A Novel Dynamic Multi-Criteria Decision Making Method Based on Generalized Dynamic Interval-Valued Neutrosophic Set
Abstract
:1. Introduction
2. Preliminary
2.1. Multi-Criteria Decision-Making Model Based on History
2.2. Dynamic Interval-Valued Neutrosophic Set and Hesitant Fuzzy Set
3. Generalized Dynamic Interval-Valued Neutrosophic Set
- (i)
- Addition
- (ii)
- Multiplication
- (iii)
- Scalar Multiplication
- (iv)
- Power
4. Dynamic TOPSIS Method
5. Applications
5.1. ASK Model for Ranking Students
5.2. Application Model
5.3. Comparison with the Related Methods
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Language Labels | Values |
---|---|
Very Poor | ([0.1, 0.26], [0.4, 0.5], [0.63, 0.76]) |
Poor | ([0.26, 0.38], [0.47, 0.6], [0.51, 0.6]) |
Medium | ([0.38, 0.5], [0.4, 0.61], [0.44, 0.55]) |
Good | ([0.5, 0.65], [0.36, 0.5], [0.31, 0.48]) |
Very Good | ([0.65, 0.8], [0.1, 0.2], [0.12, 0.2]) |
Language Labels | Values |
---|---|
Unimportant | ([0.1, 0.19], [0.32, 0.47], [0.64, 0.8]) |
Slightly Important | ([0.2, 0.38], [0.46, 0.62], [0.36, 0.55]) |
Important | ([0.45, 0.63], [0.41, 0.53], [0.2, 0.42]) |
Very Important | ([0.66, 0.8], [0.3, 0.39], [0.22, 0.32]) |
Absolutely Important | ([0.8, 0.94], [0.18, 0.29], [0.1, 0.2]) |
Criteria | Students | ||
---|---|---|---|
([0.463, 0.606], [0.018, 0.054], [0.014, 0.044]) | ([0.38, 0.5], [0.022, 0.082], [0.029, 0.059]) | ([0.43, 0.577], [0.021, 0.053], [0.017, 0.048]) | |
([0.488, 0.632], [0.005, 0.025], [0.008, 0.021]) | ([0.419,0.578], [0.011,0.037], [0.011,0.026]) | ([0.419,0.578], [0.011,0.037], [0.011,0.026]) | |
([0.463, 0.606], [0.018, 0.054], [0.014, 0.044]) | ([0.463, 0.606], [0.018, 0.054], [0.014, 0.044]) | ([0.423, 0.556], [0.02, 0.066], [0.02, 0.051]) | |
([0.423, 0.556], [0.02, 0.066], [0.02, 0.051]) | ([0.463, 0.606], [0.018, 0.054], [0.014, 0.044]) | ([0.388, 0.523], [0.023, 0.065], [0.024, 0.056]) | |
([0.523, 0.673], [0.005, 0.021], [0.005, 0.018]) | ([0.423, 0.556], [0.02, 0.066], [0.02, 0.051]) | ([0.43, 0.577], [0.021, 0.053], [0.017, 0.048]) | |
([0.43, 0.577], [0.021, 0.053], [0.017, 0.048]) | ([0.38, 0.5], [0.022, 0.082], [0.029, 0.059]) | ([0.342, 0.463], [0.026, 0.081], [0.034, 0.065]) | |
([0.38, 0.5], [0.022, 0.082], [0.029, 0.059]) | ([0.388, 0.523], [0.023, 0.065], [0.024, 0.056]) | ([0.342, 0.463], [0.026, 0.081], [0.034, 0.065]) | |
([0.26, 0.38], [0.036, 0.078], [0.046, 0.078]) | ([0.38, 0.5], [0.022, 0.082], [0.029, 0.059]) | ([0.38, 0.5], [0.022, 0.082], [0.029, 0.059]) | |
([0.463, 0.606], [0.018, 0.054], [0.014, 0.044]) | ([0.523, 0.673], [0.005, 0.021], [0.005, 0.018]) | ([0.463, 0.606], [0.018, 0.054], [0.014, 0.044]) | |
([0.5, 0.65], [0.016, 0.044], [0.01, 0.038]) | ([0.38, 0.5], [0.022, 0.082], [0.029, 0.059]) | ([0.43, 0.577], [0.021, 0.053], [0.017, 0.048]) | |
([0.463, 0.606], [0.018, 0.054], [0.014, 0.044]) | ([0.302, 0.423], [0.03, 0.079], [0.04, 0.071]) | ([0.38, 0.5], [0.022, 0.082], [0.029, 0.059]) |
Criteria | Importance Aggregated Weights |
---|---|
([0.963, 0.996], [0.022, 0.06], [0.004, 0.027]) | |
([0.908, 0.968], [0.041, 0.094], [0.017, 0.056]) | |
([0.758, 0.89], [0.077, 0.174], [0.014, 0.097]) | |
([0.648, 0.816], [0.087, 0.204], [0.026, 0.127]) | |
([0.604, 0.794], [0.06, 0.154], [0.046, 0.185]) | |
([0.963, 0.992], [0.022, 0.06], [0.004, 0.027]) | |
([0.834, 0.925], [0.069, 0.149], [0.008, 0.074]) | |
([0.758, 0.89], [0.077, 0.174], [0.014, 0.097]) | |
([0.758, 0.89], [0.077, 0.174], [0.014, 0.097]) | |
([0.936, 0.975], [0.037, 0.081], [0.01, 0.043]) | |
([0.897, 0.959], [0.05, 0.11], [0.009, 0.056]) |
Students | Weighted Ratings |
---|---|
([0.368, 0.409], [0.069, 0.168], [0.03, 0.114]) | |
([0.34, 0.382], [0.071, 0.181], [0.035, 0.12]) | |
([0.338, 0.377], [0.072, 0.178], [0.035, 0.121]) |
Students | ||
---|---|---|
0.364193 | 0.773329 | |
0.380989 | 0.763987 | |
0.382736 | 0.763579 |
Students | Closeness Coefficients | Ranking Order |
---|---|---|
0.679837 | 1 | |
0.667251 | 2 | |
0.666116 | 3 |
Criteria | Students | |||
---|---|---|---|---|
([0.699, 0.83], [0.001, 0.005], [0, 0.002]) | ([0.566, 0.75], [0.001, 0.009], [0.001, 0.003]) | ([0.637, 0.759], [0.001, 0.007], [0.001, 0.003]) | ([0.5, 0.6], [0.022, 0.046], [0.009, 0.022]) | |
([0.707, 0.852], [0.001, 0.007], [0, 0.002]) | ([0.686, 0.834], [0.001, 0.008], [0, 0.003]) | ([0.72, 0.862], [0.001, 0.006], [0, 0.002]) | ([0.498, 0.6], [0.023, 0.049], [0.009, 0.023]) | |
([0.709, 0.848], [0.003, 0.016], [0, 0.005]) | ([0.643, 0.783], [0.003, 0.018], [0, 0.006]) | ([0.603, 0.767], [0.003, 0.019], [0.001, 0.006]) | ([0.56, 0.669], [0.008, 0.029], [0.004, 0.016]) | |
([0.598, 0.766], [0.004, 0.022], [0.001, 0.007]) | ([0.639, 0.782], [0.004, 0.021], [0.001, 0.007]) | ([0.634, 0.793], [0.004, 0.02], [0.001, 0.008]) | ([0.506, 0.643], [0.009, 0.034], [0.006, 0.021]) | |
([0.721, 0.866], [0.002, 0.012], [0.001, 0.015]) | ([0.651, 0.823], [0.002, 0.013], [0.002, 0.016]) | ([0.616, 0.765], [0.002, 0.014], [0.002, 0.017]) | ([0.461, 0.604], [0.013, 0.042], [0.009, 0.035]) | |
([0.685, 0.81], [0.001, 0.005], [0, 0.002]) | ([0.623, 0.803], [0.001, 0.007], [0, 0.002]) | ([0.546, 0.72], [0.001, 0.009], [0.001, 0.004]) | ([0.3, 0.5], [0.022, 0.08], [0.022, 0.044]) | |
([0.62, 0.802], [0.002, 0.013], [0, 0.004]) | ([0.618, 0.769], [0.002, 0.013], [0.001, 0.005]) | ([0.543, 0.72], [0.002, 0.015], [0.001, 0.006]) | ([0.438, 0.569], [0.024, 0.061], [0.012, 0.03]) | |
([0.491, 0.648], [0.005, 0.025], [0.002, 0.013]) | ([0.686, 0.862], [0.004, 0.02], [0, 0.006]) | ([0.499, 0.709], [0.005, 0.025], [0.001, 0.009]) | ([0.43, 0.567], [0.026, 0.071], [0.012, 0.033]) | |
([0.702, 0.847], [0.004, 0.021], [0, 0.007]) | ([0.761, 0.891], [0.004, 0.019], [0, 0.006]) | ([0.682, 0.828], [0.004, 0.022], [0, 0.007]) | ([0.488, 0.598], [0.026, 0.062], [0.009, 0.027]) | |
([0.687, 0.8], [0.002, 0.01], [0, 0.003]) | ([0.663, 0.836], [0.001, 0.008], [0, 0.003]) | ([0.718, 0.842], [0.001, 0.008], [0, 0.003]) | ([0.534, 0.636], [0.012, 0.032], [0.006, 0.018]) | |
([0.608, 0.751], [0.001, 0.009], [0.001, 0.003]) | ([0.557, 0.722], [0.001, 0.01], [0.001, 0.006]) | ([0.565, 0.75], [0.001, 0.011], [0.001, 0.004]) | ([0.499, 0.6], [0.023, 0.048], [0.009, 0.023]) | |
([0.36, 0.533], [0.043, 0.12], [0.021, 0.06]) | ([0.4, 0.516], [0.049, 0.11], [0.023, 0.065]) | ([0.463, 0.606], [0.033, 0.089], [0.012, 0.047]) | ([0.258, 0.439], [0.049, 0.133], [0.037, 0.087]) | |
([0.229, 0.373], [0.05, 0.119], [0.055, 0.108]) | ([0.229, 0.373], [0.05, 0.119], [0.055, 0.108]) | ([0.43, 0.568], [0.038, 0.095], [0.017, 0.047]) | ([0.43, 0.568], [0.038, 0.095], [0.017, 0.047]) | |
([0.284, 0.408], [0.083, 0.167], [0.046, 0.123]) | ([0.284, 0.408], [0.083, 0.167], [0.046, 0.123]) | ([0.269, 0.486], [0.071, 0.179], [0.03, 0.098]) | ([0.431, 0.592], [0.061, 0.137], [0.017, 0.076]) |
Criteria | Importance Aggregated Weights |
---|---|
([0.999, 1], [0, 0.003], [0, 0.001]) | |
([0.997, 1], [0.001, 0.006], [0, 0.002]) | |
([0.985, 0.998], [0.003, 0.014], [0, 0.004]) | |
([0.978, 0.997], [0.003, 0.016], [0, 0.005]) | |
([0.959, 0.993], [0.002, 0.011], [0.001, 0.015]) | |
([0.999, 1], [0, 0.003], [0, 0.001]) | |
([0.993, 0.999], [0.002, 0.009], [0, 0.002]) | |
([0.975, 0.997], [0.004, 0.019], [0, 0.005]) | |
([0.975, 0.997], [0.004, 0.019], [0, 0.005]) | |
([0.996, 1], [0.001, 0.006], [0, 0.002]) | |
([0.998, 1], [0.001, 0.005], [0, 0.001]) | |
([0.963, 0.996], [0.022, 0.06], [0.004, 0.027]) | |
([0.977, 0.998], [0.016, 0.044], [0.005, 0.02]) | |
([0.897, 0.973], [0.05, 0.11], [0.009, 0.056]) |
Students | Weighted Ratings |
---|---|
([0.605, 0.76], [0.004, 0.02], [0.001, 0.009]) | |
([0.594, 0.761], [0.004, 0.02], [0.001, 0.009]) | |
([0.581, 0.744], [0.004, 0.021], [0.001, 0.009]) | |
([0.458, 0.588], [0.022, 0.058], [0.011, 0.031]) |
Students | ||
---|---|---|
0.188874 | 0.901553 | |
0.192392 | 0.900405 | |
0.200641 | 0.896588 | |
0.279475 | 0.848118 |
Students | Closeness Coefficients | Ranking Order |
---|---|---|
0.826789 | 1 | |
0.823945 | 2 | |
0.817138 | 3 | |
0.752149 | 4 |
Criteria | Students | ||||
---|---|---|---|---|---|
([0.794, 0.9], [0, 0], [0, 0]) | ([0.51, 0.75], [0, 0.006], [0, 0.002]) | ([0.764, 0.893], [0, 0], [0, 0]) | ([0.711, 0.822], [0, 0.003], [0, 0.001]) | ([0.441, 0.569], [0.022, 0.053], [0.012, 0.027]) | |
([0.871, 0.951], [0, 0], [0, 0]) | ([0.675, 0.818], [0, 0.002], [0, 0.001]) | ([0.881, 0.96], [0, 0], [0, 0]) | ([0.788, 0.891], [0, 0.001], [0, 0]) | ([0.441, 0.569], [0.022, 0.053], [0.012, 0.027]) | |
([0.829, 0.918], [0, 0.001], [0, 0]) | ([0.608, 0.785], [0.001, 0.005], [0, 0.001]) | ([0.728, 0.884], [0, 0.001], [0, 0]) | ([0.817, 0.91], [0, 0.001], [0, 0]) | ([0.569, 0.67], [0.005, 0.016], [0.004, 0.012]) | |
([0.711, 0.875], [0, 0.001], [0, 0]) | ([0.608, 0.785], [0.001, 0.005], [0, 0.001]) | ([0.816, 0.922], [0, 0.001], [0, 0]) | ([0.663, 0.795], [0, 0.002], [0, 0.001]) | ([0.569, 0.67], [0.005, 0.016], [0.004, 0.012]) | |
([0.81, 0.912], [0, 0.001], [0, 0.001]) | ([0.635, 0.804], [0, 0.003], [0, 0.001]) | ([0.777, 0.889], [0, 0.001], [0, 0.001]) | ([0.751, 0.872], [0, 0.001], [0, 0.001]) | ([0.48, 0.608], [0.011, 0.032], [0.008, 0.021]) | |
([0.832, 0.923], [0, 0], [0, 0]) | ([0.608, 0.785], [0, 0.004], [0, 0.001]) | ([0.744, 0.902], [0, 0], [0, 0]) | ([0.482, 0.69], [0.001, 0.006], [0.001, 0.003]) | ([0.536, 0.637], [0.011, 0.026], [0.006, 0.016]) | |
([0.689, 0.86], [0, 0.001], [0, 0]) | ([0.591, 0.759], [0.001, 0.004], [0, 0.002]) | ([0.682, 0.86], [0, 0.001], [0, 0]) | ([0.586, 0.733], [0.001, 0.004], [0.001, 0.002]) | ([0.441, 0.569], [0.022, 0.053], [0.012, 0.027]) | |
([0.751, 0.898], [0, 0.001], [0, 0]) | ([0.662, 0.822], [0, 0.003], [0, 0.001]) | ([0.732, 0.89], [0, 0.001], [0, 0]) | ([0.699, 0.83], [0, 0.004], [0, 0.001]) | ([0.268, 0.441], [0.027, 0.079], [0.033, 0.062]) | |
([0.874, 0.95], [0, 0.002], [0, 0]) | ([0.749, 0.861], [0, 0.002], [0, 0.001]) | ([0.889, 0.963], [0, 0.002], [0, 0]) | ([0.743, 0.853], [0, 0.003], [0, 0.001]) | ([0.418, 0.578], [0.011, 0.039], [0.011, 0.026]) | |
([0.757, 0.891], [0, 0], [0, 0]) | ([0.636, 0.804], [0, 0.003], [0, 0.001]) | ([0.837, 0.926], [0, 0], [0, 0]) | ([0.712, 0.818], [0, 0.002], [0, 0.001]) | ([0.5, 0.6], [0.022, 0.044], [0.009, 0.022]) | |
([0.753, 0.88], [0, 0], [0, 0]) | ([0.521, 0.71], [0.001, 0.005], [0.001, 0.003]) | ([0.696, 0.875], [0, 0.001], [0, 0]) | ([0.651, 0.769], [0.001, 0.004], [0, 0.002]) | ([0.569, 0.67], [0.005, 0.015], [0.004, 0.012]) | |
([0.753, 0.884], [0.001, 0.007], [0, 0.002]) | ([0.53, 0.662], [0.002, 0.011], [0.001, 0.007]) | ([0.778, 0.903], [0.001, 0.007], [0, 0.002]) | ([0.544, 0.72], [0.002, 0.013], [0.001, 0.005]) | ([0.534, 0.636], [0.012, 0.032], [0.006, 0.018]) | |
([0.677, 0.845], [0, 0.002], [0, 0.001]) | ([0.338, 0.521], [0.001, 0.006], [0.003, 0.009]) | ([0.759, 0.881], [0, 0.002], [0, 0]) | ([0.699, 0.83], [0.001, 0.004], [0, 0.001]) | ([0.374, 0.536], [0.022, 0.065], [0.016, 0.035]) | |
([0.688, 0.837], [0.001, 0.005], [0, 0.001]) | ([0.407, 0.555], [0.002, 0.008], [0.002, 0.008]) | ([0.777, 0.916], [0.001, 0.005], [0, 0.001]) | ([0.699, 0.826], [0.001, 0.007], [0, 0.002]) | ([0.44, 0.569], [0.023, 0.057], [0.012, 0.029]) |
Criteria | Importance Aggregated Weights |
---|---|
([0.99999, 1], [0, 0.00009], [0, 0.00001]) | |
([0.99995, 1], [0.00001, 0.00019], [0, 0.00002]) | |
([0.99964, 1], [0.00005, 0.00062], [0, 0.00009]) | |
([0.99912, 0.99998], [0.00009, 0.00097], [0, 0.00018]) | |
([0.99776, 0.99995], [0.00004, 0.00077], [0.00001, 0.00053]) | |
([0.99999, 1], [0, 0.00006], [0, 0]) | |
([0.99985, 1], [0.00003, 0.00039], [0, 0.00005]) | |
([0.99907, 0.99999], [0.00009, 0.00114], [0, 0.00015]) | |
([0.99842, 0.99996], [0.00014, 0.00154], [0, 0.00024]) | |
([0.99991, 1], [0.00002, 0.00029], [0, 0.00004]) | |
([0.99997, 1], [0.00001, 0.00016], [0, 0.00001]) | |
([0.99615, 0.99988], [0.00112, 0.00657], [0.00004, 0.00152]) | |
([0.99969, 1], [0.00016, 0.00145], [0.00001, 0.00026]) | |
([0.99762, 0.99993], [0.00082, 0.00483], [0.00004, 0.00116]) |
Students | Weighted Ratings |
---|---|
([0.78, 0.901], [0, 0.001], [0, 0]) | |
([0.589, 0.759], [0.001, 0.004], [0, 0.002]) | |
([0.785, 0.91], [0, 0.001], [0, 0]) | |
([0.693, 0.822], [0, 0.003], [0, 0.001]) | |
([0.476, 0.599], [0.014, 0.037], [0.009, 0.022]) |
Students | ||
---|---|---|
0.37844 | 0.776416 | |
0.352522 | 0.752181 | |
0.381797 | 0.777005 | |
0.358066 | 0.764391 | |
0.325366 | 0.738391 |
Students | Closeness Coefficients | Ranking Order |
---|---|---|
0.672305 | 4 | |
0.680890 | 3 | |
0.670525 | 5 | |
0.680998 | 2 | |
0.694135 | 1 |
Time Period | The Method in [14] | The Proposed Method |
---|---|---|
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