period, in production-inventory-customer systems with exponential machines, finite inven- tory, and random demand is provided. Using this method, the ...
LEAN BUFFERING IN PRODUCTION SYSTEMS: A QUANTITATIVE APPROACH
by
Emre Enginarlar
A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Electrical Engineering: Systems) in The University of Michigan 2003
Doctoral Committee: Professor Semyon M. Meerkov, Chair Assistant Professor Sudheer Gupta Professor Pierre T. Kabamba Professor Daniel E. Koditschek
ABSTRACT LEAN BUFFERING IN PRODUCTION SYSTEMS: A QUANTITATIVE APPROACH by Emre Enginarlar
Chair: Professor Semyon M. Meerkov This thesis addresses the problem of buffer capacity allocation in serial production lines. Both in-process and finished goods buffers are considered. The goal is to develop methods to calculate the smallest, i.e., lean, buffer capacities necessary to ensure the desired production rate or customer satisfaction level. In the case of in-process buffering, a new dimensionless parameter, referred to as the level of buffering, is introduced in order to measure the buffer capacity in terms of machine downtime and to show that the level of buffering may be estimated without knowledge of the distributions of the uptime and downtime, but based only on their coefficient of variation (
). This implies that the level of buffering is practically the same for all uptime
and downtime distributions, as long as their coefficients of variation are equal. Based on this observation, an empirical law is formulated and verified according to which the level of buffering can be upper bounded by a piece-wise linear function. Specifically, for
, this upper bound is the product of the level of buffering for exponential of the machines in question; and, for , it is a machines ( ) and the . Since can be evaluated and can be identified on the constant equal to
factory floor, the method developed provides a simple and practical tool for designing lean buffering in serial production lines.
For finished goods buffers, an analytical method for evaluating due-time performance (
), i.e., the probability of producing a certain number of parts during a given shipping
period, in production-inventory-customer systems with exponential machines, finite inventory, and random demand is provided. Using this method, the degradation of function of demand variability is quantified. In addition, it is shown that
as a
is practi-
cally independent of a particular type of demand distribution, as long as its coefficient of variation (
) remains fixed.
The results presented in this dissertation are for serial production lines with unreliable machines having identical efficiency, identical cycle time and time-dependent failures. The future work will extend these results to non-identical machines and buffers, machines with non-identical cycle time, machines with random processing time, and assembly lines.
To my family
ii
ACKNOWLEDGEMENTS
I would like to thank my advisor, Professor Semyon M. Meerkov, for his guidance, insights and encouragement throughout this work. It was his experience, wisdom and enthusiasm which guided my research in this new and exciting field. I also would like to express my appreciation to the other members of my committee, Professors Sudheer Gupta, Pierre T. Kabamba and Daniel E. Koditschek. I would like to acknowledge the contributions of Dr. Jingshan Li from the GM Research and Development Center to our research team. This work was heavily influenced by the generosity with which he shared his time and expertise in manufacturing. I would like to thank several of my colleagues for many valuable discussions. These include Professor Rachel Q. Zhang, Professor Demosthenis Teneketzis, Professor Alexander O. Ganago, Professor Selin Aviyente, Dr. H¨useyin C. Akin, Dr. Chadi Elias El Chemali, Dr. ¨ ur Yılmaz, Mr. Yongsoon Eun and Mr. Haldun Komsuoglu. Choon Yik Tang, Dr. Ali Ozg¨ The continuous support, care and love of my family is the source and encouragement of this work. I would like to thank my father, Professor H¨usn¨u Enginarlar, and my mother, Elizabeth Jean Enginarlar, from the bottom of my heart. I feel extremely lucky to have such wonderful parents who have made many sacrifices over the years to ensure that their children receive a high quality education. My brother, Destan Cem Enginarlar, has also had a tremendous positive influence on my life. It was his motivation and enthusiasm that led me to pursue a career in engineering. I am grateful to all my relatives and friends in Turkey and in the United States for their love and support over the years. Finally, I would like to acknowledge the financial support from the National Science Foundation, Department of Electrical Engineering and Computer Science, and the Scientific and Technical Research Council of Turkey. iii
TABLE OF CONTENTS
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii LIST OF APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii CHAPTERS 1
INTRODUCTION . . . . . . . . . . . . . . . . . . . 1.1 Motivation . . . . . . . . . . . . . . . . . . . 1.2 Problems Addressed . . . . . . . . . . . . . . 1.3 Literature Review . . . . . . . . . . . . . . . 1.3.1 Literature on in-process buffering . . . 1.3.2 Literature on finished goods buffering 1.4 Original Contributions . . . . . . . . . . . . .
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1 1 2 3 3 6 6
2
IN-PROCESS BUFFERING: EXPONENTIAL MACHINES . 2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Problem Formulation . . . . . . . . . . . . . . . . . 2.3 Methods of Analysis . . . . . . . . . . . . . . . . . . 2.4 Solution . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Lean buffering in two-machine lines . . . . . 2.4.2 Lean buffering in three-machine lines . . . . 2.4.3 Lean buffering in -machine lines . . .
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8 9 11 11 14 14 17 18
IN-PROCESS BUFFERING: NON-EXPONENTIAL MACHINES 3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . 3.4 Methods of Analysis . . . . . . . . . . . . . . . . . . . . . 3.4.1 Erlang machines . . . . . . . . . . . . . . . . . . .
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25 27 30 31 31 33
3
iv
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3.5 3.6 3.7
3.8
3.4.2 Other machine models . . . . . . . . . . . 3.4.3 Evaluation of . . . . . . . . . . . . . System Parameters Selected for Simulations . . . . Systems Considered . . . . . . . . . . . . . . . . . . . . . . in Serial Lines with 3.7.1 Simulation results . . . . . . . . . . . . . . 3.7.2 Empirical law: analytical expression . . . . 3.7.3 Verification of the empirical law . . . . . . . . . . . in Serial Lines with +* 3.8.1 System ! "$#%'& )( 3.8.2 Empirical law: analytical expression . . . . 3.8.3 Verification of the empirical law . . . . . . ', " . -, , "0/1* 3.8.4 System -,
4
FINISHED GOODS BUFFERING . . . . . . . . . 4.1 Model and Problem Formulation . . . . . . 4.2 Calculation . . . . . . . . . . . . . . 4.3 Degradation as a Function of Demand Variability . . . . . . . . . . . . . . . . . . . . . . 4.4 as a Function of Demand
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36 37 37 38 38 38 39 42 43 43 43 45 46
. . . . . . . . . . . . 49 . . . . . . . . . . . . 50 . . . . . . . . . . . . 52 . . . . . . . . . . . . 55 . . . . . . . . . . . . 58
5
CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6
FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
v
LIST OF TABLES
Table 2.1 2.2 2.3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 4.1 4.2 4.3 4.4 B.1
Accuracy of approximating by (percent of error). . . . . . . . . . . . Accuracy of approximating by (percent of error). . . . . . . . . . Rule-of-thumb for selecting Level of Buffering in serial production lines with 10 or more exponential machines. . . . . . . . . . . . . . . . . . . . Comparison of exponential and non-exponential cases. . . . . . . . . . . Expected value, variance, and coefficient of variation of up- and downtime distributions considered. . . . . . . . . . . . . . . . . . . . . . . . . . . Downtime distributions considered. . . . . . . . . . . . . . . . . . . . . for system % ! # & ( ( * with versus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . for system % ! # & ( ( * with versus +( . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The maximum and minimum percentage errors for all cases considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . for system ' ! #%'& +( * with ( . . versus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . for 10-machine lines with Weibull reliability model. . . . . . . . . . for 10-machine lines with gamma reliability model. . . . . . . . . . for 10-machine lines with log-normal reliability model. . . . . . . . The maximum and minimum percentage errors for all cases considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
versus for Weibull machines. . . . . . . . . . . . . . . versus for gamma machines. . . . . . . . . . . . . . . log-normal machines. . . . . . . . . . . . . versus for Uniform s considered. . . . . . . . . . . . . . . . . . . . . . . . . Systems analyzed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of demand randomness on degradation ( ) . . . . + ( for different distributions of the demand ( ). . . . Characteristics of the Erlang distribution with parameters and . . . . .
vi
. 21 . 23 . 24 . 26 . 28 . 32 . 40 . 40 . 41
. . . .
42 44 44 45
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46 47 47 48 55 56 59 59 73
LIST OF FIGURES
Figure 1.1 2.1 2.2 2.3 2.4 2.5
Classification of the literature on the buffer capacity allocation problem. . Serial production line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lean level of buffering as a function of machine efficiency. . . . . . . . . Lean level of buffering as a function of line efficiency. . . . . . . . . . . . The behavior of functions and versus . . . . . . . . . . . . . . . . . Lean level of buffering as a function of , approximated (solid lines) and exact (dashed lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Serial production line. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Different distributions with identical coefficients of variation ( 3.3 The phase representation of the Erlang model for one machine. . . . . . . 4.1 Production-inventory-customer system . . . . . . . . . . . . . . . . . . . 4.2 degradation as a function of demand randomness: System 1, low case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 degradation as a function of demand randomness: System 2, medium case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . degradation as a function of demand randomness: System 3, high 4.4 case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Probability mass functions considered. . . . . . . . . . . . . . . . . . . . : System 1, low case. . . . . . . . . 4.6 as function of demand : System 2, medium case. . . . . . . 4.7 as function of demand : System 3, high case. . . . . . . . . 4.8 as function of demand 6.1 Assembly production line. . . . . . . . . . . . . . . . . . . . . . . . . . B.1 The phases of an Erlang distribution of order . . . . . . . . . . . . . . . ( states. . . . . . . . . . . . . . . . . . . . . . C.1 A Markov chain with
vii
. 3 . 9 . 15 . 16 . 19 . . ). . .
22 27 33 34 49
. 57 . 57 . . . . . . . .
58 60 61 61 62 67 72 74
LIST OF APPENDICES
APPENDIX A PROOFS FOR CHAPTER 2 . . . . . . . . . . . B THE ERLANG DISTRIBUTION . . . . . . . . C PHASE-TYPE DISTRIBUTIONS . . . . . . . . D THE INFINITESIMAL GENERATOR MATRIX E THE POWER METHOD . . . . . . . . . . . . . F PROOFS FOR CHAPTER 4 . . . . . . . . . . .
viii
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69 72 74 76 78 80
CHAPTER 1 INTRODUCTION
1.1 Motivation Production lines with unreliable machines usually contain in-process and finished goods buffers. The former are used to decouple the machines, thereby reducing mutual interference due to machine breakdowns. The latter are used to filter out random fluctuations of production and, thus, ensure timely delivery of shipments to the customer. To eliminate waste of resources, both in-process and finished goods buffers should be as small as possible, i.e., lean. But how lean can lean be? The goal of this research is to derive an answer to this question by providing simple and practical rules for determining buffer capacities necessary to achieve a desired production rate or customer satisfaction level. The problem is approached from a systems perspective, using mathematical models based on realistic assumptions. Manufacturing systems are complicated due to their stochastic nature. Machine downtime and random customer demand contribute to this complexity. Downtime may lead to a loss of throughput in manufacturing systems. To minimize this loss, in-process buffers are used. If these buffers have very large capacity, the machines are practically decoupled, and
the system production rate, (the average number of parts produced by the last machine per unit of time), is maximized. Obviously, large buffers may lead to excessive inventory, long part-in-process time, low quality, and other production problems resulting in a waste of resources. A well-established systems theory for designing buffer levels in production 1
lines could eliminate this waste. The current literature does not offer analytical results for such in-process and finished goods buffer capacity design. Therefore, it is of interest to determine the “just right” level of buffering based on data readily available on the factory floor.
1.2 Problems Addressed In any manufacturing environment, the aim is to satisfy customer demand on time while keeping production costs low. The production rate,
, can be used to measure the per-
formance of a production system. Maximizing while keeping the buffer levels as low as possible will reduce the costs. The probability of producing a certain (fixed or random) number of parts during a given shipping period, referred to as “due-time performance” (
), is the measure used for customer satisfaction. The capacity of the finished goods
buffer (,
) directly affects the value of
.
The following two problems are considered here: 1. In-Process Buffering: Determine the smallest in-process buffer capacities necessary to attain the desired
.
2. Finished Goods Buffering: Determine the smallest finished goods buffer capacity necessary to achieve the desired
.
The remainder of this thesis is structured as follows: A review of the literature is given in the next section. The in-process buffering problem for exponential and non-exponential machines are discussed in Chapters 2 and 3, respectively. The finished goods buffering problem is investigated in Chapter 4. Conclusions and future work are formulated in Chapters 5 and 6. Proofs and background information on some of the mathematical techniques used are included in the Appendix.
2
1.3 Literature Review Buffer capacity allocation in production lines has been studied quantitatively for over 50 years and numerous publications are available. The remarks below are intended to place current research in the framework of this literature. The literature related to leanness in in-process buffers and finished goods buffers is reviewed separately.
1.3.1 Literature on in-process buffering The literature on in-process buffer capacity allocation can be classified as shown in Figure 1.1. There are two different formulations. Formulation 1 seeks the optimal allocation of an a-priori given total buffer capacity, 2 seeks the smallest
, so that throughput is maximized. Formulation
and its allocation so that an a-priori given desired throughput is
achieved (if at all possible). Each of these formulations may have two methods of solution: algorithmic and rule-based. Algorithmic methods lead to a computer code that provides a solution to a corresponding formulation. Rule-based methods give simple rules for either the best or good (i.e., suboptimal) solution of each formulation. The algorithmic and rule-based solutions of Formulation 1 can be found in [1]-[13] and [14]-[23], respectively.
Buffer capacity allocation problem
Formulation 1: Find optimal * allocation of given N so that throughput is maximized
Solution 1.1: Algorithmic
Formulation 2: * Find the smallest N and its optimal allocation so that the desired throughput is achieved
Solution 2.1:
Solution 1.2: Rule−based
Algorithmic
Solution 2.2: Rule−based
Figure 1.1: Classification of the literature on the buffer capacity allocation problem.
3
Formulation 2 has been studied much less than Formulation 1. In fact, only a few recent papers propose its algorithmic solution (e.g., [24]-[26]). There seems to be only one paper, [27], devoted to the rule-based approach to Formulation 2. The research presented here is intended to contribute to this area. With respect to the machines, production lines can be classified into two groups: unreliable machines with fixed cycle time and reliable machines with random processing time. This work addresses the first group. With respect to the machine efficiency, production lines with unreliable machines can further be divided into two groups: balanced (i.e., the machines have identical up- and downtime distributions) and unbalanced. In this work the balanced case is addressed. Buffer capacity allocation in production lines, similar to those considered in this work, was first discussed in the classic papers [28]-[31]. A review of the early work in this area is given in [32]. In particular, [31] shows that the coefficient of variation of the downtime strongly affects the efficacy of buffering. In addition, [31] associates the buffer capacity allocation with the average downtime and states that buffering beyond five-downtime can hardly be justified. These results are confirmed and further quantified in the present work. Reference [33] also connects buffer allocation with downtime. It shows that onedowntime buffering is sufficient to regain about 50% of production losses if the downtime is constant (deterministic). It suggests that random (exponential) downtime may require twice the capacity to result in comparable gains. Another line of research on buffer capacity allocation is related to the so-called storage bowl phenomenon [34]. According to this phenomenon, more buffering should be assigned to middle machines in balanced lines. It can be shown, however, that unbalancing the buffering in lines with a downtime coefficient of variation less than 1 results in only 1-3% of throughput improvement, if at all. (For further details, see [35] where it is proved that optimal buffers are of equal capacity, if the work is distributed according to the optimal bowl.) Since this improvement is quite small, the present study does not consider bowltype storage allocations and assigns equal capacity to all buffers. Finally, there exists a large body of literature on numerical algorithms that calculate the optimal buffer allocation (see [36]-[38] for representative publications). The current work 4
does not address this issue. Thus, for in-process buffers, this research follows references [31] and [33] and provides additional results on rules-of-thumb for buffer capacity allocation necessary to accommodate downtime and achieve the desired production rate of serial production lines. One of the main purposes of this study is to generalize the results using several different machine models, such as those based on the exponential, gamma, log-normal, and Weibull distributions. Up to now, very few researchers have developed manufacturing system models based on distributions without the so called “memoryless” property. The main reason for this is the complexity and computational inefficiency of such models. In the current literature, these models typically use phase-type distributions [39]-[41] instead of the exponential or Bernoulli models. A chapter of [40] is devoted to probability distributions of phase-type. The basic structure, properties and special cases are explained in detail. This book is a common reference in most papers related to such distributions. A phase-type approach is proposed in [42] to derive optimal inspection and replacement policies for semi-Markovian deteriorating systems. In this study, the general sojourn time distributions of a semi-Markovian maintenance model are approximated by phase-type distributions. The paper contains several useful numerical examples. A new algorithm to compute the internal availability distribution for systems having only one operational state was introduced in [43]. Interval availability is defined as the fraction of time during which a system is in operation over a finite observation period. The case of exponential failures and phase-type repairs is considered. The two-machine case was considered in [44] where service times for the two machines are Erlang while failure and repair times are assumed to be exponential random variables. The paper presents an efficient method to solve analytically the steady state probabilities of the system. Altiok [45] considers a serial production line where the operation and repair times of the machines are assumed to be phase-type. Time until a breakdown occurs is exponentially distributed. A numerical solution technique for the production rate is presented and a two-machine example is included. An approximate analysis of queues in series with phase5
type service times is presented in [46]. A decomposition approach for analyzing production lines with general service times and finite buffers is studied in [47]. The method is computationally highly efficient and it has an acceptable error level.
1.3.2 Literature on finished goods buffering Production/inventory systems have been considered in numerous publications for over two decades (see monographs [48]-[50] and articles [51]-[64]). In these studies, the demand is almost always assumed to be random. Production is either deterministic or random and, in many models, instantaneous. The finished goods buffer is always assumed (perhaps, tacitly) to be infinite. The problems considered typically center on optimal replenishment policies and often use queueing theory methods. Although this literature offers many important results and insights, the issue of Due-Time Performance for random demand has not been addressed. The study of the Due-Time Performance problem begins with [37]. The case of a single machine with Markovian reliability statistics is analyzed both exactly and asymptotically (with respect to the length of the shipping period). Some properties of the function of system parameters are analyzed. For longer lines, the
as a
is investigated in
[65] and [66]. Paper [67] studies the estimation of the distribution and variance of time to produce a fixed lot size with a single unreliable machine. The production systems analyzed in these references have no finished goods buffers.
Although production systems with ,
s have received considerable attention in the liter-
ature, the emphasis has been mostly on production control and scheduling, see, for instance, [68] and [69]. Analytical methods for evaluating the
for a given production system
and specified constant demand (analysis problem) were introduced in [70]-[74].
1.4 Original Contributions The novel contributions of this dissertation are: Normalization of the buffer capacities by the average machine downtime to obtain a
6
simple dimensionless parameter referred to as the level of buffering. Derivation of closed formulas for calculating the lean level of buffering in serial production lines with exponential machines. Introduction of empirical laws for calculating the lean level of buffering in serial lines with non-exponential machines. Quantification of an analytical method for calculationg
in production-inventory-
customer systems with random demand. Analysis of
degradation as a function of demand variability.
7
CHAPTER 2 IN-PROCESS BUFFERING: EXPONENTIAL MACHINES
In this chapter, serial production lines with unreliable machines and exponentially distributed up- and downtime are considered. The capacity of the buffer is quantified in units of average downtime. For instance, “ -downtime buffer” denotes the capacity of a buffer
capable of storing the number of parts produced during average downtimes. The number
is referred to as the Level of Buffering ( ). The overall line efficiency,
as the ratio of the production rate of the line with infinite
ciency
equal to
. The question addressed is: How small can
, is quantified
to that of the line with
be so that the desired line effi-
is achieved? The smallest, i.e., lean, that guarantees
referred to as the Lean Level of Buffering ( ).
is denoted as
and is
Quantitative methods for calculating lean level of buffering in serial production lines with machines having exponentially distributed up- and downtime are developed. More specifically, closed formulas, which quantify
as a function of system parameters are
derived. Along with quantitative results, several insights concerning lean buffering in serial lines with identical exponential machines are provided. These include: Lean systems are just-in-time ( where
and
), i.e., need no buffering, if and only if
are the machine and line efficiencies, respectively, and
number of machines in the system. 8
, is the
depends on three variables: machine efficiency, line efficiency, and the number
of machines in the system; it does not depend on the average up- and downtime of the machines explicitly.
is a concave function of the machine efficiency reaching its maximum at around .
growth rate, approaching infinity as
is a monotonically increasing function of the line efficiency, with increasing
(
.
is a monotonically increasing function of the number of machines in the sys-
tem but with a decreasing growth rate. As a result, practically sufficient for lines with any
(.
for 10-machine lines is
2.1 Model The block diagram of the production system considered is shown in Figure 2.1, where the circles represent the machines and the rectangles are the buffers. Although the development reported here can be carried out using various assumptions on the machines and buffers, and similar results can be obtained, the so-called flow (or continuous flow, [76]) model with deterministic processing time is adopted, where the material processed is viewed as the flow of a fluid and processing of one part requires a fixed time (referred to as the machine cycle time). This model is appropriate when the average up- and downtime of the machines are much larger than the cycle times. This often takes place in machining and assembly lines in large volume manufacturing systems.
m1
b1
m2
b2
m M-2
b M-2
m M-1
b M-1
mM
Figure 2.1: Serial production line. Specifically, the production line under consideration is defined by the following assumptions: 9
(i). Each machine
#, & (
, has two states: up and down. When up, the
machine is capable of processing one part per cycle time. The cycle times of all machines are the same. (ii). The up- and downtimes of each machine are random variables distributed exponentially with breakdown and repair rates and , respectively and independently of the buffer occupancy. Both and are in units of ( (cycle time). (iii). Each buffer is capable of storing
parts,
. The capacity
implies that the amount of work produced during infinitesimal interval (in units of cycle time) is immediately transferred to the subsequent machine, if it is up. The capacity
implies that the work processed by a machine during is placed in the subsequent buffer, if it is not full. (iv). Machine is empty at time
# , & , is starved at time if it is up at time , buffer # . and # . does not place any work in this buffer at time . Machine .
cannot be starved.
# , & ( ( , is blocked at time if it is up at time , buffer # and # . fails to take any work from this buffer at time . Machine /
(v). Machine is full at time
cannot be blocked. Remarks: Assumptions (i)–(iii) imply that all machines are identical and all buffers are of equal capacity. These assumptions are made in order to provide a compact characterization of lean buffering. Assumption (ii) implies that time-dependent failures, rather than operation-dependent failures, are considered. Although the latter are, perhaps, more prevalent in practice, the former lead to simpler analysis. Since for most practical values of machine and buffer parameters the two failure modes exhibit similar performance [48], the simpler version is adopted here. Assumption (ii) implies, in particular, that the average up- and downtime of the machines are measured in units of cycle time, and the efficiency of each machine in
10
isolation is:
2.2 Problem Formulation As mentioned earlier, it is convenient to represent the buffer capacity in units of average downtime. This representation, referred to as the Level of Buffering ( ), is defined by
(2.1)
Another normalization used in this work is referred to as the Line Efficiency, introduced as follows: Let
. It is
denote the production rate (i.e., the average number of
parts produced by the last machine per cycle time) of the line when the level of buffering is
. Let denote the production rate of the same line with infinite capacity
buffering. Then
Obviously,
( . The smallest
(2.2)
necessary to ensure the desired
is referred to
.
as the Lean Level of Buffering ( ) and denoted as
The problem addressed in this work is: Given the production line defined by assumptions (i)–(v), develop a method for calculating
.
A solution to this problem is given in Section 2.4. The results obtained in this research are due to the parametrization introduced here.
2.3 Methods of Analysis The analysis described in this work is based on a technique for production rate evaluation developed in [81]. This technique is applicable to serial lines defined by assumptions (i)–(v) even if the machines and buffers are non-identical, i.e., each machine downtime distributed exponentially with parameters and each buffer # has the capacity
#,& (
outlined (see [81] for details). 11
#
and # , respectively, &
(
#
has up- and
(
,
. Below, this technique is briefly
In the case of two-machine lines,
is calculated according to the following formula:
where # , &
(
1 . ' . ' . "
(
. (
1 ' .' . . "
, 2, is the machine efficiency,
# and function
(2.3)
.' . ' .
(2.4)
. . .
#
is given by,
1 . . ' .
#
#
if
. ( ( . % . . . ' . . . Function 1 ' . . . is obtained from .' . ' .
if
(2.5)
(2.6)
by replacing all quanti-
ties with subscript 1 (respectively, 2) by the similar quantities with subscript 2 (respectively, 1).
For
-machine lines,
is evaluated using an iterative aggregation procedure
based on (2.3)–(2.6). This procedure consists of the so-called backward and forward aggregation. In the backward aggregation, the last two machines are aggregated in a single machine "!/ . , defined by parameters #!/ . and $ /! . . Then %!/ . is aggregated with
/
to result in
!/ , which is then aggregated with
until all the machines are aggregated in
/
& to give
! . . In the forward aggregation,
!/ & , and so on
.
is aggregated
with "! to produce (' , which is then aggregated with )!& to give (&' and so on until all
the machines are aggregated in *'/ . Then the process is repeated anew.
Formally, this recursive procedure can be represented as follows:
(
#! + #! +
(
(
( 1 #! . + ' # ! . ,+ ( #' + ' $# ' , + # ( ( & .
.
. / .021 43 65 07. 1 43 65 08 43 65 08 93 5 0 -
(
12
(
&
( (2.7)
(
#' + # ' +
(
( 1 #' . , + ' $ # ' . ,+ ( #! + ( # ! + ( 08 43 . 65 08 93 . 5 /.0 93 . 5 0. 43 . 65 0 .
(
-
( !# . &
&
with boundary conditions
'. + .' + !/ + /! + )+ (
(2.8)
and initial conditions
#' # ' & where function
(
(2.9)
is defined by (2.5).
Recursion (2.7)–(2.9) is convergent and the following limits exist:
#' ,+ #'
3
$#' + # '
3
3
3
#! + #!
#! ,+ #! & +(
(2.10)
Using these limits, the estimate, , of the production rate is defined as follows:
. ' . ' / ' / . / . '/ !.
#' ( (
#! . ' #! . #' # ' # " #! . ( #' ' # ' #! . ' #! . # " #! . #! . #' ' # ' # " ( # ' . # ' . #! ' #! # . "
(2.11)
where '#
#'
$# '
#! #! & # ! #! #'
(
(2.12)
It is shown in [81] that (2.11) provides an accurate estimate of the production rate, typically within
(
of its value determined by simulations.
Expressions (2.3)–(2.12) are used below for evaluating
13
.
2.4 Solution 2.4.1 Lean buffering in two-machine lines
.
When
and
.
, expression (2.5) simplifies to
1 . ' . ' . 1 '
Therefore, as it follows from (2.3), the equation that defines ity necessary and sufficient to ensure line efficiency
Solving for
results in
Since
(
"
, i.e., the lean buffer capac-
, can be written as follows:
1 ' "
. .
if
otherwise
(2.13)
, this implies:
Theorem 2.1 [27] In two-machine lines defined by assumptions (i)–(v), the lean level of buffering is given by
.
if
otherwise
Figures 2.2a and 2.3a show the behavior of
(2.14)
as a function of and
, respectively.
The following observations follow from these figures and expression (2.14):
is a quadratic function of , increasing for
and decreasing for
.
If
does not depend explicitly on
and
.
, the lean system must contain buffering. For instance, if
,
is approximately 3 downtimes. 14
and
20
20
kexp E
15
10
5
0
10
5
0
0.2
0.4
0.6
0.8
0
1
0
0.2
0.4
e
(a)
0.6
0.8
e
(b)
20
15
kexp E
kexp E
15
E = 0.85 E = 0.90 E = 0.95
10
5
0
0
0.2
0.4
0.6
0.8
1
e
(c)
Figure 2.2: Lean level of buffering as a function of machine efficiency.
15
1
20
20
kexp E
15
10
5
0 0.8
10
5
0.85
0.9
0 0.8
0.95
0.85
E
(a)
0.9 E
(b)
20
15
kexp E
kexp E
15
e = 0.75 e = 0.85 e = 0.95
10
5
0 0.8
0.85
0.9
0.95
E
(c)
Figure 2.3: Lean level of buffering as a function of line efficiency.
16
0.95
If
, the lean system is just-in-time (
example, if
,
and
), i.e., no buffering is necessary. For
operation is acceptable.
is a monotonically increasing function of
(
approaching infinity when
with hyperbolically increasing rate,
.
2.4.2 Lean buffering in three-machine lines Since no closed formula for
in three-machine lines is available,
can-
not be evaluated using arguments similar to those above. However, based on the recursive aggregation procedure (2.7)–(2.9), the following can be proved: Theorem 2.2 In three-machine serial lines defined by assumptions (i)–(v), the lean level of buffering is given by
. . . .
. . . . .
if
otherwise
Proof: See Appendix A. Figures 2.2b and 2.3b show the behavior of
(2.15)
as a function of
and
,
respectively. These figures and (2.15) lead to the following observations: Although
is not quadratic, its qualitative behavior is similar to a parabola,
reaching its maximum around
If
again is independent of
and
.
explicitly.
is roughly 6. If
and
, is close to 1; note that for
, a lean system must be buffered. For example, if
and
, ,
this case required no buffering. If
, lean system is
region of ’s where
. Since
(
and, therefore,
is acceptable is smaller than that for 17
.
, the
is monotonically increasing as a function of
, with an increasing rate.
-machine lines
2.4.3 Lean buffering in
Unlike the two- and three-machine cases, no exact closed formula for
can be derived. However, based on the recursive aggregation procedure (2.7)–(2.9), the following can be proved:
-machine serial lines defined by assumptions (i)–(v), the lean level
Theorem 2.3 In
of buffering is given by
.
-
- . - - - - - - - -
. - . - .
-
if
(2.16)
otherwise
where
1 '/
/'
is defined by (2.5), (2.6) and '/
' /'
!/ . ' /! .
!/ . ' /! .
(2.17)
are the steady states of the aggrega-
tion procedure (2.7)–(2.9). Proof: See Appendix A. As it follows from [81], function (2.17) represents the approximation of the probability that buffer / is empty. In general, this function cannot be evaluated in a closed form and, therefore, approximations are necessary. They are carried out as follows: When
, function (2.17) can be evaluated in the closed form as (see the proof of
Theorem 2.2):
+(
(2.18)
, function (2.17) can be calculated using the recursive procedure (2.7)–(2.9). Based on these calculations, the behavior of for is illustrated in Figure 2.4 as a For
function of for several values of
and
(solid lines). This behavior can be captured by 18
0.15
0.15 actual approximation
Q
0.1
Q
0.1
0.05
0.05
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
e
(a)
e
,
(b)
,
0.15
0.1 Q
0
0.05
0
0.2
0.4
0.6
0.8
1
e
(c)
,
Figure 2.4: The behavior of functions
19
and
versus .
0.8
1
the expression:
where , , , and
are functions of
and approximates well 1
function 1
for all other values of
and
;- :
-
(
for
(2.19)
) is characterized in Table 2.1. As one can see, it
. Since this is commensurable with the accuracy of the data available in most
, i.e.
.
-
: - : - : . - : - -: : =- : - : - : . - : - : =- :
. ;- : - : .
if
(2.20)
-machine case.
in the
, expressions (2.19), (2.20) coincide with (2.15). Expressions (2.19), (2.20) define as a function of three variables:
Note that for
is substituted
otherwise
provide a closed, however approximate, formula for
a fixed
. Using trial and error,
practical applications, it follows that expressions (2.19) and (2.16), where for
is shown in Figure 2.4 by broken lines. Quantitatively, the accuracy
of this approximation (i.e.,
(
3 54 6,798
to be selected so that
The behavior of
