A picture fuzzy similarity measure based on direct operations and novel multi-attribute decision-making method

Abstract

Picture fuzzy set (PFS) is a direct generalization of the fuzzy sets (FSs) and intuitionistic fuzzy sets (IFSs) and is quite powerful in modelling the situations that involve more answers of the type yes, no, abstain, and refuse. In this study, we introduce a novel picture fuzzy (PF) similarity measure on the basis of direct operation on the function of membership, the function of non-membership, the function of neutrality, the function of refusal, and the upper bound of the function of membership of two PFSs instead on the basis of distance measure or the association between the functions of membership, non-membership, and neutrality. The comparison of the proposed PF similarity measure with the existing PF similarity measures reveals that it does not give unreasonable results and also overcomes the drawbacks of the existing PF similarity measures. The application of the proposed measure in pattern recognition is also discussed. Moreover, we also introduce a new multi-attribute decision-making (MADM) method using the proposed PF similarity measure that overcomes a major drawback of the technique for order preference by similarity to ideal solution (TOPSIS). Finally, we contrast the performance of the proposed MADM method with several existing MADM methods in the PF environment.

Appendix: proof of Theorem 1

Proof

To show that $S_{GS} \left( {G, H} \right)$ is a PF similarity measure, we show that it satisfies the four properties given in Definition 5.

(1) For each $q, r \in \left[ {0, + \infty } \right]$, we have $\sqrt {qr} \le \frac{q + r}{2}$. So, for $0 \le m_{G} \left( {t_{k} } \right) \le 1$, $0 \le n_{G} \left( {t_{k} } \right) \le 1$, $0 \le h_{G} \left( {t_{k} } \right) \le 1,$ $0 \le e_{G} \left( {t_{k} } \right) \le 1$, $0 \le 1 - m_{G} \left( {t_{k} } \right) - n_{G} \left( {t_{k} } \right) \le 1$, $0 \le 1 - m_{G} \left( {t_{k} } \right) - h_{G} \left( {t_{k} } \right) \le 1$ and $0 \le 1 - n_{G} \left( {t_{k} } \right) - h_{G} \left( {t_{k} } \right) \le 1$ we get

$$\begin{aligned} 0 & \le 3\sqrt {m_{G} \left( {t_{k} } \right)m_{H} \left( {t_{k} } \right)} + 3\sqrt {n_{G} \left( {t_{k} } \right)n_{H} \left( {t_{k} } \right)} + 3\sqrt {h_{G} \left( {t_{k} } \right)h_{H} \left( {t_{k} } \right)} \\ & \quad + \sqrt {e_{G} \left( {t_{k} } \right)e_{H} \left( {t_{k} } \right)} + \sqrt {\left( {1 - m_{G} \left( {t_{k} } \right) - n_{G} \left( {t_{k} } \right)} \right)\left( {1 - m_{H} \left( {t_{k} } \right) - n_{H} \left( {t_{k} } \right)} \right)} \\ & \quad + \sqrt {\left( {1 - m_{G} \left( {t_{k} } \right) - h_{G} \left( {t_{k} } \right)} \right)\left( {1 - m_{H} \left( {t_{k} } \right) - h_{H} \left( {t_{k} } \right)} \right)} \\ & \quad + \sqrt {\left( {1 - n_{G} \left( {t_{k} } \right) - h_{G} \left( {t_{k} } \right)} \right)\left( {1 - n_{H} \left( {t_{k} } \right) - h_{H} \left( {t_{k} } \right)} \right)} \\ & \le \frac{3}{2}\left( {m_{G} \left( {t_{k} } \right) + m_{H} \left( {t_{k} } \right)} \right) + \frac{3}{2}\left( {n_{G} \left( {t_{k} } \right) + n_{H} \left( {t_{k} } \right)} \right) \\ & \quad + \frac{3}{2}\left( {h_{G} \left( {t_{k} } \right) + h_{H} \left( {t_{k} } \right)} \right) + \frac{{e_{G} \left( {t_{k} } \right) + e_{H} \left( {t_{k} } \right)}}{2} + \frac{{\left( {1 - m_{G} \left( {t_{k} } \right) - n_{G} \left( {t_{k} } \right)} \right) + \left( {1 - m_{H} \left( {t_{k} } \right) - n_{H} \left( {t_{k} } \right)} \right)}}{2} \\ & \quad + \frac{{\left( {1 - m_{G} \left( {t_{k} } \right) - h_{G} \left( {t_{k} } \right)} \right) + \left( {1 - m_{H} \left( {t_{k} } \right) - h_{H} \left( {t_{k} } \right)} \right)}}{2} \\ & \quad + \frac{{\left( {1 - n_{G} \left( {t_{k} } \right) - h_{G} \left( {t_{k} } \right)} \right) + \left( {1 - n_{H} \left( {t_{k} } \right) - h_{H} \left( {t_{k} } \right)} \right)}}{2} \\ & = 3 + \frac{{m_{G} \left( {t_{k} } \right) + n_{G} \left( {t_{k} } \right) + h_{G} \left( {t_{k} } \right) + e_{G} \left( {t_{k} } \right)}}{2} + \frac{{m_{H} \left( {t_{k} } \right) + n_{H} \left( {t_{k} } \right) + h_{H} \left( {t_{k} } \right) + e_{H} \left( {t_{k} } \right)}}{2} \\ & = 3 + \frac{1}{2} + \frac{1}{2} = 4. \\ \end{aligned}$$

So,

$$0 \le \frac{1}{4l}\mathop \sum_{k = 1}^{l} \left[ {\begin{array}{*{20}c} {3\sqrt {m_{G} \left( {t_{k} } \right)m_{H} \left( {t_{k} } \right)} + 3\sqrt {n_{G} \left( {t_{k} } \right)n_{H} \left( {t_{k} } \right)} } \\ { + 3\sqrt {h_{G} \left( {t_{k} } \right)h_{H} \left( {t_{k} } \right)} + \sqrt {e_{G} \left( {t_{k} } \right)e_{H} \left( {t_{k} } \right)} } \\ { + \sqrt {\begin{array}{*{20}c} {\left( {1 - m_{G} \left( {t_{k} } \right) - n_{G} \left( {t_{k} } \right)} \right)} \\ { \times \left( {1 - m_{H} \left( {t_{k} } \right) - n_{H} \left( {t_{k} } \right)} \right)} \\ \end{array} } } \\ { + \sqrt {\begin{array}{*{20}c} {\left( {1 - m_{G} \left( {t_{k} } \right) - h_{G} \left( {t_{k} } \right)} \right)} \\ { \times \left( {1 - m_{H} \left( {t_{k} } \right) - h_{H} \left( {t_{k} } \right)} \right)} \\ \end{array} } } \\ { + \sqrt {\begin{array}{*{20}c} {\left( {1 - n_{G} \left( {t_{k} } \right) - h_{G} \left( {t_{k} } \right)} \right)} \\ { \times \left( {1 - n_{H} \left( {t_{k} } \right) - h_{H} \left( {t_{k} } \right)} \right)} \\ \end{array} } } \\ \end{array} } \right] \le 1$$

Therefore, we have $0 \le S_{GS} \left( {G, H} \right) \le 1$.

(2) $S_{GS} \left( {G, H} \right) = S_{GS} \left( {H, G} \right)$ is obvious due to the cumutativeness of the expression of $S_{GS} \left( {G, H} \right)$.

(3) Since $\sqrt {qr}$ achieves its maximum value $\frac{q + r}{2}$ when $q = r$. So, we have

$$\begin{aligned} & S_{GS} \left( {G, H} \right) = 1 \\ & \quad \Leftrightarrow \left[ {\begin{array}{*{20}c} {3\sqrt {m_{G} \left( {t_{k} } \right)m_{H} \left( {t_{k} } \right)} + 3\sqrt {n_{G} \left( {t_{k} } \right)n_{H} \left( {t_{k} } \right)} } \\ { + 3\sqrt {h_{G} \left( {t_{k} } \right)h_{H} \left( {t_{k} } \right)} + \sqrt {e_{G} \left( {t_{k} } \right)e_{H} \left( {t_{k} } \right)} } \\ { + \sqrt {\begin{array}{*{20}c} {\left( {1 - m_{G} \left( {t_{k} } \right) - n_{G} \left( {t_{k} } \right)} \right)} \\ { \times \left( {1 - m_{H} \left( {t_{k} } \right) - n_{H} \left( {t_{k} } \right)} \right)} \\ \end{array} } } \\ { + \sqrt {\begin{array}{*{20}c} {\left( {1 - m_{G} \left( {t_{k} } \right) - h_{G} \left( {t_{k} } \right)} \right)} \\ { \times \left( {1 - m_{H} \left( {t_{k} } \right) - h_{H} \left( {t_{k} } \right)} \right)} \\ \end{array} } } \\ { + \sqrt {\begin{array}{*{20}c} {\left( {1 - n_{G} \left( {t_{k} } \right) - h_{G} \left( {t_{k} } \right)} \right)} \\ { \times \left( {1 - n_{H} \left( {t_{k} } \right) - h_{H} \left( {t_{k} } \right)} \right)} \\ \end{array} } } \\ \end{array} } \right] = 4 \\ & \quad \Leftrightarrow m_{G} \left( {t_{k} } \right) = m_{H} \left( {t_{k} } \right), n_{G} \left( {t_{k} } \right) = n_{H} \left( {t_{k} } \right), h_{G} \left( {t_{k} } \right) \\ & \quad = h_{H} \left( {t_{k} } \right), e_{G} \left( {t_{k} } \right) = e_{H} \left( {t_{k} } \right),\left( {1 - m_{G} \left( {t_{k} } \right) - n_{G} \left( {t_{k} } \right)} \right) \\ & \quad = \left( {1 - m_{H} \left( {t_{k} } \right) - n_{H} \left( {t_{k} } \right)} \right),\left( {1 - m_{G} \left( {t_{k} } \right) - h_{G} \left( {t_{k} } \right)} \right) \\ & \quad = \left( {1 - m_{H} \left( {t_{k} } \right) - h_{H} \left( {t_{k} } \right)} \right),\left( {1 - n_{G} \left( {t_{k} } \right) - h_{G} \left( {t_{k} } \right)} \right) \\ & \quad = \left( {1 - n_{H} \left( {t_{k} } \right) - h_{H} \left( {t_{k} } \right)} \right) \Leftrightarrow G = H. \\ \end{aligned}$$

So, $S_{GS} \left( {G, H} \right) = 1$ if and only if $G = H$.

(4) Let $I \in PFS\left( U \right)$ be another PFS such that $G \subseteq H \subseteq I$. Then we have, $m_{G} \left( {t_{k} } \right) \le m_{H} \left( {t_{k} } \right) \le m_{I} \left( {t_{k} } \right)$, $n_{I} \left( {t_{k} } \right) \le n_{H} \left( {t_{k} } \right) \le n_{G} \left( {t_{k} } \right)$ and $h_{G} \left( {t_{k} } \right) \le h_{H} \left( {t_{k} } \right) \le h_{I} \left( {t_{k} } \right)$.

For $a, b, c \in \left[ {0, 1} \right], a + b + c \le 1$, we define a function $f$ as:

$$\begin{aligned} & f\left( {q, r, s} \right) = 3\sqrt {aq} + 3\sqrt {br} + 3\sqrt {cs} \\ & \quad + \sqrt {\left( {1 - a - b - c} \right)\left( {1 - q - r - s} \right)} \\ & \quad + \sqrt {\left( {1 - a - b} \right)\left( {1 - q - r} \right)} \\ & \quad + \sqrt {\left( {1 - a - c} \right)\left( {1 - q - s} \right)} + \sqrt {\left( {1 - b - c} \right)\left( {1 - r - s} \right)} \\ \end{aligned}$$

where $q, r, s \in \left[ {0, 1} \right], q + r + s \in \left[ {0, 1} \right]$.

Now,

$$\begin{aligned} \frac{\partial f}{{\partial q}} & = \frac{3\sqrt a }{{2\sqrt q }} - \frac{{\sqrt {1 - a - b - c} }}{{2\sqrt {1 - q - r - s} }} - \frac{{\sqrt {1 - a - b} }}{{2\sqrt {1 - q - r} }} - \frac{{\sqrt {1 - a - c} }}{{2\sqrt {1 - q - s} }} \\ & = \left( {\frac{\sqrt a }{{2\sqrt q }} + \frac{\sqrt a }{{2\sqrt q }} + \frac{\sqrt a }{{2\sqrt q }}} \right) - \frac{{\sqrt {1 - a - b - c} }}{{2\sqrt {1 - q - r - s} }} \\ & \quad - \frac{{\sqrt {1 - a - b} }}{{2\sqrt {1 - q - r} }} - \frac{{\sqrt {1 - a - c} }}{{2\sqrt {1 - q - s} }} \\ & = \left( {\frac{\sqrt a }{{2\sqrt q }} - \frac{{\sqrt {1 - a - b - c} }}{{2\sqrt {1 - q - r - s} }}} \right) + \left( {\frac{\sqrt a }{{2\sqrt q }} - \frac{{\sqrt {1 - a - b} }}{{2\sqrt {1 - q - r} }}} \right) \\ & \quad + \left( {\frac{\sqrt a }{{2\sqrt q }} - \frac{{\sqrt {1 - a - c} }}{{2\sqrt {1 - q - s} }}} \right) \\ & = \frac{1}{2}\left( {\frac{{\sqrt {a\left( {1 - q - r - s} \right)} - \sqrt {q\left( {1 - a - b - c} \right)} }}{{\sqrt {q\left( {1 - q - r - s} \right)} }}} \right) \\ & \quad + \frac{1}{2}\left( {\frac{{\sqrt {a\left( {1 - q - r} \right)} - \sqrt {q\left( {1 - a - b} \right)} }}{{\sqrt {q\left( {1 - q - r} \right)} }}} \right) \\ & \quad + \frac{1}{2}\left( {\frac{{\sqrt {a\left( {1 - q - s} \right)} - \sqrt {q\left( {1 - a - c} \right)} }}{{\sqrt {q\left( {1 - q - s} \right)} }}} \right) \\ & = \frac{1}{2}\left( {\frac{{a\left( {1 - q - r - s} \right) - q\left( {1 - a - b - c} \right)}}{{\sqrt {q\left( {1 - q - r - s} \right)} \left( {\sqrt {a\left( {1 - q - r - s} \right)} + \sqrt {q\left( {1 - a - b - c} \right)} } \right)}}} \right) \\ & \quad + \frac{1}{2}\left( {\frac{{a\left( {1 - q - r} \right) - q\left( {1 - a - b} \right)}}{{\sqrt {q\left( {1 - q - r} \right)} \left( {\sqrt {a\left( {1 - q - r} \right)} + \sqrt {q\left( {1 - a - b} \right)} } \right)}}} \right) \\ & \quad + \frac{1}{2}\left( {\frac{{a\left( {1 - q - s} \right) - q\left( {1 - a - c} \right)}}{{\sqrt {q\left( {1 - q - s} \right)} \left( {\sqrt {a\left( {1 - q - s} \right)} + \sqrt {q\left( {1 - a - c} \right)} } \right)}}} \right) \\ & = \frac{1}{2}\left( {\frac{{a\left( {1 - r - s} \right) - q\left( {1 - b - c} \right)}}{{\sqrt {q\left( {1 - q - r - s} \right)} \left( {\sqrt {a\left( {1 - q - r - s} \right)} + \sqrt {q\left( {1 - a - b - c} \right)} } \right)}}} \right) \\ & \quad + \frac{1}{2}\left( {\frac{{a\left( {1 - r} \right) - q\left( {1 - b} \right)}}{{\sqrt {q\left( {1 - q - r} \right)} \left( {\sqrt {a\left( {1 - q - r} \right)} + \sqrt {q\left( {1 - a - b} \right)} } \right)}}} \right) \\ & \quad + \frac{1}{2}\left( {\frac{{a\left( {1 - s} \right) - q\left( {1 - c} \right)}}{{\sqrt {q\left( {1 - q - s} \right)} \left( {\sqrt {a\left( {1 - q - s} \right)} + \sqrt {q\left( {1 - a - c} \right)} } \right)}}} \right) \\ & = \frac{1}{2}\left( {\frac{{\left( {a - q} \right)\left( {1 - b - c} \right)}}{{\sqrt {q\left( {1 - q - r - s} \right)} \left( {\sqrt {a\left( {1 - q - r - s} \right)} + \sqrt {q\left( {1 - a - b - c} \right)} } \right)}}} \right) \\ & \quad + \frac{1}{2}\left( {\frac{{\left( {a - q} \right)\left( {1 - b} \right)}}{{\sqrt {q\left( {1 - q - r} \right)} \left( {\sqrt {a\left( {1 - q - r} \right)} + \sqrt {q\left( {1 - a - b} \right)} } \right)}}} \right) \\ & \quad + \frac{1}{2}\left( {\frac{{\left( {a - q} \right)\left( {1 - c} \right)}}{{\sqrt {q\left( {1 - q - s} \right)} \left( {\sqrt {a\left( {1 - q - s} \right)} + \sqrt {q\left( {1 - a - c} \right)} } \right)}}} \right)\;{\text{at}}\;r = b \\ & \quad {\text{and}}\;s = c. \\ \end{aligned}$$

$$\begin{aligned} & \Rightarrow \frac{\partial f}{{\partial q}} = \frac{1}{2}\left( {\frac{{\left( {a - q} \right)\left( {1 - b - c} \right)}}{{\sqrt {q\left( {1 - q - r - s} \right)} \left( {\sqrt {a\left( {1 - q - r - s} \right)} + \sqrt {q\left( {1 - a - b - c} \right)} } \right)}}} \right) \\ & \quad + \frac{1}{2}\left( {\frac{{\left( {a - q} \right)\left( {1 - b} \right)}}{{\sqrt {q\left( {1 - q - r} \right)} \left( {\sqrt {a\left( {1 - q - r} \right)} + \sqrt {q\left( {1 - a - b} \right)} } \right)}}} \right) \\ & \quad + \frac{1}{2}\left( {\frac{{\left( {a - q} \right)\left( {1 - c} \right)}}{{\sqrt {q\left( {1 - q - s} \right)} \left( {\sqrt {a\left( {1 - q - s} \right)} + \sqrt {q\left( {1 - a - c} \right)} } \right)}}} \right) \\ \end{aligned}$$

Similarly, we have

$$\begin{aligned} \frac{\partial f}{{\partial r}} & = \frac{1}{2}\left( {\frac{{\left( {b - r} \right)\left( {1 - a - c} \right)}}{{\sqrt {r\left( {1 - q - r - s} \right)} \left( {\sqrt {b\left( {1 - q - r - s} \right)} + \sqrt {r\left( {1 - a - b - c} \right)} } \right)}}} \right) \\ & \quad + \frac{1}{2}\left( {\frac{{\left( {b - r} \right)\left( {1 - a} \right)}}{{\sqrt {r\left( {1 - q - r} \right)} \left( {\sqrt {b\left( {1 - q - r} \right)} + \sqrt {r\left( {1 - a - b} \right)} } \right)}}} \right) \\ & \quad + \frac{1}{2}\left( {\frac{{\left( {b - r} \right)\left( {1 - c} \right)}}{{\sqrt {r\left( {1 - r - s} \right)} \left( {\sqrt {b\left( {1 - r - s} \right)} + \sqrt {r\left( {1 - b - c} \right)} } \right)}}} \right) \\ \frac{\partial f}{{\partial s}} & = \frac{1}{2}\left( {\frac{{\left( {c - s} \right)\left( {1 - a - b} \right)}}{{\sqrt {s\left( {1 - q - r - s} \right)} \left( {\sqrt {c\left( {1 - q - r - s} \right)} + \sqrt {s\left( {1 - a - b - c} \right)} } \right)}}} \right) \\ & \quad + \frac{1}{2}\left( {\frac{{\left( {c - s} \right)\left( {1 - a} \right)}}{{\sqrt {s\left( {1 - q - s} \right)} \left( {\sqrt {c\left( {1 - q - s} \right)} + \sqrt {s\left( {1 - a - c} \right)} } \right)}}} \right) \\ & \quad + \frac{1}{2}\left( {\frac{{\left( {c - s} \right)\left( {1 - b} \right)}}{{\sqrt {s\left( {1 - r - s} \right)} \left( {\sqrt {c\left( {1 - r - s} \right)} + \sqrt {s\left( {1 - b - c} \right)} } \right)}}} \right). \\ \end{aligned}$$

For $a \le q \le 1, b \le 1, c \le 1$, we have $\frac{\partial f}{{\partial q}} \le 0$, so $f$ is a decreasing function of $q$ for $q \ge a$. For $0 \le q \le a, b \le 1, c \le 1$, we have $\frac{\partial f}{{\partial q}} \ge 0$, so $f$ is an increasing function of $q$ for $q \le a$. Similarly, for $b \le r \le 1, a \le 1, c \le 1$, we have $\frac{\partial f}{{\partial r}} \le 0$ and for $0 \le r \le b, a \le 1, c \le 1$, we have $\frac{\partial f}{{\partial r}} \ge 0$. This means that $f$ is a decreasing function of $r$ when $r \ge b$ and an increasing function of $r$ when $r \le b$. Also, for $c \le s \le 1, a \le 1, b \le 1$, we have $\frac{\partial f}{{\partial s}} \le 0$ and for $0 \le s \le c, a \le 1, b \le 1$, we have $\frac{\partial f}{{\partial s}} \ge 0$. This means that $f$ is a decreasing function of $s$ when $s \ge c$ and an increasing function of $s$ when $s \le c$.

Now, for $a = m_{G} \left( {t_{k} } \right), b = n_{G} \left( {t_{k} } \right), c = h_{G} \left( {t_{k} } \right)$ and two triplets $\left( {m_{H} \left( {t_{k} } \right), n_{H} \left( {t_{k} } \right), h_{H} \left( {t_{k} } \right)} \right)$, $\left( {m_{I} \left( {t_{k} } \right), n_{I} \left( {t_{k} } \right), h_{I} \left( {t_{k} } \right)} \right)$ satisfying $a = m_{G} \left( {t_{k} } \right) \le m_{H} \left( {t_{k} } \right) \le m_{I} \left( {t_{k} } \right), n_{I} \left( {t_{k} } \right) \le n_{H} \left( {t_{k} } \right) \le n_{G} \left( {t_{k} } \right) = b$ and $c = h_{G} \left( {t_{k} } \right) \le h_{H} \left( {t_{k} } \right) \le h_{I} \left( {t_{k} } \right)$, we have

$$\begin{aligned} & f\left( {m_{I} \left( {t_{k} } \right), n_{I} \left( {t_{k} } \right), h_{I} \left( {t_{k} } \right)} \right) \le f\left( {m_{H} \left( {t_{k} } \right), n_{I} \left( {t_{k} } \right), h_{I} \left( {t_{k} } \right)} \right) \\ & \quad \le f\left( {m_{H} \left( {t_{k} } \right), n_{H} \left( {t_{k} } \right), h_{I} \left( {t_{k} } \right)} \right) \\ \end{aligned}$$

and

$$\begin{aligned} & f\left( {m_{I} \left( {t_{k} } \right), n_{I} \left( {t_{k} } \right), h_{I} \left( {t_{k} } \right)} \right) \le f\left( {m_{I} \left( {t_{k} } \right), n_{I} \left( {t_{k} } \right), h_{H} \left( {t_{k} } \right)} \right) \\ & \quad \le f\left( {m_{I} \left( {t_{k} } \right), n_{H} \left( {t_{k} } \right), h_{H} \left( {t_{k} } \right)} \right). \\ \end{aligned}$$

So,

$$\begin{aligned} & \left[ {\begin{array}{*{20}c} {3\sqrt {m_{G} \left( {t_{k} } \right)m_{I} \left( {t_{k} } \right)} + 3\sqrt {n_{G} \left( {t_{k} } \right)n_{I} \left( {t_{k} } \right)} } \\ { + 3\sqrt {h_{G} \left( {t_{k} } \right)h_{I} \left( {t_{k} } \right)} + \sqrt {e_{G} \left( {t_{k} } \right)e_{I} \left( {t_{k} } \right)} } \\ { + \sqrt {\begin{array}{*{20}c} {\left( {1 - m_{G} \left( {t_{k} } \right) - n_{G} \left( {t_{k} } \right)} \right)} \\ { \times \left( {1 - m_{I} \left( {t_{k} } \right) - n_{I} \left( {t_{k} } \right)} \right)} \\ \end{array} } } \\ { + \sqrt {\begin{array}{*{20}c} {\left( {1 - m_{G} \left( {t_{k} } \right) - h_{G} \left( {t_{k} } \right)} \right)} \\ { \times \left( {1 - m_{I} \left( {t_{k} } \right) - h_{I} \left( {t_{k} } \right)} \right)} \\ \end{array} } } \\ { + \sqrt {\begin{array}{*{20}c} {\left( {1 - n_{G} \left( {t_{k} } \right) - h_{G} \left( {t_{k} } \right)} \right)} \\ { \times \left( {1 - n_{I} \left( {t_{k} } \right) - h_{I} \left( {t_{k} } \right)} \right)} \\ \end{array} } } \\ \end{array} } \right] \\ & \quad \le \left[ {\begin{array}{*{20}c} {3\sqrt {m_{G} \left( {t_{k} } \right)m_{H} \left( {t_{k} } \right)} + 3\sqrt {n_{G} \left( {t_{k} } \right)n_{H} \left( {t_{k} } \right)} } \\ { + 3\sqrt {h_{G} \left( {t_{k} } \right)h_{H} \left( {t_{k} } \right)} + \sqrt {e_{G} \left( {t_{k} } \right)e_{H} \left( {t_{k} } \right)} } \\ { + \sqrt {\begin{array}{*{20}c} {\left( {1 - m_{G} \left( {t_{k} } \right) - n_{G} \left( {t_{k} } \right)} \right)} \\ { \times \left( {1 - m_{H} \left( {t_{k} } \right) - n_{H} \left( {t_{k} } \right)} \right)} \\ \end{array} } } \\ { + \sqrt {\begin{array}{*{20}c} {\left( {1 - m_{G} \left( {t_{k} } \right) - h_{G} \left( {t_{k} } \right)} \right)} \\ { \times \left( {1 - m_{H} \left( {t_{k} } \right) - h_{H} \left( {t_{k} } \right)} \right)} \\ \end{array} } } \\ { + \sqrt {\begin{array}{*{20}c} {\left( {1 - n_{G} \left( {t_{k} } \right) - h_{G} \left( {t_{k} } \right)} \right)} \\ { \times \left( {1 - n_{H} \left( {t_{k} } \right) - h_{H} \left( {t_{k} } \right)} \right)} \\ \end{array} } } \\ \end{array} } \right] \\ & \quad \Rightarrow S_{GS} \left( {G, H} \right) \ge S_{GS} \left( {G, I} \right). \\ \end{aligned}$$

Now, if we let $a = m_{I} \left( {t_{k} } \right), b = n_{I} \left( {t_{k} } \right), c = h_{I} \left( {t_{k} } \right)$ and two triplets $\left( {m_{G} \left( {t_{k} } \right), n_{G} \left( {t_{k} } \right), h_{G} \left( {t_{k} } \right)} \right)$, $\left( {m_{H} \left( {t_{k} } \right), n_{H} \left( {t_{k} } \right), h_{H} \left( {t_{k} } \right)} \right)$ satisfying $m_{G} \left( {t_{k} } \right) \le m_{H} \left( {t_{k} } \right) \le m_{I} \left( {t_{k} } \right) = a, b = n_{I} \left( {t_{k} } \right) \le n_{H} \left( {t_{k} } \right) \le n_{G} \left( {t_{k} } \right)$ and $h_{G} \left( {t_{k} } \right) \le h_{H} \left( {t_{k} } \right) \le h_{I} \left( {t_{k} } \right) = c$, we have

$$\begin{aligned} & f\left( {m_{G} \left( {t_{k} } \right), n_{G} \left( {t_{k} } \right), h_{G} \left( {t_{k} } \right)} \right) \le f\left( {m_{H} \left( {t_{k} } \right), n_{G} \left( {t_{k} } \right), h_{G} \left( {t_{k} } \right)} \right) \\ & \quad \le f\left( {m_{H} \left( {t_{k} } \right), n_{H} \left( {t_{k} } \right), h_{G} \left( {t_{k} } \right)} \right) \\ \end{aligned}$$

and

$$\begin{aligned} & f\left( {m_{G} \left( {t_{k} } \right), n_{G} \left( {t_{k} } \right), h_{G} \left( {t_{k} } \right)} \right) \le f\left( {m_{G} \left( {t_{k} } \right), n_{G} \left( {t_{k} } \right), h_{H} \left( {t_{k} } \right)} \right) \\ & \quad \le f\left( {m_{G} \left( {t_{k} } \right), n_{H} \left( {t_{k} } \right), h_{H} \left( {t_{k} } \right)} \right). \\ \end{aligned}$$

So,

$$\begin{aligned} & \left[ {\begin{array}{*{20}c} {3\sqrt {m_{G} \left( {t_{k} } \right)m_{I} \left( {t_{k} } \right)} + 3\sqrt {n_{G} \left( {t_{k} } \right)n_{I} \left( {t_{k} } \right)} } \\ { + 3\sqrt {h_{G} \left( {t_{k} } \right)h_{I} \left( {t_{k} } \right)} + \sqrt {e_{G} \left( {t_{k} } \right)e_{I} \left( {t_{k} } \right)} } \\ { + \sqrt {\begin{array}{*{20}c} {\left( {1 - m_{G} \left( {t_{k} } \right) - n_{G} \left( {t_{k} } \right)} \right)} \\ { \times \left( {1 - m_{I} \left( {t_{k} } \right) - n_{I} \left( {t_{k} } \right)} \right)} \\ \end{array} } } \\ { + \sqrt {\begin{array}{*{20}c} {\left( {1 - m_{G} \left( {t_{k} } \right) - h_{G} \left( {t_{k} } \right)} \right)} \\ { \times \left( {1 - m_{I} \left( {t_{k} } \right) - h_{I} \left( {t_{k} } \right)} \right)} \\ \end{array} } } \\ { + \sqrt {\begin{array}{*{20}c} {\left( {1 - n_{G} \left( {t_{k} } \right) - h_{G} \left( {t_{k} } \right)} \right)} \\ { \times \left( {1 - n_{I} \left( {t_{k} } \right) - h_{I} \left( {t_{k} } \right)} \right)} \\ \end{array} } } \\ \end{array} } \right] \\ & \quad \le \left[ {\begin{array}{*{20}c} {3\sqrt {m_{H} \left( {t_{k} } \right)m_{I} \left( {t_{k} } \right)} + 3\sqrt {n_{H} \left( {t_{k} } \right)n_{I} \left( {t_{k} } \right)} } \\ { + 3\sqrt {h_{H} \left( {t_{k} } \right)h_{I} \left( {t_{k} } \right)} + \sqrt {e_{H} \left( {t_{k} } \right)e_{I} \left( {t_{k} } \right)} } \\ { + \sqrt {\begin{array}{*{20}c} {\left( {1 - m_{H} \left( {t_{k} } \right) - n_{H} \left( {t_{k} } \right)} \right)} \\ { \times \left( {1 - m_{I} \left( {t_{k} } \right) - n_{I} \left( {t_{k} } \right)} \right)} \\ \end{array} } } \\ { + \sqrt {\begin{array}{*{20}c} {\left( {1 - m_{H} \left( {t_{k} } \right) - h_{H} \left( {t_{k} } \right)} \right)} \\ { \times \left( {1 - m_{I} \left( {t_{k} } \right) - h_{I} \left( {t_{k} } \right)} \right)} \\ \end{array} } } \\ { + \sqrt {\begin{array}{*{20}c} {\left( {1 - n_{H} \left( {t_{k} } \right) - h_{H} \left( {t_{k} } \right)} \right)} \\ { \times \left( {1 - n_{I} \left( {t_{k} } \right) - h_{I} \left( {t_{k} } \right)} \right)} \\ \end{array} } } \\ \end{array} } \right] \\ & \quad \Rightarrow S_{GS} \left( {H, I} \right) \ge S_{GS} \left( {G, I} \right). \\ \end{aligned}$$

Thus if $G \subseteq H \subseteq I$, then $S_{GS} \left( {G, I} \right) \le S_{GS} \left( {G, H} \right)$ and $S_{GS} \left( {G, I} \right) \le S_{GS} \left( {H, I} \right)$.

Hence $S_{GS} \left( {G, H} \right)$ is a PF similarity measure.□