Some dependent aggregation operators with 2-tuple linguistic ...
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Expert Systems with Applications 39 (2012) 5881–5886
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Some dependent aggregation operators with 2-tuple linguistic information and their application to multiple attribute group decision making Guiwu Wei ⇑, Xiaofei Zhao Institute of Decision Sciences, Chongqing University of Arts and Sciences, Yongchuan, Chongqing 402160, PR China
a r t i c l e
i n f o
Keywords: Multiple attribute group decision making (MAGDM) 2-Tuple linguistic information Power average operator Dependent 2-tuple ordered weighted averaging (D2TOWA) operator The dependent 2-tuple ordered weighted geometric (D2TOWG) operator
a b s t r a c t We investigate the multiple attribute group decision making (MAGDM) problems in which the attribute values take the form of 2-tuple linguistic information. Motivated by the ideal of dependent aggregation [Xu, Z. S. (2006). Dependent OWA operators. Lecture Notes in Artificial Intelligence, 3885, 172–178], in this paper, we develop some dependent 2-tuple linguistic aggregation operators: the dependent 2-tuple ordered weighted averaging (D2TOWA) operator and the dependent 2-tuple ordered weighted geometric (D2TOWG) operator, in which the associated weights only depend on the aggregated 2-tuple linguistic arguments and can relieve the influence of unfair 2-tuple linguistic arguments on the aggregated results by assigning low weights to those ‘‘false’’ and ‘‘biased’’ ones and then apply them to develop some approaches for multiple attribute group decision making with 2-tuples linguistic information. Finally, some illustrative examples are given to verify the developed approach and to demonstrate its practicality and effectiveness. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The ordered weighted aggregation (OWA) operator (Yager, 1988) as an aggregation technique has received more and more attention since its appearance (Merigó & Casanovas, 2009; Merigó & Gil-Lafuente, 2009; Merigó, 2010; Wei, 2008a, 2008b, 2008c, 2009a, 2009b, 2010a, 2010b, 2010c, 2010d; Wei, Zhao, & Lin, 2010; Wei, 2011a, 2011b, 2011c; Wei, Wang, Lin, & Zhao, 2011a, 2011b, Wei, Wang, & Lin, 2011c; Wei, 2011d, 2011e, 2011f, 2011g; Wei & Zhao, 2012; Zhang & Chu, 2009; Xu & Da, 2003). One important step of the OWA operator is to determine its associated weights. Many authors have focused on this issue, and developed some useful approaches to obtaining the OWA weights. We classify all these approaches into the following two categories: argument-independent approaches (Filev & Yager, 1998; Fullér & Majlender, 2001; Xu, 2005; Yager, 1993; Yager & Filev, 1994) and argument-dependent approaches (Filev & Yager, 1998; Xu & Da, 2002; Xu, 2005, 2006, 2008; Yager, 1993). The weights derived by the argument-independent approaches are associated with particular ordered positions of the aggregated arguments, and have no connection with the aggregated arguments, while the argument-dependent approaches determine the weights based on the input arguments. With respect to the argument-dependent approaches, Yager (1993) introduced some families of the OWA weights, including the ideal of aggregate dependent weights. Filev and Yager (1998) developed two proce⇑ Corresponding author. Tel./fax: +86 23 49891870. E-mail address: [email protected] (G. Wei). 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.11.120
dures to obtain the OWA weights, the first one learns the weights from a collection of samples with their aggregated value, and the second one calculates the weights for a given level of orness. Xu and Da (2002) established a linear objective-programming model for obtaining the weights of the OWA operator by utilizing the given arguments under partial weight information. Xu (2005) developed a new dependent OWA (DOWA) operator and developed a new argument-dependent approach to determining the OWA weights, which can relieve the influence of unfair arguments on the aggregated results. Xu (2006) developed some dependent uncertain ordered weighted aggregation operators, including dependent uncertain ordered weighted averaging (DUOWA) operators and dependent uncertain ordered weighted geometric (DUOWG) operators, in which the associated weights only depend on the aggregated interval arguments and can relieve the influence of unfair interval arguments on the aggregated results by assigning low weights to those ‘‘false’’ and ‘‘biased’’ ones. However, in many situations, the input arguments take the form of 2-tuples linguistic variables (Fan & Liu, 2010; Herrera & Martínez, 2000, 2001; Herrera, Martínez, & Sánchez, 2005; Herrera, Herrera-Viedma, & Martínez, 2008; Merigó, Casanovas, & Martínez, 2010; Tai & Chen, 2009; Wang, 2009a, 2009b; Wei, 2010c, 2010d, 2011a; Wei, Lin, Zhao, & Wang, 2010a, Wei, Lin, Zhao, & Wang, 2010b) because of time pressure, lack of knowledge, and people’s limited expertise related with problem domain. In this paper, we will pay attention on the second category, and develop some argument-dependent approach to determining the OWA weights with 2-tuples linguistic information. The remainder of this paper
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G. Wei, X. Zhao / Expert Systems with Applications 39 (2012) 5881–5886
is set out as follows. In the next section, we introduce some basic concepts and operational laws of 2-tuple linguistic variables. In Section 3 we develop some dependent 2-tuple linguistic aggregation operators: the dependent 2-tuple ordered weighted averaging (D2TOWA) operator and the dependent 2-tuple ordered weighted geometric (D2TOWG) operator, in which the associated weights only depend on the aggregated 2-tuple linguistic arguments and can relieve the influence of unfair 2-tuple linguistic arguments on the aggregated results by assigning low weights to those ‘‘false’’ and ‘‘biased’’ ones. In Section 4 we develop an approach to multiple attribute group decision making based on these dependent aggregation operators with 2-tuple linguistic information, which is straightforward and has no loss of information. In Section 5, we give an illustrative example to verify the developed approach and to demonstrate its feasibility and practicality. In Section 6 we conclude the paper and give some remarks. 2. Preliminaries Let S = {siji = 1, 2, . . . , t} be a linguistic term set with odd cardinality. Any label, si represents a possible value for a linguistic variable, and it should satisfy the following characteristics (Herrera & Martínez, 2000, 2001): (1) The set is ordered: si > sj, if i > j; (2) Max operator: max (si, sj) = si, if si P sj; (3) Min operator: min (si, sj) = si, if si 6 sj. For example, S can be defined as
S ¼ fs1 ¼ extremely poor; s2 ¼ v ery poor; s3 ¼ poor; s4 ¼ medium; s5 ¼ good; s6 ¼ v ery good; s7 ¼ extremely goodg Herrera and Martínez (2000, 2001) developed the 2-tuple fuzzy linguistic representation model based on the concept of symbolic translation. It is used for representing the linguistic assessment information by means of a 2-tuple (si, ai), where si is a linguistic label from predefined linguistic term set S and ai is the value of symbolic translation, and ai 2 [(0.5, 0.5)]. Definition 1. Let b be the result of an aggregation of the indices of a set of labels assessed in a linguistic term set S, i.e., the result of a symbolic aggregation operation, b 2 [1, t], being tthe cardinality of S. Let i = round(b) and a = b i be two values, such that, i 2 [1, t] and a 2 [(0.5, 0.5)] then a is called a Symbolic Translation (Herrera & Martínez, 2000, 2001). Definition 2. Let S = {s1, s2, . . . , st} be a linguistic term set and b 2 [1, t] is a number value representing the aggregation result of linguistic symbolic. Then the function D used to obtain the 2-tuple linguistic information equivalent to b is defined as:
D : ½1; t ! S ½0:5; 0:5Þ;
DðbÞ ¼
si ;
i ¼ roundðbÞ;
a ¼ b i; a 2 ½0:5; 0:5Þ;
ð1Þ ð2Þ
where round (.) is the usual round operation, si has the closest index label to b and a is the value of the symbolic translation (Herrera & Martínez, 2000, 2001). Definition 3. Let S = {s1, s2, . . . , st} be a linguistic term set and (si, ai) be a 2-tuple. There is always a function D1 can be defined, such that, from a 2-tuple (si, ai) it return its equivalent numerical value b 2 [1, t] R, which is (Herrera & Martínez, 2000, 2001).
D1 : S ½0:5; 0:5Þ ! ½1; t; 1
D ðsi ; aÞ ¼ i þ a ¼ b:
Definition 4. Let (sk, ak) and (sl, al) be two 2-tuple, they should have the following properties (Herrera & Martínez, 2000, 2001). (1) If k < l then (sk, ak) is smaller than (sl, al). (2) If k = l then, (a) if ak = al, then (sk, ak), (sl, al) represents information, (b) if ak < al then (sk, ak) is smaller than (sl, al), (c) if ak > al then (sk, ak) is bigger than (sl, al),
same
3. Some dependent aggregation operators with 2-tuple linguistic information Yager (1988) provides a definition of the ordered weighted averaging (OWA) operator as follows: Definition 5. An OWA operator of dimension n is a mapping, OWA: Rn ? R, that has an associated weighting vector w = (w1, w2, . . . , wn)T P with the properties wj 2 [0, 1], and nj¼1 wj ¼ 1, such that
OWAw ða1 ; a2 ; . . . ; an Þ ¼
n X
wj apðjÞ ;
ð5Þ
j¼1
where (p(1), p(2), . . . , p(n)) is a permutation of (1, 2, . . . , n), such that ap(j1) P ap(j) for all j = 2, . . . , n. Since its appearance in 1988, the OWA operator has been used in a wide range of applications, such as engineering, neural networks, data mining, decision making, image process, expert systems, etc. Consider that the OWA operator aggregates only the exact inputs having been reordered, Herrera and Martínez (2000, 2001) extend WA operator and OWA operator to accommodate the situations where the input arguments are linguistic variables. Definition 6. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2-tuples and x = (x1, x2, . . . , xn)T be the weighting vector of 2-tuples (rj, aj) Pn (j = 1, 2, . . . , n) and xj 2 ½0; 1; j ¼ 1; 2; . . . ; n; j¼1 xj ¼ 1. The 2tuple weighted average is
2TWAx ððr 1 ; a1 Þ; ðr 2 ; a2 Þ; . . . ; ðr n ; an ÞÞ ¼ D
n X
! 1
xj D ðrj ; aj Þ :
ð6Þ
j¼1
Definition 7. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2-tuples and w = (w1, w2, . . . , wn)T be an associated weighting vector, P wj 2 [0, 1], j = 1, 2, . . . , n and nj¼1 wj ¼ 1. The 2-tuple OWA for linguistic 2-tuples is computed as
2TOWAw ððr 1 ; a1 Þ; ðr2 ; a2 Þ; . . . ; ðrn ; an ÞÞ ! n X 1 ¼D wj D r pðjÞ ; apðjÞ ;
ð7Þ
j¼1
where (p(1), p(2), . . . , p(n)) is a permutation of (1, 2, . . . , n), such that (sp(j1), ap(j1)) P sp(j), ap(j)) for all j = 2, . . . , n. Recently, Jiang and Fan (2003) extended the WG operator and OWG operator to accommodate the situations where the input arguments are linguistic variables. Definition 8. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2-tuple and x = (x1, x2, . . . , xn)T be the weighting vector of 2-tuple (rj, aj) Pn (j = 1, 2, . . . , n) and xj 2 ½0; 1; j¼1 xj ¼ 1, The 2-tuple weighted geometric operator is (Jiang & Fan, 2003)
ð3Þ ð4Þ
the
2TWGx ððr1 ; a1 Þ; ðr2 ; a2 Þ; . . . ; ðrn ; an ÞÞ ¼ D
n Y j¼1
! ðD1 ðr j ; aj ÞÞxj :
ð8Þ
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Definition 9. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2-tuple, A 2-tuple ordered weighted geometric operator of dimension n is a mapping TOWG: Rn?R that has an associated vector w = P (w1, w2, . . . , wn)T such that wj > 0 and nj¼1 wj ¼ 1. Furthermore,
2TOWGw ððr 1 ;a1 Þ; ðr 2 ; a2 Þ; . .. ; ðrn ; an ÞÞ ¼ D
n Y
wj
ðD1 ðr pðjÞ ; apðjÞ ÞÞ
;
ð9Þ
Definition 10. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2tuples, the 2-tuple arithmetic mean is computed as
2 ½0:5; 0:5Þ: a
ð10Þ
Definition 11. Let (ri, ai) and (rj, aj) be two 2-tuples, then we call
dððr i ; ai Þ; ðrj ; aj ÞÞ ¼
jD1 ðr i ; ai Þ D1 ðr j ; aj Þj ; t
ð11Þ
the distance between (ri, ai) and (rj, aj). Definition 12. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2Þ the arithmetic mean of these 2-tuples, then tuples, and let ðr ; a we define the standard deviation of these 2-tuples as
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u X u1 n ÞÞ2 : r¼t dððr j ; aj Þ; ðr ; a n j¼1
j ¼ 1; 2; . . . ; n:
ð13Þ
the degree of similarity between the jth largest 2-tuples linguistic Þ, where (p(1), p(2), . . . , p(n)) variables (rp(j), ap(j)) and the mean ðr ; a is a permutation of (1, 2, . . . , n), such that (rp(j1), ap(j1)) P (rp(j), ap(j)) for all j = 2, . . . , n. In real-life situations, the 2-tuples (r1, a1), (r2, a2), . . . , (rn, an) usually take the form of a collection of n preference values provided by n different individuals. Some individuals may assign unduly high or unduly low preference values to their preferred or repugnant objects. In such a case, we shall assign very low weights to these ‘‘false’’ or ‘‘biased’’ opinions, that is to say, the closer a preference value (argument) is to the mid one(s), the more the weight. As a result, based on (13), we define the 2TOWA weights as
j ¼ 1; 2; . . . ; n
ð14Þ
P Obviously, wj P 0, j = 1, 2, . . . , n and nj¼1 wj ¼ 1. Especially, if (sk, ak) = (sl, al), for all i, j = 1, 2, . . . , n, then by (14), we have wj ¼ 1n, for all j = 1, 2, . . . , n. By (7), we have
2TOWAððr 1 ; a1 Þ; ðr2 ; a2 Þ; . . . ; ðrn ; an ÞÞ ¼D
j¼1
Since
n X
ÞÞD1 ðr j ; aj Þ simððr j ; aj Þ; ðr ; a
and n X
ÞÞ ¼ simððr pðjÞ ; apðjÞ Þ; ðr ; a
j¼1
n X
ÞÞ simððr j ; aj Þ; ðr; a
j¼1
then we replace (15) by
2TOWAððr 1 ; a1 Þ; ðr 2 ; a2 Þ; . . . ; ðr n ; an ÞÞ ! Pn 1 j¼1 simððr j ; aj Þ; ðr ; aÞÞD ðr j ; aj Þ Pn : ¼D j¼1 simððr j ; aj Þ; ðr ; aÞÞ
ð16Þ
We call (16) a dependent 2-tuple ordered weighted averaging (D2TOWA) operator, which is a generalization of the dependent ordered weighted averaging (DOWA) operator (Xu, 2006). Consider that the aggregated value of the D2TOWA operator is independent of the ordering, thus it is also a neat operator. Based on the 2TOWG and Eq.(14), we developed the dependent 2-tuple ordered weighted geometric (D2TOWG) operator as follows:
D2TOWGððr1 ; a1 Þ; ðr2 ; a2 Þ; . . . ; ðrn ; an ÞÞ 0 1 n P ÞÞ= ÞÞ n simððr j ;aj Þ;ðr;a simððrj ;aj Þ;ðr ;a Y 1 j¼1 @ A: ¼D ðD ðrj ; aj ÞÞ
ð17Þ
j¼1
Similar to Xu (2006, 2008), we have the following result:
ÞÞ dððr ; a Þ; ðr ; a ÞÞ ¼ 1 Pn pðjÞ pðjÞ ; simððr pðjÞ ; apðjÞ Þ; ðr; a j¼1 dððr pðjÞ ; apðjÞ Þ; ðr ; aÞÞ
n X
¼
ð12Þ
Definition 13. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2Þ the arithmetic mean of these 2-tuples, then tuples, and let ðr ; a we call
ÞÞ simððr pðjÞ ; apðjÞ Þ; ðr ; a ; wj ¼ Pn ÞÞ r; a simððr ; a Þ; ð p ðjÞ p ðjÞ j¼1
j¼1
j¼1
where (p(1), p(2), . . . , p(n)) is a permutation of (1, 2, . . . , n), such that (sp(j1), ap(j1)) P (sp(j), ap(j)) for all j = 2, . . . , n.
r 2 S;
ÞÞD1 ðr pðjÞ ; apðjÞ Þ simððr pðjÞ ; apðjÞ Þ; ðr ; a
!
j¼1
X n 1 1 Þ ¼ D ðr; a D ðr ; a Þ ; j j j¼1 n
n X
ÞÞ P simððr pðjÞ ; apðjÞ Þ; ðr ; a ÞÞ; simððr pðiÞ ; apðiÞ Þ; ðr ; a
ð15Þ
then wi P wj :
The normal distribution is one of the most commonly observed and is the starting point for modeling many natural process, it is usually found in events that are the aggregation of many smaller, but independent random events. Xu (2008) introduced a normal distribution based method to determine some dependent uncertain ordered weighted aggregation operators, in which the associated weights only depend on the aggregated arguments. Motivated by the idea, in the following, we give another method for deriving the D2TOWA weights: 2
ÞÞÞ ðdððr pðjÞ ;apðjÞ Þ;ðr ;a 1 2r2 wj ¼ pffiffiffiffiffiffiffi e ; 2pr
j ¼ 2; . . . ; n;
ð18Þ
Þ and r are the arithmetric mean and the standard deviawhere ðr; a tion of these 2-tuples arguments variables {(r1, a1), (r2, a2), . . . , (rn, an)}, respectively, (p(1), p(2), . . . , p(n)) is a permutation of (1, 2, . . . , n), such that (rp(j1), ap(j1)) P (rp(j), ap(j)) for all j = 2, . . . , n. P Consider that wj P 0, j = 1, 2, . . . , n and nj¼1 wj ¼ 1, then by (18), we have
wj ¼
!
ÞÞ simððr pðjÞ ; apðjÞ Þ; ðr ; a Pn D1 ðr pðjÞ ; apðjÞ Þ : ÞÞ r ; a simððr ; a Þ; ð pðjÞ pðjÞ j¼1
Theorem 1. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2-tuples, Þ the arithmetic mean of these 2-tuples, and let ðr ; a (p(1), p(2), . . . , p(n)) is a permutation of (1, 2, . . . , n), such that (rp(j1), ap(j1)) P (rp(j), ap(j)) for all j = 2, . . . , n. If
1 pffiffiffiffi e 2p r
ðdððrpðjÞ ;apðjÞ Þ;ðr ;aÞÞÞ2
Pn
1 pffiffiffiffi j¼1 2pr e
2r2 ÞÞÞ2 ðdððrpðjÞ ;apðjÞ Þ;ðr;a
¼ 1; 2; . . . ; n; then by (7), we have
2r 2
¼
e Pn
ÞÞÞ2 ðdððr pðjÞ ;apðjÞ Þ;ðr;a 2r2
j¼1 e
ÞÞÞ2 ðdððrpðjÞ ;apðjÞ Þ;ðr;a
;
j
2r2
ð19Þ
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G. Wei, X. Zhao / Expert Systems with Applications 39 (2012) 5881–5886
2TOWAððr1 ; a1 Þ; ðr2 ; a2 Þ; . . . ; ðrn ; an ÞÞ 0 n BP ¼ D@
j¼1
0 B ¼ D@
ðdððrpðjÞ ;apðjÞ Þ;ðr ;aÞÞÞ2 2r2 e
Pn j¼1
Pn j¼1
e
e
ÞÞÞ2 ðdððr pðjÞ ;apðjÞ Þ;ðr ;a
j¼1
C D1 ðr pðjÞ ; apðjÞ ÞA
2r2
ðdððspðjÞ ;apðjÞ 2r2
Pn
Step 2. Utilize the decision information given in matrix Rk, and the 2TWA operator
1
ð20Þ
1
ÞÞÞ2 Þ;ðr;a
D1 ðr pðjÞ ;apðjÞ ÞC
n xj Y ðkÞ ðkÞ ðkÞ zi ¼ ri ; ai D1 r ðkÞ ¼D ij ; 0
Since n X
e
ÞÞÞ2 ðdððr pðjÞ ;apðjÞ Þ;ðr ;a 2r2
D1 ðrpðjÞ ; apðjÞ Þ ¼
n X
e
ÞÞÞ2 ðdððr j ;aj Þ;ðr ;a 2r2
ðkÞ
and
e
ð24Þ
to derive the individual overall preference value zi tive Ai. Step 3. Utilize the D2TOWA operator:
D1 ðr j ; aj Þ
j¼1
!
j¼1
j¼1
n X
ð23Þ
j¼1
or the 2TWG operator
A:
ÞÞÞ2 ðdððrpðjÞ ;apðjÞ Þ;ðr;a 2r2 e
! n X ðkÞ ðkÞ ðkÞ zi ¼ ri ; ai xj D1 rijðkÞ ; 0 ¼D
ÞÞÞ2 ðdððr pðjÞ ;apðjÞ Þ;ðr ;a 2r2
j¼1
¼
n X
e
zi ¼ ðri ; ai Þ 0Pt
ÞÞÞ2 ðdððr j ;aj Þ;ðr;a 2r2
¼ D@
j¼1
of the alterna-
1 ðkÞ ðkÞ ðkÞ i Þ D1 r ðkÞ r i ; ai ; ðr i ; a i ; ai A Pt ðkÞ ðkÞ i Þ r i ; ai ; ðri ; a k¼1 sim
k¼1 sim
then, (20) can be rewritten as
ð25Þ
or the D2TOWG operator:
2TOWAððr1 ; a1 Þ; ðr 2 ; a2 Þ; . . . ; ðrn ; an ÞÞ 0 1 Pn ðdððrj ;aj Þ;ð2r;aÞÞÞ2 1 2r e D ðr ; a Þ j j B j¼1 C ¼ D@ A: ÞÞÞ2 r ;a Pn ðdððrj ;aj Þ;ð 2 2r j¼1 e
ð21Þ
zi ¼ ðri ; ai Þ 0 1 t P ðkÞ ðkÞ ðkÞ n simððrðkÞ i ÞÞ= i ÞÞ ;ai Þ;ðr i ;a simððr i ;ai Þ;ðri ;a Y i ðkÞ A k¼1 ¼ D@ D1 r ðkÞ i ; ai j¼1
Obviously, (21) is also a neat and dependent 2TOWA (D2TOWA) operator. Based on the 2TOWG and Eq.(19), we developed the dependent 2-tuple ordered weighted geometric (D2TOWG) operator as follows:
D2TOWGððr 1 ; a1 Þ; ðr2 ; a2 Þ; . . . ; ðr n ; an ÞÞ 1 0 ÞÞÞ2 ÞÞÞ2 ðdððr j ;aj Þ;ðr;a ðdððrj ;aj Þ;ðr;a n P 2r 2 2r 2 n e = e C BY 1 j¼1 C: ¼ DB A @ ðD ðr j ; aj ÞÞ
ð22Þ
ð26Þ to derive the collective overall preference P values zi (i = 1, 2, . . . , m) of ðkÞ i Þ ¼ D 1t tk¼1 D1 rðkÞ . the alternative Ai, where ðr i ; a ; a i i Step 4. Rank all the alternatives Ai (i = 1, 2, . . . , m) and select the best one(s) in accordance with the collective overall preference values zi (i = 1, 2, . . . , m). Step 5. End. 5. Illustrative example
j¼1
From (14) and (19), we know that all the associated weights of the D2TOWA operator and D2TOWG operator only depend on the aggregated 2-tuples variables, and can relieve the influence of unfair arguments on the aggregated results by assigning low weights to those ‘‘false’’ and ‘‘biased’’ ones, and thus make the aggregated results more reasonable in the practical applications. 4. An approach to multiple attribute group decision making based on 2-tuple linguistic information For the multiple attribute group decision making problems, in which both the attribute weights and the expert weights take the form of real numbers, and the attribute preference values take the form of 2-tuple linguistic variables, we shall develop an approach based on the TWA and D2TOWA operators to multiple attribute group decision making based on 2-tuple linguistic information processing. Let A = {A1, A2, . . . , Am} be a discrete set of alternatives, G = {G1, G2, . . . , Gn} be the set of attributes, x = (x1, x2, . . . , xn) is the weighting vector of the attributes Gj (j = 1, 2, . . . , n), where P xj 2 ½0; 1; nj¼1 xj ¼ 1; D ¼ fD D2 ; . . . ; Dt g be the set of decision 1; ðkÞ ðkÞ makers. Suppose that Rk ¼ r ij is the, where rij 2 e S is a mn preference values, which take the form of variable, given by the decision maker Dk 2 D, for the alternative Ai 2 A with respect to the attribute Gj 2 G. ðkÞ Step 1. Transform linguistic decision matrix Rk ¼ r ij into 2 mn ðkÞ tuple linguistic decision matrix Rk ¼ r ij ; 0 . mn
Let us suppose there is an investment company, which wants to invest a sum of money in the best option (adapted from Herrera & Herrera-Viedma (2000)). There is a panel with five possible alternatives to invest the money: rA1 is a car company; sA2 is a food company; tA3 is a computer company; uA4 is a chemical company; vA5 is a TV company. The investment company must take a decision according to the following four attributes: rG1 is the risk analysis; sG2 is the growth analysis; tG3 is the social-political impact analysis; uG4 is the environmental impact analysis. The five possible alternatives Ai (i = 1, 2, . . . , 5) are to be evaluated using the linguistic term set
S ¼ fs1 ¼ extremely poor; s2 ¼ v ery poor; s3 ¼ poor; s4 ¼ medium; s5 ¼ good; s6 ¼ v ery good; s7 ¼ extremely goodg by the three decision makers Dk(k = 1, 2, 3) under the above four attributes (whose weighting vector x = (0.3, 0.1, 0.2, 0.4)), and construct, respectively, the linguistic decision matrices are shown in Tables 1–3: In the following, we shall utilize the proposed approach in this paper getting the most desirable alternative(s): Step 1. Transforming linguistic decision information given in Tables 1–3 into 2-tuple linguistic decision matrices which are given in Tables 4–6. Step 2. Utilize the decision information given in matrix Rk, and the 2TWA operator and 2TWG operator (Let x = (0.3, 0.1, 0.2, 0.4)) to derive the individual overall preference value ðkÞ zi of the alternative Ai. The results are shown in Tables 7 and 8.
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G. Wei, X. Zhao / Expert Systems with Applications 39 (2012) 5881–5886 Table 1 e1. Decision matrix R
A1 A2 A3 A4 A5
Table 7 The individual overall preference value by utilizing 2TWA operator. G1
G2
G3
G4
s4 s3 s5 s6 s7
s5 s2 s4 s3 s1
s3 s4 s5 s3 s2
s3 s3 s1 s5 s4
A1 A2 A3 A4 A5
D1
D2
D3
(s4, 0.50) (s3, 0.10) (s3, 0.30) (s5, 0.30) (s4, 0.20)
(s3, 0.50) (s4, 0.30) (s5, 0.10) (s4, 0.30) (s3, 0.30)
(s5, 0.40) (s4, 0.30) (s4, 0.20) (s4, 0.30) (s4, 0.20)
Table 8 The individual overall preference value by utilizing 2TWG operator. Table 2 e2. Decision matrix R
A1 A2 A3 A4 A5
G1
G2
G3
G4
s3 s2 s4 s7 s3
s4 s1 s5 s2 s2
s2 s5 s3 s2 s4
s2 s5 s7 s4 s2
Table 3 e3. Decision matrix R
A1 A2 A3 A4 A5
G1
G2
G3
G4
s5 s2 s6 s5 s4
s3 s5 s2 s6 s2
s2 s3 s5 s7 s4
s6 s5 s3 s2 s5
A1 A2 A3 A4 A5
D1
D2
D3
(s3, 0.44) (s3, 0.05) (s3,-0.43) (s5, 0.47) (s4, 0.41)
(s2, 0.42) (s3, 0.23) (s5, 0.17) (s4, 0.16) (s3, 0.41)
(s4, 0.25) (s3, 0.43) (s4, 0.07) (s4, 0.22) (s4, 0.08)
Table 9 The overall preference values of the alternatives.
2TWA and D2TOWA 2TWG and D2TOWG
A1
A2
A3
A4
A5
(s4, 0.48)
(s4, 0.45)
(s4, 0.20)
(s4, 0.40)
(s4, 0.17)
(s3, 0.40)
(s3, 0.24)
(s4, 0.16)
(s4, 0.01)
(s3, 0.49)
Table 10 Ordering of the alternative. Ordering 2TWA and D2TOWA 2TWG and D2TOWG
Table 4 e1. Decision matrix R
A1 A2 A3 A4 A5
G1
G2
G3
G4
(s4, 0) (s3, 0) (s5, 0) (s6, 0) (s7, 0)
(s5, 0) (s2, 0) (s4, 0) (s3, 0) (s1, 0)
(s3, 0) (s4, 0) (s5, 0) (s3, 0) (s2, 0)
(s3, 0) (s3, 0) (s1, 0) (s5, 0) (s4, 0)
Table 5 e2. Decision matrix R
A1 A2 A3 A4 A5
Step 4. According to the aggregating results shown in Table 9, the ordering of the alternatives are shown in Table 10. Note that?means ‘‘preferred to’’. As we can see, depending on the aggregation operators used, the ordering of the alternatives is slightly different. Therefore, depending on the aggregation operators used, the results may lead to different decisions. However, the best alternative is A4. 6. Conclusion
G1
G2
G3
G4
(s3, 0) (s2, 0) (s4, 0) (s7, 0) (s3, 0)
(s4, 0) (s1, 0) (s5, 0) (s2, 0) (s2, 0)
(s2, 0) (s5, 0) (s3, 0) (s2, 0) (s4, 0)
(s2, 0) (s5, 0) (s7, 0) (s4, 0) (s2, 0)
G1
G2
G3
G4
(s5, 0) (s2, 0) (s6, 0) (s5, 0) (s4, 0)
(s3, 0) (s5, 0) (s2, 0) (s6, 0) (s2, 0)
(s2, 0) (s3, 0) (s5, 0) (s7, 0) (s4, 0)
(s6, 0) (s5, 0) (s3, 0) (s2, 0) (s5, 0)
Table 6 e3. Decision matrix R
A1 A2 A3 A4 A5
A4 > A3 > A5 > A2 > A1 A4 > A3 > A5 > A1 > A2
In this paper, we have investigated the multiple attribute group decision making (MAGDM) problems in which the attribute values take the form of 2-tuple linguistic information. Motivated by the ideal of dependent aggregation (Xu, 2006, 2008), we develop some dependent 2-tuple linguistic aggregation operators: the dependent 2-tuple ordered weighted averaging (D2TOWA) operator and the dependent 2-tuple ordered weighted geometric (D2TOWG) operator, in which the associated weights only depend on the aggregated 2-tuple linguistic arguments and can relieve the influence of unfair 2-tuple linguistic arguments on the aggregated results by assigning low weights to those ‘‘false’’ and ‘‘biased’’ ones and then apply them to develop some approaches for multiple attribute group decision making with 2-tuples linguistic information. Finally, some illustrative examples are given to verify the developed approach. In the future, we shall continue working in the extension and application of the developed operators to other domains. Acknowledgments
Step 3. Utilize the D2TOWA operator and D2TOWG operator to derive the overall preference values zi (i = 1, 2, 3, 4, 5) of the alternative Ai. The results are shown in Table 9.
The author is very grateful to the editor and the anonymous referees for their insightful and constructive comments and
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suggestions, which have been very helpful in improving the paper. The work was supported by the National Natural Science Foundation of China under Grant No. 61174149, Natural Science Foundation Project of CQ CSTC of the People’s Republic of China (No. CSTC, 2011BA0035), the Humanities and Social Sciences Foundation of Ministry of Education of the People’s Republic of China under Grant No. 11XJC630011 and the China Postdoctoral Science Foundation under Grant 20100480269. References Fan, Z. P., & Liu, Y. (2010). A method for group decision-making based on multigranularity uncertain linguistic information. Expert Systems with Applications, 37(5), 4000–4008. Fullér, R., & Majlender, P. (2001). An analytic approach for obtaining maximal entropy OWA operator weights. Fuzzy Sets and Systems, 124, 53–57. Filev, D. P., & Yager, R. R. (1998). On the issue of obtaining OWA operator weights. Fuzzy Sets and Systems, 94, 157–169. Herrera, F., & Herrera-Viedma, E. (2000). Linguistic decision analysis: Steps for solving decision problems under linguistic information. Fuzzy Sets and Systems, 115, 67–82. Herrera, F., & Martínez, L. (2000). A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Transactions on Fuzzy Systems, 8, 746–752. Herrera, F., & Martínez, L. (2001). A model based on linguistic 2-tuple for dealing with multigranular hierarchical linguistic contexts in multi-expert decisionmaking. IEEE Transactions on Systems, Man, and Cybernetics, 31, 227–234. Herrera, F., Martínez, L., & Sánchez, P. J. (2005). Managing non-homogeneous information in group decision making. European Journal of Operational Research, 166(1), 115–132. Herrera, F., Herrera-Viedma, E., & Martínez, L. (2008). A fuzzy linguistic methodology to deal with unbalanced linguistic term sets. IEEE Transactions on Fuzzy Systems, 16(2), 354–370. Jiang, Y. P., & Fan, Z. P. (2003). Property analysis of the aggregation operators for 2tuple linguistic information. Control and Decision, 18(6), 754–757. Merigó, J. M., & Gil-Lafuente, A. M. (2009). The induced generalized OWA operator. Information Sciences, 179, 729–741. Merigó, J. M., & Casanovas, M. (2009). Induced aggregation operators in decision making with the Dempster–Shafer belief structure. International Journal of Intelligent Systems, 24, 934–954. Merigó, J. M. (2010). Fuzzy decision making with immediate probabilities. Computers & Industrial Engineering, 58(4), 651–657. Merigó, J. M., Casanovas, M., & Martínez, L. (2010). Linguistic aggregation operators for linguistic decision making based on the Dempster–Shafer theory of evidence. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 18(3), 287–304. Tai, W. S., & Chen, C. T. (2009). A new evaluation model for intellectual capital based on computing with linguistic variable. Expert Systems with Applications, 36(2), 3483–3488. Wang, W. P. (2009a). Evaluating new product development performance by fuzzy linguistic computing. Expert Systems with Applications, 36(6), 9759–9766. Wang, W. P. (2009b). Toward developing agility evaluation of mass customization systems using 2-tuple linguistic computing. Expert Systems with Applications, 36, 3439–3447. Wei, G. W. (2008a). Maximizing deviation method for multiple attribute decision making in intuitionistic fuzzy setting. Knowledge-based Systems, 21(8), 833–836. Wei, G. W. (2008b). Interval-valued intuitionistic fuzzy Choquet integral operator. Journal of Information and Computational Science, 5(6), 2545–2551. Wei, G. W. (2008c). Induced intuitionistic fuzzy ordered weighted averaging operator and its application to multiple attribute group decision making. Lecture Notes in Artificial Intelligence, 5009, 124–131. Wei, G. W. (2009a). Uncertain linguistic hybrid geometric mean operator and its application to group decision making under uncertain linguistic environment. International Journal of Uncertainty Fuzziness, Knowledge-Based Systems, 17(2), 251–267. Wei, G. W. (2009b). Some geometric aggregation functions and their application to dynamic multiple attribute decision making in intuitionistic fuzzy setting. International Journal of Uncertainty Fuzziness and Knowledge-Based Systems, 17(2), 179–196. Wei, G. W. (2010a). GRA method for multiple attribute decision making with incomplete weight information in intuitionistic fuzzy setting. Knowledge-based Systems, 23(3), 243–247.
Wei, G. W. (2010b). Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making. Applied Soft Computing, 10(2), 423–431. Wei, G. W. (2010c). A method for multiple attribute group decision making based on the ET-WG and ET-OWG operators with 2-tuple linguistic information. Expert Systems with Applications, 37(12), 7895–7900. Wei, G. W. (2010d). Extension of TOPSIS method for 2-tuple linguistic multiple attribute group decision making with incomplete weight information. Knowledge and Information Systems, 25, 623–634. Wei, G. W., Zhao, X. F., & Lin, R. (2010). Some induced aggregating operators with fuzzy number intuitionistic fuzzy information and their applications to group decision making. International Journal of Computational Intelligence Systems, 3(1), 84–95. Wei, G. W., Lin, R., Zhao, X. F., & Wang, H. J. (2010a). Models for multiple attribute group decision making with 2-tuple linguistic assessment information. International Journal of Computational Intelligence Systems, 3(3), 315–324. Wei, G. W., Lin, R., Zhao, X. F., & Wang, H. J. (2010b). TOPSIS-based linearprogramming methodology for multiple attribute decision making with incomplete weight information in intuitionistic fuzzy setting. Information: An International Journal, 13(5), 1249–1257. Wei, G. W. (2011a). Grey relational analysis method for 2-tuple linguistic multiple attribute group decision making with incomplete weight information. Expert Systems with Applications, 38(5), 4824–4828. Wei, G. W. (2011b). FIOWHM operator and its application to multiple attribute group decision making. Expert Systems with Applications, 38(4), 2984–2989. Wei, G. W. (2011c). Gray relational analysis method for intuitionistic fuzzy multiple attribute decision making. Expert Systems with Applications, 38(9), 11671–11677. Wei, G. W. (2011d). A novel efficient approach for interval-valued intuitionistic fuzzy multiple attribute decision making with incomplete weight information. Information: An International Interdisciplinary Journal, 14(1), 97–102. Wei, G. W. (2011e). Some generalized aggregating operators with linguistic information and their application to multiple attribute group decision making. Computers & Industrial Engineering, 61(1), 32–38. Wei, G. W. (2011f). Grey relational analysis model for dynamic hybrid multiple attribute decision making. Knowledge-based Systems, 24(5), 672–679. Wei, G. W. (2011g). GRA-based linear-programming methodology for multiple attribute group decision making with 2-tuple linguistic assessment information. Information: An International Interdisciplinary Journal, 14(4), 1105–1110. Wei, G. W., Wang, H. J., Lin, R., & Zhao, X. F. (2011a). Grey relational analysis method For intuitionistic fuzzy multiple attribute decision making with preference information on alternatives. International Journal of Computational Intelligence Systems, 4(2), 164–173. Wei, G. W., & Zhao, X. F. (2011b). Minimum deviation models for multiple attribute decision making in intuitionistic fuzzy setting. International Journal of Computational Intelligence Systems, 4(2), 174–183. Wei, G. W., Wang, H. J., & Lin, R. (2011c). Application of correlation coefficient to interval-valued intuitionistic fuzzy multiple attribute decision making with incomplete weight information. Knowledge and Information Systems, 26(2), 337–349. Wei, G. W., & Zhao, X. F. (2012). Some induced correlated aggregating operators with intuitionistic fuzzy information and their application to multiple attribute group decision making. Expert Systems with Applications, 39(2), 2026–2034. Xu, Z. S., & Da, Q. L. (2002). The uncertain OWA operator. International Journal of Intelligent Systems, 17, 569–575. Xu, Z. S., & Da, Q. L. (2003). An overview of operators for aggregating information. International Journal of Intelligent Systems, 18, 953–969. Xu, Z. S. (2005). An overview of methods for determining OWA weights. International Journal of Intelligent Systems, 20, 843–865. Xu, Z. S. (2006). Dependent OWA operators. Lecture Notes in Artificial Intelligence, 3885, 172–178. Xu, Z. S. (2008). Dependent uncertain ordered weighted aggregation operators. Information Fusion, 9(2), 310–316. Yager, R. R. (1993). Families of OWA operators. Fuzzy Sets and Systems, 59, 125–148. Yager, R. R., & Filev, D. P. (1994). Parameterized ‘‘andlike’’ and ‘‘orlike’’ OWA operators. International Journal of General Systems, 22, 297–316. Yager, R. R. (1988). On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Transactions on Systems Man and Cybernetics, 18, 183–190. Zhang, Z. F., & Chu, X. N. (2009). Fuzzy group decision-making for multi-format and multi-granularity linguistic judgments in quality function deployment. Expert Systems with Applications, 36(5), 9150–9158.
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Contents lists available at SciVerse ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Some dependent aggregation operators with 2-tuple linguistic information and their application to multiple attribute group decision making Guiwu Wei ⇑, Xiaofei Zhao Institute of Decision Sciences, Chongqing University of Arts and Sciences, Yongchuan, Chongqing 402160, PR China
a r t i c l e
i n f o
Keywords: Multiple attribute group decision making (MAGDM) 2-Tuple linguistic information Power average operator Dependent 2-tuple ordered weighted averaging (D2TOWA) operator The dependent 2-tuple ordered weighted geometric (D2TOWG) operator
a b s t r a c t We investigate the multiple attribute group decision making (MAGDM) problems in which the attribute values take the form of 2-tuple linguistic information. Motivated by the ideal of dependent aggregation [Xu, Z. S. (2006). Dependent OWA operators. Lecture Notes in Artificial Intelligence, 3885, 172–178], in this paper, we develop some dependent 2-tuple linguistic aggregation operators: the dependent 2-tuple ordered weighted averaging (D2TOWA) operator and the dependent 2-tuple ordered weighted geometric (D2TOWG) operator, in which the associated weights only depend on the aggregated 2-tuple linguistic arguments and can relieve the influence of unfair 2-tuple linguistic arguments on the aggregated results by assigning low weights to those ‘‘false’’ and ‘‘biased’’ ones and then apply them to develop some approaches for multiple attribute group decision making with 2-tuples linguistic information. Finally, some illustrative examples are given to verify the developed approach and to demonstrate its practicality and effectiveness. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The ordered weighted aggregation (OWA) operator (Yager, 1988) as an aggregation technique has received more and more attention since its appearance (Merigó & Casanovas, 2009; Merigó & Gil-Lafuente, 2009; Merigó, 2010; Wei, 2008a, 2008b, 2008c, 2009a, 2009b, 2010a, 2010b, 2010c, 2010d; Wei, Zhao, & Lin, 2010; Wei, 2011a, 2011b, 2011c; Wei, Wang, Lin, & Zhao, 2011a, 2011b, Wei, Wang, & Lin, 2011c; Wei, 2011d, 2011e, 2011f, 2011g; Wei & Zhao, 2012; Zhang & Chu, 2009; Xu & Da, 2003). One important step of the OWA operator is to determine its associated weights. Many authors have focused on this issue, and developed some useful approaches to obtaining the OWA weights. We classify all these approaches into the following two categories: argument-independent approaches (Filev & Yager, 1998; Fullér & Majlender, 2001; Xu, 2005; Yager, 1993; Yager & Filev, 1994) and argument-dependent approaches (Filev & Yager, 1998; Xu & Da, 2002; Xu, 2005, 2006, 2008; Yager, 1993). The weights derived by the argument-independent approaches are associated with particular ordered positions of the aggregated arguments, and have no connection with the aggregated arguments, while the argument-dependent approaches determine the weights based on the input arguments. With respect to the argument-dependent approaches, Yager (1993) introduced some families of the OWA weights, including the ideal of aggregate dependent weights. Filev and Yager (1998) developed two proce⇑ Corresponding author. Tel./fax: +86 23 49891870. E-mail address: [email protected] (G. Wei). 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.11.120
dures to obtain the OWA weights, the first one learns the weights from a collection of samples with their aggregated value, and the second one calculates the weights for a given level of orness. Xu and Da (2002) established a linear objective-programming model for obtaining the weights of the OWA operator by utilizing the given arguments under partial weight information. Xu (2005) developed a new dependent OWA (DOWA) operator and developed a new argument-dependent approach to determining the OWA weights, which can relieve the influence of unfair arguments on the aggregated results. Xu (2006) developed some dependent uncertain ordered weighted aggregation operators, including dependent uncertain ordered weighted averaging (DUOWA) operators and dependent uncertain ordered weighted geometric (DUOWG) operators, in which the associated weights only depend on the aggregated interval arguments and can relieve the influence of unfair interval arguments on the aggregated results by assigning low weights to those ‘‘false’’ and ‘‘biased’’ ones. However, in many situations, the input arguments take the form of 2-tuples linguistic variables (Fan & Liu, 2010; Herrera & Martínez, 2000, 2001; Herrera, Martínez, & Sánchez, 2005; Herrera, Herrera-Viedma, & Martínez, 2008; Merigó, Casanovas, & Martínez, 2010; Tai & Chen, 2009; Wang, 2009a, 2009b; Wei, 2010c, 2010d, 2011a; Wei, Lin, Zhao, & Wang, 2010a, Wei, Lin, Zhao, & Wang, 2010b) because of time pressure, lack of knowledge, and people’s limited expertise related with problem domain. In this paper, we will pay attention on the second category, and develop some argument-dependent approach to determining the OWA weights with 2-tuples linguistic information. The remainder of this paper
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is set out as follows. In the next section, we introduce some basic concepts and operational laws of 2-tuple linguistic variables. In Section 3 we develop some dependent 2-tuple linguistic aggregation operators: the dependent 2-tuple ordered weighted averaging (D2TOWA) operator and the dependent 2-tuple ordered weighted geometric (D2TOWG) operator, in which the associated weights only depend on the aggregated 2-tuple linguistic arguments and can relieve the influence of unfair 2-tuple linguistic arguments on the aggregated results by assigning low weights to those ‘‘false’’ and ‘‘biased’’ ones. In Section 4 we develop an approach to multiple attribute group decision making based on these dependent aggregation operators with 2-tuple linguistic information, which is straightforward and has no loss of information. In Section 5, we give an illustrative example to verify the developed approach and to demonstrate its feasibility and practicality. In Section 6 we conclude the paper and give some remarks. 2. Preliminaries Let S = {siji = 1, 2, . . . , t} be a linguistic term set with odd cardinality. Any label, si represents a possible value for a linguistic variable, and it should satisfy the following characteristics (Herrera & Martínez, 2000, 2001): (1) The set is ordered: si > sj, if i > j; (2) Max operator: max (si, sj) = si, if si P sj; (3) Min operator: min (si, sj) = si, if si 6 sj. For example, S can be defined as
S ¼ fs1 ¼ extremely poor; s2 ¼ v ery poor; s3 ¼ poor; s4 ¼ medium; s5 ¼ good; s6 ¼ v ery good; s7 ¼ extremely goodg Herrera and Martínez (2000, 2001) developed the 2-tuple fuzzy linguistic representation model based on the concept of symbolic translation. It is used for representing the linguistic assessment information by means of a 2-tuple (si, ai), where si is a linguistic label from predefined linguistic term set S and ai is the value of symbolic translation, and ai 2 [(0.5, 0.5)]. Definition 1. Let b be the result of an aggregation of the indices of a set of labels assessed in a linguistic term set S, i.e., the result of a symbolic aggregation operation, b 2 [1, t], being tthe cardinality of S. Let i = round(b) and a = b i be two values, such that, i 2 [1, t] and a 2 [(0.5, 0.5)] then a is called a Symbolic Translation (Herrera & Martínez, 2000, 2001). Definition 2. Let S = {s1, s2, . . . , st} be a linguistic term set and b 2 [1, t] is a number value representing the aggregation result of linguistic symbolic. Then the function D used to obtain the 2-tuple linguistic information equivalent to b is defined as:
D : ½1; t ! S ½0:5; 0:5Þ;
DðbÞ ¼
si ;
i ¼ roundðbÞ;
a ¼ b i; a 2 ½0:5; 0:5Þ;
ð1Þ ð2Þ
where round (.) is the usual round operation, si has the closest index label to b and a is the value of the symbolic translation (Herrera & Martínez, 2000, 2001). Definition 3. Let S = {s1, s2, . . . , st} be a linguistic term set and (si, ai) be a 2-tuple. There is always a function D1 can be defined, such that, from a 2-tuple (si, ai) it return its equivalent numerical value b 2 [1, t] R, which is (Herrera & Martínez, 2000, 2001).
D1 : S ½0:5; 0:5Þ ! ½1; t; 1
D ðsi ; aÞ ¼ i þ a ¼ b:
Definition 4. Let (sk, ak) and (sl, al) be two 2-tuple, they should have the following properties (Herrera & Martínez, 2000, 2001). (1) If k < l then (sk, ak) is smaller than (sl, al). (2) If k = l then, (a) if ak = al, then (sk, ak), (sl, al) represents information, (b) if ak < al then (sk, ak) is smaller than (sl, al), (c) if ak > al then (sk, ak) is bigger than (sl, al),
same
3. Some dependent aggregation operators with 2-tuple linguistic information Yager (1988) provides a definition of the ordered weighted averaging (OWA) operator as follows: Definition 5. An OWA operator of dimension n is a mapping, OWA: Rn ? R, that has an associated weighting vector w = (w1, w2, . . . , wn)T P with the properties wj 2 [0, 1], and nj¼1 wj ¼ 1, such that
OWAw ða1 ; a2 ; . . . ; an Þ ¼
n X
wj apðjÞ ;
ð5Þ
j¼1
where (p(1), p(2), . . . , p(n)) is a permutation of (1, 2, . . . , n), such that ap(j1) P ap(j) for all j = 2, . . . , n. Since its appearance in 1988, the OWA operator has been used in a wide range of applications, such as engineering, neural networks, data mining, decision making, image process, expert systems, etc. Consider that the OWA operator aggregates only the exact inputs having been reordered, Herrera and Martínez (2000, 2001) extend WA operator and OWA operator to accommodate the situations where the input arguments are linguistic variables. Definition 6. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2-tuples and x = (x1, x2, . . . , xn)T be the weighting vector of 2-tuples (rj, aj) Pn (j = 1, 2, . . . , n) and xj 2 ½0; 1; j ¼ 1; 2; . . . ; n; j¼1 xj ¼ 1. The 2tuple weighted average is
2TWAx ððr 1 ; a1 Þ; ðr 2 ; a2 Þ; . . . ; ðr n ; an ÞÞ ¼ D
n X
! 1
xj D ðrj ; aj Þ :
ð6Þ
j¼1
Definition 7. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2-tuples and w = (w1, w2, . . . , wn)T be an associated weighting vector, P wj 2 [0, 1], j = 1, 2, . . . , n and nj¼1 wj ¼ 1. The 2-tuple OWA for linguistic 2-tuples is computed as
2TOWAw ððr 1 ; a1 Þ; ðr2 ; a2 Þ; . . . ; ðrn ; an ÞÞ ! n X 1 ¼D wj D r pðjÞ ; apðjÞ ;
ð7Þ
j¼1
where (p(1), p(2), . . . , p(n)) is a permutation of (1, 2, . . . , n), such that (sp(j1), ap(j1)) P sp(j), ap(j)) for all j = 2, . . . , n. Recently, Jiang and Fan (2003) extended the WG operator and OWG operator to accommodate the situations where the input arguments are linguistic variables. Definition 8. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2-tuple and x = (x1, x2, . . . , xn)T be the weighting vector of 2-tuple (rj, aj) Pn (j = 1, 2, . . . , n) and xj 2 ½0; 1; j¼1 xj ¼ 1, The 2-tuple weighted geometric operator is (Jiang & Fan, 2003)
ð3Þ ð4Þ
the
2TWGx ððr1 ; a1 Þ; ðr2 ; a2 Þ; . . . ; ðrn ; an ÞÞ ¼ D
n Y j¼1
! ðD1 ðr j ; aj ÞÞxj :
ð8Þ
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Definition 9. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2-tuple, A 2-tuple ordered weighted geometric operator of dimension n is a mapping TOWG: Rn?R that has an associated vector w = P (w1, w2, . . . , wn)T such that wj > 0 and nj¼1 wj ¼ 1. Furthermore,
2TOWGw ððr 1 ;a1 Þ; ðr 2 ; a2 Þ; . .. ; ðrn ; an ÞÞ ¼ D
n Y
wj
ðD1 ðr pðjÞ ; apðjÞ ÞÞ
;
ð9Þ
Definition 10. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2tuples, the 2-tuple arithmetic mean is computed as
2 ½0:5; 0:5Þ: a
ð10Þ
Definition 11. Let (ri, ai) and (rj, aj) be two 2-tuples, then we call
dððr i ; ai Þ; ðrj ; aj ÞÞ ¼
jD1 ðr i ; ai Þ D1 ðr j ; aj Þj ; t
ð11Þ
the distance between (ri, ai) and (rj, aj). Definition 12. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2Þ the arithmetic mean of these 2-tuples, then tuples, and let ðr ; a we define the standard deviation of these 2-tuples as
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u X u1 n ÞÞ2 : r¼t dððr j ; aj Þ; ðr ; a n j¼1
j ¼ 1; 2; . . . ; n:
ð13Þ
the degree of similarity between the jth largest 2-tuples linguistic Þ, where (p(1), p(2), . . . , p(n)) variables (rp(j), ap(j)) and the mean ðr ; a is a permutation of (1, 2, . . . , n), such that (rp(j1), ap(j1)) P (rp(j), ap(j)) for all j = 2, . . . , n. In real-life situations, the 2-tuples (r1, a1), (r2, a2), . . . , (rn, an) usually take the form of a collection of n preference values provided by n different individuals. Some individuals may assign unduly high or unduly low preference values to their preferred or repugnant objects. In such a case, we shall assign very low weights to these ‘‘false’’ or ‘‘biased’’ opinions, that is to say, the closer a preference value (argument) is to the mid one(s), the more the weight. As a result, based on (13), we define the 2TOWA weights as
j ¼ 1; 2; . . . ; n
ð14Þ
P Obviously, wj P 0, j = 1, 2, . . . , n and nj¼1 wj ¼ 1. Especially, if (sk, ak) = (sl, al), for all i, j = 1, 2, . . . , n, then by (14), we have wj ¼ 1n, for all j = 1, 2, . . . , n. By (7), we have
2TOWAððr 1 ; a1 Þ; ðr2 ; a2 Þ; . . . ; ðrn ; an ÞÞ ¼D
j¼1
Since
n X
ÞÞD1 ðr j ; aj Þ simððr j ; aj Þ; ðr ; a
and n X
ÞÞ ¼ simððr pðjÞ ; apðjÞ Þ; ðr ; a
j¼1
n X
ÞÞ simððr j ; aj Þ; ðr; a
j¼1
then we replace (15) by
2TOWAððr 1 ; a1 Þ; ðr 2 ; a2 Þ; . . . ; ðr n ; an ÞÞ ! Pn 1 j¼1 simððr j ; aj Þ; ðr ; aÞÞD ðr j ; aj Þ Pn : ¼D j¼1 simððr j ; aj Þ; ðr ; aÞÞ
ð16Þ
We call (16) a dependent 2-tuple ordered weighted averaging (D2TOWA) operator, which is a generalization of the dependent ordered weighted averaging (DOWA) operator (Xu, 2006). Consider that the aggregated value of the D2TOWA operator is independent of the ordering, thus it is also a neat operator. Based on the 2TOWG and Eq.(14), we developed the dependent 2-tuple ordered weighted geometric (D2TOWG) operator as follows:
D2TOWGððr1 ; a1 Þ; ðr2 ; a2 Þ; . . . ; ðrn ; an ÞÞ 0 1 n P ÞÞ= ÞÞ n simððr j ;aj Þ;ðr;a simððrj ;aj Þ;ðr ;a Y 1 j¼1 @ A: ¼D ðD ðrj ; aj ÞÞ
ð17Þ
j¼1
Similar to Xu (2006, 2008), we have the following result:
ÞÞ dððr ; a Þ; ðr ; a ÞÞ ¼ 1 Pn pðjÞ pðjÞ ; simððr pðjÞ ; apðjÞ Þ; ðr; a j¼1 dððr pðjÞ ; apðjÞ Þ; ðr ; aÞÞ
n X
¼
ð12Þ
Definition 13. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2Þ the arithmetic mean of these 2-tuples, then tuples, and let ðr ; a we call
ÞÞ simððr pðjÞ ; apðjÞ Þ; ðr ; a ; wj ¼ Pn ÞÞ r; a simððr ; a Þ; ð p ðjÞ p ðjÞ j¼1
j¼1
j¼1
where (p(1), p(2), . . . , p(n)) is a permutation of (1, 2, . . . , n), such that (sp(j1), ap(j1)) P (sp(j), ap(j)) for all j = 2, . . . , n.
r 2 S;
ÞÞD1 ðr pðjÞ ; apðjÞ Þ simððr pðjÞ ; apðjÞ Þ; ðr ; a
!
j¼1
X n 1 1 Þ ¼ D ðr; a D ðr ; a Þ ; j j j¼1 n
n X
ÞÞ P simððr pðjÞ ; apðjÞ Þ; ðr ; a ÞÞ; simððr pðiÞ ; apðiÞ Þ; ðr ; a
ð15Þ
then wi P wj :
The normal distribution is one of the most commonly observed and is the starting point for modeling many natural process, it is usually found in events that are the aggregation of many smaller, but independent random events. Xu (2008) introduced a normal distribution based method to determine some dependent uncertain ordered weighted aggregation operators, in which the associated weights only depend on the aggregated arguments. Motivated by the idea, in the following, we give another method for deriving the D2TOWA weights: 2
ÞÞÞ ðdððr pðjÞ ;apðjÞ Þ;ðr ;a 1 2r2 wj ¼ pffiffiffiffiffiffiffi e ; 2pr
j ¼ 2; . . . ; n;
ð18Þ
Þ and r are the arithmetric mean and the standard deviawhere ðr; a tion of these 2-tuples arguments variables {(r1, a1), (r2, a2), . . . , (rn, an)}, respectively, (p(1), p(2), . . . , p(n)) is a permutation of (1, 2, . . . , n), such that (rp(j1), ap(j1)) P (rp(j), ap(j)) for all j = 2, . . . , n. P Consider that wj P 0, j = 1, 2, . . . , n and nj¼1 wj ¼ 1, then by (18), we have
wj ¼
!
ÞÞ simððr pðjÞ ; apðjÞ Þ; ðr ; a Pn D1 ðr pðjÞ ; apðjÞ Þ : ÞÞ r ; a simððr ; a Þ; ð pðjÞ pðjÞ j¼1
Theorem 1. Let x = {(r1, a1), (r2, a2), . . . , (rn, an)} be a set of 2-tuples, Þ the arithmetic mean of these 2-tuples, and let ðr ; a (p(1), p(2), . . . , p(n)) is a permutation of (1, 2, . . . , n), such that (rp(j1), ap(j1)) P (rp(j), ap(j)) for all j = 2, . . . , n. If
1 pffiffiffiffi e 2p r
ðdððrpðjÞ ;apðjÞ Þ;ðr ;aÞÞÞ2
Pn
1 pffiffiffiffi j¼1 2pr e
2r2 ÞÞÞ2 ðdððrpðjÞ ;apðjÞ Þ;ðr;a
¼ 1; 2; . . . ; n; then by (7), we have
2r 2
¼
e Pn
ÞÞÞ2 ðdððr pðjÞ ;apðjÞ Þ;ðr;a 2r2
j¼1 e
ÞÞÞ2 ðdððrpðjÞ ;apðjÞ Þ;ðr;a
;
j
2r2
ð19Þ
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G. Wei, X. Zhao / Expert Systems with Applications 39 (2012) 5881–5886
2TOWAððr1 ; a1 Þ; ðr2 ; a2 Þ; . . . ; ðrn ; an ÞÞ 0 n BP ¼ D@
j¼1
0 B ¼ D@
ðdððrpðjÞ ;apðjÞ Þ;ðr ;aÞÞÞ2 2r2 e
Pn j¼1
Pn j¼1
e
e
ÞÞÞ2 ðdððr pðjÞ ;apðjÞ Þ;ðr ;a
j¼1
C D1 ðr pðjÞ ; apðjÞ ÞA
2r2
ðdððspðjÞ ;apðjÞ 2r2
Pn
Step 2. Utilize the decision information given in matrix Rk, and the 2TWA operator
1
ð20Þ
1
ÞÞÞ2 Þ;ðr;a
D1 ðr pðjÞ ;apðjÞ ÞC
n xj Y ðkÞ ðkÞ ðkÞ zi ¼ ri ; ai D1 r ðkÞ ¼D ij ; 0
Since n X
e
ÞÞÞ2 ðdððr pðjÞ ;apðjÞ Þ;ðr ;a 2r2
D1 ðrpðjÞ ; apðjÞ Þ ¼
n X
e
ÞÞÞ2 ðdððr j ;aj Þ;ðr ;a 2r2
ðkÞ
and
e
ð24Þ
to derive the individual overall preference value zi tive Ai. Step 3. Utilize the D2TOWA operator:
D1 ðr j ; aj Þ
j¼1
!
j¼1
j¼1
n X
ð23Þ
j¼1
or the 2TWG operator
A:
ÞÞÞ2 ðdððrpðjÞ ;apðjÞ Þ;ðr;a 2r2 e
! n X ðkÞ ðkÞ ðkÞ zi ¼ ri ; ai xj D1 rijðkÞ ; 0 ¼D
ÞÞÞ2 ðdððr pðjÞ ;apðjÞ Þ;ðr ;a 2r2
j¼1
¼
n X
e
zi ¼ ðri ; ai Þ 0Pt
ÞÞÞ2 ðdððr j ;aj Þ;ðr;a 2r2
¼ D@
j¼1
of the alterna-
1 ðkÞ ðkÞ ðkÞ i Þ D1 r ðkÞ r i ; ai ; ðr i ; a i ; ai A Pt ðkÞ ðkÞ i Þ r i ; ai ; ðri ; a k¼1 sim
k¼1 sim
then, (20) can be rewritten as
ð25Þ
or the D2TOWG operator:
2TOWAððr1 ; a1 Þ; ðr 2 ; a2 Þ; . . . ; ðrn ; an ÞÞ 0 1 Pn ðdððrj ;aj Þ;ð2r;aÞÞÞ2 1 2r e D ðr ; a Þ j j B j¼1 C ¼ D@ A: ÞÞÞ2 r ;a Pn ðdððrj ;aj Þ;ð 2 2r j¼1 e
ð21Þ
zi ¼ ðri ; ai Þ 0 1 t P ðkÞ ðkÞ ðkÞ n simððrðkÞ i ÞÞ= i ÞÞ ;ai Þ;ðr i ;a simððr i ;ai Þ;ðri ;a Y i ðkÞ A k¼1 ¼ D@ D1 r ðkÞ i ; ai j¼1
Obviously, (21) is also a neat and dependent 2TOWA (D2TOWA) operator. Based on the 2TOWG and Eq.(19), we developed the dependent 2-tuple ordered weighted geometric (D2TOWG) operator as follows:
D2TOWGððr 1 ; a1 Þ; ðr2 ; a2 Þ; . . . ; ðr n ; an ÞÞ 1 0 ÞÞÞ2 ÞÞÞ2 ðdððr j ;aj Þ;ðr;a ðdððrj ;aj Þ;ðr;a n P 2r 2 2r 2 n e = e C BY 1 j¼1 C: ¼ DB A @ ðD ðr j ; aj ÞÞ
ð22Þ
ð26Þ to derive the collective overall preference P values zi (i = 1, 2, . . . , m) of ðkÞ i Þ ¼ D 1t tk¼1 D1 rðkÞ . the alternative Ai, where ðr i ; a ; a i i Step 4. Rank all the alternatives Ai (i = 1, 2, . . . , m) and select the best one(s) in accordance with the collective overall preference values zi (i = 1, 2, . . . , m). Step 5. End. 5. Illustrative example
j¼1
From (14) and (19), we know that all the associated weights of the D2TOWA operator and D2TOWG operator only depend on the aggregated 2-tuples variables, and can relieve the influence of unfair arguments on the aggregated results by assigning low weights to those ‘‘false’’ and ‘‘biased’’ ones, and thus make the aggregated results more reasonable in the practical applications. 4. An approach to multiple attribute group decision making based on 2-tuple linguistic information For the multiple attribute group decision making problems, in which both the attribute weights and the expert weights take the form of real numbers, and the attribute preference values take the form of 2-tuple linguistic variables, we shall develop an approach based on the TWA and D2TOWA operators to multiple attribute group decision making based on 2-tuple linguistic information processing. Let A = {A1, A2, . . . , Am} be a discrete set of alternatives, G = {G1, G2, . . . , Gn} be the set of attributes, x = (x1, x2, . . . , xn) is the weighting vector of the attributes Gj (j = 1, 2, . . . , n), where P xj 2 ½0; 1; nj¼1 xj ¼ 1; D ¼ fD D2 ; . . . ; Dt g be the set of decision 1; ðkÞ ðkÞ makers. Suppose that Rk ¼ r ij is the, where rij 2 e S is a mn preference values, which take the form of variable, given by the decision maker Dk 2 D, for the alternative Ai 2 A with respect to the attribute Gj 2 G. ðkÞ Step 1. Transform linguistic decision matrix Rk ¼ r ij into 2 mn ðkÞ tuple linguistic decision matrix Rk ¼ r ij ; 0 . mn
Let us suppose there is an investment company, which wants to invest a sum of money in the best option (adapted from Herrera & Herrera-Viedma (2000)). There is a panel with five possible alternatives to invest the money: rA1 is a car company; sA2 is a food company; tA3 is a computer company; uA4 is a chemical company; vA5 is a TV company. The investment company must take a decision according to the following four attributes: rG1 is the risk analysis; sG2 is the growth analysis; tG3 is the social-political impact analysis; uG4 is the environmental impact analysis. The five possible alternatives Ai (i = 1, 2, . . . , 5) are to be evaluated using the linguistic term set
S ¼ fs1 ¼ extremely poor; s2 ¼ v ery poor; s3 ¼ poor; s4 ¼ medium; s5 ¼ good; s6 ¼ v ery good; s7 ¼ extremely goodg by the three decision makers Dk(k = 1, 2, 3) under the above four attributes (whose weighting vector x = (0.3, 0.1, 0.2, 0.4)), and construct, respectively, the linguistic decision matrices are shown in Tables 1–3: In the following, we shall utilize the proposed approach in this paper getting the most desirable alternative(s): Step 1. Transforming linguistic decision information given in Tables 1–3 into 2-tuple linguistic decision matrices which are given in Tables 4–6. Step 2. Utilize the decision information given in matrix Rk, and the 2TWA operator and 2TWG operator (Let x = (0.3, 0.1, 0.2, 0.4)) to derive the individual overall preference value ðkÞ zi of the alternative Ai. The results are shown in Tables 7 and 8.
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G. Wei, X. Zhao / Expert Systems with Applications 39 (2012) 5881–5886 Table 1 e1. Decision matrix R
A1 A2 A3 A4 A5
Table 7 The individual overall preference value by utilizing 2TWA operator. G1
G2
G3
G4
s4 s3 s5 s6 s7
s5 s2 s4 s3 s1
s3 s4 s5 s3 s2
s3 s3 s1 s5 s4
A1 A2 A3 A4 A5
D1
D2
D3
(s4, 0.50) (s3, 0.10) (s3, 0.30) (s5, 0.30) (s4, 0.20)
(s3, 0.50) (s4, 0.30) (s5, 0.10) (s4, 0.30) (s3, 0.30)
(s5, 0.40) (s4, 0.30) (s4, 0.20) (s4, 0.30) (s4, 0.20)
Table 8 The individual overall preference value by utilizing 2TWG operator. Table 2 e2. Decision matrix R
A1 A2 A3 A4 A5
G1
G2
G3
G4
s3 s2 s4 s7 s3
s4 s1 s5 s2 s2
s2 s5 s3 s2 s4
s2 s5 s7 s4 s2
Table 3 e3. Decision matrix R
A1 A2 A3 A4 A5
G1
G2
G3
G4
s5 s2 s6 s5 s4
s3 s5 s2 s6 s2
s2 s3 s5 s7 s4
s6 s5 s3 s2 s5
A1 A2 A3 A4 A5
D1
D2
D3
(s3, 0.44) (s3, 0.05) (s3,-0.43) (s5, 0.47) (s4, 0.41)
(s2, 0.42) (s3, 0.23) (s5, 0.17) (s4, 0.16) (s3, 0.41)
(s4, 0.25) (s3, 0.43) (s4, 0.07) (s4, 0.22) (s4, 0.08)
Table 9 The overall preference values of the alternatives.
2TWA and D2TOWA 2TWG and D2TOWG
A1
A2
A3
A4
A5
(s4, 0.48)
(s4, 0.45)
(s4, 0.20)
(s4, 0.40)
(s4, 0.17)
(s3, 0.40)
(s3, 0.24)
(s4, 0.16)
(s4, 0.01)
(s3, 0.49)
Table 10 Ordering of the alternative. Ordering 2TWA and D2TOWA 2TWG and D2TOWG
Table 4 e1. Decision matrix R
A1 A2 A3 A4 A5
G1
G2
G3
G4
(s4, 0) (s3, 0) (s5, 0) (s6, 0) (s7, 0)
(s5, 0) (s2, 0) (s4, 0) (s3, 0) (s1, 0)
(s3, 0) (s4, 0) (s5, 0) (s3, 0) (s2, 0)
(s3, 0) (s3, 0) (s1, 0) (s5, 0) (s4, 0)
Table 5 e2. Decision matrix R
A1 A2 A3 A4 A5
Step 4. According to the aggregating results shown in Table 9, the ordering of the alternatives are shown in Table 10. Note that?means ‘‘preferred to’’. As we can see, depending on the aggregation operators used, the ordering of the alternatives is slightly different. Therefore, depending on the aggregation operators used, the results may lead to different decisions. However, the best alternative is A4. 6. Conclusion
G1
G2
G3
G4
(s3, 0) (s2, 0) (s4, 0) (s7, 0) (s3, 0)
(s4, 0) (s1, 0) (s5, 0) (s2, 0) (s2, 0)
(s2, 0) (s5, 0) (s3, 0) (s2, 0) (s4, 0)
(s2, 0) (s5, 0) (s7, 0) (s4, 0) (s2, 0)
G1
G2
G3
G4
(s5, 0) (s2, 0) (s6, 0) (s5, 0) (s4, 0)
(s3, 0) (s5, 0) (s2, 0) (s6, 0) (s2, 0)
(s2, 0) (s3, 0) (s5, 0) (s7, 0) (s4, 0)
(s6, 0) (s5, 0) (s3, 0) (s2, 0) (s5, 0)
Table 6 e3. Decision matrix R
A1 A2 A3 A4 A5
A4 > A3 > A5 > A2 > A1 A4 > A3 > A5 > A1 > A2
In this paper, we have investigated the multiple attribute group decision making (MAGDM) problems in which the attribute values take the form of 2-tuple linguistic information. Motivated by the ideal of dependent aggregation (Xu, 2006, 2008), we develop some dependent 2-tuple linguistic aggregation operators: the dependent 2-tuple ordered weighted averaging (D2TOWA) operator and the dependent 2-tuple ordered weighted geometric (D2TOWG) operator, in which the associated weights only depend on the aggregated 2-tuple linguistic arguments and can relieve the influence of unfair 2-tuple linguistic arguments on the aggregated results by assigning low weights to those ‘‘false’’ and ‘‘biased’’ ones and then apply them to develop some approaches for multiple attribute group decision making with 2-tuples linguistic information. Finally, some illustrative examples are given to verify the developed approach. In the future, we shall continue working in the extension and application of the developed operators to other domains. Acknowledgments
Step 3. Utilize the D2TOWA operator and D2TOWG operator to derive the overall preference values zi (i = 1, 2, 3, 4, 5) of the alternative Ai. The results are shown in Table 9.
The author is very grateful to the editor and the anonymous referees for their insightful and constructive comments and
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G. Wei, X. Zhao / Expert Systems with Applications 39 (2012) 5881–5886
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