and
otherwise,
it is unimprovable with respect to BC and WF simultaneously.
Criteria for improvability in terms of each of these definitions are given below.
5.1.2 Improvability with respect to WF
Necessary and sufficient conditions: Below we provide both theoretical and practical conditions of improvability with respect to WF.
Theorem 5.1 A Bernoulli line defined by assumptions (a)-(e) of Subsection 4.2.1 is unimprovable with respect to WF if and only if
where 
are the steady states (4.35) of the recursive aggregation procedure (4.30).
Proof: See Section 20.1.
Recall that, as it has been shown in Chapter 4, for each buffer
bi, an M-machine line can be represented as a two-machine system with virtual machines
characterized by
(see Figure 4.10). Thus, Theorem 5.1 implies that the necessary and sufficient condition of unimprovability with respect to WF is that both virtual machines are identical. Using the property (4.21) of two-machine lines with identical machines, we obtain
Corollary 5.1 Under condition (5.9),
Since 0 <
< 1, expression (5.10) implies that
Based on this, we formulate the practical
WF-Improvability Indicator 5.1: A Bernoulli line is practically unimprovable with respect to WF if each buffer is, on the average, close to being half full.
Although this indicator may seem somewhat unexpected, it is, in retrospect, quite natural. Indeed, buffer
bi is intended to protect mi from blockage and
mi+1 from starvation. From the point of view of
mi, buffer bi should be all the time empty; from the point of view of
mi+1 it should be full. The compromise is – the buffer is half full (with some correction depending on the machines' efficiency
as indicated in (5.10) by
).
Unimprovable allocation of pi: To characterize the unimprovable (i.e., optimal)
pi's, introduce the notation
Clearly,
is the largest production rate, which can be attained in the system
under constraint (5.2). It turns out that
can be calculated as follows:
Consider the recursive procedure:
Theorem 5.2 Assume
. Then recursive procedure (5.13) iis convergent and its steady state, x*, is:
Proof: See Section 20.1.
Thus, the maximum production rate can be calculated without the knowledge of the optimal WF allocation. Moreover, this
can be used to calculate
the optimal allocation of pi's, as shown below.
Theorem 5.3 The sequence
, which renders the
serial production line with Bernoulli machines unimprovable with respect to WF, is given by
Proof: See Section 20.1.
Thus, for any given sequence N1, ...,
NM−1 and p* of (5.2), the optimal allocation of work can be calculated using (5.13)-(5.17).
Taking into account that
< 1, expressions (5.15)-(5.17) lead to a
"flat" inverted bowl phenomenon:
Corollary 5.2 If all buffers are of equal capacity, i.e., Ni =: N, i =1, ..., M − 1, then
Although the flat inverted bowl allocation of optimal
pi's does take place, it does not lead, as it was mentioned before, to a significantly larger
PR as compared with the uniform allocation. Indeed, consider a system with
five Bernoulli machines, Ni = 2, i = 1,..., 4, and
p* = (0.9)5=0.5905. For this system, the optimal pi allocation, according to (5.15)-(5.17) is
leading to
= 0.8110 . If pi's were distributed in the uniform manner, i.e.,
pi = 0.9, i =1,...,5,
= 0.8050, which is only 0.74% less than the production rate for the inverted bowl allocation. To reinforce this conclusion, consider one more example:
M = 10, p = (0.95)10 = 0.5987, and Ni = 3,
i =1,..., 9. Then the optimal allocation (5.15)-(5.17) becomes
and the optimal
= 0.9132, while the uniform allocation
pi = 0.95, i =1,...,10
results in
= 0.9104, which is only 0.31% less than
.
Continuous improvement by WF re-allocation: While expressions (5.15)-(5.17) can be used for design of new production systems, continuous improvement projects for existing operations can be designed using WF-Improvability Indicator 5.1. This can be carried out as follows:
WF-Continuous Improvement Procedure 5.1:
(1) By calculations off-line (using the aggregation procedure (4.30) and formula (4.37)) or by measurements on the factory
floor, evaluate the average buffer occupancy, WIPi,
i = 1, ..., M − 1.
- (2) Determine the buffer for which |WIPi − (Ni + 1)/2| is the largest. Assume this is buffer
k.
- (3) If WIPk − (Nk
+ 1)/2 is positive, re-allocate a sufficiently small amount of
work, εpk, from mk to
mk+1; if WIPk − (Nk + 1)/2 is negative, re-allocate
ŠĆpk+1 from
mk+1 to mk (observing, of course, the constraint
pkpk+1 = const).
- (4) Return to step (1).
-
(5)
Continue this process until max|WIPi − (Ni + 1)/2| is sufficiently small.
To illustrate this procedure, consider a four-machine Bernoulli line with
Ni = 5 and p = [0.9675, 0.9225, 0.8780, 0.8372]. The
of this line is 0.8281. Running WF-Continuous Improvement Procedure 5.1 with
ŠĆ = 0.01, we arrive at the following WF allocation
p = [0.8875, 0.9125, 0.9163, 0.8841]. The resulting
= 0.8707, which is 5.14% improvement.
5.1.3 Improvability with respect to WF and BC simultaneously
Necessary and sufficient conditions: For the case of simultaneous improvability with respect to WF and BC, these conditions can be formulated as follows:
Theorem 5.4 A Bernoulli line defined by assumptions (a)-(e) of Subsection 4.2.1 is unimprovable with respect to WF and BC simultaneously if and only if
Proof: See Section 20.1.
Corollary 5.3 Under condition (5.19),
In addition,
and
Proof: See Section 20.1.
Thus, if a line is unimprovable with respect to WF and BC simultaneously, all internal machines are starved and blocked with equal frequency, the
first machine is blocked as frequently as the last machine is starved, all buffers are of equal capacity, have equal steady state occupancy, and this occupancy can be evaluated using (5.23). Clearly,
Unimprovable allocation of pi and
Ni: To characterize the unimprovable (i.e., optimal)
pi and Ni allocations, introduce
Theorem 5.5 Let N* be a multiple of M − 1. Then the sequences
and 
, which render the serial production line with Bernoulli
machines unimprovable with respect to WF and BC simultaneously, are given by
Proof: See Section 20.1.
Thus, based on the capacity
, the value of
can be calculated using
(5.13), and then 
's can be evaluated from (5.26) and (5.27). This result can be useful at the design stage of serial production lines. Note that the optimal
pi allocation is again a
flat inverted bowl, while optimal Ni's are uniform.
5.1.4 Improvability with respect to BC
Necessary and sufficient conditions:
Theorem 5.6 A Bernoulli line defined by assumptions (a)-(e) of Subsection 4.2.1 is unimprovable with respect to BC if and only if the quantity
is maximized over all sequences
such that 
.
Proof: See Section 20.1.
Unfortunately, this result is of little practical importance, since its interpretation and utilization are much less obvious than those of (5.9) and (5.19). However, numerical experiments with (5.28) show that in most cases the unimprovable
sequence
,
, can be determined by a criterion with a clearer interpretation:
Numerical Fact 5.1 The production rate ensured by the buffer capacity allocation defined by Theorem 5.6 is almost always the same as the production rate defined by the allocation that minimizes
over all sequences
such that
.
This implies that a production line is practically unimprovable with respect to BC if the occupancy of each buffer
bi−1 is as close to the availability of buffer
bi as possible. In other words, bi−1 and
bi offer practically equal protection of machine
mi against starvations and blockages, respectively. This is illustrated in Figure 5.1 where the occupancy of
bi−1 (i.e.,
) and the availability of bi (i.e.,
) are indicated by shaded areas. Based on the above, we formulate the practical
Figure 5.1: Illustration of Numerical Fact 5.1
BC-Improvability Indicator 5.1: A Bernoulli line is practically unimprovable with respect to BC if the average occupancy of each buffer is as close to the average availability of its downstream buffer as possible.
Continuous improvement by BC re-allocation: Similar to WF, the above improvability indicator can be used in the following
BC-Continuous Improvement Procedure 5.1:
(1) By calculations off-line (using (4.30) and (4.37)) or by measurements on the factory
floor, evaluate the average occupancy of each buffer, WIPi,
i = 1, ..., M − 1.
(2) Determine the buffer for which |WIPi −(Ni+1 −WIPi+1)|,
i =1,..., M −2, is the largest. Assume this is buffer k.
(3) If WIPk − (Nk+1 −
WIPk+1) is positive, transfer a unit of capacity from bk to
bk+1; if
WIPk − (Nk+1 − WIPk+1) is negative, re-allocate a unit of capacity from
bk+1 to bk.
(4) Return to step (1).
(5) Continue this process until arriving at a limit cycle and choose the buffer capacity allocation on the limit cycle, which maximizes
PR.
As an illustration, consider an 11-machine serial line with
pi =0.8, i ≠ 6 and p6 =0.6. Assume that the total buffer capacity is 24 and determine the unimprovable buffer capacity allocation. Based on the above procedure and calculating
using (4.30), (4.37), the following result is obtained:
This allocation ensures
= 0.5843, which is almost the maximum possible in this system. Note that if, following Goldratt's
Theory of Constraints, all available buffer capacity were placed in front of the
"bottleneck" machine m6, i.e.,
the resulting
= 0.427, which is 27% lower than that of the unimprovable allocation. If, keeping in mind the property of reversibility (see Section 4.3), the
"bottleneck" machine is protected on both sides, i.e., the allocation is
the resulting
= 0.491, which is still 16% lower than that of the unimprovable allocation obtained using BC-Continuous Improvement Procedure 5.1.
As a final note, it should be pointed out that in some cases, the unimprovable allocation may not appear in the limit cycle, but a few steps before the limit cycle is reached. However, the production rate obtained using BC-Continuous Improvement Procedure 5.1 is typically very close to the production rate ensured by the unimprovable allocation.
PSE Toolbox: The unimprovable allocations of WF and BC, as well as WF-and BC-Continuous Improvement Procedures, are implemented in the
Continuous Improvement function of the toolbox. For a description and illustration of these tools, see Subsections 19.4.1–19.4.4.
5.2 Unconstrained Improvability
5.2.1 Definitions
Bottleneck machine: Consider a serial production line with
M Bernoulli machines defined by parameters pi,
i = 1,..., M, and M − 1 buffers with capacities
Ni, i =1,..., M − 1. Assume that the line operates according to conventions (a)-(e) of Subsection 4.2.1.
Let, as before, PR, denote the production rate of the system, i.e.,
Definition 5.2 Machine mi, i ∈{1,..., M}, is the bottleneck machine (BN-m) of a Bernoulli line if
Due to the monotonicity properties of PR
with respect to pi's (see Section 4.3), both
derivatives in (5.30) are positive. Thus, Definition 5.2 implies
that mi is the BN-m if its infinitesimal
improvement leads to the largest increase of the production rate, as
compared with a similar improvement of any other machine in the
system.
A machine with the smallest pi
is not necessarily the BN-m in the sense of Definition 5.2. Indeed,
consider the production lines shown in Figure 5.2, where the numbers
in the circles and the rectangles are pi and Ni,
respectively, and the row of numbers under the machines represent
the estimates of partial derivatives
evaluated by numerical
simulations. Clearly, the bottleneck machines are m2
(in Figure 5.2(a)) and m4 (in Figure 5.2(b)), none
of which corresponds to the worst machine (i.e., the machine with
the smallest pi). In fact, m2 in
Figure 5.2(a) is the best machine in the system.
Figure 5.2: Examples of bottleneck machines in Bernoulli lines
Similarly, a machine with the largest work-in-process in front of it is not necessarily the bottleneck. An example is given in Figure 5.2(b), where
m2 has the largest WIP to be processed, while the BN-m is
m4.
While Definition 5.2 provides a formal characterization of the BN-m, its practical application is not straightforward because the partial derivatives involved could be neither measured on the factory
floor nor calculated analytically with any acceptable accuracy. Indeed, while PR itself is often measured on the factory
floor, its sensitivity to pi's is not and could hardly be expected to be measured, since it would require increasing efficiency of each machine and evaluating the resulting increase in PR. Such a procedure is hardly possible in most practical situations. Although the analytical calculations, described in Chapter 4, lead to acceptable estimates
, no analytical methods for evaluating
are available. Therefore, to make Definition 5.2 practical, it has to be reformulated in terms of quantities, which are either available through measurements on the factory
floor or through analytical calculations or both. It is shown in Subsections 5.2.2 and 5.2.3 that this can be accomplished using the probabilities of blockages and starvations,
BLi and STi, defined in Chapter 3 and evaluated in Chapter 4.
The above arguments notwithstanding, the machine with the smallest
pi is, in fact, the bottleneck in the sense of (5.30) if the production line is WF-unimprovable. This follows from the following
Theorem 5.7 In WF-unimprovable Bernoulli lines defined by assumptions (a)-(e) of Subsection 4.2.1,
Proof: See Section 20.1.
Thus, the BN-m in WF-unimprovable lines can be identified quite easily.
Bottleneck buffer: While the term "bottleneck machine" is widely used in practice, the term
"bottleneck buffer" is not. This happens, perhaps, because thinking locally, one usually pays more attention to the efficiency of the machines, as part-producing devices, and less attention to buffers, as
"shock absorbers" of perturbations. Nevertheless, the notion of a bottleneck buffer could and, moreover, must be introduced in order to explore all means of system improvements.
Definition 5.3 Buffer bi, i ∈{1,..., M −1}, is the bottleneck buffer (BN-b) of a Bernoulli line if
In other words, BN-b is the buffer, which leads to the largest increase of the
PR if its capacity is increased by 1, as compared with increasing any other buffer in the system.
The buffer with the smallest capacity is not necessarily the BN-b. An example is shown in Figure 5.3, where the numbers under each buffer correspond to the
PR of the system obtained by simulations when the capacity of this buffer is increased by one. Clearly, the BN-b is
b1 while the smallest buffer is b3.
To identify the BN-b using Definition 5.3, one would have to experiment with the system by increasing each buffer and measuring the resulting production rate, which is hardly possible in practice. It turns out that this is also unnecessary: as it is shown below,
BLi and STi can be used to identify not only the BN-m but also the BN-b.
Figure 5.3: Example of bottleneck buffer in Bernoulli line
5.2.2 Identification of bottlenecks in two-machine lines
Theorem 5.8 For a two-machine Bernoulli line defined by assumptions (a)-(e) of Subsection 4.1.1, the inequality
takes place if and only if
There are three benefits offered by this theorem. First, it provides a relationship between the
"non-measurable" and "non-calculable" partial derivatives of
PR and
"measurable" and "calculable" probabilities of blockages and starvations. Second, it offers a possibility of identifying the BN-m without even knowing parameters of the machines and buffer, but just by measuring
ST2 and BL1. Third, it offers a simple graphical way of representing the BN-m. To illustrate this, consider the production line of Figure 5.4, where the two rows of numbers under the machines represent
STi and BLi. Place an appropriate inequality sign between
ST2 and BL1 and turn the inequality into an arrow by adding a line within the sign of the inequality. According to Theorem 5.8, the machine, to which the arrow is pointed, is the BN-m. As it turns out, this procedure can be extended to
M > 2-machine lines as well.
Figure 5.4: Arrow-based method of bottleneck identification
Note that for the case of M = 2,
• the problem of BN-b does not arise;
• the machine with the smallest pi is the BN-m since, as it follows
from the results of Section 4.1, ST2 > BL1,
if and only if p1 < p2.
5.2.3 Identification of bottlenecks in M
> 2-machine lines
Bottleneck Indicator: As it was discussed
in Chapter 4, STi and BLi in
lines with M > 2 machines cannot be calculated exactly, and
only estimates,
and
,
are available. Therefore, the rules for BN identification in M
> 2-machine lines are formulated either in terms of STi
and BLi, which may be available from factory floor
measurements, or in terms of
and
,
which may be calculated using (4.30) and (4.39), (4.40)). Note that
the application of the former requires no knowledge of the machine
and buffer parameters.
Consider the production lines shown in Figures 5.5
and 5.6 with two rows of numbers under each machine, the first one
indicating STi and the second BLi.
Place arrows directed from one machine to another in the same manner
as in Subsection 5.2.2, i.e., according to the following
Figure 5.5: Illustration of a Bernoulli line with a single bottleneck machine
Figure 5.6: Illustration of a Bernoulli line with multiple bottleneck machines
Arrow Assignment Rule 5.1: If BLi >
STi+1, assign the arrow pointing form
mi to mi+1. If BLi <
STi+1, assign the arrow pointing from
mi+1 to mi.
Bottleneck Indicator 5.1: In a Bernoulli line with M > 2 machines,
- if there is a single machine with no emanating arrows, it is the BN-m;
- if there are multiple machines with no emanating arrows, the one with the largest severity is the Primary BN-m (PBN-m), where the severity of each
(local) BN-m is defined by
- the BN-b is the buffer immediately upstream of the BN-m (or PBN-m) if it is more often starved than blocked, or immediately downstream of the BN-m (or PBN-m) if it is more often blocked than starved.
Thus, according to this indicator, m4 and
b4 are the bottlenecks in Figure 5.5, and m2 and
b2 are the PBN-m and BN-b in Figure 5.6.
Bottleneck Indicator 5.1 is justified below using both numerical and analytical approaches.
Numerical justification: It is carried out by calculations and simulations. The calculation approach consists of calculating
and
, identifying BN-m and BN-b using Bottleneck Indicator 5.1, and then verifying the conclusions using the calculated quantities
and Definitions 5.2 and 5.3. The simulation approach is carried out analogously but using
STi, BLi and ΔPR/Δpi,
PR(Ni + 1) evaluated numerically based on Simulation Procedure 4.1. In both calculation and simulation approaches,
Δpi was selected as 0.03 and
was evaluated as
In the majority of cases analyzed, Bottleneck Indicator 5.1 identified BNs correctly. Typical examples are shown in Figures 5.7 (single BN case) and 5.8 (multiple BN case). Some counterexamples, however, have also been discovered. Two of them are shown in Figures 5.9 and 5.10 for single and multiple BN cases, respectively.
To investigate the "frequency" of the counterexamples, the following statistical experiment was carried out: 5000
five-machine serial lines were constructed by selecting pi's and Ni's randomly and equiprobably from the sets
pi ∈{0.75, 0.80, 0.85, 0.90.0.95},
Ni ∈{1, 2, 3},
respectively. For each of these lines, BNs were identified using the calculation and simulation approaches, and the percent of correct and incorrect BN identifications were evaluated. The results are shown in Figures 5.11 and 5.13 for calculation and simulation approaches, respectively. Figure 5.11(a) indicates that among the 5000 lines, investigated by calculations, about 80% had a single BN-m. Within those, Bottleneck Indicator 5.1 identified the BN-m and BN-b correctly in over 95% and 87% of cases, respectively (Figures 5.11(b) and (c)).
Figure 5.7: Examples of bottleneck identification using Bottleneck Indicator 5.1; single bottleneck case
Figure 5.8: Examples of bottleneck identification using Bottleneck Indicator 5.1; multiple bottlenecks case
Figure 5.9: Counterexample for Bottleneck Indicator 5.1, single bottleneck case
Figure 5.10: Counterexample for Bottleneck Indicator 5.1, multiple bottlenecks case
The lower accuracy of BN-b identification is, perhaps, due to the fact that, unlike pi's, buffer capacity cannot be increased infinitesimally. This conjecture is supported by Figure 5.11(d), which shows that the BN-b, identified by Definition 5.3, is one of the buffers around the BN-m in over 91% of the cases. Thus, Figures 5.11(b)-(d) indicate that Bottleneck Indicator 5.1 is a sufficiently reliable tool for BN-m and BN-b identification. Note that, as it is shown in Figure 5.12(a), the BN-m is the worst machine of the system in only 62% of the cases analyzed; thus, assuming that the worst machine is the bottleneck leads to a much lower frequency of correct BN identification.
The frequency of correct PBN identification using Bottleneck Indicator 5.1 for multiple bottlenecks case is illustrated in Figure 5.11(e). Although it is relatively low (about 72%), the true PBN is within the set of local BN-m's in over 94% of cases (Figure 5.11(f)). The frequency of correct BN-b identification is illustrated in Figures 5.11(g) and (h)). Clearly, accuracy of PBN-m identification is lower than that of just bottleneck. This is, perhaps, due to the fact that the bottleneck severity (5.34) has been defined in an ad-hoc manner and better definitions might be possible. Nevertheless, we conclude that Bottleneck Indicator 5.1 is a useful tool for bottleneck identification, especially taking into account that in only 42% of cases analyzed the worst machine was indeed the PBN (see Figure 5.12(b)).
Similar results were obtained using the simulation approach. These results are illustrated in Figures 5.13 and 5.14.
Thus, the conclusion from this investigation is that Bottleneck Indicator 5.1 provides a sufficiently accurate method for identifying BNs using either measured quantities
STi and BLi or calculated ones
and
.
We also remark that, in our experience, even when Bottleneck Indicator 5.1 leads to an incorrect BN identification, the true BN has just a slightly higher
ΔPR/Δpi and PR(Ni + 1) than those identified by the indicator. Based on the above, we formulate
BN-Continuous Improvement Procedure 5.1:
(1) By off-line calculations (using (4.30), (4.39) and (4.40)) or by measurements on the factory
floor, evaluate the probabilities (or frequencies) of blockages and starvations of each machine.
(2) Using Bottleneck Indicator 5.1, identify the BN-m (or PBN-m) and BN-b.
(3) Take actions to increase the efficiency, pi, of this machine (for instance, by improved preventative maintenance, assigning additional work force and the like).
(4) If the above, for one reason or another, is impossible, increase the BN-b or both buffers around the BN-m (or PBN-m).
(5) Return to step (1).
Our industrial experience, gained through numerous case studies, indicates that
BN-Continuous Improvement Procedure 5.1 (and its generalization for production lines with exponential and general models of machine reliability - see
Parts III and IV) is one of the most efficient ways of managing production systems.
Figure 5.11: Accuracy of Bottleneck Indicator 5.1 using calculation data
Figure 5.12: Frequency of worst machine being BN-m using calculation data
Figure 5.13: Accuracy of Bottleneck Indicator 5.1 using simulation data
Figure 5.14: Frequency of worst machine being BN-m using simulation data
Analytical justification: An analytical justification is available only for the case of a single bottleneck machine. It is based on the following
Hypothesis 5.1 Inequalities
BLj−1 > STj , j =2,..., M
and
BLj < STj+1 , j =1,..., M − 1
imply, respectively, that
and
The following lemma states that Hypothesis 5.1 indeed holds, at least for
Nj sufficiently large:
Lemma 5.1 In a Bernoulli line defined by assumptions (a)-(e), for any
, there exists N* such that if Nj
>N*, j = 1,..., M − 1, then
Theorem 5.9 Under Hypothesis 5.1, the BN-m is downstream of mj if
and upstream of mj if
.
Proof: See Section 20.1.
Thus, this theorem confirms Bottleneck Indicator 5.1 as far as a single BN-m is concerned.
PSE Toolbox: The method of BN-m and BN-b identification, using both calculated and measured data, is implemented in the
Bottleneck Identification function of the toolbox. For a description of these tools, see Subsections 19.5.1 and 19.5.3.
5.2.4 Potency of buffering
As it has been shown above, the worst machine often is not the BN of the system. Why does this happen? Clearly, this is because of an inappropriate buffer capacity allocation. To formalize the notion of buffering quality, introduce
Definition 5.4 The buffering of a production system is
- weakly potent if the BN-m is the worst machine in the system (i.e., the machine with the smallest efficiency); otherwise, it is not potent;
- potent if it is weakly potent and its production rate is sufficiently close to the BN-m efficiency (e.g., within 5% of the BN machine efficiency in isolation);
- strongly potent if it is potent and the system
has the smallest possible total buffer capacity (i.e.,
is the smallest possible to ensure the
desired production rate).
For serial lines with Bernoulli machines, one can determine if the buffering is weakly potent or not using the method of BN-m identification described in this section. To determine if it is potent, the method of
PR calculation, described in Chapter 4, can be used. To investigate the notion of strong potency, methods that allow one to calculate the smallest total buffers capacity, which is necessary to ensure the desired production rate, must be available; these methods are described in Chapter 6.
For serial lines with exponential and general models of machine reliability, similar techniques are discussed in Part III, and for assembly systems in Part
IV.
Along with its practical utility, the notion of buffering potency has conceptual significance. Indeed, production systems consist of two distinct entities: machines and buffers. The quality of the machines is characterized by their efficiency. In practice, machine efficiency is often monitored, and continuous improvement efforts are largely centered on its modification. In contrast, the quality of buffering is rarely monitored and even more rarely viewed as a resource for continuous improvement. The quantification provided by Definition
5.4 brings buffering to the same level of monitoring potential as that for machine efficiency.
To illustrate the importance of buffering potency, consider the automotive
ignition module assembly system described in Sections 3.2 and 3.10.
Its throughput losses due to machines and due to MHS are analyzed in
Table 16.4. Here, the losses due to machines are evaluated as the
difference between the nominal throughput (600 parts/h) and the
isolation throughput of the worst machine; the losses due to MHS are
evaluated as the difference between the isolation throughput of the
worst machine and the actual throughput of the system. As follows
from these data, out of roughly 240 parts/h lost, 80 parts/h are
attributed to the machines and 160 parts/h to MHS. Clearly, this MHS
is non-potent. (Chapter 16 shows that the worst machine is not the
bottleneck of the system.) Interestingly, this 1:2 ratio has been
observed in other production systems as well. Thus, ensuring potency
of MHS is an important resource of production system improvements. A
detailed analysis of the automotive ignition module assembly system
is described in Chapter 16.
5.2.5 Designing continuous improvement projects
Based on the methods developed in this chapter, the design of continuous improvement (CI) projects can be carried out following the procedure illustrated in Figure 5.15. As it is shown in this
figure, after modeling of the production system at hand and model validation (as described in Chapter 3), methods of constrained and/or unconstrained improvability (Chapter 5) can be used to determine possible avenues for system improvement, and the most efficient one is identified (using the performance evaluation techniques of Chapter 4). In this manner, rigorous improvement projects with quantitatively predicted results can be designed. Following their implementation and evaluation on the factory
floor, the process must be repeated anew, in a never ending quest for improvement.
Figure 5.15: Procedure for designing continuous improvement projects
denote the production
rate of the system, calculated using (4.30)-(4.36).
such that
= 476.7 parts/hour. In addition, increasing the
efficiencies of m9-10 by 10% leads to
