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Volume 12, Issue 2, 2019, Pages 1339 - 1352

On Consistency and Priority Weights for Uncertain 2-Tuple Linguistic Preference Relations

Authors
Peng Wu1, Jiaming Zhu1, Ligang Zhou1, 2, *, Huayou Chen1, Yu Chen1
1School of Mathematical Sciences, Anhui University, Hefei 230601, Anhui, P. R. China
2China Institute of Manufacturing Development, Nanjing University of Information Science and Technology, Nanjing 210044, Jiangsu, P. R. China
*Corresponding author. Email: shuiqiaozlg@126.com
Corresponding Author
Ligang Zhou
Received 8 November 2018, Accepted 15 October 2019, Available Online 22 November 2019.
DOI
10.2991/ijcis.d.191104.001How to use a DOI?
Keywords
Group decision-making; Uncertain 2-tuple linguistic preference relations; Additive consistency; Uncertain 2-tuple linguistic priority weights
Abstract

Consistency and priority weights of preference relations are two important phases of decision-making process since the decision-making solutions are determined by them. Therefore, it is meaningful to investigate consistency and priority weights for preference relations. In this paper, consistency and uncertain 2-tuple linguistic priority weights of uncertain 2-tuple linguistic preference relations (U2TLPRs) are investigated. First, based on the additive consistency, an additive consistency index is developed to measure the additive consistency level of U2TLPRs. Second, a goal programming model is proposed to adjust the unacceptable additive consistent U2TLPR until it satisfies acceptable additive consistency. Furthermore, an optimization model is developed to derive the uncertain 2-tuple linguistic priority weights from an U2TLPR. Meanwhile, in group decision-making (GDM) problems, similarities and confidence degrees of decision makers (DMs) are defined to determine DMs' weights. Subsequently the properties of collective U2TLPR are discussed. Finally, the proposed methods are implemented in two examples including a GDM problem to verify the validity of the proposed methods.

Copyright
© 2019 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

In decision-making process, the preference relations, which consist of pairwise comparison information provided by decision makers (DMs), are popular and powerful tools to model DMs preferences regarding decision-making problems. In recent years, various preference relations have been investigated, including multiplicative preference relation [1], fuzzy preference relation [2], interval multiplicative preference relation [3,4], interval fuzzy preference relation [5,6], triangular fuzzy reciprocal preference relation [7], triangular fuzzy additive reciprocal preference relations [8,9], multiplicative trapezoidal fuzzy preference relation [10–12], additive trapezoidal fuzzy preference relation [13], intuitionistic fuzzy preference relation [14] and hesitant fuzzy preference relation [15,16]. These preference relations have been mainly studied from two perspectives, which are as follows:

  • Consistency measure: It is a critical step for any kinds of preference relations since consistency directly affects the rationality of final decision-making results.

  • Priority weights: The other pivotal phase is to obtain priority weights from preference relations because they are utilized to select the optimal alternative(s).

The aforementioned preference relations utilized numerical values to assess pairs of alternatives. However, in the uncertainty of decision-making environment, DMs prefer to give their assessments using linguistic terms. Nevertheless, the traditional fuzzy linguistic approach does not capture all available information and may lead to the loss of information [17]. Therefore, to overcome the limitations of traditional fuzzy linguistic approach, an useful 2-tuple linguistic representation model was introduced by Herrera and Martínez [17] and the merit of 2-tuple linguistic representation model is that allows one to compute with words (CWWs) without loss of information. Since the 2-tuple linguistic representation model was put forward, research on the measurement theories [18,19], information fusion theories [20,21], preference relation theories [22] and applications [23,24] have made great achievements. Some comprehensive overviews on 2-tuple linguistic representation model have been provided [25,26].

Uncertain 2-tuple linguistic representation model [27] (Interval 2-tuple linguistic representation model or Interval-valued 2-tuple linguistic representation model in the sense of some literatures) as an extension of 2-tuple linguistic representation model has attracted many scholars' attention since it can well express DMs' uncertain qualitative preferences when the uncertainty of decision problems and/or the lack of relevant experience with the evaluated alternatives. For example, a DM may give his/her assessment on an alternative as “between good and very good,” which is a typical form of uncertain 2-tuple linguistic variable. Much work of uncertain 2-tuple linguistic variable is summarized from theories and applications for uncertain 2-tuple linguistic variable.

  • (1)

    Theories of uncertain 2-tuple linguistic variables

    • (1.1)

      Uncertain 2-tuple linguistic fusion theory: It plays a vital role in decision-making problems since the group opinion is obtained by it. Some information aggregation operators are developed, including uncertain 2-tuple weighted averaging operator [27], uncertain 2-tuple order weighted averaging operator [27], uncertain 2-tuple ordered weighted harmonic operator and uncertain 2-tuple ordered weighted quadratic operator [28], uncertain 2-tuple correlated averaging operator and uncertain 2-tuple correlated geometric operator [29], uncertain 2-tuple linguistic Bonferroni mean operator [30], interval 2-tuple linguistic induced continuous ordered weighted averaging [31], uncertain 2-tuple linguistic Choquet integral operator [32].

    • (1.2)

      Uncertain 2-tuple linguistic measurement theory: Since measurement theory is foundation of decision-making methods, it plays an important role in decision-making theory. Some distance measurements are introduced such as uncertain 2-tuple linguistic induced continuous ordered weighted distance measure [31], generalized uncertain 2-tuple linguistic interval distance measures [33], generalized uncertain 2-tuple linguistic Shapley weighted uncertain distance measures [33].

    • (1.3)

      Uncertain 2-tuple linguistic decision-making theory: For decision-making problems with uncertain 2-tuple linguistic environment, some different decision-making methods are proposed. For example, uncertain 2-tuple linguistic VIKOR method [34,35], uncertain 2-tuple linguistic MULTIMOORA method [36], uncertain 2-tuple ELECTRE II [37].

    • (1.4)

      Uncertain 2-tuple linguistic preference relations theory: In order to introduce uncertain 2-tuple linguistic variable into decision-making process, Zhang and Guo [38] defined uncertain 2-tuple linguistic preference relation (U2TLPR). For incomplete U2TLPRs, two algorithms, which contain an iterative algorithm and an optimization-based algorithm, were developed to estimate the incomplete elements. Next, Zhang and Guo [39] investigated the consistency and consensus of U2TLPR by designing consistency improving algorithm and consensus reaching algorithm. Yao and Hu [40] studied the consistency and weighting vector of U2TLPR. In addition, a consistency improving algorithm was developed to derive U2TLPR with acceptable consistency from U2TLPR with unacceptable consistency.

  • (2)

    Applications of uncertain 2-tuple linguistic variables

    The applications of uncertain 2-tuple linguistic variable have made great achievements in different fields, which include material selection [34], supplier selection [35,37], healthcare waste treatment technology evaluation and selection [36], emergency response capacity evaluation [41], failure mode and effect analysis [42], tacit knowledge [43], robot evaluation and selection [44], human-machine function allocation method for aircraft cockpit [45], energy planning [46], evaluating the risk of healthcare failure modes [47].

Theoretical research of uncertain 2-tuple linguistic variables not only has a strong theoretical research value, but also has wide application prospects in practice. Therefore, it is significant for us to investigate the decision-making problems with uncertain 2-tuple linguistic variables environment. Based on the above reviews, we find that the theory of uncertain 2-tuple linguistic variables is gradually completely and the applications of uncertain 2-tuple linguistic variables have wide scope. However, despite significant progress over the past years on decision-making methods under uncertain 2-tuple linguistic variables environment, only a few attempts have been made to deal with the consistency measure and priority weights of U2TLPRs. Moreover, the existing consistency measures are unstable since they were defined based on different consistent U2TLPRs derived from an original U2TLPR. Meanwhile, the real priority weights of U2TLPR may cause loss of uncertain 2-tuple linguistic information. Motivated by these, in this paper, we mainly discuss the consistency measure and priority weights of U2TLPRs. To accomplish these goals, the following points are considered:

  • An additive consistency index is developed to measure the additive consistency level of U2TLPRs. The additive consistency index is different from the other indices [39,40]. It is reliable and stable since it only depends on the uncertain 2-tuple linguistic information of original U2TLPRs.

  • For an U2TLPR with unacceptable additive consistency, an optimization-based model, which is a goal programming model, is presented to derive acceptable additive consistent U2TLPR from the U2TLPR with unacceptable additive consistency.

  • To obtain the uncertain 2-tuple linguistic priority weights of U2TLPR, an optimization model is developed. The uncertain 2-tuple linguistic priority weighting vector is composed of uncertain 2-tuple linguistic variables. Being different with the crisp weights [40], the uncertain 2-tuple linguistic priority weights can keep the integrity of final decision-making information derived from the original decision-making information.

  • In group decision-making (GDM) problems, the DMs' weights are determined based on the confidence degree, which is defined by the similarities among the DMs. Finally, a new method is proposed to solve GDM with U2TLPRs.

The remainder of the paper is processed as follows. In detail, Section 2 mainly reviews some preliminaries which are utilized in the following discussion. Section 3 first defines an additive consistency index of U2TLPR. Then, for unacceptable additive consistent U2TLPRs, a goal programming model is proposed to obtain U2TLPRs with acceptable additive consistency. In Section 4, an optimization model is developed to derive the uncertain 2-tuple linguistic priority weights. Section 5 provides a method to determine the DM's weights. Meanwhile, a method for addressing GDM problems is proposed. In Section 6, the application of our proposed methods is illustrated by using some examples and some comparisons are discussed simultaneously. Finally, Section 7 outlines the main work of this paper and the future research is given.

2. PRELIMINARIES

In this section, to introduced our work, some preliminaries are reviewed.

2.1. The 2-Tuple Linguistic Variable and 2-Tuple Linguistic Preference Relation (2TLPR)

Let S={s0,s1,⋯,sτ} be a linguistic term set with granularity τ+1. To CWWs, Herrera and Martínez [17] utilized a pair of values (si,α) called linguistic 2-tuple to propose the concept of 2-tuple linguistic representation model, where si∈S represents the central value of the ith linguistic term and α∈[−0.5,0.5) indicates the deviation to the central value of the ith linguistic term.

In traditional 2-tuple linguistic representation model, there is β∈[0,τ] represents the results of an aggregation of the indices of a set of labels in S. However, the range of β has a restriction in multi-granularity linguistic term sets. To overcome the restriction, Tai and Chen [48] proposed a generalized 2-tuple linguistic model and translation functions, which are defined as Definition 1.

Definition 1.

[48] Let S be as before and β∈[0,1] be a value representing the result of a symbolic aggregation operation. To obtain the 2-tuple linguistic variable equivalent to β, the generalized translation function Δ : [0,1]→S×[−12τ,12τ) is defined as follows:

Δ(β)=(si,α)withsi,i=round(β⋅τ)α=β−iτ,α∈[−12τ,12τ).

Conversely, there is a function Δ−1 : S×[−12τ,12τ)→[0,1] used to transform 2-tuple linguistic variable into its equivalent crisp β∈[0,1] as follows:

Δ−1(si,α)=iτ+α=β.

In particular, a linguistic term si can be converted into a 2-tuple linguistic variable (si,0).

Definition 2.

[22,49] Let S be as before and L=(lij)n×n be a linguistic matrix. For i,j=1,2,⋯,n, if Δ(Δ−1(lij)+Δ−1(lji))=(sτ,0),lii=(sτ/2,0), then L is called 2TLPR, where lij=(sij,αij), sij∈S and αij∈[−12τ,12τ).

2.2. The Uncertain 2-Tuple Linguistic Variable and U2TLPR

Definition 3.

[27] Let S be as before. An uncertain 2-tuple linguistic variable [(si,α1),(sj,α2)] with (si,α1)≤(sj,α2) [39] is composed of two 2-tuple linguistic variables (si,α1) and (sj,α2). An uncertain 2-tuple linguistic variable expressing the equivalent information to an interval value [β1,β2] (β1≤β2∈[0,1],β1≤β2) is obtained by the following function:

Δ[β1,β2]=[(si,α1),(sj,α2)],
in which i=round(β1⋅τ), j=round(β2⋅τ), α1=β1−iτ,α1∈[−12τ,12τ) and α2=β2−jτ,α2∈[−12τ,12τ).

Meanwhile, there exists a function Δ−1 used to transform uncertain 2-tuple linguistic variable [(si,α1),(sj,α2)] into an interval value [β1,β2] as follows:

Δ−1[(si,α1)(sj,α2)]=iτ+α1,jτ+α2=[β1,β2].

Specially, an interval [si,sj] can be transformed into an uncertain 2-tuple linguistic variable [(si,0),(sj,0)].

Definition 4.

[37] Let ã=[(si,α1),(sj,α2)] and b̃=[(sk,β1),(sl,β2)] are two uncertain 2-tuple variables. The Manhattan distance between ã and b̃ is defined as

dM(a˜,b˜)=12(|Δ−1(si,α1)−Δ−1(sk,β1)|  +|Δ−1(sj,α2)−Δ−1(sl,β2)|).

In order to rank the uncertain 2-tuple linguistic variables, Wan et al. [37] proposed the possibility degree of uncertain 2-tuple linguistic variables. Let ũi=[ui−,ui+](i=1,2,⋯,n) be uncertain 2-tuple linguistic variable, the possibility degree between ũi and ũj is defined as

p(ũi≥ũj)=121+DENU,
where DE=Δ−1ui+−Δ−1uj++Δ−1ui−−Δ−1uj−, NU=Δ−1ui+−Δ−1uj++Δ−1ui−−Δ−1uj−+lūiūj, lũiũj represents the length of intersection of interval values Δ−1(ũi) and Δ−1(ũj). The possibility degree has the features: (1) 0≤p(ũi≥ũj)≤1; (2) p(ũi≥ũj)+p(ũj≥ũi)=1.

Let pij=p(ũi≥ũj), matrix P=(pij)n×n is called possible degree matrix. It is obvious that P is a fuzzy preference relation. By summing each row of P, we can obtain qi=∑j=1npij. Then, by p¯ij=qi−qj2(n−1)+12, we can obtain a consistent fuzzy preference relation P̄=(p̄ij)n×n.

Then, by normalizing of the sum of each row for P̄, the dominant index of uncertain 2-tuple linguistic variable ũi is defined as follows:

DIi=1n(n−1)(∑j=1n p(ũi≥ũj)+n2−1).

It represents that the average possible degree of ũi preferred to ũj(j=1,2,⋯,n). The bigger the value of dominant index DIi, the larger the uncertain 2-tuple linguistic variable ũi. Therefore, based on the dominant indices DIi, uncertain 2-tuple linguistic variables can be ranked.

Definition 5.

[39] An U2TLPR is defined as Ũ=(ũij)n×n=([ũij−,ũij+])n×n. For ∀i,j=1,2,⋯,n, Ũ satisfies ũii+=ũii−=(sτ/2,0), Δ(Δ−1(ũij−)+Δ−1(ũji+))=Δ(Δ−1(ũji−)+Δ−1(ũij+))=(sτ,0) and ũij−≤ũij+. Where ũij is an uncertain 2-tuple linguistic variable, which indicates the interval linguistic preference degree of the alternative xi over xj.

Particularly, if ũij−=ũij+, then an U2TLPR degenerates into a 2TLPR.

3. ADDITIVE CONSISTENCY OF U2TLPR

In this section, the additive consistency index of an U2TLPR is presented based on the additive consistency measure. Furthermore, a goal programming model is constructed to improve the additive consistency of an U2TLPR with unacceptable additive consistency until it satisfies acceptable additive consistency.

3.1. Additive Consistency Index of U2TLPR

Motivated by the definition of interval fuzzy preference relations [50], the additive consistency of an U2TLPR is defined as Definition 6.

Definition 6.

Let Ũ=(ũij)n×n with ũij=[ũij−,ũij+] be an U2TLPR. For i,j,k=1,⋯,n, if Ũ satisfies the following condition:

Δ−1(ũij†)+Δ−1(ũjk†)+Δ−1(ũki†)= Δ−1(ũkj†)+Δ−1(ũji†)+Δ−1(ũik†),
then Ũ is an additive consistent U2TLPR for †=+,−.

According to Definition 6, we have the following theorem.

Theorem 1.

An U2TLPR Ũ=(ũij)n×n is defined as before. Ũ has additive consistency if and only if

Δ−1(ũij+)+Δ−1(ũij−)+Δ−1(ũjk+)+Δ−1(ũjk−) +Δ−1(ũki+)+Δ−1(ũki−)=3Δ−1(sτ,0).

Proof.

Sufficiency (⇒). Since Ũ is an additive consistent U2TLPR, as Definition 6, we have

Δ−1(ũij+)+Δ−1(ũjk+)+Δ−1(ũki+)= Δ−1(ũkj+)+Δ−1(ũji+)+Δ−1(ũik+),
Δ−1(ũij−)+Δ−1(ũjk−)+Δ−1(ũki−)=Δ−1(ũkj−)+Δ−1(ũji−)+Δ−1(ũik−).

Based on Definition 5, we obtain Δ−1(ũij−)+Δ−1(ũji+)=Δ−1(sτ,0). Then, according to Eqs. (10) and (11), we have

Δ−1(ũij+)+Δ−1(ũij−)+Δ−1(ũjk+)+Δ−1(ũjk−)+Δ−1(ũki+)+Δ−1(ũki−)=3Δ−1(sτ,0).

Necessity (⇐). As Δ−1(ũij+)+Δ−1(ũij−)+Δ−1(ũjk+)+Δ−1(ũjk−)+Δ−1(ũki+)+Δ−1(ũki−)=3Δ−1(sτ,0) and Δ−1(ũij−)+Δ−1(ũji+)=Δ−1(sτ,0), we have

Δ−1(ũij+)+Δ−1(ũjk+)+Δ−1(ũki+)= Δ−1(ũkj+)+Δ−1(ũji+)+Δ−1(ũik+),
Δ−1(ũij−)+Δ−1(ũjk−)+Δ−1(ũki−)= Δ−1(ũkj−)+Δ−1(ũji−)+Δ−1(ũik−).

It means that Δ−1(ũij+)+Δ−1(ũij−)+Δ−1(ũjk+)+Δ−1(ũjk−)+Δ−1(ũki+)+Δ−1(ũki−)=3Δ−1(sτ,0), i.e., Ũ has additive consistency.

Theorem 2.

Let Ũ=(ũij)n×n be an additive consistent U2TLPR. Then, the following conditions are equivalent:

  1. Δ−1(ũij+)+Δ−1(ũij−)+Δ−1(ũjk+)+Δ−1(ũjk−)+Δ−1(ũki+)+Δ−1(ũki−)=3Δ−1(sτ,0),i,j,k=1,2,⋯,n;

  2. Δ−1(ũij+)+Δ−1(ũij−)+Δ−1(ũjk+)+Δ−1(ũjk−)+Δ−1(ũki+)+Δ−1(ũki−)=3Δ−1(sτ,0),i<j<k.

Proof.

It is obvious that (I) ⇒ (II).

(II) ⇒ (I). Based on Definition 6, we know that (I) holds true when any two or three of indices i, j, k are equal. Therefore, we only consider that i≠j≠k, which contains six possible index orders.

  1. i<j<k. Obviously, (I) holds true.

  2. i<k<j. We have Δ−1(ũik+)+Δ−1(ũik−)+Δ−1(ũkj+)+Δ−1(ũkj−)+Δ−1(ũji+)+Δ−1(ũji−)=3Δ−1(sτ,0). Since Δ−1(ũij−)+Δ−1(ũji+)=Δ−1(sτ,0), we obtain

    Δ−1(ũij+)+Δ−1(ũij−)+Δ−1(ũjk+)+Δ−1(ũjk−)+Δ−1(ũki+)+Δ−1(ũki−)=3Δ−1(sτ,0).

    Similarity, (I) holds true in the rest of four cases: j<k<i, j<i<k, k<i<j and k<j<i. Then, we have Thus, we have (II) ⇒ (I). Based on the above analyses, we obtain (I) ⇔ (II).

Theorem 2 provides two ways to check the additive consistency level of an U2TLPR. That is to say, any one of (I) and (II) can be used to judge whether U2TLPR has additive consistency. Considering the computational complexity of (I) and (II), we select (II) to define the additive consistency index of U2TLPR.

In fact, the U2TLPRs provided by DMs do not satisfy additive consistency since the lack of cognition of DMs. Namely, based on (II), there is the deviation between Δ−1(ũij+)+Δ−1(ũij−)+Δ−1(ũjk+)+Δ−1(ũjk−)+Δ−1(ũki+)+Δ−1(ũki−) and 3Δ−1(sτ,0).

The total deviation can be defined by means of Manhattan distance as

MD(Ũ)=∑i<j<kn|Δ−1(ũij+)+Δ−1(ũij−)  + Δ−1(ũjk+)+Δ−1(ũjk−)+Δ−1(ũki+)  + Δ−1(ũki−)−3Δ−1(sτ,0)|.

In the following, the additive consistency index of an U2TLPR Ũ is defined as Definition 7.

Definition 7.

Based on the total deviation MD(Ũ), the additive consistency index of an U2TLPR Ũ is defined as follows:

ACI(Ũ)=2n(n−1)(n−2)∑i<j<kn|Δ−1(ũij+)  +Δ−1(ũij−)+Δ−1(ũjk+)+Δ−1(ũjk−)  +Δ−1(ũki+)+Δ−1(ũki−)−3Δ−1(sτ,0)|.

The additive consistency index indicates the reliability of the uncertain 2-tuple linguistic preference information. The smaller the value of ACI(Ũ), the more consistent and reliable the uncertain preference information in U2TLPR Ũ.

Remark 1.

Zhang and Guo [39] introduced the consistency index of an U2TLPR by calculating the deviation between an U2TLPR and its corresponding consistent U2TLPR. Yao and Hu [40] defined a consistency index of an U2TLPR based on the distance between an U2TLPR and its corresponding consistent U2TLPR. In general, different methods would derive various different consistent U2TLPRs, which would lead to diverse consistency indices of an U2TLPR. In this paper, the additive consistency index of an U2TLPR defined in Definition 7 is reliable and stable since it only depends on the uncertain preference information of original U2TLPR.

Example 1.

[39]. Let X={x1,x2,x3,x4,x5,x6} be a set of alternatives, a DM provides his preference information using U2TLPR, which is shown as follows:

 Ã=([(s4,0), (s4,0)][(s4,0), (s5,0)][(s2,0), (s4,0)][(s3,0), (s4,0)][(s4,0), (s4,0)][(s2,0), (s3,0)][(s4,0), (s6,0)][(s5,0), (s6,0)][(s4,0), (s4,0)][(s2,0), (s3,0)][(s0,0), (s2,0)][(s3,0), (s4,0)][(s2,0), (s4,0)][(s4,0), (s6,0)][(s3,0), (s5,0)][(s5,0), (s6,0)][(s3,0), (s4,0)][(s1,0), (s2,0)][(s5,0), (s6,0)][(s4,0), (s6,0)][(s2,0), (s3,0)][(s6,0), (s8,0)][(s2,0), (s4,0)][(s4,0), (s5,0)][(s4,0), (s5,0)][(s3,0), (s5,0)][(s6,0), (s7,0)][(s4,0), (s4,0)][(s5,0), (s6,0)][(s5,0), (s6,0)][(s2,0), (s3,0)][(s4,0), (s4,0)][(s4,0), (s5,0)][(s2,0), (s3,0)][(s3,0), (s4,0)][(s4,0), (s4,0)]).

By Eq. (12), the additive consistency of à is calculated as ACI(Ã)=0.1792.

In Zhang and Guo [39], the method of constructing the consistent U2TLPR was developed. Using the method, the consistent U2TLPR is obtained

ÃZhang∗=[(s4,0), (s4,0)][(s4,0), (s5,0)][(s2,0), (s4,0)][(s3,0), (s4,0)][(s4,0), (s4,0)][(s2,0), (s3,0)][(s4,0), (s6,0)][(s5,0), (s6,0)][(s4,0), (s4,0)][(s3,0), (s6,0)][(s4,0), (s6,0)][(s3,0), (s4,0)][(s1,0), (s5,0)][(s2,0), (s5,0)][(s1,0), (s3,0)][(s0,0), (s5,0)][(s1,0), (s5,0)][(s0,0), (s3,0)][(s2,0), (s5,0)][(s3,0), (s7,0)][(s3,0), (s8,0)][(s2,0), (s4,0)][(s3,0), (s6,0)][(s3,0), (s7,0)][(s4,0), (s5,0)][(s5,0), (s7,0)][(s5,0), (s8,0)][(s4,0), (s4,0)][(s5,0), (s6,0)][(s5,0), (s7,0)][(s2,0), (s3,0)][(s4,0), (s4,0)][(s4,0), (s5,0)][(s1,0), (s3,0)][(s3,0), (s4,0)][(s4,0), (s4,0)].

Then, based on the concept of consistency index [39], we obtain CI(Ã)Zhang=d(Ã,ÃZhang∗)=0.1375.

In Yao and Hu [40], another method for deriving the consistent U2TLPR was presented. According to their method, the corresponding consistent U2TLPR ÃYao∗ is obtained as follows:

ÃYao∗=[(s4,0), (s4,0)][(s4,0), (s5,0)][(s2,0), (s4,0)][(s3,0), (s4,0)][(s4,0), (s4,0)][(s2,0), (s3,0)][(s4,0), (s6,0)][(s5,0), (s6,0)][(s4,0), (s4,0)][(s2,0), (s4,0)][(s0,0), (s4,0)][(s3,0), (s4,0)][(s2,0), (s4,0)][(s4,0), (s6,0)][(s3,0), (s5,0)][(s3,0), (s6,0)][(s3,0), (s4,0)][(s1,0), (s2,0)][(s4,0), (s6,0)][(s4,0), (s6,0)][(s2,0), (s5,0)][(s4,0), (s8,0)][(s2,0), (s4,0)][(s4,0), (s5,0)][(s4,0), (s5,0)][(s3,0), (s5,0)][(s6,0), (s7,0)][(s4,0), (s4,0)][(s4,0), (s6,0)][(s5,0), (s6,0)][(s2,0), (s4,0)][(s4,0), (s4,0)][(s4,0), (s5,0)][(s2,0), (s3,0)][(s3,0), (s4,0)][(s4,0), (s4,0)].

Then, we have CI(Ã)Yao=d(Ã,ÃYao∗)=0.0250.

It is worthwhile mentioning that CI(Ã)Zhang≠CI(Ã)Yao and U2TLPRs ÃZhang∗ and ÃYao∗ derived from the same original U2TLPR Ã are consistent. It means that different methods for obtaining consistent U2TLPRs would derive diverse additive consistency based estimated U2TLPR. However, the consistency level of U2TLPR should be not change with its corresponding consistent U2TLPR. Because the additive consistency index defined in Definition 7 only depends on the uncertain 2-tuple linguistic information of original U2TLPR, it is reliable and stable.

Based on Definition 7, the following theorem is apparent.

Theorem 3.

Let U2TLPR Ũ be as before. Then,

  1. ACI(Ũ)∈[0,1];

  2. ACI(Ũ)=0 if and only if is an additive consistent U2TLPR.

Providing a predefined threshold CI¯∈[0,1], the concept of acceptable additive consistent U2TLPR is defined as follows:

Definition 8.

Let Ũ be an U2TLPR and CI¯∈[0,1] be a predefined consistency threshold. If

ACI(Ũ)≤CI¯,
then Ũ has acceptable additive consistency. Otherwise, Ũ has unacceptable additive consistency.

In general, the consistency threshold CI¯ is determined on the basis of the practical decision-making problems. As Wan et al. [51] said, if the decision-making problem is significant, the consistency threshold should be allocated a small value; if the decision-making problem is urgent, a lager value should be assigned.

3.2. Obtaining U2TLPR with Acceptable Additive Consistency

Due to the complexity of decision-making environment and the diversity of DMs' cognition, it is hard for DMs to provide U2TLPRs that satisfy additive consistency. Hence, for an unacceptable additive consistent U2TLPR, a goal programming model will be developed to derive an U2TLPR with acceptable additive consistency from the unacceptable additive consistent U2TLPR. Based on the minimum deviation between an acceptable additive consistent U2TLPR Ũ∗=(ũij∗)n×n=([ũij−∗,ũij+∗])n×n and an U2TLPR Ũ=(ũij)n×n=([ũij−,ũij+])n×n with unacceptable additive consistency, a mathematical optimization model (M-1) is constructed as follows:

(M−1)min∑i<jndevijs.t.2n(n−1)(n−2)∑i<j<knDevijk∗≤CI¯,0≤Δ−1(ũij−∗)≤Δ−1(ũij+∗)≤1,i,j=1,⋯,n,i<j,
where devij=(|Δ−1(ũij−∗)−Δ−1(ũij−)|+|Δ−1(ũij+∗)−Δ−1(ũij+)|) and Devijk∗=Δ−1(ũij+∗)+Δ−1(uij−∗)+Δ−1(ũjk+∗)+Δ−1(ũjk−∗)+Δ−1(ũki+∗)+Δ−1(ũki−∗)−3Δ−1(sτ,0). The first constraint condition ensures the obtained U2TLPR Ũ∗ is acceptable additive consistency. Meanwhile, the second condition guarantees the elements located in the upper triangular of Ũ∗ are uncertain 2-tuple linguistic variables.

To better solve model (M-1), it can be transformed into a goal programming model by introducing some parameters. Suppose that

εij=Δ−1(ũij−∗)−Δ−1(ũij−),δij=Δ−1(ũij+∗)−Δ−1(ũij+),
εij+={εij,εij≥00,εij<0,εij−={0,εij≥0−εij,εij<0,
δij+=δij,δij≥00,δij<0,δij−=0,δij≥0−δij,δij<0,
λijk=Δ−1(ũij+∗)+Δ−1(ũij−∗)+Δ−1(ũjk+∗)+Δ−1(ũjk−∗)+ Δ−1(ũki+∗)+Δ−1(ũki−∗)−3Δ−1(sτ,0),
λijk+=λijk,λijk≥00,λijk<0,λijk−=0,λijk≥0−λijk,λijk<0,

Then, |εij|=εij++εij−, |δij|=δij++δij−, |λijk|=λijk++λijk−, εij=εij+−εij−, δij=δij+−δij− and λijk=λijk+−λijk−.

Afterward, a goal programming model (M-2) can be constructed as follows:

(M−2)min∑i<jn(εij++εij−+δij++δij−)s.t.2n(n−1)(n−2)∑i<j<kn(λijk++λijk−)≤CI¯Δ−1(ũij+∗)+Δ−1(ũij−∗)+Δ−1(ũjk+∗)+Δ−1(ũjk−∗)+Δ−1(ũki+∗)+Δ−1(ũki−∗)−3Δ−1(sτ,0)=λijk+−λijk−,0≤Δ−1(ũij−∗)≤Δ−1(ũij+∗)≤1,i,j=1,⋯,n;i<j.

Solving the model (M-2) yields the optimal solutions ũij+∗ and ũij−∗ for all i,j=1,2,⋯,n and i<j. According to Definition 5, the acceptable additive consistent U2TLPR Ũ∗ derived from the original U2TLPR Ũ can be generated by

ũij∗=[ũij−∗,ũij+∗]i<j,[(sτ/2,0),(sτ/2,0)]i=j,[Δ(1−Δ−1(ũij+∗)),Δ(1−Δ−1(ũij−∗))]i>j.

4. DERIVING THE UNCERTAIN 2-TUPLE LINGUISTIC PRIORITY VECTOR FROM U2TLPR

This section develops an optimization model to derive the uncertain 2-tuple linguistic priority weights from U2TLPR.

Let V˜=(v˜1,v˜2,⋯,v˜n)T be a weighting vector in which v˜i=[vi−,vi+] is an uncertain 2-tuple linguistic variable, then V˜ is called uncertain 2-tuple linguistic priority vector if the following conditions hold:

Δ−1(vi+)+∑i≠j, j=1nΔ−1(vj−)≤1,
Δ−1(vi−)+∑i≠j, j=1nΔ−1(vj+)≥1,
0≤Δ−1(vi−)≤Δ−1(vi+)≤1.

Note that the above inequations are inspired by Sugihara et al. [52].

For an uncertain 2-tuple linguistic priority vector V˜ defined as before, let

ũij=[ũij−,ũij+]=[(sτ2,0),(sτ2,0)]i=j,[vl,vu]i≠j,
where vl=ΔΔ−1(sτ2,0)+Δ−1(sτ2,0)(Δ−1(vi−)−Δ−1(vj+)) and vu=ΔΔ−1(sτ2,0)+Δ−1(sτ2,0)(Δ−1(vi+)−Δ−1(vj−)).

Based on Eq. (18), the following theorem is obtained and it is shown as follows:

Theorem 4.

Let Ũ=(ũij)n×n be a matrix, where ũij is defined by Eq. (18), then

  1. Ũ=(ũij)n×n is an U2TLPR;

  2. Ũ=(ũij)n×n has additive consistency.

Proof.

  • (1)

    Based on Eq. (18), we have Δ−1(ũij−)=Δ−1(sτ/2,0)+Δ−1(sτ/2,0)(Δ−1(vi−)−Δ−1(vj+)) and Δ−1(ũji+)=Δ−1(sτ/2,0)+Δ−1(sτ/2,0)(Δ−1(vj+)−Δ−1(vi−)). Then, we have Δ−1(ũij−)+Δ−1(ũji+)=Δ−1(sτ,0). Namely, Δ(Δ−1(ũij−)+Δ−1(ũji+))=(sτ,0). Since ũii+=ũii−=(sτ/2,0) and ũij−≤ũij+. Therefore, we obtain Ũ=(ũij)n×n is an U2TLPR.

  • (2)

    According to Eq. (18), we have Δ−1(ũij+)+Δ−1(ũij−)+Δ−1(ũjk+)+Δ−1(ũjk−)+Δ−1(ũki+)+Δ−1(ũki−)=3Δ−1(sτ,0). It means that Ũ=(ũij)n×n is of additive consistency.

In fact, the Theorem 4 indicates that U2TLPR Ũ=(ũij)n×n with additive consistency can be constructed based on uncertain 2-tuple linguistic priority vector V˜ and Eq. (18).

For i,j=1,2,⋯,n, i≠j, Eq. (18) is equivalent to the following formulas:

Δ−1(ũij−)=Δ−1(sτ/2,0)+Δ−1(sτ/2,0)(Δ−1(vi−)−Δ−1(vj+)),
Δ−1(ũij+)=Δ−1(sτ/2,0)+Δ−1(sτ/2,0)(Δ−1(vi+)−Δ−1(vj−)).

In practical decision-making problems, the U2TLPRs provided by DMs are not additive consistency. Namely, there are deviation between the left and right of Eqs. (19) and (20), respectively. Then, the deviation dij+ and dij− are introduced as follows:

dij−=Δ−1(sτ2,0)+Δ−1(sτ2,0)  × (Δ−1(vi−)−Δ−1(vj+))−Δ−1(ũij−),
dij+=Δ−1(sτ2,0)+Δ−1(sτ2,0)  × (Δ−1(vi+)−Δ−1(vj−))−Δ−1(ũij+),

The smaller the deviation dij+ and dij−, the better additive consistency of U2TLPR Ũ. Motivated by the idea, the uncertain 2-tuple linguistic priority weights of U2TLPR can be obtained by solving the following the optimization model (M-3):

(M−3)min∑i<jn(|dij+|+|dij−|)s.t.dij−=Δ−1(sτ/2,0)+Δ−1(sτ/2,0)(Δ−1(vi−)−Δ−1(vj+))−Δ−1(uij−),dij+=Δ−1(sτ/2,0)+Δ−1(sτ/2,0)(Δ−1(vi+)−Δ−1(vj−))−Δ−1(uij+),Δ−1(vi+)+∑j=1,i≠jnΔ−1(vj−)≤1,Δ−1(vi−)+∑j=1,i≠jnΔ−1(vj+)≥1,0≤Δ−1(vi−)≤Δ−1(vi+)≤1.

For model (M-3), suppose that

ϕij+={dij+,dij+≥00,dij+<0; ϕij−={0,dij+≥0−dij+,dij+<0;
φij+={dij−,dij−≥00,dij−<0; φij−={0,dij−≥0−dij−,dij−<0.

Then

|dij+|=ϕij++ϕij−,dij+=ϕij+−ϕij−,
|dij−|=φij++φij−,dij−=φij+−φij−.

Based on the above transformations, the model (M-3) is equivalent to a linear programming model (M-4)

(M−4)min∑i<jn(ϕij++ϕij−+φij++φij−)s.t.φij+−φij−=Δ−1(sτ/2,0)+Δ−1(sτ/2,0)(Δ−1(vi−)−Δ−1(vj+))−Δ−1(uij−),ϕij+−ϕij−=Δ−1(sτ/2,0)+Δ−1(sτ/2,0)(Δ−1(vi+)−Δ−1(vj−))−Δ−1(uij+),Δ−1(vi+)+∑j=1,i≠jnΔ−1(vj−)≤1,Δ−1(vi−)+∑j=1,i≠jnΔ−1(vj+)≥1,0≤Δ−1(vi−)≤Δ−1(vi+)≤1.

By solving model (M-4), the optimal solutions V˜=(v˜1∗,⋯,v˜n∗)T=([v1−∗,v1+∗],⋯,[vn−∗,vn+∗])T can be obtained.

5. METHOD FOR GDM WITH U2TLPRs

In this section, the DM's weights are determined by the similarity measure of DMs. Then, a new method is proposed to solve GDM problems with U2TLPRs.

5.1. Deriving the Weights of DMs in GDM with U2TLPRs

Consider a GDM where the X={x1,x2,⋯,xn} and D={d1,d2,⋯,dm} are set of the alternatives and set of DMs, respectively. Let W=(w1,w2,⋯,wm)T be DM's weighting vector in which wi≥0 and ∑i=1mwi=1. Assume that Ũ(l)=(ũij(l))n×n with ũij(l)=[ũij(l)−,ũij(l)+]=[(uij(l)−,αij(l)−),(uij(l)+,αij(l)+)] is the individual U2TLPR given by DM dl.

In GDM, the key of solving GDM problems with preference relations is how to allocate the weights to DMs. Once the weights are assigned, all individual preference relations can be aggregated into a collective one.

For each pair of DM (dl,dp), similarity degree between two DMs dl and dp can be measured by Ũ(l)=(ũij(l))n×n and Ũ(p)=(ũij(p))n×n, which is defined as follows:

Slp=1−2n(n−1)∑i<jndMũij(l),ũij(p),
where dM(ũij(l),ũij(p))=12(|Δ−1(uij(l)−,αij(l)−)−Δ−1(uij(p)−,αij(p)−)| + |Δ−1(uij(l)+,αij(l)+)−Δ−1(uij(p)+,αij(p)+)|) is defined by using Eq. (5).

The larger the value of Slp, the higher the degree of similarity between dl and dp.

Based on Eq. (23), the following theorem is apparent.

Theorem 5.

Let Ũ(l)=(ũij(l))n×n and Ũ (p)=(ũij(p))n×n be as before. Then,

  1. 0≤Slp≤1;

  2. Slp=Spl;

  3. Sll=1.

In line with the similarity, the confidence degree CDl of DM dl can be defined as follows:

CDl=∑q=1,q≠lmSlq.

The confidence degree reflects the support degree of a DM from the others. The higher the confidence degree of a DM, the larger the support degree of a DM from the other DMs. Therefore, DM with higher confidence degree should be assigned a larger weight; otherwise, DM should be assigned a smaller weight. Thus, the DMs' weights can be determined by

wl=CDl∑l=1mCDl,l=1,2,⋯,m.

Based on the uncertain 2-tuple linguistic weighted averaging (ULWA2−tuple) [53], the collective matrix Ũc of Ũ(l)(l=1,2,⋯,m) is defined as Definition 9.

Definition 9.

Let Ũ(l)=([ũij(l)−,ũij(l)+])n×n be as before. Then, the the collective matrix Ũc is defined by

Ũc=([ũijc−,ũijc+])n×n,
where ũijc−=Δ( ∑l=1mwlΔ−1(ũij(l)−)), ũijc+=Δ( ∑l=1mwlΔ−1(ũij(l)+)) and ∑k=1mwl=1 and wl≥0 defined in Eq. (25).

From Eq. (26), the following theorem is obtained.

Theorem 6.

Let Ũc and Ũ(l) be as before. Then, the Ũc have the following properties:

  1. Ũc is an U2TLPR.

  2. If Ũ(l)(l=1,2,⋯,m) are additive consistency, then the Ũc is additive consistency.

  3. If Ũ(l)(l=1,2,⋯,m) are acceptable additive consistency, then the Ũc is acceptable additive consistency.

Proof.

  • (1)

    Based on the Definition 9, we have Δ−1ũijc−=∑l=1mwkΔ−1ũij(l)− and Δ−1ũjic+=∑l=1mwlΔ−1ũji(l)+. For i≠j, it follows that Δ(Δ−1ũijc−+Δ−1ũjic+)=Δ(∑l=1mwl(Δ−1ũij(l)−+Δ−1ũji(l)+))=(sτ,0). On the other hand, for i=j, we have ũiic−=ũiic+=(sτ/2,0). Thus, the collective matrix Ũc is an U2TLPR.

  • (2)

    Assume Ũ(l)(l=1,2,⋯,m) are additive consistency. Then, we have the following:

    Δ−1ũij(l)†+Δ−1ũjk(l)†+Δ−1ũki(l)†=Δ−1ũkj(l)†+Δ−1ũji(l)†+Δ−1ũik(l)†.

    It follows that

    Δ−1(ũijc†)+Δ−1(ũjkc†)+Δ−1(ũkic†) −Δ−1(ũkjc†)−Δ−1(ũjic†)−Δ−1(ũikc†)=∑l=1mwl(Δ−1(ũij(l)†)+Δ−1(ũjk(l)†)+Δ−1(ũki(l)†)  −Δ−1(ũkj(l)†)−Δ−1(ũji(l)†)−Δ−1(ũik(l)†))=0.

    Therefore, Δ−1(ũijc†)+Δ−1(ũjkc†)+Δ−1(ũkic†)=Δ−1(ũkjc†)+Δ−1(ũjic†)+Δ−1(ũikc†). It means that Ũc has additive consistency.

  • (3)

    Assume Ũ(l)(l=1,2,⋯,m) are acceptable additive consistency. According to Definition 9, we have the following:

    ACI(Ũc)=2n(n−1)(n−2)∑i<j<knΔ−1(ũijc+)+ Δ−1(ũijc−)+Δ−1(ũjkc+)+Δ−1(ũjkc−)+ Δ−1(ũkic+)+Δ−1(ũkic−)−3Δ−1(sτ,0)=2n(n−1)(n−2)∑i<j<kn∑l=1mwlΔ−1(ũij(l)+)+ Δ−1(ũij(l)−)+Δ−1(ũjk(l)+)+Δ−1(ũjk(l)−)+ Δ−1(ũki(l)+)+Δ−1(ũki(l)−)−3Δ−1(sτ,0)≤2n(n−1)(n−2)∑i<j<kn ∑l=1mwlΔ−1(ũij(l)+)+ Δ−1(ũij(l)−)+Δ−1(ũjk(l)+)+Δ−1(ũjk(l)−)+ Δ−1(ũki(l)+)+Δ−1(ũki(l)−)−3Δ−1(sτ,0)=∑l=1mwl2n(n−1)(n−2)∑i<j<knΔ−1(ũij(l)+)+ Δ−1(ũij(l)−)+Δ−1(ũjk(l)+)+Δ−1(ũjk(l)−)+ Δ−1(ũki(l)+)+Δ−1(ũki(l)−)−3Δ−1(sτ,0)2n(n−1)(n−2)∑i<j<kn≤∑l=1mwlCI¯=CI¯

5.2. New Approach for GDM with U2TLPRs

Based on the above analyses, a new method for GDM with U2TLPRs is graphically depicted in Figure 1.

Figure 1

Decision-making process of group decision-making (GDM) problems with uncertain 2-tuple linguistic preference relations (U2TLPRs).

The GDM approach with U2TLPRs based on our proposed approaches can be formally described in Algorithm 1.

Remark 2.

The aforesaid Algorithm 1 used to deal with the GDM problems with U2TLPRs. If there is one DM in decision-making problems, then the process for deriving DMs' weights and collective U2TLPR is omitted. Therefore, Algorithm 1 is reduced to the method for individual decision-making with an U2TLPR. In other words, Algorithm 1 can not only be implemented to deal with GDM with U2TLPRs, but also be used to solve the individual decision-making with an U2TLPR.

6. ILLUSTRATIVE EXAMPLES

In this section, the new approach proposed in Section 5.2 is applied to an individual decision-making problem and a GDM problem with U2TLPRs, respectively. Meanwhile, some comparative analyses between our proposed methods and other existing methods are provided.

Algorithm 1. The algorithm for the GDM approach with U2TLPRs.

Input: The U2TLPR Ũ(l)(l=1,2,⋯,m) provided by DMs in GDM problem and the predefined consistency

threshold CI¯ and an empty set E.

Output: The best alternative(s).

Begin:

fork=1,2,⋯,m do

ACI(Ũ(l)) ← Eq. (12).

ifACI(Ũ(l))>CI¯ then

E={Ũ(l)}.

Ũ∗(l) ← model (M-2) and Eq. (14).

else

Ũ∗(l) ← Ũ(l).

end

end

DMs' weightswl(l=1,2,⋯,m) ← Eqs. (23–25).

The collective U2TLPRŨc ← Eq. (26).

The uncertain 2-tuple linguistic priority vectorV˜ ← model (M-4).

fori=1,2,⋯,n do

DIi ← Eqs. (6–7).

end

By ranking the dominant indices DIi(i=1,2,⋯,n), the best alternative(s) is (are) obtained.

End

6.1. Application to Individual Decision-Making with U2TLPR

Example 1.

This example is taken from Yao and Hu [40] about risk management in construction projects. Based on the linguistic term set SExample = {s0 = extremely low, s1 = very low, s2 = low, s3 = slightly low, s4 = fair, s5 = slightly high, s6 = high, s7 = very high, s8 = extremely highg}, a DM provide his/her uncertain preference opinion by utilizing U2TLPR, which is listed as follows:

U~=[(s4,0), (s4,0)][(s1,0), (s2,0)][(s6,0), (s7,0)][(s4,0), (s4,0)][(s1,0), (s2,0)][(s4,0), (s5,0)][(s3,0), (s4,0)][(s5,0), (s6,0)][(s2,0), (s3,0)][(s1,0), (s2,0)][(s6,0), (s7,0)][(s4,0), (s5,0)][(s5,0), (s6,0)][(s3,0), (s4,0)][(s2,0), (s3,0)][(s6,0), (s7,0)][(s4,0), (s4,0)][(s6,0), (s7,0)][(s4,0), (s5,0)][(s1,0), (s2,0)][(s4,0), (s4,0)][(s2,0), (s3,0)][(s3,0), (s4,0)][(s5,0), (s6,0)][(s4,0), (s4,0)].

Obviously, this example is an individual decision-making problem. Therefore, Algorithm 1 proposed in Section 5.2 omitted the process for deriving DMs' weights and collective U2TLPR is applied to solve this example.

Set consistency threshold CI¯=0.08 [39] and the U2TLPR U~ has been provided.

Based on Eq. (12), the additive consistency index ACI(U~)=0.2667. We have ACI(U~)>CI¯, it means that U2TLPR U~ has unacceptable additive consistency.

Based on (M-2), a goal programming model is constructed as follows:

minε12++ε12−+δ12++δ12−+ε13++ε13−+δ13++δ13−+ ε14++ε14−+δ14++δ14−+ε15++ε15−+δ15++δ15−+ ε23++ε23−+δ23++δ23−+ε24++ε24−+δ24++δ24−+ ε25++ε25−+δ25++δ25−+ε34++ε34−+δ34++δ34−+ ε35++ε35−+δ35++δ35−+ε45++ε45−+δ45++δ45−s.t.130λ123++λ123−+λ124++λ124−+λ125++λ125−+λ134++λ134−+λ135++λ135−+λ145++ λ145−+λ234++λ234−+λ235++λ235−+ λ245++λ245−+λ345++λ345−≤CI¯,δ14+−δ14−=Δ−1(u~14+∗)−Δ−1(u~14+),δ15+−δ15−=Δ−1(u~15+∗)−Δ−1(u~15+),δ23+−δ23−=Δ−1(u~23+∗)−Δ−1(u~23+),δ24+−δ24−=Δ−1(u~24+∗)−Δ−1(u~24+),δ25+−δ25−=Δ−1(u~25+∗)−Δ−1(u~25+),δ34+−δ34−=Δ−1(u~34+∗)−Δ−1(u~34+),δ35+−δ35−=Δ−1(u~35+∗)−Δ−1(u~35+),δ45+−δ45−=Δ−1(u~45+∗)−Δ−1(u~45+),
s.t.Δ−1(u~12+∗)+Δ−1(u~12−∗)+Δ−1(u~23+∗)+Δ−1(u~23−∗)+Δ−1(u~31+∗)+Δ−1(u~31−∗)−3Δ−1(sτ,0)=λ123+−λ123−,Δ−1(u~12+∗)+Δ−1(u~12−∗)+Δ−1(u~24+∗)+Δ−1(u~24−∗)+Δ−1(u~41+∗)+Δ−1(u~41−∗)−3Δ−1(sτ,0)=λ124+−λ124−,Δ−1(u~12+∗)+Δ−1(u~12−∗)+Δ−1(u~25+∗)+Δ−1(u~25−∗)+Δ−1(u~51+∗)+Δ−1(u~51−∗)−3Δ−1(sτ,0)=λ125+−λ125−,Δ−1(u~13+∗)+Δ−1(u~13−∗)+Δ−1(u~34+∗)+Δ−1(u~34−∗)+Δ−1(u~41+∗)+Δ−1(u~41−∗)−3Δ−1(sτ,0)=λ134+−λ134−,Δ−1(u~13+∗)+Δ−1(u~13−∗)+Δ−1(u~35+∗)+Δ−1(u~35−∗)+Δ−1(u~51+∗)+Δ−1(u~51−∗)−3Δ−1(sτ,0)=λ135+−λ135−,Δ−1(u~14+∗)+Δ−1(u~14−∗)+Δ−1(u~45+∗)+Δ−1(u~45−∗)+Δ−1(u~51+∗)+Δ−1(u~51−∗)−3Δ−1(sτ,0)=λ145+−λ145−,Δ−1(u~23+∗)+Δ−1(u~23−∗)+Δ−1(u~34+∗)+Δ−1(u~34−∗)+Δ−1(u~42+∗)+Δ−1(u~42−∗)−3Δ−1(sτ,0)=λ234+−λ234−,Δ−1(u~23+∗)+Δ−1(u~23−∗)+Δ−1(u~35+∗)+ Δ−1(u~35−∗)+Δ−1(u~52+∗)+Δ−1(u~52−∗)−3Δ−1(sτ,0)=λ235+−λ235−,Δ−1(u~24+∗)+Δ−1(u~24−∗)+Δ−1(u~45+∗)+ Δ−1(u~45−∗)+Δ−1(u~52+∗)+Δ−1(u~52−∗)−3Δ−1(sτ,0)=λ245+−λ245−,Δ−1(u~34+∗)+Δ−1(u~34−∗)+Δ−1(u~45+∗)+Δ−1(u~45−∗)+Δ−1(u~53+∗)+Δ−1(u~53−∗)−3Δ−1(sτ,0)=λ345+−λ345−,0≤Δ−1(u~12−∗)≤Δ−1(u~12+∗)≤1,0≤Δ−1(u~13−∗)≤Δ−1(u~13+∗)≤1,0≤Δ−1(u~14−∗)≤Δ−1(u~14+∗)≤1,0≤Δ−1(u~15−∗)≤Δ−1(u~15+∗)≤1,0≤Δ−1(u~23−∗)≤Δ−1(u~23+∗)≤1,0≤Δ−1(u~24−∗)≤Δ−1(u~24+∗)≤1,0≤Δ−1(u~25−∗)≤Δ−1(u~25+∗)≤1,0≤Δ−1(u~34−∗)≤Δ−1(u~34+∗)≤1,0≤Δ−1(u~35−∗)≤Δ−1(u~35+∗)≤1,0≤Δ−1(u~45−∗)≤Δ−1(u~45+∗)≤1.

Solving the model by LINGO 11.0, the optimal solutions u~ij+∗(i<j) and u~ij−∗(i<j) are derived. Then, according to Eq. (14), we obtain the acceptable additive consistent

U~∗=[(s4,0), (s4,0)][(s1,0), (s2,0)][(s0,0), (s7,0)][(s6,0), (s7,0)][(s4,0), (s4,0)][(s3,0), (s4,0)][(s1,0), (s8,0)][(s4,0), (s5,0)][(s4,0), (s4,0)](s2,0.0625),(s2,0.0625)(s2,0.0625),(s2,0.0625)(s4,−0.025),(s4,−0.025)[(s2,0), (s3,0)][(s1,0), (s2,0)][(s3,0), (s4,0)][(s6,−0.0625), (s6,−0.0625)][(s5,0), (s6,0)][(s6,−0.0625), (s6,−0.0625)][(s6,0), (s7,0)][(s4,−0.0625), (s4,−0.0625)][(s4,0), (s5,0)][(s4,0), (s4,0)][(s2,0), (s3,0)][(s5,0), (s6,0)][(s4,0), (s4,0)].

Solving (M-3), the uncertain 2-tuple linguistic priority weights v~i of U~ is obtained as

v~1=[(s2,0),(s4,−0.0175)],v~2=(s4,−0.0625),(s4,−0.0625),v~3=[(s0,0.0175),(s2,0)],v~4=[(s1,−0.0625),(s1,−0.0625)],v~5=[(s0,0),(s0,0)].

Based on Eqs. (6) and (7), the dominant indices DIi of xi are generated as

DI1=0.2597,DI2=0.2903,DI3=0.1903,DI4=0.1597,DI5=0.1000.

In accordance with the DIi, the ranking order of alternatives is produced as x2≻x1≻x3≻x4≻x5.

With different values of consistency thresholds CI¯, the ranking results is derived and shown in Table 1.

CI¯ Ranking Orders of Alternatives Optimal Alternative
0.03 x2≻x1≻x3≻x5≻x4 x2
0.04 x2≻x3≻x1≻x5≻x4 x2
0.05 x3≻x2≻x1≻x4≻x5 x3
0.06 x3≻x1≻x2≻x4≻x5 x3
0.07 x1≻x3≻x2≻x4≻x5 x1
0.08 x2≻x1≻x3≻x4≻x5 x2
Table 1

Ranking orders of alternatives with different consistency thresholds.

As seen from Table 1, the raking order of alternatives is different based on different consistency thresholds. When CI¯=0.03, the raking order of alternatives is x2≻x1≻x3≻x5≻x4 provided by the proposed method for individual decision-making, which is different from x2≻x3≻x1≻x4∼x5 that obtained by Yao and Hu [40].

Compared with the method proposed by Yao and Hu [40], some differences are analyzed in the following:

  • The additive consistency index of U2TLPR is developed based on the uncertain preference information of original U2TLPR. It is different from the consistency index [40] of U2TLPR constructed by computing the deviation between original U2TLPR and its corresponding consistent U2TLPR.

  • In this paper, in order to keep balance between consistency improvement and uncertain preference preservation, a goal programming model is proposed. However, the consistency improving algorithm proposed in Yao and Hu [40] was based on the relationship between U2TLPR and its crisp priority weights.

  • The optimization model is proposed to derive the uncertain 2-tuple linguistic priority vector from U2TLPR. The uncertain 2-tuple linguistic priority vector is composed of uncertain 2-tuple linguistic variables. However, the model was developed in Yao and Hu [40] to obtain the priority vector, which is composed of crisp priority weights. The uncertain 2-tuple linguistic priority vector can keep the integrity of final decision-making information.

6.2. Application to GDM with U2TLPRs

In this subsection, an investment problem, which is taken from Zhang and Guo [39], is addressed. An investment company wants to invest a sum of money in the best option(s). To reduce the risks involved in making decisions in this uncertain and highly competitive environment, the leader of the company invites a group of experts to participate in the decision and hopes to achieve a consensus solution. A panel with four alternatives is given as follows: A1 is a car industry, A2 is a food company, A3 is a computer company and A4 is an arms manufacturer. There are four DMs, d1, d2, d3, d4 from four consultancy departments: the risk analysis department, the growth analysis department, the social political analysis department and the environmental impact analysis department. These DMs provide their preferences over the alternatives using U2TLPRs as follows:

L̃(1)=[(s4,0),(s4,0)][(s3,0),(s4,0)][(s4,0),(s5,0)][(s4,0),(s4,0)][(s5,0),(s6,0)][(s4,0),(s6,0)][(s2,0),(s3,0)][(s2,0),(s4,0)][(s2,0),(s3,0)][(s5,0),(s6,0)][(s2,0),(s4,0)][(s4,0),(s6,0)][(s4,0),(s4,0)][(s6,0),(s7,0)][(s1,0),(s2,0)][(s4,0),(s4,0)],
L̃(2)=[(s4,0),(s4,0)][(s0,0),(s1,0)][(s7,0),(s8,0)][(s4,0),(s4,0)][(s5,0),(s6,0)][(s3,0),(s4,0)][(s4,0),(s5,0)][(s2,0),(s4,0)][(s2,0),(s3,0)][(s3,0),(s4,0)][(s4,0),(s5,0)][(s4,0),(s6,0)][(s4,0),(s4,0)][(s4,0),(s5,0)][(s3,0),(s4,0)][(s4,0),(s4,0)],
L̃(3)=[(s4,0),(s4,0)][(s4,0),(s5,0)][(s3,0),(s4,0)][(s4,0),(s4,0)][(s2,0),(s4,0)][(s3,0),(s4,0)][(s6,0),(s7,0)][(s4,0),(s6,0)][(s4,0),(s6,0)][(s1,0),(s2,0)][(s4,0),(s5,0)][(s2,0),(s4,0)][(s4,0),(s4,0)][(s2,0),(s3,0)][(s5,0),(s6,0)][(s4,0),(s4,0)],
L̃(4)=[(s4,0),(s4,0)][(s5,0),(s7,0)][(s1,0),(s3,0)][(s4,0),(s4,0)][(s0,0),(s2,0)][(s2,0),(s3,0)][(s2,0),(s4,0)][(s4,0),(s5,0)][(s6,0),(s8,0)][(s4,0),(s6,0)][(s5,0),(s6,0)][(s3,0),(s4,0)][(s4,0),(s4,0)][(s1,0),(s3,0)][(s5,0),(s7,0)][(s4,0),(s4,0)].

Set consistency threshold CI¯=0.09 [39] and the U2TLPR L̃(k)(k=1,2,3,4) have been provided.

Based on Eq. (12), the additive consistency indices ACI(L̃(k))(k=1,2,3,4) of U2TLPR L̃(k) are obtained as follows:

ACI(L̃(1))=0.0417,ACI(L̃(2))=0.0833,ACI(L̃(3))=0.0833,ACI(L̃(4))=0.0208.

Since ACI(L̃(k))<CI¯, k=1,2,3,4, all individual U2TLPR L̃(k) have acceptable additive consistency.

By Eqs. (23–25), the weights wk(k=1,2,3,4) of four DMs are obtained:

w1=0.2482,
w2=0.2530,
w3=0.2553,
w4=0.2435.

According to Eq. (26), the collective U2TLPR L̃c is obtained as follows:

[(s4,0),(s4,0)][(s3,− 0.0021),(s4,0.0284)][(s4,− 0.0284),(s5,0.0021)][(s4,0),(s4,0)][(s3,0.0021),(s5,− 0.0606)][(s3,0.0006),(s4,0.0316)][(s4,− 0.0591),(s5,− 0.0287)][(s3,− 0.0003),(s5,− 0.0307)][(s3,0.0606),(s5,− 0.0021)][(s3,0.0287),(s4,0.0591)][(s4,− 0.0316),(s5,− 0.0006)][(s3,0.0307),(s5,0.0003)][(s4,0),(s4,0)][(s3,0.0319),(s4,0.0624)][(s4,− 0.0624),s5,− 0.0319][(s4,0),(s4,0)].

Solving model (M-4), the uncertain 2-tuple linguistic priority weights v~i(i=1,2,3,4) are obtained:

v~1=[(s1,−0.0023),(s2,−0.0049)],v~2=[(s2,−0.0617),(s3,0.0019)],v~3=[(s1,0.0031),(s2,0.0015)],v~4=[(s1,0.0017),(s2,−0.0597)].

Based on Eqs. (6) and (7), the dominant indices DIi of xi(i=1,2,3,4) as generated as follows:

DI1=0.1978,DI2=0.3339,DI3=0.2090,DI4=0.2593.

In accordance with the DIi(i=1,2,3,4), the ranking order of alternatives is

x2≻x4≻x3≻x1.

Based on the method in Zhang and Guo [39], the ranking of the four alternatives is x2≻x4≻x3≻x1, which is the same with the ranking results provided by our proposed method. Compared with the method proposed by Zhang and Guo [39], we observe:

  • In Zhang and Guo [39], a method for constructing a consistent U2TLPR from the original U2TLPR was introduced. Then, a consistency index of U2TLPR was defined by calculating the deviation between an U2TLPR and its consistent U2TLPR. However, the additive consistency index presented in this paper is developed based on the uncertain preference information of original U2TLPR, which is reliable and stable.

  • The consistency improving algorithm [39] was developed, which takes the consistent U2TLPR as the object of the improvement. The algorithm is automatic and iterative. But in this paper, to adjust the U2TLPR with unacceptable additive consistency into U2TLPR with acceptable additive consistency, a goal programming model is constructed, which can guarantee the minimum deviation between the original U2TLPR and adjusted U2TLPR.

  • In this paper, the DMs' weights are defined by the confidence degree. Nevertheless, the DMs' weights are given by subjective assignment with equal weights [39]. On the other hand, the final decision-making results depend on the uncertain 2-tuple linguistic priority vector given by a optimization model. However, the uncertain 2-tuple linguistic weighed averaging (ULWA2−tuple) [39] operator was used to obtain the ranking order of alternatives.

7. CONCLUSIONS

For decision-making problems with U2TLPRs, the additive consistency and uncertain 2-tuple linguistic priority weights are investigated. The main contributions of this paper are summarized as follows:

  1. Based on the uncertain 2-tuple linguistic preference information of the original U2TLPR, an additive consistency index is defined to check the additive consistency level of U2TLPR.

  2. For improving the additive consistency of the U2TLPRs, a goal programming model is developed to derive the U2TLPRs with acceptable additive consistency from the unacceptable additive consistent U2TLPRs.

  3. To keep the integrity of final decision-making information, an optimization model is constructed to obtain the uncertain 2-tuple linguistic priority weights of U2TLPR.

  4. To determine the DMs' weights in GDM, the similarities and confidence degrees are defined, respectively. Combing the similarities and confidence degrees, the method for determining the DMs' weights is provided.

For an unacceptably additive consistent U2TLPR with large dimension, the solving process of deriving the revised U2TLPR with acceptably additive consistency may be time-consuming. Hence, it is interesting future study to utilize some intelligence algorithms with less time complexity to solve the additive consistency improvement process. In fact, these methods proposed in this paper can be extended to GDM with other decision-making information environment. In addition, future research will be concentrated on the consensus of U2TLPRs and dealing with the consistency level and priority vectors of incomplete U2TLPRs.

CONFLICT OF INTEREST

The authors have no competing financial, professional, or personal interests from other parties that are related to this paper.

AUTHORS' CONTRIBUTIONS

Peng Wu and Ligang Zhou conceived the study and were responsible for the design and development of the data analysis. Jiaming Zhu, Huayou Chen and Yu Chen responsible for data interpretation. Peng Wu and Ligang Zhou wrote the first draft of the article.

ACKNOWLEDGMENTS

The authors would like to thank the editors and the anonymous reviewers for their insightful and constructive comments and suggestions that have led to this improved version of the paper. The work was supported by National Natural Science Foundation of China (Nos. 71771001, 71871001, 71701001, 71501002, 71901001, 71901088), Natural Science Foundation for Distinguished Young Scholars of Anhui Province (No. 1908085J03), The Academic and Technical Leaders Reserve Talents Research Activities Funding Project of Anhui Province (No. 2018H179), College Excellent Youth Talent Support Program (gxyq2019236) and Key Research Project of Humanities and Social Sciences in Colleges and Universities of Anhui Province (SK2019A0013).

REFERENCES

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Journal
International Journal of Computational Intelligence Systems
Volume-Issue
12 - 2
Pages
1339 - 1352
Publication Date
2019/11/22
ISSN (Online)
1875-6883
ISSN (Print)
1875-6891
DOI
10.2991/ijcis.d.191104.001How to use a DOI?
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risenwbib
TY  - JOUR
AU  - Peng Wu
AU  - Jiaming Zhu
AU  - Ligang Zhou
AU  - Huayou Chen
AU  - Yu Chen
PY  - 2019
DA  - 2019/11/22
TI  - On Consistency and Priority Weights for Uncertain 2-Tuple Linguistic Preference Relations
JO  - International Journal of Computational Intelligence Systems
SP  - 1339
EP  - 1352
VL  - 12
IS  - 2
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.191104.001
DO  - 10.2991/ijcis.d.191104.001
ID  - Wu2019
ER  -