12516084
Normalizing rank aggregation method for priority of a fuzzy preference relation and its effectiveness
The aim of this paper is to show that the normalizing rank aggregation method can not only be used to derive the priority vector for a multiplicative preference relation, but also for the additive transitive fuzzy preference relation. To do so, a simple functional equation between fuzzy preference's element and priority weight is derived firstly, then, based on the equation, three methods are proposed to prove that the normalizing rank aggregation method is simple and effective for deriving the priority vector. Finally, a numerical example is used to illustrate the proposed methods.
124731955
The Analytic Hierarchy Process
This chapter provides an overview of Analytic Hierarchy Process (AHP), which is a systematic procedure for representing the elements of any problem hierarchically. It organizes the basic rationality by breaking down a problem into its smaller constituent parts and then guides decision makers through a series of pair-wise comparison judgments to express the relative strength or intensity of impact of the elements in the hierarchy. These judgments are then translated to numbers. The AHP includes procedures and principles used to synthesize the many judgments to derive priorities among criteria and subsequently for alternative solutions. It is useful to note that the numbers thus obtained are ratio scale estimates and correspond to so-called hard numbers. Problem solving is a process of setting priorities in steps. One step decides on the most important elements of a problem, another on how best to repair, replace, test, and evaluate the elements, and another on how to implement the solution and measure performance.
14539228
Integrating multiplicative preference relations in a multipurpose decision-making model based on fuzzy preference relations
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.
121062942
Eigenweight vectors and least-distance approximation for revealed preference in pairwise weight ratios
A new eigenweight vector is derived for the data of pairwise weight ratios. The well-known eigenweight vector derived by Saaty is then compared and contrasted in the light of least-distance approximation models. It is shown that the new eigenweight vector commands advantages over Saaty's, including less rigid assumptions on the error terms, robustness of solution, in addition to the fact that the new eigenweight vector can be computed very easily. The reader can construct other types of eigenweight vectors and least-distance approximation models using the framework of this article.
10488002
A goal programming approach to group decision making based on multiplicative preference relations and fuzzy preference relations
Abstract This paper proposes a goal programming approach to solve group decision-making (GDM) problems where the preference information on alternatives provided by decision makers is represented in two different formats, i.e. multiplicative preference relations and fuzzy preference relations. In order to narrow the gap between the collective opinion and each decision maker’s opinion, a linear goal programming model is constructed to integrate the two different formats of preference relations and to compute the collective ranking values of the alternatives. Thus, the ranking of alternatives or selection of the most desirable alternative(s) is obtained directly from the computed collective ranking values. A numerical example is also used to illustrate the applicability of the proposed approach.
10555463
Multiperson decision-making based on multiplicative preference relations
Abstract A multiperson decision-making problem, where the information about the alternatives provided by the experts can be presented by means of different preference representation structures (preference orderings, utility functions and multiplicative preference relations) is studied. Assuming the multiplicative preference relation as the uniform element of the preference representation, a multiplicative decision model based on fuzzy majority is presented to choose the best alternatives. In this decision model, several transformation functions are obtained to relate preference orderings and utility functions with multiplicative preference relations. The decision model uses the ordered weighted geometric operator to aggregate information and two choice degrees to rank the alternatives, quantifier guided dominance degree and quantifier guided non-dominance degree. The consistency of the model is analysed to prove that it acts coherently.
31994980
Interval estimation of priorities in the AHP
This paper extends and modifies the Analytic Hierarchy Process (AHP) and the Synthetic Hierarchy Method (SHM) of priority estimation to accommodate random data in the pairwise comparison matrices. It employs a Cauchy distribution to describe the pairwise comparison of alternatives in Saaty matrices, and shows how to modify these matrices in order to handle random data. The use of random data yields Saaty matrices that are not reciprocally symmetrical. Several variants of the AHP are then modified (i) to accommodate reciprocally asymmetric matrices, and (ii) to allow each priority estimate to be expressed on an interval of possible values, rather than as a single discrete point. The merits of interval estimation are illustrated by an example.
17593088
Prioritization of organizational capital measurement indicators using fuzzy AHP
Organizational capital is a sub-dimension of the intellectual capital which is the sum of all assets that make the creative ability of the organization possible. To control and manage such an important force, the companies must measure it first. This study aims at defining a methodology to improve the quality of prioritization of organizational capital measurement indicators under uncertain conditions. To do so, a methodology based on the extent fuzzy analytic hierarchy process (AHP) is applied. Within the model, three main attributes; deployment of the strategic values, investment to the technology and flexibility of the structure; their sub-attributes and 10 indicators are defined. To define the priority of each indicator, preferences of experts are gathered using a pair-wise comparison based questionnaire. The results of the study show that ''deployment of the strategic values'' is the most important attribute of the organizational capital.
14721296
A linear goal programming priority method for fuzzy analytic hierarchy process and its applications in new product screening
Fuzzy analytic hierarchy process (AHP) has been widely used for a variety of applications such as supplier selection, customer requirements assessment and the like. The vast majority of the applications, however, were found avoiding the use of sophisticated approaches for fuzzy AHP such as fuzzy least squares method while using a simple extent analysis for the sake of simplicity. The extent analysis proves to be incorrect and may lead to a wrong decision being made. This paper proposes a sound yet simple priority method for fuzzy AHP which utilizes a linear goal programming (LGP) model to derive normalized fuzzy weights for fuzzy pairwise comparison matrices. The proposed LGP priority method is tested with three numerical examples including an application of fuzzy AHP to new product development (NDP) project screening decision making.
619463
Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations
Abstract The purpose of this paper is to study a fuzzy multipurpose decision making problem, where the information about the alternatives provided by the experts can be of a diverse nature. the information can be represented by means of preference orderings, utility functions and fuzzy preference relations, and our objective is to establish a general model which cover all possible representations. Firstly, we must make the information uniform, using fuzzy preference relations as uniform preference context. Secondly, we present some selection processes for multiple preference relations based on the concept of fuzzy majority. Fuzzy majority is represented by a fuzzy quantifier, and applied in the aggregation, by means of an OWA operator whose weights are calculated by the fuzzy quantifier. We use two quantifier guided choice degrees of alternatives, a dominance degree used to quantify the dominance that one alternative has over all the others, in a fuzzy majority sense, and a non dominance degree, that generalises Orlovski's non dominated alternative concept. The application of the two above choice degrees can be carried out according to two different selection processes, a sequential selection process and a conjunction selection process.
9621926
A Sequential Selection Process in Group Decision Making with a Linguistic Assessment Approach
In this paper, a sequential selection process in group decision making under linguistic assessments is presented, where a set of linguistic preference relations represents individuals preferences. A collective linguistic preference is obtained by means of a defined linguistic ordered weighted averaging operator whose weights are chosen according to the concept of fuzzy majority, specified by a fuzzy linguistic quantifier. Then we define the concepts of linguistic non-dominance, linguistic dominance, and strict dominance degrees as parts of the sequential selection process. The solution alternative(s) is obtained by applying these concepts.
154286442
Non-conventional Preference Relations in Decision Making
Normative theories of decision making under risk and under uncertainty.- Partial structures of preference.- A generalisation of probabilistic consistency: linearity conditions for valued preference relations.- Fuzzy preference relations in group decision making.- {P,Q,I} - preference structures.- Identifying noninferior decision alternatives based on fuzzy binary relations.- Effective convolutions of vector preference relations in decision making problems.- Choice functions associated with fuzzy preference relations.- Fuzzy possibility graphs and their application to ranking fuzzy numbers.- On measuring consensus in the setting of fuzzy preference relations.- Assumptions of individual preferences in the theory of voting procedures.
122697740
Approaches to collective decision making with fuzzy preference relations
Abstract Recent experimental studies show that the predictive accuracy of many of the solution concepts derived from the collective decision making theory leaves much to be desired. In a previous paper the author attempted to explain some of the inaccuracies in terms of the fuzzy indifference regions of the individuals participating in the voting game. This paper gives straightforward generalizations of the solutions concepts in terms of the fuzzy social or individual preference relations. It turns out that some of these new solution concepts cotain their nonfuzzy counterparts as subsets. Others, in turn, are subsets of their nonfuzzy counterparts. We also discuss a method of aggregating individual nonfuzzy preferences so as to get a fuzzy social preference relation and, furthermore, a nonfuzzy social choice set.
122848934
Fuzzy preference orderings in group decision making
Abstract In this paper, some use of fuzzy preference orderings in group decision making is discussed. First, fuzzy preference orderings are defined as fuzzy binary relations satisfying reciprocity and max-min transitivity. Then, particularly in the case where individual preferences are represented by utility functions (utility values), group fuzzy preference orderings of which fuzziness is caused by differences or diversity of individual opinions are defined. Those orderings might be useful for proceeding the group decision making process smoothly, in the same manner as the extended contributive rule method.
154286442
Non-conventional Preference Relations in Decision Making
Normative theories of decision making under risk and under uncertainty.- Partial structures of preference.- A generalisation of probabilistic consistency: linearity conditions for valued preference relations.- Fuzzy preference relations in group decision making.- {P,Q,I} - preference structures.- Identifying noninferior decision alternatives based on fuzzy binary relations.- Effective convolutions of vector preference relations in decision making problems.- Choice functions associated with fuzzy preference relations.- Fuzzy possibility graphs and their application to ranking fuzzy numbers.- On measuring consensus in the setting of fuzzy preference relations.- Assumptions of individual preferences in the theory of voting procedures.
140716562
On Group Decision Making under Fuzzy Preferences
In this paper some results on group decision making under fuzzy preferences are reviewed. Since preferences of men are often “fuzzy” and the direct interpersonal comparison of preferences is not easy, the above topic is inevitable and quite important. First we discuss how to represent fuzzy preferences. They might be represented by fuzzy choice sets, fuzzy binary relations or fuzzy utility functions. Secondly group decision making situations are classified according to the above levels in representing both individual and group preferences. Some methods are explained in each category.
30095328
Group decision analysis based on fuzzy preference relations: Logarithmic and geometric least squares methods
Logarithmic and geometric least squares methods (LLSM and GLSM) are, respectively, applied to deal with the group decision analysis problems with fuzzy preference relations, where multiplicative preference relations, if any, are transformed into fuzzy preference relations through proper transformation technique. Distance between any two fuzzy preference relations and the average distance from one fuzzy preference relation to all the others are defined and used to measure the relative importance of each fuzzy preference relation. A numerical example involving multiple fuzzy and multiplicative preference relations is examined using the proposed methods. It is shown that LLSM and GLSM provide two analytical and effective ways of modelling multiple fuzzy preference relations.
35415292
A chi-square method for obtaining a priority vector from multiplicative and fuzzy preference relations
Abstract Decision makers (DMs)’ preferences on decision alternatives are often characterized by multiplicative or fuzzy preference relations. This paper proposes a chi-square method (CSM) for obtaining a priority vector from multiplicative and fuzzy preference relations. The proposed CSM can be used to obtain a priority vector from either a multiplicative preference relation (i.e. a pairwise comparison matrix) or a fuzzy preference relation or a group of multiplicative preference relations or a group of fuzzy preference relations or their mixtures. Theorems and algorithm about the CSM are developed. Three numerical examples are examined to illustrate the applications of the CSM and its advantages.
6142906
Multiple attribute decision making based on fuzzy preference information on alternatives: Ranking and weighting
This paper investigates the multiple attribute decision making (MADM) problem with fuzzy preference information on alternatives and proposes an eigenvector method to rank them. Three optimization models are introduced to assess the relative importance weights of attributes in a MADM problem, which integrate subjective fuzzy preference relations and objective information in different ways. Since the decisions these models lead to are based on both subjective and objective information, they are more balanced and credible than those that consider only one type of information. An example is provided to illustrate the application of the proposed models.
23206750
A least deviation method to obtain a priority vector of a fuzzy preference relation
Based on the transfer relationship between fuzzy preference relation and multiplicative preference relation, this paper proposes a least deviation method to obtain a priority vector of a fuzzy preference relation and studies its desirable properties. A simple convergent iterative algorithm is given and its practicality is shown by two numerical examples.
124183540
Fuzzy Analytical Hierarchy Process
Firstly the paper pointed out the defects of AHP. Then,the paper introduced the concept of fuzzy consistent judgement matrix,and studied the properties of fuzzy consistent judgement matrix and the rationality to denote the important comparision of elements by fuzzy consistent judgement matrix,and the relation between the fuzzy consistent judgement matrix denoting the important comparision and the weigtht denoting the level of importance of element. On the basis of the research,the paper gave the principle and procedure of fuzzy analytical hierarchy process.
117565292
Generalized chi square method for the estimation of weights
This paper presents a generalized chi square method for the estimation of weights in the analytic hierarchy process and studies its desirable properties such as invariance under transpose, harmony, and rank preservation. A convergent iterative algorithm is given and its practicality is shown by a numerical example.
7530712
Some issues on consistency of fuzzy preference relations
In decision making, in order to avoid misleading solutions, the study of consistency when the decision makers express their opinions by means of preference relations becomes a very important aspect in order to avoid misleading solutions. In decision making problems based on fuzzy preference relations the study of consistency is associated with the study of the transitivity property. In this paper, a new characterization of the consistency property defined by the additive transitivity property of the fuzzy preference relations is presented. Using this new characterization a method for constructing consistent fuzzy preference relations from a set of n � 1 preference data is proposed. Applying this method it is possible to assure better consistency of the fuzzy preference relations provided by the decision makers, and in such a way, to avoid the inconsistent solutions in the decision making processes. Additionally, a similar study of consistency is developed for the case of multiplicative preference relations. 2002 Elsevier B.V. All rights reserved.
15818771
A Note on the Estimation of Missing Pairwise Preference Values: a Uninorm Consistency Based Method
Dealing with incomplete information is an important problem in decision making. In this paper, we present a short discussion on this topic and a new estimation method of missing values in an incomplete fuzzy preference relation which is based on the modelling of consistency of preferences via a representable uninorm.
13943613
Group Decision-Making Model With Incomplete Fuzzy Preference Relations Based on Additive Consistency
In decision-making problems there may be cases in which experts do not have an in-depth knowledge of the problem to be solved. In such cases, experts may not put their opinion forward about certain aspects of the problem, and as a result they may present incomplete preferences, i.e., some preference values may not be given or may be missing. In this paper, we present a new model for group decision making in which experts' preferences can be expressed as incomplete fuzzy preference relations. As part of this decision model, we propose an iterative procedure to estimate the missing information in an expert's incomplete fuzzy preference relation. This procedure is guided by the additive-consistency (AC) property and only uses the preference values the expert provides. The AC property is also used to measure the level of consistency of the information provided by the experts and also to propose a new induced ordered weighted averaging (IOWA) operator, the AC-IOWA operator, which permits the aggregation of the experts' preferences in such a way that more importance is given to the most consistent ones. Finally, the selection of the solution set of alternatives according to the fuzzy majority of the experts is based on two quantifier-guided choice degrees: the dominance and the nondominance degree
1114617
Goal programming models for obtaining the priority vector of incomplete fuzzy preference relation
In this paper, we first define the concepts of incomplete fuzzy preference relation, additive consistent incomplete fuzzy preference relation and multiplicative consistent incomplete fuzzy preference relation, and then propose two goal programming models, based on additive consistent incomplete fuzzy preference relation and multiplicative consistent incomplete fuzzy preference relation respectively, for obtaining the priority vector of incomplete fuzzy preference relation. These two goal programming models are also extended to obtain the collective priority vector of several incomplete fuzzy preference relations. Finally, two illustrative numerical examples are given to verify the developed models.
156961274
Incomplete Complementary Judgement Matrix
The concepts of incomplete complementary judgement matrix, triangle consistent incomplete complementary judgement matrix, weak consistent incomplete complementary judgement matrix, additive consistent incomplete complementary judgement matrix, multiplicative consistent incomplete complementary judgement matrix and acceptable incomplete complementary judgement matrix, etc., are defined. Some of the properties of incomplete complementary judgement matrix are studied, and a priority method of incomplete complementary judgement matrix is also presented. Finally, an analysis is given to the situation where the judgement information about the alternatives is completely unknown.