Models Of Uncertain-Random Programming

The article proposes the models of optimization with constraints under conditions of parametric mixed uncertainty ‒ aleatory and epistemic. We model parameters with aleatory uncertainty by random values with probability distribution functions obtained from statistical data. We model parameters with epistemic uncertainty by uncertain values introduced in the uncertainty theory of Liu B. Experts define the uncertainty distribution functions. We model a function of random and uncertain parameters by uncertain-random value, interpreted as epistemic value parameterized by random values. Optimization criteria (deterministic duplicates of objective functions) are combination of different characteristics of random and uncertain values, which allows both to average objective functions and to take into account risks or reliability arising from the variability of random and uncertain values. Using the proposed models of uncertain-random programming, we formalized as a two-criterion optimization problem with constraints and solved the task of preliminary aerodynamic design in the conditions of parametric mixed uncertainty ‒ calculation of aircraft weight parameters. The uncertainty theory makes possible under certain conditions (for sufficiently wide class of functions) to obtain analytical expressions for characteristics of uncertain functions, that significantly reduces computational costs. To calculate weight parameters of aircraft, we use multicriteria genetic algorithm and statistical modeling. We investigate the dependence of the optimization result on the given probability levels for random values and the expert belief degree for epistemic values reflecting the reliability of the obtained solution. As result of applying the proposed models for calculating the weight parameters of the aircraft, we obtained the Pareto fronts shown in the figures.