Design Improvement for Complex Systems with Uncertainty
Abstract
:1. Introduction
2. Motivation and Problem Statement
2.1. Motivation
- (a)
- Figure 2a shows a solution hyper-box with the maximum volume. Both R and are changed so as to achieve the design goals.
- (b)
- Figure 2b shows a solution hyper-box that includes . To achieve the design goals, only R needs to be modified. The solution hyper-box is obtained by requiring a minimum safety margin of for . This is essential, because is subject to uncertainty and cannot be controlled exactly. The realized lower safety margin is larger than the required minimum safety margin, thus more tolerance to uncertainty is provided.
- (c)
- Figure 2c shows a solution hyper-box that includes . To realize the design goals, only needs to be modified. The same minimum safety margin as in Scenario (b) is required.
2.2. Problem Statement
3. Preliminaries
3.1. Divide-the-Best Algorithm
3.2. Particle Swarm Optimization (PSO)
4. The Proposed Particle Swarm Optimization Divide-the-Best Algorithm
Algorithm 1 The particle swarm optimization divide-the-best algorithm (PSO-Divide-Best) |
|
- (1)
- PSO-Divide-Best has great possibility to reach the globally maximum solution hyper-box satisfying the constraints.
- (2)
- Due to the discrete nature of the trial points in the Divide-the-Best algorithm, the PSO-Divide-Best method can be applied to both analytically known and black-box performance functions.
- (3)
- PSO-Divide-Best guarantees that any point selected within the obtained hyper-box is a good design provided that the performance function is continuous.
5. Case Studies
5.1. Vehicle Structure Design Problem
5.2. The Power-Shift Steering Transmission Control System (PSSTCS)
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
PSO | Particle Swarm Optimization |
PSSTCS | Power-Shift Steering Transmission Control System |
PSO-Divide-Best | Particle Swarm Optimization Divide-the-Best algorithm |
EPCP | Approach in [28] including exploration phase and consolidation phase |
IA-CES | Method in [24] which combines interval arithmetic with cellular evolutionary strategies |
Appendix A
Bad Design | Good Design | Bad Design | Good Design | ||||
---|---|---|---|---|---|---|---|
EPCP | PSO-Divide-Best | EPCP | PSO-Divide-Best | ||||
0.9996 | 0.9998 | 0.9999 | 0.9996 | 0.9996 | 0.9996 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9997 | 0.9999 | ||
0.9996 | 0.9996 | 0.9998 | 0.9996 | 0.9996 | 0.9996 | ||
0.9996 | 0.9997 | 0.9997 | 0.9996 | 0.9996 | 0.9996 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9998 | 0.9996 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | ||
0.9996 | 0.9997 | 0.9998 | 0.9996 | 0.9998 | 0.9999 | ||
0.9996 | 0.9998 | 0.9998 | 0.9996 | 0.9996 | 0.9996 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9997 | 0.9999 | ||
0.9996 | 0.9997 | 0.9998 | 0.9996 | 0.9996 | 0.9996 | ||
0.9996 | 0.9997 | 0.9999 | 0.9996 | 0.9996 | 0.9996 | ||
0.9996 | 0.9997 | 0.9998 | 0.9996 | 0.9996 | 0.9996 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | ||
0.9996 | 0.9997 | 0.9999 | 0.9996 | 0.9998 | 0.9999 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9998 | 0.9998 | ||
0.9996 | 0.9998 | 0.9998 | 0.9996 | 0.9996 | 0.9996 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9998 | 0.9999 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9998 | 0.9998 | ||
0.9996 | 0.9997 | 0.9999 | 0.9996 | 0.9998 | 0.9998 | ||
0.9996 | 0.9998 | 0.9999 | 0.9996 | 0.9997 | 0.9999 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9997 | 0.9998 | ||
0.9996 | 0.9998 | 0.9997 | 0.9996 | 0.9998 | 0.9999 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9997 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9997 | ||
0.9996 | 0.9997 | 0.9999 | 0.9996 | 0.9997 | 0.9998 | ||
0.9996 | 0.9997 | 0.9995 | 0.9996 | 0.9997 | 0.9998 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | ||
0.9996 | 0.9997 | 0.9997 | 0.9996 | 0.9996 | 0.9996 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9995 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9997 | 0.9996 | ||
0.9996 | 0.9997 | 0.9997 | 0.9996 | 0.9997 | 0.9998 | ||
0.9996 | 0.9997 | 0.9997 | 0.9996 | 0.9996 | 0.9998 | ||
0.9996 | 0.9997 | 0.9998 | 0.9996 | 0.9997 | 0.9995 | ||
0.9996 | 0.9997 | 0.9997 | 0.9996 | 0.9996 | 0.9996 | ||
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9997 | 0.9996 | ||
0.9996 | 0.9996 | 0.9995 | 0.9996 | 0.9998 | 0.9998 | ||
0.9856 | 0.9912 | 0.9940 |
Lower Bound | EPCP | PSO-Divide-Best | Upper Bound | EPCP | PSO-Divide-Best |
---|---|---|---|---|---|
0.9997 | 0.9997 | 0.9999 | 1.0000 | ||
0.9995 | 0.9994 | 0.9998 | 1.0000 | ||
0.9996 | 0.9991 | 0.9998 | 1.0000 | ||
0.9995 | 0.9993 | 0.9998 | 1.0000 | ||
0.9995 | 0.9995 | 0.9998 | 1.0000 | ||
0.9997 | 0.9997 | 0.9998 | 1.0000 | ||
0.9995 | 0.9995 | 0.9997 | 1.0000 | ||
0.9995 | 0.9992 | 0.9998 | 1.0000 | ||
0.9996 | 0.9994 | 0.9998 | 1.0000 | ||
0.9995 | 0.9999 | 0.9999 | 1.0000 | ||
0.9990 | 0.9990 | 1.0000 | 1.0000 | ||
0.9990 | 0.9990 | 1.0000 | 1.0000 | ||
0.9991 | 0.9990 | 1.0000 | 1.0000 | ||
0.9997 | 0.9998 | 0.9999 | 1.0000 | ||
0.9995 | 0.9992 | 0.9998 | 1.0000 | ||
0.9996 | 0.9994 | 0.9998 | 1.0000 | ||
0.9997 | 0.9994 | 0.9999 | 1.0000 | ||
0.9995 | 0.9995 | 0.9999 | 1.0000 | ||
0.9995 | 0.9996 | 0.9999 | 1.0000 | ||
0.9997 | 0.9998 | 0.9999 | 1.0000 | ||
0.9997 | 0.9997 | 0.9999 | 1.0000 | ||
0.9997 | 0.9998 | 0.9999 | 1.0000 | ||
0.9990 | 0.9990 | 1.0000 | 1.0000 | ||
0.9995 | 0.9999 | 0.9998 | 1.0000 | ||
0.9995 | 0.9995 | 0.9999 | 1.0000 | ||
0.9997 | 0.9998 | 0.9999 | 1.0000 | ||
0.9995 | 0.9998 | 0.9999 | 1.0000 | ||
0.9990 | 0.9990 | 1.0000 | 1.0000 | ||
0.9995 | 0.9995 | 0.9999 | 1.0000 | ||
0.9997 | 0.9995 | 0.9999 | 1.0000 | ||
0.9990 | 0.9990 | 1.0000 | 1.0000 | ||
0.9990 | 0.9990 | 1.0000 | 1.0000 | ||
0.9995 | 0.9997 | 0.9999 | 1.0000 | ||
0.9997 | 0.9999 | 0.9999 | 1.0000 | ||
0.9990 | 0.9990 | 1.0000 | 1.0000 | ||
0.9991 | 0.9990 | 1.0000 | 1.0000 | ||
0.9991 | 0.9990 | 1.0000 | 1.0000 | ||
0.9992 | 0.9990 | 1.0000 | 1.0000 | ||
0.9990 | 0.9990 | 1.0000 | 1.0000 | ||
0.9997 | 0.9995 | 0.9999 | 1.0000 | ||
0.9997 | 0.9997 | 0.9999 | 1.0000 | ||
0.9995 | 0.9995 | 0.9999 | 1.0000 | ||
0.9997 | 0.9997 | 0.9999 | 1.0000 | ||
0.9997 | 0.9999 | 0.9999 | 1.0000 | ||
0.9998 | 0.9998 | 0.9999 | 1.0000 | ||
0.9997 | 0.9995 | 0.9999 | 1.0000 | ||
0.9995 | 0.9998 | 0.9999 | 1.0000 | ||
0.9997 | 0.9995 | 0.9999 | 1.0000 | ||
0.9997 | 0.9999 | 0.9999 | 1.0000 | ||
0.9995 | 0.9998 | 0.9999 | 1.0000 | ||
0.9996 | 0.9995 | 0.9998 | 1.0000 | ||
0.9995 | 0.9995 | 0.9998 | 1.0000 | ||
0.9997 | 0.9995 | 0.9999 | 1.0000 | ||
0.9997 | 0.9998 | 0.9999 | 1.0000 | ||
0.9997 | 0.9995 | 0.9999 | 1.0000 | ||
0.9997 | 0.9995 | 0.9999 | 1.0000 | ||
0.9996 | 0.9992 | 0.9998 | 1.0000 | ||
0.9995 | 0.9994 | 0.9998 | 1.0000 | ||
0.9996 | 0.9994 | 0.9998 | 1.0000 | ||
0.9995 | 0.9994 | 0.9998 | 1.0000 | ||
0.9996 | 0.9997 | 0.9998 | 1.0000 | ||
0.9996 | 0.9996 | 0.9998 | 1.0000 | ||
0.9995 | 0.9990 | 0.9998 | 1.0000 | ||
0.9996 | 0.9996 | 0.9998 | 1.0000 | ||
0.9996 | 0.9995 | 0.9998 | 1.0000 | ||
0.9995 | 0.9995 | 0.9998 | 1.0000 | ||
0.9995 | 0.9994 | 0.9998 | 1.0000 | ||
0.9996 | 0.9994 | 0.9998 | 1.0000 | ||
0.9996 | 0.9990 | 0.9998 | 1.0000 | ||
0.9995 | 0.9990 | 0.9998 | 1.0000 | ||
0.9995 | 0.9990 | 0.9998 | 1.0000 | ||
0.9995 | 0.9995 | 0.9998 | 1.0000 | ||
0.9996 | 0.9992 | 0.9998 | 1.0000 | ||
0.9995 | 0.9991 | 0.9998 | 1.0000 | ||
0.9995 | 0.9994 | 0.9998 | 1.0000 | ||
0.9996 | 0.9995 | 0.9998 | 1.0000 | ||
0.9995 | 0.9993 | 0.9998 | 1.0000 | ||
0.9995 | 0.9995 | 0.9998 | 1.0000 | ||
0.9996 | 0.9995 | 0.9998 | 1.0000 | ||
0.9996 | 0.9990 | 0.9998 | 1.0000 | ||
0.9996 | 0.9994 | 0.9998 | 1.0000 | ||
0.9996 | 0.9993 | 0.9998 | 1.0000 | ||
0.9996 | 0.9996 | 0.9998 | 1.0000 | ||
0.9996 | 0.9992 | 0.9998 | 1.0000 | ||
0.9995 | 0.9990 | 0.9998 | 1.0000 | ||
0.9997 | 0.9995 | 0.9999 | 1.0000 |
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(a) Both and Are Key Parameters | (b) Only Is Key Parameter | (c) Only Is Key Parameter | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
kN | kN | |||||||||||
kN | kN | |||||||||||
exact | EPCP | IA-CES | PSO-Divide-Best | exact | EPCP | IA-CES | PSO-Divide-Best | exact | EPCP | IA-CES | PSO-Divide-Best | |
(kN) | 294.80 | 290.74 | 284.08 | 296.15 | 386.20 | 384.30 | 392.53 | 386.20 | 250.00 | 249.14 | 245.05 | 250.00 |
(kN) | 516.40 | 516.31 | 527.30 | 515.06 | 425.00 | 422.47 | 422.78 | 425.00 | 561.20 | 556.32 | 552.28 | 561.47 |
(kN) | 516.40 | 515.81 | 535.46 | 515.06 | 425.00 | 422.06 | 422.78 | 425.00 | 561.20 | 563.67 | 575.21 | 561.47 |
(kN) | 627.20 | 622.59 | 626.95 | 627.20 | 627.20 | 623.69 | 619.36 | 627.19 | 627.20 | 627.48 | 626.38 | 627.19 |
volume (kN) | 2.4553 | 2.4086 | 2.2253 | 2.4548 | 0.7845 | 0.7696 | 0.5756 | 0.7844 | 2.0539 | 1.9601 | 1.5721 | 2.0537 |
(%) | ____ | 1.38 | 3.64 | 0.46 | ____ | 0.49 | 1.64 | 0.00 | ____ | 0.34 | 1.59 | 0.00 |
(%) | ____ | 0.02 | 2.11 | 0.26 | ____ | 0.60 | 0.75 | 0.00 | ____ | 0.87 | 1.98 | 0.05 |
(%) | ____ | 0.11 | 3.69 | 0.26 | ____ | 0.69 | 1.25 | 0.00 | ____ | 0.44 | 2.50 | 0.05 |
(%) | ____ | 0.74 | 0.04 | 0.00 | ____ | 0.56 | 0.52 | 0.01 | ____ | 0.04 | 0.13 | 0.00 |
error (%) | ____ | 1.90 | 9.29 | 0.02 | ____ | 1.90 | 26.63 | 0.01 | ____ | 4.57 | 23.46 | 0.01 |
Method | EPCP | PSO-Divide-Best |
---|---|---|
Log-volume | −698.6824 | −656.9503 |
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Chen, Y.; Shi, J.; Yi, X.-J. Design Improvement for Complex Systems with Uncertainty. Mathematics 2021, 9, 1173. https://doi.org/10.3390/math9111173
Chen Y, Shi J, Yi X-J. Design Improvement for Complex Systems with Uncertainty. Mathematics. 2021; 9(11):1173. https://doi.org/10.3390/math9111173
Chicago/Turabian StyleChen, Yue, Jian Shi, and Xiao-Jian Yi. 2021. "Design Improvement for Complex Systems with Uncertainty" Mathematics 9, no. 11: 1173. https://doi.org/10.3390/math9111173
APA StyleChen, Y., Shi, J., & Yi, X.-J. (2021). Design Improvement for Complex Systems with Uncertainty. Mathematics, 9(11), 1173. https://doi.org/10.3390/math9111173