vazsonyi scheduling in job shop type production feb57

8/12/2019 Vazsonyi Scheduling in Job Shop Type Production Feb57

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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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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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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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12

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

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L U .

>~. .J

:::l

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MANUFACTU R ING B NDS

v/ / "/ / . / vII'

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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

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w x xa

z4 0

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x3 lL X X0

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x::> x

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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 - - ; - - . , - -

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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

I I t , , I . I I , I r I I I II , I I , I , I I ~ J r - - - - - = I ~ ; . . . . 1.1 I I

I f IOOl l8Ol 120 I 8 5 l I I1 ~ 1 - f - T - 1I . I j I I2 0 ~1 1 ~ : ~ : c t t J l: : : I Ih - : h - ~ I ~ Ih i I I I I I~ I ~ I ~ : I I I I :I I I I I I I I , I I I I I I

I ~ o l l ~ I ~ I I I I I II I I f I I I i I I I I I II 2 I 1 4 I I 6 I I 8 I 11 l 112 I .l14 I 116 1

I118


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I 2 3 4 5 6 7 8 9

a QI QI QI RI Sl R2 R2 Q2 Q2 Q2 Q2 S2f)

WZ

P QI QI QI QI Sl Sl RI Q2 2 Q2 R2 Q2 S2 S2J:U:E Sl Sl RI M M S2R2 R2Y

8 p i p i p i p i p i p i p i p i p2 p2 p2 p2 p2 p2 p2 p2 p2

QI

f3 P T a a a T Y

YT A

A A

RI ~ W W a T y T W p Tf)

I -a I

T T P P T AS r y a0Z p i

u D 8 8 8 8 8 8 8 8 T A Aulf)

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f)

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S2 oW W y T W W a T ra P T

;

p 2 0 0 0 8 8 8 8 8 8 8 8 T

FIG II REPRESENTATION OF SIMULATED PRODCTION PLAN


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,

OPERATIONSHEET ' I

W ~

MAINTENANCESTATISTICS

SI MULATED PRODUCTION LATNESS REPORTPL N

--

...SHIPPING

,

REQUIREMENTS '- SET BACKTIME PART P INVENTORY LEVEL..... ASSIGNMENT MACHINE a -.

STRUCTURE~ -

SCHEDULE PART QMACHINE- PART R

M CHINE UTILiZ TION ....MACHINE ., - .

A ~ PART S

DECISIONRULE

GENERATIONOF

RANDOMNUMBERS

FIG.12 BLOCK DIAGRAM OF SIMUL TION

vazsonyi scheduling in job shop type production feb57

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