Managing incomplete preference relations in decision making: A review and future trends
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paper_title: A Cumulative Belief Degree-Based Approach for Missing Values in Nuclear Safeguards Evaluation

paper_content:

Nuclear safeguards are a set of activities to verify that a State is living up to its international undertakings not to use nuclear programs for nuclear weapons purposes. Nuclear safeguards experts of International Atomic Energy Agency (IAEA) evaluate indicators by benefitting from several sources such as State declarations, on-site inspections, the IAEA databases, and other open sources. The IAEA expert evaluations are aggregated to make a final decision, which usually incomplete because of over 900 indicators, lack of expertise, and unavailability of information sources. In this study, a cumulative belief degree approach is introduced based on the belief structure. It is used to aggregate the incomplete expert evaluations that are represented with fuzzy linguistic terms. Moreover, a reliability index is employed to find the trustworthiness of the final result depending on the available evaluations. A numerical example illustrates the applicability of the proposed methodology.
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paper_title: Individual and social strategies to deal with ignorance situations in multi-person decision making

paper_content:

Multi-person decision making problems involve the preferences of some experts about a set of alternatives in order to find the best one. However, sometimes experts might not possess a precise or sufficient level of knowledge of part of the problem and as a consequence that expert might not give all the information that is required. Indeed, this may be the case when the number of alternatives is high and experts are using fuzzy preference relations to represent their preferences. In the literature, incomplete information situations have been studied, and as a result, procedures that are able to compute the missing information of a preference relation have been designed. However, these approaches usually need at least a piece of information about every alternative in the problem in order to be successful in estimating all the missing preference values.In this paper, we address situations in which an expert does not provide any information about a particular alternative, which we call situations of total ignorance. We analyze several strategies to deal with these situations. We classify these strategies into: (i) individual strategies that can be applied to each individual preference relation without taking into account any information from the rest of experts and (ii) social strategies, that is, strategies that make use of the information available from the group of experts. Both individual and social strategies use extra assumptions or knowledge, which could not be directly instantiated in the experts preference relations. We also provide an analysis of the advantages and disadvantages of each one of the strategies presented, and the situations where some of them may be more adequate to be applied than the others.
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paper_title: When conflict induces the expression of incomplete preferences

paper_content:

Multicriteria conflict arises in pairwise comparisons, where each alternative outperforms the other one on some criterion, which imposes a trade-off. Comparing two alternatives can be difficult if their respective advantages are of high magnitude (the attribute spread is large). In this paper, we investigate to which extent conflict in a comparison situation can lead decision makers to express incomplete preferences, that is, to refuse to compare the two alternatives, or to be unable to compare them with confidence. We report on an experiment in which subjects expressed preferences on pairs of alternatives involving varying conflicts. Results show that depending on whether the participants are allowed to express incomplete preferences or not, attribute spread has a different effect: a large attribute spread increases the frequency of incomparability statements, when available, while it increases the use of indifference statements when only indifference and preference answers are permitted. These results lead us to derive some implications for preference elicitation methods involving comparison tasks.
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paper_title: An interactive procedure for multiple criteria group decision making with incomplete information

paper_content:

This paper presents an interactive procedure for multiple attribute group decision making (MAGDM) problem with incomplete information. Theoretic models are formally described for establishing group's pairwise dominance relations with group's utility range by using a separable linear programming technique.
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paper_title: A Learning Procedure to Estimate Missing Values in Fuzzy Preference Relations Based on Additive Consistency

paper_content:

In decision-making, information is usually provided by means of fuzzy preference relations. However, there may be cases in which experts do not have an in-depth knowledge of the problem to be solved, and thus their fuzzy preference relations may be incomplete, i.e. some values may not be given or may be missing. In this paper we present a procedure to find out the missing values of an incomplete fuzzy preference relation using the values known. We also define an expert consistency measure, based on additive consistency property. We show that our procedure to find out the missing values maintains the consistency of the original, incomplete fuzzy preference relation provided by the expert. Finally, to illustrate all this, an example of the procedure is presented.
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paper_title: A Consensus Model for Group Decision Making With Incomplete Fuzzy Preference Relations

paper_content:

Two processes are necessary to solve group decision making problems: A consensus process and a selection process. The consensus reaching process is necessary to obtain a final solution with a certain level of agreement between the experts; and the selection process is necessary to obtain such a final solution. In a previous paper, we present a selection process to deal with group decision making problems with incomplete fuzzy preference relations, which uses consistency measures to estimate the incomplete fuzzy preference relations. In this paper we present a consensus model. The main novelty of this consensus model is that of being guided by both consensus and consistency measures. Also, the consensus reaching process is guided automatically, without moderator, through both consensus and consistency criteria. To do that, a feedback mechanism is developed to generate advice on how experts should change or complete their preferences in order to reach a solution with high consensus and consistency degrees. In each consensus round, experts are given information on how to change their preferences, and to estimate missing values if their corresponding preference relation is incomplete. Additionally, a consensus and consistency based induced ordered weighted averaging operator to aggregate the experts' preferences is introduced, which can be used in consensus models as well as in selection processes. The main improvement of this consensus model is that it supports the management of incomplete information and it allows to achieve consistent solutions with a great level of agreement.
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paper_title: A Monte Carlo investigation of incomplete pairwise comparison matrices in AHP

paper_content:

The Analytic Hierarchy Process (AHP) is a decision analysis technique used to evaluate complex multiattribute alternatives among one or more decision makers. It imposes a hierarchical structure on any complex multicriterion problem. However, a major drawback of the AHP is that a large number of pairwise comparisons is needed to calibrate the hierarchy. When there are a few levels and sublevels, the AHP can be applied in a straightforward manner to derive the weights (relative preference for each alternative). As the size of the hierarchy increases, the number of pairwise comparisons increases rapidly. It is well established in the marketing and consumer behavior literature that in a very long interview, even under the best circumstances, the respondent is likely to suffer from information overload. Recognition of this problem was the motivation which led to the investigation of a modification of AHP which required less data collection, i.e., a reduction in the threat of information overload. The first question to be answered is the effect on AHP weights due to different patterns of missing data likely to result from reallife data collection. In this study, a Monte Carlo simulation was conducted, which uses the Incomplete Pairwise Comparisons (IPC) algorithm [14], to investigate the effect of reduced sets of pairwise comparisons in the AHP. Data for the study were generated with known structure and comparisons made between complete and incomplete matrices. The results of the simulation suggest that incomplete sets of pairwise comparison matrices can capture the attribute level weights without significant loss of accuracy and independent of decision model (form and amount of error) considered.
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paper_title: Fuzzy Preference Modelling and Multicriteria Decision Support

paper_content:

Note: Index. Bibliogr. : p. 239-251 Reference Record created on 2004-09-07, modified on 2016-08-08
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paper_title: Group Decision-Making Model With Incomplete Fuzzy Preference Relations Based on Additive Consistency

paper_content:

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
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paper_title: The Effectiveness of Alternative Preference Elicitation Methods in the Analytic Hierarchy Process

paper_content:

This study compares alternative preference elicitation methods that are currently available in software implementations of the analytic hierarchy process. For the simple problem used in this study, the elicitation methods may be sorted from least to most accurate as follows: (1) direct estimation, (2) graphical multiple bars, (3) graphical pairwise, (4) numeric pairwise and (5) verbal pairwise. However, the most accurate methods are not the easiest to use. Furthermore, actual accuracy differed substantially from users’ perceptions, indicating a need to assist the users in the selection of an appropriate elicitation method. © 1997 John Wiley & Sons, Ltd.
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paper_title: Incomplete Information, Inferences, and Individual Differences: The Case of Environmental Judgments

paper_content:

Abstract A model for inference of missing information is explicated and tested in two studies (Ns = 74, 76) of judgments about two environmental issues (endangered species reintroduction and the siting of a waste processing facility). Participants made judgments of scenarios in which information relevant to the judgment was varied orthogonally and, in some cases, relevant information was missing. The results showed individual differences—as well as intraindividual differences—in the assumptions participants made about missing information and in the tendency to infer missing cues. Reported assumptions about missing information predicted some aspects of the judgments. The data for only a small minority (15%) of the participants were consistent with the inferred values model. Participants may use different methods for dealing with missing information at different times or may not generally follow either an inference or averaging model in such contexts. Less favorable judgments were given for scenarios with incomplete information (the “penalty” effect), and this effect showed individual and intraindividual variation that was related to reported assumptions about missing information. We discuss the implications of these results for societal conflicts over controversial issues and for understanding the sources of individual differences in judgments.
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paper_title: A fuzzy relation space for group decision theory

paper_content:

In this paper reciprocal relations over n alternatives are defined, and their efficacy as a model of the group decision process is investigated. Several measures of individual preference and group consensus are defined and analyzed, and are used to generate associated measures of distance to consensus. Various decision oriented goals are identified, and we characterize these goals geometrically as subsets of a convex subset of a hyperplane in Rn. An application of this model for assessing degree of consensus is given.
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paper_title: Utility theory for decision making

paper_content:

Abstract : The book presents a concise yet mathematically complete treatment of modern utility theories that covers nonprobabilistic preference theory, the von Neumann-Morgenstern expected-utility theory and its extensions, and the joint axiomatization of utility and subjective probability.
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paper_title: The Effectiveness of Alternative Preference Elicitation Methods in the Analytic Hierarchy Process

paper_content:

This study compares alternative preference elicitation methods that are currently available in software implementations of the analytic hierarchy process. For the simple problem used in this study, the elicitation methods may be sorted from least to most accurate as follows: (1) direct estimation, (2) graphical multiple bars, (3) graphical pairwise, (4) numeric pairwise and (5) verbal pairwise. However, the most accurate methods are not the easiest to use. Furthermore, actual accuracy differed substantially from users’ perceptions, indicating a need to assist the users in the selection of an appropriate elicitation method. © 1997 John Wiley & Sons, Ltd.
---
paper_title: Utility theory for decision making

paper_content:

Abstract : The book presents a concise yet mathematically complete treatment of modern utility theories that covers nonprobabilistic preference theory, the von Neumann-Morgenstern expected-utility theory and its extensions, and the joint axiomatization of utility and subjective probability.
---
paper_title: A fuzzy relation space for group decision theory

paper_content:

In this paper reciprocal relations over n alternatives are defined, and their efficacy as a model of the group decision process is investigated. Several measures of individual preference and group consensus are defined and analyzed, and are used to generate associated measures of distance to consensus. Various decision oriented goals are identified, and we characterize these goals geometrically as subsets of a convex subset of a hyperplane in Rn. An application of this model for assessing degree of consensus is given.
---
paper_title: Fuzzy Preference Modelling and Multicriteria Decision Support

paper_content:

Note: Index. Bibliogr. : p. 239-251 Reference Record created on 2004-09-07, modified on 2016-08-08
---
paper_title: Integrating multiplicative preference relations in a multipurpose decision-makingmodel based on fuzzy preference relations

paper_content:

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.
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paper_title: Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions

paper_content:

(NOTE: Each chapter concludes with Exercises.) I: PRELIMINARIES. 1. Introduction. Rule-Based FLSs. A New Direction for FLSs. New Concepts and Their Historical Background. Fundamental Design Requirement. The Flow of Uncertainties. Existing Literature on Type-2 Fuzzy Sets. Coverage. Applicability Outside of Rule-Based FLSs. Computation. Supplementary Material: Short Primers on Fuzzy Sets and Fuzzy Logic. Primer on Fuzzy Sets. Primer on FL. Remarks. 2. Sources of Uncertainty. Uncertainties in a FLS. Words Mean Different Things to Different People. 3. Membership Functions and Uncertainty. Introduction. Type-1 Membership Functions. Type-2 Membership Functions. Returning to Linguistic Labels. Multivariable Membership Functions. Computation. 4. Case Studies. Introduction. Forecasting of Time-Series. Knowledge Mining Using Surveys. II: TYPE-1 FUZZY LOGIC SYSTEMS. 5. Singleton Type-1 Fuzzy Logic Systems: No Uncertainties. Introduction. Rules. Fuzzy Inference Engine. Fuzzification and Its Effect on Inference. Defuzzification. Possibilities. Fuzzy Basis Functions. FLSs Are Universal Approximators. Designing FLSs. Case Study: Forecasting of Time-Series. Case Study: Knowledge Mining Using Surveys. A Final Remark. Computation. 6. Non-Singleton Type-1 Fuzzy Logic Systems. Introduction. Fuzzification and Its Effect on Inference. Possibilities. FBFs. Non-Singleton FLSs Are Universal Approximators. Designing Non-Singleton FLSs. Case Study: Forecasting of Time-Series. A Final Remark. Computation. III: TYPE-2 FUZZY SETS. 7. Operations on and Properties of Type-2 Fuzzy Sets. Introduction. Extension Principle. Operations on General Type-2 Fuzzy Sets. Operations on Interval Type-2 Fuzzy Sets. Summary of Operations. Properties of Type-2 Fuzzy Sets. Computation. 8. Type-2 Relations and Compositions. Introduction. Relations in General. Relations and Compositions on the Same Product Space. Relations and Compositions on Different Product Spaces. Composition of a Set with a Relation. Cartesian Product of Fuzzy Sets. Implications. 9. Centroid of a Type-2 Fuzzy Set: Type-Reduction. Introduction. General Results for the Centroid. Generalized Centroid for Interval Type-2 Fuzzy Sets. Centroid of an Interval Type-2 Fuzzy Set. Type-Reduction: General Results. Type-Reduction: Interval Sets. Concluding Remark. Computation. IV: TYPE-2 FUZZY LOGIC SYSTEMS. 10. Singleton Type-2 Fuzzy Logic Systems. Introduction. Rules. Fuzzy Inference Engine. Fuzzification and Its Effect on Inference. Type-Reduction. Defuzzification. Possibilities. FBFs: The Lack Thereof. Interval Type-2 FLSs. Designing Interval Singleton Type-2 FLSs. Case Study: Forecasting of Time-Series. Case Study: Knowledge Mining Using Surveys. Computation. 11. Type-1 Non-Singleton Type-2 Fuzzy Logic Systems. Introduction. Fuzzification and Its Effect on Inference. Interval Type-1 Non-Singleton Type-2 FLSs. Designing Interval Type-1 Non-Singleton Type-2 FLSs. Case Study: Forecasting of Time-Series. Final Remark. Computation. 12. Type-2 Non-Singleton Type-2 Fuzzy Logic Systems. Introduction. Fuzzification and Its Effect on Inference. Interval Type-2 Non-Singleton Type-2 FLSs. Designing Interval Type-2 Non-Singleton Type-2 FLSs. Case Study: Forecasting of Time-Series. Computation. 13. TSK Fuzzy Logic Systems. Introduction. Type-1 TSK FLSs. Type-2 TSK FLSs. Example: Forecasting of Compressed Video Traffic. Final Remark. Computation. 14. Epilogue. Introduction. Type-2 Versus Type-1 FLSs. Appropriate Applications for a Type-2 FLS. Rule-Based Classification of Video Traffic. Equalization of Time-Varying Non-linear Digital Communication Channels. Overcoming CCI and ISI for Digital Communication Channels. Connection Admission Control for ATM Networks. Potential Application Areas for a Type-2 FLS. A. Join, Meet, and Negation Operations For Non-Interval Type-2 Fuzzy Sets. Introduction. Join Under Minimum or Product t-Norms. Meet Under Minimum t-Norm. Meet Under Product t-Norm. Negation. Computation. B. Properties of Type-1 and Type-2 Fuzzy Sets. Introduction. Type-1 Fuzzy Sets. Type-2 Fuzzy Sets. C. Computation. Type-1 FLSs. General Type-2 FLSs. Interval Type-2 FLSs. References. Index.
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paper_title: Non-dominance and attitudinal prioritisation methods for intuitionistic and interval-valued intuitionistic fuzzy preference relations

paper_content:

A novel intuitionistic fuzzy set (IFS) score function and an intuitionistic fuzzy preference relation (IFPR) quantifier guided non-dominance based prioritisation method are introduced. Based on Yager's continuous OWA (COWA) operator, the interval-valued intuitionistic fuzzy COWA (IVIF-COWA) operator is defined, and a new attitudinal expected score function for interval-valued intuitionistic fuzzy numbers (IVIFNs) is introduced. The novelty of this attitudinal expected score function is that it allows the comparison of IVIFNs by taking into account of the decision makers' attitudinal character. Moreover, we show that the new attitudinal expected score function extends: (i) the IFS score function introduced in this paper, which is mathematically equivalent to Chen and Tan's score function (Chen and Tan, 1994); and (ii) Xu and Chen's score function for IVIFNs (Xu and Chen, 2007). Using the proposed score functions, a method is developed to construct FPRs from a given IFPR and IVIFPR, respectively. When the hesitancy degree function is null, we prove that the score FPRs coincide with their respective IFPR and IVIFPR. Finally, a ranking sensitivity analysis of the attitudinal expected score function with respect to the attitudinal parameter is provided.
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paper_title: Intuitionistic fuzzy sets

paper_content:

A definition of the concept 'intuitionistic fuzzy set' (IFS) is given, the latter being a generalization of the concept 'fuzzy set' and an example is described. Various properties are proved, which are connected to the operations and relations over sets, and with modal and topological operators, defined over the set of IFS's.
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paper_title: A risk attitudinal ranking method for interval-valued intuitionistic fuzzy numbers based on novel attitudinal expected score and accuracy functions

paper_content:

This article investigates new score and accuracy functions for ranking interval-valued intuitionistic fuzzy numbers (IVIFNs). The novelty of these functions is that they allow the comparison of IVIFNs by taking into account of the decision makers' attitudinal character. The new attitudinal expected score and accuracy functions extend Xu and Chen's score and accuracy degree functions, and verify the following set of properties: (1) boundedness; (2) monotonicity; (3) commutativity; and (4) symmetry. These novel functions are used to propose a total order on the set of IVIFNs, and to develop an interval-valued intuitionistic fuzzy multi-attribute decision making selection process in which the final result depends on the decision maker's risk attitude. In addition, a ranking sensitivity analysis with respect to the risk attitude is provided.
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paper_title: Intuitionistic preference relations and their application in group decision making

paper_content:

Intuitionistic fuzzy set, characterized by a membership function and a non-membership function, was introduced by Atanassov [Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87-96]. In this paper, we define the concepts of intuitionistic preference relation, consistent intuitionistic preference relation, incomplete intuitionistic preference relation and acceptable intuitionistic preference relation, and study their various properties. We develop an approach to group decision making based on intuitionistic preference relations and an approach to group decision making based on incomplete intuitionistic preference relations respectively, in which the intuitionistic fuzzy arithmetic averaging operator and intuitionistic fuzzy weighted arithmetic averaging operator are used to aggregate intuitionistic preference information, and the score function and accuracy function are applied to the ranking and selection of alternatives. Finally, a practical example is provided to illustrate the developed approaches.
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paper_title: A web based consensus support system for group decision making problems and incomplete preferences

paper_content:

Reaching a high level of consensus among experts is critical in group decision making problems. Usually, it is the moderator task to assure that the consensus process is carried out properly and, if possible, to offer recommendations to the expert in order to change their opinions and narrow their differences. In this paper we present an implemented web based consensus support system that is able to help, or even replace, the moderator in a consensus process where experts are allowed to provide their preferences using one of many types (fuzzy, linguistic and multi-granular linguistic) of incomplete preference relations. This system is based on both consistency and consensus measures and it has been designed to provide advice to the experts to increase group consensus level while maintaining the individual consistency of each expert. The consistency measures are characterized by and computed using uninorm operators. When appropriate, the system also helps experts to reduce the incompleteness of their preference relations. The web interface allows to carry out distributed consensus processes and thus, experts do not necessarily need to physically meet together.
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paper_title: A linguistic consensus model for Web 2.0 communities

paper_content:

Web 2.0 communities are a quite recent phenomenon which involve large numbers of users and where communication between members is carried out in real time. Despite of those good characteristics, there is still a necessity of developing tools to help users to reach decisions with a high level of consensus in those new virtual environments. In this contribution a new consensus reaching model is presented which uses linguistic preferences and is designed to minimize the main problems that this kind of organization presents (low and intermittent participation rates, difficulty of establishing trust relations and so on) while incorporating the benefits that a Web 2.0 community offers (rich and diverse knowledge due to a large number of users, real-time communication, etc.). The model includes some delegation and feedback mechanisms to improve the speed of the process and its convergence towards a solution of consensus. Its possible application to some of the decision making processes that are carried out in the Wikipedia is also shown.
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paper_title: Individual and social strategies to deal with ignorance situations in multi-person decision making

paper_content:

Multi-person decision making problems involve the preferences of some experts about a set of alternatives in order to find the best one. However, sometimes experts might not possess a precise or sufficient level of knowledge of part of the problem and as a consequence that expert might not give all the information that is required. Indeed, this may be the case when the number of alternatives is high and experts are using fuzzy preference relations to represent their preferences. In the literature, incomplete information situations have been studied, and as a result, procedures that are able to compute the missing information of a preference relation have been designed. However, these approaches usually need at least a piece of information about every alternative in the problem in order to be successful in estimating all the missing preference values.In this paper, we address situations in which an expert does not provide any information about a particular alternative, which we call situations of total ignorance. We analyze several strategies to deal with these situations. We classify these strategies into: (i) individual strategies that can be applied to each individual preference relation without taking into account any information from the rest of experts and (ii) social strategies, that is, strategies that make use of the information available from the group of experts. Both individual and social strategies use extra assumptions or knowledge, which could not be directly instantiated in the experts preference relations. We also provide an analysis of the advantages and disadvantages of each one of the strategies presented, and the situations where some of them may be more adequate to be applied than the others.
---
paper_title: A consistency-based procedure to estimate missing pairwise preference values

paper_content:

In this paper, we present a procedure to estimate missing preference values when dealing with pairwise comparison and heterogeneous information. This procedure attempts to estimate the missing information in an expert's incomplete preference relation using only the preference values provided by that particular expert. Our procedure to estimate missing values can be applied to incomplete fuzzy, multiplicative, interval-valued, and linguistic preference relations. Clearly, it would be desirable to maintain experts' consistency levels. We make use of the additive consistency property to measure the level of consistency and to guide the procedure in the estimation of the missing values. Finally, conditions that guarantee the success of our procedure in the estimation of all the missing values of an incomplete preference relation are given. © 2008 Wiley Periodicals, Inc.
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paper_title: A method based on linguistic aggregation operators for group decision making with linguistic preference relations

paper_content:

In this paper, we define some operational laws of linguistic variables and develop some new aggregation operators such as linguistic geometric averaging (LGA) operator, linguistic weighted geometric averaging (LWGA) operator, linguistic ordered weighted geometric averaging (LOWGA) operator and linguistic hybrid geometric averaging (LHGA) operator, etc., which can be utilized to aggregate preference information taking the form of linguistic variables, and then study some desirable properties of the operators. Based on the LGA and the LHGA operators, we propose a practical method for group decision making with linguistic preference relations. The method is straightforward and has no loss of information. Finally, an illustrative numerical example is also given.
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paper_title: Direct approach processes in group decision making using linguistic OWA operators

paper_content:

In a linguistic framework, several group decision making processes by direct approach are presented. These processes are designed using the linguistic ordered weighted averaging (LOWA) operator. To do so, first a study is made of the properties and the axiomatic of LOWA operator, showing the rationality of its aggregation way. And secondly, we present the use of LOWA operator to solve group decision making problems from individuals linguistic preference relations.
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paper_title: Linguistic majorities with difference in support

paper_content:

Graphical abstractDisplay Omitted HighlightsA new linguistic aggregation rule that extends numerical majorities based on difference in support is introduced.Linguistic majorities with difference in support are formalised for fuzzy set and 2-tuples.Both representations are proved to be mathematically isomorphic.A set of normative properties have been demonstrated to hold for the new linguistic majorities. In social choice voting, majorities based on difference of votes and their extension, majorities based on difference in support, implement the crisp preference values (votes) and the intensities of preference provided by voters when comparing pairs of alternatives, respectively. The aim of these rules is declaring which alternative is socially preferred and to that, they require the winner alternative to reach a certain positive difference in its social valuation with respect to the one reached by the loser alternative. This paper introduces a new aggregation rule that extends majorities based on difference of votes from the context of crisp preferences to the framework of linguistic preferences. Under linguistic majorities with difference in support, voters express their intensities of preference between pairs of alternatives using linguistic labels and an alternative defeats another one when a specific support, fixed before the election process, is reached. There exist two main representation methodologies of linguistic preferences: the cardinal one based on the use of fuzzy sets, and the ordinal one based on the use of the 2-tuples. Linguistic majorities with difference in support are formalised in both representation settings, and conditions are given to guarantee that fuzzy linguistic majorities and 2-tuple linguistic majorities are mathematically isomorphic. Finally, linguistic majorities based on difference in support are proved to verify relevant normative properties: anonymity, neutrality, monotonicity, weak Pareto and cancellativeness.
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paper_title: Group decision making with incomplete fuzzy linguistic preference relations

paper_content:

The aim of this paper is to propose a procedure to estimate missing preference values when dealing with incomplete fuzzy linguistic preference relations assessed using a two-tuple fuzzy linguistic approach. This procedure attempts to estimate the missing information in an individual incomplete fuzzy linguistic preference relation using only the preference values provided by the respective expert. It is guided by the additive consistency property to maintain experts' consistency levels. Additionally, we present a selection process of alternatives in group decision making with incomplete fuzzy linguistic preference relations and analyze the use of our estimation procedure in the decision process. © 2008 Wiley Periodicals, Inc.
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paper_title: A two-stage logarithmic goal programming method for generating weights from interval comparison matrices

paper_content:

A two-stage logarithmic goal programming (TLGP) method is proposed to generate weights from interval comparison matrices, which can be either consistent or inconsistent. The first stage is devised to minimize the inconsistency of interval comparison matrices and the second stage is developed to generate priorities under the condition of minimal inconsistency. The weights are assumed to be multiplicative rather than additive. In the case of hierarchical structures, a nonlinear programming method is used to aggregate local interval weights into global interval weights. A simple yet practical preference ranking method is investigated to compare the interval weights of criteria or rank alternatives in a multiplicative aggregation process. The proposed TLGP is also applicable to fuzzy comparison matrices when they are transformed into interval comparison matrices using @a-level sets and the extension principle. Six numerical examples including a group decision analysis problem with a group of comparison matrices, a hierarchical decision problem and a fuzzy decision problem using fuzzy comparison matrix are examined to show the applications of the proposed methods. Comparisons with other existing procedures are made whenever possible.
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paper_title: Some issues on consistency of fuzzy preference relations

paper_content:

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.
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paper_title: Fuzzy preference orderings in group decision making

paper_content:

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.
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paper_title: Cardinal Consistency of Reciprocal Preference Relations: A Characterization of Multiplicative Transitivity

paper_content:

Consistency of preferences is related to rationality, which is associated with the transitivity property. Many properties suggested to model transitivity of preferences are inappropriate for reciprocal preference relations. In this paper, a functional equation is put forward to model the ldquocardinal consistency in the strength of preferencesrdquo of reciprocal preference relations. We show that under the assumptions of continuity and monotonicity properties, the set of representable uninorm operators is characterized as the solution to this functional equation. Cardinal consistency with the conjunctive representable cross ratio uninorm is equivalent to Tanino's multiplicative transitivity property. Because any two representable uninorms are order isomorphic, we conclude that multiplicative transitivity is the most appropriate property for modeling cardinal consistency of reciprocal preference relations. Results toward the characterization of this uninorm consistency property based on a restricted set of (n-1) preference values, which can be used in practical cases to construct perfect consistent preference relations, are also presented.
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paper_title: A consistency-based procedure to estimate missing pairwise preference values

paper_content:

In this paper, we present a procedure to estimate missing preference values when dealing with pairwise comparison and heterogeneous information. This procedure attempts to estimate the missing information in an expert's incomplete preference relation using only the preference values provided by that particular expert. Our procedure to estimate missing values can be applied to incomplete fuzzy, multiplicative, interval-valued, and linguistic preference relations. Clearly, it would be desirable to maintain experts' consistency levels. We make use of the additive consistency property to measure the level of consistency and to guide the procedure in the estimation of the missing values. Finally, conditions that guarantee the success of our procedure in the estimation of all the missing values of an incomplete preference relation are given. © 2008 Wiley Periodicals, Inc.
---
paper_title: Utility theory for decision making

paper_content:

Abstract : The book presents a concise yet mathematically complete treatment of modern utility theories that covers nonprobabilistic preference theory, the von Neumann-Morgenstern expected-utility theory and its extensions, and the joint axiomatization of utility and subjective probability.
---
paper_title: The Analytic Hierarchy Process

paper_content:

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.
---
paper_title: Incomplete fuzzy linguistic preference relations under uncertain environments

paper_content:

When addressing decision-making problems, decision-makers typically express their opinions using fuzzy preference relations. In some instances, decision-makers may have to deal with the problems in which only partial information is available. Consequently, decision-makers embody their preferences as incomplete fuzzy preference relations. The values of incomplete fuzzy preference relations have been considered crisp in recent studies. To allow decision-makers to provide vague or imprecise responses, this study proposes a novel method, called incomplete fuzzy linguistic preference relations, that uses fuzzy linguistic assessment variables instead of crisp values of incomplete fuzzy preference relations to ensure comparison consistency. The proposed method reflects an environment in which some uncertainty or vagueness exists. Examples are included that illustrate the effectiveness of the proposed method.
---
paper_title: Direct approach processes in group decision making using linguistic OWA operators

paper_content:

In a linguistic framework, several group decision making processes by direct approach are presented. These processes are designed using the linguistic ordered weighted averaging (LOWA) operator. To do so, first a study is made of the properties and the axiomatic of LOWA operator, showing the rationality of its aggregation way. And secondly, we present the use of LOWA operator to solve group decision making problems from individuals linguistic preference relations.
---
paper_title: A New Consensus Model for Group Decision Making Problems With Non-Homogeneous Experts

paper_content:

In the literature, we find that the consensus models proposed for group decision making problems are guided by consensus degrees and/or similarity measures and/or consistency measures . When we work in heterogeneous group decision making frameworks, we have importance degrees associated with the experts by expressing their different knowledge levels on the problem. Usually, the importance degrees are applied in the weighted aggregation operators developed to solve the decision situations. In this paper, we study another application possibility, i.e., to use heterogeneity existing among experts to guide the consensus model. Thus, the main goal of this paper is to present a new consensus model for heterogeneous group decision making problems guided also by the heterogeneity criterion. It is also based on consensus degrees and similarity measures, but it presents a new feedback mechanism that adjusts the amount of advice required by each expert depending on his/her own relevance or importance level.
---
paper_title: A fuzzy relation space for group decision theory

paper_content:

In this paper reciprocal relations over n alternatives are defined, and their efficacy as a model of the group decision process is investigated. Several measures of individual preference and group consensus are defined and analyzed, and are used to generate associated measures of distance to consensus. Various decision oriented goals are identified, and we characterize these goals geometrically as subsets of a convex subset of a hyperplane in Rn. An application of this model for assessing degree of consensus is given.
---
paper_title: Consistency-Driven Automatic Methodology to Set Interval Numerical Scales of 2-Tuple Linguistic Term Sets and Its Use in the Linguistic GDM With Preference Relation

paper_content:

The 2-tuple linguistic modeling is a popular tool for computing with words in decision making. In order to deal with the linguistic term sets that are not uniformly and symmetrically distributed, the numerical scale model has been developed to generalize the 2-tuple linguistic modeling. In the numerical scale model, the key task of the 2-tuple based models is the definition of a numerical scale function that establishes a one to one mapping between the linguistic information and numerical values. In this paper, we propose a consistency-driven automatic methodology to set interval numerical scales of 2-tuple linguistic term sets in the decision making problems with linguistic preference relations. This consistency-driven methodology is based on a natural premise regarding the consistency of preference relations. If linguistic preference relations provided by experts are of acceptable consistency, the corresponding transformed numerical preference relations by the established interval numerical scale are also consistent. Compared with the existing approach based on canonical characteristic values, the consistency-driven methodology provides a new way to set the interval numerical scale without the need of the semantics defined by interval type-2 fuzzy sets. Meanwhile, interval multiplicative preference relations are used in the pairwise comparisons method and the presented theory can be utilized in the pairwise comparisons method as it provides a novel approach to automatic construct interval multiplicative preference relations. Finally, we present the framework for the use of the consistency-driven automatic methodology in linguistic group decision making problems and two numerical examples are given to illustrate the feasibility and validity of this proposal.
---
paper_title: When conflict induces the expression of incomplete preferences

paper_content:

Multicriteria conflict arises in pairwise comparisons, where each alternative outperforms the other one on some criterion, which imposes a trade-off. Comparing two alternatives can be difficult if their respective advantages are of high magnitude (the attribute spread is large). In this paper, we investigate to which extent conflict in a comparison situation can lead decision makers to express incomplete preferences, that is, to refuse to compare the two alternatives, or to be unable to compare them with confidence. We report on an experiment in which subjects expressed preferences on pairs of alternatives involving varying conflicts. Results show that depending on whether the participants are allowed to express incomplete preferences or not, attribute spread has a different effect: a large attribute spread increases the frequency of incomparability statements, when available, while it increases the use of indifference statements when only indifference and preference answers are permitted. These results lead us to derive some implications for preference elicitation methods involving comparison tasks.
---
paper_title: The Analytic Hierarchy Process

paper_content:

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.
---
paper_title: Integrating multiplicative preference relations in a multipurpose decision-makingmodel based on fuzzy preference relations

paper_content:

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.
---
paper_title: A new linguistic computational model based on discrete fuzzy numbers for computing with words

paper_content:

In recent years, several different linguistic computational models for dealing with linguistic information in processes of computing with words have been proposed. However, until now all of them rely on the special semantics of the linguistic terms, usually fuzzy numbers in the unit interval, and the linguistic aggregation operators are based on aggregation operators in [0,1]. In this paper, a linguistic computational model based on discrete fuzzy numbers whose support is a subset of consecutive natural numbers is presented ensuring the accuracy and consistency of the model. In this framework, no underlying membership functions are needed and several aggregation operators defined on the set of all discrete fuzzy numbers are presented. These aggregation operators are constructed from aggregation operators defined on a finite chain in accordance with the granularity of the linguistic term set. Finally, an example of a multi-expert decision-making problem in a hierarchical multi-granular linguistic context is given to illustrate the applicability of the proposed method and its advantages.
---
paper_title: Fuzzy Preference Modelling and Multicriteria Decision Support

paper_content:

Note: Index. Bibliogr. : p. 239-251 Reference Record created on 2004-09-07, modified on 2016-08-08
---
paper_title: Empowering difficult classes with a similarity-based aggregation in multi-class classification problems

paper_content:

One-vs-One strategy divides the original multi-class problem into as many binary classification problems as pairs of classes. Then, independent base classifiers are learned to face each problem, whose outputs are combined to predict a single class label. This way, the accuracy of the baseline classifiers without decomposition is usually enhanced, aside from enabling the usage of binary classifiers, i.e., Support Vector Machines, to solve multi-class problems. This paper analyzes the fact that existing aggregations favor easily recognizable classes; hence, the accuracy enhancement mainly comes from the higher correct classification rates over these classes. Using other evaluation criteria, the significant improvements of One-vs-One are diminished, showing a weakness due to the presence of difficult classes. Difficult classes can be defined as those obtaining a lower correct classification rate than that obtained by the other classes in the problem. After studying the problem of difficult classes in this framework and aiming to empower these classes, a novel similarity-based aggregation is presented, which generalizes the well-known weighted voting. The experimental analysis shows that the new methodology is able to increase the recognition of difficult classes, obtaining a more balanced performance over all classes, which is a desirable behavior. The methodology is tested within several Machine Learning paradigms and is compared with the state-of-the-art on aggregations for One-vs-One strategy. The results are contrasted by the proper statistical tests, as suggested in the literature.
---
paper_title: The Effectiveness of Alternative Preference Elicitation Methods in the Analytic Hierarchy Process

paper_content:

This study compares alternative preference elicitation methods that are currently available in software implementations of the analytic hierarchy process. For the simple problem used in this study, the elicitation methods may be sorted from least to most accurate as follows: (1) direct estimation, (2) graphical multiple bars, (3) graphical pairwise, (4) numeric pairwise and (5) verbal pairwise. However, the most accurate methods are not the easiest to use. Furthermore, actual accuracy differed substantially from users’ perceptions, indicating a need to assist the users in the selection of an appropriate elicitation method. © 1997 John Wiley & Sons, Ltd.
---
paper_title: A Mobile Decision Support System for Dynamic Group Decision-Making Problems

paper_content:

The aim of this paper is to present a decision support system model with two important characteristic: 1) mobile technologies are applied in the decision process and 2) the set of alternatives is not fixed over time to address dynamic decision situations in which the set of solution alternatives could change throughout the decision-making process. We implement a prototype of such mobile decision support system in which experts use mobile phones to provide their preferences anywhere and anytime. To get a general system, experts' preferences are assumed to be represented by different preference representations: 1) fuzzy preference relations; 2) orderings; 3) utility functions; and 4) multiplicative preference relations. Because this prototype incorporates both selection and consensus processes, it allows us to model group decision-making situations. The prototype incorporates a tool for managing the changes on the set of feasible alternatives that could happen throughout the decision process. This way, the prototype provides a new approach to deal with dynamic group decision-making situations to help make decisions anywhere and anytime.
---
paper_title: A novel fuzzy multi-criteria decision framework for sustainable supplier selection with incomplete information

paper_content:

Both academic and corporate interest in sustainable supply chains has increased in recent years. Supplier selection process is one of the key operational tasks for sustainable supply chain management. This paper examines the problem of identifying an effective model based on sustainability principles for supplier selection operations in supply chains. Due to its multi-criteria nature, the sustainable supplier evaluation process requires an appropriate multi-criteria analysis and solution approach. The approach should also consider that decision makers might face situations such as time pressure, lack of expertise in related issue, etc., during the evaluation process. The paper develops a novel approach based on fuzzy analytic network process within multi-person decision-making schema under incomplete preference relations. The method not only makes sufficient evaluations using the provided preference information, but also maintains the consistency level of the evaluations. Finally, the paper analyzes the sustainability of a number of suppliers in a real-life problem to demonstrate the validity of the proposed evaluation model.
---
paper_title: A method based on linguistic aggregation operators for group decision making with linguistic preference relations

paper_content:

In this paper, we define some operational laws of linguistic variables and develop some new aggregation operators such as linguistic geometric averaging (LGA) operator, linguistic weighted geometric averaging (LWGA) operator, linguistic ordered weighted geometric averaging (LOWGA) operator and linguistic hybrid geometric averaging (LHGA) operator, etc., which can be utilized to aggregate preference information taking the form of linguistic variables, and then study some desirable properties of the operators. Based on the LGA and the LHGA operators, we propose a practical method for group decision making with linguistic preference relations. The method is straightforward and has no loss of information. Finally, an illustrative numerical example is also given.
---
paper_title: Logarithmic least squares method to priority for group decision making with incomplete fuzzy preference relations

paper_content:

The aim of this paper is to present a logarithmic least squares method (LLSM) to priority for group decision making with incomplete fuzzy preference relations. We give a reasonable definition of multiplicative consistent for incomplete fuzzy preference relation. We develop the acceptable fuzzy consistency ratio (FCR for short), which is simple and similar to Saaty’s consistency ratio CR for multiplicative fuzzy preference relations. We also extend the LLSM method to the case of individual preference relation with complete information. Finally, some examples are illustrated to show that our method is simple, efficient, and can be performed on computer easily.
---
paper_title: Incomplete pairwise comparison and consistency optimization

paper_content:

This paper proposes a new method for calculating the missing elements of an incomplete matrix of pairwise comparison values for a decision problem. The matrix is completed by minimizing a measure of global inconsistency, thus obtaining a matrix which is optimal from the point of view of consistency with respect to the available judgements. The optimal values are obtained by solving a linear system and unicity of the solution is proved under general assumptions. Some other methods proposed in the literature are discussed and a numerical example is presented.
---
paper_title: Least square completion and inconsistency repair methods for additively consistent fuzzy preference relations

paper_content:

In this paper, we explore the group decision making (GDM) problems with incomplete additively consistent fuzzy preference relations. Some properties of additively consistent fuzzy preference relations are also discussed. A sufficient and necessary condition is proposed to keep the additive consistency of fuzzy preference relations. Methods for determining the priority weights of fuzzy preference relations are provided. Least square completion and inconsistency repair methods are developed to deal with incomplete and inconsistent fuzzy preference relations. Some numerical examples are also given to illustrate the proposed approaches.
---
paper_title: Goal programming models for obtaining the priority vector of incomplete fuzzy preference relation

paper_content:

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.
---
paper_title: A NOTE ON THE ESTIMATION OF MISSING PAIRWISE PREFERENCE VALUES: A UNINORM CONSISTENCY BASED METHOD

paper_content:

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.
---
paper_title: Group decision-making procedure based on incomplete reciprocal relations

paper_content:

Xu (Int J Approx Reason 36:261---270, 2004) introduced the concepts of incomplete reciprocal relation and additive consistent incomplete reciprocal relation. The aim of this paper is to develop a novel procedure for group decision making with incomplete reciprocal relations. The procedure utilizes each given incomplete reciprocal relation to construct an auxiliary reciprocal relation based on additive transitivity, and then aggregates directly these auxiliary reciprocal relations into a collective auxiliary reciprocal relation. After that, based on the collective auxiliary reciprocal relation, a simple linear system of equations is established for ranking alternatives. Finally, a numerical example is given to illustrate the developed procedure.
---
paper_title: A consistency-based procedure to estimate missing pairwise preference values

paper_content:

In this paper, we present a procedure to estimate missing preference values when dealing with pairwise comparison and heterogeneous information. This procedure attempts to estimate the missing information in an expert's incomplete preference relation using only the preference values provided by that particular expert. Our procedure to estimate missing values can be applied to incomplete fuzzy, multiplicative, interval-valued, and linguistic preference relations. Clearly, it would be desirable to maintain experts' consistency levels. We make use of the additive consistency property to measure the level of consistency and to guide the procedure in the estimation of the missing values. Finally, conditions that guarantee the success of our procedure in the estimation of all the missing values of an incomplete preference relation are given. © 2008 Wiley Periodicals, Inc.
---
paper_title: Algorithms for estimating missing elements of incomplete intuitionistic preference relations

paper_content:

Preference relations as simple and efficient information description tools have been widely used in practical decision-making problems. Intuitionistic preference relation, which is often met in real problems, is usually utilized to provide the experts' vague or fuzzy opinions over objects under uncertain circumstances. Owing to the limitations of the experts' professional knowledge and experience, the provided preferences in an intuitionistic preference relation are usually incomplete. Consequently, how to estimate the missing information in an expert's incomplete intuitionistic preference relation becomes a necessary step in a decision-making process. In this paper, we define the concept of multiplicative consistent intuitionistic preference relation and develop two estimation algorithms. The first algorithm is used to estimate the missing elements using only the known preference values in an acceptable incomplete intuitionistic fuzzy preference relation with the least judgments. The second one is given for the estimation of missing elements of the acceptable incomplete intuitionistic fuzzy preference relations with more known judgments. The advantages of the developed algorithms over the existing one are detailedly analyzed, and some examples are provided to illustrate the solution processes of the algorithms and to verify their practicality and superiority. © 2011 Wiley Periodicals, Inc. © 2011 Wiley Periodicals, Inc.
---
paper_title: An interactive procedure for multiple criteria group decision making with incomplete information

paper_content:

This paper presents an interactive procedure for multiple attribute group decision making (MAGDM) problem with incomplete information. Theoretic models are formally described for establishing group's pairwise dominance relations with group's utility range by using a separable linear programming technique.
---
paper_title: A Consensus Model for Group Decision Making With Incomplete Fuzzy Preference Relations

paper_content:

Two processes are necessary to solve group decision making problems: A consensus process and a selection process. The consensus reaching process is necessary to obtain a final solution with a certain level of agreement between the experts; and the selection process is necessary to obtain such a final solution. In a previous paper, we present a selection process to deal with group decision making problems with incomplete fuzzy preference relations, which uses consistency measures to estimate the incomplete fuzzy preference relations. In this paper we present a consensus model. The main novelty of this consensus model is that of being guided by both consensus and consistency measures. Also, the consensus reaching process is guided automatically, without moderator, through both consensus and consistency criteria. To do that, a feedback mechanism is developed to generate advice on how experts should change or complete their preferences in order to reach a solution with high consensus and consistency degrees. In each consensus round, experts are given information on how to change their preferences, and to estimate missing values if their corresponding preference relation is incomplete. Additionally, a consensus and consistency based induced ordered weighted averaging operator to aggregate the experts' preferences is introduced, which can be used in consensus models as well as in selection processes. The main improvement of this consensus model is that it supports the management of incomplete information and it allows to achieve consistent solutions with a great level of agreement.
---
paper_title: Alternative modes of questioning in the analytic hierarchy process

paper_content:

The standard mode of questioning in the Analytic Hierarchy Process (AHP) requires the decision maker to complete a sequence of positive reciprocal matrices by answering n(n − 1)2 questions for each matrix, each entry being an approximation to the ratio of the weights of the n items being compared. This paper presents two extensions of the eigenvector approach of the AHP which allows the decision maker to say “I don't know” or “I'm not sure” to some of the questions being asked, and to approximate nonlinear functions of the ratios of the weights. In this way, the questioning process can be substantially shortened and better representations of the responses to certain stimuli may be derived.
---
paper_title: Group decision making with incomplete fuzzy preference relations based on the additive consistency and the order consistency

paper_content:

In this paper, we present a new method for group decision making with incomplete fuzzy preference relations based on the additive consistency and the order consistency. We estimate unknown preference values based on the additive consistency and then construct the consistency matrix which satisfies the additive consistency and the order consistency simultaneously for aggregation. The existing group decision making methods may not satisfy the order consistency for aggregation in some situations. The proposed method can overcome the drawback of the existing methods. It provides us with a useful way for group decision making with incomplete fuzzy preference relations based on the additive consistency and the order consistency.
---
paper_title: Least-square method to priority of the fuzzy preference relations with incomplete information

paper_content:

The priority problem of incomplete preference relations is investigated. Using the transformation relation between multiplicative preference relation and fuzzy preference relation, we develop a least-square model to obtain the collective priority vector of the incomplete preference relations presented by multiple decision makers, with the existence condition of the solution being developed. Meanwhile, we extend this model to the cases of the fuzzy preference relations with complete information presented by multiple decision makers and the fuzzy preference relation with complete information presented by one decision maker. Finally, it is illustrated by a numerical example that the method proposed is feasible and effective.
---
paper_title: A Note on Two Methods for Estimating Missing Pairwise Preference Values

paper_content:

This note analyzes two methods for calculating missing values of an incomplete reciprocal fuzzy preference relation. The first method by Herrera-Viedma appeared in the IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics [vol. 37, no. 1 (2007) 176-189], while the second one by Fedrizzi and Giove appeared later in the European Journal of Operational Research [vol. 183 (2007) 303-313]. The underlying concept driving both methods is the additive consistency property. We show that both methods, although different, are very similar. Both methods derive the same estimated values for the independent-missing-comparison case, while they differ in the dependent-missing-comparison case. However, it is shown that a modification of the first method coincides with the second one. Regarding the total reconstruction of an incomplete preference relation, it is true that the second method performs worse than the first one. When Herrera-Viedma 's method is unsuccessful, Fedrizzi-Giove's method is as well. However, in those cases when Fedrizzi-Giove's method cannot guarantee the successful reconstruction of an incomplete preference relation, we have that Herrera-Viedma 's method can. These results lead us to claim that both methods should be seen as complementary rather than competitors in their application, and as such, we propose a reconstruction policy of incomplete fuzzy preference relations using both methods. By doing this, the only unsuccessful reconstruction case is when there is a chain of missing pairwise comparisons involving each one of the feasible alternatives at least once.
---
paper_title: Interactive group decision making procedure based on uncertain multiplicative linguistic preference relations

paper_content:

Group decision making problems are investigated with uncertain multiplicative linguistic preference relations. An unbal- anced multiplicative linguistic label set is introduced, which can be used by the experts to express their linguistic preference informa- tion over alternatives. The uncertain linguistic weighted geometric mean operator is utilized to aggregate all the individual uncer- tain multiplicative linguistic preference relations into a collective one, and then a simple approach is developed to determining the experts' weights by utilizing the consensus degrees among the individual uncertain multiplicative linguistic preference relations and the collective uncertain multiplicative linguistic preference re- lations. Furthermore, a practical interactive procedure for group decision making is proposed based on uncertain multiplicative lin- guistic preference relations, in which a possibility degree formula and a complementary matrix are used to rank the given alterna- tives. Finally, the proposed procedure is applied to solve the group decision making problem of a manufacturing company searching the best global supplier for one of its most critical parts used in assembling process.
---
paper_title: Interactive Approach Based on Incomplete Complementary Judgement Matrices to Group Decision Making

paper_content:

Group decision-making problem based on incomplete complementary judgement matrices is investigated. Firstly, each individual incomplete complementary judgement matrix is extended to a complete complementary judgement matrix by utilizing multiplicative consistency. All these extended matrices are then aggregated into the collective complementary judgement matrix. Secondly, based on the dead line of deviations between the collective complementary judgement matrix and individual extended complementary judgement matrices, the decision makers interact in order to obtain the highest degree of consensus. Finally, an interactive approach based on incomplete complementary judgement matrices to group decision making is developed, and an illustrative example is also given.
---
paper_title: Group Decision-Making Model With Incomplete Fuzzy Preference Relations Based on Additive Consistency

paper_content:

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
---
paper_title: Linear optimization modeling of consistency issues in group decision making based on fuzzy preference relations

paper_content:

Highlights? This paper focuses on several consistency issues of fuzzy preference relations, such as individual consistency construction, consensus model and management of incomplete fuzzy preference relations. ? This paper proposes linear optimization models for solving these consistency issues on fuzzy preference relations. ? Our proposal optimally preserves original preference information in constructing individual consistency and reaching consensus, and maximizes the consistency level in calculating the missing values of incomplete fuzzy preference relations. The consistency measure is a vital basis for group decision making (GDM) based on fuzzy preference relations, and includes two subproblems: individual consistency and consensus consistency. This paper proposes linear optimization models for solving some issues on consistency of fuzzy preference relations, such as individual consistency construction, consensus model and management of incomplete fuzzy preference relations. Our proposal optimally preserves original preference information in constructing individual consistency and reaching consensus (in Manhattan distance sense), and maximizes the consistency level of fuzzy preference relations in calculating the missing values of incomplete fuzzy preference relations. Linear optimization models can be solved in very little computational time using readily available softwares. Therefore, the results in this paper are also of simplicity and convenience for the application of consistent fuzzy preference relations in GDM problems.
---
paper_title: A new incomplete preference relations based approach to quality function deployment

paper_content:

Quality function deployment (QFD) is a widely-used methodology for developing a design quality aimed at satisfying the customer and translating the customer's demand into design targets. As QFD methodology is based on the voices of customers, it requires a group of individuals, or decision makers (DMs), to express their preferences. However, there can be a bottleneck since DMs may not have a perfect knowledge of the problem to be solved. Thereby, the aim of this study is to improve the effectiveness of the evaluation process in QFD. The QFD methodology is extended by introducing a new group decision making (GDM) approach that takes into account incomplete information of DMs by means of the fuzzy set theory. The proposed approach is applied to a collaborative software development process to demonstrate its potential.
---
paper_title: A parametric GP model dealing with incomplete information for group decision-making

paper_content:

Abstract We consider a group decision-making problem where preferences given by the experts are articulated into the form of pairwise comparison matrices. In many cases, experts are not able to efficiently provide their preferences on some aspects of the problem because of a large number of alternatives, limited expertise related to some problem domain, unavailable data, etc., resulting in incomplete pairwise comparison matrices. Our goal is to develop a computational method to retrieve a group priority vector of the considered alternatives dealing with incomplete information. For that purpose, we have established an optimization problem in which a similarity function and a parametric compromise function are defined. Associated to this problem, a logarithmic goal programming formulation is considered to provide an effective procedure to compute the solution. Moreover, the parameters involved in the method have a clear meaning in the context of group problems.
---
paper_title: Integrating multiplicative preference relations in a multipurpose decision-makingmodel based on fuzzy preference relations

paper_content:

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.
---
paper_title: A novel fuzzy multi-criteria decision framework for sustainable supplier selection with incomplete information

paper_content:

Both academic and corporate interest in sustainable supply chains has increased in recent years. Supplier selection process is one of the key operational tasks for sustainable supply chain management. This paper examines the problem of identifying an effective model based on sustainability principles for supplier selection operations in supply chains. Due to its multi-criteria nature, the sustainable supplier evaluation process requires an appropriate multi-criteria analysis and solution approach. The approach should also consider that decision makers might face situations such as time pressure, lack of expertise in related issue, etc., during the evaluation process. The paper develops a novel approach based on fuzzy analytic network process within multi-person decision-making schema under incomplete preference relations. The method not only makes sufficient evaluations using the provided preference information, but also maintains the consistency level of the evaluations. Finally, the paper analyzes the sustainability of a number of suppliers in a real-life problem to demonstrate the validity of the proposed evaluation model.
---
paper_title: A NOTE ON THE ESTIMATION OF MISSING PAIRWISE PREFERENCE VALUES: A UNINORM CONSISTENCY BASED METHOD

paper_content:

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.
---
paper_title: Cardinal Consistency of Reciprocal Preference Relations: A Characterization of Multiplicative Transitivity

paper_content:

Consistency of preferences is related to rationality, which is associated with the transitivity property. Many properties suggested to model transitivity of preferences are inappropriate for reciprocal preference relations. In this paper, a functional equation is put forward to model the ldquocardinal consistency in the strength of preferencesrdquo of reciprocal preference relations. We show that under the assumptions of continuity and monotonicity properties, the set of representable uninorm operators is characterized as the solution to this functional equation. Cardinal consistency with the conjunctive representable cross ratio uninorm is equivalent to Tanino's multiplicative transitivity property. Because any two representable uninorms are order isomorphic, we conclude that multiplicative transitivity is the most appropriate property for modeling cardinal consistency of reciprocal preference relations. Results toward the characterization of this uninorm consistency property based on a restricted set of (n-1) preference values, which can be used in practical cases to construct perfect consistent preference relations, are also presented.
---
paper_title: A consistency-based procedure to estimate missing pairwise preference values

paper_content:

In this paper, we present a procedure to estimate missing preference values when dealing with pairwise comparison and heterogeneous information. This procedure attempts to estimate the missing information in an expert's incomplete preference relation using only the preference values provided by that particular expert. Our procedure to estimate missing values can be applied to incomplete fuzzy, multiplicative, interval-valued, and linguistic preference relations. Clearly, it would be desirable to maintain experts' consistency levels. We make use of the additive consistency property to measure the level of consistency and to guide the procedure in the estimation of the missing values. Finally, conditions that guarantee the success of our procedure in the estimation of all the missing values of an incomplete preference relation are given. © 2008 Wiley Periodicals, Inc.
---
paper_title: Group decision making with incomplete fuzzy preference relations based on the additive consistency and the order consistency

paper_content:

In this paper, we present a new method for group decision making with incomplete fuzzy preference relations based on the additive consistency and the order consistency. We estimate unknown preference values based on the additive consistency and then construct the consistency matrix which satisfies the additive consistency and the order consistency simultaneously for aggregation. The existing group decision making methods may not satisfy the order consistency for aggregation in some situations. The proposed method can overcome the drawback of the existing methods. It provides us with a useful way for group decision making with incomplete fuzzy preference relations based on the additive consistency and the order consistency.
---
paper_title: Interactive Approach Based on Incomplete Complementary Judgement Matrices to Group Decision Making

paper_content:

Group decision-making problem based on incomplete complementary judgement matrices is investigated. Firstly, each individual incomplete complementary judgement matrix is extended to a complete complementary judgement matrix by utilizing multiplicative consistency. All these extended matrices are then aggregated into the collective complementary judgement matrix. Secondly, based on the dead line of deviations between the collective complementary judgement matrix and individual extended complementary judgement matrices, the decision makers interact in order to obtain the highest degree of consensus. Finally, an interactive approach based on incomplete complementary judgement matrices to group decision making is developed, and an illustrative example is also given.
---
paper_title: Group Decision-Making Model With Incomplete Fuzzy Preference Relations Based on Additive Consistency

paper_content:

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
---
paper_title: A new incomplete preference relations based approach to quality function deployment

paper_content:

Quality function deployment (QFD) is a widely-used methodology for developing a design quality aimed at satisfying the customer and translating the customer's demand into design targets. As QFD methodology is based on the voices of customers, it requires a group of individuals, or decision makers (DMs), to express their preferences. However, there can be a bottleneck since DMs may not have a perfect knowledge of the problem to be solved. Thereby, the aim of this study is to improve the effectiveness of the evaluation process in QFD. The QFD methodology is extended by introducing a new group decision making (GDM) approach that takes into account incomplete information of DMs by means of the fuzzy set theory. The proposed approach is applied to a collaborative software development process to demonstrate its potential.
---
paper_title: A NOTE ON THE ESTIMATION OF MISSING PAIRWISE PREFERENCE VALUES: A UNINORM CONSISTENCY BASED METHOD

paper_content:

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.
---
paper_title: Cardinal Consistency of Reciprocal Preference Relations: A Characterization of Multiplicative Transitivity

paper_content:

Consistency of preferences is related to rationality, which is associated with the transitivity property. Many properties suggested to model transitivity of preferences are inappropriate for reciprocal preference relations. In this paper, a functional equation is put forward to model the ldquocardinal consistency in the strength of preferencesrdquo of reciprocal preference relations. We show that under the assumptions of continuity and monotonicity properties, the set of representable uninorm operators is characterized as the solution to this functional equation. Cardinal consistency with the conjunctive representable cross ratio uninorm is equivalent to Tanino's multiplicative transitivity property. Because any two representable uninorms are order isomorphic, we conclude that multiplicative transitivity is the most appropriate property for modeling cardinal consistency of reciprocal preference relations. Results toward the characterization of this uninorm consistency property based on a restricted set of (n-1) preference values, which can be used in practical cases to construct perfect consistent preference relations, are also presented.
---
paper_title: Interactive Approach Based on Incomplete Complementary Judgement Matrices to Group Decision Making

paper_content:

Group decision-making problem based on incomplete complementary judgement matrices is investigated. Firstly, each individual incomplete complementary judgement matrix is extended to a complete complementary judgement matrix by utilizing multiplicative consistency. All these extended matrices are then aggregated into the collective complementary judgement matrix. Secondly, based on the dead line of deviations between the collective complementary judgement matrix and individual extended complementary judgement matrices, the decision makers interact in order to obtain the highest degree of consensus. Finally, an interactive approach based on incomplete complementary judgement matrices to group decision making is developed, and an illustrative example is also given.
---
paper_title: A consensus reaching process under incomplete multiplicative preference relations

paper_content:

Multiplicative preference relation is the most common tool used to express decision makers' preferences over pairs of criteria. Due to the decision makers' own personal experience, learning, situation, state of mind, or the complexity of the considered decision-making problems, in many cases, the decision makers cannot or are unwilling to provide their preferences over some pairs of criteria, and thus may construct incomplete multiplicative preference relations in which some of the elements are missing, we call these problems group decision-making problems with incomplete multiplicative preference relations. In this paper, we first investigate the approach to extending each incomplete multiplicative preference relation to a complete multiplicative preference relation by making use of the multiplicative transitivity properties. Then, we present some deviation measures to evaluate the consensus among the individual multiplicative preference relations and their collective multiplicative preference relation. On the basis of the above results, finally, we develop a consensus reaching process of group decision making with incomplete multiplicative preference relations, which establishes a sound theoretical basis for the priorities of criteria.
---
paper_title: Correspondence between Incomplete Fuzzy Preference Relation and Its Priority Vector

paper_content:

Fuzzy preference relations are frequently adopted by decision makers to express their preference tendency toward alternatives. Due to the lack of expertise of knowledge, decision makers may not be able to specify complete preference relation. To deal with incomplete fuzzy preference relations, Xu [26] proposed prioritization methods for incomplete fuzzy preference relations where he postulated a correspondence between priority vector and additive consistent incomplete fuzzy preference relation. In this paper, we are going to prove the correspondence does not always hold.
---
paper_title: Logarithmic least squares method to priority for group decision making with incomplete fuzzy preference relations

paper_content:

The aim of this paper is to present a logarithmic least squares method (LLSM) to priority for group decision making with incomplete fuzzy preference relations. We give a reasonable definition of multiplicative consistent for incomplete fuzzy preference relation. We develop the acceptable fuzzy consistency ratio (FCR for short), which is simple and similar to Saaty’s consistency ratio CR for multiplicative fuzzy preference relations. We also extend the LLSM method to the case of individual preference relation with complete information. Finally, some examples are illustrated to show that our method is simple, efficient, and can be performed on computer easily.
---
paper_title: Incomplete pairwise comparison and consistency optimization

paper_content:

This paper proposes a new method for calculating the missing elements of an incomplete matrix of pairwise comparison values for a decision problem. The matrix is completed by minimizing a measure of global inconsistency, thus obtaining a matrix which is optimal from the point of view of consistency with respect to the available judgements. The optimal values are obtained by solving a linear system and unicity of the solution is proved under general assumptions. Some other methods proposed in the literature are discussed and a numerical example is presented.
---
paper_title: Least square completion and inconsistency repair methods for additively consistent fuzzy preference relations

paper_content:

In this paper, we explore the group decision making (GDM) problems with incomplete additively consistent fuzzy preference relations. Some properties of additively consistent fuzzy preference relations are also discussed. A sufficient and necessary condition is proposed to keep the additive consistency of fuzzy preference relations. Methods for determining the priority weights of fuzzy preference relations are provided. Least square completion and inconsistency repair methods are developed to deal with incomplete and inconsistent fuzzy preference relations. Some numerical examples are also given to illustrate the proposed approaches.
---
paper_title: Goal programming models for obtaining the priority vector of incomplete fuzzy preference relation

paper_content:

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.
---
paper_title: Group decision-making procedure based on incomplete reciprocal relations

paper_content:

Xu (Int J Approx Reason 36:261---270, 2004) introduced the concepts of incomplete reciprocal relation and additive consistent incomplete reciprocal relation. The aim of this paper is to develop a novel procedure for group decision making with incomplete reciprocal relations. The procedure utilizes each given incomplete reciprocal relation to construct an auxiliary reciprocal relation based on additive transitivity, and then aggregates directly these auxiliary reciprocal relations into a collective auxiliary reciprocal relation. After that, based on the collective auxiliary reciprocal relation, a simple linear system of equations is established for ranking alternatives. Finally, a numerical example is given to illustrate the developed procedure.
---
paper_title: The Analytic Hierarchy Process

paper_content:

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.
---
paper_title: A note on group decision-making procedure based on incomplete reciprocal relations

paper_content:

In a very recent paper by Xu and Chen (Soft Comput 12:515–521, 2008), a novel procedure for group decision making with incomplete reciprocal relations was developed. In this note, we examine the function between the fuzzy preference relation and its corresponding priority vector developed by Xu and Chen with a numerical example and show that the function does not hold in general cases. Then, we deduce an exact function between the additive transitivity fuzzy preference relation and its corresponding priority vector. Based on this, we develop a procedure for the decision making with an incomplete reciprocal relation and also develop a procedure for the group decision making with incomplete reciprocal relations. In order to compare the performances of our method with Xu and Chen’s method in fitting the reciprocal relation, we introduce some criteria. Theoretical analysis and numerical examples have shown that the function deduced by us is more reasonable and effective than Xu and Chen’s.
---
paper_title: Alternative modes of questioning in the analytic hierarchy process

paper_content:

The standard mode of questioning in the Analytic Hierarchy Process (AHP) requires the decision maker to complete a sequence of positive reciprocal matrices by answering n(n − 1)2 questions for each matrix, each entry being an approximation to the ratio of the weights of the n items being compared. This paper presents two extensions of the eigenvector approach of the AHP which allows the decision maker to say “I don't know” or “I'm not sure” to some of the questions being asked, and to approximate nonlinear functions of the ratios of the weights. In this way, the questioning process can be substantially shortened and better representations of the responses to certain stimuli may be derived.
---
paper_title: Least-square method to priority of the fuzzy preference relations with incomplete information

paper_content:

The priority problem of incomplete preference relations is investigated. Using the transformation relation between multiplicative preference relation and fuzzy preference relation, we develop a least-square model to obtain the collective priority vector of the incomplete preference relations presented by multiple decision makers, with the existence condition of the solution being developed. Meanwhile, we extend this model to the cases of the fuzzy preference relations with complete information presented by multiple decision makers and the fuzzy preference relation with complete information presented by one decision maker. Finally, it is illustrated by a numerical example that the method proposed is feasible and effective.
---
paper_title: A Note on Two Methods for Estimating Missing Pairwise Preference Values

paper_content:

This note analyzes two methods for calculating missing values of an incomplete reciprocal fuzzy preference relation. The first method by Herrera-Viedma appeared in the IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics [vol. 37, no. 1 (2007) 176-189], while the second one by Fedrizzi and Giove appeared later in the European Journal of Operational Research [vol. 183 (2007) 303-313]. The underlying concept driving both methods is the additive consistency property. We show that both methods, although different, are very similar. Both methods derive the same estimated values for the independent-missing-comparison case, while they differ in the dependent-missing-comparison case. However, it is shown that a modification of the first method coincides with the second one. Regarding the total reconstruction of an incomplete preference relation, it is true that the second method performs worse than the first one. When Herrera-Viedma 's method is unsuccessful, Fedrizzi-Giove's method is as well. However, in those cases when Fedrizzi-Giove's method cannot guarantee the successful reconstruction of an incomplete preference relation, we have that Herrera-Viedma 's method can. These results lead us to claim that both methods should be seen as complementary rather than competitors in their application, and as such, we propose a reconstruction policy of incomplete fuzzy preference relations using both methods. By doing this, the only unsuccessful reconstruction case is when there is a chain of missing pairwise comparisons involving each one of the feasible alternatives at least once.
---
paper_title: Group Decision-Making Model With Incomplete Fuzzy Preference Relations Based on Additive Consistency

paper_content:

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
---
paper_title: Linear optimization modeling of consistency issues in group decision making based on fuzzy preference relations

paper_content:

Highlights? This paper focuses on several consistency issues of fuzzy preference relations, such as individual consistency construction, consensus model and management of incomplete fuzzy preference relations. ? This paper proposes linear optimization models for solving these consistency issues on fuzzy preference relations. ? Our proposal optimally preserves original preference information in constructing individual consistency and reaching consensus, and maximizes the consistency level in calculating the missing values of incomplete fuzzy preference relations. The consistency measure is a vital basis for group decision making (GDM) based on fuzzy preference relations, and includes two subproblems: individual consistency and consensus consistency. This paper proposes linear optimization models for solving some issues on consistency of fuzzy preference relations, such as individual consistency construction, consensus model and management of incomplete fuzzy preference relations. Our proposal optimally preserves original preference information in constructing individual consistency and reaching consensus (in Manhattan distance sense), and maximizes the consistency level of fuzzy preference relations in calculating the missing values of incomplete fuzzy preference relations. Linear optimization models can be solved in very little computational time using readily available softwares. Therefore, the results in this paper are also of simplicity and convenience for the application of consistent fuzzy preference relations in GDM problems.
---
paper_title: A parametric GP model dealing with incomplete information for group decision-making

paper_content:

Abstract We consider a group decision-making problem where preferences given by the experts are articulated into the form of pairwise comparison matrices. In many cases, experts are not able to efficiently provide their preferences on some aspects of the problem because of a large number of alternatives, limited expertise related to some problem domain, unavailable data, etc., resulting in incomplete pairwise comparison matrices. Our goal is to develop a computational method to retrieve a group priority vector of the considered alternatives dealing with incomplete information. For that purpose, we have established an optimization problem in which a similarity function and a parametric compromise function are defined. Associated to this problem, a logarithmic goal programming formulation is considered to provide an effective procedure to compute the solution. Moreover, the parameters involved in the method have a clear meaning in the context of group problems.
---
paper_title: Interval multiplicative transitivity for consistency, missing values and priority weights of interval fuzzy preference relations

paper_content:

In this paper, the concept of multiplicative transitivity of a fuzzy preference relation, as defined by Tanino T. Tanino, Fuzzy preference orderings in group decision-making, Fuzzy Sets and Systems 12 (1984) 117-131, is extended to discover whether an interval fuzzy preference relation is consistent or not, and to derive the priority vector of a consistent interval fuzzy preference relation. We achieve this by introducing the concept of interval multiplicative transitivity of an interval fuzzy preference relation and show that, by solving numerical examples, the test of consistency and the weights derived by the simple formulas based on the interval multiplicative transitivity produce the same results as those of linear programming models proposed by Xu and Chen Z.S. Xu, J. Chen, Some models for deriving the priority weights from interval fuzzy preference relations, European Journal of Operational Research 184 (2008) 266-280. In addition, by taking advantage of interval multiplicative transitivity of an interval fuzzy preference relation, we put forward two approaches to estimate missing value(s) of an incomplete interval fuzzy preference relation, and present numerical examples to illustrate these two approaches.
---
paper_title: Dominance-based rough set approach to incomplete interval-valued information system

paper_content:

Since preference order is a crucial feature of data concerning decision situations, the classical rough set model has been generalized by replacing the indiscernibility relation with a dominance relation. The purpose of this paper is to further investigate the dominance-based rough set in incomplete interval-valued information system, which contains both incomplete and imprecise evaluations of objects. By considering three types of unknown values in the incomplete interval-valued information system, a data complement method is used to transform the incomplete interval-valued information system into a traditional one. To generate the optimal decision rules from the incomplete interval-valued decision system, six types of relative reducts are proposed. Not only the relationships between these reducts but also the practical approaches to compute these reducts are then investigated. Some numerical examples are employed to substantiate the conceptual arguments.
---
paper_title: A consistency-based procedure to estimate missing pairwise preference values

paper_content:

In this paper, we present a procedure to estimate missing preference values when dealing with pairwise comparison and heterogeneous information. This procedure attempts to estimate the missing information in an expert's incomplete preference relation using only the preference values provided by that particular expert. Our procedure to estimate missing values can be applied to incomplete fuzzy, multiplicative, interval-valued, and linguistic preference relations. Clearly, it would be desirable to maintain experts' consistency levels. We make use of the additive consistency property to measure the level of consistency and to guide the procedure in the estimation of the missing values. Finally, conditions that guarantee the success of our procedure in the estimation of all the missing values of an incomplete preference relation are given. © 2008 Wiley Periodicals, Inc.
---
paper_title: Algorithms for estimating missing elements of incomplete intuitionistic preference relations

paper_content:

Preference relations as simple and efficient information description tools have been widely used in practical decision-making problems. Intuitionistic preference relation, which is often met in real problems, is usually utilized to provide the experts' vague or fuzzy opinions over objects under uncertain circumstances. Owing to the limitations of the experts' professional knowledge and experience, the provided preferences in an intuitionistic preference relation are usually incomplete. Consequently, how to estimate the missing information in an expert's incomplete intuitionistic preference relation becomes a necessary step in a decision-making process. In this paper, we define the concept of multiplicative consistent intuitionistic preference relation and develop two estimation algorithms. The first algorithm is used to estimate the missing elements using only the known preference values in an acceptable incomplete intuitionistic fuzzy preference relation with the least judgments. The second one is given for the estimation of missing elements of the acceptable incomplete intuitionistic fuzzy preference relations with more known judgments. The advantages of the developed algorithms over the existing one are detailedly analyzed, and some examples are provided to illustrate the solution processes of the algorithms and to verify their practicality and superiority. © 2011 Wiley Periodicals, Inc. © 2011 Wiley Periodicals, Inc.
---
paper_title: Intuitionistic preference relations and their application in group decision making

paper_content:

Intuitionistic fuzzy set, characterized by a membership function and a non-membership function, was introduced by Atanassov [Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87-96]. In this paper, we define the concepts of intuitionistic preference relation, consistent intuitionistic preference relation, incomplete intuitionistic preference relation and acceptable intuitionistic preference relation, and study their various properties. We develop an approach to group decision making based on intuitionistic preference relations and an approach to group decision making based on incomplete intuitionistic preference relations respectively, in which the intuitionistic fuzzy arithmetic averaging operator and intuitionistic fuzzy weighted arithmetic averaging operator are used to aggregate intuitionistic preference information, and the score function and accuracy function are applied to the ranking and selection of alternatives. Finally, a practical example is provided to illustrate the developed approaches.
---
paper_title: Incomplete interval-valued intuitionistic fuzzy preference relations

paper_content:

The aim of this paper is to investigate decision making problems with interval-valued intuitionistic fuzzy preference information, in which the preferences provided by the decision maker over alternatives are incomplete or uncertain. We define some new preference relations, including additive consistent incomplete interval-valued intuitionistic fuzzy preference relation, multiplicative consistent incomplete interval-valued intuitionistic fuzzy preference relation and acceptable incomplete interval-valued intuitionistic fuzzy preference relation. Based on the arithmetic average and the geometric mean, respectively, we give two procedures for extending the acceptable incomplete interval-valued intuitionistic fuzzy preference relations to the complete interval-valued intuitionistic fuzzy preference relations. Then, by using the interval-valued intuitionistic fuzzy averaging operator or the interval-valued intuitionistic fuzzy geometric operator, an approach is given to decision making based on the incomplete interval-valued intuitionistic fuzzy preference relation, and the developed approach is applied to a practical problem. It is worth pointing out that if the interval-valued intuitionistic fuzzy preference relation is reduced to the real-valued intuitionistic fuzzy preference relation, then all the above results are also reduced to the counterparts, which can be applied to solve the decision making problems with incomplete intuitionistic fuzzy preference information.
---
paper_title: Group Decision-Making Model With Incomplete Fuzzy Preference Relations Based on Additive Consistency

paper_content:

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
---
paper_title: Group decision making with incomplete fuzzy linguistic preference relations

paper_content:

The aim of this paper is to propose a procedure to estimate missing preference values when dealing with incomplete fuzzy linguistic preference relations assessed using a two-tuple fuzzy linguistic approach. This procedure attempts to estimate the missing information in an individual incomplete fuzzy linguistic preference relation using only the preference values provided by the respective expert. It is guided by the additive consistency property to maintain experts' consistency levels. Additionally, we present a selection process of alternatives in group decision making with incomplete fuzzy linguistic preference relations and analyze the use of our estimation procedure in the decision process. © 2008 Wiley Periodicals, Inc.
---
paper_title: A Selection Process Based on Additive Consistency to Deal with Incomplete Fuzzy Linguistic Information

paper_content:

In group decision making situations, there may be cases in which experts do not have an in-depth knowledge of the problem to be solved and, as a result, they may present incomplete information. In this paper, we present a new selection process to deal with incomplete fuzzy linguistic information. As part of it, we use an iterative procedure to estimate the missing information. This procedure is guided by the additive consistency property and only uses the preference values provided by the experts. In addition, the additive consistency property is also used to measure the level of consistency of the information provided by the experts. The main novelties of this selection process are both the possibility to manage decision situations under incomplete fuzzy linguistic information and the importance of the experts' preferences in the aggregation processes is modeled by means of the experts' consistency.
---
paper_title: FUZZY LINMAP METHOD FOR MULTIATTRIBUTE GROUP DECISION MAKING WITH LINGUISTIC VARIABLES AND INCOMPLETE INFORMATION

paper_content:

The aim of this paper is to develop a fuzzy linear programming technique for multidimensional analysis of preference (FLINMAP) in multiattribute group decision making problems with linguistic variables and incomplete preference information. In this paper, linguistic variables are used to assess an alternative on qualitative attributes using fuzzy ratings corresponding to some triangular fuzzy numbers. Each alternative is assessed on the basis of its distance to a fuzzy positive ideal solution (FPIS) which is unknown a priori. The FPIS and the weights of attributes are calculated by constructing a new linear programming model based on the group consistency and inconsistency indices defined on the basis of preferences between alternatives given by the decision makers. The distance of each alternative to the FPIS can be calculated to determine the ranking order of all alternatives. The implementation process of this methodology is demonstrated with an example.
---
paper_title: A consistency-based procedure to estimate missing pairwise preference values

paper_content:

In this paper, we present a procedure to estimate missing preference values when dealing with pairwise comparison and heterogeneous information. This procedure attempts to estimate the missing information in an expert's incomplete preference relation using only the preference values provided by that particular expert. Our procedure to estimate missing values can be applied to incomplete fuzzy, multiplicative, interval-valued, and linguistic preference relations. Clearly, it would be desirable to maintain experts' consistency levels. We make use of the additive consistency property to measure the level of consistency and to guide the procedure in the estimation of the missing values. Finally, conditions that guarantee the success of our procedure in the estimation of all the missing values of an incomplete preference relation are given. © 2008 Wiley Periodicals, Inc.
---
paper_title: Incomplete linguistic preference relations and their fusion

paper_content:

Various linguistic preference relations, including incomplete linguistic preference relation, consistent incomplete linguistic preference relation and acceptable incomplete linguistic preference relation, are introduced. Some desirable properties of the incomplete linguistic preference relation are studied. Based on the operational laws of the linguistic evaluation scale, and the acceptable incomplete linguistic preference relation with the least judgments, we develop a simple and practical method for constructing a consistent complete linguistic preference relation by using the additive transitivity property. The method not only relieves the decision maker of time pressure and makes sufficiently using of the provided preference information, but also maintains the decision maker's consistency level and avoids checking the consistency of linguistic preference relation. Furthermore, an approach to multi-person decision making based on incomplete linguistic preference relations is developed. The approach fuses all the consistent complete linguistic preference relations, constructed by using the individual acceptable incomplete linguistic preference relations with the least judgments, into a collective complete linguistic preference relation, and then the overall information corresponding to each decision alternative is aggregated. Finally, an illustrative example is given.
---
paper_title: Linear programming techniques for multidimensional analysis of preferences

paper_content:

This paper offers a new methodology for analyzing individual differences in preference judgments with regard to a set of stimuli prespecified in a multidimensional attribute space. The individual is modelled as possessing an “ideal point” denoting his most preferred stimulus location in this space and a set of weights which reveal the relative saliences of the attributes. He prefers those stimuli which are “closer” to his ideal point (in terms of a weighted Euclidean distance measure). A linear programming model is proposed for “external analysis”i.e., estimation of the coordinates of his ideal point and the weights (involved in the Euclidean distance measure) by analyzing his paired comparison preference judgments on a set of stimuli, prespecified by their coordinate locations in the multidimensional space. A measure of “poorness of fit” is developed and the linear programming model minimizes this measure overall possible solutions. The approach is fully nonmetric, extremely flexible, and uses paired comparison judgments directly. The weights can either be constrained nonnegative or left unconstrained. Generalizations of the model to consider ordinal or interval preference data and to allow an orthogonal transformation of the attribute space are discussed. The methodology is extended to perform “internal analysis,”i.e., to determine the stimuli locations in addition to weights and ideal points by analyzing the preference judgments of all subjects simultaneously. Computational results show that the methodology for external analysis is “unbiased”—i.e., on an average it recovers the “true” ideal point and weights. These studies also indicate that the technique performs satisfactorily even when about 20 percent of the paired comparison judgments are incorrectly specified.
---
paper_title: Incomplete fuzzy linguistic preference relations under uncertain environments

paper_content:

When addressing decision-making problems, decision-makers typically express their opinions using fuzzy preference relations. In some instances, decision-makers may have to deal with the problems in which only partial information is available. Consequently, decision-makers embody their preferences as incomplete fuzzy preference relations. The values of incomplete fuzzy preference relations have been considered crisp in recent studies. To allow decision-makers to provide vague or imprecise responses, this study proposes a novel method, called incomplete fuzzy linguistic preference relations, that uses fuzzy linguistic assessment variables instead of crisp values of incomplete fuzzy preference relations to ensure comparison consistency. The proposed method reflects an environment in which some uncertainty or vagueness exists. Examples are included that illustrate the effectiveness of the proposed method.
---
paper_title: Multi-criteria Decision Making with Incomplete Linguistic Preference Relations

paper_content:

When there are n criteria or alternatives in a decision matrix, a pairwise comparison methodology of analytic hierarchy process (AHP) with the time of n(n-1)/2 is frequently used to select, evaluate or rank the neighboring alternatives. But while the number of criteria or comparison level increase, the efficiency and consistency of a decision matrix decrease. To solve such problems, this study therefore uses horizontal, vertical and oblique pairwise comparisons algorithm to construct multi-criteria decision making with incomplete linguistic preference relations model (InLinPreRa). The use of pairwise comparisons will not produce the inconsistency, even allows every decision maker to choose an explicit criterion or alternative for index unrestrictedly. When there are n criteria, only n-1 pairwise comparisons need to be carried out, then one can rest on incomplete linguistic preference relations to obtain the priority value of alternative for the decision maker's reference. The decision making assessment model that constructed by this study can be extensively applied to every field of decision science and serves as the reference basis for the future research.
---
paper_title: Multiplicative consistency-based decision support system for incomplete linguistic preference relations

paper_content:

The experts may have difficulty in expressing all their preferences over alternatives or criteria, and produce the incomplete linguistic preference relation. Consistency plays an important role in estimating unknown values from an incomplete linguistic preference relation. Many methods have been developed to obtain a complete linguistic preference relation based on additive consistency, but some unreasonable values may be produced in the estimation process. To overcome this issue, we propose a new characterisation about multiplicative consistency of the linguistic preference relation, present an algorithm to estimate missing values from an incomplete linguistic preference relation, and establish a decision support system for aiding the experts to complete their linguistic preference relations in a more consistent way. Some examples are also given to illustrate the proposed methods.
---
paper_title: Group Decision-Making Model With Incomplete Fuzzy Preference Relations Based on Additive Consistency

paper_content:

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
---
paper_title: Group decision-making procedure based on incomplete reciprocal relations

paper_content:

Xu (Int J Approx Reason 36:261---270, 2004) introduced the concepts of incomplete reciprocal relation and additive consistent incomplete reciprocal relation. The aim of this paper is to develop a novel procedure for group decision making with incomplete reciprocal relations. The procedure utilizes each given incomplete reciprocal relation to construct an auxiliary reciprocal relation based on additive transitivity, and then aggregates directly these auxiliary reciprocal relations into a collective auxiliary reciprocal relation. After that, based on the collective auxiliary reciprocal relation, a simple linear system of equations is established for ranking alternatives. Finally, a numerical example is given to illustrate the developed procedure.
---
paper_title: Individual and social strategies to deal with ignorance situations in multi-person decision making

paper_content:

Multi-person decision making problems involve the preferences of some experts about a set of alternatives in order to find the best one. However, sometimes experts might not possess a precise or sufficient level of knowledge of part of the problem and as a consequence that expert might not give all the information that is required. Indeed, this may be the case when the number of alternatives is high and experts are using fuzzy preference relations to represent their preferences. In the literature, incomplete information situations have been studied, and as a result, procedures that are able to compute the missing information of a preference relation have been designed. However, these approaches usually need at least a piece of information about every alternative in the problem in order to be successful in estimating all the missing preference values.In this paper, we address situations in which an expert does not provide any information about a particular alternative, which we call situations of total ignorance. We analyze several strategies to deal with these situations. We classify these strategies into: (i) individual strategies that can be applied to each individual preference relation without taking into account any information from the rest of experts and (ii) social strategies, that is, strategies that make use of the information available from the group of experts. Both individual and social strategies use extra assumptions or knowledge, which could not be directly instantiated in the experts preference relations. We also provide an analysis of the advantages and disadvantages of each one of the strategies presented, and the situations where some of them may be more adequate to be applied than the others.
---
paper_title: Least-square method to priority of the fuzzy preference relations with incomplete information

paper_content:

The priority problem of incomplete preference relations is investigated. Using the transformation relation between multiplicative preference relation and fuzzy preference relation, we develop a least-square model to obtain the collective priority vector of the incomplete preference relations presented by multiple decision makers, with the existence condition of the solution being developed. Meanwhile, we extend this model to the cases of the fuzzy preference relations with complete information presented by multiple decision makers and the fuzzy preference relation with complete information presented by one decision maker. Finally, it is illustrated by a numerical example that the method proposed is feasible and effective.
---
paper_title: Group Decision-Making Model With Incomplete Fuzzy Preference Relations Based on Additive Consistency

paper_content:

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
---
paper_title: Incomplete pairwise comparison and consistency optimization

paper_content:

This paper proposes a new method for calculating the missing elements of an incomplete matrix of pairwise comparison values for a decision problem. The matrix is completed by minimizing a measure of global inconsistency, thus obtaining a matrix which is optimal from the point of view of consistency with respect to the available judgements. The optimal values are obtained by solving a linear system and unicity of the solution is proved under general assumptions. Some other methods proposed in the literature are discussed and a numerical example is presented.
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paper_title: Group Decision-Making Model With Incomplete Fuzzy Preference Relations Based on Additive Consistency

paper_content:

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
---
paper_title: Multiplicative consistency-based decision support system for incomplete linguistic preference relations

paper_content:

The experts may have difficulty in expressing all their preferences over alternatives or criteria, and produce the incomplete linguistic preference relation. Consistency plays an important role in estimating unknown values from an incomplete linguistic preference relation. Many methods have been developed to obtain a complete linguistic preference relation based on additive consistency, but some unreasonable values may be produced in the estimation process. To overcome this issue, we propose a new characterisation about multiplicative consistency of the linguistic preference relation, present an algorithm to estimate missing values from an incomplete linguistic preference relation, and establish a decision support system for aiding the experts to complete their linguistic preference relations in a more consistent way. Some examples are also given to illustrate the proposed methods.
---
paper_title: Group Decision-Making Model With Incomplete Fuzzy Preference Relations Based on Additive Consistency

paper_content:

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
---
paper_title: Cardinal Consistency of Reciprocal Preference Relations: A Characterization of Multiplicative Transitivity

paper_content:

Consistency of preferences is related to rationality, which is associated with the transitivity property. Many properties suggested to model transitivity of preferences are inappropriate for reciprocal preference relations. In this paper, a functional equation is put forward to model the ldquocardinal consistency in the strength of preferencesrdquo of reciprocal preference relations. We show that under the assumptions of continuity and monotonicity properties, the set of representable uninorm operators is characterized as the solution to this functional equation. Cardinal consistency with the conjunctive representable cross ratio uninorm is equivalent to Tanino's multiplicative transitivity property. Because any two representable uninorms are order isomorphic, we conclude that multiplicative transitivity is the most appropriate property for modeling cardinal consistency of reciprocal preference relations. Results toward the characterization of this uninorm consistency property based on a restricted set of (n-1) preference values, which can be used in practical cases to construct perfect consistent preference relations, are also presented.
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paper_title: Consistency Measures for Hesitant Fuzzy Linguistic Preference Relations

paper_content:

Hesitant fuzzy linguistic term sets (HFLTSs) are used to deal with situations in which the decision makers (DMs) think of several possible linguistic values or richer expressions than a single term for an indicator, alternative, variable, etc. Compared with fuzzy linguistic approaches, they are more convenient and flexible to reflect the DMs' preferences in decision making. For further applications of HFLTSs to decision making, we develop a concept of hesitant fuzzy linguistic preference relations (HFLPRs) as a tool to collect and present the DMs' preferences. Due to the importance of the consistency measures using preference relations in decision making, we develop some consistency measures for HFLPRs to ensure that the DMs are being neither random nor illogical. A consistency index is defined to establish the consistency thresholds of HFLPRs to measure whether an HFLPR is of acceptable consistency. For HFLPRs with unacceptable consistency, we develop two optimization methods to improve the consistency until they are acceptable. Several illustrative examples are given to validate the consistency measures and the optimization methods.
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paper_title: The Collapsing Method of Defuzzification for Discretised Interval Type-2 Fuzzy Sets

paper_content:

This paper proposes a new approach for defuzzification of interval type-2 fuzzy sets. The collapsing method converts an interval type-2 fuzzy set into a type-1 representative embedded set (RES), whose defuzzified values closely approximates that of the type-2 set. As a type-1 set, the RES can then be defuzzified straightforwardly. The novel representative embedded set approximation (RESA), to which the method is inextricably linked, is expounded, stated and proved within this paper. It is presented in two forms: Simple RESA: this approximation deals with the most simple interval FOU, in which a vertical slice is discretised into 2 points. Interval RESA: this approximation concerns the case in which a vertical slice is discretised into 2 or more points. The collapsing method (simple RESA version) was tested for accuracy and speed, with excellent results on both criteria. The collapsing method proved more accurate than the Karnik-Mendel iterative procedure (KMIP) for an asymmetric test set. For both a symmetric and an asymmetric test set, the collapsing method outperformed the KMIP in relation to speed.
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paper_title: Type-Reduction of General Type-2 Fuzzy Sets: The Type-1 OWA Approach

paper_content:

For general type-2 fuzzy sets, the defuzzification process is very complex and the exhaustive direct method of implementing type-reduction is computationally expensive and turns out to be impractical. This has inevitably hindered the development of type-2 fuzzy inferencing systems in real-world applications. The present situation will not be expected to change, unless an efficient and fast method of deffuzzifying general type-2 fuzzy sets emerges. Type-1 ordered weighted averaging (OWA) operators have been proposed to aggregate expert uncertain knowledge expressed by type-1 fuzzy sets in decision making. In particular, the recently developed alpha-level approach to type-1 OWA operations has proven to be an effective tool for aggregating uncertain information with uncertain weights in real-time applications because its complexity is of linear order. In this paper, we prove that the mathematical representation of the type-reduced set (TRS) of a general type-2 fuzzy set is equivalent to that of a special case of type-1 OWA operator. This relationship opens up a new way of performing type reduction of general type-2 fuzzy sets, allowing the use of the alpha-level approach to type-1 OWA operations to compute the TRS of a general type-2 fuzzy set. As a result, a fast and efficient method of computing the centroid of general type-2 fuzzy sets is realized. The experimental results presented here illustrate the effectiveness of this method in conducting type reduction of different general type-2 fuzzy sets.
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paper_title: Intuitionistic preference relations and their application in group decision making

paper_content:

Intuitionistic fuzzy set, characterized by a membership function and a non-membership function, was introduced by Atanassov [Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87-96]. In this paper, we define the concepts of intuitionistic preference relation, consistent intuitionistic preference relation, incomplete intuitionistic preference relation and acceptable intuitionistic preference relation, and study their various properties. We develop an approach to group decision making based on intuitionistic preference relations and an approach to group decision making based on incomplete intuitionistic preference relations respectively, in which the intuitionistic fuzzy arithmetic averaging operator and intuitionistic fuzzy weighted arithmetic averaging operator are used to aggregate intuitionistic preference information, and the score function and accuracy function are applied to the ranking and selection of alternatives. Finally, a practical example is provided to illustrate the developed approaches.
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paper_title: The sampling method of defuzzification for type-2 fuzzy sets: Experimental evaluation

paper_content:

For generalised type-2 fuzzy sets the defuzzification process has historically been slow and inefficient. This has hampered the development of type-2 Fuzzy Inferencing Systems for real applications and therefore no advantage has been taken of the ability of type-2 fuzzy sets to model higher levels of uncertainty. The research reported here provides a novel approach for improving the speed of defuzzification for discretised generalised type-2 fuzzy sets. The traditional type-reduction method requires every embedded type-2 fuzzy set to be processed. The high level of redundancy in the huge number of embedded sets inspired the development of our sampling method which randomly samples the embedded sets and processes only the sample. The paper presents detailed experimental results for defuzzification of constructed sets of known defuzzified value. The sampling defuzzifier is compared on aggregated type-2 fuzzy sets resulting from the inferencing stage of a FIS, in terms of accuracy and speed, with other methods including the exhaustive and techniques based on the @a-planes representation. The results indicate that by taking only a sample of the embedded sets we are able to dramatically reduce the time taken to process a type-2 fuzzy set with very little loss in accuracy.
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paper_title: Defuzzification of the discretised generalised type-2 fuzzy set: Experimental evaluation

paper_content:

The work reported in this paper addresses the challenge of the efficient and accurate defuzzification of discretised generalised type-2 fuzzy sets as created by the inference stage of a Mamdani Fuzzy Inferencing System. The exhaustive method of defuzzification for type-2 fuzzy sets is extremely slow, owing to its enormous computational complexity. Several approximate methods have been devised in response to this defuzzification bottleneck. In this paper we begin by surveying the main alternative strategies for defuzzifying a generalised type-2 fuzzy set: (1) Vertical Slice Centroid Type-Reduction; (2) the sampling method; (3) the elite sampling method; and (4) the a-planes method. We then evaluate the different methods experimentally for accuracy and efficiency. For accuracy the exhaustive method is used as the standard. The test results are analysed statistically by means of the Wilcoxon Nonparametric Test and the elite sampling method shown to be the most accurate. In regards to efficiency, Vertical Slice Centroid Type-Reduction is demonstrated to be the fastest technique.
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paper_title: Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions

paper_content:

(NOTE: Each chapter concludes with Exercises.) I: PRELIMINARIES. 1. Introduction. Rule-Based FLSs. A New Direction for FLSs. New Concepts and Their Historical Background. Fundamental Design Requirement. The Flow of Uncertainties. Existing Literature on Type-2 Fuzzy Sets. Coverage. Applicability Outside of Rule-Based FLSs. Computation. Supplementary Material: Short Primers on Fuzzy Sets and Fuzzy Logic. Primer on Fuzzy Sets. Primer on FL. Remarks. 2. Sources of Uncertainty. Uncertainties in a FLS. Words Mean Different Things to Different People. 3. Membership Functions and Uncertainty. Introduction. Type-1 Membership Functions. Type-2 Membership Functions. Returning to Linguistic Labels. Multivariable Membership Functions. Computation. 4. Case Studies. Introduction. Forecasting of Time-Series. Knowledge Mining Using Surveys. II: TYPE-1 FUZZY LOGIC SYSTEMS. 5. Singleton Type-1 Fuzzy Logic Systems: No Uncertainties. Introduction. Rules. Fuzzy Inference Engine. Fuzzification and Its Effect on Inference. Defuzzification. Possibilities. Fuzzy Basis Functions. FLSs Are Universal Approximators. Designing FLSs. Case Study: Forecasting of Time-Series. Case Study: Knowledge Mining Using Surveys. A Final Remark. Computation. 6. Non-Singleton Type-1 Fuzzy Logic Systems. Introduction. Fuzzification and Its Effect on Inference. Possibilities. FBFs. Non-Singleton FLSs Are Universal Approximators. Designing Non-Singleton FLSs. Case Study: Forecasting of Time-Series. A Final Remark. Computation. III: TYPE-2 FUZZY SETS. 7. Operations on and Properties of Type-2 Fuzzy Sets. Introduction. Extension Principle. Operations on General Type-2 Fuzzy Sets. Operations on Interval Type-2 Fuzzy Sets. Summary of Operations. Properties of Type-2 Fuzzy Sets. Computation. 8. Type-2 Relations and Compositions. Introduction. Relations in General. Relations and Compositions on the Same Product Space. Relations and Compositions on Different Product Spaces. Composition of a Set with a Relation. Cartesian Product of Fuzzy Sets. Implications. 9. Centroid of a Type-2 Fuzzy Set: Type-Reduction. Introduction. General Results for the Centroid. Generalized Centroid for Interval Type-2 Fuzzy Sets. Centroid of an Interval Type-2 Fuzzy Set. Type-Reduction: General Results. Type-Reduction: Interval Sets. Concluding Remark. Computation. IV: TYPE-2 FUZZY LOGIC SYSTEMS. 10. Singleton Type-2 Fuzzy Logic Systems. Introduction. Rules. Fuzzy Inference Engine. Fuzzification and Its Effect on Inference. Type-Reduction. Defuzzification. Possibilities. FBFs: The Lack Thereof. Interval Type-2 FLSs. Designing Interval Singleton Type-2 FLSs. Case Study: Forecasting of Time-Series. Case Study: Knowledge Mining Using Surveys. Computation. 11. Type-1 Non-Singleton Type-2 Fuzzy Logic Systems. Introduction. Fuzzification and Its Effect on Inference. Interval Type-1 Non-Singleton Type-2 FLSs. Designing Interval Type-1 Non-Singleton Type-2 FLSs. Case Study: Forecasting of Time-Series. Final Remark. Computation. 12. Type-2 Non-Singleton Type-2 Fuzzy Logic Systems. Introduction. Fuzzification and Its Effect on Inference. Interval Type-2 Non-Singleton Type-2 FLSs. Designing Interval Type-2 Non-Singleton Type-2 FLSs. Case Study: Forecasting of Time-Series. Computation. 13. TSK Fuzzy Logic Systems. Introduction. Type-1 TSK FLSs. Type-2 TSK FLSs. Example: Forecasting of Compressed Video Traffic. Final Remark. Computation. 14. Epilogue. Introduction. Type-2 Versus Type-1 FLSs. Appropriate Applications for a Type-2 FLS. Rule-Based Classification of Video Traffic. Equalization of Time-Varying Non-linear Digital Communication Channels. Overcoming CCI and ISI for Digital Communication Channels. Connection Admission Control for ATM Networks. Potential Application Areas for a Type-2 FLS. A. Join, Meet, and Negation Operations For Non-Interval Type-2 Fuzzy Sets. Introduction. Join Under Minimum or Product t-Norms. Meet Under Minimum t-Norm. Meet Under Product t-Norm. Negation. Computation. B. Properties of Type-1 and Type-2 Fuzzy Sets. Introduction. Type-1 Fuzzy Sets. Type-2 Fuzzy Sets. C. Computation. Type-1 FLSs. General Type-2 FLSs. Interval Type-2 FLSs. References. Index.
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paper_title: Accuracy and complexity evaluation of defuzzification strategies for the discretised interval type-2 fuzzy set

paper_content:

The work reported in this paper addresses the challenge of the efficient and accurate defuzzification of discretised interval type-2 fuzzy sets. The exhaustive method of defuzzification for type-2 fuzzy sets is extremely slow, owing to its enormous computational complexity. Several approximate methods have been devised in response to this bottleneck. In this paper we survey four alternative strategies for defuzzifying an interval type-2 fuzzy set: 1. The Karnik-Mendel Iterative Procedure, 2. the Wu-Mendel Approximation, 3. the Greenfield-Chiclana Collapsing Defuzzifier, and 4. the Nie-Tan Method. We evaluated the different methods experimentally for accuracy, by means of a comparative study using six representative test sets with varied characteristics, using the exhaustive method as the standard. A preliminary ranking of the methods was achieved using a multi-criteria decision making methodology based on the assignment of weights according to performance. The ranking produced, in order of decreasing accuracy, is 1. the Collapsing Defuzzifier, 2. the NieTan Method, 3. the Karnik-Mendel Iterative Procedure, and 4. the Wu-Mendel Approximation. Following that, a more rigorous analysis was undertaken by means of the Wilcoxon Nonparametric Test, in order to validate the preliminary test conclusions. It was found that there was no evidence of a significant difference between the accuracy of the Collapsing and Nie-Tan Methods, and between that of the Karnik-Mendel Iterative Procedure and the Wu-Mendel Approximation. However, there was evidence to suggest that the collapsing and Nie-Tan Methods are more accurate than the Karnik-Mendel Iterative Procedure and the Wu-Mendel Approximation. In relation to efficiency, each method’s computational complexity was analysed, resulting in a ranking (from least computationally complex to most computationally complex) as follows: 1. the Nie-Tan Method, 2. the Karnik-Mendel Iterative Procedure (lowest complexity possible), 3. the Greenfield-Chiclana Collapsing Defuzzifier, 4. the Karnik-Mendel Iterative Procedure (highest complexity possible), and 5. the Wu-Mendel Approximation.
---
paper_title: A statistical comparative study of different similarity measures of consensus in group decision making

paper_content:

An essential aim in group decision making (GDM) problems is to achieve a high level of consensus among experts. Consensus is defined as general or widespread agreement, and it is usually modelled mathematically via a similarity function measuring how close experts' opinions or preferences are. Similarity functions are defined based on the use of a metric describing the distance between experts' opinions or preferences. In the literature, different metrics or distance functions have been proposed to implement in consensus models, but no study has been carried out to analyse the influence the use of different distance functions can have in the GDM process. This paper presents a comparative study of the effect of the application of some different distance functions for measuring consensus in GDM. By using the nonparametric Wilcoxon matched-pairs signed-ranks test, it is concluded that different distance functions can produce significantly different results. Moreover, it is also shown that their application also has a significant effect on the speed of achieving consensus. Finally, these results are analysed and used to derive decision support rules, based on a convergent criterion, that can be used to control the convergence speed of the consensus process using the compared distance functions.
---
paper_title: Reconstruction Methods for Incomplete Fuzzy Preference Relations: A Numerical Comparison

paper_content:

In this paper we compare, by means of numerical simulations, seven different methods for reconstructing incomplete fuzzy preference relations. We consider the case of highly inconsistent preference relations as well as the case of preference relations close to consistency. We compare the numerical results on the basis of the consistency of the reconstructed preference relations.
---
paper_title: A linguistic consensus model for Web 2.0 communities

paper_content:

Web 2.0 communities are a quite recent phenomenon which involve large numbers of users and where communication between members is carried out in real time. Despite of those good characteristics, there is still a necessity of developing tools to help users to reach decisions with a high level of consensus in those new virtual environments. In this contribution a new consensus reaching model is presented which uses linguistic preferences and is designed to minimize the main problems that this kind of organization presents (low and intermittent participation rates, difficulty of establishing trust relations and so on) while incorporating the benefits that a Web 2.0 community offers (rich and diverse knowledge due to a large number of users, real-time communication, etc.). The model includes some delegation and feedback mechanisms to improve the speed of the process and its convergence towards a solution of consensus. Its possible application to some of the decision making processes that are carried out in the Wikipedia is also shown.
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paper_title: A Framework for Adopting Collaboration 2.0 Tools for Virtual Group Decision Making

paper_content:

Decision making in virtual teams is gaining momentum due to globalization, mobility of employees, and the need for collective and rapid decision making by members who are in different locations. These factors resulted in a proliferation of virtual team software support tools for decision making, the latest of which is social software (also known as collaboration 2.0), which includes tools such as wikis, blogs, microblogs, discussion forums, and social networking platforms. This paper describes the potential use of collaboration 2.0 software for improving the process and the specific tasks in virtual group decision making. The paper proposes a framework for exploring the fitness between social software and the major activities in the group decision making process and how such tools can be successfully adopted. Specifically, we use a fit-viability model to help assessing whether social software fit a decision task and what organizational factors are important for such tools to be effective. Representative research issues related to the use of such tools are also presented.
---

Title: Managing incomplete preference relations in decision making: A review and future trends

Section 1: Introduction
Description 1: Introduce the topic of group decision making (GDM) and the issues arising from incomplete preference relations.

Section 2: Preference Relations in Decision Making
Description 2: Review the principal types of preference relations used in decision making, including a description of their consistency.

Section 3: Numeric Preferences
Description 3: Analyze the main types of numeric preference relations such as crisp, additive, multiplicative, interval-valued, and intuitionistic preference relations.

Section 4: Linguistic Preferences
Description 4: Discuss how linguistic preferences are used in decision making, including methodologies based on cardinal and ordinal representation.

Section 5: Consistency of Preferences
Description 5: Detail the various consistency properties of preference relations and their importance in decision making.

Section 6: Advantages and Drawbacks of Preference Relations
Description 6: Outline the advantages and drawbacks of using different types of preference relations in decision making.

Section 7: Decision Making Approaches with Incomplete Preferences
Description 7: Review the main techniques developed to handle incomplete information in decision making, divided into iterative and optimization approaches.

Section 8: Managing Missing Preference Values in APRs and MPRs
Description 8: Describe specific methods to manage incomplete additive and multiplicative preference relations.

Section 9: Managing Missing Preference Values in IVPRs and IFPRs
Description 9: Analyze methods to handle incomplete interval-valued and intuitionistic preference relations.

Section 10: Managing Missing Preference Values in LPRs
Description 10: Discuss methodologies to deal with incomplete linguistic preference relations.

Section 11: Processes Dealing with Ignorance Situations in GDM
Description 11: Present strategies to handle situations where experts provide no information about certain alternatives in GDM.

Section 12: Advantages and Drawbacks of Ignorance Strategies
Description 12: Evaluate the advantages and drawbacks of different ignorance handling strategies in group decision making.

Section 13: Trends and Future Work
Description 13: Outline current trends and future research directions in managing incomplete preferences in decision making, including hesitant and type-2 preference relations and GDM approaches in Web 2.0 contexts.

Section 14: Conclusions
Description 14: Summarize the findings and emphasize the importance of developing methodologies to handle incomplete preference relations in decision making.

Section 15: Acknowledgments
Description 15: Acknowledge the funding and support received for the research.