JURNAL OF DECISION MAKING MODEL

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.

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.