Introduction
Preliminaries
Cq-ROFSs and their operational laws
Complex q-rung orthopair fuzzy Einstein arithmetic aggregation operators
Complex q-rung orthopair fuzzy Einstein weighted averaging operator
Complex q-rung orthopair fuzzy Einstein ordered weighted averaging operator
Generalized complex q-rung orthopair fuzzy Einstein weighted averaging operator
Generalized complex q-rung orthopair fuzzy Einstein ordered weighted averaging operator
MAGDM approach based on proposed operators
Description of decision-making problem
Symbols | \({{{G}}}_{1}\) | \({{{G}}}_{2}\) | \(\ldots\ldots\ldots\ldots\) | \({{{G}}}_{{{n}}}\) |
---|---|---|---|---|
\({{{x}}}_{1}\) | \(\left({\mathcal{T}}_{11}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{T}}_{11}}},{\mathcal{N}}_{11}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{N}}_{11}}}\right)\) | \(\left({\mathcal{T}}_{12}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{T}}_{12}}},{\mathcal{N}}_{12}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{N}}_{12}}}\right)\) | \(\ldots\ldots\ldots\ldots\) | \(\left({\mathcal{T}}_{1n}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{T}}_{1n}}},{\mathcal{N}}_{1n}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{N}}_{1n}}}\right)\) |
\({{{x}}}_{2}\) | \(\left({\mathcal{T}}_{21}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{T}}_{21}}},{\mathcal{N}}_{21}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{N}}_{21}}}\right)\) | \(\left({\mathcal{T}}_{22}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{T}}_{22}}},{\mathcal{N}}_{22}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{N}}_{22}}}\right)\) | \(\ldots\ldots\ldots\ldots\) | \(\left({\mathcal{T}}_{2n}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{T}}_{2n}}},{\mathcal{N}}_{2n}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{N}}_{2n}}}\right)\) |
\({{{x}}}_{3}\) | \(\left({\mathcal{T}}_{31}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{T}}_{31}}},{\mathcal{N}}_{31}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{N}}_{31}}}\right)\) | \(\left({\mathcal{T}}_{32}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{T}}_{32}}},{\mathcal{N}}_{32}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{N}}_{32}}}\right)\) | \(\ldots\ldots\ldots\ldots\) | \(\left({\mathcal{T}}_{3n}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{T}}_{3n}}},{\mathcal{N}}_{3n}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{N}}_{3n}}}\right)\) |
\(\ldots\ldots\) | \(\ldots\ldots\) | \(\ldots\ldots\) | \(\ldots\ldots\) | \(\ldots\ldots\) |
\({{{x}}}_{{{m}}}\) | \(\left({\mathcal{T}}_{m1}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{T}}_{m1}}},{\mathcal{N}}_{m1}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{N}}_{m1}}}\right)\) | \(\left({\mathcal{T}}_{m2}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{T}}_{m2}}},{\mathcal{N}}_{m2}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{N}}_{m2}}}\right)\) | \(\ldots\ldots\ldots\ldots\) | \(\left({\mathcal{T}}_{mn}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{T}}_{mn}}},{\mathcal{N}}_{mn}{\text{e}}^{i2\pi {\mathcal{W}}_{{\mathcal{N}}_{mn}}}\right)\) |
Illustrative numerical example
Symbols | \({{{x}}}_{1}\) | \({{{x}}}_{2}\) | \({{{x}}}_{3}\) | \({{{x}}}_{4}\) | \({{{x}}}_{5}\) |
---|---|---|---|---|---|
Representations | Computer company | Furniture company | Car company | Chemical company | Food company |
Symbols | \({{{G}}}_{1}\) | \({{{G}}}_{2}\) | \({{{G}}}_{3}\) | \({{{G}}}_{4}\) |
---|---|---|---|---|
Representations | Technical ability | Accepted benefits | Competitive power in market | Management capability |
Symbols | \({{{G}}}_{1}\) | \({{{G}}}_{2}\) | \({{{G}}}_{3}\) | \({{{G}}}_{4}\) |
---|---|---|---|---|
\({{{x}}}_{1}\) | \(\left(0.22{\text{e}}^{i2\pi \left(0.31\right)},0.4{\text{e}}^{i2\pi \left(0.45\right)}\right)\) | \(\left(0.35{\text{e}}^{i2\pi \left(0.40\right)},0.33{\text{e}}^{i2\pi \left(0.3\right)}\right)\) | \(\left(0.5{\text{e}}^{i2\pi \left(0.45\right)},0.23{\text{e}}^{i2\pi \left(0.11\right)}\right)\) | \(\left(0.21{\text{e}}^{i2\pi \left(0.26\right)},0.19{\text{e}}^{i2\pi \left(0.33\right)}\right)\) |
\({{{x}}}_{2}\) | \(\left(0.24{\text{e}}^{i2\pi \left(0.31\right)},0.41{\text{e}}^{i2\pi \left(0.46\right)}\right)\) | \(\left(0.36{\text{e}}^{i2\pi \left(0.41\right)},0.34{\text{e}}^{i2\pi \left(0.31\right)}\right)\) | \(\left(0.51{\text{e}}^{i2\pi \left(0.46\right)},0.24{\text{e}}^{i2\pi \left(0.12\right)}\right)\) | \(\left(0.22{\text{e}}^{i2\pi \left(0.27\right)},0.20{\text{e}}^{i2\pi \left(0.23\right)}\right)\) |
\({{{x}}}_{3}\) | \(\left(0.25{\text{e}}^{i2\pi \left(0.33\right)},0.42{\text{e}}^{i2\pi \left(0.47\right)}\right)\) | \(\left(0.37{\text{e}}^{i2\pi \left(0.42\right)},0.35{\text{e}}^{i2\pi \left(0.32\right)}\right)\) | \(\left(0.52{\text{e}}^{i2\pi \left(0.47\right)},0.25{\text{e}}^{i2\pi \left(0.13\right)}\right)\) | \(\left(0.23{\text{e}}^{i2\pi \left(0.28\right)},0.21{\text{e}}^{i2\pi \left(0.24\right)}\right)\) |
\({{{x}}}_{4}\) | \(\left(0.27{\text{e}}^{i2\pi \left(0.34\right)},0.43{\text{e}}^{i2\pi \left(0.48\right)}\right)\) | \(\left(0.38{\text{e}}^{i2\pi \left(0.43\right)},0.36{\text{e}}^{i2\pi \left(0.33\right)}\right)\) | \(\left(0.53{\text{e}}^{i2\pi \left(0.48\right)},0.26{\text{e}}^{i2\pi \left(0.14\right)}\right)\) | \(\left(0.24{\text{e}}^{i2\pi \left(0.29\right)},0.22{\text{e}}^{i2\pi \left(0.25\right)}\right)\) |
\({{{x}}}_{5}\) | \(\left(0.35{\text{e}}^{i2\pi \left(0.35\right)},0.44{\text{e}}^{i2\pi \left(0.49\right)}\right)\) | \(\left(0.39{\text{e}}^{i2\pi \left(0.44\right)},0.37{\text{e}}^{i2\pi \left(0.34\right)}\right)\) | \(\left(0.54{\text{e}}^{i2\pi \left(0.49\right)},0.27{\text{e}}^{i2\pi \left(0.15\right)}\right)\) | \(\left(0.25{\text{e}}^{i2\pi \left(0.30\right)},0.23{\text{e}}^{i2\pi \left(0.26\right)}\right)\) |
Symbols | \({{{G}}}_{1}\) | \({{{G}}}_{2}\) | \({{{G}}}_{3}\) | \({{{G}}}_{4}\) |
---|---|---|---|---|
\({{{x}}}_{1}\) | \(\left(0.4{\text{e}}^{i2\pi \left(0.45\right)},0.22{\text{e}}^{i2\pi \left(0.31\right)}\right)\) | \(\left(0.35{\text{e}}^{i2\pi \left(0.40\right)},0.33{\text{e}}^{i2\pi \left(0.3\right)}\right)\) | \(\left(0.5{\text{e}}^{i2\pi \left(0.45\right)},0.23{\text{e}}^{i2\pi \left(0.11\right)}\right)\) | \(\left(0.21{\text{e}}^{i2\pi \left(0.26\right)},0.19{\text{e}}^{i2\pi \left(0.33\right)}\right)\) |
\({{{x}}}_{2}\) | \(\left(0.41{\text{e}}^{i2\pi \left(0.46\right)},0.24{\text{e}}^{i2\pi \left(0.31\right)}\right)\) | \(\left(0.36{\text{e}}^{i2\pi \left(0.41\right)},0.34{\text{e}}^{i2\pi \left(0.31\right)}\right)\) | \(\left(0.51{\text{e}}^{i2\pi \left(0.46\right)},0.24{\text{e}}^{i2\pi \left(0.12\right)}\right)\) | \(\left(0.22{\text{e}}^{i2\pi \left(0.27\right)},0.20{\text{e}}^{i2\pi \left(0.23\right)}\right)\) |
\({{{x}}}_{3}\) | \(\left(0.42{\text{e}}^{i2\pi \left(0.47\right)},0.25{\text{e}}^{i2\pi \left(0.33\right)}\right)\) | \(\left(0.37{\text{e}}^{i2\pi \left(0.42\right)},0.35{\text{e}}^{i2\pi \left(0.32\right)}\right)\) | \(\left(0.52{\text{e}}^{i2\pi \left(0.47\right)},0.25{\text{e}}^{i2\pi \left(0.13\right)}\right)\) | \(\left(0.23{\text{e}}^{i2\pi \left(0.28\right)},0.21{\text{e}}^{i2\pi \left(0.24\right)}\right)\) |
\({{{x}}}_{4}\) | \(\left(0.43{\text{e}}^{i2\pi \left(0.48\right)},0.27{\text{e}}^{i2\pi \left(0.34\right)}\right)\) | \(\left(0.38{\text{e}}^{i2\pi \left(0.43\right)},0.36{\text{e}}^{i2\pi \left(0.33\right)}\right)\) | \(\left(0.53{\text{e}}^{i2\pi \left(0.48\right)},0.26{\text{e}}^{i2\pi \left(0.14\right)}\right)\) | \(\left(0.24{\text{e}}^{i2\pi \left(0.29\right)},0.22{\text{e}}^{i2\pi \left(0.25\right)}\right)\) |
\({{{x}}}_{5}\) | \(\left(0.44{\text{e}}^{i2\pi \left(0.49\right)},0.35{\text{e}}^{i2\pi \left(0.35\right)}\right)\) | \(\left(0.39{\text{e}}^{i2\pi \left(0.44\right)},0.37{\text{e}}^{i2\pi \left(0.34\right)}\right)\) | \(\left(0.54{\text{e}}^{i2\pi \left(0.49\right)},0.27{\text{e}}^{i2\pi \left(0.15\right)}\right)\) | \(\left(0.25{\text{e}}^{i2\pi \left(0.30\right)},0.23{\text{e}}^{i2\pi \left(0.26\right)}\right)\) |
Method | Getting values |
---|---|
\( r_{1} = {\text{GCq - ROFOWA}}\left( {r_{{11}} ,r_{{12}} , \ldots ,r_{{1n}} } \right) \) | \(\left(0.317{\text{e}}^{i2\pi \left(0.319\right)},0.082{\text{e}}^{i2\pi \left(0.073\right)}\right)\) |
\( r_{2} = {\text{GCq - ROFOWA}}\left( {r_{{21}} ,r_{{22}} , \ldots ,r_{{2n}} } \right) \) | \(\left(0.321{\text{e}}^{i2\pi \left(0.323\right)},0.088{\text{e}}^{i2\pi \left(0.069\right)}\right)\) |
\( r_{3} = {\text{GCq - ROFOWA}}\left( {r_{{31}} ,r_{{32}} , \ldots ,r_{{3n}} } \right) \) | \(\left(0.325{\text{e}}^{i2\pi \left(0.327\right)},0.094{\text{e}}^{i2\pi \left(0.075\right)}\right)\) |
\( r_{4} = {\text{GCq - ROFOWA}}\left( {r_{{41}} ,r_{{42}} , \ldots ,r_{{4n}} } \right) \) | \(\left(0.329{\text{e}}^{i2\pi \left(0.331\right)},0.1{\text{e}}^{i2\pi \left(0.08\right)}\right)\) |
\( r_{5} = {\text{GCq - ROFOWA}}\left( {r_{{51}} ,r_{{52}} , \ldots ,r_{{5n}} } \right) \) | \(\left(0.333{\text{e}}^{i2\pi \left(0.335\right)},0.116{\text{e}}^{i2\pi \left(0.085\right)}\right)\) |
Score values of the alternatives |
---|
\(\mathcal{S}\left({{{x}}}_{1}\right)=\mathcal{S}\left({{{r}}}_{1}\right)=\mathcal{S}\left({\mathbb{E}}_{1}\right)=0.241\) |
\(\mathcal{S}\left({{{x}}}_{2}\right)=\mathcal{S}\left({{{r}}}_{2}\right)=\mathcal{S}\left({\mathbb{E}}_{2}\right)=0.244\) |
\(\mathcal{S}\left({{{x}}}_{3}\right)=\mathcal{S}\left({{{r}}}_{3}\right)=\mathcal{S}\left({\mathbb{E}}_{3}\right)=0.242\) |
\(\mathcal{S}\left({{{x}}}_{4}\right)=\mathcal{S}\left({{{r}}}_{4}\right)=\mathcal{S}\left({\mathbb{E}}_{4}\right)=0.239\) |
\(\mathcal{S}\left({{{x}}}_{5}\right)=\mathcal{S}\left({{{r}}}_{5}\right)=\mathcal{S}\left({\mathbb{E}}_{5}\right)=0.233\) |
Effect of parameter \({{\delta}}\)
Value of \({{\delta}}\) | Score values of the methods | Rankings of the methods |
---|---|---|
\({{\delta}}=2\) | \(\mathcal{S}\left({\mathbb{E}}_{1}\right)=0.336,\mathcal{S}\left({\mathbb{E}}_{2}\right)=0.339,\mathcal{S}\left({\mathbb{E}}_{3}\right)=0.331,\mathcal{S}\left({\mathbb{E}}_{4}\right)=0.321,\mathcal{S}\left({\mathbb{E}}_{5}\right)=0.295\) | \({\mathbb{E}}_{5}\le {\mathbb{E}}_{4}\le {\mathbb{E}}_{3}\le {\mathbb{E}}_{1}\le {\mathbb{E}}_{2}\) |
\({{\delta}}=3\) | \(\mathcal{S}\left({\mathbb{E}}_{1}\right)=0.282,\mathcal{S}\left({\mathbb{E}}_{2}\right)=0.284,\mathcal{S}\left({\mathbb{E}}_{3}\right)=0.181,\mathcal{S}\left({\mathbb{E}}_{4}\right)=-0.72,\mathcal{S}\left({\mathbb{E}}_{5}\right)=-0.23\) | \({\mathbb{E}}_{4}\le {\mathbb{E}}_{5}\le {\mathbb{E}}_{3}\le {\mathbb{E}}_{1}\le {\mathbb{E}}_{2}\) |
\({{\delta}}=5\) | \(\mathcal{S}\left({\mathbb{E}}_{1}\right)=0.001,\mathcal{S}\left({\mathbb{E}}_{2}\right)=0.002,\mathcal{S}\left({\mathbb{E}}_{3}\right)=0.0019,\mathcal{S}\left({\mathbb{E}}_{4}\right)=0.002,\mathcal{S}\left({\mathbb{E}}_{5}\right)=0.001\) | \({\mathbb{E}}_{5}\le {\mathbb{E}}_{1}\le {\mathbb{E}}_{3}\le {\mathbb{E}}_{4}\le {\mathbb{E}}_{2}\) |
\({{\delta}}=7\) | \(\mathcal{S}\left({\mathbb{E}}_{1}\right)=0.0026,\mathcal{S}\left({\mathbb{E}}_{2}\right)=0.0030,\mathcal{S}\left({\mathbb{E}}_{3}\right)=0.0029,\mathcal{S}\left({\mathbb{E}}_{4}\right)=0.0029,\mathcal{S}\left({\mathbb{E}}_{5}\right)=0.0021\) | \({\mathbb{E}}_{5}\le {\mathbb{E}}_{4}\le {\mathbb{E}}_{1}\le {\mathbb{E}}_{3}\le {\mathbb{E}}_{2}\) |
\({{\delta}}=10\) | \(\mathcal{S}\left({\mathbb{E}}_{1}\right)=0.0031,\mathcal{S}\left({\mathbb{E}}_{2}\right)=0.0036,\mathcal{S}\left({\mathbb{E}}_{3}\right)=0.0036,\mathcal{S}\left({\mathbb{E}}_{4}\right)=0.0035,\mathcal{S}\left({\mathbb{E}}_{5}\right)=0.0024\) | \({\mathbb{E}}_{5}\le {\mathbb{E}}_{1}\le {\mathbb{E}}_{3}\le {\mathbb{E}}_{4}\le {\mathbb{E}}_{2}\) |
\({{\delta}}=15\) | \(\mathcal{S}\left({\mathbb{E}}_{1}\right)=-0.004,\mathcal{S}\left({\mathbb{E}}_{2}\right)=-0.003,\mathcal{S}\left({\mathbb{E}}_{3}\right)=-0.004,\mathcal{S}\left({\mathbb{E}}_{4}\right)=-0.005,\mathcal{S}\left({\mathbb{E}}_{5}\right)=-0.008\) | \({\mathbb{E}}_{5}\le {\mathbb{E}}_{4}\le {\mathbb{E}}_{3}\le {\mathbb{E}}_{1}\le {\mathbb{E}}_{2}\) |
\({{\delta}}=20\) | \(\mathcal{S}\left({\mathbb{E}}_{1}\right)=-0.004,\mathcal{S}\left({\mathbb{E}}_{2}\right)=-0.004,\mathcal{S}\left({\mathbb{E}}_{3}\right)=-0.005\) \(,\mathcal{S}\left({\mathbb{E}}_{4}\right)=-0.006,\mathcal{S}\left({\mathbb{E}}_{5}\right)=-0.009\) | \({\mathbb{E}}_{5}\le {\mathbb{E}}_{4}\le {\mathbb{E}}_{3}\le {\mathbb{E}}_{1}\le {\mathbb{E}}_{2}\) |
Effect of parameter \({{q}}\)
Value of \({{q}}\) | Score values of the methods | Rankings of the methods |
---|---|---|
\({{q}}=2\) | \(\mathcal{S}\left({\mathbb{E}}_{1}\right)=0.0082,\mathcal{S}\left({\mathbb{E}}_{2}\right)=0.0088,\mathcal{S}\left({\mathbb{E}}_{3}\right)=0.0084,\mathcal{S}\left({\mathbb{E}}_{4}\right)=0.0078,\mathcal{S}\left({\mathbb{E}}_{5}\right)=0.0059\) | \({\mathbb{E}}_{5}\le {\mathbb{E}}_{4}\le {\mathbb{E}}_{1}\le {\mathbb{E}}_{3}\le {\mathbb{E}}_{2}\) |
\({{q}}=3\) | \(\mathcal{S}\left({\mathbb{E}}_{1}\right)=-0.0021,\mathcal{S}\left({\mathbb{E}}_{2}\right)=-0.0021,\mathcal{S}\left({\mathbb{E}}_{3}\right)=-0.0025,\mathcal{S}\left({\mathbb{E}}_{4}\right)=-0.003,\mathcal{S}\left({\mathbb{E}}_{5}\right)=-0.0039\) | \({\mathbb{E}}_{5}\le {\mathbb{E}}_{4}\le {\mathbb{E}}_{3}\le {\mathbb{E}}_{1}\le {\mathbb{E}}_{2}\) |
\({{q}}=4\) | \(\mathcal{S}\left({\mathbb{E}}_{1}\right)=-0.00058,\mathcal{S}\left({\mathbb{E}}_{2}\right)=-0.00053,\mathcal{S}\left({\mathbb{E}}_{3}\right)=-0.00077,\mathcal{S}\left({\mathbb{E}}_{4}\right)=-0.00093,\mathcal{S}\left({\mathbb{E}}_{5}\right)=-0.00095\) | \({\mathbb{E}}_{5}\le {\mathbb{E}}_{4}\le {\mathbb{E}}_{3}\le {\mathbb{E}}_{1}\le {\mathbb{E}}_{2}\) |
\({{q}}=5\) | \(\mathcal{S}\left({\mathbb{E}}_{1}\right)=-0.00020,\mathcal{S}\left({\mathbb{E}}_{2}\right)=-0.000015,\mathcal{S}\left({\mathbb{E}}_{3}\right)=-0.00019,\mathcal{S}\left({\mathbb{E}}_{4}\right)=-0.00024,\mathcal{S}\left({\mathbb{E}}_{5}\right)=-0.00036\) | \({\mathbb{E}}_{5}\le {\mathbb{E}}_{4}\le {\mathbb{E}}_{1}\le {\mathbb{E}}_{3}\le {\mathbb{E}}_{2}\) |
Advantages and comparative analysis
Methods | Score values of the methods | Rankings of the methods |
---|---|---|
Garg and Rani [40] | \(\mathcal{S}\left({\mathbb{E}}_{1}\right)=0.231,\mathcal{S}\left({\mathbb{E}}_{2}\right)=0.234,\mathcal{S}\left({\mathbb{E}}_{3}\right)=0.232,\mathcal{S}\left({\mathbb{E}}_{4}\right)=0.229,\mathcal{S}\left({\mathbb{E}}_{5}\right)=0.222\) | \({\mathbb{E}}_{5}\le {\mathbb{E}}_{4}\le {\mathbb{E}}_{1}\le {\mathbb{E}}_{3}\le {\mathbb{E}}_{2}\) |
Rani and Garg [42] | \(\mathcal{S}\left({\mathbb{E}}_{1}\right)=0.251,\mathcal{S}\left({\mathbb{E}}_{2}\right)=0.254,\mathcal{S}\left({\mathbb{E}}_{3}\right)=0.252,\mathcal{S}\left({\mathbb{E}}_{4}\right)=0.249,\mathcal{S}\left({\mathbb{E}}_{5}\right)=0.243\) | \({\mathbb{E}}_{5}\le {\mathbb{E}}_{4}\le {\mathbb{E}}_{1}\le {\mathbb{E}}_{3}\le {\mathbb{E}}_{2}\) |
Proposed Method for CPFS | \(\mathcal{S}\left({\mathbb{E}}_{1}\right)=0.0082,\mathcal{S}\left({\mathbb{E}}_{2}\right)=0.0088,\mathcal{S}\left({\mathbb{E}}_{3}\right)=0.0084,\mathcal{S}\left({\mathbb{E}}_{4}\right)=0.0078,\mathcal{S}\left({\mathbb{E}}_{5}\right)=0.0059\) | \({\mathbb{E}}_{5}\le {\mathbb{E}}_{4}\le {\mathbb{E}}_{1}\le {\mathbb{E}}_{3}\le {\mathbb{E}}_{2}\) |
Proposed methods for Cq-ROFS | \(\mathcal{S}\left({\mathbb{E}}_{1}\right)=-0.0021,\mathcal{S}\left({\mathbb{E}}_{2}\right)=-0.0021,\mathcal{S}\left({\mathbb{E}}_{3}\right)=-0.0025,\mathcal{S}\left({\mathbb{E}}_{4}\right)=-0.003,\mathcal{S}\left({\mathbb{E}}_{5}\right)=-0.0039\) | \({\mathbb{E}}_{5}\le {\mathbb{E}}_{4}\le {\mathbb{E}}_{3}\le {\mathbb{E}}_{1}\le {\mathbb{E}}_{2}\) |
First numerical example for comparison
Symbols | \({{{G}}}_{1}\) | \({{{G}}}_{2}\) | \({{{G}}}_{3}\) | \({{{G}}}_{4}\) |
---|---|---|---|---|
\({{{x}}}_{1}\) | \(\left(0.7{\text{e}}^{i2\pi \left(0.65\right)},0.22{\text{e}}^{i2\pi \left(0.31\right)}\right)\) | \(\left(0.61{\text{e}}^{i2\pi \left(0.7\right)},0.33{\text{e}}^{i2\pi \left(0.3\right)}\right)\) | \(\left(0.8{\text{e}}^{i2\pi \left(0.94\right)},0.23{\text{e}}^{i2\pi \left(0.11\right)}\right)\) | \(\left(0.84{\text{e}}^{i2\pi \left(0.7\right)},0.19{\text{e}}^{i2\pi \left(0.33\right)}\right)\) |
\({{{x}}}_{2}\) | \(\left(0.71{\text{e}}^{i2\pi \left(0.66\right)},0.24{\text{e}}^{i2\pi \left(0.31\right)}\right)\) | \(\left(0.62{\text{e}}^{i2\pi \left(0.71\right)},0.34{\text{e}}^{i2\pi \left(0.31\right)}\right)\) | \(\left(0.81{\text{e}}^{i2\pi \left(0.93\right)},0.24{\text{e}}^{i2\pi \left(0.12\right)}\right)\) | \(\left(0.83{\text{e}}^{i2\pi \left(0.72\right)},0.20{\text{e}}^{i2\pi \left(0.23\right)}\right)\) |
\({{{x}}}_{3}\) | \(\left(0.72{\text{e}}^{i2\pi \left(0.67\right)},0.25{\text{e}}^{i2\pi \left(0.33\right)}\right)\) | \(\left(0.63{\text{e}}^{i2\pi \left(0.72\right)},0.35{\text{e}}^{i2\pi \left(0.32\right)}\right)\) | \(\left(0.82{\text{e}}^{i2\pi \left(0.92\right)},0.25{\text{e}}^{i2\pi \left(0.13\right)}\right)\) | \(\left(0.82{\text{e}}^{i2\pi \left(0.73\right)},0.21{\text{e}}^{i2\pi \left(0.24\right)}\right)\) |
\({{{x}}}_{4}\) | \(\left(0.73{\text{e}}^{i2\pi \left(0.68\right)},0.27{\text{e}}^{i2\pi \left(0.34\right)}\right)\) | \(\left(0.64{\text{e}}^{i2\pi \left(0.73\right)},0.36{\text{e}}^{i2\pi \left(0.33\right)}\right)\) | \(\left(0.83{\text{e}}^{i2\pi \left(0.91\right)},0.26{\text{e}}^{i2\pi \left(0.14\right)}\right)\) | \(\left(0.81{\text{e}}^{i2\pi \left(0.74\right)},0.22{\text{e}}^{i2\pi \left(0.25\right)}\right)\) |
\({{{x}}}_{5}\) | \(\left(0.74{\text{e}}^{i2\pi \left(0.69\right)},0.35{\text{e}}^{i2\pi \left(0.35\right)}\right)\) | \(\left(0.65{\text{e}}^{i2\pi \left(0.74\right)},0.37{\text{e}}^{i2\pi \left(0.34\right)}\right)\) | \(\left(0.84{\text{e}}^{i2\pi \left(0.9\right)},0.27{\text{e}}^{i2\pi \left(0.15\right)}\right)\) | \(\left(0.8{\text{e}}^{i2\pi \left(0.76\right)},0.23{\text{e}}^{i2\pi \left(0.26\right)}\right)\) |
Score values of the alternatives |
---|
\(\mathcal{S}\left({{{x}}}_{1}\right)=\mathcal{S}\left({{{r}}}_{1}\right)=\mathcal{S}\left({\mathbb{E}}_{1}\right)=0.099\) |
\(\mathcal{S}\left({{{x}}}_{2}\right)=\mathcal{S}\left({{{r}}}_{2}\right)=\mathcal{S}\left({\mathbb{E}}_{2}\right)=0.096\) |
\(\mathcal{S}\left({{{x}}}_{3}\right)=\mathcal{S}\left({{{r}}}_{3}\right)=\mathcal{S}\left({\mathbb{E}}_{3}\right)=0.094\) |
\(\mathcal{S}\left({{{x}}}_{4}\right)=\mathcal{S}\left({{{r}}}_{4}\right)=\mathcal{S}\left({\mathbb{E}}_{4}\right)=0.093\) |
\(\mathcal{S}\left({{{x}}}_{5}\right)=\mathcal{S}\left({{{r}}}_{5}\right)=\mathcal{S}\left({\mathbb{E}}_{5}\right)=0.089\) |
Methods | Score values of the methods | Rankings of the methods |
---|---|---|
Garg and Rani [40] | \(\text{Cannot be Calculated}\) | \(\text{Cannot be Calculated}\) |
Rani and Garg [42] | \(\text{Cannot be Calculated}\) | \(\text{Cannot be Calculated}\) |
Proposed Method for CPFS | \(\mathcal{S}\left({\mathbb{E}}_{1}\right)=0.069,\mathcal{S}\left({\mathbb{E}}_{2}\right)=0.068,\mathcal{S}\left({\mathbb{E}}_{3}\right)=0.066\) \(,\mathcal{S}\left({\mathbb{E}}_{4}\right)=0.064,\mathcal{S}\left({\mathbb{E}}_{5}\right)=0.061\) | \({\mathbb{E}}_{5}\le {\mathbb{E}}_{4}\le {\mathbb{E}}_{3}\le {\mathbb{E}}_{2}\le {\mathbb{E}}_{1}\) |
Proposed methods for Cq-ROFS | \(\mathcal{S}\left({\mathbb{E}}_{1}\right)=0.099,\mathcal{S}\left({\mathbb{E}}_{2}\right)=0.096,\mathcal{S}\left({\mathbb{E}}_{3}\right)=0.094\) \(,\mathcal{S}\left({\mathbb{E}}_{4}\right)=0.093,\mathcal{S}\left({\mathbb{E}}_{5}\right)=0.089\) | \({\mathbb{E}}_{5}\le {\mathbb{E}}_{4}\le {\mathbb{E}}_{3}\le {\mathbb{E}}_{2}\le {\mathbb{E}}_{1}\) |
Second numerical example for comparison
Symbols | \({{{G}}}_{1}\) | \({{{G}}}_{2}\) | \({{{G}}}_{3}\) | \({{{G}}}_{4}\) |
---|---|---|---|---|
\({{{x}}}_{1}\) | \(\left(0.7{\text{e}}^{i2\pi \left(0.65\right)},0.61{\text{e}}^{i2\pi \left(0.64\right)}\right)\) | \(\left(0.61{\text{e}}^{i2\pi \left(0.7\right)},0.51{\text{e}}^{i2\pi \left(0.69\right)}\right)\) | \(\left(0.8{\text{e}}^{i2\pi \left(0.94\right)},0.79{\text{e}}^{i2\pi \left(0.9\right)}\right)\) | \(\left(0.84{\text{e}}^{i2\pi \left(0.7\right)},0.19{\text{e}}^{i2\pi \left(0.33\right)}\right)\) |
\({{{x}}}_{2}\) | \(\left(0.71{\text{e}}^{i2\pi \left(0.66\right)},0.62{\text{e}}^{i2\pi \left(0.65\right)}\right)\) | \(\left(0.62{\text{e}}^{i2\pi \left(0.71\right)},0.52{\text{e}}^{i2\pi \left(0.7\right)}\right)\) | \(\left(0.81{\text{e}}^{i2\pi \left(0.93\right)},0.8{\text{e}}^{i2\pi \left(0.91\right)}\right)\) | \(\left(0.83{\text{e}}^{i2\pi \left(0.72\right)},0.20{\text{e}}^{i2\pi \left(0.23\right)}\right)\) |
\({{{x}}}_{3}\) | \(\left(0.72{\text{e}}^{i2\pi \left(0.67\right)},0.63{\text{e}}^{i2\pi \left(0.66\right)}\right)\) | \(\left(0.63{\text{e}}^{i2\pi \left(0.72\right)},0.53{\text{e}}^{i2\pi \left(0.71\right)}\right)\) | \(\left(0.82{\text{e}}^{i2\pi \left(0.92\right)},0.81{\text{e}}^{i2\pi \left(0.92\right)}\right)\) | \(\left(0.82{\text{e}}^{i2\pi \left(0.73\right)},0.21{\text{e}}^{i2\pi \left(0.24\right)}\right)\) |
\({{{x}}}_{4}\) | \(\left(0.73{\text{e}}^{i2\pi \left(0.68\right)},0.64{\text{e}}^{i2\pi \left(0.67\right)}\right)\) | \(\left(0.64{\text{e}}^{i2\pi \left(0.73\right)},0.54{\text{e}}^{i2\pi \left(0.72\right)}\right)\) | \(\left(0.83{\text{e}}^{i2\pi \left(0.91\right)},0.82{\text{e}}^{i2\pi \left(0.89\right)}\right)\) | \(\left(0.81{\text{e}}^{i2\pi \left(0.74\right)},0.22{\text{e}}^{i2\pi \left(0.25\right)}\right)\) |
\({{{x}}}_{5}\) | \(\left(0.74{\text{e}}^{i2\pi \left(0.69\right)},0.65{\text{e}}^{i2\pi \left(0.68\right)}\right)\) | \(\left(0.65{\text{e}}^{i2\pi \left(0.74\right)},0.55{\text{e}}^{i2\pi \left(073\right)}\right)\) | \(\left(0.84{\text{e}}^{i2\pi \left(0.9\right)},0.83{\text{e}}^{i2\pi \left(0.79\right)}\right)\) | \(\left(0.8{\text{e}}^{i2\pi \left(0.76\right)},0.23{\text{e}}^{i2\pi \left(0.26\right)}\right)\) |
Score values of the alternatives |
---|
\(\mathcal{S}\left({{{x}}}_{1}\right)=\mathcal{S}\left({{{r}}}_{1}\right)=\mathcal{S}\left({\mathbb{E}}_{1}\right)=-0.00357\) |
\(\mathcal{S}\left({{{x}}}_{2}\right)=\mathcal{S}\left({{{r}}}_{2}\right)=\mathcal{S}\left({\mathbb{E}}_{2}\right)=-0.00234\) |
\(\mathcal{S}\left({{{x}}}_{3}\right)=\mathcal{S}\left({{{r}}}_{3}\right)=\mathcal{S}\left({\mathbb{E}}_{3}\right)=-0.00287\) |
\(\mathcal{S}\left({{{x}}}_{4}\right)=\mathcal{S}\left({{{r}}}_{4}\right)=\mathcal{S}\left({\mathbb{E}}_{4}\right)=-0.00283\) |
\(\mathcal{S}\left({{{x}}}_{5}\right)=\mathcal{S}\left({{{r}}}_{5}\right)=\mathcal{S}\left({\mathbb{E}}_{5}\right)=-0.00224\) |
Methods | Score values of the methods | Rankings of the methods |
---|---|---|
Garg and Rani [40] | \(\text{Cannot be calculated}\) | \(\text{Cannot be calculated}\) |
Rani and Garg [42] | \(\text{Cannot be calculated}\) | \(\text{Cannot be calculated}\) |
Proposed Method for CPFS | \(\text{Cannot be calculated}\) | \(\text{Cannot be calculated}\) |
Proposed methods for Cq-ROFS | \(\mathcal{S}\left({\mathbb{E}}_{1}\right)=-0.00357,\mathcal{S}\left({\mathbb{E}}_{2}\right)=-0.00234,\mathcal{S}\left({\mathbb{E}}_{3}\right)=-0.00287\) \(,\mathcal{S}\left({\mathbb{E}}_{4}\right)=-0.00283,\mathcal{S}\left({\mathbb{E}}_{5}\right)=-0.00224\) | \({\mathbb{E}}_{1}\le {\mathbb{E}}_{3}\le {\mathbb{E}}_{4}\le {\mathbb{E}}_{2}\le {\mathbb{E}}_{5}\) |
Defense budget
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Pakistan navy budget: 0.11%;
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Pakistan’s inter-service budget: 0.20%.
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Pakistan investment deposit account budget: 0.8%;
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Pakistan’s other accounts budget: 0.5%;
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Pakistan net receipts budget: 0.13%.
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Pakistan zakat collection account budget: 0.25%;
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Pakistan civil and criminal deposit amount budget: 0.16%;
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Pakistan’s personal deposit budget: 0.11%;
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Defense budget of Pakistan: 0.249.
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National saving scheme of Pakistan: 0.4
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Deposit and reserves of Pakistan: 0.155.
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Development expenditure and revenue account for Pakistan: 0.1925.