SwarmIntelligence Optimization Method for Dynamic Optimization Problem
Abstract
:1. Introduction
 A universal swarmintelligence dynamic optimization method is summarized and proposed, which lays a theoretical foundation for subsequent research on using the swarmintelligence technique to solve dynamic optimization problems.
 A novel modified SSA is implemented from the application side and utilized to improve the efficiency and accuracy of typical dynamic optimization problems.
 Other wellknown swarmintelligence techniques for dynamic optimization are further investigated under a universal optimization framework.
2. Preliminaries
2.1. Dynamic Optimization Problem Description
2.2. CVP Strategy
2.3. SwarmIntelligence Dynamic Optimization Method Based on CVP Strategy
 (1)
 Through the CVP strategy, $u(t)$ is transformed into $\sigma (t)$, and the dynamic optimization problem shown in Equation (1) is transformed into the static optimization problem form shown in Equation (3).
 (2)
 Set relevant parameters, such as population size, the maximum number of iterations, and algorithm parameters.
 (3)
 Initialize the population.
 (4)
 Evaluate and sort the fitness values of individuals in the population and record the current optimal value.
 (5)
 According to the evolution strategy of the algorithm, a new population is generated.
 (6)
 Compare the fitness value of the new solution and replace it if it is better than the current value.
 (7)
 Determine whether the current condition meets the stop criterion; if so, terminate the algorithm and output the optimal solution. Otherwise, return to (4) and continue to execute, and set t = t + 1.
3. Mathematical Models and Algorithms
3.1. Sparrow Search Algorithm
3.2. MultiStrategy Improved Hybrid SwarmIntelligence Optimization Algorithm
3.2.1. GoodPoint Set Theory
3.2.2. Hybrid Algorithm Strategy
3.2.3. Stagnation Disturbance Strategy Based on Lévy Flight
3.2.4. Early Warning Process Based on Student’s tDistribution Mutation Factor
Algorithm 1: The framework of CMHSSA 
Input: Max_Iter: the maximum iteration; N: the population size; PD: the proportion of producers; SD: the proportion of early warning sparrows; c_{s}, c_{e}: the inertia weight adjustment parameters. Output:X_{best}: the optimal individual location; f_{g}: the fitness value of the optimal individual. /* Initialization*/
/*Iterative search*/
/*Algorithm terminated*/

3.3. Benchmark Function Experiments
3.3.1. Parameter Settings
3.3.2. Statistical Result Comparison
4. Case Studies in Dynamic Optimization
4.1. Problem 1: Batch Reactor Consecutive Reaction
4.2. Problem 2: Catalyst Mixing Reaction in Tubular Reactor
4.3. Problem 3: Parallel Reactions in Tubular Reactor
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Benchmark Function  Formula  Range  Opt 

Sphere Model  ${F}_{1}(x)={\displaystyle \sum _{i=1}^{n}{x}_{i}{}^{2}}$  [−100, 100]  0 
Schwefel’s problem 2.22  ${F}_{2}(x)={\displaystyle \sum _{i=1}^{n}\left{x}_{i}\right+}{\displaystyle \prod _{i=1}^{n}\left{x}_{i}\right}$  [−10, 10]  0 
Schwefel’s problem 1.2  ${F}_{3}(x)={{\displaystyle \sum _{i=1}^{n}\left({\displaystyle \sum _{j=1}^{i}{x}_{j}}\right)}}^{2}$  [−100, 100]  0 
Schwefel’s problem 2.21  ${F}_{4}(x)={\mathrm{max}}_{i}\left\{\left{x}_{i}\right,1\le i\le n\right\}$  [−100, 100]  0 
Generalized Schwefel’s problem 2.26  ${F}_{5}(x)={\displaystyle \sum _{i=1}^{n}{x}_{i}\mathrm{sin}\sqrt{\left{x}_{i}\right}}$  [−500, 500]  −4.18.9829D 
Generalized Rastrigin’s Function  ${F}_{6}(x)={\displaystyle \sum _{i=1}^{n}[{x}_{i}^{2}10\mathrm{cos}(2\pi {x}_{i})+10]}$  [−5.12, 5.12]  0 
Ackley’s Function  ${F}_{7}(x)=20\mathrm{exp}\left(0.2\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{x}_{i}{}^{2}}}\right)\mathrm{exp}\left(\frac{1}{n}{\displaystyle \sum _{i=1}^{n}\mathrm{cos}\left(2\pi {x}_{i}\right)}\right)+20+e$  [−32, 32]  0 
Generalized Griewank Function  ${F}_{8}(x)=\frac{1}{4000}{\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}{\displaystyle \prod _{i=1}^{n}\mathrm{cos}\frac{{x}_{i}}{\sqrt{i}}+1}}$  [−600, 600]  0 
Branin Function  ${F}_{9}(x)={\left({x}_{2}\frac{5.1}{4{\pi}^{2}}{x}^{2}+\frac{5}{\pi}{x}_{1}6\right)}^{2}+10\left(1\frac{1}{8\pi}\right)\mathrm{cos}{x}_{1}+10$  [−5, 5]  0.398 
Goldstein–Price Function  $\begin{array}{c}{F}_{10}(x)=[1+{({x}_{1}+{x}_{2}+1)}^{2}(1914{x}_{1}+3{x}_{1}^{2}14{x}_{2}+6{x}_{1}{x}_{2}+3{x}_{2}^{2})]\times \\ [30+{(2{x}_{1}3{x}_{2})}^{2}(1832{x}_{1}+12{x}_{1}^{2}+48{x}_{2}36{x}_{1}{x}_{2}+27{x}_{2}^{2})]\end{array}$  [−2, 2]  3 
Function  Result  WOA  MPA  HHO  SSA  SGO  CMHSSA 

${F}_{1}$  Mean  3.5706 × 10^{−10}  1.9437  1.3052 × 10^{−20}  2.9595 × 10^{−33}  4.3773 × 10^{−135}  0 
Std.  7.1180 × 10^{−10}  1.0336  5.8268 × 10^{−20}  1.3235 × 10^{−33}  2.9367 × 10^{−136}  0  
TIC/TOC  0.075297  0.260806  0.122167  0.098561  0.121907  0.102957  
${F}_{2}$  Mean  9.7066 × 10^{−9}  9.6357 × 10^{−2}  4.3809 × 10^{−13}  2.0208 × 10^{−21}  1.5103 × 10^{−68}  0 
Std.  2.1815 × 10^{−8}  3.3799 × 10^{−2}  1.0991 × 10^{−12}  8.9306 × 10^{−21}  1.6423 × 10^{−69}  0  
TIC/TOC  0.060907  0.197253  0.121605  0.090216  0.127647  0.096940  
${F}_{3}$  Mean  9.8067 × 10^{4}  2.1566 × 10^{2}  1.8472 × 10^{−13}  4.1637 × 10^{−33}  2.2632 × 10^{−135}  0 
Std.  2.8622 × 10^{4}  1.8756 × 10^{2}  8.2479 × 10^{−13}  1.8621 × 10^{−32}  9.9703 × 10^{−136}  0  
TIC/TOC  0.098011  0.306055  0.232791  0.116329  0.246341  0.148943  
${F}_{4}$  Mean  5.6341 × 10^{1}  4.5856 × 10^{−1}  3.6428 × 10^{−13}  3.9747 × 10^{−21}  1.0950 × 10^{−68}  0 
Std.  2.8796 × 10^{1}  1.0924 × 10^{−1}  6.1075 × 10^{−13}  1.7745 × 10^{−20}  5.3688 × 10^{−70}  0  
TIC/TOC  0.060370  0.203315  0.113611  0.106351  0.126110  0.117772  
${F}_{5}$  Mean  −8.6688 × 10^{3}  −7.2695 × 10^{3}  −1.2356 × 10^{4}  −6.2868 × 10^{3}  −6.9435 × 10^{3}  −1.06 × 10^{4} 
Std.  1.0522 × 10^{3}  4.7419 × 10^{2}  7.9240 × 10^{2}  1.6650 × 10^{3}  6.5873 × 10^{2}  7.8299 × 10^{2}  
TIC/TOC  0.067170  0.251228  0.157786  0.079425  0.111361  0.098025  
${F}_{6}$  Mean  1.2998 × 10^{−8}  8.6017  0  0  0  0 
Std.  5.3053 × 10^{−8}  7.2342  0  0  0  0  
TIC/TOC  0.060027  0.217248  0.177083  0.076583  0.130681  0.106733  
${F}_{7}$  Mean  3.7669 × 10^{−7}  8.9223 × 10^{−2}  7.3576 × 10^{−12}  1.0658 × 10^{−15}  8.8818 × 10^{−16}  8.8818 × 10^{−16} 
Std.  5.5884 × 10^{−7}  2.8663 × 10^{−2}  2.2196 × 10^{−12}  7.9441 × 10^{−16}  0  0  
TIC/TOC  0.071849  0.177886  0.127309  0.079305  0.122893  0.098612  
${F}_{8}$  Mean  9.1942 × 10^{−1}  2.7695 × 10^{−1}  0  0  0  0 
Std.  2.8307 × 10^{−1}  1.4355 × 10^{−1}  0  0  0  0  
TIC/TOC  0.073628  0.198219  0.159379  0.077422  0.131028  0.115157  
${F}_{9}$  Mean  4.0011 × 10^{−1}  3.9789 × 10^{−1}  3.9853 × 10^{−1}  3.9789 × 10^{−1}  3.9789 × 10^{−1}  3.9789 × 10^{−1} 
Std.  3.7678 × 10^{−3}  5.0943 × 10^{−11}  1.1419 × 10^{−3}  6.3089 × 10^{−7}  2.3781 × 10^{−8}  0  
TIC/TOC  0.051275  0.173310  0.137616  0.098129  0.096289  0.083473  
${F}_{10}$  Mean  8.4292  3.0201  3.0231  3.0023  3.0001  3.0000 
Std.  1.1139 × 10^{1}  1.8833 × 10^{−10}  1.4497 × 10^{−4}  2.7527 × 10^{−7}  1.9782 × 10^{−7}  2.2017 × 10^{−15}  
TIC/TOC  0.056070  0.183948  0.155738  0.071500  0.107347  0.081171 
Function  CMHSSA vs. WOA  CMHSSA vs. MPA  CMHSSA vs. HHO  CMHSSA vs. SSA  CMHSSA vs. SGO 

${F}_{1}$  8.0065 × 10^{−9}  8.0065 × 10^{−9}  8.0065 × 10^{−9}  8.0065 × 10^{−9}  8.0065 × 10^{−9} 
${F}_{2}$  8.0065 × 10^{−9}  8.0065 × 10^{−9}  8.0065 × 10^{−9}  8.0065 × 10^{−9}  8.0065 × 10^{−9} 
${F}_{3}$  8.0065 × 10^{−9}  8.0065 × 10^{−9}  8.0065 × 10^{−9}  2.992 × 10^{−8}  8.0065 × 10^{−9} 
${F}_{4}$  8.0065 × 10^{−9}  8.0065 × 10^{−9}  8.0065 × 10^{−9}  8.0065 × 10^{−9}  8.0065 × 10^{−9} 
${F}_{5}$  2.6609 × 10^{−6}  6.7004 × 10^{−8}  6.1833 × 10^{−4}  6.8341 × 10^{−7}  6.7004 × 10^{−8} 
${F}_{6}$  2.9868 × 10^{−8}  8.0065 × 10^{−9}  N/A  N/A  N/A 
${F}_{7}$  8.0065 × 10^{−9}  8.0065 × 10^{−9}  1.0433 × 10^{−7}  3.4211 × 10^{−4}  N/A 
${F}_{8}$  8.0065 × 10^{−9}  8.0065 × 10^{−9}  N/A  N/A  N/A 
${F}_{9}$  1.1597 × 10^{−4}  6.7956 × 10^{−8}  1.0581 × 10^{−4}  8.0065 × 10^{−9}  5.0209 × 10^{−5} 
${F}_{10}$  8.0065 × 10^{−9}  8.0065 × 10^{−9}  2.1025 × 10^{−7}  4.0137 × 10^{−8}  1.9299 × 10^{−3} 
Method  Mean  Std.  TIC/TOC 

WOA  0.60718532  5.9120 × 10^{−4}  359.5657 
MPA  0.61070726  7.8608 × 10^{−4}  344.6441 
HHO  0.61047035  1.9521 × 10^{−3}  1092.0483 
SSA  0.61077333  2.4912 × 10^{−7}  351.2811 
SGO  0.60584429  9.7315 × 10^{−4}  767.8168 
CMHSSA  0.61079200  2.9799 × 10^{−7}  347.2429 
Method  NE  J/(mol/L) 

OC [50]    0.61 
SQP [51]  80  0.610775 
IDP [52]  80  0.610775 
PSOCVP [12]    0.6105359 
IKEA [53]  10  0.6101 
20  0.610426  
100  0.610781–0.610789  
HIGA [54]  10  0.61007 
20  0.61046  
IKBCA [17]  10  0.6101 
20  0.610454  
100  0.610779–0.610787  
EBSO [13]  10  0.610558922 
20  0.61064758  
80  0.61078114  
MSFO [16]  50  0.610771–0.610785 
ISOA [15]  30  0.61059223 
CVPPSO [3]    0.6107847 
CVPAPSO [3]    0.6107850 
This work (CMHSSA)  100  0.61079200 
Method  Mean  Std.  TIC/TOC 

WOA  0.47625678  5.8233 × 10^{−4}  426.7259 
MPA  0.47742011  6.4257 × 10^{−4}  1034.2457 
HHO  0.47478338  1.7305 × 10^{−3}  1213.1210 
SSA  0.47744034  9.2403 × 10^{−4}  503.7366 
SGO  0.47530289  3.8821 × 10^{−4}  1161.0388 
CMHSSA  0.47770179  2.7368 × 10^{−5}  457.1058 
Methods  NE  J/(mol/L) 

IDP [52]  20  0.47527 
40  0.47695  
ACO [14]    0.47615 
IKEA [53]  10  0.475 
20  0.4757  
100  0.47761–0.47768  
IKBCA [17]  20  0.4753 
100  0.47768–0.47770  
EBSO [13]  10  0.47502183 
20  0.47627191  
40  0.47697288  
MSFO [16]  20  0.47562 
70  0.477544–0.47760  
ISOA [15]  40  0.47721 
This work (CMHSSA)  100  0.47770179 
Method  Mean  Std.  TIC/TOC 

WOA  0.56836465  1.4475 × 10^{−3}  364.3671 
MPA  0.57349880  1.0338 × 10^{−3}  381.3942 
HHO  0.57152795  2.9227 × 10^{−3}  1062.2493 
SSA  0.57269740  8.7921 × 10^{−3}  349.0882 
SGO  0.55138595  3.5277 × 10^{−4}  685.3328 
CMHSSA  0.57355371  4.2218 × 10^{−6}  376.5377 
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Liu, R.; Mo, Y.; Lu, Y.; Lyu, Y.; Zhang, Y.; Guo, H. SwarmIntelligence Optimization Method for Dynamic Optimization Problem. Mathematics 2022, 10, 1803. https://doi.org/10.3390/math10111803
Liu R, Mo Y, Lu Y, Lyu Y, Zhang Y, Guo H. SwarmIntelligence Optimization Method for Dynamic Optimization Problem. Mathematics. 2022; 10(11):1803. https://doi.org/10.3390/math10111803
Chicago/Turabian StyleLiu, Rui, Yuanbin Mo, Yanyue Lu, Yucheng Lyu, Yuedong Zhang, and Haidong Guo. 2022. "SwarmIntelligence Optimization Method for Dynamic Optimization Problem" Mathematics 10, no. 11: 1803. https://doi.org/10.3390/math10111803