Table Of ContentGraduate Texts in Operations Research
Series Editors: Richard Boucherie · Johann Hurink
Éric D. Taillard
Design of Heuristic
Algorithms
for Hard
Optimization
With Python Codes for the Travelling
Salesman Problem
Graduate Texts in Operations Research
SeriesEditors
RichardBoucherie,UniversityofTwente,Enschede,TheNetherlands
JohannHurink,UniversityofTwente,Enschede,TheNetherlands
This series contains compact volumes on the mathematical foundations of Oper-
ations Research, in particular in the areas of continuous, discrete and stochastic
optimization. Inspired by the PhD course program of the Dutch Network on the
Mathematics of Operations Research (LNMB), and by similar initiatives in other
territories,thevolumesinthisseriesofferanoverviewofmathematicalmethodsfor
post-masterstudentsandresearchersinOperationsResearch.Booksintheseriesare
basedontheestablishedtheoreticalfoundationsinthediscipline,teachtheneeded
practicaltechniquesandprovideillustrativeexamplesandapplications.
Éric D. Taillard
Design of Heuristic
Algorithms for Hard
Optimization
With Python Codes for the Traveling
Salesman Problem
ÉricD.Taillard
UniversityofAppliedSciencesofWestern
Switzerland,HEIG-VD
Yverdon-les-Bains,Switzerland
Theopen access publication ofthis book hasbeen published with thesupport ofthe SwissNational
ScienceFoundation.
ISSN2662-6012 ISSN2662-6020 (electronic)
GraduateTextsinOperationsResearch
ISBN978-3-031-13713-6 ISBN978-3-031-13714-3 (eBook)
https://doi.org/10.1007/978-3-031-13714-3
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Preface
This book is intended for students, teachers, engineers and scientists wishing
to become familiar with metaheuristics. Frequently, metaheuristics are seen as
an iterative master process guiding and modifying the operations of subordinate
heuristics. As a result, the works in the field are organized into chapters, each
presenting a metaheuristic, such as simulated annealing,tabu search, artificial ant
coloniesorgeneticalgorithms,tonameonlythebestknown.
Thisbookaddressesmetaheuristicsfromanewangle.Itpresentsthemasasetof
basicprinciplesthatarecombinedwitheachothertodesignaheuristicalgorithm.
Heuristics andMetaheuristics
When addressing a new problem, we try to solve it by exploiting the knowledge
acquiredbyexperience.Iftheproblemseemspeculiarlydifficult,asolutionthatis
notnecessarilythebestpossibleisaccepted.Thematteristodiscoverthesolution
with a reasonable computationaleffort. Such a resolution method is then called a
heuristic.
Byanalysingawholemenagerieofmetaheuristicsproposedintheliterature,we
identifiedfivemajorbasicprinciplesleadingtothedesignofanewalgorithm:
1. Problem modeling The most delicate phase when confronted with a new
problem is its modeling. Indeed, if a problem is taken by the “wrong end,”
its resolution can be largely compromised. Naturally, this phase is not the
prerogativeofmetaheuristics.
2. Decomposition into sub-problem When one has to solve a complex problem
or an instance of large size, it is necessary to decompose it into simpler or
smaller sub-problems.These may themselvesbe difficult. Hence, they must be
approachedbyanappropriatetechnique,forexampleametaheuristic.
3. Building a solution When a suitable modelis found, it becomeseasy to build
asolutiontotheproblem,evenifitisnotgoodoreveninapplicableinpractice.
v
vi Preface
Oneofthemostcommonconstructionmethodsisagreedyalgorithm,whichmay
evenprovideexactsolutionsforsimpleproblemssuchastheminimumspanning
treeortheshortestpath.
4. ModifyingasolutionThenextsteptriestoimproveasolutionbyapplyingslight
modifications.Thisapproachcanbeseenasatranslationtothediscreteworldof
gradientmethodsfordifferentiableoptimization.
5. Randomization and learning Finally, the repetition of constructions or mod-
ifications makes it possible to improve the quality of the solutions produced,
providedthatarandomcomponentand/oralearningprocessareinvolved.
Table1 Contextofapplicationofaheuristicmethodandametaheuristicframework
Heuristics Metaheuristic
Areaofapplication Agenericoptimizationproblem Combinatorialoptimization
Knowledgetoinclude Specifictotheproblem Heuristicoptimizationmethods
Datatoprovide Numericalvaluesofaproblem Agenericoptimizationproblem
instance
Result Aheuristicsolutiontothe Aheuristicalgorithm
instance
Metaheuristics have become an essential tool to tackle difficult optimization
problems, even if they have sometimes been decried, especially in the 1980s by
people who opposedexactand heuristic methods.Since then, it has been realized
that many exact methods embed several heuristic procedures and did guarantee
optimalityonlywithlimitedprecision!
BookStructure
This book is divided into three parts. The first part recalls some basics of linear
programming,graphtheoryand complexitytheory,and presents some simple and
intractablecombinatorialoptimizationproblems.Theaimofthisfirstpartistomake
the field intelligible to a readerwith no particularknowledgeaboutcombinatorial
optimization.
The second part deals with the fundamentalbuilding blocks of metaheuristics:
the construction and improvementof solutions as well as the decomposition of a
probleminto sub-problems.Primitive metaheuristicsassembling these ingredients
andusingtheminiterativeprocessesrunningwithoutmemoryarealsoincorporated.
The third part presents more advanced metaheuristics exploiting forms of
memoryandlearningthatallowthedevelopmentofmoreelaborateheuristics.The
exploitationofamemorycanbedoneindifferentforms.Onecantrylearninghowto
buildgoodsolutionsdirectlyonthebasisofstatisticsgatheredfromprevioustrials.
Preface vii
Anotherpossibility isto try to exploitmemoriesto moveintelligentlythroughthe
solutionspace.Finally,onecanstoreawholesetofsolutionsandcombinethem.
Thebookconcludeswithachapterprovidingsomeadviceondesigningheuris-
tics,anappendixprovidingsourcecodefortestingvariousmethodsdiscussedinthe
bookandsolutionstotheexercisesgivenattheendofeachchapter.
Chapter1“ElementsofGraphsandComplexityTheory” Beforeconsidering
developing a heuristic for a problem, it is necessary to ensure the problem is
difficultandthatthereisnotanefficientalgorithmtosolveitexactly.Thischapter
includes a very brief introductionto two techniquesfor modeling optimization
problems, linear programming and graph theory. Formulating a problem as a
linearprogramdescribesitformallyandunambiguously.Onceexpressedinthis
form and if the problemdata is nottoo large,automatic solverscan be used to
solveit.SomesolversareindeedbuiltintospreadsheetsofOfficesuites.Witha
littleluck,thereisnoneedtodesignaheuristic!
Manycombinatorialoptimizationproblemscanbe“drawn”andthusrepresented
intuitively and relatively naturally by a graph. This book is illustrated by
numerous examples of problems from graph theory. The traveling salesman
problemismostlikelythebestknownandhasservedasaguidelineinthewriting
ofthisbook.
Some elements of complexity theory are also presented in this introductory
chapter. This area deals with the classification of problems according to their
difficulty. Some simple techniques are given to show a problem is intractable.
Thishelpstojustifywhyitisessentialtoturntothedesignofheuristics.
Chapter2“AShortListofCombinatorialOptimizationProblems” This
chapterreviewsanumberofclassicalproblemsincombinatorialoptimization.It
illustratesthesometimesnarrowboundarybetweenaneasyproblem,forwhich
anefficientalgorithmisknown,andanintractableproblemwhichdiffersmerely
inasmalldetailthatmayseemtrivialatfirstsight.
Likewise,itallowsthereaderwhoisnotfamiliarwithcombinatorialoptimization
todiscoverabroadvarietyofproblemsinvariousdomains:optimalpaths,travel-
ingsalesman,vehiclerouting,assignment,networkflow,scheduling,clustering,
etc.
Chapter3“ProblemModeling” This chapter begins by describing techniques
forsimplifyingthe treatmentofconstraints,notablybytransformingtheobjec-
tive into a fitness function. Then, it gives a brief overview of multi-objective
optimization.Finally,itprovidessomepracticalapplicationsofclassicalcombi-
natorialoptimizationproblems.Itgivesexamplesofdatatransformationstodeal
withapplicationsthatareatfirstsightfarfromclassicaldescriptions.
Chapter4“ConstructiveMethods” This chapter presents methods for con-
structing solutions, starting with a reminder of the separation and evaluation
methods,widelyusedforthedesignofexactalgorithms.Next,twobasicmethods
are presented,randomconstructionandgreedyconstruction.Thelattersequen-
tiallyselectstheelementstobeaddedtoapartialsolution,neverquestioningthe
choicesthathavebeenmade.Thismethodcanbeimprovedbyevaluatingmore
viii Preface
deeplytheconsequencesofchoosinganelement.Thebeamsearchandthepilot
methodarepartofthese.Theconstructionofasolutionconstitutesthefirststep
inthedesignofaheuristic.
Chapter5“LocalSearch” The further step is to improve an existing solution
by searching for minor changes that improve it. Local searches constitute the
backboneofmostmetaheuristics.Thesemethodsarebasedonthedefinitionofa
setofneighborsolutions,foranysolutionoftheproblem.Thedefinitionofthis
setnaturallydependsonthemodelingoftheproblem.Dependingonthelatter,a
naturallyexpressedneighborhoodmaybe toosmallto leadto qualitysolutions
or,onthecontrary,toolarge,leadingtoprohibitivecomputationaltimes.Various
methodshavebeenproposedtoenlargetheneighborhood,suchasfilterandfan
orejectionchains,ortoreduceit,suchasgranularsearchorcandidatelist.
Chapter6“DecompositionMethods” Intheprocessofdevelopinganewalgo-
rithm, this chapter should logically have been placed after the one devoted to
problem modeling. However, decomposition methods are only used when the
sizeofthedatatobeprocessedislarge.Itis,therefore,anoptionalphase,which
thereadercanignorebeforemovingontostochasticandlearningmethods.This
is the reason why it is placed at the end of the first part of this book, devoted
to the key ingredients of metaheuristics. In this chapter, we consider methods
likePOPMUSICormoregeneralmethodssuchaslargeneighborhoodsearchor
fix-and-optimize.
Chapter7“RandomizedMethods” Thischapterisdevotedtomethodsrepeat-
ingrandomlyandwithoutmemoryconstructionsormodificationsofsolutions.
Among the most popular techniques, we find GRASP, which integrates two
basic bricks of metaheuristics: a randomized greedy construction and a local
search. Four randomized local searches are presented in this chapter, showing
thatwiththesameclassicrecipe,differentheuristicscanbeobtained:simulated
annealing, threshold accepting, great deluge and the noising methods. The
variableneighborhoodsearchequallyfindsitsplaceinthischapter.
Chapter8“ConstructionLearning” Following the order in which the key
ingredients of metaheuristics are presented, one can first seek to improve
the solution building process. Having constructed many solutions, one can
collect statistics on their structure and exploit this data to construct new
solutions.Artificialantcoloniesrepresentatypicalexample.Anothertechnique,
vocabularybuilding,isalsodiscussedinthischapter.
Chapter9“LocalSearchLearning” If local searches constitute the backbone
of metaheuristics, the taboo search, which seeks to learn how to iteratively
modify a solution, can be considered as the master of metaheuristics. The
term “metaheuristic”wascoinedby its inventor.Thischapterwillfocusonthe
ingredientsthatcanbeconsideredasthebasisoftaboosearch,namelytheuseof
memoriesandsolutionexplorationstrategies.Otheringredientsoftaboosearch
proposed by its inventor, such as candidate lists, ejection chains or vocabulary
building,findamorelogicalplaceinotherchapters.
Chapter10“PopulationManagement” When one has a population of solu-
tions, one can try learning how to combine them and how to manage this
Preface ix
population. The most popular method in this field is undoubtedly genetic
algorithms.However,geneticalgorithmsarealessadvancedmetaheuristicthan
scattersearch,whichprovidesstrategiesformanagingapopulationofsolutions.
GRASP method with path relinking shows how to design a simple heuristic
integratingseveralbasicbricksofmetaheuristics,rangingfromrandomizedcon-
struction to population exploitation through local searches. Ultimately, among
the latest metaheuristics, we find particle swarm methods, which seem to be
adapted to continuousoptimization.It should be noted thatthe spreadsheetsof
Officesuitesdirectlyintegratethistypeofheuristicmethodsamongtheproposed
solvers.
Chapter11“HeuristicsDesign” Theconcludingchapterofthebookdispenses
some advice on designing heuristics. It returns to the difficulty that can be
encounteredwhenmodelingtheproblemandgivesanexampleofdecomposition
intoachainofsub-problemsforeasyhandling.Next,itproposesanapproachfor
thedevelopmentofaheuristic.Finally,sometechniquesfortheparametertuning
andcomparingtheefficiencyofalgorithmsarereviewed.
SourceCodes for theTraveling SalesmanProblem
Oneofthereasonsforthepopularityofmetaheuristicsisthattheyallowaddressing
difficult problems with simple codes. This book contains several pieces of code
illustrating how to implementthe basic methods discussed. Due to a certain level
of abstraction, these principles could be perceived as a sculpture on clouds. The
codeseliminateallambiguityontheinevitableinterpretationsthatcanbedonewhen
presentingametaheuristicframework.Thecomputerscientistwishingtodevelopa
heuristicmethodforaparticularproblemcanbeinspiredbythesecodes.
As a source code is useful only when one wants to know all the details of a
method, but that it is of little interest when reading. So, we have simplified and
shortened the codes, trying to limit them to a single page. These codes come
in addition to the text and not the opposite. The reader with little interest in
programmingornotfamiliarwiththeprogramminglanguageusedcanskipthem.
These codes have been tested and are relatively efficient. They addressed the
emblematic traveling salesman problem. The latter is pedagogically interesting
becauseitssolutioncanbegraphicallydrawn.Certainly,thesecodesarenot“horse
race,”buttheycontainthequintessenceofmethods,andtheirextremebrevityshould
allowthereadertounderstandthem.Morethanadozendifferentmethodshavebeen
implemented,whilejointlytakinglessthanone-tenthofthenumberoflinesofcode
ofoneofthefastestimplementations.However,wehadtocomplywiththisbrevity.
The codes are somewhat compact, succinctly commented and sometimes placing
severalinstructionsonthesameline.So,weaskthereaderusedtosparsercodesto
beindulgent.
