Compatibility measurement-based group decision making with interval fuzzy preference relations

Abstract

In this paper, we put forward a ratio-based compatibility degree between any two ]0,1[-valued interval numbers to measure how proximate they approach to each other. A compatibility measurement is presented to evaluate the compatibility degree between a pair of ]0,1[-valued interval fuzzy preference relations (IFPRs). By employing the geometric mean, a measurement formula is proposed to calculate how close one interval fuzzy preference relation is to all the other interval fuzzy preference relations in a group. We devise an induced interval fuzzy ordered weighted geometric (IIFOWG) operator to aggregate ]0,1[-valued interval numbers, and apply the induced interval fuzzy ordered weighted geometric operator to fuse interval fuzzy preference relations into a collective one. Based on the compatibility measurement between two interval fuzzy preference relations, a notion of acceptable consensus of interval fuzzy preference relations is introduced to check the consensus level between an individual interval fuzzy preference relation and a collective interval fuzzy preference relation, and a novel procedure is developed to handle group decision-making problems with interval fuzzy preference relations. A numerical example with respect to the evaluation of e-commerce websites is provided to illustrate the proposed procedure.

Introduction

In group decision making, preference relations such as multiplicative preference relations ¹ and fuzzy preference relations (FPRs) ² are common tools for decision makers to express their opinions over alternatives or criteria. However, when dealing with complicated matters, as a decision maker may not have a good knowledge of the problem domain, he/she could not express his/her judgments with crisp numbers but is able to provide interval-valued judgments. For this reason, interval fuzzy preference relations (IFPRs)^3–5 have been used to express decision makers' judgments with vagueness and uncertainty,^6–9 and extensively applied in solving group decision-making problems.^10–12

The compatibility measurement of preference relations is a critical issue for the consensus reaching of group decision making. Herrera-Viedma et al. ¹³ developed a compatibility measurement for FPRs. Alonso et al. ¹⁴ used this compatibility measurement to establish a support system for group consensus decision making with incomplete FPRs. Xu et al. ¹⁵ defined an individual to group consensus index (ICI) for evaluating the deviation between an individual FPR and a collective FPR. Also, they introduced a group consensus index to measure the overall consensus of the group and identify if the iterative procedure of the improving consensus should terminate. By making use of the Euclidean distance measurement and the weighted averaging operator, Xu ¹⁶ developed a consensus method for solving group decision making with interval intuitionistic preference relations. Xu ³ defined the compatibility degree of two interval fuzzy numbers, and introduced the concept of compatibility degree of two IFPRs to measure how close an IFPR is to the leading one. It is clear that the aforesaid compatibility measurement methods are based on the deviation between two preference relations. In this paper, we introduce the indifference ratio between two ]0,1[-valued interval numbers to measure the compatibility degree for IFPRs. A method is proposed to determine the geometric averaging compatibility degree of an IFPR to all the other IFPRs in a group.

Aggregation of preference relations is another critical issue for solving group decision-making problems. Some aggregation operators have been developed and utilized in group decision making, such as the induced ordered weighted averaging (IOWA) operator proposed by Yager and Filev ¹⁷ and the induced ordered weighted geometric (IOWG) operator presented by Chiclana et al., ¹⁸ which are the generalized form of the ordered weighted averaging (OWA) operator ¹⁹ and the ordered weighted geometric (OWG) operator. ²⁰ The IOWA operator employs the arithmetic mean to synthesize preference information, and the IOWG operator is based on the geometric mean. Based on the IOWA operator, Chiclana et al. ²¹ introduced three operators whose order inducing variables are the importance of the decision information, the consistency of the decision information, and the preference of the decision information, respectively. Bottom on the IOWG operator, Zhou et al. ²² established several aggregation operators to deal with group decision-making problems with linguistic preference relations. Liu et al. ²³ developed a generalized operator to aggregate interval multiplicative reciprocal matrices on the basis of consistency. This paper extends the IOWG operator to devise an induced interval fuzzy ordered weighted geometric (IIFOWG) operator for aggregating individual IFPRs into a group IFPR. By using the developed compatibility measurement for IFPRs, we introduce a notion of acceptable consensus of IFPRs to check whether an individual IFPR reaches the consensus level. A six-step procedure is then put forward to solve group decision-making problems with IFPRs.

The remaining parts are set out as follows. The next section reviews some basic concepts with regard to the FPR, IFPR, and the IOWG operator. Ratio-based compatibility measurements are introduced to respectively compute the compatibility degree of two interval fuzzy numbers, and the compatibility degree of two ]0,1[-valued IFPRs. Besides, the IIFOWG operator is developed in Section ‘Compatibility measurements and aggregation of ]0,1[-valued IFPRs’. Section ‘Group decision making with IFPRs’ proposes a notion of acceptable consensus of IFPRs and presents a six-step procedure. Then, the next section gives a practical example about the evaluation of e-commerce websites to illustrate the proposed models. In the end, conclusions are drawn.

Preliminaries

Consider a multi-criteria decision-making problem with an alternative set $X=\{x1,x2,\dots ,xn\}$ , where $xi(i=1,2,\dots ,n)$ represents the i th alternative. A decision maker compares each pair of alternatives in X and provides his/her opinions of one alternative over others with an FPR defined below.

Definition 1.

² An FPR on the set X is characterized by an additively reciprocal relation $R=(r\mathrm{ij})n\times n$ with

$$0\le r\mathrm{ij}\le 1,r\mathrm{ij}+r\mathrm{ji}=1,r\mathrm{ii}=0.5;i,j=1,2,\dots ,n$$

(1) where $r\mathrm{ij}$ stands for the preference of x_i over x_j.

From the ratio viewpoint, $r\mathrm{ij}$ is given on the bipolar]0,1[scale, and $r\mathrm{ij}/r\mathrm{ji}$ is referred to as the ratio of preference intensity for x_i to x_j. If $r\mathrm{ij}=0.5$ , one has $r\mathrm{ij}/r\mathrm{ji}=1$ , meaning that there is no difference between x_i and x_j. If $0.5<r\mathrm{ij}<1$ , then $r\mathrm{ij}/r\mathrm{ji}>1$ , showing that x_i is preferred to x_j with a ratio $r\mathrm{ij}/r\mathrm{ji}$ . If $0<r\mathrm{ij}<0.5$ , then $r\mathrm{ij}/r\mathrm{ji}<1$ , indicating that x_i is inferior to x_j with a ratio $r\mathrm{ij}/r\mathrm{ji}$ .

All judgments in an FPR are crisp values. However, sometimes, it is hard for decision makers to provide crisp judgments due to the complexity of real-world decision problems. To better model and structure decision makers’ preferences with vagueness and uncertainty, Xu ³ introduced the notion of IFPRs below.

Definition 2.

³ An IFPR on the set X is defined by an interval additively reciprocal relation $\overline{R}=(\overline{r}\mathrm{ij})n\times n\subset X\times X$ satisfying

$$0\le r_{\mathrm{ij}}^{-}\le r_{\mathrm{ij}}^{+}\le 1,r_{\mathrm{ii}}^{-}=r_{\mathrm{ii}}^{+}=0.5,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{r}_{\mathrm{ij}}^{-}+r_{\mathrm{ji}}^{+}=1,i,j=1,2,\dots ,n$$

(2) where $\overline{r}\mathrm{ij}=[r_{\mathrm{ij}}^{-},r_{\mathrm{ij}}^{+}]$ represents the interval-valued preference of x_i over x_j, $r_{\mathrm{ij}}^{-}$ and $r_{\mathrm{ij}}^{+}$ are the lower and upper bounds of $\overline{r}\mathrm{ij}$ , respectively.

For two positive interval numbers $\overline{r}1=[r_1^{-},r_1^{+}]$ and $\overline{r}2=[r_2^{-},r_2^{+}]$ , their operations are provided as follows.

$\overline{r}1+\overline{r}2=[r_1^{-}+r_2^{-},r_1^{+}+r_2^{+}]$

$\overline{r}1-\overline{r}2=[r_1^{-}-r_2^{+},r_1^{+}-r_2^{-}]$

$\overline{r}1\times \overline{r}2=[r_1^{-}{r}_2^{-},r_1^{+}{r}_2^{+}]$

$\frac{\overline{r}1}{\overline{r}2}=[\frac{r_1^{-}}{r_2^{+}},\frac{r_1^{+}}{r_2^{-}}]$

Based on interval arithmetic, the additive reciprocity of $r_{\mathrm{ij}}^{-}+r_{\mathrm{ji}}^{+}=1,i,j=1,2,\dots ,n$ can be equivalently expressed as $\overline{r}\mathrm{ji}=1-\overline{r}\mathrm{ij}$ . Moreover, for an IFPR $\overline{R}=(\overline{r}\mathrm{ij})n\times n=([r_{\mathrm{ij}}^{-},r_{\mathrm{ij}}^{+}])n\times n$ satisfying $0<r_{\mathrm{ij}}^{-},r_{\mathrm{ij}}^{+}<1,\forall i,j=1,2,\dots ,n$ , $\frac{\overline{r}\mathrm{ij}}{\overline{r}\mathrm{ji}}=[\frac{r_{\mathrm{ij}}^{-}}{r_{\mathrm{ji}}^{+}},\frac{r_{\mathrm{ij}}^{+}}{r_{\mathrm{ji}}^{-}}]$ can be considered as an interval ratio of the preference intensity for x_i to x_j.

With the development of the social economy, the process of making a decision is generally concerned with a cluster of decision makers. Thus, aggregating single opinions into a collective one is essential. A lot of aggregation operators have been developed by many researchers. Chiclana et al. ¹⁸ proposed the IOWG operator as follows.

Definition 3

¹⁸ Let $\Re +$ be the positive real line, then an IOWG operator of dimension m is a mapping $\mathrm{IOWG}:(\Re +)m\to \Re +$ , which has an associated weighting vector $W=(\omega 1,\omega 2,\dots ,\omega m)T$ , with $\omega l\in [0,1](l=1,2,\dots ,m)$ and $\sum _{l=1}^m\omega l=1$ , expressed as

$$\mathrm{IOWG}(〈\mu 1,p1〉,〈\mu 2,p2〉,\dots ,〈\mu m,pm〉)=\underset{l=1}{\overset{m}{\Pi }}(p\sigma (l))\omega l$$

(3) where σ: $\{1,\dots ,m\}\to \{1,\dots ,m\}$ is a permutation such that $p\sigma (l)\ge p\sigma (l+1)$ , for $\forall l=1,\dots ,m-1$ .

In equation (3), $\mu l(l=1,2,\dots ,m)$ are usually named the order inducing variables, and $pl(l=1,2,\dots ,m)$ often act as the argument variables. Besides, $(p\sigma (1),p\sigma (2),\dots ,p\sigma (m))$ is a reordering of $(p1,p2,\dots ,pm)$ according to the ordering of values $\mu 1,\mu 2,\dots ,\mu m$ .

Compatibility measurements and aggregation of ]0,1[-valued IFPRs

This section puts forward a ratio-based formula to measure the compatibility degree between a pair of ]0,1[-valued interval numbers, and introduces a notion of ratio-based compatibility for IFPRs. Additionally, a new aggregation operator for synthesizing IFPRs is presented.

Compatibility measurements

Definition 4

Let $\overline{r}1=[r_1^{-},r_1^{+}]$ and $\overline{r}2=[r_2^{-},r_2^{+}]$ be a pair of ]0,1[-valued interval numbers, then the compatibility degree between $\overline{r}1$ and $\overline{r}2$ is defined as

$$\mathrm{CD}(\overline{r}1,\overline{r}2)=\sqrt{(\frac{min\{r_1^{-},r_2^{-}\}}{max\{r_1^{-},r_2^{-}\}})(\frac{min\{r_1^{+},r_2^{+}\}}{max\{r_1^{+},r_2^{+}\}})}$$

(4)

Obviously, $0<\mathrm{CD}(\overline{r}1,\overline{r}2)\le 1$ and $\mathrm{CD}(\overline{r}1,\overline{r}2)=\mathrm{CD}(\overline{r}2,\overline{r}1)$ . The larger the value of $\mathrm{CD}(\overline{r}1,\overline{r}2)$ , the more compatible $\overline{r}1$ to $\overline{r}2$ . Moreover, $\mathrm{CD}(\overline{r}1,\overline{r}2)=1$ , if and only if $\overline{r}1$ and $\overline{r}2$ are perfectly compatible, i.e. $\overline{r}1=\overline{r}2$ .

Equation (4) provides a way to measure compatibility of any two ]0,1[-valued intervals. This measurement is established from the viewpoint of the ratio-based idea. Moreover, it is noticed that Definition 4 differs from the notion of compatibility for two interval fuzzy numbers in Xu, ³ where the compatibility measurement is defined by the average distance of endpoints of two intervals.

Definition 5

Let $\overline{R}1=(\overline{r}1\mathrm{ij})n\times n=([r_{1\mathrm{ij}}^{-},r_{1\mathrm{ij}}^{+}])n\times n$ and $\overline{R}2=(\overline{r}2\mathrm{ij})n\times n=([r_{2\mathrm{ij}}^{-},r_{2\mathrm{ij}}^{+}])n\times n$ be two IFPRs with $0<r_{1\mathrm{ij}}^{-},r_{1\mathrm{ij}}^{+},r_{2\mathrm{ij}}^{-},r_{2\mathrm{ij}}^{+}<1\forall i,j=1,2,\dots ,n$ , then we call

$$\mathrm{CD}r(\overline{R}1,\overline{R}2)\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}=(\underset{i\ne j}{\Pi }(\frac{min\{r_{1\mathrm{ij}}^{-},r_{2\mathrm{ij}}^{-}\}}{max\{r_{1\mathrm{ij}}^{-},r_{2\mathrm{ij}}^{-}\}})(\frac{min\{r_{1\mathrm{ij}}^{+},r_{2\mathrm{ij}}^{+}\}}{max\{r_{1\mathrm{ij}}^{+},r_{2\mathrm{ij}}^{+}\}}))\frac{1}{2(n2-n)}$$

(5)

the compatibility degree between $\overline{R}1$ and $\overline{R}2$ .

Evidently, the greater the value of $\mathrm{CD}r(\overline{R}1,\overline{R}2)$ , the better the compatibility of $\overline{R}1$ and $\overline{R}2$ .

It should be noted that Definition 5 differs from the compatibility degree defined by Xu, ³ which is established based on the sum of absolute deviation between all intervals in the two IFPRs.

By equation (5), properties of the compatibility degree $\mathrm{CD}r(\overline{R}1,\overline{R}2)$ are directly obtained as follows.

Theorem 1

Let $\mathrm{CD}r(\overline{R}1,\overline{R}2)$ be defined by equation (5), then

$0<\mathrm{CD}r(\overline{R}1,\overline{R}2)\le 1$

$\mathrm{CD}r(\overline{R}1,\overline{R}2)=1$ , if and only if $\overline{R}1=\overline{R}2$

$\mathrm{CD}r(\overline{R}1,\overline{R}2)=\mathrm{CD}r(\overline{R}2,\overline{R}1)$

Property (1) indicates that the compatibility degree $\mathrm{CD}r(\overline{R}1,\overline{R}2)$ lies in (0, 1). Property (2) means that $\overline{R}1$ and $\overline{R}2$ are perfectly compatible if and only if $\mathrm{CD}r(\overline{R}1,\overline{R}2)=1$ . Property (3) shows that the compatibility degree $\mathrm{CD}r(\overline{R}i,\overline{R}j)$ is independent of the order of the indices $i,j$ .

Based on property (2), one can directly obtain the following result.

Corollary 1

Let $\overline{R}l=(\overline{r}\mathrm{lij})n\times n=([r_{\mathrm{lij}}^{-},r_{\mathrm{lij}}^{+}])$ (l =1, 2, 3) be three IFPRs with $0<r_{\mathrm{lij}}^{-}\le r_{\mathrm{lij}}^{+}<1$ for all $i,j=1,2,\dots ,n,$ $l=1,2,3$ , if $\mathrm{CD}r(\overline{R}1,\overline{R}2)=1$ and $\mathrm{CD}r(\overline{R}2,\overline{R}3)=1$ , then $\mathrm{CD}r(\overline{R}1,\overline{R}3)=1$ .

Theorem 2

Let $\overline{R}l=(\overline{r}\mathrm{lij})n\times n=([r_{\mathrm{lij}}^{-},r_{\mathrm{lij}}^{+}])(l=1,2)$ be two IFPRs with $0<r_{\mathrm{lij}}^{-}\le r_{\mathrm{lij}}^{+}<1$ for all $i,j=1,2,\dots ,n,$ and ${\overline{R}}_l^c=({\overline{r}}_{\mathrm{lij}}^c)n\times n=([r_{\mathrm{lij}}^{c-},r_{\mathrm{lij}}^{c+}])n\times n=(1-[r_{\mathrm{lij}}^{-},r_{\mathrm{lij}}^{+}])n\times n$ be the complement of $\overline{R}l$ , then $\mathrm{CD}r(\overline{R}1,\overline{R}2)=\mathrm{CD}r({\overline{R}}_1^c,{\overline{R}}_2^c)$ .

Proof

As per equation (5) and the additive reciprocity of IFPRs, we have

$$\mathrm{CD}r(\overline{R}1,\overline{R}2)=(\underset{i\ne j}{\Pi }(\frac{min\{r_{1\mathrm{ij}}^{-},r_{2\mathrm{ij}}^{-}\}}{max\{r_{1\mathrm{ij}}^{-},r_{2\mathrm{ij}}^{-}\}})(\frac{min\{r_{1\mathrm{ij}}^{+},r_{2\mathrm{ij}}^{+}\}}{max\{r_{1\mathrm{ij}}^{+},r_{2\mathrm{ij}}^{+}\}}))\frac{1}{2(n2-n)}=(\underset{i\ne j}{\Pi }(\frac{min\{1-r_{1\mathrm{ji}}^{+},1-r_{2\mathrm{ji}}^{+}\}}{max\{1-r_{1\mathrm{ji}}^{+},1-r_{2\mathrm{ji}}^{+}\}})(\frac{min\{1-r_{1\mathrm{ji}}^{-},1-r_{2\mathrm{ji}}^{-}\}}{max\{1-r_{1\mathrm{ji}}^{-},1-r_{2\mathrm{ji}}^{-}\}}))\frac{1}{2(n2-n)}=(\underset{i\ne j}{\Pi }(\frac{min\{r_{1\mathrm{ji}}^{c-},r_{2\mathrm{ji}}^{c-}\}}{max\{r_{1\mathrm{ji}}^{c-},r_{2\mathrm{ji}}^{c-}\}})(\frac{min\{r_{1\mathrm{ji}}^{c+},r_{2\mathrm{ji}}^{c+}\}}{max\{r_{1\mathrm{ji}}^{c+},r_{2\mathrm{ji}}^{c+}\}}))\frac{1}{2(n2-n)}=\mathrm{CD}r({\overline{R}}_1^c,{\overline{R}}_2^c)$$

To obtain a rational result in group decision making, considering the compatibility between one IFPR and all the others is required. In what follows, by combining Definition 5 and the geometric averaging operator, we establish a formula to compute the geometric average compatibility degree of a ]0,1[-valued IFPR.

Definition 6

Let $\overline{R}l=(\overline{r}\mathrm{lij})n\times n=([r_{\mathrm{lij}}^{-},r_{\mathrm{lij}}^{+}])n\times n$ $(l=1,2,\dots ,m)$ be m IFPRs with $0<r_{\mathrm{lij}}^{-}\le r_{\mathrm{lij}}^{+}<1$ for $i,j=1,2,\dots ,n,$ then the geometric average compatibility degree of the lth IFPR $\overline{R}l$ is defined as

$$\mathrm{GACD}(\overline{R}l)=\sqrt[m-1]{\underset{k=1,k\ne l}{\overset{m}{\Pi }}\mathrm{CD}r(\overline{R}l,\overline{R}k)}$$

(6)

It is obvious that $0<\mathrm{GACD}(\overline{R}l)\le 1$ . If $\mathrm{GACD}(\overline{R}l)=1$ , $\overline{R}l=\overline{R}k$ for all $k=1,2,\dots ,m$ . In other words, all IFPRs are identical. What’s more, the larger the value of $\mathrm{GACD}(\overline{R}l)$ , the more proximate $\overline{R}l$ to the others.

Aggregation of ]0,1[-valued IFPRs

In many group decision situations, a result is generally made by a cluster of decision makers. Thus, in the process of group decision making, it is important to aggregate single decision maker’s opinions into a collective one.

By making use of the IOWG operator recalled in Section ‘Compatibility measurements and aggregation of ]0,1[-valued IFPRs’, one could easily obtain the equation that $\mathrm{IOWG}(〈\mu 1,p1〉,〈\mu 2,p2〉,\dots ,〈\mu m,pm〉)\mathrm{IOWG}(〈\mu 1,p_1^c〉,〈\mu 2,p_2^c〉,\dots ,〈\mu m,p_m^c〉)=1$ where $p_l^c(l=1,2,\dots ,m)$ are the multiplicative reciprocal of $pl(l=1,2,\dots ,m)$ , say, $p_l^{c}pl=1$ . Obviously, if the IOWG operator is applied to the aggregation of m IFPRs, then the obtained matrix has no additive reciprocity. Therefore, we cannot employ the IOWG operator to directly aggregate ratio-scale-based preferences in group decision making with IFPRs.

Let $\overline{\Re }]0,1[$ be the set of all ]0,1[-valued intervals, i.e.

$$\overline{\Re }]0,1[=\{[x-,x+]|0<x-\le x+<1\}$$

(7)

Next, we propose an aggregation method bottom on the and-like representable cross ratio uninorm. ²⁴

Definition 7

An IIFOWG operator of dimension m is an aggregation function $\overline{U}:(\overline{\Re }]0,1[)m\to \overline{\Re }]0,1[$ , which is expressed by the following form

$$\overline{U}(〈\mu 1,\overline{r}1〉,〈\mu 2,\overline{r}2〉,\dots ,〈\mu m,\overline{r}m〉)\hspace{1em}=\overline{U}(〈\mu 1,[r_1^{-},r_1^{+}]〉,〈\mu 2,[r_2^{-},r_2^{+}]〉,\dots ,〈\mu m,[r_m^{-},r_m^{+}]〉)=[r_a^{-},r_a^{+}]\hspace{1em}=[\frac{\underset{l=1}{\overset{m}{\Pi }}(r_{\sigma (l)}^{-})\omega l}{\underset{l=1}{\overset{m}{\Pi }}(r_{\sigma (l)}^{-})\omega l+\underset{l=1}{\overset{m}{\Pi }}(1-r_{\sigma (l)}^{-})\omega l},\frac{\underset{l=1}{\overset{m}{\Pi }}(r_{\sigma (l)}^{+})\omega l}{\underset{l=1}{\overset{m}{\Pi }}(r_{\sigma (l)}^{+})\omega l+\underset{l=1}{\overset{m}{\Pi }}(1-r_{\sigma (l)}^{+})\omega l}]$$

(8) where $W=(\omega 1,\omega 2,\dots ,\omega m)T$ is an associated weighting vector satisfying $\omega l\in [0,1]$ and $\sum _{l=1}^m\omega l=1$ , and σ: $\{1,\dots ,m\}\to \{1,\dots ,m\}$ is a permutation such that $p\sigma (l)\ge p\sigma (l+1)$ , for $\forall l=1,\dots ,m-1$ .

If $\omega l=1/m$ for all $l=1,2,\dots ,m$ , then equation (8) can be simplified as

$$\overline{U}(\overline{r}1,\overline{r}2,\dots ,\overline{r}m)\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}=\overline{U}([r_1^{-},r_1^{+}],[r_2^{-},r_2^{+}],\dots ,[r_m^{-},r_m^{+}])=[r_a^{-},r_a^{+}]\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}=[\frac{\sqrt[m]{\underset{l=1}{\overset{m}{\Pi }}{r}_l^{-}}}{\sqrt[m]{\underset{l=1}{\overset{m}{\Pi }}{r}_l^{-}}+\sqrt[m]{\underset{l=1}{\overset{m}{\Pi }}(1-r_l^{-})}},\frac{\sqrt[m]{\underset{l=1}{\overset{m}{\Pi }}{r}_l^{+}}}{\sqrt[m]{\underset{l=1}{\overset{m}{\Pi }}{r}_l^{+}}+\sqrt[m]{\underset{l=1}{\overset{m}{\Pi }}(1-r_l^{+})}}]$$

(9)

It is easy to prove the following corollary.

Corollary 2

For any associated weighting vector $W=(\omega 1,\omega 2,\dots ,\omega m)T$ and order inducing values $\mu l(l=1,2,\dots ,m)$ , there is

$$\overline{U}(〈\mu 1,[0.5,0.5]〉,〈\mu 2,[0.5,0.5]〉,\dots ,〈\mu m,[0.5,0.5]〉)\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}=[0.5,0.5]$$

(10)

This corollary reveals that the neutral element 0.5 on the bipolar ]0,1[scale is maintained by the proposed operator.

Theorem 3

Let $\overline{r}l=[r_l^{-},r_l^{+}](l=1,2,\dots ,m)$ be m interval fuzzy numbers with $0<r_l^{-}\le r_l^{+}<1$ , then

$$\overline{U}(〈\mu 1,{\overline{r}}_1^c〉,〈\mu 2,{\overline{r}}_2^c〉,\dots ,〈\mu m,{\overline{r}}_m^c〉)\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}=1-\overline{U}(〈\mu 1,\overline{r}1〉,〈\mu 2,\overline{r}2〉,\dots ,〈\mu m,\overline{r}m〉)$$

(11) where ${\overline{r}}_l^c$ is the additive reciprocal of $\overline{r}l$ , say, ${\overline{r}}_l^c=1-\overline{r}l$ .

Proof

According to the interval arithmetic described in Section ‘Compatibility measurements and aggregation of ]0,1[-valued IFPRs’, it is easy to obtain ${\overline{r}}_l^c=1-\overline{r}l=[1-r_l^{+},1-r_l^{-}]$ for all $l=1,2,\dots ,m$ . Then by equation (8), we have

$$\overline{U}(〈\mu 1,{\overline{r}}_1^c〉,〈\mu 2,{\overline{r}}_2^c〉,\dots ,〈\mu m,{\overline{r}}_m^c〉)=[\frac{\underset{l=1}{\overset{m}{\Pi }}(1-r_{\sigma (l)}^{+})\omega l}{\underset{l=1}{\overset{m}{\Pi }}(1-r_{\sigma (l)}^{+})\omega l+\underset{l=1}{\overset{m}{\Pi }}(r_{\sigma (l)}^{+})\omega l},\frac{\underset{l=1}{\overset{m}{\Pi }}(1-r_{\sigma (l)}^{-})\omega l}{\underset{l=1}{\overset{m}{\Pi }}(1-r_{\sigma (l)}^{-})\omega l+\underset{l=1}{\overset{m}{\Pi }}(r_{\sigma (l)}^{-})\omega l}]=1-[\frac{\underset{l=1}{\overset{m}{\Pi }}(r_{\sigma (l)}^{-})\omega l}{\underset{l=1}{\overset{m}{\Pi }}(1-r_{\sigma (l)}^{-})\omega l+\underset{l=1}{\overset{m}{\Pi }}(r_{\sigma (l)}^{-})\omega l},\frac{\underset{l=1}{\overset{m}{\Pi }}(r_{\sigma (l)}^{+})\omega l}{\underset{l=1}{\overset{m}{\Pi }}(1-r_{\sigma (l)}^{+})\omega l+\underset{l=1}{\overset{m}{\Pi }}(r_{\sigma (l)}^{+})\omega l}]=1-\overline{U}(〈\mu 1,\overline{r}1〉,〈\mu 2,\overline{r}2〉,\dots ,〈\mu m,\overline{r}m〉)$$

Therefore, Theorem 3 is completely proved.

Theorem 4

Let $\overline{R}l=(\overline{r}\mathrm{lij})n\times n=([r_{\mathrm{lij}}^{-},r_{\mathrm{lij}}^{+}])n\times n$ $(l=1,2,\dots ,m)$ be m IFPRs with $0<r_{\mathrm{lij}}^{-}\le r_{\mathrm{lij}}^{+}<1$ , then

$$\overline{R}c=({\overline{r}}_{\mathrm{ij}}^c)n\times n=([r_{\mathrm{ij}}^{c-},r_{\mathrm{ij}}^{c+}])n\times n=(\overline{U}(〈\mu 1,\overline{r}1\mathrm{ij}〉,〈\mu 2,\overline{r}2\mathrm{ij}〉,\dots ,〈\mu m,\overline{r}\mathrm{mij}〉))n\times n$$

(12)

is an additively reciprocal IFPR.

Proof

According to equation (8) and Definition 2, we have

$$r_{\mathrm{ij}}^{c-}+r_{\mathrm{ji}}^{c+}=\frac{\underset{l=1}{\overset{m}{\Pi }}(1-r_{\mathrm{lij}}^{+})\omega l}{\underset{l=1}{\overset{m}{\Pi }}(r_{\mathrm{lij}}^{+})\omega l+\underset{l=1}{\overset{m}{\Pi }}(1-r_{\mathrm{lij}}^{+})\omega l}+\frac{\underset{l=1}{\overset{m}{\Pi }}(1-r_{\mathrm{lji}}^{-})\omega l}{\underset{l=1}{\overset{m}{\Pi }}(r_{\mathrm{lji}}^{-})\omega l+\underset{l=1}{\overset{m}{\Pi }}(1-r_{\mathrm{lji}}^{-})\omega l}=\frac{\underset{l=1}{\overset{m}{\Pi }}(1-r_{\mathrm{lij}}^{+})\omega l}{\underset{l=1}{\overset{m}{\Pi }}(r_{\mathrm{lij}}^{+})\omega l+\underset{l=1}{\overset{m}{\Pi }}(1-r_{\mathrm{lij}}^{+})\omega l}+\frac{\underset{l=1}{\overset{m}{\Pi }}(r_{\mathrm{lij}}^{+})\omega l}{\underset{l=1}{\overset{m}{\Pi }}(r_{\mathrm{lij}}^{+})\omega l+\underset{l=1}{\overset{m}{\Pi }}(1-r_{\mathrm{lij}}^{+})\omega l}=1$$

As per Definition 2, $\overline{R}c$ is an additively reciprocal IFPR.

In our group decision-making context, the order inducing values should be determined in the process of group decision making. Since the GACD of each individual IFPR reflects the compatibility degree of each IFPR to all the other IFPRs in a group, we take the GACD of IFPRs as order inducing variables, and introduce a geometric averaging compatibility degree-based IIFOWG (GACD-IIFOWG) operator as follows.

Definition 8

Let $\overline{R}l=(\overline{r}\mathrm{lij})n\times n=([r_{\mathrm{lij}}^{-},r_{\mathrm{lij}}^{+}])n\times n(l=1,2,\dots ,m)$ be m IFPRs with $0<r_{\mathrm{lij}}^{-}\le r_{\mathrm{lij}}^{+}<1$ , then a GACD-IIFOWG operator of dimension m is a function with the following form

$$\overline{I}(\overline{R}1,\overline{R}2,\dots ,\overline{R}m)\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}=(\overline{U}(〈\mathrm{GACD}(\overline{R}1),\overline{r}1\mathrm{ij}〉,〈\mathrm{GACD}(\overline{R}2),\overline{r}2\mathrm{ij}〉,\dots ,〈\mathrm{GACD}(\overline{R}m),\overline{r}\mathrm{mij}))n\times n$$

(13)

Obviously, this proposed GACD-IIFOWG operator is an IIFOWG operator whose order inducing values are the geometric average compatibility degree of individual IFPRs $\mathrm{GACD}(\overline{R}l)(l=1,2,\dots ,m)$ .

Group decision making with IFPRs

Based on the compatibility measurement between two IFPRs, this section puts forward a method for measuring the consensus degree between an individual IFPR and an aggregated one. Also, a procedure for tackling group decision-making problems with IFPRs is presented.

Definition 9

Let $\overline{R}l=(\overline{r}\mathrm{lij})n\times n=([r_{\mathrm{lij}}^{-},r_{\mathrm{lij}}^{+}])n\times n(l=1,2,\dots ,m)$ be m IFPRs with $0<r_{\mathrm{lij}}^{-}\le r_{\mathrm{lij}}^{+}<1$ , then the consensus degree between a single IFPR $\overline{R}l=(\overline{r}\mathrm{lij})n\times n=([r_{\mathrm{lij}}^{-},r_{\mathrm{lij}}^{+}])n\times n$ and a group one $\overline{R}G=({\overline{r}}_{\mathrm{ij}}^G)n\times n=([r_{\mathrm{ij}}^{G-},r_{\mathrm{ij}}^{G+}])$ is defined as

$$C(\overline{R}l,\overline{R}G)\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}=(\underset{i\ne j}{\Pi }(\frac{min\{r_{\mathrm{lij}}^{-},rG-\}}{max\{r_{\mathrm{lij}}^{-},r_{\mathrm{ij}}^{G-}\}})(\frac{min\{r_{\mathrm{lij}}^{+},r_{\mathrm{ij}}^{G+}\}}{max\{r_{\mathrm{lij}}^{+},r_{\mathrm{ij}}^{G+}\}}))\frac{1}{2(n2-n)}$$

(14) where $\overline{R}l$ is an IFPR given by the lth decision maker in a group.

It is obvious that the larger the value of $C(\overline{R}l,\overline{R}G)$ is, the closer $\overline{R}l$ is to $\overline{R}G$ , such that the decision maker d_l has a better consensus with the group.

In the perspective of the value of $C(\overline{R}l,\overline{R}G)$ , we provide two definitions as follows.

Definition 10

If $C(\overline{R}l,\overline{R}G)=1$ , then $\overline{R}l=(\overline{r}\mathrm{lij})n\times n$ is of perfect consensus with the collective IFPR $\overline{R}G=({\overline{r}}_{\mathrm{ij}}^G)n\times n$ .

Indeed, due to the complexity of real-world decision environment and the diverse preferences of decision makers, it is always difficult to reach a full and unanimous agreement. To make it easier and faster to deal with real decision problems, the following definition is introduced.

Definition 11

If $C(\overline{R}l,\overline{R}G)\ge \lambda$ , then $\overline{R}l=(\overline{r}\mathrm{lij})n\times n$ is of acceptable consensus with the collective IFPR $\overline{R}G=({\overline{r}}_{\mathrm{ij}}^G)n\times n$ .

where λ is the threshold value of acceptable consensus. On one hand, since it is hardly possible for a crowd of decision makers to reach a full agreement, we impose the restriction: $\lambda <1$ ; On the other hand, mostly when no less than 50% decision makers approve of the decision-making output, it is always deemed as an acceptable result, thus the restriction: $\lambda \ge 0.5$ is proposed. Conclusively, $\lambda \in [0.5,1)$ is taken into account in realistic applications, and the closer λ to 1, the more accredited the decision result.

Based on the above measurements, now we develop a systematical procedure to deal with group decision-making problems with IFPRs.

Step 1: For a group decision-making problem, let $X=\{x1,x2,\dots ,xn\}$ be a finite set of alternatives, and $D=\{d1,d2,\dots ,dm\}$ be a fixed set of decision makers whose importance weights are given by a weighting vector $W=(\omega 1,\omega 2,\dots ,\omega m)T$ . Every decision maker $dl(l=1,2,\dots ,m)$ compares each pair of alternatives and constructs an IFPR $\overline{R}l=(\overline{r}\mathrm{lij})n\times n=([r_{\mathrm{lij}}^{-},r_{\mathrm{lij}}^{+}])$ satisfying $0<r_{\mathrm{lij}}^{-}\le r_{\mathrm{lij}}^{+}<1$ .

Step 2: Utilize equation (5) to calculate compatibility degrees between each pair of individual IFPRs, then apply equation (6) to figure out the geometric average compatibility degree of each IFPR, which determines the sequence of IFPRs.

Step 3: Employ the GACD-IIFOWG operator to fuse all individual IFPRs into a collective IFPR, then calculate the consensus degree $C(\overline{R}l,\overline{R}G)(l=1,2,\dots ,m)$ between an IFPR and the collective IFPR through equation (14).

Step 4: Predefine the value of the threshold λ, and check whether or not all $C(\overline{R}l,\overline{R}G)\ge \lambda (l=1,2,\dots ,m)$ . If all $C(\overline{R}l,\overline{R}G)(l=1,2,\dots ,m)$ satisfy the inequality, i.e. an acceptable consensus result of the group is obtained, then go to the next step; if not, say, at least one $C(\overline{R}l,\overline{R}G)<\lambda$ exists, then utilize equation (4) to compute $\mathrm{CD}(\overline{r}\mathrm{lij},{\overline{r}}_{\mathrm{ij}}^G)$ between each pair of corresponding elements $(\overline{r}\mathrm{lij},{\overline{r}}_{\mathrm{ij}}^G)$ in $\overline{R}l$ and $\overline{R}G$ , respectively. Subsequently, return $\overline{R}l$ with $\overline{R}G$ and several elements which make little contribution to the group consensus to the decision maker d_l for adjustment, and then go back to Step 2. Repeat this proceeding till all $C(\overline{R}l,\overline{R}G)\ge \lambda (l=1,2,\dots ,m)$ or the proceeding must stop on account that the repetition time beyond the restriction set up by the group. Finally, output the re-constructed collective IFPR.

Step 5: Plug the collective IFPR exported from Step 4 into the goal programming model (4.13) in Xu ³ to obtain the relative weights of alternatives

Step 6: Apply the following formula introduced in Wang et al., ²⁵ to compute the possibility of $\overline{\upsilon }i\ge \overline{\upsilon }j$ and then rank the alternatives.

$$p(\overline{\upsilon }i\ge \overline{\upsilon }j)=\frac{max\{0,{\upsilon }_i^{+}-{\upsilon }_j^{-}\}-max\{0,{\upsilon }_i^{-}-{\upsilon }_j^{+}\}}{{\upsilon }_i^{+}-{\upsilon }_i^{-}+{\upsilon }_j^{+}-{\upsilon }_j^{-}}$$

(16)

A practical example

With the rapid development of the commercialization of the Internet, the e-commerce website gradually becomes the entry by which enterprises attach to the virtual world of the Internet. When a traditional company first marches towards the area of e-commerce, to set up a new website and operate it well in a short time is neither easy nor inexpensive. Participating in a reliable website that has already been managed successfully may be a good choice for this traditional company. However, there are not a few e-commerce websites. Thus, the selection of a suitable and available website becomes an inevitable decision-making problem. Here, we give a simplified estimation process by making application of our procedure.

For the sake of tractability, it is presumed that a committee of four managers $D=\{d1,d2,d3,d4\}$ from different departments, whose weight vector is $W=(0.2914,0.2517,0.2379,0.2190)T$ are invited to select the best e-commerce website from a set of alternatives $X=\{x1,x2,x3,x4\}$ . Each manager d_l is demanded to furnish his/her comprehensive assessment on the four candidates by means of an IFPR $\overline{R}l=(\overline{r}\mathrm{lij})4\times 4$ .

$$\overline{R}1=[[0.50,0.50][0.33,0.57][0.25,0.42][0.49,0.65][0.43,0.67][0.50,0.50][0.22,0.39][0.53,0.62][0.58,0.75][0.61,0.78][0.50,0.50][0.61,0.78][0.35,0.51][0.38,0.47][0.22,0.39][0.50,0.50]]\overline{R}2=[[0.50,0.50][0.38,0.61][0.27,0.40][0.51,0.62][0.39,0.62][0.50,0.50][0.25,0.41][0.57,0.62][0.60,0.73][0.59,0.75][0.50,0.50][0.60,0.75][0.38,0.49][0.38,0.43][0.25,0.40][0.50,0.50]]$$

$$\overline{R}3=[[0.50,0.50][0.29,0.45][0.30,0.39][0.50,0.61][0.55,0.71][0.50,0.50][0.19,0.31][0.43,0.69][0.61,0.70][0.69,0.81][0.50,0.50][0.59,0.70][0.39,0.50][0.31,0.57][0.30,0.41][0.50,0.50]]\overline{R}4=[[0.50,0.50][0.41,0.61][0.30,0.45][0.45,0.63][0.39,0.59][0.50,0.50][0.25,0.41][0.51,0.61][0.55,0.70][0.59,0.75][0.50,0.50][0.59,0.78][0.37,0.55][0.39,0.49][0.22,0.41][0.50,0.50]]$$

By utilizing equation (5), the compatibility degrees between each two IFPRs are calculated as follows

$$\mathrm{CD}r(\overline{R}1,\overline{R}2)=0.9433,\mathrm{CD}r(\overline{R}1,\overline{R}3)=0.8822,\mathrm{CD}r(\overline{R}1,\overline{R}4)=0.9360,\mathrm{CD}r(\overline{R}2,\overline{R}3)=0.8723,\mathrm{CD}r(\overline{R}2,\overline{R}4)=0.9538,\mathrm{CD}r(\overline{R}3,\overline{R}4)=0.8610.$$

then GACDs of each individual IFPR are computed by equation (6)

$$\mathrm{GACD}(\overline{R}1)=\sqrt[3]{0.9433\times 0.8822\times 0.9360}=0.9201,\mathrm{GACD}(\overline{R}2)=\sqrt[3]{0.9433\times 0.8723\times 0.9538}=0.9224,\mathrm{GACD}(\overline{R}3)=\sqrt[3]{0.8975\times 0.9005\times 0.8888}=0.8956,\mathrm{GACD}(\overline{R}4)=\sqrt[3]{0.9360\times 0.9538\times 0.8610}=0.9160$$

Markedly, one can obtain $\mathrm{GACD}(\overline{R}2)>\mathrm{GACD}(\overline{R}1)>\mathrm{GACD}(\overline{R}4)>\mathrm{GACD}(\overline{R}3)$ .

According to the descending order, a collective IFPR $\overline{R}G=({\overline{r}}_{\mathrm{ij}}^G)n\times n=([r_{\mathrm{ij}}^{-G},r_{\mathrm{ij}}^{+G}])n\times n$ is obtained by utilizing the GACD-IIFOWG operator as

$$\overline{R}G=[[0.5000,0.5000][0.3535,0.5656][0.2782,0.4146][0.4885,0.6279][0.4344,0.6465][0.5000,0.5000][0.2284,0.3821][0.5150,0.6291][0.5854,0.7218][0.6179,0.7716][0.5000,0.5000][0.5980,0.7550][0.3721,0.5115][0.3709,0.4850][0.2450,0.4020][0.5000,0.5000]]$$

Based on the four individual IFPRs and the collective IFPR, consensus degrees $C(\overline{R}l,\overline{R}G)(l=1,2,3,4)$ are computed through equation (14)

$$C(\overline{R}1,\overline{R}G)=0.9687,C(\overline{R}2,\overline{R}G)=0.9556,C(\overline{R}3,\overline{R}G)=0.8987,C(\overline{R}4,\overline{R}G)=0.9496.$$

In this case, we suppose that the threshold value $\lambda =0.92$ . Clearly, $C(\overline{R}l,\overline{R}G)>0.92(l=1,2,4)$ , yet $C(\overline{R}3,\overline{R}G)=0.8987<0.92$ . Then we use equation (4) to calculate the compatibility degrees $\mathrm{CD}(\overline{r}3\mathrm{ij},{\overline{r}}_{\mathrm{ij}}^G)(i,j=1,2,3,4)$ between each pair of corresponding elements $(\overline{r}3\mathrm{ij},{\overline{r}}_{\mathrm{ij}}^G)$ in $\overline{R}3$ and $\overline{R}G$

$$\mathrm{CD}(\overline{r}312,{\overline{r}}_{12}^G)=0.8079,\mathrm{CD}(\overline{r}313,{\overline{r}}_{13}^G)=0.9340,\mathrm{CD}(\overline{r}314,{\overline{r}}_{14}^G)=0.9742\mathrm{CD}(\overline{r}321,{\overline{r}}_{21}^G)=0.8480,\mathrm{CD}(\overline{r}323,{\overline{r}}_{23}^G)=0.8215,\mathrm{CD}(\overline{r}324,{\overline{r}}_{24}^G)=0.8725\mathrm{CD}(\overline{r}331,{\overline{r}}_{31}^G)=0.9647,\mathrm{CD}(\overline{r}332,{\overline{r}}_{32}^G)=0.9236,\mathrm{CD}(\overline{r}334,{\overline{r}}_{34}^G)=0.9147\mathrm{CD}(\overline{r}341,{\overline{r}}_{41}^G)=0.9657,\mathrm{CD}(\overline{r}342,{\overline{r}}_{42}^G)=0.8433,\mathrm{CD}(\overline{r}343,{\overline{r}}_{43}^G)=0.8948$$

According to the above results, $\overline{R}3$ together with $\overline{R}G$ and several interval numbers like $\overline{r}312$ and $\overline{r}321$ should be returned to the manager d₃ for re-evaluation. Assume the values are improved as $\overline{r}312=[0.37,0.55]$ and $\overline{r}321=[0.45,0.63]$ . Therefore, the re-evaluated IFPR is

$${\overline{R}}_3^n=[[0.50,0.50][0.37,0.55][0.30,0.39][0.50,0.61][0.45,0.63][0.50,0.50][0.19,0.31][0.43,0.69][0.61,0.70][0.69,0.81][0.50,0.50][0.59,0.70][0.39,0.50][0.31,0.57][0.30,0.41][0.50,0.50]]$$

Then by equation (5), the compatibility degrees between ${\overline{R}}_3^n$ and other IFPRs are respectively computed again as

$$\mathrm{CD}r(\overline{R}1,{\overline{R}}_3^n)=0.8975,\mathrm{CD}r(\overline{R}2,{\overline{R}}_3^n)=0.9005,\mathrm{CD}r({\overline{R}}_3^n,\overline{R}4)=0.8888.$$

Likewise, geometric average compatibility degrees of each IFPR are altered as

$$\mathrm{GACD}(\overline{R}1)\text{'}=\sqrt[3]{0.9433\times 0.8975\times 0.9360}=0.9254\mathrm{GACD}(\overline{R}2)\text{'}=\sqrt[3]{0.9433\times 0.9005\times 0.9538}=0.9322\mathrm{GACD}({\overline{R}}_3^n)=\sqrt[3]{0.8975\times 0.9005\times 0.8888}=0.8956\mathrm{GACD}(\overline{R}4)\text{'}=\sqrt[3]{0.9360\times 0.9538\times 0.8888}=0.9258$$

It is evident that $\mathrm{GACD}(\overline{R}2)\text{'}>\mathrm{GACD}(\overline{R}4)\text{'}>\mathrm{GACD}(\overline{R}1)\text{'}>\mathrm{GACD}({\overline{R}}_3^n)$ .

By the GACD-IIFOWG operator, a collective IFPR is reconstructed as

$$\overline{R}\mathrm{nG}=[[0.5000,0.5000][0.3735,0.5873][0.2794,0.4152][0.4880,0.6275][0.4127,0.6265][0.5000,0.5000][0.2297,0.3833][0.5147,0.6328][0.5848,0.7206][0.6167,0.7703][0.5000,0.5000][0.5977,0.7537][0.3725,0.5120][0.3672,0.4853][0.2463,0.4023][0.5000,0.5000]]$$

Then based on IFPRs $\overline{R}1,\overline{R}2,{\overline{R}}_3^n,\overline{R}4$ and the reconstructed collective IFPR $\overline{R}\mathrm{nG}$ , consensus degrees $C(\overline{R}l,\overline{R}\mathrm{nG})(l=1,2,4)$ and $C({\overline{R}}_3^n,\overline{R}\mathrm{nG})$ are computed through equation (14)

$$C(\overline{R}1,\overline{R}\mathrm{nG})=0.9617,C(\overline{R}2,\overline{R}\mathrm{nG})=0.9639,C({\overline{R}}_3^n,\overline{R}\mathrm{nG})=0.9216,C(\overline{R}4,\overline{R}\mathrm{nG})=0.9524.$$

It is obvious that $C(\overline{R}l,\overline{R}\mathrm{nG})>0.92(l=1,2,4)$ and $C({\overline{R}}_3^n,\overline{R}\mathrm{nG})>0.92$ .

Thus, the group reaches an acceptable consensus.

Let $\overline{V}=(\overline{\upsilon }1,\overline{\upsilon }2,\overline{\upsilon }3,\overline{\upsilon }4)T=([{\upsilon }_1^{-},{\upsilon }_1^{+}],[{\upsilon }_2^{-},{\upsilon }_2^{+}],[{\upsilon }_3^{-},{\upsilon }_3^{+}],[{\upsilon }_4^{-},{\upsilon }_4^{+}])T$ be the associated weight vector of the alternative websites, as per equation (15), one has

$$\overline{\upsilon }1=[0.1691,0.2393],\overline{\upsilon }2=[0.1681,0.2458],\overline{\upsilon }3=[0.3375,0.4373],\overline{\upsilon }4=[0.1421,0.1775]$$

To rank the overall importance weights, equation (16) is employed to calculate the possibility of $\overline{\upsilon }i\ge \overline{\upsilon }j(i,j=1,2,3,4,i\ne j)$ , such that a crisp possibility matrix is shaped as

$$P=[0.50.482700.92160.51730.500.9221110.510.07840.077900.5]$$

in which the final ranking is pointed out: $\overline{\upsilon }3\succ 100\%\overline{\upsilon }2\succ 51.73\%\overline{\upsilon }1\succ 92.16\%\overline{\upsilon }4$ . This signifies that alternative x₃ is totally preferred to x₂, x₂ is preferred to x₁ to the degree of 51.73%, x₁ is preferred to x₄ to the degree of 92.16%. Without hesitation, the best choice is x₃.

Conclusion

In this work, a ratio-based compatibility measurement between two ]0,1[-valued interval numbers is introduced, and it is further applied to decide the compatibility degree between any two ]0,1[-scale IFPRs. Then, by employing the geometric mean, a method for calculating the geometric average compatibility degree of an IFPR is proposed to measure how close one IFPR is to all the other IFPRs in a group. In order to generate a collective preference relation, an aggregation operator named IIFOWG operator is put forward on the basis of the representable cross ratio uninorm. Based on the compatibility measurement between two IFPRs, we put forward a notion of acceptable consensus of IFPRs. Besides, an iterative consensus achieving procedure for tackling group decision-making problems with IFPRs is developed. Finally, a numerical example concerning the estimation of e-commerce websites is furnished to demonstrate the effectiveness of our proposed measures.

In the future, we will concentrate on consensus measures for group decision-making problems with IFPRs based on multiplicative consistency.