Generalized interaction aggregation operators in intuitionistic fuzzy multiplicative preference environment and their application to multicriteria decision-making

Abstract

The main objective of this manuscript is to present a new preference relation called the intuitionistic fuzzy multiplicative preference relation. Under this, some series of new aggregation operators, by overcoming the shortcomings of some existing operators, have been defined. As most of the aggregation operators have been constructed under the intuitionistic fuzzy preference relation which deals with the conditions that the attribute values grades are symmetrical and uniformly distributed. In this manuscript, these assumptions have been relaxed by distributing the attribute grades to be asymmetrical around 1 and hence under it, some series of aggregation operators, namely intuitionistic fuzzy multiplicative interactive weighted, ordered weighted and hybrid weighted averaging operators have been proposed. Various desirable properties of these operators have also been discussed in details. A group decision-making method has been presented, based on the proposed operators, for ranking the different alternatives. A real example is taken to demonstrate the applicability and validity of the proposed methodology.

Appendix

Proof Property 4 As α _i , β ∈ IMNs, so

$$\alpha_{i} \oplus \beta = \left<\frac{(1+2\mu_{i})(1+2\mu_{\beta})-1}{2}, \frac{2\left\{1-(1-\mu_{i}\nu_{i})(1-\mu_{\beta}\nu_{\beta}) \right\}}{(1+2\mu_{i})(1+2\mu_{\beta})-1} \right>$$

Therefore,

$$\begin{array}{@{}rcl@{}} \text{IFMIWA}(\alpha_{1}\oplus\beta, \alpha_{2}\oplus \beta, \ldots, \alpha_{n} \oplus \beta) &=& \left<\frac{\prod\limits_{i=1}^{n} \left\{(1+2\mu_{1})(1+2\mu_{\beta})\right\}^{\omega_{i}}-1}{2}, \frac{2\left\{1-\prod\limits_{i=1}^{n} \left\{(1-\mu_{i}\nu_{i})(1-\mu_{\beta}\nu_{\beta})\right\}^{\omega_{i}} \right\}}{\prod\limits_{i=1}^{n} \left\{(1+2\mu_{i})(1+2\mu_{\beta})\right\}^{\omega_{i}}-1} \right> \\ &=& \left< \frac{\prod\limits_{i=1}^{n} (1+2\mu_{i})^{\omega_{i}}(1+2\mu_{\beta})-1}{2}, \frac{2\left\{1-\prod\limits_{i=1}^{n} (1-\mu_{i}\nu_{i})^{\omega_{i}} (1-\mu_{\beta}\nu_{\beta}) \right\}}{\prod\limits_{i=1}^{n} (1+2\mu_{i})^{\omega_{i}}(1+2\mu_{\beta})-1} \right> \\ &=& \left< \frac{\prod\limits_{i=1}^{n} (1+2\mu_{i})^{\omega_{i}}-1}{2}, \frac{2\left\{1-\prod\limits_{i=1}^{n} (1-\mu_{i}\nu_{i})^{\omega_{i}}\right\}}{\prod\limits_{i=1}^{n} (1+2\mu_{i})^{\omega_{i}}-1} \right> \oplus \langle \mu_{\beta}, \nu_{\beta}\rangle \\ &=& \text{IFMIWA}(\alpha_{1},\alpha_{2}\ldots, \alpha_{n}) \oplus \beta \end{array} $$

Hence, IFMIWA(α ₁ ⊕ β, α ₂ ⊕ β,…, α _n ⊕ β) = IFMIWA(α ₁, α ₂…, α _n ) ⊕ β.

Proof of Property 6: Since α _i = 〈μ _i , ν _i 〉∈ IMNs for i = 1, 2,…, n. Therefore, for β > 0, we have

$$\beta \alpha_{i} = \left< \frac{(1+2\mu_{i})^{\beta}-1}{2}, \quad \frac{2\left\{1-(1-\mu_{i}\nu_{i})^{\beta}\right\}}{(1+2\mu_{i})^{\beta}-1} \right>$$

Therefore,

$$\begin{array}{@{}rcl@{}} \text{IFMIWA}(\beta \alpha_{1}, \beta \alpha_{2}, \ldots, \beta \alpha_{n}) &=&\left\langle \frac{\prod\limits_{i=1}^{n} \left( 1+2\frac{(1+2\mu_{i})^{\beta}-1}{2} \right)^{\omega_{i}}-1}{2}, \frac{2\left\{1-\prod\limits_{i=1}^{n} (1-\mu_{i}\nu_{i})^{\beta\omega_{i}}\right\}}{\prod\limits_{i=1}^{n} \left( 1+2\frac{(1+2\mu_{i})^{\beta}-1}{2} \right)^{\omega_{i}}-1} \right\rangle\\ &=& \left< \frac{\prod\limits_{i=1}^{n} \left( (1+2\mu_{i})^{\beta}\right)^{\omega_{i}}-1}{2}, \frac{2\left\{1-\prod\limits_{i=1}^{n} (1-\mu_{i}\nu_{i})^{\beta\omega_{i}}\right\}}{\prod\limits_{i=1}^{n} \left( (1+2\mu_{i})^{\beta}\right)^{\omega_{i}}-1} \right> \\ &=& \left< \frac{\left( \prod\limits_{i=1}^{n} (1+2\mu_{i})^{\omega_{i}}\right)^{\beta}-1}{2}, \frac{2\left\{1-\left( \prod\limits_{i=1}^{n}(1-\mu_{i}\nu_{i})^{\omega_{i}}\right)^{\beta} \right\}}{\left( \prod\limits_{i=1}^{n} (1+2\mu_{i})^{\omega_{i}}\right)^{\beta}-1} \right> \\ &=& \beta \, \left<\frac{\prod\limits_{i=1}^{n} (1+2\mu_{i})^{\omega_{i}}-1}{2}, \frac{2\left\{1-\prod\limits_{i=1}^{n} (1-\mu_{i}\nu_{i})^{\omega_{i}}\right\}}{\prod\limits_{i=1}^{n} (1+2\mu_{i})^{\omega_{i}}-1} \right> \\ &=& \beta \text{IFMIWA}(\alpha_{1},\alpha_{2}\ldots,\alpha_{n}) \end{array} $$

Hence, IFMIWA(βα ₁, βα ₂,…, βα _n ) = β IFMIWA(α ₁, α ₂…, α _n )

Proof of Property 6: As $\alpha _{i}=\langle \mu _{\alpha _{i}}, \nu _{\alpha _{i}}\rangle$ and $\beta =\langle \mu _{\beta _{i}}, \nu _{\beta _{i}}\rangle (i=1,2,\ldots ,n)$ be two collections of IMNs, then

$$\alpha_{i} \oplus \beta_{i} = \left< \frac{(1+2\mu_{\alpha_{i}})(1+2\mu_{\beta_{i}})-1}{2}, \frac{2\left[ 1- (1-\mu_{\alpha_{i}}\nu_{\alpha_{i}})(1-\mu_{\beta_{i}}\nu_{\beta_{i}}) \right]}{(1+2\mu_{\alpha_{i}})(1+2\mu_{\beta_{i}})-1} \right>$$

Therefore,

$$\begin{array}{@{}rcl@{}} && \text{IFMIWA}(\alpha_{1}\oplus \beta_{1}, \alpha_{2}\oplus\beta_{2}, \ldots, \alpha_{n}\oplus \beta_{n}) \\ &=&\left< \frac{\prod\limits_{i=1}^{n} \left( 1+2\frac{(1+2\mu_{\alpha_{i}})(1+2\mu_{\beta_{i}})-1}{2}\right)^{\omega_{i}}-1}{2}, \frac{2\left[1-\prod\limits_{i=1}^{n}\left\{(1-\mu_{\alpha_{i}}\nu_{\alpha_{i}})(1-\mu_{\beta_{i}}\nu_{\beta_{i}})\right\}^{\omega_{i}} \right]}{\prod\limits_{i=1}^{n} \left( 1+2\frac{(1+2\mu_{\alpha_{i}})(1+2\mu_{\beta_{i}})-1}{2}\right)^{\omega_{i}}-1} \right> \\ &=& \left< \frac{\prod\limits_{i=1}^{n} \{(1+2\mu_{\alpha_{i}})(1+2\mu_{\beta_{i}})\}^{\omega_{i}}-1}{2}, \frac{2\left[1-\prod\limits_{i=1}^{n}\left\{(1-\mu_{\alpha_{i}}\nu_{\alpha_{i}})(1-\mu_{\beta_{i}}\nu_{\beta_{i}})\right\}^{\omega_{i}} \right]}{\prod\limits_{i=1}^{n} \{(1+2\mu_{\alpha_{i}})(1+2\mu_{\beta_{i}})\}^{\omega_{i}}-1} \right> \\ &=& \left<\frac{\prod\limits_{i=1}^{n} (1+2\mu_{\alpha_{i}})^{\omega_{i}}\prod\limits_{i=1}^{n} (1+2\mu_{\beta_{i}})^{\omega_{i}}-1}{2}, \frac{2\left[1-\prod\limits_{i=1}^{n} (1-\mu_{\alpha_{i}}\nu_{\alpha_{i}})^{\omega_{i}} \prod\limits_{i=1}^{n} (1-\mu_{\beta_{i}}\nu_{\beta_{i}})^{\omega_{i}} \right]}{\prod\limits_{i=1}^{n} (1+2\mu_{\alpha_{i}})^{\omega_{i}}\prod\limits_{i=1}^{n} (1+2\mu_{\beta_{i}})^{\omega_{i}}-1} \right> \\ &=& \left<\frac{\prod\limits_{i=1}^{n} (1+2\mu_{\alpha_{i}})^{\omega_{i}}-1}{2}, \frac{2\left[1-\prod\limits_{i=1}^{n} (1-\mu_{\alpha_{i}}\nu_{\alpha_{i}})^{\omega_{i}}\right]}{\prod\limits_{i=1}^{n} (1+2\mu_{\alpha_{i}})^{\omega_{i}}-1} \right> \oplus \left< \frac{\prod\limits_{i=1}^{n} (1+2\mu_{\beta_{i}})^{\omega_{i}}-1}{2}, \frac{2\left[1-\prod\limits_{i=1}^{n} (1-\mu_{\beta_{i}}\nu_{\beta_{i}})^{\omega_{i}}\right]}{\prod\limits_{i=1}^{n} (1+2\mu_{\beta_{i}})^{\omega_{i}}-1} \right> \\ &=& \text{IFMIWA}(\alpha_{1},\alpha_{2}\ldots,\alpha_{n}) \oplus \text{IFMIWA}(\beta_{1},\beta_{2}\ldots,\beta_{n}) \end{array} $$

Hence, IFMIWA(α ₁ ⊕ β ₁,…, α _n ⊕ β _n ) = IFMIWA(α ₁,…, α _n ) ⊕IFMIWA(β ₁,…, β _n )