## 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.

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Communicated by Marcos Eduardo Valle.

## Appendix

### 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

Then for all \(\kappa _i\in X_1\),

and for all \(\kappa _i\in X_2\),

Using (4.8),

Similarly,

**Corollary:** For any PFS \(\vartheta \) and its complement \(\vartheta ^c\), we have

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Joshi, R. A new picture fuzzy information measure based on Tsallis–Havrda–Charvat concept with applications in presaging poll outcome.
*Comp. Appl. Math.* **39, **71 (2020). https://doi.org/10.1007/s40314-020-1106-z

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### Keywords

- Picture fuzzy set
- Tsallis–Havrda–Charvat picture fuzzy entropy
- Hamming distance
- VIKOR
- TOPSIS

### Mathematics Subject Classification

- 94A15
- 94A24
- 26D15