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.

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.

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.

Option Pricing Method in a Market Involving Interval Number Factors

The method for pricing the option in a market with interval number factors is proposed. The no-arbitrage principle in the interval number valued market and the rule to judge the reasonability of a price interval are given. Using the method, the price interval where the riskless interest and the volatility under B-S setting is given. The price interval from binomial tree model when the key factors u, d, R are all interval numbers is also discussed.

A Method of Synthetical Appraisal With Interval Numbers

In this paper,A method of synthetical appraisal with interval numbers is given, which combines qualitative analysis with quantitative analysis,and take quantitative analysis as the dominant factor.It is flexible and conveniente in use,It can be used in synthetical sequence for appraisal objects.

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.

Recommend trustworthy services using interval numbers of four parameters via cloud model for potential users

How to discover the trustworthy services is a challenge for potential users because of the deficiency of us- age experiences and the information overload of QoE （qual- ity of experience） evaluations from consumers. Aiming to the limitations of traditional interval numbers in measuring the trustworthiness of service, this paper proposed a novel ser- vice recommendation approach using the interval numbers of four parameters （INF） for potential users. In this approach, a trustworthiness cloud model was established to identify the eigenvalue of INF via backward cloud generator, and a new formula of INF possibility degree based on geometrical anal- ysis was presented to ensure the high calculation precision. In order to select the highly valuable QoE evaluations, the similarity of client-side feature between potential user and consumers was calculated, and the multi-attributes trustwor- thiness values were aggregated into INF by the fuzzy ana- lytic hierarchy process method. On the basis of ranking INF, the sort values of trustworthiness of candidate services were obtained, and the trustworthy services were chosen to recommend to potential user. The experiments based on a real-world dataset showed that it can improve the recommendation accuracy of trustworthy services compared to other approaches, which contributes to solving cold start and information overload problem in service recommendation.

Fusion Method for Interval Number Multi-sensor Information

The interval numbers are used to types and observation of sensors, a new fusion represent the characteristic values of object method for multi-sensor object recognition is proposed from the viewpoint of decision making theory. The method defines the distance matrix and grey association matrix between all object types and unknown object. After solving the optimization problem of maximizing the standard deviations for all attributes, the weights of the attributes are obtained. Thus, the result of recognition for the unknown object is given by the grey association degree. This method avoids the subjectivity of selecting attributes weights. It is straightforward and can be performed on computer easily. The simulated example demonstrates the feasibility and effectiveness of the proposed method.

MPC-based interval number optimization for electric water heater scheduling in uncertain environments

In this paper,interval number optimization and model predictive control are proposed to handle the uncertain-but-bounded parameters in electric water heater load scheduling.First of all,interval numbers are used to describe uncertain parameters including hot water demand,ambient temperature,and real-time price of electricity.Moreover,the traditional thermal dynamic model of electric water heater is transformed into an interval number model,based on which,the day-ahead load scheduling problem with uncertain parameters is formulated,and solved by interval number optimization.Different tolerance degrees for constraint violation and temperature preferences are also discussed for giving consumers more choices.Furthermore,the model predictive control which incorporates both forecasts and newly updated information is utilized to make and execute electric water heater load schedules on a rolling basis throughout the day.Simulation results demonstrate that interval number optimization either in day-ahead optimization or model predictive control format is robust to the uncertain hot water demand,ambient temperature,and real-time price of electricity,enabling customers to flexibly adjust electric water heater control strategy.

The Column Summation Contrary Method of Calculation Interval Number Weights

In the uncertain type and group of AHP, the element of judgement matrix can be an interval number. This paper expands the Column Summation Contrary Method (i. e. CSCM) of general AHP, gives the CSCM of calculation interval number weights, discusses the property of rank preservation, illustrates that CSCM is more reasonable and convenient from theory to reality.

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.

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.

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.

Time variant multi-objective linear fractional interval-valued transportation problem

This paper studies a time-variant multi-objective linear fractional transportation problem. In reality, transported goods should reach in destinations within a specific time. Considering the importance of time, a time-variant multi-objective linear fractional transportation problem is formulated here. We take into account the parameters as cost, supply and demand are interval valued that involved in the proposed model, so we treat the model as a multi-objective linear fractional interval transportation problem. To solve the formulated model, we first convert it into a deterministic form using a new transformation technique and then apply fuzzy programming to solve it. The applicability of our proposed method is shown by considering two numerical examples. At last, conclusions and future research directions regarding our study is included.

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.

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.

Application of Interval Multi-attribute Decision-Making Method to Aeroengine Performance Ranking

In view of the uncertainty of the monitored performance parameters of aeroengines, the fluctuating scope of the monitored infurmation during a period is taken as interval numbers, and the interval multi-attribute decision-making method is employed to predict the performance of aeroengine, The synthetic weights of interval numbers are obtained by calculating deviation degree and possibility degree. As an example of application, 5 performance parameters monitored on 10 CF6 aeroengines of China Eastern Airlines Co., Ltd are adopted as decision attributes to verify the algorithm. The obtained synthetic ranking result shows the effectiveness and rationality of the proposed method in reflecting the performance stares of aeroengins.

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.

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.

Application of uncertain type of AHP to condition assessment of cable-stayed bridges

In order to guarantee the safety service and life-span of long-span cable-stayed bridges, the uncertain type of analytic hierarchy process （AHP） method is adopted to access the bridge condition. The correlative theory and applied objects of uncertain type of AHP are introduced, and then the optimal transitive matrix method is chosen to calculate the interval number judgment matrix, which makes the weights of indices more reliable and accurate. Finally, with Harbin Songhua River Cable-Stayed Bridge as an example, an index system and an assessment model are proposed for the condition assessment of this bridge, and by using uncertain type of AHP, the weights of assessment indices are fixed and the final assessment results of the bridge are calculated, which proves the feasibility and practicability of this method. The application of this assessment method can provide the scientific basis for maintenance and management of long-span cable-stayed bridges.