Economic selection of mean value for a filling process under quadratic quality loss

Consider a filling process where containers are filled with an important ingredient in a character. All containers are inspected, and the containers satisfying to meet the predetermined specification limits are sold in a regular market for a fixed price, and failing to meet them are emptied and refilled by the same filling process after some reprocessing. We assume that reprocessing cost is proportional to the quantity of the ingredients in a container that is not changed after reprocessing. An economic model is constructed on the basis of the selling price and the costs of production, inspection, reprocessing, and quality. We assume that the quality cost function is a quadratic function of the deviation from target and the quantity of the ingredients in a container is normally distributed with a known variance. Method for finding the optimum process mean is presented and a numerical example is given.