# A Guideline for Implementing a Robust Optimization of a Complex Multi-Stage Manufacturing Process
## 3 The Methodology for Continuous Improvement: Robust Process Optimization
### The Statistical Modeling  
When considering the application of statistical methods specifically defined for the robust process optimization, undoubtedly the two main key-points are the two-step procedure, and the concept of robust design, initially defined by [PERSON] [17]. When dealing with statistical methods and modeling for a robust process optimization, the most widely applied methodology is the response surface methodology (RSM) [55], essentially of the I and II order. RSM was firstly suggested as an alternative method with respect to [PERSON]'s parameter design [16], in particular considering two seminal studies: (i) the dual response approach [56], and in 1992 the proposal of the combined array [57], where the robust design concept found the first and effective valid implementation. Moreover, starting from 1990s and more recently, all literature on this subject has been developed considering: (i) the problem of heteroscedasticity, and/or the inclusion of noises and/or random effects, among the others [58; 59]; and (ii) the specific planning of experimental designs that are the most suitable for evaluating fixed and random effects for a robust design modeling and optimization, such as the split-plot design [20; 60]. In this specific case study, we consider a theoretical surface:  
\[\eta=\text{f}(\text{x}_{1},\ldots,\text{x}_{i},\ldots,\text{x}_{k}), \tag{2}\]  
where \(\eta\) is a surface function of \(k\) input or independent variables \(\text{x}_{i}\) (\(\text{i}=1,\ldots,\text{k}\)). Starting from Formula (2), the local approximation in Taylor Series within an I(x\({}_{0}\)) allows for obtaining (and estimating) the response surface models of \(d=1 st\) and \(d=2 nd\) order, by assuming that the function \(f\) is continuous and derivable of order \(\text{d}+1\), to this end all the x\({}_{i}\) must be quantitative and continuous variables. It must be noted that the 2 nd order RSM is expounded with respect to the 1 st order model by adding the first order interaction terms (interaction including two factors) and the quadratic terms for all the independent variables \(\text{x}_{i}\) (\(\text{i}=1,\ldots,\text{k}\)). Let us now consider a dependent and quantitative variable (\(\gamma\)) assumed as the measure of the quality of the process; let (x\({}_{1}\),..., x\({}_{i}\),..., x\({}_{k}\)) be the set of factors/variables involved in the study; obviously we are assuming that all the x\({}_{i}\) have an influence on the response variable (Y). The 2 nd order RS model for the general u-th experimental observation (u = 1,..., n), and for K = 2, is as follows:  
\[y_{u}=\ \beta_{0}\ +\ \beta_{1}x_{1}+\ \beta_{2}\ x_{2}+\beta_{12}\ x_{12}+\beta_{11 }\ x_{1}^{2}+\beta_{22}\ x_{2}^{2}+\epsilon_{u}\ \text{u}\ =\ 1,\ \ldots,\ \text{n} \tag{3}\]  
In Formula (3), \(\beta_{1}\) and \(\beta_{2}\) are the two unknown coefficients related to the linear effects of the model, related to the variables x\({}_{1}\) and x\({}_{2}\); \(\beta_{12}\) is the coefficient related to the 1 st order interaction, while \(\beta_{11}\) and \(\beta_{22}\) are related to the quadratic effects. It must be noted that all the unknown 2 nd order coefficients, \(\beta_{12}\), \(\beta_{11}\) and \(\beta_{22}\), are the elements of the array **B** = (2 \(\times\) 2) that plays a fundamental role for the optimization; \(\epsilon_{u}\) (u = 1,..., n) is the random error. The logistic regression is now briefly illustrated in order to outline the fundamental elements for the case study; further details related to the logit model can be found in [61]. In this kind of statistical modeling, the main feature is the nature of the response variable; in fact, in this model the dependent variable Y is binary: Y = 0,1. The set of input (independent) variables, also called explicative variables, can be formed by qualitative and/or quantitative variables. Therefore, through the logit model we can estimate the probability of the event \"success\", usually coded by \"1\". Thus, P(Y = 1!