1 Introduction

One of the main ideas in the analytic hierarchy process (AHP) (Saaty 1980) is to pairwisely compare alternatives in relative measurements and express the judgements of decision makers as multiplicative reciprocal preference relations. Based on the given preference relations, the priority weights of alternatives are determined and the best alternative(s) is(are) chosen. Up to now, the format of multiplicative reciprocal matrices in the typical AHP has been extended to various preference representation structures, such as additive reciprocal matrices (Tanino 1984; 1988), interval-valued preference relations (Saaty and Vargas 1987; Xu and Chen 2008; Li 2010; Xu and Cai 2012; Cavallo and Brunelli 2018; Yeni and Özçelik 2019), triangular fuzzy preference relations (Van Laarhoven and Pedrycz 1983) and type-2 fuzzy preference relations (Qin 2019) and others.

In order to capture the logical consistency of decision makers, the consistency of preference relations is an important issue (Saaty 1980; Koczkodaj 1993). The inconsistency degree of multiplicative reciprocal matrices in the typical AHP is usually characterized by an inconsistency index, such as the Consistency Index (CI) Saaty (1980, 2013), the Geometric Consistency Index (GCI) (Aguarón and Moreno-Jiménez 2013), the Harmonic Consistency Index (HCI) (Stein and Mizzi 2007), the Statistical Index (SI) (Lin et al. 2013), the Cosine Consistency Index (CCI) (Kou and Lin 2014) and others. A systemic study about the axiomatic properties of the consistency indexes was further investigated (Brunelli and Fedrizzi 2015; Brunelli 2017; Koczkodaj and Szwarc 2014, 2018). Similarly, for additive reciprocal matrices, the consistency definitions and indexes have also attracted great attention (Tanino 1984; Herrera-Viedma et al. 2004; Chiclana et al. 2009a). In particular, when considering the uncertainty experienced by decision makers in comparing alternatives, it is noted that additive reciprocal matrices have been extended to interval additive reciprocal matrices. The consistency definitions of interval additive reciprocal matrices have been reviewed in Liu et al. (2018a), where the new concept of additive approximation-consistency has been proposed by considering the randomness of decision makers in comparing alternatives. Moreover, some consistency indexes have been proposed to quantify the inconsistency degree of interval additive reciprocal matrices. For example, by developing the method in Herrera-Viedma et al. (2007a), Liu and Zhang (2014a) gave a consistency measure of interval additive reciprocal matrices. Considering all possible additive reciprocal matrices related to interval ones, the average-case consistency measure was presented by Dong et al. (2016), where a random distribution of the associated additive reciprocal matrices has been used. Recently, Wan et al. (2018) proposed the geometric consistency index to measure the inconsistency degree of interval additive reciprocal matrices, where the risk attitude of decision makers has been incorporated. It is worth noting that the invariance under permutations of alternatives is considered to be an axiomatic property of the inconsistency indexes of multiplicative reciprocal matrices in Brunelli and Fedrizzi (2015). In fact, the permutations of alternatives are related to the randomness experienced by decision makers in comparing alternatives (Liu et al. 2017b). The property of the invariance under permutations of alternatives reflects the invariance under the random behavior of choosing alternatives to compare. The random behaviour of decision makers in choosing alternatives may affect the inconsistency indexes of interval additive reciprocal matrices (Liu and Zhang 2014a; Dong et al. 2016; Wan et al. 2018). An inconsistency index for interval multiplicative reciprocal matrices has been proposed and the application to group decision making has been addressed (Liu et al. 2018a). Motivated by this consideration, some known consistency indexes of interval additive reciprocal matrices are analyzed in the present study. A new consistency index of the interval-valued matrices is proposed by incorporating the uncertainty and the randomness experienced by decision makers in the pairwise comparison of alternatives.

On the other hand, it is always considered that the wisdom of a group of experts is superior to any individual in public policies, political forecasting, and other organizational tasks. A great number of group decision making methods have been developed under the considerations of various theories and uncertain environments (Lu et al. 2007; Dong and Xu 2016; Zhang et al. 2013, 2016; Xia and Chen 2015; Chen and Zhou 2012; Li et al. 2019). An important phase in group decision making is the aggregation of individual preference relations (Chiclana et al. 1998; Herrera et al. 2001; Zhou and Chen 2011; Rademaker and De Baets 2011; Chen et al. 2015). One of the main aggregation methods of individual comparison matrices is the application of aggregation operators (Yager 1988, 1993, 1999; Xu and Da 2003; Liu 2006; Chiclana et al. 2007b). For example, the weighted geometric averaging (WGA) operator was proposed to aggregate multiplicative reciprocal preference relations (Aczél and Saaty 1983). The WGA operator was further extended to the ordered weighted geometric averaging (OWGA) operator (Herrera et al. 2001), the induced ordered weighted geometric averaging (IOWA) operator, and the induced ordered weighted geometric averaging (IOWGA) operator (Xu and Da 2003). Herrera-Viedma et al. (2007a) utilized an additive-consistency measure as the inducing variable to introduce a new IOWA operator, which is called the AC-IOWA operator. The way of aggregation using the AC-IOWA operator is that the more importance is given to additive reciprocal preference relations with the more additive consistency (Herrera-Viedma et al. 2007b). Following the ideas in Herrera-Viedma et al. (2007a, ??b), Liu et al. (2014b, 2006) proposed the CIOWGA operator and the GCI-IOWGA to aggregate interval multiplicative reciprocal matrices where the order-inducing variable is the defined consistency index and geometric consistency index, respectively. In our study, as we will offer a new consistency index of interval additive reciprocal matrices, a novel IOWA operator is proposed correspondingly to aggregate the interval-valued matrices. The properties of the collective interval additive reciprocal matrix will be addressed in detail.

The structure of the paper is organized as follows. In Section 2, some definitions associated with interval additive reciprocal preference relations are recalled. Some indexes for measuring the inconsistency degree of interval additive reciprocal matrices are reviewed. Section 3 gives a new consistency index of interval additive reciprocal matrices and some comparisons with the existing ones. In Section 4, a novel aggregation operator is proposed and its properties are studied in detail. Section 5 presents a new algorithm of solving group decision making problems with interval additive reciprocal matrices. Numerical examples are carried out to illustrate the proposed index and algorithm. The main conclusions are covered in Section 6.

2 Preliminaries

It is significant to recall some definitions related to additive reciprocal matrices and interval additive reciprocal matrices, respectively. Assume that \(X=\{x_{1}, x_{2},\dots , x_{n}\}\) is the set of alternatives to be compared. A series of pairwise comparisons over X are made and they are expressed as the following fuzzy binary relation:

$$ \mu_{R}:X\times X\rightarrow [0,1]. $$
(1)

In what follows, we focus on additive reciprocal matrices and interval additive reciprocal matrices, thus the methods of defining their consistency and measuring their inconsistency degree.

2.1 Some definitions

Under the assumption of rij = μR(xi, xj) (xi ∈ X), one has the definition of additive reciprocal matrices as follows:

Definition 1

Tanino (1984) R = (rij)n×n is called an additive reciprocal matrix, if rij is the preference intensity of alternative xi over alternative xj with rij + rji = 1, 0 ≤ rij ≤ 1, and rii = 0.5 for \(i,j=1,2,\dots n\).

According to Definition 1, the additive property rij + rji = 1 is the main reason to call R = (rij)n×n as an additive reciprocal matrix. Moreover, the relations among the entries of R = (rij)n×n reveal the logical properties of the judgements provided by decision makers. When considering the additive transitivity, we have the definition of additive reciprocal matrices with additive consistency:

Definition 2

Tanino (1984) An additive reciprocal matrix R = (rij)n×n is considered to be additively consistent, if

$$ r_{ij}=r_{ik}-r_{jk}+0.5, $$
(2)

for \(\forall i,j,k\in \{ 1,2,{\dots } ,n\}.\)

When considering the multiplicative transitivity, an additive reciprocal matrix with multiplicative consistency is defined as follows:

Definition 3

Tanino (1984) An additive reciprocal matrix R = (rij)n×n is with multiplicative consistency, if

$$ \frac{r_{ij}}{r_{ji}}\cdot\frac{r_{jk}}{r_{kj}}=\frac{r_{ik}}{r_{ki}}, $$
(3)

for all \(i,j,k\in \{ 1,2,{\dots } ,n\}\).

Generally, it is feasible to write the relations of the entries in R = (rij)n×n with a consistency as a function. For example, the functional consistency of additive reciprocal matrices is expressed as Chiclana et al. (2009a)

$$ r_{ij}=g(r_{ik},r_{jk}),\forall i,j,k\in \{ 1,2,{\dots} ,n\}, $$
(4)

where g is a function from [0,1] × [0,1] to [0,1]. Clearly, the formulas in Eqs. 2 and 3 are the particular ones of Eq. 4. The relations in Eqs. 2 and 3 can be used as the standard to estimate the missing entries and measure the inconsistency degree of additive reciprocal matrices Herrera-Viedma et al. (2007a, ??b). In this study, the additive consistency of additive reciprocal matrices is used.

Furthermore, in order to model the uncertainty experienced by decision makers in the pairwise comparison of alternatives, interval-valued preference relations are used extensively (Saaty and Vargas 1987; Wan and Dong 2015; Meng and Tang 2017; Wu and Liu 2016, 2018). It is seen that an additive reciprocal matrix R = (rij)n×n is extended to an interval additive reciprocal matrix defined as follows:

Definition 4

Xu and Chen (2008) An interval additive reciprocal matrix \(\tilde {R}=(\tilde {r}_{ij})_{n\times n}\) is represented as

$$ \tilde{R}=(\tilde{r}_{ij})_{n \times n} = \left( \begin{array}{cccc} \left[0.5,0.5\right] & [r^{-}_{12},r^{+}_{12}] & {\cdots} & [r^{-}_{1n},r^{+}_{1n}] \\ \left[r^{-}_{21},r^{+}_{21}\right]&[0.5,0.5]&\cdots&[r^{-}_{2n},r^{+}_{2n}]\\ \vdots&\vdots&\ddots&\vdots\\ \left[r^{-}_{n1},r^{+}_{n1}\right]&[r^{-}_{n2},r^{+}_{n2}]&\cdots&[0.5,0.5] \end{array} \right), $$
(5)

where \(\tilde {r}_{ij}=[r^{-}_{ij}, r^{+}_{ij}]\) is the interval preference degree of alternative xi over alternative xj. The non-negative real numbers \(r^{-}_{ij}\) and \(r^{+}_{ij}\) satisfy \(r^{-}_{ij}\le r^{+}_{ij}\), \(r^{-}_{ii}=r^{+}_{ii}=0.5\), \(r^{-}_{ij} + r^{+}_{ji}=1\) and \(r^{+}_{ij} + r^{-}_{ji}=1\) for \(\forall i,j=1,2,\dots n.\)

Recently, the consistency definitions of \(\tilde {R}=(\tilde {r}_{ij})_{n \times n}\) have been reviewed (Liu et al. 2018a; Krejčí 2017, 2019) and some shortcomings have been shown. In particular, it is noted that interval additive reciprocal matrices are considered to be inconsistent in nature and the concept of additive approximation-consistency is proposed by Liu et al. (2018b). That is, we define a one-to-one mapping σ from I = {1,2,⋯ , n} to I with

$$ \sigma: k\rightarrow i_{k}, \forall k, i_{k}\in\{1,2,\cdots,n\}. $$
(6)

For the sake of convenience, the mapping σ also stands for a permutation of I = {1,2,⋯ , n} expressed as σ = (i1, i2,⋯ , in). Moreover, an interval additive reciprocal matrix with permutation σ is written as \(\tilde {R}^{\sigma }=(\tilde {r}^{\sigma }_{ij})_{n \times n}\) with \(\tilde {r}^{\sigma }_{ij}=[r^{-}_{\sigma (i) \sigma (j)},r^{+}_{\sigma (i) \sigma (j)}].\) It is seen that \(\tilde {R}^{\sigma }=(\tilde {r}^{\sigma }_{ij})_{n \times n}\) is obtained by applying the mapping σ to \(\tilde {R}=(\tilde {r}_{ij})_{n \times n}.\) In addition, two additive reciprocal preference relations \(P_{\sigma }=(p_{ij}^{\sigma })_{n\times n}\) and \(Q_{\sigma }=(q_{ij}^{\sigma })_{n\times n}\) are constructed, where

$$ p_{ij}^{\sigma}=\left\{ \begin{array}{cc} r^{-}_{\sigma(i) \sigma(j)} , &i<j,\\ 0.5, &i=j,\\ r^{+}_{\sigma(i) \sigma(j)} , &i>j, \end{array} \right. q_{ij}^{\sigma}=\left\{ \begin{array}{cc} r^{+}_{\sigma(i) \sigma(j)} , &i<j,\\ 0.5, &i=j,\\ r^{-}_{\sigma(i) \sigma(j)} , &i>j. \end{array} \right. $$
(7)

Hence, the definition of additive approximation-consistency of interval additive reciprocal matrices is given as follows:

Definition 5

Liu et al. (2018b) It is assumed that \(\tilde R^{\sigma }\) is an interval-valued matrix determined by applying a permutation σ to \(\tilde {R}=(\tilde {r}_{ij})_{n \times n}.\) If the two matrices Pσ and Qσ determined by Eq. 7 are all additively consistent under a permutation σ (Definition 2), \(\tilde {R}\) is of additive approximation-consistency.

It is found from Definition 5 that the reciprocity and the permutations of alternatives have been considered in defining the additive approximation-consistency of interval additive reciprocal matrices. Furthermore, the inconsistency degree of \(\tilde {R}^{\sigma }=(\tilde {r}^{\sigma }_{ij})_{n \times n}\) and \(\tilde {R}^{\sigma }=(\tilde {r}^{\sigma }_{ij})_{n \times n}\) should be quantified. In the next subsection, we will review the consistency indexes of interval additive reciprocal matrices and show some of their shortcomings.

2.2 Reviews on consistency indexes of interval additive reciprocal matrices

Prior to reviewing the consistency indexes of interval additive reciprocal matrices, it is convenient to recall the consistency indexes of additive reciprocal matrices. One can see from Definition 2 that the application of Eq. 2 leads to the additive reciprocity rij + rji = 1. Then the additive transitivity property was only considered to propose the consistency index of an additive reciprocal matrix in Herrera-Viedma et al. (2007a). However, when an additive reciprocal matrix R = (rij)n×n is not additively consistent, we cannot determine the reciprocity rij + rji = 1. In other words, the reciprocity rij + rji = 1 is the basic property of additive reciprocal matrices, which is in accordance with the findings in Saaty (1980) for the axiomatic properties of AHP. It is further found that three different ways to estimate the values of rij in Herrera-Viedma et al. (2007a, ??b) can yield the same result when applying the reciprocity rij + rji = 1 (Liu et al. 2014c). So the consistency index of an additive reciprocal matrix in Herrera-Viedma et al. (2007a, ??b) can be simplified and one has the following definition:

Definition 6

Herrera-Viedma et al. (2007a, ??b) Let R = (rij)n×n is an additive reciprocal matrix. The consistency index CI(R) is expressed as

$$ \begin{array}{@{}rcl@{}} CI(R)&=&1-\frac{4}{n(n-1)(n-2)}\sum\limits_{i=1}^{n}\sum\limits_{j=i+1}^{n}\sum\limits_{k=j+1}^{n}\left|r_{ij}\right.\\ &&\left.+r_{jk}-r_{ik}-0.5\right|. \end{array} $$
(8)

It is seen that the consistency index CI(R) in Liu and Zhang (2014a) was written in the following form:

$$ \begin{array}{@{}rcl@{}} CI(R)&=&\frac{1}{n^{2}-n} \sum\limits_{i=1}^{n}\sum\limits_{j=1,j\ne i}^{n}\\&&\times\left[1-\frac{2}{3}\left( \frac{1}{n-2}\sum\limits_{k=1,k\ne i,j}^{n}\left|r_{ij}+r_{jk}-r_{ik}-0.5\right|\right)\right]. \end{array} $$
(9)

Some computations reveal that Eq. 9 is in accordance with Eq. 8. Moreover, according to Definition 6, one can obtain 0 ≤ CI(R) ≤ 1. When CI(R) = 1, it means that R = (rij)n×n is additively consistent in virtue of Definition 2. The consistency index CI(R) in Eq. 8 is an average deviation of R from an additive reciprocal matrix with additive consistency. The larger the value of CI(R) is, the more consistent R is. In addition, it is found that the consistency index CI(R) in Definition 6 is invariant under the permutations of alternatives.

In what follows, we review the known consistency indexes of interval additive reciprocal matrices. For convenience, it is assumed that all additive reciprocal matrices R = (rij)n×n with \(r_{ij}\in [r_{ij}^{-}, r_{ij}^{+}]\) form a set denoted by P(R). Based on the consistency definition of interval additive reciprocal matrices in Xu and Chen (2008), a consistency index is proposed as follows:

Definition 7

Xu and Chen (2008), Dong et al. (2016), and Fedrizzi and Pereira (1995) Let \(\tilde {R}=([r^{-}_{ij}, r^{+}_{ij}])_{n\times n}\) be an interval additive reciprocal matrix. If there is a matrix R = (rij)n×n such that CI(R) = 1, then \(\tilde {R}\) is considered to be of additive consistency. The classical consistency index (CCI) is defined as follows:

$$ CCI(\tilde{R})=\max\limits_{R\in P(R)}CI(R). $$
(10)

It is seen from Eq. 10 that CCI is only dependent on an additive reciprocal matrix associated with \(\tilde {R}=([r^{-}_{ij}, r^{+}_{ij}])_{n\times n}.\) The additive reciprocity of interval additive reciprocal matrix is not considered and the inconsistency degree cannot be characterized perfectly.

Moreover, for a particular permutation σ = (1,2,⋯ , n), Liu et al. (2014c) defined a consistency index of interval additive reciprocal matrices as follows:

Definition 8

Liu and Zhang (2014a) The consistency level of an interval additive reciprocal preference relation \(\tilde {R}=([r^{-}_{ij}, r^{+}_{ij}])_{n\times n}\) is defined as

$$ LCI(\tilde{R})=\frac{CI(P)+CI(Q)}{2}, $$
(11)

where P and Q are obtained from \(\tilde {R}\) through (7) under the permutation σ = (1,2,⋯ , n). CI(P) and CI(Q) stand for the values of the consistency index in Eq. 8.

One can see from Definition 8 that two boundary additive reciprocal matrices together with the reciprocity of \(\tilde {R}\) have been considered in the consistency index. However, it is found that when changing the permutation of alternatives, the values of the consistency index (11) are changed. In other words, the invariance under the permutations of alternative cannot hold, which can also be derived from the findings in Liu et al. (2018a).

It is worth noting that the consistency indexes in Definitions 6 and 7 are related to particular additive reciprocal matrices. Dong et al. (2016) considered that the consistency index should be dependent on all additive reciprocal matrices R ∈ P(R). Hence, the average-case consistency index is proposed as in the following definition:

Definition 9

Dong et al. (2016) Assume that \(\tilde R\) is an interval additive reciprocal matrix. The average-case consistency index of \(\tilde R\) is defined as

$$ ACI(\tilde R)=E(CI(R)), $$
(12)

where R ∈ P(R) is the randomly created additive reciprocal matrix and E(CI(R)) denotes the expected value of the consistency index of R.

When the consistency index in Eq. 6 is used, the average-case consistency index is computed as (Dong et al. 2016):

$$ \begin{array}{@{}rcl@{}} ACI(\tilde R)&=&1-\frac{4}{n(n-1)(n-2)}\sum\limits_{i=1}^{n}\sum\limits_{j=i+1}^{n}\sum\limits_{k=j+1}^{n}E\left|r_{ij}\right.\\ &&\left.+r_{jk}-r_{ik}-0.5\right|. \end{array} $$
(13)

We can find from the results in Dong et al. (2016) that if and only if \(\tilde {R}\) degenerates to an additive reciprocal matrix with additive consistency, we have \(ACI(\tilde R)=1.\) This implies that the average-case consistency index has reflected the derivation degree of an interval additive reciprocal matrix from an additive reciprocal matrix with additive consistency. However, as shown in Dong et al. (2016), in order to compute the values of the average-case consistency index, a complex computation procedure should be implemented.

In addition, Wan et al. (2018) defined a geometric consistency index for an interval additive reciprocal matrix as follows:

Definition 10

Wan et al. (2018) The geometric consistency index for an interval additive reciprocal matrix \(\tilde {R}=([r^{-}_{ij}, r^{+}_{ij}])_{n\times n}\) is defined as

$$ GCI(\tilde R) = \delta MCI(\tilde R) + (1 - \delta )SCI(\tilde R), $$
(14)

where δ ∈ [0,1] is the altitude parameter. \(MCI(\tilde R)\) is the min-consistency index measuring the level of geometric consistency of an interval additive reciprocal matrix \(\tilde R\) by the Chebyshev distance, which is given as

$$ \begin{array}{@{}rcl@{}} MCI(\tilde{R})&=&\frac{1}{{C_{n}^{3}}}\sum\limits_{i=1}^{n}\sum\limits_{j=i+1}^{n}\sum\limits_{k=j+1}^{n}\left|[\ln r_{ij}^{-} + \ln r_{ij}^{+}+\ln r_{jk}^{-}+\ln r_{jk}^{+}\right.\\ &&+\ln(1 - r_{ik}^{-})+\ln(1 - r_{ik}^{+})]-[\ln r_{ik}^{-} + \ln r_{ik}^{+}+\ln(1 - r_{jk}^{+})\\ &&+\left.\ln(1 - r_{jk}^ - ) + \ln (1 - r_{ij}^ + ) + \ln (1 - r_{ij}^ - )]\right|. \end{array} $$

\(SCI(\tilde R)\) stands for the max-consistency index measuring the level of geometric consistency of an interval additive reciprocal matrix \(\tilde R\) by the Hamming distance defined as follows

$$ \begin{array}{@{}rcl@{}} SCI(\tilde R)&=&\max_{1\le i<j<k \le n} \{|[\ln r_{ij}^{-}+\ln r_{ij}^{+}+\ln r_{jk}^{-}+\ln r_{jk}^{+}+\ln(1-r_{ik}^{-})\\ &&+\ln(1-r_{ik}^{+})]-[\ln r_{ik}^{-}+\ln r_{ik}^{+}+\ln(1-r_{jk}^{-})\\ &&+\ln(1-r_{jk}^{+})+ \ln(1 - r_{ij}^ - ) + \ln(1 - r_{ij}^ - )]|\}. \end{array} $$

The geometric consistency index reflects the reliability of the information provided by the decision maker. Clearly, the smaller the value of \(GCI(\tilde R)\) is, the more consistent and reliable the information in interval additive reciprocal matrix \(\tilde R\) is. However, it is found that the geometric consistency index presented in Eq. 14 does not take the permutations of alternatives into account. The following example shows that the geometric consistency index (14) is dependent on the permutations of alternatives.

Example 1

Consider the following interval additive reciprocal matrix

$$\tilde{R}_{1} = \left( {\begin{array}{*{20}{c}} {\left[ {0.50,0.50} \right]}&{\left[ {0.45,0.55} \right]}&{\left[ {0.30, 0.65} \right]}&{\left[ {0.50, 0.75} \right]}\\ {\left[ {0.45,0.55} \right]}&{\left[ {0.50,0.50} \right]}&{\left[ {0.60, 0.85} \right]}&{\left[ {0.25, 0.65} \right]}\\ {\left[ {0.35,0.70} \right]}&{\left[ {0.15,0.40} \right]}&{\left[ {0.50,0.50} \right]}&{\left[ {0.35, 0.65} \right]}\\ {\left[ {0.25,0.50} \right]}&{\left[ {0.45,0.75} \right]}&{\left[ {0.35,0.65} \right]}&{\left[ {0.50,0.50} \right]} \end{array}} \right).$$

As an example, we choose δ = 0.5 to compute the geometric consistency index of \(\tilde {R}_{1}.\) The obtained results are given in Table 1, where the symbol \(\tilde {R}_{1}^{\sigma }\) denotes the interval-valued matrix derived by applying σ to \(\tilde {R}_{1}.\) One can see from Table 1 that the values of the geometric consistency index are changed with respect to the permutations of alternatives. This means that the inconsistency degree of \(\tilde {R}_{1}\) is changed under the permutations of alternatives.

Table 1 The values of the geometric consistency index \(\tilde {R}_{1}^{\sigma }\) with δ = 0.5 for all permutations of alternatives
Full size table

It is seen from the above analysis that the consistency index of interval additive reciprocal matrices has attracted some attention. According to the idea in Liu et al. (2018b), interval additive reciprocal matrices are inconsistent in nature due to the uncertainty. The permutations of alternatives correspond to the randomness experienced by decision makers in the pairwise comparison of alternatives. In order to measure the inconsistency degree, it is suitable to consider interval additive reciprocal matrices as the softened versions of additive reciprocal matrices with additive consistency. In what follows, we propose a novel consistency index of interval additive reciprocal matrices to incorporate the reciprocity and the permutations of alternatives.

3 A weak-consistency index of interval additive reciprocal matrices

In order to consider all permutations of alternatives in the consistency index of interval additive reciprocal matrices, we extend the method in Definition 10 and give the following definition:

Definition 11

Suppose that \(\tilde {R}=(\tilde {r}_{ij})_{n\times n }\) is an interval additive reciprocal matrix and σ is a permutation of {1,2,⋯ , n}. Pσ and Qσ are constructed from \(\tilde {R}_{\sigma }=(\tilde {r}_{\sigma (i)\sigma (j)})_{n\times n }\) in terms of Eq. 7. The inconsistency degree of \(\tilde {R}=(\tilde {r}_{ij})_{n\times n}\) is quantified by the following weak-consistency index

$$ WCI(\tilde{R})= \frac{1}{n!}{\sum}_{\sigma}\frac{CI(P_{\sigma})+CI(Q_{\sigma})}{2}, $$
(15)

where CI(Pσ) and CI(Qσ) are defined as in (8).

Here the terminology of weak-consistency index is used to distinguish that of the consistency index. The main reason is that we consider interval additive reciprocal matrices be inconsistent in essence (Liu et al. 2018a). One can see that all permutations of alternatives are involved in the weak-consistency index (15). It is easy to verify that \(0\leq WCI(\tilde {R})\leq 1\) since CI(Pσ) ∈ [0,1] and CI(Qσ) ∈ [0,1]. The larger the value of \(WCI(\tilde {R})\) is, the more approximate \(\tilde {R}\) is to an additive reciprocal matrix with additive consistency.

In what follows, let us further investigate the properties of the proposed weak-consistency index and give the following results.

Theorem 1

The interval-valued matrix \(\tilde {R}=(\tilde {r}_{ij})_{n\times n }\) degenerates to an additive reciprocal one with additive consistency, if and only if the value of \(WCI(\tilde {R})\) equals to 1.

Proof

When the value of the weak-consistency index \(WCI(\tilde {R})\) is equal to 1, according to Eq. 15, one has CI(Pσ) = CI(Qσ) = 1 for any permutation σ. It means that Pσ and Qσ are additively consistent for any permutation σ. According to the findings in Liu et al. (2018b), \(\tilde {R}=(\tilde {r}_{ij})_{n\times n }\) degenerates to an additive reciprocal preference relation with additive consistency.

Conversely, when \(\tilde {R}=(\tilde {r}_{ij})_{n\times n }\) degenerates to an additive reciprocal preference relation with additive consistency R = (rij)n×n, we have Pσ = Qσ for any permutation σ. Consequently, one has \(WCI(\tilde {R})=CI(R)=1\) and the proof is complete. □

It is seen from Theorem 1 that the weak-consistency index (15) can be used to describe the deviation degree of interval additive reciprocal matrices from additive reciprocal matrices with additive consistency. When the value of the weak-consistency index (15) approaches 1, then \(\tilde {R}=(\tilde {r}_{ij})_{n\times n}\) is approximately equal to an additive reciprocal preference relation with additive consistency. The above observation is in accordance with that obtained by the average-case consistency index in Eq. 13. Generally, for an interval additive reciprocal matrix, the values of the weak-consistency index (15) are less than 1 and greater than zero. In other words, the values of the weak-consistency index quantify the degree to which interval additive reciprocal matrices differ from additive reciprocal matrices with additive consistency. Furthermore, we find that the weak-consistency index is invariant for any permutation σ. That is, we have the following result:

Theorem 2

Let \(\tilde {R}=(\tilde {r}_{ij})_{n\times n }\) be an interval additive reciprocal matrix. The weak-consistency index \(WCI(\tilde {R})\) is invariant with respect to the permutations of alternatives.

Proof

It is easy to see that all permutations of alternatives have been considered in Eq. 15. In fact, \(WCI(\tilde {R})=WCI(\tilde {R}^{\sigma })\) means that \(WCI(\tilde {A})\) is invariant with respect to any permutation. □

On the other hand, in order to compute the value of the weak-consistency index (15), one should carry out n! pairs of CI(Pσ) and CI(Qσ). As shown in Liu et al. (2018b), the times of computing CI(Pσ) and CI(Qσ) can be reduced by a half. Now we introduce a lemma as follows (Liu et al., 2018):

Lemma 1

It is assumed that \(I=\{1,2,\dots ,n\}\) and ∀i, j, m, k ∈ I for i≠j and m≠k. \(\tilde {\sigma }\) and \(\hat {\sigma }\) are two permutations of \(\{1,2,\dots ,n\}\) with \(\tilde {\sigma }(i)=\hat {\sigma }(m)\) and \(\tilde {\sigma }(j)=\hat {\sigma }(k)\). \(\tilde {R}^{\tilde {\sigma }}\) and \(\tilde {R}^{\hat {\sigma }}\) are derived from \(\tilde {R}\) under the applications of \(\tilde {\sigma }\) and \(\hat {\sigma },\) respectively. The entries above the diagonal of \(R^{\tilde {\sigma }}\) are the same as those below the diagonal of \(\tilde {R}^{\hat {\sigma }}\), and the entries below the diagonal of \(\tilde {R}^{\hat {\sigma }}\) are identical to those above the diagonal of \(\tilde {R}^{\tilde {\sigma }},\) if and only if (i − j)(m − k) < 0.

Furthermore, we have the following result:

Theorem 3

Suppose that \(\tilde {R}^{\tilde {\sigma }}\) and \(\tilde {R}^{\hat {\sigma }}\) are two interval additive reciprocal matrices obtained by applying \(\tilde {\sigma }=(i_{1}, i_{2},\dots ,i_{n})\) and \(\hat {\sigma }=(i_{n},i_{n-1},\dots ,i_{1})\) to \(\tilde {R}=(\tilde {r}_{ij})_{n\times n},\) respectively. The values of \(LCI(\tilde {R}^{\tilde {\sigma }})\) and \(LCI(\tilde {R}^{\hat {\sigma }})\) (Definition 8) are identical.

Proof

The proof is straightforward by considering Theorem 4 in Liu et al. (2018a) and the consistency index of additive reciprocal matrices presented in Definition 6. The details are omitted here. □

According to Theorem 3, we only need to compute n!/2 pairs of CI(Pσ) and CI(Qσ). The n!/2 pairs of Pσ and Qσ can be formed through the method given in Liu et al. (2018a).

In order to illustrate the difference from the existing inconsistency indexes (11), (13) and (14), some theoretical and numerical investigations should be offered. From the viewpoint of theoretical analysis, the new inconsistency index (15) is the improved version of Eq. 11 by incorporating the effects of permutations of alternatives. Moreover, it is seen that the two inconsistency indexes (14) and (15) are all based on the boundary values of interval additive reciprocal matrices. The advantage of the proposed index (15) is to overcome the existing shortcoming of Eq. 14. The ideas underlying both (13) and (15) are different. The inconsistency index (13) is based on the contribution of a random variable in an interval. The proposed index (15) is to capture the uncertainty of an interval by using the boundary points. When the values of the two indexes (13) and (15) are tending to a limiting one, the additive reciprocal matrix with additive consistency is retrieved. But the inconsistency index (15) is more powerful to distinguish different interval-valued comparison matrices. The above phenomena can be observed by carrying out the following example.

Example 2

Consider the four interval additive reciprocal matrices \(\tilde {V}_{1}\), \(\tilde {V}_{2}\), \(\tilde {V}_{3}\) and \(\tilde {V}_{4}\) (Dong et al. 2016):

$$\tilde {V}_{1} = \left[ {\begin{array}{*{20}{c}} {\left[ {0.5,0.5} \right]}&{\left[ {0.7,1.0} \right]}&{\left[ {0.1,0.4} \right]}&{\left[ {0.7,1.0} \right]}\\ {\left[ {0.0,0.3} \right]}&{\left[ {0.5,0.5} \right]}&{\left[ {0.2,1.0} \right]}&{\left[ {0.2,0.5} \right]}\\ {\left[ {0.6,0.9} \right]}&{\left[ {0,0.8} \right]}&{\left[ {0.5,0.5} \right]}&{\left[ {0.4,0.8} \right]}\\ {\left[ {0,0.3} \right]}&{\left[ {0.5,0.8} \right]}&{\left[ {0.2,0.6} \right]}&{\left[ {0.5,0.5} \right]} \end{array}} \right],$$
$$\tilde {V}_{2} = \left[ {\begin{array}{*{20}{c}} {\left[ {0.5,0.5} \right]}&{\left[ {0.4,1.0} \right]}&{\left[ {0.2,0.4} \right]}&{\left[ {0,0.5} \right]}\\ {\left[ {0.0,0.6} \right]}&{\left[ {0.5,0.5} \right]}&{\left[ {0.5,1.0} \right]}&{\left[ {0.6,0.8} \right]}\\ {\left[ {0.6,0.8} \right]}&{\left[ {0,0.5} \right]}&{\left[ {0.5,0.5} \right]}&{\left[ {0.6,1.0} \right]}\\ {\left[ {0.5,1.0} \right]}&{\left[ {0.2,0.4} \right]}&{\left[ {0,0.4} \right]}&{\left[ {0.5,0.5} \right]} \end{array}} \right],$$
$$\tilde {V}_{3} = \left[ {\begin{array}{*{20}{c}} {\left[ {0.5,0.5} \right]}&{\left[ {0,1} \right]}&{\left[ {0.4,0.6} \right]}&{\left[ {0.4,0.6} \right]}\\ {\left[ {0.0,1.0} \right]}&{\left[ {0.5,0.5} \right]}&{\left[ {0.2,0.8} \right]}&{\left[ {0.3,0.7} \right]}\\ {\left[ {0.4,0.6} \right]}&{\left[ {0.2,0.8} \right]}&{\left[ {0.5,0.5} \right]}&{\left[ {0.3,0.7} \right]}\\ {\left[ {0.4,0.6} \right]}&{\left[ {0.3,0.7} \right]}&{\left[ {0.3,0.7} \right]}&{\left[ {0.5,0.5} \right]} \end{array}} \right],$$
$$\tilde {V}_{4} = \left[ {\begin{array}{*{20}{c}} {\left[ {0.5,0.5} \right]}&{\left[ {0,0.6} \right]}&{\left[ {0.1,0.2} \right]}&{\left[ {0.3,0.4} \right]}\\ {\left[ {0.4,1.0} \right]}&{\left[ {0.5,0.5} \right]}&{\left[ {0,0.8} \right]}&{\left[ {0.3,0.7} \right]}\\ {\left[ {0.8,0.9} \right]}&{\left[ {0.2,1.0} \right]}&{\left[ {0.5,0.5} \right]}&{\left[ {0.3,0.7} \right]}\\ {\left[ {0.6,0.7} \right]}&{\left[ {0.3,0.7} \right]}&{\left[ {0.3,0.7} \right]}&{\left[ {0.5,0.5} \right]} \end{array}} \right].$$

Using Eqs. 11, 14 and 15, we obtain the values of LCI, GCI and WCI, respectively. In addition, combining with the values of CCI and ACI in Dong et al. (2016), the computing results are given in Table 2, where the values of GCI are obtained by assuming δ = 0.5 in Eq. 14. It is seen from Table 2 that the values of CCI for \(\tilde {V}_{i}\) (i = 1,2,3,4) all equal to 1. This means that the differences among \(\tilde {V_{i}}(i=1,2,3,4)\) cannot be revealed by using CCI. In fact, CCI only depends on a particular additive reciprocal matrix R ∈ P(R). The value of 1 shows that there exists an additive reciprocal matrix with additive consistency. Moreover, it is found from \(\tilde {V}_{3}\) that \(CCI(\tilde {V}_{3})=ACI(\tilde {V}_{3})=1.\) According to Corollary 1 in Dong et al. (2016), \(ACI(\tilde {V}_{3})=1\) if and only if \(\tilde {V}_{3}\) is an additive reciprocal matrix with additive consistency. However, \(\tilde {V}_{3}\) is not an additive reciprocal matrix but an interval additive reciprocal matrix. There is a shortcoming here. In addition, it is seen that the values of GCI for \(\tilde {V}_{i}(i=1,2,3,4)\) are all null. The reason is that there are zero boundary values in the interval entries of \(\tilde {V_{i}}(i=1,2,3,4)\) and \(\ln 0\) is undefined. This means that GCI is not suitable to quantify the inconsistency degrees of \(\tilde {V}_{i}(i=1,2,3,4).\) From the values of LCI and WCI, one can see that four interval-valued matrices can be distinguished. All the values are less than 1, meaning that \(\tilde {V_{i}}(i=1,2,3,4)\) are the softened versions of additive reciprocal matrices with additive consistency and they are inconsistent. The WCI is an extension of the LCI under the consideration of the randomness experienced by decision makers. In conclusion, WCI is more suitable than LCI to quantify the inconsistency degree of interval additive reciprocal matrices.

Table 2 The values of CCI, ACI, GCI, LCI and WCI for \(\tilde {V}_{i}\) (i = 1,2,3,4) respectively
Full size table

4 Application to group decision making

In this section, we apply the weak-consistency index of interval multiplicative reciprocal matrices to a group decision making problem. Suppose that there are a set of alternatives X = {x1, x2,…, xn} and a group of experts E = {e1, e2,…, em}. The experts eτ(τ = 1,2,…, m) express their opinions as interval additive reciprocal matrices \(\tilde {R}^{\tau }\) by pairwisely comparing alternatives in X. It is seen that one of the important issues in group decision making problems is the aggregation of individual decision information (Yager 1988, 1993; Xu and Da 2003; Liu 2006; Herrera-Viedma et al. 2007a). When additive reciprocal matrices are used to express the judgements of decision makers, the additive-consistency based IOWA operator was proposed in Herrera-Viedma et al. (2007a). The main idea in Herrera-Viedma et al. (2007a) is to reorder the preference relations according to their values of the consistency index. The more importance is assigned to the decision maker who provides more consistent decision information under the assumption of heterogeneous group decision making problems. Here the weak-consistency index is proposed to measure the inconsistency degree of interval additive reciprocal matrices. Similar to the method in Herrera-Viedma et al. (2007a), the WCI-based IOWA operator should be developed.

4.1 The aggregation operator

We first recall the IOWA operator as follows:

Definition 12

Yager and Filev (1999) An IOWA operator of dimension m is a function

$${{\Phi}_{\nu}}: {\left( {\mathbf{R}\times \mathbf{R}}\right)^{m}}\to \mathbf{R}$$

with the real number set R such that

$$ {{\Phi}_{\nu} }\left( {\left\langle{{\mu_{1}},{p_{1}}} \right\rangle , {\cdots} ,\left\langle {{\mu_{m}},{p_{m}}} \right\rangle } \right)={\sum}_{i=1}^{m} {{\nu_{i}}}\cdot{p_{\sigma \left( i \right)}}. $$
(16)

where ν = (ν1, ν2⋯ , νm) is the associated weight vector with \({\nu _{i}} \in \left [ {0,1} \right ]\) and \(\sum \limits _{i = 1}^{n} {{\nu _{i}} = 1} \). \(\left \langle {{\mu _{\sigma \left (i \right )}},{p_{\sigma \left (i \right )}}} \right \rangle \) is the pair with the i th highest value \({\mu _{\sigma \left (i \right )}}\) in the set \(\left \{ {{\mu _{1}},{\mu _{2}}, {\cdots } ,{\mu _{m}}} \right \}\). σ is a permutation of {1,2,⋯ , n} such that \({\mu _{\sigma \left (i\right )}} \ge {\mu _{\sigma \left ({i+1}\right )}}\) where i = 1,2,⋯ , m − 1.

In order to aggregate interval additive reciprocal matrices, the IOWA operator has been used to construct the ideal interval-valued matrix and determine the collective one in Liu and Zhang (2014a). Here based on the weak-consistency index (15), the WCI-IOWA operator is proposed as follows:

Definition 13

Suppose that ν = (ν1, ν2,…, νm) is the associated exponential weight vector with νi ∈ [0,1] and \({\sum }^{m}_{i=1}\nu _{i}=1.\) \(\tilde {R}^{\tau }=(\tilde {r}_{ij}^{\tau })_{n\times n}=([r_{ij}^{-\tau }, r_{ij}^{+\tau }])_{n\times n} \) (τ = 1,2,…, m) are interval additive reciprocal matrices and \(\tilde {R}^{c}=(\tilde {r}_{ij}^{c})_{n\times n}\) is a collective matrix. The order-inducing variable set is

$$\{WCI(\tilde{R}^{1}), WCI(\tilde{R}^{2}), \ldots, WCI(\tilde{R}^{m})\}$$

and the argument variable is \(\tilde {r}_{ij}^{\tau }.\) The WCI-IOWA operator is defined as

$$ \begin{array}{@{}rcl@{}} &&f(\langle WCI(\tilde{R}^{1}), \tilde{r}_{ij}^{1}\rangle, \langle WCI(\tilde{R}^{2}), \tilde{r}_{ij}^{2}\rangle, \ldots,\\ &&\langle WCI(\tilde{R}^{m}), \tilde{r}_{ij}^{m}\rangle)=\tilde{r}_{ij}^{c}, \end{array} $$
(17)

where \(\tilde {r}_{ij}^{c}={\sum }^{m}_{k=1}{\nu _{k}}(\tilde {\upbeta }_{ij}^{k})=[{\sum }^{m}_{\tau =1}\nu _{\tau } ({\upbeta }_{ij}^{- \tau }),\) \({\sum }^{m}_{\tau =1}{\nu _{\tau }}({\upbeta }_{ij}^{+ \tau })]\) (i, j = 1,2,…, n) and \(WCI(\tilde {R}^{\tau })\) stands for the WCI of \(\tilde {R}^{\tau }.\) Moreover, \(\tilde {\upbeta }_{ij}^{k}\) is the term \(\tilde {r}_{ij}^{\tau }\) in \(\langle WCI(\tilde {R}^{\tau }),\)\(\tilde {r}_{ij}^{\tau }\rangle \) such that \(WCI(\tilde {R}^{k})\) is the k th-highest value in the set as follows:

$$\{WCI(\tilde{R}^{1}), WCI(\tilde{R}^{2}), \ldots, WCI(\tilde{R}^{m})\}.$$

As shown in Definition 13, the matrices \(\tilde {R}^{\tau }\) (τ = 1,2,⋯ , m) are reordered and aggregated by their values of the weak-consistency index. The larger the value of the weak-consistency index is, the more forward the order of the interval additive reciprocal matrix is.

Moreover, the investigation of the WCI-IOWA operator leads to the following result:

Theorem 4

The collective interval additive reciprocal matrix \(\tilde {R}^{c}=(\tilde {r}_{ij}^{c})_{n\times n}\) is of additive approximation-consistency, if there is a permutation σ such that individual interval-valued preference relations \(\tilde {R}^{\tau }(\tau =1,2,\ldots ,m)\) are of additive approximation-consistency.

Proof

In virtue of Eq. 7, a series of additive reciprocal matrices can be constructed from \(\tilde {R}^{c}\) and \(\tilde {R}^{\tau }(\tau =1,2,\ldots ,m).\) They are written as \(P^{c}_{\sigma }=(p_{ij}^{c})_{n \times n},\) \(Q^{c}_{\sigma }=(q_{ij}^{c})_{n\times n},\)\(P^{\tau }_{\sigma }=(p_{\sigma (i)\sigma (j)}^{\tau })_{n\times n}\) and \(Q^{\tau }_{\sigma }=(q_{\sigma (i)\sigma (j)}^{\tau })_{n\times n},\) respectively. Since there exists a permutation σ such that all interval additive reciprocal matrices \(\tilde {R}^{\tau }(\tau =1,2,\ldots ,m)\) are of additive approximation-consistency, the corresponding matrices \(P^{\tau }_{\sigma }\) and \(Q^{\tau }_{\sigma }\) (τ = 1,2,…, m) are additively consistent by virtue of Definition 5. That is, we have the following relations

$$p_{\sigma(i)\sigma(j)}^{\tau}=p_{\sigma(i)\sigma(k)}^{\tau}+p_{\sigma(k)\sigma(j)}^{\tau}-0.5,$$
$$q_{\sigma(i)\sigma(j)}^{\tau}=q_{\sigma(i)\sigma(k)}^{\tau}+q_{\sigma(k)\sigma(j)}^{\tau}-0.5,$$

for ∀i, j, k = 1,2,⋯ , n. Then, using Definition 13, it follows that

$$ \begin{array}{@{}rcl@{}} p_{\sigma(i)\sigma(j)}^{c}&=&{\sum}_{\tau=1}^{m} {\nu_{\tau}}\cdot p_{\sigma(i)\sigma(j)}^{\tau} \\ &=&{\sum}_{\tau=1}^{m}\nu_{\tau}(p_{\sigma(i)\sigma(k)}^{\tau}+p_{\sigma(k)\sigma(j)}^{\tau}-0.5)\\ &=&{\sum}_{\tau=1}^{m}\nu_{\tau}\cdot p_{\sigma(i)\sigma(k)}^{\tau} + {\sum}_{\tau=1}^{m}\nu_{\tau}\cdot p_{\sigma(k)\sigma(j)}^{\tau}-0.5\\ &=&p_{\sigma(i)\sigma(k)}^{c}+p_{\sigma(k)\sigma(j)}^{c}-0.5, \end{array} $$
$$ \begin{array}{@{}rcl@{}} q_{\sigma(i)\sigma(j)}^{c}&=&{\sum}_{\tau=1}^{m} {\nu_{\tau}}\cdot q_{\sigma(i)\sigma(j)}^{\tau} \\ &=&{\sum}_{\tau=1}^{m}\nu_{\tau}(q_{\sigma(i)\sigma(k)}^{\tau}+q_{\sigma(k)\sigma(j)}^{\tau}-0.5)\\ &=&{\sum}_{\tau=1}^{m}\nu_{\tau}\cdot q_{\sigma(i)\sigma(k)}^{\tau} + {\sum}_{\tau=1}^{m}\nu_{\tau}\cdot q_{\sigma(k)\sigma(j)}^{\tau}-0.5\\ &=&q_{\sigma(i)\sigma(k)}^{c}+q_{\sigma(k)\sigma(j)}^{c}-0.5. \end{array} $$

This implies that \(P^{c}=(p_{ij}^{c})_{n\times n}\) and \(Q^{c}=(q_{ij}^{c})_{n\times n}\) are of additive consistency for the permutation σ. According to Definition 5, \(\tilde {R}^{c}\) is of additive approximation-consistency and the proof is completed. □

Theorem 4 shows that when all individual matrices \(\tilde {R}^{\tau }(\tau =1,2,\ldots ,m)\) are additive approximation-consistency for a permutation σ, the property of additive approximation-consistency holds for the collective one.

4.2 The associated weights

Furthermore, when using the IOWA operator, an important issue is how to determine the associated weight vector ν = (ν1, ν2⋯ , νm). Following the idea in Herrera-Viedma et al. (2007a), we offer the more importance to the bigger value of the weak-consistency index. It is assumed that

$$T={\sum}_{\tau=1}^{m}WCI(\tilde{R}^{\tau}),$$

which stands for the total sum of the importance of experts. Since \(\tilde {R}^{\tau }\) (τ = 1,2,…, m) are interval additive reciprocal matrices, the value of T is always greater than zero according to Theorem 1. Then the normalized weak-consistency index is defined as

$$ \overline{WCI}(\tilde{R}^{\tau})=\frac{WCI(\tilde{R}^{\tau})}{T},\tau=1,2,\ldots, m. $$
(18)

Moreover, it is supposed that the order of \(\tilde {R}^{\tau }\) (τ = 1,2,…, m) is rearranged from large to small with respect to the values of \(\overline {WCI}(\tilde {R}^{\tau })\) as \(\tilde {R}^{\bar {\sigma }(i)}\) (i = 1,2,…, m). Hereafter \(\bar {\sigma }\) stands for the permutation of {1,2,⋯ , m}. Based on the suggestions in Yager (1996) for the weight vector associated with the OWA operator, the weight \(\nu _{\bar {\sigma }(i)}\) corresponding to the expert \(e_{\bar {\sigma }(i)}\) can be further determined as

$$ \nu_{\bar{\sigma}(i)}=\left( {\sum}_{\tau=1}^{i}\overline{WCI}(\tilde{R}^{\bar{\sigma}(\tau)})\right)^{\alpha}- \left( {\sum}_{\tau=1}^{i-1}\overline{WCI}(\tilde{R}^{\bar{\sigma}(\tau)})\right)^{\alpha}, $$
(19)

where 0 < α < 1 and 2 ≤ i ≤ m. When i = 1, one has the value of \(\nu _{\bar {\sigma }(1)}\) as follows

$$ \nu_{\bar{\sigma}(1)}=\overline{WCI}(\tilde{R}^{\bar{\sigma}(1)}). $$
(20)

It is found from Eqs. 19 and 20 that \(0<\nu _{\bar {\sigma }(i)}\leq 1\) (i = 1,2,…, m) and

$$ \nu_{\bar{\sigma}(1)}+\nu_{\bar{\sigma}(2)}+\cdots+\nu_{\bar{\sigma}(m)}= \left( {\sum}_{\tau=1}^{m}\overline{WCI}(\tilde{R}^{\bar{\sigma}(\tau)})\right)^{\alpha}=1. $$
(21)

Moreover, under the assumption of

$$ t_{i}={\sum}_{\tau=1}^{i}\overline{WCI}(\tilde{R}^{\bar{\sigma}(\tau)}), $$

we have ti+ 1 > ti and

$$ \nu_{\bar{\sigma}(i+1)}-\nu_{\bar{\sigma}(i)}=2\left( \frac{t_{i+1}^{\alpha}+t_{i-1}^{\alpha}}{2}-t_{i}^{\alpha}\right)<0, $$

where the convexity property of the function f(t) = tα (0 < α < 1) has been used. It is easy to show that \(\nu _{\bar {\sigma }(1)}>\nu _{\bar {\sigma }(2)}>\cdots >\nu _{\bar {\sigma }(m)}\). Since \(\overline {WCI}(\tilde {R}^{\bar {\sigma }(i)})>\overline {WCI}(\tilde {R}^{\bar {\sigma }(i+1)}),\) it means that the more importance is given to that with more consistency by the method of determining the weight vector, which is similar to known findings in the literature (Herrera et al. 2001; Herrera-Viedma et al. 2007a; Liu et al. 2014a, 2006).

When determining the collective interval additive reciprocal matrix \(\tilde {R}^{c},\) the remaining issue is how to obtain the priority weights of alternatives and choose the best alternative(s). Usually, one can directly apply \(\tilde {R}^{c}\) to derive the priority weights of alternatives and give the ranking. As shown in Liu et al. (2018b), under a permutation σ, we obtain the interval priority weight of the alternative xk as follows:

$$\omega_{k}^{\sigma}=[\min\{\omega_{k}(P_{\sigma}^{c}),\omega_{k}(Q_{\sigma}^{c})\},\max\{\omega_{k}(P_{\sigma}^{c}),\omega_{k}(Q_{\sigma}^{c})\}].$$

Since the randomness experienced by decision makers in comparing alternatives pairwise is considered in the study, the mean value of the interval priority weights under all the permutations should be computed to obtain the final weight (Liu et al. 2017a, 2018a). That is, it gives

$$\omega_{k}=\frac{1}{n!}{\sum}_{\sigma}\omega_{k}^{\sigma},$$

and the interval weight vector

$$ \omega=(\omega_{1},\omega_{2},\dots,\omega_{n}). $$
(22)

Finally, it should be pointed out that the collective interval additive reciprocal matrix \(\tilde {R}^{c}\) should be adjusted to that with additive approximation-consistency. The basic reason is due to enhancing the logical consistency of decision making. The additive approximation-consistency of \(\tilde {R}^{c}\) can be archived by using the direct approach such as that in Liu et al. (2018a) and Theorem 4, respectively. On the other hand, the sensitivity analysis with respect to the values of the weak-consistency index can also be performed to investigate the effects on the ranking of alternatives (Wan et al. 2018). In our study, Theorem 4 is applied to ensure the additive approximation-consistency of \(\tilde {R}^{c}.\) Numerical examples are carried out to illustrate the proposed model and some comparisons are offered in the next section.

5 The algorithm and discussion

For the purpose of illustrating the proposed index and model, we outline an algorithm for group decision making with interval additive reciprocal matrices.

  • Step 1: Assume that there are a set of alternatives X = {x1, x2,…, xn} and a group of experts eτ(τ = 1,2,…, m) in a group decision making problem. Interval additive reciprocal matrices \(\tilde {R}^{\tau }(\tau =1,2,\ldots ,m)\) are given through comparing pairwise the alternatives in X.

  • Step 2: The values of the weak-consistency index (15) for \(\tilde {R}^{\tau }\) (τ = 1,2,…, m) are computed.

  • Step 3: The associated exponential weight vector ν = (ν1, ν2,…, νm) is obtained through (19) and (20). The sensitivity analysis is made under different values of the parameter α.

  • Step 4: Additive approximation-consistency of \(\tilde {R}^{\tau }\) (τ = 1,2,…, m) is checked by testing the additive consistency of \(P_{\sigma }^{\tau } \) and \(Q_{\sigma }^{\tau } .\) If there is a permutation σ such that \(P_{\sigma }^{\tau } \) and \(Q_{\sigma }^{\tau }\) (τ = 1,2,…, m) are all of additive consistency, one can go directly to the next step. Otherwise, any one of them can be adjusted to that with additive consistency by the consistency improving method (Ma et al. 2006).

  • Step 5: Based on the aggregation operator (17), a collective interval additive reciprocal matrix \(\tilde {R}^{c}\) is determined.

  • Step 6: Using Eq. 22, the interval priority weights of alternatives are obtained.

  • Step 7: By applying the possibility degree formula in Liu (2009), the possibility degree matrix is formed.

  • Step 8: A simple row-column elimination method in Wang et al. (2005) is used to generate the ranking vector from the obtained possibility degree matrix.

  • Step 9: End.

In what follows, an example in Wan et al. (2018) is recomputed to illustrate the proposed group decision making model and compare it with the existing ones. The sensitivity analysis is carried out by considering the variations of the parameter α.

Example 3

With the increasing development of business, a new enterprise resource planning (ERP) system should be installed in a firm. Four ERP systems named as x1, x2, x3 and x4 are considered to be the alternatives after preliminary screening. A committee with three decision makers e1, e2 and e3 are invited to make a final decision. through pairwisely comparing alternatives, and all decision makers express their opinions as interval additive reciprocal matrices \(\tilde {R}^{\tau }(\tau =1,2,3)\) (Wan et al. 2018):

$$\tilde{R}^{1}=\left[ {\begin{array}{*{20}{c}} {\left[ {0.50,0.50} \right]}&{\left[ {0.65,0.80} \right]}&{\left[ {0.50,0.60}\right]}&{\left[{0.30,0.55}\right]}\\ {\left[ {0.20,0.35} \right]}&{\left[ {0.50,0.50} \right]}&{\left[ {0.30,0.55} \right]}&{\left[ {0.50,0.65} \right]}\\ {\left[ {0.40,0.50} \right]}&{\left[ {0.45,0.70} \right]}&{\left[ {0.50,0.50} \right]}&{\left[ {0.55,0.70} \right]}\\ {\left[ {0.45,0.70} \right]}&{\left[ {0.35,0.50} \right]}&{\left[ {0.30,0.45} \right]}&{\left[ {0.50,0.50} \right]} \end{array}} \right],$$
$$\tilde{R}^{2}=\left[ {\begin{array}{*{20}{c}} {\left[ {0.50,0.50} \right]}&{\left[ {0.65,0.70} \right]}&{\left[ {0.60,0.80}\right]}&{\left[{0.75,0.90}\right]}\\ {\left[ {0.30,0.35} \right]}&{\left[ {0.50,0.50} \right]}&{\left[ {0.50,0.60} \right]}&{\left[ {0.70,0.75}\right]}\\ {\left[ {0.20,0.40} \right]}&{\left[ {0.40,0.50} \right]}&{\left[ {0.50,0.50} \right]}&{\left[ {0.60,0.75}\right]}\\ {\left[ {0.10,0.25}\right]} &{\left[ {0.25,0.30} \right]}&{\left[ {0.25,0.40} \right]}&{\left[ {0.50,0.50}\right]} \end{array}} \right],$$
$$\tilde{R}^{3}=\left[ {\begin{array}{*{20}{c}} {\left[ {0.50,0.50} \right]}&{\left[ {0.45,0.55} \right]}&{\left[ {0.30,0.65}\right]}&{\left[{0.50,0.75}\right]}\\ {\left[ {045,0.55} \right]}&{\left[ {0.50,0.50} \right]}&{\left[ {0.60,0.85} \right]}&{\left[ {0.25,0.55}\right]}\\ {\left[ {0.35,0.70} \right]}&{\left[ {0.15,0.40}\right]}&{\left[ {0.50,0.50} \right]}&{\left[ {0.35,0.65}\right]}\\ {\left[ {0.25,0.50} \right]}&{\left[ {0.45,0.75} \right]}&{\left[ {0.35,0.65} \right]}&{\left[ {0.50,0.50}\right]} \end{array}} \right].$$

In virtue of Eq. 15, the values of WCI for \(\tilde {R}^{1},\) \(\tilde {R}^{2}\) and \(\tilde {R}^{3}\) are computed as \(WCI(\tilde {R}^{1})=0.1389\), \(WCI(\tilde {R}^{2})=0.0472\) and \(WCI(\tilde {R}^{3})=0.1611,\) respectively. Then the associated weights of the WCI-IOWA operator can be obtained according to Eqs. 19 and 20. By choosing α = 0.1,0.3,0.5,0.7 and 0.9, the values of ν = (ν1, ν2, ν3) are given in Table 3. It is seen from Table 3 that one has ν3 > ν1 > ν2 for any value of α, meaning that the variations of α have no influence on the ranking of the associated weights. In other words, the ranking of the associated weights is insensitive to the parameter α. In addition, with the increasing of α, the values of ν1 and ν2 are increasing, and the values of ν3 are decreasing. This means that if one wants to emphasize the expert with more consistency, the value of α should be small.

Table 3 The associated exponential weight vector in Example 3
Full size table

Moreover, let us check the additive approximation-consistency of \(\tilde {R}^{1},\) \(\tilde {R}^{2}\) and \(\tilde {R}^{3},\) respectively. Based on Definition 5, if we adopt the permutation σ = (1,2,3,4), the additive reciprocal matrices Pσ and Qσ constructed from \(\tilde {R}^{3}\) are additively consistent. This implies that \(\tilde {R}^{3}\) is of additive approximation-consistency. Similarly, \(\tilde {R}^{1}\) and \(\tilde {R}^{2}\) are proved not to be with additive approximation consistency and they should be adjusted. As shown in Theorem 4, in order to ensure the collective matrix \(\tilde {R}^{c}\) be additive approximation-consistency, the permutation σ = {1,2,3,4} is chosen to adjust \(\tilde {R}_{1}\) and \(\tilde {R}_{2}\) to those with additive approximation-consistency. Applying the consistency improving method in Ma et al. (2006), the adjusted interval additive reciprocal matrices are determined as follows:

$$ \begin{array}{@{}rcl@{}}\tilde{R}^{1}_{a} = \left[ {\begin{array}{*{20}{c}} {\left[ {0.5000,0.5000} \right]}&{\left[ {0.6425,0.8000} \right]}&{\left[ {0.4925,0.6000}\right]}&{\left[{0.3150,0.5500}\right]}\\ {\left[ {0.2000,0.3575} \right]}&{\left[ {0.5000,0.5000} \right]}&{\left[ {0.3025,0.5500} \right]}&{\left[ {0.4875,0.6500} \right]}\\ {\left[ {0.4000,0.5075} \right]}&{\left[ {0.4500,0.6975} \right]}&{\left[ {0.5000,0.5000} \right]}&{\left[ {0.5475,0.7000} \right]}\\ {\left[ {0.4500,0.6850} \right]}&{\left[ {0.3500,0.5125} \right]}&{\left[ {0.3000,0.4525} \right]}&{\left[ {0.5000,0.5000} \right]} \end{array}} \right],\\ \tilde{R}^{2}_{a}=\left[ {\begin{array}{*{20}{c}} {\left[ {0.5000,0.5000} \right]}&{\left[ {0.6463,0.7000} \right]}&{\left[ {0.6025,0.8000}\right]}&{\left[{0.7512,0.9000}\right]}\\ {\left[ {0.3000,0.3537} \right]}&{\left[ {0.5000,0.5000} \right]}&{\left[ {0.5012,0.6000} \right]}&{\left[ {0.6950,0.7500}\right]}\\ {\left[ {0.2000,0.3975} \right]}&{\left[ {0.4000,0.4988} \right]}&{\left[ {0.5000,0.5000} \right]}&{\left[ {0.6038,0.7500}\right]}\\ {\left[ {0.1000,0.2488}\right]} &{\left[ {0.2500,0.3050} \right]}&{\left[ {0.2500,0.3962} \right]}&{\left[ {0.5000,0.5000}\right]} \end{array}} \right]. \end{array} $$

By using the aggregation operator (17), the collective interval multiplicative reciprocal matrices \(\tilde {R}^{c}_{\alpha }\) (α = 0.1,0.3,0.5,0.7,0.9) are obtained as follows:

$$ \begin{array}{@{}rcl@{}}\tilde {R}^{c}_{0.1} = \left[{\begin{array}{*{10}{c}} {\left[ {0.5000,0.5000} \right]}&{\left[ {0.4643,0.5670} \right]}&{\left[ {0.3185,0.6492}\right]}&{\left[{0.4927,0.7403}\right]}\\ {\left[ {0.4330,0.5357} \right]}&{\left[ {0.5000,0.5000} \right]}&{\left[ {0.5810,0.8286} \right]}&{\left[ {0.2706,0.5588}\right]}\\ {\left[ {0.3508,0.6842} \right]}&{\left[ {0.1714,0.4190} \right]}&{\left[ {0.5000,0.5000} \right]}&{\left[ {0.3564,0.6544}\right]}\\ {\left[ {0.2597,0.5073}\right]} &{\left[ {0.4412,0.7294} \right]}&{\left[ {0.3456,0.6346} \right]}&{\left[ {0.5000,0.5000}\right]} \end{array}} \right],\\ \tilde {R}^{c}_{0.3} = \left[ {\begin{array}{*{20}{c}} {\left[ {0.5000,0.5000} \right]}&{\left[ {0.4898,0.5972} \right]}&{\left[ {0.3443,0.6483}\right]}&{\left[{0.4806,0.7239}\right]}\\ {\left[ {0.4028,0.5102} \right]}&{\left[ {0.5000,0.5000} \right]}&{\left[ {0.5477,0.7904} \right]}&{\left[ {0.3078,0.5749}\right]}\\ {\left[ {0.3517,0.6557} \right]}&{\left[ {0.2096,0.4523} \right]}&{\left[ {0.5000,0.5000} \right]}&{\left[ {0.3931,0.6624}\right]}\\ {\left[ {0.2761,0.5194}\right]} &{\left[ {0.4251,0.6922} \right]}&{\left[ {0.3376,0.6069} \right]}&{\left[ {0.5000,0.5000}\right]} \end{array}} \right],\\ \tilde {R}^{c}_{0.5} = \left[ {\begin{array}{*{20}{c}} {\left[ {0.5000,0.5000} \right]}&{\left[ {0.5117,0.6227} \right]}&{\left[ {0.3692,0.6482}\right]}&{\left[{0.4718,0.7110}\right]}\\ {\left[ {0.3774,0.4884} \right]}&{\left[ {0.5000,0.5000} \right]}&{\left[ {0.5198,0.7579} \right]}&{\left[ {0.3404,0.5890}\right]}\\ {\left[ {0.3519,0.6309} \right]}&{\left[ {0.2422,0.4803} \right]}&{\left[ {0.5000,0.5000} \right]}&{\left[ {0.4170,0.6695}\right]}\\ {\left[ {0.2891,0.5283}\right]} &{\left[ {0.4111,0.6597} \right]}&{\left[ {0.3306,0.5831} \right]}&{\left[ {0.5000,0.5000}\right]} \end{array}} \right],\\ \tilde {R}^{c}_{0.7} = \left[ {\begin{array}{*{20}{c}} {\left[ {0.5000,0.5000} \right]}&{\left[ {0.5304,0.6442} \right]}&{\left[ {0.3907,0.6487}\right]}&{\left[{0.4655,0.7009}\right]}\\ {\left[ {0.3558,0.4696} \right]}&{\left[ {0.5000,0.5000} \right]}&{\left[ {0.4964,0.7301} \right]}&{\left[ {0.3689,0.6013}\right]}\\ {\left[ {0.3514,0.6093} \right]}&{\left[ {0.2699,0.5036} \right]}&{\left[ {0.5000,0.5000} \right]}&{\left[ {0.4376,0.6757}\right]}\\ {\left[ {0.2991,0.5345}\right]} &{\left[ {0.3987,0.6311} \right]}&{\left[ {0.3244,0.5624} \right]}&{\left[ {0.5000,0.5000}\right]} \end{array}} \right],\\ \tilde {R}^{c}_{0.9} = \left[ {\begin{array}{*{20}{c}} {\left[ {0.5000,0.5000} \right]}&{\left[ {0.5465,0.6625} \right]}&{\left[ {0.4096,0.6496}\right]}&{\left[{0.4614,0.6933}\right]}\\ {\left[ {0.3376,0.4534} \right]}&{\left[ {0.5000,0.5000} \right]}&{\left[ {0.4769,0.7064} \right]}&{\left[ {0.3940,0.6122}\right]}\\ {\left[ {0.3503,0.5903} \right]}&{\left[ {0.2935,0.5230} \right]}&{\left[ {0.5000,0.5000} \right]}&{\left[ {0.4554,0.6810}\right]}\\ {\left[ {0.3066,0.5385}\right]} &{\left[ {0.3877,0.6059} \right]}&{\left[ {0.3189,0.5445} \right]}&{\left[ {0.5000,0.5000}\right]} \end{array}} \right]. \end{array} $$

According to Eq. 22, the interval priority weights of alternatives x1, x2, x3 and x4 can be determined as shown in Table 4. Based on the methods for ranking interval numbers in Liu (2009) and alternatives in Wang et al. (2005), the ranking of alternatives are given in Table 4. It is seen from Table 4 that the best alternative is x1, which is in agreement with that in Wan et al. (2018). As compared to the model in Wan et al. (2018), here we proposed a new consistency index to quantify the inconsistency degree of interval additive reciprocal matrices. The randomness experienced by decision makers in comparing alternatives is considered to overcome the shortcomings existing in the methods described in the literature so far. In addition, it is seen from Table 4 that when various values of the parameter α are used, the best alternative is not changed. This implies that the best solution of the group decision making problem is insensitive to the parameter α. But the ranking of the four alternatives is sensitive to the the parameter α, since there are x4 ≻ x3 for α = 0.1,0.3,0.5 and x3 ≻ x4 for α = 0.7,0.9, respectively.

Table 4 Interval priority weights of alternatives and the ranking of alternatives in Example 3
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6 Conclusions and further study

In order to cope with the uncertainty experienced by decision makers in comparing alternatives pairwise, interval additive reciprocal matrices have been extensively used to express the judgements. It is seen that interval-valued preference relations are inconsistent in nature and the inconsistency degree should be quantified. The known consistency indexes of interval additive reciprocal matrices have been analyzed and some shortcomings have been shown. By considering the randomness exhibited in comparison matrices, the weak-consistency index has been proposed. Some comparative studies reveal that the proposed weak-consistency index can be used to characterize the derivation degree of interval additive reciprocal matrices from additive reciprocal matrices with additive consistency. The existing shortcomings in the known consistency indexes have been resolved. Furthermore, a group decision making model has been proposed when the opinions of decision makers are expressed as interval additive reciprocal matrices. The WCI-IOWA operator has been proposed to aggregate individual interval additive reciprocal matrices. The properties of the aggregation operator have been studied. A new algorithm has been presented and an example to illustrate it has been reported. As compared to the existing methods, the proposed model has incorporated the uncertainty and the randomness experienced by decision makers in providing their judgements on alternatives.

In future work, we plan to use the weak-consistency index to deal with incomplete interval additive reciprocal matrices. Moreover, by considering the effects of the permutations of alternatives, some novel inconsistency indexes for various fuzzy-valued comparison matrices could be developed. Some group decision making models could be proposed under an environment of triangular fuzzy numbers, type-2 fuzzy numbers, intuitionistic fuzzy numbers and others.