2.1. Linguistic Information and 2-Tuple Linguistic Representation Model

Suppose that S = {s₀, s₁, …, s_g−1} is a linguistic term set with odd cardinality g, such as 5 or 7, where each label s_i represents a possible value of a linguistic variable. Usually, it requires that the linguistic term set S satisfies the following characteristics: 1) the set of the linguistic terms is ordered: s_i ≥ s_j if i ≥ j; 2) there exists a negation operator “Neg” such that Neg (s_i) = s_g−i; 3) the maximization operator “max”: max (s_i, s_j) = s_i if s_i ≥ s_j; and 4) the minimization operator “min”: min (s_i, s_j) = s_i if s_i ≤ s_j.

In order to avoid information loss in computing with words, Herrera and Martínez [4] introduced the 2-tuple linguistic representation model on the basis of the symbolic translation. In this model, a 2-tuple (s_i, α_i) is used to represent linguistic information, where s_i is a linguistic term pertaining to the predefined linguistic term set and α_i ∈ [−0.5, 0.5) represents the symbolic translation. Specifically, the 2-tuple linguistic representation model is formally defined as follows:

2.2. Linguistic Distribution Assessments

In [7], Zhang et al. defined the concept of distribution assessments in a linguistic term set. In this subsection, we review some related knowledge regarding linguistic distribution assessment.

Definition 3.1.

[7]: Let m1=sk,βk1|k=0,1,…,g−1 and m2=sk,βk2|k=0,1,…,g−1 be two linguistic distribution assessments of a linguistic term set S, then the distance between m₁ and m₂ is defined as

The drawback of the distance measure defined by Eq. (4) is shown with Example 1.

Example 1.

Let S^example = {s₀, s₁, s₂, s₃, s₄, s₅, s₆} be a linguistic term set and there are three linguistic distribution assessments: m₁ = {(s₀, 0.5), (s₁, 0.5), (s₂, 0), (s₃, 0), (s₄, 0), (s₅, 0), (s₆, 0)}, m₂ = {(s₀, 0), (s₁, 0), (s₂, 0), (s₃, 0.5), (s₄, 0.5), (s₅, 0), (s₆, 0)}, m₃ = {(s₀, 0), (s₁, 0), (s₂, 0), (s₃, 0), (s₄, 0), (s₅, 0.5), (s₆, 0.5)}. By Definition 3.1, we can obtain dm1,m2=120.5+0.5+0+0.5+0.5+0+0=1 and dm1,m3=120.5+0.5+0+0+0+0.5+0.5=1 which means that the distance between m₁ and m₂ is equal to that between m₁ and m₃. Obviously it is unreasonable.

Zhang et al. [14] pointed out that the distance measure defined by Eq. (4) just calculates the deviation between symbolic proportions and ignores the linguistic terms. To overcome the drawback, Zhang et al. [14] proposed a new distance measure as follows:

Definition 3.2.

[14]: Let m1=sk,βk1|k=0,1,…,g−1 and m2=sk,βk2|k=0,1,…,g−1 be two linguistic distribution assessments of a linguistic term set S, then the distance between m₁ and m₂ is defined as

Reconsider Example 1. By Definition 3.2, we can obtain d (m₁, m₂) = 0.50 and d (m₁, m₃) = 0.83. Indeed, the distance measure defined by Eq. (5) overcomes the limitation of the distance measure defined by Eq. (4) to a certain extent. However, the distance measure defined by Eq. (5) also has limitation, which can be illustrated by the following Example 2:

Example 2.

Let S^example = {s₀, s₁, s₂, s₃, s₄, s₅, s₆} be a linguistic term set and there are three linguistic distribution assessments: m₁ = {(s₀, 0), (s₁, 0), (s₂, 0.1), (s₃, 0.8), (s₄, 0.1), (s₅, 0), (s₆, 0)}, m₂ = {(s₀, 0), (s₁, 0.1), (s₂, 0.2), (s₃, 0.4), (s₄, 0.2), (s₅, 0.1), (s₆, 0)}. Obviously, we can see that m₁ and m₂ are two different linguistic distribution assessments, that is, m₁ ≠ m₂. However, by definition 3.2, we obtain d (m₁, m₂) = 0. According to definition 3.2, it is not difficult to infer that d (m₁, m₂) = 0 if E (m₁) = E(m₂). We argue that an ideal distance measure for linguistic distribution assessment should possess the basic property: d (m₁, m₂) = 0 if and only if m₁ = m₂.

Recently, Yu et al. [15] defined a new distance measure between linguistic distribution assessments as follows:

Definition 3.3.

[15]: Let m1=s1k1,β1k1|k1=1,2,…,♯m1 and m2=s2k2,β2k2|k1=1,2,…,♯m2 be two linguistic distribution assessments of a linguistic term set S, where ♯m_i is the number of linguistic terms in m_i, βiki>0 and ∑ki=1♯miβiki=1, i=1,2. The generalized distance measure between two linguistic distribution assessments m₁ and m₂ can be defined by Eq. (6), where f is the linguistic scale function presented in Yu et al. [15] and r > 0.

The advantage of the distance measure defined by Eq. (6) can measure the distance between some unequal linguistic distribution assessments which previous distance measurement cannot measure [15]. However, it is not free from drawbacks.

Example 3.

Let S^example = {s₀, s₁, s₂, s₃, s₄, s₅, s₆} be a linguistic term set and there are three linguistic distribution assessments: m₁ = {(s₀, 0), (s₁, 0), (s₂, 0)), (s₃, 0), (s₄, 0), (s₅, 0.2), (s₆, 0.8)}, m₂ = {(s₀, 0), (s₁, 0), (s₂, 0.5), (s₃, 0.5), (s₄, 0), (s₅, 0), (s₆, 0)}, m₃ = {(s₀, 0), (s₁, 0), (s₂, 0), (s₃, 0.5), (s₄, 0.5), (s₅, 0), (s₆, 0)}. By definition 3.3 (r = 1), we can obtain dm1,m2=12120+0.55+120+0.0833=0.1583 and dm1,m3=12120.0833+0.4667+120.0833+0.1667=0.2000 which means that the distance between m₁ and m₂ is smaller to that between m₁ and m₃. Obviously it is unreasonable.

Based on the above analysis, this paper attempts to develop a new distance measure between distribution assessments by making full use of the proportion distribution information of linguistic distribution assessment. Similar to the cumulative probability distribution, the concept of cumulative proportion distribution can be derived for linguistic distribution assessment. In particular, the vector ∑r=00βr,∑r=01βr,…,∑r=0kβr,…,∑r=0g−1βr is defined as the cumulative proportion distribution of linguistic distribution assessment m = {(s_k, β_k) |k = 0, 1, …, g − 1}. Using the cumulative proportion distribution, we can define a distance measure between linguistic distribution assessments m₁ and m₂ by computing their difference in cumulative proportion distributions. The proposed distance measure is given below.

Definition 3.4.

Let m1=sk,βk1|k=0,1,…,g−1 and m2=sk,βk2|k=0,1,…,g−1 be two linguistic distribution assessments of a linguistic term set S, then the distance between two linguistic distribution assessments m₁ and m₂ is defined as

Example 4.

Reconsider the linguistic distribution assessments in Example 2: m₁ = {(s₀, 0), (s₁, 0), (s₂, 0.1), (s₃, 0.8), (s₄, 0.1), (s₅, 0), (s₆, 0)}, m₂ = {(s₀, 0), (s₁, 0.1), (s₂, 0.2), (s₃, 0.4), (s₄, 0.2), (s₅, 0.1), (s₆, 0)}. For m₁ and m₂, we can calculate their cumulative proportion distributions CP₁ = (0, 0, 0.1, 0.9, 1, 1, 1) and CP₂ = (0, 0.1, 0.3, 0.7, 0.9, 1, 1). By using Eq. (7), we have d (m₁, m₂) = 0.1. It is worth noting that the proposed distance measure defined by Eq. (7) has very clear geometric meaning. Figure 1 shows the proportion distribution of linguistic distribution assessments m₁ and m₂ (for convenience, the subscripts of linguistic term are taken as the values of variable). The distance d (m₁, m₂) between two linguistic distribution assessments m₁ and m₂ can be explained as the ratio of intersecting area of the corresponding cumulative proportion distributions and g − 1. From Figure 1, we can see that the distance d (m₁, m₂) equals the ratio of the sum of area of S1, S2, S3, S4, and the area of rectangle OABC. Clearly, the larger the intersecting area between the two cumulative proportion distributions is, the farther the distance between the two linguistic distribution assessments is.

Figure 1

The geometric interpretation of the proposed distance measure.

In order to analyze the performance of the proposed distance measure, we conduct a comparative analysis by using numerical examples. Reconsider Example 1, Example 2, and Example 3. We calculate the values of the distance by applying the proposed distance measure defined by Equation (7) and the existing distance measures defined by Equations. (4–6). Table 1 shows the comparison results. As we can see from Table 1 on the one hand, the existing distance measures show some limitations, which has been explained in detail in section 3. On the other hand, we can also see that the proposed distance measure in this paper not only can distinguish linguistic distribution assessments as shown in Example 1, in which the positive proportions are distributed on different linguistic terms, but also can distinguish linguistic distribution assessments with the same expectation values. Moreover, for the linguistic distribution assessments in Example 3, the results calculated by the proposed distance measure are more reasonable than that calculated by the distance measure proposed in [15]. From above analysis, we can conclude that the proposed distance measure for linguistic distribution assessments exhibits better performance.

For the proposed distance measure, we can obtain the following properties.

Theorem 3.1.

Let m1=sk,βk1|k=0,1,…,g−1, m2=sk,βk2|k=0,1,…,g−1 and m3=sk,βk3=|k=0,1,…,g−1 be any three linguistic distribution assessments of a linguistic term set S.

Then the distance measure defined by Eq. (7) satisfies the following properties:

1. d (m₁, m₂) ≥ 0;

2. d (m₁, m₂) = d (m₂, m₁);

3. d (m₁, m₂) = 0 if and only if m₁ = m₂, i.e., βk1=βk2k=0,1,…,g−1;

   d (m₁, m₂) = 1 if and only if m₁ = m_min (or m_max), m₂ = m_max (or m_min)

   where m_min = {(s₀, 1), …, (s_{g − 1}, 0)} and m_max = {(s₀, 0), …, (s_{g − 1}, 1)};

4. d (m₁, m₂) ≤ d (m₁, m₃) + d (m₃, m₂).

Proof.

(1) and (2) are obvious.

(3). According to Definition 3.4, d (m₁, m₂) = 0 holds if and only if 1g−1∑k=0g−1|∑r=0kβ−∑r=0kβr2|=0, which means that |∑r=0kβr1−∑r=0kβr2|=0,k=0,1,…,g−1. For k = 0, we can obtain β01=β02; For k = 0, 1, we can obtain β11=β12; …; For k = 0, 1, …, g−1, we can obtain βg−11=βg−12. Therefore, d (m₁, m₂) = 0 if and only if m₁ = m₂. For the second part, on the one hand, if m₁ = {(s₀, 1), …, (s_g−1, 0)}, m₂ = {(s₀, 0), …, (s_{g − 1}, 1)} or m₁ = {(s₀, 0), …, (s_{g − 1}, 1)}, m₂ = {(s₀, 1), …, (s_{g − 1}, 0)}, one can get d (m₁, m₂) = 1 by using the distance measure in Definition 3.4. On the other hand, if d (m₁, m₂) = 1, we know that 1g−1∑k=0g−1|∑r=0kβr1−∑r=0kβr2|=1. Since 0≤|∑r=0kβr1−∑r=0kβr2|≤1, we can obtain |∑r=0kβr1−∑r=0kβr2|=1,k=0,1,…,g−1. Notice that 0≤∑r=0kβr1≤1, 0≤∑r=0kβr2≤1,k=0,1,…,g−1. Hence, we have ∑r=0kβr1=1 and ∑r=0kβr1=0, k=0,1,…,g−1 or ∑r=0kβr1=0 and ∑r=0kβr1=1, k=0,1,…,g−1. Since ∑r=0kβr1≥∑r=0k−1βr1, ∑r=0kβr2≥∑r=0k−1βr2, k = 1, …, g − 1 we can conclude that β01=βg−12=1,β11=…=βg−11=β02=…=βg−22=0 or βg−11=β02=1,β01=…=βg−21=β12=…=βg−12=0, i.e., m₁ = m_min (or m_max), m₂ = m_max (or m_min).

(4) Notice that d(m1,m2) = 1g−1∑k=0g−1|∑r=0kβr1−∑r=0kβr2| = 1g−1∑k=0g−1|∑r=0kβr1 − ∑r=0kβr3+∑r=0kβr3−∑r=0kβr2| ≤1g−1∑k=0g−1(|∑r=0kβr1−∑r=0kβr3|+|∑r=0kβr3−∑r=0kβr2|) =1g−1∑k=0g−1|∑r=0kβr1−∑r=0kβr3|+1g−1∑k=0g−1|∑r=0kβr3−∑r=0kβr2|=d(m1,m3) +d(m3,m2) , i.e., d(m₁, m₂) ≤ d (m₁, m₃) + d (m₃, m₂).

This completes the proof of Theorem 3.1.

Definition 3.5.

Let m1={sk,βk1|k=0,1,…,g−1} and m2={sk,βk2|k=0,1,…,g−1} be two linguistic distribution assessments of a linguistic term set S, then the similarity degree between two linguistic distribution assessments m₁ and m₂ can be defined as

Obviously, 0 ≤ s (m₁, m₂) ≤ 1. The closer s (m₁, m₂) is to 1, the more similar m₁ is to m₂, while the closer s (m₁, m₂) is to 0, the more distant m₁ is from m₂.

4.1. The Framework of the Proposed Consensus Support Model

Consider a MAGDM problem with linguistic distribution assessments. Let X = {x₁, x₂, …, x_m} be a discrete set of m (m ≥ 2) potential alternatives, C = {c₁, c₂, …, c_n} be the set of n (n ≥ 2) attributes or criteria. Suppose that W = {w₁, w₂, …, w_n}^T is the weight vector of attributes, such that ∑j=1nwj=1,wj≥0,j∈1,2,…,n and w_j denotes the weight of attribute c_j. There are a group of experts, E = {e₁, e₂, …, e_T}. Assume λ=λ1,λ2,…,λTT is the weight vector of the experts, where 0 ≤ λ_t ≤ 1, t ∈ {1, 2, …, T} and ∑t=1Tλt = 1. Suppose that M_t = (m_ij,t)_m×n is the linguistic distribution assessment decision matrix given by the expert e_t ∈ E, where mij,t=〈sk,pij,tk〉|k=0,1,…,g−1 represents the performance of the alternative x_i with respect to the attribute c_j. The aim of MAGDM is to select or prioritize these finite alternatives with a consensus.

A typical resolution method for a GDM problem consists of two different processes: the consensus process and the selection process. The consensus process refers to how to obtain the maximum degree of consensus or agreement among the experts on the solution alternatives, while the selection process consists in how to obtain the solution set of alternatives from the opinions on the alternatives given by the experts. Inspired by these two processes, we propose the framework of the MAGDM with linguistic distribution assessments. The details of the framework are described in Figure 2. As we know, the feedback mechanism is the core part of a consensus reaching model. There are three main steps in the proposed feedback mechanism. In the first step, an IR is implemented to locate the experts who need to reevaluate his/her assessment values and the position where he/she need to modify. In the second step, an optimization model is established to minimize the deviation between the adjusted values and initial preferences. Simultaneously, additional preference information provided by the experts is also incorporated into the optimization model. By solving the constructed optimization model, adjustment advices are generated in the third step. The details of the consensus process are introduced in the following subsections.

Figure 2

The resolution framework for multi-attribute group decision- making (MAGDM) problem with LDA.

4.2. The Consensus Measure

The weighted averaging operator DAWA_λ can be used to transform individual linguistic distribution assessment decision matrix M_t = (m_ij,t)_m×n, t ∈ {1, 2, …, T} into the following collective decision matrix M_col = (m_ij,col)_m×_n:

where

Based on the similarity degree of two linguistic distribution assessments, for each expert e_t (t = 1, 2, …, T), a similarity matrix St,col=sijt,colm×n can be defined by

where sijt,col is the similarity between the expert e_t and the group in their assessments of the position (i, j), i = 1, 2, …, m; j = 1, 2, …, n. That is,

There are T similarity matrices. By aggregating all the similarity matrices S^t,col (t = 1, 2, …, T) associated with each expert e_t, we can obtain a consensus matrix Scol=sijcolm×n as follows:

The aggregation is conducted using an weighted aggregation operator Agg,

In the following, based on the consensus matrix, the group consensus index can be defined.

4.3. The Feedback Mechanism

When comparing the current consensus level GCI with the predefined consensus threshold value GCI¯, if GCI≥GCI¯, then the CRP ends; otherwise, a new interaction should be conducted. The feedback mechanism generates personalized suggestions to the decision makers according to the consensus criteria. It helps decision makers to modify their preference values with respect to attributes. The proposed feedback mechanism consists of two advice rules: IRs and recommendation rule (RR).

(a) Identification rules: The IRs are utilized to identify attribute values provided by the experts that are contributing less to the attainment of a high consensus level. In the CRP for MAGDM, the IR are given to locate which expert need to reevaluate his/her assessment values and in which specific position he/she need to modify. Here, we define the set IDE that contains 3-tuples (t, i, j) symbolizing attribute values m_ij,t that should be changed because they affect badly to that consensus state. To compute IDE, we apply a two step identification process that uses the consensus measures previously defined.

1. Identification of position (i, j):

   (16)POS=i,j|sijcol=minsijcol|1≤i≤m,1≤j≤n

2. Identification of expert-position (t, i, j):

   (17)IDE=t,i,j|sijt,col<GCI¯∧i,j∈POS

(b) Recommendation rules: The RRs are to generate personalized advices to help experts to change their evaluation matrices. For each (t, i, j) ∈ IDE, the RRs will give recommendations for adjusting the corresponding attribute value. The RRs of the proposed model is based on an optimization model. At each iteration of our method, an optimization model is established for each (i, j) ∈ POS. By solving the optimization model, suggestions will be generated for all experts who need to adjust preferences in the position (i, j). The model proposed in this paper has two prominent characteristics. Firstly, the adjustment of attribute values takes into account the additional preference information from the experts. Secondly, under the condition that the group consensus level is improved, the optimization model protects the expert initial preference information by minimizing the adjustments.

Let EijM=t|sijt,col<GCI¯ be the set of subscripts of experts who need to modify the values in the position (i, j) ∈ POS. Let ♯EijM denote the number of elements contained in the set ♯EijM. At each iteration, there may be one or more experts who need to modify their preferences, that is, ♯EijM≥1 for (i, j) ∈ POS. Since the attribute values is in the form of linguistic distribution assessments, the modification of attribute value mij,t=〈sk,pij,tk〉|k=0,1,…,g−1 is essentially to adjust the distribution assessment pij,t=pij,t0,pij,t1,…,pij,tg−1. Let dij,t−=dij,t0,−,dij,t1,−,…,dij,tg−1,− and dij,t+=dij,t0,+,dij,t1,+,…,dij,tg−1,+ be the vector of negative deviations and positive deviations, respectively. The proposed RRs are based on the following optimization model (M − 1):

where devmij,t′,mij,t is the deviation between the adjusted linguistic distribution assessment mij,t′ and the original linguistic distribution assessment m_ij,t, GCI′ and GCI are the group consensus indices after and beforeadjustment, respectively, and Fij,t− and Fij,t+ are the feasible sets of dij,t− and dij,t+. In model (M − 1), the objective is to minimize the sum of deviations, so that the original preference information of the experts can be retained to the largest possible extent. The purpose of imposing constraint condition, GCI′ > GCI, on model (M − 1) is to improve the group consensus level. Notice that (i, j) ∈ POS means GCI=sijcol, which means that the increase of sijcol will lead to the improvement of group consensus level. In order to handle the constraint of improvement effectively, we transform the model (M − 1) into the following model (M − 2) by increasing the group similarity in the position (i, j): where △_ij is a proper parameter for controlling the group similarity level in the position (i, j) after adjustment. To ensure an adequate improvement of the group consensus level, we set the value of the parameter △_ij as where min2sijcol is the second smallest value in the consensus matrix S^col, and ε is a small positive number.

Δ_ij determined by the Equation (17) can guarantee that after the adjustment of preference values, the group consensus level in the position (i, j) can either reach the threshold GCI¯ or exceed the second smallest value in the consensus matrix S^col. In other words, it ensures that the group consensus level can be adequately improved by using Δ_ij determined by the Equation (17) at each iteration. We also note that the positive number ε is an important parameter in CRP. Under the premise that the model has solutions, a larger ε will reduce the number of iterations to reach consensus. But on the other hand, when ε is too large, there may be no solution to the optimization problem. Therefore, there is a trade-off between iterative speed and solvability in determining the value of ε.

In order to reflect the real preferences of decision makers, the proposed consensus reaching model allows decision makers to provide additional preference information at each iteration. It is supposed in our model that decision makers are willing to provide preference information about deviations dij,tk,− or/and dij,tk,+. For (i, j) ∈ POS, denote

and where Bij,tk,−=max0,pij,tk−pij,colkk=0,1,…,g−1 and Bij,tk,+=max0,pij,colk−pij,tkk=0,1,…,g−1;t∈EijM.

Clearly, the variables dij,tk,− and dij,tk,+ have to meet 0≤dij,tk,−≤Bij,tk,− and 0≤dij,tk,+≤Bij,tk,+ in order to improve the consensus level. Therefore, the vectors Bij,t− and Bij,t+ can be used to elicit additional preference information from the decision maker e_t. In particular, Bij,t− and Bij,t+ can be regarded as the upper bounds for determining the ranges of dij,t0k,LB and dij,t0k,UB. Suppose that the decision maker e_t provides the maximum adjustment dij,t0k,LB∈0,Bi0j0,tk,− for the decision variable dij,t0k,−, as well as the maximum adjustment dij,t0k,UB∈0,Bi0j0,tk,+ for the decision variable dij,t0k,+. By specifying the constraints on the variables dij,tk,− and dij,tk,+, we can obtain the following model (M − 3):

In model (M − 3), the objective is to minimize the weighted sum of all deviation variables for t∈TijM. The second constraint guarantees that the adjusted distribution still satisfies the normalization condition, that is, ∑k=0g−1pij,tk−dij,tk,−+dij,tk,+=1. The third and the fourth constraints are the restrictions on the additional preference information provided by decision makers, as well as the nonnegativity of the adjusted distribution.

4.4. An Iterative Algorithm for CRP

For a MAGDM problem, let M_t = (m_ij,t) _m×n be the decision matrix given by the expert e_t ∈ E, where mij,t=〈sk,pij,tk〉|k=0,1,…,g−1 is linguistic distribution assessment representing the performance of the alternative x_i with respect to the attribute c_j. Suppose that W = {w₁, w₂, …, w_n}^T is the weight vector of attributes, such that ∑j=1nwj=1,wj≥0,j∈1,2,…,n and λ = {λ₁, λ₂, cdots, λ_T}^T is the weight vector of the experts, where 0 ≤ λ_t ≤ 1, t ∈ {1, 2, …, T} and ∑t=1Tλt=1. Assume Round_max is the maximum number of iterative times, GCI¯ is the threshold of group consensus level. Let ε be a small positive number. Let l be the number of the iterative times. To reach a predefined consensus level GCI¯, we propose the following algorithm.

The proposed Algorithm 1 is an iterative process which can improve the group consensus level. Regarding convergence of Algorithm 1, we can prove the following Theorem 4.1.

Therefore, if GCI¯<min2sij,lcol, we have GCIl+1≥GCI¯; otherwise, we have GCIl+1≥min2sij,lcol>Minsij,lcol=GCIl, which completes the proof of the Theorem 4.1.

Theorem 4.1 shows that the group consensus level GCI_(l) is increasing in the process of iteration. Since the parameter ε is positive, we can always obtain an improved decision matrix for each expert, which satisfies the predefined consensus level GCI¯ after finite times of implementing Algorithm 1.