Consensus-based framework to MCGDM under multi-granular uncertain linguistic environment

Abstract

This article puts forward a consensus framework for multi-criteria group decision-making (MCGDM) with multi-granular uncertain linguistic information, where different decision-makers employ different linguistic scales to express their opinions on alternatives. In this framework, we first establish a relative projection model for two interval linguistic 2-tuples denoted by linguistic term sets with different granularities, and then extend it to multi-granular interval 2-tuple linguistic matrices. Based on this, the similarity measurement between two individual matrices as well as the consensus degree of an individual matrix to a group one is defined. Furthermore, the similarity degrees are applied to determine the importance weights of decision-makers. Besides, a concept of acceptable consensus is introduced to assist in devising an iterative procedure for the purpose of promoting the group consensus and reaching a satisfactory agreement. Finally, to demonstrate the validity and advantages of our proposed framework, an illustrative example which has been examined is furnished.

Keywords

MCGDM Multi-granular uncertain linguistic information similarity measurement acceptable consensus relative projection model

1 Introduction

Multi-criteria group decision-making (MCGDM) is usually concerned with a fixed set of decision-makers, several criteria and a collection of alternatives. Since MCGDM is very close to real-life, it has been extensively studied [1 –4]. Generally, decision-makers are required to give assessments over alternatives with respect to different criteria. The assessments can be expressed by various forms: intuitionistic fuzzy numbers [5], single-valued trapezoidal neutrosophic numbers [6, 7], type-2 fuzzy numbers [8, 9], interval type-2 fuzzy numbers [10, 11], extended grey numbers [12, 13], intuitionistic linguistic fuzzy numbers [14, 15], uncertain linguistic values [16], 2-tuple linguistic values [17], 2-tuple linguistic intuitionistic fuzzy numbers [18], interval 2-tuple linguistic values [19], etc.

Owing to the complexity and vagueness of real-world decision environment and the limited knowledge of decision-makers, giving qualitative assessments like uncertain linguistic values is more appropriate, precise and flexible for decision-makers to express judgments on alternatives. For this reason, this paper primarily focuses on MCGDM with uncertain linguistic variables which are further transformed into interval linguistic 2-tuples.

In light of the reality that the educational backgrounds, professional knowledge and experiences of different decision-makers are quite distinctive, they may employ multi-granular linguistic term sets to evaluate alternatives. This is considered as one kind of MCGDM problems with multi-granular linguistic context. Recently, many researchers have paid attention to this MCGDM from different angles. For instance, Zhang [20, 21] established several aggregation operators to handle multi-granular linguistic MCGDM problems. Liu et al. [22] put forward a VIKOR (VIsekriterijumska optimizacija iKOmpromisno Resenj) method based on interval linguistic 2-tuples to deal with multi-granular MCGDM problems and furnished a case on personnel selection. Xu et al. [23] developed a four-way procedure to estimate missing values for multi-granular incomplete fuzzy linguistic preference relations.

Consensus is a main issue that should be concerned in MCGDM. To measure the group consensus, the group opinion should be derived at first. To do this, the generation of decision-makers’ weights is imperative to be addressed. It is well-known that the more approval the decision maker is considered by others, the more importance he/she is. This can be reflected by calculating the importance weight of the decision-maker on the basis of the similarity between a decision-maker and all the others. Up to now, the similarity measurement has been researched extensively, e.g., Liu and Zhang [24] defined a similarity degree between interval fuzzy preference relations (IFPRs) by calculating the average value of absolute deviation of each pair of corresponding elements in two IFPRs. Hwang [25] considered the use of the Sugeno integral to define the similarity measurement of intuitionistic fuzzy sets.

Regarding the group consensus, consensus reaching methods are imperative and have also already received great attention [26 –33]. Among the published literatures, Xu et al. [28] developed a consensus reaching process for uncertain linguistic setting to reach a satisfactory consensus in a group. Dong et al. [29] made a research on the consensus of the GDM problem based on multi-granular unbalanced 2-tuple linguistic preference relations.

Summarizing the above-stated researches, the motivations and contributions of this paper are:

To the best of our knowledge, the notion of similarity degree for multi-granular interval linguistic 2-tuples has not been studied yet. Among the measurements for calculating the similarity for two elements, the projection is a relatively significant tool as it can describe the distance as well as the included angle between two objects. Therefore, it can better reflect the difference between two elements and has been applied to various areas [34 –36]. Following this, we intend to introduce the relative projection model for two interval 2-tuple linguistic matrices denoted by different linguistic scales to further define the similarity measurement for two individual matrices as well as the consensus measurement for an individual matrix to a group one.

A common point in the aforementioned works [20 –23] is that different decision-makers adopt different linguistic scales to express assessments. Meanwhile, the afore-mentioned consensus methods [26 –33] are based on interval linguistic context with one linguistic term set or are not suitable for interval linguistic settings. Therefore, the consensus problem for MCGDM based on multi-granular uncertain linguistic context has not been taken into consideration yet. This creates the need to put forward a consensus framework to tackle MCGDM problems with multi-granular uncertain linguistic assessments.

The rest of this paper is arranged as follows. Section 2 reviews some basic concepts. In Section 3, relative projection models are established for multi-granular interval linguistic 2-tuples and multi-granular interval 2-tuple linguistic matrices, respectively. Also, a similarity measurement and consensus definitions are introduced. Section 4 presents a consensus-based framework for MCGDM with multi-granular uncertain linguistic information. In Section 5, a practical example is furnished to verify the validity of the proposed framework. In the end, Section 6 draws some conclusions.

2 Some basic concepts

Since 1975, Zadeh [37] introduced the concept of linguistic variables, it has been extensively applied to represent decision-makers’ opinions. Generally, the linguistic terms belong to a scale, termed a linguistic term set which can be expressed by various forms. In this paper, we employ the expression S = {s_τ|τ = 0, 1, …, g}. The following characteristics are required to be satisfied for two arbitrary linguistic terms [17]:

s_τ
₁ < s_τ
₂, iff τ₁ < τ₂;

There is a negation operator: Neg (s_τ
₁) = s_τ
₂, so that τ₂ = g - τ₁;

min(s_τ
₁, s_τ
₂) = s_τ
₁, iff s_τ
₁ ≤ s_τ
₂;

max(s_τ
₁, s_τ
₂) = s_τ
₁, iff s_τ
₁ ≥ s_τ
₂.

Regarding the 2-tuple linguistic modelling whose emergence is aiming to computing with linguistic values, different computational techniques are established [17 , 38–41]. To cope with MCGDM with multi-granular uncertain linguistic information, we are going to employ the method in Ref. [38], which has been extended to interval linguistic 2-tuples [20].

Definition 1. [20] Let S = {s_τ|τ = 0, 1, …, g} be a linguistic term set, and [δ^-, δ⁺] is an interval number, then an interval linguistic 2-tuple [(s_{τ
^-}, μ^-) , (s_{τ
⁺}, μ⁺)] that represents identical information to [δ^-, δ⁺] can be generated with: $\begin{matrix} Λ ([δ^{-}, δ^{+}]) = [(s_{τ^{-}}, μ^{-}), (s_{τ^{+}}, μ^{+})], \\ with {\begin{matrix} s_{τ^{-}}, & τ^{-} = round (δ^{-} \cdot g) \\ s_{τ^{+}}, & τ^{+} = round (δ^{+} \cdot g) \\ μ^{-} = δ^{-} - \frac{τ^{-}}{g}, & μ^{-} \in [- 0.5 / g, 0.5 / g] \\ μ^{+} = δ^{+} - \frac{τ^{+}}{g}, & μ^{+} \in [- 0.5 / g, 0.5 / g] \end{matrix} . \end{matrix}$ (1)

On the contrary, an equivalent interval number [δ^-, δ⁺] can be returned from [(s_{τ
^-}, μ^-) , (s_{τ
⁺}, μ⁺)] by a converse function Λ^-1 defined as: $\begin{matrix} Λ^{- 1} ([(s_{τ^{-}}, μ^{-}), (s_{τ^{+}}, μ^{+})]) \\ = [\frac{τ^{-}}{g} + μ^{-}, \frac{τ^{+}}{g} + μ^{+}] = [δ^{-}, δ^{+}] . \end{matrix}$ (2)

Similar to the idea of converting linguistic terms to linguistic 2-tuples [17], the conversion of an uncertain linguistic variable to an interval linguistic 2-tuple can be implemented by separately adding a value 0 to the upper and lower bounds of the linguistic interval as the corresponding symbolic translations,i.e., $Λ ([s_{τ^{-}}, s_{τ^{+}}]) = [(s_{τ^{-}}, 0), (s_{τ^{+}}, 0)] .$ (3)

Through Equation (3), the drawbacks that uncertain linguistic variables represent discrete information and virtual linguistic terms [42] which are continuous but can only appear in operation can be overcame.

In group decision-making, an important issue to be addressed is aggregation. In the following, we recall the interval-valued 2-tuple ordered weighted average (IVTOWA) operator.

Definition 2. [20] Let S = {s_τ|τ = 0, 1, …, g}, and $\tilde{V} = {\tilde{v} = [(s_{τ^{-}}, μ^{-}), (s_{τ^{+}}, μ^{+})] | s_{τ^{+}} \in S = {s_{0}, s_{1}, \dots, s_{g}}, μ^{-},$ μ⁺∈ [-0.5/g, 0.5/g) } is a set of interval linguistic 2-tuples. An IVTOWA operator is a function: $IVTOWA : (\tilde{V})^{n} \to \tilde{V}$ , expressed with $\begin{matrix} IVTOWA ({\tilde{v}}_{1}, {\tilde{v}}_{2}, \dots, {\tilde{v}}_{n}) \\ = ω_{1} {\tilde{v}}_{o (1)} \oplus ω_{2} {\tilde{v}}_{o (2)} \oplus \dots \oplus ω_{n} {\tilde{v}}_{o (n)} \\ = Λ [\sum_{p = 1}^{n} ω_{p} Λ^{- 1} (s_{τ_{o (p)}^{-}, μ_{o (p)}^{-})}, \\ \sum_{p = 1}^{n} ω_{p} Λ^{- 1} (s_{τ_{o (p)}^{+}}, μ_{o (p)}^{+})] \end{matrix}$ (4) where o : {1, 2, …, n} → {1, 2, …, n} is any permutation such that $[(s_{τ_{o_{(p)}}^{-}}, μ_{o_{(p)}}^{-}), (s_{τ_{o_{(p)}}}^{+}, μ_{o_{(p)}}^{+})] \geq [(s_{τ_{o_{(p + 1)}}}^{-}, μ_{o_{(p + 1)}}^{-}),$ $(s_{τ_{o_{(p + 1)}}^{+}}, μ_{o_{(p + 1)}}^{+})]$ and ϖ = (ω₁, ω₂, …, ω_n) ^T is the associated weight vector of the unordered interval 2-tuples, satisfying $\sum_{p = 1}^{n} ω_{p} = 1$ and ω_p ∈ [0, 1].

3 Relative-projection-based similarity and consensus measurements

This section firstly concentrates on developing the relative projection models for multi-granular interval context. Based on this, the similarity and consensus between interval 2-tuple linguistic matrices expressed with multi-granular linguistic term sets are defined.

3.1 Relative projection model for multi-granular interval linguistic 2-tuples

Based on the fundamental principle of the projection, Wei [43] put forward a projection method for linguistic 2-tuples. Insufficiently, there are some deficiencies existing in that method. On one hand, it could not precisely describe how close one vector approaches to another because of the thought that the larger the closer is not suitable to all situations [35]. On the other hand, the employed transformation functions between a linguistic 2-tuple and a crisp number [17] cannot afford to reflect the difference between two seemingly same linguistic terms which may be from different linguistic scales. In other words, it is only applicable for linguistic variables assessed in one linguistic term set. To overcome these limitations, we combine Definition 1 with the relative projection [35] to develop a relative projection model for multi-granular interval linguistic 2-tuples.

Definition 3. Let $S^{(l)} = {s_{τ}^{(l)} | τ = 0, 1, \dots, g_{l}} (l = 1, 2)$ be two linguistic term sets with different granularities, ${\tilde{v}}_{l} = [(s_{τ_{l}^{-}}^{(1)}, μ_{l}^{-}), (s_{τ_{l}^{+}}^{(1)}, μ_{l}^{+})] (l = 1, 2)$ are two interval linguistic 2-tuples with $s_{τ_{l}^{-}}^{(l)}, s_{τ_{l}^{+}}^{(l)} \in {s_{0}^{(l)}, s_{1}^{(l)}, \dots, s_{g_{l}}^{(l)}}$ and $μ_{l}^{-}, μ_{l}^{+} \in [- 0.5 / g_{l}, 0.5 / g_{l})$ , then the relative projection of ${\tilde{v}}_{1}$ over ${\tilde{v}}_{2}$ is defined by $\begin{matrix} \bar{{\tilde{P}}_{{\tilde{v}}_{2}} ({\tilde{v}}_{1})} \\ = \frac{{\tilde{P}}_{{\tilde{v}}_{2}} ({\tilde{v}}_{1})}{| {\tilde{v}}_{2} |} \\ = \frac{(\frac{τ_{1}^{-}}{g_{1}} + μ_{1}^{-}) (\frac{τ_{2}^{-}}{g_{2}} + μ_{2}^{-}) + (\frac{τ_{1}^{+}}{g_{1}} + μ_{1}^{+}) (\frac{τ_{2}^{+}}{g_{2}} + μ_{2}^{+})}{\sqrt{{(\frac{τ_{2}^{-}}{g_{2}} + μ_{2}^{-})}^{2} + {(\frac{τ_{2}^{+}}{g_{2}} + μ_{2}^{+})}^{2}} \cdot | {\tilde{v}}_{2} |} \\ = \frac{(\frac{τ_{1}^{-}}{g_{1}} + μ_{1}^{-}) (\frac{τ_{2}^{-}}{g_{2}} + μ_{2}^{-}) + (\frac{τ_{1}^{+}}{g_{1}} + μ_{1}^{+}) (\frac{τ_{2}^{+}}{g_{2}} + μ_{2}^{+})}{{(\frac{τ_{2}^{-}}{g_{2}} + μ_{2}^{-})}^{2} + {(\frac{τ_{2}^{+}}{g_{2}} + μ_{2}^{+})}^{2}} \end{matrix}$ (5) where $| {\tilde{v}}_{2} | = \sqrt{{(\frac{τ_{2}^{-}}{g_{2}} + μ_{2}^{-})}^{2} + {(\frac{τ_{2}^{+}}{g_{2}} + μ_{2}^{+})}^{2}}$ stands for the module of ${\tilde{v}}_{2}$ .

From Definition 3, it can be pointed out that when $\bar{{\tilde{P}}_{{\tilde{v}}_{2}} ({\tilde{v}}_{1})} = 1$ , ${\tilde{v}}_{1}$ is identical to ${\tilde{v}}_{2}$ . Therefore, the closer the value of $\bar{{\tilde{P}}_{{\tilde{v}}_{2}} ({\tilde{v}}_{1})}$ approaches to 1, the more proximate ${\tilde{v}}_{1}$ is to ${\tilde{v}}_{2}$ .

In what follows, we will employ this relative projection model to define the similarity and consensus measurements for two interval 2-tuple linguistic matrices assessed in different linguistic scales.

3.2 The similarity measurement

Consider an MCGDM problem under multi-granular uncertain linguistic environment. LetX = {x₁, x₂, …, x_n} be a finite set of alternatives and x_i (i = 1, 2, …, n) represents the ith alternative; A = {a₁, a₂, …, a_m} is a pre-selected set of criteria, whose weights are w = (w₁, w₂, …, w_m), and a_j (j = 1, 2, …, m) stands for the jth criterion. Assume D = {d¹, d², …, d^q} is a fixed set of q decision makers whose weights are unknown. As per different linguistic term sets $S^{(k)} = {s_{τ}^{(k)} | τ = 0, 1, \dots, g_{k}} (k = 1, 2, \dots, q)$ where g_k + 1 refers to the granularity of S^(k), each decision maker provides his/her assessments for each alternative on different criteria with an uncertain linguistic matrix $R_{(k)} = (r_{kij})_{n \times m} = {(s_{τ_{kij}^{-}}^{(k)}, s_{τ_{kij}^{+}}^{(k)})}_{n \times m} .$

In this paper, we apply the thought that transforming linguistic terms into linguistic 2-tuples to implement the process of computing with words. Following this, the original decision matrices R_(k) (k = 1, 2, …, q) are converted into interval 2-tuple linguistic matrices as ${\tilde{R}}_{(k)} = ({\tilde{r}}_{kij})_{n \times m} = ([(s_{τ_{kij}^{-}}^{(k)}, μ_{kij}^{-}), {(s_{τ_{kij}^{+}}^{(k)}, μ_{kij}^{+})])}_{n \times m} (k = 1, 2, \dots, q)$ by means of Equation (3).

Definition 4. Let ${\tilde{R}}_{(k)} = ({\tilde{r}}_{kij})_{n \times m} = ((s_{τ_{kij}^{-}}^{(k)}, μ_{kij}^{-}), (s_{τ_{kij}^{+}}^{(k)}, {μ_{kij}^{+}))}_{n \times m} (k = 1, 2)$ be anytwo interval 2-tuple linguistic matrices, then $\begin{matrix} \bar{{\tilde{P}}_{{\tilde{R}}_{(2)}} ({\tilde{R}}_{(1)})} = \frac{{\tilde{P}}_{{\tilde{R}}_{(2)}} ({\tilde{R}}_{(1)})}{| {\tilde{R}}_{(2)} |} \\ = \frac{\sum_{i = 1}^{n} \sum_{j = 1}^{m} ((\frac{τ_{1 ij}^{-}}{g_{1}} + μ_{1 ij}^{-}) (\frac{τ_{2 ij}^{-}}{g_{2}} + μ_{2 ij}^{-}) + (\frac{τ_{1 ij}^{+}}{g_{1}} + μ_{1 ij}^{+}) (\frac{τ_{2 ij}^{+}}{g_{2}} + μ_{2 ij}^{+}))}{\sqrt{\sum_{i = 1}^{n} \sum_{j = 1}^{m} ({(\frac{τ_{2 ij}^{-}}{g_{2}} + μ_{2 ij}^{-})}^{2} + {(\frac{τ_{2 ij}^{+}}{g_{2}} + μ_{2 ij}^{+})}^{2})} \cdot | {\tilde{R}}_{(2)} |} \\ = \sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{((\frac{τ_{1 ij}^{-}}{g_{1}} + μ_{1 ij}^{-}) (\frac{τ_{2 ij}^{-}}{g_{2}} + μ_{2 ij}^{-}) + (\frac{τ_{1 ij}^{+}}{g_{1}} + μ_{1 ij}^{+}) (\frac{τ_{2 ij}^{+}}{g_{2}} + μ_{2 ij}^{+}))}{({(\frac{τ_{2 ij}^{-}}{g_{2}} + μ_{2 ij}^{-})}^{2} + {(\frac{τ_{2 ij}^{+}}{g_{2}} + μ_{2 ij}^{+})}^{2})} \end{matrix}$ (6) is defined as the relative projection of ${\tilde{R}}_{(1)}$ on ${\tilde{R}}_{(2)}$ , where $\begin{matrix} | {\tilde{R}}_{(2)} | \\ = \sqrt{\sum_{i = 1}^{n} \sum_{j = 1}^{m} ({(\frac{τ_{2 ij}^{-}}{g_{2}} + μ_{2 ij}^{-})}^{2} + {(\frac{τ_{2 ij}^{+}}{g_{2}} + μ_{2 ij}^{+})}^{2})} \end{matrix}$ is the module of ${\tilde{R}}_{(2)}$ . Similarly, the closer the value of $\bar{{\tilde{P}}_{{\tilde{R}}_{(2)}} ({\tilde{R}}_{(1)})}$ to 1, the closer ${\tilde{R}}_{(1)}$ is to ${\tilde{R}}_{(2) .}$

Based on Equation (6), we define the following similarity degree of two interval 2-tuple linguistic decision matrices whose linguistic variables are expressed with multi-granular linguistic term sets.

Definition 5. Let ${\tilde{R}}_{(k)} = ({\tilde{r}}_{kij})_{n \times m} = ((s_{τ_{kij}^{-}}^{(k)}, μ_{kij}^{-}), (s_{τ_{kij}^{+}}^{(k)}, μ_{kij}^{+}))_{n \times m} (k = 1, 2)$ , then the similarity degree between ${\tilde{R}}_{(1)}$ and ${\tilde{R}}_{(2)}$ is defined with the following equation $SD ({\tilde{R}}_{(1)}, {\tilde{R}}_{(2)}) = \frac{1}{2} (\bar{{\tilde{P}}_{{\tilde{R}}_{(2)}} ({\tilde{R}}_{(1)})} + \bar{{\tilde{P}}_{{\tilde{R}}_{(1)}} ({\tilde{R}}_{(2)})}) .$ (7)

In accordance with Definition 4, the closer the value of $SD ({\tilde{R}}_{(1)}, {\tilde{R}}_{(2)})$ approaches to 1, the more similar relation exists between ${\tilde{R}}_{(1)}$ and ${\tilde{R}}_{(2)}$ . Moreover, it is obvious that $SD ({\tilde{R}}_{(1)}, {\tilde{R}}_{(2)}) = SD ({\tilde{R}}_{(2)}, {\tilde{R}}_{(1)})$ , and $SD ({\tilde{R}}_{(1)}, {\tilde{R}}_{(2)}) = 1$ iff ${\tilde{R}}_{(1)} = {\tilde{R}}_{(2)}$ .

3.3 The consensus measurements

Generally, consensus measurements are classified into two categories: distance to the collective preference and distances between decision-makers [44]. While some scholars researches on the former definition [45, 46], some follows the latter one [47, 48]. In this paper, we employ the former one for the consensus measurement.

Definition 6. Let ${\tilde{R}}_{(k)} = ({\tilde{r}}_{kij})_{n \times m} = ((s_{τ_{kij}^{-}}^{(k)}, μ_{kij}^{-}), (s_{τ_{kij}^{+}}^{(k)}, μ_{kij}^{+}))_{n \times m} (k = 1, 2, \dots, q)$ , which are aggregated into a collective one as ${\tilde{R}}_{(c)} = ({\tilde{r}}_{cij})_{n \times m} = ((s_{τ_{cij}^{-}}^{(*)}, μ_{cij}^{-}), (s_{τ_{cij}^{+}}^{(*)}, {μ_{cij}^{+}))}_{n \times m}$ satisf-ying $s_{τ_{cij}^{-}}^{(*)}, s_{τ_{cij}^{+}}^{(*)} \in S^{(*)} = {s_{0}^{(*)}, s_{1}^{(*)}, \dots, s_{g_{*}}^{(*)}}$ and g_* = max {g₁, g₂, …, g_q}. Then the consensus degree of an individual matrix ${\tilde{R}}_{(k)}$ to the group matrix ${\tilde{R}}_{(c)}$ is defined as ${CD}_{(k)} = | \bar{{\tilde{P}}_{{\tilde{R}}_{(c)}} ({\tilde{R}}_{(k)})} - 1 |$ .

According to the thought of the relative projection, the value of $\bar{{\tilde{P}}_{{\tilde{R}}_{(c)}} ({\tilde{R}}_{(k)})}$ can be smaller or greater than 1, and the closer the better. Thus, $| \bar{{\tilde{P}}_{{\tilde{R}}_{(c)}} ({\tilde{R}}_{(k)})} - 1 |$ refers to the deviation of ${\tilde{R}}_{(k)}$ to ${\tilde{R}}_{(c)}$ , and the smaller the value of CD_(k), the more consistent an individual is with the group, such that the more consensus an individual possesses.

As is stated above, we introduce the following two definitions regarding group consensus.

Definition 7. If CD_(k) = 0, then ${\tilde{R}}_{(k)}$ is of perfect consensus with ${\tilde{R}}_{(c)}$ .

Indeed, it is always difficult and unrealistic to reach a full and unanimous agreement due to the distinctive academic and empirical backgrounds of different decision-makers. Considering this practical situation, an acceptable consensus is defined to assure the group to reach a satisfactory agreement eventually.

Definition 8. If CD_(k) ≤ θ, then ${\tilde{R}}_{(k)}$ is of acceptable consensus with ${\tilde{R}}_{(c)}$ .

Herein, the threshold θ is predefined by the group. Certainly, as per different cases, the value of θ may be varying in a large range. The smaller the value of θ, the more rigorous the group requires, such that the higher the consensus degree is.

4 Consensus reaching framework

In this section, we put forward a consensus reaching framework for MCGDM problems based on multi-granular uncertain linguistic information. The framework is depicted in Fig. 1 and the details are given in the following subsections.

Fig.1

The flowchart of the proposed consensus framework.

4.1 Formulating the decision problem and processing the original data

This paper considers the kind of multi-granular linguistic MCGDM problems depicted in Section 3.2. To computing with the original decision data, apply Equation (3) to form multi-granular interval 2-tuple linguistic matrices.

4.2 Calculating similarity degrees and generating weight information of the decision-maker

It is well-known that the more admissive a decision-maker is evaluated by others, the more important he/she is considered. To fuse this common sense into group decision-making, we first make use of Equations (6) and (7) to figure out the similarity degree of one individual matrix to another. Then, the following equation is established to determine the importance weight of each decision-maker: $λ_{k} = \frac{{SD}_{o (k)}}{\sum_{k = 1}^{q} {SD}_{(k)}} = \frac{{SD}_{o (k)}}{\sum_{k = 1}^{q} \sum_{\begin{matrix} K = 1, \\ K \neq k \end{matrix}}^{q} SD ({\tilde{R}}_{(k)}, {\tilde{R}}_{(K)})}$ (8) where SD_(k) refers to the similarity degree of ${\tilde{R}}_{(k)}$ to the others, and o : {1, 2, …, q} → {1, 2, …, q} is the descending order of SD_(k), such that SD_o(k) is the kth largest value among SD_(k) (k = 1, 2, …, q).

By this proceeding, the weights of decision-makers are generated with a weight vector λ = (λ₁, λ₂, …, λ_q), satisfying $\sum_{k = 1}^{q} λ_{k} = 1$ and λ_k ∈ [0, 1].

4.3 Aggregating individual decision information

In the proceeding of MCGDM, the integration of individual opinions is in need of being addressed. Considering the practical situation where decision-makers select multi-granular linguistic term sets to express his/her judgements on alternatives, we adjust the afore-mentioned IVTOWA operator to an aggregating formula as $\begin{matrix} {\tilde{r}}_{cij} & = & [(s_{τ_{cij}^{-}}^{(*)}, μ_{cij}^{-}), (s_{τ_{cij}^{+}}^{(*)}, μ_{cij}^{+})] \\ τ_{cij}^{-} & = & round (g_{*} \cdot \sum_{k = 1}^{q} λ_{k} \cdot (\frac{τ_{o (k) ij}^{-}}{g_{o (k)}} + μ_{o (k) ij}^{-})), \\ μ_{cij}^{-} & = & \sum_{k = 1}^{q} λ_{k} \cdot (\frac{τ_{o (k) ij}^{-}}{g_{o (k)}} + μ_{o (k) ij}^{-}) - \frac{τ_{cij}^{-}}{g_{*}} \\ τ_{cij}^{+} & = & round (g_{*} \cdot \sum_{k = 1}^{q} λ_{k} \cdot (\frac{τ_{o (k) ij}^{+}}{g_{o (k)}} + μ_{o (k) ij}^{+})), \\ μ_{cij}^{+} & = & \sum_{k = 1}^{q} λ_{k} \cdot (\frac{τ_{o (k) ij}^{+}}{g_{o (k)}} + μ_{o (k) ij}^{+}) - \frac{τ_{cij}^{+}}{g_{*}} \end{matrix}$ (9) where $s_{τ_{cij}^{-}}^{(*)}, s_{τ_{cij}^{+}}^{(*)} \in S^{(*)} = {s_{0}^{(*)}, s_{1}^{(*)}, \dots, s_{g_{*}}^{(*)}}$ with g_* = max {g₁, g₂, …, g_q}. To be illustrated, the values $τ_{o (k) ij}^{-}, μ_{o (k) ij}^{-}, τ_{o (k) ij}^{+}, μ_{o (k) ij}^{+}$ and g_o(k) all belong to the individual interval 2-tuple linguistic matrix whose similarity degree is ranked at the kth position.

By this process, the multi-granular individual data is simultaneously normalized and synthesized into the collective data in which the linguistic terms are belonging to one linguistic term set.

4.4 Checking and improving the consensus of an individual to the group

In order to make an approbatory and satisfactory decision, the consensus degree of the individual information to the group information should be measured. As per Definition 6, we first calculate the consensus degree of an individual interval 2-tuple linguistic matrix to the collective one. Next, to check and promote the consensus of the decision-maker who is the most distant from the group, an iterative process is designed: Predefine the threshold value θ, and check whether or not all CD_(k) ≤ θ (k = 1, 2, …, q). If all consensus degrees CD_(k) (k = 1, 2, …, q) fulfill the inequality, i.e., results of acceptable consensus are obtained, then go on; if not, say, at least one CD_(k′) > θ (k′ ∈ {1, 2, …, q}) exists, then utilize the following strategy to revise the corresponding decision matrix ${\tilde{R}}_{(k^{'})}$ . $\begin{matrix} r_{(k^{'}, η + 1) ij} & = & [(s_{τ_{(k^{'}, η + 1) ij}^{-}}^{(k^{'})}, μ_{(k^{'}, η + 1) ij}^{-}), (s_{τ_{(k^{'}, η + 1) j}^{+}}^{(k^{'})}, μ_{(k^{'}, η + 1) ij}^{+})] \\ τ_{(k^{'}, η + 1) ij}^{-} & = & round (g_{k^{'}} \cdot (σ (\frac{τ_{(k^{'}, η) ij}^{-}}{g_{k^{'}}} + μ_{(k^{'}, η) ij}^{-}) + (1 - σ) (\frac{τ_{(c, η) ij}^{-}}{g_{*}} + μ_{(c, η) ij}^{-}))), \\ μ_{(k^{'}, η + 1) ij}^{-} & = & σ (\frac{τ_{(k^{'}, η) ij}^{-}}{g_{k^{'}}} + μ_{(k^{'}, η) ij}^{-}) + (1 - σ) (\frac{τ_{(c, η) ij}^{-}}{g_{*}} + μ_{(c, η) ij}^{-}) - \frac{τ_{(k^{'}, η + 1) ij}^{-}}{g_{k^{'}}} \end{matrix}$ (10) $\begin{matrix} τ_{(k^{'}, η + 1) ij}^{+} & = & round (g_{k^{'}} \cdot (σ (\frac{τ_{(k^{'}, η) ij}^{+}}{g_{k^{'}}} + μ_{(k^{'}, η) ij}^{+}) + (1 - σ) (\frac{τ_{(c, η) ij}^{+}}{g_{*}} + μ_{(c, η) ij}^{+}))), \\ μ_{(k^{'}, η + 1) ij}^{+} & = & σ (\frac{τ_{(k^{'}, η) ij}^{+}}{g_{k^{'}}} + μ_{(k^{'}, η) ij}^{+}) + (1 - σ) (\frac{τ_{(c, η) ij}^{+}}{g_{*}} + μ_{(c, η) ij}^{+}) - \frac{τ_{(k^{'}, η + 1) ij}^{+}}{g_{k^{'}}} \end{matrix}$ where 0 < σ < 1. After this revision, set η = η + 1 (η refers to the number of iterations) and go back to Section 4.1.

By this feedback mechanism, the consensus of the group ought to be improved. Consequently, output the revised individual interval 2-tuple linguistic matrix ${\tilde{R}}_{(k^{'}, η)} = ({\tilde{r}}_{(k^{'}, η) ij})_{n \times m} = {((s_{τ_{(k^{'}, η) ij}^{-}}^{(k^{'})}, μ_{(k^{'}, η) ij}^{-}), (s_{τ_{(k^{'}, η) ij}^{+}}^{(k^{'})}, μ_{(k^{'}, η) ij}^{+}))}_{n \times m}$ andthe re-constructed group interval 2-tuple linguistic matrix ${\bar{R}}_{(c)} = ({\bar{r}}_{cij})_{n \times m} = {((s_{{\bar{τ}}_{cij}^{-}}^{(*)}, {\bar{μ}}_{cij}^{-}), (s_{{\bar{τ}}_{cij}^{+}}^{(*)}, {\bar{μ}}_{cij}^{+}))}_{n \times m}$ . To be noted, if the group has reached a high consensus at the first round, ${\bar{R}}_{(c)}$ equals to ${\tilde{R}}_{(c)}$ .

To show the convergence of this mechanism, we provide the following theorem.

Theorem 1. Let ${\tilde{R}}_{(k^{'}, η)} = ({\tilde{r}}_{(k^{'}, η) ij})_{n \times m} = ((s_{τ_{(k^{'}, η) ij}^{-}}^{(k)}, μ_{(k^{'}, η) ij}^{-}), {(s_{τ_{(k^{'}, η) ij}^{+}}^{(k)}, μ_{(k^{'}, η) ij}^{+}))}_{n \times m} (k^{'} \in {1, 2, \dots, q})$ be the individual interval 2-tuple linguistic matrices which are revised, then we have CD_(k′,η+1) < CD_(k′,η).

Proof. Only some steps are given in this proof. $\begin{matrix} | {CD}_{(k^{'}, η + 1)} - 1 | \\ = | \sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{(\begin{matrix} (σ (\frac{τ_{(k^{'}, η) ij}^{-}}{g_{k^{'}}} + μ_{(k^{'}, η) ij}^{-}) + (1 - σ) (\frac{τ_{(c, η) ij}^{-}}{g_{*}} + μ_{(c, η) ij}^{-})) (\frac{τ_{(c, η + 1) ij}^{-}}{g_{*}} + μ_{(c, η + 1) ij}^{-}) \\ + (σ (\frac{τ_{(k^{'}, η) ij}^{+}}{g_{k^{'}}} + μ_{(k^{'}, η) ij}^{+}) + (1 - σ) (\frac{τ_{(c, η) ij}^{+}}{g_{*}} + μ_{(c, η) ij}^{+})) (\frac{τ_{(c, η + 1) ij}^{+}}{g_{*}} + μ_{(c, η + 1) ij}^{+}) \end{matrix})}{{(\frac{τ_{(c, η + 1) ij}^{-}}{g_{*}} + μ_{(c, η + 1) ij}^{-})}^{2} + {(\frac{τ_{(c, η + 1) ij}^{+}}{g_{*}} + μ_{(c, η + 1) ij}^{+})}^{2}} - 1 | \\ < | \sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{(\frac{τ_{(k^{'}, η) ij}^{-}}{g_{k^{'}}} + μ_{(k^{'}, η) ij}^{-}) (\frac{τ_{(c, η + 1) ij}^{-}}{g_{*}} + μ_{(c, η + 1) ij}^{-}) + (\frac{τ_{(k^{'}, η) ij}^{+}}{g_{k^{'}}} + μ_{(k^{'}, η) ij}^{+}) (\frac{τ_{(c, η + 1) ij}^{+}}{g_{*}} + μ_{(c, η + 1) ij}^{+})}{{(\frac{τ_{(c, η + 1) ij}^{-}}{g_{*}} + μ_{(c, η + 1) ij}^{-})}^{2} + {(\frac{τ_{(c, η + 1) ij}^{+}}{g_{*}} + μ_{(c, η + 1) ij}^{+})}^{2}} - 1 | \\ < | \sum_{i = 1}^{n} \sum_{j = 1}^{m} \frac{((\frac{τ_{(k^{'}, η) ij}^{-}}{g_{k^{'}}} + μ_{(k^{'}, η) ij}^{-}) (\frac{τ_{(c, η) ij}^{-}}{g_{*}} + μ_{(c, η) ij}^{-}) + (\frac{τ_{(k^{'}, η) ij}^{+}}{g_{k^{'}}} + μ_{(k^{'}, η) ij}^{+}) (\frac{τ_{(c, η) ij}^{+}}{g_{*}} + μ_{(c, η) ij}^{+}))}{{(\frac{τ_{(c, η) ij}^{-}}{g_{*}} + μ_{(c, η) ij}^{-})}^{2} + {(\frac{τ_{(c, η) ij}^{+}}{g_{*}} + μ_{(c, η) ij}^{+})}^{2}} - 1 | \\ = | {CD}_{(k^{'}, η)} - 1 | . \end{matrix}$

4.5 Obtaining comprehensive scores of alternatives and ranking

Since an acceptable collective opinion is formed, this subsection addresses on how to select the best alternative. To do so, an intuitive way is to compare the comprehensive scores of alternatives. For ease of computation, we calculate the comprehensive score of each alternative by synthesizing each row elements of ${\bar{R}}_{(c)}$ by the following equation. $\begin{matrix} B_{x_{i}} & = & [\sum_{j = 1}^{m} w_{j} \cdot (\frac{{\bar{τ}}_{cij}^{-}}{g_{*}} + {\bar{μ}}_{cij}^{-}), \\ \sum_{j = 1}^{m} w_{j} \cdot (\frac{{\bar{τ}}_{cij}^{+}}{g_{*}} + {\bar{μ}}_{cij}^{+})] . \end{matrix}$ (11)

Then, apply the comparison rule of interval numbers [49] to derive the possibility degrees of one alternative over another, that is, $\begin{matrix} p (x_{i} ≻ x_{j}) \\ = \frac{max {0, b_{i}^{+} - b_{j}^{-}} - max {0, b_{i}^{-} - b_{j}^{+}}}{b_{i}^{+} - b_{i}^{-} + b_{j}^{+} - b_{j}^{-}} . \end{matrix}$ (12)

By Equation (12), a possibility matrix for alternatives can be constructed so as to rank the alternatives.

5 Example of application

In this section, we adopt the examined example about an investment problem [21] to demonstrate the validity of our framework.

Suppose an investment company plans to invest plenty of money in the optimal option. Four possible alternatives are chosen: x₁ is a car industry, x₂ is a food company, x₃ is a computer company, and x₄ is an arms industry. By discussion, this investment company decides four criteria: a₁ is the risk analysis, a₂ is the growth analysis, a₃ is the social-political analysis, and a₄ is the environmental impact analysis. The weight vector of the criteria is w = (0.3, 0.1, 0.2, 0.4). Three decision-makers d^p (p = 1, 2, 3) are invited to evaluate alternatives over each criterion. As per different linguistic term sets, decision-makers provide assessments for each alternative on different criteria: d¹ uses a linguistic term set with 9 labels $S^{(1)} = {s_{τ}^{(1)} | τ = 0, 1, \dots, 8}$ , d² uses a linguistic term set with 7 labels $S^{(2)} = {s_{τ}^{(2)} | τ = 0, 1, \dots, 6}$ and d³ uses a linguistic term set with 5 labels $S^{(3)} = {s_{τ}^{(3)} | τ = 0, 1, \dots, 4}$ . The assessments are separately expressed by corresponding uncertain linguistic matrices $R_{(k)} = (r_{kij})_{4 \times 4} = {(s_{τ_{kij}^{-}}^{(k)}, s_{τ_{kij}^{+}}^{(k)})}_{4 \times 4} (k = 1, 2, 3)$ as $\begin{matrix} R_{(1)} & = & [\begin{matrix} [s_{7}^{(1)}, s_{7}^{(1)}] & [s_{1}^{(1)}, s_{1}^{(1)}] & [s_{1}^{(1)}, s_{3}^{(1)}] & [s_{3}^{(1)}, s_{5}^{(1)}] \\ [s_{5}^{(1)}, s_{7}^{(1)}] & [s_{4}^{(1)}, s_{4}^{(1)}] & [s_{3}^{(1)}, s_{4}^{(1)}] & [s_{4}^{(1)}, s_{6}^{(1)}] \\ [s_{4}^{(1)}, s_{7}^{(1)}] & [s_{3}^{(1)}, s_{4}^{(1)}] & [s_{3}^{(1)}, s_{6}^{(1)}] & [s_{1}^{(1)}, s_{1}^{(1)}] \\ [s_{3}^{(1)}, s_{3}^{(1)}] & [s_{0}^{(1)}, s_{2}^{(1)}] & [s_{4}^{(1)}, s_{4}^{(1)}] & [s_{3}^{(1)}, s_{4}^{(1)}] \end{matrix}] \\ R_{(2)} & = & [\begin{matrix} [s_{3}^{(2)}, s_{3}^{(2)}] & [s_{5}^{(2)}, s_{5}^{(2)}] & [s_{1}^{(2)}, s_{3}^{(2)}] & [s_{3}^{(2)}, s_{4}^{(2)}] \\ [s_{5}^{(2)}, s_{5}^{(2)}] & [s_{4}^{(2)}, s_{5}^{(2)}] & [s_{1}^{(2)}, s_{3}^{(2)}] & [s_{3}^{(2)}, s_{5}^{(2)}] \\ [s_{2}^{(2)}, s_{4}^{(2)}] & [s_{4}^{(2)}, s_{5}^{(2)}] & [s_{1}^{(2)}, s_{3}^{(2)}] & [s_{0}^{(2)}, s_{1}^{(2)}] \\ [s_{4}^{(2)}, s_{5}^{(2)}] & [s_{1}^{(2)}, s_{2}^{(2)}] & [s_{3}^{(2)}, s_{4}^{(2)}] & [s_{3}^{(2)}, s_{5}^{(2)}] \end{matrix}] \\ R_{(3)} & = & [\begin{matrix} [s_{2}^{(3)}, s_{3}^{(3)}] & [s_{1}^{(3)}, s_{2}^{(3)}] & [s_{1}^{(3)}, s_{3}^{(3)}] & [s_{3}^{(3)}, s_{3}^{(3)}] \\ [s_{3}^{(3)}, s_{3}^{(3)}] & [s_{2}^{(3)}, s_{3}^{(3)}] & [s_{2}^{(3)}, s_{3}^{(3)}] & [s_{1}^{(3)}, s_{1}^{(3)}] \\ [s_{3}^{(3)}, s_{3}^{(3)}] & [s_{0}^{(3)}, s_{2}^{(3)}] & [s_{2}^{(3)}, s_{3}^{(3)}] & [s_{2}^{(3)}, s_{2}^{(3)}] \\ [s_{1}^{(3)}, s_{1}^{(3)}] & [s_{2}^{(3)}, s_{3}^{(3)}] & [s_{1}^{(3)}, s_{3}^{(3)}] & [s_{3}^{(3)}, s_{3}^{(3)}] \end{matrix}] . \end{matrix}$

By using Equation (3), R_(k) (k = 1, 2, 3) are transformed into interval 2-tuple linguistic matrices ${\tilde{R}}_{(k)} = ({\tilde{r}}_{kij})_{4 \times 4}$ , $= ([(s_{τ_{kij}^{-}}^{(k)}, 0), (s_{τ_{kij}^{+}}^{(k)}, 0)])_{4 \times 4} (k = 1, 2, 3)$ , e.g., ${\tilde{R}}_{(1)} = [\begin{matrix} [\begin{matrix} (s_{7}^{(1)}, 0), \\ (s_{7}^{(1)}, 0) \end{matrix}] & [\begin{matrix} (s_{1}^{(1)}, 0), \\ (s_{1}^{(1)}, 0) \end{matrix}] & [\begin{matrix} (s_{1}^{(1)}, 0), \\ (s_{3}^{(1)}, 0) \end{matrix}] & [\begin{matrix} (s_{3}^{(1)}, 0), \\ (s_{5}^{(1)}, 0) \end{matrix}] \\ [\begin{matrix} (s_{5}^{(1)}, 0), \\ (s_{7}^{(1)}, 0) \end{matrix}] & [\begin{matrix} (s_{4}^{(1)}, 0), \\ (s_{4}^{(1)}, 0) \end{matrix}] & [\begin{matrix} (s_{3}^{(1)}, 0), \\ (s_{4}^{(1)}, 0) \end{matrix}] & [\begin{matrix} (s_{4}^{(1)}, 0), \\ (s_{6}^{(1)}, 0) \end{matrix}] \\ [\begin{matrix} (s_{4}^{(1)}, 0), \\ (s_{7}^{(1)}, 0) \end{matrix}] & [\begin{matrix} (s_{3}^{(1)}, 0), \\ (s_{4}^{(1)}, 0) \end{matrix}] & [\begin{matrix} (s_{3}^{(1)}, 0), \\ (s_{6}^{(1)}, 0) \end{matrix}] & [\begin{matrix} (s_{1}^{(1)}, 0), \\ (s_{1}^{(1)}, 0) \end{matrix}] \\ [\begin{matrix} (s_{3}^{(1)}, 0), \\ (s_{3}^{(1)}, 0) \end{matrix}] & [\begin{matrix} (s_{0}^{(1)}, 0), \\ (s_{2}^{(1)}, 0) \end{matrix}] & [\begin{matrix} (s_{4}^{(1)}, 0), \\ (s_{4}^{(1)}, 0) \end{matrix}] & [\begin{matrix} (s_{3}^{(1)}, 0), \\ (s_{4}^{(1)}, 0) \end{matrix}] \end{matrix}] .$

Then, employ Equations (6) and (7) to figure out the similarity degrees between one individual interval 2-tuple linguistic matrix and the other ones as $\begin{matrix} SD ({\tilde{R}}_{(1)}, {\tilde{R}}_{(2)}) = 0.8994, SD ({\tilde{R}}_{(1)}, {\tilde{R}}_{(3)}) = 0.8894, \\ SD ({\tilde{R}}_{(2)}, {\tilde{R}}_{(3)}) = 0.8464 . \end{matrix}$

Subsequently, according to Equation (8), the weight vector of decision-makers is formed as $λ = (0.3437, 0.3331, 0.3232) .$

Thus, the collective decision matrix can be generated via Equation (9) as below. ${\tilde{R}}_{(c)} = [\begin{matrix} [\begin{matrix} (s_{5}^{(*)}, - 0.0001), \\ (s_{6}^{(*)}, - 0.0443) \end{matrix}] & [\begin{matrix} (s_{3}^{(*)}, 0.0339), \\ (s_{4}^{(*)}, - 0.0103) \end{matrix}] & [\begin{matrix} (s_{1}^{(*)}, 0.0547), \\ (s_{4}^{(*)}, 0.0392) \end{matrix}] & [\begin{matrix} (s_{4}^{(*)}, 0.0392), \\ (s_{5}^{(*)}, 0.0547) \end{matrix}] \\ [\begin{matrix} (s_{6}^{(*)}, - 0.0130), \\ (s_{7}^{(*)}, - 0.0547) \end{matrix}] & [\begin{matrix} (s_{4}^{(*)}, 0.0573), \\ (s_{6}^{(*)}, - 0.0546) \end{matrix}] & [\begin{matrix} (s_{3}^{(*)}, - 0.0312), \\ (s_{5}^{(*)}, - 0.0442) \end{matrix}] & [\begin{matrix} (s_{3}^{(*)}, 0.0442), \\ (s_{5}^{(*)}, - 0.0080) \end{matrix}] \\ [\begin{matrix} (s_{4}^{(*)}, 0.0235), \\ (s_{6}^{(*)}, 0.0130) \end{matrix}] & [\begin{matrix} (s_{3}^{(*)}, - 0.021), \\ (s_{5}^{(*)}, - 0.0104) \end{matrix}] & [\begin{matrix} (s_{3}^{(*)}, - 0.0312), \\ (s_{5}^{(*)}, 0.0391) \end{matrix}] & [\begin{matrix} (s_{2}^{(*)}, - 0.0468), \\ (s_{2}^{(*)}, 0.0105) \end{matrix}] \\ [\begin{matrix} (s_{3}^{(*)}, 0.0598), \\ (s_{4}^{(*)}, - 0.0079) \end{matrix}] & [\begin{matrix} (s_{2}^{(*)}, - 0.0311), \\ (s_{4}^{(*)}, - 0.0598) \end{matrix}] & [\begin{matrix} (s_{3}^{(*)}, 0.0442), \\ (s_{5}^{(*)}, 0.0131) \end{matrix}] & [\begin{matrix} (s_{4}^{(*)}, 0.0392), \\ (s_{6}^{(*)}, - 0.0546) \end{matrix}] \end{matrix}] .$

In the following, as per Definition 6, calculate the consensus degree of ${\tilde{R}}_{(k)}$ to ${\tilde{R}}_{(c)}$ and obtain ${CD}_{(1)} = 0.0884, {CD}_{(2)} = 0.0568, {CD}_{(3)} = 0.0307 .$

Generally, the threshold value is predefined as 0.1. Thus, it is obvious that CD_(k) ≤ 0.1 (k = 1, 2, 3). This indicates that the group has reached a consensus degree under the assumption that the group permits the error of 10 percent.

Herein, to illustrate how to apply the iterative procedure in promoting the consensus degree of an individual decision-maker, we suppose the threshold value is 0.085. Obviously, CD_(k) ≤ 0.085 (k = 2, 3) and CD₍₁₎ > 0.085. In this situation, we need to utilize Equation (10) to revise ${\tilde{R}}_{(1)}$ . Without loss of generally, we assume σ = 0.9. After one iteration, one has CD₍₁₎ = 0 . 0824, CD₍₂₎ = 0 . 0543, CD₍₃₎ = 0 .0315. This implies that all CD_(k) ≤ 0.085 (k = 1, 2, 3) hold. Therefore, for this case, we stop at the fist iteration and conclude that the group has reached an acceptable consensus, such that the final decision can be approval. Consequently, output ${\tilde{R}}_{(1, 1)}$ and ${\bar{R}}_{(c)}$ as ${\tilde{R}}_{(1, 1)} = [\begin{matrix} [\begin{matrix} (s_{7}^{(1)}, - 0.0250), \\ (s_{7}^{(1)}, - 0.0169) \end{matrix}] & [\begin{matrix} (s_{1}^{(1)}, 0.0284), \\ (s_{1}^{(1)}, 0.0365) \end{matrix}] & [\begin{matrix} (s_{1}^{(1)}, 0.0055), \\ (s_{3}^{(1)}, 0.0164) \end{matrix}] & [\begin{matrix} (s_{3}^{(1)}, 0.0164), \\ (s_{5}^{(1)}, 0.0055) \end{matrix}] \\ [\begin{matrix} (s_{5}^{(1)}, 0.0112), \\ (s_{7}^{(1)}, - 0.0055) \end{matrix}] & [\begin{matrix} (s_{4}^{(1)}, 0.0057), \\ (s_{4}^{(1)}, 0.0195) \end{matrix}] & [\begin{matrix} (s_{3}^{(1)}, - 0.0031), \\ (s_{4}^{(1)}, 0.0081) \end{matrix}] & [\begin{matrix} (s_{4}^{(1)}, - 0.0081), \\ (s_{6}^{(1)}, - 0.0133) \end{matrix}] \\ [\begin{matrix} (s_{4}^{(1)}, 0.0024), \\ (s_{7}^{(1)}, - 0.0112) \end{matrix}] & [\begin{matrix} (s_{3}^{(1)}, - 0.0021), \\ (s_{4}^{(1)}, 0.0115) \end{matrix}] & [\begin{matrix} (s_{3}^{(1)}, - 0.0031), \\ (s_{6}^{(1)}, - 0.0086) \end{matrix}] & [\begin{matrix} (s_{1}^{(1)}, 0.0078), \\ (s_{1}^{(1)}, 0.0136) \end{matrix}] \\ [\begin{matrix} (s_{3}^{(1)}, 0.0060), \\ (s_{3}^{(1)}, 0.0117) \end{matrix}] & [\begin{matrix} (s_{0}^{(1)}, 0.0219), \\ (s_{2}^{(1)}, 0.0190) \end{matrix}] & [\begin{matrix} (s_{4}^{(1)}, - 0.0081), \\ (s_{4}^{(1)}, 0.0138) \end{matrix}] & [\begin{matrix} (s_{3}^{(1)}, 0.0164), \\ (s_{4}^{(1)}, 0.0195) \end{matrix}] \end{matrix}],$ ${\bar{R}}_{(c)} = [\begin{matrix} [\begin{matrix} (s_{5}^{(*)}, - 0.0058), \\ (s_{6}^{(*)}, - 0.0458) \end{matrix}] & [\begin{matrix} (s_{3}^{(*)}, 0.0349), \\ (s_{4}^{(*)}, - 0.0051) \end{matrix}] & [\begin{matrix} (s_{1}^{(*)}, 0.0567), \\ (s_{4}^{(*)}, 0.0410) \end{matrix}] & [\begin{matrix} (s_{4}^{(*)}, 0.0452), \\ (s_{5}^{(*)}, 0.0567) \end{matrix}] \\ [\begin{matrix} (s_{6}^{(*)}, - 0.0112), \\ (s_{7}^{(*)}, - 0.0567) \end{matrix}] & [\begin{matrix} (s_{4}^{(*)}, 0.0571), \\ (s_{6}^{(*)}, - 0.0509) \end{matrix}] & [\begin{matrix} (s_{3}^{(*)}, - 0.0288), \\ (s_{5}^{(*)}, - 0.0400) \end{matrix}] & [\begin{matrix} (s_{3}^{(*)}, 0.0400), \\ (s_{5}^{(*)}, - 0.0164) \end{matrix}] \\ [\begin{matrix} (s_{4}^{(*)}, 0.0279), \\ (s_{6}^{(*)}, 0.0112) \end{matrix}] & [\begin{matrix} (s_{3}^{(*)}, - 0.0276), \\ (s_{5}^{(*)}, - 0.0109) \end{matrix}] & [\begin{matrix} (s_{3}^{(*)}, - 0.0288), \\ (s_{5}^{(*)}, 0.0394) \end{matrix}] & [\begin{matrix} (s_{2}^{(*)}, - 0.0404), \\ (s_{2}^{(*)}, 0.0167) \end{matrix}] \\ [\begin{matrix} (s_{3}^{(*)}, 0.0574), \\ (s_{4}^{(*)}, - 0.0106) \end{matrix}] & [\begin{matrix} (s_{2}^{(*)}, - 0.0230), \\ (s_{4}^{(*)}, - 0.0516) \end{matrix}] & [\begin{matrix} (s_{3}^{(*)}, 0.0400), \\ (s_{5}^{(*)}, 0.0170) \end{matrix}] & [\begin{matrix} (s_{4}^{(*)}, 0.0452), \\ (s_{6}^{(*)}, - 0.0509) \end{matrix}] \end{matrix}] .$

Finally, plug ${\bar{R}}_{(c)}$ and $\bar{λ}$ into Equation (11) to derive the comprehensive score of each alternativeas $\begin{matrix} B_{x_{1}} = [0.4812, 0.6416], B_{x_{2}} = [0.5126, 0.6758], \\ B_{x_{3}} = [0.3462, 0.5293], B_{x_{4}} = [0.4535, 0.5997] . \end{matrix}$

Evidently, the intervals B_{x
_i} (i = 1, 2, 3, 4) have intersected relationship with each other. Therefore, there is a necessity to utilize Equation (12) to calculate the possibility of one score over another and thus build up a possibility matrix as $Y = [\begin{matrix} 0.5 & 0.3986 & 0.8600 & 0.6135 \\ 0.6014 & 0.5 & 0.9518 & 0.7185 \\ 0.1400 & 0.0482 & 0.5 & 0.2302 \\ 0.3865 & 0.2815 & 0.7698 & 0.5 \end{matrix}] .$

From the above matrix, it can be pointed out that the ranking of alternatives is $x_{2} \overset{60.14 %}{≻} x_{1} \overset{61.35 %}{≻} x_{4} \overset{76.98 %}{≻} x_{3}$ , which is basically consistent with the result presented in the example in [21]. This validates the feasibility and effectiveness of our proposed framework.

6 Conclusions

In summary, this paper presents a consensus framework for multi-granular uncertain linguistic MCGDM with unknown weight information. The advantages and disadvantages are listed below.

Firstly, the weight of the decision-maker in our framework is not subjectively given, which is more accordance with realistic and complex decision environment. To determine the weights, we calculate the similarity degree of one decision-maker to all the others. This thought is coincident with the common sense that the more reliable the decision-maker, the more important he/she is.

Secondly, this framework considers the consensus problem based on multi-granular uncertain linguistic context, which has not been taken into consideration for the group decision-making process. The consensus degree is measured by fully utilizing the advantage of the established relative projection model which could reflect the relationship between two interval 2-tuple matrices denoted by different linguistic scales.

Thirdly, a feedback mechanism is devised to revise the individual opinions which are not consistent with those of the group as well as to improve the group consensus, such that the collision between decision-makers can be avoided.

Finally, an illustrative example is furnished and verifies the validity and feasibility of the proposed framework for practical decision problems.

However, this framework only focuses on the consensus for the MCGDM problems where different decision-makers use different linguistic term sets to express assessments on alternatives. In the future, we may research on the consensus of other kinds of multi-granular linguistic MCGDM problems such as the attribute values of different alternatives are expressed with different linguistic scales.

Conflict of interests

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

Footnotes

Acknowledgments

The authors thank the editors and anonymous reviewers for their helpful comments and suggestions. This work is supported by the National Natural Science Foundation of China (No. 71571193).

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