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Top1. Introduction
Response surface methodology (RSM) consists of a group of techniques used in empirical study between a response and a number of input variables. The researcher attempts to find the optimal setting for the input variables that either maximizes or minimizes the response (Myers et al., 2009). RSM assumes that variance of the response is constant and focuses on optimizing the mean of the response. However, the constant variance assumption might not be valid in practice. In such cases, not only the mean response, but also the standard deviation of the response should be considered in determining the optimum conditions for the input variables.
Dual-response optimization (DRO) attempts to optimize both the mean and standard deviation of the response. The conventional approach to DRO requires building statistical models for mean and standard deviation of the response, by fitting response surface models (usually second-order models) to experimental data. Then, an optimal setting for the input variables is obtained by analyzing the statistical models (Lee et al., 2010; Lee & Kim, 2012).
Usually, RSM focuses on optimization of a single response. However, it is very common to consider several response variables in product and process development. This is called a multi-response problem. Some typical problems include the tire tread compound problem (Derringer & Suich, 1980), the injection molding of a washing machine agitator problem (Reddy et al., 1997), the rapid prototyping system problem (Palmer et al., 2006), and semiconductor manufacturing process problems (Lee & Kim, 2012; Lee & Kim 2013). Similar to DRO, multi-response optimization (MRO) usually builds statistical models to analyze the means of multiple responses by fitting response surface models (usually second-order models) to experimental data. Then, an optimal setting for the input variables is obtained by analyzing the statistical models (Lee & Kim, 2012).
There have been many methods for MRO. However, most of them consider only the mean of multiple responses, like the traditional RSM. As previously mentioned, the constant variance assumption might not be valid in practice. Thus, methods for optimizing both mean and standard deviation of multiple responses are needed. We call this research area extended MRO (EMRO). Figure 1 shows the relationships among RSM, DRO, MRO, and EMRO. In spite of the importance of EMRO, limited studies have been performed.
Figure 1. Relationships among RSM, DRO, MRO, and EMRO
One possible approach to EMRO is to utilize the conventional modeling methods of RSM, DRO, and MRO. However, these approaches have two practical limitations. First, efforts for conducting experiments can be a burden for process engineers. In order to build the statistical models, experiments are conducted and experimental data are gathered. In particular, in the case of DRO, a large number of experiments are usually conducted, because replicate runs at each design point are needed to fit the statistical model for standard deviation of the response, whereas replicate runs are not mandatory in RSM.
Second, the optimal settings for the input variables obtained from the conventional approaches have uncertainty (Lee & Lee, 2016). This is because the optimal setting is obtained from the prediction models. Care must be taken regarding uncertainty when the prediction capability of the models is poor. As the prediction model varies from the true model, the resulting optimal setting might be quite far from optimal (Xu & Albin, 2003). Also, it is reported that the prediction capability of the models is often poor when the models are fitted to operation data from manufacturing lines, whose volume is typically large (Lee & Kim, 2008).