Table Of ContentNEE E etworks in Action
International Series in Operations
Research & Management Science
Volume 144044040 0
SeriesEditor:
FrederickS.Hillier
StanfordUniversity, CA, USA
Forothertitlespublishedinthisseries,goto
www.springer.com/series/6161
Gerard Sierksma • Diptesh Ghosh
Networks in Action
Text and Computer Exercises in Network
Optimization
Gerard Sierksma Diptesh Ghosh
Fac. of Economics and Business Department of Production and
University of Groningen Quantitative Methods
9700 AV Groningen Indian Institute of Management
The Netherlands Ahmedabad-380015
[email protected] Vastrapura, India
[email protected]
ISSN0884-8289
ISBN978-1-4419-5512-8 e-ISBN978-1-4419-5513-5
DOI10.1007/978-1-4419-5513-5
Springer New York Dordrecht Heidelberg London
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Preface
One of the most well-known of all network optimization problems is the shortest
pathproblem,whereashortestconnectionbetweentwolocationsinaroadnetwork
is to be found. This problem is the basis of route planners in vehicles and on the
Internet. Networks are very common structures; they consist primarily of a finite
number of locations (points, nodes), together with a number of links (edges, arcs,
connections) between the locations. Very often a certain number is attached to the
links,expressingthedistanceorthecostbetweentheendpointsofthatconnection.
Networksoccurinanextremelywiderangeofapplications,amongthemare:
roadnetworks; cablenetworks;
humanrelationsnetworks; projectschedulingnetworks;
productionnetworks; distributionnetworks;
neuralnetworks; networksofatomsinmolecules.
Inallthesecasesthereare“objects”and“relations”betweentheobjects.Anet-
work optimization problem is actually nothing else than the problem of finding a
subset of the objects and the relations, such that a certain optimization objective is
satisfied.
Whyabookwithcomputerexercisesonnetworkoptimization?Firstofall,net-
work problems in practice are mostly very large and extremely complicated. Only
withtheassistanceofcomputershighqualitysolutionscanbeobtained.Mostbooks
in this field only offer exercises that support the understanding of the theory. The
case studies, that are discussed in the literature, are usually not suitable for class-
roomanalysis,sincethedatasetsaremissing,oraremuchtoolargetohandleina
classroomsetting.
Thisbookcontainsawiderangeofnottoolargenetworkoptimizationproblems,
that need to be analyzed and solved by using the computer. The emphasis of the
book is not on solution techniques. The reader may find her favorite software on
VI Preface
the Internet. Some suggestions for associated readings and Internet sites related to
optimizationareprovidedinChapterCofthebook.
Eventhefirsttheoreticalchapteriswritteninthespiritofthebook.Thereader
learns the basic concepts within the framework of networks and computer tech-
niques.
Theprerequisitesforthisbookareminimal.Alltheoreticalconceptsareclearly
explained. However, the reader is highly recommended to use this book in com-
bination with a textbook on the subject. Our suggestion is Graphs, Networks, and
AlgorithmsbyD.Jungnickel[18].Thisbookcontainsacollectionofmajornetwork
optimizationproblemstogetherwiththemainsolutiontechniques.Italsocontainsa
widerangeofinterestingexamples.Thebookisclearlywrittenandveryaccessible.
(FurthersuggestionsfortextbooksaregivenintheChapterCofthebook.)Thelitera-
turelistcontainssuggestionsonthissubjectaswell.Weadvicethereadertoexamine
someintroductorygraphtheoryandlinearprogramming/optimizationtheory.
What actually is the subject that this book refers to? An Internet search on the
term“networkoptimization”doesnotresultinmanytextbooks.Thereasonisobvi-
ous.Networkoptimizationisrelatedtoabroadrangeofdisciplines,including:
quantitativelogistics; supplychainlogistics;
combinatorialoptimization; networktheory;
integerprogramming; integerlinearprogramming;
operationsresearch; managementscience;
discretemathematics; finitemathematics;
algorithmicgraphtheory; computerscience.
The closest related disciplines to network optimization are combinatorial opti-
mizationandquantitativelogistics.Combinatorialoptimizationdealswithproblems
wheresolutionsarecombinationsofthe“objects”andthe“relations”intheproblem.
The number of such solutions is finite but typically very large. A decision maker
needs to choose a solution that satisfies a pre-specified objective. Quantitative lo-
gistics deals with logistics and supply chain management problems that have been
formulatedinmathematicalterms.Soliteratureonthesubjectsofcombinatorialop-
timizationandquantitativelogisticsareusuallythebestchoicestoknowmoreabout
networkoptimization.
In order to avoid that the book would become a disconnected set of exercises,
weusethefictitiouscompanyGlobalTelecomCompany(GTC)asacommontheme
throughout.Theinternationalmarketfortelecommunicationisstilloneofthefastest
growing markets in the world. There is an increasing demand for more and more
customer-specificproductsandservices.Alsointhismarket,thesupplychainsfrom
product-suppliersviamanufacturers,distributioncenters,warehousesandretailersto
thefinalcustomershavemadeashiftfrom“make-to-stock”tothecustomer-specific
“make-to-order” concept. Outsourcing and focusing on core-business have become
the key success factors. Demand-supply chain management, where all links in the
production network cooperate, is a key issue for surviving and for obtaining the
necessary competitive advantage over other supply networks. Finding the balance
betweenononehandmaximizingthecustomerservice,andontheotherhandmin-
Preface VII
imizing the total costs, wastes, and inventories is done on a supply chain/network
scale.GTCoperatesinthisscenario.
GTChasbusinessesinvariouscountriesacrosstheworld.Thecompanymanages
alargeworldwidetelecommunicationsystem,andproducessophisticatedequipment
for telecommunication. GTC operations are divided into a number of largely in-
dependent operation divisions, such as “Research & Development”, “Cables”, and
“Services”.OneofthemaingoalsofGTCistoextenditspositionasanimportant
supplieroftelecommunicationservices,withanemphasisonacontinuousimprove-
mentoftheprice-qualityrelationsofitsproductsandservices.
These facts justify a look at network optimization from a telecommunication
pointofview.Theproblemshavebeencarefullyselectedandformulatedandreflect
ahighdegreeofrealism,althoughtherealityisnotalwaysreachedcompletely.When
facedwiththechoicebetweenrealismononehandanddidacticjustificationonthe
other, we have often chosen for the latter. We have formulated problems with data
sets that are manageable and surveyable for the student, but too large to be solved
“byinspection”.
The exercises in the book have been extensively tested in classroom settings.
Manyofourstudents’suggestionstoimprovetheclarityofthequestionshavebeen
implemented.However,weareeagertoobtainfurthersuggestionsonimprovingthe
book.Thesemaybesentbyemailtotheauthors’addresses(g.sierksma@rug.
nl and [email protected]). Nevertheless, some questions may not
becompletelycleartothereader.Insuchcases,thereaderischallangedtoformulate
her/his own interpretation of the question. Even a number of alternative scenarios
maybeconsidered.Inpracticalsituations,whenthemanagementisnotcompletely
clear about the formulation of a problem, a pro-active approach of proposing al-
ternative scenarios maybe effective, especially in understandingthe problem more
clearly.
Manypersonshavecontributedtothefinalversionofthisbook.Wewouldespe-
ciallyliketomentionthecontributionsofMatthijsStreutker,whoprovidedthefirst
versionofthe chapteronfacilitylocation, anddevisedthesoftware forsolvingthe
problems,andMarc-JanMenkhorstforprovidingthefirstversionofthechapteron
matching.Anyerrorsthatremaininthebookareofcoursetheresponsibilityofthe
authors.
Groningen,GerardSierksma
Ahmedabad,DipteshGhosh
Contents
Preface............................................................ V
A TheModelingandImplementingProcess ......................... 1
B NetworkTheory ............................................... 3
B.1 GraphsandNetworks ....................................... 3
B.2 SolutionTechniques ........................................ 7
B.3 GraphRepresentations ...................................... 9
C ReferenceswithComments...................................... 11
1 ShortestPaths ................................................. 17
1.1 Introduction ............................................... 17
1.2 Applications............................................... 18
1.2.1 Schedulingtruckmovementthroughcities ............... 18
1.2.2 Makinginvestmentplans.............................. 18
1.3 ALinearProgrammingFormulation........................... 20
1.4 AlgorithmsforShortestPathProblems......................... 22
1.4.1 Dijkstra’salgorithm .................................. 22
1.4.2 Bellman-Ford’salgorithm ............................. 26
1.5 OtherPathProblems........................................ 27
1.5.1 Thelongestpathproblem ............................. 27
1.5.2 Thebottleneckshortestpathproblem.................... 28
1.5.3 Thehop-constrainedshortestpathproblem............... 28
1.5.4 TheHamiltonianpathproblem......................... 29
1.5.5 Thestochasticshortestpathproblem .................... 29
1.6 ExercisesonShortestPathProblems........................... 30
2 MinimumSpanningTrees....................................... 37
2.1 Introduction ............................................... 37
2.2 Applications............................................... 39
X Contents
2.2.1 Designingnetworksfordisasters ....................... 39
2.2.2 Grouptechnology.................................... 40
2.2.3 Storinglargebutsimilardata .......................... 40
2.3 LinearProgrammingFormulations ............................ 40
2.4 AlgorithmsforMinimumSpanningTrees ...................... 43
2.4.1 Prim’salgorithm..................................... 44
2.4.2 Kruskal’salgorithm .................................. 47
2.5 OtherTreeProblems........................................ 48
2.5.1 TheSteinertreeproblem .............................. 48
2.5.2 Thecapacitatedminimumspanningtreeproblem ......... 49
2.5.3 Thedegreeconstrainedminimumspanningtreeproblem ... 50
2.5.4 Themostreliableminimumspanningtreeproblem ........ 50
2.6 ExercisesonMinimumSpanningTreeProblems ................ 50
3 NetworkFlows................................................. 61
3.1 Introduction ............................................... 61
3.2 Applications............................................... 63
3.2.1 Productionplanning.................................. 63
3.2.2 Touristreservationsystem............................. 64
3.2.3 Staffallocation ...................................... 65
3.3 LinearProgrammingFormulations ............................ 67
3.4 AlgorithmsforNetworkFlowProblems........................ 72
3.4.1 Ford-Fulkerson’salgorithm............................ 72
3.5 OtherNetworkFlowProblems ............................... 75
3.5.1 Themulticommiodityflowproblem..................... 75
3.5.2 Thereliablenetworkflowproblem...................... 75
3.5.3 Thenetworkcutproblem.............................. 76
3.6 ExercisesonNetworkFlowProblems ......................... 77
4 Matchings..................................................... 87
4.1 Introduction ............................................... 87
4.2 Applications............................................... 90
4.2.1 Constructinguniversitytimetables ...................... 90
4.2.2 Constructing3-dimensionalmodelsfrom2-dimensionaldata 90
4.2.3 Crewpairing ........................................ 91
4.3 LinearProgrammingFormulations ............................ 91
4.3.1 Themaximumcardinalitymatchingproblem ............. 92
4.3.2 Themaximumweightmatchingproblem ................ 92
4.3.3 Thenon-bipartitematchingproblem .................... 93
4.3.4 Thebottleneckmatchingproblem ...................... 95
4.4 AlgorithmsforMatchings ................................... 96
4.5 OtherMatchingProblems.................................... 98
4.5.1 Thestablemarriageproblem........................... 98
4.5.2 The3-dimensionalstablematchingproblem.............. 99
4.6 ExercisesonMatchingProblems.............................. 100
Contents XI
5 FacilityLocation ............................................... 117
5.1 Introduction ............................................... 117
5.2 Applications............................................... 121
5.2.1 Clusteranalysis...................................... 122
5.2.2 Locatingundesirablefacilities ......................... 122
5.3 LinearProgrammingFormulations ............................ 122
5.3.1 Theuncapacitatedfacilitylocationproblem .............. 122
5.3.2 Thecapacitatedfacilitylocationproblem ................ 124
5.3.3 The p-medianproblem ............................... 126
5.3.4 The p-centerproblem................................. 128
5.4 AlgorithmsforLocationProblems ............................ 130
5.4.1 Anexactalgorithm:Branchandbound .................. 130
5.4.2 Anon-exactalgorithm:Greedyheuristic................. 133
5.5 OtherFacilityLocationProblems ............................. 136
5.5.1 Thecompetitivefacilitylocationproblem................ 136
5.5.2 Themulti-objectivefacilitylocationproblem ............. 136
5.6 ExercisesonFacilityLocationProblems ....................... 137
6 CyclicRoutingonNetworks..................................... 143
6.1 Introduction ............................................... 143
6.2 Applications............................................... 148
6.2.1 Manufacturingofprintedcircuitboards.................. 148
6.2.2 Orderpickinginwarehouses........................... 149
6.2.3 Postaldeliveryrouting................................ 149
6.3 (Integer)LinearProgrammingFormulations .................... 150
6.3.1 Thetravelingsalesmanproblem........................ 150
6.3.2 TheChinesepostmanproblem ......................... 155
6.4 AlgorithmsforRoutingProblems ............................. 157
6.4.1 Travelingsalesmanproblem ........................... 157
6.4.2 Chinesepostmanproblem ............................. 167
6.5 OtherRoutingProblems..................................... 169
6.5.1 Thebottlenecktravelingsalesmanproblem............... 169
6.5.2 Theprize-collectingtravelingsalesmanproblem .......... 170
6.5.3 Thegeneralizedtravelingsalesmanproblem.............. 171
6.5.4 Theruralpostmanproblem ............................ 171
6.6 ExercisesonNetworkRoutingProblems ....................... 171
References..................................................... 181
Erratum........................................................... E1
Index ............................................................. 183
