JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2024, Vol. 59 ›› Issue (3): 118-126.doi: 10.6040/j.issn.1671-9352.7.2023.4667

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A new probabilistic hesitant fuzzy multi-attribute group decision making method based on improved distance measures

Mengdi LIU1,2(),Xianyong ZHANG1,2,*(),Zhiwen MO1,2   

  1. 1. School of Mathematical Sciences, Sichuan Normal University, Chengdu 610066, Sichuan, China
    2. Institute of Intelligent Information and Quantum Information, Sichuan Normal University, Chengdu 610066, Sichuan, China
  • Received:2023-04-26 Online:2024-03-20 Published:2024-03-06
  • Contact: Xianyong ZHANG E-mail:719867875@qq.com;xianyongzh@sina.com

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Abstract:

Aiming at the multi-attribute group decision making problem with known attribute weights under probabilistic hesitant fuzzy environments, hesitation degrees of probabilistic hesitant fuzzy sets are considered, and thus a new method of probabilistic hesitant fuzzy multi-attribute group decision making is proposed based on improved distance measures. Firstly, combining the traditional probabilistic hesitant fuzzy distance measures, improved probabilistic hesitant fuzzy distance measures with hesitation degrees are defined through information fusion, including the Hamming distance, Euclidean distance, and generalized Euclidean distance. These new measures depend on combination coefficients to achieve the theoretical expansion and fusion optimization, and size relationships and parameter monotonicity of distance measures are studied. Secondly, according to the improved distance measures, a new method of multi-attribute group decision making is constructed by using the technique for order preference by similarity to ideal solution(TOPSIS) method, and an example of company location is used to make decision selection. The effectiveness of the proposed method is revealed by parameter analysis and decision comparison. Related researches systematically deepen probabilistic hesitant fuzzy distance measures, and effectively enrich multi-attribute group decision-making methods.

Key words: multi-attribute group decision making, probabilistic hesitant fuzzy set, hesitation degree, information fusion, distance measure

CLC Number: 

  • O159

Fig.1

The three-dimensional drawing of the distance measure given the parameter"

Table 1

Probabilistic hesitant fuzzy decision matrix for company location selection"

Ai C1 C2 C3 C4
A1 {0.3|0.4, 0.75|0.6} {0.25|0.3, 0.55|0.7} {0.5|0.4, 0.65|0.6} {0.6|0.2, 0.85|0.8}
A2 {0.45|0.5, 0.6|0.5} {0.3|0.6, 0.4|0.4} {0.45|0.2, 0.5|0.8} {0.25|0.4, 0.45|0.6}
A3 {0.5|0.2, 0.6|0.8} {0.4|0.55, 0.6|0.45} {0.6|0.5, 0.7|0.5} {0.4|0.3, 0.6|0.7}
A4 {0.4|0.6, 0.8|0.4} {0.25|0.7, 0.5|0.3} {0.7|0.7, 0.9|0.3} {0.45|0.5, 0.7|0.5}

Table 2

The distance between scheme Ai and V+, V-"

α D1+ D2+ D3+ D4+ D1- D2- D3- D4-
1 0.099 8 0.193 4 0.091 2 0.100 5 0.153 6 0.011 0 0.108 4 0.128 1
2 0.103 7 0.224 2 0.106 4 0.122 6 0.183 0 0.014 0 0.121 6 0.140 0
4 0.125 4 0.265 7 0.138 4 0.140 4 0.212 3 0.017 0 0.138 1 0.163 7
5 0.117 9 0.270 2 0.126 6 0.136 5 0.220 0 0.017 8 0.142 8 0.172 1

Table 3

The closeness value Ωi and the sorting result"

α Ω1 Ω2 Ω3 Ω4 排序结果
1 0.606 0 0.053 5 0.543 2 0.560 3 A1>A4>A3>A2
2 0.638 2 0.058 7 0.533 3 0.554 3 A1>A4>A3>A2
4 0.628 7 0.060 0 0.499 5 0.538 3 A1>A4>A3>A2
5 0.651 1 0.062 7 0.530 1 0.557 7 A1>A4>A3>A2

Table 4

The influence value of parameters on scheme results and the ranking result"

α μ1, μ2 Ω1 Ω2 Ω3 Ω4 排序结果
1 μ1=1, μ2=0 0.626 1 0.022 9 0.537 5 0.537 1 A1>A3>A4>A2
μ1=0.8, μ2=0.2 0.617 8 0.028 3 0.540 5 0.537 9 A1>A3>A4>A2
μ1=0.4, μ2=0.6 0.600 4 0.039 6 0.547 1 0.539 4 A1>A3>A4>A2
μ1=0, μ2=1 0.580 4 0.008 2 0.545 9 0.544 4 A1>A3>A4>A2
2 μ1=1, μ2=0 0.649 7 0.031 1 0.529 4 0.544 3 A1>A4>A3>A2
μ1=0.8, μ2=0.2 0.644 8 0.037 4 0.530 3 0.541 4 A1>A4>A3>A2
μ1=0.4, μ2=0.6 0.634 2 0.048 8 0.532 2 0.534 5 A1>A4>A3>A2
μ1=0, μ2=1 0.622 2 0.059 8 0.534 6 0.525 0 A1>A3>A4>A2
4 μ1=1, μ2=0 0.655 5 0.043 3 0.529 8 0.552 8 A1>A4>A3>A2
μ1=0.8, μ2=0.2 0.938 8 0.087 1 0.643 3 0.604 6 A1>A3>A4>A2
μ1=0.4, μ2=0.6 0.646 0 0.056 0 0.528 4 0.539 0 A1>A4>A3>A2
μ1=0, μ2=1 0.638 5 0.063 0 0.526 8 0.519 8 A1>A3>A4>A2
5 μ1=1, μ2=0 0.655 4 0.047 3 0.529 9 0.555 7 A1>A4>A3>A2
μ1=0.8, μ2=0.2 0.652 7 0.051 1 0.529 4 0.552 3 A1>A4>A3>A2
μ1=0.4, μ2=0.6 0.653 9 0.057 4 0.539 3 0.550 2 A1>A4>A3>A2
μ1=0, μ2=1 0.639 7 0.063 2 0.525 9 0.519 8 A1>A3>A4>A2

Table 5

Probabilistic hesitant fuzzy group decision matrix"

xi a1 a2 a3
x1 {0.55|0.15, 0.65|0.25, 0.76|0.1, 0.8|0.5} {0.2|0.05, 0.3|0.325, 0.4|0.125, 0.65|0.25, 0.75|0.25} {0.55|0.5, 0.75|0.25, 0.8|0.15, 0.94|0.1}
x2 {0.4|0.25, 0.58|0.25, 0.69|0.25, 0.95|0.25} {0.35|0.25, 0.6|0.075, 0.65|0.25, 0.7|0.35, 0.8|0.075} {0.45|0.25, 0.55|0.125, 0.56|0.25, 0.66|0.125, 0.85|0.25}
x3 {0.3|0.1, 0.5|0.35, 0.6|0.3, 0.68|0.25} {0.45|0.25, 0.55|0.125, 0.56|0.25, 0.66|0.125, 0.85|0.25} {0.45|0.25, 0.55|0.25, 0.68|0.25, 0.75|0.25}
x4 {0.15|0.1, 0.37|0.15, 0.4|0.25, 0.6|0.25, 0.73|0.25} {0.48|0.4, 0.55|0.25, 0.62|0.1, 0.66|0.25} {0.38|0.25, 0.5|0.125, 0.7|0.125, 0.75|0.25, 0.85|0.25}

Table 6

Proximity value and ranking result"

α μ1, μ2 Ω1 Ω2 Ω3 Ω4 排序结果
1 μ1=1, μ2=0 0.760 1 0.354 3 0.457 1 0.320 6 x1>x3>x2>x4
μ1=0.8, μ2=0.2 0.755 9 0.353 2 0.462 9 0.328 2 x1>x3>x2>x4
μ1=0.6, μ2=0.4 0.778 0 0.323 8 0.457 8 0.339 6 x1>x3>x4>x2
μ1=0.4, μ2=0.6 0.748 1 0.347 9 0.470 0 0.338 5 x1>x3>x2>x4
μ1=0.2, μ2=0.8 0.744 5 0.345 3 0.473 4 0.343 2 x1>x3>x2>x4
μ1=0, μ2=1 0.741 0 0.342 8 0.476 7 0.347 6 x1>x3>x4>x2
2 μ1=1, μ2=0 0.790 4 0.312 3 0.437 2 0.304 0 x1>x3>x2>x4
μ1=0.8, μ2=0.2 0.787 8 0.308 1 0.439 9 0.310 9 x1>x3>x4>x2
μ1=0.6, μ2=0.4 0.785 2 0.303 7 0.442 5 0.316 9 x1>x3>x4>x2
μ1=0.4, μ2=0.6 0.782 6 0.299 2 0.445 2 0.322 3 x1>x3>x4>x2
μ1=0.2, μ2=0.8 0.780 2 0.294 4 0.440 7 0.327 2 x1>x3>x4>x2
μ1=0, μ2=1 0.777 7 0.289 3 0.450 3 0.331 6 x1>x3>x4>x2
4 μ1=1, μ2=0 0.817 9 0.322 3 0.432 4 0.295 0 x1>x3>x2>x4
μ1=0.8, μ2=0.2 0.803 0 0.315 0 0.431 3 0.303 0 x1>x3>x2>x4
μ1=0.6, μ2=0.4 0.804 7 0.306 5 0.430 1 0.308 3 x1>x3>x4>x2
μ1=0.4, μ2=0.6 0.806 7 0.295 8 0.428 9 0.312 0 x1>x3>x4>x2
μ1=0.2, μ2=0.8 0.808 7 0.281 2 0.427 5 0.314 2 x1>x3>x4>x2
μ1=0, μ2=1 0.811 0 0.254 5 0.426 0 0.315 2 x1>x3>x4>x2
5 μ1=1, μ2=0 0.657 4 0.421 9 0.651 1 0.391 3 x1>x3>x2>x4
μ1=0.8, μ2=0.2 0.798 4 0.330 5 0.434 0 0.309 0 x1>x3>x2>x4
μ1=0.6, μ2=0.4 0.677 7 0.405 6 0.648 0 0.413 0 x1>x3>x4>x2
μ1=0.4, μ2=0.6 0.806 8 0.309 8 0.428 2 0.325 5 x1>x3>x4>x2
μ1=0.2, μ2=0.8 0.812 4 0.246 0 0.383 2 0.263 4 x1>x3>x4>x2
μ1=0, μ2=1 0.819 3 0.324 3 0.639 9 0.436 0 x1>x3>x4>x2
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