IIE Transactions (1999) 31, 99±111
CONWIP-based production lines with multiple bottlenecks:
performance and design implications
E.M. DAR-EL1 , Y.T. HERER2 and M. MASIN 2
1
Faculty of Industrial Engineering and Management, Technion ± Israel Institute of Technology, Haifa 32000, Israel
Department of Industrial Engineering, Tel-Aviv University, Ramat-Aviv 69978, Israel
E-mail: yale@eng.tau.ac.il
2
Received June 1996 and accepted January 1998
This research focuses on CONWIP, a closed production control system where all containers traverse a circuit incorporating the
entire production line. We develop estimates, for an important level of work in process inventory, for four important performance
measures: the means and variances of time between departures and ¯owtime. We develop our estimates through the concept of a
``conceptual bottleneck machine''. This concept enables us to develop an analogy between deterministic and stochastic systems.
This concept also allows us to handle migrating bottlenecks, an issue generally neglected. The model is widely applicable, assuming
only ®nite means and variances of the processing time distributions. We test our model computationally, both against existing
models and on a wide range of randomly generated problems. Finally we detail insights, obtained from our analytical model, into
how CONWIP production systems operate. These insights enable us to explain the sources of the values of our performance
measures, thus aiding system design and modi®cation.
1. Introduction
An important aspect for operating any manufacturing
system is to predict the system's performance. The best
performance measure, the one every company is concerned with, is Net Present Value (NPV). However, NPV
is dicult to measure directly. Thus, on a day-to-day
basis one normally uses surrogate operational measures
such as throughput rate, ¯owtime, and inventory levels,
for evaluating system performance. The correct de®nition
and accurate estimation of these operational parameters
are important for estimating the behavior of monetary
criteria. In addition to monetary considerations, accurate
estimates of operational measures of system performance
can aid management in the level of service they provide to
customers.
In this work we: (1) develop estimates for four important performance measures: the means and variances
of time between departures and ¯owtime; (2) computationally test our model, both against existing models and
on a wide range of randomly generated problems; and
(3) provide insights, derived from our model, into how
the production system operates. We show how these
insights can aid management in system design and
modi®cation.
Most papers which address the performance of systems
concentrate on ®rst order performance measures (means).
However, to fully understand how any system operates
0740-817X
Ó
1999 ``IIE''
one needs estimates of the second order performance
measures (variances). This information can be used in
models such as the one developed by Hopp et al. [1] which
help determine production quotas in pull manufacturing
systems. Variances are important because they describe
uncertainty in the system and the presence of large variances is sometimes more costly than sub-optimal mean
values. A detailed example demonstrating that control of
variances is sometimes more important than reducing
mean values is discussed in Section 5.
Analytical models for predicting mean and variance
parameters are useful for planning purposes and are superior to simulation models (as used by Hendricks and
McClain [2] for bu€ered production lines). Both analytical and simulation models can describe the system's behavior, but only analytical models can explain and
highlight the sources of this behavior. The knowledge
thus gained can aid management in system design and
modi®cation (discussed in Section 5).
Analytical models for estimating the variances of system performance measures are considered by some authors. These models can be separated into two groups.
The ®rst group addresses bu€ered serial production lines.
Miltenberg [3] has estimated the variance of the number
of units produced on a serial line; however his computational times are prohibitively large. Hendricks [4,5] has
studied serial lines in which machines have exponential
distributions.
100
The second group addresses CONWIP (CONstant
Work In Process inventory) production systems. Duenyas
and Hopp [6] have estimated the system throughput
variance assuming exponential processing times. Duenyas
et al. [7] have considered the throughput variance assuming constant processing times on unreliable machines
(down and up times are again exponential). Our research
belongs to this category.
Performance means in CONWIP lines have also received attention in the literature. Hopp and Spearman [8]
have estimated the mean throughput assuming constant
processing times on unreliable machines (exponential
failures). Duenyas and Hopp [9] have extended this estimate to CONWIP assembly systems (i.e., an assembly
station fed by multiple CONWIP lines). The problems
considered by Duenyas et al. [7] and Hopp and Spearman
[8] are very similar to ours, and we compare our method
to theirs in Section 4.
1.1. The system
CONWIP is a new production management technique,
proposed by Spearman et al. [10] and Spearman and
Zazanis [11]. CONWIP is a closed production management system in which a ®xed number of containers (or
cards) traverse a circuit that includes the entire production line (see Fig. 1). When a container reaches the end of
the line the ®nished product is removed. The container is
then sent back to the beginning of the line where it waits
in a queue to receive another batch of items. During each
container's cycle all items in the container are of the same
type. The amount of material put into the container is set
by a predetermined transfer lot size.
Since CONWIP systems are closed manufacturing
systems, as is Kanban, they have the following advantages over open systems: easier control, smaller variances,
and smaller average Work In Process (WIP) levels (and
thus also shorter ¯owtimes) for the same throughput.
They are also self-regulating. In addition, as described in
Spearman and Zazanis [11], CONWIP systems have the
following advantages over Kanban: (1) they are very robust regarding changes in the production environment
and are easier to forecast; (2) they easily handle the introduction of new products and changes in the product
mix; (3) they cope with ¯owshop operations with large
Fig. 1. An illustration of a CONWIP system.
Dar-El et al.
set-up times and permit a large product mix; and (4)
CONWIP systems also yield larger throughput than
Kanban Systems for the same number of containers
(maximum inventory) [12,13], even for systems with yield
losses [14].
1.2. Performance measures
Operational parameters need to be estimated to enable us
to predict job completion times in order to aid management in the optimization of monetary criteria, customer
service, and system design. In this work we estimate two
operational parameters: Time Between Departures (TBD)
and ¯owtime. We use TBD since it allows us to calculate
completion times and ®x realistic due dates. Mean TBD is
the inverse of the mean throughput rate and therefore
may be used in its place. Flowtime is another important
parameter that characterizes the system's performance
and is thus of interest to management. The variance of
TBD and the variance of ¯owtime are included in our
measures of system performance.
It is worth noting that the measure, mean TBD, is
`equivalent' to the criterion of mean ¯owtime for a given
average WIP level (Little's Law, Little [15]). However, the
TBD variance and ¯owtime variance are only correlated;
one variance value does not determine the other.
1.3. The desired level of WIP
What is the role of WIP in a manufacturing system? WIP
ensures continuity of production by bu€ering the bottleneck resources. As WIP increases so does throughput,
up to the maximum capacity of the manufacturing system. But WIP has a cost and too much of it is simply
wasteful. For example, WIP increases the mean and
variance of ¯owtime resulting in long lead times, poor
forecasting, and late feedback.
Generally, we want as small a WIP as possible that
allows us to approach the maximum throughput of the
system. In other words the desired level of WIP is somewhere just past the ``knee'' of the TBD versus WIP curve.
That is, the mean TBD decreases quickly until the WIP is
raised to the desired level and decreases much more
slowly afterwards. In this paper we ®nd such a WIP level
and estimate the system performance for this WIP level.
Our model is based on an analogy to the deterministic
system (with no machine breakdowns). In Section 2 we
examine how the deterministic system operates, and then
use these results to de®ne a conceptual bottleneck machine. The concepts are then used in Section 3 to explain
the behavior of, and to gain insights into, stochastic
systems. In Section 3 we also develop our estimates of the
performance measures. We describe and present the results of our computational tests in Section 4. Finally we
present design implications of our model in Section 5 and
our conclusions in Section 6.
CONWIP-based production lines with multiple bottlenecks
2. Production environment: a deterministic analogy
E€ective operation of a closed manufacturing system
requires the selection of an appropriate WIP level
(number of containers). The CONWIP system is no exception to this requirement [10]. Choosing a WIP level
determines both the TBD and ¯owtime. From Little's law
[15] we know that for a given mean WIP level, the criterion of mean TBD is equivalent to the criterion of mean
¯owtime. However, when selecting a WIP level, there is a
trade o€ between TBD and ¯owtime. A low WIP level
yields short ¯owtimes and high TBD (insucient
throughput) whereas a high WIP level yields short TBD
(sucient throughput), but enormously long ¯owtimes
(resulting in high inventory holding costs).
For a CONWIP production system with in®nite demand, the average WIP level is equal to the maximum
WIP level. To gain insights into the system and establish a
desirable WIP level we ®rst consider the amount of WIP
(containers) needed in a deterministic system. In such
systems we can achieve the
ideal situation: the bottleneck machine works continuously, without a queue before it or in any other part of
the system.
The bottleneck machine is the machine with the largest
(deterministic) processing time. Since the bottleneck machine works continuously the WIP level needed to achieve
the ideal situation would also give us maximal throughput. For deterministic production lines these conditions
are satis®ed when the number of containers n^ † is equal
to:
n^ ˆ
M
X
X^i =X^BN ;
1†
iˆ1
where M is the number of machines, X^i is the processing
time on machine i, and X^BN
ÿ is the processing time
on
the bottleneck machine X^BN ˆ maxi2f1;...;Mg X^i [8].
Throughout this paper we use the symbol ^ to indicate
values associated
system. We note
P with the deterministic
PM
^ ^
^
^
here that n^ ˆ M
iˆ1 Xi =XBN iˆ1 XBN =XBN ˆ M.
In general, n^ will not be integral, whereas in practice
the number of containers must be an integer. For WIP
levels above n^ there is only one queue in the system: the
one in front of the bottleneck machine. For such WIP
levels (say n^), a container arriving at the bottleneck machine waits n^ ÿ n^ †X^BN units of time in the queue in front
of the bottleneck machine. If the number of containers is
n e, then
equal to the smallest integer greater than n^ , d^
the bottleneck machine works continuously and there are
minimal queues in the system.
We now turn to estimating our performance measures
in deterministic systems in order to use the insights gained
in stochastic systems. We ®rst estimate TBD. Consider
two containers, one immediately following the other
101
through a CONWIP production line in the ideal situation.
Start a stopwatch when the ®rst container leaves the
bottleneck machine. Since, by assumption, queues form
only in front of the bottleneck machine, this ®rst container proceeds to the end of the production line without
waiting in queues. Hence, the time when the ®rst container exits the system is equal to the sum of its processing
times on all of the machines following the bottleneck
machine. Since the bottleneck machine works continuously the second container begins processing on the
bottleneck machine immediately after the ®rst container
leaves. Therefore, the time when the second container
exits the system is equal to the sum of its processing time
on the bottleneck machine and on all subsequent machines. Since TBD is by de®nition equal to the completion time of the second container minus the completion
time of the ®rst container we have,
!
!
M
M
X
X
00
00
0
^
^
^
^
X
X ;
ÿ
2†
T ˆ X ‡
BN
jˆBN‡1
j
jˆBN‡1
j
where X^ represents the processing times for the second
0
container, and X^ represents the processing times for the
®rst container. Since processing times are deterministic,
Equation (2), and thus TBD, are equal to a deterministic
X^BN .
Now consider the ¯owtime in our deterministic setting.
obtainable, then the ¯owtime
If the WIP level of n^ is P
^
^ X^BN (from Equawould be a deterministic M
iˆ1 Xi ˆ n
tion 1). This expression can be generalized to WIP levels
above n^ . The derivation which follows can be simpli®ed
for the deterministic setting; however, since we will use it
in the next section when analyzing the stochastic environment, we present the derivation in a form appropriate
to the stochastic setting.
Consider the `th part through the production line, and
de®ne X^i` to be its processing time on machine i. The
¯owtime of the `th part is comprised of: (1) its processing
time before the bottleneck machine; (2) its waiting time
before the bottleneck machine; and (3) its processing
time on the bottleneck machine and on all machines
thereafter.
PMThe `sum of the ®rst and third components is
^
clearly
jˆ1 Xj . To calculate the second component
consider starting a stopwatch when part ` ÿ n^ leaves the
bottleneck machine. Note that part ` ÿ n^ is the part that
previously occupied the container now carrying part `.
The container (now carrying part `) arrives at the queue
in
the bottleneck machine at time
PM front`ÿ^n of PBNÿ1
^ ‡
^`
X
BN‡1 j
jˆ1 Xj ; however, before the bottleneck
machine can work on this part it must process all of the
parts in the n^ ÿ 1 remaining
containers, (i.e., parts
P`ÿ1
^k
` ÿ n^ ‡ 1; . . . ; ` ÿ 1) taking
kˆ`ÿ^
n‡1 XBN time units.
Hence, the queue
of the bottleneck
ÿP
P machine
P`ÿ1 time ink front
M
^ ÿ
^ `ÿ^n ‡ BNÿ1 X^ ` .
X
X
for part ` is kˆ`ÿ^
j
n‡1 BN
BN‡1 j
jˆ1
Adding all there parts together we obtain a total ¯owtime of
00
102
L^ ˆ
Dar-El et al.
M
X
jˆ1
ˆ
X^j` ‡
X̀
kˆ`ÿ^
n‡1
`ÿ1
X
kˆ`ÿ^
n‡1
k
‡
X^BN
k
X^BN
ÿ
M
X
jˆBN‡1
M
X
BN‡1
X^j` ÿ
X^j`ÿ^n ‡
M
X
jˆBN‡1
BNÿ1
X
X^j`ÿ^n :
jˆ1
!
X^j`
3†
Since processing times are deterministic, Equation (3) is
equal to n^X^BN .
Thus far, the analysis is valid for deterministic processing times only; however, processing times may not be
deterministic for any number of reasons. For example,
machines may breakdown from time to time, many parts
may be produced on the same line, and/or the processing
time for the parts may be truly stochastic. All these situations can be modeled (jointly or separately) as a single
part having a stochastic processing time. In this stochastic
situation, the problem of identifying the bottleneck machine arises. The question, ``What is the bottleneck machine for this stochastic environment?'' does not have a
clear answer. The naive choice, the machine with maximum mean processing time, could be a poor one. Generally the bottleneck ``¯oats'' dynamically from machine
to machine; the machine with maximum mean processing
time would be the unique bottleneck machine only with
extremely large WIP levels.
To deal with the question, ``What is the bottleneck
machine?'' we de®ne a conceptual machine that constrains the system's performance. We will think of this
conceptual machine as corresponding to the bottleneck
machine in the deterministic case and we call it the conceptual BottleNeck machine (BN). Using this analogy we
forecast the production system's behavior without making assumptions about the form of the processing time
distributions (except ®nite mean and variance).
In a stochastic environment we view all machines as
potentially contributing to the BN. We de®ne the distribution of the processing time on the BN to be the
distribution of the maximum processing time taken over
all machines. We must remember that the BN represents
a concept, i.e., a virtual, not real, machine. The BN can
be thought of as the machine that currently constrains
the system's operation, in the same way that the real and
unique bottleneck machine constrains the system's operation in the deterministic environment. Of course the
BN is not a machine, but our model is built on the
insights obtained by considering it to be a machine.
Hence we write about `the queue in front of the BN',
when we actually mean `the queue in front of the machine that corresponds to the conceptual bottleneck
machine'.
We realize that the BN concept is simplistic. However,
it is our opinion that simple models are better than
complicated ones. Furthermore, we feel that the contribution of a model should be judged on its performance
compared to other existing models (Section 4) and on its
contribution to aiding our understanding of the system
(Section 5).
We now rede®ne X^i to be the random variable representing the processing time on machine i, and denote it by
Xi . In mathematical terms we de®ne the processing time
on the BN as:
XBN max
i2f1;...;Mg
Xi †:
If we say that Xi is distributed with cumulative distribution function Fi t†, then the
Q cumulative distribution
function of XBN is FBN t† ˆ M
iˆ1 Fi t†. Machine i will be
considered the BN with probability equal to the probability that machine i has the longest processing time for a
given part. That is, with probability
!
Z1 Y
Fj t† fi t†dt;
ai ˆ P fXj < Xi for all j 6ˆ ig ˆ
0
j6ˆi
where fi t† represents the probability density function of
the processing time on machine i.
We now draw a direct analogy to the deterministic
system discussed above. Just as in the deterministic environment, we want the
ideal situation: the BN works continuously, without a
queue before it or in any other part of the system.
We also assume that as WIP grows, containers accumulate only in front of the BN, just as they accumulate only
in front of the bottleneck machine in the deterministic
case. We recognize that in stochastic systems queues may
occur simultaneously before more than one machine, thus
contradicting this assumption; however, this probability
is small for our choice of WIP level.
3. The CBN (Conceptual BottleNeck) model
In this section we develop an analytical model for predicting our performance measures: mean TBD, variance
of TBD, mean ¯owtime, and variance of ¯owtime. In
developing this model we assume only a ®nite mean and
variance of the processing time distributions; we make no
other assumptions of their form. Our analytical model,
which we call the Conceptual BottleNeck (CBN) model,
is a straightforward consequence of our assumptions
from the previous section: (1) there is a conceptual bottleneck machine called the BN that works continuously;
and (2) queues form only in front of the BN. We use the
following notation (recall that we have already de®ned Xi ,
XBN , ai , M, and n):
T = true steady-state TBD of the system;
T = estimate of T based on an analogy
with the deterministic system;
Ti = estimate of T when considering machine i to be the BN;
CONWIP-based production lines with multiple bottlenecks
L = true steady-state ¯owtime of the system;
L = estimate of L based on an analogy
with the deterministic system;
Li = estimate of L when considering machine i to be the BN;
XBNi , X:BNi = processing time on machine i when it
is (respectively, is not) considered the
BN;
n = number of containers needed to approximate the ideal situation in our
model.
In addition the following quantities are also de®ned:
li , r2i = mean and variance of Xi ;
lBN , r2BN = mean and variance of XBN ;
l:BNi , r2:BNi = mean and variance of X:BNi .
F:BNi t† ˆ P fXi < t jXi is not the BNg
! ,
Zt
Y
Fk s† ds
1 ÿ ai †:
ˆ fi s† 1 ÿ
k6ˆi
Having de®ned the cumulative distribution functions,
the probability
density functions areQ fBNi t†ˆ
Q
fi t†
k6ˆi Fk t† =ai and f:BNi t† ˆ fi t† 1 ÿ
k6ˆi Fk t† =
Q
M
d
F
t†
iˆ1 i
d FBN t††
ˆ
dt
dt
!
M
M
Y
X
X
fi t†
Fk t† ˆ
ai fBNi t†:
ˆ
fBN t† ˆ
iˆ1
k6ˆi
M
X
iˆ1
2
r2BN ˆ E‰XBN
Š ÿ l2BN ˆ
M
X
iˆ1
Put into words, Equation (4) says that the density of the
processing time distribution on the BN is simply the sum
of the densities of the processing times of each machine
when they are considered the BN, multiplied by the
probability that they are considered the BN. Recall that
M
X
iˆ1
2
ai E‰XBN
Š ÿ l2BN
i
We ®rst analyze the time between departures in the
stochastic environment. In the deterministic environment
we saw that T^ ˆ X^BN . This expression was derived from
Equation (2); this derivation does not hold here because
the processing times on the machines are stochastic.
However, we can use the same argument that was used
to derive Equation (2) to obtain our estimate T of T
which is based on an analogy with the deterministic
system:
00
‡
T ˆ XBN
M
X
jˆBN‡1
00
X:BN
ÿ
j
M
X
jˆBN‡1
0
X:BN
:
j
5†
If we condition on machine i being considered the BN, we
obtain the following formula for our estimate of T when
considering machine i to be the BN:
M
X
jˆi‡1
00
X:BN
ÿ
j
M
X
jˆi‡1
0
X:BN
:
j
6†
When developing our formulas for the mean and
variance of Ti we assume that all random variables in
Equations (5) and (6) are independent. This assumption is
not completely justi®ed for XBNi and X:BNj (i 6ˆ j); however, if we consider the BN as an independent machine
currently placed at the location of machine i this assumption seems reasonable. In any case the dependency is
small. We can now determine the mean and variance of
Ti .
M
M
X
X
l:BNj ÿ
l:BNj ˆ lBNi ;
lTi ˆ lBNi ‡
jˆi‡1
4†
ai lBNi
ai r2BNi ‡ l2BNi † ÿ l2BN :
00
Ti ˆ XBN
‡
i
k6ˆi
1 ÿ ai †. Note that
lBN ˆ
iˆ1
FBNi t† ˆ P fXi < t jXi is the BNg
! ,
Zt
Y
Fk s† ds ai ;
ˆ fi s†
0
the BN is not a machine, rather an idea. Hence, this result
and the other results that follow should be viewed as a
means of developing a meaningful analogy to the deterministic system so as to obtain accurate estimates of our
performance measures.
We present here expressions for lBN and r2BN that will
be used below.
ˆ
In addition to what is de®ned above, we use the symbols l and r2 to indicate the mean and variance of particular quantities. Thus, lT and r2T represent the mean
and variance of T .
In the CBN model we use processing time distributions
consistent with the concept of the virtual bottleneck
machine. We de®ne the conditional distributions FBNi t†
and F:BNi t†, respectively, as the cumulative distribution
function of the processing time on machine i given that
machine i is the BN, and the cumulative distribution
function of the processing time on machine i given that
machine i is not the BN. Thus,
0
103
r2Ti ˆ r2BNi ‡
jˆi‡1
M
X
jˆi‡1
ˆ r2BNi ‡ 2
r2:BNj ‡
M
X
jˆi‡1
M
X
jˆi‡1
r2:BNj
r2:BNj :
Using the law of total probability we determine the mean
and the variance of our estimate of T as follows:
104
Dar-El et al.
lT ˆ
r2T
ˆ
"
M
X
iˆ1
ˆ
M
X
iˆ1
M
X
iˆ1
ai lTi ˆ
M
X
iˆ1
#
ai E‰Ti2 Š
ai r2BNi ‡
ÿ lT † ˆ
M
X
ai 2
iˆ1
jˆ2
r2:BNj
M
X
iˆ1
M
X
jˆi‡1
M
X
iˆ1
ˆ r2BN ‡ 2
"
2
‡
M
X
ai lBNi ˆ lBN ;
‡
!
M
X
iˆ1
#
ai l2Ti
ÿ l2T
r2:BNj
ai l2BNi
jÿ1
X
ai r2Ti
7†
ÿ
Whereas both of the methods given above for developing
an estimate for TBD yield the same expressions for lT
and r2T , they yield di€erent expressions for r2L (they yield
the same expressions for lL ). After computationally
comparing the expressions resulting from both methods
we found that the second method of derivation yields a
more accurate estimate. Hence, we derive the expressions
directly from Equation (10)
lL ˆ nlBN :
l2BN
r2L ˆ nr2BN ‡ 2
!
ai :
8†
M
X
jˆBN‡1
11†
r2:BNj
ˆ r2T ‡ n ÿ 1†r2BN :
iˆ1
12†
13†
As expected, our estimate of the mean TBD is equal to
the mean processing time on the BN. This result
strengthens our analogy to the deterministic case. The
mean of our estimate of the TBD in the deterministic
environment (respectively, stochastic environment), is
equal to the processing time on the bottleneck machine
(respectively, the BN).
Our estimate
Pjÿ1of the variance of TBD is very interesting.
ai can be interpreted as the probability
Note that iˆ1
that machine j follows the BN. Hence, we see that our
model predicts that the machines following the BN impact
on the TBD variance twice as much as the BN itself, while
machines before the BN have no impact. Hence, if we want
to reduce the TBD variance, our model suggests reducing
the processing time variance on the post-BN machines. We
will consider this point in detail in Section 5.
In retrospect we could have used a more direct method
to derive Equations (7) and (8). If we consider Equation
(5) directly we see that lT ˆ lBN as in Equation (7) and
we obtain
M
X
r2:BNj :
9†
r2T ˆ r2BN ‡ 2
Equation (13) is obtained directly from Equation (12) by
using Equation (9).
Equation (13) explicitly describes how our model views
the relationship between TBD and ¯owtime variance. In
particular we see that our model predicts that the ¯owtime variance will be strictly larger than the TBD variance. This relationship also shows us that, as was the case
with TBD variance, we predict that the ¯owtime variance
will depend on machines after the BN and not on machines before the BN. But unlike the TBD variance our
model predicts that the ¯owtime variance will depend
heavily on the variance of the BN itself.
Using Little's law [15] and our analogy to the deterministic system, we can calculate the number of containers, n , needed to approximate the ideal situation (i.e.,
the BN works continuously with no queues in the system)
in our model. If the ideal situation were truly obtained (an
impossible situation in a stochastic P
environment) then
our estimate of the ¯owtime is simply M
jˆ1 Xj , which can
be shown to be equal to
Since the BN is not an actual machine the last term of
Equation (9) must be modi®ed. Instead of summing over
all machines after the BN, we must sum over all machines
at all stages, multiplying each term by the probability that
the machine is after the BN; this is Equation (8).
We now turn our attention to estimating the ¯owtime.
Again, we use the analysis from Section 2 and extend it to
the stochastic environment. In the deterministic environment we found that L^ ˆ n^X^BN . This expression was
derived from Equation (3), but since processing times
here are stochastic, this derivation, as well, no longer
holds. Following the same argument that was used to
derive Equation (3) we obtain our estimate L of L which
is based on an analogy with the deterministic system:
and therefore,
XBN ‡
X̀
kˆ`ÿn‡1
k
XBN
‡
M
X
jˆBN‡1
Xj` ÿ
M
X
jˆBN‡1
Xj`ÿn :
10†
1 ÿ ai †X:BNi ;
14†
iˆ1
jˆBN‡1
Lˆ
M
X
lL ˆ lBN ‡
M
X
iˆ1
1 ÿ ai †l:BNi :
15†
Equation (15), and not Equation (11), is used when applying Little's law because when trying to calculate n we
cannot use an expression containing n . Equation (14)
can be interpreted as saying that our estimate of the
¯owtime is the processing time on the BN plus the processing time on all other machines when they are not the
BN. Finally, computing n we obtain:
P
lBN ‡ M
1
jˆ1 1 ÿ aj †l:BNj
n ˆ lL ˆ
lBN
lT
PM ÿ
lBN ‡ jˆ1 1 ÿ aj lBN
ˆ M:
16†
lBN
CONWIP-based production lines with multiple bottlenecks
We feel that it is uninformative to examine the case
where the actual number of containers is less than n . In
such a system our model predicts that the BN would not
work continuously, and thus the system would be underutilized. On the other hand, if the number of containers,
n, would be greater than n , then our model predicts, as
we explained with respect to the deterministic system, a
queue of n ÿ n containers would form in front of the
conceptual bottleneck machine. Clearly this queue will
not be a constant physical presence, but it is a useful
component of our estimates.
Any inaccuracy in our model is likely to be caused by
the system's WIP level. When the number of containers
is equal to one, P
the mean TBD is equal to its upper
bound, which is M
iˆ1 li . As the number of containers is
increased the mean TBD decreases to its lower bound,
which is maxi li †. This lower bound is achieved when
the WIP (the number of containers) is in®nite. Our
model works best when the number of containers is n .
When this is not the case (this happens frequently because the number of containers must be integral
whereas n is an integer only in degenerate cases), we
expect errors to occur. In particular, we would expect
the mean TBD to be greater or smaller than the model's
lT depending on whether the number of containers is
smaller or greater than n . As stated previously, we are
not interested in the case when n < n ; when the number of containers is only slightly greater than n , say
dn e, we would expect the above mentioned deviation to
be small.
Even though the model is built around a single WIP
level, we feel that this WIP level is of particular interest.
We observed from our experiments that if the coecients
of variation of the processing times are less than one, then
n lies just past the ``knee'' of the TBD versus n curve.
That is, the mean TBD decreases quickly until the number of containers is raised to n and decreases much more
slowly when the number of containers is raised beyond n .
For all of these reasons we focus our computational investigations, which are detailed in the next section, on
CONWIP production systems using dn e containers. Also
note that dn e is the unique WIP level that yields the
minimum TBD (i.e., maximum throughput) with minimal
WIP in deterministic systems.
4. Simulation experiments
Our model for predicting the operational parameters of
CONWIP production systems is based on the concept of
BN and the corresponding analogy to deterministic systems. When developing our model we assumed that the
BN works continuously and that queues form only in
front of the BN. We now present the details of a simulation experiment designed to both test this concept and
the usefulness of our model.
105
The experiment is divided into two parts. The ®rst part
compares our model to existing models that compute
similar operational parameters for CONWIP production
lines with very speci®c processing time distributions. The
second part is a wide-ranging test on randomly generated
problems. In both parts we report the percentage error
(our model's estimate minus the simulated value, divided
by the simulated value) for all four of our operational
parameters. Since the percentage error for mean TBD
and mean ¯owtime are always identical, we report the
number only once (under the heading l's). Since the units
for mean and standard deviation are the same, the error
in estimating the variance information is reported in
terms of the standard deviation.
Before detailing our experiments we note that another
method for estimating the mean values of our operational
parameters neither mentioned above nor tested below
exists. This is Mean Value Analysis (MVA). MVA was
developed by Reiser and Lavenberg [15] and it is a popular technique for ®nding mean values in tandem queuing
networks. MVA gives exact values for CONWIP systems
with machines that have exponentially distributed processing times. Our initial tests indicated that our method
clearly out-performed MVA for estimating lT and lL
when the machines had non-exponential processing time
distributions. In addition, the results of Hopp and
Spearman [8] also indicate that their method out-performed MVA. More importantly, MVA cannot be used
to estimate the variance of the operational measures.
4.1. Comparison with existing models
In this section, subject to the explanation below, we
compare our method for estimating performance measures to two existing models, that of Duenyas et al. [7]
and that of Hopp and Spearman [8]. The purpose of these
experiments is to compare the quality of our model's
estimates to existing models. Note that the existing
models were developed for systems whose machines are
characterized by deterministic processing times and random (exponential) breakdowns. In addition the models of
Duenyas et al. [7] and Hopp and Spearman [8] estimate
throughput information; our does not estimate throughput. Rather, we estimate time between departures and
¯owtime. We do not compare our model directly to the
one found in Duenyas and Hopp [6] who consider
CONWIP lines in which each machine has an exponential
processing time distribution. However, we investigate
these systems in Section 4.2.
There are two ways to model the time between machine
failures: (1) failures occur in relationship to the amount of
clock time since the last failure (as used in Duenyas et al.
[7] and Hopp and Spearman [8]); and (2) failures occur in
relationship to the amount of time the machine has been
working since the last failure (as used in this paper). The
main di€erence is that under the ®rst assumption, as
106
Dar-El et al.
opposed to the second, machines can fail even when they
are not working on a part. We feel both assumptions are
reasonable for di€erent types of machines. We estimate
our performance measures by building the machine
breakdown distribution into the processing time distributions.
We took all nine problems investigated computationally in Hopp and Spearman [8] and tested our estimates
on these problems. The results are given in Table 1. We
do not repeat the descriptions of the CONWIP lines because of the space this would require. However, we note
that the CONWIP lines associated with the ®rst two
problems contain four machines, whereas in the remainder of the problems the CONWIP lines contain three
machines. The ®rst column of Table 1 contains the
abbreviation used in Hopp and Spearman [8] to describe
the problems. The second column contains the number of
containers used in the CONWIP line. The third, fourth,
and ®fth columns are the percentage error of our model's
estimates. The sixth column is the percentage error of
Hopp and Spearman's [8] estimate of the mean
throughput for the same number of containers. The values in this column were calculated based on the graphs
that appear in Hopp and Spearman [8] and thus are likely
to contain some error. For this reason as well we only
report these errors to the closest whole number. The value
for dn e was equal to the number of machines in all but
four cases. For these cases dn e was equal to the number
of machines minus one. Since the values of dn e were so
small for these cases, we also report the results for
dn e ‡ 1 in the last part of Table 1.
Duenyas et al. [7] developed a model for estimating the
throughput variance of a CONWIP line with deterministic processing times and random (exponential) breakdowns. We took all 13 problems they investigated
Table 1. Percent error of our model's estimates and the
estimates of Hopp and Spearman [8]
Case
n
l's
rT
Comparison using dn e containers
case I
3
)12.2
0.4
case II
3
)14.0
35.9
EE
3
)0.2
3.5
GE
3
0.2
5.7
GG1
3
)2.3
)9.8
GG2
3
1.0
6.7
BE
2
)6.2
)9.6
BG1
2
5.3
40.9
BG2
3
2.3
7.7
Comparison using dn e ‡ 1 containers
in cases where n is small
case I
4
1.2
2.4
case II
4
)1.2
3.3
BE
3
5.2
)1.3
BG1
3
3.5
1.8
rL
lthroughput
5.2
28.4
3.7
5.7
)9.6
6.4
)7.5
34.4
4.5
)3
0
5
1
)6
)1
)20
)8
1
4.9
3.8
2.5
3.9
6
9
)8
1
computationally and tested our estimates on these problems. With the risk of comparing apples, oranges, and
mangos we present the results in Table 2. We feel it is
appropriate to present this table because no other method
exists for computing TBD and ¯owtime variances. Again,
because of the space it would require we do not repeat the
descriptions of the CONWIP lines; we note that cases
1±12 are three-machine examples, whereas case 13 is a sixmachine example. The ®rst ®ve columns are as described
above for Table 1. The last column is the percentage error
as reported by Duenyas et al. [7]. For the ®rst 12 systems
the value of dn e is always three (except in cases 3 and 6
where it is two). Since for these 12 problems Duenyas
et al. [7] do not report the percentage error of their model
for two containers, but do report the percentage error for
three containers, we also report the results for three
containers. For case 13 we report our results for dn e (i.e.,
®ve containers) and we report the results of Duenyas et al.
[7] for the lowest number of containers for which they
reported a value (i.e., seven containers).
We see from Tables 1 and 2 our model compares quite
favorably to the existing models. In Table 1 our model's
errors are greatest for case I, case II, and BG1. All three
of these cases are characterized by small values of dn e,
and in all four of the cases with small values of dn e our
model signi®cantly improved when examining dn e ‡ 1
containers. Though the quality of our model's estimates,
as compared to those of Duenyas et al. and Hopp and
Spearman [8] are not uniformly superior, they are comparable. This is encouraging when we consider that their
estimates were developed for very specialized systems,
and our estimates are applicable to any processing time
distributions.
4.2. Testing on random problems
We tested our model on production lines having between
two and 12 machines, several distribution types (uniform,
Table 2. Percent error of the estimates of our model and that of
Duenyas et al. [7]
Case
n
l's
1
2
3
4
5
6
7
8
9
10
11
12
13
3
3
3
3
3
3
3
3
3
3
3
3
5
0.5
0.2
)0.9
)0.2
2.3
4.5
)0.2
)1.2
)2.2
0.1
1.4
2.8
)0.8
rT
)14.1
)5.5
)3.1
12.7
)17.6
)10.4
0.1
)2.6
)2.2
19.2
)0.5
)2.0
4.9
rL
)5.6
)2.9
)2.1
12.5
)10.2
)4.4
)0.1
)2.1
)1.8
8.3
2.0
0.9
14.6
rthroughput
1.0
3.6
4.3
1.3
)2.1
1.2
0.1
)2.7
)5.9
7.2
7.4
)2.0
)7.0
CONWIP-based production lines with multiple bottlenecks
107
Table 3(a and b). Description of the distributions used in the computational tests on random problems
(a)
Name
Type
U1
U2
N1
N2
Exp
D1
Uniform
Uniform
Normal
Normal
Exponential
Discrete
D2
Discrete
Parameter description
Mean
c.v. = r=l
Low
High
Low
High
Low
High
U(0.65,1.35)
U(0.5,1.5)
N(1,0:22 )
N(1,0:32 )
E(1)
0.83 p = 0.5
1.00 p = 0.2
1.27 p = 0.3
0.80 p = 0.3
0.97 p = 0.5
1.36 p = 0.2
U(0.8,1.6)
U(0.6,1.8)
N(1.2,0:242 )
N(1.2,0:362 )
E(1.2)
1.00 p = 0.5
1.20 p = 0.2
1.54 p = 0.3
0.90 p = 0.2
1.00 p = 0.3
1.43 p = 0.5
1
1
1
1
1
1
1.2
1.2
1.2
1.2
1.2
1.2
0.2
0.3
0.2
0.3
1
0.2
0.2
0.3
0.2
0.3
1
0.2
1
1.2
0.2
0.2
(b)
Name
M1
M2
Type
Mixed
Mixed
Machine
1
2
3
4
5
6
E
N2
N1
N1
U2
U2
D1
D1
N2
E
N1
N1
normal, exponential, discrete, and a mix of these), and
many di€erent con®gurations. In total we tested our algorithm on 56 random problems (in addition to the 22
problems examined in Section 1). The distributions used
are detailed in Table 3(a and b). For the types of distributions with controllable coecient of variation (c.v.) we
examine the e€ects of the distribution with a c.v. of both
0.2 and 0.3. For each distribution type and for each c.v.
the mean processing time can take on one of two values, a
low and a high value. When the distribution takes on its
higher value it is termed a ``bottleneck''. We write ``bottleneck'' with quotation marks since a machine with a low
form of the distribution may temporarily become the
bottleneck machine, constraining the production system
performance for a short period of time. For example,
consider a line having machines with processing times of
the U1 type. The actual processing time on a machine
with a low form of the distribution may be greater than
1.2 for several parts, while the actual processing time on a
machine with a high form of the distribution may be less
than 0.9 for several parts. Thus the slower machine may
become the bottleneck for a limited period of time.
Our ®rst set of experiments is on CONWIP lines containing three machines. The results of this set of experiments are contained in Table 4. The ®rst column indicates
the position of the ``bottleneck'' machine(s). A `?' indicates that we use the high form of the distribution and a `-'
indicates that we use the low form of the distribution.
The number of containers used in the CONWIP line as
well as the error in our performance measures (our
model's estimate minus the simulated value, divided by
the simulated value) are reported for lines with a single
``bottleneck'' machine in each of the three possible positions. We also report these errors for CONWIP lines with
two ``bottleneck'' machines and for lines where all three
of the machines are ``bottlenecks''. These errors are reported for six di€erent distribution families (two uniform,
two normal, one discrete, and one exponential). The results for CONWIP lines with machines having exponentially distributed processing times are reported for lines
both where the number of containers is equal to dn e and
to dn e ‡ 1, because dn e was equal to three for all distribution families except the exponential family for which
it was equal to two.
The results are again quite encouraging. The percentage error in estimating the means of our performance measures is on average 4.6%. Furthermore, for
lines having machines with a processing time c.v. of 0.2,
the average is only 2.9% with a maximum value of
3.4%. For lines having machines with a processing time
c.v. of 0.3 the average is 4.5% with a maximum value of
5.3%. These values increase for lines with machines that
have exponential processing time distributions (c.v.
equal to 1.0). Hence we see that our estimates of the
means of our performance measures tend to deteriorate
as the coecient of variation of the processing time
distributions grow. The estimates of the variances of the
performance measures do not seem to have the same
trend. The average error in estimating the standard deviation of TBD (rT ) is 4.1% and the average error in
)7.7
)7.9
)8.7
)7.5
)8.8
)12.9
)13.4
)14.6
)13.1
)14.3
)8.3
)8.1
)8.1
)8.2
)7.8
2
2
2
2
2
)3.1
)3.8
)3.8
)2.8
)3.4
)5.3
)6.4
)7.7
)4.8
)6.8
10.1
9.2
9.1
10.2
10.2
3
3
3
3
3
2.9 )1.9 )6.9
2.8 )5.2 )7.3
2.8 )2.3 )5.0
3.1
1.2 )2.1
2.6 11.8 1.2
3
3
3
3
3
4.8
6.7 )2.8
4.1 )1.0 )4.8
4.1 )2.1 )7.1
4.5
1.5 )4.1
5.3
5.2 )1.3
3
3
3
3
3
)5.6
)6.7
)7.8
)5.1
)8.0
2.7
)3.3
)2.0
)0.4
)2.6
2.8
2.5
2.7
3.3
3.0
3
3
3
3
3
4.2 4.8 )3.8
4.1 )4.2 )1.9
4.5 )8.3 )13.0
4.7 3.1 )4.4
5.0 3.6 )7.2
3
3
3
3
3
5.0 )4.6
)4.7 )8.7
)6.0 )9.7
)0.5 )6.6
)2.3 )10.2
2.8
2.8
2.8
3.4
3.1
3
3
3
3
3
?--?--?
?????
rT
l's
rT
l's
rT
l's
rT
l's
rT
l's
rT
l's
rT
l's
n
U1(dn e)
rL
n
U2(dn e)
rL
n
N1(dn e)
rL
n
N2(dn e)
rL
n
D1(dn e)
rL
n
E1(dne ‡ 1)
rL
n
E1(dn e)
rL
Dar-El et al.
Table 4. Percentage error of our model's estimates on randomly generated three machine CONWIP lines. ? indicates a high processing time (``bottleneck''), - indicates
a low processing time
108
estimating the standard deviation of the ¯owtime (rL ) is
5.4%.
Our second set of experiments was carried out on
CONWIP lines containing six machines. The results of
this set of experiments are contained in Table 5. The ®rst
column again indicates the position of the ``bottleneck''
machine(s).
The number of containers used in the CONWIP line as
well as the errors in our performance measures are reported for two of the six con®gurations of lines with a
single ``bottleneck'' machine. We also report these errors
for two con®gurations of lines with three ``bottleneck''
machines and for lines where all six of the machines are
``bottlenecks''. These errors are reported for four di€erent
distribution families (one uniform, one discrete, and two
mixed). The results for CONWIP lines in which the machines have processing time distributions from various
distributions (mixed) are reported for lines both where
the number of containers is equal to dn e and to dn e ‡ 1,
because dn e was equal to ®ve for all distribution families
except the mixed families for which it was equal to four.
The results of this second set of experiments are similar
to the results for CONWIP lines containing three machines. The percentage error in estimating the means of
our performance measures is on average 3.2%. This is
even better than the average for three machines. The
average error in estimating the standard deviation of
TBD (rT ) is 7.5% and the average error in estimating the
standard deviation of the ¯owtime (rL ) is 6.4%.
Our third and ®nal set of experiments was carried out
on variable length CONWIP lines. We examined lines
ranging in length from two to 15 machines. What remained constant was the number of ``bottlenecks''. All
lines regardless of length had three ``bottlenecks'', one at
the very beginning, one at the very end, and one in the
middle. Of course, the CONWIP line with two machines
had only two ``bottlenecks''. The results are contained in
Table 6. The ®rst column again indicates the position of
the ``bottleneck'' machines. The second column emphasizes the number of machines in the CONWIP line. The
last four columns give the results for the U1 distribution
family always using dn e containers.
The results are at the same time encouraging and disappointing. The quality of our model's estimates of the
mean of our performance measures and of the standard
deviation of the ¯owtime (rL ) seems to be insensitive to
the length of the CONWIP line. However, the quality of
our model's estimate of the standard deviation of TBD
(rT ) clearly deteriorates as the length of the CONWIP
line grows.
Before concluding this section it is important to make
the following comments. First of all, while our estimates
may not be as accurate as we would like, they are robust.
Our estimates are e€ective on a wide variety of distributions (including machine breakdowns). The errors in the
estimates of the means are accurate enough to be used for
)16.1
0.3
)13.0
4.2
)13.7
0.1
)19.6 )1.6
)15.6
2.0
)9.8
)8.6
)10.0
)10.6
)10.7
4
4
4
4
4
2.2
0.8
3.0
0.4
1.1
4.8
4.4
4.6
4.1
3.6
8.3
7.8
7.1
6.0
8.4
?--------??-?--?
-???-??????
5
5
5
5
5
)2.4 17.1
5.8
)2.4 )4.5 )9.6
)1.6 13.5
)2.7
)1.6
3.7
)4.8
)4.3 )0.5 )12.4
5
5
5
5
5
)1.1
14.7
)1.1 )10.4
)1.1
3.5
)0.3
3.8
)4.2
6.4
)0.5
)17.3
)13.2
)7.3
)9.5
5
5
5
5
5
4.7
4.6
5.3
3.9
3.8
10.0
11.4
13.4
9.5
11.2
4
4
4
4
4
)8.8
)9.6
)8.2
)10.6
)10.6
)6.0
)10.2
)5.8
)11.3
)8.0
5.2
2.7
4.6
)1.5
4.4
5
5
5
5
5
)0.7
)3.1
4.3
)6.1
)1.7
rL
rT
l's
l's
rL
n
l's
rT
rL
n
l's
rT
rL
n
l's
rT
n
l's
rT
rL
n
rT
rL
n
M2(dn e)
M2(dn e ‡ 1)
M1(dn e)
M1(dn e ‡ 1)
D2(dn e)
U1(dn e)
Table 5. Percentage error of our model's estimates on randomly generated six machine CONWIP lines. ? indicates a high processing time (``bottleneck''), - indicates a
low processing time
CONWIP-based production lines with multiple bottlenecks
109
Table 6. Percentage error of our model's estimates on randomly
generated CONWIP lines of di€erent lengths. ? indicates a high
processing time (``bottleneck''), - indicates a low processing
time
U1(dn e)
M
??
???
??-?
?-?-?
?-?--?
?---?---?
?----?-----?
?------?------?
2
3
4
5
6
9
12
15
n
l's
rT
rL
2
3
4
4
5
7
9
11
)0.1
3.1
5.7
)6.0
)1.6
)3.9
)5.0
)5.8
)11.0
)2.3
16.7
)7.1
13.5
20.8
33.5
41.6
)7.2
)10.2
)7.5
)6.7
)2.7
)0.5
6.6
8.0
planning purposes and/or as a guide for a detailed simulation study of the system. This is also true of the estimates of the standard deviations. Even though they may
not be as accurate as our estimates of the means, they do
not need to be. A rough idea of the standard deviation is
all that is needed for planning purposes. In addition, we
concur with the view of Duenyas et al. [7] when they
write, ``It is important to note that estimating second
moments is considerably more dicult than estimating
®rst moments. Given that no other method for estimating
r m† exists, 8% error is not unreasonable.''
5. Insights
When management has to make a machine procurement/
replacement decision it can use the CBN concept as a
guideline on where to invest money. For example, take
the simple manufacturing environment detailed in the
®rst line of Table 7. The question facing management is
one of machine replacement. We assume that management has enough funds to replace one of the existing
machines with a machine having a deterministic processing time of nine. Traditional evaluation methods indicate the second machine should be replaced so as to
achieve a balanced line. We ask ourselves, ``Does it make
sense to have a line with two machines having deterministic processing times of respectively nine and ten,
when we could have a line with both machines having an
average processing time of nine?''. The answer is yes. This
improves the BN's performance.
Our model correctly identi®es (see Table 7) that a
balanced line is sub-optimal. The reason for this is clear.
With the unbalanced line the second machine works
continuously because there is no variability in the system.
With the balanced line, the ®rst machine's stochastic
processing time introduces variability that causes the
second machine to be starved from time to time.
110
Dar-El et al.
Table 7. An example of a machine replacement decision and its consequences
Machine 1
Machine 2
From CBN model
lT ˆ lBN
Original
Option 1
Option 2
U(4,14)
U(4,14)
Deter. 9
Deter. 10
Deter. 9
Deter. 10
10.8
10.25
10
From simulation
rT
rL
1.22
1.62
0
1.73
2.29
0
Often a manufacturing line does not sit in isolation, but
rather is part of a larger manufacturing environment. Just
as a machine processing time variance can cause a fast
machine to become the BN from time to time, a high
variance in the TBD can cause the CONWIP line to become the bottleneck in the overall system. Thus, our decision of machine replacement will be in¯uenced by the
resulting variances in the CONWIP line. Using the same
example as above, the preferred option of replacement
based on mean TBD is also preferred based on TBD
variance, and our model correctly identi®es this preference. The results regarding the mean ¯owtime are identical to those of the mean TBD [15]. The results for
¯owtime variance are also predicted by our model.
Since CONWIP systems can be viewed as closed
queuing networks, one may mistakenly view the system as
a loop (having no beginning nor end). This allows one to
`cut' the line at any point in order to evaluate its performance. This approach, as recognized by our model, is
valid for mean performance measures (Equations (7) and
(11)), but very mistaken for the variance of the performance measures (Equations (11) and (13)). Take as a
simple example the two machine line labeled `original' in
Table 7 with machine 1 having a processing time distributed uniformly between ®ve and ten. The time between departures is clearly a constant ten. If, however, the
system is viewed as a loop and `cut' between machines 1
and 2, we see that the resulting production line would
have a larger TBD variance.
According to our model the TBD variance (Equation (8)) does not depend on machines before the BN.
This is because, by assumption, the BN works continuously. The TBD variance does depend on the BN and the
machines after the BN. In fact our model predicts that the
machines after the BN contribute to the TBD variance
twice as much as the BN itself. Our model's estimate of
the ¯owtime variance (Equation (13)) also depends only
on the variance of the processing time on the BN and the
machines after the BN. The weighting is, however, different. The BN processing time variance has a weight of n
and the machines following the BN have a weight of two.
Furthermore, the CBN model predicts that the ¯owtime
variance will never be less than the TBD variance. This
was indeed the case in all our simulation runs.
The CBN model thus indicates that when management
wishes to reduce variances, the order of the machines
mean TBD
var. TBD
var. ¯owtime
10.8
10.25
10
1.22
1.61
0
1.73
2.28
0
must be considered. In fact, simply changing the order of
the machines can actually reduce the variances. A good
rule of thumb is to put variable machines at the start of
the line and `bottleneck' machines at the end. In addition,
when management has a limited budget for machine replacement/improvement it should concentrate on variable
machines at the end of the line.
6. Conclusion
An analytical model (CBN) was developed for predicting
the mean and variance of TBD and ¯owtime. We introduced the concept of a virtual bottleneck machine which
allowed us to employ analogies between deterministic and
stochastic systems. This concept enabled us to handle
migrating bottlenecks, an issue that is generally neglected.
The results of our simulation experiments show that our
analytical model is very accurate for predicting the mean
TBD and mean ¯owtime, and suciently accurate for
predicting the standard deviations of TBD and ¯owtime.
The simulation experiments also showed that the analytical models are much quicker than simulations. Since
we did not constrain the type of the processing time
distributions when we developed our models, the in¯uence of machine breakdowns was also considered by including them into the processing time distributions.
Finally we used our model to gain insights into how the
machine characteristics a€ect the values of our performance measures.
Before concluding we wish to point out that some of
the information that we estimate can be estimated by
modeling our system as a closed queuing network and
then using a general queuing network analyzer. However,
we feel that this approach would be unsatisfactory. First,
and most important, the insights obtained in Section 5
would be unavailable. Second, we feel that the unique
way our model is developed is of interest.
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111
Biographies
Ezey M. Dar-El, B.Mech.E. 1953, M.Sc. 1960, Ph.D. 1970 (University
of Melbourne). Ezey Dar-El holds the Harry Lebensfeld Chair in Industrial Engineering. Prof. Dar-El is on the Editorial Board of the ®ve
learned journals, is a fellow of the Institute of Industrial Engineers, an
Executive Board Member of the IFPR (International Federation of
Production Research), and Director of the Israeli branch of the world
Council of Productivity Sciences. He is also a member of ORSIS,
Human Factors Society, the Israel Ergonomics Society and the Society
of Manufacturing Engineers. He has published extensively on Assembly Line Design; job shop and ¯ow shop scheduling; project control;
project scheduling; FMS scheduling; industrial safety; work measurement; R&D Quality and Productivity; Gain Sharing; productivity development and industrial learning. Prof. Dar-El has written the
de®nitive work on Gain Sharing and is recognized as the leading researcher in this area. Prof. Dar-El is an internationally known consultant, having worked with over 60 companies, variously located in :
Australia, Israel, England, U.S.A., Italy, The Netherland Antilles and
Peru.
Yale T. Herer, B.Sc. (1986), M.Sc. (1990), Ph.D. (1990), - Cornell
University. Yale joined the Faculty of Industrial Engineering and
Management at the Technion ± Israel Institute of Technology in 1990
immediately after the completion of his graduate studies. Currently
Yale is visiting the Department of Industrial Engineering at Tel-Aviv
University. He has worked for several industrial concerns, both as a
consultant and as an advisor to project groups. Yale is a member of the
Institute for Operations Research and Management Sciences (INFORMS), Institute of Industrial Engineers (IIE), and the Operations
Research Society of Israel (ORSIS). His research interests include inventory control, especially when integrated with routing and/or distribution. He is also interested in production control and production
system design.
Michael Masin, M.Sc. 1992, Ph.D. 1998 (Industrial Engineering,
Technion, Israel), M.Sc. 1987 (Mechanical Engineering, Moscow State
University of Railway Transport, Russia). Michael is visiting the Department of Industrial Engineering at Tel-Aviv University. During his
Ph.D. studies he was employed by the Israeli Defense Forces for four
years, working in the ®elds of Operations Research and Systems
Analysis. Michael is a member of the Institute for Operations Research
and the Management Sciences (INFORMS), and Institute of Industrial
Engineers (IIE). His research interests include production control
systems, advanced manufacturing systems, system analysis and integration.