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Some interval-valued 2-tuple linguistic aggregation operators and application in multiattribute group decision making
Applied Mathematical Modelling 37 (2013) 4269–4282
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Some interval-valued 2-tuple linguistic aggregation operators and application in multiattribute group decision making Huimin Zhang ⇑ School of Management, Henan University of Technology, Zhengzhou 450001, China
a r t i c l e
i n f o
Article history: Received 26 January 2012 Received in revised form 10 August 2012 Accepted 11 September 2012 Available online 21 September 2012 Keywords: MAGDM 2-tuple Interval-valued 2-tuple Aggregation operator Degree of precision
a b s t r a c t This paper deals with multiattribute group decision making (MAGDM) problems with interval-valued 2-tuple linguistic information. First, we introduce some new aggregation operators, such as the interval-valued 2-tuple weighted geometric (IVTWG) operator, the interval-valued 2-tuple ordered weighted geometric (IVTOWG) operator, the generalized interval-valued 2-tuple weighted average (GIVTWA) operator and the generalized interval-valued 2-tuple ordered weighted average (GIVTOWA). Then, we discuss their desired properties and relationships among them. Furthermore, we put forward a new method to determine the weight vector of interval-valued 2-tuple aggregation operator based on the concept of degree of precision. Finally, a numerical example is provided to illustrate the efficiency of the proposed method in dealing with interval-valued 2-tuple linguistic information under multi-granular linguistic contexts. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction The 2-tuple fuzzy linguistic representation model, composed by a linguistic term and a real number, was developed by Herrera and Martínez [1] based on the concept of symbolic translation. Since the 2-tuple linguistic model can express any counting of information in the universe of discourse and avoid the information loss in the process of linguistic information processing, it has been widely studied and applied in decision making [2–10]. As a useful computational approach for computing with words (CW) (more detailed literature about CW is available in [11,12]), the 2-tuple linguistic model improves the accuracy and facilitate the processes of CW. Herrera and Herrera-Viedma [13] pointed out the solution scheme in the linguistic decision analysis must be formed by three steps: the choice of the linguistic term set with its semantic, the choice of the aggregation operator of linguistic information and the choice of the best alternatives, which is carried out in two phases, i.e., aggregation phase of linguistic information and exploitation phase. In light of the fact that information aggregation always plays an important role in decision making process, many 2-tuple aggregation operators have been proposed to aggregate information. Herrera and Martínez [1] put forward the 2-tuple arithmetic mean, weighted average and ordered weighted average (OWA) operator. Jiang and Fan [14] presented the 2-tuple OWA operator and the 2-tuple ordered weighted geometric (OWG) operator. Zhang and Fan [15] developed the extended linguistic ordered weighted averaging (ELOWA) operator. Wang [16] proposed a modified linguistic OWA operator based on entropy maximization. Wang and Hao [17] introduced the quantifier-guided OWA aggregation operator and anchoring value-based OWA aggregation operator for 2-tuples. Dong et al. [18] developed the 2-tuple OWA operator and the extended OWA operator. Wei [19] proposed the extended 2-tuple weighted geometric operator and extended 2-tuple OWG operator. By using the Choquet integral, Yang and ⇑ Address: School of Management, Henan University of Technology, Lianhua Road, Zhengzhou 450001, China. Tel.: +86 13607672706; fax: +86 0371 67756389. E-mail address: [email protected] 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.09.033
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Chen [20] developed the 2-tuple correlated averaging operator, the 2-tuple correlated geometric operator and the generalized 2-tuple correlated averaging operator. Based on the ideal of dependent aggregation of Xu [21], Wei and Zhao [22] proposed the dependent 2-tuple OWA operator and the dependent 2-tuple OWG operator. In the foregoing literature about 2-tuple aggregation operator based decision making method, the original decision information represent by 2-tuple linguistic variables is always derived from a given linguistic term set. In other words, all the decision makers are supposed to provide their assessments about all the alternatives on each attribute with only a certain linguistic term from the same linguistic term set. In reality, it is somewhat unrealistic for every decision maker to express his or her decision information fully and correctly under such constraint. From this perspective, the experts may find the cardinality of the term set is too small to fully express their professional judgments on some attributes and so big that the evaluations on some other attributes are out of their ability [23]. To ensure the decision makers’ fully expressing their opinions, there are at least two approaches worth trying. On one hand, if all decision makers have the option to choose their own linguistic term sets, i.e., the decision information is derived from several linguistic term sets with different granularity, then they can no doubt better provide their professional knowledge and experience. At this point, however, there is the need to unify multi-granularity linguistic information before aggregation operation, which always demands lots of calculation work. For example, Herrera et al. [24] present a fusion approach of multi-granularity linguistic information with the basic linguistic term. Based on the linguistic hierarchies [25] and the 2-tuple linguistic representation model [1], Espinilla et al. [26] presented the extended linguistic hierarchies for managing multi-granular linguistic scales. On the other hand, by using extended 2-tuple fuzzy linguistic representation model, decision makers can also naturally provide their decision information. For instance, Wang and Hao [27] present the proportional 2-tuple fuzzy linguistic representation model and put forward some aggregation operators for proportional 2-tuples. Lin et al. [28] proposed the definition of interval 2-tuple linguistic variable to better express decision information. On the basis of hesitant fuzzy sets [29], Rodriguez et al. [30] proposed the concept of hesitant fuzzy linguistic term set to suit the modeling of quantitative settings. But, in [27,28,30], all decision information given by different decision makers is also derived from the predefined (fixed) linguistic term set. And it is actually not easy to choose only an appropriate linguistic term set meeting the requirements of all the decision makers. To overcome above limitations, Zhang [23] put forward the interval-valued 2-tuple linguistic representation model based on the definition of [28,31], which can be regarded as standardized interval 2-tuple linguistic model and is suitable for dealing with MAGDM problems under multi-granular linguistic contexts. By using interval-valued 2-tuple linguistic representation model, decision information can not only be fully expressed but also be unified easily under multi-granular linguistic contexts. In view of key role aggregation operator plays in CW process, in this paper, we focus on some new aggregation operators for interval-valued 2-tuple together with their desired properties. Furthermore, a new method to determine weight vector of interval-valued 2-tuple aggregation operator is proposed. The remainder of this paper is set out as follows: Section 2 presents a brief introduction to the basic knowledge of 2-tuple and interval-valued 2-tuple linguistic model. Section 3 introduces some new interval-valued 2-tuple aggregation operators and their properties. Based on the concept of degree of precision, Section 4 presents the method to determine the weight vector of aggregation operator and a MAGDM method on the basis of interval-valued 2-tuple aggregation operators. Section 5 provides a numerical example to illustrate the efficiency of the proposed method. Section 6 gives some concluding remarks. 2. Preliminaries In this section, some relevant concepts and basic knowledge are briefly illustrated. Le S = {s0, s1, . . . , sg} be a finite linguistic term set with odd cardinality, where si represents a possible linguistic term for a linguistic variable. For example, a set of seven terms S can be expressed as follows:
S ¼ fs0 ¼ NðnoneÞ; s1 ¼ VLðvery lowÞ; s5 ¼ VHðvery highÞ; s6 ¼ PðperfectÞg:
s2 ¼ LðlowÞ;
s3 ¼ MðmediumÞ;
s4 ¼ HðhighÞ;
It is required that the linguistic term set should satisfy the following characteristics [1,18,25,32]: (1) (2) (3) (4)
The set is ordered: si > sj, if and only if i > j. There is a negation operator: Neg(si) = sj such that j = g i. Max operator: max (si, sj) = si, if and only if i P j. Min operator: min (si, sj) = si, if and only if i 6 j.
The 2-tuple fuzzy linguistic representation model [1] is based on the concept of symbolic translation. A 2-tuple (si, a) is used to express the linguistic information, where the term si is a linguistic label of the predefined linguistic term set S and a (a 2 [0.5, 0.5]) is a numerical value representing the value of symbolic translation. Definition 1 ([1]). Let b be the result of an aggregation of the indexes of a set of labels assessed in a linguistic term set S, i.e., the result of a symbolic aggregation operation.b 2 [0,g] and g + 1 is the cardinality of S. Let i = round (b) and a = b i be two values such that i 2 [0,g] and, then a is called a symbolic translation, where round () is the usual round operation.
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Definition 2 ([1]). Let S = {s0, s1, . . . , sg} be an ordered linguistic term set and b 2 [0, g] be a value supporting result of a symbolic aggregation operation, then the 2-tuple that expresses the equivalent information to b is obtained with the following function:
D : ½0; g ! S ½0:5; 0:5Þ; DðbÞ ¼ ðsi ; aÞ; with
si ;
ð1Þ i ¼ roundðbÞ;
ð2Þ
a ¼ b i; a 2 ½0:5; 0:5Þ;
where si has the closest index label to b and a is the value of the symbolic translation. Definition 3 ([1]). Let S = {s0,s1, . . . ,sg} be an ordered linguistic term set and (si,a) be a 2-tuple. There is always a D1 function such that from a 2-tuple it returns its equivalent numerical value b 2 ½0; g Rþ , where
D1 : S ½0:5; 0:5Þ ! ½0; g;
ð3Þ
1
D ðsi ; aÞ ¼ i þ a ¼ b:
ð4Þ
It is obvious that the conversion of a linguistic term si(si 2 S) into a linguistic 2-tuple consist of adding a value zero as symbolic translation: D(si) = (si, 0), i = 0, 1, 2, . . ., g. Definition 4 ([1]). The comparison of linguistic information represented by 2-tuples is carried out according to an ordinary lexicographic order. Let (sk, a1) and (sl, a2) be two 2-tuples, with each one representing a linguistic assessment as follows: (1) If k < l, then (sk, a1) is smaller than (sl, a2); (2) If k = l, then if a1 = a2, then (sk, a1) and (sl, a2) represent the same information; if a1 < a2, then (sk, a1) is smaller than (sl, a2); if a1 > a2, then (sk, a1) is bigger than (sl, a2).
Definition 5 ([1]). The negation operator over 2-tuples is defined as
Negððsi ; aÞÞ ¼ Dðg ðD1 ðsi ; aÞÞÞ;
ð5Þ
where g + 1 is the cardinality of S, S = {s0, s1, . . . , sg}. Chen and Tai [31] put forward a generalized 2-tuple linguistic model and translation function.
Definition 6 ([31]). Suppose S = {s0, s1, . . . , sg} be an ordered linguistic term set, any crisp value b 2 [0, g] can be transformed into one 2-tuple linguistic variable by the following function:
DðbÞ ¼ ðsi ; aÞ; with
si ;
i ¼ roundðb gÞ
a ¼ b i=g; a 2 ½0:5=g; 0:5=gÞ
ð6Þ
On the contrary, any 2-tuple can be converted into a crisp value b 2 [0, 1] as follows:
D1 ðsi ; aÞ ¼ i=g þ a ¼ b:
ð7Þ
Based on the definition of [28,31], Zhang [23] put forward a new interval-valued 2-tuple linguistic representation model. Definition 7 ([23]). Suppose S = {s0, s1, . . . ,sg} be an ordered linguistic term set. An interval-valued 2-tuple is composed of two linguistic terms and two crisp numbers, denoted by ((si, a1), (sj, a2)), where i 6 j and a1 6 a2 if i = j, si(sj) and a1(a2) represent the linguistic label of the predefined linguistic term set S and symbolic translation, respectively. The interval-valued 2-tuple that expresses the equivalent information to an interval value [b1, b2](b1, b2 2 [0, 1], b1 6 b2) is derived by the following function:
8 si ; > > > a 1 ¼ b1 i=g; > > : a2 ¼ b2 j=g;
i ¼ roundðb1 gÞ; j ¼ roundðb2 gÞ;
a1 2 ½0:5=g; 0:5=gÞ; a2 2 ½0:5=g; 0:5=gÞ:
ð8Þ
Conversely, there is always a D1 function such that an interval-valued 2-tuple can be converted into an interval value [b1, b2](b1, b2 2 [0, 1], b1 6 b2) as follows:
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D1 ððsi ; a1 Þ; ðsj ; a2 ÞÞ ¼ ½i=g þ a1 ; j=g þ a2 ¼ ½b1 ; b2 :
ð9Þ
The 2-tuple based CW models for multi-granular linguistic information allow expressing the results (obtained at computational phase) in the initial expression domain used by each expert [33]. Similarly, by Eq. (8), the interval-valued 2-tuple linguistic model can easily express the final results (derived in aggregation operation) in the initial expression domains chosen by each expert. Specially, if si = sj and a1 = a2, then the interval-valued 2-tuple reduces to a 2-tuple. The negation operator over intervalvalued 2-tuple is defined as follows:
Negðððsi ; a1 Þ; ðsj ; a2 ÞÞÞ ¼ Dð½1 D1 ðsj ; a2 Þ; 1 D1 ðsi ; a1 ÞÞ;
ð10Þ
where g + 1 is the cardinality of S, S = {siji = 0,1,2, . . . ,g}. From Definitions 6 and 8, we have 0 6 b 6 1, which provides convenience for comparison and operation among 2-tuples (interval-valued 2-tuples) under multi-granular linguistic contexts, without converting multi-granularity linguistic term sets into the basic linguistic term set based on linguistic transformation function [24]. Definition 8 ([23]). Suppose S = {s0, s1, . . . , sg} be an ordered linguistic term set. For an interval-valued 2-tuple A = ((si, a1), (sj, a2)), the score function is expressed by the following formula
SðAÞ ¼ ði þ jÞ=ð2gÞ þ ða1 þ a2 Þ=2:
ð11Þ
The accuracy function is expressed by the following function
HðAÞ ¼ ðj iÞ=g þ a2 a1 :
ð12Þ
It is easy to prove that 0 6 S(A) 6 1 and 0 6 H(A) 6 1. The procedure to compare any two interval-valued 2-tuples is listed as follows. Theorem 1 ([23]). Let S = {s0, s1, . . . , sg} be an ordered linguistic term set, A = ((si, a1), (sj, a2)) and B = ((sk, a3), (sl, a4)) be two interval-valued 2-tuples: If S(A) > S(B), then A > B; If S(A) < S(B), then A < B; If S(A) = S(B), then: (1) If H(A) > H(B), then A < B; (2) If H(A) < H(B), then A > B; (3) If H(A) = H(B), then A = B.
Definition 9 ([23]). Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . ,((rn, an), (ln, en))} be a set of interval-valued 2-tuples and w = P (w1, w2, . . . , wn)T be their associated weights, with wi 2 [0, 1] and ni¼1 wi ¼ 1. The interval-valued 2-tuple weighted average (IVTWA) operator is defined as
IVTWAðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ ¼ D
" # n n X X wi D1 ðri ; ai Þ; wi D1 ðli ; ei Þ i¼1
ð13Þ
i¼1
Definition 10 ([23]). Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))} be a set of interval-valued 2-tuples and w = P (w1, w2, . . . ,wn)T be their associated weights, with wi 2 [0, 1] and ni¼1 wi ¼ 1. The interval-valued 2-tuple ordered weighted average (IVTOWA) operator is defined as
IVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððrn ; an Þ; ðln ; en ÞÞÞ ¼ D
" # n n X X wi D1 ðr rðiÞ ; arðiÞ Þ; wi D1 ðlrðiÞ ; erðiÞ Þ ; i¼1
ð14Þ
i¼1
where [(rr(i), ar(i)), (lr(i), er(i))] is the ith largest of ((rj, aj), (lj, ej)). 3. Some aggregation operators with interval-valued 2-tuple linguistic information In this section, we introduce some new interval-valued 2-tuple aggregation operators together with their typical properties. Definition 11. Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))} be a set of interval-valued 2-tuples and w = P (w1, w2, . . . , wn)T be their associated weights, with wi 2 [0, 1] and ni¼1 wi ¼ 1. The interval-valued 2-tuple weighted geometric (IVTWG) operator is defined as
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IVTWGðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ ¼ D
" n Y
ðD1 ðr i ; ai ÞÞwi ;
i¼1
n Y
# ðD1 ðli ; ei ÞÞwi :
ð15Þ
i¼1
The desired properties of the IVTWG operator can be listed as follows. Theorem 2. Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))} be a set of interval-valued 2-tuples, w = (w1, w2, P be their associated weights, with wi 2 [0, 1] and ni¼1 wi ¼ 1, then, for IVTWG operator, we have
T 1, w2, . . . , wn)
(1) Idempotency. If all ((rj, aj), (lj, ej)) (j = 0, 1, 2, . . . , n) are equal, i.e., ((rj, aj), (lj, ej)) = ((r, a), (l, e)), then IVTWG (((r1, a1), (l1, e1)), ((r2, a2), (l2, 1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))) = ((r, a), (l, e)). (2) Monotonicity. 0 0 0 0 0 Let X0 ¼ r 01 ; a01 ; l1 ; e01 ; r 02 ; a02 ; l2 ; e02 ;. . . ; r 0n ;a0n ; ln ; e0n be a set ofinterval-valued 0 0 2-tuples. If ri ; ai P ðr i ; ai Þ 0 0 0 0 0 0 0 0 0 0 0 0 and li ; ei P ðli ; ei Þ, for any i, then IVTWG r1 ; a1 Þ; l1 ; e1 Þ; r2 ; a2 ; ðl2 ; e2 Þ ; . . . ; r n ; an ; ln ; en Þ P IVTWGðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ; for any w. (3) Boundary.
Dð½minðr i ; ai Þ; minðli ; ei ÞÞ 6 IVTWGðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ i
i
6 Dð½maxðr i ; ai Þ; maxðli ; ei ÞÞ: i
i
Proof (1) Since ((rj, aj), (lj, ej)) = ((r, a), (l, e)), j = 0, 1, 2, . . ., n, then
IVTWGðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ ¼ D
" n Y
ðD1 ðr; aÞÞwi ;
i¼1 n X
"
n Y
# ðD1 ðl; eÞÞwi
i¼1
n X
wi
¼ D ðD1 ðr; aÞÞ i¼1 ; ðD1 ðl; eÞÞ i¼1
wi
¼ ððr; aÞ; ðl; eÞÞ:
(2) If r 0i ; a0i P ðri ; ai Þ, for any i, then we have n n Y Y wi D1 r0i ; a0i P D1 ðr i ; ai Þ ) D1 r0i ; a0 P ðD1 ðr i ; ai ÞÞwi ) ðD1 ðr0i ; a0 ÞÞwi P ðD1 ðr i ; ai ÞÞwi : i¼1
i¼1
0
Similarly, when ðli ; e0i Þ P ðli ; ei Þ for any i, we get n Y
n Y 0 wi D1 li ; e0i P ðD1 ðli ; ei ÞÞwi :
i¼1
i¼1
Based on Theorem 1, we obtain
D
" # " # n n n n Y Y Y wi Y 0 wi PD D1 r0i ; a0i ; D1 li ; e0i ðD1 ðr i ; ai ÞÞwi ; ðD1 ðli ; ei ÞÞwi ; i¼1
i¼1
i¼1
i¼1
0 0 0 i:e:; IVTWG r 01 ; a01 ; l1 ; e01 ; r 02 ; a02 ; l2 ; e02 ; . . . ; r 0n ; a0n ; ln ; e0n P IVTWGðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ: (3) Since maxi(ri, ai) P (ri, ai) P mini(ri, ai) and mini(li, ei) 6 (li, ei) 6 maxi(li, ei) for any i, then, based on the Monotonicity of Theorem 2, we derive
Dð½minðr i ; ai Þ; minðli ; ei ÞÞ 6 IVTWGðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ i
i
6 Dð½maxðr i ; ai Þ; maxðli ; ei ÞÞ: i
i
Definition 12. Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))} be a set of interval-valued 2-tuples and w = (w1, P w2, . . . , wn)T be their associated weights, with wi 2 [0, 1] and ni¼1 wi ¼ 1. The interval-valued 2-tuple ordered weighted geometric (IVTOWG) operator is defined as
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IVTOWGðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ " # n n Y Y wi wi 1 1 ¼D ðD ðr rðiÞ ; arðiÞ ÞÞ ; ðD ðlrðiÞ ; erðiÞ ÞÞ i¼1
ð16Þ
i¼1
where [(rr(i), ar(i)), (lr(i), er(i))] is the ith largest of ((rj, aj), (lj, ej)). Similarly, the desired properties of the IVTOWG operator can be listed as follows. Theorem 3. Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))} be a set of interval-valued 2-tuples, w = (w1, w2, P . . . ,wn)T be their associated weights, with wi 2 [0, 1] and ni¼1 wi ¼ 1, then, for IVTOWG operator, we have (1) Idempotency. If all ((rj, aj), (lj, ej)) (j = 0, 1, 2, . . . , n) are equal, i.e., ((rj, aj), (lj, ej)) = ((r, a), (l, e)), then IVTOWG (((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))) = ((r, a), (l, e)). (2) Monotonicity. 0 0 0 0 0 0 0 0 0 Let X 0 ¼ r 1 ; a1 ; l1 ; e1 ; r2 ; a2 ; l2 ; e2 ; . . . ; r0n ; a0n ; ln ; e0n be a set of interval-valued 2-tuples. If r0i ; a0i P ðr i ; ai Þ 0 and ðli ; e0i ÞP ðli ; ei Þ, for any i, then 0 0 IVTOWG r01 ; a01 ; l1 ; e01 ; r02 ; a02 ; l2 ; e02 ; 0 0 0 0 . . . ; rn ; an ; ln ; en Þ P IVTOWGðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððrn ; an Þ; ðln ; en ÞÞÞ; for any w. (3) Boundary.
Dð½minðr i ; ai Þ; minðli ; ei ÞÞ 6 IVTOWGðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððrn ; an Þ; ðln ; en ÞÞÞ i
i
6 Dð½maxðr i ; ai Þ; maxðli ; ei ÞÞ: i
i
(4) Commutativity. 0 0 0 0 0 0 0 0 0 If r 1 ; a1 ; l1 ; e1 ; r 2 ; a2 ; l2 ; e2 ; . . . ; r 0n ; a0n ; ln ; e0n is any permutation of (((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))), then for any w 0
0
0
IVTOWGðððr 01 ; a01 Þ; ðl1 ; e01 ÞÞ; ððr 02 ; a02 Þ; ðl2 ; e02 ÞÞ; . . . ; ððr 0n ; a0n Þ; ðln ; e0n ÞÞÞ ¼ IVTOWGðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððrn ; an Þ; ðln ; en ÞÞÞ: Definition 13. Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))} be a set of interval-valued 2-tuples and w = (w1, P w2, . . . , wn)T be their associated weights, with wi 2 [0, 1] and ni¼1 wi ¼ 1. The interval-valued 2-tuple ordered weighted harmonic (IVTOWH) operator is defined as
IVTOWHðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ 2 !1 !1 3 n n X X wi wi 4 5; ; ¼D 1 1 i¼1 D ðr rðiÞ ; arðiÞ Þ i¼1 D ðlrðiÞ ; erðiÞ Þ
ð17Þ
where [(rr(i), ar(i)), (lr(i), er(i))] is the ith largest of ((rj, aj), (lj, ej)). Definition 14. Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))} be a set of interval-valued 2-tuples and w = (w1, P w2, . . . , wn)T be their associated weights, with wi 2 [0, 1] and ni¼1 wi ¼ 1. The interval-valued 2-tuple ordered weighted quadratic (IVTOWQ) operator is defined as
IVTOWQðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ 2 !0:5 !0:5 3 n n X X 2 2 1 1 5; wi ðD ðr rðiÞ ; arðiÞ ÞÞ ; wi ðD ðlrðiÞ ; erðiÞ ÞÞ ¼ D4 i¼1
ð18Þ
i¼1
where [(rr(i), ar(i)), (lr(i), er(i))] is the ith largest of ((rj, aj), (lj, ej)). It is easy to prove that both IVTOWH operator and IVTOWQ operator are idempotent, monotonic, bounded and commutative. Motivated by Yager [34], we develop the generalized interval-valued 2-tuple weighted average (GIVTWA) operator and generalized interval-valued 2-tuple ordered weighted average (GIVTOWA) operator.
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Definition 15. Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))} be a set of interval-valued 2-tuples and w = (w1, P w2, . . . , wn)T be their associated weights, with wi 2 [0, 1], ni¼1 wi ¼ 1 and a parameter k 2 (1, 0) [ (0, +1). The generalized interval-valued 2-tuple weighted average (GIVTWA) operator is defined as
GIVTWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ
X 1k X 1k n n : ¼D wi ðD1 ðr i ; ai ÞÞk ; wi ðD1 ðli ; ei ÞÞk i¼1
ð19Þ
i¼1
If each ((rj, aj), (lj, ej)) is degenerated to 2-tuple (rj, aj), the GIVTWA operator will become the generalized 2-tuple weighted average operator. Definition 16. Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))} be a set of interval-valued 2-tuples and w = (w1, P w2, . . . , wn)T be their associated weights, with wi 2 [0, 1], ni¼1 wi ¼ 1 and a parameter k 2 (1, 0) [ (0, +1). The generalized interval-valued 2-tuple ordered weighted average (GIVTOWA) operator is defined as
GIVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ 2 !1k !1k 3 n n X X k k 1 1 4 wi ðD ðr rðiÞ ; arðiÞ ÞÞ ; wi ðD ðlrðiÞ ; erðiÞ ÞÞ 5; ¼D i¼1
ð20Þ
i¼1
where [(rr(i), ar(i)), (lr(i), er(i))] is the ith largest of ((rj, aj), (lj, ej)). If each ((rj, aj), (lj, ej)) is reduced to 2-tuple (rj, aj), the GIVTOWA operator will become the generalized 2-tuple ordered weighted average operator. And if ((ri, ai), (li, ei)) = [(rr(i), ar(i)), (lr(i), er(i))] for all i, then the GIVTOWA operator will become the GIVTWA operator. Similarly, the desired properties of the GIVTOWA operator are listed as follows. Theorem 4. Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))} be a set of interval-valued 2-tuples, w = (w1, Pn w2, . . . , wn)T be their associated weights, with wi 2 [0, 1], i¼1 wi ¼ 1 and k 2 (1, 0) [ (0, +1), then, for GIVTOWA operator, we have (1) Idempotency. If all ((rj, aj), (lj, ej)) (j = 0, 1, 2, . . . , n) are equal, i.e., ((rj, aj), (lj, ej)) = ((r, a), (l, e)), then GIVTOWA (((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))) = ((r, a), (l, e)). (2) Monotonicity. 0 0 0 0 0 0 0 0 0 Let X 0 ¼ r 1 ; a1 ; l1 ; e1 ; r 2 ; a2 ; l2 ; e2 ; . . . ; r 0n ; a0n ; ln ; e0n be aset of interval-valued 2-tuples. If r0i ; a0i P ðr i ; ai Þ 0 0 0 and ðli ; e0i Þ P ðli ; ei Þ, for any i, then. GIVTOWA r01 ; a01 ; l1 ; e01 ; r02 ; a02 ; l2 ; e02 ; . . . ; r0n ; a0n ; 0 0 ln ; en ÞÞ P GIVTOWAðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ; for any w. (3) Boundary.
Dð½minðr i ; ai Þ; minðli ; ei ÞÞ 6 GIVTOWAðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ i
i
6 Dð½maxðr i ; ai Þ; maxðli ; ei ÞÞ: i
i
0 0 0 0 0 0 0 0 0 (4) Commutativity. If r1 ; a1 ; l1 ; e1 ; r 2 ; a2 ; l2 ; e2 ; . . . ; r0n ; a0n ; ln ; e0n is any permutation of (((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))), then for any w
0 0 0 GIVTOWA r 01 ; a01 ; l1 ; e01 ; r 02 ; a02 ; l2 ; e02 ; . . . ; r0n ; a0n ; ln ; e0n ¼ GIVTOWAðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ: Proof (1) Since ((rj, aj), (lj, ej)) = ((r, a), (l, e)), j = 0, 1, 2, . . ., n, then
2
n X GIVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ ¼ D4 wi ðD1 ðr; aÞÞk i¼1 1
k
1 k
1
k
1 k
¼ D½ððD ðr; aÞÞ Þ ; ððD ðl; eÞÞ Þ ¼ ððr; aÞ; ðl; eÞÞ: (2) If r 0i ; a0i P ðri ; ai Þ, for any i, then we have
!1k
n X ; wi ðD1 ðl; eÞÞk i¼1
!1k 3 5
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D1 ðr 0i ; a0i Þ P D1 ðr i ; ai Þ ) wi D1 ðr 0i ; a0i Þ P wi D1 ðr i ; ai Þ )
8 n n X X > > > ðwi D1 ðr 0i ; a0i ÞÞk P ðwi D1 ðr i ; ai ÞÞk ; for k > 0; > > < i¼1 i¼1 n n > X X > > 1 0 0 k > ðw D ðr ; a ÞÞ 6 ðwi D1 ðr i ; ai ÞÞk ; for k < 0; > i i i : i¼1
)
8 !1k n > X > 1 0 > 0 k > ðw D ðr ; a ÞÞ P > i i > < i¼1
n X
i¼1
!1k ðwi D1 ðr i ; ai ÞÞk
; for k > 0;
i¼1
!1k > > n n > X X 1 > k 1 0 > 0 > ðwi D ðr i ; a ÞÞ P ð ðwi D1 ðr i ; ai ÞÞk Þk ; for k < 0: : i¼1
i¼1
0 Similarly, when li ; e0i P ðli ; ei Þ for any i, we can get n X
0 k wi D1 li ; e0
!1k
n X
P
i¼1
!1k 1
k
wi ðD ðli ; ei ÞÞ
for k 2 ð1; 0Þ [ ð0; þ1Þ
i¼1
, Based on Theorem 1, we obtain
0 0 0 GIVTOWA r 01 ; a01 ; l1 ; e01 ; r 02 ; a02 ; l2 ; e02 ; . . . ; r0n ; a0n ; ln ; e0n P GIVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ;
for any w:
(3) Since maxi(ri, ai) P (ri, ai) P mini(ri, ai) and mini(li, ei) 6 (li, ei) 6 maxi(li, ei) for any i, then, based on the Monotonicity of Theorem 4, we derive
Dð½minðr i ; ai Þ; minðli ; ei ÞÞ 6 GIVTWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ i
i
6 Dð½maxðr i ; ai Þ; maxðli ; ei ÞÞ: i
i
(4) Trivial from the definition of GIVTOWA operator. If we take different parameter k into the GIVTOWA operator, we can derive a series of aggregation operators, such as follows.
Remark 1. If k = 1, the GIVTOWA operator is degenerated to the IVTOWA operator.
GIVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ ¼ D
" # n n X X wi D1 ðr rðiÞ ; arðiÞ Þ; wi D1 ðlrðiÞ ; erðiÞ Þ i¼1
i¼1
¼ IVTOWAðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ: Remark 2. If k ? 0, the GIVTOWA operator is degenerated to the IVTOWG operator. By L’Hopital’s rule, we get
lim log GIVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððrn ; an Þ; ðln ; en ÞÞÞ k!0 " ! !# n n X X 1 1 wi ðD1 ðr rðiÞ ; arðiÞ ÞÞk ; lim log wi ðD1 ðlrðiÞ ; erðiÞ ÞÞk ¼ D lim log k!0 k k!0 k i¼1 i¼1 " # Pn Pn k k 1 1 1 1 w ð D ðr ; a ÞÞ log D ðr ; a Þ i r ðiÞ r ðiÞ r ðiÞ r ðiÞ i¼1 i¼1 wi ðD ðlrðiÞ ; erðiÞ ÞÞ log D ðlrðiÞ ; erðiÞ Þ ¼ D lim ; lim Pn Pn k k 1 1 k!0 k!0 i¼1 wi ðD ðr rðiÞ ; arðiÞ ÞÞ i¼1 wi ðD ðlrðiÞ ; erðiÞ ÞÞ "P # " # Pn n 1 1 n n Y Y wi wi 1 1 i¼1 wi log D ðr rðiÞ ; arðiÞ Þ i¼1 wi log D ðlrðiÞ ; erðiÞ Þ Pn Pn ¼ D log ðD ðr rðiÞ ; arðiÞ ÞÞ ; log ðD ðlrðiÞ ; erðiÞ ÞÞ : ¼D ; i¼1 wi i¼1 wi i¼1 i¼1 That is, if k ? 0, then,
GIVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ ¼ IVTOWGðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððrn ; an Þ; ðln ; en ÞÞÞ:
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Remark 3. If k = 1, the GIVTOWA operator is degenerated to the IVTOWH operator.
GIVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ 2 !1 !1 3 n n X X wi wi 4 5 ; ¼D 1 1 i¼1 D ðr rðiÞ ; arðiÞ Þ i¼1 D ðlrðiÞ ; erðiÞ Þ ¼ IVTOWHðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ: Remark 4. If k = 2, the GIVTOWA operator is degenerated to the IVTOWQ operator.
GIVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ 2 !0:5 !0:5 3 n n X X 2 2 1 1 5 wi ðD ðr rðiÞ ; arðiÞ ÞÞ ; wi ðD ðlrðiÞ ; erðiÞ ÞÞ ¼ D4 i¼1
i¼1
¼ IVTOWQðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððrn ; an Þ; ðln ; en ÞÞÞ: Remark 5. If k ? +1, the GIVTOWA operator reduces to the Max operator, and if k ? 1, the GIVTOWA operator reduces to the Min operator, i.e., GIVTOWA (((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))) = maxi((ri, ai), (li, ei)), for k ? +1; GIVTOWA (((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))) = mini((ri, ai), (li, ei)), for k ? 1. If k ? +1, then according to the monotonicity and idempotency of the GIVTOWA operator, we have
GIVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððrn ; an Þ; ðln ; en ÞÞÞ 2 !1k !1k 3 n n X X k k 1 1 4 6D wi ðD ðr rð1Þ ; arð1Þ ÞÞ ; wi ðD ðlrð1Þ ; erð1Þ ÞÞ 5 ¼ maxððr i ; ai Þ; ðli ; ei ÞÞ: i¼1
i
i¼1
Since k > 0, we get
GIVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððrn ; an Þ; ðln ; en ÞÞÞ 2 !1k !1k 3 n n h i X X k k 1 1 4 5 P D ðw1 ðD1 ðr rð1Þ ; arð1Þ ÞÞk Þ1k ; ðw1 ðD1 ðlrð1Þ ; erð1Þ ÞÞk Þ1k ¼D wi ðD ðr rðiÞ ; arðiÞ ÞÞ ; wi ðD ðlrðiÞ ; erðiÞ ÞÞ i¼1
i¼1
h 1 1 i h 1 1 i 1 1 þ1 ¼ D w1k ðD1 ðr rð1Þ ; arð1Þ ÞÞ ; w1k ðD1 ðlrð1Þ ; erð1Þ ÞÞ ¼ D wþ1 1 ðD ðr rð1Þ ; arð1Þ ÞÞ ; w1 ðD ðlrð1Þ ; erð1Þ ÞÞ h i ¼ D ðD1 ðr rð1Þ ; arð1Þ ÞÞ; ðD1 ðlrð1Þ ; erð1Þ ÞÞ ¼ maxððr i ; ai Þ; ðli ; ei ÞÞ: i
In consequence, we get maxi((ri, ai), (li, ei)) 6 GIVTOWA(((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))) 6 maxi((ri, ai), (li, ei)), i.e., if k ? +1, we get the Max operator. A similar proof can be given for k ? 1. 4. An approach to MAGDM based on the GIVTWA operator The interval-valued 2-tuple aggregation operators can be widely applied in solving decision making problems, where the values of attributes take the form of interval-valued 2-tuples. In this section, we will analyze the process to follow in MAGDM problem by using the GIVTWA operator. Since we should first determine the weight vector for the aggregation operator before the aggregation operation, in what follows, we put forward a method to determine the weight vector of interval-valued 2-tuple aggregation operator based on the concept of degree of precision. 4.1. A method to determine the weight vector of interval-valued 2-tuple aggregation operator In this paper, the decision information is expressed by interval-valued 2-tuple, such as ((si, a1), (sj, a2)) represented that the performance of one alternative on an attribute is between 2-tuples (si, a1) and (sj, a2). Obviously, the closer between (si, a1) and (sj, a2), the more precise information the decision maker can give with the predefined linguistic term set. So we can represent the degree of precision of the decision information by the difference between D1(si, a1) and D1(sj, a2). For a given alternative and attribute, decision makers may have different professional knowledge (experience or ability) and apply different interval-valued 2-tuples to express the assessments. Thus, we should aggregate such interval-valued 2-tuple linguistic
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information for the ranking of alternatives. Theoretically speaking, we should assign bigger weights to the interval-valued 2tuples with higher degree of precision, and vice versa. However, according to above analysis, an interval-valued 2-tuple provided by the decision maker who choose a linguistic term with smaller cardinality incline to have higher degree of precision. To avoid such limitation, we define the degree of precision as follows. Definition 17. Suppose S = {s0, s1, . . . , sg} be an ordered linguistic term set. An interval-valued 2-tuple ((si, a1), (sj, a2)) is composed of two linguistic terms from S. The degree of precision of ((si, a1), (sj, a2)) is formulated as follows.
DPððsi ; a1 Þ; ðsj ; a2 ÞÞ ¼ 1
D1 ðsj ; a2 Þ D1 ðsi ; a1 Þ þ 1=g : 1 þ 1=g
ð21Þ
It is easy to prove 0 6 DP ((si, a1), (sj, a2)) 6 g/(1 + g). If D1(si, a1) = D1(sj, a2), we can get DP ((si, a1), (sj, a2)) = g/(1 + g). In this situation, ((si, a1), (sj, a2)) is reduced to (si, a1) and the degree of precision reaches its maximum, which is smaller than 1. In essence, a 2-tuple is not a crisp number and its degree of precision should be smaller than 1 unless the cardinality of S (i.e.,1 + g) approaches to positive infinity. On the contrary, we have DP ((si, a1), (sj, a2)) = 0 if D1(si, a1) = 0 and D1(sj, a2) = 1, i.e., (si, a1) = (s0, 0) and (sj, a2) = (sg, 0). This is another extreme case, which means this assessment cannot provide any useful information for decision making. Based on these ideas, we can determine the interval-valued 2-tuple aggregation operator weights in MAGDM as follows. For a given alternative Ai with respect to the attribute Cj, suppose the decision makers dk (1 6 k 6 t) provide their ij ij ij . Then, the weight vector assessment information as: r ij1 ; aij1 ; l1 ; eij1 ; rij2 ; aij2 ; l2 ; eij2 ; . . . ; r ijt ; aijt ; lt ; eijt ij ij ij xij ¼ x1 ; x2 ; . . . ; xt of the aggregation operator can be determined as follows.
ij DP r ijk ; aijk ; lk ; eijk : x ¼ Pt ij r ijh ; aijh ; lh ; eijh h¼1 DP ij k
ð22Þ
P It is easy to prove that xijk 2 ½0; 1 and tk¼1 xijk ¼ 1. On that basis, the overall performance of the alternative on the attribute can be derived by the interval-valued 2-tuple aggregation operator. Next, we will present the application of above-mentioned theories, taking the GIVTWA operator for example. 4.2. The process of MAGDM based on the GIVTWA operator In this section, we will analyze the process to follow in MAGDM problem by using the GIVTWA operator, where the weights of attributes are known and the performance values take the form of interval linguistic variables. Suppose A = (A1, A2, . . . , Am) be a discrete set of m feasible alternatives and C = (C1, C2, . . . , Cn) be a finite set of attributes, P whose weight vector is w = (w1, w2, . . . , wn), with wi 2 [0, 1] and ni¼1 wi ¼ 1. Let d = (d1, d2, . . . , dt) be the set of decision mak ers. Assume that each decision maker dk provides his own decision matrix Ek ¼ eijk ðk ¼ 0; 1; 2; . . . ; tÞ, where eijk repremn
sents the preference value of alternative Ai with respect to attribute Cj.eijk takes the form of interval linguistic variable [sa, sb], where sa and sb are derived from S = {s0, s1, . . . , sg}, with 0 6 a 6 b 6 g. In addition, decision makers may use different linguistic term sets S to express the preference values. The process with the GIVTWA operator in MAGDM involves the following steps. Step 1. The decision makers choose their own linguistic term sets and provide their evaluations about the alternative Ai under the attribute Cj, denoted by interval linguistic decision matrix Ek ¼ eijk . mn Step 2. Transform the interval linguistic decision matrix Ek ¼ eijk into interval-valued 2-tuple linguistic decision mn ij ij matrix T k ¼ tk ¼ ððsa ; 0Þ; ðsb ; 0ÞÞk . mn mn Step 3. Utilize Eqs. (21) and (22) to obtain the weight vector of the GIVTWA operator, xij ¼ xij1 ; xij2 ; . . . ; xijt . Step 4. Aggregate all the decision matrices Tk (k = 1, 2, . . . , t) into a collective decision matrix T = (tij)mn based on GIVTWA operator, t ij ¼ GIVTWA tij1 ; tij2 ; . . . ; tijt , whose weight vector is xij ¼ xij1 ; xij2 ; . . . ; xijt . Step 5. Utilize the GIVTWA operator ti = GIVTWA (ti1, ti2, . . . , tin) to yield the collective overall preference values ti for each alternative Ai(i = 0,1, 2, . . ., m), where the weight vector is w = (w1, w2, . . . , wn). Step 6. Rank the alternatives in accordance with ti, referring to Theorem 1. Step 7. End. 5. Illustrative example In the following, we consider an example adapted from Herrera et al. [24], Herrera and Herrera-Viedma [13].
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H. Zhang / Applied Mathematical Modelling 37 (2013) 4269–4282 Table 1 The decision matrix E1 given by D1.
A1 A2 A3 A4
C1
C2
C3
C4
[x7, x7] [x5, x7] [x4, x7] [x3, x3]
[x1, x1] [x4, x4] [x3, x4] [x0, x2]
[x1, x3] [x3, x4] [x3, x6] [x4, x4]
[x3, x5] [x4, x6] [x1, x1] [x3, x4]
Table 2 The decision matrix E2 given by D2.
A1 A2 A3 A4
C1
C2
C3
C4
[y3, y3] [y5, y5] [y2, y4] [y4, y5]
[y5, y5] [y4, y5] [y4, y5] [y1, y2]
[y1, y3] [y1, y3] [y1, y3] [y3, y4]
[y3, y4] [y3, y5] [y0, y1] [y3, y5]
Table 3 The decision matrix E3 given by D3. C1
C2
C3
C4
[z2, z3] [z3, z3] [z3, z3] [z1, z1]
[z1, z2] [z2, z3] [z0, z2] [z2, z3]
[z1, z3] [z2, z3] [z2, z3] [z1, z3]
[z3, z3] [z1, z1] [z2, z2] [z3, z3]
C1
C2
C3
C4
((x7, 0), (x7, 0)) ((x5, 0), (x7, 0)) ((x4, 0), (x7, 0)) ((x3, 0), (x3, 0))
((x1, 0), (x1, 0)) ((x4, 0), (x4, 0)) ((x3, 0), (x4, 0)) ((x0, 0), (x2, 0))
((x1, 0), (x3, 0)) ((x3, 0), (x4, 0)) ((x3, 0), (x6, 0)) ((x4, 0), (x4, 0))
((x3, 0), (x5, 0)) ((x4, 0), (x6, 0)) ((x1, 0), (x1, 0)) ((x3, 0), (x4, 0))
A1 A2 A3 A4
Table 4 The decision matrix T1.
A1 A2 A3 A4
Suppose an investment company is looking for an optimal investment. There are four possible alternatives to invest the money: A1 is a car industry; A2 is a food company; A3 is a computer company; A4 is an arms industry. After careful review of the information, the company selects four criteria: C1 is the risk analysis; C2 is the growth analysis; C3 is the social-political impact analysis; C4 is the environmental impact analysis. The weight vector of attributes is w = (0.3, 0.1, 0.2, 0.4). Three experts are invited to provide their preferences for each alternative on each attribute using different linguistic term sets:D1 provides his preferences in the set of 9 labels,X = {x0, x1, x2, x3, x4, x5, x6, x7, x8}; D2 provides his preferences in the set of 7 labels, Y = {y0, y1, y2, y3, y4, y5, y6}; D3 provides his preferences in the set of 5 labels, Z = {z0, z1, z2, z3, z4}. Step 1. The decision makers choose their own linguistic term sets as mentioned above and give their decision matrices ðk ¼ 1; 2; 3Þ as shown in Tables 1–3, respectively. Ek ¼ eijk 44 Step 2. Transform decision matrix Ek ¼ ðeijk Þ44 into interval-valued 2-tuple linguistic decision matrix T k ¼ t ijk ¼ 44 ij ððsa ; 0Þ; ðsb ; 0ÞÞk . Taking R1 as an example, we can get the interval-valued 2-tuple linguistic decision matrix 44
T1 as shown in Table 4.
Step 3. Utilize Eqs. (21) and (22) to obtain the GIVTWA weights,xij ¼ xij1 ; xij2 ; xij3 . For example, based on the assessment information (((x7,0), (x7,0)), ((y3,0), (y3,0)), ((z2,0), (z3, 0))) provided by three decision makers, we can derive ¼ ð0:3789; 0:3654; 0:2558Þ. Step 4. Aggregate the decision matrices Tk (k = 1,2,3) into a collective decision matrix T = (tij)44 by GIVTWA operator, and let k = 1, the decision matrix T = (tij)44 is shown in Table 5. Step 5. Utilize the GIVTWA operator ti = GIVTWA (ti1, ti2, ti3, ti4)(k = 1) to yield the collective overall preference values ti for each alternative Ai (i = 1, 2, 3, 4), where the weight vector is w = (0.3, 0.1, 0.2, 0.4). The collective overall preference values are shown in Table 6. 11 11 x11 ¼ x11 1 ; x2 ; x3
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Table 5 The collective decision matrix T.
A1 A2 A3 A4
C1
C2
C3
C4
[0.6421, 0.7061] [0.7450, 0.8167] [0.5544, 0.7613] [0.4201, 0.4696]
[0.4158, 0.4798] [0.5540, 0.6761] [0.4058, 0.6258] [0.2112, 0.4315]
[0.1701, 0.5102] [0.3524, 0.5770] [0.3495, 0.6673] [0.4501, 0.6094]
[0.5535, 0.6845] [0.4019, 0.5771] [0.2127, 0.2622] [0.5478, 0.6817]
Table 6 The collective overall preference values and the rankings of alternatives. t1
t2
t3
t4
Ranking
D([0.4896, 0.6357]) ((x4, 0.0104), (x5, 0.0107))
D([0.5101, 0.6589]) ((x4, 0.0101), (x5, 0.0339))
D([0.3619, 0.5293]) ((x3, 0.0131), (x4, 0.0293))
D([0.4563, 0.5786]) ((x4, 0.0437), (x5, 0.0464))
A2 > A1 > A4 > A3
The values of score function
0.8 0.7 0.6 C
0.5
B
0.4
A A1 A2 A3 A4
0.3 0.2 -100
-80
-60
-40
-20
0
20
40
60
80
100
λ value Fig. 1. The values of score function with k 2 [100, 100].
Table 7 The rankings of alternatives with k 2 [100, 100].
Ranking
100 6 k < kA
kA 6 k 6 kB
kB < k 6 kC
kC < k 6 100
A2 > A1 > A4 > A3
A2 > A4 > A1 > A3
A2 > A1 > A4 > A3
A2 > A1 > A3 > A4
In addition, by Eq. (8), we can express the final results (interval values) in the initial expression domain used by each expert. Taking the expert D1 as an example, the final results can be expressed by interval-valued 2-tuples derived from the linguistic term set X with 9 labels, which are listed in Table 6. Step 6. By ranking ti (i = 1, 2, 3, 4) based on Theorem 1, the priorities of the alternatives can be obtained, which are listed in Table 6. Given the collective decision matrix T = (tij)44 in this example, we next consider what happens when the values of k change. Depending on the steps of our proposed aggregation operators with different k values, we can obtain the collective overall preference values ri for each alternative Ai together with their score functions as shown in Fig. 1, where k is set to [100, 100]. As we can see, it is obvious that the values of score function are non-decreasing with respect to k. In addition, the ranking order of alternatives may be different with the variation of parameter k. However,A2 is always the optimal choice despite the change of k value. According to Fig. 1, there are three k values changing the ranking order of the alternatives, i.e.,kA = 9.7247, kB = 1.3348 and kC = 5.0701. Furthermore, we can further determine the ranking when k = kA, k = kB and k = kC, respectively. According to accuracy function defined in Definitions 8 and Theorem 1, if k = kA, we get H(A1) = 0.3553 > H(A4) = 0.2218, i.e.,A4 > A1. Similarly, we have A4 > A1 if k = kB and A4 > A3 if k = kC. Consequently, the ranking orders can be shown in Table 7.
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Since the variation of k value may lead to different ranking orders, decision makers may have difficulty in selecting the optimal alternative with different k values. In other words, it is necessary for the decision maker to set k value before information aggregation. In general, the more pessimistic of the decision maker, the larger k value he or she may set, which means each alternative is associated with a higher performance value (score function). On the contrary, the more optimistic of the decision maker, the smaller k value he or she may set. If the decision maker does not give his or her subjective preference, the most commonly used value k = 1 can be taken. 6. Conclusion In this paper, we present a series of interval-valued 2-tuple aggregation operators, together with their properties and the relationships among them. In addition, we develop a method to determine the weight vector of interval-valued 2-tuple aggregation operator based on the concept of degree of precision. This method assigns bigger weights to the interval-valued 2-tuples with higher degree of precision, and vice versa. As a result, in MAGDM, the decision information given by the decision maker who chooses the appropriate linguistic term set inclines to play a big role in information aggregation. In addition, the illustrative example demonstrates that the decision makers can naturally provide and easily aggregate their assessment information by interval-valued 2-tuple linguistic approach under multi-granular linguistic contexts. Acknowledgment The author thank the anonymous referees for the valuable comments and suggestions which are very helpful to improve this paper. The work was supported by Humanities and Social Science Research Planning Project of Henan Provincial Education Department (2011-GH-0052). Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/ j.apm.2012.09.033. References [1] F. 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Herrera, Computing with words in decision support systems: an overview on models and applications, Int. J. Comput. Intell. Syst. 3 (2010) 382–395. [13] F. Herrera, E. Herrera-Viedma, Linguistic decision analysis: Steps for solving decision problems under linguistic information, Fuzzy Sets Syst. 115 (2000) 67–82. [14] Y.P. Jiang, Z.P. Fan, Property analysis of the aggregation operators for 2-tuple linguistic information, Control Decision 18 (2003) 754–757. [15] Y. Zhang, Z.P. Fan, An approach to linguistic multiple attribute decision-making with linguistic information based on ELOWA operator, Syst. Eng. 4 (2006) 98–101. [16] S.Y. Wang, Applying 2-tuple multigranularity linguistic variables to determine the supply performance in dynamic environment based on productoriented strategy, IEEE Trans. Fuzzy Syst. 16 (2008) 29–39. [17] J.H. Wang, J.Y. Hao, An approach to aggregation of ordinal information in multi-criteria multi-person decision making using Choquet integral of Fubini type, Fuzzy Optim. Decision Making 8 (2009) 365–380. [18] Y.C. Dong, Y.F. Xu, H.Y. Li, B. Feng, The OWA-based consensus operator under linguistic representation models using position indexes, Eur. J. Oper. Res. 203 (2010) 455–463. [19] G.W. Wei, A method for multiple attribute group decision making based on the ET-WG and ET-OWG operators with 2-tuple linguistic information, Expert Syst. Appl. 37 (2010) 7895–7900. [20] W. Yang, Z.P. Chen, New aggregation operators based on the Choquet integral and 2-tuple linguistic information, Expert Syst. Appl. 39 (2012) 2662– 2668. [21] Z.S. Xu, Dependent OWA operators, Lect. Notes Artif. Intel. 3885 (2006) 172–178. [22] G.W. Wei, X.F. Zhao, Some dependent aggregation operators with 2-tuple linguistic information and their application to multiple attribute group decision making, Expert Syst. 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[24] F. Herrera, E. Herrera-Viedma, L. Martı´nez, A fusion approach for managing multi-granularity linguistic terms sets in decision making, Fuzzy Sets Syst. 114 (2000) 43–58. [25] F. Herrera, L. Martı´nez, A model based on linguistic 2-tuple for dealing with multi-granular hierarchical linguistic contexts in multi-expert decisionmaking, IEEE Trans. Syst. Man Cybernet. 31 (2001) 227–234. [26] M. Espinilla, J. Liu, L. Martnez, An extended hierarchical linguistic model for decision-making problems, Comput. Intell. 27 (2011) 489–512. [27] J.H. Wang, J.Y. Hao, A new version of 2-tuple fuzzy linguistic representation model for computing with words, IEEE Trans. Fuzzy Syst. 14 (2006) 435– 445. [28] J. Lin, J.B. Lan, Y.H. Lin, Multi-attribute group decision-making method based on the aggregation operators of interval 2-tuple linguistic information, J. Jilin Normal Univ. 1 (2009) 5–9. [29] V. Torra, Hesitant fuzzy sets, Int. J. Intell. Syst. 25 (2010) 529–539. [30] R.M. Rodríguez, L. Martínez, F. Herrera, Hesitant fuzzy linguistic term sets for decision making, IEEE Trans. Fuzzy Syst. 20 (2012) 109–119. [31] C.T. Chen, W.S. Tai, Measuring the intellectual capital performance based on 2-tuple fuzzy linguistic information, in: Proceedings of the 10th Annual Meeting of Asia Pacific Region of Decision Sciences Institute, Taiwan, 2005. [32] F. Herrera, L. Martı´nez, P.J. Sanchez, Managing non-homogeneous information in group decision-making, Eur. J. Oper. Res. 166 (2005) 15–132. [33] L. Martínez, F. Herrera, An overview on the 2-tuple linguistic model for computing with words in decision making: extensions, applications and challenges, Inform. Sci. 207 (2012) 1–18. [34] R.R. Yager, Generalized OWA aggregation operators, Fuzzy Optim. Decision Making 3 (2004) 93–107.
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Some interval-valued 2-tuple linguistic aggregation operators and application in multiattribute group decision making Huimin Zhang ⇑ School of Management, Henan University of Technology, Zhengzhou 450001, China
a r t i c l e
i n f o
Article history: Received 26 January 2012 Received in revised form 10 August 2012 Accepted 11 September 2012 Available online 21 September 2012 Keywords: MAGDM 2-tuple Interval-valued 2-tuple Aggregation operator Degree of precision
a b s t r a c t This paper deals with multiattribute group decision making (MAGDM) problems with interval-valued 2-tuple linguistic information. First, we introduce some new aggregation operators, such as the interval-valued 2-tuple weighted geometric (IVTWG) operator, the interval-valued 2-tuple ordered weighted geometric (IVTOWG) operator, the generalized interval-valued 2-tuple weighted average (GIVTWA) operator and the generalized interval-valued 2-tuple ordered weighted average (GIVTOWA). Then, we discuss their desired properties and relationships among them. Furthermore, we put forward a new method to determine the weight vector of interval-valued 2-tuple aggregation operator based on the concept of degree of precision. Finally, a numerical example is provided to illustrate the efficiency of the proposed method in dealing with interval-valued 2-tuple linguistic information under multi-granular linguistic contexts. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction The 2-tuple fuzzy linguistic representation model, composed by a linguistic term and a real number, was developed by Herrera and Martínez [1] based on the concept of symbolic translation. Since the 2-tuple linguistic model can express any counting of information in the universe of discourse and avoid the information loss in the process of linguistic information processing, it has been widely studied and applied in decision making [2–10]. As a useful computational approach for computing with words (CW) (more detailed literature about CW is available in [11,12]), the 2-tuple linguistic model improves the accuracy and facilitate the processes of CW. Herrera and Herrera-Viedma [13] pointed out the solution scheme in the linguistic decision analysis must be formed by three steps: the choice of the linguistic term set with its semantic, the choice of the aggregation operator of linguistic information and the choice of the best alternatives, which is carried out in two phases, i.e., aggregation phase of linguistic information and exploitation phase. In light of the fact that information aggregation always plays an important role in decision making process, many 2-tuple aggregation operators have been proposed to aggregate information. Herrera and Martínez [1] put forward the 2-tuple arithmetic mean, weighted average and ordered weighted average (OWA) operator. Jiang and Fan [14] presented the 2-tuple OWA operator and the 2-tuple ordered weighted geometric (OWG) operator. Zhang and Fan [15] developed the extended linguistic ordered weighted averaging (ELOWA) operator. Wang [16] proposed a modified linguistic OWA operator based on entropy maximization. Wang and Hao [17] introduced the quantifier-guided OWA aggregation operator and anchoring value-based OWA aggregation operator for 2-tuples. Dong et al. [18] developed the 2-tuple OWA operator and the extended OWA operator. Wei [19] proposed the extended 2-tuple weighted geometric operator and extended 2-tuple OWG operator. By using the Choquet integral, Yang and ⇑ Address: School of Management, Henan University of Technology, Lianhua Road, Zhengzhou 450001, China. Tel.: +86 13607672706; fax: +86 0371 67756389. E-mail address: [email protected] 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.09.033
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Chen [20] developed the 2-tuple correlated averaging operator, the 2-tuple correlated geometric operator and the generalized 2-tuple correlated averaging operator. Based on the ideal of dependent aggregation of Xu [21], Wei and Zhao [22] proposed the dependent 2-tuple OWA operator and the dependent 2-tuple OWG operator. In the foregoing literature about 2-tuple aggregation operator based decision making method, the original decision information represent by 2-tuple linguistic variables is always derived from a given linguistic term set. In other words, all the decision makers are supposed to provide their assessments about all the alternatives on each attribute with only a certain linguistic term from the same linguistic term set. In reality, it is somewhat unrealistic for every decision maker to express his or her decision information fully and correctly under such constraint. From this perspective, the experts may find the cardinality of the term set is too small to fully express their professional judgments on some attributes and so big that the evaluations on some other attributes are out of their ability [23]. To ensure the decision makers’ fully expressing their opinions, there are at least two approaches worth trying. On one hand, if all decision makers have the option to choose their own linguistic term sets, i.e., the decision information is derived from several linguistic term sets with different granularity, then they can no doubt better provide their professional knowledge and experience. At this point, however, there is the need to unify multi-granularity linguistic information before aggregation operation, which always demands lots of calculation work. For example, Herrera et al. [24] present a fusion approach of multi-granularity linguistic information with the basic linguistic term. Based on the linguistic hierarchies [25] and the 2-tuple linguistic representation model [1], Espinilla et al. [26] presented the extended linguistic hierarchies for managing multi-granular linguistic scales. On the other hand, by using extended 2-tuple fuzzy linguistic representation model, decision makers can also naturally provide their decision information. For instance, Wang and Hao [27] present the proportional 2-tuple fuzzy linguistic representation model and put forward some aggregation operators for proportional 2-tuples. Lin et al. [28] proposed the definition of interval 2-tuple linguistic variable to better express decision information. On the basis of hesitant fuzzy sets [29], Rodriguez et al. [30] proposed the concept of hesitant fuzzy linguistic term set to suit the modeling of quantitative settings. But, in [27,28,30], all decision information given by different decision makers is also derived from the predefined (fixed) linguistic term set. And it is actually not easy to choose only an appropriate linguistic term set meeting the requirements of all the decision makers. To overcome above limitations, Zhang [23] put forward the interval-valued 2-tuple linguistic representation model based on the definition of [28,31], which can be regarded as standardized interval 2-tuple linguistic model and is suitable for dealing with MAGDM problems under multi-granular linguistic contexts. By using interval-valued 2-tuple linguistic representation model, decision information can not only be fully expressed but also be unified easily under multi-granular linguistic contexts. In view of key role aggregation operator plays in CW process, in this paper, we focus on some new aggregation operators for interval-valued 2-tuple together with their desired properties. Furthermore, a new method to determine weight vector of interval-valued 2-tuple aggregation operator is proposed. The remainder of this paper is set out as follows: Section 2 presents a brief introduction to the basic knowledge of 2-tuple and interval-valued 2-tuple linguistic model. Section 3 introduces some new interval-valued 2-tuple aggregation operators and their properties. Based on the concept of degree of precision, Section 4 presents the method to determine the weight vector of aggregation operator and a MAGDM method on the basis of interval-valued 2-tuple aggregation operators. Section 5 provides a numerical example to illustrate the efficiency of the proposed method. Section 6 gives some concluding remarks. 2. Preliminaries In this section, some relevant concepts and basic knowledge are briefly illustrated. Le S = {s0, s1, . . . , sg} be a finite linguistic term set with odd cardinality, where si represents a possible linguistic term for a linguistic variable. For example, a set of seven terms S can be expressed as follows:
S ¼ fs0 ¼ NðnoneÞ; s1 ¼ VLðvery lowÞ; s5 ¼ VHðvery highÞ; s6 ¼ PðperfectÞg:
s2 ¼ LðlowÞ;
s3 ¼ MðmediumÞ;
s4 ¼ HðhighÞ;
It is required that the linguistic term set should satisfy the following characteristics [1,18,25,32]: (1) (2) (3) (4)
The set is ordered: si > sj, if and only if i > j. There is a negation operator: Neg(si) = sj such that j = g i. Max operator: max (si, sj) = si, if and only if i P j. Min operator: min (si, sj) = si, if and only if i 6 j.
The 2-tuple fuzzy linguistic representation model [1] is based on the concept of symbolic translation. A 2-tuple (si, a) is used to express the linguistic information, where the term si is a linguistic label of the predefined linguistic term set S and a (a 2 [0.5, 0.5]) is a numerical value representing the value of symbolic translation. Definition 1 ([1]). Let b be the result of an aggregation of the indexes of a set of labels assessed in a linguistic term set S, i.e., the result of a symbolic aggregation operation.b 2 [0,g] and g + 1 is the cardinality of S. Let i = round (b) and a = b i be two values such that i 2 [0,g] and, then a is called a symbolic translation, where round () is the usual round operation.
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Definition 2 ([1]). Let S = {s0, s1, . . . , sg} be an ordered linguistic term set and b 2 [0, g] be a value supporting result of a symbolic aggregation operation, then the 2-tuple that expresses the equivalent information to b is obtained with the following function:
D : ½0; g ! S ½0:5; 0:5Þ; DðbÞ ¼ ðsi ; aÞ; with
si ;
ð1Þ i ¼ roundðbÞ;
ð2Þ
a ¼ b i; a 2 ½0:5; 0:5Þ;
where si has the closest index label to b and a is the value of the symbolic translation. Definition 3 ([1]). Let S = {s0,s1, . . . ,sg} be an ordered linguistic term set and (si,a) be a 2-tuple. There is always a D1 function such that from a 2-tuple it returns its equivalent numerical value b 2 ½0; g Rþ , where
D1 : S ½0:5; 0:5Þ ! ½0; g;
ð3Þ
1
D ðsi ; aÞ ¼ i þ a ¼ b:
ð4Þ
It is obvious that the conversion of a linguistic term si(si 2 S) into a linguistic 2-tuple consist of adding a value zero as symbolic translation: D(si) = (si, 0), i = 0, 1, 2, . . ., g. Definition 4 ([1]). The comparison of linguistic information represented by 2-tuples is carried out according to an ordinary lexicographic order. Let (sk, a1) and (sl, a2) be two 2-tuples, with each one representing a linguistic assessment as follows: (1) If k < l, then (sk, a1) is smaller than (sl, a2); (2) If k = l, then if a1 = a2, then (sk, a1) and (sl, a2) represent the same information; if a1 < a2, then (sk, a1) is smaller than (sl, a2); if a1 > a2, then (sk, a1) is bigger than (sl, a2).
Definition 5 ([1]). The negation operator over 2-tuples is defined as
Negððsi ; aÞÞ ¼ Dðg ðD1 ðsi ; aÞÞÞ;
ð5Þ
where g + 1 is the cardinality of S, S = {s0, s1, . . . , sg}. Chen and Tai [31] put forward a generalized 2-tuple linguistic model and translation function.
Definition 6 ([31]). Suppose S = {s0, s1, . . . , sg} be an ordered linguistic term set, any crisp value b 2 [0, g] can be transformed into one 2-tuple linguistic variable by the following function:
DðbÞ ¼ ðsi ; aÞ; with
si ;
i ¼ roundðb gÞ
a ¼ b i=g; a 2 ½0:5=g; 0:5=gÞ
ð6Þ
On the contrary, any 2-tuple can be converted into a crisp value b 2 [0, 1] as follows:
D1 ðsi ; aÞ ¼ i=g þ a ¼ b:
ð7Þ
Based on the definition of [28,31], Zhang [23] put forward a new interval-valued 2-tuple linguistic representation model. Definition 7 ([23]). Suppose S = {s0, s1, . . . ,sg} be an ordered linguistic term set. An interval-valued 2-tuple is composed of two linguistic terms and two crisp numbers, denoted by ((si, a1), (sj, a2)), where i 6 j and a1 6 a2 if i = j, si(sj) and a1(a2) represent the linguistic label of the predefined linguistic term set S and symbolic translation, respectively. The interval-valued 2-tuple that expresses the equivalent information to an interval value [b1, b2](b1, b2 2 [0, 1], b1 6 b2) is derived by the following function:
8 si ; > > >
i ¼ roundðb1 gÞ; j ¼ roundðb2 gÞ;
a1 2 ½0:5=g; 0:5=gÞ; a2 2 ½0:5=g; 0:5=gÞ:
ð8Þ
Conversely, there is always a D1 function such that an interval-valued 2-tuple can be converted into an interval value [b1, b2](b1, b2 2 [0, 1], b1 6 b2) as follows:
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D1 ððsi ; a1 Þ; ðsj ; a2 ÞÞ ¼ ½i=g þ a1 ; j=g þ a2 ¼ ½b1 ; b2 :
ð9Þ
The 2-tuple based CW models for multi-granular linguistic information allow expressing the results (obtained at computational phase) in the initial expression domain used by each expert [33]. Similarly, by Eq. (8), the interval-valued 2-tuple linguistic model can easily express the final results (derived in aggregation operation) in the initial expression domains chosen by each expert. Specially, if si = sj and a1 = a2, then the interval-valued 2-tuple reduces to a 2-tuple. The negation operator over intervalvalued 2-tuple is defined as follows:
Negðððsi ; a1 Þ; ðsj ; a2 ÞÞÞ ¼ Dð½1 D1 ðsj ; a2 Þ; 1 D1 ðsi ; a1 ÞÞ;
ð10Þ
where g + 1 is the cardinality of S, S = {siji = 0,1,2, . . . ,g}. From Definitions 6 and 8, we have 0 6 b 6 1, which provides convenience for comparison and operation among 2-tuples (interval-valued 2-tuples) under multi-granular linguistic contexts, without converting multi-granularity linguistic term sets into the basic linguistic term set based on linguistic transformation function [24]. Definition 8 ([23]). Suppose S = {s0, s1, . . . , sg} be an ordered linguistic term set. For an interval-valued 2-tuple A = ((si, a1), (sj, a2)), the score function is expressed by the following formula
SðAÞ ¼ ði þ jÞ=ð2gÞ þ ða1 þ a2 Þ=2:
ð11Þ
The accuracy function is expressed by the following function
HðAÞ ¼ ðj iÞ=g þ a2 a1 :
ð12Þ
It is easy to prove that 0 6 S(A) 6 1 and 0 6 H(A) 6 1. The procedure to compare any two interval-valued 2-tuples is listed as follows. Theorem 1 ([23]). Let S = {s0, s1, . . . , sg} be an ordered linguistic term set, A = ((si, a1), (sj, a2)) and B = ((sk, a3), (sl, a4)) be two interval-valued 2-tuples: If S(A) > S(B), then A > B; If S(A) < S(B), then A < B; If S(A) = S(B), then: (1) If H(A) > H(B), then A < B; (2) If H(A) < H(B), then A > B; (3) If H(A) = H(B), then A = B.
Definition 9 ([23]). Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . ,((rn, an), (ln, en))} be a set of interval-valued 2-tuples and w = P (w1, w2, . . . , wn)T be their associated weights, with wi 2 [0, 1] and ni¼1 wi ¼ 1. The interval-valued 2-tuple weighted average (IVTWA) operator is defined as
IVTWAðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ ¼ D
" # n n X X wi D1 ðri ; ai Þ; wi D1 ðli ; ei Þ i¼1
ð13Þ
i¼1
Definition 10 ([23]). Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))} be a set of interval-valued 2-tuples and w = P (w1, w2, . . . ,wn)T be their associated weights, with wi 2 [0, 1] and ni¼1 wi ¼ 1. The interval-valued 2-tuple ordered weighted average (IVTOWA) operator is defined as
IVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððrn ; an Þ; ðln ; en ÞÞÞ ¼ D
" # n n X X wi D1 ðr rðiÞ ; arðiÞ Þ; wi D1 ðlrðiÞ ; erðiÞ Þ ; i¼1
ð14Þ
i¼1
where [(rr(i), ar(i)), (lr(i), er(i))] is the ith largest of ((rj, aj), (lj, ej)). 3. Some aggregation operators with interval-valued 2-tuple linguistic information In this section, we introduce some new interval-valued 2-tuple aggregation operators together with their typical properties. Definition 11. Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))} be a set of interval-valued 2-tuples and w = P (w1, w2, . . . , wn)T be their associated weights, with wi 2 [0, 1] and ni¼1 wi ¼ 1. The interval-valued 2-tuple weighted geometric (IVTWG) operator is defined as
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IVTWGðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ ¼ D
" n Y
ðD1 ðr i ; ai ÞÞwi ;
i¼1
n Y
# ðD1 ðli ; ei ÞÞwi :
ð15Þ
i¼1
The desired properties of the IVTWG operator can be listed as follows. Theorem 2. Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))} be a set of interval-valued 2-tuples, w = (w1, w2, P be their associated weights, with wi 2 [0, 1] and ni¼1 wi ¼ 1, then, for IVTWG operator, we have
T 1, w2, . . . , wn)
(1) Idempotency. If all ((rj, aj), (lj, ej)) (j = 0, 1, 2, . . . , n) are equal, i.e., ((rj, aj), (lj, ej)) = ((r, a), (l, e)), then IVTWG (((r1, a1), (l1, e1)), ((r2, a2), (l2, 1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))) = ((r, a), (l, e)). (2) Monotonicity. 0 0 0 0 0 Let X0 ¼ r 01 ; a01 ; l1 ; e01 ; r 02 ; a02 ; l2 ; e02 ;. . . ; r 0n ;a0n ; ln ; e0n be a set ofinterval-valued 0 0 2-tuples. If ri ; ai P ðr i ; ai Þ 0 0 0 0 0 0 0 0 0 0 0 0 and li ; ei P ðli ; ei Þ, for any i, then IVTWG r1 ; a1 Þ; l1 ; e1 Þ; r2 ; a2 ; ðl2 ; e2 Þ ; . . . ; r n ; an ; ln ; en Þ P IVTWGðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ; for any w. (3) Boundary.
Dð½minðr i ; ai Þ; minðli ; ei ÞÞ 6 IVTWGðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ i
i
6 Dð½maxðr i ; ai Þ; maxðli ; ei ÞÞ: i
i
Proof (1) Since ((rj, aj), (lj, ej)) = ((r, a), (l, e)), j = 0, 1, 2, . . ., n, then
IVTWGðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ ¼ D
" n Y
ðD1 ðr; aÞÞwi ;
i¼1 n X
"
n Y
# ðD1 ðl; eÞÞwi
i¼1
n X
wi
¼ D ðD1 ðr; aÞÞ i¼1 ; ðD1 ðl; eÞÞ i¼1
wi
¼ ððr; aÞ; ðl; eÞÞ:
(2) If r 0i ; a0i P ðri ; ai Þ, for any i, then we have n n Y Y wi D1 r0i ; a0i P D1 ðr i ; ai Þ ) D1 r0i ; a0 P ðD1 ðr i ; ai ÞÞwi ) ðD1 ðr0i ; a0 ÞÞwi P ðD1 ðr i ; ai ÞÞwi : i¼1
i¼1
0
Similarly, when ðli ; e0i Þ P ðli ; ei Þ for any i, we get n Y
n Y 0 wi D1 li ; e0i P ðD1 ðli ; ei ÞÞwi :
i¼1
i¼1
Based on Theorem 1, we obtain
D
" # " # n n n n Y Y Y wi Y 0 wi PD D1 r0i ; a0i ; D1 li ; e0i ðD1 ðr i ; ai ÞÞwi ; ðD1 ðli ; ei ÞÞwi ; i¼1
i¼1
i¼1
i¼1
0 0 0 i:e:; IVTWG r 01 ; a01 ; l1 ; e01 ; r 02 ; a02 ; l2 ; e02 ; . . . ; r 0n ; a0n ; ln ; e0n P IVTWGðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ: (3) Since maxi(ri, ai) P (ri, ai) P mini(ri, ai) and mini(li, ei) 6 (li, ei) 6 maxi(li, ei) for any i, then, based on the Monotonicity of Theorem 2, we derive
Dð½minðr i ; ai Þ; minðli ; ei ÞÞ 6 IVTWGðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ i
i
6 Dð½maxðr i ; ai Þ; maxðli ; ei ÞÞ: i
i
Definition 12. Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))} be a set of interval-valued 2-tuples and w = (w1, P w2, . . . , wn)T be their associated weights, with wi 2 [0, 1] and ni¼1 wi ¼ 1. The interval-valued 2-tuple ordered weighted geometric (IVTOWG) operator is defined as
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IVTOWGðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ " # n n Y Y wi wi 1 1 ¼D ðD ðr rðiÞ ; arðiÞ ÞÞ ; ðD ðlrðiÞ ; erðiÞ ÞÞ i¼1
ð16Þ
i¼1
where [(rr(i), ar(i)), (lr(i), er(i))] is the ith largest of ((rj, aj), (lj, ej)). Similarly, the desired properties of the IVTOWG operator can be listed as follows. Theorem 3. Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))} be a set of interval-valued 2-tuples, w = (w1, w2, P . . . ,wn)T be their associated weights, with wi 2 [0, 1] and ni¼1 wi ¼ 1, then, for IVTOWG operator, we have (1) Idempotency. If all ((rj, aj), (lj, ej)) (j = 0, 1, 2, . . . , n) are equal, i.e., ((rj, aj), (lj, ej)) = ((r, a), (l, e)), then IVTOWG (((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))) = ((r, a), (l, e)). (2) Monotonicity. 0 0 0 0 0 0 0 0 0 Let X 0 ¼ r 1 ; a1 ; l1 ; e1 ; r2 ; a2 ; l2 ; e2 ; . . . ; r0n ; a0n ; ln ; e0n be a set of interval-valued 2-tuples. If r0i ; a0i P ðr i ; ai Þ 0 and ðli ; e0i ÞP ðli ; ei Þ, for any i, then 0 0 IVTOWG r01 ; a01 ; l1 ; e01 ; r02 ; a02 ; l2 ; e02 ; 0 0 0 0 . . . ; rn ; an ; ln ; en Þ P IVTOWGðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððrn ; an Þ; ðln ; en ÞÞÞ; for any w. (3) Boundary.
Dð½minðr i ; ai Þ; minðli ; ei ÞÞ 6 IVTOWGðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððrn ; an Þ; ðln ; en ÞÞÞ i
i
6 Dð½maxðr i ; ai Þ; maxðli ; ei ÞÞ: i
i
(4) Commutativity. 0 0 0 0 0 0 0 0 0 If r 1 ; a1 ; l1 ; e1 ; r 2 ; a2 ; l2 ; e2 ; . . . ; r 0n ; a0n ; ln ; e0n is any permutation of (((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))), then for any w 0
0
0
IVTOWGðððr 01 ; a01 Þ; ðl1 ; e01 ÞÞ; ððr 02 ; a02 Þ; ðl2 ; e02 ÞÞ; . . . ; ððr 0n ; a0n Þ; ðln ; e0n ÞÞÞ ¼ IVTOWGðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððrn ; an Þ; ðln ; en ÞÞÞ: Definition 13. Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))} be a set of interval-valued 2-tuples and w = (w1, P w2, . . . , wn)T be their associated weights, with wi 2 [0, 1] and ni¼1 wi ¼ 1. The interval-valued 2-tuple ordered weighted harmonic (IVTOWH) operator is defined as
IVTOWHðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ 2 !1 !1 3 n n X X wi wi 4 5; ; ¼D 1 1 i¼1 D ðr rðiÞ ; arðiÞ Þ i¼1 D ðlrðiÞ ; erðiÞ Þ
ð17Þ
where [(rr(i), ar(i)), (lr(i), er(i))] is the ith largest of ((rj, aj), (lj, ej)). Definition 14. Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))} be a set of interval-valued 2-tuples and w = (w1, P w2, . . . , wn)T be their associated weights, with wi 2 [0, 1] and ni¼1 wi ¼ 1. The interval-valued 2-tuple ordered weighted quadratic (IVTOWQ) operator is defined as
IVTOWQðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ 2 !0:5 !0:5 3 n n X X 2 2 1 1 5; wi ðD ðr rðiÞ ; arðiÞ ÞÞ ; wi ðD ðlrðiÞ ; erðiÞ ÞÞ ¼ D4 i¼1
ð18Þ
i¼1
where [(rr(i), ar(i)), (lr(i), er(i))] is the ith largest of ((rj, aj), (lj, ej)). It is easy to prove that both IVTOWH operator and IVTOWQ operator are idempotent, monotonic, bounded and commutative. Motivated by Yager [34], we develop the generalized interval-valued 2-tuple weighted average (GIVTWA) operator and generalized interval-valued 2-tuple ordered weighted average (GIVTOWA) operator.
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Definition 15. Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))} be a set of interval-valued 2-tuples and w = (w1, P w2, . . . , wn)T be their associated weights, with wi 2 [0, 1], ni¼1 wi ¼ 1 and a parameter k 2 (1, 0) [ (0, +1). The generalized interval-valued 2-tuple weighted average (GIVTWA) operator is defined as
GIVTWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ
X 1k X 1k n n : ¼D wi ðD1 ðr i ; ai ÞÞk ; wi ðD1 ðli ; ei ÞÞk i¼1
ð19Þ
i¼1
If each ((rj, aj), (lj, ej)) is degenerated to 2-tuple (rj, aj), the GIVTWA operator will become the generalized 2-tuple weighted average operator. Definition 16. Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))} be a set of interval-valued 2-tuples and w = (w1, P w2, . . . , wn)T be their associated weights, with wi 2 [0, 1], ni¼1 wi ¼ 1 and a parameter k 2 (1, 0) [ (0, +1). The generalized interval-valued 2-tuple ordered weighted average (GIVTOWA) operator is defined as
GIVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ 2 !1k !1k 3 n n X X k k 1 1 4 wi ðD ðr rðiÞ ; arðiÞ ÞÞ ; wi ðD ðlrðiÞ ; erðiÞ ÞÞ 5; ¼D i¼1
ð20Þ
i¼1
where [(rr(i), ar(i)), (lr(i), er(i))] is the ith largest of ((rj, aj), (lj, ej)). If each ((rj, aj), (lj, ej)) is reduced to 2-tuple (rj, aj), the GIVTOWA operator will become the generalized 2-tuple ordered weighted average operator. And if ((ri, ai), (li, ei)) = [(rr(i), ar(i)), (lr(i), er(i))] for all i, then the GIVTOWA operator will become the GIVTWA operator. Similarly, the desired properties of the GIVTOWA operator are listed as follows. Theorem 4. Let X = {((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))} be a set of interval-valued 2-tuples, w = (w1, Pn w2, . . . , wn)T be their associated weights, with wi 2 [0, 1], i¼1 wi ¼ 1 and k 2 (1, 0) [ (0, +1), then, for GIVTOWA operator, we have (1) Idempotency. If all ((rj, aj), (lj, ej)) (j = 0, 1, 2, . . . , n) are equal, i.e., ((rj, aj), (lj, ej)) = ((r, a), (l, e)), then GIVTOWA (((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))) = ((r, a), (l, e)). (2) Monotonicity. 0 0 0 0 0 0 0 0 0 Let X 0 ¼ r 1 ; a1 ; l1 ; e1 ; r 2 ; a2 ; l2 ; e2 ; . . . ; r 0n ; a0n ; ln ; e0n be aset of interval-valued 2-tuples. If r0i ; a0i P ðr i ; ai Þ 0 0 0 and ðli ; e0i Þ P ðli ; ei Þ, for any i, then. GIVTOWA r01 ; a01 ; l1 ; e01 ; r02 ; a02 ; l2 ; e02 ; . . . ; r0n ; a0n ; 0 0 ln ; en ÞÞ P GIVTOWAðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ; for any w. (3) Boundary.
Dð½minðr i ; ai Þ; minðli ; ei ÞÞ 6 GIVTOWAðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ i
i
6 Dð½maxðr i ; ai Þ; maxðli ; ei ÞÞ: i
i
0 0 0 0 0 0 0 0 0 (4) Commutativity. If r1 ; a1 ; l1 ; e1 ; r 2 ; a2 ; l2 ; e2 ; . . . ; r0n ; a0n ; ln ; e0n is any permutation of (((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))), then for any w
0 0 0 GIVTOWA r 01 ; a01 ; l1 ; e01 ; r 02 ; a02 ; l2 ; e02 ; . . . ; r0n ; a0n ; ln ; e0n ¼ GIVTOWAðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ: Proof (1) Since ((rj, aj), (lj, ej)) = ((r, a), (l, e)), j = 0, 1, 2, . . ., n, then
2
n X GIVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ ¼ D4 wi ðD1 ðr; aÞÞk i¼1 1
k
1 k
1
k
1 k
¼ D½ððD ðr; aÞÞ Þ ; ððD ðl; eÞÞ Þ ¼ ððr; aÞ; ðl; eÞÞ: (2) If r 0i ; a0i P ðri ; ai Þ, for any i, then we have
!1k
n X ; wi ðD1 ðl; eÞÞk i¼1
!1k 3 5
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D1 ðr 0i ; a0i Þ P D1 ðr i ; ai Þ ) wi D1 ðr 0i ; a0i Þ P wi D1 ðr i ; ai Þ )
8 n n X X > > > ðwi D1 ðr 0i ; a0i ÞÞk P ðwi D1 ðr i ; ai ÞÞk ; for k > 0; > > < i¼1 i¼1 n n > X X > > 1 0 0 k > ðw D ðr ; a ÞÞ 6 ðwi D1 ðr i ; ai ÞÞk ; for k < 0; > i i i : i¼1
)
8 !1k n > X > 1 0 > 0 k > ðw D ðr ; a ÞÞ P > i i > < i¼1
n X
i¼1
!1k ðwi D1 ðr i ; ai ÞÞk
; for k > 0;
i¼1
!1k > > n n > X X 1 > k 1 0 > 0 > ðwi D ðr i ; a ÞÞ P ð ðwi D1 ðr i ; ai ÞÞk Þk ; for k < 0: : i¼1
i¼1
0 Similarly, when li ; e0i P ðli ; ei Þ for any i, we can get n X
0 k wi D1 li ; e0
!1k
n X
P
i¼1
!1k 1
k
wi ðD ðli ; ei ÞÞ
for k 2 ð1; 0Þ [ ð0; þ1Þ
i¼1
, Based on Theorem 1, we obtain
0 0 0 GIVTOWA r 01 ; a01 ; l1 ; e01 ; r 02 ; a02 ; l2 ; e02 ; . . . ; r0n ; a0n ; ln ; e0n P GIVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ;
for any w:
(3) Since maxi(ri, ai) P (ri, ai) P mini(ri, ai) and mini(li, ei) 6 (li, ei) 6 maxi(li, ei) for any i, then, based on the Monotonicity of Theorem 4, we derive
Dð½minðr i ; ai Þ; minðli ; ei ÞÞ 6 GIVTWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ i
i
6 Dð½maxðr i ; ai Þ; maxðli ; ei ÞÞ: i
i
(4) Trivial from the definition of GIVTOWA operator. If we take different parameter k into the GIVTOWA operator, we can derive a series of aggregation operators, such as follows.
Remark 1. If k = 1, the GIVTOWA operator is degenerated to the IVTOWA operator.
GIVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ ¼ D
" # n n X X wi D1 ðr rðiÞ ; arðiÞ Þ; wi D1 ðlrðiÞ ; erðiÞ Þ i¼1
i¼1
¼ IVTOWAðððr1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ: Remark 2. If k ? 0, the GIVTOWA operator is degenerated to the IVTOWG operator. By L’Hopital’s rule, we get
lim log GIVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððrn ; an Þ; ðln ; en ÞÞÞ k!0 " ! !# n n X X 1 1 wi ðD1 ðr rðiÞ ; arðiÞ ÞÞk ; lim log wi ðD1 ðlrðiÞ ; erðiÞ ÞÞk ¼ D lim log k!0 k k!0 k i¼1 i¼1 " # Pn Pn k k 1 1 1 1 w ð D ðr ; a ÞÞ log D ðr ; a Þ i r ðiÞ r ðiÞ r ðiÞ r ðiÞ i¼1 i¼1 wi ðD ðlrðiÞ ; erðiÞ ÞÞ log D ðlrðiÞ ; erðiÞ Þ ¼ D lim ; lim Pn Pn k k 1 1 k!0 k!0 i¼1 wi ðD ðr rðiÞ ; arðiÞ ÞÞ i¼1 wi ðD ðlrðiÞ ; erðiÞ ÞÞ "P # " # Pn n 1 1 n n Y Y wi wi 1 1 i¼1 wi log D ðr rðiÞ ; arðiÞ Þ i¼1 wi log D ðlrðiÞ ; erðiÞ Þ Pn Pn ¼ D log ðD ðr rðiÞ ; arðiÞ ÞÞ ; log ðD ðlrðiÞ ; erðiÞ ÞÞ : ¼D ; i¼1 wi i¼1 wi i¼1 i¼1 That is, if k ? 0, then,
GIVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ ¼ IVTOWGðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððrn ; an Þ; ðln ; en ÞÞÞ:
H. Zhang / Applied Mathematical Modelling 37 (2013) 4269–4282
4277
Remark 3. If k = 1, the GIVTOWA operator is degenerated to the IVTOWH operator.
GIVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ 2 !1 !1 3 n n X X wi wi 4 5 ; ¼D 1 1 i¼1 D ðr rðiÞ ; arðiÞ Þ i¼1 D ðlrðiÞ ; erðiÞ Þ ¼ IVTOWHðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ: Remark 4. If k = 2, the GIVTOWA operator is degenerated to the IVTOWQ operator.
GIVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððr n ; an Þ; ðln ; en ÞÞÞ 2 !0:5 !0:5 3 n n X X 2 2 1 1 5 wi ðD ðr rðiÞ ; arðiÞ ÞÞ ; wi ðD ðlrðiÞ ; erðiÞ ÞÞ ¼ D4 i¼1
i¼1
¼ IVTOWQðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððrn ; an Þ; ðln ; en ÞÞÞ: Remark 5. If k ? +1, the GIVTOWA operator reduces to the Max operator, and if k ? 1, the GIVTOWA operator reduces to the Min operator, i.e., GIVTOWA (((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))) = maxi((ri, ai), (li, ei)), for k ? +1; GIVTOWA (((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))) = mini((ri, ai), (li, ei)), for k ? 1. If k ? +1, then according to the monotonicity and idempotency of the GIVTOWA operator, we have
GIVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððrn ; an Þ; ðln ; en ÞÞÞ 2 !1k !1k 3 n n X X k k 1 1 4 6D wi ðD ðr rð1Þ ; arð1Þ ÞÞ ; wi ðD ðlrð1Þ ; erð1Þ ÞÞ 5 ¼ maxððr i ; ai Þ; ðli ; ei ÞÞ: i¼1
i
i¼1
Since k > 0, we get
GIVTOWAðððr 1 ; a1 Þ; ðl1 ; e1 ÞÞ; ððr 2 ; a2 Þ; ðl2 ; e2 ÞÞ; . . . ; ððrn ; an Þ; ðln ; en ÞÞÞ 2 !1k !1k 3 n n h i X X k k 1 1 4 5 P D ðw1 ðD1 ðr rð1Þ ; arð1Þ ÞÞk Þ1k ; ðw1 ðD1 ðlrð1Þ ; erð1Þ ÞÞk Þ1k ¼D wi ðD ðr rðiÞ ; arðiÞ ÞÞ ; wi ðD ðlrðiÞ ; erðiÞ ÞÞ i¼1
i¼1
h 1 1 i h 1 1 i 1 1 þ1 ¼ D w1k ðD1 ðr rð1Þ ; arð1Þ ÞÞ ; w1k ðD1 ðlrð1Þ ; erð1Þ ÞÞ ¼ D wþ1 1 ðD ðr rð1Þ ; arð1Þ ÞÞ ; w1 ðD ðlrð1Þ ; erð1Þ ÞÞ h i ¼ D ðD1 ðr rð1Þ ; arð1Þ ÞÞ; ðD1 ðlrð1Þ ; erð1Þ ÞÞ ¼ maxððr i ; ai Þ; ðli ; ei ÞÞ: i
In consequence, we get maxi((ri, ai), (li, ei)) 6 GIVTOWA(((r1, a1), (l1, e1)), ((r2, a2), (l2, e2)), . . . , ((rn, an), (ln, en))) 6 maxi((ri, ai), (li, ei)), i.e., if k ? +1, we get the Max operator. A similar proof can be given for k ? 1. 4. An approach to MAGDM based on the GIVTWA operator The interval-valued 2-tuple aggregation operators can be widely applied in solving decision making problems, where the values of attributes take the form of interval-valued 2-tuples. In this section, we will analyze the process to follow in MAGDM problem by using the GIVTWA operator. Since we should first determine the weight vector for the aggregation operator before the aggregation operation, in what follows, we put forward a method to determine the weight vector of interval-valued 2-tuple aggregation operator based on the concept of degree of precision. 4.1. A method to determine the weight vector of interval-valued 2-tuple aggregation operator In this paper, the decision information is expressed by interval-valued 2-tuple, such as ((si, a1), (sj, a2)) represented that the performance of one alternative on an attribute is between 2-tuples (si, a1) and (sj, a2). Obviously, the closer between (si, a1) and (sj, a2), the more precise information the decision maker can give with the predefined linguistic term set. So we can represent the degree of precision of the decision information by the difference between D1(si, a1) and D1(sj, a2). For a given alternative and attribute, decision makers may have different professional knowledge (experience or ability) and apply different interval-valued 2-tuples to express the assessments. Thus, we should aggregate such interval-valued 2-tuple linguistic
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information for the ranking of alternatives. Theoretically speaking, we should assign bigger weights to the interval-valued 2tuples with higher degree of precision, and vice versa. However, according to above analysis, an interval-valued 2-tuple provided by the decision maker who choose a linguistic term with smaller cardinality incline to have higher degree of precision. To avoid such limitation, we define the degree of precision as follows. Definition 17. Suppose S = {s0, s1, . . . , sg} be an ordered linguistic term set. An interval-valued 2-tuple ((si, a1), (sj, a2)) is composed of two linguistic terms from S. The degree of precision of ((si, a1), (sj, a2)) is formulated as follows.
DPððsi ; a1 Þ; ðsj ; a2 ÞÞ ¼ 1
D1 ðsj ; a2 Þ D1 ðsi ; a1 Þ þ 1=g : 1 þ 1=g
ð21Þ
It is easy to prove 0 6 DP ((si, a1), (sj, a2)) 6 g/(1 + g). If D1(si, a1) = D1(sj, a2), we can get DP ((si, a1), (sj, a2)) = g/(1 + g). In this situation, ((si, a1), (sj, a2)) is reduced to (si, a1) and the degree of precision reaches its maximum, which is smaller than 1. In essence, a 2-tuple is not a crisp number and its degree of precision should be smaller than 1 unless the cardinality of S (i.e.,1 + g) approaches to positive infinity. On the contrary, we have DP ((si, a1), (sj, a2)) = 0 if D1(si, a1) = 0 and D1(sj, a2) = 1, i.e., (si, a1) = (s0, 0) and (sj, a2) = (sg, 0). This is another extreme case, which means this assessment cannot provide any useful information for decision making. Based on these ideas, we can determine the interval-valued 2-tuple aggregation operator weights in MAGDM as follows. For a given alternative Ai with respect to the attribute Cj, suppose the decision makers dk (1 6 k 6 t) provide their ij ij ij . Then, the weight vector assessment information as: r ij1 ; aij1 ; l1 ; eij1 ; rij2 ; aij2 ; l2 ; eij2 ; . . . ; r ijt ; aijt ; lt ; eijt ij ij ij xij ¼ x1 ; x2 ; . . . ; xt of the aggregation operator can be determined as follows.
ij DP r ijk ; aijk ; lk ; eijk : x ¼ Pt ij r ijh ; aijh ; lh ; eijh h¼1 DP ij k
ð22Þ
P It is easy to prove that xijk 2 ½0; 1 and tk¼1 xijk ¼ 1. On that basis, the overall performance of the alternative on the attribute can be derived by the interval-valued 2-tuple aggregation operator. Next, we will present the application of above-mentioned theories, taking the GIVTWA operator for example. 4.2. The process of MAGDM based on the GIVTWA operator In this section, we will analyze the process to follow in MAGDM problem by using the GIVTWA operator, where the weights of attributes are known and the performance values take the form of interval linguistic variables. Suppose A = (A1, A2, . . . , Am) be a discrete set of m feasible alternatives and C = (C1, C2, . . . , Cn) be a finite set of attributes, P whose weight vector is w = (w1, w2, . . . , wn), with wi 2 [0, 1] and ni¼1 wi ¼ 1. Let d = (d1, d2, . . . , dt) be the set of decision mak ers. Assume that each decision maker dk provides his own decision matrix Ek ¼ eijk ðk ¼ 0; 1; 2; . . . ; tÞ, where eijk repremn
sents the preference value of alternative Ai with respect to attribute Cj.eijk takes the form of interval linguistic variable [sa, sb], where sa and sb are derived from S = {s0, s1, . . . , sg}, with 0 6 a 6 b 6 g. In addition, decision makers may use different linguistic term sets S to express the preference values. The process with the GIVTWA operator in MAGDM involves the following steps. Step 1. The decision makers choose their own linguistic term sets and provide their evaluations about the alternative Ai under the attribute Cj, denoted by interval linguistic decision matrix Ek ¼ eijk . mn Step 2. Transform the interval linguistic decision matrix Ek ¼ eijk into interval-valued 2-tuple linguistic decision mn ij ij matrix T k ¼ tk ¼ ððsa ; 0Þ; ðsb ; 0ÞÞk . mn mn Step 3. Utilize Eqs. (21) and (22) to obtain the weight vector of the GIVTWA operator, xij ¼ xij1 ; xij2 ; . . . ; xijt . Step 4. Aggregate all the decision matrices Tk (k = 1, 2, . . . , t) into a collective decision matrix T = (tij)mn based on GIVTWA operator, t ij ¼ GIVTWA tij1 ; tij2 ; . . . ; tijt , whose weight vector is xij ¼ xij1 ; xij2 ; . . . ; xijt . Step 5. Utilize the GIVTWA operator ti = GIVTWA (ti1, ti2, . . . , tin) to yield the collective overall preference values ti for each alternative Ai(i = 0,1, 2, . . ., m), where the weight vector is w = (w1, w2, . . . , wn). Step 6. Rank the alternatives in accordance with ti, referring to Theorem 1. Step 7. End. 5. Illustrative example In the following, we consider an example adapted from Herrera et al. [24], Herrera and Herrera-Viedma [13].
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A1 A2 A3 A4
C1
C2
C3
C4
[x7, x7] [x5, x7] [x4, x7] [x3, x3]
[x1, x1] [x4, x4] [x3, x4] [x0, x2]
[x1, x3] [x3, x4] [x3, x6] [x4, x4]
[x3, x5] [x4, x6] [x1, x1] [x3, x4]
Table 2 The decision matrix E2 given by D2.
A1 A2 A3 A4
C1
C2
C3
C4
[y3, y3] [y5, y5] [y2, y4] [y4, y5]
[y5, y5] [y4, y5] [y4, y5] [y1, y2]
[y1, y3] [y1, y3] [y1, y3] [y3, y4]
[y3, y4] [y3, y5] [y0, y1] [y3, y5]
Table 3 The decision matrix E3 given by D3. C1
C2
C3
C4
[z2, z3] [z3, z3] [z3, z3] [z1, z1]
[z1, z2] [z2, z3] [z0, z2] [z2, z3]
[z1, z3] [z2, z3] [z2, z3] [z1, z3]
[z3, z3] [z1, z1] [z2, z2] [z3, z3]
C1
C2
C3
C4
((x7, 0), (x7, 0)) ((x5, 0), (x7, 0)) ((x4, 0), (x7, 0)) ((x3, 0), (x3, 0))
((x1, 0), (x1, 0)) ((x4, 0), (x4, 0)) ((x3, 0), (x4, 0)) ((x0, 0), (x2, 0))
((x1, 0), (x3, 0)) ((x3, 0), (x4, 0)) ((x3, 0), (x6, 0)) ((x4, 0), (x4, 0))
((x3, 0), (x5, 0)) ((x4, 0), (x6, 0)) ((x1, 0), (x1, 0)) ((x3, 0), (x4, 0))
A1 A2 A3 A4
Table 4 The decision matrix T1.
A1 A2 A3 A4
Suppose an investment company is looking for an optimal investment. There are four possible alternatives to invest the money: A1 is a car industry; A2 is a food company; A3 is a computer company; A4 is an arms industry. After careful review of the information, the company selects four criteria: C1 is the risk analysis; C2 is the growth analysis; C3 is the social-political impact analysis; C4 is the environmental impact analysis. The weight vector of attributes is w = (0.3, 0.1, 0.2, 0.4). Three experts are invited to provide their preferences for each alternative on each attribute using different linguistic term sets:D1 provides his preferences in the set of 9 labels,X = {x0, x1, x2, x3, x4, x5, x6, x7, x8}; D2 provides his preferences in the set of 7 labels, Y = {y0, y1, y2, y3, y4, y5, y6}; D3 provides his preferences in the set of 5 labels, Z = {z0, z1, z2, z3, z4}. Step 1. The decision makers choose their own linguistic term sets as mentioned above and give their decision matrices ðk ¼ 1; 2; 3Þ as shown in Tables 1–3, respectively. Ek ¼ eijk 44 Step 2. Transform decision matrix Ek ¼ ðeijk Þ44 into interval-valued 2-tuple linguistic decision matrix T k ¼ t ijk ¼ 44 ij ððsa ; 0Þ; ðsb ; 0ÞÞk . Taking R1 as an example, we can get the interval-valued 2-tuple linguistic decision matrix 44
T1 as shown in Table 4.
Step 3. Utilize Eqs. (21) and (22) to obtain the GIVTWA weights,xij ¼ xij1 ; xij2 ; xij3 . For example, based on the assessment information (((x7,0), (x7,0)), ((y3,0), (y3,0)), ((z2,0), (z3, 0))) provided by three decision makers, we can derive ¼ ð0:3789; 0:3654; 0:2558Þ. Step 4. Aggregate the decision matrices Tk (k = 1,2,3) into a collective decision matrix T = (tij)44 by GIVTWA operator, and let k = 1, the decision matrix T = (tij)44 is shown in Table 5. Step 5. Utilize the GIVTWA operator ti = GIVTWA (ti1, ti2, ti3, ti4)(k = 1) to yield the collective overall preference values ti for each alternative Ai (i = 1, 2, 3, 4), where the weight vector is w = (0.3, 0.1, 0.2, 0.4). The collective overall preference values are shown in Table 6. 11 11 x11 ¼ x11 1 ; x2 ; x3
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Table 5 The collective decision matrix T.
A1 A2 A3 A4
C1
C2
C3
C4
[0.6421, 0.7061] [0.7450, 0.8167] [0.5544, 0.7613] [0.4201, 0.4696]
[0.4158, 0.4798] [0.5540, 0.6761] [0.4058, 0.6258] [0.2112, 0.4315]
[0.1701, 0.5102] [0.3524, 0.5770] [0.3495, 0.6673] [0.4501, 0.6094]
[0.5535, 0.6845] [0.4019, 0.5771] [0.2127, 0.2622] [0.5478, 0.6817]
Table 6 The collective overall preference values and the rankings of alternatives. t1
t2
t3
t4
Ranking
D([0.4896, 0.6357]) ((x4, 0.0104), (x5, 0.0107))
D([0.5101, 0.6589]) ((x4, 0.0101), (x5, 0.0339))
D([0.3619, 0.5293]) ((x3, 0.0131), (x4, 0.0293))
D([0.4563, 0.5786]) ((x4, 0.0437), (x5, 0.0464))
A2 > A1 > A4 > A3
The values of score function
0.8 0.7 0.6 C
0.5
B
0.4
A A1 A2 A3 A4
0.3 0.2 -100
-80
-60
-40
-20
0
20
40
60
80
100
λ value Fig. 1. The values of score function with k 2 [100, 100].
Table 7 The rankings of alternatives with k 2 [100, 100].
Ranking
100 6 k < kA
kA 6 k 6 kB
kB < k 6 kC
kC < k 6 100
A2 > A1 > A4 > A3
A2 > A4 > A1 > A3
A2 > A1 > A4 > A3
A2 > A1 > A3 > A4
In addition, by Eq. (8), we can express the final results (interval values) in the initial expression domain used by each expert. Taking the expert D1 as an example, the final results can be expressed by interval-valued 2-tuples derived from the linguistic term set X with 9 labels, which are listed in Table 6. Step 6. By ranking ti (i = 1, 2, 3, 4) based on Theorem 1, the priorities of the alternatives can be obtained, which are listed in Table 6. Given the collective decision matrix T = (tij)44 in this example, we next consider what happens when the values of k change. Depending on the steps of our proposed aggregation operators with different k values, we can obtain the collective overall preference values ri for each alternative Ai together with their score functions as shown in Fig. 1, where k is set to [100, 100]. As we can see, it is obvious that the values of score function are non-decreasing with respect to k. In addition, the ranking order of alternatives may be different with the variation of parameter k. However,A2 is always the optimal choice despite the change of k value. According to Fig. 1, there are three k values changing the ranking order of the alternatives, i.e.,kA = 9.7247, kB = 1.3348 and kC = 5.0701. Furthermore, we can further determine the ranking when k = kA, k = kB and k = kC, respectively. According to accuracy function defined in Definitions 8 and Theorem 1, if k = kA, we get H(A1) = 0.3553 > H(A4) = 0.2218, i.e.,A4 > A1. Similarly, we have A4 > A1 if k = kB and A4 > A3 if k = kC. Consequently, the ranking orders can be shown in Table 7.
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Since the variation of k value may lead to different ranking orders, decision makers may have difficulty in selecting the optimal alternative with different k values. In other words, it is necessary for the decision maker to set k value before information aggregation. In general, the more pessimistic of the decision maker, the larger k value he or she may set, which means each alternative is associated with a higher performance value (score function). On the contrary, the more optimistic of the decision maker, the smaller k value he or she may set. If the decision maker does not give his or her subjective preference, the most commonly used value k = 1 can be taken. 6. Conclusion In this paper, we present a series of interval-valued 2-tuple aggregation operators, together with their properties and the relationships among them. In addition, we develop a method to determine the weight vector of interval-valued 2-tuple aggregation operator based on the concept of degree of precision. 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