A New Approach for Multiple-Response Optimization

The desirability of a response that increases in a linear manner from the lower specification limit to the target value and decreases in a linear manner from the target value to the upper specification limit results in a breakpoint (nondifferentiable point) at the target value of its desirability function. Thus, modified desirability functions were proposed to solve this breakpoint problem so that the generalized reduced gradient can be applied. The modified desirability functions require the fitting of a quartic polynomial in the vicinity of the target. However, the fitting of the quartic polynomial needs to satisfy five conditions and involves numerous computations. Therefore, an approach with less computations and yet that is simple to implement is proposed in this article. This approach is applicable when the lower specification limit, the target value, and the upper specification limit of a response are available. In addition, this approach also allows practitioners to assign different degrees of importance to the responses. An example from the original paper of modified desirability functions is used to illustrate this approach.