Decision making with the generalized intuitionistic fuzzy power interaction averaging operators

Abstract

Considering the interactions between different intuitionistic fuzzy sets, this paper extends the power averaging operators to intuitionistic fuzzy environments and develops some intuitionistic fuzzy power interaction aggregation operators, including the generalized intuitionistic fuzzy power interaction averaging operator, the weighted generalized intuitionistic fuzzy power interaction averaging operator and the generalized intuitionistic fuzzy power ordered weighted interaction averaging operator. The properties of these aggregation operators are investigated. The key advantages of these operators are that they not only accommodate situations in which the input arguments are intuitionistic fuzzy numbers (IFNs) and take the interactions of different IFNs into consideration, but also consider the decision situations that the relationships between the IFNs are fused. Moreover, we apply the new proposed aggregation operators to multiple attributes decision making and examples are illustrated to show the validity and feasibility of the new approaches.

Appendices

Appendix A

Proof

With the mathematical induction on n, we have

• (1) When $n=1$, ${w_i ( {1+T( {A_i })})} /{\sum \nolimits _{j=1}^1 {w_j ( {1+T(A_j )})} }=1$,

  $$\begin{aligned}&\mathrm{WIFPIA}(A_1 )=A_1 =\langle {u_{A_1 } ,v_{A_1 } } \rangle \nonumber \\&\quad =\langle {1-( {1-u_{A_1 } })^1,1-u_{A_1 } ^1-( {1-(u_{A_1 } +v_{A_1 } )})^1} \rangle . \end{aligned}$$

  Thus, Eq. (15) holds.

• (2) If Eq. (15) is established for $n=k$, i.e.,

  $$\begin{aligned}&\mathrm{WIFPIA}(A_1 ,\ldots ,A_k )\nonumber \\&\quad =\left\langle {1-\prod \limits _{i=1}^k {( {1-u_{A_i } })^{{w_i ( {1+T( {A_i })})} /{\sum \nolimits _{j=1}^k {w_j ( {1+T(A_j )})} }}} ,} \right. \\&\quad \quad \left. \prod \limits _{i=1}^k {( {1-u_{A_i } })^{{w_i ( {1+T( {A_i })})} / {\sum \nolimits _{j=1}^k {w_j ( {1+T(A_j )})} }}}\right. \nonumber \\&\quad \quad \left. -\prod \limits _{i=1}^k {( {1-(u_{A_i } +v_{A_i } )})^{{w_i ( {1+T( {A_i })})}/{\sum \nolimits _{j=1}^k {w_j ( {1+T(A_j )})} }}} \right\rangle \end{aligned}$$

  Then for $n=k+$1, by the improved addition operation, i.e., Eq. (5), we get

  $$\begin{aligned}&\mathrm{WIFPIA}(A_1 ,\ldots A_{k+1} )=\mathop {\hat{\oplus }}\limits _{i=1}^{k+1} w_i A_i=\mathrm{WIFPIA}(A_1 ,\ldots A_k )\\&\quad \hat{\oplus }\left( {w_{k+1} ( {1+T( {A_{k+1} })})A_{k+1} }/ {\sum \limits _{j=1}^{k+1} {w_j ( {1+T(A_j )})} }\right) \\&\quad =\left\langle 1-\prod \limits _{i=1}^k {( {1-u_{A_i } })^{{w_i ( {1+T( {A_i })})} /{\sum \nolimits _{j=1}^k {w_j ( {1+T(A_j )})} }}},\right. \\&\quad \quad \left. \prod \limits _{i=1}^k {( {1-u_{A_i } })^{{w_i ( {1+T( {A_i })})} /{\sum \nolimits _{j=1}^k {w_j ( {1+T(A_j )})} }}}\right. \\&\quad \quad \left. -\prod \limits _{i=1}^k {( {1-(u_{A_i } +v_{A_i } )})^{{w_i ( {1+T( {A_i })})} /{\sum \nolimits _{j=1}^k {w_j ( {1+T(A_j )})} }}} \right\rangle \\&\quad \quad \hat{\oplus }\left\langle 1-( {1-u_{A_{k+1} } })^{{w_{k+1} ( {1+T( {A_{k+1} })})} / {\sum \nolimits _{j=1}^{k+1} {w_j ( {1+T(A_j )})} }},\right. \\&\quad \quad \left. ( {1-u_{A_{k+1} } })^{{w_{k+1} ( {1+T( {A_{k+1} })})} / {\sum \nolimits _{j=1}^{k+1} {w_j ( {1+T(A_j )})} }}\right. \\&\quad \quad \left. -( {1-(u_{A_{k+1} } +v_{A_{k+1} } )})^{{w_{k+1} ( {1+T( {A_{k+1} })})} / {\sum \nolimits _{j=1}^{k+1} {w_j ( {1+T(A_j )})} }} \right\rangle \\&\quad =\left\langle 1-\prod \limits _{i=1}^{k+1} {( {1-u_{A_i } })^{{w_i ( {1+T( {A_i })})} /{\sum \nolimits _{j=1}^{k+1} {w_j ( {1+T(A_j )})} }}},\right. \\&\quad \quad \left. \prod \limits _{i=1}^{k+1} {( {1-u_{A_i } })^{{w_i ( {1+T( {A_i })})} /{\sum \nolimits _{j=1}^{k+1} {w_j ( {1+T(A_j )})} }}}\right. \\&\quad \quad \left. -\prod \limits _{i=1}^{k+1} {( {1-(u_{A_i } +v_{A_i } )})^{{w_i ( {1+T( {A_i })})} /{\sum \nolimits _{j=1}^{k+1} {w_j ( {1+T(A_j )})} }}} \right\rangle . \end{aligned}$$

  Thus, Eq. (15) holds for $n=k+$1. Therefore, by using mathematic induction on n, Eq. (15) holds for n. Besides, according to Definition 1, we have $\mathrm{WIFPIA}(A_1,\ldots ,A_n )\in \mathrm{IFNs}(X)$. Therefore, Theorem 1 holds. $\square$

Appendix B

Proof

By using mathematical induction on n, we prove Eq. (20) as follows:

• (1) When $n=1$, ${( {1+T( {A_1 })})} / {( {\sum \nolimits _{j=1}^1 {1+T( {A_j })} })}=1$, we have

  $$\begin{aligned}&{\mathop {\hat{\oplus }}\limits _{i=1}^1 ( {1+T( {A_i })})A_i ^\lambda } /{\sum \limits _{j=1}^1 {( {1+T( {A_j })})} }\\&\quad =A_1^\lambda =\langle ( {1-v_{A_1 } })^\lambda \\&\qquad -( {1-( {u_{A_1 } +v_{A_1 } })})^\lambda ,1-( {1-v_{A_1 } })^\lambda \rangle \\&\quad =\langle 1-( {1-( {1-v_{A_1 } })^\lambda +( {1-( {u_{A_1 } +v_{A_1 } })})^\lambda })^1,\\&\quad \quad ( {1-( {1-v_{A_1 } })^\lambda +( {1-( {u_{A_1 } +v_{A_1 } })})^\lambda })^1\\&\qquad -( {1-( {u_{A_1 } +v_{A_1 } })})^{1\cdot \lambda } \rangle \\&\quad =\langle 1-( 1-( {1-v_{A_1 } })^\lambda \\&\qquad +( {1-( {u_{A_1 } +v_{A_1 } })})^\lambda )^{{( {1+T( {A_1 })})} / {( {\sum \nolimits _{j=1}^1 {1+T( {A_j })} })}},\\&\quad \quad ( 1-( {1-v_{A_1 } })^\lambda \\&\qquad +( {1-( {u_{A_1 } +v_{A_1 } })})^\lambda )^{{( {1+T( {A_1 })})} / {( {\sum \nolimits _{j=1}^1 {1+T( {A_j })} })}}\\&\quad \quad -( {1-( {u_{A_1 } +v_{A_1 } })})^{\lambda \cdot {\left( {1+T( {A_1 })}\right) } /{( {\sum \nolimits _{j=1}^1 {1+T( {A_j })} })}} \rangle \end{aligned}$$

  Thus, Eq. (20) is established for $n=1$.

• (2) If Eq. (20) holds for $n=k$. Then, $n=k+1$, by inductive assumption and Eq. (5), we get

  $$\begin{aligned}&\frac{\mathop {\hat{\oplus }}\nolimits _{i=1}^{k+1} ( {1+T( {A_i })})A_i ^\lambda }{\sum \nolimits _{j=1}^{k+1} {( {1+T( {A_j })})} }\\&\quad =\left( {\frac{\mathop {\hat{\oplus }}\nolimits _{i=1}^k ( {1+T( {A_i })})A_i ^\lambda }{\sum \nolimits _{j=1}^{k+1} {( {1+T( {A_j })})} }}\right) \hat{\oplus }\frac{( {1+T( {A_{k+1} })})A_{k+1} ^\lambda }{\sum \nolimits _{j=1}^{k+1} {( {1+T( {A_j })})} }\\&\quad =\left\langle 1-\prod \limits _{i=1}^k ( 1-( {1-v_{A_i } })^\lambda \right. \\&\qquad +\,( {1-( {u_{A_i } +v_{A_i } })})^\lambda )^{{( {1+T( {A_i })})} /{\sum \nolimits _{j=1}^{k+1} {1+T( {A_j })} }},\\&\quad \quad \prod \limits _{i=1}^k ( 1-( {1-v_{A_i } })^\lambda \\&\qquad +\,( 1-( {u_{A_i } +v_{A_i } }))^\lambda )^{{( {1+T( {A_i })})} /{\sum \nolimits _{j=1}^{k+1} {1+T( {A_j })} }}\\&\quad \quad \left. -\,\prod \limits _{i=1}^k ( {1-( {u_{A_i } +v_{A_i } })})^{\lambda {( {1+T( {A_i })})} /{\sum \nolimits _{j=1}^{k+1} {1+T( {A_j })} }} \right\rangle \\&\quad \quad \hat{\oplus }\left\langle 1-( 1-( {1-v_{A_{k+1} } })^\lambda \right. \\&\qquad +\,( {1{-}( {u_{A_{k+1} } +v_{A_{k+1} } })})^\lambda )^{{( {1+T( {A_{k+1} })})} /{\sum \nolimits _{j=1}^{k+1} {1+T( {A_j })} }},\\&\quad \quad ( 1-( {1-v_{A_{k+1} } })^\lambda \\&\qquad +\,( {1-( {u_{A_{k+1} } +v_{A_{k+1} } })})^\lambda )^{{( {1+T( {A_{k+1} })})} /{\sum \nolimits _{j=1}^{k+1} {1+T( {A_j })} }}\\&\quad \quad \left. -\,( {1{-}( {u_{A_{k+1} } +v_{A_{k+1} } })})^{\lambda \cdot {( {1+T( {A_{k+1} })})} /{\sum \nolimits _{j=1}^{k+1} {1+T( {A_j })} }} \right\rangle \\&\quad =\left\langle 1-\prod \limits _{i=1}^{k+1} ( 1-( {1-v_{A_i } })^\lambda \right. \\&\qquad +\,( {1-( {u_{A_i } +v_{A_i } })})^\lambda )^{{( {1+T( {A_i })})}/{\sum \nolimits _{j=1}^{k+1} {1+T( {A_j })} }} ,\\&\quad \quad \prod \limits _{i=1}^{k+1} ( 1-( {1-v_{A_i } })^\lambda \\&\qquad +\,( {1-( {u_{A_i } +v_{A_i } })})^\lambda )^{{( {1+T( {A_i })})} /{\sum \nolimits _{j=1}^{k+1} {1+T( {A_j })} }}\\&\quad \quad \left. -\,\prod \limits _{i=1}^{k+1} {( {1-( {u_{A_i } +v_{A_i } })})^{\lambda {( {1+T( {A_i })})} /{\sum \nolimits _{j=1}^{k+1} {1+T( {A_j })} }}} \right\rangle \end{aligned}$$

  i.e. Eq. (20) is established for $n=k+$1.

Thus, Eq. (20) holds for all n with the mathematical induction on n. $\square$