Improved Grey Wolf Optimization Algorithm and Application
<p>Grey wolf class system.</p> "> Figure 2
<p>Prey tracing map.</p> "> Figure 3
<p>Chaotic mapping curve. (<b>a</b>) Tent; (<b>b</b>) improved tent.</p> "> Figure 4
<p>Convergence factor.</p> "> Figure 5
<p>Convergence curves of algorithms on test function. (<b>a</b>) f1 function; (<b>b</b>) f2 function; (<b>c</b>) f3 function; (<b>d</b>) f4 function; (<b>e</b>) f5 function; (<b>f</b>) f7 function; (<b>g</b>) f8 function; (<b>h</b>) f9 function; (<b>i</b>) f10 function; (<b>j</b>) f11 function; (<b>k</b>) f12 function; (<b>l</b>) f14 function.</p> "> Figure 5 Cont.
<p>Convergence curves of algorithms on test function. (<b>a</b>) f1 function; (<b>b</b>) f2 function; (<b>c</b>) f3 function; (<b>d</b>) f4 function; (<b>e</b>) f5 function; (<b>f</b>) f7 function; (<b>g</b>) f8 function; (<b>h</b>) f9 function; (<b>i</b>) f10 function; (<b>j</b>) f11 function; (<b>k</b>) f12 function; (<b>l</b>) f14 function.</p> "> Figure 6
<p>Path planning results. (<b>a</b>) Improved GWO; (<b>b</b>) MGWO; (<b>c</b>) NGWO; (<b>d</b>) GWO-fuzzy; (<b>e</b>) GWO-EPD; (<b>f</b>) convergence curves.</p> "> Figure 6 Cont.
<p>Path planning results. (<b>a</b>) Improved GWO; (<b>b</b>) MGWO; (<b>c</b>) NGWO; (<b>d</b>) GWO-fuzzy; (<b>e</b>) GWO-EPD; (<b>f</b>) convergence curves.</p> ">
Abstract
:1. Introduction
- An improved GWO algorithm based on a multi-strategy hybrid is proposed.
- The improved GWO algorithm is applied to the path planning of mobile robot.
- The performance of the proposed approach is compared with standard GWO, Sparrow Search Algorithm (SSA), Mayfly Algorithm (MA), Modified Grey Wolf Optimization Algorithm (MGWO) [10], Novel Grey Wolf Optimization Algorithm (NGWO) [11], A Fuzzy Hierarchical Operator in the Grey Wolf Optimizer Algorithm (GWO-fuzzy) [12], and Evolutionary population dynamics and grey wolf optimizer (GWO-EPD) [13].
2. Related Work
2.1. Research Situation
2.2. GWO Algorithm
3. Improved GWO Algorithm
3.1. Wolf Pack Initialization
- Produce random initial values y0 in (0, 1) with i = 0.
- Calculate iteratively using Equation (9) to produce the sequence.
- Stop iterating when the iteration reaches the maximum value and saves the sequence.
3.2. Nonlinear Convergence Factor
3.3. Dynamic Proportional Weighting Strategy
Algorithm 1: Pseudo Code of Improved GWO | |
1 | Initialize (Xi (i = 1, 2…, n)) t, Tmax, a, A, C |
2 | Initialize Tent map x0 |
3 | Calculate the fitness of each wolf |
4 | Xa = best wolf. Xβ = second wolf. Xw = third wolf. |
5 | While t < Tmax |
6 | Sort fitness of each wolf |
7 | Update chaotic number, a |
8 | for each search agent |
9 | Update position current wolf using |
10 | end |
11 | Calculate fitness of each wolf |
12 | Update Xa, Xβ, Xw |
13 | t = t + 1 |
14 | end |
4. Result
4.1. Comparison with GWO and Other Improvement GWO
4.1.1. Convergence Accuracy Analysis
4.1.2. Convergence Speed Analysis
4.2. Comparison with Other Intelligent Optimization Algorithms
4.3. Path Planning Application
4.3.1. Problem Description
- Maximum cornering angle constraint
- 2.
- Threat area constraints
4.3.2. Path Planning
- Establish the search space according to the actual environment, and set the starting point and target point.
- Initialize the parameters of grey wolf algorithm, including the number of wolves, the maximum number of iterations, tent mapping parameters, and upper and lower bounds for parameter values.
- Initialize the grey wolf’s position and objective function according to the utilization mapping.
- Calculate each grey wolf’s fitness and select the top three grey wolves as wolf α, wolf β, and wolf w for the fitness ranking.
- Compare with the objective function to update the position and the objective function.
- Update the convergence factor at each iteration.
- Calculate the next position of other wolves according to the positions of wolf α, wolf β, and wolf w.
- Reach the maximum number of iterations and output the optimal path.
5. Conclusions
Author Contributions
Funding
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Parameter Symbols | Meaning | Take Value |
---|---|---|
N | Population size | 30 |
Tmax | Maximum Iteration | 500 |
a1 | Initial value of convergence factor | 2 |
a2 | Final value of convergence factor | 0 |
Function | Dim | Scope | Solution |
---|---|---|---|
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 | |
30 | [−5.12, 5.12] | 0 | |
30 | [−32, 32] | 0 | |
30 | [−600, 600] | 0 | |
30 | [−50, 50] | 0.398 | |
30 | [−50, 50] | 3 | |
30 | [−10, 10] | 0 | |
30 | [−100, 100] | 0 | |
30 | [−15, 15] | 0 |
Function | Algorithm | Average Value | Standard Deviation |
---|---|---|---|
f1 | GWO | 4.389 × 10−27 | 1.056 × 10−27 |
Improved GWO | 0 | 0 | |
MGWO | 5.996 × 10−199 | 0 | |
NGWO | 9.939 × 10−49 | 4.754 × 10−48 | |
GWO-fuzzy | 9.887 × 10−40 | 4.977 × 10−40 | |
GWO-EPD | 1.501 × 10−31 | 2.289 × 10−30 | |
f2 | GWO | 2.167 × 10−5 | 3.958 × 10−6 |
Improved GWO | 0 | 0 | |
MGWO | 1.617 × 10−102 | 2.154 × 10−102 | |
NGWO | 2.133 × 10−26 | 1.143 × 10−26 | |
GWO-fuzzy | 1.572 × 10−24 | 1.374 × 10−23 | |
GWO-EPD | 1.893 × 10−19 | 2.358 × 10−20 | |
f3 | GWO | 1.115 × 10−7 | 3.463 × 10−5 |
Improved GWO | 0 | 0 | |
MGWO | 6.982 × 10−166 | 0 | |
NGWO | 1.015 × 10−33 | 3.789 × 10−31 | |
GWO-fuzzy | 5.981 × 10−8 | 3.753 × 10−7 | |
GWO-EPD | 4.505 × 10−8 | 2.456 × 10−6 | |
f4 | GWO | 8.423 × 10−7 | 4.583 × 10−7 |
Improved GWO | 0 | 0 | |
MGWO | 5.368 × 10−90 | 9.664 × 10−89 | |
NGWO | 4.414 × 10−20 | 1.104 × 10−19 | |
GWO-fuzzy | 4.995 × 10−9 | 8.259 × 10−7 | |
GWO-EPD | 3.395 × 10−7 | 7.652 × 10−6 | |
f5 | GWO | 2.706 × 101 | 6.824 × 10−1 |
Improved GWO | 2.867 × 101 | 2.611 × 10−2 | |
MGWO | 2.761 × 101 | 3.917 × 10−1 | |
NGWO | 2.719 × 101 | 5.836 × 10−1 | |
GWO-fuzzy | 2.855 × 101 | 8.518 × 10−1 | |
GWO-EPD | 2.818 × 101 | 8.075 × 10−1 | |
f6 | GWO | 1.013 | 2.816 × 10−1 |
Improved GWO | 6.533 × 10−1 | 2.860 × 10−1 | |
MGWO | 5.261 | 6.381 × 10−1 | |
NGWO | 1.829 | 3.763 × 10−1 | |
GWO-fuzzy | 2.324 | 5.052 × 10−1 | |
GWO-EPD | 1.238 | 4.725 × 10−1 | |
f7 | GWO | 1.154 × 10−3 | 1.226 × 10−3 |
Improved GWO | 2.961 × 10−7 | 2.373 × 10−7 | |
MGWO | 1.914 × 10−4 | 1.369 × 10−4 | |
NGWO | 1.347 × 10−3 | 2.747 × 10−4 | |
GWO-fuzzy | 1.744 × 10−3 | 1.047 × 10−3 | |
GWO-EPD | 1.646 × 10−3 | 1.031 × 10−3 | |
f8 | GWO | 6.934 × 10−12 | 4.701 |
Improved GWO | 0 | 0 | |
MGWO | 0 | 0 | |
NGWO | 5.684 × 10−14 | 2.017 × 10−1 | |
GWO-fuzzy | 6.130 × 10−1 | 1.657 × 10−1 | |
GWO-EPD | 1.715 × 10−13 | 3.852 | |
f9 | GWO | 1.103 × 10−13 | 1.633 × 10−14 |
Improved GWO | 8.811 × 10−16 | 1.164 × 10−16 | |
MGWO | 4.440 × 10−15 | 6.486 × 10−15 | |
NGWO | 2.930 × 10−14 | 2.420 × 10−15 | |
GWO-fuzzy | 2.930 × 10−14 | 3.923 × 10−15 | |
GWO-EPD | 4.352 × 10−14 | 6.4963 × 10−15 | |
f10 | GWO | 7.558 × 10−3 | 1.412 × 10−2 |
Improved GWO | 0 | 0 | |
MGWO | 0 | 0 | |
NGWO | 0 | 0 | |
GWO-fuzzy | 7.2159 × 10−4 | 3.0047 × 10−3 | |
GWO-EPD | 5.6751 × 10−3 | 5.7892 × 10−3 | |
f11 | GWO | 3.8124 × 10−1 | 6.7824 × 10−2 |
Improved GWO | 2.1331 × 10−3 | 6.8945 × 10−3 | |
MGWO | 5.3122 × 10−1 | 3.1121 × 10−2 | |
NGWO | 1.1021 × 101 | 3.0031 | |
GWO-fuzzy | 1.3811 | 8.3221 | |
GWO-EPD | 1.2254 × 10−2 | 4.2214 × 10−1 | |
f12 | GWO | 7.3712 | 4.1077 × 10−1 |
Improved GWO | 1.2922 × 10−2 | 7.6012 × 10−2 | |
MGWO | 8.3211 | 3.2454 × 10−1 | |
NGWO | 1.6722 × 101 | 3.1207 | |
GWO-fuzzy | 6.1545 × 10−1 | 4.5512 | |
GWO-EPD | 8.21475 × 102 | 8.1542 × 102 | |
f13 | GWO | 4.5214 × 10−3 | 2.5784 × 10−3 |
Improved GWO | 2.4457 × 10−6 | 6.3641 × 10−6 | |
MGWO | 7.7541 × 10−5 | 8.2231 × 10−4 | |
NGWO | 2.1441 × 101 | 8.1601 | |
GWO-fuzzy | 1.2215 × 101 | 2.2232 × 101 | |
GWO-EPD | 1.2014 × 10−2 | 1.2424 × 101 | |
f14 | GWO | 1.4125 × 10−2 | 2.3622 × 10−3 |
Improved GWO | 3.1337 × 10−3 | 1.1184 × 10−3 | |
MGWO | 4.3221 × 10−3 | 1.4752 × 10−3 | |
NGWO | 4.8842 × 10−1 | 2.4821 × 10−3 | |
GWO-fuzzy | 1.3315 × 10−2 | 2.4774 × 10−1 | |
GWO-EPD | 3.9454 × 10−1 | 1.7424 × 10−1 | |
f15 | GWO | 1.2547 × 10−10 | 7.2242 × 10−11 |
Improved GWO | 2.4467 × 10−13 | 1.0871 × 10−14 | |
MGWO | 7.2101 × 10−4 | 7.9945 × 10−5 | |
NGWO | 1.5547 × 101 | 9.0141 | |
GWO-fuzzy | 2.4875 × 10−13 | 1.0401 × 101 | |
GWO-EPD | 7.2154 × 102 | 9.4012 × 101 |
Function | Algorithm | Average Value | Standard Deviation |
---|---|---|---|
f1 | Improved GWO | 0 | 0 |
PSO | 3.125 × 10−2 | 2.716 × 10−2 | |
SSA | 1.891 × 10−257 | 0 | |
MA | 1.711 × 10−43 | 4.254 × 10−43 | |
f2 | Improved GWO | 0 | 0 |
PSO | 1.416 × 10−1 | 3.581−1 | |
SSA | 1.435 × 10−93 | 8.487 × 10−93 | |
MA | 2.255 × 102 | 8.183 × 102 | |
f3 | Improved GWO | 0 | 0 |
PSO | 7.225 × 10−2 | 5.331 × 10−1 | |
SSA | 2.821 × 10−180 | 0 | |
MA | 7.318 × 10−5 | 5149 × 10−4 | |
f4 | Improved GWO | 0 | 0 |
PSO | 9.225 × 10−2 | 1.153 × 10−1 | |
SSA | 1.354 × 10−93 | 6.81 × 10−93 | |
MA | 8.154 × 10−7 | 6.518 × 10−5 | |
f5 | Improved GWO | 2.867 × 101 | 2.611 × 10−2 |
PSO | 1.314 × 102 | 1.795 × 102 | |
SSA | 2.327 × 10−3 | 2.189 × 10−3 | |
MA | 4.501 × 10−1 | 5.587 × 10−1 | |
f6 | Improved GWO | 6.533 | 2.801 × 10−1 |
PSO | 8.792 × 105 | 9.782 × 105 | |
SSA | 1.047 × 101 | 4.772 | |
MA | 3.128 × 101 | 8.791 × 102 | |
f7 | Improved GWO | 2.961 × 10−7 | 2.373 × 10−7 |
PSO | 2.561 × 10−1 | 7.844 × 10−1 | |
SSA | 1.144 × 10−4 | 3.581 × 10−3 | |
MA | 3.254 × 10−2 | 4.358 × 10−1 | |
f8 | Improved GWO | 0 | 0 |
PSO | 3.015 | 2.641 | |
SSA | 8.161 × 10−185 | 1.254 × 10−186 | |
MA | 2.271 × 10−45 | 5.174 × 10−44 | |
f9 | Improved GWO | 8.881 × 10−16 | 1.604 × 10−16 |
PSO | 3.712 × 10−2 | 2.816 × 10−1 | |
SSA | 8.881 × 10−16 | 0 | |
MA | 4.213 × 10−10 | 1.576 × 10−9 | |
f10 | Improved GWO | 0 | 0 |
PSO | 5.001 × 10−3 | 2.655 × 10−1 | |
SSA | 4.114 × 10−210 | 3.241 × 10−211 | |
MA | 5.260 × 10−140 | 0 | |
f11 | Improved GWO | 2.1331 × 10−3 | 6.8945 × 10−3 |
PSO | 1.8741 | 4.4411 | |
SSA | 1.496 × 10−2 | 2.106 × 10−2 | |
MA | 2.714 × 10−1 | 1.954 × 10−17 | |
f12 | Improved GWO | 1.292 × 10−2 | 7.6012 × 10−2 |
PSO | 8.4152 | 8.3372 | |
SSA | 7.346 × 10−1 | 1.355 × 10−2 | |
MA | 8.214 | 1.245 × 10−2 | |
f13 | Improved GWO | 2.4457 × 10−6 | 6.3641 × 10−6 |
PSO | 1.052 × 102 | 1.2362 | |
SSA | 1.232 × 10−3 | 1.571 × 10−4 | |
MA | 3.247 × 10−3 | 5.014 × 10−3 | |
f14 | Improved GWO | 3.1337 × 10−3 | 1.1184 × 10−3 |
PSO | 3.958 × 10−1 | 1.541 × 10−2 | |
SSA | 9.001 × 10−2 | 0 | |
MA | 3.971 × 10−1 | 6.051 × 10−1 | |
f15 | Improved GWO | 2.4467 × 10−13 | 1.0871 × 10−14 |
PSO | 7.1522 | 9.142 × 101 | |
SSA | 4.701 × 10−7 | 3.147 × 10−8 | |
MA | 5.445 × 10−2 | 4.401 × 10−2 |
Function | GWO | MGWO | NGWO | GWO-Fuzzy | GWO-EPD | SSA | MA | PSO | |
---|---|---|---|---|---|---|---|---|---|
f1 | P | 6.52 × 10−12 | 8.78 × 10−8 | 5.05 × 10−12 | 6.52 × 10−12 | 6.52 × 10−12 | 6.01 × 10−5 | 6.52 × 10−12 | 6.52 × 10−12 |
R | + | + | + | + | + | + | + | + | |
f2 | P | 2.07 × 10−11 | 1.40 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 |
R | + | + | + | + | + | + | + | + | |
f3 | P | 3.77 × 10−10 | 6.52 × 10−12 | 6.52 × 10−12 | 6.52 × 10−12 | 6.52 × 10−12 | 3.77 × 10−10 | 6.52 × 10−12 | 6.52 × 10−12 |
R | + | + | + | + | + | + | + | + | |
f4 | P | 6.52 × 10−12 | 5.05 × 10−11 | 6.52 × 10−12 | 6.52 × 10−12 | 6.52 × 10−12 | 3.77 × 10−11 | 6.52 × 10−12 | 6.52 × 10−12 |
R | + | + | + | + | + | + | + | + | |
f5 | P | 4.60 × 10−3 | 1.20 × 10−5 | 6.01 × 10−3 | 1.09 × 10−2 | 1.68 × 10−4 | 2.05 × 10−2 | 4.23 × 10−1 | 1.20 × 10−6 |
R | + | + | + | + | + | - | - | + | |
f6 | P | 2.07 × 10−11 | 1.41 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 |
R | + | + | + | + | + | + | + | + | |
f7 | P | 3.01 × 10−11 | 5.24 × 10−9 | 3.01 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 |
R | + | + | + | + | + | + | + | + | |
f8 | P | 6.52 × 10−12 | NaN | 6.52 × 10−12 | 6.52 × 10−12 | 6.52 × 10−12 | 2.07 × 10−11 | 6.52 × 10−12 | 6.52 × 10−12 |
R | + | = | + | + | + | + | + | + | |
f9 | P | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 3.77 × 10−10 | 2.07 × 10−11 | 2.07 × 10−11 |
R | + | + | + | + | + | + | + | + | |
f10 | P | 6.52 × 10−12 | NaN | NaN | 6.52 × 10−12 | 6.52 × 10−12 | 2.07 × 10−11 | 2.07 × 10−11 | 6.52 × 10−12 |
R | + | = | = | + | + | + | + | + | |
f11 | P | 6.52 × 10−12 | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 6.52 × 10−12 | 2.07 × 10−11 | 2.07 × 10−11 | 6.52 × 10−12 |
R | + | + | + | + | + | + | + | + | |
f12 | P | 6.52 × 10−12 | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 6.52 × 10−12 | 2.07 × 10−11 | 2.07 × 10−11 | 6.52 × 10−12 |
R | + | + | + | + | + | + | + | + | |
f13 | P | 6.52 × 10−12 | 1.20e−06 | 6.52 × 10−12 | 6.52 × 10−12 | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 6.52 × 10−12 |
R | + | + | + | + | + | + | + | + | |
f14 | P | 6.52 × 10−12 | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 2.07 × 10−11 | 6.52 × 10−12 |
R | + | + | + | + | + | + | + | + | |
f15 | P | 2.07 × 10−11 | 6.52 × 10−12 | 6.52 × 10−12 | 2.07 × 10−11 | 6.52 × 10−12 | 6.52 × 10−12 | 6.52 × 10−12 | 6.52 × 10−12 |
R | + | + | + | + | + | + | + | + |
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Hou, Y.; Gao, H.; Wang, Z.; Du, C. Improved Grey Wolf Optimization Algorithm and Application. Sensors 2022, 22, 3810. https://doi.org/10.3390/s22103810
Hou Y, Gao H, Wang Z, Du C. Improved Grey Wolf Optimization Algorithm and Application. Sensors. 2022; 22(10):3810. https://doi.org/10.3390/s22103810
Chicago/Turabian StyleHou, Yuxiang, Huanbing Gao, Zijian Wang, and Chuansheng Du. 2022. "Improved Grey Wolf Optimization Algorithm and Application" Sensors 22, no. 10: 3810. https://doi.org/10.3390/s22103810