On Weighted Possibilistic Mean,Variance and Correlation of Interval-valued Fuzzy Numbers

In this paper, the concept of weighted possibilistic mean of interval- valued fuzzy number is first introduced. Further, the notions of weighted possibilistic variance, covariance and correlation of interval-valued fuzzy numbers are presented. Meantime, some important properties of them and relationships between them are studied.

On AP-Henstock Integrals of Interval-Valued Functions and Fuzzy-Number-Valued Functions

In 2000, Wu and Gong [1] introduced the thought of the Henstock integrals of inter-valvalued functions and fuzzy-number-valued functions and obtained a number of their properties. The aim of this paper is to introduce the thought of the AP- Henstock integrals of interval-valued functions and fuzzy-number-valued functions which are extensions of [1] and investigate a number of their properties.

On Henstock-Stieltjes Integrals of Interval-Valued Functions and Fuzzy-Number-Valued Functions

In this paper we introduce the notion of the Henstock-Stieltjes (HS) integrals of interval-valued functions and fuzzy-number-valued functions and discuss some of their properties.

ENTROPY MEASUREMENT OF INTERVAL GREY SEQUENCE AND ITS APPLICATION

The uncertainty measurement method for grey information theory and the metric formula are established, and its application in decision-making is researched. The entropy measurement of grey sequence based on the limited interval grey number sequence is different from the Shannon probability entropy. The measurement formula of grey number and its properties are studied, such as the invariance, the applicable conditions, and the grey entropy of union and intersection of two grey numbers, and so on. Finally, the algorithm for interval grey sequence and an example are given to show the effectiveness of the method.

Prediction model of interval grey number based on DGM(1,1)

In grey system theory,the studies in the field of grey prediction model are focused on real number sequences,rather than grey number ones.Hereby,a prediction model based on interval grey number sequences is proposed.By mining the geometric features of interval grey number sequences on a two-dimensional surface,all the interval grey numbers are converted into real numbers by means of certain algorithm,and then the prediction model is established based on those real number sequences.The entire process avoids the algebraic operations of grey number,and the prediction problem of interval grey number is usefully solved.Ultimately,through an example＇s program simulation,the validity and practicability of this novel model are verified.

Multi-attributed decision making for mining methods based on grey system and interval numbers

In the case of unknown weights, theories of multi-attributed decision making based on interval numbers and grey related analysis were used to optimize mining methods. As the representative of independence for the indicator, the smaller the correlation of indicators is, the greater the weight is. Hence, the weights of interval numbers of indicators were determined by using correlation coefficient. Relative closeness based on positive and negative ideal methods was calculated by introducing distance between interval numbers, which made decision making more rational and comprehensive. A new method of ranking interval numbers based on normal distribution was proposed for the optimization of mining methods, whose basic properties were discussed. Finally, the feasibility and effectiveness of this method were verified by theories and practice.

Mathematical Modeling and Evaluation of Reliability Parameters Based on Survival Possibilities under Uncertain Environment

In this article,mathematical modeling for the evaluation of reliability is studied using two methods.One of the methods,is developed based on possibility theory.The performance of the reliability of the system is of prime concern.In view of this,the outcomes for the failure are required to evaluate with utmost care.In possibility theory,the reliability information data determined from decision-making experts are subjective.The samemethod is also related to the survival possibilities as against the survival probabilities.The other method is the one that is developed using the concept of approximation of closed interval including the piecewise quadratic fuzzy numbers.In this method,a decision-making expert is not sure of his/her estimates of the reliability parameters.Numerical experiments are performed to illustrate the efficiency of the suggested methods in this research.In the end,the paper is concluded with some future research directions to be explored for the proposed approach.

Combining TOPSIS and GRA for supplier selection problem with interval numbers

Supplier selection can be regarded as a typical multiple attribute decision-making problem. In real-world situation, the values of the alternative attributes and their weights are always being nondeterministic, and as a result of this, the values are considered interval numbers. In addition, the common approach to measure the similarity between alternatives through their distance suffers from some minor shortcomings. To address these problems, this study develops a novel hybrid decision-making method by combining the technique for order preference by similarity to an ideal solution （TOPSIS） with grey relational analysis （GRA） for supplier selection with interval numbers. By introducing the intervals theory, the extensions of Euclidean distance and grey relational grade are defined. And then a new comprehensive closeness coefficient is constituted for supplier alternatives evaluation based on the interval Euclidean distance and the interval grey relational grade, which could indicate the distance-based similarity and the shape-based similarity simultaneously. A mtmerical example is taken to validate the flexibility of the proposed method, and result shows that this method can tackle the uncertainty in real-world supplier selection and also help decision makers to effectively select optimal suppliers.

Mersenne Numbers, Recursive Generation of Natural Numbers, and Counting the Number of Prime Numbers

A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: MN = 2N – 1, is used to count the number of prime numbers within the precise Mersenne natural number intervals: [0;MN]. This permits the formulation of an extended twin prime conjecture. Moreover, it is found that the prime numbers subsets contained in Mersenne intervals have cardinalities strongly correlated with the corresponding Mersenne numbers.

Selection Method of Evaluation Indicators with Three-Parameter Interval Grey Number

The evaluation problem with three-parameter interval grey number (T-PIGN) widely exists in real world. To select effective evaluation indicators of the problem, this paper puts forward evaluation index system selection principle of T-PIGN based on distance entropy model, and gives out evaluation index system selection judgment criterion of T-PIGN. Furthermore, for the redundancy of evaluation index system with T-PIGN, a selection method of evaluation index system with T-PIGN is proposed. Finally, the applicability of the proposed method is verified by concrete examples.

An improved interval model updating method via adaptive Kriging models

Interval model updating(IMU)methods have been widely used in uncertain model updating due to their low requirements for sample data.However,the surrogate model in IMU methods mostly adopts the one-time construction method.This makes the accuracy of the surrogate model highly dependent on the experience of users and affects the accuracy of IMU methods.Therefore,an improved IMU method via the adaptive Kriging models is proposed.This method transforms the objective function of the IMU problem into two deterministic global optimization problems about the upper bound and the interval diameter through universal grey numbers.These optimization problems are addressed through the adaptive Kriging models and the particle swarm optimization(PSO)method to quantify the uncertain parameters,and the IMU is accomplished.During the construction of these adaptive Kriging models,the sample space is gridded according to sensitivity information.Local sampling is then performed in key subspaces based on the maximum mean square error(MMSE)criterion.The interval division coefficient and random sampling coefficient are adaptively adjusted without human interference until the model meets accuracy requirements.The effectiveness of the proposed method is demonstrated by a numerical example of a three-degree-of-freedom mass-spring system and an experimental example of a butted cylindrical shell.The results show that the updated results of the interval model are in good agreement with the experimental results.

Alternative Method of Constructing Granular Neural Networks

Utilizing granular computing to enhance artificial neural network architecture, a newtype of network emerges—thegranular neural network (GNN). GNNs offer distinct advantages over their traditional counterparts: The ability toprocess both numerical and granular data, leading to improved interpretability. This paper proposes a novel designmethod for constructing GNNs, drawing inspiration from existing interval-valued neural networks built uponNNNs. However, unlike the proposed algorithm in this work, which employs interval values or triangular fuzzynumbers for connections, existing methods rely on a pre-defined numerical network. This new method utilizesa uniform distribution of information granularity to granulate connections with unknown parameters, resultingin independent GNN structures. To quantify the granularity output of the network, the product of two commonperformance indices is adopted: The coverage of numerical data and the specificity of information granules.Optimizing this combined performance index helps determine the optimal parameters for the network. Finally,the paper presents the complete model construction and validates its feasibility through experiments on datasetsfrom the UCIMachine Learning Repository. The results demonstrate the proposed algorithm’s effectiveness andpromising performance.

INTERVAL ANALYSIS OF FUZZY-RANDOM HEAT CONDUCTION IN COMPOSITE TUBES

During the analysis of stability heat conduction in the composite tubes, firstly, when the temperature boundary conditions are the random conditions, equations of the mean values and variances of the random thermal function are transformed. Secondly, when the heat conduct parameters are the fuzzy numbers and the temperature boundary conditions are the random numbers, interval equations of the heat conduction are presented. Thirdly, by comparison of the interval results, the result in the interval analysis is larger than that in the confidence interval. Moreover the error expecting equation is presented. Finally, with upper （lower） approximation in rough set theory, a new method of the interval analysis to deal with the stability heat conduction is presented.

CALCULATION FOR KERNEL OF INTERVAL GREY NUMBER BASED ON BARYCENTER APPROACH

The kernel of interval grey number is most likely the real number,which can be used to represent whitenization value of interval grey number.A novel method for calculating kernel of interval grey number is constructed based on the geometric barycenter of whitenization weight function in the two-dimensional coordinate plane,and the calculation of kernel is converted to the calculation of barycenter in geometric figures.The method fully considers the effect of all information contained in whitenization weight function on the calculation result of kernel,and is the extension and perfection of the existing methods in the scope of application.

Sustainability Evaluation of Modern Photovoltaic Agriculture Based on Interval Type-2 Fuzzy AHP-TOPSIS and Least Squares Support Vector Machine Optimized by Fireworks Algorithm

Photovoltaics(PV)has been combined with many other industries,such as agriculture.But there are many problems for the sustainability of PV agriculture.Timely and accurate sustainability evaluation of modern photovoltaic agriculture is of great significance for accelerating the sustainable development of modern photovoltaic agriculture.In order to improve the timeliness and accuracy of evaluation,this paper proposes an evaluation model based on interval type-2 Fuzzy AHP-TOPSIS and least squares support vector machine optimized by fireworks algorithm.Firstly,the criteria system of modern photovoltaic agriculture sustainability is constructed from three dimensions including technology sustainability,economic sustainability and social sustainability.Then,analytic hierarchy process(AHP)and technique for order preference by similarity to an ideal solution(TOPSIS)methods are improved by using interval type-2 fuzzy theory,and the traditional evaluation model based on interval type-2 Fuzzy AHP-TOPSIS is obtained,and the improved model is used for comprehensive evaluation.After that,the optimal parameters of least squares support vector machine(LSSVM)model are obtained by Fireworks algorithm(FWA)training,and the intelligent evaluationmodel for the sustainability of modern photovoltaic agriculture is constructed to realize fast and intelligent calculation.Finally,an empirical analysis is conducted to demonstrate the scientificity and accuracy of the proposed model.This study is conducive to the comprehensive evaluation of the sustainability of modern photovoltaic agriculture,and can provide decision-making support for more reasonable development model in the future of modern photovoltaic agriculture.

NOVEL GENERALIZED GREY INCIDENCE MODEL BASED ON INTERVAL GREY NUMBERS

To extend the traditional generalized grey incidence model, a novel grey incidence model based on inter- val grey numbers is constructed. Considering the numerical information of indexes cannot be accurately obtained and can be defined as interval grey numbers, the interval grey numbers are defined as standard interval grey num- bers which are split in white part and grey part. The absolute degree of incidence and relative degree of incidence based on the interval grey numbers are constructed and their arithmetic are given. Finally, an example about commercial aircraft index selection illuminates the effectiveness of the model. The results show that the model can sort indexes better and can extend the grey incidence models significantly.

A New Method of Ranking Fuzzy Number Based on α-Cut

This paper presents a new method of comparing or ranking of interval and fuzzy numbers. Based on another way of understanding fuzzy concept, the paper proposes the concept of number meaning of interval and the comparison method of interval. The paper defines the new ranking method by the α cut intervals of fuzzy number, and proves that the new inequality relation of fuzzy numbers is transitive. Our approach is a simple way of considering all the existence levels and can be calculated at different exact level according to different applications.

Interval grey number sequence prediction by using non-homogenous exponential discrete grey forecasting model

This paper aims to study a new grey prediction approach and its solution for forecasting the main system variable whose accurate value could not be collected while the potential value set could be defined. Based on the traditional nonhomogenous discrete grey forecasting model（NDGM）, the interval grey number and its algebra operations are redefined and combined with the NDGM model to construct a new interval grey number sequence prediction approach. The solving principle of the model is analyzed, the new accuracy evaluation indices, i.e. mean absolute percentage error of mean value sequence（MAPEM） and mean percent of interval sequence simulating value set covered（MPSVSC）, are defined and, the procedure of the interval grey number sequence based the NDGM（IG-NDGM） is given out. Finally, a numerical case is used to test the modelling accuracy of the proposed model. Results show that the proposed approach could solve the interval grey number sequence prediction problem and it is much better than the traditional DGM（1,1） model and GM（1,1） model.

New approach to multiple attribute decision making with interval numbers

In an ambiguous decision domain, the evaluation values of alternatives against attributes would be interval numbers because of the inherent, uncertain property of the problems. By using a number of linear programming models, Bryson and Mobolurin propose an approach to compute attribute weights and overall values of the alternatives in the form of interval numbers. The intervals of the overall values of alternatives are then transformed into points or crisp values for comparisons among the alternatives. However, the attribute weights are different because of the use of linear programming models in Bryson and Mobolurin＇s approach. Thus, the alternatives are not comparable because different attribute weights are employed to calculate the overall values of the alternatives. A new approach is proposed to overcome the drawbacks of Bryson and Mobolurin＇s approach. By transforming the decision matrix with intervals into the one with crisp values, a new linear programming model is proposed, to calculate the attribute weights for conducting alternative ranking.

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.