lr- a              PRODUCTION PLANNING PROBLEMS                IN ENGINEERING INDUSTRY                   (A GOAL          ...
t?                                   CERT        _rJ J-.C-A-T-E-                                           that the disser...
_l_.c_F_N_o-,$rJL G E M E N T S                                                ED          I have great pleasure          ...
tvrrI                              _C_.OJIIJ_E-N-T-S-                                                                     ...
r                                                                                  tj                                     ...
LI ST OI. NOTA I ONSb.             GoaI set bY decision                  maker. 1               The cost      for      ove...
Nurnber of        goals.M             Number of         decision            varlablesn             Overtime hours in Perio...
t;                                                     _A_B_S T R A C T                            In this      dissertati...
ffi                                                                                                                  1?   ...
I            !            .l @                                                                                            ...
3           So th e cornplexity ln tfr e aggregate pro chrc                                                         tion p...
. l                                                                                                   4It          1.2    ...
5   I                                                   t j r e s e q u e n c eo f     process must consider the cost effe...
.l                                                                                                                        ...
This model is         very floclble      inLnvento rY , & d w o r k f o r c e ;                                to a partic...
B(tqZf ) using both discrete                        and contlnuous           events slmul-ation.An important              ...
aw                                                                                                                        ...
3 . 1 . .              frrnI                                                                                              ...
ll    .1                                                             propo ses a linear             pmgtarffning          ...
T2      2.2.1 .2     Heuristic Mo el s:                               d      (a)        The production parametric planning...
Si                                                                                             l;l                        ...
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I*-1l    EN 1 I.:                                                                                                         ...
I7    1 t h e s u b g o a l w t r i c h a c q u i r e s m a x i m u md i f f e r e n t i a l   weight wiII   besatisfied  ...
tfr                                                9.U.AP_IEE-                    IV-                 GOAL PROGMI4MING A M...
fr                                                                                                                        ...
2f,lreco rd     i t    in to       RFISco lumn .cj                  Variable                 R .H . S .         d;..      ...
F    l.-                                                                                                                  ...
2,)             avalues at          higher priority                      l . e v e l i n t t r e s d m ec o l u m n .     ...
f                                                                                                              2:l        ...
24             TliE I-rvL SIMPLEXSOLUTION             (a)     The Riqht Hand side                           Thls    shows ...
2{t        It     presents   the constants of only         the basic choic evarl abl es.(f)   Analvsls     of   the Obiect...
|*                                                                                                                        ...
j,tl            I                                                                                                         ...
2B   5 .2             (            PRT.ORITYI    SALES REALISATII}.I              E q n . ( t ) rep re sen t s a gen eraL ...
2lf          For convenience,                       Iet u s put                       tr*              =            oa*   ...
;i0               pg = 13 *S3               12  Fmm ( q) and (s)               Pg = (oa*- D;) *s3                         ...
3t     1--)                Pct n Pc2             ot, + D^^       vz                                                       ...
J]?                     DJt DZt              Deviation al           vari ables                     Pit                   P...
:i3      5,4        PRIORITY (III1      to ttrr:,tt rne cost (Rs.1asgoctRteowttlt      IIWENTORYLEVEL To SPECIFIED.4{vlCx....
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
Pankaj Chandna MTech Dissertation
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Pankaj Chandna MTech Dissertation

  1. 1. lr- a PRODUCTION PLANNING PROBLEMS IN ENGINEERING INDUSTRY (A GOAL APPROACH) PROGRAMT||}|G A EDISSEFT1nAITION SUBMI.TTED IN PARTIAL FULFILMBNT OF THE REQUIREMENTSFOR THE AWARD OF THE DEGREE OF fflagter o[ 6,e*lnologP in ,ff[erhantuI g S,ngineerin BY A PAilKAfCHAtlDlf ttzltt U nder the gui danco of Prof. S.K.SHARMA rtment o[ Sler[anital @ngineerfng @epa Begional@ngtneertng 6otlege - &uruh*tletra 132ttg
  2. 2. t? CERT _rJ J-.C-A-T-E- that the dissertation entitred rt is certified },ROIICTION PLPJ{I.II}.G IN INruSTRY PITOBLE]IS ENGIIIEERING t by A G.AL pRocRAl/$rNGAppRoAcH i. s being submitted partial fuif ilment of MTech Panka.i char:cina 7B2f Bg, i.n , of Kunrkshetra in l{echanic a} Brgin eering Degree course o f h i s e w T lw o r k c a r r i e d u n i v e r s i t y , K u r u ks he t r a i s a r e c o r d out bY h:-m under mY guidanc e ernbo ed in di tJr i. s di s sertation ha s no t been Th e matter previou sl y f or t[ e award of any otir er degree sutrnltted Plac e Ktrruk shetra Dated g311 GitrreY ( s. K. 9tarma ) Assistant Profes$cr-t Itechanical Engg. DeparLnerrt Regional thgin eeri19 ColIe-Q€, f.unrk shetra-132 1 1 9. --1-
  3. 3. _l_.c_F_N_o-,$rJL G E M E N T S ED I have great pleasure in xecording my profound gratitude SharTna,Assistant Prof essor, Mechanical Rrgin eeringto prnf . s.K. CoIlege, Kurukshetra for hisDepartrnent, Regional Engineeringlnvaluab}eguidanC€lconstantencouragernentandimmensehelpgiven r a o k , w hi c h r e v e a r s h l s rat each and every stage of persuing this of Production Planning. His inclslvevast knowledge in the fierd discussions and valuable suggestions arwayscomments, fruitfuledif ied me vrith j est to carryout my work f irmly am very thankful to Prof . B.s. Gillr chairmanl Departrnent Iof Mechanicar Engineering, Regionar Engineering college t facilities to carryout this work.Kurukshetra for providing t h a n k s a r e d u e to Er . L.M. Sain i r Er. Rai e sh Jan 9ra" becial f o r th eir kind heJP during mY Er. R . S . B h a t i a a n d E l . D . K Jain computer lab. work. In addition I am highly thankful to aII my friends who helped e s p e ci a l l Y to Arvind Rajender, Vinod and Rajiv out mY dissertation work. me a lot in carrying PIace : Kunrkshetra Dated z 8 Z t2tl i) t n,ffcHAI{D}JA 7 8 2 /B e -1r-
  4. 4. tvrrI _C_.OJIIJ_E-N-T-S- Paqe 1 CERT ICATE IF 11 ACKNCT{L EDGEMENTS 111 COIITENTS Y LIST OF NOTATIONS vl1 ABSTRrcT CHAPTER I INTROUJCTION 1 1.1 AGGREGATEPRODUCTIONPLN{NING 2 ( cEttERAL Fonlvt) 1;2 SMPLEST STRTJCTURE AGGRECATE oF PLAI.INING PROBLF{ 1.3 MULTI STAGE AGGREGATEPLANINING SYSTEM CHAPTER I1 LIT ERATUREREVI ET{ 6 2.1 DESCRIPTIVE MODELS 6 2.1.1 Th e Management Coefficient lvlodel 6 2.1 .2 The Sequential ModeJ, of Gordon 7 2.1 .3 Simulation Models 7 I{ODEL NORT1ATIVE S I 2.2;1 Aggregate Pfanning Models E 2.2. 1:1 Exact lvtodels I 2 .2 .1 .2 H zu ri stlc Mocie.Is L2 -1i i-
  5. 5. r tj TO GOAL PRG RAtvlMING INT RODLJCTION CHAPTES_:--III COI{CEPT L5 3.1 THE GOAL PROGRAI{MING GOAL 3.2 OBJECTIVE zuNCTlON IN t6 PRGRAI/tlvtING OF MULTIPLE 3.3 RAIIKING AllD WEIC+{ING !6 CSALS cHAPTE_&_:--IV CoALPR0GMJ{I4INGAsAMATI{EIVIATICAL 18 TOOL USED 18 MODEL 4.1 GBERAL MATTIEMATICAL 4.2 STEPSoFTHESIMPLEXMETHoDoFG0AL t9 PROGRAMIVIING OF @AL CCI!1zuTERBASED SOLUTION 22 PRGRA[[MIIIG 2t 4.4 AI{ALYSIS OF THE COIPUTER0JTPUT - CHAPTER v FOII},ATJLATION PROBLE}/ OF 26 5.1 GEIIERAL 2E 5.2 PRroRrrY ( r) ( rr) 1t 5.3 PRToRTTY J 5.4 P r l r O R t t Yr r r ) ( 58 PRToRITY rv ) ( 5.5 39 5.6 CChISTRATNTS 59 5.6-1 Productive hours constralnt q1 5 .6.2 6vertime C o ns t r a i n t ) DI SCUSSION Of-- RESULT @ 8 APPEIDIX b2 ES REFERET.JC -trOO-
  6. 6. LI ST OI. NOTA I ONSb. GoaI set bY decision maker. 1 The cost for overtime hour.ci Standard variable co st of prodttcing one unit of product i.c? C os t i n c u m e d for cauying one unit of product i.,-10 Cost incurred for one unit of product ibackordered"i per peri-od. +Dit Finished goods inventory of pnrduct i in period t;Dit Backorder quantity of product i in period t. +Dzt ,- Nunber of workers in excess of the desired maximum.Dit Number of workexs less than the desired maximum. +DotD6t Deviational variabJes. +DztDzt Deviation aI variable s.rt In ven to ry at th e en d of t th Period.Tt +t In ven to ry during t th Peri o d.^tT- Shortage during t Ur Peri-od.rt-1 - Inventory at the end of (t-t)Ul perioci.k Numl:erof priori ties.
  7. 7. Nurnber of goals.M Number of decision varlablesn Overtime hours in Period tot produc tion rat6 f or ith type of motor duringPit tth period (aecision variable) forPj The ple-emPtlve weiqht i Ievel for pnoduction rate co stsPnct Managenren target tPt Productionrateduringttl:Iperiod M a x i m u r nd e s i r e d c h a n g e i n w or k f o r c e } e v e l Qt st Sales in t tjr Period for one unit of motir i Ti Hours required Efficiency coeff icient for old work€rso T1 Efficlency coefflcient for n e u rw o r k e l s . T2 Efficiency coefficient during ovel time hours T3 during t th peri-od. vlt size of work force ( t-t ) trt period. vtt- t size o f w or k f o r c e d u r i n g D e ci s i o n v a r i a b l e t o be found xj xt ChangeinthenumberofworkersinperiodIt. -O(rO- - v1-
  8. 8. t; _A_B_S T R A C T In this dissertation an attempt has been made to anaryse the aggregate production pranning of ABc ( tne actual, nane has been disguised) optimally. T h e d e n r a n do f the nrotors with diff erent specificaticns vrere not constant ciuring the pranning horizon of one year i.e. lgg8-89, conslsting of three planning perlods. To meet the fluctu- ation in d e r n a n da g g r e g a t e p l a n n i n g model wBs formulated, wttich conc en trates on determi-nin g which cornblnation of t1.re clecision varjables like production rate, inventory, back- ordering, o vertime etc. should be utilised in order t,o optirnally adj ust th e dernand f Luctuations within the con straints if "ny-. The aggregate planning model was formulated in the form of goals with different priorities. The problem was tii en soL.ied by usinc{ Computerized technique of S.[:, Lee to soire the goal proqraruning problemst. Tne decision variabLes ltereobtalned for arr the planrring periods. -OoO- -vi-- $-i.i"tt, r.,$ s $
  9. 9. ffi 1? CHAPTER - - - F I INTRODUCTION - to plan and con trol operatlon s at llo st managers want level thmugh some klnd of agglegate plannlng tJre broadest of lndividuar products and detaired that by passes detalrs sch edr.rrlng of f ac irlties and personn el. Managernent wourd deal wlur baslc relevant decisions of programmlng the use of resources. This is accomplished by revlevrlng pnoiected ancl by settlng activlty rates that can be emplolm€rrt ievels wlth ln a glven ernproyment rever by varylng hours worked- varied decisions have been made for the firce these baslc upcomlng period, detailed schedulinE can ploceed at a lowel the broad pran. Finalry ra st Iever wittr ln the con strain ts of activlty levels need to be made with the minute changes ln realisation of thelr possible effects on the cost of changing production level and on inventory co sts if th ey are a part of th e sy st,em.
  10. 10. I ! .l @ 2 i 1.1 AC€ EC"ATEPROIqIION PLAI{NING GENERAL FORM The aggregate prodtrctlon plannlng pmblem tn lts most general form can be stated as follows z A set of forecasts of denrandfor each period 1s glven - (a) The size of work force Tlt ( b) The rate of Production Pt (c) The quantitY striPPed St The resultlng |n ventory per monti can be determln ed as follows - rt It_t +Pt- St. The Problsn is usually tesolved analytically by mininizing th e exp ec ted total cost ovel a given plannlng horizon conslsting of some o r all of tfr e f o lloning co st component s.g.,}.j :$,.$ (a) The cost of regular pay-roIl anci over-time-rrfis (r ) Th e co st of ch anglng tJr e p ro duc tion rate f rom$* one period to tJre next. inventotY..,ry (c ) The cost of carrYing#r IP (o) Co st of, sho rtag e s re su I tlng f rom no t meeti.ngfr,# :l$ ri, th e dernanci. # i,! ".! I i :i Th e soluiion to tli e p robl sn i s simpl if ied lf a verage ir d e r n a n co v e r i the planrring horizon is expected t,o be constant.
  11. 11. 3 So th e cornplexity ln tfr e aggregate pro chrc tion plannlngppoblem arlses frrrm the fact that ln most sltrrations demandper period i s not constant but are subj ected to substantlalf 1uctuatiop s. The question arises as to how tfrese f luctuationsshould be absorbed. Assuming tjr at th ere ar€ no pr,oblem lnrecelvlng a constant supply of raw material and labour at af lx ed vjage rate , th e problen may be seen by considering ttr esepure alternatlves of responding to such fluctuations.(a) A increase in orders is met by hiring and a decrease ln orders is accompllshed by lay-offs. (b) Mai6tenance of constant work force, adjustlng production rate to orders by wo rking o vertinre or undertime acco rdingly . (c ) Maintenance of a c o n s t a n t v l o r k f o r c e a nd c o n s t a n t tro duc tion rate, dllor^ring inventorie s and order bac klog s to fluctuate. ( d) Mainten anc e of con stan t wo rk f orc e and meet th e f luc tu- a tion in dern ci th ro ugh p I ann ed b ac k log s o r* by subcon t- an ra ting d. exc e s s dernan In gmera] none of t.|re above alternatives will prove best but some cornbination of then can cio. Order f.luctuations showed in g eneral be ab so rbed partly by in vento ry , partly by o vertirre and partly by hiring and layof f s anci the optimum ernphasis on the se f actcrs wiII d e p e n c lu p o n t h e c o s t s i n a n y p a r t i c u l a r f acto ly.
  12. 12. . l 4It 1.2 STRUCTURE SIIV1PLEST OFjTSGREGATEPLAIININ9 PROBL4I The structure of the aggregate planning problem ls represented by the single stage sy stqn 1; e; the plannlng horlzon ls only one perlod ahead. The stage of the system at the end of period ls def in ed by Ho, Po and Io , the aggre- gate work f orce si zer prcduction ox activity rate and inven- tory level respectively. The ending state conditions become the initj.al condition s for the upcoming period. Wehave a forecast of the requirements for the upcoming period through s o m ep r c c e s s . The decision mademay call for hiring or laylng of f personnel, tJrus expanding or contracting the ef f ectlve capacity of tJre pro duction systern. The work force size together wi th th e ciec slon on ac tivlty i rate during th e perlod th en deter- min es th e *requi red amount of o vertiffi€ r in ventory level s or back orderlng whether or not a shift must be addedor deleted and other posslble changes ln operatlng pmcedure. 1 .3 .PLAI.ININGSYSTEMS MULTISTAGE AGGREGATE In this type of planning system, our obj ectlve ls to make the declsions concerning the work force slze and production rate f or the upcoming periods. In doing so, howeverr w€ conslder the sequence of projected decisions in relation to forecasts and their cost effects. The decislon for the upcorning period is to be affected by the future period forecasts and the decision
  13. 13. 5 I t j r e s e q u e n c eo f process must consider the cost effects of decisrons. The connecting rlnks between the severar stages at the end of one p.eriod are the lrfr P and I Values tJrat are and the beglnning of the next. The feedback roop frorn tjre proc edure to obtain decision process may invorve some lterative be a sotutloD. The sequential nature of tjre decislons should or wxong onry in terms kept in mind. Arr decisions are right a period of time o f t h e s e q u e n c eo f d e c i s i o n s o v e r -OO0-jII Ia1IttI
  14. 14. .l t; g_u.a8.tgE.- Ir LITERATURE REVIEN - t-^- , The pro duc tion planning problenr i s conc erned with the optimal quantlties to be prcduced in order spec if ying a sp ec if ied planning hori zon. Many model s t to meet denand for lll pros and cons, have been deveroped to each of which has its help to solve thls Ploblem in the llterature differ ln These rnodels introduced and methodorogy. Howevert their orientatiorl r scope, contents we can classify these models ln two maln catagorles ciescrjPtlve and normative 2.1 pEqpRrPTrVE MODELS models aim of describing the plocess by Descriptlve .in practic e. The maln example whichr procluction are determined of such mode} s are z 2.1 .1 T h e M a n a g e r n e n tC o e f f i c i e n t Model / 1/ intro clr.rc by Boran ( 1 963 ) and exten ded by Kumren ed Ther ( t loo; , this mocier assumes th at manager behave ef f icientry to rec ent d.r average, but suf f er f rom in--con si stency and biases regression is used to develop decision rules events. Linear and rrork force decisrons utiriz.ing inde- for acr,uar production such as past sales arrcirogged prociurction r)endtnt variables
  15. 15. This model is very floclble inLnvento rY , & d w o r k f o r c e ; to a particular functional behaviour ofbeing not restrlctedthe cost elements involved. th e p r c c e d u r e i s the essentiallY A s eriou s drawbac k of of t h e f o r m of tjr e rule.subj ective selection ( 1966f2,1 .2 Trre sequential Model of C€rdon Themainideaofthlsmodellstopxoceedinsequence rarge of inventory t startlng f rom a prespec if led acc ep tabre andsetaccordlnglytjneline-shiftlevelsofwork-folce.Thus to the range of lnventory deviatlon from adjust tJrese according deviation s occur too f requently, tien lts permi ssj.ble range. rf inventory ranges are subject to adjustrnent- the acceptabre lever 2.1 .3 Siriulation wro els d out ln ttrls fierd using F;terrsive work has been carried stati stlc al tlc aI apprc ach e s lnc rudlng an d matjr erna dif f erent MonteCar}osampllng,andcomputerana}ogue.Inthismode} ( 1966) , th e simuration starts with a 1n troduc ed by Virgln exper5,ence of ttre form and productlon pran based on tJre past emproyment rever r ov€xtimet then changes are introduced ln untir a minirrrun local lnventorles, sub_Contracting and so fc,rth, opetatlng cost is achieved. Otjrer simulatlon models in ttris qzo) and by Naylor t and sisson (t regard are de.ieloped by Enshof
  16. 16. B(tqZf ) using both discrete and contlnuous events slmul-ation.An important f eature of simulation 1s that stochastlc demandpattern can be incorporated in t-he model. This permlts theanalysis of the forecast error on strategy development.2.2 E NORT4ATIV MOELS T h e c o m m o nf o c u s in normative models is on what pmductionplanners should do. Mode1s of this category are further classi-fled into classes;2;2.1 Aqqreqate Plannlnq llodels Th ei r common o bj ec tlve i s to determin e th e op timalprodtrction quantity to prcduce and work force level to use inaggtegate for t}le next planning hori zon. l{ocie}s J.n this cla ssare elthJr exact or heuristlc.2.2. 1.1 6xact ,Models : Transportation method fo unulatlon ofBowan ( t gSO) / 1/ propo sed the di stribution model of linearprcgrarnming for aggregate planning. thl s model f ocussed on theobjectlve of assigning units of productive capacity so thatproduction plus sto rage co sts were minimi sed and sales denandwas rnet witi in the con straints of avaiiable capaclty. Thismodel does not account for prodrction charge co st s. Such ashiring and layoff of personnel , and tirere is not cost penalty f or back ordering or l - os t sales.
  17. 17. aw it ,.] The simplex method of linear prcgranming makes it posslble to include prod,rction level . Change costs and in vento ry shortage co sts in the model . Han ssrnan and Hess /2/ developed a simplex rnodel using work fo rc e and production rate as independent decision variables and in terms of the components of the costs model. AII cost functions axe considered linear. : I!.l I One of the baslc weakness of llnear progranrmi-ng approaches I IIII and most other aggregate planning technique is the assumptlon ofII determlnl stlc dernand. Anoth er sho rt coming of th e lin eat prograrnmj,ng model is the requirement of linear co st f unction s. However, tJre posslbility of plee wi se llnearity lmproves tJre validity. HoIt llodigliani and Simon /3/ gave tfre weII known rnodel in which tiey minimi se a quadratic co st function and come up with a llnear decision rule that solves for optimal aggregate pro duc tion rate and wo rk f orc e si ze f or aII tJre periods ovell the planning horizon. L.D.R. hasnany advantages. First the model 1 s op tiroi zing an d th e two dec i sion nrl es onc e derl ved are simple to apply. In addi tion tlr e model is dynamic and representative of the multistage klnd of system. But quadratic cost structure may have severe limitation and probably does not adequately represent the co st struc ture of ally organizatlon. Bergstrom and Snith / 4/ extended the capabillties of the L.l).R. lrtodel in two n6rJ directions. Becauseof the
  18. 18. 3 . 1 . . frrnI l0 rI I ir * aggregate nature of L.D. R. it is not po ssible to solve directly lc l: I t for the optfunum prod.,rction rates for indivldual pxockrcts. The development and application of the M.D.R. model suggests that it ls now operationally feasible to temove tJre requirement of an aggxegate production dimension in planning models. FurtherTnore, given the avail-ability of revenue curves for each product in each time period the M.D.R. model can deter- mlne optlrnal prcduction, sa1es, inventory a n d w or k - f o r c e level s so a s to maximi se prof 1t over a spec if ied time horl zorro Larvrenc e and Burbridge /5/ presented a multiple goal Iin ear programming mociel consldering commonly occurl-ng goals of the firm in coordinating prcductj-on and logistic planning. The solutlon technique fo r thi s model will b e a c o m p u t e r j -z e d m u l t i p l e obj ectivq. analogue of th e revi sed si.mplex method. Coodnan /6/ presented goaJ. prograniming apploach to solve non-Ilnear aggregate planning models. If actual costs (niring and firing co st, overtime and idletime, lnventory and shortage cost) can not be satisfactorily represented quadrati- c al l; , th en th e so lu tlon b ecomes more compl ex . One app ro ach to hanCle these mote contplex rnociels is to atternpt formulation of an approx j,rnati-ng linear model to the original non-llnear co st terms and to apply some variate of the siml:Iex metliod. This appro ach offers the net acivantage of at Least providing an optinral solution tc tJre nroieJ used ano is based upon tf,e goal prograr:rring.
  19. 19. ll .1 propo ses a linear pmgtarffning Tang and Abdulbhan /7 / aggregate prodtrctron pranning pnoblem ln thefo rmuration of heavy manufacturing lndustry A baslc model 1scontext of co st of p ro duc tion wh lchf i r st deverop ed to mln imi se th e to tal llnear. the baslc model ls thenis assumed to be piece-ryise a llneat proglamming model to seek an optlrnaltransf erred lnto a series of pranning periods witJrln tlr e planningsolution f orho rl zon . Jaa skalain€ss r V /B/ has propo seci a go al prcgrammingmodel for the sch eduling of produc tion , employment and lnvento- requirement ovex a f inite time rl es to satl sf y known demand or separate ard lncomplete goars, hori_Zo.. Thls model sets three the level of, prcduction, errrployment and inventorles; formulated a rnulti-objective Thornas and HlIl /9/ p r o d t r ct i o n pranning moder as a goar pxogram which capitarlzes goar-prograrnming ln incorporating rnurtipre on the strength of into the anarysis. Thls paper lncrudes economic considerati.ons aspectsr ignored by cco&nan /6/ and Jaakelalnen /B/ the has attempted to plovlde a Jarnes, P. Ignizio /1o/ very n 6^ f ield of go al p rogrammlng brlef bcok at th e reratl struc tu re As such th e gen eral rm der e p I e-{5npti ve p rio ri ty is viewed as a pxactical goal- prcgrarruning model presented naturar rerrresentation of a wide variety rearlstic and rather of many real world Problems
  20. 20. T2 2.2.1 .2 Heuristic Mo el s: d (a) The production parametric planning model by Jones ( tgZS): This model assumes tjre exl stence of two basic decision nrles addressing work force anci pxoduction levels respec- tively, each of which is expressed as a weighted s-trm f o rates required to meet future sales drrring the planning1l.,1 I ho ri zoo . I I t (b) A switrh rule prcpo sed by Elmaleh and Eiton (tgt +) z They specify three inventory leve1 s and three prc cLrction levels to be obtained by various combination of control parameters over a historical demand series.and chooslng th e set f or wh ich pro dr.rction i s limited to discrete level s such as food and chenricalsi -O OO-
  21. 21. Si l;l q.H.&P-TEE ur INTROqJCTION T9 GOAL PROGRATTTTING vary according to the charac- organisational objectives philosophy of management md particular teristics, types, the organization There is no single conditlons of envlronmental univelsalgoalforallotganizations.Intodaytsdynamicbusl- put great €rnphasis on ocial xesponsjbi- ness errvlronment firms public relations and indurstrial Iities, social contributions, relatlons etc and labour Ifwegranttjratmanagenerrthasmultiplcconf}icting dec i sion c riteria shourd a} so be mul ti - ob j ec t1 ve s to ach 1e ve t]r e that whsr a deci sion invorves multiple dimen sioqar . This impries multiple shourd be capabre of handling goals the technique used technique has a decision criterla The linear programming invorvlng multipre goalsi limlted varue for problems Theprimarydifficu}tywithlinearprcgrammingisnotits lnabllitytoreflectcomplexreality.Ratheritllesinthe which requires cost the obj ective f unction unidimen sj.onarlty of to obtain that is of ten armo st impo ssibre or prof it info rmation of the obj ective f unction un idimen sionarity To o vercome ur e Iequiredinthelinearprogranulingeffortshavebeenmadeto convertvariousgealsrcostsor-valuemeasureintoonecriterion * * * * ft,.,.*.il*
  22. 22. ,:,| l4 namely utllltY. Howeverr €Xact rneasurement f utllity o is not slmple. So decislon making tirough llnear programrning via a utittty function is only feasible 1n theoretical sense. Croal pxogramming i s a modif ic atlon and extm sion of Ilnear pDograrnming. The goal programmlng approach ls a tech- nlque that is capable of handling decislon problems that deal wlth a single goal witjr multlple s u b g o a l s r E s w e I I a s r p r o b l e ms with multiple goals wlth multiple subgoals. We can soJve these problems using llnear programming wlth multiple obj ectj.ves. We may lntroduce other obj ectlve f unc tion s a s model con stra int s . But tJr1s model require s th at the optlrnal solutlon must sati sfy alI constraints. Furtherrnore, 1t is assumed tJrat equal importance is attached to various obJectives. However, such assumption are absurd. It 1s quite po ssible that all the constraints of the problem can not be satisfled. Such a problsn is called infeasibLe. Secondly aII constraints Co not have equal importance. Therefore goal programming which rsnoves all such difflcultles is used to solve such ProbI€fns.
  23. 23. |:l a l5 : t, 3.1 CONICEPT THE GOAL PROGRATIMING rec eiving much attention a s a powel- croaI prcgramming ls multi-objective decision maklng probrern. ful toor for analysing introduced by A- charnes The concept of goal prcgranrning was flrst to resorve infeaslble linear prcgraurming and lt.lt..cooper as a tool reflned by Y. rjlrr and probrerns. Thls technique has been further tre popurarity of GP s.Mi Lee and ot^ers. The maln reason of sumstobeassociatedwithtJreawarenessofthemanagernentscience orientation towards multl-goal or techniques and very natural and uses The goals set by the multi-obj ective formulation only at the expense of otier management are often achlevable goals are in commensurabrei-€. they goars. Furt,reqnor€r these unit Scare. Thus there is a need cannot be measured on tJre sane conf lic ting for establlshing a hlerarchy of lmportance among tjrese - orly after the goals are considered. goars so that row order goars are satisfied or have reached the higher orders priority improvenrent is deslrabre- Hence point beyond which no further by goar programming lf the managem the problem can be solved th eir ranklng of the goals in tenms of can prCIvide tJre ordinal o f t h e m o c i e l. r t i s n o t a l w a y s importance and all rerationship goal f urry to the extent desired po ssible to achieve th e every or without programmihg r tJ.Ie managel by managernent. Thu s with attachesacertalnprioritytptieachieverrrentofaparticular goal proglarnmint] is therefore lles goal. The tnre value of j-nvorving multiple conflicting goals in the sorution of probrerns*isx{;x
  24. 24. I*-1l EN 1 I.: l{; I 1 I I I t I acco rdlng to tJr e Manager I s priori I ty struc ture. i 3.2 QBJECTIVE zutCTIOt{ IN GOA! PRCMI4IIING In goal programming lnstead of trylng to maxirnise or minlnise the objective criterion directly a s in lin ea r p ro g rarnm-i lng r lt trie s to min imi se th e devi a tion s ariong the go als wi tJr in I t I the given sets of constraints. The obj ective func tion i s tJr e I I t t minlmisati.on of these deviations b a s e d o n t h e relative impo rt,arrc e I I t i or priority assigred to them. 3.3 RANKTNG Arlp_nEIcHfINq_oF_wI.TIpLE coALs In order to achieve the ordinal solutlon that i s to achieve the goals according to th eir importance negative or posltlve deviations about the gcal must be rarrked according to f tpre-€niptivet pr5-orlty factors. rn this way the row-order goals are considered only after hiqher-order goals are achleved Bs desired. The pre-entptive priority f actors have the relation ship of Pi)))Pi JJ +1 which lmplies that the multiplicatlon of n however rarge it may be cannot make pj greater than or equar to p5. *t The next step to be con sidered in t h e g oa l p r o E r a m m i n g is the weighing of deviational variables at the sane priority Level. It any goal invo.Ives many deviationa-l- variables and we want to give priority to one over the other. This can be achi-eved by assigning different weights to these deviationaL variables a t t h e s a n r ep r i o r i t y - l - e v e L . A t t h e s a r n ep r l o r i t y l-evel
  25. 25. I7 1 t h e s u b g o a l w t r i c h a c q u i r e s m a x i m u md i f f e r e n t i a l weight wiII besatisfied flrst and then it qo to next. The criterla fordetermining t|.re different weights of deviatlonal variable couldbe the minimization of opportunity cost. Therefor€r devlationalvarlables o n t h e s a m ep r l o r i t y level must be commensurable,aldrough deviation s that are on tfre dif f erent prlorlty level sneed not be commensurable. -OOO-
  26. 26. tfr 9.U.AP_IEE- IV- GOAL PROGMI4MING A MATHEMATICAL AS TOOLUSED 4.1 MODEL GENERALI4ATHEh4ATICAI. The goal prograrnming was originally proposed by Charnes and Cooper f or a lln ear model which has been f urther developed by many others. A preferred solutlon is one which minimises the deviations from the set goals. Thus a simple llnear goal progranr.ning probl em f ormulation i s sfrolvn belovr z k (o- + *) lvlin imi z e : Pj d. ] j=1 n Subj ec t to : b. for 1 = 1....ID. 1 j=1 *J,or*, dr-V o foralliandj + wh ere d. x d.- 11 x. Decision variable to be found J k Nurnberof prioriti es n N u r n b e ro f decision variables m Number of goal s l^ Goal set by the decision maker ]- DJ . . The pre-anptive weights such that P >>> nj +r r ! It?G*IE)lfif;#
  27. 27. fr i l ll) In addition to setting goals for the obj ectives, the decisicn maker must also be able to give an ordj,nal ranking to the obj ectives. The ranking can aJso be f oundout by paired comparison method which prcvides some check on tJre consistency in the value judgenrent of the decision maker. In g^ris method the decision maker is asked to compare the goars taken two at a timeand indicate which goal is the more important in the paj-r. Thisprocedure is applied to al.r combinations of goar pairs. Thisanalysis results in a complete ordinaL ranking o f , . _t h e g o a l s 1 nt errns o f th eir impo r tanc e . The go al prog rannmin util g i ses th e simplex method ofso Jving Iin ear prog ramming plcoblern. Horr.ever several modif r ic ation sa r e r e q u i r e d a n c i i s o f t e n r e f e r r e d a s fr n o d i f i e d s i m p l e x method| .4.2 SIF.PS OF TILE SIUPLE(-UFTHOD OF GOAL PROGRAIIMII.JGStep - 1 set up th e ini tial table f rrrm goa-r programming f ormuratj.on.We assume that the initia] solution is at origin. Therefore, alrthe negative deviationaf var:-abLes in tf,re modeL constrain t mustenter the solution base initially prepare a table a s s f r o w nb e l o w .Firr up this table i.e. all arj and bi values. The cj corumn willcontain ttr€ coefficient of deviational" variabJe because thesevarjables onJ.y enter tl-re solution fj.rst. In il^re (rj a:) matrixl-ist tl,e priority .Ievel in l j r e v a r l a b L e c o J u m n f r o m . L o l v e s ta t t h etop of the hicyhest at tfre bottom. C a l - c u r L a t et f r e , j values and
  28. 28. 2f,lreco rd i t in to RFISco lumn .cj Variable R .H . S . d;.. . oi" xj..a bi cijZ. cj P5 J P4 P.., J P2 P1Step-2l. Determin e th e Nerv D:lterlnq Varl_ab]g Find th e high est priority Jevel, that has no t been attain edcompletely b y e x a m i n i n gJ Z , J values in the R.li.5. column. Afterdete rrnj-n in g t j r i s fi-nd out the highest Z. JJ Ci entry column. Thevariable of t h i s c o l u r n n w i 1 1 e nt e r t h e s ol u t i o n b as e i n th e nex t i tera tion . In c a se or ti e, c l :e c k t h e n e x t prio ri ty level and sef ect tt^,e coluntr that has the greater value.
  29. 29. F l.- ?l yariable from the Solution Base ltep-3: Determine tne leavin D i v i d e t h e R . H .S . v a l u e s b y t h e c o e f f l c i e n t s in the keY column. This will g i v e t h e n q i l F [ . H .S . v a l u e s . Select the q)r, w h i c h h a s t h e m i n i m u mn o n - n e g a t i v e v a l u e . The variable in that column ln the row wlll be replaced by the varj,able ln the key If tjrere exists a tie, find the row that has the next iteration. variable with the higher priority factor. In tnis way tlre higher order goals will be attained first and thereby reduces the nunber of iteration s. Step 4 2 D ete rmin e th e Nsr Solu tion - f ind the net, R.H.S. values and coef f icient of the key First old values by the pivot elsnent i. e. the element row by dividing at the infersec tion of the key row anci key column. Then f ind the ne$, varues for alr otjrer rov"s by using calculation. (oro varue ( intersectional eI snen t of that row X Nerrvvalue in the the same column)). Norv compLete the table by flnding tj key row in and ,j Cj values for the PrioritY rolvs Determin e wh etn er So]ution i s tirnal or Not ? Step-5: Analyse t1re goal attainment fevel of each goal by cttecking rovJ If th e Z: value s are al-I zero th e Z: v a l uJ se f vo r e- a- c h p r i o r i t y . - - Y - - | J J is a optimal solution tjrere are positi ve (2. Therr if tj) this J (2, valu e s in th e rov,r d€termin e wh eth er th ere ale n e g a t i v e , J tj),ti
  30. 30. 2,) avalues at higher priority l . e v e l i n t t r e s d m ec o l u m n . If thereis negative (zj a: ) value at a higher priority revel for thepositive (z: a-:) value in the row of interest then the solutionis opt5-maI. Finally if there exists a positive (Z; C*) value J Jat a certain priority level and there is no negative (Z; C* ) JJva lu e at a h igh er priority Jevel in th e sarne co rumn , tJrJ.s is no tan optimal solution. H e n ce r e t u r n to step 2 and continue.4.3 COI/IRTER B45ED SOLUTION OF GOAL 88etr8At4tu1ING rn order for g o a r p r o g r a m m i n g t o b e a u s e f u l mdnagernen tscience techni-que for decision analysis, a c o m l - r u t e rb a s e d s o l u t i o n1s an essential requiremento After suitabre m od i f i c a t i o n s the computer based solutionproc edure of goal progranrming presented by Lee can be u sed tosorve problems- The prccess of finding computer sorution conslstsof data input, calcul-ating the resul-ts and printing out the results.DATA INP9T First of all the fol,Iorving data is to be fed tothe computer through the key board PROB NROWS IWAR NPRT Th en input i s th e di rec tlon of unc ertain ty B for B ot h direc tion s L for Less than E for Exactly equal G f or Grea ter tfr srr
  31. 31. f 2:l then tJre gbjective function ln input is given in the followlng manner. devi atlon row in whlch p rio rity wei ght -ve/lve dev. occurs Then the d a t a a b o u t t e c h n o l o g i c a l coefficient of the choice variable is entered lik e Row ln wh ic h Colurnn ln which Value of appeared apPeared tiJ "tj "tj Then the rlght hand side value of aI] the eqns. are e nt e r e d . 4.4 AI{ALYSI S OF THE COMRJIER OUTRJT Computer solution of goal programming pllovides the following outPut - Computer print out of input data (tne right hand slcie, rates, and tjre objective function) and final the substitution solution tabl-e ( inc luding tj Cj matrix a nd e v a l u a t i o n simplex f unction) , slack anallrsis, varlable analysis and of obj ective the anal.ysis of the objective. I j I 1 I+2.t !
  32. 32. 24 TliE I-rvL SIMPLEXSOLUTION (a) The Riqht Hand side Thls shows the right hand side varues of the variabre (d evi a tion a 1 a n d d e c i s i o n T h e n u m b e r s o n . t h e r e f t h a nd s l d e ). I i are vari able numbers for the basica l it I varlabres. The real values i on th e righ t h a n d s i d e r e p r e s e n t c o n s t a n t s I I of the basi,c variabres. I ( n) rh e (rj_jt Matrix This shows the (Z: cj ) *" trix o f th e la st i, tera tion . (c ) This evar.uation simpry represents the tj values of goals. rn othur-*ords, the values present the r"der attalned portion of goal g. (d) The Slr:ck Anal-vsis RL}{ AVAILABL E POS- SLK N EG-g.K It presents the values of the right hand side and aJ so value of ttre negative anci po s i ti ve vari able s fo r each equation. ( u) Variabl_e Ana]ysls VARIABLL /tioLilJT
  33. 33. 2{t It presents the constants of only the basic choic evarl abl es.(f) Analvsls of the Obiective It presents the t j values for the goals. These valuesrepresent the under attained portion of goaI5. PRIORITY UNDERrcHIEVEIJIENT
  34. 34. |* 2$ 9.U-AP_ ER T V- FORMTULATIONOF T H E PROBL E4 5.1 GENERAL1 l1 { ABC Company produces the motors of several kinds which I I I I I differ from each other in severaL aspects like frame size, horse I l :l t I I povJerr R.P.lvlo, nurnber of poles etc. It forecasted the demandof total horse power, to be produced for the year 19BB-89. Manage- m e nt e s t i m a t e d a cumulative grovrth of 15% in the demand of horse povrer. The demand e.f horse power wds dif f erent for every period (four months). Hence an atternpt is made to meet tjre demand for every perioci in an optimal way con sidering production rat€, inventory., back ordering, overtime etc. This also had the demand record of every type of motor (:-n numbers) for the year l gBB-89 - gi ven in Appendix ( table 1). ttith th e knowledge of the Last year r e c o r c i , t h e d e r n a n df o r every kind of motor j-s assessed quarterly for the complete year 19BB-89 (nppendix Table 2). An attempt is also made to meet rvith the ffuctuations in demandfor every kind of motor in an optimal way. For each f ranre size, there were f urt-|er many klnds of motors with dif f eren t specif ications. Therefore, only tt:e representative member of each frarre size was consicereci. The types of motor vrere still too many to make tne problern as a wnole very large to be deal-t with. Hence th ose type of motor v;hich dici not s f r o wm u c h v a r i a t i o n s in their machini.g
  35. 35. j,tl I 27 I times were cJubed together r€drcnably. It was realised that this problenr can be solved by making aggregate planning mode.1 w h i c h c o n c e nt r a t e s on determining rrrhich combination of th e decision variable should be utilized in order to optimally adjust t h e d e r n a n df l u c t u a t i o n s within tfre constraints if doy. M a n a g e m e r r to f the company also desired to incorporate other re-levant aspects such as possibly stable employment for the workers m a n a g e m e n tp o l i c i e s or goals relative to inventory a nd w o r k e r s a t i s f a c t l o n a nd p e r f o r m a n c e . T he s e a r e a l s o incorporated in the problsn formuLation. The overall cost func tion wa s segregated in to maj o r components i . e. pro duc tion rate cost and irr ventory co sts so that m a n a g e m e n c r l t - rl - . a v e a c t d i t i o n a l t flexibil it; in penali zLng deviation s from the various types of co st s. The mocie] optinizes the aggregate production variable ds well as detennining the opt,irnal procuction rate. The cornplete probfsn 1s formulated in the form of goal.s anci is uren sol-ved by u s i n g c o r n p r u t e rb a s e d s o l u t i o n tecl:nique of g oa f p r o g r a m m i n g /12/ . The followirrg goals are incorporated in the problcrn 1n order or priori ty l ( a) S a Je s r e a J i s a t : . o r r ( b) To lir::iL the cost associated witit production rate to a sp ec .i f i c,ci srirc L. rlh (c) T o I i ; : l t t t l r e c o st ? s s o c i a t e d !rrtir irrven tory _l-evels Lo a sFiec f ierJ ar!rorjn i t. ( d) [c p . r r o m o t e . i , c . r - ] l e r S f r o Lv a t i o n r j tf rrc;t,rghLaiX)r for.ce stal,j.J.1ty. l { I i,]a ;
  36. 36. 2B 5 .2 ( PRT.ORITYI SALES REALISATII}.I E q n . ( t ) rep re sen t s a gen eraL rel. a tion sh ip . rt-r +Pt = st +rt .... (r) where rt-t = rnventory at the end of t-r tf, period rt = lnventory at the errd of t t,l period Pt = pqr duc tion rate during t th period st = Sales in t tn period. Let (t . L)/ Inventory during t th period ( r. ) Srortage during t th period The + and sign above the parantJreses mean that ^- the quantities inside the parantheses can have onJ_y * or ve values respec tively. B y u s i n g tran sfo rrnat ion ..Let + = a lal a 77 O = 0 otherwise a la l a O othenvise + =aTlt en aaT he r e f o r e .t+ *t 1t 1t (:)and Tt+ - It-r 1 It-r (:)
  37. 37. 2lf For convenience, Iet u s put tr* = oa* rt- Dt- and -t rl -1 = oJ-t rLr oa-t E q ns . (2) a nd ( g ) c a n b e r e w r i t t e n ds oa* - Dt- = rt . ... (q) oi-l - ot-l = rt-r .... (s) F r o m e q n s . ( 1) , (4) and (S) Pt = st+(oJ-o.)-(oJ_, DLr) .... (6) T-=T= -r.! L-tI l- o (oJ-, D+ 1) Zeto (z) Frorn (6) and (z) p1 = (q* Di) +s1 (B) e Pz= Iz + 52 It From (+) ancJ (s) Pz . . ( g) Frorn (B) and ( e) Fz+ Fr Y1 I (oJ- q) +(s, +sr) ( 1c ) , ..1.;,i*.,".il. E* ,3
  38. 38. ;i0 pg = 13 *S3 12 Fmm ( q) and (s) Pg = (oa*- D;) *s3 (D; - D;) "" (tt 1 From (t o) and (il 1 P, + P ^ + p-^ 3 = (oa+ D ; ) + s g + s z * I z *s1 . ... (lz1 Thus for each type of motor there are tJrree eqnso 8, 1 0 and 12 for tJrree planning perlods respectively. For F;<arnple z Type A motor PRt = D R t+ + Dnt = sRt aaoa (t:1 PAt + Paz t; uA2 J- r set + sez aaaa ( 14) PRt + Fez + Prc n- rLJ sRt +sRz +sag o... (t:1Type B motor Pgt - ofi r ou = sut (te1 .... Ptt + Psz i, +ou, = s g t + sez .... (1?) P n t + P,3z + Pa: ui + D,r: Sst +sirz +seg ..o (ts1T y pe C m o t o r Pct ,-+ trl ua., sn1 z l .... (1a)
  39. 39. 3t 1--) Pct n Pc2 ot, + D^^ vz = sct + scz .... (zo1 Pct * Pc2 * Pca tJ. * Dfs sct + scz * sca .... .2l1 Type D motor o? ojt + fo1 = sot . . .. (zz7 Pot * Pp oJ, + Db * so2 o.. o (zs; not + P m + pD 3 oi spt + s P + sog . ... Q+1 Type E motor PEt tJt {r sgt . ... (zs; + PEt P-^ Dez + D a set * sE2 ,,... (2a1 P-. +P-^ +P,-^ ^+ E3 + DE: = set * sE2 * sE3 Et cz tr,J .... Ql7 Simiiar t:,pe of e q ns . c a n b e w r i t t e n for F, G, H, I & Jt y p e o f m o t o r s a n d w e r e gi ven th e ecn s. number f rom (ZA to 42) .5.3 pRrontry( rr rTO LJIII_Irr{E cosr (r ASSOCIATED WITH PRODUCTION RATE Pit x ci * cTot + Dot Jt = PRct .... (+s1wh ere a Standard variable cost p ro cfuc 1 of ing on e unit of product I The cost per overtime hour "l . h l a n a g e m e n tI s t a r g e t Je.veJ for prochrction RCt rate costs.
  40. 40. J]? DJt DZt Deviation al vari ables Pit Prodrction rate for ith type of motor during t th period (Oecision variable) ot Overtime hours in period t In the piesent problen, idl e time vva n o t a 1 l o w e d . s The cost for producing one unit of every type of motor is given in Appendix (tante 5). The eqn. (+e; for three planning periods can bewritten as follow5 ,For t = 1 11€2 Pat 3553 Pet 662C Pct l OZl q pOt . 12675 PEt 16533 Pr 2443t Oo1 3 0 e 11 P H r 4 6 80 0 p l t r 7 A2cO p-, t Bot + DZr _+ = 61 24266000 ,,... (qq)For t- ) 1 4 8 2 P , e . + 3 5 5 3 P e z + 6 6 2 0 Pcz + 1021 4 Poz + 12675 PE2 + 16 5 3 3 P r z + 2 4 4 3 1 P o r + 3 O g 1 PH2* 46800 1 PtZ + 20200 plZ + uoz *D62 DOZ - 24266C00 .... (45)For t - 3 14€2 p^^ + 3553 Ps: + 662C Pcg l{J 1O21 Po: 4 12675 Pe: 1 6 5 3 3 P - - + 24431 p* + 30C)1 P,,^ 1 rJ tlJ 468 CCrpl: 7 0 2C 0 F ; : BC^ + D.- ,Ja = 2.1266 c)oc < |-< V J . ... (+o1
  41. 41. :i3 5,4 PRIORITY (III1 to ttrr:,tt rne cost (Rs.1asgoctRteowttlt IIWENTORYLEVEL To SPECIFIED.4{vlCx.JNT Inventory costs are anotJrer important component of total aggregate planning costs and for finished goods include carryingt- costs, and back order costs.1.1!4#i In general form 2 t.i D i t + + c i -0 D i t ) + , ^1 n- %t rct o... (+ty q w he r e ti cost incurred for carrying one unit of product cl 0 1 cost incurred for one unit of product i, back- ordered per period oi; - Fini shed goods in ventory of product i in period t Di. = B a ck o r d e r q u a nt i t y of product i in perio d t Dit anci Devia tion aI "i. variables. 1n T h e v a l , u e s o f C ? a n d Ci f o r e very t)p e of moto r are gi ven in 1 a p pe nd i x ( tante 4) . The final equations are as gi ven beLow 2 For t = 1 1 E ; 2 . 4 D ; J + 4 1i . 2 ( o J ) + 8 1 4 . 6 ( D J l) + 1257 (D; ( ) 1360 toi ) + 573.9 (oi) +3006.6 (D;) + 3804.4 (oJ) + 57c0 cni I + E 6 4 c ( o _ i .) + z 2 B ( o o . ,) + 5 1 4 ( n o . ,) + 1 0 1 8 ( o J . , , )

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