Robust Averaging–Geometric Aggregation Operators for Complex Intuitionistic Fuzzy Sets and Their Applications to MCDM Process

Abstract

This paper presents aggregation operators which encapsulate the interaction among the criteria and preferences under complex intuitionistic fuzzy (CIF) conditions. The current extensions of fuzzy set theory handle only the uncertain data by representing the satisfaction and dissatisfaction degrees as real values and hence lose some information. A modification to these, CIF sets are portrayed by complex-valued membership degrees and can handle the data concurrently using additional terms, called phase terms, which usually give knowledge related to periodicity. Motivated by the features of the CIF model, this paper studies some aggregation operators, weighted averaging and geometric, for CIF sets and investigates their properties. To ease with the possible application, we explore the decision-making (DM) process in the CIF set environment and present an algorithm to solve the multiple criteria DM problems. Finally, a practical example is presented to demonstrate the DM process based on the proposed operators and compared their performance with some similar approaches.

Appendix

Proof of Theorem 1

Proof

The first part of the result follows from Definition 7. To validate Eq. (7), we use the principle of mathematical induction (PMI) on n.

Step 1:

For $n=2$, we have $\mathcal {A}_1=\Big (\big (\zeta _1,w_{\zeta _1}\big ),\big (\vartheta _1,w_{\vartheta _1}\big )\Big )$ and $\mathcal {A}_2=\Big (\big (\zeta _2,w_{\zeta _2}\big ),\big (\vartheta _2,w_{\vartheta _2}\big )\Big )$. Thus, by the operation of CIFNs, we get

$$\begin{aligned}&\xi _1\mathcal {A}_1\\&\quad =\left( \left( 1-\big (1-\zeta _1 \big )^{\xi _1} , 1-\left( 1-w_{\zeta _1}\right) ^{\xi _1} \right) , \right. \\&\qquad \left. \left( \big (\vartheta _1\big )^{\xi _1}, \big (w_{\vartheta _1}\big )^{\xi _1} \right) \right) \\&\xi _2\mathcal {A}_2\\&\quad =\left( \left( 1-\big (1-\zeta _2 \big )^{\xi _2} , 1-\left( 1-w_{\zeta _2}\right) ^{\xi _2} \right) , \right. \\&\qquad \left. \left( \big (\vartheta _2\big )^{\xi _2}, \big (w_{\vartheta _2}\big )^{\xi _2} \right) \right) \end{aligned}$$

and

$$\begin{aligned}&\text {CIFWA}(\mathcal {A}_1, \mathcal {A}_2) \\&\quad = \xi _1 \mathcal {A}_1 \oplus \xi _2 \mathcal {A}_2 \\&\quad =\left( \left( 1-\big (1-\zeta _1 \big )^{\xi _1} , 1-\left( 1-w_{\zeta _1} \right) ^{\xi _1} \right) ,\right. \\&\qquad \left. \left( \big (\vartheta _1\big )^{\xi _1}, \big (w_{\vartheta _1}\big )^{\xi _1} \right) \right) \\&\qquad \oplus \left( \left( 1-\big (1-\zeta _2 \big )^{\xi _2} , 1-\left( 1-w_{\zeta _2}\right) ^{\xi _2} \right) , \right. \\&\qquad \left. \left( \big (\vartheta _2\big )^{\xi _2}, \big (w_{\vartheta _2}\big )^{\xi _2} \right) \right) \\&\quad = \left( \left( 1-\prod \limits _{j=1}^2 \big (1-\zeta _j \big )^{\xi _j} , 1-\prod \limits _{j=1}^2 \left( 1-w_{\zeta _j}\right) ^{\xi _j} \right) , \right. \\&\qquad \left. \left( \prod \limits _{j=1}^2 \big (\vartheta _j\big )^{\xi _j}, \prod \limits _{j=1}^2 \big (w_{\vartheta _j}\big )^{\xi _j} \right) \right) \end{aligned}$$

Thus, result is true.

Step 2:

Suppose that result holds for $n=m>2$, then

$$\begin{aligned}&\text {CIFWA}(\mathcal {A}_1, \mathcal {A}_2, \ldots ,\mathcal {A}_m) \\&\quad =\left( \left( 1-\prod \limits _{j=1}^m \big (1-\zeta _j \big )^{\xi _j}, 1-\prod \limits _{j=1}^m \big (1-w_{\zeta _j} \big )^{\xi _j} \right) , \right. \\&\qquad \left. \left( \prod \limits _{j=1}^m \big (\vartheta _j\big )^{\xi _j}, \prod \limits _{j=1}^m \big (w_{\vartheta _j}\big )^{\xi _j} \right) \right) . \end{aligned}$$

Further, for $n=m+1$, we have

$$\begin{aligned}&\text {CIFWA}(\mathcal {A}_1, \mathcal {A}_2, \ldots ,\mathcal {A}_{m+1}) \\&\quad = \text {CIFWA}(\mathcal {A}_1, \mathcal {A}_2, \ldots ,\mathcal {A}_m) \oplus \xi _{m+1}\mathcal {A}_{m+1} \\&\quad = \left( \left( 1-\prod \limits _{j=1}^m \big (1-\zeta _j \big )^{\xi _j}, 1-\prod \limits _{j=1}^m \big (1-w_{\zeta _j} \big )^{\xi _j}\right) ,\right. \\&\qquad \left. \left( \prod \limits _{j=1}^m \big (\vartheta _j\big )^{\xi _j}, \prod \limits _{j=1}^m \big (w_{\vartheta _j}\big )^{\xi _j} \right) \right) \\&\qquad \oplus \left( \left( 1-\big (1-\zeta _{m+1} \big )^{\xi _{m+1}}, 1-\big (1-w_{\zeta _{m+1}} \big )^{\xi _{m+1}}\right) , \right. \\&\qquad \left. \left( \big (\vartheta _{m+1}\big )^{\xi _{m+1}}, \big (w_{\vartheta _{m+1}}\big )^{\xi _{m+1}}\right) \right) \\&\quad = \left( \left( 1-\prod \limits _{j=1}^{m+1} \big (1-\zeta _j \big )^{\xi _j}, 1-\prod \limits _{j=1}^{m+1} \big (1-w_{\zeta _j} \big )^{\xi _j} \right) , \right. \\&\qquad \left. \left( \prod \limits _{j=1}^{m+1} \big (\vartheta _j\big )^{\xi _j}, \prod \limits _{j=1}^{m+1} \big (w_{\vartheta _j}\big )^{\xi _j}\right) \right) \end{aligned}$$

which is true for $n=m+1$. Therefore, by the PMI, result holds $\forall n\in \mathbb {Z}^+$.

$\square$