Mathematical Models and Methods of Effective Estimation in Multi-objective Optimization Problems Under Uncertainties

Abstract

The selection of the optimal criteria for solutions and sought values of the objective functions (multi-objective decision making) is a complex task. It becomes a real challenge when prior data are uncertain. In this paper you will find a new approach to solve this task. The new method uses the updated method of getting the scalar convolutions of the criteria for the described task. The updated method references Ashby`s law of Requisite Variety, Kolmogorov’s concept of power averages and Tikhonov`s ideas of regularization.

The obtained set of the scalar convolutions was evaluated from the maximum likelihood principle. The set can be used for synthesis of robust meta-models, mathematical model identification, and robust optimal engineering.

The scalar convolutions for the criteria were obtained from Student`s statistics. It served as a criterion to check the equality of distribution centers for representative samples from two multidimensional general t populations. Student`s statistics also played a role of multidimensional analogue of Romanovsky criterion Ro to check the hypothesis about the equality of covariance matrices Ro or statistics H (H is the mutual information) instead of statistics Ro.

The paper contains the mathematical formulations and computational methods for the synthesis of quasi-solutions of stochastic optimization problems with mixed conditions. The implementation of the research results will provide the developers with the robust estimations of the sought values even if the prior data are uncertain.

The paper also deals with a new probability-based method to solve the direct problem of dimensional engineering networks. In accordance with the probability-based method the mathematical expectations and confidence intervals of the control variables of functional elements are obtained from the mathematical expectations and confidence intervals of the decision selection criteria or from the phase variables of considered systems or processes.

To make the processing time several times less one proposed the memetic algorithm with the consistent application of advanced real coded evolutionary method, decremental neighborhoods method, randomized path relinking method.

At the end of the paper you can observe the interactive decision support system «Concept_Pro_St^®» that focuses on a wide range of users in the fields of: engineering, project management, data-monitoring oriented management and production supervision to ensure product quality (Design for Six Sigma), industrial safety, environmental, pharmaceuticals, medicine, etc., working on issues of construction of robust meta-models (formal mathematical models in the form of regression equations), robust optimal design and diagnostics of systems and processes.

To validate the method regarding to some particular object one solved the problem of robust optimal designing of centrifugal impeller fitted with backward curved blades in the conditions of stochastic nature of the input data.