Weight computation of criteria in a decision-making problem by knowledge measure with intuitionistic fuzzy set and interval-valued intuitionistic fuzzy set

The aim of this study is to propose an objective method for determining weights of criteria (also called attributes) based on a new measure of intuitionistic fuzzy information, called knowledge measure, in a real-world multi-criteria decision-making problem under intuitionistic fuzzy and interval-valued intuitionistic fuzzy environment. To address this issue, we first analyze the existing entropy measures and show that their use in objective weight determination process may lead us to produce unreliable weights of criteria by citing appropriate examples. Then we analyze important properties of knowledge measure of intuitionistic fuzzy set (IFS) and also define knowledge measure for interval-valued intuitionistic fuzzy set. Then a new method to determine the weights of criteria is developed on the basis of knowledge measure where information about criteria weights is completely unknown and partly known. A real-life example is presented to illustrate the proposed weight determination method and a comparative analysis is carried out to indicate the practicality and effectiveness of knowledge-based weight-generation method under both intuitionistic fuzzy and interval-valued intuitionistic fuzzy environment. Finally, we formulate the axioms for knowledge measure associated with IFSs and we also propose families (classes) of knowledge measures.     

Characterization of rough set approximations in Atanassov intuitionistic fuzzy set theory

The primitive notions in rough set theory are lower and upper approximation operators defined by a fixed binary relation and satisfying many interesting properties. Many types of generalized rough set models have been proposed in the literature. This paper discusses the rough approximations of Atanassov intuitionistic fuzzy sets in crisp and fuzzy approximation spaces in which both constructive and axiomatic approaches are used. In the constructive approach, concepts of rough intuitionistic fuzzy sets and intuitionistic fuzzy rough sets are defined, properties of rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are examined. Different classes of rough intuitionistic fuzzy set algebras and intuitionistic fuzzy rough set algebras are obtained from different types of fuzzy relations. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are defined by axioms. Different axiom sets of upper and lower intuitionistic fuzzy set-theoretic operators guarantee the existence of different types of crisp/fuzzy relations which produce the same operators.     

A revised method for ranking fuzzy numbers using maximizing set and minimizing set

A large number of methods have been proposed for ranking fuzzy numbers in the last few decades. Nevertheless, none of these methods can always guarantee a consistent result for every situation. Some of them are even non-intuitive and not discriminating. Chen proposed a ranking method in 1985 to overcome these limitations and simplify the computational procedure based on the criteria of total utility through maximizing set and minimizing set. However, there were some shortcomings associated with Chen’s ranking method. Therefore, we propose a revised ranking method that can overcome these shortcomings. Instead of considering just a single left and a single right utility in the total utility, the proposed method considers two left and two right utilities. In addition, the proposed method also takes into account the decision maker’s optimistic attitude of fuzzy numbers. Several comparative examples and an application demonstrating the usage, advantages, and applicability of the revised ranking method are presented. It can be concluded that the revised ranking method can effectively resolve the issues with Chen’s ranking method. Moreover, the revised ranking method can be used to differentiate different types of fuzzy numbers.     

Group decision making in medical system: An intuitionistic fuzzy soft set approach

In medical system, there may be many critical diseases, where experts do not have sufficient knowledge to handle those problems. For these cases, experts may provide their opinion only about certain aspects of the disease and remain silent for those unknown features. Feeling the need of prioritizing different experts based on their given information, this article uses a novel concept for assigning confident weights to different experts which are mainly based on their provided information. Experts provide their opinions about various symptoms using intuitionistic fuzzy soft matrix (IFSM). In this article, we propose an algorithmic approach based on intuitionistic fuzzy soft set (IFSS) which explores a particular disease reflecting the agreement of all experts. This approach is guided by the group decision making (GDM) model and uses cardinals of IFSS as novel concept. We have used choice matrix (CM) as an important parameter which is based on choice parameters of individual expert. This article has also validated the proposed approach using distance measurements and consents of the majority of experts. The effectiveness of the proposed approach is demonstrated using a suitable case study.     

Resource service optimal-selection based on intuitionistic fuzzy set and non-functionality QoS in manufacturing grid system

In manufacturing grid (MGrid) system, according to functional requirements of a task, there exist a lot of resource services which have similar functional characteristics. Multiple resource services with similar functional characteristics raise the concern over resource service optimal-selection (RSOS). It is important to select the optimal resource service according to their non-functionality characteristics or quality of service (QoS). However, QoS attributes are not easy to measure due to their complexity and involvement of ill-structured information. In this study, user’s feeling is taken into account in RSOS in an MGrid system. The non-functionality QoS evaluation of resource services is based on users’ feeling and transaction experiences using intuitionistic fuzzy set (IFS). Furthermore, the dynamics of non-functionality QoS is considered, and a time-decay function is introduced into non-functionality QoS evaluation. A new method is proposed for RSOS based on IFS and non-functionality QoS, and the procedures are presented in detail. A practice case study is used to illustrate the proposed method and procedure. The performance and advantage of the proposed method are discussed.     

A new measure using intuitionistic fuzzy set theory and its application to edge detection

In this paper, a new attempt has been made using Attanassov's intuitionistic fuzzy set theory for image edge detection. Intuitionistic fuzzy set takes into account the uncertainty in assignment of membership degree known as hesitation degree. Also a new distance measure, called intuitionistic fuzzy divergence, has been proposed. With this proposed distance measure, edge detection is carried out, and the results are found better with respect to the previous methods.     

Fabrication process suitability ranking for micro-electro-mechanical systems using a fuzzy inference system

In this paper we present the development of a system to evaluate alternatives for manufacturing process steps for micro-electro-mechanical systems (MEMS). We explain in detail a formal process flow definition for MEMS fabrication. Then, we develop a fuzzy inference system which allows MEMS developers to capture users’ preferences to rank the alternatives available to complete the process steps required to fabricate a device. In the last part to this work, we present two case studies: alternatives evaluation for impurity doping and for lead zirconate titanate (PZT) patterning. Using an assortment of user preference data to rate a variety of criteria for potential alternatives, our approach produces a clear preference for one specific alternative in each case, which exemplifies the usefulness of the system proposed and illustrates how effective this methodology is towards improving the fabrication process for MEMS.     

A rank ordered filter for medical image edge enhancement and detection using intuitionistic fuzzy set

This paper gives a novel scheme using intuitionistic fuzzy set theory to enhance the edges of medical images. Medical images contain lots of uncertainties, as they are poorly illuminated and fuzzy/vague in nature. So, direct segmentation techniques will not produce better results. There are lots of researches on edge enhancement starting from non-fuzzy to fuzzy set, but proper enhancement (highlighting important structures) is not obtained. Enhancement of edges helps in recovering the important structures that are not visible properly. Even minute pathological blood vessels/cells are not visible properly and in that case edge enhancement will enhance these blood vessels/cells. Intuitionistic fuzzy set theory is found suitable in medical image processing as it considers more (two) uncertainties as compared to fuzzy set theory. In the processing phase, image is initially converted to intuitionistic fuzzy image and intuitionistic fuzzy entropy is used to obtain the optimum value of the parameter in the membership and non-membership functions. Then it computes the total variation of the pixels with respect to the median value of the image window (rank order filtering). This enhances the borders or the edges of the image. The resulting image is then segmented (edge detected) using standard Canny's edge detector, when simply using Canny's edge detector does not give better result. From the result it is observed that on comparing with non-fuzzy and fuzzy methods, the proposed method gives better information about the images, which is helpful to the pathologists in accurate diagnosing of diseases.     

用直觉模糊Petri网构建的供应链可靠性诊断模型

针对不断变化的供应链系统内外部环境因素,围绕供应链系统可靠性诊断问题,将直觉模糊集引入模糊Petri网建模,用直觉模糊数表示库所状态、变迁阈值和变迁输出置信度,构建了基于直觉模糊Petri网的供应链可靠性诊断模型。对直觉模糊产生式规则按照变迁激发前后变迁和库所之间的与或关系,将供应链可靠性诊断模型模糊推理规则划分为四种类型,得到了变迁触发前和触发后的与或直觉模糊推理规则。同时提出了相应的模糊推理算法,并通过实例验证了模型和算法的有效性,能够及时发现供应链系统故障。     

Intuitionistic nil radicals of intuitionistic fuzzy ideals and Euclidean intuitionistic fuzzy ideals in rings

The notion of intuitionistic nil radicals of intuitionistic fuzzy ideals in rings is introduced, and related properties are investigated. The notion of semiprime intuitionistic fuzzy ideals is provided, and its characterization is established. The concept of Euclidean intuitionistic fuzzy ideals is also introduced, and its characterization is established.     

基于区间直觉模糊集的Sinarchy超矩阵构造方法改进

针对网络分析法（ANP）中Sinarchy结构的超矩阵构造问题,提出了一种基于区间直觉模糊集的超矩阵构造改进方法。从Sinarchy结构中具有反馈关系的最后两层元素集的固有属性及特征出发,分析了采用传统方法构造方案对准则的影响矩阵（IMAC）的局限性,提出了应结合整体论与还原论、并遵循钱学森倡导的从定性到定量的综合集成法作为ANP分析解决问题的思路。通过引入区间直觉模糊集代替传统点估计值反映专家偏好进而重新构造IMAC。具体实例应用表明,该方法针对Sinarchy结构的排序问题具有较好的实际应用可行性。     

基于新的区间直觉模糊集相似性测度的模式识别

有关区间值直觉模糊数（集）的相似性研究较少,并且现有的方法在处理实际问题时效果较差。针对这个问题,提出了区间值直觉模糊数（集）相似性测度的新方法,包含了隶属度,非隶属度,犹豫度,以及后者对前两者的影响,将隶属度,非隶属度,犹豫度的相似度表示成三元组的形式,用TOPSIS的思想处理该三元组,得出一种新的有效的相似性度,证明其合理性。将其应用到模式识别实例中,验证其有效性。     

Medical diagnosis with the aid of using fuzzy logic and intuitionistic fuzzy logic

The objective of the present study is to develop/establish a web-based medical diagnostic support system (MDSS) by which health care support can be provided for people living in rural areas of a country. In this respect, this research provides a novel approach for medical diagnosis driven by integrating fuzzy and intuitionistic fuzzy (IF) frameworks. Subsequently, based on the proposed approach a web-based MDSS is developed. The proposed MDSS comprises of a knowledge base (KB) and intuitionistic fuzzy inference system (IFIS). Based on the observation that medical data cannot be described with both precision and certainty, a medical KB is constructed in the form of a set of if-then decision rules by employing both fuzzy and IF logics. After constructing the medical KB, a new set of patients is considered for diagnosing the diseases. For each patient, linguistic values of the patients’ symptoms are considered as inputs of the proposed IFIS and modeled by using the generalized triangular membership functions. Subsequently, integrated fuzzy and IF rule-based inference system is used to find a valid conclusion for the new set of patients. In a nutshell, in this paper fuzzy rule-based and IFS based inference systems are combined for better and more realistic representation of uncertainty of the medical diagnosis problem and for more accurate diagnostic result. The method is composed of following four steps: (1) the modeling of antecedent part of the rules, which consist of linguistic assessments of the patients’ symptoms provided by the doctors/medical experts with their corresponding confidence levels, by using generalized fuzzy numbers; (2) the modeling of consequent part, which reveals the degree of association and the degree of non-association of diseases into the patient, by using IFSs; (3) the use of IF aggregation operator in inference process; (4) the application of relative closeness function to find the final crisp output for a given diagnosis. Finally, the applicability of the proposed approach is illustrated with a suitable case study. This article has also justified the proposed approach by using similarity measurement.     

On the position of intuitionistic fuzzy set theory in the framework of theories modelling imprecision

Intuitionistic fuzzy sets [K.T. Atanassov, Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia (deposed in Central Science-Technical Library of Bulgarian Academy of Science, 1697/84), 1983 (in Bulgarian)] are an extension of fuzzy set theory in which not only a membership degree is given, but also a non-membership degree, which is more or less independent. Considering the increasing interest in intuitionistic fuzzy sets, it is useful to determine the position of intuitionistic fuzzy set theory in the framework of the different theories modelling imprecision. In this paper we discuss the mathematical relationship between intuitionistic fuzzy sets and other models of imprecision.     

语言真值直觉模糊命题逻辑系统的推理规则

提出了一种基于语言真值直觉模糊代数的直觉模糊命题逻辑系统。基于语言真值格蕴涵代数生成语言真值直觉模糊代数,可同时处理具有可比性或不可比性信息。该方法可以同时处理不确定性问题的正面证据和反面证据。研究了语言真值直觉模糊命题逻辑系统LP(S)的性质,得到了其公理及推理规则,也获得了LP(S)中的证明与定理。实例说明,该方法在处理同时具有可比性和不可比性的直觉模糊决策问题中更灵活、更有效。     

Representation of spatial data in an OODB using roughand fuzzy set modeling

This paper introduces and formally defines a fuzzy rough object-oriented database (OODB) model based on a formal framework using an algebraic type system and formally defined constraints. This generalized model incorporates both rough set and fuzzy set uncertainty, while remaining compliant with object-oriented database standards set forth by the Object Database Management Group. Rough and fuzzy set uncertainty enhance the OODB model so that it can more accurately model real world applications. Spatial databases have a particular need for uncertainty management that can be achieved through rough and fuzzy techniques.     

Behavior-modulation technique in mobile robotics using fuzzy discrete event system

This paper presents a novel behavior-modulation technique using a fuzzy discrete event system (FDES) for behavior-based robotic control. The method exploits the multivalued feature of fuzzy logic (FL) and event-driven property of a discrete event system (DES) to generate the activity of a behavior using fuzzy state vectors. State-based prediction of an activity is accomplished using fuzzily defined event matrices. A central arbiter employs priority-based arbitration among the activity state vectors and generates new event matrices to modify the activity states of the behaviors. The method combines aspects of both command fusion and behavior arbitration. Furthermore, the proposed approach has the ability to define state-based observability and controllability to handle sensory uncertainty and environmental dynamics. Observability describes decision vagueness associated with sensory data, whereas controllability specifies undesirable state-reach within the observed environment. Real-time results of FDES-based mobile robot navigation are presented and compared against four different modulation methods to validate its superior performance.     

Rating and ranking of multiple-aspect alternatives using fuzzy sets

A method is proposed to deal with multiple-alternative decision problems under uncertainty. It is assumed that all the alternatives in the choice set can be characterized by a number of aspects, and that information is available to assign weights to these aspects and to construct a rating scheme for the various aspects of each alternative. The method basically consists of computing weighted final ratings for each alternative and comparing the weighted final ratings. The uncertainty that is assumed to be inherent in the assessments of the ratings and weights is accounted for by considering each of these variables as fuzzy quantities, characterized by appropriate membership functions. Accordingly, the final evaluation of the alternatives consists of a degree of membership in the fuzzy set of alternatives ranking first. A practical method is given to compute membership functions of fuzzy sets induced by mappings, and applied to the problem at hand. A number of examples are worked out. The method is compared to another one proposed by Kahne who approaches the problem probabilistically.     

Credit scoring analysis using a fuzzy probabilistic rough set model

Credit scoring analysis is an important activity, especially nowadays after a huge number of defaults has been one of the main causes of the financial crisis. Among the many different tools used to model credit risk, the recent development of rough set models has proved effective. The original development of rough set theory has been widely generalized and combined with other approaches to uncertain reasoning, especially probability and fuzzy set theories. Since coherent conditional probability assessments cope well with the problem of unifying these different approaches, a merging of fuzzy rough set theory with this subjectivist approach is proposed. Specifically, expert partial probabilistic evaluations are encompassed inside a gradual decision rule structure, with coherence of the conclusion as a guideline. In line with Bayesian rough set models, credibility degrees of multiple premises are introduced through conditional probability assessments. Nonetheless, discernibility with this method remains too fine. Therefore, the basic partition is coarsened by equivalence classes based on the arity of positively, negatively and neutrally related criteria. A membership function, which grades the likelihood of default, is introduced by a peculiar choice of t-norms and t-conorms. To build and test the model, real data related to a sample of firms are used.