l r- a \

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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\8

b2

- t r O O -

b .1

ci

c?,-1 0" i

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

g.,}.j

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rrfis

$*.,ry

#rIPfr,#:l$ri,

#i,!

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2

I!

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i

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

l l\' l

t;

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

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f,i 1 7. t1t111666569. t-rt-rt_rt_rrJ

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Jt i . tJr-r t j t - r rJ8(i. (J(j(j(J(:)

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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

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1 9 7 5 . 6 1 . 9 8 f ,1 4 . : f , 7 9 6

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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

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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 -

,