Integrating
multiplicative preference relations in a multipurpose decision-making model
based on fuzzy preference relations
F. Chiclana, F.
Herrera ∗, E. Herrera-Viedma
Department of Computer Science and
Arti cial Intelligence, University of Granada, 18071 –Granada, Spain
Received 23 July
1998; received in revised form 8 October 1999; accepted 8 December 1999
Abstract
The aim of this paper is to study
the integration of multiplicative preference relation as a preference
representation structure in fuzzy multipurpose decision-makingproblems.
Assumingfuzzy multipurpose decision-makingproblems under di4erent preference
representation structures (ordering, utilities and fuzzy preference relations)
and using the fuzzy preference relations as uniform representation elements,
the multiplicative preference relations are incorporated in the decision
problem by means of a transformation function between multiplicative and fuzzy
preference relations. A consistency study of this transformation function,
which demonstrates that it does not change the informative content of
multiplicative preference relation, is shown. As a consequence, a selection
process based on fuzzy majority for multipurpose decision-makingproblems under
multiplicative preference relations is presented. To design it, an aggregation
operator of information, called ordered weighted geometric operator, is
introduced, and two choice degrees, the quanti7er-guided dominance degree and
the quanti7er-guided non-dominance degree, are de7ned for multiplicative
preference relations. c 2001 Elsevier Science B.V. All rights reserved.
Keywords: Multipurpose
decision-making; Fuzzy preference relations; Multiplicative preference relations;
Fuzzy majority; Selection process
1.
Introduction
Decision
makingin situations with multiple criteria and=or persons is a prominent area
of research in normative decision theory. This topic has been widely studied
[2,8,11,21,23]. We do not distinguish between “persons” and “criteria”, and
interpret the decision process in the fuzzy framework of mul-tipurpose
decision-making(MPDM) [4], assuming that the fuzzy property of human decisions
can be satisfactorily modeled by fuzzy sets theory as in [8,11,13,14]. In an
MPDM problem, we have a set of alternatives to be analyzed accordingto di4erent
purposes in order to select the best one(s). For each purpose a set of
evaluations about the alternatives is known. Then, a classical choice scheme
for an MPDM problem follows two steps before it achieves a 7nal decision
[4,6,19]: “aggregation” and “exploitation”. The aggregation phase de7nes an
(outranking) relation which indicates the global preference between every
ordered pair of alternatives, takinginto consideration the di4erent purposes.
The exploitation phase transforms the global information about the alternatives
into a global rankingof them. This can be done in di4erent ways, the most
common one beingthe use of a ranking method to obtain a score function [18]. In
[4], we consider MPDM problems where, for each purpose (expert or criterion),
the information about the alternatives could be supplied in di4erent ways. With
a view to build a more Iexible framework and to give more freedom degree to
represent the evaluations, we assumed that they could be provided in any of
these three ways: (i) as a preference ordering of the alternatives, (ii) as a
fuzzy preference relation and (iii) as a utility function. There we presented a
decision process to deal with this decision situation which, before applyingthe
classical choice scheme, made the information uniform, usingfuzzy preference
relations as the main element of the uniform representation of the evaluations,
and then obtained the solution by means of a selection process based on the
concept of fuzzy majority [10] and on the Ordered Weighted Averaging (OWA)
operator [30]. In this paper, we increase the Iexibility degree of our decision
model proposed in [4]. We give a new possibility for representingthe evalutions
about the alternatives, i.e., to use multiplicative preference relations. This
representation structure of evaluations has been widely used (see
[9,20–22,27,29]). In [20,21] Saaty designed a choice scheme, called Analytic
Hierarchy Process (AHP), for dealingwith decision problems where the
evaluations about the alternatives are provided by means of the multiplicative
preference relations. We incorporate the multiplicative preference relations in
our decision model presentinga transformation mechanism between multiplicative
and fuzzy preference relations and analyzingits consistency. Then, as a
consequence, we propose an alternative choice scheme to the classical one
designed by Saaty. Followingour selection process given in [4], we design a new
choice scheme using the concept of fuzzy majority and a new aggregation
operator, called ordered weighted geometric (OWG) operator. In order to do
this, the paper is set out as follows. The MPDM problem under four evaluation
structures is presented in Section 2. A transformation mechanism between
multiplicative and fuzzy preference relations is proposed in Section 3. Section
4 is devoted to presentingthe OWG operator and to design the new scheme choice
for dealingwith decision problems under multiplicative preference relations. In
Section 5 some concludingremarks are pointed out. Finally, the fuzzy majority
concept and the OWA operator are presented in Appendix A.
2. A multiplicative selection model based
on fuzzy majority
In
the AHP it is assumed that we have a set of m + 1 individual multiplicative
preference relations, {A1; A2;:::;Am; B}; where B is the importance matrix.
Followingthe scheme of the selection process given in Section 2, we present a
selection process based on fuzzy majority to choose the best alternatives from
multiplicative preference relations. With a view to design it, we introduce a
new aggregation operator guided by fuzzy majority in Section 4.1. In the
following subsections we show how to apply this aggregation operator to solve
the MPDM problem under multiplicative preference relations representingthe
experts’ preferences. 4.1. The ordered weighted geometric operator If we have a
set of m multiplicative preference relations, {A1; A2;:::;Am}; to be
aggregated, normally, the collective multiplicative preference relation, Ac ,
which expresses the opinion of the group, is derived by means of the geometric
mean, i.e., Ac = [ac ij]; ac ij = m k=1 (ak ij) 1=m: In this context, we can
de7ne the ordered weighted geometric (OWG) operator, which provides a family of
aggregators having the “and” operator at one extreme, the “or” operator at the
other extreme, and the geometric mean as a particular case. The (OWG) operator
is based on the OWA operator [30] and on the geometric mean, therefore, it is a
special case of OWA operator. It is applied in our selection process to
calculate a collective multiplicative preference relation and the
quanti7er-guided dominance and non-dominance choice degrees from multiplicative
preference relations.
3.
Concluding remarks
In
this paper, we have studied how to integrate the multiplicative preference
relations in fuzzy MPDM models under di4erent preference representation
structures (orderings, utilities and fuzzy preference relations). We have given
a consistent method using the fuzzy preference relations as uniform
representation element. This study together with our fuzzy MPDM model presented
in [4] provides a more Iexible framework to manage di4erent structures of
preferences, constitutingan approximate decision model to real decision
situations with experts of di4erent knowledge areas. Later, we have provided an
alternative choice process to the classical AHP for dealingwith MPDM problems
under multiplicative preference relations. The aim of the multiplicative
selection model is that it is based on fuzzy majority represented by a fuzzy
linguistic quanti7er. Futhermore, to design it, we have introduced a new
aggregation operator based on the OWA operators to aggregate multiplicative
preference relations, and have extended quanti7er-guided dominance and
non-dominance degrees to act with multiplicative preference relations.
A. Appendix: Fuzzy majority and OWA operator
The majority is traditionally
de7ned as a threshold number of individuals. Fuzzy majority is a soft majority
concept expressed by a fuzzy quanti7er, which is manipulated via a fuzzy
logic-based calculus of linguistically quanti7ed propositions. In this appendix
we present the fuzzy quanti7ers, used for representingthe fuzzy majority, and
the OWA operators, used for aggregating information. The OWA operator reIects
the fuzzy majority calculatingits weights by means of the fuzzy quanti7ers.
A.1. Fuzzy majority Quanti7ers can be used to represent the amount of items
satisfying a given predicate. Classic logic is restricted to the use of the two
quanti7ers, there exists and for all, that are closely related, respectively,
to the or and and connectives. Human discourse is much richer and more diverse
in its quanti7ers, e.g. about 5, almost all, a few, many, most, as many as
possible, nearly half, at least half. In an attempt to bridge the gap between
formal systems and natural discourse and, in turn, to provide a more Iexible
knowledge representation tool, Zadeh introduced the concept of fuzzy quanti7ers
[31]. Zadeh suggested that the semantic of a fuzzy quanti7er can be captured by
usingfuzzy subsets for its representation. He distinguished between two types
of fuzzy quanti7ers, absolute and relative. Absolute quanti7ers are used to
represent amounts that are absolute in nature such as about 2 or more than 5.
These absolute linguistic quanti7ers are closely related to the concept of the
count or number of elements. He
Pk (xi; xj) = pk ij denotes the preference degree or intensity of the alternative xi over xj [10,12,14,25]: pk ij = 1 2 indicates indi4erence between xi and xj, pk ij = 1 indicates that xi is absolutely preferred to xj, and pk ij¿1 2 indicates that xi is preferred to xj. In this case, the preference matrix, Pk , is assumed additive reciprocal, i.e., by de7nition [17,25] pk ij+ pk ji = 1 and pk ii = 1 2 . de7ned these quanti7ers as fuzzy subsets of the nonnegative real numbers, R+. In this approach, an absolute quanti7er can be represented by a fuzzy subset Q, such that for any r ∈ R+ the membership degree of r in Q, Q(r), indicates the degree to which the amount r is compatible with the quanti7er represented by Q. Relative quanti7ers, such as most, at least half, can be represented by fuzzy subsets of the unit interval, [0,1]. For any r ∈[0; 1], Q(r) indicates the degree to which the proportion r is compatible with the meaning of the quanti7er it represents. Any quanti7er of natural language can be represented as a relative quanti7er or given the cardinality of the elements considered, as an absolute quanti7er. Functionally, fuzzy quanti- 7ers are usually of one of three types, increasing, decreasing, and unimodal. An increasing-type quanti7er is characterized by the relationship Q(r1)¿Q(r2) if r1¿r2.
Pk (xi; xj) = pk ij denotes the preference degree or intensity of the alternative xi over xj [10,12,14,25]: pk ij = 1 2 indicates indi4erence between xi and xj, pk ij = 1 indicates that xi is absolutely preferred to xj, and pk ij¿1 2 indicates that xi is preferred to xj. In this case, the preference matrix, Pk , is assumed additive reciprocal, i.e., by de7nition [17,25] pk ij+ pk ji = 1 and pk ii = 1 2 . de7ned these quanti7ers as fuzzy subsets of the nonnegative real numbers, R+. In this approach, an absolute quanti7er can be represented by a fuzzy subset Q, such that for any r ∈ R+ the membership degree of r in Q, Q(r), indicates the degree to which the amount r is compatible with the quanti7er represented by Q. Relative quanti7ers, such as most, at least half, can be represented by fuzzy subsets of the unit interval, [0,1]. For any r ∈[0; 1], Q(r) indicates the degree to which the proportion r is compatible with the meaning of the quanti7er it represents. Any quanti7er of natural language can be represented as a relative quanti7er or given the cardinality of the elements considered, as an absolute quanti7er. Functionally, fuzzy quanti- 7ers are usually of one of three types, increasing, decreasing, and unimodal. An increasing-type quanti7er is characterized by the relationship Q(r1)¿Q(r2) if r1¿r2.
The OWA operators 7ll the gap between
the operators Min and Max. It can be immediately veri- 7ed that OWA operators
are commutative, increasing monotonous and idempotent, but in general not
associative. A natural question in the de7nition of the OWA operator is how to
obtain the associated weighting vector. In [30], Yager proposed two ways to
obtain it. The 7rst approach is to use some kind of learning mechanism
usingsome sample data; and the second approach is to try to give some semantics
or meaning to the weights. The 7nal possibility has allowed multiple
applications on areas of fuzzy and multi-valued logics, evidence theory, design
of fuzzy controllers, and the quanti7er-guided aggregations. We are interested
in the area of quanti7er-guided aggregations. Our idea is to calculate weights
for the aggregation operations (made by means of the OWA operator)
usinglinguistic quanti7ers that represent the concept of fuzzy majority. In
[30], Yager suggested an interestingway to compute the weights of the OWA
aggregation operator using fuzzy quanti7ers, which, in the case of a
non-decreasingrelative quanti7er Q, is given by the expression.
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Latar
belakang Model proses pengambilan keputusan
Beberapa atribut
pengambilan keputusan masalah yang dihadapi dalam berbagai situasi di mana
sejumlah alternatif dan tindakan atau calon harus dipilih berdasarkan satu set
atribut. Ketika kita mempertimbangkan satu set diskrit alternatif digambarkan
oleh beberapa atribut, ada tiga jenis analisis yang dapat dilakukan untuk memberikan
dukungan yang signifikan untuk pengambil keputusan:
• Pastikan bahwa pembuat keputusan
mengikuti "rasional" perilaku (pilihan normatif) - Fungsi nilai,
teori utilitas, jarak ke ideal.
• Berikan beberapa saran berdasarkan
akal (tapi tidak terbantahkan) aturan – Perancis Sekolah.
• Cari solusi yang lebih disukai dari
keputusan hipotesis parsial - metode interaktif. Analisis pengambilan keputusan
disiplin muncul, setelah ada nama hanya sejak Howard (1966). Sangat menarik
untuk merenungkan pandangan dari tiga pendiri analisis, karena setiap (; Keeney
dan Raiffa 1976 Howard, 1966) pengambilan keputusan. Membandingkan alternatif
adalah kunci untuk membuat keputusan. Namun, dalam kasus yang bertentangan alternatif,
pembuat keputusan juga harus mempertimbangkan data tidak tepat atau ambigu, yang
adalah norma dalam jenis masalah pengambilan keputusan.
Metode
Model proses pengambilan keputusan
Multi-atribut
metode pengambilan keputusan yang rawan dengan informasi tepat ditentukan. Salah
satu metode tersebut adalah metode COPRAS.
3.1. Metode COPRAS umum
1. Pemilihan set yang tersedia atribut
yang paling penting, yang menggambarkan alternatif.
2. Mempersiapkan dari pengambilan
keputusan matriks X
3. Menentukan bobot atribut qi (Kendall,
1970; Zavadskas, 1987).
4. Normalisasi dari? X matriks
pengambilan keputusan. Nilai-nilai normalisasi ini
Matriks
5. Perhitungan tertimbang dinormalisasi
pengambilan keputusan matriks? X. tertimbang
nilai normal xji
6. Menghitung jumlah Pj dari nilai-nilai
atribut, yang nilainya lebih besar
7. Menghitung jumlah Rj nilai atribut,
yang nilainya lebih kecil
8. Menentukan nilai minimal Rj
9. Menghitung bobot masing-masing Qj
alternatif
10. Menentukan kriteria optimalitas K
11. Menentukan prioritas proyek.
Kesimpulan
Model proses pengambilan keputusan
Dalam kehidupan
nyata pemodelan multi-atribut masalah penilaian multi-alternatif atribut
nilai-nilai, yang berkaitan dengan masa depan, dapat diungkapkan dalam
interval. COPRAS-G baru saja metode yang dikembangkan untuk penilaian
alternatif oleh multipleattribute nilai-nilai yang ditentukan dalam interval.
Pendekatan ini dimaksudkan untuk mendukung proses pengambilan keputusan dan
meningkatkan efisiensi proses penyelesaian. Metode COPRAS-G dapat diterapkan
untuk solusi dari berbagai multiattribute diskrit masalah penilaian dalam
konstruksi.