A new consensus model for group decision making using fuzzy linguistic preference relations with heterogeneous experts

Abstract

Group decision making (GDM) problems consist of finding the most acceptable solution based on a group of experts’ preferences. Sometimes, the experts experience difficulty expressing their preferences using crisp values or making comparisons between each pair of alternatives. Meanwhile, because of their different backgrounds or knowledge concerning a specific problem, the opinions of different experts may carry different weights along the decision process. Thus, a new consensus model is proposed to solve these problems. First, experts are required to express their opinions using fuzzy linguistic preference relations, and then, a new method is proposed for classifying the experts into three different importance levels according to these opinions. Then, consistency measures and proximity measures are used to guide the decision-making process. A new feedback mechanism that generates advice for experts according to their different levels is proposed. The importance degrees of experts are taken into consideration throughout the process, which is one of the main novelties of this model. Finally, a numerical example is conducted to illustrate the utilization and compared results are also presented to check the feasibility of the proposed model.

Keywords

Group decision making fuzzy linguistic preference relation consensus process feedback mechanism

1 Introduction

A Group Decision Making (GDM) problem is defined as a decision problem in which multiple individuals interact to select a decision from a set of alternatives. Usually the experts come from different fields and have different knowledge and preferences. Therefore, the aim is to reconcile all their different options to reach a consensus and find an alternative that is acceptable to the entire group [1].

However, it may be difficult or impossible in many situations for an expert to assign crisp assessments to alternatives. Thus, a more realistic approach is to use linguistic assessments as a means of communicating information that is more natural than numerical values [2]. Many GDM problems provide fuzzy linguistic preference relations for alternatives [3 –5]. Meanwhile, in some situations, experts are unable to express their preference in each pair of alternatives because of a lack of knowledge, time pressure and their limited expertise in the problem domain, in which case incomplete fuzzy preference relations are generated. GDM with incomplete preference relations has been receiving increasing attention. For instance, Herrera-Viedma et al. [6] introduced a selection process to handle incomplete fuzzy preference relations. Wang and Li [7] presented a goal programming framework to solve GDM problems with incomplete interval additive reciprocalcomparison matrices. Alonso et al. [8] proposed a procedure to compute missing values and developed a Web-based consensus support system for GDM problems. Usually, the resolution method for a GDM problem consists of two processes: consensus and selection [4]. The former one focuses on reaching a final agreement with a certain level among the experts. The latter one focuses on selecting the best solution based on the group preference. In the literature, different consensus approaches according to different criteria can be found. The first kind of approach is based on the reference domain used for computing the soft consensus measures [9], while the second kind is based on the coincidence method used for computing the soft consensus measures according to three different approaches [10]. The third kind of approach is based on the generation method of recommendations supplied to the experts. In the early consensus approaches, the consensus-reaching process was guided by a moderator [9] who provided advice to the experts. Later, some consensus approaches incorporating a feedback mechanism [11] were proposed. It is noteworthy that the current consensus trends focus on developing automated feedbackmechanism.

Normally, in GDM problems, the experts’ opinions are considered equally important in most situations. In existing studies in the literature, most consensus models provided advice to the experts so that they could reach an agreement; however, these models did not consider the different preferential levels and ranks of individual decision makers’ assessments [12]. The most common approach included in the literature reflects the relevance of the experts and uses some operators to compute a weighted aggregation of their preferences to obtain a collective group preference [13]. In some studies [14], the weights of decision makers are treated as the same. However, experts come from different fields and have different knowledge or backgrounds. Thus, they may hold quite different preferences concerning a specific problem and it may be reasonable to handle this information differently. A new consensus model for GDM problems involving heterogeneous experts was proposed in [15], but it cannot detect or guarantee the consistency of the experts [16].

To fill the lacuna left by prior studies in the literature, this study focused on a new consensus model for GDM problems. In this model, fuzzy linguistic preference relations are used by experts to make comparisons between alternatives. Then, the experts are assigned different weights based on their preferences. The model is based on two consensus criteria to guide the consensus-reaching process. A new feedback mechanism is proposed guided by experts’ levels of importance.

The rest of this paper is organized as follows. In the following section, the introduction of the GDM problems and fuzzy linguistic preference relations used in this paper are described. A new consensus model is proposed in Section 3. In Section 4, a numerical example is presented to illustrate the practicality and feasibility of the proposed model. Our conclusions are discussed in Section 5.

2 The GDM problem with fuzzy linguistic preference relations

2.1 GDM problems

GDM is currently regarded as a main part of modern decision science and operational research. Because of the complexity of the socio-economic environment, GDM has recently become a critical issue [17, 18].

Following tradition, let X = {x₁, x₂, …, x_n} be a finite set of alternatives, and a group of experts be E = {e₁, e₂, …, e_m} characterized by their unique backgrounds and knowledge. The GDM process consists of finding the best alternative based on the experts’ preferences {P₁, P₂, …, P_m}.

GDM problems can be roughly classified into two groups, homogeneous and heterogeneous [19], and different representation formats can be used to express experts’ opinions. In this paper, fuzzy linguistic preference relations with heterogeneous groups (i.e., experts’ opinions are treated differently) are focused on because of their effectiveness as a tool for modeling the decision-making process.

2.2 Fuzzy linguistic preference relations

Fuzzy set theory was introduced by Zadeh [20] for dealing with imprecision and uncertainty related to information in a complex environment. Fuzzy set theory treats vague data as possibility distributions in terms of set memberships [21].

Definition 1. A triangular fuzzy number (TFN) $\tilde{a}$ on R can be expressed as a triplet (a, b, c) if its membership function $μ_{\tilde{a}} (x) : R \to [0, 1]$ is equal to: $μ_{\tilde{a}} (x) = {\begin{matrix} 0, & x \leq a, \\ \frac{x - a}{b - a}, & a < x \leq b, \\ \frac{c - x}{c - b}, & b < x \leq c, \\ 0, & x > c, \end{matrix}$ (1)where a and c are the lower and upper bounds of the fuzzy number, respectively, and b is the median value.

Definition 2. Let ${\tilde{a}}_{1} = (a_{1}, b_{1}, c_{1})$ and ${\tilde{a}}_{2} = (a_{2},$ b₂, c₂) be TFNs; then, the vertex method is defined to calculate the distance between them: $D ({\tilde{a}}_{1}, {\tilde{a}}_{2}) = \sqrt{1 / 3 [(a_{1} - a_{2})^{2} + (b_{1} - b_{2})^{2} + (c_{1} - c_{2})^{2}]}$ (2)

As aforementioned, an expert may be unable to express his/her preference degrees using exact crisp numeric values [22]. Therefore, a more effective method of showing their preferences is to use linguistic assessments [23]. In this study, seven linguistic assessment variables with their respective TFNs were used to construct the fuzzy linguistic preference relations. The fuzzy linguistic preference relation is represented by a n * n matrix $\tilde{P} = ({\tilde{p}}_{ij})$ , where ${\tilde{p}}_{ij} = \tilde{P} (x_{i}, x_{j})$ , for all i, j ∈ {1, 2, …, n}.

In a complete n * n comparison matrix, decision makers need to make all judgments on each pair of alternatives. In reality, the decision maker is sometimes unable or unwilling to express all these opinions because of a lack of knowledge, time pressure or limited expertise. In this situation, the result is an incomplete comparison matrix [24, 25]. However, it is crucial to have consistent fuzzy preference relations to achieve a robust ranking. In this study, the fuzzy linguistic preference relations proposed by Wang and Chen [21] were applied to conduct a pairwise comparison matrix with additive reciprocal property and consistency. The remaining values are calculated based on the reciprocity and consistency properties. This method yields consistent decision ranking from only n-1 pairwise comparisons. Some important propositions are given below.

Proposition 1. If a fuzzy reciprocal linguistic preference relation, $\tilde{P} = ({\tilde{p}}_{ij})$ with ${\tilde{p}}_{ij} \in [0, 1]$ , verifies the additive reciprocal, then the following statements are equivalent [21]. $p_{ij}^{L} + p_{ji}^{R} = 1$ (3) $p_{ij}^{M} + p_{ji}^{M} = 1$ (4) $p_{ij}^{R} + p_{ji}^{L} = 1$ (5)

Proposition 2. A reciprocal fuzzy linguistic preference relation $\tilde{P} = ({\tilde{p}}_{ij}) = (p_{ij}^{L}, p_{ij}^{M}, p_{ij}^{R})$ is consistent if [21]: $p_{ij}^{L} + p_{jk}^{L} + p_{ki}^{R} = \frac{3}{2}, \forall i < j < k$ (6) $p_{ij}^{M} + p_{jk}^{M} + p_{ki}^{M} = \frac{3}{2}, \forall i < j < k$ (7) $p_{ij}^{R} + p_{jk}^{R} + p_{ki}^{L} = \frac{3}{2}, \forall i < j < k$ (8) $p_{i (i + 1)}^{L} + \dots + p_{(j - 1) j}^{L} + p_{ji}^{R} = \frac{j - i + 1}{2}, \forall i < j$ (9) $p_{i (i + 1)}^{M} + \dots + p_{(j - 1) j}^{M} + p_{ji}^{M} = \frac{j - i + 1}{2}, \forall i < j$ (10) $p_{i (i + 1)}^{R} + \dots + p_{(j - 1) j}^{R} + p_{ji}^{L} = \frac{j - i + 1}{2}, \forall i < j$ (11)

If, after the fuzzy linguistic preference relation decision matrix has been constructed and calculated, the values of some elements of the aggregated matrix do not have a value between zero and one, the following transformations should be applied to transform the elements to the interval [0,1], i.e., f : [- c, 1 + c] → [0, 1], where c is the maximum amount of violation from interval [0,1] among elements of $\tilde{P}$ . $f (p_{ij}^{L}) = \frac{p_{ij}^{L} + c}{1 + 2 c}$ (12) $f (p_{ij}^{M}) = \frac{p_{ij}^{M} + c}{1 + 2 c}$ (13) $f (p_{ij}^{R}) = \frac{p_{ij}^{R} + c}{1 + 2 c}$ (14)

3 New model for GDM with fuzzy linguistic preference relations

Various consensus models for GDM were proposed in the past decades, most of which use consistency measures and/or proximity measures to guide the feedback mechanism. However, one of their main drawbacks is that they either overlook the experts’ different levels of knowledge or simply treat them as the same, which is undoubtedly unreasonable in reality. Thus, it’s of great importance to distinguish the experts’ opinions due to their different levels of understanding to a specific problem throughout the decision-making process. In this section, a new consensus model for GDM problems involving heterogeneous experts under a fuzzy environment is proposed. Experts express their preferences by means of linguistic fuzzy preference relations. Then, they are assigned different weights. A new method is put forward to classify them into three levels. Finally, a new feedback mechanism that considers the experts’ weights is generated based on consistency and proximity measures. To the best of our knowledge, this is the first time that such a framework, which can provide more insight into the consensus-reaching process, has been proposed.

The new consensus model consists of several stages as shown in Fig. 1. Typically, the consensus process can be summarized as follows. First, the problem is put to the experts, along with a given set of alternatives, and all the experts state their individual opinions about these alternatives, based on their knowledge, in a particular format. Some consensus measures are computed reflecting all the experts’ preferences, and then, whether the level of agreement is reached is determined. If yes, the consensus process stops and the exploitation process begins; otherwise, some advice is generated for the experts so that they can change their opinions optionally until a certain level of agreement is reached.

In the following subsections, different stages are described in detail.

3.1 Compute missing values

In the first step, all the experts belonging to the decision-making group make their own evaluations of different alternatives based on their experience and knowledge. Fuzzy linguistic preference relations are used to construct pairwise comparison matrices with additive reciprocal property and consistency. As aforementioned, it is necessary for each individual to make only n-1 pairwise comparisons using the fuzzy linguistic assessment variables. Note that each expert can make comparisons of those alternatives about which he or she is more sure or certain, based on his or her unique knowledge.

After obtaining the incomplete fuzzy pairwise comparison matrix $\tilde{P} = ({\tilde{p}}_{ij})$ , where ${\tilde{p}}_{ij}$ is a fuzzy linguistic variable or its equivalent triangular fuzzy number to show the experts’ preferences of alternative i over alternative j, the variables are converted to their corresponding fuzzy numbers. The remaining cells can be calculated using Equations (3–14) based on the reciprocity and consistency properties of a positive additive matrix. The completed comparison matrix $P^{k} = (p_{ij}^{k})$ which shows a consensus, can be finally obtained.

3.2 Compute consistency measures

This measure is used to evaluate the agreement of all the experts and to guide the consensus process until a final decision is made. The consistency measures are calculated in the following steps [15, 26].

1. First, a similarity matrix ${SM}^{kl} = ({sm}_{ij}^{kl})$ is calculated for each pair of experts (e^k, e^l) (k = 1, …, m - 1 ; l = k + 1, …, m) as ${sm}_{ij}^{kl} = 1 - D (p_{ij}^{k}, p_{ij}^{l})$ (15)where $D (p_{ij}^{k}, p_{ij}^{l})$ is the distance between $p_{ij}^{k}$ and $p_{ij}^{l}$ using Equation (2).

2. Then, a consensus matrix CM = (cm_ij) is obtained by aggregating all the similarity matrices, using the arithmetic mean as the aggregation function [15] as ${cm}_{ij} = 2 \sum_{k = 1}^{m - 1} \sum_{l = k + 1}^{m} {sm}_{ij}^{kl} / m (m - 1)$ (16)

Apparently, 0 ≤ cm_ij ≤ 1. A complete consensus for the preference p_ij if cm_ij = 1 is obtained.

3. When CM has been determined, the consensus degree at three different levels is defined:

Consensus degree on preference p_ij

{cd}_{ij} = {cm}_{ij}

(17)

Consensus degree in the alternative x_i

{CD}_{i} = \sum_{j = 1, j \neq i}^{n} {cd}_{ij} / (n - 1)

(18)

Consensus degree on the relation

CD = \sum_{i = 1}^{n} {CD}_{i} / n

(19)

Then a consensus threshold CR is determined. If CD ≥ CR, then the consensus process stops and the exploitation process begins; otherwise, it means that the decision-making process is unacceptable, i.e., there exists a wide discrepancy between the experts’ opinions and the feedback mechanism is activated.

A maximum number of rounds should also be determined [27] in case the global consensus measure does not converge to the minimum required level.

3.3 Feedback mechanism

The feedback mechanism generates advice for the experts based on the consistency and proximity measures. It helps experts to change their preferences to reach a harmonious state. However, in a heterogeneous situation, it is also important to develop different strategies when advising experts having different weights. It is reasonable to think that experts with lower weights need more advice and make more modifications than those with higher weights. Therefore, in this study different recommendations for experts are generated based on their weights in order to reach a more reasonable agreement in the subsequent round.

Before the details of the operation of the feedback mechanism are provided, the calculations of the experts’ weights and their classification into different levels, as well as the calculation of the proximity measures used in the feedback mechanism, are explained in detail in the following subsection.

3.3.1 Weights of decision makers

Because of their lack of knowledge or limited expertise in relation to the increasing uncertainty and complexity of practical problems, the experts’ weights are usually completely unknown or cannot be determined in advance but rather are obtained directly from the preferences expressed by the experts. According to Chen and Yang [28], a preference value close to the mean value should be assigned a large weight and vice versa.

Suppose that $p_{ij}^{k}$ is the preference for alternative x_i over alternative x_j expressed by expert e_k. Then the mean value of the preference ${\bar{p}}_{ij}$ of alternative x_i over alternative x_j can be computed as ${\bar{p}}_{ij} = \frac{1}{k} \sum_{k = 1}^{m} p_{ij}^{k} i = 1, 2, \dots, n; j = 1, 2, \dots, n$ (20)

Then, the similarity between each expert’s preferences and the mean preference can be computed as $S_{k} = 1 - \sum_{i = 1}^{n} \sum_{j = 1}^{n} D (p_{ij}^{k}, {\bar{p}}_{ij}) / (n \times n)$ (21)where $D (p_{ij}^{k}, {\bar{p}}_{ij})$ is the distance between $p_{ij}^{k}$ and ${\bar{p}}_{ij}$ using Equation (2).

Then, the weight of decision maker e_k can be computed as $w_{k} = S_{k} / \sum_{i = 1}^{m} S_{k}$ (22)

However, there are some other methods to determine weights of decision makers [29 –31].

In this study, the experts are classified into three different importance levels: low, medium and high. This classification is performed as follows.

First, all the weight values need to be ranked in ascending order w = [w₁, w₂, …, w_m], the middle value w_M can be obtained as $w_{M} = {\begin{matrix} w_{\frac{1 + m}{2}} & m is odd \\ (w_{\frac{m}{2}} + w_{\frac{m}{2} + 1}) / 2 & m is even \end{matrix}}$ (23)

It is reasonable to think that the corresponding expert is more likely to be classified into the medium-importance level when the weight is closer to w_M, into the high-importance level when the weight is greater than w_M, and into the low-importance level when the weight is smaller than w_M. Then, a parameter ɛ is defined as $ɛ = \min {\frac{w_{M} - w_{min}}{2}, \frac{w_{max} - w_{M}}{2}}$ (24)

The threshold value of w_Low and w_High can be calculated as $w_{Low} = w_{M} - ɛ; w_{High} = w_{M} + ɛ$ (25)

Note that w_min < w_Low < w_M < w_High < w_max. Therefore, the experts can always be classified into three different levels as $\begin{matrix} E_{Low} = {e_{i} | w_{i} \leq w_{Low}} \\ E_{Med} = {e_{i} | w_{i} \in (w_{Low}, w_{High})} \\ E_{High} = {e_{i} | w_{i} \geq w_{High}} \end{matrix}$ (26)

3.3.2 Aggregation phase

In the decision-making process, all the experts form their own opinions about the alternatives based on their backgrounds. However, some may form unrealistic and unscientific opinions about alternatives because of their lack of knowledge or limited expertise. Thus, the opinions expressed by the expert with a higher weight should carry more weight when gathering all the experts’ preferences into a global one.

Numerous aggregation operators have been put forward to aggregate all the experts’ comparison matrices into one matrix [32 –35], the most common of which are the weighted arithmetic and the geometric mean. Here, in this study, the weighted arithmetic mean was employed, considering different weights of the experts.

A collective comparison matrix P = (p_ij) is obtained as: $p_{ij} = \sum_{k = 1}^{m} w_{k} p_{ij}^{k}$ (27)

3.3.3 Compute proximity measures

Proximity measures are used to evaluate the agreement between the individuals in the group and to guide the discussion phase of the consensus process. The process is continued in order to calculate the proximity measures.

First, the collective group preference P = (p_ij) summarizing all the experts’ preferences needs to be collected, as described in Section 3.3.2. Then, the proximity measure can be computed as follows.

Proximity measure on preferences of each expert ${PM}^{k} = ({pm}_{ij}^{k})$

{pm}_{ij}^{k} = 1 - D (p_{ij}^{k}, p_{ij})

(28)where

D (p_{ij}^{k}, p_{ij})

is the distance between

p_{ij}^{k}

and p_ij using Equation (2).

Proximity measures on alternatives:

P M_{i}^{k} = \sum_{j = 1, j \neq i}^{n} p m_{i j}^{k} / (n - 1)

(29)

Proximity measures on the relation

{PM}^{k} = \sum_{i = 1}^{n} {PM}_{i}^{k} / n

(30)

3.3.4 Generate different advice strategies

Three different strategies are proposed for generating the advice given to the experts of three different importance levels after all these related parameters have been calculated and obtained.

a) Low-importance experts

For experts in this subset, very significant changes in their preferences are required to reach an agreement level in the subsequent round. They need to modify their preference for alternatives depending on both the consensus degree and proximity measures. A vector value β = [β_i] is determined, where $β_{i} = \sum_{k = 1}^{m} {PM}_{i}^{k} / m$ (i = 1, 2, …, n).

Then, the preference values that need to be changed are identified as: $R_{L o w}^{k} = \{(i, j) |C D_{i} < C D \cup P M_{i}^{k} < β_{i}|\}$ (31)

b) Medium-importance experts

In this case, it is reasonable to reduce the number of changes as compared with the low-importance experts. Therefore, only the preference values for alternatives in disagreement with the whole group preference based on the proximity measures are discussed. The preference values that need to be changed areidentified as $R_{Med}^{k} = {(i, j) | {PM}_{i}^{k} < β_{i}} i = 1, 2, \dots, n$ (32)

c) High-importance experts

In this situation, high-importance experts’ opinions should change less or remain stable so that they can exert an influence on the others. It is preferable that the whole group reaches an agreement as soon as possible. Therefore, the preference values that need to be changed are identified as $R_{H i g h}^{k} = \{(i, j) |C D_{i} < C D \cap P M_{i}^{k} < β_{i}|\}$ (33)

It is noteworthy that in this study, experts need to make only n-1 comparisons between the alternatives, and therefore, only some important preference values are estimated. When different i are identified using the above Equations (31–33), experts need to change only the specific preferences they first formed at the beginning of the problem instead of all their preference values, which will no doubt be very convenient and accelerate the pace of reaching a final agreementlevel.

For each preference value that needs to be modified, the model suggests the expert increase the preference if $p_{ij}^{k} < p_{ij}$ or decrease the current preference if $p_{ij}^{k} > p_{ij}$ .

3.4 Exploitation process

When an acceptable agreement level is reached among the group of experts and the global information of all the experts’ preferences has been collected, the selection process is initiated. Its objective is to rank a given set of alternatives and select the best one according to all the experts’ opinions. To achieve this goal, in this study, selection functions [4] are defined using the dominance concept [36].

The dominance degree px_i of each alternative x_i is calculated using the group fuzzy preference relations as ${px}_{i} = \sum_{j = 1, j \neq i}^{n} p_{ij}^{L} + p_{ij}^{M} + p_{ij}^{H}$ (34)

Thus, the ranking order of all alternatives can be determined and the best with the maximum px_i among the alternatives can be selected.

4 Example

4.1 Our method

In this section, an experiment conducted in order to clearly illustrate the utilization and procedure of the proposed model and to demonstrate its effectiveness is described.

Assuming that four experts from different fields were invited to make a selection from four alternatives. The experts’ preferences using the fuzzy linguistic assessment variables shown in Table 1 were

$\begin{matrix} E^{1} = [\begin{matrix} - & VP & - & - \\ - & - & P & - \\ - & - & - & G \\ - & - & - & - \end{matrix}] \\ E^{2} = [\begin{matrix} - & P & - & - \\ - & - & MG & - \\ - & - & - & VG \\ - & - & - & - \end{matrix}] \\ E^{3} = [\begin{matrix} - & MP & - & - \\ - & - & P & - \\ - & - & - & G \\ - & - & - & - \end{matrix}] \\ E^{4} = [\begin{matrix} - & M & - & - \\ - & - & MP & - \\ - & - & - & M \\ - & - & - & - \end{matrix}] \end{matrix}$

The following steps were executed:

1) Computation of missing values

The variables were converted to their corresponding fuzzy numbers. The remaining values were then calculated using Equations (3–11). The completed preference matrix of each expert was finallyobtained: $\begin{matrix} {\tilde{p}}^{1} = [\begin{matrix} (0.5, 0.5, 0.5) (0.0, 0.0, 0.1) (- 0.5, - 0.4, - 0.1) (- 0.3, 0, 0.4) \\ (0.9, 1.0, 1.0) (0.5, 0.5, 0.5) (0.0, 0.1, 0.3) (0.2, 0.5, 0.8) \\ (1.1, 1.4, 1.5) (0.7, 0.9, 1.0) (0.5, 0.5, 0.5) (0.7, 0.9, 1.0) \\ (0.6, 1.0, 1.3) (0.2, 0.5, 0.8) (0.0, 0.1, 0.3) (0.5, 0.5, 0.5) \end{matrix}] \\ {\tilde{p}}^{2} = [\begin{matrix} (0.5, 0.5, 0.5) (0, 0.1, 0.3) (0, 0.3, 0.7) (0.4, 0.8, 1.2) \\ (0.7, 0.9, 1.0) (0.5, 0.5, 0.5) (0.5, 0.7, 0.9) (0.9, 1.2, 1.4) \\ (0.3, 0.7, 1.0) (0.1, 0.3, 0.5) (0.5, 0.5, 0.5) (0.9, 1.0, 1.0) \\ (- 0.2, 0.2, 0.6) (- 0.4, - 0.2, 0.1) (0, 0, 0.1) (0.5, 0.5, 0.5) \end{matrix}] \\ {\tilde{p}}^{3} = [\begin{matrix} (0.5, 0.5, 0.5) (0.1, 0.3, 0.5) (- 0.4, - 0.1, 0.3) (- 0.2, 0.3, 0.8) \\ (0.5, 0.7, 0.9) (0.5, 0.5, 0.5) (0.0, 0.1, 0.3) (0.2, 0.5, 0.8) \\ (0.7, 1.1, 1.4) (0.7, 0.9, 1.0) (0.5, 0.5, 0.5) (0.7, 0.9, 1.0) \\ (0.2, 0.7, 1.2) (0.2, 0.5, 0.8) (0.0, 0.1, 0.3) (0.5, 0.5, 0.5) \end{matrix}] \\ {\tilde{p}}^{4} = [\begin{matrix} (0.5, 0.5, 0.5) (0.3, 0.5, 0.7) (- 0.1, 0.3, 0.7) (- 0.3, 0.3, 0.9) \\ (0.3, 0.5, 0.7) (0.5, 0.5, 0.5) (0.1, 0.3, 0.5) (- 0.1, 0.3, 0.7) \\ (0.3, 0.7, 1.1) (0.5, 0.7, 0.9) (0.5, 0.5, 0.5) (0.3, 0.5, 0.7) \\ (0.1, 0.7, 1.3) (0.3, 0.7, 1.1) (0.3, 0.5, 0.7) (0.5, 0.5, 0.5) \end{matrix}] \end{matrix}$ (35)

As some elements fell outside the interval [0,1], Equations (12–14) were used to transfer the elements into the interval, where c = 0.5, 0.4, 0.4, and 0.3 for the four matrices, respectively. Finally, the final preference results were $\begin{matrix} P^{1} = [\begin{matrix} (0.5, 0.5, 0.5) (0.25, 0.25, 0.3) (0, 0.05, 0.2) (0.1, 0.25, 0.45) \\ (0.7, 0.75, 0.75) (0.5, 0.5, 0.5) (0.25, 0.3, 0.4) (0.35, 0.5, 0.65) \\ (0.8, 0.95, 1.0) (0.6, 0.7, 0.75) (0.5, 0.5, 0.5) (0.6, 0.7, 0.75) \\ (0.55, 0.75, 0.9) (0.35, 0.5, 0.65) (0.25, 0.3, 0.4) (0.5, 0.5, 0.5) \end{matrix}] \\ P^{2} = [\begin{matrix} (0.5, 0.5, 0.5) (0.22, 0.28, 0.39) (0.22, 0.39, 0.61) (0.44, 0.68, 0.89) \\ (0.61, 0.72, 0.78) (0.5, 0.5, 0.5) (0.5, 0.61, 0.72) (0.72, 0.89, 1.0) \\ (0.39, 0.61, 0.78) (0.28, 0.39, 0.5) (0.5, 0.5, 0.5) (0.72, 0.78, 0.78) \\ (0.11, 0.33, 0.56) (0, 0.11, 0.28) (0.22, 0.22, 0.28) (0.5, 0.5, 0.5) \end{matrix}] \\ P^{3} = [\begin{matrix} (0.5, 0.5, 0.5) (0.28, 0.39, 0.5) (0, 0.17, 0.39) (0.11, 0.39, 0.67) \\ (0.5, 0.61, 0.72) (0.5, 0.5, 0.5) (0.22, 0.28, 0.39) (0.33, 0.5, 0.67) \\ (0.61, 0.83, 1.0) (0.61, 0.72, 0.78) (0.5, 0.5, 0.5) (0.61, 0.72, 0.78) \\ (0.33, 0.61, 0.89) (0.33, 0.5, 0.67) (0.22, 0.28, 0.39) (0.5, 0.5, 0.5) \end{matrix}] \\ P^{4} = [\begin{matrix} (0.5, 0.5, 0.5) (0.375, 0.5, 0.625) (0.125, 0.375, 0.625) (0, 0.375, 0.75) \\ (0.375, 0.5, 0.625) (0.5, 0.5, 0.5) (0.25, 0.375, 0.5) (0.125, 0.375, 0.625) \\ (0.375, 0.625, 0.875) (0.5, 0.625, 0.75) (0.5, 0.5, 0.5) (0.375, 0.5, 0.625) \\ (0.25, 0.625, 1.0) (0.375, 0.625, 0.875) (0.375, 0.5, 0.625) (0.5, 0.5, 0.5) \end{matrix}] \end{matrix}$

All the computations were performed using MATLAB. Other software such as Microsoft 2007 or LINGO can also be used.

2) Consistency measure computation

The consistency measures at three different levels were obtained using Equations (15–19).

On preference

CD = [\begin{matrix} 1.0 0.73 0.59 0.58 \\ 0.73 1.0 0.69 0.56 \\ 0.59 0.69 1.0 0.75 \\ 0.58 0.56 0.75 1.0 \end{matrix}]

On alternatives

{CD}_{1} = 0.63; {CD}_{2} = 0.66; {CD}_{3} = 0.68; {CD}_{4} = 0.63

On relations

CD = \sum_{i = 1}^{4} {CD}_{i} / 4 = 0.65

Then, CD was compared with a given threshold CL = 0.7, and CD < CL. It seemed unacceptable to complete the decision-making process. Then the feedback mechanism was activated.

3) Operation of feedback mechanism

- Weights of decision makers

First, the weights of all the experts were determined using Equations (20–22). The results were: $w_{1} = 0.252, w_{2} = 0.237, w_{3} = 0.263, w_{4} = 0.248 .$

Then, these four experts were classified into three levels using Equations (23–26). The middle value was w_M = (0.248 + 0.252)/2 = 0.25.

The parameter was $ɛ = \min {\frac{0.25 - 0.237}{2}, \frac{0.263 - 0.25}{2}} = 0.0065 .$

Then, the threshold values of w_Low and w_High were calculated as $\begin{matrix} w_{Low} = w_{M} - ɛ = 0.25 - 0.0065 = 0.2435; \\ w_{High} = w_{M} + ɛ = 0.25 + 0.0065 = 0.2565 \end{matrix}$

Thus, E_Low = {e₂} 1pt E_Med = {e₁, e₄} ; E_High = {e₃}.

–Aggregation phase

When all the experts’ weights and their corresponding preferences were obtained over different alternatives, a global final comparison matrix was obtained using Equation (27). $P = [\begin{matrix} (0.5, 0.5, 0.5) (0.28, 0.36, 0.45) (0.08, 0.24, 0.45) (0.16, 0.42, 0.69) \\ (0.55, 0.64, 0.72) (0.5, 0.5, 0.5) (0.30, 0.39, 0.50) (0.38, 0.56, 0.73) \\ (0.55, 0.76, 0.92) (0.50, 0.61, 0.70) (0.5, 0.5, 0.5) (0.58, 0.67, 0.73) \\ (0.31, 0.58, 0.84) (0.27, 0.44, 0.62) (0.27, 0.33, 0.42) (0.5, 0.5, 0.5) \end{matrix}]$

–Proximity measure computation

The proximity measures of each expert were calculated using Equations (28–30).

On preference:

\begin{matrix} {PM}^{1} = [\begin{matrix} 1.0 0.89 0.81 0.83 \\ 0.89 1.0 0.92 0.94 \\ 0.81 0.92 1.0 0.98 \\ 0.83 0.94 0.98 1.0 \end{matrix}] \\ {PM}^{2} = [\begin{matrix} 1.0 0.93 0.85 0.75 \\ 0.93 1.0 0.79 0.69 \\ 0.85 0.79 1.0 0.89 \\ 0.75 0.69 0.89 1.0 \end{matrix}] \\ {PM}^{3} = [\begin{matrix} 1.0 0.97 0.93 0.96 \\ 0.97 1.0 0.9 0.94 \\ 0.93 0.9 1.0 0.96 \\ 0.96 0.94 0.96 1.0 \end{matrix}] \\ {PM}^{4} = [\begin{matrix} 1.0 0.86 0.87 0.90 \\ 0.86 1.0 0.97 0.81 \\ 0.87 0.97 1.0 0.83 \\ 0.90 0.81 0.83 1.0 \end{matrix}] \end{matrix}

On alternatives:

\begin{matrix} {PM}_{1}^{1} = 0.84; {PM}_{2}^{1} = 0.92; {PM}_{3}^{1} = 0.90; {PM}_{4}^{1} = 0.91 \\ {PM}_{1}^{2} = 0.85; {PM}_{2}^{2} = 0.80; {PM}_{3}^{2} = 0.84; {PM}_{4}^{2} = 0.78 \\ {PM}_{1}^{3} = 0.95; {PM}_{2}^{3} = 0.94; {PM}_{3}^{3} = 0.93; {PM}_{4}^{3} = 0.95 \\ {PM}_{1}^{4} = 0.88; {PM}_{2}^{4} = 0.88; {PM}_{3}^{4} = 0.89; {PM}_{4}^{4} = 0.85 \end{matrix}

On relations:

\begin{matrix} {PM}^{1} = 0.894; {PM}^{2} = 0.817; \\ {PM}^{3} = 0.943; {PM}^{4} = 0.873 \end{matrix}

–Generation of different advice strategies

A vector β was determined as $\begin{matrix} β_{1} = 0.88; β_{2} = 0.885; \\ β_{3} = 0.89; β_{4} = 0.8725 \end{matrix}$ According to the measures and rules, the preferences that should be modified were identified as follows.

a) For the low-importance expert e₂ $\begin{matrix} R_{Low}^{2} = {(i, j) | {CD}_{i} < 0.65 \cup {PM}_{i}^{2} < β_{i}} \\ = {(i, j) | i = 1, 4 \cup i = 1, 2, 3, 4} \\ = {(i, j) | i = 1, 2, 3, 4} \\ = {(1, 2); (2, 3); (3, 4)} \end{matrix}$

b) For the medium-importance experts e₁ and e₄: $\begin{matrix} R_{Med}^{1} = {(i, j) | {PM}_{i}^{1} < β_{i}} = {(1, 2)} \\ R_{Med}^{4} = {(i, j) | {PM}_{i}^{4} < β_{i}} = {(2, 3)} \end{matrix}$

c) For the high-importance expert e₃: $\begin{matrix} R_{High}^{3} = {(i, j) | {CD}_{i} < CD \cap {PM}_{i}^{3} < β_{i}} \\ = {(i, j) | i = 1, 4 \cap \emptyset} = \emptyset \end{matrix}$

It is meaningless when i = 4 in this problem because expert e₃ did not give a fuzzy linguistic preference value for alternative 4 on any other alternative.

These experts were required to change their opinions based on the recommendations: $\begin{matrix} R^{1} = [\begin{matrix} + \\ = \\ = \end{matrix}] R^{2} = [\begin{matrix} + \\ - \\ - \end{matrix}] \\ R^{3} = [\begin{matrix} = \\ = \\ = \end{matrix}] R^{4} = [\begin{matrix} = \\ + \\ = \end{matrix}] \end{matrix}$

where r_ij = +/-/= means the expert should increase/decrease/retain his or her preference accordingly.

It is noteworthy that the weights of the experts can be changed in each consensus round based on their preferences. Suppose after some discussions and in the subsequent consensus round, all the experts reached a final global preference that satisfied the consensus level with CD = 0.738> CL, which is $\begin{matrix} E^{1} = [\begin{matrix} - P - - \\ - - P - \\ - - - G \\ - - - - \end{matrix}] E^{2} = [\begin{matrix} - MP - - \\ - - M - \\ - - - G \\ - - - - \end{matrix}] \\ E^{3} = [\begin{matrix} - MP - - \\ - - P - \\ - - - G \\ - - - - \end{matrix}] E^{4} = [\begin{matrix} - M - - \\ - - M - \\ - - - M \\ - - - - \end{matrix}] \end{matrix}$

The aggregated group preference was: $P = [\begin{matrix} (0.5, 0.5, 0.5) (0.27, 0.39, 0.52) (0.06, 0.28, 0.54) (0.11, 0.46, 0.80) \\ (0.48, 0.61, 0.73) (0.5, 0.5, 0.5) (0.29, 0.39, 0.52) (0.34, 0.57, 0.79) \\ (0.46, 0.72, 0.94) (0.48, 0.61, 0.71) (0.5, 0.5, 0.5) (0.55, 0.68, 0.76) \\ (0.20, 0.54, 0.89) (0.21, 0.43, 0.66) (0.24, 0.32, 0.45) (0.5, 0.5, 0.5) \end{matrix}]$

–Exploitation process

Finally, the dominance degree of each alternative was obtained from the collective group preference using Equation (34). ${px}_{1} = 3.41; {px}_{2} = 4.73; {px}_{3} = 5.91; {px}_{4} = 3.95$

Thus, the ranking list was px₁ < px₄ < px₂ < px₃ and therefore, x₃ was chosen as the recommended solution.

4.2 Comparison with other methods

Using the same data as in the above example, three other consensus models dealing with incomplete fuzzy preference relations were also studied: (1) a goal programming method [37]; (2) a selection process based on additive consistency [38]; and (3) a classical consensus model [26].

In the first method, it was not necessary to compute or recover missing preference values. In the two latter models, the missing values needed to be estimated and then the best alternative based on complete preference values needed to be selected. Finally, three different ranking lists were obtained

a) x₃ > x₂ > x₄ > x₁

b) x₂ > x₃ > x₄ > x₁

c) x₃ > x₄ > x₂ > x₁

The first model yielded the same results as those of our method. In the second model, the ranking of alternative x₂ and x₃ was different. The reason may be that the methods to determine the weights of the experts in the decision-making process are different. However, based on the high-importance level expert e₃, x₃ was the best alternative. As compared with the method in [26], the ranking of alternative x₂ and x₄ was different; however, the best and worst alternatives remained the same and fewer consensus rounds were required to reach a satisfactory consensus level using our method. Moreover, the variation in the approaches to compute the missing values can also influence the outcomes.

Therefore, with our consensus model, a more realistic solution, i.e., x₃ > x₂ > x₄ > x₁ was obtained. Thus, in practice, the experts should be assigned different importance levels and their preference values should be treated differently in the consensus-reaching process.

5 Conclusions

In this paper, a new consensus model for GDM problems was proposed. In general, methods based on fuzzy sets theory seem to be more reasonable. Therefore, experts use the fuzzy linguistic assessment variables characterized by TFNs to express their preference on alternatives instead of crisp data in order to handle the vagueness and uncertainty that exist in real life. To achieve a robust ranking, consistent fuzzy preference relations are essential [39]. The method proposed in this paper can yield consistent priority ranking and requires fewer pairwise comparisons. Then, experts are assigned different importance levels based on their expressed preference during the decision-making problem. As compared with the method in [15], the experts’ weights are not determined in advance and a new method is proposed to classify the experts into three different levels. It is noteworthy that the experts’ weights are not necessarily the same in each consensus round. The consistency and proximity measures are considered simultaneously to help the experts’ opinions reach a final consistency. In general, the opinions of an expert with a high weight should be considered superior throughout the decision-making process. Thus, the concept of the experts’ importance levels should not be overlooked and can be exploited to guide the decision-making process. Therefore, in this paper a new feedback mechanism that considers the experts’ importance levels when generating advice was proposed. Normally, experts with a lower weight need more advice than those with a higher weight. In this model, experts need to make fewer comparisons and fewer modifications in the entire consensus reaching process, which saves time and very significantly accelerates the pace at which a consensus level is achieved using our model as compared with other consensus models.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China: Reasoning and Learning Approach to Evidential Network with Application under Grant nos. 71201168.

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