An Improved Hybrid Aquila Optimizer and Harris Hawks Algorithm for Solving Industrial Engineering Optimization Problems
Abstract
:1. Introduction
2. Preliminaries
2.1. Aquila Optimizer (AO)
- Step 1:
- Expanded exploration (X1): high soar with a vertical stoop
- Step 2:
- Narrowed exploration (X2): contour flight with short glide attack
- Step 3:
- Expanded exploitation (X3): low flight with a slow descent attack
- Step 4:
- Narrowed exploitation (X4): walking and grabbing prey
2.2. Harris’s Hawks Optimizer (HHO)
2.2.1. Exploration Phase
2.2.2. Transition from Exploration to Exploitation Phase
2.2.3. Exploitation Phase
- Soft besiege
- Hard besiege
- Soft besiege with progressive rapid dives
- Hard besiege with progressive rapid dives
2.3. Nonlinear Escaping Energy Parameter
2.4. Random Opposition-Based Learning (ROBL)
3. The Proposed IHAOHHO Algorithm
3.1. The Detail Design of IHAOHHO
Algorithm 1 Pseudo-code of IHAOHHO. |
1: Set initial values of the population size N and the maximum number of iterations T 2: Initialize positions of the population X 3: While t < T 4: For i = 1 to N 5: Check if the position goes out of the search space boundary, and bring it back. 6: Calculate the fitness of Xi 7: Update Xbest 8: End for 9: Update x, y, QF, G1, G2, E1 10: For i = 1 to N 11: Update E using Equation (24) % Nonlinear escaping energy parameter 12: If |E| ≥ 1 % Exploration part of AO 13: If rand < 0.5 14: Update the position of Xnewi using Equation (1) 15: If f(Xnewi) < f(Xi) 16: Xi = Xnewi 17: End if 18: Else 19: Update the position of Xnewi using Equation (3) 20: If f(Xnewi) < f(Xi) 21: Xi = Xnewi 22: End if 23: End if 24: Else % Exploitation part of HHO 25: If r ≥ 0.5 and |E| ≥ 0.5 26: Update the position of Xi using Equation (12) 27: End if 28: If r ≥ 0.5 and |E| < 0.5 29: Update the position of Xi using Equation (15) 30: End if 31: If r < 0.5 and |E| ≥ 0.5 32: Update the position of Xnewi using Equation (16) 33: If f(Xnewi) < f(Xi) 34: Xi = Xnewi 35: Else 36: Update the position of Xnewi using Equation (17) 37: If f(Xnewi) < f(Xi) 38: Xi = Xnewi 39: End if 40: End if 41: End if 42: If r < 0.5 and |E| < 0.5 43: Update the position of Xnewi using Equation (21) 44: If f(Xnewi) < f(Xi) 45: Xi = Xnewi 46: Else 47: Update the position of Xnewi using Equation (22) 48: If f(Xnewi) < f(Xi) 49: Xi = Xnewi 50: End if 51: End if 52: End if 53: Update the position of Xnewi using Equation (26) % ROBL 54: If f(Xnewi) < f(Xi) 55: Xi = Xnewi 56: End if 57: End if 58: t = t + 1 59: End for 60: End while 61: Return Xbest |
3.2. Computational Complexity of IHAOHHO
4. Results and Discussion
4.1. Benchmark Function Experiments
4.1.1. Evaluation of Exploitation Capability (Functions F1–F7)
4.1.2. Evaluation of Exploration Capability (Functions F8–F23)
4.1.3. Analysis of Convergence Behavior
4.1.4. The Wilcoxon Test
4.1.5. Computation Time
4.2. Experiments on Industrial Engineering Design Problems
4.2.1. Pressure Vessel Design Problem
4.2.2. Speed Reducer Design Problem
4.2.3. Tension/Compression Spring Design Problem
4.2.4. Three-Bar Truss Design Problem
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Function | Dim | Range | Fmin |
---|---|---|---|
30 | (−100, 100) | 0 | |
30 | (−10, 10) | 0 | |
30 | (−100, 100) | 0 | |
30 | (−100, 100) | 0 | |
30 | (−30,30) | 0 | |
30 | (−100, 100) | 0 | |
30 | (−1.28, 1.28) | 0 |
Function | Dim | Range | Fmin |
---|---|---|---|
30 | (−500, 500) | −418.9829 × 30 | |
30 | (−5.12, 5.12) | 0 | |
30 | (−32, 32) | 0 | |
30 | (−600, 600) | 0 | |
30 | (−50, 50) | 0 | |
30 | (−50, 50) | 0 |
Function | Dim | Range | Fmin |
2 | (−65, 65) | 1 | |
4 | (−5, 5) | 0.00030 | |
2 | (−5, 5) | −1.0316 | |
2 | (−5, 5) | 0.398 | |
2 | (−2, 2) | 3 | |
3 | (−1, 2) | −3.86 | |
6 | (0, 1) | −3.32 | |
4 | (0, 10) | −10.1532 | |
4 | (0, 10) | −10.4028 | |
4 | (0, 10) | −10.5363 |
Algorithm | Parameters |
---|---|
AO | U = 0.00565; r1 = 10; ω = 0.005; α = 0.1; δ = 0.1; G1 ∈ [−1, 1]; G2 = [2, 0] |
HHO | q ∈ [0, 1]; r ∈ [0, 1]; E0 ∈ [−1, 1]; E1 = [2, 0]; E ∈ [−2, 2]; |
SMA | z = 0.03 |
SSA | c1 = [1, 0]; c2 ∈ [0, 1]; c3 ∈ [0, 1] |
WOA | a1 = [2, 0]; a2 = [−1, −2]; b = 1 |
GWO | a = [2, 0] |
PSO | c1 = 2; c2 = 2; vmax = 6 |
F | IHAOHHO | AO | HHO | SMA | SSA | WOA | GWO | PSO | |
---|---|---|---|---|---|---|---|---|---|
F1 | Avg | 0.0000 × 100 | 2.5120 × 10−128 | 1.7359 × 10−98 | 6.7559 × 10−287 | 2.0918 × 10−7 | 7.0172 × 10−75 | 2.7553 × 10−27 | 1.7920 × 10−4 |
Std | 0.0000 × 100 | 1.3759 × 10−127 | 3.8748 × 10−98 | 0.0000 × 100 | 2.5521 × 10−7 | 2.0985 × 10−74 | 7.4745 × 10−27 | 2.1473 × 10−4 | |
F2 | Avg | 3.1773 × 10−283 | 3.0714 × 10−51 | 3.6162 × 10−49 | 1.7722 × 10−136 | 2.1400 × 100 | 2.1103 × 10−49 | 7.2224 × 10−17 | 2.2676 × 10−1 |
Std | 0.0000 × 100 | 1.6823 × 10−50 | 1.9747 × 10−48 | 9.7069 × 10−136 | 1.5737 × 100 | 1.1221 × 10−48 | 4.3158 × 10−17 | 2.0215 × 10−2 | |
F3 | Avg | 0.0000 × 100 | 2.3884 × 10−101 | 7.9368 × 10−70 | 2.7958 × 10−305 | 1.5707 × 103 | 4.8346 × 104 | 1.9688 × 10−5 | 8.7992 × 101 |
Std | 0.0000 × 100 | 9.262 × 10−101 | 4.3417 × 10−69 | 0.0000 × 100 | 1.0057 × 103 | 1.5295 × 104 | 8.5080 × 10−5 | 3.7192 × 101 | |
F4 | Avg | 1.1105 × 10−281 | 1.0656 × 10−53 | 1.2768 × 10−49 | 1.0217 × 10−160 | 1.1623 × 101 | 5.4222 × 101 | 9.2533 × 10−7 | 1.0783 × 100 |
Std | 0.0000 × 100 | 5.8309 × 10−53 | 4.4293 × 10−49 | 5.5961 × 10−160 | 3.3373 × 100 | 2.9852 × 101 | 9.1688 × 10−7 | 2.1854 × 10−1 | |
F5 | Avg | 2.8203 × 10−3 | 6.4303 × 10−3 | 1.1390 × 10−2 | 9.4019 × 100 | 3.1709 × 102 | 2.7969 × 101 | 2.7412 × 101 | 1.0424 × 102 |
Std | 4.4716 × 10−3 | 9.1289 × 10−3 | 1.2058 × 10−2 | 1.2466 × 101 | 8.0601 × 102 | 4.5551 × 10−1 | 8.8086 × 10−1 | 9.9130 × 101 | |
F6 | Avg | 4.2411 × 10−6 | 1.1861 × 10−4 | 1.1430 × 10−4 | 5.2584 × 10−3 | 3.5188 × 10−7 | 3.6078 × 10−1 | 8.0826 × 10−1 | 1.1828 × 10−4 |
Std | 6.2092 × 10−6 | 2.1625 × 10−4 | 1.4084 × 10−4 | 3.1160 × 10−3 | 7.3563 × 10−7 | 1.8848 × 10−1 | 3.3042 × 10−1 | 1.3013 × 10−4 | |
F7 | Avg | 7.1381 × 10−5 | 9.2969 × 10−5 | 1.4408 × 10−4 | 2.2317 × 10−4 | 1.7310 × 10−1 | 2.6756 × 10−3 | 2.2547 × 10−3 | 1.8040 × 10−1 |
Std | 7.6852 × 10−5 | 1.1466 × 10−4 | 1.5482 × 10−4 | 1.6750 × 10−4 | 7.8997 × 10−2 | 2.3949 × 10−3 | 1.1317 × 10−3 | 7.5627 × 10−2 | |
F8 | Avg | −12,447.8654 | −7073.9882 | −12,568.7811 | −12,568.9426 | −7591.3246 | −10,430.3986 | −6049.3246 | −5317.3115 |
Std | 4.5359 × 102 | 3.5511 × 103 | 1.3999 × 100 | 4.0261 × 10−1 | 6.9106 × 102 | 1.9097 × 103 | 8.0214 × 102 | 1.5005 × 103 | |
F9 | Avg | 0.0000 × 100 | 0.0000 × 100 | 0.0000 × 100 | 0.0000 × 100 | 5.5253 × 101 | 1.8948 × 10−15 | 4.8419 × 100 | 5.6659 × 101 |
Std | 0.0000 × 100 | 0.0000 × 100 | 0.0000 × 100 | 0.0000 × 100 | 1.9037 × 101 | 1.0378 × 10−14 | 6.2042 × 100 | 1.5111 × 101 | |
F10 | Avg | 8.8818 × 10−16 | 8.8818 × 10−16 | 8.8818 × 10−16 | 8.8818 × 10−16 | 2.7561 × 100 | 3.9672 × 10−15 | 1.0356 × 10−13 | 2.0903 × 10−1 |
Std | 0.0000 × 100 | 0.0000 × 100 | 0.0000 × 100 | 0.0000 × 100 | 1.9773 × 100 | 2.4210 × 10−15 | 2.1323 × 10−14 | 4.4871 × 10−1 | |
F11 | Avg | 0.0000 × 100 | 0.0000 × 100 | 0.0000 × 100 | 0.0000 × 100 | 1.7030 × 10−2 | 5.9385 × 10−3 | 2.5384 × 10−3 | 4.9459 × 10−3 |
Std | 0.0000 × 100 | 0.0000 × 100 | 0.0000 × 100 | 0.0000 × 100 | 1.9430 × 10−2 | 3.2527 × 10−2 | 8.7348 × 10−3 | 9.4682 × 10−3 | |
F12 | Avg | 5.3164 × 10−7 | 4.6513 × 10−6 | 9.2636 × 10−6 | 5.0331 × 10−3 | 7.0564 × 100 | 2.5778 × 10−2 | 3.7730 × 10−2 | 6.9126 × 10−3 |
Std | 9.6698 × 10−7 | 8.9371 × 10−6 | 1.2911 × 10−5 | 6.3463 × 10−3 | 3.0595 × 100 | 2.0942 × 10−2 | 1.8369 × 10−2 | 2.6301 × 10−2 | |
F13 | Avg | 1.1694 × 10−5 | 3.3938 × 10−5 | 1.2604 × 10−4 | 7.3800 × 10−3 | 1.7887 × 101 | 5.8549 × 10−1 | 6.1135 × 10−1 | 4.4120 × 10−3 |
Std | 1.7961 × 10−5 | 3.2363 × 10−5 | 1.5375 × 10−4 | 8.9329 × 10−3 | 1.5307 × 101 | 2.9719 × 10−1 | 1.7136 × 10−1 | 6.6275 × 10−3 | |
F14 | Avg | 1.7919 × 100 | 1.5940 × 100 | 1.1635 × 100 | 9.9800 × 10−1 | 1.1637 × 100 | 5.0748 × 100 | 5.2681 × 100 | 3.5906 × 100 |
Std | 9.1746 × 10−1 | 2.1763 × 100 | 4.5784 × 10−1 | 1.1156 × 10−12 | 3.7678 × 10−1 | 4.4603 × 100 | 4.6022 × 100 | 2.904 × 100 | |
F15 | Avg | 3.5291 × 10−4 | 5.5590 × 10−4 | 4.0350 × 10−4 | 5.1576 × 10−4 | 2.8218 × 10−3 | 6.6118 × 10−4 | 6.3719 × 10−3 | 9.3864 × 10−4 |
Std | 4.8766 × 10−5 | 1.1640 × 10−4 | 2.3353 × 10−4 | 3.0066 × 10−4 | 5.9580 × 10−3 | 7.1226 × 10−4 | 1.2424 × 10−2 | 2.6081 × 10−4 | |
F16 | Avg | −1.0316 × 100 | −1.0311 × 100 | −1.0316 × 100 | −1.0316 × 100 | −1.0316 × 100 | −1.0316 × 100 | −1.0316 × 100 | −1.0316 × 100 |
Std | 1.0379 × 10−10 | 3.7614 × 10−4 | 2.5745 × 10−9 | 4.3934 × 10−10 | 2.0489 × 10−14 | 6.1164 × 10−10 | 1.4772 × 10−8 | 6.4539 × 10−16 | |
F17 | Avg | 3.9789 × 10−1 | 3.9812 × 10−1 | 3.9790 × 10−1 | 3.9789 × 10−1 | 3.9789 × 10−1 | 3.9789 × 10−1 | 3.9789 × 10−1 | 3.9789 × 10−1 |
Std | 5.4022 × 10−7 | 2.2378 × 10−4 | 2.4237 × 10−5 | 2.4814 × 10−8 | 1.4663 × 10−14 | 8.5493 × 10−6 | 8.9987 × 10−7 | 0.0000 × 100 | |
F18 | Avg | 3.0000 × 100 | 3.0439 × 100 | 3.0000 × 100 | 3.0000 × 100 | 3.0000 × 100 | 3.0000 × 100 | 3.0000 × 100 | 3.0000 × 100 |
Std | 2.714 × 10−7 | 6.4693 × 10−2 | 1.6198 × 10−7 | 4.7705 × 10−10 | 9.5042 × 10−14 | 2.6269 × 10−4 | 4.7607 × 10−5 | 1.639 × 10−15 | |
F19 | Avg | −3.8628 × 100 | −3.8539 × 100 | −3.8616 × 100 | −3.8628 × 100 | −3.8628 × 100 | −3.8597 × 100 | −3.8593 × 100 | −3.8628 × 100 |
Std | 1.8351 × 10−4 | 6.0669 × 10−3 | 1.7013 × 10−3 | 3.0254 × 10−7 | 8.1972 × 10−13 | 3.1652 × 10−3 | 4.2427 × 10−3 | 2.6823 × 10−15 | |
F20 | Avg | −3.1298 × 100 | −3.1572 × 100 | −3.0533 × 100 | −3.2425 × 100 | −3.2215 × 100 | −3.2391 × 100 | −3.2442 × 100 | −3.2665 × 100 |
Std | 1.1264 × 10−1 | 1.0448 × 10−1 | 1.1671 × 10−1 | 5.7177 × 10−2 | 5.1720 × 10−2 | 1.3596 × 10−1 | 9.0427 × 10−2 | 6.0328 × 10−2 | |
F21 | Avg | −1.0152 × 101 | −1.0142 × 101 | −5.5370 × 100 | −1.0152 × 101 | −7.3774 × 100 | −9.0891 × 100 | −9.1419 × 100 | −6.7868 × 100 |
Std | 5.6352 × 10−4 | 1.8288 × 10−2 | 1.484 × 100 | 2.2592 × 10−3 | 2.9079 × 100 | 2.0545 × 100 | 2.3491 × 100 | 3.2622 × 100 | |
F22 | Avg | −1.0402 × 101 | −1.0388 × 101 | −5.2528 × 100 | −1.0402 × 101 | −8.1232 × 100 | −7.5395 × 100 | −1.0401 × 101 | −8.1542 × 100 |
Std | 6.3272 × 10−4 | 2.4782 × 10−2 | 9.3628 × 10−1 | 7.5981 × 10−4 | 3.3371 × 100 | 3.1570 × 100 | 8.9128 × 10−4 | 3.2898 × 100 | |
F23 | Avg | −1.0535 × 101 | −1.0525 × 101 | −5.2858 × 100 | −1.0535 × 101 | −7.6861 × 100 | −6.6213 × 100 | −1.0535 × 101 | −1.0087 × 101 |
Std | 9.8617 × 10−4 | 6.9516 × 10−3 | 8.8012 × 10−1 | 1.3006 × 10−3 | 3.6004 × 100 | 3.0127 × 100 | 9.0143 × 10−4 | 1.7472 × 100 |
F | IHAOHHO vs. AO | IHAOHHO vs. HHO | IHAOHHO vs. SMA | IHAOHHO vs. SSA | IHAOHHO vs. WOA | IHAOHHO vs. GWO | IHAOHHO vs. PSO |
---|---|---|---|---|---|---|---|
F1 | 6.1035 × 10−5 | 6.1035 × 10−5 | N/A | 6.1035 × 10−5 | 6.1035 × 10−5 | 6.1035 × 10−5 | 6.1035 × 10−5 |
F2 | 6.1035 × 10−5 | 6.1035 × 10−5 | 6.1035 × 10−5 | 6.1035 × 10−5 | 6.1035 × 10−5 | 6.1035 × 10−5 | 6.1035 × 10−5 |
F3 | 6.1035 × 10−5 | 6.1035 × 10−5 | N/A | 6.1035 × 10−5 | 6.1035 × 10−5 | 6.1035 × 10−5 | 6.1035 × 10−5 |
F4 | 6.1035 × 10−5 | 6.1035 × 10−5 | 1.2207 × 10−4 | 6.1035 × 10−5 | 6.1035 × 10−5 | 6.1035 × 10−5 | 6.1035 × 10−5 |
F5 | 6.7877 × 10−1 | 6.3867 × 10−1 | 6.1035 × 10−5 | 6.1035 × 10−5 | 6.1035 × 10−5 | 6.1035 × 10−5 | 6.1035 × 10−5 |
F6 | 1.5076 × 10−2 | 8.5449 × 10−4 | 6.1035 × 10−5 | 8.5449 × 10−4 | 6.1035 × 10−5 | 6.1035 × 10−5 | 6.1035 × 10−5 |
F7 | 8.0396 × 10−1 | 4.2725 × 10−3 | 3.0518 × 10−4 | 6.1035 × 10−5 | 1.2207 × 10−4 | 6.1035 × 10−5 | 6.1035 × 10−5 |
F8 | 1.0699 × 10−3 | 5.5359 × 10−3 | 6.7139 × 10−3 | 8.5449 × 10−4 | 7.2998 × 10−2 | 6.1035 × 10−5 | 6.1035 × 10−5 |
F9 | N/A | N/A | N/A | 6.1035 × 10−5 | N/A | 6.1035 × 10−5 | 6.1035 × 10−5 |
F10 | N/A | N/A | N/A | 6.1035 × 10−5 | 4.8828 × 10−4 | 6.1035 × 10−5 | 6.1035 × 10−5 |
F11 | N/A | N/A | N/A | 6.1035 × 10−5 | N/A | 6.2500 × 10−2 | 6.1035 × 10−5 |
F12 | 9.7797 × 10−1 | 2.7686 × 10−1 | 6.1035 × 10−5 | 6.1035 × 10−5 | 6.1035 × 10−5 | 6.1035 × 10−5 | 5.2448 × 10−1 |
F13 | 8.9038 × 10−1 | 3.5339 × 10−2 | 6.1035 × 10−5 | 6.1035 × 10−5 | 6.1035 × 10−5 | 6.1035 × 10−5 | 2.1545 × 10−2 |
F14 | 3.5339 × 10−2 | 3.8940 × 10−2 | 1.2207 × 10−4 | 4.7913 × 10−2 | 6.1035 × 10−4 | 1.0699 × 10−1 | 2.1545 × 10−2 |
F15 | 3.3569 × 10−3 | 7.1973 × 10−1 | 4.7913 × 10−2 | 6.1035 × 10−5 | 1.1597 × 10−3 | 2.1545 × 10−2 | 6.1035 × 10−5 |
F16 | 6.1035 × 10−5 | 3.0151 × 10−2 | 8.5449 × 10−4 | 4.0283 × 10−3 | 4.2725 × 10−3 | 6.1035 × 10−5 | 1.2207 × 10−4 |
F17 | 6.1035 × 10−5 | 3.0280 × 10−1 | 1.0254 × 10−2 | 6.1035 × 10−5 | 6.7139 × 10−3 | 2.5574 × 10−2 | 6.1035 × 10−5 |
F18 | 6.1035 × 10−5 | 8.3618 × 10−3 | 8.3618 × 10−3 | 3.0518 × 10−4 | 1.2207 × 10−4 | 6.1035 × 10−5 | 6.1035 × 10−5 |
F19 | N/A | N/A | N/A | N/A | N/A | N/A | 6.1035 × 10−5 |
F20 | 7.2998 × 10−2 | 1.8762 × 10−1 | 7.2998 × 10−2 | 1.0699 × 10−2 | 2.7686 × 10−1 | 1.0254 × 10−2 | 3.3569 × 10−3 |
F21 | 1.8762 × 10−1 | 6.1035 × 10−5 | 4.8871 × 10−1 | 4.2120 × 10−1 | 8.5449 × 10−4 | 5.9949 × 10−3 | 2.5574 × 10−2 |
F22 | 4.7913 × 10−2 | 6.1035 × 10−5 | 1.8066 × 10−2 | 8.0396 × 10−1 | 1.2207 × 10−4 | 2.0776 × 10−1 | 8.3618 × 10−3 |
F23 | 6.1035 × 10−5 | 6.1035 × 10−5 | 5.5359 × 10−2 | 8.3252 × 10−2 | 6.1035 × 10−5 | 8.3252 × 10−2 | 8.3252 × 10−2 |
F | IHAOHHO | AO | HHO | SMA | SSA | WOA | GWO | PSO |
---|---|---|---|---|---|---|---|---|
F1 | 2.8539 × 10−1 | 2.3253 × 10−1 | 1.3713 × 10−1 | 8.8997 × 10−1 | 8.5420 × 10−2 | 7.5875 × 10−2 | 1.1491 × 10−1 | 6.5132 × 10−2 |
F2 | 2.8946 × 10−1 | 2.5214 × 10−1 | 1.4672 × 10−1 | 9.1203 × 10−1 | 1.0346 × 10−1 | 1.1982 × 10−1 | 1.2761 × 10−1 | 7.3814 × 10−2 |
F3 | 1.6030 × 100 | 9.2890 × 10−1 | 9.3324 × 10−1 | 1.2673 × 100 | 4.6382 × 10−1 | 3.9400 × 10−1 | 4.2700 × 10−1 | 3.9204 × 10−1 |
F4 | 2.8070 × 10−1 | 1.9787 × 10−1 | 1.5712 × 10−1 | 9.5399 × 10−1 | 8.2341 × 10−2 | 7.3767 × 10−2 | 1.1442 × 10−1 | 6.4915 × 10−2 |
F5 | 3.3725 × 10−1 | 2.2214 × 10−1 | 2.2123 × 10−1 | 1.0204 × 100 | 9.8470 × 10−2 | 8.7778 × 10−2 | 1.2667 × 10−1 | 7.8503 × 10−2 |
F6 | 2.7707 × 10−1 | 2.0399 × 10−1 | 1.7800 × 10−1 | 9.0977 × 10−1 | 8.2725 × 10−2 | 7.4251 × 10−2 | 1.1248 × 10−1 | 6.5708 × 10−2 |
F7 | 5.0109 × 10−1 | 3.0078 × 10−1 | 2.8662 × 10−1 | 9.5443 × 10−1 | 1.3880 × 10−1 | 1.2862 × 10−1 | 1.6701 × 10−1 | 1.1976 × 10−1 |
F8 | 3.9395 × 10−1 | 2.3581 × 10−1 | 2.3276 × 10−1 | 9.7695 × 10−1 | 1.0531 × 10−1 | 9.7443 × 10−2 | 1.3674 × 10−1 | 9.1720 × 10−2 |
F9 | 3.2379 × 10−1 | 1.9907 × 10−1 | 1.9594 × 10−1 | 9.5132 × 10−1 | 9.5204 × 10−2 | 7.9254 × 10−2 | 1.1801 × 10−1 | 7.4441 × 10−2 |
F10 | 3.5602 × 10−1 | 2.3037 × 10−1 | 2.3125 × 10−1 | 9.4870 × 10−1 | 1.0399 × 10−1 | 9.0064 × 10−2 | 1.2725 × 10−1 | 8.3986 × 10−2 |
F11 | 4.0659 × 10−1 | 2.4303 × 10−1 | 2.4198 × 10−1 | 9.3026 × 10−1 | 1.1382 × 10−1 | 1.0089 × 10−1 | 1.3566 × 10−1 | 9.2499 × 10−2 |
F12 | 1.0131 × 100 | 6.0006 × 10−1 | 6.9400 × 10−1 | 1.1939 × 100 | 2.6401 × 10−1 | 2.5229 × 10−1 | 3.4237 × 10−1 | 2.4517 × 10−1 |
F13 | 1.0300 × 100 | 5.6112 × 10−1 | 6.1205 × 10−1 | 1.1549 × 100 | 2.7393 × 10−1 | 2.7208 × 10−1 | 3.3915 × 10−1 | 2.4746 × 10−1 |
F14 | 2.3159 × 100 | 1.2173 × 100 | 1.5168 × 100 | 8.9676 × 10−1 | 5.9818 × 10−1 | 6.0722 × 10−1 | 5.9328 × 10−1 | 5.5450 × 10−1 |
F15 | 2.6135 × 10−1 | 1.7086 × 10−1 | 1.7031 × 10−1 | 3.4136 × 10−1 | 9.9034 × 10−2 | 7.5482 × 10−2 | 6.4104 × 10−2 | 4.2546 × 10−2 |
F16 | 2.0719 × 10−1 | 1.4146 × 10−1 | 1.3859 × 10−1 | 2.7170 × 10−1 | 5.9081 × 10−2 | 4.9666 × 10−2 | 6.0033 × 10−2 | 4.1193 × 10−2 |
F17 | 1.8138 × 10−1 | 1.3529 × 10−1 | 1.5833 × 10−1 | 2.7311 × 10−1 | 5.2979 × 10−2 | 4.1321 × 10−2 | 4.1556 × 10−2 | 2.3066 × 10−2 |
F18 | 1.8108 × 10−1 | 1.3183 × 10−1 | 1.2693 × 10−1 | 2.7041 × 10−1 | 5.4471 × 10−2 | 4.0487 × 10−2 | 4.1752 × 10−2 | 2.2830 × 10−2 |
F19 | 3.5119 × 10−1 | 2.4635 × 10−1 | 2.4016 × 10−1 | 3.4125 × 10−1 | 9.8544 × 10−2 | 8.5903 × 10−2 | 9.2049 × 10−2 | 7.0253 × 10−2 |
F20 | 3.7106 × 10−1 | 2.2549 × 10−1 | 2.4656 × 10−1 | 4.0519 × 10−1 | 1.0270 × 10−1 | 9.0294 × 10−2 | 9.8020 × 10−2 | 7.2095 × 10−2 |
F21 | 5.8451 × 10−1 | 3.2169 × 10−1 | 3.6385 × 10−1 | 4.1713 × 10−1 | 1.4920 × 10−1 | 1.3664 × 10−1 | 1.3885 × 10−1 | 1.2170 × 10−1 |
F22 | 7.2861 × 10−1 | 3.9414 × 10−1 | 4.3943 × 10−1 | 4.9071 × 10−1 | 1.8789 × 10−1 | 1.7154 × 10−1 | 1.7638 × 10−1 | 1.5215 × 10−1 |
F23 | 9.4549 × 10−1 | 4.9412 × 10−1 | 5.7464 × 10−1 | 4.9527 × 10−1 | 2.3551 × 10−1 | 2.7717 × 10−1 | 2.2549 × 10−1 | 2.0513 × 10−1 |
Algorithm | Optimum Variables | Optimum Cost | |||
---|---|---|---|---|---|
Ts | Th | R | L | ||
IHAOHHO | 0.8363559 | 0.4127868 | 45.08462 | 142.9202 | 5932.3392 |
AO [55] | 1.0540 | 0.182806 | 59.6219 | 38.8050 | 5949.2258 |
HHO [42] | 0.81758383 | 0.4072927 | 42.09174576 | 176.7196352 | 6000.46259 |
SMA [41] | 0.7931 | 0.3932 | 40.6711 | 196.2178 | 5994.1857 |
WOA [38] | 0.8125 | 0.4375 | 42.0982699 | 176.638998 | 6059.7410 |
GWO [32] | 0.8125 | 0.4345 | 42.0892 | 176.7587 | 6051.5639 |
MVO [23] | 0.8125 | 0.4375 | 42.090738 | 176.73869 | 6060.8066 |
GA [3] | 0.8125 | 0.4375 | 42.097398 | 176.65405 | 6059.94634 |
ES [6] | 0.8125 | 0.4375 | 42.098087 | 176.640518 | 6059.74560 |
CPSO [65] | 0.8125 | 0.4375 | 42.091266 | 176.7465 | 6061.0777 |
Algorithm | Optimum Variables | Optimum Weight | ||||||
---|---|---|---|---|---|---|---|---|
x1 | x2 | x3 | x4 | x5 | x6 | x7 | ||
IHAOHHO | 3.49924 | 0.7 | 17 | 7.3 | 7.8191 | 3.35006 | 5.28531 | 2996.0935 |
AO [55] | 3.5021 | 0.7 | 17 | 7.3099 | 7.7476 | 3.3641 | 5.2994 | 3007.7328 |
PSO [26] | 3.5001 | 0.7 | 17.0002 | 7.5177 | 7.7832 | 3.3508 | 5.2867 | 3145.922 |
AOA [25] | 3.50384 | 0.7 | 17 | 7.3 | 7.72933 | 3.35649 | 5.2867 | 2997.9157 |
MFO [66] | 3.49745 | 0.7 | 17 | 7.82775 | 7.71245 | 3.35178 | 5.28635 | 2998.9408 |
GA [3] | 3.51025 | 0.7 | 17 | 8.35 | 7.8 | 3.36220 | 5.28772 | 3067.561 |
SCA [24] | 3.50875 | 0.7 | 17 | 7.3 | 7.8 | 3.46102 | 5.28921 | 3030.563 |
HS [67] | 3.52012 | 0.7 | 17 | 8.37 | 7.8 | 3.36697 | 5.28871 | 3029.002 |
FA [68] | 3.50749 | 0.7001 | 17 | 7.71967 | 8.08085 | 3.35151 | 5.28705 | 3010.13749 |
MDA [69] | 3.5 | 0.7 | 17 | 7.3 | 7.67039 | 3.54242 | 5.24581 | 3019.58336 |
Algorithm | Optimum Variables | Optimum Weight | ||
---|---|---|---|---|
d | D | N | ||
IHAOHHO | 0.055883 | 0.52784 | 4.7603 | 0.011144 |
AO [55] | 0.0502439 | 0.35262 | 10.5425 | 0.011165 |
HHO [42] | 0.051796393 | 0.359305355 | 11.138859 | 0.012665443 |
SSA [39] | 0.051207 | 0.345215 | 12.004032 | 0.0126763 |
WOA [38] | 0.051207 | 0.345215 | 12.004032 | 0.0126763 |
GWO [32] | 0.05169 | 0.356737 | 11.28885 | 0.012666 |
PSO [26] | 0.051728 | 0.357644 | 11.244543 | 0.0126747 |
MVO [23] | 0.05251 | 0.37602 | 10.33513 | 0.012790 |
GA [3] | 0.051480 | 0.351661 | 11.632201 | 0.01270478 |
HS [67] | 0.051154 | 0.349871 | 12.076432 | 0.0126706 |
Algorithm | Optimum Variables | Optimum Weight | |
---|---|---|---|
x1 | x2 | ||
IHAOHHO | 0.79002 | 0.40324 | 263.8622 |
AO [55] | 0.7926 | 0.3966 | 263.8684 |
HHO [42] | 0.788662816 | 0.408283133832900 | 263.8958434 |
SSA [39] | 0.78866541 | 0.408275784 | 263.89584 |
AOA [25] | 0.79369 | 0.39426 | 263.9154 |
MVO [23] | 0.78860276 | 0.408453070000000 | 263.8958499 |
MFO [66] | 0.788244771 | 0.409466905784741 | 263.8959797 |
GOA [70] | 0.788897555578973 | 0.407619570115153 | 263.895881496069 |
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Wang, S.; Jia, H.; Abualigah, L.; Liu, Q.; Zheng, R. An Improved Hybrid Aquila Optimizer and Harris Hawks Algorithm for Solving Industrial Engineering Optimization Problems. Processes 2021, 9, 1551. https://doi.org/10.3390/pr9091551
Wang S, Jia H, Abualigah L, Liu Q, Zheng R. An Improved Hybrid Aquila Optimizer and Harris Hawks Algorithm for Solving Industrial Engineering Optimization Problems. Processes. 2021; 9(9):1551. https://doi.org/10.3390/pr9091551
Chicago/Turabian StyleWang, Shuang, Heming Jia, Laith Abualigah, Qingxin Liu, and Rong Zheng. 2021. "An Improved Hybrid Aquila Optimizer and Harris Hawks Algorithm for Solving Industrial Engineering Optimization Problems" Processes 9, no. 9: 1551. https://doi.org/10.3390/pr9091551