Yogesh Saxena MTech Disseration, IIT Delhi

, ,f,,f+[i:l*trP ql r..rl:.1?"i:::lJt,l.;?fdJ:, #ii,f,i*+ rllt{^tii:: ,;:,"8:;i,.:,i:..:t.-, tt ,ACr, -i., r+ l) ir l:J,?i ;icl;;k:"| r ; :i^"i: :, "1.:,:nL, ,,;,;:,,;;i:ii11,., ;fiij.:;lf ;;i1;1 {#ffitrt:i iiii,ii#ilii::;ii ltir,.ri Ii::i; ;, St.l . r i " - . t a l .r,i;";iTij|,:s.;;r- - l j:ti1;;|i; i5;:;,: iiiiiiilJr i*ir:. ffiffiffi

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,id r0 TiIs DEPART}MVT MECHANICAI OF M{G]NEERING ? rNDrAlIrNsTrrutB oF TEcHIIOIocy, DEIHI 1982

This is to certily thatIlrr Yogesh Sarcenaworked. for hisi{. Tecirr proS ec t rGoal prog ransdngApproaci:. to Aegregate productlonPlanning : A ease strdyrr r:nd.errV sup ervi sion in the i,iechanic aLEngineering Depar tiuento Ind.ianIns titute o f Technologyr Del jrl e I further certify thattn-ls proJ ect has no t been takenup before for the award. of anydegr€er ,i ( DF. I,i. SIIIGH) . Deptt. of iieeh. Engg. I.I.TrDellti.

g 9-$J E N.p A C_$_N. br_,L D_ 0 I aJngrcattry ind.ebt€d. to Dr. N.Singhmy pro j ect supervl sor and. express ryg rati tud.e for his af fec tionate and encourag tngguld,anceo During the year in wlrieh I worked.uncler hiln I forrnd. hls invaluable adl.Lce ofg reat he1P. Thanlrs are also d.ue to I4r. GDrSardanaeGen, r"ianas K .Ganpathy r iianager, llalru- "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 nngineerand. other staff of llj-nd.rrstan BronrnBovffr. Thanks are al-so due tc the s taf f o fC o n r p u t e rC e n t r e , I . I . T . DelhlrI . I. TrDelhl =)*l-f<^^ 19E2. (YoGESHSAlGliA)

LB_S T R A C_T In this thesls an attempt has been mad.etoanaly s e the Agg reg ate Proclrrction p] annlng o fHindustan Brown govd, Far idabad, op tirnally. The denand of the noicr s w:tth d.ifferent specificatlonsve re no t the c ons tant during the planning horizon of on eyear io€r 1982-83? Consisting of three plaruring period.s. To meet wltir the fl-uc fuations in demand.eAg g regate Plannlng mo,iel was formirlated., which concerr-trate s on d.etermining lrhich comblnatton of the d.eclsionvariables J.il<e prodirction Taie, inventoryl backord.eringover tiile etc. should be r-rtilis ed. in order to optimallyacUus t tlre demand fluc tuation s wi th-tn the con sl"raln tsi f BnX. The AggregaLe planrring moder-was formulated. inthe form of goal s wi thr dlf f erent prloritieso Theproblen was then solved by uslng rCoraputerlsed.techniqueo f S .ii. Lee to solve the CoaJ- Prog ramnrlng ProbL errls The rr,decis icn variabl es were obtained for all the planningp eriods.

C O N T E N T-S Page 1r INTRODUCTION 1.1 Oeneral t fl 1 oz $eg reg at e prod.uctlon p1 annlng i L General Form tl 1 o3 lirqlest structure of Aggregate Prod.uctlon plannlne 1.4 IIul tl s tag e AgSreg ate pl annlng Sys tem 5 1r I rO Intpor tance of loal prog raJxalng G 1.6 The Goal Prograuxning Concept a 1o7 0 bJ ec tive Func tion ln Goa-l p rog ra$ming 3 1.8 Rankire & weighr.Lng of iiul tlpl e g oal s 3 2. IJTEiiATUiiE nnVIS^I 11 3o GOAL PnOGRAiIiIIliG A I,,.,A,[E,.1ATICAL AS ICOI L3.1 General iiath einatical ;,iocieI LI 3o2 Step s o f the Siuplex method of Goal L2- Prog rarunlng3.3 Computer based Solution of Goal L6 Prog raminlng3o4 Flow Di aeraul4. PROCIE;.ISTAIE.i.fiI{ T 3L4.1 General 3L4o2 Data Oollection Tabl_es 3G5o GOAL PRG RAi,li,iN,IG}-IJrd,ir.JLArIOl,I ql6. SOLUTIUI] At,iDO iiEjitS 557. S UGGESTIuIjS FOn FIJliIiG.,t yjr,,rrK 58Bo REFEREJ]CBS 539. APPENDIX

CHAPTER I - rNT,Rg.pucTIoN 1 .1 GHIEF4,.L : ilost manag ers want to plan and, control operatlons at the broades t revelthro ugh some rrrnd. of aggreg ate plannlng that by passes detalls of indlrridual prod.ucts and detailed schedrrllng of facillties and. personirelo iianagemer: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 -

1oZ AC.CRECATF g s The Aggregate prod.uctl0n planing 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.

The compl-exlty ln the Aggregate productionPlannlng problen arlses fr"on the fact that tn mostsituations d.enrand. fer perlod. ls not ccmstant but aresubJ ec t to subs tantlal ff-uc baablon and the ques tlonatLs es as to how the se func tions should. b e absorb€d..Assunlng that there are no problems ln receivjng aconstant supply of raw materials and. labour at a fixed.wage rate, the problem ouy be seen by eonsld.erlngthr ee Pure al ternative ways o f r e spondlng to suchfluc fuations oa) A lnciease 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-lternattvesldlscus s ed w111 prove be s t, but rather some courblnation

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 te Aggregate plannfrg problen is represented. by tlre slngle stag e sys tem trer the planlng horlzon ls only one period aead.r the state of the system at ttre ed. of perlod. 1s d,efined. by wo, Pe and ror the Asgregate work force slzel productlonor ac tlvibl rate and. jnventory leve1, respectlvely.rhe endlng 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

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 conslder 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 decision 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€

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 teehnlquehas a llelted value for problems hvolvlng rruLttplet oal sr The primary dlfflcurty i.rrth llnear progsamm{ngts not its inablllty to refrect connplexreallty. Rather,lts dlfficuJ-ty lles rn the unldlmensl0nallby of theobJ ective Jnctionl vrhleh requtres cost or proflt fuifor-matl0n that 1s often alnost lnposslble to obtalnr To

.: ,! 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 fIence1 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 LP The goal progra-mrnlns approach ls a technlque tha t t s capable of handlrg 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 trts 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 lsassumed here that equal lnportarrce 1s attached. to varlousobJ ecttves r However in reall wr such assurnp t10n areobsurdo trtrst of arl, it 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 therefore 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 lntroduced by A Charnes & l^lol{oCooper as a tool to resoLve lnjeasible 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 declslonp robl ens with a slngle g oal or uul tlpl e g oal s o Thegoals set by tlre ttanagenent are often achlevable onlyat the erpense of other goals. zurfher-no!€ theseg oal s are ln couunensurable i o€. they cannot be measured.on the same unlt scsl€r Thus there 1s a need. fcrestablishlng a hlerarcly of tnportance aupng theseconfllctlng goal s so that low ord.er goals a.re consld.ered.only afLer the hrgher orders prlorlty goals aresatisfied or have reached. the point beyond wlrlch nofurtlrer lqprovement j.s deslrableo Hence the problencan be solved. by goal pfogrenryr{ng tif the uuaagementcan provide the ordtnal ranklng of the goals tn tenms

si*?.rt.1. of thetr tuportance & arl relattonshlp of the rcd.elr Econonlcaily 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 ianagement. 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 probleSl lnvolrnlng !rutttp1e, confltet,,g goals accoulng to tlre iianag 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

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 consldered only after higher- ord.er goals are achleved as deslred. The IPreerrytlvet 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 varlables 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 tlary nodelse each of vblch has lts pros cons, have been d. and. evel0ped to help to solve trrls probl em. Productdon nlanlng 1s of a hlerarchical naturee since each level of the organl zatLon jr[erar.;*tlc1_-p8 tes rrr t he planlng process wlth d.lfferent braphaslsr scoPer and planning hortz6n. Those operattrng at the strategtc level are prlnarlly concerned. v,*ft the 10ng_ rnge plans of the org anLzatl0n as a whoJe. This requlres slnrl 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 tactlcaland opela tlonal levels r planntng horlzon d.ecreaseand ttre degree of uncertatntby Ceereases. However, thed ep en dence b e bween the frnc t10na1 ac t1v:Ltl e s t sbyplcaly coordlnated. more at the tactical level than ; Ir

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

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 fexlble ln belne not restrlcted. to a parttcular frrnctl0nal beharrour of ttre cost elernents 1nvo1ved.. t, A Serl0us d.rawbackof the proeedre 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 rarrge 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),

tFre slurrlatton starts wlth a productlon plan basirtion past e{perlence of the flrn, and, then cLrangessreln troduced. 1n enployment levele ov€rtlne1 lnventorles ,sub -contractjng r and so forttrl untll a loca] opexst:lgcos t mlrrlmunr ls achiwed.r 0 ther slnrrlatlon nocleJ.slnbhl.s regard. axe developed by Enshoff and Sisson ( 1g?0)rand by tlayior ( 19?1) r using both discrete, and contlnuousevents sinnrlation. An lryortant feature of slurulstionls that stoehasttc d.ernand pattern can be lncorporiltedln the uodel o Thls p erml ts the analysls of the forecasterror on strategy developme:t. No_rILE tlv.e_ liosl el s : a Tire corunon focus 1n normative rrcCels ls on wiratprod.uctlon planners should dor i,lodels of thjs categoryare f:r ther clas sl fied. into class€sr(1.) Aggregate PLannirg irpdels; Ilrelr -- - - comrnonobj ec tlve ls to d,eteruilne the optlmalproduction quarttlty to produee anci r,rork force leve] tous 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 0f llnear progra-urring 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. las 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 usrg work force an. prod.uctl0n rate as lnclependrentdee1s10nvarlables ancl in terus of the coiliponents of the cos tmoderr arl cost frure tdons are consrclered, rlnear. One of the basic wealaress of llnear prograurmlng3pproaches ( ana rcst oQrer aggregate planriing technlqees)is the assLnrption of d,eterud.nls tic demand.o Anothershort-contrrg of tlre llnear progranunlrrg urod,el ls the I t il

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

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{irig ct firing cost, overtime & lclebl,ne,rnventory & shortag e cos t) can not L. satisfactorlly erepres entad quafuatically, then the solution b eeornesu}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)roxirnatingl_lnea"r mod.el to the originaL non llnear cost teruisan d to apply souie vari ate o f the s iunplex metirod.. 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. 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 .

Ja.raesPo Ignlzlo t, 5 f tras atterpted to provlde loo}<, at the relatlvelJ nev field of goala brief under a preemptlve priorlry structureprogrammlng goal prograd-ng raodel presentedAs such, the general realistlc and rather n:fruralls vlewed as a practlcall worldrepresentatlon of a wtd,e varj-ety of nany realprobl ens r ileuristlcs Models 3 paralnetrlc plarrrdrry nodel 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

(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 l11-1 Iie defined. C1g1 as a frarction of (ittt 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 declslon 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

ii!- 0ITAPTFR rrr . cOrq,.t 4g-4 liATHEl,la_TIcAt USIE .p&w54l0fiNc 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 -+ ilinlnlze Z = 2 pJ (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

In addl tlon to s e ttlng g o aJ. for s the obJ ec tlves 1the decision maker must also be able to glve an or-d,lnal ranking to the obJ ectives. The ranking eanalso be fotmd. out by paired colnparison nrethod whlchprovid,es some check on the consi-stency ln the valueJ udg ement of the decision makerr In tf s nethodthe d.eclslon maker ls asked to compare the goals twoat a same tlme and. indicate r*htch goal is the upreinportant ln the palr. Thls procedure is appllecl toall combjnations of goal pairs. Thls analyslsresults ln a complete ordlnal ranking of the goalsln terms of their lnPortancer t The goal prograunlng ut1-llses tbe siunplex nethodo f solving the linear prog ramrnlrrg probl en. !{or,revers everal modifications are required and that lswhy the slmplex rnethod. of goal progranralng is oftenref erred, to as the tmodifled slmplex methodorr3 o2 srEts ,0LIlE-F.r]/IplF,ic ; 9L,cOlI,,3n09n4l.t,isi9 lF3j{OeS-J: Set up the |nltial table flora goal progra.nningfornulation. We assume that the lnttlal solutlon1 s at orr€ j3e Therefore all the neg ative deviationalvariables in tlre mod.el constralnt nrmst enter the

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 ttreprl0rlty level ln the variabre columrr fbon l0west atthe Gop to the hlghest at the bottomr Calcr.&ate the Z1values ac1 record. it into the RHScorruorl carc*late

the ZJ - C3 Values for eacb columr and.record. lt ln theapproprlate colu.umoS tep 2 : 4e.tsrgml.ne _bhe ne.v SnteJ:l,pg Vali,ablg: Flrxl the highest prlorlty Level that has notbeen attalned coryletely by exaurlning the ZJ values lnthe nHS columro After d.eternlnlng this, f1nd. out thehlghest zJ -cJ entry columrr rle variable of thiscolurn wILl enter the solutlon bas e ln tlre n ext i teration . In cas e of tie, cLreck the next lower prlorityLevel- and s el ec t the colun:I that has the g reatervalueo If at thls stage, the tle carrrot be lbr.oken,choose one on an arbitrary basis The other columr willbe chosen in subsequent lterations. rhis is Imor,aras 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 coefftcientsln the key colrurr r Thls wlll- g lve the nelr ruIS val-ue s oSelect 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 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 tt: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 oSiep O : Analyse tne goal attain:rent revel of eacjr goalb1- checkig ;ire zJ value for eacrr pr"lority Tou. rfrhe zJ values are all zero, u.nis is tJre optimal soLutlonrIhenr lf there are posi_tive Zj _ Cj va]ues in the rowld.eternrlne 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 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 solutioo 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 ror t dlscusses the data input for the con-outer so1ut10n, the lnput proc ess the proe r 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; card and. defines the ::-;":: ;:il"::: H:. numbf 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 ale possibleot , !- ,, Iess th3.rrr, tt iU rExactJ.y Egual .r, tr /) tt r,Sreater tltap.r,0n e or. i t/,,i!- :evlational- varlable s Af a cons tan t rrnrs tapp ear l./ 7.-e obi eetive flrctionr If nelther d,evlationCvar Lab I rt Q ?" ar s in the obJ ective frnc tlon, it 1spos sLttl,, Enzz both deviational varlables nay end. upln tho tru-T and. the - cons tralnt -s d; . d1 = 0wLLl fittl, be neto3. 1I: ,l,l rtse c ards are t pre fac ed. by a n ae c ard wlthtrO&l-rf puuChedo

!,,ix 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.

2?r=i5. The .3iFlt-Han$-S i4e:g args The flrst eard. is punched with trre word trRIGHTtronlyr Rest card.s are punched with the values ofRight hand side of al-J- the equatlons r Angir sl s o f the_9ornprrler 0! tpgli The Computer soLutlon of goal prograrn providesthe 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 matrixan 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 5IiiAI SIIP,L,E{ SOtqTIOlia) 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

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 crCeroc) THEZJ - CJ i.iATRIX ThLs shows the ZJ - CJ matrj_x of the last i teratlon od) 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 goalsoe) Tin SLACK .q.NAIXS IS d,U AVAI L{3IE ,pOS U( .I,iE0 -S IJ{r rj -S rt presents the varues of the rlght hand. side and also varues of the negatlve and positlve varj-ables for each equationof) VAJ1IABIE Ai{AIXSIS vA.lrABLE,AI,IO{I{T It presents Ure constants of only the basic chotce variables,

nr.,.. .rSItrAIVAI.YSfS TT{E oF OBJECTI ru Itpresents the ZJ values for theBo&lso These values refresent theattalned portd.on und.er of go&lsrPnr0luTY UIDERAC}IrEI&l,IHlT

ffit 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. iIB.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..

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 lay ._ 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 weie clubed. together reasonably,. It was real_ised. t::a : :iris problen can be solvetL by ,iraking Aggp€gratePlan:-ans uodel, which concentrates on d.eterininlrrg wSichc 3 -f,:::at:on of the d.ecision variabl es si:oul-d b e utili s eclin o: iel to op timally adj us t the d.e,.,land. tuations flucvr -;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 posslblys tac- e eurployurent for the workersl manageinent pollcieso r 8qa1s rel atLve to lnven tory and vorker s ati sf ac tion1Ttc erforuarlCs o J Therefore these obJ ec tlves were also

5l- 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 ad 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.theproblem; 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

1o Foundary Sec tion 2o Iiachinlng Sec tton 3o i^Iin*ing Seetlon 4. Asserrtbly Sectton 5. Shaft Processing Sectlono ;,anagerr ltanufacturing 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 hcus - rhe sorkers efflciency coef ficlen t for old. ^-crker & new worker ( rf hlred.) ancl for norrnal & overtinreuoiking :::urs wer e J<nor*nfrom the l,ianag er, i,lanufacturingse rvlc es ..,irrision and are given below: - ,l Eier i, rl hrs. - 4r.g:- - tl3f fi-c i ency :, 1rOO 018 1.00Coefficiit, 1.O0 ,! I I l { lir l ,t,

-t fl tr)Lr:: t OLLE CT osr7 PDrn Table 1 Fra.me rri.se d.emando f notors for 1982-83 1. 80 1.0 2600 2. 90 2.O 3500 3o 1oo 3.0 4000 112 5.0 6000 4. 132 10.0 650o 5r 6. 160 15.0 6ooo 180 2 5r 0 1475 7o 200 40.0 500 8 225 60.0 350 9. 250 75.0 75 10. 280 1oo.o 1 2 0. 11. 315 13 0 , 0 BO 12. 35s 27OrO 30 13. J r6o 15 250 14. 180 40 180 15. , 200 50 230 16. 22s ntr, 8o 17. 250 125 40 18. 315 270 15 19. g, 180 25 25 20. 200 40 40 2 1. 225 75 30 22. 250 100 30 23.

TABIE 7 Denand. of motors on quarterly basl sSo 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 11182o 90 809 1 237 14543. 100 1425 e46 1 62e4. 112 1904 19 3 S 21585. 132 2982 2073 19956. 160 2 0 33 1972 3937. 180 515 56? 231B. 200 106 163 919r 225 110 1qe, 29 10. 250 19 27 5311. 280 23 44 50 12. 315 B 22 5013. 355 B 4 18 -g 14. 160 & 75 12115. 1Bo s6 74 5016. 2oo 74 114 o917. 225 29 26 2518. 250 4 22 1419. 315 4 6 5 S Tso20c I 16 1 2 1. 200 1B 1a 4 22c 225 6 10 14 23o 250 6 17

Table 5 Frame rLn Slze 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 31333i 2oo 4 . 1 S 2 I) [) 132s 2oo 4 . 2 O 7 T) 96 4 .197a 225 4e882 ) )s 225 4 rB82 ) 145 13s 130 ) 4.996Qu zzs 5.226 )qztu 5o903 I Ics 250 5.903 I 31 6 , 0 41 3 IQu zso 6 r31B IQu 2BO 7.979 ) )e 315 I 1435 ) 53 8.207Qu 315 11 .395 I 22 50 1 1 oBgSQu 35S 1 30 5 6 5 x 4 1 8 1B. a 6 s

Table q Inven C os t (Rs.)A 182.4 228B 411.2 514 B14oB 1018.6 1257 1571.4 1560 1950 573 o9 717.39 3OO6.B 3 7 5 8r 6E 3804 o4 4755.5 5?60 7200i , __ _ g&o _ _ , 10poo_ _ _ iacLe 5 Productj.on Cost (Jsr) for every type of ,-torSoNoo GrquB----1. A fI 11322o ts aqqe3. 66204o D loztq5. E 126756. F 165337. G 244318. TI t-t 30e1 19. I 4 6800 10. J 70200

g I I I IJ .rl{J +J .rl ct Fl fp to ro to r-.yA g. m.O n.n Ito (U 5 l@coD- cj -_t- C-u r{o -tAO N F{tr e.q e5qqa, Ae L F N c? +t F{ Fjto &-6il toSbo l T F S, a o o : { o c o o o - 39, r-r-..B .i;d;dJid ::fffiS ;$$3 I N dP r I I (D! I B r:t tr lt*u,.oro o (d H L1 *)F{ oo IQA666Pa$rHEv9339$*au? .o Fl HP. . or,cri+S$ i o o.o o o. b N i.r V. V i i. o , o. . o . . 3$$$ a Jo , . . ... fu-.-. r I fi*l $ Ur. H I (D F. lEEq8,8,3 - lr-f ,-l r-l r{ -{ RRg.,.999 o.o.o.o.o.o. o c:oo O . ;i ; ; rt 1-t Ft r{,_i r{ r-F I F H oacoo ec L.) ho I pE.jq ppsi! ( loocnc,o,o,oRP1l:tr: t t . o orf r{Nce .adol -..a-r Fr .rl l oq I +) v, lcoc)o)cccDc0o? do *rF{ s:t A Hq:Eqqg."."*ppp 8888pp 88Ep oQ . . . . o . . . . . r . . . . . o . . . l. o I o I so ccrrro@ I cttcr)oa lcccr{Lcco x qi1tln to.t;ac.o. Orrr_rce yTTl,- | .r{ fr{ l,, | , I r,,.i,irTy I x (nA ol o t_ ct.l o &1s H a rn | | | | tc? | | . i-l-l . .qqggq sss$gg ss$s . . . ? .. . .. . { ^ . 4 lt ? . G + polgqra()Nto@ r-t(c) 5.) h0 tCDol Ot9C-mdf-to$ru: iri dho -f -f -f Ct C,tC{ Ol m crJ$t t! !O AQeflqrp u;Ooot Fld aoaaaoaaaaaaaao.a.aoao tl C;:V + fj i- C{ V, ${ u) -{ -1 r-{ rt F{g ,:) I ofl p. ultO tO tt)tr)LOtOLOU)LOrr1TO F{(d . . . . . . . | . o . . _ !O!O!OA LOtl?LD mF{ OOCOOOOOOCICIO nU)Nc-Nt< urr-ND.: . . .. . . . . f . uob Ejs€jRSEr-r@o)oi ga(9r d .r,l E{ .lJ ooob;;qB{8fr$ 3EIs8n .SXSn ^ QqF aaaaaa.araio.aaarraaa. €o oE1 F{. I Iro ro L/)ro tf) rf).ro to u) u).ro tJ?tr) o rJ)totr)U)ulLrl CDA . . o . . .. . .. aaaaaaaaaa U)U)tr)u) l. lqt p gg ro e to ocro co ., cDc -_ (). rtQ CCttt H XBgsYtqrQq IERR . . Cnr-{(.(or-fo (/i a a a . . to r{@ 6q F{O?@;i . o s l. . o r rrlOlC.! CCd{tO o r{ r{ r{ Ci C0 a a rt -t __l Ci .r.l o I lto url Le tr) Le F{ Er O. |Q q t9 t9 to tt trl tr; $t t str Sr $l U)rOtotr)tr)tr) |.r)Lr)rOu) Ftl (D Ia-oaaaaaaaoaaa aaaaQaaaoa l-l .-f Fl r-{ r-r rf r{ r-l r-f rl r-f r-{ r-f r-{ r{ r-f r-l r{ r{ r-l r-f .-f ri I I l.s,c1q .,.foo.orou,o r{)mtr) +)l lleSSXgRRl.!tqq dpglTq o . . . .. orlr{F{40C0 r pq1j 3, I .r-l C, t() l^ ^ _ rfr o rl r-l rl t-{ o rf r-f r-t lto Q tO to .to tO tr) .tO C- p -rOrO ro pppppp pppp 3H l_:_:_? l^ o . o .N[- -. i . * C - C tC C C t O r O . r i-f oaaaaaoooa o Eo CdN I t{.rl 5 hoo iEEgflfiggRRRRHH*88R8 888fr8p -lrlNCINCC<l)-td:NN- !"f*lsirFqlqs,

{1. ctwTEtl -_g Goft L PRs)GR{ r"r}1Ncn,PnoB IEr,,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 theIire + and - slEr: above tjre parantheses mean that, thequaritr r,los ilislcie the paran theses can have onr-y + or _ veval-ues rcripecblvely. By uslng transforrnation: Let a>o "*=fal O otherwis e a lal a< o 4 =Q otherrrise I

t4L" + f; - a-Then a = a lttherefore It* - It - = rt It il (2 - and l.lt rt-i = rt-1 (3)F o r c o n v e n i e n e e , L e t u s put lr rr* + = D..- (/ T lr t,t = - D . il tj tl tl .l! tl d{ + + --and. r t-1 " t-1 ^ TN - at1 t-l il tlEqns (2) & (3) can be rewrltterr as li it l! + Dt- oi= -t T (4) I i: - Di-t o11 - t- *t-1 (5) ii li, Frorn eqns ( 1) (a> 8-( 5 ) i; ii + + P t r = s t + ( D t - D ; ) - (tt:1 - u611l it (6) i: {gg-Ligs-! pgrufu -t =-1 + It=1=Io =(Dt-1-Dt-1) = Zerc (z> +Fror (6) & (Z> P1 = (D1-D1) + St (B)Igr_Seqe4 Pz = T? + sz-r1 +- +- From (4) & (5) PZ = (b2-D2 - (Df-Dt) (e) ) + s2

+-From (8) & (9) Pe+P1= (D?-Dz ) +(sz*sr) ( 10)Foq ,tFlgl Pegio$. t -=- ! P3=13f$g-Tz + +Frors (4) & (5) P 3 = ( D g tgl + S3-(Dz-Dz) (11)From q10) & (11) +- = (De+Dg)+ g* SZ*SI S (12)PttZ*Pg33us for eaclr type of mo tor there are three eQrrsr3 r 10, 12 fOr three planrrlng periods I€srectively._=Of EXarnple: +:ype A nptor oir= P1,1-DA1+ slt ({3) +- - Pl,1+ PAZ D12+Da2 = S A t f S R e ( 14) + P.q.t + PgZ + Plg - DAg *DAg =SA1*Sir*S.l,g ( 15) lype B motor Pgt-DSt+Dg1=SE1 ( 16) Pet+PBZ{3Z +Dfl2= Sgt+SBz ( 17 Pg1*Pgz+ Psg{gg +Dgg= sg1*9gz+sgg ( 18) +- Type C motor- PC1- DCI+ DC.t= SCt ( 1e) +- PCt+PgZ - DCZ+ DCZ = ta, + SCZ ( 20) P C t + P C Z + P C a - D C e + D C g = S C t + S g 2 t S g g( 2 1 )

l}pe D notor +- PDl- bt+ Dp1 = sot ( 227 +Po2 Ppt -d + ooJ= so *soe t ( aal + PD1+PD2+ PDg- Dog + Dpg = SO1 +Ste+SOa ( 2+7 Type E motor Ppl - pir* orr = snt ( zsl +- Pnt + Pue - Dna+one = -cJ | I)Z "EZ- DnZ Snl+SnZ ( zo7 Pgt + Pnz + PEa- =sE1 +$na+sEa ( zz7 "ul-tr, Slmllar type of O Q D S c a n o b e lrrltten for f, G, H, I &J type of rootes & were g iven t h e € Q n s r numbe from (28 to 4Z>.P5ro1g.ry( rrl J,I$IT qU_C-uS.T T9- JBsI ASgjcrsl Prr x cl + c1ot + oit_ ol, = pRct (as1w h et e : ci = stanciard variable cost of prod.uclng one rurlt of product 1 CZ = flee cos t per overtime hour PnCt = I,ianag ementr s turg ut Level for produc tlon r a t e c o st s .

+a D6t, D6t = Deviati-onal Variables rctor Ptt = Productlon rate for lth tYPe of duri.ng ttfr Period ( Decln{on varlable) 01 = overtlme horrrs ln Perlod t was not allowed In the present probleml Idle tlne every type of motor The cost for producing one unlt of E ls given ln Table !. eqg> (.4SD The€QIIr(4alfortnreeplarrnlngpel{odscanbe wrltten as followstfuJ:r PCt + 10214Ppt +12625Pnt + 1482 P^0.1+3553 + 66ZO Pel + 46800 Pl1 + ?0200 PJ1* 16533 Pr.r + ?A4g1Pc1 + 30911Pnt -: + + D6t - D61 = 242 , 650OO G4> BOtu4 + PpZ + 126?5 Pne + 1482 P.o,z 3553 Pnz + 66zo PCZ loZNq + + 46800 PtZ + 70200 PtZ 16533 Pre t 244g1PcZ + 30911 Pg1Z + ( 45) 802 + D6Z - D6z = 2426600ctr <r-r =3 t + 10214 Pog + 12675 Pss + 14Bz P.e,g assg pse + 6620Pc# + + 46800 Plg + ?0200 Ple 1 6 5 3 3 P p g + 24431Pcg + 30911 Pgg - + (46) 803 + D6g - D6g = 24266000

-l*,PEIoRTJY 3fg 11g$.t the Fst. (RFr)-.asgoci-q.tgd wl.th-Invepl9rv tLeJgl-to Epecifi e9-gmoufr : Inventory costs are arrotner lmportant coryonento f tota-l Aggreg ate outplannjng costs and.for fjnlshedg oods lnclud.e carrfing cos ts 1 and back order co sts .In gener al form: o+10-+ ( Ci Dit + ci D i t ) + D z t D ? E= I c t G7>where Cr9 = Cost incurred for ca.rrying one unit ofproduct i t 1o Ci = Cost tn curred for one unit of Product ir backorderedPer p e r i o d t + Dtt = Finlshed goods inventory of Prodr-ct I in Period t. pit = Quantity of product i in Baclcorcl.er p erlod t 1+ DzterldDzt=DeviationalVar-iables. 9 10 The values of Ci and C1 for every type of motor are given ln table q o

-lf are as given belowThe fina]. equatlons + ++ + (ol)t + 412 (Dgt) 814 B(Dct) + o 1s2,.4{:I}J + + +++ (Dnl) + 5 ? 3 o e( D r t ) + g o o 5 . 8( D c t ) lzs1 (Dp1 ) + 1560 + +++ + sz6o tcit) + 8&o (Dlt) + 223 (D..qt)B8o4o4(hr) * tprr) + ?17tuir) 1?50 tolr ) + 1E?1 st.tos;l + 1o1B teirl* 3?58(Dnr)+4755(Dii1)+?2OO(Dr1)+1c800(Dlt)+ - + (48) aanrAA- =22r0OOOO DZl -DZ1 + + (DCZ) + 182o4 411.2 (Daz) 8148 (D,q,.e)+ r = z + *, (Dce) + (Dnz) + 573e(Dre) + 3996.8 1zs?tpizl + 1560 ?, + a&o (D;z) + 228(Dne) + + 38o4.4crliz) 57cotplzl + 15?t(of ) + 1e5o (DEe)+ 717(Dp2)+ st.(pnz ) + 1O1B tncz) raootP;r ) + + 72oo(Drz) + 1c ( pnz) 3?bBcpoi) + 4755 ( 4e) -. D72= 22rO0ooo D Ze + + + 8148 (Dce ) + 1g2,4 ( D l g ) n 411 ( Pne) q)+ .2 For t = 3 ++ 1257(Dug)+1560(Dge)+5?319(Dr.g)+3996.8(Dcg)+ (olg ) + (D; + 1t + 5760 ( DrA) + s6+o (D,o) + 22s J3 3Bo4 14 ( Dge) (Dre)+ + 1.71 cu]31 + 1 9 5 0( D n g) + ? 1 ? 5 1 4 c p g 3 + 1 o 1 Bt u c i ) ) + lo8oo (Dls) + sz58 (Dcs) + 4755(Dne) + ?zootol,g) r 1bo) UZi - DZg = 22tO0O0Oo

q8Eqns. ( +g) to ( 50) does not contaln any choic evariables, it is tryosslble to prepare seleetlon $tueard.so In such a cas€r lle can treat (Did ancl --+.( Ora) as if they were a choice varlables say (Uft) &( Vf d respectj-velyoIherefore tlre above eens; for t = 11 2t & 3 car beexor es s ed. as b elow:192.4 URt + 411 uet + 814.8 uct + 125? rt o2 + 1560 un1 +SZBo9 Uu.. + 3OO6.8P UC + 3804.4 Ug.t + 5?60 Ult + 8640 U.l1n .t f +228 VA1 + 514 Vg1 + 1018 VCt + 15?1 Vp1 + 19pOVnt + 717 Vpts?s3 Vcl + AtssVill + ?zOOVtt + 10800V;t + oZi ol.1 =22 tOOOOO. ( 51) 192.4 URe+ 411.2 Une + 31418 UCz+ 125? Upe + 1560 %, + + +5?3o9 U32 + 3006.8 UCa + 3804.4 U;1Z 5760 UtZ A6+OU,lZ+ +ZZg Y*Z + 514 Vge + 1018 VCZ+ 15?1 VUZ+ 1950 VgZ?1? VIZ + 3758 VCZ + 4?55 VnZ + 72OOVtZ + 10800 YtZ + -+D ? Z - D Z Z= 2 2 r 0 O O O O (52) + 1560 up3 + 1g2.4 U.g,g 411 Ltgg + 814.8 ucg + 125? uog + .2 + 5?60 Utg + B&O ul3 + 5?3o9 UF3 + 3006.8 Lb6 + 38044 Q6 + + 1950 Uug + 228 Vl.{, + 514 Vgg + 1018 VCg 15?1 Vpg

q3 lllr ir i 717 vrg + 3758 VCg + 4?55 Vna + ?2OOVIS + IOSOOVOg + li. - + Dra - Dzg = 22 t00000 ( SS1 slnce f ail ) an<t, a-l ) are present tn rhe eQrls of r first goal ( sales xed*x realisation) also. Therefore, tbe first goar eQnso ( la) to (42) are also expressed. rn terrns of utt and. vtt and, are giver.rrbelor,r: IVpS JA)...motor PAt + Vnt - UOt = ?12O 54 Prfl + Pte - U-qA VtZ = 13314 + 55 { Pa1 + Ptz + Pag URg+VAg=20COO s6(ts) P s l + v,:1 ur1 3277 57 P s 1 + Pgz 4:e + v:e = 6569 5B Pel + Pne p -83 LIeg + Vng = looZS 5e(c) Pc1 + vct uc1 110 60 Pct p -c2 ucz vce = 259 61 Pc1 Pcz Pcg ucg+vcg=430 6z(D) P p t + vpl up1 114 63 P p l + Ppe -uoe+ VpZ = 293 & POl + Ppe + POg - Upg + UOg= 525 6s

5o)e (E) Pn1 +Vnt-Ugl=92 66. Pnt + Png -. Une* Vng = n4 6? Pnq + Pnz * Eng - Qng+ vng = 320 6a)e ( F) Pn1 +Vnl -Ufl = 145 6g Pf1 + Ppg - UfZ + Vi.,g = 330 70 Pf.l + Pfe + Png - Uf,g + Vfg = q6 71)e (c) PCl + VC1 - tiCt = 30 72 Pqt+PoZaUCz+Vce=85 73 PO1 + PCe + PCg - UCg + VCg = 145 74pe (H) Pgt+Vnt-UHl =23 75 PUt+Pge-Uge+VUe=Bz ?6 PU1 + PHZ + PHg - UUg + Vng = 135 77yp e( I) Pft + Vft - Ul1 = I 7B Prl+Ple-Urr+Vte=30 79 Pft + Pte + Ptg + Vtg - Ulg = 80 80Ype(J) P.fl + V,ft 3. U.ft = B 81 P.l1 + PIZ - + V;Z = 12 82 %Z Pl 1 + P.lZ + P.lg - U,lg + V;a = 30 83.

5l PEIOR_rTg Q> t Employee notrvatlonr p€rforniance on the Job, and. s a tls fac tion derlved. by workers ar e all enharrc ed. when workers perceive a stabre enploynent environrnerit. Further tire flrm may fbel that its image in the labor force is entranced. throwh the effort to maintain vork force stabilityr In generaf l x6+uJt-ol.= Qt ( ael where3 x*=Change in the number of workers in t, p eriod + tO I I.ianagernent dld not alLow firing of the works rTherefor€, Xt represents only tne number T of workers hlred.oDzt & Dzt = tire nlti;iber of workers less iiran or ln excesso f tlre desired. maxlniuml resp ec tivery. Q; = tra>cinrundesired change in vrork force level.For three planning period.s, tire eQr.rr( a:1 can be wri ttenas Delow: _ +tror t = 1 , x1 + D21- Dz1=2 ( a+1 +For t = Z, xZ + Dzz - Dzz = 2 ( 35) +tro r t = 3r *g + Dig - D e g = 1 ( ao1 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 Generalr L --:I _ P i t = T 1 ( i i t - 1 ) x ( t i . u . h r s ) + I" xt X (lrirdrhrs)+ c) T3 og ( 8?; i.iif ef e T1 = hours re qulred for one urri t o f motor I 1l = Ef .-iciency coefficient for olcl worl<€rs o Tz = coefficlency coefficient for new work€rso ? T- = Ef rici ency Co eff iclen t d.urjng over tlme hours X6 = Number of workers irired ln t6 Period.. Tn e follolrlnc r ecur sive rel atlonship j. s al so requir ed.o Ll. - * r ^ E--l a.t = Ht * . N owtt qt n! o*F.i.^1 ho*^r5 l

,s showsI t that the labor force size in period t wil1equal to the labor force size of the prevlous periodpl-us the increas e 1n workers durlng period. t.Fort=1 l{1 = l{o + X1Fort=2 WZ= W1 + Xz or {Z =Uo+X1 +hFort=3 1^lB=W2+Xa or Wg = Ido + X1 + XZ + XBBy using the values of Tl, giverr ln Table G, the€Qrio ( aZ1 is r,ml tten beLolr for Effee p erlod.s r o748Plt + 1.48 PB1 + 2.65 PCt + 3 o33 Ppt + !=]4.19P81 + 4199 Pf1 + 6104 PC1 + BtZ Pn1 + 11.3g Pl1 +1 3 . 3 6 P . l t = 1 x 5 x 1 6 16 + rB /- 1616 x (Xl) + 01. + o748 P.0,1 1e48 Pe1 + 2.65 Pct r 333 Po1 + 4.19 PEt9I4o99 Pft + 6.Oq PCt + 8e2 Pitl + 11.39 Pf + 18.36 p;1 - l1292 oB x 1 - O 1 = 8 O 8 O (aA1For t = 2, + o748 P.O,e 1o48 Pn2* 2.65 PCZ+ BoB3Ppe4.19 Pge + 4o99 Pfe 6 . 0 4 P c e + 8 o Z P n Z+ t 1 . 3 9 P t e +1 3 . 3 6 P I Z - 1 6 O OX 1 1280 - oz = B0o0 (s9) xz

.2l Fort=3 + c748 P.O,g 1 r48 Png + 2,65 PCe+ 3.33 POg + 4 . 1 9 P p g + 4 o99 Prg 6c,q pcg + ?o2 PHg+ 1 1. 3 9 P l g + 1 3 . 3 6 P , l A - 1616X1 1616xZ - 1292t8 x 3 -Og = 8080 ..( 90)ii) o.ltrRIJlr$wJsJIi$].rJ : The nanagerr manufacturing serric es Division, - a]-lowed the overtirne but not more tnan 10 percent of the normal rrrork hours o Tirereforer the over tlrne constrajnts for three p eriods are Biven belovr: Fort=1 01 +UO1 =B0B fify - For t= 2 O2 + %Z = 8OO (OZy Fort=3 0g+%S = 808 ( e3) Thus tire obJ ec tive of the prcblen is to rniniurlze tire d evi- a t,ional ve.riable arrd. i s forinul a t:d below: I.lin Z =-p1 .E 1.2S (oit) +3+ + 1r o o ( D i t ) + P z Z ( D 6 d + rt CltoJ t=1 t= 1 to 3 P3 Z- ( Dzt) + Pq L (D2g) E1 G1 S u bj to ; Eqns ( 1A) to (eg)r already Bivenr

bD SOLINION - The problem formr:rated 1n the last chapterhas been solved. by the conput€rr rhe completeresults are sl:orcr in Append.lx. The inain r esults aredi s euss ed belolr:Ai{ EVALUATI0N I}IE O&IrcrIUE Frn,tCTION 0F4 0.0003 801728.002 0 "0001 0.000This shor s ti:at the Is t, 2nd & ALn goaLs are acirj-eved.fu11y whiJ-e tnird goal is not. ih1s 1s due to thatbhe estimated. targ et cost of production is less thanthe actual cos t of Produc tion. The variabl-e Analysj-s,g iven ln App endix r is explained. belol:VAT,IASI,E DESCI].IPTIOI; Ai/rOUjJ T37 14g,oo6z 3 5 4 3 . Oo26 4452.oo52 59 . go7 259. oo78 +60.oo1 2 O m . 7o61 5040 o .295 . 5694? rs€rB6 I .oO

27 2?.4O6s 1692.1566 5198O? ,1e 85oo14 1 3 2 .o o4 2975 t6211 14,2412 232.oo40 1&,755 1 9 q 1. o o21 ef,)o o22 gN.oo59 18 oo2 z 6go.5 g91 872o5cso 171.o o3 1a229,5315 95oo10 26g-oo9A 30. oo56 2,2 .5925 . 8oc)49 5 5 . oo77 329C o24 53,0q?6 14s oo64 301.85 {e 92.oo This ta.rl e gi-ves tiie a:-ralysis of the obj ectivei r € o ainoun of t eacir devi sion variabl e. T.ilrd thinqis to discuss s]ack-analysis lririch is also rrFrareproduced below:nou AVAIIAB[g P0S..S LK I{ EG-S IJ(1 7 129 00o o.oo2 13 3 1 4 o.o o O.c)o 200000 Og cl O.Cfo4 32775 656e o.oo oo o6 1oozg c).oo o c )o7 110 0, o o o. o oE 259 o.ou o. o o 430 Oo tJ o. o ov,10 114 o.o o o .oo o. o 0 o . OO

ROW AVAIIABI,E -S POS LK -S NEG TK Ooo Ooo11. 293 0.oo o.oo o .oo1213 525 92 o oo O glo14 224 0 Cto o .oo15 320 ooo 0oo1(r 145 330 o oo17 0oo oo o. oo 0 oo1B 450,19 30 o 2o 21 B5 145 0.00 0oo e 33 22 23 E..ro 23 B2 o00 ooo 24 135 25 I ooo o.)0 26 30 n oo o oo 2,7 23 80 I 6 .oo O o .oo OO 29 12 0.c)o . oo 30 0oo o 30 31 24266000 0 c)o o , gO , Oo 32 2.4266000 Oc)o o .C)o 33 242c6oOO O.oo O.oo tQ o .c)o 34 2200000 C ,. o o o , Ot) aq IJ 2200000 o .oo o .c)o o LJ 35 2 200000 BO1?%1.60 .oc) 37 3B H BOSO BOO0 o oo o o Cf o . oo QO BO80 ooo - o Frnoqo o 60 4.. -t 4A 2 1.12 l.JaJ 41 2 .00 0oo 1 .13053 lr2 1 00 0 .oo t.@0 43 808 0oo 907.12 44 800 799 o4.2 Aq =.J 808 3:33 307r99 The table isself explained. T h - L sr a b l - 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. t h e a b o v e s t a t e d ( R . H . S . ) g o a l i r € o P O S - S L !o r i t (

was und.erachl eved i oe o -S .NEG tK from the zJ -cJ matrlx one can verify the optinality of the Rrs&k probl€ffio Thls shols negatlve esttries at Ist & flnd and rvth priorlty Lever. posltlve enteri es are there but at thlrd. priority r ever. That reans the s olution is optinal. SU.GGESTT : In the ab sence of profi t d.ata, ( cl,.:eto the Secrecy) one of the import,arrt goal of the organieation to .:ralce u:axlmumprofit or to a definlt{i,il"t couJd not befu1ly incorporated.. AL thor:gir it l^/as trl ed. to incorporatei t, indirec t1y b) fixjr:f: produc tlon rai;e cos t 1,o a (predecided J-irnito For s ame uro tors e s tarrcrard. tir::e d.ata !ree 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 tlvtire problern couLcl have been better t:ra thi_s.

RE_FER4_IcJ€1. Bor*manl E.rloe 19s6, production scheduling by the transportation of Ltnear prograJnmlng, methrod. 0 p s o R e s e a r c hi 4 .2t B erg s troin, Gary L. and. suxlth, E. , Iiul ti-ltem Production Plennlng - An exLenslon of the HMI4S R u l e e i i a n a g e n r e ns c i e n c e l v o l r t 16, ior1g, Jure519?0.3. Goodnan 5 D rd o ( 1g?4, Goal p rog ramialng appro ach to aggregats Plannirg of prouuctlon and. work Force. ilglnto Scl. 2 0 t 1 5 6 9- 1 S Z S .4. llans smann, F. And lle ss: t^l rA Llnear prog ramrclng .1 Approach to Prod.ucU-on and Jnrployraent sched.uLing, r, ilanagemenb eclurologyr volr T 1, j,io.1 (January 1960) .5. rgnlzlor Jamesr p., A -ilevi-ew f Goal progrannrlng : o A Tool for IfultiobJ ectlve analysis I Journal of o p e r a t i o n f r , e . S o c l e t y V o l , Z g , 11 1 9 2 g . s 16. Jaakelalnen, v; ( 1969) A goaf prograrunlng mod.el of aggregate Prodttction Planning r Swedlsit J r of Economics, 2 t 14-27.7. L a w r e n c e , K . D . a n d .B u r b r i d g e , J . J . , A multlple 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. R E S E A R C I I1 9 ? 6 2 V o I . 1 4 , I { o e 2 o ,8. Tang e John c os o Adulbhon and Zubairl Tahrlr I An aggregate production plannlng for a heary manufacturtng ind.ustryr fnt. Jro of Froductlon Research.

-Ifiornozr, t.A9r & Hi tt new mod.el for Aggregate output plannl n{, Omega5Vol.63 No.3.1o. Declsion systems For rnventory l,ianagementAnd Productlon Pranning by Rein peterson ci Ed.ward, Ao Si-lver, John Wiley & Sons. i,lewyork.11. Ifodern Produc tlon/Qp erations t,ianag eiuen by t jJ-ey & Sonsr N ew york. E.S o Buf fa; Joirn I,l12o Plannlng Productlon, Inventories and. i.Jork Force by Holtr iiodigllantr l"ruth and si.rnon - prentice Hallo13o Goal Prog raronrlng for deci sion Analyrsls by S ang . ii. Lee .14. Irlnear optlnrlzatlon for .,.an&g einent by s , i.i. Dee.15r rn tro cuc ti on to De ci s ion s c iene e b.| Lee and. Mooreo

r l Y |-f f ENrrll- I I0RIlAil tr.e.7(tzglJE FORTRI{I TXT t ! g 9 F ^ t ( G A L r( 6 ( ) -kEr IItElIlil o$t iifElgigt( {8[i,,, !I!ElgIci Y(( BIIEtgtgt tFij;rf, iifr;;ir, iiiEl$iil qIIElrsIcl, DLr,r25fl Iillli5,zs., iiiElslgl rilEl Itii;il,i,, 8l8r. (HARTCTEI c f 9 l ! t R c ( r A r , r iG tl3l:ln; i )t r v t Lxr tLY,FRtI I { H f1 , R F C K , r EP T , r E S T ) 7.1 f 6tl=tJ ) (J) Iq-?t-I=1,N atJ l(I)=I 15 r CR t l T ( I : qI,7f.,.i) i (* IT ISniilttllc::i t8 ,?[=i;h ( , , =! e5 I6frl= [ [ e , | I x k, r ) I I lb I tflil!*tn f hEhrARrreLrs c (fLCtLAT ttl C0ntRIttTICf 0t Er(H V, 31 [ 1=C TI A E L E ( R V L r ( K , J ) ) 7? r J:1.11 13 IF(Kl-1) 6C|.,tttrA( 1n 90 C( K=i--Kl- f U 0 [ J = , . 1 , - St,t(F:C- q1 iiyiIyl;j(].,,r.,) IUfP:suftt+F (ChlIhtrE Fv L x ( K , J ) = s L f t - v A I x ( t , J ) 6l (CItI ]NUE JTER=ITEtrl !PIlSrln (x;) t c l0 9t, J=1-F 9Z J ( =K= +- l _ ) 5 2 , 7 ( , 7 t Ir f ([ . 3 1 cG 91 K=f 4 -L tf (ttLx rl - iil 9r,,s1,g1 91 (0NIIlttiE 71 I F ( R f L X ( 1 3 , J ) - Z t r , A)) S ( ,9(,gt)

ol I rl lUl{ lfr AN . ll.t.ft t6, ( v q P L r I I 0 l r ; 1 Ee , r t FORlRII"IEXI80 linl=nvL) tK3,i) [?=J90 toti ltuF ,,95c lfiitloillrlltcitlgtflEr.,vEDrlE E,sIs Fncr, vAnIAELE I TIh( Arl rct ErcHEAsrttoc It,E!bol!rfli o11C liliEtllUI ;l8rti91i G0 r( 83( I F ( q! L K i ) , 1 :12C13C ttrtle=iic0 , 1 i 0 , 1 ( 014C t t lTI_!( )u P t DT ( I i / C ( I , K Z ) (ox lN : E15Cc SELE(1 SIALLt tT F(SIllVELIi{lIIl,C Atl1 I=116C !F (ArT(r)) 1i(,21(.21117C I=I+1 tt(IrN) 1C,Cr1("1,1[0lEC IgrlE,ftis) ,rT(h)a1c ZfIt=AfiT(l) f1=lztl I = I+ 1 It(I-h) .lE,;itJ,:[0c r,,z,2t i i iiiiriii iil:t4r2t:22 1,,30c Y(Xl)=x(fe) t 0 3 1 | " K = 1 , 1 !A LY(K1,1):!f I X (K,K2)31C (0hIlhtrEc (TL(I,LATI NEI RIGIT-TND SIDTS D0 4(C 1:1,h t fiDI ( I)=f F DI( I ) -Zl, IN, ( (l,f Z)qUL C0l,lIIttrE FRCI(K1):1tr,Itc (lLCtLATt fiEt StJEfTIItTICt RITES Dc !(q J=1,lt t0 5C0 I=1.N t ;Jl- cuqt,J)r r,Kz, (K1,r (c( t( i))(zlr tliiil;E, t0 51il J=1,1 c (XlrJ)=( ( K1,. ) I ((Kl,t 2)51C (0titItUE [3J?7,1) i:1;l ((Ir )=D(lrJ52C (ONl INUEc I RI 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 .Tc6 , )[ JlfiloEl,lcrflctt( t0 61ll l=1.h hRIrE (6,131 t I I),tRDl( I)61C (ONIIhUE ir !ll l ll r I il

elt:tt lUl(flAN ll.E.(lrabt (unPrLrII0tr "1rE,2tt O R T R N1 E X I . B 0- 6 ? A l = 1 , h IR IrE !6;12, (t (I rJl,J= lei)6,2c [8*llnlE ]ExI tohEFPFloRtlYtEvEtlrt ItyErI? r0 T( 32c hRIIE FIIAL f,tStrttS F808 lnII [ [2,.11r.11] TE -,r:, ERArI0ir-----lBll [8f;ft{tltl}!I] E( 6 . 5 h RI r f ,C [ ) ,50c0 r0RfitT(5),rIlt sttPLt) sctuTIcN) rFItE(6.iIC1) HA]t srcE)Aii!t [8*[g,i]h!^rtent (I) r.r1), PRtl81c tsllI[t€13)50ca IStllftilEal,tsslrTUrI0N FATts)811 t0-8 12 f:1,h h Hr I E ( 6 . , 2 ) ( ( t I r . r ) , J : 1 r t r t )812 (0NrtNUE-- _ hFIIE(6-:r..t3) r) ?J-ci t{,r r} Ra{lt [8*[il,;= I :[ ( t v L x ( K r . l ) , tltIrE(6."2) J=1rt )814 (0ttilINlJEc TvAtLATE CEJt(Tl!t FtI,CTICN [0 EZt, K=1,1 iUlL(K)={.- t0 8ZC.I=1,hA-- t!ALlKllitrAL(l )+FFBT(1)*!rLY(I,|)62C (0NIIhtrE h RI T E ( 6 - : ! C 4 )5ilc4 rcRFfr(rrr tt,Lultr0t 0F cBJEcTITE rt (TIch) Do 821 K:1,1 I tK = L - K IF(ItSl.[C.1.( )GC T0 {9 KK=KF+189 rRIrl^te,15) lK,iAL(f )8?1 coNl INgg- ( n L L F I N I S h ( Fl S 1 , F R D l , V A L Y , L . N P C I ,Y T i T K E F I T T E S I )83C STOF END

a t i I - ? , ! Y . a - t - - I iRIRT{ lTXT . .l;".. . :1f. " g U B R c g Il l F S l R I ( i R 0 l t , f i l r R r l F R l , C , V t L Xr t t L J r R H S ,F n S ; r F C X , , . - , 1 Irl I f EPr;IESl) I F E - S T A R l S L E fO U l I N E I S I E S I G I E I T O - 1, K E I N F c R T r o1 . t i l i 1 sp I C {t N . i 0 n r r r A r D r F r N s r c R Ht r l t { T 0 , 1 s t R I E s 0 F u S A E L s , t A l l t l E S , , IEC IENL j.zrs, t tfiEl Utt:|tcl:i,v Lt1 5, 2 5 ( ) 3l8lE g !I q 0 t , $xl ( 1 r r t fe l q t g l I L It lIHEiSlCt XEFl(6t,) DlllEislcl Rht.1 6[) ( rtrErSICl Rlt (60) CHARICIETT4,IAHE l.V=250 trt=6[ f 0RptT(At,St!.i. rlG r Iili ilit i rffi /Ist - " TATA OBJ/3QEJI inln FRCEtFt(Er/ tAlA B lr-l rL / I iili [=Fsfi t:n," t ES i . [ 1r(i[i rEIO THE FICEIEIl (ART FOT THT NIIEER (t FC!lS VAl,ITFLES . 1(] I f; I ?6ll}iil n[:IH6(l rtlIFnr f? t I :fif5rl IelltI(:1.15,:I5) LISF=trPRl+1 lrtil|fi1:tE:tll8 ltiE 88 DiI IEIl ixlllE::h i *8!,loul E:t eCI1c t D TliE 5ICl, CAR[-cc tT hlLL (Ct,ilf lN ciE (l ilf f OR EAUATS E F(,LL(tIhc TETIERS FCR EtCli FCtr Ii tl.1c f0R LESS lHAt 0tr t0Url, T( L rti rIC. f CF GREAlTFI}AN (R TGUAL TO ( Itlc TOR EOTH DE!]TICIS B ;{i (lil , AL ( I ),I =,1.R0h s s) { lii11 iEf,FI? : 5 {1, (6. ( i.ill l lqc h f rI T t ) !q! UALSI) r I=1, NR(tlS) {,1 f cRFtI(5r,1Cr: ) ,.* n t!RJ=tJ h (01ft1 THt hLf.I ER ( F T( SII VE S L ( K V, F I A E L E S t t L D S = l J I g0 lZ L=,,, tFCtS {r I I t (t6LALt i f ) . 1 c . t ) N F Lt S = t t L D S { 1 t12c I F ( E G U A L t( tEsT t0R slzt I ) . t Q . G ) N F t t S = t rF L D S { 1 ii t I I ,S I Z E = h F LI S r h t 0 1 iS { N V r t l IF(hFohl s..GT.ir) IF(tSIZE.CT.ll) c( T( 911 G( T( 911 l (LEAF ALI I.IItIc,TS 11 UD=hpRl{ 1 D f0 1C J=1rhsl?E

Ivr) |llf "r"ill:, 1, , ,. 1 ) . .". r . ; . . . 1 i{ , I i l 1 :; t , " I :"",i. -,r|i{ "; , JR1Rl,{,tlt,E11, i i" Io :1i IFtl.:li:{DuD)-Go ,;. !!1|fi$lltll;:;r;i, xPCF=0 r tlltil*iril! l;:tl^:ltR"l*^^ (EJEClIVE fUI{[IIOh IO f,EEI IHt iriErriliiiE:!E! i? iii i ii scro 1E 14 iItiiil6iiiii:iq:Ei t;lx(K,J)=1( 1 ttAnl=lAR1{ r9 E1i 1s Ii:[ri[!ii; l- iii;ii;=l i=olt(KJ!=1 t tf gl:l:[t* (o TC 13 l I 1a ii[[iltfiirI, i i IFii];r:" 13 tliliiii:!Fii Ft(TIct iE:" z) JI tl 6n iEili{t;f :lt:lllc,z( :il ig I=Q t E::t j iliiri$E:: E!] E!iE,6 .tl { .l r i. a i zi: iEil +loi]z: 5,, iislitlllfieirl*litIl lr:l ! t ii l 1i[l:i[1rioii v. { z1 i?ti li [: lt.li. !lii ?i i=LISP-r, i irif;iiEitilii G c T c ?p rE!:louiii I I I ? t rt(tlltrE.id:;ds a

*, nccHrJ)ZDrr , . I . . . . l0ntf AH t1..E..7C12S) . . r, ;fl . ",t - . 1 E NUTIBER tf, ffo c0 1c ?7 tiil t73 c,o J=I I6.f[*1il)=rErr *g c> JsKEFI(I] (0 I T I X q P T ( ! ) _ . E e * . 0 ) l c 1 (? 6 itd !!LX(K,J)=TErt t77 !78 !79 z7 c c.7r,ez( iirltoF9r, l I F r n i i l x t l l :8tl i,fF1 iZ 30 rEr?airi Eir i? iili iilii !EiP.IHE.AA lfil:!E:f ccfii ,63 I F (AiAl.iE..E 4A :8.4 ] Toroer .87 F2 iil!51:$[Ff". G0 t( 3rl l iff s( t t]f DE- Ir8 89 Is fEi8,l:E, I,!ll P-!I rs) : rl( r e 9tl I l i ! ii(,-,l l_([1) ; llI t cus) 91 Yd, tz [8F[!lt;irf lf;ili,ilirt!!, RrsuLr s ?3 c t4 B? 51 5 1 [8tIttflt ; ilil s*t(HT A]rt I D E - I I P U I I ) t S $I 42 tfi{tliIIitttl,4z,t3 1f 43 [filt I {l;11 i,Rh(r) t{] r t1 i1 1111 41 igni o , ( z c , r.: ilft(r,r3,2r,11) l3 tRIrt( :4 62C f0RfitT(1|1) r5 hF:IIt(6.j11(,) r6 T,51,) I oi!"TI1urlC N 5 f l1 6 t7 t8 [8*Ti i]i FITES-Itriil.Tt) hkIIE(6.ar1g I t9 251e f 0 R FI T ( I ) , . R C r , I : ) ilr3 ! 8III{fu}.l;l:1 ),.=1, s I u E ) l It 17 i l,J t 13 ifiil[[8:ti], I t :it1[1,;]l i *!"Ecr vEf U N ( 1 I ( 1 r - I h F t I ) 1 l4 5ij17 l5 l6 i=LISP-X t7 tfiIIt(6,i15c) ft t8 u1 5 0 t9 1r 14 i liiil r;:i illilirlil,.,, ,J=IrNSIZE) iFIiEtS;!ttl, t a t l 2Y t2 5,1 8 1 lCBllT(tt,,St t tfAF Ot INFLT IhF(tHAMn?1 i3 l1 lHii;iSlitr;r hRcbs- I.PR1,NIT 1) z *.1 17 lrtf ( f cRr x ,rt t m6tFOiIf8f S rI5rtrIOXrrNLfiBEF c t vtRItE

vlltt "4{BE :,F0R IRtttllteXtl,,.i*::i:ilt:il.],, l!t-.:lt-.,r;r5;/;:JGrli.iri,i:ilt"0FFRrcRrTtrs i[$iil:1,:;,fii;;;;;;;_; , _. 1 . 1.1 ., , I ...,,rt5;t;itc;x;i ;.r;;;l# 9 1C 914 nt, rnrilr,rs fx.inE, n,rH,6in,n, 1090 1091 {g1lf*;;iillft*r$ru ril,rf f 0 R r { r r ( l r p n c i t n E tA7 t C 0r [ ,t l f i l c R r 0 , r T bvl r r F I r I T I 6 N) s9_Ig-gg. t 92C 921 c,FD riIiiti{fu1li,lFdt{IIIIr0hIItTHE t,n6.r. xEaIicrer ,fr! As rxitiDtiiiiiofiil;i-EEEr rs ztxilt xe 1020 G0rc ,rirdidiYll9rs 10?1 !TIIE .(Q,1CA1: 8[[8,lg*[r*HF$ithEfpRt(Rrrre tf Hf{fiS;z[[ t,g s cAmror i Er 1822 (6,1q;.il IRIIE 1023 cctint tALtt ot pRrcRrry I r!filiilii; v,ruErs rarnr r0 0RLEss I iLt! co rc 99! | 911 ! RI t r ( 6 , 5 1 2 I q12 ttt rttirEI-r li8:iil.li, t filHEli* f-brRrfgII!-ryFr g,i[!["noe Frc(FA iliili$F; cED-r9 iti i;8iiEl "-ll !qFprIF_urs r n, intn tEl ifii ; cI 14 2 6 Mlql q,iLzi) 1$ 2 7 . IS ItADtro rritilIzt N0hExrsrrrT i tiEUiliir.lTltMFI pcsrTr!E I 1Q?4 1q25 ftrif ?tlc.:] t t N c l I 0 N F R I c R r n Enx L L UE Dsrr fr T E Dr * F F c | I t[fi][ilf;;tqiricIIE u t c E i r sr Eu htFEhF gg:iti|? Ct 9 r l1 u.Iq(Q.|i?)i t t ( i 902 t R i l e T ( r FR E t , ( A !R t t {rIr SrS r i l . v rt ! r I S t t r r c F E D r ) " ! " r s 0 ",lJlLiluFtD) I .CQ-II -?91 i 926 ! ! I : l F J6 e $ 2 7 ) 927 I -i qFJEqllv! -sEcrI0l,l-!EFrtED!rLLr rcni lilE:lEj;,f rf itPuitTIII 2ipFriE rbIi r-cr! i E ]3II, q [f ! !i i iiliioFI :ftE i;IiiiI IiL f fiII f; Efi -id{ilftl:iig,llll-yttgls ARE-!I:rALr(bED rHEFrGrr HArn 0N !sIDE. CCt zE-lp,qi-l ,t*rr-ElBir0ri rliGirTrrE Ilf ll ln[3rer I fi[.], rpii roni., I G0 Tr 999 i 999 tETUf,h I TND I

., .; .ir l,;;-1).f 1 .i:1;;"" i t"tl;-Eitjf f O R R l N , I tI) x T , . ,, . , . ; , : ,1 . , i f _A _ ,,,-r" 0 r s, K E r r E s L l i : , : I U E R C TI i E , f I cu I r r s h( R H s , R hs , v r L y r r p R rr l pc r , Y r [ R 1 F i - i fi tifiE,l : " i t1;ri DIiETI . lIllEh: iiiEiiiiitiil TI i l E I , lrll :iilitI:ii illiHir:s!! Fi:! .![!si?irniiiiiirs isi Hii DIllEIt FHSl THE EI , i l D rHE$EsLL,] I !i i3:FLirr.D-rNii_rli_iTttorqrnleEl t c c0LUr(N. PIG sLrcx ct lHE . IHE FEr-rArhDE T 0 rr r H E A L t i S i R E i n i i i t o . c i - t E S r JT S . It . SLACTANLYSIT r bFITE cK, r) .sL, NAtYsrs ; tSFlt Ii,!,;, liit h RI I E : h RI T T ] iIt8;l t 0 Rl4T ( 1 ( x - . F ( t l t lI , 61 t ( ,.f AIt rBLE r lTX,rt s-5 L K , 1 2 X t t E G - SL K ) , t f i I l I tt( 6 , 1 ) i 0019 I=lrhRCb! ) l,EGSt F0sst it[=f:b D 0 11 . J = r r , t r t ( t S l1 l=Y(. I F (I. ; I t (n.illlrltllr* 11,1 ( QNII I t U E z,t1 11 ] e0 Tl I il I[*lit t ,.,, tf e SI 12 G0 1( FOSSI It*1i,, h RI I I r,.,, rK EGs ossLKt 13 11 t 0FF,iiii;i i iii : tt:i :lr ( 0 t rItI h U E 19 15 C V AR Itiiillnlir+1,15:) f ORtr bnIt t t,5(x,!ARITTLE f NALYSIs) 4 t 0 R l!{?,+i|,rr, ,, r r nll touNl.,t t 45 t0 Rril!{tt,i,,rt t ,tRIrt LE t 0 4 1 I = " 1, N R ( S 17 h1 hcHc r FII (gbl iIitlil i:ii:::i,, I r (h illi illtHla, r RI T 7? T C R Ft T ( 1 1 1 ) irr 5 ri t F11 [ 0 Rt { b i , i n r , lli y s l t 0 ! T H t 0 B t E qllY E l r1 c x , P A c E 8 t, t t t 1cX 1 . F RiIt F i i i i , i t i ; i L t o E t A c l t E v E r E t lt) t052 K=1.NEh] 52 I v AL 51 [0 5 ZVlt (r.K)rnxs lii:i::;lll I i*n,uY r,) L I SF=hPRl{1 I

c0|I I tl J,l t -1. {, *1t",f rYCG",. f-/alRtN ltXT KK=LTSPI I F ( T T S T . I G ,f,.(I GC 52 tr, XK ctt t RT-lt - ((78 lIfttrSIrlI I, t iAA:LrnTGI F5r (2t A L ) ) t 0 R l { l T( t , l G x t T .5? [f; 1t,[1tu :1Y,:!l:l ) ]il ?0.5?1 [8ntilll" FETtrFt{ tilD

,ftPry.I {.16CLg(3), fi AX_$ C_[EN(1 : E H ? 5 C B Y I TSOE t 45 ., t l sc -, l EBBEI E EB E EBBET, E EB F{ TEgBE t BF EEEEE E LL L 1 1 1,25 { c t 1 1 1 1:il 1.25 K 1I 1.,i5 i 1 1.25 i a 1 1 1 :i? 1.25 1, 1 1,?5 1/ -i l 1 1 7,,i2 1,25 1_1 1 1J 1 1.7i 1: 1 1 ,75 1t 1 1,2:, t, 1 1,t5 1i 1 1,25 1e t 1 1 1 1:i? 1 ,? 5 71 ts /-t 1 1 .c 7i (,( 1 4 1 I 7..i7 1 .7:, )t ,| f .i: ,I I 1.2,- ::! /l 1,i5 I ?$ I 1,i1 3r l I I 1 I i:[i ; ? 1:it 1.i i ; 1.r C 1, ( ) -t 1.i j t 1.t i ? 1.i ! l1 1 . t )^ r 1l t: .rO-e ).O fr.$- r:l (vN

;t (3 4 -l.ltil : ? l.!Ut, 74 t 1..[0 t, C4 r ( 4 1.[0 5 1..[0 -1.(il f ( t5 (5 7 d l+ tr l:tfi 1.I|:I ( 6 1,Ir] ( 36 -1.(fl i C6 1.Ii-l ! 7 1.([], ! 77 -1.(:.1 I (7 1:( 0 i 7 1.{tl a t c .x - 1 .. { tJ 1 f l ( ix 1.( rj 5 ( 7 1.(:l x 1.1l 7 q 1.[,J ; j9 -1.Ii) r f9 1. (,11,1 1) 1. ( J1, d ,t - 1. (i1,: 7"l 1.( iJ11 1.! 1 . i l11 11 1.[ ]11 t,1 -1.Iil11 ?1 1.(il 1t1;J l - l{1: 11 1a 1:i{ 1.d !1i t/ -1.Iu1r. 1: ? , 1. [,]1: t ) 1.[t] .1 " LJ 3 -1.iii 1: ( I . t ll 1r 1j 1.IiJ 1! t 1( L4 -tr:[:i 1i 1: 1l 1i fl..iI ,J l i i 1: 1lt 1.(,1 1: 15 1 . I tj 1: (5 -1.L,i t: i5 1.(il lo 1.I I lr l Lir -1,f,j ,i 1i i6 1.(,..! 11 1f, 1 . (,1 17 r i l 11 lr t7 -1ri t, ll 1: 1r i7 16 17 1:ti 1. ( , l 1 [ J 1l: 1i 1il r8 - 1 ..{ , , 1 1; | ) 1.C l 1! - 1 . t ,1,i 1q 1,; Lq, 1. ( 1t ie 1. (l 19 1. [ ,l 2;l tl 1.[ ) ); (il -1.ri ai. I ri 1.IrJ 1 I1 L 19 1.(r"l ?1 cll 1 . Cr l 1) 1 c 21 :1 1.tq --l t U 1t F1 Q) It ?2 1 L I- l l l. . l (d :z -1.[,] 7Z LV 1.(] (: i2 l,(0 ?ri 7 t (,J !3 -1:rr ?1 gt 1.(il <C , 1:(,1 ! i 1l "L 23 . 24 -I 1. [ l _1:Cr :4 1.f"1 :q. {:t,i -1:til ^.i ::

. .i,.j :l;,^l"t";.rl .,. ii* rii:;?nii C.d ;ii ,l.*,lit e l!.fg ,:yi,:ir:}! .i ii l,1,itjil;: I,iii*ffi A ,:,. .,:. :;,. . ;, .r;1.*,i,.71 . .,, ,:,9f -:. IB-, -l:t8 .,,.iU :..i;,;1"ii;.i;* .,;. ,"" . . .. ,. f z r ,,7E i 2 ?.[. ,1 28 -!s l,:[fl -1..f0 Eg 1..c0 ?5 ?s ?s 75 ze lq I :tB - 1. I iJ er t9 1.[il if in I:tB 3f ii 1[ ft -l:tlJ 1.Itl 11 1 51 31 t, 7 !ff3:tlt (620.(fl 31 lrf 1(214.(rJ 3{l " 1J 1 2 6 75 . f r t i6 31 l{ =) 19 t,( lt|il:tii 3t911.(;l ii c) 2r, .[ i tl,lijI f l l 31 91 8. I; 7 32i l$lt:l;:i it 7i , 1L ( 0 2 1 1. - c g 1( 214. ( l J; 3; 17 l) ligll:fi1 7t431 .(,1 ii,ij l(911.Iil 3; ro ( ( l . , ! j ., l 3, 1? ,o i. s? (. i t L ,}i: ;l i: i 3i Itl?.( l 6 -) i 5. {,iJ,37-9 (? tl ,l) ii tt|lf : I ij 1?675.(,) Il 1r 1,il,ttl ] ii i], :t l(911.i,j li i!, ftS,ili:i,j i;l !i [, . ( i-] 1aA.Z;) -i j/ 14 4 1 1 . 7 -, 1 :7 81.,. (;r :.t t, t t:i l i5)C i t. .{ i.!i l 1 ? ,11 t6 :lr ,t .q E? ,ill:?ri ti.l { :t it 15 :g iioi,:ti Io4,r.i.l 3,t (1 VT :1 EL cl ltl:t:i 1 l1g . [ ,l i1 7t l,l, 1571.(,J 195J.(,) :1 i8 -21?.(), 3{ 3t l? E5 ilil:trl 7?ltr.(ii Zr fg 1(8rit.l.Ir1 ?r( ii 3: i3 ltt:i t -z8 815.It]- il it 3t 3: Zlr tl 1t3[:til 573.5iI ?c c./l 1: r{ iHjt:Ii 3: 5d - ( 7 i , : :i ( , 1 ii i7 *i3i:Ii ?; lq, 11 i( c td 7, 1 i1l:[ { 1571.(rl i?. ?r i4 iZ 7?l:1il I i I ,^-"

36 411.t 0 . ?9 8 1 &.J I 12 1t 3 fr:r: t U f I 15 6 ? :t 0 L. 15 ( llr l I -tn 5 i 3,.9 0 .! s1 :4 I ? 6 .I o ! 6 .( 0 :1 En . 1 il (4 i 7 6 0 . b rl l .1 :7 Ci) i $ r 0 .L 0 4 .1 (3 z78 . 1 {.1 i ( (6 5 14 ,..I 1 1 1f . i. ( r) d9 tc 75 II: 1 7 1..r r) I 1 9 5 ( 1 t ,4 t l, I . ( tl , 7 ..-.(I(f l I t,l i8, z r1 x r- a t i ili /l t4 15. t i-l 87 i l I t J .( ,J[ . : (I , ) !n . 5 ,l 1 ( l J r. l ( ;5 t .l 4 1 ..l , r 8 I 1 i.) 7 I ?..(cj . ),.. j?! 15 4 r. 19 f 19 l 16 4 ,. tq9 r ;t 1ct 6 . tt. . l4 t ( ;i g . ;,(.l . i -c CJ 1 1 . ?a; 9 i.l 9 t 2i 1 3r .. l c ) a 5tr q1 1 re( 2r :r tjttiIlFr1l t4:f r / 1 j. i: tJ r , : 1.., lL l3 l 4 c) ? .,! :5 ) C - a C 3. l3 ll., i i 9 11 1t, ?r 1 + I 17 L ,, ( t Q t. c 6 ,, L! tt , a f ZJ Y . i1 , l ( (t{ I -.i ,|l - a l - 1 ,1 . t Y 7 /h 1 , It . l l . c 1,1 : I , ! l l l l( I: . , 94 55 a ^ >1 :: 9 tt J l.lJ lf l)-rt , . I l -1 L I : 1 ,. [ ] a ! I a i:i 1 r/ . i l rI a itZ t; 1. , l l a - () 1/. ii, , , ,j I etr 1T ,t, )1 9 +. L l ota I I CO lii . ?1 6 ., l ( 1 () t Itl 2 r,! 4 . J a a I ,C |7 1 l(. l e jr a 1 . 1:- ? 6 : , ) a I 94 116, 16 . r( 6 a a -c5 11i, 1 o . ! , J r( t . 1 7 , 9 ., l-.! 1 ;, c iJ i I 96 - 1 .- i a 1 1 ,ti i :r ! 4 . a tL, a s:, 1 ,. ( r , l , I qi to 1 ,, I[ l . a (r! 1 ,- t I+t 4l t1 c,l. 9! 1,(: 1 .I,][ a a a :tll ( . ( _ ) ( I ) l , lr ( 5 69 . t) [.[ 1 3 31 4 . ( . zt:l )t.r 5 ; 7 t . tt 1 0[ 7 5 . ( [.I s: 9 . ( t: lt:,t 11 t :It, 5 t3i:fl iti:t trl:A ii:I ltt. C d,q.t ),t.r; g 5.( 14 : . ) 2 3 . (( 3 L.t 8(. i 6 . It ,! t i . c z t 2 6i r ;12{frl i l f . . | 2 Z , ll ( t I f :f 22 rt ( r l 0 [ . 0 2 7c 0 l 2( ( . tol:I l-.t. t ( It:t 8[ .,i z u.0l). , t L[.( I il[ . l

), iLi.:-r i-?,, !i 6:" ;lI JrG a r1 t:::j;l :, t ;il aY,r1,;l -il I i l;r i.i!J1 t1 l1:1i f-lr .i 4iri,!:lr t"ltj i l , ., td I I . . . h .r a :.,: a . i:t ,i a I I t a l L al a f I I l ;l ,+ t ,I I t., I )l rl ) ll l l, a .. a ,l a a a r1 n ) a .l ..1 i:li ! lc t it I l9 l. a a 1 a b !t i,ll i$ a t 01 t a a I I ,- I : l It ! I "lCl -l rt I :Ir1 rI - 33 l ! i t ) 1 a a I I -1.,Z lrlitr I I i i it l . 3 , :t il:iIn a I tl rf tii rl:lIr il I ll,1 [,iI irI dl. l ( .l tl . ili r) r l . ( ( . ) :l n I I ll: trii it: ll f. l r). t(, tr1 I il ;l li il ] tt il 8: i"l , tf lf . :lI tl . i l tl :l::iiI itl fr ti :l: itl :i! t I :Tt(l :1il It{ i a a a LI ! -[ Itt (u t!l (tt a a Il .FT 0f rtil ll ,.i r rt:ll Itrl li 00 d tt 0 Ir cr 0 c ( I 0t il It ftt iil I ffi It 6 T t:I fft I fi[ :l fiil r$l ffi[ ;:l ,01 0cl ,Cl :.1 :l i[ti , . l : : l rl ,CC ill .ll Il) r i; ttt ti I::tI I ii fr r t :8t t: lft ll :fl tI [[8 $: [ I [; t[8 8: t I:lI t::8t [ :I !i lJ r fr- r!(

,:,.,f +iift*riii: u:,ttt; ."ii._.,1; ,,,...:i.,,,-,i1.;:..iffi:ur.;l: , . - . ;, , . , , , , r i , i i : ) l + i i:lf;1 : :i" r . . . ,i:;;.",rl r i j : t,,ri1 ,jitiit;, ,,.:i ,,,,; i -i ,:;; i.i1i :.,i ,,1:!: ;,:" .-,. ,. tr ; ,. ,. ;r,., i. ."...l.:.."i...;.;.i,.iii::i:.:1.1:,....:;,.,. r.i. {.,rlh.,|.,(JA-.L.,tds.J4,|i^/|.,L,!^.,|r.|r..4.,,.,,t.1 .,i. JLr|irr,lrrtJvt.JJl-,r.,L,JLJ|r,. .r-rirr"ru-r !t-trJL|1.Jr-r!.l., (-FJI.J ULj)t r-, trJuLJtI. rJJCJ- LI-rLJLr,-4.rG.rlJ-, cFl|t-a/l i qrclgr(Lr r-iQr(J--.r..rr.rrl(-r grslrlll L.Lt(.r-t LJgtG.rlr+J |;li r:l-, LtJLrt-.xz-r(-:tj *1r:: fl gr(r grJ-, q,142 ct(I.) tl q)L, qr) r:r-. c:(-rLt-l tf EJcrt",c, (1, glt :r."u t ,LG, I t|iJ- yj: iJ,-la-r,.r.r -c-r r..ru:lr,.r.ru^.1-rl-rJrr.rr.t-rt 4..;LrrJ-rtr.r, clJ!L,iJ.rrJrs;rl-rt a-t Ff l-:1-r1 i lal r-r t | | t | | r | | t a . t r a | | ! t I a | . | | | | Io . I | | ,tl t t 0 ti. alr,,, ,.it^;C*;Urr,r.r*c*eul*rl-lrrJGrr.l{.rrJerl|1,grirpr;OgGJ3(.rrJrt..,t,!g|.l+erll Jlta rrrf-lvt IrJ .U 3 g-a-rJ sAJ- -rl 1J Llt !-n-Jsr-J*-t rrr{-rr . |JL,:.aJt-lt .rt*-, r-tLc-Aru l-a.rLJLt"rrr r-r! (...:-:;:Tr:alc-r--rr- -rl(*),-=rr:::=c-;r:t::a?cf crsr:rloirre;o:lcr1Jcr:::::i -a-tf:?1 -.-!:. -.-?-Ft!:t:r-?:rr<r15 1;r:1g?r3C.tlrl(,|TftCitt?:f,?1E:ar:"i! ,:-1 JvJvtt r-brJ.t.rLJtJlrrJl-r|JJ- rtlhr--t..;Jl<ltet|Jt"t.Jr"J"AJ-JVrfJrJttrJ t,.& r.-Jr-a-r.^t rt-rrar.-rL., uJr.a-r:AJvrJJ,rr Ar *!rJ trl - d ba .a-, .ryvw {,.A- | | r | | | | | t | | | l r | ! a a r r a . I t | | | o. a I a r a I o a a t - f,: .-nrlt.lt g4g1ggcpF !:a?C:(.r.- - " r.-- -.;q,. r-.1r:.i..--a C.*S (ltTr ri21f L:()T tFmf x (, ) ta tr ,!F I (3 U, .q, (J TL t LrJbrr^-rg bLJ!.* .r J-v -"JbL --tv*Jt rJtJ ba-".4Jrf-rgrJt"aJJrA"rAi- -f - A---r-* :--:y:c-:-r-=c)-rr:r:ccc::c(:?q?t:c:" " .-.:an=a --- -.- --- -ilf-:-a.:<:-:f ?3::.::- -c?::elffi::: LJL/r..-,r..4.,r.,,t,vt..,|- g.- J... Yi " .,U Jd.- |,. ./-...^-,..,J..Jai.,.4. JvuVJ .LJ.J.9.. -r ,J-.rr.lrrJ.r).-rtr.v.rr.t.ttJ--r"tJ-Jr-lir*lllLl!r-ttuF+Ja-lr,iJbir-L,JufJu:{ | t I t I | | r r r o | ,1, | . I t | | t | | | a a I t I a t t t fv(rr" J :-.t!a;v?,-^,:,t-: -?l1Lnr.t:ur:tcruatrtlnJtnstr:::LcnJ:-?;?r-:??f E srrtrt- rtY,!ttYaFrV{lt^,l r!{Fl lrlw Tt"+Y-:t::* taa ryr yl*rrt- -!l rF li.A/"t"1-tt-)LJ { t+nftJtJUt..FlrrrrF(vtrl t,Fr?{rl{ F ,l,i-"J-u-sr^ lsfrlJlNr.J J.t{t*I-J rr! ,t1g,11rlrvr{ r( ilf a I Vl wt a (( a,IJnJ l lt|/lltfr!tr-rvf{l{t11l:t^/(}v-tt-t .t FtVnl{.rfcrl!.rs^$Flvrl|utt|,b(AgtrsFtlT (lrl r3"ti sl nfrr {}J t|.l4 fF |. F frt^ r- t F?- F. F r-rF Ff ns.li.l NN J .f .:l. ,-.-rur . rJ, v......, f| , t,, r:- - .. i):- J /.r: -r{ I, rr f .) l su .t ":

"llli;*f ;191,., ft : Pf t F (? YAtttEtE ^rtttsls vIRITELF. A)( t ifI t? (,? 7C 1i T tl 1 f)l :li r I ot (! t It 1!, ( lr li i ?.: t-( t C ()L I 1 t! tl ti 5i (t - , i t i ti ;, t, 7( il 1l l?TALYSI! 0l ll.{t ( t:Ji(TItt i .l I I F,rlr)f,ItY l i . o e r - r c l l E v t r E r .I It I I I a ?,litti,lJfft I . 1 . 1; )r l [ 0 .!Fl ARIlFT(IAL n . ( c 0 ([

Yogesh Saxena MTech Disseration, IIT Delhi

by IndiaEducationrReview on Jul 06, 2012

Accessibility

Categories

Upload Details

Usage Rights

Statistics

Yogesh Saxena MTech Disseration, IIT Delhi Document Transcript