Group Decision Making With Consistency of Intuitionistic Fuzzy Preference Relations Under Uncertainty
 IEEE/CAA Journal of Automatica Sinica  2018, Vol. 5 Issue(3): 741-748 PDF
Group Decision Making With Consistency of Intuitionistic Fuzzy Preference Relations Under Uncertainty
Yang Lin1, Yingming Wang2
1. Institute of Decision Science, Fuzhou University, Fuzhou 350116, China, and also with the School of Economics, Fujian Normal University, Fuzhou 350116, China;
2. Institute of Decision Science, Fuzhou University, Fuzhou 350116, China
Abstract: Intuitionistic fuzzy preference relation (IFPR) is a suitable technique to express fuzzy preference information by decision makers (DMs). This paper aims to provide a group decision making method where DMs use the IFPRs to indicate their preferences with uncertain weights. To begin with, a model to derive weight vectors of alternatives from IFPRs based on multiplicative consistency is presented. Specifically, for any IFPR, by minimizing its absolute deviation from the corresponding consistent IFPR, the weight vectors are generated. Secondly, a method to determine relative weights of DMs depending on preference information is developed. After that we prioritize alternatives based on the obtained weights considering the risk preference of DMs. Finally, this approach is applied to the problem of technical risks assessment of armored equipment to illustrate the applicability and superiority of the proposed method.
Key words: Group decision making (GDM)     intuitionistic fuzzy preference relation (IFPR)     intuitionistic fuzzy set (IFS)     multiplicative consistency     risk preference     uncertain weights
Ⅰ. INTRODUCTION

Group decision making (GDM) is one of the common activities in human daily life, which consists of ranking a given set of alternatives and finding the most preferred one by a group of DMs. During group decision making, each DM is usually asked to provide his/her preference over alternatives and then the preference relations (PRs) are generated automatically [1]. In some situations, however, due to the complexity and ambiguity of human mind, fuzzy judgments are often easier than precise ones to make, and DMs usually provide their uncertain and rough PRs in GDM.

As a practical yet valid theory for dealing with uncertain and vague information, intuitionistic fuzzy set (IFS) [2] has attracted many scholars' attention [3]-[6]. The advantage of IFS is the capability of describing fuzzy judgments and the capability of representing positive, negative and hesitative viewpoint through membership function [7]. When the membership of preference relations is characterized by intuitionistic fuzzy values (IFV, which are the basic components of IFS), the IFPR is generated. Namely, an IFPR is a matrix of values that are created by pairwise comparisons over the given alternatives, and each value implies the preference degree of one alternative over another [8]. IFPR has found huge application in various aspects of group decision making [3]-[7], [9]-[15] and increasing attention has been paid to IFPR in recent years. The formal definition of IFPR was given by Szmidt and Kacprzyk [10], who investigated the mechanism of consensus reaching, and examined the extent of agreement in a group of DMs as well [11]. Xu and Yager [12] introduce a similarity measure between IFSs and apply this measure in group decision making with consensus analysis based on IFPR. Gong et al. [13] employed the IFPR to study and evaluate the industry meteorological service for Meteorological Bureau of China. Liao and Xu [9] proposed some fractional models for group decision making with IFPRs and applied these models for ranking the main factors of electronic learning.

The consistency, the basic property of IFPR [14], ensures a DM's judgment yields no self-contradiction during pairwise comparisons with alternatives. How to derive and rank priority weights from an IFPR is considered to be a major issue of use of consistency [15]. Generally, existing consistency of IFPR can be classified into two categories, the multiplicative consistency and additive consistency. Numerous ways for acquiring priority weights have been suggested relying on the condition of consistency. Liao and Xu [16] pointed out there is a flaw in additive consistency because it conflicts with the [0, 1] scale when used as preference value. In real decision-making problems, however, it is impractical and even impossible for a DM to provide a consistent IFPR due to the limitations of brain and inherent complexity in realistic environments. Since an inconsistent IFPR may lead to reasonable results, it is natural to take the consistency condition into consideration for deriving priority weights.

As to GDM analysis, determining the weights of all DMs is crucial during the process of decision making. Once the weight vector is confirmed, the IFPR of each DM can be directly aggregated to form a collective opinion [17]. Xu et al. [3] developed an approach to GDM based on IFPR as well as an approach to GDM based on incomplete IFPR, respectively, where two types of intuitionistic fuzzy averaging operator were defined and employed to aggregate the intuitionistic fuzzy information. Li et al. [18] proposed the intuitionistic fuzzy set generalized ordered weighted averaging (OWA) operator to solve the GDM problem. Liao and Xu [9] develop some algorithms for GDM with multiplicative IFPR via deriving the weight of each DM directly from the individual IFPR. What should be pointed out, however, is that the relative weights of DMs under IFPR environment are determined subjectively in existing literatures. Subjective weights are determined only by DM's expertise and judgment, otherwise objective weights are obtained via mathematical calculation. The methods of objective weights determination are particularly applicable in cases where reliable subjective weights are not available.

The methods mentioned above have succeeded in solving many GDM problems with IFPR information, however, some limitations in these methods still exist:

1) The relative weights of DMs under intuitionistic fuzzy environments in current literatures are regarded as the same or assigned by subjective weighting methods, which sound somewhat unpractical or even unreasonable in some situations.

2) Current methods [4], [6]-[9], [12], [13], [15], [16] for ranking alternatives in the form of IFPRs are without taking DM's risk preference (attitude) into account. In practice, various DMs have different preference for risk. That is to say, results may vary in terms of DM's risk preference for same decision-making problems.

To overcome these drawbacks, in this paper, we developed a new approach for intuitionistic fuzzy GDM based on IFPR with uncertain weights. The motivation of this paper is threefold. First, a new method is suggested for objective weight determination to aggregate each DM's individual IFPR into a collective one. In this method, relative weights of DMs are derived mathematically from the given preference information and have nothing to do with DMs' subjectivity. Second, we introduce an approach for group decision making by analyzing the multiplicative consistent IFPR. Besides, we review a method that ranks intuitionistic fuzzy weights with considering risk preference of each DM.

The remainder of this paper is organized as follows. In Section Ⅱ, we briefly review some basic knowledge about the IFS, the IFPR and so on. In Section Ⅲ, a nonlinear programming model is proposed for exploiting intuitionistic fuzzy weights based multiplicative consistency of individual and group IFPR, respectively; Moreover, a method for deriving relative weights of DMs from IFPRs is developed as well. Section Ⅳ investigates two numerical examples using the proposed method and compares three different methods. The paper concludes in Section Ⅴ.

Ⅱ. PRELIMINARIES A. Intuitionistic Fuzzy Preference Relation

Owing to the increasing complexity of the decision-making environment, it is hard and even impossible for DMs, or experts to provide accurate preferences on the pairwise comparison of alternatives. In other words, DMs may not have full confidence in their judgments. In this case, intuitionistic fuzzy sets appear to be a suitable and effective way to deal with such uncertainty and vagueness [16].

Definition 1 [2]: Let $X=(x_{1}, x_{2}, \ldots, x_{n})$ be a fixed non-empty set, an IFS $A$ in $X$ is defined as

 $A = \{\langle x, u_{A} (x), v_{A} (x)\rangle\left| {x\in X} \right. \}$ (1)

which is characterized by a membership function $u_{A}: X\to [0, 1]$ and non-membership function $v_{A}: X\to [0, 1]$ with the condition $0\leq u_{A} (x)+v_{A} (x)\leq 1$, $\exists x\in X$. The value, $\pi_{A} (x)=$ $1-u_{A} (x)-v_{A} (x)$ is called the indeterminacy degree or hesitation degree of element $x$ in set $A$. Particularly, if $\pi_{A} (x)=$ 0, then the IFS $A$ is reduced to a common fuzzy set.

Definition 2 [3]: An IFPR $\tilde{{R}}$ on the set $X= \{x_{1}$, $x_{2}, \ldots, x_{n} \}$ is characterized by an intuitionistic fuzzy judgments matrix $\tilde{{R}}$ $=$ $(\tilde{{r}}_{ij})_{n\times n} \subset X\times X$ with $\tilde{{r}}_{ij} =(u_{ij}, v_{ij})$, where

 \begin{align} & u_{ij} +v_{ij} \leq 1\notag\\ & u_{ij} =v_{ji}\notag\\ & u_{ii} =v_{ii} =0.5\notag\\ &\qquad\qquad u_{ij}, v_{ij} \in [0, 1], \ \ i, j= 1, 2, \ldots, n \end{align} (2)

$u_{ij}$ is the degree up to which $x_{i}$ is preferred over $x_{j}$, $v_{ij}$ is the degree to which $x_{i}$ is non-preferred to $x_{j}$, and $\pi_{ij} =1-u_{ij} -v_{ij}$ is expressed as the indeterminacy degree to which $x_{i}$ is preferred to $x_{j}$.

Definition 3 [4]: Let $\alpha = (u_{\alpha}, v_{\alpha}$) be an IFV, the score function of $\alpha$ is defined as

 $S(\alpha)=u_{\alpha} -v_{\alpha}.$ (3)

Inspired by the score function, Wang and Luo [19] introduced a formula to rank IFVs, which was in the restricted form as

 $S_{\lambda} (\alpha)=(u_{\alpha} -v_{\alpha})(1+\pi_{\alpha} )+\lambda \pi_{\alpha} ^{2}$ (4)

where $\pi_{\alpha}$ is the hesitation degree of element $\alpha$; and $\lambda \in [-1, 1]$ is the risk parameter given by the DMs in consensus, reflecting a DM's attitude towards risk. A smaller value of $\lambda$ is accompanied with higher levels of risk aversion. On the contrary, a bigger value means DMs are risk seeking. When $\lambda$ is close to 0, it indicates that DMs are risk-neutral and risk aversion (seeking) vanishes. Based on function (4), a ranking method for any two IFV $\alpha$ and $\beta$ is as below:

1) if $S_{\lambda} (\alpha)<S_{\lambda} (\beta)$, then $\alpha <\beta$;

2) if $S_{\lambda} (\alpha)=S_{\lambda} (\beta)$, then $\alpha =\beta$.

B. Multiplicative Consistency of IFPR

Definition 4 [16]: An intuitionistic fuzzy preference relation $\tilde{{R}}=(\tilde{{r}}_{ij})_{n\times n}$ with $\tilde{{r}}_{ij} =$ ($u_{ij}, v_{ij}$) is called multiplicative consistent if satisfying the following condition

 \begin{align} &u_{ij} \cdot u_{jk} \cdot u_{ki} =v_{ij} \cdot v_{jk} \cdot v_{ki} \nonumber\\ &\qquad\qquad\qquad\forall\, i, j, k = 1, 2, \ldots, n. \end{align} (5)

As $u_{ij} =v_{ji}$ for any IFPR according to Definition 2, (5) can be rewritten as

 \begin{align} &u_{ij} \cdot u_{jk} \cdot u_{ki} =u_{ji} \cdot u_{kj} \cdot u_{ik}\nonumber\\ &\qquad\qquad\qquad\forall\, i, j, k = 1, 2, \ldots, n. \end{align} (6)

Equation (6) contains only membership degrees of an IFPR, which facilitate our discussion later.

Definition 5 [19]: An intuitionistic fuzzy weight vector ${\tilde{\pmb w}}$ $=$ $(\tilde{{w}}_{1}, \tilde{{w}}_{2}, \ldots, \tilde{{w}}_{n}$) with $\tilde{{w}}_{i} = (w_{i}^{u}, w_{i}^{v}$) is said to be normalized if it satisfies the following conditions:

 \begin{align} \begin{cases} \displaystyle \sum\limits_{j=1, j\ne i}^n {w_{j}^{u}} \leq w_{i}^{v} \\[4mm] \displaystyle \sum\limits_{j=1, j\ne i}^n {w_{j}^{v}} \leq w_{i}^{u} +n-2 \\ \end{cases}\qquad \forall i= 1, 2, \ldots, n. \end{align} (7)

Motivated by the multiplicative consistent FPR and (7), we suppose that

 \begin{align} \tilde{{p}}_{ij} =(p_{ij}^{u}, p_{ij}^{v})=\begin{cases} (0.5, 0.5), &\text{if}\;i=j \\ ((w_{i}^{u} w_{j}^{v})^{\frac{1}{2}}, (w_{i}^{v} w_{j}^{u} )^{\frac{1}{2}}), &\text{if}\;i\ne j \\ \end{cases} \end{align} (8)

where $0\leq w_{i}^{u}$, $w_{i}^{v} \leq 1$, $w_{i}^{u} +w_{i}^{v} \leq 1$, $\sum_{j=1, j\ne i}^n {w_{j}^{u}} \leq w_{i}^{v}$, and $\sum_{j=1, j\ne i}^n {w_{j}^{v}} \leq w_{i}^{u} +n-2$, for $i= 1, 2, \ldots, n$. then we have the following theorem.

Theorem 1: Assume that the elements of $\tilde{{P}}=(\tilde{{p}}_{ij})_{n\times n}$ are defined by, then $\tilde{{P}}$ is a multiplicative consistent IFPR.

Proof: It is apparent that $p_{ij}^{u} =p_{ji}^{v}$, $p_{ij}^{v} =p_{ji}^{u}$ for $i, j=1, 2,$ $\ldots,$ $n$. Since $0\leq w_{i}^{u}$, $w_{i}^{v} \leq 1$ and $w_{i}^{u} +w_{i}^{v} \leq 1$, we have

 $(w_{i}^{u} w_{j}^{v})^{\frac{1}{2}}+(w_{i}^{v} w_{j}^{u})^{\frac{1}{2}} \leq \frac{w_{i}^{u} +w_{j}^{v}} {2}+ \frac{w_{i}^{v} +w_{j}^{u}} {2} \leq 2 \times \frac{1}{2}=1.$

As per Definition 2, $\tilde{{P}}=(\tilde{{p}}_{ij})_{n\times n}$ is an IFPR. Moreover, by (8), this gives

 \begin{align*} &p_{ij}^{u} \cdot p_{jk}^{u} \cdot p_{ki}^{u} = (w_{i}^{u} w_{j}^{v})^{\frac{1}{2}}\times (w_{j}^{u} w_{k}^{v})^{\frac{1}{2}} \\ &(w_{k}^{u} w_{i}^{v})^{\frac{1}{2}}= (w_{i}^{u} w_{j}^{v} w_{j}^{u} w_{k}^{v} w_{k}^{u} w_{i}^{v} )^{\frac{1}{2}} \end{align*}

and

 \begin{align*} &p_{ij}^{v} \cdot p_{jk}^{v} \cdot p_{ki}^{v} = (w_{i}^{v} w_{j}^{u})^{\frac{1}{2}}\times (w_{j}^{v} w_{k}^{u})^{\frac{1}{2}} \\ & (w_{k}^{v} w_{i}^{u})^{\frac{1}{2}}= (w_{i}^{v} w_{j}^{u} w_{j}^{v} w_{k}^{u} w_{k}^{v} w_{i}^{u} )^{\frac{1}{2}}. \end{align*}

Apparently, $p_{ij}^{u} \cdot p_{jk}^{u} \cdot p_{ki}^{u} =p_{ij}^{v} \cdot p_{jk}^{v} \cdot p_{ki}^{v}$. As per Definition 4, it is confirmed that $\tilde{{P}}=(\tilde{{p}}_{ij})_{n\times n}$ is a multiplicative consistent IFPR.

Based on Theorem 1, one can easily obtain the corollary as follows.

Corollary 1: Let $\tilde{{P}}=(\tilde{{p}}_{ij})_{n\times n}$ be an IFPR, if there exists a normalized intuitionistic fuzzy weight vector ${\tilde{{\pmb w}}}= (\tilde{{w}}_{1}, \tilde{{w}}_{2} ,$ $\ldots,$ $\tilde{{w}}_{n}$) such that

 \begin{align} \tilde{{p}}_{ij} =(p_{ij}^{u}, p_{ij}^{v})=\begin{cases} (0.5, 0.5), &\text{if}\;=j \\ ((w_{i}^{u} w_{j}^{v})^{\frac{1}{2}}, (w_{i}^{v} w_{j}^{u} )^{\frac{1}{2}}), &\text{if}\;i\ne j \\ \end{cases} \end{align} (9)

then $\tilde{{P}}=(\tilde{{p}}_{ij})_{n\times n}$ is a multiplicative consistent IFPR.

Ⅲ. GENERATE THE INTUITIONISTIC FUZZY WEIGHTS BASED ON IFPR

In this section, we propose nonlinear goal programming models for deriving intuitionistic fuzzy weight vector from individual and group IFPRs, respectively.

A. Individual Decision Making With IFPR

Every IFPR built by DMs are expected to be consistent, is the basis for reasonable prioritization as mentioned in Section Ⅰ. Multiplicative consistency guarantees a DM's judgment is logical and understandable rather than random. However, in real decision making scenario, it is harsh or sometime impossible for a DM to provide such multiplicative consistent IFPR. Under this case, it is expected that the absolute difference between given IFPRs and the multiplicative consistent IFPRs $\tilde{{P}}$, which yielded by (9), should be as small as possible. So, we introduce the following deviation variables $\phi_{ij}$ and $\varphi_{ij}$ to gauge the difference.

 \begin{align} \begin{cases} \varphi_{ij} =(w_{i}^{u} w_{j}^{v})^{\frac{1}{2}}-u_{ij}, &i, j=1, 2, \ldots, n;\ i\ne j \\ {\phi}_{ij} =(w_{i}^{v} w_{j}^{u})^{\frac{1}{2}}-v_{ij}, &i, j=1, 2, \ldots, n;\ i\ne j.\\ \end{cases} \end{align} (10)

The smaller the absolute difference, the better the results will be produced. This leads to an objective function such that,

 $\min J=\sum\limits_{i=1}^n {\sum\limits_{j=1}^n {\left| {\varphi_{ij}} \right|}} +\left| {{\phi}_{ij}} \right|.$ (11)

As $u_{ij} =v_{ji}$, $v_{ij} =u_{ji}$, one can get $\varphi_{ij} =(w_{i}^{u} w_{j}^{v})^{\frac{1}{2}}-u_{ij} =$ $(w_{j}^{v} w_{i}^{u} )^{\frac{1}{2}}-v_{ji} ={\phi}_{ji}$ for $i$, $j= 1, 2, \ldots, n$. Hence, (11) is equivalent to consider only the upper diagonal elements. Besides, we find (11) contains the absolute value, similar to the disposition in Wang et al. [20]. Let

 $\varphi_{ij}^{+} =\frac{1}{2}(\left| {\varphi_{ij}} \right|+\varphi_{ij} ) \;\;\text{and}\;\;\varphi_{ij}^{-} =\frac{1}{2}(\left| {\varphi_{ij}} \right|-\varphi_{ij})\nonumber\\ \qquad\qquad\qquad i= 1, 2, \ldots, n-1;\ j=i+ 1, 2, \ldots, n$ (12)
 ${\phi}_{ij}^{+} =\frac{1}{2}(\left| {{\phi}_{ij}} \right|+{\phi}_{ij} ) \;\;\text{and}\;\; {\phi}_{ij}^{-} =\frac{1}{2}(\left| {{\phi}_{ij}} \right|-{\phi} _{ij})\nonumber\\ \qquad\qquad\qquad i= 1, 2, \ldots, n-1;\ j=i+ 1, 2, \ldots, n$ (13)

then $| {\varphi_{ij}} |=\varphi_{ij}^{+} +\varphi_{ij}^{-}$, $\varphi _{ij} =\varphi_{ij}^{+} -\varphi_{ij}^{-}$, $| {{\phi}_{ij}} |={\phi}_{ij}^{+} +{\phi}_{ij}^{-}$, ${\phi}_{ij}=$ ${\phi} _{ij}^{+} -{\phi}_{ij}^{-}$; where $\varphi_{ij}^{+}$, $\varphi_{ij}^{-}$, ${\phi}_{ij}^{+}$, ${\phi}_{ij}^{-} \geq$ 0, and

 $\varphi_{ij}^{+} \cdot \varphi_{ij}^{-} =0, \quad {\phi}_{ij}^{+} \cdot {\phi}_{ij}^{-} =0.$

As a result, a nonlinear goal programming model can be built to derive the intuitionistic fuzzy weights as follow.

 \begin{align} &\min J=\sum\limits_{i=1}^{n-1} {\sum\limits_{j=i+1}^n {(\varphi_{ij}^{+} +\varphi_{ij}^{-} +{\phi}_{ij}^{+} +{\phi}_{ij}^{-})}}\nonumber\\ &{\rm s.t.} \begin{cases} (w_{i}^{u} w_{j}^{v})^{\frac{1}{2}}-u_{ij} -\varphi_{ij}^{+} +\varphi_{ij}^{-} =0\\ \qquad\qquad\qquad i=1, 2, \ldots, n-1; \ \, j=i+1, \ldots, n \\ (w_{i}^{v} w_{j}^{u})^{\frac{1}{2}}-v_{ij} -{\phi}_{ij}^{+} +{\phi} _{ij}^{-} =0\\ \qquad\qquad\qquad i=1, 2, \ldots, n-1;\ \, j=i+1, \ldots, n \\ \varphi_{ij}^{+} \geq 0;\ \ \varphi_{ij}^{-} \geq 0;\ \ {\phi}_{ij}^{+} \geq 0;\ \ {\phi}_{ij}^{-} \geq 0\\ \qquad\qquad\qquad i=1, 2, \ldots, n-1;\ \, j=i+1, \ldots, n \\ 0\leq w_{i}^{u} \leq 1;\ \ 0\leq w_{i}^{v} \leq 1;\ \ w_{i}^{u} +w_{i}^{v} \leq 1\\ \qquad\qquad\qquad i=1, 2, \ldots, n \\ \sum\limits_{j=1, j\ne i}^n {w_{j}^{u}} \leq w_{i}^{v};\ \ \sum\limits_{j=1, j\ne i}^n {w_{j}^{v}} \leq w_{i}^{u} +n-2\\ \qquad\qquad\qquad i=1, 2, \ldots, n. \\ \end{cases} \end{align} (14)

By using some popular optimization tools like MATLAB, WinSQB and so on, model (14) can be solved and an optimal intuitionistic fuzzy weight vector for $\tilde{{P}}$ is obtained. The optimal weight vector is denoted as ${\tilde{{\pmb w}}}^{\ast} = (\tilde{{w}}_{1}^{\ast}, \tilde{{w}}_{2}^{\ast}, \ldots, \tilde{{w}}_{n}^{\ast}$) where $\tilde{{w}}_{i}^{\ast} = (w_{i}^{u\ast}, w_{i}^{v\ast}$).

Apparent that when $J^{\ast} =$ 0, we have $\varphi_{ij}^{+} =\varphi_{ij}^{-} ={\phi}_{ij}^{+} =$ ${\phi}_{ij}^{-}$ $=$ 0. It implies the IFPR $\tilde{{P}}$ provided by a DM is multiplicative consistent, and thus, the obtained weight vector is credible.

B. GDM With IFPR Under Uncertain Weights

In real-world situations, decisions are usually made by a group of DMs (or experts) rather than an individual. Hence, group decision making is a more significant topic in current management science that has attracted considerable attention [14], [16], [18], [21], [22].

Let ${\pmb e}= \{e_{1}, e_{2}, \ldots, e_{s} \}$ be the set of DMs who are invited to express their opinions on alternatives ${\pmb X}= \{x_{1}, x_{2}, \ldots, x_{n} \}$, given that the IFPR $\tilde{{P}}^{k}$ given by DM $e_{k}$, $k= 1, 2, \ldots, s$ is denoted as $\tilde{{P}}^{k}=(\tilde{{p}}_{ij}^{k} )_{n\times n}$, where $\tilde{{p}}_{ij}^{k} =(u_{ij}^{k}, v_{ij}^{k})$, $i, j= 1, 2, \ldots, n$. The set of weight vector of DMs is ${\pmb c}=$ ($c_{1}, c_{2}, \ldots, c_{s}$), where $c_{k}$ is the $k$th DM's weight and satisfying $c_{k} >$ 0 and $\sum\nolimits_{k=1}^s {c_{k}} =1$, $k$ $=$ $1, 2, \ldots, s$. To determine the weights of each DM is the prerequisite for any GDM problems. A simple way to do that is average assignment if there are no special differences among them. However, a DM may not be able to grasp all aspects of a problem but on some parts of it for which the person is capable [23]. So it is natural and reasonable to assume each DM should have a different weight, which is uncertain beforehand and needs to be determined.

In order to determine the weights $c_{k}$ numerically, consider the score function of $\tilde{{p}}_{ij}^{k}$

 $S(\tilde{{p}}_{ij}^{k}) =u_{ij}^{k} -v_{ij}^{k}, \quad i, j= 1, 2, \ldots, n.$ (15)

It is noted that $S(\tilde{{p}}_{ij}^{k})$ is increasing with $u_{ij}^{k}$, the membership degree of $\tilde{{p}}_{ij}^{k}$, nevertheless decreasing with $v_{ij}^{k}$, the non-membership degree of $\tilde{{p}}_{ij}^{k}$. Therefore values of score function associated to alternatives can be understood as a sort of preference by DMs.

Let

 $t_{ik} =\sum\limits_{j=1, j\ne i}^n {S(\tilde{{p}}_{ij}^{k})}, \quad i= 1, 2, \ldots, n; \ \, k= 1, 2, \ldots, s.$ (16)

As aforementioned, since the $S(\tilde{{p}}_{ij}^{k})$ can be interpreted as the degree of preference of $x_{i}$ over $x_{j}$ by DM $e_{k}$, accordingly $t_{ik}$ can be seen as the overall degree of preference of $x_{i}$ over all the other ($n-1$) alternatives $x_{j}$ $(j= 1, 2, \ldots, n,$ $j\ne i$). Obviously, greater values of $t_{ik}$ are associated with higher levels of preference on $x_{i}$ by a DM [17]. Thus, the overall degree of preference of all DMs can be concisely expressed in the matrix format as below:

 \begin{align} {\pmb T}=(t_{ik})_{n\times s} = \begin{array}{l} \quad e_{1}\quad \quad e_{2}\quad \cdots\quad e_{s} \\ \left({{\begin{array}{*{20}c} {t_{11}} & {t_{12}} & \cdots & {t_{1s}} \\ {t_{21}} & {t_{22}} & \cdots & {t_{2s}} \\ \vdots & \vdots & \ddots & \vdots \\ {t_{n1}} & {t_{n2}} & \cdots & {t_{ns}} \\ \end{array}}} \right) \\ \end{array} \end{align} (17)

For the IFPR $\tilde{{P}}^{k}=(\tilde{{p}}_{ij}^{k})_{n\times n}$ provided by $e_{k}$, we define an index, the standard deviation between $t_{ik}$ and $\bar{{t}}_{k}$ given by

 $\sigma_{k} =\sqrt{\frac{1}{n}\sum\limits_{i=1}^n {(t_{ik} -\bar{{t}}_{k} )^{2}}}$ (18)

where $\bar{{t}}_{k} =\frac{1}{n}\sum_{i=1}^n {t_{ik}}$, $k= 1, 2, \ldots, s$. Since $t_{ik} =\sum_{j=1, j\ne i}^n {S(\tilde{{p}}_{ij}^{k} )}$, we have $\bar{{t}}_{k} =\frac{1}{n}\sum_{i=1}^n {\sum_{j=1, j\ne i}^n {S(\tilde{{p}}_{ij}^{k})}} =$ 0.

Then (18) can be equivalently written as

 $\sigma_{k} =\sqrt{\frac{1}{n}\sum\limits_{i=1}^n {t_{ik}^{2}}}.$ (19)

Note that $\sigma_{k} =$ 0 if $t_{ik} =$ 0, $i= 1, 2, \ldots, n$. In this case, the removal of $t_{ik}$ $(i= 1, 2, \ldots, n)$ from matrix ${\pmb T}$ has little effect on final prioritization. That is to say, the IFPR $\tilde{{P}}^{k}$ can be removed without little impact on group decision making. Therefore, it should be assigned a relatively small weight. Conversely, the greater the $\sigma _{k}$, the bigger difference among $t_{ik} (i= 1, 2, \ldots, n)$, which implies a stronger preference of the $e_{k}$ due to (18). From the DM's perspective, a bigger $\sigma_{k}$ indicates it is more important for group decision making [24], and naturally be assigned a relatively big weight.

Based upon the above analysis, we concluded that greater values of weights should be assigned to those preferences with big deviation. It is reasonable that the relative weights going to be determined should maximize the sum of deviation of the $s$ overall degree of preferences. Thus, a self-evident optimization model to determine the weights of DMs is constructed as follows:

 \begin{align} &\max Z=\sum\limits_{k=1}^s {\sigma_{k} c_{k}} =\sum\limits_{k=1}^s {c_{k} \left(\sqrt{\frac{1}{n}\sum\limits_{i=1}^n {t_{ik}^{2}}}\right)}\nonumber\\ &{\rm s. t.}\; \sum\limits_{k=1}^s {c_{k}} =1, \ \, c_{k} >0, \quad k= 1, 2, \ldots, s. \end{align} (20)

Clearly, model (20) is a single linear optimization problem, which can be easily solved by the Simplex method or some mathematical optimization toolkits.

Determination of the DMs' relative weights is a key issue in any group decision making. Let us now consider a more general version of weight vector ${\pmb c}=(c_{1}, c_{2}, \ldots, c_{s}$), and denote it as ${\tilde{{\pmb c}}}=(\tilde{{c}}_{1}, \tilde{{c}}_{2}, \ldots, \tilde{{c}}_{s}$) which satisfies

 $\sum\limits_{k=1}^s {\tilde{{c}}_{k}^{\alpha}} =1, \quad \alpha >1$ (21)

instead of the normalized weight constraint $\sum\nolimits_{k=1}^s {c_{k} =1}$, where $\alpha$ $>$ $1$ be a positive parameter that offers flexible choice of weights for DMs. Thus, model (20) can be converted into the following model:

 \begin{align} &\max Z=\sum\limits_{k=1}^s {\sigma_{k} \tilde{{c}}_{k}} =\sum\limits_{k=1}^s {\tilde{{c}}_{k} \left(\sqrt{\frac{1}{n}\sum\limits_{i=1}^n {t_{ik}^{2}}}\right)}\nonumber\\ &{\rm s. t.}\; \sum\limits_{k=1}^s {\tilde{{c}}_{k}^{\alpha}} =1, \ \, \tilde{{c}}_{k} >0, \quad k= 1, 2, \ldots, s. \end{align} (22)

Regarding this optimization model, we have the following theorems.

Theorem 2: Let ${\tilde{{\pmb c}}}^{\ast} = (\tilde{{c}}_{1}^{\ast}, \tilde{{c}}_{2}^{\ast}, \ldots, \tilde{{c}}_{s}^{\ast}$) be the optimal solution to model (22), then

 \begin{align} \tilde{{c}}_{k}^{\ast} =\frac{\displaystyle\left(\sum\limits_{i=1}^n {t_{ik}^{2}} \right)^{\textstyle\frac{1}{\alpha -1}}}{\displaystyle\sum\limits_{l=1}^s {\left(\sum\limits_{i=1}^n {t_{il}^{2}}\right)^{\textstyle\frac{1}{\alpha -1}}}}, \quad k= 1, 2, \ldots, s. \end{align} (23)

Proof: Since ${\tilde{{\pmb c}}}^{\ast} = (\tilde{{c}}_{1}^{\ast}, \tilde{{c}}_{2}^{\ast}, \ldots, \tilde{{c}}_{s}^{\ast}$) is a bounded vector and $Z$ is a continuous function of $\tilde{{c}}_{k}$, $k= 1, 2, \ldots, s$, there must exist a maximum point that model (22) holds. To obtain the optimal solution, we using the Lagrangian multiplier method, and the Lagrangian function derived from (22) can be formulated as below:

 $L({\tilde{{\pmb c}}}, \lambda)=\sum\limits_{k=1}^s {\tilde{{c}}_{k} \left(\sqrt{\frac{1}{n}\sum\limits_{i=1}^n {t_{ik}^{2}}}\right)} -\lambda \left(\sum\limits_{k=1}^s {\tilde{{c}}_{k}^{2}} -1\right)$ (24)

taking the partial derivatives of $L$ within $\tilde{{c}}_{k}$ and letting them be 0, yields that

 $\frac{\delta L({\tilde{{\pmb c}}}, \lambda)}{\delta \tilde{{c}}_{k} }=\sqrt{\frac{1}{n}\sum\limits_{i=1}^n {t_{ik}^{2}}} -\lambda \alpha \tilde{{c}}_{k}^{\alpha -1} =0, \quad k= 1, 2, \ldots, s.$ (25)

Thus, one can get $\tilde{{c}}_{k} =\left( {\frac{\frac{1}{n}\sum\nolimits_{i=1}^n {t_{ik}^{2}}} {\alpha \lambda}} \right)^{\textstyle\frac{1}{\alpha -1}}$; due to the fact that $\sum\nolimits_{k=1}^s {\tilde{{c}}_{k}^{\alpha}}$ $=1$, $\tilde{{c}}_{k} \geq 0$, we have

 $\sum\limits_{k=1}^s {\left({\frac{\displaystyle\frac{1}{n}\sum\limits_{i=1}^n {t_{ik}^{2}}} {\alpha \lambda}} \right)^{\textstyle\frac{\alpha} {\alpha -1}}} =1, \quad \lambda >0.$ (26)

It follows that

 $\lambda = \frac{1}{\alpha} \left[{\sum\limits_{k=1}^s {\left( {\frac{1}{n}\sum\limits_{i=1}^n {t_{ik}^{2}}} \right)}^{\textstyle\frac{\alpha-1}{\alpha}}} \right]^{\textstyle\frac{\alpha} {\alpha -1}}.$ (27)

Hence, we obtain

 \begin{align} &\tilde{{c}}_{k}^{\ast} =\left({\frac{\displaystyle\frac{1}{n}\sum\limits_{i=1}^n {t_{ik}^{2}}} {\displaystyle\left[{\sum\limits_{k=1}^s {(\frac{1}{n}\sum\limits_{i=1}^n {t_{ik}^{2}})}^{\textstyle\frac{\alpha-1}{\alpha}}} \right]^{\textstyle\frac{\alpha} {\alpha -1}}}} \right)^{\textstyle\frac{1}{\alpha -1}}\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad k= 1, 2, \ldots, s. \end{align} (28)

After normalization, (28) becomes,

 \begin{align} \tilde{{c}}_{k}^{\ast} =\frac{\displaystyle\left(\sum\limits_{i=1}^n {t_{ik}^{2}} \right)^{\textstyle\frac{1}{\alpha -1}}}{\displaystyle\sum\limits_{l=1}^s {\left(\sum\limits_{i=1}^n {t_{il}^{2}}\right)^{\textstyle\frac{1}{\alpha -1}}}}, \qquad k= 1, 2, \ldots, s. \end{align} (29)

From (29), we find that $\tilde{{c}}_{k}^{\ast}$ is an exponential function of factor $\alpha$. We denoted $\alpha$ as the weight assignment factor. When $\alpha$ is approaching to $+\infty$, then $\tilde{{c}}_{k}^{\ast} ={1}/{s}$, $k= 1, 2, \ldots s.$

Once we obtain the weight vector of DMs, the IFPR given by each DM can be aggregated for building a collective goal programming model. As aforementioned in Section Ⅳ-A, we expect that the deviation between the given IFPRs, as well as the multiplicative consistent IFPRs approach to zero. This idea yields the following deviation variables:

 \begin{align} \begin{cases} \varphi_{ij}^{k} =(w_{i}^{u} w_{j}^{v})^{\frac{1}{2}}-u_{ij}^{k}, &i=1, 2, \ldots, n-1;\ \, j=i+1, \ldots, n \\ {\phi}_{ij}^{k} =(w_{i}^{v} w_{j}^{u})^{\frac{1}{2}}-v_{ij}^{k}, &i=1, 2, \ldots, n-1;\ \, j=i+1, \ldots, n. \\ \end{cases} \end{align} (30)

Thus, a group goal programming model was constructed in a similar way of model (14) to derive an intuitionistic fuzzy weight

 \begin{align} \min J=\sum\limits_{k=1}^s {\sum\limits_{i=1}^{n-1} {\sum\limits_{j=i+1}^n {c_{k} (\varphi_{ij}^{k+} +\varphi_{ij}^{k-} +{\phi} _{ij}^{k+} +{\phi}_{ij}^{k-})}}} \end{align} (31)
 \begin{align} {\rm s. t.} \begin{cases} (w_{i}^{u} w_{j}^{v})^{\frac{1}{2}}-u_{ij}^{k} -\varphi_{ij}^{k+} +\varphi _{ij}^{k-} =0\\ \qquad i=1, 2, \ldots, n-1; \ \, j=i+1, \ldots, n; \ \, k=1, 2, \ldots, s \\ (w_{i}^{v} w_{j}^{u})^{\frac{1}{2}}-v_{ij}^{k} -{\phi}_{ij}^{k+} +{\phi}_{ij}^{k-} =0 \\ \qquad i=1, 2, \ldots, n-1;\ \, j=i+1, \ldots, n;\ \, k=1, 2, \ldots, s \\ \varphi_{ij}^{k+} \geq 0;\; \varphi_{ij}^{k-} \geq 0; \;{\phi}_{ij}^{k+} \geq 0;\; {\phi}_{ij}^{k-} \geq 0\\ \qquad i=1, 2, \ldots, n-1; \ \, j=i+1, \ldots, n;\ \, k=1, 2, \ldots, s \\ 0\leq w_{i}^{u} \leq 1;\; 0\leq w_{i}^{v} \leq 1; \;w_{i}^{u} +w_{i}^{v} \leq 1\\ \qquad i=1, 2, \ldots, n \\ \sum\limits_{j=1, j\ne i}^n {w_{j}^{u}} \leq w_{i}^{v}; \;\sum\limits_{j=1, j\ne i}^n {w_{j}^{v}} \leq w_{i}^{u} +n-2\\ \qquad i=1, 2, \ldots, n. \\ \end{cases} \end{align} (32)

Note that $(w_{i}^{u} w_{j}^{v})^{\frac{1}{2}}-u_{ij}^{k} -\varphi_{ij}^{k+} +\varphi_{ij}^{k-} =$ 0, for all $i= 1, 2,$ $\ldots,$ $n-1$, $j=i+1, \ldots, n$, $k= 1, 2, \ldots, s$. If we multiply $c_{k}$ on both sides of this formula, we get

 \begin{align} c_{k} (w_{i}^{u} w_{j}^{v})^{\frac{1}{2}}-c_{k} u_{ij}^{k} -c_{k} \varphi _{ij}^{k+} +c_{k} \varphi_{ij}^{k-} =0, \quad k= 1, 2, \ldots, s. \end{align} (33)

Since $\sum\nolimits_{k=1}^s {c_{k}} =1$, adding all these $s$ formulas together, it yields

 \begin{align} &c_{k} (w_{i}^{u} w_{j}^{v})^{\frac{1}{2}}-\sum\limits_{k=1}^s {c_{k}} u_{ij}^{k} -\sum\limits_{k=1}^s {c_{k}} \varphi_{ij}^{k+}+\sum\limits_{k=1}^s {c_{k}} \varphi_{ij}^{k-} =0\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad k= 1, 2, \ldots, s. \end{align} (34)

Likewise, we can obtain

 \begin{align} &c_{k} (w_{i}^{u} w_{j}^{v})^{\frac{1}{2}}-\sum\limits_{k=1}^s {c_{k}} v_{ij}^{k} -\sum\limits_{k=1}^s {c_{k}} {\phi}_{ij}^{k+}+\sum\limits_{k=1}^s {c_{k}} {\phi}_{ij}^{k-} =0 \nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad k= 1, 2, \ldots, s. \end{align} (35)

Let $\tilde{{\phi}} _{ij}^{k+} =\sum\nolimits_{k=1}^s {c_{k}} \varphi _{ij}^{k+}$, $\tilde{{\phi}} _{ij}^{k-} =\sum\nolimits_{k=1}^s {c_{k}} \varphi_{ij}^{k-}$, ${\phi}_{ij}^{k+} =\sum\nolimits_{k=1}^s {c_{k}} {\phi}_{ij}^{k+}$, and ${\phi}_{ij}^{k-} =\sum\nolimits_{k=1}^s {c_{k}} {\phi}_{ij}^{k-}$. The model (31) can be transformed into the following optimization model naturally,

 \begin{align} &\min J=\sum\limits_{i=1}^{n-1} {\sum\limits_{j=i+1}^n {(\tilde{{\varphi }}_{ij}^{k+} +\tilde{{\phi}} _{ij}^{k-} +\tilde{{{\phi}}} _{ij}^{k+} +\tilde{{{\phi}}} _{ij}^{k-})}}\nonumber\\ &{\rm s. t.}\; \begin{cases} (w_{i}^{u} w_{j}^{v})^{\frac{1}{2}}-\sum\limits_{k=1}^s {c_{k}} u_{ij}^{k} -\tilde{{\phi}} _{ij}^{k+} +\tilde{{\phi}} _{ij}^{k-} =0\\ \qquad i=1, 2, \ldots, n-1;\ \, j=i+1, \ldots, n; \ \, k=1, 2, \ldots, s \\ (w_{i}^{v} w_{j}^{u})^{\frac{1}{2}}-\sum\limits_{k=1}^s {c_{k}} v_{ij}^{k} -\tilde{{{\phi}}} _{ij}^{k+} +\tilde{{{\phi}}} _{ij}^{k-} =0\\ \qquad i=1, 2, \ldots, n-1;\ \, j=i+1, \ldots, n; \ \, k=1, 2, \ldots, s \\ \tilde{{\phi}} _{ij}^{k+} \geq 0;\; \tilde{{\phi}} _{ij}^{k-} \geq 0; \;\tilde{{{\phi}}} _{ij}^{k+} \geq 0; \;\tilde{{{\phi}}} _{ij}^{k-} \geq 0\\ \qquad i=1, 2, \ldots, n-1;\ \, j=i+1, \ldots, n; \ \, k=1, 2, \ldots, s \\ 0\leq w_{i}^{u} \leq 1;\; 0\leq w_{i}^{v} \leq 1; \; w_{i}^{u} +w_{i}^{v} \leq 1\\ \qquad i=1, 2, \ldots, n \\ \sum\limits_{j=1, j\ne i}^n {w_{j}^{u}} \leq w_{i}^{v}; \;\sum\limits_{j=1, j\ne i}^n {w_{j}^{v}} \leq w_{i}^{u} +n-2\\ \qquad i=1, 2, \ldots, n. \\ \end{cases} \end{align} (36)

By solving model (35), it gives an overall intuitionistic fuzzy weight vector ${\tilde{{\pmb w}}}^{\ast} = (\tilde{{w}}_{1}^{\ast}, \tilde{{w}}_{2}^{\ast}, \ldots, \tilde{{w}}_{n}^{\ast}$), where $\tilde{{w}}_{i}^{\ast} = (w_{i}^{u\ast}, w_{i}^{v\ast}$), for GDM with $\tilde{{P}}^{k}=(\tilde{{p}}_{ij}^{k})_{n\times n}$, $k= 1, 2, \ldots, s$. Obviously, model (35) has less computational complexity as it can be accomplished in linear time.

Ⅳ. NUMERICAL ILLUSTRATION

To illustrate the proposed method, two numerical examples are examined to show how to apply it to generate the intuitionistic fuzzy weights from IFPR for prioritization. Meanwhile, a comparison analysis of the obtained solutions with other methods is carried out in this section.

A. Description and Decision Model

Example 1: Consider a DM with risk-attitude representing his/her IFPR over a set of alternatives {$x_{1}$, $x_{2}$, $x_{3}$}, which are taken from Wang [15].

 \begin{align*} \tilde{{P}}=&\ (\tilde{{p}}_{ij})_{3\times 3} \nonumber\\ =&\, \begin{pmatrix} {(0.5, 0.5)} & {(0.065, 0.935)} & {(0.48, 0.52)} \\ {(0.935, 0.065)} & {(0.5, 0.5)} & {(0.915, 0.085)} \\ {(0.52, 0.48)} & {(0.085, 0.915)} & {(0.5, 0.5)} \\ \end{pmatrix}. \end{align*}

It can be checked via (6) that $\tilde{{P}}=(\tilde{{p}}_{ij})_{3\times 3}$ does not satisfy the condition of multiplicative consistent. According to model (14), the goal programming model is built as follow:

 \begin{align*} \min J=&\ (\varphi_{12}^{+} +\varphi_{12}^{-} +{\phi}_{12}^{+} +{\phi} _{12}^{-})+(\varphi_{13}^{+} +\varphi_{13}^{-}\\ & \, +{\phi}_{13}^{+} +{\phi} _{13}^{-})+(\varphi_{23}^{+} +\varphi_{23}^{-} +{\phi}_{23}^{+} +{\phi} _{23}^{-})\\ &{\rm s. t.} \begin{cases} \sqrt{w_{1}^{u} w_{2}^{v}} -0.065-\varphi_{12}^{+} +\varphi_{12}^{-} =0\\ \sqrt{w_{1}^{v} w_{2}^{u}} -0.935-{\phi}_{12}^{+} +{\phi} _{12}^{-} =0\\ \sqrt{w_{1}^{u} w_{3}^{v}} -0.48-\varphi_{13}^{+} +\varphi_{13}^{-} =0\\ \sqrt{w_{1}^{v} w_{3}^{u}} -0.52-{\phi}_{13}^{+} +{\phi} _{13}^{-} =0\\ \sqrt{w_{2}^{u} w_{3}^{v}} -0.915-\varphi_{23}^{+} +\varphi_{23}^{-} =0\\ \sqrt{w_{2}^{v} w_{3}^{u}} -0.085-{\phi}_{23}^{+} +{\phi}_{23}^{-} =0\\ \varphi_{12}^{+} \geq 0, \ \, \varphi _{12}^{-} \geq 0, \ \, {\phi} _{12}^{+} \geq 0, \ \, {\phi} _{12}^{-} \geq 0, \ \, \varphi_{13}^{+} \geq 0, \ \, \varphi_{13}^{-} \geq 0\\ {\phi}_{13}^{+} \geq 0, \ \, {\phi}_{13}^{-} \geq 0, \ \, \varphi_{23}^{+} \geq 0, \ \, \varphi_{23}^{-} \geq 0, \ \, {\phi}_{23}^{+} \geq 0, \ \, {\phi}_{23}^{-} \geq 0\\ 0\leq w_{1}^{u} \leq 1, \ \, 0\leq w_{2}^{u} \leq 1, \ \, 0\leq w_{3}^{u} \leq 1, \ \, 0\leq w_{1}^{v} \leq 1\\ 0\leq w_{2}^{v} \leq 1, \ \, 0\leq w_{3}^{v} \leq 1\\ w_{1}^{u} +w_{1}^{v} \leq 1, \ \, w_{2}^{u} +w_{2}^{v} \leq 1, \ \, w_{3}^{u} +w_{3}^{v} \leq 1\\ w_{2}^{u} +w_{3}^{u} \leq w_{1}^{v}, \ \, w_{1}^{u} +w_{3}^{u} \leq w_{2}^{v}\\ w_{1}^{u} +w_{2}^{u} \leq w_{3}^{v}, \ \, w_{1}^{u} +1\geq w_{2}^{v} +w_{3}^{v}\\ w_{2}^{u} +1\geq w_{1}^{v} +w_{3}^{v}, \ \, w_{3}^{u} +1\geq w_{1}^{v} +w_{2}^{v}. \\ \end{cases} \end{align*}

By using Lingo 11 to solve this model, we have the following results:

 \begin{align*} &w_{1}^{u} =0.0394, \ \, w_{1}^{v} =0.9605, \ \, w_{2}^{u} =0.8929, \ \, w_{2}^{v} =0.1070\\ &w_{3}^{u} =0.0675, \ \, w_{3}^{v} =0.9324, \ \, \varphi_{12}^{+} =0.0, \ \, \varphi_{12}^{-} =0.0\\ &\varphi_{13}^{+} =0.0, \ \, \varphi_{13}^{-} =0.2881, \ \, \varphi_{23}^{+} =0.0, \ \, \varphi_{23}^{-} =0.0025\\ &{\phi}_{12}^{+} =0.0, \ \, {\phi}_{12}^{-} =0.009, \ \, {\phi}_{13}^{+} =0.0, \ \, {\phi}_{13}^{-} =0.2653\\ &{\phi}_{23}^{+} = 0.0, \ \, {\phi}_{23}^{-} = 0.0. \end{align*}

Therefore, the optimal intuitionistic fuzzy weight vector ${\tilde{{\pmb w}}}^{\ast}$ $=$ ($\tilde{{w}}_{1}^{\ast}, \tilde{{w}}_{2}^{\ast}, \tilde{{w}}_{3}^{\ast})= ((0.039, 0.961),$ $(0.893, 0.107),$ $(0.068,$ $0.932))$. As we know $\lambda = 0$, and calculated by (4), we get $S(\tilde{{w}}_{1}^{\ast})$ $=$ $-0.921,$ $S(\tilde{{w}}_{2}^{\ast})= 0.786$, $S(\tilde{{w}}_{3}^{\ast})= -0.865$ which gives the ranking of $x_{2} \succ x_{3} \succ x_{1}$.

Wang [15] used the additive consistency-based method to derive a priority weight vector, and Liao and Xu [16] constructed a fractional programming model to extract priority weights based on multiplicative consistency-based method. Their findings are listed in Table Ⅰ which led to the same ranking: $x_{2} \succ x_{3} \succ x_{1}$ as our method does, but with slightly different membership degree of preference.

Table Ⅰ
COMPARISON FOR PRIORITY WEIGHTS WITH DIFFERENT METHODS

With the development of weapon and the requirement of modern warfare, many state-of-the-art technologies, Such as Markovian jumping systems, dynamic feedback control [25] etc., have been applied to military weapons and equipment. In pursuit of high performance of modern weapons, however, there is more risk in their application. That is to say, military high-techs are generally accompanied with higher risk. They are often lacking of necessary and precious data for evaluating associated technological risk, which may cause a serious effect on finalization of the military and industrial products. In this situation, intuitionistic fuzzy value as well as preference relation is a powerful tool in estimating underlying technical risk by experts.

A research institute of Nanjing military region in China planned to have an evaluation of technical risk of a tentative armored vehicle. Three experts $e_{1}$, $e_{2}$ and $e_{3}$ were invited to join this plan. The research institute had hesitation in determining the weight of each expert owing to lack of cooperation basis. There are four potential risk factors identified by experts for further estimating, which include maturity ($x_{1}$), complexity ($x_{2}$), reliability ($x_{3}$) and prospective ($x_{4}$) of technology. Each expert $e_{k}$ $(k= 1, 2, 3)$, is asked to have a pair-wise comparison for these factors, resulting in the following intuitionistic fuzzy preference relation $\tilde{{P}}^{k}$ $(k= 1, 2, 3)$.

 \begin{align*} &\tilde{{P}}^{1}=(\tilde{{p}}_{ij}^{1})_{4\times 4} \\ &=\begin{pmatrix} {(0.50, 0.50)} & {(0.55, 0.40)} & {(0.69, 0.31)} & {(0.30, 0.41)} \\ {(0.40, 0.55)} & {(0.50, 0.50)} & {(0.52, 0.34)} & {(0.35, 0.60)} \\ {(0.31, 0.69)} & {(0.34, 0.52)} & {(0.50, 0.50)} & {(0.34, 0.66)} \\ {(0.41, 0.30)} & {(0.60, 0.35)} & {(0.66, 0.34)} & {(0.50, 0.50)} \\ \end{pmatrix}\\[3mm] &\tilde{{P}}^{2}=(\tilde{{p}}_{ij}^{2})_{4\times 4} \\ &=\begin{pmatrix} {(0.50, 0.50)} & {(0.66, 0.21)} & {(0.63, 0.29)} & {(0.45, 0.28)} \\ {(0.21, 0.66)} & {(0.50, 0.50)} & {(0.54, 0.41)} & {(0.42, 0.34)} \\ {(0.29, 0.63)} & {(0.41, 0.54)} & {(0.50, 0.50)} & {(0.38, 0.54)} \\ {(0.28, 0.45)} & {(0.34, 0.42)} & {(0.54, 0.38)} & {(0.50, 0.50)} \\ \end{pmatrix}\\[3mm] &\tilde{{P}}^{3}=(\tilde{{p}}_{ij}^{3})_{4\times 4} \\ &=\begin{pmatrix} {(0.50, 0.50)} & {(0.46, 0.39)} & {(0.63, 0.22)} & {(0.28, 0.52)} \\ {(0.39, 0.46)} & {(0.50, 0.50)} & {(0.50, 0.42)} & {(0.33, 0.47)} \\ {(0.22, 0.63)} & {(0.42, 0.50)} & {(0.50, 0.50)} & {(0.27, 0.51)} \\ {(0.52, 0.28)} & {(0.47, 0.33)} & {(0.51, 0.27)} & {(0.50, 0.50)} \\ \end{pmatrix}. \end{align*}

By using (16) and (17), the overall degree of preference of all the experts is denoted by matrix ${\pmb T}$, as follows:

 ${\pmb T}=(t_{ik})_{4\times 3} = \begin{pmatrix} {0.42} & {0.96} & {0.24} \\ {-0.22} & {-0.24} & {-0.13} \\ {-0.88} & {-0.63} & {-0.73} \\ {0.68} & {-0.09} & {0.62} \\ \end{pmatrix}.$

Based on ${\pmb T}$, the priority weights vector of experts can be derived by (29). Suppose the weight assignment parameter $\alpha$ $= 2.0$, thus the relative weight of each expert can be obtained as: $c_{1} = 0.381$, $c_{2} = 0.361$, $c_{3} = 0.258$. According to model (34), a nonlinear programming model for GDM can be constructed, and solving this model by using Lingo 11 software toolkit, we obtained the optimal intuitionistic fuzzy weights ${\tilde{{\pmb w}}}^{\ast} =(\tilde{{w}}_{1}^{\ast}, \tilde{{w}}_{2}^{\ast}, \tilde{{w}}_{3}^{\ast}, \tilde{{w}}_{4}^{\ast}$): $\tilde{{w}}_{1}^{\ast} = (0.396, 0.598)$, $\tilde{{w}}_{2}^{\ast} =$ $(0.191,$ $0.809)$, $\tilde{{w}}_{3}^{\ast} = (0.130, 0.869)$, $\tilde{{w}}_{4}^{\ast} = (0.277, 0.717)$.

Consider the experts are all risk-neutral, leads to $\lambda_{k} = 0$, $k$ $=$ $1, 2, 3$. As per (4), we can easily get $S(\tilde{{w}}_{1}^{\ast })=-0.203$, $S(\tilde{{w}}_{2}^{\ast} )$ $=$ $-0.618$, $S(\tilde{{w}}_{3}^{\ast} )=-0.739$, $S(\tilde{{w}}_{4}^{\ast})=-0.443$, which gives the ranking of $x_{1} \succ x_{4} \succ x_{2} \succ x_{3}$. Hence, the maturity of technology ($x_{1}$), should be given first priority during the process of technical risk evaluation of armored equipment.

From the results we have, the factor of maturity ($x_{1}$) ranks first in the risk control process. This is somewhat in conformity with our intuition in that the factor of maturity in any modern military equipment should be placed in a fundamental position.

Remark 1: We compare the results, by varying DMs' risk preference from risk-neutral to risk aversion, and risk seeking as well. Assume that the DMs feel an aversion to factors $x_{1}$ and $x_{2}$, but in favor of risk-seeking to the two others yet. Given that the risk parameter for each factor after negotiation be $\lambda_{1}$ $=$ $-0.5$, $\lambda_{2} =-0.75$, $\lambda_{3} = 0.8$, $\lambda_{4} = 0.35$. As per (4), it yields that $S(\tilde{{w}}_{1}^{\ast})=-0.709$, $S(\tilde{{w}}_{2}^{\ast} )=-1.368$, $S(\tilde{{w}}_{3}^{\ast})= 0.062$, $S(\tilde{{w}}_{4}^{\ast })$ $=$ $-0.088$, respectively.

As can be seen from Table Ⅱ, if DMs vary their risk preference, the priorities of these factors are also change. This shows that it is reasonable and necessary to bring the risk parameter into consideration during a GDM course. Besides, it is deserving to point out that these existing methods without considering risk preference are just a special case of our method when risk parameters are all equal to zero.

Table Ⅱ
RANKING RESULTS WITH DIFFERENT RISK PARAMETERS
B. Comparison Analysis

A comparative study was conducted to contrast between our method and other ones. As the same problem of Example 2, we use several different approaches to tackle this problem. To facilitate our analysis, we suppose that all DMs are risk-neutral. We use Xu et al. normalizing rank summation method [5] firstly. For simplicity, suppose each DM has the same relative weight and similarly hereinafter. The intuitionistic fuzzy weights produced by this method are $\tilde{{w}}_{1} = (0.232,$ $0.661)$, $\tilde{{w}}_{2} = (0.195, 0.697)$, $\tilde{{w}}_{3} = (0.169, 0.755)$, $\tilde{{w}}_{4} = (0.220,$ $0.658)$, and the values of the corresponding score function are, $S(\tilde{{w}}_{1} )$ $=$ $-0.4748$, $S(\tilde{{w}}_{2})= -0.5566$, $S(\tilde{{w}}_{3})= -0.6307$, $S(\tilde{{w}}_{4})$ $=$ $-0.4911$.

Secondly, we use the approach in Liao and Xu [16] to derivate the weights of alternative. Then we obtain optimal relative weights such that $\tilde{{w}}_{1} =(0.432, 0.257)$, $\tilde{{w}}_{2} = (0.076,$ $0.678)$, $\tilde{{w}}_{3} = (0.013, 0.913)$, $\tilde{{w}}_{4}= (0.180, 0.508)$. Using (4), we have $S(\tilde{{w}}_{1})= 0.2290$, $S(\tilde{{w}}_{2})=-0.7492$, $S(\tilde{{w}}_{3})=-0.9888$, $S(\tilde{{w}}_{4})$ $=$ $-0.4301$.

Moreover, we employ Gong et al. goal-programming-based model [13], which need to transform the given IFPRs into interval FPR before (for more details refer to [13]). By building and solving a goal programming model, we obtain $w_{1}$ $=[0.242, 0.376]$, $w_{2} = [0.208, 0.234]$, $w_{3} = [0.166, 0.189]$, $w_{4}$ $=[0.206, 0.362]$. Through the equation $w_{ij}^{-} =u_{ij}$, $w_{ij}^{+} =$ $1-v_{ij}$, where $w^{+}$ and $w^{-}$ are upper and lower bounds of a range, these interval-value weights can be transformed into intuitionistic fuzzy weights as $\tilde{{w}}_{1} = (0.242, 0.624)$, $\tilde{{w}}_{2} =(0.208, 0.766)$, $\tilde{{w}}_{3}$ $=$ $(0.166, 0.810)$, $\tilde{{w}}_{4} = (0.206, 0.638)$, and it gives $S(\tilde{{w}}_{1})=-0.4327$, $S(\tilde{{w}}_{2} )=-0.5719$, $S(\tilde{{w}}_{3} )=-0.6599$, $S(\tilde{{w}}_{4} )$ $=$ $-0.50$. We show in Table Ⅲ the ranking order of alternatives obtained by these methods.

Table Ⅲ
COMPARISON ANALYSIS
Ⅴ. CONCLUSIONS

In this paper, we have put forward an efficient approach for decision making where preference information on alternatives is IFPRs and is extended to GDM surrounding. The main idea of this method is first to minimize the absolute deviation between the given IFPRs and the converted consistent IFPRs, then the prioritization of alternatives is obtained based on multiplicative consistent constraint. It is necessary to point out that the proposed method is simple and does not need to solve the fractional programming model as Liao and Xu [16] does. Thus, the standard deviation, an index of overall degree of preferences, was defined and adopted to measure the importance of DMs in a group. By minimizing this index, the weights of DMs were determined. Thus, the model was extended from individual to group application by IFPRs aggregation. Specifically, the risk preference of each DM was considered for rank alternatives which makes this approach more general and flexible. In the future, we will improve our approach and apply it to correlated multi-attribute, and dynamic hybrid multi-attribute GDM problems with IFPRs.

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