4. Process Modeling 4.1. Introduction to Process Modeling 4.1.3. What are process models used for? 4.1.3.3. Optimization
More on Optimization	As mentioned earlier, the goal of optimization is to determine the necessary process input values to obtain a desired output. Like calibration, optimization involves substitution of an output value for the response variable and solving for the associated predictor variable values. The process model is again the link that ties the inputs and output together. Unlike calibration and prediction, however, successful optimization requires a cause and effect relationship between the predictors and the response variable. Designed experiments, run in a randomized order, must be used to ensure that the process model represents a cause and effect relationship between the variables. Typically quadratic models are used, along with standard calculus for finding minimums and maximums, to carry out an optimization. Other techniques can also be used, however. The example discussed below includes a graphical depiction of the optimization process.
Example	In a manufacturing process which requires a chemical reaction to take place, the temperature and pressure under which the process is carried out can affect reaction time. To maximize the through put of this process, an optimization experiment has been carried out in the neighborhood of the conditions felt to be best using a central composite design with 13 runs. Calculus is used to determine the input values associated with local extremes in the regression function. The plot below shows the quadratic surface that was fit to the data and conceptually how the input values associated with the maximum throughput are found.
	As for prediction and calibration, randomness in the data and the need to sample data from the process affects the results. If the optimization experiment were carried out again under identical conditions, the optimal input values computed using the model would be slightly different. As a result, it is important to understand how much random variability there is in the results to interpret the results correctly.
Optimization Result from Repeated Experiment
Optimization Uncertainty	Like prediction and calibration, the uncertainty in the input values estimated to maximize through put can also be computed from the data used to fit the model. Unlike prediction or calibration, however, optimization almost always involves simultaneous estimation of several quanties, the values of each process input. As a result, we will compute a joint confidence region for all of the input values, rather than separate uncertainty intervals for each input. This confidence region will contain the complete set of true process inputs that will maximize through put with high probability. The plot below shows the contours of equal through put on a map of various possible input value combinations. The solid contours show through put while the dashed contour in the center encloses the plausible combinations of input values that yield optimum results. The "+" marks the estimated optimum value. The dashed region is a 95% joint confidence region for the two process inputs. In this region the through put of the process will be approximately 217 units/hour.
Contour Plot, Estimated Optimum & Confidence Region
More Info	Computational details for optimization are primarily presented in Chapter 5: Process Improvement along with material on appropriate experiment designs experiments for optimization. Section 6 specifically focuses on optimization methods and their associated uncertainties.