Fuzzy least absolute linear regression

The distance between triangular fuzzy numbers is an important research topic in many fields. In this paper, we introduce a new distance between triangular fuzzy numbers, merge least absolute deviation method with the new distance and propose fuzzy regression model. We also investigate the properties and model algorithm of fuzzy least absolute linear regression model in detail by transforming this model into linear programming. Further, we use three numerical examples to illustrate our proposed model reasonable and make some comparisons with some existing fuzzy regression models. Finally, we investigate the robust property of our proposed model and apply our model in the missing data set to verify model data.     

MAGDM Linear-Programming Models With Distinct Uncertain Preference Structures

Group decision making with preference information on alternatives is an interesting and important research topic which has been receiving more and more attention in recent years. The purpose of this paper is to investigate multiple-attribute group decision-making (MAGDM) problems with distinct uncertain preference structures. We develop some linear-programming models for dealing with the MAGDM problems, where the information about attribute weights is incomplete, and the decision makers have their preferences on alternatives. The provided preference information can be represented in the following three distinct uncertain preference structures: 1) interval utility values; 2) interval fuzzy preference relations; and 3) interval multiplicative preference relations. We first establish some linear-programming models based on decision matrix and each of the distinct uncertain preference structures and, then, develop some linear-programming models to integrate all three structures of subjective uncertain preference information provided by the decision makers and the objective information depicted in the decision matrix. Furthermore, we propose a simple and straightforward approach in ranking and selecting the given alternatives. It is worth pointing out that the developed models can also be used to deal with the situations where the three distinct uncertain preference structures are reduced to the traditional ones, i.e., utility values, fuzzy preference relations, and multiplicative preference relations. Finally, we use a practical example to illustrate in detail the calculation process of the developed approach.      

Compatibility of interval fuzzy preference relations with the COWA operator and its application to group decision making

We develop a new compatibility for the interval fuzzy preference relations based on the continuous ordered weighted averaging (COWA) operator and use it to determine the weights of experts in group decision making (GDM). We define some concepts of the compatibility degree and the compatibility index for the two interval fuzzy preference relations based on the COWA operator. We study some desirable properties of the compatibility index and investigate the relationship between the each expert’s interval fuzzy preference relation and the synthetic interval fuzzy preference relation. The prominent characteristic of the compatibility index based on the COWA operator is that it can deal with the compatibility of all the arguments by using a controlled parameter considering the attitude of decision maker rather than the compatibility of the simply two points in intervals. To determine the experts’ weights in the GDM with the interval fuzzy preference relations, we propose an optimal model based on the criterion of minimizing the compatibility index. In the end, we give a numerical example to develop the new approach to GDM with interval fuzzy preference relations.     

Multi-criteria group decision making with incomplete hesitant fuzzy preference relations

In order to simulate the hesitancy and uncertainty associated with impression or vagueness, a decision maker may give her/his judgments by means of hesitant fuzzy preference relations in the process of decision making. The study of their consistency becomes a very important aspect to avoid a misleading solution. This paper defines the concept of additive consistent hesitant fuzzy preference relations. The characterizations of additive consistent hesitant fuzzy preference relations are studied in detail. Owing to the limitations of the experts’ professional knowledge and experience, the provided preferences in a hesitant fuzzy preference relation are usually incomplete. Consequently, this paper introduces the concepts of incomplete hesitant fuzzy preference relation, acceptable incomplete hesitant fuzzy preference relation, and additive consistent incomplete hesitant fuzzy preference relation. Then, two estimation procedures are developed to estimate the missing information in an expert's incomplete hesitant fuzzy preference relation. The first procedure is used to construct an additive consistent hesitant fuzzy preference relation from the lowest possible number, (n − 1), of pairwise comparisons. The second one is designed for the estimation of missing elements of the acceptable incomplete hesitant fuzzy preference relations with more known judgments. Moreover, an algorithm is given to solve the multi-criteria group decision making problem with incomplete hesitant fuzzy preference relations. Finally, a numerical example is provided to illustrate the solution processes of the developed algorithm and to verify its effectiveness and practicality.     

Group decision making with incomplete fuzzy preference relations based on the additive consistency and the order consistency

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.     

Multiple-attribute group decision making with different formats of preference information on attributes.

Interval utility values, interval fuzzy preference relations, and interval multiplicative preference relations are three common uncertain-preference formats used by decision-makers to provide their preference information in the process of decision making under fuzziness. This paper is devoted in investigating multiple-attribute group-decision-making problems where the attribute values are not precisely known but the value ranges can be obtained, and the decision-makers provide their preference information over attributes by three different uncertain-preference formats i.e., 1) interval utility values; 2) interval fuzzy preference relations; and 3) interval multiplicative preference relations. We first utilize some functions to normalize the uncertain decision matrix and then transform it into an expected decision matrix. We establish a goal-programming model to integrate the expected decision matrix and all three different uncertain-preference formats from which the attribute weights and the overall attribute values of alternatives can be obtained. Then, we use the derived overall attribute values to get the ranking of the given alternatives and to select the best one(s). The model not only can reflect both the subjective considerations of all decision-makers and the objective information but also can avoid losing and distorting the given objective and subjective decision information in the process of information integration. Furthermore, we establish some models to solve the multiple-attribute group-decision-making problems with three different preference formats: 1) utility values; 2) fuzzy preference relations; and 3) multiplicative preference relations. Finally, we illustrate the applicability and effectiveness of the developed models with two practical examples.     

基于粒计算的犹豫模糊多准则决策方法*

提出基于粒计算的犹豫模糊多准则决策方法.给出各个准则下对应的犹豫模糊集中犹豫模糊元的大于可能度定义,并构造相应准则下的加性一致的模糊偏好矩阵.根据各准则的模糊偏好矩阵对应的预序熵及预序粒结构相似度确定属性的权重,对各个准则下模糊偏好矩阵的排序向量加权平均得到最终的排序向量.文中方法以评价数据序信息量及准则序与整体之间的关系确定准则权重,通过计算加权两两比较下的排序向量得到最终的排序决策结果.最后运用实例验证算法的有效性及可行性.     

Intelligent data analysis with fuzzy decision trees

Intelligent data analysis has gained increasing attention in business and industry environments. Many applications are looking not only for solutions that can automate and de-skill the data analysis process, but also methods that can deal with vague information and deliver comprehensible models. Under this consideration, we present an automatic data analysis platform, in particular, we investigate fuzzy decision trees as a method of intelligent data analysis for classification problems. We present the whole process from fuzzy tree learning, missing value handling to fuzzy rules generation and pruning. To select the test attributes of fuzzy trees we use a generalized Shannon entropy. We discuss the problems connected with this generalization arising from fuzzy logic and propose some amendments. We give a theoretical comparison on the fuzzy rules learned by fuzzy decision trees with some other methods, and compare our classifiers to other well-known classification methods based on experimental results. Moreover, we show a real-world application for the quality control of car surfaces using our approach.     

Preference relations based on hesitant-intuitionistic fuzzy information and their application in group decision making

Preference relations are a powerful quantitative decision approach that assists decision makers in expressing their preferences over alternatives. In real-life applications, decision makers may not be able to provide exact preference information with crisp numbers. To solve this problem, a hesitant-intuitionistic fuzzy number (Hesitant-IFN) is proposed in this paper, and a proposal for the hesitant-intuitionistic fuzzy preference relation (Hesitant-IFPR) and its complementary form (Hesitant-IFCPR) for uncertain preference information are presented. Compared with other preference relations, the proposed relations use hesitant fuzzy elements (HFEs) to express the priority intensities of decision makers and produce the corresponding non-priority intensities by a conversion formula. In addition, we have deduced the operational laws and comparative methods of Hesitant-IFNs and used such information to investigate the corresponding aggregation operators and the approximate consistency tests. Next, we have constructed a group decision-making approach under a hesitant-intuitionistic fuzzy environment. Finally, two case studies are presented to illustrate the preference relations, the approximate consistency tests and the group decision method.     

Estimating incomplete information in group decision making: A framework of granular computing

A general assumption in group decision making scenarios is that of all individuals possess accurate knowledge of the entire problem under study, including the abilities to make a distinction of the degree up to which an alternative is better than other one. However, in many real world scenarios, this may be unrealistic, particularly those involving numerous individuals and options to choose from conflicting and dynamics information sources. To manage such a situation, estimation methods of incomplete information, which use own assessments provided by the individuals and consistency criteria to avoid discrepancy, have been widely employed under fuzzy preference relations. In this study, we introduce the information granularity concept to estimate missing values supporting the objective of obtaining complete fuzzy preference relations with higher consistency levels. We use the concept of granular preference relations to form each missing value as a granule of information in place of a crisp number. This offers the flexibility that is required to estimate the missing information so that the consistency levels related to the complete fuzzy preference relations are as higher as possible.     

Group decision making with incomplete fuzzy linguistic preference relations

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.     

A method based on preference degrees for handling hybrid multiple attribute decision making problems

In this paper, we investigate hybrid multiple attribute decision making problems with various forms of attribute values (real numbers, linguistic labels, interval numbers, intuitionistic fuzzy numbers and interval intuitionistic fuzzy numbers). We propose a method based on preference degrees which may take the forms of fuzzy numbers, intuitionistic fuzzy numbers and interval intuitionistic fuzzy numbers. The method first normalizes various forms of attribute values into preference degrees, and then uses a preference degree-based weighted averaging operator to aggregate the normalized preference degrees. Meanwhile, for convenience of calculation, a new linguistic representation model is presented, whose feasibility is verified by comparing it with the traditional 4-tuple linguistic representation model, and from our model, the mapping relationship between interval intuitionistic fuzzy numbers and linguistic labels can be constructed. Finally, we illustrate the rationality and practicality of the proposed method by an application example.     

Incomplete interval-valued intuitionistic fuzzy preference relations

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.     

Computing the Numerical Scale of the Linguistic Term Set for the 2-Tuple Fuzzy Linguistic Representation Model

When using linguistic approaches to solve decision problems, we need the techniques for computing with words (CW). Together with the 2-tuple fuzzy linguistic representation models (i.e., the Herrera and MartÍnez model and the Wang and Hao model), some computational techniques for CW are also developed. In this paper, we define the concept of numerical scale and extend the 2-tuple fuzzy linguistic representation models under the numerical scale. We find that the key of computational techniques based on linguistic 2-tuples is to set suitable numerical scale with the purpose of making transformations between linguistic 2-tuples and numerical values. By defining the concept of the transitive calibration matrix and its consistent index, this paper develops an optimization model to compute the numerical scale of the linguistic term set. The desired properties of the optimization model are also presented. Furthermore, we discuss how to construct the transitive calibration matrix for decision problems using linguistic preference relations and analyze the linkage between the consistent index of the transitive calibration matrix and one of the linguistic preference relations. The results in this paper are pretty helpful to complete the fuzzy 2-tuple representation models for CW.      

Fuzzy query processing for document retrieval based on extendedfuzzy concept networks

In this paper, we present a new method for fuzzy query processing for document retrieval based on extended fuzzy concept networks. In an extended fuzzy concept network, there are four kinds of fuzzy relationships between concepts, i.e., fuzzy positive association, fuzzy negative association, fuzzy generalization, and fuzzy specialization. An extended fuzzy concept network can be modeled by a relation matrix and a relevance matrix, where the elements in a relation matrix represent the fuzzy relationships between concepts, and the elements in a relevance matrix indicate the degrees of relevance between concepts. The implicit fuzzy relationships between concepts can be inferred by the transitive closure of the relation matrix. The implicit degrees of relevance between concepts also can be inferred by the transitive closure of the relevance matrix. The proposed method allows the users to perform positive queries, negative queries, generalization queries, and specialization queries. The proposed method allows the users to perform fuzzy queries in a more flexible and more intelligent manner.     

Elicitation strategies for soft constraint problems with missing preferences: Properties,algorithms and experimental studies

We consider soft constraint problems where some of the preferences may be unspecified. This models, for example, settings where agents are distributed and have privacy issues, or where there is an ongoing preference elicitation process. In this context, we study how to find an optimal solution without having to wait for all the preferences. In particular, we define algorithms, that interleave search and preference elicitation, to find a solution which is necessarily optimal, that is, optimal no matter what the missing data will be, with the aim to ask the user to reveal as few preferences as possible. We define a combined solving and preference elicitation scheme with a large number of different instantiations, each corresponding to a concrete algorithm, which we compare experimentally. We compute both the number of elicited preferences and the user effort, which may be larger, as it contains all the preference values the user has to compute to be able to respond to the elicitation requests. While the number of elicited preferences is important when the concern is to communicate as little information as possible, the user effort measures also the hidden work the user has to do to be able to communicate the elicited preferences. Our experimental results on classical, fuzzy, weighted and temporal incomplete CSPs show that some of our algorithms are very good at finding a necessarily optimal solution while asking the user for only a very small fraction of the missing preferences. The user effort is also very small for the best algorithms.     

Enhancing PROMETHEE method with intuitionistic fuzzy soft sets

The notion of intuitionistic fuzzy soft sets (IFSSs) provides an effective tool for solving multiple attribute decision making with intuitionistic fuzzy information. The most crucial issue in decision making based on IFSSs is how to derive the ranking of alternatives from the information quantified in terms of intuitionistic fuzzy values. In this study, we propose a new extension of the preference ranking organization method for enrichment evaluation (PROMETHEE), by taking advantage of IFSSs. In addition to presenting a myriad of new notions, such as intuitionistic fuzzy membership (or nonmembership) deviation matrices, intuitionistic fuzzy membership (or nonmembership) preference matrices, and aggregated intuitionistic fuzzy preference matrices, we put more emphasis on the construction of three distinct preference structures and related utility functions on the corresponding weakly ordered sets by considering the positive, negative, and net flows of the alternatives based on the aggregated intuitionistic fuzzy preference matrix. We present a new algorithm for solving multiple attribute decision-making problems with the extended PROMETHEE method based on IFSSs. Moreover, a benchmark problem concerning risk investment is investigated to give a comparative analysis and show the feasibility of our approach.     

An alternative to fuzzy methods in decision-making problems

In this work we present a construction method for Atanassov’s intuitionistic fuzzy preference relations starting from fuzzy preference relations and taking into account the ignorance of the expert in the construction of the latter. Moreover, we propose two generalizations of the weighted voting strategy to work with Atanassov’s intuitionistic fuzzy preference relations. An advantage of these algorithms is that they start from fuzzy preference relations and their results can be compared with those of any other decision-making algorithm based on fuzzy sets theory. We verify that our proposal is able to provide a unique solution in some cases in which the voting strategy is not able to order the alternatives.     

Hesitant fuzzy information measures and their applications in multi-criteria decision making

Hesitant fuzzy set (HFS) is a powerful decision tool to express uncertain information more flexibly and comprehensively. The aim of this paper is to propose more reasonable information measures for HFSs in comparison with the existing ones. First, a series of distance measures is suggested for hesitant fuzzy element and hesitant fuzzy sets. These measures are directly calculated from hesitant fuzzy elements without judging the decision-makers’ risk preference and adding any values into the hesitant fuzzy element with the smaller number of elements. Then, some similarity and entropy measures are proposed based on the transforming relationship among the information measures. Additionally, based on the proposed information measures, a TOPSIS method for hesitant fuzzy information is provided. Finally, some numerical examples are used in order to illustrate the proposed decision method and a comparative analysis is made to demonstrate that the suggested measures are more objective and feasible in certain cases.