Apparatus and methods for delivering and placing proppant to a subterranean formation fracture including identifying control variables and uncertain parameters of the proppant delivery and placement, optimizing a performance metric of the proppant delivery and placement under uncertainty, calculating sensitivity indices and ranking parameters according to a relative contribution in total variance for an optimized control variable, and updating a probability distribution for parameters, repeating optimizing comprising the probability distribution, and evaluating a risk profile of the optimized performance metric using a processor. Some embodiments may deliver proppant to the fracture using updated optimized values of control variables.