Consensus models based on distance for interval fuzzy and multiplicative preference relations

Abstract

This paper presents consensus models based on distance for group decision making problems under interval fuzzy and multiplicative preference relations. First, some quadratic programming models based upon the idea of minimizing the sum of squared distances between all pairs of weighted interval fuzzy or multiplicative preference judgments are developed to obtain the weights of experts. Then, two indices, an individual to group consensus index (ICI) and a group consensus index (GCI) are defined. Furthermore, iterative consensus algorithms are proposed and the processes stop until both the ICI and GCI are less than predefined thresholds or reaching the maximum number of iteration. Finally, two illustrative examples are given to demonstrate the feasibility and effectiveness of the proposed methods.

Keywords

Group decision making interval fuzzy preference relation interval multiplicative preference relation consensus iterative algorithms

1 Introduction

Group decision-making (GDM) problems consist of choosing the best alternative(s) from a set of alternatives according to the preference relations given by a group of experts. Before the final choice is identified, two processes are usually carried out: (1) a consensus process and (2) a selection process. The first process is used to obtain a maximum degree of consensus or agreement among the decision-makers (DMs) over the alternative set, while the second process handles the derivation of the alternative set based on the DMs’ individual judgment on alternatives [14].

Consensus is a hot topic in group decision making and many literatures have put forward consensus models [3–5 , 33]. Pérez, et al. [22] developed a prototype of model with two important characteristic based on mobile technologies. Xu and Wu [32] presented a discrete model to support the consensus reaching process for multiple attribute group decision making (MAGDM) problems, where the basic idea is to shorten the range of viewpoints amongst experts. Palomares, et al. [23] proposed a consensus model in which DMs can express their opinions by using different types of information. Xu and Wu [33] developed an approach to solve consensus problems as expert preference information is in the form of uncertain linguistic preference relations. Pérez, et al. [21] proposed a new consensus model that has been designed to model GDM frameworks with heterogeneous experts. While Cabrerizo, et al. [3] showed several challenges on consensus approaches that have to be faced.

In GDM problems, the experts’ preferences on decision alternatives are often expressed by fuzzy or multiplicative preference relations [8 , 47]. In many real-life situations, however, the DM may not estimate his/her preference with exact judgments, but with interval number due to the complexity of decision making problems and incomplete information, and the limitations of DM’s domain knowledge. In such situations, interval fuzzy or multiplicative preference relations are very universal for describing the DM’s uncertain preference information.

Up to now, considerable research has been conducted on consensus reaching process under interval fuzzy preference relations (IFPRs) [30 , 44] or interval multiplicative preference relations (IMPRs) [1]. Jiang [18] applied the similarity index to check the group opinion’s consistency, then a practical approach to group decision making in terms of interval fuzzy preference relation was developed based on the error-propagation principle. García,et al. [12] developed a consensus model for GDM problems with linguistic interval fuzzy preference relations in the basis of two consensus criteria, consensus measures and proximity measures, and a feedback mechanism. In [31], the experts’ weights were determined by the relative projections of individual preference relations on the collective one. Then consensus algorithms with IFPRs and IMPRs based on the relative projections was designed. Chen, et al. [6] presented a new method for GDM using group recommendations based on interval fuzzy preference relations and consistency matrices. Xu [43] established a quadratic programming model from the angle of making all the deviations of individual IFPRs and collective IFPR smallest to derive the experts’ weights. However, there is an evident disadvantage exits for their quadratic programming models, as we can see in Section 2 of this paper, the derived weight vector is always the same for each DM. Furthermore, for existing paper about the GDM problem, the consensus models with IFPRs or IMPRs have not been widely investigated. It is often the case that the final adjusted preference relations significantly differ from the DMs’ original judgment information. In order to obtain a reliable solution, the consensus decision model should retain the DMs’ opinions as much as possible. GDM should utilize the DMs’ opinions amply to find a solution. Thus, to address these deficiencies, we provide new interactive algorithms to help DMs modify an IFPR or IMPR.

In this paper, we put forward a consensus model based on distance for IFPRs. First, some quadratic programming models are constructed in order to derive the experts’ weight. Then an Uncertain Weighted Arithmetic Averaging aggregation operator is used to obtain a collective IFPR. An individual to group consensus index (ICI) and a group consensus index (GCI) are subsequently introduced, followed by an iterative algorithm for consensus reaching with a stoppage condition when both ICI and GCI are lower than predefined thresholds. The model and algorithm are then extended to IMPRs.

This paper is structured in the following way. Section 2 briefly reviews quadratic programming model introduced by Xu [43] with comments on their drawbacks, then a consensus model based on distance for IFPRs is put forward, which can determine DMs’ weights, then a consensus reaching algorithm is developed. Section 3 extends the model and algorithm to settle IMPRs’ consensus problems. In Section 4, we give examples to illustrate the proposed method and compare the results with those existing approaches. Concluding remarks are furnished in Section 5.

2 Distance-based group consensus models for interval fuzzy preference relations

In the real life situations, a decision is usually made by a group of the experts e_k (k = 1, 2, …, m), whose weight vector w = (w₁, w₂, …, w_m)^T is to be determined. These experts provide their preferences over a collection of alternatives x_i (i = 1, 2, …, n), and the preferences are expressed in interval numbers instead of exact numerical values. That is, each expert e_k compares each pair of the objects x_i (i = 1, 2, …, n), and constructs an interval fuzzy preference relation P_k = (p_ij,k) _n×n, where $p_{ij, k} = [p_{ij, k}^{-}, p_{ij, k}^{+}]$ is a preference value, which indicates that the object x_i is as important as the object x_j between $p_{ij, k}^{-}$ and $p_{ij, k}^{+}$ times, and satisfies $\begin{matrix} p_{ij, k}^{-} + p_{ji, k}^{+} = p_{ij, k}^{+} + p_{ji, k}^{-} = 1, \\ p_{ij, k}^{+} \geq p_{ij, k}^{-} \geq 0 \\ p_{ii, k}^{-} = p_{ii, k}^{+} = 0.5, i, j = 1, 2, \dots, n \end{matrix}$ (1)

The experts’ weight vector w satisfies $\sum_{k = 1}^{m} w_{k} = 1, w_{k} \geq 0, k = 1, 2, \dots, m$ (2)

To obtain a collective judgment for the group, we aggregate all P_k = (p_ij,k) _n×n (k = 1, 2, …, m) into the collective preference relation P = (p_ij) _n×n by the Uncertain Weighted Arithmetic Averaging (UWAA) operator: $\begin{matrix} p_{ij} = [p_{ij}^{-}, p_{ij}^{+}] = \sum_{k = 1}^{m} w_{k} p_{ij, k} \\ = [\sum_{k = 1}^{m} w_{k} p_{ij, k}^{-}, \sum_{k = 1}^{m} w_{k} p_{ij, k}^{+}], i, j = 1, 2, \dots, n \end{matrix}$ (3) then by (1), P = (p_ij) _n×n is also an IFPR.

Apparently, an important step for using Equation (3) is to determine the weight vector w. If all P_k = (p_ij,k) _n×n are the same as P = (p_ij) _n×n, then

$\begin{matrix} p_{ij, k} = [p_{ij, k}^{-}, p_{ij, k}^{+}] = p_{ij} = [p_{ij}^{-}, p_{ij}^{+}] \\ = [\sum_{l = 1}^{m} w_{l} p_{ij, l}^{-}, \sum_{l = 1}^{m} w_{l} p_{ij, l}^{+}] \\ i, j = 1, 2, \dots, n, k = 1, 2, \dots, m \end{matrix}$ (4)

However, Equation (4) does not always hold. Let $\begin{matrix} ξ_{ij, k}^{-} = | p_{ij, k}^{-} - \sum_{l = 1}^{m} w_{l} p_{ij, l}^{-} |, \\ ξ_{ij, k}^{+} = | p_{ij, k}^{+} - \sum_{l = 1}^{m} w_{l} p_{ij, l}^{+} |, \\ i, j = 1, 2, \dots, n, k = 1, 2, \dots, m \end{matrix}$ (5) where $ξ_{ij, k}^{-}$ , $ξ_{ij, k}^{+}$ (i, j = 1, 2, …, n, k = 1, 2, …, m) are the absolute deviation between individual and collective IFPRs. And then the square deviations among all P_k = (p_ij,k) _n×n (k = 1, 2, …, m) and P = (p_ij) _n×n are defined as:

$\begin{matrix} ξ (w) & = & \sum_{k = 1}^{m} \sum_{i = 1}^{n} \sum_{j = 1}^{n} ({(ξ_{ij, k}^{-})}^{2} + {(ξ_{ij, k}^{+})}^{2}) \\ = & \sum_{k = 1}^{m} \sum_{i = 1}^{n} \sum_{j = 1}^{n} ({(p_{ij, k}^{-} - \sum_{l = 1}^{m} w_{l} p_{ij, l}^{-})}^{2} \\ + {(p_{ij, k}^{+} - \sum_{l = 1}^{m} w_{l} p_{ij, l}^{+})}^{2}) \end{matrix}$ (6)

From the viewpoint of maximizing group consensus, Xu [43] established the following quadratic programming model: $\begin{matrix} (M - 1) min ξ (w) = min \\ \sum_{k = 1}^{m} \sum_{i = 1}^{n} \sum_{j = 1}^{n} ({(p_{ij, k}^{-} - \sum_{l = 1}^{m} w_{l} p_{ij, l}^{-})}^{2} \\ + {(p_{ij, k}^{+} - \sum_{l = 1}^{m} w_{l} p_{ij, l}^{+})}^{2}) \\ s . t . w_{l} \geq 0, l = 1, 2, \dots, m, \sum_{l = 1}^{m} w_{l} = 1 \end{matrix}$

The solution to this model generates a weight vector for all DMs e_k (k = 1, 2, …, m) and can be derived as follows [43]: $w = \frac{D^{- 1} e (1 - e^{T} D^{- 1} p)}{e^{T} D^{- 1} e} + D^{- 1} p$ (7) Where

$\begin{matrix} p = (\sum_{i = 1}^{n} \sum_{j = i}^{n} \sum_{k = 1}^{m} (p_{ij, k}^{-} p_{ij, 1}^{-} + p_{ij, k}^{+} p_{ij, 1}^{-}), \sum_{i = 1}^{n} \sum_{j = 1}^{n} \sum_{k = 1}^{m} (p_{ij, k}^{-} p_{ij, 2}^{-} + p_{ij, k}^{+} p_{ij, 2}^{-}), \\ {\dots, \sum_{i = 1}^{n} \sum_{j = 1}^{n} \sum_{k = 1}^{m} (p_{ij, k}^{-} p_{ij, m}^{-} + p_{ij, k}^{+} p_{ij, m}^{-}))}^{T} \end{matrix}$ (8) $D = {(\begin{matrix} \sum_{i = 1}^{n} \sum_{j = 1}^{n} m ({(p_{ij, 1}^{-})}^{2} + {(p_{ij, 1}^{+})}^{2}) & \sum_{i = 1}^{n} \sum_{j = 1}^{n} m (p_{ij, 1}^{-} p_{ij, 2}^{-} + p_{ij, 1}^{+} p_{ij, 2}^{+}) & \dots & \sum_{i = 1}^{n} \sum_{j = 1}^{n} m (p_{ij, 1}^{-} p_{ij, m}^{-} + p_{ij, 1}^{+} p_{ij, m}^{+}) \\ \sum_{i = 1}^{n} \sum_{j = 1}^{n} m (p_{ij, 1}^{-} p_{ij, 2}^{-} + p_{ij, 1}^{+} p_{ij, 2}^{+}) & \sum_{i = 1}^{n} \sum_{j = 1}^{n} m ({(p_{ij, 2}^{-})}^{2} + {(p_{ij, 2}^{+})}^{2}) & \dots & \sum_{i = 1}^{n} \sum_{j = 1}^{n} m (p_{ij, 2}^{-} p_{ij, m}^{-} + p_{ij, 2}^{+} p_{ij, m}^{+}) \\ \dots & \dots & \dots & \dots \\ \sum_{i = 1}^{n} \sum_{j = 1}^{n} m (p_{ij, m}^{-} p_{ij, 1}^{-} + p_{ij, m}^{+} p_{ij, 1}^{+}) & \sum_{i = 1}^{n} \sum_{j = 1}^{n} m (p_{ij, m}^{-} p_{ij, 2}^{-} + p_{ij, m}^{+} p_{ij, 2}^{+}) & \dots & \sum_{i = 1}^{n} \sum_{j = 1}^{n} m ({(p_{ij, m}^{-})}^{2} + {(p_{ij, m}^{+})}^{2}) \end{matrix})}_{m \times m}$ (9)

and w = (w₁, w₂, …, w_m)^T is an optimal weight vector of the experts.

In the following, a further analysis is conducted for the model (M-1).

Theorem 1.For IFPRsP_k = (p_ij,k) _n×n, where $p_{ij, k} = [p_{ij, k}^{-}, p_{ij, k}^{+}]$ , k = 1, 2, …, m, the optimal solution to (M-1) model is $w = (1 / m, 1 / m, \dots, 1 / m)^{T}$ (10)

Proof. According to Equations (8) and (9), the relation between p and D can be expressed as follows: $p = \frac{De}{m}$ (11)

Plugging (11) into (7), one has $\begin{matrix} w = \frac{D^{- 1} e (1 - e^{T} D^{- 1} p)}{e^{T} D^{- 1} e} + D^{- 1} p \\ = \frac{D^{- 1} e (1 - \frac{e^{T} D^{- 1} De}{m})}{e^{T} D^{- 1} e} \\ + \frac{D^{- 1} De}{m} \\ = \frac{D^{- 1} e (1 - \frac{e^{T} e}{m})}{e^{T} D^{- 1} e} + \frac{e}{m} \\ = \frac{D^{- 1} e (1 - 1)}{e^{T} D^{- 1} e} + \frac{e}{m} \\ = {(\frac{1}{m}, \frac{1}{m}, \dots, \frac{1}{m})}^{T} \end{matrix}$ (12)

This result indicates that (M-1) always generates an equal weight of 1/m for each DM as long as there does not reach absolutely consensus among the group. It’s also the reason the numerical examples in [43] always were given an equal weight of 1/m for all DMs.

We can see some limitations for the model in Xu [43] from foregoing analysis: Xu [43] utilized the quadratic programming model (M-1) to determine an optimal weight vector w = (w₁, w₂, …, w_m)^T, Theorem 1 shows that the optimal weight vector is always w = (1/m, 1/m, …, 1/m)^T, which means that all the DMs’ IFPRs play an equal role in the consensus reaching process and makes the modelredundant.

To address the aforesaid deficiencies, a new model and algorithm will be developed below for reaching acceptable levels of consensus in GDM withIFPRs.

To reach a group consensus, the method in Xu [43] adjusts IFPRs P_k to make them as close to the collective IFPR P as possible. In this paper, we consider from a different perspective, it is highly likely that individual IFPRs are largely dispersed if their weights are not considered. Therefore, the weights should be incorporated into each IFPR. In order to achieve maximum consensus, the weighted IFPRs should come closer to each other. Based upon this idea, an optimization model based on distance is proposed to integrate different DMs’ preferencerelations.

We define the squared distance between each pair of individual IFPRs (P_k, P_l) as $\begin{matrix} d^{2} (w_{k} P_{k}, w_{l} P_{l}) \\ = \sum_{i = 1}^{n} \sum_{j = 1}^{n} ({(w_{k} p_{ij, k}^{-} - w_{l} p_{ij, l}^{-})}^{2} \\ + {(w_{k} p_{ij, k}^{+} - w_{l} p_{ij, l}^{+})}^{2}) \end{matrix}$ (13)

Based upon this definition, the following optimization model is established to make the sum of squared distances between all pairs of weighted interval fuzzy preference judgments smallest: $\begin{matrix} (M - 2) min \\ J_{1} = \sum_{k = 1}^{m} \sum_{l = 1, l \neq k}^{m} \sum_{i = 1}^{n} \sum_{j = 1}^{n} ({(w_{k} p_{ij, k}^{-} - w_{l} p_{ij, l}^{-})}^{2} \\ + {(w_{k} p_{ij, k}^{+} - w_{l} p_{ij, l}^{+})}^{2}) \end{matrix}$ (14) $s . t . \sum_{l = 1}^{m} w_{l} = 1$ (15) $w_{l} \geq 0, l = 1, 2, \dots, m$ (16)

Theorem 2.Model (M-2) is equivalent to (M-3) below in a matrix form $(M - 3) min J_{1} = w^{T} Gw$ (17) $s . t . e^{T} w = 1$ (18) $w \geq 0$ (19) where w = (w₁, w₂, …, w_m)^T, e = (1, 1, …, 1)^T, $G = {(g_{kl})}_{m \times m} = 2 [\begin{matrix} (m - 1) (\sum_{i = 1}^{n} \sum_{j = 1}^{n} (\begin{matrix} {(p_{ij, 1}^{-})}^{2} \\ + {(p_{ij, 1}^{+})}^{2} \end{matrix})) & - \sum_{i = 1}^{n} \sum_{j = 1}^{n} (\begin{matrix} p_{ij, 1}^{-} p_{ij, 2}^{-} \\ + p_{ij, 1}^{+} p_{ij, 2}^{+} \end{matrix}) & \dots & - \sum_{i = 1}^{n} \sum_{j = 1}^{n} (\begin{matrix} p_{ij, 1}^{-} p_{ij, m}^{-} \\ + p_{ij, 1}^{+} p_{ij, m}^{+} \end{matrix}) \\ - \sum_{i = 1}^{n} \sum_{j = 1}^{n} (\begin{matrix} p_{ij, 2}^{-} p_{ij, 1}^{-} \\ + p_{ij, 2}^{+} p_{ij, 1}^{+} \end{matrix}) & (m - 1) (\sum_{i = 1}^{n} \sum_{j = 1}^{n} (\begin{matrix} {(p_{ij, 2}^{-})}^{2} \\ + {(p_{ij, 2}^{+})}^{2} \end{matrix})) & \dots & - \sum_{i = 1}^{n} \sum_{j = 1}^{n} (\begin{matrix} p_{ij, 2}^{-} p_{ij, m}^{-} \\ + p_{ij, 2}^{+} p_{ij, m}^{+} \end{matrix}) \\ \dots & \dots & \dots & \dots \\ - \sum_{i = 1}^{n} \sum_{j = 1}^{n} (\begin{matrix} p_{ij, m}^{-} p_{ij, 1}^{-} \\ + p_{ij, m}^{+} p_{ij, 1}^{+} \end{matrix}) & - \sum_{i = 1}^{n} \sum_{j = 1}^{n} (\begin{matrix} p_{ij, m}^{-} p_{ij, 2}^{-} \\ + p_{ij, m}^{+} p_{ij, 2}^{+} \end{matrix}) & \dots & (m - 1) (\sum_{i = 1}^{n} \sum_{j = 1}^{n} (\begin{matrix} {(p_{ij, m}^{-})}^{2} \\ + {(p_{ij, m}^{+})}^{2} \end{matrix})) \end{matrix}]$ (20)

Proof. $\begin{matrix} J_{1} = \sum_{k = 1}^{m} \sum_{l = 1, l \neq k}^{m} \sum_{i = 1}^{n} \sum_{j = 1}^{n} ({(w_{k} p_{ij, k}^{-} - w_{l} p_{ij, l}^{-})}^{2} \\ + {(w_{k} p_{ij, k}^{+} - w_{l} p_{ij, l}^{+})}^{2}) \\ = \sum_{k = 1}^{m} \sum_{l = 1, l \neq k}^{m} \sum_{i = 1}^{n} \sum_{j = 1}^{n} [{(w_{k} p_{ij, k}^{-})}^{2} + {(w_{l} p_{ij, l}^{-})}^{2} \\ + {(w_{k} p_{ij, k}^{+})}^{2} + {(w_{l} p_{ij, l}^{+})}^{2}] \\ - 2 \sum_{k = 1}^{m} \sum_{l = 1, l \neq k}^{m} \sum_{i = 1}^{n} \sum_{j = 1}^{n} w_{k} w_{l} (p_{ij, k}^{-} p_{ij, l}^{-} + p_{ij, k}^{+} p_{ij, l}^{+}) \\ = \sum_{k = 1}^{m} [2 (m - 1) \sum_{i = 1}^{n} \sum_{j = 1}^{n} ({(p_{ij, k}^{-})}^{2} + {(p_{ij, k}^{+})}^{2})] w_{k}^{2} \\ + \sum_{k = 1}^{m} \sum_{l = 1, l \neq k}^{m} \sum_{i = 1}^{n} \sum_{j = 1}^{n} (- 2 p_{ij, k}^{-} p_{ij, l}^{-} - 2 p_{ij, k}^{+} p_{ij, l}^{+}) w_{k} w_{l} \end{matrix}$ (21)

As for J₁ represented by Equation (17), we have

$\begin{matrix} J_{1} = w^{T} Gw = \sum_{k = 1}^{m} \sum_{l = 1}^{m} g_{kl} w_{k} w_{l} \\ = \sum_{k = 1}^{m} g_{kk} w_{k}^{2} + \sum_{k = 1}^{m} \sum_{l = 1, l \neq k}^{m} g_{kl} w_{k} w_{l} \end{matrix}$ (22)

Comparing (21) and (22), we obtain (20).

Analogous to Definition 7 in [33], based on Equation (5) and the optimal weight vector w, the deviation (referred to as an individual to group consensus index ICI in this paper) between the individual IFPR P_k and the collective IFPR P can be defined as

$\begin{matrix} ICI (P_{k}) = d (P_{k}, P) = \frac{2}{n (n - 1)} \sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} \\ (| p_{ij, k}^{-} - \sum_{l = 1}^{m} w_{l} p_{ij, l}^{-} | + | p_{ij, k}^{+} - \sum_{l = 1}^{m} w_{l} p_{ij, l}^{+} |) \end{matrix}$ (23)

And the weighted sum of all the deviations d (P_k, P) (k = 1, 2, …, m) (referred to as a group consensus index GCI) can be defined as $GCI = Δ_{1} = \sum_{k = 1}^{m} w_{k} d (P_{k}, P)$ (24)

Theorem 3.For the model (M-3), if for anyi, j, kandl, there exists at least one inequalityp_ij,k ≠ p_ij,l, then matrixGdetermined by (20) is positive definite and, hence, non-singular and invertible.

Proof. Apparently, J₁ = w^TGw ≥ 0. Now, we prove that J₁ ≠ 0 if there exists at least one inequality p_ij,k ≠ p_ij,l.

Assume that there exists a weight vector w, for all i, j, k and l, such that J₁ = 0. Then,

$w_{k} p_{ij, k}^{-} = w_{l} p_{ij, l}^{-}$ and $w_{k} p_{ij, k}^{+} = w_{l} p_{ij, l}^{+}$

(where $p_{ij, k} = [p_{ij, k}^{-}, p_{ij, k}^{+}]$ , $p_{ij, l} = [p_{ij, l}^{-}, p_{ij, l}^{+}]$ )

Thus, we can obtain $\begin{matrix} \frac{w_{k}}{w_{l}} = \frac{p_{ij, l}^{-}}{p_{ij, k}^{-}} = \frac{p_{ji, l}^{+}}{p_{ji, k}^{+}} = \frac{1 - p_{ij, l}^{-}}{1 - p_{ij, k}^{-}} = \frac{p_{ij, l}^{+}}{p_{ij, k}^{+}} = \frac{p_{ji, l}^{-}}{p_{ji, k}^{-}} \\ = \frac{1 - p_{ij, l}^{+}}{1 - p_{ij, k}^{+}} \end{matrix}$

thus

p_ij,k = p_ij,l, for all i, j, k and l

This contradicts with the assumption that there exists at least one inequality p_ij,k ≠ p_ij,l. Therefore, J₁ > 0 and the symmetry of matrix G and the definition of positive definiteness confirm that G is positive definite, and, hence, nonsingular and invertible, i.e., G^-1 exists. This completes the proof of Theorem 3.

Theorem 4.As for the model (M-3), if for anyi, kandl, there exists at least one of the inequalities $p_{ij, k}^{-} \neq p_{ij, l}^{-}$ or $p_{ij, k}^{+} \neq p_{ij, l}^{+}$ , then the optimal solution is $w^{*} = \frac{G^{- 1} e}{e^{T} G^{- 1} e}$

Proof. To solve (M-3), the following Lagrangian function is constructed by neglecting the non-negativity constraint (19): $L (w, λ) = w^{T} Gw + 2 λ (e^{T} w - 1)$ (25) where λ is the Lagrangian multiplier. Let ∂L/∂w = 0 and ∂L/∂λ = 0, then $Gw + λ e = 0$ (26) $e^{T} w = 1$ (27)

By Theorem 3, matrix G is invertible. Thus, solutions to (26) and (27) are given as $w^{*} = \frac{G^{- 1} e}{e^{T} G^{- 1} e}$ (28) $λ^{*} = - \frac{1}{e^{T} G^{- 1} e}$ (29)

The Hessian matrix for Lagrangian function L (w, λ) with regard to w is given by $H (w, λ) = \nabla^{2} L (w, λ) = 2 G$ (30)

Since matrix G is positive definite, H (w, λ) is also positive definite, Therefore, the Lagrangian function L (w, λ) has the global minimum value. Consequently, vector w^* determined by (28) is the optimal solution.

Below, we construct an iterative consensus reaching algorithm for IFPRs based on the quadratic programming model (M-2).

Algorithm 1

Input:P_k = (p_ij,k) _n×n ( $p_{ij, k} = [p_{ij, k}^{-}, p_{ij, k}^{+}]$ , k = 1,2, ... , m), the maximum iteration number t^*, the thresholds α₁ and λ₁ for ICI and GCI, respectively.

Output: Improved IFPRs ${\bar{P}}_{k}$ (k = 1, 2, …, m), the iteration step t, individual consensus index $ICI ({\bar{P}}_{k})$ (k = 1, 2, …, m) and group consensus index GCI.

Step 1. Let t = 0, $P_{k}^{(0)} = P_{k}$ (k = 1, 2, …, m).

Step 2. Utilize the UWAA operator (3) to aggregate all the individual IFPR $P_{k}^{(t)} = (p_{ij, k}^{(t)})_{n \times n} (k = 1, 2, \dots, m)$ into the collective IFPR $P^{(t)} = (p_{ij}^{(t)})_{n \times n}$ , where

$\begin{matrix} p_{ij, k}^{(t)} = [p_{ij, k}^{- (t)}, p_{ij, k}^{+ (t)}] (k = 1, 2, \dots, m) \\ p_{ij}^{(t)} = [p_{ij}^{- (t)}, p_{ij}^{+ (t)}] . \end{matrix}$

Step 3. Apply the quadratic programming model (M-2) to determine the optimal weighting vector $w^{(t)} = (w_{1}^{(t)}, w_{2}^{(t)}, \dots, w_{m}^{(t)})^{T}$ of $P_{k}^{(t)} = (p_{ij, k}^{(t)})_{n \times n}$ (k = 1, 2, …, m), and then obtain the collective IFPR P^(t) by Equation (3).

Step 4. Compute individual consensus index $ICI (P_{k}^{(t)}) = d (P_{k}^{(t)}, P^{(t)}) (k = 1, 2, \dots, m)$ and the group consensus index Δ₁ (t) using Equations (23) and (24), respectively. If Δ₁ (t) ≤ λ₁ and $ICI (P_{k}^{(t)}) \leq α_{1}$ (for all k = 1, 2, …, m) or t = t^*, go to Step 6. Otherwise, find the IFPR $P_{k}^{(t)}$ such that $ICI (P_{k}^{(t)}) > λ_{1}$ . Go to Step 5.

Step 5. Find the position of the elements $d_{i_{τ} j_{τ}, k}^{(t)}$ for DM e_k if $ICI (P_{k}^{(t)}) > λ_{1}$ , where $d_{i_{τ} j_{τ}, k}^{(t)} = {max}_{i, j} (d_{i_{τ} j_{τ}, k}^{- (t)} + d_{i_{τ} j_{τ}, k}^{+ (t)})$ , $d_{i_{τ} j_{τ}, k}^{- (t)} =$ $| p_{ij, k}^{- (t)} - p_{ij}^{- (t)} |$ , $d_{i_{τ} j_{τ}, k}^{+ (t)} = | p_{ij, k}^{+ (t)} - p_{ij}^{+ (t)} |$ , modify DM e_k’s IFPR. Let $P_{k}^{(t + 1)} = (p_{ij, k}^{(t + 1)})_{n \times n}$ , where $p_{ij, k}^{- (t + 1)} = {\begin{matrix} p_{ij}^{- (t)}, if i = i_{τ}, j = j_{τ} \\ p_{ij, k}^{- (t)}, otherwise \end{matrix},$ $p_{ij, k}^{+ (t + 1)} = {\begin{matrix} p_{ij}^{+ (t)}, if j = j_{τ}, i = i_{τ} \\ p_{ij, k}^{+ (t)}, otherwise \end{matrix}$ and t = t + 1. Then go to Step 2.

Step 6. Let ${\bar{P}}_{k} = P_{k}^{(t)}$ . Output the modified IFPRs ${\bar{P}}_{k}$ (k = 1, 2, …, m), the individual consensus index $ICI (P_{k}^{(t)}) (k = 1, 2, \dots, m)$ , the group consensus index GCI, and the number of iterations t.

Remark 1. [37] Generally, for the two thresholds α₁ and λ₁, it is sensible to set α₁ > λ₁. Otherwise, if α₁ ≤ λ₁, and ICI (P_k) ≤ α₁ ≤ λ₁, it follows that $GCI = Δ_{1} = \sum_{k = 1}^{m} w_{k} ICI (P_{k}) \leq \sum_{k = 1}^{m} w_{k} α_{1} = α_{1} \leq λ_{1}$ . By setting α₁ > λ₁, the individual to group consensus index (ICI (P_k)) is allowed to be a little larger than the group consensus index (GCI), which allows the individual to deviate a certain degree from the group judgment. These parameters are often subjectively given by the experts in the group or by a super expert [16]. While there is no specific rule to determine the threshold values, they can generally be specified by a trial-and-error process. The two thresholds provide a flexible choice for the group to control the decision process.

Remark 2. Xu and Wu [33] adopted Equation (23) to measure the group consensus assuming that a consensus is reached if all DMs’ uncertain linguistic preference relations are sufficiently close to the group preference relation (deviations are smaller than a given threshold). As commented in Remark 1, this treatment is same as setting α₁ ≤ λ₁, and, hence, can be viewed as a special case of the presented measure. In addition, Xu and Wu [32] utilized Step 5 in Algorithm 1 to identify which decision maker should change his/her preference values. The proposed method extends the relevant research reported by Xu and Wu [32, 33].

Remark 3. This algorithm automatically updates the experts’ preference values in order to reach a group consensus. This treatment helps to relieve the experts from the burden of constantly adjusting their judgments. On the other hand, if the experts are willing to reevaluate their preferences, the algorithm can serve as an invaluable aid to the expert in identifying which preference values to change so that the highest degree of consensus can be reached expeditiously.

3 Group consensus models for interval multiplicative preference relations

If DM e_k compares each pair of alternatives in X = {x₁, x₂, …, x_n} (n ≥ 2) and provides an interval multiplicative preference relation (IMPR) A_k = (a_ij,k) _n×n, where $a_{ij, k} = [a_{ij, k}^{-}, a_{ij, k}^{+}]$ and satisfies

$\begin{matrix} a_{ji, k}^{-} = \frac{1}{a_{ij, k}^{+}}, a_{ji, k}^{+} = \frac{1}{a_{ij, k}^{-}}, a_{ij, k}^{+} \geq a_{ij, k}^{-} > 0 \\ a_{ii, k}^{+} = a_{ii, k}^{-} = 1, i, j = 1, 2, \dots, n \end{matrix}$ (31)

Let v = (v₁, v₂, …, v_m)^T be the implied weight vector of IMPRs A_k = (a_ij,k) _n×n, (k = 1, 2, …, m) where $a_{ij, k} = [a_{ij, k}^{-}, a_{ij, k}^{+}]$ , v_k ≥ 0 and $\sum_{k = 1}^{m} v_{k} = 1$ . To obtain a collective opinion, we employ the Uncertain Weighted Geometric Average (UWGA) operator: $\begin{matrix} a_{ij} = [a_{ij}^{-}, a_{ij}^{+}] = \prod_{k = 1}^{m} (a_{ij, k})^{v_{k}} \\ = [\prod_{k = 1}^{m} (a_{ij, k}^{-})^{v_{k}}, \prod_{k = 1}^{m} (a_{ij, k}^{+})^{v_{k}}], i, j = 1, 2, . . ., n \end{matrix}$ (32)

to aggregate individual IMPRs A_k = (a_ij,k) _n×n ( $a_{ij, k} = [a_{ij, k}^{-}, a_{ij, k}^{+}]$ , k = 1, 2, …, m) into a collective preference relation A = (a_ij) _n×n $(a_{ij} = [a_{ij}^{-}, a_{ij}^{+}])$ .

It is easy to certify that A satisfies (31), and is thus an IMPR as well.

As per Proposition 2.1 in [15], a MPR can be transformed into a FPR by the following formula: $p_{ij} = \frac{1}{2} (1 + {log}_{9} (a_{ij}))$ (33)

Similarly, an IMPR can be transformed into an IFPR by the following formula: $p_{ij}^{-} = \frac{1}{2} (1 + {log}_{9} (a_{ij}^{-})), p_{ij}^{+} = \frac{1}{2} (1 + {log}_{9} (a_{ij}^{+}))$ (34)

Analogous to model (M-2), a squared weighted distance between each pair of individual IMPRs (A_k, A_l) can be defined as $\begin{matrix} d^{2} (v_{k} A_{k}, v_{l} A_{l}) \\ = \sum_{i = 1}^{n} \sum_{j = 1}^{n} ({(v_{k} . \frac{1}{2} (1 + {log}_{9} (a_{ij, k}^{-})) - v_{l} . \frac{1}{2} (1 + {log}_{9} (a_{ij, l}^{-})))}^{2} \\ + {(v_{k} . \frac{1}{2} (1 + {log}_{9} (a_{ij, k}^{+})) - v_{l} . \frac{1}{2} (1 + {log}_{9} (a_{ij, l}^{+})))}^{2}) \\ = \frac{1}{4} \sum_{i = 1}^{n} \sum_{j = 1}^{n} ({(v_{k} (1 + {log}_{9} (a_{ij, k}^{-})) - v_{l} (1 + {log}_{9} (a_{ij, l}^{-})))}^{2} \\ + {(v_{k} (1 + {log}_{9} (a_{ij, k}^{+})) - v_{l} (1 + {log}_{9} (a_{ij, l}^{+})))}^{2}) \end{matrix}$ (35)

Following this definition, an optimization model is constructed to minimize the sum of squared weighted distances between all pairs of IMPRs: $\begin{matrix} (M - 4) min J_{2} = \frac{1}{4} \sum_{i = 1}^{n} \sum_{j = 1}^{n} ((v_{k} (1 + {log}_{9} (a_{ij, k}^{-})) \\ {- v_{l} (1 + {log}_{9} (a_{ij, l}^{-})))}^{2} \\ + {(v_{k} (1 + {log}_{9} (a_{ij, k}^{+})) - v_{l} (1 + {log}_{9} (a_{ij, l}^{+})))}^{2}) \end{matrix}$ (36) $s . t . \sum_{l = 1}^{m} v_{l} = 1$ (37) $v_{l} \geq 0, l = 1, 2, \dots, m$ (38) Similar to the case of IFPRs, (M-4) can be rewritten in a matrix form.

Theorem 5.Model (M-4) is equivalent to (M-5) below in a matrix form $(M - 5) min J_{2} = v^{T} Bv$ (39) $s . t . e^{T} v = 1$ (40) $v \geq 0$ (41) where v = (v₁, v₂, …, v_m)^T, e = (1, 1, …, 1)^T.

The elements in matrix B are

$\begin{matrix} b_{kk} = \frac{(m - 1)}{2} \sum_{i = 1}^{n} \sum_{j = 1}^{n} ({(1 + {log}_{9} (a_{ij, k}^{-}))}^{2} \\ + {(1 + {log}_{9} (a_{ij, k}^{+}))}^{2}) \end{matrix}$ (42) $\begin{matrix} b_{kl} = - \frac{1}{2} \sum_{i = 1}^{n} \sum_{j = 1}^{n} ((1 + {log}_{9} (a_{ij, k}^{-})) (1 + {log}_{9} (a_{ij, l}^{-})) \\ + (1 + {log}_{9} (a_{ij, k}^{+})) (1 + {log}_{9} (a_{ij, l}^{+}))), \\ k, l = 1, 2, \dots, m, k \neq l \end{matrix}$ (43)

Similarly, the Lagrangian multiplier method is employed to solve (M-4) as follows $v^{*} = \frac{B^{- 1} e}{e^{T} B^{- 1} e}$ (44)

$λ^{*} = - \frac{1}{e^{T} B^{- 1} e}$ (45)

Based on the aforesaid models, similar to Algorithm 1, a consensus algorithm is devised for GDM with IMPRs.

Algorithm 2

Input: IMPRs A_k = (a_ij,k) _n×n (k = 1, 2, …, m), the maximum number of iterations t^*, the thresholds α₂, λ₂ for individual and group consensus indices, respectively. Generally, α₂ > λ₂.

Output: Improved IMPRs ${\bar{A}}_{k}$ (k = 1, 2, …, m), terminal iterative step t, individual consensus index $ICI (A_{k}^{(t)}) (k = 1, 2, \dots, m)$ and group consensus degree GCI.

Step 1. Let t = 0, $A_{k}^{(0)} = A_{k}$ (k = 1, 2, …, m).

Step 2. Utilize the UWAA operator (3) to aggregate all the individual IMPRs $A_{k}^{(t)} = (a_{ij, k}^{(t)})_{n \times n} (k = 1, 2, \dots, m)$ into a collective IMPR $A^{(t)} = (a_{ij}^{(t)})_{n \times n}$ , where $a_{ij, k}^{(t)} = [a_{ij, k}^{- (t)}, a_{ij, k}^{+ (t)}]$ (k = 1, 2, …, m), $a_{ij}^{(t)} = [a_{ij}^{- (t)}, a_{ij}^{+ (t)}]$ .

Step 3. Apply the quadratic programming model (M-4) to determine the optimal weight vector $v^{(t)} = (v_{1}^{(t)}, v_{2}^{(t)}, \dots, v_{m}^{(t)})^{T}$ of $A_{k}^{(t)} = (a_{ij, k}^{(t)})_{n \times n} (k = 1, 2, \dots, m)$ , and then get the collective IMPR A^(t) using Equation (32).

Step 4. Calculate individual consensus index $ICI (A_{k}^{(t)})$ by the following formula: $\begin{matrix} ICI (A_{k}) = \frac{1}{n (n - 1)} \sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} (| {log}_{9} (a_{ij, k}^{-}) - \sum_{l = 1}^{m} v_{l} {log}_{9} (a_{ij, l}^{-}) | + \\ | {log}_{9} (a_{ij, k}^{+}) - \sum_{l = 1}^{m} v_{l} {log}_{9} (a_{ij, l}^{+}) |) \end{matrix}$ (46)

and $GCI = Δ_{2} = \sum_{k = 1}^{m} v_{k} d (A_{k}, A)$ (47) respectively. If Δ₂ (t) ≤ λ₂ and $ICI (A_{k}^{(t)}) \leq α_{2}$ (for all k = 1, 2, …, m) or t = t^*, go to Step 6. Otherwise, find the interval multiplicative preference relation $A_{k}^{(t)}$ such that $ICI (A_{k}^{(t)}) > λ_{2}$ . Go to Step 5.

Step 5. Find the position of the elements $d_{i_{τ} j_{τ}, k}^{(t)}$ for DM e_k such that $ICI (A_{k}^{(t)}) > λ_{2}$ , where $d_{i_{τ} j_{τ}, k}^{(t)} = max_{i, j} (d_{i_{τ} j_{τ}, k}^{- (t)} + d_{i_{τ} j_{τ}, k}^{+ (t)})$ , $d_{i_{τ} j_{τ}, k}^{- (t)} = | {log}_{9} (a_{ij, k}^{- (t)}) - {log}_{9} (a_{ij}^{- (t)}) |$ , $d_{i_{τ} j_{τ}, k}^{+ (t)} = | {log}_{9} (a_{ij, k}^{+ (t)}) - {log}_{9} (a_{ij}^{+ (t)}) |$ , modify DM e_k’s interval multiplicative preference relation. Let $A_{k}^{(t + 1)} = (A_{ij, k}^{(t + 1)})_{n \times n}$ , where

$a_{ij, k}^{- (t + 1)} = {\begin{matrix} a_{ij}^{- (t)}, if i = i_{τ}, j = j_{τ} \\ a_{ij, k}^{- (t)}, otherwise \end{matrix},$

$a_{ij, k}^{+ (t + 1)} = {\begin{matrix} a_{ij}^{+ (t)}, if i = i_{τ}, j = j_{τ} \\ a_{ij, k}^{+ (t)}, otherwise \end{matrix}$ and t = t + 1. Then go to Step 2.

Step 6. Let ${\bar{A}}_{k} = A_{k}^{(t)}$ . Derive the modified IMPRs ${\bar{A}}_{k} (k = 1, 2, \dots, m)$ , the individual consensus index $ICI (A_{k}^{(t)}) (k = 1, 2, \dots, m)$ , the group consensus index GCI, and the number of iterations t.

4 Illustrative examples

In the following, we give two illustrative examples on GDM problems with interval fuzzy preference relations and interval multiplicative preference relations to demonstrate the feasibility and effectiveness of the proposed models and methods.

Example 1. Considering a GDM problem under uncertainty as shown in Xu and Liu [31]. Fire system is a dynamic system achieved by collocating and allocating various firearms involved in an appropriate way. A committee comprising of three experts e_k (k = 1, 2, 3) has been set up to provide assessment information on five factors x_i (i = 1, 2, 3, 4, 5). The experts e_k (k = 1, 2, 3) compare each pair of the factors x_i (i = 1, 2, 3, 4, 5) (here both the weight vector ω = (ω₁, ω₂, ω₃, ω₄, ω₅)^T of the factors and the weight vector w = (w₁, w₂, w₃)^T of the experts are to be determined) and construct the interval fuzzy preference relations P_k = (p_ij,k) _5×5( $p_{ij, k} = [p_{ij, k}^{-}, p_{ij, k}^{+}]$ , i = 1, 2, 3, 4, 5, k = 1, 2, 3): $\begin{matrix} P_{1} = (\begin{matrix} [0.5, 0.5] & [0.6, 0.8] & [0.7, 1.0] & [0.2, 0.3] & [0.4, 0.5] \\ [0.2, 0.4] & [0.5, 0.5] & [0.4, 0.6] & [0.7, 0.8] & [0.3, 0.5] \\ [0.0, 0.3] & [0.4, 0.6] & [0.5, 0.5] & [0.6, 0.9] & [0.4, 0.7] \\ [0.7, 0.8] & [0.2, 0.3] & [0.1, 0.4] & [0.5, 0.5] & [0.3, 0.4] \\ [0.5, 0.6] & [0.5, 0.7] & [0.3, 0.6] & [0.6, 0.7] & [0.5, 0.5] \end{matrix}) \\ P_{2} = (\begin{matrix} [0.5, 0.5] & [0.5, 0.7] & [0.8, 0.9] & [0.3, 0.5] & [0.3, 0.6] \\ [0.3, 0.5] & [0.5, 0.5] & [0.6, 0.7] & [0.5, 0.6] & [0.4, 0.5] \\ [0.1, 0.2] & [0.3, 0.4] & [0.5, 0.5] & [0.7, 0.9] & [0.6, 0.7] \\ [0.5, 0.7] & [0.4, 0.5] & [0.1, 0.3] & [0.5, 0.5] & [0.5, 0.6] \\ [0.4, 0.7] & [0.5, 0.6] & [0.3, 0.4] & [0.4, 0.5] & [0.5, 0.5] \end{matrix}) \\ P_{3} = (\begin{matrix} [0.5, 0.5] & [0.7, 0.9] & [0.8, 1.0] & [0.4, 0.5] & [0.3, 0.4] \\ [0.1, 0.3] & [0.5, 0.5] & [0.6, 0.7] & [0.4, 0.6] & [0.4, 0.6] \\ [0.0, 0.2] & [0.3, 0.4] & [0.5, 0.5] & [0.7, 0.8] & [0.5, 0.8] \\ [0.5, 0.6] & [0.4, 0.6] & [0.2, 0.3] & [0.5, 0.5] & [0.4, 0.7] \\ [0.6, 0.7] & [0.4, 0.6] & [0.2, 0.5] & [0.3, 0.6] & [0.5, 0.5] \end{matrix}) \end{matrix}$ Algorithm 1 is employed to solve the GDM problem. Suppose that the maximum number of iteration t^*= 10, the individual consensus degree threshold α₁ = 0.065, the group consensus degree threshold is set at λ₁1pt = 0.05.

Step 1. Initiate the algorithm by setting t = 0 and $P_{k}^{(t)} = P_{k}$ .

Step 2. Utilizing the quadratic program (M-3) or Equation (28) to determine the optimal weight

vector $w^{(0)} = (w_{1}^{(0)}, w_{2}^{(0)}, w_{3}^{(0)})^{T}$ of the experts e_k (k = 1, 2, 3): $w^{(0)} = (0.3298, 0.3385, 0.3318)^{T}$

Step 3. Using Equation (3) to aggregate all P_k (k = 1, 2, 3) into the collective IFPR P = (p_ij) _5×5 ( $p_{ij} = [p_{ij}^{-}, p_{ij}^{+}]$ , i, j = 1, 2, 3, 4, 5) as shown in Table 1.

Step 4. Calculating $ICI (P_{k}^{(0)})$ (k = 1, 2, 3) and GCI (0) based on Equations (23) and (24): $\begin{matrix} ICI (P_{1}^{(0)}) = 0.1509, ICI (P_{2}^{(0)}) = 0.1090, \\ ICI (P_{3}^{(0)}) = 0.1266, GCI (0) = 0.1287 . \end{matrix}$

Step 5. Now that GCI (0) =0.1287 > 0.05, $ICI (P_{k}^{(0)}) > 0.065$ (k = 1, 2, 3), we need to find the posi-tion of elements $d_{i_{τ} j_{τ}, k}^{(0)} (k = 1, 2, 3)$ , where $d_{i_{τ} j_{τ}, k}^{(0)} = max_{i, j} (d_{i_{τ} j_{τ}, k}^{- (0)} + d_{i_{τ} j_{τ}, k}^{+ (0)})$ , $d_{i_{τ} j_{τ}, k}^{- (0)} = | p_{ij, k}^{- (0)} - p_{ij}^{- (0)} |$ , $d_{i_{τ} j_{τ}, k}^{+ (0)} = | p_{ij, k}^{+ (0)} - p_{ij}^{+ (0)} |$ . For $P_{1}^{(0)}$ , since $d_{24, 1}^{(0)} = d_{42, 1}^{(0)} = max_{i, j} (d_{24, 1}^{- (0)} + d_{24, 1}^{+ (0)}) = max_{i, j} (| p_{ij, 1}^{- (0)} -$ $p_{ij}^{- (0)} | + | p_{ij, 1}^{+ (0)} - p_{ij}^{+ (0)} |) = 0.1672 + 0.134 = 0.3012$ , replacing these two preference values with the corresponding elements in the collective IFPR P⁽⁰⁾, $p_{24, 1}^{(0)} = [p_{24, 1}^{- (0)}, p_{24, 1}^{+ (0)}] = p_{24}^{(0)} = [p_{24}^{- (0)}, p_{24}^{+ (0)}] = [0.5328, 0.6660]$ , $p_{42, 1}^{(0)} =$ $[p_{42, 1}^{- (0)}, p_{42, 1}^{+ (0)}] = p_{42}^{+ (0)} = [p_{42}^{- (0)}, p_{42}^{+ (0)}] = [0.3340, 0.4672]$ . Similarly, the same procedure is used to update the other two DMs’ IFPRs. $\begin{matrix} p_{12, 2}^{(0)} = p_{12}^{(0)} = [0.5994, 0.7994], \\ p_{21, 2}^{(0)} = p_{21}^{(0)} = [0.2006, 0.4006], \\ p_{12, 3}^{(0)} = p_{12}^{(0)} = [0.5994, 0.7994], \\ p_{21, 3}^{(0)} = p_{21}^{(0)} = [0.2006, 0.4006] . \end{matrix}$

Let t = 1, then go to step 2.

This procedure terminates after 6 iterations, and the detailed iterative processes are depicted in Table 1.

The final improved individual IFPRs ${\bar{P}}_{k} (k = 1, 2, 3)$ and group IFPR $\bar{P}$ are $\begin{matrix} {\bar{P}}_{1} = (\begin{matrix} [0.5, 0.5] & [0.6, 0.8] & [0.7667, 0.9666] & [0.2668, 0.4116] & [0.4, 0.5] \\ [0.2, 0.4] & [0.5, 0.5] & [0.5335, 0.6667] & [0.5328, 0.6660] & [0.3, 0.5] \\ [0.0334, 0.2333] & [0.3333, 0.4665] & [0.5, 0.5] & [0.6, 0.9] & [0.4668, 0.7444] \\ [0.5884, 0.7332] & [0.3340, 0.4672] & [0.1, 0.4] & [0.5, 0.5] & [0.4005, 0.5671] \\ [0.5, 0.6] & [0.5, 0.7] & [0.2556, 0.5332] & [0.4329, 0.5995] & [0.5, 0.5] \end{matrix}) \end{matrix}$ $\begin{matrix} {\bar{P}}_{2} = (\begin{matrix} [0.5, 0.5] & [0.5994, 0.7994] & [0.8, 0.9] & [0.2668, 0.4116] & [0.3333, 0.5003] \\ [0.2006, 0.4006] & [0.5, 0.5] & [0.6, 0.7] & [0.5, 0.6] & [0.4, 0.5] \\ [0.1, 0.2] & [0.3, 0.4] & [0.5, 0.5] & [0.7, 0.9] & [0.5005, 0.7333] \\ [0.5884, 0.7332] & [0.4, 0.5] & [0.1, 0.3] & [0.5, 0.5] & [0.5, 0.6] \\ [0.4997, 0.6667] & [0.5, 0.6] & [0.2667, 0.4995] & [0.4, 0.5] & [0.5, 0.5] \end{matrix}) \end{matrix}$ $\begin{matrix} {\bar{P}}_{3} = (\begin{matrix} [0.5, 0.5] & [0.5994, 0.7994] & [0.8, 1.0] & [0.3002, 0.4338] & [0.3333, 0.5003] \\ [0.2006, 0.4006] & [0.5, 0.5] & [0.6, 0.7] & [0.4, 0.6] & [0.3667, 0.5332] \\ [0.0, 0.2] & [0.3, 0.4] & [0.5, 0.5] & [0.7, 0.8] & [0.5, 0.8] \\ [0.5662, 0.6998] & [0.4, 0.6] & [0.2, 0.3] & [0.5, 0.5] & [0.4337, 0.6223] \\ [0.4997, 0.6667] & [0.4668, 0.6333] & [0.2, 0.5] & [0.3777, 0.5663] & [0.5, 0.5] \end{matrix}) \end{matrix}$ $\begin{matrix} \bar{P} = (\begin{matrix} [0.5, 0.5] & [0.5996, 0.7996] & [0.7888, 0.9555] & [0.2779, 0.4189] & [0.3557, 0.5002] \\ [0.2004, 0.4004] & [0.5, 0.5] & [0.5776, 0.6888] & [0.4779, 0.6222] & [0.3553, 0.5110] \\ [0.0445, 0.2112] & [0.3112, 0.4224] & [0.5, 0.5] & [0.6664, 0.8669] & [0.4890, 0.7591] \\ [0.5811, 0.7221] & [0.3778, 0.5221] & [0.1331, 0.3336] & [0.5, 0.5] & [0.4446, 0.5963] \\ [0.4998, 0.6443] & [0.4890, 0.6447] & [0.2409, 0.5110] & [0.4037, 0.5554] & [0.5, 0.5] \end{matrix}) \end{matrix}$

The corresponding $ICI (P_{k}^{(t)})$ (k = 1, 2, 3) for the final modified IFPRs and GCI (t) are:

$\begin{matrix} ICI (P_{1}^{(6)}) = 0.0536, ICI (P_{2}^{(6)}) = 0.0405 \\ ICI (P_{3}^{(6)}) = 0.0472, GCI (6) = 0.0471, t = 6 \end{matrix}$

Table 1 shows that after four iterations (i.e., t = 4), $ICI (P_{2}^{(4)}) = 0.0455 < 0.05$ , indicating that DM e₂’s modified IFPR has reached an acceptable level of consensus with the collective FPR at this step. Therefore, $P_{2}^{(4)}$ will not be further updated, in this way, the DM’s original judgments can be by and large retained. Similarly, at t = 5, $ICI (P_{3}^{(5)}) = 0.0481 < 0.05$ , the updating of P₃ will be stopped. When t = 6, the group consensus index GCI (6) =0.0471 < 0.05, and all individual to group consensus indices are less than the threshold 0.065, so the iteration process terminates. The updated IFPRs ${\bar{P}}_{1}$ , ${\bar{P}}_{2}$ and ${\bar{P}}_{3}$ are considered to reach an acceptable consensus level, and a normalizing rank aggregation method of interval numbers [45] $ω_{i} = [\frac{\sum_{j = 1}^{n} p_{ij}^{-}}{\sum_{i = 1}^{n} \sum_{j = 1}^{n} p_{ij}^{+}}, \frac{\sum_{j = 1}^{n} p_{ij}^{+}}{\sum_{i = 1}^{n} \sum_{j = 1}^{n} p_{ij}^{-}}]$

is applied to derive an interval priority vector for the collective IFPR $\bar{P}$ as follows $\begin{matrix} ω = ([0.1778, 0.2935], [0.1488, 0.2517], [0.1418, 0.2552], \\ {[0.1436, 0.2473], [0.1504, 0.2640])}^{T} \end{matrix}$

Then compare each ω_i with all ω_i (i = 1, 2, 3, 4, 5) using the possibility degree formula [46]: $\begin{matrix} p (ω_{i} \geq ω_{j}) = max {1 - max [\frac{ω_{j}^{+} - ω_{i}^{-}}{L (ω_{i}) + L (ω_{j})}, 0], 0} \\ (ω_{i} = [ω_{i}^{-}, ω_{i}^{+}], ω_{j} = [ω_{j}^{-}, ω_{j}^{+}] and L (ω_{i}) = ω_{i}^{+} - ω_{i}^{-}, \\ L (ω_{j}) = ω_{j}^{+} - ω_{j}^{-}) \end{matrix}$

and get the possibility degree matrix: $F = [\begin{matrix} 0.5 & 0.6619 & 0.6622 & 0.6832 & 0.6241 \\ 0.3381 & 0.5 & 0.5081 & 0.5232 & 0.4679 \\ 0.3378 & 0.4919 & 0.5 & 0.5140 & 0.4617 \\ 0.3168 & 0.4768 & 0.4860 & 0.5 & 0.4459 \\ 0.3759 & 0.5321 & 0.5383 & 0.5541 & 0.5 \end{matrix}]$

Finally, the normalizing rank aggregation method [35] ${\bar{ω}}_{i} = \frac{2}{n^{2}} \sum_{j = 1}^{n} f_{ij}$ is adopted to derive a priority vector for the collective IFPR P as follows

$\bar{ω} = {(0.2505, 0.1870, 0.1844, 0.1780, 0.2000)}^{T}$

Rank all the alternatives x_i (i = 1, 2, 3, 4, 5) according to ${\bar{ω}}_{i} (i = 1, 2, 3, 4, 5)$ : $x_{1} ≻ x_{5} ≻ x_{2} ≻ x_{3} ≻ x_{4} .$

As we can see, the weight of each expert is different but very close. After our careful calculating, this is because that the three experts’ interval fuzzy preference relations are not significantly different from each other. As such, one would expect that the derived expert weights should not differ too much. In summary, this paper proposes a different model to determine expert weights and a distinct algorithm that aims to retain as much as possible the experts’ original preference values in adjusting interval preference relations in order to reach a group consensus. If we set all the experts’ weights are equal, then the ranking order is same: x₁ ≻ x₅ ≻ x₂ ≻ x₃ ≻ x₄. Although the ranking order is identical, it can’t reflect the importance degree of experts if all the experts’ weights are equal. If the preference relations provided by experts have a very wide gap, the difference between the weights of experts will be reflected and the result should be very different.

The ranking order given by Xu [43] is: x₁ ≻ x₂ ≻ x₅ ≻ x₃ ≻ x₄, the preference order of the alternatives obtained by Xu and Liu [31] and Chen, et al. [6] is the same: x₁ ≻ x₅ ≻ x₃ ≻ x₂ ≻ x₄. x₁ is the best option for all the methods.

Compared with the approaches proposed in [6 , 43], the idea in this paper differs in several sides:

The proposed quadratic programming models in this paper can be used to obtain expert weights automatically. Although the purpose of Xu [43] constructed quadratic models is to determine the weights of experts, however, our theoretic analysis and their illustrative examples demonstrate that the expert weights derived from his model are always 1/m for each DM (m is the number of DMs in the GDM problem). As for Chen, et al. [6], individual IFPRs are transformed into real number preference relations at first, then derive the experts’ weights based on the consistency degrees of the experts, which leads to information losing.

In this paper, ICI and GCI are defined, which are used to check not only the consistency of individual preferences but also the consensus level, Xu [43] only check the individual consistency. In addition, separate thresholds α₁, λ₁ are set for individual and group consensus indices, this can allow each expert to express his/her judgments slightly different from the group opinion. While Chen, et al. [6] assuming that a consensus is reached if the group consensus degree CD (where $CD = \frac{\sum_{i = 1}^{n} \sum_{j = 1, j \neq i}^{n} \sum_{k = 1}^{m} c_{ij}^{k}}{m \times (n^{2} - n)}$ , $c_{ij}^{k} =$ $1 - \frac{1}{2} | u_{ij}^{*} - p_{ij}^{k} | = 1 - \frac{1}{2} (| u_{ij}^{- *} - p_{ij}^{- k} | +$ $| u_{ij}^{+ *} - p_{ij}^{+ k} |)$ , $C^{k} = {(c_{ij}^{k})}_{n \times n}$ is the consensus relation, $U^{*} = {(u_{ij}^{*})}_{n \times n}$ is the group collective preference relation, $P^{k} = {(p_{ij}^{k})}_{n \times n}$ is the IFPR given by expert e_k, m is the number of expert, n is the number of alternative) is larger than or equal to the predefined threshold value (the overall deviation is smaller than the predefined threshold value), without considering individual deviation, this treatment may lead to undesirable situations. For instance, if deviations of some DMs e_k (k = 1, 2, …, l, l < m) are negligible, but remaining deviations of DMs e_k (k = l + 1, … m) are very high. In this case, as long as the sum of all the deviations is smaller than the predefined threshold value, Chen, et al. [6] still considered the group reaches an acceptable consensus. However, those large deviation variables indicate that some DMs e_l+1, …, e_m still hold preferences far away from the group consensus. Therefore, it is reasonable to impose a threshold for individual deviations as well.

The proposed iterative algorithm can update DMs’ preference values automatically. Furthermore, only one pair of judgments, in each DM’s individual IFPR that deviate the most from the corresponding elements in the collective IFPR are adjusted in the proposed consensus reaching process at each iteration. The purpose is to retain each DM’s original preference information. In Xu and Liu [31], if the group does not reach an acceptable level of consensus, some DMs need to modify some elements in their preferences relations to reach a predefined consensus level, while the iterative algorithm for consensus reaching process only help DMs identify which reference values to be revised without giving a specific revise method. In Xu [43] the preference information are largely modified for each DM, this makes the final modified IFPRs often remarkably differ from the primitive judgments assessed by the DMs.

Example 2. Suppose a core enterprise of the VE has to select a partner for a sub-project. The partner selection decision is made on the basis of five main attributes including Cost (x₁), Time (x₂), Trust (x₃), Risk (x₄) and Quality (x₅). The first problem needing to solve is to rank and select the critical attribute to direct the selection of the partner. So a committee of three DMs e_k (k = 1, 2, 3) (whose weight vector is unknown) has set up to prioritize these attributes x_i (i = 1, 2, 3, 4, 5). The DMs compare each pair of these attributes and provide their interval multiplicative preference relations A₁, A₂, A₃ as follows (Adapted from Xu and Liu [31]): $\begin{matrix} A_{1} = (\begin{matrix} [1, 1] & [6, 7] & [\frac{1}{7}, \frac{1}{5}] & [7, 8] & [\frac{1}{6}, \frac{1}{5}] \\ [\frac{1}{7}, \frac{1}{6}] & [1, 1] & [6, 7] & [6, 7] & [\frac{1}{7}, \frac{1}{5}] \\ [5, 7] & [\frac{1}{7}, \frac{1}{6}] & [1, 1] & [6, 7] & [7, 8] \\ [\frac{1}{8}, \frac{1}{7}] & [\frac{1}{7}, \frac{1}{6}] & [\frac{1}{7}, \frac{1}{6}] & [1, 1] & [6, 7] \\ [5, 6] & [5, 7] & [\frac{1}{8}, \frac{1}{7}] & [\frac{1}{7}, \frac{1}{6}] & [1, 1] \end{matrix}) \\ A_{2} = (\begin{matrix} [1, 1] & [6, 7] & [\frac{1}{2}, 1] & [7, 8] & [\frac{1}{2}, 1] \\ [\frac{1}{7}, \frac{1}{6}] & [1, 1] & [6, 7] & [6, 7] & [\frac{1}{7}, \frac{1}{6}] \\ [1, 2] & [\frac{1}{7}, \frac{1}{6}] & [1, 1] & [5, 7] & [7, 8] \\ [\frac{1}{8}, \frac{1}{7}] & [\frac{1}{7}, \frac{1}{6}] & [\frac{1}{7}, \frac{1}{5}] & [1, 1] & [6, 7] \\ [1, 2] & [6, 7] & [\frac{1}{8}, \frac{1}{7}] & [\frac{1}{7}, \frac{1}{6}] & [1, 1] \end{matrix}) \end{matrix}$

$\begin{matrix} A_{3} = (\begin{matrix} [1, 1] & [6, 7] & [\frac{1}{2}, 1] & [5, 6] & [\frac{1}{8}, \frac{1}{7}] \\ [\frac{1}{7}, \frac{1}{6}] & [1, 1] & [6, 8] & [6, 7] & [\frac{1}{7}, \frac{1}{6}] \\ [1, 2] & [\frac{1}{8}, \frac{1}{6}] & [1, 1] & [7, 8] & [6, 7] \\ [\frac{1}{6}, \frac{1}{5}] & [\frac{1}{7}, \frac{1}{6}] & [\frac{1}{8}, \frac{1}{7}] & [1, 1] & [7, 8] \\ [7, 8] & [6, 7] & [\frac{1}{7}, \frac{1}{6}] & [\frac{1}{8}, \frac{1}{7}] & [1, 1] \end{matrix}) \end{matrix}$

Now, Algorithm 2 is applied to solve the problem. Assume that the maximum number of iteration t^*= 10, the individual consensus degree threshold α₂ = 0.055, and the group consensus degree threshold λ₂ = 0.05. The iterations terminate after 3 steps. Table 2 lists the iteration time t along with the weight vector v^(t), the individual to group consensus degree $ICI (A_{k}^{(t)})$ and the group consensus index GCI (t) at each iteration.

The terminal improved individual IMPRs ${\bar{A}}_{k}$ (k = 1, 2, 3) and group IMPR $\bar{A}$ are

$\begin{matrix} {\bar{A}}_{1} = (\begin{matrix} [1, 1] & [6, 7] & [0.333, 0.590] & [7, 8] & [\frac{1}{6}, \frac{1}{5}] \\ [\frac{1}{7}, \frac{1}{6}] & [1, 1] & [6, 7] & [6, 7] & [\frac{1}{7}, \frac{1}{5}] \\ [1.694, 3.006] & [\frac{1}{7}, \frac{1}{6}] & [1, 1] & [6, 7] & [7, 8] \\ [\frac{1}{8}, \frac{1}{7}] & [\frac{1}{7}, \frac{1}{6}] & [\frac{1}{7}, \frac{1}{6}] & [1, 1] & [6, 7] \\ [5, 6] & [5, 7] & [\frac{1}{8}, \frac{1}{7}] & [\frac{1}{7}, \frac{1}{6}] & [1, 1] \end{matrix}) \\ {\bar{A}}_{2} = (\begin{matrix} [1, 1] & [6, 7] & [\frac{1}{2}, 1] & [7, 8] & [0.219, 0.310] \\ [\frac{1}{7}, \frac{1}{6}] & [1, 1] & [6, 7] & [6, 7] & [\frac{1}{7}, \frac{1}{6}] \\ [1, 2] & [\frac{1}{7}, \frac{1}{6}] & [1, 1] & [5, 7] & [7, 8] \\ [\frac{1}{8}, \frac{1}{7}] & [\frac{1}{7}, \frac{1}{6}] & [\frac{1}{7}, \frac{1}{5}] & [1, 1] & [6, 7] \\ [3.226, 4.567] & [6, 7] & [\frac{1}{8}, \frac{1}{7}] & [\frac{1}{7}, \frac{1}{6}] & [1, 1] \end{matrix}) \\ {\bar{A}}_{3} = (\begin{matrix} [1, 1] & [6, 7] & [\frac{1}{2}, 1] & [5, 6] & [0.219, 0.310] \\ [\frac{1}{7}, \frac{1}{6}] & [1, 1] & [6, 8] & [6, 7] & [\frac{1}{7}, \frac{1}{6}] \\ [1, 2] & [\frac{1}{8}, \frac{1}{6}] & [1, 1] & [7, 8] & [6, 7] \\ [\frac{1}{6}, \frac{1}{5}] & [\frac{1}{7}, \frac{1}{6}] & [\frac{1}{8}, \frac{1}{7}] & [1, 1] & [7, 8] \\ [3.226, 4.567] & [6, 7] & [\frac{1}{7}, \frac{1}{6}] & [\frac{1}{8}, \frac{1}{7}] & [1, 1] \end{matrix}) \end{matrix}$

$\begin{matrix} \bar{A} = (\begin{matrix} [1, 1] & [6, 7] & [0.437, 0.841] & [6.258, 7.269] & [0.200, 0.268] \\ [\frac{1}{7}, \frac{1}{6}] & [1, 1] & [6, 7.318] & [6, 7] & [\frac{1}{7}, 0.177] \\ [1.190, 2.288] & [0.137, \frac{1}{6}] & [1, 1] & [5.939, 7.318] & [6.650, 7.652] \\ [0.138, 0.160] & [\frac{1}{7}, \frac{1}{6}] & [0.137, 0.168] & [1, 1] & [6.316, 7.318] \\ [3.728, 4.997] & [5.650, 7] & [0.131, 0.150] & [0.137, 0.158] & [1, 1] \end{matrix}) \end{matrix}$

Derive the interval weight of $\bar{A}$ using the interval eigenvector method (IEM) in [26] $\begin{matrix} ϖ = ([0.1367, 0.1450], [0.1355, 0.1467], \\ [0.1455, 0.1637], \\ {[0.0824, 0.0874], [0.1259, 0.1412])}^{T} \end{matrix}$

Finally, we get the alternatives’ weight vector as follows: $γ = {(0.2445, 0.2612, 0.3164, 0.04, 0.1378)}^{T}$

The ranking result of five alternatives is x₃ ≻ x₂ ≻ x₁ ≻ x₅ ≻ x₄, which is same as the ranking order given by Xu and Liu [31], while the ranking order in Wu, et al. [27] and Chen, et al. [6] is the same: x₃ ≻ x₁ ≻ x₂ ≻ x₅ ≻ x₄.

Compared with the methods in [6 , 31], the differences of our approach are as follows:

In Xu and Liu [31], the weights of experts are obtained by the relative projections of individual preference relations on the initial collective one, however, as analyzed by Chen, et al. [6], this process has some drawbacks. While Wu,et al. [27] and Chen, et al. [6]’s method needs to transform individual IMPRs into real number preference relations at first in order to determine the experts’ weights, which will lead to information losing. In this paper, we propose a new model, which can be used to determine the optimal weights for experts. Besides, the weights of DMs would change when DMs adjust their preference values in the consensus reaching stage. This can use the DMs’ information sufficiently.

When we have obtained collective IMPR, based on the idea of comparison of interval numbers, we propose a possibility degree formula to compare two interval numbers. The formula is very simple and effective. While Xu and Liu [31]’s method needs to transform the interval collective preference relation into real number preference relations, this may distort the information.

5 Conclusions

In this paper, a consensus model based on distance for group decision making with IFPRs and IMPRs have been proposed. According to the presented quadratic programming models, the expert weights can be determined automatically. We include an individual to group consensus index (ICI) and a group consensus index (GCI,). An ICI evaluates how far an individual judgment is from the collective judgment and is used to determine whether an individual should adjust his/her judgments in the consensus achieving phase. A GCI, measures the group’s overall consensus degree and is employed to judge whether the group should continue to the next consensus improving stage. If the consensus level does not achieve the predefined consensus threshold, two iterative algorithms are devised to attain acceptable consensus level. Comparing with existing consensus models, the consensus models proposed in this paper have the following novels:

By introducing the ICI and GCI, consensus models based on distance for IFPRs and IMPRs have been presented.

From a new perspective, we establish quadratic programming models which can determine the weights of experts automatically. The weights of DMs would change when DMs adjust their preference values in the consensus reaching stage. This makes the whole consensus reaching process more reliable.

Iterative algorithms are raised to reach the group consensus, during each iteration, if the level of consensus are not reach predefined standards, that is to say, an individual’s consensus index or group’s consensus index are larger than predefined thresholds, we only revise one pair of his/her preference values with the maximum deviation from the corresponding group preference values.

In the future research, we intend to extend the proposed approach to support other formats of preference information, e.g. linguistic preference relation [2 , 42], linguistic interval fuzzy preference relations [12], incomplete fuzzy preference relations [39, 41], hesitant preference relations [11, 19], and application in mobile decision support systems [22], and other similarity consensus measures [7].

Footnotes

Acknowledgments

The authors are very grateful to Associate Editor and the anonymous reviewers for their constructive comments and suggestions that have further helped to improve the quality of this paper. This work was partly supported by the National Natural Science Foundation of China (NSFC) under Grants No. 71471056 and 71433003, the Fundamental Research Funds for the Central Universities (No. 2015B23014), Program for Excellent Talents in Hohai University, sponsored by Qing Lan Project of Jiangsu Province.

References

Arbel

, Approximate articulation of preference and priority derivation, European Journal of Operational Research43 (1989), 317–326.

Cabrerizo

F.J.

, Alonso

and Herrera-Viedma

, A consensus model for group decision making problems with unbalanced fuzzy linguistic information, International Journal of Information Technology & Decision Making8 (2009), 109–131.

Cabrerizo

F.J.

, Chiclana

, Al-Hmouz

, Morfeq

, Balamash

A.S.

and Herrera-Viedma

, Fuzzy decision making and consensus: Challenges, Journal of Intelligent & Fuzzy Systems29 (2015), 1109–1118.

Cabrerizo

F.J.

, Moreno

J.M.

, Pérez

I.J.

and Herrera-Viedma

, Analyzing consensus approaches in fuzzy group decision making: Advantages and drawbacks, Soft Computing14 (2010), 451–463.

Cabrerizo

F.J.

, Ureña

, Pedrycz

and Herrera-Viedma

, Building consensus in group decision making with an allocation of information granularity, Fuzzy Sets and Systems255 (2014), 115–127.

Chen

S.M.

, Cheng

S.H.

and Lin

T.E.

, Group decision making systems using group recommendations based on interval fuzzy preference relations and consistency matrices, Information Sciences298 (2015), 555–567.

Chiclana

, García

J.M.T.

, Moral

M.J.D.

and Herrera-Viedma

, A statistical comparative study of different similarity measures of consensus in group decision making, Information Sciences221 (2013), 110–123.

Chiclana

, Herrera-Viedma

, Herrera

and Alonso

, Some induced ordered weighted averaging operators and their use for solving group decision making problems based on fuzzy preference relations, European Journal of Operational Research182 (2007), 383–399.

Chiclana

, Herrera

and Herrera-Viedma

, Integrating multiplicative preference relations in a multipurpose decision-making model based on fuzzy preference relations, Fuzzy Sets and Systems122 (2001), 277–291.

10.

Dong

Y.C.

, Li

C.C.

and Herrera

, An optimization-based approach to adjusting unbalanced linguistic preference relations to obtain a required consistency level, Information Sciences292 (2015), 27–38.

11.

Farhadinia

, Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets, Information Sciences240 (2013), 129–144.

12.

García

J.M.T.

, Moral

M.J.D.

, Martínez

M.A.

and Herrera-Viedma

, A consensus model for group decision making problems with linguistic interval fuzzy preference relations, Expert Systems with Applications39 (2012), 10022–10030.

13.

Herrera-Viedma

, Cabrerizo

F.J.

, Kacprzyk

and Pedrycz

, A review of soft consensus models in a fuzzy environment, Information Fusion17 (2014), 4–13.

14.

Herrera-Viedma

, Herrera

and Chiclana

, A consensus model for multiperson decision making with different preference structures, IEEE Transactions on Systems Man and Cybernetics, Part A: Systems and Humans32 (2002), 394–402.

15.

Herrera-Viedma

, Herrera

, Chiclana

and Luque

, Some issues on consistency of fuzzy preference relations, European Journal of Operational Research154 (2004), 98–109.

16.

Herrera-Viedma

, Martínez

, Mata

and Chiclana

, A consensus support system model for group decision-making problems with multigranular linguistic preference relations, IEEE Transactions on Fuzzy Systems13 (2005), 644–658.

17.

Herrera

, Herrera-Viedma

and Chiclana

, Multiperson decision-making based on multiplicative preference relations, European Journal Operational Research129 (2001), 372–385.

18.

Jiang

Y.L.

, An approach to group decision making based on interval fuzzy preference relations, Journal of Systems Science and Systems Engineering16 (2007), 113–120.

19.

Liao

H.C.

, Xu

Z.S.

and Xia

M.M.

, Multiplicative consistency of hesitant fuzzy preference relation and its application in group decision making, International Journal of Information Technology & Decision Making13 (2014), 47–76.

20.

Orlovsky

S.A.

, Decision-making with a fuzzy preference relation, Fuzzy Sets and Systems1 (1978), 155–167.

21.

Pérez

I.J.

, Cabrerizo

F.J.

, Alonso

and Herrera-Viedma

, A new consensus model for group decision making problems with non-homogeneous experts, IEEE Transactions on Systems, Man, and Cybernetics: Systems44 (2014), 494–498.

22.

Pérez

I.J.

, Cabrerizo

F.J.

and Herrera-Viedma

, A mobile decision support system for dynamic group decision-making problems, IEEE Transactions on Systems Man and Cybernetics - Part A Systems and Humans40 (2010), 1244–1256.

23.

Palomares

, Rodríguez

R.M.

and Martínez

, An attitude-driven web consensus support system for heterogeneous group decision making, Expert Systems with Applications40 (2013), 139–149.

24.

Saaty

T.L.

and Vargas

L.G.

, Uncertainty and rank order in the analytic hierarchy process, European Journal of Operational Research32 (1987), 107–117.

25.

Wang

Y.M.

, Fan

Z.P.

and Hua

Z.S.

, A chi-square method for obtaining a priority vector from multiplicative and fuzzy preference relations, European Journal of Operational Research182 (2007), 356–366.

26.

Wei

Y.Q.

, Liu

J.S.

and Wang

X.Z.

, Concept of consistence and weights of the judgement matrix in the uncertain type of AHP, Systems Engineering-Theory & Practice14 (1994), 16–22.

27.

, Li

J.C.

, Li

and Duan

W.Q.

, The induced continuous ordered weighted geometric operators and their application in group decision making, Computers & Industrial Engineering56 (2009), 1545–1552.

28.

Z.B.

and Xu

J.P.

, A concise consensus support model for group decision making with reciprocal preference relations based on deviation measures, Fuzzy Sets & Systems206 (2012), 58–73.

29.

Z.B.

and Xu

J.P.

, Managing consistency and consensus in group decision making with hesitant fuzzy linguistic prefenence relations, Omega (2015). doi: 10.1016/j.omega

30.

Xia

M.M.

and Xu

Z.S.

, Interval weight generation approaches for reciprocal relations, Applied Mathematical Modelling38 (2014), 828–838.

31.

G.L.

and Liu

, An approach to group decision making based on interval multiplicative and fuzzy preference relations by using projection, Applied Mathematical Modelling37 (2013), 3929–3943.

32.

J.P.

and Wu

Z.B.

, A discrete consensus support model for multiple attribute group decision making, Knowledge-Based Systems24 (2011), 1196–1202.

33.

J.P.

and Wu

Z.B.

, A maximizing consensus approach for alternative selection based on uncertain linguistic preference relations, Computers & Industrial Engineering64 (2013), 999–1008.

34.

Y.J.

and Da

Q.L.

, Standard and mean deviation methods for linguistic group decision making and their applications, Expert Systems with Applications37 (2010), 5905–5912.

35.

Y.J.

, Da

Q.L.

and Liu

L.H.

, Normalizing rank aggregation method for priority of a fuzzy preference relation and its effectiveness, International Journal of Approximate Reasoning50 (2009), 1287–1297.

36.

Y.J.

, Da

Q.L.

and Liu

X.W.

, Some properties of linguistic preference relation and its ranking in group decision making, Systems Engineering and Electronics21 (2010), 244–249.

37.

Y.J.

, Li

K.W.

and Wang

H.M.

, Distance-based consensus models for fuzzy and multiplicative preference relations, Information Sciences253 (2013), 56–73.

38.

Y.J.

, Li

K.W.

and Wang

H.M.

, Incomplete interval fuzzy preference relations and their applications, Computers & Industrial Engineering67 (2014), 93–103.

39.

Y.J.

and Patnayakuni

, Logarithmic least squares method to priority for group decision making with incomplete fuzzy preference relations, Applied Mathematical Modelling37 (2013), 2139–2152.

40.

Y.J.

, Patnayakuni

and Wang

H.M.

, The ordinal consistency of a fuzzy preference relation, Information Sciences224 (2013), 152–164.

41.

Y.J.

and Wang

H.M.

, Eigenvector method, consistency test and inconsistency repairing for an incomplete fuzzy preference relation, Applied Mathematical Modelling37 (2013), 5171–5183.

42.

Y.J.

and Wang

H.M.

, Power geometric operators for group decision making under multiplicative linguistic preference relations, International Journal of Uncertainty Fuzziness and Knowledge-Based Systems20 (2012), 139–159.

43.

Z.S.

, Consistency of interval fuzzy preference relations in group decision making, Applied Soft Computing11 (2011), 3898–3909.

44.

Z.S.

, On compatibility of interval fuzzy preference matrices, Fuzzy Optimization and Decision Making3 (2004), 217–225.

45.

Z.S.

, A practical method for priority of interval number complementary judgment matrix, Operations Research and Management Science10 (2001), 16–19.

46.

Z.S.

and Da

Q.L.

, Research on method for ranking interval numbers, Systems Engineering6 (2001).

47.

Yan

H.B.

and Ma

T.J.

, A group decision-making approach to uncertain quality function deployment based on fuzzy preference relation and fuzzy majority, European Journal of Operational Research241 (2015), 815–829.