Upper and Lower Bounds of the α-Universal Triple I Method for Unified Interval Implications

Theα-universal triple I(α-UTI)method is a recognized scheme in the field of fuzzy reasoning,whichwas proposed by our research group previously.The robustness of fuzzy reasoning determines the quality of reasoning algorithms to a large extent,which is quantified by calculating the disparity between the output of fuzzy reasoning with interference and the output without interference.Therefore,in this study,the interval robustness(embodied as the interval stability)of theα-UTI method is explored in the interval-valued fuzzy environment.To begin with,the stability of theα-UTI method is explored for the case of an individual rule,and the upper and lower bounds of its results are estimated,using four kinds of unified interval implications(including the R-interval implication,the S-interval implication,the QL-interval implication and the interval t-norm implication).Through analysis,it is found that theα-UTI method exhibits good interval stability for an individual rule.Moreover,the stability of theα-UTI method is revealed in the case of multiple rules,and the upper and lower bounds of its outcomes are estimated.The results show that theα-UTI method is stable for multiple rules when four kinds of unified interval implications are used,respectively.Lastly,theα-UTI reasoning chain method is presented,which contains a chain structure with multiple layers.The corresponding solutions and their interval perturbations are investigated.It is found that theα-UTI reasoning chain method is stable in the case of chain reasoning.Two application examples in affective computing are given to verify the stability of theα-UTImethod.In summary,through theoretical proof and example verification,it is found that theα-UTImethod has good interval robustness with four kinds of unified interval implications aiming at the situations of an individual rule,multi-rule and reasoning chain.

Interval Type-2 Fuzzy Model for Intelligent Fire Intensity Detection Algorithm with Decision Making in Low-Power Devices

Local markets in East Africa have been destroyed by raging fires,leading to the loss of life and property in the nearby communities.Electrical circuits,arson,and neglected charcoal stoves are the major causes of these fires.Previous methods,i.e.,satellites,are expensive to maintain and cause unnecessary delays.Also,unit-smoke detectors are highly prone to false alerts.In this paper,an Interval Type-2 TSK fuzzy model for an intelligent lightweight fire intensity detection algorithm with decision-making in low-power devices is proposed using a sparse inference rules approach.A free open–source MATLAB/Simulink fuzzy toolbox integrated into MATLAB 2018a is used to investigate the performance of the Interval Type-2 fuzzy model.Two crisp input parameters,namely:FIT and FIG��are used.Results show that the Interval Type-2 model achieved an accuracy value of FIO�=98.2%,MAE=1.3010,MSE=1.6938 and RMSE=1.3015 using regression analysis.The study shall assist the firefighting personnel in fully understanding and mitigating the current level of fire danger.As a result,the proposed solution can be fully implemented in low-cost,low-power fire detection systems to monitor the state of fire with improved accuracy and reduced false alerts.Through informed decision-making in low-cost fire detection devices,early warning notifications can be provided to aid in the rapid evacuation of people,thereby improving fire safety surveillance,management,and protection for the market community.

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.

Long-term Traffic Volume Prediction Based on K-means Gaussian Interval Type-2 Fuzzy Sets

This paper uses Gaussian interval type-2 fuzzy se theory on historical traffic volume data processing to obtain a 24-hour prediction of traffic volume with high precision. A K-means clustering method is used in this paper to get 5 minutes traffic volume variation as input data for the Gaussian interval type-2 fuzzy sets which can reflect the distribution of historical traffic volume in one statistical period. Moreover, the cluster with the largest collection of data obtained by K-means clustering method is calculated to get the key parameters of type-2 fuzzy sets, mean and standard deviation of the Gaussian membership function.Using the range of data as the input of Gaussian interval type-2 fuzzy sets leads to the range of traffic volume forecasting output with the ability of describing the possible range of the traffic volume as well as the traffic volume prediction data with high accuracy. The simulation results show that the average relative error is reduced to 8% based on the combined K-means Gaussian interval type-2 fuzzy sets forecasting method. The fluctuation range in terms of an upper and a lower forecasting traffic volume completely envelopes the actual traffic volume and reproduces the fluctuation range of traffic flow.

New approach to determine the priorities from interval fuzzy preference relations

An approach is proposed to solve the problem how to obtain the priorities from interval fuzzy preference relations. Firstly, another expression of interval numbers is given. Then, some basic definitions on consistency and weak transitivity of real and interval fuzzy preference relations are described. Based on these definitions, a two-phase process for determining the priorities from interval fuzzy preference relations is presented. Finally, two exam- ples are used to illustrate the use of the proposed approach.

Parameter Optimization of Interval Type-2 Fuzzy Neural Networks Based on PSO and BBBC Methods

Interval type-2 fuzzy neural networks(IT2FNNs)can be seen as the hybridization of interval type-2 fuzzy systems(IT2FSs) and neural networks(NNs). Thus, they naturally inherit the merits of both IT2 FSs and NNs. Although IT2 FNNs have more advantages in processing uncertain, incomplete, or imprecise information compared to their type-1 counterparts, a large number of parameters need to be tuned in the IT2 FNNs,which increases the difficulties of their design. In this paper,big bang-big crunch(BBBC) optimization and particle swarm optimization(PSO) are applied in the parameter optimization for Takagi-Sugeno-Kang(TSK) type IT2 FNNs. The employment of the BBBC and PSO strategies can eliminate the need of backpropagation computation. The computing problem is converted to a simple feed-forward IT2 FNNs learning. The adoption of the BBBC or the PSO will not only simplify the design of the IT2 FNNs, but will also increase identification accuracy when compared with present methods. The proposed optimization based strategies are tested with three types of interval type-2 fuzzy membership functions(IT2FMFs) and deployed on three typical identification models. Simulation results certify the effectiveness of the proposed parameter optimization methods for the IT2 FNNs.

MATHEMATICAL EXPECTATION ABOUT DISCRETE RANDOM VARIABLE WITH INTERVAL PROBABILITY OR FUZZY PROBABILITY

The character and an algorithm about DRVIP（ discrete random variable with interval probability） and the secured kind DRVFP （discrete random variable with crisp event-fuzzy probability） are researched. Using the fuzzy resolution theorem, the solving mathematical expectation of a DRVFP can be translated into solving mathematical expectation of a series of RVIP. It is obvious that solving mathematical expectation of a DRVIP is a typical linear programming problem. A very functional calculating formula for solving mathematical expectation of DRVIP was obtained by using the Dantzig＇s simplex method. The example indicates that the result obtained by using the functional calculating formula fits together completely with the result obtained by using the linear programming method, but the process using the formula deduced is simpler.

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.

IFCEM based recognition method for target with interval-overlapped hybrid attributes

When the attributes of unknown targets are not just numerical attributes,but hybrid attributes containing linguistic attributes,the existing recognition methods are not effective.In addition,it is more difficult to identify the unknown targets densely distributed in the feature space,especially when there is interval overlap between attribute measurements of different target classes.To address these problems,a novel method based on intuitionistic fuzzy comprehensive evaluation model(IFCEM)is proposed.For numerical attributes,targets in the database are divided into individual classes and overlapping classes,and for linguistic attributes,continuous interval-valued linguistic term set(CIVLTS)is used to describe target characteristic.A cloud modelbased method and an area-based method are proposed to obtain intuitionistic fuzzy decision information of query target on numerical attributes and linguistic attributes respectively.An improved inverse weighted kernel fuzzy c-means(IWK-FCM)algorithm is proposed for solution of attribute weight vector.The possibility matrix is applied to determine the identity and category of query target.Finally,a case study composed of parameter sensitivity analysis,recognition accuracy analysis.and comparison with other methods,is taken to verify the superiority of the proposed method.

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.

Nonsingular Terminal Sliding Mode Control With Ultra-Local Model and Single Input Interval Type-2 Fuzzy Logic Control for Pitch Control of Wind Turbines

As wind energy is becoming one of the fastestgrowing renewable energy resources,controlling large-scale wind turbines remains a challenging task due to its system model nonlinearities and high external uncertainties.The main goal of the current work is to propose an intelligent control of the wind turbine system without the need for model identification.For this purpose,a novel model-independent nonsingular terminal slidingmode control(MINTSMC)using the basic principles of the ultralocal model(ULM)and combined with the single input interval type-2 fuzzy logic control(SIT2-FLC)is developed for non-linear wind turbine pitch angle control.In the suggested control framework,the MINTSMC scheme is designed to regulate the wind turbine speed rotor,and a sliding-mode(SM)observer is adopted to estimate the unknown phenomena of the ULM.The auxiliary SIT2-FLC is added in the model-independent control structure to improve the rotor speed regulation and compensate for the SM observation estimation error.Extensive examinations and comparative analyses were made using a real-time softwarein-the-loop(RT-SiL)based on the dSPACE 1202 board to appraise the efficiency and applicability of the suggested modelindependent scheme in a real-time testbed.

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.

Electric Multiple Unit Bogie Maintainability Allocation with Interval Analysis and Fuzzy Evaluation Method

Based on the failure rate and design features allocation method,considering the multiple influential factors which affect electric multiple unit( EMU) bogies,maintainability allocation on EMU bogie was presented by interval analytic hierarchy analysis and fuzzy comprehensive assessment. The maintainability allocation model was established. Weight based on the influence degree of each factor on maintenance was assigned. Fuzzy interval numbers were used to substitute real numbers and express uncertain information.The maintenance weighting factors for each subsystem were calculated by fuzzy comprehensive assessment. Then the allocation method was applied to EMU bogie. The results show that the method is feasible. The problem difficult to quantify for EMU bogie maintenance allocation is solved effectively.

Non-probabilistic fuzzy reliability analysis of pile foundation stability by interval theory

Randomness and fuzziness are among the attributes of the influential factors for stability assessment of pile foundation. According to these two characteristics, the triangular fuzzy number analysis approach was introduced to determine the probability-distributed function of mechanical parameters. Then the functional function of reliability analysis was constructed based on the study of bearing mechanism of pile foundation, and the way to calculate interval values of the functional function was developed by using improved interval-truncation approach and operation rules of interval numbers. Afterwards, the non-probabilistic fuzzy reliability analysis method was applied to assessing the pile foundation, from which a method was presented for non- probabilistic fuzzy reliability analysis of pile foundation stability by interval theory. Finally, the probability distribution curve of non- probabilistic fuzzy reliability indexes of practical pile foundation was concluded. Its failure possibility is 0.91%, which shows that the pile foundation is stable and reliable.

Fully implicational methods for approximate reasoning based on interval-valued fuzzy sets

The aim of this paper is to discuss the approximate rea- soning problems with interval-valued fuzzy environments based on the fully implicational idea. First, this paper constructs a class of interval-valued fuzzy implications by means of a type of impli- cations and a parameter on the unit interval, then uses them to establish fully implicational reasoning methods for interval-valued fuzzy modus ponens （IFMP） and interval-valued fuzzy modus tel- lens （IFMT） problems. At the same time the reversibility properties of these methods are analyzed and the reversible conditions are given. It is shown that the existing unified forms of α-triple I （the abbreviation of triple implications） methods for FMP and FMT can be seen as the particular cases of our methods for IFMP and IFMT.

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.

Traffic Flow Data Forecasting Based on Interval Type-2 Fuzzy Sets Theory

This paper proposes a long-term forecasting scheme and implementation method based on the interval type-2 fuzzy sets theory for traffic flow data. The type-2 fuzzy sets have advantages in modeling uncertainties because their membership functions are fuzzy. The scheme includes traffic flow data preprocessing module, type-2 fuzzification operation module and long-term traffic flow data forecasting output module, in which the Interval Approach acts as the core algorithm. The central limit theorem is adopted to convert point data of mass traffic flow in some time range into interval data of the same time range (also called confidence interval data) which is being used as the input of interval approach. The confidence interval data retain the uncertainty and randomness of traffic flow, meanwhile reduce the influence of noise from the detection data. The proposed scheme gets not only the traffic flow forecasting result but also can show the possible range of traffic flow variation with high precision using upper and lower limit forecasting result. The effectiveness of the proposed scheme is verified using the actual sample application. © 2014 Chinese Association of Automation.

Interval-valued intuitionistic fuzzy aggregation operators

The notion of the interval-valued intuitionistic fuzzy set （IVIFS） is a generalization of that of the Atanassov＇s intuitionistic fuzzy set. The fundamental characteristic of IVIFS is that the values of its membership function and non-membership function are intervals rather than exact numbers. There are various averaging operators defined for IVlFSs. These operators are not monotone with respect to the total order of IVIFS, which is undesirable. This paper shows how such averaging operators can be represented by using additive generators of the product triangular norm, which simplifies and extends the existing constructions. Moreover, two new aggregation operators based on the t.ukasiewicz triangular norm are proposed, which are monotone with respect to the total order of IVIFS. Finally, an application of the interval-valued intuitionistic fuzzy weighted averaging operator is given to multiple criteria decision making.

Linguistic Dynamic Modeling and Analysis of Psychological Health State Using Interval Type-2 Fuzzy Sets

The study of psychological health state is helpful to build appropriate models and take effective intervention strategies, and the results benefit the intervened released from psychological distress within the shortest possible time. In this paper, interval type-2 fuzzy sets and fuzzy comprehension evaluation are applied in the analysis of mental health status and crisis intervention. A closed-loop linguistic dynamic intervention model for psychological health state is built. Linguistic dynamic systems based on interval type-2 fuzzy sets are used to describe and analyze the evolutionary process of psychological health status. © 2014 Chinese Association of Automation.