1 Introduction
2 Statement of the Problem
3 Basics of Differential Evolution
4 Proposed Algorithm: ATMDE
4.1 Improved DE
4.2 Adaptive Trade-Off Model
4.2.1 Infeasible Situation
4.2.2 Semi-feasible Situation
4.2.3 Feasible Situation
4.3 Shrinking Space Technique
4.4 Framework of ATMDE
5 Benchmark Test Functions
Function | No. of variables
n
| Type of function | Ration
ρ/ % | No. of constraints
N
|
---|---|---|---|---|
g01 | 13 | Quadratic | 0.01 | 9 |
g02 | 20 | Nonlinear | 99.9 | 2 |
g03 | 10 | Polynomial | 0.00 | 1 |
g04 | 5 | Quadratic | 52.1 | 6 |
g05 | 4 | Cubic | 0.00 | 5 |
g06 | 2 | Cubic | 0.01 | 2 |
g07 | 10 | Quadratic | 0.00 | 8 |
g08 | 2 | Nonlinear | 0.86 | 2 |
g09 | 7 | Polynomial | 0.51 | 4 |
g10 | 8 | Linear | 0.00 | 6 |
g11 | 2 | Quadratic | 0.00 | 1 |
g12 | 3 | Quadratic | 4.48 | 1 |
g14 | 10 | Nonlinear | 0.00 | 3 |
g15 | 3 | Quadratic | 0.00 | 2 |
g16 | 5 | Nonlinear | 0.02 | 38 |
g18 | 9 | Quadratic | 0.00 | 13 |
g19 | 15 | Nonlinear | 33.4 | 5 |
g24 | 2 | Linear | 79.6 | 2 |
5.1 Parameter Settings
5.2 General Performance of ATMDE
Function | Optimal solution
f
*
| Best solution
f
best
| Median solution
f
median
| Mean solution
μ
f
| Worst solution
f
worst
| Standard deviation
σ
f
|
---|---|---|---|---|---|---|
g01 | −15.000 | −15.000 | −15.000 | −15.000 | −15.000 | 0 |
g02 | −0.803 619 | −0.803 617 | −0.803 617 | −0.803 617 | −0.803 610 | 1.238 9 × 10−6
|
g03 | −1.000 50 | −1.005 00 | −1.005 00 | −1.005 00 | −1.005 00 | 2.081 6 × 10−9
|
g04 | −30 665.538 6 | −30 665.538 6 | −30 665.538 6 | −30 665.538 6 | −30 665.538 6 | 1.110 × 10−11
|
g05 | 5126.496 71 | 5126.496 71 | 5126.496 71 | 5126.496 71 | 5126.496 71 | 1.013 3 × 10−12
|
g06 | −6961.813 87 | −6961.813 87 | −6961.813 87 | −6961.813 87 | −6961.813 87 | 1.850 × 10−12
|
g07 | 24.306 209 | 24.306 209 | 24.306 209 | 24.306 209 | 24.306 209 | 2.211 4 × 10−8
|
g08 | −0.095 825 | −0.095 825 | −0.095 825 | −0.095 825 | −0.095 825 | 2.564 1 × 10−17
|
g09 | 680.630 05 | 680.630 05 | 680.630 05 | 680.630 05 | 680.630 05 | 4.634 8 × 10−13
|
g10 | 7049.248 02 | 7049.248 02 | 7049.248 02 | 7049.248 02 | 7049.24802 | 8.700 × 10−7
|
g11 | 0.749 90 | 0.749 90 | 0.749 90 | 0.749 90 | 0.749 90 | 1.011 2 × 10−7
|
g12 | −1.000 00 | −1.000 00 | −1.000 00 | −1.000 00 | −1.000 00 | 0 |
g14 | −47.764 888 | −47.764 888 | −47.764 888 | −47.764 888 | −47.764 888 | 1.953 9 × 10−10
|
g15 | 961.715 022 | 961.715 022 | 961.715 022 | 961.715 022 | 961.715 022 | 6.937 8 × 10−13
|
g16 | −1.905 155 | −1.905 155 | −1.905 155 | −1.905 155 | −1.905 155 | 6.775 2 × 10−16
|
g18 | −0.866 025 | −0.866 025 | −0.866 025 | −0.866 025 | −0.866 025 | 7.454 9 × 10−10
|
g19 | 32.655 59 | 32.655 63 | 32.655 86 | 32.656 00 | 32.657 25 | 3.753 8 × 10−4
|
g24 | −5.508 013 | −5.508 013 | −5.508 013 | −5.508 013 | −5.508 013 | 3.735 5 × 10−15
|
5.3 ATMDE Compared with AATM
Function | Best solution f
best
| Mean solution μ
f
| Worst solution f
worst
| Standard deviation σ
f
| ||||
---|---|---|---|---|---|---|---|---|
ATMDE | AATM | ATMDE | AATM | ATMDE | AATM | ATMDE | AATM | |
g01 | −15.000 | −15.000 | −15.000 | −15.000 | −15.000 | −15.000 | 0 | 3.1 × 10−7
|
g02 | −0.803 617 | −0.803 38 | −0.803 617 | −0.791 21 | −0.803 61 | −0.767 | 1.2 × 10−6
| 8.6 × 10−3
|
g03 | −1.005 00 | −1.00 | −1.005 00 | −1.00 | −1.005 00 | −1.00 | 2.1 × 10−9
| 3.5 × 10−4
|
g04 | −30 665.539 | −30 665.5 | −30 665.539 | −30 665.5 | −30 665.5 | −30 665.5 | 1.1 × 10−11
| 1.0 × 10−4
|
g05 | 5 126.496 7 | 5 126.498 | 5 126.496 71 | 5 126.714 | 5 126.496 7 | 5 128.824 | 1.0 × 10−12
| 4.3 × 10−1
|
g06 | −6 961.814 | −6 961.81 | −6 961.814 | −6 961.81 | −6 961.81 | −6 961.81 | 1.6 × 10−12
| 7.1 × 10−12
|
g07 | 24.306 209 | 24.307 | 24.306 209 | 24.317 | 24.306 209 | 24.356 | 2.2 × 10−8
| 1.3 × 10−2
|
g08 | −0.095 825 | −0.095 82 | −0.095 825 | −0.095 82 | −0.095 82 | −0.095 82 | 2.6 × 10−17
| 5.8 × 10−
18
|
g09 | 680.630 | 680.630 | 680.630 05 | 680.639 4 | 680.630 05 | 680.646 | 4.6 × 10-13
| 4.5 × 10−3
|
g10 | 7 049.248 | 7 049.603 | 7 049.2480 2 | 7 077.477 | 7 049.248 | 7 183.295 | 8.7 × 10−7
| 3.1 × 101
|
g11 | 0.74990 | 0.75 | 0.7499 | 0.75 | 0.7499 | 0.75 | 1.0 × 10−7
| 3.8 × 10−6
|
g12 | −1.000 | −1.000 | −1.000 | −1.000 | −1.000 | −1.000 | 0 | 0 |
g14 | −47.764 888 | −47.762 | −47.764 888 | −47.750 | −47.764 8 | −47.712 | 1.9 × 10−10
| 1.0 × 10−2
|
g15 | 961.715 | 961.715 | 961.715 | 961.715 | 961.715 02 | 961.716 | 6.9 × 10−13
| 3.0 × 10−4
|
g16 | −1.905 155 | −1.905 15 | −1.905 155 | −1.905 15 | −1.905 15 | −1.905 15 | 6.8 × 10−16
| 2.4 × 10−14
|
g18 | −0.866 025 | −0.866 02 | −0.866 025 | −0.865 95 | −0.866 02 | −0.864 84 | 7.5 × 10−10
| 2.1 × 10−4
|
g19 | 32.655 63 | 32.725 | 32.655 86 | 32.952 | 32.657 25 | 33.243 | 3.8 × 10−4
| 1.4 × 10−1
|
g24 | −5.508 01 | −5.508 01 | −5.508 01 | −5.508 01 | −5.508 01 | −5.508 01 | 3.7 × 10−15
| 1.8 × 10−15
|
w/t/l
| 8/10/0 | 11/7/0 | 11/7/0 | 15/1/2 |
5.4 Effectiveness of the “DE/rand/best/1” Strategy
Function | Method | Best solution
f
best
| Median solution
f
median
| Mean solution
μ
f
| Worst solution
f
worst
| Standard deviation
σ
f
|
---|---|---|---|---|---|---|
g01 | ATMDE | −15.000 | −15.000 | −15.000 | −15.0000 | 0 |
ATMDE1 | −14.999 9 | −14.999 9 | −14.999 9 | −14.999 9 | 8.98 × 10−7
| |
g02 | ATMDE | −0.803 617 | −0.803 617 | −0.803 617 | −0.803 610 | 1.24 × 10−6
|
ATMDE1 | −0.802 125 | −0.802 125 | −0.802 124 | −0.802 092 | 6.14 × 10−6
| |
g03 | ATMDE | −1.005 00 | −1.005 00 | −1.005 00 | −1.005 00 | 2.08 × 10−9
|
ATMDE1 | −1.005 00 | −1.005 00 | −0.985 8 | −0.798 4 | 4.93 × 10−2
| |
g04 | ATMDE | −30 665.53 | −30 665.53 | −30 665.53 | −30 665.5 | 1.11 × 10−11
|
ATMDE1 | −30 665.53 | −30 665.53 | −30 665.53 | −30 665.53 | 1.85 × 10−11
| |
g05 | ATMDE | 5126.496 71 | 5 126.496 71 | 5126.496 71 | 5 126.496 71 | 1.01 × 10−12
|
ATMDE1 | 5126.496 71 | 5 126.496 71 | 5126.496 71 | 5 126.496 71 | 2.95 × 10−9
| |
g06 | ATMDE | −6961.813 | −6961.813 | −6961.813 | −6961.81 | 1.85 × 10−12
|
ATMDE1 | −6961.813 | −6961.813 | −6961.813 | −6961.81 | 2.78 × 10−12
| |
g07 | ATMDE | 24.306 209 | 24.306 209 | 24.306 209 | 24.306 209 | 2.21 × 10−8
|
ATMDE1 | 24.3062 497 | 24.306 2497 | 24.306 253 | 24.306 364 | 2.09 × 10−5
| |
g08 | ATMDE | −0.095 825 | −0.095 825 | −0.095 825 | −0.095 825 | 2.56 × 10−17
|
ATMDE1 | −0.095 825 | −0.095 825 | −0.095 825 | −0.095 825 | 2.82 × 10−17
| |
g09 | ATMDE | 680.630 05 | 680.630 05 | 680.630 05 | 680.630 05 | 4.63 × 10−13
|
ATMDE1 | 680.630 05 | 680.630 05 | 680.630 05 | 680.630 05 | 4.85 × 10−13
| |
g10 | ATMDE | 7 049.248 02 | 7 049.248 02 | 7 049.248 02 | 7 049.248 02 | 8.70 × 10−7
|
ATMDE1 | 7 049.339 29 | 7 049.699 7 | 7 049.800 6 | 7 051.246 3 | 4.59 × 10−1
| |
g11 | ATMDE | 0.749 90 | 0.749 90 | 0.749 90 | 0.749 90 | 1.01 × 10−7
|
ATMDE1 | 0.749 90 | 0.749 90 | 0.749 90 | 0.749 90 | 1.12 × 10−16
| |
g12 | ATMDE | −1.000 00 | −1.000 00 | −1.000 00 | −1.000 00 | 0 |
ATMDE1 | −1.000 00 | −1.000 00 | −1.000 00 | −1.000 00 | 0 | |
g14 | ATMDE | −47.764 888 | −47.764 888 | −47.764 888 | −47.764 88 | 1.95 × 10−10
|
ATMDE1 | −47.764 888 | −47.764 888 | −47.764 888 | −47.764 88 | 1.67 × 10−8
| |
g15 | ATMDE | 961.715 022 | 961.715 022 | 961.715 022 | 961.715 022 | 6.94 × 10−13
|
ATMDE1 | 961.715 022 | 961.715 022 | 961.715 022 | 961.715 022 | 6.94E × 10−13
| |
g16 | ATMDE | −1.905 155 | −1.905 155 | −1.905 155 | −1.905 155 | 6.78 × 10−16
|
ATMDE1 | −1.905 102 | −1.905 102 | −1.905 102 | −1.905 102 | 6.78 × 10−16
| |
g18 | ATMDE | −0.866 025 | −0.866 025 | −0.866 025 | −0.866 025 | 7.45 × 10−10
|
ATMDE1 | −0.866 025 | −0.866 025 | −0.866 025 | −0.866 025 | 4.49 × 10−6
| |
g19 | ATMDE | 32.655 63 | 32.655 86 | 32.656 00 | 32.657 25 | 3.75 × 10−4
|
ATMDE1 | 32.676 38 | 32.702 88 | 32.704 75 | 32.774 96 | 2.16 × 10−2
| |
g24 | ATMDE | −5.508 013 | −5.508 013 | −5.508 013 | −5.508 013 | 3.74 × 10−15
|
ATMDE1 | −5.508 013 | −5.508 013 | −5.508 013 | −5.508 013 | 4.52 × 10−15
| |
w/t/l
| 6/12/0 | 6/12/0 | 7/11/0 | 7/11/0 | 14/3/1 |
5.5 Four Mechanical Benchmark Engineering Designs
Engineering benchmark | No. of variables
n
| Ration
ρ/ % | No. of constrains
N
|
---|---|---|---|
Weld-beam design | 4 | 37.625 | 5 |
Spring design | 3 | 0.732 3 | 4 |
Speed reducer design | 7 | 23.015 2 | 11 |
Three-bar truss design | 2 | 21.870 6 | 3 |
Engineering problems | Method | Best solution
f
best
| Mean solution
μ
f
| Worst solution
f
worst
| Standard deviation
σ
f
|
---|---|---|---|---|---|
Weld-beam design | ATMDE | 2.380 956 | 2.380 956 | 2.380 956 | 5.88 × 10−11
|
AATM | 2.382 326 | 2.386 976 | 2.391 592 | 2.20 × 10−3
| |
Spring design | ATMDE | 0.012 665 | 0.012 665 | 0.012 665 | 1.05 × 10−15
|
AATM | 0.012 668 | 0.012 708 | 0.012 861 37 | 4.50 × 10−5
| |
Speed reducer | ATMDE | 2994.473 6 | 2994.474 4 | 2994.474 45 | 1.18 × 10−5
|
AATM | 2994.516 7 | 2994.585 4 | 2994.659 79 | 3.30 × 10−2
| |
Three-bar truss design | ATMDE | 263.895 84 | 263.895 84 | 263.895 843 | 2.87 × 10−13
|
AATM | 263.895 84 | 263.896 6 | 263.900 41 | 1.10 × 10−3
|
Engineering problems | Method | Best design variable
x
best
| Best function values
f
best
|
---|---|---|---|
Weld-beam design | ATMDE | 0.244 368 975, 6.217 519 715, 8.291 471 390, 0.244 368 975 | 2.380 956 580 |
AATM | 0.244 106 586, 6.220 903 633, 8.298 161 229,0.244 382 231 | 2.382 326 | |
Spring design | ATMDE | 0.356 717 739, 0.051 689 061, 11.288 965 783 04 | 0.012 665 232 |
AATM | 0.359 690 411, 0.051 813 095, 11.119 252 680 | 0.012 668 261 | |
Speed reducer design | ATMDE | 3.50, 0.7, 17, 7.309 819 903, 7.715 173 384 44, 3.350 233 018 67, 5.286 521 228 48 | 2 994.473 624 |
AATM | 3.500 016 221, 0.700 001 177, 17.000 029 883, 7.300 297 290, 7.716 049 465, 3.350 239 798, 5.286 660 476 6 | 2 994.516 778 | |
Three-bar truss design | ATMDE | 0.788 675 135, 0.408 248 289 | 263.895 843 |
AATM | 0.788 681 755, 0.408 229 565 | 263.895 843 |