Table Of ContentStudies in Fuzziness and Soft Computing
Cengiz Kahraman
İrem Otay Editors
Fuzzy Multi-criteria
Decision-Making
Using Neutrosophic
Sets
Studies in Fuzziness and Soft Computing
Volume 369
Series editor
Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences,
Warsaw, Poland
e-mail: [email protected]
The series “Studies in Fuzziness and Soft Computing” contains publications on
various topics in the area of soft computing, which include fuzzy sets, rough sets,
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More information about this series at http://www.springer.com/series/2941
İ
Cengiz Kahraman rem Otay
(cid:129)
Editors
Fuzzy Multi-criteria
Decision-Making Using
Neutrosophic Sets
123
Editors
CengizKahraman İrem Otay
Department ofIndustrial Engineering Department ofIndustrial Engineering
Istanbul TechnicalUniversity Istanbul OkanUniversity
Macka, Istanbul,Turkey Akfirat-Tuzla, Istanbul,Turkey
ISSN 1434-9922 ISSN 1860-0808 (electronic)
Studies in FuzzinessandSoft Computing
ISBN978-3-030-00044-8 ISBN978-3-030-00045-5 (eBook)
https://doi.org/10.1007/978-3-030-00045-5
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I dedicate this book to my children;
my daughter Nazlı Ece Kahraman;
my son Yunus Emre Kahraman;
and
my son Onur Kahraman.
Prof. Cengiz Kahraman
I dedicate this book to my family;
my dear grandparents Nazmiye Fero and
Bayram Fero and my aunt Melek Fero;
my lovely parents Cavit, Güzide and
my brother Mustafa Yiğit.
Asst. Prof. İrem Otay
Preface
Multiple-criteria decision-making(MCDM)isamodelingandmethodologicaltool
fordealingwithcomplexengineeringproblems.Therearetwomainapproachesto
MCDM problems: Multiple-attribute decision-making (MADM) and multiple-
objective decision-making (MODM). MADM refers to making selections among
some courses of action in the presence of multiple, usually conflicting, attributes.
MADM problems are assumed to have a predetermined, finite number of decision
alternatives. InMODM problems, thenumber of alternatives is effectively infinite,
and the tradeoffs among the considered criteria are typically described by contin-
uous functions.
Very often in MCDM problems, data are imprecise and fuzzy. Fuzzy set
approaches are especially suitable when modeling human knowledge is required.
WhenordinaryfuzzysetswerefirstintroducedbyZadehin1965,thedefinitionsof
membershipandnonmembershipfunctionshadbeendefinedsothattheirsumwas
equal to 1.0. Later, these concepts became an attractive research area that many
researchers focused on. These researchers determined the direction of the progress
ofthefuzzysettheory.Intuitionisticfuzzysetsremovingthenecessitythatthesum
must be equal to 1.0 have become the most attractive extension of ordinary fuzzy
sets.Intuitionisticfuzzysetsletdecisionmakersincorporatetheirhesitancytotheir
decisions so that the sum might be at most 1.0. Later, intuitionistic fuzzy sets of
second type (IFS2) were introduced (Atanassov, 1999), providing a wider area for
the assignment of membership and nonmembership degrees. In IFS2, the sum of
squared membership and nonmembership degrees is at most equal to 1.0.
Smarandache (1998) introduced neutrosophic sets as a generalization of intuition-
istic fuzzy sets, incorporating a new parameter to the definition of a membership
andnonmembershipconcepts.Thesumoftheindependentparameterscomposedof
truth, indeterminacy, and falsity degrees can be between 0 and 3 in these sets.
Especially after 2015, neutrosophic sets have attracted the interest of many
researchers. Within the past 3 years, neutrosophic sets have been significantly
improved by both theoretical works and practical works in the literature.
vii
viii Preface
Thisbookinvolvestotally27chaptersunder4mainparts.Thefirstpartpresents
anintroductiontoneutrosophicsets,whilethesecondpartisonthefundamentalsof
neutrosophic set theory. Third part gives the preliminaries for neutrosophic
multi-criteria decision-making and the last section includes representative theoret-
ical and practical studies on neutrosophic multi-criteria decision-making.
Chapter “A State-of-the-Art Review of Neutrosophic Sets and Theory” aims at
classifying all these publications and to exhibiting the place of neutrosophic sets
andlogicintheliterature.Tabularandgraphicalillustrationsareusedtosummarize
the review results. Chapter “Arithmetic Operations of Neutrosophic Sets, Interval
Neutrosophic Sets and Rough Neutrosophic Sets” presents the basic concepts of
neutrosophic sets as well as some of their hybrid structures. Interval-valued neu-
trosophic sets and rough neutrosophic set are also studied with some of their
properties.
Chapter “Power Harmonic Weighted Aggregation Operator on Single-Valued
Trapezoidal Neutrosophic Numbers and Interval-Valued Neutrosophic Sets”
introducesnewaggregationoperatorscalledpowerharmonicweightedaggregation
operatorwithsingle-valuedtrapezoidalneutrosophicnumberandpowerharmonic
weighted aggregation operator with interval-valued neutrosophic set. They are
tested in MCDM and the results are compared. Chapter “Linear and Non-linear
Neutrosophic Numbers” introduces neutrosophic numbers from different points of
view. It defines different types of linear and nonlinear generalized neutrosophic
numbers, which are very important for uncertainty theory. The different properties
of that type of numbers are also discussed. Chapter “Rough Neutrosophic
AggregationOperatorsforMulti-criteria Decision-Making”presentsabriefreview
of decision-making in rough neutrosophic environment. It proposes two aggrega-
tion operators namely, a rough neutrosophic arithmetic mean operator and a rough
neutrosophic geometric mean operator and establishes the basic properties of the
proposed operators. It develops four new neutrosophic multi-criteria decision-
making methods by defining a cosine function to obtain the unknown criteria
weights. Chapter “On Single Valued Neutrosophic Refined Rough Set Model and
Its Application” introduce single-valued neutrosophic refined rough sets by com-
bining single-valued neutrosophic refined sets with rough sets and further studies
the hybrid model from two perspectives—constructive viewpoint and axiomatic
viewpoint.Chapter “BipolarNeutrosophic Graphs”presentstheconcept ofbipolar
neutrosophic graphs and discusses operations on bipolar neutrosophic graphs. It
presentsthecertaincharacterizationofbipolarneutrosophicgraphsbylevelgraphs
and their application to decision-making. Chapter “Properties of Interval-Valued
Neutrosophic Graphs” introduces the properties of neutrosophic graphs for han-
dling uncertainty and vagueness in attributes. It introduces the notion of
interval-valued neutrosophic sets as a generalization of intuitionistic fuzzy sets,
interval-valued fuzzy sets, interval-valued intuitionistic fuzzy sets, and single-
valued neutrosophic sets. Chapter “Laplacian Energy of a Complex Neutrosophic
Graph” extends the concept of energy of fuzzy graph, intuitionistic fuzzy graph,
single-valued neutrosophic graph to the energy of complex neutrospohic graph. It
defines the adjacency matrix of complex neutrosophic graph. The lower and upper
Preface ix
boundsfortheenergyofcomplexneutrosophicgrapharederived.Chapter“Matrix
GameswithSimplifiedNeutrosophicPayoffs”aimsatdevelopingsomemodelsfor
games,wherethepayoffsarerepresentedwithsimplifiedneutrosophicsets.Itgives
anapplicationofsimplifiedneutrosophicsetstotwo-personzero-summatrixgames
and introduces three solutions which are called neutrosophic saddle point method,
neutrosophic upper and neutrosophic lower value method, and neutrosophic elim-
ination method.
Chapter “Similarity Measures in Neutrosophic Sets-I” aims to consider the
various similarity measures such as distance-based similarity measure, tangent
similarity measure, and vector similarity measure for single-valued neutrosophic
sets (SVNS). In Chapter “Similarity Measures in Neutrosophic Sets-II”, SVNS are
shifted to the interval-valued neutrosophic sets and the quadripartitioned single-
valued neutrosophic sets. The chapter provides a complemental overview of the
similaritymeasuresexistentintheoverallgeneralizedneutrosophictheory.Chapter
“CorrelationCoefficientofNeutrosophicSetsandItsApplicationsinDecision-Making”
presents the methods of correlation coefficient measures between two neutro-
sophic sets, two interval-neutrosophic sets, and two neutrosophic refined sets.
Furthermore, it presents some applications of these methods in multi-criteria
decision-making problems. Chapter “A New Approach in Content-Based Image
Retrieval Neutrosophic Domain” presents texture features for images embedded
in the neutrosophic domain with hesitancy degrees. A hesitancy degree is the
fourthcomponentofneutrosophicsets.Thegoalofthechapteristoextractasetof
features to represent the content of each image in the training database to be used
for the purpose of retrieving images from the database similar to the image under
consideration. Chapter “Pareto Solution in Neutrosophic Set Setting for Multiple
Criteria Decision Making Problems” investigates the multiple-criteria group
decision-making problem with neutrosophic linguistic preference relations.
A generalization of an aggregation operator in the decision-making problem is
defined. Then, a neutrosophic Pareto solution is presented for the problems with
linguisticpreferencerelations.Chapter“Three–WayBipolarNeutrosophicConcept
Lattice” proposes a method for precise representation of bipolar information using
the properties of bipolar neutrosophic sets. The hierarchical-order visualization of
generated bipolar neutrosophic concepts and its interpretation are also discussed
with an illustrative example.
Chapter “Interval-Valued Neutrosophic Numbers with WASPAS” introduces
interval-valued trapezoidal neutrosophic numbers for weighted aggregated sum
product assessment (WASPAS). A simple case of classification of athletes in
Paralympicsisrepresentedusinginterval-valuedtrapezoidalneutrosophicnumbers.
Chapter “Interval-Valued Neutrosophic EDAS Method: An Application to
Prioritization of Social Responsibility Projects” extends ordinary fuzzy evaluation
based on distance from average solution (EDAS) method to interval-valued neu-
trosophic EDAS for reflecting decision makers’ views to the truthiness (T),
falsity (F), and indeterminacy (I). The proposed method is applied to the
prioritization of multi-criteria and multi-expert social responsibility projects and a
sensitivity analysis is conducted to check the robustness of the given decisions.
x Preface
Chapter“COPRASMethodwithNeutrosophicSets”developsanextendedformof
complex proportional assessment (COPRAS) method used for solving the
decision-makingproblemsinwhichallthedatapresentedbydecisionmakersarein
theformofinterval neutrosophicnumbers.Inorder toaccomplishthisgoal,anew
score function and an accuracy function that consider the decision maker’s risk
attitudearedefined.Then,basedontheideaofMaclaurinsymmetricmeanoperator
that can capture the interrelationships among multi-input arguments, some aggre-
gation operators are defined, such as interval neutrosophic Maclaurin symmetric
mean operator and interval neutrosophic-weighted Maclaurin symmetric mean
operator. Chapter “Analytic Network Process with Neutrosophic Sets” develops
analytic network process with neutrosophic sets. Based on the inner and outer
dependencies, the weights of criteria and alternatives are calculated from neutro-
sophic pairwise comparison matrices. Super matrix and limit matrix are also
obtained.Chapter“NeutrosophicTOPSISwithGroupDecisionMaking”presentsa
general overview about the development of technique for order preference by
similarity to ideal solution (TOPSIS) under neutrosophic environment. It extends
TOPSISmethodtosolvemulti-attributegroupdecision-makingproblemsbasedon
single-valued neutrosophic sets and interval neutrosophic sets. Chapter “VIKOR
Method for Decision Making Problems in Interval Valued Neutrosophic
Environment” discusses the VIKOR method for solving MCDM problem with
interval-valued neutrosophic numbers. It develops INNWAA and INNWGA
operators under interval-valued neutrosophic environment. Chapter “Multiple
Attribute Projection Methods with Neutrosophic Sets” presents the general pro-
jection measure (PM) and bidirectional PM between two simplified neutrosophic
sets(SNS),andthenaharmonicaveragingPMofSNSsisfurtherintroducedbased
on two (bidirectional) projections. Chapter “An Integrated AHP & DEA
Methodology with Neutrosophic Sets” proposes a new neutrosophic analytic
hierarchy process (NAHP). Then, neutrosophic AHP is integrated with neutro-
sophic DEA for bringing solutions to performance measurement problems. The
inputs and outputs of DEA method are weighted by neutrosophic AHP. Chapter
“Simple Additive Weighting and Weighted Product Methods Using Neutrosophic
Sets” extends simple additive weighting (SAW) and weighted product methods
(WPM) to their fuzzy versions by using neutrosophic sets. These sets not only
handle the vagueness but also clarify indeterminacy of decision makers’ opinions.
Chapter “Bipolar Complex Neutrosophic Sets and Its Application in Decision
Making Problem” focuses on measuring uncertainty and its fluctuation using the
propertiesofcomplexneutrosphicsets.Italsointroducesconnectionamongbipolar
and complex neutrosophic sets to compute the similarity among bipolar complex
neutrosophic sets. Chapter “Development of Fuzzy-Single Valued Neutrosophic
MADM Technique to Improve Performance in Manufacturing and Supply Chain
Functions”developsanewapproachformulti-attributedecision-making(MADM),
whichworkswithconversiononcrisp/fuzzysetintosingle-valuedneutrosophicset.
Theproposedapproachisappliedtoacase studyofselectionofautomatedguided
vehicle (AGV) for flexible manufacturing cell in a given industrial application.
