The Equilibrium Optimizer (EO) algorithm is a novel meta-heuristic algorithm based on the strength of physics. To achieve better global search capability, a Parallel Equilibrium Optimizer algorithm, named PEO, is proposed in this paper. PEO is inspired by the idea of parallelism and adopts two different communication strategies between groups to improve EO. The first strategy is used to speed up the convergence rate and the second strategy promotes the algorithm to search for a better solution. These two kinds of communication strategies are used in the early and later iterations of PEO respectively. To check the optimization effect of the proposed PEO algorithm, it is tested on 23 benchmark functions and compared with the Particle Swarm Optimization (PSO), Grey Wolf Optimizer (GWO), Parallel Particle Swarm Optimization (PPSO), and EO as well. The empirical study demonstrates that the abilities of exploration and exploitation of PEO are superior to the above four algorithms in most benchmark functions. Finally, we apply PEO to solve the Capacitated Vehicle Routing Problem (CVRP) in the field of transportation. Experimental results show that PEO can achieve a better driving route.

Traditional optimization algorithms usually show better performance in solving single-extreme value optimization problems. But in fact, the practical engineering problems are multiple, extreme and complex. The traditional optimization algorithms are difficult to solve these problems. The emergence of intelligent optimization algorithms breaks the above-mentioned bottleneck. When solving complex problems, they can find approximately optimal solutions within an acceptable time and improve efficiency. Therefore, the research of intelligent optimization algorithms is becoming more and more important.

Intelligent optimization algorithms mostly belong to heuristic algorithms. They are mainly used to solve optimization problems, and their inspiration comes from existing phenomena or things [

The EO is enlightened by the model that controls the mass-volume balance and adopts a random exploration mechanism. In EO, each particle updates its position through a candidate particle randomly selected from the equilibrium pool, and the adaptive value of control parameters in the algorithm can reduce the moving speed of particles. The abilities of exploration and exploitation are largely due to this random update strategy and the control parameters in the algorithm. And the tests of EO in benchmark functions and engineering problems prove the practicability and effectiveness of the algorithm [

As a new algorithm, the convergence and global search ability of EO need to be further improved. Its balance mechanism leads to the problem of premature convergence. To further advance global search capability and avoid its premature convergence, this paper improves the EO based on parallel ideas. There are many improvement strategies to improve the performance of the meta-heuristic algorithm, like parallel, compact, multi-strategy and space vector improvement [

To test the ability of PEO in solving practical problems, the Capacitated Vehicle Routing Problem (CVRP) is treated as an application example of PEO. At present, many scholars have done a lot of research in the field of intelligent transportation information [

The rest of this paper is organized as follows. Section 2 gives a brief retrospect to the basic concept of EO. Section 3 formally proposes our PEO algorithm with details. Section 4 evaluates the algorithm using extensive experiments. Further application analysis of the proposed PEO algorithm in the CVRP is demonstrated in Section 5. The last section describes the conclusions about the presented work.

In EO, each particle (solution) and its position (concentration) are used as a search agent. During the iteration process, the search agent randomly updates its current position (concentration) according to the equilibrium candidates until the end of the iteration to get the optimal result (equilibrium state). The initialization of EO is similar to the general meta-heuristic algorithm. The initial population consists of randomly distributed particles in the search space, as shown in

The EO expects the final equilibrium state to be globally optimum. To heighten the capabilities of exploration and exploitation, an equilibrium pool is established in EO to update the particles. The five particles in the equilibrium pool are called equilibrium candidates. These five particles are the four particles with the best fitness and their average. The four best candidates are beneficial to improve the exploration ability and their average is beneficial to heighten the exploitation ability. Besides, the number of equilibrium candidates is not fixed and can be adjusted for different application problems, which is similar to the selection of three wolves in GWO [

To maintain a balance between exploration and exploitation, the parameter

In

The parameter

where:

In

In the

In this section, we propose a parallel Equilibrium Optimizer algorithm, named PEO, to improve the optimization capability of EO. In PEO, a multi-group structure is adopted, and each group search strategy is consistent with the original EO. Regularly, particles from different groups communicate with each other to increase cooperation among groups. Two communication strategies are used in PEO. The first strategy of inter-group communication is used for individual mutation of poor particles, which can heighten the convergence speed of the algorithm. The second strategy is to replace the poor particles in each group, which can ameliorate the exploitation capacity of the algorithm. The two strategies are respectively applied in the early and late stages of the algorithm to further improve the convergence results. Besides, the

In the initialization phase, the groups are divided according to the predetermined total number of particles. In each group, the fitness values of all particles are calculated and sorted, to get the optimal solution and four equilibrium candidates in the group. Comparing all groups, we can get the global optimal solution

The first inter-group communication strategy is shown in

where

The second inter-group communication strategy is shown in

To check the optimization effectiveness of PEO, we test it on 23 benchmark functions and compare it with the PSO, PPSO, GWO, and EO algorithms as well. Among the 23 benchmark functions, F1~F7 represent unimodal functions, F8~F13 represent multimodal functions, and F14~F23 represent fixed-dimensional functions. The detailed description of these functions can be obtained from [

Algorithm | Parameter name and value |
---|---|

PSO | |

PPSO | |

GWO | |

EO | |

PEO |

Functions | PSO | PPSO | GWO | EO | PEO | |||||
---|---|---|---|---|---|---|---|---|---|---|

Ave | Std | Ave | Std | Ave | Std | Ave | Std | Ave | Std | |

F1 | 1.28E-37 | 2.11E-37 | 6.10E-151 | 1.30E-150 | 1.08E-198 | 0 | 5.42E-285 | 0 | 0 | |

F2 | 2.86E-16 | 5.04E-16 | 6.86E-77 | 9.65E-77 | 3.18E-112 | 3.6E-112 | 8.53E-156 | 1.04E-155 | 0 | |

F3 | 6.85E-02 | 5.34E-02 | 4.22E-134 | 5.98E-134 | 1.15E-68 | 3.58E-68 | 6.26E-97 | 1.00E-96 | 1.65E-149 | |

F4 | 9.86E-03 | 5.26E-03 | 2.79E-72 | 6.15E-72 | 7.99E-52 | 7.74E-52 | 5.06E-75 | 1.20E-74 | 2.71E-102 | |

F5 | 51.0647 | 29.05343 | 25.6475 | 0.35362 | 26.0433 | 0.69419 | 21.7378 | 0.20643 | 0.76870 | |

F6 | 9.18E-32 | 2.55E-31 | 1.38E-07 | 1.27E-07 | 1.00E-01 | 1.74E-01 | 0 | 5.18E-31 | 1.55E-30 | |

F7 | 1.10E-02 | 4.49E-03 | 6.02E-05 | 6.26E-05 | 6.79E-05 | 5.40E-05 | 6.76E-05 | 3.33E-05 | 3.24E-05 | |

F8 | −6946.93 | 691.2639 | −7261.07 | 908.612 | −6457.95 | 1012.11 | −8829.28 | 602.889 | 447.560 | |

F9 | 2.19E+01 | 7.777942 | 0 | 0 | 0 | 0 | ||||

F10 | 1.37E-14 | 2.48E-15 | 0 | 7.99E-15 | 0 | 4.44E-15 | 0 | 0 | ||

F11 | 3.55E-01 | −8 | 1.87E-15 | −7.96808 | 9.52E-02 | −8 | 1.87E-15 | −8 | 1.87E-15 | |

F12 | 4.12E-34 | 7.81E-07 | 4.78E-07 | 1.31E-02 | 8.73E-03 | 7.65E-32 | 1.92E-31 | 7.77E-32 | 1.92E-31 | |

F13 | 2.32E-32 | 1.58E-32 | 3.21E-03 | 7.16E-03 | 5.09E-02 | 5.38E-02 | 2.88E-48 | 2.07E-01 | 1.89E-01 | |

F14 | 0 | 1.39562 | 0.51331 | 1.79164 | 1.02E+00 | 0 | 0 | |||

F15 | 4.40E-04 | 2.79E-04 | 4.91E-04 | 3.86E-04 | 5.28E-04 | 4.72E-04 | 2.31E-03 | 6.34E-03 | 2.90E-04 | |

F16 | 0 | 0 | 5.10E-10 | 0 | 7.40E-17 | |||||

F17 | 0 | 0 | 2.35E-08 | 0 | 0 | |||||

F18 | 1.12E-15 | 1.04E-15 | 4.90E-08 | 4.44E-16 | 7.40E-16 | |||||

F19 | 0 | 0 | 2.03E-07 | 0 | 0 | |||||

F20 | 4.44E-16 | 0 | −3.26255 | 0.08407 | −3.26255 | 0.08407 | 0 | |||

F21 | −7.62698 | 3.572606 | 0 | 1.23E-05 | 0 | 0 | ||||

F22 | 0 | 0 | 2.20E-05 | 1.77E-15 | 0 | |||||

F23 | −10.0003 | 1.69522 | 1.96E-15 | 2.81E-05 | 1.87E-15 | 2.13E-15 |

It can be known from the bold data in

To more intuitively observe the experimental results of the five algorithms, 6 groups are selected from 23 groups of experimental graphs to display.

In this section, we introduce the mathematical model of the Capacitated Vehicle Routing Problem (CVRP) and test the proposed PEO algorithm on the CVRP.

In this paper, the parameter

In the mathematical model of the CVRP,

Firstly, a set of simple data is used to test. The task point coordinate and demand are shown in

Sequence | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

Coordinate | (18,54) | (22,60) | (58,69) | (71,71) | (83,46) | (91,38) | (24,42) | (18,40) |

Requirement | 0 | 89 | 14 | 28 | 33 | 21 | 41 | 57 |

In this set of simple test cases, 0 is the central warehouse, the maximum load of the car is set to 100, the task distribution is completed by 3 cars, and the known optimal path is 217.8. In the test, the number of iterations is 200, and the particle swarm size is 180, divided into 6 groups. The test result of PEO reveals in

To further test the effect of PEO in the CVRP, we select five groups of test data in the CVRP international standard example VRPLIB and compare the results with EO, MMSCA [

PSO-route | MMSCA-route | EO-route | PEO-route | |
---|---|---|---|---|

A-n32-k5 | 1137 | 1080 | 1226 | |

A-n33-k5 | 1177 | 1011 | 977 | |

A-n33-k6 | 1155 | 1037 | 1081 | |

A-n44-k7 | 1577 | 1446 | 1486 | |

A-n46-k7 | 1746 | 1507 | 1631 |

Analyzing the data in the

In this paper, we developed a parallel paradigm for the EO algorithm, called PEO. Two efficient communication strategies are designed and applied to the set of groups in PEO. Experiments show that our approach outperforms existing approaches in most of 23 benchmark functions. Furthermore, applying the PEO algorithm on CVRP can get a shorter distance than PSO, EO, and MMSCA algorithms, which further testify the effectiveness and superiority of the proposed PEO algorithm.