Extension of TOPSIS method for group decision-making under triangular linguistic neutrosophic cubic sets

Abstract

In this paper, we define the new concept of interval-valued triangular linguistic neutrosophic fuzzy set and operational laws. We discuss the IVTLNFWAA operator of IVTLNFS, and then the score function of IVTLNFS is introduced. We define the hamming distance of IVTLNFS. We define the triangular linguistic neutrosophic cubic fuzzy number and operational laws. We also discuss the triangular linguistic neutrosophic cubic fuzzy number, such as the TLNCFWAA operator. We define the score function and accuracy of TLNCFNs. We also discuss the hamming distance of TLNCFNs. We discuss some basic properties of the proposed operator, including idempotency and commutativity. The new concept of the triangular linguistic neutrosophic cubic fuzzy TOPSIS method is introduced. Furthermore, we extend the MCDM method based on the triangular linguistic neutrosophic cubic fuzzy TOPSIS method. Finally, an illustrative example is given to verify and demonstrate the practicality and effectiveness of the proposed method.

Appendices

Appendix A

Proof

Since $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b}_{j} = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b} ,$ then TLNCFWAA $(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b}_{1} ,\,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b}_{2} , \ldots ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b}_{n} ) =$$\left\{ {\begin{array}{*{20}c} {s_{{\mathop \prod \limits_{j = 1}^{n} \theta_{j}^{{\mathop \sum \limits_{j = 1}^{n} w_{j} }} }} ,[\mathop \prod \limits_{j = 1}^{n} r_{j}^{{\mathop \sum \limits_{j = 1}^{n} w_{j} }} ,\,\mathop \prod \limits_{j = 1}^{n} s_{j}^{{\mathop \sum \limits_{j = 1}^{n} w_{j} }} ,\,\mathop \prod \limits_{j = 1}^{n} t_{j}^{{\mathop \sum \limits_{j = 1}^{n} w_{j} }} ],} \\ {\langle [1 - \mathop \prod \limits_{j = 1}^{n} (1 - F_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{n}_{j} }}^{ - } )^{{\mathop \sum \limits_{j = 1}^{n} w_{j} }} ,\,1 - \mathop \prod \limits_{j = 1}^{n} (1 - F_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{n}_{j} }}^{ + } )^{{\mathop \sum \limits_{j = 1}^{n} w_{j} }} ],} \\ {1 - \mathop \prod \limits_{j = 1}^{n} (1 - F_{{n_{j} }} )^{{\mathop \sum \limits_{j = 1}^{n} w_{j} }} \rangle ,} \\ {\langle [\mathop \prod \limits_{j = 1}^{n} (G_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{n}_{j} }}^{ - } )^{{\mathop \sum \limits_{j = 1}^{n} w_{j} }} ,\,\mathop \prod \limits_{j = 1}^{n} (G_{{n_{j} }}^{ + } )^{{\mathop \sum \limits_{j = 1}^{n} w_{j} }} ],\,\mathop \prod \limits_{j = 1}^{n} G_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{n}_{j} }}^{{\mathop \sum \limits_{j = 1}^{n} w_{j} }} \rangle ,} \\ {\langle [\mathop \prod \limits_{j = 1}^{n} (I_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{n}_{j} }}^{ - } )^{{\mathop \sum \limits_{j = 1}^{n} w_{j} }} ,\,\mathop \prod \limits_{j = 1}^{n} (I_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{n}_{j} }}^{ + } )^{{\mathop \sum \limits_{j = 1}^{n} w_{j} }} ],\,\mathop \prod \limits_{j = 1}^{n} I_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{n}_{j} }}^{{\mathop \sum \limits_{j = 1}^{n} w_{j} }} \rangle } \\ \end{array} } \right\}$ $= \left\{ {\begin{array}{*{20}c} {s_{{\theta_{j} }} ,[r_{j} ,\,s_{j} ,\,t_{j} ];} \\ {\langle [F_{j}^{ - } ,\,F_{j}^{ + } ],\,F_{j} \rangle ,} \\ {\langle [G_{j}^{ - } ,\,G_{j}^{ + } ],\,G_{j} \rangle ,} \\ {\langle [I_{j}^{ - } ,\,I_{j}^{ + } ],\,I_{j} \rangle } \\ \end{array} } \right\} = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b} .$□

Appendix B

Proof

Let TLNCFWAA $(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b}_{1} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b}_{2} , \ldots ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b}_{n} ) = \mathop \sum \limits_{j = 1}^{n} w_{j} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b}_{j} ,$ TLNCFWAA $(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b}_{1}^{ * } ,\,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b}_{2}^{ * } , \ldots ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b}_{n}^{ * } ) = \mathop \sum \limits_{j = 1}^{n} w_{j} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b}_{j}^{ * } .$□

Since $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b}_{j} \le \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b}_{j}^{ * }$ for all $j,$ then we have TLNCFWAA $(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b}_{1} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b}_{2} , \ldots ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b}_{n} ) \le$ TLNCFWAA $(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b}_{1}^{ * } ,\,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b}_{2}^{ * } , \ldots ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b}_{n}^{ * } )$.

Reviewer #1: No. What I meant was, the paper needs a theorem to prove that the solutions of original fuzzy models and embedded to interval form are the same. Otherwise it is rejected.

Response: We have added Theorem 2.6 and written below

Theorem 2.6

Every fuzzy set (FS) can be extended to an IVFS.

Proof

Let $X$ be any universe of discourse and $A \subseteq X.$ Then, an FS in $A$ is a mapping $\mu_{A} \;\;:\;\;A \to [0,\,1].$□

The value $\mu_{A} (a)$ is said to be a degree of membership of $a$ in $A.$

Define a multi-interval $S$ of $[0,\,1]$ such that $S = [\mu_{A} (a),\,\mu_{A} (a) + c],\forall a \in A$where $c$ is a rational number such that $0 \le \mu_{A} (a) \le \mu_{A} (a) + c \le 1.$

Now, we have an IVFS $\mu_{A} (a)$ in $A$ defined by $\mu_{A} (a) = [\mu_{A} (a),\,\mu_{A} (a) + c],\forall a \in A$

It completes the proof.□