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The new algorithms for the dynamic flowshop scheduling
THE NEW ALGORlTHMS FOR THE DYNAMIC FLOWSHOP SCHEDU ...
14th World Congress ofIFAC
A-Ic-03-2
Copyright© 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China
THE NEW ALGORITHMS FOR THE DYNAMIC FLOWSHOP SCHEDULING'
Jiyin Liu land Lixin Tang 2t
1
Department ofIndustrial Engineering and Engineering Management, The Hong Kong University of Science and Technology, HONG KONG· 2
Institute a/Systems Engineering, Northeastern University,P.R. CHINA
Abstract: Conventional scheduling theory and algorithms have not solved the practical needs of factory management. Because of the complexity of modem manufacturing systems and the fact that they operate in a stochastic and dynamic environment, dynamic scheduling is often required. Dynamic flowshop scheduling is frequently met in practical situations. In this paper we apply an adaptive optimization framework and develop a dynamic flowshop scheduling model with the objective of minimizing the mean of flowtime of jobs arriving as a Poisson process. Static heuristic algorithms are embedded in control model as scheduling controller. The most commonly used dispatching rule method is also implemented and compared with the new model through simulation. Analysis is done for different shop load levels and different numbers of machines. The results show that the new model performs significantly better than the dispatching rule method .. Copyright © 19991FAC. Keywords: dynamic flow shop, adaptive control model, static embedded algorithms
1. INTRODUCTION
Production scheduling can be defmed generally as the allocation of the resources in a production system over time to perform the operations needed to transform raw materials into products. An effective and efficient scheduling system is necessary to well achieve the potentials of a production facility. Production scheduling problems are extremely
complex. The complexity is mainly due to the following two features of the problem (Liu, 1995). Interconnected Decisions: The components of a production system, e.g., machines, material handling devices and storage buffers, are related to each other. The operations of and the constraints on one part of the system affect the other parts. Therefore, when a decision is to be made of which operation should be performed next, all available information should be considered including how this operation may affect
• The Project is Supported by HKUST Direct Allocation GrantDAG95196.E08, National Natural Science Foundation of China Through Approved No 79700006, and National 863/CIMS of China through Approved No. 863-511-708-009, t To whom correspondence should be addressed.
307
Copyright 1999 IF AC
ISBN: 0 08 043248 4
14th World Congress oflFAC
THE NEW ALGORITHMS FOR THE DYNAMIC FLOWSHOP SCHEDU ...
future operations in the whole production system. This is to say that production scheduling is an overall optimization problem for the whole system. Dynamic Process: A production system is a dynamic system. From time to time, new orders come and fmished products leave the system. Scheduling the activities in such a system is therefore a dynamic process in nature. As White (1990) indicated, the scheduling problem encountered in production is really one of rescheduling. Many practical situations are dynamic in nature, the great challenges from the increasingly competitive world market and the quickly changing customer needs faced by manufacturing companies require more effective and more flexible production scheduling and shop floor control strategies. Because the conditions and environments in dynamic and static scheduling problems are different, study on the scheduling methods which more suit to dynamic environment is important and challenging. Dynamic scheduling problems are often addressed using dispatching rule based approaches. These approaches emphasize the dynamic nature of the system. Simple dispatching rules such as First-InFirst-Out (FIFO) and Shortest Processing Time (SPT) are often used to each machine independently. Over a hundred such rules have been developed; classification and comparison of these rules have been made (Panwalkar and Iskander, 1977; Blackstone, et al., 1982). While these rules can handle dynamic system changes and are easy to implement, they have a major disadvantage of being myopic, i.e., a decision is made only based on the situation on a single machine. The effects of the decision on other parts of the system are not considered. Therefore the over,ll system performance is not optimized. Simulation methods have been used to improve the performance by trying a number of rules and selecting the best (Park, 1988). As the system becomes complex and more rules are involved, the computation time needed to perform a valid comparison of the rules may dissolve the dynamic merit of the dispatching rules. Artificial intelligence (AI) methods have also been used to improve the performance of rules (Pierreval, 1992). They try to incorporate expert and domain knowledge in selection of rules so that not all the rules need to be tried every time. The performance of these methods reply heavily on the quality of the knowledge incorporated which is not easy to obtain. To address both features of the dynamic scheduling problem, Liu (1995) proposed an adaptive optimization framework taking advantages of both above types of methods. In this framework static algorithms for global optimization run dynamically to make the schedule adaptive to production system changes. To suit the dynamic running, the static algorithms .must balance optimality and computation time very well. Trying to keep advantages of
optimization approaches but avoid their computation burden, good heuristic procedures need to be used. . In this paper we implement this framework for the flowshop dynamic scheduling problem. In this problem, all jobs are processed through m different machines in the same routing path. The arrival times of the jobs are unknown in advance and job arrivals follow a Poisson process. Once a job arrives, the processing times on all the machines of the job are assumed known. The objective is to minimize the mean job completion time". The next section describes the adaptive control model implemented. Section 3 reports the simulation study comparing this model with dispatching rule method. Section 4 concludes the paper.
2.
THE ADAPTIVE CONTROL MODEL AND ALGORITHMS
We describe here the dynamic scheduling model which embeds an static heuristic algorithm, taking its advantage for considering overall optimization, into an dynamic framework to handle the dynamic changes.
2.1. The overall structure of the model Control theory is a very useful tool to deal with dynamic performance of a system. A general feedback control system model is shown in Figure 1. Con1rol Requirem~t
+
Controller
Signal
Feedback Figllre I" A general control system model
However, the model cannot be applied directly to production scheduling problems since it is developed for continuously changing processes. In production scheduling problems, the system state at any time is a collect of states of all the components in the system at that time. We can defme the state of a machine or material handling device as whether it is performing a processing or transport operation, the state of a storage buffer as the number of workpieces being stored in it. Then it is obvious that the system state changes at only certain time points. These changes are referred to as events. Three types of events may be identified: (1) directly controlled events - those must be directly determined in the scheduling problem (start of operations); (2) expected events - those are expected to happen once the directly controlled events are scheduled (end of operations); (3) unpredictable events Those happen randomly (new jobs, machine breakdowns, expected events happen before
308
Copyright 1999 IF AC
ISBN: 0 08 043248 4
14th World Congress oflFAC
THE NEW ALGORITHMS FOR THE DYNAMIC FLOWSHOP SCHEDU ...
scheduled or delay). Based on these discussions, a d)mamic model for scheduling problem is implemented following the framework as shown in Figure 2. Unpred icta ble
The adaptive optimization framework,
Even\s
Figure 2. An implemented framework of the dynamic model for scheduling problem. The requirement in the model represents the production orders. The global scheduler is a static scheduling model or algorithm which solves the static scheduling problem from a global optimization point of view. Reactive controller monitors' the changes of requirement and system state, and trigs the running of the global scheduler accordingly. Because the static scheduler is used as a block of the dynamic model, it will be run very frequently. Therefore, it must satisfy the following two requirements. (1) The current state of the system must be taken into account along with the job requirement. (2) The computation time for generating a schedule must be short to suit the real-time dynamic environment.
2.2 Implementation
2.2.1. The reactive controller
This part is the core of the control mechanism. It works in the following way. The controller monitors the changes of requirement and system state and decides whether the global scheduler needs to be rerun. For expected changes over time, the controller simply keeps track of them. It trigs the re-run of the static scheduler only when unexpected changes happen such as new job arrivals and machine breakdowns. In our implementation here for the f1owshop problem, we do not consider machine breakdowns and focuses the normal changes, i.e. job arrivals.
2.2.2 The global scheduler For the static scheduling problem, heuristic algorithms taking a global view of the system should be applied as the global scheduler since they can give reasonably good solutions while taking relatively short computation time. When re-running the algorithm, the existing operations in the system need to be considered because they affect the availability of the system resources. This approach decomposes the dynamic problem into a series of static problems.
A static problem is generated at each occurrence of a non-deterministic even in system, then solve entirely, and the solution is implemented on a rolling horizon basis. In this procedure the entire system is considered at each instance of the static problem, in contrast to priority rules which consider only one machine at a time. Of course, when compared with priority rules, this approach requires greater computational effort, but also can lead to significantly better system performance. Taking into account the computing power available today, this cost seems to be worth to pay. For static flowshop scheduling problem with objective of minimizing mean completion time, Rajendran's algorithm (1993) is a very effective heuristic method. We use this static heuristic algorithm within the dynamic flowshop scheduling model in the following three ways. (l) Heuristic 1: A static problem is fonnulated at each occurrence of a new job arrival. The static algorithm is applied to the set of jobs consists of the newly arrived job and the jobs queuing on the fIrst machine. Those jobs which are being processed on all the machines and queued on other machines (not first machine) are not considered in set of sequencing jobs. Their effect on the current static problem is not considered. When trrst machine become available, the job of first position in the sequence generated from the static algorithm is chosen for processing. In order to hold permutation sequence of the static algorithm, machlnes following the fust machine select jobs according to the FIFO rule. (2) Heuristic 2: Analogous to heuristic 1. Heuristic 2 differs from heuristic 1 in that it considers the state of in-processing jobs. The set of jobs considered consist of the newly arrived job and the jobs which are being processed and which are queuing on the machines. The munber of existing jobs is asswned nl. Their current sequence is regarded as the subsequence of the first nl jobs in sequence of static scheduling problem. The static scheduling problem is the problem in which sequence the fIrst r11 jobs is known. For each of these fIrst nt jobs in the static problem, the processing times on the machines which have already processed the job are zero, the processing time on the machine which is processing the job is the remaining processing time on this machine, the processing time on the machines which will process the job are the original processing times on these machines. When the first machine is freed, the job in the tITst position of the sequence generated from static algorithm is chosen. In order to hold permutation sequence of static algorithms, the machines after the fIrst machine select jobs according to the FIFO rule. (3) Heuristic 3: Analogous to heuristic 2. Heuristic 3 differs from heuristic 2 in that it is implemented on a rolIing horizon basis. In this procedure the static scheduling problem is solved after each preset time period to schedule the new jobs arrived in the period. Heuristic 3 is applied only
309
Copyright 1999 IFAC
ISBN: 0 08 043248 4
THE NEW ALGORITHMS FOR THE DYNAMIC FLOWSHOP SCHEDU. ..
when the dynamic system reaches steady state. Before system reaches steady state, heuristic 2 is used in simulation.
3. COMPUTATIONAL SIMULATION The simulation model was developed using C Language. The commonly used dispatching rule FIFO was used as a benchmark to compare with the new framework and the algorithms embedded.
14th World Congress ofIFAC
By continuous increasing the number of jobs procegsed, it was determined that analytical steadystate level were reached at around the completion of 2000 jobs which corresponded to about 10000 jobs in a whole simulation run. Each simulation replication was performed as fonows: Simulation size (jobs): 10000 Steady state (jobs): > 2000 Non-steady state (jobs) < 2000
3.2 Computational results
3.1 Related parameter setting 3.1.1
Job arrival pattern
A Poisson arrival model is used in simulating job arrivals. This distribution provides a good approximation for the job arrival times if the generating sources can be assumed to be independent. The arrival rate of the jobs is assumed to be equal to the ratio of the capacity of the shop to the average amount of work required. The arrival rate, denoted by A, which is derived from the inter-arrival time information is then given as A,= "lIP
where P = average operation time. T! = machine load capacity utilization. Various arrival rates can be obtained by adjusting the machine load capacity factor 17. By changing the arrival rate, we can control the degree of congestion in the shop, each set of simulations was carried out until a total of 10000 jobs had arrived and been scheduled and completed.
3.1.2
Because none of the all algorithms can guarantee optimal solution to the problem, we cannot find an absolute measurement for the optimality performance of the algorithms. Therefore, we used a relative optimality measurement C/C*, where C* is the best among the results for a problem instance given by the four methods and C is the result for the instance given by the method being evaluated. All the four methods were used to solve the problem in all the scenarios described above. The average optimality performances of these methods against different problem structures and against different problem sizes are shown in tables 1, and 2 respectively. Tables 3 and 4 show the average running times of the algorithms against different problem structures and problem sizes, respectively. All the times shown are in seconds on aPentium PC. Table J Ayerage optimality performances with djfferent shpp load levels
0.8 0.85 0.90 0.95
Shop loading level
The shop load level, repregenting the utilization of the facility, has probably a greater effect on system performance than the job arrival pattern. The system was analyzed under four different shop load level: 80%, 85%, 90% and 95%. These shop load levels are conunonly used in dynamic scheduling simulation.
3.1.3
Iahl!i: 2
HI 1.0508 1.0767 1.1391 1.2536
A~[illi:!i:
H2 1.0764 1.1383 1.1929 1.4290
H3 Ll056 1.1323 1.1867 1.4214
QPtimality Il~IfQ[IDWl!;;~ ll:ith
different number Qima!;;bines
4
Processing time range and distribution
6 8 10
Problem scenarios of different sizes and structures were generated for the experiment. The processing times were randomly generated from uniform distributions. The range of processing times was set to be from I to 50 for all problem instances. The number of machines was chosen at four levels: 4, 6, 8 and 10. Considering both shop load level and number of machines, we had sixteen different problem gcenarios. Ten replications for each scenario were run. Totally, 160 replications were run in the simulation.
3.1.4
FIFO 1.1690 1.2252 1.3050 1.6073
FIFO 1.3801 1.3503 1.3098 1.2663
Iabl!:: 3 AYe:(agl:
0.8 0.85 0.90 0.95
FIFO 1.925 2.275 2.850 4.725
HI 1.1048 1.1295 1.1629 1.1230
H2
1.2225 1.2339 1.1785 1.2018
H3 1.2163 1.2238 1.2173 1.1885
~omputation tiwl:s with different shop IQad l~els
HI 9.300 17.575 53.650 294.450
H2 5.125 8.950 16.825 62.100
H3 3.000 4.675 10.800 51.525
Steady state control
310
Copyright 1999 IF AC
ISBN: 008 0432484
THE NEW ALGORITHMS FOR THE DYNAMIC FLOWSHOP SCHEDU. ..
Table 4 Average computation times of algorithms with different machjne sizes FIFO
4 6 8 10
1.425
2.250 3.425 4.675
HI 30.825 55.200 131.450 157.500
H2 14.275 14.625 26.300 37.800
H3 10.075 13.125 24.500 22.300
3.3 Discussion and analysis ofresults From the figures in the above tables, we can see that: (1) For dynamic flowshop environment, the performances of the proposed three heuristic algoritlnns which are embedded in the adaptive control model are better than that of the FIFO dispatching rule. (2) The test on shop load level showed the mean flow time at 80% loading level was significantly smaller than the mean flow time at 95% loading level. This is to be expected since higher utilization levels can only be achieved by more jobs waiting to be processed at each machine; consequently the time a job spends in the system is higher. (3) The heuristics and the dispatching rule are less affected by the machine size. (4) Although proposed three algorithms take more time than FIFO, they are useful for on-line scheduling where there are enough available time to produce schedules.
14th World Congress ofIFAC
REFERENCES
Blackstone, J. H. Jr., D. T., Philiphs, and G. L. Hogg (1982). A state-of-the-art survey of dispatching rules for manufacturing job shop operations, International Journal of Production Research, 20,27-45. Liu, J., (1995). An adaptive optimization model for scheduling in dynamic environments, Proceedings of the 13th International Conference of Production Research, Jeraslum, 612-614. Panwa\kar, S. S. and W. Iskander (1977). A survey of scheduling, Operations Research, 25, 45-61. Park Y. B. (1988). An evaluation of static flowshop scheduling heuristics in dynamic flowshop model via a computer simulation. Computers & Industrial Engineering, 14, 103-112. Pierreval, H. (1992). Expert system for selecting priority rules in flexible manufacturing systems, Expert Systems with Applications,S, 51-57. Rajendran, C. (1993). Heuristic algorithm for scheduling in a flowshop to minimize total fiowtime, International Journal of Production Economics, 29, 65-93. White, Jr., K. P. (1990). Advances in the theory and practice of production scheduling, Control and Dynamic Systems, 37,115-157.
4. CONCLUSIONS
In this paper we have developed an adaptive control system model for dynamic scheduling problem with the performance measure ofminirnizing the mean flowtime. Three static heuristic algorithms are used as the static scheduler embedded in the dynamic control system model. The performance of the model and the algorithms were compared with the FIFO dispatching rule through simulation. For all scenarios with different shop loading level and different number of machines simulated, the proposed three heuristic algorithms performed superior to the benchmark FIFO dispatching rule method.
311
Copyright 1999 IF AC
ISBN: 008 0432484
THE NEW ALGORITHMS FOR THE DYNAMIC FLOWSHOP SCHEDU. ..
14th World Congress ofIFAC
Control Signal
Requirement
Response
------~~--iController f---=--~
+ Feedback Figure 1. A general control system model
:, The adaptive optimization framework:I
Unpredictable Events ,-------1-------,
Re
System state
I I -
- - - - - - - - - - - - - - - - - - - - - - - - ______ 1
Feedback -
-------~--~~---------------.
312
Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress ofIFAC
A-Ic-03-2
Copyright© 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China
THE NEW ALGORITHMS FOR THE DYNAMIC FLOWSHOP SCHEDULING'
Jiyin Liu land Lixin Tang 2t
1
Department ofIndustrial Engineering and Engineering Management, The Hong Kong University of Science and Technology, HONG KONG· 2
Institute a/Systems Engineering, Northeastern University,P.R. CHINA
Abstract: Conventional scheduling theory and algorithms have not solved the practical needs of factory management. Because of the complexity of modem manufacturing systems and the fact that they operate in a stochastic and dynamic environment, dynamic scheduling is often required. Dynamic flowshop scheduling is frequently met in practical situations. In this paper we apply an adaptive optimization framework and develop a dynamic flowshop scheduling model with the objective of minimizing the mean of flowtime of jobs arriving as a Poisson process. Static heuristic algorithms are embedded in control model as scheduling controller. The most commonly used dispatching rule method is also implemented and compared with the new model through simulation. Analysis is done for different shop load levels and different numbers of machines. The results show that the new model performs significantly better than the dispatching rule method .. Copyright © 19991FAC. Keywords: dynamic flow shop, adaptive control model, static embedded algorithms
1. INTRODUCTION
Production scheduling can be defmed generally as the allocation of the resources in a production system over time to perform the operations needed to transform raw materials into products. An effective and efficient scheduling system is necessary to well achieve the potentials of a production facility. Production scheduling problems are extremely
complex. The complexity is mainly due to the following two features of the problem (Liu, 1995). Interconnected Decisions: The components of a production system, e.g., machines, material handling devices and storage buffers, are related to each other. The operations of and the constraints on one part of the system affect the other parts. Therefore, when a decision is to be made of which operation should be performed next, all available information should be considered including how this operation may affect
• The Project is Supported by HKUST Direct Allocation GrantDAG95196.E08, National Natural Science Foundation of China Through Approved No 79700006, and National 863/CIMS of China through Approved No. 863-511-708-009, t To whom correspondence should be addressed.
307
Copyright 1999 IF AC
ISBN: 0 08 043248 4
14th World Congress oflFAC
THE NEW ALGORITHMS FOR THE DYNAMIC FLOWSHOP SCHEDU ...
future operations in the whole production system. This is to say that production scheduling is an overall optimization problem for the whole system. Dynamic Process: A production system is a dynamic system. From time to time, new orders come and fmished products leave the system. Scheduling the activities in such a system is therefore a dynamic process in nature. As White (1990) indicated, the scheduling problem encountered in production is really one of rescheduling. Many practical situations are dynamic in nature, the great challenges from the increasingly competitive world market and the quickly changing customer needs faced by manufacturing companies require more effective and more flexible production scheduling and shop floor control strategies. Because the conditions and environments in dynamic and static scheduling problems are different, study on the scheduling methods which more suit to dynamic environment is important and challenging. Dynamic scheduling problems are often addressed using dispatching rule based approaches. These approaches emphasize the dynamic nature of the system. Simple dispatching rules such as First-InFirst-Out (FIFO) and Shortest Processing Time (SPT) are often used to each machine independently. Over a hundred such rules have been developed; classification and comparison of these rules have been made (Panwalkar and Iskander, 1977; Blackstone, et al., 1982). While these rules can handle dynamic system changes and are easy to implement, they have a major disadvantage of being myopic, i.e., a decision is made only based on the situation on a single machine. The effects of the decision on other parts of the system are not considered. Therefore the over,ll system performance is not optimized. Simulation methods have been used to improve the performance by trying a number of rules and selecting the best (Park, 1988). As the system becomes complex and more rules are involved, the computation time needed to perform a valid comparison of the rules may dissolve the dynamic merit of the dispatching rules. Artificial intelligence (AI) methods have also been used to improve the performance of rules (Pierreval, 1992). They try to incorporate expert and domain knowledge in selection of rules so that not all the rules need to be tried every time. The performance of these methods reply heavily on the quality of the knowledge incorporated which is not easy to obtain. To address both features of the dynamic scheduling problem, Liu (1995) proposed an adaptive optimization framework taking advantages of both above types of methods. In this framework static algorithms for global optimization run dynamically to make the schedule adaptive to production system changes. To suit the dynamic running, the static algorithms .must balance optimality and computation time very well. Trying to keep advantages of
optimization approaches but avoid their computation burden, good heuristic procedures need to be used. . In this paper we implement this framework for the flowshop dynamic scheduling problem. In this problem, all jobs are processed through m different machines in the same routing path. The arrival times of the jobs are unknown in advance and job arrivals follow a Poisson process. Once a job arrives, the processing times on all the machines of the job are assumed known. The objective is to minimize the mean job completion time". The next section describes the adaptive control model implemented. Section 3 reports the simulation study comparing this model with dispatching rule method. Section 4 concludes the paper.
2.
THE ADAPTIVE CONTROL MODEL AND ALGORITHMS
We describe here the dynamic scheduling model which embeds an static heuristic algorithm, taking its advantage for considering overall optimization, into an dynamic framework to handle the dynamic changes.
2.1. The overall structure of the model Control theory is a very useful tool to deal with dynamic performance of a system. A general feedback control system model is shown in Figure 1. Con1rol Requirem~t
+
Controller
Signal
Feedback Figllre I" A general control system model
However, the model cannot be applied directly to production scheduling problems since it is developed for continuously changing processes. In production scheduling problems, the system state at any time is a collect of states of all the components in the system at that time. We can defme the state of a machine or material handling device as whether it is performing a processing or transport operation, the state of a storage buffer as the number of workpieces being stored in it. Then it is obvious that the system state changes at only certain time points. These changes are referred to as events. Three types of events may be identified: (1) directly controlled events - those must be directly determined in the scheduling problem (start of operations); (2) expected events - those are expected to happen once the directly controlled events are scheduled (end of operations); (3) unpredictable events Those happen randomly (new jobs, machine breakdowns, expected events happen before
308
Copyright 1999 IF AC
ISBN: 0 08 043248 4
14th World Congress oflFAC
THE NEW ALGORITHMS FOR THE DYNAMIC FLOWSHOP SCHEDU ...
scheduled or delay). Based on these discussions, a d)mamic model for scheduling problem is implemented following the framework as shown in Figure 2. Unpred icta ble
The adaptive optimization framework,
Even\s
Figure 2. An implemented framework of the dynamic model for scheduling problem. The requirement in the model represents the production orders. The global scheduler is a static scheduling model or algorithm which solves the static scheduling problem from a global optimization point of view. Reactive controller monitors' the changes of requirement and system state, and trigs the running of the global scheduler accordingly. Because the static scheduler is used as a block of the dynamic model, it will be run very frequently. Therefore, it must satisfy the following two requirements. (1) The current state of the system must be taken into account along with the job requirement. (2) The computation time for generating a schedule must be short to suit the real-time dynamic environment.
2.2 Implementation
2.2.1. The reactive controller
This part is the core of the control mechanism. It works in the following way. The controller monitors the changes of requirement and system state and decides whether the global scheduler needs to be rerun. For expected changes over time, the controller simply keeps track of them. It trigs the re-run of the static scheduler only when unexpected changes happen such as new job arrivals and machine breakdowns. In our implementation here for the f1owshop problem, we do not consider machine breakdowns and focuses the normal changes, i.e. job arrivals.
2.2.2 The global scheduler For the static scheduling problem, heuristic algorithms taking a global view of the system should be applied as the global scheduler since they can give reasonably good solutions while taking relatively short computation time. When re-running the algorithm, the existing operations in the system need to be considered because they affect the availability of the system resources. This approach decomposes the dynamic problem into a series of static problems.
A static problem is generated at each occurrence of a non-deterministic even in system, then solve entirely, and the solution is implemented on a rolling horizon basis. In this procedure the entire system is considered at each instance of the static problem, in contrast to priority rules which consider only one machine at a time. Of course, when compared with priority rules, this approach requires greater computational effort, but also can lead to significantly better system performance. Taking into account the computing power available today, this cost seems to be worth to pay. For static flowshop scheduling problem with objective of minimizing mean completion time, Rajendran's algorithm (1993) is a very effective heuristic method. We use this static heuristic algorithm within the dynamic flowshop scheduling model in the following three ways. (l) Heuristic 1: A static problem is fonnulated at each occurrence of a new job arrival. The static algorithm is applied to the set of jobs consists of the newly arrived job and the jobs queuing on the fIrst machine. Those jobs which are being processed on all the machines and queued on other machines (not first machine) are not considered in set of sequencing jobs. Their effect on the current static problem is not considered. When trrst machine become available, the job of first position in the sequence generated from the static algorithm is chosen for processing. In order to hold permutation sequence of the static algorithm, machlnes following the fust machine select jobs according to the FIFO rule. (2) Heuristic 2: Analogous to heuristic 1. Heuristic 2 differs from heuristic 1 in that it considers the state of in-processing jobs. The set of jobs considered consist of the newly arrived job and the jobs which are being processed and which are queuing on the machines. The munber of existing jobs is asswned nl. Their current sequence is regarded as the subsequence of the first nl jobs in sequence of static scheduling problem. The static scheduling problem is the problem in which sequence the fIrst r11 jobs is known. For each of these fIrst nt jobs in the static problem, the processing times on the machines which have already processed the job are zero, the processing time on the machine which is processing the job is the remaining processing time on this machine, the processing time on the machines which will process the job are the original processing times on these machines. When the first machine is freed, the job in the tITst position of the sequence generated from static algorithm is chosen. In order to hold permutation sequence of static algorithms, the machines after the fIrst machine select jobs according to the FIFO rule. (3) Heuristic 3: Analogous to heuristic 2. Heuristic 3 differs from heuristic 2 in that it is implemented on a rolIing horizon basis. In this procedure the static scheduling problem is solved after each preset time period to schedule the new jobs arrived in the period. Heuristic 3 is applied only
309
Copyright 1999 IFAC
ISBN: 0 08 043248 4
THE NEW ALGORITHMS FOR THE DYNAMIC FLOWSHOP SCHEDU. ..
when the dynamic system reaches steady state. Before system reaches steady state, heuristic 2 is used in simulation.
3. COMPUTATIONAL SIMULATION The simulation model was developed using C Language. The commonly used dispatching rule FIFO was used as a benchmark to compare with the new framework and the algorithms embedded.
14th World Congress ofIFAC
By continuous increasing the number of jobs procegsed, it was determined that analytical steadystate level were reached at around the completion of 2000 jobs which corresponded to about 10000 jobs in a whole simulation run. Each simulation replication was performed as fonows: Simulation size (jobs): 10000 Steady state (jobs): > 2000 Non-steady state (jobs) < 2000
3.2 Computational results
3.1 Related parameter setting 3.1.1
Job arrival pattern
A Poisson arrival model is used in simulating job arrivals. This distribution provides a good approximation for the job arrival times if the generating sources can be assumed to be independent. The arrival rate of the jobs is assumed to be equal to the ratio of the capacity of the shop to the average amount of work required. The arrival rate, denoted by A, which is derived from the inter-arrival time information is then given as A,= "lIP
where P = average operation time. T! = machine load capacity utilization. Various arrival rates can be obtained by adjusting the machine load capacity factor 17. By changing the arrival rate, we can control the degree of congestion in the shop, each set of simulations was carried out until a total of 10000 jobs had arrived and been scheduled and completed.
3.1.2
Because none of the all algorithms can guarantee optimal solution to the problem, we cannot find an absolute measurement for the optimality performance of the algorithms. Therefore, we used a relative optimality measurement C/C*, where C* is the best among the results for a problem instance given by the four methods and C is the result for the instance given by the method being evaluated. All the four methods were used to solve the problem in all the scenarios described above. The average optimality performances of these methods against different problem structures and against different problem sizes are shown in tables 1, and 2 respectively. Tables 3 and 4 show the average running times of the algorithms against different problem structures and problem sizes, respectively. All the times shown are in seconds on aPentium PC. Table J Ayerage optimality performances with djfferent shpp load levels
0.8 0.85 0.90 0.95
Shop loading level
The shop load level, repregenting the utilization of the facility, has probably a greater effect on system performance than the job arrival pattern. The system was analyzed under four different shop load level: 80%, 85%, 90% and 95%. These shop load levels are conunonly used in dynamic scheduling simulation.
3.1.3
Iahl!i: 2
HI 1.0508 1.0767 1.1391 1.2536
A~[illi:!i:
H2 1.0764 1.1383 1.1929 1.4290
H3 Ll056 1.1323 1.1867 1.4214
QPtimality Il~IfQ[IDWl!;;~ ll:ith
different number Qima!;;bines
4
Processing time range and distribution
6 8 10
Problem scenarios of different sizes and structures were generated for the experiment. The processing times were randomly generated from uniform distributions. The range of processing times was set to be from I to 50 for all problem instances. The number of machines was chosen at four levels: 4, 6, 8 and 10. Considering both shop load level and number of machines, we had sixteen different problem gcenarios. Ten replications for each scenario were run. Totally, 160 replications were run in the simulation.
3.1.4
FIFO 1.1690 1.2252 1.3050 1.6073
FIFO 1.3801 1.3503 1.3098 1.2663
Iabl!:: 3 AYe:(agl:
0.8 0.85 0.90 0.95
FIFO 1.925 2.275 2.850 4.725
HI 1.1048 1.1295 1.1629 1.1230
H2
1.2225 1.2339 1.1785 1.2018
H3 1.2163 1.2238 1.2173 1.1885
~omputation tiwl:s with different shop IQad l~els
HI 9.300 17.575 53.650 294.450
H2 5.125 8.950 16.825 62.100
H3 3.000 4.675 10.800 51.525
Steady state control
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THE NEW ALGORITHMS FOR THE DYNAMIC FLOWSHOP SCHEDU. ..
Table 4 Average computation times of algorithms with different machjne sizes FIFO
4 6 8 10
1.425
2.250 3.425 4.675
HI 30.825 55.200 131.450 157.500
H2 14.275 14.625 26.300 37.800
H3 10.075 13.125 24.500 22.300
3.3 Discussion and analysis ofresults From the figures in the above tables, we can see that: (1) For dynamic flowshop environment, the performances of the proposed three heuristic algoritlnns which are embedded in the adaptive control model are better than that of the FIFO dispatching rule. (2) The test on shop load level showed the mean flow time at 80% loading level was significantly smaller than the mean flow time at 95% loading level. This is to be expected since higher utilization levels can only be achieved by more jobs waiting to be processed at each machine; consequently the time a job spends in the system is higher. (3) The heuristics and the dispatching rule are less affected by the machine size. (4) Although proposed three algorithms take more time than FIFO, they are useful for on-line scheduling where there are enough available time to produce schedules.
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REFERENCES
Blackstone, J. H. Jr., D. T., Philiphs, and G. L. Hogg (1982). A state-of-the-art survey of dispatching rules for manufacturing job shop operations, International Journal of Production Research, 20,27-45. Liu, J., (1995). An adaptive optimization model for scheduling in dynamic environments, Proceedings of the 13th International Conference of Production Research, Jeraslum, 612-614. Panwa\kar, S. S. and W. Iskander (1977). A survey of scheduling, Operations Research, 25, 45-61. Park Y. B. (1988). An evaluation of static flowshop scheduling heuristics in dynamic flowshop model via a computer simulation. Computers & Industrial Engineering, 14, 103-112. Pierreval, H. (1992). Expert system for selecting priority rules in flexible manufacturing systems, Expert Systems with Applications,S, 51-57. Rajendran, C. (1993). Heuristic algorithm for scheduling in a flowshop to minimize total fiowtime, International Journal of Production Economics, 29, 65-93. White, Jr., K. P. (1990). Advances in the theory and practice of production scheduling, Control and Dynamic Systems, 37,115-157.
4. CONCLUSIONS
In this paper we have developed an adaptive control system model for dynamic scheduling problem with the performance measure ofminirnizing the mean flowtime. Three static heuristic algorithms are used as the static scheduler embedded in the dynamic control system model. The performance of the model and the algorithms were compared with the FIFO dispatching rule through simulation. For all scenarios with different shop loading level and different number of machines simulated, the proposed three heuristic algorithms performed superior to the benchmark FIFO dispatching rule method.
311
Copyright 1999 IF AC
ISBN: 008 0432484
THE NEW ALGORITHMS FOR THE DYNAMIC FLOWSHOP SCHEDU. ..
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Control Signal
Requirement
Response
------~~--iController f---=--~
+ Feedback Figure 1. A general control system model
:, The adaptive optimization framework:I
Unpredictable Events ,-------1-------,
Re
System state
I I -
- - - - - - - - - - - - - - - - - - - - - - - - ______ 1
Feedback -
-------~--~~---------------.
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Copyright 1999 IF AC
ISBN: 008 0432484