We consider the NPhard problem of minimizing makespan for
This research considers the problem of scheduling
Chen and Vestjens [
To the best of our knowledge, no research has yet been published that develops an efficient algorithm to minimize makespan for unrelated parallel machines with release dates. The rest of this paper is organized as follows. Section
We propose two straightforward and easily implementable lower bounds for the studied problem.
To illustrate the proposed lower bounds, we consider example 1, which has 2 machines and 7 jobs. The matrix of processing times for example 1 is given in Table
The matrix of processing times for example 1.










15  29  40  32  46  8  44 

41  29  40  32  31  24  49 
 

27  4  3  9  13  17  1 
PSO was first introduced by Kennedy and Eberhart [
A population of
PSO has been successfully applied to a variety of continuous nonlinear optimization problems. In recent years, considerable effort has been expended on solving scheduling problems by PSO algorithms. Articles [
A coding scheme developed in [
Representation of a particle’s solution.
The completed schedule is thus jobs 7, 6, 1, and 4 on machine 1; jobs 3, 2, and 5 on machine 2. This coding scheme specifies not only which jobs are assigned to which machine but also the order of the jobs on each machine. These pieces of information are important, since we are scheduling unrelated parallel machines with release dates.
In order to give the PSO algorithm good initial solutions and to increase the chances of getting closer to regions that yield good objective functions, we propose a heuristic, named SRD_Reassign. The proposed heuristic SRD_Reassign is described as follows.
Let
Arrange the jobs in the order of the shortest release date (SRD) first, and then assign the job to machine
Let
Identify machine
Select machine
Terminate the procedure.
The first two particles are generated by firstcome, firstserve (FCFS) rule and SRD_Reassign. The remaining particles are generated by applying local search to the solution found by SRD_Reassign. For FCFS rule, we consider all unscheduled jobs and schedule each one on the first available machine according to FCFS. Local search is done by randomly choosing two jobs
Kashan and Karimi [
The subtraction operator (
The subtraction operator (
The multiplication operator (
The multiplication operator (
The addition operator (
The addition operator (
During the execution of the
It is well known that evolutionary memetic algorithms can be improved by hybridization with local search. For each particle
Initially, set
Identify machine
Randomly choose two jobs
If
Again, during the execution of LSP, whenever a complete schedule generates a better solution than the best current solution
We studied the effects of five important parameters (
In this section, we present several computational results of the proposed PSO algorithm. We compare our proposed PSO algorithm with a mixed integer programming (MIP) model developed in our previous research [
We compared the proposed heuristic SRD_Reassign with the optimal solutions obtained from the MIP model [
The performance of heuristic SRD_Reassign for small problem instances.
4 
MIP  SRD_Reassign  FCFS  


Mean  Avg. time  Mean  Avg. time  Mean  Avg. time 
0.10  1  1176.32  1.11  0.003  1.46  0.000 
0.25  1  239.19  1.09  0.002  1.43  0.000 
0.50  1  58.02  1.04  0.002  1.34  0.000 
 
Average  1  491.18  1.08  0.002  1.41  0.000 
Mean = average ratios of heuristic/MIP obtained from 20 instances.
The performance of heuristic SRD_Reassign for large problem instances.
10 
SRD_Reassign  FCFS  


Mean  Avg. time  Mean  Avg. time 
0.10  1.43  0.005  2.03  0.001 
0.25  1.17  0.003  1.62  0.000 
0.50  1.05  0.005  1.36  0.000 
 
Average  1.22  0.004  1.67  0.000 
Mean = average ratios of heuristic/LB obtained from 20 instances.
We compared the proposed PSO with an existing metaheuristic, namely, the version of simulated annealing (SA) described by Lee et al. [
The performance of PSO for small problem instances.
4 
MIP  PSO  SA  PSO/SA  


Mean  Avg. time  Mean  Avg. time  Mean  Avg. time  
0.10  1  1176.32  1.00  11.73  1.05  12.79  11/0 
0.25  1  239.19  1.00  10.69  1.06  12.17  18/0 
0.50  1  58.02  1.00  8.69  1.03  10.28  10/0 
 
Average  1  491.18  1.00  10.37  1.05  11.75  13/0 
Mean = average ratios of algorithm/MIP obtained in 20 instances.
The performance of PSO for large problem instances.
10 
PSO  SA  PSO/SA 




Mean  Avg. time  Mean  Avg. time  
0.10  1.20  82.61  1.32  84.1  20/0  0/20 
0.25  1.03  49.37  1.11  85.54  20/0  10/10 
0.50  1.00  25.34  1.01  55.18  10/0  20/0 
 
Average  1.08  52.44  1.15  74.94  16.7/0  10/10 
Mean = average ratios of algorithm/LB obtained in 20 instances.
Computational results show that the proposed PSO outperformed the SA in terms of makespan. For small problem instances, the PSO found optimal solutions at all three
For large problem instances, the average PSO makespan was 1.08times greater than the LB, and the average SA makespan was 1.15 times the LB. The last column in Table
Next, since the proposed PSO effectively incorporates a number of ideas (initial solutions, SRD, ECT, and LSP), we examine which parts are essential to its functionality. We examine these effects by disabling a single component, running the proposed PSO without that component, and observing performance. We choose to study large problems. These experiments are described as follows.
Table
The effects of the proposed PSO.
10 
PSOinitial heuristics  PSOSRD  PSOECT  PSOLSP 
PSOLSP+ LS [ 


Mean  Mean  Mean  Mean  Mean 
0.10  1.054  1.017  1.055  1.019  1.023 
0.25  1.001  1.001  1.004  1.002  1.003 
0.50  1.001  1.001  1.001  1.001  1.001 
 
Average  1.019  1.006  1.020  1.007  1.009 
 
Ave. time  52.55  51.14  51.29  48.44  1624.31 
Mean = average ratios of PSOvariant/standardPSO obtained in 20 instances.
Moreover, we compared our proposed PSO with another existing PSO. The closest existing PSO that we were able to find was the hybridized discrete PSO (HDPSO) proposed in [
Comparison between two PSOs.
Problem  Initial heuristic 

Local search  

HDPSO 

n/a  Coding scheme and LPT rule are not suitable for the 
Local search algorithm 
 
PSO 

SRD_Reassign  Coding scheme, SRD, and ECT are designed for the 
LSP 
We studied the problem of scheduling jobs on unrelated parallel machines with release dates to minimize makespan. In this research, we proposed two lower bounds for the studied problem. We also proposed a heuristic, SRD_Reassign, and a metaheuristic, PSO, to tackle the problem. Computational results showed that SRD_Reassign outperformed the commonly used heuristic, FCFS, in terms of makespan. The proposed PSO outperformed a comparable variant of SA in terms of makespan. Future work can extend our approach for other performance criteria or even for multiobjective parallel machine scheduling problems.
We studied the effects of five important parameters (
Factors and levels of PSO parameter study.
Factor  Design units  

−1  +1  

0.1  0.9 

0.1  0.9 

10  50 

10  100 

1000  5000 
Results of 2_{III}^{5–21} factorial design for parameters of PSO.
Run 







1  0.1  0.9  50  10  1000  118.65 
2  0.9  0.1  50  10  5000  115.45 
3  0.9  0.9  10  100  1000  115.00 
4  0.9  0.1  10  10  1000  116.30 
5  0.1  0.1  10  100  5000  116.35 
6  0.9  0.9  50  100  5000  114.45 
7  0.1  0.9  10  10  5000  117.10 
8  0.1  0.1  50  100  1000  118.10 
Next, we used the method of steepest descent (Myers et al. [
The path of steepest descent for PSO.
Run  Coded units  Natural units 

Improv.  Time  





Pop. size  Iteration  
0  Base  0  0  0  0.5  55  3000  117.30  —  50.68 
Increment = 
1  0.40  0.52  0.4  18  1044.4  —  —  —  











2  Base + 2 
2  0.80  1.04  0.90  91  5089  114.40  0.024  129.06 
3  Base + 3 
3  1.20  1.57  0.90  109  6133  114.95  0.002  193.28 