Multivariate optimization by visual inspection

In this paper we present a user-friendly multivariate optimization methodology based on visual inspection of contour plots of selected optimization criteria. Several possibilities exist when choosing optimization criteria, but we focus on the overall closeness to target (quadratic form), its robustness, the maximum individual deviation from target (i.e. the worst optimized response) and the empirical variance of the individual deviations from target. The approach assumes that there exist models relating the input variables to the responses, and that the responses have certain desirable target values. The methodology is general but particularly useful when the number of control variables is low and the responses to be optimized are many. In cases with several control variables, conditional contour plots can be used. We demonstrate the use of the methodology with two examples from the food industry. The examples show that the different criteria do not necessarily predict the same optimum and that weighing of the individual quadratic deviations from target (using an inverse covariance matrix) can be important if the responses are strongly correlated.