Are MCDA Methods Benchmarkable? A Comparative Study of TOPSIS, VIKOR, COPRAS, and PROMETHEE II Methods
Abstract
:1. Introduction
2. Literature Review
2.1. MCDA State of the Art
2.1.1. MCDA Foundations
2.1.2. Operational Point of View and Preference Aggregation Techniques
- The use of a single synthesized criterion: In this approach, the result of the variants’ comparisons is determined for each criterion separately. Then the results are synthesized into a global assessment. The full order of variants is obtained here [65];
- The synthesis of the criteria based on the relation of outranking: Due to the occurrence of incomparability relations, this approach allows for obtaining the partial order of variants [65];
2.1.3. American School-Based MCDA Methods
2.1.4. European School-Based MCDA Methods
2.1.5. Mixed and Rule-Based Methods
2.2. MCDA Methods Selection and Benchmarking Problem
3. Preliminaries
3.1. MCDA Methods
3.1.1. TOPSIS
3.1.2. VIKOR
3.1.3. COPRAS
3.1.4. PROMETHEE II
3.2. Normalization Methods
3.3. Weighting Methods
3.3.1. Equal Weights
3.3.2. Entropy Method
3.3.3. Standard Deviation Method
3.4. Correlation Coefficients
3.4.1. Spearman’s Rank Correlation Coefficient
3.4.2. Weighted Spearman’s Rank Correlation Coefficient
3.4.3. Rank Similarity Coefficient
4. Study Case and Numerical Examples
- Step 1.
- Calculate 3 vectors of weights, using equations described in Section 3.3;
- Step 2.
- Split criteria into profit and cost criteria: Assuming we have n criteria, first are considered to be profit criteria and the rest ones are considered to be cost;
- Step 3.
- Compute 3 rankings using MCDA methods listed in Section 3.1 and three different weighting vectors.
Algorithm 1 Research algorithm |
|
4.1. Decision Matrices
4.2. TOPSIS
4.3. VIKOR
4.4. PROMETHEE II
4.5. Different Methods
4.6. Summary
5. Results and Discussion
5.1. TOPSIS
5.2. VIKOR
5.3. PROMETHEE II
5.4. Comparison of the MCDA Methods
5.5. Dependence of Ranking Similarity Coefficients on the Distance between Weight Vectors
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
TOPSIS | Technique for Order of Preference by Similarity to Ideal Solution |
VIKOR | VlseKriterijumska Optimizacija I Kompromisno Resenje (Serbian) |
COPRAS | Complex Proportional Assessment |
PROMETHEE | Preference Ranking Organization Method for Enrichment of Evaluation |
MCDA | Multi Criteria Decision Analysis |
MCDM | Multi Criteria Decision Making |
PIS | Positive Ideal Solution |
NIS | Negative ideal Solution |
SD | Standard Deviation |
Appendix A. Figures
Appendix B. Tables
0.947 | 0.957 | 0.275 | |
0.018 | 0.631 | 0.581 | |
0.565 | 0.295 | 0.701 | |
0.423 | 0.602 | 0.509 | |
0.664 | 0.637 | 0.786 | |
0.333 | 0.333 | 0.333 | |
0.678 | 0.172 | 0.151 | |
0.442 | 0.303 | 0.255 |
Minmax | Max | Sum | Vector | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
(a) | (b) | (c) | (a) | (b) | (c) | (a) | (b) | (c) | (a) | (b) | (c) | ||||
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||
4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | ||||
5 | 3 | 4 | 4 | 3 | 3 | 4 | 3 | 3 | 4 | 3 | 3 | ||||
2 | 4 | 3 | 3 | 4 | 4 | 3 | 4 | 4 | 3 | 4 | 4 | ||||
3 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
None | Minmax | Max | Sum | Vector | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
(a) | (b) | (c) | (a) | (b) | (c) | (a) | (b) | (c) | (a) | (b) | (c) | (a) | (b) | (c) | |||||
3 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |||||
5 | 5 | 5 | 4 | 5 | 5 | 4 | 5 | 5 | 4 | 5 | 5 | 4 | 5 | 5 | |||||
4 | 3 | 4 | 5 | 3 | 4 | 5 | 3 | 4 | 5 | 3 | 4 | 5 | 3 | 4 | |||||
2 | 4 | 3 | 2 | 4 | 2 | 2 | 4 | 2 | 2 | 4 | 3 | 2 | 4 | 2 | |||||
1 | 2 | 1 | 3 | 2 | 3 | 3 | 2 | 3 | 3 | 2 | 2 | 3 | 2 | 3 |
Usual | U-Shape | V-Shape | Level | V-Shape 2 | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
(a) | (b) | (c) | (a) | (b) | (c) | (a) | (b) | (c) | (a) | (b) | (c) | (a) | (b) | (c) | |||||
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |||||
4 | 5 | 5 | 4 | 5 | 5 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | |||||
5 | 3 | 4 | 4 | 4 | 4 | 5 | 4 | 4 | 4 | 4 | 4 | 4 | 3 | 4 | |||||
3 | 4 | 3 | 2 | 3 | 2 | 2 | 3 | 2 | 2 | 3 | 3 | 2 | 4 | 3 | |||||
2 | 2 | 2 | 3 | 2 | 3 | 3 | 2 | 3 | 3 | 2 | 2 | 3 | 2 | 2 |
TOPSIS | VIKOR | PROM. II | COPRAS | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
(a) | (b) | (c) | (a) | (b) | (c) | (a) | (b) | (c) | (a) | (b) | (c) | ||||
1 | 1 | 1 | 3 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | ||||
4 | 5 | 5 | 5 | 5 | 5 | 4 | 5 | 5 | 5 | 5 | 5 | ||||
5 | 3 | 4 | 4 | 3 | 4 | 5 | 3 | 4 | 4 | 3 | 4 | ||||
2 | 4 | 3 | 2 | 4 | 3 | 3 | 4 | 3 | 3 | 4 | 3 | ||||
3 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
Norm | Weighting Method | 2 Criteria | 3 Criteria | 4 Criteria | 5 Criteria | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Alternatives | Alternatives | Alternatives | Alternatives | ||||||||||||||||||
3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | ||
(a) | equal | 0.828 | 0.923 | 0.977 | 0.999 | 1.000 | 0.755 | 0.897 | 0.966 | 0.999 | 1.000 | 0.744 | 0.887 | 0.966 | 0.998 | 1.000 | 0.735 | 0.871 | 0.964 | 0.998 | 1.000 |
entropy | 0.979 | 0.986 | 0.991 | 0.999 | 1.000 | 0.973 | 0.980 | 0.989 | 0.999 | 1.000 | 0.977 | 0.974 | 0.989 | 0.999 | 1.000 | 0.973 | 0.973 | 0.988 | 0.999 | 1.000 | |
std | 0.916 | 0.960 | 0.985 | 0.999 | 1.000 | 0.913 | 0.949 | 0.977 | 0.999 | 1.000 | 0.904 | 0.938 | 0.976 | 0.998 | 1.000 | 0.897 | 0.933 | 0.974 | 0.998 | 1.000 | |
(b) | equal | 0.740 | 0.808 | 0.821 | 0.815 | 0.807 | 0.651 | 0.754 | 0.805 | 0.825 | 0.821 | 0.596 | 0.679 | 0.714 | 0.733 | 0.735 | 0.571 | 0.658 | 0.702 | 0.742 | 0.748 |
entropy | 0.916 | 0.895 | 0.851 | 0.812 | 0.804 | 0.905 | 0.878 | 0.847 | 0.826 | 0.820 | 0.878 | 0.796 | 0.755 | 0.731 | 0.733 | 0.865 | 0.796 | 0.766 | 0.742 | 0.745 | |
std | 0.836 | 0.844 | 0.832 | 0.813 | 0.805 | 0.823 | 0.815 | 0.819 | 0.824 | 0.820 | 0.783 | 0.735 | 0.727 | 0.732 | 0.734 | 0.762 | 0.722 | 0.720 | 0.741 | 0.747 | |
(c) | equal | 0.803 | 0.900 | 0.957 | 0.994 | 0.997 | 0.703 | 0.856 | 0.934 | 0.991 | 0.996 | 0.696 | 0.839 | 0.929 | 0.989 | 0.995 | 0.701 | 0.818 | 0.922 | 0.989 | 0.995 |
entropy | 0.972 | 0.977 | 0.984 | 0.996 | 0.998 | 0.961 | 0.969 | 0.976 | 0.993 | 0.997 | 0.963 | 0.959 | 0.973 | 0.992 | 0.996 | 0.960 | 0.958 | 0.967 | 0.991 | 0.996 | |
std | 0.899 | 0.942 | 0.970 | 0.994 | 0.997 | 0.890 | 0.921 | 0.951 | 0.991 | 0.996 | 0.875 | 0.903 | 0.947 | 0.990 | 0.996 | 0.874 | 0.893 | 0.938 | 0.989 | 0.995 | |
(d) | equal | 0.870 | 0.849 | 0.827 | 0.813 | 0.806 | 0.842 | 0.829 | 0.821 | 0.824 | 0.821 | 0.785 | 0.747 | 0.732 | 0.732 | 0.733 | 0.782 | 0.741 | 0.723 | 0.741 | 0.747 |
entropy | 0.925 | 0.895 | 0.850 | 0.810 | 0.802 | 0.913 | 0.882 | 0.847 | 0.824 | 0.819 | 0.883 | 0.802 | 0.756 | 0.729 | 0.732 | 0.873 | 0.802 | 0.769 | 0.740 | 0.744 | |
std | 0.895 | 0.864 | 0.833 | 0.811 | 0.804 | 0.880 | 0.850 | 0.827 | 0.822 | 0.819 | 0.833 | 0.763 | 0.734 | 0.731 | 0.733 | 0.839 | 0.763 | 0.731 | 0.739 | 0.746 | |
(e) | equal | 0.966 | 0.971 | 0.980 | 0.995 | 0.998 | 0.926 | 0.950 | 0.965 | 0.992 | 0.996 | 0.917 | 0.941 | 0.960 | 0.991 | 0.996 | 0.923 | 0.934 | 0.956 | 0.990 | 0.995 |
entropy | 0.986 | 0.986 | 0.989 | 0.996 | 0.998 | 0.976 | 0.980 | 0.983 | 0.994 | 0.997 | 0.974 | 0.972 | 0.977 | 0.993 | 0.996 | 0.970 | 0.964 | 0.972 | 0.992 | 0.996 | |
std | 0.975 | 0.979 | 0.983 | 0.995 | 0.998 | 0.970 | 0.963 | 0.971 | 0.992 | 0.996 | 0.954 | 0.952 | 0.966 | 0.991 | 0.996 | 0.947 | 0.943 | 0.960 | 0.990 | 0.996 | |
(f) | equal | 0.868 | 0.850 | 0.829 | 0.812 | 0.806 | 0.861 | 0.846 | 0.833 | 0.826 | 0.822 | 0.788 | 0.750 | 0.729 | 0.732 | 0.733 | 0.795 | 0.751 | 0.731 | 0.742 | 0.748 |
entropy | 0.922 | 0.897 | 0.849 | 0.808 | 0.801 | 0.925 | 0.890 | 0.852 | 0.824 | 0.819 | 0.886 | 0.803 | 0.755 | 0.727 | 0.730 | 0.874 | 0.808 | 0.771 | 0.740 | 0.744 | |
std | 0.899 | 0.870 | 0.835 | 0.811 | 0.804 | 0.888 | 0.859 | 0.837 | 0.825 | 0.821 | 0.830 | 0.767 | 0.734 | 0.731 | 0.733 | 0.848 | 0.764 | 0.736 | 0.741 | 0.747 |
Norm | Weighting Method | 2 Criteria | 3 Criteria | 4 Criteria | 5 Criteria | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Alternatives | Alternatives | Alternatives | Alternatives | ||||||||||||||||||
3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | ||
(a) | equal | 0.864 | 0.931 | 0.980 | 0.999 | 1.000 | 0.837 | 0.910 | 0.967 | 0.999 | 1.000 | 0.826 | 0.900 | 0.966 | 0.999 | 1.000 | 0.821 | 0.886 | 0.962 | 0.998 | 1.000 |
entropy | 0.983 | 0.986 | 0.988 | 0.999 | 1.000 | 0.978 | 0.979 | 0.987 | 0.999 | 1.000 | 0.982 | 0.973 | 0.985 | 0.999 | 1.000 | 0.979 | 0.972 | 0.984 | 0.999 | 1.000 | |
std | 0.939 | 0.961 | 0.985 | 0.999 | 1.000 | 0.934 | 0.946 | 0.974 | 0.999 | 1.000 | 0.925 | 0.940 | 0.974 | 0.999 | 1.000 | 0.921 | 0.936 | 0.971 | 0.999 | 1.000 | |
(b) | equal | 0.812 | 0.842 | 0.865 | 0.926 | 0.944 | 0.774 | 0.813 | 0.857 | 0.915 | 0.932 | 0.748 | 0.776 | 0.810 | 0.868 | 0.885 | 0.740 | 0.767 | 0.809 | 0.869 | 0.886 |
entropy | 0.936 | 0.908 | 0.889 | 0.927 | 0.945 | 0.928 | 0.898 | 0.887 | 0.917 | 0.932 | 0.908 | 0.843 | 0.836 | 0.873 | 0.887 | 0.897 | 0.841 | 0.838 | 0.873 | 0.889 | |
std | 0.889 | 0.871 | 0.874 | 0.926 | 0.944 | 0.878 | 0.852 | 0.865 | 0.916 | 0.932 | 0.852 | 0.805 | 0.816 | 0.869 | 0.886 | 0.840 | 0.801 | 0.815 | 0.870 | 0.886 | |
(c) | equal | 0.847 | 0.912 | 0.964 | 0.998 | 0.999 | 0.805 | 0.881 | 0.945 | 0.996 | 0.999 | 0.800 | 0.868 | 0.939 | 0.995 | 0.998 | 0.803 | 0.852 | 0.932 | 0.993 | 0.998 |
entropy | 0.977 | 0.976 | 0.981 | 0.998 | 0.999 | 0.969 | 0.969 | 0.975 | 0.996 | 0.999 | 0.970 | 0.957 | 0.970 | 0.995 | 0.998 | 0.968 | 0.957 | 0.965 | 0.995 | 0.998 | |
std | 0.927 | 0.946 | 0.972 | 0.998 | 0.999 | 0.919 | 0.923 | 0.955 | 0.996 | 0.999 | 0.906 | 0.910 | 0.949 | 0.995 | 0.998 | 0.909 | 0.905 | 0.944 | 0.994 | 0.998 | |
(d) | equal | 0.909 | 0.871 | 0.868 | 0.925 | 0.944 | 0.889 | 0.865 | 0.865 | 0.915 | 0.932 | 0.853 | 0.809 | 0.818 | 0.867 | 0.885 | 0.852 | 0.812 | 0.816 | 0.869 | 0.886 |
entropy | 0.943 | 0.909 | 0.891 | 0.927 | 0.945 | 0.936 | 0.902 | 0.890 | 0.917 | 0.932 | 0.913 | 0.847 | 0.839 | 0.873 | 0.887 | 0.906 | 0.844 | 0.840 | 0.873 | 0.889 | |
std | 0.924 | 0.884 | 0.875 | 0.926 | 0.944 | 0.914 | 0.878 | 0.871 | 0.916 | 0.932 | 0.883 | 0.823 | 0.821 | 0.868 | 0.886 | 0.890 | 0.824 | 0.821 | 0.869 | 0.886 | |
(e) | equal | 0.972 | 0.969 | 0.980 | 0.998 | 0.999 | 0.945 | 0.951 | 0.967 | 0.997 | 0.999 | 0.939 | 0.943 | 0.960 | 0.995 | 0.998 | 0.943 | 0.937 | 0.956 | 0.994 | 0.998 |
entropy | 0.989 | 0.984 | 0.987 | 0.998 | 0.999 | 0.983 | 0.979 | 0.981 | 0.997 | 0.999 | 0.979 | 0.969 | 0.976 | 0.996 | 0.998 | 0.978 | 0.963 | 0.971 | 0.995 | 0.998 | |
std | 0.980 | 0.978 | 0.983 | 0.998 | 0.999 | 0.977 | 0.963 | 0.971 | 0.996 | 0.999 | 0.964 | 0.950 | 0.963 | 0.996 | 0.999 | 0.960 | 0.945 | 0.960 | 0.994 | 0.998 | |
(f) | equal | 0.906 | 0.870 | 0.848 | 0.825 | 0.821 | 0.897 | 0.866 | 0.839 | 0.791 | 0.773 | 0.857 | 0.800 | 0.774 | 0.745 | 0.734 | 0.852 | 0.801 | 0.770 | 0.730 | 0.724 |
entropy | 0.942 | 0.909 | 0.872 | 0.821 | 0.816 | 0.942 | 0.904 | 0.865 | 0.781 | 0.769 | 0.913 | 0.841 | 0.802 | 0.740 | 0.731 | 0.905 | 0.843 | 0.802 | 0.725 | 0.721 | |
std | 0.926 | 0.885 | 0.854 | 0.823 | 0.821 | 0.917 | 0.879 | 0.846 | 0.788 | 0.774 | 0.881 | 0.817 | 0.782 | 0.744 | 0.733 | 0.891 | 0.813 | 0.774 | 0.730 | 0.722 |
Norm | Weighting Method | 2 Criteria | 3 Criteria | 4 Criteria | 5 Criteria | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Alternatives | Alternatives | Alternatives | Alternatives | ||||||||||||||||||
3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | ||
(a) | equal | 0.083 | −0.003 | −0.029 | −0.028 | −0.030 | 0.237 | 0.317 | 0.313 | 0.344 | 0.338 | 0.043 | 0.058 | 0.066 | 0.067 | 0.069 | 0.162 | 0.223 | 0.238 | 0.255 | 0.255 |
entropy | 0.010 | −0.030 | −0.036 | −0.016 | −0.026 | 0.310 | 0.345 | 0.336 | 0.361 | 0.330 | 0.031 | 0.006 | 0.037 | 0.053 | 0.066 | 0.211 | 0.192 | 0.260 | 0.236 | 0.233 | |
std | 0.022 | −0.033 | −0.040 | −0.027 | −0.032 | 0.270 | 0.329 | 0.304 | 0.345 | 0.329 | 0.038 | 0.014 | 0.033 | 0.057 | 0.065 | 0.199 | 0.175 | 0.226 | 0.237 | 0.246 | |
(b) | equal | 0.083 | −0.003 | −0.029 | −0.028 | −0.030 | 0.237 | 0.317 | 0.313 | 0.344 | 0.338 | 0.043 | 0.058 | 0.066 | 0.067 | 0.069 | 0.162 | 0.223 | 0.238 | 0.255 | 0.255 |
entropy | 0.010 | −0.030 | −0.036 | −0.016 | −0.026 | 0.310 | 0.345 | 0.336 | 0.361 | 0.330 | 0.031 | 0.006 | 0.037 | 0.053 | 0.066 | 0.211 | 0.192 | 0.260 | 0.236 | 0.233 | |
std | 0.022 | −0.033 | −0.040 | −0.027 | −0.032 | 0.270 | 0.329 | 0.304 | 0.345 | 0.329 | 0.038 | 0.014 | 0.033 | 0.057 | 0.065 | 0.199 | 0.175 | 0.226 | 0.237 | 0.246 | |
(c) | equal | 0.088 | −0.033 | −0.020 | 0.222 | 0.358 | 0.250 | 0.265 | 0.231 | 0.377 | 0.468 | 0.055 | 0.018 | −0.029 | 0.121 | 0.235 | 0.159 | 0.210 | 0.129 | 0.237 | 0.335 |
entropy | 0.022 | 0.003 | 0.046 | 0.277 | 0.386 | 0.315 | 0.356 | 0.370 | 0.499 | 0.532 | 0.026 | −0.006 | 0.048 | 0.173 | 0.248 | 0.194 | 0.181 | 0.240 | 0.305 | 0.356 | |
std | 0.017 | −0.014 | 0.014 | 0.242 | 0.367 | 0.266 | 0.305 | 0.290 | 0.426 | 0.487 | −0.002 | −0.044 | −0.025 | 0.094 | 0.186 | 0.164 | 0.108 | 0.141 | 0.220 | 0.290 | |
(d) | equal | 0.083 | −0.003 | −0.029 | −0.028 | −0.030 | 0.237 | 0.317 | 0.313 | 0.344 | 0.338 | 0.043 | 0.058 | 0.066 | 0.067 | 0.069 | 0.162 | 0.223 | 0.238 | 0.255 | 0.255 |
entropy | 0.010 | −0.030 | −0.036 | −0.016 | −0.026 | 0.310 | 0.345 | 0.336 | 0.361 | 0.330 | 0.031 | 0.006 | 0.037 | 0.053 | 0.066 | 0.211 | 0.192 | 0.260 | 0.236 | 0.233 | |
std | 0.022 | −0.033 | −0.040 | −0.027 | −0.032 | 0.270 | 0.329 | 0.304 | 0.345 | 0.329 | 0.038 | 0.014 | 0.033 | 0.057 | 0.065 | 0.199 | 0.175 | 0.226 | 0.237 | 0.246 | |
(e) | equal | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
entropy | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |
std | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |
(f) | equal | 0.950 | 0.889 | 0.851 | 0.791 | 0.760 | 0.964 | 0.915 | 0.865 | 0.826 | 0.812 | 0.925 | 0.903 | 0.843 | 0.793 | 0.778 | 0.925 | 0.918 | 0.852 | 0.805 | 0.795 |
entropy | 0.938 | 0.901 | 0.865 | 0.795 | 0.759 | 0.937 | 0.909 | 0.884 | 0.846 | 0.821 | 0.892 | 0.846 | 0.813 | 0.767 | 0.748 | 0.909 | 0.857 | 0.849 | 0.793 | 0.772 | |
std | 0.917 | 0.880 | 0.847 | 0.791 | 0.758 | 0.922 | 0.901 | 0.867 | 0.833 | 0.814 | 0.873 | 0.839 | 0.806 | 0.765 | 0.749 | 0.896 | 0.849 | 0.836 | 0.784 | 0.769 | |
(g) | equal | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
entropy | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |
std | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |
(h) | equal | 0.950 | 0.889 | 0.851 | 0.791 | 0.760 | 0.964 | 0.915 | 0.865 | 0.826 | 0.812 | 0.925 | 0.903 | 0.843 | 0.793 | 0.778 | 0.925 | 0.918 | 0.852 | 0.805 | 0.795 |
entropy | 0.938 | 0.901 | 0.865 | 0.795 | 0.759 | 0.937 | 0.909 | 0.884 | 0.846 | 0.821 | 0.892 | 0.846 | 0.813 | 0.767 | 0.748 | 0.909 | 0.857 | 0.849 | 0.793 | 0.772 | |
std | 0.917 | 0.880 | 0.847 | 0.791 | 0.758 | 0.922 | 0.901 | 0.867 | 0.833 | 0.814 | 0.873 | 0.839 | 0.806 | 0.765 | 0.749 | 0.896 | 0.849 | 0.836 | 0.784 | 0.769 | |
(i) | equal | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
entropy | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |
std | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |
(j) | equal | 0.950 | 0.889 | 0.851 | 0.791 | 0.760 | 0.964 | 0.915 | 0.865 | 0.826 | 0.812 | 0.925 | 0.903 | 0.843 | 0.793 | 0.778 | 0.925 | 0.918 | 0.852 | 0.805 | 0.795 |
entropy | 0.938 | 0.901 | 0.865 | 0.795 | 0.759 | 0.937 | 0.909 | 0.884 | 0.846 | 0.821 | 0.892 | 0.846 | 0.813 | 0.767 | 0.748 | 0.909 | 0.857 | 0.849 | 0.793 | 0.772 | |
std | 0.917 | 0.880 | 0.847 | 0.791 | 0.758 | 0.922 | 0.901 | 0.867 | 0.833 | 0.814 | 0.873 | 0.839 | 0.806 | 0.765 | 0.749 | 0.896 | 0.849 | 0.836 | 0.784 | 0.769 |
Norm | Weighting Method | 2 Criteria | 3 Criteria | 4 Criteria | 5 Criteria | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Alternatives | Alternatives | Alternatives | Alternatives | ||||||||||||||||||
3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | ||
(a) | equal | 0.458 | 0.487 | 0.475 | 0.369 | 0.340 | 0.570 | 0.596 | 0.621 | 0.601 | 0.567 | 0.537 | 0.521 | 0.533 | 0.468 | 0.441 | 0.570 | 0.582 | 0.610 | 0.592 | 0.575 |
entropy | 0.599 | 0.526 | 0.489 | 0.392 | 0.355 | 0.686 | 0.662 | 0.643 | 0.613 | 0.570 | 0.570 | 0.528 | 0.519 | 0.457 | 0.435 | 0.623 | 0.590 | 0.613 | 0.570 | 0.554 | |
std | 0.549 | 0.493 | 0.474 | 0.374 | 0.343 | 0.622 | 0.624 | 0.621 | 0.605 | 0.567 | 0.540 | 0.511 | 0.520 | 0.463 | 0.441 | 0.593 | 0.571 | 0.599 | 0.584 | 0.571 | |
(b) | equal | 0.458 | 0.487 | 0.475 | 0.369 | 0.340 | 0.570 | 0.596 | 0.621 | 0.601 | 0.567 | 0.537 | 0.521 | 0.533 | 0.468 | 0.441 | 0.570 | 0.582 | 0.610 | 0.592 | 0.575 |
entropy | 0.599 | 0.526 | 0.489 | 0.392 | 0.355 | 0.686 | 0.662 | 0.643 | 0.613 | 0.570 | 0.570 | 0.528 | 0.519 | 0.457 | 0.435 | 0.623 | 0.590 | 0.613 | 0.570 | 0.554 | |
std | 0.549 | 0.493 | 0.474 | 0.374 | 0.343 | 0.622 | 0.624 | 0.621 | 0.605 | 0.567 | 0.540 | 0.511 | 0.520 | 0.463 | 0.441 | 0.593 | 0.571 | 0.599 | 0.584 | 0.571 | |
(c) | equal | 0.470 | 0.507 | 0.564 | 0.747 | 0.829 | 0.571 | 0.588 | 0.644 | 0.814 | 0.878 | 0.540 | 0.514 | 0.542 | 0.700 | 0.790 | 0.566 | 0.580 | 0.595 | 0.746 | 0.823 |
entropy | 0.607 | 0.560 | 0.593 | 0.763 | 0.836 | 0.691 | 0.686 | 0.712 | 0.844 | 0.891 | 0.580 | 0.550 | 0.580 | 0.710 | 0.780 | 0.628 | 0.614 | 0.650 | 0.761 | 0.818 | |
std | 0.558 | 0.531 | 0.580 | 0.754 | 0.832 | 0.627 | 0.639 | 0.675 | 0.828 | 0.883 | 0.533 | 0.515 | 0.549 | 0.675 | 0.755 | 0.588 | 0.564 | 0.606 | 0.726 | 0.789 | |
(d) | equal | 0.458 | 0.487 | 0.475 | 0.369 | 0.340 | 0.570 | 0.596 | 0.621 | 0.601 | 0.567 | 0.537 | 0.521 | 0.533 | 0.468 | 0.441 | 0.570 | 0.582 | 0.610 | 0.592 | 0.575 |
entropy | 0.599 | 0.526 | 0.489 | 0.392 | 0.355 | 0.686 | 0.662 | 0.643 | 0.613 | 0.570 | 0.570 | 0.528 | 0.519 | 0.457 | 0.435 | 0.623 | 0.590 | 0.613 | 0.570 | 0.554 | |
std | 0.549 | 0.493 | 0.474 | 0.374 | 0.343 | 0.622 | 0.624 | 0.621 | 0.605 | 0.567 | 0.540 | 0.511 | 0.520 | 0.463 | 0.441 | 0.593 | 0.571 | 0.599 | 0.584 | 0.571 | |
(e) | equal | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
entropy | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |
std | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |
(f) | equal | 0.959 | 0.897 | 0.890 | 0.946 | 0.962 | 0.970 | 0.916 | 0.884 | 0.937 | 0.957 | 0.942 | 0.909 | 0.874 | 0.931 | 0.952 | 0.944 | 0.923 | 0.876 | 0.923 | 0.949 |
entropy | 0.950 | 0.914 | 0.903 | 0.948 | 0.963 | 0.951 | 0.924 | 0.909 | 0.945 | 0.960 | 0.917 | 0.875 | 0.868 | 0.911 | 0.930 | 0.929 | 0.882 | 0.883 | 0.915 | 0.933 | |
std | 0.933 | 0.896 | 0.889 | 0.946 | 0.962 | 0.944 | 0.909 | 0.890 | 0.940 | 0.958 | 0.907 | 0.863 | 0.858 | 0.914 | 0.935 | 0.926 | 0.871 | 0.867 | 0.912 | 0.933 | |
(g) | equal | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
entropy | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |
std | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |
(h) | equal | 0.959 | 0.897 | 0.890 | 0.946 | 0.962 | 0.970 | 0.916 | 0.884 | 0.937 | 0.957 | 0.942 | 0.909 | 0.874 | 0.931 | 0.952 | 0.944 | 0.923 | 0.876 | 0.923 | 0.949 |
entropy | 0.950 | 0.914 | 0.903 | 0.948 | 0.963 | 0.951 | 0.924 | 0.909 | 0.945 | 0.960 | 0.917 | 0.875 | 0.868 | 0.911 | 0.930 | 0.929 | 0.882 | 0.883 | 0.915 | 0.933 | |
std | 0.933 | 0.896 | 0.889 | 0.946 | 0.962 | 0.944 | 0.909 | 0.890 | 0.940 | 0.958 | 0.907 | 0.863 | 0.858 | 0.914 | 0.935 | 0.926 | 0.871 | 0.867 | 0.912 | 0.933 | |
(i) | equal | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
entropy | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |
std | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |
(j) | equal | 0.954 | 0.897 | 0.874 | 0.874 | 0.876 | 0.971 | 0.917 | 0.878 | 0.870 | 0.869 | 0.943 | 0.910 | 0.869 | 0.873 | 0.876 | 0.945 | 0.922 | 0.873 | 0.865 | 0.869 |
entropy | 0.950 | 0.913 | 0.889 | 0.885 | 0.884 | 0.950 | 0.922 | 0.900 | 0.892 | 0.885 | 0.919 | 0.872 | 0.858 | 0.856 | 0.853 | 0.930 | 0.881 | 0.873 | 0.866 | 0.866 | |
std | 0.934 | 0.894 | 0.872 | 0.877 | 0.878 | 0.944 | 0.908 | 0.882 | 0.879 | 0.875 | 0.906 | 0.863 | 0.849 | 0.862 | 0.865 | 0.928 | 0.869 | 0.865 | 0.866 | 0.865 |
Type | k | Weighting Method | 2 Criteria | 3 Criteria | 4 Criteria | 5 Criteria | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Alternatives | Alternatives | Alternatives | Alternatives | |||||||||||||||||||
3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | |||
(a) | 0.25/0.5 | equal | 0.967 | 0.930 | 0.959 | 0.992 | 0.996 | 0.951 | 0.929 | 0.956 | 0.994 | 0.997 | 0.952 | 0.929 | 0.957 | 0.994 | 0.997 | 0.944 | 0.923 | 0.958 | 0.993 | 0.997 |
entropy | 0.971 | 0.937 | 0.955 | 0.992 | 0.996 | 0.965 | 0.932 | 0.958 | 0.993 | 0.997 | 0.958 | 0.926 | 0.954 | 0.993 | 0.997 | 0.968 | 0.927 | 0.956 | 0.993 | 0.997 | ||
std | 0.967 | 0.933 | 0.957 | 0.992 | 0.996 | 0.956 | 0.928 | 0.954 | 0.993 | 0.997 | 0.952 | 0.919 | 0.956 | 0.993 | 0.997 | 0.957 | 0.924 | 0.958 | 0.993 | 0.997 | ||
0.25/1 | equal | 0.946 | 0.885 | 0.938 | 0.990 | 0.995 | 0.934 | 0.875 | 0.928 | 0.989 | 0.995 | 0.931 | 0.874 | 0.928 | 0.988 | 0.995 | 0.931 | 0.860 | 0.925 | 0.988 | 0.994 | |
entropy | 0.969 | 0.898 | 0.931 | 0.989 | 0.995 | 0.945 | 0.877 | 0.927 | 0.988 | 0.995 | 0.948 | 0.874 | 0.924 | 0.988 | 0.994 | 0.952 | 0.873 | 0.924 | 0.988 | 0.994 | ||
std | 0.951 | 0.887 | 0.934 | 0.990 | 0.995 | 0.932 | 0.870 | 0.926 | 0.989 | 0.995 | 0.929 | 0.865 | 0.925 | 0.988 | 0.994 | 0.931 | 0.862 | 0.926 | 0.988 | 0.994 | ||
0.5/1 | equal | 0.958 | 0.906 | 0.946 | 0.991 | 0.995 | 0.950 | 0.897 | 0.941 | 0.991 | 0.996 | 0.944 | 0.900 | 0.940 | 0.990 | 0.996 | 0.944 | 0.888 | 0.939 | 0.990 | 0.995 | |
entropy | 0.973 | 0.915 | 0.943 | 0.991 | 0.995 | 0.952 | 0.902 | 0.941 | 0.990 | 0.996 | 0.956 | 0.899 | 0.935 | 0.990 | 0.995 | 0.961 | 0.894 | 0.939 | 0.990 | 0.995 | ||
std | 0.964 | 0.906 | 0.945 | 0.991 | 0.995 | 0.944 | 0.892 | 0.941 | 0.991 | 0.996 | 0.949 | 0.889 | 0.937 | 0.990 | 0.996 | 0.946 | 0.889 | 0.939 | 0.991 | 0.995 | ||
(b) | 0.25/0.5 | equal | 1.000 | 0.991 | 0.994 | 0.999 | 1.000 | 0.996 | 0.990 | 0.994 | 0.999 | 1.000 | 0.997 | 0.992 | 0.994 | 0.999 | 1.000 | 0.993 | 0.985 | 0.994 | 0.999 | 1.000 |
entropy | 0.996 | 0.993 | 0.995 | 0.999 | 1.000 | 0.997 | 0.992 | 0.995 | 0.999 | 1.000 | 0.997 | 0.992 | 0.994 | 0.999 | 1.000 | 0.994 | 0.991 | 0.994 | 0.999 | 1.000 | ||
std | 0.995 | 0.991 | 0.995 | 0.999 | 1.000 | 0.996 | 0.991 | 0.994 | 0.999 | 1.000 | 0.994 | 0.990 | 0.994 | 0.999 | 1.000 | 0.995 | 0.990 | 0.994 | 0.999 | 1.000 | ||
0.25/1 | equal | 1.000 | 0.980 | 0.989 | 0.998 | 0.999 | 0.995 | 0.980 | 0.984 | 0.998 | 0.999 | 0.992 | 0.980 | 0.985 | 0.998 | 0.999 | 0.987 | 0.977 | 0.986 | 0.998 | 0.999 | |
entropy | 0.992 | 0.985 | 0.989 | 0.998 | 0.999 | 0.986 | 0.982 | 0.988 | 0.998 | 0.999 | 0.990 | 0.982 | 0.986 | 0.998 | 0.999 | 0.986 | 0.980 | 0.986 | 0.998 | 0.999 | ||
std | 0.995 | 0.982 | 0.989 | 0.998 | 0.999 | 0.984 | 0.982 | 0.987 | 0.998 | 0.999 | 0.990 | 0.980 | 0.986 | 0.998 | 0.999 | 0.985 | 0.978 | 0.986 | 0.998 | 0.999 | ||
0.5/1 | equal | 1.000 | 0.987 | 0.993 | 0.999 | 0.999 | 0.993 | 0.988 | 0.989 | 0.998 | 0.999 | 0.991 | 0.985 | 0.990 | 0.998 | 0.999 | 0.987 | 0.986 | 0.990 | 0.999 | 0.999 | |
entropy | 0.992 | 0.989 | 0.992 | 0.999 | 0.999 | 0.986 | 0.988 | 0.992 | 0.999 | 0.999 | 0.991 | 0.987 | 0.991 | 0.999 | 0.999 | 0.987 | 0.986 | 0.990 | 0.999 | 0.999 | ||
std | 0.994 | 0.986 | 0.993 | 0.999 | 1.000 | 0.986 | 0.987 | 0.992 | 0.999 | 0.999 | 0.990 | 0.988 | 0.990 | 0.999 | 0.999 | 0.983 | 0.985 | 0.990 | 0.998 | 0.999 | ||
(c) | 0.25/0.5 | equal | 0.936 | 0.930 | 0.967 | 0.994 | 0.997 | 0.930 | 0.935 | 0.964 | 0.995 | 0.998 | 0.940 | 0.935 | 0.962 | 0.995 | 0.998 | 0.929 | 0.929 | 0.962 | 0.995 | 0.998 |
entropy | 0.954 | 0.938 | 0.962 | 0.993 | 0.997 | 0.949 | 0.934 | 0.964 | 0.994 | 0.998 | 0.947 | 0.937 | 0.960 | 0.994 | 0.997 | 0.944 | 0.935 | 0.960 | 0.994 | 0.997 | ||
std | 0.948 | 0.937 | 0.965 | 0.994 | 0.997 | 0.943 | 0.937 | 0.963 | 0.995 | 0.997 | 0.944 | 0.935 | 0.962 | 0.994 | 0.998 | 0.930 | 0.932 | 0.964 | 0.995 | 0.998 | ||
0.25/1 | equal | 0.854 | 0.888 | 0.943 | 0.992 | 0.996 | 0.858 | 0.876 | 0.934 | 0.988 | 0.995 | 0.870 | 0.880 | 0.930 | 0.987 | 0.994 | 0.859 | 0.864 | 0.928 | 0.988 | 0.993 | |
entropy | 0.942 | 0.898 | 0.935 | 0.991 | 0.996 | 0.917 | 0.886 | 0.931 | 0.988 | 0.996 | 0.901 | 0.878 | 0.924 | 0.987 | 0.993 | 0.900 | 0.873 | 0.926 | 0.987 | 0.993 | ||
std | 0.904 | 0.889 | 0.939 | 0.991 | 0.996 | 0.871 | 0.877 | 0.930 | 0.988 | 0.995 | 0.873 | 0.876 | 0.929 | 0.987 | 0.994 | 0.867 | 0.868 | 0.928 | 0.987 | 0.993 | ||
0.5/1 | equal | 0.862 | 0.909 | 0.952 | 0.993 | 0.996 | 0.876 | 0.899 | 0.946 | 0.991 | 0.996 | 0.879 | 0.903 | 0.945 | 0.990 | 0.995 | 0.874 | 0.891 | 0.941 | 0.990 | 0.995 | |
entropy | 0.941 | 0.917 | 0.948 | 0.993 | 0.996 | 0.922 | 0.909 | 0.944 | 0.990 | 0.996 | 0.912 | 0.899 | 0.940 | 0.990 | 0.995 | 0.907 | 0.902 | 0.939 | 0.990 | 0.995 | ||
std | 0.913 | 0.911 | 0.950 | 0.993 | 0.996 | 0.884 | 0.900 | 0.945 | 0.990 | 0.996 | 0.883 | 0.896 | 0.944 | 0.990 | 0.995 | 0.885 | 0.894 | 0.940 | 0.990 | 0.995 | ||
(d) | 0.25/0.5 | equal | 0.962 | 0.965 | 0.986 | 0.997 | 0.999 | 0.970 | 0.970 | 0.985 | 0.998 | 0.999 | 0.974 | 0.975 | 0.985 | 0.998 | 0.999 | 0.972 | 0.973 | 0.984 | 0.998 | 0.999 |
entropy | 0.969 | 0.970 | 0.983 | 0.997 | 0.999 | 0.976 | 0.970 | 0.985 | 0.998 | 0.999 | 0.977 | 0.975 | 0.983 | 0.998 | 0.999 | 0.980 | 0.973 | 0.985 | 0.998 | 0.999 | ||
std | 0.973 | 0.971 | 0.986 | 0.997 | 0.999 | 0.979 | 0.974 | 0.986 | 0.998 | 0.999 | 0.974 | 0.979 | 0.986 | 0.998 | 0.999 | 0.973 | 0.973 | 0.987 | 0.998 | 0.999 | ||
0.25/1 | equal | 0.922 | 0.933 | 0.973 | 0.995 | 0.998 | 0.932 | 0.937 | 0.968 | 0.995 | 0.998 | 0.948 | 0.941 | 0.967 | 0.995 | 0.997 | 0.936 | 0.938 | 0.965 | 0.995 | 0.997 | |
entropy | 0.939 | 0.942 | 0.966 | 0.995 | 0.998 | 0.946 | 0.935 | 0.968 | 0.995 | 0.998 | 0.943 | 0.944 | 0.964 | 0.994 | 0.997 | 0.951 | 0.940 | 0.966 | 0.994 | 0.997 | ||
std | 0.940 | 0.940 | 0.972 | 0.995 | 0.998 | 0.946 | 0.936 | 0.968 | 0.995 | 0.998 | 0.942 | 0.945 | 0.966 | 0.995 | 0.997 | 0.946 | 0.938 | 0.966 | 0.995 | 0.997 | ||
0.5/1 | equal | 0.937 | 0.957 | 0.983 | 0.997 | 0.999 | 0.948 | 0.958 | 0.978 | 0.997 | 0.999 | 0.960 | 0.959 | 0.978 | 0.996 | 0.998 | 0.956 | 0.953 | 0.977 | 0.996 | 0.998 | |
entropy | 0.963 | 0.964 | 0.978 | 0.997 | 0.999 | 0.964 | 0.956 | 0.978 | 0.997 | 0.999 | 0.962 | 0.959 | 0.976 | 0.996 | 0.998 | 0.968 | 0.960 | 0.976 | 0.996 | 0.998 | ||
std | 0.961 | 0.961 | 0.980 | 0.997 | 0.999 | 0.960 | 0.956 | 0.977 | 0.997 | 0.998 | 0.961 | 0.960 | 0.977 | 0.996 | 0.998 | 0.964 | 0.958 | 0.976 | 0.996 | 0.998 |
Type | k | Weighting Method | 2 Criteria | 3 Criteria | 4 Criteria | 5 Criteria | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Alternatives | Alternatives | Alternatives | Alternatives | |||||||||||||||||||
3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | |||
(a) | 0.25/0.5 | equal | 0.969 | 0.942 | 0.967 | 0.994 | 0.997 | 0.952 | 0.940 | 0.965 | 0.993 | 0.997 | 0.961 | 0.940 | 0.965 | 0.993 | 0.997 | 0.944 | 0.928 | 0.962 | 0.993 | 0.996 |
entropy | 0.970 | 0.940 | 0.963 | 0.993 | 0.997 | 0.955 | 0.929 | 0.961 | 0.993 | 0.996 | 0.944 | 0.923 | 0.958 | 0.993 | 0.996 | 0.957 | 0.923 | 0.957 | 0.993 | 0.996 | ||
std | 0.958 | 0.934 | 0.961 | 0.994 | 0.997 | 0.942 | 0.927 | 0.957 | 0.993 | 0.997 | 0.935 | 0.915 | 0.958 | 0.993 | 0.996 | 0.942 | 0.916 | 0.959 | 0.993 | 0.996 | ||
0.25/1 | equal | 0.939 | 0.892 | 0.937 | 0.986 | 0.992 | 0.924 | 0.871 | 0.923 | 0.982 | 0.990 | 0.926 | 0.871 | 0.925 | 0.982 | 0.989 | 0.924 | 0.842 | 0.919 | 0.982 | 0.989 | |
entropy | 0.965 | 0.895 | 0.929 | 0.985 | 0.991 | 0.925 | 0.864 | 0.918 | 0.982 | 0.989 | 0.929 | 0.852 | 0.915 | 0.981 | 0.989 | 0.938 | 0.849 | 0.909 | 0.981 | 0.989 | ||
std | 0.940 | 0.881 | 0.930 | 0.985 | 0.991 | 0.907 | 0.852 | 0.917 | 0.982 | 0.990 | 0.900 | 0.842 | 0.917 | 0.981 | 0.989 | 0.905 | 0.833 | 0.914 | 0.981 | 0.989 | ||
0.5/1 | equal | 0.954 | 0.916 | 0.952 | 0.989 | 0.994 | 0.945 | 0.902 | 0.943 | 0.988 | 0.993 | 0.943 | 0.902 | 0.943 | 0.987 | 0.993 | 0.939 | 0.885 | 0.942 | 0.987 | 0.993 | |
entropy | 0.968 | 0.918 | 0.946 | 0.989 | 0.994 | 0.938 | 0.899 | 0.938 | 0.987 | 0.993 | 0.941 | 0.884 | 0.935 | 0.987 | 0.993 | 0.947 | 0.879 | 0.933 | 0.987 | 0.993 | ||
std | 0.955 | 0.906 | 0.946 | 0.989 | 0.994 | 0.925 | 0.883 | 0.939 | 0.987 | 0.993 | 0.928 | 0.876 | 0.937 | 0.987 | 0.993 | 0.927 | 0.875 | 0.936 | 0.987 | 0.993 | ||
(b) | 0.25/0.5 | equal | 1.000 | 0.992 | 0.996 | 1.000 | 1.000 | 0.995 | 0.991 | 0.996 | 1.000 | 1.000 | 0.997 | 0.993 | 0.996 | 0.999 | 1.000 | 0.992 | 0.987 | 0.996 | 0.999 | 1.000 |
entropy | 0.995 | 0.994 | 0.997 | 1.000 | 1.000 | 0.996 | 0.993 | 0.996 | 0.999 | 1.000 | 0.997 | 0.993 | 0.996 | 0.999 | 1.000 | 0.993 | 0.992 | 0.996 | 0.999 | 1.000 | ||
std | 0.994 | 0.992 | 0.996 | 1.000 | 1.000 | 0.995 | 0.992 | 0.996 | 0.999 | 1.000 | 0.993 | 0.991 | 0.996 | 1.000 | 1.000 | 0.994 | 0.991 | 0.996 | 0.999 | 1.000 | ||
0.25/1 | equal | 1.000 | 0.982 | 0.991 | 0.999 | 1.000 | 0.994 | 0.982 | 0.989 | 0.999 | 0.999 | 0.990 | 0.981 | 0.990 | 0.999 | 0.999 | 0.984 | 0.979 | 0.989 | 0.999 | 0.999 | |
entropy | 0.991 | 0.987 | 0.993 | 0.999 | 1.000 | 0.983 | 0.983 | 0.991 | 0.999 | 0.999 | 0.988 | 0.983 | 0.990 | 0.999 | 0.999 | 0.983 | 0.981 | 0.989 | 0.999 | 0.999 | ||
std | 0.994 | 0.984 | 0.991 | 0.999 | 1.000 | 0.981 | 0.984 | 0.991 | 0.999 | 0.999 | 0.988 | 0.981 | 0.990 | 0.999 | 0.999 | 0.981 | 0.981 | 0.990 | 0.999 | 0.999 | ||
0.5/1 | equal | 1.000 | 0.989 | 0.995 | 0.999 | 1.000 | 0.992 | 0.989 | 0.993 | 0.999 | 1.000 | 0.990 | 0.987 | 0.994 | 0.999 | 1.000 | 0.984 | 0.987 | 0.993 | 0.999 | 1.000 | |
entropy | 0.990 | 0.990 | 0.995 | 0.999 | 1.000 | 0.983 | 0.989 | 0.994 | 0.999 | 1.000 | 0.989 | 0.988 | 0.994 | 0.999 | 1.000 | 0.983 | 0.988 | 0.993 | 0.999 | 1.000 | ||
std | 0.992 | 0.988 | 0.995 | 0.999 | 1.000 | 0.982 | 0.989 | 0.994 | 0.999 | 1.000 | 0.987 | 0.989 | 0.993 | 0.999 | 1.000 | 0.979 | 0.987 | 0.993 | 0.999 | 1.000 | ||
(c) | 0.25/0.5 | equal | 0.940 | 0.941 | 0.974 | 0.995 | 0.998 | 0.930 | 0.943 | 0.971 | 0.995 | 0.997 | 0.944 | 0.945 | 0.970 | 0.995 | 0.997 | 0.926 | 0.935 | 0.968 | 0.995 | 0.997 |
entropy | 0.947 | 0.941 | 0.969 | 0.995 | 0.997 | 0.932 | 0.936 | 0.968 | 0.995 | 0.997 | 0.932 | 0.935 | 0.965 | 0.994 | 0.997 | 0.926 | 0.932 | 0.965 | 0.994 | 0.997 | ||
std | 0.933 | 0.940 | 0.969 | 0.995 | 0.998 | 0.926 | 0.937 | 0.967 | 0.995 | 0.997 | 0.926 | 0.935 | 0.966 | 0.994 | 0.997 | 0.902 | 0.927 | 0.967 | 0.994 | 0.997 | ||
0.25/1 | equal | 0.802 | 0.893 | 0.944 | 0.985 | 0.990 | 0.827 | 0.867 | 0.930 | 0.981 | 0.987 | 0.841 | 0.874 | 0.929 | 0.980 | 0.986 | 0.812 | 0.848 | 0.924 | 0.980 | 0.986 | |
entropy | 0.924 | 0.894 | 0.935 | 0.983 | 0.989 | 0.884 | 0.871 | 0.923 | 0.979 | 0.986 | 0.856 | 0.859 | 0.919 | 0.979 | 0.986 | 0.858 | 0.849 | 0.914 | 0.979 | 0.985 | ||
std | 0.865 | 0.882 | 0.937 | 0.984 | 0.989 | 0.815 | 0.860 | 0.924 | 0.980 | 0.987 | 0.815 | 0.858 | 0.923 | 0.979 | 0.986 | 0.806 | 0.841 | 0.920 | 0.979 | 0.986 | ||
0.5/1 | equal | 0.807 | 0.918 | 0.958 | 0.989 | 0.993 | 0.854 | 0.902 | 0.948 | 0.987 | 0.992 | 0.855 | 0.904 | 0.948 | 0.987 | 0.991 | 0.844 | 0.888 | 0.943 | 0.987 | 0.991 | |
entropy | 0.921 | 0.918 | 0.951 | 0.988 | 0.993 | 0.893 | 0.902 | 0.944 | 0.986 | 0.991 | 0.877 | 0.888 | 0.940 | 0.986 | 0.991 | 0.873 | 0.889 | 0.937 | 0.986 | 0.991 | ||
std | 0.876 | 0.908 | 0.952 | 0.988 | 0.993 | 0.834 | 0.895 | 0.945 | 0.987 | 0.991 | 0.831 | 0.887 | 0.944 | 0.986 | 0.991 | 0.830 | 0.881 | 0.940 | 0.986 | 0.991 | ||
(d) | 0.25/0.5 | equal | 0.966 | 0.970 | 0.988 | 0.999 | 0.999 | 0.971 | 0.973 | 0.988 | 0.998 | 0.999 | 0.973 | 0.978 | 0.988 | 0.998 | 0.999 | 0.968 | 0.976 | 0.988 | 0.998 | 0.999 |
entropy | 0.965 | 0.974 | 0.988 | 0.999 | 0.999 | 0.970 | 0.973 | 0.989 | 0.998 | 0.999 | 0.972 | 0.977 | 0.988 | 0.998 | 0.999 | 0.975 | 0.976 | 0.989 | 0.998 | 0.999 | ||
std | 0.972 | 0.974 | 0.989 | 0.999 | 0.999 | 0.975 | 0.976 | 0.989 | 0.999 | 0.999 | 0.967 | 0.981 | 0.988 | 0.998 | 0.999 | 0.966 | 0.975 | 0.990 | 0.998 | 0.999 | ||
0.25/1 | equal | 0.924 | 0.935 | 0.976 | 0.996 | 0.998 | 0.924 | 0.937 | 0.972 | 0.995 | 0.997 | 0.941 | 0.942 | 0.970 | 0.995 | 0.997 | 0.921 | 0.938 | 0.970 | 0.995 | 0.997 | |
entropy | 0.933 | 0.946 | 0.973 | 0.995 | 0.997 | 0.932 | 0.935 | 0.972 | 0.995 | 0.997 | 0.929 | 0.945 | 0.968 | 0.994 | 0.997 | 0.936 | 0.939 | 0.970 | 0.994 | 0.997 | ||
std | 0.934 | 0.943 | 0.975 | 0.996 | 0.998 | 0.935 | 0.936 | 0.972 | 0.995 | 0.997 | 0.927 | 0.946 | 0.970 | 0.994 | 0.997 | 0.928 | 0.937 | 0.971 | 0.995 | 0.997 | ||
0.5/1 | equal | 0.929 | 0.958 | 0.985 | 0.998 | 0.999 | 0.936 | 0.959 | 0.983 | 0.997 | 0.998 | 0.950 | 0.961 | 0.982 | 0.997 | 0.998 | 0.945 | 0.955 | 0.982 | 0.997 | 0.998 | |
entropy | 0.958 | 0.967 | 0.984 | 0.997 | 0.999 | 0.954 | 0.959 | 0.982 | 0.997 | 0.998 | 0.953 | 0.960 | 0.981 | 0.997 | 0.998 | 0.960 | 0.960 | 0.980 | 0.997 | 0.998 | ||
std | 0.956 | 0.965 | 0.984 | 0.997 | 0.999 | 0.950 | 0.958 | 0.982 | 0.997 | 0.998 | 0.952 | 0.962 | 0.981 | 0.997 | 0.998 | 0.953 | 0.960 | 0.981 | 0.997 | 0.998 |
Method | Weighting Method | 2 Criteria | 3 Criteria | 4 Criteria | 5 Criteria | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Alternatives | Alternatives | Alternatives | Alternatives | ||||||||||||||||||
3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | ||
(a) | equal | 0.060 | −0.009 | −0.024 | −0.025 | −0.029 | 0.181 | 0.261 | 0.272 | 0.300 | 0.296 | 0.030 | −0.019 | −0.003 | −0.014 | −0.011 | 0.135 | 0.133 | 0.148 | 0.165 | 0.170 |
entropy | 0.018 | −0.024 | −0.033 | −0.010 | −0.023 | 0.307 | 0.330 | 0.317 | 0.326 | 0.294 | 0.011 | −0.025 | −0.002 | −0.014 | −0.004 | 0.182 | 0.156 | 0.207 | 0.159 | 0.157 | |
std | 0.024 | −0.034 | −0.032 | −0.021 | −0.030 | 0.246 | 0.304 | 0.277 | 0.307 | 0.290 | 0.011 | −0.028 | −0.013 | −0.015 | −0.008 | 0.166 | 0.128 | 0.167 | 0.158 | 0.167 | |
(b) | equal | 0.891 | 0.889 | 0.934 | 0.985 | 0.992 | 0.896 | 0.879 | 0.919 | 0.982 | 0.990 | 0.881 | 0.876 | 0.925 | 0.983 | 0.990 | 0.860 | 0.872 | 0.922 | 0.983 | 0.991 |
entropy | 0.900 | 0.898 | 0.936 | 0.980 | 0.988 | 0.850 | 0.866 | 0.913 | 0.969 | 0.983 | 0.837 | 0.848 | 0.899 | 0.967 | 0.980 | 0.790 | 0.831 | 0.883 | 0.965 | 0.980 | |
std | 0.848 | 0.865 | 0.925 | 0.983 | 0.990 | 0.789 | 0.852 | 0.907 | 0.977 | 0.988 | 0.798 | 0.836 | 0.904 | 0.977 | 0.987 | 0.771 | 0.821 | 0.903 | 0.978 | 0.988 | |
(c) | equal | 0.754 | 0.832 | 0.853 | 0.873 | 0.872 | 0.735 | 0.803 | 0.852 | 0.884 | 0.885 | 0.751 | 0.851 | 0.906 | 0.944 | 0.948 | 0.760 | 0.835 | 0.905 | 0.941 | 0.946 |
entropy | 0.919 | 0.900 | 0.874 | 0.869 | 0.868 | 0.914 | 0.880 | 0.867 | 0.877 | 0.880 | 0.912 | 0.889 | 0.910 | 0.938 | 0.944 | 0.900 | 0.884 | 0.903 | 0.934 | 0.941 | |
std | 0.839 | 0.860 | 0.859 | 0.871 | 0.870 | 0.847 | 0.840 | 0.855 | 0.881 | 0.884 | 0.854 | 0.875 | 0.909 | 0.942 | 0.947 | 0.855 | 0.860 | 0.909 | 0.939 | 0.944 | |
(d) | equal | 0.173 | 0.044 | −0.011 | −0.028 | −0.029 | 0.209 | 0.259 | 0.266 | 0.293 | 0.292 | 0.084 | 0.004 | −0.003 | −0.018 | −0.017 | 0.169 | 0.150 | 0.149 | 0.159 | 0.164 |
entropy | 0.019 | −0.019 | −0.031 | −0.016 | −0.025 | 0.269 | 0.310 | 0.298 | 0.313 | 0.288 | 0.010 | −0.039 | −0.004 | −0.017 | −0.012 | 0.167 | 0.127 | 0.180 | 0.150 | 0.152 | |
std | 0.004 | −0.017 | −0.026 | −0.025 | −0.030 | 0.197 | 0.271 | 0.264 | 0.298 | 0.287 | −0.006 | −0.021 | −0.020 | −0.017 | −0.014 | 0.144 | 0.110 | 0.155 | 0.153 | 0.162 | |
(e) | equal | 0.014 | −0.018 | 0.002 | 0.085 | 0.124 | 0.113 | 0.171 | 0.218 | 0.305 | 0.324 | −0.003 | −0.039 | −0.005 | −0.009 | −0.003 | 0.115 | 0.080 | 0.113 | 0.138 | 0.142 |
entropy | 0.005 | −0.010 | 0.014 | 0.098 | 0.130 | 0.289 | 0.305 | 0.306 | 0.345 | 0.337 | 0.002 | −0.040 | 0.002 | −0.007 | 0.003 | 0.181 | 0.127 | 0.177 | 0.138 | 0.139 | |
std | 0.006 | −0.021 | 0.003 | 0.089 | 0.125 | 0.233 | 0.247 | 0.248 | 0.317 | 0.325 | −0.022 | −0.041 | −0.011 | −0.007 | 0.000 | 0.133 | 0.098 | 0.133 | 0.135 | 0.142 | |
(f) | equal | 0.776 | 0.820 | 0.853 | 0.882 | 0.883 | 0.742 | 0.778 | 0.832 | 0.883 | 0.889 | 0.764 | 0.818 | 0.879 | 0.940 | 0.949 | 0.741 | 0.793 | 0.871 | 0.935 | 0.945 |
entropy | 0.895 | 0.877 | 0.877 | 0.885 | 0.884 | 0.857 | 0.836 | 0.859 | 0.887 | 0.890 | 0.855 | 0.847 | 0.890 | 0.942 | 0.950 | 0.811 | 0.838 | 0.883 | 0.938 | 0.946 | |
std | 0.811 | 0.826 | 0.854 | 0.882 | 0.883 | 0.767 | 0.793 | 0.836 | 0.884 | 0.889 | 0.774 | 0.826 | 0.878 | 0.940 | 0.949 | 0.758 | 0.799 | 0.871 | 0.936 | 0.945 |
Method | Weighting Method | 2 Criteria | 3 Criteria | 4 Criteria | 5 Criteria | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Alternatives | Alternatives | Alternatives | Alternatives | ||||||||||||||||||
3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | 3 | 5 | 10 | 50 | 100 | ||
(a) | equal | 0.463 | 0.465 | 0.464 | 0.374 | 0.339 | 0.561 | 0.569 | 0.592 | 0.590 | 0.561 | 0.528 | 0.489 | 0.495 | 0.442 | 0.420 | 0.562 | 0.546 | 0.563 | 0.566 | 0.558 |
entropy | 0.596 | 0.520 | 0.490 | 0.403 | 0.357 | 0.672 | 0.652 | 0.638 | 0.614 | 0.569 | 0.569 | 0.527 | 0.512 | 0.452 | 0.424 | 0.611 | 0.588 | 0.611 | 0.560 | 0.542 | |
std | 0.540 | 0.477 | 0.466 | 0.380 | 0.342 | 0.611 | 0.604 | 0.598 | 0.596 | 0.561 | 0.519 | 0.494 | 0.490 | 0.443 | 0.423 | 0.573 | 0.544 | 0.571 | 0.559 | 0.555 | |
(b) | equal | 0.873 | 0.876 | 0.933 | 0.990 | 0.995 | 0.903 | 0.878 | 0.924 | 0.988 | 0.994 | 0.884 | 0.876 | 0.924 | 0.986 | 0.993 | 0.875 | 0.879 | 0.926 | 0.985 | 0.992 |
entropy | 0.919 | 0.905 | 0.933 | 0.990 | 0.995 | 0.885 | 0.881 | 0.924 | 0.987 | 0.994 | 0.880 | 0.872 | 0.914 | 0.985 | 0.993 | 0.851 | 0.857 | 0.906 | 0.983 | 0.992 | |
std | 0.876 | 0.879 | 0.932 | 0.990 | 0.995 | 0.847 | 0.873 | 0.923 | 0.988 | 0.994 | 0.859 | 0.860 | 0.916 | 0.986 | 0.993 | 0.842 | 0.854 | 0.916 | 0.985 | 0.992 | |
(c) | equal | 0.822 | 0.857 | 0.881 | 0.941 | 0.955 | 0.821 | 0.840 | 0.879 | 0.935 | 0.947 | 0.830 | 0.873 | 0.912 | 0.958 | 0.965 | 0.836 | 0.861 | 0.911 | 0.954 | 0.962 |
entropy | 0.938 | 0.910 | 0.900 | 0.942 | 0.955 | 0.935 | 0.898 | 0.897 | 0.937 | 0.948 | 0.933 | 0.903 | 0.918 | 0.958 | 0.966 | 0.922 | 0.896 | 0.914 | 0.955 | 0.963 | |
std | 0.890 | 0.879 | 0.887 | 0.941 | 0.955 | 0.891 | 0.867 | 0.882 | 0.935 | 0.947 | 0.891 | 0.890 | 0.915 | 0.957 | 0.965 | 0.891 | 0.877 | 0.914 | 0.955 | 0.963 | |
(d) | equal | 0.492 | 0.505 | 0.523 | 0.510 | 0.507 | 0.567 | 0.586 | 0.642 | 0.730 | 0.745 | 0.532 | 0.500 | 0.522 | 0.506 | 0.508 | 0.556 | 0.549 | 0.586 | 0.632 | 0.644 |
entropy | 0.626 | 0.544 | 0.520 | 0.512 | 0.506 | 0.674 | 0.639 | 0.651 | 0.725 | 0.737 | 0.570 | 0.515 | 0.525 | 0.508 | 0.508 | 0.601 | 0.564 | 0.601 | 0.624 | 0.635 | |
std | 0.578 | 0.506 | 0.519 | 0.511 | 0.507 | 0.595 | 0.609 | 0.642 | 0.730 | 0.744 | 0.521 | 0.508 | 0.520 | 0.507 | 0.509 | 0.567 | 0.549 | 0.590 | 0.630 | 0.643 | |
(e) | equal | 0.481 | 0.545 | 0.604 | 0.701 | 0.741 | 0.547 | 0.595 | 0.682 | 0.806 | 0.838 | 0.518 | 0.516 | 0.570 | 0.627 | 0.657 | 0.562 | 0.553 | 0.612 | 0.703 | 0.730 |
entropy | 0.604 | 0.564 | 0.598 | 0.700 | 0.740 | 0.676 | 0.670 | 0.706 | 0.809 | 0.836 | 0.573 | 0.540 | 0.575 | 0.625 | 0.653 | 0.619 | 0.594 | 0.642 | 0.701 | 0.728 | |
std | 0.562 | 0.547 | 0.600 | 0.702 | 0.741 | 0.617 | 0.634 | 0.690 | 0.808 | 0.838 | 0.531 | 0.520 | 0.570 | 0.627 | 0.658 | 0.579 | 0.566 | 0.622 | 0.703 | 0.730 | |
(f) | equal | 0.777 | 0.831 | 0.882 | 0.945 | 0.958 | 0.804 | 0.824 | 0.875 | 0.940 | 0.951 | 0.811 | 0.845 | 0.898 | 0.959 | 0.967 | 0.807 | 0.836 | 0.894 | 0.956 | 0.965 |
entropy | 0.919 | 0.893 | 0.898 | 0.945 | 0.958 | 0.893 | 0.864 | 0.892 | 0.942 | 0.952 | 0.890 | 0.870 | 0.905 | 0.960 | 0.967 | 0.861 | 0.861 | 0.899 | 0.957 | 0.966 | |
std | 0.870 | 0.850 | 0.880 | 0.944 | 0.958 | 0.837 | 0.836 | 0.877 | 0.940 | 0.951 | 0.840 | 0.854 | 0.898 | 0.958 | 0.967 | 0.834 | 0.843 | 0.893 | 0.957 | 0.965 |
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