PRODUCTION PLANNING & CONTROL, 2002, VOL. 13, NO. 1, 56–65
Performance evaluation of re-entrant lines with multi-class jobs and multi-server workstations YOUNGSHIN PARK, SOOYOUNG KIM, and CHI-HYUCK JUN
Keywords re-entrant line, mean value analysis, multi-server workstations, batch machines Abstract. An approximation method is proposed for estimating the performance measures of re-entrant lines with multiserver workstations based on the Mean Value Analysis (MVA) technique. The system of interest can be found in the semiconductor wafer fabrication line in which several circuit types are manufactured through re-entrant processes at both the single-job and the batch workstations. Each workstation may have several identical machines. Multi-class jobs are assumed to be processed in a predetermined routing, in which some processes may utilize the same workstation several times in the re-entrant fashion. The performance measures of interest are the steady-state average of the cycle time of each job class, the queue length of each bu¶ er, and the throughput of the system. The system may not be modelled by a product form queueing network due to the inclusion of batch machines, multi-class jobs with di¶ erent processing times, and multi-server workstations.
Thus, a methodology is proposed for analysing such a reentrant line approximately using the iterative procedures based upon the MVA and some heuristic adjustments. Results of numerical tests are provided to show the performance of the proposed approach against the simulation results.
1. Introduction In this paper, an approximate approach is proposed for analysing the performance of the re-entrant line with multi-server workstations and multi-class jobs using the mean value analysis (MVA) technique. In the re-entrant line, jobs may visit a certain workstation more than once at di¶ erent stages of processing. Each workstation may consist of several identical machines, which is therefore called a multi-server workstation. Two types of machines are considered: single-job machines and batch machines.
Authors: Youngshin Park, Sooyoung Kim, and Chi-Hyuck Jun, Division of Mechanical and Industrial Engineering, Pohang University of Science and Technology, San 31 Hyojadong, Pohang, 790-784, Korea (South), E-mail:
[email protected],
[email protected] and
[email protected]. YoungshinPark is a post-doctoral researcher of Industrial Engineering Department at Pohang University of Science and Technology in Korea. She received BS, MS and PhD degrees in Industrial Engineering from Pohang University of Science and Technology. Her research interests are in performance analysis of production systems and various issues in supply chain management.
Sooyoung Kim is Associate Professor of the department of Industrial Engineering at Pohang University of Science and Technology (‘POSTECH’) in Korea. He received his BS in Mechanical Engineering from Seoul National University, MS in Manufacturing Engineering from Korea Advanced Institute of Science and Technology, and his PhD in IE from University of California at Berkeley, USA in 1988. From 1989 to 1993, he was Assistant Professor of Industrial Engineering at Rutgers University in New Jersey, USA. His research interests are in production planning, scheduling and control, especially in semiconductor manufacturing processes.
Production Planning & Control ISSN 0953–7287 print/ISSN 1366–5871 online # 2002 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/0953728011006158 4
Performance evaluation of re-entrant lines
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Chi-Hyuck Jun is head and professor in Industrial Engineering Department at Pohang University of Science and Technology. He received BS in Mineral and Petroleum Engineering from Seoul National University in 1977, MS in Industrial Engineering from Korea Advanced Institute of Science and Technology in 1979 and PhD in Operations Research from University of California, Berkeley, in 1986. He is interested in performance analysis of communication systems and production systems. He has published in Probability in the Engineering and Informational Sciences, Microelectronics & Reliability, IEICE Transactions on Communications, etc.
A single-job machine processes one job at a time and a batch machine processes several jobs as a batch at a time. The number of jobs to be processed together is referred to as the batch size. A single-job machine can be regarded as a batch machine with the batch size of one. The workstations having one or more single-job machines are called the single-job workstations and the workstations having one or more batch machines are called the batch workstations. Each workstation is assumed to have only one type of machines, either single-job or batch machines. A well-known example of the re-entrant manufacturing system is the semiconductor manufacturing process, especially wafer fabrication line (‘fab’). In the semiconductor fab line, almost every workstation has a set of identical machines, where the photolithograph y workstation is a typical example. As examples of batch processing, the deposition process or the furnaces wait for jobs in the queue until the number of jobs reaches the predetermined batch size and then process jobs together as a batch of predetermined size. Mixture of these batch workstations and single-job workstations complicates the exact analysis of the system and moreover the ow of the multi-class jobs also requires an elaborate model for analysis. Unfortunately, such a system does not lend itself to be modelled as a product-form queueing network, and thus it is often analysed using simulation in practice. However, the simulation of a large-scale system, such as a fab line producing the commodity semiconductor devices, often requires excessive time for modelling and running it. Managers and engineers who are interested in nding a quick answer for the system performance under certain changes in product mix or system con guration often need an alternative method. In lieu of the simulation, an approximate MVA approach is presented to analyse the system based on the works of Narahari and Khan (1996, 1998) and Park et al. (2000a, 2000b). Narahari and Khan (1996) rst proposed the application of MVA to the re-entrant line with a single-class job and single-job machines (i.e., no batch machines considered). Park et al. (2000a, 2000b) extended the approach and solved the cases of the single-class job and the multiclass jobs with mixture of both single-job and batch
machines. However, all the models in the previous works assume the single-server workstations where each workstation holds only one machine. Such an assumption of the single-server workstation limits the applicability of the proposed approaches to the real-world problems. Thus, the previous research is extended to consider the case of the multi-class jobs and the multi-server workstations. Even for the case of single-class jobs, re-entrant ow with batch machines makes the system impossible to be modelled as a product-form queueing network. Extending the case to include the multi-class jobs and multi-server workstations further complicates the model, and thus an approximate analysis technique is proposed in this paper. The system is assumed to be a closed network, in which the total work-in-process is kept constant. The detailed description of the problem is to be given in the following section. Re-entrant lines are explained in Kumar (1993) in detail, and the research on scheduling problems and stability of re-entrant lines with single class of jobs are presented in Lu and Kumar (1991), Kumar (1995), Kumar and Meyn (1995) and Dai and Weiss (1996). Connors et al. (1996) proposed an approximate queueing model for semiconductor manufacturing systems with reentrant characteristics and analysed using decomposition based approach. Dai et al. (1997) proposed the QNET method to nd the performance measures of re-entrant lines with single-class jobs. Regarding the batch processing, Chaudhry and Templeton (1983) summarized the general theory concerning batch arrivals and batch service queues, and Neuts (1967), Powell (1986) studied several rules and control policies of batch processing. Chaudhry et al. (1992), Wagner (1997), Laevens and Bruneel (1995, 1998) and Kao and Wilson (1999) analysed the multi-server systems with various environments using the analytical method such as Markov model. MVA is used in Reiser and Lavenberg (1980), Suri and Hildebrant (1984), Shalev-Oren et al. (1985) and Schmidt (1997) in order to analyse the multi-server systems. The approaches using MVA except Schmidt (1997) used the method of dividing the waiting time in a queue by the number of machines to extend the single server case to the multi-server case. Schmidt (1997) proposed the concept of inter-departure time of jobs in a
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multi-server workstation. This method of inter-departure time of jobs is adopted to the analysis for the multi-server workstations. In the following section, the problem and the assumptions are explained with an example. The proposed approach is described in section 2. In section 2.1, the method of obtaining the average waiting time of jobs at each bu¶ er of the workstations is explained. Section 2.2 presents the procedure for computing the mean cycle time and the other measures of the system. To enhance the performance of the proposed MVA method, some heuristic adjustment methods are added in section 2.3. Section 3 presents the test results obtained by comparing the results from the proposed approach and those of the simulation experiments for some sample cases. Finally, the summary of the paper and discussions of the further research issues are given.
2. Model and MVA approach Multi-class jobs are assumed to be owing through a deterministic route and the routes can be di¶ erent for di¶ erent job classes. Jobs may visit the same workstation more than once at di¶ erent stages of processing, thus creating a re-entrant ow. Each workstation has several queues, each representing di¶ erent stage for di¶ erent job class, and a separate bu¶ er is assumed per each queue. Each bu¶ er is shared among all the machines in the workstation. In case of the batch workstations, a batch is assumed to consist of jobs from only one bu¶ er (i.e., no mixture of di¶ erent bu¶ ers). The batch size is assumed to be xed and it is assumed that all jobs have to be processed in full batch size. In practice, this assumption is true because a batch machine normally has a big bu¶ er in front of it as we observed at most of the batch machines in a semiconductor manufacturing system. The processing time of a job of class j visiting workstation i on its lth visit is assumed as an independent exponentially distributed random variable. This assumption is not necessary for the MVA analysis, but is used for simulation purpose. Figure 1 is an example for such a system, which will be used later for numerical experiments in section 3. It depicts a system that has 2 classes of jobs and 3 workstations each of which may have multiple single-job or batch machines. Each job enters workstation 1, ows in a predetermined route as depicted by arrows, and nally exits from workstation 3. Dispatching or scheduling policies to decide which job to process next when a processing machine becomes available are factors that may have a signi cant e¶ ect on the performance of re-entrant lines. In the real world, there are many di¶ erent kinds of queueing policies utilized. However, when one applies the MVA to a com-
Workstation 1
Workstation 2
Workstation 3
Figure 1. Example system of re-entrant line.
plicated system, it is quite diµ cult to consider the sophisticated rules. For example, the LBFS (Last Bu¶ er First Served Rule) analysed in Narahari and Khan (1996) was easy to be modelled in the single job-class case, but not in the present case. There are a few bu¶ er-priority scheduling policies having proven stability as in Lu and Kumar (1991) for applying to the multi-class cases. The policies can make unique ordering of bu¶ ers only if there is the priority of classes. In the present approach, the simple First Come First Served (FCFS) policy is assumed with no consideration of the bu¶ er priorities as in Park et al. (2000b) . However, if the ordering of bu¶ ers is predetermined, this approach can be applied to that ordering easily by considering what the higher priority bu¶ ers than current bu¶ er’s are. The FCFS policy is assumed in this paper because it is a basic and easy rule and generally used in practical environment. The setup time that may be needed when changing the class of job at a workstation is also ignored. Workstation s are assumed to be free of failures or disturbances. The original MVA is an iterative method for analysing a system in steady-stat e based on the arrival theorem (Reiser and Lavenberg 1980) and Little’s law (Little, 1961). These principles hold when the system can be described as a closed and product-form queueing network. Although the present system is not this case, the MVA is adopted as an approximation methodology. Using these principles, the performance measures of system when there are k ‡ 1 jobs can be evaluated by the value of those when there are k jobs. The performance measures to be considered are the mean cycle time, mean queue length of the bu¶ er, and the steady-state throughput rate for a given amount of WIP (Work In Process) in the system. In the following approach, the term ‘cycle time’ is used for representing the total time spent for a job from the beginning of the process to the end. For the time taken by
Performance evaluation of re-entrant lines a job from entering a bu¶ er at a workstation until it leaves the station, we use the term ‘waiting time’. First the mean waiting time at each bu¶ er is computed and then the mean cycle time in the system is computed. Using the mean cycle time in the system, the throughput rate of system and the queue length at each bu¶ er are obtained. In the following two subsections, how the MVA procedure is applied for calculating the mean waiting time, the mean cycle time and the throughput is explained. The third subsection then explains how the results from the MVA calculation steps can be modi ed to reduce the errors. In the last subsection, the overall calculation steps are summarized. The notations used are as follows: i j k l M P N vij rj
index for workstation, i ˆ 1; . . . ; M index for job class, j ˆ 1; . . . ; P amount of WIP in system, k ˆ 1; . . . ; N index for visit count, l ˆ 1; . . . ; vij number of workstations in the system number of job classes in the system maximum number of WIP to be considered number of visits of class j jobs to workstation i proportion of class j jobs in the system, P X j ˆ 1; . . . ; P; rj ˆ 1 jˆ1
kj number of class j WIP when the number of total P X WIP in the system is k, kj ˆ k jˆ1
bijl bu¶ er at workstation i with class j jobs of lth visit 1=·ij mean processing time of a job (at single-job machine) or a batch (at batch machine) in buffer bijl (i.e., ·ijl is the mean processing rate) Bi batch size at workstation i P X ni number of bu¶ ers at workstation i, ¸ij ˆ ni jˆ1 s span of moving average ci number of machines at workstation i …k† Lijl mean number of jobs in bijl when the system has k jobs Wijl …k† mean waiting time of a job in bijl when the system has k jobs Wj …k† mean cycle time of a class j job when the system has k jobs ¶j …k† throughput rate of class j when the system has k jobs 1 HPijl number of batches already being made when a job arrives at bu¶ er bijl of the batch workstation 2 HPijl number of batches made during the time until the arriving job is included in a batch when the job is at the bu¶ er bijl of the batch workstation
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1 Wijl processing time of the jobs already being made in the batches when a job arrives at the bu¶ er bijl of the batch workstation 2 Wijl time that the arriving job should wait until it is included in a batch when the job is at the bu¶ er bijl of the batch workstation RBijl number of remaining jobs in bu¶ er bijl that is not formed in the batch when the job arrives at the bu¶ er of the batch workstation WBijl time that an arriving job should wait until it is formed in a batch when a job is at the bu¶ er bijl of the batch workstation
2.1. Mean waiting time at each bu· er In the following steps, the mean waiting time of each class of job at each bu¶ er is calculated. The case of singlejob workstation is considered rst and then the case of batch workstation explained. When a job arrives at some bu¶ er in a single-job workstation, the job waits until all the jobs already in the bu¶ ers of the same workstation are processed due to the FCFS policy. In bu¶ er bijl , an arriving job when there are total k jobs in the system, according to the arrival theorem, would see L ixy …k ¡ 1† jobs in the bu¶ ers bixy ; x ˆ 1; . . . ; P; y ˆ 1; . . . ; vix . Under the FCFS policy, the number of jobs P with P ixhigher priority than that of the arriving job is Pxˆ1 vyˆ1 Lixy …k ¡ 1†. If the number of jobs is smaller than the number of machines in the PP Pvix …k ¡ workstation, i.e., L 1† < ci , the mean xˆ1 yˆ1 ixy waiting time of the arriving job is the processing time of itself. PP Pvix Otherwise, if ci , the job xˆ1 yˆ1 Lixy …k ¡ 1† waits until one of the machines becomes idle. The number jobs that have to be processed is P P of ix … Pxˆ1 vyˆ1 Lixy …k ¡ 1† ¡ ci ‡ 1† until one of the machines becomes idle. If the number of machines is 1, the waiting of the arriving job is the processing time P Ptime ix of … Pxˆ1 vyˆ1 Lixy …k ¡ 1† ¡ ci ‡ 1† jobs. When ci > 1, however, the waiting time should be evaluated di¶ erently. The interdeparture time is utilized that is presented in Schmidt (1997). The interdeparture time is the time interval between successive job completion times in multi-server workstation. The mean interdeparture time is obtained from the processing time of all higher priority jobs divided by the number of machines and by the number of higher priority jobs. Then the mean interdeparture time of single server workstation is the mean processing time of the higher priority jobs. Thus, the mean waiting time of a job in bijl which visits a single-job workstation is as follows:
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8 ¸ix P X X > 1 > > if Lixy …k ¡ 1† < ci > > ·ijl > > xˆ1 yˆ1 > > > > ¸ix P X X > 1 > > > Lixy …k ¡ 1† > > · > ijl 1 xˆ1 yˆ1 > > > ‡ …1† > ¸ix P X < ·ijl X ci Lixy …k ¡ 1† Wijl …k† ˆ > > xˆ1 yˆ1 > > Á ! > > ¸ix P X > X > > > Lixy …k ¡ 1† ¡ ci ‡ 1 > > > > xˆ1 yˆ1 > > > > ¸ix P X X > > > > if Lixy …k ¡ 1† ci :
1 HPijl
…2†
Bi
xˆ1 yˆ1
While the arriving job waits until it is formed in a batch, some jobs may arrive at the other bu¶ ers and they can be made in batches. These batches have also higher priority. To nd how many they are, consider the number of remaining jobs in bu¶ er bijl that is not formed in batches, RBijl , and the time that an arriving job should wait until it is formed in a batch, WBijl . Then, RBijl ˆ L ijl …k ¡ 1† ¡
xˆ1 yˆ1
In case of the batch workstation, the work unit is batch, not job. Thus, it is necessary to consider the number of batches for this case. When a job arrives at a bu¶ er of batch workstation, the job may see some batches and some jobs that have not been formed in batches at the bu¶ ers. For example, consider the case in gure 2. It represents a batch workstation with one machine having 3 bu¶ ers. Let the batch size be 2. The left bu¶ er has 3 jobs, the middle one has 2, and the right one has 1. When a new job – black one – arrives at the middle bu¶ er, it rst waits during the processing time of two batches (one in the left bu¶ er and the other in the middle). Then, this job waits until another job arrives at the middle bu¶ er. If one job arrives at the right bu¶ er beforehand, for example, then the two jobs at that bu¶ er become a batch and it has a higher priority than the black one under the FCFS rule. That is, under the FCFS rule, a batch formed rst gets the service rst. Also, the arrival time of a batch is that of the last job that forms the batch. To compute the mean waiting time at a batch workstation, rst consider the number of the higher priority batches that a job sees on its arrival at bu¶ er bijl , which is given by
ˆ
¸ ¸ix · P X X Lixy …k ¡ 1†
WBijl ˆ
·
Lijl …k ¡ 1† Bi
¸
Bi
max‰0; Bi ¡ …RBijl ‡ 1†Š ¶j …k ¡ 1†
…3† …4†
As a result, the number of the higher priority batches is as follows: 2 HP ijl
¸ ¸ix · P X X WBijl ¶x …k ¡ 1† ‡ RBixy ˆ Bi xˆ1 yˆ1
…5†
1 2 The processing times of HPijl and HPijl are denoted by 1 2 Wijl and Wijl , respectively. They are calculated as follows:
1 ˆ Wijl
2 Wijl
¸ ¸ix · P X X L ixy …k ¡ 1† xˆ1 yˆ1
Bi
1 ·ixy
…6†
¸ ¸ix · P X X WB ijl ¶x …k ¡ 1† ‡ RBixy 1 ˆ …7† B · i ixy xˆ1 yˆ1
The total processing time of higher priority batches dur1 2 ‡ Wijl ing which the arriving job should wait is Wijl , and the total number of higher priority batches seen by the 1 2 ‡ HPijl arriving job is HPijl . Thus, the mean waiting time of an arriving job in bijl at a batch workstation is as follows:
Arriving Job
Batch Workstation Figure 2. A batch workstation with 3 bu¶ ers.
Performance evaluation of re-entrant lines 8 1 1 2 > > ‡ HPijl if HP ijl < ci > > · > ijl > > > > < 1 Wijl1 ‡ Wijl2 ‡ ‡ WB ijl …k† ˆ W ijl 2 1 ‡ ·ijl † ci …HP ijl HPijl > > > > 1 2 > > …HPijl ‡ HPijl ci ‡ 1† > > > : 1 2 ‡ HPijl if HPijl ci
…8†
In case of Bi ˆ 1, equation (8) is reduced to equation (1) of single-job workstation. It can be noted that the arrival rate is involved in calculating WBijl . Since the initial values for the arrival rate and the queue length 2 are all zeros (i.e., the system is empty), Wijl cannot be 2 obtained by an iterative method. Further, Wijl becomes unreasonably large when the arrival rate and k are small. The waiting time for forming a batch can be even in nite if k is less than the minimum batch size. Thus, it is proposed to use 0 for WBijl (i.e., ignore the batch-forming 2 time) for calculating Wijl when k is less than the minimum batch size. Since the only interest is the case where there is a relatively large number of WIP in the system, it is necessary to propose a correction scheme to reduce the error in the nal result from this assumption.
2.2. Mean cycle time of each class The mean cycle time of a job in class j is the sum of the waiting times in bu¶ ers over all workstations and visits in the entire system. Therefore, it is obtained as below when there are k jobs in the system. Wj …k† ˆ
¸ij M X X iˆ1 lˆ1
Wijl …k†;
j ˆ 1; . . . ; P
…9†
If the mean cycle time of each class in the system is computed, the arrival rate or the throughput rate of each class as well as the mean queue length of each bu¶ er is obtained by applying Little’s Law (Little 1961) as follows: ¶j …k† ˆ
kj W j …k†
L ijl …k† ˆ ¶j …k†Wijl …k†
…10† …11†
Note that since the arrival rate of the system should be same as the throughput rate of the system for a closed network, it remains same for all the workstation s and for all the bu¶ ers. The initial conditions of iteration are:
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Lijl …0† ˆ 0;
W ijl …0† ˆ 0;
l ˆ 1; . . . ; ¸ij ;
j ˆ 1; . . . ; P; i ˆ 1; . . . ; M ¶j …0† ˆ 0;
j ˆ 1; . . . ; P
Using the initial conditions above and the recursive relationships of Wijl …k†, ¶j …k†, Lijl …k†, the mean cycle time of each class in the system, the average queue length of the bu¶ er and the throughpu t rate of each class for k ˆ 1; 2; . . . ; N, where N denotes the speci ed total WIP of the system can be computed. Wj …N ) and ¶j …N† can be nally obtained using equations (9) and (10), respectively.
2.3. Modi cations of mean cycle time – use of smoothing and cuto· As mentioned earlier, the MVA is employed as an approximation technique to analyse a non-product form network. Therefore, some error is inevitable in computing the mean waiting times, and the use of recursion for the evaluation of the performance measures may increase the error. Moreover, it has been observed that the performance value calculated uctuates during the iterations, which is not a desirable pattern. Park et al. (2000a, 2000b) proposed the following modi cation scheme, and the same technique is modi ed as follows. In order to solve a uctuation problem rst, an adjustment of the MVA calculations is proposed using a moving-average smoothing technique. With the moving span size of s, revising Wj …k† as follows is suggested: new ¡ Wj …k† ˆ
1 s
k X
lˆk¡s‡1
Wj …l†;
j ˆ 1; . . . ; P
…12†
If the system has batch machines and 0 for the initial WBijl is assumed, especially for small k, the mean cycle time obtained by equation (12) might be wrong. Therefore it is more reasonable to ignore these values for relatively small k until some ‘cuto¶ point’ is reached, and to take the values and apply smoothing only after this cuto¶ point. From the experimental results, the following cuto¶ point, K, seems to work properly: K ˆ P max j Kj Kj ˆ …PB j ‡ s†
…13† 2;
j ˆ 1; . . . ; P
…14†
where PB j is the number of bu¶ ers for class j jobs at batch workstations. As Kj is the value for job class j, cuto¶ point of the system is given by the number of classes multiplied by the maximum value of Kj .
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Finally, the mean cycle time and the throughput rate of each class of jobs in the system with the WIP of N should be obtained as follows: mean cycle time of class j jobs ˆ new ¡ W2 …N†
…15†
N rj new ¡ Wj …N†
…16†
throughput rate of class j jobs ˆ
The overall steps to calculate the mean cycle time and the throughput rate are now summarized: Step 0. Set the values of the variables needed. M; P; ¸ij ; N; s; rj ;
1 ; Bi ; ci ; l ˆ 1; . . . ; ¸ij ; ·ijl
j ˆ 1; . . . ; P; i ˆ 1; . . . ; M Step 1. Initialize variables and determine the cuto¶ point K. Lijl …0† ˆ 0; Wijl …0† ˆ 0; l ˆ 1; . . . ; ¸ij ; j ˆ 1; . . . ; P; i ˆ 1; . . . ; M ¶j …0† ˆ 0; j ˆ 1; . . . ; P kj ˆ k r j kˆ0 Step 2. Increase k by 1. Step 3. Compute Wijl …k† using equations (1) and (8) and compute Wj …k† using equation (9). Step 4. If k is larger than K of equation (13), modify Wj …k† using equation (12) and set Wj …k† ˆ new ¡ Wj …k†. Step 5. Compute ¶j …k† and Lijl …k† using equation (10) and Eq.(11). Step 6. If k is N , go to step 7, else go to step 2. Step 7. Find the nal mean cycle time using equation (15) and the nal throughput rate using equation (16).
jCycle time from MVA†¡ …Cycle time from simulation†j ARE ˆ Cycle time from simulation …17†
100…%†
3.1. System I System I has 3 workstations and 2 classes of jobs. The mean processing times of each class on di¶ erent visits are shown in table 1. The same ratio of class 1 and class 2 is assumed, that is, r1 ˆ r2 ˆ 0:5. The processing times of System I are exponentially distributed with means in table 1. Two cases are considered in this system. Case 1-1 assumes that all workstations are single-job workstations and Case 1-2 considers that Workstation 1 and Workstation 3 are single-job workstations and Workstation 2 is batch workstation. The batch size of Workstation 2 is 3 and the number in parenthesis of table 1 is the mean processing time of a batch machine. The value 5 is used for s, the span of moving average, and 200 for N. Table 2 shows the number of machines in each workstation of System I. Table 3 shows the comparison results with simulation in terms of ARE. The AREs of Case 1-1 are about 1%. This means that the values of mean cycle time computed Table 1. Mean processing time of System I. Class 1
Class 2
1st visit 2nd visit 1st visit 2nd visit 3rd visit Workstation 1 0.7 0.5 Workstation 2 1.2(2.7) 0.6(1.5) Workstation 3 0.9
1 0.5(1) 1
0.7 0.9(1.8) 0.5
0.9 1.2(2)
Note: the number in parenthesis is mean processing time for a batch machine.
Table 2. Number of machines in System I.
3. Numerical experiments
Number of machines
To see how closely the proposed approach can estimate the performance measures compared to the actual performance of the re-entrant lines, two example systems in gure 1 (System I) and gure 4 (System II) were tested. In the following two subsections are presented the results of the computational tests for two systems, comparing the results of the proposed approach against the simulation results. A commercial software package named ‘Simul 8’ was used for simulation. As a measure for comparison between the MVA and simulation, the following measure of ‘absolute relative error’ (ARE) was used.
Workstation 1 Workstation 2 Workstation 3
2 3 1
Table 3. AREs in System I. Cycle time
Throughput rate
Case
Class 1
Class 2
Class 1
Class 2
Case1-1 Case1-2
0.49% 3.59%
1.27% 1.89%
0.21% 1.56%
0.97% 2.83%
Performance evaluation of re-entrant lines Cycle time of Case 1- 2
class 2
class 1
350
63 class 3
class 4
300
Workstation 1
Cycle time
250 C1-MVA
200
C1-Simul. C2-MVA
150
Workstation 2
C2-Simul.
100
Workstation 3
50 0 30
50
70
90
110
130
150
170
190
Workstation 4
No. of Jobs
Figure 3. Cycle times of Case 1-2. Workstation 5 Throughput rate of Case 1- 2
Workstation 6
0.7 0.6
Workstation 7
Throughput
0.5 C1-MVA
0.4
C1-Simul C2-MVA
0.3
C2-Simul.
Workstation 8
0.2 0.1
Workstation 9
0 30
50
70
90
110
130
150
170
190
No. of Jobs
Workstation 10
Figure 4. Throughput rates of Case 1-2.
by the proposed method are close to the results of the simulation. The ARE of Class 1 in Case 1-2 is 3.59%, which is worse than Case 1-1 but still a reasonable approximation error. Figure 3 shows the cycle times and gure 4 shows the throughput rates according to the number of jobs in Case 1-2. These graphs show the closeness of the results of MVA and the results of the simulation.
3.2. System II System II of gure 5 is a larger example than System I with 10 workstations and 4 classes of jobs. The case is adopted from Park et al. (2000b), and modi ed to allow the multi-server workstations. N ˆ 1500 is assumed, and rj chosen according to the proportion of the bu¶ ers of class j in the system. Total number of bu¶ ers of System II is 72 and the numbers of bu¶ ers of class 1, class 2, class 3 and class 4 are 11, 21, 15 and 25, respectively. Thus, r1 ˆ 11=72 ˆ 0:1528, r2 ˆ 0:2917, r3 ˆ 0:2083 and r4 ˆ 0:3472. Again, 5 is used for the value of s.
Figure 5. System II : 10 workstations and 4 classes of jobs.
Table 4 shows the characteristics of System II. Column 2 in table 4 shows the processing times when the workstations have single-job machines. The processing times of the batch workstations are assumed to take the value of column 2 multiplied by the batch sizes. The third column of table 4 shows the number of machines in each workstation. The number of machines is arranged by considering the processing times. Three cases are tested on System II with changing the number of batch workstations, and table 5 shows the arrangement of batch workstations. Table 6 shows the comparison results with simulation in terms of ARE. The AREs except for the results of job class 1 are not so di¶ erent from the AREs of System I. For job class 1, AREs are relatively big, and it is because its relative volume owing through the system is small. We observe that as the total WIP gets smaller for a job class it is more likely for the proposed MVA approximatio n to have the larger error. Especially, as the number of batch workstations is increasing, such tendency is obvious as shown in the result of Case 2-3. This is due to the fact
Y. Park et al.
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Table 4. Characteristics of System II. No. of visits for each class
Workstation Workstation Workstation Workstation Workstation Workstation Workstation Workstation Workstation Workstation
1 2 3 4 5 6 7 8 9 10
Proc. time
No. of m/c
Class 1
Class 2
Class 3
Class 4
0.5 0.7 1.2 3 0.8 0.5 1 1.7 0.3 0.9
1 4 3 5 2 2 4 2 1 1
1 2 2 1 1 1 1 0 1 1
2 3 2 3 3 2 2 2 1 1
2 3 2 1 1 1 2 1 1 1
3 4 3 2 4 3 2 2 1 1
Table 5. Arrangement of batch workstations in System II. No. of batch workstation
Arrangement of batch workstation (batch size)
0 2 3
2(5), 6(3) 2(5), 6(3), 7(4)
Case 2-1 Case 2-2 Case 2-3
Table 6. AREs of cycle time in System II.
Case 2-1 Case 2-2 Case 2-3
Class 1
Class 2
Class 3
Class 4
3.78% 6.33% 6.55%
0.95% 1.46% 3.31%
0.74% 0.61% 4.00%
1.34% 1.02% 1.41%
4. Conclusions In this paper, an approximation methodology is proposed based on the MVA for estimating the performance of the re-entrant lines with multi-server workstations and multiple classes of jobs. The proposed approach is an extension of the earlier work by Park et al. (2000a, 2000b) to accommodate the multi-server case, which is a more general situation to be found in the real world. The concept of the interdeparture time proposed by Schmidt (1997) was adopted, which is the time interval between successive job completion times at a multi-server workstation. Modi cations to the estimated values are made applying the same scheme as proposed in Park et al. (2000b). From the computationa l tests, we found that the mean cycle time and the throughput can be estimated with the relative error of 7% or less for the sample cases. It is concluded that the proposed methodology may be applied to a real system to achieve reasonably good estimations of the average cycle time and the throughput quickly. In the model presented the ideal shop conditions were assumed, such as no breakdowns of the machines, no random reworks, or no yield losses. More realistic and/ or extended conditions may be included in both modelling and computational procedure in the further research.
Figure 6. Cycle times of Case 2-2.
that the major source of errors is batch processing and the approximation does not quite work as the WIP size becomes smaller relatively. Figure 6 shows the cycle time versus the total number of jobs in the system for Case 2-2. The two higher lines are of class 2, not of class 4. It is observed that the class with longer processing steps does not always have a larger cycle time.
A cknowledgements We would like to thank an anonymous reviewer for valuable comments. The work by the third author was supported in part by KOSEF through Statistical Research Center for Complex Systems at Seoul National University.
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