Table Of ContentDECISION AND DISCRETE MATHEMATICS
maths for decision-making in business and industry
Mathematicspossessesnotonlytruth,butsupremebeauty-abeauty
coldandaustere, likethatofsculpture,andcapableofstemperfection,
suchasonlygreatart canshow.
BertrandRussellinThePrinciplesofMathematics
Etymologia Scientifica
Algorism-(re·lgoriz'm). ME. [a. OFr. au
gorisme, algorisme, ad. med.L, algorismus, f.
vi
Arab. al-Khouiaraemi, i, e. native Khiva sur
I
name ofan Arab. mathematician. Cf. Euclid=
plane geornetry.'] The Arabic, or decimal sys
tem of nurnerauon : lienee, arithmetic. Also
attrib.
Corruptlye written..Augrim for algorisrne, as the
Arabians sounde it RECORD£. Hence Algorismlc
a. arithmetical.
Algorithm, erron. refash. of ALGORISM.
(Shorter OxfordEnglish Dictionaryon HistoricalPrinciples)
About the Spode Group
The SpodeGroup was formed in 1980by a number ofdedicated teachers ofmathematics who were
resolved to improve the standard of mathematics teaching and learning throughout schools and
universities. They planned to produce relevant and stimulating mathematical material for use in
teaching,inorderto demonstratehowmathematics can be usedto solvepractical problems in real life
a
situations. Aswellas producing a wide range ofmathematics curriculum material, which included
strong element of software, they initiated such projects as mathematical modelling for teachers,
project and practical work in statistics, computing, and course work for the General Certificate of
SchoolEducation(GCSE).
They metregularlyintheir "own time" inExeter and other places for weekendseminars. The Spade
Group was then unknown and not acknowledged by the authorities. Their only reward was that of
knowingtheywere improvingtheteaching standards oftheir subject, recognisingthe importancethat
mathematicsplays inthelivesoftheir students and everyone throughout their workingandsocial life.
This small nucleus ofenthusiasts, all ofwhom have now moved to higher academic responsibility,
was at that time directed by Dr.David Burghes now Professor ofEducation at Exeter University and
Director ofthe Centre for Innovation in Mathematics Teaching, in close rapport with Dr John Berry
now Professor of Mathematics in Plymouth University, and Dr Ian Huntley now Director of
Continuing Education in The University ofBristol, as Associated Directors. With great foresight
they envisagedand initiated a new module for mathematics teaching closely related to working and
business life,as distinctfromthe pure ortheapplied mathematics papers ofthe A levelsyllabuses..
In due course they collaborated to write Decision Mathematics, the forerunner ofour present book
which became the recommendedset text for the Oxford University Delegacy ofLocal Examinations
for the Advanced Level mathematics option. Published in 1986 by Ellis Horwood Limited the text
was most successful, widely used in secondary schools, but is now "out of print". Since that early
breakthrough, teaching mathematics for business life has become nationally accepted by all A-level
examination boards. This second edition, rewritten, updated and retitled Decision and Discrete
Mathematics, again providesa coverageforthe Discrete Mathematics module.
TheSpodeGroup has sincegainedsupport and iswidelyacknowledgedfor its significant contribution
to mathematics teaching. As the publisher privileged to launch both versions ofthis mathematical
dichotomyin two successive editions, I have enjoyed the friendship ofmembers ofthe Spade Group
throughout allthoseyears. Iamsensitiveoftheir trustand supportand ofour continuingassociation
ofAlbionPublishingLimitedas theirpublishers, for whichhonourIshallalways begrateful.
EllisHorwood, the publisher at AlbionPublishing
Chichester June
1996
DECISION AND DISCRETE MATHEMATICS
maths for decision-making in business and industry
Written on behalfofThe Spode Group by
Ian Hardwick, MA (Oxon)
Head ofDepartment ofMathematics
Truro School
Cornwall
Edited by
Nigel Price, BSc (Aston), MPhil (Exeter)
Innovative Mathematics Teaching Centre
University ofExeter School ofEducation
Exeter
WP
WOODHEAD
PUBLISHING
Oxford Cambridge Philadelphia NewDelhi
PublishedbyWoodhead Publishing Limited,
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First published byAlbion PublishingLimited, 1996
ReprintedbyWoodhead PublishingLimited, 2011
c
TheSpode Group, 1996
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Contents
Preface vii
Acknowlegements vii
Chapter1 An introductionto networks
1. 1 Terminology 1.4 Chinesepostmanproblem
1.2 Investigations 1.5 Travellingsalesmanproblem
1.3 Minimumconnectorproblem 1.6 Notes
Chapter2 Recursion 17
2. I Definition 2.4 Highest commonfactors
2. 2 Investigation 2.5 Notes
2. 3 Divisibility
Chapter3 Shortestroute 21
3. I Investigations 3.4 Chinesepostmanproblem
3. 2 Dijkstra'salgorithm 3. 5 Arcs withnegativevalues
3. 3 Delays at nodes 3.6 Notes
Chapter4 Dynamicprogramming 31
4. 1 Investigations 4. 3 Applicationsofdynamic
4.2 The method ofdynamic programming
programming 4.4 Notes
Chapter5 Flowsinnetworks 45
5. 1 Investigations 5.6 Restrictionsonnodes
5.2 Terminology 5.7 Several sourcesand/orsinks
5.3 Cutsets 5. 8 flowaugmentationsystem
5.4 Maximumflow, minimum cut 5.9 Networkswitharcs having lower
theorem capacities
5. 5 Cutsetsindirectednetworks 5.10 Notes
Chapter6 Criticalpathanalysis 63
6. 1 Activityon arc: investigations 6. 6 Activity on node: procedure
6. 2 Activityon arc: definitions 6. 7 Activity onnode: total float
and conventions 6. 8 Activity on arc : total float
6. 3 Activity on arc: procedure 6. 9 Ganttcharts
6.4 Activity on node: investigations 6.10 Resourcelevelling
6. 5 Definitionsandconventions 6.11 Notes
Chapter7 Linearprogramming(graphical) 82
7. 1 Investigation 7 . 3 Drawingtheobjectivefunction
7. 2 Graphicalrepresentation 7 . 4 Notes
Chapter8 Linearprogramming:simplexmethod 92
8. 1 Investigation 8. 5 Three dimensions
8. 2 Simplexmethod 8. 6 Minimisationproblems
8. 3 The simplextableau 8. 7 Notes
8. 4 ~ constraints
Chapter9 Thetransportation problem 107
9. 1 Investigations 9. 5 Non-uniqueoptimalsolutions
9. 2 Thetransportationarray 9. 6 Degeneracy
9. 3 Maximisationproblems 9.7 Notes
9. 4 Unbalancedproblems
vi
Chapter10 Matchingand assignment problems 121
10.1 Investigations 10. 6 Impossibleassignments
10.2 Hall'smarriagetheorem 10. 7 Maximisingproblems
10.3 Matching improvementalgorithm 10. 8 Non-uniquesolutions
10.4 Investigations 10. 9 Unbalancedproblems
10.5 Hungarianalgorithm 10.10 Notes
Chapter11 Gametheory 140
11.1 Investigation 11. 7 Minimaxmixedstrategy
11.2 Theminimax theory 11. 8 Games withknown values
11.3 What isazero-sumgame? 11. 9 Usinglinearprogramming
11.4 Stablesolutions 11.10 Miscellaneousexercises
11.5 Rowandcolumn domination 11.11 Notes
=
11.6 Expectation expectedpay-off
Chapter12 Recurrencerelations 156
12.1 Whatisarecurrence relation? 12. 6 Inhomogeneous: a= 1
12.2 Investigations 12. 7 Secondorder linear
12.3 Firstorder linearequations 12. 8 Secondorderhomogeneous
12.4 Complementaryfunctions 12. 9 Summary
andparticularsolutions 12.10 Miscellaneousexercises
12.5 Inhomogeneousuo+\ = au.+A: 12.11 Notes
Chapter13 Simulation 177
13.1 Introduction 13. 3 Useofrandom numbers
13.2 Randomnumbers 13. 4 Notes
Chapter14 Iterativeprocesses 184
14.1 Introduction 14. 6 Intervalbisection
14.2 Investigation 14. 7 Problems withintervalbisection
14.3 Convergentsequences 14. 8 Iterationformula vinterval bisection
14.4 Graphicalrepresentation 14. 9 Hero'smethod forsquareroots
14.5 Rootsofpolynomials 14.10 Notes
Chapter IS Sortingand packing 199
15.1 Investigation 15. 6 Quicksort(super-pointerversion)
15.2 Sorting 15. 7 Firsthalfofsortprogram
15.3 Bubblesorting 15. 8 (Bin)packing
15.4 Shuttlesort 15. 9 Packingalgorithms
15.5 Shellsort 15.10 Notes
Chapter16 Algorithms 216
16.1 Whatisanalgorithm? 16.4 Theefficiencyofalgorithms
16.2 Investigations 16. 5 Useofrecurrencerelations
16.3 Simplealgorithms 16. 6 Notes
Glossary 228
Answers 230
Index 254
vii
Preface
Recentdecadeshaveseenavastincreaseinthedevelopmentsandapplicationsofmathematics
tosolveproblemsrequiringdiscretemathematics.
Theprocessofsolvingsuchproblemsisoftenreferredtoasoperationalresearch,and
employstechniquesthathavebeendevelopedtosolveparticularclassesofproblems,often
relatedtotheefficientuseofresources.
Asecondclassofproblemshasdevelopedoutoftheneedforsolvingproblemsrelated
totheITrevolution- manyoftheseproblemsrequiremathematicsintheirsolution,butitis
aratherdifferenttypeofmathematicsfromthefamiliarcontinuoustheories.
Bothclassesofproblemhave,attheirheart,theuseofdecision(ordiscrete)mathematics,
andthisisthefocusofthistext.
Manyexam boards,at A-level,havenowincludedtopicsinDiscreteMathsin their
modularcourses.Thistext,basedontheearlyOxfordBoardASsyllabus,hasbeenexpanded,
andwillbeofinteresttostudents(andteachers)takingDiscreteMathscoursesforanyexam
board.
ThistotallyrewrittenversionoftheoriginalSpadeGrouptexton'DecisionMathematics'
hasbeen writtenby Ian Hardwickandeditedby NigelPrice. We aregratefulfor their
dedicationandhardwork,anddelightedthatthenewversionofourbookisnowavailable.
DavidBurghes
onbehalfofThe Spode Group
Acknowledgements
Wearegratefulforhelpwiththisbookto
MarkCarroll
JoeChan
SimonCollinge
BenFerrett
JamesKnight
JoMooney
anddelightedthatEllisHorwoodisagainpublishingourtext.
Finally,wearemostgratefultoAnnTylisczukfortypingafirstdraftandtoLizHolland
fortypesettingthefinalversion.
1
An introduction to networks
1.1 TERMINOLOGY
Anetworkisasetofpointsandlineseachofwhichhasitsendsatapointorpoints. Various
equivalentterms exist and maybemet; these are
network - graph
node - vertex - point
arc - edge line
region - face area
(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) Fig. 1.1
1.2 INVESTIGATIONS *
Investigation 1
A pointwhere lines meet is calleda node.
Theorderofthenodeisthenumberofends Gnode
6node
oflines thatmeet there.
Draw, if possible, networks with the
specificationsgiven inthe table. I node 3nodes 4 nodes 5nodes
(a) 0 2 I 0
(b) 1 1 1 0
(c) 1 2 2 0
Canyougivearuletodescribethosenetworks
(d) 3 3 1 0
thatcannotbedrawn?
(e) 0 3 1 1
(t) 1 1 0 1
Theorder ofthenode mayalso bereferred toas thevertexdegreeor valency.
