Multi-Job Production Systems: Definition, Problems, Analysis, and Product-Mix Performance Portrait of Serial Lines P. Alavian a , P. Denno b , and S. M. Meerkov a a Department of Electrical Engineering and Computer Science University of Michigan Ann Arbor, MI, 48109-2122 b Systems Integration Division, Engineering Laboratory National Institute of Standards and Technology Gaithersburg, MD 20899-1070 Abstract Multi-job production (MJP) is a class of flexible manufacturing systems intended to produce dif- ferent products (job-types) according to a given product-mix and build-schedule. In MJP systems, all job-types are processed by the same sequence of manufacturing operations, but with different processing times at some or all machines. To characterize MJP, we introduce the work-based (rather than the traditional part-based) model of production systems, which is “insensitive” to whether a single- or multi-job manufacturing takes place. Using this model, we develop a method for per- formance analysis of MJP serial lines with the emphasis on their throughput and bottlenecks as functions of the product-mix. We show, in particular, that for the so-called conflicting job-types, there exists a range of product-mix, where the throughput of MJP is larger than that of any individ- ual job-type involved. To characterize the global behavior of MJP systems, we introduce the notion of Product-Mix Performance Portrait, which represents the system throughput and bottlenecks for all feasible product-mixes. Finally, we apply the results obtained to a section of the underbody assembly system at an automotive assembly plant, calculate its performance portrait, and evaluate the efficacy of potential continuous improvement projects. Keywords: Multi-product manufacturing; Flexible production systems; Product-mix; Serial lines; Exponential machines; Finite buffers; Work-based model; Throughput; Bottleneck; Performance por- trait; Automotive assembly.
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Multi-Job Production Systems: Definition,Problems, Analysis, and Product-MixPerformance Portrait of Serial Lines
P. Alaviana, P. Dennob, and S. M. Meerkova
aDepartment of Electrical Engineering and Computer ScienceUniversity of Michigan
Ann Arbor, MI, 48109-2122bSystems Integration Division, Engineering Laboratory
National Institute of Standards and TechnologyGaithersburg, MD 20899-1070
Abstract
Multi-job production (MJP) is a class of flexible manufacturing systems intended to produce dif-
ferent products (job-types) according to a given product-mix and build-schedule. In MJP systems,
all job-types are processed by the same sequence of manufacturing operations, but with different
processing times at some or all machines. To characterize MJP, we introduce the work-based (rather
than the traditional part-based) model of production systems, which is “insensitive” to whether a
single- or multi-job manufacturing takes place. Using this model, we develop a method for per-
formance analysis of MJP serial lines with the emphasis on their throughput and bottlenecks as
functions of the product-mix. We show, in particular, that for the so-called conflicting job-types,
there exists a range of product-mix, where the throughput of MJP is larger than that of any individ-
ual job-type involved. To characterize the global behavior of MJP systems, we introduce the notion
of Product-Mix Performance Portrait, which represents the system throughput and bottlenecks for
all feasible product-mixes. Finally, we apply the results obtained to a section of the underbody
assembly system at an automotive assembly plant, calculate its performance portrait, and evaluate
the efficacy of potential continuous improvement projects.
Keywords: Multi-product manufacturing; Flexible production systems; Product-mix; Serial lines;
Proof: Follows the proof of formulas (11.13)-(11.17) in [57, Section 11.1] and is omitted here. The
expression for WIP pwq can also be derived, which turns out to be the same as (11.14) in [57], but
with N wW instead of N .
Discussion: (a) Using (3.4), it is possible to show that function Q is monotonically deceasing in w.
Thus, as it follows from (3.1), PR is an increasing function of w. Similar result can be obtained for
two-machine synchronous lines with non-identical machine work-capacities and jobs having different
work-requirements, as long as (2.2) is satisfied.
(b) Using the aggregation procedures of [57, Section 11.1] with Q from (3.4), it is possible to
show that the above effect takes place for synchronous lines with M ¡ 2 as well (see Figure 3.1
for an illustration). Thus, the work-based model demonstrates that serial lines are more effective
producing parts with larger work-requirements than those with smaller ones. This phenomenon takes
place because the machine downtime in units of job cycle time is smaller for jobs with larger work-
requirements.
(c) The above conclusion does not necessarily imply that TP is an increasing function of w (since
TP is proportional to Ww PR pwq, and, therefore, TP may be decreasing even if PR is increasing).
However, the efficacy of buffers, as a tool for disturbance rejection, is indeed increasing with w.
10
w
1 2 3 4 5
PR(w
)
0.82
0.83
0.84
0.85
0.86
0.87
0.88
Figure 3.1: Production rate of 5-machine synchronous exponential SJP line as a function of job work-requirement pλi � 0.1, µi � 0.9,Wi � 1, i � 1, � � � , 5 and N � r2, 2, 2, 2sq
(d) While Theorem 3.1 addresses synchronous lines, a similar theorem can be proved for asyn-
chronous lines as well. In this case, the performance of two-machine SJP line is characterized by
expressions (11.40), (11.41), (11.43), (11.45) of [57] with ci replaced by Wiwi
.
(e) The performance of SJP asynchronous lines with more than two machines, can be analyzed
using the aggregation procedure of [57, Section 11.2], leading to a conclusion similar to that of item
(b) above.
Note that all performance metrics of SJP synchronous and asynchronous lines in the framework of
work-based model can be evaluated using the PSE Toolbox, [58]. Indeed, although this toolbox has
been developed for the part-based model, it can be used for the work-based model as well by modifying
Ni into NiwiWi
for synchronous lines and using Wiwi
instead of ci for asynchronous ones.
3.2 Bottlenecks and their identification
In [57, Section 13.2], the bottleneck machine (BN) of an SJP serial line has been defined as the machine,
mi, having the largest effect on the system throughput quantified as
BTP
Bci¡
BTP
Bcj, @j � i, (3.6)
where ck � 1{τk is the capacity of machine mk, and τk is its cycle time.
Since in the work-based model τk � wkWk
, and the only variable, which characterizes the machine is
Wk, expression (3.6) becomes
wiBTP
BWi¡ wj
BTP
BWj, @j � i, (3.7)
This implies that the bottleneck is defined not only by the machines (i.e., their work capacity), but by
Figure 3.2: Bottlenecks of SJP line as a function of job work-requirements.
the job work-requirements as well. In other words, in the same system, different job-types may have
different bottlenecks.
It turns out, however, that for a given job-type, the BN identification in SJP systems can be carried
out using the same procedure as that of [57, Section 13.2] for the part-based model. This procedure
consists of the following:
• Calculate or measure on the factory floor blockages and starvations of all machines in the system,
BLi, i � 1, � � � ,M � 1, and STi, i � 2, � � � ,M .
• Place arrows between machines mi and mi�1 according to the following rule: If BLi ¡ STi�1,
the arrow is directed from mi to mi�1, i � 1, � � � ,M � 1; if BLi STi�1, the arrow is directed
from mi�1 to mi.
• If there is a single machine with no emanating arrows, it is the BN. If there are multiple machines
with no emanating arrows, the one with the largest severity is the BN, where the severity of the
bottleneck, Si, is defined as
S1 � |BL1 � ST2|,
Si � |BLi�1 � STi| � |BLi � STi�1|, i � 2, � � � ,M � 1, (3.8)
SM � |BLM�1 � STM |.
Example 3.1. Consider a five-machine SJP line with λi � 0.1, µi � 0.9, i � 1, � � � , 5, W �
r1.2, 1.0, 1.1, 1.05, 1.2s, and N � r3, 5, 1, 1s. If this line produces job J1 with work-requirements
w1 � r1.0, 1.2, 1.1, 1.0, 1.2s, the bottleneck is machine m2 (see Figure 3.2(a)); if job J2 with work-
requirement w2 � r1.0, 1.0, 1.1, 1.2, 1.2s is manufactured, the bottleneck is machinem4 (Figure 3.2(b)).
This illustrates the effect of w on the bottleneck position (as quantified in (3.7)).
12
4 Performance and Bottleneck Analysis of MJP Serial Lines Using
Work-based Model
4.1 Approach
The approach to performance analysis of MJP serial lines, developed in this paper, is based on reducing
MJP to SJP. This is accomplished by introducing the notion of virtual job and analyzing the resulting
virtual SJP line (denoted as SJPv) using the method of Section 3.
Specifically, given an MJP serial line defined by model (i)-(vi), this approach is described by the
following procedure:
Stage 1: Calculate the work-requirement of the virtual job,
wi,v :�S
j�1rjwij , (4.1)
and introduce the virtual SJP line (denoted as SJPv) consisting of the original machines and
buffers, but manufacturing the virtual job.
Stage 2: Estimate the performance metrics of SJPv line, using the method of Section 3 (based on the
aggregation procedure of [57, Section 11.2]), i.e., evaluateyTP v,{WIP i,v, xST i,v, andyBLi,v.Stage 3: Estimate the performance metrics of the original MJP line according to:
yTP j � rjyTP v, j � 1, � � � , S,
{WIP i �{WIP i,v, i � 1, � � � ,M � 1,
xST i � xST i,v, i � 2, � � � ,M, (4.2)
yBLi �yBLi,v, i � 1, � � � ,M � 1.
Each of these stages may introduce errors in the performance metrics evaluation. These errors are
investigated next.
13
Figure 4.1: Flow diagram of MJP performance evaluation and the corresponding errors
4.2 Accuracy
4.2.1 Preliminaries
Figure 4.1 illustrates the above procedure by the three-stage diagram. As mentioned, using (4.1),
Stage I transfers the original MJP line (with the performance metrics TP �°Sj�1 TPj , WIPi, STi,
BLi) into the virtual line SJPv (with the performance metrics TPv, WIPi,v, STi,v, BLi,v). Using
the recursive aggregation procedure of [57, Section 11.2], Stage II transfers SJPv into ySJPv (with
the performance metrics yTP v, {WIP i,v, xST i,v, yBLi,v). Finally, using (4.2), Stage III transfers ySJPvinto zMJP (with the performance metrics yTP �
°Sj�1yTP j , {WIP i, xST i, yBLi). We denote the errors
introduced by each stage as εIX , εII
X , and εIIIX (see Figure 4.1), where the subscript ‘X’ stands for one
of the four performance metrics. In order to quantify the errors introduced by Stage III, we evaluate
the errors denoted as εTX (see Figure 4.1), where ‘T’ stands for the total error between the metrics of
MJP and zMJP. For Stage I, these errors are defined as follows:
εITP �
|TP � TPv|
TP� 100%,
εIWIP �
1M � 1
M�1¸i�1
|WIPi �WIPi,v|
Ni� 100%,
εIST �
1M � 1
M
i�2|STi � STi,v|, (4.3)
εIBL �
1M � 1
M�1¸i�1
|BLi �BLi,v|.
Errors εIIX , εIII
X , and εTX are defined similarly. In this section, we evaluate these errors using simulations
for Stage I and simulations/calculations for Stages II and T; then εIX , εII
X , and εTX , provide information
about the errors of Stage III.
14
In addition to evaluating errors in performance metrics, we evaluate the discrepancies in bottleneck
identification of the first three systems in Figure 4.1. (Note that, as it follows from (4.2), the bottleneck
of the forth system is the same as that of the third one.) This is accomplished as follows: Denote
these bottlenecks as BN, BNv, andyBNv, respectively, identify them using the arrow-based method of
Section 3, and quantify the discrepancies among them by:
εIBN �
°Kk�1 I
Ipkq
K� 100%,
εIIBN �
°Kk�1 I
IIpkq
K� 100%, (4.4)
εTBN �
°Kk�1 I
Tpkq
K� 100%,
where k P t1, � � � ,Ku is the index of the line analyzed and IIpkq is the indicator function taking value
0 when BN � BNv and 1 otherwise. The indicator functions IIIpkq and ITpkq are defined similarly
in terms of the discrepancies between BNv and yBNv and between BN and yBNv, respectively. The
analysis of these discrepancies is also carried out in this section.
Specific MJP lines, for which these evaluations are carried out, have been constructed as follows:
The values of M and S have been selected from the sets
M P t2, 3, 4, 5, 10u, S P t2, 3, 4u. (4.5)
For each pair pM,Sq from these sets, 400 MJP serial lines have been constructed by selecting their
parameters randomly and equiprobably from:
Tup,i P r20, 100s, ei P r0.80, 0.99s,Wi P r0.75, 1.25s, wij P r1.0, 1.5s,
Ni � tkiWiTdown,iu� 1, where ki P t1, 2, 3, 4, 5u, (4.6)
rj P r0.1, 0.9s so thatS
j�1rj � 1,
where ki � NiµiWi
represents the number of average downtimes the buffer of capacity Ni protects
machine i. Thus, the total of 6000 lines have been constructed and evaluated using the following
simulation procedure: In each simulation run, the first 20,000 units of time were considered as warm-
up period, and the subsequent 180,000 units of time were used to statistically evaluate TP sj , WIP si ,
15
ST si , and BLsi , where s is the index of the simulation run. For each line, 20 simulation runs have been
carried out, leading to the expected values denoted as TPj , WIPi, STi, and BLi, with 95% confidence
intervals less than 0.002 for TP , less than 0.1 for WIPi, and less than 0.005 for STi and BLi.
The results of these analyses for performance metrics and bottlenecks are presented in Subsections
4.2.2 and 4.2.3, respectively.
4.2.2 Accuracy of performance metrics evaluation
The average values of εIX , εII
X , and εTX are shown in Tables 4.1-4.3. Examining these data, we arrive
at the following:
Observation 4.1.
• Stage I induces practically no errors in all four performance metrics for all M and S considered.
• Stage II does introduce errors in all performance metrics. The errors in TP are two-to-four
times smaller than those in WIP . The errors in BL and ST are practically identical. All the
errors are increasing functions of M and practically independent of S. We note that these errors
are similar to those observed in evaluating asynchronous SJP lines (see [57, Section 11.2]).
• Stage III introduces practically no errors. This follows from the fact that the values of εIIX and
εTX are almost the same.
4.2.3 Accuracy of bottleneck identification
The values of εIBN , εII
BN , and εTBN are shown in Table 4.4. Part (a) of this table considers all 6000 lines
mentioned in Subsection 4.2.1. In some of these systems there is only one machine with no emanating
arrows (i.e., a single BN). In others there are multiple machines with no emanating arrows and, thus,
Table 4.1: Average errors and confidence intervals of Stage I
(a) Without starvation of Op. 7 (b) Without starvation of Op.7 and Op. 1 cycle time reduced
Figure 7.2: PP of improved Line MA
investigate the effect of improvement of subassembly delivery and loading, we have calculated PP
with ST sub7 � 0. The corresponding PP is shown in Figure 7.2(a). As one can see from this PP
(which is quite similar to that of Figure 7.1), eliminating ST sub7 allows for meeting the daily target
with product-mix r P r0.25, 0.42s. To meet the daily target for r P p0.42, 0.50s, one must improve
the corresponding bottleneck, which is Op. 1. Since Op. 1 has high efficiency and no starvation by
subassemblies, the only venue of improvement is to reduce its cycle time. Reducing the cycle time for
J1 of Op. 1 by 15% allows the system to satisfy its daily target for all product-mixes, as shown in
Figure 7.2(b). Note that cycle time of Op. 1 for J2 does not need to be reduced.
The above recommendations have been communicated to the plant management and found their
favorable acceptance.
8 Conclusions and Future Work
This paper addressed a class of flexible manufacturing systems referred to as Multi-job Production
(MJP). In MJP systems all job-types are processed by the same sequence of machines (operations),
31
but with job-dependent processing time at some or all machines. A characteristic feature of MJP
systems is that their performance depends not only on the machine, buffer, and job parameters, but
also on the product-mix. Therefore, the emphasis of this paper is on investigating the throughput and
bottlenecks of MJP as functions of the product-mix. The systems addressed are MJP serial lines with
exponential machines and infinite or finite buffers. Specific results obtained are:
• To characterize the MJP operation, a work-based model is introduced; unlike the traditional
part-based approach, this model is insensitive to whether a single- or multi-job production takes
place.
• Using this model, it is shown that buffers are more effective in protecting against downtime for
jobs with larger work-requirements; this phenomenon is due to the fact that the downtime in
units of the machine cycle time is smaller for jobs requiring more work.
• Performance and bottleneck analysis methods for MJP lines are developed; this is carried out
by converting an MJP line into a Single-job Production (SJP) line manufacturing a virtual
job, whose work-requirement is defined as weighted by the product-mix average of the work-
requirements of the constituent jobs.
• Using these methods, the throughput and the bottlenecks of MJP lines are investigated as
functions of the product-mix. In particular, it is shown that for the so-called conflicting jobs,
there exists a range of product-mixes, where the throughput of MJP is larger than that of SJP
of any constituent job-type; this takes place because SJP overloads the respective bottlenecks,
whereas MJP with the“right” product-mix leads to a more balanced work allocation.
• To represent the performance of MJP lines as a function of the product-mix, the so-called
product-mix performance portrait is introduced and a software tool for its calculation and user-
friendly representation is developed; this portrait is intended to help managing MJP lines having
frequent changes of product-mix.
• The methods developed have been applied to a section of the underbody assembly system at
an automotive assembly plant. Analyzing the throughput part of the resulting performance
portrait, it has been shown that the system cannot meet the daily production target for any
product-mix. Analyzing the bottleneck part of the performance portrait, improvement measures
have been suggested, resulting in the desired system operation; these suggestions have been
32
favorably accepted by the plant management.
Numerous problems related to MJP systems still remain open. These include:
• Analysis and improvement methods for hybrid SJP/MJP systems. Such systems, where some
machines operate in SJP and others in MJP regimes, are often encountered in practice.
• Analysis and improvement methods for MJP assembly systems. (Note that the underbody
assembly system analyzed in Section 7 is, in fact, an assembly system; however, in the study
reported here we reduced it to a serial lines using the measured probabilities of its starvation by
subassemblies.) Development of methods, which explicitly take into account interactions between
the main line and the subassembly lines, is an important practical and theoretical problem.
• Development of a theory for closed MJP systems. (Note that the system described in Section 7
was, in fact, a closed line; we treated it here as an open one based on the observation that no
starvation and blockages by carriers took place.) This problem also has a substantial practical
and theoretical significance.
• Robustness properties of MJP assembly systems. As illustrated in Figure 1.2, there may be
different configurations of subassembly lines supplying the main assembly. They may be SJP or
MJP, have dedicated or non-dedicated buffers, use job release for the sequence or for the buffer,
etc. Which one of these structures is the most robust with respect to various perturbations, e.g.,
machine downtime, release errors, etc.? Answering this question would lead to novel approaches
to MJP assembly systems design.
• Additional problems of importance include the issues of leanness, transients, and product quality,
which have been investigated for SJP systems (see, for instance, [57]), but not addressed yet in
MJP setting.
Solutions of the problems mentioned above will lead to a relatively complete and practical theory
of MJP systems analysis, design, and continuous improvement.
33
Appendix
Proof of Theorem 5.1: For N � 8, BN-machine for each job j, is the machine with the smallest
stand-alone throughput, given by
tpij �eiWi
wij, i � 1, � � � ,M, j � 1, 2. (A.1)
Since mk is the common BN for J1 and J2,
tpk1 tpi1, tpk2 tpi2, @i � k.
Substituting (A.1) into the above inequalities and inverting them, we have
wk1ekWk
¡wi1eiWi
, @i � k, (A.2)wk2ekWk
¡wi2eiWi
, @i � k. (A.3)
Multiplying both sides of (A.2) by r, and (A.3) by p1 � rq, and adding the inequalities, we obtain:
rwk1 � p1 � rqwk2ekWk
¡rwi1 � p1 � rqwi2
eiWi, @i � k, r P r0, 1s. (A.4)
Since the numerators in (A.4) are the work contents of the virtual jobs (see (4.1)), the above inedquality
can be rewritten aswk,vprq
ekWk¡wi,vprq
eiWi, @i � k, r P r0, 1s, (A.5)
which implies that
tpk,vprq tpi,vprq,@i � k, r P r0, 1s. (A.6)
Clearly, expression (A.6) shows that mk is the BN machine of the virtual job for all r P r0, 1s. This
proves part (a).
34
Using (5.1) and (5.2), part (b) is proved as follows:
TPvprq � min1¤i¤M
ttpi,vprqu � min1¤i¤M
"eiWi
wi,vprq
*�
ekWk
wk,vprq
�ekWk
rwk1 � p1 � rqwk2�
1r wk1ekWk
� p1 � rq wk2ekWk
�1
rTPJ1
� 1�rTPJ2
.
Monotonicity properties of part (c) follow directly from the derivative of (5.6) with respect to r:
BTPvprq
Br�
TPJ1 � TPJ2
TPJ1TPJ2
�r
TPJ1� 1�r
TPJ2
2 . (A.7)
For N � 0, the bottleneck and throughput are evaluated using the same steps as above, replacing
tpij by smcij �Wi{wij and using (5.3) and (5.4). �
Proof of Theorem 5.2: For N � 8, BNpr � 1q � m1 implies that tp1,vp1q tp2,vp1q. Similarly,
BNpr � 0q � m2 implies tp1,vp0q ¡ tp2,vp0q. Since, as it follows from (5.2), tpi,vprq is a continuous
function of r P r0, 1s, there exist r�, such that tp1,vpr�q � tp2,vpr
�q. Solving for r�, we obtain
r� �w22e2W2
� w12e1W1
w11�w12e1W1
� w22�w21e2W2
, (A.8)
which is unique as long as w11 � w12 or w21 � w22. Thus, TP is characterized by
TP prq � minrttpi,vprqu �
$'&'%
tp1,vprq, if r� ¤ r ¤ 1,
tp2,vprq, if 0 ¤ r r�.(A.9)
Using (3.7), it follows from (A.9) that BN of the line is m1 for r P pr�, 1s, and m2, for r P r0, r�q. This
proves part (a).
To prove part (b), we generalize (5.2) for any r1 r2 (rather than r1 � 0 and r2 � 1). This is
35
accomplished as follows
tpi,vprq �eiWi
rwi1 � p1 � rqwi2�
1rtpi1
� 1�rtpi2
�r2 � r1
rpr2�r1qtpi1
� p1�rqpr2�r1qtpi2
�r2 � r1
rr2�rr1tpi1
� r2�r1�rr2�rr1tpi2
�r2 � r1
rr2�rr1�r1r2�r1r2tpi1
� r2�r1�rr2�rr1�r�r�r1r2�r1r2tpi2
�r2 � r1
pr � r1q�r2tpi1
� 1�r2tpi2
� pr2 � rq
�r1tpi1
� 1�r1tpi2
�
r2 � r1r�r1
tpi,vpr2q� r2�r
tpi,vpr1q
. (A.10)
The last equality in (A.10) is obtained by taking into account that tpi,vpr1q �1
r1{tpi1�p1�r1q{tpi2and
tpi,vpr2q �1
r2{tpi1�p1�r2q{tpi2.
Each tpi,vprq in (A.9) can be rewritten using (A.10). Specifically, for tp1,vprq, set r1 � r� and
r2 � 1, and for tp2,vprq, set r1 � 0 and r2 � r�. This will prove part (b).
For part (c) differentiating (5.2) with respect to r gives us
Btpi,vprq
Br�
eiWipwi1 � wi2q
prwi1 � p1 � rqwi2q2. (A.11)
As a result, if wi1 ¡ wi2, then tpi,vprq is a decreasing function of r. Similarly, if wi1 wi2, then tpi,vprq
is an increasing function of r. Finally, if wi1 � wi2, then tpi,vprq is a constant function. With conditions
given in the first bullet of part (c), for all r P r0, r�s, TPvprq � tp2,vprq is an increasing function of
r, similarly for all r P rr�, 1s, TPvprq � tp1,vprq is decreasing. Therefore, TPvprq is non-monotonic.
Furthermore, r� yields maximum throughput, because of the monotonicity of each constituent part of
TPvprq. This proves bullet one. Other bullets can be proved similarly.
For N � 0, the proof is similar to the above with substituting tpi,vprq by smci,vprq. Switch point
in this case will be
r� �w22W2
� w12W1
w11�w12e1W1
� w22�w21W2
. (A.12)
�
To prove Theorem 5.3, we need the following auxiliary statements:
Lemma A.1. The stand-alone throughput tpi,vprq and system modified capacity scmi,vprq, i � 1, � � � ,M ,
are continuous functions of r for r P r0, 1s.
36
Proof of Lemma A.1: According to (5.2),
tpi,vprq �eiWi
rwi1 � p1 � rqwi2,
which is either a constant (when wi1 � wi2), or a hyperbolic function of r. The hyperbolic function is
continuous on all of its domain, except for r � wi2wi2�wi1
. If wi1 wi2, the discontinuity takes place for
r ¡ 1; if wi1 ¡ wi2, the discontinuity is for r 0. Thus, tpi,vprq is continuous on r P r0, 1s.
Proof for smci,vprq follows the same steps. �
Lemma A.2. Every pair of functions ptpi,vprq, tpj,vprqq, defined by (5.2), has at most one intersection
on r P r0, 1s, unless the two functions are identical in the sense that tpi,vprq � tpj,vprq,@r P r0, 1s.
Proof of Lemma A.2: If tpi1 tpj1 and tpi2 tpj2, then for all r P p0, 1q, rtpi1
¡ rtpi2
and1�rtpi1
¡ 1�rtpi2
. Thus, tpi,vprq tpj,vprq, and, therefore, tpi,vprq and tpj,vprq do not intersect in (0,1).
Similar result takes place when tpi1 ¡ tpj1 and tpi2 ¡ tpj2.
If tpi1 tpj1 and tpi2 ¡ tpj2, then solving tpi,vprq � tpj,vprq yields
r
tpi1�
1 � r
tpi2�
r
tpj1�
1 � r
tpj2, (A.13)
which has a unique solution r� � tp�1j2 �tp�1
i2tp�1i1 �tp�1
i2 �tp�1j1 �tp�1
j2. If tpi1 ¡ tpj1 and tpi2 tpj2, solving tpi,vprq �
tpj,vprq yields similar results. �
Lemma A.3. The equality tpi,vprq � minttp1,vprq, � � � , tpM,vprqu takes place on at most one interval
of r P[0,1].
Proof of Lemma A.3: As follows from Lemma A.2, since every pair of functions tpi,vprq and tpj,vprq
intersect only once, at, say, r�, one of the following can happen:
(α) tpi,vprq tpa,vprq, for r P r0, r�q, or
(β) tpi,vprq tpb,vprq, for r P pr�, 1s.
If (α) takes place, tpi,vprq cannot be the minimum for any r P pr�, 1s, because at least tpa,vprq has
smaller values in this range. Similar statement holds for (β). Now, consider machine i with tpi,vprq,
for which no machine l with tpl,vprq tpi,vprq,@r P r0, 1s exists (if such machine exists, then tpi,vprq is
37
never the minimum). Let tpa1prq, � � � , tpasprq be tp functions that intersect with tpi,v at ra1 � � � ras ,
and satisfy item (α) above. Similarly, let tpb1prq, � � � , tpbtprq be tp functions that intersect with tpi,v
at rb1 � � � rbt , and satisfy item (β). Then, as stated earlier, tpi,v cannot be the minimum in�sj�1praj , 1s � pra1 , 1s. It also cannot be the minimum in
�tk�1r0, rbkq � r0, rbtq. Thus, if rbt ra1 ,
then tpi,vprq � minttp1,vprq � � � , tpM,vprqu only at the interval prbt , ra1q. �
Proof of Theorem 5.3: As it follows from Lemma A.3, there can be no more than M intervals in
which different machines are the bottlenecks. Thus, at most M � 1 switch points exist. This proves
part (a).
For part (b), let r1 � � � rK be product-mixes at which bottlenecks switch and i0, � � � , iK the
indices of the corresponding bottleneck machines (i.e., K switches and K � 1 bottlenecks). Defining
r0 � 0, rK�1 � 1, the bottleneck is given by
BNprq �
$'''''''&'''''''%
mi0 , if r0 ¤ r r1,
mi1 , if r1 r r2,
� � �
miK , if rK r ¤ rK�1,
which implies the throughput function given by
TPvprq �
$'''''''&'''''''%
tpi0prq, if r0 ¤ r r1,
tpi1prq, if r1 r r2,
� � �
tpiK prq, if rK r ¤ rK�1.
Since, as stated in Lemma A.1, each tpiprq is continuous on [0,1], and at r � rk, k � 1, � � � ,K,
tpikprq � tpik�1prq, TPvprq is also continuous. Differentiability of tpiprq on [0,1] implies piecewise
differentiability of TPvprq. This proves the first bullet of part (b).
Under conditions of the second bullet of part (b), tpi0,vprq is increasing on r0, r1s and tpik,vprq is
decreasing on rrK , 1s. If TPJ1 TPJ2, then for any r P p0, r1q, TPvprq ¡ TPJ2 ¡ TPJ1. Similarly,
if TPJ1 ¡ TPJ2, then for any r P prK , 1q, TPvprq ¡ TPJ1 ¡ TPJ2. Finally, if TPJ1 � TPJ2, for any
r P p0, r1q�prK , 1q, TPvprq ¡ TPJ1 � TPJ2. This proves the second bullet of part (b). �
38
Acknowledgement
This work has been supported, in part, by the National Institute of Standards and Technology
under the Award Number 70NANB16H017.
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