, ,f,,f+[i':l*t

#ii, f,i*' , ; : , " 8 : ; i , . ' : , i : . . : t . - ,t t ,ACr , - i . , r ' + l )ir l:'J,?i ;ic'l;;k :'"'|r ; :i^"i:

"1.:,:'nL,' :,

,, ;,;:,,;;i:ii'11,.,; f i i j ' . : ; l f ' ; ; i1;1

{#ffit'rt:iiiii,ii#ilii::;iiltir,'.ri St.l Ii::i; ;,'

. r i " - ' . t a l

. r , i ; " ; iT i j | , :s . ; ; r - - l

j:ti1;;\|\i; i5;:;',:iiiiii'ilJr 'i*ir:.

ffiffiffi

r P q l r '. . r l : .1?" i:::'lJt,l.;

?fdJ:,+ rllt{^

t i i : :

GOAI PROGRAMMING AIPNOACE TO

ACGREGAM PRODUCTIOH PI,AI{NING

I A CA,SE SIIIDY

A Thesls submltted.

In Parttal l\rlfilnent of the

Reqrrlrements for the Degree of

}IASMR OF TECIINOIOGY

BY

YOGESH SA:GNA

r0 TiIs

DEPART}MVT OF MECHANICAI M{G ]NEERING?

rNDrAlI rNsTrrutB oF TEcHIIOIocy, DEIHI

1982

, i ' d

This is to certily that

I' lrr Yogesh Sarcena worked. for his

i{. Tecirr proS ec t r'Goal prog ransdng

Approaci:. to Aegreg ate 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 Technology r D el jrl e

I further certify that

tn-ls proJ ect has no t been taken

up before for the award. of any

degr€er

( DF.' . I,i . SIIIG H)Dept t . of i ' ieeh. Engg.I . I .T rDe l l t i .

\,i

A C_$_N. 0 br_,L g D_ 9-$J E N. p

I aJn grcattry 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.

facturing Senrices, for provioj-ng roe inva^]-uable

h elp and. sugg estions . '

I aJ so acls"Ioi^IJ e€ e '*[ th t]rank s the he]p

extended by llr. Sond|rl l Indlts trial nngineer

and. other staff of llj-nd.rrstan Bro'nrn Bovffr.

Thanks are al- so due tc the s taf f o f

Conrputer Cent re , I . I .T . De lh l r

Thanlrs are

G en, r"ianas "l;pt?;

I . I . TrDelh l

19E2.

also d.ue to I4r. G'DrSardanae

K .G anp athy r i'ianag er, l'lalru-

=)*'l-f<^^

(YoGESH SAlGliA)

LB_S T R A C_T

In this thesls an attempt has been mad.e to

analy s e the Ag g reg ate Proclrrc tion 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? Consist ing of three p larur ing per iod.s.

To mee t 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 ti i le etc. should be r-rti l is ed. in order to optimally

acU us t tlre demand fluc tuation s wi th-tn the con sl"raln ts

i f BnX.

The Aggreg aLe planrring moder- was formulated. in

the form of goal s wi thr dlf f erent prlori t ieso The

problen was then solved by uslng r 'Coraputerlsed. technique

o f S .i'i. Lee to solve the C oaJ- P rog ramnrlng P robL errls rr, The

decis icn variabl es were obtained for all the planning

p e r iods .

C O N T E N T-S

1 r INTRODUCTION

1.1 Oenera l

1 oz $eg reg at e prod.uc tlon p1 annlng iG eneral Form

1 o3 l i rq lest s t ructure of AggregateProd.uc tlon plannlne

tfl

L

P age

tl

5

Ga33

11

L\

LI

L2-

L6

3L3L

3Gql

55

5853

1.41 rI r O

1.6

1o7

1.8

2.

3 o

3.1

3o2

3.3

3 o 4

4.

4 .1

4 o 2

5 o

6.

7 .

B o

9.

I'Iul tl s tag e Ag S reg ate pl annlng Sys tem

Intpor tance of 'loal prog raJxalng

The Goal Prograuxning Concept

0 bJ ec tive Func tion ln Go a-l p rog ra$ming

Rankire & weighr.Lng of i'iul tlpl e g oal s

IJTEiiATUiiE nnVIS^I

GOAL PnOGRAi'Ii'IIliG AS A I,',.,A,[E,.1ATICAL ICOI

General i'iath einatical ;,iocieI

Step s o f the Siuplex me thod of G oalProg rarunlng

Computer based Solut ion of GoalP rog raminlng

Flow Di ae raul

PROCIE;.I S TA'IE.i.fiI{ T

G eneral

Data Oollect ion Tabl_es

GOAL PRG RAi,li,iN,IG }-IJrd,ir.JLArIOl,I

SOLUTIUI\] At,iD O iiEjitS

S UGGESTIuI'jS FOn FIJliIiG.,t yjr,,rrK

REFEREJ\]CBS

APPENDIX

CHAPTER I-

rNT,Rg.pucTIoN

1 .1 GH'IEF4,.L :

i'lost manag ers want to plan and, control operatlonsat the broades t revel thro ugh some rrrnd. of agg reg atepl'annlng that by passes detalls of indlrridual prod.ucts

and detailed schedrrllng of facillt ies and. personireloi'ianagemer:t would. tr.eal $r:ith baslc relevant d,ecisionsof progra.uaud.ngg tne use of resourC€sr Thls ts &ccon_

-pllshed by reviewirrg proJ ected. enpl0yment levels andby setting actlvity rates trat can be varied. wlth rna Blven errploynent level b7 varytng hours ruorked,

( worklng overtirne or rrnd.ertiiie) r

Once ilres e basic d.ecislons have been mad.e forthe 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 levelsneed to be uade with the real isat lon of thelr posslble

ef fects on ttre cos t of clnnghg prod.uctton level,and on lnventory costs lf they are a part of thesJ,:S temo \

-

1oZ AC.CRECATF g s

The Aggregate prod.uctl0n plan'ing problen lnlts 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{tb) Ihe rate of prpduc tlon, pt

c) The QuantltY shlpped, Str

The resrrrtrng lnventory per month can be deter_nlned. as f ; = I t -1 + pt - St.

The problen ls 'sua{y resolved analytlcarly bymlnlntzlng the e :rpected to tar cos t over a g i.ven Flann-

-lng horrzon consrsting of soc* or arl of the fouowfuscost coqponents:

Ihe Cost of regular payroll and, over tlneThe cost of chanelng the productl0n ratefrom one perlod. to the next

The cost of carrfing jnventory

Cost of shortag es resul tlng frour notmeettrg the d.ennand.

The solutron to ttre problem ls greatly srncpll_-fled' lf average d.emand over the prannlng horlzon isexpeeteti. to be constant.

a)

b)

c)

d)

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 baa blon 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.

wag e 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.ecreas e 1n orders ls accotapll sned b1' .layoff s.

b) i.iain tenance of constant work force, adJ us 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,ia jn tenance of c snstant wor k force and mee t the

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

o f ttrem. ord,er flue ttratl0ns showed. In g eneral l beabsorbed, partly bD' inventorxr parily by overtlme,and partJ.y by fririne and, Iayof,f,s and the opttuuneqphasls of these factors lnlll depend. upon the costsln any parttcular factoryr

uPRosrFS :

The structure of t'e Aggregate plannfrg problenis represented. by tlre slngle stag e sys tem trer theplan'lng horlzon ls only one period a'ead.r the stateof the system at ttre e'd. of perlod. 1s d,efined. by wo,Pe and ror the Asgregate work force slzel productlonor ac tlvibl' rate and. jnventory leve1, respectlvely.'rhe end'lng state c qrd.lttons beeoure the 1nltlal condtttonsfor the upcourlng perrod. 'rle have a forecast of therequlrements for the upconlng perlod.s through sogeproc €ss o Deelsions are nad,e that set the slze of thework force and' prod.uetron rate for the up-cond.ng perlod..The d,eclsions ma,ie uray call- for hlrlng or layj_rrg offpersonnelt thus expand.lng or contracttng ttre effcctlvecapacity of the productJ.ua system, The uork forceslzel together lrrth the d,ec1slon on actlvrty rate durc-nsthe pertodl tlren d.eterrnrnes the requlred. arrcunt of

1.3

5

overtimel lnventory levels or back ordering r whetheror not a shlft nust be added or deleted. and otherposstble changes tn operating procedur€o

1o4 :

Ftg . shows a mrrl tl s tag e ag g reg a te pLanntngsys teun vhere the horlzon has been expand.ed, w"l th for e _cas ts for eac' perl0d.o u*" obJ ec tive 1s to nake thedeclsions eoncernlng the work force slze and. producttonra te for the upconing p erlo d., In clolng so r howeverwe consid,er the sequenee of proJ ected decisions lnrelat ion to forecas ts and their cos i effectso Thedeclsion for the upcorntng perl0d, ls to be arf,ected. bythe futr*e perl0d. forecasts a:d. the declsl0n processnnrs t consld'er the cost effects of the sequence of d,eclslons.Tir e conn ec tlng rtnks b e tween the s everal s tag es ar ethe w, pr and. r values that a^re at the end. of oneperlod and the beglnnlrrg of the nextr The feedback loopfrour the d'ecision process ru4y lnvolve some lterativeprocedure to obtaln a solutl0no The seguentlal natureo f the declsions should. be kep t 1n mlnd.. All d. eclsionsare rlght 01' wrong only ln terrns of the sequence ofdeclstons over a perlod. of t lne.

h€

1o5 :

Organizattonal obJectlves vary aecord.ing tothe elraracteristicse typesr FhtlosoptXr of &anageuentlso 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 esocial contrlbutlons, publlc relatlons r lndustrlaland 1abor relat lonsl €tcr

rf we grant that roanagerent has m.[tlple conffls_tlng obJ eetives to achi€ver the d.eclslon crltertashould arso be nru' trdlnensl0nal0 ,rh1s

tupr_les thatwhen a decislon lnvoLves nultlple goa_1sr the_quantltatlvetecl:nlque used. should. be eapable of hand*ne muLtlpledectsj-on criterlao The llnear programudrg teehnlquehas a llelted value for problems hvolvlng rruLttplet oal sr

The primary dlfflcurty i.rrth llnear progsamm{ngts not i ts inabl l l ty to refrect connplex real l ty. Rather,lts dlfficuJ-ty lles rn the unldlmensl0nallby of theobJ ective J\nctionl vrhleh requtres cost or proflt fuifor-matl0n that 1s often alnost lnposslble to obtalnr To

. : , !

1

overcone the urld,lnenstonallty of the obJ ecttvef*rrctton requlred. ln the llnear prograrrrnilng, effortshave been natr"e to convert varl0us goals, costs, orvalue neasure lnto one crlterton, nanely utlllty.

However exact neasurernent of uttllty ls no t aslropl e ma ttere f'Ienc e1 d.ec1s10n naklng throug h llnearprogrammtng vla a uullw fraretl0n is onry feaslbleln a theorettcal serseo

Goal progra^urrd.ng ts a mod.ification and. extensl0nof L'P' ' The goal progra-mrnlns approach ls a technlquetha t t s capable of handlr'g deelslon probleros thatdealtvlth a slngle goal wtth nrrltlple subgoals., aswell asi problerus wlt' multJ.ple 80a1s wlth n*ltlptesub goa lso

irle can solve the se prdblerns us jng Lrp r \^rbthj{ul tlple obJ ee tives o For t'rts r w€ nay ln trod.uce o therthan the obJ ective fr:nctlon, as rod.el constralnts.The l.p- rccel r equires ttrat the cptlnum soluttonnrrs t satlsf! all constralrrts. Furttrermore, lt lsassumed here that equal lnportarrce 1s attached. to varlousobJ ecttves r However in reall wr such assurnp t10n areobsurdo trtrst of arl , i t ls quite posslbre that arltl:e constratnts of the problem can not be satisfied..

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 lR0GRAtl},IryG q)i,tgEF.T :

The concept of goal progyarnr4ing was 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 r\rrther reflned. by yorJ lri & s rl,lrlee and.

o thers r Goal progran,ulng wnd.ch is 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

si*?.rt.1.

of thetr tuportance & arl relattonshlp of the rcd.elrEcononl'caily spealclngr the msnager faces the problenof the allocatlon of scrace resourc€so ft ls notalways posslble to achleve ttre wery goar f*lly tothe extent d.esrred. by i'anagement. Thus, wrth orwl thout Plogramnlng , the manag er attaches a c er taln prtor _

-1ty to the achreveinent of a partlcurar goal. the truevalue of goar- progrannrins ir, there-or.€1 the sorutronof proble'Sl lnvolrnlng !rutttp1e, confltet,,'g goalsacco'ulng to tlre i'ianag er r s pr10r1ty s truc tur.e.

1 .? :

rr: goal programmrpg rnstead. of try1ne to haxroriseor nlnlnlae the obJ ec tive crlterlcm dlreetly as lnrlnear progranndng, 1t trles to nlnfudze the d.errrattonsanong the goaLs and wl th ln Lhe g lven sets of cons tralnts.rhe devlatlonar vartable is Tepresented. ln twodimsrsl0ns 1n the obJ ecttve 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 relatlvelnpor tance or prlorlty as srgned. to then.

1 .8 0AIS :

in order to achleve the ord.lnal soLutlon-that

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 'Preerryt lvet ' pr iori ty factors have

the relat ionship of pJ

the multlplicatlon of De however large lt may be,

cartuot rnake pJ+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.Ielg hts to tl:e s e deviational variabl-es at the

sarne prlorl ty leveI. At the sarne prlorl ty levelr the

subgoal which acguires manrfuouui dlfferenttal qeight w111

be satlsf led f irst & then l t 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.evlat lonal 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 -#

&'

:" -: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 and.cons, have been d. evel0ped to help to solve trrlsprobl em.

'Productd'on nlan'lng 1s of a hlerarchical natureesince 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 thestrategtc level are prlnarlly concerned. v,*ft the 10ng_r''nge plans of the org anLzatl0n as a whoJe. Thisrequlres sl'nrl taneous consld.eratlon of the dlfferentfunc tional policles and tirelr coordlnatlon so that tLref trnt s frarc tlonal s trateg ies b e consls tent r*rth eachotherr As we go from the top level to t|re tactlcal

and opela tlonal levels r planntng horlzon d.ecreas eand ttre degree of uncertatntby Ceereases. However, thed ep en d'enc e b e bwe en the f\rnc t10na1 ac t1v:L tl e s t sbyplcal\y coordlnated. more at the tactical level than

;Ir

- ' L , , t s i ' ' - { ' ,\ lz:

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 ifythes e models ln two r.raln categor.i es ; deserlp tlve &normative.

Dggglpttve i,rod ef,S 3

Descrlpt lve nodels

by whlch procluctlon plans

The rnaln examples of such

alm pf descrlblng the process

are determlned ln practice.

rnodels are!

1 ) :

t lo] rntrod.uced br Bownan ( 1gfu) and extend.edby Kumren ther ( 1969) , thls nod.el assunes that manag erbehave efflci entry an average, but suf fer frora 1n-

- cons ls tency and. blas es to recent events o LrnearFRE8 regresslon ls used. to d.evelop decislon ruJ.esfor actual productlon ancl vork force oeclstons uttlizlng

r ':.' i - ..r l.*;i 1

lnd.epend.ent vartables such as pas t sq,les and. loggedproduetlon, tnventoryr ard. work forceo lhts nod.e1ls very f 'exlble ln belne not restr lcted. to a partt-cular frrnctl0nal beharrour of ttre cost elernents1nvo1ved..

t, I A Serl0us d.rawback of the proeed're lst ire essential ly subJ ective selectfq of the form ofthe ruler rt very easi ly can be sereete. ln co*ectlyo

i.1) ljre-s ):

Ti:e marn id.ea of thl s model is to proeeed insequence s tart lng from a prespecif led. acceptablerange of inventoryr and set accordtngly the llne_shlftlevels of ruork forceo rhen ad.Just these accordingto the rar'rge of lnventory d.eviatlon frorn lts pernlsdlbler8'.g e r r J' devlatl0ns occur too frequentlyl then theacc ep tabl e Level inven tory rang es ar e subJ ec t to ad.J us t-

- i lent r

r11) :

c ] Extensrve work has been c a*ied. out rnthls fleld' uslng dlfferent statlstlcal and. mathenatlcalapproaehes lncludlng vronte carl0r saryll,,g, and. conputeranal0gu€o rn t, he nodell introd.uced. by vlrgln ( 1966),

tFre slurrlatton starts wlth a productlon plan basirti

on past e{perlence of the flrn, and, then cLranges sre

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.s ln

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.

N o_rILE tlv.e_ liosl el s :

Tire corunon focus 1n

prod.uctlon planners should

are f\:r ther clas sl fied. into

a

normative rrcCels ls on wirat

dor i,lodels of thjs category

c lass€sr

(1.) Aggregate PLannirg i rpdels; I l re l r - -

- - comrnon obj 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.

T!

E{Acr }rQgJ$ :

tarrsportatl0n I'{ethod foruulatlon of tsowraan( 10s61 L 1 l proposed. the dis trlbutl0n rnod.el 0fllnear progra-ur'ring fo:: Asgreg ate planning , Th[s mod.elf ocus s ed' on tJ:e obJ ee ttve of as s lgnlng units ofproduc tive capact ty, s o that procluction plus s tora€ ecos ts were ,u''.luc-sed. and, sales d.ernand. l'as met with iJlthe cons tralnts of avaiJ.abL e capaci ty. Thls nrodeld.oes not aceornrt for prod.uctlon ehange cos tsr Suchas hirlng & layoff of personnell and. there is noco s t p enal ty for baekor,J.erlng or 10 s t sal es .

The slrnplex iuethod. of llnear prograoro,lng urakes1t possib le to inc lu3.e prod"uct ion level change costsand inventory shortag e costs in ihe r.,roclel. Iianssnanand' lless a+r d.ever-oped a slrrplex *rodel usr'g workforce an. prod.uctl0n rate as lnclependrent dee1s10nvarlables ancl in terus of the coiliponents of the cos tmoderr arl cost frure tdons are consrclered, rlnear.

One of the basic wealaress of l lnear prograurmlng3pproaches ( ana rcst oQrer aggregate planriing technlqees)is the assL'nrption of d,eterud.nls tic demand.o Anothershort-contrrg of tlre llnear progranunlrrg urod,el ls the

I

t

il

regutrenent of llnear cost f\rrctloDso 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 Age reg ate prod.uc tion 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 dynami g

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 . mo d.el 1n two new dlr ec tlons . Be -

-c&u.s€ of the a€gregate natr r re of L.D.R. tE t t 1s

not posslble to solve dlrectly for the optlnrnrm prod.uctlon

ra t e s for lnd.ilrldual produc ts . The d,evelopnren t and.

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

Further-toorer glven ttrre availr,b1llty of rev€nue curresfor each product in each tlme perlod. the MDR nrcd.elcan d.eterrntne optlnal prod.uctionl sales1 rnventoryrand work force levels so as to raaxLrd-ze proflt overa specified. tlme horlzon.

i l nnpence & Burbridge CZf presented. a uult i ;olegoal llneal programrnlng moclel consld.ering comrrcrrlyoccurlng goals of tlre firin 1n coord.lnatlng prodrrctionand 1og is tic planning . Tlre solsflon technique for l,'-Lstnodel I^ILll- ]:c a cci-rl-Jute':rze:I .rr1 bi.i1 c coj:cLir,: i l;r.o.,l-oj._r.'rc f the revisecl simplex methoC.

G ood.man C a f presented. goal prog"u*.,[rre approachto soLving non-lrnear agtregate plannlng iocr.els. rfactual cos ts ( i { i r i 'g ct f i r ing cost, overt ime & lclebl,ne,rnventory &' shortag e cos t) can not L. e satisfactorllyrepres entad quafu'atically, then the solution b eeornesu}cre conplex. One approachr to i:andllng these inoie corr-pl ex moclel s i s to atLet:pt fon:u:latlon o f arr apl)roxirnatingl_lnea"r mod.el to the originaL non llnear cost teruisan d' to apply souie vari ate o f the s iunpl ex me tirod.. Thi sapproaclr offers the re*' advantage of at least provid,tngan optlual solutton to the mocler usecl and. ls b a^d.ed.

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. to be piece-

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.

Jaaskala iness, v t 6) has proposed. a goal

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_ete goals, the Level of

productlonr einployment and. lnventories r

Thornas and 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 C a I and. Jaakelalnenf 61 .

Ja.raes Po Ignlzlo t, 5 f tras atterpted' to provld'e

a brief loo}<, at the relatlvelJ nev field of goal

programmlng under a preemptlve priorlry structure'

As such, the general goal prograd-ng raodel presented'

ls vlewed as a practlcall realistlc and' rather n:fr' ural

representatlon of a wtd,e varj-ety of nany real world

probl ens r

(a )

( b )

i leuristlcs Models 3

The Procluctlon paralnetrlc plarrrdrry nod'el b)'

J one s ( 1 9?5) . TtrL s model as sume s the exis tence

of tvro basic declsion rules addrosSlng work force

and. productlon levels respeetivelyl each of

whlch 1s expressed' a*s a welghted suin of rates

required. to meet f\rtr8 e sal es durlng the plannins

horizon.

A Swltch rule proposed. by Elmaleh and' Ellon 019?4)'

Theyspec l f y t l r ree inven to ry leve ls ,a r rd . t i r ree

prod.uctlon 1eveIs, to be obtalned' by various

combjnatlons of control parameters over a hlstori-

-ca1 dernand series, and chooslng the set for w$orl

production ls linlted to dlscrete levels, such

as food' and. chemica-ls '

(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'

Iie defined. C1g1 as a frarction of (i 'ttt Ptt l '11-1 I

0 t) and. then ldentified. the values within

CtOt bY the folIowlng veetors:

Declslon Veotors = Pt, Wt

S tag e Veetor = H t-1, It- l I

Paraneter Vector

at t imee t

= Cost Coef f i c len ts

SDR searcires d.lrectly for d'eclslon vectors trrat

red.uce CIOT. Couiputer search routi-nes atterrpt to

q&x op tlniz e all s tag es sinirl tarre ously g ene ra ting trial

d.ecisions per l ierat lon. The search procedure terruinates

when successlve tterations resr:J-t in sna-ll reduc tlon

in Cf0T'

i i ! -

3o1

. 0ITAPTFR rrr

' '

cOrq,.t .p&w54l0'fiNc 4g-4 liATHEl,la_TIcAt I09IL USIE

' ,

G4{ER4'.I, },tAIrEuj[TI.Cg, ]'{oDE} :

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 ' l in ln lze Z = 2 p 'J (q+q- )

J t = 1

SubJ ec t to

n

z

J=1

*J+

,d i

wnere E xq. =0

arJ t xr + d,lJ A

+- q - = b 1 f O f i = l r o o o l l l

, d ;

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

In addl tlon to s e ttlng g o aJ. s for 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 lnPortancert

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 lF3j{Oe 9L,cOl\I,,3n09n4'l.t',is'i9 ;

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 ne g ative deviational

variables in tlre mod.el constralnt nrmst enter the

so].utlon base lnltiallJ. Preare a table as shown below:

c1

Variabl e RI{S d, o o r l d i o o r l X 1 o o r+

bi CU

'J - cJ PS

D' 4

P3

P2

P1

Fill up ttrl s tabre 1r € r all .i J &b+ .

The cJ colum wtlr contaln the coeffleient of d.evlationalvarlabre because thes e vartables only enter thes oJ.utlon firs to fn the (ZJ _ Cj ) matrlxr l1s t ttreprl0rlty level ln the variabre columrr fbon l0west atthe Gop to the hlghest at the bottomr Calcr.&ate the Z1values a'c1 record. it into the RHS corruorl carc*late

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 t ie, cLreck the next lower prlori ty

Level- and s el ec t the colun:I that has the g reater

valueo If at thls stage, the t le carrrot be lbr.oken,

choose one on an arbitrary basis' The other columr will

be chosen in subsequent l terat ions. rhis is Imor,ar

as key colttur.

S tep 3 3 ])elg rrxfn e- tbg_ !egvlps_yari-+19

S olutl_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

there exts ts a tle, f,Lnd the ro*r that has thevariable with the higher prlority faetor. rn thisway tire higher order goals \nilt be attained. firstand 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 threkey row by d.ivid.lng old values by the plvot elementi r €o the erernent at tl.e lnrersec tion of the key rowand key colunr. Then fina the new var-ues for allo:irer rol/s qr usi::g the c:j-c-r-,._ai.-o;.t :j..,oce ,.._;Je c.f :

( ui- . i "r

t ' t :e - ( Intersectlonal element of that now x i , leuvaLue ill the Key row iJr the sarire coluriur) ) . lrlow courple tetire tacle by find.jns ZJ and Zj _ Cj vali:es for ilrep r io r i V ro\rs o

S iep O :

Analyse tne goal attain:rent revel of eacjr goalb1- checki'g ;ire zJ value for eacrr pr"lority Tou. rfrhe zJ values are all zero, u.nis is tJre optimal soLutlonrIhenr l f there are posi_t ive Zj _ Cj va]ues in the rowld.e ternrlne whether there ar. e neg ative ZJ _ CJ valuesa t a hlgher prlorl ty leveL r', the sarfle eolunnrr

"26',

I f there i s n eg ative ZJ {J value at a higherprlorlbf level for the poslttve ZJ _ CJ value fu therow o f tntere str the solutlon is op tlnal. F1nally1 tfthere exlsts a positlve ZJ{.J value at a certalnprlority 1evel and. there ls no neg ative ZJ r CJ value-at' a hlgher pfloriw level ln the sai^,e cor-urnn, tiris isno t an optlmal solutio'o Hence return to step 2 and,con tlnue.

Flg ot.deptc ts ttre slnpl ex solutton proc edure forg oal progra"umr:Lng problems ',' the form of Jf-ow ci:art,

3 .3 @ :

{In ord.er for goal prograrnrntng to be a usefUl

manag enent sclence technlque for d.ecision analysis, acompuLer-based, solut lon 1s an e ss en t ial reguireuren t.

Lee t 13 ] presented. a colxputer-based solut ionprocedure of Goal Progranmj.ng 'rrhich can be used. tosolve the problem after sultable mod.ificationsrThe l-l sttng o f the prog rauup is shourn in App elndtx r or t dlscusses the data input for the con-outer so1ut10n,the lnput proc ess r the proe es s for careuratlng theresultsl and, flnally ure proced.ure for prrnt out ofthe 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,

z-lk

1. ;*-= ?r.o hl eul g.ar4;

numb'f ::-;":: ;:il":':: H:.o f pt i r,'/1= 3s as slrown belov:

/ a: i{Rows 1-IVA.R

card and. defines the

varlables and. nurnber

NPRT

2. ;-e S:qn Carg:

. ' :-? s scond card descrlb es

s tralr t ?,*, o

the direct ion of con-

a l 'e poss ib leo t ', t . H

,' !-'

t t i U

tr /) t t

" both direct ions

,, Iess th3.rrr',

r'Exac tJ.y Egual .r,

r ,Sreater t l tap.r,

0n e or. i t/,,i!- :evlational- varlable s Af a cons tan t rrnrs t

app ear l./ ' 7.-e obi eetive f\lrctionr If nelther d,evlationC

var Lab I rt Q ?" ar s in the obJ ec tive f\rnc tlon, it 1s

pos sLttl,, E'nzz both deviational varlables nay end. up

ln tho t ru-T -s and. the - cons tralnt d; . d1 = 0

wLLl f i t t l , be neto

3. 1I:

,l,l t rt se c ards are pre fac e d. by a n ae' c ard wlthtrO&l- rf puuChedo

! , , i

x

All other gard.s are punehed. ln the folr.owingrn=rrY]gro

f ernlation Rov jn whlchieqlationa_Dpeared

P r iorlty Welg ht

ir -trlj

t-l

These carc.s sp eclf! the technclog ical cterricientsoi 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. o .- ix wlfl chaij app eared

Colunnn ln uhl chaif appeared.

Value of aU

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 r lght hand slde,

the substj. brtion rates, and the obJ ec tive f\rnctlon) ,

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 obJ ective.

The lmpor tant ones are elaborated bel-ow :

T:Ii 5Ii'iAI SII'P,L,E{ SOtqTIOli

a) TIIE :iIGiIT HAND S IDE

This shor s the rigbt hand side values of

the variable ( Ceviational- and. d ecision) . ' l-he

nurc:r er s on the lef t-hand sd.de are varl abl e

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

b)

c)

TrI1s

iterat lon.+

o f dT, di,

THE ZJ - CJ

ThLs

i teratlon o

f)

TTIE SUBSTIIUTTON RATAS

shows the vaj:es of aU of last

It ls based, c:1 the colurrr arran€rement

xJ r ln that crCero

i'.iATRIX

shows the ZJ - CJ matrj_x of the last

d) Aii EVALUATf 0i'[ 0F 0B.IECT: rE FU]{CTION

Thr-Ls evaLuatlon s!p1y represents ilre Zj

value of goarsr rn other vord.s, the values

present t l re under attalneJ, port lon of goalso

e ) Tin S LACK .q.NAIXS IS

d,U rj AVAI L{3IE ,pOS -S U( .I,i E0 -S IJ{ r

rt presents the va'rues of the r lght hand.

side and also varues of the negatlve and positlve

varj-ables for each equationo

VAJ1IABIE Ai{AIXSIS

vA.lrABLE, AI,IO{I{T

It presents Ure constants of only the

basic chotce variables,

nr., . . .rSItr

AIVAI.YSfS oF TT{E OBJECTI ru

It 'presents the ZJ values for theBo&lso These values refresent the und.erattalned portd.on of go&lsr

Pnr0luTYU\IDERAC}IrEI&l,IH\lT

ffit

IV

EEQBI4:M SrAgEi,q,rI

4 t1 qmElui!

Hinclus tan . Boown.Boverl. ( 3ariclabad.) Is aprominent org anisatton for proaucinS the el-ectrici i rotors. i I 'B.8. produces the t rcrcrs of several k lndswhich dif fer from each other in several aspects

l lke f rame s ize, I {orse powere i . ,p . i , i . r l :u^urber of poles

e tc .

H rilo Jo forecasted. the d.e:iand- of the t,otal I{orsepol'/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 i lors e power. Ihe clemand. o f sors e Dower l/as d.iff -

er en t for every period..+ Frenc e ar a ttenp t is rnade toiaeet the denrand. for every pericl 1n ar] optinal way

consldering procruction rater fnr-entory, Backorderingr

overt ime etc. H. B.B. a lso had the de. , iand. record. of

ever? type of uio tor ( iJI number s) for hlre year 1g8hg31g i-ven in Tabl-e I . Wlth the imowledg e of the Las t

Four nron ths a,re taken a s one planniry p eriod..

CEAPTER%

)

tear record, the d.emand. for e\rery k1nd. of motor lsaS -e gS S ed., O.*,tqV1j.6 o' -t , for the c ou{) le te ye ar 1 g g2 _BB, Tob Q.e Z

a]1 atteunc t ls also rnad,e to rneet r,rith the fluctuationsin ceuand. for errcry khd. of notor 1n an opttmal l'ay ._3cl each frame rlzer there were frrther rrany kirrd.s cf:rc ; i is \ ' r i th dif ferent specif icat iorfs r Therefor ec:l-; che representative rnernber of the each frame sizeua s cons idered. af ter the dl scus s ion wi th ,,ianag q r-'-aru jac turi:rg services Divi sion. The types of nnotorr=:e s tilL too many to make the problem as a whcleYer:r larg e to dealt with. I ience those types of notor,tr;i c-: ,ti-d not show nuch variation in thej_n rnachiningt j - :=s wei 'e c lubed. together reasonably, . I t was real_ ised.t::a : :iris problen can be solvetL by ,iraking Agg p€g ratePlan:-'ans uodel, which concentrates on d.eterininlrrg wSichc 3 -f,:::at:'on of the d.ecision variabl es si:oul-d b e util i s eclin o: iel' to op timally adj us t the d.e,.,land. fluc tuationsvr -;ri-n the con s traj-nts l f &rf, e

j,lanag ement of ilre conpany al so deslre d. to 1n _

corpc:ate other re levant aspects such as posslb lys tac- e eurployurent for the workersl manageinent pollcies

o r 8qa1s rel atLve to lnven tory and vorker s ati sf ac tion1'Ttc' J erforuarlCs o Therefore these obJ ec tlves were also

5'l-

incorporated, ln the problen fornnrJ-atton. TLre overaLlcos t functl0n was segreg ated. lnto inal or componer ts1o €e Productlon 'rate and. rnventory costs so that r,uJ.,Eg e-inent can have adclitionar fr-exlbirity ln penari z.'tgdevlations fro m the v,:rious typ es of cos ts a'd uianagementr sp ercep t ion of tradaoffs among the cost conponents.

The rnodel optl iaizes t jre ASgregate procluct ion

variabr es as well as ce terrnlning the op tirual p roduc tmix r The cornpl ete prcbl- era i s forrnulatecl in the form ofgoals and is then soLved. b), uslng coriiputer based. solu_tion technique of goal prograrruir lng f lb I .

The forlouing 3oa1s are lncorporated in.theprob lem; in o - rc \e { " t

p^ r io - , \ ,

( a )

(b )

Sales . tea l l sa t ion

I To I i tndt the cos t associatecl wi th prod.uetlon

rate to a sp ec: f:-ed. a-roo,mt,

To l1mit the cost associated. with rnventonr

L evel s to a sp e c if ie ci arooun t.

ro prono te vorkers irc tivation tirroug h rabor forces tabj-lityo

There were f ive sect lons 1n I I .3.g. r lke:

(c )

( d )

tITt !

il

iI

ii i

iii li l

il

1o

2o

3o

4 .

5 .

Foundary Sec tion

I'iachinlng Sec tton

i^Iin*ing Seetlon

Asserrtbly Sectton

Shaf t Processing Sect lono

;',anag err l 'tanufacturing Services DiuLsion sugg es ted.tnat the ,iacirlnLrB Seetion was the only crucial Sectionto be considerech Stand.ard. t tmes require4 for variousop erations, per-forned. in the raachinlng section and.o ;her s ec tions were co.llec teC from the fnciustriaLlngineering Departuent and are r-rsted. in Tabr_e c" .

rnventory carrnng cost and. Backord.erlng costf or every repre sentative mo tor were also }crown from- lar:a; eiler: t and are 8 iven in table q . The over tlrue1{3s alloved but not ncre tharr 1o:4 of the normal worklnghcu's - rhe 'sorkers eff lciency coef f iclen t for old.'^-crker & new worker ( rf hlred.) ancl for norrnal & overtinreuoiking :::urs wer e J<nor*n from the l,ianag er, i,lanufac turingse rvlc es ..,irrision and are given below:

Eier -

hrs. - 4r.g:- -

, l

i ,r lt l

: ,3f fi-c i encyCoe f f i c i i t ,

1 rOO 0 1 8 1 .00 1 .O0

, !

II'l{lil r, t ,

r'7 PD'rn t OLLE CT \os\- t

f l t r ) L r : :

Table

Fra.me rri.se d.emand' notors for 1982-83

1.0

2.O

3.0

5.0

10.0

15.0

25 r0

40.0

60.0

75 .0

1oo.o1 30,0

27O rO

15

40

50

ntr,

125

270

25

40

75

100

1

of

1.

2 .

3 o

4.

5 r

6.

7 o

8 '

9 .

10.

11.

12.

13.

80

90

1oo

112

132

160

180

200

225

250

280

315

35s

Jr6o

, 180

200

22s

250

315

g,

180

200

225

250

2600

3 500

4000

6000

650o

6ooo1475

500

350

75

120 .

BO

30

14.

15.

16.

17.

18.

19.

250

180

230

8o

40

15

20.

21 .

22.

23.

25

40

30

30

Denand. of

TABIE 7

motors on quarterly basl s

SoNoo

1.

2o

3.

4 .

5 .

6.

7 .

B.

9 r

10.

11.

12.

13.

tr'!Hnes iae

Tg--80

90

100

112

132

160

180

200

225

250

280

315

355

-g160

1Bo

2oo

225

250

315S'Tso

200

225

250

{ H*trAus o

729

809

1425

1 904

2982

203 3

515

106

110

19

23

&

s6

74

29

4

4

aXrJ unet P e p l o l 0 C t r lNovr eDec o

d i a r r o l . F t s O o ,ApriJ- | 83.rg2

B

B

753

1 237

e46

1 93S

2073

1972

56?

163

1 qe,

27

44

22

4

1118

1454

1 62e

2158

1 995

393

231

91

29

53

50

50

18

14.

15.

16.

17.

18.

19.

20c

21 .

2 2 c

2 3 o

75

74

114

26

22

6

16

1a

10

6

121

50

o 9

25

14

5

1

4

14

17

I

1B

6

Table 5

FrameSl'ze Group Isb

p erl-- gd_

rLnUn1t IInd.

n.:t"1

61e4

flfrd

.n."to:; ' )

).?175 )

)o7415 )

).8005 )

1.31?

11485

1 o5O4

2 o533 ))

2.88 )

I

lBtI

712o (SA6 ,?482s

3277 3292 2904 1 e4ggs

110 149 171 e. cs5g

Qu 90

Qu loo

au 114

Qu 1gz

Qu 15o

Qu Bo

Qu 13O

e 160

e 1Bo

iu zoo

I 1Bo

i 2oo

s 2oo

a 225

s 225

Qu zzsqztu

s 250

Qu zso

Qu 2BO

e 315

Qu 315

Qu 35S

3.109

3.357

4.1S2

4.2O7

4e882

4 rB82

5 .2 ' 26

5o903

5.903

6 r31B

7.979 ))

I 1435 )

1 1 .395

13 0565

IIl

I )[ )T))))))

114 232 31333

132 96 4 .197

145 13s 130 4 .996

31 6 ,04 13

53 8.207

50 1 1 oBgS

18 1 B .a6s

IIcIII

I

x22

4

Table q

InvenCo st ( Rs . )

A

B

228

514

1018.6

1571.4

1 950

717 .39

3758 r 6

4755.5

7200

i , __ _ g&o _ _ , 10poo_ _ _

i 'acLe 5

Product j .on Cost (Jsr) for every type of , - tor

S o N o o G rquB----

182.4

411.2

B14oB

1257

1560

573 o9

3OO6.B

3804 o4

5?60

E

AfI

ts

1.

2 o

3.

4 o

5.

6.

7.

8 .

9 .

10 .

D

E

F

G

T It - t

I

J

1132

a q q e

6620

loz tq

12675

1 6533

24431

30e1 1

4 6800

70200

!"f*lsirFqlqs,

-

IIIfp n.nIto to ro to r-. yA g. m.O e.q cj -_ t- C-u Ae L F N c?l@ coD- r{o '- tAO N F{tr Fjto e5qqa, &-6i l toSb'olTFSNBaoo' : {ocooo-r-r-. , . . i;d;dJ id ::fffiS ;$$3IIIIl t *u , .o roIQA'666Pa$rHEv93 39$*au? .oiooooobNi rVVi io ,o \ r ,c r ' \ i '+S$ 3$$$J o . . . o . . . . o . . . , . . . ' . ' . f u ' - . - . rIIlEEq8,8,3 RRg.,.999 o.o.o.o.o.o. o c:oolr-f ,-l r-l r{ '-{ - . ;i ; ; rt 1-t Ft r{,_i r{I

I oacoo ec L . )loocnc,o,o,oRP1' l : t r : pE. jq ppsi !l '

' t ' ' t . o o r f r { N c e ' . a d o l - . ' . a - r

Il c o c ) o ) c c c D c 0 o ?

Hq:Eqqg."."*ppp 8888pp 88Epl .

o Q . . . . o . . . . . r . . . . . o . . .II s o a l c c c r { L c c o c c r r r o @I c t t \ c r ) o \ t o . t ; a c . o . O r r r _ r c el,, | , I r,,.i, irTy qi1tln yTTlIt_| . .qqggq sss$gg ss$sl t

| | | t c ? | | . i - l - l . . . ? . . . . . ? .

IIro ro L/) ro tf) rf) ro to u) u) ro tJ? tr)l .

. . o . . . . . . . . .

lqt p gg ro e to oc ro co .', cD cIERR XBgsYtqrQql . . . o . r r r l O l C \ . ! C C d { t O

Ilto url Le tr) Le|Q q t9 t9 to \tt trl tr; $t \t' str Sr $lI a - o a a a a a a a o a a a

l-l .-f Fl r-{ r-r' rf r{ r-l r-f rl r-f r-{ r-fI

l . s , c1q . , . foo .o rou ,ol leSSXgRRl.!tqqt ( )

o . . . . . o r l r { F { 4 0 C 0

l ^ ^ _ r f rlto Q tO to to tO tr) tO C- p -rO rO rol _ : _ : _ ? o . . o . . N [ - i .

l ^ * C- \ r C \ t C ' C C\ tOrO .

- . i - f

IiEEgflfiggRRRRHH*

gIJ

. r l { J+J .rl

ctFl 5(U+t F{

39,

dr P(D!

r:'t tr(d* ) F {ooH P .

fi*l

p o l g q \ r a ( ) N t o @ r - t ( c )t \ C D o l O t 9 C - m d f - t o $ r u : A Q e \ f l q r p u ; O o ' o t'-f '-f '-f Ct C,t C{ Ol m crJ $t t! !O tl C;: V + fj i- C{ V, ${ u)a o a a a o a a a a a a a a o . a . a o a o

u l t O t O t t ) t r ) L O t O L O U ) L O r r 1 T O _ ! O ! O ! O A L O t l ? L D. . . . . . . | . o . . n U ) N c - N t < u r r - N D . :O O C O O O O O O C I C I O . . . . . . . . f .

Ejs€ j 'RSEr - r@o)o i ga (9 r \ ^ .QqFooob;;qB{8fr$ 3EIs8n SXSna a a a a a . a r a i o . a a a r r a a a .

or J ) t o t r ) U ) u l L r l U ) U ) t r ) u )

a a a a a a a a a a

-_ (). rtQ CC ttt to r{@ 6qC n r - { ( ' . ( o r - f o F { O ? @ ; i( / i a a a . . a . o ao r{ r{ r{ Ci C0 rt -t __l Ci

U)rOto t r ) t r ) t r ) | . r )L r ) rOu)a a a a Q a a a o a

r-{ r{ r-f r-l r{ r{ r-l r-f .-f ri

r { ) m t r )

dpglTq pq1' jr o rl r-l rl t-{ o rf r-f r-t

pppppp ppppo a a a a a o o o a

+h0d h o

F l dr-{ 'rtF { gofl p.F { ( dm F {

uobd \

.r,l E{

.lJ€ oo E 1

F { .

C D A

Hs.r.l oF{ ErO .Ftl (D

I+) l3, I

.r-l C,

3H

888fr8p 88R8' - l r l N C \ I N C C < l ) - t d : N N

oE oC d Nt{.r l 5h o o

ho(

Fr .rlo q+) v,d o* r F {s:t A

oox.r{fr{

ol( n A\ o&1 s

o

Ha

rn

BoHL1F'la$Ur.H(DF.\

Or-FF H

, - |xct.l

^ . 4. \ {G5.)i r i

-{'-1,:)' I

Pn o B I'E r,,l .

( 1 )

t-1 th p eriod.

f . h no rJ n , - lI7 v4 - rv, \.*

a

4

I

ctwTEtl -_gGoft L PRs) GR{\ r"r}1\Ncn,

PRI9SI,T"Y ( 1) :

s A LE Si_ IirA,tI SAT r0l!

Eqn. ( 1 ) rep resen ts a genera l re la t i onsh ip .

r t -1 * Pt = $t + r t

Where I t -1 = Inventory at the encl of

I t = Inventory ab the end of t

p t

= ProCuc tion ra te cluring t th -o erlod

gt = Saies tn t th per iod. .

Le t ( I t ) * = Inventory dur in { t th per lo ,J .

-( I g) = shor tag e clur irrg t trr p e' ioci the

I i re + and - s lEr : above t j re parantheses mean that , thequari tr r,los il islcie the paran theses can have onr-y + or _ veval-ues rcr ipec 'b lvely.

By uslng transforrnation:

Let "*=fa l a>o

O otherwis e

la l a< o

=Q otherrr ise

{1 .

Rrtr.vruL.sTrsN

t4L"

f;ltIt

illriltjt lt l

. l !d {

tlilt ll ii tl !

Ii:iil i,i ;i ii ti :

Then

therefore I t * - I t

and l.lt

For convenienee, Let us

= rt-

r t - i = r t -1

+ -a -a = a

-

(2 '

(3)

and.

rr*+

r t -1

+= D..-(/

+" t-1

put

-Tl r = D .t , t

- -T N^ t-l - at1

Eqns (2 ) (3) can be rewr l t terr as&

+Dt-

o1- t -1

Frorn eqns ( 1)

Ptr = s t

oi=- Di-t

(a> 8-+

+ (D t

T- t

* t -1

(5)

- D; ) - - u611l

( 4 )

(5 )

+( t t :1 (6)

{gg-Ligs-! pgrufu -t =-1

+I t=1= Io = (D t -1 -D t -1 ) = Ze rc

+Fror (6) & (Z> P1 = (D1-D1) + S t

(z>

(B )

Igr_Seqe4

Pz = T? + sz-r1

From (4 ) & (5 )+-

PZ = (b2-D2 )

+-- (D f -D t ) (e )+ s2

+-From (8) & (9) Pe+P1 = (D?-Dz

Foq ,tFlgl Pegio$. t -=- !

P3=13f$g-Tz

+P3 = (Dg

) + (sz*sr) ( 10)

++ S3-(Dz-Dz) (11)Frors (4) & (5) tgl

From q10) & (11)

+-Pt tZ*Pg = (De+Dg)+ Sg* SZ*SI (12)

33us for eaclr type of mo tor there are three eQrrsr

3 r 10, 12 fOr three planrrlng periods I€s'rec t ively.

_=Of EXarnple: +

:ype A nptor P1,1-DA1+

+-Pl,1+ PAZ - D1 2+Da2

P.q.t + PgZ + Plg -

lype B motor

oir= slt ({3)

= SAt f SRe ( 14)

+DAg *DAg =SA1*Sir*S.l,g ( 15)

( 16)

( 17'

Pgt -DSt+Dg1=SE1

Pet+PBZ{3Z +Dfl2 = Sgt+SBz

Pg1*Pgz + Psg{gg +Dgg = sg1*9gz+sgg ( 18)

+-Type C motor- PC1- DCI+ DC.t = SCt

+-PCt+PgZ - DCZ + DCZ = ta, + SCZ

( 1e)

( 20)

PCt + PCZ + PCa - DCe + DCg = SCt+Sg2tSgg (21)

l}pe D notor

+-PDl -

bt+ Dp1 = sot

Ppt +Po2 -d + ooJ =+

PD1+PD2 + PDg- Dog +

Type E motor

Ppl - pir* orr = snt+-

Slmllar type of

H , I &J t ype o f roo tes

from (28 to 4Z>.P5ro1g.ry( rrl

so t *soe

Dpg = SO1 +Ste+SOa

Pnt + Pue - Dna+ one-cJ | ' I)Z "EZ- DnZ = Snl+SnZ

Pgt + Pnz + PEa - "ul-tr,

=sE1 +$na+sEa

OQDS o can be

& were g iven

lrrltten for f, G,

the €Qnsr numbe

T9- J,I$IT qU_C-uS.T JBsI ASgj crsl

Prr x cl + c1ot + oit_ ol, = pRct

wh e t ' e :

ci = stanciard variable cost of prod.uclngof p roduc t 1

CZ = flee cos t per overtime hour

PnCt = I,ianag ementr s turg ut Level forra te cos ts .

produc t lon

( 227

( aal

( 2+7

( zsl

( zo7

( zz7

( as1

\

on e rur lt

+ a

D6t, D6t = Deviati-onal Variables

Ptt = Productlon rate for lth tYPe

duri.ng ttfr Period ( Decln{on

of rctor

varlable)

01 = overtlme horrrs ln Perlod' t

In the present probleml Idle tlne was not allowed'

The cost for producing one unlt of E every type of motor

ls given ln Table !. eqg> (.4SD

The€QI I r (4a l fo r tn reep la r rn lngpe l {odscanbe

wrltten as followst

fuJ:r1482 P^0.1+3553 Pel + 66ZO PCt + 10214 Ppt +12625 Pn't +

16533 Pr.r + ?A4g1 Pc1 + 30911 Pnt + 46800 Pl1 + ?0200 PJ1*- : +

BOt + D6t - D61 = 242 , 650OO G4>

u41482 P.o,z +

1 6533 Pre

802 + D6Z

tr <r-r t =3

14Bz P.e,g +

16533 Ppg-

803 + D6g

3553 Pnz + 66zo PCZ + loZNq PpZ + 126?5 Pne +

t 244g1 PcZ + 30911 Pg1Z + 46800 PtZ + 70200 PtZ

+- D6z = 2426600c ( 45)

assg pse + 6620 Pc# + 10214 Pog + 12675 Pss +

+ 24431 Pcg + 30911 Pgg + 46800 Plg + ?0200 P'le

+- D6g = 24266000 (46)

- l * ,

PEIoRTJY 3

fg 11g$.t the Fst. (RFr)-.asgoci-q.tgd wl.th-Invepl'9rv

LeJgl-to Ep ecifi e9-gmoufr t :

Inventory costs are arrotner lmportant coryonent

o f tota-l Agg reg ate outplannjng co sts and. for fjnlshed

g oods lnclud.e carrfing co s ts 1 and back order co s ts .

In gener al form:

o+10-+( Ci Di t + c i Di t ) + Dzt ' D?E = Ic t G7>

where Cr9 = Cost incurred'

product i

1oCi = Cost tn curred'

backordered Per

+Dtt = Finlshed goods

in Period t .

for ca.rrying one unit of

t

for one unit of Product ir

per iodt

inventory of Prodr'-ct I

pit = Baclcorcl.er Quantity of product i in

p erlod t

1 +

Dzte r ldDz t=Dev ia t iona lVar - iab les .

9 10The values of Ci and C1 for every type of motor are

given ln table q o

- l ' f

The fina]. equatlons are as given below

++

{:I}J 1s2,.4 (ol)t + 4\1 '2 (Dgt) +

+++lzs1 (Dp1 ) + 1560 (Dnl ) + 5?3oe (Dr t )

+++

B8o4o4 (hr) + sz6o tcit) + 8&o (Dlt) + 223 (D..q' 't) +

st. tos; l + 1o1B tolr ) + 1E?1 teir l* 1?50 tprr) + ?17 tuir) *

3?58(Dnr )+4755(D i i1 )+?2OO(Dr1)+1c800(D l t )+- + '

a a n r A A - ( 4 8 )

DZl -DZ1 =22r0OOOO'

+

*, r = z 182o4 (D,q,.e)+ 411.2 (Daz) + 814'8 (DCZ ) +

+

+814 oB (Dct) +

++ goo5.8 (Dct) +

+ 3996.8 (Dce) +

?,

( 4e)

+

J3

1zs? tpizl + 1560 (Dnz) + 573'e (Dre)

38o4.4crliz) + 57co tplzl + a&o (D;z) + 228 (Dn'e ) +

st.(pnz ) + 1O1B tncz) + 15?t ( 'of ) + 1e5o (DEe) + 717 (Dp2)+

raoo tP;r ) +

3?bB c po i) + 4755 ( pnz) + 72oo (Drz) + 1c

D Ze -. D72= 22 rO0ooo '

For t = 3 1g2,4 (Dlg)n+q) + 411 .2 ( Pne) + 814 '8 (Dce ) +

+3Bo4 14 ( Dge)

++

1257 (Dug )+1560 (Dge )+5?319 (Dr .g )+3996 .8 (Dcg )+

1t + s6+o (D,o) + 22s (olg ) +(D;+ 5760 ( DrA)

514cpg3 ) + 1o1B tuci ) + 1.71 cu]31 + 1950 (Dng ) + ?1? (Dre)+

sz58 (Dcs) + 4755 (Dne)'r

+ ?zoo tol,g) + lo8oo (Dls) +

UZi - DZg = 22 tO0O0Oo1 bo)

'q8

Eqns. ( +g) to ( 50) does not contaln any choic e

variables, i t is tryosslble to prepare $tu

seleetlon

eard.so In such a cas€r l le can treat (Did ancl--+.

( Ora) as if they were a choice varlables say (Uft) &

( Vf d respectj-velyo

Iherefore t lre above eens; for t = 11 2t & 3 car be

ex'o r es s ed. as b elow:

192.4 URt + 411 o2 uet + 814.8 uct + 125? r't + 1560 un1 +

SZB o9 Uu.. .t + 3OO6.8P UC f + 3804.4 Ug.t + 5?60 Ult + 8640 U.l1n

( 51)

228 VA1 + 514 Vg1 + 1018 VCt + 15?1 Vp1 + 19pO Vnt + 717 Vpt +

s?s3 Vcl + AtssVi l l + ?zOOVtt + 10800 V;t + oZi ol .1 =

22 tOOOOO.

192.4 URe + 411.2 Une + 31418 UCz + 125? Upe + 1560 %, +

5?3o9 U32 + 3006.8 UCa + 3804.4 U;1Z + 5760 UtZ + A6+O U,lZ+

ZZg Y*Z + 514 Vge + 1018 VCZ + 15?1 VUZ + 1950 VgZ +

?1? VIZ + 3758 VCZ + 4?55 VnZ + 72OO VtZ + 10800 YtZ +

-+D?Z - DZZ = 22r0OOOO (52)

1g2.4 U.g,g + 411 .2 Ltgg + 814.8 ucg + 125? uog + 1560 up3 +

5?3o9 UF3 + 3006.8 Lb6 + 3804'4 Q6 + 5?60 Utg + B&O ul3 +

228 Vl.{, + 514 Vgg + 1018 VCg + 15?1 Vpg + 1950 Uug +

q3

lllri r i

li.717

-

Dra

vrg 3758 VCg + 4?55 Vna + ?2OO VIS + IOSOO VOg +

= 22 t00000 ( SS1

slnce f ail ) an<t , a-l ) are present tn rhe eQrls r off irst goal ( sales xed*x real isat ion) also. Therefore,

tbe f i rs t goar eQnso ( la) to (42) are a lso expressed.

rn terrns of utt and. vtt and, are giver.rrbelor,r:

IVpS JA)...motor

PAt + Vnt - UOt = ?12O 54

55{

s6

57

5B

5e

Prf l + Pte - U-qA + VtZ = 13314

+

+- Dzg

+ Ptz + Pag

( ts)

(c)

Pa1

Psl

Ps1

+ v,:1

+ Pgz

ur1

4:e

p-83

uc1

ucz

Pcg

up1

Pel +

Pc1 +

Pct

Pc1

Ppt +

Ppl +

Pne

vct

p-c2

Pcz

vpl

Ppe

URg+VAg=20COO

3277

+ v:e = 6569

LIeg + Vng = looZS

110

vce = 259

ucg+vcg=430

114

VpZ = 293

60

61

6z

63

&

6s

-uoe+

(D)

POl + Ppe + POg - Upg + UOg = 525

5o

)e ( F)

)e (c)

)e (E) Pn1 +Vnt-Ugl=92

Pnt + Png -. Une * Vng = n4

Pnq + Pnz * Eng - Qng + vng = 320

Pn1 +Vnl -Uf l = 145

Pf''1 + Ppg - UfZ + Vi.,g = 330

Pf.l + Pfe + Png - Uf,g + Vfg = q6

PCl + VC1 - t iCt = 30

' pe (H)

Pqt+PoZaUCz+Vce=85

PO1 + PCe + PCg - UCg + VCg = 145

Pgt+Vnt -UHl =23

PUt+Pge-Uge+VUe=Bz

PU1 + PHZ + PHg - UUg + Vng = 135

Pf t + Vf t - U l1 = I

Pr l+Ple-Urr+Vte=30

Pft + Pte + Ptg + Vtg - Ulg = 80

P.f l + V,f t 3. U.f t = B

P.l 1 + PIZ - %Z

+ V;Z = 12

Pl 1 + P.lZ + P.lg - U,lg + V;a = 30

yp e( I)

66.

6?

6a

6g

70

71

75

?6

72

73

74

77

7B

79

80

82

83.

81Ype(J)

5l

For three planning period.s,

as Delow:

PEIOR_rTg Q>

_t = 1 , x1 + D21-

tire eQr.rr ( a:1 can be wri tten

( a+1

( 35)

( ao1

t

tr'or

For

tr'o r

Dzz -

Dig -

Employee notrvatlonr p€rforniance on the Job, and.s a tls fac tion d'erlved. by workers ar e all enharrc ed. whenworkers perceive a stabre enploynent environrnerit .Further tire flrm may fbel that its image in the laborforce is entranced. throwh the effort to maintain vorkforce stabi l i tyr In generaf l

x6+uJt-ol.=

the number of workers in

Qt ( ael

where3 x*=Changet,

p er iodI

I.ianag ernent dld not alLow firing of the works rTherefor€, Xt represents only tne number of workers h l red.oT

Dzt & Dzt = t ire nlt i ; iber of workers less i i ran or ln excesso f tlre desired. maxlniuml resp ec tivery.

Q; = t 'ra>cinrun desired change in vrork force level.

in

+t O

t = Z, xZ +

t = 3r *g +

+Dz1= 2

+Dzz = 2

+Deg=1

I

C9IIS-RA.q,l,Tg

( 1) lgductige holg'.g_-Qgnstraint

The hours required for the production of vartous

klnd of no tors shor&d be equal to the eif ec tlve j:ours

avalla ble. rn case the hours required are l ess than

the hours available, we can g o for ov*' tlrne as well

as can increas e tire work f orc e d.uring the norsiaL

worklng hours.

In Genera l r

Pi t = T1 ( i i t -1) x( t i .u.hrs) +

T3 og

i . i i f ef e

X6 = Number of workers ir ired ln

Tn e follolrlnc r ecur sive rel atlonship j. s

Ll . - * \r^ E--l a.t =

T1 = hours re qulred for one urri t o f motor I

1l = Ef .- ' iciency coefficient for olcl worl<€rs o

c)' I " xt X ( lr i r 'drhrs)+

( 8?;

L - :_-I

Tz = coef f ic lency coef f ic ient for new work€rso

?T- = Ef r'ici ency Co eff iclen t d.urjng over tlme hours

t6 Period..

al so requir ed.o

Ht

* '. N owtt q't \n! o*F.i.^1 ho*^r5\l

,s

I t 'shows

equal to

pl-us the

For t=1

For t=2

1292 oB x 1

For t = 2 ,

4 .19 Pge + 4o99 P fe

13 .36 P IZ - 16OO X1

or \ {Z =Uo+X1 +h

For t=3 1^ lB=W2+Xa

or Wg = Ido + X1 + XZ + XB

By using the values of Tl, giverr ln Table G, the

€Qrio ( aZ1 is r,ml tten beLolr for Effee p erlod.s r

!= ] o748 P l t + 1 .48 PB1 + 2 .65 PCt + 3 o33 Pp t +

4 .19P81 + 4199 P f '1 + 6104 PC1 + B tZ Pn1 + 11 .3g P l1 +

13 .36 P . l t = 1 x 5 x 161 6 + rB / - 1616 x (X l ) + 01 .

9 I o748 P.0 ,1 + 1e48 Pe1 + 2 .65 Pc t r 3 '33 Po1 + 4 .19 PEt

'4o99 P f t + 6 .Oq PCt + 8e2 P i t l + 11 .39 P f l + 18 .36 p ;1 -

that the labor force size in period t wil'1

the labor force size of the prevlous period

increas e 1n workers durlng period. t.

l{1 = l{o + X1

WZ = W1 + Xz

-O1 = 8O8O (aA1

o748 P.O,e + 1o48 Pn2* 2.65 PCZ + BoB3 Ppe

6.04 Pce + 8oZ PnZ + t 1 .39 Pte +

1280 xz - oz = B0o0 (s9)

.2' l

For t=3

4.19 Ppg +

13.36 P, lA

c748

4 o99

- 1616

P3

2,65 PCe +

?o2 PHg +

1292t8 x 3

P.O,g + 1 r48 Png +

6c,q pcg +

1616 xZ -

3.33 POg +

11 .39 P lg +

-Og = 8080 . . ( 90)

Prg

X1

i i ) o.ltrRIJlr$ wJsJIi$].rJ :

The nanag err manufacturing serric es Division, -a]-lowed the overtirne but not more tnan 10 percent of

the normal rrrork hours o

Tirereforer the over t lrne constrajnts for three

p eriods are Biven belovr:

For t=1 01 +UO1 =B0B f i fy-

For t= 2 O2 + %Z = 8OO (OZy

For t=3 0g+%S =

Thus tire obJ ec tive

d evi- a t,ional ve.r iable arr d.

I . l in Z =- p1 .E 1.2S (oit)\ r t

Cl toJ

t= 1 to 3

808 ( e3)

of the prcblen is to rniniurlze tire

i s forinul a t:d below:+3++1 roo (D i t ) + PzZ (D6d +

t=1

Z- ( Dzt) + Pq L (D2 g)E1 G1

Sub j to ; Eqns ( 1A) to (eg) r a l ready B ivenr

bD

SOLINION-

The problem formr:rated 1n the last chapter

has been solved. by the conput€rr 'rhe complete

results are sl:orcr in Append.lx. The inain r esults are

di s eus s ed belolr:

Ai{

4

3

2

1

VAT,IASI,E

376z265277816195B6

DESCI].IPTIOI'; Ai/rOUj'J T

14g ,oo3543 .O o4452 .oo59 . go

259 . oo+60.oo

2Om.7 o5040 .2 o

. 5694? rs€rI . oO

EVALUATI0N 0F 'I}IE O&IrcrIUE Frn,tCTION

0.000

801 728.00

0 "0000.000

This shor s ti:at the Is t, 2nd & ALn goaLs are acirj-eved.

fu11y whiJ-e tnird goal is not. ' ih1s 1s due to that

bhe est imated. targ et cos t of produc t ion is less than

the actual cos t of Produc t ion. The variabl-e Analysj-s,

g iven ln App endix r is explained. belol:

276s661e1441112405212259291o

315109 A

5625497724?664{e

2?.4O1692.155198 ,O?

8 5 ' o o132. oo

297 5 t6214,24

232 .oo1&,75

19q1 .o oef,)' o ogN.oo

18 ooz 6go.5 g

'872o5 'cs171.o o

1a229,5395 'oo

26g-oo30 . oo2,2 .59. 8 ' o c )

55.o o329 'C o

53 ,0q14s ' oo301 .85

92 .oo

This ta.rl e gi-ves ti ie a:-ralysis of the obj ective

i r € o ainoun t of eacir devi sion variabl e. T.'ilrd thinq

is to discuss s]ack-analysis lr i r ich is also rrFra

reproduced below:

nou12

456

7Ev,

10

AVAIIAB[g

7 1291 3314

2000003277656e

1 oozg110

259430114

P0S..S LK

0'0oo.o oO'g c lo.ooc).oo0, o oo.ouO'o tJo.o oo. o 0

I{ EG -S IJ(o.ooO.c)oO . C f o

o'o oo' c) oo.o.o.oo

oooooo.oo. O O

ROW

11.121314151(r171B192o2122232425262,723293031323334a qLJ IJ

35373BQ O4.. -t

4A41l r2

4344A q= . J

AVAIIA BI,E

29352592

224320145330450,

30B5

14523B2

135I

3080

I1230

242660002.4266000242c6oOO2 2000002 2000002 200000

H BOSOBOO0BO80

22 .001 '00808800808

POS -S LK

O'oo0.ooo 'oo0 ' Ct o

0'oo0'oo0 'oo0.000'ooo'00o'oo

NEG -S TKO'ooo .ooo .ooO 'g l o

o .ooo'ooo 'ooo. ooo 'oo

e 33E..roo'ooo' . )0o 'oon 'oo

6 .oo0.c)o0'oo0 ' c) 'oO'c )oO.ooO.ooC, . ooo .oo

BO1?%1.60o 'ooo'ooo

- 60

Oooooo

t Qoooooo

1.12

.oo' O O. oo, g O, Oo.C)o.c)o, O t ). c )o.oc )

' C f o. oo' oo

Frn ql . J a J

0 'oo0 .oo0 'oo

3:33

1 .13053t.@0

907.12799 o4.2307 r99

The table issel f expla ined. Th-Ls rabl -e

snovls for e acl: and eve ry rowe hovl mucl: !I as the rigirt

hand side and wheti:er tire final solution has exeeeded.

the above s ta ted (R .H.S . ) goa l i r€o POS-SL! ( o r i t

was und.erachl eved i o e o .NEG -S tK from the zJmatrlx one can verify the optinality of theprobl€ff io Thls shols negatlve esttr ies at Istand rvth priorlty Lever. poslt lve enteri esbut at thlrd. priority r ever. That reans theis opt inal .

S U.GGESTT :

-cJ

Rrs&k

& flnd

are there

s olution

In the ab sence of profi t d.ata, ( cl,.:e to the S ecrecy)one of the import,arrt goal of the organieation to .:ralceu:axlmum profi t or to a definlt{ i , i l " t couJd not befu1ly i 'ncorporated.. AL thor:g ir it l^/as trl ed. to incorporatei t, indirec t1y b)' fixjr:f: produc tlon rai;e cos t 1,opredecided J-irnito

For s ame uro tors e s tarrcrard. tir::e d.ata !re'e no 1,itr the record', * ol' the coinpany & vere tol-ci by judgeuient.

Had all the s tantlarcl ti nre d.ata be en provioed. exac tlvt ire problern couLcl have been better t :ra' thi_s.

a(

1.

RE_FER4_IcJ€

Bor*manl E.rloe 19s6, production scheduling by

the transportation methrod. of Ltnear prograJnmlng,

0pso Research i 4 .

B erg s troin, G ary L. and. suxl th, E. , I'iul ti-ltem

Production Plennlng - An exLenslon of the HMI4S

Rulee i ' i anagenrent sc ience l vo l r 16 , ' i o r1g , Jure519?0.

G oodnan 5 D rd o ( 1g?4, Goal p rog ramialng app ro ach to

aggregats Plannirg of prouuctlon and. work Force.

i lg lnto Scl . 20 t 1569 -1SZS.

llans smann , F. And lle ss: t^l .1 r'A Llnear prog ramrclng

Approach to Prod.ucU-on and Jnrployraent sched.uLing, r,

i ' lanagemenb Tecluro logyr vol r 1, j , io .1 (January 1960) .

rgnlz lor Jamesr p. , A - i lev i -ew of Goal progrannr lng :

A Tool for I 'fultiobJ ectlve analysis I Journal of

opera t ion f r ,e s . Soc le ty Vo l , Zg , 1 1 1 192g.

Jaakelalnen, v; ( 1969) A goaf prograrunlng mod.el of

aggregate Prodttc t ion Planning r Swedlsit J r of Economics,

2 t 14-27.

Lawrence, K.D. and. Burbr idge, J .J. , A mul t lp le

g oar Lin ear proir rarunlng niod.el for coord,lnated.

produc tion and Iogi s tic s pl annlrrg I II,IT. J . pR0D.

RESEARCII , 19?62 VoI . 14, I {oe2o

Tang e John c os o Adulbhon and Zubairl Tahrlr I Anaggregate production plannlng for a heary manufacturtngind.ustryr fnt. Jro of Froductlon Research.

2 t

3.

4 .

5.

6.

7.

8.

9 r

1o.

-Ifiornozr, & Hi tt t.A new mod.el for Aggregate

output plannl n{, Omega5 Vol.63 No.3.

Declsion systems For rnventory l,ianagement And

Productlon Pranning by Rein peterson ci Ed.ward,

Ao Si- lver, John Wiley & Sons. i , lew york.

I'fodern Produc tlon/Qp erations t,ianag eiuen t by

E.S o Buf fa; Joirn I,l jJ-ey & Sons r N ew york.

Plannlng Productlon, Inventories and. i.Jork Force

by Holtr i ' iodigllantr l"ruth and si.rnon - prentice

Hal lo

Goal Prog raronrlng for deci sion Analyrs ls by

S ang . i'i. Lee .

Irlnear optlnrlzatlon for .,.an&g einent by s , i. i. Dee.

rn tro cuc ti on to D e ci s ion s c iene e b.| Lee an d.

Mooreo

11.

12o

13o

14.

15r

r l \ Y | - f f ENrr l l - I

FORTR ' I { I TXT

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(RVLr (K , J ) )

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t !g9F^t (GALI I tE l I l i l

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iifElgigt o$t {8[i,,,! I !E lg Ic i Y( ( (BIIEtgtgt tFij;rf,'iiiEl$iil iifr;;ir,qI IE l rs Ic l , DLr , r25f liiiElslgl Iillli5,zs.,r ilEl 8l8r. Iti i ;il,i',,(HARTCTEIf 9 l ! tRc( r Ar , r i Gf 6t l t

t l3 l : ln; i ) t ' r 'v t Lx ' r tL Y, F Rt I ') ( J ) = JIq -? t - I=1 ,Nl ( I ) = Ir C R t l T ( I :

i ( * q I , 7 f . , . i )ISniilttl lc::it8 ,? [=i;hI6 frl , [ [ e , , = ! | I x ( k , r )I I l b = I

t f l i l ! * tn hEh rARrreLrs( f LC tLAT f t \ t l C0n tR I t tT IC f \ 0 t E r (H[ 1=Cr J :1 .11IF (K l -1 ) 6C | . , t t t rA (90 C ( K= i - -K l -f U 0 [ J= ' , ' . 1 , -S t , t (F :C -

'

i iyiIyl;j(].,,r.,)IU fP :su f t t+F( C h l I h t r EF v L x ( K , J ) = s L f t - v A I x ( t , J )( C I t I ] N U EJ T E R = I T E t r l

!PIlSrln t (x;)l 0 9 t , J = 1 - FJf ( [= . - l_ ) 52,7( ,7 tI r ( =K 3 + 1cG 91 K= f 4 - Lt f ( t t Lx r l - i i l 9 r , , s1 ,g1(0N I I l t t iE 'I F (R f LX (13 , J ) - Z t r ,A ) ) S ( , 9 ( , g t )

c

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a t J15

V , T I A E L Ec317 ?131n

q 1

6 l

c

917 '1

! l lirl lr l

I

il

o l I r l

FORlR ' I I " IEX I

l U l { l f r A N . l ' l . t . f t \ t 6 , ( v q 'P r L r I I0 l r ; ' 1 t E e ,

l in l=nvL) tK3, i )[?=Jtot i l tuF , ,l f i i t loil lr l l tcit lgtflEr.,vED Fncr, rlE E,sIsI t ,E!bol ! r f l i I TIh( Ar l rct ErcH EAsrt vAnIAELE

l i l iEtllU I ;'l 8rti9'1i oG0 r ( 83(IF (q !LKi ) , 1 : 0 ,1 i0 ,1(0'ttrtle=i'ict t ' l T_ ( ! ) :P t D T ( I i /C ( I ,KZ )(ox I lNuESELE(1 S IALL t tT F (S I l lVE 'L I i { l I I l ,C A t l 1I=1!F (ArT( r ) ) 1 i ( ,21( .211I = I+1t t ( I rN ) 1C ,Cr1 ( "1 ,1 [0

IgrlE,ft is) ,rT(h)Z f I t \ = A f i T ( l )f 1 = lI = I + 1I t ( I - h ) . l E , ; i t J , : [ 0

i i iiii ri i i iil :t4r2t:22 r,,z 1,,Y(X l )=x ( fe )t0 31 | " ' K=1 ,1! A L Y ( K 1 , 1 ) : ! f I X ( K , K 2 )( 0 h I l h t r E( T L ( ' I , L A T I N E I R I G I T - T ' N D S I D T SD 0 4 ( C 1 : 1 , ht f i D I ( I ) = f F D I ( I ) - Z l , I N , ( ( l , f Z )C 0 l , l I I t \ t r EF R C I ( K 1 ) : 1 t r , I t( l L C t L A T t f i E t S t J E f T I I t T I C t ' R I T E SDc ! (q J=1 , l tt0 5C0 I=1 .N

t l i i i l ;E, t ;J l- c uqt,J)r (c( r ,Kz, t( (K1,r i ))

t 0 51 i l J=1 ,1c (X l r J )= ( ( K1 , . ) I ( (K l , t 2 )(0 t i t I t \UE

[3 ?7,1 i:1;l( ( I r J ) = D ( l r J )( O N l I N U EI R I I T A L t 1 ' T I E S ( R J t S T C P T T T A L T F B I . T

JlfiloEl,lcrflctt(t 0 6 1 l l l = 1 . hh R I r E ( 6 , 1 3 1 t I I ) , t R D l ( I )( O N I I h U E

80

11C

9095ctoc

12C13C14C15Cc16C17C

lEC

a1czt l

,2tc30c

31Cc

q U L

c

( z l r

51C

61C

52Ccc6 , ) [

l U l ( f l A N l l . E . ( l r a b t ( unPrL r I I 0 t r " 1 rE ,2 t te l t : t t

r t (T Ich ' )

ORTR'N .1 EXI

B0 -6?A l= 1 ,hIR I rE !6 ;12 , ( t ( I r J l , J= le i )6,2c [8*l lnlE

l r t I tyErI? ]ExI tohEF PFloRt lY tEvEt

r0 T( 32c hR I IE F I IAL f , tS t r t tS808 ln II [ [2,.11r.11]' TE FlBl l [8f;ft{t l t l} ' ! I ] ERArI0ir----- - ' ,r:,

h R I r E (6 .5 f , C [ , )50c0 r0Rf i tT (5 ) , r I l t s t tPL t ) sc tuT IcN ' )

rF I tE (6 . i IC1)

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81c tsl l I [ t€13) r .r 1), PRtl ( I)

50ca ISt l l f t ' i lEal, tsslrTUrI0N FATts')811 t 0 -8 12 f : 1 ,h

h H r I E ( 6 . ' , 2 ) ( ( t I r . r ) , J : 1 r t r t )812 (0Nr tNUE- - _

hF I IE (6 - : r . . t 3 )

a{l t [8*[ i l , ;= I : [ ? J-ci t{ , ' r R r} r)t l t I rE (6 . "2 ) ( t vLx (K r . l ) , J=1 r t )

814 (0 t t i l IN l JEc TvA tLATE CEJ t (T l ! t F t I ,CT ICN

[0 EZ t , K=1 ,1iU lL (K )= { . -t 0 8ZC . I=1 ,h

A - - t !AL lK l l i t rAL ( l ) +FFBT(1 ) * ! r LY ( I , | )62C (0N I Ih t rE

h R IT E (6 - : ! C4 )5 i l c4 r cRF f r ( r r r t t , Lu l t r 0 t 0F cBJEcT ITE

Do 821 K :1 ,1I t K=L -KIF ( I tS l . [C .1 . ( )GC T0 {9KK=KF+1

89 rR I r l ^ t e ,15 ) lK , i \AL ( f )8?1 coN l INgg -

(nLL F IN ISh (F l S1 ,FRD l , VALY ,L .NPCI ,83C STOF

END

Y T i T K E F I T T E S I )

a - - I t i I - ? , ! Y . a - t

iR IR 'T { l TXT

Iit l . 1r t ir II t l

; { il i ii l l'.i'l l{,1,.*nhI{ r

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: 1 f . . l ; " . ' .

L X r t tLJ rRHS, FnS 1 ; rFCX, , . - ,, KE INFcRT{t1 roN . t i l ' i '1 . ' ' spI0F uSAELs, tAl l t lES, ,

( t F C ! ' l S ' V A l , . I T F L E S '

T E T I E R S F C R E t C l i F C t r

{

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gUBRcg I l l F S l ' R I ( i R0 l t , f i l rR r lFR l ,C ,V tf EP r ; IES l )IFE -STAR l SLE f OU l INE IS IES IG I \E I TO-1i0nr r r ArD rFrNsrcRH t r l t {T0,1 s tRIEsIENL IECt tfiEl 3l8l Utt I :|tcl:i, 'x t1 j.zrs,r r t f e lq tg l Eg! I L q I t 0 t , $ vL l ( 15 ,25( )l IHE iS lC t XEF l ( 6 t , )D l l lE i s l c l Rh t . 1 ( 6 [ ' )r t rE rS IC l R l t ( 60 )CHARIC IETT4 , IAHEl .V=250t r t=6 [f 0Rp tT (A t ,S t ! . i .

Ii l i i l it i rf fi Is t ' ' 'r ' l r G

"TATA OBJ /3QEJ I ' / -i n l n FRCEt 'F t (E r /tA lA B l ' r - ' l

i i l i Fsfi 1r(i[ ' i t:n," rL I /

t E S ' [ = i . [rE IO THE F ICE IE I l (ART FOT THT N I IEER

I f; f? I ?6ll t }ii l I n[:IH6(l :fif rtlIFnrIe l l t I ( :1 .15, : I5 )L I S F = t r P R ' l + 1

lrtil|fi1:tE:tl 88 l8 ltiEI I ixll lE::h E:t i *8! ,loul DiI eCI1t E l D T l i E 5 I C l , C A R [ -t T h l L L ( C t , i l f l N c i E ( l i l ' f F ( , L L ( t I h cf O R E A U A T S Ef 0 R L E S S l H A t \ 0 t r t 0 U r l , T ( Lf C F G R E A l T F ' I } A N ( R T G U A L T O (T O R E O T H D E ! ] ' T I C I ' S B

iE f,FI? ( l i l {1, , AL s ( I ) , I =\ ,1.R0h s)hf r IT t (6 . :5 ) !q ! UALS ( I ) r I=1 , NR( t lS )f cRFt I (5 r ,1Cr : )t \ ! R J = t J( 0 1 f t 1 T H t h L f . I E R ( F T ( S I ' I ' V E S L ' ( K Vt \ t L D S = l Jg 0 l Z L = , , , t \ F C t SI t ( t 6 L A L t i f ) . 1 c . t ) N F L t S = t \ t L D S { 1I F ( E G U A L t ( I ) . t Q . G ) N F t t S = t r F L D S { 1tEsT t0R s lz tI , S I Z E = h F L I S r h t 0 1 i l S { N V r tI F ( h F o h l s . . G T . i r ) c ( T ( 9 1 1I F ( t \ S I Z E ' . C T . l ' l ) G ( T ( 9 1 1( L E A F A L I I . I ' I t I c , T S1 1 D U D = h p R l { 1f 0 1 C J = 1 r h s l ? E

I

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sc ro 1E'14

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