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The purpose of this paper is to focus on the influences of the uncertain dynamic responses on the reconstruction of loads.

Based on the assumption of unknown-but-bounded (UBB) noise, a time-domain approach to estimate the uncertain time-dependent external loads is presented by combining the inverse system method in modern control theory and interval analysis in interval mathematics. Inspired by the concept of set membership identification in control theory, an interval analysis model of external loads time history, which is indeed a region or feasible set containing all possible loads being consistent with the bounded structural acceleration responses is established and further solved by two interval algorithms.

Unlike traditional loads identification methods which only give a point estimation, an interval estimation of external loads time history, which is a region containing all the possible loads being consistent with the uncertain structural responses, is determined. The correlation characteristics among the responses of acceleration, velocity, and displacement are also discussed in consideration of the UBB uncertainty.

For one hand, the solution of the inverse problem in original system is transformed to the solution of the direct problem in inverse system; for another, the authors deal with the uncertainty by use of interval analysis method, and the identified interval process, which contains any possible external loads time history being consistent with the bounded structural responses can be approximately obtained.

This paper aims to propose a novel statistic energy analysis method with fuzzy parameters to study the dynamic and acoustic responses of coupled system with fuzzy parameters, which can expand the applied range of statistic energy analysis method in engineering to some extent.

On the basis of the property of membership level, the uncertain fuzzy parameters are expressed as the interval forms. Interval mathematics and interval expansion principle are adopted to solve the problem with interval parameters. At last, two numerical examples, which include a two-plate coupled system and a single-partition sound-insulation system, are carried out to demonstrate the feasibility and validity of the presented method.

Interval mathematics and interval expansion principle are adopted to solve the problem with interval parameters.

By integrating the interval analysis, optimization technique and Taylor expansion method, two non-probabilistic, set-theoretical statistical energy analyses are proposed for predicting the dynamical and acoustical response of the complex coupled system with uncertain parameters in high-frequency domain.

The purpose of this paper is to formulate the reliability optimization problem in stochastic and interval domain and also to solve the same under different stochastic set up.

Stochastic programming technique has been used to convert the chance constraints into deterministic form and the corresponding problem is transformed to mixed-integer constrained optimization problem with interval objective. Then the reduced problem has been converted to unconstrained optimization problem with interval objective by Big-M penalty technique. The resulting problem has been solved by advanced real coded genetic algorithm with interval fitness, tournament selection, intermediate crossover and one-neighbourhood mutation.

A new optimization technique has been developed in stochastic domain and the concept of interval valued parameters has been integrated with the stochastic setup so as to increase the applicability of the resultant solution to the interval valued nonlinear optimization problems.

The concept of probability distribution with interval valued parameters has been introduced. This concept will motivate the researchers to carry out the research in this new direction.

The application of genetic algorithm is extended to solve the reliability optimization problem in stochastic and interval domain.

Uncertainty is ubiquitous in practical engineering and scientific research. The uncertainties in parameters can be treated as interval numbers. The prediction of upper and lower bounds of the response of a system including uncertain parameters is of immense significance in uncertainty analysis. This paper aims to evaluate the upper and lower bounds of electric potentials in an electrostatic system efficiently with interval parameters.

The Taylor series expansion is proposed for evaluating the upper and lower bounds of electric potentials in an electrostatic system with interval parameters. The uncertain parameters of the electrostatic system are represented by interval notations. By performing Taylor series expansion on the electric potentials obtained using the equilibrium governing equation and by using the properties of interval mathematics, the upper and lower bounds of the electric potentials of an electrostatic system can be calculated.

To evaluate the accuracy and efficiency of the proposed method, the upper and lower bounds of the electric potentials and the computation time of the proposed method are compared with those obtained using the Monte Carlo simulation, which is referred to as a reference solution. Numerical examples illustrate that the bounds of electric potentials of this method are consistent with those obtained using the Monte Carlo simulation. Moreover, the proposed method is significantly more time-saving.

This paper provides a rapid computational method to estimate the upper and lower bounds of electric potentials in electrostatics analysis with interval parameters. The precision of the proposed method is acceptable for engineering applications, and the computation time of the proposed method is significantly less than that of the Monte Carlo simulation, which is the most widely used method related to uncertainties. The Monte Carlo simulation requires a large number of samplings, and this leads to significant runtime consumption.

Many analysis and design problems in engineering and science involve uncertainty to varying degrees. This paper is concerned with the structural vibration problem involving uncertain material or geometric parameters, specified as fuzzy parameters. The requirement is to propagate the parameter uncertainty to the eigenvalues of the structure, specified as fuzzy eigenvalues. However, the usual approach is to transform the fuzzy problem into several interval eigenvalue problems by using the α-cuts method. Solving the interval problem as a generalized interval eigenvalue problem in interval mathematics will produce conservative bounds on the eigenvalues. The purpose of this paper is to investigate strategies to efficiently solve the fuzzy eigenvalue problem.

Based on the fundamental perturbation principle and vertex theory, an efficient perturbation method is proposed, that gives the exact extrema of the first-order deviation of the structural eigenvalue. The fuzzy eigenvalue approach has also been improved by reusing the interval analysis results from previous α-cuts.

The proposed method was demonstrated on a simple cantilever beam with a pinned support, and produced very accurate fuzzy eigenvalues. The approach was also demonstrated on the model of a highway bridge with a large number of degrees of freedom.

This proposed Vertex-Perturbation method is more efficient than the standard perturbation method, and more general than interval arithmetic methods requiring the non-negative decomposition of the mass and stiffness matrices. The new increment method produces highly accurate solutions, even when the membership function for the fuzzy eigenvalues is complex.

The purpose of this paper is to propose a novel non-probabilistic reliability-based topology optimization (NRBTO) method for continuum structural design under interval uncertainties of load and material parameters based on the technology of 3D printing or additive manufacturing.

First, the uncertainty quantification analysis is accomplished by interval Taylor extension to determine boundary rules of concerned displacement responses. Based on the interval interference theory, a novel reliability index, named as the optimization feature distance, is then introduced to construct non-probabilistic reliability constraints. To circumvent convergence difficulties in solving large-scale variable optimization problems, the gradient-based method of moving asymptotes is also used, in which the sensitivity expressions of the present reliability measurements with respect to design variables are deduced by combination of the adjoint vector scheme and interval mathematics.

The main findings of this paper should lie in that new non-probabilistic reliability index, i.e. the optimization feature distance which is defined and further incorporated in continuum topology optimization issues. Besides, a novel concurrent design strategy under consideration of macro-micro integration is presented by using the developed RBTO methodology.

Uncertainty propagation analysis based on the interval Taylor extension method is conducted. Novel reliability index of the optimization feature distance is defined. Expressions of the adjoint vectors between interval bounds of displacement responses and the relative density are deduced. New NRBTO method subjected to continuum structures is developed and further solved by MMA algorithms.

This paper attempts to establish the general formula for computing the inverse of grey matrix, and the results are applied to solve grey linear programming. The inverse of a grey matrix and grey linear programming plays an important role in establishing a grey computational system.

Starting from the fact that missing information often appears in complex systems, and therefore that true values of elements are uncertain when the authors construct a matrix, as well as calculate its inverse. However, the authors can get their ranges, which are called the number-covered sets, by using grey computational rules. How to get the matrix-covered set of inverse grey matrix became a typical approach. In this paper, grey linear programming was explained in detail, for the point of grey meaning and the methodology to calculate the inverse grey matrix can successfully solve grey linear programming.

The results show that the ranges of grey value of inverse grey matrix and grey linear programming can be obtained by using the computational rules.

Because the matrix and the linear programming have been widely used in many fields such as system controlling, economic analysis and social management, and the missing information is a general phenomenon for complex systems, grey matrix and grey linear programming may have great potential application in real world. The methodology realizes the feasibility to control the complex system under uncertain situations.

The paper successfully obtained the ranges of uncertain inverse matrix and linear programming by using grey system theory, when the elements of matrix and the coefficients of linear programming are intervals and the results enrich the contents of grey mathematics.

The purpose of this paper is to propose a straightforward model for selecting slightly non‐homogeneous vendors.

In this paper the use of the interval data envelopment analysis (DEA) is suggested. The bounds of intervals are constant and can be obtained by various estimation techniques. The interval DEA model provides for the decision making units (DMUs) with missing values a lower and an upper bound of their efficiency score corresponding to their most favorable and unfavorable option.

Employing the proposed method for selecting slightly non‐homogeneous vendors largely reduced practical difficulties for vendor selection. This method does not exclude any vendor from the selection problem. For all the vendors it provides bounds of the efficiency scores depended on the particular data values that the vendors with missing data assign within the intervals so to maximize their efficiency score.

The proposed model considers a slightly non‐homogeneous situation for vendor selection. The proposed approach is driven by multiple criteria. The joint consideration of multiple criteria in a slightly non‐homogeneous environment helps managers select vendors using a comprehensive approach that goes beyond just purchase costs.

This paper is believed to be the first to discuss the problem of slightly non‐homogeneous vendor selection with respect to interval mathematics.

In recent years there has been a significant acceleration in the market growth of social impact investing. Policy makers, regulatory bodies and national decision-makers should base their decision-making processes on multiple criteria. These criteria are, by nature, imprecise, ambiguous and uncertain. The purpose of this paper is to provide decision-makers with a mathematical tool which aids them in their decision-making processes identifying the degree of appropriateness of less developed countries in terms of potential success of investment in vaccination campaigns.

In this work, the authors have developed a decision-making tool within the framework of multiple criteria decision making and Fuzzy Logic, which aims to aid decision-makers for vaccinations campaigns in less developed countries. In particular, the authors have proposed a Technique for Order Preference by Similarity to Ideal Solution-based method which is able to work in fuzzy environment in order to assess and rank countries based on their fuzzy degree of appropriateness for impact investing in vaccines.

The impact investing market provides capital from private sources to address many pressing global challenges such as access to basic services as health. Governments have, therefore, an essential role in supporting the development of this market by improving the risk/return profile of investments through access to credit facilities, tax credits or subsidies or defining the regulation of the supply of investments, provision of technical assistance to investing private companies and co-financing. The proposed framework permits funding decision making taking into account the degree of preparedness and adequacy for impact investing in vaccines of the selected countries.

Impact investing can play a key role in the reduction of immunization gap offering suitable strategies for both, governments and private investors for the achievement of United Nations Sustainable Development Goals (SDGs). However, in order to make good financial decisions managers should take into account not only health, income, education and other social criteria but also the degree of basic preparedness of the countries in order to ensure the success of the immunization campaigns which means taking into account availability of basic infrastructures, access to electricity, political stability among other criteria.

However, in order to make good financial decisions managers should take into account not only health, income, education and other social criteria but also the degree of basic preparedness of the countries in order to ensure the success of the immunization campaigns which means taking into account availability of basic infrastructures, access to electricity, political stability among other criteria.

The proposed model will allow public and private decision makers to make better investment decisions in terms of effectiveness as the provided ranking of countries candidates for the investments is more realistic and takes into account more decision dimensions.

This paper discusses the possibility of automating the design of electromagnetic devices. Several paradigms currently exist to accelerate the design process, search the design space and examine the effects of tolerances on various parameters. Amongst these are semantic networks, response surfaces, interval mathematics and sensitivity analysis. This paper explores all of these and also suggests what is needed in the future in order to create a true computer based design system.