Taking decisions by using fuzzy models

In the most specialty papers, a mathematical model for establishing the best decisional alternatives is a couple M=(R, K) made up of a matrix R with m lines and n columns and a column vector K with m components. The mathematical modeling of an uncertain information evaluates its most probable value am and the minus possible values as and plus ad. The three real values form an ordered triplet named fuzzy triangular number. A fuzzy model for establishing the best decisional alternative is a couple formed by the results matrix having fuzzy triangular numbers as elements and the importance coefficients vector K. These models, (fuzzy and/or classical) are equivalent models related to a method of hierarchy of decisional alternatives if the two models have the same criteria and alternatives and by using the method to both models, the same hierarchy is obtained.uncertainty, fuzzy triangular numbers, taking decisions, fuzzy models

Bayesian Updating, Model Class Selection and Robust Stochastic Predictions of Structural Response

A fundamental issue when predicting structural response by using mathematical models is how to treat both modeling and excitation uncertainty. A general framework for this is presented which uses probability as a multi-valued conditional logic for quantitative plausible reasoning in the presence of uncertainty due to incomplete information. The fundamental probability models that represent the structure’s uncertain behavior are specified by the choice of a stochastic system model class: a set of input-output probability models for the structure and a prior probability distribution over this set that quantifies the relative plausibility of each model. A model class can be constructed from a parameterized deterministic structural model by stochastic embedding utilizing Jaynes’ Principle of Maximum Information Entropy. Robust predictive analyses use the entire model class with the probabilistic predictions of each model being weighted by its prior probability, or if structural response data is available, by its posterior probability from Bayes’ Theorem for the model class. Additional robustness to modeling uncertainty comes from combining the robust predictions of each model class in a set of competing candidates weighted by the prior or posterior probability of the model class, the latter being computed from Bayes’ Theorem. This higherlevel application of Bayes’ Theorem automatically applies a quantitative Ockham razor that penalizes the data-fit of more complex model classes that extract more information from the data. Robust predictive analyses involve integrals over highdimensional spaces that usually must be evaluated numerically. Published applications have used Laplace's method of asymptotic approximation or Markov Chain Monte Carlo algorithms

Unified polynomial expansion for interval and random response analysis of uncertain structure–acoustic system with arbitrary probability distribution

© 2018 Elsevier B.V. For structure–acousticsystem with uncertainties, the interval model, the random model and the hybrid uncertain model have been introduced. In the interval model and the random model, the uncertain parameters are described as either the random variable with well defined probability density function (PDF) or the interval variable without any probability information, whereas in the hybrid uncertain model both interval variable and random variable exist simultaneously. For response analysis of these three uncertain models of structure–acoustic problem involving arbitrary PDFs, a unified polynomial expansion method named as the Interval and Random Arbitrary Polynomial Chaos method (IRAPCM) is proposed. In IRAPCM, the response of the structure–acoustic system is approximated by APC expansion in a unified form. Particularly, only the weight function of polynomial basis is required to be changed to construct the APC expansion for the response of different uncertain models. Through the unified APC expansion, the uncertain properties of the response of three uncertain models can be efficiently obtained. As the APC expansion can provide a free choice of the polynomial basis, the optimal polynomial basis for the random variable with arbitrary PDFs can be obtained by using the proposed IRAPCM. The IRAPCM has been employed to solve a mathematical problem and a structure–acoustic problem, and the effectiveness of the unified IRAPCM for response analysis of three uncertain models is demonstrated by fully comparing it with the hybrid first-order perturbation method and several existing polynomial chaos methods

A little about models

We discuss several aspects of creation of adequate mathematical models in other sciences. In particular, many difficulties stem from great complexity of the source systems and the presence of a variety of uncertain factors. We illustrate the effect of uncertainty on the known consumer demand model. We conclude that not every uncertainty can be represented by a random variable, and that these concepts are not equivalent. We discuss also the role of different information concepts in mathematical models. We give additional illustrative examples of models of quite complex systems.Comment: 15 page

Second order adjoints for solving PDE-constrained optimization problems

Inverse problems are of utmost importance in many fields of science and engineering. In the variational approach inverse problems are formulated as PDE-constrained optimization problems, where the optimal estimate of the uncertain parameters is the minimizer of a certain cost functional subject to the constraints posed by the model equations. The numerical solution of such optimization problems requires the computation of derivatives of the model output with respect to model parameters. The first order derivatives of a cost functional (defined on the model output) with respect to a large number of model parameters can be calculated efficiently through first order adjoint sensitivity analysis. Second order adjoint models give second derivative information in the form of matrix-vector products between the Hessian of the cost functional and user defined vectors. Traditionally, the construction of second order derivatives for large scale models has been considered too costly. Consequently, data assimilation applications employ optimization algorithms that use only first order derivative information, like nonlinear conjugate gradients and quasi-Newton methods. In this paper we discuss the mathematical foundations of second order adjoint sensitivity analysis and show that it provides an efficient approach to obtain Hessian-vector products. We study the benefits of using of second order information in the numerical optimization process for data assimilation applications. The numerical studies are performed in a twin experiment setting with a two-dimensional shallow water model. Different scenarios are considered with different discretization approaches, observation sets, and noise levels. Optimization algorithms that employ second order derivatives are tested against widely used methods that require only first order derivatives. Conclusions are drawn regarding the potential benefits and the limitations of using high-order information in large scale data assimilation problems

Evaluation of Corporate Sustainability

As a consequence of an increasing demand in sustainable development for business organizations, the evaluation of corporate sustainability has become a topic intensively focused by academic researchers and business practitioners. Several techniques in the context of multiple criteria decision analysis (MCDA) have been suggested to facilitate the evaluation and the analysis of sustainability performance. However, due to the complexity of evaluation, such as a compilation of quantitative and qualitative measures, interrelationships among various sustainability criteria, the assessor’s hesitation in scoring, or incomplete information, simple techniques may not be able to generate reliable results which can reflect the overall sustainability performance of a company. This paper proposes a series of mathematical formulations based upon the evidential reasoning (ER) approach which can be used to aggregate results from qualitative judgments with quantitative measurements under various types of complex and uncertain situations. The evaluation of corporate sustainability through the ER model is demonstrated using actual data generated from three sugar manufacturing companies in Thailand. The proposed model facilitates managers in analysing the performance and identifying improvement plans and goals. It also simplifies decision making related to sustainable development initiatives. The model can be generalized to a wider area of performance assessment, as well as to any cases of multiple criteria analysis