# A Guideline for Implementing a Robust Optimization of a Complex Multi-Stage Manufacturing Process
## 1 Introduction  
Advanced robust process optimization techniques are widely adopted in the manufacturing field, in order to enhance the productivity and to improve the quality of the processes by limiting the effect of fluctuations, noises, and scraps. At the same time, the more and more competitive and stringent industrial scenario requires: (i) keeping production costs to a minimum, and simultaneously, (ii) the maximization of the products' quality [1]. Every existing company must be able to maintain and increase the efficiency and the effectiveness with the aim to maintain their competency and competitiveness athigh level [2]. Quality control methodologies, such as Six Sigma, created at Motorola in the 1980s, have been adopted by leading companies throughout the world for quality and process improvement aiming to enhance an organization's competitiveness. Six Sigma [3; 4; 5] is a quality methodology which reduces variations in a process, and it helps to lower the cost of product as well as process. In the last three decades, the Six Sigma approach has been implemented by many industries and most of them obtained fruitful benefits [6]. Nevertheless, in the Industry 4.0 scenario, and for a complex multi-stage production process, the application of the Six Sigma methodology needs to be expounded and integrated by also considering other techniques. Moreover, the engineers, aiming towards the continuous improvement of the manufacturing, should consider the risk level mitigation for each complex production stage by identifying and understanding the weak points. The support of advanced statistical methods must be integrated [7; 8]. To this end, it must be noted that, more recently, the statistical methods suggested and applied for a robust process optimization have been widely extended by considering: (i) the wide range of experimental designs [9], including optimal designs; (ii) the possibility to evaluate heteroscedasticity and random effects (not only limited to noises) by applying, in the modeling step, a class of statistical models specifically used to treat/study complex situations [10]; (iii) achieving the optimum of the response and the corresponding optimal setting (optimal solution) of the experimental variables by analytical optimization [11], or, alternatively, by comparing several optimal solutions in a decision process that can also include costs and other economic issues. In [12], the decomposition of the mean square error (MSE) is performed also according to several error components related to external elements, such as costs or consumer loss. To this end, in this paper we aim to address a guideline (also called procedure), in which the whole quality process improvement is outlined by considering engineering issues and statistical methodologies, also evaluating features and recent developments in the literature. These days, robust design optimization (RDO) methods have obtained a great deal of attention for improving engineering performances and manufacturing processes [13]. The concept of robust design was primarily introduced by [PERSON] [14; 15; 16; 17; 18]. Robust design allows us to make a product and/or a process insensitive to environmental noises and/or other internal or external sources of variability [19]. The key issue for obtaining a RDO resides in setting the control factor levels to be insensitive to noise variables, often involved in the studies as random effects [20]. There are many potential uncertainties that can arise and influence the practical application in a manufacturing scenario by including process failures, material properties variability, environmental variability (noises), and human-factor dependency [21]. In addition to uncertainty and noise effects, the choice of the right statistical model must consider other issues, which might be imposed in problems under real operational conditions such as dynamic situations, multi-response cases, discrete and continuous data, physical constraints to design factors, performance costs, computational complexity [10; 11; 12].