An $(R',S')$-norm fuzzy relative information measure and its applications in strategic decision-making

Abstract

In this communication, we propose two quantities named ‘An $(R', S')$-norm directed divergence measure’ and ‘$(R', S')$-norm fuzzy directed divergence measure’, establish their validity and discuss their properties. Further, the performance of proposed divergence measure is compared with some existing measures through a numerical example. Finally, the application of proposed fuzzy directed divergence measure is given in strategic decision-making and the outcome is compared with other methods in literature.

Appendix: Proof of properties

Proof

Consider the sets

$$\begin{aligned} X_1=\{z_i \in X/\mu _P (z_i) \ge \mu _Q (z_i)\} \end{aligned}$$

(8.1)

and

$$\begin{aligned} X_2=\{z_i\in X/\mu _P (z_i)<\mu _Q (z_i)\}. \end{aligned}$$

(8.2)

Using the notations given in the ‘Preliminaries’ section, i.e., $\square$

In set $X_1$,

$$\begin{aligned} P\cup Q&=\mathrm {Union~ of}~ P\quad \mathrm {and}\quad Q \Leftrightarrow \mu _ {P\cup Q} (z_i)=\max \{\mu _P (z_i), \mu _Q (z_i)\} =\mu _P (z_i),\nonumber \\ P\cap Q&= \mathrm {Intersection~ of}~ P\quad \mathrm {and}\quad Q \Leftrightarrow \mu _{P\cap Q} (z_i)=\min \{\mu _P (z_i),\mu _Q (z_i)\}=\mu _Q (z_i). \end{aligned}$$

(8.3)

In set $X_2$,

$$\begin{aligned} P\cup Q&=\mathrm {Union~of}~P~\mathrm {and}~ Q \Leftrightarrow \mu _{P\cup Q} (z_i)=\max \{\mu _P (z_i), \mu _Q (z_i)\}=\mu _Q(z_i),\nonumber \\ P\cap Q&=\mathrm {Intersection~ of}~ P\mathrm {and}~ Q \Leftrightarrow \mu _{P\cap Q} (z_i)=\min \{\mu _P(z_i),\mu _Q (z_i)\}=\mu _P (z_i). \end{aligned}$$

(8.4)

(i) Consider

$$\begin{aligned}&I_{R'}^{S'} (P\cup Q, P)+I_{R'}^{S'} (P\cap Q, P) \nonumber \\&=\frac{R'\times S'}{n (S'-R')}\sum _{i=1}^n \left( \begin{array}{c} \left( (\mu _{P\cup Q} (z_i))^{S'}(\mu _P (z_i))^{1-S'}+(1-\mu _{P\cup Q}(z_i))^{S'}(1-\mu _P (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _{P\cup Q} (z_i))^{R'}(\mu _P (z_i))^{1-R'}+(1-\mu _{P\cup Q}(z_i))^{R'}(1-\mu _P (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) \nonumber \\&\quad +\frac{R'\times S'}{n (S'-R')}\sum _{i=1}^n \left( \begin{array}{c} \left( (\mu _{P \cap Q} (z_i))^{S'}(\mu _P (z_i))^{1-S'}+(1-\mu _{P\cap Q} (z_i))^{S'} (1-\mu _P (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _{P\cap Q} (z_i))^{R'}(\mu _P (z_i))^{1-R'}+(1-\mu _{P\cap Q} (z_i))^{R'} (1-\mu _P (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) . \end{aligned}$$

(8.5)

Dividing X into $X_1$ and $X_2$, we get

$$\begin{aligned} =\frac{R'\times S'}{n (S'-R')}\sum _{X_1} \left( \begin{array}{c} \left( (\mu _{P\cup Q} (z_i))^{S'}(\mu _P (z_i))^{1-S'}+(1-\mu _{P\cup Q}(z_i))^{S'}(1-\mu _P (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _{P\cup Q} (z_i))^{R'}(\mu _P (z_i))^{1-R'}+(1-\mu _{P\cup Q} (z_i))^{R'}(1-\mu _P (z_i))^{1-R'}\right) ^\frac{1}{R'}\\ +\left( (\mu _{P \cap Q} (z_i))^{S'} (\mu _P (z_i))^{1-S'}+(1-\mu _{P\cap Q} (z_i))^{S'} (1-\mu _P (z_i))^{1-S'}\right) ^\frac{1}{S'}\nonumber \\ -\left( (\mu _{P\cap Q} (z_i))^{R'}(\mu _P (z_i))^{1-R'}+(1-\mu _{P\cap Q} (z_i))^{R'}(1-\mu _P (z_i))^{1-R'}\right) ^\frac{1}{R'}\nonumber \end{array} \right) \\ \quad +\frac{R'\times S'}{n (S'-R')}\sum _{X_2} \left( \begin{array}{c} \left( (\mu _{P\cup Q} (z_i))^{S'}(\mu _P (z_i))^{1-S'}+(1-\mu _{P\cup Q} (z_i))^{S'} (1-\mu _P (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _{P\cup Q} (z_i))^{R'}(\mu _P (z_i))^{1-R'}+(1-\mu _{P \cup Q} (z_i))^{R'} (1-\mu _P (z_i))^{1-R'}\right) ^\frac{1}{R'}\\ +\left( (\mu _{P \cap Q} (z_i))^{S'}(\mu _P (z_i))^{1-S'}+(1-\mu _{P\cap Q} (z_i))^{S'} (1-\mu _P (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _{P\cap Q} (z_i))^{R'} (\mu _P (z_i))^{1-R'}+(1-\mu _{P\cap Q} (z_i))^{R'} (1-\mu _P (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) . \end{aligned}$$

(8.6)

Using (8.3) and (8.4) and simplifying, we get

$$\begin{aligned}&I_{R'}^{S'}(P\cup Q, P)+I_{R'}^{S'} (P \cap Q, P) \nonumber \\&=\frac{R'\times S'}{n (S'-R')} \sum _{X_1} \left( \begin{array}{c} \left( (\mu _Q (z_i))^{S'}(\mu _P (z_i))^{1-S'}+(1-\mu _Q (z_i))^{S'}(1-\mu _P (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _Q (z_i))^{R'}(\mu _P (z_i))^{1-R'}+(1-\mu _Q (z_i))^{R'}(1-\mu _P (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) \nonumber \\&\quad +\frac{R'\times S'}{n (S'-R')}\sum _{X_2} \left( \begin{array}{c} \left( (\mu _Q (z_i))^{S'}(\mu _P (z_i))^{1-S'}+(1-\mu _Q (z_i))^{S'}(1-\mu _P (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _Q (z_i))^{R'}(\mu _P (z_i))^{1-R'}+(1-\mu _Q (z_i))^{R'}(1-\mu _P (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) . \end{aligned}$$

(8.7)

Therefore,

$$\begin{aligned}&I_{R'}^{S'} (P\cup Q, P)+I_{R'}^{S'} (P\cap Q, P) \nonumber \\&\quad =\frac{R'\times S'}{n (S'-R')}\sum _{i=1}^n \left( \begin{array}{c} \left( (\mu _Q (z_i))^{S'}(\mu _P (z_i))^{1-S'}+(1-\mu _Q (z_i))^{S'}(1-\mu _P (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _Q (z_i))^{R'}(\mu _P (z_i))^{1-R'}+(1-\mu _Q (z_i))^{R'}(1-\mu _P (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) \nonumber \\&\quad =I_{R'}^{S'} (Q, P). \end{aligned}$$

(8.8)

Hence, $I_{R'}^{S'}(P\cup Q, P)+I_{R'}^{S'}(P \cap Q, P)=I_{R'}^{S'} (Q, P)$.

(ii) We have to prove $I_{R'}^{S'} (P \cup Q, C)+I_{R'}^{S'} (P \cap Q, C)=I_{R'}^{S'} (P, C)+I_{R'}^{S'} (Q, C)$.

For this, consider

$$\begin{aligned}&I_{R'}^{S'} (P \cup Q, C)+I_{R'}^{S'} (P \cap Q, C) \nonumber \\&=\frac{R'\times S'}{n (S'-R')}\sum _{i=1}^n \left( \begin{array}{c} \left( (\mu _{P \cup Q} (z_i))^{S'}(\mu _C (z_i))^{1-S'}+(1-\mu _{P \cup Q}(z_i))^{S'}(1-\mu _C(z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _{P\cup Q} (z_i))^{R'}(\mu _C (z_i))^{1-R'}+(1-\mu _{P \cup Q}(z_i))^{R'} (1-\mu _C (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) \nonumber \\&\quad +\frac{R'\times S'}{n (S'-R')}\sum _{i=1}^n \left( \begin{array}{c} \left( (\mu _{P \cap Q} (z_i))^{S'}(\mu _C (z_i))^{1-S'}+(1-\mu _{P \cap Q}(z_i))^{S'} (1-\mu _C (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _{P\cap Q} (z_i))^{R'}(\mu _C (z_i))^{1-R'}+(1-\mu _{P \cap Q} (z_i))^{R'} (1-\mu _C (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) . \end{aligned}$$

(8.9)

Dividing X into $X_1$ and $X_2$, we get

$$\begin{aligned}&=\frac{R'\times S'}{n (S'-R')}\sum _{X_1} \left( \begin{array}{c} \left( (\mu _{P \cup Q} (z_i))^{S'}\mu _C (z_i)^{1-S'}+(1-\mu _{P \cup Q}(z_i))^{S'}(1-\mu _C (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _{P\cup Q} (z_i))^{R'}(\mu _C (z_i))^{1-R'}+(1-\mu _{P\cup Q} (z_i))^{R'} (1-\mu _C (z_i))^{1-R'}\right) ^\frac{1}{R'}\\ +\left( (\mu _{P\cap Q} (z_i))^{S'}(\mu _C (z_i))^{1-S'}+(1-\mu _{P\cap Q} (z_i))^{S'} (1-\mu _C(z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _{P\cap Q} (z_i))^{R'}(\mu _C (z_i))^{1-R'}+(1-\mu _{P \cap Q} (z_i))^{R'} (1-\mu _C (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) \nonumber \\&\quad +\frac{R'\times S'}{n (S'-R')}\sum _{X_2} \left( \begin{array}{c} \left( (\mu _{P\cup Q} (z_i))^{S'}(\mu _C (z_i))^{1-S'}+(1-\mu _{P \cup Q} (z_i))^{S'}(1-\mu _C (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _{P \cup Q} (z_i))^{R'}(\mu _C (z_i))^{1-R'}+(1-\mu _{P \cup Q} (z_i))^{R'} (1-\mu _C (z_i))^{1-R'}\right) ^\frac{1}{R'}\\ +\left( (\mu _{P\cap Q} (z_i))^{S'}(\mu _C (z_i))^{1-S'}+(1-\mu _{P\cap Q} (z_i))^{S'} (1-\mu _C (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _{P\cap Q} (z_i))^{R'}(\mu _C (z_i))^{1-R'}+(1-\mu _{P \cap Q} (z_i))^{R'}(1-\mu _C (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) . \end{aligned}$$

(8.10)

Using (8.3) and (8.4) and simplifying, we get

$$\begin{aligned}&I_{R'}^{S'} (P\cup Q, C)+I_{R'}^{S'} (P\cap Q, C) \nonumber \\&=\frac{R'\times S'}{n (S'-R')}\sum _{i=1}^n \left( \begin{array}{c} \left( (\mu _P (z_i))^{S'}(\mu _C (z_i))^{1-S'}+(1-\mu _P (z_i))^{S'}(1-\mu _C (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _P (z_i))^{R'}(\mu _C (z_i))^{1-R'}+(1-\mu _P (z_i))^{R'}(1-\mu _C (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) \nonumber \\&\quad +\frac{R'\times S'}{n (S'-R')}\sum _{i=1}^n \left( \begin{array}{c} \left( (\mu _P (z_i))^{S'}(\mu _C (z_i))^{1-S'}+(1-\mu _P (z_i))^{S'}(1-\mu _C (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _P (z_i))^{R'}(\mu _C (z_i))^{1-R'}+(1-\mu _P (z_i))^{R'}(1-\mu _C (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) , \end{aligned}$$

(8.11)

$=I_{R'}^{S'} (P, C)+I_{R'}^{S'} (Q, C)$. Thus proved.

(iii) We have to prove that $I_{R'}^{S'}\overline{(P\cup Q},\overline{P\cap Q)}=I_{R'}^{S'}(\bar{P}\cap \bar{Q}, \bar{P}\cup \bar{Q})$.

To prove this, consider

$$\begin{aligned}&I_{R'}^{S'}(\overline{P \cup Q}, \overline{P \cap Q}) \qquad \qquad \qquad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \nonumber \\&\quad =\frac{R'\times S'}{n (S'-R')}\sum _{i=1}^n \left( \begin{array}{c} \left( (\mu _{\overline{P \cup Q}} (z_i))^{S'}(\mu _{\overline{P \cap Q}} (z_i))^{1-S'}+(1-\mu _{\overline{P \cup Q}} (z_i))^{S'} (1-\mu _{\overline{P\cap Q}} (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _{\overline{P \cup Q}} (z_i))^{R'}(\mu _{\overline{P \cap Q}} (z_i))^{1-R'}+(1-\mu _{\overline{P \cup Q}} (z_i))^{R'}(1-\mu _{\overline{P \cap Q}} (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) . \end{aligned}$$

(8.12)

Using definition (3.2), we get

$$\begin{aligned} =\frac{R'\times S'}{n (S'-R')}\sum _{i=1}^n \left( \begin{array}{c} \left( (1-\mu _{P\cup Q} (z_i))^{S'}(1-\mu _{P \cap Q} (z_i))^{1-S'}+(\mu _{P \cup Q} (z_i))^{S'}(\mu _{P\cap Q} (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (1-\mu _{P\cup Q} (z_i))^{R'}(1-\mu _{P \cap Q} (z_i))^{1-R'}+(\mu _{P \cup Q} (z_i))^{R'}(\mu _{P\cap Q} (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) . \end{aligned}$$

(8.13)

Bifurcating X into $X_1$ and $X_2$ and applying (8.3) and (8.4), we get

$$\begin{aligned}&=\frac{R'\times S'}{n (S'-R')}\sum _{X_1} \left( \begin{array}{c} \left( (1-\mu _{P} (z_i))^{S'} (1-\mu _Q (z_i))^{1-S'}+(\mu _P (z_i))^{S'}(\mu _{Q} (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (1-\mu _{P} (z_i))^{R'}(1-\mu _{Q} (z_i))^{1-R'}+(\mu _P (z_i))^{R'}(\mu _Q (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) \nonumber \\&\quad +\frac{R'\times S'}{n (S'-R')}\sum _{X_2} \left( \begin{array}{c} \left( (1-\mu _{Q} (z_i))^{S'}(1-\mu _{P} (z_i))^{1-S'}+(\mu _{Q} (z_i))^{S'}(\mu _{P} (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (1-\mu _{Q} (z_i))^{R'}(1-\mu _{P} (z_i))^{1-R'}+(\mu _{Q} (z_i))^{R'}(\mu _{P} (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) . \end{aligned}$$

(8.14)

Now consider,

$$\begin{aligned}&I_{R'}^{S'}(\bar{P}\cap \bar{Q}, \bar{P}\cup \bar{Q}) \nonumber \\&\quad =\frac{R'\times S'}{n (S'-R')}\sum _{i=1}^n \left( \begin{array}{c} \left( (\mu _{\bar{P}\cap \bar{Q}} (z_i))^{S'}(\mu _{\bar{P}\cup \bar{Q}} (z_i))^{1-S'}+(1-\mu _{\bar{P}\cap \bar{Q}} (z_i))^{S'} (1-\mu _{\bar{P}\cup \bar{Q}} (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _{\bar{P}\cap \bar{Q}} (z_i))^{R'}(\mu _{\bar{P}\cup \bar{Q}} (z_i))^{1-R'}+(1-\mu _{\bar{P}\cap \bar{Q}}(z_i))^{R'}(1-\mu _{\bar{P}\cup \bar{Q}} (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) . \end{aligned}$$

(8.15)

Dividing X into $X_1$ and $X_2$ and simplifying using (8.3) and (8.4), we get

$$\begin{aligned}&=\frac{R'\times S'}{n (S'-R')}\sum _{X_1} \left( \begin{array}{c} \left( (\mu _{\bar{P}} (z_i))^{S'}(\mu _{\bar{Q}} (z_i))^{1-S'}+(1-\mu _{\bar{P}}(z_i))^{S'}(1-\mu _{\bar{Q}} (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _{\bar{P}} (z_i))^{R'}(\mu _{\bar{Q}} (z_i))^{1-R'}+(1-\mu _{\bar{P}} (z_i))^{R'} (1-\mu _{\bar{Q}} (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) \nonumber \\&\quad +\frac{R'\times S'}{n (S'-R')}\sum _{X_2} \left( \begin{array}{c} \left( (\mu _{\bar{Q}} (z_i))^{S'}(\mu _{\bar{P}} (z_i))^{1-S'}+(1-\mu _{\bar{Q}} (z_i))^{S'}(1-\mu _{\bar{P}}(z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _{\bar{Q}} (z_i))^{R'}(\mu _{\bar{P}} (z_i))^{1-R'}+(1-\mu _{\bar{Q}} (z_i))^{R'}(1-\mu _{\bar{P}} (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) . \end{aligned}$$

(8.16)

Using definition (3.2), we get

$$\begin{aligned}&=\frac{R'\times S'}{n (S'-R')}\sum _{X_1} \left( \begin{array}{c} \left( (1-\mu _{P} (z_i))^{S'}(1-\mu _{Q} (z_i))^{1-S'}+(\mu _{P} (z_i))^{S'}(\mu _{Q} (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (1-\mu _{P} (z_i))^{R'} (1-\mu _{Q} (z_i))^{1-R'}+(\mu _{P} (z_i))^{R'}(\mu _{Q} (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) \nonumber \\&\quad +\frac{R'\times S'}{n (S'-R')}\sum _{X_2} \left( \begin{array}{c} \left( (1-\mu _{Q} (z_i))^{S'} (1-\mu _{P} (z_i))^{1-S'}+(\mu _{Q} (z_i))^{S'}(\mu _{P} (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (1-\mu _{Q} (z_i))^{R'} (1-\mu _{P} (z_i))^{1-R'}+(\mu _{Q} (z_i))^{R'}(\mu _{P} (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) , \end{aligned}$$

(8.17)

$=I_{R'}^{S'}(\overline{P \cup Q}, \overline{P \cap Q})$.

Therefore, (iii) holds.

(iv) Consider

$$\begin{aligned}&I_{R'}^{S'} (P, \bar{P}) \nonumber \\&\quad =\frac{R'\times S'}{n (S'-R')}\sum _{i=1}^n \left( \begin{array}{c} \left( (\mu _P (z_i))^{S'}(\mu _{\bar{P}} (z_i))^{1-S'}+(1-\mu _P (z_i))^{S'}(1-\mu _{\bar{P}} (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _P (z_i))^{R'}(\mu _{\bar{P}} (z_i))^{1-R'}+(1-\mu _P (z_i))^{R'}(1-\mu _{\bar{P}} (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) . \end{aligned}$$

(8.18)

Using notations in definition (3.2), we get

$$\begin{aligned}&I_{R'}^{S'} (P, \bar{P}) \nonumber \\&\quad =\frac{R'\times S'}{n (S'-R')}\sum _{i=1}^n \left( \begin{array}{c} \left( (1-\mu _{\bar{P}} (z_i))^{S'} (1-\mu _P (z_i))^{1-S'}+(\mu _{\bar{P}} (z_i))^{S'}(\mu _P (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (1-\mu _{\bar{P}} (z_i))^{R'} (1-\mu _{P} (z_i))^{1-R'}+(\mu _{\bar{P}} (z_i))^{R'}(\mu _P (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) , \end{aligned}$$

(8.19)

$=I_{R'}^{S'}(\bar{P}, P)$.

(v) Consider

$$\begin{aligned}&I_{R'}^{S'}(\bar{P}, \bar{Q}) \nonumber \\&\quad =\frac{R'\times S'}{n (S'-R')}\sum _{i=1}^n \left( \begin{array}{c} \left( (\mu _{\bar{P}} (z_i))^{S'}(\mu _{\bar{Q}} (z_i))^{1-S'}+(1-\mu _{\bar{P}}(z_i))^{S'} (1-\mu _{\bar{Q}} (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _{\bar{P}} (z_i))^{R'}(\mu _{\bar{Q}} (z_i))^{1-R'}+(1-\mu _{\bar{P}} (z_i))^{R'}(1-\mu _{\bar{Q}} (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) . \end{aligned}$$

(8.20)

Using notations in definition (3.2) and simplifying, we get

$$\begin{aligned} =\frac{R'\times S'}{n (S'-R')}\sum _{i=1}^n \left( \begin{array}{c} \left( (\mu _P (z_i))^{S'}(\mu _Q (z_i))^{1-S'}+(1-\mu _{P} (z_i))^{S'}(1-\mu _Q (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _P (z_i))^{R'}(\mu _Q (z_i))^{1-R'}+(1-\mu _P (z_i))^{R'}(1-\mu _Q (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) , \end{aligned}$$

(8.21)

$=I_{R'}^{S'} (P, Q)$.

(vi) Consider

$$\begin{aligned}&I_{R'}^{S'}(P, \bar{Q}) \nonumber \\&\quad =\frac{R'\times S'}{n (S'-R')}\sum _{i=1}^n \left( \begin{array}{c} \left( (\mu _P (z_i))^{S'}(\mu _{\bar{Q}} (z_i))^{1-S'}+(1-\mu _{P}(z_i))^{S'}(1-\mu _{\bar{Q}} (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (\mu _P (z_i))^{R'}(\mu _{\bar{Q}} (z_i))^{1-R'}+(1-\mu _P (z_i))^{R'} (1-\mu _{\bar{Q}} (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) . \end{aligned}$$

(8.22)

Using notations in definition (3.2) and simplifying, we get

$$\begin{aligned} =\frac{R'\times S'}{n (S'-R')}\sum _{i=1}^n \left( \begin{array}{c} \left( (1-\mu _{\bar{P}} (z_i))^{S'} (1-\mu _{Q} (z_i))^{1-S'}+(\mu _{\bar{P}} (z_i))^{S'}(\mu _Q (z_i))^{1-S'}\right) ^\frac{1}{S'}\\ -\left( (1-\mu _{\bar{P}}(z_i))^{R'}(1-\mu _Q (z_i))^{1-R'}+(\mu _{\bar{P}} (z_i))^{R'}(\mu _Q (z_i))^{1-R'}\right) ^\frac{1}{R'} \end{array} \right) , \end{aligned}$$

(8.23)

$=I_{R'}^{S'}(\bar{P}, Q)$.

(vii) It follows directly from (b) and (c).