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Job-shop scheduling using neural networks
Control Eng. Practice, Vol. 1. No. 6, pp. 957-961,1993 Printedin GreatBritain.All rightsreserved.
0967-0661/93S6.00+ 0.00 © 1993PergamonPressLtd
JOB-SHOP SCHEDULING USING NEURAL NETWORKS T. Watanabe, H. Tokumaru and Y. Hashimoto Department of Computer Science and Systems Engineering, Ritzumeikan University, Kyoto 603-77. Japan
Abstract..lob-shop scheduhng cannot easily be analyticall) accomplished, so, it is done by computer simulation using heuristic priority rules. The SLACK rule for calculatingthe margins of jobs to their due-dates is effective in meeting the due-dates However, the calculated margins are not precise because the actual margm is shortened due to conflicts with other jobs. The authors propose a method for estimating the margins by using a neural network. It is found that the method is effective for improving the average lateness to duedates but not the maximum lateness. This paper proposes a method for adding a second neural network for judging the reliability of the estimated marg~us composed to the first one and for switching to the margins calculated by the SLACK rule when the reliability is low. The proposed method is verified by scheduhng simulationsto he effective in decreasingthe maximumlateness to due-dates as much as the average lateness Key Words. Automation; Computer simulation; Computer software. Flexible manufacturing; Management systems; Neural nets; Scheduling;.lob shop scheduling
to their due-dates on schedules simulated by using the SLACK rule are learned by the neural network having inputs of theoretical margins, the number of jobs and machines, and the mean operation number for each job, among others. It was found that the proposed scheduling method is effective for improving the average lateness to due-dates, but not the maximum lateness. This is due to the estimation error of the neural network under some unidentified conditions.
1. INTRODUCTION Job-shop type production, in which each job has a different work-flow, is adopted in factories to produce various kinds of products in small volumes. Efficient management is required for job-shop type production because its work-flows are very complicated. A computer simulation method using heuristic rules, such as SPT, LPT, EDD, SLACK or Look Ahead, among others, is ordinarily used for the scheduling of job-shops (Watanabe and Sakamoto, 1984, 1986; Watanabe and Fujii, 1988; Watanabe et al, 1989, 1990, Watanabe, 1990). This is because analytical or artificial intelligence methods, such as the branch-andbound method, are not practical due to problem complexity.
Therefore, a method for adding another neural network to the system to evaluate the reliability of the estimation of margins is proposed in this paper. It is only when a high reliability is indicated by the second neural network, that the margin estimated by the first one is adopted instead of that calculated by the SLACK rule. The improvement of performance indices for evaluating schedules made by the proposed method is investigated by computer simulations.
SLACK is one of the most effective heuristic rules for making good job-shop schedules that meet due-dates. It gives priority to a job having the shortest margin to its due-date. The margin is calculated by subtracting the operation time of the job from the remained time to the due-date without taking into account the interference of other jobs (called "theoretical margin" in this paper). However, the actual margin on real schedules becomes shorter than this calculated margin due to the interference.
2. THE ESTIMATION OF MARGINS TO DUE-DATES Job-shop scheduling for n jobs processed by m machines is discussed in this paper. Each job is divided into several operations with a smaller number than m. It is supposed that each operation for each job is processed by a different machine. The numbers of jobs and machines, the number of operations of each job, the order for processing the operations for each job, the processing time of each operation, and the
The authors proposed a method for estimating actual margins to the due-date under the existence of an interference from jobs by using a neural network (Watanabe et al, 1992a, 1992b). The margins of jobs 957
958
T. Watanabe et al.
due-dates are determined by using random numbers. The neural network shown in the upper pan of Fig. 1 estimates the margin of a job to its due-date, and the lower one evaluates the reliability of the estimated margin. The function of the upper neural network is discussed in this section and the other in the next, respectively. The discussed neural network has three layers: the first is an input layer of ten neurons, the second is a hidden layer of twenty neurons, and the last one is an output layer of one neuron to produce the estimated margin to the due-date of a job. The inputs to the first layer are as follows:
p~
: the number of operations of job i
di
: the due-date of job i
Mthi : the theoretical margin of job i Tar , : the average operation time of job i n
Mav = ~ Mth~ : the average of theoretical ,=~ margins to due-dates n : the number of jobs m : the number of machines Prai : (the number of the operations of job i ) / (the number of total operations) Tra , : (the sum of operation times of job i ) / (the sum of operation times of the all jobs) Dr, n : (the number of the operations of job i ) / Mth , "
Ordinary scheduling simulation for fifty scheduling studies is performed by using the SLACK rule in order to prepare teaching information. The number of machines m is selected randomly at between 5 and 20, and the number of jobs n between 20 and 100. for each study by using random numbers. The numbers of operations, operation times, and the due-dates are also set by using random numbers. The inputs of each job are given to the neural network. The output of the neural network is compared to the margin of the job to its due-date on the schedule introduced by the simulation. The difference is used to modify the weights on the inputs of the neurons of the network based on the back-propagation algorithm. The learning is performed ten thousand times.
Pi
d~
eral
D,o, n m
I ! I P,
a,
Ma,
L~ ~v
Reliability
Pral Dral n m
Fig. 1. Neural networks for estimating the margin of job i to its due-date, and for evaluating the reliability of the estimated margin
After the learning process, the simulation done to estimate the margins to the due dates by using the neural network is performed for ten kinds of problems: (m, n ) = (5, 20), (I0, I0), (tO, 30), (10, _50), (20, 5), and (20, 100). Figure 2 shows the effectiveness of the learning of the neural network. The horizontal axis means the margin on the schedules introduced by using the SLACK rule. The vertical axis represents the theoretical margin calculated by the SLACK rule in the ordinary scheduling and margin forecast by the neural network in Fig. 2 (a) and (b), respectively. The number and total time of the remaining operations are used as p, and Ta,., in the estimation process, respectively. It is shown that the theoretical margins calculated by the SLACK rule arelonger than the actual margins on the schedules introduced by using the theoretical ones, as shown in Fig.2 (a), because the theoretical margins are calculated by neglecting the interruptions of other jobs as mentioned in the INTRODUCTION. On the other hand, the margins estimated by the neural net agree well, on the average, with those on the schedules introduced by using the SLACK rule as shown in Fig. 2 (b). Therefore, better schedules can be expected by using the neural network. However, the stochastic
959
Job-Shop Scheduling error is also shown in Fig. 2 (b). That is, scheduling using this neural network for estimating margins might be effective for i m p r o v i n g its average characteristics, such as the average lateness to duedates but not the maximum lateness to due dates due to the error.
(the estimation error) = I (the margin estimated by the neural network) (the actual margin on the schedules given by using the SLACK rule ) [
4. SCHEDULING SIMULATION 3. THE EVALUATION OF RELIABILITY In this paper, a method is proposed for evaluating the reliability of the estimated margin by the other neural network and only for using the estimated value instead of that calculated by the S L A C K rule when the reliability is high. The neural network shown in the lower part in Fig. 1 estimates the reliability. The composition of the neural network is almost the same as the upper one. The difference between the two is that the lower network has eleven inputs, including the signal from the output layer of the upper one.
Scheduling simulation using the proposed method is performed, and the developed schedules are compared with those developed by using the ordinary method. Fifty scheduling problems are solved for each pair of
(m. n ) = (5, 20), (I0, IO), (I0, 30), (I0. 50), (20, 5), and (20, I00). The following performance indices are used in order to evaluate the schedules: TT : total processing time ML : the maximum lateness to due-dates TL : total lateness to due-dates AL : average lateness to due-dates NL : the number of jobs generating lateness to due-dates MWR : machine working rate MLR : the maximum lateness rate to due-dates ( lateness rate of j o b i ) = (completion time of job i ) / (due-date of job i ) ALR : average lateness rate to due-dates
A hundred examples are selected for the learning of the neural network to evaluate the reliability. Fifty of them give a good estimation, and the rest are bad. One is given for the good estimation examples, and zero for the bad ones as the teacher signal for the network. The judge as to whether the estimation is good or not is determined as; teacher signal = I for the estimation error < 10 h teacher signal = 0 for the estimation error > 10 h
The comparison with those of schedules using the ordinary SLACK rule is done by using the following improvement ratio:
where the estimation error is calculated as:
.,c
.t=
g /
• .-.~..7. • ..:~,....
..., ~..?x~'.'" •z, ,,,,
• .'3¢*-~
t~
/i
i, t-
.-.
O. = _t n
/
/
t~
.
/
/
/
g '-I00
J 0 Margin to due-date on schedule made by using SLACK rule
100h
m= 10. n= l0 (a) The theoretical margin of the ordinary SLACK rule Fig. 2. Estimation of the margin to the due-date
'-I00
100 h Margin to due-date on schedule made by using SLACK rule rn= 10. n= 10
(b) Estimation of the margin to the due-date by using the neural network
960
T. Watanabe et al. Table ! Improvement-ratios of performance indices for the proposed method compared to those for the ordinary SLACK rule L.D.D. : Lateness to due-date (re, n ) I.R. of the Maximum L.D.D. I.R. of the Total L.D.D. I.R. of Number of Jobs Generating L.D.D. Rate of Using Neural Network
(5,20) (10.10) 7.6 % 7.9 % 9.8 % 10.6 % 1.0 % 2.7 % 28.8 % 25.0 %
(m, n )
(5, 20) (10, 50) (20, 100) -88.8 % -39.1% -31.9 % -70.0 % -15.2 % 8.3 % 7.5 %
(10, 50) (20, 5) (20, 100) 5 . 1 % 14.2% 0.9% 0 . 0 % 14.5% 3.0% -0.5 % 0.0 % 0.0 % 23.6 % 35.5 % 15.3 %
is clearly shown that the lateness to due-dates (negative margins to due-dates) is significantly reduced.
Table 2 The improvement-ratios when the neural network for reliability is not used
I.R. of the Maximum L.D.D. I.R. of the Total L.D.D. I.R. of Number of Jobs Generating L.D.D.
(10, 30) 3.7% 0.7% 0.0 % 24.4 %
-7.3 % 4.6 %
(I.R. : Improvement Ratio) = S X
I ( Performance Index of the Schedule given by the Proposed Method) - ( Performance Index of the Schedule given by the Ordinary Method) I ( Performance Index of the Schedule given by the Ordinary Method) S = + 1 for improvement - I for no-improvement Figures 3 and 4 show Gantt charts given by the simulation using the SLACK rule and the proposed neural networks for (m. n ) = (5. 20). respectively. It
Table I shows the i m p r o v e m e n t - r a t i o s of the performance indices of the schedules using the proposed method c o m p a r e d with those of the schedules using the ordinary method for (m. n ) = (5, 20), (10, 10), (10, 30), (10, 50), (20, 5), and (20, 100). The rate to use the neural networks instead of the SLACK rule is also shown on the bottom line. Table I shows that the performance indices (the total flow-time, the maximum lateness to due-dates, the total lateness to due-dates, and the operational rate of machines) improve significantly when the proposed neural networks are used. In particular, the maximum lateness shows an improvement of 14 - 0.9 %. Table 2 shows the improvement ratios when the neural network for reliability is not used. It is clear that the maximum lateness does not improve without the neural network to estimate the reliability. The improvement ratios decrease as the scale parameters m and n of the studies increase. However, even a very small improvement in schedules contributes greatly to the increase in the profit:; of companies.
!
5
,3 = io
5 ~0
e-
= 15
15
20 m 0
20
I 10
Machine number: No. 1 [ ] No. 2
20
Time
30
40
h
0
10
20
30
40
Time Machine number;
[ ] No. 3 ~ No.4
N No.5
I : Due-date Fig. 3. The Gantt chart given by using the ordinary SLACK rule
- . ~ o . 1 NNo. 2 [ ] . ~ o 3
~o.4
@No5
l : Due-date Fig. 4. The Gantt chart given by using the proposed neural networks
Job-Shop Scheduling 5. CONCLUSIONS A method for improving the job-shop scheduling algorithm using neural networks was proposed in this paper. The margin of a job to its due-date was estimated by using a neural network in the proposed method. The reliability of the estimation was evaluated by the other neural network. Scheduling was performed by selecting the job first which has the shortest estimated margin to its due-date instead of simply calculating the margin by subtracting the operation time from the remaining time to due-date as in the ordinary SLACK rule, when the reliability of the estimation is judged to be high. It is verified by scheduling simulation that the proposed method is very effective for improving the average and maximum lateness to due-dates. In the paper, the targets given to the neural network to estimate the margins are those on the schedules introduced by the ordinary scheduling method using the SLACK rule. The next step in the research is to adopt the margins of the introduced schedules by using the neural network as the targets of learning. Though the convergency of the learning when this process is repeated is not guaranteed, the achievement of it is going to be tried anyway.
6. REFERENCES Watanabe, T. and M. Sakamoto (1984). On-line scheduling for adaptive control machine tools in FMS. In: Preprints of IFAC 9th World Congress. VoI.VI-08.3/C-3, pp. 169-174. Pergamon Press, Oxford. Watanabe, T. and M. Sakamoto (1986). On-line scheduling for adaptive control machine tools in FMS. Robotics and Computer Integrated Manufacturing, 2-1, 56-64. Watanabe, T. and R. Fuji (1988). Determining the job operation speed and schedule for machine tools in F'MS by forecasting using a simulator and a system performance index. In: Proc. of the U.S.A.-Japan Symposium on Flexible Automation (M. Donath, Ed. ), pp.751-756. ASME, New York. Watanabe, T., R. Fuji, K. Kido and K. Inagami (1989). Self modification of scheduling in production. In: Proc. of IFAC Int. Workshop on Decisional Structures in Automated Manufacturing (A. Villa, Ed.), pp.71-80. IFAC. Watanabe, T.. K. Kido and R. Fuji (1990). The automatic improvement of job-shop and new dispatching rules. In: Proc. 1990 Japan-U.S.A. Symposium on Flexible Automation (T. Watanabe, Ed.), pp. 1133 - 1138. ISCIE, Kyoto. Watanabe, T. (1990). Job-shop scheduling using fuzzy logic in a computer integrated manufacturing environment. In: Preprints of 6th International
961
Conference of Advanced Systems Researches, lnformatics and Cybernetics (G.E. Lasker, Ed.), pp. 1 - 8), IASRIC. Watanabe, T., H. Tokumaru, and Y. Hashimoto (1992a). An analysis of dynamic seheAuling for job-shop scheduling and its improvement using intelligent algorithms. In: Proc. of IMACS/SICE Int. Syrup. Robotics, Mechatronics and Manufacturing Systems '92 Kobe (K. Tsuchiya, Ed.), pp.516-522. SICE, Tokyo. Watanabe, T., H. Tokumaru, Y. Hashimoto, and Y. Hirose (199"2b). Job-shop scheduling based on the estimation of margins to due-dates by using a neural network. In: Proc. of Pacific Conference on Manufacturing (H. Ohta, Ed.), pp.516-522. PCM.
0967-0661/93S6.00+ 0.00 © 1993PergamonPressLtd
JOB-SHOP SCHEDULING USING NEURAL NETWORKS T. Watanabe, H. Tokumaru and Y. Hashimoto Department of Computer Science and Systems Engineering, Ritzumeikan University, Kyoto 603-77. Japan
Abstract..lob-shop scheduhng cannot easily be analyticall) accomplished, so, it is done by computer simulation using heuristic priority rules. The SLACK rule for calculatingthe margins of jobs to their due-dates is effective in meeting the due-dates However, the calculated margins are not precise because the actual margm is shortened due to conflicts with other jobs. The authors propose a method for estimating the margins by using a neural network. It is found that the method is effective for improving the average lateness to duedates but not the maximum lateness. This paper proposes a method for adding a second neural network for judging the reliability of the estimated marg~us composed to the first one and for switching to the margins calculated by the SLACK rule when the reliability is low. The proposed method is verified by scheduhng simulationsto he effective in decreasingthe maximumlateness to due-dates as much as the average lateness Key Words. Automation; Computer simulation; Computer software. Flexible manufacturing; Management systems; Neural nets; Scheduling;.lob shop scheduling
to their due-dates on schedules simulated by using the SLACK rule are learned by the neural network having inputs of theoretical margins, the number of jobs and machines, and the mean operation number for each job, among others. It was found that the proposed scheduling method is effective for improving the average lateness to due-dates, but not the maximum lateness. This is due to the estimation error of the neural network under some unidentified conditions.
1. INTRODUCTION Job-shop type production, in which each job has a different work-flow, is adopted in factories to produce various kinds of products in small volumes. Efficient management is required for job-shop type production because its work-flows are very complicated. A computer simulation method using heuristic rules, such as SPT, LPT, EDD, SLACK or Look Ahead, among others, is ordinarily used for the scheduling of job-shops (Watanabe and Sakamoto, 1984, 1986; Watanabe and Fujii, 1988; Watanabe et al, 1989, 1990, Watanabe, 1990). This is because analytical or artificial intelligence methods, such as the branch-andbound method, are not practical due to problem complexity.
Therefore, a method for adding another neural network to the system to evaluate the reliability of the estimation of margins is proposed in this paper. It is only when a high reliability is indicated by the second neural network, that the margin estimated by the first one is adopted instead of that calculated by the SLACK rule. The improvement of performance indices for evaluating schedules made by the proposed method is investigated by computer simulations.
SLACK is one of the most effective heuristic rules for making good job-shop schedules that meet due-dates. It gives priority to a job having the shortest margin to its due-date. The margin is calculated by subtracting the operation time of the job from the remained time to the due-date without taking into account the interference of other jobs (called "theoretical margin" in this paper). However, the actual margin on real schedules becomes shorter than this calculated margin due to the interference.
2. THE ESTIMATION OF MARGINS TO DUE-DATES Job-shop scheduling for n jobs processed by m machines is discussed in this paper. Each job is divided into several operations with a smaller number than m. It is supposed that each operation for each job is processed by a different machine. The numbers of jobs and machines, the number of operations of each job, the order for processing the operations for each job, the processing time of each operation, and the
The authors proposed a method for estimating actual margins to the due-date under the existence of an interference from jobs by using a neural network (Watanabe et al, 1992a, 1992b). The margins of jobs 957
958
T. Watanabe et al.
due-dates are determined by using random numbers. The neural network shown in the upper pan of Fig. 1 estimates the margin of a job to its due-date, and the lower one evaluates the reliability of the estimated margin. The function of the upper neural network is discussed in this section and the other in the next, respectively. The discussed neural network has three layers: the first is an input layer of ten neurons, the second is a hidden layer of twenty neurons, and the last one is an output layer of one neuron to produce the estimated margin to the due-date of a job. The inputs to the first layer are as follows:
p~
: the number of operations of job i
di
: the due-date of job i
Mthi : the theoretical margin of job i Tar , : the average operation time of job i n
Mav = ~ Mth~ : the average of theoretical ,=~ margins to due-dates n : the number of jobs m : the number of machines Prai : (the number of the operations of job i ) / (the number of total operations) Tra , : (the sum of operation times of job i ) / (the sum of operation times of the all jobs) Dr, n : (the number of the operations of job i ) / Mth , "
Ordinary scheduling simulation for fifty scheduling studies is performed by using the SLACK rule in order to prepare teaching information. The number of machines m is selected randomly at between 5 and 20, and the number of jobs n between 20 and 100. for each study by using random numbers. The numbers of operations, operation times, and the due-dates are also set by using random numbers. The inputs of each job are given to the neural network. The output of the neural network is compared to the margin of the job to its due-date on the schedule introduced by the simulation. The difference is used to modify the weights on the inputs of the neurons of the network based on the back-propagation algorithm. The learning is performed ten thousand times.
Pi
d~
eral
D,o, n m
I ! I P,
a,
Ma,
L~ ~v
Reliability
Pral Dral n m
Fig. 1. Neural networks for estimating the margin of job i to its due-date, and for evaluating the reliability of the estimated margin
After the learning process, the simulation done to estimate the margins to the due dates by using the neural network is performed for ten kinds of problems: (m, n ) = (5, 20), (I0, I0), (tO, 30), (10, _50), (20, 5), and (20, 100). Figure 2 shows the effectiveness of the learning of the neural network. The horizontal axis means the margin on the schedules introduced by using the SLACK rule. The vertical axis represents the theoretical margin calculated by the SLACK rule in the ordinary scheduling and margin forecast by the neural network in Fig. 2 (a) and (b), respectively. The number and total time of the remaining operations are used as p, and Ta,., in the estimation process, respectively. It is shown that the theoretical margins calculated by the SLACK rule arelonger than the actual margins on the schedules introduced by using the theoretical ones, as shown in Fig.2 (a), because the theoretical margins are calculated by neglecting the interruptions of other jobs as mentioned in the INTRODUCTION. On the other hand, the margins estimated by the neural net agree well, on the average, with those on the schedules introduced by using the SLACK rule as shown in Fig. 2 (b). Therefore, better schedules can be expected by using the neural network. However, the stochastic
959
Job-Shop Scheduling error is also shown in Fig. 2 (b). That is, scheduling using this neural network for estimating margins might be effective for i m p r o v i n g its average characteristics, such as the average lateness to duedates but not the maximum lateness to due dates due to the error.
(the estimation error) = I (the margin estimated by the neural network) (the actual margin on the schedules given by using the SLACK rule ) [
4. SCHEDULING SIMULATION 3. THE EVALUATION OF RELIABILITY In this paper, a method is proposed for evaluating the reliability of the estimated margin by the other neural network and only for using the estimated value instead of that calculated by the S L A C K rule when the reliability is high. The neural network shown in the lower part in Fig. 1 estimates the reliability. The composition of the neural network is almost the same as the upper one. The difference between the two is that the lower network has eleven inputs, including the signal from the output layer of the upper one.
Scheduling simulation using the proposed method is performed, and the developed schedules are compared with those developed by using the ordinary method. Fifty scheduling problems are solved for each pair of
(m. n ) = (5, 20), (I0, IO), (I0, 30), (I0. 50), (20, 5), and (20, I00). The following performance indices are used in order to evaluate the schedules: TT : total processing time ML : the maximum lateness to due-dates TL : total lateness to due-dates AL : average lateness to due-dates NL : the number of jobs generating lateness to due-dates MWR : machine working rate MLR : the maximum lateness rate to due-dates ( lateness rate of j o b i ) = (completion time of job i ) / (due-date of job i ) ALR : average lateness rate to due-dates
A hundred examples are selected for the learning of the neural network to evaluate the reliability. Fifty of them give a good estimation, and the rest are bad. One is given for the good estimation examples, and zero for the bad ones as the teacher signal for the network. The judge as to whether the estimation is good or not is determined as; teacher signal = I for the estimation error < 10 h teacher signal = 0 for the estimation error > 10 h
The comparison with those of schedules using the ordinary SLACK rule is done by using the following improvement ratio:
where the estimation error is calculated as:
.,c
.t=
g /
• .-.~..7. • ..:~,....
..., ~..?x~'.'" •z, ,,,,
• .'3¢*-~
t~
/i
i, t-
.-.
O. = _t n
/
/
t~
.
/
/
/
g '-I00
J 0 Margin to due-date on schedule made by using SLACK rule
100h
m= 10. n= l0 (a) The theoretical margin of the ordinary SLACK rule Fig. 2. Estimation of the margin to the due-date
'-I00
100 h Margin to due-date on schedule made by using SLACK rule rn= 10. n= 10
(b) Estimation of the margin to the due-date by using the neural network
960
T. Watanabe et al. Table ! Improvement-ratios of performance indices for the proposed method compared to those for the ordinary SLACK rule L.D.D. : Lateness to due-date (re, n ) I.R. of the Maximum L.D.D. I.R. of the Total L.D.D. I.R. of Number of Jobs Generating L.D.D. Rate of Using Neural Network
(5,20) (10.10) 7.6 % 7.9 % 9.8 % 10.6 % 1.0 % 2.7 % 28.8 % 25.0 %
(m, n )
(5, 20) (10, 50) (20, 100) -88.8 % -39.1% -31.9 % -70.0 % -15.2 % 8.3 % 7.5 %
(10, 50) (20, 5) (20, 100) 5 . 1 % 14.2% 0.9% 0 . 0 % 14.5% 3.0% -0.5 % 0.0 % 0.0 % 23.6 % 35.5 % 15.3 %
is clearly shown that the lateness to due-dates (negative margins to due-dates) is significantly reduced.
Table 2 The improvement-ratios when the neural network for reliability is not used
I.R. of the Maximum L.D.D. I.R. of the Total L.D.D. I.R. of Number of Jobs Generating L.D.D.
(10, 30) 3.7% 0.7% 0.0 % 24.4 %
-7.3 % 4.6 %
(I.R. : Improvement Ratio) = S X
I ( Performance Index of the Schedule given by the Proposed Method) - ( Performance Index of the Schedule given by the Ordinary Method) I ( Performance Index of the Schedule given by the Ordinary Method) S = + 1 for improvement - I for no-improvement Figures 3 and 4 show Gantt charts given by the simulation using the SLACK rule and the proposed neural networks for (m. n ) = (5. 20). respectively. It
Table I shows the i m p r o v e m e n t - r a t i o s of the performance indices of the schedules using the proposed method c o m p a r e d with those of the schedules using the ordinary method for (m. n ) = (5, 20), (10, 10), (10, 30), (10, 50), (20, 5), and (20, 100). The rate to use the neural networks instead of the SLACK rule is also shown on the bottom line. Table I shows that the performance indices (the total flow-time, the maximum lateness to due-dates, the total lateness to due-dates, and the operational rate of machines) improve significantly when the proposed neural networks are used. In particular, the maximum lateness shows an improvement of 14 - 0.9 %. Table 2 shows the improvement ratios when the neural network for reliability is not used. It is clear that the maximum lateness does not improve without the neural network to estimate the reliability. The improvement ratios decrease as the scale parameters m and n of the studies increase. However, even a very small improvement in schedules contributes greatly to the increase in the profit:; of companies.
!
5
,3 = io
5 ~0
e-
= 15
15
20 m 0
20
I 10
Machine number: No. 1 [ ] No. 2
20
Time
30
40
h
0
10
20
30
40
Time Machine number;
[ ] No. 3 ~ No.4
N No.5
I : Due-date Fig. 3. The Gantt chart given by using the ordinary SLACK rule
- . ~ o . 1 NNo. 2 [ ] . ~ o 3
~o.4
@No5
l : Due-date Fig. 4. The Gantt chart given by using the proposed neural networks
Job-Shop Scheduling 5. CONCLUSIONS A method for improving the job-shop scheduling algorithm using neural networks was proposed in this paper. The margin of a job to its due-date was estimated by using a neural network in the proposed method. The reliability of the estimation was evaluated by the other neural network. Scheduling was performed by selecting the job first which has the shortest estimated margin to its due-date instead of simply calculating the margin by subtracting the operation time from the remaining time to due-date as in the ordinary SLACK rule, when the reliability of the estimation is judged to be high. It is verified by scheduling simulation that the proposed method is very effective for improving the average and maximum lateness to due-dates. In the paper, the targets given to the neural network to estimate the margins are those on the schedules introduced by the ordinary scheduling method using the SLACK rule. The next step in the research is to adopt the margins of the introduced schedules by using the neural network as the targets of learning. Though the convergency of the learning when this process is repeated is not guaranteed, the achievement of it is going to be tried anyway.
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