Economic selection of sub-process mean values for a mixture production process

The problem of determining the most profitable sub-process means is considered for a production process where multiple ingredients are mixed. The quantities of the ingredients are controlled through the corresponding sub-processes. The values of the sub-process means directly affect the fraction defective and the costs of production, reworking, and scrap of the mixture process. Every item accepted through screening inspection is sold to a regular market at a fixed price, and any rejected item is scrapped. It is assumed that the quantities of the multiple ingredients are normally distributed with known variances. An economic model is constructed, which involves the selling price, production costs, inspection cost, and scrap and reprocessing costs. The goal is to determine the optimum sub-process mean values maximizing expected profit function. A solution procedure is provided, and an illustrative example for the case with two main ingredients is given.