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Bottlenecks in Bernoulli Serial Lines With Rework Stephan Biller, Member, IEEE, Jingshan Li, Senior Member, IEEE, Samuel P. Marin, Semyon M. Meerkov, Fellow, IEEE, and Liang Zhang, Member, IEEE
Abstract—The bottleneck (BN) of a production system is a machine with the strongest effect on the system’s throughput. In this paper, a method for BN identification in serial lines with rework and Bernoulli machines is developed. The method can be applied using either calculated or measured data on blockages and starvations of the machines. For the case of calculated data, a technique for evaluating performance measures of Bernoulli lines with rework is developed. Along with these quantitative contributions, the paper provides three qualitative results. First, it shows that Bernoulli lines with rework do not observe the property of reversibility. Second, it demonstrates that downstream machines may have a larger effect on the throughput than upstream ones. Third, it demonstrates that BNs may be shifting not only because of changes in machine and buffer parameters but also due to changes in quality of parts produced. Note to Practitioners—This paper shows how one can identify the bottleneck of a serial line with rework using data on blockages and starvations of the machines. Also, it demonstrates that the BN shifts when the quality buy rate is fluctuating. Finally, it shows that downstream machines in production lines with rework have a larger effect on the throughput than those upstream (which is unlike the serial lines with no rework where the effect of a machine is independent of its position in the system). Index Terms—Bernoulli machines, bottleneck identification, serial line with rework.
I. INTRODUCTION ERIAL production lines with rework are used in large volume manufacturing, for instance, in the automotive industry, when parts produced may have defects, which can be repaired and “reworked.” The block diagram of such a line machines and is shown in Fig. 1. Here, the main line has buffers, while the rework loop includes machines buffers where the defective parts are repaired. It and is assumed that each part (referred to interchangeably as a job) is nondefective with probability at the output of machine
S
Manuscript received September 19, 2008; revised February 12, 2009. First published August 18, 2009; current version published April 07, 2010. This paper was recommended for publication by Associate Editor F. Chen and Editor Y. Narahari upon evaluation of the reviewers’ comments. S. Biller is with GM R&D and Planning, General Motors Corporation, Warren, MI 48090-9055 USA. J. Li is with the Department of Electrical and Computer Engineering, Center for Manufacturing, University of Kentucky, Lexington, KY 40506 USA (e-mail:
[email protected]). S. P. Marin, deceased, was with GM R&D and Planning, General Motors Corporation, Warren, MI 48090-9055 USA. S. M. Meerkov is with Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109-2122 USA (e-mail:
[email protected]). L. Zhang is with Department of Industrial and Manufacturing Engineering, University of Wisconsin, Milwaukee, WI 53201 USA (e-mail: liangzh@eecs. umich.edu). Digital Object Identifier 10.1109/TASE.2009.2023463
and defective with probability . The nondefective jobs continue to be processed along the main line, while the defective ones are routed into the rework loop and reenter the main . To prevent deadlocks, the repaired line through machine jobs have a higher priority than new ones, i.e., takes a job only if buffer is empty. from buffer Although production systems have been investigated using analytical tools for almost 50 years (see, for instance, pioneering papers [1] and [2] and monographs [3]–[8]), production lines with rework have been addressed only in a few recent publications [10]–[15]. Mostly performance analysis problems have been addressed. Specifically, analytical techniques for evaluating their production rates and probabilities of machine blockages and starvations have been derived under the assumption that the machines obey the exponential reliability model. In addition, since production lines with rework can be viewed as a part of the general field of reentrant lines [16], [17], issues of deadlock and stability have also been investigated [18]. However, bottlenecks in such systems have not been analyzed. The current paper is intended to contribute to this end. More precisely, the primary goal of this work is to develop a method for bottleneck (BN) identification in serial lines with obeys the Bernoulli reliability rework where each machine model, i.e., produces a part in a cycle time with probability and fails to do so with probability . Such a reliability model is applicable to operations where the unscheduled downtime is, on the average, close to the machine cycle time. This often happens in automotive painting and assembly operations, where the downtime is primarily due to quality problems rather than machine breakdowns (see [19]–[22] for examples). Since methods for performance analysis of Bernoulli lines with rework are not available in the literature and since the BN identification technique developed here requires data on machine blockages and starvations, the secondary goal of this paper is to present a method for performance analysis of Bernoulli serial lines with rework. The outline on this paper is as follows. Section II formulates the model under consideration and the problems addressed. Sections III and IV are devoted to performance analysis and BN identification. Section V presents the conclusions. II. SYSTEM MODEL AND PROBLEM FORMULATION A. Model Consider a production line shown in Fig. 1. Assume that it operates according to the following assumptions. i) The main line consists of machines, , and buffers, ; the repair part of the rework loop consists of machines , and buffers, . The set of all machines is , ; the denoted as
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Fig. 1. Bernoulli serial production line with rework.
set of all buffers is denoted as , . Machines and are referred to as the merge and split machines, respectively. ii) The machines have identical cycle time . The time axis is slotted with the slot duration . The status of the machines (up or down) is determined at the beginning of each time slot. iii) The machines obey the Bernoulli reliability model, i.e., , , being neither blocked nor starved, produces a part during a time slot with probability and fails to do . Parameter is referred to as so with probability . the efficiency of iv) Each buffer , , is characterized by its capacity, , where . The state of the buffer (i.e., the number of parts in it) is determined at the end of each time slot. , is starved during a time slot if buffer v) Machine , is empty at the beginning of the time slot. Machine is starved if both and are empty. Machine is never starved. , is blocked during a time slot if buffer vi) Machine , has parts at the beginning of the time slot and mafails to take a part during this time slot. Machine chine is blocked by the main line if is full and does not take a part during this time slot; it is blocked by is full and does not take a part the rework loop if is never blocked. during this time slot. Machine are nondefective and defective vii) Parts at the output of , respectively. Parameter with probability and is referred to as the quality buy rate. Defective and nondefective parts form a sequence of independent random variables. Nondefective and defective parts are routed to , respectively. buffers and viii) The repaired parts from the rework loop have a higher priority than those from the main line. In other words, does not take a part from unless is empty. Note that assumptions vi) and vii) imply, respectively, that the blocked before service convention [23] is used and that the contains a quality control operation. Note also split machine that the defective parts may be produced by all machines prior , but they are identified as such only at . to B. Problems Addressed In the framework of the above model, this paper addresses the following problems. 1) Performance Analysis Problem: Given the machine and buffer parameters and the quality buy rate, evaluate the producand the probabilities of tion rate of the line with rework blockages and starvations of each machine in
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Fig. 2. Bottleneck identification in open serial lines: single bottleneck case.
Fig. 3. Bottleneck identification in open serial lines: multiple bottleneck case.
has two upstream buffers, and , the system. Since has two downstream buffers, and , their starvaand tions and blockages are denoted as follows:
A solution to this problem is given in Section III. 2) Bottleneck Identification Problem: The bottleneck machine of a production line has been defined in [24] as the machine that has the largest effect on the system production rate. In the framework of a production line with rework, this definition , is the BN if implies that , (1) Because this definition is difficult to apply in practice (since the partial derivatives involved cannot be evaluated analytically and are difficult to obtain from factory floor measurements), reference [24] offered an indirect method applicable to open serial lines (i.e., lines without rework). This method, illustrated in Figs. 2 and 3, consists of calculating analytically or measuring , and on the factory floor the probabilities of blockages, , of all machines in the system and assigning starvations, to if and from an arrow directed from to if . If there is a single machine with no emanating arrows (see Fig. 2), it is the BN in the sense of , instead of (1) (with the production rate of the open line, ). If there are multiple machines with no emanating arrows (see Fig. 3), the one with the largest severity is the primary bottleneck (PBN), where the severity of the bottleneck (i.e., each machine with no emanating arrows) is defined as
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(2)
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2) Aggregation of Bernoulli Serial Lines: It has been shown in [25] that the performance measures of a serial line consisting Bernoulli machines with efficiencies, , and of buffers with capacities, , can be evaluated using the following. Recursive Procedure 1:
(3) with initial conditions
and boundary conditions
Fig. 4. Overlapping decomposition of serial production lines with rework.
where if if
With a rework loop added, this method is not applicable due to the merge and split operations. The main goal of this paper is to extend this method to Bernoulli lines with rework. This is carried out in Section IV. III. PERFORMANCE ANALYSIS A. Approach Due to the complexity of Markov chains involved in their description, direct analysis of Bernoulli lines with rework is all but impossible. Therefore, a simplification is necessary. In this paper we use two simplification techniques: overlapping decomposition and recursive aggregation. The method of overlapping decomposition, developed in [10] and [12], represents a complex production system as several open serial lines. The recursive aggregation technique, proposed in [25], allows for analytical performance evaluation of open serial lines. Below, these two simplification techniques are briefly reviewed. 1) Overlapping Decomposition: This method is based on representing the line with rework as four overlapping serial lines and four virtual serial lines shown in Fig. 4(a) and (b), respectively. The overlapping lines include the overlapping machines and , each belonging to three lines. The virtual lines do not contain overlapping machines; instead, they include six vir, , and , , , efficiencies of tual machines, which are selected so as to represent the effect of the rest of the system on a particular virtual line. Calculating appropriately the and , , , of the virtual machines, efficiencies, , , allows one to analyze the performance of the original line with rework. For the exponential model of machine reliability, this has been carried out in [12]. For Bernoulli machines, this approach is developed in Section III-B.
This procedure results in two sequences and , , which are proved to be convergent [25], and the following limits exist: (4) In terms of these limits, the performance measures of a Bernoulli line are evaluated as follows [25]:
(5) (6) (7) In addition, the overlapping decomposition of Section III-B is either blocked requires the probability that is full while is empty; we denote these or down and the probability that probabilities as and , respectively. As it follows from (6) and (7), they can be evaluated as (8) and
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(9)
BILLER et al.: BOTTLENECKS IN BERNOULLI SERIAL LINES WITH REWORK
Recursive Procedure 1 and the estimates of the performance measures (5)–(9) are used below for analysis of the virtual serial lines of Fig. 4(b). B. Recursive Procedure for Bernoulli Serial Lines With Rework
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Then, perform Recursive Procedure 1 on virtual line 1 , , and, using (5)–(9), calculate and . Step 4: Consider virtual line 2 of Fig. 4(b) and update the and as follows: efficiencies of machines
Consider the virtual lines of Fig. 4(b) and introduce the following notations: Production rate of virtual line machine in virtual line is blocked machine
in virtual line is starved
is full and is empty is full and is full and is empty
is either down or blocked is empty is either down or blocked is either down or blocked
The estimates of these probabilities, which allow us to evalof the virtual mauate the parameters , , and , , chines , , and , , , can be calculated using a recursive procedure described below.
Then, perform Recursive Procedure 1 on virtual line 2 , , and using (5)–(9), calculate , and . Step 5: If the stopping rule (10) is satisfied, the procedure is terminated; otherwise return to Step 1. Denote the limits of this procedure as
Recursive Procedure 2: , and Step 0: Select the initial conditions randomly and equiprobably from the interval (0,1). Step 1: Consider virtual line 4 of Fig. 4(b) and update the and as follows: efficiencies of machines
Then, perform Recursive Procedure 1 on virtual line 4 and, using (5)–(9), calculate , , , and . Step 2: Consider virtual line 3 of Fig. 4(b) and update the as follows: efficiency of machine
Then perform Recursive Procedure 1 on virtual line 3 and, using (5)–(9), calculate , , and . Step 3: Consider virtual line 1 of Fig. 4(b) and update the as follows: efficiency of machine
(11) Unfortunately, the existence of and the convergence to these limits cannot be proved analytically (due to a nonmonotonic , ). Therefore, it has been inbehavior of vestigated numerically. A total of 5 000 000 lines with rework have been analyzed, and in every case the convergence [with in (10)] took place. The details of this investigation are as follows: Justification: A total of 5 000 000 lines have been gener, ’s, ’s and randomly and ated by selecting , , , equiprobably from the sets (12) (13) (14) (15) (16) (17) (18) (19) Note that the efficiencies of the machines in the rework loop are selected lower than those of the main line because in practice the capacity of the repair part of the system is typically smaller than that of the main line.
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For each of the lines, we ran Recursive Procedure 2 and observed the convergence in all cases studied, with the convergence taking place within a second using a standard laptop with a Pentium M 1.60 GHz processor and 1.23 GB RAM. Thus, we conclude that this procedure can be used for analysis of production lines with rework defined by assumptions i)–viii). Concluding this subsection, we point out the following relationships among the production rates of the virtual lines 1–4. Theorem 1: The production rates of the virtual lines are related as follows:
Proof: The production rates of Lines 2, 3, and 4 can be expressed as
The accuracy of these estimates has been investigated by simulations. Since an analogous approach is used elsewhere in this paper, we describe it below as a standard procedure. Numerical Simulation Procedure 1: • A C++ code to simulate the production system defined by assumptions i)–viii) is constructed. • The initial status of each machine is selected up with probability and down with probability , . • • For each line with rework under consideration, 20 runs of the simulation code are carried out. • In each run, the first 20 000 time slots are used as a warm-up period and the subsequent 200 000 time slots are used to statistically evaluate the performance measures of interest. with 95% confidence • This results in the estimate of and with intervals 0.001 and estimates of 95% confidence interval 0.002. , , and For the system parameters , we constructed 100 000 lines with ’s, ’s and ’s selected randomly and equiprobably from sets (16)–(19). The following metrics were used to evaluate the accuracy of the estimates: (24)
This leads to
(25) On the other hand,
(26) Therefore,
and
C. Performance Measure Estimates and Their Accuracy In Section II, the production rate of the line with rework was and the probabilities of blockages and stardenoted as and . Based on the limits (11) of Revations as cursive Procedure 2, their estimates are introduced as follows: (20) (21)
(22)
(23)
where , and are obtained by Numerical and , are defined in Simulation Procedure 1 and , Section II-B. Among the 100 000 lines studied, the average of was up to 3.97%, with very few extreme cases resulting in 20.4%. This accuracy is comparable with that obtained in [10] for the case of exponential machines with similar parameters. and were both less than 0.01. ThereThe average of fore, we conclude that Recursive Procedure 2 provides an effective tool for performance evaluation of serial lines with rework defined by assumptions i)-viii). D. System Properties of Lines With Rework Using Recursive Procedure 2, we establish two system-theoretic properties described next. As it is well known, serial production lines observe the property of reversibility [26]–[29]: the production rates of a serial line and its reverse (i.e., when the parts flow from the last machine to the first) are the same. This property certainly holds for open and closed Bernoulli lines. However, as we show below, reversibility does not hold for Bernoulli lines with rework. Indeed, consider a serial line and its reverse shown in Fig. 5(a) and (b), respectively. Using Recursive Procedure
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Fig. 5. Production system with rework and its reverse.
2, we determine that the production rate of the reverse line is not the same as that of the original one; this conclusion is also supported by simulations (see the data of Fig. 5). Thus, reversibility is violated. The lack of reversibility constitutes a fundamental difference between the usual (i.e., open) Bernoulli lines and those with rework. In addition, comparing the data of Fig. 5, we observe that placing more efficient machines towards the end of the line results in higher production rate than placing them upstream. This is also qualitatively different from serial lines with no rework where the position of a machine does not indicate its importance for performance of the system. The reasons for the loss of reversibility in Bernoulli lines with rework can be explained by the asymmetric routing of jobs at the merge and the split machines. Indeed, at the input of the merge machine, the repaired jobs have higher priority than the new ones. Also, after the split machine, the routing of the jobs is based on the quality buy rate. Finally, the merge machine is and are empty, while, due to the starved when both blocked before service convention, the split machine does not or by . As produce a part when it is blocked either by a result, the flow of jobs in Bernoulli lines with rework is no longer reversible, and the downstream machines, which process reworked parts, have a larger effect on the production rate than the upstream ones. IV. BOTTLENECK IDENTIFICATION A. Approach The method of Section III reduces a line with rework to four usual serial lines, i.e., the virtual lines of Fig. 4(b). Their BNs can be identified using and , ,2,3,4, evaluated according to Recursive Procedure 2. This, however, would identify not the BN of the line with rework but the BNs of the serial lines, where the rest of the system is represented by virtual machines , , and , , . In other words, each of these BNs would be a machine with the strongest effect on the overall production rate from the point of view of an individual virtual line. Although these bottlenecks may be of interest in some applications, the goal of this paper is to identify the true BN of the line with rework in the sense of (1).
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Fig. 6. LBNs identification.
Therefore, to ensure high accuracy of system bottleneck identification, rather than using the virtual lines, at the first stage of the identification procedure, we determine BNs of the four overlapping lines of Fig. 4(a); we refer to them as local bottlenecks (LBNs). Then, at the second stage, we identify the overall bottleneck of the line with rework, which is referred to as the global bottleneck (GBN). We describe below how LBNs can be identified (using either calculated or measured data) and show that one of them is practically always the GBN. B. Local Bottlenecks Identification Consider the line with rework of Fig. 4(a). Represent its overlapping lines as shown in Fig. 6. Clearly, each overlapping machine enters three of them. Assume that these lines are in isolation, i.e., the first machines are not starved and the last ones are not blocked. The probabilities of blockages and starvations of all other machines can be either calculated or measured during normal system operation. Specifically, when the parameters of the machines and buffers, as well as the quality buy rate, are known, and can be calculated using Recursive Procedure 2 and expressions (21)–(23). Based on these data, the BN of each overlapping line can be identified using the arrow method of Section II-B2. When the parameters of the machines and buffers or the and can be quality buy rate are not available, evaluated during normal system operation, keeping in mind and can be starved and blocked, respectively, by that two buffers. Based on these data, again the four LBNs can be identified using the arrow method of Subsection II-B2. Clearly, the same can be carried out when a simulation model is available. Thus, based on either Recursive Procedure 2 and expressions (21)–(23) or factory floor measurements or simulations, one can identify four local bottlenecks of a Bernoulli line with rework. C. Global Bottleneck Identification The foundation for GBN identification is based on the following. 1) Numerical Fact 1: In serial lines with rework defined by assumptions i)–viii), the GBN is practically always one of the four LBNs.
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Although a justification of this fact (including the quantification of the term “practically always”) is given later on (along with other numerical facts formulated below), its application is clear: To identify the GBN, one must test the effect of each LBN on the production rate of the system; the LBN with the largest effect is the GBN. While this process is somewhat involved—due to four, rather than one, BNs to be investigated—it can be facilitated by the following. 2) Numerical Fact 2: For serial lines with rework defined by assumptions i)–viii) (a) if an overlapping machine is the LBN in three of the overlapping lines, then it is practically always the GBN; (b) if an overlapping machine is the LBN in only one of the overlapping lines, then it is practically never the GBN. 3) Numerical Fact 3: In a serial line with rework defined by assumptions i)-viii) and with the quality buy rate , if its GBN is a nonoverlapping machine of line 1, then ; this machine is practically always the GBN for all if its GBN is a nonoverlapping machine of line 3, then ; this machine is practically always the GBN for all if its GBN is a nonoverlapping machine of line 4, then ; this machine is practically always the GBN for all if its GBN is a nonoverlapping machine of line 2 or line 4, then the GBN is practically always in ether line 2 or line 4 for . all Justification: The justification of Numerical Facts 1–3 has been carried out as follows: A total of 100 000 lines have been , , , and and paramgenerated with eters of machines and buffers selected randomly and equiprobably from sets (16)–(19). For the calculation-based approach, each of these lines has been analyzed using the Recursive Procedure 2 and the LBNs have been identified by calculating blockages and starvations using (21)–(23). Then, the partial derivatives have been estimated using
Fig. 7. GBN identification of Example 1 for q
= 0:8.
holds in 96.7% of cases using calculations — Claim and 95.4% of cases using measurements. — Claim holds in 97.0% of cases using calculations and 97.2% of cases using measurements. Based on these data, we conclude that Numerical Facts 1–3 indeed take place. Below, two examples illustrating this twostage procedure for GBN identification are presented. D. Example 1
with . For the measurement-based approach, each of these lines has been analyzed using Numerical Simulation Procedure 1 and the partial derivatives have been estimated via simulation using
with . Based on this procedure, the following results, quantifying the term “practically always” have been obtained: • Numerical Fact 1 holds in 97.7% of cases using calculations and 93.3% of cases using measurements. • Numerical Fact 2: — Claim (a) holds in 88.1% of cases using calculations and 86.9% of cases using measurements. — Claim (b) holds in 95.3% of cases using calculations and 94.9% of cases using measurements. • Numerical Fact 3: holds in 91.3% of cases using calculations — Claim and 92.2% of cases using measurements. — Claim holds in 94.6% of cases using calculations and 94.0% of cases using measurements.
Consider the line with rework shown in Fig. 7 along with the corresponding overlapping lines 1–4. Their local bottlenecks, identified both by calculations and the measurement-based approaches (using simulations), are also indicated. Since is the LBN in only one of the virtual lines, according to Numerical Fact 2 (b) it is not the GBN. Thus, the candidates are and . Increasing their efficiencies by 0.01, we determine that is the GBN. According to Numerical Fact 3 , this machine remains the GBN for all quality buy rate less than 0.8. Note that in this case (as well as in all other cases considered below), the two-stage procedure identifies correctly the GBN , by 0.01 and evaluating (verified by increasing each , the resulting effect on the ). Assume now that the quality buy rate in this system increases to 0.85. Then, as it follows from Fig. 8 and Numerical Fact 2 or . Increasing their efficiencies (b), the GBN is either is GBN. Note that it is by 0.01 leads to the conclusion that not the worst machine of the main line. and , Further, Figs. 9 and 10 show that the GBNs are if the quality buy rates are 0.9 and 0.95, respectively. This example indicates that bottlenecks may be shifting not only because of changes in the machine and buffer parameters but also due to changes in the quality buy rates. Thus, BNs must be tracked when the product quality is fluctuating.
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Fig. 8. GBN identification of Example 1 for q
= 0:85.
Fig. 10. GBN identification of Example 1 for q
= 0:9.
Fig. 11. GBN identification of Example 2.
Fig. 9. GBN identification of Example 1 for q
E. Example 2 Consider the system shown in Fig. 11. Assume first that the quality buy rate is 1, i.e., the rework loop is not activated. Then, since the machine efficiencies are allocated symmetrically and
= 0:95.
the buffers are identical, machines and are the bottlenecks, and they have identical effect on the system’s production rate. , for instance, , the On the other hand, when . This data of Fig. 11 show that the GBN is one: machine discrepancy between lines with and without the rework loop is
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due to the lack of reversibility in the former ones and the fact that downstream machines have a larger effect on the production rate than those upstream. V. CONCLUSION This paper provided tools for analysis and improvement of serial lines with Bernoulli machines and with rework. As far as the analysis is concerned, the method of overlapping decomposition has been extended to Bernoulli machines; the average accuracy of the resulting production rate estimate is within 4%. As far as the improvement is concerned, it has been shown how four local bottlenecks of the overlapping lines can be identified, one of which is practically always the global bottleneck of the overall system; the accuracy of the global bottleneck identification is above 90%. In addition, it has been shown that the property of reversibility—one of the fundamental properties of open and closed serial lines—does not hold in serial lines with rework. Based on this, it has been demonstrated that the machines downstream have a stronger effect on the production rate than those upstream. As topics for future work on production systems with rework, the following can be mentioned. • Extensions of the bottleneck identification technique to other than Bernoulli reliability models (e.g., exponential, Weibull, and, perhaps, general). • Extensions to assembly systems with rework. • Extensions to systems with machines having different cycles times (i.e., the asynchronous case). • Extensions to systems where the quality buy rate is a function of machine parameters. • Analysis of transients in serial lines and assembly systems with rework. REFERENCES [1] B. A. Sevast’yanov, “Influence of storage bin capacity on the average standstill time of a production line,” Theory Prob. Appl., vol. 7, no. 4, pp. 429–438, 1962. [2] J. A. Buzacott, “Automatic transfer lines with buffer stocks,” Int. J. Prod. Res., vol. 5, no. 3, pp. 183–200, 1967. [3] N. Viswanadham and Y. Narahari, Performance Modeling of Automated Manufacturing System. Englewood Cliffs, NJ: Prentice-Hall, 1992. [4] R. Askin and C. R. Standridge, Modeling and Analysis of Manufacturing Systems. New York: Wiley, 1993. [5] J. A. Buzacott and J. G. Shantikumar, Stochastic Models of Manufacturing Systems. Englewood Cliffs, NJ: Prentice-Hall, 1993. [6] H. T. Papadopoulos, C. Heavey, and J. Browne, Queueing Theory in Manufacturing Systems Analysis and Design. London, U.K.: Chapman & Hall, 1993. [7] S. B. Gershwin, Manufacturing Systems Engineering. Englewood Cliffs, NJ: Prentice-Hall, 1994. [8] H. G. Perros, Queueing Networks with Blocking. Oxford, U.K.: Oxford Univ. Press, 1994. [9] T. Altiok, Performance Analysis of Manufacturing Systems. New York: Springer, 1997. [10] J. Li, “Performance analysis of production systems with rework loops,” IIE Trans., vol. 36, no. 8, pp. 755–765, 2004. [11] J. Li, “Throughput analysis in automotive paint shops: A case study,” IEEE Trans. Autom. Sci. Eng., vol. 1, no. 1, pp. 90–98, 2004. [12] J. Li, “Overlapping decomposition: A system-theoretic method for modeling and analysis of complex manufacturing systems,” IEEE Trans. Autom. Sci. Eng., vol. 2, no. 1, pp. 40–53, 2005. [13] J. Li, D. E. Blumenfeld, and S. P. Marin, “Manufacturing system design to improve quality buy rate: An automotive paint shop application study,” IEEE Trans. Autom. Sci. Eng., vol. 4, no. 1, pp. 75–79, 2007.
[14] A. Korugan and O. F. Hancer, “On the quality information feedback and the rework loop,” in Proc. 6th Int. Conf. Anal. Manuf. Syst., 2007, pp. 7–14. [15] D. Borgh, M. Colledani, F. Simone, and T. Tolio, “Integrated analysis of production logistics and quality performance in transfer lines with rework,” in Proc. 6th Int. Conf. Anal. Manuf. Syst., 2007, pp. 15–20. [16] P. R. Kumar, “Re-entrant lines,” Queueing Syst. Theory Appl., vol. 13, no. 1–3, pp. 87–110, 1993. [17] J. R. Morrison and P. R. Kumar, “On the guaranteed throughput and efficiency of closed re-entrant lines,” Queueing Syst. Theory Appl., vol. 28, no. 1, 1998. [18] S. Kumar and P. R. Kumar, “Fluctuation smoothing policies are stable for stochastic re-entrant lines,” Discrete Event Dyn. Syst., Theory Applicat., vol. 6, no. 4, pp. 361–370, 1996. [19] J.-T. Lim, S. M. Meerkov, and F. Top, “Homogeneous, asymptotically reliable serial production lines: Theory and a case study,” IEEE Trans. Automat. Contr., vol. 35, pp. 524–534, 1990. [20] J.-T. Lim and S. M. Meerkov, “On asymptotically reliable closed serial production lines,” Contr. Eng. Pract., vol. 1, no. 1, pp. 147–152, 1993. [21] S.-Y. Chiang, C.-T. Kuo, J.-T. Lim, and S. M. Meerkov, “Improvability of assembly systems II: Improvability indicators and case study,” Math. Prob. Eng., vol. 6, pp. 359–393, 2000. [22] J. Li and S. M. Meerkov, “Customer demand satisfaction in production systems: A due-time performance approach,” IEEE Trans. Robot. Automat., vol. 17, no. 4, pp. 474–482, 2001. [23] Y. Dallery and S. B. Gershwin, “Manufacturing flow line systems: A review of models and analytical results,” Queueing Syst. Theory Appl., vol. 12, no. 1–2, pp. 3–94, 1992. [24] C.-T. Kuo, J.-T. Lim, and S. M. Meerkov, “Bottlenecks in serial production lines: A system-theoretic approach,” Math. Prob. Eng., vol. 2, no. 3, pp. 233–276, 1996. [25] D. Jacobs and S. M. Meerkov, “A system-theoretic property of serial production lines—Improvability,” Int. J. Syst. Sci., vol. 26, no. 5, pp. 755–785, 1995. [26] G. Yamazaki and H. Sakasegawa, “Property of duality in tandem queueing systems,” Ann. Inst. Statist. Math., vol. 27, pp. 201–212, 1975. [27] G. Yamazaki and H. Sakasegawa, “Property of duality in tandem queueing systems,” Manage. Sci., vol. 31, no. 1, pp. 78–83, 1985. [28] X.-G. Liu and J. A. Buzacott, “The reversibility of cyclic queues,” Oper. Res. Lett., vol. 11, no. 4, pp. 233–242, 1992. [29] D. W. Cheng, “Line reversibility of tandem queues with general blocking,” Manage. Sci., vol. 41, no. 5, pp. 864–873, 1995. Stephan Biller (M’07) received the Dipl.-Ing. degree in electrical engineering from the RWTH Aachen, Germany, the Ph.D. degree in industrial engineering and management science from Northwestern University, Chicago, IL, and the M.B.A. degree from the University of Michigan, Ann Arbor. He is currently a Group Manager with GM R&D and Planning, General Motors Corporation, Warren, MI, where he has responsibility for innovations in plant floor systems and controls. He is currently focusing on the digital factory, the real-time information enterprise, and the interoperability of the two.
Jingshan Li (S’97–M’00–SM’06) received the B.S. degree in automation from Tsinghua University, Beijing, China, the M.S. degree in automation from the Chinese Academy of Sciences, Beijing, and the Ph.D. degree in electrical engineering systems from the University of Michigan, Ann Arbor, in 1989, 1992, and 2000, respectively. From 2000 to 2006, he worked in the Manufacturing Systems Research Lab, General Motors Research & Development Center, Warren, MI. He joined the University of Kentucky in 2006 as a joint faculty member in electrical engineering and the Center for Manufacturing. His research area is in system and control with applications to manufacturing and service systems modeling, analysis, control, lean system design, and supply chain management. He is a coauthor (with S. M. Meerkov) of the textbook Production Systems Engineering (New York: Springer 2009), and he has about 70 referred journal and conference publications.
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BILLER et al.: BOTTLENECKS IN BERNOULLI SERIAL LINES WITH REWORK
Dr. Li is a senior member of the IIE. He received the 2005 IEEE Transactions on Automation Science and Engineering Best Paper Award, the 2006 IEEE Early Industry/Government Career Award in Robotics and Automation, and the 2009 IIE Transactions Best Application Paper Award. He was also a finalist for the Best Automation Paper Award in the 2005 IEEE International Conference on Robotics and Automation. He is currently an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, and has served as Associate Editor for Mathematical Problems in Engineering, the IEEE International Conference on Robotics and Automation, and the IEEE International Conference on Automation Science and Engineering.
Samuel P. Marin received the Ph.D. degree in mathematics from Carnegie Mellon University, Pittsburgh, PA, in 1978. He was with the General Motors Research and Development Center since 1978. He was a Research Fellow and Laboratory Group Manager in the Manufacturing Systems Research Laboratory, and conducted and managed research programs to develop new mathematical modeling and analysis tools for application to GM’s engineering, manufacturing, and design operations. Dr. Marin was a member of the Board of Governors of the Institute for Mathematics and Its Applications at the University of Minnesota, and also served as Co-Director of the General Motors/University of Michigan Collaborative Research Laboratory in Advanced Vehicle Manufacturing. He was a member of SIAM and Sigma Xi. He passed away in May 2008.
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Semyon M. Meerkov (M’78–SM’83–F’90) received the M.S.E.E. degree from the Polytechnic of Kharkov, Kharkov, Ukraine, in 1962 and the Ph.D. degree in systems science from the Institute of Control Sciences, Moscow, Russia, in 1966. He was with the Institute of Control Sciences until 1977. From 1979 to 1984, he was with the Department of Electrical and Computer Engineering, Illinois Institute of Technology, Chicago. Since 1984, he has been a Professor at the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor. He has held visiting positions at the University of California, Los Angeles (1978–1979), Stanford University (1991), Technion, Israel (1997–1998, 2008), and Tsinghua, China (2008). He is a coauthor (with J. Li) of the recent textbook Production Systems Engineering (New York: Springer, 2009). His research interests are in systems and control with applications to production systems and communication networks.
Liang Zhang (S’04–M’09) received the B.E. and M.E. degrees from the Center for Intelligent Networks and Systems (CFINS), Department of Automation, Tsinghua University, Beijing, China, in 2002 and 2004, respectively, and the Ph.D. degree in electrical engineering systems from the University of Michigan, Ann Arbor, in 2009. He is currently an Assistant Professor with the Department of Industrial and Manufacturing Engineering, University of Wisconsin, Milwaukee. His research interests include modeling, analysis, and continuous improvement and design of manufacturing and service systems.
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