Interval-valued intuitionistic linguistic multi-criteria group decision-making method based on the interval 2-tuple linguistic information

1 Introduction

In the real multi-criteria decision-making (MCDM) situation, it is too complex or ill-defined to describe quantitative expressions. Therefore, it will be more suitable to provide the assessment by means of linguistic values rather than numerical ones. Recent years, many methods have been proposed to deal with linguistic information, which can mainly be divided into five categories. The first converts the linguistic information into the triangular fuzzy numbers or other kinds of fuzzy numbers by means of a membership function [1, 2]. Fuzzy numbers, such as triangular fuzzy numbers and trapezoidal fuzzy numbers, can express the linguistic assessment information objectively and have advantages in operation and understanding. However, this method leads to information distortion and may cause some information missing in the process of transformation. The second computes the index of the linguistic terms [3–6, 33]. It is easy to operate, but the aggregated result is hard to be referred to the pre-established discrete linguistic information. The third, based on cloud model [7], the cloud model can describe the fuzziness and randomness of linguistic terms, and avoid the problem of defuzzification. However, the transformation and operation are complex. The forth, based on the outranking approach [8, 9], is appropriate to solve MCDM problems because of its simple logic, but the operation is complex. The fifth, based on 2-tuple linguistic representation model, avoids information distortion and loss which occur previously in the linguistic information processing [10–12].

Herrera and Martínez [11] first proposed the 2-tuple linguistic representation model in 2000. Recently, many studies about the 2-tuple linguistic model focused on the following two aspects: (1) the 2-tuple linguistic aggregation operators [13–15]; (2) the existing MCDM method mixed with the 2-tuple linguistic method [16]. The advantages of 2-tuple linguistic representation model in processing linguistic information are obvious. Due to the complexity of decision-making problems, decision makers’ evaluation information can be categorized into two linguistic grades. Therefore, the results aggregated by the 2-tuple linguistic operators may be between two 2-tuples. In order to solve this problem, the interval 2-tuple is proposed [17, 39]. At present, some studies [18, 39] have been conducted on the interval 2-tuple.

Atanassov [20] introduced intuitionistic fuzzy sets (IFSs). Despite the fact that it is a successful application for fuzzy multi-criteria decision-making (MCDM) [21–25], there are some limitations of IFSs such as it only describes whether a criterion belongs to a fuzzy concept or not. Linguistic values, same as IFSs, implicates that criterion member of the linguistic value is 1. In other words, linguistic values could not describe the non-membership and hesitation of the decision maker.

Afterward, Wang and Li [26] proposed intuitionistic linguistic number (ILN), which is a linguistic term closest to the evaluation information with the membership degree and non-membership degree to this linguistic term. ILN not only fits decision maker’s language expression habit, but also takes the membership degree and non-membership degree into consideration. The expected values and similarity measure of ILN were introduced [26], and some operators of ILN were presented [28, 29]. Yang et al. [30] proposed a fuzzy matrix game model and obtained the weights of criteria by solving a linear programming, then established the decision-making models based on ILN operators. ILNs have some unique advantages: (1) It is more accurate to depict fuzziness than uncertain linguistic variable. For example, given a linguistic set H = {h₀, h₁, h₂, h₃, h₄, h₅, h₆} = {extremely poor, very poor, poor, fair, good, very good, extremely good}, perhaps, the performance evaluation result is then thought to be higher than “good” and lower than “very good”. If it is expressed with uncertain linguistic variables, it will be [h₄, h₅], while the preference degree of h₄ or h₅ is not clear. When ILNs are used, the performance evaluation will be expressed as 〈h₅, (0.8, 0.0) 〉, which is absolutely more precise than uncertain linguistic variables. (2) Intuitionistic linguistic variables are more flexible to express fuzzy information than Atanassov’s intuitionistic fuzzy numbers (IFNs). With regard to an IFN, it represents the membership degree and non-membership degree of elements belonging to a specific qualitative concept, while the specific qualitative concept may not be suitable for describing all performance evaluations, but ILNs will not be limited to a specific qualitative concept. It is applied to a linguistic term set, which makes the expressions with fuzzy information much more flexible. Inspired by Atanassov’s interval-valued intuitionistic fuzzy numbers, there exists such situation that the membership degree and non-membership degree are interval values corresponding to ILNs, which are defined as the interval-valued intuitionistic linguistic numbers (IVILNs) [31].

Regarding MCDM problems based on ILNs or IVILNs, we may take advantage of traditional linguistic MCDM methods aforementioned, but the deficiencies of them will pass down as well. For instance, Liu [32] presented some generalized dependent aggregation operators with ILNs and Liu [34] developed some geometric operators of interval-valued intuitionistic uncertain linguistic variables. Both of them dealt with linguistic terms according to the linguistic symbolic model, and thus the rough descriptions of the uncertainties of qualitative information are inevitable, which may lead to information distortion. Fortunately, the interval 2-tuple linguistic representation model is a good way for linguistic computation without the shortcomings of aforementioned methods.

Based on the previous work, we proposed a novel ranking method for the interval 2-tuples, and then developed an MCDM method with IVILN information based on the interval 2-tuples in this paper. The rest of the paper is organized as follows. In Section 2, some basic notions related to interval 2-tuples are presented. In Section 3, some existing ranking methods are introduced, and then a novel ranking approach is developed. In Section 4, the IVILNs and the related concepts are presented. Then a method of conversion of the IVILN into the interval 2-tuple is established. In Section 5, an illustrative example is provided and subsequently, some comparisons are made. Finally, we conclude this paper in Section 6.

2 Interval 2-tuple and related concepts

Let H = {h _i |i = 0, 1, …, 2g} be a finite linguistic term set with odd cardinality, where represents a possible value for a linguistic variable. It satisfies the following characteristics [35]:

The set is ordered: h _i  > h _j , if i > j;

The 2-tuple linguistic representation model is based on the concept of symbolic translation [11, 36]. It represents the linguistic assessment information by means of a 2-tuple (h _i , α _i ), where h _i is a linguistic term from a predefined linguistic term set H and α _i  ∈ [-0.5, 0.5) is the so-called symbolic translation.

Definition 1. [11] Let H = {h₀, h₁, …, h_2g} be a linguistic term set and θ ∈ [0, 2g] be a value representing the result of a symbolic aggregation, then the 2-tuple that expresses the equivalent information to θ can be obtained by the following function δ, where { Δ : [ 0 , 2 g ] → H × [ - 0.5 , 0.5 ) Δ ( θ ) = ( h i , α i ) = { h i , i = round ( θ ) α i = θ - i

In this formula, Round (θ) denotes a round number of θ.

Proposition 1. [11] Let H = {h₀, h₁, …, h_2g} be a linguistic assessment set and (h _i , α _i ) be a 2-tuple. There is a function Δ^-1, so that the 2-tuple (h _i , α _i ) can return to its equivalent numerical value θ ∈ [0, 2g], the function Δ^-1 is an inverse function of Δ defined in Definition 1, and can be described as follows: { Δ - 1 : H × [ - 0.5 , 0.5 ) → [ 0 , 2 g ] Δ - 1 ( h i , α i ) = θ = i + α i

Definition 2. [17] Given two 2-tuples (h _i , α _i ) and ( h i ′ , α i ′ ) , then an interval 2-tuple can be defined as [ ( h i , α i ) , ( h i ′ , α i ′ ) ] , satisfying h i , h i ′ ∈ H , α i , α i ′ ∈ [ - 0.5 , 0.5 ) and ( h i , α i ) ≤ ( h i ′ , α i ′ ) . Specially, if h i = h i ′ and α i = α i ′ , the interval 2-tuple reduces to a 2-tuple.

For example, the interval 2-tuple [(h₂, 0.4) , (h₂, 0.4)] reduces to 2-tuple (h₂, 0.4).

Definition 3. [40] Given two interval 2-tuples h ˜ 1 = [ ( h 1 , α 1 ) , ( h 1 ′ , α 1 ′ ) ] and h ˜ 2 = [ ( h 2 , α 2 ) , ( h 2 ′ , α 2 ′ ) ] , then their operations are defined as ( 1 ) h ˜ 1 ⊕ h ˜ 2 = [ Δ ( Δ - 1 ( h 1 , α 1 ) + Δ - 1 ( h 2 , α 2 ) ) , Δ ( Δ - 1 ( h 1 ′ , α 1 ′ ) + Δ - 1 ( h 2 ′ , α 2 ′ ) ) ] , ( 2 ) λ h ˜ 1 = [ Δ ( λ Δ - 1 ( h 1 , α 1 ) ) , Δ ( λ Δ - 1 ( h 1 ′ , α 1 ′ ) ) ] , λ ≥ 0 .

Definition 4. [17] Let h ˜ i = [ ( h i , α i ) , ( h i ′ , α i ′ ) ] 1 pt 1 pt ( i = 1 , 2 , … , n ) be a set of interval 2-tuples, then the weighted arithmetic averaging (IT-WAA) operator is computed as

IT - WAA ( h ˜ 1 , h ˜ 2 , … , h ˜ n ) = ω 1 [ ( h 1 , α 1 ) , ( h ′ 1 , α ′ 1 ) ] ⊕ ω 2 [ ( h 2 , α 2 ) , ( h ′ 2 , α ′ 2 ) ] ⊕ ⋯ ⊕ ω n [ ( h n , α n ) , ( h ′ n , α ′ n ) ] = Δ [ ∑ i = 1 n ω i Δ - 1 ( h i , α i ) , ∑ i = 1 n ω i Δ - 1 ( h ′ i , α ′ i ) ] ,(1) where ω = (ω₁, ω₂, …, ω _n ) is the weight vector of ( h ˜ 1 , h ˜ 2 , … , h ˜ n ) with ω _i  ∈ [0, 1] and ∑ i = 1 n ω i = 1 .

Definition 5. [17] Let h ˜ i = [ ( h i , α i ) , ( h i ′ , α i ′ ) ] ( i = 1 , 2 , … , n ) be a set of interval 2-tuples, then the ordered weighted averaging (IT-OWA) operator is defined as IT - OWA ( h ˜ 1 , h ˜ 2 , … , h ˜ n ) = w 1 [ ( h φ 1 , α φ 1 ) , ( h ′ φ 1 , α ′ φ 1 ) ] 1 pt ⊕ w 2 [ ( h φ 2 , α φ 2 ) , ( h ′ φ 2 , α ′ φ 2 ) ] ⊕ ⋯ ⊕ w n [ ( h φ n , α φ 2 ) , ( h ′ φ 2 , α ′ φ 2 ) ] = Δ [ ∑ i = 1 n w i Δ - 1 ( h φ i , α φ i ) , ∑ i = 1 n w i Δ - 1 ( h ′ φ i , α ′ φ i ) ] ,(2)

where w = (w₁, w₂, …, w _n ) is the associated weight vector of ( h ˜ φ 1 , h ˜ φ 2 , … , h ˜ φ n ) , w _i  ∈ [0, 1] and ∑ i = 1 n w i = 1, (φ₁, φ₂, …, φ _n ) is a permutation of (1, 2, …, n), such that h ˜ φ i ≥ h ˜ φ i + 1 for all i = 1, 2, …, n - 1.

In this section, we review some existing ranking methods for the interval 2-tuples, and analyze the limitations of these methods. Then, we propose a novel ranking method of the interval 2-tuples.

Zhang [39] developed a ranking method to compare two interval 2-tuples based on the score and accuracy functions.

Definition 6. [39] Let h ˜ = [ ( h i , α i ) , ( h j ′ , α j ′ ) ] be an interval 2-tuple, then its score function is defined as Sf ( h ˜ ) = ( i + j ) / ( 4 g ) + ( α i + α ′ j ) / 2(3)

Definition 7. [39] Let h ˜ = [ ( h i , α i ) , ( h j ′ , α j ′ ) ] be an interval 2-tuple, then its accuracy function is defined as Af ( h ˜ ) = ( j - i ) / ( 2 g ) + ( α j ′ - α i )(4)

Definition 8. [39] Given two interval 2-tuples h ˜ 1 = [ ( h i 1 , α i 1 ) , ( h j 1 ′ , α j 1 ′ ) ] and h ˜ 2 = [ ( h i 2 , α i 2 ) , ( h j 2 ′ , α j 2 ′ ) ] , then

If Sf ( h ˜ 1 ) > Sf ( h ˜ 2 ) , then h ˜ 1 > h ˜ 2 ,

Example 1. Suppose that H = {h₀, h₁, …, h₄} (g = 2) is a linguistic term set, h ˜ 1 = [ ( h 2 , 0.4 ) , ( h 3 , 0.1 ) ] and h ˜ 2 = [ ( h 3 , 0.025 ) , ( h 3 , 0.225 ) ] are two interval 2-tuples, by applying equation (3) and (4), we have Sf ( h ˜ 1 ) = 0.875 , Sf ( h ˜ 2 ) = 0.875 , Af ( h ˜ 1 ) = 0.2 , Af ( h ˜ 2 ) = 0.2 . However, the result is obviously wrong, because it is clear that h ˜ 2 is better than h ˜ 1 .

Lin et al. [17] proposed a ranking method based on the possibility degree:

Definition 9. [17] Given two interval 2-tuples h ˜ 1 = [ ( h i 1 , α i 1 ) , ( h j 1 ′ , α j 1 ′ ) ] and h ˜ 2 = [ ( h i 2 , α i 2 ) , ( h j 2 ′ , α j 2 ′ ) ] , the possibility degree formula for the comparison between h ˜ 1 and h ˜ 2 can be represented as follows: p ( h ˜ 1 ≥ h ˜ 2 ) = max { 1 - max ( Δ - 1 ( h ′ j 2 , α ′ j 2 ) - Δ - 1 ( h i 1 , α i 1 ) l h ˜ 1 + l h ˜ 2 , 0 ) , 0 }(5)

where l h ˜ 1 = Δ - 1 ( h j 1 ′ , α j 1 ′ ) - Δ - 1 ( h i 1 , α i 1 ) ,

      l h ˜ 2 = Δ - 1 ( h j 2 ′ , α j 2 ′ ) - Δ - 1 ( h i 2 , α i 2 ) .

Assume that there are n interval 2-tuples h ˜ i ( i ∈ n ) , and we compare each interval 2-tuple h ˜ i with all interval 2-tuples h ˜ j ( j ∈ n ) by using Equation (5), then a complementary matrix can be set up as follows: P = [ p 11 p 12 ⋯ p 1 n p 21 p 22 ⋯ p 2 n ⋮ p n 1 p n 2 ⋯ p nn ] where p _ij  ≥ 0, p _ij  + p _ji  = 1 and p _ii  = 1/2.

According to Ref. [37], Theorem 2 can be obtained as follows.

Theorem 2. [37] Let P = (p _ij ) _n×n be a fuzzy complementary judgment matrix, then its priority vector can be obtained as follows: v i = ∑ j = 1 n p ij n 2 / 2(6)

Definition 10. Given two interval 2-tuples h ˜ 1 = [ ( h i 1 , α i 1 ) , ( h j 1 ′ , α j 1 ′ ) ] and h ˜ 2 = [ ( h i 2 , α i 2 ) , ( h j 2 ′ , α j 2 ′ ) ] , the elements of their priority vectors are (v₁, v₂), then the order is defined as follows:

if v₁ > v₂, then h ˜ 1 > h ˜ 2 ,

Example 2. Let h ˜ 1 = [ ( h 1 , 0.2 ) , ( h 2 , 0.4 ) ] and h ˜ 2 = [ ( h 1 , 0.4 ) , ( h 2 , 0.2 ) ] be two interval 2-tuples, then use equation (5) and (6) to obtain that v₁ = v₂ = 0.5. Thus we cannot get which one is better.

Definition 11. Let h ˜ 1 = [ ( h i 1 , α i 1 ) , ( h j 1 ′ , α j 1 ′ ) ] , h ˜ 2 = [ ( h i 2 , α i 2 ) , ( h j 2 ′ , α j 2 ′ ) ] and h ˜ 0 = [ ( h 0 , 0 ) , ( h 2 g , 0 ) ] be three interval 2-tuples, then the order is defined as follows:

if p ( h ˜ 1 ≥ h ˜ 0 ) > p ( h ˜ 2 ≥ h ˜ 0 ) , then h ˜ 1 > h ˜ 2 ,

Example 3. Assume that H = {h₀, h₁, …, h₆} (g = 3) is a linguistic term set, h ˜ 1 = [ ( h 2 , 0.4 ) , ( h 3 , 0.1 ) ] and h ˜ 2 = [ ( h 3 , 0.025 ) , ( h 3 , 0.225 ) ] are two interval 2-tuples, by Definition 11, we can have p ( h ˜ 1 ≥ h ˜ 0 ) = 0.463 , p ( h ˜ 2 ≥ h ˜ 0 ) = 0.520 . Thus, h ˜ 2 ≻ h ˜ 1 .

Example 4. Assume that H = {h₀, h₁, …, h₆} (g = 3) is a linguistic term set, h ˜ 1 = [ ( h 1 , 0.2 ) , ( h 2 , 0.4 ) ] and h ˜ 2 = [ ( h 1 , 0.4 ) , ( h 2 , 0.2 ) ] are two interval2-tuples, we can obtain p ( h ˜ 1 ≥ h ˜ 0 ) = 0.333 , p ( h ˜ 2 ≥ h ˜ 0 ) = 0.324 . So h ˜ 1 ≻ h ˜ 2 .

China Railway Wuxin Industry is a large enterprise in Changsha, and mainly engages in the design and production of moulds and engineering equipment. In the past several years, its business has achieved great success and over 40 kinds of products have been designed. Recently, the design department designed a new type of mould, and the managers decided to produce them on a larger scale. The purchasing department had to select a supplier to purchase a new kind of raw material from several companies based on market surveys and analysis. After a pre-evaluation, the following four companies were chosen to take into consideration: (1) Changsha Lyrun New Material Co., Ltd. (x₁), (2) Zhongxing Technology Co., Ltd. (x₂), (3) Times Technology Co., Ltd. (x₃), (4) Kelite Co., Ltd. (x₄). As the differences between the prices of raw materials in each company were rather small, so the following criteria were mainly taken into considerate: Quality of material (c₁), supply capacity (c₂) and response to changes (c₃). The weights vector of criteria is (0.4, 0.3, 0.3).

The purchasing department invited three experts to evaluate these four alternatives based on the following established linguistic term set: H = {h₀ = Very Poor, h₁ = Poor, h₂ = Medium Poor, h₃ = Fair, h₄ = Medium Good, h₅ = Good, h₆ = Very Good}. The evaluation information was provided in the form of IVILNs, which is shown in Tables 1–3.

In this paper, we define the certainty interval of interval-valued intuitionistic linguistic numbers and a novel ranking method for interval 2-tuples. Under the background of interval-valued intuitionistic linguistic decision-making problems, this paper makes three contributions with respect to the existing studies. Firstly, a novel ranking method to compare the interval 2-tuples makes great sense. The ranking method can avoid information distortion and loss which occur in the linguistic information processing. Secondly, the conversion of IVILNs into interval 2-tuple can be realized by applying the certainty interval of IVILNs. Lastly, an MCDM method with IVIL information based on the interval 2-tuple linguistic aggregation operators is developed, thus it can provide solutions for uncertain linguistic decision-making. Our method can be used to solve the multi-criteria group decision-making problems in which the criteria values are IVILNs, such as supplier selection and investment return evaluation. The validity and feasibility of this method have been demonstrated by an applied example together with the corresponding comparison analysis with three other methods.

However, this research also has some limitations, which can be considered in future. (1) To extend the application area of the proposed method, it can be utilized to more real world problems to demonstrate its practicality and efficiency. (2) Due to the complexity and uncertainty of MCDM problems, there exists the situation that large-scale experts may be involved in the decision-making process, for example, emergency decision-making. So some new models will be needed to address large group decision-making problems [41]. (3) Some researchers proposed different kinds of complex linguistic preferences [42], such as hesitant fuzzy linguistic term set [43], linguistic distribution model [44], which can be related to deal with MCDM problems in an IVIL environment. Thus, it will be interesting to propose some new approaches based on complex linguistic preferences.

Conflict of interests

The authors declare that there is no conflict of interests regarding the publication of this paper.