A new multi-criteria decision-making method based on intuitionistic fuzzy information and its application to fault detection in a machine

Abstract

Intuitionistic Fuzzy Sets (IFSs) introduced by Atanassov are well suitable to deal with hesitancy and vagueness. In this communication, a new bi-parametric exponential information measure based on IFSs is introduced. Besides establishing its validity, some of its major properties are also discussed. Further, a new multi-criteria decision-making method based on the proposed IF measure and weighted correlation coefficients is introduced. The proposed method is utilized in detecting the fault in a machine that is not working properly through a numerical example.

Appendix

Now, we prove the Theorems (5) and (6).

Proof of Theorem (5):

To prove the theorem, we bifurcate the universe of discourse $X=(o_1, o_2, \ldots , o_n)$ as follows:

$$\begin{aligned} X_1&=\{o_i\in X| D(o_i)\subseteq E(o_i)\}~\text {and}\nonumber \\ X_2&=\{o_i\in X| D(o_i)\supseteq E (o_i)\}. \end{aligned}$$

(53)

This implies that for all $o_i\in X_1$, $\mu _D (o_i)\le \mu _E (o_i)$ and for all $o_i\in X_2$, $\mu _E (o_i)\le \mu _D (o_i)$, where $\mu _D (o_i)$ and $\mu _E (o_i)$ denote the membership degrees of D and B respectively. This gives

$$\begin{aligned}&\text {For all} ~ o_i\in X_1; ~\mu _{D\cup E} (o_i)=\max {(\mu _D (o_i), \mu _E (o_i))}=\mu _E (o_i);\nonumber \\&\quad \text {and}~ \mu _{D\cap E} (o_i)=\min {(\mu _D (o_i), \mu _E (o_i))}=\mu _D (o_i). \end{aligned}$$

(54)

Similarly,

$$\begin{aligned}&\text {For all} ~o_i\in X_2; \mu _{D\cup E} (o_i)=\max {(\mu _D (o_i), \mu _E (o_i))}=\mu _D (o_i);\nonumber \\&\quad ~\text {and}~ \mu _{D\cap E} (o_i)=\min {(\mu _D (o_i), \mu _E (o_i))}=\mu _E (o_i). \end{aligned}$$

(55)

Now, to prove theorem (5), consider

$$\begin{aligned}&H_\rho ^\varsigma (D\cup E)+H_\rho ^\varsigma (D\cap E)=\sum _{i=1}^n\frac{1}{n\left( 2^{1-\rho }e^{1-2^{-\rho }}-2^{1-\varsigma }e^{1-2^{-\varsigma }}\right) } \nonumber \\&\quad \times \left[ \begin{array}{c} \left( \begin{array}{c} \left( \left( \frac{\mu _{D\cup E} (o_i)+1-\nu _{D\cup E} (o_i)}{2}\right) ^\rho e^{1-\left( \frac{\mu _{D\cup E} (o_i)+1-\nu _{D\cup E} (o_i)}{2}\right) ^\rho }\right. \\ \left. +\left( \frac{\nu _{D\cup E} (o_i)+1-\mu _{D\cup E} (o_i)}{2}\right) ^\rho e^{1-\left( \frac{\nu _{D\cup E} (o_i)+1-\mu _{D\cup E} (o_i)}{2}\right) ^\rho }\right) \\ -\left( \left( \frac{\mu _{D\cup E} (o_i)+1-\nu _{D\cup E} (o_i)}{2}\right) ^\varsigma e^{1-\left( \frac{\mu _{D\cup E} (o_i)+1-\nu _{D\cup E} (o_i)}{2}\right) ^\varsigma }\right. \\ \left. +\left( \frac{\nu _{D\cup E} (o_i)+1-\mu _{D\cup E} (o_i)}{2}\right) ^\varsigma e^{1-\left( \frac{\nu _{D\cup E} (o_i)+1-\mu _{D\cup E} (o_i)}{2}\right) ^\varsigma }\right) \end{array} \right) \\ +\left( \begin{array}{c} \left( \left( \frac{\mu _{D\cap E} (o_i)+1-\nu _{D\cap E} (o_i)}{2}\right) ^\rho e^{1-\left( \frac{\mu _{D\cap E} (o_i)+1-\nu _{D\cap E} (o_i)}{2}\right) ^\rho }\right. \\ \left. +\left( \frac{\nu _{D\cap E} (o_i)+1-\mu _{D\cap E} (o_i)}{2}\right) ^\rho e^{1-\left( \frac{\nu _{D\cap E} (o_i)+1-\mu _{D\cap E} (o_i)}{2}\right) ^\rho }\right) \\ -\left( \left( \frac{\mu _{D\cap E} (o_i)+1-\nu _{D\cap E} (o_i)}{2}\right) ^\varsigma e^{1-\left( \frac{\mu _{D\cap E} (o_i)+1-\nu _{D\cap E} (o_i)}{2}\right) ^\varsigma }\right. \\ \left. +\left( \frac{\nu _{D\cap E} (o_i)+1-\mu _{D\cap E} (o_i)}{2}\right) ^\varsigma e^{1-\left( \frac{\nu _{D\cap E} (o_i)+1-\mu _{D\cap E} (o_i)}{2}\right) ^\varsigma }\right) \end{array} \right) \end{array} \right] . \end{aligned}$$

(56)

Using (53), (54) and (55), (56) gives

$$\begin{aligned}&H_\rho ^\varsigma (D\cup E)+H_\rho ^\varsigma (D\cap E)=\sum _{i=1}^n\frac{1}{n\left( 2^{1-\rho }e^{1-2^{-\rho }}-2^{1-\varsigma }e^{1-2^{-\varsigma }}\right) } \nonumber \\&\quad \times \left[ \begin{array}{c} \sum _{X_1}\left( \begin{array}{c} \left( \left( \frac{\mu _E (o_i)+1-\nu _E (o_i)}{2}\right) ^\rho e^{1-\left( \frac{\mu _E (o_i)+1-\nu _E (o_i)}{2}\right) ^\rho }\right. \\ \left. +\left( \frac{\nu _E (o_i)+1-\mu _E (o_i)}{2}\right) ^\rho e^{1-\left( \frac{\nu _E (o_i)+1-\mu _E (o_i)}{2}\right) ^\rho }\right) \\ -\left( \left( \frac{\mu _E (o_i)+1-\nu _E (o_i)}{2}\right) ^\varsigma e^{1-\left( \frac{\mu _E (o_i)+1-\nu _E (o_i)}{2}\right) ^\varsigma }\right. \\ \left. +\left( \frac{\nu _E (o_i)+1-\mu _E (o_i)}{2}\right) ^\varsigma e^{1-\left( \frac{\nu _E (o_i)+1-\mu _E (o_i)}{2}\right) ^\varsigma }\right) \end{array} \right) \\ +\sum _{X_2}\left( \begin{array}{c} \left( \left( \frac{\mu _D (o_i)+1-\nu _D (o_i)}{2}\right) ^\rho e^{1-\left( \frac{\mu _D (o_i)+1-\nu _D (o_i)}{2}\right) ^\rho }\right. \\ \left. +\left( \frac{\nu _D (o_i)+1-\mu _D (o_i)}{2}\right) ^\rho e^{1-\left( \frac{\nu _D (o_i)+1-\mu _D (o_i)}{2}\right) ^\rho }\right) \\ -\left( \left( \frac{\mu _D (o_i)+1-\nu _D (o_i)}{2}\right) ^\varsigma e^{1-\left( \frac{\mu _D (o_i)+1-\nu _D (o_i)}{2}\right) ^\varsigma }\right. \\ \left. +\left( \frac{\nu _D (o_i)+1-\mu _D (o_i)}{2}\right) ^\varsigma e^{1-\left( \frac{\nu _D (o_i)+1-\mu _D (o_i)}{2}\right) ^\varsigma }\right) \end{array} \right) \\ +\sum _{X_1}\left( \begin{array}{c} \left( \left( \frac{\mu _D (o_i)+1-\nu _D (o_i)}{2}\right) ^\rho e^{1-\left( \frac{\mu _D (o_i)+1-\nu _D (o_i)}{2}\right) ^\rho }\right. \\ \left. +\left( \frac{\nu _D (o_i)+1-\mu _D (o_i)}{2}\right) ^\rho e^{1-\left( \frac{\nu _D (o_i)+1-\mu _D (o_i)}{2}\right) ^\rho }\right) \\ -\left( \left( \frac{\mu _D (o_i)+1-\nu _D (o_i)}{2}\right) ^\varsigma e^{1-\left( \frac{\mu _D (o_i)+1-\nu _D (o_i)}{2}\right) ^\varsigma }\right. \\ \left. +\left( \frac{\nu _D (o_i)+1-\mu _D (o_i)}{2}\right) ^\varsigma e^{1-\left( \frac{\nu _D (o_i)+1-\mu _D (o_i)}{2}\right) ^\varsigma }\right) \end{array} \right) \\ +\sum _{X_2}\left( \begin{array}{c} \left( \left( \frac{\mu _E (o_i)+1-\nu _E (o_i)}{2}\right) ^\rho e^{1-\left( \frac{\mu _E (o_i)+1-\nu _E (o_i)}{2}\right) ^\rho }\right. \\ \left. +\left( \frac{\nu _E (o_i)+1-\mu _E (o_i)}{2}\right) ^\rho e^{1-\left( \frac{\nu _E (o_i)+1-\mu _E (o_i)}{2}\right) ^\rho }\right) \\ -\left( \left( \frac{\mu _E (o_i)+1-\nu _E (o_i)}{2}\right) ^\varsigma e^{1-\left( \frac{\mu _E (o_i)+1-\nu _E (o_i)}{2}\right) ^\varsigma }\right. \\ \left. +\left( \frac{\nu _E (o_i)+1-\mu _E (o_i)}{2}\right) ^\varsigma e^{1-\left( \frac{\nu _E (o_i)+1-\mu _E (o_i)}{2}\right) ^\varsigma }\right) \end{array} \right) \end{array} \right] . \end{aligned}$$

(57)

On computing (57), we get

$$\begin{aligned} H_\rho ^\varsigma (D\cup E)+ H_\rho ^\varsigma (D\cap E)= H_\rho ^\varsigma (D)+ H_\rho ^\varsigma (E). \end{aligned}$$

(58)

$\square$

Corollary

Proof follows directly from the proof of theorem (5) by taking$E=D^c$.

Proof of Theorem (6):

First we prove that $H_\rho ^\varsigma (D)$ is independent of $\rho$ when D is most fuzzy set, that is, $\mu _D (o_i)=\nu _D (o_i)$ for all $o_i\in X$. Therefore, substituting $\mu _D (o_i)=\nu _D (o_i)$ in (10), we get

$$\begin{aligned} H_\rho ^\varsigma (D)= \frac{n\left( 2^{1-\rho }e^{1-2^{-\rho }}-2^{1-\varsigma }e^{1-2^{-\varsigma }}\right) }{n\left( 2^{1-\rho }e^{1-2^{-\rho }}-2^{1-\varsigma }e^{1-2^{-\varsigma }}\right) }=1; \end{aligned}$$

(59)

which is independent of $\rho$ and $\varsigma$.

Similarly, if D is least fuzzy set, that is, taking  $\mu _D (o_i)=1, \nu _D (o_i)=0$ or $\mu _D (o_i)=0, \nu _D (o_i)=1$ in (10), we find that $H_\rho ^\varsigma (D)=0$ which is again free of $\rho$ and $\varsigma$. This proves the theorem.