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PRODUCTION PLANNING PROBLEMSIN ENGINEERING INDUSTRY
(A GOAL PROGRAMT'||}|G APPROACH)
A EDISSEFT1nAITI 'ON
SUBMI.TTED IN PARTIAL FULFILMBNT OF THE
REQUIREMENTS FOR THE AWARD OF THE DEGREE
OF
fflagter o[ 6,e*lnologPin
,ff[erhantuI S,ngineerin g
BY
PAilKAf CHAtlDlf A ttzltt
Under the guidanco of
Prof. S.K. SHARMA
@epa rtment o[ Sler[anital @ngineerfng
Begional @ngtneertng 6otle ge
&uruh*tletra - 132 ttg
t ?
CERT
r t i s cer t i f ied tha t the d isser ta t ion en t i t red '
' },ROIICTION PLPJ{I.II}.G PITOBLE]\IS IN ENGII'IEERING INruSTRY
A G.AL pRocRAl/$rNG AppRoAcH t i. s being submitted by
Panka.i char:cina , 7B2f Bg , i .n part ial fuif i lment of M'Tech '
in l{ech an ic a} Brgin eering Degree course of Kunrk shetra
univers i ty , Kuruk sh etra is a record of h is ewTl work carr ied
out bY h:-m under mY guidanc e'
Th e ma tter ernbo di ed in tJr i. s di s sertation ha s no t been
sutrnl t ted previou sl y f or t [ e award of any ot i r er degree'
Ktrruk shetra
g'3 ' \11\
_rJ J-.C-A-T-E-
P l a c e
Dated Gitrre\''Y( s . K. 9 tarma )
Ass i s tan t P ro fes$c r - tI t echan ica l Engg . DeparLner r t 'Regional thgin eer i19 ColIe-Q€,f .unrk shetra-132 1 1 9.
- - 1 -
EDGEMENTS
I have g rea t p leasu re i n xeco rd ing my p ro found g ra t i t ude
to prn f . s .K. SharTna, Ass is tant Prof essor , Mechanica l Rrg in eer ing
Depar t rnent , Regiona l Eng ineer ing CoI lege, Kurukshet ra ' fo r h is
l nva luab }egu idanC€ l cons tan tencou rage rnen tand immensehe lpg i ven
at each and every s tage o f persu ing th is rao rk , wh ich revear s h ls
vast knowledge in the f ie rd o f Product ion P lann ing. H is inc ls lve
comments , f ru i t f u l d i scuss ions and va luab le sugges t i ons a rways
edi f ied me vr i th j es t to car ryout my work f i rmly '
I am very thankfu l to Prof . B .s . Gi l l r cha i rmanl Depar t rnent
o f Mechanicar Engineer ing, Regionar Engineer ing co l lege t
Ku rukshe t ra f o r p rov id i ng f ac i l i t i e s t o ca r r you t t h i s wo rk .
_l_.c_F_N_o-,$rJL
bec ia l t hanks a re due
Er . R .S . Bha t i a and E l . D .K '
compu te r l ab . wo rk .
In add i t ion ' I
espec i a l lY to A rv ind '
me a l o t i n ca r r y i ng
to Er . L.M. Sain i r Er. Rai e sh Jan 9ra"
Jain f o r th eir k ind heJ'P dur ing mY
am h igh l y thank fu l t o a I I my f r i ends
Ra jender , V inod and Ra j i v who he lped
ou t mY d i sse r ta t i on work .
P Iace : Kun rkshe t ra
Da ted z 8 Z t2 t l i)
t n\,ffcHAI{D}JA782/ Be
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CERT IF ICAT E
ACKNC'T{L EDGEMENTS
COI'ITENTS
LIST OF NOTATIONS
ABSTRrcT
CHAPTER I
1.1
1 ;2
1 .3
CHAPTER I1
_C_.OJIIJ_E-N-T-S-
INTROUJCTION
AGGREGATE PRODUCTION PLN{NING
( cEt'tERAL Fonlvt)
SMPLEST STRTJCTURE oF AGGREC'ATE
PLAI.INING PROBL F{
MULTI STAGE AGGREGATE PLANINING
SYSTEM
LIT ERATURE REVI ET{
DESCRIPTIV E MODELS
Th e Management Coeff ic ient lv lodel
The Sequent ial ModeJ, of Gordon
Simula t ion Models
NORT1ATIVE I{ODEL S
Aggregate Pfann ing Models
2 .2 . 1 :1 Exac t l v t ode l s
2 .2 .1 .2 H zu r i st lc Mo cie. I s
Paqe
1
11
111
Y
vl1
1
2
\
2.1
2 .1 .1
2 .1 .2
2 .1 .3
'2 .2 ;1
6
6
6
7
7
I
E
I
L 2
- 1 i i -
'r
CHAPTES_:--III INT RODLJCTION TO GOAL PRG RAtvlMI'NG
3.1
3 .2
3 .3
cHAPTE_&_:--IV
4.1
4 .2
t j
L5
t6
!6
18
18
t9
THE GOAL PROGRAI{MING COI{CEPT
OBJECTIVE zuNCTlON IN GOAL
PRGRAI/tlvtING
RAI'IKING Al'lD WEIC+{ING OF MULTIPLE
CSALS
CoALPR0GMJ\{I4INGAsAMATI{EIVIATICAL
TOOL USED
GB\ERAL MATTIEMATICAL MODEL
STEPSoFTHESIMPLEXMETHoDoFG0AL
PROGRAMIVIING
CCI!1zuTER BASED SOLUTION OF @AL
PRGRA[[MII'IG
AI{ALYSIS OF THE CO\IPUTER 0'JTPUT
FOII},ATJLATION OF PROBL E}/'
GEI'I ERAL
PRroRrrY ( r)
PRToRTTY ( rr)
Pr l rORt ' tY ( r r r )
PRToRITY ( rv )
CChISTRATNTS
Productive hours constralnt
6ver t ime Con s t ra in t
DI SCU SSION Of-- RESULT
22
2t4 .4
CHAPTER - v
5 .1
5 .2
5 .3
5 .4
5 .5
5 .6
5 .6 -1
5 . 6 .2
@APP EIDIX
REF ERET.JC ES
26
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58
39
59
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LI ST OI. NOTA I ONS
GoaI set bY dec is ion maker .
The cos t f o r ove r t ime hou r .
Standard var iab le co s t o f pro 'd t tc ing one un i t
o f p roduc t i .
Co s t i ncumed fo r cauy ing one un i t o f p roduc t i .
Cos t i ncu r red fo r one un i t o f p roduc t i backo rde red
per pe r i -od .
F in i shed goods i nven to ry o f pn rduc t i i n pe r iod t ;
Backo rde r quan t i t y o f p roduc t i i n pe r iod t .
Nunber o f wo rke rs i n excess o f t he des i red max imum.
Number o f workexs less than the des i red maximum.
Dev ia t i ona l va r i abJes .
Dev ia t i on a I va r i ab le s .
In ven to ry at th e en d of t th Per iod.
In ven to ry dur ing t th Peri o d.
Sho r tage du r i ng t U r Pe r i - od .
I nven to r y a t t he end o f ( t - t )U l pe r i oc i .
Num l :e r o f p r i o r i t i e s .
+Di t
Di t
+Dzt ,-
Dit
+Dot 'D6 t
+Dzt 'Dzt
r t
T +t t
T -^ t
r t -1 -
k
M
n
ot
Pi t
Nurnber o f go a ls .
Number o f dec i s ion va r lab les '
Over t ime hou rs i n Pe r iod t '
produc t ion rat6 f or i th type of motor dur ing
t t h pe r i od (aec i s i on va r i ab le ) '
The p l e -emPt l ve we iqh t f o r i '
Managenren t target Ievel for pnoduct ion rate co sts '
P roduc t i on ra tedu r i ng t t l : I pe r i od '
Max imurn des i red change in wo rk forc e }eve l '
Sa les in t t j r Per iod '
Hou rs requ i red f o r one un i t o f mo t i r i '
E f f i c i ency coe f f i c i en t f o r o l d wo rk€ rso
E f f i c l ency coe f f l c i en t f o r neu r wo rke l s .
E f f i c i ency coe f f i c i en t du r i ng ove l t ime hou rs '
s ize o f work force dur ing t th per i -od.
s ize o f wo rk forc e dur ing ( t - t ) t r t per iod.
Dec i s i on va r iab l e to be found '
Change in thenumbero fwo rke rs i npe r i od I t ' .
Pj
Pnct
Pt
Qt
st
T i
T1
T2
T3
vlt
vtt- t
xj
xt
- O ( r O -
- v1-
t;_A_B_S TRACT
In th is d isser ta t ion an a t tempt has been made to
anaryse the aggregate product ion prann ing o f ABc ( tne
actua l , nane has been d isgu ised) opt imal ly . The denrand o f
the nro tors wi th d i f f e rent spec i f ica t icns vrere not constant
c iu r i ng the p rann ing ho r i zon o f one yea r i . e . l gg8 -89 ,
cons l s t i ng o f t h ree p lann ing pe r lods . To mee t t he f l uc tu -
a t ion in dernand aggregate p lann ing model wBs formula ted,
wtt ich conc en trates on determi-nin g which cornblnat ion of t '1.re
c lec i s ion va r jab les l i ke p roduc t i on ra te , i nven to ry , back -
o rde r ing , o ve r t ime e tc . shou ld be u t i l i sed i n o rde r t , o
opt i rna l ly ad j us t th e dernand f Luc tuat ions wi th in the con s t ra in ts
if "ny-.
The agg rega te p lann ing mode l was fo rmu la ted i n t he
fo rm o f goa l s w i t h d i f f e ren t p r i o r i t i e s . The p rob lem was
t i i en soL. ied by us inc{ 'Computer ized techn ique o f S. [ ' : , Lee to
so i r ' e t he goa l p roq ra run ing p rob lems t . Tne dec i s i on va r i abLes
l ' t ' e re ob ta lned fo r a r r t he p lan r r i ng pe r iods .
- O o O -
-vi--
$-i.i"tt, r.,$s$
ffi'1?
I
INTRODUCTION-
llo st managers want to plan and con trol operatlon s at
tJre broadest level thmugh some klnd of agglegate plannlng
that by passes deta l rs o f lnd iv iduar products and deta i red
sch edr.rrlng of f ac ir lt ies and personn el. Managernent wourd
deal w lur bas lc re levant dec is ions o f programmlng the use o f
resou rces . Th i s i s accomp l i shed by rev lev r l ng pno iec ted
emplo lm€rr t ieve ls anc l by set t lng ac t iv l ty ra tes that can be
varied wlth ln a glven ernproyment rever by varylng hours worked-
f i rce these bas lc dec is ions have been made for the
upcomlng per iod, deta i led schedul inE can p loceed a t a lowel
Iever w i t t r ln the con s t ra in ts o f the broad pran. F ina l ry ra s t
m inu te changes l n ac t i v l t y l eve l s need to be made w i th the
rea l i sa t i on o f t he l r poss ib l e e f f ec t s on t he cos t o f chang ing
product ion leve l and on inventory co s ts i f th ey are a par t o f
th e sy st,em .
CHAPTER- - - F
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1.1 AC€ EC"ATE PROI'qIION PLAI{NING GENERAL FORM
The aggregate prodtrct lon plannlng pmblem tn l ts most
general form can be stated as fo l lows z
A set o f fo recasts o f denrand for each per iod 1s g lven -
(a ) The s i ze o f work fo rce ' T l t
( b) The rate of Product ion ' Pt
(c ) The quant i tY s t r iPPed ' St
The resu l t lng |n ventory per mont i can be determln ed as
fo l l ows -
r t
The Prob lsn
th e exp ec ted total
o f some o r a l l o f
I t_ t +Pt - S t .
i s usua l l y t eso l ved ana ly t i ca l l y by m in in i z ing
cos t ove l a g i ven p lann lng ho r i zon cons l s t i ng
tfr e f o l loning co st compon en t s.
( a )
(r )
( c )
( o )
The cos t o f regu la r pay - ro I l anc i ove r - t ime-
Th e co st of ch anglng tJr e p ro duc tion rate f rom
one pe r iod to t J re nex t .
The cos t o f ca r rY ing i nven to tY .
Co st of, sho rtag e s re su I t lng f rom no t m eeti.ng
th e dernan ci.
Th e so lu i ion to t l i e p robl sn i s s impl i f ied l f a verage
dernanc i ove r t he p lan r r i ng ho r i zon i s expec ted t , o be cons tan t .
3
So th e cornplexity ln t fr e aggregate pro chrc t ion plannlng
ppoblem ar lses f r r rm the fact that ln most s l t r rat ions demand
per per iod i s not constant but are subj ected to substant la l
f 1uctuat iop s. The quest ion ar ises as to how t f rese f luctuat ions
should be abso rbed. 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 con sidering ttr ese
pure a l ternat lves of responding to such f luctuat ions.
A inc rease i n o rde rs i s me t by h i r i ng and a dec rease l n
o rde rs i s accomp l l shed by l ay -o f f s .
( a )
(b )
( c )
( d )
Mai6 tenance o f cons tan t work fo rce , ad jus t l ng p roduc t i on
rate to orders by wo rking o vert inre or undert ime acco rdingly .
Ma in tenance o f a cons tan t v lo rk f o rce an d cons tan t
t ' ro duc t ion rate, dl lor^r ing inventor ie s and order bac klog s
to f l uc tua te .
Mainten anc e of con stan t wo rk f orc e and meet th e f luc tu-
a tion in dern an ci th ro ugh p I ann ed b ac k log s o r* by subcon t-
ra t ing exc e s s dernan d.
In gmera] none o f t . | re abo ve a l ternat ives wi l l p rove best
but some cornb inat ion o f then can c io . Order f . luc tuat ions showed
in g eneral be ab so rbed part ly by in vento ry , part ly by o vert i r re
and par t ly by h i r ing and layof f s anc i the opt imum ernphas is on
the se f ac t c rs w i I I depenc l upon the co s t s i n any pa r t i cu la r f ac to l y .
It
. l
4
1.2 SIIV1PLEST STRUCTURE OFjTSGREGATE PLAIININ9 PROBL4I
The structure of the aggregate planning problem ls
represented by the single stage sy stqn 1; e; the plannlng
hor lzon ls only one per lod 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 s i zer prcduct ion ox act iv i ty rate and inven-
tory level respect ively. The ending state condi t ions become
the in i t j .a l condi t ion s for the upcoming per iod. ' We have a
forecast of the requirements for the upcoming per iod through
some prccess . The dec is ion made may ca l l fo r h i r ing or lay lng
of f personnel, tJrus expanding or contracting the ef f ect lve
capacity of tJre pro duction systern. The work force size together
wi th th e ciec i slon on ac t ivl ty rate du r ing th e perlod th en deter-
min es th e *requi red amount of o vert i f f i€ r in ventory level s or back
order lng whether o r no t a sh i f t must be added or de le ted and
other posslb le changes ln operat lng pmcedure.
1 .3 MULTISTAGE AGGREGATE .PLAI.INING SYSTEMS
In th i s t ype o f p lann ing sys tem, ou r ob j ec t l ve l s t o
make the dec l s ions conce rn ing the work fo rce s l ze and p roduc t i on
ra te f o r t he upcoming pe r iods . I n do ing so , however r w€ cons lde r
the sequence o f p ro jec ted dec i s ions i n re la t i on to fo recas ts and
the i r cos t e f f ec t s . The dec i s l on f o r t he upco rn ing pe r i od i s t o
be a f f ec ted by t he f u tu re pe r i od f o recas t s and t he dec i s i on
5I
process must cons ider the cos t e f fec ts o f t j re sequence o f
decisrons. The connect ing r lnks between the severar stages
are the lr f r P and I Values tJrat are at the end of one p.eriod
and the beglnning of the next . The feedback roop f rorn t j re
dec ision process may invorve some lterat ive proc edure to obtain
a sotut loD. The sequent ia l nature of t j re decis lons should be
kept in mind. Arr decis ions are r ight or wxong onry in terms
of the sequence of decis ions over a per iod of t ime'
- O O 0 -
j
II
Ia 1IttI
. l
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g_u.a8.tgE.- Ir
LITERATURE REVI EN- t-^- ,
The pro duc t ion planning problenr i s conc erned with
spec i f y ing the opt imal quant l t ies to be prcduced in order
to meet denand for a sp ec i f ied planning hor i zon. Many mo del s t
each o f wh ich has i ts pros and cons, have been deveroped to
he lp to so lve th ls P lob lem'
These rnodels in t roduced in the l l te ra ture d i f fe r ln
the i r or ienta t io r l r scope, contents and methodorogy. However t
we can c lass i fy these models ln two maln catagor les
c iesc r jP t l ve and no rma t i ve ' '
pEqpRrPTrVE MODELS2.1
2 .1 .1
Desc r ip t l ve mode ls a im o f desc r ib ing the p locess by
whichr procluct ion are determined . in pract ic e. The maln example
o f such mo de} s are z
The Managernent Coef f ic ient Model
/ 1 / intro clr.rc ed by Bo\ran ( 1 963 ) and exten ded by Kumren
Ther ( t l oo ; , t h i s moc ie r assumes th a t manager behave e f f i c i en t r y
d. r average, but suf f e r f rom in- -con s i s tency and b iases to rec ent
even t s . L i nea r r eg ress ion i s used t o deve lop dec i s i on ru l es
fo r ac r , ua r p roduc t i on and r r o r k f o r ce dec i s rons u t i r i z . i ng i nde -
r )end tn t va r i ab les such as pas t sa les a r r c i r ogged p roc iu r c t i on '
Ln vento rY
be ing no t
the co s t
, &d work fo rce ; Th i s mode l i s ve ry f l oc lb le i n
res t r lc ted to a par t icu lar funct iona l behav iour o f
e lements invo lved.
A s eriou s drawbac k of th e
sub j ec t ive se lec t ion o f the fo rm
prccedure i s t he essen t i a l lY
of t j r e rule.
2,1 .2 Trre sequent ia l Model of C€rdon ( 1966' f
Thema in ideao f t ' h l smode l l s t opxoceed insequence
start lng f rom a prespec i f led acc ep tabre rarge of inventory t
andse tacco rd lng l y t j ne l i ne -sh i f t l e ve l so fwo rk - f o l ce .Thus
adjust tJrese according to the range of lnventory deviat lon from
l ts permi ss j .b le range. r f dev ia t ion s occur too f requent ly , t ien
the acceptabre lever inventory ranges are sub jec t to ad jus t rnent -
2.1 .3 Sir iu lat ion wro dels
F; te r r s i ve work has been ca r r i ed ou t l n t t r l s f i e rd us ing
dif f erent stati stlc al an d matjr erna tlc aI apprc ach e s lnc rudlng
Mon teCar }o ' samp l l ng ,andcompu te rana }ogue . I n th i smode } '
1n troduc ed by Virgln ( 1 966) , th e simurat ion starts with a
product lon pran based on tJ re past exper5,ence o f t t re form and
then changes are in t roduced ln emproyment rever r ov€xt imet
lnventor les , sub_Cont rac t ing and so fc , r th , unt i r a min i r r run loca l
ope ta t l ng cos t i s ach ieved . O t j re r s imu la t l on mode ls i n t t r i s
rega rd a re de . i e loped by Ensho f t and s i sson ( t qzo ) and by Nay lo r
B
( t qZ f ) us ing bo th d i sc re te and con t l nuous even ts s lmu l -a t i on .
An important f eature of s imulat ion 1s that stochast lc demand
pat tern can be incorporated in t -he model . Th is perml ts the
analys is o f the forecast er ror on s t ra tegy deve lopment .
2.2 NORT4ATIV E MOELS
The common focus in normat ive models is on what pmduct ion
p lanners shou ld do . Mode1s o f t h i s ca tego ry a re fu r the r c lass i -
f l ed i n to c l asses ;
2 ;2 .1 Aqqreqate P lann lnq l ' lode l s
Th ei r common o bj ec tlve i s to determin e th e op timal
prodtrct ion quant i ty to prcduce and work force level to use in
aggtegate for t } le next p lann ing hor i zon. l ' {oc ie}s J .n th is c la ss
a re e l t hJ r exac t o r heu r i s t l c .
2 .2 . 1 .1 6xact ,Models : Transpor ta t ion method fo unu la t lon o f
Bowan ( t gSO ) / 1 / propo sed the di str ibut ion model of l inear
prcgrarnming for aggregate p lann ing. th l s model f ocussed on the
ob jec t l ve o f ass ign ing un i t s o f p roduc t i ve capac i t y ' so t ha t
product ion p lus s to rage co s ts were min imi sed and sa les de 'nand
was rne t w i t i i n t he con s t ra in t s o f ava i i ab le capac l t y . Th i s
mode l does no t accoun t f o r p rod rc t i on cha rge co s t s . Such as
h i r i ng and l ayo f f o f pe rsonne l , and t i r e re i s no t cos t pena l t y
f o r back o rde r i ng o r l - o s t sa l es .
:I
!I
. l
II
IIII
I
a wi t
, . ]
The s implex method o f l inear prcgranming makes i t
poss lb le to inc lude prod, rc t ion leve l . 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 product ion rate
as independent dec is ion var iab les and in terms of the components
of the costs model . AI I cost funct ions axe cons idered l inear .
One of the baslc weakness of l lnear progranrmi-ng approaches
and most other aggregate planning technique is the assumptlon of
determlnl stlc dernan d. Anoth er sho rt coming of th e lin eat
prograrnmj,ng model is the requirement of l inear co st f unct ion s.
However , tJ re poss lb i l i ty o f p lee wi se l lnear i ty lmproves tJ re
va l i d i t y .
HoIt l lodigl iani and Simon /3/ gave t f re weII known
rnodel in which t iey minimi se a quadrat ic co st funct ion and come
up wi th a l lnear dec is ion ru le that so lves for opt imal aggregate
pro duc t ion rate and wo rk f orc e si ze f or aI I tJr e per iods ovel l
t he p l ann ing ho r i zon . L .D .R . hasnany advan tages . F i r s t t he
mo del 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 t ion t l r e model is dynamic and
rep resen ta t i ve o f t he mu l t i s tage k lnd o f sys tem. Bu t quadra t i c
cos t s t ruc tu re may have seve re l im i ta t i on and p robab ly does no t
adequate ly represent the co s t s t ruc ture o f a l ly organ izat lon.
Bergst rom and Sni th / 4 / ex tended the capabi l l t ies o f
the L . l ) .R . l r t ode l i n two n6 rJ d i rec t i ons . Because o f t he
3 . 1 . . 'f ' ' ' '
r
rnI
I
rIir*lcl:
It
l0
agg rega te na tu re o f L .D . R . i t i s no t po ss i b l e t o so l ve d i r ec t l y
for the opt funum prod. , rc t ion ra tes for ind iv ldua l pxockrc ts . The
deve lopmen t and app l i ca t i on o f t he M.D .R . mode l sugges ts tha t i t
l s now operat iona l ly feas ib le to temove tJ re requ i rement o f an
aggxegate product ion d imens ion in p lann ing models .
Fur therTnore, g iven the ava i l -ab i l i ty o f revenue curves
for each product in each t ime per iod the M.D.R. model can deter -
mlne opt l rna l prcduct ion, sa1es, inventory and wo rk- force leve l s
so a s to maximi se prof 1t over a spec i f ied t ime hor l zorr o
Larvrenc e and Burbr idge /5 / presented a mul t ip le goa l
I in ear programming mocie l cons lder ing commonly occur l -ng goa ls o f
the f i rm in coord inat ing prcduct j -on and log is t ic p lann ing. The
so lu t lon techn ique fo r th i s model w i l l be a computer j - zed mul t ip le
obj ect ivq. analogue of th e revi sed si .mplex method.
C'oodnan /6/ presented goaJ. prograniming apploach to
so l ve non - I l nea r agg rega te p lann ing mode ls . I f ac tua l cos ts
(n i r i ng and f i r i ng co s t , ove r t ime and i d le t ime , l nven to ry and
sho r tage cos t ) can no t be sa t i s f ac to r i l y r ep resen ted quad ra t i -
c al l ; ' , th en th e so lu t lon b ecome s mo re compl ex . On e app ro ach to
hanCle these mote contp lex rnoc ie ls is to a t ternpt formula t ion o f an
approx j , rnat i -ng l inear model to the or ig ina l non- l lnear co s t te rms
and to app ly some var ia te o f the s iml : Iex met l ' iod . Th is appro ach
o f f e r s t he ne t ac i van tage o f a t Leas t p rov id i ng an op t i n ra l
so lu t i on t c t J re n ro ieJ used ano i s based upon t f , e goa l p rog ra r : r r i ng .
l l .1
Tang and Abdulbhan /7 / propo ses a l inear pmgtarf fning
fo rmurat ion of aggregate prodtrctron pranning pnoblem ln the
context of heavy manufactur ing lndustry ' A baslc model 1s
f i r st deverop ed to mln imi se th e to tal co st of p ro duc tion wh lch
is assumed to be piece-ryise l lnear. the baslc model ls then
transf erred lnto a l lneat proglamming model to seek an optlrnal
solut ion f or a ser ies of pranning per iods witJr ln t l r e planning
ho rl zon .
Jaa skalain€ss r V /B/ has propo seci a go al prcgramming
model for the sch edul ing of produc t ion , employment and lnvento-
r l es to sat l sf y known demand or requirement ovex a f in i te t ime
hor i_Zo. . Th ls model sets three separate ard lncomplete goars ,
the level of , prcduct ion, errrployment and inventor les;
Thornas and HlI l /9/ formulated a rnul t i -object ive
pro d t rc t ion prann ing moder as a go ar pxogram which cap i tar lzes
on the strength of goar-prograrnming ln incorporat ing rnurt ipre
economic cons ide ra t i . ons i n to the ana rys i s . Th l s paper l nc rudes
the aspectsr ignored by cco&nan /6 / and Jaake la lnen /B/ '
Ja rnes , P . I gn i z io /1o / has a t tempted to p lov lde a
br lef bcok at th e rerat l very n 6^' f ie ld of go al p rogrammlng
rm der e p I e-{5np ti ve p rio ri ty struc tu re ' As such th e gen eral
goa l - prcgrar run ing model presented is v iewed as a pxact ica l '
r ea r l s t i c and ra the r na tu ra r re r r resen ta t i on o f a w ide va r ie t y
o f many rea l wo r ld P rob lems '
1 l. , 1
III
t
T2
2.2 .1 .2 Heur i s t i c Mo de l s :
(a) The product ion parametr ic p lanning model by Jones ( tgZS):
This model assumes t jre exl stence of two basic decision
nr les addressing work force anci pxoduct ion levels respec-
t ive ly, each of which is expressed as a weighted s- t rm of
rates required to meet future sales drrr ing the planning
ho ri zoo .
( b ) A switrh rule prcpo sed by Elmaleh and Eiton ( ' tgt +) z
Th ey spec i fy three inventory leve1 s and three prc cLrc t ion
leve l s to be ob ta ined by va r ious comb ina t i on o f con t ro l
pa ramete rs ove r a h i s to r i ca l demand se r ies .and choos lng
th e set f or wh ich pro dr.rct ion i s l imited to discrete level s
such as food and chenr i ca l s i
- O O O -
Si
q.H.&P-TEE
l ; l
ur
INTROqJCTION T9 GOAL PROGRATTTTING
organisat iona l ob jec t ives vary accord ing to the charac-
te r i s t i cs , t ypes , ph i l osophy o f managemen t md pa r t i cu la r
env l ronmenta l cond i t lons o f t 'he organ izat ion ' There is no s ing le
un i ve l sa lgoa l f o ra l l o t gan i za t i ons . I n today t sdynamicbus l -
ness er rv l ronment f i rms put great €rnphas is on oc ia l xespons j 'b i -
I i t ies , soc ia l cont r ibut ions, pub l ic re la t ions and indurs t r ia l
and labour re la t lons e tc '
I fweg ran t t j r a tmanagene r r t hasmu l t i p l ccon f } i c t i ng
ob j ec t1 ve s to ach 1e ve t]r e dec i sion c riteria shourd a} so be mul t i -
d imen s ioqar . Th is impr ies that wh sr a dec i s ion invorves mul t ip le
goa ls the techn ique used shourd be capabre o f hand l i ng mu l t i p le
dec i s ion c r i t e r l a ' The l i nea r p rog ramming techn ique has a
l im l ted va rue fo r p rob lems invo rv lng mu l t i p re goa ls i
Thep r ima ryd i f f i cu } t yw i t h l i nea rp rcg ramming i sno t i t s
l nab l l i t y t o re f l ec t comp lex rea l i t y .Ra the r i t l l e s i n the
unid imen s j .onar l ty o f the ob j ec t ive f unct ion which requ i res cost
or prof i t info rmat ion that is of ten armo st impo ssibre to obtain '
To o vercome ur e un id imen s ionar i ty o f the ob j ec t ive f unct ion
Iequ i red in the l i nea rp rog ranu l i nge f f o r t shavebeenmade to
conve r t va r i ousg 'ea l s r cos t ' so r - va luemeasu re in toonec r i t e r i on
***
*ft,.
,.*.
il*
|
,':,l4
namely u t l l l tY .
Howeverr €Xact rneasurement of ut l l i ty is not s lmple.
So decislon making t irough l lnear programrning via a ut i t t ty
func t ion is on ly feas ib le 1n theore t ica l sense.
Croal pxogramming i s a modif ic at lon and extm sion of
I lnear pDograrnming. The goal programmlng approach ls a tech-
nlque that is capable of handl ing decis lon problems that deal
wl th a s ingle goal wi t j r mul t lp le subgoals r Es weI I as r problem s
wi th mu l t ip le goa ls w l th mu l t ip le subgoa ls .
We can soJve these prob lems us ing l lnear programming
wl th mul t ip le ob j ec t j .ves. We may ln t roduce o ther ob j ec t lve
f unc t ion s a s model con stra int s . But tJr 1s mo del require s th at
the op t l rna l so lu t l on mus t sa t i s f y a l I cons t ra in t s . Fu r the r rno re ,
1 t is assumed tJ ra t equa l impor tance is a t tached to var ious
obJec t i ves . However , such assumpt ion a re absu rd . I t 1s qu i te
po ss ib le tha t a l l t he cons t ra in t s o f t he p rob lem can no t be
sa t i s f l ed .
Such a p rob l sn i s ca l l ed i n feas ibLe . Second l y a I I
cons t ra in t s Co no t have equa l impor tance . The re fo re goa l
p rog ramming wh ich r snoves a l l such d i f f l cu l t l es i s used to
so l ve such P rob I€ fns .
| : la '
l5: t' ,
3.1 THE GOAL PROGRAT'IMING CONICEPT
cro aI prcgramming ls rec eiv ing much attent ion a s a powel-
fu l toor for ana lys ing mul t i -ob jec t ive dec is ion maklng probrern .
The concept o f goa l prcgranrn ing was f l rs t in t roduced by A- charnes
and \ l t . l t . .cooper as a too l to resorve in feas lb le l inear prcgraurming
probrerns. Th ls techn ique has been fur ther re f lned by Y. r j l r r and
s.Mi Lee and o t^ers . The maln reason o f t ' re popurar i ty o f GP
sums tobeassoc ia tedw i t h t J reawarenesso f t hemanage rnen tsc i ence
techn iques and very natura l or ienta t ion towards mul t l -goa l or
mu l t i - ob j ec t i ve fo rmu la t i on and uses ' The goa ls se t by the
managemen t a re o f ten ach levab le on l y a t t he expense o f o t i e r
goa rs . Fu r t , reqnor€ r t hese goa ls a re i n commensurab re i -€ . t hey
cannot be measured on tJ re sane un i t Scare. Thus there is a need
for es tab l lsh ing a h lerarchy o f lmpor tance among t j rese conf l i c t ing
-goa rs so tha t row o rde r goa ls a re cons ide red . o r l y a f te r t he
h ighe r o rde rs p r i o r i t y goa rs a re sa t i s f i ed o r have reached the
poin t beyond which no fur ther improvenrent is des l rabre- Hence
the prob lem can be so lved by goar programming l f the managem
can prCIv ide tJ re ord ina l rank lng o f the goa ls in tenms of th e i r
impor tance and a l l r e ra t i onsh ip o f t he moc ie l . r t i s no t a lways
po ss ib le to ach ieve th e every goa l f u r ry to the extent des i red
by managernent. Thu s with or wi thout programmihg r tJ. Ie managel
a t t achesace r ta l np r i o r i t y t p t i each ieve r r ren to fapa r t i cu l a r
goa l . The tn re va lue o f goa l p rog la rnmin t ] i s ' t he re fo re ' l l es
in the so ru t i on o f p rob re rns j -nvo rv ing mu l t i p le con f l i c t i ng goa ls '
*i
sx{;x
1l
1.:
I*'-ENII1
III
t
II
i
l { ;
' iIItIIt
t
IIit
acco rdlng to tJr e Manager I s pr ior i ty struc ture.
3.2 QBJECTIVE zut\CTIOt{ IN GOA! PRCMI4I'IING
In goa l programming lns tead o f t ry lng
min ln i se the ob jec t i ve c r i t e r i on d i rec t l y as
lng r lt tr ie s to min imi se th e devi a t ion s ariong
the g i ven se ts o f cons t ra in t s . The ob j ec t i ve
min lm isa t i . on o f t hese dev ia t i ons based on the
o r p r i o r i t y ass ig red to them.
to maxirnise or
in lin ea r p ro g rarnm-
the go als wi tJr in
func tion i s tJr e
relative impo rt,arrc e
3.3 RANKTNG Arlp_nEIcHfINq_oF_wI.TIpLE coAL s
In order to ach ieve the ord ina l so lu t lon that i s to
ach ieve the goa ls acco rd ing to th e i r impor tance nega t i ve o r
pos l t l ve dev ia t i ons abou t t he gca l mus t be ra r r ked acco rd ing tof
tp re-€n ip t ive t pr5-or l ty fac tors . rn th is way the row-order goa ls
a re cons ide red on l y a f te r h iqhe r -o rde r goa ls a re ach leved Bs
des i red . The p re -en tp t i ve p r i o r i t y f ac to rs have the re la t i on sh ip
o f P i ) ) )P i +1 wh ich lmp l i es tha t t he mu l t i p l i ca t l on o f n howeverJ J
ra rge i t may be canno t make p j * t g rea te r t han o r equar to p5 .
The next s tep to be con s idered in the go a l proEramming
is the we igh ing o f dev ia t i ona l va r i ab les a t t he sane p r i o r i t y
' Leve l . I t any goa l i nvo . I ves many dev ia t i ona - l - va r i ab les and we
wan t t o g i ve p r i o r i t y t o one ove r t he o the r . Th i s can be
ach i -eved by ass ign ing d i f f e ren t we igh t s t o t hese dev ia t i onaL
va r i ab les a t t he san re p r i o r i t y - l - eveL . A t t he sa rne p r l o r i t y l - eve l
1 'I7
the subgoal wtr ich acquires maximum di f ferent ia l weight wi I I be
sat is f ied f l rs t and then i t qo to next . The cr i ter la for
determining t | .re dif ferent weights of deviat lonal variable could
be the minimizat ion of opportuni ty cost . Therefor€r devlat ional
var lables on the same pr lor i ty level must be commensurable,
aldrough deviat ion s that are on tfre dif f erent prlorl ty level s
need no t be commensurab le .
- O O O -
!I
t
?GI*E)l
fif;#
tfr
9.U.AP_IEE- IV-
GOAL PROGMI4MING AS A MATHEMATICAL TOOL USED
4.1 GENERAL I4ATHEh4ATICAI. MODEL
The goa l p rog ra rnming was o r i g ina l l y p roposed by Charnes
and Cooper f o r a l ln ear model which has been f ur ther deve loped
by many o the rs . A p re fe r red so lu t l on i s one wh ich m in im ises the
dev ia t i ons f rom the se t goa ls . Thus a s imp le l l nea r goa l
progranr.ning probl em f ormulation i s sfrolvn belovr z
lvlin imi z e
Subj ec t to
wh ere
x .J
k
n
m
l^\]-
D .. J
Pj ' (o ' - +k
:
j = 1
d .]'
*)
f o r 1 = 1 . . . . I D .
*J ,o r * , d r -V o fo ra l l i and j
d . + x d . -1 1
Dec is ion va r iab le to be found
Nurnber o f pr io r i t i es
Nurnber o f dec i s ion va r iab les
Number o f go a l s
Goa l se t by the dec i s ion maker
The p re -anp t i ve we igh t s such t ha t
P r' >>> nj +r
n
:j =1
b .1
fr\i l
l l )
I n add i t i on to se t t i ng goa ls fo r t he ob j ec t i ves , t he
dec is icn maker must a lso be ab le to g ive an ord j ,na l rank ing to
the ob j ec t i ves . The rank ing can aJso be f oundou t by pa i red
compar i son me thod wh ich p rcv ides some check on t J re cons i s tency
in the va lue judgenrent o f the dec is ion maker . In g^r is method the
dec i s ion maker i s asked to compare the goa rs taken two a t a t ime
and ind i ca te wh ich goa l i s t he more impor tan t i n t he pa j - r . Th i s
p rocedure i s app l i ed to a l . r comb ina t i ons o f goa r pa i r s . Th i s
ana lys i s resu l t s i n a comp le te o rd inaL rank ing o f , . _ t he goa ls 1n
t errn s o f th eir impo r tanc e .
Th e go al prog rannmin g ut i l i ses th e simplex method of
so Jving I in ear prog ramming plcoble'rn. Horr. 'ever r several mo di f ic at ion sare requ i red anc i is o f ten re f er red as f rnod i f ied s implex method | .
4.2 SIF.PS OF TILE SIUPLE(-UFTHOD OF GOAL PROGRAIIMII.JG
Step - 1
set up th e in i t ia l table f r rrm goa-r programming f ormurat j .on.
We assume tha t t he i n i t i a ] so l u t i on i s a t o r i g i n . The re fo re , a l r
t he nega t i ve dev ia t i ona f va r : -abLes i n t f , r e modeL cons t ra in t mus t
en te r t he so lu t i on base i n i t i a l l y p repa re a tab le as s f rown be low .
F i r r up t h i s t ab le i . e . a l l a r j and b i va l ues . The c j co rumn w i l l
con ta i n t t r € coe f f i c i en t o f dev ia t i ona l " va r i abJe because t hese
va r j ab les onJ . y en te r t l - r e so lu t i on f j . r s t . I n i l ^ r e ( r j a : ) ma t r i x
l - i s t t l , e p r i o r i t y . I eve l i n l j r e va r labLe coJumn f rom .Lo l ves t a t t he
top o f t he h i cyhes t a t t f r e bo t tom. Ca l - cu rLa te t f re , j va lues and
2f, l
reco rd i t in to RFIS co lumn .
cj
Var i ab le R . H . S . d ; . . . oi"' x j . .a
bi ci j
Z .J
cj P5
P4
P..,J
P2
P1
S t e p - 2 l .
F ind
comp le te l y
dete rrn j-n in g
va r i ab le o f
i t e ra t i on .
Determin e th e Nerv D: l ter lnq Varl_ab]g
th e h igh es t p r i o r i t y Jeve l , t ha t has no t been a t ta in ed
by exam in ing Z , va lues i n t he R . l i . 5 . co l umn . A f t e rJ J
t j r i s f i - n d o u t t h e h i g h e s t Z . C i e n t r y c o l u m n . T h eJ J
th i s co lu rnn wi 11 en ter th e so lu t ion ba se in th e nex t
I n c a s e o r t i e , c l ' : e c k t h e n e x t p r i o r i t y l e v e l a n d s e f e c t
t t ^ , e c o l u n t r t h a t h a s t h e g r e a t e r v a l u e .
Fl . -
?l
l t ep -3 : Determine tne leavin yar iab le f rom the Solu t ion Base
Div ide the R.H. S . va lues by the coef f l c ien ts in the keY
column. This wi l l g ive the nqi l F[ .H. S. values. Select the q) \ r ,
which has the minimum non-negat ive value. The var iable in that
row wll l be replaced by the varj ,able ln the key column ln the
next i terat ion. I f t j rere exists a t ie , f ind the row that has the
var iable wi th the h igher pr ior i ty factor . In tn is way t l re h igher
order goals wi l l be at ta ined f i rs t and thereby reduces the nunber
o f i te ra t ion s .
Step 4 2- D ete rmin e th e N sr So lu tion
F i r s t f i nd the ne t , R .H . S . va lues and coe f f i c i en t o f t he key
row by d iv id ing o ld va lues by the p ivot e lsnent i . e . the e lement
at the infersec t ion of the key row anci key column. Then f ind the
ne$, varues for a l r o t j rer rov"s by us ing ca lcu la t ion.
( oro varue
key row in
and ,j Cj
S tep -5 :
( intersect ional eI snen t of that row X Nerrv value in the
the same co lumn)) . Norv compLete the tab le by f lnd ing t j
va lues fo r t he P r io r i tY ro l vs '
Determin e wh etn er So ]ut ion i s t i rnal or Not ?
Ana lyse t1 re goa l a t ta inmen t f eve l o f each goa l by c t teck ing
th e Z: va lu e fo r each pr io r i ty rovJ ' I f th e Z:J
J s - v - Y - - - - | . J
t h i s i s a op t ima l so lu t i on ' The r r i f t j r e re a re
valu e s in th e rov,r , d€termin e wh eth er th ere ale
va lue s are a l - I zero
po s i t i ve (2 .J
nega t i ve (2 ,J
t j )
t j )
i,t
2',)
va lues a t 'a
h igher p r io r i t y l .eve l in t t re sdme co lumn. I f there
is negat ive (z j a : ) va lue a t a h igher p r io r i t y reve l fo r theposi t ive (z: a- : ) value in the row of in terest then the solut ionis op t5-maI . F ina l l y i f there ex is ts a pos i t i ve (Z ; C* ) va lue
J J 'a t a ce r t a i n p r i o r i t y l eve l and t he re i s no nega t i ve (Z ; C* )J Jva lu e at a h igh er priority Jevel' in th e sarne co rumn , tJr J. s is no tan opt imal so lu t ion. Henc e re turn to s tep 2 and cont inue.
4.3 COI/IR'TER B45ED SOLUTION OF GOAL 88etr8At\4tu1ING
rn o rde r f o r goa r p rog ramming to be a use fu l
sc ience techn i -que fo r dec i s ion ana lys i s , a coml - ru te r
1s an essen t i a l r equ i remen to
mdnagernen t
based so lu t i on
A f te r su i t ab re mo d i f i ca t i on s the compu te r based so lu t i on
proc edure o f goa l progranrming presented by Lee can be u sed to
so rve p rob lems- The p rccess o f f i nd ing compu te r so ru t i on cons l s t s
o f da ta i npu t , ca l cu l - a t i ng t he resu l - t s and p r i n t i ng ou t t he resu l t s .
DATA INP9T F i rs t o f a l l the fo l , Io rv ing 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 t lon of unc ertain ty
B fo r Bo th d i rec t i on s
L f o r L e s s t h a n
E f o r E x a c t l y e q u a l
G f o r Grea ter t fr srr
f'2:l
t hen t J re gb jec t i ve
manner .
funct ion ln input is g iven in the fo l lowlng
devi at lon-ve/' l 've
row in whlchdev . occurs
p rio rity wei gh t
Then the
cho i ce va r i ab le
data about
is entered
technolog ica l coef f ic ient o f the
l ik e
Row ln wh ic h
" t j appeared
Colurnn ln which
" t j apPeared
Va lue o ft iJ
Then the r l gh t hand s ide va lue o f a I ] t he eqns . a re
en te red .
4 .4 AI{ALYSI S OF THE COMRJIER OUTRJT
Computer so lu t ion o f goa l programming p l lov ides the
fo l l ow ing ou tPu t ' -
Compu te r p r i n t ou t o f i npu t da ta ( t ne r i gh t hand s l c ie ,
t he subs t i t u t i on ra tes , and t j r e ob jec t i ve f unc t i on ) and f i na l
s implex so lu t ion tab l -e ( inc lud ing t j C j mat r ix an d eva luat ion
o f ob j ec t i ve f unc t i on ) , s l ack ana l l r s i s , va r l ab le ana l ys i s and
the ana l . ys i s o f t he ob jec t i ve .
Ij
I
1I
+2. t
!
I
ia l\ i
I' tiIII
24
TliE I-rv\L SIMPLEX SOLUTION
(a ) The R iqh t Hand s ide
shows the r ight hand s ide varues of the var iabreand deci s ion ) . The numbers on. th e ref t han d s ldenumbers for the basic var labres. The real valueshand s ide represent cons tan ts o f the bas i ,c var iabres .
( n) rh e (rj_jt Matrix
Th is shows the (Z : ) *" trix o f th e la st i, tera tion .
Th l s
(d evi a t ion a 1
a re va r i ab le
on th e r igh t
cj
( c )
Th is eva r .ua t i on s impry rep resen ts the
rn o thu r - *o rds , t he va lues p resen t t he r " ' de r
g o a l g .
t j v a l u e s o f g o a l s .
a t t a l n e d p o r t i o n o f
( d ) T h e S l r : c k A n a l - v s i s
RL}{
I t p resen t s
o f t t r e nega t i ve anc i
AVAILABL E
t h e v a l u e s o f t h e r i g h t
po s i t i ve va r i ab le s fo r
N EG-g.K
h a n d s i d e a n d a J s o v a l u e
e a c h e q u a t i o n .
POS- SLK
( u ) Var i ab l_e Ana ]ys l s
VARIABL L /t'ioLilJT
2{t
I t p resen t s t he con s tan t s o f on l y t he bas i c cho i c e
va r l ab l e s .
( f ) Ana l vs l s o f t he Ob iec t i ve
I t p resen t s t he t j va l ues f o r t he goa l s .
rep resen t t he under a t ta ined po r t i on o f goa I5 .
PRIORITY UNDERrcHIEVEIJIENT
These va lues
| *
2$
9.U-AP_TER V-
1 1' l' {IIIII
I' l:ltII
FORMTULATIONOF THE PROBL E4
5.1 G EN ERAL
ABC Company produces the motors o f severa l k inds which
d i f f e r f r ' ' om each o the r i n seve raL aspec ts l i ke f rame s i ze , ho rse
povJe r r R .P . l v l o , nu rnbe r o f po les e t c . I t f o recas ted the demand o f
to ta l ho rse power , t o be p roduced fo r t he yea r 19BB-89 . Manage-
men t es t imated a cumulat ive grovr th o f 15% in the demand of horse
povrer . The demand e. f horse power wd s d i f f e rent for every per iod
( fou r mon ths ) . Hence an a t te rnp t i s made to mee t t j r e demand fo r
eve ry pe r ioc i i n an op t ima l way con s ide r ing p roduc t i on ra t€ ,
i nven to r y . , back o rde r i ng , ove r t ime e t c . Th i s a l so had t he demand
reco rd o f eve ry t ype o f mo to r ( : -n numbers ) f o r t he yea r l gBB-89
gi ven in Appendix ( tab le 1 ) . t t i th - th e knowledge o f the Last year
reco rc i , t he de rnand f o r eve ry k i nd o f mo to r j - s assessed qua r te r l y
f o r t he comp le te yea r ' 19BB-89 (nppend i x Tab le 2 ) . An a t t emp t i s
a l so made to mee t r v i t h t he f f uc tua t i ons i n demand fo r eve ry k ind
o f mo to r i n an op t ima l way . Fo r each f r an re s i ze , t he re we re
f u r t - | e r many k lnds o f mo to rs w i th d i f f e ren t spec i f i ca t i ons .
The re fo re , on l y t t : e r ep resen ta t i ve member o f each f r a r re s i ze was
cons i ce rec i . The t ypes o f mo to r v re re s t i l l t oo many t o make t ne
p rob le rn as a wno le ve ry l a rge t o be dea l - t w i t h . Hence t h ose t ype
o f mo to r v ; h i ch d i c i no t s f r ow much va r i a t i ons i n t he i r mach in i . g
j,tl
II
l
{I
i
, ] a ;
27
t imes we re cJubed t oge the r r €d rcnab l y . I t was rea l i sed t ha t
th i s p rob len r can be so l ved by mak ing agg rega te p lann ing mode .1
which conc en t ra tes on determin ing r r rh ich combinat ion o f th e
dec i s ion va r iab le shou ld be u t i l i zed i n o rde r t o op t ima l l y
ad jus t t he de rnand f l uc tua t i ons w i th in t f r e cons t ra in t s i f doy .
Managemerr t o f the company a lso des i red to incorporate
o the r re - l ' evan t aspec ts such as poss ib l y s tab le emp loymen t f o r
the worke rs ' managemen t po l i c i es o r goa ls re la t i ve to i nven to ry
an d work er sat i s fac t lon an d per formanc e . Th ese are a lso
inco rpo ra ted i n t he p rob l sn fo rmuLa t ion . The ove ra l l cos t
func t ion wa s segregated in to maj o r compon ents i . e. pro duc t ion rate
cost and i r r ventory co s ts so that managemen t c r l t - r l - .ave actd i t iona l
f l ex ib i l i t ; ' i n pena l i zLng dev ia t i on s f rom the va r ious t ypes o f
co s t s .
The moc ie ] op t i n i zes the agg rega te p roduc t i on va r iab le
ds we l l as de tenn in ing the op t , i r na l p rocuc t i on ra te . The co rnp le teprobfsn 1s formula ted in the form of goa l .s anc i is uren so l -ved by
us ing co rnp ru te r based so lu t i on tec l : n ique o f go a f p rog ramming /12 / .The fo l l ow i r rg goa ls a re i nco rpo ra ted i n t he p rob l c rn 1n o rde r o rp r i o r i t y l
( a )
( b )
( c )
( d )
SaJ es reaJ i sa t : . o r r
To l i r : : i L t he cos t
sp ec .i f i c,ci srirc rlh L.
To I i ; : l t t t l r e co s t
sFiec i f ierJ ar!ror jn t .
a s s o c i a t e d w i t i t p r o d u c t i o n r a t e t o a
? s s o c i a t e d ! ' ' r r t i r i r r v e n t o r y _ l - e v e l s L o a
' [ c p . r romote . \ i , c . r - ' ] l e rS r f ro 'L j va t ion t f r r c ; t , rgh La iX) r fo r . ce s ta l , j . J .1 ty .
5 .2 PRT.ORITY ( I \
SALES REALISATII}.I
Eqn. ( t
wh ere r t - t
r t
Pt
st
Le t
and ^ -
s ign above the pa ran tJ reses mean
the pa ran theses can have onJ_y * o r
2B
t ha t t he quan t i t i es
ve va lues respec t i ve l y .
( : )
( : )
) rep re sen t s a gen eraL rel. a tion sh ip .
r t - r +Pt = s t +r t . . . . ( r )
= rnventory a t the end o f t - r t f , per iod
= lnventory at the errd of t t , ' l per iod
= pqr duc t ion rate dur ing t th per iod
= Sa les i n t t n pe r iod .
Inventory dur ing t th per iod
Sror tage dur ing t th per iod
( t ). L /
( r . )
The +
ins i de
Le t
Tlt en
Th e re fo re
and
By us ing
+a
+a a
. t +* t
T +' t - 1
tran sfo rrna t ion ..
= la l a 77 O
= 0 o therw ise
l a l a
O o thenv i se
=a
a
1t
I t - r
1 t
I t - r
For conven ience ,
t r* =
and r l =- t -1
Ie t u s pu t
oa*
oJ-t
rt-
rLr
Dt-
oa-t
2lf
( 1 c )
Eqn s . (2) an d ( g ) c an be rewr i t ten d s
oa* - Dt- = rt
oi-l - ot-l = rt-r
From eqns . ( 1 ) , ( 4 ) and (S )
Pt = st+(oJ-o.)-(oJ_,
. . . . ( q )
. . . . (s)
DLr ) . . . . ( 6 )
1) Zeto (z)
(B )+s1
. . ( g )
T - = T =- r . ! I l -L - t o( oJ-,
= (q*
D+
Di)Fro rn (6 ) and ( z ) p1
ePz=
F r o m ( + ) a n c J
Pz
F r o r n ( B ) a n d
Fz+
Iz + 52 I t
(oJ- q) +(s, +sr)
(s)
( e)
FrY 1
I
,..1
.;,i*.,".il.
E
*,3
pg = 13 *S3 12
Fmm ( q ) and (s )
Pg = (oa* - D ; ) *s3
From ( t o ) and ( i l 1
(D ; - D ; )
; i0
" " ( t t 1
. . . . ( lz1
and
P, +P^ +p^I z - 3
Thus f o r each t ype o f
12 fo r t J r ree p lann ing
For F;<arnple z
Type A mo to r
D;) +sg +sz *s1
motor there are tJ r ree eqnso
pe r l ods respec t i ve l y .
= (oa+ *
8 , 10
PR't
PAt +
PRt +
moto r
Pgt -
Pt't +
Pn t +
m o t o r
Pct
+=DRt+
Paz t;
Fez + Prc
Dnt =
J- r'\' uA2
sRt
set
a a o a
a a a a
( t :1
( 1 4 )
n-rLJ
+ sez
sRt +sRz +sag o . . . ( t : 1
Type B
Typ e C
ofi r ou'
Psz 'i,
P,3z + Pa:
= su't
+ou, =
ui + D,r:
. . . . ( te1
. . . . ( 1? )sg t + sez
Sst +s i rz +seg
, - +t r l ua., s n 1
\ z l
. . o ( t s1
. . . . ( 1a )
3t 1--)
Pct n Pc2 ot, sct + scz . . . . (zo1
Pct * Pc2 * Pca tJ. * Dfs sct + scz * sca . . . . .2 l1
+ D^^ =vz
Type D
Type E
motor
o?'
not +
motor
PEt
ojt +
Pm+
tJt
P-^
fo1 =
oJ, +
p'D3
{r+Dez
* so2
spt +
. . . . (zz7
o . . o ( zs ;
. . . . Q+1
Pot * Pp
sot
Db
oi sP+ sog
set * sE2
+ DE: = set * sE2 * sE3P- . +P -^ +P , - ^Et cz t r , J
S im i i a r t : , pe o f
t ype o f mo to rs and we re
sgt . . . . (zs;
, , . . . (2a1
. . . . Q l7
PEt +Da
^+'E3
eqn s .
gi ven
can be w r i t t en f o r F , G , H , I & J
th e ecn s. number f rom (ZA to 42) .
5.3
TO
pRrontry ( r r r
LJIII_I rr{E cosr (r' ASSOCIATED WITH PRODUCTIONRATE
w h e r e S t a n d a r d v a r i a b l e
u n i t o f p r o d u c t I
T h e c o s t p e r o v e r t i m e h o u r
h lanageme 'n t I s ta rge t Je .veJ
PRct
p ro cfuc ing on e
. . . . (+s1
f o r p r o c h r c t i o n r a t e c o s t s .
'Jt =
c o s t o f
"l
. RCt
Pi t x c i * cTot + Dot
a1
DJt' DZt
Pi t
Dev ia t i on a l va r i a b les
Prod rc t i on ra te fo r i t h t ype
du r i ng t t h pe r i od (Oec i s i on
Over t ime hou rs i n pe r iod t
id l e t ime vva s
eve ry t ype o f
J]?
o f mo to r
va r i ab le )
no t a1 lowed .
motor is g iven
ot
I n t he p i esen t p rob len ,
The cos t f o r p roduc ing one un i t o f
i n Append i x ( t an te 5 ) .
The eqn . (+e ; f o r t h ree p l ann ing pe r i ods can bewr i t t en as f o l l ow5 ,
Fo r t = 1
11€2 Pat
1 6533 Pr r
Bot + DZr
3553 Pet
2443 t Oo1
_ +'61 =
662C Pct
3 0e1 1 PHr
l OZl q pOt
468 00 p l t
24266000
662C Pcg
3 0 C ) 1 1 P , , ^t l J
Pcz + 1021 4
PH2 * 468 00
. 12675 PEt
7 A2cO p-, t
, , . . . ( q q )
Poz + 12675 PE2 +
PtZ + 20200 p lZ +
. . . . ( 45 )
12675 Pe:
7 02 C0 F; :
. . . . (+o1
For t - )
Fo r t - 3
1482 P,e. + 3553 Pez + 6620
1 6533 Prz + 24431 Por+ 3Og1 1
uoz *D62 DOZ - 24266C00
14€2 p^^ +l{J
1 6 5 3 3 P - - +r J
B C ^ + D . -< | - \ <
V J
3553 Ps: +
24431 p* +
1O21 4 Po:
468 CCr pl :
,Ja = 2.1266 c)oc
: i3
5 ,4 PR IORITY ( I I I 1
to ttrr:,tt rne cost (Rs.1 asgoctRteo wttltIIWENTORY LEVEL To SPECIFIED .4{vlCx.JNT
I nven to ry cos ts a re ano tJ re r
agg rega te p lann ing cos ts and fo r
cos t s , and back o rde r cos t s .
impor tant component o f to ta l
f i n i shed goods i nc lude ca r ry ing
t -
1.1!4
#'i
wh ere
In genera l
t.i
form 2
toi )
cni I
qt i
cl1
+Dit
0
, ^1 0 n-+ c i - Di t ) + %t
"i.
Bac k o rder quan t i ty
Dev ia t i on a I
rct
o f p r o d u c t i i n
v a r i a b l e s .
o. . . (+ ty
i i n pe r iod t
pe r io d t
1257 (D; )
(oJ' ) +
+ 1018(oJ. , )
cos t i ncu r red f o r ca r r y i ng one un i t o f p roduc t
cos t i ncu r red f o r one un i t o f p roduc t i , back -
o rde red pe r pe r i od
F in i shed goods i n ven to ry o f p roduc toi; -
Di. =
Dit an ci
The va l ,ues o f C? an d1
app en d i x ( t an te 4 ) .
1 nCi
' ' f o r e very t) 'p e of mo to r are gi ven in
Fo r t = 1
1360
57c0
+ 573 .9 (o i ) +3006 .6 (D ; ) + 3804 .4
+ E64c (o_i . , ) + z2B (oo. , ) + 514 (no. , )
T h e f i n a l e q u a t i o n s a r e a s g i v e n b e L o w 2
1E;2.4 (D;J + 41 i .2 (oJ ' ) + 814.6 (DJ l ) +
{r
[fITIrilit:ttlrJt, l
,1'irt
iI
A
{ t
; i4
22,00000 .
182 .4 to [ )
, l l
:J
ifr,I
,f;.rl:;l: f
1571 (oor )
72CO ( or , )+
n
\z " lz 22 , 00000 .
573.e t {. I
8640 (o_i.) +
o . . . ( 4 9 )
+ 1257 to$l
. . . . (so1
cho i c e
+
+
+
+ 3006.8 (o&) + 3804.4
228 (orc) + 514 (o-r .)
+ 717 3758 (n[. )
22 , 00000 .
( { . ) +
(oJa) as i f they were
respec t i veJ_y .
1521 (Der) +1e50 (Dur) +717 (or ' ) +3?58 (0E. , ) +
4755 ( o[, ) + 72oo (oI ' ) + 10800 (oJ , ) +
+ 411.2 to j r l + 814.8 to&l
1560 (oL) + 573.e (+) + 3006.8 (DJr) + 3804.4 (D;) +
+ 514 (D ;2 )5?60 (o i ) + 8640 (D; ) + 2zB (o_) + 1 o1B(ofr ) +
+ 1e50 (DE2) + 717 (o i r ) + 3?58 (%) + 47s5(or r )+
+ 1 CrB 00 (fr) +
. 182.4 (o i . ) + 411.2 (o i ) + 814.8 to i t
1 560 toi l
5?60 toi I
157r (o f . ) + 1q5o (oo. )
72oo (o i . ) + 1 0B0o (oJ . )
I n ou r case we t r ea t (O r ta ) and
va r j . ab les say (U ra ) and (V ra )
T h e r e f o r e t h e a b o v e e q n s , f o r t _ j , 2 a n d 3 c a n b e
e x p r e s s e d a s b e l o r v t
qt
. . . , . (4s)
+ 1257 (oJr) +
(o,i. ) +
+ 1 018(of . ) +
+ 4755( o[. ) +
;r5
182.4 ue t + 411 .2 Ue t + 81 4 .8 Uc1 + 1257 Uot + 1560 Uet +
573.9 Unt + 3006.8 uc t + 38 O4.4 UHt + 5760 u l l + 8640 Ut t +
228 Ve' t + 511 Vgt + 1018 Vct + 1571 Vpt +
3758 Vot + 4755 Vut + 72oe Vt t + 1osoo Jvt
22 r00000.. . - . . . (st1
182.4 UeZ + 411 .2 UeZ + Bl 4.8 Ucz + 12s7 UOZ + 1560 UeZ +
+ 3006.6 UCZ + 3804.4 UUZ + 5760
514 ygZ + 1 01 B VCZ + 1571 VOZ +
+ 4755 yp .Z + 7200 y tZ + 10800 y lZ
1950 Ve t + 717 V f t
*4r 4t =
utz + 8640 ulz +
1950 vp +717 Vrz +
+42 t, =
. . . . (sz1
573.9 Urz
228 VRZ +
3758 VcZ
22 ,00000 .
1 92 .4 URg + 41 1 .2 UA3 + 81 4 .8 UCg + 1257 Ua: + 1560 U:s +
573 .9 u rg + 3006 .E Ucs + 3804 .4 UHg + 5760
228 Veg + 514 Ve : + 10 tB VC3 + 1571 Vpg +
3758 VC: + 4755 Vng + ?200 V l : + l CAOO Vt :
22 , 00000 .
utg + 864o u.rg +
1 950 Ves + 717 Vfs +
+4s Dz*. =
. . . . ( : :1
p resen t i n t he eqns . o f f i r s t goa l
The re fo re , t he f i r s t goa l eqns . ( t : )
i n t e rms o f U i t and V i t anc l a re g i ven
S i n c e
( s a l e s
to (a2)
b e L o w
+(o r . ) and (o r . ) a re
r ea l i sa t i on ) a l so .
a re aJso exp ressed
a
i
I
?-,., ;t r;
oo., + Vet URt = 7120
Uez +vM = 13314
+ P,t3 UA3 * VA3 = 200C0
Pnt + Prz
Pnt + Paz
IJp. ' ( g )
out + ur t
Pt t + Pr ,
ou l + Pgz
Tvpe (C t
PCt * VC1
oa, + Fcz
oa't + P cz
T-rcs-lelPot + Vot
ool + P o,
Pnt + no,
utt
uuz +
+ P ^ ^TJJ
-^ 'a t
'cz = 159
Vcg = 43C
o a a a
a a a a
. a a a
( s+1
(ss1
(so1
(sz;
(ss1
(so1
(oc;
(ot 1
(oz1
3277
V gz = 6569
Ug: *Vg3 = 10c75
'it
11C
+ vcza a a a
a a a c
+ Pc: Ucg +
uot
urrz +
+ rog
= 114
Yoz = 293
Uog * UD3 52't
a a a a
a a a a
a a a a
( o:1
( 64 )
( o:1
Tvpe ( E l
Pe' ' *ua ' t Uer q2 (ar,)
F 37
. . .o (oz1
. . . . (oe1
. . . . (oe1
.. . . (zo;
. . . . (zt 1
. . . . ( lz1
. . . . (zs)
. . . . ( t+1
.. . . (zs1
.. . . (zo1
. . . . ( l t \
. . . . (ze1
.. . . (zs1
. . . . ( B r )
P- .t r l
Pet
Prt
Prt
Prt
Yrz 330
Tvpe (F \
+ P-- tF 'L_.L
+ P..^ +r-z
*VF1 -
+ Prz
+Prz+
Urt
urz +
p' F3 urg + Vrg
30
+ Pcz
+ Pcz *,
* u*t
+ PHz
+Puz+
UH,t
ugz +
p'H3 UHg +
23
Vnz B2
Vua 135
Pt t
nt t
ot ' '
B
Ytz
* Vr1 - U l t
30
Ur:
Ulz +
P l g + V l g
; rr-_
irus-ls)oot
oot
Fct
PHt
nn''
PHt
* Vcl 'ot
uez * Y F2 224
PE3eue3+ Veg
145
Ucz * Vc2 B5
Pc: Ucg * VG3 145
320
460
BO
1, \i 6. . I
3B
tvpe ( ; ' f
n,r t +v*-1 = u; ' t = B
P. l t *PJ2 -u lz*YJz = 12
P.tt * PJ2 * PJ3 u.rg * VJg = 30
5.5 PRIORITY ( IV \
LABO.JR FORCE SfABILITY GOAL z
. . o . ( 8 1 )
. . . . (92)
. . . . (as)
. . . . (gs)
Bnployee mot iva t ion, per for rnanc e on th e job, and
sat is f ac t ion der ived by vo rkers are a l l enhanced vuh en. workers
perce ive a s tab le snp loyment env i ronment . Fur t t rer the f inm may
fee l that 1 ts lmage ln the labor force is enhanced t f r rough t ] re
e f fo r t t o ma ln ta in work fo rce s tab i l i t y . ID genera l ?
*t *Dit oi , = et
wh ere r * t = Ch ange ln th e number of work ers in
p e r i o d t .
Manage{nen t d id no t a l l ow h l r i ng o f t he worke rs .
Th e re f o re * t rep resen ts o n l y t he number o f wo rke rs h l red .
DZ t and OJa = the number o f wo rke rs Jess than o r i n excess
o f t he des i r ed max imum respec t i ve l y .
Q t = Max i rnum des i red change in work fo rce Jeve l .
i
t3ff
For t h ree p l ann ing pe r i ods , t he eqn .
a s be.Iow z
( 83 ) c an be wr i t ten
Fo r
Fo r
Fo r
*2+
*3+
Dzt
"22
%,
D)t
n +u22
+'23
1 ,
2 ,
3 ,
t 1 . . . . ( 84 )
. . . . ( 85 )
. . . . ( 86 )
5 .6 CONSTRAINTS
5.6 . 1 Prgduc t i ve Hours Cons t ra in t
The hou rs r equ i red f o r t he p roduc t i on o f va r i ous k i nd
o f mo to rs shou ld be equa l t o t he e f f ec t i ve hou rs ava i Jab l . e .
r n case t he hou rs r equ i red a re Less t j r an t he hou rs ava i l ab le ,
we can go f o r ove r t ime as we l l as can i nc rease t he wo rk f o r cedur ing t f re *no r rna l wo rk ing hou rs .
In Gen era j. z
wh erc.
*3I
+x
rt- -t ' J o r m a l _ w o r k i n g h o u r s .
. . . . (B?)
un i t o f moto r i .
f o r o L d w o r k e l t s .
f o r n e v J \ ^ b r k e r s .
T i P i t = T1 (wa- . , ) x ( t t . vJ . h rs ) * + T2 S t x (N .v i . h rs ) * +
?T" O,
T,
hou rs re ( l u i r ed f o r one
e f f i c l ency coe f f i c i en t
e f f i c i ency coe f i - i c i en t
e f f i c i ency coe f f i c i en t c i u r i ng ove r t ime hou rs .
nun - r i : e r o f wo rke rs h i r ed i n t t n pe r i od .
T.l_
1T
')T*
rn , f0
{I
tl rlj{^
$Iit :
. l tj t i, t j. b ri t i
$̂{
tit
l '{'irlrf :
iiI
tI:I
:I .
t .l i
. iri 'Iir! r:.,l !ll
; 't;FItt .
ir1"l::,
il'
n 'sf.t
:'..;
- . 'i1
The fo l rorv lng recurs ive re lat tonship 1s ar-so required.
wt- l +Xt = t t
I t shows tha t the labor fo rce s ize in per iod t w i l r equa lto the la 'bor force s ize of tJre previous per iod p lus tJrej.n c rea se in wo rk ex s durin g p eriod t .
For t
For t
v{ l =wo +x1
wz =v{1 +\
or wz =wo +\ +\
For t J w3 =w2 +L
"iws=%+
By us ing the va lues o f T l ,
eqn . ( 87 ) i s w r i t t en be low
The wo rke rs e f f i c i ency coe f f i c i en t f o r o rd and( i f h i r ed ) we re known f r om th i s
Xr *\ +L
g iven i n append ix ( f "b - l e 6 ) , t he
for tJ r ree per iods.
n e$/ vlo rk er s
Ol- d Vrro rk er Nevr V[orker No rmalh r s .
Over t lm eh r s .
E f f i c i e r r cy
Coe f f i c i en t1 .00 o.B 1 .00 1:00
C
I
1
Iia
II
t i
iIIt
For t = 1 .79 pa, t
4 .19 Pg t + 4 .gg p f t +
13 .36 Pr i = 1 x 5 x 1J I
Or .7 g pRt
+ 1 .48 Pg t + 2 .65
6 .04 PCt + 8 .2 pHt
616 + .B x 1616 x
Pct + 3 .33 Ppt
+ 11 .39 p t t +
(x . , ) * 01.
+ 2 .65 Pc t + 3 .33 pp t + 4 .1g
6 .04 PCt + 8 .2 pHt + 1 1 .39 p t t + 1 3 .36 P l t
+ 7 .48 Pe1
4l
P- . +t r l
4 . gg P - .r l
1292 .6 X l o1 BOBO.
For t = 2 .748 p , lZ
4 .19 PEZ + 4 .99 p fZ +
+ 1 '48 Pgz + 2 .65 Pe .
6 . C4 PCZ + 8 .2 pUZ + 1
. . (BB)
+ 3 .33 POZ +
1 '39 P tz +
13 '36 Prz 16co x1 12Bo xz 02 = Booo . . . . (s9)
For t=3 ' -748 Prc +1 .4 pa3 +2 .65 pcg +3 .33p0 : +
4 ' 19 PE: + 4 '99 Png + 6 -a4 Pcg + 8 .2 p 'g + 11 .39 p lg +
13 .36P- l g 161 6 \ 161 6 \ 1292 .8x3 O :
5 .6 .2 o/ ERTII, i: COIJ STRAIT,jT
fh e manager , manu l .ac tu r i ng
the ove r t ime bu t no t mo re t han 1 O
ho u t s .
8080
. . . o ( so1
s e r v i c e s d i v i s i o n , d l l o w e d
p e r c e n t o f t h e n o r m a l w o r k
iI
, lI
II'lF'r! ,12
There fo r€ r t h e o ve r t ime con s t ra in t s f o r i h ree pe r iods
a re g i ven be low z
For t =1 01 +d6Z = B0B . . . . (91 )
i -A toJ
t=1 to3
For t = 2 02 +O6Z = 800 . . . . (92)
For t = 3 Og +o5s = B0B . . . . (gS)
Thus t J re ob j ec t l ve o f t he p rob lem i s t o m in i rn i ze the
dev ia t l ona l va r i ab le and i s f o rmu la ted be low z
Min z = p1 : 1 .25 (Di t ) + 1. c0 tof . ) * p2 g to j . ) +t
i r t r v 4 r ' < -
t = l \ " ' u
33Ps i (%*.) + P4 E coJ.l
t=1 t=l
Sub jec t to : Eqns ( tg ) to (9g) , dJ ready g iven-
- o o o -
i
llI
- \i ' j
a . lr f r 1
a.H.aBlEB-y,t
DISCUSSION OF RL9ULTS
The p rob rem fo rmura ted i n t he l as t chap te r has been
. so l ved by the compu te r . The comp le te resu l t s a re shown in
: Append i x ' Th e ma in resu r t s a re d i scussed be row t
.i AI.I EVALUATION oF THE ozuECTIVE zuI\cTIoN
:, 4 o.ooo'
3 BO1?2B.oo
0 .000
0 .000
Th j- s show s th at tJr e 1 st , 2nd and 4th go aI s are ach ieved ful lywh i l e t h i r d goa l i s no t . Th i s i s due t o t ha t t he es t ima ted
ta rge t cos t o f * p roduc t i on i s l ess t han t he ac tua l cos t o fp roduc t i on - The va r i ab le ana rys i s , g i ven i n Append i x , i sexp la ined be -Low z
VA.RJAi]LE DESCRIPTIOI,I
37
62
26
52
7
76
1
AI{OTJNT
1 49 . 00
3543 . O0
4459. 00
59 . 00
259. 00
464. 00
207 9 .7 C
L _ - -
rl1 1' lI
'aA
I
?IJ
t' !
,,
44
i
61
95
BB
27
65
66
19
14
4
11
12
40
5
21
22
59
2
94
9
3
15
10
29
56
25
504C .2O
.56947
B. oo
2 '7 .40
1692 .15
51 98 .07
85 ; 00
132.00
2975.-62
14 .24
232;00
154 .?5
1901 .00
60 . 00
92 . 00
18 . 00
7690 ;50
.87205
171.00
0229.53
96 .00
268. 00
3o. oo
22.59
B. 00
?
fr#wir&qd
i{t:,.'ZTV
r
! * ,
.A;i1.4
&wF'r*.- { j -. . :*'sg :
faY.lt-*rc'
Fftii i r r
fi
i'l'H$+ff&
tri*gr.
$',ii
;.,r"
k(tl
t t
I
II
I
45
49
77
24
76
64
13
55. 00
329.00
53 . 00
1 45 . 00
301 .35
92. 00
Thi s tabLe
o f each dec i s i ,on
ana l ys i s r vh i ch i s
g i ves t he ana l ys i s o f t he
va r iab le . Th i rd th ing i s
aI so reproduc ed below z
obj ec t i ve i . e. Enroun t
to d i scuss s lack-
NEG-s-K
0 .00
o: oo
0. 00
0 .00
o. oo
o. oo
0.00
o; oo
0 .00
0 .00
0 .00
o; oo
0 . 00
0 . 00
ROttl
1
2
3 r
4
5
6
7
B
9
10
11
12
13
14
AVAI LABL E
7120
13314
2 0000 0
3277
6569
10m5
110
250
430
114
293
52-5
92
224
POS.-SLK
o; oo
o. oo
0.00
o. oo
0.00
0 .00
0 .00
0 .00
0" 00
0 .00
0 .00
o" oc
0" 00
0 . o0
"l'. l [ ;
15
16
17
18
19
2O
2i
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
+o
320
145
330
460
30
B5
145
23
82
135
B
30
BO
B
12
3C
24266 000
24266 0 0 0
242660 00
2200000
220 0000
2200000
8080
BOCO
B C € O
0.00
o; oo
0.00
0. 00
o- oo
0.00
0" 00
0 . 00
o; oo
c.00
0.00
o. oo
0.00
0 .00
0 . 00
0 . o0
o. oo
0 . 00
0 . 00
0 .00
0 .00
B01 728 .60
0 .00
0 .00
0. oo
o; oo
0; 00
0 .00
0 .00
o: oo
0.00
o. oo
0. 0o
0. 00
o; oo
o: oo
0.00
o; oo
0" 00
0 . 00
0 .00
o: oo
0.00
o. oo
0 ,00
0 . 00
0 . 00
0 . 00
o" oo
0" 00
1 . 12- t 950" ( rC
i ')r
47
41
42
43
44
45
2. 00
J-t ooI eoa
):r -+ .g o0
BOB
0; 00
0 . 00
o; oo
o: oo
o: oo
1 .43053
-1 .000
.BA7 .72
7 gg .42
3 0./ ;99
This table 1s sel f expla lned. This table shows for eachand every row, how much was the r ight hand s ide and whether thef ina l so lu t ion has exceeded the above s ta ted (R .H.S . ) goa l 1 .e ;Pos' -sLK or l t wa s under achieved i . e . NEG-SLK. FrDm the t j * jmatr ix one can veri fy t tre opti-marlty of U^re problern. Thi.s showsnegat ive en t r ies a t 1 s t and 2nd and 4 th p r io r i t y rever . pos i t i veen t r ies arce there but at th i rd pr ior i ty level . That means theso lu t lon 1s op t ima l .
SUGGESTIOTI FOR RJRTHER TTORK
rn the absence o f p ro f i t da ta , ( due to the sec recy ) oneof the lmportant goar of the organ . 'zat lon to make maximum prof l to r t o a de f i n i t e f i xed ta rge t cou ld no t be fu l l y i nco rpo ra ted .A l though i t was t r ied to incorporate i t ind i rec t ly by f ix ingproduct lon ra te cost to a predec ided l imi t . For sarne motors ,s tandard t ime da ta were no t i n t he reco rd o f t he company and wereto ld by j udgernent ; Had a l r the s tandard t lme data been prcv ldedexac t l y t he p rcb r -em cou rd have been be t te r t han th i s .
H
- O o O -
"t r)! ,l 8
APPENDIX
TABLE - 1
Frame-wl se dernan d of Mo to rs fo r I 9BB-89
S.No l Frame si ze H. P . /Motor Quan ti ty ln Numbers
1 .
2 .
3 .
4 .
5 .
6 .
7 .
B .
9 ;
10 .
11 .
12 .
13 .
as,BO
90
100
112
132
160
180
200
225
250
280
315
355
1.0
2 .0
3 .0
5 .0
10 ;0
15 ; 0
25 : 0
4o; o
6o; o
75. 0
100 .0
180 :0
270 .O
15
40
50
75
125
2600
35 00
4000
6 000
65 00
6000
1475
500
350
75
120
BO
30
250
180
280
BO
40
a.14 .
15 .
16 .
17 .
1E .
160
180
200
225
250
t'
Iteh
4t|
19.
20,
21 .
22.
23.
315
180
200
225
250
270
25
40
75
100
25
40
30
30
15
g
Tab le - 2
Denrand of motors on Quarter ly BasisI'
ji
s{il.t
tHR*Fs
rii l
iiEtD
S.No .
Framesl ze
May l June tJuIy r Aug.
IBB
Sep t . rOc t ; t\lrc v. I Dec .
IBB
Jan . lF€b . 1Max. ,Apr l
rBg
1 .
2 .
3 ;
4 .
q,- r a
6.
7 .
B .
9 .
10 ;
11 .
t 1 .
ag.BO
90
100
112
132
160
180
200
221
250
280
315
355
720
809
1 425
1904
n82
2 033
515
106
110
19
23
B
B
753
1237
946
1 938
2073
1 972
567
163
149
27
44
22
4
1118
1454
1629
21 58
1 995
393
231
91
29
53
50
50
1B
I l , !
. i n50
a.14.
15 .
16 .
17 .
18 .
19 .
20 .
21 .
22.
23,-
160
180
200
2%
250
315
180
200
225
250
54
56
74
29
4
4
B
1B
6
7
75
74
114
26
22
6
121
50
92
25
14
5
1
4
14
17
s
16
1B
10
6
FrameSl ze
Trme/Un 1t
Dernan dGmup 1s t
p erio d2nd
p eriod3rd
p erio d
Averagetime/un it
Qu 90
Qu 100
Qu 112
eu 132
160
BO
140
.7 13
.7 775
.7 415
.8005
1 .317
1 :4B5
1.504
Qu
Qu
Qu
A 7120 61 94 6686 :'7 4825
3277 3292 29A4 1 .4885
I
r r | :t ) l
a 1 60 2 .533
c i 180 2 .88
Qu 200 3. 1 09
s 180 3 .357
c 110 149 171 2 .656€
D 114 17 9 232 3 .333
a 200 4 .187E 92 132 96 4.1q7
s 200 4.207t:i
i q 225 4.882
s 225 4 .482 F 145 185 130 4 .996
Qu 225 5.226
a 250 5 .903
s 250 5 .903 G 11 28 31 6 .0413
Qu 25 0 5- .31 B
Qu 280 7 .Y79H 23 59 53 8.20-7
a 31 5 8 .435
Qu 315 1 i .395 I B 22 50 11"395
Qu 355 13 .565 J B 4 18 13 .365
L
f '; 52
Tab le 4
Group Inventory Carry ing Cost(Bs; )
Co st of Shortage(ns; )
A
B
cn
tr
F
G
H
I
J
182.4
411 .2
B 14 .8
1257
1560
573.9
3 006 .8
38 04. 4
5760
8640
228
514
1018.6
15? 1 .4
1950
717 .39
3758 . 6
4755 .5
7200
1 0800
Tab le - 5
Pm duc tion Co st (n . ) fo r e very typ e of Mo tor
1 .
2 .
3 .
4 .
5 .
6 .
7 -
B .
9 .
10 .
A
B
C
D
E
F
G
H
I
J
11€2
3553
6620
loz tq
12675
1 6533
24431
3C91 1
468 C0
70200
- - . - l p g 1 \ l l l
1888l"RHI
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| .1.
I foooo l r , * r l I. . . . l o 1 O H l O ,( t ( r g n g r l F t f O F l
l'[FFl il:::: RF"i"| ; FEsdE l.t .l
$ F'' 'lllr#l l ll
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e
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sbbbl *3l -dd dd
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ANALYS I S
71io, fi,Y$,lhABLE1f, f , 14 . t lOt-r tJO?QUt-rr_r. rJi]rl,:Jo
f,i 1 7. t1t111666569. t-rt-rt_rt_rrJ
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f,:9. t_rc-rr-rt-ttJ9?. r l t i t : r t ) r i
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Jt i . tJr-r t j t - r rJ8(i. (J(j(j(J(:)
1. 45 . rJntjot-ri.f . tlt:trjt:rt-rB?. tJt- t tJtJtJ
1JE. tJt--rrJt)r_r$, t.tr-tt-rt-ltJ
Ir_r. r)r_rr_lr_ir_iBr-r . r j t - r t - r r - l i_r
E. i - l - r t_ t t_r r - l42. t_r t j r_r t_r t_rl l t - r . t_r r ' r t ) t - l r_ l
? 4 : j , !r 6r_rr.rr_r " t_t i- l t_lr_tt-r: 4. ? 5 cr(:i rJ r_r . i--rt-it_rr_r r_r?. 4 :i 6 6.-r r-t r-i . t) r-rr-r L-r t_r
" :;! 6 ot-tt_r . {i r-it-ltl i-r
r?6or- r r_r . r ) r_r r_r rJr_r:?e,6( j ( : r . r . r r - r i - r r - r r - r
g(:) g ( j , i_r t_rt_rt_rr-rBt - rBr j . i . t r - l r_r r - r t_r8r_rFr- r . i ) r_r r - r i - i t_ r
] " t - t t - i , - t t - r t _ rI " i_ r r_r t_r r_r i - r. ] " i _ r t_ l i - t i - r i - t1 _ t_ t i , l c - t i _ r t_ i
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t-tOt-rt-rr_lt_rr-rrJOCr(l(looot j t . l t jOc_r(:!O(:r{j(_)(J(Jr:,O(:l(J(:)(:)(:) i:rt_tt_tt..ti-rt_lt_-r r_r rll r_r r_rt-t t_lt_lr j t l ri:r (:) (--) (:) (:)t.tt-rt-tt-rt_rt-r r_l r-r t-r t_1r-t t_r t_l t_t t_r(:)(: f( : ,( :){ j
r_tr j r j t - r t - rr_jtjr_lr-rt-rt_r t-r t_r r-r rlrr-r r_i r.l tlr t-tt_t t-r tlt r-l t_rt._1r.-tt._l(lt.-lr_t t_r t_i t_l t_tt j r j t_t r - r r - rr - i r ' r t_ l r - t t - i= t 7 : , q4 f , r iSei-rr_rr_rrlrt.t
t_t r_t t_l r_t t-ti l t (- i i - t (-r i - t{_ l t_ t r - t i - r l - ,
t_i i_l r-t i - t t ' ri-t t_ i i-t t- t i_r
i ' t t - t t _ l t _ t t - t
t- t t- t t- l r-r r- i( :) (-! (_, {-t r-rr-1 t-t r-t i-t t- ti - i i - l i - t t- t '
- i
t- t t- l ' - l f- l ' ' - i
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1 9 7 5 . 6 1 . 9 8 f ,1 4 . : f , 7 9 6
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6 t - t " t _ t r - r ( - r t - t t . rg r-. . rJr_r A7 41 B. r- t t_rr-rr ' r i - r
A I v A L Y S I S n F
F F i t O R I T ' l
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. o g : 1 : j5 ? F q c ,r_ r i:l (-l (-) (- )r. 'tr-r 4 4 4i j t _ r t - t t ' r i - lr: a'1 /-\ a /
" ' l ? r - 1 7 { :, i - r t - r t - t 4 t r
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T l - J F n F . I E r T 1 c , , F
LlFln E Fj-,:rf--H I F rrEf.1 E I, l r
f_r " t-t r-l t-t t- t t- r8t" t 1 ; l rg. r_tr_rr_tr - r t_r
t - i . i _ l t _ l t _ t t j t - ti - t . i_ l i - t r l l i - r i_ l
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R-EFERENCES- - - - - - -
Bowan, E.H. , 1956, Pmduct ion Schedul lng by the
Tran sportat ion Method of Linear programming, opsi
Research 4 .
Hanpsnan, F . and Hess, w. , t4 L lnear programmlng Appmach
to Product ion and Bnployment Schedul lngf.
P lann ing, Prb ch. rc t ion, Inventor ies and l tb rk Force by HoI t ,
Mo di gl l an i , lvtuth an d Simon-pren tic e Hall
Berg strom r Gang L. and gni th , E. , Multi-ttem pro dtrc t lon
Prannlng An F-xtenslon of the HMMS h-rre, Managernent
Sc lence r VoJ . 16 , No . 10 , June , 19?0 .
Lawrence r K .O . and Burb r idge , J . J . , A Mu l t i p le Goa l L inea r
Prog ramming Mo del f or Coordj-n ated Pro duc tion and Logi stic s
P rann ing . I n t . J . P rod . Resea rch , 1976 , Vo t , 14 , No . 2 .
Goodnan , D .A . (19?4) , Cca I p rog ramrn ing Approach to
Aggregate P lann lng o f Product ion and l to rk Force. Mgrnt ;
Sc l . 20 , 1969 -1V15 .
Tang , John c . s. , Adurbhom an d Zubai. r , Tah i r , An Aggregate
Pnrduc t i on P lann ing fo r a t {eavy Manu fac tu r i ng Indus t r y ,
In t . J r . o f Pro duc t lon Research .
Jaake ra i "nen , v . (1969) , A Goa l p rog ran r rn ing Mode l o f
Aggrega te P roduc t i on P lann ing r S r . red i sh J . o f Econon ics ,
1 4-27 "
.1,*.-- . :-- . : . , ; i -- .
j " i i
63
9' Thomas and Hi I l , fA Ner, r Model for Aggregate output
P lann ing t , Onega , Vo I . 16 , No . 3 .
10; Ignlz io, James P. , A Revlery of Coal Programming z A
TooI for Mul t iobj ect ive Analys is; Journal of Operat ion
Res . Soc ie ty , Vo l . 29 , 11 , 19?8 .
11; Decis ion Systenrs for Inventory Managernent and product lon
Pranning by Reln peterson & Edruard A. sirvexr John u|lley
& Son s , Nerrv Yo rk .
12. GoaI Programming for Deci s ion Analys ls by Sang, M. Lee;
13. L ln€r opt in izat lon for Managernent by s. [ i . Leei
14. Int roduct ion to Decis ion Science by Lee and Moore;
- O O O -
,