Introduction
Literature review
Some Fuzzy sets | Truth grade | Falsity grade | Phase term | Power full condition |
---|---|---|---|---|
FSs | \(\surd \) | \(\times \) | \(\times \) | \(\times \) |
IFSs | \(\surd \) | \(\surd \) | \(\times \) | \(\times \) |
PFSs | \(\surd \) | \(\surd \) | \(\times \) | \(\times \) |
Q-ROFSs | \(\surd \) | \(\surd \) | \(\times \) | \(\surd \) |
CFSs | \(\surd \) | \(\times \) | \(\surd \) | \(\times \) |
CIFSs | \(\surd \) | \(\surd \) | \(\surd \) | \(\times \) |
CPFSs | \(\surd \) | \(\surd \) | \(\surd \) | \(\times \) |
CQ-ROFSs | \(\surd \) | \(\surd \) | \(\surd \) | \(\surd \) |
Motivation and advantages
Main contributions
Preliminaries
Symbol | Meaning | Symbol | Meaning | Symbol | Meaning |
---|---|---|---|---|---|
\({\widetilde{\beth }}_{{\varvec{z}}}\) | Positive integers | \(X\) | Universal set | \(\Lambda \ge 1\) | Positive number |
\(\widetilde{\beth }\) | CQ-ROF set | \(x\) | Element of the universal set | \({\nabla }_{SV}\) | Score values |
\(\overline{\overline{{\Delta }_{\widetilde{\beth }}}}\) | Complex truth grade | \(\overline{\overline{{\Delta }_{\widetilde{rp}}}}\) | The real part of truth grade | \(\overline{\overline{{\Delta }_{\widetilde{ip}}}}\) | The imaginary part of truth grade |
\(\overline{\overline{{\Xi }_{\widetilde{\beth }}}}\) | Complex falsity grade | \(\overline{\overline{{\Xi }_{\widetilde{rp}}}}\) | The real part of falsity grade | \(\overline{\overline{{\Xi }_{\widetilde{ip}}}}\) | The imaginary part of falsity grade |
\({\Pi }_{rg}\) | Complex neutral grade | \({\Pi }_{rp}\) | The real part of the neutral grade | \({\Pi }_{ip}\) | The imaginary part of the neutral grade |
\({\nabla }_{AV}\) | Accuracy values | \(\varphi \) | Positive number |
Power Yager aggregation operators for CQ-ROFNs
MADM methods based on the proposed operators
\({C}_{1}\) | \({C}_{2}\) | \({C}_{3}\) | \({C}_{4}\) | |
---|---|---|---|---|
\({\widetilde{\beth }}_{1}\) | \(\left(\left(\mathrm{0.8,0.8}\right),\left(\mathrm{0.3,0.6}\right)\right)\) | \(\left(\left(\mathrm{0.81,0.81}\right),\left(\mathrm{0.31,0.61}\right)\right)\) | \(\left(\left(\mathrm{0.82,0.82}\right),\left(\mathrm{0.32,0.62}\right)\right)\) | \(\left(\left(\mathrm{0.83,0.83}\right),\left(\mathrm{0.33,0.63}\right)\right)\) |
\({\widetilde{\beth }}_{2}\) | \(\left(\left(\mathrm{0.7,0.6}\right),\left(\mathrm{0.5,0.6}\right)\right)\) | \(\left(\left(\mathrm{0.71,0.61}\right),\left(\mathrm{0.51,0.61}\right)\right)\) | \(\left(\left(\mathrm{0.72,0.62}\right),\left(\mathrm{0.52,0.62}\right)\right)\) | \(\left(\left(\mathrm{0.73,0.63}\right),\left(\mathrm{0.53,0.63}\right)\right)\) |
\({\widetilde{\beth }}_{3}\) | \(\left(\left(\mathrm{0.8,0.2}\right),\left(\mathrm{0.8,0.7}\right)\right)\) | \(\left(\left(\mathrm{0.81,0.21}\right),\left(\mathrm{0.81,0.71}\right)\right)\) | \(\left(\left(\mathrm{0.82,0.22}\right),\left(\mathrm{0.82,0.72}\right)\right)\) | \(\left(\left(\mathrm{0.83,0.23}\right),\left(\mathrm{0.83,0.73}\right)\right)\) |
\({\widetilde{\beth }}_{4}\) | \(\left(\left(\mathrm{0.8,0.7}\right),\left(\mathrm{0.8,0.8}\right)\right)\) | \(\left(\left(\mathrm{0.81,0.71}\right),\left(\mathrm{0.81,0.81}\right)\right)\) | \(\left(\left(\mathrm{0.82,0.72}\right),\left(\mathrm{0.82,0.82}\right)\right)\) | \(\left(\left(\mathrm{0.83,0.73}\right),\left(\mathrm{0.83,0.83}\right)\right)\) |
\({\widetilde{\beth }}_{5}\) | \(\left(\left(\mathrm{0.7,0.5}\right),\left(\mathrm{0.4,0.5}\right)\right)\) | \(\left(\left(\mathrm{0.71,0.51}\right),\left(\mathrm{0.41,0.51}\right)\right)\) | \(\left(\left(\mathrm{0.72,0.52}\right),\left(\mathrm{0.42,0.52}\right)\right)\) | \(\left(\left(\mathrm{0.73,0.53}\right),\left(\mathrm{0.43,0.53}\right)\right)\) |
Comprehensive values for the different alternatives | \(CQ-ROFPYA\) | \(CQ-ROFPYG\) |
---|---|---|
\({\widetilde{\beth }}_{1}\) | \(\left(\left(\mathrm{0.8164,0.8164}\right),\left(\mathrm{0.000001,0.0008}\right)\right)\) | \(\left(\left(\mathrm{0.0139,0.0139}\right),\left(\mathrm{0.3186,0.6169}\right)\right)\) |
\({\widetilde{\beth }}_{2}\) | \(\left(\left(\mathrm{0.7166,0.6169}\right),\left(\mathrm{0.0001,0.0008}\right)\right)\) | \(\left(\left(\mathrm{0.036,0.0008}\right),\left(\mathrm{0.5173,0.6169}\right)\right)\) |
\({\widetilde{\beth }}_{3}\) | \(\left(\left(\mathrm{0.8164,0.2199}\right),\left(\mathrm{0.0139,0.0036}\right)\right)\) | \(\left(\left(\mathrm{0.0139,0.0002}\right),\left(\mathrm{0.8164,0.7166}\right)\right)\) |
\({\widetilde{\beth }}_{4}\) | \(\left(\left(\mathrm{0.8164,0.2199}\right),\left(\mathrm{0.0139,0.0139}\right)\right)\) | \(\left(\left(\mathrm{0.0139,0.000002}\right),\left(\mathrm{0.8164,0.8164}\right)\right)\) |
\({\widetilde{\beth }}_{5}\) | \(\left(\left(\mathrm{0.7166,0.5173}\right),\left(0.0.\mathrm{00004,0.0001}\right)\right)\) | \(\left(\left(\mathrm{0.0036,0.0001}\right),\left(\mathrm{0.4178,0.5173}\right)\right)\) |
Score Values | \(CQ-ROFPYA\) | \(CQ-ROFPYG\) |
---|---|---|
\({\widetilde{\beth }}_{1}\) | 0.816 | − 0.4539 |
\({\widetilde{\beth }}_{2}\) | 0.6663 | − 0.5649 |
\({\widetilde{\beth }}_{3}\) | 0.5095 | − 0.7596 |
\({\widetilde{\beth }}_{4}\) | 0.7527 | − 0.8077 |
\({\widetilde{\beth }}_{5}\) | 0.6169 | − 0.4656 |
Methods | \(\mathrm{Ranking values}\) |
---|---|
\(\mathrm{CQ}-\mathrm{ROFPYA operator}\) | \({\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{4}>{\widetilde{\beth }}_{2}>{\widetilde{\beth }}_{5}>{\widetilde{\beth }}_{3}\) |
\(CQ-\mathrm{ROFPYG operator}\) | \({\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{5}>{\widetilde{\beth }}_{2}>{\widetilde{\beth }}_{3}>{\widetilde{\beth }}_{4}\) |
\({C}_{1}\) | \({C}_{2}\) | \({C}_{3}\) | \({C}_{4}\) | |
---|---|---|---|---|
\({\widetilde{\beth }}_{1}\) | \(\left(\left(\mathrm{0.6,0.7}\right),\left(\mathrm{0.5,0.6}\right)\right)\) | \(\left(\left(\mathrm{0.61,0.71}\right),\left(\mathrm{0.51,0.61}\right)\right)\) | \(\left(\left(\mathrm{0.62,0.72}\right),\left(\mathrm{0.52,0.62}\right)\right)\) | \(\left(\left(\mathrm{0.63,0.73}\right),\left(\mathrm{0.53,0.63}\right)\right)\) |
\({\widetilde{\beth }}_{2}\) | \(\left(\left(\mathrm{0.6,0.6}\right),\left(\mathrm{0.6,0.5}\right)\right)\) | \(\left(\left(\mathrm{0.61,0.61}\right),\left(\mathrm{0.61,0.51}\right)\right)\) | \(\left(\left(\mathrm{0.62,0.62}\right),\left(\mathrm{0.62,0.52}\right)\right)\) | \(\left(\left(\mathrm{0.63,0.63}\right),\left(\mathrm{0.63,0.53}\right)\right)\) |
\({\widetilde{\beth }}_{3}\) | \(\left(\left(\mathrm{0.8,0.7}\right),\left(\mathrm{0.3,0.4}\right)\right)\) | \(\left(\left(\mathrm{0.81,0.71}\right),\left(\mathrm{0.31,0.41}\right)\right)\) | \(\left(\left(\mathrm{0.82,0.72}\right),\left(\mathrm{0.32,0.42}\right)\right)\) | \(\left(\left(\mathrm{0.83,0.73}\right),\left(\mathrm{0.33,0.43}\right)\right)\) |
\({\widetilde{\beth }}_{4}\) | \(\left(\left(\mathrm{0.7,0.8}\right),\left(\mathrm{0.2,0.3}\right)\right)\) | \(\left(\left(\mathrm{0.71,0.81}\right),\left(\mathrm{0.21,0.31}\right)\right)\) | \(\left(\left(\mathrm{0.72,0.82}\right),\left(\mathrm{0.22,0.32}\right)\right)\) | \(\left(\left(\mathrm{0.73,0.83}\right),\left(\mathrm{0.23,0.33}\right)\right)\) |
\({\widetilde{\beth }}_{5}\) | \(\left(\left(\mathrm{0.7,0.5}\right),\left(\mathrm{0.4,0.5}\right)\right)\) | \(\left(\left(\mathrm{0.71,0.51}\right),\left(\mathrm{0.41,0.51}\right)\right)\) | \(\left(\left(\mathrm{0.72,0.52}\right),\left(\mathrm{0.42,0.52}\right)\right)\) | \(\left(\left(\mathrm{0.73,0.53}\right),\left(\mathrm{0.43,0.53}\right)\right)\) |
comprehensive values for the different alternatives | \(CQ-ROFPYA\) | \(CQ-ROFPYG\) |
---|---|---|
\({\widetilde{\beth }}_{1}\) | \(\left(\left(\mathrm{0.6153,0.7153}\right),\left(\mathrm{0.1428,0.2115}\right)\right)\) | \(\left(\left(\mathrm{0.2115,0.3008}\right),\left(\mathrm{0.5154,0.6153}\right)\right)\) |
\({\widetilde{\beth }}_{2}\) | \(\left(\left(\mathrm{0.6153,0.6153}\right),\left(\mathrm{0.2115,0.1428}\right)\right)\) | \(\left(\left(\mathrm{0.2115,0.2115}\right),\left(\mathrm{0.6153,0.5154}\right)\right)\) |
\({\widetilde{\beth }}_{3}\) | \(\left(\left(\mathrm{0.8152,0.7153}\right),\left(\mathrm{0.051,0.0902}\right)\right)\) | \(\left(\left(\mathrm{0.4202,0.3008}\right),\left(\mathrm{0.3156,0.4155}\right)\right)\) |
\({\widetilde{\beth }}_{4}\) | \(\left(\left(\mathrm{0.7153,0.8152}\right),\left(\mathrm{0.0234,0.051}\right)\right)\) | \(\left(\left(\mathrm{0.3008,0.4202}\right),\left(\mathrm{0.2159,0.3156}\right)\right)\) |
\({\widetilde{\beth }}_{5}\) | \(\left(\left(\mathrm{0.7153,0.5154}\right),\left(\mathrm{0.0902,0.1428}\right)\right)\) | \(\left(\left(\mathrm{0.3008,0.1428}\right),\left(\mathrm{0.4155,0.5154}\right)\right)\) |
Score values | \(\mathrm{CQ}-\mathrm{ROFPYA}\) | \(\mathrm{CQ}-\mathrm{ROFPYG}\) |
---|---|---|
\({\widetilde{\beth }}_{1}\) | 0.4881 | − 0.3092 |
\({\widetilde{\beth }}_{2}\) | 0.4382 | − 0.3539 |
\({\widetilde{\beth }}_{3}\) | 0.6947 | − 0.005 |
\({\widetilde{\beth }}_{4}\) | 0.728 | 0.0948 |
\({\widetilde{\beth }}_{5}\) | 0.4988 | − 0.2436 |
Methods | \(\mathrm{Ranking values}\) |
---|---|
\(\mathrm{CQ}-\mathrm{ROFPYA operator}\) | \({\widetilde{\beth }}_{4}>{\widetilde{\beth }}_{3}>{\widetilde{\beth }}_{5}>{\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{2}\) |
\(\mathrm{CQ}-\mathrm{ROFPYG operator}\) | \({\widetilde{\beth }}_{4}>{\widetilde{\beth }}_{3}>{\widetilde{\beth }}_{5}>{\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{2}\) |
\({C}_{1}\) | \({C}_{2}\) | \({C}_{3}\) | \({C}_{4}\) | |
---|---|---|---|---|
\({\widetilde{\beth }}_{1}\) | \(\left(\left(\mathrm{0.5,0.4}\right),\left(\mathrm{0.1,0.2}\right)\right)\) | \(\left(\left(\mathrm{0.51,0.41}\right),\left(\mathrm{0.11,0.21}\right)\right)\) | \(\left(\left(\mathrm{0.52,0.42}\right),\left(\mathrm{0.12,0.22}\right)\right)\) | \(\left(\left(\mathrm{0.53,0.43}\right),\left(\mathrm{0.13,0.23}\right)\right)\) |
\({\widetilde{\beth }}_{2}\) | \(\left(\left(\mathrm{0.6,0.2}\right),\left(\mathrm{0.1,0.2}\right)\right)\) | \(\left(\left(\mathrm{0.61,0.21}\right),\left(\mathrm{0.11,0.21}\right)\right)\) | \(\left(\left(\mathrm{0.62,0.22}\right),\left(\mathrm{0.12,0.22}\right)\right)\) | \(\left(\left(\mathrm{0.63,0.23}\right),\left(\mathrm{0.13,0.23}\right)\right)\) |
\({\widetilde{\beth }}_{3}\) | \(\left(\left(\mathrm{0.7,0.1}\right),\left(\mathrm{0.1,0.2}\right)\right)\) | \(\left(\left(\mathrm{0.71,0.11}\right),\left(\mathrm{0.11,0.21}\right)\right)\) | \(\left(\left(\mathrm{0.72,0.12}\right),\left(\mathrm{0.12,0.22}\right)\right)\) | \(\left(\left(\mathrm{0.73,0.13}\right),\left(\mathrm{0.13,0.23}\right)\right)\) |
\({\widetilde{\beth }}_{4}\) | \(\left(\left(\mathrm{0.6,0.6}\right),\left(\mathrm{0.2,0.2}\right)\right)\) | \(\left(\left(\mathrm{0.61,0.61}\right),\left(\mathrm{0.21,0.21}\right)\right)\) | \(\left(\left(\mathrm{0.62,0.62}\right),\left(\mathrm{0.22,0.22}\right)\right)\) | \(\left(\left(\mathrm{0.63,0.63}\right),\left(\mathrm{0.23,0.23}\right)\right)\) |
\({\widetilde{\beth }}_{5}\) | \(\left(\left(\mathrm{0.7,0.5}\right),\left(\mathrm{0.1,0.3}\right)\right)\) | \(\left(\left(\mathrm{0.71,0.51}\right),\left(\mathrm{0.11,0.31}\right)\right)\) | \(\left(\left(\mathrm{0.72,0.52}\right),\left(\mathrm{0.12,0.32}\right)\right)\) | \(\left(\left(\mathrm{0.73,0.53}\right),\left(\mathrm{0.13,0.33}\right)\right)\) |
comprehensive values for the different alternatives | \(CQ-ROFPYA\) | \(CQ-ROFPYG\) |
---|---|---|
\({\widetilde{\beth }}_{1}\) | \(\left(\left(\mathrm{0.5151,0.5142}\right),\left(\mathrm{0.1149,0.2149}\right)\right)\) | \(\left(\left(\mathrm{0.5149,0.4149}\right),\left(\mathrm{0.1155,0.2153}\right)\right)\) |
\({\widetilde{\beth }}_{2}\) | \(\left(\left(\mathrm{0.6151,0.2153}\right),\left(\mathrm{0.1149,0.2149}\right)\right)\) | \(\left(\left(\mathrm{0.6148,0.2149}\right),\left(\mathrm{0.1155,0.2153}\right)\right)\) |
\({\widetilde{\beth }}_{3}\) | \(\left(\left(\mathrm{0.7151,0.1155}\right),\left(\mathrm{0.1149,0.2149}\right)\right)\) | \(\left(\left(\mathrm{0.7148,0.1149}\right),\left(\mathrm{0.1155,0.2153}\right)\right)\) |
\({\widetilde{\beth }}_{4}\) | \(\left(\left(\mathrm{0.6151,0.6151}\right),\left(\mathrm{0.2149,0.2149}\right)\right)\) | \(\left(\left(\mathrm{0.6148,0.6148}\right),\left(\mathrm{0.2153,0.2153}\right)\right)\) |
\({\widetilde{\beth }}_{5}\) | \(\left(\left(\mathrm{0.7151,0.5151}\right),\left(\mathrm{0.1149,0.3149}\right)\right)\) | \(\left(\left(\mathrm{0.7148,0.5149}\right),\left(\mathrm{0.1155,0.3152}\right)\right)\) |
Score Values | \(CQ-ROFPYA\) | \(CQ-ROFPYG\) |
---|---|---|
\({\widetilde{\beth }}_{1}\) | 0.3002 | 0.2995 |
\({\widetilde{\beth }}_{2}\) | 0.2503 | 0.2495 |
\({\widetilde{\beth }}_{3}\) | 0.2504 | 0.2494 |
\({\widetilde{\beth }}_{4}\) | 0.4002 | 0.3995 |
\({\widetilde{\beth }}_{5}\) | 0.4002 | 0.3995 |
Methods | \(\mathrm{Ranking values}\) |
---|---|
\(\mathrm{CQ}-\mathrm{ROFPYA operator}\) | \({\widetilde{\beth }}_{5}>{\widetilde{\beth }}_{4}>{\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{3}>{\widetilde{\beth }}_{2}\) |
\(\mathrm{CQ}-\mathrm{ROFPYG operator}\) | \({\widetilde{\beth }}_{5}>{\widetilde{\beth }}_{4}>{\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{3}>{\widetilde{\beth }}_{2}\) |
Comparative analysis
\(\mathrm{Methods}\) | \(\mathrm{Score values}\) | \(\mathrm{Ranking values}\) |
---|---|---|
Xu [12] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Xu and Yager [13] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Xu [14] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Rahman et al. [15] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Wei and Lu [16] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Shahzadi and Akram [17] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Liu and Wang [18] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Riaz et al. [19] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Akram and Shahzadi [20] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Garg and Rani [21] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Garg and Rani [22] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Rani and Garg [23] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Akram et al. [24] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Garg et al. [25] | 0.351,0.101,− 0.2483,− 0.0491,0.1511 | \({\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{5}>{\widetilde{\beth }}_{2}>{\widetilde{\beth }}_{4}>{\widetilde{\beth }}_{3}\) |
CQ-ROFPYA operator | 0.816,0.6663,0.5095,0.7527,0.6169 | \({\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{4}>{\widetilde{\beth }}_{2}>{\widetilde{\beth }}_{5}>{\widetilde{\beth }}_{3}\) |
CQ-ROFPYG operator | − 0.4539,− 0.5649,− 0.7596,− 0.8077,− 0.4656 | \({\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{5}>{\widetilde{\beth }}_{2}>{\widetilde{\beth }}_{3}>{\widetilde{\beth }}_{4}\) |
Methods | Score values | Ranking values |
---|---|---|
Xu [12] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Xu and Yager [13] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Xu [14] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Rahman et al. [15] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Wei and Lu [16] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Shahzadi and Akram [17] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Liu and Wang [18] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Riaz et al. [19] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Akram and Shahzadi [20] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Garg and Rani [21] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Garg and Rani [22] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Rani and Garg [23] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Akram et al. [24] | 0.4862,0.4362,0.6909,0.7236,0.4959 | \({\widetilde{\beth }}_{4}>{\widetilde{\beth }}_{3}>{\widetilde{\beth }}_{5}>{\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{2}\) |
Garg et al. [25] | 0.1004,0.0503,0.4005,0.5006,0.1504 | \({\widetilde{\beth }}_{4}>{\widetilde{\beth }}_{3}>{\widetilde{\beth }}_{5}>{\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{2}\) |
CQ-ROFPYA operator | 0.4881,0.4382,0.6947,0.728,0.4988 | \({\widetilde{\beth }}_{4}>{\widetilde{\beth }}_{3}>{\widetilde{\beth }}_{5}>{\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{2}\) |
CQ-ROFPYG operator | − 0.3092,− 0.3539,− 0.005,0.0948,− 0.2436 | \({\widetilde{\beth }}_{4}>{\widetilde{\beth }}_{3}>{\widetilde{\beth }}_{5}>{\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{2}\) |
Methods | Score values | Ranking values |
---|---|---|
Xu [12] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Xu and Yager [13] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Xu [14] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Rahman et al. [15] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Wei and Lu [16] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Shahzadi and Akram [17] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Liu and Wang [18] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Riaz et al. [19] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Akram and Shahzadi [20] | \(\mathrm{Restricted}/\mathrm{Limited}\) | \(\mathrm{Restricted}/\mathrm{Limited}\) |
Garg and Rani [21] | 0.3004,0.2504,0.2504,0.4003,0.4004 | \({\widetilde{\beth }}_{5}>{\widetilde{\beth }}_{4}>{\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{3}>{\widetilde{\beth }}_{2}\) |
Garg and Rani [22] | 0.2998,0.2498,0.2497,0.3999,0.3999 | \({\widetilde{\beth }}_{5}>{\widetilde{\beth }}_{4}>{\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{2}>{\widetilde{\beth }}_{3}\) |
Rani and Garg [23] | 0.3005,0.2505,0.2506,0.4005,0.4005 | \({\widetilde{\beth }}_{5}>{\widetilde{\beth }}_{4}>{\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{3}>{\widetilde{\beth }}_{2}\) |
Akram et al. [24] | 0.3002,0.2502,0.2503,0.4001,0.4001 | \({\widetilde{\beth }}_{5}>{\widetilde{\beth }}_{4}>{\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{3}>{\widetilde{\beth }}_{2}\) |
Garg et al. [25] | 0.3005,0.2505,0.2506,0.4005,0.4005 | \({\widetilde{\beth }}_{5}>{\widetilde{\beth }}_{4}>{\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{3}>{\widetilde{\beth }}_{2}\) |
CQ-ROFPYA operator | 0.3002,0.2503,0.2504,0.4002,0.4002 | \({\widetilde{\beth }}_{5}>{\widetilde{\beth }}_{4}>{\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{3}>{\widetilde{\beth }}_{2}\) |
CQ-ROFPYG operator | 0.2995,0.2495,0.2494,0.3995,0.3995 | \({\widetilde{\beth }}_{5}>{\widetilde{\beth }}_{4}>{\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{2}>{\widetilde{\beth }}_{3}\) |