Þ:

4 X j¼1

D1 ðlj ; gj Þ D1 ðr0rðjÞ ; a0rðjÞ Þ P4 1 j¼1 D ðlj ; gj Þ

!!

Remark 2. If the correlated 2-tuple linguistic weighted vector L = ((l1, g1), (l2, g2), . . . , (ln, gn))T in Definitions 11 and 12 is reduced to linguistic weighted vector L = (l1, l2, . . . ,ln)T, we can converted L = (l1, l2, . . . , ln)T to 2-tuple linguistic weighted vector L = ((l1, g1), (l2, g2), . . . , (ln, gn))T by using Eq. (3), then Definitions 11 and 12 are still validated. Remark 3. Xu [33] proposed the linguistic hybrid arithmetic averaging operator (i.e., LHAA) for virtual linguistic variables. In the LHAA operator, the weight vector x = (x1, x2, . . . , xn)T of the virtual linguistic variables and the associated weight vector w = (w1, w2, . . . , wn)T are all just in the form of real numbers rather than linguistic variables or 2-tuples. Whereas, for the proposed T-HLWA operator in this paper, the associated weight vector L = ((l1, g1), (l2, g2), . . . , (ln, gn))T takes the form of 2-tuples; for the proposed ET-

HLWA operator in this paper, the associated weight vector L = ((l1, g1), (l2, g2), . . . , (ln, gn))T and the weight vector of 2-tuples C = ((c1, b1), (c2, b2), . . . , (cn, bn))T take the form of 2-tuples. The THWA, T-HLWA and ET-HLWA operators aim at the arguments of 2-tuples while the LHAA operator aims at the arguments of virtual linguistic variables. If 2-tuple linguistic variable is equivalent to the corresponding virtual linguistic term as stated by Dong et al. [24], then the THWA operator proposed in this paper is equivalent to the LHAA operator, but the LHAA operator can not deal with the weight vectors in the form of linguistic variables or 2-tuples while the T-HLWA and ET-HLWA operators can effectively solve this issue. From this point of view, the THLWA and ET-HLWA operators generalized the LHAA operator.

442

4. MAGDM method with 2-tuple linguistic assessments

457

4.1. MAGDM problem description using 2-tuple linguistic assessments

458

This section describes the MAGDM problem with 2-tuple linguistic assessments. Let A = {A1, A2, . . . , Am} be a discrete set of m possible alternatives and F = {a1, a2, . . . , an} be a finite set of n attributes, where Ai denotes the ith alternative and aj denotes the jth attribute. Let D = {D1, D2, . . . , Dt} be a finite set of t experts, where Dk denotes the kth expert. The expert Dk provides his/her assessment information of an   alternative Ai on an attribute aj as a 2-tuple rkij ¼ skij ; akij according

459

to a predefined linguistic term set S, where skij 2 S; akij 2 ½0:5; 0:5Þði ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; nÞ. Thus, the experts’ assessment information can be represented by the 2-tuple   linguistic decision matrices Rk ¼ rkij ðk ¼ 1; 2; . . . ; tÞ.

468

Suppose that both attribute weights and expert weights also can be represented by the 2-tuple linguistic information. Let W = ((w1, h1), . . . , (wn, hn))T be the 2-tuple linguistic weight vector of the attributes aj(j = 1, 2, . . . , n) and C = ((c1, b1), (c2, b2), . . . , (ct, bt))T be the 2-tuple linguistic weight vector of the experts Dk(k = 1, 2, . . . , t), where wj 2 S, ck 2 S, hj 2 [0.5, 0.5) and bk 2 [0.5, 0.5).

472

The problem concerned in this paper is how to rank alternatives or select the most desirable alternative(s) among the finite set A on the basis of the 2-tuple linguistic decision matrices and the 2-tuple linguistic weight information of attributes and experts.

478

4.2. The decision method with 2-tuple linguistic assessments

482

In this section, we propose a new method to solve the MAGDM problems with 2-tuple linguistic assessments. An algorithm and process of the MAGDM problems with 2-tuple linguistic assessments may be given as follows.

483

Step 1. Utilizing the decision matrix Rk and the ET  WA operator,   the individual overall preference value zki ¼ ski ; aki of the alternative Ai is derived as follows:

487

mn

444 445 446 447 448 449 450 451 452 453 456 455 454

460 461 462 463 464 465 466 467

469 470 471 473 474 475 476 477 479 480 481

484 485 486

488 489

490

      zki ¼ ski ; aki ¼ ET  WAW ski1 ; aki1 ; . . . ; skin ; akin ! n   X D1 ðwj ; hj Þ ¼D D1 skij ; akij ; ski 2 S; aki Pn 1 j¼1 j¼1 D ðwj ; hj Þ 2 ½0:5; 0:5Þ;

443

ð12Þ

492

where W = ((w1, h1), . . . , (wn, hn)) be the 2-tuple linguistic weight vector of the attributes aj (j = 1, 2, . . . , n). Step 2. Using the ET  HLWA operator to integrate   all the individual overall preference value zki ¼ ski ; aki ðk ¼ 1; 2; . . . ; tÞ of alternative Ai, the collective overall preference value zi = (si, ai) of alternative Ai is obtained as follows:

493

T

Q1 Please cite this article in press as: S.-P. Wan, J.-Y. Dong, 2-Tuple linguistic hybrid arithmetic aggregation operators and application to multi-attribute group decision making, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.02.002

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 1

  zi ¼ ðsi ; ai Þ ¼ ET  HLWAL;C s1i ; ai ; . . . ; sti ; ati ! t   X D1 ðlj ; gj Þ 0rðjÞ 0rðjÞ 1 ; D s ; a ¼D Pt i i 1 j¼1 j¼1 D ðlj ; gj Þ

S ¼ fs0 ¼ extremely poor; s1 ¼ very poor; s2 ¼ poor; s3 ¼ slightly poor; s4 ¼ fairðmediumÞ; ð13Þ

where L = ((l1, g1), (l2, g2), . . . , (lt, gt))T is the 2-tuple linguistic 0rðjÞ 0rðjÞ weighted vector correlating with ET  HLWG; ðsi ; ai Þ is the jth  0k 0k  largest 2-tuples si ; ai ðk ¼ 1; 2; . . . ; tÞ with  0k 0k  2-tuple of   si ; ai ¼ tðck ; bk Þ ski ; aki , and C = ((c1, b1), (c2, b2), . . . , (ct, bt))T is the 2-tuple linguistic weight vector of experts Dk(k = 1, 2, . . . , t). Step 3. Rank all the alternatives Ai(i = 1, 2, . . . , m) and select the best one(s) in accordance with the 2-tuple zi = (si, ai) (i = 1, 2, . . . , m). If any alternative has the highest zi value, then, it is the best alternative. Remark 4. In Step 2, we suppose that the weight vector of experts is in the form of 2-tuples C = ((c1, b1), (c2, b2), . . . , (ct, bt))T, so the ET  HLWA operator is used to integrate the individual overall preference values of alternative into the collective one. Even if the weight vector of experts is in the form of linguistic labels, the ET  HLWA operator can also be used to obtain the collective one since the linguistic term can be readily transformed into 2-tuple by Eq. (3). But if the weight vector of experts is in the form of real numbers, then the T  HLWA operator (i.e., (10)) can be used to obtain the collective one. Remark 5. Obviously, real number form and 2-tuple form are quite different for the weight vector of experts. In order to obtain the collective overall preference values of alternatives, only the ET  HLWA operator could be used for the latter while only the T  HLWA operator for the former.

532

Remark 6. In this paper, we take the weight vector of experts as 2tuple form. If the 2-tuples for the weight vector of experts and for the assessment of attribute values come from different linguistic term sets, they should be firstly converted into the fuzzy sets defined in the basic linguistic term set as stated in Remark 1. This solves the normalization problem of the 2-tuple weight vector of experts.

533

5. A real application to a personnel selection problem

534

In this section, a real personnel selection problem is used to illustrate the proposed method in this paper. Ahead Software Company Limited was registered in Nanchang, Jiangxi of China. It is a key national project software enterprise and key national high-tech enterprise. Established in 1994, it specializes in research and develop of platform software and trade application software and selling. The company desires to hire a system analyst from national recruitment. The expert panel consists of two board members D1 and D2, Company chairman D3 and Company vice chairman D4. Since Company chairman D3 has engaged in human resource management for many years and accumulated rich experience, Company chairman D3 is named as the group leader which is responsible for the whole recruitment work. After preliminary screening, five candidates (i.e., alternatives) Ai (i = 1, 2, . . . , 5) remain for further evaluation. Generally, many attributes should be used to evaluate these candidates. To improve the efficiency and rapidly make decision, three attributes are chosen by the four experts. These attributes are oral communication skills a1, emotional steadiness a2 and past experience a3, respectively. Since these attributes are all qualitative attributes, it is reasonable for the experts to use linguistic variables or 2-tuples to represent the evaluation information on the candidates with respective to the attributes. Consequently, the five candidates are to be evaluated using the 2-tuple linguistic information according to the linguistic term set:

526 527 528 529 530 531

535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558

559

s5 ¼ slightly goodðimportantÞ; s6 ¼ goodðimportantÞ; s7 ¼ very goodðimportantÞ; s8 ¼ extremely goodðimportantÞg

561

by the four experts under these three attributes. The 2-tuple linguistic decision matrices provided by each expert are respectively as follows:

562

0

ðs3 ;0:2Þ ðs1 ;0:4Þ ðs4 ;0:3Þ ðs5 ;0:4Þ ðs8 ;0:1Þ

ðs0 ;0:4Þ B B ðs4 ; 0:3Þ B R1 ¼ B B ðs2 ; 0:2Þ B @ ðs1 ; 0:3Þ ðs7 ;0:2Þ 0

ðs4 ; 0:3Þ B B ðs3 ; 0:4Þ B R3 ¼ B B ðs1 ; 0:3Þ B @ ðs5 ; 0:1Þ ðs7 ;0:2Þ

ðs2 ;0:4Þ ðs2 ;0:1Þ ðs4 ;0:3Þ ðs8 ;0:3Þ ðs7 ;0:4Þ

1

0

1

0

ðs8 ;0:1Þ ðs2 ;0:1Þ B C ðs7 ; 0:2Þ C B ðs5 ;0:3Þ B C 2 B ðs6 ;0:3Þ C C; R ¼ B ðs2 ;0:2Þ B C ðs7 ;0:2Þ A @ ðs2 ;0:3Þ ðs0 ; 0:1Þ ðs6 ;0:2Þ ðs7 ;0:3Þ ðs1 ;0:3Þ C B ðs5 ; 0:2Þ C B ðs3 ;0:3Þ C B 4 B ðs6 ;0:3Þ C C; R ¼ B ðs1 ;0:2Þ C B ðs7 ;0:2Þ A @ ðs1 ;0:4Þ ðs2 ;0:4Þ ðs6 ;0:3Þ

ðs4 ;0:2Þ ðs3 ;0:1Þ ðs7 ; 0:3Þ ðs1 ;0:4Þ ðs7 ; 0:1Þ

ðs0 ;0:4Þ ðs5 ;0:4Þ ðs6 ;0:2Þ ðs5 ;0:3Þ ðs3 ;0:1Þ

1

ðs6 ;0:1Þ C ðs6 ;0:2Þ C C ðs6 ;0:3Þ C C; C ðs7 ;0:2Þ A ðs8 ;0:1Þ 1

ðs7 ;0:1Þ C ðs8 ; 0:2Þ C C ðs8 ;0:3Þ C C: C ðs8 ; 0:2Þ A ðs1 ;0:3Þ

563 564

565

567 568

570

With ever increasing complexity in real human resource management, it is very difficult to give precisely the linguistic assessment information on the expert weights and attribute weights according to the given linguistic term set in advance. For example, the experts think that the past experience a3 is important and the weight may be s6 but less than s6, thus the weight of attribute a3 can be represented using the linguistic 2-tuple (w3, h3) = (s6, 0.2). After the negotiation and investigation of the experts, they determine the 2-tuple linguistic weight vector W = ((w1, h1), (w2, h2), (w3, h3))T of the attributes, where (w1, h1) = (s8,  0.4), (w2, h2) = (s1, 0.3) and (w3, h3) = (s6,  0.2). As the stated earlier, Company chairman D3, named as the group leader, has rich experience, knowledge and speciality in human resource management. Obviously, his importance degree is extremely high and may be s8 but less than s8, therefore, the weight of Company chairman D3 can be represented using the linguistic 2-tuple (c3, b3) = (s8, 0.1). Analogously, the 2-tuple linguistic weight vector C = ((c1, b1), (c2, b2), (c3, b3), (c4, b4))T of the experts can be obtained, where (c1, b1) = (s5, 0.1), (c2, b2) = (s1, 0.2), (c3, b3) = (s8, 0.1) and (c4, b4) = (s3, 0.4). Next, we adopt the proposed method to solve the above personnel selection example.

571

Step 1. Combining the decision matrix R1 and the 2-tuple linguistic weight vector of attributes W = ((w1, h1), (w2, h2), (w3, h3))T with the ET  WA operator, the individual overall   preference value z11 ¼ s11 ; a11 of candidate A1 is derived as follows:

593



 1

z11 ¼ s11 ; a1 ¼ ET  WAW ¼D

3 X j¼1



D1 ðwj ; hj Þ P3 1 j¼1 D ðwj ; hj Þ

     s111 ; a111 ; s112 ; a112 ; s113 ; a113 !   ¼ ðs4 ; 0:3143Þ: D1 s11j ; a11j

Likewise, we have

z12



s12 ;

1 2



573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590

591 592

594 595 596 597

598

600 601

ðs5 ; 0:0299Þ; z13



s13 ;

1 3



602

ðs4 ; 0:0034Þ; z14

a ¼ ¼ a ¼  1 1  1 1 1 ¼ s4 ; a4 ¼ ðs4 ; 0:0803Þ; z5 ¼ s5 ; a5 ¼ ðs4 ; 0:2714Þ: ¼

572

604 605

z21



  ¼ a ¼ ¼ s22 ; a22 ¼ ðs5 ; 0:1503Þ; z23     ¼ s23 ; a23 ¼ ðs4 ; 0:2156Þ; z24 ¼ s24 ; a24 ¼ ðs4 ; 0:1537Þ; z25   ¼ s25 ; a25 ¼ ðs7 ; 0:0116Þ: s21 ;

2 1



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ðs4 ; 0:1361Þ; z22

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  z31 ¼ s31 ; a1 ¼ ðs5 ; 0:3156Þ; z32 ¼ s32 ; a32 ¼ ðs4 ; 0:1626Þ;     z33 ¼ s33 ; a33 ¼ ðs4 ; 0:4619Þ; z34 ¼ s34 ; a34 ¼ ðs6 ; 0:1585Þ;   z35 ¼ s35 ; a35 ¼ ðs5 ; 0:1170Þ:

611

 3

612

613

    z41 ¼ s41 ; a41 ¼ ðs4 ; 0:4912Þ; z42 ¼ s42 ; a42 ¼ ðs5 ; 0:2612Þ;     z43 ¼ s43 ; a43 ¼ ðs4 ; 0:4435Þ; z44 ¼ s44 ; a44 ¼ ðs4 ; 0:2701Þ;   z45 ¼ s45 ; a45 ¼ ðs4 ; 0:0442Þ:

615 616 617 618 619 620 621 622 623 624 625

626

Step 2. Assume that the correlated 2-tuple weighted vector with ET  HLWA operator is L = ((l1, g1), (l2, g2), (l3, g3), (l4, g4))T, where (l1, g1) = (s2, 0.2), (l2, g2) = (s5, 0.1), (l3, g3) = (s7, 0.2), and (l4, g4) = (s6, 0.3). Using the 2-tuple linguistic weight vector of experts C = ((c1, b1), (c2, b2), (c3, b3), (c4, b4))T and the ET  HLWA operator to integrate all the individual   overall preference values zk1 ¼ sk1 ; ak1 (k = 1, 2, 3, 4) of candidate A1, the collective overall preference value of candidate A1 is thus calculated as follows:

    z1 ¼ ET  HLWAL;C s11 ; a11 ; . . . ; s41 ; a41 ! 4   X D1 ðlj ; gj Þ ¼ ðs4 ; 0:2189Þ: D1 s10rðjÞ ; a01rðjÞ ¼D P4 1 j¼1 j¼1 D ðlj ; gj Þ

628 629

    z2 ¼ ET  HLWAL;C s12 ; a12 ; . . . ; s42 ; a42 ¼D

4 X j¼1

632 633

1

z4 ¼ ET  HLWAL;C ¼D

4 X j¼1

638

1 4



a ;...;



s44 ;

4 4

a

P4

j¼1 D

1

ðlj ; gj Þ

4 X j¼1

1



s15 ;

1 5

¼ ðs5 ; 0:2661Þ;



a ;...;

D ðlj ; gj Þ P4 1 j¼1 D ðlj ; gj Þ



s45 ;

4 5

a

¼ ðs5 ; 0:1673Þ;



!   ¼ ðs5 ; 0:1682Þ: D1 s50rðjÞ ; a05rðjÞ

Step 3. Since z5 > z4 > z3 > z1 > z2, the ranking result of the candidates is A5  A4  A3  A1  A2 and therefore the best candidate is A5, which will be recommended to Ahead Software Company Limited.

648

6. Comparison analyses of the results obtained

649

6.1. Comparison with the approach to MAGDM with linguistic power average operators

651 652 653 654 655

as

0 14

Sup

Xu et al. [25] proposed four approaches to MAGDM with linguistic power average operators. To illustrate the superiorities of the proposed method, we use Approach I of [25] to solve the above personnel selection problem, and then conduct a comparison analysis. The following symbols Supkh, Tk and Vk see [25] in detail.

656 657

658

1

0:7875 0:8750 0:7500 C B B 0:9500 0:7875 0:9250 C C B 12 21 B Sup ¼ Sup ¼ B 1:0000 0:7000 1:0000 C C; C B @ 0:8750 0:6000 1:0000 A 0:9250 0:8500 0 1 0 0:5125 0:9000 0:9000 C B B 0:8875 0:9125 0:7500 C C B C; Sup13 ¼ Sup31 ¼ B 0:8875 1:0000 1:0000 C B C B @ 0:5250 0:6125 1:0000 A 1:0000 0:9125 0:7125

41

¼ Sup

0:8875

0:6500

0:8750

660

1

661

C B B 0:8750 0:5000 0:8750 C C B C ¼B B 0:8750 0:7625 0:7500 C; C B @ 0:9875 0:9125 0:9250 A 0:9375 0:3750 0:8500 1 0:7250 0:7750 0:8500 C B B 0:8375 0:8750 0:8250 C C B C ¼B B 0:8875 0:7000 1:0000 C; C B @ 0:6500 0:2125 1:0000 A 0:9250 0:9375 0:2875 0

Sup23 ¼ Sup32

Sup

42

¼ Sup

Sup34 ¼ Sup43

0

647

650

Supkh(k, h = 1, 2, 3, 4, k – h)

0:9000

0:5250 0:8750

1

C B B 0:8250 0:7125 0:8000 C C B C ¼B B 0:8750 0:9375 0:7500 C; C B @ 0:8875 0:5125 0:9250 A 0:9875 0:5250 0:1500 1 0 0:6250 0:7500 0:9750 C B B 0:9875 0:5875 0:6250 C C B B ¼ B 0:9875 0:7625 0:7500 C C: C B @ 0:5375 0:7000 0:9250 A 0:9375 0:4625 0:8625

Step 2: Calculate the matrices Tk(k = 1, 2, 3, 4) as follows:

!

D1 ðs40rðjÞ ; a40rðjÞ Þ



matrices

0

24

and

¼D

646

s14 ;

D ðlj ; gj Þ

z5 ¼ ET  HLWAL;C

645



1

640

644

!

D ðlj ; gj Þ D1 ðs20rðjÞ ; a20rðjÞ Þ P4 1 j¼1 D ðlj ; gj Þ

636

643

the

0

    z3 ¼ ET  HLWAL;C s13 ; a13 ; . . . ; s43 ; a43 ! 4   X D1 ðlj ; gj Þ 0rðjÞ 0rðjÞ 1 D s3 ; a3 ¼ ðs4 ; 0:0373Þ; ¼D P4 1 j¼1 j¼1 D ðlj ; gj Þ

635

642

Step 1: Calculate follows:

Similarly, we have

630

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609

2:1875 2:4250 2:5250

B B 2:7125 B T1 ¼ B B 2:7625 B @ 2:3875 2:8625 0 2:4125 B B 2:6125 B T2 ¼ B B 2:7625 B @ 2:4125

2:2000 2:4625 2:1250 2:1375 2:1750 2:3750 2:3375 1:3250

2:8375 2:3125 0

1

C 2:5500 C C 2:7500 C C; C 2:9250 A 1:5625 1 2:4750 C 2:5500 C C 2:7500 C C; C 2:9250 A 0:4375

1:8625 2:4250 2:7250

1

663 664

666 667

668

670 671

C B B 2:7125 2:3750 2:2000 C C B C T3 ¼ B B 2:7625 2:4625 2:7500 C; C B @ 1:7125 1:5250 2:9250 A 2:8625 2:3125 1:8625 1 2:4125 1:9250 2:7250 C B B 2:6875 1:8000 2:3000 C C B B T4 ¼ B 2:7375 2:4625 2:2500 C C: C B @ 2:4125 2:1250 2:7750 A 2:8625 1:3625 1:8625 0

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Suppose that the weight vector of experts is w = (0.2898, 0.0682, 0.4489, 0.1932)T. Utilize w to calculate the matrices Vk(k = 1, 2, 3, 4) as follows:

0

0:2979 B B 0:2907 B V1 ¼ B B 0:2901 B @ 0:3176 0:2899 0 0:0701 B B 0:0684 B V2 ¼ B B 0:0683 B @ 0:0747

682 683 684 685

686

689

690 691 692

693 695 696 697 698 699 700

701 702 703 704 705 706 707 708 709 710 711 712 713 714 715

0:2905 0:3233 0:2953 0:0705 0:0679 0:0684 0:0761

C 0:3076 C C 0:2974 C C; C 0:2919 A 0:2772 1 0:0659 C 0:0724 C C 0:0700 C C; C 0:0687 A 0:0652 1

0

0:4615 0:4643 0:4335 C B B 0:4503 0:4470 0:4764 C C B B V3 ¼ B 0:4494 0:4500 0:4607 C C; C B @ 0:4919 0:5008 0:4522 A 0:4491 0:4575 0:4294 1 0 0:1986 0:1998 0:1866 C B B 0:1938 0:1924 0:2050 C C B C V4 ¼ B B 0:1934 0:1937 0:1983 C: C B @ 0:2117 0:2155 0:1946 A 0:1933 0:1969 0:1848

0

s2:5089

s6:6654

s2:4495 s2:5921 s4:8426 s6:5916 s6:8677

s7:1575

1

C s6:4264 C C s6:8631 C C: C s7:3703 A s1:8271

Step 4: Suppose that the weight vector of attributes is x = (0.5170, 0.0884, 0.3946)T. Utilize x, R and the LWA operator to derive the collective overall preference values zi(i = 1, 2, 3, 4, 5) of the alternatives as follows:

z1 ¼ s4:3380 ; z5 ¼ s4:7741 :

z2 ¼ s4:6995 ;

z3 ¼ s3:9660 ;

716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740

For example, suppose that the weight vector of experts is w = (0.2898, 0.0682, 0.4489, 0.1932)T, then using the ET  HLWA operator (i.e., Eq. (10)), the collective overall preference values of alternatives are obtained as follows:

Step 3: Utilize the LPWA operator to aggregate all the individual decision matrixes into the collective decision matrix as follows:

B B s3:7419 B R¼B B s1:6049 B @ s3:3898 688

0:2886

0:0682 0:0695

679 680

0:2997 0:2798

1

where the weight vectors of attributes and experts are all real numbers rather than linguistic variables or 2-tuples. Whereas, the method proposed in this paper can deal with three cases: linguistic variables, 2-tuples and numerical values for the weight information of attributes and experts (see Subsection 6.2 in detail), which is the most difference between the method [25] and the method of this paper. (1) If the weight information of experts is given by linguistic variables, the linguistic variables can be easily transformed into 2-tuples by using Eq. (3), then the ET  HLWA operator can still be used to integrate the individual overall preference values of alternatives to derive the collective ones of alternatives (see the third line of Table 1 in Subsection 6.2). (2) If the weight information of experts is given by 2-tuples, the ET  HLWA operator can be directly used to integrate the individual overall preference values of alternatives to derive the collective ones of alternatives (see the example of Section 5). (3) If the weight information of experts is given by the real numbers, we can use the T  HLWA operator to replace the ET  HLWA operator to derive the collective overall preference values of alternatives.

z4 ¼ s5:2436 ;

Thus, the ranking result obtained by [25] is A4  A5  A2  A1  A3, which is remarkably different from that obtained by this paper. The best alternative by the former is A4 while that by the latter is A5. The worst alternative by the former is A3 while that by the latter is A2. The main reasons and comparison analysis are made as following: (A) Xu et al. [25] proposed Approaches I and II to MAGDM based on the LPWA and LPOWA operators, respectively. However, different experts assess the alternatives according to the same extended continuous linguistic term set in these approaches. In real-life decision problems, different experts may express their opinions from different granularity linguistic term sets. These approaches in [25] did not discuss this case while the proposed method in this paper can be used to solving multi-granularity linguistic MAGDM (as stated in Remarks 1 and 6). (B) The weighted vectors for LPWA and LPOWA operators are only in the form of real numbers. That is to say, the four approaches to MAGDM with linguistic power average operators proposed in [25] can only deal with the situation

z1 ¼ ðs4 ; 3953Þ; z2 ¼ ðs5 ; 0:4525Þ; z3 ¼ ðs4 ; 0:1060Þ; z4 ¼ ðs5 ; 0:0082Þ; z5 ¼ ðs5 ; 0:2061Þ:

741 742 743 744

745 747

Hence, the ranking order of the alternatives is A4  A5  A2  A1  A3, which is accordance with that obtained by [25]. In sum, the above discussion demonstrates that the proposed method in this paper is of universality and flexibility.

748

6.2. Comparison with the best related 2-tuple MAGDM method

752

Wei [22] proposed a MAGDM method based on the ET-WG and ET-OWG operators with 2-tuple linguistic information. In the following, to further illustrate the superiorities of the proposed method, we use the method of this paper to solve the investment selection problem of [22], and then conduct a comparison analysis. An investment company wants to invest a sum of money in the best option. There is a panel with five possible alternatives to invest the money: a car company A1, a food company A2, a computer company A3, an arms company A4 and a TV company A5. The investment company must take a decision according to the four attributes: the risk analysis a1, the growth analysis a2, the socialpolitical impact analysis a3 and the environmental impact analysis a4. The five possible alternatives Ai(i = 1, 2, 3, 4, 5) are to be evaluated using the linguistic term set S = {s1 = extremely poor (EP); s2 = very poor (VP); s3 = poor (P); s4 = medium (M); s5 = good (G); s6 = very good (VG); s7 = extremely good (EG)} by three experts Dk (k = 1, 2, 3) under the above four attributes. They respectively   construct the decision matrices Rk ¼ ~r kij ðk ¼ 1; 2; 3Þ as follows:

753

0

M

G

B B P VP B B R1 ¼ B B G M B B VG P @ EG

EP

P M G P VP

P

1

C P C C C EP C C; C GC A M

54

0

P

M

B B VP EP B B R2 ¼ B BM G B B EG VP @ P

VP

VP G P VP M

VP

1

750 751

754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770

771

C G C C C EG C C; C MC A VP

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775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831

1

G P VP VG C B P G C B VP G C B R3 ¼ B P C C: B VG VP G C B @ G VG EG VP A M VP M G In [22], the linguistic weight vector of the attributes is H ¼ ðs03 ; s01 ; s02 ; s04 Þ using the linguistic term set S0 ¼ fs01 ¼ extremely important; s02 ¼ very important; s03 ¼ important; s04 ¼ medium; s05 ¼ bad; s06 ¼ very bad; s07 ¼ extremely badg. For the ET  OWG operator [22], the correlated linguistic weighted vector  T is taken as V ¼ s06 ; s04 ; s02 .  T We suppose that the weight vector of experts is x ¼ s04 ; s04 ; s05 0 according to the linguistic term set S . In addition, for the ET  HLWA operator of this paper, we also take the correlated linguistic  T weighted vector as V ¼ s06 ; s04 ; s02 . Applying the proposed method in this paper, the above linguistic decision matrices, the linguistic weight vectors of the attributes and experts, and the correlated linguistic weighted vector should be firstly transformed into 2-tuple linguistic forms by using Eq. (3). Then, repeating the same steps as in Section 5, the collective overall preference values of alternatives can be obtained. Table 1 lists the collective overall preference values of alternatives obtained by the method [22] and method in this paper. It is easily seen from Table 1 that the ranking results obtained by the method [22] and the method proposed in this paper are slightly different. The difference is the ranking order of A4 and A3, i.e., A4  A3 by the former while A3  A4 by the latter. The worst alternative is A2 by both methods, but the best alternative by the former is A4, while the best alternative by the latter is A3. Compared with the former, the main advantages of the latter mainly lie in the following: (i) The latter sufficiently considers the importance degrees of different experts. Before utilizing the ET  HLWA operator, the individual overall preference values of alternatives should be first weighted by the expert weights and then the collective ones of alternatives can be obtained. However, the former is based on the ET  WG and ET  OWG operators, which does not consider the importance degrees of different experts at all. Namely, the former supposed that the expert weights are absolutely unknown and used the ET  OWG operators to integrate the individual overall preference values of alternatives into the collective ones. In real-life decision problems, different experts usually act as different roles in the decision process (such as the expert D3 in Section 5). Some experts may assign unduly high or unduly low uncertain preference values to their preferred or repugnant objects. To relieve the influence of these unfair arguments on the decision results and reflect the importance degrees of all the experts, the latter first weights each individual overall preference value by using the corresponding expert weight, and then utilizes the ET  HLWA operator to aggregate all the individual weighted overall preference values of each alternative into the collective ones of alternatives. Therefore, the ET  HLWA or T  HLWA operator can make the decision results more reasonable through assigning low weights to those ‘‘false’’ or ‘‘biased’’ arguments. These advantages can not be reflected in the former. (ii) The former is only suitable for the case where the weight information of attributes is the form of the linguistic variables, whereas the latter can deal with the three cases: linguistic variables, 2-tuples and numerical values for the

9

weight information of attributes and experts (see Subsection 6.1 in detail), which also shows that the latter is more universal and flexible than the former.

832 833 834 835

6.3. Comparison with other normal linguistic MAGDM methods

836

Ma et al. [31] developed a fuzzy multi-criteria group decisionmaking (MCGDM) support system, which is called a Decider. By means of existing works on linguistic methods, Ma et al. [31] can deal with subjective and objective information at the same time. The Comparison analyses between [31] and this paper are conducted from four aspects.

837

(a) The research focuses of both papers are quite different. The former constructed a MCGDM model and developed a fuzzy MCGDM support system under multi-level criteria and multi-level evaluators, while the latter focuses on developing some new 2-tuple linguistic hybrid arithmetic aggregation operators. (b) Although the subjective information in the former may be in the form of linguistic terms, the linguistic assessment information is all transformed into fuzzy numbers to deal with. Any transformation process between linguistic terms and fuzzy numbers may easily result in information losses and distortions to some degree. However, the latter utilizes the proposed 2-tuple linguistic hybrid arithmetic aggregation operators to integrate the linguistic assessment information, which needs not such transformation between linguistic terms and fuzzy numbers. (c) The former selected the existing aggregation operators rather than developed news aggregation operators to integrate in the MCGDM model, whereas the latter developed some new 2-tuple linguistic hybrid arithmetic aggregation operators and applied to the MAGDM. (d) The former constructed the MCGDM model with multi-level hierarchies of criteria and evaluators, while the MAGDM model in the latter has only one-level hierarchy of attributes. Though the former can be applied to many decision problems, such as fabric material ranking, strategy evaluation and nonwoven product assessment, the latter also has some prominent advantages as stated in Subsection 6.2 in detail.

843

838 839 840 841 842

844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871

7. Conclusion

872

This paper defines the operation laws for 2-tuples and studies the related properties of the operation laws. After reviewing the existing 2-tuple linguistic arithmetic aggregation operators, some hybrid arithmetic aggregation operators with 2-tuple linguistic information are developed including THWA, T  HLWA, and ET  HLWA operators. The THWA operator generalizes both the TWA and TOWA operators. The ET  OWA operator is a special case of the T  HLWA operator. A new decision method is proposed to solve the MAGDM problem with 2-tuple linguistic information. The method is based on ET  WA and ET  HLWA operators which can sufficiently consider the importance degrees of different experts and thus relieve the influence of those unfair arguments on the decision results. The proposed hybrid arithmetic aggregation operators with 2-tuple linguistic information enlarge the research content on 2-tuple linguistic information and enrich the ideas for solving fuzzy MAGDM problems with linguistic information. Although the developed method is illustrated using a personnel selection problem, it is expected to be applicable to decision problems in many areas,

873

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especially in situations where multiple experts are involved and the weights of attributes and experts are represented by linguistic variables or 2-tuples instead of real numbers, such as the enterprise project selection and water environment assessment, partner choice in supply chain, and so on. Furthermore, the developed method can also deal with the expert weights in the form of real numbers only if we use the T  HLWA operator to replace the ET  HLWA operator, which indicates that the method proposed in this paper is of universality and flexibility. In this paper we do not make any conclusion about the determining method of the linguistic (or 2-tuple linguistic) weighted vector correlating with 2-tuple linguistic hybrid arithmetic aggregation operators and how to effectively determine the expert weights in the form of the linguistic or 2-tuples, which will be investigated in the near future. In addition, 2-tuple linguistic hybrid geometric aggregation operators are also worthy of consideration for future research.

909

Acknowledgments

910

921

This work was partially supported by the National Natural Science Foundation of China (Nos. 71061006, 61263018, 71171055 and 70871117), the Program for New Century Excellent Talents in University (the Ministry of Education of China, NCET-10-0020), the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20113514110009), the Humanities Social Science Programming Project of Ministry of Education of China (No. 09YGC630107), the Natural Science Foundation of Jiangxi Province of China (No. 20114BAB201012) and the Science and Technology Project of Jiangxi province educational department of China (No. GJJ12265) and the Excellent Young Academic Talent Support Program of Jiangxi University of Finance and Economics.

922

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