# Representing uncertainty about fuzzy membership grade

01 Sep 2020-Vol. 24, Iss: 17, pp 12691-12707

TL;DR: A novel uncertainty representation framework is introduced based on the inter-linkage between the inherent fuzziness and the agent's confusion in its representation to take into consideration the DM’s individualistic bias in the representation of the underlying fuzziness.

Abstract: A novel uncertainty representation framework is introduced based on the inter-linkage between the inherent fuzziness and the agent’s confusion in its representation. The measure of fuzziness and this confusion is considered to be directly related to the lack of distinction between membership and non-membership grades. We term the proposed structure as confidence fuzzy set (CFS). It is further generalized as generalized CFS, quasi CFS and interval-valued CFS to take into consideration the DM’s individualistic bias in the representation of the underlying fuzziness. The operations on CFSs are investigated. The usefulness of CFS in multi-criteria decision making is discussed, and a real application in supplier selection is included.

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TL;DR: In this article , a new set named doubt fuzzy set was developed and a new defuzzification method by introducing the concept of power of difficulty/ opportunity power which are complementary to each other.

Abstract: This article deals with the novel application of complex neutrosophic set in a production inventory model. First of all, we develop a new set named doubt fuzzy set the real and imaginary parts of which are the membership functions of fuzzy variables. The real part of the set is coined as “true or sure” and that for complex part it is defined as “hesitation or suspect” membership functions or it may be defined as “optimism” and “pessimism” respectively in the psychological view point. Then, we give some definitions of doubt fuzzy set in a rectangular complex plane. Subsequently, we give novel defuzzification method by introducing the concept of power of difficulty/ opportunity power which are complementary to each other. Secondly, we develop a backlogging economic production quantity (EPQ) model the demand function of which is disrupted due to the presence of shortages. Assuming the demand function as complex in nature, we develop four types of doubt fuzzy sets namely proper doubt, harmful doubt, depressive doubt and confident doubt respectively and split the model into four sub-models accordingly. Based on new defuzzification method, we have introduced a new solution algorithm named dynamical doubt fuzzy optimization algorithm (DDFOA). By this new approach we have shown that with the application of learning vector by means of opportunity power/ fitness of various test functions the decision maker can achieve and avail the financial benefit as (s)he wishes to adopt. However, the concept of robust intelligent decision making has been discussed extensively through the numerical illustrations. Finally, sensitivity analysis, graphical illustrations are made to justify the proposed approach.

8 citations

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TL;DR: In this article, a new set named doubt fuzzy set was developed and a new defuzzification method by introducing the concept of power of difficulty/ opportunity power which are complementary to each other.

Abstract: This article deals with the novel application of complex neutrosophic set in a production inventory model. First of all, we develop a new set named doubt fuzzy set the real and imaginary parts of which are the membership functions of fuzzy variables. The real part of the set is coined as “true or sure” and that for complex part it is defined as “hesitation or suspect” membership functions or it may be defined as “optimism” and “pessimism” respectively in the psychological view point. Then, we give some definitions of doubt fuzzy set in a rectangular complex plane. Subsequently, we give novel defuzzification method by introducing the concept of power of difficulty/ opportunity power which are complementary to each other. Secondly, we develop a backlogging economic production quantity (EPQ) model the demand function of which is disrupted due to the presence of shortages. Assuming the demand function as complex in nature, we develop four types of doubt fuzzy sets namely proper doubt, harmful doubt, depressive doubt and confident doubt respectively and split the model into four sub-models accordingly. Based on new defuzzification method, we have introduced a new solution algorithm named dynamical doubt fuzzy optimization algorithm (DDFOA). By this new approach we have shown that with the application of learning vector by means of opportunity power/ fitness of various test functions the decision maker can achieve and avail the financial benefit as (s)he wishes to adopt. However, the concept of robust intelligent decision making has been discussed extensively through the numerical illustrations. Finally, sensitivity analysis, graphical illustrations are made to justify the proposed approach.

8 citations

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TL;DR: This study combines the relative measure with the lower approximation of FRSs to propose a relative uncertainty measure which can address the above-mentioned problem and designs a fuzzy rough feature selection algorithm to test the effectiveness and efficiency of the proposed measure.

Abstract: Uncertainty measure is an important tool for data analysis. In practical applications, the collected data are subject to different probability distributions. This requires that the uncertainty measure has generalization performance. Fuzzy rough set (FRS) theory is a popular mathematical tool for uncertainty measure, but the theory does not work well for some data distributions. For example, when the class density difference of the data set is large, FRS theory cannot effectively evaluate the classification uncertainty of samples. In this study, we combine the relative measure with the lower approximation of FRSs to propose a relative uncertainty measure which can address the above-mentioned problem. Furthermore, a fuzzy rough feature selection algorithm is designed, and it is mainly used to test the effectiveness and efficiency of the proposed measure. Experimental results demonstrate that the proposed feature selection algorithm has good performance. It indirectly proves that the relative uncertainty measure is effective and efficient in classification tasks.

1 citations

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TL;DR: In this paper , an evaluation framework for the safety resilience of existing high-speed railway tunnels is constructed, and a system resilience evaluation model based on TOPSIS improved fuzzy matter-element is constructed to determine the classification criteria of resilience.

Abstract: Once the high-speed railway tunnel is put into use, its resilience will determine the possibility of permanent safety of the tunnel due to the closure of the structural space of the high-speed railway tunnel in service. Resilience theory is introduced into a risk analysis of operating high-speed rail tunnels to improve the ability to respond to risks in operating high-speed rail tunnels and to relieve the aging phenomenon caused by changes in the tunnel with time. First, an evaluation framework for the safety resilience of existing high-speed railway tunnels is constructed. Starting from the attributes of resilience such as resistance, adaptability, and resilience, and considering the characteristics of high-speed railway tunnels, protective measures, emergency management measures, and other factors, we fit the risk factors and probability of accident type of the high-speed railway tunnel and establish a tunnel safety resilience evaluation index system with 10 indexes. Secondly, the method of information fusion is used to combine subjective weighting and objective weighting. Then, the comprehensive weight of the evaluation index is obtained based on the principle of minimum discriminant information. Thirdly, the system resilience evaluation model based on the TOPSIS improved fuzzy matter-element is constructed to determine the classification criteria of resilience. On this basis, based on the temporal and spatial variability of the ductile tunnel, the concepts of ductile transition and ductile attenuation are introduced and the tunnel toughness optimization model is established to suppress the attenuation situation, enhance the transition ability, and then improve the system resilience level. On this basis, an optimal lifting scheme is obtained. Finally, taking Ai-Min tunnel of Ha-Mu high-speed railway as the engineering background, the flexibility of the resilience system is calculated, and the resilience grade (3) of the rock system surrounding the tunnel is obtained. Combined with the numerical model, improvement measures for specific tunnel facilities are proposed. The results show that the Ai-Min tunnel system has a general ability to resist external intrusion and prevent disasters, and the resilience level is general. It should focus on improving the resilience level of the transition index. The resilience evaluation results of the evaluation model are consistent with the actual situation of the project.

##### References

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01 Aug 1996

TL;DR: A separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.

Abstract: A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.

52,705 citations

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TL;DR: 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.

Abstract: 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.

13,376 citations

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TL;DR: Much of what constitutes the core of scientific knowledge may be regarded as a reservoir of concepts and techniques which can be drawn upon to construct mathematical models of various types of systems and thereby yield quantitative information concerning their behavior.

Abstract: One of the fundamental tenets of modern science is that a phenomenon cannot be claimed to be well understood until it can be characterized in quantitative terms.l Viewed in this perspective, much of what constitutes the core of scientific knowledge may be regarded as a reservoir of concepts and techniques which can be drawn upon to construct mathematical models of various types of systems and thereby yield quantitative information concerning their behavior.

12,530 citations

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01 Jan 1973TL;DR: By relying on the use of linguistic variables and fuzzy algorithms, the approach provides an approximate and yet effective means of describing the behavior of systems which are too complex or too ill-defined to admit of precise mathematical analysis.

Abstract: The approach described in this paper represents a substantive departure from the conventional quantitative techniques of system analysis. It has three main distinguishing features: 1) use of so-called ``linguistic'' variables in place of or in addition to numerical variables; 2) characterization of simple relations between variables by fuzzy conditional statements; and 3) characterization of complex relations by fuzzy algorithms. A linguistic variable is defined as a variable whose values are sentences in a natural or artificial language. Thus, if tall, not tall, very tall, very very tall, etc. are values of height, then height is a linguistic variable. Fuzzy conditional statements are expressions of the form IF A THEN B, where A and B have fuzzy meaning, e.g., IF x is small THEN y is large, where small and large are viewed as labels of fuzzy sets. A fuzzy algorithm is an ordered sequence of instructions which may contain fuzzy assignment and conditional statements, e.g., x = very small, IF x is small THEN Y is large. The execution of such instructions is governed by the compositional rule of inference and the rule of the preponderant alternative. By relying on the use of linguistic variables and fuzzy algorithms, the approach provides an approximate and yet effective means of describing the behavior of systems which are too complex or too ill-defined to admit of precise mathematical analysis.

8,547 citations

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TL;DR: It is proved that the envelope of the hesitant fuzzy sets is an intuitionistic fuzzy set, and it is proved also that the operations proposed are consistent with the ones of intuitionist fuzzy sets when applied to the envelope.

Abstract: Several extensions and generalizations of fuzzy sets have been introduced in the literature, for example, Atanassov's intuitionistic fuzzy sets, type 2 fuzzy sets, and fuzzy multisets. In this paper, we propose hesitant fuzzy sets. Although from a formal point of view, they can be seen as fuzzy multisets, we will show that their interpretation differs from the two existing approaches for fuzzy multisets. Because of this, together with their definition, we also introduce some basic operations. In addition, we also study their relationship with intuitionistic fuzzy sets. We prove that the envelope of the hesitant fuzzy sets is an intuitionistic fuzzy set. We prove also that the operations we propose are consistent with the ones of intuitionistic fuzzy sets when applied to the envelope of the hesitant fuzzy sets. © 2010 Wiley Periodicals, Inc.

2,232 citations