**3. Similarity between Two** LMFSS *s* **Based on Set Theoretic Approach**

In this section, we introduce the concept of similarity measure of two LMFSS *s* and further results on similarity measure of two LMFSS *s*. In all the following context, denote *a* ∧ *b* to be the minimum of *a* and *b*, denote *a* ∨ *b* to be the maximum of *a* and *b*.

**Definition 16.** *Let* (<sup>R</sup>, <sup>Γ</sup>),(<sup>Q</sup>, <sup>Γ</sup>) <sup>∈</sup> LMF*J*S S (*U*)*.*

*The similarity measure of* (<sup>R</sup>, <sup>Γ</sup>) *and* (<sup>Q</sup>, <sup>Γ</sup>) *over U is defined as*

$$\mathbf{S}((\vec{\mathcal{R}},\Gamma),(\vec{\mathcal{Q}},\Gamma)) = \frac{\sum\_{j \in J} \mathbf{S}\_j((\mathcal{R},\Gamma),(\vec{\mathcal{Q}},\Gamma))}{|J|}$$

*where*

$$S\_j((\mathcal{R}, \Gamma), (\bar{\mathcal{Q}}, \Gamma)) = \frac{\sum\_{\gamma \in \Gamma} \vee\_{\mathfrak{x} \in \mathcal{U}} \{\mu\_{\bar{\mathcal{R}}(\gamma), j}(\mathfrak{x}) \wedge \mu\_{\bar{\mathcal{Q}}(\gamma), j}(\mathfrak{x})\}}{\sum\_{\gamma \in \Gamma} \vee\_{\mathfrak{x} \in \mathcal{U}} \{\mu\_{\bar{\mathcal{R}}(\gamma), j}(\mathfrak{x}) \vee \mu\_{\bar{\mathcal{Q}}(\gamma), j}(\mathfrak{x})\}}$$

**Example 2.** *Let U*<sup>0</sup> = {*x*1, *x*2, *x*3, *x*4}*. Let J*<sup>0</sup> = {1, 2, 3}*.*

*Let* Γ<sup>0</sup> = {*γ*1, *γ*2, *γ*3, *γ*4}*, for which γ*<sup>1</sup> *γ*<sup>2</sup> *γ*<sup>3</sup> *γ*4*. Let* (<sup>R</sup>0, <sup>Γ</sup>0),(<sup>Q</sup>0, <sup>Γ</sup>0) <sup>∈</sup> LMF*J*0S S (*U*0) *which are defined as follows:*

$$\left(\widetilde{\mathcal{R}}\_{0}, \Gamma\_{0}\right) = \left\{\widetilde{\mathcal{R}}\_{0}(\gamma\_{1}) = \left\{\frac{x\_{1}}{(0.2, 0.1, 0.3)}, \frac{x\_{2}}{(0.1, 0.2, 0.4)}, \frac{x\_{3}}{(0.18, 0.23, 0.45)}, \frac{x\_{4}}{(0.3, 0.6, 0.2)}\right\}, \ldots\right\}$$

$$\begin{array}{l} \widetilde{\mathcal{R}}\_{0}(\gamma\_{2}) = \{ \frac{\mathbf{x}\_{1}}{(0.28, 0.49, 0.5)}, \frac{\mathbf{x}\_{2}}{(0.33, 0.42, 0.63)}, \frac{\mathbf{x}\_{3}}{(0.29, 0.33, 0.52)}, \frac{\mathbf{x}\_{4}}{(0.4, 0.68, 0.3)} \}, \\ \widetilde{\mathcal{R}}\_{0}(\gamma\_{3}) = \{ \frac{\mathbf{x}\_{1}}{(0.6, 0.72, 0.69)}, \frac{\mathbf{x}\_{2}}{(0.47, 0.73, 0.8)}, \frac{\mathbf{x}\_{3}}{(0.5, 0.62, 0.8)}, \frac{\mathbf{x}\_{4}}{(0.5, 0.7, 0.4)} \}, \\ \widetilde{\mathcal{R}}\_{0}(\gamma\_{4}) = \{ \frac{\mathbf{x}\_{1}}{(0.8, 0.85, 0.93)}, \frac{\mathbf{x}\_{2}}{(0.52, 0.8, 0.9)}, \frac{\mathbf{x}\_{3}}{(0.6, 0.7, 0.9)}, \frac{\mathbf{x}\_{4}}{(0.7, 0.9, 1.0)} \} \} \end{array}$$

*and*

$$
\begin{array}{c}
\left(\widetilde{\mathcal{Q}}\_{0},\Gamma\_{0}\right) = \left\{\begin{array}{l}
\widetilde{\mathcal{Q}}\_{0}(\gamma\_{1}) = \left\{\frac{x\_{1}}{(0.1,0.4,0.2)},\frac{x\_{2}}{(0.3,0.1,0.2)},\frac{x\_{3}}{(0.2,0.1,0.3)},\frac{x\_{4}}{(0.1,0.4,0.3)}\right\}, \\
\widetilde{\mathcal{Q}}\_{0}(\gamma\_{2}) = \left\{\frac{x\_{1}}{(0.23,0.5,0.4)},\frac{x\_{2}}{(0.4,0.25,0.36)},\frac{x\_{3}}{(0.32,0.4,0.5)},\frac{x\_{4}}{(0.32,0.5,0.4)}\right\}, \\
\widetilde{\mathcal{Q}}\_{0}(\gamma\_{3}) = \left\{\frac{x\_{1}}{(0.43,0.8,0.7)},\frac{x\_{2}}{(0.5,0.6,0.7)},\frac{x\_{3}}{(0.4,0.71,0.6)},\frac{x\_{4}}{(0.4,0.6,2.0,5)}\right\}, \\
\widetilde{\mathcal{Q}}\_{0}(\gamma\_{4}) = \left\{\frac{x\_{1}}{(0.7,0.82,1.0)},\frac{x\_{2}}{(0.8,0.75,1.0)},\frac{x\_{3}}{(0.8,0.9,0.95)},\frac{x\_{4}}{(0.8,0.7,0.6)}\right\}, \\
\end{array}
$$

*By Definition 16, we obtain,*

$$S\_1( (\bar{\mathcal{R}}, \Gamma), (\bar{\mathcal{Q}}, \Gamma) ) = $$

$$\begin{split} \mathcal{Q} &= \frac{\sum\_{\mathbf{y}\in\Gamma} \vee\_{\mathbf{x}\in\mathcal{U}} \{\mu\_{\widetilde{\mathcal{K}}(\gamma),\mathfrak{j}\_{\mathbf{l}}}(\mathbf{x}) \wedge \mu\_{\widetilde{\mathcal{Q}}(\gamma),\mathfrak{j}\_{\mathbf{l}}}(\mathbf{x})\}}{\sum\_{\mathbf{y}\in\Gamma} \vee\_{\mathbf{x}\in\mathcal{U}} \{\mu\_{\widetilde{\mathcal{K}}(\gamma),\mathfrak{j}\_{\mathbf{l}}}(\mathbf{x}) \vee \mu\_{\widetilde{\mathcal{Q}}(\gamma),\mathfrak{j}\_{\mathbf{l}}}(\mathbf{x})\}} \\ &= \frac{(0.1 \vee 0.1 \vee 0.18 \vee 0.1) + (0.23 \vee 0.33 \vee 0.29 \vee 0.32) + (0.43 \vee 0.47 \vee 0.4 \vee 0.4) + (0.7 \vee 0.52 \vee 0.6 \vee 0.7)}{0.2 \vee 0.3 \vee 0.2 \vee 0.3 + 0.28 \vee 0.4 \vee 0.32 \vee 0.4 + 0.6 \vee 0.5 \vee 0.5 + 0.8 \vee 0.8 \vee 0.8 \vee 0.8} \\ \end{split}$$

$$=\begin{array}{c}0.8+0.33+0.47+0.7\\0.3+0.4+0.6+0.8\end{array}$$

= 0.8

$$S\_2((\bar{\mathcal{R}}, \Gamma), (\bar{\mathcal{Q}}, \Gamma)) = $$

$$=\frac{\sum\_{\boldsymbol{\lambda}\in\Gamma}\bigvee\_{\boldsymbol{x}\in\boldsymbol{\Omega}}\{\mu\_{\widehat{\mathsf{K}}(\boldsymbol{\lambda}),\boldsymbol{\upbeta}}(\boldsymbol{\omega})\wedge\mu\_{\widehat{\mathsf{Q}}(\boldsymbol{\lambda}),\boldsymbol{\upbeta}}(\boldsymbol{\omega})\}}{\sum\_{\boldsymbol{\lambda}\in\Gamma}\bigvee\_{\boldsymbol{x}\in\boldsymbol{\upbeta}}\{\mu\_{\widehat{\mathsf{K}}(\boldsymbol{\lambda}),\boldsymbol{\upbeta}}(\boldsymbol{\omega})\vee\mu\_{\widehat{\mathsf{Q}}(\boldsymbol{\chi}),\boldsymbol{\upbeta}}(\boldsymbol{\omega})\}}$$


= 0.82

*<sup>S</sup>*3((<sup>R</sup>, <sup>Γ</sup>),(<sup>Q</sup>, <sup>Γ</sup>)) =

$$\begin{split} \mathcal{I} &= \frac{\sum\_{\gamma \in \Gamma} \vee\_{\mathcal{K} \cap \mathcal{U}} \{\mu\_{\widetilde{\mathcal{K}}(\gamma), \widetilde{\mu}}(x) \wedge \mu\_{\widetilde{\mathcal{Q}}(\gamma), \widetilde{\mu}}(x)\}}{\sum\_{\gamma \in \Gamma} \vee\_{\mathcal{K} \cap \mathcal{U}} \{\mu\_{\widetilde{\mathcal{K}}(\gamma), \widetilde{\mu}}(x) \vee \mu\_{\widetilde{\mathcal{Q}}(\gamma), \widetilde{\mu}}(x)\}} \\ &= \frac{(0.2 \vee 0.2 \vee 0.3 \vee 0.2) + (0.4 \vee 0.36 \vee 0.5 \vee 0.3) + (0.69 \vee 0.7 \vee 0.6 \vee 0.4) + (0.93 \vee 0.9 \vee 0.9 \vee 0.6)}{0.3 \vee 0.4 \vee 0.45 \vee 0.3 + 0.5 \vee 0.63 \vee 0.52 \vee 0.4 + 0.7 \vee 0.8 \vee 0.8 \vee 0.5 + 1.0 \vee 1.0 \vee 0.95 \vee 1.0}} \\ &= \frac{0.3 + 0.5 + 0.7 + 0.93}{0.45 + 0.7 + 0.9 + 1.0} \end{split}$$

$$=\sqrt{0.45 + 0.63 + 0.8 + 1.0}$$

= 0.84

*Then the similarity between* (<sup>R</sup>0, <sup>Γ</sup>0) *and* (<sup>Q</sup>0, <sup>Γ</sup>0) *over U*<sup>0</sup> *is thus* **<sup>S</sup>**((<sup>R</sup>0, <sup>Γ</sup>0),(<sup>Q</sup>0, <sup>Γ</sup>0)) = <sup>∑</sup>*j*∈*<sup>J</sup>* <sup>0</sup> *Sj*((<sup>R</sup>0,Γ0),(<sup>Q</sup>0,Γ0)) <sup>|</sup>*J*0<sup>|</sup> <sup>=</sup> 0.80+0.82+0.84 <sup>3</sup> <sup>=</sup> 0.82*.* **Theorem 1.** *Let* (<sup>R</sup>, <sup>Γ</sup>),(<sup>Q</sup>, <sup>Γ</sup>),(<sup>P</sup>, <sup>Γ</sup>) <sup>∈</sup> LMF*J*S S (*U*)*. Then the following holds:*


**Proof.** (i) For

$$\begin{split} \mathcal{S}\_{\bar{j}}((\tilde{\mathcal{R}},\Gamma),(\tilde{\mathcal{Q}},\Gamma)) &= \ \frac{\sum\_{\gamma\in\Gamma} \vee\_{\boldsymbol{x}\in\boldsymbol{\mathcal{U}}} \{\mu\_{\tilde{\mathcal{R}}(\gamma),\boldsymbol{j}}(\mathbf{x}) \wedge \mu\_{\tilde{\mathcal{Q}}(\gamma),\boldsymbol{j}}(\mathbf{x})\}}{\sum\_{\boldsymbol{\gamma}\in\Gamma} \vee\_{\boldsymbol{x}\in\boldsymbol{\mathcal{U}}} \{\mu\_{\tilde{\mathcal{R}}(\gamma),\boldsymbol{j}}(\mathbf{x}) \vee \mu\_{\tilde{\mathcal{Q}}(\gamma),\boldsymbol{j}}(\mathbf{x})\}} \\ &= \ \frac{\sum\_{\boldsymbol{\gamma}\in\Gamma} \vee\_{\boldsymbol{x}\in\boldsymbol{\mathcal{U}}} \{\mu\_{\tilde{\mathcal{Q}}(\gamma),\boldsymbol{j}}(\mathbf{x}) \wedge \mu\_{\tilde{\mathcal{R}}(\gamma),\boldsymbol{j}}(\mathbf{x})\}}{\sum\_{\boldsymbol{\gamma}\in\Gamma} \vee\_{\boldsymbol{x}\in\boldsymbol{\mathcal{U}}} \{\mu\_{\tilde{\mathcal{Q}}(\gamma),\boldsymbol{j}}(\mathbf{x}) \vee \mu\_{\tilde{\mathcal{R}}(\gamma),\boldsymbol{j}}(\mathbf{x})\}} \\ &= \ \mathcal{S}\_{\bar{j}}((\tilde{\mathcal{Q}},\Gamma),(\tilde{\mathcal{R}},\Gamma)) \end{split}$$

we have,

$$\begin{aligned} \mathbf{S}((\tilde{\mathcal{R}},\Gamma),(\tilde{\mathcal{Q}},\Gamma)) &=& \frac{\sum\_{\tilde{j}\in I} \mathbf{S}\_{\tilde{j}}((\tilde{\mathcal{R}},\Gamma),(\tilde{\mathcal{Q}},\Gamma))}{|I|} \\ &=& \frac{\sum\_{\tilde{j}\in I} \mathbf{S}\_{\tilde{j}}((\tilde{\mathcal{Q}},\Gamma),(\tilde{\mathcal{R}},\Gamma))}{|I|} \\ &=& \mathbf{S}((\tilde{\mathcal{Q}},\Gamma),(\tilde{\mathcal{R}},\Gamma)) \end{aligned}$$

(ii) Proof of this condition is trivally followed from the Definition 16. (iii)

$$\begin{split} \mathsf{S}((\overline{\mathcal{R}},\Gamma),(\overline{\mathcal{Q}},\Gamma)) &= 1 \quad \Rightarrow \quad \frac{\sum\_{j\in I} \mathsf{S}\_{j}((\overline{\mathcal{R}},\Gamma),(\overline{\mathcal{Q}},\Gamma))}{|I|} = 1 \\ &\Rightarrow \quad \mathsf{S}\_{\uparrow}((\overline{\mathcal{R}},\Gamma),(\overline{\mathcal{Q}},\Gamma)) = 1 \\ &\Rightarrow \quad \frac{\sum\_{\gamma\in\Gamma} \vee\_{\mathrm{x\in\mathcal{U}}} \{\mu\_{\underline{\mathcal{K}}(\gamma),\downarrow}(x) \wedge \mu\_{\underline{\mathcal{Q}}(\gamma),\downarrow}(x)\}}{\sum\_{\gamma\in\Gamma} \vee\_{\mathrm{x\in\mathcal{U}}} \{\mu\_{\underline{\mathcal{K}}(\gamma),\downarrow}(x) \vee \mu\_{\underline{\mathcal{Q}}(\gamma),\downarrow}(x)\}} = 1 \\ &\Rightarrow \quad \sum\_{\gamma\in\Gamma} \vee\_{\mathrm{x\in\mathcal{U}}} \{\mu\_{\widetilde{\mathcal{K}}(\gamma),\downarrow}(x) \wedge \mu\_{\underline{\mathcal{Q}}(\gamma),\downarrow}(x)\} = \sum\_{\gamma\in\Gamma} \vee\_{\mathrm{x\in\mathcal{U}}} \{\mu\_{\widetilde{\mathcal{K}}(\gamma),\downarrow}(x) \vee \mu\_{\underline{\mathcal{Q}}(\gamma),\downarrow}(x)\} \\ &\Rightarrow \quad \mu\_{\widetilde{\mathcal{K}}(\gamma),\downarrow}(x) = \mu\_{\widetilde{\mathcal{Q}}(\gamma),\downarrow}(x) \neq \mathcal{Q} \\ &\Rightarrow \quad (\overline{\mathcal{R}},\Gamma) = (\overline{\mathcal{Q}},\Gamma) \neq \mathcal{Q} \end{split}$$

(iv)

$$\begin{split} \mathbf{S}((\bar{\mathcal{R}},\Gamma),(\bar{\mathcal{Q}},\Gamma)) &= 0 \quad \Rightarrow \quad \frac{\sum\_{j\in I} S\_{j}((\mathcal{R},\Gamma),(\underline{\mathcal{Q}},\Gamma))}{|I|} = 0 \\ &\Rightarrow \quad S\_{j}((\bar{\mathcal{R}},\Gamma),(\bar{\mathcal{Q}},\Gamma)) = 0 \\ &\Rightarrow \quad \frac{\sum\_{\gamma\in\Gamma} \vee\_{\mathrm{x}\in\mathcal{U}} \{\mu\_{\tilde{\mathcal{K}}(\gamma),\boldsymbol{\zeta}}(\boldsymbol{x}) \wedge \mu\_{\tilde{\mathcal{Q}}(\gamma),\boldsymbol{\zeta}}(\boldsymbol{x})\}}{\sum\_{\gamma\in\Gamma} \vee\_{\mathrm{x}\in\mathcal{U}} \{\mu\_{\tilde{\mathcal{K}}(\gamma),\boldsymbol{\zeta}}(\boldsymbol{x}) \vee \mu\_{\tilde{\mathcal{Q}}(\gamma),\boldsymbol{\zeta}}(\boldsymbol{x})\}} = 0 \\ &\Rightarrow \quad \sum\_{\gamma\in\Gamma} \vee\_{\mathrm{x}\in\mathcal{U}} \{\mu\_{\tilde{\mathcal{K}}(\gamma),\boldsymbol{\zeta}}(\boldsymbol{x}) \wedge \mu\_{\tilde{\mathcal{Q}}(\gamma),\boldsymbol{\zeta}}(\boldsymbol{x})\} = 0 \\ &\Rightarrow \quad \mu\_{\tilde{\mathcal{K}}(\gamma),\boldsymbol{\zeta}}(\boldsymbol{x}) \cap \mu\_{\tilde{\mathcal{Q}}(\gamma),\boldsymbol{\zeta}}(\boldsymbol{x}) = \mathcal{Q} \\ &\Rightarrow \quad (\bar{\mathcal{R}},\Gamma) \cap (\bar{\mathcal{Q}},\Gamma) = \mathcal{Q} \end{split}$$

(v) Since (<sup>R</sup>, <sup>Γ</sup>) <sup>⊆</sup> (<sup>Q</sup>, <sup>Γ</sup>) <sup>⊆</sup> (<sup>P</sup>, <sup>Γ</sup>),

<sup>⇒</sup> *<sup>μ</sup>*<sup>R</sup>(*γ*),*<sup>j</sup>* (*x*) <sup>≤</sup> *<sup>μ</sup>*<sup>Q</sup>(*γ*),*<sup>j</sup>* (*x*) <sup>≤</sup> *<sup>μ</sup>*<sup>P</sup>(*γ*),*<sup>j</sup>* (*x*) <sup>⇒</sup> <sup>∑</sup>*γ*∈<sup>Γ</sup> <sup>∨</sup>*x*∈*U*{*μ*<sup>R</sup>(*γ*),*<sup>j</sup>* (*x*) <sup>∧</sup> *<sup>μ</sup>*<sup>P</sup>(*γ*),*<sup>j</sup>* (*x*)} <sup>∑</sup>*γ*∈<sup>Γ</sup> <sup>∨</sup>*x*∈*U*{*μ*<sup>R</sup>(*γ*),*<sup>j</sup>* (*x*) <sup>∨</sup> *<sup>μ</sup>*<sup>P</sup>(*γ*),*<sup>j</sup>* (*x*)} <sup>≤</sup> <sup>∑</sup>*γ*∈<sup>Γ</sup> <sup>∨</sup>*x*∈*U*{*μ*<sup>Q</sup>(*γ*),*<sup>j</sup>* (*x*) <sup>∧</sup> *<sup>μ</sup>*<sup>P</sup>(*γ*),*<sup>j</sup>* (*x*)} <sup>∑</sup>*γ*∈<sup>Γ</sup> <sup>∨</sup>*x*∈*U*{*μ*<sup>Q</sup>(*γ*),*<sup>j</sup>* (*x*) <sup>∨</sup> *<sup>μ</sup>*<sup>P</sup>(*γ*),*<sup>j</sup>* (*x*)} <sup>⇒</sup> *Sj*((<sup>R</sup>, <sup>Γ</sup>),(<sup>P</sup>, <sup>Γ</sup>)) <sup>≤</sup> *Sj*((<sup>Q</sup>, <sup>Γ</sup>),(<sup>P</sup>, <sup>Γ</sup>)) <sup>⇒</sup> <sup>∑</sup>*j*∈*<sup>J</sup> Sj*((<sup>R</sup>, <sup>Γ</sup>),(<sup>P</sup>, <sup>Γ</sup>)) <sup>|</sup>*J*<sup>|</sup> <sup>≤</sup> <sup>∑</sup>*j*∈*<sup>J</sup> Sj*((<sup>Q</sup>, <sup>Γ</sup>),(<sup>P</sup>, <sup>Γ</sup>)) |*J*| <sup>⇒</sup> **<sup>S</sup>**((<sup>R</sup>, <sup>Γ</sup>),(<sup>P</sup>, <sup>Γ</sup>)) <sup>≤</sup> **<sup>S</sup>**((<sup>Q</sup>, <sup>Γ</sup>),(<sup>P</sup>, <sup>Γ</sup>)).

Next, we discuss about the application of similarity measure.

### **4. Application of** LMFSS **Using Similarity Measure in Decision Making**

In this section, an application for the decision making by using the similarity measure of two LMFSS *s* to analyse the rainfall in 2016 and 2017 with expected rainfall.

Let *U*<sup>0</sup> = {*x*1, *x*2, *x*3, *x*4} be the universal set, where *x*1, *x*2, *x*3, *x*<sup>4</sup> stands for the set of four cities in India. Let Γ<sup>0</sup> = {*γ*1, *γ*2, *γ*3, *γ*4} as the parameters which is consider as the set of rainfalls in rainy season, where *γ*<sup>1</sup> stands for "Hour-wise rainfall" which includes 1 h, 2 h, and 3 h, *γ*<sup>2</sup> stands for "Day-wise rainfall" which includes 2 day, 3 day and 4 day, *γ*<sup>3</sup> stands for "Week-wise rainfall" which includes 2 week, 3 week, 4 week and *γ*<sup>4</sup> stands for "Month-wise rainfall" which includes 3 month, 4 month, 5 month respectively.

In this example, suppose (<sup>R</sup>0, <sup>Γ</sup>0) <sup>∈</sup> LMFSS (*U*0) represents the expected rainfall in India, defined as follows:

$$\begin{array}{l} (\widetilde{\mathcal{R}}\_{0},\Gamma\_{0}) = \{\mathcal{R}\_{0}(\gamma\_{1}) = \{\frac{x\_{1}}{(0,0,0.12)},\frac{x\_{2}}{(0,1,0,0)},\frac{x\_{3}}{(0,0.13,0)},\frac{x\_{4}}{(0,0,0)}\},\\ \mathcal{R}\_{0}(\gamma\_{2}) = \{\frac{x\_{1}}{(0,1,0.12,0.13)},\frac{x\_{2}}{(0,15,0.1,0.13)},\frac{x\_{3}}{(0,16,0.12,0.1)},\frac{x\_{4}}{(0,10,13,0.1)}\},\\ \mathcal{R}\_{0}(\gamma\_{3}) = \{\frac{x\_{1}}{(0,15,0.14,0.16)},\frac{x\_{2}}{(0,16,0.12,0.14)},\frac{x\_{3}}{(0,18,0.19,0.2)},\frac{x\_{4}}{(0,13,0.14,0.11)}\},\\ \mathcal{R}\_{0}(\gamma\_{4}) = \{\frac{x\_{1}}{(0,18,0.21)},\frac{x\_{2}}{(0,17,0.15,0.19)},\frac{x\_{3}}{(0,19,0.2,0.21)},\frac{x\_{4}}{(0,15,0.16,0.11)}\}.\end{array}$$

Tabulation of (<sup>R</sup>0, <sup>Γ</sup>0) is given in Table <sup>5</sup> and Figure 1.

**Figure 1.** The expected rainfall in 2016 and 2017.


**Table 5.** Tabulation of (<sup>R</sup>0, <sup>Γ</sup>0) Representing the Expected Rainfall.

Now let (<sup>H</sup>1, <sup>Γ</sup>0),(<sup>H</sup>2, <sup>Γ</sup>0) <sup>∈</sup> LMFSS (*U*0) represents the recorded rainfall in India for the year 2016 and 2017 respectively, defined as follows:

(<sup>H</sup>1, <sup>Γ</sup>0) = {H1(*γ*1) = { *<sup>x</sup>*<sup>1</sup> (0,0,0), *<sup>x</sup>*<sup>2</sup> (0,0,0.1), *<sup>x</sup>*<sup>3</sup> (0,0,0), *<sup>x</sup>*<sup>4</sup> (0.11,0,0) }, <sup>H</sup>1(*γ*2) = { *<sup>x</sup>*<sup>1</sup> (0,0.13,0), *<sup>x</sup>*<sup>2</sup> (0.1,0,0.14), *<sup>x</sup>*<sup>3</sup> (0,0,0.2), *<sup>x</sup>*<sup>4</sup> (0.15,0,0.2) }, <sup>H</sup>1(*γ*3) = { *<sup>x</sup>*<sup>1</sup> (0.1,0.15,0), *<sup>x</sup>*<sup>2</sup> (0.15,0.1,0.17), *<sup>x</sup>*<sup>3</sup> (0.1,0,0.21), *<sup>x</sup>*<sup>4</sup> (0.2,0.1,0.21) } <sup>H</sup>1(*γ*4) = { *<sup>x</sup>*<sup>1</sup> (0.3,0.18,0.1), *<sup>x</sup>*<sup>2</sup> (0.17,0.12,0.2), *<sup>x</sup>*<sup>3</sup> (0.17,0.21,0.23 , *<sup>x</sup>*<sup>4</sup> (0.22,0.13,0.23) }}, (<sup>H</sup>2, <sup>Γ</sup>0) = {H2(*γ*1) = { *<sup>x</sup>*<sup>1</sup> (0.1,0,0.13), *<sup>x</sup>*<sup>2</sup> (0,0.12,0.1), *<sup>x</sup>*<sup>3</sup> (0.1,0.14,0.1), *<sup>x</sup>*<sup>4</sup> (0.1,0.11,0.13) }, <sup>H</sup>2(*γ*2) = { *<sup>x</sup>*<sup>1</sup> (0.13,0.1,0.15), *<sup>x</sup>*<sup>2</sup> (0.1,0.14,0.12), *<sup>x</sup>*<sup>3</sup> (0.13,0.16,0.13), *<sup>x</sup>*<sup>4</sup> (0.2,0.18,0.18) }, <sup>H</sup>2(*γ*3) = { *<sup>x</sup>*<sup>1</sup> (0.14,0.13,0.17), *<sup>x</sup>*<sup>2</sup> (0.15,0.17,0.18), *<sup>x</sup>*<sup>3</sup> (0.17,0.18,0.16), *<sup>x</sup>*<sup>4</sup> (0.25,0.31,0.26) } <sup>H</sup>2(*γ*4) = { *<sup>x</sup>*<sup>1</sup> (0.16,0.18,0.3), *<sup>x</sup>*<sup>2</sup> (0.19,0.18,0.2), *<sup>x</sup>*<sup>3</sup> (0.2,0.23,0.3), *<sup>x</sup>*<sup>4</sup> (0.28,0.34,0.28) }}.

Tabulations of (<sup>H</sup>1, <sup>Γ</sup>0) and (<sup>H</sup>2, <sup>Γ</sup>0) are presented as in Tables <sup>6</sup> and <sup>7</sup> and Figures <sup>2</sup> and <sup>3</sup> respectively.

**Table 6.** Tabulation of (<sup>H</sup>1, <sup>Γ</sup>0) Representing the Rainfall 2016.


**Table 7.** Tabulation of (<sup>H</sup>2, <sup>Γ</sup>0) Representing the Rainfall in 2017.


**Figure 2.** The recorded rainfall in India for the year 2016.

**Figure 3.** The recorded rainfall in India for the year 2017.

In order to make the decision of whether the rainfall in 2016 or the rainfall in 2017 is the expected rainfall in India, we use similarity measure on LMFSS *s* to calculate the similarity between the expected rainfall and the rainfall in 2016 (**S**((<sup>R</sup>0, <sup>Γ</sup>0),(<sup>H</sup>1, <sup>Γ</sup>0))); the similarity between the expected rainfall and the rainfall in 2017 (**S**((<sup>R</sup>0, <sup>Γ</sup>0),(<sup>H</sup>2, <sup>Γ</sup>0))). Comparing the obtained results, the higher similarity means the closer to expected rainfall.

First we calculate the similarity measure between (<sup>R</sup>0, <sup>Γ</sup>0) and (<sup>H</sup>1, <sup>Γ</sup>0):

*<sup>S</sup>*1((<sup>R</sup>0, <sup>Γ</sup>0),(<sup>H</sup>1, <sup>Γ</sup>0)) <sup>=</sup> (<sup>0</sup> <sup>∨</sup> <sup>0</sup> <sup>∨</sup> <sup>0</sup> <sup>∨</sup> <sup>0</sup>)+(<sup>0</sup> <sup>∨</sup> 0.1 <sup>∨</sup> <sup>0</sup> <sup>∨</sup> 0.1)+(0.1 <sup>∨</sup> 0.15 <sup>∨</sup> 0.1 <sup>∨</sup> 0.13)+(0.18 <sup>∨</sup> 0.17 <sup>∨</sup> 0.17 <sup>∨</sup> 0.15) 0 ∨ 0.1 ∨ 0 ∨ 0.11 + 0.1 ∨ 0.15 ∨ 0.16 ∨ 0.15 + 0.15 ∨ 0.16 ∨ 0.18 ∨ 0.2 + 0.3 ∨ 0.17 ∨ 0.19 ∨ 0.22 <sup>=</sup> <sup>0</sup> <sup>+</sup> 0.1 <sup>+</sup> 0.15 <sup>+</sup> 0.18 0.1 <sup>+</sup> 0.18 <sup>+</sup> 0.2 <sup>+</sup> 0.18 <sup>=</sup> 0.39 *<sup>S</sup>*2((<sup>R</sup>0, <sup>Γ</sup>0),(<sup>H</sup>1, <sup>Γ</sup>0)) <sup>=</sup> (<sup>0</sup> <sup>∨</sup> <sup>0</sup> <sup>∨</sup> <sup>0</sup> <sup>∨</sup> <sup>0</sup>)+(0.12 <sup>∨</sup> <sup>0</sup> <sup>∨</sup> <sup>0</sup> <sup>∨</sup> <sup>0</sup>)+(0.14 <sup>∨</sup> 0.1 <sup>∨</sup> <sup>0</sup> <sup>∨</sup> 0.1)+(0.18 <sup>∨</sup> 0.12 <sup>∨</sup> 0.2 <sup>∨</sup> 0.13) 0 ∨ 0 ∨ 0 ∨ 0 + 0.13 ∨ 0.1 ∨ 0.17 ∨ 0.13 + 0.15 ∨ 0.12 ∨ 0.19 ∨ 0.14 + 0.2 ∨ 1 ∨ 0.21 ∨ 0.16 <sup>=</sup> <sup>0</sup> <sup>+</sup> 0.12 <sup>+</sup> 0.14 <sup>+</sup> 0.2 <sup>0</sup> <sup>+</sup> 0.17 <sup>+</sup> 0.19 <sup>+</sup> <sup>1</sup> <sup>=</sup> 0.33 *<sup>S</sup>*3((<sup>R</sup>0, <sup>Γ</sup>0),(<sup>H</sup>1, <sup>Γ</sup>0)) <sup>=</sup> (<sup>0</sup> <sup>∨</sup> <sup>0</sup> <sup>∨</sup> <sup>0</sup> <sup>∨</sup> <sup>0</sup>)+(<sup>0</sup> <sup>∨</sup> 0.13 <sup>∨</sup> 0.1 <sup>∨</sup> 0.1)+(<sup>0</sup> <sup>∨</sup> 0.14 <sup>∨</sup> 0.2 <sup>∨</sup> 0.11)+(0.1 <sup>∨</sup> 0.19 <sup>∨</sup> 0.21 <sup>∨</sup> 0.12) 0.12 ∨ 0.1 ∨ 0 ∨ 0 + 0.13 ∨ 0.14 ∨ 0.2 ∨ 0.2 + 0.16 ∨ 0.17 ∨ 0.21 ∨ 0.21 + 1 ∨ 0.2 ∨ 0.23 ∨ 0.23 <sup>=</sup> <sup>0</sup> <sup>+</sup> 0.13 <sup>+</sup> 0.2 <sup>+</sup> 0.21 0.12 <sup>+</sup> 0.2 <sup>+</sup> 0.21 <sup>+</sup> 1.0 <sup>=</sup> 0.35

Hence **<sup>S</sup>**((<sup>R</sup>0, <sup>Γ</sup>0),(<sup>H</sup>1, <sup>Γ</sup>0)) = 0.39+0.33+0.35 <sup>3</sup> <sup>=</sup> 0.36. Next we calculate the similarity measure between (<sup>R</sup>0, <sup>Γ</sup>0) and (<sup>H</sup>2, <sup>Γ</sup>0):


$$\mathop{S\_2}((\mathcal{R}\_0, \Gamma\_0), (\mathcal{H}\_{2'}\Gamma\_0))$$


$$\mathcal{S}\_{\mathfrak{Z}}((\bar{\mathcal{R}}\_0, \Gamma\_0), (\bar{\mathcal{H}}\_{2\prime}\Gamma\_0))$$


Hence **<sup>S</sup>**((<sup>R</sup>0, <sup>Γ</sup>0),(<sup>H</sup>2, <sup>Γ</sup>0)) = 0.59+0.691+0.395 <sup>3</sup> <sup>=</sup> 0.559.

It is clear from the above results, that (<sup>H</sup>2, <sup>Γ</sup>0) has significantly greater similarity to (<sup>R</sup>0, <sup>Γ</sup>0), as compared with (<sup>H</sup>1, <sup>Γ</sup>0) to (<sup>R</sup>0, <sup>Γ</sup>0). So we conclude that the rainfall in 2016 is not an expected rainfall and the rainfall in 2017 is an expected rainfall in India.

#### **5. Discussion**

In this paper, our motivation to introduce the concept of similarity between two LMFSS is achieved. This similarity measure satisfies the good properties of similarity measures. Advantages of similarity measure on lattice ordered multi-fuzzy soft set include:


The disadvantage of the proposed similarity measure is that it is only applicable to lattice ordered structures and does not work for other fuzzy structures.

Some properties of proposed measure are stated and proved by a theorem. Apart from that, an application for the decision making by using the similarity measure of two LMFSS to analyse the rainfall is obtained in this research. This application shows that our proposed measure is worth to use.

### **6. Conclusions**

Multi-fuzzy soft set and its extensions are used in many different applications in decision making. The similarity measure on complex multi-fuzzy soft set has been proposed. LMFSS was applied in solving forecast problems, but the similarity on LMFSS was not introduced. In this paper, the concept of similarity measure of LMFSS is introduced. The numerical examples are presented in detail to illustrate the proposed similarity measure. We also define some properties of similarity measure on two LMFSS *s*. These properties are proved by Theorem 3.3. Finally, an application of this similarity measure in decision making is presented.

In further works, we are going to extend the operations and properties of LMFSS using similarity measure. Besides, the using of this similarity measure in solving other real life problems will be studied.

**Author Contributions:** Methodology: V.J. and T.T.N., writing—original draft preparation: S.B.S. and R.S.; software: V.J. and S.B.S.; validation: T.T.N.; writing—review and editing: T.T.N. and Ganeshsree Selvachandran. All authors have read and agreed to the published version of the manuscript.

**Funding:** The article has been written with the joint financial support of RUSA-Phase 2.0 grant sanctioned vide letter No.F 24-51/2014-U, Policy (TN Multi-Gen), Dept. of Edn. Govt. of India, Dt. 09.10.2018, UGC-SAP (DRS-I) vide letter No.F.510/8/DRS-I/2016(SAP-I) Dt. 23.08.2016, DST-PURSE 2nd Phase programme vide letter No. SR/PURSE Phase 2/38 (G) Dt. 21.02.2017 and DST (FST - level I) 657876570 vide letter No.SR/FIST/MS-I/2018/17 Dt. 20.12.2018.

**Acknowledgments:** The authors would like to thank the Editor-in-Chief and the anonymous reviewers for their valuable comments and suggestions.

**Conflicts of Interest:** The authors declare no conflict of interest.
