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.

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

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

which is characterized by a membership function

*Definition 2 [3]:* An IFPR

$ \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) |

*Definition 3 [4]:* Let

$ 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

1) if

2) if

*B. Multiplicative Consistency of IFPR*

*Definition 4 [16]:* An intuitionistic fuzzy preference relation

$ \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

$ \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

$ \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

*Theorem 1:* Assume that the elements of

*Proof:* It is apparent that

$ (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,

$ \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,

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

*Corollary 1:* Let

$ \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

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

$ \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 *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}^{+} \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

Apparent that when

*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

In order to determine the weights

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

It is noted that

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

$ \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

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

where

Then (18) can be equivalently written as

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

Note that

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

$ \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

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

instead of the normalized weight constraint

$ \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

$ \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

$ 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

$ \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

$ \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

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

$ \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

$ \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

$ \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

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 {

$ \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

$ \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

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:

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

$ \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}=(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

Consider the experts are all risk-neutral, leads to

From the results we have, the factor of maturity (

*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

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.

*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

Secondly, we use the approach in Liao and Xu [16] to derivate the weights of alternative. Then we obtain optimal relative weights such that

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

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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