CHAPTER 4 The Proposed Algorithm
4.8 An Example
In this section, an example is given to illustrate the proposed scheduling algorithm.
This is a simple example to show how the proposed algorithm can be used to find the scheduling with the minimum total tardiness under the mold constraint. Assume a scheduling requirement to be solved is that eight jobs are assigned on three machines to process while using four molds. Table 4.7 shows the profiles of the eight jobs to be scheduled. Table 4.8 shows the profiles of the four molds and table 4.9 shows the
Table 4.7 The property of the eight jobs
Job 1 2 3 4 5 6 7 8
Processing Time 5 4 6 4 7 3 6 4
Mold 3,4 2,3,4 1,4 1,3 1,2 1,2,4 2,3 3
Table 4.8 The amount of each mold
Mold 1 2 3 4
Number of molds 1 2 2 1
Table 4.9 The adoptable molds for each machine
Machine A B C
Adoptable molds 1, 3 1, 3, 4 2,3,4
STEP 1: P individuals are randomly generated as the initial population according to the
mentioned encoding scheme. Each individual thus represents a possible schedule. For example, assume p is set at six for simplicity. Assume the six initially generated individuals are shown in Table 4.10.
STEP 2: Adjust the individuals by the designed adjustment operators for improving the
structures of chromosomes. The result is shown in Table 4.11.
Table 4.10 The six initial chromosomes in the example
ID Chromosomes
C1 4A1-7B3-3C4-1A3-8B3-6C2-5B1-2A3 C2 6A1-1B4-2C3-5A2-4C3-7B3-8C3-3C4 C3 8A3-3B4-5C2-1A3-4B1-2C3-7A3-6B1 C4 2A3-6B4-4C2-7B3-8A3-3C4-1A3-5B1 C5 1A3-3B1-7C2-4A3-2B4-5C2-8A3-6A1 C6 3A1-1B4-7C2-4B3-5A2-6C4-2B4-8C3
Table 4.11 The improved chromosomes by the designed adjustment operators
ID Adjusted Chromosomes
C1 4A1-1B3-5C2-7A3-8B3-6C2-3B4-2A3 C2 6A1-1B4-2C3-5A2-4C3-7B3-8C3-3C4 C3 8A3-3B1-5C2-1A3-4B1-2C3-7A3-6B1 C4 7A3-6B1-5C2-4B1-8A3-2B3-3C4-1A3 C5 1A3-3B1-7C2-4A3-5B1-2C4-8A3-6B1 C6 3A1-1B3-4C2-6C2-2B3-5A1-8C3-7B3
STEP 3: The fitness value of each chromosome is then calculated for the makespan
schedule. Take chromosome C1 as an example. The chromosome C1 is shown in Figure 4.14. The makespan of chromosome C1 is calculated as 21. The fitness values of all the chromosomes in the population are calculated with their results shown in Table 4.12.
Figure 4.14 The schedule presented by chromosome C1
Table 4.12 The fitness values of all the chromosomes in the population
ID L
C1 21
C2 24
C3 18
C4 19
C5 19
C6 18
STEP 4: Execute the adopted crossover operation on the population. Assume that the
two chromosomes C1 and C2 are randomly chosen by the crossover operator to
generate offsprings. Assume that the crossover point a is randomly set at 1,
STEP 5: The reverse mutation is executed to generate possible offspring. Assume that
the chromosome C5 is selected to execute the mutation operator. The following offspring chromosome is generated as follows.
The selected chromosome:
C5: 1A3-3B1-7C2-4A3-5B1-2C4-8A3-6B1 The chromosome generated from mutation:
C5t’:6B1, 8A3, 2C4, 5B1, 4A3, 7C2, 3B1, 1A3
STEP 6: The offspring chromosomes are adjusted by the designed adjustment operator.
Assume that there are six newly generated offspring chromosomes reproduced from STEP 4 and STEP 5. The newly generated offspring chromosomes are shown in Table 4.13 and their fitness values are then evaluated.
Table 4.13 The generated offspring chromosomes
ID Newly generated chromosomes L
C1t’ 4A3-2B4-5C2-1A3-6B4-3B4-7C3-8A3 16
STEP 7 to 9: The best chromosomes are selected as the next generation. The same
procedure is then executed again until the termination criterion is satisfied.
The best chromosome in the last generation is then outputted as the best scheduling result.
CHAPTER 5
Experimental Results
This chapter describes the experiments that were made to show the performance of the proposed GA-based algorithm for scheduling on unrelated parallel machines with the mold constraints. The following Table 5.1 shows the environment of the given experiments. Three parameters are considered to define the problem, number of jobs, number of machines, and number of molds. The experiments show the comparison of the proposed GA-based scheduling algorithms with and without the adjustment operators.
Table 5.1 The environment of the given experiments CPU Intel(R) Core(TM) i5-3450 @3.10GHz 3.50GHz
RAM 8.0 GB
OS Microsoft windows 7 (64 bits) Program tools Microsoft visual C#
Some related parameters for the proposed GA-based scheduling algorithm are shown in Table 5.2 as follows.
Table 5.2 The parameters for the proposed GA-based scheduling algorithm
Generation 100
Population 50
Mutation probability pm 0.03
Setup time 5
Since it is difficult to find a proper real-world dataset to exam the proposed scheduling algorithm in the following experiments, the synthetic datasets are thus considered as being adopted in the experiments. For this reason, a schedule dataset generator is then made to produce the synthetic datasets of the following experiments.
A screen shot of the schedule dataset generator is shown in Figure 5.1. The properties of the synthetic datasets can be set such as the attributes of a project, the mutation modes, and the attributes of machines, jobs and molds.
Figure 5.1 The schedule dataset generator
The experiments are implemented by Microsoft Visual C#, and the screen shot of the scheduling program executing our proposed scheduling algorithm and the simple GA without the adjustment operators is shown in Figure 5.2.
Figure 5.2 The screen shot of the scheduling program after the running the proposed algorithm
In the first experiment, the job numbers are set at 20, 40, 60, 80 and 100, the machine number and the mold number are both set at 10. A set of molds between one to eight and the respective processing time are then randomly assigned to each job. Both methods are run 100 times and their makespan are kept. The averages of the makespan of the two GA-based scheduling algorithms for different numbers of jobs are shown in Figure 5.3.
Figure 5.3 The average makespan of the schedules by the two GA-based scheduling algorithms for different numbers of jobs
From Figure 5.3, it can be observed that the more number of jobs the higher the makespan to complete the assigned jobs. Moreover, it is easily seen that the proposed algorithm with the adjustment operator achieved a better result than without the adjustment operator. The proposed adjustment operators can thus obviously improve the scheduling results.
In the second experiment, we devote time on the comparison with the makespan of the two GA-based scheduling algorithms for different numbers of machines, and the results are shown in Figure 5.4. In the experiments, the job number is set at 50, and the mold number is set at 10. The makespan of both methods are tested for different
one to eight molds and its processing time. Both methods are run 100 times and their makespan are kept. The averages of the makespan of the two GA-based scheduling algorithms for different numbers of machines are then shown in Figure 5.4.
Figure 5.4 The makespan of the schedules by the two GA-based scheduling algorithms for different numbers of machines
From Figure 5.4, it can also be easily seen that the proposed algorithm with the adjustment operator acquired a better result than that without the adjustment operator for any number of machines.
Next, the makespan of the two GA-based scheduling algorithms for different numbers of molds are shown in Figure 5.5. In the experiments, the job number is set at
0
50, the machine number is set at 10and the mold numbers are set at 5, 10, 15 and 20.
Each job is then randomly assigned a set of one to eight molds and its processing time.
Figure 5.5 The makespan of the schedules by the two GA-based scheduling algorithms for different numbers of molds
From the comparison of results, it is easily observed that the proposed algorithm with the adjustment operator has the best makespan than that without the adjustment operator for the number of molds.
From Figures 5.3, 5.4 and 5.5, we find the results with the adjustment operators are better than those without it. The adjustment operators do obviously improve the scheduling results on the machines with the mold constraints.
0
experiment is made for comparing the GA-based scheduling algorithms with, and without, the adjustment operators along different ratios of the mold set-up time, and the average job processing time. The properties of the experiment are shown in Table 5.3.
Table 5.3 The parameter settings of data set
Parameter Value
The number of job 25
The number of machine 4
The number of mold 5
A job average the process time 31.2 A job average the number of using molds 2.82 A machine average the suitable molds 2.5
The average of the number of molds 2.2
Figure 5.6 shows the results of the GA-based scheduling algorithms with and without the adjustment operators. The Y-axis represents the average makespan of the 100 times experiment, and X-axis represents the ratios of the mold setup time, and the average job processing time is set around 31.2.When the mold set-up time is set at 300, the ratio is thus as 1:10. It can be observed that the adjustment operators are more effective to reduce the makespan as the mold set-up time is much larger than the job processing time.
Figure 5.6 Comparisons along different ratios of the mold set-up time and the average job processing time
0 200 400 600 800 1000 1200 1400
10:1 5:1 1:1 1:5 1:10
with Adjustment without Adjustment
CHAPTER 6
Conclusions and Future Work
In this thesis, we have attempted to solve the scheduling problems that jobs are assigned to multiple unrelated machines with mold constraints. A GA-based scheduling algorithm is proposed for finding the minimum makespan of these problems. In the proposed algorithm, an encoding schema is introduced, and the adjustment operators are also given for improving fitness values of the chromosomes when they are generated. The proposed approach can be easily extended to solve the same problem, as well as the fitness function of the total complete time. Finally, the experimental results show that the adjustment operators are effective in avoiding the mold conflicts and are then able to receive better scheduling results. Thus, it can be shown that the scheduling results with the adjustment operators are much better than those without it.
In the future, we will consider solving more related scheduling problems on tardiness issues by using different strategies. We will continue our research in other related fields.
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