NC-Cross Entropy Based MADM Strategy in Neutrosophic Cubic Set Environment
Abstract
:1. Introduction
1.1. Research Gap: NC-Cross Entropy-Based MADM Strategy in NCS Environments
- Is it possible to introduce an NC-cross entropy measure in NCS environments?
- Is it possible to introduce a weighted cross entropy measure in NCS environments?
- Is it possible to develop a novel MADM strategy based on weighted NC-cross entropy?
1.2. Motivation
- To introduce a NC-cross entropy measure and establish its basic properties in an NCS environment.
- To introduce a weighted NC- cross measure and establish its basic properties in NCS environments.
- To develop a novel MADM strategy based on weighted NC-cross entropy measure in NCS environments.
2. Preliminaries
- 1.
- 2.
3. NC-Cross-Entropy Measure in NCS Environment
- (i)
- .
- (ii)
- iff , , , , , and for all
- (iii)
- ,
- (iv)
- .
- (i)
- For all values of , , , , , , , , , , .Then,Similarly,Again,Similarly, we can show thatAdding Equation (20) to Equation (28), we obtain .
- (ii)
- Again,From, Equation (29) to Equation (37), we obtain iff , , , , and for all
- (iii)
- Using Definition (1), Definition (4) and Definition (10), we obtain the following expression:
- (iv)
- Since , for a single valued neutrosophic part, we obtain:, , , , , .Then,, ,, , , ,For the interval neutrosophic part, we obtain , , , , , .Then, we obtain, , , , , ,Similarly,, , , , , , then , , , , , ,Thus, . ☐
- (i)
- .
- (ii)
- iff , , , , and
- (iii)
- .
- (iv)
- .
- (i)
- For all values of , , , , , , , , , , .Then,Similarly,Again,Similarly, we can show thatAdding Equation (39) to Equation (47), and using , we have . Hence, this completes the proof. ☐
- (ii)
- Again,Using Equation (48) to Equation (56) and , , we have iff , , , , and for all ☐
- (ii)
- Using Definition (20), Definition (4), and Definition (10), we obtain the following expression:
- (iv)
- Since , for single valued parts, we obtain:, , , , , .Then, we obtain, , , ,,For interval neutrosophic part, we have, , , , , .Then, we obtain, , , , ,,Similarly,, , ,, , , then,,In addition, , .Thus, hence completing the proof. ☐
4. MADM Strategy Using Proposed NC-Cross Entropy Measure in the NCS Environment
5. Illustrative Example
- (1)
- Automobile company (A1)
- (2)
- Military manufacturing enterprise (A2)
- (3)
- TV media company (A3)
- (4)
- Food enterprises (A4)
- (5)
- Computer software company (A5)
- (1)
- Social and political factor (G1)
- (2)
- The environmental factor (G2)
- (3)
- Investment risk factor (G3)
- (4)
- The enterprise growth factor (G4).
6. Contributions of the Paper
- We have introduced an NC-cross entropy measure and proved its basic properties in NCS environments.
- We have introduced a weighted NC-cross entropy measure and proved its basic properties in NCS environments.
- We have developed a novel MADM strategy based on weighted NC- cross entropy to solve MADM problems.
- We solved an illustrative example of MADM problem using proposed strategies.
7. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
References
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Pramanik, S.; Dalapati, S.; Alam, S.; Smarandache, F.; Roy, T.K. NC-Cross Entropy Based MADM Strategy in Neutrosophic Cubic Set Environment. Mathematics 2018, 6, 67. https://doi.org/10.3390/math6050067
Pramanik S, Dalapati S, Alam S, Smarandache F, Roy TK. NC-Cross Entropy Based MADM Strategy in Neutrosophic Cubic Set Environment. Mathematics. 2018; 6(5):67. https://doi.org/10.3390/math6050067
Chicago/Turabian StylePramanik, Surapati, Shyamal Dalapati, Shariful Alam, Florentin Smarandache, and Tapan Kumar Roy. 2018. "NC-Cross Entropy Based MADM Strategy in Neutrosophic Cubic Set Environment" Mathematics 6, no. 5: 67. https://doi.org/10.3390/math6050067
APA StylePramanik, S., Dalapati, S., Alam, S., Smarandache, F., & Roy, T. K. (2018). NC-Cross Entropy Based MADM Strategy in Neutrosophic Cubic Set Environment. Mathematics, 6(5), 67. https://doi.org/10.3390/math6050067