Rapid sequence optimization of spot welds for improved geometrical quality using a novel stepwise algorithm

Figures & data

Figure 1. Schematic view of a self-compensating assembly line.

Figure 2. Three sheet metal reference assemblies. (a) Assembly II, (b) Assembly III.

Table 1. Optimal sequence, assembly deviation (in millimetres) and evaluation time of the reference assemblies from the exhaustive search.

Assembly	Optimal sequence	Assembly deviation (D)	Single Eval. Time (s)	Total Eval. Time (h)
I	{1-4-2-5-3-6-7}	0.32	7.2	10.08
II	{3-5-7-2-4-6-1}	0.08	3.73	5.22
III	{3-4-6-7-2-5-1-(8, 9)}^a	0.655	12.35	17.29

${}^a$ The sequence of welds are represented by the weld numbers, where {W1-W4-W2-W5-W3-W6-W7} is represented by {1-4-2-5-3-6-7}. In Assembly III, welds W8 and W9 are welded simultaneously at the end of the sequence, represented by (8,9).

Figure 3. Mean number of function evaluations (NFEs) for different population sizes in 100 trials.

Figure 4. Mean number of function evaluations (NFEs) (left column) and assembly deviation range (right column) for different population sizes in 100 trials between the compared algorithms for Assembly I.

Figure 5. First step of the stepwise algorithm with one and two elements compared to the exhaustive search.

Figure 6. Optimization steps using the stepwise algorithm.

Table 2. Optimal sequence and the assembly deviation (in millimetres) of the reference assemblies retrieved from the stepwise algorithm with s = 1 and s = 2.

	Optimal sequence		Assembly deviation (D)
Assembly	s = 1	s = 2	s = 1	s = 2
I	{1,4,2,5,6,7,3}	{1,4,2,6,7,5,3}	0.32	0.32
II	{3,5,2,7,4,6,1}	{3,5,7,2,4,6,1}	0.08	0.08
III	{6,2,1,5,4,3,7,(89)}	{3,4,5,2,6,7,1,(89)}	0.667	0.655

Table 3. Evaluation time comparison.

	Assembly I		Assembly II		Assembly III
Algorithm	NFEs	Time (s)	NFEs	Time (s)	NFEs	Time (s)
GA	70.1	504.7	62.2	232	107.8	1331.3
PSO	64	460.8	60	223.8	111	1370.8
Hybrid GAPSO	307	2363.9	216	805.7	370	4569.5
Rule-based	56	403.2	42	156.7	21	259.4
Stepwise (s = 1)	27	194.4	27	100.7	27	333.45
Stepwise (s = 2)	56	403.2	56	208.9	56	691.6

Table 4. Accuracy comparison.

	Assembly I		Assembly II		Assembly III
Algorithm	Mean D	Range D	Mean D	Range D	Mean D	Range D
GA	0.33	0.03	0.08	0.01	0.657	0.012
PSO	0.33	0.10	0.08	0.02	0.657	0.012
Hybrid GAPSO	0.33	0.03	0.08	0.02	0.656	0.007
Rule-based	0.33	0.01	0.08	0.01	0.656	0.005
Stepwise (s = 1)	0.32	0	0.08	0	0.667	0
Stepwise (s = 2)	0.32	0	0.08	0	0.655	0