The consistency measures of multi-granularity linguistic group decision making

Abstract

When different experts have different uncertainty degrees on the phenomenon, several linguistic term sets with different granularity of uncertainty are necessary. In this paper, we deal with the consistency problem with multi-granularity linguistic term sets applied to group decision making. Firstly, we develop a transformation model to maintain the uncertainty degrees of different granularity associated with each expert. We use the uncertain linguistic variables as the unified form. Some computational rules about uncertain linguistic variables are given. Then consistency index of linguistic preference relations based on the distance of a multi-granularity linguistic preference relation is defined. Chi-square statistic is used to establish the consistency thresholds. Finally two numerical examples are used to show the process and effects of our new proposed method.

Keywords

Multi-granularity linguistic term sets group decision making (GDM)uncertain linguistic variables consistency index

1 Introduction

To convey the information we intend, we continually are faced with natural language to describe objects and instances in our everyday life. Linguistic variables are the appropriate tools to describe vague concepts. Linguistic symbolic computational model is a type of important research method [1 –10]. Since the 2-tuple linguistic computational model [3] was proposed, many extensions to the 2-tuple linguistic model have been developed. Wang and Hao [6] extended the 2-tuple fuzzy linguistic representation model to proportional 2-tuples. Xu [11] extended a discrete term set to a continuous term set by virtual linguistic terms which can be obtained and manipulated to avoid loss of information. Fan, Yue, Feng and Liu [12] modeled the linguistic uncertainty directly by a fuzzy relation on the set of linguistic labels.

Linguistic information as preferences in decision is an important application. In any linguistic approach, an important parameter to determine is the “granularity of uncertainty”, i.e., the cardinality of the linguistic term set is used to express the information [2]. According to the uncertainty degree that an expert faces in group decision making, several linguistic term sets with different granularities of uncertainty are necessary. Some methods of the management of multi-granularity linguistic term sets applied to decision making problems have been proposed [2 , 14].

The consistency measure is a very important problem in decision making which uses preference relations. It is performed to ensure that the decision maker is being neither random nor illogical in his or her pairwise comparisons. Similar to the numerical preference relations, some definitions to measure consistency of linguistic preference relations have also been researched.However, these consistency definitions in [14 –17] are unable to deal with multi-granularity linguistic term sets. The method proposed by Herrera-Viedma, Mata, Martínez, Chiclana and Pérez [18] can deal with multi-granularity linguistic term sets, but it is unable to identify whether the consistency degree is acceptable.

In this paper we focus on the problem of how to measure the degree of consistency with multi-granularity linguistic term sets, and when to accept the consistency degree. The main aim of this paper is to present a consistency index of multi-granularity linguistic preference relations, and develop a more flexible consistency measure method. Using this method, we can measure whether it is a satisfactory result of a group decision.

The rest of this paper is organized as follows. Section 2 introduces some basic notations of consistency measurement of multi-granularity linguistic GDM problems. Section 3 develops a computational model of multi-granularity linguistic preference relations. On the basis of the computational model, we propose the distance measure. Section 4 shows the consistency measures of multi-granularity linguistic preference relations. In Section 5, two illustrative examples are provided. Concluding remarks are made in Section 6.

2 Preliminaries

2.1 Consistency and transitivity in fuzzy preference relations

Preference relation is the most common representation of information used in decision making problems. In a preference relation an expert associates with every pair of alternatives a value that reflects some degree of preference of the first alternative over the second one. In a fuzzy context, an expert expresses his/her opinions using fuzzy preference relations.

Definition 1. [19, 20] A fuzzy preference relation P on a set of alternatives X is a fuzzy set on the product set X × X, i.e., it is characterized by a membership function μ _p : X × X → [0, 1]. The preference relation may be conveniently represented by the n × n matrix P = (p _ij) _n×n.

The previous Definition 1 of a fuzzy preference relation does not imply any kind of consistency property. In fact, preference values of a fuzzy preference relation can be contradictory. For making a consistent choice, a set of consistency properties have to be satisfied. Some of the suggested properties are given [21]. Many important decision models have been developed mainly using two kinds of preference relations. One is Saaty’s consistency property for multiplicative preference relations [22]. The other is the additive transitivity for fuzzy preference relations which was given by Tetsuzo [23]. Therefore, the additive transitivity is the only property that we will assume throughout this paper. We will use the term additive consistency to refer to consistency for fuzzy preference relations based on the additive transitivity property.

Property 1. A reciprocal fuzzy preference relation P = (p_ij) _n×n is consistent if $p_{ij} + p_{jk} + p_{ki} = \frac{3}{2} \forall i, j, k$ (1)

The preference value p _ik (i ≠ k) can be estimated in three different ways [24]: ${cp}_{ik}^{j 1} = p_{ij} + p_{jk} - 0.5$ (2) ${cp}_{ik}^{j 2} = p_{jk} - p_{ji} + 0.5$ (3) ${cp}_{ik}^{j 3} = p_{ij} - p_{kj} + 0.5$ (4)

However, because experts are not always fully consistent, the information given by an expert may not be same as the results computed by expressions (2)–(4). We can compare the information given by an expert with the results computed by expressions (2)–(4) to get $ɛ p_{ik}^{1}$ , $ɛ p_{ik}^{2}$ and $ɛ p_{ik}^{3}$ . Finally we can get the global distance ɛp _ik. $ɛ p_{ik}^{1} = \frac{\sum_{j = 1 j \neq i, k}^{n} ({cp}_{ik}^{j 1} - p_{ik})}{n - 2}$ (5) $ɛ p_{ik}^{2} = \frac{\sum_{c j = 1 j \neq i, k}^{n} ({cp}_{ik}^{j 2} - p_{ik})}{n - 2}$ (6) $ɛ p_{ik}^{3} = \frac{\sum_{c j = 1 j \neq i, k}^{n} ({cp}_{ik}^{j 3} - p_{ik})}{n - 2}$ (7) $ɛ p_{ik} = \frac{2}{3} \cdot \frac{ɛ p_{ik}^{1} + ɛ p_{ik}^{2} + ɛ p_{ik}^{3}}{3}$ (8)

The consistency level associated with a preference value p _ik can be defined as ${CL}_{ik} = 1 - ɛ p_{ik}$ (9)

When CL _ik = 1, there is no inconsistency at all. The lower the value of CL _ik, the more inconsistent is p _ik with respect to the rest of the information.

2.2 The consistency measure of multi-granularity linguistic GDM problems

A GDM problem based on linguistic preference relations may be defined as follows: consider a group decision making problem with linguistic preference relations. Let X ={ x ₁, x ₂, …, x _n } be the set of alternatives, D ={ d ₁, d ₂, …, d _m } be the set of decision makers, and λ ={ λ ₁, λ ₂, …, λ _m } be the weight vector of decision makers, where $λ_{k} > 0, \sum_{k = 1}^{m} λ_{k} = 1$ . Let A _k be the linguistic preference relations provided by decision makers d _k, k = 1, 2, …, m, where $A_{k} = {(a_{ij}^{k})}_{n \times n}, k = 1, 2, \dots, m; i, j = 1, 2, \dots, n$ . Each expert d _k provides his/her preferences on X by means of a linguistic preference relation, $a_{ij}^{k} : X \times X \to S$ , where S ={ s ₀, s ₁, …, s _g } is a linguistic term set characterized by its cardinality or granularity, # (S) = g + 1.

Then denote $\bar{A} = {(\bar{a_{ij}})}_{n \times n} = λ_{1} A_{1} \oplus λ_{2} A_{2} \oplus \dots \oplus λ_{m} A_{m}$ as the collective linguistic preference relation of GDM, where $\bar{a_{ij}} = λ_{1} a_{ij}^{1} \oplus λ_{2} a_{ij}^{2} \oplus \dots \oplus λ_{m} a_{ij}^{m}; i, j = 1, 2, \dots, n$ . The ideal situation in GDM problems in a linguistic context would be one where all the experts use the same linguistic term set S to provide their opinions. However, experts may belong to distinct research areas and have different levels of knowledge of the alternatives. A consequence of this is that the expression of preferences will be based on linguistic term sets with different granularity. In another words, the granularity # (S) of expert d _k’s preferences is different from that of expert d _l’s preferences. So we deal with the consistency measure of multi-granularity linguistic GDM problems. We will consider a fuzzy preference relation to be “additive consistent”.

However, it is hard to obtain perfect consistency using the linguistic preference relations, especially when the number of alternatives is large. A consistency index of linguistic preference relations isneeded.

The measurement of consensus in multi-granularity linguistic GDM problems is therefore carried out in three steps: (i) making the linguistic information uniform, (ii) computation of consensus degrees and (iii) computation of proximity measures [18].

As we assume multi-granularity linguistic context, Herrera-Viedma, Mata, Martínez, Chiclana and Pérez [18] give the first step of obtaining a uniform representation of the preferences. Many methods use distance functions $d (a_{ij}^{k}, a_{ij}^{l})$ to measure the proximity between the linguistic preferences $a_{ij}^{k}$ and $a_{ij}^{l}$ given by experts d _k and d _l [18 , 26]. When using distance functions, we can obtain some consistency index to measure the additional consistency.

3 Computational model of multi-granularity linguistic preference relations

3.1 The transformation of multi-granularity linguistic preference

It seems natural to allow decision makers to provide their preferences using multi-granularity linguistic term sets. For example, in a grading system a decision maker could choose to use a linguistic term set S ={ Low, Medium, High } and another decision maker could prefer a linguistic term set with higher granularity as S ={ VeryLow, Low, Medium, High, VeryHigh } (the second one may better discriminate his preferences). Firstly, we give the definition of a linguistic term set whose granularity is # (S ^g) = g + 1.

Definition 2. Let $S^{g} = {s_{0}^{g}, \dots, s_{g}^{g}}$ be a linguistic term set where $s_{0}^{g} < s_{1}^{g} < \dots < s_{g}^{g}$ . The linguistic term is represented as $s_{i}^{g} \in S^{g}$ , where superscript $\sup (s_{i}^{g}) = g$ measures the uncertainty; subscript $sub (s_{i}^{g}) = i$ is the value of the term to measure order in the set.

We need to transform multi-granularity linguistic value into a unified form to maintain the uncertainty degrees associated with each one of the possible domains. We use the uncertain linguistic variables as the unified form.

Definition 3. [27] Let $\tilde{s} = [s^{-}, s^{+}]$ , where s ^-, s ⁺ ∈ S ^g, s ⁺ ≥ s ^-, $S^{g} = {s_{0}^{g}, s_{1}^{g}, \dots, s_{g}^{g}}$ , s ^- and s ⁺ are the lower and the upper limits, respectively. We then call $\tilde{s}$ the uncertain linguistic variable.

In particular, uncertain linguistic variable will degenerate to common linguistic variable when s ^- = s ⁺ in $\tilde{s} = [s^{-}, s^{+}]$ .

For multi-granularity linguistic value has different semantics, S _T is a basic linguistic term set with a larger number of terms than the number of terms that a person is able to discriminate [2]. The first step is to identify S _T. We look for a S _T whose number (g) is the lowest common multiple (L.C.M.) of (g ₁, g ₂, …, g _m). So $S_{T} = (s_{0}^{g}, s_{1}^{g}, \dots, s_{g}^{g})$ .

Once S _T has been selected, the following multi-granularity transformation function is applied to transform every linguistic value into an uncertain linguistic variable composed by S _T.

Definition 4. Let $S = (s_{0}^{g}, s_{1}^{g}, \dots, s_{g}^{g})$ and $S_{T} = (s_{0}^{cg}, s_{1}^{cg}, \dots, s_{cg}^{cg})$ be two linguistic term sets. Then, a multi-granularity transformation function is defined as $τ : S \to \tilde{s}$ $s_{i}^{g} = {\begin{matrix} [s_{0}^{cg}, s_{c (i + 1)}^{cg}], & i - 1 < 0 \\ [s_{c (i - 1)}^{cg}, s_{c (i + 1)}^{cg}], & i - 1 \geq 0, i + 1 \leq g \\ [s_{c (i - 1)}^{cg}, s_{cg}^{cg}], & i + 1 > g, i - 1 < g \end{matrix}$

If a multi-granularity linguistic value $s_{i}^{cg}$ is needed to represent in the uncertain linguistic form, then we can present it as a degenerated uncertain linguistic variable, which is $s_{i}^{cg} = [s_{i}^{cg}, s_{i}^{cg}]$ .

Their operational laws are defined as $s_{i}^{g} \oplus s_{j}^{g} = s_{i + j}^{g}$ (10) $λ \cdot s_{i}^{g} = s_{λ \cdot i}^{g}, where λ \in [0, 1]$ (11) $[s_{a^{-}}^{g}, s_{a^{+}}^{g}] \oplus [s_{b^{-}}^{g}, s_{b^{+}}^{g}] = [s_{a^{-} + b^{-}}^{g}, s_{a^{+} + b^{+}}^{g}]$ (12) $λ \cdot [s_{a^{-}}^{g}, s_{a^{+}}^{g}] = [s_{λ \cdot a^{-}}^{g}, s_{λ \cdot a^{+}}^{g}], where λ \in [0, 1]$ (13)

If $[s_{b^{-}}^{g}, s_{b^{+}}^{g}]$ is a degenerated uncertain linguistic variable which means b ^- = b ⁺ = b, the Equation (12) is $[s_{a^{-}}^{g}, s_{a^{+}}^{g}] \oplus [s_{b}^{g}, s_{b}^{g}] = [s_{a^{-} + b}^{g}, s_{a^{+} + b}^{g}]$ (14)

Since previous operational laws are widely accepted, our operational laws should not be in contradiction with previous operational laws. Now we try to prove the internally consistence through formula derivation. Let us take the operational laws of Xu [26] for example.

For any two labels s _a, s _b ∈ S, S = (s ₀, s ₁, …, s _g), Xu [26] defines their operational laws as follows: $\begin{matrix} s_{α} \oplus s_{β} = s_{α + β} \\ λ \cdot s_{α} = s_{λ α}, where λ \in [0, 1] \end{matrix}$

For s _a, s _b have the same granularity, we can replace s _a, s _b ∈ S with the $s_{α}^{g}, s_{β}^{g} \in S^{g}$ . Then we can get the Equations (10, 11).

Xu [11] defines uncertain linguistic variables ${\tilde{s}}_{1} = [s_{α_{1}}, s_{β_{1}}]$ , ${\tilde{s}}_{2} = [s_{α_{2}}, s_{β_{2}}]$ . Their operational laws are defined as $\begin{matrix} {\tilde{s}}_{1} \oplus {\tilde{s}}_{2} = [s_{α_{1} + α_{2}}, s_{β_{1} \oplus β_{2}}] \\ λ {\tilde{s}}_{1} = [s_{λ α_{1}}, s_{λ β_{1}}], where λ \in [0, 1] \end{matrix}$

We can replace ${\tilde{s}}_{1} = [s_{α_{1}}, s_{β_{1}}]$ , ${\tilde{s}}_{2} = [s_{α_{2}}, s_{β_{2}}]$ with the $[s_{a^{-}}^{g}, s_{a^{+}}^{g}], [s_{b^{-}}^{g}, s_{b^{+}}^{g}]$ . Then we can get the Equations (12, 13). So we can see the operational laws based on the unified forms transformed from multi-granularity linguistic value are reasonable.

3.2 Distance measure

We will present a consistency index of linguistic preference relations, by defining the distance of a multi-granularity linguistic preference relation. This consistency index can be used to measure the acceptable degree of linguistic preference relations in GDM. Initially, we define the distance between $s_{i}^{g}$ and $s_{j}^{g}$ through extending the deviation degree measure of Xu [26] and get the definition 5.

Definition 5. Let $s_{i}^{g}$ and $s_{j}^{g} \in S^{g}$ , Then the distance measure is defined as $d (s_{i}^{g}, s_{j}^{g}) = \frac{| sub (s_{i}^{g}) - sub (s_{j}^{g}) |}{g} = \frac{| j - i |}{g}$ (15)

However sometimes we need to compute the deviation degree measure of linguistic variables with different semantics. Then we give the definition 6.

Definition 6. Let $s_{a}^{g_{1}}$ and $s_{b}^{g_{2}}$ be two multi-granularity linguistic variables. They are transformed to the uniform, $s_{a}^{g_{1}} = [s_{a^{-}}^{g}, s_{a^{+}}^{g}]$ , $s_{b}^{g_{2}} = [s_{b^{-}}^{g}, s_{b^{+}}^{g}]$ . Then the distance measure is defined as

$\begin{matrix} d (s_{a}^{g_{1}}, s_{b}^{g_{2}}) \\ = \frac{| sub (s_{a^{-}}^{g}) - sub (s_{b^{-}}^{g}) | + | sub (s_{a^{+}}^{g}) - sub (s_{b^{+}}^{g}) |}{2 g} \\ = \frac{1}{2 g} [| a^{-} - b^{-} | + | a^{+} - b^{+} |] \end{matrix}$ (16)

The definition of distance measure should be proven reasonable in theory through satisfying common axioms which have long been taken as a widely acknowledged truth. Various metric distances have been proposed before. A metric distance d in a set A is a real function d : A × A → R, which satisfies the following three axioms for x, y, z ∈ A:

d (x, y) =0 ⇔ x = y,

Proof.

\begin{matrix} d (s_{a}^{g_{1}}, s_{b}^{g_{2}}) & = & \frac{1}{2 g} [| a^{-} - b^{-} | + | a^{+} - b^{+} |] \\ = & 0 \Leftrightarrow | a^{-} - b^{-} | = 0, | a^{+} - b^{+} | = 0 \\ \Leftrightarrow a^{-} = b^{-}, a^{+} = b^{+} \Leftrightarrow s_{a}^{g_{1}} = s_{b}^{g_{2}} \end{matrix}

d (x, y) = d (y, x) (symmetric) ,

Proof.

\begin{matrix} d (s_{a}^{g_{1}}, s_{b}^{g_{2}}) & = & \frac{1}{2 g} [| a^{-} - b^{-} | + | a^{+} - b^{+} |] \\ = & \frac{1}{2 g} [| b^{-} - a^{-} | + | b^{+} - a^{+} |] \\ \Leftrightarrow d (s_{a}^{g_{1}}, s_{b}^{g_{2}}) = d (s_{b}^{g_{2}}, s_{a}^{g_{1}}) \end{matrix}

d (x, z) + d (z, y) ≥ d (x, y) (triangleinequality) .

Proof.

\begin{matrix} d (s_{a}^{g_{1}}, s_{b}^{g_{2}}) + d (s_{b}^{g_{2}}, s_{c}^{g_{3}}) \geq d (s_{a}^{g_{1}}, s_{c}^{g_{3}}) \\ | a^{-} - c^{-} | & = & | (a^{-} - b^{-}) + (b^{-} - c^{-}) | \\ \leq | a^{-} - b^{-} | + | b^{-} - c^{-} | \\ | a^{+} - c^{+} | & = & | (a^{+} - b^{+}) + (b^{+} - c^{+}) | \\ \leq | a^{+} - b^{+} | + | b^{+} - c^{+} |} \\ \Leftrightarrow \frac{1}{2 g} [| a^{-} - b^{-} | + | a^{+} - b^{+} | \\ + | b^{-} - c^{-} | + | b^{+} - c^{+} |] \\ \geq \frac{1}{2 g} (| a^{-} - c^{-} | + | a^{+} - c^{+} |) \\ \Leftrightarrow d (s_{a}^{g_{1}}, s_{b}^{g_{2}}) + d (s_{b}^{g_{2}}, s_{c}^{g_{3}}) \geq d (s_{a}^{g_{1}}, s_{c}^{g_{2}}) \end{matrix}

It shows that the presented definition 6 satisfies a number of “reasonable axioms”. We can regard Equation (16) as a reasonable distance measure.

4 Consistency measures of multi-granularity linguistic preference relations

4.1 The consistency index of multi-granularity linguistic preference relations

Consistency is a stronger condition than transitivity. However, it is hard to obtain perfect consistency using the linguistic preference relations, especially when the experts’ preferences are expressed as multi-granularity linguistic variables. To make the measurement of consensus in multi-granularity GDM problems possible and easier, we need to define consistency index based on the unified forms $\tilde{s} = [s^{-}, s^{+}]$ . Xu [26], Dong, Xu and Li [25] defined the consistency index of a linguistic term set, but cannot solve different linguistic term sets problems. Consensus in group decision-making with different linguistic term sets (multi-granularity linguistic information) is very common. Herrera-Viedma, Mata, Martínez, Chiclana and Pérez [18] defined this type consistency index, but the unified forms of different linguistic term sets are complicated. Compared with the traditional consistency definitions, we try to define a new consistency index which is more flexible.

Here we use two indexes. One is individual additive consistency index which is used to evaluate the agreement in different alternatives of an expert. The other is group additive consistency index which is used to evaluate the agreement between the individual experts’ opinions and the group opinion.

Individual Additive Consistency Index

Definition 7. [25] $A_{k} = (a_{ij}^{k})_{n \times n}$ is called a consistent linguistic preference relation if there exists $a_{il}^{k} \oplus a_{lj}^{k} = a_{ij}$ for i, j, l = 1, 2, …, n.

Theorem 1. [25] $A_{k} = (a_{ij}^{k})_{n \times n}$ is a transitive linguistic preference relation under the condition that A _k is a consistent linguistic preference relation.

Expression (1) can be used to calculate an estimated value of a preference degree using other preference degrees in a fuzzy preference relation. We can extend expression (1) to apply into a linguistic preference relation. For the preferences of one expert are with the same granularity (g), we suppose the middle linguistic variable is $s_{g / 2}^{g}$ .

$a_{ij}^{l} = a_{ik}^{l} + a_{kj}^{l} - s_{g / 2}^{g}, k \neq i, j$

We know the preference value $a_{ij}^{l} (i \neq j)$ can be estimated in three different ways [24]. Then we can get the matrix $B^{l} = (b_{ij}^{l})_{n \times n}$ $b_{ij}^{l} = \frac{1}{n - 2} \sum_{c k = 1 k \neq i, j}^{n} a_{ik}^{l} + a_{kj}^{l} - s_{g / 2}^{g}, k \neq i, j$ (17)

So there may be inconsistency. We define $ϑ (a_{ij}^{l})$ to reflect the possible inconsistency of expert e _l according to the preference of alternative i over j. $ϑ_{ij}^{l} = d (a_{ij}^{l}, b_{ij}^{l}) = \frac{| sub (a_{ij}^{l}) - sub (b_{ij}^{l}) |}{\sup (a_{ij}^{l})}$ (18)

Here the distance measure is Equation (15) for $a_{ij}^{l}$ in individual preference matrix has the same granularity and the same form. We can see $[ϑ_{ij}^{l}]_{n \times n}$ is a symmetric matrices, for $ϑ_{ij}^{l} = ϑ_{ji}^{l} = d (a_{ij}^{l}, b_{ij}^{l}) = \frac{| sub (a_{ij}^{l}) - sub (b_{ij}^{l}) |}{\sup (a_{ij}^{l})}$ . For symmetric matrices, it is necessary to store only the upper triangular half of the matrix (upper triangular format). In a new matrix of $[ϑ_{ij}^{l}]_{4 \times 4}$ , the elements of $ϑ_{12}^{l}$ , $ϑ_{13}^{l}$ , $ϑ_{14}^{l}$ , $ϑ_{23}^{l}$ , $ϑ_{24}^{l}$ , $ϑ_{34}^{l}$ are sufficient to describe the consistency index. These elements are independent, normally distributed with mean 0 and standard deviation σ, namely $ϑ_{ij}^{l} \sim N (0, σ^{2})$ . $ϑ_{ij}^{l}$ is between 0 and 1. The smaller the $ϑ_{ij}^{l}$ is, the more acceptable the expert’s preference is. The Chi Square Distribution is the distribution of the sum of squared standard normal deviates. The degree of freedom of the distribution is equal to the number of standard normal deviates being summed. We think individual additive consistency can be described as Chi Square Distribution.

So for expert e _l, we define the Individual Additive Consistency Index CI (A ^l) $CI (A^{l}) = \sqrt{\frac{2}{n (n - 1)} \sum_{i = 1}^{n} \sum_{j = i + 1}^{n} (ϑ_{ij}^{l})^{2}}$ (19)

For some experts, they can keep consistency. But not all experts can always keep consistency.

Global Additive Consistency Index

Firstly, we aggregate individual experts’ preference and get the global preference A ^* $A^{*} = (a_{ij}^{*})_{n \times n} = {(\sum_{l = 1}^{m} a_{ij}^{l} \cdot w_{l})}_{n \times n}$ (20)

$ɛ_{ij}^{l} = d (a_{ij}^{l}, a_{ij}^{*}) = \frac{| sub (a_{ij}^{l -}) - sub (a_{ij}^{* -}) | + | sub (a_{ij}^{l +}) - sub (a_{ij}^{* +}) |}{2 \sup (a_{ij}^{l -})}$ (21)

We can get the matrix $ɛ^{l} = (ɛ_{ij}^{l})_{n \times n}$ . The theory is similar to individual consistency. We still think global consistency can be described as Chi Square Distribution. So we define the global consistency index $CI (A^{l}, A^{*}) = \sqrt{\frac{2}{n (n - 1)} \sum_{i = 1}^{n} \sum_{j = i + 1}^{n} (ɛ_{ij}^{l})^{2}}$ (22)

In Equation (21), the distance measure is Equation (16). For different decision-makers’ linguistic variables have different granularity, we need to unify them into uncertain linguistic variables.

We set CI (A ^l, A ^*) as the group consistency index of the expert l. Obviously, the smaller the value of CI (A ^l, A ^*), the more consistent the group linguistic preference relation A ^*. If CI (A ^l, A ^*) =0, then the group preference relation is a consistent linguistic preference relation. But for a multi-granularity linguistic group decision problem, it’s almost impossible. So we need to decide when the group preference relation is acceptable.

4.2 Establishing the consistency thresholds

Thresholds for Individual Additive Consistency

Individual Consistency CI (A ^l) reflects the consistent degree of an expert. So at first we require the expert to keep consistent individually. Then we try to keep the group decision consistent. De Jong [28] thinks the decision makers often have certain consistency tendency in making pairwise comparisons.

As we assume individual consistency fits with Chi Square Distribution. We can use the following theorem.

Theorem 2. $\frac{n (n - 1)}{2} (\frac{1}{σ} \times CI (A^{l})))^{2}$ is a chi-square distribution with $\frac{n (n - 1)}{2}$ degrees of freedom, namely $\frac{n (n - 1)}{2} (\frac{1}{σ} \times CI (A^{l}))^{2} \sim χ^{2} (\frac{n (n - 1)}{2})$ .

The Proof of Theorem 2 is provided in Appendix.

If we further assume that $σ^{2} = σ_{0}^{2}$ , namely $ϑ_{ij}^{l'} \sim N (0, σ_{0}^{2})$ , then the consistency measure is to test hypothesis H ₀ versus hypothesis H ₁: $\begin{matrix} H_{0} : σ^{2} \leq σ_{0}^{2}; \\ H_{1} : σ^{2} < σ_{0}^{2} \end{matrix}$

The degrees of freedom of the estimator $χ^{2} = \sum_{i = 1}^{n} \sum_{j = i + 1}^{n} (\frac{ϑ_{ij}^{l}}{σ_{0}})^{2}$ is $\frac{n (n - 1)}{2}$ . We suppose it is a one-sided right-tailed test, and we can get the critical value λ _α of the χ ² distribution at the significance level α. In this way, we have that $\bar{CI} = σ_{0} \sqrt{\frac{2 λ_{α}}{n (n - 1)}}$

Then we can get:

$CI (A^{l}) \leq \bar{CI}$ , we accept the consistency level of A ^l;

$CI (A^{l}) > \bar{CI}$ , we don’t accept the consistency level of A ^l.

According to the actual situation, the decision makers can set different values for α and σ ₀. Table 1 shows the values of $\bar{CI}$ for different n when setting α = 0.1 and σ ₀ = 0.3

Thresholds for Global Additive Consistency

CI (A ^l, A ^*) reflects the deviation degree between individual’s preference relation A ^l and the global linguistic preference relation A ^*. As we assume global consistency fits with Chi Square Distribution. We can use the following theorem.

Theorem 3. $\frac{n (n - 1)}{2} (\frac{1}{σ} \times CI (A^{l}, A^{*}))^{2}$ is a chi-square distribution with $\frac{n (n - 1)}{2}$ degrees of freedom, namely $\frac{n (n - 1)}{2} (\frac{1}{σ} \times CI (A^{l}, A^{*}))^{2} \sim χ^{2} (\frac{n (n - 1)}{2})$ .

The Proof of Theorem 3 is provided in Appendix.

We have the same assumption with individual consistency. The degrees of freedom of the estimator $χ^{2} = \sum_{i = 1}^{n} \sum_{j = i + 1}^{n} (\frac{ɛ_{ij}^{l}}{σ_{0}})^{2}$ is $\frac{n (n - 1)}{2}$ . We suppose it is a one-sided right-tailed test, and we can get the critical value λ _α of the χ ² distribution at the significance level α. In this way, we have that $\bar{CI} = σ_{0} \sqrt{\frac{2}{n (n - 1)} λ_{α}}$

According to the actual situation, the decision makers can set different values for α and σ ₀ to determine $\bar{CI}$ to decide whether the group preference relation is of acceptable consistency.

5 Illustrative examples

5.1 Example 1

In order to show how these theoretical results work in practice, let us consider the following two examples. In the first example, two experts supply linguistic preference relations A ^l: $A^{1} = (\begin{matrix} s_{4}^{8} & s_{3}^{8} & s_{6}^{8} & s_{7}^{8} \\ s_{6}^{8} & s_{4}^{8} & s_{7}^{8} & s_{7}^{8} \\ s_{3}^{8} & s_{2}^{8} & s_{4}^{8} & s_{6}^{8} \\ s_{3}^{8} & s_{2}^{8} & s_{1}^{8} & s_{4}^{8} \end{matrix}), A^{2} = (\begin{matrix} s_{3}^{6} & s_{1}^{6} & s_{4}^{6} & s_{5}^{6} \\ s_{5}^{6} & s_{3}^{6} & s_{5}^{6} & s_{6}^{6} \\ s_{5}^{6} & s_{2}^{6} & s_{3}^{6} & s_{5}^{6} \\ s_{3}^{6} & s_{1}^{6} & s_{2}^{6} & s_{3}^{6} \end{matrix}) .$

We can get $B^{1} = (\begin{matrix} s_{4}^{8} & s_{0.5}^{8} & s_{1}^{8} & s_{3}^{8} \\ s_{2}^{8} & s_{4}^{8} & s_{2}^{8} & s_{5}^{8} \\ s_{0.5}^{8} & s_{1}^{8} & s_{4}^{8} & s_{1.5}^{8} \\ s_{2}^{8} & s_{3.5}^{8} & s_{1}^{8} & s_{4}^{8} \end{matrix}) B^{2} = (\begin{matrix} s_{3}^{6} & s_{0}^{6} & s_{0.5}^{6} & s_{2}^{6} \\ s_{3.5}^{6} & s_{3}^{6} & s_{2.5}^{6} & s_{4}^{6} \\ s_{1.5}^{6} & s_{0}^{6} & s_{3}^{6} & s_{3}^{6} \\ s_{0.5}^{6} & s_{2}^{6} & s_{0.5}^{6} & s_{3}^{6} \end{matrix})$

Here we set α = 0.1 and σ ₀ = 0.3. From Table 1, we get $\bar{CI} = 0.182$ . $\begin{matrix} CI (A^{1}) = \sqrt{\frac{2}{n (n - 1)} \sum_{i = 1}^{n} \sum_{j = i + 1}^{n} (ϑ_{ij}^{l})^{2}} = 0.150 \\ CI (A^{2}) = \sqrt{\frac{2}{n (n - 1)} \sum_{i = 1}^{n} \sum_{j = i + 1}^{n} (ϑ_{ij}^{l})^{2}} = 0.123 \end{matrix}$

By using our consistency measure method, we find $CI (A^{l}) \leq \bar{CI}$ , which shows that the preference A ^l is acceptable. The decision maker does not need to adjust the elements in A ^l in order to improve the consistency.

5.2 Example 2

Let us consider the example in which there are two experts $A^{1} = (\begin{matrix} s_{4}^{8} & s_{3}^{8} & s_{6}^{8} & s_{7}^{8} \\ s_{6}^{8} & s_{4}^{8} & s_{7}^{8} & s_{7}^{8} \\ s_{3}^{8} & s_{2}^{8} & s_{4}^{8} & s_{6}^{8} \\ s_{3}^{8} & s_{2}^{8} & s_{1}^{8} & s_{4}^{8} \end{matrix}) and A^{2} = (\begin{matrix} s_{3}^{6} & s_{1}^{6} & s_{4}^{6} & s_{5}^{6} \\ s_{5}^{6} & s_{3}^{6} & s_{5}^{6} & s_{6}^{6} \\ s_{5}^{6} & s_{2}^{6} & s_{3}^{6} & s_{5}^{6} \\ s_{3}^{6} & s_{1}^{6} & s_{2}^{6} & s_{3}^{6} \end{matrix})$

From example1 we get the preferences of A ¹ and A ² are acceptable.

Then, the two matrices are transformed to uniform through the function $τ : S \to \tilde{s}$ in definition 4. $\begin{matrix} A^{1} = (\begin{matrix} (s_{9}^{24}, s_{15}^{24}) & (s_{6}^{24}, s_{12}^{24}) & (s_{15}^{24}, s_{21}^{24}) & (s_{18}^{24}, s_{24}^{24}) \\ (s_{15}^{24}, s_{21}^{24}) & (s_{9}^{24}, s_{15}^{24}) & (s_{18}^{24}, s_{24}^{24}) & (s_{18}^{24}, s_{24}^{24}) \\ (s_{6}^{24}, s_{12}^{24}) & (s_{3}^{24}, s_{9}^{24}) & (s_{9}^{24}, s_{15}^{24}) & (s_{15}^{24}, s_{21}^{24}) \\ (s_{6}^{24}, s_{12}^{24}) & (s_{3}^{24}, s_{9}^{24}) & (s_{0}^{24}, s_{6}^{24}) & (s_{9}^{24}, s_{15}^{24}) \end{matrix}) \\ A^{2} = (\begin{matrix} (s_{8}^{24}, s_{16}^{24}) & (s_{0}^{24}, s_{8}^{24}) & (s_{12}^{24}, s_{20}^{24}) & (s_{16}^{24}, s_{24}^{24}) \\ (s_{16}^{24}, s_{24}^{24}) & (s_{8}^{24}, s_{16}^{24}) & (s_{16}^{24}, s_{24}^{24}) & (s_{20}^{24}, s_{24}^{24}) \\ (s_{16}^{24}, s_{24}^{24}) & (s_{4}^{24}, s_{12}^{24}) & (s_{8}^{24}, s_{16}^{24}) & (s_{16}^{24}, s_{24}^{24}) \\ (s_{8}^{24}, s_{16}^{24}) & (s_{0}^{24}, s_{8}^{24}) & (s_{4}^{24}, s_{12}^{24}) & (s_{8}^{24}, s_{16}^{24}) \end{matrix}) \end{matrix}$

The collective linguistic preference relation with average operator $\begin{matrix} A^{*} = (\begin{matrix} (s_{8.5}^{24}, s_{15.5}^{24}) & (s_{3}^{24}, s_{10}^{24}) & (s_{13.5}^{24}, s_{20.5}^{24}) & (s_{17}^{24}, s_{24}^{24}) \\ (s_{15.5}^{24}, s_{22.5}^{24}) & (s_{8.5}^{24}, s_{15.5}^{24}) & (s_{17}^{24}, s_{24}^{24}) & (s_{19}^{24}, s_{24}^{24}) \\ (s_{11}^{24}, s_{18}^{24}) & (s_{3.5}^{24}, s_{10.5}^{24}) & (s_{8.5}^{24}, s_{15.5}^{24}) & (s_{15.5}^{24}, s_{22.5}^{24}) \\ (s_{7}^{24}, s_{14}^{24}) & (s_{1.5}^{24}, s_{8.5}^{24}) & (s_{2}^{24}, s_{9}^{24}) & (s_{8.5}^{24}, s_{15.5}^{24}) \end{matrix}) \end{matrix}$

Here we set α = 0.1 and σ ₀ = 0.3. From Table 1, we get $\bar{CI} = 0.182$ . $\begin{matrix} ɛ^{1} = [\begin{matrix} - & 0.104 & 0.042 & 0.021 \\ 0.042 & - & 0.021 & 0.021 \\ 0.229 & 0.042 & - & 0.042 \\ 0.063 & 0.042 & 0.104 & - \end{matrix}] \\ ɛ^{2} = [\begin{matrix} - & 0.104 & 0.042 & 0.021 \\ 0.042 & - & 0.021 & 0.021 \\ 0.229 & 0.042 & - & 0.042 \\ 0.063 & 0.042 & 0.104 & - \end{matrix}] \end{matrix}$ $CI (A^{1}, A^{*}) = \sqrt{\frac{2}{n (n - 1)} \sum_{i = 1}^{n} \sum_{j = i + 1}^{n} (ɛ_{ij}^{l})^{2}} = 0.051$

$CI (A^{1}, A^{*}) = CI (A^{2}, A^{*}) < \bar{CI}$ , which shows that the decision maker don’t need to adjust the elements in A ¹ or A ² in order to improve the consistency.

6 Conclusion

The linguistic performance values given to the alternatives by the different sources are represented in linguistic term sets with different granularity and/or semantic. How to deal with multi-granularity linguistic information is very important. The consistency measure is a basic problem in GDM using preference relations. This paper mainly contributes to developing a transformation model to maintain the uncertainty degrees. We use the uncertain linguistic variables as the unified form. Consistency index of linguistic preference relations based on the distance of a multi-granularity linguistic preference relation is defined. Chi-square statistic is used to establish the consistency thresholds.

There are two main characteristics of this manuscript. First, the consistency measure laws based on the uncertain linguistic variable are in accord with previous operational laws. But our extension model can deal with multi-experts decision problems where different

experts may have different levels of knowledge about a problem and multi-granularity linguistic information can be used to express their opinions. Second, we combine linguistic computational model with statistics methods to measure consistency in GDM based on the assumption that expert is being neither random nor illogical in his or her pairwise comparisons. In the future research, how to adjust the preferences of experts to improve the consistency is needed. More consistency optimization models and analysis will be done to improve the result of group decisions.

7 Appendix

Proof of Theorem 2. Since $ϑ_{ij}^{l}$ is independent normally distributed with mean 0 and standard deviation 1, we have that $\begin{matrix} CI (A^{l}) & = & \sqrt{\frac{2}{n (n - 1)} \sum_{i = 1}^{n} \sum_{j = i + 1}^{n} (ϑ_{ij}^{l})^{2}} \\ \Rightarrow \frac{n (n - 1)}{2} {(\frac{1}{σ} \times CI (A^{l}))}^{2} \\ \sim χ^{2} (\frac{n (n - 1)}{2}) \end{matrix}$ This completes the proof of Theorem 2.

Proof of Theorem 3. A ^* is a consistent linguistic preference relation on the condition that all of A ¹, A ²,… A ^m are consistent linguistic preference relations.

Theorem 2 tells us that Individual additive consistency can be described as Chi Square Distribution. Then $CI (A^{l}) = \sqrt{\frac{2}{n (n - 1)} \sum_{i = 1}^{n} \sum_{j = i + 1}^{n} (ϑ_{ij}^{l})^{2}} \leq \bar{CI}$

Let $A^{*} = (a_{ij}^{*})_{n \times n} = {(\sum_{l = 1}^{m} a_{ij}^{l} \cdot w_{l})}_{n \times n}, and ɛ_{ij}^{l} = d (a_{ij}^{l}, a_{ij}^{*})$

let $y_{ij}^{k} = (a_{ij}^{k} - \frac{1}{n} \sum_{c = 1}^{n} (a_{ic}^{l} + a_{cj}^{l}))^{2}$ $\begin{matrix} CI (A^{l}, A^{*}) & = & \sqrt{\frac{2}{n (n - 1)} \sum_{i = 1}^{n} \sum_{j = i + 1}^{n} (ɛ_{ij}^{l})^{2}} = \sqrt{\frac{2}{n (n - 1)} \sum_{i = 1}^{n} \sum_{j = i + 1}^{n} (\sum_{k = 1}^{m} (w_{k}^{2} y_{ij}^{k}) + 2 \sum_{k < l} (w_{k} w_{l} \sqrt{y_{ij}^{k} y_{ij}^{l}}))} \\ \leq \sqrt{\frac{2}{n (n - 1)} \sum_{i = 1}^{n} \sum_{j = i + 1}^{n} (\sum_{k = 1}^{m} (w_{k}^{2} y_{ij}^{k}) + \sum_{k < l} (w_{k} w_{l} (y_{ij}^{k} + y_{ij}^{l})))} \\ = & \sqrt{\frac{2}{n (n - 1)} (\sum_{i = 1}^{n} (w_{k}^{2} \sum_{j = i + 1}^{n} \sum_{k = 1}^{m} y_{ij}^{k}) + \sum_{k < l} (w_{k} w_{l} (y_{ij}^{k} + y_{ij}^{l})))} \end{matrix}$

we know $CI (A^{l}, A^{*}) \leq \bar{CI} \sqrt{\sum_{k = 1}^{m} w_{k}^{2} + 2 \sum_{k < l} (w_{k} w_{l})} = \bar{CI}$ which completes the proof of Theorem 3.

Footnotes

Acknowledgments

The authors are grateful to the editors and the three anonymous reviewers for their insightful comments and suggestions. In addition, this research was partly supported by the National Natural Science Foundation of China (71401078, 71171115, 71273139, 71101073), the General Project of Philosophy and Social Science Research in Colleges and Universities in Jiangsu Province (2014SJB059), the reform Foundation of Postgraduate Education and Teaching in Jiangsu Province (JGKT10034), Qing Lan Project, Natural Science Foundation of Higher Education of Jiangsu Province of China under Grant (08KJD630002).

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