Introduction
Preliminaries
FS | Fuzzy set |
SFS | Spherical fuzzy set |
SFN | Spherical fuzzy number |
T-SFS | T-spherical fuzzy set |
\({\mathfrak {X}}\) | Initial universe |
s(x) (or s) | Membership degree (MD) |
i(x) (or i) | Abstinence degree (AD) |
d(x) (or d) | Non-membership degree (NMD) |
HT-SFS | Hesitant T-spherical fuzzy set |
HT-SFE | Hesitant T-spherical fuzzy element |
SV\(({\mathfrak {h}})\) | Score value of hesitant T-spherical fuzzy element \({\mathfrak {h}}\) |
AV\(({\mathfrak {h}})\) | Accuracy value of hesitant T-spherical fuzzy element \({\mathfrak {h}}\) |
\(\ell _{{\mathfrak {h}}}\) | Length of hesitant T-spherical fuzzy element \({\mathfrak {h}}\) |
\({\mathfrak {h}}^{-}\) | Lower bound of hesitant T-spherical fuzzy element \({\mathfrak {h}}\) |
\({\mathfrak {h}}^{+}\) | Upper bound of hesitant T-spherical fuzzy element \({\mathfrak {h}}\) |
\(\triangleleft \) | Hesitant T-spherical fuzzy subset |
\(\Cup \) | Union operation of HT-SFSs |
\(\Cap \) | Intersection operation of HT-SFSs |
\({\mathbb {T}}^c\) | Complement of HT-SFSs \({\mathbb {T}}\) |
\(f\otimes g\) | Dombi t-norm of \(f \text {and} g\) |
\(f\oplus g\) | Dombi t-conorm of \(f \text {and} g\) |
\(\gamma \) | Parameter in Dombi operator |
\(\omega =(w_1,w_2,\ldots ,w_m)\) | Weight vector with m component |
HTSDFWAA | Hesitant T-spherical Dombi fuzzy weighted arithmetic averaging |
HTSDFWGA | Hesitant T-spherical Dombi fuzzy weighted geometric averaging |
HTSDFOWAA | Hesitant T-spherical Dombi fuzzy ordered weighted arithmetic averaging |
HTSDFOWGA | Hesitant T-spherical Dombi fuzzy ordered weighted geometric averaging |
Hesitant T-spherical fuzzy sets
\(x_1\) | \(x_2\) | \(x_3\) | |
---|---|---|---|
SV\(({\mathfrak {h}}_1)\) | 0.126 | 0.132 | 0.273 |
SV\(({\mathfrak {h}}_2)\) | 0.244 | 0.270 | 0.378 |
Set-theoretical operations of HT-SFSs
Hesitant T-spherical Dombi fuzzy aggregation operators
Dombi t-norm and t-conorm
Dombi operations of HT-SFEs
Hesitant T-spherical Dombi fuzzy weighted arithmetic averaging operator
Hesitant T-spherical Dombi fuzzy weighted geometric averaging (HTSDFWGA) operator
Hesitant T-spherical Dombi fuzzy ordered weighted arithmetic averaging (HTSDFOWAA) operator
Hesitant T-spherical Dombi fuzzy ordered weighted geometric averaging (HTSDFOWGA) operator
Multiple criteria group decision-making (MCGDM) method under HT-SF information
\(\kappa =\{\kappa _1,\kappa _2,\ldots , \kappa _l\}\) | \(\text {Set of alternatives}\) |
\(\epsilon =\{\epsilon _1,\epsilon _2, \ldots ,\epsilon _s\}\) | \(\text {Set of criteria}\) |
\(\partial =\{\partial _1,\partial _2,\ldots ,\partial _t\}\) | \(\text {Set of decision-makers}\) |
\(D_{\kappa _i}\) | \(\text {For alternative } \kappa _i \text {decision matrix }\) |
HTSF\(_i\) | Collection of i column elements of \(D_{\kappa _i} \text {matrix} \) |
\(\zeta _{yj}\) | \(\text {Element of }\) \(D_{\kappa _i}\) corresponding to y row and j column |
\({\mathfrak {A}}_i\) | \(\text {HT-SFE corresponding to alternative } \kappa _i\) |
Illustrative example
Grades | HT-SFNs |
---|---|
Very poor (VP) | (0.100, 0.700, 0.900) |
Poor (P) | (0.233, 0.634, 0.767) |
Medium poor (MP) | (0.367, 0.567, 0.634) |
Fairly (F) | (0.500, 0.500, 0.500) |
Medium good (MG) | (0.633, 0.436, 0.367) |
Good (G) | (0.764, 0.370, 0.234) |
Very good (VG) | (0.900, 0.300, 0.100) |
\(\epsilon _1\) | \(\epsilon _2\) | \(\epsilon _3\) | ||
---|---|---|---|---|
\(D_{\kappa _1}\) | \(\partial _1\) | (0.633, 0.436, 0.367) | (0.233, 0.634, 0.767) | * |
\(\partial _2\) | (0.100, 0.700, 0.900) | (0.900, 0.300, 0.100) | (0.500, 0.500, 0.500) | |
\(\partial _3\) | * | (0.764, 0.370, 0.234) | * | |
\(D_{\kappa _2}\) | \(\partial _1\) | * | (0.233, 0.634, 0.767) | * |
\(\partial _2\) | (0.233, 0.634, 0.767) | (0.367, 0.567, 0.634) | (0.500, 0.500, 0.500) | |
\(\partial _3\) | (0.764, 0.370, 0.234) | (0.100, 0.700, 0.900) | (0.633, 0.436, 0.367) | |
\(D_{\kappa _3}\) | \(\partial _1\) | (0.367, 0.567, 0.634) | (0.633, 0.436, 0.367) | (0.100, 0.700, 0.900) |
\(\partial _2\) | (0.900, 0.300, 0.100) | (0.764, 0.370, 0.234) | * | |
\(\partial _3\) | (0.633, 0.436, 0.367) | (0.100, 0.700, 0.900) | (0.500, 0.500, 0.500) | |
\(D_{\kappa _4}\) | \(\partial _1\) | (0.900, 0.300, 0.100) | (0.500, 0.500, 0.500) | (0.100, 0.700, 0.900) |
\(\partial _2\) | (0.900, 0.300, 0.100) | * | (0.500, 0.500, 0.500) | |
\(\partial _3\) | * | (0.100, 0.700, 0.900) | (0.500, 0.500, 0.500) | |
\(D_{\kappa _5}\) | \(\partial _1\) | (0.367, 0.567, 0.634) | (0.500, 0.500, 0.500) | (0.100, 0.700, 0.900) |
\(\partial _2\) | (0.100, 0.700, 0.900) | (0.367, 0.567, 0.634) | (0.633, 0.436, 0.367) | |
\(\partial _3\) | (0.900, 0.300, 0.100) | (0.100, 0.700, 0.900) | (0.500, 0.500, 0.500) | |
\(D_{\kappa _6}\) | \(\partial _1\) | (0.500, 0.500, 0.500) | (0.633, 0.436, 0.367) | (0.764, 0.370, 0.234) |
\(\partial _2\) | * | (0.367, 0.567, 0.634) | (0.633, 0.436, 0.367) | |
\(\partial _3\) | * | (0.233, 0.634, 0.767) | (0.500, 0.500, 0.500) | |
\(D_{\kappa _7}\) | \(\partial _1\) | (0.500, 0.500, 0.500) | (0.633, 0.436, 0.367) | (0.500, 0.500, 0.500) |
\(\partial _2\) | (0.367, 0.567, 0.634) | (0.100, 0.700, 0.900) | (0.233, 0.634, 0.767) | |
\(\partial _3\) | (0.233, 0.634, 0.767) | (0.100, 0.700, 0.900) | * |
HTSDFWAA | HTSDFWGA | |
---|---|---|
\({\mathfrak {A}}_1\) | {(0.542, 0.488, 0.441),(0.776, 0.382, 0.141),(0.653, 0.429, 0.309) | {(0.322, 0.488, 0.614), (0.578, 0.382, 0.423), (0.572, 0.429, 0.425), |
(0.404, 0.571, 0.601),(0.761, 0.401, 0.141),(0.606, 0.466, 0.324)} | (0.129, 0.571, 0.820), (0.130, 0.401, 0.800), (0.130, 0.466, 0.800) | |
\({\mathfrak {A}}_2\) | {(0.408, 0.561, 0.592), (0.519, 0.510, 0.453), (0.421, 0.549, 0.578), | {(0.263, 0.561, 0.711), (0.264, 0.510, 0.704), (0.291, 0.549, 0.674), |
(0.525, 0.503, 0.449), (0.405, 0.568, 0.599), (0.518, 0.514, 0.454), | (0.293, 0.503, 0.664), (0.140, 0.568, 0.799), (0.140, 0.514, 0.796), | |
(0.644, 0.440, 0.300), (0.673, 0.423, 0.291), (0.647, 0.437, 0.299), | (0.323, 0.440, 0.609), (0.326, 0.423, 0.593), (0.466, 0.437, 0.515), | |
(0.675, 0.420, 0.291), (0.644, 0.442, 0.300) (0.673, 0.425, 0.291)} | (0.490, 0.420, 0.481), (0.141, 0.442, 0.761), (0.141, 0.425, 0.756)} | |
\({\mathfrak {A}}_3\) | {(0.476, 0.538, 0.496), (0.523, 0.495, 0.460), (0.588, 0.483, 0.328), | {(0.126, 0.538, 0.821), (0.438, 0.495, 0.548), (0.126, 0.483, 0.821), |
(0.611, 0.456, 0.323), (0.284, 0.636, 0.750), (0.423, 0.551, 0.578), | (0.441, 0.456, 0.543), (0.111, 0.636, 0.868), (0.141, 0.551, 0.776), | |
(0.804, 0.373, 0.130), (0.808, 0.365, 0.130), (0.816, 0.362, 0.129), | (0.126, 0.373, 0.812), (0.586, 0.365, 0.417), (0.126, 0.362, 0.811), | |
(0.820, 0.355, 0.129), (0.795, 0.384, 0.130), (0.800, 0.375, 0.130), | (0.601, 0.355, 0.404), (0.111, 0.384, 0.863), (0.141, 0.375, 0.761), | |
(0.566, 0.484, 0.415), (0.593, 0.457, 0.399), (0.636, 0.449, 0.314), | (0.126, 0.484, 0.813), (0.560, 0.457, 0.437), (0.126, 0.449, 0.812), | |
(0.653, 0.429, 0.309) (0.501, 0.533, 0.471), (0.541, 0.492, 0.443)} | (0.572, 0.429, 0.425), (0.111, 0.533, 0.863), (0.141, 0.492, 0.762)} | |
\({\mathfrak {A}}_4\) | {(0.798, 0.378, 0.130), (0.798, 0.378, 0.130), (0.798, 0.378, 0.130), | {(0.126, 0.378, 0.814), (0.126, 0.378, 0.814), (0.126, 0.378, 0.814), |
(0.798, 0.378, 0.130), (0.803, 0.370, 0.130), (0.803, 0.370, 0.130), | (0.126, 0.378, 0.814), (0.550, 0.370, 0.452), (0.550, 0.370, 0.452), | |
(0.800, 0.375, 0.130), (0.800, 0.375, 0.130), (0.800, 0.375, 0.130), | (0.141, 0.375, 0.761), (0.141, 0.375, 0.761), (0.141, 0.375, 0.761), | |
(0.800, 0.375, 0.130), (0.800, 0.375, 0.130), (0.800, 0.375, 0.130)} | (0.141, 0.375, 0.761), (0.141, 0.375, 0.761), (0.141, 0.375, 0.761)} | |
\({\mathfrak {A}}_5\) | {(0.388, 0.577 0.620), (0.549, 0.482, 0.434), (0.467, 0.519, 0.531), | {(0.126, 0.577, 0.823), (0.444, 0.482, 0.538), (0.428, 0.519, 0.562), |
(0.324, 0.605, 0.694), (0.533, 0.493, 0.444), (0.437, 0.535, 0.561), | (0.125, 0.605, 0.827), (0.409, 0.493, 0.575), (0.399, 0.535, 0.594), | |
(0.284, 0.636, 0.750), (0.526, 0.504, 0.449), (0.423, 0.551, 0.578), | (0.111, 0.636, 0.868), (0.141, 0.504, 0.772), (0.141, 0.551, 0.776), | |
(0.358, 0.612, 0.664), (0.540, 0.496, 0.440), (0.452, 0.539, 0.550), | (0.107, 0.612, 0.877), (0.130, 0.496, 0.800), (0.130, 0.539, 0.803), | |
(0.261, 0.652, 0.781), (0.523, 0.509, 0.451), (0.417, 0.559, 0.586), | (0.107, 0.652, 0.879), (0.130, 0.509, 0.805), (0.130, 0.559, 0.808), | |
(0.100, 0.700, 0.900), (0.516, 0.521, 0.457), (0.402, 0.579, 0.608), | (0.100, 0.700, 0.900), (0.114, 0.521, 0.855), (0.114, 0.579, 0.857), | |
(0.798, 0.378, 0.130), (0.811, 0.362, 0.130), (0.803, 0.370, 0.130), | (0.126, 0.378, 0.814), (0.614, 0.362, 0.392), (0.550, 0.370, 0.452), | |
(0.796, 0.381, 0.130), (0.809, 0.364, 0.130), (0.801, 0.372, 0.130), | (0.126, 0.381, 0.818), (0.494, 0.364, 0.479), (0.469, 0.372, 0.514), | |
(0.795, 0.384, 0.130), (0.808, 0.367, 0.130), (0.800, 0.375, 0.130)} | (0.111, 0.384, 0.863), (0.141, 0.367, 0.756), (0.141, 0.375, 0.761)} | |
\({\mathfrak {A}}_6\) | {(0.683, 0.415, 0.284), (0.599, 0.454, 0.394), (0.546, 0.479, 0.444), | {(0.589, 0.415, 0.410), (0.567, 0.454, 0.430), (0.521, 0.479, 0.477), |
(0.660, 0.430, 0.290), (0.555, 0.478, 0.430), (0.477, 0.513, 0.521), | (0.470, 0.430, 0.510), (0.462, 0.478, 0.520), (0.444, 0.513, 0.547), | |
(0.658, 0.433, 0.291), (0.550, 0.484, 0.433), (0.469, 0.521, 0.529)} | (0.323, 0.433, 0.607), (0.322, 0.484, 0.612), (0.319, 0.521, 0.627)} | |
\({\mathfrak {A}}_7\) | {(0.546, 0.479, 0.444), (0.510, 0.508, 0.470), (0.467, 0.526, 0.533), | {(0.521, 0.479, 0.477), (0.289, 0.508, 0.663), (0.141, 0.526, 0.766), |
(0.393, 0.575, 0.612), (0.467, 0.526, 0.533), (0.393, 0.575, 0.612), | (0.139, 0.575, 0.803), (0.141, 0.526, 0.766), (0.139, 0.575, 0.803), | |
(0.523, 0.495, 0.460), (0.479, 0.529, 0.492), (0.423, 0.551, 0.578), | (0.438, 0.495, 0.548), (0.283, 0.529, 0.686), (0.141, 0.551, 0.776), | |
(0.295, 0.617, 0.721), (0.423, 0.551, 0.578), (0.295, 0.617, 0.721), | (0.139, 0.617, 0.810), (0.141, 0.551, 0.776), (0.139, 0.617, 0.810), | |
(0.515, 0.504, 0.466), (0.467, 0.542, 0.500), (0.405, 0.568, 0.599), | (0.298, 0.504, 0.650), (0.250, 0.542, 0.732), (0.140, 0.568, 0.799), | |
(0.217, 0.648, 0.791), (0.405, 0.568, 0.599), (0.217, 0.648, 0.791)} | (0.138, 0.648, 0.826), (0.140, 0.568, 0.799), (0.138, 0.648, 0.826)} |
HTSDFWAA | HTSDFWGA | |
---|---|---|
SV\(({\mathfrak {A}}_1)\) | 0.156 | \(-\) 0.208 |
SV\(({\mathfrak {A}}_2)\) | 0.076 | \(-\) 0.217 |
SV\(({\mathfrak {A}}_3)\) | 0.165 | \(-\) 0.285 |
SV\(({\mathfrak {A}}_4)\) | 0.409 | \(-\) 0.305 |
SV\(({\mathfrak {A}}_5)\) | 0.063 | \(-\) 0.335 |
SV\(({\mathfrak {A}}_6)\) | 0.088 | \(-\) 0.035 |
SV\(({\mathfrak {A}}_7)\) | \(-\) 0.104 | \(-\) 0.319 |
Ordering | |
---|---|
HTSDFWAA | SV\(({\mathfrak {A}}_4)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_5)>\mathrm{{SV}}({\mathfrak {A}}_7)\) |
HTSDFWGA | SV\(({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_4)>\mathrm{{SV}}({\mathfrak {A}}_7)>\mathrm{{SV}}({\mathfrak {A}}_5)\) |
\(\gamma \) | SV\(({\mathfrak {A}}_1)\) | SV\(({\mathfrak {A}}_2)\) | SV\(({\mathfrak {A}}_3)\) | SV\(({\mathfrak {A}}_4)\) | SV\(({\mathfrak {A}}_5)\) | SV\(({\mathfrak {A}}_6)\) | SV\(({\mathfrak {A}}_7)\) | Ranking order |
---|---|---|---|---|---|---|---|---|
1 | 0.273 | 0.148 | 0.265 | 0.573 | 0.156 | 0.134 | \(-\) 0.049 | SV\(({\mathfrak {A}}_4)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_5)>\mathrm{{SV}}({\mathfrak {A}}_7)\) |
2 | 0.237 | 0.125 | 0.235 | 0.530 | 0.130 | 0.118 | \(-\) 0.067 | SV4\(({\mathfrak {A}}_4)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_5)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_7)\) |
3 | 0.273 | 0.148 | 0.265 | 0.573 | 0.156 | 0.134 | \(-\) 0.049 | SV\(({\mathfrak {A}}_4)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_5)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_7)\) |
4 | 0.292 | 0.161 | 0.281 | 0.595 | 0.170 | 0.143 | \(-\) 0.040 | SV\(({\mathfrak {A}}_4)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_5)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_7)\) |
5 | 0.305 | 0.169 | 0.291 | 0.607 | 0.178 | 0.149 | \(-\) 0.033 | SV\(({\mathfrak {A}}_4)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_5)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_7)\) |
6 | 0.313 | 0.175 | 0.298 | 0.615 | 0.183 | 0.153 | \(-\) 0.029 | SV\(({\mathfrak {A}}_4)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_5)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_7)\) |
7 | 0.319 | 0.179 | 0.303 | 0.621 | 0.187 | 0.157 | \(-\) 0.026 | SV\(({\mathfrak {A}}_4)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_5)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_7)\) |
8 | 0.323 | 0.182 | 0.307 | 0.626 | 0.190 | 0.159 | \(-\) 0.023 | SV\(({\mathfrak {A}}_4)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_5)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_7)\) |
9 | 0.327 | 0.185 | 0.310 | 0.629 | 0.192 | 0.161 | \(-\) 0.022 | SV\(({\mathfrak {A}}_4)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_5)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_7)\) |
10 | 0.330 | 0.187 | 0.312 | 0.632 | 0.194 | 0.162 | \(-\) 0.020 | SV\(({\mathfrak {A}}_4)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_5)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_7)\) |
\(\gamma \) | SV\(({\mathfrak {A}}_1)\) | SV\(({\mathfrak {A}}_2)\) | SV\(({\mathfrak {A}}_3)\) | SV\(({\mathfrak {A}}_4)\) | SV\(({\mathfrak {A}}_5)\) | SV\(({\mathfrak {A}}_6)\) | SV\(({\mathfrak {A}}_7)\) | Ranking order |
---|---|---|---|---|---|---|---|---|
1 | \(-\) 0.316 | \(-\) 0.333 | \(-\) 0.392 | \(-\) 0.465 | \(-\) 0.435 | \(-\) 0.099 | \(-\) 0.434 | \(\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_4)>\mathrm{{SV}}({\mathfrak {A}}_7)>\mathrm{{SV}}({\mathfrak {A}}_5)\) |
2 | \(-\) 0.284 | \(-\) 0.297 | \(-\) 0.363 | \(-\) 0.422 | \(-\) 0.406 | \(-\) 0.076 | \(-\) 0.395 | \(\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_7)>\mathrm{{SV}}({\mathfrak {A}}_5)>\mathrm{{SV}}({\mathfrak {A}}_4)\) |
3 | \(-\) 0.316 | \(-\) 0.333 | \(-\) 0.392 | \(-\) 0.465 | \(-\) 0.435 | \(-\) 0.099 | \(-\) 0.434 | \(\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_7)>\mathrm{{SV}}({\mathfrak {A}}_5)>\mathrm{{SV}}({\mathfrak {A}}_4)\) |
4 | \(-\) 0.333 | \(-\) 0.352 | \(-\) 0.408 | \(-\) 0.486 | \(-\) 0.449 | \(-\) 0.113 | \(-\) 0.455 | \(\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_5)>\mathrm{{SV}}({\mathfrak {A}}_7)>\mathrm{{SV}}({\mathfrak {A}}_4)\) |
5 | \(-\) 0.344 | \(-\) 0.364 | \(-\) 0.417 | \(-\) 0.498 | \(-\) 0.458 | \(-\) 0.123 | \(-\) 0.468 | \(\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_5)>\mathrm{{SV}}({\mathfrak {A}}_7)>\mathrm{{SV}}({\mathfrak {A}}_4)\) |
6 | \(-\) 0.350 | \(-\) 0.372 | \(-\) 0.423 | \(-\) 0.507 | \(-\) 0.464 | \(-\) 0.129 | \(-\) 0.477 | \(\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_5)>\mathrm{{SV}}({\mathfrak {A}}_7)>\mathrm{{SV}}({\mathfrak {A}}_4)\) |
7 | \(-\) 0.355 | \(-\) 0.378 | \(-\) 0.427 | \(-\) 0.512 | \(-\) 0.468 | \(-\) 0.133 | \(-\) 0.484 | \(\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_5)>\mathrm{{SV}}({\mathfrak {A}}_7)>\mathrm{{SV}}({\mathfrak {A}}_4)\) |
8 | \(-\) 0.359 | \(-\) 0.383 | \(-\) 0.431 | \(-\) 0.517 | \(-\) 0.472 | \(-\) 0.137 | \(-\) 0.489 | \(\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_5)>\mathrm{{SV}}({\mathfrak {A}}_7)>\mathrm{{SV}}({\mathfrak {A}}_4)\) |
9 | \(-\) 0.362 | \(-\) 0.386 | \(-\) 0.433 | \(-\) 0.520 | \(-\) 0.474 | \(-\) 0.140 | \(-\) 0.492 | \(\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_5)>\mathrm{{SV}}({\mathfrak {A}}_7)>\mathrm{{SV}}({\mathfrak {A}}_4)\) |
10 | \(-\) 0.364 | \(-\) 0.389 | \(-\) 0.435 | \(-\) 0.523 | \(-\) 0.476 | \(-\) 0.142 | \(-\) 0.495 | \(\mathrm{{SV}}({\mathfrak {A}}_6)>\mathrm{{SV}}({\mathfrak {A}}_1)>\mathrm{{SV}}({\mathfrak {A}}_2)>\mathrm{{SV}}({\mathfrak {A}}_3)>\mathrm{{SV}}({\mathfrak {A}}_5)>\mathrm{{SV}}({\mathfrak {A}}_7)>\mathrm{{SV}}({\mathfrak {A}}_4)\) |
n (degrees of component) | k (length) | Condition | ||||
---|---|---|---|---|---|---|
1 | 2 | \(>2\) | 1 | \(>1\) | ||
Fuzzy set [1] | \(\checkmark \) | \(\times \) | \(\times \) | \(\checkmark \) | \(\times \) | \(0\le s_k+d_k=1\), \(i_k=0\) |
Intuitionistic fuzzy set [4] | \(\checkmark \) | \(\times \) | \(\times \) | \(\checkmark \) | \(\times \) | \(s_k+i_k+d_k=1\) |
Pythagorean fuzzy set [5] | \(\checkmark \) | \(\checkmark \) | \(\times \) | \(\checkmark \) | \(\times \) | \(0\le s_k^2+d_k^2\le 1\) |
Picture fuzzy set [7] | \(\checkmark \) | \(\times \) | \(\times \) | \(\checkmark \) | \(\times \) | \(0\le s_k+i_k+d_k\le 1\) |
Spherical fuzzy set [25] | \(\checkmark \) | \(\checkmark \) | \(\times \) | \(\checkmark \) | \(\times \) | \(0\le s_k^2+i_k^2+d_k^2\le 1\) |
q-rung orthopair fuzzy set [6] | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\times \) | \(0\le s_k^n+d_k^n\le 1\) |
T-spherical fuzzy set [27] | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\times \) | \(0\le s_k^n+i_k^n+d_k^n\le 1\) |
\(\checkmark \) | \(\times \) | \(\times \) | \(\checkmark \) | \(\checkmark \) | \(0\le s_k+d_k=1\), \(i_k=0\) | |
Intuitionistic hesitant fuzzy set [75] | \(\checkmark \) | \(\times \) | \(\times \) | \(\checkmark \) | \(\checkmark \) | \(0\le s_k+d_k=1\) |
Hesitant Pythagorean fuzzy set [76] | \(\checkmark \) | \(\checkmark \) | \(\times \) | \(\checkmark \) | \(\checkmark \) | \(0\le s_k^2+d_k^2\le 1\) |
q-rung Orthopair hesitant fuzzy set [77] | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(0\le s_k^n+d_k^n\le 1\) |
Picture hesitant fuzzy set [59] | \(\checkmark \) | \(\checkmark \) | \(\times \) | \(\checkmark \) | \(\checkmark \) | \(0\le s_k+i_k+d_k\le 1\) |
Hesitant T-spherical fuzzy set | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(0\le s_k^n+i_k^n+d_k^n\le 1\) |