Abstract
There are many engineering problems which are reduced to the mathematical programming problem. The constraints and/or the objective function established are sometimes subjected to errors due to experiments or estimations, and thus are probabilistic in nature. In such a case, a stochastic approach must be adopted to make the program realistic by treating the constraints and/or the objective function as random variables. Thus we set up the problems 1) to minimize the expected value of the objective function under the chance-constraints on the constraints and 2) to minimize the expected value of the objective function under the chance-constraint on the objective function as well as on the constraints.
The constraints and the objective function are random variables, the distributions of which are not predetermined. Thus, the chance-constraints are not to be calculated directly. In this paper, a new approach is employed to transform the chance-constraints into the equivalent deterministic nonlinear constraints. Validity of this transformation is proved by using Tchebychev inequality. A possible algorithm to solve the problems is proposed and numerical examples are also provided to illustrate the given method.
