MAY ON

EvOU C'E I EVHLII G

CENTRE FOR NEWFOUNDLAND STUDIES

TOTAL OF 1 PAGES ON Y MAY BE EROXED

(Without Author's Permission)

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RESOURCE LEVELLING

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A T~E~IS. SUBMITTED

qy

'Yasuhiko _',ramura '

I 1

IN PARTIAL 'FULFILMENT_ OF'

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THE DEGREE.QF ^I:

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M.Eng.

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· vi ^{. .}

^TABLE' OF CONTENTS

List of Tables

...

^(iv)

List of F.igures '-···~·· ^(iv) -~ Acknowledgements •• •••••••••••••••.••••.••••• •··.. (vi) Abstract

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(vii)

1 ,,

Chapter I I~troduction

1.1. Introductiop . . . .. • . . . · . . . · . . . . ~ 1

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1'.2 1.3 l:.4 '1. 5 1.6

Two Approaches to Resource _. Lev~1-H.ng •.•••

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·Methods o.!.,. Resource Leve1lihg ·

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Present State of the Art

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Problem Statement • •• ~.

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Design Criteria

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2 5. 7

12 13

--~---~---~~~

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Chapter r'I··Mathematical· Approach to Resource , Levelling. . . . .

2.1 Theoreti'qp·l ·Approach to. Resource

// •

/

· /--Levelling Problem . · . . . ~ . . . ~ .. • . : ..

/ / . .

2r1.1 The Concept of the Obje9tive

Function ·••••••• •••..••• tl ~ •••.•.••• ' I

2,.1. 2 Formulation of Objecti_ve Function •

2.1~3 Opt~~izati~~

of

Resour~e

~ . . ·a·

.• - :,£

2 •1. 4 Num~riqal _. , . Example ^{• • •} ' ^{I• •}

Cha.nges ••

... . ...

-.?~ 2

.-...

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^{\ .}

Project Model based on CPM. N~twork 1.

Analysis ••••••••••••••••• ·.,-·~ -: ·. ···.····~·

. ~. I• .

2. 2 •· 1 Optimum ·Resource Level¹l~ng ) · . . : on a Simple CPM' Netwqrk . ••••••••••

(ii) _.

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37

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. 2. 3.

2.2.2: Impracticality of Achieving

the-Opt_imti~ Resour9·e Profile'

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Heuristic· Algorithm· for Large N~tworks. · ...

, D . . I

2.3.1 Activity chain _~

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2.3.2 Numerical Example

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Chapter III Computer .~rogramme of the

3.1 3.2

3.3-

3. 4"

Heuristic Algori ~hm . •..•••.••••••••

Initialization Phase·

.•... ...•.... . •...

S9he~~l~r Phase.I Scheduler Ph~II

Re~scheduler Phase

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3. 5 Final· Phase • ,• • • • • • " . . . • •• • a • • • • • • • • •

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Chapter IV N~~rical Example using

-'Cqmpu-t;.er: Programme . • ••••••.••

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44

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62 64 67

·73

76

87

89

99

Bibliograph~ .. - . . : 102

• • • • • • • • .... • a; • • • ~ • • • • • • • • • • • • • • • • • • • •

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'Appendix A

·Appendix B

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·. Appendi ces _{. {.} .>,

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List of 'Tabl~s

.Table r;r:...l

. ^\ . ·' ^Resou~ce Scheduling ···~···

, .Tab.le -II-2 . Optimum Resource·qchedule

. . .·

...

Jable·II-3 Generalized Expressions . , .. fo;r· Resource Change ,•. ~ . . . ·• ~--.••• ·

·Table.· II,-4,

• • ' I

-Table IJ:-.5 Table II-6

Bar Chart,; ••.•.•••..•••.•••.••

Optimum Distribution .• _. ... · •••

Optimuin- Schedule •••..••••••. ^{! • • •} Table II-7 \Activ.ityJchains -••••••••• •.· .•• .'.

Table .II-8 ' Table Iv-1·

Li~t

. r:rf.·

·,Figures , Figu~e · I-1· Figure

•

II~.l

Op~imal Resource Schedule • .' .•••

Optimal Schedule

... ...

Rectangular and Paraboiic

Resource Allocation _I. ···~···

·Res~urce ' Allocation P_rof_i Le____.__..___._ •

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19·

28

33'

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42 51

··61' . 98 .

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'h:ure

^II-2

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· ··Fig~re

_II-3 ^Optimal~ Resource Profile •• _. . .

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29

\ '•

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Flow Chart for BASS •.•..•••• ~ .• ³⁸ - Figure II-4 ·

Figure II-5

Numerical E~ample Network. •· •••..

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Opt1murn Resour_ce Profile ••• : •.•

39 ' .,

43. ^.'

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F_igure II-6-·.Activity Chain ••• ~ :·-. •.•• .' ••• ~."-•• ^-48

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Figure II-7 (a) Schedule with ~~me-lags .

iri Activity Chain····.:···· . . . ... · ⁴⁹~ ^{: '. .} • ^. ,. ^{. '·}.. : ^.·.J ^{: ..}

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(b)· ~che'dule, wi t~aut ··Time;- lag$·.. · · . in Activity Chain· • ~ •••.•• ,~ ••• ·. -~ .. ,

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' 49· Figure II-8 CPM Network •••• · •• ;. · •••.••••.• · ••• · 52·

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Figur~ II-9 · · Acti:vi ty Chains . . . .. ,

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Fig~re .

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^{Profiles· .•}

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Figure II-11 ·. Final Optimal Resource Profile ••

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Figure I..II-1 Figure III-.2

F~gure III-3'

III-4

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SysJtem Flow Ch~rt for BACS .•••• ._,

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Initialization Phase

Flo~· Chart . . . • . . . · . . . .

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The First Activity of the

Activity Chain . · .•.• • ..•..•..••••

Figure . Intermediate Activity of the

Ac ti vi ty Chain • . . • • • • • • • • • • . • • • • . ·

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Figure III-5 An ·Activity C!la~n- ••..••••••. · •• -·." ._,

· ... Figure III-6 Figm;e III.:.7 Fig'ure III-8 Figure III-9

Sch~dul~~ Phase ·r Flo~ Chart •• ;~

Scheduler Ehase II Flow Chart •• ;

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Resour~e Profile •••..••. ~ •• ~: .••

I •

Peak in Resource· Profile, .... _. ; ••.

Figure III-10 Activity Schedule Chart ! • • • • ~ ^{._ • •}

. F..t.gure

II:_t:! .~----·"~~~~urce

Profile ....

t~·:,...r. ^{. • •} :~.:. ^{: - . • • ·• ·•}

1Figure. III-12

i?~ak in:

^Resource

^{Pro t:.i ) _ e} _~:.·_._.~·-­

Figure. III-13 , Actf.vi 1;-Y ··scheq'ule· Chart

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Fi~ure.III-14 Re~scheduler.Phase ^Flow Chart. ~·.

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Final Pha~e Flow Chart ..•••.••••

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_Fi~ure· III-15'

Figure IV-1 Figure

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·.Gas Station

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Fi'gure IV-3· Figure IV-4- _ .. .Figl:lre

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. Figu~• IV-6 .

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Network· Information .

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'Activity Chains .• .,.., ..••••••.••••

.: Priority Value .of Each Chain .••• !' I

Ini, tiai Resourc.e Profile

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Final Optimal Resource Profile

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ACKNOWLEDGEMENTS

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I. w,if?h to express my• sinc.ere gr~t~tud'e to Profe·ssor H. N. Ahuja for

1

sugge~t~ng this -r~searcJi _problem to me· .

. . . and for h~s .guidance and constr':lc~.tive .C:ritici~m ^{.. -}Duri~g

. the ,past" two₁ year~·.Jhe has .been a ··sourceo of informed _opinion

• - ; 1 • • • • • • • •

on matters. Qoth scientific and personal. I would also ·

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··like to express my gratitude and-~ppreciation to Mr.M. Iwadart.' ..

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and . associates of the ·ta~a-shlma .T~xtile Millls Ltd. for

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prov?:-ding ~'an op-por.tun.i ty for me

to : · .

~tudy in :·can~da. I

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I wish to acknowled.ge ,Professor E.: Moore, P~ofessor· .. M.E.· El~~-~~-~ .... ry,' Professor 'p .J·. Arnaria, ·-and' Mr. rrerr'_{. .} y ' 'Buckley _{. .} I· for their "

(h'elpful distussions. F-inally_,· I ·.w9¥"ld like to thank Memorial ' . 0 :.

I - . University of Newfoundland, for

- ,

^{. .} 'aw.~:frding ^. a ·scalarship to

me, ·

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thi.s enabling...me-t~pur.sue.-m¥--Studi\:l-

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\ ABSTRACT

RESOURCE LEVELLING

.. For projects w})~re highly ' _skilled workers are · needed· whqse ·training could -take -9. l~ng t i,me, gradu~lness i·n ttie· · ·

. .

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,.

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• I '

~espurq~.profile coul~ provide the ne6essary tr~ining per~od~

"This thesis preserlt:$ _I . ./:iln heuris'tic

.- proc~dur.e

for optlmal _{• •}

~

. . .. '( . - . .

resource level.ling ~chieving :·a gra-du~l-· .( parabolic· )

. . I

resource distribution.· ·rt· a.·i.ms-at minimizing .. the ·fluctuation

•.

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. of ·day-to-day resource ·charil.e using the variand~ of reso~· ce. ·

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_: cha_\ge as the primary ind~_x_. Mathem~_~ica~ly b~'sed~_this. \ .. ~~- \ .,._

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'-t--

·- ·heuristic ~ives--a meas~re whe-reby an JfPtim,·al schedule can~· ·. .

\ . . .· )'\ • J .

•• to,. \ 0 \t • •

_ t, be· recogn i';z ed. · · . . \, . . :;::~ :·'!:. · •

~ ' t.. : :·

• . - l . . . . . . ',. f! ¹1

us~rig

·the c.omputation-preeedure

f~r ~trr~h eur1.st1.e ~Y- ·

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fi coniplJter programm: cal~;d ~~~S (B?ild-up Activ\ty f~ain

· · ..

Scheduler) has been_. -developed which· yields an optimal

~~ . .

I

schedule for la~ge . P+Oj Eicts . f.rom. : data ·obtained from CPM .

analysis ^.\.r ~tris . ~rbgramrne is. both easy to .use and' has a.

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shor.t C?~putational time.

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., ^' "' CHAP.TER. I ·

1.1 . Introduction

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p n • . '· ' • .

. The utilization 'of ~esources 1.~ .an. ·ever present

· ·-

~ . . . . ..

condition of project accomplishme'nt.· R~sou·rc:'es

mean

^{'money~ ·}

. • • - . • . . ~ . . . ·.• I , . . • " ,;

labour, equ1.pment, rnater1.als, and

.

. any other conunod1. ty . . ^~ .

( }~llich

^is

expr~ssible. i~ phys;~~l ~l.~$. ^{· . . · .. ...• . •}

·

\.~:;.•

^{, ·} Present CPM'· (Crl. tJ.cal· Path Me.thod)

sc.~edu:IJ.n'! .an~.

^{. ,}

^{. .}

' )

!resource allocation are' based ._. on 'one

of '

^three' p~s·~i.bl~

. ' . . .

I .arrangements: ^{· ... · ..}

.. .

^,'

'

• :. J... T{le a\ra.i:iability of ~e'sources for each date· is. £ix~d ^I

'

, I

and from· th}.s. the minimu~. pr6j~ct duration is determ.iped. ·~ ^..

. ·' . .

2. The

proj~ct .

duration .is f,ixed ^{. '· ,}

·~rid

^{the ,. resourcee .. .} . .

required to meet· this

d~ration

^is

d~.t~rmineci.

^{·_.. · ·}

.,. • , tf • • • • • • • .. :· ' ' 0

3. The resource availal?ili ty 'for. each day: .is,.

a."

''-~riable,

.

.

^{. ~ . ' ' .}

the. pro~ect durat~on is fiXe~,

•:d

~~tj,mization is . •tieved, , .

by. varyJ.ng the: dally res~urce leve~ ln order. t~ keep . the total

resource '•as

cl~s~

^{'to, the}

' mi~imwri r~q~·ir,~d

^as.'

·p~s~ibl~. '

<' • I '

.

^{' ,.} (\

In many projects

.

_i t is des'irable to avoid sharp,

fluctuations· in resource r·equirements. ·.

·Re~rui

ting ,· hiring,·

. . . " ' .

.and training ·personnel" .. is costly ·and. time-con'swidng.

It is not feasible to' 'hire and lay-off the ~ork-fqrc~

according up .of the

to ~he' fl,u~tuating ne.t?d::f. ·Therefore, th~ build~

., '" . ' I ·~ . • •

work

.£~-r~e . sho~ld.

^be

~;ad~~-i

^{;. that· fs, Ute. resource.}

"'. ·

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profile. should· gradually rise-st.ep by step 'to a peak and

' ' ' I

then fall gradually· towards project· co~pletion. This

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p~·oblem , kn'own, as Resource Leveliing is studied t in 'this

thesis under the c~ristraints of the thirca''type .of . ^{c . . .}

w • ' . •

arrangement, that is .the pr_oject duration is fixed and.

I

. , . . • 0.,.

··the· d,aily resource ·~evel is v'a.ried· to obtain a ste

1

p ~Y-· ·:· ·

step build.~up ~nd a~. gradual ~la~-·off k,e)Pi~_g the to"tal ..

resource requirement to ·a minimum ..

.

. · ' <

...-1.. 2 Two Approaches to 'Resource

.

^{' ')}

Much thought and st'fdY has

' . I .

Levelling

.

^{. I.}

been devoted t.o the

2

I,, .

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resource levelling· problem. ·Two approaches to resourc;:e

. -

^' ., .

l_evelling are in existence:

. .

The first approach ke~ps -the work force ·to a minilt\um

.. . . ' - t

and attempts to ·achieve a flat level profile. (see Figure

I-1 (a) )

pr~viously

^{· co·}nsidered by De-Witte (Data .Rroesing ·. '

• I I . '

sc~~n~/Engineering,2.~·

_.

,29

_. (1964) ), Levy e·t al (Navai

. . ..

' ~ : .

' •'

Res~arch

^{Logist·ics u}buarterly; 9,

l-,~-4~9-o"'2l

^{) ,}

~J3M ,....-, ("= I= BM =-~-- ,-

^•.

- - r

Prograinme Number 5734-xp4), anCI. Burgess et· al. (J.Indust. _. . Eng,i

,'~III· ,

(1962") ··) ·• ·

T~s

j,s impoj:'tant for a p:z;oject,

w~ere·

camp facilities have to be· provided for the workers 0r Where WOrk 'progresses

on

^a contiilt~OUS basis and the .size

. . . - . . • .

lf\~.

of' .the work crew is governed by peak requirements. By ke:.l..o:Dng

. . I .'t;t) 1

- ~'

the level ~f resources required 'on any d~y to the ·minimum, the cost of providing the 'camps or maintaining. ^' . a work

.

'^{0 •}

. ^~ ' . " \ '

. .

crew can be kept to the minimum. Also in the ,fj.rst

. ^' .

·approach, many resources are required on _{. .. , . .}

date ,and have to be· cut off ..

ab~uptl~

^at

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.,

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th"e ·

s

^{tarti~cj ·}

project·. completion.

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·, It is not always pOSf?ible to rn'intain 'a constant_

.

~o'rk ...

for~e

^{/ .} part'icui.arly .'.in. one-time jobs.. It is ·easier,

- .

however, to ·achie;e a

sm~oth

build up of resources tollowed

,.

I •..

by ... a gradual taperi~g off with a minimum fluctuations

r

during the proj~ct •

./ -

.

..

_'

The -second appr~ach gradually builds up the -~rk

force .to a pe~k and .then gradually lays ... them off towards

-t

project. completion (see Figure I-l(b) ) work .in this area

0 0 0 - . " • •

.

, has been done by Mcitrtino

.

(P~oject Managem~nt· 0 0 .

.

and Control Series III, (1965)). This approach is useful for projects

'"'

where highly_skilleq workers aJ:'e needed whose recrui tig

-'~nd, o~

·fraihing would .take a _{. .}

l~rlg

_{,., \ .} ^{time •. The}

·gra~lialnes~~

·in such· a case -would: provide the necessary rE!cruiting and

.,

~ · · tra~ning peri6~ •. ,.

--~~ - - ~ ··~~--~~~~~~~~~~---~ ~

.Figure .I-l (a) can be conce.iye"d

as

the' result of yerticaln

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·. cornpre~sion of the resource profile, which re·sults in a . rectangule. · Sim~l~ry_, . Figure I-1 (~)· ccrf ~-e ·co~ceiv.~d a~ •

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· the result of a horizontal . com~tression ·r~sul t·ing i.n a parabola •

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Methods ()f Resource :Levellin<}:--', ·•

I'•

" ~:t; .ese~i:~eth)'ds . :f ^reso~i:ce

1;/velling can be.

d:v~ded

^[nto

three ·a~eas: a)c Handsmoothing, \)) mathema-tics, c) heuristics·.

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-" In the hand-scrlut;.ion, use is made of the human Is '

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.·a·bility to take ,a G.ompr~hensi've look at a ·resource profile

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'"~e·tach par~i~~l~r j~o?s ca~mg'>mwan~~ that~-

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shifted· to fill tip some val1eys elsewhere in _{. '} th~. schedule.

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roo~~ comple~ ' p~oject~, ha~dsmo~thing , by

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^{and time}

cdil.surni~g.

^{just :drawi ng}

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pro.fil~s ··can/be a tedious and redrawing '. t:h~· resource

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·task

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^{~the · number of}· jobs increa,ses •· If there were several

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'···resources. to level, not' oi)ly would tpere be. inor~. resource '

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profiles to keep track of but arso ·the .-~echno:j,.ogical

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_interrelationships oi~ioqs would make simultaneous_ smoothing of _Jll~ resources.:·a .tricky propos·itiori..

.

- !.~~

^{1:0 . . '.}

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. · Mai:hem.~tl.c~t_·appr<Dacqes_ have ~o.t been· f<:n~nd practical.' .. for .

;.'u'g~ ·scal!' · p~pj ec.~s

^{• The} most recogni ze.d rna thema tical

approach · ~ tli~ Dsource

levelling problem can bE?. seen·.:in

"yariation in Activit¥. L~vel · ~.m a cy:clical Arrow D.iag:tam" ~²

,·, I

The -~pproach, ~hic;h ~a~hematic.i.ans ref~r t.o ·as ~large.: . ~

Q co~bi~atorial problem·, .. ,leads ·to a rectang.ular resource

'r:Jro'f,~ le.

^{In ·} this case, there

-~re

^a

~ ,;ery -lca:J;J : , ri~r

^of

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. co~i ~a:ti;;~~

of activity •'starif.

t~mes . Wf

^{_large ,:to. enume'}F")ite

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, even with a

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b~g comp~ter. ^{' .}

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Unfortunately, ·analytical techniques

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·are- ~·ornputat~onally ·}.~prac.tic~i !or. moet real-life proble~s ... ^'•'

of 'this ·ki~d. '

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· · several ki~of heuristic methods using ·vert~cal

-- - -- - r ' 7 .. : . • •

.and horizontal co~pre~sion have been presented for·the

r.es.ource leve~i·ng .·p:yoblem, .. some of thes~ ~ill be discussed

,- ' I

later, .ap.d~ ~~~~ral ^heur·istic ~ype computer prograrilrnes. have .,'• . 2,4,6,10,16,17

also b~en written. Their objective {a to

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\'

smooth resqurce re~q.ir

.

~~ment by shifting' ~;J-ack· ⁽ jbbs beyond

~·

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• 1

periods of" peak requirement. Some of these essentia~ly. · repl~cate what a ·h~an scheduler.would d~ if he had enough

I

· time . Othei"s make .use .. of unusu~l devices ¹.or procedures

;

depign~d esped'iq.lly for the· computer. Ail of them, ··of

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course, depend'

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for⁷their success on the _:;r ^.

.

tr~mendous capabilitiep

··of electronic "data processing macf,lines. However, no attempt

h~s .

^{• beeh}

'ina'tl~

^· to provide a rna t.hema tical ^'

· bas~:.

for thes'e heur,tstic ·ap-proaches • ,.

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