T-norms Based Fuzzy Ordering and their [616075]
T-norms Based Fuzzy Ordering and their
Applications in Fuzzy Preference Modeling .
By
Atiq-ur-Rehman
CIIT/SP12 -PMATH -001/LHR
PhD Thesis
In
Mathematics
COMSATS Institute of Information Technology
Laho re – Pakistan
Spring , 2017
ii
COMSATS Institute of Information Technology
T-norms Based Fuzzy Ordering and their
Applications in Fuzzy Preference Modeling
A Thesis Presented to
COMSATS Institute of Information Technology, Lahore
In partial fulfillment
of the requirement for the degree of
PhD (Mathematics)
By
Atiq-ur-Rehman
CIIT \SP12 -PMATH -001/LHR
Spring , 2017
iii
T-norms Based Fuzzy Ordering and their
Applications in Fuzzy Preference Modeling
A Post Graduate Thesis submitted to the Department of Mathematics as parti al
fulfillment of the requirement for the award of Degree of PhD in Mathematics.
Name
Registration Number
Atiq-ur-Rehman
CIIT/SP12 -PMATH -001/LHR
Supervisor
Dr. Samina Mazhar
Associate Professor
Department of Mathematics
COMSATS Institute of Information Technology (C IIT)
Lahore
July, 2017
iv
Certificate of Approval
This is to certify that the research work presented in this thesis entitled “T-norm s
Based Fuzzy Ordering and t heir Applications in Fuzzy Preference Modeling ” was
conducted by Mr. Atiq-ur-Rehman , CIIT/SP12 -PMATH -001/LHR , under the
supervision of Dr. Samina Mazhar . No part of this thesis has been submitted
anywhere else for any other degree. This thesis is submitted to the Department of
Mathematics, COMSATS Institute of Information Technology Lahore, in the partial
fulfillment of the requirements for the degree of Doctor of Philosophy in the field of
Mathematics.
Student: [anonimizat]: Atiq-ur-Rehman Signature: __________________
External Committee:
External Examiner 1 Name External Examiner 2 Name
(Designation & Office Address) (Designation & Office Address)
………………………………… ….……………………………..
………………………………… ….………………………… …..
Dr. Samina Mazhar Dr. Sarfraz Ahmad
Supervisor HoD, Department of Mathematics,
Department of Mathematics, CIIT, Lahore
CIIT, Lahore
Prof. Dr. Moiz ud Din Khan Prof. Dr. Arshad Saleem Bhatti
Chairperson, Department of Mathematics , Dean, Faculty of Sciences
CIIT CIIT
v
Author’s Declaration
I Atiq-ur-Rehman , CIIT /SP12 -PMATH -001/LHR , hereby state that my PhD thesis
titled “ T-norm s Based Fuzzy Ordering and t heir Applications in Fuzzy Preference
Modeling ” is my own work and has not been submitted previously by me for taking
any deg ree from this University “ COMSATS Institute of Information Technology ” or
anywhere else in the country/world.
At any time if my statement is found to be incorrect even after my Graduate the
university has the right to withdraw my PhD degree.
Date: ____ __________ __________________________
Atiq -ur-Rehman
CIIT/ SP12 -PMATH -001/LHR
vi
Plagiarism Undertaking
I solemnly declare that research work presented in the thesis titled “ T-norm s Based
Fuzzy Ordering and t heir Applications in Fuzzy Preference Modeling ” is solely my
research work with no significant contribution from any other person small
contribution/help wherever taken has been duly acknowledge and that complete thesis
has been written by me.
I understand the zero toler ance policy of HEC and COMSATS Institute of
Information Tech nology towards plagiarism . Therefore , I as an author of the above
titled thesis declare that no portion of my thesis has been plagiarized and material
used as reference is properly refer red/cited.
I undertake that if I am found guilty of any formal plagi arism in the above titled thesis
even after award of PhD degree, the U niversity reserves the right to withdraw/revoke
my PhD degree and that HEC and the University has the right to publish my name on
the HEC/University website on which names of students are placed who submitted
plagiarized thesis .
Date: ______________ __ Signature: _______________________
Atiq -ur-Rehman
CIIT/SP 12-PMATH -001/LHR
vii
Cert ificate
It is certified that Atiq-ur-Rehman , CIIT /SP12 -PMATH -001/LHR has carried out all
the work related to this thesis under my supervision at the Department of
Mathematics, COMSATS Institute of Information Technology, Lahore campus and
the work fulfil ls the requirement for award of PhD degree.
Date: _________________
Supervisor:
___________________________
Dr. Samina Mazhar
Associate Professor
Head of Department:
____________________________
Dr. Sarfraz Ahmed
Associate Professor
Department of Mathematics .
viii
DEDICATION
This effort is dedicated to my swee t parents,
wife and children.
ix
Acknowledgements
Above all, I thank to Allah Almighty, Who is the only creator of universe, for his
munificent blessings to human beings in understanding the universe by the kn owledge
bequeath by Him, and for inspiring me with confidence, self believe, and in particular,
for maintaining a ray of hope in times of helplessness during the intensive research
journey.
I really believe, without the dedicated, sympathetic, and extended guidance and
supervision of Dr. Samina Mazhar the realization of my PhD dream would not have
been possible. It is especially pleasant for me to express my whole hearted thanks to
Dr. Sarfraz Ahmad , Head Department of Mathematics for his guidance and kindn ess
during the course of study.
In the most challenging and laborious phase of my academic life, I really consider
myself lucky to have professionals’ support from all corners of my surroundings. I
would like to express my sincere thanks to my family. My p arents and wife have
always been extremely supportive during the whole course of my studies, with their
sincere prayers, deep love, immeasurable care and financial help.
Finally, I would like to thank all of my friends and colleagues especially Dr. Kashif
Ali, Dr. Muhammad Hussain, Dr. Muhammad Younas, Dr. Tahir Raza Rizvi, Dr.
Tajammal Hussain, Mr. Imran Zulfiqar Cheema and Mr. Hamid Mahmood ,
COMSATS Institute of Information Technology, Lahore, who always motivated and
supported me to complete this projec t.
May Allah bless us with bright success in our life and after this life. ( Amin )
Atiq -ur-Rehman
CIIT/ SP12 -PMATH -001/LHR
x
ABSTRACT
T-norms Based Fuzzy Ordering and their
Applications in Fuzzy Preference Modeling
This research work focuses on T-transitive fuzzy orderings, their mathematical
structures and representation results on these orderings . We have shown that the
consistent nature of a fuzzy preference relation has direct relationship with its
being more or less T-transitive. After handling this task, some work is done on
application of T-transitive fuzzy orderings in the area of preference modeling.
Here we have presented some new algorithms to complete an incomplete fuzzy
preference relation. These algorithms are based on T-transitivity of a given fuzzy
relation. This work has its roots in multi criteria and multi agent decision making.
The target is to achieve a ranking among alternatives while incomplete information
is provided by the judges about criteria of section and pair wise preferences
amongst alternatives. We have further extended these results to their interval
valued versions where the consistent and consensus degree s are also
accommodated. The theory and algorithms provided so far are then used to solve real
life problems of industry i.e., the problem of supplier selection in the Supply Chain
Management where the successful implementation of the results is demonstrated.
xi
TABLE OF CONTENT
1 Introduction ………………………….. ………………………….. ………………………….. …. 1
1.1 Fuzzy Sets ………………………….. ………………………….. ………………………….. … 2
1.1.1 Basic History ………………………….. ………………………….. ……………….. 2
1.1.2 Fuzzy Relations and Fuzzy Orderings ………………………….. …………. 3
1.2 Decision Making ………………………….. ………………………….. …………………… 6
1.2.1 General Structure of GDM ………………………….. …………………………. 7
1.2.2 Supplier Selection ………………………….. ………………………….. ……….. 10
1.2.3 Preference Relations ………………………….. ………………………….. ……. 11
1.2.4 Interval -Valued Fuzzy Preference Relations ………………………….. .. 13
1.2.5 Incomplete Fuzzy Preference Relation s ………………………….. ……… 14
1.3 Goals and Format of Thesis ………………………….. ………………………….. …… 14
2 Preliminaries ………………………….. ………………………….. ………………………….. . 17
3 T-transitive Fuzzy Orderings ………………………….. ………………………….. …… 31
3.1 Fuzzy Partial Order Relations ………………………….. ………………………….. . 32
3.2 Fuzzy Lattices ………………………….. ………………………….. …………………….. 37
4 New Results on t he Measures of Transitivity ………………………….. ………… 42
4.1 Measures of T-transitivity ………………………….. ………………………….. ……. 44
4.1.1 The Inequalities Related to R-implicators ………………………….. ….. 44
4.1.2 The Inequalit ies for S-implicators ………………………….. …………….. 48
4.2 Measures of S -transitivity ………………………….. ………………………….. ……. 50
4.2.1
-Transitivity Related to R-implicators ………………………….. ……. 51
4.2.2
-Transitivity Related to S-implicators ………………………….. …….. 53
5 Group Decision Making by us ing Incomplete Fuzzy Preference R elations
………………………….. ………………………….. ………………………….. ……………….. 55
5.1 Estimation of Missing Preference Value ………………………….. …………….. 57
5.2 Iterative Procedure for GDM Based on Consistency Relations ………….. 63
5.2.1 Estimating Missing Preferences ………………………….. ……………….. 64
5.2.2 Consistency Measures ………………………….. ………………………….. … 65
5.2.3 Allocating Weights to Experts ………………………….. ………………….. 65
xii
5.2.4 Consensus Measures ………………………….. ………………………….. …… 66
5.2.5 Feedback Mecha nism ………………………….. ………………………….. …. 67
5.2.6 Accumulation Phase ………………………….. ………………………….. …… 68
5.2.7 Selection Phase ………………………….. ………………………….. ………….. 69
5.3 Numerical Example ………………………….. ………………………….. ……………… 70
6 Group Decision Making by using Incomplete Interval -valued Fuzzy
Preference Relations ……………………………………………………… 76
6.1 Method to Repair an IIVFPR ………………………….. ………………………….. … 79
6.2 IIVFPRs Based Group Decision Making ………………………….. …………….. 82
6.2.1 Determine Unknown Preferences ………………………….. ……………… 83
6.2.2 Consistency Measures ………………………….. ………………………….. … 83
6.2.3 Assigning Weights to Experts ………………………….. ………………….. 84
6.2.4 Collective IVFPR ………………………….. ………………………….. ………. 84
6.2.5 Consensus Level ………………………….. ………………………….. ………… 84
6.2.6 Selection Phase ………………………….. ………………………….. ………….. 85
6.3 Numerical Example ………………………….. ………………………….. ……………… 87
7 Supplier Selection Based on T-transitive Fuzzy Preference
Relations ………… …………………………………………………… …94
7.1 Estimation of Missing Values and Ranking of Alternatives ………………. 95
7.1.1 Consistency Measures ………………………….. ………………………….. … 98
7.1.2 Consensus Measures ………………………….. ………………………….. …… 98
7.2 GDM by Using Incomplete AHP ………………………….. ……………………….. 99
7.2.1 Estimate the Criteria’s Priority Weights ………………………….. …….. 99
7.2.2 Priority Ratings of Each Alternative Regarding to Each
Criterion ………………………….. ………………………….. ………………….. 100
7.2.3 Priority Weight of Each Alternative ………………………….. ………… 101
7.3 Numerical Example ………………………….. ………………………….. ……………. 101
8 Conclusion ………………………….. ………………………….. ………………………….. … 108
9 References ………………………….. ………………………….. ………………………….. … 111
xiii
LIST OF FIGURES
Figure 1 .1: Consensus Process Scheme ………………………….. ………………………….. … 8
Figure 1.2: Selection Process Scheme ………………………….. ………………………….. …… 9
Figure 2.1: Minimum t -norm ………………………….. ………………………….. ……………….. 19
Figure 2.2: Maximum t -conorm ………………………….. ………………………….. ……………. 19
Figure 2.3: Lukasiewicz t-norm ………………………….. ………………………….. ……………. 20
Figure 2.4: Bounded Sum t -conorm ………………………….. ………………………….. ………. 20
Figure 2.5: Product t -norm ………………………….. ………………………….. ………………….. 20
Figure 2.6: Probabilistic Sum t -conorm ………………………….. ………………………….. ….. 20
Figure 5.1: Reso lution Process for GDM ………………………….. …………………………. 69
Figure 6.1: Structure of Resolution Process ………………………….. …………………….. 86
Figure 7.1: Hierarchical structure of the decision problem ………………………….. . 102
xiv
LIST OF TABLES
Table 7.1: Collective Consistent FPR for Five Criteria and its Priority Weights 105
Table 7.2: Collective Consistent FPRs of Four Suppliers with Respect to Each
Criterion and the ir Priority Ratings ………………………….. ………………. 106
Table 7.3: Final Priority Weights of Four Suppliers ………………………….. ………… 107
xv
LIST OF ABBREVIATIONS
FBPR Fuzzy binary preference relation
MPR Multiplicative preference relation
FPR Fuzzy preference re lation
GDM Group D ecision Making
MCDM Multi -Criteria Decision Making
IFPR Incomplete Fuzzy P reference Relation
IVFPR Interval -valued Fuzzy P reference Relation
IIVFPR Incomplete Interval -valued Fuzzy P reference Relation
TCI T-consistency Index
CoD Consensu s Degree
AHP Analytical Hierarchy P rocess
SCM Supply Chain Management
TQM Total Quality Management
poset Partial Ordered S et
fpor Fuzzy Partial Order R elation
Epsilon
Beta
Chapter 1
Introduction
1.1 Fuzzy Sets
1.1.1 Basic History
A sharp, crisp, and vague division exists between a member and a non-member
for any well-de
ned set of individuals in "Classical Set Theory". There is a very
strict and clear boundary to indicate if an object belongs (or does not belong) to
a set. Consequently, numerous real-world problems cannot be handled by classical
set theory. On contrary, "Fuzzy Logic" deals with situations arising from com-
putational contexts as well as cognition, and hence, uncertain, imprecise, vague,
partially true, or without sharp boundaries sets are brought into existence. In fact,
fuzzy logic presents a very powerful method for con ict resolution of multiple cri-
teria by providing exible and hence better assessment of options. Furthermore,
new computing methods based on fuzzy logic can be used in the development of in-
telligent systems for decision making, identi
cation, optimization, and automated
control [4, 61].
Historically, the concept of "Fuzzy Set" was introduced by Prof. Lot
A.
Zadeh [126] in 1965 by his seminal paper, where in
nite degrees of membership are
accepted, and are designated with a number between 0and 1. The generalization
of the evaluation set from the pair of numbers f0;1gto all the numbers in [0;1]
opens the way for fuzzy sets. In fact a fuzzy set is [0,1]-valued mapping commonly
known as membership function and is symbolized as A(x), for the fuzzy set A
in the universe X. Membership functions are mathematical tools for indicating
exible membership to a set, modeling and quantifying the meaning of symbols.
They can represent a subjective notion of a vague class, such as set of young people
in some population, the set of trustable suppliers in a supply chain, and the set
of brilliant students etc. If Xis the universe of discourse, and xis a particular
element of X, then a fuzzy set Ade
ned on Xmay be written as a collection of
ordered pairs: A=f(x; A(x))g; x2X, where each pair (x; A(x))is a singleton.
In a crisp set singletons are only x, but in fuzzy sets it is two things; xandA(x).
A fuzzy set Ais supported by a set, known as support set of elements, that have
a membership function di¤erent from zero. Alternative notations for the fuzzy sets
are summations or integrals to indicate the union of the fuzzy set, depending if
2
the universe of discourse is discrete or continuous. The notation of a fuzzy set
with a discrete universe of discourse is A=X
xi2XA(xi)
xiwhich is the union of all the
singletons. For a continuous universe of discourse we write the set as A=Z
XA(x)
x,
where the integral sign indicates the union of allA(x)
xsingletons.
The notion of a fuzzy set is completely non-statistical in nature and the concept
of fuzzy set provides a natural way of dealing with problems in which the source
of imprecision is the absence of sharply de
ned criteria of class membership rather
than the presence of random variables. Since the concept of fuzzy set theory is
introduced, several advances have been made in di¤erent parts of the world. Since
the 1970s Japanese researchers have been the primary force in the implementation
of fuzzy theory and now have thousands of patents in the area. The world response
to fuzzy logic has been varied. On the one hand, western cultures are mired with
the yes or no , guilty or not guilty , of the binary Aristotelian logic world and
their interpretation of the fuzziness causes a con ict because they are given a
negative connotation. On the other hand, Eastern cultures easily accommodate
the concept of fuzziness because it does not imply disorganization and imprecision
in their languages as it does in english.
1.1.2 Fuzzy Relations and Fuzzy Orderings
After the emergence of fuzzy set theory in 1965, the simple task of looking at
relations was initiated by Lot
A. Zadeh himself in 1971 [168]. In that seminal
paper, he introduced the concept of a fuzzy relation, de
ned the notion of similarity
as a generalization of the notion of equivalence, and presented the concept of fuzzy
ordering. He de
ned a fuzzy relation Ron a set Xas a fuzzy subset of XX,
characterized by its membership function XX![0;1]and symbolized by
R(x; x).
The use of fuzzy relations to overcome imprecision, ambiguity or intensities
in human preferences o¤ers a much realistic attitude to decision problems than
classical models [44, 84]. Fuzzy relations introduced in [168] capture the idea
that objects may be more or less related and when they model human decision
making, transitivity is the foremost consistency assumption in the fuzzy tactic
3
to decision theory [43]. In the fuzzy contexts, an extensive class of transitivity
conditions generalizes the classical property [13, 16, 115, 140, 141, 168]. Amongst
these, T-transitivity (as we have previously mentioned) is the most famous and
generalized form de
ned with the help of Ta t-norm [89], while a t-norm is a two
place function which is assumed to be the best
t model for fuzzy conjunction.
The concept of fuzzy ordering was introduced by generalizing the notions of
re exivity, anti-symmetry, and transitivity.
Re exivity: A fuzzy relation is re exive if for all x,R(x; x) = 1 .
Antisymmetry: In [168], a relation is called anti-symmetric if R(x; y)>0and
R(y; x)>0together imply x=y. Another de
nition seen in the literature is
to require that for x6=y,R(x; y) =R(y; x)implies that R(x; y) =R(y; x) = 0 .
According to this de
nition an anti-symmetric relation Rcan have R(x; y) = 0 :9,
andR(y; x) = 1 , but cannot have R(x; y) = R(y; x) = 0 :4. For x6=y, we
will require that if R(x; y) = 1 , then R(y; x) = 0 . In fact we call a relation
anti-symmetric if and only if R(x; y) +R(y; x)>1implies x=y. A relation
satisfying the anti-symmetric condition of [168] is called perfectly anti-symmetric.
If a relation is perfectly anti-symmetric, then clearly it is anti- symmetric.
Transitivity: Transitivity is the most subtle of the properties we require for
the fuzzy ordering. Again, Zadeh is the
rst one to generalize the concept of
transitivity to fuzzy relations. Though, he used the so called max-min transitivity
to de
ne the fuzzy order, and suggested several di¤erent possible de
nitions of
transitivity [16]. In 1981, Trillas generalized di¤erent forms of transitivity under
the name of T-transitive relation [143] which is still contains gaps.
The emergence of fuzzy orderings naturally opened ways for new ideas on fuzzy
upper and lower bounds and hence paved the way for fuzzy chains and lattices.
In [166], Yuan and Wu familiarized the ideas of fuzzy sublattices and fuzzy ideals
of a lattice. In [1], Ajmal and Thomas de
ned a fuzzy lattice as a fuzzy algebra
and described fuzzy sub-lattices. In [39], Chon presented excellent results on fuzzy
orderings which were based upon the max-min transitivity.
Albeit only scarcely recognized, fuzzy orderings of crisp or fuzzy alternatives
play central role in areas associated to fuzzy systems and fuzzy control, where the
orderings of a numerical domain is most frequently used. It is easy to observe that
4
there might only be a marginal number of fuzzy systems or controllers in which
expressions like small , medium , or large do not occur. This ordering, is
even more crucial as soon as automatic tuning measures are concerned, which are
supposed to give interpretable, i.e. reasonable, results [9, 21]. Similar queries are
to be faced in linguistic estimation [53] which may be measured as a kind of inverse
process to
nd a linguistic label for a given fuzzy set. A third application state is
rule interpolation [92] which is concerned with having conclusions for observations
which are not covered by any precursor in a fuzzy rule base.
In 1977, Baas and Kwakernaak put forward a canonical approach to extend the
natural ordering of real numbers to fuzzy numbers [8]. Since then several authors
have been studying fuzzy relations and orderings [22], but a general theory of fuzzy
ordered sets still has gapes. On the other hand, the Baas-Kwakernaak index is still
acknowledged as "very natural in the framework of fuzzy set theory" [51]. On the
whole, the Baas-Kwakernaak index de
nes a non-transitive fuzzy binary relation
on a set of fuzzy numbers and is called an induced fuzzy ordering. S. V. Ovchin-
nikov and M. I. Migdal proved in 1987 that the strict part of the induced fuzzy
ordering is a transitive fuzzy binary relation [116], and the result was strengthened
by M. Roubens and P. Vincke in 1988 [117]. Consequently, intransitivity of the
induced ordering is basically due to intransitivity of its symmetric part. Lot
A.
Zadeh himself noticed that there are no goodmax-min transitive symmetric rela-
tions (similarity relations) on the set of real numbers and to resolve this di¢ culty,
he suggested there to substitute max-min transitivity by max-product transitivity.
S. V. Ovchinnikov, generalized the Baas-Kwakernaak index by replacing the
min operation with t-norm and some properties for induced fuzzy ordering are
established. He also introduced the fuzzy binary relations and their properties,
and established T-transitivity properties of fuzzy strict preference relations and
indi¤erence relation associated with a fuzzy T-transitive preference relation [117].
Since the start of fuzzy theory, it has been understood to be bene
cial to
deal with nebulousness or uncertainty, and has made signi
cant development.
Explicitly, it highly contributes to systems science as one of essential mathematical
tools, because individuals are basic elements in systems science. In the
eld of
decision making, fuzzy theory is very helpful in dealing with fuzziness of human
5
judgment quantitatively, and initially, a number of results have been published
(see Dubois and Prade [50], IV-3). Some applications of fuzzy theory to group
decision making have been initiated by Blin [19], Fung and Fu [58], Bezdek et al.
[17, 18], Kuz min and Ovchinnikov [94, 95], Nurmi [111] and others. Presently, a
lot of research regarding decision making has been conductiong.
1.2 Decision Making
Decision Making is the intellectual process which deals with selecting the best
option(s) from among multiple di¤erent options, it begins when we need to do
something but we do not know what. In our framework we say that we have
a predetermined set X=fx1; x2; :::; x ng; n2of possible alternatives for the
problem and selection of the best alternative(s) to solve the problem is the goal.
Decision making situations are very common in every person s daily life: usual
examples include shopping, deciding what to eat, and deciding whom or what to
vote for in an election or referendum, and can be categorized in several di¤erent
groups according to certain characteristics as the source(s) for the information and
the preference representation formats that are used to solve the decision problem.
Single Criteria Decision Making
Single criteria decision making carries a situation where we have only one source
of information (or criteria) to solve a decision problem. Thus, in this kind of
situations, the solution of the problem comes directly and exclusively from the
information provided. Perceptibly, consensus process is not necessary, and even
in the selection process to obtain the
nal solution of the problem we do not need
to aggregate any information. Then, the
nal solution for the problem is found
by applying the exploitation step of the selection process.
Multicriteria Decision Making
Multicriteria decision making include the decision making situations where the
information about the alternatives comes from di¤erent sources (or multiple crite-
ria) [30, 32, 55, 75]. Generally, the consensus process may or may not be required
6
in these situations, since the information obtained by the di¤erent sources may
not be possible to change to produce a more consensued solution (for example,
both the output of the diagnosis machine and the historical data are immutable).
Though, to handle these situations we have to use some aggregation step to com-
bine all the statistics obtained from the di¤erent sources prior to the application
of an exploitation step where we will select the solution of the problem.
Group Decision Making
In a decision making procedure, an expert mostly needs to compare a
nite set
of alternatives xi(i= 1;2; :::; n ) and construct a preference relations. However,
decision making is not only the case for a single expert, whereas some decision
problems have to be explained by a group of experts which work together to
nd
the best alternative(s) from a set of feasible alternatives under a particular sit-
uation. This nature of decision making with multiple experts is called as group
decision making (GDM) [33, 94, 95, 107, 113, 141, 142, 155, 161] which is also
known as multiperson decision making. The presence of multiple experts in a
decision process may produce some additional di¢ culties to select the best alter-
native(s). For example, each expert has exclusive characteristics regarding to the
knowledge, abilities, experience and nature of preferences, which suggests that dif-
ferent experts may provide their evaluations by means of di¤erent representation
format and, hence, opinion about the alternative(s) can be di¤erent. Therefore,
to have some kind of agreement among experts is compulsory preceding to the
actual selection of the best alternative(s).
1.2.1 General Structure of GDM
To solve a GDM problem appropriately, two main processes have to be carried out;
the consensus process and the selection process. Both have been widely studied
by di¤erent authors and in di¤erent group decision making contexts [28, 55].
Consensus Process
A consensus process is an iterate process which is composed by several consensus
rounds, where the experts accept to change their preferences following the advice
7
given by a moderator. The moderator knows the agreement in each moment of
the consensus process by means of the computation of some consensus measures.
As aforementioned, most of the consensus models are guided and controlled by
means of consensus measures [4, 28, 72, 121].
The consensus process can be divided in several steps which are graphically
depicted in the following
gure:
Fig 1.1. Consensus Process Scheme
1. First of all, the problem to be solved is presented to the experts, along with
the di¤erent alternatives among they have to choose the best one(s).
2. Then, experts can discuss and share their knowledge about the problem
and alternatives in order to facilitate the process of latterly express their
opinions.
3. Experts provide their preferences about the alternatives in a particular pref-
erence representation format.
4. The moderator receives all the expertspreferences and computes some con-
sensus measures that will allow to identify if a consensus enough state has
been reached or not.
5. If a consensual enough state has been reached the consensus process stops
and the selection process begins. Otherwise, we can apply an advice genera-
tion step where the moderator, with all the information that he/she has (all
preferences expressed by experts, consensus measures and so on) can prepare
some guidance and advice for experts to more easily reach consensus. Note
that this step is optional and is not present in every consensus model.
8
6. Finally, the advice is given to the experts and the
rst round of consensus is
nished. Again, experts must discuss their opinions and preferences in order
to approach their points of view (step 2).
Selection Process
Once the consensus process has been carried out, (that is, expertsopinions are
close enough) the selection process takes place. This process main aim is to select
the
nal solution set of alternatives for the problem from the preferences given by
the experts. This process is shown in the following
gure:
Fig 1.2. Selection Process Scheme
The selection process can be splitted in two di¤erent phases:
Accumulation Phase
In this phase all preferences given by the experts must be accumulated into only
one preference structure, it is usually carried out by means of particular accumu-
lation operators that are usually de
ned for this purpose. This step can be more
complicated if we have an heterogeneous decision making situation (not equally
important experts or di¤erent preference representation formats), as some kind of
homogenization must be carried out to transform all di¤erent preference represen-
tation for mats into a particular one which acts as a base for the accumulation,
and the accumulation operator must be able to handle the weights assigned to
the experts (that is, giving more importance to some expertspreferences than
others).
Exploitation Phase
This
nal step uses the information produced in the aggregation phase to identify
the solution set of alter- natives for the problem. To do so we must apply some
9
mechanism to obtain a partial order of the alternatives and thus select the best
alternative(s). There are several di¤erent ways to do this, but a usual one is
to associate a certain utility value to each alternative (based on the aggregated
information), thus producing a natural order of the alternatives.
1.2.2 Supplier Selection
A supply chain consists of units involved in designing new products and services,
purchasing raw materials, giving them semi-
nished and
nished shapes and sup-
plying them to end customers. The process of evaluation and selection of suitable
suppliers plays a signi
cant role in the long term commitments and performance
of supply chains in manufacturing organizations. Therefore, a systematical and
e¤ective technique or process to select the most suitable supplier e¤ectively leads
to the reduction in purchase risk and increases the number of available just-in-
time suppliers and improves the total quality management (TQM) of production.
Swaminathan and Tayur [138] describe major issues in traditional supply chain
management (SCM) and provide an overview of relevant analytical models to be
used in the area of e-business and supply chain. Later literature may be found fo-
cussing on how to choose a proper and e¢ cient method to evaluate the alternative
suppliers [23, 32, 46, 75, 93, 109]
A well-established industrial organization carries a team of experts to select
the appropriate suppliers for purchasing the raw material and essential parts for
new products. The experts consider a number of criteria to evaluate the best
supplier, the philosophy of the criteria may change for di¤erent types of business.
Which criteria are appropriate and should be used for the assessment of suppliers
in GDM problems for an organization is critical. The history of literature on
criteria to select suppliers goes back to Dickson [47] who in 1966 investigated and
presented twenty-three criteria to select the vendor including, the quality, delivery
and performance history as the most signi
cant factors than others. Weber et
al [152], Swift [139], Choi & Hartley [38], Barbarosoglu & Yazgac [11], Go¢ n,
Szwejczewski & New [62], Muralidharan, Anantharaman & Deshmukh [107] are
the major contributors in this area. The preceding literatures, more or less, cover
all the common criteria to select the suitable supplier in SCM for several types
10
of business. Furthermore, di¤erent corporations may have di¤erent organizational
and social backgrounds which may also a¤ect the process of supplier selection.
Intrinsically, the supplier selection problem is considered as a multi-criteria
decision making (MCDM) problem in a group decision making (GDM) environ-
ment [29, 109]. The quality of the solution of MCDM problems is in uenced by
the preferences given by the experts. However, the human brain interprets vague
and partial sensual information delivered by sensitive organs, the preferences are
habitually based on insu¢ cient information or personal judgments which result in
unsuitable and biased decisions. The fuzzy MCDM approach can explicate more
suitably expert s evaluation of available alternatives to select the best solution
when critical criteria have subjective perceptions. Several approaches have been
developed to solve fuzzy MCDM problems [31, 49, 69, 113, 137, 155].
1.2.3 Preference Relations
The information of most of the decision making processes is based on the pref-
erence relations because it is a useful source to model decision processes. In
preference modelling, an agent, expert or system assigns a preference to any pair
of alternatives in a certain universe. The set or system consisting of all these
assignations is then called a preference relation. Di¤erent types of preference re-
lations are being used according to the domain used to evaluate the intensity of
preference. A preference relation Aon a universe Xis characterized by its mem-
bership function A:XX!D;where Dis the domain of representation of
preference degree. When cardinality of Xis small, the preference relation may be
conveniently represented by an nnmatrix A= (aij)nnbeing aij=A(xi; xj)
for all i; j2 f1;2;3; :::; ng:
If the preferences are expressed as yes or no , then for any two elements
in the universe, four assignations are possible: an expressed preference for any of
the two alternatives, an indi¤erence between the two alternatives or an incompa-
rability of the two alternatives. There are several ways to model such preference
relations [120]. Classically, such a preference relation can be represented by a crisp
relation that, for two alternatives xandy, contains either the couple (x; y)or the
couple (y; x)in case of a strict preference, both couples in case of indi¤erence or
11
neither of the couples in case of incomparability. In the absence of incomparable
alternatives, a preference relation can be represented by a complete crisp relation.
A disadvantage of this crisp representation is that the information on both couples,
(x; y)and(y; x), is needed to learn whether there is a preference or an indi¤erence
between the alternatives, which creates the gap for other representation formates
for decision making. In literature, two types of preference relations have been
widely used to develop the decision models; fuzzy preference relations (FPRs) and
multiplicative preference relations (MPRs) [36, 55, 127, 142].
Fuzzy Preference Relations
In FPRs, the intensity of preference is usually measured using a di¤erence scale
[0;1], a preference relation Aon a set Xof alternatives is characterized by a
membership function A:XX![0;1], where every value aijrepresents the
preference degree of alternative xioverxj:aij=1
2indicates indi¤erence between
alternatives xiandxj(xixj);aij= 1speci
es that alternative xiis completely
preferred to xj;aij>1
2indicates that alternative xiis preferred to xj(xi xj).
Multiplicative Preference Relations
While in MPRs, the preference represents the ratio of the preference intensity
between the alternatives, Saaty suggests measuring every value using ratio scale,
precisely the 1 9scale [127, 128, 129]. A multiplicative preference relation Bon
a set Xof alternatives is chracterized by a membership function B:XX!
[1;1=9].
The following meanings are associated to numbers:
1 equally important
3 weakly more important
5 strongly more important
7 demonstrably or very strongly more important
9 absolutely more important
2,4,6,8 compromise between slightly di¤ering judgments.
GDM is occasionally studied discretely as process and outcome. Process raises
the connections among the experts that lead to the best choice of a speci
c course
12
of action while an outcome is the signi
cance of the choice. It is convenient to
separate process and outcome because it supports to clarify that a good outcome
is not guaranteed by a good decision process, and that a good process is not
presupposed by a good outcome. Actually, a critical phase for GDM is the ability
to conclude a choice.
To model GDM situations appropriately, some aspects have to be considered
such as:
(i) Representation formats: The representation layouts used by the experts to
express their preferences and estimations can a¤ect the whole decision procedure
because each expert may have exclusive inspirations or objectives and a di¤erent
decision process.
(ii) Lack of knowledge: Even though, it is necessary for the experts to have an
extensive and comprehensive information about the alternatives while facing the
decision problem, this is a prerequisite that is not usually satis
ed. In decision
making, there are several diverse social and individual aspects which lead to the
situations for the experts to have incomplete information. For instance, if the set
of feasible alternatives is large, then experts may not be able to distinguish some
similar alternatives appropriately.
(iii) Absence of consistency: To have some useful discussion and study of
decision problem, contradiction (inconsistency) in the estimations of the di¤erent
experts is often required while inconsistency in the individual ideas of the experts
is not fruitful. Actually, a person inclines to be ignored by the others in every real
life situation when self-contradictory points of view are expressed by him.
1.2.4 Interval-Valued Fuzzy Preference Relations
Interval-valued fuzzy set theory (apparently introduced independently in the mid-
seventies by Grattan-Guinness, Jahn and Zadeh [67, 78, 169]) is an increasingly
popular extension of fuzzy set theory where traditional [0, 1]-valued membership
degrees are replaced by intervals in [0, 1] that approximate the (partially unknown)
exact degrees. Hence, not only vagueness (lack of sharp class boundaries), but also
a feature of uncertainty (lack of information) can be addressed intuitively.
To develop convincing and consistent GDM models and procedures, the study
13
of above said aspects plays a central role. But there arise some situations during
pair-wise comparison in which experts face di¢ culty in expressing the preferences
with exact numerical values while there is enough information as to estimate the
intervals [71, 80, 158, 160], and therefore, they construct Interval-Valued fuzzy
preference relations (IVFPRs) as the alternatives to FPRs.
1.2.5 Incomplete Preference Relation
The above mentioned researches focused on IVFPRs with complete information,
however, in decision making problems such situations are unavoidable in which
an expert does not have comprehensive information of the problem because of
time constraint, lack of knowledge and the expert s limited expertise within the
problem domain [3, 36, 54, 96, 97, 159, 161, 162]. Consequently, the expert may
not be able to give his/her opinion about speci
c traits of the problem, and hence
an incomplete fuzzy preference relation (IFPR) would be constructed, missing
information in GDM environment is a problem that we have to cope with because
usual decision-making measures assume that experts are able to provide preference
degrees after pair-wise comparison of alternatives, which is not always possible.
In literature, researches based on IFPRs have been given, but there are only few
researches in GDM related to incomplete IVFPRs [163].
1.3 Goals and Format of Thesis
The central aim of the work is twofold: Firstly to study the theoretical aspects
ofT-transitivity and then to present the applications of T-transitivity in GDM
situations particularly in the contexts of handling incomplete information in Sup-
ply Chain Management (SCM) in industry. As an extended work one aim is to
develop GDM models with incomplete information (incomplete fuzzy preference
relations (IFPRs) and incomplete interval-valued fuzzy preference relations (IIVF-
PRs)) based on triangular norms (t-norms) which come to deal with evaluation
and selection of alternatives in decision problems. The main goal can be apart
into the following points:
Develop some new results in the form of inequalities connecting transitivity
14
of the given fuzzy relation with its clustering nature. The additive and multi-
plicative generators of t-norms and t-conorms play the key role in developing
these results.
Extend some existing results on max-mintransitive fuzzy orderings to a class
of results on the T-transitive fuzzy orderings.
Develop an iterative scheme to estimate missing information in IFPRs based
on t-norms and transitivity properties of FPRs.
Construct an aggregation and selection procedure for FPRs based on the
consistency measure which allows the resolution of GDM problems with
incomplete data.
Extend and modify the whole procedure for multi-criteria decision making
with IFPRs by using analytical hierarchy process.
This thesis is organized into two parts (Part-I & Part-II) :
In Part-I, some theoratical aspects of T-transitivity based on the additive and mul-
tiplicative generators of t-norms and t-conorms (de
ned preliminaries) are given
while in Part-II, application perspectives in GDM particularly in the contexts of
dealing with incomplete information are presented.
Several chapters regarding the above mentioned contents and the structure are
organized as follows:
Part-I
Chapter 1: In this chapter (present chapter), we have mentioned the main
objectives after approaching the problem.
Chapter 2: Some important preliminaries accompanied by historic survey of
transitivity properties will be discussed in this chapter.
Chapter 3: We extend some existing results on max-mintransitive fuzzy
orderings to a class of results on the T-transitive fuzzy orderings. We de
ne
a fuzzy partial order relation by using T-transitive nature and
nd su¢ cient
conditions for the inverse image of a fuzzy partial order relation to be a fuzzy
15
partial order relation. Develop some basic properties of fuzzy lattices by
using T-transitivity and show that a fuzzy totally ordered set is a distributive
fuzzy latticez.
Chapter 4: It presents some new results in the form of inequalities connecting
transitivity of the given fuzzy relation with its clustering nature. These
results are established by using additive and multiplicative generators of
t-norms and t-conorms. Among several other achievements 2-transitivity
presents a connection between non-transitive and T-transitive relations using
their degrees of transitivity.
Part-II
Chapter 5: This chapter presents a selection procedure to handle decision
making problems with IFPRs. To do so, an iterative process is developed to
estimate the unknown information in IFPRs while T-consistency and order
consistency allow the expertspreferences to be aggregated.
Chapter 6: We present a new procedure to estimate the missing information
in IIVFPRs and construct the consistent matrix based on extended t-norm
(particularly extended Lukasciwicz t-norm). Furthermore, an algorithm for
GDM with IIVFPRs is developed.
Chapter 7: We develop the modi
ed form of the procedure proposed in chap-
ter 5 to estimate the missing information from IFPRs, speci
cally, based on
Lukasciwicz t-norm. Moreover, a method for the selection of best supplier(s)
is proposed in GDM environment with IFPRs. The proposed method em-
ploys the structure of criteria in AHP model and utilizes the L-consistent
FPRs to construct the decision matrices.
Chapter 8: Final remarks are given in this chapter, we will highlight the
outcomes of this thesis along with some conclusions about them and at the
end we present some aspects of future work.
Chapter 9: This chapter contains all the references used to support the work
done.
16
Chapter 2
Preliminaries
The Fuzzy Logic was historically introduced by Lot
Zadeh in 1960s, and is a
mathematical tool for dealing with vagueness. It provides a mechanism for signi-
fying linguistic constructs such as many, low, medium, often, few etc. In general,
the fuzzy logic develops an inference structure that empowers human intellectual
skills in a very appropriate manner. The fuzzy logic theory is based upon the
notion of relative graded membership and so are the functions of intellectual and
perceptive processes. Mathematically, fuzzy logic rest in fuzzy set theory, which
can be seen as a generality of traditional set theory.
In 1965, Zadeh introduced the notion of fuzzy set theory by presenting his
formative paper Fuzzy Sets [167], designated with a number between 0and 1.
This notion demonstrates that an object links more or less to the speci
c group
we want to adjust it to; that was how the idea of de
ning the membership of an
element to a set not on the Aristotelian pair f0;1gany more but on the continuous
interval [0;1]was born. In this chapter, we include some de
nitions and results
which are essential to make this work comprehendable for the reader.
De
nition 2.1. Fuzzy Set : A fuzzy set Aon the universe of discourse Xis
denoted by A=f(x; A(x))g; x2X. Where the value A(x)is called the degree of
membership of xinAi.e.,A(x) =Degree (x2A), and the map A:X![0;1]is
called a set function or membership function. The class of all fuzzy sets in Xis
denoted by F(X) .
The fuzzy sets having their values only on the boundary of [0;1]are called crisp
sets. For example, the universe Xand the empty set ;are fuzzy sets de
ned
as:X(x) = 1 and;(x) = 0 for all x2X:Moreover, the intersection, union and
complement of fuzzy sets were de
ned by Zadeh himself in [167] as, if A; B2F(X)
for all x2X:
A[B(x) = max( A(x); B(x));
A\B(x) = min( A(x); B(x));
Ac(x) = 1 A(x):
These operations follow De Morgan s, idempotency, commutativity, associativity,
absorption and distributivity laws, but the laws of contradiction and excluded
18
middle are violated to more hold. This drawback created space for new models of
fuzzy conjunction and disjunction.
In 1980, Zimmermann and Zysno [171] proved that any t-norm Tand any
t-conorm Scan be used to model fuzzy intersection and union respectively. The
notion of triangular norms was started by Menger [104] in 1942 with the paper
"Statistical metrics". Karl Menger used the idea of t-norms to construct metric
spaces in which the distance between two objects of the space is described by
using probability distributions instead of numbers. Originally, the set of axioms
regarding t-norms was signi
cantly weaker, together with other functions which
are today known as t-conorms. The present form of axioms for t-norms is provided
by Schweizer & Sklar in [131, 132, 133], and after having rede
nition of statistical
metric spaces in 1962 [135], this
eld developed rapidly. Several important results
relating t-norms were carried out during this development, most of which are
summarized in [134].
The de
nitions of t-norm and t-conorm (in Schweizer & Sklar sense) are given as
follows:
De
nition 2.2. Triangular Norm and Triangular Conorm : The triangular norm
(t-norm) Tand triangular conorm (t-conorm) Sare the [0;1]2 ! [0;1], increas-
ing, associative and commutative mappings satisfying: T(1; x) =xandS(x;0) = x
for all x2X .
Typical basic but very important choices of t-norms and their dual t-conorms are
The minimum and maximum operators: T(x; y) = min( x; y)andS(x; y) =
max( x; y);
1.00.0 x1.00.0
0.5
0.0
y0.5z0.51.0
Fig 2.1 Minimum t-norm
1.00.01.0
y0.5
x 0.50.01.0
z0.5
0.0 Fig 2.2 Maximum t-conorm
The Lukasiewicz t-norm and its dual t-conorm: T(x; y) = max( x+y 1;0)
19
andS(x; y) = min( x+y;1);
1.00.0 x1.00.0
0.5
0.0
y0.50.5z1.0
Fig 2.3 Lukasiewicz t-norm
0.01.00.5z1.0
y0.5
x0.0
1.00.50.0 Fig 2.4 Bounded sum t-conorm
The product operator and probabilistic sum: T(x; y) =xyandS(x; y) =
x+y xy:
1.00.0 x1.00.0
0.5
0.0
y0.5z0.51.0
Fig 2.5 Product operator
1.00.01.0
z0.5
1.0
y0.5
x 0.50.0
0.0 Fig 2.6 Probabilistic sum
Remark 2.1.(i) It can be deduced from De
nition 2.2 that each t-norm Tsatis
es
some extra boundary conditions for all x2[0;1]as:
T(0; x) =T(x;0) = 0 andT(1; x) =x: (2.1)
Hence, all t-norms coincideon the boundary of [0;1]2(unit square).
(ii)The increasing nature (monotonicity) of a t-norm Talongwith commutativity
leads the monotonicity in both components, i.e.,
T(x1; y1)T(x2; y2)whenever x1x2andy1y2: (2.2)
Certainly, for x1x2andy1y2;we have:
T(x1; y1)T(x1; y2) =T(y2; x1)T(y2; x2) =T(x2; y2): (2.3)
20
Klement and Mesiar [90] (continuing with t-norms and t-conorms) proposed their
additive and multiplicative generators which were further improved by Nguyen
[110].
De
nition 2.3. Additive Generator of t-norm : An additive generator fof a
t-norm Tis a strictly decreasing [0;1]![0;1]function which is also right con-
tinuous in 0and satis
es f(1) = 0 ;such that:
T(x; y) =f 1((f(x) +f(y))^f(0))for all x; y2X; (2.4)
the t-norm Tis strictly monotone if and only if f(0) = 1and nilpotent if and
only if f(0)<1 .
De
nition 2.4. Multiplicative Generator of t-norm : A multiplicative generator
gof a t-norm Tis a strictly increasing [0;1]![0;1]function which is right
continuous in 0and satis
es g(1) = 1 , such that:
T(x; y) =g 1((g(x)g(y))_g(0))for all x; y2[0;1]; (2.5)
where Tis nilpotent if and only if g(0)>0and is strict if and only if g(0) = 0 .
De
nition 2.5. Additive Generator of t-conorm : An additive generator sof a
t-conorm Sis a strictly increasing [0;1]![0;1]function which is left continuous
in1and satis
es s(0) = 0 , such that:
S(x; y) =s 1[s(1)^(s(x) +s(y))]for any x; y2[0;1]; (2.6)
where Sis nilpotent if s(1)>1and is strict if s(1) = 1:
De
nition 2.6. Multiplicative Generator of t-conorm : A multiplicative genera-
torof a t-conorm Sis a strictly decreasing [0;1]![0;1]function which is left
continuous in 1and satis
es (0) = 1 , such that:
S(x; y) = 1((x)(y)_(1))for all x; y2[0;1]; (2.7)
andSis nilpotent if (1)>0and is strict if (1) = 0 :
De
nition 2.7. Continuous and Archimedean t-norm (resp. t-conorm ): A t-
norm T(resp. a t-conorm S)is said to be:
21
(i)continuous, if T(resp. S) is continuous as a function on the unit interval;
(ii)Archimedean, if lim
n!1x(n)
T= 0(resp. lim
n!1x(n)
S= 1), for all x2]0;1[.
For any x2[0;1]and any associative binary operation Kon[0;1]; x(n)
Kdenotes
the nth power of xde
ned x(n)
K=K(x; :::; x ) .
De
nition 2.8. Negator and Complement : A negator Nis an order-reversing
[0;1]![0;1]mapping such that N(0) = 1 andN(1) = 0 :Negators are used
to model complementation in the calculus of fuzzy set theory i.e., for a given
fuzzy set Ain the universe X;its complement Acis de
ned by negator Nas:
Ac(x) =N(A(x))for all x2X:A negator is strict, if it is strictly decreasing
and continuous and it is involutive if N(N(x)) = xfor all x2[0;1]:A strict
and involutive negator is called a strong negator. A popular strong negator is the
standard negator Ns(Zadeh s complement) de
ned as: Ns(x) = 1 x:
De
nition 2.9. Triplet : A triplet (T; S; N ), where Tis a t-norm, Sis a t-conorm
andNis a strict negation, is called a De Morgan triplet if for all x; y2[0;1]:
T(x; y) =N 1(S(N(x); N(y))); S(x; y) =N 1(T(N(x); N(y))): (2.8)
De
nition 2.10. Fuzzy Implicator : A fuzzy implicator is a binary operation
on[0;1]with order reversing
rst partial mappings and order preserving second
partial mappings such that: I(0;0) = I(0;1) = I(1;1) = 1 ;andI(1;0) = 0 [44]:
Following is the list of some examples of fuzzy implicators:
i. Kleene-Dienes: Ib(x; y) = max(1 x; y);
ii. Reichenbach: Ir(x; y) = 1 x+xy;
iii. × ukasiewicz: IL(x; y) = min(1 x+y;1);
iv. Gödel: Ig(x; y) =8
<
:1;ifxy
y;ifx > y.
Continuing with implicators, Trillas proposed in [143, 144] two types of fuzzy
implicators de
ned with the help of fuzzy negation, conjunction and disjunction.
22
De
nition 2.11. R-implicator : An R-implicator generated from a t-norm Tis
de
ned by:
I(x; y) = sup ft2[0;1]jT(x; t)ygfor all x; y2[0;1]: (2.9)
IfTis a left-continuous, then above implicator can be rewritten as:
I(x; y) =maxft2[0;1]jT(x; t)ygfor all x; y2[0;1]: (2.10)
R-implicators generated from t-norms play an important role in the literature.
Some of the R-implicators based on di¤erent t-norms are Ig(x; y); IL(x; y)de
ned
above.
Remark 2.2. [110] If Tis a continuous Archimedean t-norm and f: [0;1]!
[0;1]is an additive generator of T, then the R-implicator associated with Tcan
be obtained by:
IT(x; y) =f 1(max( f(y) f(x);0))for all x; y2X: (2.11)
Remark 2.3. [110] If Tis continuous Archimedean t-norm and gis a multi-
plicative generator of T, then the R-implicator associated with Tcan be obtained
by:
IT(x; y) =g 1
ming(y)
g(x);1
for all x; y2[0;1]: (2.12)
De
nition 2.12. S-implicator : Let Sbe a t-conorm and Nbe a fuzzy negation.
AnS-implicator (also known as ( S; N )-implicator) is de
ned as:
I(x; y) =S(N(x); y)for all x; y2[0;1] . (2.13)
Some of the S-implicators based on di¤erent t-conorms are Ib(x; y); Ir(x; y)de
ned
above.
Remark 2.4. [110] If sis an additive generator of an Archimedean conorm S,
then its S-implicator can be represented for all x; y2[0;1]by:
IS(x; y) =s 1(min( s(N(x)) +s(y); s(1))): (2.14)
23
Remark 2.5. [110] If gis the multiplicative generator of Archimedean t-conorm
S, then its S-implicator can be represented for all x; y2[0;1]as:
IS(x; y) = 1[maxf(N(x))(y); (1)g]: (2.15)
In 1971, Zadeh [168] himself presented the idea of fuzzy relations along with
equivalences, and gave the notion of fuzzy ordering. To overcome imprecision,
ambiguity or intensities in human preferences, use of fuzzy relations o¤ers a much
realistic attitude to decision problems than classical models based on ordinary
relations. Fuzzy relations introduced in [168] capture the idea that an object
corresponds more or less to the speci
c class we want to adjust it to.
De
nition 2.13. Fuzzy Binary Relation : A fuzzy binary relation (or a fuzzy
relation) Rfrom a universe Xto a universe Yis a fuzzy subset of XY, where
Rxydenotes the degree of relationship between xandy .
Order relations provide mathematical base to several application areas [59, 83,
123, 126].
De
nition 2.14. Preorder Relation : In Mathematics, especially in order theory,
a preorder or a quasiorder is a binary relation that is re exive and transitive .
De
nition 2.15. Partial Order Set : Ifenjoys antisymmetry as well, then it
is called partial order. A set Xalong with an order relation is called a partially
ordered set .
De
nition 2.16. Incomparability : If pqandqpdo not hold for a given
pair of p; qin an ordered set X;then the elements p; qare called incomparable .
De
nition 2.17. Totally Ordered Set : If there are no incomparable elements in
an ordered set S;then Sis called totally ordered set .
De
nition 2.18. Greatest Lower Bound : A subset Qof an ordered set Xis
said to be bounded from below if there exists an element p2Ssuch that pq
for any q2Q. Ifpis the greatest element of the set of lower bounds of Q, then
it is called the greatest lower bound and is denoted by Inf(Q) orglb(Q) (the glb
if it exists, is unique) .
De
nition 2.19. Least Upper Bound : A subset Zof an ordered set Sis said to
be bounded from above if there exists an element p2Ssuch that rpfor any
r2Z. The element pis called an upper bound for Zand if pis in Z;then it is
24
called the greatest element of Z. Ifpis the smallest element of the set of upper
bounds of Z, then it is called the least upper bound and is denoted by Supor
lub(Q) (the lubif it exists, is unique) .
De
nition 2.20. Bounded Set : A subset, which is both bounded from below
and above, of Sis called bounded .
Fuzzy relations when model human decision making, transitivity is the primary
consistency assumption in the fuzzy approach to decision theory. The very
rst
de
nition of fuzzy transitivity is proposed in 1971 by the founder of fuzzy sets
named as max-min transitivity.
De
nition 2.21 .max-mintransitive : A fuzzy relation Ron universe of discourse
Xis called max-min transitive, if it satis
es the following inequality.
Rxzmin(Rxy; Ryz)for all x; y; z 2X . (2.16)
The above said transitivity is criticized in various conducts, laterely, by using
di¤erent t-norms several other formulations of transitivity have been introduced
in literature by di¤erent authors which play a central role in fuzzy equivalence as
well as fuzzy orderings [45, 20]. Amongst these, T-transitivity is the most famous
and generalized form de
ned with the help of Tt-norm
Though various versions of T-transitivity have catered to large number of es-
pecial applications of equivalence and preference, but all these de
nitions have
also been censured and the space for newer models of transitivity is still there.
When we deal with decision making problems especially in GDM environment,
then there arise two main issues; (i) representation formate, and (ii) consistency. In
research, to model GDM problems all the experts, commonly, express their pref-
erences by means of the same preference representation format homogeneously.
However, it is not always possible in actual life because of the exclusive physiog-
nomies regarding to knowledge, expertise, skill and nature of each expert, due to
which di¤erent experts may present their estimations by means of di¤erent pref-
erence representation formats heterogeneously. Actually, this is an issue which
has fascinated many researchers to think for it in the area of GDM, and con-
sequently di¤erent methodologies to integrate di¤erent preference representation
formats have been suggested in literature [35, 71, 73].
25
In those methods, experts commonly select a speci
c representation format
and use it as a base to integrate the di¤erent preference structures in the problem.
Several motives are given for fuzzy preference relations (FPRs) to be selected as
the base to integrate preference structures. Amongst these reasons it is important
to note that FPRs are a valuable tool to aggregate the expertspreferences into
group preferences [55, 74, 141].
De
nition 2.22. Fuzzy Preference Relation : A fuzzy preference relation Rover
a set Xof alternatives is a fuzzy set on the product set XX, i.e., it is char-
acterized by a membership function R:XX![0;1], and on a
nite set
X=fx1; x2; x3; :::; x ngof alternatives it can be conveniently expressed by an
nnmatrix R= (rij)nn, where rijdenotes the degree of preference of alterna-
tivexiover the alternative xjwith rij2[0;1]; rij+rji= 1(additive reciprocity)
for1inand 1jn. Ifrij= 0:5;then there is no di¤erence between
the alternatives xiandxj. Ifrij>0:5;then alternative xiis preferred over the
alternative xj. Ifr= 1;then the alternative xiis de
nitely preferred over the
alternative xj .
In real life, an expert may have imprecise information for the preference degrees
of one alternative over another and it may not always be possible to estimate
his/her judgment by means of an exact numerical value. This situation results in
an interval-valued fuzzy preference relation (IVFPR) [71, 80, 158, 160].
De
nition 2.23. Interval-valued Fuzzy Set [40, 106]: An interval-valued fuzzy
setAon a universe Xis de
ned as: A=f(a;[x ; x+])ja2X;[x ; x+]2
L([0;1])gwhere L([0;1]) =f[x ; x+]j[x ; x+][0;1]with x x+g .
De
nition 2.24. Interval-valued Fuzzy Preference Relation [158]: Let R=
(rij)nnbe a fuzzy preference relation over the set of alternatives X=fx1; x2; x3; :::; x ng
where rij= [r
ij; r+
ij];0r
ijr+
ij1;rij= [1;1] rjiandrii= [0:5;0:5]for all
i; j2N, then Ris called an interval-valued fuzzy preference relation .
In GDM problems such situations are unavoidable in which an expert does not
have comprehensive information of the problem because of time constraint, lack
of knowledge and the expert s limited expertise within the problem domain [3, 36,
54, 96, 97, 158, 161, 162]. Consequently, the expert may not be able to give his/her
opinion about speci
c traits of the problem, and hence an incomplete preference
26
relation would be constructed. In literature, researches based on incomplete FPRs
have been given, but there are only few researches in GDM related to IIVFPRs
[163].
Consistency is an important issue to accept when data is provided by the ex-
perts, because the consistent information is more applicable or important than the
information having ambiguities. A set of consistency properties have been sug-
gested in literature to make a consistent choice when FPRs have been assumed.
Transitivity is one of the most signi
cant property which plays the central role re-
garding to preferences, and it represents the idea that the preference value obtained
by comparing directly two alternatives should be equal to or greater than the pref-
erence value between those two alternatives obtained using an indirect chain of
alternatives [50, 142]. Some of the suggested properties are given below and in
all these cases Ris a fuzzy relation on a universe Xi.e. it is an XX![0;1]
mapping which models the pairwise preference degrees. [74, 141, 142]:
1.Probabilistic-sum transitivity :Ris a probabilistic-sum transitive relation if
8x; y; z 2Xit holds:
R(x; y)>0; R(y; z)>0 =)R(x; z)R(x; y) +R(y; z) R(x; y):R(y; z):
(2.17)
2.Weak transitivity :Ris called weak transitive if for all x; y; z 2Xit holds:
R(x; y)0:5; R(y; z)0:5 =)R(x; z)0:5: (2.18)
3.Max-min transitivity :Ris said to be max-min transitive if for all x; y; z 2X
it holds:
min(R(x; y); R(y; z))R(x; z): (2.19)
Remark 2.6. A max-min transitive fuzzy relation wich is re exive ( R(x; x) = 1
for all x2X) and symmetric ( R(x; y) = R(y; x)for all x; y2X) is called
similarity relation.
4.Weak max-min transitivity :Ris called weak max-min transitive if 8x; y; z 2
27
Xit holds:
R(x; y)R(y; z); R(y; z)R(z; y) =)R(x; z)min(R(x; y); R(y; z)):
(2.20)
5.Restricted max-min transitivity :Ris called restricted max-min transitive if
8x; y; z 2Xit holds:
R(x; y)0:5; R(y; z)0:5 =)R(x; z)min(R(x; y); R(y; z)):(2.21)
6.Max-product transitivity :Ris called max-product transitive if 8x; y; z 2X
it holds:
R(x; z)R(x; y)R(y; z): (2.22)
7.Max-max transitivity :Ris called max-max transitive if 8x; y; z 2Xit
holds:
R(x; y)>0; R(y; z)>0 =)R(x; z)max( R(x; y); R(y; z)):(2.23)
8.Restricted max-max transitivity :Ris called restricted max-max transitive if
8x; y; z 2Xit holds:
R(x; y)0:5; R(y; z)0:5 =)R(x; z)max( R(x; y); R(y; z)):(2.24)
9.Max- transitivity :Ris called max- transitive if 8x; y; z 2Xit holds:
R(x; z)R(x; y) +R(y; z) 1: (2.25)
10.Multiplicative transitivity : A fuzzy relation RonXis called multiplicative
transitive if 8x; y; z 2Xit holds:
R(z; x)
R(x; z)=R(y; x)
R(x; y)R(z; y)
R(y; z): (2.26)
28
11.Additive transitivity :Ris called additive transitive if 8x; y; z 2Xit holds:
R(x; z) =R(x; y) +R(y; z) 0:5: (2.27)
12.Preference-sensitive transitivity :Ris called preference-sensitive transitive if
8x; y; z 2Xit holds:
R(x; y)>0; R(y; z)>0 =)R(x; z)>0: (2.28)
13.Weighted-mean transitivity :Ris called weighted-mean transitive if 8x; y; z 2
X,R(x; y)>0; R(y; z)>0 =) 9 2(0;1)such that:
R(x; z)max( R(x; y)+R(y; z))+(1 )min(R(x; y)+R(y; z)):(2.29)
14.T-transitivity :Ris said to be T-transitive if 8x; y; z 2Xit holds that [145]:
T(R(x; y); R(y; z))R(x; z): (2.30)
Remark 2.7. If t-norm in (2.30) is considered to be the × ukasiewicz t-norm then
the transitive relation is called × ukasiewicz-transitive, and there exists following
relationship between fuzzy equivalence relation and pseudo-metrics on X.
Proposition 2.1. Let t-norm in (2.30) is × ukasiewicz t-norm. Then:
i. IfRis a fuzzy equivalence relation on Xwith respect to × ukasiewicz t-norm
TLthen R= 1 Ris a pseudometric on X, and
ii. If :X!Iis a pseudometric on Xthen R(x; y) = 1 min((x; y);1)is a
fuzzy equivalence relation on Xwith respect to TL.
15.S-transitivity : A mapping R:XX![0;1]is called S-transitive such
that for all x; y; z 2Xit holds [24]:
S(R(x; y); R(y; z))R(x; z): (2.31)
29
16. -transitivity : Let ~Rbe a fuzzy subset of
. The relation ~Ris called
-transitive if for any x; y; z 2
it holds that [105]:
x~Ry ; y~Rz =)x~Rz : (2.32)
If~Ris -equivalence relation, the value of fuzzy subset ~Ron any point can
be denoted by ~R(x; y)or by x~Ry.
From the above mentioned transitivities, the weakest is L-transitivity, i.e. rik
max( rij+rjk 1;0)and is shown that it is the most appropriate notion of transi-
tivity for fuzzy ordering [146]. The weak transitivity is the minimum requirement
condition which should be satis
ed by a consistent FPR.
30
Chapter 3
T-transitive Fuzzy Orderings
In this chapter, we de
ne a fuzzy partial order relation by using T-transitive nature
and
nd su¢ cient conditions for the inverse image of a fuzzy partial order relation
to be a fuzzy partial order relation. We de
ne a fuzzy lattice as a fuzzy relation,
develop some basic properties of fuzzy lattices by using T-transitivity and show
that a fuzzy totally ordered set is a distributive fuzzy lattice.
Throughout, SandQstand for crisp universes of generic elements, a fuzzy
setAin the universe Sis a mapping from Sto[0;1]. For any p2S;the value
A(p)is called the degree of membership of pinA. The class of all fuzzy sets in
Sis denoted by F(S). Let A; B2F(S), then: (i). Ais a subset of B(in Zadeh s
sense) or ABif and only if A(p)B(p)for all p2S; (ii). AandBare equal
fuzzy sets or A=Bif and only if A(p) =B(p)for all p2S[167].
3.1 Fuzzy Partial Order Relations
Now we formalize our de
nitions and results on fuzzy orderings. As mentioned
in the introduction this work is motivated from the results presented in [39].
Chon based his results on min transitivity whereas we are formulating the re-
sults on fuzzy lattices and fuzzy chains while the ordering under consideration is
T-transitive.
De
nition 3.1. A function L:SS![0;1]is called a fuzzy partial order
relation if following axioms are satis
ed:
L(p; p) = 1 for all p2S;
L(p; q)>0andL(q; p)>0implies p=qfor all p; q2S;
L(p; r)T[L(p; q); L(q; r)]for all p; q; r 2S:
Remark 3.1. A fuzzy partial order relation Lis a fuzzy total order relation i¤
L(p; q)>0orL(q; p)>0for all p; q2S. IfLis a fuzzy partial order relation in
a set S, then (S; L)is called a fuzzy partial ordered set or a fuzzy poset. If Lis a
fuzzy total order relation, then (S; L)is called fuzzy total ordered set or a fuzzy
chain.
Proposition 3.1. IfLi
i2Iis a family of fuzzy partial order relations in a set S,
then (S;\
i2ILi)is a fuzzy partial order set (poset).
32
Proof. Since Li
i2Iis a family of partial order relations, hence it is obvious that
\
i2ILiis re exive and antisymmetric relation. Now by logic:
\
i2ILi(p; r) = min
i2ILi(p; r)for all p; r2S:
Using De
nition 3.1 , it implies for all p; q; r 2Sthat
\
i2ILi(p; r)min
i2IT[Li(p; q); Li(q; r)]
T[min
i2IfLi(p; q); Li(q; r)g]
=T[min
i2ILi(p; q);min
i2ILi(q; r)]
=T[(\
i2ILi)(p; q);(\
i2ILi)(q; r)]:
Therefore, (S;\
i2ILi)is a fuzzy partial order set (poset).
De
nition 3.2. LetLSbe a fuzzy relation in a set SandLQbe a fuzzy relation
in a set Q. Let fbe a mapping from SStoQQ, then for all (a; b)2QQ;
(p; q)2SS:
f(LS)(a; b) =8
><
>:Sup
(p;q)2f 1(a;b)LS(p; q);iff 1(a; b)6=
0 otherwise; (3.1)
LS=f 1(LQ)(p; q) =LQ(f(p; q)): (3.2)
Theorem 3.1. LetLQbe a fuzzy partial order relation in a set Q. Let f:
SS!QQ; f 1:SS!Qandf2:SS!Qbe the mappings such that:
(a)f1(p; p) =f2(p; p)for all p2S;
(b)f1(p; q) =f1(p; r)for all p; q; r 2S;
(c)f2(p; q) =f2(r; q)for all p; q; r 2S;
(d)p6=qimplies that f1(p; q)6=f1(q; p)(orf2(p; q)6=f2(q; p)),
where f(p; q) = (f1(p; q); f2(p; q)):Then f 1(LQ)is a fuzzy partial relation.
Proof. From (3.2) it implies that
f 1(LQ)(p; p) =LQ(f(p; p))for all p2S;
33
since LQis a fuzzy partial order relations, so axiom (a) implies
LQ(f(p; p)) =LQ(f1(p; p); f2(p; p)) = 1 for all p2S:
Now suppose that f 1(LQ)(p; q)>0and f 1(LQ)(q; p)>0:Then:
LQ(f(p; q)) =LQ(f1(p; q); f2(p; q))>0
and
LQ(f(q; p)) =LQ(f1(q; p); f2(q; p))>0:
Since f1(p; q) =f2(q; p)for all p; q2Sby axioms (a), (b) and (c), it further
implies that
LQ(f1(p; q); f2(p; q))>0andLQ(f2(p; q); f1(p; q))>0for all p; q2S:
AsLQis antisymmetric, therefore f1(p; q) =f2(p; q) =f1(q; p) =f2(q; p):By
axiom (d) p=q;thus f 1(LQ)is antisymmetric.
f 1(LQ)(p; r) =LQ(f(p; r)) =LQ(f1(p; r); f2(p; r))
T[LQ(f1(p; r); q); LQ(q; f2(p; r))]for all p; q; r 2S:
By using axioms (c) and (d) of our hypothesis
f 1(LQ)(p; r)T[LQ(f1(p; w); q); LQ(q; f2(w; r))]
T[LQ(f1(p; w); f2(p; w); LQ(f2(p; w); f2(w; r))]:
(b) and (c) of the hypothesis results in
f 1(LQ)(p; r)T[LQ(f1(p; w); f2(p; w); LQ(f1(w; r); f2(w; r))]
=T[LQ(f(p; w)); LQ(f(w; r))]
=T[f 1(LQ)(p; w); f 1(LQ)(w; r)]for all w; p; r 2S:
Example 3.1. LetS=f1;2;3;4gandQ=fp; q; r; s gbe the two sets and let LQ
34
be the partial order relation in Qsuch that
LQ=p
q
r
sp q r s2
66666641 0 0 0
0:416 1 0 0
0:475 0 :484 1 0
0:4 0 :45 0 :35 13
7777775:
Then we can
nd the partial order relation LSinSby using Theorem 3.1 as
follows:
(i)f1(a; a) =f2(a; a)for all a2S:Let
f1(1;1) = f2(1;1) = p;
f1(2;2) = f2(2;2) = r;
f1(3;3) = f2(3;3) = s;
f1(4;4) = f2(4;4) = q:
(ii)f1(a; b) =f1(a; c)for all a; b; c 2S:According to this condition, it follows
that:
f1(1;2) = f1(1;3) = f1(1;4) = p;
f1(2;1) = f1(2;3) = f1(2;4) = r;
f1(3;1) = f1(3;2) = f1(3;4) = s;
f1(4;2) = f1(4;2) = f1(4;3) = q:
(iii)f2(a; b) =f2(c; b)for all a; b; c 2S:According to this condition, it follows
that:
f2(2;1) = f2(3;1) = f2(4;1) = p;
f2(1;2) = f2(3;2) = f2(4;2) = r;
35
f2(1;3) = f2(2;3) = f2(4;3) = s;
f2(1;4) = f2(2;4) = f2(3;4) = q:
(iv) Clearly a6=bimplies that f1(a; b)6=f1(b; a)andf2(a; b)6=f2(b; a)for all
a; b2S. Hence f(a; b) = (f1(a; b); f2(a; b))implies that:
f(1;1) = ( f1(1;1); f2(1;1)) = ( p; p);
f(1;2) = ( f1(1;2); f2(1;2)) = ( p; r);
f(1;3) = ( f1(1;3); f2(1;3)) = ( p; s);
f(1;4) = ( f1(1;4); f2(1;4)) = ( p; q);
and
f(2;1) = ( f1(2;1); f2(2;1)) = ( r; p);
f(2;2) = ( f1(2;2); f2(2;2)) = ( r; r);
f(2;3) = ( f1(2;3); f2(2;3)) = ( r; s);
f(2;4) = ( f1(2;4); f2(2;4)) = ( r; q);
and
f(3;1) = ( f1(3;1); f2(3;1)) = ( s; p);
f(3;2) = ( f1(3;2); f2(3;2)) = ( s; r);
f(3;3) = ( f1(3;3); f2(3;3)) = ( s; s);
f(3;4) = ( f1(3;4); f2(3;4)) = ( s; q);
and
f(4;1) = ( f1(4;1); f2(4;1)) = ( q; p);
f(4;2) = ( f1(4;2); f2(4;2)) = ( q; r);
f(4;3) = ( f1(4;3); f2(4;3)) = ( q; s);
f(4;4) = ( f1(4;4); f2(4;4)) = ( q; q):
36
Therefore by (3.2), LSis given as follows:
LS=1
2
3
41 2 3 42
66666641 0 0 0
0:475 1 0 0 :484
0:4 0 :35 1 0 :45
0:416 0 0 13
7777775:
Theorem 3.2. LetSandQbe sets and LSbe a fuzzy partial order relation in
setS. Let f:SS!QQbe a mapping such that:
(a)there exists p2Sfor every q2Qimplies f(p; p) = (q; q);
(b)there exists q2Qfor every p; r2Simplies f(p; r) = (q; q):
Then f(LS)is a fuzzy partial order relation in Q:
Proof. From (a) of the hypothesis, f(LS)(q; q) = Sup
(a;b)2f 1(q;q)LS(a; b) = 1 for all
q2Q:From (b) of our hypothesis f 1(p; q) = 0 ifp6=q;thus f(LS)(p; q) = 0 :
With the contrapositive law, f(LS)(p; q)>0implies p=q. Hence f(LS)(p; q)>
0andf(LS)(q; p)>0implies p=qi.e.,f(LS)is antisymmetric. Now from
(a) and (b), f(LS)(p; r) = Sup
(u;v)2f 1(p;p)LS(u; v) = 1 and therefore, f(LS)(p; r)
T[f(LS)(p; q); f(LS)(q; r)]:
3.2 Fuzzy Lattices
De
nition 3.3. LetLSbe a fpor in set SandQS;an element a2Sis
an upper bound (strict upper bound) for subset Qsatisfying LS(q; a)LS(a; q)
(LS(q; a)> L S(a; q))for all q2Q. An upper bound a0forQis the lub for Qif
LS(a0; a)LS(a; a 0)for all aforQ. The lower bound (strict lower bound) for
subset Qis an element b2Ssuch that LS(b; q)LS(q; b) (LS(b; q)> L S(q; b))
for all q2Q:A lower bound b0forQis the glb of QifLS(b; b0)LS(b0; b).
For two point set fp; qg, we use p_qfor superemum and p^qfor in
mum. A
lattice is an ordered set whose every pair of objects has a supremum and in
mum.
By induction, it can be shown easily that every
nite set in lattice has a supremum
and an in
mum.
37
De
nition 3.4. Let(S; L S)be a fposet. (S; L S)is a fuzzy lattice i¤ p_qand
p^qexist for all p; q2S.
Proposition 3.2. LetLSbe a fuzzy partial order relation and (S; L S)be a fuzzy
lattice. Then for all p; q; r 2S:
(a)LS(p; p_q)>0; LS(q; p_q)>0; LS(p^q; p)>0; LS(p^q; q)>0:
(b)LS(p; r)>0andLS(q; r)>0implies LS(p_q; r)>0:
(c)LS(r; p)>0andLS(r; q)>0implies LS(r; p^q)>0:
(d)LS(p; q)>0i¤p_q=q:
(e)LS(p; q)>0i¤p^q=p:
(f)LS(p_q; p_r)>0andLS(p^q; p^r)>0ifLS(q; r)>0:
Proof. Since LSis a fuzzy partial order relation, therefore (a), (b) and (c) are
forthright. To prove (d) consider LS(p; q)>0:AsLS(q; q) = 1 >0by de
nition
ofLS,LS(p_q; q)>0by (b). Since LS(q; p_q)>0by (a), hence p_q=qby
the de
nition of LS:
Conversely, consider p_q=q;then LS(p; q) =LS(p; p_q)>0:
Proof of (e) is straightforward and similar as of (d). Now for (f) suppose that
LS(q; r)>0:Then T-transitivity of LSimplies that
LS(p^q; r)T(LS(p^q; w); LS(w; r))>0for all w; p; q; r 2S:
AsLS(p^q; p)>0by (a), p^qis a lower bound of fp; rg:Since p^ris the glb
offp; rg;hence LS(p^q; p^r)>0:Also
LS(q; p_r)T(LS(q; w); LS(w; p_r))>0for all w; p; q; r 2S:
Since LS(q; p_r)>0by (a), hence LS(p_q; p_r)>0by (b).
Example 3.2. LetS=fp1; p2; p3; p4gand let LSbe a fuzzy relation in Ssuch
38
that
LS=0
BBBBBB@1 0 0 0
0:4 1 0 0
0:6 0:3 1 0
0:8 0:5 0:2 11
CCCCCCA:
Then we can easily check that LSis a totally ordered relation. Also p_q=
p; p_r=p; p_w=p; q_r=q; q_w=q; r_w=r;p^q=q; p^r=r; p^w=
w; q^r=r; q^w=wandr^w=w:Therefore, (S; L S)is a fuzzy lattice.
Theorem 3.3. Let(S; L S)be a fuzzy lattice. Then the operations _and^
satis
es associative, commutative, idempotency and absorption laws for all p; q; r 2
S:
Proof. (i) Associative laws: (p_q)_r=p_(q_r);(p^q)^r=p^(q^r):
Since LS(p; p_(q_r))>0from (a) of Proposition 3.2 and
LS(q; p_(q_r))T(LS(q; w); LS(w; p_(q_r)))
T(LS(q; q_r); LS(q_r; p_(q_r)))
>0
AsLS(p_q; p_(q_r))>0from (b) of Proposition 3.2 . and
LS(r; p_(q_r))T(LS(r; w); LS(w; p_(q_r)))
T(LS(r; q_r); LS(q_r; p_(q_r)))
>0;
by (b) of Proposition 3.2 it implies that LS((p_q)_r; p_(q_r))>0:In the
same way we can show LS(p_(q_r);(p_q)_r)>0:Then by the antisymmetry
ofLS,(p_q)_r=p_(q_r):Similarly (p^q)^r=p^(q^r)can be proved.
(ii) Commutative laws: p_q=q_p; p^q=q^p:
Commutative laws are straightforward becuase for a set S,Sup(S)does not depend
on order of the elements which are listed.
(iii) Idempotency laws: p_p=p; p^p=p:
Letp; q2Sandpqthen the lubof the set fp; qguisqand the glbof the set
39
fp; qglisp:Therefore, p_q=qandp^q=p:In particular if is re exive, then
p_p=pandp^p=p:
(iv) Absorption laws: p_(p^q) =p; p^(p_q) =p:
Letfp_q; pg S:Since LS(p; p_q)>0andLSis re exive, therefore pis a lower
bound of the set fp_q; pg:Ifris the lower bound of fp_q; pg;then LS(r; p)>0:
Sopis the glboffp_q; pg;hence p^(p_q) =p. In the similar way it can be
shown that p_(p^q) =p.
Lemma 3.1. [39] Let (S; L S)be a fuzzy lattice. Then pq;p_q=q;p^q=p
are equivalent for p; q2S.
De
nition 3.5. A fuzzy lattice (S; L S)is said to be distributive if and only if
p^(q_r) = (p^q)_(p^r)and (p_q)^(p_r) =p_(q^r):
Proposition 3.3. If(S; L S)is a fuzzy lattice , then LS((p^q)_(p^r); p^(q_r))>
0andLS(p_(q^r);(p_q)^(p_r))>0forp; q; r 2S:
Proof. Since from (a) of Proposition 3.2 LS(p^q; q)>0andLS(q; q_r)>
0gives LS(p^q; q_r)>0which further with LS(p^q; p)>0implies that
LS(p^q; p^(q_r))>0by (c) of Proposition 3.2 . Also LS(p^r; r)>0and
LS(r; q_r)>0gives LS(p^r; q_r)>0which further with LS(p^r; p)>0
implies that LS(p^r; p^(q_r))>0by (c) of Proposition 3.2 . So p^(q_r)is
an upper bound of fp^q; p^rg:Since (p^q)_(p^r)isluboffp^q; p^rg;
therefore LS((p^q)_(p^r); p^(q_r))>0:In the similar way we can prove
LS((p_q)^(p_r); p_(q^r))>0:These inequalities are distributive inequalities.
Theorem 3.4. LetLSbe the total order relation. Then the fuzzy lattice (S; L S)
is distributive.
Proof. IfLSis a total order relation, then LS(p; q)>0orLS(q; p)>0for all
p; q2S:
Case-i IfLS(p; q)>0:
If we consider LS(p; q)>0;then p^q=pby (e) of Proposition 3.2 . As LS(p^
(q_r); p)>0by (a) of Proposition 3.2 and results in LS(p^(q_r); p^q)>0:
From (a) of Proposition 3.2 it implies that LS(p^q;(p^q)_(p^r))>0:Since
LSisT-transitive relation, so: LS(p^(q_r);(p^q)_(p^r))T[LS(p^(q_
r); w); LS(w;(p^q)_(p^r))]T[LS(p^(q_r); p^q); LS(p^q;(p^q)_(p^r))]>0:
40
By distributive inequalities p^(q_r) = (p^q)_(p^r):Now by absorption law,
p^(p_q) =p:Therefore, LS((p_q)^(p_r); p_(q^r)) =LS([(p_q)^p]_[(p_q)^
r]; p_(q^r)) =LS(p_[r^(p_q)]; p_(q^r)) =LS(p_[(r^p)_(r^q)]; p_(q^r)) =
LS([p_(r^p)]_(r^q)]; p_(q^r)):Since p_(r^p) =pby absorption law,
LS((p_q)^(p_r); p_(q^r)) =LS(p_(r^q); p_(q^r)) =LS(p_(q^r); p_(q^r))>
0:In the similar way we can show that LS(p_(q^r);(p_q)^(p_r))>0:Hence
due to antisymmetry of LS;it implies that (p_q)^(p_r) =p_(q^r):Thus the
fuzzy lattice (S; L S)is distributive.
Case-ii IfLS(q; p)>0:
If we consider LS(q; p)>0;then p_q=pby (d) of Proposition 3.2 . SoLS((p_q)^
(p_r); p) =LS(p^(p_r); p)>0;by (a) of proposition 3.2 LS(p; p^(p_r))>0:
Since LSisT-transitive relation, so: LS((p_q)^(p_r); p_(q^r))T[LS((p_
q)^(p_r); w); LS(w; p_(q^r))]T[LS((p_q)^(p_r); p); LS(p; p_(q^r))]>0:
By distributive inequalities (p_q)^(p_r) =p_(q^r):Now by absorption law,
p_(p^q) =p:Therefore, LS((p^q)_(p^r); p^(q_r)) =LS([(p^q)_p]_[(p^q)_
r]; p^(q_r)) =LS(p^[r_(p^q)]; p^(q_r)) =LS(p^[(r_p)^(r_q)]; p^(q_r)) =
LS([p^(r_p)]^(r_q)]; p^(q_r)):Since p^(r_p) =pby absorption law,
LS((p^q)_(p^r); p^(q_r)) =LS(p^(r_q); p^(q_r)) =LS(p^(q_r); p^(q_r))>
0:In the similar way we can show that LS(p^(q_r);(p^q)_(p^r))>0:Hence
due to antisymmetry of LS;it implies that (p^q)_(p^r) =p^(q_r):Thus the
fuzzy lattice (S; L S)is distributive.
Conclusion
In this chapter, it has been successfully characterized a fuzzy partial order relation
based on T-transitivity, and presented appropriate conditions for the image and
inverse image of a fuzzy partial order relation to be a fuzzy partial order relation.
Fuzzy lattice is de
ned as a fuzzy relation, and some basic properties of fuzzy
lattices by using T-transitivity are developed and show that a fuzzy totally ordered
set is a distributive fuzzy lattice.
41
Chapter 4
New Results on the Measures of Transitivity
In this chapter, we present some new results in the form of inequalities connecting
transitivity of the given fuzzy preference relation with its consistent behaviour.
The additive and multiplicative generators of t-norms and t-conorms play the key
role in establishing these results.
Though various versions of T-transitivity have been used in large number of
especial applications of preference, but all these de
nitions have also been censured
and the space for newer models of transitivity is still there. In [14], Beg and Ashraf
proposed a new de
nition of degree or measure of a fuzzy relation for being fuzzy T-
transitive called -transitive fuzzy relation. This new notion was based upon fuzzy
set theoretic operators, in particular the fuzzy implicator. In the former literature,
di¤erent de
nitions of fuzzy implicators have been developed by di¤erent authors.
Two types of fuzzy implicators were proposed by Trillas in [143, 144] which were
de
ned with the help of fuzzy negation, conjunction and disjunction. In [14], Beg
and Ashraf obtained several novel results using these forms of implicators in the
de
nition of -transitivity.
The notion of transitivity and consistency of a fuzzy relation are synonymous
in some sense. The major inconsistency raises in a crisp preference relation when
according to some experts opinion: ais preferable to bandbis preferable to cbut
ais not preferable to c. So nothing could be concluded from the given relation as
a structure of preference. In fuzzy preference relations, the similar phenomenon
arises but in the form of degrees of preference. As far as FPRs are concerned, there
is a list of consistency conditions required from degrees of preference in the form
of transitivity conditions. A summarized list for such conditions has been already
given in Chapter 2 : the detailed study could be found in [16, 25, 43, 89, 90, 91, 105].
Any of these conditions may be used to discuss the transitivity of a given
preference relation, but all these conditions make a strict decision that the given
relation is transitive or not. While in real life situations, we must be able to talk
about degree of transitivity of a fuzzy relation i.e., it may be more or less transitive.
This chapter makes use of the degrees of transitivity and preference to discuss
the mathematical results that re ect the e¤ect of being more or less transitive
for a fuzzy preference relation and consequently being more or less consistent.
Several new inequalities have been obtained establishing relationships between
43
the preference degrees and consistency degrees. To our knowledge this is the
rst
attempt to work out the mathematical formulations representing the nature of
preference data while given the degree of being transitive.
4.1 Measures of T-transitivity
In this section, we shall focus our attention on exploring the relationship between
the degree of being T-transitive and consistent behavior of the fuzzy preference
relation. This will be done in two ways:
rst we will explore the situation in
which the degree for being T-transitive is considered while an R-implicator is
used to calculate this degree then in the next subsection we shall investigate the
results of using degree of using an S-implicator in the same de
nition. To dispense
with all possible situations we shall workout both the additive and multiplicative
generators.
4.1.1 The Inequalities Related to R-implicators
In [14], Beg and Ashraf proposed a new de
nition of degree or measure of a fuzzy
relation for being fuzzy T-transitive called -transitive relation. This new notion
was based upon fuzzy set theoretic operators, in particular the fuzzy implicator. In
the former literature, di¤erent de
nitions of fuzzy implicators have been developed
by di¤erent [143, 144]. Beg and Ashraf obtained several novel results using these
forms of implicators in the de
nition of -transitivity [14].
De
nition 4.1. Fuzzy Ternary Relation & -Fuzzy Transitive Relation Let R
be a fuzzy relation on X. The fuzzy set of transitivity trI;T(R)is a fuzzy ternary
relation on Xfor all x; y; z 2Xand is de
ned as:
trI;T(R)(x; y; z ) =I(T(R(x; y); R(y; z)); R(x; z)): (4.1)
The transitivity function so de
ned assigns a degree of transitivity to the relation
at each point of XXX. For a given fuzzy relation Ron a universe X;a t-norm
Tand any fuzzy implicator I;the measure of transitivity of Rfor all x; y; z 2X
44
is given by:
TrI;T(R) = inf
x;y;z2X(I(T(R(x; y); R(y; z)); R(x; z))) (4.2)
ifTr(R) =;then the relation Ris called -fuzzy transitive ; Ris non-transitive if
= 0;weak fuzzy transitive if <0:5and strong fuzzy transitive if 0:5:
Theorem 4.1. If an R-implicator Iassociated with a continuous Archimedean
t-norm Twith fas its additive generator is used in the de
nition of fuzzy tran-
sitivity of an -fuzzy transitive relation R, then for all x; y; z 2X;there exists
c2(Rxz; T(Rxy; Ryz))such that
either T(Rxy; Ryz)RxzorT(Rxy; Ryz)Rxz+ f()
f (c).
Proof. Given Ris an -fuzzy transitive relation so,
inf
x;y;z2XI(T(Rxy; Ryz); Rxz) =for all x; y; z 2X;
it implies that I(T(Rxy; Ryz); Rxz) >0for all x; y; z 2X:
Using representation of I(Remark 2.2 ), we get
f 1(max( f(Rxz) f(T(Rxy; Ryz));0))for all x; y; z 2X:
Applying fon both sides with strictly decreasing nature
max( f(Rxz) f(T(Rxy; Ryz));0)f()
it further implies that
f(Rxz) f(T(Rxy; Ryz))f()and 0f()for all x; y; z 2X: (4.3)
Here arise two cases:
(a)T(Rxy; Ryz)Rxzi.e., the given relation is T-transitive. As proved in [14], the
T-transitive relations are 1-fuzzy transitive. Hence (4.3) takes the form f(Rxz)
f(T(Rxy; Ryz))0:Which trivially gives T(Rxy; Ryz)Rxz;i.e., the assumption.
(b)Rxz< T(Rxy; Ryz)i.e.,f(Rxz)> f(T(Rxy; Ryz))due to decreasing nature of
45
f:So (4.3) takes the form f(T(Rxy; Ryz)) f(Rxz) f():Due to Lagrange s
mean value theorem and properties of f, there exists a c2]Rxz; T(Rxy; Ryz)[such
that f (c)(T(Rxy; Ryz) Rxz) f():Since f (c)<0due to strictly decreasing
nature of f, we get
T(Rxy; Ryz) Rxz f()
f (c);
hence
T(Rxy; Ryz)Rxz+ f()
f (c)for all x; y; z 2X: (4.4)
Combining (4.4) with the assumption for the case (a) we
nally get for all x; y; z 2
X
either T(Rxy; Ryz)Rxz
or other wise T(Rxy; Ryz)Rxz+ f()
f (c)
This gives us a relationship which help us determine the least value of Rxzgiven the
degree of consistency of the given relation Rxy;andRyz. Since fis a monotonically
decreasing function f()decreases with increase in . Moreover, f (c)is a negative
quantity. These facts clearly indicate that as increases, T(Rxy; Ryz)approaches
Rxzi.e. the relation moves towards T-transitivity.
Corollary 4.1. If the requirements of Lagrange s Mean Value Theorem could
not be met, then the inequality (4.3) may be reworked to get another view of the
transitivity scenario. For an -equivalence relation R;if an R-implicator Iis used
in the de
nition of fuzzy transitivity related to an Archimedean t-norm T, then
for all x; y; z 2X
f(Rxz) f(Rxy) f(Ryz)f()andf(Rxz)f(0) + f();
where fis an additive generator of T. From (4.3) it follows that f(Rxz)
f(T(Rxy; Ryz))f()and 0f():Using the fact that fis an additive gen-
erator of Tthen for all x; y; z 2X:
f(Rxz) min(f(Rxy) +f(Ryz); f(0))f();
46
max ( f(Rxz) f(Rxy) f(Ryz); f(Rxz) f(0))f():
Hence
f(Rxz) f(Rxy) f(Ryz)f()andf(Rxz)f() +f(0): (4.5)
Example 4.1. We consider relations with lower and higher degrees of transitivity.
For this, we consider Lukaseiwicz t-norm Walong with it s additive generator
f(x) = 1 x:The inequality (4.4) can be used to conclude following results:
Degree of transitivity Rxyand Ryz region for Rxz
= 0:4; 0:8and 0:9implies Rxz0:1
= 0:6; 0:8and 0:9implies Rxz0:3
= 0:9; 0:8and 0:9implies Rxz0:6
Now we consider product t-norm P, along with its additive generator t(x) = lnx
sot0(x) = 1
x;so (4.5) takes the form:
ln(Rxz) + ln( Rxy) + ln( Ryz) ln()and ln(Rxz) ln() ln(0):
ln(Rxz)+ ln( Rxy) + ln( Ryz) + ln( )and ln(Rxz)+1
ln(Rxz)ln(Rxy:Ryz:)i.e.,RxzRxy:Ryz::
Degree of transitivity Rxyand Ryz region for Rxz
= 0:4; 0:8and 0:9implies Rxz0:288
= 0:6; 0:8and 0:9implies Rxz0:432
= 0:9; 0:8and 0:9implies Rxz0:4248
We conclude that the implicit relationship between degree of consistency of an
FPR and its degrees of preference is explicit now. Moreover the di¤erent forms of
transitivities behave less or more stringent.
Theorem 4.2. If an R-implicator Iis used in the de
nition of fuzzy transitivity
of an -transitive relation Rwith a continuous Archimedean t-norm Twith a
47
multiplicative generator g, then
g(Rxy)g(Ryz)g(Rxz)
g()for all x; y; z 2X:
Proof. By using Remark 2.3 , asRemark 2.2 in previous theorem, -transitivity
implies that
g 1
ming(Rxz)
g(T(Rxy; Ryz));1
for all x; y; z 2X:
Applying gon both sides with increasing nature
ming(Rxz)
g(T(Rxy; Ryz));1
g()for all x; y; z 2X
It implies that
g(Rxz)
g(T(Rxy; Ryz))g();
and
g(Rxz)g(T(Rxy; Ryz))g()for all x; y; z 2X:
Since gis a multiplicative generator, it further implies that
g(Rxz)max( g(Rxy)g(Ryz); g(0))g()for all x; y; z 2X;
and hence
g(Rxy)g(Ryz)g(Rxz)
g()for all x; y; z 2X:
4.1.2 The Inequalities For S-implicators
Theorem 4.3. LetRbe an -transitive relation which is not T-transitive. If an
S-implicator Iis used in the de
nition of fuzzy transitivity related to continuous
Archimedean t-conorm Swith an additive generator s, then
Rx;zs 1(s() s(N(Rx;y)) s(N(Ry;z)))for all x; y; z 2X:
48
Proof. Using Remark 2.4 in the de
nition of -transitivity implies that
s 1(min( s(N(T(Rx;y; Ry;z))) + s(Rx;z); s(1)))for all x; y; z 2X:
Increasing nature of simplies that
min(s(N(T(Rx;y; Ry;z))) + s(Rx;z); s(1))s()for all x; y; z 2X:
Applying De
nition 2.9 it further implies that
min(s(S(N(Rx;y); N(Ry;z))) + s(Rx;z); s(1))s();
and
s(S(N(Rx;y); N(Ry;z))) + s(Rx;z)s()for all x; y; z 2X:
Using Remark 2.4 again
min(s(N(Rx;y)) +s(N(Ry;z)); s(1)) + s(Rx;z)s()
and
s(N(Rx;y)) +s(N(Ry;z)) +s(Rx;z)s()for all x; y; z 2X: (4.6)
(4.6) can be written as
Rx;zs 1(s() s(N(Rx;y)) s(N(Ry;z)))for all x; y; z 2X:
Theorem 4.4. If an S-implicator Iis used in the de
nition of fuzzy transitivity
related to continuous Archimedean t-conorm Swith a multiplicative generator ,
then
(Rxz)()
(N(Rxy))(N(Ryz))for all x; y; z 2X:
Proof . Choosing Remark 2.5 in (4.2) we have
1[maxf(N(T(Rx;y; Ry;z)))(Rx;z); (1)g]for all x; y; z 2X:
49
Applying with decreasing nature
max[(N(T(Rx;y; Ry;z)))(Rx;z); (1)]();
which further implies that
max[(S(N(Rx;y); N(Ry;z)))(Rx;z); (1)]();
and
(S(N(Rx;y); N(Ry;z)))(Rx;z)()for all x; y; z 2X:
Now using De
nition 2.6
maxf(1); (N(Rxy))(N(Ryz))g (Rxz)();
and
(Rxz)()
(N(Rxy))(N(Ryz))for all x; y; z 2X:
4.2 Measures of S-transitivity
This section focus on exploring the relationship between the degree of being S-
transitive and consistent behavior of the fuzzy preference relation. The central
aim of the work is twofold:
rstly it explores the situation in which the degree
for being S-transitive is considered while an R-implicator is used to calculate this
degree, and secondly we investigate the results using the degree for an S-implicator
in the same de
nition.
De
nition 4.2. -Fuzzy Transitive Relation : A fuzzy relation Ris called S-
transitive to the degree or -fuzzy transitive if: = inf
y2XI(Rxz; S(Rxy; Ryz))for
allx; y; z 2X:
50
4.2.1 -Transitivity Related to R-implicators
Theorem 4.5. IfRis a -fuzzy transitive relation and an R-implicator Iis used
in the de
nition of an FPR, then
RxzS(Rxy; Ryz) f( )
f (a)for all x; y; z 2X;
where fis an additive generator of continuous Archimedean t-norm.
Proof. IfRis a -transitive relation, then
inf
y2XI(Rxz; S(Rxy; Ryz)) = ;
I(Rxz; S(Rxy; Ryz)) for all x; y; z 2X:
Now, using Remark 2.2
f 1(max( f(S(Rxy; Ryz)) f(Rxz);0)) :
Using fit implies
max( f(S(Rxy; Ryz)) f(Rxz);0)f( );
which further implies that
f(S(Rxy; Ryz)) f(Rxz)f( ):
Since the nature of fis strictly decreasing, therefore by Taylor s mean value the-
orem there exists a real number a2]Rxz; S(Rxy; Ryz)[such that
f(S(Rxy; Ryz)) f(Rxz) =f (a)[S(Rxy; Ryz) Rxz]:
It implies now
f (a)[S(Rxy; Ryz) Rxz]f( );
and
RxzS(Rxy; Ryz) f( )
f (a)for all x; y; z 2X:
51
Theorem 4.6. IfRis a -fuzzy transitive relation and an R-implicator Iis used
in the de
nition of fuzzy transitivity related to continuous Archimedean t-norm
with a multiplicative generator g, then
g(Rxz)g(Rxy)g(Ryz)
g( )for all x; y; z 2X:
Proof. Using De
nition 4.2
inf
y2XI(Rxz; S(Rxy; Ryz)) = ;
and
I(Rxz; S(Rxy; Ryz)) for all x; y; z 2X:
By using Remark 2.3
g 1
ming(S(Rxy; Ryz))
g(Rxz);1
for all x; y; z 2X:
Applying gon both sides with increasing nature
ming(S(Rxy; Ryz))
g(Rxz);1
g( )for all x; y; z 2X:
Hence, it implies that
g(S(Rxy; Ryz))
g(Rxz)g( );
and
g(S(Rxy; Ryz))g(Rxz)g( )for all x; y; z 2X:
Since gis a multiplicative generator, it implies that
max( g(Rxy)g(Ryz); g(1))g(Rxz)g( );
and
g(Rxz)g(Rxy)g(Ryz)
g( )for all x; y; z 2X:
52
4.2.2 -Transitivity Related to S-implicators
Theorem 4.7. LetRbe a fuzzy preference relation. If an S-implicator Iis used
in the de
nition of fuzzy transitivity related to continuous Archimedean t-conorm
Swith an additive generator s, then
N(Rx;z)s 1(s() s(Rx;y) s(Ry;z))for all x; y; z 2X:
Proof. Given Ris an -fuzzy transitive relation so, inf
y2XI(Rxz; S(Rxy; Ryz)) = ;
i.e.,
I(Rxz; S(Rxy; Ryz)) for all x; y; z 2X:
By using Remark 2.4
s 1(min( s(N(Rx;z)) +s(S(Rx;y; Ry;z)); s(1))) for all x; y; z 2X:
Increasing nature of simplies that
min(s(N(Rx;z)) +s(S(Rx;y; Ry;z)); s(1))s( );
and
s(N(Rx;z)) +s(S(Rx;y; Ry;z))s( )for all x; y; z 2X:
Again by using Remark 2.4
min(s(Rx;y) +s(Ry;z); s(1)) + s(N(Rx;z))s( );
s(Rx;y) +s(Ry;z) +s(N(Rx;z))s( )
and
N(Rx;z)s 1(s() s(Rx;y) s(Ry;z))for all x; y; z 2X:
Conclusion
This chapter presents some new results in the form of inequalities connecting tran-
sitivity of the given fuzzy relation with its clustering nature, here -transitivity has
been used to formulate new inequalities interconnecting with T-transitivity and
53
S-transitivity. Additive and multiplicative generators of t-norms and t-conorms
play a prime role for these inequalities. The forms obtained open new research di-
rection. The results based on the measure of the considered transitivity conditions
ensure coherence among the objects.
54
Chapter 5
Group Decision Making by using Incomplete Fuzzy
Preference Relations
The results included in this chapter have been partially published in [7].
Decision making is the rational procedure to select the best alternative(s) among
several feasible options. In our perspective, we have a
nite set of possible alter-
natives X=fx1; x2; : : : ; x ng; n2, for the problem from where we wish to get a
solution set of alternatives (the best option(s) to solve the problem).
A GDM problem is to be faced when there is a question to be solved, a set
of possible alternatives from which to choose and a set of experts which express
their opinions or preferences about the existing options. The presence of multiple
experts in a decision process may produce some additional di¢ culties to select
the best alternative(s). For example, each expert has exclusive characteristics
regarding to the knowledge, abilities, experience and nature of preferences, which
suggests that di¤erent experts may provide their evaluations by means of di¤erent
representation format and, hence, opinion about the alternative(s) can be varied.
Therefore, to have some kind of agreement among experts is compulsory preceding
to the actual selection of the best alternative(s).
Preference relation is the most common representation format used in GDM
because it is a valuable tool in modeling decision processes, when we have to com-
bine expertspreferences into group preferences [55, 127, 141]. Mainly two types
of preference relations have been used to develop the decision models; multiplica-
tive preference relations (MPRs) [36, 127], and fuzzy preference relations (FPRs)
[55, 142]. The most popular preference relations which are being used to express
an expert s opinion over alternatives after pair-wise comparison, are FPRs.
There arise some situations during pair-wise comparison in which experts do
not have comprehensive information of the problem to be explained. In such
situations, expertsmay not be able to give their preferences about speci
c as-
pects of the problem and they may present incomplete FPRs. Some techniques
to estimate the missing values have been proposed in the literature [3, 4, 159].
Consistency is an important issue to be addressed when data is provided by the
experts [37, 74], and it is strongly associated with the transitivity property as
has already been discussed in introduction. A thorough discussion on theoretical
aspects of T-transitivity, the one being used in this thesis rigorousely is presented
in
rst part of this thesis as a representative of consistency. Consistency, that
56
is, no contradiction can be associated by the transitivity, and hence, if an FPR
R= (rij)nnveri
es T-transitivity property i.e.,
rikT(rij; rjk)8i; j; k2 f1;2;3; :::; ng (5.1)
we can say that it is a T-consistent ( Tbeing a triangular norm) when for every
three alternatives in the problem xi; xj; xk2X, there associated preference degrees
rij; rjk; rikful
l expression (5.1).
In this chapter, we focus on the scenarios where incomplete information is pro-
vided in the form of IFPRs by experts included in the group for group decision
making. We present a simple and practical procedure to estimate the missing
values in the relational matrix. We further construct the consistent FPRs based
on the triangular norm T, and the process is extended for GDM in IFPRs envi-
ronment.
De
nition 5.1. LetR= (rij)nnbe a fuzzy preference relation on a universe
X, a row vector of Ribyithelement of Xis a fuzzy set RxonXde
ned as:
Ri(j) =Rij. As can be easily seen a row vetor is a fuzzy set that shows the
preference of ithelement over all other elements.
De
nition 5.2. Incomplete FPR : A fuzzy preference relation R= (rij)nnis
said to be incomplete if it contains at least one unknown preference value rijfor
which expert has no idea about the degree of preference of alternative xiover the
alternative xj[71]:
De
nition 5.3. Order Consistency : A fuzzy preference relation R= (rij)nnis
said to be order consistent if for every pair of elements iandj;with row vectors
RiandRjeither RiRjorRjRifor all i; j2 f1;2;3; :::; ng[96]:
5.1 Estimation of Missing Preference Value
In this section, we put forward a new technique to determine missing values in an
IFPR. Further, the algorithm is used to construct a consistency matrix which is
T-consistent and order consistent simultaneously. In order to determine unknown
values in an incomplete fuzzy preference relation R= (rij)nn, the pairs of alter-
natives for known and unknown preference values are represented by the following
57
sets:
K=f(i; j)jrijis known g; (5.2)
U=f(i; j)jrijis unknown g; (5.3)
where the preference value of alternative xiover xjbelongs to [0;1](i.e., rij2
[0;1]). Since rij+rji= 1;for1inand 1jn, therefore, T-transitivity
[equ. ref] can be written as:
rikT(rij; rjk);rikT(1 rji; rjk);rikT(rij;1 rkj): (5.4)
Hence, we can de
ne following sets of intermediate alternative xjwhich can be used
to determine the unknown preference value rikof alternative xiover alternative
xk:
W1
ik=fjj(i; j)2K;(j; k)2Kand (i; k)2Ug; (5.5)
W2
ik=fjj(j; i)2K;(j; k)2Kand (i; k)2Ug; (5.6)
W3
ik=fjj(i; j)2K;(k; j)2Kand (i; k)2Ug; (5.7)
for1in,1jnand 1kn. Based on (5.5),(5.6) and (5.7), we can
determine the unknown preference value rikforxioverxkas follows:
rik=r1
ik+r2
ik+r3
ik
3; (5.8)
where
r1
ik=8
>><
>>:1
jW1
ikjX
j2W1
ikT(rij; rjk);ifjW1
ikj 6= 0
0:5; otherwise; (5.9)
r2
ik=8
>><
>>:1
jW2
ikjX
j2W2
ikT(1 rji; rjk);ifjW2
ikj 6= 0
0:5; otherwise; (5.10)
58
r3
ik=8
>><
>>:1
jW3
ikjX
j2W3
ikT(rij;1 rkj);ifjW3
ikj 6= 0
0:5; otherwise; (5.11)
where jW1
ikj;jW2
ikjandjW3
ikjare the cardinalities of the sets W1
ik; W2
ikandW3
ikre-
spectively. New sets of the pairs of alternatives for known and unknown preference
values are evaluated as follows:
K0=K[ f(i; k)g; (5.12)
U0=U f(i; k)g: (5.13)
To achieve additive reciprocity (i.e., rij+rji= 1) ofR, we use the following scaling
condition:
(i) If rij+rji>1, then
(rij hi) + (rji hi) = 1 such that hi=rij+rji 1
2: (5.14)
(ii) If rij+rji<1, then
(rij+ki) + (rji+ki) = 1 such that ki=1 (rij+rji)
2: (5.15)
After having a complete FPR R= (rij)nn,T-consistent FPR eR= (erij)nnis
obtained by using following transitive closure formula:
eR= (erij)nn=
Sup
k6=i;j(rij; T(rik; rkj)
nnwitherij+erji= 1: (5.16)
Remark 5.1. During the course of this chapter, we will consider Lukasiewicz
t-norm T= max( rij+rjk 1;0)as particular triangular norm to illustrate the
numerical examples.
Example 5.1. LetR= (rij)44be an incomplete fuzzy preference relation for
59
the alternatives x1; x2; x3andx4, given as follows:
R=2
66666640:5r120:7 0:6
r210:5 0:9r24
0:3 0:1 0:5 0:7
0:4r420:3 0:53
7777775;
where r12; r21; r24andr42are unknown preference values. Now after applying
(5.2)-(5.15), we can estimate the unknown preference values for the alternative xi
overxk,1i4and 1k4;and will have complete FPR given as follows:
R=2
66666640:5 0 :35 0 :7 0 :6
0:65 0 :5 0 :9 0:6833
0:3 0 :1 0 :5 0 :7
0:4 0:3167 0 :3 0 :53
7777775:
Hence, Rbecomes a T-consistent fuzzy preference relation eRunder the use of
(5.16) and is shown as:
eR=2
66666640:5 0 :35 0 :7 0 :6
0:65 0 :5 0 :9 0:6833
0:3 0 :1 0 :5 0 :7
0:4 0:3167 0 :3 0 :53
7777775: (5.17)
From (5.17), we can observe that er12= 0:35ander13= 0:7which shows that x2is
preferred to x3:We can also observe that er42= 0:3167 ander43= 0:3which shows
thatx3is preferred to x2. Hence, complete fuzzy preference relation eRis not order
consistent. Now to ensure order consistency, we adopt following technique.
Proposition 5.1. IfR= (rij)nnis a complete fuzzy preference relation for the
set of alternatives X=fx1; x2; x3; :::; x ng,rij+rji= 1; rii= 0:5for1in
and 1jn. Then the order consistency matrix eRcan be made based on
T-consistency fuzzy preference relation eR;as follows:
eR= (erik)nn
1
nnX
j=1T(erij;erjk)!
nn: (5.18)
60
Proof. From T-transitivity, we can obtain
erikT(eri1;er1k);
erikT(eri2;er2k);
:
:
erikT(erin;ernk):
By adding all above, we get
neriknX
j=1T(erij;erjk);
erik1
nnX
j=1T(erij;erjk): (5.19)
Remark 5.2. The expression (5.19) serves as the lower bound for erik. We se-
lect the in
mum to obtain a de
nite value i.e., erik=1
nnX
j=1T(erij;erjk):It may be
noted that the consistent values of erikcan also be obtained under the use of " "
operation. An order consistent matrix eR= (erik)nnif does not satisfy the transi-
tivity property, then it can be converted into consistency matrixeeR= (eerik)nnby
repeated application of (5.16).
Proposition 5.2. IfeeR= (eerik)nnis a consistency matrix obtained by using
Proposition 5.1 andRemark 5.1 , then it satis
es the T-consistency (i.e.,eerik
T(eerij;eerjk)) and order consistency (i.e.,eerikeerilfor all 1in;where k2
f1;2;3; ; :::; n gandl2 f1;2;3; ; :::; n g):
Proof. To check additive reciprocity,
eerik+eerki=1
nnX
j=1T(erij;erjk) +1
nnX
j=1T(erkj;erji)
=1
nnX
j=1erik+1
nnX
j=1erki=1
nnX
j=1(erik+erki)
=1
n[n(erik+erki)] =erik+erki
61
=n
1
nX
j=1T(erij;erjk) +n
1
nX
j=1T(erkj;erji)
=1
nnX
j=1(erik+erki) =erik+erki= 1:
Now
eerii=1
nnX
j=1T(erij;erji) =1
nnX
j=1erii=1
nnX
j=10:5
=1
n(0:5n) = 0 :5:
We also have
eerik=1
nnX
j=1T(erij;erjk) =1
nnX
j=1erik=erik
=n
1
nX
j=1T(erij;erjk) =1
nnX
j=1erik=erik:
AseR= (erik)nnisT-consistent, therefore, without loss of generality we obtain:
eerikT(eerij;eerjk).
Now suppose thateerikeeril, thus we get
1
nnX
j=1T(erij;erjk)1
nnX
j=1T(erij;erjl);
i=timplies that
1
nnX
j=1T(ertj;erjk)1
nnX
j=1T(ertj;erjl);
eertkeertl:
Thuseerikeerilholds for all i2 f1;2;3; :::; ng;where k2 f1;2;3; :::; ngandl2
f1;2;3; :::; ng:
Example 5.2. Consider the complete fuzzy preference relation eR= (erij)44for
the alternatives x1; x2; x3andx4given by (5.17). It is clear that fuzzy prefer-
ence relation eRis not order consistent. Therefore, by using Proposition 5.1 with
62
Remark 5.2 a consistency matrix eRfromeRhas been constructed and is gven
below.
eR=2
66666640:5 0 :4271 0 :5688 0 :5792
0:5729 0 :5 0 :6417 0 :6521
0:4312 0 :3583 0 :5 0 :5104
0:4208 0 :3479 0 :4896 0 :53
7777775: (5.20)
The relationeRobtained in (5.20) is clearly a T-consistent with T= max(erij+
erjk 1;0)i.e.,erikmax(erij+erjk 1;0);and order consistent. Order consistency
means thaterikerilfor all i2 f1;2;3;4g;where k2 f1;2;3;4gandl2 f1;2;3;4g.
Aseri2eri1;eri1eri3anderi3eri4for all i2 f1;2;3;4g;hence, the ranking order
of the alternatives x1; x2; x3andx4isx2> x 1> x 3> x 4.
Remark 5.3. IfeR= (erij)nnis a consistency matrix, then the ranking order
of alternative xican be obtained by using following normalizing formula where
RV(xi)represents the ranking value of alternative xi[96].
RV(xi) =2
n2nX
j=1erij;where 1inandnX
i=1RV(xi) = 1 : (5.21)
If we apply (5.21) on consistency matrix obtained in (5.20), then the ranking
values of alternatives x1; x2; x3andx4areRV(x1) = 0 :2594; R V(x2) = 2958 ;
RV(x3) = 0 :2250 andRV(x4) = 0 :2198 respectively. It implies that the ranking
order of alternatives coincides with the order previously given by order-consistency
i.e.,x2> x 1> x 3> x 4:
5.2 Iterative Procedure for GDM Based on Con-
sistency Relations
Now we turn towards our major task of using the results constructed in Section
5.1 to GDM while IFPRs will be used. In this section, a new step-by-step pro-
cedure is presented for GDM based on T-consistency and order consistency. An
explanatory example is given to validate the anticipated technique. For ease, the
structure of the estimation process is also shown in Fig.5.1. Suppose that there
arenalternatives x1; x2; :::; x nandmexperts E1; E2; :::; E m. Let Rqbe the fuzzy
63
preference relation for the expert Eqshown as follows:
Rq=
rq
ij
nn=2
66666666640:5rq
12: : rq
1n
rq
210:5: : rq
2n
: : : :
: : : :
rq
n1rq
n2: : 0:53
7777777775;
where rq
ij2[0;1]is the preference value given by expert Eqfor alternative xi
over xj,rq
ij+rq
ji= 1;1in;1jnand 1qm. Some of the
preference matrices given by the experts may be incomplete FPRs, because of time
constraint or lack of knowledge. Following this, the T-transitivity property helps
us to estimate the missing values (working is given in Section 5.1) and measure
the degree of consistency of each expert. Trust weights are also used to measure
the degree of trust of experts. To begin with, same initial trust weights are being
allocated to all the experts. On the whole, experts with high trust weights along
with satisfactory consistency should be assigned larger weights than those with low
consistency and low trust weights. Therefore, by using this idea the weights of the
experts are estimated by merging these two trustworthy resources..The proposed
group decision making technique consists of several stages which are described as
follows:
5.2.1 Estimating Missing Preferences
To determine the missing preference values of an IFPR Rqform by the expert Eq,
initially, the sets KqandUqof pairs of alternatives for known and unknown pref-
erence values are introduced as in (5.2) and (5.3). After this, the T-transitivity
based preference values are estimated by using (5.5)-(5.15) to construct the com-
plete FPR Rq.
5.2.2 Consistency Measures
After evaluating the complete FPRs, their corresponding T-consistent FPRs eRq;1
qmcan also be obtained with the help of (5.16). We can then approximate the
64
degree of consistency of an FPR Rqbased on its similarity with the corresponding
T-transitivity based eRqby computing their distances.
1.T-consistent index ( TCI) of pair of alternatives is determined by using:
TCI (rq
ij) = 1 d(rq
ij;erq
ij) (5.22)
where d(rq
ij;erq
ij)is the error (distance) measured by "rq
ij=d(rq
ij;erq
ij) =
rq
ij erq
ij . Seemingly, the higher the value of TCI (rq
ij), the more consistent
rq
ijis with respect to the rest of the preference values involving alternatives
xiandxj.
2.TCI of alternatives xi,1in, is evaluated by:
TCI (xi) =1
2(n 1)nX
j=1(TCI (rq
ij) +TCI (rq
ji)) (5.23)
3.TCI of an FPR Rqis obtained by taking average of all TCI of alternatives
xi:
TCI (Rq) =1
nnX
i=1TCI (xi) (5.24)
Hence, consistency weights can be assigned to the experts by using the relation:
Cw(Eq) =TCI (Rq)
mX
q=1TCI (Rq)(5.25)
5.2.3 Allocating Weights to Experts
The trust weight (degree of trust of others) tw(Eq)can also be enjoyed by each
expert such thatmX
q=1tw(Eq) =m. The larger an expert s trust weight, the greater
the degree to which the expert is trusted by others. Therefore, it makes sense to
allocate higher weights to experts having larger trust weights, so that convincing
estimations can have more weight in the aggregation process. Initially, each expert
is assigned the same trust weight tw(Eq). However, note that the trust weights of
experts do not remain the same in each consensus round.
65
We assign weights to experts by combining their trust weights and consistency
weights using the following relation:
w(Eq) =tw(Eq)Cw(Eq)
mX
q=1tw(Eq)Cw(Eq)(5.26)
where the sum of all weights to the experts must be 1i.e.,mX
q=1w(Eq) = 1 :
5.2.4 Consensus Measures
After having the FPRs with complete information, it is necessary to measure the
consensus among the experts. Regarding this, similarity matrices Sqr= (sqr
ij)nn
for every pair of experts (Eq; Er) (q= 1;2; :::; m 1;r=q+ 1; :::; m )are to be
determined and de
ned as:
sqr
ij= 1 d(rq
ij; rr
ij) (5.27)
where d(rq
ij; rr
ij) = rq
ij rr
ij , and the collective similarity matrix S= (sij)nn
after aggregating all the similarity matrices by using following relation.
sij=2
m(m 1)m 1X
q=1mX
r=q+1sqr
ij (5.28)
Three di¤erent levels involve to compute the degree of consensus amongst the
experts as follows:
1. At
rst level, the consensus degree on a pair of alternative (xi; xj), denoted
bycod(rij)is de
ned to estimate the degree of consensus amonst all experts
on that pair of alternatives:
cod(rij) =sij (5.29)
2. At second level, the consensus degree on alternatives xi, denoted by CoD (xi)
is de
ned to determine the consensus degree amongst all the experts on that
66
alternative:
CoD (xi) =1
2(n 1)nX
j=1;j6=i(sij+sji) (5.30)
3. At third level, the consensus degree on the relations Rq, denoted by CoD (Rq)
is de
ned to calculate the global degree of consensus amongst all the experts
judgments:
CoD (Rq) =1
nmX
i=1CoD (xi) (5.31)
Once the global consensus level among all the experts reach, it rquire to compare
with a threshold consensus degree , generally settled in advance depending upon
the nature of problem. If CoD (R), this shows that a satisfactory level of
consensus has been obtained, and the decision process begins. Otherwise, the
consensus degree is not stable, and feedback mechanism is originated.
5.2.5 Feedback Mechanism
The central aim of feedback mechanism is to provide comprehensive knowledge
to experts, so as to change their opinions acceptably to enhance the consensus
degree. At the
rst hand, we have to identify the expert(s) whose judgment and
preference values need to be improved. As stated above, the weights of experts
are allocated based on the consistency weights and trust weights. There are four
cases arise altogether regarding this process: i.(high, high); ii.(high, low); iii.(low,
high) and iv.(low, low). Here, we always categorize these two pairs of index values
into highand lowby relating an index value to its mean. If the index value
is not smaller than the mean, then the conforming weight can be treated as high;
on the contrary, it can be treated as low.
There are four cases arise altogether regarding this process: i. (high, high); ii.
(high, low); iii. (low, high) and iv. (low, low). Here, we always categorize these
two pairs of index values into highand lowby relating an index value to its
mean. If the index value is not smaller than the mean, then the conforming weight
can be treated as high; on the contrary, it can be treated as low.
If both of the indexes are at high level, it speci
es that the judgment of the
corresponding expert is well trusted by others, and is consistent. The estimation
67
should hence remain unchanged, and hence, it can exercise an in uence on others.
In the second case, it shows that the opinion of an expert di¤ers with those of others
to a signi
cant degree, and therefore, the respective expert should have to adjust
his/her preference relation under the action of certain rule (given below). For the
third case, the corresponding expert have to regulate his/her preference relation
based more on the consistency property to meet the consistency constraint. If
both are low, the corresponding expert needs to focus more on both aspects while
modifying his/her preferences.
Procedure:
When consensus is not su¢ ciently high, then we have to identi
ed the preference
values that are to be changed, and following formula helps us in this regard:
Rq=f(i; j)jcod(rij)< CoD (R)[rq
ij=2 ;g; i; j = 1;2; :::; n: (5.32)
The system recommends that the corresponding expert has to increase value if it
is smaller than the mean value of the valuations of the rest of experts, or decrease
it if it is greater than the mean. All at once, there may be a possibility that the
minority opinion is precise [27], a con ict resolution method has been proposed
to handle such a situation which contains correct opinions against few experts
[99]. We need to
nd out all alternatives x i whose consensus measures CoD (xi)
are lower than CoD (R), and then intricate the particular pair of experts whose
opinions contradict one another on alternative xito the greatest extent, ( i.e., with
the largest deviation of Son alternatives xi, which can be measured by the sum
ofnX
j=1;j6=i(sij+sji)). With this, we can make them change their initial preferences
and, at the same time, accelerates the pace at which consensus is reached
5.2.6 Accumulation Phase
In this phase, we construct order consistent matrix eRq
=
erq
ik
nnagainst each
expert Eqbased oneRqby using following formula:
68
eRq
=
erq
ik
nn=
1
nnX
j=1T(erq
ij;erq
jk)!
nn: (5.33)
It may quite often that the preference value put forward by each expert is weighted
di¤erently. As soon as the weights of the experts have been obtained, their opinions
need to be accumulated into a global whole. Determine the collective matrix RC
against all experts, shown as follows:
RC=
rc
ij
nn= mX
q=1w(Eq)eerq
ij!
nn; (5.34)
where 1in;1jnand 1qmwitheeRq
=eerq
ij
nnbeing an order
consistent and T-consistent FPR.
5.2.7 Selection Phase
After reaching at an agreeable consensus level among all experts, we enter into
selection phase. This results in ranking all alternatives in order to pick the best
one. To this end, the order consistency property (see Example 5.2 ) is used to
rank the alternatives.
Experts’ SetIncomplete FuzzyPreference RelationsComplete FuzzyPreferenceRelationsConsistent FuzzyPreferenceRelationsEstimation ProcedureTconsistencyOrderconsistency
SelectionPhaseWeights toExpertsConsensusMeasuresTrust Weights
NotAcceptableAcceptableFeedback MechanismConsistencyWeights
Fig. 5.1. Resolution Process for GDM
69
5.3 Numerical Example
This section deals with a numerical example in order to demonstrate the process
of the proposed method and its e¤ectiveness.
Consider that three experts E1; E2,E3from di¤erent
elds are requested to
select the best alternative out of four alternatives x1; x2; x3; x4. The three experts
give their FPRs as follows:
R1=2
66666640:5r1
120:4r1
14
r1
210:5r1
230:7
0:6r1
320:5 0:2
r1
410:3 0:8 0:53
7777775; R2=2
66666640:5r2
120:6 0:2
r2
210:5 0:4r2
24
0:4 0:6 0:5 0:6
0:8r2
420:4 0:53
7777775;
R3=2
66666640:5r3
120:9r3
14
r3
210:5r3
230:9
0:1r3
320:5r3
34
r3
410:1r3
430:53
7777775:
The threshold consensus.level settled in advance is 0:80. Now, we perform the
following steps to evaluate the result:
Step-i: Estimating the missing preferences
Initially, all the missing preference values need to be determined using T-transitivity
(particularly L-transitivity) property mentioned in Section 1.
Taking R1;for example. The sets of pairs of alternatives for known and un-
known preference values are determined as follows:
K1=f(1;1);(1;3);(2;2);(2;4);(3;1);(3;3);(3;4);(4;2);(4;3);(4;4)g;
U1=f(1;2);(1;4);(2;1);(2;3);(3;2);(4;1)g:
All the missing preference are calculated under the use of (5.5)-(5.15),in this step
to complete the given IFPR. Hence, the fuzzy preference relation R1against expert
70
E1is obtained as follows:
R1=2
66666640:5 0 :6139 0 :4 0 :4074
0:3861 0 :5 0 :5651 0 :7
0:6 0 :4349 0 :5 0 :2
0:5926 0 : 0:8 0 :53
7777775:
Similarly, the complete forms of R2andR3can be obtained and given as:
R2=2
66666640:5 0 :55 0 :6 0 :2
0:45 0 :5 0 :4 0:4056
0:4 0 :6 0 :5 0 :6
0:8 0:5944 0 :4 0 :53
7777775;
R3=2
66666640:5 0 :6278 0 :9 0 :7074
0:3722 0 :5 0 :5698 0 :9
0:1 0 :4302 0 :5 0 :4862
0:2926 0 :1 0 :5138 0 :53
7777775: (5.35)
Step-ii: Consistency analysis
Consistency analysis is being conducted to allocate consistency weights to the
experts. For this purpose, all complete FPRs are to be converted into their T-
consistent forms by using (5.16), and are given below:
eR1=2
66666640:5 0 :6139 0 :4 0 :4074
0:3861 0 :5 0 :5651 0 :7
0:6 0 :4349 0 :5 0 :2
0:5926 0 :3 0 :8 0 :53
7777775;
eR2=2
66666640:5 0 :55 0 :6 0 :2
0:45 0 :5 0 :4 0:4056
0:4 0 :6 0 :5 0 :6
0:8 0:5940 0 :4 0 :53
7777775,
71
eR3=2
66666640:5 0 :6278 0 :9 0 :7074
0:3722 0 :5 0 :5698 0 :9
0:1 0 :4302 0 :5 0 :4862
0:2926 0 :1 0 :5138 0 :53
7777775.
The signi
cant TCI values of the experts are evaluated using (5.22)-(5.24), as:
TCI (R1) = 1 ; TCI (R2) = 1 ;andTCI (R3) = 1 :
Finally, the consistency weights to the experts are computed by using (5.25), as:
Cw(E1) = 1 =3; Cw (E2) = 1 =3;andCw(E3) = 1 =3:
Step-iii: Weights to experts
Primarily, all experts are assigned the same trust weights: tw(E1) = 1 ; tw(E2) = 1
andtw(E3) = 1 . Therefore, the weights of the experts remain same in the
rst
round as the consistency weights based on (5.26), as:
w(E1) = 1 =3; w(E2) = 1 =3;andw(E3) = 1 =3:
Step-iv: Consensus measures
After getting complete FPRs, a mutual similarity relation is computed by ag-
gregating the di¤erent similarity matrices among the experts using (5.27)-(5.28).
Then, the consensus measures are computed at the three levels using (5.29)-(5.31).
1.On pair of alternatives :
CoD =2
66666641 0 :9481 0 :8987 0 :8075
0:9481 1 0 :8868 0 :8704
0:8987 0 :8868 1 0 :7333
0:8075 0 :8704 0 :7333 13
7777775:
2.On alternatives :
CoD (x1) = 0 :8848; CoD (x2) = 0 :9018; CoD (x3) = 0 :8396; CoD (x4) = 0 :8037:
72
3.On relations :
CoD (Rq) = 0 :8575:
Now, the threshold consensus degree settled in advance is compared with
global consensus degree CoD of the relations; CoD (Rq)> : This indicates that
the given consensus degree is acceptable amongst the experts, and we have to
enter into accumulation phase.
Step-v: Accumulation phase
In this phase, we constructed the order consistency matrices eR1
;eR2
,eR3
fromeR1;
eR2;eR3using (5.33) against the experts E1; E2andE3respectively, as:
eR1
=2
66666640:5 0 :4713 0 :5233 0 :4661
0:5287 0 :5 0 :5520 0 :4948
0:4767 0 :4480 0 :5 0 :4428
0:5339 0 :5052 0 :5572 0 :53
7777775;
eR2
=2
66666640:5 0 :5118 0 :4688 0 :4445
0:4882 0 :5 0 :4570 0 :4328
0:5312 0 :5430 0 :5 0 :4757
0:5555 0 :5672 0 :5243 0 :53
7777775,
eR3
=2
66666640:5 0 :5492 0 :6524 0 :6661
0:4509 0 :5 0 :6032 0 :6170
0:3476 0 :3968 0 :5 0 :5138
0:3339 0 :3830 0 :4862 0 :53
7777775.
Here one can easily check that eR1
;eR2
andeR3
are consistency matrices (i.e., eR1
=
eeR1
;eR2
=eeR2
andeR3
=eeR3
), therefore, there is no need to repeat (5.16). The
collective matrix RCagainst all experts is obtained uder the use of (5.34), as:
RC=2
66666640:5 0 :5108 0 :5482 0 :5256
0:4892 0 :5 0 :5374 0 :5149
0:4518 0 :4626 0 :5 0 :4774
0:4744 0 :4851 0 :5226 0 :53
7777775: (5.36)
73
Step-vi: Selection phase
The relation RCobtained in (5.36) is clearly a T-consistent with T= max(erij+
erjk 1;0)i.e.,erikmax(erij+erjk 1;0);and order consistent. Order consistency
means thaterikerilfor all i2 f1;2;3;4g;where k2 f1;2;3;4gandl2 f1;2;3;4g.
Aseri1eri2;eri2eri4anderi4eri3for all i2 f1;2;3;4g;hence, the ranking order
of the alternatives x1; x2; x3andx4isx1> x 2> x 4> x 3.
If (5.21) is applied to estimate the ranking value RV(xi)of alternative xi,
1i4, we have:
RV(x1) =2
424X
j=1rc
1j=1
8[0:5000 + 0 :5108 + 0 :5482 + 0 :5256] = 0 :260575 ;
RV(x2) =2
424X
j=1rc
2j=1
8[0:4893 + 0 :5000 + 0 :5374 + 0 :5149] = 0 :255188 ;
RV(x3) =2
424X
j=1rc
3j=1
8[0:4518 + 0 :4626 + 0 :5000 + 0 :4774] = 0 :236475 ;
RV(x4) =2
424X
j=1rc
4j=1
8[0:4744 + 0 :4851 + 0 :5226 + 0 :5000] = 0 :247762 :
AsRV(x1)> R V(x2)> R V(x4)> R V(x3), therefore, ranking order of alternatives
x1; x2; x3andx4is:x1> x 2> x 4> x 3, where the best option is x1as obtained
by order consistency.
Conclusion
In this chapter, a new consensus based technique for GDM dealing with IFPRs
was proposed. The T-transitivity property was applied to measur the missing
preferences, being a generalized form of transitivity under the triangular norm
T.TCI was de
ned to reach the level of the consistency of every expert. The
consistency weights of the experts were derived based on the consistency analysis.
Reasonably, the experts with high consistency and good trust level should be
allocated large weights, in order for their consistent opinions to have more weights
in the accumulation phase.After getting collective FPR by all the experts, it is
converted into consistency matrix ( T-consistent and Order-consistent) to rank
74
the alternatives. Some numerical examples were given to anticipate the proposed
method.
75
Chapter 6
Group Decision Making by using Incomplete
Interval -valued Fuzzy Preference Relations
The results presented in this chapter have been partially published in [6].
In this chapter, we extend the procedure given in previous chapter to GDM in
IIVFPRs environment based on the extended t-norm Te. Obviously, the consistent
information is more applicable or important than the information having ambigu-
ities, consistency is linked with de
nite transitivity properties. Several properties
have been endorsed to model transitivity of FPRs, one of these properties is the
T-transitivity. A method based on Te-consistency is proposed to determine un-
known interval-valued preferences of one alternative over others and further, it is
extended to develop an algorithm for GDM to select the best alternative.
As we have already mentioned in previous chapter, GDM is a situation faced
when a number of experts work together to
nd the best alternative(s) from a set of
feasible alternatives. Each expert may have exclusive inspirations or objectives and
a di¤erent decision procedure, but has a common interest in approaching to select
the "best" option(s). Experts compare the alternatives pair-wise and express their
outcomes in form of preference relations, FPRs are the most popular, by assigning
a numerical value to every pair of alternatives.
In a decision making procedure, an expert mostly needs to compare a
nite
set of alternatives xi(i= 1;2; :::; n )and construct an FPR [55, 114, 141, 147].
However, an expert may have imprecise information for the preference degrees of
one alternative over another and it may not always be possible to estimate his/her
preference by means of an exact numerical value. In such a situation, an expert
constructs an IVFPR.
In 2004, Xu de
ned the notion of compatibility degree of two IVFPRs and
showed the compatible connection among individual and collective IVFPRs [158].
In 2005, Herrera et al. established an aggregation process for combining IVFPRs
with other forms of information as; numerical preference relation (NPR) and lin-
guistic preference relation (LPR) [71]. In 2007, Jiang proposed a technique to
measure the similarity degree of two IVFPRs and used the error-propagation rule
to
nd the priority vector of the accumulated IVFPRs [80]. In 2008, Xu and Chen
developed some linear programming models to derive the priority weights from
several IVFPRs [160].
All the above researches focused on the IVFPRs with complete information.
77
However, in decision making problems such situations are unavoidable in which
an expert does not have comprehensive information of the problem because of
time constraint, lack of knowledge and the expert s limited expertise within the
problem domain [3, 36, 54, 96, 97, 159, 161, 162]. Consequently, the expert may
not be able to give his/her opinion about speci
c traits of the problem, and hence
an incomplete preference relation would be constructed, missing information in
GDM environment is a problem that we have to cope with because usual decision-
making measures assume that experts are able to provide preference degrees after
pair-wise comparison of alternatives, which is not always possible. In literature,
researches based on IFPRs have been given, but there are only few researches in
GDM related to incomplete IVFPRs [163].
InChapter 2 , the de
nitions ( De
nition 2.23 andDe
nition 2.24 ) of IVFS Aand
IVFPR R= (rij)nnon a universe Xhave already been explained. However, the
following formulae can be used to perform arithmatic operations for all P; Q2
L([0;1])(P= [p ; p+]andQ= [q ; q+]) [106]:
P+Q= [p +q ; p++q+],
P Q= [p q+; p+ q ],
PQ= [min( p q ; p q+; p+q ; p+q+);max( p q ; p q+; p+q ; p+q+)],
P=Q = [p ; p+][1
q+;1
q ]if0=2[q ; q+].
De
nition 6.1. Inclusion and Equality of IVFSs : Let Xbe a universe and A
andBtwo interval-valued fuzzy sets. The inclusion of AintoBis de
ned as:
ABif and only if A(a)B(a)for all a2Xand the equality between Aand
Bis de
ned as: A=Bif and only if A(a) =B(a)for all a2X[40] .
The concept of a t-norm on [0;1]can be extended to subintervals of [0;1].
De
nition 6.2. An extended t-norm, Te, is an increasing, commutative, associa-
tive and L([0;1])L([0;1])!L([0;1])mapping that satis
es:
Te([1;1];[x ; x+]) = [ x ; x+]for all [x ; x+]2L([0;1]):
78
LetTbe a triangular norm. The mapping Tede
ned as:
Te([a ; a+];[b ; b+]) = [ T(a ; b ); T(a+; b+)]
for[a ; a+],[b ; b+]2L([0;1]);is an extended t-norm on (L([0;1]);), where
represents the crisp set inclusion. The extended interval t-norm corresponding to
theLukasiewicz operator can be computed by:
Tw([a ; a+];[b ; b+]) = [max( a +b 1;0);max( a++b+ 1;0)]: (6.1)
De
nition 6.3. An IVFPR Ris said to be Te-consistent, if for i; jandkbelonging
tof1;2;3; :::; ngit holds :
rikTe(rij;rjk); (6.2)
where r
ikT(r
ij; r
jk)andr+
ikT(r+
ij; r+
jk).
De
nition 6.4. An IVFPR relation R= (rij)nnis said to be incomplete if it
contains at least one unknown preference value rijfor which the expert has no
idea about the degree of preference of alternative xiover the alternative xj:
6.1 Method to Repair an IIVFPR
This section includes the extended technique to estimate missing values in an
IIVFPR. Further, the algorithm is used to construct a Te-consistent matrix. In
order to estimate missing preference degrees in an IIVFPR R= (rij)nn, the
pairs of alternatives for known and unknown preference values are represented in
form of sets KPandUPin the same mode as (5.2) and (5.3) in Chapter 5 . But
here, the preference value of alternative xioverxjbelongs to the family of closed
subintervals of [0;1](i.e., rij2L([0;1])). Since rij= [1;1] rji;rii= [0:5;0:5]for
1inand 1jn, therefore, (6.2) can be written as:
rikTe(rij;rjk);rikTe([1;1] rji;rjk);rikTe(rij;[1;1] rkj): (6.3)
79
Hence, the following sets can be de
ned to determine the unknown preference
value rikof alternative xiover alternative xk:
S1
ik=fjj(i; j)2KP;(j; k)2KPand (i; k)2UPg; (6.4)
S2
ik=fjj(j; i)2KP;(j; k)2KPand (i; k)2UPg; (6.5)
S3
ik=fjj(i; j)2KP;(k; j)2KPand (i; k)2UPg; (6.6)
fori=f1;2;3; :::; ng,j=f1;2;3; :::; ngandk=f1;2;3; :::; ng. Based on
(6.4),(6.5) and (6.6), we can determine the unknown preference value rikforxi
overxkas follows:
rik=r1
ik+r2
ik+r3
ik
3; (6.7)
where
r1
ik=8
><
>:1
jS1
ikjP
j2S1
ikTe(rij;rjk); ifjS1
ikj 6= 0
[0:5;0:5]; otherwise(6.8)
r2
ik=8
><
>:1
jS2
ikjP
j2S2
ikTe([1;1] rji;rjk);ifjS2
ikj 6= 0
[0:5;0:5]; otherwise(6.9)
r3
ik=8
><
>:1
jS3
ikjP
j2S3
ikTe(rij;[1;1] rkj);ifjS3
ikj 6= 0
[0:5;0:5]; otherwise(6.10)
where jS1
ikj;jS2
ikjandjS3
ikjare the cardinalities of the sets S1
ik; S2
ikandS3
ikrespec-
tively. New sets of the pairs of alternatives for known and unknown preference
values are evaluated in the same way as (5.12) and (5.13).
To accomplish additive reciprocity, following scaling conditions will be used:
Scaling conditions: Ifr
ij+r+
ji71andr+
ij+r
ji?
r
ij="
r
ij+1 (r
ij+r+
ji)
2; r+
ij+1 (r+
ij+r
ji)
2#
(6.11)
and
r
ji="
r
ji+1 (r+
ij+r
ji)
2; r+
ji+1 (r
ij+r+
ji)
2#
: (6.12)
80
Remark 6.1. Throughout this chapter, we will consider exended Lukasiewicz
t-norm TL(rij;rjk) = [max( r
ij+r
jk 1;0);max( r+
ij+r+
jk 1;0)]as particular
triangular norm to illustrate the numerical examples.
Example 6.1. LetR= (rij)44be an IIVFPR for the alternatives x1; x2; x3and
x4, given as follows:
R=2
6666664[0:5;0:5] r12 [0:4;0:6] [0 :3;0:7]
r21 [0:5;0:5] [0 :7;0:8] r24
[0:4;0:6] [0 :2;0:3] [0 :5;0:5] [0 :3;0:4]
[0:3;0:7] r42 [0:6;0:7] [0 :5;0:5]3
7777775
where r12;r21;r24andr42are unknown preference values. Now applying above
said procedure to estimate the unknown preference values for the alternative xi
overxk,1i4and 1k4, we obtain:
KP=f(1;1);(1;3);(1;4);(2;2);(2;3);(3;1);(3;2);(3;3);(3;4);(4;1);
(4;3);(4;4)g;
UP=f(1;2);(2;1);(2;4);(4;2)g:
S1
12=f3g; S2
12=f3g; S3
12=f3g;
r1
12=TL(r13;r32) = [0 ;0];r2
12=TL([1;1] r31;r32) = [0 ;0];
r3
12=TL(r13;[1;1] r23) = [0 :2;0:3];
r12=1
3(r1
12+r2
12+r3
12) = [0 :07;0:1]:
K0
P=f(1;1);(1;2);(1;3);(1;4);(2;2);(2;3);(3;1);(3;2);(3;3);(3;4);
(4;1);(4;3);(4;4)g;
U0
P=UP f(1;2)g=f(2;1);(2;4);(4;2)g:
S1
21=f3g; S2
21=f1;3g; S3
21=f2;3g;
r1
21=TL(r23;r31) = [0 :1;0:4];
r2
21=1
2[TL([1;1] r12;r11) +TL([1;1] r32;r31)] = [0 :25;0:415];
r3
21=1
2[TL(r22;[1;1] r12) +TL(r23;[1;1] r13)] = [0 :25;0:415];
r21=1
3(r1
21+r2
21+r3
21) = [0 :2;0:41]:
81
In similar way remaining missing preference values are estimated. Hence, the
complete IVFPR is
R=2
6666664[0:5;0:5] [0 :07;0:1] [0 :4;0:6] [0 :3;0:7]
[0:2;0:] [0 :5;0:5] [0 :7;0:8] [0 :03;0:24]
[0:4;0:6] [0 :2;0:3] [0 :5;0:5] [0 :3;0:4]
[0:3;0:7] [0 :058;0:159] [0 :6;0:7] [0 :5;0:5]3
7777775(6.13)
By (6.11)-(6.12) on (6.13), Rbecomes a TL-consistent IVFPReRas follows:
eR=2
6666664[0:5;0:5] [0 :33;0:45] [0 :4;0:6] [0 :3;0:7]
[0:55;0:67] [0 :5;0:5] [0 :7;0:8] [0 :44;0:59]
[0:4;0:6] [0 :2;0:3] [0 :5;0:5] [0 :3;0:4]
[0:3;0:7] [0 :41;0:56] [0 :6;0:7] [0 :5;0:5]3
7777775:
In next section, we move towards our main assignment of using the above con-
structed results to GDM while IIVFPRs will be used. A new step-by-step process
will be presented for GDM based on Te-transitivity, for ease, the structure of esti-
mation process is also shown in Fig. 6.1. At the end of next section, an explanatory
example is given to validate the proposed method.
6.2 IIVFPRs Based Group Decision Making
Suppose that there are nalternatives x1; x2; :::; x nandmexperts E1; E2; :::; E m.
LetRqbe the IVFPR for the expert Eqshown as follows:
Rq=
rq
ij
nn=2
6666666664[0:5;0:5] rq
12 : : rq
1n
rq
21 [0:5;0:5]: : rq
2n
: : : :
: : : :
rq
n1 rq
n2 : : [0:5;0:5]3
7777777775;
where rq
ij2L([0;1])is the preference value given by expert Eqfor alternative xi
over xj,rq
ij= [1;1] rq
ji;rq
ii= [0:5;0:5];1in;1jnand 1qm.
82
The proposed GDM technique consists of several stages which clearly explain the
whole process:
6.2.1 Determine Unknown Preferences
To estimate the unknown preference values of an IIVFPR Rqput forward by
the expert Eq, initially, the sets KqandUqof pairs of alternatives for known
and unknown preference values are made as in (5.2) and (5.3). After this, the Te-
transitivity based preference values are estimated by using (6.3)-(6.10) to construct
the complete IVFPR Rq.
6.2.2 Consistency Measures
After geting the complete forms of all IIVFPRs, their corresponding Te-consistent
IVFPRseRq
;1qmcan also be obtained by using following closure formula
which is based on extended t-norm Te:
eRq
= (erq
ij)nn=
Sup
k6=i;j(rq
ij; Te(rq
ik;rq
kj)
nnwitherq
ij= [1;1] erq
ji: (6.14)
Now to estimate the consistency level of IVFPR given by the expert Eq, construct
the FPR Aq=
aq
ij
nnand the consistency matrix eAq=
eaq
ij
nnfor expert Eq
using average values as:
Aq=
aq
ij
nn=1
2
aq
ij+aq+
ij
nn(6.15)
eAq=
eaq
ij
nn=1
2
eaq
ij+eaq+
ij
nn(6.16)
where preference values aq
ijandeaq
ijfall in [0;1],1in;1jnand
1qm. Now, to estimate the consistency degree of an FPR Aqbased on its
similarity with the corresponding T-transitive FPR eAq, we use three levels (5.22)-
(5.24) of T-consistency index TCI. Hence, consistency weights can be assigned to
the experts by using (5.25) i.e., Cw(Eq) =TCI (Aq)
mX
q=1TCI (Aq):
83
6.2.3 Assigning Weights to Experts
As we have already mentioned in Chapter 5 that each expert Eqcan also enjoy the
trust weight tw(Eq)from others such thatmX
q=1tw(Eq) =m. The larger an expert s
trust weight, the greater the degree to which the expert is trusted by others.
Therefore, it makes sense to allocate higher weights to experts having larger trust
weights, so that convincing estimations can have more weight in the aggregation
process. Initially, each expert is assigned the same trust weight tw(Eq), therefore,
We assign weights to experts by combining their trust weights and consistency
weights using the relation given in (5.26) withmX
q=1w(Eq) = 1 .
6.2.4 Collective IVFPR
It has previously mentioned that the preference value put forward by each expert
is weighted di¤erently. As soon as the weights of the experts have been obtained,
their opinions need to be aggregated into a collective opinion. To achieve the level,
determine the collective IVFPR RCagainst all experts, shown as follows:
RC=
rC
ij
nn= mX
q=1w(Eq)erq
ij!
nn; (6.17)
where 1in;1jnand 1qmwitheRq
=
erq
ij
nnbeing an
Te-consistent FPR.
6.2.5 Consensus Level
After getting the complete IVFPRs, it is necessary to measure the consensus
level amongst the experts. To achieve this, we use the concept of consensus
level proposed by X. Y. Zhang and Z. J. Wang in [170] i.e., if Rq=
rq
ij
nn=
([rq
ij; rq+
ij])nn(q= 1;2;3; :::; m )bemIVFPRs with 0< rq
ijrq+
ij<1, then the
consensus level between a single IVFPR Rq=
rq
ij
nn= ([rq
ij; rq+
ij])nnand a
collective one RC=
rC
ij
nn= ([rC
ij; rC+
ij])nnis de
ned as:
CL(Rq;RC) =2n(n 1)vuut
i6=j
min(rq
ij; rC
ij)
max( rq
ij; rC
ij)!
min(rq+
ij; rC+
ij)
max( rq+
ij; rC+
ij)!!
(6.18)
84
Obviously, the larger the value of CL(Rq;RC)is,the closer Rqis toRC, so that the
expert Eqhas a better consensus with the group. If CL(Rq;RC) = 1 , then IVFPR
Rq=
rq
ij
nnhas perfect consensus with the group one RC=
rC
ij
nn, but it is
always challenging to raech a full and undisputed agreement because of complexity
of real-world decison environment. To deal with real-world decision problems in
easier and faster way, settle acceptable value (threshold value) of consensus level
in advance, and if CL(Rq;RC)then Rq=
rq
ij
nnis acceptable for selection
process. Generally, it is very hard for a group of experts to have a full agreement,
therefore, we enforce the restriction: 0:5 < 1; because, mostly when less
than 50% experts accept the decision output, it is considered as a rejected result.
Convincingly, 2[0:5;1[is taken into account in real-world problems, and the
closer to1, the more validated the decision output. On the other hand, if
CL(Rq;RC)< , then use the formula
CL(rq
ij;rC
ij) =vuut
min(rq
ij; rC
ij)
max( rq
ij; rC
ij)!
min(rq+
ij; rC+
ij)
max( rq+
ij; rC+
ij)!
(6.19)
to estimate the consensus level CL(rq
ij;rC
ij)of each pair of corresponding prefer-
ences (rq
ij;rC
ij)inRqandRC, respectively [170]. Consequently, IVFPR Rqis being
returned with RCto the expert Eqto adjust several values which make little conti-
bution to the decision consensus, proceed in this way such that all CL(Rq;RC)
(q= 1;2;3; :::; m ):
6.2.6 Selection Phase
The selection procedure is initiated after reaching the acceptable consensus level,
its goal is to rank the alternatives to select the best one. This process contains
three di¤erent levels as:
1.Average degree of alternatives: To calculate the average degree A(xi)of
alternative xiover all other alternatives, apply interval normalizing formula:
A(xi) =nP
j=1rc
ij
nP
i=1nP
j=1rc
ij; i= 1;2;3; :::; n: (6.20)
85
2.Complementry matrix: Use the following formula introduce in [157] to
calculate the possibility degree dij=d(A(xi)A(xj));and then construct
the complementry matrix D= (dij)nn:
d(AiAj) = min(
max
A+
i A
j
A+
i A
j+A+
j A
i;0!
;1)
(6.21)
where dij0; dij+dji= 1; dii= 0:5; i; j = 1;2;3; :::; n:
3.Consistency matrix: To this end,
rstly, the complementry matrix D=
(dij)nnis to be converted into order consistency matrix by applying D=
dij
nn=
1
nnX
j=1T(dij; djk)!
nn(see Example 5.2 for procedure), and sec-
ondly, the order consistent matrix D=
dij
nnis transformed into T-
consistency matrixeD=edij
nn(if it is not) by repeated application of
(5.16). This will order the values (i.e.,edikedilfor all 1in;where
k2 f1;2;3; ; :::; n gandl2 f1;2;3; ; :::; n gm) which lead the ranking of al-
ternatives to select the best one.
The structure of estimation procedure is shown in following
gure.
Incomplete IVFPRsEstimatingProcedureConsistencyProcedureAggregationPhaseConsensusProcess
Best Option(s)(Solution)SelectionPhaseExperts Set
PossibilityDegreePhaseConsistentIVFPRsCollective IVFPRComplementary matrixComplete IVFPRs
Fig. 6.1. Structure of Resolution Process
86
6.3 Numerical Example
Example 6.2. A
rm produces solar water re
ners. In its production process,
the company has to buy solar panels in di¤erent sizes and voltages from di¤erent
suppliers. Presently, Japan Solar Company has four potential suppliers in four
di¤erent countries, namely, Korea, China, Italy and Turkey, signi
ed as xi(i=
1;2;3;4), respectively. A committee consisting of three experts Eq(q= 1;2;3)from
di¤erent departments has been formed to assess the four suppliers xi(i= 1;2;3;4).
Suppose that the experts Eq(q= 1;2;3)provide their assessments in the form of
following incomplete IVFPRs:
R1=2
6666664[0:5;0:5] r1
12 [0:6;0:8] [0 :4;0:6]
r1
21 [0:5;0:5] r1
23 [0:3;0:7]
[0:2;0:4] r1
32 [0:5;0:5] [0 :6;0:9]
[0:4;0:6] [0 :3;0:7] [0 :1;0:4] [0 :5;0:5]3
7777775;
R2=2
6666664[0:5;0:5] r12 [0:4;0:6] [0 :3;0:7]
r21 [0:5;0:5] [0 :7;0:8] r24
[0:4;0:6] [0 :2;0:3] [0 :5;0:5] [0 :3;0:4]
[0:3;0:7] r42 [0:6;0:7] [0 :5;0:5]3
7777775;
and
R3=2
6666664[0:5;0:5] r3
12 [0:6;0:8] r3
14
r3
21 [0:5;0:5] r3
23 [0:4;0:7]
[0:2;0:4] r3
32 [0:5;0:5] r3
34
r3
41 [0:3;0:6] r3
43 [0:5;0:5]3
7777775:
The threshold consensus level settled in advance is 0:80. Now, we evaluate the
result under the use of procedure explained above:
Step i: Estimation of unknown preferences
Initially, all the missing preference values need to be determined using Te-transitivity
(particularly TL-transitivity) property mentioned in Section 1.
Taking R1;for example. The sets of pairs of alternatives for known and un-
known preference values are determined as follows:
87
K1=f(1;1);(1;3);(2;2);(2;4);(3;1);(3;3);(3;4);(4;2);(4;3);(4;4)g;
U1=f(1;2);(1;4);(2;1);(2;3);(3;2);(4;1)g:
All the missing preference are calculated under the use of (6.4)-(6.10),in this step
to complete the given IIVFPR. Hence, the IVFPR R1against expert E1is obtained
as follows:
S11
12=f4g; S12
12=f4g; S13
12=f4g;
r11
12= [max( r
14+r
42 1;0);max( r+
14+r+
42 1;0)] = [0 ;0:3];
r12
12= [max( r
42 r+
41;0);max( r+
42+r
41;0)] = [0 ;0:3];
r13
12= [max( r
14 r+
24;0);max( r+
14 r
24;0)] = [0 ;0:3];
r1
12=1
3(r11
12+r12
12+r13
12) = [0 ;0:3]:
K10
P=f(1;1);(1;2);(1;3);(2;2);(2;4);(3;1);(3;3);(3;4);(4;2);(4;3);
(4;4)g;
U10
P=U1
P f(1;2)g=f(1;4);(2;1);(2;3);(3;2);(4;1)g:
Hence, continuing as above R1against expert E1is completed and given as:
R1=2
6666664[0:5;0:5] [0 ;0:3] [0 :6;0:8] [0 :4; :6]
[0:067;0:37] [0 :5;0:5] [0 :05;0:24] [0 :3;0:7]
[0:2;0:4] [0 :058;0:46] [0 :5;0:5] [0 :6;0:9]
[0:4;0:6] [0 :3;0:7] [0 :1;0:4] [0 :5;0:5]3
7777775;
using (6.11)-(6.12), R1satsi
es the additive reciprocity under the interval arith-
matic operation, and is as:
R1=2
6666664[0:5;0:5] [0 :32;0:62] [0 :6;0:8] [0 :4;0:6]
[0:38;0:68] [0 :5;0:5] [0 :295;0:591] [0 :3;0:7]
[0:2;0:4] [0 :409;0:705] [0 :5;0:5] [0 :6;0:9]
[0:4;0:6] [0 :3;0:7] [0 :1;0:4] [0 :5;0:5]3
7777775:
88
Likewise,IVFPRs R2andR3against the experts E2andE3are obtained:
R2=2
6666664[0:5;0:5] [0 :45;0:63] [0 :4;0:6] [0 :3;0:7]
[0:37;0:55] [0 :5;0:5] [0 :7;0:8] [0 :35;0:62]
[0:4;0:6] [0 :2;0:3] [0 :5;0:5] [0 :3;0:4]
[0:3;0:7] [0 :38;0:65] [0 :6;0:7] [0 :5;0:5]3
7777775,
R3=2
6666664[0:5;0:5] [0 :42;0:61] [0 :6;0:8] [0 :45;0:52]
[0:39;0:58] [0 :5;0:5] [0 :54;0:66] [0 :4;0:7]
[0:2;0:4] [0 :34;0:46] [0 :5;0:5] [0 :38;0:46]
[0:48;0:55] [0 :3;0:6] [0 :54;0:62] [0 :5;0:5]3
7777775.
Step ii: Consistency analysis
Consistency analysis is being conducted to allocate consistency weights to the
experts. For this purpose, all complete IVFPRs are to be converted into their
Te-consistent forms by using (6.14) and are given below, respectively:
eR1
=2
6666664[0:5;0:5] [0 :32;0:62] [0 :5;0:8] [0 :4;0:7]
[0:38;0:68] [0 :5;0:5] [0 :295;0:591] [0 :3;0:7]
[0:2;0:5] [0 :409;0:705] [0 :5;0:5] [0 :6;0:9]
[0:3;0:6] [0 :3;0:7] [0 :1;0:4] [0 :5;0:5]3
7777775;
eR2
=2
6666664[0:5;0:5] [0 :45;0:63] [0 :4;0:6] [0 :3;0:7]
[0:37;0:55] [0 :5;0:5] [0 :7;0:8] [0 :35;0:62]
[0:4;0:6] [0 :2;0:3] [0 :5;0:5] [0 :3;0:4]
[0:3;0:7] [0 :38;0:65] [0 :6;0:7] [0 :5;0:5]3
7777775,
eR3
=2
6666664[0:5;0:5] [0 :42;0:61] [0 :6;0:8] [0 :45;0:52]
[0:39;0:58] [0 :5;0:5] [0 :54;0:66] [0 :4;0:7]
[0:2;0:4] [0 :34;0:46] [0 :5;0:5] [0 :38;0:46]
[0:48;0:55] [0 :3;0:6] [0 :54;0:62] [0 :5;0:5]3
7777775.
To this end, we transform the IVFPRs RqandeRq
(q= 1;2;3)into corresponding
89
FPRs AqandeAq(q= 1;2;3)by using (6.15)-(6.16):
A1=2
66666640:5 0:47 0 :7 0 :5
0:53 0 :5 0:44 0 :5
0:3 0:56 0 :5 0:75
0:5 0 :5 0:25 0 :53
7777775andeA1=2
66666640:5 0:47 0 :65 0 :55
0:53 0 :5 0:44 0 :5
0:35 0 :56 0 :5 0:75
0:45 0 :5 0:25 0 :53
7777775;
A2=2
66666640:5 0:54 0 :5 0 :5
0:46 0 :5 0:75 0 :48
0:5 0:25 0 :5 0:35
0:5 0:52 0 :65 0 :53
7777775andeA2=2
66666640:5 0:54 0 :5 0 :5
0:46 0 :5 0:75 0 :48
0:5 0:25 0 :5 0:35
0:5 0:52 0 :65 0 :53
7777775;
A3=2
66666640:5 0:52 0 :7 0:48
0:48 0 :5 0 :6 0:55
0:3 0 :4 0 :5 0:42
0:52 0 :45 0 :58 0 :53
7777775andeA3=2
66666640:5 0:52 0 :7 0:48
0:48 0 :5 0 :6 0:55
0:3 0 :4 0 :5 0:42
0:52 0 :45 0 :58 0 :53
7777775;
where A2=eA2andA3=eA3which shows that A2andA3are fully consistent,
therefore, using (5.22)-(5.24) we can determine that TCI (A2) = 0 :9833; TCI (A2) =
1andTCI (A3) = 1 . Hence, the consistency weights to the experts are estimated
as:
Cw(E1) = 0 :3296; Cw (E2) = 0 :3352;andCw(E3) = 0 :3352
where3X
q=1Cw(Eq) = 1 .
Step iii: Assigning weights to experts
Primarily, all experts are assigned the same trust weights: tw(E1) = 1 ; tw(E2) = 1
andtw(E3) = 1 . Therefore, the weights of the experts remain same in the
rst
round as the consistency weights based on (5.26), as:
w(E1) = 0 :3296; w(E2) = 0 :3352;andw(E3) = 0 :3352
with3X
q=1w(Eq) = 1 .
Step iv: Collective IVFPR
90
The collective TL-consistent matrix against all the experts is obtained using (6.17):
RC=2
6666664[0:5;0:5] [0 :4;0:62] [0 :5;0:73] [0 :38;0:64]
[0:38;0:6] [0 :5;0:5] [0 :51;0:68] [0 :35;0:67]
[0:27;0:5] [0 :32;0:49] [0 :5;0:5] [0 :43;0:59]
[0:36;0:62] [0 :33;0:65] [0 :41;0:57] [0 :5;0:5]3
7777775
Step v: Consensus level
By utilizing equation (6.18), the consensus levels CL(Rq;RC) (q= 1;2;3)based
on the three individual IVFPRs and the collective IVFPR are computed as:
CL(R1;RC) = 0 :8125; CL (R2;RC) = 0 :8265; CL (R3;RC) = 0 :8824
clearly, CL(Rq;RC)> = 0:80 (q= 1;2;3), which shows that consensus degree is
reached at acceptable level amongst the experts. Now, we can enter into selection
phase.
Step vi: Selection process
Average degree of alternatives : The average degree A(xi);(i= 1;2;3;4)of each
alternative is derived by using (6.20), given as:
A(x1) =4P
j=1rc
1j
4P
i=14P
j=1rc
ij=[1:78;2:49]
[6:64;9:36]= [0:1902;0:3750];
A(x2) =4P
j=1rc
2j
nP
i=1nP
j=1rc
ij=[1:74;2:45]
[6:64;9:36]= [0:1859;0:3690];
A(x3) =4P
j=1rc
3j
nP
i=1nP
j=1rc
ij=[1:52;2:08]
[6:64;9:36]= [0:1624;0:3133];
A(x4) =4P
j=1rc
4j
nP
i=1nP
j=1rc
ij=[1:6;2:34]
[6:64;9:36]= [0:1709;0:3524] :
91
Complementry matrix : The complementry matrix D= (dij)44is obtained utiliz-
ing (6.21):
D= (dij)44=2
66666640:5 0 :5140 0 :6333 0 :5572
0:4860 0 :5 0 :6186 0 :5433
0:3667 0 :3814 0 :5 0 :4284
0:4428 0 :4567 0 :5716 0 :53
7777775:
Consistency matrix : In order to rank the alternatives xi;1i4;construct the
consistency matrixeD=edij
44using (5.16) and (5.33):
eD=edij
44=2
66666640:5 0 :5071 0 :5494 0 :5143
0:4929 0 :5 0 :5440 0 :5108
0:4506 0 :4560 0 :5 0 :4821
0:4857 0 :4892 0 :5179 0 :53
7777775:
The relationeD=edij
44is clearly a T-consistent with T= max(erij+erjk 1;0)
i.e.,erikmax(erij+erjk 1;0);and order consistent. Order consistency means
thaterikerilfor all i2 f1;2;3;4g;where k2 f1;2;3;4gandl2 f1;2;3;4g. As
eri1eri2;eri2eri4anderi4eri3for all i2 f1;2;3;4g;hence, the ranking order of
the alternatives x1; x2; x3andx4isx1> x 2> x 4> x 3:Therefore, x1is the best
alternative.
The numerical examples show the way to apply the proposed technique to
construct the complete IVFPR based on Tw-consistency. In general, the proposed
approach is quite easy for use in estimating unknown interval-valued preferences.
Conclusion
In this chapter, the extended t-norm was used successfully to determine the missing
values in incomplete IVFPRs and further extended to construct the T-consistent
matrix. Numerical studies show that the proposed technique can handle all type
of incomplete IVFPRs. Consequently, consistency and consensus based algorithm
was established to deal with GDM problems with incomplete IVFPRs. This
process involves several stages: the estimation of unknown interval-valued pref-
92
erences, measure the consistency level of each expert and estimate the consensus
degree amongst experts, accumulate the collective matrix, calculate the possibility
degree matrix, and the choice of the best alternative(s).
93
Chapter 7
Supplier Selection Based on T-transitive Fuzzy
Preference Relations
In this chapter, we continue to research GDM problems in IFPRs environment
and propose an improved procedure to estimate the missing values based on the
T-transitivity. The structure of criteria in AHP [127, 128], which has already given
the e¤ective results in several domains [30, 100, 112, 151, 153], is used for fuzzy
MCDM problems in consistency and consensus based environment. As consistency
is an important issue to accept when data is provided by the experts, the method
can estimate more reasonable and consistent values when an FPR carries missing
preferences. Consistency is associated with the transitivity property for which
several useful forms or conditions have been suggested in the literature of FPRs
[141]. The weakest of them is L-transitivity, i.e. rikmax( rij+rjk 1;0);and
it is the most appropriate notion of transitivity for fuzzy ordering [146], therefore,
the individual and collective FPRs obtained by this method are fully consistent
under the use of t-norm T.
This chapter is organized as follows; Section 1 presents a new procedure to
estimate the missing values in IFPRs based on T-transitivity. After completing
FPR, it is made consistent using (5.16) under T-transitivity, and to manage the
unacceptably consistent case some consistency measures are de
ned. In Section
2, we propose a new consensus and consistency based method for MCDM in an
incomplete fuzzy AHP environment. At the end, an example is used to illustrate
the proposed method for GDM in multi-person settings.
7.1 Estimation of Missing Values and Ranking
of Alternatives
In this section, a new procedure to estimate the missing preference values in an
IFPR to construct a complete T-consistent FPR based on T-transitivity is pre-
sented. Then some consistency measures are de
ned; consistency level of a pref-
erence value, consistency level of an alternative and the consistency level of FPR.
In order to determine the unknown preference values in an IFPR R= (rij)nn,
following sets can be de
ned to represent the pairs of alternatives for known and
95
unknown preference values in the same way as (5.2) and (5.3):
Kv =f(i; j)jrijis known g; (7.1)
Uv =f(i; j)jrijis unknown g; (7.2)
where rij2[0;1]shows the preference values of alternative xiover the alternative
xj,rij+rji= 1 for1inand 1jnand for simplicity it will be
noticed that rii= ;8i2 f1;2; :::; ngas an alternative cannot be compared
with itself. Therefore, the following set can be de
ned to estimate the unknown
preference value rikof alternative xiover alternative xkbased on T-transitivity
rikT(rij; rjk).
Qik=fj6=i; kj(i; j)2Kv;(j; k)2Kvand (i; k)2Uvg; (7.3)
fori; j; k2 f1;2;3; :::; ng. Based on eq. (7.3), rikis estimated by using:
rik=8
><
>:Sup
j2Qik(T(rij; rjk));ifjQikj 6= 0
0:5; otherwise; (7.4)
rki= 1 rik; (7.5)
where jQikjis the cardinality of the set Qik. Now, we de
ne two new sets K0vand
U0vas follows:
K0v=Kv[ f(i; k)g;andU0v=Uv f(i; k)g: (7.6)
Note:- After observing the FPR given by the expert, all those positions in Rfor
which rij2Kvandrji2Uvorrij2Uvandrji2Kvare to be
lled by using
additive reciprocity i.e., rij+rji= 1. For example, if an FPR is given as:
R=2
6666664 0:2 0:4 0:7
r21 0:8 0:6
0:6 0:2 r34
0:3 0:4 0:3 3
7777775
96
Here, r21andr34are the positions with unknown preference values which are
determined as follows:
r21= 1 r12= 1 0:2 = 0 :8;
r34= 1 r43= 1 0:3 = 0 :7:
After having a complete FPR R= (rij)nn, one can easily come to know that the
FPReR= (erij)nnisT-consistent, where:
eR= (erij)nn=
Sup
k6=i;j(rij; T(rij; rjk))
nnwitherij+erji= 1: (7.7)
Proposition 8.1. If there is a consistent FPR eR= (eaij)nn, then the ranking
value Rv(xi)of alternative xi,i= 1;2;3; ::; n, is de
ned by:
Rv(xi) =2
n(n 1)nX
eaij
j=1; j6=i; i= 1;2;3; :::; n
withnP
i=1Rv(xi) = 1 .
Proof. Let us take the consistent FPR eR= (eaij)nn, shown as follows:
eR= (eaij)nn=2
6666666664 er12: :er1n
er21 : :er2n
: : : :
: : : :
ern1ern2: : 3
7777777775
97
Now
nX
i=1nX
eaij
j=1; j6=i=nX
i=1(eai1+eai2+eai3+:::+eain)
= (ea21+ea31+ea41+:::+ean1) + (ea12+ea32+ea42+:::+ean2)
+(ea13+ea23+ea43+:::+ean3) + (ea14+ea24+ea34+:::+ean4)
+:::+ (ea1n+ea2n+ea3n+:::+ean(n 1))
= (ea12+ea21) + (ea13+ea31) + (ea14+ea41) +:::+ (ea1n+ean1)
+(ea23+ea32) + (ea24+ea42) + (ea25+ea52) +:::+ (ea1n+ean1)
+:::+ (ean(n 1)+ea(n 1)n)
= 1 + 1 + 1 + :::+ 1 (n(n 1)
2times ) =n(n 1)
2:
LetRv(xi) =cnPeaij
j=1; j6=iandnP
i=1Rv(xi) = 1 . Therefore, we have
nX
i=1Rv(xi) =cnX
i=1nX
eaij
j=1; j6=i= 1 =)c=1
nP
i=1nPeaij
j=1; j6=i=2
n(n 1):
Hence, Rv(xi) =2
n(n 1)nPeaij
j=1; j6=i:
7.1.1 Consistency Measures
The experts are not always able to provide the consistent information based on
the transitivity. If R= (rij)nnis a fully consistent relation, then it derives R=eR
whereeRis given in Eq. (7.7) :Hence,eRcan be used to judge the consistency level
of the FPR R= (rij)nn. Taking this into consideration, three levels (5.22)-(5.24)
are to be used to determine the consistency degree of the given FPR R= (rij)nn:
7.1.2 Consensus Measures
After having the FPRs with complete information, it is necessary to measure the
consensus among the experts. Regarding this, (5.27)-(5.31) have to use to measure
the consensus degree amongst the experts, if it is greater or equal to the prede
ned
level then we have to enter into accumulation phase, otherwise, experts have to
98
revise some preference values using (5.32) to reach the consensus.
7.2 GDM by Using Incomplete AHP
In real world, there are many decision-making processes which take place in multi-
person settings because the increase of complexity and uncertainty of the socio-
economics environment makes it less possible for a single expert to consider all
related traits of a decision-making problem. For an hierarchy problem with GDM,
let there be nalternatives x1; x2; :::; x nandmcriteria c1; c2; :::; c m;letE1; E2; :::; E l
be the experts having weight vector = ( 1; 2; 3; :::; l)T, where k0,k=
1;2;3; :::; l,lX
k=1 k= 1. The procedure for the incomplete AHP is given as follows:
7.2.1 Estimate the Criteria s Priority Weights
Step i: In the
rst step, the expert Ek(k= 1;2;3; :::; l )compares each pair of
criteria and gives his/her judgments in form of incomplete FPR R(k)= [r(k)
ij]mm
shown as follows:
R(k)= [r(k)
ij]mm=c1
c2
:
:
cmc1c2: : c m2
6666666664 r(k)
12: : r(k)
1m
r(k)
21 : : r(k)
2m
: : : :
: : : :
r(k)
m1r(k)
m2: : 3
7777777775;
where r(k)
ij2[0;1]is the preference value given by expert Ekfor criterion ciover
cj,r(k)
ij+r(k)
ji= 1;1im;1jmand 1kl.
Step ii: Utilize (7.3)-(7.7) to estimate the unknown preferences in R(k), and
construct the consistent FPR eR(k)= [er(k)
ij]mm.
Step iii: After constructing consistent FPRs, the level of consistency against each
FPR over the criteria is to be measured using (5.22)-(5.24). Normally, if its value
is greater than 0:5then relation is said to be consistent by certain intensity and
is fully consistent if level is 1. Now using (5.27)-(5.31), consensus degree amongst
the experts over criteria is to be estimated.
99
Step iv: Determine the collective matrixc
Ragainst all experts, shown as follows:
c
R=hcriji
mm="lX
k=1 ker(k)
ij#
mm; (7.8)
where 1im;1jmand 1kl:
Step v: Determine the priority weight of criteria by using the ranking aggregation
formula:
RV(ci) =2
m(m 1)mX
i=1;i6=jcrij; (7.9)
where 1imandmX
i=1RV(ci) = 1 .
7.2.2 Priority Ratings of Each Alternative Regarding to
Each Criterion
Step i: The expert Ek(k= 1;2;3; :::; l )compares each pair of alternatives with
respect to each criterion q(q= 1;2;3:::; m )and gives his/her judgments in the
form of incomplete FPR qA(k)= [qa(k)
uv]nn:
Step ii: Utilize (7.3)-(7.7) to estimate the unknown preferences in qA(k), and
construct the consistent FPR qeA(k)= [qea(k)
uv]nn.
Step iii: The consistency degree for FPRs over each criterion is to be measured us-
ing (5.22)-(5.24), and using (5.27)-(5.31) consensus level among the expert against
each criterion is measured.
Step iv: Determine the collective matrix qAcagainst all experts with respect to
each criterion, shown as follows:
qAc= [qac
ij]nn="lX
k=1 k
qea(k)
uv#
nn; (7.10)
where 1un;1vnand 1kl:
Step v: Determine the priority ratings for each alternative with respect to each
criterion by using the ranking aggregation formula:
qRV(xu) =2
n(n 1)nX
v=1;v6=u(qac
uv); (7.11)
100
where 1unandnX
u=1(qRV(xu)) = 1 .
7.2.3 Priority Weight of Each Alternative
To determine the priority weight of each alternative, multiply the matrix of priority
ratings of each alternative with respect to each criterion to the column vector
of criteria s priority weight. Let Abe the matrix of ratings of alternatives with
respect to each criterion arranged column-wise and wbe the priority weight vector
of criteria, then the priority weight pof each alternative is:
p=Aw: (7.12)
Hence, to apply the proposed method, we consider the following fuzzy MCDM
problem in an IFPRs environment.
Example A high-tech engineering company needs to select a suitable material
supplier to buy the key components of new products. After initial selection, four
candidates ( x1; x2; x3; x4) remain for further evaluation. A committee of three ex-
perts ( E1; E2; E3) whose weight vector is = (1=3;1=3;1=3)T, has been formed to
select the most suitable supplier. Five bene
t criteria are considered: (1) Technical
abilities and leadership ( c1); (2) Social responsibility ( c2); (3) Competitive pricing
(c3); (4) Quality and safety ( c4); (5) Delivery ( c5). The threshold consensus.level
settled in advance for each criterion is 0:75.
The hierarchy structure is shown in Fig.7.1 regarding to this decision problem.
Technicalabilities andleadershipSocialResponsibilityCompetitivepricingQuality &SafetyDeliveryGoal
x1x2x3x4
101
Fig. 7.1. Hierarchical structure of the decision problem
Incomplete FPRs for
ve criteria given the three experts:
R(1)=2
6666666664 0:6r1
130:4 0:7
0:4 r1
230:6 0:7
r1
31r1
32 0:3 0:4
0:6 0:4 0:7 r1
45
0:6 0:3 0:6r1
54 3
7777777775;R(2)=2
6666666664 0:4r2
130:6r2
15
0:6 0:6 0:5r2
25
r2
310:4 0:3 0:7
0:4 0:5 0:7 0:2
r2
51r2
520:3 0:8 3
7777777775;
R(3)=2
6666666664 0:3 0:4 0:7 0:7
0:7 r3
23r3
24r3
25
0:6r3
32 0:2 0:8
0:3r3
420:8 0:3
0:3r3
520:2 0:7 3
7777777775:
Incomplete FPRs for four suppliers with respect to each criterion given by the
three experts:
Technical abilities and leadership:
1A(1)=2
6666664 0:81a1
13 1a1
14
0:2 0:4 0 :6
1a1
310:6 0:7
1a1
410:4 0:3 3
7777775;1A(2)=2
6666664 0:7 1a2
130:6
0:3 0:5 0 :7
1a2
310:5 1a2
34
0:4 0:31a2
43 3
7777775;
1A(3)=2
6666664 1a3
120:7 1a3
14
1a3
21 1a3
230:6
0:3 1a3
32 0:5
1a3
410:4 0 :5 3
7777775:
102
Social responsibility:
2A(1)=2
6666664 0:7 0:52a1
14
0:3 0:4 0:6
0:5 0:6 0:8
2a1
410:4 0:2 3
7777775;2A(2)=2
6666664 0:82a2
13 2a2
14
0:2 0:3 0 :5
2a2
310:7 0:3
2a2
410:5 0:7 3
7777775;
2A(3)=2
6666664 0:92a3
130:6
0:1 0:4 0 :4
2a3
310:6 2a3
34
0:4 0:62a3
43 3
7777775:
Competitive pricing:
3A(1)=2
6666664 0:6 3a1
130:7
0:4 0:3 3a1
24
3a1
310:7 0:4
0:3 3a1
420:6 3
7777775;3A(2)=2
6666664 3a2
12 3a2
130:6
3a2
21 3a2
230:2
3a2
31 3a2
32 0:4
0:4 0 :8 0 :6 3
7777775;
3A(3)=2
6666664 3a3
120:3 0 :5
3a3
21 3a3
230:3
0:7 3a3
32 3a3
34
0:5 0 :7 3a3
43 3
7777775:
Quality and safety:
4A(1)=2
6666664 0:2 4a1
130:7
0:8 4a1
23 4a1
24
4a1
31 4a1
32 4a1
34
0:3 4a1
42 4a1
43 3
7777775;4A(2)=2
6666664 0:4 0:74a2
14
0:6 0:74a2
24
0:3 0 :3 0:6
4a2
41 4a2
420:4 3
7777775;
4A(3)=2
6666664 4a3
120:3 0 :4
4a3
21 0:2 0 :6
0:7 0 :8 4a3
34
0:6 0 :4 4a3
43 3
7777775:
103
Delivery:
5A(1)=2
6666664 5a1
12 5a1
130:1
5a1
21 0:4 5a1
24
5a1
310:6 0:7
0:9 5a1
420:3 3
7777775;5A(2)=2
6666664 0:2 5a2
130:4
0:8 5a2
230:3
5a2
31 5a2
32 0:3
5a2
410:7 0 :7 3
7777775;
5A(3)=2
6666664 0:3 0 :4 5a3
14
0:7 0:6 5a3
24
0:6 0 :4 5a3
34
5a3
41 5a3
42 5a3
43 3
7777775:
Utilization of the proposed procedure results in the collective FPR for
ve criteria
and respective priority weights as shown in Table 7.1 while the consistency level
and consensus degree for the experts are as
Cl(R1) = 1; Cl(R2) = 0 :98;Cl(R3) = 0 :95;andCoD (Rk) = 0 :7933; k= 1;2;3:
All the collective FPRs for four suppliers with respect to each criterion and respec-
tive priority ratings are shown in Table 7.2, and consistency level of each expert
in each criterion with consensus degree amongst experts are estimated as
Technical abilities and leadership :
Cl(1A1) = 1; Cl(1A2) = 1; Cl(1A3) = 1 ;andCoD (1Ak) = 0 :8189; k= 1;2;3:
Social responsibility :
Cl(2A1) = 1; Cl(2A2) = 1; Cl(2A3) = 1 ;andCoD (2Ak) = 0 :8111; k= 1;2;3:
Competitive pricing :
Cl(3A1) = 1; Cl(3A2) = 1; Cl(3A3) = 1 ;andCoD (3Ak) = 0 :8445; k= 1;2;3:
104
Quality and safety :
Cl(4A1) = 1; Cl(4A2) = 1; Cl(4A3) = 1 ;andCoD (4Ak) = 0 :7933; k= 1;2;3:
Delivery :
Cl(5A1) = 1; Cl(5A2) = 1; Cl(5A3) = 1 ;andCoD (5Ak) = 0 :7877; k= 1;2;3:
The
nal priority weights of four suppliers are given in the last column of
Table 7.3, which are p1= 0:2060; p2= 0:2071; p3= 0:2896 andp4= 0:2973. As
p4> p 3> p 2> p 1;therefore, ranking order of four suppliers x1; x2; x3andx4is:
x4> x 3> x 2> x 1:
Table 7.1. Collective consistent FPR for
ve criteria and its priority weights
c1 c2 c3 c4 c5 Priority weights
c10.50000 0.4333 0.3667 0.5667 0.4667 0.1833
c20.5667 0.5000 0.3667 0.5000 0.4667 0.1900
c30.6333 0.6333 0.5000 0.4333 0.6333 0.2333
c40.4333 0.5000 0.5667 0.5000 0.2667 0.1767
c50.5333 0.5333 0.3667 0.7333 0.5000 0.2167
105
Table 7.2. Collective consistent FPRs of four suppliers with respect to each
criterion and their priority ratings
s1 s2 s3 s4 Priority ratings
c1
s10.5000 0.6667 0.3667 0.4000 0.2389
s20.3333 0.5000 0.3667 0.6333 0.2222
s30.6333 0.6333 0.5000 0.4333 0.2833
s40.6000 0.3667 0.5667 0.5000 0.2556
c2
s10.5000 0.8000 0.3000 0.4000 0.2500
s20.2000 0.5000 0.3667 0.5000 0.1778
s30.7000 0.6333 0.5000 0.4667 0.3000
s40.6000 0.5000 0.5333 0.5000 0.2722
c3
s10.5000 0.4000 0.2667 0.6000 0.2111
s20.6000 0.5000 0.1333 0.2000 0.1556
s30.7333 0.8667 0.5000 0.3333 0.3222
s40.4000 0.8000 0.6667 0.5000 0.3111
c4
s10.5000 0.2333 0.5000 0.4667 0.2000
s20.7667 0.5000 0.4000 0.4667 0.2722
s30.5000 0.6000 0.5000 0.4000 0.2500
s40.5333 0.5333 0.6000 0.5000 0.2778
c5
s10.5000 0.3333 0.1667 0.3333 0.1389
s20.6667 0.5000 0.5000 0.1667 0.2222
s30.8333 0.5000 0.5000 0.3667 0.2833
s40.6667 0.8333 0.6333 0.5000 0.3556
106
Table 7.3. Final priority weights of four suppliers
c1 c2 c3 c4 c5 Priority weights
Criteria weights 0.1833 0.1900 0.2333 0.1767 0.2167
x1 0.2389 0.2500 0.2111 0.2000 0.1389 0.2060
x2 0.2222 0.1778 0.1556 0.2722 0.2222 0.2071
x3 0.2833 0.3000 0.3222 0.2500 0.2833 0.2896
x4 0.2556 0.2722 0.3111 0.2778 0.3556 0.2973
Conclusion
In this chapter, a simple and e¢ cient technique for selection and rating the sup-
pliers is presented. The proposed procedure utilizes the AHP for criteria and
employs consistent fuzzy FPRs for supplier evaluation. The proposed method
has three working steps: Firstly, it evaluates the priority weight vector of criteria
based consistency and consensus levels; Secondly, it evaluates the priority ratings
of each alternative against each criterion based consistency and consensus levels;
Thirdly, it
nds the priority weight vector for alternatives. At the end of this
study, authors successfully applied the proposed procedure to select the suitable
supplier in SCM by illustrating a numerical example.
107
Chapter 8
Conclusion
The work presented in this thesis has three major parts:
Part I . Initially, we study the mathematical structures representing orderings or
rankings in fuzzy contexts. In this part, we focus on the notion of t-norms and
their role in de
ning fuzzy transitivity. We have presented some work on repre-
sentation theorems of fuzzy orderings. After discussing the various forms of fuzzy
transitivity used in preference modeling so far we focus on the most recent ones
that have been explored very less. Initially, we discuss the mathematical formula-
tions of T-transitivity and the role played by them in consistency issues in fuzzy
preference relations. In this context, we make rigorous use of theory presented
in [88, 89, 90, 91] to convert the de
nition of measure of transitivity into simpler
inequalities. The additive and multiplicative generators of t-norms and t-conorms
play a prime role for these inequalities. The results are interpreted in terms of
fuzzy preference relations and it is observed that the measure of transitivity con-
ditions ensure consistency among pair wise preference data.
Part II . Next, we have proposed a new technique for group decision making
dealing with incomplete fuzzy preference relations. This technique is based on
T-consistency and order consistency. We determine unknown preference values
based on T-consistency and then construct the consistency matrix which satis
es
theT-consistency and the order consistency simultaneously for aggregation. This
technique can handle the ignorance situations of fuzzy preference relations which
do not satisfy the T-consistency or order consistency. It can be used for the in-
complete fuzzy preference relations which are not T-consistent or order consistent,
and it can overcome the drawbacks of previous techniques.
After handling the task of simple FPRs we turn towards the extended t- norms
and extended fuzzy preference relations i.e., the interval valued fuzzy preference
relations. The extended minimum t-norm has been used successfully to deter-
mine the missing values in incomplete IVFPR and further extends to construct
the min-consistent matrix. Numerical studies show that the proposed technique
can handle all type of incomplete IVFPR. Consequently, another algorithm is es-
tablished to deal with GDM problems with incomplete IVFPRs. This process
involves two stages, the estimation of unknown interval-valued preference values
and the choice of the best alternative(s).
109
Part III . In third major part of our work we have modi
ed the technique to
estimate the missing preferences, established in previous section to a practical sit-
uation in Supply Chain Management Systems i.e., the Supplier selection problem.
In this part, a simple and e¢ cient technique, based on T-consistency, for selection
and rating the suppliers is presented. The proposed procedure utilizes the AHP
hierarchy for criteria and employs consistent fuzzy FPRs for supplier evaluation.
The proposed method has three working steps: Firstly, it evaluates the priority
weight vector of criteria based consistency and consensus levels; Secondly, it evalu-
ates the priority ratings of each alternative against each criterion based consistency
and consensus levels; Thirdly, it
nds the priority weight vector for alternatives.
At the end of this study, authors successfully applied the proposed procedure to
select the suitable supplier in SCM by illustrating a numerical example.
In nutshell the thesis has covered the T-transitive fuzzy orderings from theo-
retical and computational points of views and as a major part of work has demon-
strated the application of theory and algorithms presented in a real life scenario
i.e., in supplier selection problem.
Future Directions
At the end of this work we suggest that the thesis has opened new research ques-
tions for the future researchers.
1. The theory algorithms presented may be applied to other situations other
than supplier selection problems.
2. The thesis remained restricted to additive reciprocity the results may be
formulated for multiplicative reciprocity.
3. Research may be conducted to remove some conditions for example the reci-
procity may be removed and alternative softer conditions may be explored.
110
Chapter 9
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