## 2 Materials and Methods  
### The Case Study Data  
The revisited case study is a real wastewater process optimization project that was recently published [36]. The recycling raw feed solution was paper-and-cardboard effluent from a papermill factory that received some pretreatment before this attempt to conduct a comprehensive wastewater ultrafiltration performance improvement. This case study is appealing from several perspectives: (1) it was a real application in a large ultrafiltration operation, (2) it used modern technology, (3) it promoted the \"small-data\" approach, (4) it was a lean-and-green quality enhancement project, (5) it dealt with very complex physicochemical processes, (6) it considered a multi-characteristic process, (7) it performed multifactorial/multi-parameter adjustments, (8) it investigated potential non-linearities in the relationships between inputs and outputs, (9) it \"screened-and-optimized\" an operation in a single step (Taguchi method), (10) it implemented a saturated FFD sampling scheme, and (11) it attempted a concurrent process enhancement. In brief, the three examined quality characteristics used to monitor the wastewater ultrafiltration performance status of the papermill factory were (1) the average permeate flux (\(\rm{J_{P}}\) in \(\rm{L/(m^{2}h)}\)), (2) the chemical oxygen demand rejection rate (COD in %), and (3) the cumulative flux decline (SFD). The synchronous optimization goals were to maximize the average permeate flux and the COD rejection rate, as well as to minimize the cumulative flux decline. The controlling inputs that were designated to screen and optimize the overall ultrafiltration unit operation performance were (1) the transmembrane pressure (TMP), (2) the cross-flow velocity (CFV), (3) the temperature (T), and (4) the molecular weight cut-off (MWCO). The ranges of the four controlling factors were assigned as (1) TMP (1.0-3.0 bar), (2) CFV (0.463-1.041 m/s), (3) T (15-30 \({}^{\circ}\)C), and (4) MWCO (10-100 kDa); the four controlling factors are coded for ease of reference in this work as A, B, C, and D, respectively. The recommended trial design scheme was the Taguchi-type L9(34) OA [26], which was chosen in order to also accommodate potential quadratic effects. The experiments were performed in triplicate. There are several motivating aspects behind an alternative OA dataset analysis path. First of all, it is a \"small-data\" problem. Thus, it is instructive to try other procedures in order to evaluate how different approaches approximate a final solution. This is particularly intriguing because the experiments were replicated and the repeatability of the measurements was deemed satisfactory. The authors used the Taguchi method to screen and optimize the three process characteristics, one response at a time. However, the Taguchi toolbox does not include a solver to carry out the multi-response problem. Surely, there are several successful applications showing how to resolve the multivariate problem in specific FFD datasets; there are techniques available such as MANOVA, desirability analysis, gray relational analysis, and so forth. But in the DOE research literature, there is no clear consensus about any specific treatment. Justifiably, the authors of the ultrafiltration case study used the utility concept, which is another useful option. Another motivating factor arose from an observation during a preliminary inquiry, in which a strong correlation between two characteristics was revealed: the average permeate flux and the cumulative flux decline. It was not surprising that the two characteristics were positively correlated. This occurrence would simply cause the omission of one of the two correlated characteristic responses. Then, the concurrent screening-and-optimization procedure would proceed with a simpler model that would only involve two characteristic responses, instead of three. Nevertheless, a complication emerges as the two correlated process characteristics are set up to be conditioned on opposite optimization goals.