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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

PERTURBATIONAL SOLUTIONS FOR FUZZY-STOCHASTIC FINITE ELEMENT EQUILIBRIUM EQUATIONS (FSFEEE)

In this paper, the random interval equilibrium equations (RIEE) is obtained by lambda-level cutting the fuzzy-stochastic finite element equilibrium equations (FSFEEE). Based on the relations between the variables of equilibrium equations, solving RIEE is transformed into solving two kinds of general random equilibrium equations (GREE). Then the recursive equations of evaluating the random interval displacement is derived from the small-parameter perturbation theory. The computational formulae of statistical characteristic of the fuzzy random displacements, the fuzzy random strains and the fuzzy random stresses are also deduced in detail.

Study on the extending multi-attribute decision model of grey target

Based on the distance of interval numbers and the two-stage decision methods, this paper expands the decision model of grey target into some situation under which the decision information and target weights are the interval numbers at the same time. It also gives the optimization method of weights in the grey target. We get the optimum coordinated vector utilizing the combination assigning method, based on the local optimization of various schemes. So it can shift the weights of interval number into real number form and sequence it according to the weighted off-target distance. Finally the effectiveness and practicality of the model is proved by a real project.

Environmental stress level evaluation approach based on physical model and interval grey association degree

Associating environmental stresses （ESs） with built-in test （BIT） output is an important means to help diagnose intermittent faults （IFs）. Aiming at low efficiency in association of traditional time stress measurement device （TSMD）, an association model is built. Thereafter, a novel approach is given to evaluate the integrated environmental stress （IES） level. Firstly, the selection principle and approach of main environmental stresses （MESs） and key characteristic parameters （KCPs） are presented based on fault mode, mechanism, and ESs analysis （FMMEA）. Secondly, reference stress events （RSEs） are constructed by dividing IES into three stress levels according to its impact on faults; and then the association model between integrated environmental stress event （IESE） and BIT output is built. Thirdly, an interval grey association approach to evaluate IES level is proposed due to the interval number of IES value. Consequently, the association output can be obtained as well. Finally, a case study is presented to demonstrate the proposed approach. Results show the proposed model and approach are effective and feasible. This approach can be used to guide ESs measure, record, and association. It is well suited for on-line assistant diagnosis of faults, especially IFs.

CONFLICT MEASURE MODEL FOR LARGE GROUP DECISION BASED ON INTERVAL INTUITIONISTIC TRAPEZOIDAL FUZZY NUMBER AND ITS APPLICATION

The problem of measuring conflict in large-group decision making is examined with every decision preference expressed by multiple interval intuitionistic trapezoidal fuzzy numbers （IITFNs）. First, a distance measurement between two IITFNs is given and a function of conflict between two members of the large group is proposed. Second, members of the large group are clustered. A measurement model of group conflict, which is applied to aggregating large-group preferences, is then proposed by employing the conflict measure of clusters. Finally, a simulation example is presented to validate the models. These models can deal with the preference analysis and coordination of a large-group decision, and are thus applicable to emergency group decision making.

Dynamic Multi-Attribute Decision Analysis on Three-Parameter Interval Grey Number of Upper(Lower)Deviation Quasi-Normal Distribution

Aiming at the dynamic multi-attribute decision making problem where the weight of each decision stage and attribute weight are completely unknown and the attribute value is unknown distributed three-parameter interval grey number,a threeparameter interval grey number dynamic multiattribute grey target decision making method with attribute value following quasi-normal distribution is proposed.Firstly,the position relationship between the“center of gravity”point and the kernel of the threeparameter interval grey number is discussed.According to the characteristicthat the attribute value obeys the quasi-normal distribution,anew weight isgiventothe“centerof gravity”point,and a new distance measure formula of the three-parameter interval grey number is defined.Secondly,according to the principle of maximum entropy,the objective programming model is constructed to determine the stage weight and attributeweight.Then,the schemes aresorted according to thesize of the comprehensive bull's-eye distance Finally an example is given to illustrate the effectiveness of the decision model.

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.

Grey clustering evaluation based on AHP and interval grey number

Purpose–The purpose of this paper is to propose a grey clustering evaluation model based on analytic hierarchy process(AHP)and interval grey number(IGN)to solve the clustering evaluation problem with IGNs.Design/methodology/approach–First,the centre-point triangular whitenisation weight function with real numbers is built,and then by using interval mean function,the whitenisation weight function is extended to IGNs.The weights of evaluation indexes are determined by AHP.Finally,this model is used to evaluate the flight safety of a Chinese airline.The results indicate that the model is effective and reasonable.Findings–When IGN meets certain conditions,the centre-point triangular whitenisation weight function based on IGN is not multiple-cross and it is normative.It provides a certain standard and basis for obtaining the effective evaluation indexes and determining the scientific evaluation of the grey class.Originality/value–The traditional grey clustering model is extended to the field of IGN.It can make full use of all the information of the IGN,so the result of the evaluation is more objective and reasonable,which provides supports for solving practical problems.

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.

On the Number of Finite Non-isomorphic Abelian Groups in Short Intervals

Let the arithmetic function a(n) denote the number of non-isomorphic Abeliangroups of order n;k, positive integer, and x≥0. We setA_k(x)= sum from n≤x a(n)=k to (1)andA_k(x;h) =A_k(x+h)-A_k(x). A. Ivice first investigated the distribution of the values of finite non-isomorphicAbelian groups in short intervals. E. Kratzel reduced the problem to estimate theerror term △(1, 2, 3;x) in the three-dimensional multiplicative problem, and furtherimproved Ivice’s result.