Exponential and non-Exponential Based Generalized Similarity Measures for Complex Hesitant Fuzzy Sets with Applications

Figures & data

Figure 1. The geometrical representation of the explored approach.

Table 1. Calculation of proposed SMs for $\lambda =1$ and $\left({\alpha }_{cc},{\text{β}}_{cc}\right)=\left(0.5,0.5\right)$.

Similarity measures	$E_1$	$E_2$	$E_3$	$E_4$	Ranking
$S_c^1\left(E,E_j\right)$	0.4928	0.5109	0.3218	0.4503	$E_2\ge E_1\ge E_4\ge E_3$
$S_c^2\left(E,E_j\right)$	0.5316	0.5686	0.4309	0.555	$E_2\ge E_4\ge E_1\ge E_3$
$S_c^3\left(E,E_j\right)$	0.4175	0.4281	0.2536	0.3801	$E_2\ge E_1\ge E_4\ge E_3$
$S_c^4\left(E,E_j\right)$	0.4465	0.4448	0.2529	0.5003	$E_4\ge E_2\ge E_1\ge E_3$
$S_c^5\left(E,E_j\right)$	0.4366	0.4259	0.2516	0.4667	$E_4\ge E_1\ge E_2\ge E_3$

Table 2. Calculation of proposed WSMs for Example (3) based on $\lambda =1$ and $\left({\alpha }_{cc},{\text{β}}_{cc}\right)=\left(0.5,0.5\right)$.

Similarity Measures	$E_1$	$E_2$	$E_3$	$E_4$	Ranking
$S_{c_w}^1\left(E,E_j\right)$	0.5164	0.4902	0.3065	0.4259	$E_1\ge E_2\ge E_4\ge E_3$
$S_{c_w}^2\left(E,E_j\right)$	0.5543	0.5537	0.4157	0.5253	$E_1\ge E_2\ge E_4\ge E_3$
$S_{c_w}^3\left(E,E_j\right)$	0.4368	0.4082	0.2405	0.3594	$E_1\ge E_2\ge E_4\ge E_3$
$S_{c_w}^4\left(E,E_j\right)$	0.4717	0.4514	0.2772	0.4725	$E_4\ge E_1\ge E_2\ge E_3$
$S_{c_w}^5\left(E,E_j\right)$	0.4588	0.4367	0.2776	0.432	$E_1\ge E_2\ge E_4\ge E_3$

Figure 2. Graphical representation of established SMs for Example 3.

Table 3. Calculation of proposed SMs for $\lambda =1$ and $\left({\alpha }_{cc},{\text{β}}_{cc}\right)=\left(0.5,0.5\right)$.

Similarity measures	${\mathcal{D}}_1$	${\mathcal{D}}_2$	${\mathcal{D}}_3$	${\mathcal{D}}_4$	Ranking
$S_c^1\left(\mathcal{P},{\mathcal{D}}_j\right)$	0.5396	0.5083	0.4023	0.2781	${\mathcal{D}}_1\ge {\mathcal{D}}_2\ge {\mathcal{D}}_3\ge {\mathcal{D}}_4$
$S_c^2\left(\mathcal{P},{\mathcal{D}}_j\right)$	0.6447	0.5759	0.4881	0.3554	${\mathcal{D}}_1\ge {\mathcal{D}}_2\ge {\mathcal{D}}_3\ge {\mathcal{D}}_4$
$S_c^3\left(\mathcal{P},{\mathcal{D}}_j\right)$	0.4588	0.4335	0.336	0.2173	${\mathcal{D}}_1\ge {\mathcal{D}}_2\ge {\mathcal{D}}_3\ge {\mathcal{D}}_4$
$S_c^4\left(\mathcal{P},{\mathcal{D}}_j\right)$	0.5249	0.3981	0.2614	0.1474	${\mathcal{D}}_1\ge {\mathcal{D}}_2\ge {\mathcal{D}}_3\ge {\mathcal{D}}_4$
$S_c^5\left(\mathcal{P},{\mathcal{D}}_j\right)$	0.5713	0.3956	0.2675	0.1488	${\mathcal{D}}_1\ge {\mathcal{D}}_2\ge {\mathcal{D}}_3\ge {\mathcal{D}}_4$

Table 4. Calculation of proposed WSMs for Example (4) based on $\lambda =1$ and $\left({\alpha }_{cc},{\text{β}}_{cc}\right)=\left(0.5,0.5\right)$.

Similarity measures	${\mathcal{D}}_1$	${\mathcal{D}}_2$	${\mathcal{D}}_3$	${\mathcal{D}}_4$	Ranking
$S_{c_w}^1\left(\mathcal{P},{\mathcal{D}}_j\right)$	0.5633	0.5821	0.4749	0.2692	${\mathcal{D}}_2\ge {\mathcal{D}}_1\ge {\mathcal{D}}_3\ge {\mathcal{D}}_4$
$S_{c_w}^2\left(\mathcal{P},{\mathcal{D}}_j\right)$	0.6802	0.6525	0.5662	0.3376	${\mathcal{D}}_1\ge {\mathcal{D}}_2\ge {\mathcal{D}}_3\ge {\mathcal{D}}_4$
$S_{c_w}^3\left(\mathcal{P},{\mathcal{D}}_j\right)$	0.4852	0.5013	0.4022	0.2099	${\mathcal{D}}_2\ge {\mathcal{D}}_1\ge {\mathcal{D}}_3\ge {\mathcal{D}}_4$
$S_{c_w}^4\left(\mathcal{P},{\mathcal{D}}_j\right)$	0.5106	0.4423	0.3089	0.1332	${\mathcal{D}}_1\ge {\mathcal{D}}_2\ge {\mathcal{D}}_3\ge {\mathcal{D}}_4$
$S_{c_w}^5\left(\mathcal{P},{\mathcal{D}}_j\right)$	0.5812	0.4433	0.3163	0.1327	${\mathcal{D}}_1\ge {\mathcal{D}}_2\ge {\mathcal{D}}_3\ge {\mathcal{D}}_4$

Figure 3. Graphical representation of established WSMs for Example 3.

Figure 4. Graphical representation of established WSMs for Example 4.

Figure 5. Graphical representation of established WSMs for Example 4.

Figure 6. Graphical representation of the comparison of the establish SMs with existing SMs for Example 5.

Figure 7. Graphical representation of the comparison of the establish SMs with existing SMs for Example 3.

Table 5. Comparison between Proposed SMs and Existing SMs for Example (5) and $\lambda =1$, $\left({\alpha }_{cc},{\text{β}}_{cc}\right)=\left(1,0\right)$.

Method	Score value	Ranking
Xu and Xia [25]	$s_g\left(E,E_1\right)=$0.7083, $s_g\left(E,E_2\right)=0.6042$ $s_g\left(E,E_3\right)=0.6083$, $s_g\left(E,E_4\right)=0.7$	$E_1\ge E_4\ge E_3\ge E_2$
Zeng et al. [32]	$s_h\left(E,E_1\right)=0.8375$, $s_h\left(E,E_2\right)=0.6792$ $s_h\left(E,E_3\right)=0.6167$, $s_h\left(E,E_4\right)=0.7979$	$E_1\ge E_4\ge E_2\ge E_3$
Tang et al. [33]	$s_h^{\mathrm{\prime }}\left(E,E_1\right)=0.6319$, $s_h^{\mathrm{\prime }}\left(E,E_2\right)=0.3652$ $s_h^{\mathrm{\prime }}\left(E,E_3\right)=0.2333,$ $s_h^{\mathrm{\prime }}\left(E,E_4\right)=0.5541$	$E_1\ge E_4\ge E_2\ge E_3$
Proposed SM	$S_c^1\left(E,E_1\right)=$0.6664, $S_c^1\left(E,E_2\right)=0.5127$ $S_c^1\left(E,E_3\right)=$0.3563, $S_c^1\left(E,E_4\right)=0.4819$	$E_1\ge E_4\ge E_2\ge E_3$
Proposed SM	$S_c^2\left(E,E_1\right)=0.6664$, $S_c^2\left(E,E_2\right)=$0.5127 $S_c^2\left(E,E_4\right)=0.3563$, $S_c^2\left(E,E_5\right)=0.5582$	$E_1\ge E_4\ge E_2\ge E_3$
Proposed SM	$S_c^3\left(E,E_1\right)=0.6177$, $S_c^3\left(E,E_2\right)=0.4366$ $S_c^3\left(E,E_3\right)=0.2833$, $S_c^3\left(E,E_4\right)=0.4775$	$E_1\ge E_4\ge E_2\ge E_3$
Proposed SM	$S_c^4\left(E,E_1\right)=0.6694$, $S_c^4\left(E,E_2\right)=0.4115$ $S_c^4\left(E,E_3\right)=0.2438$, $S_c^4\left(E,E_4\right)=0.4885$	$E_1\ge E_4\ge E_2\ge E_3$
Proposed SM	$S_c^5\left(E,E_1\right)=0.6694$, $S_c^5\left(E,E_2\right)=0.4115$ $S_c^5\left(E,E_3\right)=0.2438$, $S_c^5\left(E,E_4\right)=0.4885$	$E_1\ge E_4\ge E_2\ge E_3$

Table 6. Comparison between proposed SMs with existing SMs for Example (3) and $\lambda =1$.

Method	Score value	Ranking
Xu and Xia [25]	Failed	Failed
Zeng et al. [32]	Failed	Failed
Tang et al. [33]	Failed	Failed
Proposed SM	$S_c^1\left(E,E_1\right)=0.4928$,$S_c^1\left(E,E_2\right)=$0.5109 $S_c^1\left(E,E_3\right)=0.3218$, $S_c^1\left(E,E_4\right)=0.4503$	$E_2\ge E_1\ge E_4\ge E_3$
Proposed SM	$S_c^2\left(E,E_1\right)=0.5316$,$S_c^2\left(E,E_2\right)=0.5686$ $S_c^2\left(E,E_3\right)=0.4309$, $S_c^2\left(E,E_4\right)=0.555$	$E_2\ge E_4\ge E_1\ge E_3$
Proposed SM	$S_c^3\left(E,E_1\right)=0.4175$,$S_c^3\left(E,E_2\right)=0.4281$ $S_c^3\left(E,E_3\right)=0.2536$, $S_c^3\left(E,E_4\right)=0.3801$	$E_2\ge E_4\ge E_1\ge E_3$
Proposed SM	$S_c^4\left(E,E_1\right)=0.4465$,$S_c^4\left(E,E_2\right)=0.4448$ $S_c^4\left(E,E_3\right)=0.2529$, $S_c^4\left(E,E_4\right)=0.5003$	$E_4\ge E_2\ge E_1\ge E_3$
Proposed SM	$S_c^5\left(E,E_1\right)=0.4366$,$S_c^5\left(E,E_2\right)=0.4259$ $S_c^5\left(E,E_3\right)=0.2516$, $S_c^5\left(E,E_4\right)=0.4667$	$E_4\ge E_1\ge E_2\ge E_3$

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