Interval-valued Fuzzy Relations Research Articles

Analytical-solution-based interval fuzzy utility acquisition and decision making with interval-valued fuzzy preference relations based on additive consistency

Acquiring an analytical solution of interval fuzzy utility vectors (IFUVs) from interval-valued fuzzy preference relations (IVFPRs) plays a key role in improving the efficiency and performance of decision making with IVFPRs. This study devises formulas to compute uncertainty indices of interval-valued fuzzy assessments and IVFPRs, and builds uncertainty constraints among fuzzy assessments in an additively consistent IVFPR. By dividing all IVFPRs with the same size into two categories, least square models with constraints of uncertainty indices are set up and their analytical solutions are found to respectively acquire normalized IFUVs from the two categories of IVFPRs. The analytical solutions are then integrated into unified computational formulas acquiring normalized IFUVs from IVFPRs. On the basis of analytical-solution-based IFUVs, a distance based additive consistency index is devised and a novel acceptability concept is presented by taking both additive inconsistency acceptability and uncertainty acceptability into account. Subsequently, this paper proposes a multi-criteria decision making method with highlighting individual characteristics of alternatives. An illustration with one consistent IVFPR and three inconsistent IVFPRs is offered and a comparative study is implemented to validate the models presented. An annual-performance-evaluation-based outstanding teacher recommendation problem is utilized to show the practicality of the decision method proposed.

Interval-valued fuzzy discernibility pair approach for attribute reduction in incomplete interval-valued information systems

Interval-valued information systems provide rich semantic interpretation and greater flexibility compared to real-valued information systems. Meanwhile, incomplete interval-valued information systems containing missing values exist widely in reality. One of the key issues in handling incomplete interval-valued information systems is attribute reduction. Unfortunately, there are few studies on attribute reduction for incomplete interval-valued information systems. Moreover, most of the existing studies use fuzzy similarity relations to measure the similarity of objects. Due to the complexity of incomplete interval-valued information systems, traditional fuzzy similarity relations cannot adequately represent the information contained within incomplete interval-valued information systems. To address this issue, this study proposes interval-valued fuzzy min-max similarity relations for incomplete interval-valued information systems, along with two attribute reduction algorithms based on interval-valued fuzzy discernibility pairs model for incomplete interval-valued information systems. Firstly, this study defines interval-valued fuzzy min-max similarity relations using distance-based extended restricted equivalence functions. Secondly, this study presents the interval-valued fuzzy discernibility pairs and discerning ability measure for attribute sets. Furthermore, an attribute reduction algorithm is proposed for incomplete interval-valued information systems. Thirdly, taking into account decision attributes, the relative interval-valued fuzzy discernibility pairs and an attribute reduction algorithm for incomplete interval-valued decision information systems are proposed. Finally, the effectiveness of the proposed method is compared to four representative attribute reduction algorithms in experiments. The experimental results illustrate that our method is effective and has advantages over the compared algorithms.

Multiplicative consistency analysis of interval-valued fuzzy preference relations

Interval-valued fuzzy preference relations (IVFPRs) have been applied to many real-life decision-making problems. However, most definitions of consistency of IVFPRs do not satisfy invariability to compared objects’ labels. To overcome this drawback, this paper mainly focuses on the multiplicative consistency analysis of interval-valued fuzzy preference relations (IVFPRs). Firstly, this paper proposes a new multiplicative consistency of complete IVFPRs. It is proved that this new multiplicative consistency is robust and invariable to compared objects’ labels. Then, the definition of acceptable incomplete IVFPRs (In-IVFPRs) is presented. To make full use of all direct and indirect evaluations of decision-makers, an algorithm is devised to evaluate the missing elements of an acceptable incomplete In-IVFPR. To comprehensively describe the closeness between any two complete IVFPRs, the total deviation of two complete IVFPRs is defined based on the p-norm of a vector. By minimizing the total deviation of two complete IVFPRs, a programming model is built to determine an interval weight vector from a complete IVFPR. Subsequently, a novel decision-making method with an In-IVFPR is proposed. Lastly, three practical and numerical examples and simulation-based comparative analyses are provided to further validate the practicability and advantages of the proposed method.

Multiperson Decision-Making Using Consistent Interval-Valued Fuzzy Information with Application in Supplier Selection

This study describes a consistency-based approach for multiperson decision-making (MPDM) in which decision-makers’ suggestions are expressed as incomplete interval-valued fuzzy preference relations. The presented approach utilizes Lukasiewicz’s t-norm in conjunction with additive reciprocity to obtain comprehensive interval valued fuzzy preference relations from each expert, and the transitive closure formula also produces L-consistency. We would evaluate the consistency weights of the experts using consistency analysis. Experts are allocated final priority weights by combining the consistency weights and preset weights. A collective consistency matrix is then constructed from the weighted sum of preference matrices. After computing the possibility degrees, the normalization procedure is utilized to generate complimentary matrices, and the final ranking values of alternatives are derived as well. Finally, a numerical example demonstrates the efficacy of the suggested approach following a comparison analysis.

Triangular Bounded Consistency of Interval-Valued Fuzzy Preference Relations

The consistency of interval-valued fuzzy preference relations (IFPRs) is a prerequisite for the application of IFPRs in real problems. To support the application of IFPRs, various types of consistency of IFPRs have been developed. They all satisfy some fixed mathematical conditions under the assumption that decision makers are perfectly rational. In practice, this assumption is usually violated because decision makers generally have bounded rationality. Considering the bounded rationality of decision makers, this article develops a new type of consistency of IFPRs called triangular bounded consistency, which is based on the historical preferences of decision makers. A triangular framework is designed to describe the three IFPRs of any three alternatives. The strict transitivity of IFPRs is defined as the restricted max-max transitivity of IFPRs, which is reconstructed in the triangular framework. In this situation, under the assumption that the preferences of a decision maker are consistent in similar circumstances, the triangular bounded consistency of IFPRs is defined by use of the historical IFPRs of decision makers. Based on the developed consistency, the process of determining the optimal estimations of missing IFPRs in an incomplete IFPR matrix is developed. A problem of selecting suppliers of simulation systems is analyzed using the multicriteria group decision-making (MCGDM) process with the triangular bounded consistency of IFPRs to demonstrate the application of the developed consistency in MCGDM.

An improved HK model-driven consensus reaching for group decision making under interval-valued fuzzy preference relations with self-confidence

Self-confidence actually affects group decision making (GDM) because it can somehow reflect decision maker (DM)’s expertise and cognitive level. This paper proposes a new form of preference relation named interval-valued fuzzy preference relation with self-confidence (IFPR-SC). IFPR-SC requires DMs to express self-confidence levels when offering interval-valued preferences. Some operation laws regarding IFPR-SC are presented. Moreover, this paper uses an induced-ordered weighted aggregation operator to aggregate individual IFPRs-SC, then presents a novel feedback adjustment mechanism called improved Hegselmann-Kraus (IHK) model to update DMs’ IFPRs-SC. Finally, this paper offers an illustrative example and several comparisons to verify the feasibility and efficiency of IHK model.

A novel group decision making method for interval-valued pythagorean fuzzy preference relations

Interval-valued Pythagorean fuzzy preference relation (IVPFPR) plays an important role in representing the complex and uncertain information. The application of IVPFPRs gives better solutions in group decision making (GDM). In this paper, we investigate a new method to solve GDM problems with IVPFPRs. Firstly, novel multiplicative consistency and consensus measures are proposed. Subsequently, the procedure for improving consistency and consensus levels are put forward to ensure that every individual IVPFPR is of acceptable multiplicative consistency and consensus simultaneously. In the context of minimizing the deviations between the individual and collective IVPFPRs, the objective experts’ weights are decided according to the optimization model and the aggregated IVPFPR is derived. Afterwards, a programming model is built to derive the normalized Pythagorean fuzzy priority weights, then the priority weights of alternatives are identified as well. An algorithm for GDM method with IVPFPRs is completed. Finally, an example is cited and comparative analyses with previous approaches are conducted to illustrate the applicability and effectiveness of the proposed method.

Development of Granular Fuzzy Relation Equations Based on a Subset of Data

Developing and optimizing fuzzy relation equations are of great relevance in system modeling, which involves analysis of numerous fuzzy rules. As each rule varies with respect to its level of influence, it is advocated that the performance of a fuzzy relation equation is strongly related to a subset of fuzzy rules obtained by removing those without significant relevance. In this study, we establish a novel framework of developing granular fuzzy relation equations that concerns the determination of an optimal subset of fuzzy rules. The subset of rules is selected by maximizing their performance of the obtained solutions. The originality of this study is conducted in the following ways. Starting with developing granular fuzzy relation equations, an interval-valued fuzzy relation is determined based on the selected subset of fuzzy rules (the subset of rules is transformed to interval-valued fuzzy sets and subsequently the interval-valued fuzzy sets are utilized to form interval-valued fuzzy relations), which can be used to represent the fuzzy relation of the entire rule base with high performance and efficiency. Then, the particle swarm optimization (PSO) is implemented to solve a multi-objective optimization problem, in which not only an optimal subset of rules is selected but also a parameter ε for specifying a level of information granularity is determined. A series of experimental studies are performed to verify the feasibility of this framework and quantify its performance. A visible improvement of particle swarm optimization (about 78.56% of the encoding mechanism of particle swarm optimization, or 90.42% of particle swarm optimization with an exploration operator) is gained over the method conducted without using the particle swarm optimization algorithm.

Solving max-Archimedean t-norm interval-valued fuzzy relation equations

This paper discusses a new method for finding the complete set of tolerable solutions of max-Archimedean interval-valued fuzzy relation equations. According to the literature, three types of solution sets, namely; tolerable solution set, united solution set and controllable solution set can be identified with interval-valued fuzzy relation equations. The structure and the properties of the tolerable solution set are studied. The complete set of tolerable solutions can be characterized by one maximum solution and finitely many minimal solutions. An efficient method based on the concept of covering is proposed which computes all minimal solutions. The concept of covering is useful for large size problems in terms of computation. The proposed method is illustrated with some examples.

And-like-uninorm-based transitivity and analytic hierarchy process with interval-valued fuzzy preference relations

The framework of interval-valued fuzzy preference relations (IVFPRs) is adequate and effective to model human preference evaluations under indeterminacy. This paper analyzes three recently presented multiplicative transitivity models of IVFPRs and exposes their drawbacks. An and-like-uninorm-based functional transitivity equation is developed to introduce a multiplicative consistency notion for IVFPRs. Based on the transitivity logarithmic equation, a geometric-consistency index is further proposed to compute the inconsistency level of an IVFPR. The paper builds a logarithmic least squares model with row indeterminacy constraints and equivalently transforms it into a quadratic programming model for finding a closed-form solution of the normalized interval-valued fuzzy weights of IVFPRs. A novel method is subsequently presented to check the acceptability of an IVFPR by examining its acceptable consistency and acceptable indeterminacy. An approach including an and-like-uninorm-based maximization model is introduced to aggregate local interval-valued fuzzy weights and an interval fuzzy analytic hierarchy process is designed step-by-step. An illustrative example and a comparison study are utilized to demonstrate the performance and merits of the presented models. Meanwhile, an outstanding undergraduate student selection problem in international exchange is provided to show the application of the proposed decision method.

Multiple-attribute decision making based on equivalence consistency under probabilistic linguistic dual hesitant fuzzy environment

Due to the complexity of the real problems and the limitations of expert’s knowledge, it is often difficult for decision makers to give the exact decision information. How to accurately and adequately express the evaluation information of decision makers becomes the primary problem of multi-attribute decision making(MADM). Therefore, a large number of fuzzy sets(such as linguistic hesitant fuzzy sets(LHFS), probabilistic hesitant fuzzy sets(PHFS), dual hesitant fuzzy sets(DHFS)) that express decision makers’ judgment information have been widely used in multi-attribute decision making. Inspired by these fuzzy sets, this paper proposes a new set called probabilistic linguistic dual hesitant fuzzy sets(PLDHFS). Based on the PLDHFS, the probabilistic linguistic dual hesitant fuzzy preference relation(PLDHFPR) is further given. In order to obtain a reasonable ranking weight vector, the equivalence consistency between the PLDHFPR and the interval-valued fuzzy preference relation(IVFPR) is proposed, and we present the objective programming model(OPM) based on the multiplicative consistency of the IVFPR. Moreover, we establish a consistency improvement model for inconsistent PLDHFPR. Finally, an example of venture capital is analyzed to demonstrate the feasibility and effectiveness of the proposed method.

(1705-3711) UNCERTAINTY DATA CREATING INTERVAL-VALUED FUZZY RELATION IN DECISION MAKING MODEL WITH GENERAL PREFERENCE STRUCTURE

The paper introduces a new approach to preference structure, where from a weak preference relation derive the following relations:strict preference, indifference and incomparability, which by aggregations and negations are created and examined. We decomposing a preference relation into a strict preference, anindifference, and an incomparability relation.This approach allows one to quantify different types of uncertainty in selecting alternatives.In presented preference structure we use interval-valued fuzzy relations, which can be interpreted as a tool that may help to model in a better way imperfect information, especially under imperfectlydefined facts and imprecise knowledge.Preference structures are of great interest nowadays because of their applications, so we propose at the end the algorithm of decision making by use new preference structure.

Additively Consistent Interval-Valued Intuitionistic Fuzzy Preference Relations and Their Application to Group Decision Making

This paper aims to propose an innovative approach to group decision making (GDM) with interval-valued intuitionistic fuzzy (IVIF) preference relations (IVIFPRs). First, an IVIFPR is proposed based on the additive consistency of an interval-valued fuzzy preference relation (IVFPR). Then, two mathematical or adjusted programming models are established to extract two special consistent IVFPRs. In order to derive the priority weight of an IVIFPR, after taking the two special IVFPRs into consideration, a linear optimization model is constructed by minimizing the deviations between individual judgments and between the width degrees of the interval priority weights. For GDM with IVIFPRs, the decision makers’ weights are generated by combining the adjusted subjective weights with the objective weights. Subsequently, using an IVIF-weighted averaging operator, the collective IVIFPR is obtained and utilized to derive the IVIF priority weights. Finally, a practical example of a supplier selection is analyzed to demonstrate the application of the proposed method.

Multi-granulation interval-valued fuzzy probabilistic rough sets and their corresponding three-way decisions based on interval-valued fuzzy preference relations

In this paper, we study interval-valued fuzzy probabilistic rough sets (IVF-PRSs) based on multiple interval-valued fuzzy preference relations (IVFPRs) and consistency matrices, i.e., the multi-granulation interval-valued fuzzy preference relation probabilistic rough sets (MG-IVFPR-PRSs). First, in the proposed study, we convert IVFPRs into fuzzy preference relations (FPRs), and then construct the consistency matrix, the collective consistency matrix, the weighted collective preference relations, and the group collective preference relation (GCPR). Using this GCPR, four types of MG-IVFPR-PRSs are founded in terms of different constraints on parameter. Finally, to find a suitable way of explaining and determining these parameters in each model, three-way decisions are studied based on Bayesian minimum-risk procedure, i.e., the multi-granulation interval-valued fuzzy preference relation decision-theoretic rough set approach. An example is included to show the feasibility and potential of the theoretic results obtained.

A Novel Approach for Group Decision-Making from Intuitionistic Fuzzy Preference Relations and Intuitionistic Multiplicative Preference Relations

During the decision-making process, evaluation information may be given in different formats based on the decision makers’ research fields or personal customs. To address the situation that alternatives are evaluated by both intuitionistic fuzzy preference relations (IFPRs) and intuitionistic multiplicative preference relations (IMPRs), a new priority approach based on a net flow score function is proposed. First, the two preference relations above are transformed into the corresponding interval-valued fuzzy preference relations (IVFPRs) and interval-valued multiplicative preference relations (IVMPRs), respectively. Second, the net flow score functions of individual IFPRs and IMPRs are obtained. Third, according to information theory, a mean deviation maximization model is constructed to compute the weights of decision-makers objectively. Finally, the collective net flow score of each alternative is obtained to determine the ranking result. The proposed method is certified to be simple, valid, and practical with three examples.

Isomorphic Multiplicative Transitivity for Intuitionistic and Interval-Valued Fuzzy Preference Relations and Its Application in Deriving Their Priority Vectors

Intuitionistic fuzzy preference relations (IFPRs) are used to deal with hesitation, while interval-valued fuzzy preference relations (IVFPRs) are for uncertainty in multicriteria decision making (MCDM). This paper aims to explore the isomorphic multiplicative transitivity for IFPRs and IVFPRs, which builds the substantial relationship between hesitation and uncertainty in MCDM. To do that, the definition of the multiplicative transitivity property of IFPRs is established by combining the multiplication of intuitionistic fuzzy sets and Tanino's multiplicative transitivity property of fuzzy preference relations. It is proved to be isomorphic to the multiplicative transitivity of IVFPRs derived via Zadeh's extension principle. The use of the multiplicative transitivity isomorphism is twofold: 1) to discover the substantial relationship between IFPRs and IVFPRs, which will bridge the gap between hesitation and uncertainty in MCDM problems; and 2) to strengthen the soundness of the multiplicative transitivity property of IFPRs and IVFPRs by supporting each other with two different reliable sources, respectively. Furthermore, based on the existing isomorphism, the concept of multiplicative consistency for IFPRs is defined through a strict mathematical process, and it is proved to satisfy the following several desirable properties: weak transitivity, max–max transitivity, and center-division transitivity. A multiplicative consistency-based multiobjective programming (MOP) model is investigated to derive the priority vector from an IFPR. This model has the advantage of not losing information, as the priority vector representation coincides with that of the input information, which was not the case with the existing methods, where crisp priority vectors were derived as a consequence of the modeling transitivity just for the intuitionistic membership function and not for the intuitionistic nonmembership function. Finally, a numerical example concerning green supply selection is given to validate the efficiency and practicality of the proposed multiplicative consistency MOP model.

N-Reciprocity Property for Interval-Valued Fuzzy Relations with an Application to Group Decision Making Problems in Social Networks

In this paper we study interval-valued fuzzy relations. We consider preference relations, i.e. a triplet consisting of strict preference, indifference and incomparability which are defined with the use of a fuzzy negation. We analyze the preservation of the fuzzy negation based reciprocity property of interval-valued fuzzy relations by aggregation functions and by some basic interval-valued fuzzy relations. We use diverse representa-tions of aggregation functions. We also consider the connection between N-reciprocal relations and transitivity properties. We provide a numerical example where the final alternative is chosen with the use of generalized voting method, where admissible linear orders for intervals are applied.

New types of aggregation functions for interval-valued fuzzy setting and preservation of pos-B and nec-B-transitivity in decision making problems

In this contribution new types of aggregation functions for interval-valued fuzzy setting are introduced. They are called necessary and possible aggregation functions, respectively. In the monotonicity conditions for these aggregation functions the classical monotonicity for intervals is replaced with the new comparability relations. These relations follow naturally from the interpretations of interval-valued fuzzy sets and together with the classically used monotonicity for intervals, form a family of all possible approaches to define relations of comparability for intervals. Moreover, in this paper dependencies between considered families of aggregation functions are presented. Furthermore, transitivity properties of interval-valued fuzzy relations, based on these new comparability relations, are studied and preservation of them by possible and necessary aggregation functions are considered.

An intuitionistic fuzzy programming method for group decision making with interval-valued fuzzy preference relations

The paper develops a new intuitionistic fuzzy (IF) programming method to solve group decision making (GDM) problems with interval-valued fuzzy preference relations (IVFPRs). An IF programming problem is formulated to derive the priority weights of alternatives in the context of additive consistent IVFPR. In this problem, the additive consistent conditions are viewed as the IF constraints. Considering decision makers' (DMs') risk attitudes, three approaches, including the optimistic, pessimistic and neutral approaches, are proposed to solve the constructed IF programming problem. Subsequently, a new consensus index is defined to measure the similarity between DMs according to their individual IVFPRs. Thereby, DMs' weights are objectively determined using the consensus index. Combining DMs' weights with the IF program, a corresponding IF programming method is proposed for GDM with IVFPRs. An example of E-Commerce platform selection is analyzed to illustrate the feasibility and effectiveness of the proposed method. Finally, the IF programming method is further extended to the multiplicative consistent IVFPR.

Composition of interval-valued fuzzy relations using aggregation functions

In this paper we present the composition of interval-valued fuzzy relations using interval-valued aggregation functions. In particular, we propose a generalization of Zadeh’s composition rule, replacing the minimum by an interval-valued aggregation function. We analyze the preservation of different properties of interval-valued fuzzy relations by this new composition, and we include an illustrative example in approximate reasoning in order to justify our proposal.