Some hybrid weighted aggregation operators under neutrosophic set environment and their applications to multicriteria decision-making

Abstract

A single-valued and an interval neutrosophic sets are two instances of the neutrosophic set(NS), which can efficiently deal with uncertain, imprecise and inconsistent information. In the present work, the work has been done under this environment to develop some novel hybrid aggregation operators based on arithmetic and geometric aggregation operators. The preferences related to the attributes are made in the form of single-valued and interval neutrosophic numbers. Their desirable properties such as idempotency, boundedness and monotonicity are also investigated. Furthermore, a decision-making approach presents to investigate the multi-criteria decision-making problem. The effectiveness of the approach is demonstrated through a case study and a comparison analysis with some other existing methods has been done to validate the results.

Appendix

Proof of the Theorem 1:

Proof

By using the operational laws for SVNNs β_j(j = 1, 2,…,n) and a real number λ ∈ [0, 1], we have

$$\begin{array}{@{}rcl@{}} \bigoplus\limits_{j = 1}^{n}\omega_{j}\beta_{j} &=& \left( 1-\prod\limits_{j = 1}^{n}(1-\zeta_{j})^{\omega_{j}}, \prod\limits_{j = 1}^{n}(\kappa_{j})^{\omega_{j}}, \prod\limits_{j = 1}^{n}(\varphi_{j})^{\omega_{j}} \right) \\ \text{and} \qquad \prod\limits_{j = 1}^{n}\beta_{j}^{\omega_{j}} &=& \left( \prod\limits_{j = 1}^{n}(\zeta_{j})^{\omega_{j}}, 1-\prod\limits_{j = 1}^{n}(1-\kappa_{j})^{\omega_{j}}, 1-\prod\limits_{j = 1}^{n}(1-\varphi_{j})^{\omega_{j}}\right) \end{array} $$

which implies that

$$\begin{array}{@{}rcl@{}} \left( \bigoplus\limits_{j = 1}^{n}\omega_{j}\beta_{j}\right)^{\lambda} &=&\left( \begin{array}{lllll} & \left( 1-\prod\limits_{j = 1}^{n}(1-\zeta_{j})^{\omega_{j}}\right)^{\lambda}, 1-\left( 1-\prod\limits_{j = 1}^{n}(\kappa_{j})^{\omega_{j}}\right)^{\lambda}, \\ & \qquad \qquad \qquad 1-\left( 1- \prod\limits_{j = 1}^{n}(\varphi_{j})^{\omega_{j}}\right)^{\lambda} \end{array} \right) \end{array} $$

(18)

and

$$\begin{array}{@{}rcl@{}} \left( \prod\limits_{j = 1}^{n}\beta_{j}^{\omega_{j}}\right)^{1-\lambda}&=& \left( \begin{array}{lllll} & \left( \prod\limits_{j = 1}^{n}(\zeta_{j})^{\omega_{j}}\right)^{1-\lambda}, 1-\left( \prod\limits_{j = 1}^{n}(1-\kappa_{j})^{\omega_{j}}\right)^{1-\lambda}, \\ & \qquad \qquad \qquad 1-\left( \prod\limits_{j = 1}^{n}(1-\varphi_{j})^{\omega_{j}}\right)^{1-\lambda} \end{array} \right) \end{array} $$

(19)

Thus, by Definition 5, we have

$$\begin{array}{@{}rcl@{}} &&\text{H-SVNWAG}(\beta_{1},\beta_{2}, \ldots, \beta_{n}) \\ &=& \left( \begin{array}{lllll} & \left( 1-\prod\limits_{j = 1}^{n}(1-\zeta_{j})^{\omega_{j}}\right)^{\lambda}, \\ & 1-\left( 1-\prod\limits_{j = 1}^{n}(\kappa_{j})^{\omega_{j}}\right)^{\lambda},\\ & 1-\left( 1- \prod\limits_{j = 1}^{n}(\varphi_{j})^{\omega_{j}}\right)^{\lambda} \end{array} \right) \bigotimes \left( \begin{array}{lllll} & \left( \prod\limits_{j = 1}^{n}(\zeta_{j})^{\omega_{j}}\right)^{1-\lambda}, \\ & 1-\left( \prod\limits_{j = 1}^{n}(1-\kappa_{j})^{\omega_{j}}\right)^{1-\lambda}, \\ & 1-\left( \prod\limits_{j = 1}^{n}(1-\varphi_{j})^{\omega_{j}}\right)^{1-\lambda} \end{array} \right) \\ \end{array} $$

$$\begin{array}{@{}rcl@{}} &=& \left( \begin{array}{lllll} & \left( 1-\prod\limits_{j = 1}^{n}(1-\zeta_{j})^{\omega_{j}}\right)^{\lambda} \left( \prod\limits_{j = 1}^{n}(\zeta_{j})^{\omega_{j}}\right)^{1-\lambda}, \left\{1-\left( 1-\prod\limits_{j = 1}^{n}(\kappa_{j})^{\omega_{j}}\right)^{\lambda} \right\} + \left\{1-\left( \prod\limits_{j = 1}^{n}(1-\kappa_{j})^{\omega_{j}}\right)^{1-\lambda} \right\} \\ & -\left\{1-\left( 1-\prod\limits_{j = 1}^{n}(\kappa_{j})^{\omega_{j}}\right)^{\lambda}\right\} \left\{1-\left( \prod\limits_{j = 1}^{n}(1-\kappa_{j})^{\omega_{j}}\right)^{1-\lambda}\right\}, \left\{1-\left( 1-\prod\limits_{j = 1}^{n}(\varphi_{j})^{\omega_{j}}\right)^{\lambda} \right\} \\ & + \left\{1-\left( \prod\limits_{j = 1}^{n}(1-\varphi_{j})^{\omega_{j}}\right)^{1-\lambda} \right\} -\left\{1-\left( 1-\prod\limits_{j = 1}^{n}(\varphi_{j})^{\omega_{j}}\right)^{\lambda}\right\} \left\{1-\left( \prod\limits_{j = 1}^{n}(1-\varphi_{j})^{\omega_{j}}\right)^{1-\lambda}\right\} \end{array} \right)\\ &=& \left( \begin{array}{lllll} & \left( 1-\prod\limits_{j = 1}^{n}(1-\zeta_{j})^{\omega_{j}}\right)^{\lambda}\left( \prod\limits_{j = 1}^{n}{\zeta_{j}}^{\omega_{j}}\right)^{1-\lambda}, 1-\left( 1-\prod\limits_{j = 1}^{n}{\kappa_{j}}^{\omega_{j}}\right)^{\lambda} \left( \prod\limits_{j = 1}^{n}(1-\kappa_{j})^{\omega_{j}}\right)^{1-\lambda} \\ & 1-\left( 1-\prod\limits_{j = 1}^{n}{\varphi_{j}}^{\omega_{j}}\right)^{\lambda} \left( \prod\limits_{j = 1}^{n}(1-\varphi_{j})^{\omega_{j}}\right)^{1-\lambda} \end{array} \right) \end{array} $$

Hence, the result.

□

Proof of the Theorem 2:

Proof

For SVNNs β_j = (ζ_j, κ_j, φ_j)(j = 1, 2,…,n) and ω = (ω₁, ω₂,…,ω_n)^T be their normalized weight vector, then we have $\prod \limits _{j = 1}^{n}\zeta _{j}^{\omega _{j}} \leq 1-\prod \limits _{j = 1}^{n}(1-\zeta _{j})^{\omega _{j}}$ and $0 \leq \prod \limits _{j = 1}^{n}\zeta _{j}^{\omega _{j}} , 1-\prod \limits _{j = 1}^{n}(1-\zeta _{j})^{\omega _{j}} \leq 1$. Therefore, for a real number λ ∈ [0, 1] we have

$$\left( \prod\limits_{j = 1}^{n}\zeta_{j}^{\omega_{j}}\right)^{1-\lambda} \leq \left( 1-\prod\limits_{j = 1}^{n}(1-\zeta_{j})^{\omega_{j}}\right)^{1-\lambda} $$

which implies that

$$ \left( 1-\prod\limits_{j = 1}^{n}(1-\zeta_{j})^{\omega_{j}}\right)^{\lambda}\left( \prod\limits_{j = 1}^{n}\zeta_{j}^{\omega_{j}}\right)^{1-\lambda} \leq 1-\prod\limits_{j = 1}^{n}(1-\zeta_{j})^{\omega_{j}} $$

(20)

Similarly, for $\prod \limits _{j = 1}^{n}\kappa _{j}^{\omega _{j}} \geq 1-\prod \limits _{j = 1}^{n}(1-\kappa _{j})^{\omega _{j}}$ and $\prod \limits _{j = 1}^{n}\varphi _{j}^{\omega _{j}} \geq 1-\prod \limits _{j = 1}^{n}(1-\varphi _{j})^{\omega _{j}}$, we have

$$ \prod\limits_{j = 1}^{n}\kappa_{j}^{\omega_{j}} \leq 1-\left( 1-\prod\limits_{j = 1}^{n}\kappa_{j}^{\omega_{j}}\right)^{\lambda}\left( \prod\limits_{j = 1}^{n}(1-\kappa_{j})^{\omega_{j}}\right)^{1-\lambda} $$

(21)

and

$$ \prod\limits_{j = 1}^{n}\varphi_{j}^{\omega_{j}} \leq 1-\left( 1-\prod\limits_{j = 1}^{n}\varphi_{j}^{\omega_{j}}\right)^{\lambda}\left( \prod\limits_{j = 1}^{n}(1-\varphi_{j})^{\omega_{j}}\right)^{1-\lambda} $$

(22)

Thus, by using (20), (21) and (22) and the definition of score function, we get

$$\begin{array}{@{}rcl@{}} && S(\text{SVNWA}(\beta_{1},\beta_{2},\ldots, \beta_{n})) \\ &=& 1-\prod\limits_{j = 1}^{n}(1-\zeta_{j})^{\omega_{j}} - \prod\limits_{j = 1}^{n}\kappa_{j}^{\omega_{j}} - \prod\limits_{j = 1}^{n}\varphi_{j}^{\omega_{j}}\\ &\geq & \left( 1-\prod\limits_{j = 1}^{n}(1-\zeta_{j})^{\omega_{j}}\right)^{\lambda}\left( \prod\limits_{j = 1}^{n}\zeta_{j}^{\omega_{j}}\right)^{1-\lambda} \\&&- 1+\left( 1-\prod\limits_{j = 1}^{n}\kappa_{j}^{\omega_{j}}\right)^{\lambda}\left( \prod\limits_{j = 1}^{n}(1-\kappa_{j})^{\omega_{j}}\right)^{1-\lambda} \\ && - 1+\left( 1-\prod\limits_{j = 1}^{n}\varphi_{j}^{\omega_{j}}\right)^{\lambda}\left( \prod\limits_{j = 1}^{n}(1-\varphi_{j})^{\omega_{j}}\right)^{1-\lambda}\\ &=& S(\text{H-SVNWAG}(\beta_{1},\beta_{2},\ldots, \beta_{n})) \end{array} $$

Hence, H-SVNWAG(β₁, β₂,…,β_n) ≤SVNWA(β₁, β₂, …,β_n). Similarly, we can obtain that H-SVNWAG(β₁, β₂,…,β_n) ≥SVNWG(β₁, β₂,…,β_n). Thus, we get the required proof. □