Most of the existing methods for the analysis and optimization of multiple responses require some kinds of weighting of these responses, for instance in terms of cost or desirability. Particularly at the design stage, such information is hardly available or will rather be subjective. An alternative strategy uses loss functions and a penalty matrix that can be decomposed into a standardizing (data-driven) and a weight matrix. The effect of different weight matrices is displayed in joint optimization plots in terms of predicted means and variances of the response variables. In this article, we propose how to choose weight matrices for two and more responses. Furthermore, we prove the Pareto optimality of every point that minimizes the conditional mean of the loss function.
In this article, we provide a strategy for the simultaneous optimization of multiple responses. Cases are covered where a set of response variables has finite target values and depends on easy to control as well as on hard to control variables. Our approach is based on loss functions, without the need for a predefined cost matrix. For each element of a sequence of possible weights assigned to the individual responses, settings of the easy to control parameters are determined, which minimize the estimated mean of a multivariate loss function. The estimation is based on statistical models, which depend only on the easy to control variables. The loss function itself takes the value zero, if all responses are on target with zero variances. In each case, the derived parameter settings are connected to a specific compromise of the responses, which is graphically displayed to the engineer by so called joint optimization plots. The expert can thereby gain valuable insight into the production process and then decide on the most sensible parameter setting. The proposed strategy is illustrated with a data set from the literature and new data from an up to date application.
SUMMARYSheet metal spinning is a very complex forming process with a large number of quality characteristics. Within the scope of a joint project of the Department of Statistics and the Chair of Forming Technology the impact of process parameters (design factors) on important quality characteristics has been investigated both theoretically and experimentally. In the past, every response has been treated individually and uncontrollable disturbances (noise factors) have been neglected. Now this approach has been extended to robust multiresponse parameter design. For this, a review of common multivariate approaches for robust parameter design has been carried out, which also leads to the proposal of some new variants. In addition to the theoretical comparison, the methods were applied to data gained in the sheet metal spinning process. The obtained results were evaluated in terms of applicability, limitations and quality accuracy. Practical experiments confirmed the high degree of efficiency that the finally proposed method based on desirabilities promises.
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