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SCHEDULING IN JOB SHOP TYP PRODUCTION
y
Andrew VazsonyiHead Management Sciences Department
The Ramo-Wooldridge orporationLos Angeles alifornia
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BSTR CT
The Accessories Division of Thompson Products, Inc . , i s manufacturing
hundreds of d i fferent types of assemblies including a var ie ty of fUel pumps.
The bulk of these products are manufactured in appropriate l o t sizes to meet
shipping commitments. s a par t of a broad assignment to determine what role
electronic data processors should play a t Thompson Products, a study to im-
prove t h i s job-shop type production was in i t i a t ed .
The study began with a survey of the exis t ing production control system,
then the development of a mathematical model for a job-shop. type production
scheduling system followed. This mathematical model allows the computation
of s t a r t and completion dates for each operation on each parto t also es
tabl i shes purchasing, labor, machine and inventory requirements.
Accomplishments as of today, include 1) a new scheduling system u t i l i z
ing the concept of manufacturing bands has been ins ta l led , 2) a par ts
c lass i f ica t ion system has been established which balances the cost of inven
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SCHEDULING IN JOB SHOP TYPE PRODUCTION
byAndrew Vazsonyi
During the l a s t few years, considerable e f f ~ r thas been eXpended in
developing new sc ien t i f i c methods for the scheduling of manufacturing opera-t ions . This e ffo r t i s well warranted, since the product iv i ty of i ndus t r i a l
organizations i s in t imate ly connected with the a b i l i t y of scheduling pro-
duction efficientlyo In spi te of the fac t t ha t a great deal of e f f o r t has
been expended and important progress has been made in th i s f i e ld , i t i s
s t i l l f a i r to say t ha t no general theory of p ~ o d u t i o nscheduling has been
developed so f a r ,Most of the work applies to a , r e s t r i c t ed area of sche dul-
ing and does not have universal appl icabi l i ty. The paper we are to describe
here i s of the same natureo t has been suc cessful in a par t icular ::type of
production operat ion, and i t i s believed to have wide app l i cab i l i t y. tdoes not have universal appl icabi l i ty, and i t does not present a g ~ n e r a l
th f d ti h d li g
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2
a search for a be t te r scheduling system. In par t icular, ut i l i za t ion of large
scale electronic computers, and the introduction of ~ ~ wsc ien t i f i c pr incip les
of sched.uling were envisioned as leading to improvements.
, In order to describe the work t ha t has been conducted, one could take
two al ternat ive approaches. ne could describe i n de ta i l the work as i t
has been conducted t Thompson Products, or one could present resul ts in
a g e n e ~ a l i z e dfashion. The f i r s t approach has the advantage tha t i t can
be very fac tual and specific. However, the second approach of describ,ing
the method in a generalized form has the,advantage t ha t i t impl ic i t ly suggests
the a:pplicabili y to other production c o n t r o ~problems. This i s the reason
t ha t we se lec t t h i s second approach and describe the s G h e d ~ l i n g s y s t ~ mde -
veloped in terms of the job-shop type scheduling problem faced by a h y p o ~
the t i ca l firm-
e begin t h i s paper by a generalized statement of. the problem of sched-
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s shown in Figure 1. For instance, Figure I sho,V's t ha t the par t i cu la r
~ i n i s h e dp ~ o d u c tA3 i s assembled from four different a r t i c l e s A5, A4, A7,
and All . Some of these a r t i c l e s are assemblies themselves, others are fab
r ica ted par ts .
The a r t i c l e s are manufactured against sales orders tha t are considered
firm. Some of the a r t i c l e s (e .g. ' , f inished a r t i c l e s ) are manufactured on
assembly l ines j other a r t i c l e s are manufactured e i the r on assembly l ines
or in a job-shop type operat ion. The a r t i c l e s are manufactured in various
l o t s izes . (There are only a few a r t i c l e s t ha t are manufactured on a con-
t inuous l ine-f low type of operat ion, most of the a r t i c l e s are manufactured
in l o t s in a cycl ic fashion.)
At the time we s t a r t ed our'work, there was a wide-spread desire for
improvements in production control . I t was f e l t t ha t par ts were not manu-
factured in bes t quant'i t i e s and not a t the bes t t imes. There was a feel ing
3
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4
In, order to appreciate the complexity of the production problem, l e t
us consider Figure :2 where a h i s to r i ca l record of ahypothet , ical production
s i tua t ion i s described. The horizontal axis d e s c ~ i b e stime in manufacturing
days and hours. We dist inguish between four. labor clas ses, each or'tl1.em re-.
fer t ing to a par t icu lar machine or machine group l 3 n 3 C l j. l ~ . L l2 ~ l . . 1 3
describes three successive operations of the t h i rd l o ~of AI I t can be
seen t ha t the f i r s t operation 'on t h i s par t icular l o t i s to be performed on
the f i r s t g roup of machines. After th i p operat ion i s completed, the ~ e m i -
f inished l o t i s t ransported to the t h i rd gro1l:P of m c ~ i n e swhere the second
opera.tion i s performed. Final ly, the l o t i s t r a n s p o ~ t e dto the second group
of machines, where the t h i rd (or f ina l ) operat ion i s performed. This means
t h a t a t t ha t point the t h i rd l o t of A I l s completed. Similarly, one can
follow the production of the f i r s t l o t of A 3 ~ ldescribes the f i r s t
operat ion on t h i s lo t ; t h i s i s performed on the fourth group ,of machines.,
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5
par t icular l o t should,be worked upon. I f the l o t i s not avai lab le , he knows
t ha t the l o t i s l a t e and he t a k ~ scorrect ive measures.
In the speci f ic case under discussion, t was considered impract ical to
follow t h i s sor t of planning scheme. There ar9 hundreds of shippable :items
involved and thousands of par ts manufactured and purchased. The possible
number of'. combinations in t h i s "jig-saw puzzle" are astronomical. Prepara-
t i on of even a single chart o f t h i s type would be. an e x c e e d i n g ~ t i m e - c o n s u m i n
job, but even then there would be no assurance t ha t the f i r s t t r y i s an e f f i -
cient one. There i s however, a fur ther d i ff i cu l ty.
Suppose we are looking a t the plan t h i s morning and t ry ing to determine
what operation we should perform. e observe t ha t up to today we w e r ~on
schedule and we have performed the three indicated operations according to
our or ig inal char t shown in Figure 2. e recognize a t t h i s point t ha t the
second l o t of A2 has not arrived yet , and therefore opera i ~ ni t 1 cannot
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6
Furthermore, not only i s i t impractical to carry out the computations, but
there i s a conceptual diff icul ty. What i s the point of carrying out th i s very
elaborate scheduling computation, say for three months in advance, when we
know t ha t every day we wi l l have to rework the whole schedule completely?
. Suppose every morning we could determine by some magi'c the optimum -' schedule.
In what sense could such a plan be optimum i f we have to change i t tomorrow?
Ear l i e r, we found i t useful to look upon th i s problem in d i f f e r ~ n t
way.* Figure 4 shows the four production machines of our h y p o t h e ~ i c ~ p r o b. .
lem. The available semifinished l o t s are conceived as forming a wait ing l ine
in f ront of each of the machines.4
L3,2 refers to the fourth l o t of A3 . The
l s t index 2 denotes t ha t the second operation was l r e ~ yperformed. e
placed th i s semifinished lo t Lj,2 in front of machine 1 , as the th i rd opera-
t ion on the lo t i s to be performed on machine 1. As shown in Figure 4, there
are three l o t s waiting to get on the f i r s t machine, four lo t s to get on the
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e formulate our problem in production control , then, as the problem of
developing decisio? rules t h a ~wil l ins t ruct or aid the foreman in what to
do.
Let us speculate for a moment on the types of decision rules one could
conceivably have. Suppose there is a master pr io r i ty l i s t of a l l the par ts
to be manufactured, and the foreman i s instructed to take the l o t that has
the highest pr io r i y o n the l i s t This would certainly be a possible deci-
sion rule .
7
The trouble with th i s decision rule i s that par ts that are low on prior-
i t y l i s t would be pushed back more and more, and perhaps never would be, com-
p l e t e d ~Subassemblies would not be available and the manufacture of some
assemblies would stop- As new orders came into the shop, some of these lo ts. .
would bem a n u f a ~ t u r e d f i r s t
and production would get out of balance. Veryl ike ly the whole production system would collapse.
T l t l t th t th f ld h i t
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8
Development of Decision Rules
In f ac t , the people in production control did have some decision rules
of t h i s type. They had some so r t o f a feel t ha t to ld them j u s t how long i t
takes to get certain par ts through the shop. The diff icul ty, however was
t ha t th i s feel was quite uncertain, and i t varied from one person to another.
In our terminology today, we would say t ha t the decision rules were not ex
p l i c i t l y formulated and s tabi l ized. What was needed f i r s t was not so much
the development of optimum scheduling procedures as a s tabi l iza t ion of the
unwritten decision ru les .
The f i r s t step in the s tabi l iza t ion of the decision rules can be des
cribed with the aid of Figure 5 The horizontal scale i s time, the ver t i ca l
scale i s the cumulative number of ar t i c les manufactured. n the r ight-hand
side there i s a l i ne t ha t denotes the shipping schedule, which t e l l s how many
ar t i c les should be shipped t what dateo To the l e f t of the shipping sched
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9
foreman to make decisions. Let us recognize, though t ha t the concept of
manufacturing bands somehow implies that there i s a f ixed flow time for
each ar t i c le .
These ideas' so fa r are somewhat vague, and many questions can be asked.How seriously should the foreman take these indates and outdates? How should
the width of the m n u f c ~ u r i n gband be determined? Does t h i s whole scheme
, make sense?
In order to answer these questions, l e t us make our proposition more
precise Let us assume t ha t there i s a f ixed width for each manufacturing
band, and t ha t these widths, to be cal led ttmake-spans can be represented
by a setback chart as shown in Figure 7 Suppose we set up the hypothesis
t ha t the plant indeed has been operating according to a scheme of t h i s so r t .
I s there any way to ver i fy t h i s hypothesis?e could examine the records of the company and study the his tory of
h i di id l l Th ld k t t i t i l d d d i
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that the make-span in fact does not depend on the standard times, but corre
la tes well with the number of operationso Figure 8 gives a hypothetical
sca t t e r diagram showing how the make-span i s related to the number of opera
,tiona 0
On the basis of th is correlat ion an,alysis, we have decided that i t does
make good sense to use these make-spans. e prepared a , l i s t of make-spans
for each of the parts; we prepared set-back c h ~ r t ssuch as Figure 7, and we
proceeded to i n s t a l l th is production control system.t wil l be of some in te res t here to describe some of the ~ t h e m a t i c a l
deta i ls of the scheduling procedure
e denote by ~ i kthe set-back of Ai in Ak For instance, ~ 8 3i s 90
days in Figure 7. e denote by s ~ the number of ar t i c les Ai that must be
shipped in the m'th production period. Finally, we denote by x ~ t ~ number
of ar t i c les A. that must be manufactured in the m'th production period to~
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11
In conjunction with our method of scheduling, we a lso developed a method
of labor forecas t ing . One can readi ly see tha t once the make-span for every
ar t i c le i s postulated there i s a unique re la t ion to predic t labor loads.
The t o t a l labor hours required in production period m on machine type n i s
given by
Li
m.x.,
n ~ ~(2)
where T . i s the hours required to m n u f c t u r ~A On machine n, and hm i sn ~ ~ n
the t o t a l hours required on machine n in period m
The i n s t a l l a t ion of t h i s scheduling system required a great deal of
systems, and procedures work, the descript ion of which l i e s beyond the pur-
p.ose of t h i s presenta t ion . Howeyer, a f t e r the i n s t a l l a t ion of t h i s system
i t was recognized t ha t in spi te of the great ~ ~ p r o v e m e n t srea l ized , and
the large economic saving effected, there are s t i l l fur ther improvements
possible. This prompted us to continue our invest igat ion and ref ine the
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In Figure 9, we show a detai led production plan for a par t icular l o t .
The two heavy dots represent s t a r t and completion dates. Each cross on the
diagram represents a par t icular operation. For instance, i t can be seen
t ha t the f i r s t operation takes two days. The second operation i s performed
on the th i rd day, and the t h i rd operation i s performed on the t h i rd day too.
The fourth operation takes three days, as shown in Figure 9 by the three
crosses. The r e s t of the diagram shows the schedule of the r e s t of the
operat ions. In the scheduling system previously described, onlyt ~
twoheavy dots were specif ied and the production personnel, so to speak, ,maneu-
vered,between.these two end points . Now we wish to specify in de ta i l when
each operation i s to be performed..
In order. to carry out t h i s deta i led scheduling system, i t was necessary
to develop rules which specify what type of production lo t i s allowed one
day, two days, or three days, e t c . of manufacturing time. In order to
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Star t ing with t h i s class i f ica t ion system i t i s possible to develop the type
of schedules shown in Figure 9 and therefore i t i s possible to specify
s t a r t and completion dates for each l o t and each operationo
The implementation of t h i s Time Assignment Scheduling System required
profound changes in the method of production control . To specify with t h i s
d ~ g r e eof de ta i l the schedule of each operation requires an int imate know-
ledge of a l l events occurring on the production f loor. Consequently i t
did not seem advisable to attempt to i n s t a l l t h i s new scheduling system in the
ent i re Division t was thought be t t e r to design f i r s t a p i l o t ins ta l l a t ion
A re la t ive ly simple commodity was selected for. the p i l o t i n s t a l l a t ion as
i t was f e l t t ha t t h i s smaller system could be monitored with re la t ive ease
and without the expenditure of a great deal of manpowero However care was
taken tha t the commodity selected possessed a l l the s igni f icant features
of the problem so t ha t the p i lo t i n s t a l l a t ion could be considered--as a t rue
representat ion of the ent i re scheduling problem
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t i s believed tha t the scheduling system described here, represents a
very s igni f icant improvement in the operations of t h i s firm. However we do
not believe in any way tha t t h i s scheduling system i s an optimum one. The
development of optimum s ~ h e d u l i n gsystems i s an extremely di ff i cu l t problem,
and here we can report only on our plans. e are considering simulating the
production scheduling problem with the Monte Carlo Method on a large scale
electronic d i g i t a l computer. As our ideas are preliminary here, i t wi l l be
perhaps bes t to describe the proposed approach through an example.
Simulation of Scheduling
Consider a very simple production scheduling problem where four par t s ,
P, Q R and S, are to be manufactured on machines a, ~ y and o. The pro-
duction requirements are shown in Figure 10. t can be seen, for instance,
that 100 of assembly P i s required by the end of the s ix th week 80 by the
end of the ninth week 120 by the end of the twelfth week and 85 by the end
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of the f i r s t l o t of each par t ; the l ines headed by 0 ~ y and B show the
his tory of each machine. The two representat ions are somewhat redundant
and must be in agreement. e begin by producing the f i r s t l o t of Q i . e .
Ql. Figure 11 shows t ha t Ql goes on machine on day 1. Each symbol inFigure 11 denotes two hours of production, and so i t can be seen t ha t the
f i r s t operat ion on l o t Q takes four hours. During the t h i rd quarter of
day 1, Ql i s i n - t r a n s i t to machine o. This i s designated by the l e t t e r T.
Then Ql goes on machine 0 for a six-hour period. As a comparison we see in
the l ine headed a t h a t Q l i s indeed, manufactured on machine a , as indi
cated by the l e t t e r Ql. Then Ql i s i n - t r ans i t for two hours to go on
machine T where i t stays for four hours. Then Ql i s i n - t r ans i t again, and
becomes available as designated by the l e t t e r A. At the beginning of day
4 Ql i s available for assembly. Manufacturing of p l on machine 8 beginsin the second quarter of day 4. This can be seen i n the l ine headed by B
15
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denoted by the l e t t e r Mo There are many other factors that must .be i n ~ l u e
in a rea l i s t i c . s i tua t ion , but here for the sake of simplici ty, these other
factors are omitted.
In order to.carry.out, in a systematic fashion; the preparation of suchschedules, i t i s necessary to have decision rules which specify what to do
in each i n s t n t ~In par t icular, when different par ts compete for the same
machine, there i s a need for a decision rule to indicate which par t should
go on the machine f i r s t The simplest decision rule i s to take the par t
f i r s t which arr ives f i rs to More elaborate decision rules which re la te more
int imately the production sohedule to the delivery requirements have been
*uggested. by various authorso
In addit ion to the speci f ica tion of these decision rules, i t i s neces-
16
sary to know the s ta t i s t . ics of each of the factors that enter in to p reduction.The s t a t i s t i c a l dis t r ibut ion of the time i t takes to manufacture a par t ,
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17
W e ~ l l u s t r a t e ,tenta t ively in Figure 12 by a block diagram of how the
simulation process could be performedo 'The b a s i ~input i s the shipping
schedule as shown on the lef t-hand sideo The next block shows the set-back
structure that i s to be employed in the computationso Then we need to storein the memory of the computer the operations sheets which describe on what
machines the par t icular par t i s to be manufactured and in what sequence.
e need, to stoI, e information on maintenance stat is t i .cso The par t i cu la r Time
Assignment c h e d ~ l i n gSystem to be employed, must be stored in the memory 0
e need to specify the par t icular decision rule to be employed, and f ina l ly
we need to generate random numberso This i s the information from which the
computer can prepare s i ~ u l a t e dproduction plans of the type shown in Figure
11 : From et:lch simulated production plan we obtain information about the
e f f ~ ~ t i y e n e s s o fthe par t i cu la r scheduling system employedo I t i s recognized
here t h a t no single measure of effect iveness i s available yet to evaluate
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8
effect iveness, the par t icular system of scheduling s evaluated. Then pro
posed improvements in the scheduling system are introduced, new runs on the
computer are made and these proposed improvements are evaluated.
One of the important problems we plan to study, s the problem of shor
tening lead-times. Instead of using the empirical set-back ru les , Figure 8),
we plan to experiment with various proposals for shortening the lead times.
With the aid of simulation, we wi l l evaluate the va l id i ty of these new se t
back s t ruc tures , and wil l determine whether these set-back structures are
prac t i ca l in actual production.
In summary then, we can say that our point of view of looking a t the
problem of production control as a problem in decision theory, has been f r u i t
fule e have developed, ins ta l led , and tes ted , cer ta in decision rules in a
ra ther complex s i tua t ion . Our study s tar ted with the purpose of introducing
high speed electronic data processors, but a great deal of work had to be
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OZINTO GR PH
FIG I
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f ) 1wJ)
(J)2J
U
SHOP LO DING
_ 1 Q ~ V g:3 ~I \ I j \I \ I I \ \I \ I \ \ \I \ I . n ~ 3 I I I \ \I \ \ I \ \I \ / \ I / \ \I \ / I 1 \ \ \
. n ~ 1 ~ 2 - - - r 1 1 ) . n . 2 ~J \.n.3.'2. I I \ \r \ I \ \
I I \ \ \ I \ \j / ~ ~ \ \ I \ \
I I n 3 ~I I I I I . n . 2 ~ 2k \I f I \ \ \I I / \ \ \I I I \ \ \I
I 1\ \
t t f 3 J _t 3 t3' t z t t22 tSTART DATES COMPLETION DATES
MANUFACTURING DAYS ND HOURSFIG 2
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SHOP LO DING
J I l ~ I I . Q 3 3 Q 2 ~ 3\\ /\\
I2 \n 1 ~ 3
\\ /\ /\
I /
3 I n 1 ~ 2 . n 3 2 I I n 2 ~ 1 I/ 1 I
I/ I II I
4 n 3 ~ I I I n 2 ~ 2II
III
TOD Y
MODIFIC TION O SHOP LO DING
FIG. 3
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G 8 I M A C H I N E Io I A C H I N E 2
I M CHINE 3
o I M A C H I N E 4
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f )w...JU
odz
U
>
>~~u
MANUFACTURING BANDS
AUG. SEPT. OCT. NOV.
M NUF CTURING D Y
FIG.5
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nw...JU--C
LLo
oZ
L U .
>~. .J
:::l
U
MANUFACTU R ING B NDS
v/ / "/ / . / vII'
~ /v
/ IvV j / V '1X: j i
II
/, II II II
, I
I/
v Ifj
/ /I
JV
V /
I I Lv
AUG. SEPr. OCT NOV:
M A N U FACTU R NG D AYFIG.6
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SETBACK CHART
A3 iAs
I iI i AsI II r iI A 12.
i A.21-- LI i I
I I A14 I A15I L I AlI Ir AI II
~AJ3
I A orI A9L
L A7II AllL
I I I I 1 I I I I I I I I I I I I I I2 4 6 8 100 12 14 16 18 2
M A N U FACTU RING DAY
FIG 7
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5 n
a
w x xa
z4 0
.) x>00aa
x3 lL X X0
a::wCD
x::> x
2 z x
Iza
n xw
10
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I I I I I I I I I II I I 1 I I I I I I X I X
- - r - - - - - . J . - - . l . - - - r - - ~ - - , . - - - - - t J . I
I I I I I I I I I I I1 I 1 I I I I 1 I X I I
CJ)t -+ -1 - 1 _ L _ -t t T _ . J _ l f1 1 1 I I 1 1 X I X 1 I
;2 1 1 1 1 I I I X I 1 I Io - - - - - - - - - - - 1 - - - J . - - T - - - r - - - I - - - - 1 - - - - - - ~ - -i= : I : : I : I : _; __ _ _ J. -1 - - - - r - - 1 - - I - - ; - - . , - -
1 I 1 I I I I 1 Ia:: I I I X I X 1 X I I I I 1W - r- - -+--T--- .L--- - - -T---+--- t - - - t - - t - -
Q. I Ix: : I I I I I I Io - - t - - - - \ - - - r - - ~ - - - - ~ - t - - t_ 0.1. __ - -I I I I I I 1 I I 1~ -tX -- J._ ~ __ J __ - _- . __ l_
X X I I 1 I 1I 1 I 1
WEEKS
FIG.9 TIME ASSIGNMENT SCHEDULING SYSTEM
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PARTS
P
Q
R
S
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FIG II REPRESENTATION OF SIMULATED PRODCTION PLAN
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FIG.12 BLOCK DIAGRAM OF SIMUL TION