The meta-heuristic algorithm is a global probabilistic search algorithm for the iterative solution. It has good performance in global optimization fields such as maximization. In this paper, a new adaptive parameter strategy and a parallel communication strategy are proposed to further improve the Cuckoo Search (CS) algorithm. This strategy greatly improves the convergence speed and accuracy of the algorithm and strengthens the algorithm’s ability to jump out of the local optimal. This paper compares the optimization performance of Parallel Adaptive Cuckoo Search (PACS) with CS, Parallel Cuckoo Search (PCS), Particle Swarm Optimization (PSO), Sine Cosine Algorithm (SCA), Grey Wolf Optimizer (GWO), Whale Optimization Algorithm (WOA), Differential Evolution (DE) and Artificial Bee Colony (ABC) algorithms by using the CEC-2013 test function. The results show that PACS algorithm outperforms other algorithms in 20 of 28 test functions. Due to the superior performance of PACS algorithm, this paper uses it to solve the problem of the rectangular layout. Experimental results show that this scheme has a significant effect, and the material utilization rate is improved from 89.5% to 97.8% after optimization.

With the development of the meta-heuristic algorithm, researchers have paid more and more attention to it. In real life, meta-heuristic algorithms have been widely used in industry, finance, mathematics and other fields, and achieved good results. For example, Particle Swarm Optimization (PSO) is applied to heterogeneous Wireless Sensor Networks (WSN) [

Layout problems are widely used in the real world, such as stacking of items in warehouse, newspaper text layout, container packing, and so on [

The main contributions of this paper include the following four aspects:

A new adaptive parameter strategy is proposed to improve step size and rejection probability respectively and the performance of the algorithm is significantly improved.

Based on the original algorithm, a new parallel communication strategy is applied to enhance the global optimization of the population.

The new improved algorithm is compared with the original algorithm (CS) and several popular algorithms (WOA, SCA, PSO, GWO, ABC, DE) through the CEC-2013 test function.

The improved algorithm is combined with the rule of the lowest horizontal line strategy to expand the practice in the field of the rectangular layout.

The rest of this paper is organized as follows. In

Yang et al. proposed the CS algorithm in 2009 [

Cuckoo birds randomly choose a nest to lay eggs, and lay only one egg in a nest. At this time, a nest can be abstracted into a solution.

The best nests will be preserved. That means the best solution is saved for the next iteration.

The number of nests

The CS algorithm performs two population updates in one iteration: a random walk update based on the Lévy flight and an update by dropping the probability

In

In

The random step size based on Lévy distribution has the characteristics of high probability short-distance exploration and occasionally long-distance walking, as shown in

The second update strategy is as follows: CS algorithm discards partial solutions according to a certain probability of discovery and then adopts preference walk to regenerate the same number of solution.

In

Step 1: The number of nests is set to

Step 2: Bird’s Nest location update. The optimal nest location of the previous generation

Step 3: During the solution process, the egg of the cuckoo has a certain probability (

Step 4: Find the optimal nest location

This algorithm has been proved by Yang et al. through many experiments. In general,

The CS algorithm finds the bird’s nest through the Lévy flight mechanism. Lévy flight is a random walk process consisting of short-distance flight with high probability and long-distance flight with low probability. Therefore, the cuckoo’s nest finding path is easy to jump between different search areas, resulting in the poor local fine search ability of the CS algorithm. After the algorithm is decomposed, it is easy to oscillate in the area near the optimal solution, resulting in low algorithm efficiency.

The last section mentioned that the CS algorithm has some advantages over other algorithms but also has some disadvantages. To improve the performance of CS algorithm and expand the application field of CS algorithm, researchers have developed multiple enhanced versions of CS algorithm, such as binary CS algorithm used to replace discrete optimization problem, a chaotic CS algorithm used to improve the performance of CS algorithm [

Aiming at the shortcoming of CS algorithm’s low local precision search ability, in this paper, improvements are made from two aspects: on the one hand, population parallel communication strategy is adopted; on the other hand, adaptive parameter strategy is adopted.

Parallel communication strategy can improve the search ability and convergence speed of meta-heuristic optimization algorithms and find better solutions in a better way [

An example is given in

The interpopulation communication of the parallel strategy is beneficial to the algorithm to avoid falling into the local optimum and to find the global optimum or its optimal solution with high probability in multiple ranges. Simultaneously, the strategy will accelerate the optimization speed of the population to a certain extent and further improve work efficiency. This can also be proved in the algorithm comparison section.

In this paper, CS algorithm adopts an adaptive parameter strategy, which ensures a large-scale random search in the early stage and a small-scale local precision search in the later stage. In the early stage of the whole process, the CS algorithm is more conducive to finding the global optimal solution or its vicinity by large-scale global random search. In the later stage, frequent local search can more accurately converge to the global optimal solution or approximate global optimal solution, so as to ensure better compliance with the experimental requirements.

The CS algorithm used in this paper involves a few parameters, and the real influential factors are the abandonment probability

(1) For the problem of abandonment probability

(2) As for Lévy flight step size impact factor

Parallel Adaptive Cuckoo Search (PACS) algorithm pseudo-code is shown in algorithmic 3.1.

In of 9 this paper, the performance algorithms is compared through 28 benchmark test functions [

Fi | Function introduction | Optimal fitness value | |
---|---|---|---|

F1 | Sphere Function | −1400 | |

F2 | Rotated High Conditioned Elliptic Function | −1300 | |

Unimodal Functions | F3 | Rotated Bent Cigar Function | −1200 |

F4 | Rotated Discus Function | −1100 | |

F5 | Different Powers Function | −1000 | |

F6 | Rotated Rosenbrock’s Function | −900 | |

F7 | Rotated Schaffers F7 Function | −800 | |

F8 | Rotated Ackley’s Function | −700 | |

F9 | Rotated Weierstrass Function | −600 | |

F10 | Rotated Griewank’s Function | −500 | |

F11 | Rastrigin’s Function | −400 | |

Basic Multimodal Functions | F12 | Rotated Rastrigin’s Function | −300 |

F13 | Non-Continuous Rotated Rastrigin’s Function | −200 | |

F14 | Schwefe’s Function | −100 | |

F15 | Rotated Schwefel’s Function | 100 | |

F16 | Rotated Kalsuara Function | 200 | |

F17 | Lunacek Bi_Rastrigin Function | 300 | |

F18 | Rotated Luncek Bi_Rastrigin Function | 400 | |

F19 | Expanded Griewank’s plus Rosenbrock’s Function | 500 | |

F20 | Expanded affer’s F6 Function | 600 | |

F21 | Composition Function 1 (n = 5,Rotated) | 700 | |

F22 | Composition Function 2 (n = 3,Unrotated) | 800 | |

F23 | Composition Function 3 (n = 3,Rotated) | 900 | |

Composition Functions | F24 | Composition Function 4 (n = 3,Rotated) | 1000 |

F25 | Composition Function 5 (n = 3,Rotated) | 1100 | |

F26 | Composition Function 6 (n = 5,Rotated) | 1200 | |

F27 | Composition Function 7 (n = 5,Rotated) | 1300 | |

F28 | Composition Function 8 (n = 5,Rotated) | 1400 |

Combined with the above comparison

Function | CS | PCS | PACS | PSO | GWO | WOA | ABC | SCA | DE | |
---|---|---|---|---|---|---|---|---|---|---|

F1 | Mean | −1.40E+03 | −1.40E+03 | −1.34E+03 | 1.43E+03 | −1.39E+03 | 3.66E+03 | 2.68E+04 | 4.32E+03 | |

Std | 1.87E−07 | 1.29E−09 | 9.44E−09 | 2.14E+02 | 1.28E+03 | 7.98E+00 | 4.19E+02 | 4.01E+03 | 4.92E+02 | |

Best | −1.40E+03 | −1.40E+03 | −1.40E+03 | −1.40E+03 | −4.61E+02 | −1.40E+03 | 2.58E+03 | 1.84E+04 | 3.36E+03 | |

F2 | Mean | 1.75E+07 | 9.78E+06 | 8.52E+06 | 3.08E+07 | 6.54E+07 | 2.39E+0 8 | 3.42E+08 | 1.13E+09 | |

Std | 3.68E+06 | 2.81E+06 | 1.23E+06 | 5.96E+06 | 1.51E+07 | 1.22E+07 | 4.20E+07 | 1.02E+08 | 8.97E+07 | |

Best | 1.47E+07 | 7.61E+06 | 1.21E+07 | 4.53E+06 | 2.08E+07 | 5.60E+07 | 1.91E+08 | 2.49E+08 | 1.02E+09 | |

F3 | Mean | 1.91E+09 | 9.50E+08 | 9.65E+08 | 1.24E+10 | 3.65E+10 | 2.27E+13 | 9.98E+10 | 1.25E+11 | |

Std | 4.96E+08 | 2.98E+08 | 9.22E+07 | 2.62E+07 | 4.26E+09 | 5.74E+09 | 2.33E+13 | 2.18E+10 | 6.70E+09 | |

Best | 1.53E+09 | 7.64E+08 | 3.60E+07 | 9.40E+08 | 7.67E+09 | 3.08E+10 | 3.50E+12 | 7.82E+10 | 1.17E+11 | |

F4 | Mean | 1.05E+05 | 6.13E+04 | 3.25E+04 | 4.15E+04 | 4.32E+04 | 8.28E+04 | 6.50E+04 | 1.79E+05 | |

Std | 5.37E+03 | 2.37E+04 | 3.17E+03 | 2.36E+02 | 2.02E+03 | 1.19E+04 | 2.69E+03 | 4.39E+03 | 2.56E+04 | |

Best | 9.97E+04 | 4.41E+04 | 2.88E+04 | −2.13E+02 | 3.94E+04 | 3.11E+04 | 7.98E+04 | 6.05E+04 | 1.55E+05 | |

F5 | Mean | −1.00E+03 | −1.00E+03 | −9.61E+02 | −1.59E+02 | −8.54E+02 | 9.72E+02 | 2.53E+03 | −7.48E+02 | |

Std | 3.80E−05 | 3.13E−04 | 1.57E−05 | 6.60E+01 | 2.88E+02 | 2.46E+01 | 1.87E+02 | 1.25E+03 | 1.96E+01 | |

Best | −1.00E+03 | −1.00E+03 | −1.00E+03 | −1.00E+03 | −7.21E+02 | −9.17E+02 | 5.85E+02 | 1.22E+03 | −7.87E+02 | |

F6 | Mean | −8.55E+02 | −8.56E+02 | −8.20E+02 | −6.44E+02 | −6.93E+02 | −5.32E+01 | 9.02E+02 | 1.80E+02 | |

Std | 2.83E+00 | 1.50E+00 | 2.38E-01 | 2.82E+01 | 4.34E+01 | 5.50E+01 | 4.68E+01 | 1.28E+02 | 7.68E+01 | |

Best | −8.61E+02 | −8.57E+02 | −8.57E+02 | −8.51E+02 | −6.81E+02 | −7.28E+02 | −1.07E+02 | 8.10E+02 | 1.05E+02 | |

F7 | Mean | −6.49E+02 | −6.88E+02 | −6.89E+02 | −6.87E+02 | −1.82E+02 | 1.26E+03 | −6.23E+02 | −5.66E+02 | |

Std | 2.54E+00 | 8.21E+00 | 9.42E+00 | 2.95E+01 | 2.93E+01 | 4.19E+02 | 1.36E+02 | 2.00E+01 | 1.85E+01 | |

Best | −6.52E+02 | −6.96E+02 | −7.00E+02 | −7.20E+02 | −7.56E+02 | −5.32E+02 | 1.15E+03 | −6.43E+02 | −5.86E+02 | |

F8 | Mean | −6.79E+02 | −6.79E+02 | −6.79E+02 | −6.79E+02 | −6.79E+02 | −6.79E+02 | −6.79E+02 | −6.79E+02 | |

Std | 4.17E−02 | 3.00E−02 | 2.04E−02 | 5.34E−02 | 2.08E−02 | 3.82E−02 | 1.97E−02 | 2.59E−02 | 6.88E−03 | |

Best | −6.79E+02 | −6.79E+02 | −6.80E+02 | −6.79E+02 | −6.79E+02 | −6.79E+02 | −6.79E+02 | −6.79E+02 | −6.79E+02 | |

F9 | Mean | −5.43E+02 | −5.40E+02 | −5.42E+02 | −5.40E+02 | −5.31E+02 | −5.56E+02 | −5.27E+02 | −5.26E+02 | |

Std | 7.05E−02 | 4.47E−01 | 3.55E−01 | 4.40E+00 | 2.35E+00 | 1.80E+00 | 2.62E+00 | 1.78E+00 | 9.85E-01 | |

Best | −5.43E+02 | −5.40E+02 | −5.42E+02 | −5.45E+02 | −5.64E+02 | −5.32E+02 | −5.59E+02 | −5.29E+02 | −5.27E+02 | |

F10 | Mean | −4.99E+02 | −4.99E+02 | −4.12E+02 | 4.46E+01 | −2.51E+02 | 5.81E+02 | 3.19E+03 | 5.94E+03 | |

Std | 1.32E−01 | 7.74E−02 | 3.59E−01 | 9.37E+01 | 2.25E+02 | 8.87E+01 | 8.62E+01 | 7.36E+02 | 9.31E+02 | |

Best | −4.99E+02 | −4.99E+02 | −5.00E+02 | −4.72E+02 | −1.02E+02 | −3.49E+02 | 5.07E+02 | 2.72E+03 | 5.09E+03 | |

F11 | Mean | −2.27E+02 | −2.38E+02 | 1.85E+00 | −1.82E+02 | 3.51E+02 | −7.43E+01 | 2.95E+02 | 3.14E+01 | |

Std | 1.38E+01 | 1.92E+01 | 9.61E+00 | 6.64E+01 | 4.04E+01 | 8.96E+01 | 6.15E+01 | 4.25E+01 | 1.80E+01 | |

Best | −2.54E+02 | −2.71E+02 | −3.01E+02 | −9.48E+01 | −2.39E+02 | 1.66E+02 | −1.87E+02 | 2.06E+02 | −1.56E+01 | |

F12 | Mean | 6.55E+01 | 5.86E+01 | 2.40E+02 | 2.06E+01 | 5.43E+02 | 1.45E+02 | 4.25E+02 | 4.34E+02 | |

Std | 2.14E+01 | 4.73E+01 | 3.13E+01 | 1.20E+02 | 1.47E+02 | 2.23E+02 | 3.41E+01 | 6.85E+01 | 3.71E+01 | |

Best | 4.67E+01 | 5.57E+00 | −5.35E+01 | 1.32E+02 | −8.72E+01 | 3.59E+02 | 1.06E+02 | 3.59E+02 | 3.91E+02 | |

F13 | Mean | 1.71E+02 | 2.86E+02 | 4.31E+02 | 2.85E+02 | 7.71E+02 | 2.49E+02 | 5.17E+02 | 5.68E+02 | |

Std | 3.36E+01 | 3.51E+01 | 1.18E+01 | 2.59E+01 | 3.03E+01 | 1.91E+01 | 2.80E+01 | 4.46E+01 | 2.05E+01 | |

Best | 1.37E+02 | 2.47E+02 | 1.41E+02 | 4.09E+02 | 2.50E+02 | 7.54E+02 | 2.17E+02 | 4.74E+02 | 5.53E+02 | |

F14 | Mean | 6.02E+03 | 5.25E+03 | 6.25E+03 | 5.47E+03 | 9.01E+03 | 5.10E+03 | 1.32E+04 | 8.17E+03 | |

Std | 4.00E+02 | 4.55E+02 | 4.18E+02 | 6.38E+02 | 9.45E+02 | 1.19E+03 | 2.05E+02 | 4.09E+02 | 3.10E+02 | |

Best | 5.03E+03 | 4.21E+03 | 3.98E+03 | 5.07E+03 | 3.76E+03 | 5.73E+03 | 4.52E+03 | 1.22E+04 | 7.59E+03 | |

F15 | Mean | 9.27E+03 | 8.50E+03 | 8.53E+03 | 8.45E+03 | 1.06E+04 | 1.28E+04 | 1.43E+04 | 1.43E+04 | |

Std | 5.72E+02 | 1.80E+02 | 3.59E+02 | 3.43E+02 | 3.44E+02 | 1.02E+03 | 3.97E+01 | 1.40E+02 | 4.02E+02 | |

Best | 8.61E+03 | 8.33E+03 | 7.75E+03 | 8.12E+03 | 8.09E+03 | 9.66E+03 | 1.28E+04 | 1.42E+04 | 1.38E+04 | |

F16 | Mean | 2.03E+02 | 2.02E+02 | 2.03E+02 | 2.03E+02 | 2.03E+02 | 2.03E+02 | 2.03E+02 | 2.03E+02 | |

Std | 2.90E−01 | 3.88E−01 | 3.27E−01 | 4.99E−01 | 2.89E−01 | 4.71E−01 | 2.48E−01 | 2.65E−01 | 2.50E−01 | |

Best | 2.02E+02 | 2.01E+02 | 2.01E+02 | 2.01E+02 | 2.03E+02 | 2.02E+02 | 2.03E+02 | 2.03E+02 | 2.03E+02 | |

F17 | Mean | 6.09E+02 | 6.91E+02 | 7.35E+02 | 6.13E+02 | 1.41E+03 | 6.45E+02 | 1.27E+03 | 1.33E+03 | |

Std | 2.40E+01 | 5.03E+01 | 2.63E+01 | 7.12E+01 | 4.93E+01 | 1.10E+02 | 1.67E+01 | 6.94E+01 | 8.05E+01 | |

Best | 5.46E+02 | 5.84E+02 | 4.66E+02 | 6.40E+02 | 5.28E+02 | 1.17E+03 | 6.11E+02 | 1.13E+03 | 1.21E+03 | |

F18 | Mean | 7.66E+02 | 9.09E+02 | 8.88E+02 | 9.04E+02 | 1.42E+03 | 9.47E+02 | 1.40E+03 | 1.50E+03 | |

Std | 5.20E+01 | 2.35E+01 | 1.19E+01 | 1.19E+02 | 1.65E+01 | 9.33E+01 | 2.63E+01 | 6.24E+00 | 9.42E+01 | |

Best | 7.08E+02 | 8.91E+02 | 6.77E+02 | 8.12E+02 | 8.86E+02 | 1.33E+03 | 9.22E+02 | 1.39E+03 | 1.40E+03 | |

F19 | Mean | 5.22E+02 | 5.37E+02 | 6.37E+02 | 1.67E+03 | 6.52E+02 | 5.49E+04 | 3.33E+04 | 5.47E+03 | |

Std | 2.24E+00 | 7.56E+00 | 5.81E+00 | 1.89E+02 | 1.90E+03 | 4.74E+01 | 1.93E+04 | 1.47E+04 | 2.05E+03 | |

Best | 5.21E+02 | 5.28E+02 | 5.13E+02 | 5.29E+02 | 5.74E+02 | 5.97E+02 | 3.33E+04 | 1.65E+04 | 3.51E+03 | |

F20 | Mean | 6.24E+02 | 6.23E+02 | 6.22E+02 | 6.22E+02 | 6.25E+02 | 6.22E+02 | 6.24E+02 | 6.23E+02 | |

Std | 4.46E−01 | 4.50E−01 | 2.03E−01 | 2.02E+00 | 1.80E+00 | 5.61E−03 | 2.45E−01 | 1.33E−01 | 8.29E−02 | |

Best | 6.23E+02 | 6.23E+02 | 6.19E+02 | 6.20E+02 | 6.19E+02 | 6.25E+02 | 6.22E+02 | 6.24E+02 | 6.23E+02 | |

F21 | Mean | 9.06E+02 | 1.04E+03 | 1.65E+03 | 2.78E+03 | 1.67E+03 | 2.63E+03 | 4.57E+03 | 3.71E+03 | |

Std | 2.59E+00 | 2.91E+02 | 2.82E+01 | 2.79E+02 | 6.31E+02 | 2.43E+02 | 4.15E+02 | 1.28E+02 | 6.71E+02 | |

Best | 9.00E+02 | 8.12E+02 | 8.08E+02 | 9.11E+02 | 1.31E+03 | 9.58E+02 | 2.16E+03 | 4.32E+03 | 2.54E+03 | |

F22 | Mean | 8.50E+03 | 7.90E+03 | 1.03E+04 | 7.11E+03 | 1.23E+04 | 6.44E+03 | 1.52E+04 | 9.69E+03 | |

Std | 4.51E+02 | 5.46E+02 | 4.78E+02 | 1.49E+03 | 9.31E+02 | 1.54E+03 | 3.31E+02 | 4.55E+02 | 2.83E+02 | |

Best | 7.55E+03 | 6.78E+03 | 4.34E+03 | 7.79E+03 | 5.56E+03 | 9.77E+03 | 6.04E+03 | 1.45E+04 | 8.99E+03 | |

F23 | Mean | 1.17E+04 | 1.11+04 | 1.15E+04 | 1.18E+04 | 1.37E+04 | 1.41E+04 | 1.60E+04 | 1.55E+04 | |

Std | 4.11E+02 | 8.74E+02 | 7.79E+02 | 1.12E+03 | 2.65E+03 | 1.37E+03 | 3.76E+02 | 4.21E+02 | 3.78E+02 | |

Best | 1.06E+04 | 8.79E+03 | 9.93E+03 | 9.31E+03 | 6.47E+03 | 9.93E+03 | 1.31E+04 | 1.48E+04 | 1.47E+04 | |

F24 | Mean | 1.37E+03 | 1.37E+03 | 1.37E+03 | 1.38E+03 | 1.41E+03 | 1.34E+03 | 1.42E+03 | 1.39E+03 | |

Std | 6.34E+00 | 7.75E+00 | 7.09E+00 | 1.62E+01 | 1.07E+01 | 1.23E+01 | 4.30E+00 | 5.66E+00 | 4.21E+00 | |

Best | 1.36E+03 | 1.35E+03 | 1.35E+03 | 1.35E+03 | 1.28E+03 | 1.39E+03 | 1.33E+03 | 1.41E+03 | 1.38E+03 | |

F25 | Mean | 1.51E+03 | 1.50E+03 | 1.50E+03 | 1.53E+03 | 1.53E+03 | 1.46E+03 | 1.55E+03 | 1.50E+03 | |

Std | 4.28E+00 | 8.61E+00 | 8.33E+00 | 2.49E+01 | 1.00E+01 | 1.64E+01 | 6.92E+00 | 6.14E+00 | 3.25E+00 | |

Best | 1.50E+03 | 1.47E+03 | 1.48E+03 | 1.48E+03 | 1.42E+03 | 1.49E+03 | 1.45E+03 | 1.53E+03 | 1.50E+03 | |

F26 | Mean | 1.40E+03 | 1.40E+03 | 1.63E+03 | 1.59E+03 | 1.67E+03 | 1.59E+03 | 1.56E+03 | 1.68E+03 | |

Std | 3.83E−01 | 3.96E−01 | 1.54E−01 | 6.31E+01 | 1.19E+01 | 5.15E+01 | 7.26E+01 | 1.22E+02 | 7.80E+00 | |

Best | 1.40E+03 | 1.40E+03 | 1.40E+03 | 1.40E+03 | 1.58E+03 | 1.41E+03 | 1.41E+03 | 1.43E+03 | 1.66E+03 | |

F27 | Mean | 3.13E+03 | 3.22E+03 | 2.67E+03 | 3.25E+03 | 3.48E+03 | 2.98E+03 | 3.63E+03 | 3.49E+03 | |

Std | 3.07E+02 | 5.65E+01 | 6.95E+02 | 1.51E+02 | 1.23E+02 | 9.85E+01 | 7.22E+01 | 4.61E+01 | 3.28E+01 | |

Best | 2.18E+03 | 3.10E+03 | 1.70E+03 | 2.97E+03 | 2.39E+03 | 3.28E+03 | 2.81E+03 | 3.49E+03 | 3.43E+03 | |

F28 | Mean | 1.80E+03 | 1.80E+03 | 5.63E+03 | 3.27E+03 | 9.43E+03 | 2.68E+03 | 6.74E+03 | 5.13E+03 | |

Std | 3.51E−02 | 2.17E−03 | 7.37E−03 | 2.95E+03 | 1.64E+03 | 1.69E+03 | 9.66E+01 | 7.21E+02 | 1.75E+03 | |

Best | 1.80E+03 | 1.80E+03 | 1.80E+03 | 2.22E+03 | 2.30E+03 | 7.97E+03 | 2.57E+03 | 6.15E+03 | 3.11E+03 |

The first five are unimodal functions, and the performance of the PACS algorithm is the best. Where, in F1, although the final result of CS and PACS is the same, PACS converges faster than CS. In F4, PSO shows good performance. F6 to F20 are multimodal functions. PACS performance is slightly lower than CS performance in F6 and GWO performance in F7 and F9 respectively. Among the other test functions, PACS performed better. F21 to F28 are compound functions. PACS and GWO win or lose half, but the former is more stable. Overall, PACS show better performance in 20 functions, while CS, PSO and GWO all had their strengths in the remaining eight functions. The best performing algorithm in each function is highlighted in bold. Comparing the experimental data of CS and PCS, it can be seen that the parallel strategy works well in improving performance. Comparing the experimental data of PCS and PACS, it can be seen that the performance of the algorithm is significantly improved by the adaptive strategy. As can be seen from

Function | CS | PCS | PACS |
---|---|---|---|

F1 | 9.44E+00 | 5.98E+00 | 6.02E+00 |

F2 | 1.28E+01 | 8.81E+00 | 8.73E+00 |

F3 | 1.30E+01 | 9.08E+00 | 9.41E+00 |

F4 | 1.11E+01 | 7.66E+00 | 7.66E+00 |

F5 | 9.73E+00 | 6.20E+00 | 6.22E+00 |

F6 | 1.06E+01 | 7.23E+00 | 7.28E+00 |

F7 | 1.95E+01 | 1.51E+01 | 1.53E+01 |

F8 | 1.80E+01 | 1.37E+01 | 1.37E+01 |

F9 | 7.54E+01 | 7.30E+01 | 7.23E+01 |

F10 | 1.28E+01 | 9.39E+00 | 9.28E+00 |

F11 | 1.33E+01 | 9.36E+00 | 9.58E+00 |

F12 | 1.66E+01 | 1.28E+01 | 1.28E+01 |

F13 | 1.67E+01 | 1.29E+01 | 1.29E+01 |

F14 | 1.52E+01 | 1.11E+01 | 1.13E+01 |

F15 | 1.57E+01 | 1.26E+01 | 1.25E+01 |

F16 | 2.70E+01 | 2.12E+01 | 2.21E+01 |

F17 | 9.89E+00 | 7.47E+00 | 7.50E+00 |

F18 | 1.25E+01 | 9.81E+00 | 9.89E+00 |

F19 | 1.03E+01 | 7.30E+00 | 7.38E+00 |

F20 | 1.34E+01 | 1.03E+01 | 9.98E+00 |

F21 | 2.31E+01 | 1.99E+01 | 1.99E+01 |

F22 | 2.61E+01 | 2.26E+01 | 2.29E+01 |

F23 | 2.96E+01 | 2.83E+01 | 2.82E+01 |

F24 | 9.66E+01 | 9.08E+01 | 9.10E+01 |

F25 | 9.70E+01 | 9.17E+01 | 9.15E+01 |

F26 | 1.03E+02 | 1.00E+02 | 9.95E+01 |

F27 | 1.01E+02 | 9.70E+01 | 9.72E+01 |

F28 | 4.09E+01 | 3.65E+01 | 3.66E+01 |

To judge whether the experimental results are statistically significant, Wilcoxon’s rank-sum test is executed at a 5% significance level. The results are shown in

Function | CS | PCS | PSO | SCA | WOA | GWO | ABC | DE | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

F1 | 1.21E-12 | > | 1.21E-12 | > | 1.21E-12 | > | 1.21E-12 | > | 1.21E-12 | > | 1.21E-12 | > | 1.21E-12 | > | 1.21E-12 | > |

F2 | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > |

F3 | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > |

F4 | 3.02E-11 | > | 3.02E-11 | > | 6.28E-06 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > |

F5 | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > |

F6 | 7.51E-01 | = | 1.32E-04 | > | 1.07E-09 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > |

F7 | 3.02E-11 | > | 3.02E-11 | > | 4.98E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 1.36E-07 | > | 1.36E-07 | > | 3.02E-11 | > |

F8 | 1.84E-02 | > | 7.62E-03 | > | 1.44E-02 | > | 3.02E-11 | > | 2.68E-04 | > | 6.55E-04 | > | 2.68E-04 | > | 2.47E-04 | > |

F9 | 4.08E-11 | > | 6.70E-11 | > | 1.29E-06 | > | 3.02E-11 | > | 3.02E-11 | > | 1.41E-09 | > | 6.73E-11 | > | 3.50E-11 | > |

F10 | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > |

F11 | 3.02E-11 | > | 3.34E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > |

F12 | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 5.57E-10 | > | 3.02E-11 | > | 3.02E-11 | > |

F13 | 5.49E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 7.77E-09 | > | 3.02E-11 | > | 3.43E-11 | > |

F14 | 2.67E-09 | > | 7.60E-07 | > | 1.16E-07 | > | 3.02E-11 | > | 3.69E-11 | > | 2.01E-04 | > | 2.01E-04 | > | 2.01E-04 | > |

F15 | 5.00E-09 | > | 2.83E-08 | > | 5.30E-01 | = | 3.02E-11 | > | 6.12E-10 | > | 9.88E-03 | > | 5.84E-05 | > | 7.68E-03 | > |

F16 | 5.49E-11 | > | 5.97E-09 | > | 5.57E-10 | > | 3.02E-11 | > | 9.26E-09 | > | 3.69E-11 | > | 5.97E-09 | > | 5.97E-09 | > |

F17 | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.69E-11 | > | 3.02E-11 | > | 3.02E-11 | > |

F18 | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > |

F19 | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 5.49E-11 | > | 5.32E-11 | > | 4.46E-11 | > |

F20 | 3.02E-11 | > | 4.20E-10 | > | 6.52E-09 | > | 1.72E-12 | > | 3.00E-11 | > | 7.73E-01 | = | 5.49E-11 | > | 3.30E-10 | > |

F21 | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > |

F22 | 3.02E-11 | > | 3.69E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 5.46E-09 | > | 3.69E-11 | > | 3.45E-11 | > |

F23 | 6.53E-07 | > | 1.75E-05 | > | 2.53E-04 | > | 3.02E-11 | > | 2.03E-09 | > | 5.61E-05 | > | 2.53E-04 | > | 2.78E-04 | > |

F24 | 3.02E-11 | > | 3.02E-11 | > | 2.61E-10 | > | 3.02E-11 | > | 3.02E-11 | > | 1.07E-09 | > | 3.02E-11 | > | 3.32E-10 | > |

F25 | 3.02E-11 | > | 1.96E-10 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 6.05E-07 | > | 4.02E-11 | > | 3.45E-11 | > |

F26 | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > |

F27 | 4.18E-09 | > | 3.02E-11 | > | 1.46E-10 | > | 3.02E-11 | > | 3.02E-11 | > | 1.34E-05 | > | 2.34E-06 | > | 3.44E-05 | > |

F28 | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > | 3.02E-11 | > |

Researchers have conducted some similar case studies in the field of layout. In [

A group of rectangular

There are four commonly used methods about the existing rectangular layout algorithms: the BL algorithm, step down algorithm, the lowest horizontal line algorithm, and improved search algorithm based on the lowest horizontal line. Among them, the BL algorithm has severe defects, some of the best solutions cannot be found. Moreover, the phenomenon of left side height may occur. The descending step algorithm is similar to the BL algorithm, but the difference is that it is easy to produce the sensation of high right side. The lowest horizontal line algorithm generally does not show the sensation that one side is too high, but there are other drawbacks. Stay when the width of the minimum horizontal line is smaller than the width of the rectangular of a row. The method used by the algorithm is to find the current lowest horizontal line and compare the current rectangular blocks to be arranged. However, the height of the lowest level is less than right now to the width of the rectangular of a row, but that doesn’t mean it’s less than the width of all the rectangles. This can make the minimum horizontal lines above small rectangular part of the waste. The search algorithm based on the lowest horizontal line is better than the BL algorithm and the lower step algorithm. Compared with the lowest horizontal line algorithm, this algorithm can solve the waste defect of the small rectangular part above the lowest horizontal line to a certain extent. In this paper, the search algorithm based on the lowest horizontal line is used to arrange the set of rectangular in the determined order.

The specific steps are divided into four steps.

Step 1: Set the initial horizontal line queue. The horizontal line queue is arranged in an increasing height. At this time, there is only one horizontal line in the queue with a height of 0 and a length equal to the width of the raw material. Sets the queue of rectangular to be arranged in a known initial order.

Step 2: Whenever a rectangular is placed, select the section with the lowest height from the horizontal line queue. If there are several segments, select the leftmost segment to determine whether the width of the horizontal segment is greater than or equal to the width of the part to be arranged. If the horizontal segment is less than the width of the rectangular, the width of the rectangular state continues to contrast. If still cannot let go of the rectangular, continued to contrast the other block after this block, while the block after all rectangular is unable to meet the conditions. If no block matches the placement criteria, set the lowest horizontal line to the next in the horizontal line queue.

Step 3: Repeat the Step 2 process until a part can be discharged. Each time a part is discharged, the part is removed from the queue of waiting parts.

Step 4: Repeat Step 2 and Step 3 until all parts are discharged. Finally, the maximum height of the upper edge of all rectangular is the required plate’s height.

In the experiment, the coding of the solution is divided into two parts: on the one hand, each rectangular block is initially assigned with a number; on the other hand, each rectangular block has its own state (horizontal and vertical), as shown in the first rows of

Rectangular order | 1 | 3 | 5 | 2 | 6 | 4 | 7 | 8 |

Status (horizontal and vertical) | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 |

Order1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

Status | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 |

Order2 | 2 | 5 | 6 | 4 | 1 | 8 | 7 | 3 |

Status | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 0 |

Simulation experiment environment: Windows 10, MATLAB R2020a.

Raw material: width is 20, height is 200 rectangle.

The rectangular layout information is shown in

W | 3 | 4 | 6 | 4 | 2 | 6 | 4 | 4 | 9 | 4 | 6 | 4 | 9 | 4 | 2 | 8 | 9 | 6 | 6 | 2 | 9 | 3 | 8 | 3 | 7 | 6 | 7 | 8 | 8 | 5 |

H | 6 | 7 | 7 | 2 | 5 | 4 | 2 | 6 | 6 | 7 | 4 | 6 | 3 | 5 | 7 | 4 | 6 | 3 | 3 | 6 | 7 | 5 | 5 | 4 | 4 | 3 | 5 | 7 | 9 | 3 |

Order1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |

Order2 | 25 | 21 | 6 | 30 | 15 | 11 | 19 | 7 | 29 | 2 | 14 | 13 | 9 | 16 | 24 | 8 | 4 | 20 | 1 | 22 | 18 | 5 | 28 | 10 | 23 | 3 | 26 | 12 | 17 | 27 |

In order to test the superior performance of PACS in the application, this paper also combines other 8 algorithms (CS, PCS, GWO, SCA, WOA, DE, ABC, PSO) with the application to compare their performance. After 20 comparative experiments, the PACS algorithm can find the optimal solution every time, that is, the solution with a height of 43 in the experimental graph. However, other algorithms may not obtain the optimal solution. The specific experimental results are shown in

CS | PCS | PACS | PSO | SCA | WOA | GWO | ABC | DE | |
---|---|---|---|---|---|---|---|---|---|

43.2 | 43.15 | 43.15 | 43.3 | 43.25 | 43.05 | 43.3 | 43.2 | ||

44 | 44 | 44 | 45 | 44 | 44 | 45 | 45 | ||

43 | 43 | 43 | 43 | 43 | 43 | 43 | 43 | ||

16 | 17 | 17 | 16 | 15 | 19 | 16 | 17 |

According to the initial sequence, the material height used in the layout results is 47, and the material utilization rate is 841/940 = 89.5%.

The material height of the layout results obtained by combining the improved algorithm is 43, and the material utilization ratio is 841/860 = 97.8%.

In this paper, the convergence speed and accuracy of CS algorithm is improved by parallel and adaptive strategies. The performance comparison experiments of PACS, CS, PCS, PSO, GWO, SCA, ABC, DE and WOA are carried out through the CEC-2013 test function set, and the results show that the improvement effect is significant. The optimal layout is a classical NP-hard problem, and the solution set of rectangular layout is huge. In this paper, PACS algorithm is used to obtain the layout order and state of the rectangle. According to the results of application experiments, the material utilization ratio of the improved CS algorithm applied to the layout system is increased from 89.5% to 97.8% compared with the layout algorithm based on the lowest horizontal line algorithm.

In the following research work, we will study Monte Carlo theory [