Analysis of Re-entrant Lines: An Iterative Approach

Proceedings of the 13th IFAC Symposium on Information Control Problems in Manufacturing Moscow, Russia, June 3-5, 2009

Analysis of Re-entrant Lines: An Iterative Approach Yang Liu ∗ Jingshan Li ∗∗ Shu-Yin Chiang ∗∗∗ ∗

Department of Electrical and Computer Engineering and Center for Manufacturing, University of Kentucky, Lexington, KY 40506 USA (e-mail: [email protected]). ∗∗ Department of Electrical and Computer Engineering and Center for Manufacturing, University of Kentucky, Lexington, KY 40506 USA (e-mail: [email protected]) ∗∗∗ Dept. of Information and Telecommunication Engineering, Ming Chuan University, Taoyuan 333, Taiwan, ROC ([email protected]) Abstract: In this paper, we present an approximation method to estimate the production rate of re-entrant lines with exponential machine reliability models. Recursive procedures are developed and structural properties are investigated. The results show that the proposed method provides an acceptable accuracy in production rate approximation for re-entrant lines. Keywords: Re-entrant lines, production rate, serial lines, iteration procedures. 1. INTRODUCTION In many manufacturing systems, parts need to visit one or more machines multiple times. Such systems are referred to as re-entrant lines. For instance, in semiconductor manufacturing, a production process is carried out layer by layer by imprinting multiple layers of material on the wafer. In automotive powertrain manufacturing plants, in order to keep the ignition components clean during the production, they need to be washed more than one times in centralized washers. Due to its wide applications in many manufacturing industries, in particular, semiconductor and electronic manufacturing, the analysis, design and management of re-entrant lines have significant importance. Although substantial amount of research effort has been devoted to throughput analysis of production systems, from serial lines, to assembly lines and systems with complex operations, such as rework, parallel, split, merge and closed loops, etc. (see reviews Dallery and Gershwin (1992), Li et al. (2008) and monographs Viswanadham and Narahari (1992), Buzacott and Shanthikumar (1993), Gershwin (1994), Li and Meerkov (2008)), the study on performance analysis of lines with re-entrant operations is still limited. Most of the research attention in re-entrant lines focuses on studying the scheduling and control policies. Less work is devoted to performance analysis of reentrant lines, in particular, lines with unreliable machines and finite buffers. Queueing network model, fluid model and mean value analysis have been the tools in such studies (see, for example, Kumar (1993), Connors et al. (1996), Dai and Weiss (1996), Narahari and Khan (1996), and Kumar and Kumar (2001). However, in these studies, ⋆ This work is partially supported by University of Kentucky Faculty Research Grant. ⋆⋆Please send all correspondences to Prof. Jingshan Li.

978-3-902661-43-2/09/$20.00 © 2009 IFAC

504

infinite buffer capacity is typically assumed so that the blockage phenomenon is ignored. Stochastic petri-net provides another approach for analytical study of re-entrant lines (e.g., Zhou and Jeng (1998), Jeng et al. (2000), Choi and Reveliotis (2003)). Analytical formulation for scheduling policy can be obtained using these methods. State space explosion is one of the difficulties encountered in this method. In addition to analytical approaches, discrete event simulations have been used as a popular tool for re-entrant line studies to evaluate performance, analyze scheduling and control policies, etc. (for example, Wein (1988), Lu et al. (1994)). Although detailed results can be obtained using simulations, such methods suffer from long model development time and long simulation time, which limit their applicabilities. In another direction, research work to study systems with similar feature to re-entrant lines, such as multiple part types, rework loops, etc., has been carried out (Li (2004a, 2004b, 2005), Colledani et al. (2005), Li and Huang (2005)). To extend the results to be applicable to re-entrant lines, substantial effort is required. Therefore, in spite of these work, developing an efficient analytical method to estimate the performance of reentrant lines, in particular, lines with unreliable machines and finite buffers, is still needed (Shanthikumar et al. (2007), Li et al. (2009)). This paper is intended to contribute to this end. Specifically, we present an iterative approach to estimate the production rate of re-entrant lines with exponential reliability machines having identical cycle times and finite buffers. The remaining of the paper is structured as follows: Section 2 formulates the problem. The modeling and analysis method is presented in Section 3. Section 4 discusses structural properties. Finally, conclusions are given in Section 5.

10.3182/20090603-3-RU-2001.0079

13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009

2. PROBLEM FORMULATION A typical structure of a re-entrant line (with one reentrance) is shown in Figure 1, where the circles represent the machines and the rectangles are the buffers. The machines, the buffers, and their interactions are addressed by the following assumptions. N12

N11 λ 1 µ1 N20

b20

m1

λ 3 µ3

λ 2 µ2 b 11 N21

b21

N1,M−1 λM−1µM−1

b 12 m2

N22

b22

m M−1

N2,M−1

Let P R be the production rate of the system, i.e., the average number of finished parts produced by the last machine per unit of time. The problem addressed in this work is formulated as follows: Given production system 1)-7), develop a method to calculate the production rate as a function of the system parameters and investigate structural properties.

λ M µM b 1,M−1

m3

7) Machine mi , i = 2, . . . , M , is starved at time t if it is up, both buffers b1,i−1 and b2,i−1 are empty, and their upstream machines fail to send any part into them at time t. Machine m1 is never starved by the first time job.

mM

A solution to the problem is presented in Sections 3 and 4.

b2,M−1

Fig. 1. Re-entrant lines

3. MODELING AND ANALYSIS

1) The system consists of M machines and 2M − 1 buffers separating two consecutive machines. The first time jobs are processed at machines mi , i = 1, . . . , M , and buffers b1i , i = 1, . . . , M − 1. After first time processing at machine mM , all jobs are sent to buffer b20 , waiting for second time processing. Then the jobs are reprocessed at machines mi , i = 1, . . . , M , but through buffers b2i , i = 1, . . . , M − 1. Jobs leave the system after being processed at mM for the second time. 2) All machines have identical processing times, normalized as 1 unit of time. The time is slotted as cycle time. 3) The up- and downtimes of machine mi are random variables exponentially distributed with parameters λi and µi , respectively. In other words, λi and µi are failure and repair rates, respectively. 4) Each buffer bk , k = 11, 12, . . ., (1, M − 1), 20, 21, 22, . . ., (2, M − 1), has capacity Nk , 0 < Nk < ∞. 5) Machine mi , i = 1, . . . , M − 1, is blocked by the first (respectively, second) time job at time t if it is up, buffer b1i (respectively, b2i ) is full and machine mi+1 does not take part from it at time t. Machine mM is blocked by the first time job at time t if it is up, buffer b20 is full and machine m1 does not take part from b20 at time t. Machine mM is never blocked by the second time job. 6) The second time jobs have higher priorities than the first time ones. In other words, when it is up, machine mi , i = 2, . . . , M − 1, always takes part from buffer b2,i−1 if it is not empty and mi is not blocked by b2i , otherwise it will take part from buffer b1,i−1 if it is not empty and mi is not blocked by b1i . When machine m1 is up and is not blocked by b21 , it takes part from buffer b20 if it is not empty; otherwise new part will be loaded to be processed at m1 if it is not blocked by b11 . Machine mM will take part from b2,M−1 if it is not empty, otherwise mM loads from b1,M−1 if it is not empty and mM is not blocked by b20 . Remark 1. Priority scheduling policy is typical in re-entrant lines, i.e., more processed jobs have higher priority than less processed ones. Such policy is also referred to as last buffer first serve (LBFS) policy. More details about LBFS and other scheduling policies can be found in Kumar (1993).

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3.1 Idea of the Approximation The main difficulty of analyzing re-entrant lines is that the machines are used for multiple processing of jobs. In addition to the complexity caused by blocking and starving, which is typical in serial lines, more difficulties, from the allocation of machine capacity to multiple processing of jobs, the priority loading and the dedicated dispatching policies, etc., make the exact analysis of system performance impossible. Therefore, approximation method is pursued in this work. Specifically, an iterative approach is developed to estimate the production rate of a re-entrant line with exponential machine reliability models. The idea of the method is to represent a M -machine re-entrant line in Figure 1 by a 2M -machine serial line (see Figure 2 for illustration). The first M machines (denoted as m′1 − m′M ) characterize the operations dedicated for the first time jobs and the second M machines (denoted as m′′1 − m′′M ) for the second time jobs. Such an equivalence represents the path of parts flowing within the system, i.e., they start visiting m1 to mM for first time processing, then through buffer b20 , they return to m1 for second time operations. The parameters of the machines (m′i and m′′i , i = 1, . . . , M ) are modified to take into account the shared processing of the first and second time jobs on each machine. Then, an iterative procedures is introduced to update these parameters recursively. When the procedure is convergent, an estimate of system production rate can be obtained. λ’1 µ 1’

N11

m’1

b11

’ N1,M−1 λ’M µ’ λ ’M−1µ M−1 M

λ’2 µ 2’ N12 m’2

m’M−1

b 12

b1,M−1

m’M

N20

µ ’’ N21 λ’’ 1 1 m’’ 1

b21

µ ’’ N22 λ’’ 2 2 m’’ 2

b22

Fig. 2. Equivalent serial lines

b20

µ’’ µ ’’ N2,M−1 λ’’ λ ’’ M M M−1 M−1 m’’ M−1

b2,M−1 m’’M

13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009

To obtained the parameters of the modified machines, m′i and m′′i , i = 1, . . . , M , allocations of machine mi to the first and second time jobs need to be estimated. Since the second time jobs have higher priority than the first time ones, when machine is up, mi , i = 1, . . . , M − 1, will load and process parts from buffer b2,i−1 if it is not blocked by b2,i and starved by b2,i−1 . Machine mM will carry out the last operation on jobs from b2,M−1 if it is not empty and release the finished products when work is completed. However, for the first time jobs, mi , i = 2, . . . , M , will load from buffer b1,i−1 only when the operations on second time jobs are not feasible (i.e., mi is either starved or blocked by second time jobs). Similarly, a new job will be loaded by m1 only when b20 is empty or m1 is blocked by b21 . Therefore, it is equivalent to claim that mi is available to second time jobs as long as it is up, while it is available to first time jobs only when it is up and cannot work on second time jobs. Thus, machines m′′i keep the same parameters as mi , however, the parameters of m′i need to be modified by enlarging its downtimes. In other words, from the point of view of the first time jobs, the time when machine mi is working on second time jobs is viewed as downtime, since during this time, the machine is not available to the first time jobs. Since machine downtime typically has larger impacts on system performance compared with uptime, therefore, we modify downtime parameters, µi , first. µ′′i = µi ,

i = 1, . . . , M,

µ′i = µi Prob[b2,i−1 is empty or b2,i is full], i = 1, . . . , M − 1, µ′M

= µi Prob[b2,M−1 is empty]. The parameters of λi are modified so that the efficiency of the modified machine equals to µi , i = 1, . . . , M, λi + µi e′i = ei Prob[b2,i−1 is empty or b2,i is full],

e′′i = ei =

i = 1, . . . , M − 1, e′M

= ei Prob[b2,M−1 is empty]. Therefore, we obtain λ′i = λi + µi − µ′i ,

λ′′i = λi + µi − µ′′i ,

Consider an M -machine serial production line with machine parameters λi , µi , i = 1, . . . , M , and buffer parameters Ni , i = 1, . . . , M − 1, we have Procedure 1.

 µbi (l + 1) = µi − µi Q λbi+1 (l + 1), µbi+1 (l + 1), λfi (l),  µfi (l), Ni , i = 1, · · · , M − 1,

λbi (l + 1) = λi + µi − µbi (l + 1), (1)  f f f b µi (l + 1) = µi − µi Q λi−1 (l + 1), µi−1 (l + 1), λi (l + 1),  µbi (l + 1), Ni−1 , i = 2, · · · , M, λfi (l + 1) = λi + µi − µfi (l + 1),

with boundary conditions λf1 (l) = λ1 ,

µf1 (l) = µ1 ,

λbM (l) = λM ,

µbM (l) = µM ,

l = 0, 1, 2, . . . , and initial conditions λfi (0) = λi ,

µfi (0) = µi ,

i = 2, · · · , M − 1.

where l is iteration number, and Q(λ1 , µ1 , λ2 , µ2 , N ) =  (1 − e1 )(1 − φ) λ1 λ2   , if 6= ,  −βN  µ2  1  1 − φeλ (λ + λ )(µ + µ )/(λ + µ µ 1 1 2 1 2 1 1) ,  (λ + λ )(µ + µ ) + λ µ (λ + λ + µ 1 2 1 2 2 1 1 2 1 + µ2 )N    λ2 λ1   = , if µ1 µ2 e1 (1 − e2 ) φ= , (2) e2 (1 − e1 ) (λ1 + λ2 + µ1 + µ2 )(λ1 µ2 − λ2 µ1 ) β= , (λ1 + λ2 )(µ1 + µ2 ) µi , i = 1, 2. ei = λi + µi Here superscripts ‘f’ and ‘b’ represent forward and backward aggregations in the procedure. It is shown that the procedure is convergent so that

i = 1, . . . , M.

lim µfi (l) := µfi , lim µbi (l) := µbi ,

Since the probabilities of buffer empty or full are unknown, we introduce iterations. At the first step, assuming these probabilities are known, using a serial line analysis method, we analyze the 2M -machine serial line with the modified parameters and obtained the probabilities of blockages and starvations for machines m′′i . Then using these probabilities, we conduct the second iteration and continue until the procedure is convergent.

Then the line production rate is obtained as

3.2 Recursive Procedures

Introduce operator Θpr (·) to denote the procedure calculating the production rate of a serial line, i.e.,

Clearly, in order to carry out the above procedure, a serial line analysis method is needed. Such a method has been developed in Li (2004a) and Li and Meerkov (2008). To make this paper self-contained, we introduce the method below:

506

s→∞

s→∞

lim λfi (l) := λfi , lim λbi (l) := λbi .

s→∞

PR =

s→∞

µfM λfM

+

µfM

=

λb1

µb1 . + µb1

P R := Θpr (λ1 , µ1 , . . . , λM , µM , N1 , . . . , NM−1 ).

(3)

(4)

(5)

b 2i and st b 2i to approximate Using this operator, introduce bl the probabilities that buffers b2i or b2,i−1 , i = 1, . . . , M , are

13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009

b 2M = 0). full or empty, respectively (note that we assume bl Then the following procedure can be developed for reentrant lines: Procedure 2. b 2i (s) + st b 2i (s)st b 2i (s) − bl b 2i (s)), µ′i (s + 1) = µi (bl

When the procedure is convergent, the steady state equations of Procedure 2 and estimate (10) lead to a unique solution. In this way, an estimate of the production rate of a re-entrant line in steady state is obtained. 3.4 Accuracy

i = 1, . . . , M,

λ′i (s + 1) = λi + µi − µ′i (s + 1), i = 1, . . . , M, b 2i (s + 1) = 1 − Θpr (λ′ (s + 1), µ′ (s + 1), . . . , λ′ (s + 1), bl 1 1 M µ′M (s + 1), λ1 , µ1 , . . . , λM , µM , N11 , . . . , N1,M−1 , N20 , N21 , . . . , N2,M−1 )/

(6)

Θpr (λ′1 (s + 1), µ′1 (s + 1), . . . , λ′M (s + 1), µ′M (s + 1), λ1 , µ1 , . . . , λi , µi , , N11 , . . . , N1,M−1 , N20 , N21 , . . . , N2,i−1 ), i = 1, . . . , M, b 2i (s + 1) = 1 − Θpr (λ′1 (s + 1), µ′1 (s + 1), . . . , λ′M (s + 1), st

µ′M (s + 1), λ1 , µ1 , . . . , λM , µM , N11 , . . . , (7)

N1,M−1 , N20 , N21 , . . . , N2,M−1 )/Θpr (λi , µi , . . . , λM , µM , , N2i , . . . , N2,M−1 ),

i = 1, . . . , M,

and s is the iteration number.

3.3 Production Rate Estimation It turns out that Procedure 2 returns two convergent sequences for even and odd iteration numbers. Then, based on extensive numerical experiments, we obtain Numerical Fact 1. Under assumptions 1)-7), Procedure 2 results in two convergent sequences, therefore, the following limits exists:  ′ µi,even if s is even, ′ lim µ (s) := µ′i,odd if s is odd, s→∞ i  ′ λi,even if s is even, lim λ′ (s) := (8) λ′i,odd if s is odd. s→∞ i Introduce pr b even and pr b odd as the production rates with λ′i,even , µ′i,even and λ′i,odd , µ′i,odd , respectively, we have pr b even = Θpr (λ′1,even , µ′1,even , . . . , λ′M,even , µ′M,even ,

λ1 , µ1 , . . . , λM , µM , N11 , . . . , N1,M−1 , N20 , N21 , . . . , N2,M−1 ),

pr b odd =

M ∈ {2, 3, 5, 10, 20, 50} ej ∈ [0.75, 0.95], j = 1, . . . , M, 1 ∈ [1, 20], j = 1, . . . , M, (11) µj 1 1 }⌋, i = 1, 2, j = 1, . . . , M − 1, Nij ∈ ⌊k · max{ , µj µj+1 1 1 N20 ∈ ⌊k · max{ , }⌋, µ1 µM k ∈ [1, 3], i = 1, 2, j = 1, . . . , M, where ⌊x⌋ denote the largest integer less than or equal to x, and λj is calculated from ej and µj .

i = 1, . . . , M, s = 0, 1, 2, . . . , with initial conditions b 2i (0) ∈ (0, 1), st b 2i (0) ∈ (0, 1), bl

The accuracy of the approximation is investigated numerically. Specifically, we consider a total of 275 re-entrant lines by randomly and equiprobably selecting machine and buffer parameters from the following sets:

For each of these lines, both analytical method using Procedure 2 and simulation approach using Simul8 (Hauge and Paige 2002) are pursued to evaluate system production rate. In each simulation, 10,000 cycles of warmup time are assumed, which is sufficient to guarantee that the steady state is reached, and the next 100,000 cycles are used for collecting steady state statistics. 20 replications are carried out to obtain the average production rate, with 95% confidence intervals consistently ranging around ±0.001. The differences between analytical and simulation results are evaluated as d P R − PR ǫ= · 100%, (12) PR d where P R and P R are the production rates obtained by simulation and recursive procedure, respectively.

The results of this investigation are illustrated in Figure 3. Table 1 provides the average and maximum errors in all the experiments, and the percentages where errors are within 5 or 10%. It is shown that in more than 86% of cases, the difference ǫ is within 5%, with a few exceptions (less than 2% of cases) going up more than 10% (maximum to 12%). Thus, we conclude that in all the cases we studied, Procedure 2 results in an acceptable accuracy for production rate estimation.

(9)

Table 1. Accuracy

Θpr (λ′1,odd , µ′1,odd, . . . , λ′M,odd , µ′M,odd , Average |ǫ| 2.4%

λ1 , µ1 , . . . , λM , µM , N11 , . . . , N1,M−1 , N20 , N21 , . . . , N2,M−1 ).

Then, the production rate of the re-entrant line is selected d as the average of them, and denoted as P R, pr b + pr b even odd d P R= . (10) 2

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max |ǫ| 12.4%

|ǫ| < 5% 86.6%

|ǫ| < 10% 98.2%

For illustration purpose, five typical examples are provided in Table 2. From these results, we conclude that Procedure 2 provides a useful tool for design and analysis of re-entrant lines.

13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009

and estimate (10), we investigate these properties in reentrant lines. Property 1. Under assumptions 1)-7), the system producd tion rate defined by (10), P R, is monotonically increasing with respect to µi , i = 1, . . . , M , N20 and Nij , ij = 11, . . . , (1, M − 1), 20, 21, . . . , (2, M − 1), and decreasing with respect to λi , i = 1, . . . , M .

15

10

ǫ(%)

5

0

−5

d Illustrations of the monotonicity of P R with respect to λi , µi and Nij for a five-machine re-entrant line are shown in Figure 4(a), (b) and (c), respectively. In Figure 4(a), all machines are identical with failure rates λi ranging from 0.05 to 0.25, and all repair rates µi are 0.95 and all buffers have capacity Nij = 1. In Figure 4(b), machines are still identical but with all λi = 0.05, and all µi increasing from 0.7 to 0.95, and all buffer capacities are still kept at 1. In Figure 4(c), buffers are identical with Nij changing from 1 to 20, all machines have λi = 0.05 and µi = 0.95. Clearly, decreasing machine failures, improving machine repair rates or increasing buffer capacity result in improvement in system production rate. However, the improvement is gradually diminishing when buffer capacity is increased. Such results are the same as in serial line case.

−10

−15 0

50

100

150

200

250

300

Case Number

Fig. 3. Accuracy of Procedure 2 Table 2. Illustration examples λi N1

[0.0211, 0.1166] 29 N2

PR

0.3481

µi 58

Pc R

[0.0732, 0.3555] N0 13

0.3532

1.46%

ǫ

(a) M = 2

λi µi N1i

[0.0049, 0.0391, 0.0048] [0.0937, 0.1720, 0.1365] [16, 23] N2i [28, 15] N20

PR

0.4030

Pc R

0.3851

0.45

16

0.4

-4.46%

ǫ

0.35

d P R

(b) M = 3

0.3

[0.0063, 0.0047, 0.0054, 0.0066, 0.0171] [0.0759, 0.1774, 0.2674, 0.1174, 0.0652] [26, 20, 6, 9] N2i [28, 12, 8, 25]

λi µi N1i 21

N20

PR

0.3909

Pc R

0.3832

ǫ

0.25 0.2 0.05

-1.95%

N1i N2i N20

4

(a) Monotonicity with respect to λ

[0.0183, 0.0100, 0.0137, 0.0049, 0.0064 0.0056, 0.0286, 0.0085, 0.0042, 0.0058] [0.0551, 0.0569, 0.1373, 0.09554, 0.1001 0.0610, 0.0745, 0.0984, 0.5841, 0.1663] [51, 30, 34, 9, 26, 17, 45, 20, 23] [52, 33, 9, 22, 18, 26, 30, 30, 2] P R 0.3401 Pc R 0.3460 ǫ 1.71%

0.4 0.39 0.38 0.37 0.36

N1i N2i N20

0.35 0.7

[0.0058, 0.0058, 0.1105, 0.0146, 0.0076, 0.0043, 0.0042 0.0147, 0.0070, 0.0049, 0.0044, 0.0185, 0.1977, 0.0060 0.0051, 0.0785, 0.0071, 0.0147, 0.0274, 0.0060] [0.0710, 0.0973, 0.2639, 0.0744, 0.1638, 0.2261, 0.0536 0.0504, 0.0968, 0.4268, 0.2471, 0.0601, 0.0575, 0.0843 0.6829, 0.2379, 0.0847, 0.0954, 0.4501, 0.8722] [37 18 27 4 18 12 8 19 54 16 4 6 48 46 36 2 9 19 16] [37 17 6 38 11 12 48 58 31 5 5 39 19 17 2 9 31 16 5] 3

PR

0.1144

Pc R

0.1124

ǫ

0.75

0.8

µ

0.85

0.9

0.95

(b) Monotonicity with respect to µ 0.46 0.45 0.44

d P R

µi

0.25

0.41

(d) M = 10

λi

0.2

d P R

µi

0.15

λ

(c) M = 5 λi

0.1

0.43 0.42 0.41

0.4 0

-1.67%

5

10

15

20

N

(c) Monotonicity with respect to N

(e) M = 20

4. STRUCTURAL PROPERTIES

Fig. 4. Monotonicity with respect to λ, µ and N

4.1 Monotonicity 4.2 Asymptotic property It has been shown in Li and Meerkov (2008) that monotonicity holds in serial lines and assembly systems, i.e., system production rate can be improved by increasing machine reliability and/or buffer capacity. Using Procedure 2

508

It is clear that for serial production lines, the production rate converges to the efficiency of the worst machine when buffers are infinite, i.e., P R approximates to mini ei when

13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009

Nij → ∞ (Li and Meerkov (2008)). For re-entrant lines, we can show that Property 2. Under assumptions 1)-7), the system production rate defined by (10) satisfies lim Nij → ∞ ij = 11, . . . , (1, M − 1), 20, 21, . . . , (2, M − 1)

d P R= =

min

µi 2(λi + µi )

min

ei . 2

i=1,...,M

i=1,...,M

(13)

4.3 Reversibility Reversibility is also observed in serial production lines (Li and Meerkov (2008)). For re-entrant lines, let the line in Figure 5 be the reversed line of the one in Figure 1. Higher priority is still assigned to buffer b2i , i = 1, . . . , M − 1. rev d d Let P R and P R denote the production rates obtained through Procedure 2 and estimate (10). We have λM µM N20

b20

mM

λM−1µM−1 b 2,M−1 N1,M−1

b 1,M−1

N 21

N2,M−2

N2,M−1

mM−1

λM−2µM−2

b 2,M−2 N1,M−2

λ 2 µ2

λ1 µ 1 b 21

m M−2

b1,M−2

m2

N 11

m1

b 11

Fig. 5. Reversed re-entrant line Property 3. Under assumptions 1)-7), the system production rate defined by (10) satisfies rev d d P R=P R . (14) 5. CONCLUSIONS

Re-entrant lines are widely used in many manufacturing industries. In this paper, we present a method to approximate the system production rate of re-entrant lines with exponential machine reliability models. The numerical results suggest that this method can provide an acceptable precision for system production rate estimation. The future work will focus on extending the method to systems with multiple re-entrances, lines with asynchronous machine speeds, and lines with non-identical machine parameters for re-entrant jobs, etc. The successful development of such methods will provide production engineers a quantitative tool for design and continuous improvement of reentrant lines. REFERENCES J.A. Buzacott and J.G. Shantikumar, Stochastic Models of Manufacturing Systems, Prentice Hall, 1993. J.Y. Choi and S.A. Reveliotis, “A Generalized Stochastic Petri Net Model for Performance Analysis and Control of Capacitated Re-entrant Lines,” IEEE Trans. on Robot. and Autom., 19:474-480, 2003. M. Colledani, A. Matta and T. Tolio, “Performance Evaluation of Production Lines with Finite Buffer Capacity Producing Two Different Products,” OR Spec., 27:243263, 2005.

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