Introduction
Preliminaries
New similarity measures on Pythagorean fuzzy sets
Applications of the proposed SMs
Verification and the comparative analysis
Authors | Similarity measure |
---|---|
Li et al. [7] |
\(\mathcal {S}_{L}(\mathcal {M}, \mathcal {N})=1-\sqrt{\dfrac{\sum \nolimits _{i=1}^n((\mu _\mathcal {M}(x_i)-\mu _\mathcal {N}(x_i))^2+(\nu _\mathcal {M}(x_i)-\nu _\mathcal {N}(x_i))^2)}{2n}} \)
|
Chen [8] |
\(\mathcal {S}_C(\mathcal {M}, \mathcal {N})=1-\dfrac{\sum \nolimits _{i=1}^n|\mu _\mathcal {M}(x_i)-\nu _\mathcal {M}(x_i)-(\mu _\mathcal {N}(x_i)-\nu _\mathcal {N}(x_i))|}{2n} \)
|
Chen and Chang [9] |
\(\mathcal {S}_{CC}(\mathcal {M}, \mathcal {N})=1-\dfrac{1}{n}\sum \nolimits _{i=1}^n \left( |\mu _\mathcal {M}(x_i)-\mu _\mathcal {N}(x_i)|\times (1-\dfrac{\pi _\mathcal {M}(x_i)+\pi _\mathcal {N}(x_i)}{2})+ \left( \int _{0}^1|\mu _{M_{x_i}}(z)-\mu _{N_{x_i}}(z)|d_z\right) \times \left( \dfrac{\pi _\mathcal {M}(x_i)+\pi _\mathcal {N}(x_i)}{2}\right) \right) \)
|
\( \text {where}~~~ \mu _{M_{x_i}}(z)= {\left\{ \begin{array}{ll} 1,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text {if}~~z=\mu _\mathcal {M}(x_i)=1-\nu _\mathcal {M}(x_i), \\ \dfrac{1-\nu _\mathcal {M}(x_i)-z}{1-\mu _\mathcal {M}(x_i)-\nu _\mathcal {M}(x_i)},~~~~~~~~~\text {if}~~z\in [\mu _\mathcal {M}(x_i),1-\nu _\mathcal {M}(x_i)], \\ 0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text {otherwise}.\\ \end{array}\right. }\)
| |
Hung and Yang [10] |
\( \mathcal {S}_{HY1}(\mathcal {M}, \mathcal {N})=1-\dfrac{\sum \nolimits _{i=1}^n\text {max}(|\mu _\mathcal {M}(x_i)-\mu _\mathcal {N}(x_i)|,|\nu _\mathcal {M}(x_i)-\nu _\mathcal {N}(x_i)|)}{n},\mathcal {S}_{HY2}(\mathcal {M}, \mathcal {N})=\dfrac{e^{\mathcal {S}_{HY1}(\mathcal {M}, \mathcal {N})-1}-e^{-1}}{1-e^{-1}}\)
|
\(\mathcal {S}_{HY3}(\mathcal {M}, \mathcal {N})=\dfrac{\mathcal {S}_{HY1}(\mathcal {M}, \mathcal {N})}{2-\mathcal {S}_{HY1}(\mathcal {M}, \mathcal {N})}\)
| |
Hong and Kim [11] |
\(\mathcal {S}_{HK}(\mathcal {M}, \mathcal {N})=1-\dfrac{\sum \nolimits _{i=1}^n(|\mu _\mathcal {M}(x_i)-\mu _\mathcal {N}(x_i)|+|\nu _\mathcal {M}(x_i)-\nu _\mathcal {N}(x_i)|)}{2n}\)
|
Li and Cheng [12] |
\(\mathcal {S}_{LC}(\mathcal {M}, \mathcal {N})=1-\root p \of {\dfrac{\sum \nolimits _{i=1}^n|\psi _{M}(x_i)-\psi _{N}(x_i)|^p}{n}}, \)
|
\(\text {where} ~\psi _{M}(x_i)=\dfrac{\mu _\mathcal {M}(x_i)+1-\nu _\mathcal {M}(x_i)}{2},\psi _{N}(x_i)=\dfrac{\mu _\mathcal {N}(x_i)+1-\nu _\mathcal {N}(x_i)}{2}, \text {and} ~1\le p<\infty \). | |
Li and Xu [13] |
\(\mathcal {S}_{LX}(\mathcal {M}, \mathcal {N})=1-\dfrac{\sum \nolimits _{i=1}^n(|\mu _\mathcal {M}(x_i)-\nu _\mathcal {M}(x_i)-(\mu _\mathcal {N}(x_i)-\nu _\mathcal {N}(x_i))|+|\mu _\mathcal {M}(x_i)-\mu _\mathcal {N}(x_i)|+|\nu _\mathcal {M}(x_i)-\nu _\mathcal {N}(x_i)|)}{4n}\)
|
Liang and Shi [14] | \( \mathcal {S}_{LS1}(\mathcal {M}, \mathcal {N})=1-\root p \of {\dfrac{\sum \nolimits _{i=1}^n|\phi _{\mu }(x_i)+\phi _{\nu }(x_i)|}{n}},\mathcal {S}_{LS2}(\mathcal {M}, \mathcal {N})=1-\root p \of {\dfrac{\sum \nolimits _{i=1}^n|\varphi _{\mu }(x_i)+\varphi _{\nu }(x_i)|}{n}}\), |
\( \mathcal {S}_{LS3}(\mathcal {M}, \mathcal {N})=1-\root p \of {\dfrac{\sum \nolimits _{i=1}^n (\eta _1(x_i)+\eta _2(x_i)+\eta _3(x_i))^p}{3n}},\)
| |
\(\text {where} ~\phi _{\mu }(x_i)=\dfrac{|\mu _\mathcal {M}(x_i)-\mu _\mathcal {N}(x_i)|}{2},\phi _{\nu }(x_i)=\dfrac{|\nu _\mathcal {M}(x_i))-\nu _\mathcal {N}(x_i)|}{2},\varphi _{\mu }(x_i)=\dfrac{|m_{M1}(x_i)-m_{N1}(x_i)|}{2},\)
| |
\(\varphi _{\nu }(x_i)=\dfrac{|m_{M2}(x_i)-m_{N2}(x_i)|}{2},m_{M1}(x_i)= \dfrac{|\mu _{M}(x_i)+m_{M}(x_i)|}{2},m_{N1}(x_i)=\dfrac{|\mu _{N}(x_i)+m_{N}(x_i)|}{2}, \)
| |
\(m_{M2}(x_i)=\dfrac{|1-\nu _{M}(x_i)+m_{M}(x_i)|}{2},m_{N2}(x_i)=\dfrac{|1-\nu _{N}(x_i)+m_{N}(x_i)|}{2},m_{M}(x_i)=\dfrac{|\mu _{M}(x_i)+1-\nu _{M}(x_i)|}{2}, \)
| |
\(m_{N}(x_i)=\dfrac{|\mu _{N}(x_i)+1-\nu _{N}(x_i)|}{2} ,\eta _1(x_i)=\dfrac{|\mu _{M}(x_i)-\mu _{N}(x_i)|+|\nu _{M}(x_i)-\nu _{N}(x_i)|}{2},\)
| |
\( \eta _2(x_i)=\dfrac{|(\mu _{M}(x_i)-\nu _{M}(x_i))-(\mu _{N}(x_i)-\nu _{N}(x_i))|}{2}\), | |
\(\eta _3(x_i)=\text {max}\Bigg (\dfrac{\pi _\mathcal {M}(x_i)}{2},\dfrac{\pi _\mathcal {N}(x_i)}{2}\Bigg )-\text {min}\Bigg (\dfrac{\pi _\mathcal {M}(x_i)}{2},\dfrac{\pi _\mathcal {N}(x_i)}{2}\Bigg ).\)
| |
Mitchell [15] | \( \mathcal {S}_\mathcal {M}(\mathcal {M}, \mathcal {N})=\dfrac{\rho _{\mu }(\mathcal {M}, \mathcal {N})+\rho _{\nu }(\mathcal {M}, \mathcal {N})}{2}\), |
\(\text {where} ~\rho _{\mu }(\mathcal {M}, \mathcal {N})=1-\root p \of {\dfrac{\sum \nolimits _{i=1}^n|\mu _\mathcal {M}(x_i)-\mu _\mathcal {N}(x_i)|^p}{n}},\). | |
\(\rho _{\nu }(\mathcal {M}, \mathcal {N})=1-\root p \of {\dfrac{\sum \nolimits _{i=1}^n|\nu _\mathcal {M}(x_i)-\nu _\mathcal {N}(x_i)|^p}{n}}, \text {and} ~1\le p<\infty \)
| |
Ye [3] |
\(\mathcal {S}_Y(\mathcal {M}, \mathcal {N})=\dfrac{1}{n}\sum \nolimits _{i=1}^n\dfrac{\mu _\mathcal {M}(x_i)\mu _\mathcal {N}(x_i)+\nu _\mathcal {M}(x_i)\nu _\mathcal {N}(x_i)}{\sqrt{\mu _\mathcal {M}^2(x_i)+\nu _\mathcal {M}^2(x_i)} \sqrt{\mu _\mathcal {N}^2(x_i)+\nu _\mathcal {N}^2(x_i)}}\)
|
Wei and Wei [36] |
\(\mathcal {S}_W(\mathcal {M}, \mathcal {N})=\dfrac{1}{n}\sum \nolimits _{i=1}^n\dfrac{\mu _\mathcal {M}^2(x_i)\mu _\mathcal {N}^2(x_i)+\nu _\mathcal {M}^2(x_i)\nu _\mathcal {N}^2(x_i)}{\sqrt{\mu _\mathcal {M}^4(x_i)+\nu _\mathcal {M}^4(x_i)} \sqrt{\mu _\mathcal {N}^4(x_i)+\nu _\mathcal {N}^4(x_i)}}\)
|
Zhang [35] |
\(\mathcal {S}_Z(\mathcal {M}, \mathcal {N}) =\dfrac{1}{n}\sum \nolimits _{i=1}^n\dfrac{|\mu _\mathcal {M}^2(x_i)-\nu _\mathcal {N}^2(x_i)|+|\nu _\mathcal {M}^2(x_i)-\mu _\mathcal {N}^2(x_i)|+|\pi _\mathcal {M}^2(x_i)-\pi _\mathcal {N}^2(x_i)|}{|\mu _\mathcal {M}^2(x_i)-\mu _\mathcal {N}^2(x_i)|+|\nu _\mathcal {M}^2(x_i)-\nu _\mathcal {N}^2(x_i)|}\)
|
\(+|\pi _\mathcal {M}^2(x_i)-\pi _\mathcal {N}^2(x_i)|+|\mu _\mathcal {M}^2(x_i)-\nu _\mathcal {N}^2(x_i)| +|\nu _\mathcal {M}^2(x_i)-\mu _\mathcal {N}^2(x_i)|+|\pi _\mathcal {M}^2(x_i)-\pi _\mathcal {N}^2(x_i)|\)
| |
Peng et al. [49] |
\(\mathcal {S}_{P1}(\mathcal {M}, \mathcal {N})=1-\dfrac{\sum \nolimits _{i=1}^n|\mu _\mathcal {M}^2(x_i)-\nu _\mathcal {M}^2(x_i)-(\mu _\mathcal {N}^2(x_i)-\nu _\mathcal {N}^2(x_i))|}{2n} \)
|
\(\mathcal {S}_{P2}(\mathcal {M}, \mathcal {N})=\dfrac{1}{n}\sum \nolimits _{i=1}^n\dfrac{(\mu _\mathcal {M}^2(x_i)\bigwedge \mu _\mathcal {N}^2(x_i))+(\nu _\mathcal {M}^2(x_i)\bigwedge \nu _\mathcal {N}^2(x_i))}{(\mu _\mathcal {M}^2(x_i)\bigvee \mu _\mathcal {N}^2(x_i))+(\nu _\mathcal {M}^2(x_i)\bigvee \nu _\mathcal {N}^2(x_i))}\)
| |
\(\mathcal {S}_{P3}(\mathcal {M}, \mathcal {N})=\dfrac{1}{n}\sum \nolimits _{i=1}^n\dfrac{(\mu _\mathcal {M}^2(x_i)\bigwedge \mu _\mathcal {N}^2(x_i))+(1-\nu _\mathcal {M}^2(x_i))\bigwedge (1-\nu _\mathcal {N}^2(x_i))}{(\mu _\mathcal {M}^2(x_i)\bigvee \mu _\mathcal {N}^2(x_i))+(1-\nu _\mathcal {M}^2(x_i))\bigvee (1-\nu _\mathcal {N}^2(x_i))}\)
| |
Boran and Akay [16] |
\(\mathcal {S}_{BA}(\mathcal {M}, \mathcal {N})=1-\root p \of {\dfrac{\sum \nolimits _{i=1}^n \{ |t(\mu _\mathcal {M}(x_i)-\mu _\mathcal {N}(x_i))-(\nu _\mathcal {M}(x_i)-\nu _\mathcal {N}(x_i))|^p+|t(\nu _\mathcal {M}(x_i)-\nu _\mathcal {N}(x_i))-(\mu _\mathcal {M}(x_i)-\mu _\mathcal {N}(x_i))|^p \}}{2n(t+1)^p}}\)
|
Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 | |
---|---|---|---|---|---|---|
\(\mathcal {M}\)
|
\(\{(x,0.3,0.3)\} \)
|
\( \{(x,0.3,0.4)\}\)
|
\(\{(x,1,0)\} \)
|
\(\{(x,0.5,0.5)\} \)
|
\(\{(x,0.4,0.2)\} \)
|
\(\{(x,0.4,0.2)\} \)
|
\(\mathcal {N}\)
|
\(\{(x,0.4,0.4)\} \)
|
\( \{(x,0.4,0.3)\}\)
|
\( \{(x,0,0)\} \)
|
\(\{(x,0,0)\} \)
|
\( \{(x,0.5,0.3)\} \)
|
\(\{(x,0.5,0.2)\} \)
|
\(\mathcal {S}_{L} \) [7] | 0.9 | 0.9 | 0.2929 | 0.5 | 0.9 | 0.9293 |
\(\mathcal {S}_{\mathcal {R}}\) [8] |
1
| 0.9 | 0.5 |
1
|
1
| 0.95 |
\(\mathcal {S}_{CC}\) [9] | 0.9225 | 0.88 | 0.25 | 0.5 | 0.9225 | 0.8913 |
\(\mathcal {S}_{HY1}\) [10] | 0.9 | 0.9 |
0
| 0.5 | 0.9 | 0.9 |
\(\mathcal {S}_{HY2}\) [10] | 0.8495 | 0.8495 |
0
| 0.3775 | 0.8495 | 0.8495 |
\(\mathcal {S}_{HY3}\) [10] | 0.8182 | 0.8182 |
0
| 0.3333 | 0.8182 | 0.8182 |
\(\mathcal {S}_{HK}\) [11] | 0.9 | 0.9 | 0.5 | 0.5 | 0.9 | 0.95 |
\(\mathcal {S}_{LC} \) [12] |
1
| 0.9 | 0.5 |
1
|
1
| 0.95 |
\(\mathcal {S}_{LX} \) [13] | 0.95 | 0.9 | 0.5 | 0.75 | 0.95 | 0.95 |
\(\mathcal {S}_{LS1} \) [14] | 0.9 | 0.9 | 0.5 | 0.5 | 0.9 | 0.95 |
\(\mathcal {S}_{LS2} \) [14] | 0.95 | 0.9 | 0.5 | 0.75 | 0.95 | 0.95 |
\(\mathcal {S}_{LS3} \) [14] |
0.9333
| 0.9333 | 0.5 | 0.6667 | 0.9333 | 0.95 |
\(\mathcal {S}_{M} \) [15] | 0.9 | 0.9 | 0.5 | 0.5 | 0.9 | 0.95 |
\(\mathcal {S}_{Y} \) [3] |
1
| 0.96 | N/A | N/A | 0.9971 | 0.9965 |
\(\mathcal {S}_W \) [36] |
1
| 0.8546 | N/A | N/A | 0.9949 | 0.9963 |
\(\mathcal {S}_Z \) [35] | 0.5 |
0
| 0.5 | 0.5 |
0.6
| 0.7 |
\(\mathcal {S}_{P1}\) [49] |
1
| 0.93 | 0.5 |
1
| 0.98 | 0.955 |
\(\mathcal {S}_{P2}\) [49] | 0.5625 | 0.5625 |
0
|
0
| 0.5882 | 0.6897 |
\(\mathcal {S}_{P3}\) [49] | 0.8692 | 0.8692 | 0.5 | 0.6 | 0.8843 | 0.9256 |
\(\mathcal {S}_{BA}\) [16] | 0.967 | 0.9 | 0.5 | 0.8333 | 0.9667 | 0.95 |
\(\mathcal {S}_0(\text {proposed}) \)
| 0.8694 | 0.8694 | 0.3679 | 0.6065 | 0.8694 | 0.9139 |
\(\mathcal {S}_1(\text {proposed}) \)
| 0.9324 | 0.9324 | 0.6839 | 0.7788 | 0.9326 | 0.9570 |
Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 | |
---|---|---|---|---|---|---|
\(\mathcal {M}\)
|
\(\{(x,0.5,0.5)\} \)
|
\( \{(x,0.6,0.4)\}\)
|
\(\{(x,0,0.87)\} \)
|
\(\{(x,0.6,0.27)\} \)
|
\(\{(x,0.125,0.075)\} \)
|
\(\{(x,0.5,0.45)\} \)
|
\(\mathcal {N}\)
|
\(\{(x,0,0)\} \)
|
\( \{(x,0,0)\}\)
|
\( \{(x,0.28,0.55)\} \)
|
\(\{(x,0.28,0.55)\} \)
|
\( \{(x,0.175,0.025)\} \)
|
\(\{(x,0.55,0.4)\} \)
|
\(\mathcal {S}_{L} \) [7] | 0.5 | 0.4901 | 0.6993 | 0.6993 | 0.95 | 0.95 |
\(\mathcal {S}_{\mathcal {R}}\) [8] |
1
| 0.9 | 0.7 | 0.7 | 0.95 | 0.95 |
\(\mathcal {S}_{CC}\) [9] | 0.5 | 0.45 | 0.7395 | 0.7055 | 0.9125 | 0.95 |
\(\mathcal {S}_{HY1}\) [10] | 0.5 | 0.4 | 0.68 | 0.68 | 0.95 | 0.95 |
\(\mathcal {S}_{HY2}\) [10] | 0.3775 | 0.2862 | 0.5668 | 0.5668 | 0.9229 | 0.9229 |
\(\mathcal {S}_{HY3}\) [10] | 0.3333 | 0.25 | 0.5152 | 0.5152 | 0.9048 | 0.9048 |
\(\mathcal {S}_{HK}\) [11] | 0.5 | 0.5 | 0.7 | 0.7 | 0.95 | 0.95 |
\(\mathcal {S}_{LC} \) [12] |
1
| 0.9 | 0.7 | 0.7 | 0.95 | 0.95 |
\(\mathcal {S}_{LX} \) [13] | 0.75 | 0.7 | 0.7 | 0.7 | 0.95 | 0.95 |
\(\mathcal {S}_{LS1} \) [14] | 0.5 | 0.5 | 0.7 | 0.7 | 0.95 | 0.95 |
\(\mathcal {S}_{LS2} \) [14] | 0.75 | 0.75 | 0.7 | 0.7 | 0.95 | 0.95 |
\(\mathcal {S}_{LS3} \) [14] | 0.6667 | 0.6333 | 0.7933 | 0.7933 | 0.9667 | 0.9667 |
\(\mathcal {S}_{M} \) [15] | 0.5 | 0.5 | 0.7 | 0.7 | 0.95 | 0.95 |
\(\mathcal {S}_{Y} \) [3] | N/A | N/A | 0.8912 | 0.7794 | 0.9216 | 0.9946 |
\(\mathcal {S}_W \) [36] | N/A | N/A | 0.968 | 0.438 | 0.9476 | 0.9812 |
\(\mathcal {S}_Z \) [35] | 0.5 |
0.5
| 0.5989 | 0.1696 | 0.625 | 0.6557 |
\(\mathcal {S}_{P1}\) [49] |
1
| 0.9 | 0.7336 | 0.7444 | 0.99 | 0.9525 |
\(\mathcal {S}_{P2}\) [49] |
0
|
0
| 0.3621 | 0.2284 | 0.4483 | 0.8119 |
\(\mathcal {S}_{P3}\) [49] | 0.6 | 0.6176 | 0.3133 | 0.6028 | 0.9806 | 0.9168 |
\(\mathcal {S}_{BA}\) [16] | 0.8333 | 0.8333 | 0.7 | 0.7 | 0.95 | 0.95 |
\(\mathcal {S}_0(\text {proposed}) \)
| 0.6065 | 0.5945 | 0.5870 | 0.5998 | 0.9802 | 0.9094 |
\(\mathcal {S}_1(\text {proposed}) \)
| 0.7788 | 0.7749 | 0.7797 | 0.7747 | 0.9901 | 0.9536 |
Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 | |
---|---|---|---|---|---|---|
\(\mathcal {M}\)
|
\(\{(x,0.3,0.7)\} \)
|
\( \{(x,0.3,0.7)\}\)
|
\(\{(x,0.5,0.5)\} \)
|
\(\{(x,0.4,0.6)\} \)
|
\(\{(x,0.1,0.5)\} \)
|
\(\{(x,0.4,0.2)\} \)
|
\(\mathcal {N}\)
|
\(\{(x,0.4,0.6)\} \)
|
\( \{(x,0.2,0.8)\}\)
|
\( \{(x,0,0)\} \)
|
\(\{(x,0,0)\} \)
|
\( \{(x,0.2,0.3)\} \)
|
\(\{(x,0.2,0.3)\} \)
|
\(\mathcal {S}_{L} \) [7] | 0.6863 | 0.6863 | 0.5 | 0.4901 | 0.8419 | 0.8419 |
\(\mathcal {S}_{\mathcal {R}}\) [8] | 0.9 | 0.9 |
1
| 0.9 | 0.85 | 0.85 |
\(\mathcal {S}_{CC}\) [9] | 0.9 | 0.9 | 0.5 | 0.55 | 0.8438 | 0.7685 |
\(\mathcal {S}_{HY1}\) [10] | 0.9 | 0.9 | 0.5 | 0.4 | 0.8 | 0.8 |
\(\mathcal {S}_{HY2}\) [10] | 0.8494 | 0.8494 | 0.3775 | 0.2862 | 0.7132 | 0.7132 |
\(\mathcal {S}_{HY3}\) [10] | 0.8182 | 0.8182 | 0.3333 | 0.25 | 0.6667 | 0.6667 |
\(\mathcal {S}_{HK}\) [11] | 0.9 | 0.9 | 0.5 | 0.5 | 0.85 | 0.85 |
\(\mathcal {S}_{LC} \) [12] | 0.9 | 0.9 |
1
| 0.9 | 0.85 | 0.85 |
\(\mathcal {S}_{LX} \) [13] | 0.9 | 0.9 | 0.75 | 0.7 | 0.85 | 0.85 |
\(\mathcal {S}_{LS1} \) [14] | 0.9 | 0.9 | 0.5 | 0.5 | 0.85 | 0.85 |
\(\mathcal {S}_{LS2} \) [14] | 0.9 | 0.9 | 0.5 | 0.75 | 0.85 | 0.85 |
\(\mathcal {S}_{LS3} \) [14] | 0.95 | 0.95 | 0.6667 | 0.6333 | 0.8833 | 0.8833 |
\(\mathcal {S}_{M} \) [15] | 0.9 | 0.9 | 0.5 | 0.5 | 0.85 | 0.85 |
\(\mathcal {S}_{Y} \) [3] | 0.9832 | 0.9873 | N/A | N/A | 0.9249 | 0.8685 |
\(\mathcal {S}_W \) [36] | 0.9721 | 0.9929 | N/A | N/A | 0.9293 | 0.6156 |
\(\mathcal {S}_Z \) [35] | 0.7174 | 0.7857 |
0.5
|
0.5
| 0.5676 | 0.3684 |
\(\mathcal {S}_{P1}\) [49] | 0.9 | 0.9 |
1
| 0.9 | 0.905 | 0.915 |
\(\mathcal {S}_{P2}\) [49] | 0.6923 | 0.726 |
0
|
0
| 0.3448 | 0.32 |
\(\mathcal {S}_{P3}\) [49] | 0.75 | 0.6667 | 0.6 | 0.5517 | 0.8 | 0.8482 |
\(\mathcal {S}_{BA}\) [16] | 0.9 | 0.9 | 0.8333 | 0.8333 | 0.8667 | 0.8667 |
\(\mathcal {S}_0(\text {proposed}) \)
| 0.8187 | 0.8185 | 0.6065 | 0.5945 | 0.8270 | 0.8437 |
\(\mathcal {S}_1(\text {proposed}) \)
| 0.9052 | 0.9060 | 0.7788 | 0.7749 | 0.9113 | 0.9191 |
Applications related to pattern recognition
Similarity measures | Measurement values of \(\mathcal {Q}\) from | Ranking order | ||
---|---|---|---|---|
\(\mathcal {P}_1\)
|
\(\mathcal {P}_2\)
|
\(\mathcal {P}_3\)
| ||
Measure \(\text {PFC}^1\) proposed by Wei and Wei [36] | 0.96864 | 0.97113 | 0.98464 |
\(\mathcal {P}_3\succ \mathcal {P}_2\succ \mathcal {P}_1\)
|
Measure \(\text {PFC}^2\) proposed by Wei and Wei [36] | 0.62729 | 0.72373 | 0.92666 |
\(\mathcal {P}_3\succ \mathcal {P}_2\succ \mathcal {P}_1\)
|
Measure \(\text {Sm}\) proposed by [35] | 0.64018 | 0.63749 | 0.70645 |
\(\mathcal {P}_3\succ \mathcal {P}_1\succ \mathcal {P}_2\)
|
Measure \(\mathcal {S}_0\) proposed in this paper | 0.60585 | 0.63322 | 0.78205 |
\(\mathcal {P}_3\succ \mathcal {P}_2\succ \mathcal {P}_1\)
|
Measure \(\mathcal {S}_1\) proposed in this paper | 0.79017 | 0.80766 | 0.88802 |
\(\mathcal {P}_3\succ \mathcal {P}_2\succ \mathcal {P}_1\)
|
Measure \(\mathcal {S}_2\) proposed in this paper | 0.57250 | 0.58144 | 0.63113 |
\(\mathcal {P}_3\succ \mathcal {P}_2\succ \mathcal {P}_1\)
|
\(\mathcal {G}_1\)
|
\(\mathcal {G}_2\)
|
\(\mathcal {G}_3\)
|
\(\mathcal {G}_4\)
|
\(\mathcal {G}_5\)
|
\(\mathcal {G}_6\)
| |
---|---|---|---|---|---|---|
\(\mathcal {P}_1\)
| (0.2, 0.5) | (0.3, 0.8) | (0.4, 0.9) | (0.3, 0.7) | (0.2, 0.4) | (0.8, 0.4) |
\(\mathcal {P}_2\)
| (0.3, 0.5) | (0.8, 0.5) | (0.5, 0.6) | (0.5, 0.6) | (0.4, 0.7) | (0.6, 0.5) |
\(\mathcal {P}_3\)
| (0.4, 0.3) | (0.6, 0.8) | (0.6, 0.7) | (0.6, 0.8) | (0.8, 0.6) | (0.7, 0.4) |
\(\mathcal {P}_4\)
| (0.4, 0.7) | (0.6, 0.8) | (0.4, 0.7) | (0.7, 0.6) | (0.5, 0.7) | (0.5, 0.8) |
\(\mathcal {P}_5\)
| (0.6, 0.8) | (0.4, 0.7) | (0.7, 0.4) | (0.3, 0.4) | (0.7, 0.7) | (0.4, 0.7) |
Approach | Measurement values of the alternatives from \(\mathcal {P}_b\) | Ranking order | ||||
---|---|---|---|---|---|---|
\(\mathcal {P}_1\)
|
\(\mathcal {P}_2\)
|
\(\mathcal {P}_3\)
|
\(\mathcal {P}_4\)
|
\(\mathcal {P}_5\)
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Method by \(\text {PFC}^1\) proposed in [36] | 0.45365 | 0.64741 | 0.76863 | 0.42520 | 0.59741 |
\(\mathcal {P}_3\succ \mathcal {P}_2\succ \mathcal {P}_5\succ \mathcal {P}_1\succ \mathcal {P}_4\)
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Method by \(\text {PFC}^2\) proposed in [36] | 0.35854 | 0.51423 | 0.64996 | 0.39823 | 0.44496 |
\(\mathcal {P}_3\succ \mathcal {P}_2\succ \mathcal {P}_5\succ \mathcal {P}_4\succ \mathcal {P}_1\)
|
Method by \(\text {Sm}\) proposed in [35] | 0.45293 | 0.51224 | 0.533656 | 0.38015 | 0.45933 |
\(\mathcal {P}_3\succ \mathcal {P}_2\succ \mathcal {P}_5\succ \mathcal {P}_1\succ \mathcal {P}_4\)
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Method by \(\mathcal {S}_0\) proposed in the paper | 0.36337 | 0.38339 | 0.4134 | 0.27814 | 0.34522 |
\(\mathcal {P}_3\succ \mathcal {P}_2\succ \mathcal {P}_1\succ \mathcal {P}_4\succ \mathcal {P}_4\)
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Method by \(\mathcal {S}_1\) proposed in the paper | 0.60576 | 0.62673 | 0.6468 | 0.53046 | 0.59481 |
\(\mathcal {P}_3\succ \mathcal {P}_2\succ \mathcal {P}_1\succ \mathcal {P}_4\succ \mathcal {P}_4\)
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Method by \(\mathcal {S}_2\) proposed in the paper | 0.4397 | 0.45107 | 0.46275 | 0.37751 | 0.42857 |
\(\mathcal {P}_3\succ \mathcal {P}_2\succ \mathcal {P}_1\succ \mathcal {P}_4\succ \mathcal {P}_4\)
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