1. lr
- a
PRODUCTION PLANNING PROBLEMS
IN ENGINEERING INDUSTRY
(A GOAL APPROACH)
PROGRAMT'||}|G
A EDISSEFT1nAITI'ON
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. t?
CERT _rJ J-.C-A-T-E-
that the dissertation entitred'
rt is certified
' },ROIICTION PLPJ{I.II}.G IN INruSTRY
PITOBLE]IS ENGII'IEERING
t by
A G.AL pRocRAl/$rNGAppRoAcH i. s being submitted
partial fuif ilment of M'Tech'
Panka.i char:cina 7B2f Bg, i.n
,
of Kunrkshetra
in l{echanic a} Brgin eering Degree course
o f h i s e w T lw o r k c a r r i e d
u n i v e r s i t y , K u r u ks he t r a i s a r e c o r d
out bY h:-m under mY guidanc e'
ernbo ed in
di tJr i. s di s sertation ha s no t been
Th e matter
previou sl y f or t[ e award of any otir er degree'
sutrnltted
Plac e Ktrruk shetra
Dated
g'3'11 Gitrre''Y
( s. K. 9tarma )
Assistant Profes$cr-t
Itechanical Engg. DeparLnerrt'
Regional thgin eeri19 ColIe-Q€,
f.unrk shetra-132 1 1 9.
--1-

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 eering
to prnf . s.K.
CoIlege, Kurukshetra' for his
Departrnent, Regional Engineering
lnvaluab}eguidanC€lconstantencouragernentandimmensehelpgiven
r a o k , w hi c h r e v e a r s h l s
r
at each and every stage of persuing this
of Production Planning. His inclslve
vast knowledge in the fierd
discussions and valuable suggestions arways
comments, fruitful
edif ied me vrith j est to carryout my work f irmly'
am very thankful to Prof . B.s. Gillr chairmanl Departrnent
I
of 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 heJ'P 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. tv
rr
I
_C_.OJIIJ_E-N-T-S-
Paqe
1
CERT ICATE
IF
11
ACKNC'T{L
EDGEMENTS
111
COI'ITENTS
Y
LIST OF NOTATIONS
vl1
ABSTRrcT
CHAPTER I INTROUJCTION 1
1.1 AGGREGATEPRODUCTIONPLN{NING 2
( cEt'tERAL
Fonlvt)
1;2 SMPLEST STRTJCTURE AGGREC'ATE
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. 'r
tj
TO GOAL PRG RAtvlMI'NG
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 RAI'IKING Al'lD 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[[MII'IG
2t
4.4 AI{ALYSIS OF THE COIPUTER0'JTPUT
-
CHAPTER v FOII},ATJLATION PROBLE}/'
OF
26
5.1 GEI'IERAL
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. LI ST OI. NOTA I ONS
b. GoaI set bY decision maker.
1
The cost for overtime hour.
ci Standard variable co st of pro'dttcing 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.
+
Dot'D6t Deviational variabJes.
+
Dzt'Dzt 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.
^t
T- Shortage during t Ur Peri-od.
rt-1 - Inventory at the end of (t-t)Ul perioci.
k Numl:erof priori ties.

7. Nurnber of goals.
M
Number of decision varlables'
n
Overtime hours in Period t'
ot
produc tion rat6 f or ith type of motor during
Pit
tth period (aecision variable)'
for
Pj The ple-emPtlve weiqht i'
Ievel for pnoduction rate co sts'
Pnct Managenren target
t
Pt 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. 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 t'1.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
soir'e the goal proqraruning problemst. Tne decision variabLes
l't'ereobtalned for arr the planrring periods.
-OoO-
-vi--
$
-i.i"tt
, r.,$
s
$

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. I
!
.l @
2
i
1.1 AC€ EC"ATEPROI'qIION 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-
rrfi
s (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.ng
fr
,#
: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. 3
So th e cornplexity ln tfr e aggregate pro chrc
tion plannlng
ppoblem arlses frrrm the fact that ln most sltrrations demand
per period i s not constant but are subj ected to substantlal
f 1uctuatiop s. The question arises as to how tfrese f luctuations
should be absorbed. Assuming tjr at th ere ar€ no pr,oblem ln
recelvlng a constant supply of raw material and labour at a
f lx ed vjage rate , th e problen may be seen by considering ttr ese
pure 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
t'ro 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. . l
4
I
t
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. 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-
j
I
I
I
a1
It
t
I

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
ll
'l
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. This model is very floclble in
Lnvento rY , & d w o r k f o r c e ;
to a particular functional behaviour of
being not restrlcted
the 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
( 1966'f
2,1 .2 Trre sequential Model of C€rdon
Themainideaoft'hlsmodellstopxoceedinsequence
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}o'sampllng,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. B
(tqZf ) using both discrete and contlnuous events slmul-ation.
An important f eature of simulation 1s that stochastlc demand
pattern can be incorporated in t-he model. This permlts the
analysis 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 pmduction
planners should do. Mode1s of this category are further classi-
fled into classes;
2;2.1 Aqqreqate Plannlnq l'lodels
Th ei r common o bj ec tlve i s to determin e th e op timal
prodtrction quantity to prcduce and work force level to use in
aggtegate for t}le next planning hori zon. l'{ocie}s J.n this cla ss
are elthJr exact or heuristlc.
2.2. 1.1 6xact ,Models : Transportation method fo unulatlon of
Bowan ( t gSO) / 1/ propo sed the di stribution model of linear
prcgrarnming for aggregate planning. thl s model f ocussed on the
objectlve of assigning units of productive capacity' so that
production plus sto rage co sts were minimi sed and sales de'nand
was rnet witi in the con straints of avaiiable capaclty. This
model does not account for prodrction charge co st s. Such as
hiring and layoff of personnel , and tirere is not cost penalty
f or back ordering or l - os t sales.

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
I
I
I
I
and most other aggregate planning technique is the assumptlon of
I
I 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. 3' . 1 '.' . '
'
f
r
rn
I
l0
r
I
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.
C'oodnan /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 metl'iod. 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. ll .1
propo ses a linear pmgtarffning
Tang and Abdulbhan /7 /
aggregate prodtrctron pranning pnoblem ln the
fo rmuration of
heavy manufacturing lndustry ' A baslc model 1s
context of
co st of p ro duc tion wh lch
f i r st deverop ed to mln imi se th e to tal
llnear. the baslc model ls then
is assumed to be piece-ryise
a llneat proglamming model to seek an optlrnal
transf erred lnto
a series of pranning periods witJrln tlr e planning
solution f or
ho rl zon .
Jaa skalain€ss r V /B/ has propo seci a go al prcgramming
model 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. 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 planning
1l
.,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. 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,
t'he organization' There is no single
conditlons of
envlronmental
univelsalgoalforallotganizations.Intodaytsdynamicbusl-
put great €rnphasis on ocial xesponsj'bi-
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
convertvariousg'ealsrcost'sor-valuemeasureintoonecriterion
*
*
*
*
ft,.
,.*.
il
*

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. |:l
a'
l5 ': t,
3.1 CONICEPT
THE GOAL PROGRAT'IMING
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
t're 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
*
i
s
x
{
;x

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! PRCMI4I'IING
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. 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 be
satisfied flrst and then it qo to next. The criterla for
determining t|.re different weights of deviatlonal variable could
be the minimization of opportunity cost. Therefor€r devlational
varlables 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 s
need not be commensurable.
-OOO-

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'
!
I
t
?
G
*
I
E
)l
fi
f;
#

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
time
and indicate which goal is the more important in the paj-r. This
procedure is applied to al.r combinations of goar pairs. This
analysis results in a complete ordinaL ranking o f , . _t h e g o a l s 1 n
t errns o f th eir impo r tanc e .
The go al prog rannmin util
g i ses th e simplex method of
so Jving Iin ear prog ramming plcoble'rn. Horr.'ever several modif r ic ation s
a 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.JG
Step - 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, alr
the negative deviationaf var:-abLes in tf,re modeL constrain t must
enter 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 will
contain ttr€ coefficient of deviational" variabJe because these
varjables onJ.y enter tl-re solution fj.rst. In il^re (rj a:) matrix
l-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 e
top of the hicyhest at tfre bottom. C a l - c u r L a t et f r e , j values and

28. 2f,l
reco rd i t in to RFISco lumn .
cj
Variable R .H . S . d;.. . oi"' xj..a
bi cij
Z. cj P5
J
P4
P..,
J
P2
P1
Step-2l. Determin e th e Nerv D:lterlnq Varl_ab]g
Find th e high est priority Jevel, that has no t been attain ed
completely b y e x a m i n i n gJ Z ,
J values in the R.li.5. column. After
dete rrnj-n in g t j r i s fi-nd out the highest Z.
JJ
Ci entry column. The
variable 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. 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)
,t
i

30. 2',)
'a
values at higher priority l . e v e l i n t t r e s d m ec o l u m n . If there
is negative (zj a: ) value at a higher priority revel for the
positive (z: a-:) value in the row of interest then the solution
is opt5-maI. Finally if there exists a positive (Z; C*) value
J J'
at a certain priority level and there is no negative (Z; C* )
JJ
va lu e at a h igh er priority Jevel' in th e sarne co rumn , tJrJ.s is no t
an optimal solution. H e n ce r e t u r n to step 2 and continue.
4.3 COI/IR'TER 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
t
science 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 n
1s an essential requiremento
After suitabre m od i f i c a t i o n s the computer based solution
proc edure of goal progranrming presented by Lee can be u sed to
sorve problems- The prccess of finding computer sorution conslsts
of 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 to
the 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. 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/'l've 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. 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 basic
a l
i
't 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 /t'ioLilJT

33. 2{t
It presents the constants of only the basic choic e
varl abl es.
(f) Analvsls of the Obiective
It presents the t j values for the goals. These values
represent the under attained portion of goaI5.
PRIORITY UNDERrcHIEVEIJIENT

34. |*
2$
9.U-AP_ ER
T V-
FORMTULATIONOF T H E PROBL
E4
5.1 GENERAL
1 l1
'
'{ ABC Company produces the motors of several kinds which
I
I
I
I
I differ fr''om 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. 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-l'evant 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. 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
+ =a
Tlt en aa
T he r e f o r e .t+
*t 1t 1t (:)
and 'Tt+
- It-r
1 It-r (:)

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. ;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
PR't = 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:1
Type B motor
Pgt - ofi r ou' = su't
(te1
....
Pt't + Psz 'i, +ou, = s g t + sez
.... (1?)
P n t + P,3z + Pa: ui + D,r: Sst +sirz +seg ..o (ts1
T y pe C m o t o r
Pct ,-+
trl ua., sn1
z l .... (1a)

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 & J
t 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 r
TO LJIII_Irr{E cosr (r' ASSOCIATED
WITH PRODUCTION
RATE
Pit x ci * cTot + Dot 'Jt = PRct .... (+s1
wh 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. 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 be
written 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. :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 carrying
t-
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 . ,
,
)

42. A
{t
[f ;i4
I
T
r
I
il
it
:t
t
l
rJ 1521 (Der) +1e50 (Dur) +717 (or') +3?58 (0E.,) +
t ,l
,1
'i
rt
i 4755 ( o[, ) + 72oo(oI' ) + 1 0 8 0 0 ( o J , ) + qt
I {r
22,00000. ...,. (4s)
182.4 to[) + 4 1 1 . 2t o j r l + 8 1 4 . 8 t o & l + 1257(oJr) +
1 5 6 0( o L ) + 5 7 3 . e ( + ) + 3 0 0 6 . 8 ( D J r ) + 3 8 0 4 . 4( D ; ) +
5?60 (oi) + 8640 (D;) + 2zB (o_) + 514 (D;2) + 1o 1 B ( o f r +
)
,ll
1571 (oor) + 1e50(DE2) + 717 (oir) + 3?58 (%) + 47s5(orr)+
:J
i
fr
,I
,f;
7 2 C O( o r , ) + 1 CrB (fr)
00 + n
+
22, 00000.
.rl
:;l
z "lz
:f
o... (49)
. 1 8 2 . 4( o i . ) + 4 1 1 . 2( o i ) + 814.8 toit + 1 2 5 7t o $ l +
1560 toil 573.e t
{. I + 3 0 0 6 . 8 ( o & ) + 3 8 0 4 . 4 (o,i.
+ +
)
5?60toi I + 8640 (o_i.) + 228 (orc) + 514 (o-r.) + 10 1 8 ( o f . ) +
157r (of.) + 1q5o (oo.) + 717 ({.) + 3758 (n[. ) + 4755(o[. +
)
72oo (oi.) + 10 B 0 o ( o J . ) 22, 00000.
.. .. (so1
In our case we treat (Orta) and (oJa) as if they were cho ic e
varj.ables say (Ura) and (Vra) respectiveJ_y.
Therefore the above eqns, for t _ j
, 2 and 3 can be
expres sed as belorv t