A new picture fuzzy information measure based on Tsallis–Havrda–Charvat concept with applications in presaging poll outcome

Abstract

The picture fuzzy set (PFS) proposed by Cuong and Kreinovich are well suitable to capture the uncertain information in vague circumstances. The main objective of this communication is to propose a new framework as a criteria of fuzzy entropy for PFSs. Further, a new picture fuzzy information measure based on Tsallis–Havrda–Charvat entropy is proposed and validated in accordance with newly proposed framework. Besides this, some major properties of proposed information measure are also discussed. Apart from this, a new multi-criteria decision-making method using the concept of VIKOR (Vlsekriterijumska Optimizacija i Kompromisno Resenje) based on relative projection is proposed. To show the practical utility of proposed decision-making method, two numerical examples based on election forecast through opinion polls have been discussed.

Appendix

Proof of Theorem (4.3):

To prove theorem (4.3), we divide X into two parts say $X_1$ and $X_2$ such that

$$\begin{aligned} X_1=\{\kappa _i\in X: \vartheta _1\subseteq \vartheta _2\}, \quad \qquad X_2=\{\kappa _i\in X: \vartheta _1\supseteq \vartheta _2\}. \end{aligned}$$

(8.1)

Then for all $\kappa _i\in X_1$,

$$\begin{aligned} \mu _{\vartheta _1} (\kappa _i)\le \mu _{\vartheta _2} (\kappa _i),~\delta _{\vartheta _1} (\kappa _i)\le \delta _{\vartheta _2} (\kappa _i),~\nu _{\vartheta _1} (\kappa _i)\ge \nu _{\vartheta _2} (\kappa _i), \end{aligned}$$

(8.2)

and for all $\kappa _i\in X_2$,

$$\begin{aligned} \mu _{\vartheta _1} (\kappa _i)\ge \mu _{\vartheta _2} (\kappa _i),~\delta _{\vartheta _1} (\kappa _i)\ge \delta _{\vartheta _2} (\kappa _i),~\nu _{\vartheta _1} (\kappa _i)\le \nu _{\vartheta _2} (\kappa _i). \end{aligned}$$

(8.3)

Using (4.8),

$$\begin{aligned} \psi (\vartheta _1\cup \vartheta _2)=&\frac{1}{n(1-\tau )}\sum _{i=1}^n\left( \left( \mu _{\vartheta _1\cup \vartheta _2} (\kappa _i)^\tau +\delta _{\vartheta _1\cup \vartheta _2} (\kappa _i)^\tau +\nu _{\vartheta _1\cup \vartheta _2} (\kappa _i)^\tau +\pi _{\vartheta _1\cup \vartheta _2} (\kappa _i)^\tau \right) -1\right) . \end{aligned}$$

(8.4)

$$\begin{aligned} \psi (\vartheta _1\cup \vartheta _2)=&\frac{1}{n(1-\tau )}\sum _{X_1}\left( \left( \mu _{\vartheta _2} (\kappa _i)^\tau +\delta _{\vartheta _2} (\kappa _i)^\tau +\nu _{\vartheta _2} (\kappa _i)^\tau +\pi _{\vartheta _2} (\kappa _i)^\tau \right) -1\right) \nonumber \\&+\frac{1}{n(1-\tau )}\sum _{X_2}\left( \left( \mu _{\vartheta _1} (\kappa _i)^\tau +\delta _{\vartheta _1} (\kappa _i)^\tau +\nu _{\vartheta _1} (\kappa _i)^\tau +\pi _{\vartheta _1} (\kappa _i)^\tau \right) -1\right) . \end{aligned}$$

(8.5)

Similarly,

$$\begin{aligned} \psi (\vartheta _1\cap \vartheta _2)=&\frac{1}{n(1-\tau )}\sum _{X_1}\left( \left( \mu _{\vartheta _1} (\kappa _i)^\tau +\delta _{\vartheta _1} (\kappa _i)^\tau +\nu _{\vartheta _1} (\kappa _i)^\tau +\pi _{\vartheta _1} (\kappa _i)^\tau \right) -1\right) \nonumber \\&+\frac{1}{n(1-\tau )}\sum _{X_2}\left( \left( \mu _{\vartheta _2} (\kappa _i)^\tau +\delta _{\vartheta _2} (\kappa _i)^\tau +\nu _{\vartheta _2} (\kappa _i)^\tau +\pi _{\vartheta _2} (\kappa _i)^\tau \right) -1\right) . \end{aligned}$$

(8.6)

From (8.5) and (8.6), we get

$$\begin{aligned} \psi (\vartheta _1\cup \vartheta _2)+\psi (\vartheta _1\cap \vartheta _2)=\psi (\vartheta _1)+\psi (\vartheta _2). \end{aligned}$$

(8.7)

Corollary: For any PFS $\vartheta$ and its complement $\vartheta ^c$, we have

$$\begin{aligned} \psi (\vartheta )=\psi (\vartheta ^c)=\psi (\vartheta \cup \vartheta ^c)=\psi (\vartheta \cap \vartheta ^c). \end{aligned}$$

(8.8)