q阶三角犹豫模糊BM算子及其多属性决策应用

doi:10.12305/j.issn.1001-506X.2022.01.23

Abstract

Abstract:

In order to solve a wider range of fuzzy decision-making problems, and at the same time make decision information closer to human cognitive thinking, combining q-rung hesitant fuzzy sets and triangular fuzzy numbers, the concept of q-rung hesitant triangular fuzzy sets is proposed and the operation of q-rung hesitant triangular fuzzy sets is defined. In order to describe the relationship between evaluation information in the process of information integration, the Bonferroni mean operator is extend to the q-rung hesitant triangular fuzzy set, and the q-rung hesitant triangular fuzzy Bonferroni mean operator is proposed. In order to describe more correlations, the generalized Bonferroni mean operator is extend to the q-rung hesitant triangular fuzzy set, and the q-rung hesitant triangular fuzzy generalized Bonferroni mean operator is proposed. Considering that the importance of different attributes of evaluation information is different, their weighted form is proposed. In the end, a multi-attribute decision-making method under q-rung hesitant triangular fuzzy environment is proposed and verified the experimental results by an example.

Key words: q-rung hesitant triangular fuzzy set, Bonferroni mean operator, generalized Bonferroni mean operator, multiple criteria decision-making

CLC Number:

TP311

Yaojun REN, Xiujiu YUAN, Lin HUANG. q-rung hesitant triangular fuzzy BM operator and its application in multiple criteria decision making[J]. Systems Engineering and Electronics, 2022, 44(1): 181-191.

Figures/Tables 5

Fig.1

Table 1

Table 2

Table 3

Table 4

Sorting results"

算子	参数选择	得分函数	排序结果
q-HTFWBM	s=1, t=1	S(h₁)=(0.171 1, 0.220 7, 0.274 2), S(h₂)=(0.193 6, 0.250 9, 0.310 0), S(h₃)=(0.160 9, 0.207 8, 0.261 4), S(h₄)=(0.180 1, 0.229 5, 0.286 0), S(h₅)=(0.166 2, 0.215 4, 0.275 4)	(2, 4, 1, 5, 3)
q-HTFWBM	s=1, t=2	S(h₁)=(0.192 6, 0.243 4, 0.298 9), S(h₂)=(0.201 8, 0.258 5, 0.318 2), S(h₃)=(0.176 3, 0.224 4, 0.279 5), S(h₄)=(0.200 4, 0.251 2, 0.308 9), S(h₅)=(0.187 4, 0.237 8, 0.302 4)	(2, 4, 1, 5, 3)
q-HTFWBM	s=2, t=2	S(h₁)=(0.184 3, 0.235 8, 0.291 4), S(h₂)=(0.203 3, 0.258 8, 0.318 4), S(h₃)=(0.174 8, 0.223 0, 0.278 7), S(h₄)=(0.196 1, 0.247 7, 0.304 6), S(h₅)=(0.183 0, 0.233 9, 0.297 0)	(2, 4, 5, 1, 3)
q-HTFGWBM	s=1, t=1, r=1	S(h₁)=(0.374 2, 0.470 8, 0.569 0), S(h₂)=(0.389 3, 0.499 2, 0.596 7), S(h₃)=(0.338 6, 0.434 9, 0.539 9), S(h₄)=(0.389 5, 0.486 4, 0.586 8), S(h₅)=(0.370 8, 0.466 9, 0.578 9)	(2, 4, 5, 1, 3)
q-HTFGWBM	s=1, t=1, r=2	S(h₁)=(0.391 4, 0.484 9, 0.581 0), S(h₂)=(0.404 8, 0.511 6, 0.607 5), S(h₃)=(0.351 6, 0.445 8, 0.549 5), S(h₄)=(0.403 8, 0.498 5, 0.596 5), S(h₅)=(0.387 6, 0.480 9, 0.592 6)	(2, 4, 5, 1, 3)
q-HTFGWBM	s=1, t=2, r=2	S(h₁)=(0.402 0, 0.493 3, 0.587 7), S(h₂)=(0.414 2, 0.518 7, 0.613 4), S(h₃)=(0.359 5, 0.452 2, 0.554 9), S(h₄)=(0.412 5, 0.505 6, 0.601 8), S(h₅)=(0.397 9, 0.489 1, 0.600 3)	(2, 4, 5, 1, 3)
q-HTFGWBM	s=2, t=2, r=2	S(h₁)=(0.409 3, 0.498 9, 0.592 0), S(h₂)=(0.420 4, 0.523 4, 0.617 1), S(h₃)=(0.364 9, 0.456 4, 0.558 3), S(h₄)=(0.418 4, 0.510 2, 0.605 2), S(h₅)=(0.404 9, 0.494 3, 0.605 1)	(2, 4, 5, 1, 3)
q-HTFWA	—	S(h₁)=(0.393 0, 0.493 8, 0.599 2), S(h₂)=(0.407 8, 0.519 9, 0.624 7), S(h₃)=(0.407 2, 0.510 1, 0.616 1), S(h₄)=(0.425 5, 0.526 5, 0.632 3), S(h₅)=(0.393 8, 0.494 5, 0.616 7)	(4, 2, 3, 5, 1)
q-HTFWG	—	S(h₁)=(0.320 3, 0.426 9, 0.529 4), S(h₂)=(0.350 8, 0.467 0, 0.567 3), S(h₃)=(0.278 6, 0.388 6, 0.499 9), S(h₄)=(0.341 2, 0.446 6, 0.556 0), S(h₅)=(0.308 5, 0.416 4, 0.527 0)	(2, 4, 1, 5, 3)
DHq-ROFWMM^[29]	P=(1, 1, 0, 0)	S(h₁^L)=0.203 1, S(h₂^L)=0.235 8, S(h₃^L)=0.195 6, S(h₄^L)=0.212 6, S(h₅^L)=0.199 1 S(h₁^M)=0.220 7, S(h₂^M)=0.250 9, S(h₃^M)=0.207 8, S(h₄^M)=0.229 5, S(h₅^M)=0.215 4 S(h₁^R)=0.242 2, S(h₂^R)=0.267 8, S(h₃^R)=0.226 7, S(h₄^R)=0.253 5, S(h₅^R)=0.242 5	(2, 4, 1, 5, 3) (2, 4, 1, 5, 3) (2, 4, 5, 1, 3)

Table 4

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网络系统	评估因素
网络系统	M₁	M₂	M₃	M₄
A₁	〈{(0.6, 0.7, 0.8), (0.7, 0.8, 0.9)}, {(0.4, 0.5, 0.6)}〉	〈{(0.3, 0.4, 0.5)}, {(0.7, 0.8, 0.9)}〉	〈{(0.4, 0.5, 0.6)}, {(0.5, 0.6, 0.7), (0.6, 0.7, 0.8)}〉	〈{(0.3, 0.4, 0.5), (0.4, 0.5, 0.6)}, {(0.6, 0.7, 0.8), (0.7, 0.8, 0.9)}〉
A₂	〈{(0.4, 0.5, 0.6)}, {(0.4, 0.5, 0.6), (0.5, 0.6, 0.8)}〉	〈{(0.6, 0.7, 0.8), (0.7, 0.8, 0.9)}, {(0.3, 0.4, 0.5), (0.4, 0.5, 0.6)}〉	〈{(0.4, 0.6, 0.7), (0.6, 0.7, 0.8)}{(0.2, 0.3, 0.4), (0.3, 0.4, 0.5)}〉	〈{(0.3, 0.4, 0.5)}, {(0.6, 0.7, 0.8)}〉
A₃	〈{(0.5, 0.6, 0.7), (0.6.0.7.0.8)}, {(0.4, 0.5, 0.6), (0.5, 0.6, 0.7)}〉	〈{(0.1, 0.2, 0.3), (0.2, 0.3, 0.4)}, {(07, 0.8, 0.9)}〉	〈{(0.5, 0.6, 0.7)}, {(0.4, 0.5, 0, 6), (0.6, 0.7, 0.8)}〉	〈{(0.3, 0.4, 0.5), (0.4, 0.5, 0.6)}, {(0.5, 0.7, 0.8), (0.7, 0.8, 0.9)}〉
A₄	〈{(0.6, 0.7, 0.8), (0.7, 0.8, 0.9)}, {(0.4, 0.5, 0.6)}〉	〈{(0.2, 0.3, 0.5), (0.4, 0.5, 0.6)}, {(0.6, 0.7, 0.8)}〉	〈{(0.3, 0.4, 0.5), (0.4, 0.5, 0.6)}, {(0.5, 0.6, 0.7), (0.6, 0.7, 0.8)}〉	〈{(0.4, 0.5, 0.6)}, {(0.6, 0.7, 0.8), (0.7, 0.8, 0.9)}〉
A₅	〈{(0.5, 0.6, 0.8), (0.7, 0.8, 0.9)}, {(0.3, 0.4, 0.5), (0.4, 0.5, 0.6)}〉	〈{(0.3, 0.4, 0.5), (0.5, 0.6, 0.7)}, {(0.5, 0.6, 0.7), (0.6, 0.7, 0.8)}〉	〈{(0.2, 0.3, 0.4)}, {(0.6, 0.7, 0.8)}〉	〈{(0.3, 0.4, 0.5)}, {(0.7, 0.8, 0.9)}〉

综合评估	取值
h₁	〈{(0.263 6, 0.330 5, 0.401 2), (0.285 0, 0.351 8, 0.423 1), (0.285 7, 0.355 9, 0.432 4), (0.308 6, 0.378 5, 0.455 5)}, {(0.870 8, 0.904 7, 0.935 7), (0.882 5, 0.915 7, 0.946 7), (0.876 7, 0.909 9, 0.940 4), (0.888 4, 0.920 8, 0.951 5)}〉
h₂	〈{(0.271 1, 0.346 9, 0.418 5), (0.293 2, 0.360 0, 0.433 3), (0.292 2, 0.370 6, 0.448 7), (0.313 9, 0.383 4, 0.463 1)}, {(0.803 6, 0.849 2, 0.887 8), (0.814 3, 0.857 0, 0.893 9), (0.815 7, 0.859 0, 0.896 0), (0.826 3, 0.866 7, 0.902 0), (0.816 1, 0.860 5, 0.909 6), (0.826 4, 0.868 0, 0.915 1), (0.827 5, 0.869 7, 0.916 7), (0.837 5, 0.876 9, 0.922 0)}〉
h₃	〈{(0.243 1, 0.304 1, 0.369 8), (0.263 1, 0.325 0, 0.391 6), (0.246 4, 0.309 5, 0.376 8), (0.265 7, 0.329 5, 0.397 5), (0.265 9, 0.328 6, 0.397 1), (0.286 8, 0.350 4, 0.419 8), (0.269 2, 0.333 9, 0.403 9), (0.289 3, 0.354 7, 0.425 4)}, {(0.852 2, 0.899 1, 0.930 8), (0.875 7, 0.910 0, 0.941 7), (0.865 0, 0.909 9, 0.940 4), (0.888 4, 0.920 8, 0.951 5), (0.862 5, 0.907 5, 0.938 4), (0.884 3, 0.917 7, 0.948 3), (0.875 0, 0.917 8, 0.947 2), (0.896 4, 0.927 8, 0.957 1)}〉
h₄	〈{(0.269 8, 0.334 9, 0.415 1), (0.277 8, 0.343 1, 0.423 1), (0.291 0, 0.358 1, 0.429 7), (0.297 1, 0.364 7, 0.436 8), (0.293 1, 0.361 3, 0.447 5), (0.301 2, 0.369 7, 0.455 5), (0.314 9, 0.384 9, 0.462 4), (0.321 0, 0.391 6, 0.469 5)}, {(0.861 6, 0.896 4, 0.927 8), (0.873 4, 0.907 4, 0.938 6), (0.867 6, 0.901 5, 0.932 4), (0.879 3, 0.912 5, 0.943 3)}〉
h₅	〈{(0.226 3, 0.291 0, 0.384 9), (0.265 1, 0.329 0, 0.425 6), (0.269 3, 0.339 7, 0.415 7), (0.313 7, 0.381 9, 0.458 6)}, {(0.861 2, 0.896 5, 0.928 9), (0.871 4, 0.905 6, 0.937 6), (0.869 7, 0.904 0, 0.935 4), (0.879 3, 0.912 5, 0.943 3)}〉

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q-rung hesitant triangular fuzzy BM operator and its application in multiple criteria decision making

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得数函数	q-HTFWBM^{1, 1}
S(h₁)	(0.171 1, 0.220 7, 0.274 2)
S(h₂)	(0.193 6, 0.250 9, 0.310 0)
S(h₃)	(0.160 9, 0.207 8, 0.261 4)
S(h₄)	(0.180 1, 0.229 5, 0.286 0)
S(h₅)	(0.166 2, 0.215 4, 0.275 4)