An Entropy Weight-Based Lower Confidence Bounding Optimization Approach for Engineering Product Design
Abstract
:1. Introduction
2. Background
2.1. Kriging Model
2.2. Review of the Typical Adaptive Surrogate-Based Design Optimization Methods
2.2.1. The Lower Confidence Bounding Method
2.2.2. The Parameterized Lower Confidence Bounding Method
2.2.3. The Expected Improvement Method
2.2.4. The Weighted Expected Improvement Method
3. Proposed Approach
3.1. Step 1: Generate the Initial Sample Set
3.2. Steps 2 and 3: Constructing the Kriging Model and Obtaining the Current Optimal Solution
3.3. Step 4: Check the Terminal Condition
3.4. Steps 5: Update the Sample Set through the Proposed EW-LCB
3.5. Step 6: Output the Optimal Solution
4. Tested Cases
4.1. Numerical Examples
- Peaks function (PK)
- Banana function (BA)
- Sasena function (SA)
- Six-hump camp-back function (SC)
- Himmelblau function (HM)
- Goldstein–Price function (GP)
- Generalized polynomial function (GF)
- Levy 3 function (L3)
- Hartmann 3 function (H3)
- Leon (LE)
4.2. Engineering Application
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A
Methods | Variables (Rounded) | Weight (t) | ||||||
---|---|---|---|---|---|---|---|---|
t1 mm | t3 mm | t5 mm | t2 mm | t4 mm | t6 mm | Allowance 3.0327 | ||
EI | 88 | 59 | 12 | 40 | 16 | 12 | 3.945 | 3.0058 |
90 | 60 | 12 | 40 | 13 | 13 | 3.991 | 3.0151 | |
90 | 60 | 12 | 40 | 14 | 12 | 4.020 | 3.0136 | |
90 | 59 | 12 | 40 | 15 | 12 | 4.031 | 3.0125 | |
83 | 60 | 12 | 40 | 20 | 12 | 4.004 | 3.0158 | |
88 | 59 | 12 | 40 | 16 | 12 | 3.945 | 3.0058 | |
90 | 60 | 12 | 40 | 13 | 13 | 3.991 | 3.0151 | |
90 | 60 | 12 | 40 | 14 | 12 | 4.020 | 3.0136 | |
90 | 59 | 12 | 40 | 15 | 12 | 4.031 | 3.0125 | |
83 | 60 | 12 | 40 | 20 | 12 | 4.004 | 3.0158 | |
88 | 59 | 12 | 40 | 16 | 12 | 3.945 | 3.0058 | |
90 | 60 | 12 | 40 | 13 | 13 | 3.991 | 3.0151 | |
90 | 60 | 12 | 40 | 14 | 12 | 4.020 | 3.0136 | |
90 | 59 | 12 | 40 | 15 | 12 | 4.031 | 3.0125 | |
83 | 60 | 12 | 40 | 20 | 12 | 4.004 | 3.0158 | |
88 | 59 | 12 | 40 | 16 | 12 | 3.945 | 3.0058 | |
90 | 60 | 12 | 40 | 13 | 13 | 3.991 | 3.0151 | |
90 | 60 | 12 | 40 | 14 | 12 | 4.020 | 3.0136 | |
90 | 59 | 12 | 40 | 15 | 12 | 4.031 | 3.0125 | |
83 | 60 | 12 | 40 | 20 | 12 | 4.004 | 3.0158 | |
WEI | 88 | 59 | 12 | 40 | 16 | 13 | 3.919 | 3.0172 |
89 | 60 | 12 | 40 | 14 | 12 | 4.042 | 3.0047 | |
88 | 60 | 12 | 40 | 16 | 12 | 4.013 | 3.0172 | |
83 | 60 | 13 | 40 | 20 | 12 | 3.974 | 3.0266 | |
82 | 60 | 13 | 40 | 21 | 12 | 3.964 | 3.0281 | |
88 | 59 | 12 | 40 | 16 | 13 | 3.919 | 3.0172 | |
89 | 60 | 12 | 40 | 14 | 12 | 4.042 | 3.0047 | |
88 | 60 | 12 | 40 | 16 | 12 | 4.013 | 3.0172 | |
82 | 60 | 13 | 40 | 21 | 12 | 3.964 | 3.0281 | |
83 | 60 | 13 | 40 | 20 | 12 | 3.974 | 3.0266 | |
88 | 59 | 12 | 40 | 16 | 13 | 3.919 | 3.0172 | |
89 | 60 | 12 | 40 | 14 | 12 | 4.042 | 3.0047 | |
88 | 60 | 12 | 40 | 16 | 12 | 4.013 | 3.0172 | |
83 | 60 | 13 | 40 | 20 | 12 | 3.974 | 3.0266 | |
82 | 60 | 13 | 40 | 21 | 12 | 3.964 | 3.0281 | |
88 | 59 | 12 | 40 | 16 | 13 | 3.919 | 3.0172 | |
89 | 60 | 12 | 40 | 14 | 12 | 4.042 | 3.0047 | |
88 | 60 | 12 | 40 | 16 | 12 | 4.013 | 3.0172 | |
83 | 60 | 13 | 40 | 20 | 12 | 3.974 | 3.0266 | |
82 | 60 | 13 | 40 | 21 | 12 | 3.964 | 3.0281 | |
LCB | 81 | 58 | 16 | 42 | 18 | 12 | 3.808 | 3.0203 |
78 | 58 | 20 | 40 | 18 | 14 | 3.857 | 3.0457 | |
75 | 52 | 17 | 46 | 17 | 21 | 3.676 | 3.0409 | |
69 | 51 | 25 | 43 | 20 | 19 | 3.688 | 3.0617 | |
76 | 50 | 21 | 44 | 15 | 22 | 3.647 | 3.0499 | |
81 | 58 | 16 | 42 | 18 | 12 | 3.808 | 3.0203 | |
78 | 58 | 20 | 40 | 18 | 14 | 3.857 | 3.0457 | |
75 | 52 | 17 | 46 | 17 | 21 | 3.676 | 3.0409 | |
69 | 51 | 25 | 43 | 20 | 19 | 3.688 | 3.0617 | |
76 | 50 | 21 | 44 | 15 | 22 | 3.647 | 3.0499 | |
81 | 58 | 16 | 42 | 18 | 12 | 3.808 | 3.0203 | |
78 | 58 | 20 | 40 | 18 | 14 | 3.857 | 3.0457 | |
75 | 52 | 17 | 46 | 17 | 21 | 3.676 | 3.0409 | |
69 | 51 | 25 | 43 | 20 | 19 | 3.688 | 3.0617 | |
76 | 50 | 21 | 44 | 15 | 22 | 3.647 | 3.0499 | |
81 | 58 | 16 | 42 | 18 | 12 | 3.808 | 3.0203 | |
78 | 58 | 20 | 40 | 18 | 14 | 3.857 | 3.0457 | |
75 | 52 | 17 | 46 | 17 | 21 | 3.676 | 3.0409 | |
78 | 47 | 21 | 47 | 14 | 20 | 3.651 | 3.0303 | |
70 | 51 | 25 | 49 | 12 | 21 | 3.632 | 3.0461 | |
PLCB | 78 | 55 | 12 | 40 | 11 | 12 | 3.892 | 2.8243 |
84 | 47 | 17 | 46 | 12 | 21 | 3.769 | 3.0154 | |
84 | 47 | 24 | 42 | 12 | 19 | 3.748 | 3.0331 | |
45 | 51 | 33 | 46 | 35 | 17 | 3.533 | 3.1076 | |
84 | 38 | 28 | 44 | 15 | 19 | 3.684 | 3.0270 | |
78 | 55 | 12 | 40 | 11 | 12 | 3.892 | 2.8243 | |
84 | 47 | 17 | 46 | 12 | 21 | 3.769 | 3.0154 | |
84 | 47 | 24 | 42 | 12 | 19 | 3.748 | 3.0331 | |
45 | 51 | 33 | 46 | 35 | 17 | 3.533 | 3.1076 | |
84 | 38 | 28 | 44 | 15 | 19 | 3.684 | 3.0270 | |
78 | 55 | 12 | 40 | 11 | 12 | 3.892 | 2.8243 | |
84 | 47 | 17 | 46 | 12 | 21 | 3.769 | 3.0154 | |
84 | 47 | 24 | 42 | 12 | 19 | 3.748 | 3.0331 | |
45 | 51 | 33 | 46 | 35 | 17 | 3.533 | 3.1076 | |
84 | 38 | 28 | 44 | 15 | 19 | 3.684 | 3.0270 | |
78 | 55 | 12 | 40 | 11 | 12 | 3.892 | 2.8243 | |
84 | 47 | 17 | 46 | 12 | 21 | 3.769 | 3.0154 | |
84 | 47 | 24 | 42 | 12 | 19 | 3.748 | 3.0331 | |
67 | 59 | 39 | 40 | 11 | 12 | 3.645 | 3.0369 | |
45 | 51 | 35 | 47 | 33 | 16 | 3.524 | 3.1050 | |
EW-LCB | 88 | 60 | 12 | 40 | 15 | 12 | 4.015 | 3.0065 |
87 | 60 | 12 | 40 | 17 | 12 | 4.017 | 3.0187 | |
87 | 60 | 12 | 40 | 17 | 12 | 4.016 | 3.0187 | |
85 | 60 | 12 | 40 | 18 | 12 | 4.016 | 3.0118 | |
84 | 60 | 12 | 40 | 18 | 12 | 4.015 | 3.0034 | |
88 | 60 | 12 | 40 | 15 | 12 | 4.015 | 3.0065 | |
87 | 60 | 12 | 40 | 17 | 12 | 4.017 | 3.0187 | |
87 | 60 | 12 | 40 | 17 | 12 | 4.016 | 3.0187 | |
85 | 60 | 12 | 40 | 18 | 12 | 4.016 | 3.0118 | |
84 | 60 | 12 | 40 | 18 | 12 | 4.015 | 3.0034 | |
88 | 60 | 12 | 40 | 15 | 12 | 4.015 | 3.0065 | |
87 | 60 | 12 | 40 | 17 | 12 | 4.017 | 3.0187 | |
87 | 60 | 12 | 40 | 17 | 12 | 4.016 | 3.0187 | |
85 | 60 | 12 | 40 | 18 | 12 | 4.016 | 3.0118 | |
84 | 60 | 12 | 40 | 18 | 12 | 4.015 | 3.0034 | |
89 | 59 | 12 | 40 | 16 | 12 | 4.061 | 3.0143 | |
89 | 58 | 12 | 40 | 16 | 12 | 4.056 | 3.0026 | |
88 | 60 | 12 | 40 | 16 | 12 | 4.062 | 3.0172 | |
89 | 60 | 12 | 40 | 14 | 12 | 4.060 | 3.0047 | |
89 | 60 | 12 | 40 | 15 | 12 | 4.059 | 3.0155 |
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Parameter | Values |
---|---|
Population Size | 100 |
Maximum generation | 100 |
Crossover probability | 0.95 |
Mutation probability | 0.01 |
Methods | EI | WEI | LCB | PLCB | EW-LCB |
---|---|---|---|---|---|
Mean Value | 7.64 | 7.86 | 7.98 | 7.66 | 6.48 |
Standard deviations | 1.352 | 1.498 | 1.301 | 1.780 | 0.505 |
Functions | Items | EI | WEI | LCB | PLCB | EW-LCB |
---|---|---|---|---|---|---|
PK | FEmean | 29.82/3 | 30.13/4 | 29.68/2 | 31.99/5 | 26.97/1 |
FEstd | 2.435/2 | 2.884/3 | 2.068/1 | 5.107/4 | 5.34/5 | |
BA | FEmean | 33.23/3 | 32.15/2 | 33.88/4 | 34.34/5 | 26.33/1 |
FEstd | 3.989/3 | 3.777/2 | 5.664/4 | 6.144/5 | 2.78/1 | |
SA | FEmean | 32.12/3 | 36.15/5 | 34.88/4 | 31.23/2 | 27.92/1 |
FEstd | 4.674/3 | 4.865/4 | 2.813/2 | 5.241/5 | 2.722/1 | |
SC | FEmean | 39.30/4 | 40.66/5 | 39.20/3 | 36.41/2 | 33.42/1 |
FEstd | 3.965/3 | 3.634/1 | 3.785/2 | 5.292/4 | 5.320/5 | |
HM | FEmean | 45.76/4 | 46.22/5 | 44.12/3 | 41.34/2 | 35.22/1 |
FEstd | 3.456/3 | 2.973/2 | 1.894/1 | 5.157/5 | 3.49/4 | |
GP | FEmean | 117.66/5 | 115.67/4 | 105.27 /3 | 97.77/2 | 89.27/1 |
FEstd | 19.11/5 | 11.81/2 | 17.55/4 | 14.44/3 | 9.99/1 | |
GF | FEmean | Failed/5 | Failed/5 | Failed/5 | 140.42/2 | 116.67/1 |
FEstd | Failed/5 | Failed/5 | Failed/5 | 75.66/2 | 30.63/1 | |
L3 | FEmean | 300.4/3 | 534.6/4 | 540.4/5 | 167.1/1 | 199.2/2 |
FEstd | 119.1/3 | 147.0/4 | 159.4/5 | 55.21/1 | 88.89/2 | |
H3 | FEmean | 37.50/3 | 37.62/2 | 38.20/4 | 39.34/5 | 36.58/1 |
FEstd | 3.50/4 | 3.46/3 | 3.29/2 | 3.64/5 | 2.56/1 | |
H6 | FEmean | 107.03/4 | 105.13/3 | 103.67/2 | 114.1/5 | 101.16/1 |
FEstd | 44.50/2 | 50.39/5 | 48.26/3 | 49.30/4 | 43.43/1 |
Metrics | EI | WEI | LCB | PLCB | EW-LCB | |
---|---|---|---|---|---|---|
Average rank | FEmean | 3.70 | 4.00 | 3.22 | 3.10 | 1.10 |
FEstd | 2.50 | 3.40 | 2.90 | 3.70 | 2.30 |
i | Hypothesis | p-Values |
---|---|---|
1 | EI vs. EW-LCB | 0.0028 |
2 | WEI vs. EW-LCB | 0.0001 |
3 | LCB vs. EW-LCB | 0.0016 |
4 | PLCB vs. EW-LCB | 0.0056 |
Functions | Initial Sample Size | EI | WEI | LCB | PLCB | EW-LCB |
---|---|---|---|---|---|---|
SA | 27.55/4 | 28.98/5 | 27.02/3 | 24.12/2 | 23.34/1 | |
32.12/3 | 36.15/5 | 34.88/4 | 31.23/2 | 27.92/1 | ||
41.67/4 | 43.98/5 | 41.12/3 | 40.20/2 | 38.03/1 | ||
L3 | 393.6/3 | 524.5/5 | 513.4/4 | 142.5/1 | 173.2/2 | |
300.4/3 | 534.6/4 | 540.4/5 | 167.1/1 | 199.2/2 | ||
403.4/3 | 525.4/4 | 536.5/5 | 166.6/1 | 206.1/2 |
Fixed Parameters | Values |
---|---|
Elastic modulus | |
Density | |
Poisson’s ratio | 0.3 |
The length of the Hull | 12,000 mm |
The radius of the Hull | 3300 mm |
Rib space | 600 mm |
Size of the ribs’ | mm |
The radius of the base web opening | 75 mm |
Width of the base web opening | 210 mm |
Design Variables | Ranges | |
---|---|---|
Former half | The thickness of the base panel | 40–90 mm |
The thickness of the base web | 10–60 mm | |
The thickness of the base bracket | 12–40 mm | |
Remaining half | The thickness of the base panel | 40–90 mm |
The thickness of the base web | 10–60 mm | |
The thickness of the base bracket | 12–40 mm |
Methods | ||||
---|---|---|---|---|
Max | Mean | Std | Succeeded | |
EI | 4.031 | 3.998 | 0.03060 | 20/20 |
WEI | 4.042 | 3.983 | 0.04316 | 20/20 |
LCB | 3.857 | 3.733 | 0.08629 | 5/20 |
PLCB | 3.892 | 3.723 | 0.12230 | 11/20 |
EW-LCB | 4.062 | 4.027 | 0.01953 | 20/20 |
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Qian, J.; Yi, J.; Zhang, J.; Cheng, Y.; Liu, J. An Entropy Weight-Based Lower Confidence Bounding Optimization Approach for Engineering Product Design. Appl. Sci. 2020, 10, 3554. https://doi.org/10.3390/app10103554
Qian J, Yi J, Zhang J, Cheng Y, Liu J. An Entropy Weight-Based Lower Confidence Bounding Optimization Approach for Engineering Product Design. Applied Sciences. 2020; 10(10):3554. https://doi.org/10.3390/app10103554
Chicago/Turabian StyleQian, Jiachang, Jiaxiang Yi, Jinlan Zhang, Yuansheng Cheng, and Jun Liu. 2020. "An Entropy Weight-Based Lower Confidence Bounding Optimization Approach for Engineering Product Design" Applied Sciences 10, no. 10: 3554. https://doi.org/10.3390/app10103554