Fuzzy Set Pair Research Articles

A new operation over intuitionistic fuzzy pairs

The basic study of fuzzy sets theory was introduced by Lotfi Zadeh in 1965. Many authors investigated possibilities how two fuzzy sets can be compared and the most common kind of measures used in the mathematical literature are dissimilarity measures. The previous approach to the dissimilarities is too restrictive, because the third axiom in the definition of dissimilarity measure assumes the inclusion relation between fuzzy sets. While there exist many pairs of fuzzy sets, which are incomparable to each other with respect to the inclusion relation. Therefore we need some new concept for measuring a difference between fuzzy sets so that it could be applied for arbitrary fuzzy sets. We focus on the special class of so called local divergences. In the next part we discuss the divergences defined on more general objects, namely intuitionistic fuzzy sets. In this case we define the local property modified to this object. We discuss also the relation of usual divergences between fuzzy sets to the divergences between intuitionistic fuzzy sets.

Divergence measures on intuitionistic fuzzy sets

The basic study of fuzzy sets theory was introduced by Lotfi Zadeh in 1965. Many authors investigated possibilities how two fuzzy sets can be compared and the most common kind of measures used in the mathematical literature are dissimilarity measures. The previous approach to the dissimilarities is too restrictive, because the third axiom in the definition of dissimilarity measure assumes the inclusion relation between fuzzy sets. While there exist many pairs of fuzzy sets, which are incomparable to each other with respect to the inclusion relation. Therefore we need some new concept for measuring a difference between fuzzy sets so that it could be applied for arbitrary fuzzy sets. We focus on the special class of so called local divergences. In the next part we discuss the divergences defined on more general objects, namely intuitionistic fuzzy sets. In this case we define the local property modified to this object. We discuss also the relation of usual divergences between fuzzy sets to the divergences between intuitionistic fuzzy sets.

Business Model and Management Decision-Making Method of Comprehensive Apartments for the Elderly

With the development of the aging structure of the social population, ensuring the survival rights and interests of the elderly group and realizing the cultivation of the elderly are important issues that need to be solved urgently for the sake of social development. In recent years, the development of apartment industry for the elderly has provided new ideas and experience for the development of the pension system. However, at present, the development of elderly apartments in China is still in the initial stage, and there are many deficiencies to be solved. Therefore, this paper puts forward the business model and management decision-making method of comprehensive apartments for the elderly, so as to provide scientific and reasonable decision-making reference for the development of comprehensive apartments for the elderly. This paper evaluates the different combinations of operation and management decision-making modes of comprehensive old-age apartments through fuzzy set pair and fuzzy analytic hierarchy process. The experimental results show that the financing mode of comprehensive old-age apartments has a great impact on its operation and management decision-making. Through the corresponding screening and testing, among the five qualified old-age apartment combination modes and private nonenterprise team operation and management modes, financial fund policy capital combination mode is the best choice.

Traceability of Information Routing Based on Fuzzy Associative Memory Modelling in Fisheries Supply Chain

Traceability is the ability to verify the history and location of a food product, thus providing information on each supply-chain actor, who the immediate supplier is, and to whom the product was sent. The information system approach has been used to manage and integrate all such information by collecting, storing, then retrieving data and information about the product from earlier stages of the production process. Besides documentation and information sharing, such traceability information systems can also support timely resolution of customer complaints. This paper presents modelling of routing and handling time prediction using a fuzzy associative memory (FAM) method. As a first response to customers, information about the time required to resolve an issue can be provided after the source of a product defect has been traced. With regards to handling, traceability can assist with several issues, e.g., product replacement, product recalls based on retrieval of the contact numbers of affected customers on a recall list , and inspections at each production unit to ensure food safety standards. Based on such activities, it is assumed that the handling time will be affected by the size of the product inventory that can be used to replace a defective product, the amount of product that must be recalled from the market, and the time required internally for the inspection process, which is set as the FAM input variable. A FAM is a set of fuzzy-set pairs (A, B) that maps an input vector fuzzy set A to an output vector fuzzy set B. Our experiments show that, from such a FAM formulation, one can obtain 27 rules. The FAM will encode a fuzzy-set pair (A, B) to obtain matrix memories, denoted by M. As the prediction result, the matrix B can be obtained from the computational matrix A and the matrix M; For instance, in case of a contamination incident in a fish product with inventory conditions of as much as 4 tonnes, the product recall amounts to 21 tonnes and the inspection will take 25 h, while the results of the computational experiment show that the total handling time for this case will be 66 h with low error rates.

A roughness measure of fuzzy sets from the perspective of distance

Covering generalized rough set theory is an important extension of classical rough set theory. To characterize a fuzzy set in a given covering approximation space, a pair of fuzzy sets, called covering rough fuzzy lower and upper approximations, were introduced, but they do not describe well how much uncertainty is induced by the granularity of knowledge. In this paper, we first discuss the relationship between uncertainty and granularity of knowledge. Then we examine several commonly used distance measures, and indicate that some of them exhibit some limitations. Next we propose a roughness measure based on Minkowski distance, and examine some important properties of this measure. Finally, an illustrative example is provided to demonstrate the application of the roughness measure to incomplete information systems with fuzzy decision.

Exponential information measures on pairs of fuzzy sets

Exponential information measures on pairs of fuzzy sets

Similarity in fuzzy systems

This paper proposes to study the concept of similarity and its use in the design of fuzzy system. The concept of similarity relation is effectively used in fuzzification of crisp values. Similarity index is used in measuring approximate (graded) equality of fuzzy sets over a given universe of discourse. It is proposed to use such an index in modifying a fuzzy relation. Different similarity measures in the literature are elucidated, and a comparative study between different pairs of fuzzy sets is presented. One of these similarity measures has been used successfully in rule selection and modification of a fuzzy relation. In the process, a number of modification schemes based on different logic have been extensively studied for different reasoning mechanisms based on the same data and results are tabulated. A specificity-based approach to defuzzification is also presented, which is found to be suitable for similarity-based fuzzy systems. The results are illustrated with the behaviour of a direct current (DC) shunt motor.

Analysis of Water Quality Assessment Model Based on the SPA Theory

The comprehensive evaluation of water quality is an important foundation work for environmental planning and management.Because of the characteristics of fuzziness and impreciseness of river water system, the water quality assessment model based on the set pair analysis theory (WQA-SPA) for evaluating water quality of the river mainstream is proposed. The WQA-SPA model couples fuzzy logic theory and set pair analysis theory, combining identity、discrepancy and contrary (IDC). According to this model, the water level recognition results and the membership of various levels can be calculated by using the river water quality data. Research shows that the evaluation result is accurate, credible. It has a certain value in the comprehensive evaluation of various types of known systems.

Analysing Fuzzy Sets Through Combining Measures of Similarity and Distance

Reasoning with fuzzy sets can be achieved through measures such as similarity and distance. However, these measures can often give misleading results when considered independently, for example giving the same value for two different pairs of fuzzy sets. This is particularly a problem where many fuzzy sets are generated from real data, and while two different measures may be used to automatically compare such fuzzy sets, it is difficult to interpret two different results. This is especially true where a large number of fuzzy sets are being compared as part of a reasoning system. This paper introduces a method for combining the results of multiple measures into a single measure for the purpose of analysing and comparing fuzzy sets. The combined measure alleviates ambiguous results and aids in the automatic comparison of fuzzy sets. The properties of the combined measure are given, and demonstrations are presented with discussions on the advantages over using a single measure.

Double Fuzzy Point Extension of the Two-step Fuzzy Rule Interpolation Methods

The rule representation opens a new dimension for expressing changes of fuzziness in fuzzy rule-based systems. In the case of standard Fuzzy rule representations, it is difficult to describe fuzzy functions in which crisp observations are required to have fuzzy conclusions, or in which an increase in the fuzziness of observations leads to reduced fuzziness in conclusions. These problems are mainly due to a lack of information. A fuzzy point rule determines the connection between a pair of fuzzy sets taken from the domain and the range of the rule. Expressing the fuzzy function through a set of fuzzy points and fuzzy interpolation between pairs of those points, each fuzzy point can be considered as a node point with given location and fuzziness. In common, sparse rule-base definitions, these node points are usually disjunctive on the domain, defining only single antecedent-consequent fuzziness connections at the location of the fuzzy points. However, this kind of information is insufficient when the goal is to express changes in the fuzziness of a given location in the domain. One solution to this problem is the double fuzzy point rule representation concept. Double fuzzy points are pairs of fuzzy points which share the same reference locations, but have different fuzziness properties. The existence of two different fuzziness values in a single location within the domain creates new possibilities for introducing fuzzy interpolation methods capable of interpolating not only between locations, but between changes in local fuzziness values as well. The main goal of this paper is to discuss how two-step Rule Interpolation methods can be adapted to be able to handle the double fuzzy point concept. To this end, an approach referred to as the Generalized Double Point Methodology (GDFPM) is proposed.

Research on Construction Safety Evaluation Based on Fuzzy-Set Pair Analysis Model

Given factors influencing construction safety has characteristics of fuzziness and uncertainty, an evaluation model based on Fuzzy-Set Pair Analysis is put forward, which combines advantages of fuzzy comprehensive analysis method and set pair analysis method. According to characteristics of construction, the construction safety evaluation index system is constructed, the weight of each index is determined by Interval Analytic Hierarchy Process （IAHP） and connection degree is calculated, so the safety condition development and uncertainty rank of evaluated construction sites are calculated. It is shown that the model can deal with certainty and uncertainty of information in safety evaluation at the same time, thus good results can be obtained through application of the model.

A probabilistic definition of a nonconvex fuzzy cardinality

The existing methods to assess the cardinality of a fuzzy set with finite support are intended to preserve the properties of classical cardinality. In particular, the main objective of researchers in this area has been to ensure the convexity of fuzzy cardinalities, in order to preserve some properties based on the addition of cardinalities, such as the additivity property. We have found that in order to solve many real-world problems, such as the induction of fuzzy rules in Data Mining, convex cardinalities are not always appropriate. In this paper, we propose a possibilistic and a probabilistic cardinality of a fuzzy set with finite support. These cardinalities are not convex in general, but they are most suitable for solving problems and, contrary to the generalizing opinion, they are found to be more intuitive for humans. Their suitability relies mainly on the fact that they assume dependency among objects with respect to the property “to be in a fuzzy set”. The cardinality measures are generalized to relative ones among pairs of fuzzy sets. We also introduce a definition of the entropy of a fuzzy set by using one of our probabilistic measures. Finally, a fuzzy ranking of the cardinality of fuzzy sets is proposed, and a definition of graded equipotency is introduced.

Fuzzy relational equation with defuzzification – algorithm for the largest solution

This paper provides an algorithm for solving a new fuzzy relational equation including defuzzification. An input fuzzy set is first transformed into an internal fuzzy set by a fuzzy relation, and then an element giving the largest membership value is selected from the support set as an output. Our purpose is to obtain the internal fuzzy relation from pairs of input fuzzy sets and output set-elements. These relational equations are classified into two types, called Type I and Type II in the text, depending on where internal fuzzy sets are defuzzified. Discussions in this paper are done on the largest solutions of these problems. Algorithms to obtain the largest solutions are shown as well as some propositions and numerical examples.