A New Approach to Derive Priority Weights from Additive Interval Fuzzy Preference Relations Based on Logarithms

Abstract

This paper investigates the consistency definition and the weight-deriving method for additive interval fuzzy preference relations (IFPRs) using a particular characterization based on logarithms. In a recently published paper, a new approach with a parameter is developed to obtain priority weights from fuzzy preference relations (FPRs), then a new consistency definition for the additive IFPRs is defined, and finally linear programming models for deriving interval weights from consistent and inconsistent IFPRs are proposed. However, the discussion of the parameter value is not adequate and the weights obtained by the linear models for inconsistent IFPRs are dependent on alternative labels and not robust to permutations of the decision makers’ judgments. In this paper, we first investigate the value of the parameter more thoroughly and give the closed form solution for the parameter. Then, we design a numerical example to illustrate the drawback of the linear models. Finally, we construct a linear model to derive interval weights from IFPRs based on the additive transitivity based consistency definition. To demonstrate the effectiveness of our proposed method, we compare our method to the existing method on three numerical examples. The results show that our method performs better on both consistent and inconsistent IFPRs.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Since its introduction, the analytic hierarchy process (AHP) [1] method has been widely used in many applications and intensively studied by lots of researchers [2]. AHP derives priority weights from pairwise reciprocal matrix. With the increase of the complexity of decision-making problems, the classical AHP has been extended in many aspects (scales used to measure the results of pairwise comparisons [3–5], the styles in which the pairwise comparisons carried out [6–8], combinations with other methods [9, 10], uncertainty concerns [11–16], etc.). FPR which introduces fuzzy thoughts and methods into pairwise comparison is an important extension of AHP. The studies of FPR [17–20] mainly focus on the consistency issues and the derivation of priority weights. In FPR, the decision-makers (DMs) assign a real number between 0 and 1 to represent the degree of a preference relation. However, in some circumstances, the DMs would choose an interval number rather than a crisp value to represent preferences due to the uncertainty or lack of information.

IFPRs are FPRs with interval judgments. Many researchers have paid attention to the definition of consistency of IFPRs and the derivation of interval priority weights from IFPRs [21–23]. The consistency issue is the foundation of the study on IFPR. There are two kinds of consistency: additive consistency [24] and multiplicative consistency [14]. In this paper, we focus on the additive consistency, and for multiplicative consistency we refer to [14].

According to the additive consistency of FPR defined by Tanino [18], Xu et al. [25] proposed an additive consistency definition based on the feasible region. An IFPR is consistent if the feasible region is not empty. The interval priority weights are derived by minimizing and maximizing each weight in the feasible region for the consistent IFPRs and by minimizing the sum of deviations for the inconsistent IFPRs. Xu et al. [26] defined an additive transitivity based consistency, but Wang [27] pointed out that this definition was highly dependent on alternative labels and not robust to the permutations of the DMs’ judgments. Wang et al. [28] defined another additive transitivity based consistency and proposed a goal programming model to obtain the interval priority weights. Liu et al. [29] transformed the interval fuzzy preference relation into an interval multiplicative preference relation and used the method in [30] to check the consistency. But Li et al. [13] pointed out that the definition in [30] was technical deficiency and yielded contradictory results for the same judgment matrix after the alternatives were relabeled. Dong et al. [12] defined the average-case consistency index as the average consistency degree of all FPRs associated with the IFPR. Krejčí [24] reviewed the definitions of additive consistency and proposed two new definitions.

In a recent paper, Wang et al. [19] put forward a new method with a parameter using a particular characterization based on logarithms to obtain priority weights from FPR and defined a new consistency definition based on the feasible region restricted by the characterization for additive IFPR. Based on the new definition, linear programming models for deriving interval priority weights from consistent IFPRs were proposed. For inconsistent IFPRs, they proposed to revise the inconsistent IFPR to a new consistent IFPR and then use the linear models designed for consistent IFPRs to derive interval weights. However, the value of the parameter is not fully investigated and the weights obtained by the proposed method for inconsistent IFPRs are dependent on alternative labels and not robust to permutations of the DMs’ judgments. Although there are some drawbacks in [19], the basic idea is very interesting. The purpose of this paper is to illustrate the drawbacks in [19] and to improve these drawbacks. Firstly, we investigate the value of the parameter more thoroughly and we give the closed form solution for the parameter. Secondly, we show that the results obtained by the linear models are not robust to permutations of the DMs’ judgments by a numerical example. Then, we propose a new method based on logarithms and the consistency definition proposed by Wang et al. [28] to derive interval weights from IFPRs. Our proposed method is robust to permutations of the DMs’ judgments. Finally, we compare our method to the method in [19] on three numerical examples with respect to the fitted error. The results demonstrate the effectiveness of our method.

The rest of the paper is organized as follows. Some basic concepts and the main idea of Wang et al.’s method [19] are briefly reviewed in Section 2. The value of the parameter is discussed in Section 3. Section 4 illustrates the rankings derived by the linear models in [19] from inconsistent IFPRs are not robust to permutations of DMs’ judgments. A linear model is proposed to derive interval weights from IFPRs in Section 5. Section 6 shows the effectiveness of the proposed model by numerical examples. The innovations of this paper and the future research directions are concluded in Section 7.

2. Preliminaries

2.1. Basic Concepts

This section reviews some basic concepts related to FPR and IFPR.

Definition 1.

A fuzzy preference relation (FPR) [20] on a set of $n$ objects $x_{\mathrm{1}},x_{\mathrm{2}},\dots ,x_n$ is a complementary matrix $R={\left(r_{ij}\right)}_{n\times n}$, $r_{ij}\in [\mathrm{0,1}]$, and for all $i,j=\mathrm{1,2},\dots ,n$, $r_{ij}+r_{ji}=\mathrm{1}$.

The value of $r_{ij}$ represents the preference degree of $x_i$ over $x_j$. $r_{ij}>\mathrm{0.5}$ means $x_i$ is preferred to $x_j$ and $r_{ij}<\mathrm{0.5}$ indicates $x_j$ is preferred to $x_i$ and $r_{ij}=\mathrm{0.5}$ shows that $x_i$ and $x_j$ are indifferent.

Definition 2.

A FPR $R={\left(r_{ij}\right)}_{n\times n}$ is additive consistent if it satisfies the additive transitivity [20]:$$\begin{array}{}\text{(1)}& r_{ij}=r_{ik}-r_{jk}+\mathrm{0.5},\hspace{1em}\text{for\hspace{0.17em}\hspace{0.17em}all\hspace{0.17em}\hspace{0.17em}}i,j,k=\mathrm{1,2},\dots ,n.\end{array}$$

Xu [20] points out that a FPR is additively consistent if there exists a normalized weight vector $W=\left(w_{\mathrm{1}},w_{\mathrm{2}},\dots ,w_n\right)$, where ${\sum }_{i=\mathrm{1}}^n{w}_i=\mathrm{1}$ such that$$\begin{array}{}\text{(2)}& r_{ij}=\mathrm{0.5}\left(w_i-w_j+\mathrm{1}\right),\hspace{1em}\text{for\hspace{0.17em}\hspace{0.17em}all\hspace{0.17em}\hspace{0.17em}}i,j=\mathrm{1,2},\dots ,n.\end{array}$$

Definition 3.

An interval fuzzy preference relation (IFPR) [20] on a set of $n$ objects $x_{\mathrm{1}},x_{\mathrm{2}},\dots ,x_n$ is defined as $\tilde{R}={\left({\tilde{r}}_{ij}\right)}_{n\times n}$, ${\tilde{r}}_{ij}=\left[r_{ij}^{-},r_{ij}^{+}\right]$, where, for all $i,j=\mathrm{1,2},\dots ,n$, $\mathrm{0}\le r_{ij}^{-}\le r_{ij}^{+}\le \mathrm{1}$, $r_{ij}^{-}+r_{ji}^{+}=\mathrm{1}$, $r_{ii}^{-}=r_{ii}^{+}=\mathrm{0.5}$.

The interval value ${\tilde{r}}_{ij}=\left[r_{ij}^{-},r_{ij}^{+}\right]$ indicates the interval-valued preference degree of $x_i$ over $x_j$.

As the DMs usually prefer interval weights to crisp weights, we give the definition of normalized interval weights here.

Definition 4.

A normalized interval weight vector [31] is defined as $\tilde{W}=\left({\tilde{w}}_{\mathrm{1}},{\tilde{w}}_{\mathrm{2}},\dots ,{\tilde{w}}_n\right)$, ${\tilde{w}}_i=\left[w_i^L,w_i^U\right]$, where for all $i,j=\mathrm{1,2},\dots ,n$, $\mathrm{0}\le w_i^L\le w_i^U\le \mathrm{1}$, ${\sum }_{j=\mathrm{1},j\ne i}^n{w}_j^L+w_i^U\le \mathrm{1}$, ${\sum }_{j=\mathrm{1},j\ne i}^n{w}_j^U+w_i^L\ge \mathrm{1}$.

2.2. Wang et al.’s Method

Wang et al. [19] put forward a necessary and sufficient proposition for additive consistent fuzzy preference relation as defined in Definition 2.

Theorem 5.

$R={\left(r_{ij}\right)}_{n\times n}$ is an additive consistent FPR if and only if there exists a normalized weight vector $W=\left(w_{\mathrm{1}},w_{\mathrm{2}},\dots ,w_n\right)$ and a parameter $\theta >\mathrm{1}$, such that [19]$$\begin{array}{}\text{(3)}& r_{ij}=\mathrm{l}\mathrm{o}{\mathrm{g}}_{\theta }\hspace{0.17em}{w}_i-\mathrm{l}\mathrm{o}{\mathrm{g}}_{\theta }\hspace{0.17em}{w}_j+\mathrm{0.5},\hspace{1em}\text{for\hspace{0.17em}\hspace{0.17em}all\hspace{0.17em}\hspace{0.17em}}i,j=\mathrm{1,2},\dots ,n.\end{array}$$

Wang et al. [19] proved that, given the parameter $\theta$, the weight derived from a FPR can be represented as$$\begin{array}{}\text{(4)}& w_i\left(\theta \right)=\frac{{\theta }^{\left(\mathrm{1}/n\right){\sum }_{k=\mathrm{1}}^n{r}_{ik}}}{{\sum }_{l=\mathrm{1}}^n{\theta }^{\left(\mathrm{1}/n\right){\sum }_{k=\mathrm{1}}^n{r}_{lk}}},\hspace{1em}i=\mathrm{1,2},\dots ,n.\end{array}$$

In order to determine the value of $\theta$, Wang et al. [19] proposed the following constrained programming model in which $t=\ln \theta$.$$\begin{array}{c}\text{(5)}& \mathrm{m}\mathrm{i}\mathrm{n}\hspace{1em}{\displaystyle \sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\displaystyle \sum }_{j=i+\mathrm{1}}^n{\left[\frac{t}{n}\left(\sum _{l=\mathrm{1}}^n{r}_{il}-\sum _{l=\mathrm{1}}^n{r}_{jl}\right)+\mathrm{0.5}-r_{ij}\right]}^{\mathrm{2}}\mathrm{s}.\mathrm{t}.\hspace{1em}\mathrm{0}\le \frac{t}{n}\left({\displaystyle \sum }_{l=\mathrm{1}}^n{r}_{il}-{\displaystyle \sum }_{l=\mathrm{1}}^n{r}_{jl}\right)+\mathrm{0.5}\le \mathrm{1},\hspace{1em}i=\mathrm{1,2},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n,\\ \hspace{1em}t>\mathrm{0}.\end{array}$$

Based on Theorem 5, Wang et al. [19] defined the additive consistency of an IFPR $\tilde{R}={\left({\tilde{r}}_{ij}\right)}_{n\times n}$, ${\tilde{r}}_{ij}=\left[r_{ij}^{-},r_{ij}^{+}\right]$ using the feasible region $S_W$.$$\begin{array}{}\text{(6)}& S_W=\left\{\left(w_{\mathrm{1}},w_{\mathrm{2}},\dots ,w_n\right)\mid \begin{array}{c}{r}_{ij}^{-}\le \ln w_i-\ln w_j+\mathrm{0.5}\le r_{ij}^{+}\hspace{0.17em}\hspace{0.17em}i=\mathrm{1},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n\\ {\displaystyle \sum }_{i=\mathrm{1}}^n{w}_i=\mathrm{1}\hspace{0.17em}\hspace{0.17em}{w}_i\ge \mathrm{0},\hspace{0.17em}\hspace{0.17em}i=\mathrm{1,2},\dots ,n\end{array}\right\}.\end{array}$$

If $S_W$ is not empty, $\tilde{R}$ is additive consistent; otherwise, $\tilde{R}$ is inconsistent.

Wang et al. [19] proposed the following linear programming models to derive interval weights $\tilde{W}=\left({\tilde{w}}_{\mathrm{1}},{\tilde{w}}_{\mathrm{2}},\dots ,{\tilde{w}}_n\right)$, ${\tilde{w}}_i=\left[w_i^L,w_i^U\right]$ from additive consistent IFPRs.$$\begin{array}{c}\text{(7)}& \hspace{1em}{w}_i^L=\min w_{i}s.t.\hspace{1em}\ln w_i-\ln w_j+\mathrm{0.5}\ge r_{ij}^{-},\hspace{1em}i=\mathrm{1},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n\\ \hspace{1em}\ln w_i-\ln w_j+\mathrm{0.5}\le r_{ij}^{+},\hspace{1em}i=\mathrm{1},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n\\ \hspace{1em}{{\displaystyle \sum }}_{i=\mathrm{1}}^n{w}_i=\mathrm{1}\\ \hspace{1em}{w}_i\ge \mathrm{0},\hspace{1em}i=\mathrm{1,2},\dots ,n.\\ \text{(8)}& \hspace{1em}{w}_i^U=\max w_{i}s.t.\hspace{1em}\ln w_i-\ln w_j+\mathrm{0.5}\ge r_{ij}^{-},\hspace{1em}i=\mathrm{1},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n\\ \hspace{1em}\ln w_i-\ln w_j+\mathrm{0.5}\le r_{ij}^{+},\hspace{1em}i=\mathrm{1},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n\\ \hspace{1em}{{\displaystyle \sum }}_{i=\mathrm{1}}^n{w}_i=\mathrm{1}\\ \hspace{1em}{w}_i\ge \mathrm{0},\hspace{1em}i=\mathrm{1,2},\dots ,n.\end{array}$$

For an inconsistent IFPR $\tilde{R}$, Wang et al. [19] proposed to revise $\tilde{R}$ to acquire a consistent IFPR ${\tilde{R}}^{\ast }$ and derive interval weights from ${\tilde{R}}^{\ast }$.

To obtain the consistent IFPR ${\tilde{R}}^{\ast }$, Wang et al. [19] first divided $\tilde{R}$ into two crisp FPRs $R_{(\mathrm{1})}$ and $R_{(\mathrm{2})}$:$$\begin{array}{c}\text{(9)}& R_{\left(\mathrm{1}\right)}={\left(r_{ij}^{\left(\mathrm{1}\right)}\right)}_{n\times n}=\left(\begin{array}{cccc}\mathrm{0.5}& r_{\mathrm{12}}^{+}& \cdots & r_{\mathrm{1}n}^{+}\\ r_{\mathrm{21}}^{-}& \mathrm{0.5}& \cdots & r_{\mathrm{2}n}^{+}\\ ⋮& ⋮& \ddots & ⋮\\ r_{n\mathrm{1}}^{-}& r_{n\mathrm{2}}^{-}& \cdots & \mathrm{0.5}\end{array}\right),R_{\left(\mathrm{2}\right)}={\left(r_{ij}^{\left(\mathrm{2}\right)}\right)}_{n\times n}=\left(\begin{array}{cccc}\mathrm{0.5}& r_{\mathrm{12}}^{-}& \cdots & r_{\mathrm{1}n}^{-}\\ r_{\mathrm{21}}^{+}& \mathrm{0.5}& \cdots & r_{\mathrm{2}n}^{-}\\ ⋮& ⋮& \ddots & ⋮\\ r_{n\mathrm{1}}^{+}& r_{n\mathrm{2}}^{+}& \cdots & \mathrm{0.5}\end{array}\right).\end{array}$$Then two consistent FPRs $R_{(\mathrm{1})}^{\ast }={\left(r_{ij}^{\ast (\mathrm{1})}\right)}_{n\times n}$ and $R_{(\mathrm{2})}^{\ast }={\left(r_{ij}^{\ast (\mathrm{2})}\right)}_{n\times n}$ were determined as$$\begin{array}{}\text{(10)}& r_{ij}^{\ast \left(\mathrm{1}\right)}=\frac{t_{\mathrm{1}}}{n}\left({\displaystyle \sum }_{l=\mathrm{1}}^n{r}_{il}^{\left(\mathrm{1}\right)}-{\displaystyle \sum }_{l=\mathrm{1}}^n{r}_{jl}^{\left(\mathrm{1}\right)}\right)+\mathrm{0.5},\hspace{1em}\text{for\hspace{0.17em}\hspace{0.17em}all\hspace{0.17em}\hspace{0.17em}}i,j=\mathrm{1,2},\dots ,n.\text{(11)}& r_{ij}^{\ast \left(\mathrm{2}\right)}=\frac{t_{\mathrm{2}}}{n}\left({\displaystyle \sum }_{l=\mathrm{1}}^n{r}_{il}^{\left(\mathrm{2}\right)}-{\displaystyle \sum }_{l=\mathrm{1}}^n{r}_{jl}^{\left(\mathrm{2}\right)}\right)+\mathrm{0.5},\hspace{1em}\text{for\hspace{0.17em}\hspace{0.17em}all\hspace{0.17em}\hspace{0.17em}}i,j=\mathrm{1,2},\dots ,n.\end{array}$$where $t_{\mathrm{1}}$ and $t_{\mathrm{2}}$ can be obtained using model (5). Combining $R_{(\mathrm{1})}^{\ast }$ and $R_{(\mathrm{2})}^{\ast }$, the IFPR ${\tilde{R}}^{\ast }={\left({\tilde{r}}_{ij}^{\ast }\right)}_{n\times n}$, ${\tilde{r}}_{ij}^{\ast }=\left[r_{ij}^{\ast -},r_{ij}^{\ast +}\right]$, where $r_{ij}^{\ast -}=\mathrm{m}\mathrm{i}\mathrm{n}\left\{r_{ij}^{\ast (\mathrm{1})},r_{ij}^{\ast (\mathrm{2})}\right\}$, $r_{ij}^{\ast +}=\mathrm{m}\mathrm{a}\mathrm{x}\left\{r_{ij}^{\ast (\mathrm{1})},r_{ij}^{\ast (\mathrm{2})}\right\}$, can be obtained. It is obvious that ${\tilde{R}}^{\ast }={\left({\tilde{r}}_{ij}^{\ast }\right)}_{n\times n}$ is an additive consistent IFPR.

After obtaining ${\tilde{R}}^{\ast }$, the interval weights can be calculated using models (7) and (8).

3. The Value of the Parameter Used in [19]

Wang et al. [19] ended the discussion of the value of $\theta$ with the programming model (5). In this section, we continue the investigation and discuss the value of $\theta$ more thoroughly.

Let $r_i={\sum }_{k=\mathrm{1}}^n{r}_{ik}$, $q_{ij}=r_{ij}-\mathrm{0.5}$ for all $i,j=\mathrm{1,2},\dots ,n$; then the objective function of model (5) can be written as$$\begin{array}{}\text{(12)}& \frac{\mathrm{1}}{n^{\mathrm{2}}}{\displaystyle \sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\displaystyle \sum }_{j=i+\mathrm{1}}^n{\left(r_i-r_j\right)}^{\mathrm{2}}{t}^{\mathrm{2}}-\frac{\mathrm{2}}{n}{\displaystyle \sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\displaystyle \sum }_{j=i+\mathrm{1}}^n\left(r_i-r_j\right)q_{ij}t+{\displaystyle \sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\displaystyle \sum }_{j=i+\mathrm{1}}^n{\left(q_{ij}\right)}^{\mathrm{2}},\end{array}$$which is a quadratic function, and the first constraint can be written as $-\mathrm{0.5}n\le t\left(r_i-r_j\right)\le \mathrm{0.5}n$.

We first discuss the constraints. The value of $r_i-r_j$ can be either positive or negative or zero. When $r_i-r_j=\mathrm{0}$, the first constraint is always satisfied and we can get $t>\mathrm{0}$; when $r_i-r_j>\mathrm{0}$, the first ‘≤’ in the first constraint is always satisfied and we can get $\mathrm{0}<t\le \mathrm{0.5}n/{\max }_{i,j=\mathrm{1},\dots ,n}\left(r_i-r_j\right)$; when $r_i-r_j<\mathrm{0}$, the second ‘≤’ in the first constraint is always satisfied and we can get $\mathrm{0}<t\le -\mathrm{0.5}n/{\min }_{i,j=\mathrm{1},\dots ,n}\left(r_i-r_j\right)$. As $r_i$ is the sum of the elements in the ith row in the FPR, it is obvious that ${\max }_{i,j=\mathrm{1},\dots ,n}\left(r_i-r_j\right)=-{\min }_{i,j=\mathrm{1},\dots ,n}\left(r_i-r_j\right)$. From the discussion above, we can draw the conclusion that the constraints of problem (5) can be written as $\mathrm{0}<t\le \mathrm{0.5}n/{\max }_{i,j=\mathrm{1},\dots ,n}\left(r_i-r_j\right)$.

As the objective function of problem (5) is a quadratic function whose quadratic coefficient is nonnegative, the minimum will be obtained at $t=n{\sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\sum }_{j=i+\mathrm{1}}^n\left(r_i-r_j\right)q_{ij}/{\sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\sum }_{j=i+\mathrm{1}}^n{\left(r_i-r_j\right)}^{\mathrm{2}}$ without considering the constraints.

Theorem 6.

$t=n{\sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\sum }_{j=i+\mathrm{1}}^n\left(r_i-r_j\right)q_{ij}/{\sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\sum }_{j=i+\mathrm{1}}^n{\left(r_i-r_j\right)}^{\mathrm{2}}=\mathrm{1}$ .

Proof.

Since $q_{ij}=r_{ij}-\mathrm{0.5}$, then $q_{ij}+q_{ji}=r_{ij}-\mathrm{0.5}+r_{ji}-\mathrm{0.5}=\mathrm{0}$, and ${\sum }_{j=\mathrm{1}}^n{q}_{ij}=r_i-n/\mathrm{2}$, for all $i,j=\mathrm{1,2},\dots ,n$. From the definition of FPR, we can get ${\sum }_{i=\mathrm{1}}^n{r}_i=n^{\mathrm{2}}/\mathrm{2}$. According to these properties, we can make the following deductions.$$\begin{array}{c}\text{(13)}& n{\displaystyle \sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\displaystyle \sum }_{j=i+\mathrm{1}}^n\left(r_i-r_j\right)q_{ij}=n\left({\displaystyle \sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\displaystyle \sum }_{j=i+\mathrm{1}}^n{r}_i{q}_{ij}-{\displaystyle \sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\displaystyle \sum }_{j=i+\mathrm{1}}^n{r}_j{q}_{ij}\right)=n\left({\displaystyle \sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{r}_i\left({\displaystyle \sum }_{j=\mathrm{1}}^{n-\mathrm{1}}{q}_{ij}\right)-{\displaystyle \sum }_{i=\mathrm{2}}^n{r}_i\left({\displaystyle \sum }_{j=\mathrm{1}}^{i-\mathrm{1}}{q}_{ji}\right)\right)=n{\displaystyle \sum }_{i=\mathrm{1}}^n{r}_i\left({\displaystyle \sum }_{j=\mathrm{1}}^n{q}_{ij}\right)=n{\displaystyle \sum }_{i=\mathrm{1}}^n{r}_i\left(r_i-\frac{n}{\mathrm{2}}\right)=n{\displaystyle \sum }_{i=\mathrm{1}}^n{r_i}^{\mathrm{2}}-\frac{n^{\mathrm{4}}}{\mathrm{4}},{\displaystyle \sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\displaystyle \sum }_{j=i+\mathrm{1}}^n{\left(r_i-r_j\right)}^{\mathrm{2}}={\displaystyle \sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\displaystyle \sum }_{j=i+\mathrm{1}}^n\left(r_i^{\mathrm{2}}+r_j^{\mathrm{2}}-\mathrm{2}{r}_i{r}_j\right)={\displaystyle \sum }_{i=\mathrm{1}}^{n-\mathrm{1}}\left(n-i\right)r_i^{\mathrm{2}}+{\displaystyle \sum }_{i=\mathrm{2}}^n\left(i-\mathrm{1}\right)r_i^{\mathrm{2}}-\mathrm{2}{\displaystyle \sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\displaystyle \sum }_{j=i+\mathrm{1}}^n{r}_i{r}_j=\left(n-\mathrm{1}\right){\displaystyle \sum }_{i=\mathrm{1}}^n{r}_i^{\mathrm{2}}-\mathrm{2}{\displaystyle \sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\displaystyle \sum }_{j=i+\mathrm{1}}^n{r}_i{r}_j,\\ \mathrm{2}{\displaystyle \sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\displaystyle \sum }_{j=i+\mathrm{1}}^n{r}_i{r}_j=\mathrm{2}{\displaystyle \sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{r}_i\left(\frac{n^{\mathrm{2}}}{\mathrm{2}}-{\displaystyle \sum }_{j=\mathrm{1}}^i{r}_i\right)=\mathrm{2}{\displaystyle \sum }_{i=\mathrm{2}}^n{r}_i\left(\frac{n^{\mathrm{2}}}{\mathrm{2}}-{\displaystyle \sum }_{j=i}^n{r}_i\right)={\displaystyle \sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{r}_i\left(\frac{n^{\mathrm{2}}}{\mathrm{2}}-{\displaystyle \sum }_{j=\mathrm{1}}^i{r}_i\right)+{\displaystyle \sum }_{i=\mathrm{2}}^n{r}_i\left(\frac{n^{\mathrm{2}}}{\mathrm{2}}-{\displaystyle \sum }_{j=i}^n{r}_i\right)={\displaystyle \sum }_{i=\mathrm{1}}^n{r}_i\left(\frac{n^{\mathrm{2}}}{\mathrm{2}}-r_i\right)=\frac{n^{\mathrm{4}}}{\mathrm{4}}-{\displaystyle \sum }_{i=\mathrm{1}}^n{r}_i^{\mathrm{2}}.\end{array}$$

So, we can get ${\sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\sum }_{j=i+\mathrm{1}}^n{\left(r_i-r_j\right)}^{\mathrm{2}}=n{\sum }_{i=\mathrm{1}}^n{r_i}^{\mathrm{2}}-n^{\mathrm{4}}/\mathrm{4}=n{\sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\sum }_{j=i+\mathrm{1}}^n\left(r_i-r_j\right)q_{ij}$; then $t=\mathrm{1}$. This completes the proof.

Combining Theorem 6 and the constraints, we can get $t=\mathrm{m}\mathrm{i}\mathrm{n}\left\{\mathrm{1},\mathrm{0.5}n/{\max }_{i,j=\mathrm{1,2},\dots ,n}\left(r_i-r_j\right)\right\}$, and $\theta =\mathrm{m}\mathrm{i}\mathrm{n}\left\{e,e^{\mathrm{0.5}n/{\max }_{i,j=\mathrm{1,2},\dots ,n}\left(r_i-r_j\right)}\right\}$.

With this conclusion, it will be more convenient and efficient to derive weights or to revise the inconsistent FPRs by the methods in [19].

It is worth noting that the situation in which $\mathrm{0.5}n/{\max }_{i,j=\mathrm{1,2},\dots ,n}\left(r_i-r_j\right)\le \mathrm{1}$ is satisfied is very extreme and unusual. If this situation do appear, the DMs will be able to determine that the ith object is the best or the jth object is the worst intuitively and they could remove the ith or the jth object and then reconstruct the matrix to get a more usual situation in which $\mathrm{0.5}n/{\max }_{i,j=\mathrm{1,2},\dots ,n}\left(r_i-r_j\right)\ge \mathrm{1}$. In the following discussion, we will take $t=\mathrm{1}$ and $\theta =e$ acquiescently.

4. The Invalidity of the Weight-Deriving Method in [19] for Inconsistent IFPRs

This section develops a numerical example to illustrate the technical deficiency of the weight-deriving method in [19] for inconsistent IFPRs.

It is notable that Wang et al. [19] took $\theta =e$ when they defined the feasible region $S_W$ in (6), but they did not give the reason. From the discussion in Section 3 we can see that it is reasonable to set $\theta =e$.

Example 1.

Suppose an IFPR on $X=\left\{x_{\mathrm{1}},x_{\mathrm{2}},x_{\mathrm{3}},x_{\mathrm{4}}\right\}$ as follows:$$\begin{array}{}\text{(14)}& \tilde{R}=\left(\begin{array}{cccc}\left[\mathrm{0.50,0.50}\right]& \left[\mathrm{0.84,0.92}\right]& \left[\mathrm{0.43,0.80}\right]& \left[\mathrm{0.29,0.47}\right]\\ \left[\mathrm{0.08,0.16}\right]& \left[\mathrm{0.50,0.50}\right]& \left[\mathrm{0.54,0.56}\right]& \left[\mathrm{0.27,0.98}\right]\\ \left[\mathrm{0.20,0.57}\right]& \left[\mathrm{0.44,0.46}\right]& \left[\mathrm{0.50,0.50}\right]& \left[\mathrm{0.72,0.75}\right]\\ \left[\mathrm{0.53,0.71}\right]& \left[\mathrm{0.02,0.73}\right]& \left[\mathrm{0.25,0.28}\right]& \left[\mathrm{0.50,0.50}\right]\end{array}\right).\end{array}$$

It can be verified that $\tilde{R}$ is an inconsistent IFPR according to the definition 3 in [19] on page 73, so $\tilde{R}$ has to be revised first.

By dividing $\tilde{R}$ according to (9), we can get$$\begin{array}{c}\text{(15)}& R_{\left(\mathrm{1}\right)}=\left(\begin{array}{cccc}\mathrm{0.50}& \mathrm{0.92}& \mathrm{0.80}& \mathrm{0.47}\\ \mathrm{0.08}& \mathrm{0.50}& \mathrm{0.56}& \mathrm{0.98}\\ \mathrm{0.20}& \mathrm{0.44}& \mathrm{0.50}& \mathrm{0.75}\\ \mathrm{0.53}& \mathrm{0.02}& \mathrm{0.25}& \mathrm{0.50}\end{array}\right),R_{\left(\mathrm{2}\right)}=\left(\begin{array}{cccc}\mathrm{0.50}& \mathrm{0.84}& \mathrm{0.43}& \mathrm{0.29}\\ \mathrm{0.16}& \mathrm{0.50}& \mathrm{0.54}& \mathrm{0.27}\\ \mathrm{0.57}& \mathrm{0.46}& \mathrm{0.50}& \mathrm{0.72}\\ \mathrm{0.71}& \mathrm{0.73}& \mathrm{0.28}& \mathrm{0.50}\end{array}\right).\end{array}$$

From Theorem 6, we can get $t_{\mathrm{1}}=\mathrm{1}$ and $t_{\mathrm{2}}=\mathrm{1}$. Then, we can get the following two additive consistent FPRs according to (10) and (11):$$\begin{array}{c}\text{(16)}& R_{\left(\mathrm{1}\right)}^{\ast }=\left(\begin{array}{cccc}\mathrm{0.50}& \mathrm{0.64}& \mathrm{0.70}& \mathrm{0.85}\\ \mathrm{0.36}& \mathrm{0.50}& \mathrm{0.56}& \mathrm{0.70}\\ \mathrm{0.30}& \mathrm{0.44}& \mathrm{0.50}& \mathrm{0.65}\\ \mathrm{0.15}& \mathrm{0.30}& \mathrm{0.35}& \mathrm{0.50}\end{array}\right),R_{\left(\mathrm{2}\right)}^{\ast }=\left(\begin{array}{cccc}\mathrm{0.50}& \mathrm{0.65}& \mathrm{0.45}& \mathrm{0.46}\\ \mathrm{0.35}& \mathrm{0.50}& \mathrm{0.30}& \mathrm{0.31}\\ \mathrm{0.55}& \mathrm{0.70}& \mathrm{0.50}& \mathrm{0.51}\\ \mathrm{0.54}& \mathrm{0.69}& \mathrm{0.49}& \mathrm{0.50}\end{array}\right).\end{array}$$

Combining $R_{(\mathrm{1})}^{\ast }$ and $R_{(\mathrm{2})}^{\ast }$ we can get the following IFPR:$$\begin{array}{}\text{(17)}& {\tilde{R}}^{\ast }=\left(\begin{array}{cccc}\left[\mathrm{0.50,0.50}\right]& \left[\mathrm{0.64,0.65}\right]& \left[\mathrm{0.45,0.70}\right]& \left[\mathrm{0.46,0.85}\right]\\ \left[\mathrm{0.35,0.36}\right]& \left[\mathrm{0.50,0.50}\right]& \left[\mathrm{0.30,0.56}\right]& \left[\mathrm{0.31,0.70}\right]\\ \left[\mathrm{0.30,0.55}\right]& \left[\mathrm{0.44,0.70}\right]& \left[\mathrm{0.50,0.50}\right]& \left[\mathrm{0.51,0.65}\right]\\ \left[\mathrm{0.15,0.54}\right]& \left[\mathrm{0.30,0.69}\right]& \left[\mathrm{0.35,0.49}\right]& \left[\mathrm{0.50,0.50}\right]\end{array}\right).\end{array}$$

${\tilde{R}}^{\ast }$ is an additive consistent IFPR, and the interval priority weights $\tilde{W}=\left({\tilde{w}}_{\mathrm{1}},{\tilde{w}}_{\mathrm{2}},\dots ,{\tilde{w}}_n\right)$ derived from ${\tilde{R}}^{\ast }$ according to the linear models (7) and (8) are ${\tilde{w}}_{\mathrm{1}}=\left[\mathrm{0.2527,0.2952}\right]$, ${\tilde{w}}_{\mathrm{2}}=\left[\mathrm{0.2183,0.2556}\right]$, ${\tilde{w}}_{\mathrm{3}}=\left[\mathrm{0.2340,0.2748}\right]$, ${\tilde{w}}_{\mathrm{4}}=\left[\mathrm{0.2082,0.2633}\right]$.

We adopt the method proposed by Xu et al. [32] to compare the interval weights. Let $\tilde{a}=\left[a^{-},a^{+}\right]$ and $\tilde{b}=\left[b^{-},b^{+}\right]$ be any interval numbers; then the degree of possibility of $\tilde{a}\ge \tilde{b}$ is defined as$$\begin{array}{}\text{(18)}& p\left(\tilde{a}\ge \tilde{b}\right)=\mathrm{m}\mathrm{a}\mathrm{x}\left\{\mathrm{1}-\mathrm{m}\mathrm{a}\mathrm{x}\left\{\frac{b^{+}-a^{-}}{a^{+}-a^{-}+b^{+}-b^{-}},\mathrm{0}\right\},\mathrm{0}\right\}.\end{array}$$

As per (18), the preference relation matrix $P={\left(p_{ij}\right)}_{n\times n}$, $p_{ij}=p\left({\tilde{w}}_i\ge {\tilde{w}}_j\right)$ can be constructed and the preference degree ${\phi }_i={\sum }_{j=\mathrm{1}}^n{p}_{ij}$ can be used to rank the weights.

In this example, $P=\left(\begin{array}{cccc}\mathrm{0.500}& \mathrm{0.964}& \mathrm{0.735}& \mathrm{0.891}\\ \mathrm{0.036}& \mathrm{0.500}& \mathrm{0.277}& \mathrm{0.513}\\ \mathrm{0.265}& \mathrm{0.723}& \mathrm{0.500}& \mathrm{0.694}\\ \mathrm{0.109}& \mathrm{0.487}& \mathrm{0.306}& \mathrm{0.500}\end{array}\right)$, ${\phi }_{\mathrm{1}}=\mathrm{3.09}$, ${\phi }_{\mathrm{2}}=\mathrm{1.33}$, ${\phi }_{\mathrm{3}}=\mathrm{2.18}$, ${\phi }_{\mathrm{4}}=\mathrm{1.40}$, so we can get ${\tilde{w}}_{\mathrm{1}}\succ {\tilde{w}}_{\mathrm{3}}\succ {\tilde{w}}_{\mathrm{4}}\succ {\tilde{w}}_{\mathrm{2}}$.

With the same judgment information, we relabel the alternatives as $X=\left\{x_{\mathrm{3}},x_{\mathrm{2}},x_{\mathrm{1}},x_{\mathrm{4}}\right\}$ and the original IFPR $\tilde{R}$ can be rewritten as$$\begin{array}{}\text{(19)}& {\tilde{R}}^{\prime }=\left(\begin{array}{cccc}\left[\mathrm{0.50,0.50}\right]& \left[\mathrm{0.44,0.46}\right]& \left[\mathrm{0.20,0.57}\right]& \left[\mathrm{0.72,0.75}\right]\\ \left[\mathrm{0.54,0.56}\right]& \left[\mathrm{0.50,0.50}\right]& \left[\mathrm{0.08,0.16}\right]& \left[\mathrm{0.27,0.98}\right]\\ \left[\mathrm{0.43,0.80}\right]& \left[\mathrm{0.84,0.92}\right]& \left[\mathrm{0.50,0.50}\right]& \left[\mathrm{0.29,0.47}\right]\\ \left[\mathrm{0.25,0.28}\right]& \left[\mathrm{0.02,0.73}\right]& \left[\mathrm{0.53,0.71}\right]& \left[\mathrm{0.50,0.50}\right]\end{array}\right).\end{array}$$

It can be verified that ${\tilde{R}}^{\prime }$ is also an inconsistent IFPR. Adopting the same method to revise ${\tilde{R}}^{\prime }$, we can get$$\begin{array}{}\text{(20)}& {\tilde{R}}^{\prime \ast }=\left(\begin{array}{cccc}\left[\mathrm{0.50,0.50}\right]& \left[\mathrm{0.40,0.74}\right]& \left[\mathrm{0.28,0.57}\right]& \left[\mathrm{0.39,0.77}\right]\\ \left[\mathrm{0.26,0.60}\right]& \left[\mathrm{0.50,0.50}\right]& \left[\mathrm{0.18,0.52}\right]& \left[\mathrm{0.30,0.72}\right]\\ \left[\mathrm{0.43,0.72}\right]& \left[\mathrm{0.48,0.82}\right]& \left[\mathrm{0.50,0.50}\right]& \left[\mathrm{0.47,0.84}\right]\\ \left[\mathrm{0.23,0.61}\right]& \left[\mathrm{0.28,0.70}\right]& \left[\mathrm{0.16,0.53}\right]& \left[\mathrm{0.50,0.50}\right]\end{array}\right).\end{array}$$

From ${\tilde{R}}^{\prime \ast }$, we can get the interval weights ${\tilde{w}}_{\mathrm{1}}^{\prime }=\left[\mathrm{0.2355,0.2726}\right]$, ${\tilde{w}}_{\mathrm{2}}^{\prime }=\left[\mathrm{0.2146,0.2602}\right]$, ${\tilde{w}}_{\mathrm{3}}^{\prime }=\left[\mathrm{0.2548,0.2941}\right]$, ${\tilde{w}}_{\mathrm{4}}^{\prime }=\left[\mathrm{0.2088,0.2628}\right]$, and the preference relation matrix $P^{\prime }=\left(\begin{array}{cccc}\mathrm{0.500}& \mathrm{0.702}& \mathrm{0.235}& \mathrm{0.701}\\ \mathrm{0.298}& \mathrm{0.500}& \mathrm{0.064}& \mathrm{0.516}\\ \mathrm{0.765}& \mathrm{0.936}& \mathrm{0.500}& \mathrm{0.914}\\ \mathrm{0.299}& \mathrm{0.484}& \mathrm{0.086}& \mathrm{0.500}\end{array}\right)$ and the preference degrees ${\phi }_{\mathrm{1}}^{\prime }=\mathrm{2.14}$, ${\phi }_{\mathrm{2}}^{\prime }=\mathrm{1.38}$, ${\phi }_{\mathrm{3}}^{\prime }=\mathrm{3.12}$, ${\phi }_{\mathrm{4}}^{\prime }=\mathrm{1.37}$.

So, the ranking of weights obtained from ${\tilde{R}}^{\prime \ast }$ is ${\tilde{w}}_{\mathrm{3}}^{\prime }\succ {\tilde{w}}_{\mathrm{1}}^{\prime }\succ {\tilde{w}}_{\mathrm{2}}^{\prime }\succ {\tilde{w}}_{\mathrm{4}}^{\prime }$. Comparing the two rankings, we can find that though the judgment information is the same, the ranks of $x_{\mathrm{2}}$ and $x_{\mathrm{4}}$ are contradictory in the two rankings.

Example 1 demonstrates that the rankings derived by the linear models in [19] from inconsistent IFPRs are not robust to permutations of DMs’ judgments and rank reversal problem may arise when the alternatives are relabeled.

5. New Models to Derive Interval Weights from IFPR

In this section, we devise a formula to transform normalized interval weights into an additive consistent IFPR according to an additive transitivity based consistency definition proposed in Wang et al. [28] and develop a linear models to derive interval weights.

5.1. Consistency Issues

Definition 7.

An IFPR $\tilde{R}={\left({\tilde{r}}_{ij}\right)}_{n\times n}$, ${\tilde{r}}_{ij}=\left[r_{ij}^{-},r_{ij}^{+}\right]$ is additive consistent if it satisfies the following additive transitivity [28]:$$\begin{array}{}\text{(21)}& {\tilde{r}}_{ij}+{\tilde{r}}_{jk}+{\tilde{r}}_{ki}={\tilde{r}}_{kj}+{\tilde{r}}_{ji}+{\tilde{r}}_{ik},\hspace{1em}\text{for\hspace{0.17em}\hspace{0.17em}all\hspace{0.17em}\hspace{0.17em}}i,j,k=\mathrm{1,2},\dots ,n.\end{array}$$

Given a normalized interval weight vector $\tilde{W}=\left({\tilde{w}}_{\mathrm{1}},{\tilde{w}}_{\mathrm{2}},\dots ,{\tilde{w}}_n\right)$, ${\tilde{w}}_i=\left[w_i^L,w_i^U\right]$, we define an interval matrix $\tilde{P}={\left({\tilde{p}}_{ij}\right)}_{n\times n}$ as$$\begin{array}{}\text{(22)}& {\tilde{p}}_{ij}=\left[p_{ij}^L,p_{ij}^U\right]=\left\{\begin{array}{cc}\left[\mathrm{0.5,0.5}\right],\hfill & i=j,\hfill \\ \left[\ln w_i^L-\ln w_j^U+\mathrm{0.5},\ln w_i^U-\ln w_j^L+\mathrm{0.5}\right],\hfill & i\ne j.\hfill \end{array}\right.\text{(23)}& \mathrm{0}\le \ln w_i^L-\ln w_j^U+\mathrm{0.5}\le \ln w_i^U-\ln w_j^L+\mathrm{0.5}\le \mathrm{1},\hspace{1em}i\ne j.\end{array}$$

Theorem 8.

$\tilde{P}={\left({\tilde{p}}_{ij}\right)}_{n\times n}$ is an additive consistent IFPR according to Definition 7.

Proof.

From the definition of ${\tilde{p}}_{ij}$ and normalized interval weight, we can draw $\mathrm{0}\le p_{ij}^L\le p_{ij}^U\le \mathrm{1}$ and $p_{ii}^L=p_{ii}^U=\mathrm{0.5}$ for all $i,j=\mathrm{1,2},\dots ,n$ easily. Moreover, for all $i,j=\mathrm{1,2},\dots ,n$, $p_{ij}^L+p_{ji}^U=\left(\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^L-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^U+\mathrm{0.5}\right)+\left(\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^L+\mathrm{0.5}\right)=\mathrm{1}$. Then we can say that $\tilde{P}$ is an IFPR according to Definition 3.

On the other hand, as per interval arithmetic, it follows from (22) that$$\begin{array}{}\text{(24)}& {\tilde{p}}_{ij}+{\tilde{p}}_{jk}+{\tilde{p}}_{ki}=\left[\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^L-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^U+\mathrm{0.5},\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^L+\mathrm{0.5}\right]+\left[\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^L-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_k^U+\mathrm{0.5},\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_k^L+\mathrm{0.5}\right]+\left[\ln w_k^L-\ln w_i^U+\mathrm{0.5},\ln w_k^U-\ln w_i^L+\mathrm{0.5}\right]=\left[\mathrm{1.5}+\ln \frac{w_i^L{w}_j^L{w}_k^L}{w_i^U{w}_j^U{w}_k^U},\mathrm{1.5}+\ln \frac{w_i^U{w}_j^U{w}_k^U}{w_i^L{w}_j^L{w}_k^L}\right]=\left[\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^L-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_k^U+\mathrm{0.5},\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_k^L+\mathrm{0.5}\right]+\left[\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_k^L-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^U+\mathrm{0.5},\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_k^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^L+\mathrm{0.5}\right]+\left[\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^L-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^U+\mathrm{0.5},\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^L+\mathrm{0.5}\right]={\tilde{p}}_{ik}+{\tilde{p}}_{kj}+{\tilde{p}}_{ji},\hspace{1em}\forall i,j,k=\mathrm{1,2},\dots ,n\end{array}$$

As per Definition 7, $\tilde{P}$ is an additive consistent IFPR. This completes the proof.

Corollary 9.

$\tilde{R}={\left({\tilde{r}}_{ij}\right)}_{n\times n}$ is an additive consistent IFPR if there exists a normalized interval weight vector $\tilde{W}=\left({\tilde{w}}_{\mathrm{1}},{\tilde{w}}_{\mathrm{2}},\dots ,{\tilde{w}}_n\right)$, ${\tilde{w}}_i=\left[w_i^L,w_i^U\right]$ such that$$\begin{array}{}\text{(25)}& r_{ij}^{-}=\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^L-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^U+\mathrm{0.5},\hspace{1em}\text{for\hspace{0.17em}\hspace{0.17em}all\hspace{0.17em}\hspace{0.17em}}i,j=\mathrm{1,2},\dots ,n,\hspace{0.17em}\hspace{0.17em}i\ne j,\text{(26)}& r_{ij}^{+}=\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^L+\mathrm{0.5},\hspace{1em}\text{for\hspace{0.17em}\hspace{0.17em}all\hspace{0.17em}\hspace{0.17em}}i,j=\mathrm{1,2},\dots ,n,\hspace{0.17em}\hspace{0.17em}i\ne j.\end{array}$$

It is obvious that when the IFPR $\tilde{R}={\left({\tilde{r}}_{ij}\right)}_{n\times n}$ is reduced to a FPR $R={\left(r_{ij}\right)}_{n\times n}$, Corollary 9 will be equivalent to Theorem 1 in [19] with $\theta =\mathrm{1}$.

Theorem 10.

$\tilde{P}={\left({\tilde{p}}_{ij}\right)}_{n\times n}$ is an additive consistent IFPR under the definition provided by (6).

Proof.

As $\tilde{W}=\left({\tilde{w}}_{\mathrm{1}},{\tilde{w}}_{\mathrm{2}},\dots ,{\tilde{w}}_n\right)$, ${\tilde{w}}_i=\left[w_i^L,w_i^U\right]$ is a normalized interval weight vector; it is obvious that there exists a weight vector $W=(w_{\mathrm{1}},w_{\mathrm{2}},\dots ,w_n)$, such that for all $i=\mathrm{1,2},\dots ,n$, $w_i^L\le w_i\le w_i^U$, and ${\sum }_{i=\mathrm{1}}^n{w}_i=\mathrm{1}$. Then, we can get $p_{ij}^L=\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^L-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^U+\mathrm{0.5}\le \mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j+\mathrm{0.5}\le \mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^L+\mathrm{0.5}=p_{ij}^U,\hspace{0.17em}\hspace{0.17em}i\ne j.$

So, we can say the feasible region $S_W$ of $\tilde{P}$ contains $W=(w_{\mathrm{1}},w_{\mathrm{2}},\dots ,w_n)$, which means $S_W$ is not empty and $\tilde{P}$ is additive consistent under the definition provided by (6).

5.2. Deriving Interval Weights from IFPR

Equations (25) and (26) hold for additive consistent IFPRs. However, it is difficult for DMs to provide consistent IFPRs due to the subjectivity of DMs’ judgment and complexity of decision problems in many decision situations [33]. As per Theorem 8, a normalized interval weight vector can reproduce an additive consistent IFPR. In order to derive suitable decision result from an inconsistent IFPR $\tilde{R}$, we seek for a normalized interval weight vector such that the reproduced IFPR $\tilde{P}$ is close to $\tilde{R}$ as much as possible under the sense that$$\begin{array}{}\text{(27)}& r_{ij}^{-}-\mathrm{0.5}\approx \mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^L-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^U,\hspace{1em}\text{for\hspace{0.17em}\hspace{0.17em}all\hspace{0.17em}\hspace{0.17em}}i,j=\mathrm{1,2},\dots ,n,\hspace{0.17em}\hspace{0.17em}i\ne j,\text{(28)}& r_{ij}^{+}-\mathrm{0.5}\approx \mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^L,\hspace{1em}\text{for\hspace{0.17em}\hspace{0.17em}all\hspace{0.17em}\hspace{0.17em}}i,j=\mathrm{1,2},\dots ,n,\hspace{0.17em}\hspace{0.17em}i\ne j.\end{array}$$

It is obvious that the smaller the deviations between two sides of (27) and (28) are, the closer the $\tilde{P}$ is to $\tilde{R}$. Therefore, the following nonlinear programming model is established to derive normalized interval weights from $\tilde{R}$.$$\begin{array}{c}\text{(29)}& \mathrm{m}\mathrm{i}\mathrm{n}\hspace{1em}J={\displaystyle \sum }_{i=\mathrm{1}}^n{\displaystyle \sum }_{j=\mathrm{1},j\ne i}^n\left(\left|\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^L-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^U-\left(r_{ij}^{-}-\mathrm{0.5}\right)\right|+\left|\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^L-\left(r_{ij}^{+}-\mathrm{0.5}\right)\right|\right)\mathrm{s}.\mathrm{t}.\hspace{1em}\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^L-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^U+\mathrm{0.5}\ge \mathrm{0},\hspace{1em}i,j=\mathrm{1,2},\dots ,n,\hspace{0.17em}\hspace{0.17em}i\ne j,\\ \hspace{1em}\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^L+\mathrm{0.5}\le \mathrm{1},\hspace{1em}i,j=\mathrm{1,2},\dots ,n,\hspace{0.17em}\hspace{0.17em}i\ne j,\\ \hspace{1em}\mathrm{0}\le w_i^L\le w_i^U\le \mathrm{1},\hspace{1em}i=\mathrm{1,2},\dots ,n,\\ \hspace{1em}{\displaystyle \sum }_{j=\mathrm{1},j\ne i}^n{w}_j^L+w_i^U\le \mathrm{1},\hspace{1em}i=\mathrm{1,2},\dots ,n,\\ \hspace{1em}{\displaystyle \sum }_{j=\mathrm{1},j\ne i}^n{w}_j^U+w_i^L\ge \mathrm{1},\hspace{1em}i=\mathrm{1,2},\dots ,n.\end{array}$$

It is obvious that the solution of model (29) will not depend on the permutations of the DMs’ judgments. So, our nonlinear model will not suffer from the problem suffered by Wang et al.’s method.

The first two inequalities in model (29) come from the constraints in the definition of $\tilde{P}$ in (23), and the last three lines are constraints to make sure the interval weights are normalized.

Since $r_{ij}^{-}+r_{ji}^{+}=\mathrm{1}$ for all $i,j=\mathrm{1,2},\dots ,n$, we can get$$\begin{array}{}\text{(30)}& \mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^L-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^U-\left(r_{ji}^{-}-\mathrm{0.5}\right)=\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^L-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^U-\left(\mathrm{0.5}-r_{ij}^{+}\right)=-\left(\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^L-\left(r_{ij}^{+}-\mathrm{0.5}\right)\right),\hspace{1em}i,j=\mathrm{1,2},\dots ,n,\hspace{0.17em}\hspace{0.17em}i\ne j,\text{(31)}& \mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^L-\left(r_{ji}^{+}-\mathrm{0.5}\right)=\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^L-\left(\mathrm{0.5}-r_{ij}^{-}\right)=-\left(\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^L-\mathrm{l}{\mathrm{n}}_j^U-\left(r_{ij}^{-}-\mathrm{0.5}\right)\right),\hspace{1em}i,j=\mathrm{1,2},\dots ,n,\hspace{0.17em}\hspace{0.17em}i\ne j.\end{array}$$

Moreover, it is obvious that$$\begin{array}{c}\text{(32)}& \mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^L-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^U+\mathrm{0.5}\ge \mathrm{0},\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^L+\mathrm{0.5}\le \mathrm{1},\\ \hspace{1em}i,j=\mathrm{1,2},\dots ,n,\hspace{0.17em}\hspace{0.17em}i\ne j\iff \\ \mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^L-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^U+\mathrm{0.5}\ge \mathrm{0},\\ \mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^L+\mathrm{0.5}\le \mathrm{1},\\ \hspace{1em}i=\mathrm{1,2},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n.\end{array}$$

Therefore, model (29) can be simplified as follows:$$\begin{array}{c}\text{(33)}& \mathrm{m}\mathrm{i}\mathrm{n}\hspace{1em}J={\displaystyle \sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\displaystyle \sum }_{j=i+\mathrm{1}}^n\left(\left|\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^L-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^U-\left(r_{ij}^{-}-\mathrm{0.5}\right)\right|+\left|\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^L-\left(r_{ij}^{+}-\mathrm{0.5}\right)\right|\right)\mathrm{s}.\mathrm{t}.\hspace{1em}\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^L-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^U+\mathrm{0.5}\ge \mathrm{0},\hspace{1em}i=\mathrm{1,2},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n,\\ \hspace{1em}\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^L-\mathrm{0.5}\le \mathrm{0},\hspace{1em}i=\mathrm{1,2},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n,\\ \hspace{1em}\mathrm{0}\le w_i^L\le w_i^U\le \mathrm{1},\hspace{1em}i=\mathrm{1,2},\dots ,n,\\ \hspace{1em}{\displaystyle \sum }_{j=\mathrm{1},j\ne i}^n{w}_j^L+w_i^U\le \mathrm{1},\hspace{1em}i=\mathrm{1,2},\dots ,n,\\ \hspace{1em}{\displaystyle \sum }_{j=\mathrm{1},j\ne i}^n{w}_j^U+w_i^L\ge \mathrm{1},\hspace{1em}i=\mathrm{1,2},\dots ,n.\end{array}$$

For all $i=\mathrm{1,2},\dots ,n-\mathrm{1}$, $j=i+\mathrm{1},\dots ,n$, let ${\epsilon }_{ij}=\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^L-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^U-\left(r_{ij}^{-}-\mathrm{0.5}\right)$, ${\delta }_{ij}=\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^L-\left(r_{ij}^{+}-\mathrm{0.5}\right)$, ${\epsilon }_{ij}^{+}=\left(\left|{\epsilon }_{ij}\right|+{\epsilon }_{ij}\right)/\mathrm{2}$, ${\epsilon }_{ij}^{-}=\left(\left|{\epsilon }_{ij}\right|-{\epsilon }_{ij}\right)/\mathrm{2}$, ${\delta }_{ij}^{+}=\left(\left|{\delta }_{ij}\right|+{\delta }_{ij}\right)/\mathrm{2}$, ${\delta }_{ij}^{-}=\left(\left|{\delta }_{ij}\right|-{\delta }_{ij}\right)/\mathrm{2}$; then $\left|{\epsilon }_{ij}\right|={\epsilon }_{ij}^{+}+{\epsilon }_{ij}^{-}$, ${\epsilon }_{ij}={\epsilon }_{ij}^{+}-{\epsilon }_{ij}^{-}$, $\left|{\delta }_{ij}\right|={\delta }_{ij}^{+}+{\delta }_{ij}^{-}$, ${\delta }_{ij}={\delta }_{ij}^{+}-{\delta }_{ij}^{-}$ and ${\epsilon }_{ij}^{+}·{\epsilon }_{ij}^{-}=\mathrm{0}$, ${\delta }_{ij}^{+}·{\delta }_{ij}^{-}=\mathrm{0}$. Model (33) can be converted to model (34) by substituting ${\epsilon }_{ij}^{+}$, ${\epsilon }_{ij}^{-}$, ${\delta }_{ij}^{+}$, and ${\delta }_{ij}^{-}$ into (33).$$\begin{array}{c}\text{(34)}& \mathrm{m}\mathrm{i}\mathrm{n}\hspace{1em}J={\displaystyle \sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\displaystyle \sum }_{j=i+\mathrm{1}}^n\left({\epsilon }_{ij}^{+}+{\epsilon }_{ij}^{-}+{\delta }_{ij}^{+}+{\delta }_{ij}^{-}\right)\mathrm{s}.\mathrm{t}.\hspace{1em}\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^L-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^U+\mathrm{0.5}\ge \mathrm{0},\hspace{1em}i=\mathrm{1,2},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n,\\ \hspace{1em}\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^L-\mathrm{0.5}\le \mathrm{0},\hspace{1em}i=\mathrm{1,2},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n,\\ \hspace{1em}\mathrm{0}\le w_i^L\le w_i^U\le \mathrm{1},\hspace{1em}i=\mathrm{1,2},\dots ,n,\\ \hspace{1em}{\displaystyle \sum }_{j=\mathrm{1},j\ne i}^n{w}_j^L+w_i^U\le \mathrm{1},\hspace{1em}i=\mathrm{1,2},\dots ,n,\\ \hspace{1em}{\displaystyle \sum }_{j=\mathrm{1},j\ne i}^n{w}_j^U+w_i^L\ge \mathrm{1},\hspace{1em}i=\mathrm{1,2},\dots ,n,\\ \hspace{1em}{\epsilon }_{ij}^{+}-{\epsilon }_{ij}^{-}=\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^L-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^U-\left(r_{ij}^L-\mathrm{0.5}\right),\hspace{1em}i=\mathrm{1,2},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n,\\ \hspace{1em}{\delta }_{ij}^{+}-{\delta }_{ij}^{-}=\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_i^U-\mathrm{l}\mathrm{n}\hspace{0.17em}{w}_j^L-\left(r_{ij}^U-\mathrm{0.5}\right),\hspace{1em}i=\mathrm{1,2},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n,\\ \hspace{1em}{\delta }_{ij}^{+}\ge \mathrm{0},\\ \hspace{1em}{\delta }_{ij}^{-}\ge \mathrm{0},\\ \hspace{1em}{\epsilon }_{ij}^{+}\ge \mathrm{0},\\ \hspace{1em}{\epsilon }_{ij}^{-}\ge \mathrm{0},\\ \hspace{1em}i=\mathrm{1,2},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n.\end{array}$$

Solving model (34) we can obtain the optimal value $J^{\ast }$ and the corresponding interval weights. If $J^{\ast }=\mathrm{0}$, we can draw the conclusion that $\tilde{R}$ is additive consistent.

It is obvious that model (34) is a nonlinear optimization problem, and we can solve it using the MATLAB toolbox. Now we illustrate how to construct a linear model to derive the interval weights.

It is worth noting that (25) and (26) can also be written as$$\begin{array}{}\text{(35)}& e^{\left(r_{ij}^{-}-\mathrm{0.5}\right)}{w}_j^U=w_i^L,\hspace{1em}\text{for\hspace{0.17em}\hspace{0.17em}all\hspace{0.17em}\hspace{0.17em}}i,j=\mathrm{1,2},\dots ,n,\hspace{0.17em}\hspace{0.17em}i\ne j,\text{(36)}& e^{\left(r_{ij}^{+}-\mathrm{0.5}\right)}{w}_j^L=w_i^U,\hspace{1em}\text{for\hspace{0.17em}\hspace{0.17em}all\hspace{0.17em}\hspace{0.17em}}i,j=\mathrm{1,2},\dots ,n,\hspace{0.17em}\hspace{0.17em}i\ne j.\end{array}$$

Similarly, the two constraints in (23) can be written as$$\begin{array}{}\text{(37)}& w_i^L\ge e^{-\mathrm{0.5}}{w}_j^U,\hspace{1em}i,j=\mathrm{1,2},\dots ,n,\hspace{0.17em}\hspace{0.17em}i\ne j,\text{(38)}& w_i^U\le e^{\mathrm{0.5}}{w}_j^L,\hspace{1em}i,j=\mathrm{1,2},\dots ,n,\hspace{0.17em}\hspace{0.17em}i\ne j.\end{array}$$

Let ${\alpha }_{ij}^{-}=e^{\left(r_{ij}^{-}-\mathrm{0.5}\right)}$, ${\alpha }_{ij}^{+}=e^{\left(r_{ij}^{+}-\mathrm{0.5}\right)}$; then, analogous to the nonlinear model, we can construct the following goal programming model:$$\begin{array}{c}\text{(39)}& \mathrm{m}\mathrm{i}\mathrm{n}\hspace{1em}J={\displaystyle \sum }_{i=\mathrm{1}}^n\hspace{0.17em}{\displaystyle \sum }_{j=\mathrm{1},j\ne i}^n\left(\left|{\alpha }_{ij}^{-}{w}_j^U-w_i^L\right|+\left|{\alpha }_{ij}^{+}{w}_j^L-w_i^U\right|\right)\mathrm{s}.\mathrm{t}.\hspace{1em}{w}_i^L\ge e^{-\mathrm{0.5}}{w}_j^U,\hspace{1em}i,j=\mathrm{1,2},\dots ,n,\hspace{0.17em}\hspace{0.17em}i\ne j,\\ \hspace{1em}{w}_i^U\le e^{\mathrm{0.5}}{w}_j^L,\hspace{1em}i,j=\mathrm{1,2},\dots ,n,\hspace{0.17em}\hspace{0.17em}i\ne j,\\ \hspace{1em}\mathrm{0}\le w_i^L\le w_i^U\le \mathrm{1},\hspace{1em}i=\mathrm{1,2},\dots ,n,\\ \hspace{1em}{\displaystyle \sum }_{j=\mathrm{1},j\ne i}^n{w}_j^L+w_i^U\le \mathrm{1},\hspace{1em}i=\mathrm{1,2},\dots ,n,\\ \hspace{1em}{\displaystyle \sum }_{j=\mathrm{1},j\ne i}^n{w}_j^U+w_i^L\ge \mathrm{1},\hspace{1em}i=\mathrm{1,2},\dots ,n.\end{array}$$

The objective function of model (39) is to minimize the deviations between the two sides of (35) and (36). The objective function of model (29) is to minimize the deviations between the two sides of (25) and (26). As (35) and (36) are transformations of (25) and (26), the basic idea of model (39) and model (29) is the same.

Since $r_{ij}^{-}+r_{ji}^{+}=\mathrm{1}$ for all $i,j=\mathrm{1,2},\dots ,n$, we can get$$\begin{array}{}\text{(40)}& \left|{\alpha }_{ji}^{-}{w}_i^U-w_j^L\right|=\left|e^{\left(r_{ji}^{-}-\mathrm{0.5}\right)}{w}_i^U-w_j^L\right|=\left|e^{\left(\mathrm{0.5}-r_{ij}^{+}\right)}{w}_i^U-w_j^L\right|=\frac{\left|{\alpha }_{ij}^{+}{w}_j^L-w_i^U\right|}{{\alpha }_{ij}^{+}},\text{(41)}& \left|{\alpha }_{ji}^{+}{w}_i^L-w_j^U\right|=\left|e^{\left(r_{ji}^{+}-\mathrm{0.5}\right)}{w}_i^L-w_j^U\right|=\left|e^{\left(\mathrm{0.5}-r_{ij}^{-}\right)}{w}_i^L-w_j^U\right|=\frac{\left|{\alpha }_{ij}^{-}{w}_j^U-w_i^L\right|}{{\alpha }_{ij}^{-}}.\end{array}$$

Moreover, it is obvious that$$\begin{array}{c}\text{(42)}& w_i^L\ge e^{-\mathrm{0.5}}{w}_j^U,w_i^U\le e^{\mathrm{0.5}}{w}_j^L,\\ \hspace{1em}i,j=\mathrm{1,2},\dots ,n,\hspace{0.17em}\hspace{0.17em}i\ne j\iff \\ w_i^L\ge e^{-\mathrm{0.5}}{w}_j^U,\\ w_i^U\le e^{\mathrm{0.5}}{w}_j^L,\\ \hspace{1em}i,=\mathrm{1},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n.\end{array}$$

Therefore, model (39) can be simplified as follows:$$\begin{array}{c}\text{(43)}& \mathrm{m}\mathrm{i}\mathrm{n}\hspace{1em}J={\displaystyle \sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\displaystyle \sum }_{j=i+\mathrm{1}}^n\left(\frac{\mathrm{1}+{\alpha }_{ij}^{-}}{{\alpha }_{ij}^{-}}\left|{\alpha }_{ij}^{-}{w}_j^U-w_i^L\right|+\frac{\mathrm{1}+{\alpha }_{ij}^{+}}{{\alpha }_{ij}^{+}}\left|{\alpha }_{ij}^{+}{w}_j^L-w_i^U\right|\right)\mathrm{s}.\mathrm{t}.\hspace{1em}{w}_i^L\ge e^{-\mathrm{0.5}}{w}_j^U,\hspace{1em}i=\mathrm{1,2},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n,\\ \hspace{1em}{w}_i^U\le e^{\mathrm{0.5}}{w}_j^L,\hspace{1em}i=\mathrm{1,2},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n,\\ \hspace{1em}\mathrm{0}\le w_i^L\le w_i^U\le \mathrm{1},\hspace{1em}i=\mathrm{1,2},\dots ,n,\\ \hspace{1em}{\displaystyle \sum }_{j=\mathrm{1},j\ne i}^n{w}_j^L+w_i^U\le \mathrm{1},\hspace{1em}i=\mathrm{1,2},\dots ,n,\\ \hspace{1em}{\displaystyle \sum }_{j=\mathrm{1},j\ne i}^n{w}_j^U+w_i^L\ge \mathrm{1},\hspace{1em}i=\mathrm{1,2},\dots ,n.\end{array}$$

For all $i=\mathrm{1,2},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n$, let ${\epsilon }_{ij}={\alpha }_{ij}^{-}{w}_j^U-w_i^L$, ${\delta }_{ij}={\alpha }_{ij}^{+}{w}_j^L-w_i^U$, ${\epsilon }_{ij}^{+}=\left(\left|{\epsilon }_{ij}\right|+{\epsilon }_{ij}\right)/\mathrm{2}$, ${\epsilon }_{ij}^{-}=\left(\left|{\epsilon }_{ij}\right|-{\epsilon }_{ij}\right)/\mathrm{2}$, ${\delta }_{ij}^{+}=\left(\left|{\delta }_{ij}\right|+{\delta }_{ij}\right)/\mathrm{2}$, ${\delta }_{ij}^{-}=\left(\left|{\delta }_{ij}\right|-{\delta }_{ij}\right)/\mathrm{2}$, then $\left|{\epsilon }_{ij}\right|={\epsilon }_{ij}^{+}+{\epsilon }_{ij}^{-}$, ${\epsilon }_{ij}={\epsilon }_{ij}^{+}-{\epsilon }_{ij}^{-}$, $\left|{\delta }_{ij}\right|={\delta }_{ij}^{+}+{\delta }_{ij}^{-}$, ${\delta }_{ij}={\delta }_{ij}^{+}-{\delta }_{ij}^{-}$ and ${\epsilon }_{ij}^{+}·{\epsilon }_{ij}^{-}=\mathrm{0}$, ${\delta }_{ij}^{+}·{\delta }_{ij}^{-}=\mathrm{0}$.

Substituting ${\epsilon }_{ij}^{+}$, ${\epsilon }_{ij}^{-}$, ${\delta }_{ij}^{+}$, and ${\delta }_{ij}^{-}$ into (33), we can get the following linear programming model.$$\begin{array}{c}\text{(44)}& \mathrm{m}\mathrm{i}\mathrm{n}\hspace{1em}J={\displaystyle \sum }_{i=\mathrm{1}}^n{\displaystyle \sum }_{j=\mathrm{1},j\ne i}^n\left(\frac{\mathrm{1}+{\alpha }_{ij}^{-}}{{\alpha }_{ij}^{-}}\left({\epsilon }_{ij}^{+}+{\epsilon }_{ij}^{-}\right)+\frac{\mathrm{1}+{\alpha }_{ij}^{+}}{{\alpha }_{ij}^{+}}\left({\delta }_{ij}^{+}+{\delta }_{ij}^{-}\right)\right)\mathrm{s}.\mathrm{t}.\hspace{1em}{w}_i^L\ge e^{-\mathrm{0.5}}{w}_j^U,\hspace{1em}i=\mathrm{1,2},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n,\\ \hspace{1em}{w}_i^U\le e^{\mathrm{0.5}}{w}_j^L,\hspace{1em}i=\mathrm{1,2},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n,\\ \hspace{1em}{\epsilon }_{ij}^{+}-{\epsilon }_{ij}^{-}={\alpha }_{ij}^{-}{w}_j^U-w_i^L,\hspace{1em}i=\mathrm{1,2},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n,\\ \hspace{1em}{\delta }_{ij}^{+}-{\delta }_{ij}^{-}={\alpha }_{ij}^{+}{w}_j^L-w_i^U,\hspace{1em}i=\mathrm{1,2},\dots ,n-\mathrm{1},\hspace{0.17em}\hspace{0.17em}j=i+\mathrm{1},\dots ,n,\\ \hspace{1em}{\epsilon }_{ij}^{+},{\epsilon }_{ij}^{-},{\delta }_{ij}^{+},{\delta }_{ij}^{-}\ge \mathrm{0},\hspace{1em}i,j=\mathrm{1,2},\dots ,n,\hspace{0.17em}\hspace{0.17em}i\ne j,\\ \hspace{1em}\mathrm{0}\le w_i^L\le w_i^U\le \mathrm{1},\hspace{1em}i=\mathrm{1,2},\dots ,n,\\ \hspace{1em}{\displaystyle \sum }_{j=\mathrm{1},j\ne i}^n{w}_j^L+w_i^U\le \mathrm{1},\hspace{1em}i=\mathrm{1,2},\dots ,n,\\ \hspace{1em}{\displaystyle \sum }_{j=\mathrm{1},j\ne i}^n{w}_j^U+w_i^L\ge \mathrm{1},\hspace{1em}i=\mathrm{1,2},\dots ,n.\end{array}$$

It is obvious the solution of model (44) will not change when the DMs’ judgments are relabeled; i.e., our linear model will not suffer from the problem suffered by Wang et al.’s method.

Solving model (44) we can obtain the optimal value $J^{\ast }$ and the corresponding interval weights. If $J^{\ast }=\mathrm{0}$, we can draw the conclusion that $\tilde{R}$ is an additive consistent IFPR.

6. Numerical Examples

In this section, we take the three IFPRs used in [19] to demonstrate the effectiveness of the proposed linear models in Section 5. For convenience, NM, LM are used to denote Wang et al.’s method [19] and our linear model, respectively. For the sake of brevity, we only give the IFPRs used for computation. For detailed information about the three IFPRs, we refer to [19].

We adopt the fitted error [19] to measure the quality of interval weights derived by different models. The fitted error is defined as follows:$$\begin{array}{}\text{(45)}& D\left(\tilde{P},\tilde{R}\right)={\displaystyle \sum }_{i=\mathrm{1}}^n{\displaystyle \sum }_{j=\mathrm{1}}^n\left(\left|p_{ij}^{-}-r_{ij}^{-}\right|+\left|p_{ij}^{+}-r_{ij}^{+}\right|\right),\end{array}$$where $\tilde{R}$ is the original IFPR and $\tilde{P}$ is the fitted IFPR transformed from the interval weights. The transformation can be done as per (22).

Example 2.

The IFPR in this example is$$\begin{array}{}\text{(46)}& {\tilde{R}}_{\mathrm{3}}=\left(\begin{array}{cccc}\left[\mathrm{0.50,0.50}\right]& \left[\mathrm{0.39,0.41}\right]& \left[\mathrm{0.29,0.31}\right]& \left[\mathrm{0.19,0.21}\right]\\ \left[\mathrm{0.59,0.61}\right]& \left[\mathrm{0.50,0.50}\right]& \left[\mathrm{0.39,0.41}\right]& \left[\mathrm{0.29,0.31}\right]\\ \left[\mathrm{0.69,0.71}\right]& \left[\mathrm{0.59,0.61}\right]& \left[\mathrm{0.50,0.50}\right]& \left[\mathrm{0.39,0.41}\right]\\ \left[\mathrm{0.79,0.81}\right]& \left[\mathrm{0.69,0.71}\right]& \left[\mathrm{0.59,0.61}\right]& \left[\mathrm{0.50,0.50}\right]\end{array}\right).\end{array}$$

For both models, ${\tilde{R}}_{\mathrm{3}}$ is additive consistent. Table 1 gives the interval weights, the fitted error, and the ranking of the weights derived by the two models. It is worth noting that the results obtained by NM in the original paper [19] are incorrect due to computational error. Table 1 shows clearly that our model reproduces the original IFPR perfectly and performs better than NM.

Table 1

Results of Example 2.

Method	Weights	Fitted error	Ranking
NM	$${\tilde{w}}_{\mathrm{1}}=\left[\mathrm{0.2122,0.2155}\right],\hspace{0.17em}\hspace{0.17em}{\tilde{w}}_{\mathrm{2}}=\left[\mathrm{0.2345,0.2381}\right]$$ ,	0.12	$${\tilde{w}}_{\mathrm{4}}\succ {\tilde{w}}_{\mathrm{3}}\succ {\tilde{w}}_{\mathrm{2}}\succ {\tilde{w}}_{\mathrm{1}}$$
	$${\tilde{w}}_{\mathrm{3}}=\left[\mathrm{0.2593,0.2611}\right],\hspace{0.17em}\hspace{0.17em}{\tilde{w}}_{\mathrm{4}}=\left[\mathrm{0.2866,0.2907}\right].$$
LM	$${\tilde{w}}_{\mathrm{1}}=\left[\mathrm{0.2132,0.2154}\right],\hspace{0.17em}\hspace{0.17em}{\tilde{w}}_{\mathrm{2}}=\left[\mathrm{0.2356,0.2380}\right]$$ ,	0	$${\tilde{w}}_{\mathrm{4}}\succ {\tilde{w}}_{\mathrm{3}}\succ {\tilde{w}}_{\mathrm{2}}\succ {\tilde{w}}_{\mathrm{1}}$$
	$${\tilde{w}}_{\mathrm{3}}=\left[\mathrm{0.2604,0.2630}\right],\hspace{0.17em}\hspace{0.17em}{\tilde{w}}_{\mathrm{4}}=\left[\mathrm{0.2878,0.2907}\right].$$

Example 3.

The IFPR in this example is$$\begin{array}{}\text{(47)}& {\tilde{R}}_{\mathrm{4}}=\left(\begin{array}{cccc}\left[\mathrm{0.5,0.5}\right]& \left[\mathrm{0.3,0.4}\right]& \left[\mathrm{0.5,0.7}\right]& \left[\mathrm{0.4,0.5}\right]\\ \left[\mathrm{0.6,0.7}\right]& \left[\mathrm{0.5,0.5}\right]& \left[\mathrm{0.6,0.8}\right]& \left[\mathrm{0.2,0.6}\right]\\ \left[\mathrm{0.3,0.5}\right]& \left[\mathrm{0.2,0.4}\right]& \left[\mathrm{0.5,0.5}\right]& \left[\mathrm{0.4,0.8}\right]\\ \left[\mathrm{0.5,0.6}\right]& \left[\mathrm{0.4,0.8}\right]& \left[\mathrm{0.2,0.6}\right]& \left[\mathrm{0.5,0.5}\right]\end{array}\right).\end{array}$$

NM identifies ${\tilde{R}}_{\mathrm{4}}$ as additive consistent while LM identifies ${\tilde{R}}_{\mathrm{4}}$ as inconsistent. The reason of this phenomenon is that the additive transitivity based consistency definition is stricter than the feasible region based consistency definition. Table 2 gives the results of the two models on Example 3. From Table 2, we can find that the rankings are the same for the two models, but the fitted error of LM is smaller than NM’s which indicates our model captures the DM’s preference more accurately than NM.

Table 2

Results of Example 3.

Method	Weights	Fitted error	Ranking
NM	$${\tilde{w}}_{\mathrm{1}}=\left[\mathrm{0.2311,0.2494}\right],\hspace{0.17em}\hspace{0.17em}{\tilde{w}}_{\mathrm{2}}=\left[\mathrm{0.2625,0.2823}\right]$$ ,	2.10	$${\tilde{w}}_{\mathrm{2}}\succ {\tilde{w}}_{\mathrm{4}}\succ {\tilde{w}}_{\mathrm{1}}\succ {\tilde{w}}_{\mathrm{3}}$$
	$${\tilde{w}}_{\mathrm{3}}=\left[\mathrm{0.2256,0.2436}\right],\hspace{0.17em}\hspace{0.17em}{\tilde{w}}_{\mathrm{4}}=\left[\mathrm{0.2436,0.2625}\right].$$
LM	$${\tilde{w}}_{\mathrm{1}}=\left[\mathrm{0.2303,0.2532}\right],\hspace{0.17em}\hspace{0.17em}{\tilde{w}}_{\mathrm{2}}=\left[\mathrm{0.2585,0.2813}\right]$$ ,	2.00	$${\tilde{w}}_{\mathrm{2}}\succ {\tilde{w}}_{\mathrm{4}}\succ {\tilde{w}}_{\mathrm{1}}\succ {\tilde{w}}_{\mathrm{3}}$$
	$${\tilde{w}}_{\mathrm{3}}=\left[\mathrm{0.2303,0.2339}\right],\hspace{0.17em}\hspace{0.17em}{\tilde{w}}_{\mathrm{4}}=\left[\mathrm{0.2532,0.2545}\right].$$

Example 4.

The IFPR in this example is$$\begin{array}{}\text{(48)}& {\tilde{R}}_{\mathrm{5}}=\left(\begin{array}{cccc}\left[\mathrm{0.5,0.5}\right]& \left[\mathrm{0.1,0.2}\right]& \left[\mathrm{0.5,0.7}\right]& \left[\mathrm{0.4,0.5}\right]\\ \left[\mathrm{0.8,0.9}\right]& \left[\mathrm{0.5,0.5}\right]& \left[\mathrm{0.6,0.8}\right]& \left[\mathrm{0.2,0.6}\right]\\ \left[\mathrm{0.3,0.5}\right]& \left[\mathrm{0.2,0.4}\right]& \left[\mathrm{0.5,0.5}\right]& \left[\mathrm{0.4,0.8}\right]\\ \left[\mathrm{0.5,0.6}\right]& \left[\mathrm{0.4,0.8}\right]& \left[\mathrm{0.2,0.6}\right]& \left[\mathrm{0.5,0.5}\right]\end{array}\right).\end{array}$$

For both models, ${\tilde{R}}_{\mathrm{5}}$ is inconsistent. The results of the two models on Example 4 are shown in Table 3. The ranking obtained by NM is different from LM’s. As the fitted error of our model is smaller, the ranking obtained by our model is more convincing.

Table 3

Results of Example 4.

Method	Weights	Fitted error	Ranking
NM	$${\tilde{w}}_{\mathrm{1}}=\left[\mathrm{0.2182,0.2442}\right],\hspace{0.17em}\hspace{0.17em}{\tilde{w}}_{\mathrm{2}}=\left[\mathrm{0.2616,0.2961}\right]$$ ,	3.23	$${\tilde{w}}_{\mathrm{2}}\succ {\tilde{w}}_{\mathrm{4}}\succ {\tilde{w}}_{\mathrm{3}}\succ {\tilde{w}}_{\mathrm{1}}$$
	$${\tilde{w}}_{\mathrm{3}}=\left[\mathrm{0.2238,0.2510}\right],\hspace{0.17em}\hspace{0.17em}{\tilde{w}}_{\mathrm{4}}=\left[\mathrm{0.2249,0.2820}\right].$$
LM	$${\tilde{w}}_{\mathrm{1}}=\left[\mathrm{0.2306,0.2548}\right],\hspace{0.17em}\hspace{0.17em}{\tilde{w}}_{\mathrm{2}}=\left[\mathrm{0.2574,0.2817}\right]$$ ,	2.80	$${\tilde{w}}_{\mathrm{2}}\succ {\tilde{w}}_{\mathrm{4}}\succ {\tilde{w}}_{\mathrm{1}}\succ {\tilde{w}}_{\mathrm{3}}$$
	$${\tilde{w}}_{\mathrm{3}}=\left[\mathrm{0.2306,0.2329}\right],\hspace{0.17em}\hspace{0.17em}{\tilde{w}}_{\mathrm{4}}=\left[\mathrm{0.2548,0.2548}\right].$$

From Examples 2–4, we can say that our linear model performs better than the models in [19] on both consistent and inconsistent IFPRs.

7. Conclusions

There are three main innovations in this paper.

$(1)$ We discuss the value of the parameters $\theta$ and $t$ in [19] more thoroughly and draw the conclusion that $\theta =\mathrm{m}\mathrm{i}\mathrm{n}\left\{e,e^{\mathrm{0.5}n/{\max }_{i,j=\mathrm{1,2},\dots ,n}\left(r_i-r_j\right)}\right\}$ and $t=\mathrm{m}\mathrm{i}\mathrm{n}\left\{\mathrm{1},\mathrm{0.5}n/{\max }_{i,j=\mathrm{1,2},\dots ,n}\left(r_i-r_j\right)\right\}$.

$(2)$ We illustrate that the rankings derived by the method in [19] from inconsistent IFPRs are not robust to permutations of DMs’ judgments and rank reversal problem may arise when the alternatives are relabeled by numerical example.

$(3)$ We propose a linear model to derive interval weights from IFPR. Examples on both consistent and inconsistent IFPRs demonstrate that our model performs better than the models in [19].

In the future, we will investigate estimation of missing values in incomplete IFPRs based on the methods proposed in this paper. Moreover, the methods proposed in this paper can also be extended to handle group decision problems.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Natural Science Foundation of Jiangsu Province (Grant no. BK20150720).

References

[1] T. Saaty, "How to make a decision: The analytic hierarchy process," European Journal of Operational Research, vol. 18, 1994.

[2] A. Ishizaka, A. Labib, "Review of the main developments in the analytic hierarchy process," Expert Systems with Applications, vol. 38 no. 11, pp. 14336-14345, DOI: 10.1016/j.eswa.2011.04.143, 2011.

[3] X. Qi, C. Yin, K. Cheng, X. Liao, "The Interval Cognitive Network Process for Multi-Attribute Decision-Making," Symmetry, vol. 9 no. 10,DOI: 10.3390/sym9100238, 2017.

[4] K. K. F. Yuen, "The Primitive Cognitive Network Process: Comparisons With the Analytic Hierarchy Process," International Journal of Information Technology & Decision Making, vol. 10 no. 4, pp. 659-680, DOI: 10.1142/S0219622011004506, 2011.

[5] K. K. F. Yuen, "Fuzzy cognitive network process: comparisons with fuzzy analytic hierarchy process in new product development strategy," IEEE Transactions on Fuzzy Systems, vol. 22 no. 3, pp. 597-610, DOI: 10.1109/tfuzz.2013.2269150, 2014.

[6] J. Ma, J. Lu, G. Zhang, "Decider: a fuzzy multi-criteria group decision support system," Knowledge-Based Systems, vol. 23 no. 1, pp. 23-31, DOI: 10.1016/j.knosys.2009.07.006, 2010.

[7] J. Rezaei, "Best-worst multi-criteria decision-making method," Omega, vol. 53, pp. 49-57, DOI: 10.1016/j.omega.2014.11.009, 2015.

[8] H. Zhang, C. Yin, X. Qi, R. Zhang, X. Kang, "Cognitive best worst method for multiattribute decision-making," Mathematical Problems in Engineering, vol. 2017,DOI: 10.1155/2017/1092925, 2017.

[9] J. Korpela, A. Lehmusvaara, J. Nisonen, "Warehouse operator selection by combining AHP and DEA methodologies," International Journal of Production Economics, vol. 108 no. 1-2, pp. 135-142, DOI: 10.1016/j.ijpe.2006.12.046, 2007.

[10] B. Oztaysi, "A decision model for information technology selection using AHP integrated TOPSIS-Grey: the case of content management systems," Knowledge-Based Systems, vol. 70, pp. 44-54, DOI: 10.1016/j.knosys.2014.02.010, 2014.

[11] B. S. Ahn, "The analytic hierarchy process with interval preference statements," Omega, vol. 67, pp. 177-185, DOI: 10.1016/j.omega.2016.05.004, 2016.

[12] Y. Dong, C. C. Li, F. Chiclana, E. Herrera-Viedma, "Average-case consistency measurement and analysis of interval-valued reciprocal preference relations," Knowledge-Based Systems, vol. 114, pp. 108-117, DOI: 10.1016/j.knosys.2016.10.005, 2016.

[13] K. W. Li, Z.-J. J. Wang, X. Tong, "Acceptability analysis and priority weight elicitation for interval multiplicative comparison matrices," European Journal of Operational Research, vol. 250 no. 2, pp. 628-638, DOI: 10.1016/j.ejor.2015.09.010, 2016.

[14] F. Meng, C. Tan, X. Chen, "Multiplicative consistency analysis for interval fuzzy preference relations: A comparative study," Omega (United Kingdom), vol. 68, pp. 17-38, DOI: 10.1016/j.omega.2016.05.006, 2017.

[15] B. Zhu, Z. Xu, "Consistency measures for hesitant fuzzy linguistic preference relations," IEEE Transactions on Fuzzy Systems, vol. 22 no. 1, pp. 35-45, DOI: 10.1109/TFUZZ.2013.2245136, 2014.

[16] X. You, T. Chen, Q. Yang, "Approach to multi-criteria group decision-making problems based on the best-worst-method and electre method," Symmetry, vol. 8 no. 9,DOI: 10.3390/sym8090095, 2016.

[17] X. Liu, Y. Pan, Y. Xu, S. Yu, "Least square completion and inconsistency repair methods for additively consistent fuzzy preference relations," Fuzzy Sets and Systems, vol. 198,DOI: 10.1016/j.fss.2011.11.009119, 2012.

[18] T. Tanino, "Fuzzy preference orderings in group decision making," Fuzzy Sets and Systems, vol. 12 no. 2, pp. 117-131, DOI: 10.1016/0165-0114(84)90032-0, 1984.

[19] J. Wang, J. Lan, P. Ren, Y. Luo, "Some programming models to derive priority weights from additive interval fuzzy preference relation," Knowledge-Based Systems, vol. 27, pp. 69-77, DOI: 10.1016/j.knosys.2011.12.001, 2012.

[20] Z. Xu, "A survey of preference relations," International Journal of General Systems, vol. 36 no. 2, pp. 179-203, DOI: 10.1080/03081070600913726, 2007.

[21] F. Liu, Y. N. Peng, Q. Yu, H. Zhao, "A decision-making model based on interval additive reciprocal matrices with additive approximation-consistency," Information Sciences, vol. 422, pp. 161-176, DOI: 10.1016/j.ins.2017.09.014, 2018.

[22] S. Wan, F. Wang, J. Dong, "A group decision making method with interval valued fuzzy preference relations based on the geometric consistency," Information Fusion, vol. 40, pp. 87-100, DOI: 10.1016/j.inffus.2017.06.003, 2018.

[23] J. Wu, F. Chiclana, H. Liao, "Isomorphic Multiplicative Transitivity for Intuitionistic and Interval-Valued Fuzzy Preference Relations and Its Application in Deriving Their Priority Vectors," IEEE Transactions on Fuzzy Systems, vol. 26 no. 1, pp. 193-202, DOI: 10.1109/TFUZZ.2016.2646749, 2018.

[24] J. Krejčí, "On additive consistency of interval fuzzy preference relations," Computers & Industrial Engineering, vol. 107, pp. 128-140, DOI: 10.1016/j.cie.2017.03.002, 2017.

[25] Z. Xu, J. Chen, "Some models for deriving the priority weights from interval fuzzy preference relations," European Journal of Operational Research, vol. 184 no. 1, pp. 266-280, DOI: 10.1016/j.ejor.2006.11.011, 2008.

[26] Y. Xu, K. W. Li, H. Wang, "Incomplete interval fuzzy preference relations and their applications," Computers & Industrial Engineering, vol. 67, pp. 93-103, DOI: 10.1016/j.cie.2013.10.010, 2014.

[27] Z. Wang, "A note on “Incomplete interval fuzzy preference relations and their applications”," Computers & Industrial Engineering, vol. 77, pp. 65-69, DOI: 10.1016/j.cie.2014.09.011, 2014.

[28] Z. Wang, K. W. Li, "Goal programming approaches to deriving interval weights based on interval fuzzy preference relations," Information Sciences, vol. 193, pp. 180-198, DOI: 10.1016/j.ins.2012.01.019, 2012.

[29] F. Liu, W. G. Zhang, J. H. Fu, "A new method of obtaining the priority weights from an interval fuzzy preference relation," Information Sciences, vol. 185 no. 1, pp. 32-42, DOI: 10.1016/j.ins.2011.09.019, 2012.

[30] F. Liu, "Acceptable consistency analysis of interval reciprocal comparison matrices," Fuzzy Sets and Systems, vol. 160 no. 18, pp. 2686-2700, DOI: 10.1016/j.fss.2009.01.010, 2009.

[31] K. Sugihara, H. Ishii, H. Tanaka, "Interval priorities in AHP by interval regression analysis," European Journal of Operational Research, vol. 158 no. 3, pp. 745-754, DOI: 10.1016/s0377-2217(03)00418-1, 2004.

[32] Z. S. Xu, Q. L. Da, "The uncertain OWA operator," International Journal of Intelligent Systems, vol. 17 no. 6, pp. 569-575, DOI: 10.1002/int.10038, 2002.

[33] Z.-J. Wang, "A two-stage linear goal programming approach to eliciting interval weights from additive interval fuzzy preference relations," Soft Computing, vol. 20 no. 7, pp. 2721-2732, DOI: 10.1007/s00500-015-1673-x, 2016.