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ffiffiffi
2. GOAI PROGRAMMING
AIPNOACETO
ACGREGAMPRODUCTIOH
PI,AI{NING
I A CA,SESIIIDY
A Thesls submltted.
In Parttal lrlfilnent of the
Reqrrlrementsfor the Degree of
}IASMR OF TECIINOIOGY
BY
YOGESHSA:GNA
,i'd
r0 TiIs
DEPART}MVT MECHANICAI
OF M{G]NEERING
?
rNDrAlIrNsTrrutB oF TEcHIIOIocy,
DEIHI
1982
3. This is to certily that
I'lrr Yogesh Sarcenaworked. for his
i{. Tecirr proS ec t r'Goal prog ransdng
Approaci:. to Aegregate productlon
Planning : A ease strdyrr r:nd.er
rV sup ervi sion in the i,iechanic aL
Engineering Depar tiuento Ind.ian
Ins titute o f Technologyr Del jrl e
I further certify that
tn-ls proJ ect has no t been taken
up before for the award. of any
degr€er
,i
( DF.' I,i. SIIIGH)
.
Deptt. of i'ieeh. Engg.
I.I.TrDellti.
4. g 9-$J E N.p
A C_$_N. br_,L D_
0
I aJngrcattry ind.ebt€d. to Dr. N.Singh
my pro j ect supervl sor and. express ry
g rati tud.e for his af fec tionate and encourag tng
guld,anceo During the year in wlrieh I worked.
uncler hiln I forrnd. hls invaluable adl.Lce of
g reat he1P.
Thanlrs are also d.ue to I4r. G'DrSardanae
Gen, r"ianas K .Ganpathy r i'ianager, l'lalru-
"l;pt?;
facturing for provioj-ng roe inva^]-uable
Senrices,
'
h elp and. suggestions .
I aJ so acls"Ioi^IJ e '*[ th t]rank s the he]p
e€
extended by llr. Sond|rl l Indlts trial nngineer
and. other staff of llj-nd.rrstan Bro'nrnBovffr.
Thanks are al-so due tc the s taf f o f
C o n r p u t e rC e n t r e , I . I . T . Delhlr
I . I. TrDelhl
=)*'l-f<^^
19E2. (YoGESHSAlGliA)
5. LB_S T R A C_T
In this thesls an attempt has been mad.eto
analy s e the Agg reg ate Proclrrction p] annlng o f
Hindustan Brown govd, Far idabad, op tirnally.
The denand of the noicr s w:tth d.ifferent specificatlons
ve re no t the c ons tant during the planning horizon of on e
year io€r 1982-83? Consisting of three plaruring period.s.
To meet wltir the fl-uc fuations in demand.e
Ag g regate Plannlng mo,iel was formirlated., which concerr-
trate s on d.etermining lrhich comblnatton of the d.eclsion
variables J.il<e prodirction Taie, inventoryl backord.ering
over tiile etc. should be r-rtilis ed. in order to optimally
acUus t tlre demand fluc tuation s wi th-tn the con sl"raln ts
i f BnX.
The AggregaLe planrring moder-was formulated. in
the form of goal s wi thr dlf f erent prloritieso The
problen was then solved by uslng r'Coraputerlsed.technique
o f S .i'i. Lee to solve the CoaJ- Prog ramnrlng ProbL errls The
rr,
decis icn variabl es were obtained for all the planning
p eriods.
6. C O N T E N T-S
Page
1r INTRODUCTION
1.1 Oeneral t
fl
1 oz $eg reg at e prod.uctlon p1 annlng i L
General Form
tl
1 o3 lirqlest structure of Aggregate
Prod.uctlon plannlne
1.4 I'Iul tl s tag e AgSreg ate pl annlng Sys tem 5
1r
I rO Intpor tance of 'loal prog raJxalng G
1.6 The Goal Prograuxning Concept a
1o7 0 bJ ec tive Func tion ln Goa-l p rog ra$ming 3
1.8 Rankire & weighr.Lng of i'iul tlpl e g oal s 3
2. IJTEiiATUiiE nnVIS^I 11
3o GOAL PnOGRAi'Ii'IIliG A I,',.,A,[E,.1ATICAL
AS ICOI L
3.1 General i'iath einatical ;,iocieI LI
3o2 Step s o f the Siuplex method of Goal L2-
Prog rarunlng
3.3 Computer based Solution of Goal L6
Prog raminlng
3o4 Flow Di aeraul
4. PROCIE;.ISTA'IE.i.fiI{
T 3L
4.1 General 3L
4o2 Data Oollection Tabl_es 3G
5o GOAL PRG RAi,li,iN,IG}-IJrd,ir.JLArIOl,I ql
6. SOLUTIUI]
At,iDO iiEjitS 55
7. S UGGESTIuI'jS
FOn FIJliIiG.,t yjr,,rrK 58
Bo REFEREJ]CBS 53
9. APPENDIX
7. CHAPTER I
-
rNT,Rg.pucTIoN
1 .1 GH'IEF4,.L
:
i'lost manag
ers want to plan and, control operatlons
at the broades t revelthro ugh some rrrnd. of aggreg ate
pl'annlng that by passes detalls of indlrridual prod.ucts
and detailed schedrrllng of facillties and. personirelo
i'ianagemer:t would. tr.eal $r:ith baslc relevant
d,ecisions
of progra.uaud.nggtne use of resourC€sr
Thls ts &ccon_
-pllshed by reviewirrg proJ ected.
enpl0yment levels and
by setting actlvity rates trat can be varied. wlth rn
a Blven errploynent level b7 varytng hours ruorked,
( worklng overtirne or rrnd.ertiiie) r
Once ilres e basic d.ecislons have been mad.e
for
the upconlng perlod, detalled. sched.rrltng ean proceed.
a t a lorser level rvi thln the cons traln ts o f the bro ad.
pIan. Finally last rnlnute ciranges tn actlvtty levels
need to be uade with the realisatlon of thelr posslble
ef fects on ttre cos t of clnnghg prod.uctton level,
and on lnventory costs lf they are a part of the
sJ,:S
temo
-
8. 1oZ AC.CRECATF
g
s
The Aggregate prod.uctl0n plan'ing problen ln
lts most generar form ear be stated.
as forlows.
Given a set of forecasts of
d.emand,,
what shorrl.
be for each period
a) Itre size of work forcel l{t
b) Ihe rate of prpductlon, pt
c) The QuantltY shlpped, Str
The resrrrtrng lnventory per month
can be deter_
nlned. as f; = It-1 + pt - St.
The problen ls 'sua{y resolved
analytlcarly by
mlnlntzlng the e :rpected to tar
cost over a g i.ven Flann-
-lng horrzon consrsting
of soc* or arl of the fouowfus
cost coqponents:
a) Ihe Cost of regular payroll and,over tlne
b) The cost of chanelng the productl0n
rate
from one perlod. to the next
c) The cost of carrfing jnventory
d) Cost of shortag es resul tlng frour
not
meettrg the d.ennand.
The solutron to ttre problem
ls greatly srncpll_
-fled' lf average d.emand
over the prannlng horlzon is
expeeteti. to be constant.
9. The compl-exlty ln the Aggregate production
Plannlng problen arlses fr"on the fact that tn most
situations d.enrand.
fer perlod. ls not ccmstant but are
subJ ec t to subs tantlal ff-uc baablon and the ques tlon
atLs es as to how the se func tions should. b e absorb€d..
Assunlng that there are no problems ln receivjng a
constant supply of raw materials and. labour at a fixed.
wage rate, the problem ouy be seen by eonsld.erlng
thr ee Pure al ternative ways o f r e spondlng to such
fluc fuations o
a) A lnci'ease in orders ls met by hirirrg anC a
d.ecrease 1n orders ls accotapll sned b1'.layoff s.
b) i.iain tenance of constant work force, adJus ting
productlon r ate to orders by working ovelrtlme
and wrder t1 ure ac c or dlng ly .
c) i,ialntenanee of a constant work force and constant
pTo duc tlon rate r allow-tng lnventories and order
b acl0og s to fluc t,aate .
d) l,iajn tenance of c snstant wor k force and mee the
t
fluetuation i.:n demalid.through planned bacKlogs
o r by sub con trac tlng exce s s d.e
marrd,
r
In generalr none of the so-called. pure a-lternattvesl
dlscus s ed w111 prove be s t, but rather some courblnation
10. o f ttrem. ord,er flue ttratl0ns
showed. g eneral be
In l
absorbed, partly bD' inventorxr parily
by overtlme,
and partJ.y by fririne and, Iayof,f,s
and the opttuun
eqphasls of these factors lnlll depend.
upon the costs
ln any parttcular factoryr
1.3 u
PRosrFS
:
The structure of t'e Aggregate plannfrg problen
is represented. by tlre slngle
stag e sys tem trer the
plan'lng horlzon ls only one period a'ead.r
the state
of the system at ttre e'd. of perlod.
1s d,efined. by wo,
Pe and ror the Asgregate work force
slzel productlon
or ac tlvibl' rate and. jnventory leve1, respectlvely.
'rhe end'lng
state c qrd.lttons beeoure the 1nltlal condtttons
for the upcourlng perrod. 'rle
have a forecast of the
requlrements for
the upconlng perlod.s through soge
proc €ss o Deelsions are nad,e that set the slze of the
work force and' prod.uetron rate for the up-cond.ng perlod..
The d,eclsions ma,ie uray call- for hlrlng or layj_rrg off
personnelt thus expand.lng or contracttng ttre effcctlve
capacity of the productJ.ua system, The uork force
slzel together lrrth the d,ec1slon on actlvrty rate durc-ns
the pertodl tlren d.eterrnrnes the requlred. arrcunt of
11. 5
overtimel lnventory levels or back ordering
r whether
or not a shlft nust be added
or deleted. and other
posstble changes tn operating
procedur€o
1o4
:
Ftg . shows a mrrl tl s tag
e agg reg a te pLanntng
sys teun vhere the horlzon
has been expand.ed, th for _
w"l e
cas ts for eac' perl0d.o u*"
obJec tive 1s to nake the
declsions eoncernlng the work
force slze and. productton
ra te for the upconing p erlo d., In clolng
so r however
we consid,er the sequenee of proJ
ected decisions ln
relation to forecas ts and their cos i
effectso The
declsion for the upcorntng perl0d,
ls to be arf,ected. by
the futr*e perl0d. forecasts
a:d. the declsl0n process
nnrs consld'er the cost effects
t
of the sequence of d,eclslons.
Tir e conn ec tlng rtnks b e tween
the s everal s tag es ar e
the w, pr and. r values that
a^re at the end. of one
perlod and the beglnnlrrg
of the nextr The feedback loop
frour the d'ecision process
ru4ylnvolve some lterative
procedure to obtaln a
solutl0no The seguentlal nature
o f the declsions should. be
kep t 1n mlnd.. All d.eclsions
are rlght 01' wrong only ln terrns
of the sequence of
declstons over a perlod. of tlne.
h€
12. 1o5
:
Organizattonal
obJectlves vary
aecord.ing to
the elraracteristicse
typesr FhtlosoptXr
of &anageuentl
so partierrlar environuenta-l (o ndlttons
of the organr_
aat10n' There ts no slngle raelversal
goal fo.. a,.
org anrzatl0ns. rn boayr , cf,rnand.c
business envlronment,
fl*ns Flace g reat
emptrasls on soclal
responslblll ttes
social contrlbutlons, e
publlc relatlons
r lndustrlal
and 1abor relatlonsl
€tcr
rf we grant
that roanagerent has
m.[tlple conffls_
tlng obJ eetives to
achi€ver the d.eclslon
crlterta
should arso be nru' trdlnensl0nal0 ,rh1s
tupr_les that
when a decislon lnvoLves
nultlple goa_1sr the_quantltatlve
tecl:nlque used. should.
be eapable of hand*ne
muLtlple
dectsj-on criterlao
The llnear programudrg
teehnlque
has a llelted value for problems
hvolvlng rruLttple
t oal sr
The primary dlfflcurty i.rrth llnear progsamm{ng
ts not its inablllty to refrect connplexreallty.
Rather,
lts dlfficuJ-ty lles rn the unldlmensl0nallby
of the
obJ ective Jnctionl
vrhleh requtres cost
or proflt fuifor-
matl0n that 1s often
alnost lnposslble to
obtalnr To
13. .: ,!
1
overcone the urld,lnenstonallty
of the obJ ecttve
f*rrctton requlred. ln the llnear
prograrrrnilng, efforts
have been natr"eto convert
varl0us goals, costs,
or
value neasure lnto one
crlterton, nanely utlllty.
However exact neasurernent
of uttllty ls no t a
slropl e mattere f'Ience1 d.ec1s10n naklng
throug h llnear
programmtng vla a uullw
fraretl0n is onry feaslble
ln a theorettcal serseo
Goal progra^urrd.ngts
a mod.ification and. extensl0n
of L'P' ' The goal progra-mrnlns
approach ls a technlque
tha t t s capable of handlr'g
deelslon probleros that
dealtvlth a slngle goal wtth nrrltlple
subgoals., as
well asi problerus wlt'
multJ.ple 80a1s wlth n*ltlpte
sub goalso
irle can solve the se prdblerns us jng Lrp r
^rbth
j{ul tlple
obJ ee tives o For t'rts r w€ nay ln trod.uce
o ther
than the obJ ective fr:nctlon, as rod.el constralnts.
The l.p- rccel r equires
ttrat the cptlnum solutton
nrrst satlsf! all constralrrts. Furttrermore, lt ls
assumed here that equal
lnportarrce 1s attached. to
varlous
obJ ecttves r However in
reall wr such assurnp t10n are
obsurdo trtrst of arl,
it ls quite posslbre that
arl
tl:e constratnts of the problem
can not be satisfied..
14. such a problera 1s called. rlnfeasiblerro
secondly
all eonstralnts do not have equal lcportanc€o there-
fore goal progranrnlng vhech renpves al.r.
such dlffteul-
tles ls us ed. to solve such probleins.
1.o ru&_QOAt q)i,tgEF.T
lR0GRAtl},IryG :
The concept of goal progyarnr4ingwas first lntro-
d'uced by A' Charnes & l'^lol{oCooper as a tool to resoLve
lnj'easible linear prograrnrd:rg problefis o Ttrls technlque
has been rrrther reflned. by yorJ lri & s rl,lrlee and.
o thers r Goal progran,ulng wnd.chis s pecial extenslon
of llnear programrulng, ls capable of solvlng declslon
p robl ens with a slngle g oal or uul tlpl e g oal s o The
goals set by tlre ttanagenent are often achlevable only
at the erpense of other goals. zurfher-no!€ these
g oal s are ln couunensurable i o€. they cannot be measured.
on the same unlt scsl€r Thus there 1s a need. fcr
establishlng a hlerarcly of tnportance aupng these
confllctlng goal s so that low ord.er goals a.re consld.ered.
only afLer the hrgher orders prlorlty goals are
satisfied or have reached. the point beyond wlrlch no
furtlrer lqprovement j.s deslrableo Hence the problen
can be solved. by goal pfogrenryr{ng tif the uuaagement
can provide the ordtnal ranklng of the goals tn tenms
15. si
*?.
rt
.1.
of thetr tuportance & arl relattonshlp
of the rcd.elr
Econonl'caily spealclngr
the msnager faces the problen
of the allocatlon of scrace
resourc€so ft ls not
always posslble to achleve
ttre wery goar f*lly to
the extent d.esrred.by i'anagement.
Thus, wrth or
wl thout Plogramnlng the manag
, er attaches a c er taln prtor _
-1ty to the achreveinent
of a partlcurar goal. the true
value of goar- progrannrins
ir, there-or.€1 the sorutron
of proble'Sl lnvolrnlng !rutttp1e,
confltet,,'g goals
acco'ulng to tlre i'ianag r s pr10r1ty
er s truc tur.e.
1.?
:
rr: goal programmrpg rnstead.
of try1ne to haxrorise
or nlnlnlae the obJec tive crlterlcm
dlreetly as ln
rlnear progranndng, 1t trles to
nlnfudze the d.errrattons
anong the goaLs and wl th ln Lhe g
lven sets of constralnts.
rhe devlatlonar vartable is
Tepresented. two
ln
dimsrsl0ns 1n the obJecttve functl0n,
a posttlve and.
a negatlve deviatlon fr"om each
subgoal and/or con_
s trainto Then the obJ ectlve functlon becones
trre ninl-
-wLza*ton of these d,evlatlonsl
based,on the relatlve
lnpor tance or prlorlty as srgned. to then.
1 .8
0AIS :
in order to achleve the ord.lnal
soLutlon-that
16. lsr to achle ve the goals aecord.lng to thelr lryortaneel
(-) Begatlve and Sr posltlve devlatlons about the goal
must be ranlced accord.Jng to the r,prerytiver' priority
factorso In thls way the low-ord.er goals are consl-
dered only after higher- ord.er goals are achleved as
deslred. The I'Preerrytlvet' priority factors have
the relationship of pJ
the multlplicatlon of De however large lt may be,
cartuot rnakepJ+1 greater thran or equal to pJ.
The next step to be consldered tn the goal
prog ramrnlng i s the welg h-tng c.if devlatlonal variable s at
the same priorlty leve.lr rf any goal involveb many
deviational variables and lre want to glve prlorlty to
one over the other, thl.s can be achieved. by assigning
dj.ff er ent l.Ielghts to tl:e s e deviational variabl-es at the
sarne prlorlty leveI. At the sarneprlorlty levelr the
subgoal which acguires manrfuouui
dlfferenttal qeight w111
be satlsfled first & then lt wLLl go t o the next. Ihe
crlteria for ,leterinlnlng the different veights of the
devlatlonaL varlable could be the rnlnl rnlzatton of
opportwtiW cos t or regret. Therefore, d.evlatlonal varl-
ables on the s ame priorlty level must be coulrrensurable,
although deviatlons that are on the dlfferent prlority
levels need no t be conrnensurable.
i;
,.1-
#
&'
17. :" -:TT
cHAPTE&
rr
The Productlon plannlng problen
ts concerned.
vt th sp eclfylng the optlmar quantlttes
to be prod.uced.
1n or.der to rneet
d.enrand,
for a speclfled. planntng
'orlaon' t'lary nod'else each
of vblch has lts pros
cons, have been d. and.
evel0ped to help
to solve trrls
probl em.
'Productd'on
nlan'lng 1s of a hlerarchical
naturee
since each level of the organl zatLon jr[erar.;*tlc1_
-p8 tes rrr t he plan'lng process wlth d.lfferent
braphaslsr
scoPer and planning
hortz6n. Those operattrng at the
strategtc level are prlnarlly
concerned. v,*ft the
10ng_
r''nge plans of the
org anLzatl0n as a
whoJe. This
requlres sl'nrl taneous consld.eratlon
of the dlfferent
func tional policles
and tirelr coordlnatlon
so that tLre
f trnt s frarc tlonal
s trateg ies b e consls
tent r*rth each
otherr As we go from the top
level to t|re tactlcal
and opela tlonal levels
r planntng horlzon d.ecrease
and ttre degree of
uncertatntby Ceereases.
However, the
d ep en d'ence b e bween
the frnc t10na1 ac
t1v:Ltl e s t s
byplcaly coordlnated. more
at the tactical
level than
;
Ir
18. lz
-' L , ,tsi ''- {',
:
at the operatlonal levelr Thls also hints at the hlerar-
- chlcal lnfornatlon problems associatal u:tth prod.ucfi,on
plannlng slncb pl-ans at any glven l_evel
are based.
on the inforunatlon before
the factl and trren upd.ated.
? accordlng to the lnformatl0n
feed.-back after the f aet.
productlon plannlng nooels t ] lntroduced.
in the Li teratrere trffer ln thelr oriertation, scope,
co n ten ts & n ethodology. Ilowever e lre can cras s ify
thes e models ln two r.raln categor.i es ; deserlp tlve
&
normative.
Dggglpttve i,rod
ef,S 3
Descrlptlve nodels alm pf descrlblng the process
by whlch procluctlon plans are determlned ln practice.
The rnaln examples of such rnodels are!
1):
t lo]
rntrod.uced br Bownan ( 1gfu) and extend.ed
by Kumren ther ( 1969) thls
, nod.el assunes that manager
behave efflci entry an average,
but suf fer frora 1n-
- cons ls tency and. blas
es to recent events o Lrnear
FRE8 regresslon ls used. to d.evelop decislon ruJ.es
for actual productlon ancl vork force oeclstons uttlizlng
19. r ':.' i ..r
- l.*;i
1
lnd.epend.ent vartables such as pas t sq,les and.
logged
produetlon, tnventoryr ard. work forceo lhts nod.e1
ls very f'exlble ln belne not restrlcted.
to a partt-
cular frrnctl0nal beharrour of ttre cost elernents
1nvo1ved..
t,
A Serl0us d.rawbackof the proeed're
I
ls
tire essentially subJective selectfq of the form of
the ruler rt very easily can be sereete.
ln co*ectlyo
i.1) ljre-s ):
Ti:e marn id.ea of thl s model is
to proeeed in
sequence s tartlng from a prespecifled.
acceptable
range of inventoryr and set
accordtngly the llne_shlft
levels of ruork forceo rhen ad.Just
these according
to the rar'rge of lnventory d.eviatlon frorn lts pernlsdlble
r8'.g e r r J' devlatl0ns occur too frequentlyl then the
acc ep tabl e Level inven tory rang
es ar e subJ ec t to ad.J t-
us
- ilentr
r11) :
cExtensrve work has been c a*ied.
] out rn
thls fleld' uslng dlfferent statlstlcal
and.mathenatlcal
approaehes lncludlng vronte
carl0r saryll,,g, and.conputer
anal0gu€o rn t, he nodell introd.uced.
by vlrgln ( 1966),
20. tFre slurrlatton starts wlth a productlon plan basirti
on past e{perlence of the flrn, and, then cLrangessre
ln troduced. 1n enployment levele ov€rtlne1 lnventorles ,
sub -contractjng r and so forttrl untll a loca] opexst:lg
cos t mlrrlmunr ls achiwed.r 0 ther slnrrlatlon nocleJ.sln
bhl.s regard. axe developed by Enshoff and Sisson ( 1g?0)r
and by tlayior ( 19?1) r using both discrete, and contlnuous
events sinnrlation. An lryortant feature of slurulstion
ls that stoehasttc d.ernand
pattern can be lncorporilted
ln the uodel o Thls p erml ts the analysls of the forecast
error on strategy developme:t.
No_rILE
tlv.e_ liosl el s :
a
Tire corunon focus 1n normative rrcCels ls on wirat
prod.uctlon planners should dor i,lodels of thjs category
are f:r ther clas sl fied. into class€sr
(1.) Aggregate PLannirg irpdels; Ilrelr --
- - comrnonobj ec tlve ls to d,eteruilne the optlmal
production quarttlty to produee anci r,rork force leve] to
us e ln aggregate for a cordng ts plannlng horlLcut.
j.iod,els ln thls class are elther exact or lreurlstlc.
21. T!
E{Acr }rQgJ$ :
tarrsportatl0n I'{ethod foruulatlon
of tsowraan
( 10s61
L 1 l proposed. the dis trlbutl0n
rnod.el 0f
llnear progra-ur'ring
fo:: Asgreg ate planning
, Th[s mod.el
f ocus s ed' on tJ:e obJ ee
ttve of ass lgnlng units
of
produc tive capact ty,
s o that procluction plus
s tora€ e
cos ts were ,u''.luc-sed.
and, sales d.ernand. l'as met with iJl
the cons tralnts of avaiJ.abL e capaci ty.
Thls nrodel
d.oes not aceornrt for prod.uctlon
ehange cos tsr Such
as hirlng & layoff of personnell
and. there is no
co s t p enal ty for baekor,J.erlng
or 10 s t sal es .
The slrnplex iuethod. of
llnear prograoro,lng urakes
1t possible to inclu3.e prod"uction
level change costs
and inventory shortag e costs in ihe
r.,roclel. Iianssnan
and' lless a+r d.ever-oped slrrplex *rodel
a usr'g work
force an. prod.uctl0n rate as lnclependrentdee1s10n
varlables ancl in terus of the coiliponents
of the cos t
moderr arl cost frure tdons are consrclered,
rlnear.
One of the basic wealaress
of llnear prograurmlng
3pproaches ( ana rcst oQrer
aggregate planriing technlqees)
is the assL'nrption of d,eterud.nls
tic demand.o Another
short-contrrg of tlre llnear
progranunlrrg urod,el ls
the
I
t
il
22. regutrenent of llnear cost frrctloDso iloweverl ttr.e po-
sslbiLiby of piece rrrlse ltnear{.ty lnrproves the vatre}ty.
Holtr l,iodtellanl and. S1rcn t lLl gave tLre
well lceown mod.el ln whlch they mlntnlze a qua{ratlc
cost f:nctlon and come up with a llnear decision rure
that solves for op tlrnal Agereg ate prod.uction rate and.
work force size for al-l the perlod.s over tLre plannlng
horlzon. L.i).R. has nany advant&g€s o First the nod.el
ls optlmld-W and the two decislon rules, once d.erlvede
are slniple to apply. In ad.dltion the rcd.el 1s dynamig
and representattve of the unrltlstage klnd. of sys temo
But quadrattc cost structure nay have severe llmltation
and. probably d.oes not ad.equately represent the cost
s truc tur e o f any or€ ani zatlon .
tsergstron and sulth E 2 7 extended. the capabl-
- li tie s of the L. D .3 . mod.el 1n two new dlr ec tlons . Be-
-c&u.s€ of the a€gregate natrrre of L.D.R. tE tt 1s
not posslble to solve dlrectly for the optlnrnrmprod.uctlon
ra t es for lnd.ilrldual produc ts . The d,evelopnren and.
t
applicatlon of thre l.DR rnod.el-
suggests that it 1s now
operatlonalLy feaslble to remove the requlrement of
an adgregate productlon dluenslon ln plannlng mod.elso
23. Further-toorer glven ttrre availr,b1llty
of rev€nue curres
for each product in each tlme
perlod. the MDRnrcd.el
can d.eterrntne optlnal prod.uctionl
sales1 rnventoryr
and work force levels so
as to raaxLrd-ze proflt over
a specified. tlme horlzon.
il nnpence Burbridge
& CZf presented.a uulti;ole
goal llneal programrnlng
moclel consld.ering comrrcrrly
occurlng goals of tlre firin
1n coord.lnatlng prodrrction
and 1ogis tic planning . Tlre solsflon
technique for l,'-Ls
tnodel I^ILll- ]:c a cci-rl-Jute':rze:I
.rr1 bi.i1 c coj:cLir,: il;r.o.,l-oj._r.'r
c f the revisecl simplex methoC.
Good.man a f
C presented. goal prog"u*.,[rre
approach
to soLving non-lrnear agtregate
plannlng iocr.els. rf
actual cos ts ( i{iri'g ct firing cost, overtime & lclebl,ne,
rnventory &' shortag e cos t) can not L. satisfactorlly
e
repres entad quafu'atically, then the solution b eeornes
u}cre conplex. One approachr to i:andllng these inoie
corr-
pl ex moclels i s to atLet:pt
fon:u:latlon o f arr apl)roxirnating
l_lnea"r mod.el to the originaL non llnear cost teruis
an d' to apply souie vari ate o f the s iunplex
metirod.. Thi s
approaclr offers the re*' advantage
of at least provid,tng
an optlual solutton to the mocler usecl and. ls b
a^d.ed.
24. upon the goal progra.nnrtng in thls peperr
Tang and Adulbhan r B ] proposes a 11near prog -
rarunlng formul-atlon of Aggregate prod.uction planning
problem 1n the context of heaqy uianufactrrying lnd.ustry.
A bastc rrroclel is first rLevelopeci to rnd,nlrrd-ze
the
total cost of prod.uction which 1s assumed. be piece-
to
wise linear. Tire basic updel is then transforre.d.
into a llnear progra^m.:alng
inoCel to seek an optlmal
solutlon for a serj-es of plannlng periods wtthln the
pl annlng horlzon.
Jaaskalainess, v t 6) h a s p r o p o s e d .a g o a l
prograrunlng inodel for the sclied.ullng of produc tlon,
eatployment and. j-nventorj-es to s atl sf}r lcno.'nrn
d.emand.
re qulrernent over a finl te tlme horlzon. Thi s mod.el
sets tnree separaue and inconpl_etegoals, the Level of
productlonr einployment and. lnventories r
Thornasand HiJ-1 Lg I forunrlated a nmlti-obJ ectlve
pr.od.uctlon plannlng modeJ as a goaf progran which
c apt taliz es on tire s treng tirs of g oa1 progranmlng in 1n,-
- corporatln8 mul tiple behavloral and, economlc consld.erations
in to the analysl s r Thls flceurr paper lncludes the
aspectsr lgnored. by Goo,iuran a I
C and.Jaakelalnenf 61 .
25. Ja.raesPo Ignlzlo t, 5 f tras atterpted' to provld'e
loo}<, at the relatlvelJ nev field of goal
a brief
under a preemptlve priorlry structure'
programmlng
goal prograd-ng raodel presented'
As such, the general
realistlc and' rather n:fr'ural
ls vlewed as a practlcall
world
representatlon of a wtd,e varj-ety of nany real
probl ens r
ileuristlcs Models 3
paralnetrlc plarrrdrry nod'el b)'
(a) The Procluctlon
J one s ( 19?5) .TtrLs model as sume the exis tence
s
work force
of tvro basic declsion rules addrosSlng
each of
and. productlon levels respeetivelyl
suin of rates
whlch 1s expressed' a*s a welghted
durlng the plannins
required. to meet frtr8 e sal es
horizon.
(b) A Swltch rule proposed. by Elmaleh and' Ellon 019?4)'
Theyspeclfytlrreeinventorylevels,arrd.tirree
by various
prod.uctlon 1eveIs, to be obtalned'
over a hlstori-
combjnatlons of control parameters
for w$orl
-ca1 dernand series, and chooslng the set
to dlscrete levels, such
production ls linlted
as food' and. chemica-ls '
26. (c) Search Declslon Rulesl
-
taub erb, extend.ed. tJ1e computer slnoulatton metho
d.ology to lts qlti.urate ggl eralib,v by d.eveloplng
technlqugs calIed. Search Decislon Rules LlO J'
l'11-1
Iie defined. C1g1 as a frarction of (i'ttt Ptt I
0 t) and. then ldentified. the values within
CtOt bY the folIowlng veetors:
Declslon Veotors = Pt, Wt
Stag e Veetor = Ht-1, It-lI
Paraneter Vector = C o s t C o e f f i c l e n t s
at timee t
for d'eclslon vectors trrat
SDR searcires d.lrectly
red.uce CIOT. Couiputer search routi-nes atterrpt to
s tag es sinirl tarreously g enera ting trial
q&x op tlniz e all
d.ecisions per lieratlon. The search procedure terruinates
when successlve tterations resr:J-t in sna-ll reduc tlon
in Cf0T'
27. ii!-
0ITAPTFR
rrr
. ''
cOrq,.t 4g-4 liATHEl,la_TIcAt USIE
.p&w54l0'fiNc I09IL
',
3o1 G4{ER4'.I, :
]'{oDE}
},tAIrEuj[TI.Cg,
The goa.l prograrnrnlng tlas ortgtnally pDoposed.
by Chanres & Cooper for a linear mod.elo lllhlch has
been further d.eveloped W unny othersr A preferled
sol.utton ts one whlch nlnlnt zes the d.errlatlons from
the set goa1s, Ihus a sturple llnear goal prograrnmlrg
problem fb.rnulatton ls shor,nr belou:
r -+
i'linlnlze Z = 2 p'J (q+q- )
Jt = 1
n
+
SubJect to z arJ t xr
JA
+ d,l - q - = b 1 f O f i=lrooolll
J=1
+
*J ,d; ,di
wnere E xq. =0
xJ = Dectslon varlables to be found
K = Number of prlorlty
n = Nunber of declslon varlables
m = Nunrber of goals
b1 = GoaI set by ttre deelslon maker
pJ, = The Breenptlve wergbts suclr that pJ I
28. In addl tlon to s e ttlng g o aJ. for
s the obJ ec tlves 1
the decision maker must also be able to glve an or-
d,lnal ranking to the obJ ectives. The ranking ean
also be fotmd. out by paired colnparison nrethod whlch
provid,es some check on the consi-stency ln the value
J udg ement of the decision makerr In tf s nethod
the d.eclslon maker ls asked to compare the goals two
at a same tlme and. indicate r*'htch goal is the upre
inportant ln the palr. Thls procedure is appllecl to
all combjnations of goal pairs. Thls analysls
results ln a complete ordlnal ranking of the goals
ln terms of their lnPortancer
t
The goal prograunlng ut1-llses tbe siunplex nethod
o f solving the linear prog ramrnlrrg probl en. !{or,rever
s everal modifications are required and that ls
why the slmplex rnethod. of goal progranralng is often
ref erred, to as the t'modifled slmplex methodorr
3 o2 srEts ,0LIlE-F.r]/IplF,ic ;
9L,cOlI,,3n09n4'l.t',is'i9
lF3j{Oe
S-J:
Set up the |nltial table flora goal progra.nning
fornulation. We assume that the lnttlal solutlon
1 s at orr€ j3e Therefore all the neg ative deviational
variables in tlre mod.el constralnt nrmst enter the
29. so].utlon base lnltiallJ.
Preare a table as shown below:
c1
Variabl e RI{S +
d, oorl di oorl X1 oor
bi CU
'J - cJ PS
'D4
P3
P2
P1
Fill up ttrl s tabre 1r € r all
.i J&b+ .
The cJ colum wtlr contaln
the coeffleient of d.evlational
varlabre because thes
e vartables only enter
the
s oJ.utlon firs to fn the
(ZJ _ Cj ) matrlxr
l1s t ttre
prl0rlty level ln the variabre
columrr fbon l0west at
the Gop to the hlghest
at the bottomr Calcr.&ate
the Z1
values a'c1 record. it
into the RHScorruorl carc*late
30. the ZJ - C3 Values for eacb columr and.record. lt ln the
approprlate colu.umo
S tep 2 : 4e.tsrgml.ne _bhe ne.v SnteJ:l,pg Vali,ablg:
Flrxl the highest prlorlty Level that has not
been attalned coryletely by exaurlning the ZJ values ln
the nHS columro After d.eternlnlng this, f1nd. out the
hlghest zJ -cJ entry columrr rle variable of this
colurn wILl enter the solutlon bas e ln tlre n ext i teration .
In cas e of tie, cLreck the next lower prlority
Level- and s el ec t the colun:I that has the g reater
valueo If at thls stage, the tle carrrot be lbr.oken,
choose one on an arbitrary basis' The other columr will
be chosen in subsequent lterations. rhis is Imor,ar
as key colttur.
Step 3 3 ])elg rrxfn e- tbg_!egvlps_yari-+19
,,
Solutl_on- b_a,$S
Dirt:ide the values of Rits by the coefftcients
ln the key colrurr r Thls wlll- g lve the nelr ruIS val-ue s o
Select the ro,J whlch has the aininun non-o€gatlve value.
The variabre tJs that row wiJ-r- be replaeed. by bhe varl-
abre ln the key eolumr in the next lterationo rf
31. there exts ts a tle,
f,Lnd the ro*r that has the
variable with the higher prlority faetor. rn this
way tire higher order goals
nilt be attained. first
and thereby red,uces the nrrmber
of iterationso
Step 4 : Delgrn+ins tl€ nelr
.sglu!ro!:
First find. tJre ner.r?Jis and. co_€fficients
of thre
key row by d.ivid.lng old values
by the plvot element
i r €o the erernent at tl.e lnrersec
tion of the key row
and key colunr. Then fina the new var-ues for all
o:irer rol/s qr usi::g the c:j-c-r-,._ai.-o;.t
:j..,oce,.._;Je :
c.f
( ui-.i t't:e - ( Intersectlonal element of that now x i,leu
"r
vaLue ill the Key row iJr the sarire
coluriur)) . lrlow courplete
tire tacle by find.jns ZJ and Zj _ Cj vali:es for ilre
p rio ri V rors o
Siep O :
Analyse tne goal attain:rent revel of eacjr goal
b1- checki'g ;ire zJ value for eacrr pr"lority Tou. rf
rhe zJ values are all zero, u.nis is tJre optimal soLutlonr
Ihenr lf
there are posi_tive Zj _ Cj va]ues
in the rowl
d.eternrlne whether there ar.e neg
ative ZJ _ CJ values
a t a hlgher prlorl ty leveL r', the sarfle eolunnrr
32. "26',
I f there i s n egative
ZJ{J value at a higher
prlorlbf level for the poslttve
ZJ _ CJ value fu the
row o f tntere str the solutlon
is optlnal. F1nally1 tf
there exlsts a positlve
ZJ{.J value at a certaln
prlority 1evel and. there ls no neg
ative ZJ r CJ value
-at' a hlgher pfloriw level ln the sai^,e cor-urnn,tiris is
no t an optlmal solutio'o
Hence return to step 2
and,
con tlnue.
Flg ot.deptcts ttre slnpl ex
solutton proc edure for
g oal progra"umr:Lngproblems ',' the form of
Jf-ow ci:art,
3.3
@ :
{
In ord.er for
goal prograrnrntng to
be a usefUl
managenent sclence technlque
for d.ecision analysis,
a
compuLer-based,solutlon
1s an e ss ential reguireuren
t.
Lee t 13 ] presented. a
colxputer-based solution
procedure of Goal Progranmj.ng 'rrhich
can be used. to
solve the problem after
sultable mod.ificationsr
The l-l sttng o f the prog
rauup is shourn in Appelndtx
ro
r t dlscusses the data input
for the con-outer so1ut10n,
the lnput proc ess the proe
r es s for careuratlng the
resultsl and, flnally ure proced.ure for prrnt
out of
the re sul ts o The d.ata
lnput ls dl scus s ed. bel0w
and.
the corylete llst of data lnput is shown
in Append.ix II,
33. z-lk
1. ;*-= ?r.o hl eul g.ar4;
card and. defines the
::-;":: ;:il":':: H:.
numb'f varlables and. nurnber
o f pt i r,'/1= 3s as slrown belov:
/ a: i{Rows 1-IVA.R NPRT
2. ;-e S:qn Carg:
.':-? s scond card descrlb es the direction
of con-
s tralr t ?,*,
o
" both directions
,t .H
al'e possibleot'
,' !-' ,, Iess th3.rrr',
tt iU r'ExactJ.y Egual
.r,
tr
/) tt r,Sreater tltap.r,
0n e or. i t/,,i!- :evlational- varlable s Af a cons tan t rrnrs
t
app ear l./' 7.-e obi eetive flrctionr If nelther d,evlationC
var Lab I rt Q ?" ar s in the obJ ective frnc tlon, it 1s
pos sLttl,, E'nzz both deviational varlables nay end. up
ln tho tru-T and. the - cons tralnt
-s d; . d1 = 0
wLLl fittl, be neto
3. 1I:
,l,l rtse c ards are
t pre fac ed. by a n ae' c
ard wlth
trO&l-rf puuChedo
34. !,,i
x
All other gard.s are punehed. ln the
folr.owing
rn=rrY]gro
f ernlation Rov jn whlch Pr iorlty Welght
ieqlation
a_Dpeared
ir -trlj
t-l
These carc.s sp eclf! the technclog ical cterricients
oi ine choice vciables. loer ( a1J) r and are prrnched.
i - tre folloivlns rcrrr€r o The fir s t card ls punched.
vi --:: the word. ,')A.I-qrr, onlyr
.3. .- ix wlfl ch
o Colunnn ln uhl ch
aij app eared
Value of aU
aif appeared.
35. 2?r=i
5. The .3iFlt-Han$-S i4e:g args
The flrst eard. is punched with trre word trRIGHTtr
onlyr Rest card.s are punched with the values of
Right hand side of al-J- the equatlons r
Angir sl s o f the_9ornprrler 0! tpgli
The Computer soLutlon of goal prograrn provides
the folloiring output;
Computer print out of lnput dara ( the rlght hand slde,
the substj. brtion rates, and the obJec tive frnctlon) ,
the fixa-l sirrplex solutlon table ( lncLudlng Zj - CJ matrix
an d. evalua tlon o f ob j ective fr:nc tion) , slack analysls ,
varlable analysisr and the analysis of the obJective.
The lmpor tant ones are elaborated bel-ow:
T:Ii 5Ii'iAI SII'P,L,E{ SOtqTIOli
a) TIIE :iIGiIT HAND SIDE
This shor s the rigbt hand side values of
the variable ( Ceviational- and.d ecision) . 'l-he
nurc:r s on the lef t-hand sd.de are varl abl e
er
nul"ir er s f or trte basle varlabl es r The rsat
values on the r{-ghf-hand, sid.e represent constants
of tne basle varibbleso
36. b) TTIESUBSTIIUTTONRATAS
TrI1s shows the vaj:es of aU of last
iteratlon. It ls based, c:1 the colurrr arran€rement
+
o f dT, di, xJ r ln that crCero
c) THEZJ - CJ i'.iATRIX
ThLs shows the ZJ - CJ matrj_x of the last
i teratlon o
d) Aii EVALUATf 0F 0B.IECT:rE FU]{CTION
0i'[
Thr-LsevaLuatlon s!p1y represents ilre Zj
value of goarsr rn other vord.s, the values
present tlre under attalneJ, portlon of goalso
e) Tin SLACK .q.NAIXS
IS
d,U AVAI L{3IE ,pOS U( .I,iE0 -S IJ{r
rj -S
rt presents the va'rues of the rlght hand.
side and also varues of the negatlve and positlve
varj-ables for each equationo
f) VAJ1IABIE
Ai{AIXSIS
vA.lrABLE,AI,IO{I{T
It presents Ure constants of only the
basic chotce variables,
37. nr.,..
.rSItr
AIVAI.YSfS TT{E
oF OBJECTI
ru
It'presents the ZJ values
for the
Bo&lso These values
refresent the
attalned portd.on und.er
of go&lsr
Pnr0luTY
UIDERAC}IrEI&l,IHlT
38. ffi
t
CEAPTERIV
%
)
EEQBI4:M
SrAgEi,q,rI
4 t1 qmElui!
Hinclus tan . Boown.Boverl. ( 3ariclabad.) Is
a
prominent org anisatton for proaucinS
the el-ectric
iirotors. iI'B.8. produces the trcrcrs
of several klnds
which differ from each other in several aspects
llke f r a m e s i z e , I { o r s e p o w e r e i . , p . i , i . r l : u ^ u r b eo f p o l e s
r
et c .
H rilo Jo forecasted. the d.e:iand- the t,otal I{orse
of
pol'/er r to be produced. for the :/ear 1g32-g3. l,ianag
enent
es tj-mated a cuuruLative gror+tr cf 1s,,,in the d.euiand.
o f ilors e power. Ihe clemand. f sors e Dower l/as d.iff -
o
er en t for every period..+ Frenc ar a ttenp t is rnade to
e
iaeet the denrand.
for every pericl 1n ar] optinal way
consldering procruction rater fnr-entory, Backorderingr
overtime etc. H. B.B. also had the de.,iand. cord. of
re
ever? type of uiotor ( iJI numbers) for hlre year 1g8hg31
g i-ven in Tabl-e I . Wlth the imowledge of the Las t
Four nron
ths a,re taken a s one planniry p eriod..
39. tear record, the d.emand. for erery k1nd. of motor
ls
aS -egS S ed., O.*,tqV1j.6 ' o
-t
, for the c ou{) te ye ar 1g g2 _BB, Tob Q.e
le Z
a]1 atteunc t ls also rnad,e
to rneet r,rith the fluctuations
in ceuand. for errcry khd.
of notor 1n an opttmal l'ay ._
3cl each frame rlzer there
were frrther rrany kirrd.s cf
:rc ;iis 'rith different specificatiorfs r Therefor e
c:l-; che representative rnernber of the each frame size
ua s cons idered. af ter the
dl scuss ion wi th ,,ianag
q r
-'-aru jac turi:rg services Divi sion.
The types of nnotor
r=:e s tilL too many to make the problem as
a whcle
Yer:r larg e to dealt with. Iience those types of notor,
tr;i c-: ,ti-d not show nuch variation in thej_n rnachining
tj-:=s wei'e clubed. together reasonably,.
It was real_ised.
t::a : :iris problen can be solvetL by ,iraking
Aggp€grate
Plan:-'ans uodel, which concentrates on d.eterininlrrg
wSich
c 3 -f,:::at:'on of the d.ecision variabl es
si:oul-d b e utili s ecl
in o: iel' to op timally adj us t the d.e,.,land. tuations
fluc
vr -;ri-n the
con s traj-nts l f &rf, e
j,lanagement ofilre conpany al so deslre d. to 1n _
corpc:ate other relevant aspects such as posslbly
s tac- e eurployurent for the workersl manageinent
pollcies
o r 8qa1s rel atLve to lnven tory and vorker
s ati sf ac tion
1'Ttc' erforuarlCs o
J Therefore these obJ ec tlves were also
40. 5'l-
incorporated, ln the problen fornnrJ-atton. TLre overaLl
cos t functl0n was segreg ated. lnto inalor
componer ts
1o €e Productlon 'rate and.
rnventory costs so that r,uJ.,Eg
e-
inent can have adclitionar fr-exlbirity
ln penari z.'tg
devlations fro m the v,:rious typ es of cos
ts a'd uianagementr
s
p ercep tion of tradaoffs
among the cost conponents.
The rnodel optliaizes tjre ASgregate procluction
variabr es as well as ce terrnlning the op tirual p roduc t
mix r The cornplete prcbl- erai s forrnulatecl
in the form of
goals and is then soLved.
b), uslng coriiputer based. solu_
tion technique of goal prograrruirlng
f lb I .
The forlouing 3oa1s are lncorporated in.the
problem; in o-rce{ p^rio-,,
"t
(a) Sales .teallsation
(b) I To Iitndt the cos t associatecl wi th prod.uetlon
rate to a sp ec: f:-ed. a-roo,mt,
(c) To l1mit the cost associated. with rnventonr
L evel s to a sp ec if ie ci arooun
t.
(d) ro prono te vorkers irctivation tirroug h rabor force
s tabj-lityo t
I
T
t!
il
There were five sectlons 1n II.3.g. rlke:
i
I
i
ii
ii
il
il
il
41. 1o Foundary Sec tion
2o I'iachinlng Sec tton
3o i^Iin*ing Seetlon
4. Asserrtbly Sectton
5. Shaft Processing Sectlono
;',anagerr l'tanufacturing Services DiuLsion sugg
es ted.
tnat the ,iacirlnLrB Seetion was the only
crucial Section
to be considerech Stand.ard. ttmes require4
for various
op erations, per-forned. in the raachinlng section and.
o ;her s ec tions were co.llec teC from the
fnciustriaL
lngineering Departuent and are r-rsted. in Tabr_e c" .
rnventory carrnng cost and. Backord.erlng cost
f or every repre sentative motor were also }crown from
- lar:a; eiler: t and are 8 iven in table q . The over tlrue
1{3s alloved but not ncre tharr 1o:4of the normal worklng
hcu's - rhe 'sorkers efflciency coef ficlen t
for old.
'^-crker
& new worker ( rf hlred.) ancl for norrnal
& overtinre
uoiking :::urs wer e J<nor*nfrom the l,ianag
er, i,lanufacturing
se rvlc es ..,irrision and are given below:
- ,l
Eier i,
rl
hrs. - 4r.g:- -
tl
3f fi-c i ency :,
1rOO 018 1.00
Coefficiit, 1.O0
,!
I
I
'l
{
lir
l
,t,
43. TABIE 7
Denand. of motors on quarterly basl s
So tr'!Hne aXrJunet Peplol0Ctrl d iarrol .FtsOo,
s iae
{ H*trAuso ApriJ- | 83.
Noo N o v r e D e co
rg2
Tg--
1. 80 729 753 1118
2o 90 809 1 237 1454
3. 100 1425 e46 1 62e
4. 112 1904 19 3 S 2158
5. 132 2982 2073 1995
6. 160 2 0 33 1972 393
7. 180 515 56? 231
B. 200 106 163 91
9r 225 110 1qe, 29
10. 250 19 27 53
11. 280 23 44 50
12. 315 B 22 50
13. 355 B 4 18
-g
14. 160 & 75 121
15. 1Bo s6 74 50
16. 2oo 74 114 o9
17. 225 29 26 25
18. 250 4 22 14
19. 315 4 6 5
S
'Tso
20c I 16 1
2 1. 200 1B 1a 4
22c 225 6 10 14
23o 250 6 17
44. Table 5
Frame rLn
Sl'ze Un1 t Group Isb IInd. flfrd
-
p erl-
gd_ n.:t"1
.n."to:
Qu 90 ;')
)
Qu loo .?175 ) 712o 61e4 (SA6
) ,?482s
au 114 o7415 )
)
Qu 1gz .8005 )
Qu 15o 1.31? I
Qu Bo 11485 lB 3277 3292 2904 1 e4ggs
Qu 13O 1 o5O4
t
I
e 160 2 o533 )
) 110 149 171 e. cs5g
e 1Bo 2.88 )
iu zoo 3.109 I
I 114 232
I 1Bo 3.357 l 31333
i 2oo 4 . 1 S 2 I)
[) 132
s 2oo 4 . 2 O 7 T) 96 4 .197
a 225 4e882 )
)
s 225 4 rB82 ) 145 13s 130
) 4.996
Qu zzs 5.2'26 )
qztu 5o903 I
Ic
s 250 5.903 I 31 6 , 0 41 3
I
Qu zso 6 r31B I
Qu 2BO 7.979 )
)
e 315 I 1435 ) 53 8.207
Qu 315 11 .395 I 22 50 1 1 oBgS
Qu 35S 1 30 5 6 5 x 4 1 8 1B. a 6 s
45. Table q
Inven
C os t (Rs.)
A 182.4 228
B 411.2 514
B14oB 1018.6
1257 1571.4
1560 1950
573 o9 717.39
3OO6.B 3 7 5 8r 6
E 3804 o4 4755.5
5?60 7200
i , __ _ g&o _ _ , 10poo_
_ _
i'acLe 5
Productj.on Cost (Jsr) for every type of ,-tor
SoNoo GrquB----
1. A
fI 1132
2o ts aqqe
3. 6620
4o D loztq
5. E 12675
6. F 16533
7. G 24431
8. TI
t-t 30e1
1
9. I 4 6800
10. J 70200
46. g I
I
I
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ct
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l T F S, a o o ' : { o c o o o -
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ctwTEtl -_g
Goft L PRs)GR{ r"r}1Ncn,PnoB I'Er,,l Rrtr.vruL.sTrsN
.
( 1)
PRI9SI,T"Y :
s ALE IirA,tI SAT
Si_ r0l!
Eqn. ( 1) represents a general relationship.
rt-1 * Pt = $t + rt ( 1)
Where It-1 = Inventory a t t h e e n c l o f t-1 th p eriod.
It = Inventory ab the end of t f.h n ov4r J n , .*l
-
- rv,
I7
p = ProCuction ra te cluring t th erlod
t -o
gt = Saies tn t th period..
Let (It)* = Inventory durin{ t th perlo,J.
-
( I g) = shor tag e clur irrg t trr p e' ioci the
Iire + and - slEr: above tjre parantheses
mean that, the
quaritr r,los ilislcie the paran theses
can have onr-y + or _ ve
val-ues rcripec'blvely.
By uslng transforrnation:
Let a>o
"*=fal
O otherwis e
a lal a< o
4 =Q otherrrise
I
![GOAI PROGRAMMING
AIPNOACETO
ACGREGAMPRODUCTIOH
PI,AI{NING
I A CA,SESIIIDY
A Thesls submltted.
In Parttal lrlfilnent of the
Reqrrlrementsfor the Degree of
}IASMR OF TECIINOIOGY
BY
YOGESHSA:GNA
,i'd
r0 TiIs
DEPART}MVT MECHANICAI
OF M{G]NEERING
?
rNDrAlIrNsTrrutB oF TEcHIIOIocy,
DEIHI
1982](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)