## 2 Materials and Methods
### Data Analysis  
#### 2.2.1 Lenth Statistics and Belief Functions  
The FFD sampler that was shown to guide the data collection task in the published case study is a non-linear L\({}_{9}\)(3\({}^{4}\)) OA trial planner with a maximum capacity for handling four controlling factors, each examined at three preset settings, while demanding only nine specific factorial setting combinations. Its key feature is its reduced trial volume requirement, down to merely 1/9 of the demand that is otherwise necessitated by a full factorial design. This lean-and-green curvilinear-orientated design is populated with factorial settings in order to make full use of the recipe maker; i.e., it is adjusted to saturate the planner matrix. The recipe maker is an \(m\times n\) array in which the \(n\) rows designate the factorial setting combinations in their preset trial run schedules. The \(m\) columns indicate the positions of the investigated controlling factors on the OA planner. This specific design offers an opportunity to quantify quadratic effects. In the examined saturation mode, the relationship between the number of runs and the number of factors becomes \(n=2\times m+1\). The \(m\) controlling factor vectors are defined as \(\mathbf{X_{j}}\) for \(1\leq j\leq m\) (\(m\in\) N), and their associated entries (settings) are symbolized as \(\mathbf{x_{ij}}\) for \(1\leq i\leq n\) (\(n\in\) N), and \(1\leq j\leq m\). The three-level notation comprises \(\mathbf{k_{ij}}\) fixed levels for each jth factor (\(1\leq j\leq m\) and \(1\leq i\leq n\)) with \(1\leq\mathbf{k_{ij}}\leq 3\) (\(\mathbf{k_{ij}}\in\) N). The multi-characteristic response matrix is defined as \(\mathbf{R}=\left(\mathbf{r_{icd}}\right)\), with \(1\leq i\leq n\), \(1\leq c\leq L\) (L e N), and \(1\leq d\leq D\) (D e N), where L is the total number of examined characteristic responses and D is the total number of times that the OA scheme has been replicated. The proposed approach uses D times the Lenth test [48] on each individual saturated OA replicate datapoints, using the critical values that were computed by [PERSON] and [PERSON] [49] for the individual error rate (IER) and the experiment-wise error rate (EER), respectively. This means that the Lenth statistics are collected D times on the D \"unreplicated\" OA dataset screenings. If the estimated effects are \(\epsilon_{ij}\) for \(1\leq i\leq D\) and \(1\leq j\leq m\), then the pseudo standard error (PSE) is  
\[PSE_{i}=1.5\cdot median_{\left|\epsilon_{ij}\right|<2.5\epsilon_{ij}}\left| \epsilon_{ij}\right|\text{with }s_{oi}=1.5\cdot median\left|\epsilon_{ij}\right|\text{and }t_{L,i,j}=\frac{\epsilon_{ij}}{PSE_{i}}\]  
where the \(t_{L,i,j}\) is the Lenth statistic for the \(j\)th effect, as it is evaluated after processing the \(i\)th OA-replicate dataset. This work, in a practical manner, attempts to take into account influences on screening solutions due to stochastic uncertainty and epistemic uncertainty. The stochastic uncertainty contributions are fathomed by using the Lenth method on the replicate-partitioned saturated-unreplicated FFD datasets. Furthermore, the theory of belief functions is implemented to incorporate elements from the evidence theory in order to deal with the epistemic uncertainty part. By adopting Dempster-Shafer theory [59; 60; 61; 62; 63; 64], the process used to perform the fusion of uncertain information is greatly facilitated, hence aiding our understanding of the influence of uncertainty in decision-making. Each separate OA dataset replication is treated as a different evidence opportunity. Unreplicated-saturated OA datasets may be viewed as \"unmodeled\" relationships, which might lead to \"imprecise\" predictions. This is because the unexplainable error remains an unknown after each factorial screening has been completed. The state of the belief in the assertion about a group of suggested active factors will be captured by combining successive screening results using evidence combination.