Multi-criteria decision-making based on bi-parametric exponential fuzzy information measures and weighted correlation coefficients

Abstract

This paper proposes a new bi-parametric exponential fuzzy information measure. In addition to the validation of proposed fuzzy information measure, some of its major properties are also studied. Besides, the performance of proposed fuzzy information measure is demonstrated using two numerical examples. Further, based on the concept of TOPSIS (Technique for Order Preference by Similarity to Ideal Solutions) method, a new improved TOPSIS method based on weighted correlation coefficients has been introduced. Considering the importance of criteria weights in the solution of Multi-Criteria Decision-Making (MCDM) problems, two methods have been discussed for the evaluation of criteria weights. In first method, criteria weight evaluation from the partial information provided by experts is discussed. Second method proposes the criteria weight evaluation in case they are completely unknown or incompletely known. The proposed MCDM method is explained through a numerical example based on fault detection in an ill-functioning machine.

Appendices

Appendix A

Proof of Theorem (3.2)

To prove the theorem, we bifurcate the universe of discourse $X=(\ell _1, \ell _2, \ldots , \ell _n)$ as follows:

$$\begin{aligned} X_1=&\{\ell _i\in X| \odot _1(\ell _i)\subseteq \odot _2(\ell _i)\}\nonumber \\&\text {and}\, X_2=\{\ell _i\in X| \odot _1(\ell _i)\supseteq \odot _2 (\ell _i)\}. \end{aligned}$$

(58)

This implies that for all $\ell _i\in X_1$, $\mu _{\odot _1} (\ell _i)\le \mu _{\odot _2} (\ell _i)$ and for all $\ell _i\in X_2$, $\mu _{\odot _2} (\ell _i)\le \mu _{\odot _1} (\ell _i)$, where $\mu _{\odot _1} (\ell _i)$ and $\mu _{\odot _2} (\ell _i)$ denote the membership degrees of $\odot _1$ and $\odot _2$, respectively. This gives

$$\begin{aligned}&\text {For all} \, \ell _i\in X_1;\,\mu _{\odot _1\cup \odot _2} (\ell _i)=\max {(\mu _{\odot _1} (\ell _i), \mu _{\odot _2} (\ell _i))}=\mu _{\odot _2} (\ell _i);\nonumber \\&\text {and}\, \mu _{\odot _1\cap \odot _2} (\ell _i)=\min {(\mu _{\odot _1} (\ell _i), \mu _{\odot _2} (\ell _i))}=\mu _{\odot _1} (\ell _i). \end{aligned}$$

(59)

Similarly,

$$\begin{aligned}&\text {For all} \,\ell _i\in X_2; \mu _{\odot _1\cup \odot _2} (\ell _i)=\max {(\mu _\odot (\ell _i), \mu _{\odot _2} (\ell _i))}=\mu _{\odot _1} (\ell _i);\nonumber \\&\text {and}~ \mu _{\odot _1\cap \odot _2} (\ell _i)=\min {(\mu _{\odot _1} (\ell _i), \mu _{\odot _2} (\ell _i))}=\mu _{\odot _2} (\ell _i). \end{aligned}$$

(60)

Now, to prove theorem (3.2), consider

$$\begin{aligned} {}_\varsigma ^\varrho H (\odot _1\cup \odot _2)+{}_\varsigma ^\varrho H (\odot _1\cap \odot _2)=\frac{A_4+B_4}{n\left( 2^{1-\varsigma }e^{1-2^{-\varsigma }}-2^{1-\varrho }e^{1-2^{-\varrho }}\right) }, \end{aligned}$$

(61)

where

$$\begin{aligned} A_4=A_{41}-A_{42},~ B_4=B_{41}-B_{42} \end{aligned}$$

and

$$\begin{aligned} A_{41}=&\sum _{i=1}^n \left( \mu _{\odot _1\cup \odot _2} (\ell _i)^\varsigma e^{(1-\mu _{\odot _1\cup \odot _2} (\ell _i)^\varsigma )}\right. \\&\left. +(1-\mu _{\odot _1\cup \odot _2} (\ell _i))^\varsigma e^{1-{(1-\mu _{\odot _1\cup \odot _2} (\ell _i))}^\varsigma }\right) ;\\ A_{42}=&\sum _{i=1}^n\left( \mu _{\odot _1\cup \odot _2} (\ell _i)^\varrho e^{(1-\mu _{\odot _1\cup \odot _2} (\ell _i)^\varrho )}\right. \\&\left. +(1-\mu _{\odot _1\cup \odot _2} (\ell _i))^\varrho e^{1-{(1-\mu _{\odot _1\cup \odot _2} (\ell _i))}^\varrho }\right) ;\\ B_{41}=&\sum _{i=1}^n \left( \mu _{\odot _1\cap \odot _2} (\ell _i)^\varsigma e^{(1-\mu _{\odot _1\cap \odot _2} (\ell _i)^\varsigma )}\right. \\&\left. +(1-\mu _{\odot _1\cap \odot _2} (\ell _i))^\varsigma e^{1-{(1-\mu _{\odot _1\cap \odot _2} (\ell _i))}^\varsigma }\right) ;\\ B_{42}=&\sum _{i=1}^n\left( \mu _{\odot _1\cap \odot _2} (\ell _i)^\varrho e^{(1-\mu _{\odot _1\cap \odot _2} (\ell _i)^\varrho )}\right. \\&\left. +(1-\mu _{\odot _1\cap \odot _2} (\ell _i))^\varrho e^{1-{(1-\mu _{\odot _1\cap \odot _2} (\ell _i))}^\varrho }\right) . \end{aligned}$$

Using (59) and (60), we get

$$\begin{aligned}&{}_\varsigma ^\varrho H (\odot _1\cup \odot _2)+{}_\varsigma ^\varrho H (\odot _1\cap \odot _2)\nonumber \\&\quad =\frac{\left[ (A_{41}'+A_{41}'')-(A_{42}'+A_{42}'')\right] +\left[ (B_{41}'+B_{41}'')-(B_{42}'+B_{42}'')\right] }{n\left( 2^{1-\varsigma }e^{1-2^{-\varsigma }}-2^{1-\varrho }e^{1-2^{-\varrho }}\right) }, \end{aligned}$$

(62)

where

$$\begin{aligned} A_{41}'=\sum _{X_1} \left( \mu _{\odot _2} (\ell _i)^\varsigma e^{(1-\mu _{\odot _2} (\ell _i)^\varsigma )}+(1-\mu _{\odot _2} (\ell _i))^\varsigma e^{1-{(1-\mu _{\odot _2} (\ell _i))}^\varsigma }\right) ;\\ A_{41}''=\sum _{X_2}\left( \mu _{\odot _1} (\ell _i)^\varsigma e^{(1-\mu _{\odot _1} (\ell _i)^\varsigma )}+(1-\mu _{\odot _1} (\ell _i))^\varsigma e^{1-{(1-\mu _{\odot _1} (\ell _i))}^\varsigma }\right) ;\\ A_{42}'=\sum _{X_1} \left( \mu _{\odot _2} (\ell _i)^\varrho e^{(1-\mu _{\odot _2} (\ell _i)^\varrho )}+(1-\mu _{\odot _2} (\ell _i))^\varrho e^{1-{(1-\mu _{\odot _2} (\ell _i))}^\varrho }\right) ;\\ A_{42}''=\sum _{X_2}\left( \mu _{\odot _1} (\ell _i)^\varrho e^{(1-\mu _{\odot _1} (\ell _i)^\varrho )}+(1-\mu _{\odot _1} (\ell _i))^\varrho e^{1-{(1-\mu _{\odot _1} (\ell _i))}^\varrho }\right) ;\\ B_{41}'=\sum _{X_1} \left( \mu _{\odot _1} (\ell _i)^\varsigma e^{(1-\mu _{\odot _1} (\ell _i)^\varsigma )}+(1-\mu _{\odot _1} (\ell _i))^\varsigma e^{1-{(1-\mu _{\odot _1} (\ell _i))}^\varsigma }\right) ;\\ B_{41}''=\sum _{X_2}\left( \mu _{\odot _2} (\ell _i)^\varsigma e^{(1-\mu _{\odot _2} (\ell _i)^\varsigma )}+(1-\mu _{\odot _2} (\ell _i))^\varsigma e^{1-{(1-\mu _{\odot _2} (\ell _i))}^\varsigma }\right) ;\\ B_{42}'=\sum _{X_1} \left( \mu _{\odot _1} (\ell _i)^\varrho e^{(1-\mu _{\odot _1} (\ell _i)^\varrho )}+(1-\mu _{\odot _1} (\ell _i))^\varrho e^{1-{(1-\mu _{\odot _1} (\ell _i))}^\varrho }\right) ;\\ B_{42}''=\sum _{X_2}\left( \mu _{\odot _2} (\ell _i)^\varrho e^{(1-\mu _{\odot _2} (\ell _i)^\varrho )}+(1-\mu _{\odot _2} (\ell _i))^\varrho e^{1-{(1-\mu _{\odot _2} (\ell _i))}^\varrho }\right) . \end{aligned}$$

On computing (62), we get

$$\begin{aligned} {}_\varsigma ^\varrho H (\odot _1\cup \odot _2)+{}_\varsigma ^\varrho H (\odot _1\cap \odot _2)={}_\varsigma ^\varrho H (\odot _1)+{}_\varsigma ^\varrho H (\odot _2). \end{aligned}$$

(63)

Corollary

Proof follows directly from the proof of theorem (3.2) by taking $\odot _2=\odot _1^c$.

Proof of Theorem (3.3):

First, we prove that ${}_\varsigma ^\varrho H(\odot )$ is independent of $\varsigma$ when $\odot$ is most fuzzy set, that is, $\mu _\odot (\ell _i)=0.5$ for all $\ell _i\in X$. Therefore, substituting $\mu _\odot (\ell _i)=.5$ in (8), we get

$$\begin{aligned} {}_\varsigma ^\varrho H (\odot )= \frac{n\left( 2^{1-\varsigma }e^{1-2^{-\varsigma }}-2^{1-\varrho }e^{1-2^{-\varrho }}\right) }{n\left( 2^{1-\varsigma }e^{1-2^{-\varsigma }}-2^{1-\varrho }e^{1-2^{-\varrho }}\right) }=1, \end{aligned}$$

(64)

which is free of $\varsigma$ and $\varrho$.

Similarly, if $\odot$ is least fuzzy set, that is, taking  $\mu _\odot (\ell _i)=1~ \text {or}~0$ in (8), we find that ${}_\varsigma ^\varrho H(\odot )=0$ which is again free of $\varsigma$ and $\varrho$. This proves the theorem.

Appendix B

Consider an linear programming problem (LPP) of minimization type defined by

$$\begin{aligned} Z=&\min _Xf^TX;\end{aligned}$$

(65)

$$\begin{aligned}&\text {such that} {\left\{ \begin{array}{ll} M\cdot X\le ub,\\ M_{eq}\cdot X=b_{eq},\\ lb\le X\le ub, \end{array}\right. } \end{aligned}$$

(66)

where $f, X, ub, b_{eq}, lb$ are vectors and M and $M_{eq}$ are matrices.

The syntax for MATLAB code for solving above LPP is given by

$$\begin{aligned}{}[X, fval]=linprog(f,M,b,M_{eq},b_{eq},lb,ub). \end{aligned}$$

(67)

The Example discussed in Sect. 5.1 is solved as follows: $X=\left[ \begin{array}{c} w_1\\ w_2\\ w_3\\ w_4 \end{array} \right]$ represents the function to be optimized and $f=\left[ \begin{array}{c} .9601\\ .8742\\ .9221\\ .9821 \end{array} \right]$ denotes the coefficient vector. M is matrix of constraints given by

$$\begin{aligned} M= \left[ \begin{array}{cccc} 1 &{} 0 &{} 0 &{} 0\\ 0 &{} 1 &{} 0 &{} 0\\ 0 &{} 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 &{} 1 \end{array} \right] . \end{aligned}$$

(68)

$ub=(.3,.2,.5,.3)$ is the upper bound of constraints and $lb=(0,.1,.2,.1)$ represents the lower bound of constraints. Further, $M_{eq}=(1, 1, 1, 1)$, $b_{eq}=1$ and fval is the optimum value of X.