In the real world, scheduling for parallel machine problems with jobs arriving in dynamics and with sequence-dependent setup time is important, especially in bottleneck stage. To design the testing problem, we consider the characteristics of polyimide (PI) print process in the cell factory of TFT-LCD industry. The PI print process is regarded as the bottleneck stage in the cell factory because of the high ratio of setup time to the processing time.
For testing problems, we consider 5 machines and the number of jobs are set to 180. The processing time is uniform distributed over the interval [1200, 1800] with the unit in second. The arrival time is exponential distributed with the mean decided by that the expected service time divided by machine utilization. The expected service time is the sum of expected processing time and expected setup time. We keep the machine utilizations to 90% and the arrival rate can be obtained from the above statements.
The job-type feature, JR, includes three levels, 15, 30, and 60; the setup time severity feature, ST, includes three levels, 1, 2, and 4; the setup time range feature, SR, includes three levels, 0.5, 1, and 2; and finally the tightness feature, T, includes three levels determined by equation (2).
We divide the due dates of jobs into three levels, which are 20, 30 and 40 hours.
If the tightness feature, T, is 1975, that means there are 30 jobs assigned for 20 hours, 60 jobs assigned for 30 hours and 90 jobs assigned for 40 hours. If the feature T is 1775, that indicates there are 60 jobs assigned for 20, 30 and 40 hours respectively.
There are 70 jobs assigned for 20 hours, 60 jobs assigned for 30 hours and 50 jobs assigned for 40 hours when T is 1708.3. The tightness feature gets tighter as the values of T decreasing. For all combinations of different features, we will have 81 testing problems.
For all testing problems, we apply four modified parallel insertion algorithms stated in section 4 to solve these problems respectively. We set the rescheduling criterion as 10 and 20 respectively which indicates the job sequences will be rescheduled when the length of unscheduled buffer reaches the rescheduling criterion.
Parameters as and respectively used in the considered algorithms, PIA-II and PIA-III, are divided into five levels, 0.1, 0.3, 0.5, 0.7, and 0.9. We note that PIA-I
includes no parameters in the cost function and PIA-IV is the combination of PIA-I and PIA-II, which indicates PIA-IV also including parameter in the cost function.
For all problems, we apply each considered algorithm to find the best rescheduling criterion and the best value for the parameter, with which the algorithm can solve the testing problem. We use eM-Plant 4.6 to simulate the considered manufacturing environment and apply NeuroSolutions 5.0 for training data with neural network and SVM with 20% of training set for cross verification. The computational time for solving a testing problem is around 5 minutes with Pentium IV 2.4G Hz and 1,536 Ram. Table 1 and Table 2 show the summary of the number of problems with lowest makespan for various rescheduling criteria and values for parameters.
Table 1. Summary of the number of problems with lowest makespan for various rescheduling criteria
PIA-I PIA-II PIA-III PIA-IV All Algorithms
Rescheduling
Criterion 10 20 10 20 10 20 10 20 10 20
Number of
Cases* 41 40 45 36 45 36 41 40 172 152
* The total cases for each algorithm are 81 and the total cases for all algorithms are 324
Table 2. Summary of the number of problems with lowest makespan for various values for parameters
PIA-II ( ) PIA-III ( ) PIA-IV ( ) All Algorithms Value for
Parameter 0.9 Others 0.1 Others 0.9 Others 0.1 0.9 Others Number of
Cases 58 23 65 16 54 27 83* 113 77
* The total cases for all algorithms with considering parameters are 273 because the PIA-I have no parameter
For considered algorithms, the best rescheduling criterion with 10 takes 172 better cases among 324 testing problems and more than the rescheduling criterion with 20. The parameter equal to 0.9 is appropriate for PIA-II and PIA-IV; the PIA-III with equal to 0.1 will be more efficiently. Table 3 shows the number of the best among all 81 problems solved by the considered algorithms, in which the rescheduling criterion is set to 10 and values for parameters, and , in the cost function is set to 0.9 and 0.1 respectively.
Table 3. The number of the best among all problems solved by the considered algorithms
PIA-I* PIA-II
with = 0.9 PIA-III
with = 0.1 PIA-IV with = 0.9 Number of
better cases** 27 12 30 12
* For all algorithms, the scheduling criterion is set to 10
** The number of better cases for a particular algorithm among 81 testing cases
The results of Table 3 indicate that there is no specific algorithm with predetermined rescheduling criterion and cost function parameter values can always perform well for all problems which is a main reason why we develop the proposed approach for considered algorithm in a problem with specific features to determine the proper levels for the rescheduling criterion and cost function parameter.
To enable our proposed approach performs well for the problem features in various levels, we apply backpropagation neural network and support vector machine (SVM) to make the decisions to select the rescheduling criterion and the parameter in the cost function, which both are required for the considered algorithms to solve the dynamic parallel machine problem. We apply a 4-layer neural network with 10 hidden nodes in each hidden layer which is determined by testing several alternative configurations of the network model and SVM techniques to select an appropriate modified parallel algorithm for a specific problem and decide the rescheduling criterion and value for parameter of the choosing algorithm. Table 4 shows the number of correct identification for the rescheduling criterion and value for parameter by using neural network and SVM techniques. The correct identification indicates the estimated value for the rescheduling criterion and parameter by neural network and SVM equal to the best rescheduling criterion and parameter. The performance of estimating rescheduling criterion and value for parameter will be better by using neural network or SVM with higher number of correct identification.
The number of correct identification for the selecting algorithms with neural network and SVM are shown in Table 5. Applying the classifying technique with higher percentage of correct classifications will perform well for selecting considered algorithms. The percentage is the number of correct identification divided by the number of the best among all problems solved by the considered algorithms presented in Table 3.
Table 4. The number of correct identification for rescheduling criterion and value for parameter with neural network and SVM
Neural Network SVM
Considered
Algorithms Decision of Rescheduling
Criterion
Decision of parameter
Decision of Rescheduling
Criterion
Decision of parameter
PIA-I
60 (74.07%
*)
78 (96.30%)
PIA-II
55 (67.90%) 60 (74.07%) 69 (85.19%) 78 (96.30%)
PIA-III59 (72.84%) 76 (93.83%) 71 (87.65%) 81 (100%)
PIA-IV63 (77.78%) 54 (66.67%) 74 (91.36%) 69 (85.19%)
* The number of correct identification divided by 81 cases
Table 5. The number of correct identification for the selecting algorithms with neural network and SVM
PIA-I PIA-II PIA-III PIA-IV All
Algorithms Neural
Network 13 (48.15%) 7 (58.33%) 24 (80%) 9 (75%) 53 (65.43%) SVM 25 (92.59%) 9 (75%) 27 (90%) 9 (75%) 70 (86.42%) Number of
Better Cases* 27 12 30 12
* The number of better cases for a particular algorithm among 81 testing cases
We compare the performance of proposed approaches for estimating the rescheduling criteria and values for parameters with each considered algorithm with the best rescheduling criterion and the best value for parameter and summarize the results in Table 6. The NN&PIA-I~IV and SVM&PIA-I~IV mean that apply neural network and SVM to select the considered algorithms and estimate the rescheduling criteria and values for parameters to solve problems respectively. The NN&PIA-X or SVM&PIA-X versus a particular algorithm indicate that we compare the performance of applying neural network or SVM to estimate the rescheduling criterion and value for parameter of the algorithm with setting the rescheduling criterion and value for parameter by the best conditions based on the Table 1 and Table 2. For example, the SVM-PIA-II vs. PIA-II means the comparison between using SVM to obtain the
proper rescheduling criterion and value for used in the PIA-II and setting the rescheduling criterion and to 10 and 0.9 respectively. The results indicate our approaches to estimate the rescheduling criteria and values for parameters perform better than deciding arbitrarily even choosing the fixed value which perform well in most cases.
Table 6. Comparisons of performance for each considered algorithm with setting the rescheduling criterion and value for parameter by neural network or SVM and
fixed them according to the best conditions Number of
* For the two comparing approaches, the former approach can obtain the number of better cases than the later
Table 7. Comparisons of performance between applying neural network and SVM to choose the considered algorithm and estimate the rescheduling criterion and
parameter value for the selecting algorithm Number of
Table 7 shows the comparisons of performance between “MPIA with SVM” and
“MPIA with NN” which indicate that applying SVM and neural network to choose the
one of the modified parallel insertion algorithm and estimate the rescheduling criterion and value for parameter of the selected algorithm respectively. The results show that using SVM to select the one of the modified parallel insertion algorithm and estimate the rescheduling criterion and value for parameter of the selected algorithm performs much better than neural network do.
The Figure 6 shows the number of even or better cases which obtained by different approaches such as neural network and SVM to decide the rescheduling criterion and value for parameter. The line “Fixed” means that use the better rescheduling criterion and value for parameter of a particular algorithm based on Table 1, Table 2 and fix them to solve all problems.
Figure 6. The number of even or better cases for a particular algorithm with different approaches to decide the rescheduling criterion and value for parameter
The results shown in Table 6 and Figure 6 indicate that applying neural network and SVM for estimating the rescheduling criterion and value for parameter is more efficient than deciding arbitrarily even fixing them as the best conditions in the most cases. For every modified parallel insertion algorithm, using SVM and neural network to estimate the rescheduling criterion and value for parameter are excellent
PIA-I PIA-II PIA-III PIA-IV
0 10 20 30 40 50 60 70 80
The Considered Algorithm
The Number of Even or Better Cases
PIA NN&PIA SVM&PIA
approaches. Applying neural network to estimate the rescheduling criteria and values for parameters obtains the even or better cases of 86.11% for all considered algorithms and SVM accomplishes the even or better cases of 92.90%. The results shown in Table 7 display the performance of SVM is better than neural network in our testing problems, especially in the results of selecting the consider algorithm and getting the even or better cases of 88.89%.