INLAPS : AN Jf\iTEGRATED LAYOUT PLANNING - . SYSTEM

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INLAPS : AN Jf\iTEGRATED LAYOUT PLANNING - . SYSTEM

<-

Som e alg01;ithms

[ortne

FLf)l8ing'Qu a dr~tjcProgramming andStatisti~alAna.l~sjs

by'

'

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'

Vijay.Sankar B.~ech.[Hons.]. .

.

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@A thesis submitted to theSchool.ofGraduate Studies .in"part ialr~l.fill~entofthe - -' --;equirem6IttSforthe degree

or

--~-~ ^Master

. .

;;rE ngineering. •

'- Facultyof EngineeriQgead AppliedScience

-

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~emorialUuiversiryof Newfoundland JulylQ86'

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,8t:Jobn's INeWroundhmd ", Canada

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...·autbrisationB. ' te··accordee"

AIll,Bibliotheque·nationale du Ca n ad a ue microfillller .

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.L'aute~r.ltitulaire du droit- d'auteur l se'.reserve lee._"

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ex t r ai t s' d e ce Ll.e-ecf,:n e doiv ent itr e" i1nprilll4..a, ou:

autre_entrepr~duitl!lBand:BOn auto:r::ha.tion ltc'rite., ~ . "

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·~·.inis8ion

^{.bas been}

9~anted

to'the National Library of Cana d a to 1I1-crofi l m this· theaie,a nd to lend._,or sell cOP.ie ~of th~,f i lm. . .

<t.

.ABS TRACT i.

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This-thesisisccnceraedwltbt hesolutionofa.Facilities Leyout-Prcblern using matbematica i pro grammin gtech niques,heuristics ,:applied"sta tis tics,~nd speetellaed algoti£hmspeculiarto'operationsresearch andcompu te reelenee.The

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core cte nexpertsystembaSbeen developedthatcangeneratesub-optimalsolu- tions(or~h.erLPuaing a micro~mini-computer.A knowledgebaseconsi~tiilg

(.of8..construction algorithm, anImprovementnl~r!tb m,amtnlmex.algorithm ,

equipment·selectionaod mater ialhaad llngopti mizationalgorit hms ,si~ple ~Dd'^. mult iplelineb~ladcing.algorit hmsetc.:" bas,beendeveloped. Heairist lcrul~·ha.ve

.": - , " .' ", ' " . ', ' " .. -' . -- - -

-~>~-::.;

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beenused'for~evelopment,ofaU.theabovement ioned algorit hmsexcep·t~,thct..., "..;':'

minim~ ~nd m.t'~:Ib~nd~~g ~pti';';"tion ,o~tin",

^.•1'b.

M;~im~ ~!j;dthm'~

was basedon.thesolution:;of·~.generalizea.Stei~er-Weberproblem and the ."

.material·handling

oPti~iza~oIi 'w~ acbi~ved ~'8i1ig a .~eDeraii~ed ·D.di~ensi6h.s.i'

kiui.psac~

^{'problemamodel. .}

-'--~~--'-~~'~--'~

~- decis j~n' . ~upport

,.^system, ^{that'quantifies}..

'.' .

^theparameters of's.production

.

^."

~

layouta~ddeci~eswhethe r~lar o uth"astobechanged ata,'given tirheperiod

has -',

"bee~developedUS~Dgthe'principles of~tatis'iieal quali~y'control'theory:a n'd deci-

sionflilaJfsis..The algorithm~tf1jzesdat a fromth~etherp rograms mentioned'.

.above and ror.agiven time spa.n ofopeiat ionsandtherelevan t costs,gives't he

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... .' . ... .

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alternatives tothedecisionmaker or.calculat~swhether thelayout .ebe ngeisa . .-"," , • • . .. " .; .' .'. . '-I:.', "" '..

profit able,one.Th.~ea.n beconsidered to.bethe inferen ce en,p oe,or the expert

.system.•Aset,ot unlltiestha~caie.ulat e tl1e,vari~us par~met~rs.have:also been

. . . . . I ,,' _

• d~v;elop~d,They in-clude a stepwise anamulti~leregression algorith m,time series' ana'rs~,lineer~rogrammiDg.usingsimp lexmet.b.Qd'\l,nd·arandom accessQ~ta

base, _{r'" .}

The'whole systemhiLs hee.n.develop:d tC:/betightly integrated in tb.e sense

thatdata'f~om^{tm: outputof}~ne^{algorithmcanbe}us.e~ⁱⁿ~he^resi.'For'tb~^pur-,

pose,INuPS {ln tegreted LayoutPla~Ding ~yst~m',h~a modular~tr.~.cture-with' six ehelle,naniely.LAYOvr~ MATERlAt,BALMiCE, .DECISION,UTILITIEs

~ ; . ', .

ari4~LP.for':[adli~ieslayout.;n:a~~~rili.l·ha,n1:l'ling'oPtimizatio~,,line: b~lan~~~g, the

d'ec~ions,ip~ort' ~yst~ni, v'~ti~'~s'uti~it,i~ ari'd,hel~, ~esp~~iivel;',

^{Each,or:these' -}

-. r: " "-sliells.,have'~'~ri~~s

^{,modufes'that}carry out"their

~esp~cti~'e iunciions:---~I

^thepro-

'grams are menu-drivenwitha detailed help'facility,

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'1lCK,1'iOWLEDCE¥ EiNTS

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wo~ld'~iketo

th,Il.Dk'my superviso;:Prof. D.A,Friis

fO/his ex~el(ent

^gui:

dince,eccperet lcnandcareful review of thel;Jlanuscript.tgra'tefullyacknowledge the help"givenbyDr.E,~~re^{end- Dr.}~.^K.Lewisinstat~ti~1l.1^analysis.

Tba~J{$

^{aredue toDr.T.}

R.:

Chari,'

Associat~

^Deanof Engineeringforhis .e~ur~~me~t'and'm.6rals~pport, Tb~ .hel~givenbrDr.,J.s.~o.it,Dr-.C. J.

Michalski,·Dr-,F.:~.Aid~iC~and~r,G.

R :

,P~t~rsat Y\iOUS',times~.~n.gthe courseofmystudyatMemorialUn'i~ersityof.Newrou Ii~andis apprecufted,t

':o~ld lih t~-;~'~~ M'

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M;

~. S",l~",,;.~,..A·K:,"I;~. Dui. ' Jerasurya~.¥~s."

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P~ddle

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^Ii.Chafe

fOI" ~sh~tin~

^-their

idea:\~ab~ut' c~m4.

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^{.receipt.of"i}

Memorialuni~ersitY FeIlO~Shi~

^and

'~adua~e 's~ppie~e~~

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dur ing~",perio"d-^{of., bis'study}i~'gr~teful1Y'acknowledged. It,fs'also

e ck-"

nowledgedthat part;or thiS thesis

w~

^{doneVolith:'tt e}

~id

^{ofMACSYMA,a large'}

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m~nipulatro;-;~~ ~d'ev:;ioped '

^{at'the .}

Mlis~achusett~

^{Institut eof}

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Technology Labor.atorytor Computcr,Scienceandsupport ed from101 5te 1083 by-theNatipnal

A~r~na~tics

^'andSpace

'A~~inistration'

^under'grant-

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the"Office of

~aY~1 Resca~ch -:~d~r.gr~nt N~OOI4-77'~~6l hy

^{,thc ·,U.S.}

Departme ,ntor.En~rgy ~ndergrani ET·78-C-02-4687;alid-brtbeU. S.Ai'r'Force','

~~d~~:' ~ant. ~4g62()'7~~.020:

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MACSYMA isatrademark orSymboliesInc.

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TABLErOF COl'l TE NTS, L~"

J:'age

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3 ,5

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viii'

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13 ...16 ,'26

","29 'Af\STAACT

ACKNOWLEDGEMENTS ,LiST OF'FIGURES

GLOSSARY

•'CHAPTERI INTRODUCTION 1.1TheLayout 1:2 Formulatlon.ollhe.FLP 1.3.'Obj een vee.

.

CIIAPTER'~4L~MAS~~~_STR(rciuRED

^{\ ,}

2.1:'Nature:or theproblem

2.2 "TheS~iJ.tionMethodology ,._ . . 2.2.. T~eTraditional~ethodJl

21.!.2 -MathematicalProgiammmmg..Metb·ods.

~2.3Grap.bTheoretic Me.thods. ',.

2.2.4Heur.istie;Progra~~~g-.

CHAPTER 3 HEURISTICS,TIlE SOLUTION, - ,IN PRACTISE'

s.i C~~at~~eti~n'

^Aigoritbms 3-,2.Imprcvement Algorithms

,-. CHAPTER4GONCEITuAL

D~LOPr,lENT

^OF'A

, 'SYSTEMS APPROACH --:i.I'·De.Yei~pm~ni~r^anloiegr.atedAl>proacb

4.2_DeeisicuAnalysis..' I .

: : :_ ~~:~:o~o:s~::::::sHandl~g

4.5Qualitative Considfl'rati9DS

, , 4'6 Tb' ,~NW!(thO:OIOgy ,

, ' ( I

3.

3.

38 42 43' 46'. 48 SO 52 54

•Page 60

6' 6'

68 13 16 11 11, ' 18,

83'~-. 85 86 80

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.4 06 01 100 100 100. 103 llO ll4 US"

.120 120 124 125

i'

.,f' CIlAJ?TER

51NJ,Aps ,

AN IN,l'EGRATED(LAYOUT.

. ' PLANNINGSYS;EM . '

5.~LAY0t.?! :FLPOp~~zation AJ~orit~D15.

. ::~:~ ~~~:t~.~A~:t;~~~~r~h;rith~

5.1.3'Improve-An ImbrovementAlgorithm 5.2l\1ATERIAL.

Materia.lrH~ndlin~ OPtimiz~tion

,~gorith,~ "

I' .

5.2.1TheJ~Dap8aelt ~rQblem ."

5.2.2'A c.?~9trli~ti~D,1Ieu!isti~for Material Handling, OptumzatloD ..

5.2.aAnA"-~rithmrdr.MaterjaJ-Ha'ndling .Equ.ip'mentOptip;aizatioil and Selection 5;2;4EqJ)sel~ct-.Pre~aryEquipment- 5.2.5

. ~:~:~~~ ~o~~pmen~ ~~I~~D 'ror

-, ' a

Produ'e~On ~i.he

^{:. .,-' .".}

, '. '.5.2.6Conveypr.\CoD'v~orSeleericnendD~ign.'

S.3 BALANCE-Line\B'a1J nc'ing'

Ailo~ithmS .-

' .5.3.1co,Mo.l-

h ili, B~I";dDg A1g.,ritb~

, 5.3.2MU1~iple~.A'~rltip~e

poe

BalaDcing'A1g~rttbm 5.4 DECISION - The Decll!lion Support System •

5.~.i

^The

~MathemaiJcal

^Decision

Madej' '

__5.4~2 A~ru~t~re.or~is~onModule 5.5.UTILITIES· Miscellaneous Mathematicaland

. StaUstical-~odules.

1 ' \" ": . ::::~

^_

~;:;~~~ _ MM~~lilf;~~~m~~~~i~:a~e:~::~OD

5.5.3Tseries.·TimeSeries'.i\nalysis •

.,3.5,.4Sim.^p,I'X,-,LiD€!1 rrogra,mmiDSModule

j 5.5.5~aDdom.DataeeeModule .

I , " . "

,CHAPTE R 6 CONCLUSiONS , .

,f \

. 6.1DiscuMion I 6.2 Recommendations

I

^{.6.3~ussestio~for FutureWork.}

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140 126

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,144

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Th~INLAPSMlnnuJ CO~plJri~on ~it~~V~ila6tt.

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^hi/are

A Sample Pro61tm BmUOGIUIlIIY.

APPENDIX A

APPENDIX C APPENDIX B

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80 70

75 82

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LIST'OF FIGURES Title

Struct ureoftheLAYOUT module Steepest-descent pairwise interchaege procedur e Themainm?dules of INLAPS

TheDEC~ION.module TheMATERIALmod~le

[0

C

_II

Fig'.No.

6· Struc ture

cit

theiBAL~CEmodule. Struct.ute'of

i'~ ~T1L~T~ES

^{mod ule}

~_at~OOW".iJ:1NLAPS

~elationsh'ips between'the main modules'of

I NLAPy

MiniD}aJC:An~~lgo~ithmfor therectiliaee r FLPas

•as7rne~-Weberproblem - .

Construct:The corrstrucfion.algorithm

/

^"

.

12 /~=truct:The construc t ionalgori~h m...(con~in-

, [ 1

^Improve,J\J> Improvementalgori thm /

4 Mtlopt:Materialhandling cpt im iaetion-endselec-

. , tion . ' \

15 Eqpeelect-Quantific ati on.ofqualitative equipment

, selection_analys~ >

. . 16 Peodseleee...Equip'~ebtselection for production

~

^{.- ._. .17 .}Conveyer> Conveyorselecti onand design

~~" ",,;L~ .·,

";,:,:.,..,

ix

Sle pteg•Mult iple andstep wiseregression"modul e r105 StepTeg-Multipleandstepwberegression module lod (continued)

Comsoal •Computer methodfatsequencingoperf.. 03 tionsfot.."eroblylines'

.

,

Multiple-Multipleas,~inbIJline.balaneing os

.~iDreg- Linearregres!ion module 101

" 'Teeriea-

!ime

~eri,:, an~alysis-m~ule 111

Si"mplex-Linear programmingbysimplexmerbod-. "115' .

Sim~;ex

^r

Li~ear

^{programmingby}

simp~~ m~t~od ~

^...116·

(cohtinuedJ . . -

.18

JO· 20

" .

22

23.

·24 '25

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2~/ .~Random-A database managemea].systeJ? 110

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.:.X:,

... .

GLOSSARY

.

'.

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t

The-followingabbreviationsendsymbols.b~,!,ebeenused as(Iobal variables' .~ \;.

ia'

v~rtou~,' m~dules

^{...r dshells',p,q,r,}

~,'I

^.and't

.denote-intege~· v~lu~fJlucras

number

af

equipjnent,number~fdepartmental g;ves, types'ofequip~^ty'pes ofdepartmentalmoves,productionperiods, moduie

nU~beu

^{and[Iumttr'of}

v~ri.

'ousc,ogts~o~sil:l.ered ,'i.and j.repre;enttb~~uipmen~types and'interdepari.·

mental moves, or.tbe

material'~'W

^'andfacilitydat a in various

rl\od~les, 1he

I'·:

L , , '( . . , (

.algorithmscall lor a number~f um;ny arHystOt-matrixm~ni~ulations,^:c'om:

• • • •. • '4 _ " . . '~: ^{'. . ...}

parlsonaete.,andthe>: are denotedby pz,qz~n.ete.,whea.z.denoeesan integer

1~ ~.

^{3,". n: VariablenamesrJ:,{z,y),,-o{(z:y),pot(z,y),q'uo(z,y).:}

,t'tJj;, ~)

^,"

and~i"(z,y)ar~global l?~caivariabl~'thatd~nete the.~~ulh,taJ"at veeioue modul esduring.ditrerenJ processingstages.-T~y areexplainedindetailin

"

Ch'apter 5 and theparticularvalul!tdeneted bytbese variablescan befound'QU,t

" i . ' -

It '

from theflowebert e. _ .

'.1

t'~nz,"

^{pointz,e.ountz,}

't'o~tz,

^{gtt z,}

idt'nz~nd ' P".:.'~h~ ar~ " ~'co'unte

where z range:jfrom 'l to n.C~t ,tuD~ti;n9and;objectiverU1J'c~ion,are~

represented~y 'Cz and Zz respectively. lorld,lor/6,,orle etc denote the, sorted arraysend matr ices·ofthe'global variablesmentio nea:above. Someof the .

,

^"

.

.iJ;Ilore importan tabbreviat ions/v ariable:!!areexplainedbelow, Aeoq,plet e'd~crjp.

.:ti0D;and.listin'gof

~ariables

ⁱⁿeach module?rsub-module.can be

rO~d

^'at

tb~'

appropr iatesection inC~apter5.

'~

tlt'IIt')=the average speedatwhichtheequipmentperforms, cul f)=the ('!)try ing('apa~ityofequipment typei tlot(.;

y .

⁼theavailab leoperatingtimeforequipment i .~l' .

~"d .

..• .','j

., " . . -;r

' .,- ,

ttuIi)

=

~hecapita lcost ofoneunitofequipme~t trp~i ace(i)'=tbe opera t ingCO!!tincurr ed by using

eqllip~ent

^i".

/ dl(j)=~heflowbetween~epartmentpairi rddIj) =th~er':etilin~ar ~istancebetweendepartment pairi

, .

t.Ot(i ,j )=tbet~aloperat ingcostofperforTjn~movej withe~Il~~menti • )' npe {i }.

~

^the

Dumb~r

⁰¹

pi~~ o' ,~eleCted equip~ent

^{type i.:'}

, ^{" . "}

.

^{\ . ,}

~o.de(i),=:=_nU~berot.~;~~e n~9con~dere4,for' IOCa!mexime rtJ(i,j' )'.=:=relatio~~bi~"bet~een<lldj ~n~nodes".' 1"

mtll(l' ,j):=~t~1tildse'nessratingina~rix. cr>. ..

n~mtl.zU.;'j)^.Jequipmen~ Da~in Eqpselectmodule. ,;."_

.. tu~9A! (r,J ). ,:",, 'Weights' ~9~Kned

^ee

~cruipm~t ~

^{.: - _,}

I'ndez(l',j) :: indi ces assigned toeach

~qllipm~nt

^with)espect

to.:a~~·pJt~ular

^..

~

^"

.~ :~

operation.

' .~'

^{" -" , './' - '. \·....}

·~l'1' .. · . -~-

chtlr~l,j ) ⁼product ionperiodc;Dsideredwhilegelecti~gequipment,

t1pt~:(i,j

^{,k )=}

operatio~

value Cor"each

~a~hine

typeperproduct ion period for:

,68/(i) j ):= ¥.iLluesfrombel; conveyorshort·Cormtable

_ ." -rr8

(i,jr~=

^velues

rto~

^live

roller'short'Co~~

^table,

. "

-xii -

"mel(i)'::!::::m·aKim.u~cycle.ti metoreach task'

nump(i)

==

number ol'r,etl¥lb tor

ea"Cb:t~,

foWL,

^=tbtai

~~g~ls as~ign~d

to each array (

, 'x

1 . , . •

.

W~(lJill:'

":=

( eighte d mean-t"

coe}I ='ooe!ficients ol',regression equations-

;~;;

-,

,

/ .

^'

widlhe[f]

=

width between Ieame members in aconveyor trite(i) =frietioJ;1ractorsin conveyordesign Iene(i)

=

lengthor conveyor ',designvalues po.were(i)='horsepowerrequired tor the conveyor' npl

=

number of trialsneededfo~theOcmsoalelgontbm n;t.'

=

^numberof tasks-tobeperformed"

Tz,·(i )=timer~,qu~r~to~~omplete eac~task'.

I

.\

li~~one(i)=preeedsee elistingof the.various} asks cod =,coefficlept,01determi~atj9n

. ' . ,,' " .

...." =

s~mof~uare.erro...~" .

:, . ~ae. '.~ sta~da!d,d~~.i~tion. <?~ es~m~ r: .:

, ..wU.:..wei"gh ts:assign edlor~ar1abiesin\statisiicalanalysis'module{ _'_

"

errco]

=

stand ard error of coefficients

matz(i,j )=:matriceslor.sum01squares and cross'p rod ucts rwtm

==

residu~lsu~s.or.equares end.~fim^products

eor:( i) =correlationcoefficients • .

p",,,U) ='pa,ti.1 '.'r.la;;onoJ.llid.nts' ;': . '

~ '

ma!intlz(i,j ),"=inverses of tbemetrleesdesthedabove,- etrom 1to n-

. : .

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.xiii·

, ' .

,

"

;,t,rtm, (i,;.l.

=

transformations or variables to logariihms or·polynomiab..·

select edbytheappropriate pointe r. ...

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Chap te r 1.

J.

" INTRODUCTION

. .

partjt ularto abcilityereignoreddue.tothe.ASsumptionsinthealgoritb~.The.

",r 'beSO!U:ionof a

rae~iti~

^Layout

pro~l~m (F~PI

^isan

.im~;tant r~:lor

^for

increasedproduCtivity and moreeconomicalope~ati~nin aManufact ur ingatgan-• iu.tion:Statistieians ,E~giDeers^andMathesnat ieia'nshavetried to solvetbbprob-

~in

^ina

~ariet)' ~r waYS' usin~dratk Programming,

Bra nc h andBoundAlgo-' rltb me,Modular Allocat ion Techniques, Sub-Opti malbeurlerics and Gropp T.~oret~c~eebniques.'AlgorThhms thathave'beendevelopedarecomplex,mak ing' . use ofco~binatorialmathemat ic!as wellasbeur i!,tic.t~cbniq~,andgive sub-':

.optimal solutionsatc~nsider'ab~eexp\ns~.Q':lite~·orten.qU~litativeconsideratio ns \ .

..~.

objectiveofthi.!study.~to·develop anintegrated·approach.to solvethefacilities

,allocati?o·probli"!'

effi~iently:~n~ ~xp~re

^wayst?

.impfO"~

^the'

st:~ of·th~.~r~.

^{'. ,}

".,

.

, ,A~arcon;ept·ofthe layou tas.asystemis'essentialbelcredelio.ingwhat exactl yistbe·FLP.-rhemainfeatures.of.\'Iayout'-are

~~plalDec:t belowo

1.1.THE LA YOUTI

-~,.

._{; . '}.._~.:.

. -

ior

,

Alayo ut ma{be consideredas a.eyetemcomp"'~ing m~riY^different,in~ivi-,

.dual,

interact~ng

facilitics, ordep:rtmenta, e,ach'

~aci1itt

^ordepartm ent beingII

subsystem within,thewhole.Alayontcan signi6cantlyalf~ctthea~i1itYOf~he prod uction

~nit ~

"Iuncticu'efficientl y as acomplete

~y~tem.

^Hence

~ht err:teie~cy

ofa layout.ehculd be considered whendesigning a Dew layo ut

as

^{well aswhen}

decldmg'about'cbanglng tbe

ex~ting lay~ut

^An

~fficient la ;~u~Jnay.

^be

de6 ne~

~o^{. :}

~.:.

.

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~c ..(Mu t her, 1076 ):uonethatminiiniz~tranllporta tloncostllpertinentto thelay- out,e~rdinate^withother (a c to n Wh it ho'C,'Onl'ributeto makin g·t heI~yout a~

.eecno mieeland~iable pro~ition,T~~ot~er^{(attonme ntionedheremayieelude.'} t1~l"ibility,spateutilizationetc ...,we ll ueego ao mie,tfl:hn ologiealand-psytbo-

. . .

·logie oJratt~rs'.~thans:e in tb'e-:sYlItemlayoulis.appro priat ewhen theecet or e1Jeetlc S:the.thoa nge-isless thin'the,avin'p thatwould a:eeruedue~^lLa intre~ed^efficieneyreSul t ing'(rom theeban~E;:^wbee ehalls:eis advocated.011\bis 'buill,the existinglayoutm~ybeconsid eredredulldillt.

. -

^.'

. c

~uther(1976)b.a.spostulate~th e Icllowing u halliepri~eiplesIor the,l~you1:. 01pla ntendequipmen t. \ . .\~J '

a)

INTEGRATION~

^{ThatlayoutIsbeetwhleh'}

toter;rat~

^{the men,mateet-}

ab,maehiDe·rY. lu~Portingaeti vitfes"atl;dany.qthe rt~nsid~rations^ina.~a;·

•.

~ ' ~b~t r~ult3·;n t~e b~i·.~m!romise.

^{.. ...\}

'.

b)-"!?IS~ANCE MO~D :

^{Ot her}thins:iJ>eingequai ,

~hat

^{.Iayo uti5?estwhiJh,.}

_0'per mits

~hfi mat~rial

^{to movethe}

rni~imuin dista.~ee·betw~n operatio~.

e]FLOW:Otherthings being equal,

t~at'layoU:;

^{illbnt which}

~~raDg~·the ~.

wor t areaforeaeh

~perat ioll

^or

proc:~

^{-iOnthe ....meorderof}seque uee tha.t•

. '

..

^~

... .... ...

( ,, "

., .'i... d).Sp'A~Etn'IUZATION:EcOnomy'istobeo~~ainfd^by'usin~ ~frectively^. allavailab le)pae e-both vertical.andhorfzon t al.•.

e)

~AF~ _.

^;

O~he~'~bi~~ - . , '

^...

~('ing equ~l,

^the

~~Y<lut

^.'is

b~;

^{. whi;h,mak es}

wo~k

^.

satierrin g-a llq,ar~(orthew.orkers.

'rrFLEXl~ILlTY

^{:'Oth ertbi·ng, b}eing equal; tb"atlayoutlebest

~hicb c~~

^lJe;:':

, .

\~

, · f .

.

\

. .

",:" : - . . '··-11. · .:'-.''".; •.. ;,.~., 'O':..

adjust~dand rearraaged atmini~u.m^eoetand'inconvenience-;

.

^\

1.2 ~ORMULATIONOF THE FLPI

The-prima rynbject. iv eof PlantLa you t (App le, 1950)istoplanthear rnnge- meutoffacilitiesand personn el50that themanufacturing processis cur r iedout as eflect.ivelyaspossibl e.Tofullyrealizetheimport an ce of PlantLayout, inthe

I •

genera l contextofmanu f~cturingindu stry , thefollowing·fund.ll.ment nl' concep ts (Lewis&Block ,H170)havetobeexamine d.

1. ~hemajor partor prod ucti onwork isno..tprocessing ,~usuallys~pposed ,".:.. ..:

bu tm:nte ria l handling.

'" ,

.

^{; ...}

.

2. Thespeed ofp~dul::lion

in

^aplan t."is dete r m inedprfrne rilyby the'adequa cy

'of'i ts

"mat~fal-hau~'li~g

faciliti es.

3~ ^A'goai pla.utii yout

,

is

.

desi g nedto'

..

provideth

JI

e'prop~rlneilities

. .

^fO

.

^'f

.

~Dterial

. ,

bandlin~ ~"well"asproeesslti g.

,':,."

plant".

. .. ' . ~. ,

betweengoing to andcoming from alocation or.a department.

.;'

·:i\.s~uming ~.hat,

( ,

.

^, ., '

a) Botht~eE andDmatricesare symmetrical,i. e.,therelstnodistiriction T~e~n:'ost gener~1~nd.allwen'compassing.mathematiCalformulat'ionwasg~n as earlyas 1063(Armouret

al l

and

is

quoted as below.

\,Given a. physicalarea.representinglocati.on!.eveneble

~or

occupation by

~he

d~pll.rtmeDtsin the plant,!~isP~sS~b.let1~efinet"'1'atr.ices.

.r: 5PATlAL RELATIO~SmpMA't:RIX,'D:This matrix contains values

\ . -'.'

re~~~senti~g

the dlstanee

betw.ee~

\he locetione

ava~la~e

in the plant,usu-

·.ally the distance is betweenthe center point ofthelocati?ns.

~: ^CqSTRELAT i? NS.in P MA~RIX.'

E:

^This,matrix contain,S' values~

·represeefings ome cost per,unit

'd,ista~ce

bet.weenthe,departments in

a

plant,

", ..thhb~iDg.an averageo,v.ertl.stii,tableinte~~8.I,o(time:

(1.1) 0..

=, 't

E·.tj j'd"j. '·

1'_ l j_1 .

,'. '

.. . ' " 0\' •

.Thi.s·~quatio~,.defines"aquad ratic.assignm~n;~roble.m. I .

·

~se

^.ofthe'above Jormellaed l'Ioyoui:apprcaeb

req~ires ~he ac;cep~~ce ~r ~oss~.

"..assumpt~o~s.Cost and Flow'dat a ereas~umedtoe_xi:st'fo;^{conditionswhich'are'}

, ',

"""

^"

^{. '}

^{" "}

.(

~:.,.

delinitio.,palllunknown.Materiahhandlingeosu are eeumedtobelinear,

'iDC~

^{".. '}

mentaleadmigu1'6letospecificacU, ities..F\owdati withstllfhu tie',rope rtie arew~lJledtobedet~rmiDi5lieandthei~ttm'tioD~ot he rsystem

,

problems

r

' - ' withthelal outproblemisi~ored.Approachearefrequently'adopt l!doradYo- catedwith~utadequate~Uentio1ltoth~compatibilityor theproblemsil uatioa with tb.emodel assumptionsjVoUmann,lU66)

In addition,qualitative aspectsor the problemareeit.be rignoredorgi'I'D, ve·ryJittle importance.·

1.3 OBJECTIVESI-.

.:\ Methodsthat hlnbeen proposedaret.ithert~eomplieete d, iovolv.in,.

. J . . ." . ,

.expensive.CPU time,?!h~ghllsllbjecti.'l' innat~re.AlsoIhe~omplexmathem'li- .cal rottDulatiou'of,t b:au ilableaJg~~ithmS^makeiti~possible^lora.~IllUractlU.^. ~'';

,

^,.

.

^{" . . "-}

.

; , ;

.

.-.inLcn~~r'~~inputq~l.litatile eo~side"ationsp~rti~ent^{tohisown!ayouL}~Q^{: ..•}

·'additiOOall thesol~tioll!^{sener.t edan}m~the~~lica1l11 111~op~^..I.en a

lhciup._ _ ' _ ;.'

theyare~aUedopti~allolutiolli. ^Anatt'empth~~ll mad~~tb~^studlio

quan,tify

so~e

^{orthe s·ubjectin}

as~ls·

^{,orthedecision}

ma'iill~ r~ard~ng the

^....

changeorlayouta~d^{todevelop an}algorith m th. tiseconomical andeasltouse..

T h': Obi"ti""

re".IOIIoW') " ,. ' : "..

'. 'i) Tobuilda

matbemat~cal m~dellor

^the

~~yout ch~p~~,

^deelslon allalysis.

iilTo provide aDeasy·t~·use^toolfor monitoringthe efficiency of',layout

.

^'

. . ,

^.

, ·andsuggest tbeneces~ityo{eba nge,

. .

"

iii)Todeeelcp ii.liorithmsthateeare'aDoptimal/lub-o ptimalSolu.tion·fo,r

" _.

.. ..

-

,

,

'-" .

.:~,

.. .... .

theFLP.

0,:"",

\

> . ."

6,

:.'

iv)TQ.ensurethatthese algorithrJ1..!.ar~e35ilyc~ea~leinanin~~~nslve.

micro/min ieon'tputerenviro nment.. .

_v)To d,e:velopasoCt~are p~cka&:eullin.&:theaboye algorithm.. 10

~ ~anuraeiuring

orCanization,

~ost

^ofthe

produ~ti~itr' jmpro~ements

^are

I.' • ~' • •

achieved.ina~exi9tingset uP.withoutmodi~Yin!.~~yort~e srs~em~n(rain\S. Butai'temptsarei~cre8.llioglybeingmade to(:ompkins&WhitelUS4) challgeor modify.'t~e-\ys t emits elfsot~~~'jt beco mesmore~lfieient.FlexibleI~youtsand modul~r allo~atiOn't!!ch niques -aresomeofthe.resultsof thispbilcsopb y.Even .

th~ugh t~'ese tOOi; ~re 'a~aiiabl~

^toQ

pr~eli~ing hldu!ltrial"eDginee~,

^{·it·hasbe,en}

foundthat the re'llefewaidstbathelp.in"ratIottal and~cientifi;"dedsion

mak:-'

, '" /': ,,, ,',

',: ,'"

^{, '}

..

^{' ,}

" ,

iogprocess,..Thes~atist!c.1 decisio~makinglmodel'L'J proposedis~>D ewayof ,q..ntifying.,ubj",;vedeeelon.'

' ! -:-'

._._.

.B-~ther

^.;

'part~~ipate i~

^thedecision making

~~ regitdin~

^the

'~truc:

jureof•layo9-t,facilit,'plannenaimost~Iw&y'^{reactto,theneed!definedby.}

others.Enll'th(lughtbisisu~dentandable,'collllideringthecomplexity,orthe. p;oble ry-'decisi on.tools

~re

^{absolutelynecessary}tojust ifyHie need lora'

chang~

i~ ·. 'Iat~ut. H~ne,

ⁱ⁽

waS)rop~sed·th~t·.a ~tistieal 'decisi~n

makingt901 be

,~n~~

stru~ted.which wi!'inc? tporate'tbe principlesofengineer ing design. • , 1fa:n.orga nization,cQntinu~:mslyupdate~itSprcductlc nopera~ionstobe as, .r, prodhc tiveI.spossible,then the re·mus ibe

contill~ou:J,

'relayou t and rearra nge-

,. '

~eDt acti~i~1 ' in,p~gJ'~.

^Onlyit'

ra~e' sitllati~~

^can

"~ew

^{protestora new}

:....

' " " . ' " ' .

.

' "

.

. .pleeeofequipmentb~Intrcdae edinto

a

ne;~y~te~witboutdisru ptingongoing. "~. ~'.

,.- ,'

. , r ^. \

^\

,activiti.es.AsiD~lecha nge~lI.yhave a significa n timpact,on.integratedte chnolog- kat,manageme ntand personnel systems,resultingin stlb-optimlsationproblems.

Thiscanonlybe avoidedorresolved through th e applicat ionof fa{lilities design concepts...

Onceth~ d~cision

t;

cbang#ea layouthasbeen made, the planningengineer

, .

needs aneasy- to-usetoolthatwillhelphim allocatetherad tit ies to theproper

" ~ /

. .

f location s. Algo'tithlM andsolut~ons'Lhavebeenproposedbelore/ forsol~;\l1,gthis problembutmost ofth eIormul atiode arenot.practi;iLlror.regul ar use,Therea-

SO~ ^Cor

^this

iriclude~u5;, of ~'pe~~ivelcPU-time,

complicated

~n~"intr.ic;t·e nnlu~e

^•

or'

m.i~.m"k'l

pro".inm;,g;

SUb+;~'

^espeete01

tk. probl~m,

^{endlook or •}

sy_st~msi:once~~in.th~l~~:Obt:fedbyco.mp.u.te~methods.,It.~asteltt~at an algo tithm tosolv~tbe FLPusing[miemputer wo uld beo~greatJ.1~!inIi ..pract ic a.l

~,~vironment.

^An

iliterdisdplin~~appr~~cb w~

^{proposedinv,olvin}

g •.

theoret icalsup port fromthe sciences

l

_ofMathematics,engineering,Statistics,

.' . ' I

OperationsResearchand Producti on Management.

. I .

Although the teak of d;signi ng or4etcrmhi'ingthe:arran~em~ntotequ~pmeot' and rela tedwor kareasIsonly onetacetofthe.tota lfacility planningprOCellS,tbe

. ' I. . .

.develop mentoftpe best possib le tacUi.ty 4esigp.was 'consideredcentra l00'tbe

. . " ~ "

.

,

.

^,

facilityplanning activity .Jtwas decidedto proporeen-a lgorithmwhichwould integrateasJaru

p :Si~le,

^the

ditr~ren: ~;;ic~i ~ciliiies desi~ Objec~;ves. ~o~e

^•

ott~ese differe~t?bje~tivC5,whic hmay evenbeconftieti~garegiv\llb~low.

... a)"Support tbe?bject ives.oitheorganlaatloe thro ugbimpr oved 'm ateria ls' haD.dlin~, m~~ei!alscontrol,and goo d.housek eeping.

'I

i ~ - ;.

..

b)Beflexible,eff~ctivelyuse people , equipment,space,energy,invest ment

...

Anorganizedapproach to facilities

'planDi~g

^is

pr~sentecl i.h~~

^{work ao.d_}

etc and provide easy mainte na,nce.

c) Provideemployee safetyand job satisfact ion.

the'general approachtotheproblem isbased

o~

^thefamiliarengineeringdesign

I

processwhichgoesthrough tbestagesof defining'theproblem, analyzin gthe

f

piobl' ,!,••

~",ado.

elterneuv e

d"'gn,,,

evoJ>:,dng the

alt",ad~"" ,~e<t;n.

^and

!.

carrying-out-{h~design. - .

.The

te~t

^.is

~iVid~d i~to'

^six

part~. Th~

^firstchapter introduces

~~e

^problem

"

.

.Dodles~rib~'t he"o'bjeet ives,.chaptertwo"t racesthede~elopment

or

concep ts~Dd^.' /tecbniquee

tha~_

^{,areavailable'}

^{, C o;}

^the

.~Olu~ion ~r:

^tire

p~bl~fI.l:'

^\he,tblrd-

ch~P~ ~-'~

c~t.inu·es'

^with·:he

st.at~:~~ th~

^{art .; focusesmore}

~ist;nCtly

^on

th~

sto chastic•

~:~,

.probebilistieprocessesand·~p.eheuristi~methods. Chap,ter four exptcreeweyeof

i~t1!gratiDgthe materials handling proeesaesM~ellas qualitativeconside r au'ons.

in thegeDe~~ioDofthelayout plan anddescribestheconceptual developm entof

. • t . ~

thealgo r it hm.Chapt er fivedes cribestheproposedsystem and themat hematical '""

~oTmullttiOnS

or' t he individualmodules.Thesixt h

r-: ^~ompar~

^the

^·comp.~ta­

tiona}methode of thedifferentalgorithms'·availa blean~concludest~ewor k with.

, . ,

asumm a ryofIi.n~iDgsandthesco~efOr futureres~arch".'

Chapt er2

THE FLP A S AN ILL-STRUCn;RED PROBLEM

. . F adliti~~ign,

^{sometimt't:llfdplantJayouL,jj'ODeelwiul}

~tI.

ⁱⁿ

w~irh'

theiDdust~gilleerhuchosen tooperate,Madhubeeede:sign illll;thephy'si- .eelfacilitiesaround·himoverall recorded history.Differentpeople h&n"examilled the

F~P

^{from theirown}

pe"pedi~es.

^Researeli...'continues,

,i~ce

^{aneffidentlayout.}

isor the

ilt~t

^{irnp'ortancein}

increu~

^the

p~ductivitl

^orany

mallllra~turing '

^,

org:anizatioll.'S~tjgti~~showtbatmoretba ll8%or,tb~.GNPo( the&5,S.goesto:'

wards'tl).econstructionorDewJa~ilitles. ileneea lotorr!!Searcha-direct edto-

, ,

wardsth e~1~tioD

o f

this-problem.P~pl~^workingindivene u,e.M.liktArchilec:.

ture(J~h~solluno ,Leej07J,Mitchell1010,Newmlll1068, StewartlLee19't2), BusU;essAdministration(BulJ

.

a'

.

&;Vol!mallll1963,^: Love I061i1,^{, '} Ritz~aD^-1012, .,V()lImann 1968),Building~i.~lice(Klt'/ciri k1116.0,Mo~cka~.1l81,Wbitth~ad11l10, Whit~bead^&:Eldm 19&4), Ch'UEb!inemll~^{(Spillllen&) \o}'eidliDger-lWOI, .<fmputerSeiecee (Ed....ardsetal~~~or&;Ems10811,Comput~r G~Jph·

its(Banna&:Sp!tfers'IV12,Tekh'ob IV68, 11l1Z), IndustriaiEngineerins,

Mltb~mMies,OperltrouResearehand SliUstiesbav~made.contribut ionsto- wardstbesolut ionorthe FLPrr~mtheirownpe.rsperli,~.

2.1 NATUREOFTlJEPROBLEM; , ,

' ,.

^,

.

. Itbas

.b~~n

suggested(Ben,li,11r1)that an FLPcan,be

eat~l9riz~d . !-"

anilI- st ructuredproblem.,AnIll-structuredproblem,as,suggested byNewe[](106~)has som~or alloftbe.followingcbaraet~risties:

..-:'

Itcannotbedescribed exdu5in11innum~.riealnriables.

J " /

;:.: .c- ::

.. ..

.,:.:.,. :. j

':.

,(

. j

¹⁰

ii) The

go~

^{to be}

&ttain~

c,ann ot be

,,\C'1 ' d

^{byI quantitative}

o~jedive

^,

-_{funct ion.}-

,

_{. .}

^.

iii) Algor i t hnutha t..permi tthe bJt

!OIutio~

^to

b;

^{found,a ndstated}

'i~

^{eume ri-}

, eettermsd.Onotexist.

~

^{• ..}

The,~.&tioltalebeh in dcat eg orizingth~FLPasin ill-struct u red problemrome'

. . .

^~

(rom twosou~es.A,faci\itJcao.beconsideredtohavehe components(Mut her..

's:Hal~,^{19 70"narnely:}

aJ: .~^YOUT;Tb~ arran~~ment^{ofact ivi ties,}fe aturesand spaces aroundthe.- relat~on sbips~hatexist.between them, _

,

. ' .

b),~L1~:'I;hemet~1sctmcvlegproduc~,ntateriaJs,peoplea.~d equipmentbet weenvariouspoin tsi~^{tb;facilit y•}

..1:).COMMUNlCATlONS-:Themea~softransm ittingipronnation between variouspointsintbe(a.c:: i,lity,

dl.-UTILITIES:

ne

conductorsa~d ~istrib~tb?....

or

fu bstances Ii~ewa ter,.

p1:~aste,^airan~Power.

7 "

..

e).BUILDING, ;-Theform,m~terialsandtbe struetureitself.

'\'

ThesefivehaYe complex

.

^{', '} iater-relet iceehipe

.

/lud

..

beeee

.

theFLP'

.

rormalat ioo shouldbeb1sedODallortb~m,whichismathematically infeasible,Alsothe

. ' . ' .. . .

'analysis includesqua oti'ative dat a. as wellB.Squal!tative inrorm.atio n.(Mut her

, >

'070)like: )'

a)PROPUCTS/PERSONNE~:Presen tan6li kelyl\lt ureeharacterlsticaor .\ '

, '

prcduet sand materials~sise,eompcsulon,&nisb.,weight,fragilityere.,pe r-

..,

: :,: ,,,:. ;,'.

,;: .:-:

. .

^'

' \ .:.

/

c

u

soneelre q uirementssuchasski ll's, attitud es,wark in.! houn ,prIvacy

req~ire-

^:

menu,and physi ul need".

. - ' .' I

b)QUANTITIES:Presen tandhblyCu t u ,,:qua ntitiesforeachproduct or materialandtor'eathpositid'n'.

c) ROUTI NGS/P.ROCESSSE~UEN9E:Presentandproposedroutings or manu!acturingprocesstor eachproduct andm~terial.

dtSUPPO RTIN GSERVICES:Buiidingsuppor tineludingmechantealnod ele.driealsystems ,air

copditio~ing aD~ ~eDtilation,,,w~ te di9po~a~" P~tlution:

c~ntrol, Iightin g , plul'!lb ing etc.,and.P~rsonllclsuppor t like.f?o deervtees, break~nd ree~eatioQalfa cilities,parking,credit-union,firstaidet~'1 ^•

•e).

TIMING~IME~

RELATED FACTOR S:.Restrietions 'ilJnd

~sibiliti~

^.•

h:ga'rding o'+'efti me,extrashifts,extn~rkinr;^{bou n}

_.

^and_a~tiy

,

ity^penks~nd

'J

theinter- relatio n"bip5betwee!,them:,

Becaus eorthese«)m plnitiit5,wecaneceetc d ethattbeFLPisanill-structured ..

proklem,

D1-structu~ed prOble~

^{ca?behandledill avant.t)'ofw'ay,:'. .:}

t p a ) ~T~ ~~ERATlON:

^Totl:!enu meF6tio nisreasib;le ,whentheprob- lemissm aU an dthe.constraints are not many.Tbisi,the brute forc e.

app~a~h ~here

^tbe

COrtlp~ i5

^{usedasa.fa.1t}

C~l~'u~itor

^aliiall

t~e

^{poedb le}

alte rllat ive sareevalu~ted..Thist:chllique eeeacr,beu~,ed.fortheFLPIl.'IIob.e.

numberofphysically feasible alt ernatives wouldrunintobillions even fora

'small facility. ', •, ' ./

, b)

COMPUlf.ERIZED'tECHNIQUES:This levo lvesrolvioJtheproblem

b y . ...:....::..:'

,

mat~rmatie~1 ~eallS

^or

b~

,imulatiOD"t e chniques.Theeegiaeerrelates·the '

~

. .

^'.

~

-.

.'

",:"

. I c '

. : . 12

" pro~e~ .~~ ~ · aqal)'\ic~1 b~~

^.and:&S!IumPtlons aremade". ,

e~~ru~· .

eva luationofthecons t rai ntS.or th.eprob lem; S'eDsj~ivit1^{an alysiscanbe}

per-..

tormedloevaluate theext e nt~I^{in fluenceor}~b^ofthe~umptioDS^{and a} .•re~D li.bleMlptio narr ived at.

o •

clINTEIV.CTIVEPROGRAMMING:Herethet~leot5,01boththeman.

and the'ml\ch ill e areu~edinthesense tha t theprobl emsolverba nd Ies cer- tainsu bjec tiveasp~~.s^orthe problem.w h ilethequa ntitat iv e analysisU.nr·

rieq outby thecomput~r:Thedeelsiona~utthefinal~Iutionⁱ⁹let!t~the prob lem

&Olver

and the,comput erjust gives seve ralsub:o p t imalsolu t io ns.A limit ed an/o u Dtofworkusin g

il1tera ;tive .pr?gram~~g

^{app liedto}

th~

^FLP'

.

^{' .}

-.~B~DnaandSpill~t!l~~2j~oore19:1,~wart;ndLe elQ.72)hashe~n"done.• d) HEURISTICPROGRAMMING":This'in volve,'algorith msthathavethe -. "character isti c:of_.

. reau~ing

^the

am~unt.; sear~ 'req~ired . . ...

^to'find

a~ aee~pt-

•~hle-'Olut.lon.Aset of,in~ernal dec:isioDrulest.h~~ ~anin te r na llymod,ify.the.

~~irec:tion'~f^{surchbJbUii..intothe}algorit h lpand thpm~kes'beur~tic

p ro-

..g~ammingpartieul arl y.•~plic:atJI.~toiI~'huctU.tedproblems, • • "

•e)INTE LL!G ENT

MACHINEs':

The -prete.Dtday teeh.Di4 ues"'are

mostl~

ge'aredto tbe ma.Dagement.of thesy,temu i.Da.CADfCAMopera tio n.The

~ ' , ,

';designor-jnhieateaDdeo~plex;optimi'zi.Dg'algorith ms may Ieed tOanera ,_.

where

t~e ,y'te~ ~oUld b~ c:a~;ble"of.looki~g ar~er

^it.3!llf.

2.2· T~E SOLUTIO~MET B O D OLOGYl..

·T~emeth od s adopted by'var\o~~^{scientistsfor:}~he.~olutio'n ~rth.~^.pr?blem

...;..

: ... .

"'^""" ' "

- ,

T

can be studied as follows.

a) Thetraditionaimethods b) MathematicalPrOgr4?iI!lin~,methods c) Graphtheo ryand listprocessors d) Heuristi cprocedures

2.~2.1 THETRAD~T10NALMET H O DSI

:.. ."

Tr~ditional'met hods classify the FLP as manufacturing or non- manufacturing,produ ct-orprocess'andinitialurre-layout problems.The·0.011-

manufact~ri~~laYOut^is"notgive n much im'portonce~.,it^isconsideredon.n.log.o~s~

to the manutacturlngont;oUsuallyre'com~endationswere.rnatie to. adapt the tools orone totheother .P;oductlayout become s a

,

' line-balnnein~.~r~blem,~ ^here. wfth-tb e assumpti on'orstandardized methqd and.ar~iayoutis~on~ider~d00'0-,

,.cePt~~lIY~fieeenre!IS,~~i_niti~1layoutbutwith:a.d~e~c~s.traints.,T his tro:di- .tloaelap~roa:hJ~ads(Buffa.:.ID67)to the selection'ofthe initial"manufact u.ring, [ob-ehoplayout'as the-most complex case,wi~hotherl.ypesbeing somewhat ancillary'toit.:

.Traditiona.lplan!layou t toob Included graphicand,schemat ic modele,two and tbree-dimenslon ef.templates,'assemblycharts,op~J\ationprocesscharts,pro-. .ductf!tlWprecess charts., link analysis,etc.These tools wer;used to minimizethe

.material Dow'bylocat ing'C:il pari ments

i~

^{eueh'w ay}that the-·volume or

no~-

.'

.

adjacentdepartmentalBow wee mlnimhed. The criterionfor,qu,antitativelayout

,

. .

^'

modelswass.t at ed

as

the minimizationofmaterialhandlingcost; which~u

14

assumed tobe "an lncrementullinearCuncti onofthe distance s bet wee.o"rbecom- pcnenteotthesyste m under st udy.Thesemeth cda-are discussed inImmer(J050), Mallik and Gaudreau(105'1),Shubin and Ma deheim(IOSI),.Ireson(lUS2),Muther

(l OSS),Reed(1061),Moore(1062) andApple IHI53).

.

Thetr~ditionalmetho~swereint ended

l or

ananalyst'w~o^{usedhis own} knowledgeandin~uitioDor the system,to inp'utwhatev~rInspir at ionbehad. As theyrelied ontheexperienceoftheanalystthey werehighlysubjeebive,' Because' oCthe'changes in the manufacturin g processes and,methods the in-ter- ...,,_ _--Ireia tioDsbips betw; enthe various factorsmentionedabovebecame complicated

andtheneedformoreobjectiVeme t hods'arose.

.

^.'

"

.Tbeeffor~.byearlypra"ctitioDers .to ~e"rrioreobjectiv~·.led

!O

the introduc- tionor travel"char t ing, relation sbip'charts,ope~aii~,Ds .seque~ce'analysis and a.

bes tof-othermethodsdescribe dbelow.

Travel cbartins: wasintroducedqy Cemeree- (lQS2) and 'Smit h (1953) and

. .

improvedbyLundy,(1955). Llewellyn (1958)andSchneider (1960).HerjlDow pat- ternsand"volumes were esta blished",~ndapreliminarydesign made basedd'othis~

.

^/

Sta t istics were tbeu compiledtorthe,preliminarylayout.and interchangesmade tor a moreeffieie~tlayout.Muther(1961) introduced,therel~tionshipchllrt lIS

•p~rtor'his SystematicLayoutPlanning(SLP).Th~metbod Systemat ically':.l~ed

' tbe'e1Oll~ness

ratings inJhe-cl!artto

i~enti"ry

^layouts

~pbicaIlY.

Here the- areas andsbaRes.werecOD~idered and·~comparat i:vescore'calculat ed,~~ecpeeeetcce sequence'a nalysis (Buffa,1955) assumed that.all tacilit ies occupiedequalareas:' and thatlocatio~sdiagoo'a,l1yopposite"area~jaceDt.Asequenceofoperat ion!

... .

IS· were,ideotified for the productioo.ofeachitem~~dalloutiol1S10 loeatiol1Smade ODtbe basi" ofsimplifyin( tra velfor.seqeenees

or

majoritenu.Reil l.lldAnderson (IQ6111lSed theirrelatinlmpcrtaaeerad on~obtainlctura tentimatMofthe dosene5s ra tinp.

.SomematbematieaJ mcdela werealsodeYeloped'by trlLdit ional practit ionen. '. Wimmert(IOS8)deyelopedameth~d^usinglinear al(e bra tbatwas subsequently roundto be based onwroll(Msumptio~.ConwayandMa~w.ell(lOBI),proY~'an alt er nativefor mulat ionth~tcould not beimplementedrornOIHrivialpro blems

.. ,

,(Ritzman,1972).WhiteheadandElda~(HIM) describedII.progr&mmab'~algo- r~hmbyalloca.tin(r~ili.tiesto locat ions in tbeord.erof theirimport~nce,ba.sedr cetbenumb."oftripsm'ade:to tbem:Facilit ies'We~e^{dividMintoseveral}e1emc~^.

•tat units and~rti6ciallyhighjour:nel ,"clll Wl!~dC~Dedbelw~n twl~boftbe samefacilityto:IlSur~ t~~th~':eieadj.cen~^{ill.tbe ifn i}lal~ut.^.-

ThetraditionalsyStemati cmelbodsmostly dep'endOIlan analysisofard a:

• I!l •

t~~hipchu torsimilartable.nielayoutwasth eaeODS~ructedhu ed011 minimulii materii.lhaodlinr;.Unfortulla~IYI^IfIO:5t

or

themethodsdescribedaboVe .haYe thesame (lll)elfide oC1as thesch~m alicmethods utheyrelyOlltheanalyst tokeep all asped,ofth~system in mind andthen comeupwithanoptimurn '\olut ion.·T he basicreMPnS'ar e asrollowl:

."",

Authe graphicll.1and5Y5t~mit.iic;^approachesdeaerlbed. above'dependo~.^the analys isoC..ll'IaterialBow.Dat ais,a ccumUlat ed.and9rgail.,izedusingchartsor. .

~atr~cd

^{and:thenanahzed}so tbat thematerial1I0w

betw~D ~Don

^{•• dj.een;}

depar tments'ismi~imired.Tb~^{im"lies a}subjeeti'vee~aJu.tloDer-altern.tinll'>.

"';'::.- . I .'.' ·,

.'to

16 andbecause of a t.rial I\nderrorpr~eW!res^adoptedin-t he analysis, theyfail when thenumberof faciliti es or~epa;tmen~sislarge, say tenor more.

2..2.2

~TH~MAT]C~ P~GRAMMIN?

^METHOPS"

In general,allmath ematicalpr?gr~mingmetho~s .for solvingtheF~Pfall underbranch-and-b oundn~~th~.Branehing-and~boundingis oneof them~t ~ .genej-alapproachesto any.constra inedoptimizati~nproblem and.involvesan inte lligen tly struct uredse~rchofthe"pace o! all feasible solutio ns.T~espace of

.

.

all repible solutionsispar tit ioned into smallersubse~repeat edlyand a-lower boundCor thecost~r:soh~tionsineachs~bsetiscalcj.llate d(ferminimizat ion)", The eubaets,wh~sebounds.exce~dthecost ofaknownfeasibl.e solUtion'are cxclud; d Cromfu ~t.berparti tioning and part it iohing continue duntilafeasible

. :., . .. " "' :.

opt imumsolutio n isfound, The feAsible optimu msolutionisdefinedsuch'that.itS. -cost

i~ ^'rio,t

great er than

ih~

^'hound:or,any subset.

B~anch-ali~:b'oU~d

methods'een

. ' . ' . '

. '. .

-

be appliedtop~bJemsinscheduling,~ecisionprecesses,oombina to rics, integer prograni~ing,traveliag-salesmanproblems, quadraticassi~mentproblems etc.

The complete descript ion aCa branch-a nd-boun d

- - ..

al~rithm

.

^needs

.

.specificationoCther~~~thatdeterminewhichofthecurren tlyact iveboun<!ing' problems are to be b,ra nc6,dand themet hodfor

deri~i~g

^.newboun ding,prab- l.ems,Branch-and-boundalgorithm s aregroupedunde.~

a)·,S.olutlon forthe traveling-salesmai'problem b):Solut ion forthequa~raticasalgumeutproblem:

( .

~

17 asappliedtotbeFLP.

a)THE'.fRAVELING·SALESMANPROBLEM:

TbeFtP can betormulated"asalladaptation~rthe traveting-salesma.nprob·. le:ri-(C'avett and

~Iyter,

^{"lG66).T'be basic}

r~rmulatioll_

^ofthetraveling-salesman

'

^...~

..

/ ,

problem(Little et ai,Hl63) isas follows:

. ., ^.

Asalesman,st Artingin one cityIwishes tovisiteachof e-t other cities onceand only _onceandret urntp the

r:

^{Inwhat ordershonld he,visil theelt les~:}

minimize the total dist ance tra veled ! ror-dish.nce, other measuresof

_. , 4

effectivenessliketime, cost ete.,'cenbesubstitu tedasdesiredand:nll the dis-

~an«soreceteb~ween.the",ariella.,ei~ypairs are8.9su.me~tobek~o~~.The problemhasbecomefam- .-ous becauseitcombines

..

easeorst atement

.

with"difficulty'

.

^'

^.

•orsolution.

The

~aveling-s~lesman.proble~

^caD.be

geoer:li~ed

to'formulate

seve'r~'1 ~oni:-

^-".

bina~'iafproblems, oneof whichisthe'FLP:There.a'renfixedlocatioll3 to' whichnfacilities h~v.e^{to be}~signed.One and"only o.ne facility.may be lISsig:lled

, .

to onetceeuceamia ,'dis tance' -is associated witheeeh pair.oflocations,«10-

. .

sldered the «1s tor anindexorcost.Anindexofthe'Ita lJicilllen9ity'between

.

^'

. .

~

transferredbetween the twolacilit ies ormoregenerally,a measureofscree ~(

>facilities Iaessoeiated withea'eh pair,us~ally^the.ialeatwhi~h^{materi alswillbe}

depeodeueebet"een

t~~

^{'t wo}raciliti es, Tbeoo;ti

of assigning a pa;, 'offacilitiesto

"a pair

~I

^10catio.D:

~

t{ e preduct-ofthe,loeation

~disia~ce

^t!mes'

th~

facility tt:mc intenslt1

~nd

^the.cost

o~ 'a

tal'asslgnmentISthesum01these products foralt

" .

thel~atlonfaCility pair; inthassIgnment"

.

-~ -,.,.' '

.

/.'"

.r ·

-'

,

".

Gavett &lid Plyter',

.,

~Igorithmdiffersfrom the^.

- ^\

original traveli..Dg.sll.lesman probleminseYeral reepeete. nam ely.met hodof.uccessively.redu cing the cost matrnsothat't heinitialvalue of tilelower bound~ ~aximi%ed.methodofelim- inatingelements inthe,costmaJ.,rix~hat\wouldresultilla~inad missiblea.ssi~·

menf.The math ematical formul at ton as welluthealgorithmcanbe understood

f~m6~re

^{I. 4}

•Thecomp utational effortthat is1nvolved Inthis.rorrnulation n:akesit illle urr; le'for

larg~

^real-life"problems,

Gavet~

^and

Plyt~

^notethat the lar gest. Dum b erolfacilit iefthat couldbe convenien tly handledWMeight.

. , . :' .

Theconccptor'di9cr eleoptimizing'h asbee~-employed byPeg~1s(uiB6)in

r '

." . '

' . -

^{. . ,}

near-optimu~t~anling salC!Jman^{algorithm,ro"fthe-FLP,Here}sta tlsi icallJ&m~

, " ,

. .

" ' . " . ."

piing,was.emrI6y~lorlemi:'enumerationan~'oco.mputat,ionalco,tJ·abo'con~

'id~r~d wben·de~i~in·g~.~

^slop;tbe'sam pJilill-lPcedure:·

A.;and~mlY ~ra:,wD ,am -

plela>:ou~^wasiJ.~ed^toIDt~rcbange ~ectiODS^{that are'}adja~ellt^orb~ve^equal.areas ~

~crording

^iolome specific

sequf:nc~' a~.d 's~ccmive

^{drawsorsampl:}

layo~b

^and

'. ~ .

- .. '

repetitionolthe samesteps('ontiny:~Thesampling wasterminatf:dwhee 'the

aver~ge

^cost.

^{p t} l~atiDg

^{anothe r}

~al)~Ptim~~~ ;llc~,the

retur n from

~a~ing1.

bett eJ;.loealopti mdrntimestheproba bUity'ofloeating that betterlocaloptimnm' :

'Cle~rl)' th~' t~c~:~,iqUe ~ _Dot:.~ar~~.tee ~~ ~bsolute

^{optirnumandSolution.}

~e

^....

at best,~ub-.optimal.Peger _~x·periimce^wu,t bat tbe algorithm

i:'

'comp lex end r~quiresooo!lidera hle~~~ut.ing^time'.

:-'

. .

^:'

.... .. ;,,'-' .

.:

i ;

bJT\lEQUADRATICASSIG~'MENTPROBLEM,

r..'· .:

..

/

" ' ^",;'., :-.

Lawler (1Q63)defined~heoptimiut~otu kio't'oInd·ilIthe leDeralformo~

.the quadraticusipment.problemas,themioimi~ati~oattheroUov iolqoadratic. rostIunctioo.:

Minimi!e:

(2.1)

,

-,

(j= I,....,n )·

EZii=1 (1·=·l,....,nj

i '

Zij =0.or1 (i.j= l, ....," )

Th,'.' ";~d~,; ';i~' (.J1;"~"iP" "ryi";"~ ^{i to '4..b '; ',} t~~

Dumber

or

itemstor assignment)ereeeemedtobehO""D.Koopman!andBeet-

, t

'manDlUIS!) splitth~ coell''"t~enu(ij'.,intot~ sothat .-

...

~~:

,)

li ,·=-',. ·,

'di~-=d,i,

ThetwoDby0

~atrices 'i ,

^anddj ,mayabobe

~umed

^{to be}

!lymm'etrkal~

•~ . , • • J , •" ," •

.withlerosaloDIthe maiodiagonal.Thatis,

v •

', .

-,tii

-0:

dji",,:0

_ -. if "

In~beC~Dtexto!a.n

ru- ,

cii"lepreseob thecost,at/,ran!pOrtat ionIrom ..

<:

:.,--;

....

^{..<.-••••, ;}

20 [acilityi.tk>eationjtofacilitypAtlocationq.The6rstsetofconstraints ensures thatfi at tlynneJacilit,isassignedtoeach~iRan^an:t~e^secondsetof.

" c0Q51rmU resulll In

ea.eh "

facilitybtiog assignedtoexac:t1,.oae kx:atioo. 00 the other hlIld,

t :

~~J_f1I,aJao[\e(kmaIlDformulation (&11beiDterpre~asthe minimif'atlOU-ofthe-productoftWomatricesti,and dj, repm ent ingtheDumber

"

fofloads tobetransportdfromracilit~itofacility.~andtheCQ5tortrans~rtill.g

.

.

on~loadfromloc.t~oll^jtolocatiooqrespedi vely(EI.R~ylhand HoUier,IG70j.

Eachassignmentof facilitiesto availablelocation',isrepresented byan'nby.a

•

p~rmuiatioll

^matrixX

~

^{IIZ,'j IIwherezij";,,:,1 il'}(ad lityi illassignedtoloc·.- ticnjand'%,'j

=

0otb~in.^Theto~al c?S~^ofusig;or~~t^ismeasur~d^by2,.1 and optimal;Ssign~ell~·obti.ined·'Whell^«lStis~ini~ii:ed. ^.

TheCollceptuaidevdopm~lIt

&

nrlous' quadraticus~ment alp;orith~^as

.

^{. '\ . ' .}

., ." " . ..

:apr1ied..to tbeF~a:e discussed~;ow f~rtbe{lah of.ornpl~tenes!aswellasfor elaboratinp;QJi.thevalidity&ramlerc-eompeterbased sub-optimal;heuristic,~n- Itruetionalgorith~fortb,FLP.

TbeFLP as_. "q~adratic_{f .}assignmentprcblem_•bas beeb solved,or at least feasible,.sub-cptimumsioiutions pycnbyVoUmann, Nu ~andZ~ler,Hillier, ,Francis'andWhil,eetc.GilmoreandLawlerb'ave woedODdevelopingexeet

solutionproeeduree.Heuristicalgorithmsnamely,

1)",Steepest-DescentP~rwise-IQterchaDgePrU<:edure.byFrancisand White,

whichisthe~&Sisfor~RAFT&~d,~

2) TheVollmapn:Nugent,'lartletpr~ure.

f .

<,.

".:)

.

^".;.'.

·"·r

;,,: .·.J. , \ ·.->;:4

21 are aplaiDedaDd&D.a1yIedbelow.

4 AutXI:~ttoIuHotlprocedu~but'<!:Oil.the work01Gilmoreud Lawleri!also

'0

_ discussed.

.I) STEEPEST-DESCENT PMRlv\SE-lvrERCllANGEPROCEDURE, CRAf~e5ge~tiallyis&IInttnSiop 01 this algorithmIFt.ods&:White).Th~

,

^.

basicprinciplesareas'follows.Let.betheassignmentYeetOt

a= ( .(1)•• (2).;(31•....••(,)) (2.21

• ^wh~e

^ilA

^com~Dellt

^{isthethenumber}

^{or ~be l~catioD}

^towhich

^~aCm~y

^ihll.'l

·\bee.,",I,.,d. F?,allI,'".""I.",01j;,d , between0" eed n,dli., ) be

lb~

· ~staD.cebetweenl~atioD~j-eedq.FinaUylettD!ihetbeCOn!tDn~_orproper-

»,lion. lity converting thedis\l.Ilce"betweenfacility'i

a.ni

P,for alli

,P,

^iniqa

ecet,sotha! if'radli!yiisata(i)andracilitypIf.a(p)the toialeostrOtfuilities

! . . .

i

a:

^{pis w}"i f (lI (;),I1(']-The.totaJrostlot anassip~en'01facilitiesto,sitl!:!

isthen

'TC(a)".

E

0;;; ('(i)••(,11, (2.31

, ISi <r S-

·~ThechangeiDloostQbt~iDedbyill.terchaDgingtheloel,lion orr. ties u,and,fer

a

^giveo

assignni~t,

^denoted

~Y

^DTC.. (a)illgi,eo'by:

,DTC..(a)

=

f::(w;.-w;.lI; ('(i),.( 'II-d(.

(;1..

(,III

'" i_I '

-2w.;; (' (').' ('11 . (2.')

- ,

" .

a·

22

"Fig. 1.Steepest-descentpetrwise-dnterchenge procedur~

(Francis &',Whit e• .197 4), '. '

2'

.The pairwise-interchangea.'go~ithmcan be explicitlystated as follows : l. 'Collectdatafora,D:W,theassignment veetor,distll~ee^{matrixand weight,}

m~trix resp~tively .

2. ComputeTC (a)using 2,3. .

3. Set,

~

^{=0 wbicl1}

deD~tes

the grllatest'deer'easein

1 0& )

^foundso

~Ilr

^lorthe

PV;Dassignment.

4. Set i =.l andp=2.

5. Compute DT.c_ip(al.usin.s 2.4 6. UDTC;p(alisgreater thane, goto7,else8.

7. Sete toDTCip(al.setIItoiandvtop.

8. Up=n,golo g, e[se·lO.

g..Ifi=D·~,^{go'to12,else II.}

r-- 10.j;=j+t,go to 5.

'I I.i=,i+l,.j

=

iH ,gotoS.

12.Ireisp9sitive,.goto13,elseIS.

./

13. ReplaceTC(a)by TC(a l•e.

14.Reviseabyinterchanging1helocationofracHities~and v, goto3.

ISo-Stop.

.( 21THEVOLLMANN,NUGENT, ZARTLER PROCEDURE,

Thestet!~e!t-descelltpairwise-interchangeprocedure"requiresa. tremendous

, j • •

amorlnt~rcomputa tional.effortandtberollo~i~gal~erDative^WIl.8sugge:'ted by