1 Introduction
2 Methodology
2.1 Problem description
2.2 Information entropy risk measurement of decision participants’ rights distribution
2.3 Decision procedure
Department | Decision | Scene rescue |
---|---|---|
Administration of work safety | 7 | 224 |
Department of public health | 5 | 150 |
Environmental Protection Department | 5 | 123 |
The Meteorological Department | 3 | 57 |
The people’s Liberation Army | 4 | 1353 |
The armed police forces | 3 | 986 |
Firefighting force | 5 | 2124 |
Others (Explosion-proof, anti-chemical, epidemic prevention, etc.) | 8 | 534 |
Total | 40 | 5551 |
3 Numerical examples
Attribute code |
\(c_1 \)
|
\(c_2 \)
|
\(c_3 \)
|
\(c_4 \)
|
\(c_5 \)
|
---|---|---|---|---|---|
Member1 | 0.25 | 0.12 | 0.30 | 0.13 | 0.20 |
Member2 | 0.23 | 0.15 | 0.32 | 0.05 | 0.20 |
Member3 | 0.30 | 0.12 | 0.25 | 0.05 | 0.23 |
Member4 | 0.20 | 0.20 | 0.33 | 0.13 | 0.24 |
Weight | 0.245 | 0.147 | 0.300 | 0.091 | 0.217 |
Attribute code |
\(c_1 \)
|
\(c_2 \)
|
\(c_3 \)
|
\(c_4 \)
|
\(c_5 \)
|
---|---|---|---|---|---|
Degree | 0.8086 | 0.9771 | 0.3684 | 1 | 0.9349 |
Cluster code | Number of members | Detailed list | Consistency index |
---|---|---|---|
\(\varOmega _1 \)
| 11 |
\(e_1 ,e_5 ,e_{10} ,e_{12} ,e_{13} ,e_{22} ,e_{31} ,e_{32} ,e_{36} ,e_{37} ,e_{38} \)
| 0.8521 |
\(\varOmega _2 \)
| 5 |
\(e_2 ,e_9 ,e_{18} ,e_{21} ,e_{25} \)
| 0.8925 |
\(\varOmega _3 \)
| 7 |
\(e_3 ,e_7 ,e_{11} ,e_{17} ,e_{27} ,e_{30} ,e_{35} \)
| 0.9335 |
\(\varOmega _4 \)
| 8 |
\(e_4 ,e_{15} ,e_{19} ,e_{28} ,e_{33} ,e_{34} ,e_{39} ,e_{40} \)
| 0.9290 |
\(\varOmega _5 \)
| 5 |
\(e_6 ,e_{14} ,e_{20} ,e_{23} ,e_{24} \)
| 0.8539 |
\(\varOmega _6 \)
| 4 |
\(e_8 ,e_{16} ,e_{26} ,e_{29} \)
| 0.9486 |
Cluster code |
\(\varOmega _1 \)
|
\(\varOmega _2 \)
|
\(\varOmega _3 \)
|
\(\varOmega _4 \)
|
\(\varOmega _5 \)
|
\(\varOmega _6 \)
|
---|---|---|---|---|---|---|
Weight interval | [0.3, 0.4] | [0.1, 0.2] | [0.2, 0.3] | [0.2, 0.3] | [0, 0.1] | [0, 0.1] |
Cluster | Weight vector | Risk value |
---|---|---|
\(\varOmega _1 \)
|
\((0.0912, 0.0730, 0.0311, 0.0745, 0.0880, 0.0981, 0.0984, 0.0950, 0.0655, 0.0981, 0.1872)^{\mathrm{T}}\)
| 0.5070 |
\(\varOmega _2 \)
|
\((0.5576, 0.1075, 0.1976, 0.0601, 0.0772)^{\mathrm{T}}\)
| 0.2368 |
\(\varOmega _3 \)
|
\((0.0881, 0.0681, 0.0938, 0.0227, 0.1251, 0.2974, 0.3047)^{\mathrm{T}}\)
| 0.9813 |
\(\varOmega _4 \)
|
\((0.2456, 0.0052, 0.1706, 0.1188, 0.3008, 0.0123, 0.0115, 0.1352)^{\mathrm{T}}\)
| 0.5000 |
\(\varOmega _5 \)
|
\((0.1371, 0.0493, 0.5811, 0.0293, 0.2033)^{\mathrm{T}}\)
| 0.5215 |
\(\varOmega _6 \)
|
\((0.0787, 0.8617, 0.0181, 0.0415)^{\mathrm{T}}\)
| 0.0702 |
Cluster | Weight vector | Risk value |
---|---|---|
\(\varOmega _1 \)
|
\((0.0887, 0.0921, 0.0847, 0.0500, 0.0776, 0.0980, 0.0972, 0.0502, 0.0987, 0.0735, 0.1894)^{\mathrm{T}}\)
| 0.5708 |
\(\varOmega _2 \)
|
\((0.5520, 0.1010, 0.1273, 0.0386, 0.1811)^{\mathrm{T}}\)
| 0.7886 |
\(\varOmega _3 \)
|
\((0.0294, 0.0889, 0.0616, 0.0821, 0.1795, 0.2150, 0.3434)^{\mathrm{T}}\)
| 0.7008 |
\(\varOmega _4 \)
|
\((0.2106, 0.0032, 0.1252, 0.1849, 0.3019, 0.0271, 0.0322, 0.1148)^{\mathrm{T}}\)
| 0.6441 |
\(\varOmega _5 \)
|
\((0.1051, 0.0085, 0.5161, 0.1004, 0.2699)^{\mathrm{T}}\)
| 0.8766 |
\(\varOmega _6 \)
|
\((0.0614, 0.8270, 0.0452, 0.0664)^{\mathrm{T}}\)
| 0.1656 |
Cluster | Clustering results |
---|---|
\(\varOmega _1 \)
|
\(D_{\varOmega _1} =\left[ {\begin{array}{lllll} 0.4183 &{}0.4906 &{}0.6100 &{}0.1479 &{}0.8797 \\ 0.6031 &{}0.5144 &{}0.1126 &{}0.6970 &{}0.2171 \\ 0.4922 &{}0.8489 &{}0.3000 &{}0.2095 &{}0.3100 \\ 0.4000 &{}0.4927 &{}0.3000 &{}0.5804 &{}0.2470 \\ \end{array}} \right] \)
|
\(\varOmega _2 \)
|
\(D_{\varOmega _2 } =\left[ {\begin{array}{lllll} 0.6450 &{}0.4893 &{}0.3000 &{}0.1425 &{}0.7678 \\ 0.6000 &{}0.3802 &{}0.1832 &{}0.7000 &{}0.5708 \\ 0.5013 &{}0.8725 &{}0.4893 &{}0.1725 &{}0.3000 \\ 0.4000 &{}0.5000 &{}0.1000 &{}0.6000 &{}0.2000 \\ \end{array}} \right] \)
|
\(\varOmega _3 \)
|
\(D_{\Omega _3 } =\left[ {\begin{array}{lllll} 0.6137 &{}0.2000 &{}0.2634 &{}0.3074 &{}0.6297 \\ 0.8611 &{}0.1726 &{}0.5225 &{}0.8694 &{}0.4204 \\ 0.4702 &{}0.8374 &{}0.4695 &{}0.1125 &{}0.5999 \\ 0.4698 &{}0.1000 &{}0.5931 &{}0.1305 &{}0.1907 \\ \end{array}} \right] \)
|
\(\varOmega _4 \)
|
\(D_{\Omega _4 } =\left[ {\begin{array}{lllll} 0.6905 &{}0.7503 &{}0.2949 &{}0.6235 &{}0.3395 \\ 0.6429 &{}0.1971 &{}0.5936 &{}0.5337 &{}0.2000 \\ 0.3067 &{}0.2803 &{}0.4120 &{}0.1699 &{}0.5473 \\ 0.6302 &{}0.3238 &{}0.7100 &{}0.6005 &{}0.2894 \\ \end{array}} \right] \)
|
\(\varOmega _5 \)
|
\(D_{\Omega _5 } =\left[ {\begin{array}{lllll} 0.2029 &{}0.7989 &{}0.3838 &{}0.4419 &{}0.2279 \\ 0.7370 &{}0.6677 &{}0.5586 &{}0.1784 &{}0.6163 \\ 0.4838 &{}0.4050 &{}0.5030 &{}0.4620 &{}0.6635 \\ 0.2000 &{}0.1407 &{}0.2366 &{}0.1000 &{}0.6594 \\ \end{array}} \right] \)
|
\(\varOmega _6 \)
|
\(D_{\Omega _6 } =\left[ {\begin{array}{lllll} 0.4805 &{}0.4236 &{}0.7746 &{}0.2236 &{}0.3847 \\ 0.6983 &{}0.5291 &{}0.8843 &{}0.1060 &{}0.1315 \\ 0.5959 &{}0.1296 &{}0.3157 &{}0.3157 &{}0.6157 \\ 0.7587 &{}0.1921 &{}0.5607 &{}0.8370 &{}0.4236 \\ \end{array}} \right] \)
|
Cluster | Clustering results |
---|---|
\(\varOmega _1 \)
|
\(D_{\varOmega _1 } =\left[ {\begin{array}{lllll} 0.4100 &{}0.4935 &{}0.6196 &{}0.1436 &{}0.8793 \\ 0.5970 &{}0.5049 &{}0.1135 &{}0.6916 &{}0.2190 \\ 0.4874 &{}0.8496 &{}0.2976 &{}0.2050 &{}0.3113 \\ 0.4000 &{}0.4908 &{}0.3000 &{}0.5806 &{}0.2390 \\ \end{array}} \right] \)
|
\(\varOmega _2 \)
|
\(D_{\varOmega _2 } =\left[ {\begin{array}{lllll} 0.6451 &{}0.4899 &{}0.3000 &{}0.1306 &{}0.7697 \\ 0.6000 &{}0.3873 &{}0.1860 &{}0.7000 &{}0.5617 \\ 0.4845 &{}0.8759 &{}0.4899 &{}0.1692 &{}0.3000 \\ 0.4000 &{}0.5000 &{}0.1000 &{}0.6000 &{}0.2000 \\ \end{array}} \right] \)
|
\(\varOmega _3 \)
|
\(D_{\varOmega _3 } =\left[ {\begin{array}{lllll} 0.6047 &{}0.2000 &{}0.2696 &{}0.3104 &{}0.6214 \\ 0.8613 &{}0.1774 &{}0.5272 &{}0.8656 &{}0.4194 \\ 0.4785 &{}0.8359 &{}0.4656 &{}0.1179 &{}0.5999 \\ 0.4547 &{}0.1000 &{}0.5910 &{}0.1343 &{}0.1902 \\ \end{array}} \right] \)
|
\(\varOmega _4 \)
|
\(D_{\varOmega _4 } =\left[ {\begin{array}{lllll} 0.7417 &{}0.7881 &{}0.2725 &{}0.6874 &{}0.2548 \\ 0.6468 &{}0.1342 &{}0.6743 &{}0.5104 &{}0.2000 \\ 0.2874 &{}0.1626 &{}0.4447 &{}0.1698 &{}0.5874 \\ 0.6564 &{}0.3000 &{}0.7574 &{}0.6003 &{}0.3027 \\ \end{array}} \right] \)
|
\(\varOmega _5 \)
|
\(D_{\varOmega _5 } =\left[ {\begin{array}{lllll} 0.2100 &{}0.7863 &{}0.3968 &{}0.4484 &{}0.2434 \\ 0.7475 &{}0.6936 &{}0.5689 &{}0.1786 &{}0.6032 \\ 0.4968 &{}0.4008 &{}0.5100 &{}0.4604 &{}0.6698 \\ 0.2000 &{}0.1540 &{}0.2302 &{}0.1000 &{}0.6460 \\ \end{array}} \right] \)
|
\(\varOmega _6 \)
|
\(D_{\varOmega _6 } =\left[ {\begin{array}{lllll} 0.4882 &{}0.4184 &{}0.7771 &{}0.2184 &{}0.3949 \\ 0.6859 &{}0.5224 &{}0.8877 &{}0.1112 &{}0.1246 \\ 0.5934 &{}0.1296 &{}0.3123 &{}0.3123 &{}0.6123 \\ 0.7743 &{}0.1939 &{}0.5693 &{}0.8509 &{}0.4184 \\ \end{array}} \right] \)
|
W
|
R
|
---|---|
\((0.3167, 0.1085, 0.2063, 0.2099, 0.0105, 0.1481)^{\mathrm{T}}\)
| 1.5208 |
\((0.3204, 0.1732, 0.2338, 0.2094, 0.0204, 0.0428)^{\mathrm{T}}\)
| 1.4521 |
\((0.3337, 0.1588, 0.2111, 0.2681, 0.0263, 0.0021)^{\mathrm{T}}\)
| 1.4291 |
\((0.3704, 0.1017, 0.2125, 0.2924, 0.0055, 0.0175)^{\mathrm{T}}\)
| 1.4084 |
\((0.3063, 0.1309, 0.2165, 0.2079,0.0663, 0.0721)^{\mathrm{T}}\)
| 1.5405 |
\((0.3089, 0.1408, 0.2378, 0.2165, 0.0692, 0.0268)^{\mathrm{T}}\)
| 1.4998 |
\((0.3212, 0.1208, 0.2747, 0.2302, 0.0256, 0.0275)^{\mathrm{T}}\)
| 1.4415 |
\((0.3689, 0.1026, 0.2643, 0.2223, 0.0139, 0.0280)^{\mathrm{T}}\)
| 1.4109 |
\((0.3291, 0.1386, 0.2734, 0.2027, 0.0179, 0.0383)^{\mathrm{T}}\)
| 1.4329 |
\((0.3328, 0.1324, 0.2353, 0.2556, 0.0287, 0.0152)^{\mathrm{T}}\)
| 1.4447 |
W
|
R
|
---|---|
\((0.3045, 0.1801,0.2401, 0.2128, 0.0613, 0.0012)^{\mathrm{T}}\)
| 2.1749 |
\((0.3249, 0.1702, 0.2227, 0.2576, 0.0145, 0.0101)^{\mathrm{T}}\)
| 2.0957 |
\((0.3435, 0.1654, 0.2064, 0.2122, 0.0259, 0.0466)^{\mathrm{T}}\)
| 2.1593 |
\((0.3166, 0.1243, 0.2190, 0.2426, 0.0398, 0.0577)^{\mathrm{T}}\)
| 2.2512 |
\((0.3101, 0.1979, 0.2134, 0.2166, 0.0051, 0.0569)^{\mathrm{T}}\)
| 2.1102 |
\((0.3159, 0.1150, 0.2498, 0.2360, 0.0348, 0.0485)^{\mathrm{T}}\)
| 2.1988 |
\((0.3009, 0.1341, 0.2152, 0.2001, 0.0992, 0.0505)^{\mathrm{T}}\)
| 2.3402 |
\((0.3539, 0.1678, 0.2067, 0.2218, 0.0205, 0.0292)^{\mathrm{T}}\)
| 2.1161 |
\((0.3449, 0.1030, 0.2685, 0.2349, 0.0001, 0.0486)^{\mathrm{T}}\)
| 2.0969 |
\((0.3034, 0.1580, 0.2530, 0.2033, 0.0473, 0.0350)^{\mathrm{T}}\)
| 2.1730 |
Alternatives/attributes |
\(c_1 \)
|
\(c_2 \)
|
\(c_3 \)
|
\(c_4 \)
|
\(c_5 \)
| Final evaluation |
---|---|---|---|---|---|---|
\(a_1 \)
| 0.5624 | 0.5052 | 0.4143 | 0.3233 | 0.6450 | 0.5055 |
\(a_2 \)
| 0.6716 | 0.3364 | 0.3635 | 0.6730 | 0.2920 | 0.4477 |
\(a_3 \)
| 0.4360 | 0.6676 | 0.3894 | 0.1768 | 0.4473 | 0.4349 |
\(a_4 \)
| 0.4873 | 0.3534 | 0.4661 | 0.4945 | 0.2480 | 0.4102 |
Alternatives/attributes |
\(c_1 \)
|
\(c_2 \)
|
\(c_3 \)
|
\(c_4 \)
|
\(c_5 \)
| Final evaluation |
---|---|---|---|---|---|---|
\(a_1 \)
| 0.5605 | 0.4928 | 0.3740 | 0.3143 | 0.5997 | 0.4805 |
\(a_2 \)
| 0.6500 | 0.2999 | 0.3737 | 0.6493 | 0.3142 | 0.4428 |
\(a_3 \)
| 0.4192 | 0.6347 | 0.3996 | 0.1684 | 0.4432 | 0.4273 |
\(a_4 \)
| 0.4665 | 0.3323 | 0.4409 | 0.4663 | 0.2438 | 0.3910 |