Line heating process parameter forecasting method based on optimization concept

Abstract

In ship construction, line heating is a necessary process for forming ship plates with complex curvature. The determination of line heating process parameters has a significant impact on the accuracy of the process and the efficiency of shipbuilding. However, previous studies on this subject only stay at the level of predicting deformation, which has not meet the actual processing production requirements. In this paper, a new idea of forecasting line heating parameters proposed. Firstly, an Radial Basis Function (RBF) proxy model between processing parameters and deformation was constructed based on experimental data, and then the reliability of the proxy model is verified. The optimization objective for proxy model was set as the minimum difference between the final deformation and the target deformation. Meanwhile, an improved Multi-Objective Particle Swarm Optimization (MOPSO) method was employed to search the optimal-set of processing parameters with non-dominated solutions. The performance of the optimized parameter set was ranked by using the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS), and the processing parameters combination ranking first was selected as the parameter forecasting result. The results showed that the predicted processing parameters have some reference for the practical engineering.

Keywords

Line heating parametric forecasting RBF improved MOPSO TOPSIS

Introduction

During the construction of a ship, a huge amount of time is spent on forming and processing the outer plates, which are usually composed of complex hyperbolic surfaces and are formed mainly by line heating process. The process is mostly carried out manually by experienced technicians at present,¹ using gas torch as heat source, adjusting processing parameters continuously according to processing experience and target plate type, and completing the firing of target plate type from partial to whole little by little, but this empirical trial-and-error method tends to lead to high repetitiveness and low efficiency, which greatly affects the cycle time of ship building. Therefore, it is essential to perform line heating process parameters research.

To realize the systematic development of line heating process parameters, plenty of research works have been carried out at home and abroad. Shin et al.² proposed a comprehensive algorithm for an automated line-heating process, defined a new automatic wire heating process based on quantitative and computer heating information. Wang et al.³ proposed a new mathematical formula to explain the relationship between deflection and shrinkage during the forming process of wire heating sheet, which lays a foundation for the prediction of parameters of large curvature thick plate considering deflections. Feng et al.⁴ proposed a method of modeling the line heating process based on RSs and fuzzy logic, which effectively extracts the technological rules of plate formation. Biswas et al.⁵ obtained a new relationship between the input parameters and the residual deformation during the wire heating process by means of work-volume analysis. Qi et al.⁶ defined two new coupling variables to reflect the relationship between the heating parameters and the final residual deformation by numerical simulation. Seong et al.⁷ inferred the required forming parameters backwards based on the bending angle and shrinkage, etc. by establishing a database mapping of the relationship between the geometric and forming parameters. Nomoto et al.⁸ proposed a simulation method of water cooling process to improve the predictability of plate deformation caused by line heating.

Although there are many studies, they all stay at the level of predicting deformation, which is not quite in line with the actual production application requirements. In view of this, this paper proposes a new method for predicting the processing parameters of water-fire bent plates. First, based on the optimal Latin hypercube test data, an RBF proxy model between the processing parameters and the forming volume is constructed; then, the plate thickness, acetylene flow rate, heating line length, heating speed, and water-fire distance are taken as optimization variables, and a multi-objective optimization model with the minimum difference between the final angular deformation and transverse deformation and the target deformation is adopted as the optimization objective. Improved MOPSO is used to iteratively find the optimization of the established model; finally, TOPSIS is used to rank the performance of the optimized machining parameter solutions, and the solution with the first ranking is provided to the machining as the result of the forecast.

Experiment procedure and results

Experimental design

It is known from previous studies that the deformation of line heating does not correlate well with the length and width of the plates.⁹ Therefore, the length and width of the steel plate used for the study in this paper is fixed, the plate length is 800 mm, the plate width is 500 mm. The material of the plate is mild steel and the specific material properties are listed in Table 1. Flame path for experiment is perpendicular to the midpoint of the plate width, flame gun diameter, and water gun diameter is 10 mm. The plate thickness H, acetylene flow rate Q_ch, heating line length L, heating speed V_q, and water fire distance D are selected as design variables. The specific description is shown in Figure 1.

Table 1.

Material properties of the board.

Temperature(°C)	Thermal conductivity(W/m °C)	Specific heat(J/kg °C)	Coefficient of thermalexpansion (10e−6/°C)	Bergsonby
200	49	520	11	0.33
300	46	530	12	0.35
400	42	600	13	0.37
600	35	750	14	0.41
700	31	850	14	0.43
800	26	950	14	0.44

Figure 1.

Experimental figure of line heating forming: (a) processing method and (b) measurement method.

According to the processing experience, the range of values of design variable parameters is selected as shown in Table 2, and the upper and lower limits of design variables are expressed by Xi^(L) and Xi^(U), respectively. Because the deformation of the steel plate is the only criterion for evaluating the forming situation during the processing of the line heating, the minimum difference between the final corner deformation and transverse deformation of the steel plate and the target deformation is the optimization goal.

Table 2.

Range of values of design variable parameters.

Design variables	H (mm)	Q_ch (L/h)	L (mm)	V_q (mm/s)	D (mm)
Xi^(L)	8	1200	250	2.2	90
Xi^(U)	20	1800	420	3.3	130

Measurement method

In order to measure the linear heating deformation results of the plate, the experiment chose coordinate measuring machine to measure the deformation of the plate. Because the experimental heating line position is chosen in the middle line of the plate, the deformation of both sides of the plate is relatively symmetrical. When measuring, we can choose one side of the plate for the measurement of transverse shrinkage deformation and angular deformation. The final processing transverse shrinkage of the whole is two times of the unilateral shrinkage, and the angular deformation is the deformation of the unilateral side. The specific description is shown in Figure 1(b), the dashed part of the figure is the plate before heating, the bolded part is the deformed plate after heating, with the unilateral boundary point as the calibration point, the value of L is the unilateral transverse shrinkage, H is the height difference between the calibration point before and after heating, and θ is the angular deformation amount.

Sample construction and experimental results

The construction of the proxy model relies on the selection of sample points, and through scientific experimental design, the sample points selected in the experimental scheme can be more evenly distributed and representative. Common experimental design methods include orthogonal array, central combination design, Latin hypercube design, and optimal Latin hypercube design. Among them, the Latin hypercube design has the ability to fit higher-order nonlinear relationships, and has more effective space-filling ability and higher efficiency than the full factorial design¹⁰; while the optimal Latin hypercube design improves the inhomogeneity of the random Latin hypercube design, and can effectively avoid the possibility of losing some design space regions.¹¹ Therefore, in this paper, the optimal Latin hypercube method is selected for the experimental design, and the five design variables are selected, and the experimental results are shown in Table 3.

Table 3.

Experimental results of optimal Latin hypercube design.

Experimentsnumber	H (mm)	Q_ch (L/h)	L (mm)	V_q (mm/s)	D (mm)	Transverseshrinkageamount (mm)	Angulardeformationamount (°)
1	13	1221	414	2.8	104	0.5338	0.3859
2	16	1200	291	2.9	122	0.4152	0.2366
3	11	1324	256	2.5	102	0.6601	0.0812
4	19	1366	367	3.2	108	0.2882	0.3832
5	12	1800	385	2.5	100	0.5900	0.1883
6	11	1779	332	3.1	119	0.7192	0.1096
7	9	1407	361	2.4	93	0.3513	0.2828
8	15	1614	297	2.3	91	0.5016	0.1226
9	8	1283	326	3.1	107	0.8254	0.1042
10	18	1531	402	2.6	94	0.4072	0.4380
11	20	1717	350	2.4	113	0.4596	0.2405
12	19	1593	309	3.0	127	0.3517	0.2826
13	9	1634	303	2.2	112	0.8887	0.0258
14	14	1510	338	2.7	111	0.5269	0.2222
15	16	1448	285	2.2	123	0.4712	0.1516
16	14	1469	262	3.3	109	0.4692	0.1966
17	20	1428	268	2.7	105	0.3946	0.1405
18	15	1303	314	3.0	90	0.4254	0.2733
19	12	1572	391	3.2	97	0.5107	0.4349
20	13	1386	379	3.3	126	0.5305	0.4058
21	18	1345	397	2.6	124	0.3846	0.4447
22	10	1490	273	2.8	129	0.7135	0.0802
23	18	1738	320	3.0	98	0.3932	0.2671
24	8	1552	408	2.8	116	0.7611	0.2520
25	17	1697	420	2.9	118	0.4399	0.4185
26	13	1676	373	2.4	130	0.6310	0.1040
27	15	1759	250	2.6	115	0.5210	0.0817
28	10	1262	356	2.3	120	0.6647	0.1676
29	10	1655	279	2.9	96	0.6072	0.1067
30	17	1241	344	2.2	101	0.4115	0.2933

Proxy modeling of steel plate line heating forming accuracy

RBF-based deformation modeling of steel plates

The relationship between the processing deformation and design variables of the line heating needs to be determined through processing tests. The process of process parameter optimization involves multiple tests, which leads to serious time consumption and huge costs, so the approximate model technique is used to construct a mathematical model that meets the accuracy requirements and consumes little calculation to establish the mapping relationship between input and output. The subsequent optimization work on this model will greatly reduce the optimization cost and speed up the optimization process.

The neural network approximation model has a strong ability to approximate complex nonlinear functions,^12,13 and has a fast learning speed, excellent generalization ability and strong fault tolerance, so that the neural network model can obtain a good fit even in the case of a limited number of samples. Therefore, neural network is used to establish the complex nonlinear relationship between process parameters and deformation of line heating.

The RBF neural network consists of an input layer, an implicit radial basis layer and an output linear layer, and its network structure is shown in Figure 2. The radial basis functions are radially symmetric and commonly used as Gaussian functions:

R_{i} (x) = \exp (- \frac{| | x - c_{i} | |^{2}}{2 {σ_{i}}^{2}}), i = 1, 2, \dots, p

(1)

Figure 2.

Structure of RBF neural network.

Where: $x$ is the m-dimensional input vector; $c_{i}$ is the center of the ith basis function; $σ_{i}$ is the variance of the ith basis function; and $p$ is the number of perceptual units.

The input layer of the RBF network implements a nonlinear mapping from $x \to R_{i} (x)$ , and the output layer implements a linear mapping from $R_{i} (x) \to y_{R}$ . Specifically, as shown in formula (2):

y_{R} = \sum_{i = 1}^{p} ω_{ki} R_{i} (x), k = 1, 2, \dots, q

(2)

Where: $q$ is the number of output nodes and $ω_{ki}$ is the moderation weight between the kth output layer and the ith implicit layer nerve.

RBF model prediction results and analysis

The plate thickness H, acetylene flow rate Q_ch, heating line length L, heating speed V_q, and water fire distance D are selected as the input nodes of the neural network, and the transverse shrinkage and angular deformation are taken as the output layer nodes. The first 20 sets of data are used as training samples, and 21–30 sets of data are used as test data. The experimental and predicted values were compared and the results are shown in the Table 4 and the comparison line graph is shown in the Figure 3. Among them, the black “■” is the experimental value of transverse shrinkage, the red “•” represents the predicted value of transverse shrinkage by RBF mode, the blue “▴” stands for the experimental value of angular shrinkage, the green “▾” means the predicted value of angular shrinkage by RBF mode. It’s obvious, the error between the predicted and experimental values is very small when the RBF model is used.

Table 4.

Comparison of experimental values and RBF predicted values.

Serial number	Transverse shrinkage amount		Angular deformation amount
	Experimental value	Predicted value	Experimental value	Predicted value
21	0.3846	0.4316	0.4447	0.3327
22	0.7135	0.6834	0.0802	0.1273
23	0.3932	0.4445	0.2671	0.2529
24	0.7611	0.6814	0.2520	0.2378
25	0.4399	0.4945	0.4185	0.3334
26	0.6310	0.6335	0.1040	0.1771
27	0.5210	0.6112	0.0817	0.0961
28	0.6647	0.6215	0.1676	0.1912
29	0.6072	0.6444	0.1067	0.1191
30	0.4115	0.4305	0.2933	0.2519

Figure 3.

Comparison of predicted and experimental deformation values.

In order to further evaluate the prediction performance of the model. We choose two indexes: coefficient of determination ( $R^{2}$ ) and mean square error (MSE). When the $R^{2}$ value is close to 1, the higher the accuracy of the approximation model, and when the MSE value is close to 0, the smaller the deviation between the predicted and actual values. These formulas are as follows:

R^{2} = \frac{{(n \sum_{i = 1}^{n} \overset{\land}{y_{i}} \cdot y_{i} - \sum_{i = 1}^{n} \overset{\land}{y_{i}} \sum_{i = 1}^{n} y_{i})}^{2}}{(n \sum_{i = 1}^{n} {\overset{\land}{y_{i}}}^{2} - {(\sum_{i = 1}^{n} \overset{\land}{y_{i}})}^{2}) (n \sum_{i = 1}^{n} {y_{i}}^{2} - {(\sum_{i = 1}^{n} y_{i})}^{2})}

(3)

MSE = \frac{1}{n} \sum_{i = 1}^{n} (y_{i} - {\hat{y}}_{i})^{2}

(4)

Where, ${\hat{y}}_{i}$ is the predicted value of the ith sample, $y_{i}$ is the experimental value of the ith sample, $n$ is the number of samples.

After calculation, the $R^{2}$ value corresponding to the test sample is 0.99403 and the MSE value is 0.0036. The established RBF model of deformation is more accurate and meets the accuracy requirements of the agent model.

Optimization of process parameters based on improved multi-objective particle swarm algorithm

Process parameter optimization model

To optimize the process parameters of line heating, the main inquiry is to minimize the difference between the processing deformation and the target deformation, according to this and based on the constraints, the optimization model of line heating process parameters can be constructed:

{\begin{matrix} min (f_{1}, f_{2}) (H, Q_{ch}, L, V_{q}, D) \\ f_{1} = | T_{f} - T_{t} |, f_{2} = | A_{f} - A_{t} | \\ 8 mm \leq H \leq 20 mm \\ 1200 L / h \leq Q_{ch} \leq 1800 L / h \\ 250 mm \leq L \leq 420 mm \\ 2.2 mm / s \leq V_{q} \leq 3.3 mm / s \\ 90 mm \leq D \leq 130 mm \end{matrix}

(5)

Where: $H, Q_{ch}, L, V_{q}, D$ are plate thickness, acetylene flow, heating line length, heating speed, and water fire distance; $T_{f}$ and $A_{f}$ are the transverse deformation and angular deformation of steel plate water fire bending plate processing; $T_{t}$ and $A_{t}$ are the target transverse deformation and angular deformation of steel plate, respectively.

Improvement of multi-objective particle swarm algorithm

Theory of multi-objective particle swarm algorithm

The particle swarm algorithm (PSO) is an evolutionary algorithm that simulates the foraging behavior of birds in nature.¹⁴ The algorithm generates a set of initial solutions, and then the particles in the population will follow certain rules to change the speed and position in each iteration to search for the optimal solution. The updated velocity and position are calculated as follows.

\begin{matrix} V_{i}^{t + 1} = ω V_{i}^{t} + C_{1} random (0, 1) (P_{i}^{t} - X_{i}^{t}) \\ + C_{2} random (0, 1) (G^{t} - X_{i}^{t}) \end{matrix}

(6)

X_{i}^{t + 1} = X_{i}^{t} + V_{i}^{t + 1}

(7)

Where: $X_{i}^{t}$ , $V_{i}^{t}$ , $X_{i}^{t + 1}$ , $V_{i}^{t + 1}$ represent the positions and velocities of the particles in the tth and t + 1th generations, respectively; ω is the inertia factor; C₁ and C₂ are the individual learning factor and the environmental learning factor, respectively, whose value interval is generally [0, 4]; $random (0, 1)$ is the random number on [0, 1]; $P_{i}^{t}$ and $G^{t}$ are the individual optimal position and the global optimal position, respectively.

Multi-objective Particle Swarm Optimization (MOPSO) combines the most common Pareto theory in the field of multi-objective solutions, and establishes an external population to store the non-dominated solutions found in each iteration, and selects the global optimal particle gbest in the external population according to certain rules.¹⁵ During the iterative process, the external population is continuously updated, and finally the particles in the external population are output as the optimal set of solutions.

New population search formula

How to balance the full search as well as the local search capability has been an important problem to be solved in the algorithm solution process, because the inertia weight is better for local refinement search when it is small and better for global search when it is large,¹⁶ so to solve this problem, this paper proposes a new formula for the change of the inertia weight, which will change nonlinearly in time as the number of iterations t increases. Specifically, as shown in formula (8):

ω (t) = ω_{max} - (ω_{max} - ω_{min}) \times \tan (\frac{t}{t_{max}} \times \frac{π}{4})

(8)

Where: $ω_{\max}$ and $ω_{\min}$ are the maximum and minimum values of the weight coefficients, respectively; $t_{\max}$ is the maximum number of iterations.

New linearly varying learning factors as shown in formula (9):

\begin{matrix} c_{1} = (c_{1 max} - c_{1 min}) (t - t_{max}) / t_{max} + c_{1 max} \\ c_{2} = (c_{2 max} - c_{2 min}) (t_{max} - t) / t_{max} + c_{2 max} \end{matrix}

(9)

New population initialization strategy

Since the multi-objective particle swarm optimization algorithm is more sensitive to the initial value of particles, a more uniformly distributed initial population will lead to a better distribution of the obtained Pareto frontier, and the greater the probability of obtaining a solution set with diversity and good convergence. Most MOPSO algorithms are generated by randomly generating the initial population, which cannot guarantee that the initial population covers the decision space of the problem and easily falls into local optimum, thus failing to maintain the population diversity. Since the inverse trigonometric logistic mapping is more uniformly distributed on the interval $[0, π / 2]$ , this paper uses the inverse trigonometric logistic mapping to assign values to the initial population. The formula of inverse trigonometric logistic mapping is as follows:

y_{n + 1} = \arcsin \sqrt{4 x_{n} (1 - x_{n})}

(10)

Using equation (7) to generate a random number in the interval, let the particle position take the value range of $[x_{\min}, x_{\max}]$ , then the initial position of the particle as shown in formula (11):

x_{i} (0) = x_{min} + 2 r / π (x_{max} - x_{min})

(11)

This initialization method allows the initial population to be uniformly distributed in the decision space and provides a good start for the search of the algorithm.

Verification of algorithm performance

To verify the performance of the improved MOPSO algorithm, the standard test functions ZDT1 and ZDT2 were selected for testing, and GD (generation distance), SP (spatial metric), and IGD (inverse generation distance) were used as indicators for verification, and compared with the standard MOPSO algorithm under the same conditions, and the smaller the three indicators, the better the performance of the algorithm. The parameters of the improved MOPSO algorithm were set as follows: population size 100; inertia weight maximum value 0.9, minimum value 0.5; iteration number 350; individual learning factor and environmental learning factor maximum value 2, minimum value 1. To keep comparability, other algorithm parameters were set the same, and the test data of the standard MOPSO algorithm and the improved MOPSO algorithm are shown in Table 5.

Table 5.

Algorithm test data comparison table.

Test Functions	Optimization algorithm	Evaluation Indicators
		GD	SP	IGD
ZTD1	MOPSO	9.0 × 10⁻⁴	0.3 × 10⁻²	8.6 × 10⁻³
	Improved MOPSO	8.2 × 10⁻⁴	0.35 × 10⁻²	7.9 × 10⁻³
ZTD2	MOPSO	8.0 × 10⁻⁴	6.2 × 10⁻³	9.2 × 10⁻³
	Improved MOPSO	6.9 × 10⁻⁴	5.8 × 10⁻³	9.2 × 10⁻³

As can be seen from Table 4: the improved MOPSO algorithm outperforms the MOPSO algorithm. Although the standard MOPSO algorithm outperforms the improved MOPSO algorithm in terms of the SP index of the test function ZDT1, all other indexes are worse than the improved MOPSO algorithm, so the overall performance of the improved MOPSO algorithm is better than MOPSO. thus verifying that the improved MOPSO algorithm has better convergence and diversity.

Analysis of optimization results

To prove that the improved multi-objective particle swarm algorithm is suitable for solving the parameter optimization problem of line heating, the target transverse shrinkage is 0.4 mm and the target angular deformation is 0.3° at the plate thickness H = 12 mm, and the process parameters are solved optimally by the improved MOPSO algorithm. The parameters of the multi-objective particle swarm algorithm are set as follows: inertia factor $ω_{\max} = 0.9$ , $ω_{\min} = 0.5$ ; individual learning factor and environment learning factor $c_{1 \max} = c_{2 \max} = 2$ , $c_{1 \min} = c_{2 \min} = 1$ ; the number of particles is 40, the maximum number of iterations is set to 50, and the maximum flight speed is 100. The iterative optimization results are shown in Figure 4.

Figure 4.

Iterative optimization search results.

From the optimization search results, it can be seen that the algorithm used can converge well to the vicinity of the target value when the water fire bending plate processing agent model is optimally solved, and the optimization effect is very good. To further verify the accuracy of the optimization results, the solution set of process parameters (as shown in Table 6) after iteration of the algorithm was experimentally verified, and the verification results are shown in Figures 5 and 6. Among them, the black “■” is the optimization value by the improved MOPSO, the red “•” represents the processing value, the blue “▴”stands for the target deformation amount.

Table 6.

Solution set of process parameters after improved MOPSO search.

Processparameterssolving Pi	Process parameter solution set					Optimization target value
	D (mm)	H (mm)	L (mm)	Q _ch (L/h)	V _q (mm/s)	Transverse shrinkageamount (mm)	Angulardeformationamount (°)
P1	94	12	329	1200	2.87	0.4996	0.2402
P2	101	12	383	1401	2.63	0.5030	0.3013
P3	94	12	355	1234	3.03	0.5011	0.3169
P4	101	12	376	1441	2.93	0.5379	0.2996
P5	95	12	345	1317	2.94	0.5006	0.2761
P6	94	12	347	1271	2.95	0.4981	0.2924
P7	103	12	356	1272	2.71	0.5484	0.2997
P8	95	12	369	1348	3.07	0.5005	0.3541
P9	102	12	392	1427	2.49	0.5014	0.3100
P10	101	12	368	1241	2.3	0.4631	0.2994

Figure 5.

Comparison of the results of transverse shrinkage.

Figure 6.

Comparison of angular deformation results.

Analyzing the above results, the average error between the optimized value of transverse shrinkage and the actual processing value is 2.7%, and the average error between the optimized value of angular deformation and the actual processing value is 6.4%, which proves that the optimization results of this paper are reliable, and the comprehensive error between the processing value and the target value is 4.1%, so it can be concluded that the optimization effect is good.

TOPSIS-based decision making for processing parameters

In the actual processing production process, parameter forecasting often requires only one set of process parameters, so in order to obtain intuitive and clear process parameter solutions, the established transverse shrinkage and angular deformation are used as targets, and $(\min f_{1}, \min f_{2})$ is used as the process parameter solution performance evaluation index, and TOPSIS is used to perform a multi-attribute decision process on the set of process parameter solutions obtained by multi-objective particle swarm search. TOPSIS is an efficient and advanced MCDM (multi-criteria decision-making) methodology that was proposed to obtain the finest choice based on the compromise solution principle,¹⁷ the main process is shown in Figure 7.

Figure 7.

Process parameter performance ranking process diagram.

For line heating forming processing, the transverse shrinkage and angular deformation are equally important, so the weights of the two evaluation indexes are set to 0.5 respectively, and the optimized process parameter solutions are decided according to the process in Figure 7, and the set of process parameter solutions after the decision is shown in Table 7.

Table 7.

Solution set of process parameters after decision.

Process parameters solution Pi	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10
Standardized scoring	0.49	0.961	0.81	0.576	0.74	0.91	0.51	0.516	0.885	0.583
Rating Sort	10	1	4	7	5	2	9	8	3	6

According to the standardized scoring of the process parameter solutions, it is easy to see that the P2 solution {10,112,383, 1401, 2.63} is the solution closest to the target value when the transverse shrinkage and angular deformation are equally important. The analysis of the optimization results in 3.3 shows that the errors of the actual processing deformation and the target deformation of the P2 solution are 2.6%–4.8% respectively, the details are shown in Table 8, which meet the actual processing accuracy requirements .

Table 8.

Optimal solution deformation and target value error analysis.

	Targetdeformation	Optimal processingdeformation	Degreeof error
Transverseshrinkage	0.5 mm	0.5030 mm	2.6%
Angulardeformation	0.3º	0.3013º	4.8%

Conclusion

The RBF proxy model between process parameters and deformation of line heating was established, and the accuracy of the relational model was checked. The test results showed that the model was fitted well, and the feasibility of using the RBF method to establish the proxy model of process parameters and deformation was verified.

The population initialization method and population search strategy of the multi-objective particle swarm algorithm are improved, and the three evaluation indexes show that the improved MOPSO algorithm has better convergence and diversity.

Through the example verification, the improved MOPSO algorithm is applicable to the parameter optimization problem of line heating.

Based on TOPSIS, multi-attribute decision making is performed on the optimized set of process parameter solutions, and the process parameter solutions are obtained as the forecast parameters after performance evaluation, and the results meet the actual processing accuracy requirements.

The current study is the local process parameter forecasting of line heating, considering that the processing of line heating involves the overall deformation problem, our next research work is the parameter forecasting of the overall deformation.

Footnotes

Handling Editor: Chenhui Liang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Special Funding Project for Key Technologies of Ship Intelligence Manufacturing from the MIIT of China under Grant MC-201704-Z02, and thanks to the support of the RO-RO Passenger Ship Efficient Construction Process and Key Technology Research (Project No. CJ07N20) the Intelligent Methanol-Fueled New Energy Ship R&D Project (Guangdong Natural Resources Cooperation [2021] No. 44).

ORCID iDs

Zhao Wang

Binjiang Xu

Data availability

All the data used to support the findings of this study are included within the article.

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