Application of the smooth approximation of the probability function in some applied stochastic programming problems

This paper is devoted to the application of the smooth approximation of the probability function in the solution of three different stochastic optimization problems: minimization of an airstrip area under the constrained probability of successful landing, minimization of the cost of water supply system with random performance and with predefined water consumption, and determination of the set of wind speed vectors which guarantees the safe landing of an aircraft in future with the given probability. The first two problems are mathematical programming problems with probability constraint, and the third one is a problem of constructing the isoquant surface of the probability function. Smooth approximation of the probability function allows to use the gradient projection method in the constrained optimization problem and to define the isoquant surface as the solution to a partial differential equation. We provide an example for each of the considered problems and compare the results with known results previously obtained using the confidence method.