Parameters Optimization of Taguchi Method Integrated Hybrid Harmony Search Algorithm for Engineering Design Problems
Abstract
:1. Introduction
- The effect of variable run values on finding the optimum solution by employing different complex benchmark engineering design optimization problems and a real-size engineering design problem, frequently considered in optimization analyzes in the literature has been investigated;
- Taguchi method integrated hybrid harmony search algorithm (TIHHSA) has been generated based on the HSA and Taguchi method, namely the proposed hybridization can be defined as initial optimization for optimum algorithm parameter values of HSA;
- The effect of HSA parameters on the objective function and the optimum number of runs and maximum iterations with optimum HSA control parameters have been examined for different engineering optimization design problems utilizing TIHHSA;
- Whether the variation of the optimum values of the HSA parameters depending on the nature of the engineering design optimization problem has been evaluated;
- According to accomplished optimum results for engineering design optimization problems, the robustness, and the benefits of TIHHSA presented a new method have been interpreted and evaluated with previously reported studies in the literature.
2. Materials and Methods
2.1. Complex Benchmark Engineering Design Optimization Problems
2.1.1. Welded Beam Design Problem
2.1.2. Pressure Vessel Design Problem
2.1.3. Gear Train Design
2.1.4. Speed Reducer Design
2.2. Real-Size Engineering Design Optimization Problem
Input Parameters | Symbol | Value | Unit |
---|---|---|---|
Stem height | H | 4.5 | m |
Surcharge load | q | 30 | kPa |
Backfill slope | β | 0 | ° |
Internal friction angle of the retained soil and the base soil | Ør and Øb | 36 and 34 | ° |
Unit weight of retained soil, base soil, and concrete | γr, γb, and γc | 17.5, 18.5, 23.5 | kPa |
Cohesion of base and retained soils | cb and cr | 0 | kPa |
Depth of soil in front of the wall | Df | 0.75 | m |
Terzaghi bearing capacity factors for Øb = 34° [45] | Nc, Nq, Nγ | 52.64, 36.50, 38.04 | – |
The factor of safety for sliding and overturning stability | SFss and SFso | 1.50 | – |
The factor of safety for bearing capacity | SFsb | 3.00 | – |
Reinforcing steel yield strength | fy | 400 | MPa |
Concrete compressive strength | fc | 21 | MPa |
Concrete cover | cc | 0.07 | m |
Shrinkage and temperature reinforcement percentage | ρst | 0.002 | – |
Nominal strength coefficient for the flexural moment | ϕm | 0.90 | – |
Nominal strength coefficient for shear force | ϕ | 0.75 | – |
Reinforcement location factor (1.0 for concrete below < 30.48 cm) | ψt | 1.00 | – |
Coating factor (for uncoated bars) | ψe | 1.00 | – |
Lightweight aggregate concrete factor (1.0 for normal-weight conc.) | λ | 1.00 | – |
Cost of steel and concrete | Cs and Cc | 0.4 and 40 | $/kg and $/m3 |
2.3. Harmony Search Algorithm
- Step 1: HSA is initialized by determining the constant algorithm parameters (HMS, HMCR, PAR, and maximum iteration number) and generating design space with design variable values according to range limitation;
- Step 2: HM matrix is formed randomly by selecting from design space;
- Step 3: Improvisation of a new HM matrix conceiving memory consideration, random selection, and pitch adjustment mechanisms is carried out;
- Step 4: HM matrix is updated depending on whether a better solution is obtained, and then the worst solution is drawn from HM by replacing the better one;
- Step 5: Until the current iteration is reached the predefined maximum iteration number, Step 3 and Step 4 are repeated. If it is conducted HSA is ended.
2.4. Taguchi Method Background
2.5. A New Hybrid Method Based on Taguchi for Optimum Values of Algorithm Parameters
2.5.1. Forming Taguchi Design Matrix
2.5.2. Initializing HSA Process
2.5.3. Performing Taguchi Analyses
3. Design Experiments and Results
3.1. Optimization Analyses of Engineering Design Problems and Real-Size Engineering Design Optimization Problem
3.2. Taguchi Analyses
3.2.1. Part I: Investigation of Five Optimum Design Parameter Values with Effect on the Fitness Value
3.2.2. Part II: Investigation of Four Optimum Design Parameter Values with Effect on the Fitness Value
4. Discussion
5. Conclusions
- The obtained estimations have a reasonable relative error in determining optimum values of algorithm design parameters without performing many trials;
- It has been seen that the optimum values of the algorithm design parameters vary depending on the nature of the design optimization problem, which includes the number of design variables, the number of design constraints, exposure to the constraints.
- Instead of taking into account the value of the algorithm parameter proposed for characteristically different optimization problems in the literature, it has been concluded that using the optimum values yielded statistically according to the nature of the problem is an effective and prosperous manner in converging to the optimum.
- Instead of the trial-error method, which is time-consuming and exhaustive, it has been concluded that the newly proposed TIHHSA is a robust and reliable method for estimating the optimum algorithm parameter values of the harmony search metaheuristic optimization technique in a shorter time without conducting sensitivity analyses which are utilized to increase convergence rate in the solution of the design optimization problem.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
Optimum Solutions | x1(h)(in.) | x2(l) (in.) | x3(t) (in.) | x4(b) (in.) | f(x) ($) | |
---|---|---|---|---|---|---|
Literature | Ragsdell and Phillips [50] | 0.24550 | 6.1960 | 8.2730 | 0.2455 | 2.38593 |
Deb [51] | 0.2489 | 6.173 | 8.1789 | 0.2533 | 2.433116 | |
Coello [52] | 0.2088 | 3.4205 | 8.9975 | 0.21 | 1.7483 | |
Huang et al. [53] | 0.203137 | 3.542998 | 9.033498 | 0.206179 | 1.73346 | |
Yildiz [54] | 0.20573 | 3.47042 | 9.03649 | 0.205735 | 1.7248 | |
Çarbaş and Saka [55] | 0.203907 | 3.499898 | 9.063898 | 0.205594 | 1.72966 | |
Case PS | R30 | 0.206741 | 3.65285 | 8.54856 | 0.231265 | 1.85149 |
R100 | 0.171535 | 4.42418 | 8.98313 | 0.208289 | 1.80231 | |
R500 | 0.198864 | 3.66442 | 8.94678 | 0.209895 | 1.75598 | |
R1000 | 0.19823 | 3.64539 | 9.02857 | 0.206407 | 1.74026 | |
WBD-5P | 0.195872 | 3.70387 | 9.07235 | 0.205574 | 1.7455 | |
WBD-4P | 0.188171 | 3.95948 | 8.91723 | 0.21133 | 1.78312 |
Optimum Solutions | x1(Ts) (in.) | x2(Th) (in.) | x3(R) (in.) | x4(L) (in.) | f(x) ($) | |
---|---|---|---|---|---|---|
Literature | Sandgren [35] | 1.125 | 0.625 | 48.97 | 106.72 | 7982.5 |
Kannan and Kramer [56] | 1.25 | 0.625 | 50 | 120 | 7198.20 | |
Deb [57] | 0.9375 | 0.50 | 48.329 | 112.679 | 6410.381 | |
Coello [52] | 0.8125 | 0.4375 | 40.3239 | 200.0 | 6288.7445 | |
Gao et al. [58] | 0.75 | 0.375 | 38.8441 | 221.612 | 5852.639 | |
Çarbaş and Saka [55] | 0.8125 | 0.4375 | 42.09845 | 176.6366 | 6059.7143 | |
Case PS | R30 | 0.876366 | 0.434563 | 45.3293 | 140.344 | 6089.66 |
R100 | 0.915835 | 0.454142 | 47.2497 | 121.872 | 6195.1 | |
R500 | 0.833985 | 0.413952 | 43.1577 | 163.94 | 6000.09 | |
R1000 | 0.814181 | 0.403799 | 42.1533 | 176.032 | 5959.86 | |
PVD-5P | 0.84119 | 0.430252 | 43.5749 | 159.245 | 6054.14 | |
PVD-4P | 0.822121 | 0.409189 | 42.367 | 173.44 | 6005.19 |
Optimum Solutions | x1(Ta) (piece) | x2(Tb) (piece) | x3(Td) (piece) | x4 (Tf) (piece) | Gear ratio | f(x) (unitless) | |
---|---|---|---|---|---|---|---|
Literature | Zhang and Wang [59] | 43 | 16 | 19 | 49 | 0.1442 | 2.36 × 10−9 |
Deb and Goyal [60] | 33 | 14 | 17 | 50 | 0.1442 | 1.362 × 10−9 | |
Parsopoulos and Vrahatis [61] | 43 | 16 | 19 | 49 | 0.1442 | 2.701 × 10−12 | |
Gandomi [62] | 43 | 16 | 19 | 49 | 0.1442 | 2.701 × 10−12 | |
Arora et al. [63] | 43 | 16 | 19 | 49 | 0.1442 | 2.701 × 10−12 | |
Deniz [64] | 43 | 16 | 19 | 49 | 0.1442 | 2.701 × 10−12 | |
Case PS | R30 | 44 | 13 | 21 | 43 | 0.144292 | 1.54505 × 10−10 |
R100 | 43 | 16 | 19 | 49 | 0.144281 | 2.70086 × 10−12 | |
R500 | 49 | 16 | 19 | 43 | 0.144281 | 2.70086 × 10−12 | |
R1000 | 49 | 16 | 19 | 43 | 0.144281 | 2.70086 × 10−12 | |
GTD-5P | 49 | 16 | 19 | 43 | 0.144281 | 2.70086 × 10−12 | |
GTD-4P | 49 | 16 | 19 | 43 | 0.144281 | 2.70086 × 10−12 |
Optimum Solutions | x1 (cm) | x2 (cm) | x3 (piece) | x4 (cm) | x5 (cm) | x6 (cm) | x7 (cm) | f(x) (kg) | |
---|---|---|---|---|---|---|---|---|---|
Literature | Li and Papalambros [65] | 3.50 | 0.70 | 17.00 | 7.30 | 7.71 | 3.3500000 | 5.2900000 | 2996.30977 |
Kuang et al. [66] | 3.60 | 0.70 | 17.00 | 7.30 | 7.80 | 3.4000000 | 5.0000000 | 2876.22 | |
Azarm and Li [67] | 3.50 | 0.70 | 17.00 | 7.30 | 7.71 | 3.3500000 | 5.2900000 | 2996.30978 | |
Vanderplaats [68] | 3.50 | 0.70 | 17.00 | 7.30 | 7.30 | 3.3502145 | 5.2865176 | 2985.15188 | |
Ray [69] | 3.50 | 0.70 | 17.00 | 7.30 | 7.30 | 3.3502145 | 5.2865176 | 2985.15188 | |
Carbas et al. [70] | 3.50 | 0.70 | 17.00 | 7.17984 | 7.70889 | 3.35009 | 5.28668 | 2993.13917 | |
Case PS | R30 | 3.5001 | 0.700016 | 17.0017 | 7.30052 | 7.71562 | 3.35025 | 5.28667 | 2994.93 |
R100 | 3.50014 | 0.700021 | 17.0002 | 7.30117 | 7.71637 | 3.35053 | 5.28667 | 2994.79 | |
R500 | 3.50029 | 0.700019 | 17.0001 | 7.3009 | 7.71572 | 3.35053 | 5.28681 | 2994.90 | |
R1000 | 3.50025 | 0.700016 | 17.0004 | 7.30034 | 7.71652 | 3.35036 | 5.28673 | 2994.84 | |
SRD-5P | 3.50184 | 0.700073 | 17.0008 | 7.3036 | 7.72133 | 3.35036 | 5.28679 | 2995.97 | |
SRD-4P | 3.50006 | 0.700006 | 17.0034 | 7.30108 | 7.71868 | 3.3516 | 5.28677 | 2995.63 |
R30 | R100 | R500 | R1000 | 5P | 4P | R30 | R100 | R500 | R1000 | 5P | 4P | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
WBD | g1(x) | −8.821 | −0.086 | −7.948 | −7.525 | −65.745 | −32.706 | SRD | g1(x) | 0.92592 | 0.92598 | 0.92595 | 0.92595 | 0.92536 | 0.92587 |
g2(x) | −178.142 | −14.670 | −1.819 | −45.079 | −213.275 | −7.722 | g2(x) | 0.80178 | 0.80190 | 0.80188 | 0.80187 | 0.80134 | 0.80165 | ||
g3(x) | −0.025 | −0.037 | −0.011 | −0.008 | −0.010 | −0.023 | g3(x) | 0.50085 | 0.50086 | 0.50081 | 0.50079 | 0.50141 | 0.50012 | ||
g4(x) | −3.317 | −3.338 | −3.400 | −3.414 | −3.407 | −3.368 | g4(x) | 0.09535 | 0.09539 | 0.09536 | 0.09539 | 0.09555 | 0.09545 | ||
g5(x) | −0.082 | −0.047 | −0.074 | −0.073 | −0.071 | −0.063 | g5(x) | 0.99994 | 0.99944 | 0.99944 | 0.99974 | 0.99975 | 0.99752 | ||
g6(x) | −0.235 | −0.235 | −0.235 | −0.236 | −0.236 | −0.235 | g6(x) | 0.99998 | 0.99998 | 0.99982 | 0.99991 | 0.99985 | 0.99987 | ||
g7(x) | −2211.802 | −202.416 | −330.005 | −55.908 | −1.921 | −446.574 | g7(x) | 0.29754 | 0.29751 | 0.29751 | 0.29751 | 0.29755 | 0.29756 | ||
PVD | g1(x) | −0.002 | −0.002 | −0.004 | −0.001 | −0.001 | 0.000 | g8(x) | 0.99999 | 0.99999 | 0.99994 | 0.99995 | 0.99958 | 0.99999 | |
g2(x) | −0.002 | −0.002 | −0.003 | −0.002 | −0.002 | −0.015 | g9(x) | 0.41667 | 0.41667 | 0.41669 | 0.41669 | 0.41684 | 0.41667 | ||
g3(x) | −89.466 | −89.466 | −636.323 | −9.038 | −412.835 | −498.618 | g10(x) | 0.94861 | 0.94859 | 0.94862 | 0.94866 | 0.94824 | 0.94882 | ||
g4(x) | −99.656 | −99.656 | −118.128 | −76.060 | −63.968 | −80.755 | g11(x) | 0.99996 | 0.99987 | 0.99997 | 0.99986 | 0.99924 | 0.99958 |
Optimum Solutions | x1 (m) | x2 (m) | x3 (m) | x4 (m) | x5 (m) | x6 (m) | x7 (m) | x8 (cm2) | x9 (cm2) | x10 (cm2) | x11 (cm2) | x12 (cm2) | f(x) ($/kg) | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Literature | Gandomi [40] | 2.709 | 1 | 0.412 | 0.25 | 0.4 | 2.455 | 0.2 | 0.2 | 21.9911 | 11.7809 | 11.7809 | 4.7124 | 163.98 |
Gandomi [40] | 2.727 | 1.035 | 0.36 | 0.28 | 0.401 | 2.274 | 0.293 | 0.296 | 32.1699 | 13.3517 | 13.8544 | 8.6394 | 182.79 | |
Gandomi [40] | 2.816 | 0.988 | 0.447 | 0.294 | 0.422 | 2.223 | 0.367 | 0.203 | 21.9911 | 15.2681 | 15.2681 | 12.7234 | 185.05 | |
Gandomi [40] | 2.694 | 0.836 | 0.403 | 0.27 | 0.405 | 2.346 | 0.227 | 0.445 | 23.7504 | 12.7234 | 22.6195 | 26.1380 | 182.84 | |
CasePS | R30 | 3.3439 | 1.1598 | 0.3526 | 0.2504 | 0.4 | 2.5435 | 0.2002 | 0.2 | 21.2999 | 14.3212 | 18.8016 | 7.1532 | 180.082 |
R100 | 3.3431 | 1.1596 | 0.3917 | 0.25 | 0.4001 | 2.4623 | 0.2001 | 0.2002 | 18.8963 | 14.2586 | 18.9194 | 9.5172 | 179.842 | |
R500 | 3.3351 | 1.1593 | 0.4418 | 0.25 | 0.4001 | 3.0523 | 0.2004 | 0.2002 | 16.6776 | 14.3189 | 16.557 | 7.2319 | 179.449 | |
R1000 | 3.3394 | 1.1593 | 0.3916 | 0.2501 | 0.4 | 2.7275 | 0.2004 | 0.2003 | 18.691 | 14.392 | 18.7638 | 7.2583 | 179.693 | |
RCRW1−5P | 3.36761 | 1.15955 | 0.391877 | 0.250366 | 0.400094 | 2.84595 | 0.200248 | 0.201545 | 18.7689 | 14.4128 | 18.7273 | 10.0995 | 181.035 | |
RCRW1−4P | 3.34757 | 1.15992 | 0.394711 | 0.250081 | 0.400157 | 2.37294 | 0.202173 | 0.201494 | 18.7718 | 14.1765 | 18.7523 | 9.34978 | 180.301 |
Optimum Solutions | x1 (m) | x2 (m) | x3 (m) | x4 (m) | x5 (m) | x6 (m) | x7 (m) | x8 (cm2) | x9 (cm2) | x10 (cm2) | x11 (cm2) | x12 (cm2) | f(x) ($/kg) | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Literature | Gandomi [40] | 2.709 | 1 | 0.412 | 0.25 | 0.4 | 2.455 | 0.2 | 0.2 | 21.9911 | 11.7809 | 11.7809 | 4.7124 | 5668.5 |
Gandomi [40] | 2.727 | 1.035 | 0.36 | 0.28 | 0.401 | 2.274 | 0.293 | 0.296 | 32.1699 | 13.3517 | 13.8544 | 8.6394 | 6034.4 | |
Gandomi [40] | 2.816 | 0.988 | 0.447 | 0.294 | 0.422 | 2.223 | 0.367 | 0.203 | 21.9911 | 15.2681 | 15.2681 | 12.7234 | 6095.9 | |
Gandomi [40] | 2.694 | 0.836 | 0.403 | 0.27 | 0.405 | 2.346 | 0.227 | 0.445 | 23.7504 | 12.7234 | 22.6195 | 26.1380 | 6094.4 | |
CasePS | R30 | 3.342 | 1.1599 | 0.2505 | 0.2504 | 0.4 | 3.1292 | 0.2001 | 0.2001 | 35.1479 | 14.1715 | 21.5672 | 7.0299 | 5886.67 |
R100 | 3.3426 | 1.16 | 0.2501 | 0.2501 | 0.4 | 3.0709 | 0.2001 | 0.2002 | 35.1127 | 15.0607 | 21.2321 | 7.0677 | 5883.61 | |
R500 | 3.343 | 1.1581 | 0.25 | 0.25 | 0.4001 | 3.0355 | 0.2001 | 0.2002 | 35.247 | 14.4184 | 21.3267 | 7.3429 | 5883.64 | |
R1000 | 3.3434 | 1.1594 | 0.2501 | 0.25 | 0.4 | 3.1304 | 0.2001 | 0.2001 | 35.2741 | 14.1095 | 21.2558 | 9.3741 | 5884.09 | |
RCRW1−5P | 2.709 | 1 | 0.412 | 0.25 | 0.4 | 2.455 | 0.2 | 0.2 | 21.9911 | 11.7809 | 11.7809 | 4.7124 | 5668.5 | |
RCRW1−4P | 2.727 | 1.035 | 0.36 | 0.28 | 0.401 | 2.274 | 0.293 | 0.296 | 32.1699 | 13.3517 | 13.8544 | 8.6394 | 6034.4 |
Optimum Cost (RCRW1) | Optimum Weight (RCRW2) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
R30 | R100 | R500 | R1000 | 5P | 4P | R30 | R100 | R500 | R1000 | |
g1(x) | −0.0462 | −0.043 | −0.037 | −0.0421 | −0.0497 | −0.0441 | −0.0536 | −0.0537 | −0.0543 | −0.0541 |
g2(x) | −0.638 | −0.6334 | −0.6224 | −0.6305 | −0.6613 | −0.6376 | −0.6468 | −0.6472 | −0.648 | −0.6484 |
g3(x) | −2.5539 | −2.569 | −2.5824 | −2.5638 | −2.6425 | −2.5813 | −2.5106 | −2.5112 | −2.5092 | −2.5137 |
g4(x) | −0.1129 | −0.1141 | −0.0221 | −0.0199 | −1.8384 | −0.3677 | −0.0072 | −0.0142 | −0.0149 | −0.0925 |
g5(x) | −0.0001 | −0.0004 | 0.0000 | −0.0002 | −0.0009 | −0.0085 | −0.0633 | −0.0615 | −0.061 | −0.0615 |
g6(x) | −0.4023 | −0.4058 | −0.4103 | −0.4057 | −0.4115 | −0.4066 | −0.3937 | −0.4242 | −0.3951 | −0.3944 |
g7(x) | −0.0326 | −0.0701 | −0.0052 | −0.0728 | −0.0531 | −0.07 | −0.062 | −0.0482 | −0.0464 | −0.0472 |
g8(x) | −0.9964 | −0.9973 | −0.9964 | −0.9964 | −0.9974 | −0.9973 | −0.9964 | −0.9964 | −0.9964 | −0.9973 |
g9(x) | −0.3955 | −0.4637 | −0.5333 | −0.4635 | −0.464 | −0.4684 | −0.1169 | −0.1154 | −0.115 | −0.1154 |
g10(x) | −0.4032 | −0.4065 | −0.4107 | −0.4064 | −0.4096 | −0.407 | −0.3956 | −0.3955 | −0.3964 | −0.3959 |
g11(x) | −0.065 | −0.0845 | −0.1132 | −0.0858 | −0.0767 | −0.0847 | −0.0179 | −0.0175 | −0.0167 | −0.017 |
g12(x) | −0.9936 | −0.9935 | −0.9936 | −0.9935 | −0.9934 | −0.9936 | −0.9935 | −0.9935 | −0.9935 | −0.9935 |
g13(x) | −0.4189 | −0.2727 | −0.0625 | −0.2729 | −0.2724 | −0.2671 | −0.7519 | −0.7523 | −0.7524 | −0.7523 |
g14(x) | −0.0097 | −0.0095 | −0.0095 | −0.0097 | −0.0095 | −0.0093 | −0.0097 | −0.0618 | −0.0095 | −0.0097 |
g15(x) | −0.2573 | −0.2571 | −0.151 | −0.2573 | −0.2571 | −0.257 | −0.3504 | −0.3408 | −0.3406 | −0.3408 |
g16(x) | −0.0087 | −0.2569 | −0.0077 | −0.0077 | −0.3028 | −0.2492 | −0.0092 | −0.0092 | −0.0092 | −0.2569 |
g17(x) | −0.6471 | −0.7181 | −0.7813 | −0.718 | −0.7182 | −0.7202 | −0.1734 | −0.1721 | −0.1718 | −0.1721 |
g18(x) | −0.7929 | −0.793 | −0.793 | −0.7929 | −0.793 | −0.793 | −0.7929 | −0.7814 | −0.793 | −0.7929 |
g19(x) | −0.7239 | −0.724 | −0.7585 | −0.7239 | −0.724 | −0.724 | −0.6844 | −0.689 | −0.689 | −0.689 |
g20(x) | −0.7931 | −0.7241 | −0.7934 | −0.7934 | −0.7059 | −0.7269 | −0.793 | −0.793 | −0.793 | −0.7241 |
g21(x) | −0.5477 | −0.536 | −0.5199 | −0.5356 | −0.5393 | −0.5356 | −0.578 | −0.5781 | −0.5788 | −0.5784 |
g22(x) | −0.1795 | −0.2036 | −0.0247 | −0.1232 | −0.0954 | −0.2308 | −0.0038 | −0.0214 | −0.0321 | −0.0039 |
g23(x) | −0.2382 | −0.3654 | −0.3654 | −0.3652 | −0.3654 | −0.3655 | −0.3652 | −0.3652 | −0.3654 | −0.3652 |
g24(x) | −0.6364 | −0.6365 | −0.6365 | −0.6364 | −0.6365 | −0.6365 | −0.6364 | −0.4909 | −0.6365 | −0.6364 |
g25(x) | −0.6364 | −0.6365 | −0.6365 | −0.6364 | −0.6365 | −0.6365 | −0.4182 | −0.5636 | −0.5638 | −0.5636 |
g26(x) | −0.1113 | −0.3654 | −0.1115 | −0.1113 | −0.1115 | −0.3655 | −0.1113 | −0.1113 | −0.1115 | −0.3652 |
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Design Variables | Lower Bound | Upper Bound | |
---|---|---|---|
x1 (m) | 1.96 | 5.50 | |
x2 (m) | 0.65 | 1.16 | |
x3 (m) | 0.25 | 0.50 | |
x4 (m) | 0.25 | 0.50 | |
x5 (m) | 0.40 | 0.50 | |
x6 (m) | 1.96 | 5.50 | |
x7 (m) | 0.20 | 0.50 | |
x8 (m) | 0.20 | 0.50 | |
x9, x10, x11, x12 | n (piece) | 3 | 30 |
db (mm) | 10 | 30 | |
As (cm2) | 2.356 | 212.0575 |
Ld Ld(a)k | L4 | L4 | L8 | L8 | L9 | L9 | L9 | L18 | L16 | L16 | L16 | L16 | L25 | L25 | L25 | L25 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
d | 4 | 4 | 8 | 8 | 9 | 9 | 9 | 18 | 16 | 16 | 16 | 16 | 25 | 25 | 25 | 25 |
k | 2 | 2 | 4 | 5 | 2 | 3 | 4 | 5 | 2 | 3 | 4 | 5 | 2 | 3 | 4 | 5 |
a | 2 | 3 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 |
Design No | Design Parameters with Levels | DM | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
P1 | P2 | P3 | P4 | P5 | HMS | HMCR | PAR | MAXITER | RUN | |
1 | 1 | 1 | 1 | 1 | 1 | 20 | 0.80 | 0.10 | 2000 | 30 |
2 | 1 | 2 | 2 | 2 | 2 | 20 | 0.85 | 0.20 | 4000 | 100 |
3 | 1 | 3 | 3 | 3 | 3 | 20 | 0.90 | 0.30 | 6000 | 500 |
4 | 1 | 4 | 4 | 4 | 4 | 20 | 0.95 | 0.40 | 8000 | 1000 |
5 | 2 | 1 | 2 | 3 | 4 | 30 | 0.80 | 0.20 | 6000 | 1000 |
6 | 2 | 2 | 1 | 4 | 3 | 30 | 0.85 | 0.10 | 8000 | 500 |
7 | 2 | 3 | 4 | 1 | 2 | 30 | 0.90 | 0.40 | 2000 | 100 |
8 | 2 | 4 | 3 | 2 | 1 | 30 | 0.95 | 0.30 | 4000 | 30 |
9 | 3 | 1 | 3 | 4 | 2 | 40 | 0.80 | 0.30 | 8000 | 100 |
10 | 3 | 2 | 4 | 3 | 1 | 40 | 0.85 | 0.40 | 6000 | 30 |
11 | 3 | 3 | 1 | 2 | 4 | 40 | 0.90 | 0.10 | 4000 | 1000 |
12 | 3 | 4 | 2 | 1 | 3 | 40 | 0.95 | 0.20 | 2000 | 500 |
13 | 4 | 1 | 4 | 2 | 3 | 50 | 0.80 | 0.40 | 4000 | 500 |
14 | 4 | 2 | 3 | 1 | 4 | 50 | 0.85 | 0.30 | 2000 | 1000 |
15 | 4 | 3 | 2 | 4 | 1 | 50 | 0.90 | 0.20 | 8000 | 30 |
16 | 4 | 4 | 1 | 3 | 2 | 50 | 0.95 | 0.10 | 6000 | 100 |
Case | Run | Iteration | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
BRN | Best | Mean | Worst | StD | Median | BIN | Best | Mean | Worst | StD | Median | ||
WBD ($) | R30 | 14(47%) | 1.85149 | 2.62202 | 3.67302 | 0.463237 | 2.57699 | 28,383(95%) | 1.85149 | 2.10940 | 7.12472 | 0.295862 | 1.97920 |
R100 | 33(33%) | 1.80231 | 2.71590 | 4.23073 | 0.467682 | 2.51094 | 22,548(75%) | 1.80231 | 1.95674 | 7.35794 | 0.309656 | 1.82127 | |
R500 | 34(7%) | 1.75598 | 2.70736 | 4.48978 | 0.500434 | 2.61591 | 21,170(71%) | 1.75598 | 2.16143 | 4.28967 | 0.312273 | 2.32239 | |
R1000 | 276(28%) | 1.74026 | 2.69101 | 4.89958 | 0.492996 | 2.61779 | 28,817(96%) | 1.74026 | 1.76394 | 2.36984 | 0.104493 | 1.74136 | |
PVD($) | R30 | 21(70%) | 6089.66 | 6815.66 | 7428.54 | 404.478 | 6856.34 | 16,355(55%) | 6089.66 | 6342.58 | 43,582.4 | 2212.31 | 6094.59 |
R100 | 13(13%) | 6195.10 | 6970.5 | 7502.81 | 367.191 | 7038.74 | 24,997(83%) | 6195.10 | 6436.25 | 44,557.6 | 1508.82 | 6228.33 | |
R500 | 216(43%) | 6000.09 | 6865.0 | 7497.80 | 415.43 | 6898.75 | 23,513(78%) | 6000.09 | 6265.77 | 55,383.2 | 1979.37 | 6051.16 | |
R1000 | 795(80%) | 5959.86 | 6887.8 | 7557.39 | 400.214 | 6919.51 | 24,449(82%) | 5959.86 | 6417.04 | 25,980.4 | 1918.94 | 6065.6 | |
GTD (unitless) | R30 | 7 (23%) | 1.54505 × 10−10 | 4.7459 × 10−8 | 5.3303 × 10−7 | 1.0461 × 10−7 | 1.6106 × 10−8 | 512 (2%) | 1.54505 × 10−10 | 9.0999 × 10−8 | 4.7841 × 10−5 | 2.0475 × 10−6 | 1.5451 × 10−10 |
R100 | 77 (77%) | 2.70086 × 10−12 | 5.8049 × 10−8 | 7.7986 × 10−7 | 1.2036 × 10−7 | 1.3531 × 10−8 | 12,552 (42%) | 2.70086 × 10−12 | 8.2076 × 10−7 | 6.1311 × 10−3 | 7.0792 × 10−5 | 2.7009 × 10−12 | |
R500 | 34 (7%) | 2.70086 × 10−12 | 4.9250 × 10−8 | 1.3811 × 10−6 | 1.0699 × 10−7 | 1.8274 × 10−8 | 210 (1%) | 2.70086 × 10−12 | 2.4972 × 10−5 | 1.0236 × 10−2 | 4.8766 × 10−4 | 2.7009 × 10−12 | |
R1000 | 198 (20%) | 2.70086 × 10−12 | 4.5731 × 10−8 | 1.0883 × 10−6 | 9.0341 × 10−8 | 1.8274 × 10−8 | 10 (0%) | 2.70086 × 10−12 | 2.7009 × 10−12 | 2.7009 × 10−2 | 7.5126 × 10−26 | 2.7009 × 10−12 | |
SRD (kg) | R30 | 5(17%) | 2994.93 | 2999.97 | 3090.64 | 17.1989 | 2996.53 | 22,048(73%) | 2994.93 | 3008.39 | 4933.98 | 66.3806 | 2996.80 |
R100 | 59(59%) | 2994.79 | 2996.69 | 3001.84 | 1.21818 | 2996.37 | 23,877(80%) | 2994.79 | 3002.48 | 3260.32 | 28.4670 | 2997.22 | |
R500 | 244(49%) | 2994.90 | 2997.30 | 3098.08 | 6.27332 | 2996.60 | 25,612(85%) | 2994.90 | 3010.85 | 5137.55 | 91.5543 | 2995.26 | |
R1000 | 112(11%) | 2994.84 | 2997.23 | 3092.00 | 4.29835 | 2996.55 | 20,879(70%) | 2994.84 | 3006.64 | 5430.07 | 88.7313 | 3000.99 |
Case | Run | Iteration | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
BRN | Best | Mean | Worst | StD | Median | BIN | Best | Mean | Worst | StD | Median | ||
RCRW1 ($/m) | R30 | 11(37%) | 180.082 | 185.85 | 194.525 | 4.52073 | 185.329 | 29,447(98%) | 180.082 | 186.376 | 627.563 | 21.5973 | 181.305 |
R100 | 76(76%) | 179.842 | 186.153 | 198.7 | 4.39706 | 186.333 | 29,975(99%) | 179.842 | 186.275 | 468.5 | 21.6613 | 181.156 | |
R500 | 359(72%) | 179.449 | 186.049 | 200.756 | 4.72074 | 185.306 | 29,405(98%) | 179.449 | 184.779 | 480.697 | 21.1682 | 180.267 | |
R1000 | 379(38%) | 179.693 | 186.064 | 200.572 | 4.6882 | 185.019 | 23,496(78%) | 179.693 | 184.553 | 462.683 | 19.8148 | 179.699 | |
RCRW2 (kg/m) | R30 | 4(13%) | 5886.67 | 5964.16 | 6411.61 | 128.141 | 5898.14 | 26,196(87%) | 5886.67 | 5987.62 | 9578.21 | 348.742 | 5894.12 |
R100 | 38(38%) | 5883.61 | 5962.77 | 6302.54 | 108.939 | 5910.18 | 25,444(85%) | 5883.61 | 5964.5 | 11,125.5 | 358.102 | 5884.06 | |
R500 | 319(64%) | 5883.64 | 5958.45 | 6764.28 | 126.388 | 5903.05 | 22,311(74%) | 5883.64 | 6007 | 9570.89 | 401.738 | 5892.82 | |
R1000 | 735(74%) | 5884.09 | 5955.51 | 6966.85 | 115.767 | 5901.38 | 28,442(96%) | 5884.09 | 6005.54 | 9984.8 | 449.616 | 5884.82 |
Design No | f(x) | S/N | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
WBD ($) | PVD ($) | GTD (Unitless) | SRD (kg) | RCRW1 ($/kg) | WBD | PVD | GTD | SRD | RCRW1 | |
1 | 2.1880 | 6595.36 | 9.94 × 10−11 | 3002.24 | 196.841 | −6.8010 | −76.38 | 200.052 | −69.5489 | −45.882 |
2 | 1.8766 | 6315.66 | 2.70 × 10−12 | 2996.59 | 183.264 | −5.4672 | −76.01 | 231.37 | −69.5325 | −45.262 |
3 | 1.8656 | 6040.75 | 2.70 × 10−12 | 2996.09 | 181.325 | −5.4165 | −75.62 | 231.37 | −69.5311 | −45.169 |
4 | 1.8073 | 5985.52 | 2.70 × 10−12 | 2996.03 | 180.652 | −5.1405 | −75.54 | 231.37 | −69.5309 | −45.137 |
5 | 1.8383 | 6039.20 | 2.70 × 10−12 | 2995.6 | 183.881 | −5.2881 | −75.62 | 231.37 | −69.5297 | −45.291 |
6 | 1.8528 | 6014.33 | 2.70 × 10−12 | 2995.81 | 181.395 | −5.3564 | −75.58 | 231.37 | −69.5303 | −45.173 |
7 | 2.1053 | 6116.35 | 2.31 × 10−11 | 3002.82 | 189.297 | −6.4661 | −75.73 | 212.736 | −69.5506 | −45.543 |
8 | 2.5046 | 6389.99 | 2.70 × 10−12 | 3001.70 | 187.417 | −7.9747 | −76.11 | 231.37 | −69.5474 | −45.456 |
9 | 2.1041 | 6257.68 | 2.70 × 10−12 | 2996.93 | 185.566 | −6.4615 | −75.93 | 231.37 | −69.5335 | −45.370 |
10 | 2.0103 | 6385.19 | 9.94 × 10−11 | 2998.29 | 186.013 | −6.0654 | −76.10 | 200.052 | −69.5375 | −45.391 |
11 | 1.7921 | 6105.73 | 2.70 × 10−12 | 2997.21 | 183.491 | −5.0673 | −75.71 | 231.37 | −69.5343 | −45.272 |
12 | 2.0951 | 6286.05 | 2.70 × 10−12 | 3000.60 | 188.117 | −6.4240 | −75.97 | 231.37 | −69.5442 | −45.489 |
13 | 1.9126 | 6136.63 | 2.70 × 10−12 | 2998.17 | 190.651 | −5.6324 | −75.76 | 231.37 | −69.5371 | −45.605 |
14 | 2.0021 | 6169.29 | 2.70 × 10−12 | 3002.63 | 195.777 | −6.0298 | −75.80 | 231.37 | −69.550 | −45.835 |
15 | 1.9833 | 6186.53 | 2.70 × 10−12 | 2998.66 | 183.548 | −5.9477 | −75.83 | 231.37 | −69.5386 | −45.275 |
16 | 2.1366 | 6147.29 | 2.70 × 10−12 | 2997.24 | 183.776 | −6.5943 | −75.77 | 231.37 | −69.5344 | −45.286 |
η | 2.0047 | 6198.22 | 1.61 × 10−11 | 2998.50 | 186.313 | −6.0083 | −75.84 | 226.29 | −69.5382 | −45.402 |
Optimization Problem | Evaluation Criteria | Design Parameter | ||||
---|---|---|---|---|---|---|
HMS | HMCR | PAR | MAXITER | RUN | ||
WBD | SS | 0.6492 | 1.7416 | 1.2039 | 1.1445 | 4.0607 |
ν | 0.216309 | 0.580439 | 0.401491 | 0.381525 | 1.35359 | |
R | 5 | 2 | 3 | 4 | 1 | |
PVD | SS | 0.0754 | 0.0865 | 0.0188 | 0.1542 | 0.4473 |
ν | 0.02515 | 0.028833 | 0.006269 | 0.051393 | 0.149109 | |
R | 4 | 3 | 5 | 2 | 1 | |
GTD | SS | 164.432 | 164.432 | 456.416 | 456.416 | 654.917 |
ν | 54.8032 | 54.8032 | 152.065 | 152.065 | 218.268 | |
R | 4 | 5 | 2 | 3 | 1 | |
SRD | SS | 4.44 × 10−5 | 9.58 × 10−6 | 4.53 × 10−5 | 6.15 × 10−4 | 1.36 × 10−4 |
ν | 1.48 × 10−5 | 3.21 × 10−6 | 1.51 × 10−5 | 2.05 × 10−4 | 4.55 × 10−5 | |
R | 4 | 5 | 3 | 1 | 2 | |
RCRW1 | SS | 0.052 | 0.1184 | 0.0349 | 0.4879 | 0.0535 |
ν | 0.0173 | 0.0395 | 0.0116 | 0.1626 | 0.0179 | |
R | 4 | 2 | 5 | 1 | 3 |
Optimization Problem | Optimum Parameter Combination | fmin (ηprediction) |
---|---|---|
WBD | HMS1-HMCR3-PAR2-MAXITER4-RUN4 | $1.63817 |
PVD | HMS2-HMCR3-PAR4-MAXITER4-RUN4 | $5813.73 |
GTD | HMS4-HMCR4-PAR2-MAXITER2-RUN3 | 2.60398 × 10−12 |
SRD | HMS1-HMCR1-PAR2-MAXITER3-RUN3 | 2994.16 kg |
RCRW1 | HMS1-HMCR3-PAR2-MAXITER4-RUN3 | $177.724/m |
Case | Run | Iteration | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
BRN | Best (ηreal) | Mean | Worst | StD | Median | BIN | Best (ηreal) | Mean | Worst | StD | Median | |
WBD | 202/1000(20%) | 1.7455 | 2.8743 | 5.10816 | 0.54932 | 2.8190 | 2598/8000 (32%) | 1.7455 | 1.82 | 9.44763 | 0.418665 | 1.7455 |
PVD | 897/1000(89%) | 6054.14 | 7000.4 | 8272.08 | 427.637 | 7032.49 | 7828/8000 (98%) | 6054.14 | 7153.31 | 42,950.5 | 3947.25 | 6180.14 |
GTD | 2/500 (0.4%) | 2.70086 × 10−12 | 5.4247 × 10−8 | 1.38114 × 10−6 | 1.34484 × 10−7 | 1.31252 × 10−8 | 312/4000 (8%) | 2.70086 × 10−12 | 2.99998 × 10−5 | 5.5068 × 10−3 | 3.7964 × 10−3 | 2.70086 × 10−12 |
SRD | 399/500 (80%) | 2995.97 | 3004.43 | 3022.88 | 4.62559 | 3003.84 | 5746/6000 (96%) | 2995.97 | 3062.32 | 5322.68 | 296.145 | 2996.6 |
RCRW1 | 425/500 (85%) | 181.035 | 191.431 | 206.659 | 5.36727 | 191.363 | 7740/8000 (97%) | 181.035 | 199.648 | 386.834 | 38.1441 | 189.368 |
DM | f(x) | S/N | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Design No | HMS | HMCR | PAR | MAXITER | RUN | |||||||||||
WBD PVD | GTD SRDR CRW1 | WBD ($) | PVD ($) | GTD (Unitless) | SRD (kg) | RCRW1 ($/kg) | WBD | PVD | GTD | SRD | RCRW1 | |||||
1 | 20 | 0.80 | 0.10 | 2000 | 1000 | 500 | 1.8162 | 6105.21 | 2.70 × 10−12 | 3000.31 | 190.785 | −5.1831 | −75.71 | 231.37 | −69.54 | −45.611 |
2 | 20 | 0.85 | 0.20 | 4000 | 1000 | 500 | 1.7882 | 6029.48 | 2.70 × 10−12 | 2996.04 | 183.739 | −5.0485 | −75.61 | 231.37 | −69.53 | −45.284 |
3 | 20 | 0.90 | 0.30 | 6000 | 1000 | 500 | 1.8059 | 5978.28 | 2.70 × 10−12 | 2996.04 | 182.141 | −5.1340 | −75.53 | 231.37 | −69.53 | −45.208 |
4 | 20 | 0.95 | 0.40 | 8000 | 1000 | 500 | 1.8331 | 6033.37 | 2.70 × 10−12 | 2996.22 | 180.155 | −5.2637 | −75.61 | 231.37 | −69.53 | −45.113 |
5 | 30 | 0.80 | 0.20 | 6000 | 1000 | 500 | 1.7931 | 6055.08 | 2.70 × 10−12 | 2996.34 | 182.660 | −5.0721 | −75.64 | 231.37 | −69.53 | −45.233 |
6 | 30 | 0.85 | 0.10 | 8000 | 1000 | 500 | 1.7869 | 5999.42 | 2.70 × 10−12 | 2996.01 | 181.975 | −5.0420 | −75.56 | 231.37 | −69.53 | −45.200 |
7 | 30 | 0.90 | 0.40 | 2000 | 1000 | 500 | 1.8419 | 6127.22 | 2.70 × 10−12 | 3000.97 | 189.144 | −5.3055 | −75.75 | 231.37 | −69.55 | −45.536 |
8 | 30 | 0.95 | 0.30 | 4000 | 1000 | 500 | 1.7892 | 6052.60 | 2.70 × 10−12 | 2997.24 | 183.061 | −5.0532 | −75.64 | 231.37 | −69.53 | −45.252 |
9 | 40 | 0.80 | 0.30 | 8000 | 1000 | 500 | 1.9165 | 6003.01 | 2.70 × 10−12 | 2996.23 | 184.049 | −5.6501 | −75.57 | 231.37 | −69.53 | −45.299 |
10 | 40 | 0.85 | 0.40 | 6000 | 1000 | 500 | 1.8280 | 6044.31 | 2.70 × 10−12 | 2996.86 | 182.456 | −5.2397 | −75.63 | 231.37 | −69.53 | −45.223 |
11 | 40 | 0.90 | 0.10 | 4000 | 1000 | 500 | 1.8945 | 6098.92 | 2.70 × 10−12 | 2997.09 | 182.774 | −5.5498 | −75.71 | 231.37 | −69.53 | −45.238 |
12 | 40 | 0.95 | 0.20 | 2000 | 1000 | 500 | 2.1363 | 6034.29 | 2.70 × 10−12 | 3002.13 | 189.270 | −6.5932 | −75.61 | 231.37 | −69.55 | −45.542 |
13 | 50 | 0.80 | 0.40 | 4000 | 1000 | 500 | 1.9349 | 6087.48 | 2.70 × 10−12 | 2997.00 | 190.329 | −5.7333 | −75.69 | 231.37 | −69.53 | −45.590 |
14 | 50 | 0.85 | 0.30 | 2000 | 1000 | 500 | 1.9848 | 6213.94 | 2.70 × 10−12 | 3001.98 | 199.838 | −5.9544 | −75.87 | 231.37 | −69.55 | −46.014 |
15 | 50 | 0.90 | 0.20 | 8000 | 1000 | 500 | 1.8826 | 6073.32 | 2.70 × 10−12 | 2995.93 | 180.585 | −5.4950 | −75.67 | 231.37 | −69.53 | −45.134 |
16 | 50 | 0.95 | 0.10 | 6000 | 1000 | 500 | 2.0482 | 6091.36 | 2.70 × 10−12 | 2997.44 | 182.249 | −6.2276 | −75.69 | 231.37 | −69.54 | −45.213 |
η | 1.8800 | 6064.21 | 2.70 × 10−12 | 2997.74 | 185.326 | −5.4716 | −75.66 | 231.37 | −69.54 | −45.356 |
Optimization Problem | Evaluation Criteria | Design Parameter | |||
---|---|---|---|---|---|
HMS | HMCR | PAR | MAXITER | ||
WBD | SS | 1.803926 | 0.53773 | 0.061213 | 0.45212 |
ν | 0.60131 | 0.17924 | 0.020407 | 0.150711 | |
p | 0.100039 | 0.353229 | 0.901733 | 0.405384 | |
R | 1 | 2 | 4 | 3 | |
PVD | SS | 0.031716 | 0.001684 | 0.003575 | 0.040544 |
ν | 0.0105718 | 0.000560958 | 0.00119166 | 0.0135146 | |
p | 0.405366 | 0.971217 | 0.921773 | 0.332366 | |
R | 2 | 4 | 3 | 1 | |
SRD | SS | 1.98 × 10−5 | 1.33 × 10−6 | 1.20 × 10−6 | 5.93 × 10−6 |
ν | 6.57677 × 10−6 | 4.40671 × 10−6 | 4.02287 × 10−7 | 1.97717 × 10−4 | |
p | 0.046127 | 0.077733 | 0.658534 | 0.000333 | |
R | 3 | 2 | 4 | 1 | |
RCRW1 | SS | 0.094191 | 0.092708 | 0.050628 | 0.598937 |
ν | 0.0314006 | 0.0309005 | 0.0168764 | 0.199651 | |
p | 0.141871 | 0.144468 | 0.272054 | 0.012293 | |
R | 2 | 3 | 4 | 1 |
Optimization Problem | Optimum Parameter Combination | fmin (ηprediction) |
---|---|---|
WBD | HMS2-HMCR2-PAR4-MAXITER2 | $1.7291 |
PVD | HMS1-HMCR4-PAR2-MAXITER4 | $5973.13 |
GTD | HMS1-HMCR1-PAR1-MAXITER1 | 2.70086 × 10−12 |
SRD | HMS1-HMCR1-PAR2-MAXITER4 | 2995.11 kg |
RCRW1 | HMS1-HMCR3-PAR2-MAXITER4 | $177.842/m |
Case | Run | Iteration | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
BRN (%) | Best (ηreal) | Mean | Worst | StD | Median | BIN (%) | Best (ηreal) | Mean | Worst | StD | Median | |
WBD | 373/1000(37%) | 1.78312 | 1.88853 | 6.22216 | 0.320969 | 1.82613 | 3561/4000 (89%) | 1.78312 | 2.82897 | 4.61973 | 0.492371 | 2.74472 |
PVD | 881/1000 (88%) | 6005.19 | 7125.64 | 8728.03 | 503.107 | 7169.81 | 7870/8000 (%98) | 6005.19 | 7442.32 | 67288.7 | 3342.35 | 6395.62 |
GTD | 89/500 (%18) | 2.70086 × 10−12 | 5.4247 × 10−8 | 1.38114 × 10−6 | 1.34484 × 10−7 | 1.31252 × 10−8 | 1250/2000 (63%) | 2.70086 × 10−12 | 2.99998 × 10−5 | 5.5068 × 10−3 | 3.7964 × 10−3 | 2.70086 × 10−12 |
SRD | 195/500 (40%) | 2995.63 | 3002.75 | 3275.14 | 13.0823 | 3001.31 | 7187/8000 (90%) | 2995.63 | 3037.28 | 5194.7 | 234.639 | 2998.37 |
RCRW1 | 460/500 (92%) | 180.301 | 190.495 | 206.994 | 5.28013 | 190.4 | 7841/8000 (98%) | 180.301 | 204.729 | 385.939 | 48.688 | 188.459 |
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Uray, E.; Carbas, S.; Geem, Z.W.; Kim, S. Parameters Optimization of Taguchi Method Integrated Hybrid Harmony Search Algorithm for Engineering Design Problems. Mathematics 2022, 10, 327. https://doi.org/10.3390/math10030327
Uray E, Carbas S, Geem ZW, Kim S. Parameters Optimization of Taguchi Method Integrated Hybrid Harmony Search Algorithm for Engineering Design Problems. Mathematics. 2022; 10(3):327. https://doi.org/10.3390/math10030327
Chicago/Turabian StyleUray, Esra, Serdar Carbas, Zong Woo Geem, and Sanghun Kim. 2022. "Parameters Optimization of Taguchi Method Integrated Hybrid Harmony Search Algorithm for Engineering Design Problems" Mathematics 10, no. 3: 327. https://doi.org/10.3390/math10030327
APA StyleUray, E., Carbas, S., Geem, Z. W., & Kim, S. (2022). Parameters Optimization of Taguchi Method Integrated Hybrid Harmony Search Algorithm for Engineering Design Problems. Mathematics, 10(3), 327. https://doi.org/10.3390/math10030327