The design of any system contemplates the elaboration of a prototype of the entire system or some parts, before the manufacturing phase. Nowadays, rapid prototyping (RP) is widely used by the designers. This paper addresses the multi-objective factors optimization of the fused deposition modelling (FDM) technology. The problem is converted into a single one using the weighted-sum method and then solved by resorting to two nature-inspired computing techniques, namely particle swarm optimization (PSO) and differential evolution (DE). The results obtained are compared.
Research Article
Multi-Objective Factors Optimization in Fused Deposition Modelling with Particle Swarm Optimization and Differential Evolution
https://doi.org/10.21203/rs.3.rs-777037/v1
This work is licensed under a CC BY 4.0 License
Journal Publication
published 25 Mar, 2022
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rapid prototyping
fused deposition modelling
multi-objective optimization
weighted-sum method
particle swarm optimization
differential evolution
For many years, the industry has been continuously evolving due to the increasingly great demand from consumers, and machining processes are widely used, including conventional and nonconventional processes [1, 2]. One of the most important things in the industry is the prototype, which is a miniature or a scaled model of a product to be created. In previous years, the prototype was manually made by molding and it took a long time to manufacture; then with the rapid development of technology, rapid prototyping (RP) can make complex 3D shapes that are made using a Computer Aided Design (CAD) with different processes, to reduce time, minimize cost and achieve complex shapes that cannot be manufactured by machining. Nowadays, various techniques are used, such as 3D printing, or virtual and augmented reality [3, 4]. To obtain a high quality object, it is necessary to optimize the processes. The prototyping processes can be found in several fields, such as medicine, aerospace/aeronautics, and robotics.
Nowadays, the optimization of the prototyping processes remains a challenge for the researchers. Most of the optimization approaches used to solve engineering problems are nature-inspired due to their effectiveness [5]. This paper provides a nonexecutive review of the literature. Comprehensive reviews can found in Refs. [6–14]. Lee et al. [15] used the Taguchi method to optimize the fused deposition modeling (FDM) to produce the acrylonitrile butadiene styrene. Udroiu and Nedelcu [16] investigated the inkjet printing and polymer jetting using CATIA software, but the results were not compared to those achieved by alternate methods. Wang et al. [17] used the Taguchi method with the Gray relational analysis to optimize the FDM. The results were compared to the prediction. Sood et al. [18] proposed empirical models for the FDM to improve the tensile, flexural and impact strength responses by considering five process parameters: layer thickness, orientation, raster angle, raster width, and air gap. Variance analysis was used to test the model. Rao and Rai [13] used the teaching-learning-based optimization (TLBO) and the non-dominated sorting TLBO algorithm to solve single and multi-objective prototyping processes, including the model developed by Sood et al. [18]. The Pareto was generated to find the set of optimal solutions. However, selecting one single solution from the Pareto set led to conflicting conclusions. Shirmohammadi et al. [19] implemented the artificial neural network (ANN) with the particle swarm optimization (PSO) to minimize the surface roughness of a 3D printing process. They concluded that the proposed hybrid approach reduced the error.
In this paper, the multi-objective FDM process is optimized with the weighted-sum method and the optimal parameters are provided using the PSO and the differential evolution (DE). The remainder of the paper is organized as follows: Sect. 2 describes the problem. Sections 3 and 4 give the description of the implemented PSO and DE, respectively. Section 5 provides the results with a discussion. Finally, the last section concludes the paper with remarks and suggested directions for future work.
The FDM is a RP process that uses a moving nozzle to extrude a polymeric material fiber [18, 20–23]. Based on empirical models, the multi-objective optimization problem of the FDM is given as follows [13, 18]:
Subject to the following design variables:
where Ts [MPa] is the tensile strength, Fs [MPa] is the flexural strength, Is [MJ/m2] is the impact strength, A [mm] is the layer thickness, B [degree] is the orientation, C [degree] is the raster angle, D [mm] is the raster width, and E [mm] is the air gap.
In this paper, the multi-objective problem described in Equations (1)-(3) is converted into a single objective problem using the weighted-sum method [24–27] as follows:
where Z is the one-scaled objective function and wi (i = 1, 2, 3; w1 + w2 + w3 = 1) are the weight factors for each objective function.
The particle swarm optimization (PSO) is a nature-inspired optimization algorithm inspired by the moving mechanism of swarms, such as flocks of birds and schools of fishes. It was initially developed by Kennedy and Eberhart [28]. It is based on the position and the velocity of the particles of the swarm. Details on the PSO can be found in Refs. [28–30]. The pseudo-code of the implemented PSO to solve the above problem is given in Algorithm 1.
Algorithm 1. Pseudo-code of the implemented PSO.
Input the parameters;
Generate random solutions;
While No. of iterations ≤ max No. of iterations Do
Evaluate the one-scaled objective function and constraint handling;
Move the particles;
Update the position and the velocity;
Find best position and velocity;
End while
Display the results: Z, Ts, Fs, Is, A, B, C, D, and E.
The differential evolution (DE) is a nature-inspired optimization algorithm inspired by the population evolution mechanisms. It was developed by Storn and Price [31] and is based on the genetic algorithms [32] but is characterized by more perturbations during iterations. More details on the DE can be found in Refs. [31, 33–35]. The pseudo-code of the implemented DE to solve the above problem is given in Algorithm 2.
Algorithm 2. Pseudo-code of the implemented DE.
Input of the parameters;
Generate random solutions;
While No. of iterations ≤ max No. of iterations Do
Evaluate the one-scaled objective function and constraint handling;
Perform evolution mechanisms;
Perturbations during iterations;
End while
Display the results: Z, Ts, Fs, Is, A, B, C, D, and E.
The problem descried has been solved using the above two algorithms and implemented using MATLAB 2017 and run on a PC with the following characteristics: i5 of 2.50 GHz with 4 GB. Each algorithm has been run over ten independent runs. The population size is 20 and the maximum number of iterations is 100. These parameters have been fixed by trial-and-error and based on experience. The weight factors are considered equal, i.e. (w1= w2=w3=0.3333).
Tables 1 and 2 report the results obtained by the PSO and the DE over the ten runs, respectively. The values of the one-scaled objective (Z), the three process performances (Ts, Fs and Is), design variables, number of function evaluations (NFE), CPU time, and standard deviation (σ) are included.
From Table 1, it can be observed that the best value of Z obtained by the PSO is 98.8849 for all runs, except #3, #6 and #7. The fewer NFE corresponds to #5 (1,220) with 12.23 s of CPU time. The optimal values are TS=170.2408 MPa, Fs=125.3889 MPa, Is=1.3162 MJ/m2, whereas the decision variables are A=0.1270 mm, B=7.6880°, C=60°, D=0.4064 mm, and E=0.008 mm. The standard deviation of the ten runs is 2.8E-04.
From Table 2, it can be observed that the best value of Z obtained by the DE is 98.8849 for all runs. The lower NFE corresponds to #9 (440) with 9.89 s of CPU time. The optimal values are TS=170.3954 MPa, Fs=125.2397 MPa, Is=1.3166 MJ/m2, whereas the decision variables are A=0.1270 mm, B=7.6923°, C=60°, D=0.4064 mm, and E=0.008 mm. The standard deviation of the ten runs is 0.
Table 3 summarizes the best results obtained by the PSO and the DE. It can be observed that both algorithms obtained the same value of the one-scale objective (Z=98.8849). However, the performances of the DE have outperformed those of the PSO in terms of number of function evaluations, CPU time, and standard deviation.
The goal of this paper was to investigate the multi-objective fused deposition modelling (FDM) optimization problem which is a rapid prototyping process (RP). The considered objectives were the tensile strength, the flexural strength, and the impact strength. The multi-objective problem was converted to a single one using the weighted-sum method in order to avoid the disadvantages of the Pareto set. The particle swarm optimization (PSO) and differential evolution (DE) were implemented with constraint handling to solve the problem. It was shown that both algorithms provides the same value of the one-scaled objective, but the performances of the PSO were better. Future work will be devoted to the development of a hybrid approach to improve the results and solve other rapid prototyping processes.
Authors’ contributions Mohamed Arezki Mellal conceived of the study, designed the study, supervised the study, and wrote the manuscript. Chahinaze Laifaouia and Fahima Ghezal investigated the study, programmed the study, and collected the results. Edward J. Williams designed the study and revised the manuscript.
Funding This study has not received funds.
Data availability The data sets supporting the results of this article are included within the article.
Consent to publish All authors agree to transfer copyright of this article to the Publisher. Competing interests The authors declare no competing interests.
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Table 1. Results obtained by the PSO.
|
Run # |
Ts [MPa] |
Fs [MPa] |
Is [MJ/m2] |
Z |
A [mm] |
B [°] |
C [°] |
D [mm] |
E [mm] |
NFE |
CPU[s] |
σ |
|
1 |
170.2548 |
125.3745 |
1.3162 |
98.8849 |
0.1270 |
7.6880 |
60 |
0.4064 |
0.008 |
1,660 |
17.28 |
2.8E-04 |
|
2 |
170.4601 |
125.1358 |
1.3167 |
98.8849 |
0.1270 |
7.7105 |
59.9944 |
0.4064 |
0.008 |
1,960 |
20.23 |
|
|
3 |
170.5864 |
125.0318 |
1.3170 |
98.8845 |
0.127 |
7.7331 |
60 |
0.4064 |
0.007 |
1,980 |
12.18 |
|
|
4 |
170.3061 |
124.9912 |
1.3161 |
98.8849 |
0.127 |
7.7191 |
59.9466 |
0.4064 |
0.008 |
1,840 |
11.90 |
|
|
5 |
170.2408 |
125.3889 |
1.3162 |
98.8849 |
0.1270 |
7.6609 |
60 |
0.4064 |
0.008 |
1,220 |
12.23 |
|
|
6 |
170.2634 |
125.3570 |
1.3162 |
98.884 |
0.1270 |
7.6671 |
59.9989 |
0.4065 |
0.008 |
2,000 |
12.38 |
|
|
7 |
170.3064 |
125.3283 |
1.3163 |
98.8848 |
0.127 |
7.6741 |
60 |
0.4064 |
0.008 |
1,720 |
11.57 |
|
|
8 |
170.3662 |
125.1776 |
13165 |
98.8849 |
0.1274 |
7.6934 |
59.9956 |
0.4065 |
0.008 |
2,000 |
11.81 |
|
|
9 |
170.4134 |
125.2184 |
1.3166 |
98.8849 |
0.127 |
7.6967 |
60 |
0.4064 |
0.008 |
1,980 |
11.74 |
|
|
10 |
170.3698 |
125.2216 |
1.3165 |
98.8849 |
0.127 |
7.6931 |
59.9936 |
0.4065 |
0.008 |
1,680 |
11.95 |
Table 2. Results obtained by the DE.
|
Run # |
Ts [MPa] |
Fs [MPa] |
Is [MJ/m2] |
Z |
A [mm] |
B [°] |
C [°] |
D [mm] |
E [mm] |
NFE |
CPU[s] |
σ |
|
1 |
170.3954 |
125.2397 |
1.3166 |
98.8849 |
0.1270 |
7.6923 |
60 |
0.4064 |
0.008 |
600 |
15.37 |
0 |
|
2 |
170.3954 |
125.2397 |
1.3166 |
98.8849 |
0.1270 |
7.6923 |
60 |
0.4064 |
0.008 |
500 |
9.90 |
|
|
3 |
170.3954 |
125.2397 |
1.3166 |
98.8849 |
0.1270 |
7.6923 |
60 |
0.4064 |
0.008 |
620 |
9.78 |
|
|
4 |
170.3954 |
125.2397 |
1.3166 |
98.8849 |
0.1270 |
7.6923 |
60 |
0.4064 |
0.008 |
640 |
9.38 |
|
|
5 |
170.3954 |
125.2397 |
1.3166 |
98.8849 |
0.1270 |
7.6923 |
60 |
0.4064 |
0.008 |
500 |
9.52 |
|
|
6 |
170.3954 |
125.2397 |
1.3166 |
98.8849 |
0.1270 |
7.6923 |
60 |
0.4064 |
0.008 |
620 |
9.43 |
|
|
7 |
170.3954 |
125.2398 |
1.3166 |
98.8849 |
0.1270 |
7.6923 |
60 |
0.4064 |
0.008 |
660 |
9.83 |
|
|
8 |
170.3954 |
125.2397 |
1.3166 |
98.8849 |
0.1270 |
7.6923 |
60 |
0.4064 |
0.008 |
560 |
9.18 |
|
|
9 |
170.3954 |
125.2397 |
1.3166 |
98.8849 |
0.1270 |
7.6923 |
60 |
0.4064 |
0.008 |
440 |
9.89 |
|
|
10 |
170.3954 |
125.2398 |
1.3166 |
98.8849 |
0.1270 |
7.6923 |
60 |
0.4064 |
0.008 |
660 |
9.58 |
Table 3. Comparison of PSO vs DE.
|
|
Z |
NFE |
CPU[s] |
σ |
|
PSO |
98.8849 |
1,980 |
12.23 |
2.8E-04 |
|
DE |
98.8849 |
440 |
9.89 |
0 |