1 Introduction

Many practical tasks could be simplified as a decision-making problem in a sense such as economic planning, resource allocation, scheme selection, conflict analysis, enterprise management, environmental protection and so on (Arrow 1963; Saaty 1980). The aim of a decision-making procedure is usually to find the best or most acceptable solution from a finite set of alternatives. Many methods have been proposed such as Arrow’s social choice theory (Arrow 1963) and Saaty’s analytic hierarchy process (AHP) (Saaty 1980). Within the framework of AHP (Saaty 1980), a complex decision making problem is divided into multiple levels involving goal, criteria and alternatives belonging to an orderly hierarchy structure. The AHP model has attracted a continuous attention with application to various fields (Li and Wang 2019; Wu et al. 2019; Chen 2020; İç and Yurdakul 2021; Lan et al. 2022; Seikh and Mandal 2022). The technique of pairwisely comparing alternatives and criteria is used to form a series of multiplicative reciprocal matrices (MRMs) under the 1–9 scale. When the scale is assumed to be [0, 1] by following the idea of fuzzy set theory (Zadeh 1965), the preference intensities of the decision maker (DM) are expressed as fuzzy preference relations (FPRs), which have been widely used to construct decision making models (Orlovsky 1978; Tanino 1984; Chiclana et al. 1998; Herrera-Viedma et al. 2004; Wang and Fan 2007; Xu et al. 2009, 2013; Li et al. 2019).

There are two important issues to be addressed for FPRs. One is the consistency measurement and the other is the method of eliciting the priorities. It is seen that the concept of additive transitivity (consistency) was proposed by Tanino (1984), which was further extended to the functional consistency of FPRs (Chiclana et al. 2009). When an FPR is additively consistent, the relation between the priorities and the entries has been investigated (Xu et al. 2009; Liu et al. 2012). In a practical case, DMs always provide inconsistent judgements when pairwisely comparing a set of alternatives (Saaty 1980). It is important to propose a consistency test method and an inconsistency index to quantify the inconsistency degree of FPRs (Xu and Da 2003, 2005; Xia and Xu 2014; Brunelli 2017; Koczkodaj and Urban 2018). For example, based on the deviation from additively consistent FPRs, a method of measuring the consistency level (CL) of inconsistent FPRs was proposed (Herrera-Viedma et al. 2007a, b; Cabrerizo et al. 2014; Li et al. 2019). Wu and Xu (2012) defined a new consistency index by considering the distance between an FPR and a consistent one. Xu and his coworkers (Xu et al. 2018; Xu and Herrera 2019) constructed a consistent matrix, which was used to test the consistency level of inconsistent FPRs. The proposed consistency index of FPRs in Xu et al. (2018) is called as the additive consistency index (ACI). In addition, multiplicative consistency of FPRs was also important and defined similar to the consistency of MRMs (Tanino 1984). The multiplicative consistency level of FPRs was measured by extending the geometric consistency index of MRMs in Xia et al. (2013).

Moreover, it is considered that a reasonable priority vector of alternatives is dependent on acceptable consistency level of pairwise comparison matrices (Saaty 1980). When an FPR is acceptable, the derived priority vector of alternatives could be reliable and convincing. In order to elicit the priority vector from acceptable FPRs, some methods have been proposed. For example, as compared to the eigenvector method in the AHP method (Saaty 1980), Chiclana et al. (2001) proposed the method of obtaining the quantifier-guided dominance/nondominance degrees of alternatives such that the best one can be selected, which has been further developed (Herrera-Viedma et al. 2007a, b). Fan et al. (2001, 2002) constructed some quadratic and linear programming problems to find the priority vector from FPRs. Xu (2004) established the multi-objective optimization problem by using the weights of alternatives to determine the ranking. Xu and Da (2005) obtained the priority vector from FPRs through a convergent iterative algorithm. Xu (2005) constructed and solved an equation system (EM) of the weight vector by using the elements in FPRs. Wang et al. (2007) proposed a chi-square method to find the priority vector from FPRs. By using a functional equation between the entries in FPRs and the weights of alternatives, Xu et al. (2009) gave the normalizing rank aggregation method to derive the priority vector. Fedrizzi and Brunelli (2010) proposed a simple averaging method for computing the priority vector from FPRs.

It is noted from the above-mentioned works that the existing consistency indexes of FPRs are mainly related to a consistent matrix and the defined distance. Here we develop a novel method of defining additive/multiplicative consistency index of FPRs and discuss the thresholds of an acceptable FPR. Then based on the idea exhibited in the proposed additive consistency index, an optimization problem is constructed to derive the priority vector from an acceptable FPR. The remaining of the paper is constructed as follows. In Sect. 2, we introduce the basic definitions of FPRs and some indexes of measuring the inconsistency degree. Section 3 focuses on the derivation of a novel additive consistency index of FPRs and the simulation analysis of acceptable consistency standard. The proposed additive consistency index is based on the relations of two row or column vectors in an FPR. By the way, the concept of transformation-based multiplicative consistency index of FPRs is proposed. In Sect. 4, an optimization model for determining the priority vector is established. The optimal solution is analyzed and a novel algorithm for obtaining the priority vector is proposed. Section 5 reports some numerical results and comparisons. The observations reveal the advantages and novelty of the proposed consistency index and the prioritization method. The conclusions and future research directions are covered in Sect. 6.

2 Preliminaries

It is supposed that the DM compares the alternatives in pairs over a finite set of alternatives \(X=\{x_1, x_{2}, \ldots , x_n\}\) and provides the preference intensities \(b_{ij}\in [0,1]\) for \(i,j\in I=\{1,2,\ldots ,n\}.\) When \(b_{ij}>0.5\), it means that alternative \(x_i\) is more important than alternative \(x_j.\) When \(b_{ij}=0.5,\) it implies that alternative \(x_{i}\) is as equally important as alternative \(x_j.\) When \(b_{ij}<0.5,\) it shows that alternative \(x_i\) is less important than alternative \(x_{j}.\) In order to analyze preference intensities, it is convenient to combine the values \(b_{ij}\) into a preference matrix. That is, one has the following definition:

Definition 1

(Tanino 1984) Matrix \(B=\left( b_{ij}\right) _{n\times n}\) is an FPR, where

$$\begin{aligned} b_{ij}+b_{ji}=1, \quad b_{ii}=0.5,\quad 0 \le b_{ij} \le 1,\quad \forall \quad i, j\in I. \end{aligned}$$
(1)

Moreover, the transitivity properties of an FPR have attracted much interest (Tanino 1984; Xu et al. 2013; De Baets and De Meyer 2005, 2008). There are two kinds of the strongest transitivity for FPRs: additive transitivity and multiplicative transitivity (Tanino 1984). The additive transitivity of FPRs is also called additive consistency and defined as follows:

Definition 2

(Tanino 1984; Herrera-Viedma et al. 2004) An FPR \(B=\left( b_{ij}\right) _{n\times n}\) is with additive consistency (transitivity) if it satisfies the following relation:

$$\begin{aligned} b_{ij}=b_{ik}-b_{jk}+0.5, \quad \forall \quad i, j, k \in I. \end{aligned}$$
(2)

The concept of multiplicative transitivity is analogous to the consistency of MRMs, which is also called multiplicative consistency and defined below:

Definition 3

(Tanino 1984) An FPR \(B=\left( b_{ij}\right) _{n\times n}\) is of multiplicative consistency (transitivity) when the following relation is satisfied:

$$\begin{aligned} \frac{b_{ij}}{b_{ji}}\cdot \frac{b_{jk}}{b_{kj}}=\frac{b_{ik}}{b_{ki}},\quad b_{ij}>0,\quad \forall \quad i, j, k \in I. \end{aligned}$$
(3)

One can see that the additive/multiplicative consistency of \(B=\left( b_{ij}\right) _{n\times n}\) is only with an ideal situation. It is important to give an index to quantify the inconsistency degree of \(B=\left( b_{ij}\right) _{n\times n}\) (Herrera-Viedma et al. 2007b; Xu et al. 2018). When considering the situation of multiplicative consistency, an FPR can be transformed into a MRM and the consistency indexes of MRMs can be used similarly (Xia et al. 2013). For the case of additive consistency, there are mainly two typical ideas to propose an inconsistency index of \(B=\left( b_{ij}\right) _{n\times n}.\) One is based on the relation in (2) to compute the deviation values from the consistent cases. As shown in Herrera-Viedma et al. (2007b), the inconsistency degree of the entry \(b_{ij}\) is computed from the following three ways:

$$\begin{aligned} cb_{ij}^{k1}= & {} \, b_{ik}+b_{kj}-0.5,\\ cb_{ij}^{k2}= & {} \, b_{kj}-b_{ki}+0.5,\\ cb_{ij}^{k3}= & {} \, b_{ik}-b_{jk}+0.5. \end{aligned}$$

The inconsistency level of \(B=(b_{ij})_{n\times n}\) can be quantified as:

$$\begin{aligned} CL(B)=\frac{1}{n}\sum _{i=1}^{n}cd_{i}, \end{aligned}$$
(4)

where \(cd_{i}\) stands for the inconsistency degree of a particular alternative \(x_{i}\):

$$\begin{aligned} cd_{i}=\frac{1}{2(n-1)}\sum _{j=1,j\ne i}^{n}\left( cd_{ij}+cd_{ji}\right) , \end{aligned}$$

with

$$\begin{aligned} cd_{ij}=1-\frac{2}{3}\left|\frac{1}{3(n-2)}\sum ^{n}_{k=1,k\ne i,j}\left( cb_{ij}^{k1}+cb_{ij}^{k2}+cb_{ij}^{k3}\right) -b_{ij}\right|. \end{aligned}$$

When the additively reciprocal property is used, it is found that the values of \(cb_{ij}^{k1},\) \(cb_{ij}^{k2}\) and \(cb_{ij}^{k3}\) are identical and \(cd_{ij}=cd_{ji}\) (Liu and Zhang 2014). Then for computing the inconsistency level of an FPR \(B=(b_{ij})_{n\times n},\) the formula (4) can be simplified as:

$$\begin{aligned} CL(B)=\frac{1}{n(n-1)}\left[ \sum ^{n}_{i=1}\sum ^{n}_{j=1,j\ne i}\left( 1-\frac{2}{3}\left|\frac{1}{n-2}\sum ^{n}_{k=1,k\ne i,j} cb_{ij}^{k}-b_{ij}\right|\right) \right] , \end{aligned}$$
(5)

where

$$\begin{aligned} cb_{ij}^{k}=b_{ik}+b_{kj}-0.5. \end{aligned}$$

It is seen that \(CL\in [0,1]\) and \(CL=1\) for an FPR with additive consistency. The more the value of CL(B) is, the higher the consistency level of B is.

The other is based on the distance of \(B=(b_{ij})_{n\times n}\) to a constructed matrix with additive consistency. For instance, the additive consistency index has been given as follows (Xu et al. 2018; Xu and Herrera 2019):

Definition 4

(Xu et al. 2018) Let \(B=(b_{ij})_{n \times n}\) be an FPR and \({\bar{B}} = ({\bar{b}}_{ij})_{n \times n}\) be the consistent matrix corresponding to B,  where

$$\begin{aligned} {\bar{b}}_{ij}=\frac{1}{n}\sum _{k=1}^{n}(b_{ik}+b_{kj})-0.5. \end{aligned}$$
(6)

The additive consistency index (ACI) of B is computed as:

$$\begin{aligned} ACI(B)=\sqrt{\frac{2}{n(n-1)}\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}(b_{ij}-{\bar{b}}_{ij})^{2}}. \end{aligned}$$
(7)

Furthermore, the threshold of ACI(B) for an acceptable matrix is defined as

$$\begin{aligned} {\overline{ACI}}=\sigma _{0}\sqrt{\frac{2}{n(n-1)}\lambda _{\alpha }}, \end{aligned}$$

where \(\lambda _{\alpha }\) is the critical value of the \(\chi ^{2}\) distribution as the significance level is \(\alpha .\) The values of \({\overline{ACI}}\) are shown in Table 1 by selecting \(\alpha =0.1\) and \(\sigma _{0} = 0.2\) for \(n=3, 4, \ldots , 9,\) respectively. If \(ACI(B) \le {\overline{ACI}},\) B is of acceptable additive consistency; otherwise, B is unacceptable. According to Definition 2, we have \(ACI(B)=0\) if and only if FPR B is additively consistent. In addition, as shown in Xu and Herrera (2019), the additive consistency index in (7) is redefined as:

$$\begin{aligned} DCI(B)=\frac{2}{n(n-1)}\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}|b_{ij}-{\bar{b}}_{ij}|. \end{aligned}$$
(8)

The difference of (7) and (8) is only the adopted distances between an FPR and the constructed consistent matrix.

Table 1 Values of \({\overline{ACI}}\) with \(\alpha =0.1\) and \(\sigma _{0}=0.2\) for \(n=3,4,\ldots , 9,\) respectively
Full size table

Moreover, in order to capture the importance of alternatives in \(X = \{x_1, x_{2}, \ldots , x_n\},\) the normalized priority \(\omega _{i}\) is always associated with alternative \(x_{i}\) satisfying

$$\begin{aligned} \sum _{i=1}^{n}\omega _{i}=1,\quad \omega _{i}\in \left[ 0, 1\right] ,\quad \forall \; i\in I. \end{aligned}$$
(9)

When \(B=(b_{ij})_{n\times n}\) is given, of much importance is to elicit the priorities of alternatives. It is worth noting that when the preferences of the DM are given by the utility values \(u(x_{i})\) \((i\in I),\) the following relation has been constructed for \(B=(b_{ij})_{n\times n}\) with additive consistency (Tanino 1984):

$$\begin{aligned} b_{ij}=0.5+0.5[u(x_{i})-u(x_{j})], \quad i, j \in I, \end{aligned}$$
(10)

where the following condition is required:

$$\begin{aligned} \max _{i}u(x_{i})-\min _{i}u(x_{i})\le 1,\quad i\in I. \end{aligned}$$
(11)

When the utility values \(u(x_{i})\) \((i\in I)\) are replaced by the priorities of alternatives, the relation (10) is changed as (Liu et al. 2012; Fedrizzi and Brunelli 2009):

$$\begin{aligned} b_{ij}=0.5+0.5(\omega _{i}-\omega _{j}), \quad i, j \in I. \end{aligned}$$
(12)

By considering the normalized condition (9), it is seen that the condition similar to (11) holds with

$$\begin{aligned} \max _{i}\omega _{i}-\min _{i}\omega _{i}\le 1. \end{aligned}$$

In general, for an additively consistent matrix \(B=(b_{ij})_{n\times n},\) it has been proved that there is a positive number \(\beta\) such that (Xu et al. 2009):

$$\begin{aligned} b_{ij}=0.5+\beta (\omega _{i}-\omega _{j}), \quad i, j \in I. \end{aligned}$$
(13)

When \(B=(b_{ij})_{n\times n}\) is of multiplicative consistency, the following relation holds (Lipovetsky and Conklin 2002):

$$\begin{aligned} b_{ij}=\frac{\omega _{i}}{\omega _{i}+\omega _{j}},\quad i,j\in I. \end{aligned}$$
(14)

When the given matrix \(B=(b_{ij})_{n\times n}\) is inconsistent, the approximate values of the priorities should be computed. In the present study, an optimization-based method for eliciting priorities from FPRs is proposed, where novel consistency indexes of \(B=(b_{ij})_{n\times n}\) are reported.

3 Novel consistency indexes of FPRs

It is worth noting that the cosine similarity measure between the row vectors of MRMs has been used to propose the consistency index in Kou and Lin (2014). Then, the row and column vectors of MRMs were considered simultaneously to give the double cosine similarity consistency index in Liu et al. (2020). Here, we investigate the consistency index of FPR \(B=(b_{ij})_{n\times n}\) according to multiplicative and additive consistency, respectively. Due to the difference of MRMs and FPRs, the cosine similarity measure of two vectors can not be used directly. The inherent property of FPRs should be considered to propose consistency indexes of FPRs.

3.1 Variance-based additive consistency index and its properties

Let us consider the inherent relation between row/column vectors of FPR \(B=(b_{ij})_{n\times n}\) with additive consistency to propose a novel additive consistency index. It is assumed that the row and column vectors of \(B=(b_{ij})_{n\times n}\) are written as:

$$\begin{aligned} \vec {b}_{i\cdot }=(b_{i1}, b_{i2}, \ldots , b_{in}),\quad \vec {b}_{\cdot j}=(b_{1j}, b_{2j}, \ldots , b_{nj})^{T}, \end{aligned}$$
(15)

for \(i,j\in I\). The difference vector between the \(i\hbox {th}\) and \(j\hbox {th}\) row vectors is obtained as

$$\begin{aligned} \vec {r}_{ij}=\vec {b}_{i\cdot }-\vec {b}_{j\cdot }=(r^{1}_{ij}, r^{2}_{ij}, \ldots , r^{n}_{ij}), \end{aligned}$$
(16)

with \(r^{k}_{ij}=b_{ik}-b_{jk}\) for \(i,j,k\in I.\) Similarly, the difference vector between the \(i\hbox {th}\) and \(j\hbox {th}\) column vectors is computed as

$$\begin{aligned} \vec {c}_{ij}=\vec {b}_{\cdot i}-\vec {b}_{\cdot j}=(c^{1}_{ij}, c^{2}_{ij}, \ldots , c^{n}_{ij})^{T}, \end{aligned}$$
(17)

where \(c^{k}_{ij}=b_{ki}-b_{kj}\) for \(i,j,k\in I.\) It is interesting to give the following result:

Theorem 1

An FPR \(B=\left( b_{ij}\right) _{n\times n}\) is additively consistent if and only if the following equality is satisfied:

$$\begin{aligned} r_{ij}^{1}=r_{ij}^{2}=\cdots =r_{ij}^{n},\quad or\quad c_{ij}^{1}=c_{ij}^{2}=\cdots =c_{ij}^{n},\quad \end{aligned}$$
(18)

for \(i,j,k\in I.\)

Proof

Let us first consider the relation between row vectors of \(B=(b_{ij})_{n\times n}.\) Based on the concept of additive consistency in Definition 2, the application of (13) leads to

$$\begin{aligned} r^{k}_{ij}=b_{ik}-b_{jk}=\beta (w_{i}-w_{j}),\quad i,j, k \in I. \end{aligned}$$
(19)

This means that the condition of \(r_{ij}^{1}=r_{ij}^{2}=\cdots =r_{ij}^{n}\) is satisfied.

On the contrary, when the condition of \(r_{ij}^{1}=r_{ij}^{2}=\cdots =r_{ij}^{n}\) is given, we have

$$\begin{aligned} b_{ik}-b_{jk}=b_{is}-b_{js},\quad i,j,k,s\in I. \end{aligned}$$
(20)

Letting \(s=j,\) one has

$$\begin{aligned} b_{ij}=b_{ik}-b_{jk}+0.5,\quad i,j,k\in I, \end{aligned}$$
(21)

where the condition of \(b_{jj}=0.5\) has been used. According to Definition 2, \(B=(b_{ij})_{n\times n}\) is additively consistent. In addition, when we consider the relation of column vectors, the similar proof procedure can be given and the details have been omitted. The proof is completed. \(\square\)

Theorem 1 shows that when the condition in (18) is not satisfied, \(B=(b_{ij})_{n\times n}\) is not additively consistent. This means that the inconsistency level of \(B=(b_{ij})_{n\times n}\) can be measured by quantifying the deviation degree from the equality in (18). Similar to the variance similarity measure of non-reciprocal FPRs in Liu et al. (2022), let us define the mean values of \(r_{ij}^{k}\) and \(c_{ij}^{k}\) with respect to k as the following forms:

$$\begin{aligned} {\bar{r}}_{ij}=\frac{1}{n}\sum _{k=1}^{n}r_{ij}^{k},\quad {\bar{c}}_{ij}=\frac{1}{n}\sum _{k=1}^{n}c_{ij}^{k},\quad i,j\in I. \end{aligned}$$
(22)

The variances are further computed as:

$$\begin{aligned} v_{ij}^{r}=\frac{1}{n}\sum _{k=1}^{n}\left( r_{ij}^{k}-{\bar{r}}_{ij}\right) ^{2},\quad v_{ij}^{c}=\frac{1}{n}\sum _{k=1}^{n}\left( c_{ij}^{k}-{\bar{c}}_{ij}\right) ^{2},\quad i,j\in I. \end{aligned}$$
(23)

Based on the result in Liu et al. (2022), due to additively reciprocal property of FPR \(B=(b_{ij})_{n\times n},\) it gives \(v_{ij}^{r}=v_{ij}^{c}\) for \(\forall i,j \in I.\) This implies that for an FPR \(B=(b_{ij})_{n \times n},\) the variances in (23) are identical. That is, one of the two variances in (23) only needs to consider for the construction of a novel additive consistency index of FPRs. Moreover, the following observations are easily reached:

$$\begin{aligned} v_{ij}^{r}=v_{ji}^{r},\quad v_{ij}^{c}=v_{ji}^{c}, \quad v_{ii}^{r}=v_{ii}^{c}=0,\quad \forall i,j\in I. \end{aligned}$$
(24)

Then according to the constructed variances in (23), an index to quantify the inconsistency degree of FPRs is proposed:

Definition 5

For an FPR \(B=\left( b_{ij}\right) _{n\times n},\) the variance-based additive consistency index (VACI) is defined as:

$$\begin{aligned} VACI(B)= \frac{2}{ n(n-1)} \sum _{i<j}^{n} v_{ij}^{r}, \end{aligned}$$
(25)

or

$$\begin{aligned} VACI(B)= \frac{2}{n(n-1)}\sum _{i<j}^{n} v_{ij}^{c}, \end{aligned}$$
(26)

where \(v_{ij}^{r}\) and \(v_{ij}^{c}\) have been defined in (23).

The index in (25) or (26) is the simplified version of that for non-reciprocal FPRs (Liu et al. 2022). This is attributed to the additively reciprocal property of FPRs. As compared to the existing methods (Herrera-Viedma et al. 2007b; Xu et al. 2018; Xu and Herrera 2019), the approach to VACI(B) is based on the inherent relation between row/column vectors of FPRs with additive consistency.

In what follows, we investigate the properties of the novel consistency index (25) or (26) and give the following results:

Theorem 2

For an FPR \(B=\left( b_{ij}\right) _{n \times n}\) \((n\ge 2),\) we have \(0\le VACI\left( B\right) \le 1.\)

Proof

Using (23), we get \(v_{ij}^{r,c}\ge 0\) for \(\forall i,j\in I,\) meaning that \(VACI(B)\ge 0.\) In addition, it is computed that

$$\begin{aligned} v_{ij}^{r}= & {} \, \frac{1}{n}\sum _{k=1}^{n}\left[ (b_{ik}-b_{jk}) -\frac{1}{n}\sum _{t=1}^{n}(b_{it}-b_{jt})\right] ^2\nonumber \\= & {} \, \frac{1}{n}\sum _{k=1}^{n}\left[ (b_{ik}-b_{jk})^2+\frac{1}{n^2} \left( \sum _{t=1}^n(b_{it}-b_{jt})\right) ^{2}\right. \nonumber \\{} & {} \left. -\frac{2}{n}(b_{ik}-b_{jk})\sum _{t=1}^{n}(b_{it}-b_{jt})\right] \nonumber \\= & {} \, \frac{1}{n}\left[ \sum _{k=1}^{n}(b_{ik}-b_{jk})^2+\frac{1}{n} \left( \sum _{t=1}^n(b_{it}-b_{jt})\right) ^{2}\right. \nonumber \\{} & {} \left. -\frac{2}{n}\left( \sum _{k=1}^{n}(b_{ik}-b_{jk})\right) ^2\right] \nonumber \\= & {} \, \frac{1}{n}\left[ \sum _{k=1}^{n}(b_{ik}-b_{jk})^2-\frac{1}{n} \left( \sum _{k=1}^n(b_{ik}-b_{jk})\right) ^{2}\right] \nonumber \\\le & {} \frac{1}{n}\sum _{k=1}^{n}(b_{ik}-b_{jk})^2. \end{aligned}$$
(27)

In virtue of \(0\le b_{ij}\le 1,\) it follows \((b_{ik}-b_{jk})^{2}\le 1\) and \(v_{ij}^{r}\le 1.\) Moreover, the condition of \(i<j\) means that the number of \(v_{ij}^{r}\) is \(n(n-1)/2.\) A direct computation can lead to the following inequality:

$$\begin{aligned} 0 \le \frac{2}{n(n-1)}\sum _{i<j}v_{ij}^{r}\le 1. \end{aligned}$$

Hence, the result \(0\le VACI(B)\le 1\) is satisfied and the proof is completed. \(\square\)

Theorem 3

For an FPR \(B=(b_{ij})_{n\times n},\) \(VACI(B)=0\) if and only if B is additively consistent.

Proof

On one hand, when \(B=(b_{ij})_{n\times n}\) is additively consistent, the application of (13) leads to

$$\begin{aligned} v_{ij}^{r}= & {} \, \frac{1}{n}\sum _{k=1}^{n}\left( (b_{ik}-b_{jk})-\frac{1}{n}\sum _{t=1}^{n}(b_{it}-b_{jt})\right) ^2\\= & {} \, \frac{1}{n}\sum _{k=1}^{n}\left( \beta (\omega _{i}-\omega _{j})-\frac{n}{n}\beta (w_{i}-w_{j})\right) ^2\\= & {} \, 0. \end{aligned}$$

This means that \(VACI(B)=0\) by virtue of Definition 5.

On the other hand, the application of \(VACI(B)=0\) yields \(v_{ij}^{r}=0.\) By considering (23), we have \(r_{ij}^{k}={\bar{r}}_{ij}\) for \(\forall k\in I.\) It is seen from Theorem 1 that \(B=(b_{ij})_{n\times n}\) is additively consistent. \(\square\)

Theorem 4

It is assumed that \(\rho\) is a one-to-one mapping from I to I. An FPR \(B_{\rho }=\left( b_{\rho (i)\rho (j)}\right) _{n\times n}\) is determined by applying \(\rho\) to \(B=(b_{ij})_{n\times n}.\) Then it follows \(VACI(B)=VACI(B_{\rho }).\)

Proof

Suppose that \(\rho (i)=k_{i}\in I\) for \(\forall i\in I.\) According to (23), we have

$$\begin{aligned} v_{\rho (i)\rho (j)}^{r}=\frac{1}{n}\sum _{k=1}^{n}\left( r_{\rho (i)\rho (j)}^{k}-{\bar{r}}_{\rho (i)\rho (j)}\right) ^{2}=v_{k_{i}k_{j}}^{r}. \end{aligned}$$

Since the mapping \(\rho\) is one-to-one from I to I,  the application of Definition 5 yields

$$\begin{aligned} {\text {VACI}}(B)={\text {VACI}}(B_{\rho }). \end{aligned}$$

The proof is completed. \(\square\)

As shown in Theorems 2 and 3, the smaller the value of \({\text {VACI}}(B)\in [0,1]\) is, the higher the consistency level of B is. Theorem 4 reveals that the value of \({\text {VACI}}(B)\) is invariant with respect to the permutations of alternatives. The observed property is in agreement with the finding in Liu et al. (2020) about the consistency index of MRMs. This means that the behavior of randomly choosing alternatives to compare does not change the value of the proposed additive consistency index.

3.2 Transformation-based multiplicative consistency index

It is seen that the constructed index \({\text {VACI}}(B)\) is based on additive consistency of B. When considering multiplicative consistency of B,  the corresponding index should be restudied. As shown in Definition 3, since multiplicative consistency of FPRs follows the idea of the consistency of MRMs, the transformation from FPRs to MRMs can be considered for constructing a multiplicative consistency index of FPRs. Let us assume that MRM \(A=(a_{ij})_{n\times n}\) is derived from FPR \(B=(b_{ij})_{n\times n}\) by the following transformation formula (Xu and Da 2003):

$$\begin{aligned} a_{ij}=\frac{b_{ij}}{b_{ji}},\quad i,j\in I. \end{aligned}$$
(28)

The geometric consistency index of A has been used to form a multiplicative consistency index of B as follows (Xia et al. 2013):

$$\begin{aligned} {\text {GCI}}(B)=\frac{2}{(n-1)(n-2)}\sum _{i<j}(\ln b_{ij}-\ln b_{ji}-\ln \omega _{i}+\ln \omega _{j})^{2}, \end{aligned}$$
(29)

where \(\omega _{i}\) is the weight of alternative \(x_{i}\) derived from B for \(i\in I.\) For convenience, the index of B such as \({\text {GCI}}(B)\) is called transformation-based multiplicative consistency index (TMCI).

The capability of TMCI in quantifying the inconsistency level of B is based on the good property of the transformation formula. That is, when MRM A is obtained from FPR B using formula (28), B is of multiplicative consistency if and only if A is consistent (Xu and Da 2003). Similar to GCI(B),  some other TMCIs can be given for FPRs. Here, we consider the inherent relation of row/column vectors of A with consistency, the double cosine similarity consistency index in Liu et al. (2020) is utilized to construct the multiplicative consistency index of B. We write the row and column vectors in \(A=(a_{ij})_{n\times n}\) as \(\vec {a}_{i\cdot }=(a_{i1}, a_{i2}, \ldots , a_{in})\) and \(\vec {a}_{\cdot i}= (a_{1i }, a_{2i}, \ldots , a_{n i})^{T}\) for \(i\in I,\) respectively. The cosine similarity measures of two row/column vectors are computed as:

$$\begin{aligned}{} & {} r_{ij}={\text {CSM}}(\vec {a}_{i\cdot }, \vec {a}_{j\cdot })=\frac{\sum _{k=1}^{n} a_{ik}a_{jk}}{\sqrt{\sum _{k=1}^{n}a_{ik}^2} \sqrt{\sum _{k=1}^{n}a_{jk}^2}}, \end{aligned}$$
(30)
$$\begin{aligned}{} & {} c_{ij}={\text {CSM}}(\vec {a}_{\cdot i}, \vec {a}_{\cdot j})=\frac{\sum _{k=1}^{n} a_{ki}a_{k j}}{\sqrt{\sum _{k=1}^{n}a_{ki}^2} \sqrt{\sum _{k=1}^{n}a_{k j}^2}}. \end{aligned}$$
(31)

Inserting (28) into (30) and (31), the values of \(r_{ij}\) and \(c_{ij}\) can be determined by using \(b_{ij}\) for \(i,j\in I.\) Then following the method in Liu et al. (2020), the multiplicative consistency index of B can be constructed:

Definition 6

Assume that MRM \(A=\left( a_{ij}\right) _{n\times n}\) is derived from FPR \(B=(b_{ij})_{n\times n}\) according to (28). The double cosine similarity consistency index (DCSCI) is defined as:

$$\begin{aligned} {\text {DCSCI}}(B)= \frac{1}{ n\left( n-1 \right) } \sum _{i<j}^n (2-r_{ij}-c_{ij}). \end{aligned}$$
(32)

Moreover, some properties of \({\text {DCSCI}}(B)\) are given as follows:

Theorem 5

For an FPR \(B=\left( b_{ij}\right) _{n \times n}\) \((n\ge 2),\) the multiplicative consistency index is given as (32). We have the following results:

  • \(0\le {\text {DCSCI}}(B)<1;\)

  • \({\text {DCSCI}}(B)=0\) if and only if B is of multiplicative consistency;

  • For a one-to-one mapping \(\rho\) from I to I,  it gives \({\text {DCSCI}}(B)={\text {DCSCI}}(B_{\rho }).\)

Proof

Based on transformation formula (28) and the results in Liu et al. (2020), the proof of the above results is straightforward and the detailed procedures have been omitted. \(\square\)

As shown in Theorem 5, the smaller the value of \({\text {DCSCI}}(B)\) is, the stronger the multiplicative consistency of B is. The observation is similar to the finding of \({\text {VACI}}(B)\) in Definition 5.

3.3 Thresholds and comparisons

It is found that the provided pairwise comparison matrices are always inconsistent in a practical case (Saaty 1980). The acceptable consistency of matrices should be investigated by determining the thresholds of consistency indexes (Saaty 1980; Xu et al. 2018, 2021). In the present study, similar to the idea in Liu et al. (2022, 2020), the simulation method is applied to give the thresholds of VACI(B) and \({\text {DCSCI}}(B)\) for capturing acceptable additive/multiplicative consistency. The simulation procedure is given as follows:

  • Using the scale 0–1 and the chosen order n,  we randomly generate 100, 000 FPRs.

  • The values of VACI and DCSCI for the generated 100, 000 FPRs are computed.

  • Under a percentile of the computed values of VACI and DCSCI, the thresholds of acceptable additive and multiplicative consistencies are determined.

Based on the above procedure, the thresholds for VACI and DCSCI are shown in Tables 2 and 3, where the orders of FPRs are chosen as \(3,4,\ldots ,9.\)

Table 2 Thresholds of VACI for randomly generated 100,000 FPRs under a percentile
Full size table
Table 3 Thresholds of DCSCI for randomly generated 100,000 FPRs under a percentile
Full size table

The results in Tables 2 and 3 show that when different percentiles are considered, the values of VACI and DCSCI are changed correspondingly for an FPR. In what follows, we attempt to give a reasonable critical value for an acceptable FPR. It is found that by following the idea of Saaty’s acceptable consistency (Saaty 1980), the critical percentage for the consistency index values of randomly generated matrices is \(22.086\%.\) Therefore, here we still choose \(22.086\%\) as the critical percentage for determining the thresholds of acceptable FPRs. The computing results are shown in Tables 4 and 5. It is obvious that they are different from the results in Liu et al. (2022) for non-reciprocal FPRs. It is seen from Tables 4 and 5 that with the increasing of n,  the critical values of VACI and DCSCI for acceptable FPRs are increasing. The observations are in agreement with the known findings (Saaty 1980; Xu et al. 2018; Xu and Herrera 2019; Liu et al. 2020).

Table 4 Thresholds of VACI under 22.086%
Full size table
Table 5 Thresholds of DCSCI under 22.086%
Full size table

Furthermore, let us compare the existing consistency indexes of FPRs through numerical computations. We further compute the critical values of the consistency indexes DCI, CL and GCI for an acceptable FPR under the critical percentage \(22.086\%\). As an illustration, we choose \(n=4\) to compute and some critical values of the consistency indexes for \(n=4\) are given in Table 6. It is seen from Table 6 that when the adopted consistency indexes are different, the critical values for an acceptable FPR are different.

Table 6 Thresholds of various consistency indexes for \(n=4\)
Full size table

In addition, for the sake of comparisons, we compute the values of the consistency indexes of four FPRs \(B_{1}\)–\(B_{4}:\)

$$\begin{aligned} B_{1}= & {} \, \left( \begin{array}{llll} {0.500} &{} {0.087} &{} {0.776} &{} {0.282} \\ {0.913} &{} {0.500} &{} {0.870} &{} {0.419} \\ {0.224} &{} {0.130} &{} {0.500} &{} {0.347} \\ {0.718} &{} {0.581} &{} {0.653} &{} {0.500}\\ \end{array} \right) ,\\ B_{2}= & {} \, \left( \begin{array}{cccc} {0.500} &{} {0.578} &{} {0.823} &{} {0.534} \\ {0.422} &{} {0.500} &{} {0.953} &{} {0.895} \\ {0.177} &{} {0.047} &{} {0.500} &{} {0.349} \\ {0.466} &{} {0.105} &{} {0.651} &{} {0.500} \end{array} \right) ,\\ B_{3}= & {} \, \left( \begin{array}{cccc} {0.500} &{} {0.208} &{} {0.735} &{} {0.506} \\ {0.792} &{} {0.500} &{} {0.602} &{} {0.236} \\ {0.265} &{} {0.398} &{} {0.500} &{} {0.234} \\ {0.495} &{} {0.764} &{} {0.766} &{} {0.500} \end{array} \right) ,\\ B_{4}= & {} \, \left( \begin{array}{rrrr} {0.500} &{} {0.040} &{} {0.466} &{} {0.007} \\ {0.960} &{} {0.500} &{} {0.646} &{} {0.709} \\ {0.534} &{} {0.355} &{} {0.500} &{} {0.942} \\ {0.993} &{} {0.105} &{} {0.651} &{} {0.500} \end{array} \right) . \end{aligned}$$

The determined results are given in Table 7. It is noted from Table 7 that \(B_{1}\) and \(B_{2}\) are acceptably consistent according to VACI, DCI, CL and DCSCI. The matrices \(B_{1},\) \(B_{2}\) and \(B_{3}\) are acceptably consistent according to GCI. Based on ACI, only the matrix \(B_{2}\) is acceptable. The above phenomenon shows that the acceptable consistency of an FPR is dependent on the chosen consistency indexes.

Table 7 Values of various consistency indexes for \(B_{1}\)–\(B_{4}\)
Full size table

4 An optimization approach to eliciting priorities

It is worth noting that mathematical optimization is widely used and being developed as one of the foundations of various fields (Sun and Yuan 2006). In this section, let us consider the optimization methods for eliciting priorities from FPR \(B=(b_{ij})_{n\times n}.\) When considering multiplicative consistency of B and formula (28), an optimization model for deriving priorities from B can be established similar to the method in Liu et al. (2020). Moreover, when the relation (14) is used, the optimization model of estimating priorities has been addressed in Lipovetsky and Conklin (2002). Hence, we only consider additive consistency of B and follow the idea of proposing the VACI to give an optimization approach to the priority vector. An algorithm is elaborated on to derive a convincing priorities of alternatives, where the consistency improving method of FPRs is incorporated. It is assumed that the priority vector \(\vec {w}=(\omega _{1}, \omega _{2}, \ldots , \omega _{n})\) is elicited from an FPR \(B=(b_{ij})_{n \times n},\) where \(\sum _{i=1}^{n}\omega _{i}=1\) and \(\omega _{i}\in [0,1] (i \in I).\) By using the relation (13), a matrix with additive consistency can be constructed as follows:

$$\begin{aligned} {\bar{B}}=\left( \begin{array}{cccc} {0.5+\beta \left( \omega _{1}-\omega _{1}\right) } &{} {0.5+\beta \left( \omega _{1}-\omega _{2}\right) }&{} {\cdots } &{} {0.5+\beta \left( \omega _{1}-\omega _{n}\right) } \\ {0.5+\beta \left( \omega _{2}-\omega _{1}\right) } &{} {0.5+\beta \left( \omega _{2}-\omega _{2}\right) } &{}{\cdots } &{} {0.5+\beta \left( \omega _{2}-\omega _{n}\right) } \\ {\vdots } &{} {\vdots }&{} {\ddots } &{} {\vdots } \\ {0.5+\beta \left( \omega _{n}-\omega _{1}\right) } &{} {0.5+\beta \left( \omega _{n}-\omega _{2}\right) } &{} {\cdots } &{} {0.5+\beta \left( \omega _{n}-\omega _{n}\right) } \end{array}\right) . \end{aligned}$$
(33)

According to (33), the n row vectors are written as follows:

$$\begin{aligned} \vec {m}_{i.}=\beta (\omega _{i},\omega _{i}, \ldots ,\omega _{i}) +\left( 0.5-\beta \omega _{1}, 0.5-\beta \omega _{2}, \ldots , 0.5-\beta \omega _{n} \right) ,i\in I. \end{aligned}$$
(34)

One can see from (34) that the row vectors are commonly dependent on the following one:

$$\begin{aligned} \vec {v}= & {} \, (v_{1}, v_{2}, \ldots , v_{n})\nonumber \\= & {} \, \left( 0.5-\beta \omega _{1}, 0.5-\beta \omega _{2}, \ldots , 0.5-\beta \omega _{n} \right) . \end{aligned}$$
(35)

According to the method of computing the variances in (23), we have

$$\begin{aligned} R_{i}(\vec {v},\vec {b}_{i\cdot })=\frac{1}{n}\sum _{k=1}^{n}\left[ (v_{k}-b_{ik}) -\frac{1}{n}\sum _{t=1}^{n}(v_{t}-b_{it})\right] ^2, \end{aligned}$$
(36)

for \(\forall i \in I,\) where \(\vec {b}_{i\cdot }\) has been defined in (15). Then we give an interesting result:

Theorem 6

An FPR \(B=(b_{ij})_{n \times n}\) is additively consistent if and only if \(R_{i}(\vec {v},\vec {b}_{i\cdot })=0\) for \(\forall i\in I,\) where the two vectors \(\vec {b}_{i.}\) and \(\vec {v}\) are defined in (34) and (35), respectively.

Proof

When \(B=(b_{ij})_{n \times n}\) is additively consistent, it gives \(B={\bar{B}}\) and \(\vec {b}_{i\cdot }=\vec {m}_{i}\) for \(\forall i\in I.\) Then one has

$$\begin{aligned} v_{k}-b_{ik}=-\beta \omega _{i}, \end{aligned}$$

and

$$\begin{aligned} R_{i}(\vec {v},\vec {b}_{i\cdot })=\frac{1}{n}\sum _{k=1}^{n}\left[ -\beta \omega _{i}-\left( -\beta \omega _{i}\right) \right] ^2=0,\quad i\in I. \end{aligned}$$

On the contrary, when \(R_{i}(\vec {v},\vec {b}_{i\cdot })=0\) for \(\forall i\in I,\) one has

$$\begin{aligned} (v_{k}-b_{ik})=\frac{1}{n}\sum _{t=1}^{n}(v_{t}-b_{it}),\quad \forall i,k\in I. \end{aligned}$$

By selecting any \(s\in I,\) it follows:

$$\begin{aligned} v_{k}-b_{ik}=v_{s}-b_{is}. \end{aligned}$$

When \(s=i,\) it gives

$$\begin{aligned} v_{k}-b_{ik}=v_{i}-0.5. \end{aligned}$$

This means that

$$\begin{aligned} b_{ik}= & {} \, v_{k}-v_{i}+0.5\\= & {} \, 0.5+\beta (\omega _{i}-\omega _{k}),\quad \forall i,k\in I, \quad \beta > 0. \end{aligned}$$

Therefore, \(B=(b_{ij})_{n \times n}\) is additively consistent and the proof is completed. \(\square\)

The observation in Theorem 6 urges us to construct an optimization problem such that the priority vector can be derived from an FPR \(B=(b_{ij})_{n \times n}.\) Let us define the objective function as follows:

$$\begin{aligned} R(\vec {w})=\sum _{i=1}^{n}R_{i}(\vec {v},\vec {b}_{i\cdot }). \end{aligned}$$
(37)

By minimizing R,  we obtain the following optimization problem:

$$\begin{aligned}{} & {} \min \quad F=R(\vec {w}) \nonumber \\{} & {} {\text {s.t.}} \left\{ {\begin{array}{*{20}{c}} &{}{\sum }_{i=1}^{n}\omega _{i}=1, \\ &{}\omega _{i}\in [0,1], {i\in I}. \\ \end{array}}\right. \end{aligned}$$
(38)

When an FPR \(B=(b_{ij})_{n\times n}\) is additively consistent, we have \(F=0.\) When an FPR \(B=(b_{ij})_{n \times n}\) is not additively consistent, the optimal priority vector is determined. Since the variance (36) has been used, the approach of eliciting priorities from an FPR \(B=(b_{ij})_{n\times n}\) is called the variance minimization method (VMM). In what follows, we focus on the optimal solution to the optimization problem (38) and report the following result:

Theorem 7

For an FPR \(B=(b_{ij})_{n\times n},\) if

$$\begin{aligned} \beta \ge \frac{n}{2}-\sum _{k=1}^{n}b_{ik},\quad \forall i\in I, \end{aligned}$$
(39)

the optimal solution to the optimization problem (38) and the optimal value F are given as follows:

$$\begin{aligned} \omega _{i}= & {} \, -\frac{1}{2\beta }+\frac{1}{n}+\frac{1}{n\beta }\sum _{k=1}^{n}b_{ik},\quad i\in I, \end{aligned}$$
(40)
$$\begin{aligned} F= & {} \, \frac{1}{n}\sum _{i=1}^{n}\sum _{k=1}^{n}\left( (0.5-\beta \omega _{k}-b_{ik}) -\frac{1}{n}\sum _{t=1}^{n}(0.5-\beta \omega _{t}-b_{it})\right) ^2. \end{aligned}$$
(41)

Proof

It is convenient to construct the following Lagrange function:

$$\begin{aligned} L(\vec {w},\lambda )=R(\vec {w})+\lambda \left( \sum _{i=1}^{n}\omega _{i}-1\right) . \end{aligned}$$
(42)

In terms of (36), we have

$$\begin{aligned} R(\vec {w})=\frac{1}{n}\sum _{i=1}^{n}\sum _{k=1}^{n}\left( \left( 0.5-\beta \omega _{k}-b_{ik}\right) -\frac{1}{n}\sum _{t=1}^{n}\left( 0.5-\beta \omega _{t}-b_{it}\right) \right) ^{2}. \end{aligned}$$
(43)

It is noted that the function \(R(\vec {w})\) is the sum of squares of linear functions. This means that \(R(\vec {w})\) is strictly convex, and there is a unique optimal solution. Letting

$$\begin{aligned} \frac{\partial L}{\partial \omega _{i}}=0,\quad i\in I, \end{aligned}$$

we get

$$\begin{aligned} \frac{1}{n}\left( 2\beta ^{2}\omega _{i}+\beta -\frac{2}{n}\beta ^{2}-\frac{2}{n}\beta \sum _{k=1}^{n}b_{ik}\right) +\lambda =0. \end{aligned}$$
(44)

By summing with respect to i,  it follows:

$$\begin{aligned} \frac{1}{n}\left( n\beta -\frac{2\beta }{n}\sum _{i=1}^{n}\sum _{k=1}^{n}b_{ik}\right) +n\lambda =0. \end{aligned}$$
(45)

As shown in the known result (Xu et al. 2009), it gives

$$\begin{aligned} \sum _{i=1}^{n}\sum _{k=1}^{n}b_{ik}=\frac{n^{2}}{2}. \end{aligned}$$
(46)

Substituting (46) into (45), one has \(\lambda =0.\) Then the application of (44) leads to the following results:

$$\begin{aligned} \omega _{i}= & {} \, -\frac{1}{2\beta }+\frac{1}{n}+\frac{1}{n\beta }\sum _{k=1}^{n}b_{ik},\quad i\in I , \\ F= & {} \, \frac{1}{n}\sum _{i=1}^{n}\sum _{k=1}^{n}\left( (0.5-\beta \omega _{k}-b_{ik})-\frac{1}{n} \sum _{t=1}^{n}(0.5-\beta \omega _{t}-b_{it})\right) ^2. \end{aligned}$$

When the condition \(\omega _{i} \in [0,1]\) is considered, one has the inequality (39). The proof is completed. \(\square\)

Theorem 7 shows that the priorities of alternatives can be expressed as the explicit forms under the condition (39). In other words, the application of \(\omega _{i}\ge 0\) \((\forall i\in I)\) leads to

$$\begin{aligned} \beta \ge \frac{n}{2}-\sum _{k=1}^{n}b_{ik},\quad \forall \, i\in I, \end{aligned}$$

and give the result (41). Moreover, it is worth noting that the same finding in Theorem 7 was also reported in Xu et al. (2014) when \(B=(b_{ij})_{n\times n}\) is additively consistent. Different to the consideration in Xu et al. (2014), here we construct an optimization model to derive the priority vector from a generalized FPR in Theorem 7. The result shows that the derived priority vector in Theorem 7 is the optimal estimation of theoretical priorities of alternatives. Naturally, when matrix B is additively consistent, the finding in Theorem 7 still holds.

In the following, we further investigate the effects of the values of \(\beta\) on the priority vector. One can see that the values of \(\beta\) have been discussed in Xu et al. (2009), where \(\beta =n/2\) was used and \(\beta =0.5, 1\) were considered to be unreasonable. In fact, one can see that \(\beta =n/2\) always satisfies the inequality (39), meaning that the derived priorities in (40) are always bigger than zero. On the other hand, when there is a positive integer i such that

$$\begin{aligned} \beta <\frac{n}{2}-\sum _{k=1}^{n}b_{ik},\quad \forall i\in I, \end{aligned}$$
(47)

the priority \(\omega _{i}\) cannot be expressed as that in (40) due to \(\omega _{i}<0.\) This is the underlying reason of the statement that \(\beta =0.5\) and 1 are unreasonable (Xu et al. 2009). However, when the values of \(\beta =0.5\) and 1 satisfy the inequality (39), the priorities in (40) are reasonable and agree with the finding in Liu et al. (2012). In addition, the condition (47) only means that the priorities cannot be written as those in (40), and it does not mean that the priorities cannot be elicited from FPRs. Therefore, the values of \(\beta\) can be chosen as 0.5,  1,  n/2 and others. In particular, it is worth noting that the existence of the normalized priorities has been addressed for \(\beta =0.5\) in Liu et al. (2012). Based on the proposed method, when the condition (39) is satisfied, the priorities are expressed as those in (40). When the condition (47) is satisfied, a numerical method should be used to solve the optimization problem (38), for example, the optimization toolbox in some softwares.

In general, if we can relax the condition of \(\omega _{i}\ge 0\) to consider a real number \(\omega _{i}\in {\mathcal {R}}.\) Based on the proof procedure in Theorem 7, the result (39) is still satisfied. Then, we have

$$\begin{aligned} \omega _{i}-\omega _{j}=\frac{1}{n\beta }\left( \sum _{k=1}^{n}b_{ik}-\sum _{k=1}^{n}b_{jk}\right) . \end{aligned}$$
(48)

For a positive real number \(\beta ,\) the sign of \((\omega _{i}-\omega _{j})\) is only dependent on the difference of the sums of the entries in the ith and jth rows of \(B=\left( b_{ij}\right) _{n\times n}.\) This implies that the value of \(\beta >0\) does not influence the final ranking of alternatives.

In what follows, we give an algorithm to derive the priority vector from FPRs. By the way, the consistency improving method is considered by following the idea in Ma et al. (2006).

  • Step 1. Considering an FPR \(B=\left( b_{ij}\right) _{n\times n},\) the value of VACI is computed.

  • Step 2. When matrix \(B=\left( b_{ij}\right) _{n\times n}\) is unacceptable, the consistent matrix is constructed as \({\bar{B}}=({\bar{b}}_{ij})_{n\times n}\) by using the formula (6).

  • Step 3. Choosing a maximum of iteration as \(M>0\) such that \(M\cdot \Delta t=1,\) we construct the iteration formula as follows:

    $$\begin{aligned} B_{k}=(1-k\Delta t)B+k\Delta t{\bar{B}}, \end{aligned}$$
    (49)

    where k is an integer with \(0\le k\le M.\)

  • Step 4. The initial value of k is chosen as zero. The value of VACI according to \(B_{k}\) is determined.

  • Step 5. When matrix \(B_{k}\) is acceptable, the iteration procedure is stopped.

  • Step 6. Using condition (39) to choose a value of \(\beta\), the priorities \(\omega _{i}\) (\(i\in I\)) of \(B_{k}\) are obtained according to Theorem 7.

  • Step 7. Outputting the weights \(\omega _{i}>0\) for \(i\in I.\)

In order to illustrate the above algorithm, let us investigate the following FPR:

$$\begin{aligned} B_{5} = \left( \begin{array}{cccccc} {0.5} &{} {0.4} &{} {0.8} &{} {0.9} &{} {0.4} &{} {0.2} \\ {0.6} &{} {0.5} &{} {0.2} &{} {0.3} &{} {0.9} &{} {0.5} \\ {0.2} &{} {0.8} &{} {0.5} &{} {0.9} &{} {0.1} &{} {0.2} \\ {0.1} &{} {0.7} &{} {0.1} &{} {0.5} &{} {0.6} &{} {0.9} \\ {0.6} &{} {0.1} &{} {0.9} &{} {0.4} &{} {0.5} &{} {0.9} \\ {0.8} &{} {0.5} &{} {0.8} &{} {0.1} &{} {0.1} &{} {0.5} \\ \end{array} \right) . \end{aligned}$$

We compute that \({\text {VACI}}(B_{5})=0.1773>0.08081,\) implying that \(B_{5}\) is unacceptable. Then \(B_{5}\) is adjusted by using the iteration algorithm for \(\Delta t=0.001\) and \(k=29.\) The adjusted matrix \(B^{*}_{5}\) with acceptable consistency is obtained as follows:

$$\begin{aligned}{} & {} B^{*}_{5} = \left( \begin{array}{cccccc} 0.5000 &{} 0.4448 &{} 0.7271 &{} 0.7823 &{} 0.4224 &{} 0.3233\\ 0.5552 &{} 0.5000 &{} 0.3177 &{} 0.3729 &{} 0.7431 &{} 0.5112\\ 0.2729 &{} 0.6823 &{} 0.5000 &{} 0.7543 &{} 0.1953 &{} 0.2953\\ 0.2177 &{} 0.6271 &{} 0.2457 &{} 0.5000 &{} 0.5383 &{} 0.7711\\ 0.5776 &{} 0.2569 &{} 0.8047 &{} 0.4617 &{} 0.5000 &{} 0.7991\\ 0.6767 &{} 0.4888 &{} 0.7047 &{} 0.2289 &{} 0.2009 &{} 0.5000 \\ \end{array} \right) . \end{aligned}$$
Fig. 1
figure 1

Variations of the priorities versus \(\beta\) in virtue of formula (40) for \(B^{*}_{5}\)

Full size image

According to formula (40), the variations of the priorities versus \(\beta\) are depicted in Fig. 1 for \(B^{*}_{5}.\) One can find from Fig. 1 that when \(\beta >0.2999,\) the derived priorities are bigger than zero. Based on the derived priorities, the ranking of alternatives is independent of the values of \(\beta .\) For example, when \(\beta =0.5,\) we have the priority vector:

$$\begin{aligned} \vec {\omega }_{1}=(0.2333, 0.1667, 0.0667, 0.1333, 0.3000, 0.1000), \end{aligned}$$

and the ranking of alternatives:

$$\begin{aligned} x_{5}\succ x_{1}\succ x_{2} \succ x_{4} \succ x_{6}\succ x_{3}. \end{aligned}$$

Moreover, when \(\beta <0.2999,\) the negative weights could occur. That is, formula (40) cannot be used directly when considering \(\omega _{i}\ge 0\) for \(i\in I.\) As an illustration, we choose \(\beta =0.25\) to solve the optimization problem (38), and the priorities are determined as:

$$\begin{aligned} \vec {\omega }_{2}=(0.2932,0.1601,0.0001,0.0933,0.4266,0.0267). \end{aligned}$$

It is found that the determined ranking of alternatives is in agreement with that for \(\beta >0.2999.\) In addition, when the condition of non-negative priority value can be relaxed to a real number, the same ranking of alternatives can be determined according to the observation in Fig. 1. The above phenomenon reveals that the values of \(\beta\) have great influence on the values of the priorities, but have no influence on the ranking of alternatives.

5 Comparison and discussion

In this section, let us recall some existing methods for deriving priorities from FPRs and give some comparisons.

5.1 Some existing prioritization methods

Let \(B=(b_{ij})_{n \times n}\) be an FPR and \(\omega =(\omega _{1}, \omega _{2}, \ldots , \omega _{n})^T\) with \(\omega _{i}\ge 0 (i\in I)\) be the priority vector derived from B using a prioritization method. In the following, we recall some existing methods.

  • The normalizing rank aggregation method (NRAM) (Xu et al. 2009)

    It is considered that when \(\beta =n/2,\) the priorities of alternatives are expressed as follows:

    $$\begin{aligned} \omega _{i}=\frac{2}{n^2}\sum \limits _{j=1}^{n}b_{ij},\quad i\in I. \end{aligned}$$

  • The eigenvalue method (EM) (Xu 2005)

    Let us construct the following algebraic equations:

    $$\begin{aligned} b_{i1}(\omega _{i}+\omega _{1})+b_{i2}(\omega _{i}+\omega _{2}) + \cdots +b_{in}(\omega _{i}+\omega _{n})=\lambda _{max}\omega _{i},\quad i\in I. \end{aligned}$$
    (50)

    Then, the priorities \(\omega _{i}\) \((i\in I)\) are determined by solving (50).

  • Simple averaging method (SAM) (Fedrizzi and Brunelli 2010).

    The weights of alternatives are computed by using the averaging method as follows:

    $$\begin{aligned} \omega _{i}=\frac{2}{n}\sum _{j=1}^{n}b_{ij},\quad i\in I. \end{aligned}$$
    (51)

  • The parametric model method (PMM) (Wang et al. 2012)

    The priorities \(\omega _{i}\) \((i\in I)\) of alternatives are determined by using the following constrained programming model:

    $$\begin{aligned}{} & {} \min F_{1}=\sum _{i=1}^{n}\sum _{j=1}^{n}\left( \log _{\theta }\omega _{i}-\log _{\theta }\omega _{j}+0.5-b_{ij}\right) ^2\nonumber \\{} & {} s.t. \left\{ {\begin{array}{*{20}{c}} &{}{\sum }_{i=1}^{n}\omega _{i}=1, \\ &{}\omega _{i}\in [0,1], i \in I. \\ \end{array}}\right. \end{aligned}$$
    (52)

    Then one has

    $$\begin{aligned} \omega _{i}(\theta )=\frac{\theta ^{\frac{1}{n}\sum _{k=1}^{n}b_{ik}}}{\sum _{l=1}^{n}\theta ^{\frac{1}{n}\sum _{k=1}^{n}b_{lk}}}, \end{aligned}$$

    where \(\theta >0.\) In the following computations, we choose \(\theta =10.\)

5.2 Numerical results and comparisons

In what follows, we carry out some numerical examples to show the effectiveness of the proposed method. Moreover, the Euclidean distance (ED) (Kou and Lin 2014; Golany and Kress 1993) and the Manhattan distance (MD) (Zhang et al. 2021) are used to assess the performance of the above-mentioned methods. That is, we have the following criteria:

$$\begin{aligned} ED=\left\{ \sum _{i=1}^{n}\sum _{j=1}^{n}\left[ b_{ij}-0.5-\beta (\omega _{i}-\omega _{j})\right] ^{2}\right\} ^{1/2}, \end{aligned}$$

and

$$\begin{aligned} MD=\sum _{i=1}^{n}\sum _{j=1}^{n}|b_{ij}-0.5-\beta (\omega _{i}-\omega _{j})|. \end{aligned}$$

The smaller the values of ED and MD are, the better the prioritization method is.

Example 1

We consider the following FPR:

$$\begin{aligned} B_{7}=\left( \begin{array}{cccc} {0.500} &{} {0.087} &{} {0.776} &{} {0.282} \\ {0.913} &{} {0.500} &{} {0.870} &{} {0.419} \\ {0.224} &{} {0.130} &{} {0.500} &{} {0.347} \\ {0.718} &{} {0.581} &{} {0.653} &{} {0.500} \end{array} \right) . \end{aligned}$$
(53)

It is computed that \({\text {VACI}}(B_{7})=0.0032<0.03470,\) meaning that \(B_{7}\) is acceptable. According to (39), one can see that when \(\beta \ge 0.799,\) the positive priorities are expressed as (40) in the present study. Hence, we select \(\beta =0.8\) to compute by using the methods of VMM, PMM, NRAM, EM and SAM, respectively. As shown in Table 8, the rankings of priorities are the same as \(\omega _{2}>\omega _{4}>\omega _{1} >\omega _{3}.\) Furthermore, let us compute the ED and MD values of different methods and give them in Table 9. It is found from Table 9 that when VMM is used, the values of ED and MD are the smallest. The above observation reveals that the proposed method is effective and exhibit some advantages. In particular, the variations of ED and MD versus \(\beta\) are shown in Figs. 2 and 3 for five prioritization methods. It is found from Figs. 2 and 3 that the values of ED and MD for VMM are always smaller than those for the other methods. The above phenomenon reveals that the performance of the proposed prioritization method is better than the others regardless of the values of \(\beta .\)

Table 8 Priority vectors and rankings for Example 1
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Table 9 Performances of various prioritization methods for Example 1
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Fig. 2
figure 2

Variations of ED versus \(\beta\) for five prioritization methods using \(B_{7}\)

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Fig. 3
figure 3

Variations of MD versus \(\beta\) for five prioritization methods using \(B_{7}\)

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Example 2

Let us consider the following matrix:

$$\begin{aligned} B_{8}= \left( \begin{array}{cccc} {0.5} &{} {0.4} &{} {0.55} &{} {0.7}\\ {0.6} &{} {0.5} &{} {0.4} &{} {0.55}\\ {0.45} &{} {0.6} &{} {0.5} &{} {0.48}\\ {0.3} &{} {0.45} &{} {0.52} &{} {0.5} \end{array} \right) . \end{aligned}$$

It is calculated that \({\text {VACI}}(B_{8})=0.0315<0.03470,\) implying that \(B_{8}\) is acceptable. As compared to \(B_{7},\) the value of \({\text {VACI}}(B_{8})\) is more closer to the threshold 0.03470. Based on the different methods, the priorities and the rankings of alternatives are provided in Table 10 for \(\beta =0.3.\) The observation shows that the determined rankings are identical and the proposed prioritization method is effective. In addition, the values of ED and MD for different methods are computed and given in Table 11. It is seen that the performance of VMM is better than those of PMM, NRAM and EM. The result reveals that the present method is good to derive a convincing priority vector.

Table 10 Priority vectors and rankings for Example 2
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Table 11 Performances of various prioritization methods for Example 2
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Example 3

Let us further consider the following matrix:

$$\begin{aligned} B_{9}= \left( \begin{array}{cccccccc} {0.5000} &{} {0.1059} &{} {0.8629} &{} {0.3413} &{} {0.1847} &{} {0.8002}\\ {0.8941} &{} {0.5000} &{} {0.5169} &{} {0.7470} &{} {0.0484} &{} {0.5759}\\ {0.1371} &{} {0.4831} &{} {0.5000} &{} {0.5176} &{} {0.5060} &{} {0.4689}\\ {0.6587} &{} {0.2530} &{} {0.4824} &{} {0.5000} &{} {0.6107} &{} {0.6525}\\ {0.8153} &{} {0.9516} &{} {0.4940} &{} {0.3893} &{} {0.5000} &{} {0.3450}\\ {0.1998} &{} {0.4241} &{} {0.5311} &{} {0.3475} &{} {0.6550} &{} {0.5000}\\ \end{array} \right) . \end{aligned}$$

It is computed that \({\text {VACI}}(B_{9})=0.1597>0.08081,\) meaning that \(B_{9}\) is unacceptable according to the standard in Table 4. Then using the iteration algorithm for \(\Delta t=0.001\) and \(k=15,\) the matrix \(B_{9}\) is adjusted to \(B^{*}_{9}\) with acceptable additive consistency:

$$\begin{aligned} B^{*}_{9}= \left( \begin{array}{cccccccc} {0.5000} &{} {0.1372} &{} {0.8296} &{} {0.3511} &{} {0.2046} &{} {0.7724}\\ {0.8628} &{} {0.5000} &{} {0.5264} &{} {0.7244} &{} {0.0901} &{} {0.5787}\\ {0.1704} &{} {0.4736} &{} {0.5000} &{} {0.5067} &{} {0.4907} &{} {0.4713}\\ {0.6489} &{} {0.2756} &{} {0.4933} &{} {0.5000} &{} {0.5940} &{} {0.6456}\\ {0.7954} &{} {0.9099} &{} {0.5093} &{} {0.4060} &{} {0.5000} &{} {0.3745}\\ {0.2276} &{} {0.4213} &{} {0.5287} &{} {0.3544} &{} {0.6255} &{} {0.5000}\\ \end{array} \right) . \end{aligned}$$
Table 12 Priority vectors and rankings according to \(B^{*}_{9}\)
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Table 13 Performances of various prioritization methods according to \(B^{*}_{9}\)
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By considering (39) and \(B^{*}_{9},\) the value of \(\beta =0.4\) is chosen to compute and the priorities of alternatives are shown in Table 12. It can be seen from Table 12 that except for EM, the rankings are identical. For all the adopted methods, the best alternative is the same. In addition, the values of ED and MD for different methods are computed and given in Table 13. It can be seen from Table 13 that we have the smallest value of \(ED=1.0826\) for VMM. The observation shows that based on the criterion ED,  the proposed method is effective.

6 Conclusions

It is important to quantify the consistency degree of fuzzy preference relations (FPRs) and derive the priorities of alternatives. In this study, a new additive consistency index of FPRs has been proposed by using the variances of the difference vectors determined by any two row/column vectors. A transformation-based multiplicative consistency index of FPRs has been considered. The properties of the proposed indexes have been investigated and the thresholds for an acceptable FPR has been discussed. Then following the idea in the proposed additive consistency index, an optimization method has been given to derive the priority vector. The optimal solution has been discussed and some comparisons have been offered. Some novel findings are covered as follows:

  • The inherent relation between row/column vectors of FPRs is effective to propose the variance-based additive consistency index (VACI).

  • Multiplicative consistency level of FPRs is reasonably quantified using the transformation-based multiplicative consistency index.

  • The variance minimization method (VMM) is useful to elicit the priorities of alternatives from FPRs. It is found that although the values of the priorities are dependent on the introduced parameter \(\beta ,\) the ranking of alternatives is independent of the value of \(\beta .\)

In the future, the proposed consistency index could be used to address group decision making problems, propose optimization models for completing the missing values in incomplete FPRs and others.