THEOREMS OF INTERVAL FUZZY SET AND ITS OPERATION RULES

Although the concept of interval fuzzy set and its properties have been defined, its three theorems and their effectiveness are not proved. Therefore, the knowledge presentation and its operation rules of interval fuzzy set are studied firstly, and then the cut set of interval fuzzy set is proposed. Moreover, the decomposition theo- rem, the representation theorem and the extension theorem of interval fuzzy set are presented. Finally, examples are given to demonstrate that the classical fuzzy set is a special case of interval fuzzy set and interval fuzzy set is an effective expansion of the classical fuzzy set.

Fuzzy interval linguistic sets with applications in multi-attribute group decision making

Uncertain and hesitant information, widely existing in the real-world qualitative decision making problems, brings great challenges to decision makers. Hesitant fuzzy linguistic term sets(HFLTSs), an effective linguistic computational tool in modeling and eliciting such information, have hence aroused many scholars’ interests and some extensions have been introduced recently.However, these methods are based on the discrete linguistic term framework with the limited expression domain, which actually depict qualitative information using several single values. Therefore,it is hard to ensure the integrity of the semantics representation and the accuracy of the computation results. To deal with this problem, a semantics basis framework called complete linguistic term set(CLTS) is designed, which adopts a separation structure of linguistic scale and expression domain, enriching semantics representation of decision makers. On this basis the concept of fuzzy interval linguistic sets(FILSs) is put forward that employs the interval linguistic term with probability to increase the flexibility of eliciting and representing uncertain and hesitant qualitative information. For practical applications, a fuzzy interval linguistic technique for order preference by similarity to ideal solution(FILTOPSIS) method is developed to deal with multi-attribute group decision making(MAGDM) problems. Through the cases of movie and enterprise resource planning(ERP) system selection, the effectiveness and validity of the proposed method are illustrated.

Cartesian product over interval valued intuitionistic fuzzy sets

The intuitionistic fuzzy set(IFS) based on fuzzy theory,which is of high efficiency to solve the fuzzy problem, has been introduced by Atanassov. Subsequently, he pushed the research one step further from the IFS to the interval valued intuitionistic fuzzy set(IVIFS). On the basis of fuzzy set(FS), the IFS is a generalization concept. And the IFS is generalized to the IVIFS.In this paper, the definition of the sixth Cartesian product over IVIFSs is first introduced and its some properties are explored.We prove some equalities based on the operation and the relation over IVIFSs. Finally, we present one geometric interpretation and a numerical example of the sixth Cartesian product over IVIFSs.

Study of Interval Type-2 Fuzzy Controller for the Twin-tank Water Level System

For dealing with large static error due to poor immunity of the traditional fuzzy control, a novel interval type-2 fuzzy control system is proposed. By extending the typical membership functions to interval type-2 membership functions, the proposed control system can efficiently reduce the uncertain disturbance from real environment without increasing the design complexity. The simulation results on the water tank level control system showed that the proposed method succeeded in better static and dynamic control with stronger robust performance than the traditional fuzzy control method.

A Novel Green Supplier Selection Method Based on the Interval Type-2 Fuzzy Prioritized Choquet Bonferroni Means

In view of the environment competencies,selecting the optimal green supplier is one of the crucial issues for enterprises,and multi-criteria decision-making(MCDM)methodologies can more easily solve this green supplier selection(GSS)problem.In addition,prioritized aggregation(PA)operator can focus on the prioritization relationship over the criteria,Choquet integral(CI)operator can fully take account of the importance of criteria and the interactions among them,and Bonferroni mean(BM)operator can capture the interrelationships of criteria.However,most existing researches cannot simultaneously consider the interactions,interrelationships and prioritizations over the criteria,which are involved in the GSS process.Moreover,the interval type-2 fuzzy set(IT2FS)is a more effective tool to represent the fuzziness.Therefore,based on the advantages of PA,CI,BM and IT2FS,in this paper,the interval type-2 fuzzy prioritized Choquet normalized weighted BM operators with fuzzy measure and generalized prioritized measure are proposed,and some properties are discussed.Then,a novel MCDM approach for GSS based upon the presented operators is developed,and detailed decision steps are given.Finally,the applicability and practicability of the proposed methodology are demonstrated by its application in the shared-bike GSS and by comparisons with other methods.The advantages of the proposed method are that it can consider interactions,interrelationships and prioritizations over the criteria simultaneously.

PSO type-reduction method for geometric interval type-2 fuzzy logic systems

In a special case of type-2 fuzzy logic systems （FLS）, i.e. geometric inteIval type-2 fuzzy logic systems （GIT-2FLS）, the crisp output is obtained by computing the geometric center of footprint of uncertainly （FOU） without type-reduction, but the defuzzifying method acts against the corner concepts of type-2 fuzzy sets in some cases. In this paper, a PSO type-reduction method for GIT-2FLS based on the particle swarm optimization （PSO） algorithm is presented. With the PSO type-reduction, the inference principle of geometric interval FLS operating on the continuous domain is consistent with that of traditional interval type-2 FLS operating on the discrete domain. With comparative experiments, it is proved that the PSO type-reduction exhibits good performance, and is a satisfactory complement for the theory of GIT-2FLS.

Multi-criteria decision making method based on improved cosine similarity measure with interval neutrosophic sets

Purpose–The fuzziness and complexity of evaluation information are common phenomenon in practical decision-making problem,interval neutrosophic sets(INSs)is a power tool to deal with ambiguous information.Similarity measure plays an important role in judging the degree between ideal and each alternative in decision-making process,the purpose of this paper is to establish a multi-criteria decision-making method based on similarity measure under INSs.Design/methodology/approach–Based on an extension of existing cosine similarity,this paper first introduces an improved cosine similarity measure between interval neutosophic numbers,which considers the degrees of the truth membership,the indeterminacy membership and the falsity membership of the evaluation values.And then a multi-criteria decision-making method is established based on the improved cosine similarity measure,in which the ordered weighted averaging(OWA)is adopted to aggregate the neutrosophic information related to each alternative.Finally,an example on supplier selection is given to illustrate the feasibility and practicality of the presented decision-making method.Findings–In the whole process of research and practice,it was realized that the application field of the proposed similarity measure theory still should be expanded,and the development of interval number theory is one of further research direction.Originality/value–The main contributions of this paper are as follows:this study presents an improved cosine similarity measure under INSs,in which the weights of the three independent components of an interval number are taken into account;OWA are adopted to aggregate the neutrosophic information related to each alternative;and a multi-criteria decision-making method using the proposed similarity is developed under INSs.

Type-2 Fuzzy Control for Driving State and Behavioral Decisions of Unmanned Vehicle

In this paper, interval type-2 fuzzy sets, fuzzy comprehensive evaluation and the fuzzy control rules are synthesized to realize the control of unmanned vehicle in driving state and behavioral decisions. Compared to the type-1 fuzzy set, type-2 fuzzy sets have more advantages in handling the model based on uncertainties, linguistic information because the membership functions are fuzzy sets. Different membership functions are established for each factor when the unmanned vehicle is driving at different speed intervals. In addition, a new evaluation method is developed to analyze unmanned vehicle’s driving state. Finally, a set of dynamic fuzzy rules are sorted out, which can be applied to the unmanned vehicle’s behavioral decision-making and provide a new idea to related research.

MCGDM model using uncertain linguistic consistency sets of single andinterval linguistic term multivalued sets

This paper presents hybrid linguistic expressions and operations for the hybridlinguistic multiple criteria group decision making (MCGDM) issue withidentical and/or different single and interval linguistic term values. First, wepropose a single and interval linguistic term multivalued set/element (SILTMS/SILTME) and develop a consistency measure of SILTMEs based on Shannonentropy to measure the consistency degree of single and interval linguistic termvalues in SILTME. Second, we converse SILTMS/SILTME into an uncertainlinguistic consistency set/element (ULCS/ULCE) in terms of the mean andconsistency measure of SILTMEs to reasonably perform operations betweendifferent information forms/sequence lengths in SILTMSs. Third, we definesome operations of ULCEs and the expected values and sorting rules ofULCEs. Fourth, we present the ULCE weighted mean and geometric operatorsand their characteristics. Finally, we develop a MCGDM model using theweighted mean operation of the two operators and apply it in the mine safetyassessment.

A consensus model for group decision making under interval type-2 fuzzy environment

We propose a new consensus model for group decision making(GDM) problems, using an interval type-2 fuzzy environment. In our model, experts are asked to express their preferences using linguistic terms characterized by interval type-2 fuzzy sets(IT2 FSs), because these can provide decision makers with greater freedom to express the vagueness in real-life situations. Consensus and proximity measures based on the arithmetic operations of IT2 FSs are used simultaneously to guide the decision-making process. The majority of previous studies have taken into account only the importance of the experts in the aggregation process, which may give unreasonable results. Thus, we propose a new feedback mechanism that generates different advice strategies for experts according to their levels of importance. In general, experts with a lower level of importance require a larger number of suggestions to change their initial preferences. Finally, we investigate a numerical example and execute comparable models and ours, to demonstrate the performance of our proposed model. The results indicate that the proposed model provides greater insight into the GDM process.

Out-of-Bounds Array Access Fault Model and Automatic Testing Method Study

Out-of-bounds array access(OOB) is one of the fault models commonly employed in the object- oriented programming language. At present, the technology of code insertion and optimization is widely used in the world to detect and fix this kind of fault. Although this method can examine some of the faults in OOB programs, it cannot test programs thoroughly, neither to find the faults correctly. The way of code insertion makes the test procedures so inefficient that the test becomes costly and time-consuming. This pa- per, uses a kind of special static test technology to realize the fault detection in OOB programs. We first es- tablish the fault models in OOB program, and then develop an automatic test tool to detect the faults. Some experiments have exercised and the results show that the method proposed in the paper is efficient and feasible in practical applications.