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Swarm intelligence algorithms are a class of bionic probabilistic heuristic search methods that are inspired by the collective behaviors of biological agents. In this paper, a multigroup cooperative evolutionary optimization algorithm is proposed by referring to the interaction behaviors of species diversity and stability in the ecosystem. First, the group updating mechanism of the traditional seeking and tracking mode with a dynamic population update mechanism is adopted. The multi-population interactive update group and the quantum entanglement update group are introduced to guide the algorithm to gradually approach the global optimal solution. Second, the proposed bionic algorithm is extended for cross-field applications. The algorithm is applied to solve the function optimization problems, as well as problems in four distinct application fields, including robot routing optimization of grid maps, vehicle scheduling optimization of dairy enterprises, location optimization of logistics centers, and plasma trajectory planning optimization. The proposed multigroup cooperative evolutionary optimization algorithm achieves competitive results in these application fields, thus demonstrating its versatility and robustness.
Introduction
Swarm intelligence optimization algorithms are bionic random probability heuristic search algorithm without central control (Blum and Li 2008). Therefore, the performance of the whole population in solving a problem is not affected when one or several individuals perform poorly. Compared with traditional algorithms, swarm intelligence optimization algorithms more easily jump out of locally optimal solutions (Blum and Li 2008). Through appropriate cooperation and competition strategies among individuals and a heuristic intelligent bionics update mechanism among individuals, great improvements have been made in function optimization (Tang et al. 2021). In combination with the swarm intelligence bionic strategy within the population, cooperative evolution strategies, such as cooperation, competition, parasitism and mutualism among different populations, are particularly important for effectively solving the function optimization and combinatorial optimization problems of engineering design, improving the prediction accuracy and controlling the prediction error (Dorigo et al. 2007). As shown in Fig. 1, these algorithms can be roughly divided into biological evolutionary algorithms, human behavior simulation algorithms (Ray and Liew 2003) and animal behavior simulation algorithms (He et al. 2009) according to the different types of simulated individual behaviors.
[See PDF for image]
Fig. 1
The overview of swarm intelligence optimization algorithms
The genetic algorithm (GA) (Lambora et al. 2019) is a method that simulates selection, crossover, and mutation behaviors in the natural selection process of biological evolution theory. The differential evolution (DE) (Price et al. 2006) algorithm considers the differences between two individuals. The selection and parameter adjustment process for artificial numerical genetic operators used by the DE algorithm is based on a time-consuming trial process. The success rate of the Bezier search differential (BeSD) (Civicioglu and Besdok 2020) algorithm is not sensitive to the structure or internal parameters of the genetic operators.
Optimization algorithms simulate human interaction behaviors, including rescue, education, games, competition, and ancient and modern war behaviors. The seeker optimization algorithm (SOA) (Dai et al. 2009) studies the intelligent behaviors of communication, collaboration, memory, reasoning, learning knowledge and experience that are used by human beings in random rescue seeking, and optimizes the solutions of the problems by simulating the experience gradient and uncertain reasoning of human search. The teaching learning based optimization (TLBO) (Rao and More 2015; Kundu and Garg 2022) simulates the single classroom teaching environment. The school based optimization (SBO) (Farshchin et al. 2018; Degertekin et al. 2021) algorithm simulates multiple classrooms with multiple teachers, and allows better teachers to be selected from other classrooms to jump out of the local best state of the current classroom. The chaos game optimization (CGO) (Talatahari and Azizi 2020, 2021) algorithm simulates human game behaviors on the basis of chaos theory. The rider optimization algorithm (ROA) (Binu and Kariyappa 2018; Alazab et al. 2021) simulates the competitive behaviors of human beings, and simulates the processes in which a rider guides the whole team to succeed through different identity divisions of detour, following, overtaking, and sprinting. The imperialist competition algorithm (ICA) (Atashpaz-Gargari and Lucas 2007) simulates modern human war behaviors, such as assimilation and revolution, intra empire competition, and inter empire competition in modern wars. The war strategy optimization (WSO) (Ayyarao and Kumar 2022) simulates the ancient war behaviors of human beings. The algorithm executes the attack strategies and defense strategies according to the battle drum instructions of the kings and the commanders, and uses a new weight update mechanism and a weak soldier relocation strategy.
Algorithms that simulate animal behaviors are divided into algorithms that simulate aerial animals, land animals, and marine animals. The pigeon inspired optimization (PIO) (Duan and Qiao 2014; Chen et al. 2019) simulates the homing behaviors of pigeons flying in the air. The algorithm distinguishes a general direction according to the sun and geomagnetic field, and then uses the geomorphic scene to correct the current direction. The artificial hummingbird algorithm (AHA) (Zhao et al. 2022) simulates three foraging strategies—guided foraging, territorial foraging and migratory foraging, and three flight strategies—axial, diagonal and omnidirectional flight. The gray wolf optimization (GWO) (Mirjalili et al. 2014; Faris et al. 2018) simulates four social hierarchies of gray wolves in land animals. Leader is responsible for all decisions of hunting and food distribution; Think tank team members are responsible for assisting in making decisions. A wolf is responsible for investigation, sentry and nursing; and another wolf is responsible for balancing internal relations. The augmented grey wolf optimization (AGWO) (Qais et al. 2018) algorithm modifies the coefficient updating mechanism from the original linear decreasing mode to the nonlinear fluctuation decreasing mode. The improved grey wolf optimization (IGWO) (Nadimi-Shahraki et al. 2021) algorithm benefits from a Dimension Learning based Hunting (DLH) strategy and the update of each dimension refers to random neighbor positions to realize information sharing. Among marine animals, the great white shark optimization (WSO) (Braik et al. 2022) algorithm simulates the abnormal hearing and smell of the great white shark while it is navigating and foraging. According to the frequency of the waves in the process of movement and turbulence, it can accurately determine the position and size of prey, and it can randomly search for prey in the deep ocean and hunt near small fish shoals.
Although swarm intelligence optimization algorithms are relatively mature, and simulate various behaviors in real life and nature, relatively few optimization algorithms consider multiple swarm intelligence interaction behaviors (Tang et al. 2021). However, real ecosystems are multi-populations, and the collective behaviors of swarm intelligence have the paradox of stability and diversity (Dalziel et al. 2021). If the number of prey in the food chain increases, the number of predators will increase due to the abundance of food. However, having too many predators will lead to a decrease in the number of prey, which will result in a decrease in the number of predators owing to a lack of food, and the predator and prey populations will show periodic fluctuations. The multi-population optimization algorithm simulates the cooperative relationship between predators and prey in an ecosystem. The predatory behaviors among populations correspond to the exploitation and utilization of optimization algorithms (Hammouri et al. 2020), and the regeneration and growth behaviors of predators and prey correspond to the exploration and innovation of algorithms (Zhou et al. 2022). The periodic fluctuations of different species in the food chain of an ecosystem maintain the ecological balance, ecological stability and diversity. With the interaction of exploitation and exploration among various groups, the swarm intelligence optimization algorithm gradually jumps out of locally optimal solutions and tends to a globally optimal solution. Cats are in the middle of the food chain and have prey and predators. In dynamic cat swarm optimization (Ahmed et al. 2021), the seeking and tracking behaviors of cats are simulated, and the interaction behaviors downstream of the food chain are considered, whereas the interaction behaviors upstream of the food chain and the quantum entanglement behavior between different cats are neglected. Although research on function optimization and single engineering applications such as robot navigation (Zeng et al. 2020) and neural signal processing (Duan et al. 2024; Lian et al. 2021), has achieved good results, relatively little research has been conducted on cross-field applications. Research on cross-field application algorithms with wide applicability and relatively good effects can promote the cross-integration of bioecology, mathematics, engineering applications, and information science; promote the transformation of scientific theory into practical products; and provide more convenient services for people’s lives.
In this work, inpired by the interactive behaviors of species that underlie the diversity and stability of an ecosystem, a multigroup cooperative evolutionary optimization algorithm was developed. The group updating mechanisms of the traditional seeking mode and tracking mode were adopted, and a dynamic population update mechanism was introduced. The multi-population interactive update group and the quantum entanglement update group in the proposed method guide the algorithm to gradually approach the globally optimal solution. In addition, the proposed bionic algorithm was extended to four cross-field applications, and competitive results were achieved.
The remainder of this paper is organized as follows. The related works are described in Sect. 2. The proposed multigroup cooperative cross-field evolutionary optimization algorithm in combination with quantum entanglement is described in Sect. 3. The experimental results are compared with those of other methods in Sect. 4. The main findings of this study are presented in Sect. 5.
Related works
Swarm intelligence algorithm
Most swarm intelligence algorithms gradually approach the optimal solution through iterative updates between individuals in the population; however, they differ in terms of the biomimetic behavior or natural laws that are referenced by iterative updates between individuals. The particle swarm optimization (PSO) (Kennedy and Eberhart 1995) is inspired by bird predation behavior, and the positions of individuals are updated by weighting their historical and global best solutions. The satin bowerbird optimization (SatinBO) (Moosavi and Bardsiri 2017; Chen et al. 2021) simulates the courtship behaviors of the adult male satin bowerbird in nature. The male individuals spend most of their time building courtship booths, and they make a squeaking hissing sound and attract the female individuals who pass by with bright objects. The carnivorous Plant Algorithm (CPA) (Ong et al. 2021) is inspired by the growth and reproductive behavior of carnivorous plants, and refers to the plant growth exploration behavior of individuals updated with the optimal food within the group and the reproductive behavior of individuals updated with the globally optimal plant. The turbulent flow of water-based optimization (TFWO) (Ghasemi et al. 2020) is inspired by the features of vortices, such as the centripetal and centrifugal forces of vortices, the influence of vortices on other objects inside a vortex, and the interactions between vortices. The Giza Pyramids construction (GPC) (Harifi et al. 2021) was inspired by the combined force of external forces exerted by workers during the Pyramid construction process and the gravity of stones on the slope, which affects the acceleration and position of individuals. The Henry gas solubility optimization (HGSO) (Hashim et al. 2019) is inspired by Henry’s law, which states that temperature and pressure affect the solubility of gases in liquids, and solubility affects the renewal of individual positions. The atom search optimization (ASO) (Zhao et al. 2019) is inspired by molecular dynamics, and the motion parameters of individuals, such as velocity and acceleration are influenced by the forces and potential energies between different atoms.
Dynamic cat swarm optimization
The cat swarm optimization (CSO) is a bionic swarm intelligence algorithm that simulates cat predation behaviors. Cats spend most of their time at rest, which is called the seeking mode, and the second mode is called the tracking mode, which is the state when cats track prey. represents the proportion of cats in the tracking mode. is smaller than 0.5, and cats spend only a small part of their time tracking prey. The relative fitness is calculated by scaling and normalizing the fitness according to the maximum and minimum values. In the seeking mode, the first individual whose relative fitness value is greater than a random number is selected.
The dynamic cat swarm optimization (DCSO) (Ahmed et al. 2021) overcomes the premature convergence of the cat swarm algorithm, modifies the selection mechanism, and adopts the dynamic parameter . The scaling range coefficient of the dimension changes from a fixed value to a random value . The dimension-level random value is modified to an individual-level random value, and the random values for different dimensions of the same individual are the same. The best cat in the seeking mode is selected for scaling, and follow the best rat in tracking mode is followed.
1
Proposed method
The proposed framework of multigroup cooperative cross-field evolutionary optimization algorithm in combination with quantum entanglement is shown in Fig. 2. The proposed framework includes two parts: bionic quantum multigroup DCSO (QMDCSO) algorithm and cross-field applications.
A multigroup cooperative evolutionary optimization algorithm QMDCSO was developed by referring to the interaction behaviors of species diversity and stability in the ecosystem. The group updating mechanism of the traditional seeking mode and tracking mode with the dynamic population update mechanism were adopted. Furthermore, multi-population interactive update group were also introduced, and a quantum entanglement update group was used to guide the algorithm to gradually approach the globally optimal solution.
The proposed bionic QMDCSO algorithm was extended to cross-field applications. The algorithm was not only applied to 33 test functions of open CEC problems, but also extended to robot routing optimization of grid maps, vehicle scheduling optimization of dairy enterprises, location optimization of logistics centers, and plasma trajectory planning optimization, and competitive results were obtained.
[See PDF for image]
Fig. 2
Overall framework of the proposed QMDCSO method
Multi-population interactive group
As shown in Fig. 3, there are separation behaviors, aggregation behaviors, alignment behaviors, predatory behaviors, and distract attention behaviors in the multi-population interactive group. The separation behaviors keep the cats away from the center of the neighbors when updating their locations. The neighbors are marked with blue edges and the center of the neighbors is represented by the small blue dot in the graph. The aggregation behaviors make a cat update its position toward the center of the neighbors. The alignment behaviors make a cat update the motion direction toward the sum direction of the neighbor motion direction vectors. The small black arrow represents the neighbor motion direction, the small red arrow represents the sum direction of the neighbor motion direction vectors, and the large red arrow represents the new direction of the current cat. Predatory behavior causes the cat to update its position toward prey, such as rats. Distractive attention behavior make the cat away from its natural enemies, such as jungle cats.
The separation behaviors of the cat population were simulated. The cats are separated from each other to avoid the individual distances being too close, which would reduce the optimization efficiency. If cat i has neighbors, is the opposite of the cumulative sum of the difference between neighbor and the current position ; otherwise it is 0. is the number of neighbors around cat i.
2
The alignment behaviors of cats were simulated. If a cat has neighbors, is the average speed of the neighbors, which is the sum of speed vectors divided by the number of neighbors; otherwise, it is the current speed.3
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Fig. 3
The figure of multi-population interactive group
The aggregation behaviors of cats were simulated, and some cats tend to approach nearby centers. is similar to , the former is related to position, and the latter is related to speed.
4
The predatory behaviors of cats were also simulated. The speed was updated according to the differences between the positions of the cats near the rat and the best individual rat in the current iteration. is the number of cats within the prey range around the rat. The distance measurement is the distance in each dimension between each cat and the rat, which is less than . If , is updated as follows:5
The behaviors of cats to distract attention from the natural enemies, such as jungle cat, leopard cat or strange human, were simulated. The positions of the cats near the enemies were updated with reference to the position of the natural enemy with the worst solution in the current iteration. If , the is updated as follows:6
Considering the separation behavior , alignment behavior , aggregation behavior , predatory behavior, and attention distraction behavior of multiple populations and the inertia factors caused by the historical speed , the speed and the position of individual i are updated as follows.7
8
Quantum entanglement group
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Fig. 4
Quantum entanglement group
Quantum entanglement is a phenomenon in which particles interact in a system that is composed of two or more particles (Chen et al. 2019). It describes the entanglement of two particles with each other. Even if they are far apart, a change in the state of one particle will affect the state of the other. As shown in Fig. 4, a quantum entanglement phenomenon occurs also between cats and tracked preys such as rats.
Some cats are encoded with quantum entanglement phenomenon and are updated on the basis of a quantum rotation gate and the best individuals. The initialization quantum is expressed in two states, and the initial probability value matrix of each state is .
The update direction of quantum individuals is determined by a normal distribution related to the current optimal solution and the quantum probability value matrix . is a random number between 0 and 1. and are the maximum and minimum values of individual in different dimensions for normalization reference.
9
10
The new speed and position of the quantum individual are updated. is the speed before the update of quantum individual i. , i is the index of the rat tracked by the current cat i, and is the current iteration number.
11
The probability value matrix of the quantum individual i is updated through revolving gate. If the best fitness after the update is better than the best fitness before the update, the quantum probability value matrix is initially a matrix with the same probability of two states . If is not better than , the update of depends on the variable angle , and . is .
12
13
Seeking group
As shown in Fig. 5, the cats with the randomly scaled updated positions and cat i at the last iteration position are put into the memory pool in the seeking group, and the best position in the memory pool will be selected as the new position of the current seeking cat. The cats with the randomly scaled updated positions are marked with red edges, and the cat with the last iteration position is marked with a black edge.
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Fig. 5
Traditional seeking and tracking group
The function can generate a random integer vector with an -dimensional value between 1 and . is the number of change dimensions. is the dimension index vector to be changed. For different dimensions of the same individual, the same random number is used. If the random number , then ; otherwise .
14
Tracking group
As shown in Fig. 5, each cat tracks the best rat . The variables i, , , , and are the index, speed before update, speed after update, position before update and position after update of the tracked individual. decreases linearly with iteration from 0.9 to 0.4 and is 2.05.
15
16
Dynamic update mechanism
The weight coefficient in multi-population interactive group is updated dynamically as follows.
17
The numbers of individuals in different groups are updated dynamically. The numbers of individuals in multi-population interactive group , quantum entanglement group , seeking group , and tracking group are updated dynamically with iteration .18
The pseudo code for the QMDCSO algorithm is as follows.[See PDF for image]
Algorithm 1
QMDCSO algorithm
Results
Owing to the randomness of the experimental results, the experimental was run randomly ten times on the open CEC problems, and the mean and variance of the ten runs were calculated. In the four cross-field expansion applications, the random seed was fixed to the same default value of 1 by referring to the strategy of the AFO algorithm (Yang et al. 2021). The parameters was 0.5 and was 0.3. In the case of the CEC problems, was 0.2, in the other cases, was 0.1. was 100 for F1-23 problems and was 50 for the standard CEC 2019 problems. In the robot routing optimization application, was 10 and was 20. In the vehicle scheduling optimization and location optimization application of the logistics center, was 20 and was 50. In plasma trajectory planning optimization, was 20 and was 100 (Fig. 6).
CEC problems
The parameters and descriptions of the test functions F1-F23 are shown in Tables 1 and 3 (Mirjalili and Lewis 2016), respectively. cecF1-cecF10 are the “100-digit challenge” of standard CEC 2019 optimization problems (Price et al. 2018), and their visualized parameter spaces under 2D input are shown in Fig. 7. The standard CEC 2019 optimization problems are described in Table 2.
Table 1. The parameters in F1-23 problems
Fun | Parameters | Fun | Parameters |
|---|---|---|---|
F1 | F13 | ||
F2 | F14 | ||
F3 | F15 | ||
F4 | F16 | ||
F5 | F17 | ||
F6 | F18 | ||
F7 | F19 | ||
F8 | F20 | ||
F9 | F21 | ||
F10 | F22 | ||
F11 | F23 | ||
F12 |
Table 2. The descriptions of standard CEC 2019 optimization
Fun | Parameters | |
|---|---|---|
cecF1 | Storn’s Chebyshev Polynomial Fitting Problem | |
cecF2 | Inverse Hilbert Matrix Problem | |
cecF3 | Lennard–Jones minimum Energy Cluster Problem | |
cecF4 | Rastrigin Function Problem | |
cecF5 | Griewangk Function Optimization Problem | |
cecF6 | Weierstrass Function Optimization Problem | |
cecF7 | Modified Schwefel Function | |
cecF8 | Expanded Schaffer Optimization Problem | |
cecF9 | Happy Cat Function | |
cecF10 | Ackley Function |
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Fig. 6
The average fitness curves of different swarm intelligence algorithms for F1-F23 test functions
Table 3. The F1 to F23 of function optimization problems (Mirjalili and Lewis 2016)
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Fig. 7
The visualized parameter space of standard CEC 2019 optimization problems under 2D input
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Fig. 8
The average fitness curves for standard CEC 2019 problems
Table 4. The average results for F1-F23 and standard CEC 2019 problems
Fun | QMDCSO | DCSO (Ahmed et al. 2021) | Compared | QMO | Compared |
|---|---|---|---|---|---|
F1 | 4.8110E-202 | 4.6310E-204 | −1 | 9.1949E-24 | 1 |
F2 | 1.0305E-109 | 7.1505E-108 | 1 | 1.8514E-01 | 1 |
F3 | 2.5966E-193 | 1.9031E-197 | −1 | 1.0726E-01 | 1 |
F4 | 1.8857E-96 | 6.6919E-100 | −1 | 1.2995E-12 | 1 |
F5 | 4.4333E+00 | 5.5379E+00 | 1 | 8.6019E+00 | 1 |
F6 | 1.3194E-07 | 3.3419E-06 | 1 | 7.4131E-01 | 1 |
F7 | 1.6819E-04 | 6.6492E-05 | −1 | 7.2423E-05 | −1 |
F8 | −3.6241E+03 | −3.4157E+03 | 1 | −3.9006E+03 | −1 |
F9 | 0.0000E+00 | 0.0000E+00 | 0 | 8.2675E-01 | 1 |
F10 | 8.8818E-16 | 8.8818E-16 | 0 | 2.5616E-11 | 1 |
F11 | 0.0000E+00 | 0.0000E+00 | 0 | 0.0000E+00 | 0 |
F12 | 1.0483E-06 | 7.3250E-06 | 1 | 1.8227E-01 | 1 |
F13 | 9.7635E-03 | 2.9578E-02 | 1 | 2.6296E-01 | 1 |
F14 | 1.1968E+00 | 1.5921E+00 | 1 | 6.3179E+00 | 1 |
F15 | 3.0749E-04 | 3.0787E-04 | 1 | 3.1266E-04 | 1 |
F16 | −1.0316E+00 | −1.0316E+00 | 1 | −1.0316E+00 | 1 |
F17 | 3.0425E-01 | 3.0425E-01 | 1 | 3.0433E-01 | 1 |
F18 | 3.0000E+00 | 3.0000E+00 | 1 | 3.3226E+00 | 1 |
F19 | −3.8625E+00 | −3.8619E+00 | 1 | −3.8626E+00 | −1 |
F20 | −3.3217E+00 | −3.2973E+00 | 1 | −3.3083E+00 | 1 |
F21 | −6.5837E+00 | −5.5650E+00 | 1 | −9.6133E+00 | −1 |
F22 | −5.6184E+00 | −5.6192E+00 | −1 | −1.0384E+01 | −1 |
F23 | −6.2096E+00 | −6.7509E+00 | −1 | −1.0516E+01 | −1 |
cecF1 | 1.0000E+00 | 1.0000E+00 | 0 | 1.0000E+00 | 1 |
cecF2 | 4.8547E+00 | 4.5082E+00 | −1 | 5.1103E+00 | 1 |
cecF3 | 2.4665E+00 | 3.8639E+00 | 1 | 7.4360E+00 | 1 |
cecF4 | 3.0647E+01 | 3.6256E+01 | 1 | 6.5558E+01 | 1 |
cecF5 | 1.5533E+00 | 1.9085E+00 | 1 | 5.5566E+01 | 1 |
cecF6 | 5.9030E+00 | 5.0457E+00 | −1 | 9.5158E+00 | 1 |
cecF7 | 1.1291E+03 | 1.2543E+03 | 1 | 7.9032E+02 | −1 |
cecF8 | 4.2260E+00 | 4.3131E+00 | 1 | 4.6933E+00 | 1 |
cecF9 | 1.2365E+00 | 1.2607E+00 | 1 | 2.9304E+00 | 1 |
cecF10 | 2.0988E+01 | 2.1061E+01 | 1 | 2.1075E+01 | 1 |
Count | – | – | 21:8 | – | 25:7 |
Table 5. The average results Avg comparison with QMDCSO and other algorithms for F1-F23 and standard CEC 2019 problems
Fun | QMDCSO | SatinBO (Moosavi and Bardsiri 2017) | TFWO (Ghasemi et al. 2020) | CPA (Ong et al. 2021) | GPC (Harifi et al. 2021) | Best |
|---|---|---|---|---|---|---|
F1 | 4.8110E-202 | 2.6983E-03 | 2.6899E-23 | 1.1099E-14 | 7.3211E-22 | QDCSO |
F2 | 1.0305E-109 | 8.5971E-03 | 9.3716E-14 | 5.4735E-09 | 4.4235E-12 | QDCSO |
F3 | 2.5966E-193 | 9.1034E+00 | 9.8351E-08 | 7.4044E-04 | 1.7463E-21 | QDCSO |
F4 | 1.8857E-96 | 1.5583E-01 | 3.3439E-04 | 4.2025E-05 | 1.4084E-11 | QDCSO |
F5 | 4.4333E+00 | 1.5105E+02 | 1.6398E+00 | 2.9425E+00 | 7.5112E+00 | TFWO |
F6 | 1.3194E-07 | 1.4753E-03 | 2.3592E-21 | 1.5512E-14 | 5.9401E-01 | TFWO |
F7 | 1.6819E-04 | 1.4355E-02 | 5.5792E-03 | 4.5623E-03 | 4.3785E-05 | GPC |
F8 | −3.6241E+03 | −2.3816E+03 | −4.0477E+03 | −4.0714E+03 | −2.6107E+03 | CPA |
F9 | 0.0000E+00 | 1.1642E+01 | 3.0772E-01 | 6.0693E+00 | 0.0000E+00 | QDCSO |
F10 | 8.8818E-16 | 1.4888E-02 | 1.9064E-11 | 3.9440E-08 | 7.8108E-12 | QDCSO |
F11 | 0.0000E+00 | 2.1883E-01 | 5.1553E-02 | 4.4032E-02 | 0.0000E+00 | QDCSO |
F12 | 1.0483E-06 | 3.6928E-05 | 8.1247E-22 | 9.8039E-16 | 9.2657E-02 | TFWO |
F13 | 9.7635E-03 | 1.5077E-04 | 1.7365E-22 | 2.5340E-15 | 4.7575E-01 | TFWO |
F14 | 1.1968E+00 | 5.2355E+00 | 9.9800E-01 | 1.1964E+00 | 2.8147E+00 | TFWO |
F15 | 3.0749E-04 | 1.8633E-03 | 3.9906E-04 | 3.0749E-04 | 5.6851E-04 | QDCSO |
F16 | −1.0316E+00 | −1.0316E+00 | −1.0316E+00 | −1.0316E+00 | −1.0242E+00 | SatinBO |
F17 | 3.0425E-01 | 1.9620E-01 | 3.0425E-01 | 3.0425E-01 | 3.3275E-01 | SatinBO |
F18 | 3.0000E+00 | 3.0000E+00 | 3.0000E+00 | 3.0000E+00 | 4.4707E+00 | TFWO |
F19 | −3.8625E+00 | −3.8628E+00 | −3.8628E+00 | −3.8628E+00 | −3.6465E+00 | TFWO |
F20 | −3.3217E+00 | −3.2507E+00 | −3.2744E+00 | −3.2744E+00 | −2.4489E+00 | QDCSO |
F21 | −6.5837E+00 | −5.6710E+00 | −7.3754E+00 | −6.8971E+00 | −1.6667E+00 | TFWO |
F22 | −5.6184E+00 | −8.3035E+00 | −9.7351E+00 | −8.4440E+00 | −1.7501E+00 | TFWO |
F23 | −6.2096E+00 | −7.3745E+00 | −7.8889E+00 | −9.7249E+00 | −2.1078E+00 | CPA |
cecF1 | 1.0000E+00 | 4.5673E+06 | 3.9915E+05 | 3.1736E+05 | 1.0000E+00 | QDCSO |
cecF2 | 4.8547E+00 | 1.8625E+03 | 5.3073E+02 | 5.4574E+02 | 5.0284E+00 | QDCSO |
cecF3 | 2.4665E+00 | 3.5682E+00 | 4.9508E+00 | 2.8784E+00 | 7.0772E+00 | QDCSO |
cecF4 | 3.0647E+01 | 3.5755E+01 | 1.8299E+01 | 1.0751E+01 | 5.7901E+01 | CPA |
cecF5 | 1.5533E+00 | 1.4928E+00 | 1.2172E+00 | 1.0391E+00 | 2.8859E+01 | CPA |
cecF6 | 5.9030E+00 | 5.9587E+00 | 4.1619E+00 | 1.1122E+00 | 6.9294E+00 | CPA |
cecF7 | 1.1291E+03 | 1.0916E+03 | 9.0752E+02 | 3.8815E+02 | 1.2290E+03 | CPA |
cecF8 | 4.2260E+00 | 4.6359E+00 | 4.0937E+00 | 3.9026E+00 | 4.5297E+00 | CPA |
cecF9 | 1.2365E+00 | 1.3191E+00 | 1.3169E+00 | 1.2069E+00 | 1.4834E+00 | CPA |
cecF10 | 2.0988E+01 | 2.1496E+01 | 2.1359E+01 | 2.1511E+01 | 2.1064E+01 | QDCSO |
Count | 13 | 2 | 9 | 8 | 1 | – |
Table 6. The standard deviation Std comparison with QMDCSO and other algorithms for F1-F23 and standard CEC 2019 problems
Fun | QMDCSO | SatinBO (Moosavi and Bardsiri 2017) | TFWO (Ghasemi et al. 2020) | CPA (Ong et al. 2021) | GPC (Harifi et al. 2021) | Best |
|---|---|---|---|---|---|---|
F1 | 0.0000E+00 | 2.6221E-03 | 3.9880E-23 | 9.8162E-15 | 1.3353E-21 | QDCSO |
F2 | 2.1212E-109 | 2.3895E-03 | 1.1253E-13 | 2.3687E-09 | 5.3952E-12 | QDCSO |
F3 | 0.0000E+00 | 4.5264E+00 | 1.9031E-07 | 6.4252E-04 | 3.8353E-21 | QDCSO |
F4 | 4.1862E-96 | 4.2567E-02 | 4.3336E-04 | 3.3331E-05 | 1.9692E-11 | QDCSO |
F5 | 4.5170E-01 | 4.3549E+02 | 1.1797E+00 | 7.4819E-01 | 1.7889E-01 | GPC |
F6 | 1.4645E-07 | 1.2455E-03 | 6.6082E-21 | 1.9283E-14 | 1.4333E-01 | TFWO |
F7 | 1.0627E-04 | 8.6048E-03 | 3.1742E-03 | 2.1632E-03 | 4.4185E-05 | GPC |
F8 | 2.9333E+02 | 3.1966E+02 | 7.4907E+01 | 1.5792E+02 | 1.5495E+02 | GPC |
F9 | 0.0000E+00 | 4.3258E+00 | 6.6877E-01 | 3.7654E+00 | 0.0000E+00 | QDCSO |
F10 | 0.0000E+00 | 4.9417E-03 | 2.3875E-11 | 1.9886E-08 | 9.8967E-12 | QDCSO |
F11 | 0.0000E+00 | 9.0642E-02 | 2.1773E-02 | 2.8693E-02 | 0.0000E+00 | QDCSO |
F12 | 1.2817E-06 | 3.7768E-05 | 1.9048E-21 | 1.1897E-15 | 3.0139E-02 | TFWO |
F13 | 3.0786E-02 | 2.2782E-04 | 4.4440E-22 | 3.4798E-15 | 6.7357E-02 | TFWO |
F14 | 4.1912E-01 | 2.2192E+00 | 7.4015E-17 | 6.2743E-01 | 5.2941E-01 | TFWO |
F15 | 1.1677E-18 | 2.1093E-03 | 2.8957E-04 | 3.6585E-13 | 7.0533E-05 | QDCSO |
F16 | 5.7294E-07 | 1.8986E-15 | 5.6689E-11 | 0.0000E+00 | 6.3732E-03 | CPA |
F17 | 0.0000E+00 | 4.4166E-02 | 0.0000E+00 | 0.0000E+00 | 6.3365E-03 | QDCSO |
F18 | 5.2274E-15 | 1.2071E-06 | 1.2648E-15 | 1.0777E-15 | 2.6925E+00 | CPA |
F19 | 4.2077E-04 | 5.0938E-08 | 9.3622E-16 | 9.3622E-16 | 2.1225E-01 | TFWO |
F20 | 4.1553E-04 | 6.1396E-02 | 6.1396E-02 | 6.1396E-02 | 3.1421E-01 | QDCSO |
F21 | 2.4611E+00 | 3.8577E+00 | 3.0185E+00 | 3.5420E+00 | 1.0949E+00 | GPC |
F22 | 1.6786E+00 | 3.3904E+00 | 2.1120E+00 | 3.2036E+00 | 1.1909E+00 | GPC |
F23 | 2.2794E+00 | 4.0843E+00 | 3.4595E+00 | 2.5661E+00 | 6.3906E-01 | GPC |
cecF1 | 0.0000E+00 | 3.2802E+06 | 3.5894E+05 | 5.6631E+05 | 2.7717E-12 | QDCSO |
cecF2 | 3.0582E-01 | 9.9878E+02 | 3.7371E+02 | 2.5368E+02 | 4.8807E-02 | GPC |
cecF3 | 8.4598E-01 | 2.3011E+00 | 2.9527E+00 | 1.8102E+00 | 7.8425E-01 | GPC |
cecF4 | 8.6485E+00 | 1.1538E+01 | 6.1406E+00 | 3.6572E+00 | 6.7215E+00 | CPA |
cecF5 | 8.0747E-01 | 1.8646E-01 | 1.0590E-01 | 2.7968E-02 | 8.8586E+00 | CPA |
cecF6 | 1.6679E+00 | 1.2292E+00 | 1.5659E+00 | 3.0953E-01 | 5.1496E-01 | CPA |
cecF7 | 3.2671E+02 | 3.2230E+02 | 3.4310E+02 | 2.8836E+02 | 1.5886E+02 | GPC |
cecF8 | 4.1689E-01 | 3.5729E-01 | 3.7172E-01 | 3.4050E-01 | 2.1704E-01 | GPC |
cecF9 | 7.1973E-02 | 1.6059E-01 | 1.0766E-01 | 7.6648E-02 | 7.5883E-02 | QDCSO |
cecF10 | 1.0359E-02 | 1.4902E-01 | 1.5110E-01 | 1.3842E-01 | 2.3124E-02 | QDCSO |
Count | 13 | 0 | 6 | 5 | 9 | – |
Table 7. The average results Avg comparison with QMDCSO, PSO, HGSO, and ASO for F1-F23 and standard CEC 2019 problems
Fun | QDCSO | PSO (Kennedy and Eberhart 1995) | Compared | HGSO (Hashim et al. 2019) | Compared | ASO (Zhao et al. 2019) | Compared |
|---|---|---|---|---|---|---|---|
F1 | 4.8110E-202 | 3.7088E-03 | 1 | 3.3911E-88 | 1 | 2.6009E-18 | 1 |
F2 | 1.0305E-109 | 1.2962E-01 | 1 | 2.7023E-46 | 1 | 5.3004E-09 | 1 |
F3 | 2.5966E-193 | 2.6566E-01 | 1 | 4.4098E-87 | 1 | 1.0083E+00 | 1 |
F4 | 1.8857E-96 | 1.4028E-01 | 1 | 4.0354E-46 | 1 | 1.8902E-09 | 1 |
F5 | 4.4333E+00 | 1.8456E+01 | 1 | 7.5153E+00 | 1 | 7.5315E+00 | 1 |
F6 | 1.3194E-07 | 5.3248E-03 | 1 | 2.8900E-01 | 1 | 3.7331E-18 | −1 |
F7 | 1.6819E-04 | 5.2097E-02 | 1 | 1.3243E-04 | −1 | 5.0999E-03 | 1 |
F8 | −3.6241E+03 | −2.5263E+03 | 1 | −4.6979E+06 | −1 | -3.0093E+03 | 1 |
F9 | 0.0000E+00 | 2.2144E+01 | 1 | 0.0000E+00 | 0 | 2.8854E+00 | 1 |
F10 | 8.8818E-16 | 1.3183E-01 | 1 | 8.8818E-16 | 0 | 2.3475E-09 | 1 |
F11 | 0.0000E+00 | 1.4549E-01 | 1 | 0.0000E+00 | 0 | 1.9717E-03 | 1 |
F12 | 1.0483E-06 | 2.0702E-04 | 1 | 6.0760E-02 | 1 | 1.4051E-19 | −1 |
F13 | 9.7635E-03 | 2.9486E-03 | −1 | 2.4926E-01 | 1 | 6.4124E-19 | −1 |
F14 | 1.1968E+00 | 1.3986E+00 | 1 | 1.3126E+00 | 1 | 1.1375E+00 | −1 |
F15 | 3.0749E-04 | 9.7259E-04 | 1 | 3.5584E-04 | 1 | 2.1200E-03 | 1 |
F16 | −1.0316E+00 | −1.0316E+00 | −1 | −1.0316E+00 | 1 | −1.0316E+00 | −1 |
F17 | 3.0425E-01 | −2.9823E+23 | −1 | −3.3021E+08 | −1 | 3.0566E-01 | 1 |
F18 | 3.0000E+00 | 3.0000E+00 | 1 | 3.0001E+00 | 1 | 3.0000E+00 | 0 |
F19 | −3.8625E+00 | −3.8628E+00 | −1 | −3.8557E+00 | 1 | −3.8628E+00 | −1 |
F20 | −3.3217E+00 | −3.2744E+00 | 1 | −3.0344E+00 | 1 | −3.3220E+00 | −1 |
F21 | −6.5837E+00 | −8.3952E+00 | −1 | −4.5969E+00 | 1 | −6.6121E+00 | −1 |
F22 | −5.6184E+00 | −5.9391E+00 | −1 | −5.2915E+00 | 1 | −1.0403E+01 | −1 |
F23 | −6.2096E+00 | −8.3917E+00 | −1 | −4.6873E+00 | 1 | −1.0022E+01 | −1 |
cecF1 | 1.0000E+00 | 7.9586E+08 | 1 | 1.0000E+00 | 1 | 7.9277E+06 | 1 |
cecF2 | 4.8547E+00 | 2.8176E+04 | 1 | 4.5871E+00 | −1 | 4.3213E+03 | 1 |
cecF3 | 2.4665E+00 | 1.0532E+01 | 1 | 7.5737E+00 | 1 | 4.7951E+00 | 1 |
cecF4 | 3.0647E+01 | 4.0314E+01 | 1 | 5.8695E+01 | 1 | 1.1248E+01 | −1 |
cecF5 | 1.5533E+00 | 1.6348E+00 | 1 | 1.5652E+01 | 1 | 1.0042E+00 | −1 |
cecF6 | 5.9030E+00 | 3.5147E+00 | −1 | 8.0950E+00 | 1 | 1.1500E+00 | −1 |
cecF7 | 1.1291E+03 | 1.0788E+03 | −1 | 1.7229E+03 | 1 | 9.9092E+02 | −1 |
cecF8 | 4.2260E+00 | 4.1370E+00 | −1 | 4.8683E+00 | 1 | 4.5878E+00 | 1 |
cecF9 | 1.2365E+00 | 1.2752E+00 | 1 | 1.7053E+00 | 1 | 1.0930E+00 | −1 |
cecF10 | 2.0988E+01 | 2.1403E+01 | 1 | 2.1293E+01 | 1 | 2.1000E+01 | 1 |
Count | 23:10 | 26:4 | 17:15 |
Contrast experiments were conducted to verify the performance of multigroup cooperative evolutionary optimization algorithm in combination with quantum entanglement (Table 3).
First, the average results of different swarm intelligence algorithms for F1-F23 and the standard CEC 2019 problems with are presented in Table 4. The loss curves for different algorithms for F1-F23 and the standard CEC 2019 problems are shown in Figs. 6 and 8. The curves of QMDCSO are represented by red lines and those of DCSO are represented by blue lines in the CEC problems. Avg is the average value of the losses, and Std is the standard deviation of losses. The average results of QMDCSO are superior to those of DCSO for 21 test functions, and are inferior to those of DCSO for 8 test functions. The traditional seeking behavior is the random scaling update on the basis of the best cat, and the traditional tracking group is the follow-up update based on best rat. The multi-population interactive group considers factors such as escaping from the worst natural enemies and approaching the best prey in the field of vision, whereas the quantum entanglement group refers to the quantum rotation and best individual. Owing to the increase in individual selectable mechanisms, a better balance may be achieved in the exploration and development of candidate solutions in some cases, thus leading to a faster approach to the global optimum. Multi-population interactive update group and the quantum entanglement update group in the proposed method refer to the mechanism by which the interaction behaviors of species diversity and stability in the ecosystem may guide the algorithm to gradually approach the global optimal solution more easily. QMO is an optimization algorithm that uses only the multi-population interactive group and quantum entanglement group, and . The average results of QMDCSO are superior to those of QMO for 25 test functions, and are inferior to those of QMO for 7 test functions. Because many factors can be referenced in multiple groups, they may sometimes serve as auxiliaries, but they may sometimes also cause interference, so other groups also need to be combined. The quantum entanglement group ignores the scaling update near the best individual and the following update with reference to the best individual (Table 5).
Second, comparisons of QMDCSO and other swarm intelligence algorithms in terms of the average results Avg and the standard deviation Std are presented in Tables 6 and 7, respectively. In CPA and TFWO, was a multiple of 3 and was set to 102 in the F1-F23 problems and 51 in the standard CEC 2019 problems. In the other algorithms, was set to 100 in F1-F23 problems and set to 50 in the standard CEC 2019 problems. The average results of QMDCSO, TFWO (Ghasemi et al. 2020), CPA (Ong et al. 2021), SatinBO (Moosavi and Bardsiri 2017) and GPC (Harifi et al. 2021) algorithms were superior to those of the other 4 algorithms for 13, 9, 8, 2 and 1 test functions, respectively. The standard deviations of QMDCSO, GPC, TFWO, CPA and SatinBO algorithms were superior to those of the other 4 algorithms for 13, 9, 6, 5 and 0 test functions, respectively. In terms of the average results, QMDCSO outperformed PSO (Kennedy and Eberhart 1995), HGSO (Hashim et al. 2019) and ASO (Zhao et al. 2019) for 23, 26 and 17 test functions, and was inferior to PSO, HGSO and ASO for 10, 4 and 15 test functions, respectively.
Table 8. The average results for function optimization when and
Fun | Avg | Std | ||||
|---|---|---|---|---|---|---|
QMDCSO | DCSO | Compared | QMDCSO | DCSO | Compared | |
F1 | 0.0000E+00 | 0.0000E+00 | 1 | 0.0000E+00 | 0.0000E+00 | 1 |
F2 | 7.7510E-265 | 4.9254E-263 | 1 | 0.0000E+00 | 0.0000E+00 | 1 |
F3 | 0.0000E+00 | 0.0000E+00 | 1 | 0.0000E+00 | 0.0000E+00 | 1 |
F4 | 1.3734E-235 | 2.9944E-239 | 0 | 0.0000E+00 | 0.0000E+00 | 1 |
F5 | 2.5042E+01 | 2.5641E+01 | 1 | 9.1728E-01 | 4.2761E-01 | 0 |
F6 | 6.1953E-02 | 8.2241E-02 | 1 | 1.0169E-01 | 2.4490E-01 | 1 |
F7 | 2.3794E-05 | 5.1360E-05 | 1 | 2.1926E-05 | 7.5198E-05 | 1 |
F8 | −3.8855E+03 | −3.1922E+03 | 1 | 1.9011E+02 | 3.5360E+02 | 1 |
F9 | 0.0000E+00 | 0.0000E+00 | 1 | 0.0000E+00 | 0.0000E+00 | 1 |
F10 | 8.8818E-16 | 8.8818E-16 | 1 | 0.0000E+00 | 0.0000E+00 | 1 |
F11 | 0.0000E+00 | 0.0000E+00 | 1 | 0.0000E+00 | 0.0000E+00 | 1 |
F12 | 1.1684E-02 | 2.0748E-02 | 1 | 9.2638E-03 | 1.3891E-02 | 1 |
F13 | 2.1863E+00 | 2.0928E+00 | 0 | 1.8670E-01 | 2.8313E-01 | 1 |
Count | – | – | 11 | – | – | 12 |
Table 9. The average results for function optimization when
Fun | ||||||
|---|---|---|---|---|---|---|
QMDCSO | DCSO | Compared | QMDCSO | DCSO | Compared | |
F1 | 0.0000E+00 | 0.0000E+00 | 1 | 0.0000E+00 | 0.0000E+00 | 1 |
F2 | 2.3762E-259 | 2.1741E-258 | 1 | 0.0000E+00 | 0.0000E+00 | 1 |
F3 | 0.0000E+00 | 0.0000E+00 | 1 | 0.0000E+00 | 0.0000E+00 | 1 |
F4 | 4.2366E-233 | 3.1156E-234 | 0 | 0.0000E+00 | 0.0000E+00 | 1 |
F5 | 9.6441E+01 | 9.7342E+01 | 1 | 9.6393E+01 | 9.5995E+01 | 0 |
F6 | 5.7049E+00 | 6.1670E+00 | 1 | 3.2135E+00 | 4.9604E+00 | 1 |
F7 | 2.9249E-05 | 3.6350E-05 | 1 | 1.3234E-05 | 1.5012E-05 | 1 |
F8 | −3.3588E+04 | −2.1065E+04 | 1 | −3.5475E+04 | −2.3657E+04 | 1 |
F9 | 0.0000E+00 | 0.0000E+00 | 1 | 0.0000E+00 | 0.0000E+00 | 1 |
F10 | 8.8818E-16 | 8.8818E-16 | 1 | 8.8818E-16 | 8.8818E-16 | 1 |
F11 | 0.0000E+00 | 0.0000E+00 | 1 | 0.0000E+00 | 0.0000E+00 | 1 |
F12 | 9.5888E-02 | 1.2083E-01 | 1 | 5.4802E-02 | 8.7199E-02 | 1 |
F13 | 9.4106E+00 | 9.4426E+00 | 1 | 9.3146E+00 | 9.3539E+00 | 1 |
Count | – | – | 12 | – | – | 12 |
Third, the results for other numbers of dimensions, namely, and , are shown in Tables 8 and 9, respectively. To obtain the results in Table 8, was set to 30 and was set to 500. The average results of QMDCSO were superior to or equivalent to those of DCSO for 11 test functions, and inferior to those of DCSO for 2 test functions. The standard deviations of QMDCSO are superior to or equivalent to those of DCSO for 12 test functions, and inferior to those of DCSO for 1 test functions. To obtain the results in Table 9, was set to 100, and was set to 500. The average results of QMDCSO were superior to or equivalent to those of DCSO for 12 test functions, and inferior to those of DCSO for 1 test functions.
Finally, the results under other numbers of iterations, such as , are shown in Table 9. The average results of QMDCSO were superior to or equivalent to those of DCSO for 12 test functions, and inferior to those of DCSO for 1 test functions.
QMDCSO achieved competitive results in CEC problems of function optimization.
Extended experiments
Table 10. The overall comparison of different algorithms in 4 applications
GMRRO | LLC | VSO | PTPO | |
|---|---|---|---|---|
SatinBO (Moosavi and Bardsiri 2017) | 71.3553 | 3.5433E+05 | 4.1631E+06 | 903.3460 |
CPA (Ong et al. 2021) | 57.1127 | 3.4705E+05 | 3.9230E+06 | 851.1543 |
TFWO (Ghasemi et al. 2020) | 47.4558 | 3.4974E+05 | 4.1247E+06 | 886.5535 |
AFO (Yang et al. 2021) | 41.2132 | 3.4960E+05 | 4.0844E+06 | 885.8526 |
DCSO (Ahmed et al. 2021) | 38.3848 | 3.4861E+05 | 4.0124E+06 | 841.1044 |
QMDCSO | 36.9706 | 3.4705E+05 | 3.8745E+06 | 822.5995 |
In addition, comparative experiments were also conducted in 4 engineering applications. An overall comparison of the algorithms is shown in Table 10, and QMDCSO achieved competitive results in 4 applications.
Grid map - robot routing optimization
First, the connected areas of a grid map were marked, and different labels were set for each connected region. For each connected region, the nodes with a distance between 1 and 2 from the nearest obstacle were selected as landmark nodes. The distances of all 273 feasible nodes were calculated, and the indices of distances less than the step size in were found. Then, the corresponding pairs of nodes were added to the landmark points, and the nodes with feasible distances that meet the step requirements were added to sparse net. Since walking up, down, left, right, and diagonally are allowed but not crossing, the step distance was set to .
The swarm intelligence individual is the priority weight of the landmark points, and controls whether the nodes of the landmark points are selected to join the path. The individual fitness is the distance value of the shortest path from the entrance and the exit node in the grid map.
The path of the grid map was generated. First, the initial entrance node and the exit node were selected to join the path; then the landmark points with the top 10 priorities were selected as the intermediate representative nodes of the path. Representative points on the map path that complied with rules, such as not crossing obstacles and having step sizes within the specified range, were connected. The shortest path Path and distance Dist of two intermediate representative nodes in the graph were calculated through . The path distance set path1 of representative points and the complete path path2 were obtained from the entrance to the exit. denotes the number of intermediate representative nodes.
19
The individual fitness is defined as follows:20
[See PDF for image]
Fig. 9
The convergence curves and optimized paths obtained by different swarm intelligence algorithms in GM-RRO
The convergence curve results and optimized paths of different swarm intelligence algorithms in grid map - robot routing optimization (GMRRO) are shown in Fig. 9. The x-axis represents the iteration and the Y-axis represents the path length, which is also the individual fitness or loss. In Fig. 9, the red line represents the optimized path, the black blocks are the obstacles, the ‘+’ blocks represent the landmark nodes of each connected area, and the blank blocks represent other feasible nodes. The results of different algorithms in the GM-RRO application are shown in Table 10. The optimized path lengths of the SatinBO (Moosavi and Bardsiri 2017), CPA (Ong et al. 2021), TFWO (Ghasemi et al. 2020), AFO (Yang et al. 2021), DCSO and QMDCSO algorithms were 71.3553, 57.1127, 47.4558, 41.2132, 38.3848 and 36.9706 in descending order. The QMDCSO achieved competitive results in the GM-RRO application.
Location optimization of logistics center
In the considered problem, 44 cities, each city needs 3 types of steel, and each kind of steel is produced in a different city. Six cities with higher priority were selected from 14 logistics centers as the main logistics centers, and the rest were reserved as standby logistics centers.
The swarm intelligence individual is a 58D vector that corresponds to the priorities of the 14 logistics centers and 44 demand locations. The logistics centers with the top 6 priorities were selected. The demand with highest priority should be satisfied first. The individual fitness is the total cost of logistics transportation under fixed, variable, repertory and punishment constraints.
The transportation cost of logistics is as follows. is the transportation costs between the logistics centers and the demand locations, and is the transportation costs between the logistics centers and the producing cities.
21
is the distance between the specified demand location and logistics center, and is the distance between the specified producing city and logistics center. is the demand quantity of a single record. , , , , and . The sum of the variable costs and repertory costs required by the demand location is defined as follows.22
Whether all demands are met, namely, is determined, and solutions that fail to meet the demands, namely, , are punished. is the penalty factor.23
The individual fitness is defined as follows:24
The convergence curve results and optimized paths of different swarm intelligence algorithms in location optimization of logistics centers (LLC) are shown in Fig. 10. The x-axis represents the iteration and the Y-axis represents the logistics cost, which is also the best individual fitness or loss. In Fig. 10, the three blue stars represent the producing cities, the six red polygons represent main logistics centers, the six green blocks represent alternative logistics centers, and the yellow dots represent 44 demand locations. The black lines represent the paths from the producing cities to the logistics centers, and the lines with different colors represent the paths from different logistics centers to the demand locations. The results of different algorithms in LLC application are shown in Table 10. The optimized total costs of the SatinBO (Moosavi and Bardsiri 2017), TFWO (Ghasemi et al. 2020), AFO (Yang et al. 2021) and DCSO algorithms were 354326.5683, 349738.2105, 349597.0107 and 348606.1629 in descending order. The CPA and QMDCSO had the minimum total cost of 347045.6023. QMDCSO achieved competitive results in the LLC application.[See PDF for image]
Fig. 10
The convergence curves and optimized paths obtained by different swarm intelligence algorithms in LLC
Vehicle scheduling optimization for a dairy enterprise
The 42D vector of swarm intelligence individual corresponds to the indices of 37 receiving places and the number of divisions, which is equal to the number of vehicles, namely, 6 minus 1.
It was decoded to obtain the path of each vehicle. S is the sorted index vector of individual . When the value of S exceeded 38, the vehicle was divided and the previous nodes were added to the vehicle. The individual fitness corresponds to the sum of the depreciation cost , the maintenance cost , the labor time cost , the fuel consumption cost , the refrigeration cost , the product cost and the punishment constraint . Owing to the need to return, the path of each vehicle consists of the delivery center at departure, the receiving places, and the delivery center at return.
is the transportation distance from the initial node to the current node of path i. and v are the transportation time consumption and speed, respectively. The load at node of path i is the weight of milk and packaging boxes required by all nodes of path i, minus the weight of milk and packaging boxes that are unloaded from the initial node to the current node of path i, plus the returned milk box weight from the initial node to the current node of path i.
25
26
The cost of each vehicle path is calculated. The depreciation cost is the product of the depreciation cost per mile and the mileage . The maintenance cost is the product of the maintenance cost per unit mile and the mileage . The labor time cost is the product of the labor cost per unit time and the labor time consumption of the vehicles. and are the transportation time consumption and other time consumption, such as the time spent on loading, unloading and packaging. The fuel consumption cost includes the no-load fuel consumption and the load fuel consumption. is the fuel consumption per unit mass of load, and is the unit price of fuel. The refrigeration cost includes the refrigeration cost during transportation and the refrigeration cost during loading and unloading. is the unit price of refrigerant. is a variable related to the internal and external area and material of the refrigerator vehicle. is a variable related to door opening and interior capacity. The product cost is multiplied by the quantity and unit product price .27
28
The punishment constraints include a penalty for the time consumption exceeding the limit , a penalty for the load exceeding the limit , and a penalty for the number of customers exceeding the limit . is the penalty factor.29
The individual fitness is defined as follows:30
[See PDF for image]
Fig. 11
The convergence curves and optimized paths obtained by different swarm intelligence algorithms in VSO
The convergence curve results and optimized paths of different swarm intelligence algorithms in vehicle scheduling optimization (VSO) are shown in Fig. 11. The x-axis represents the iteration and the y-axis represents the fitness or loss of the best individual. In Fig. 11, the red star represents the parking point and delivery center, the green dots represent the customers and receiving places, and the lines of different colors represent the paths of different vehicles. The results of different algorithms in VSO applications are shown in Table 10. The optimized total costs of the SatinBO (Moosavi and Bardsiri 2017), TFWO (Ghasemi et al. 2020), AFO (Yang et al. 2021), DCSO (Ahmed et al. 2021), CPA (Ong et al. 2021) and QMDCSO algorithms were 4163084.3093, 4124677.5937 4084415.2475, 4012360.6181, 3922965.3275 and 3874476.5871 in descending order. QMDCSO achieved competitive results in the VSO application.
Plasma trajectory planning optimization
Plasma trajectory planning optimization aims to find the path with the smallest ring while traversing all nodes. The swarm intelligence individual corresponds to the priorities of the 30 randomly generated nodes, and the ring with the minimum distance was selected to traverse all nodes. The individual fitness is the sum of the distance after traversing all nodes.
The node with the minimum value in was selected as the starting node and labeled 1. The remaining 29 nodes were traversed. The node with the smallest product of distance and priority among the remaining nodes was recursively selected to join the path, and labeled 1. The distance was accumulated, and the sequence set of other nodes was obtained. Then, the smallest node was added to the path as the end point, and the distance from the last point to the first node was increased. The individual fitness is defined as follows:
31
[See PDF for image]
Fig. 12
The convergence curves and optimized paths obtained by different swarm intelligence algorithms in PTPO
The convergence curve results and optimized paths of different swarm intelligence algorithms in plasma trajectory planning optimization (PTPO) are shown in Fig. 12. The x-axis represents the iteration and the Y-axis represents the path length, which is also the best individual fitness or loss. In Fig. 12, the green dots represent the 30 randomly generated nodes, and the lines are the paths of PTPO. The results of the different algorithms in the PTPO application are shown in Table 10. The optimized results of SatinBO (Moosavi and Bardsiri 2017), TFWO (Ghasemi et al. 2020), AFO (Yang et al. 2021), CPA (Ong et al. 2021), DCSO (Ahmed et al. 2021) and QMDCSO were 903.3460, 886.5535, 885.8526, 851.1543, 841.1044 and 822.5995 respectively in descending order. QMDCSO achieved competitive results in the PTPO application.
Conclusion
In this paper, a multigroup cooperative evolutionary optimization algorithm was proposed. First, the group updating mechanism of the traditional seeking mode and tracking mode with a dynamic population update mechanism were adopted. A multi-population interactive update group and a quantum entanglement update group were introduced to guide the algorithm to gradually approach the global optimal solution. Second, the proposed bionic algorithm was extended to cross-field applications. The algorithm was applied to 33 test functions of open CEC problems, and its application fields were extended to robot routing optimization of grid maps, vehicle scheduling optimization of dairy enterprises, location optimization of logistics centers, and plasma trajectory planning optimization, where it achieved competitive results. In the future, we will combine graph networks with different scales and swarm intelligence algorithms to solve cross-field optimization problems.
Acknowledgements
This work was supported by National Science and Technology Innovation 2030 Major Program of China (2022ZD0205005).
Author contributions
Corresponding author Bailu Si provides guidance and revisions and the first author Zhaoyang Lian conducts experiments and writes manuscripts.
Declarations
Conflict of interest
Zhaoyang Lian and Bailu Si declare that they have no conflict of interest with respect to this paper.
Informed consent
Informed consent was not required, as no human beings or animals were involved in this study.
Human and animal rights
This article does not contain any studies with human or animal participants performed by any of the authors.
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