INFORMATICA Informatica 1822-88440868-49520868-4952 Vilnius University INFO1183 10.15388/Informatica.2018.170 Research Article Interval-Valued 2-Tuple Linguistic Induced Continuous Ordered Weighted Distance Measure and Its Application to Multiple Attribute Group Decision Making LiuXiliuxi5137@126.com12

X. Liu is a lecturer of School of Mathematics and Statistics, Hefei Normal University, China. She received a PhD degree in School of Mathematical Sciences from Anhui University. She has contributed several journal articles to professional journals. Her current research interests include decision making theory, forecasting, information fusion, fuzzy statistics and fuzzy mathematics.

 HanBingice05013861001@163.com2

B. Han is a lecturer of School of Mathematical Sciences, Anhui University, China. She received a PhD degree in School of Mathematical Sciences from Anhui University. She has contributed over 10 journal articles to professional journals such as Knowledge-Based Systems and Expert Systems with Applications. Her current research interests include aggregation operators, group decision making and combined forecasting.

 ChenHuayouhuayouc@126.com2∗

H. Chen is a professor of School of Mathematical Sciences, Anhui University, China. He received a PhD degree in operational research from University of Science Technology of China in 2002. He graduated from Nanjing University for 2 years postdoctoral research work in 2005. He has published a book: The Efficient Theory of Combined Forecasting and Applications (Science Press, Beijing, 2008) and has contributed over 120 journal articles to professional journals, such as Fuzzy Sets and Systems, Information Sciences, Group Decision and Negotiation, etc. His current research interests include information fusion, multi-criteria decision making, aggregation operators and combined forecasting.

 ZhouLigangshuiqiaozlg@126.com23

L. Zhou is a professor of School of Mathematical Sciences, Anhui University. He received a PhD degree in operations research from Anhui University in 2013. He has contributed over 40 journal articles to professional journals, such as Fuzzy Sets and Systems, Applied Mathematical Modelling, Applied Soft Computing, Group Decision and Negotiation, Expert Systems with Applications, etc. His current research interests include group decision making, aggregation operators and combined forecasting.

 School of Mathematics and Statistics, Hefei Normal University, Hefei, Anhui 230011, China School of Mathematical Sciences, Anhui University, Hefei, Anhui 230601, China China Institute of Manufacturing Development, Nanjing University of Information Science and Technology, Nanjing 210044, China Corresponding author. 20181120182923213521201752018 © 2018 Vilnius University2018 Open access article under the CC BY license.

This paper aims to propose a new distance measure, the interval-valued 2-tuple linguistic induced continuous ordered weighted distance (IT-ICOWD) measure, which consists of the interval-valued 2-tuple linguistic induced continuous ordered weighted averaging (IT-ICOWA) operator and the ordered weighted distance (OWD) measure. In these operators, we consider the risk attitude of decision maker. Furthermore, we discuss some desired properties and various special cases of the IT-ICOWD measure. Additionally, a method of multiple attribute group decision making (MAGDM) in interval-valued 2-tuple linguistic environment is developed on the basis of the IT-ICOWD measure. Through this method, we obtain three simple and exact formulae to determine the order-inducing variables of the IT-ICOWD measure, the weighting vector of decision makers and the weighting vector of attributes, respectively. At last, a numerical example is presented to illustrate the practicability and feasibility of proposed method.

 group decision making, distance measure interval-valued 2-tuple linguistic information IOWA operator COWA operator Introduction

MAGDM is a crucial branch of decision theory. It is used to select the most highly preferred alternative(s) from a finite alternatives set. This process is accorded with experts’ preference information, who are required to give their preferences based on multiple attributes. Bellman and Zadeh (1970) first studied the decision making under fuzzy environment: they took time pressure into consideration to have events uncertainty limited. After that, Zadeh (1965) proposed the traditional fuzzy environment, for instance, the interval-valued fuzzy set (Moore, 1966), the intuitionistic fuzzy set (Atanassov, 1986), the hesitant fuzzy set (Torra, 2010) and the type-2 fuzzy set (Mendel, 2007). These fuzzy sets have been widely studied and applied to decision making procedures (Zadeh, 1965; Torra, 2010; Mendel, 2007; Atanassov, 2012; Sengupta and Pal, 2009; Zhou et al., 2014a). Aiming to express subjective evaluations of the decision makers, Zadeh (1975a, 1975b, 1975c) then introduced the concept of linguistic variable. Moreover, the linguistic variables have also been deeply studied and widely used to evaluate decision information, because the quantitative information is not feasible to all decision cases.

Afterwards, several linguistic representation models were proposed to fit different subjective situations. For example, 2-tuple linguistic representation model was developed by Herrera and Martinez (2000) to avoid information loss in the aggregation process of linguistic labels; the notion of linguistic intervals was introduced by Chen and Lee (2010) to describe linguistic information uncertainty; the concept of hesitant fuzzy linguistic term sets was proposed by Rodriguez et al. (2011) to express the hesitancy when linguistic labels are used; the unbalanced linguistic term set was put forward by Herrera et al. (2008), in which the linguistic labels around the centred linguistic label are not distributed symmetrically; an alternative form of uncertain linguistic variable was developed by Xu (2004) in order to demonstrate the uncertainty in linguistic information; multi-granular fuzzy linguistic modelling and fuzzy entropy methods were introduced by Morente-Molinera et al. (2017) to transform the training data in ways that represent their inner meaning more precisely; a linguistic computational model based on discrete fuzzy numbers whose support is a subset of consecutive natural numbers was presented by Massanet et al. (2014); a new method based on linguistic granular computing to solve group decision making problems defined in heterogeneous contexts was developed by Cabrerizo et al. (2013).

Decision making problem with interval-valued linguistic variable was utilized to deal with practical situation such as the health-care waste treatment technology evaluation and selection (Liu et al., 2014a), the failure mode and effects analysis (Liu et al., 2014b), searching for an optimal investment (Zhang, 2013), etc. Aggregation techniques are essential part in these applications. There are two ways to realize the aggregation: using the aggregation operators directly (Zhang, 2013, 2012), and combining aggregation operators with information measures such as the weighted distance measure (Liu et al., 2014b; Zhou et al., 2013, 2014b; Liao et al., 2014; Xu and Wang, 2011). Yager (1993) introduced the ordered weighted averaging (OWA) operator in 1993; since then, various aggregation operators were proposed and are combined with linguistic information (Meng et al., 2016; Liu et al., 2014a, 2014b; Zhang, 2013).

In addition, the distance measures are uniformly distributed on the corresponding interval variables, they are often utilized to deal with the aggregation information is denoted by exact numbers or defined by endpoints of intervals. Obviously, it varies among group decision making problems under uncertain environment. To solve this problem, Zhou et al. (2013, 2014b, 2016), developed the continuous intuitionistic fuzzy ordered weighted distance (C-IFOWD) measure, the continuous ordered weighted distance (COWD) measure and the linguistic continuous ordered weighted distance (LCOWD) measure, respectively. These three aggregation operators combine the C-IFOWA operator (or COWA operator/the LCOWA operator) with the ordered weighted distance (OWD) measure, considering the risk attribute of decision makers under interval variables environment.

Motivated by the work in Zhou et al. (2013), Zhou et al. (2014b), the aim of this paper is to develop a new distance measure named as interval-valued 2-tuples linguistic induced continuous ordered weighted distance (IT-ICOWD) measure, which is based on the ICOWA operator and the OWD measure with the interval-valued 2-tuples linguistic information. We also study some desirable properties and different families of the IT-ICOWD measure. Additionally, we extend the IT-ICOWD measure and obtain the quasi-IT-ICOWD measure. We also propose a new approach to MAGDM by using the IT-ICOWD measure. By this approach, we obtain promising formulae, which can determine the order-inducing variables in the IT-ICOWD measure, the weighting vector of decision makers and the weighting vector of attributes.

The rest of this paper is structured as follows. We briefly review some fundamental concepts about various relevant operators and measures in Section 2. In Section 3, we present the IT-COWD and the IT-ICOWD measure, and discuss some properties and families of the IT-ICOWD measure. We also develop some extensions of the IT-ICOWD measure. Section 4 provides an approach based on the IT-ICOWD measure for multiple attribute group decision making with interval-valued 2-tuple linguistic information and give a real-life example to illustrate the efficiency of the proposed method. At last, we give some further explanations in Section 5.

 Preliminaries

In this section, we briefly review basic concepts about the 2-tuple linguistic, the OWA operator, the IOWA operator, the GOWA operator, the COWA operator, the ICOWA operator, the distance measure and the OWD measure.

The 2-Tuple Fuzzy Linguistic Representation Model Processing

Zadeh firstly introduced linguistic method in Zadeh (1975a, 1975b, 1975c) as an approximate technique representing qualitative information by means of linguistic labels.

Let S = { s i ∣ i = 0 , 1 , … , g } be a linguistic term set with odd cardinality, each term  s i represents a possible value for a linguistic variable; for example, we can define S as follows: S = { s 0 = neither ( N ) , s 1 = very low ( V L ) , s 2 = low ( L ) , s 3 = medium ( M ) , s 4 = high ( H ) , s 5 = very high ( V H ) , s 6 = perfect ( P ) } , where the mid-linguistic term s 3 represents “approximately 0.5” as an assessment, and the rest of the terms placed symmetrically around it. It should be clarified that term sets should satisfy the following characteristics:

Ordered set: s i > s j ⇔ i > j{s_{j}}\Leftrightarrow i>j$]]>;

Negation operator: Neg ( s i ) = s g − i ( g + 1 is the cardinality);

Minimum operator: min ( s i , s j ) = s i ⇔ s i ⩽ s j ;

Maximum operator: max ( s i , s j ) = s i ⇔ s i ⩾ s j .

According to symbolic translation, Herrera and Martinez (2000) originally proposed the 2-tuple linguistic representation model for dealing with linguistic information. Being continuous in the domain is the primary advantage of this representation. A 2-tuple ( s i , α i ) is a 2-tuple linguistic representation model, where s i is a linguistic label of predefined linguistic term set S and α i is a numerical value representing the value of symbolic translation.

(<italic>See</italic> Herrera and Martinez, <xref ref-type="bibr" rid="j_info1183_ref_009">2000</xref>).

Let S = { s i ∣ i = 0 , 1 , … , g } be a set of finite linguistic terms, and β ∈ [ 0 , g ] is the numerical value demonstrating the result of a symbolic aggregation operation, then the function Δ denotes to obtain the 2-tuple linguistic information being equivalent to β, which can be defined as follows: Δ : [ 0 , g ] → S × [ − 0.5 , 0.5 ) , Δ ( β ) = ( s i , α i ) with s i , i = round ( β ) , α i = β − i , α i ∈ [ − 0.5 , 0.5 ) , where round ( β ) is the usual round operation, s i has the closest index label of β and α i is the value of the symbolic translation.

 (<italic>See</italic> Herrera and Martinez, <xref ref-type="bibr" rid="j_info1183_ref_009">2000</xref>).

Let S = { s i ∣ i = 0 , 1 , … , g } be a linguistic terms collection and ( s i , α i ) be a linguistic 2-tuple term. There is always a function Δ − 1 such that the value which returns from 2-tuple is an equivalent numerical value β ∈ [ 0 , g ], where Δ − 1 : S × [ − 0.5 , 0.5 ) → [ 0 , g ] , Δ − 1 ( s i , α i ) = i + α i = β , and β ∈ [ 0 , g ].

In the past few decades, multiple 2-tuple linguistic aggregation operators have been proposed in order to aggregate 2-tuple linguistic. However, these 2-tuple linguistic aggregation operators all simply focused on the usual 2-tuple. In another word, 2-tuple linguistic from different linguistic term set with different granularities cannot be aggregated directly. To overcome this obstacle, Chen and Tai (2005) put forward a generalized 2-tuple linguistic representation model and translation function.

(<italic>See</italic> Chen and Tai, <xref ref-type="bibr" rid="j_info1183_ref_005">2005</xref>).

Let S = { s i ∣ | i = 0 , 1 , … , g } be an ordered linguistic term set, and crisp value β ∈ [ 0 , 1 ] can be transformed into one 2-tuple linguistic representation model through the following function: Δ : [ 0 , 1 ] → S × [ − 1 2 g , 1 2 g ) , Δ ( β ) = ( s i , α i ) with s i , i = round ( β · g ) ; α i = β − i g , α i ∈ [ − 1 2 g , 1 2 g ) . Conversely, the 2-tuple can be converted into a crisp β ∈ [ 0 , 1 ] as follows: Δ − 1 : S × [ − 1 2 g , 1 2 g ) → [ 0 , 1 ] , Δ − 1 ( s i , α i ) = i g + α i = β , where β ∈ [ 0 , 1 ] according to Definition 3. Therefore, in this method the 2-tuple linguistic representation model is standardized, so it is easier to compare 2-tuple linguistic terms with different multiple granularity linguistic term sets. In this paper, unless mentioned explicitly, the 2-tuple linguistic representation model is generalized 2-tuple representation model in Definition 3.

Considering merits of Definition 3, a new concept of the interval-valued 2-tuple linguistic representation model was introduced by Zhang (2012) with some aggregation operators with interval-valued 2-tuple linguistic information. (<italic>See</italic> Zhang, <xref ref-type="bibr" rid="j_info1183_ref_039">2012</xref>).

Let S = { s i ∣ i = 0 , 1 , … , g } be a set of ordered linguistic terms. An interval-valued 2-tuple consists of two linguistic terms and two numbers, denoted as [ ( s i , α i ) , ( s j , α j ) ], where i ⩽ j, and α i ⩽ α j , s i , s j ∈ S, α i , α j are crisp numbers. The interval-valued 2-tuple that expresses the equivalent information to an interval-value [ β 1 , β 2 ] ( β 1 , β 2 ∈ [ 0 , 1 ] , β 1 ⩽ β 2 ) as follows: Δ ( [ β 1 , β 2 ] ) = [ ( s i , α i ) , ( s j , α j ) ] with s i , i = round ( β 1 · g ) ; s j , j = round ( β 2 · g ) ; α i = β 1 − i g , α i ∈ [ − 1 2 g , 1 2 g ) ; α j = β 2 − j g , α j ∈ [ − 1 2 g , 1 2 g ) .

There always exists the inverse function Δ − 1 satisfying that for each interval-valued 2-tuple, it returns corresponding interval value [ β 1 , β 2 ] ( β 1 , β 2 ∈ [ 0 , 1 ] , β 1 ⩽ β 2 ) as follows: Δ − 1 ( [ ( s i , α i ) , ( s j , α j ) ] ) = [ i g + α i , j g + α j ] = [ β 1 , β 2 ] .

(<italic>See</italic> Zhang, <xref ref-type="bibr" rid="j_info1183_ref_039">2012</xref>).

If [ ( s i , α i ) , ( s j , α j ) ] and [ ( s k , α k ) , ( s t , α t ) ] are any two interval-valued 2-tuple linguistic information, l , l 1 , l 2 ∈ [ 0 , 1 ], then basic operational laws can be defined as follows:

[ ( s i , α i ) , ( s j , α j ) ] ⊕ [ ( s k , α k ) , ( s t , α t ) ] = [ Δ ( min { Δ − 1 ( s i , α i ) + Δ − 1 ( s k , α k ) , 1 } ) , Δ ( min { Δ − 1 ( s j , α j ) + Δ − 1 ( s t , α t ) , 1 } ) ] ; 2 ) l ⊗ [ ( s i , α i ) , ( s j , α j ) ] = [ Δ ( l Δ − 1 ( s i , α i ) ) , Δ ( l Δ − 1 ( s j , α j ) ) ] ;

( l 1 ⊗ [ ( s i , α i ) , ( s j , α j ) ] ) ⊕ ( l 2 ⊗ [ ( s i , α i ) , ( s j , α j ) ] ) = ( l 1 + l 2 ) ⊗ [ ( s i , α i ) , ( s j , α j ) ] ;

l ⊗ ( [ ( s i , α i ) , ( s j , α j ) ] ⊕ [ ( s k , α k ) , ( s t , α t ) ] ) = ( l ⊗ [ ( s i , α i ) , ( s j , α j ) ] ) ⊕ ( l ⊗ [ ( s k , α k ) , ( s t , α t ) ] ) .

Aiming to compare two interval-valued 2-tuple linguistic terms, Zhang (2012) proposed the concept of the score and accuracy.

 (<italic>See</italic> Zhang, <xref ref-type="bibr" rid="j_info1183_ref_039">2012</xref>).

For an interval-valued 2-tuple A ˜ = [ ( s i , α i ) , ( s j , α j ) ], its score function can be defined as ( A ˜ ) = i + j 2 g + α i + α j 2 , and the accuracy function can be defined as H ( A ˜ ) = j − i g + α j − α i , where S = { s i ∣ i = 0 , 1 , … , g } is an ordered linguistic term set with g + 1 linguistic labels. Obviously, 0 ⩽ S ( A ) ⩽ 1, and 0 ⩽ H ( A ) ⩽ 1.

 (<italic>See</italic> Zhang, <xref ref-type="bibr" rid="j_info1183_ref_039">2012</xref>).

Let A ˜ = [ ( s i , α i ) , ( s j , α j ) ] and B ˜ = [ ( s k , α k ) , ( s t , α t ) ] be two interval-valued linguistic 2-tuples. It follows that:

If S ( A ˜ ) < S ( B ˜ ), then A ˜ < B ˜;

If S ( A ˜ ) = S ( B ˜ ), then: if H ( A ˜ ) < H ( B ˜ ) , then A ˜ > B ˜ ; if H ( A ˜ ) = H ( B ˜ ) , then A ˜ = B ˜ . \tilde{B};\\ {} \displaystyle \text{if}\hspace{2.5pt}H(\tilde{A})=H(\tilde{B}),\hspace{1em}\text{then}\hspace{2.5pt}\tilde{A}=\tilde{B}.\end{array}\]]]>

The OWA Operator, the IOWA Operator and the GOWA Operator

The OWA operator (Yager, 1988) provides a parameterized family of aggregation operators including the maximum, the minimum, and the average.

(<italic>See</italic> Yager, <xref ref-type="bibr" rid="j_info1183_ref_030">1988</xref>).

The n-dimensional OWA operator is a mapping OWA : R n → R , where W = ( w 1 , w 2 , … , w n ) is a weighting vector, ∑ j = 1 n w j = 1 and w j ∈ [ 0 , 1 ] satisfying OWA ( a 1 , a 2 , … , a n ) = ∑ j = 1 n w j b j , where b j is the j-th largest value among arguments a 1 , a 2 , … , a n .

The IOWA operator (Yager and Filev, 1999) can be regarded as extension of the OWA operator.

 (<italic>See</italic> Yager and Filev, <xref ref-type="bibr" rid="j_info1183_ref_034">1999</xref>).

An n-dimensional IOWA operator is a mapping IOWA : R n × R n → R , where W = ( w 1 , w 2 , … , w n ) is a weighting vector, ∑ j = 1 n w j = 1 and w j ∈ [ 0 , 1 ] satisfying IOWA ( ⟨ v 1 , a 1 ⟩ , ⟨ v 2 , a 2 ⟩ , … , ⟨ v n , a n ⟩ ) = ∑ j = 1 n w j a σ ( j ) , where a σ ( j ) is the a i value of the IOWA pair ⟨ v i , a i ⟩ having the j-th largest v i .

Yager (2004a) proposed the generalized ordered weighted averaging (GOWA) operator.

 (<italic>See</italic> Yager, <xref ref-type="bibr" rid="j_info1183_ref_032">2004a</xref>).

An n-dimensional GOWA operator is a mapping GOWA : R n → R where W = ( w 1 , w 2 , … , w n ) is a weighting vector, ∑ j = 1 n w j = 1 and w j ∈ [ 0 , 1 ] satisfying GOWA ( a 1 , a 2 , … , a n ) = ( ∑ j = 1 n w j b j λ ) 1 / λ , where λ ∈ ( − ∞ , ∞ ), λ ≠ 0, b j is the j-th largest of the arguments among a 1 , a 2 , … , a n .

A group of special cases can be obtained through giving different values to parameter λ in the GOWA operator.

 The <italic>COWA</italic> Operator and the <italic>ICOWA</italic> Operator

To deal with the case that the given argument is a continuous valued interval, Yager (2004b) introduced continuous ordered weighted averaging (COWA) operator.

(<italic>See</italic> Yager, <xref ref-type="bibr" rid="j_info1183_ref_033">2004b</xref>).

A COWA operator is a mapping F : Θ → R + F Q ( a ˜ ) = F Q ( [ a L , a U ] ) = ∫ 0 1 d Q ( y ) d y ( a U − y ( a U − a L ) ) d y , where Θ is the set of all nonnegative interval arguments and a ˜ = [ a L , a U ] ∈ Θ, and Q is a basic unit monotonic (BUM) function.

Let λ = ∫ 0 1 Q ( y ) d y be the attitudinal character of Q, then the general formulation of F Q ( a ˜ ) is as follows: F Q ( a ˜ ) = F Q ( [ a L , a U ] ) = λ a U + ( 1 − λ ) a L .

As we can see, the COWA operator F Q ( a ˜ ) is the weighted arithmetical mean of end points according to the attitudinal character. The interval a ˜ = [ a L , a U ] can be replaced by F Q ( a ˜ ).

In Zhou et al. (2010) proposed induced continuous OWA (ICOWA) operator.

(<italic>See</italic> Zhou <italic>et al.</italic>, <xref ref-type="bibr" rid="j_info1183_ref_041">2010</xref>).

Let [ a L 1 , a U 1 ] , [ a L 2 , a U 2 ] , … , [ a L n , a U n ] be an interval number set. An ICOWA operator is a mapping ICOWA: R n × Θ n → R + satisfying ICOWA ( ⟨ u 1 , [ a L 1 , a U 1 ] ⟩ , ⟨ u 2 , [ a L 2 , a U 2 ] ⟩ , … , ⟨ u n , [ a L n , a U n ] ⟩ ) = ICOWA ( ⟨ u 1 , F Q ( [ a L 1 , a U 1 ] ) ⟩ , ⟨ u 2 , F Q ( [ a L 2 , a U 2 ] ) ⟩ , … , ⟨ u n , F Q ( [ a L n , a U n ] ) ⟩ ) = ∑ j = 1 n w j F Q ( [ a L δ ( j ) , a U δ ( j ) ] ) , where W = ( w 1 , w 2 , … , w n ) is an associated weighting vector with ∑ j = 1 n w j = 1, δ ( 1 ) , δ ( 2 ) , … , δ ( n ) is any permutation of ( 1 , 2 , … , n ), ( u 1 , u 2 , … , u n ) is a set of order inducing variables satisfying u δ ( j − 1 ) ⩾ u δ ( j ) , j = 2 , 3 , … , n , and F Q ( [ a L δ ( j ) , a U δ ( j ) ] is the F Q ( [ a L i , a U i ] value of the ICOWA pair ⟨ u i , [ a L i , a U i ] ⟩ having the j-th largest u i , F Q ( [ a L i , a U i ] is calculated by Eq. (17).

Similarly, we can obtain the interval-valued 2-tuple linguistic continuous OWA (IT-COWA) operator and the interval-valued 2-tuple linguistic induced continuous OWA (IT-ICOWA) operator. Let Ω be the set of all interval-valued 2-tuple linguistic information.

If A ˜ = [ ( s i , α i ) , ( s j , α j ) ] ∈ Ω, and f Q ( A ˜ ) = f Q ( [ ( s i , α i ) , ( s j , α j ) ] ) = ( s k , α k ), where Δ − 1 ( s k , α k ) = F Q ( [ Δ − 1 ( s i , α i ) , Δ − 1 ( s j , α j ) ] ) = ∫ 0 1 d Q ( y ) d y ( Δ − 1 ( s j , α j ) − y ( Δ − 1 ( s j , α j ) − Δ − 1 ( s i , α i ) ) ) d y , then f is the interval-valued 2-tuples linguistic continuous OWA (IT-COWA) operator, where Q is the BUM function.

Especially, if λ = ∫ 0 1 Q ( y ) d y is the attitudinal character of Q, then the IT-COWA operator can be rewritten as follows: f Q ( A ˜ ) = Δ ( F Q ( [ Δ − 1 ( s i , α i ) , Δ − 1 ( s j , α j ) ] ) ) = Δ ( λ Δ − 1 ( s j , α j ) + ( 1 − λ ) Δ − 1 ( s i , α i ) ) ,

Based on Eq. (20), we can see that the IT-COWA operator may be determined by the attitudinal character λ. For convenience, f λ ( A ˜ ) denotes f Q ( A ˜ ), i.e. f λ ( A ˜ ) = Δ ( F Q ( [ Δ − 1 ( s i , α i ) , Δ − 1 ( s j , α j ) ] ) ) = Δ ( λ Δ − 1 ( s j , α j ) + ( 1 − λ ) Δ − 1 ( s i , α i ) ) .

An n-dimensional IT-ICOWA measure is a mapping IT-ICOWA R n × Ω n → Ω , which is associated weighting vector W = ( w 1 , w 2 , … , w n ), ∑ j = 1 n w j = 1 and w j ∈ [ 0 , 1 ], satisfying IT - ICOWA ( ⟨ u 1 , [ ( s 1 , α 1 ) , ( s 1 ′ , α 1 ′ ) ] ⟩ , … , ⟨ u n , [ ( s n , α n ) , ( s n ′ , α n ′ ) ] ⟩ ) = IT - ICOWA ( ⟨ u 1 , f λ ( [ ( s 1 , α 1 ) , ( s 1 ′ , α 1 ′ ) ] ) ⟩ , … , ⟨ u n , f λ ( [ ( s n , α n ) , ( s n ′ , α n ′ ) ] ) ⟩ ) = Δ ( ∑ j = 1 n w j Δ − 1 ( f λ ( [ ( s δ ( j ) , α δ ( j ) ) , ( s δ ( j ) ′ , α δ ( j ) ′ ) ] ) ) ) , where δ ( 1 ) , δ ( 2 ) , … , δ ( n ) is any permutation of ( 1 , 2 , … , n ), ( u 1 , u 2 , … , u n ) is a set of order inducing variables, such that u δ ( j − 1 ) ⩾ u δ ( j ) , j = 2 , 3 , … , n , and f λ ( [ ( s δ ( j ) , α δ ( j ) ) , ( s δ ( j ) ′ , α δ ( j ) ′ ) ] ) is the f λ ( [ ( s i , α i ) , ( s i ′ , α i ′ ) ] ) value of the IT-ICOWA pair ⟨ u i , ( s i , α i ) , ( s i ′ , α i ′ ) ⟩ with the j-th largest u i , where f λ ( [ ( s i , α i ) , ( s i ′ , α i ′ ) ] ) can be determined by Eq. (21).

 Distance Measure (<italic>See</italic> Zhou <italic>et al.</italic>, <xref ref-type="bibr" rid="j_info1183_ref_042">2013</xref>, <xref ref-type="bibr" rid="j_info1183_ref_044">2014b</xref>).

Let A 1 , A 2 , A 3 be elements of a set. A distance measure D should satisfy properties as follows:

Nonnegativity: D ( A 1 , A 2 ) ⩾ 0;

Commutativity: D ( A 1 , A 2 ) = D ( A 2 , A 1 );

Reflexivity: D ( A 1 , A 1 ) = 0;

Triangle inequality: D ( A 1 , A 2 ) + D ( A 1 , A 3 ) ⩾ D ( A 2 , A 3 ).

 Note that different D can derive different types of distance measures.

On the basis of Liu et al. (2014b), Zhou et al. (2014b), Li et al. (2014), Xu (2005), now we present several interval-valued linguistic distance measures including the interval-valued linguistic distance measure, the continuous interval-valued linguistic distance measure and the interval-valued 2-tuple linguistic distance measure.

Interval-valued linguistic distance measure: d ( A 1 , A 2 ) = d ( [ s i , s j ] , [ s k , s t ] ) = s 1 g ( | i − k | + | j − t | 2 ) , where [ s i , s j ] and [ s k , s t ] are two interval-valued linguistic variables, respectively.

Continuous interval-valued linguistic distance measure: d ′ ( A 1 , A 2 ) = d ′ ( [ s i , s j ] , [ s k , s t ] ) = s | F Q ( [ i , j ] ) − F Q ( [ k , t ] ) | g = s | λ j + ( 1 − λ ) i − ( λ t + ( 1 − λ ) k ) | g , where Q is the BUM function, λ = ∫ 0 1 Q ( y ) d y.

Interval-valued 2-tuple linguistic distance measure: d ″ ( A ˜ 1 , A ˜ 2 ) = d ″ ( [ ( s i , α i ) , ( s j , α j ) ] , [ ( s k , α k ) , ( s t , α t ) ] ) = Δ ( | Δ − 1 ( s i , α i ) − Δ − 1 ( s k , α k ) | + | Δ − 1 ( s j , α j ) − Δ − 1 ( s t , α t ) | 2 ) , where [ ( s i , α i ) , ( s j , α j ) ] and [ ( s k , α k ) , ( s t , α t ) ] are two interval-valued 2-tuple linguistic information, respectively.

The Ordered Weighted Distance Measure

Xu and Chen (2008) developed the ordered weighted distance (OWD) measure.

(<italic>See</italic> Xu and Chen, <xref ref-type="bibr" rid="j_info1183_ref_028">2008</xref>).

An n-dimensional OWD measure is a mapping OWD: R + n × R + n → R + satisfying: OWD ( α , β ) = ( ∑ j = 1 n w j ( d ( a σ ( j ) , b σ ( j ) ) ) λ ) 1 / λ , where W = ( w 1 , w 2 , … , w n ) is associated with weighting vector, ∑ j = 1 n w j = 1 and w j ∈ [ 0 , 1 ], σ ( 1 ) , σ ( 2 ) , … , σ ( n ) is any permutation of ( 1 , 2 , … , n ), d ( a σ ( j − 1 ) , b σ ( j − 1 ) ) ⩾ d ( a σ ( j ) , b σ ( j ) ), j = 1 , 2 , … , n , d ( a j , b j ) = | a j − b j | is the distance of a j and b j . α = ( a 1 , a 2 , … , a n ) and β = ( b 1 , b 2 , … , b n ) are two collections of arguments, and λ > 00$]]>.

The OWD measure is monotonic, commutative, idempotent and bounded. We can obtain a group of special cases when considering as many as values of the parameter λ in the OWD measure. For instance, the ordered weighted Hamming distance (OWHD) measure, the ordered weighted Euclidean distance (OWED) measure and the ordered weighted Geometric (OWGD) measure can be determined in the following way:

The OWHD measure is obtained if λ = 1.

The OWED measure is obtained if λ = 2.

The OWGD measure is obtained if λ → 0.

The Interval-Valued 2-Tuples Linguistic Induced Continuous Ordered Weighted Distance Measure

In this section, we introduce the interval-valued 2-tuples linguistic continuous ordered weighted distance (IT-COWD) measure and the interval-valued 2-tuples linguistic induced continuous ordered weighted distance (IT-ICOWD) measure.

The <italic>IT</italic>-<italic>COWD</italic> Measure and the <italic>IT</italic>-<italic>ICOWD</italic> Measure

Let A ˜ 1 = [ ( s 1 , α 1 ) , ( s 1 ′ , α 1 ′ ) ] ∈ Ω and A ˜ 2 = [ ( s 2 , α 2 ) , ( s 2 ′ , α 2 ′ ) ] ∈ Ω. If d λ ( A ˜ 1 , A ˜ 2 ) = Δ ( | Δ − 1 ( f λ ( A ˜ 1 ) ) − Δ − 1 ( f λ ( A ˜ 2 ) ) | ) , then d λ ( A ˜ 1 , A ˜ 2 ) is called the distance between A ˜ 1 and A ˜ 2 based on the IT-COWA operator, where f λ ( A ˜ 1 ) and f λ ( A ˜ 2 ) can be calculated by Eq. (21).

According to Eq. (27), d λ ( A ˜ 1 , A ˜ 2 ) can be defined as follows: d λ ( A ˜ 1 , A ˜ 2 ) = Δ ( | Δ − 1 ( f λ ( A ˜ 1 ) ) − Δ − 1 ( f λ ( A ˜ 2 ) ) | ) = Δ ( | λ Δ − 1 ( s 1 ′ , α 1 ′ ) + ( 1 − λ ) Δ − 1 ( s 1 , α 1 ) − ( λ Δ − 1 ( s 2 ′ , α 2 ′ ) + ( 1 − λ ) Δ − 1 ( s 2 , α 2 ) ) | ) where λ = ∫ 0 1 Q ( y ) d y is the attitudinal character of Q.

Let S 7 = { s i 7 ∣ i = 0 , 1 , … , 6 } be a linguistic term set, A 1 = [ s 2 7 , s 5 7 ] and A 2 = [ s 3 7 , s 4 7 ] be two interval-valued linguistic variables, and Q ( y ) = y 3 , so λ = ∫ 0 1 Q ( y ) d y = 1 4 . Therefore, A 1 , A 2 can be rewritten as interval-valued 2-tuple linguistic information A ˜ 1 = [ ( s 2 7 , 0 ) , ( s 5 7 , 0 ) ] and A ˜ 2 = [ ( s 3 7 , 0 ) , ( s 4 7 , 0 ) ]. By Eq. (23)–(25), (28), we have

d ( A 1 , A 2 ) = d ( [ s 2 7 , s 5 7 ] , [ s 3 7 , s 4 7 ] ) = s 1 6 ( | 2 − 3 | + | 5 − 4 | 2 ) = s 1 6 ,

d ′ ( A 1 , A 2 ) = d ′ ( [ s 2 7 , s 5 7 ] , [ s 3 7 , s 4 7 ] ) = s | 5 4 + ( 1 − 1 4 ) 2 − 4 4 − ( 1 − 1 4 ) 3 ) | 6 = s 1 12 ,

d ″ ( A ˜ 1 , A ˜ 2 ) = d ″ ( [ ( s 2 7 , 0 ) , ( s 5 7 , 0 ) ] , [ ( s 3 7 , 0 ) , ( s 4 7 , 0 ) ] ) = Δ ( | Δ − 1 ( s 2 7 , 0 ) − Δ − 1 ( s 3 7 , 0 ) | + | Δ − 1 ( s 5 7 , 0 ) − Δ − 1 ( s 4 7 , 0 ) | 2 ) = ( s 1 7 , 0 ) ,

d λ ( A ˜ 1 , A ˜ 2 ) = d λ ( [ ( s 2 7 , 0 ) , ( s 5 7 , 0 ) ] , [ ( s 3 7 , 0 ) , ( s 4 7 , 0 ) ] ) = Δ ( | 1 4 Δ − 1 ( s 5 7 , 0 ) + ( 1 − 1 4 ) Δ − 1 ( s 2 7 , 0 ) − 1 4 Δ − 1 ( s 4 7 , 0 ) − ( 1 − 1 4 ) Δ − 1 ( s 3 7 , 0 ) ) | ) = ( s 1 7 , − 1 12 ) .

As we can see, the results do not match any of the initial linguistic terms in the aggregation process of the distance measure d ( A 1 , A 2 ) and d ′ ( A 1 , A 2 ), but these two distance measures can express the results in the initial expression domain. Moreover, they are able to deal with the situations where the input arguments are represented with interval-valued 2-tuples linguistic information.

From Definition 17, we can get the following theorems:

If A ˜ 1 = [ ( s 1 , α 1 ) , ( s 1 ′ , α 1 ′ ) ] ∈ Ω, A ˜ 2 = [ ( s 2 , α 2 ) , ( s 2 ′ , α 2 ′ ) ] ∈ Ω and A ˜ 3 = [ ( s 3 , α 3 ) , ( s 3 ′ , α 3 ′ ) ] ∈ Ω, then

Nonnegativity: d λ ( A ˜ 1 , A ˜ 2 ) ⩾ 0;

Commutativity: d λ ( A ˜ 1 , A ˜ 2 ) = d λ ( A ˜ 2 , A ˜ 1 );

Reflexivity: d λ ( A ˜ 1 , A ˜ 1 ) = 0;

Triangle inequality: d λ ( A ˜ 1 , A ˜ 2 ) + d λ ( A ˜ 1 , A ˜ 3 ) ⩾ d λ ( A ˜ 2 , A ˜ 3 ).

If A ˜ 1 = [ ( s 1 , α 1 ) , ( s 1 ′ , α 1 ′ ) ] ∈ Ω, A ˜ 2 = [ ( s 2 , α 2 ) , ( s 2 ′ , α 2 ′ ) ] ∈ Ω, then Δ − 1 ( d λ ( A ˜ 1 , A ˜ 2 ) ) ⩽ 1.

The proofs of theorems are straightforward, thus omitted.

Suppose that A ˜ = ( A ˜ 1 , A ˜ 2 , … , A ˜ n ) ∈ Ω n and B ˜ = ( B ˜ 1 , B ˜ 2 , … , B ˜ n ) ∈ Ω n , we can define the interval-valued 2-tuple linguistic continuous ordered weighted distance (IT-COWD) measure as follows:

An n-dimensional IT-COWD measure is a mapping IT - COWD : Ω n × Ω n → Ω , satisfying: IT - COWD ( A ˜ , B ˜ ) = Δ ( ( ∑ j = 1 n w j [ Δ − 1 ( d λ ( A ˜ σ ( j ) , B ˜ σ ( j ) ) ) ] τ ) 1 / τ ) , where σ ( 1 ) , σ ( 2 ) , … , σ ( n ) is any permutation of ( 1 , 2 , … , n ) such that d λ ( A ˜ σ ( j − 1 ) , B ˜ σ ( j − 1 ) ) ⩾ d λ ( A ˜ σ ( j ) , B ˜ σ ( j ) ), j = 2 , 3 , … , n , and d λ ( A ˜ j , B ˜ j ) is the distance between A ˜ j and B ˜ j based on the IT-COWA operator and the parameter τ > 00$]]>, W = ( w 1 , w 2 , … , w n ) is associated with vector, ∑ j = 1 n w j = 1 and w j ∈ [ 0 , 1 ].

Here we present a simple numerical example showing how to use the IT-COWD measure in an aggregation process.

Let A ˜ = ( A ˜ 1 , A ˜ 2 , A ˜ 3 , A ˜ 4 ) = ( [ ( s 3 7 , 0.05 ) , ( s 5 7 , 0.01 ) ] , [ ( s 2 7 , 0.02 ) , ( s 4 7 , − 0.06 ) ] , [ ( s 4 7 , − 0.04 ) , ( s 5 7 , − 0.07 ) ] , [ ( s 1 7 , − 0.02 ) , ( s 2 7 , 0.03 ) ] ) and B ˜ = ( B ˜ 1 , B ˜ 2 , B ˜ 3 , B ˜ 4 ) = ( [ ( s 4 7 , 0.02 ) , ( s 5 7 , − 0.06 ) ] , [ ( s 2 7 , 0.04 ) , ( s 3 7 , 0.03 ) ] , [ ( s 4 7 , − 0.02 ) , ( s 6 7 , − 0.08 ) ] , [ ( s 1 7 , − 0.05 ) , ( s 2 7 , 0.01 ) ] ) be two collections of interval-valued 2-tuples linguistic information, and Q ( y ) = y 3 , so λ = ∫ 0 1 Q ( y ) d y = 1 4 . From Eq. (28), we have d λ ( A ˜ 1 , B ˜ 1 ) = Δ ( | 1 4 Δ − 1 ( s 5 7 , 0.01 ) + ( 1 − 1 4 ) Δ − 1 ( s 3 7 , 0.05 ) − ( 1 4 Δ − 1 ( s 5 7 , − 0.06 ) + ( 1 − 1 4 ) Δ − 1 ( s 4 7 , 0.02 ) ) | ) = Δ ( 0.0875 ) = ( s 1 7 , − 0.0792 ) , d λ ( A ˜ 2 , B ˜ 2 ) = Δ ( 0.005 ) = ( s 0 7 , 0.005 ) , d λ ( A ˜ 3 , B ˜ 3 ) = Δ ( 0.055 ) = ( s 0 7 , 0.055 ) , d λ ( A ˜ 4 , B ˜ 4 ) = Δ ( 0.01 ) = ( s 0 7 , 0.01 ) . So, d l ( A ˜ s ( 1 ) , B ˜ s ( 1 ) ) = ( s 1 7 , − 0.0792 ) , d l ( A ˜ s ( 2 ) , B ˜ s ( 2 ) ) = ( s 0 7 , 0.055 ) , d l ( A ˜ s ( 3 ) , B ˜ s ( 3 ) ) = ( s 0 7 , 0.01 ) , d l ( A ˜ s ( 4 ) , B ˜ s ( 4 ) ) = ( s 0 7 , 0.005 ) .

Let W = ( 0.3 , 0.2 , 0.4 , 0.1 ). By Eq. (29), we determine the distances of τ, which are shown in Table 1.

Aggregation result.

τ →0+ 1 2
IT-COWD ( A ˜ , B ˜ ) ( s 0 7 , 0.0252 ) ( s 0 7 , 0.0418 ) ( s 0 7 , 0.0543 )
τ 3 4 5
IT-COWD ( A ˜ , B ˜ ) ( s 0 7 , 0.0617 ) ( s 0 7 , 0.0664 ) ( s 0 7 , 0.0697 )
τ 6 7 8
IT-COWD ( A ˜ , B ˜ ) ( s 0 7 , 0.0682 ) ( s 0 7 , 0.0739 ) ( s 0 7 , 0.0754 )
τ 9 10 15
IT-COWD ( A ˜ , B ˜ ) ( s 0 7 , 0.0766 ) ( s 0 7 , 0.0776 ) ( s 0 7 , 0.0808 )

From Table 1, it is demonstrated that the aggregation result IT-COWD ( A ˜ , B ˜ ) increases as the parameter τ steadily increases.

We define the interval-valued 2-tuples linguistic induced continuous ordered weighted distance (IT-ICOWD) measure:

An n-dimensional IT-ICOWD measure is a mapping IT − ICOWD : R n × Ω n × Ω n → Ω , which is associated with the weighting vector W = ( w 1 , w 2 , … , w n ), ∑ j = 1 n w j = 1 and w j ∈ [ 0 , 1 ], satisfying: IT - ICOWD ( ⟨ u 1 , A ˜ 1 , B ˜ 1 ⟩ , ⟨ u 2 , A ˜ 2 , B ˜ 2 ⟩ , … , ⟨ u n , A ˜ n , B ˜ n ⟩ ) = Δ ( [ ∑ j = 1 n w j [ Δ − 1 ( d λ ( A ˜ δ ( j ) , B ˜ δ ( j ) ) ) ] τ ] 1 / τ ) , where δ ( 1 ) , δ ( 2 ) , … , δ ( n )is any permutation of ( 1 , 2 , … , n ), ( u 1 , u 2 , … , u n ) is a set of order inducing variables, such that u δ ( j − 1 ) ⩾ u δ ( j ) , j = 2 , 3 , … , n, and d λ ( A ˜ δ ( j ) , B ˜ δ ( j ) ) is the d λ ( A ˜ i , B ˜ i ) value of the IT-ICOWD pair ⟨ u i , A ˜ i , B ˜ i ⟩ having the j t h largest u i , d λ ( A ˜ j , B ˜ j ) can be calculated by Eq. (28) and the parameter τ > 00$]]>.

In the following, we present a simple numerical example showing how to use the IT-ICOWD measure in an aggregation process.

Let A ˜ = ( A ˜ 1 , A ˜ 2 , A ˜ 3 , A ˜ 4 ) = ( [ ( s 3 7 , 0.05 ) , ( s 5 7 , 0.01 ) ] , [ ( s 2 7 , 0.02 ) , ( s 4 7 , − 0.06 ) ] , [ ( s 4 7 , − 0.04 ) , ( s 5 7 , − 0.07 ) ] , [ ( s 1 7 , − 0.02 ) , ( s 2 7 , 0.03 ) ] ) and B ˜ = ( B ˜ 1 , B ˜ 2 , B ˜ 3 , B ˜ 4 ) = ( [ ( s 4 7 , 0.02 ) , ( s 5 7 , − 0.06 ) ] , [ ( s 2 7 , 0.04 ) , ( s 3 7 , 0.03 ) ] , [ ( s 4 7 , − 0.02 ) , ( s 6 7 , − 0.08 ) ] , [ ( s 1 7 , − 0.05 ) , ( s 2 7 , 0.01 ) ] ) be two interval-valued 2-tuples linguistic information collections, and Q ( y ) = y 2 , so λ = ∫ 0 1 Q ( y ) d y = 1 3 . From Eq. (28), we have d λ ( A ˜ 1 , B ˜ 1 ) = Δ ( | 1 3 Δ − 1 ( s 5 7 , 0.01 ) + ( 1 − 1 3 ) Δ − 1 ( s 3 7 , 0.05 ) − ( 1 3 Δ − 1 ( s 5 7 , − 0.06 ) + ( 1 − 1 3 ) Δ − 1 ( s 4 7 , 0.02 ) ) | ) = Δ ( 0.07 ) = ( s 0 7 , 0.07 ) , d λ ( A ˜ 2 , B ˜ 2 ) = ( s 0 7 , 0.0133 ) , d λ ( A ˜ 3 , B ˜ 3 ) = ( s 0 7 , 0.0667 ) , d λ ( A ˜ 4 , B ˜ 4 ) = ( s 0 7 , 0.0067 ) .

Let both sets be of the same order inducing variables: ( u 1 , u 2 , u 3 , u 4 ) = ( 5 , 1 , 7 , 3 ). Thus, d λ ( A ˜ δ ( 1 ) , B ˜ δ ( 1 ) ) = ( s 0 7 , 0.0667 ) , d λ ( A ˜ δ ( 2 ) , B ˜ δ ( 2 ) ) = ( s 0 7 , 0.07 ) , d λ ( A ˜ δ ( 3 ) , B ˜ δ ( 3 ) ) = ( s 0 7 , 0.0067 ) , d λ ( A ˜ δ ( 4 ) , B ˜ δ ( 4 ) ) = ( s 0 7 , 0.0133 ) .

Let W = ( 0.3 , 0.2 , 0.4 , 0.1 ). By Eq. (30), we have the distances of parameter τ, which are shown in Table 2.

Aggregation result.

τ → 0 1 2
IT-ICOWD ( A ˜ , B ˜ ) ( s 0 7 , 0.0228 ) ( s 0 7 , 0.038 ) ( s 0 7 , 0.0485 )
IT-IOWD ( A ˜ , B ˜ ) ( s 0 7 , 0.048 ) ( s 0 7 , 0.061 ) ( s 0 7 , 0.071 )
τ 3 4 5
IT-ICOWD ( A ˜ , B ˜ ) ( s 0 7 , 0.054 ) ( s 0 7 , 0.0572 ) ( s 0 7 , 0.0593 )
IT-IOWD ( A ˜ , B ˜ ) ( s 0 7 , 0.0775 ) ( s 0 7 , 0.0818 ) ( s 1 7 , − 0.082 )
τ 6 7 8
IT-ICOWD ( A ˜ , B ˜ ) ( s 0 7 , 0.0607 ) ( s 0 7 , 0.0617 ) ( s , 0.0625 )
IT-IOWD ( A ˜ , B ˜ ) ( s 1 7 , − 0.0798 ) ( s 1 7 , − 0.0782 ) ( s 1 7 , − 0.0768 )
τ 9 10 15
IT-ICOWD ( A ˜ , B ˜ ) ( s 0 7 , 0.0631 ) ( s 0 7 , 0.0636 ) ( s 0 7 , 0.0652 )
IT-IOWD ( A ˜ , B ˜ ) ( s 1 7 , − 0.0757 ) ( s 1 7 , − 0.0748 ) ( s 1 7 , − 0.0715 )

From Table 2, the aggregation result IT - ICOWD ( A ˜ , B ˜ ) increases as the parameter τ steadily increases.

 Properties of the <italic>IT</italic>-<italic>ICOWD</italic> Measure

The IT-ICOWD measure is of desirable properties. Now we discuss them through following theorems.

(Monotonicity-distance measure).

If A ˜ = ( A ˜ 1 , A ˜ 2 , … , A ˜ n ) ∈ Ω n , B ˜ = ( B ˜ 1 , B ˜ 2 , … , B ˜ n ) ∈ Ω n , C = ( C ˜ 1 , C ˜ 2 , … , C ˜ n ) ∈ Ω n , and d λ ( A ˜ j , B ˜ j ) ⩽ d λ ( A ˜ j , C ˜ j ) for all j, then IT - ICOWD ( A ˜ , B ˜ ) ⩽ IT - ICOWD ( A ˜ , C ˜ ) .

(Monotonicity-parameter <italic>τ</italic>).

If A ˜ = ( A ˜ 1 , A ˜ 2 , … , A ˜ n ) ∈ Ω n , B ˜ = ( B ˜ 1 , B ˜ 2 , … , B ˜ n ) ∈ Ω n , and τ 1 ⩽ τ 2 , then IT - ICOWD τ 1 ( A ˜ , B ˜ ) ⩽ IT - ICOWD τ 2 ( A ˜ , B ˜ ) .

(Idempotency).

If A ˜ = ( A ˜ 1 , A ˜ 2 , … , A ˜ n ) ∈ Ω n , B ˜ = ( B ˜ 1 , B ˜ 2 , … , B ˜ n ) ∈ Ω n , and d λ ( A ˜ j , B ˜ j ) = d for all j, then IT - ICOWD ( A ˜ , B ˜ ) = d .

(Boundedness).

If A ˜ = ( A ˜ 1 , A ˜ 2 , … , A ˜ n ) ∈ Ω n , B ˜ = ( B ˜ 1 , B ˜ 2 , … , B ˜ n ) ∈ Ω n , and max j d λ ( A ˜ j , B ˜ j ) = d max , min j d λ ( A ˜ j , B ˜ j ) = d min , then d min ⩽ IT - ICOWD ( A ˜ , B ˜ ) ⩽ d max .

(Commutativity-<italic>GOWA</italic> aggregation).

If A ˜ = ( A ˜ 1 , A ˜ 2 , … , A ˜ n ) ∈ Ω n , B ˜ = ( B ˜ 1 , B ˜ 2 , … , B ˜ n ) ∈ Ω n , and ( ( A ˆ 1 , B ˆ 1 ) , ( A ˆ 2 , B ˆ 2 ) , … , ( A ˆ n , B ˆ n ) ) is a permutation of ( ( A ˜ 1 , B ˜ 1 ) , ( A ˜ 2 , B ˜ 2 ) , … , ( A ˜ n , B ˜ n ) ), then IT - ICOWD ( A ˆ , B ˆ ) = IT - ICOWD ( A ˜ , B ˜ ) , where A ˆ = ( A ˆ 1 , A ˆ 2 , … , A ˆ n ) ∈ Ω n , B ˆ = ( B ˆ 1 , B ˆ 2 , … , B ˆ n ) ∈ Ω n .

 (Commutativity-distance measure).

If A ˜ = ( A ˜ 1 , A ˜ 2 , … , A ˜ n ) ∈ Ω n , B ˜ = ( B ˜ 1 , B ˜ 2 , … , B ˜ n ) ∈ Ω n , then IT - ICOWD ( A ˜ , B ˜ ) = IT - ICOWD ( B ˜ , A ˜ ) .

(Nonnegativity).

If A ˜ = ( A ˜ 1 , A ˜ 2 , … , A ˜ n ) ∈ Ω n , B ˜ = ( B ˜ 1 , B ˜ 2 , … , B ˜ n ) ∈ Ω n , then IT - ICOWD ( A ˜ , B ˜ ) ⩾ 0 .

(Reflexivity).

If A ˜ = ( A ˜ 1 , A ˜ 2 , … , A ˜ n ) ∈ Ω n , then IT - ICOWD ( A ˜ , A ˜ ) = 0 .

The proofs of theorems above are straightforward, thus omitted.

 Families of the <italic>IT</italic>-<italic>ICOWD</italic> Measure

Now we discuss families of the IT-ICOWD measure by using different τ and weighting vectors to get different types of distance measure.

If τ = 1, the IT-ICOWD measure reduces to the IT-ICOWHD measure: IT - ICOWHD ( A ˜ , B ˜ ) = Δ ( ∑ j = 1 n w j [ Δ − 1 ( d λ ( A ˜ δ ( j ) , B ˜ δ ( j ) ) ) ] ) .

If τ = 2, the IT-ICOWD measure reduces to the IT-ICOWED measure: IT - ICOWED ( A ˜ , B ˜ ) = Δ ( ( ∑ j = 1 n w j [ Δ − 1 ( d λ ( A ˜ δ ( j ) , B ˜ δ ( j ) ) ) ] 2 ) 1 / 2 ) .

If τ → 0 + , we obtain IT-ICOWGD measure: IT - ICOWGD ( A ˜ , B ˜ ) = Δ ( ∏ j = 1 n [ Δ − 1 ( d λ ( A ˜ δ ( j ) , B ˜ δ ( j ) ) ) ] w j ) .

Some other special measures can be obtained as follows:

The IT-ICMAXD measure: w l = 1 and w j = 0 for all j ≠ l, and d λ ( A ˜ δ ( l ) , B ˜ δ ( l ) ) = max { d λ ( A ˜ i , B ˜ i ) }, i = 1 , 2 , … , n.

The IT-ICMIND measure: w l = 1 and w j = 0 for all j ≠ l, and d λ ( A ˜ δ ( l ) , B ˜ δ ( l ) ) = min { d λ ( A ˜ i , B ˜ i ) }, i = 1 , 2 , … , n.

Step-IT-ICOWD measure: w k = 1 and w j = 0 for all j ≠ k.

The IT-ICND measure: w j = 1 / n for all j; specially, if τ = 1, we get the IT-ICNHD measure. If τ = 2, we get the IT-ICNED measure. If τ → 0 + , we get the IT-ICNGD measure.

Median IT-ICOWD measure: w ( n + 1 ) / 2 = 1 and w j = 0 for all j ≠ ( n + 1 ) / 2, n is odd; or w n / 2 = w ( n / 2 ) + 1 = 1 2 and w j = 0 for all j ≠ n / 2 , ( n / 2 ) + 1, n is even.

Based on Yager (1993), we can get families of IT-ICOWD measure. For instance:

The Olympic IT-ICOWD measure: w 1 = w n = 0 and w j = 1 / ( n − 2 ) for all j ≠ 1 , … , n.

The general Olympic IT-ICOWD measure: w j = 0 for all j = 1 , 2 , … , k , n , n − 1 , …, n − k + 1 a 2 + b 2 and for all others w j = 1 / ( n − 2 k ), where k < n / 2.

The Window IT-ICOWD measure: w j = 1 / m for k ⩽ j ⩽ k + m − 1, or w j = 0 for j ⩾ k + m and j < k.

The generalized S-IT-ICOWD measure: w k = ( 1 − ( α + β ) / n ) + α, w t = ( 1 − ( α + β ) / n ) + β and w j = 1 − ( α + β ) / n for all j ≠ k , t, where A ˜ k = max i { A ˜ i }, A ˜ t = min i { A ˜ i }, and α + β ⩽ 1 with α , β ∈ [ 0 , 1 ].

Extensions of the <italic>IT</italic>-<italic>ICOWD</italic> Measure

We can develop the extension of the IT-COWD measure through the quasi-arithmetic means, which can be named as Quasi-IT-COWD measure. The primary merit of this measure is providing a more complete generalization including a lot of particular cases that are not included in the IT-COWD measure.

An n-dimensional Quasi-IT-COWD measure is a mapping Quasi - IT - COWD : Ω n × Ω n → Ω , which is associated with the weighting vector W = ( w 1 , w 2 , … , w n ), ∑ j = 1 n w j = 1 and w j ∈ [ 0 , 1 ], satisfying: Quasi - IT - COWD ( A ˜ , B ˜ ) = Δ ( g − 1 ( ∑ j = 1 n w j g [ Δ − 1 ( d λ ( A ˜ σ ( j ) , B ˜ σ ( j ) ) ) ] ) ) , where g is a strictly continuous monotonic function, σ ( 1 ) , σ ( 2 ) , … , σ ( n ) is any permutation of ( 1 , 2 , … , n ) such that d λ ( A ˜ σ ( j − 1 ) , B ˜ σ ( j − 1 ) ) ⩾ d λ ( A ˜ σ ( j ) , B ˜ σ ( j ) ), j = 2 , 3 , … , n, and d λ ( A ˜ j , B ˜ j ) is the distance of A ˜ j and B ˜ j based on the IT-COWA operator and the parameter.

Note that the IT-COWD measure is a particular case of the Quasi-IT-COWD measure when g ( x ) = x τ .

Similarly, we can obtain the Quasi-IT-ICOWD measure.

An n-dimensional Quasi-IT-ICOWD measure is a mapping Quasi - IT - ICOWD : Ω n × Ω n → Ω , which is associated with the weighting vector W = ( w 1 , w 2 , … , w n ), ∑ j = 1 n w j = 1 and w j ∈ [ 0 , 1 ], satisfying Quasi - IT - ICOWD ( ⟨ u 1 , A ˜ 1 , B ˜ 1 ⟩ , ⟨ u 2 , A ˜ 2 , B ˜ 2 ⟩ , … , ⟨ u n , A ˜ n , B ˜ n ⟩ ) = Δ ( g − 1 [ ∑ j = 1 n w j g ( Δ − 1 ( d λ ( A ˜ δ ( j ) , B ˜ δ ( j ) ) ) ) ] ) , where g is a strictly continuous monotonic function, δ ( 1 ) , δ ( 2 ) , … , δ ( n ) is any permutation of ( 1 , 2 , … , n ), ( u 1 , u 2 , … , u n ) is a set of order inducing variables such that u δ ( j − 1 ) ⩾ u δ ( j ) , j = 2 , 3 , … , n, and d λ ( A ˜ δ ( j ) , B ˜ δ ( j ) ) is the d λ ( A ˜ i , B ˜ i ) value of the IT-ICOWD pair ⟨ u i , A ˜ i , B ˜ i ⟩ having the j-th largest u i , d λ ( A ˜ j , B ˜ j ) is calculated by Eq. (28), λ > 00$]]>.

 An Approach to 2-Tuple Linguistic Multiple Attribute Group Decision Making The Process of <italic>MAGDM</italic> Based on the <italic>IT-ICOWD</italic> Measure

The IT-ICOWD measure is of high feasibility in a broad range of situations, especially in solving multiple attribute group decision making where the attribute assessment values are represented by interval-valued 2-tuple linguistic information.

Here we propose an approach to MAGDM with interval-valued 2-tuple linguistic information by the IT-ICOWD measure. Moreover, we can determine the order-inducing variables of the IT-ICOWD measure, the weighting vector of decision makers and the weighting vector of attributes.

Let X = { X 1 , X 2 , … , X m } be a discrete set of alternatives, C = { C 1 , C 2 , … , C n } be an attributes set, and w = ( w 1 , w 2 , … , w n ) T be the weighting vector satisfying w j ∈ [ 0 , 1 ], associated with weight of C j , and ∑ j = 1 n w j = 1. Let E = { e 1 , e 2 , … , e t } be a decision maker collection, and ω = ( ω 1 , ω 2 , … , ω t ) T be the weighting vector of decision makers, where ∑ k = 1 t ω k = 1, ω k ∈ [ 0 , 1 ]. The decision makers e k ( k = 1 , 2 , … , t ) are required to give his/her assessment values of alternative X i with respect to attribute C j in linguistic term sets S T k ( S T k may have different granularity), therefore, the decision matrix R ˜ k can be built as R ˜ k = ( r ˜ i j k ) m × n = ( [ r i j k , r i j ′ k ] ) m × n , where r ˜ i j k is linguistic variable s i j k , s i j k ∈ S T k , S T k = { s i T k ∣ i ∈ { 0 , 1 , … , T k − 1 } }. The decision makers also establish the ideal alternative by giving the ideal levels of each characteristic, which is shown in Table 3, where φ ˜ k is the ideal alternative and φ ˜ j k = [ φ j k , φ j ′ k ] is the j-th ideal characteristic of  φ ˜ k .

Ideal alternative.

C 1 C 2 … C j … C n
φ ˜ k φ ˜ 1 k φ ˜ 2 k … φ ˜ j k … φ ˜ n k

The process with the IT-ICOWD measure in MAGDM involves the following steps.

Step 1. Transform the decision matrix R ˜ k = ( r ˜ i j k ) m × n ( k = 1 , 2 , … , t ) and the ideal alternative ϕ ˜ k into interval-valued 2-tuple linguistic decision matrix R ˆ k = ( r ˆ i j k ) m × n = ( [ ( r i j k , 0 ) , ( r i j ′ k , 0 ) ] ) m × n ( k = 1 , 2 , … , t ) and interval-valued 2-tuple linguistic the ideal alternative ϕ ˆ k = ( ϕ ˆ j k ) 1 × n = ( [ ( ϕ 1 k , 0 ) , ( ϕ 1 ′ k , 0 ) ] , [ ( ϕ 2 k , 0 ) , ( ϕ 2 ′ k , 0 ) ] , … , [ ( ϕ n k , 0 ) , ( ϕ n ′ k , 0 ) ] ) respectively.

Step 2. Calculate the distance between each assessment value r ˆ i j k provided by the decision maker e k and his/her ideal assessment value ϕ ˆ j k by Eq. (28): d λ ( r ˆ i j k , ϕ ˆ j k ) = Δ ( | ( λ Δ − 1 ( r i j ′ k , 0 ) + ( 1 − λ ) Δ − 1 ( r i j k , 0 ) ) − ( λ Δ − 1 ( ϕ j ′ k , 0 ) + ( 1 − λ ) Δ − 1 ( ϕ j k , 0 ) ) | ) where k = 1 , 2 , … , t, i = 1 , 2 , … , m, j = 1 , 2 , … , n, λ = ∫ 0 1 Q ( y ) d y is the attitudinal character of Q.

Step 3. Let d ¯ i j = Δ ( 1 t ∑ k = 1 t Δ − 1 ( d λ ( r ˆ i j k , ϕ ˆ j k ) ) ), i.e. ( d ¯ i j ) m × n is the mean distance matrix of d λ ( r ˆ i j k , ϕ ˆ j k ), k = 1 , 2 , … , t, and ( Λ ( d λ ( r ˆ i j k , ϕ ˆ j k ) , d ¯ ) ) m × n = ( Δ ( | Δ − 1 ( d λ ( r ˆ i j k , ϕ ˆ j k ) ) − Δ − 1 ( d ¯ ) | ) ) m × n is the absolute distance matrix between d λ ( r ˆ i j k , ϕ ˆ j k ) and d ¯ i j . Then, the similarity measure can be defined as follows: Sim k = 1 − ∑ i = 1 m ∑ j = 1 n Δ − 1 ( Λ ( d λ ( r ˆ i j k , ϕ ˆ j k ) , d ¯ i j ) ) ∑ k = 1 t ∑ i = 1 m ∑ j = 1 n Δ − 1 ( Λ ( d λ ( r ˆ i j k , ϕ ˆ j k ) , d ¯ i j ) ) .

The closer Sim k is to 1, the more representative and reliable the information provided by the k-th expert is. That is the absolute distance matrix with the more similarity measure should be more important. Thus, we can use the similarity measure Sim k as the order-inducing variables of the assessment values to be aggregated in the process of group decision making. Thus, the weighting vector ω = ( ω 1 , ω 2 , … , ω t ) T can be determined by the following formula: ω k = Sim k ∑ k = 1 t Sim k , k = 1 , 2 , … , t .

Moreover, according to the principle that the closer a preference value is to the mid one(s), the more the weight, the weighting vector w = ( w 1 , w 2 , … , w n ) T can be determined by the following formula: w j = Sim j ′ ∑ j = 1 n Sim j ′ = ( 1 − ∑ k = 1 t ∑ i = 1 m Δ − 1 ( Λ ( d λ ( r ˆ i j k , ϕ ˆ j k ) , d ¯ i j ) ) ∑ k = 1 t ∑ i = 1 m ∑ j = 1 n Δ − 1 ( Λ ( d λ ( r ˆ i j k , ϕ ˆ j k ) , d ¯ i j ) ) ) / ∑ j = 1 n ( 1 − ∑ k = 1 t ∑ i = 1 m Δ − 1 ( Λ ( d λ ( r ˆ i j k , ϕ ˆ j k ) , d ¯ i j ) ) ∑ k = 1 t ∑ i = 1 m ∑ j = 1 n Δ − 1 ( Λ ( d λ ( r ˆ i j k , ϕ ˆ j k ) , d ¯ i j ) ) ) .

Step 4. Utilize the IT-ICOWD measure r ˜ i j = ( r i j , a i j ) = IT - ICOWD ⟨ Sim 1 , [ ( r i j 1 , 0 ) , ( r i j ′ 1 , 0 ) ] , [ ( ϕ j 1 , 0 ) , ( ϕ j ′ 1 , 0 ) ] ⟩ , ⋯ , ⟨ Sim n , [ ( r i j t , 0 ) , ( r i j ′ t , 0 ) ] , [ ( ϕ j t , 0 ) , ( ϕ j ′ t , 0 ) ] ⟩ = Δ ( [ ∑ k = 1 t ω k ( Δ − 1 [ d λ ( r ˆ δ ( j ) k , ϕ ˆ δ ( j ) k ) ] ) τ ] 1 / τ ) to aggregate all the 2-tuple linguistic distance matrices into the collective 2-tuple linguistic distance matrix R ˜ = ( r ˜ i j ) m × n = ( ( r i j , a i j ) ) m × n , where ω = ( ω 1 , ω 2 , … , ω t ) T is the weighting vector of decision makers. Here, it should be mentioned that r i j ∈ S T k and a i j ∈ [ − 1 2 T k , 1 2 T k ).

Step 5. Utilize the T-GOWA operator (Liu et al., 2011) r ˜ i = ( r i , a i ) = T - GOWA ( r ˜ i 1 , r ˜ i 2 , … , r ˜ i n ) = T - GOWA ( ( r i 1 , a i 1 ) , ( r i 2 , a i 2 ) , … , ( r i n , a i n ) ) to aggregate all of the preference values r ˜ i j ( j = 1 , 2 , … , n ) in the i-th line of R ˜, and then derive the collective overall preference values r ˜ i = ( r i , a i ) ( i = 1 , 2 , … , m ) of the alternative X i ( i = 1 , 2 , … , m ), where w = ( w 1 , w 2 , … , w n ) T is the weighting vector of attribute.

Step 6. According to the comparison law, rank the r ˜ i = ( r i , a i ) ( i = 1 , 2 , … , m ) in descending order.

Step 7. Rank all of the alternatives X i ( i = 1 , 2 , … , m ), and then select the best one(s) in accordance with the collective overall preference values r ˜ i = ( r i , a i ) ( i = 1 , 2 , … , m ). The best choice is the one with the smallest distance.

Step 8. End.

 Illustrative Example

In this section, we employ a practical MAGDM problem to illustrate the efficiency of the proposed method in dealing with problems of interval-valued 2-tuple linguistic information. Suppose that an investment company wants to find an optimal investment. There are four possible alternatives to invest the money:

X 1 : car industry; X 2 : food company; X 3 : computer company X 4 : arms industry.

The investment company must make a decision according to the following four attributes:

C 1 : risk analysis; C 2 : growth analysis; C 3 : social-political impact analysis; C 4 : environment impact analysis.

In order to eliminate influence among them, three decision makers are invited to provide their preferences for each possible alternative on each attributes in anonymity and in different linguistic term sets respectively, which are seven terms: S 7 = { s 0 7 , s 1 7 , s 2 7 , s 3 7 , s 4 7 , s 5 7 , s 6 7 }, five terms: S 5 = { s 0 5 , s 1 5 , s 2 5 , s 3 5 , s 4 5 } and nine terms: S 9 = { s 0 9 , s 1 9 , s 2 9 , s 3 9 , s 4 9 , s 5 9 , s 6 9 , s 7 9 , s 8 9 }.

The linguistic decision matrices R ˜ k = ( r ˜ i j k ) 4 × 4 ( k = 1 , 2 , 3 ) and the ideal alternative ϕ ˜ k are provided as follows:

Linguistic decision matrix R ˜ 1 provided by D 1

Linguistic decision matrix R ˜ 2 provided by D 2

Linguistic decision matrix R ˜ 3 provided by D 3

Ideal alternative.

C 1 C 2 C 3 C 4
ϕ ˜ 1 ( s 4 7 , s 6 7 ) ( s 4 7 , s 5 7 ) ( s 3 7 , s 4 7 ) ( s 4 7 , s 5 7 )
ϕ ˜ 2 ( s 3 5 , s 4 5 ) ( s 2 5 , s 4 5 ) ( s 2 5 , s 3 5 ) ( s 3 5 , s 4 5 )
ϕ ˜ 3 ( s 6 9 , s 8 9 ) ( s 5 9 , s 7 9 ) ( s 6 9 , s 7 9 ) ( s 5 9 , s 6 9 )

Then, we utilize the method developed to obtain the best alternative(s).

Step 1. Transform the decision matrix R ˜ k = ( r ˜ i j k ) 4 × 4 , k = 1 , 2 , 3 and the ideal alternative ϕ ˜ k into interval-valued 2-tuple linguistic decision matrix R ˆ k = ( r ˆ i j k ) 4 × 4 = ( [ ( r i j k , 0 ) , ( r i j ′ k , 0 ) ] ) 4 × 4 and interval-valued 2-tuple linguistic ideal alternative ϕ ˆ k = ( ϕ ˆ j k ) 1 × 4 = ( [ ( ϕ 1 k , 0 ) , ( ϕ 1 ′ k , 0 ) ] , [ ( ϕ 2 k , 0 ) , ( ϕ 2 ′ k , 0 ) ] , … , [ ( ϕ 4 k , 0 ) , ( ϕ 4 ′ k , 0 ) ] ), shown as follows:

Interval-valued 2-tuple linguistic decision matrix R ˆ 1 provided by D 1

Interval-valued 2-tuple linguistic decision matrix R ˆ 2 provided by D 2

Interval-valued 2-tuple linguistic decision matrix R ˆ 3 provided by D 3

Interval-valued 2-tuple linguistic ideal alternative ϕ ˆ k ( k = 1 , 2 , 3 )

Step 2. Calculate the distance of each assessment value r ˆ i j k provided by the decision maker e k and his/her ideal assessment value ϕ ˆ j k by Eq. (44), where Q ( y ) = y 2 , λ = 1 3 . The results are listed as follows:

Distance matrix of decision maker D 1

Distance matrix of decision maker D 2

Distance matrix of decision maker D 3

Step 3. Calculate the mean distance matrix d ¯ and the absolute distance matrix Λ ( d λ ( r ˆ i j k , ϕ ˆ j k ) , d ¯ ), k = 1 , 2 , 3, where d ¯ chooses the linguistic term sets S 5 . The results are shown as follows:

The mean distance matrix d ¯ ( d ¯ i j ) 4 × 4 = ( s 2 5 , 0.0278 ) ( s 1 5 , 0.0278 ) ( s 2 5 , − 0.0972 ) ( s 1 5 , 0.0927 ) ( s 1 5 , 0.1112 ) ( s 2 5 , − 0.0324 ) ( s 1 5 , 0.0232 ) ( s 2 5 , − 0.1158 ) ( s 1 5 , − 0.0509 ) ( s 1 5 , 0.0046 ) ( s 1 5 , 0.0787 ) ( s 1 5 , 0.0278 ) ( s 2 5 , 0.0741 ) ( s 1 5 , − 0.0647 ) ( s 1 5 , 0.051 ) ( s 2 5 , − 0.0277 ) .

The absolute distance matrix of decision maker D 1 ( Λ ( d λ ( r ˆ i j 1 , ϕ ˆ j 1 ) , d ¯ i j ) ) 4 × 4 = ( s 0 7 , 0.0278 ) ( s 1 7 , 0.0556 ) ( s 1 7 , − 0.0694 ) ( s 1 7 , 0.0647 ) ( s 1 7 , 0.0276 ) ( s 0 7 , 0.0232 ) ( s 0 7 , 0.0117 ) ( s 0 7 , 0.0601 ) ( s 1 7 , − 0.0232 ) ( s 1 7 , − 0.0326 ) ( s 1 7 , − 0.0323 ) ( s 1 7 , 0.0556 ) ( s 1 7 , − 0.0184 ) ( s 0 7 , 0.0373 ) ( s 2 7 , 0.0161 ) ( s 1 7 , − 0.028 ) .

The absolute distance matrix of decision maker D 2 Λ ( d λ ( r ˆ i j 2 , ϕ ˆ j 2 ) , d ¯ i j ) 4 × 4 = ( s 0 5 , 0.0278 ) ( s 0 5 , 0.0278 ) ( s 0 5 , 0.0972 ) ( s 0 5 , 0.074 ) ( s 0 5 , 0.1112 ) ( s 0 5 , 0.0324 ) ( s 0 5 , 0.1066 ) ( s 0 5 , 0.0324 ) ( s 1 5 , − 0.0324 ) ( s 0 5 , 0.0788 ) ( s 0 5 , 0.0047 ) ( s 0 5 , 0.0278 ) ( s 1 5 , − 0.0741 ) ( s 0 5 , 0.102 ) ( s 1 5 , − 0.1157 ) ( s 1 5 , − 0.0557 ) .

The absolute distance matrix of decision maker D 3 ( Λ ( d λ ( r ˆ i j 3 , ϕ ˆ j 3 ) , d ¯ i j ) ) 4 × 4 = ( s 0 9 , 0.0556 ) ( s 2 9 , − 0.0556 ) ( s 0 9 , 0.0556 ) ( s 1 9 , 0.0323 ) ( s 2 9 , 0.0554 ) ( s 0 9 , 0.0092 ) ( s 2 9 , − 0.0254 ) ( s 1 9 , − 0.0324 ) ( s 1 9 , − 0.0509 ) ( s 2 9 , − 0.0317 ) ( s 0 9 , 0.037 ) ( s 2 9 , − 0.0556 ) ( s 3 9 , − 0.0509 ) ( s 1 9 , − 0.0603 ) ( s 1 9 , 0.0393 ) ( s 0 9 , 0.0557 ) .

Then, we can get the similarity measure by Eq. (45) and the weighting vector ω by Eq. (46): Sim 1 = 0.6386 , Sim 2 = 0.7471 , Sim 3 = 0.6143 , ω 1 = 0.3193 , ω 2 = 0.3736 , ω 3 = 0.3071.

Moreover, we can obtain the weighting vector w by Eq. (47): w 1 = 0.228 , w 2 = 0.2669 , w 3 = 0.2582 , w 4 = 0.2469.

Step 4. Utilize the IT-ICOWD measure to aggregate 2-tuple linguistic distance matrix into the collective 2-tuple linguistic distance matrix R ˜ = ( r ˜ i j ) 4 × 4 = ( ( r i j , a i j ) ) 4 × 4 , where τ = 3. We can use the similarity measure Sim k as the order-inducing variables. Note that r i j ∈ S T k and a i j ∈ [ − 1 2 T k , 1 2 T k ).

The collective 2-tuple linguistic distance matrix R ˜

Step 5. Utilize the T-GOWA operator to derive the collective overall preference value r ˜ i = ( r i , a i ) ( i = 1 , 2 , 3 , 4 ) of the alternative X i ( i = 1 , 2 , 3 , 4 ): r ˜ 1 = ( s 2 5 , − 0.0671 ) , r ˜ 2 = ( s 2 5 , − 0.0807 ) , r ˜ 3 = ( s 1 5 , 0.0821 ) , r ˜ 4 = ( s 2 5 , − 0.0055 ) .

Assume that the parameter τ in the T-GOWA operator is equal to the parameter τ in the IT-ICOWD measure.

Step 6. According to the comparison law, rank the r ˜ i = ( r i , a i ) ( i = 1 , 2 , 3 , 4 ) in descending order: r ˜ 4 > r ˜ 1 > r ˜ 2 > r ˜ 3 . {\tilde{r}_{1}}>{\tilde{r}_{2}}>{\tilde{r}_{3}}.\]]]>

Step 7. Rank all of the alternatives X i ( i = 1 , 2 , 3 , 4 ) as follows: X 3 ≻ X 2 ≻ X 1 ≻ X 4 , and the best alternative is thus X 3 . i.e. the best alternative is the computer company.

Furthermore, it is possible to analyse how the different particular cases of the IT-ICOWD measure influence for the aggregation results. Here we consider the IT-ICOWHD measure, the IT-ICOWED measure, the IT-ICOWGD measure, the IT-ICMAXD measure, the IT-ICMIND measure, the Step-IT-ICOWD measure ( k = 2), the IT-ICND measure, the IT-ICNHD measure, the IT-ICNED measure, the IT-ICNGD measure, the Median IT-ICOWD measure and the Olympic IT-ICOWD measure. The results are clearly demonstrated in Table 5. Now, we are able to propose the order of the companies for each case. The results are clearly demonstrated in Table 6. Note that the best and the most optimal investment is the one possessing the lowest distance.

Aggregated results.

IT-ICOWHD IT-ICOWED IT-ICOWGD IT-ICMAXD
X 1 ( s 2 5 , − 0.1167 ) ( s 2 5 , − 0.0864 ) ( s 1 5 , 0.0839 ) ( s 2 5 , 0.0214 )
X 2 ( s 1 5 , 0.118 ) ( s 2 5 , − 0.1042 ) ( s 1 5 , 0.0864 ) ( s 2 5 , 0.0287 )
X 3 ( s 1 5 , 0.0179 ) ( s 1 5 , 0.0572 ) ( s 1 5 , − 0.0431 ) ( s 2 5 , − 0.072 )
X 4 ( s 2 5 , − 0.122 ) ( s 2 5 , − 0.0558 ) ( s 1 5 , 0.0519 ) ( s 2 5 , 0.1233 )
IT-ICMIND Step-IT-ICOWD IT-ICND IT-ICNHD
X 1 ( s 1 5 , 0.0888 ) ( s 2 5 , − 0.0977 ) ( s 2 5 , − 0.0657 ) ( s 2 5 , − 0.1151 )
X 2 ( s 1 5 , 0.0565 ) ( s 1 5 , 0.1106 ) ( s 2 5 , − 0.0792 ) ( s 1 5 , 0.1179 )
X 3 ( s 1 5 , − 0.0779 ) ( s 1 5 , 0.033 ) ( s 1 5 , 0.0783 ) ( s 1 5 , 0.014 )
X 4 ( s 1 5 , 0.021 ) ( s 2 5 , − 0.0204 ) ( s 2 5 , − 0.0044 ) ( s 2 5 , − 0.12 )
IT-ICNED IT-ICNGD Median IT-ICOWD Olympic IT-ICOWD
X 1 ( s 2 5 , − 0.0849 ) ( s 1 5 , − 0.0853 ) ( s 2 5 , − 0.0887 ) ( s 2 5 , − 0.0887 )
X 2 ( s 2 5 , − 0.1036 ) ( s 1 5 , 0.086 ) ( s 2 5 , − 0.0933 ) ( s 2 5 , − 0.0933 )
X 3 ( s 1 5 , 0.0532 ) ( s 1 5 , − 0.0465 ) ( s 2 5 , − 0.1233 ) ( s 2 5 , − 0.1233 )
X 4 ( s 2 5 , − 0.054 ) ( s 1 5 , 0.0544 ) ( s 2 5 , − 0.375 ) ( s 2 5 , − 0.0375 )

Ordering of the companies.

Ordering Ordering
IT-ICOWHD X 3 ≻ X 2 ≻ X 4 ≻ X 1 IT-ICND X 3 ≻ X 2 ≻ X 1 ≻ X 4
IT-ICOWED X 3 ≻ X 2 ≻ X 1 ≻ X 4 IT-ICNHD X 3 ≻ X 2 ≻ X 4 ≻ X 1
IT-ICOWGD X 3 ≻ X 4 ≻ X 1 ≻ X 2 IT-ICNED X 3 ≻ X 2 ≻ X 1 ≻ X 4
IT-ICMAXD X 3 ≻ X 1 ≻ X 2 ≻ X 4 IT-ICNGD X 3 ≻ X 4 ≻ X 1 ≻ X 2
IT-ICMIND X 3 ≻ X 4 ≻ X 2 ≻ X 1 Median IT-ICOWD X 3 ≻ X 2 ≻ X 1 ≻ X 4
Step-IT-ICOWD X 3 ≻ X 2 ≻ X 1 ≻ X 4 Olympic IT-ICOWD X 3 ≻ X 2 ≻ X 1 ≻ X 4

Variations of the aggregation results with parameter τ.

As we can see, the company order varies with category of IT-ICOWD measures.

Moreover, we can also analyse how different parameter τ affects the aggregation results. Considering different values of parameter τ ∈ ( 0 , 20 ) provided by the decision makers, here we take λ = 1 3 . The results are shown in Fig. 1.

Similarly, the company order varies with parameter τ. From Fig. 1, we can conclude that

when τ ∈ ( 0 , 0.398 ], alternative’s rank is X 3 ≻ X 4 ≻ X 1 ≻ X 2 , and the best alternative is X 3 ;

when τ ∈ ( 0.398 , 1.006 ], alternative’s rank is X 3 ≻ X 4 ≻ X 2 ≻ X 1 , and the best alternative is X 3 ;

when τ ∈ ( 1.006 , 1.265 ], alternative’s rank is X 3 ≻ X 2 ≻ X 4 ≻ X 1 , and the best alternative is X 3 .

when τ ∈ ( 1.265 , 4.979 ], alternative’s rank is X 3 ≻ X 2 ≻ X 1 ≻ X 4 , and the best alternative is X 3 ;

when τ ∈ ( 4.979 , 20 ], alternative’s rank is X 3 ≻ X 1 ≻ X 2 ≻ X 4 , and the best alternative is X 3 .

 Discussion of Comparative Analysis

In order to evaluate the performance and effectiveness of the proposed distance measure with the existing one, this paper conducts a comparative study in this subsection, including the comparison with well-applied distance measures and algorithm-based aggregation tools.

(1) Comparison with the existing distance measure.

Compared with the previous distance measure, we can conclude that the IT-ICOWD measure is very useful to deal with deviated problem in aggregation on continuous valued interval 2-tuple linguistic information. The prominent characteristic of the IT-ICOWD measure is that it combines the GOWA operator with the distance measure and the IT-ICOWA operator in the same formula. The decision maker is able to consider the MAGDM problem more clearly according to his/her risk attitude in aggregation process because the parameter λ, which lies in the interval [ 0 , 1 ], can be considered as the measure of the decision maker’s attitudinal character.

(2) Comparison with the aggregation tool based algorithm.

In Liu et al. (2014b), Liu et al. proposed the interval 2-tuple hybrid weighted distance (IT-HWD) measure to aggregate the interval-valued 2-tuple linguistic information. To facilitate a comparison with the proposed approach, we adopt the algorithm proposed by Liu et al. (2014b) and solve the same illustrative example described above. The steps are as follows:

Step 1. Interval-valued 2-tuple linguistic decision matrix R ˆ k = ( r ˆ i j k ) 4 × 4 = ( [ ( r i j k , 0 ) , ( r i j ′ k , 0 ) ] ) 4 × 4 ( k = 1 , 2 , 3 ) and interval-valued 2-tuple linguistic ideal alternative ϕ ˆ k = ( ϕ ˆ j k ) 1 × 4 = ( [ ( ϕ 1 k , 0 ) , ( ϕ 1 ′ k , 0 ) ] , [ ( ϕ 2 k , 0 ) , ( ϕ 2 ′ k , 0 ) ] , … , [ ( ϕ 4 k , 0 ) , ( ϕ 4 ′ k , 0 ) ] ) are listed as shown in Section 4.2.

Step 2. According to the interval 2-tuple linguistic distance (Liu et al., 2014b) between two interval 2-tuple linguistic information, the distance of each assessment value r ˆ i j k provided by the decision maker e k and his/her ideal assessment value ϕ ˆ j k are calculated. The results are listed as follows:

Distance matrix of decision maker D 1

Distance matrix of decision maker D 2

Distance matrix of decision maker D 3

Step 3. Calculate the collective 2-tuple linguistic distance matrix R ˜ = ( r ˜ i j ) 4 × 4 = ( ( r i j , a i j ) ) 4 × 4 by using the IT-HWD measure, which is proposed by Liu’s algorithm. Here, in order to eliminate the unnecessary impacts, we also suppose the objective weight vector ω = ( 0.3193 , 0.3736 , 0.3071 ) T , the subjective weight vector W = ( 0.3 , 0.4 , 0.3 ) T and the parameter λ = 3.

The collective 2-tuple linguistic distance matrix R ˜

It is noted that the balancing coefficient is n = 3 and r i j ∈ S 5 .

Step 4. Utilize the T-GOWA operator to derive the collective overall preference value r ˜ i = ( r i , a i ) ( i = 1 , 2 , 3 , 4 ) of the alternative X i ( i = 1 , 2 , 3 , 4 ): r ˜ 1 = ( s 2 5 , − 0.0613 ) , r ˜ 2 = ( s 2 5 , − 0.0878 ) , r ˜ 3 = ( s 1 5 , 0.0967 ) , r ˜ 4 = ( s 2 5 , 0.0198 ) .

Assume that the parameter λ in the T-GOWA operator is equal to the parameter λ in the IT-HWD measure.

Step 5. According to the comparison law, rank the r ˜ i = ( r i , a i ) ( i = 1 , 2 , 3 , 4 ) in descending order: r ˜ 4 > r ˜ 1 > r ˜ 2 > r ˜ 3 . {\tilde{r}_{1}}>{\tilde{r}_{2}}>{\tilde{r}_{3}}.\]]]>

Step 6. Rank all of the alternatives X i ( i = 1 , 2 , 3 , 4 ) as follows: X 3 ≻ X 2 ≻ X 1 ≻ X 4 .

Therefore, the best alternative is X 3 , i.e. the best alternative is the computer company.

From the comparison with interval 2-tuple linguistic distance of Liu et al. (2014b), the newly proposed approach and Liu’s have their own merits. For one thing, the interval 2-tuple linguistic distance (Liu et al., 2014b) is defined by endpoints of interval-valued 2-tuple linguistic information. This measure varies with the uncertain linguistic environment in GDM and makes it more flexible in diverse circumstances. For another, the approach proposed in this paper is able to provide more decision-related information such as order-inducing variables in the IT-ICOWD measure, the weighting vector of decision makers and the weighting vector of attributes. This enables decision-making process more evidential and reliable, and these intermediate results can be applied for multiple times when necessary. However, this benefit requires more efforts in computation, opposed to Liu’s approach. Therefore, decision makers who request a deterministic answer as well as reasonable and solid evidence would prefer the novel IT-ICOWD measure regardless of computational complexity.

 Conclusion

In this paper, we introduced the interval-valued 2-tuple linguistic induced continuous ordered weighted distance (IT-ICOWD) measure. Comparing with existing methods of aggregating interval-valued linguistic variables, we firstly demonstrated that the IT-ICOWD measure is of high practicality to uncertain cases when the decision maker is only able to express preference information in interval-valued 2-tuples linguistic terms. Furthermore, we discussed several desirable properties and different families of the IT-ICOWD measure. At last, feasibility and practicability of proposed approach were illustrated by a numerical example.

In future research, we look forward to applying the IT-ICOWD measure to different decision making application, such as dynamic decision making (Pérez et al., 2010), consensus reaching process (Dong et al., 2016), social media (Dong et al., 2017), heterogeneous information merging process (Liu et al., 2017). Simultaneously, we are going to develop further extensions of the IT-ICOWD measure to other types of distance measure and decision information.

 Acknowledgements

The authors would like to thank the editor and the anonymous referees for their valuable comments and suggestions for improving the paper. The work was supported by Anhui Provincial Natural Science Foundation (No. 1808085QG211), Statistics and Science Research Foundation of China (No. 2017LZ11), Doctoral Research Start-up Funds Projects of Hefei Normal University (No. 2017rcjj03), Open Project of School of Mathematical Sciences, Anhui University, Provincial Natural Science Research Project of Anhui Colleges (No. KJ2015A379), Provincial Natural Science Research Project of Anhui Colleges (No.KJ2017A026), National Natural Science Foundation of China (Nos. 71301001, 71371011, 11426033, 11501005), Project of Anhui Province for Excellent Young Talents.

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