A Multi-Objective Scheduling Optimization Method for PCB Assembly Lines Based on the Improved Spider Monkey Algorithm ^†

Abstract

For assembly lines of circuit printing board (PCBs), this study investigated an integrated scheduling problem of the component allocation problem (CAP) and component placement sequence problem (CPSP) with the fixed interval preventive maintenance to minimize the maximum completion time (${\mathrm{C}}_{\max }$), the total energy consumption (TEC), and the total maintenance time (TMT) simultaneously. An improved spider monkey optimization (ISMO) algorithm is proposed with selecting the local leader (LL) and the global leader (GL) using a parallel lattice coordinate system. We compared the proposed ISMO algorithm with the classical optimizations, including SMO, NSGA-Ⅲ, DE, and PSO, by the production data of an enterprise; the results showed that the ISMO algorithm can obtain pareto solutions with better convergence and diversity.

1. Introduction

With the transformation and upgrading of the manufacturing process to be digital and intelligent, the PCB manufacturing industry has developed rapidly. The scheduling of PCB assembly lines directly affects the benefits of the PCB manufacturing enterprise.

Researchers have studied the scheduling problem of PCB assembly lines. The early research was mainly based on single CAP and CPSP problems. However, the two processes of component allocation and sorting are closely related, and the solution results of CAP have a considerable influence on the solution results of CPSP; thus, the current research is mainly based on the integration of CAP and CPSP. Typical studies on CAP are as follows: Ji et al. studied the CAP problem with minimizing cycle time, and designed a genetic algorithm to solve it [1]; Ho et al. studied the CAP problem with minimizing the total distance and proposed a hybrid genetic algorithm [2]. Typical studies on CPSP problems are as follows: Lin et al. studied the CPSP with maximizing production capacity, minimizing total assembly time, and head movement distance [3]; Grunow et al. proposed a three-stage heuristic method to solve CAP and CPSP problems [4].

Typical studies on integrating CAP and CPSP are as follows: Luo et al. constructed a heuristic algorithm for solving the CPSP problem that combines a genetic algorithm and a tabu search [5]; Mumtaz et al. proposed a hybrid SMO algorithm for the multi-level planning and scheduling of PCB assembly line [6]; Gao et al. proposed a hierarchical multi-objective heuristic algorithm for optimizing PCB assembly [7].

After analyzing the literature, we can find that the current research on the scheduling problem of the PCB assembly line focuses on a single objective; the maintenance of the machine and the energy consumption of the assembly line are rarely considered. Therefore, this study designed an ISMO algorithm for the integrated scheduling problem of CAP and CPSP, aiming to minimize ${\mathrm{C}}_{\max }$, TEC, and TMT at the same time.

2. Problem Formulation

The problem studies in this paper can be described as follows: if $CN$ is the number of PCB orders, then m number of machines for t types of n component patches are required. Each machine will require maintenance at regular intervals to ensure the normal state of machines, i.e., the continuous placement time of the machine cannot exceed the maintenance interval time UT. The focus is to determine the optimum distribution and placement sequence of n components on the m machine to minimize ${\mathrm{C}}_{\max }$, $\mathrm{TEC}$, and TMT simultaneously. The energy consumption of machines is mainly composed of three parts: processing energy consumption, maintenance energy consumption, and idle energy consumption.

Assumptions of this paper include the following: (1) all machines on a PCB assembly line are identical; (2) a machine can only place one component at a time on the PCB; (3) the same type of components can be assigned to multiple machines on the PCB assembly line; (4) there is no priority constraint on component placement; (5) each machine has a feeder, and multiple feed slots can be placed in each feeder, and only one type of component can be placed in each feed slot; (6) the machine cannot stop for maintenance during the placement process.

3. ISMO Algorithm

3.1. Basic Flow of the ISMO Algorithm

SMO is a proposed global optimization algorithm: the main feature is that it can increase the ability to search optimal solutions. The ISMO algorithm is proposed to solve multi-objective problems. The process of the proposed ISMO algorithm is shown in Figure 1.

3.2. Solution Based on Parallel Coordinates

The proposed problem is multi-objective optimization problem; therefore, the Pareto solution set is normalized by parallel coordinates to determine the advantages and disadvantages of each solution. The r^th target value corresponding to the ($n^s$)^th solution in the Pareto solution $f_{n^s,r}$ set to a $\left(N^S\times R\right)$ two-dimensional planar mesh ($N^S$ is the number of Pareto solutions, R represents the number of target values), using Equation (1) to obtain the lattice coordinate component $L_{n^s,r}$:

$$L_{n^s,r}=N^S\times \frac{f_{n^s,r}-f_r^{min}}{f_r^{max}-f_r^{min}},$$

(1)

where $f_r^{max}=max{f}_{n^s,r}$ and $f_r^{min}=min{f}_{n^s,r}$ are the maximum and minimum values of the r^th target of the Pareto solution, respectively. If $f_{n^s,r}=f_r^{min}$, then $L_{n^s,r}$=1.

The density distance $D\left(S{M}_u\right)$ of $S{M}_u$ can be calculated using Equation (2).

$$D\left(S{M}_u\right)={{\displaystyle \sum }}_{j=1,j\ne i}^{N^S}\frac{PCD{\left(S{M}_u,S{M}_v\right)}^2}{N^S},$$

(2)

where $S{M}_v$ is the individual in the solution set except $S{M}_u$, and $PCD\left(S{M}_u,S{M}_v\right)$ can be calculated using Equation (3).

$$PCD\left(S{M}_u,S{M}_v\right)=\{\begin{array}{c}0.5, \forall r,L_{u,r}=L_{v,r}\\ {{\displaystyle \sum }}_{r=1}^R\left|L_{u,r}-L_{v,r}\right|,\exists r,L_{u,r}\ne L_{v,r}\end{array},$$

(3)

The individual with the largest density distance in the Pareto solution set is selected as the optimal solution.

3.3. Local Leader Phase (LLP)

In the LLP, the population is divided into groups, and then the individual $S{M}_u$ is updated in the group as follows:$S{M}_u$ is crossed with $LL$ first, and the obtained offspring individuals are crossed with the random individual in the group; then, the best individual is selected as the new individual.

The crossover process is illustrated below. The crossover of the components part adopts the two-point crossover method, and the crossover of the machine part adopts the single-point crossover method. The example of crossover is shown in Figure 2.

4. Numerical Example and Analysis

4.1. Parameter Settings

In order to evaluate the solution efficiency and quality of the ISMO, SMO, NSGA-Ⅲ, PSO and DE were selected as the comparison algorithms. Based on the production data of a certain enterprise [8], the experimental problem scale and parameter range are shown in

Table 1. Experimental problem scale and parameter range.

Parameters	Value			Parameters	Value	Parameters	Value
n	10	20	30	MT	300	$FP$	U [1, 3]
$m$	2	3	4	$v^m$	100	$PP$	U [1, 9]
ct	4	5	6	$d$	U [300, 350]	$MP$	U [1, 99]
CN	10,000	20,000	30,000	UT	14,400

, and the algorithm parameters are shown in

Table 2. Parameter values for algorithms.

Instance	m = 2, n = 10					m = 3, n = 20					m = 4, n = 30
Algorithm	ISMO	SMO	NSGA-Ⅲ	PSO	DE	ISMO	SMO	NSGA-Ⅲ	PSO	DE	ISMO	SMO	NSGA-Ⅲ	PSO	DE
popsize	150	100	150	150	150	150	150	150	150	150	150	150	150	150	150
maxgen	150	100	150	200	150	200	150	150	200	200	200	150	200	200	200
$p_c$	0.7	0.7	0.75	/	/	0.7	0.7	0.75	/	/	0.75	0.75	0.8	/	/
$p_m$	0.1	0.1	0.2	/	/	0.1	0.1	0.15	/	/	0.15	0.15	0.2	/	/

. n, m, ct, and CN are the number of components, machines, component types, and PCB orders, respectively; UT and MT are the maintenance interval time and the maintenance time; $v^m$ is the velocity of the machine head; $d$ is the distance covered as the machine head moves; and $FP$, $PP$, and $MP$ are the idle power, processing power, and maintenance power of the machine, respectively.

4.2. Computational Experiments and Discussion

In this study, three problem instances were tested, with each problem instance run 10 times, in order to compare different algorithms in terms of operational efficiency and operational quality. The results are shown in

Table 3. Performance indicator results for algorithms.

Algorithm	m = 2, n = 10				m = 3, n = 20				m = 4, n = 30
	CT	Nd	IGD	NR	CT	Nd	IGD	NR	CT	Nd	IGD	NR
ISMO	13.4198	3.2	1646.5110	0.7861	41.4082	5.5	623,020.0337	0.7231	53.9259	6.8	1,892,077.6829	0.6624
SMO	12.6523	1.5	44,995.7403	0.0111	38.2206	2.4	842,868.3469	0.0000	50.7701	2.5	2,043,876.5376	0.0676
NSGA-Ⅲ	22.8764	1.3	8391.4363	0.3139	27.2215	1.2	1,595,181.4112	0.2269	45.0490	1.1	5,047,418.3441	0.2699
PSO	12.1930	1.6	22,905.7269	0.0000	14.0099	2.5	1,062,318.0880	0.0125	16.2862	3.0	4,656,473.8169	0.0000
DE	5.1382	1.8	56,670.4167	0.0000	8.3881	2.9	1,075,758.9253	0.0375	10.2588	3.2	4,453,947.1130	0.0000

. CT is the running time of the algorithm; N_d is the average number of Pareto solutions obtained by the algorithm; IGD is inverse generation distance; and NR is non-dominant rate.

It can be seen from Table 3 that the CT of DE and PSO algorithms is shorter, and the CT of ISMO, SMO, and NSGA-III algorithm is longer. The DE and PSO are basic algorithms, and the principle and running steps are relatively simple; therefore, the CT is significantly shorter than the other three algorithms. For the number of Pareto solutions obtained by different algorithms, the average order from large to small is: ISMO, DE, PSO, SMO, NSGA-III. When using IGD, the ISMO algorithm is obviously the best, followed by SMO, and the PSO and DE are poor, whereas the NSGA-III has better results in n = 10 problem instances, but worse result in the n = 20 and n = 30 problem instances. When using NR, the ISMO algorithm is obviously the best, followed by NSGA-III, and the SMO, PSO, and DE are worse. In general, compared with the other four algorithms, the ISMO algorithm has obvious advantages, but its advantages decrease with the increase in scale.

5. Conclusions

For the integrated CAP and CPSP scheduling problem considering machine preventive maintenance, this paper has proposed an ISMO algorithm based on the parallel lattice coordinate, which minimizes ${\mathrm{C}}_{\max }$, TEC, and TMT at the same time. The experimental results show that the ISMO algorithm can effectively increase the diversity and convergence of the obtained solutions.