A Group Decision Making Approach Based on Newly Defined Additively Consistent Interval-Valued Intuitionistic Preference Relations

Abstract

Because of the complexity of interval-valued intuitionistic fuzzy set, there are few literatures investigating the additive consistency of the interval-valued intuitionistic preference relation. This paper proposes a group decision making approach with interval-valued intuitionistic preference relations. Firstly, two additive reciprocal fuzzy preference relations and two fuzzy non-preferred relations are obtained by dividing an interval-valued intuitionistic preference relation in order to avoid the operations of interval-valued intuitionistic fuzzy numbers. Here, the fuzzy non-preferred relation is interpreted as the non-preferred intensity of one alternative over another one. Moreover, the rational concept of additive consistency of interval-valued intuitionistic preference relation is defined. Secondly, optimizing models to derive priority weight vector from interval-valued intuitionistic preference relation are constructed. In particular, the priority weight vector is in terms of interval-valued intuitionistic fuzzy numbers, which is rational under the interval-valued intuitionistic fuzzy environment. Thirdly, an extended approach to address group decision making (GDM) problems is proposed. Each individual expert’s consistency index is applied to measure his/her degree of importance. Finally, an illustrative example of supplier selection is provided to illustrate the proposed approach addressing GDM problems. The feasibility to derive priority weight vector in terms of interval-valued intuitionistic fuzzy numbers using the developed approach is verified. And the proposed approach is compared with the other conventional existing ones to show its advantages.

Appendix

The Procedure of Constructing Model (34)

We can also introduce the following deviation variables to gauge the difference similarly to Eqs. (17) and (18).

$$\begin{aligned} {{\varepsilon _{ij,D}}}= & {} {d_{ij}} - 0.5\left( {{{\omega _{i,D}}} - {{\omega _{j,D}}}} \right) - 0.5,\ i,j = 1,2, \ldots ,n,\;i \ne j \end{aligned}$$

(48)

$$\begin{aligned} {{\varepsilon _{ij,E}}}= & {} {e_{ij}} - 0.5\left( {{{\omega _{i,E}}} - {{\omega _{j,E}}}} \right) - 0.5,\ i,j = 1,2, \ldots ,n,\;i \ne j \end{aligned}$$

(49)

We can construct the following optimization model to obtain the optimal interval weight vector in the same way as shown in Eq. (19).

$$\begin{aligned} \begin{array}{l} \min \;{J_2} = \sum \limits _{i = 1}^n {\sum \limits _{j = 1}^n {\left( {\left| {{{\varepsilon _{ij,D}}}} \right| + \left| {{{\varepsilon _{ij,E}}}} \right| } \right) } } \\ {\mathrm{s.t.}}\;\left\{ \begin{array}{l} {{\varepsilon _{ij,D}}} - {d_{ij}} + 0.5\left( {{{\omega _{i,D}}} - {{\omega _{j,D}}}} \right) + 0.5 = 0, \quad i,j = 1,2, \ldots ,n,\;i \ne j\\ {{\varepsilon _{ij,E}}} - {e_{ij}} + 0.5\left( {{{\omega _{i,E}}} - {{\omega _{j,E}}}} \right) + 0.5 = 0, \quad i,j = 1,2, \ldots ,n,\;i \ne j\\ 0 \le {{\omega _{i,D}}} \le 1,\;0 \le {{\omega _{i,E}}} \le 1,\quad i = 1,2, \ldots ,n\\ \sum \limits _{i = 1}^n {{\omega _{i,R}^L} } + \mathop {\max }\limits _j \left( {{\omega _{i,R}^U} - {\omega _{i,R}^L} } \right) \le 1, \quad j = 1,2, \ldots ,n\\ \sum \limits _{i = 1}^n {{\omega _{i,R}^U} } - \mathop {\max }\limits _j \left( {{\omega _{i,R}^U} - {\omega _{i,R}^L} } \right) \ge 1, \quad j = 1,2, \ldots ,n \end{array} \right. \end{array} \end{aligned}$$

(50)

where, ${\omega _{i,R}^L} = \min \left\{ {{{\omega _{i,D}}},\;{{\omega _{i,E}}}\;} \right\}$ and ${\omega _{i,R}^U} = \max \left\{ {{{\omega _{i,D}}},\;\;{{\omega _{i,E}}}\;} \right\}$ .

We introduce the following variables by the same method as employed above.

$$\begin{aligned} { \varepsilon _{ij,D}^ + } =\frac{{\left| {{{\varepsilon _{ij,D}}}} \right| + {{\varepsilon _{ij,D}}}}}{2}\ \ \text {and}\ \ {\varepsilon _{ij,D}^ - } = \dfrac{{\left| {{{\varepsilon _{ij,D}}}} \right| - {{\varepsilon _{ij,D}}}}}{2},\ i = 1,2, \ldots ,n,\;j = i + 1, \ldots ,n \end{aligned}$$

(51)

$$\begin{aligned} { \varepsilon _{ij,E}^ + } =\frac{{\left| {{{\varepsilon _{ij,E}}}} \right| + {{\varepsilon _{ij,E}}}}}{2},\ \ \text {and}\ \ {\varepsilon _{ij,E}^ - } = \frac{{\left| {{{\varepsilon _{ij,E}}}} \right| - {{\varepsilon _{ij,E}}}}}{2},\ \ i = 1,2, \ldots ,n,\;j = i + 1, \ldots ,n \end{aligned}$$

(52)

Then, we have

$$\begin{aligned} {{\varepsilon _{ij,D}}} = { \varepsilon _{ij,D}^ + } - {\varepsilon _{ij,D}^ - },\ \left| {{{\varepsilon _{ij,D}}}} \right| = { \varepsilon _{ij,D}^ + } + {\varepsilon _{ij,D}^ - },\ i = 1,2, \ldots ,n,\;j = i + 1, \ldots ,n \end{aligned}$$

(53)

where ${ \varepsilon _{ij,D}^ + } \cdot {\varepsilon _{ij,D}^ - } = 0$ for $i = 1,2, \ldots ,n,\;j = i + 1, \ldots ,n$ .

Similarly, ${ \varepsilon _{ij,E}^ + }$ and $\left| {{{\varepsilon _{ij,E}}}} \right|$ can be expressed as

$$\begin{aligned} {{\varepsilon _{ij,E}}} = { \varepsilon _{ij,E}^ + } - {\varepsilon _{ij,E}^ - },\ \left| {{{\varepsilon _{ij,E}}}} \right| = { \varepsilon _{ij,E}^ + } + {\varepsilon _{ij,E}^ - },\ i = 1,2, \ldots ,n,\;j = i + 1, \ldots ,n \end{aligned}$$

(54)

where ${ \varepsilon _{ij,E}^ + } \cdot {\varepsilon _{ij,E}^ - } = 0$ for $i = 1,2, \ldots ,n,\;j = i + 1, \ldots ,n$ .

The same to Eqs. (23) and (23), we have

$$\begin{aligned} {\omega _{i,R}^L}= & {} \min \left\{ {{{\omega _{i,D}}},\;{{\omega _{i,E}}}\;} \right\} = \frac{{\left( {{{\omega _{i,D}}} + {{\omega _{i,E}}}} \right) - \left| {{{\omega _{i,D}}} - {{\omega _{i,E}}}} \right| }}{2} \end{aligned}$$

(55)

$$\begin{aligned} {\omega _{i,R}^U}= & {} \max \left\{ {{{\omega _{i,D}}},\;\;{{\omega _{i,E}}}\;} \right\} = \frac{{\left( {{{\omega _{i,D}}} + {{\omega _{i,E}}}} \right) + \left| {{{\omega _{i,D}}} - {{\omega _{i,E}}}} \right| }}{2} \end{aligned}$$

(56)

Consequently, the model (50) can be transformed into model (34).