Multiple Criteria Decision Aid. Methods, Examples and Python Implementations

179 Pages·2018·2.076 MB·english

by  Jason Papathanasiou, Nikolaos Ploskas

Jason Papathanasiou • Nikolaos Ploskas
Multiple Criteria Decision
Aid
Methods, Examples and
Python Implementations
123
JasonPapathanasiou NikolaosPloskas
DepartmentofBusinessAdministration CarnegieMellonUniversity
UniversityofMacedonia Pittsburgh,PA,USA
Thessaloniki,Greece
ISSN1931-6828 ISSN1931-6836 (electronic)
SpringerOptimizationandItsApplications
ISBN978-3-319-91646-0 ISBN978-3-319-91648-4 (eBook)
https://doi.org/10.1007/978-3-319-91648-4
LibraryofCongressControlNumber:2018942889
MathematicsSubjectClassification:90,90C29,90C70,90C90
©SpringerInternationalPublishingAG,partofSpringerNature2018
Foreword
I am particularly happy to write this foreword, for several reasons. First, I appre-
ciate the work done by Jason Papathanasiou and Nikolaos Ploskas to propagate
multicriteriadecisionaidinclasses,tomakestudentsthinkabouttheconsequences
oftheirdecisions,andtopromotetheuseofmulticriteriamethodsinactualdecision
problems.InspiredbythefamousquotebytheFrenchwriterandpoliticianAndré
Malraux, I am sure that the twenty-first century will be ethical or it will not be.
Indeed,ethicsisessentialforthesurvivalofourworld.Decisionsshapeourfuture.
Decisions are often difficult to make. Decisions can be good or bad. Decisions
have often been poorly made, especially during the last decades. Today, we can
observe the resulting crises all around the world. Many decisions are bad because
theyrelyonasingle,ofteneconomic(costorprofit)criterion.Theycanbemadeon
a qualitative basis (experience, expertise, etc.) or using unicriterion optimization
models and methods. Anyway, they usually fail because they are shortsighted
and biased: they do not take into account all the stakeholders, all the objectives
that are essential for our future. As a crucial example, achieving sustainable
development is impossible with a unicriterion approach. It calls for a multicriteria
approach encompassing economic as well as social and environmental objectives.
Multicriteriadecisionaidcanhelpustoachievebetter,moresustainabledecisions.
Thisbookisanimportantsteptowardamorewidespreaduseofmulticriteriamodels
andmethods.
A second point is that the authors do not focus on a single method but rather
reviewdifferentmulticriteriamethods.Itisimportanttounderstandthatthereexists
noidealmulticriteriadecision-makingmethod.Insteadmanydifferentmethodshave
beenproposedoverthelast50years.Eachofthemhasadvantagesandlimits.Each
ofthemismakingspecificassumptionsaboutthetypeofthedecisionproblemand
about the preferences of the decision makers. Choosing the “right” method also
dependsontheproblemathand:theavailabilityofdata,thequalityofdata,thetype
andnumberofdecisionmakers,therequirementsofthedifferentstakeholders,etc.
Alltheseparametershaveanimpactonthechoiceofanappropriatemethod.With
thisinmind,theauthorspresenttheprinciplesandthecharacteristicsofsixfamilies
ofmethods.Thissetofmethodsisnotexhaustive;otherinterestingmethodsexistas
well.However,thechoicethattheyhavemadereflectsthedifferenttypesofmethods
currentlyavailableasgoodaspossible.
TOPSIS and VIKOR are two compensatory aggregation methods that have
been widely used during the recent years. They are simple to use and rather
straightforwardtounderstand.PROMETHEEisawell-knownoutrankingmethod.
It allows for partial compensation between criteria and has many extensions
including among others advanced sensitivity analysis tools and the GAIA visual
descriptivemodel.Itthusprovidesdecisionmakerswithamuchricherinformation
attheexpenseofamorecomplexpreferencemodeling.TheSIRmethodisanother
interesting extension of PROMETHEE. On the other hand, AHP is quite different
as it is designed to work with qualitative input data and it is based on a very
specific pairwise comparison principle. Many people have been critical about the
theoreticalbasisofAHP.Yet,itisoneofthemostwidelyusedmulticriteriadecision
aid methods. Finally, goal programming methods are based on the mathematical
programmingmodel.Theyareanappropriatechoicewhendecisionsarerelatedto
decisionvariableswhosebestvaluesshouldbedeterminedunderasetofconstraints.
The lecture of this book will provide the reader with a quite good overview of
existingmulticriteriadecision-makingmethodsandwithguidelinesforselectingan
appropriate method for each decision problem. It is helpful for the practitioner as
wellasfortheresearcherwishingtocomparedifferentapproachesandmethods.
Finally, the authors provide reusable Python computer code for each of the
methodspresentedinthebook.Theavailabilityofcomputersolutionsisessentialfor
theuseofmulticriteriamethods.WhenIstartedworkingonmulticriteriamethods
back in the 1980s, people had to rely on programming mainframe computers to
implementmethods.Itwasdifficult,timeconsuming,andonlypossibleforpeople
withconfirmedcomputerskills.Interactionwasverylimitedandcomputergraphics
were nearly nonexistent. The advent of personal computers started a new era.
Around1990,interactivecomputersoftware,suchasthePROMCALCimplementa-
tionofPROMETHEE,appeared.Theymadeitpossibleforindividualuserswithout
programming skills to apply multicriteria decision aid methods in actual decision
processes.Progressively,moreapplicationsappearedandmethodsevolvedbasedon
thefeedbackfromactualapplications.Attheturnofthecentury,graphicalinterfaces
becamewidelyavailableandmoreadvanced,anduser-friendlysoftwareappeared,
boosting even more the use of multicriteria methods. Today, most multicriteria
methods are supported by specific software such as Visual PROMETHEE or
Expert Choice (for AHP). Practitioners have powerful tools available to perform
comprehensiveanddetailedanalysis.However,thesespecializedpiecesofsoftware
do not make it easy to perform comparisons between different methods, to adapt
methodstospecialdecisionproblems,ortotestnewdevelopmentsorextensionsof
the methods. The Python code provided by the authors is an elegant and original
solutiontotheseshortcomings.Itisopensource,canbemodifiedeasily,andrelies
on a modern and easy-to-learn language available on multiple platforms. Only
moderate computer skills are necessary to take advantage of it. Students will be
able to use it to better understand the inner works of the methods and to develop
moreadvancedprojects.Practitionerswillhaveawaytoadaptthemethodstotheir
specific needs. Researchers will have the basis for analyzing the characteristics of
themethods,forvalidatingtheoreticalexperiences,orfordevelopingnewmodules
ormethods.
In the compact format of this book, the authors manage to transmit essential
informationneededtolearn,understand,andapplymulticriteriadecisionaid.They
also provide interested readers with the means to practice, to test, and to validate
theiracquiredknowledge.
Πολλές ευχαριστίες στους Ιάσωνα και Νικόλαο για αυτό το υπέροχο βιβλίο!
(ManythankstoJasonandNikolaosforthiswonderfulbook!)
Brussels,Belgium BertrandMareschal
May2017
Preface
The Multiple Criteria Decision Aid (MCDA) discipline is growing rapidly as the
numberofpublicationsintheliteraturesoaredduringtherecentyears.Itissuccess-
fullyappliedinallkindsofscientificdomainsandnumerousMCDAmethodologies
exist with an even larger number of variations available. The rationale behind this
book is something that the authors were able to identify at a quite early stage on
theirengagementinthefieldofMCDA.Andthatisthat,asalreadysaid,thereisa
lotofresearchavailableintheliterature;howevermuchofitisdifficulttointerpret
or be exploited by the nonexpert researcher, practitioner, or academic. This is an
applied field of research and people need to be able to fully understand each step
ofthemethodologiesandimplementthemeasily,coherently,andmoreimportantly
correctly.
ThemainfeatureofthisbookisthepresentationofavarietyofMCDAmethods.
This book includes the thorough theoretical and computational presentation of six
MCDAmethods:
• TOPSIS
• VIKOR
• PROMETHEE
• SIR
• AHP
• GoalProgramming
The novelty of this book is that the presentation of each method is focused on
threeaspects:
• Initially, the theoretical background is presented for each method including
variantsthathavebeenproposedintheliterature.
• Secondly,athoroughnumericalexampleispresentedforeachmethod.
• Finally,aPythoncodeispresentedtofullycoverthepresentationofeachmethod.
The Python implementations that are presented in this book are as simple as
possible.
This book is addressed to students, scientists, and practitioners. Students will
learnvariousMCDAmethodsthroughillustrativeexamples,whiletheywillbeable
tosolvetheexamplesusingthePythoncodesgiveninthisbook.Thisbookcovers
thoroughly a course on Multicriteria Decision Analysis whether or not Python is
used.Scientistsandpractitionerswillhaveabookintheirlibrarythatpresentsmany
differentMCDAmethodsandtheirvariants.
The book is organized in six chapters. A simple, yet important method in the
MCDA pantheon is presented in the first chapter, namely TOPSIS. The following
chapter is about a very similar method called VIKOR, a method that recently
has gained much popularity. The algorithms in both cases have a number of
distinctstepsandvariousothertechniquesthatcanbeusedtomodifytheclassical
versions of the methods, which are also naturally presented. The fuzzy variations
areincludedtoo,aswellasthegroupdecision-makingextensions.Thethirdchapter
focuses on PROMETHEE, a well-known member of the outranking family of
methods.PROMETHEEIandIIarefullyanalyzedandthenthetextproceedswith
PROMETHEE V as well as short references to the GAIA method and the group
decision-making PROMETHEE procedure. SIR is next, a method that integrates
both TOPSIS and PROMETHEE ideas and could be actually considered as a
generalization of PROMETHEE. This is a relative new member of the MCDA
portfolio, but the authors feel that SIR could in time make a difference. A book
about MCDA cannot be complete if it does not include the AHP method. Despite
itssomewhatcontroversialstatuswithinthebroaderMCDAcommunity,AHPhas
proven to be a very popular method, perhaps the most popular. This is a reason
enough for the authors to include a chapter on AHP in this volume including a
discussiononthevariationsandsomeshortcomingsofthemethod.Thefinalchapter
is about goal programming, which is based actually on the linear programming
principles; the classical approach is complemented in this book by the weighted,
lexicographic, and Chebyshev versions. Appendix includes the revised Simos
methodforthecalculationoftheweightsassociatedwiththecriteriainanMCDA
model, a sometimes rather difficult step during the model formulation. Important
methodologies(likeELECTRE,MAUT,ANP,andmultipleobjectiveprogramming)
are not included and the authors fully acknowledge this shortcoming; if all goes
well,thisisexpectedtoberectifiedinafuture,secondeditionofthepresentvolume.
VarioussoftwarepackagesexistformanyMCDAmethods,bothproprietaryand
free.Inallthechaptersofthisbook,codeisprovidedforeachandeverystepofthe
procedures,andthisisalsoreadilyavailableonthebookwebsite.Theprogramming
languageusedisPython,amodernlanguagewithanumerousandrobustuserbase.
The authors assume a basic knowledge of programming principles and especially
of the Python language by the reader; however, they have tried to keep the code
as simple as possible. Early on, they decided that important capabilities of the
languageliketheobject-orientedprogrammingfeatures,decoratorsoriterators,are
not actually important for the purpose of this book. This is a book intended not
only for computer science specialists but also for researchers from every field of
science that need tools in order to model first and then solve a particular problem
thattheyneedtotackleusinganMCDAmethod.Pythoniswellsuitedforthistask;
the authors pondered at the beginning of this endeavor using another fully object-
orientedprogramminglanguagelikeJava,butthiswouldneedlesslycomplicatethe
code. The code is kept as simple as possible but is not simplistic; it is divided
in modules and the reader is indeed required to have some knowledge of Python
anditsparticularanduniquefeatures.Commentsareincludedinthecode,aswell
as commentary on the main text and this combined with the ability of Python
to produce very clearly defined and readable code should result in easily read,
understood,andappliedcodelistings.Moreover,thereadersareverywelcomeand
actually are encouraged to use, modify, and extend the code to better suit their
particulardemands.
All of the Python packages included in the book are free and open source as is
thelanguageitself;thiswasanotherprerequisitefirmlysetbytheauthorsearlyon.
Thereisnoproprietarycodeanywhere;theideaisthateveryoneinterestedshouldbe
abletousethecodeinthisbookwithoutrestrictions.ThePythonpackagesincluded
andnecessaryforsomeofthecodelistingsinthisbookare:
• graphviz:alibraryforcreatingandrenderinggraphdrawings
• Matplotlib:alibraryusedforplottingtheresultsoftheMCDAmethods
• PuLP: an open-source Python-based linear programming solver that allows the
usertooptimizemathematicalprogrammingmodels
• Pyomo:anopen-sourcecollectionofPythonsoftwarepackagesforformulating
optimizationmodels
Theauthorswouldliketothankthepublisher’steamfortheirhelpandpatience
during this endeavor and Professor Bertrand Mareschal for agreeing to write the
foreword to this book. They also hope that the international research community
will find the book interesting and of high standards. For any mistakes in the text
the authors, of course, acknowledge exclusive and full responsibility. All codes
andaccompaniedmaterialofthisbookcanbedownloadedfromhttps://github.com/
springer-math/Multiple-Criteria-Decision-Aid/.
Thessaloniki,Greece JasonPapathanasiou
Pittsburgh,PA,USA NikolaosPloskas
April2017
Contents
1 TOPSIS........................................................................ 1
1.1 Introduction............................................................. 1
1.2 Methodology............................................................ 2
1.2.1 NumericalExample ........................................... 4
1.2.2 PythonImplementation ....................................... 6
1.2.3 RankReversal ................................................. 12
1.3 FuzzyTOPSISforGroupDecisionMaking .......................... 14
1.3.1 Preliminaries:ElementsofFuzzyNumbersTheory......... 14
1.3.2 FuzzyTOPSISMethodology ................................. 16
1.3.3 NumericalExample ........................................... 20
1.3.4 PythonImplementation ....................................... 22
References..................................................................... 28
2 VIKOR........................................................................ 31
2.1 Introduction............................................................. 31
2.2 Methodology............................................................ 32
2.2.1 NumericalExample ........................................... 35
2.2.2 PythonImplementation ....................................... 38
2.3 FuzzyVIKORforGroupDecisionMaking........................... 40
2.3.1 Preliminaries:TrapezoidalFuzzyNumbers.................. 40
2.3.2 FuzzyVIKORMethodology.................................. 42
2.3.3 NumericalExample ........................................... 45
2.3.4 PythonImplementation ....................................... 46
References..................................................................... 54
3 PROMETHEE ............................................................... 57
3.1 Introduction............................................................. 57
3.2 Methodology............................................................ 58
3.2.1 NumericalExample ........................................... 65
3.2.2 PythonImplementation ....................................... 71
3.2.3 VisualDecisionAid:TheGAIAMethod .................... 77
3.3 PROMETHEEGroupDecisionSupportSystem ..................... 78
3.4 PROMETHEEV ....................................................... 80
3.4.1 NumericalExample ........................................... 85
3.4.2 PythonImplementation ....................................... 86
References..................................................................... 87
4 SIR............................................................................. 91
4.1 Introduction............................................................. 91
4.2 Methodology............................................................ 91
4.2.1 NumericalExample ........................................... 98
4.2.2 PythonImplementation ....................................... 101
References..................................................................... 107
5 AHP............................................................................ 109
5.1 Introduction............................................................. 109
5.2 Methodology............................................................ 110
5.2.1 NumericalExample ........................................... 114
5.2.2 PythonImplementation ....................................... 117
5.2.3 RankReversal ................................................. 124
References..................................................................... 127
6 GoalProgramming .......................................................... 131
6.1 Introduction............................................................. 131
6.2 ClassicalGoalProgramming........................................... 132
6.2.1 NumericalExample ........................................... 138
6.2.2 PythonImplementation ....................................... 143
6.3 WeightedGoalProgramming.......................................... 146
6.3.1 NumericalExample ........................................... 148
6.3.2 PythonImplementation ....................................... 149
6.4 LexicographicGoalProgramming..................................... 151
6.4.1 NumericalExample ........................................... 152
6.4.2 PythonImplementation ....................................... 155
6.5 ChebyshevGoalProgramming ........................................ 158
6.5.1 NumericalExample ........................................... 159
6.5.2 PythonImplementation ....................................... 161
References..................................................................... 163
Appendix:RevisedSimos ....................................................... 165
A.1 Introduction............................................................. 165
A.2 Methodology............................................................ 165
A.2.1 NumericalExample ........................................... 167
A.2.2 PythonImplementation ....................................... 168
References..................................................................... 169
Index............................................................................... 171