## 2 Materials and Method
### The Systematic Task-Based Assessment Method (STBAM)
#### 2.3.3 Fundamental Processes of an ST-Fuzzy Inference System (ST-FIS)  
Fuzzification  
In the general FIS structure, through the fuzzification process, the input values are transformed into fuzzy values through the fuzzification process, based on fuzzy sets predefined by their corresponding membership functions. This step allows the FIS to handle uncertainty and subjective reasoning by mapping exact inputs to degrees of membership in linguistic terms. Adapting an FIS for the ST device involves transforming the crisp evaluation metrics \(z(j)_{L}\), \(z(j)_{P}\) and \(z(j)_{A}\), as given by Equation (54), into a fuzzy-based decision framework. This allows for subtle grading, especially in scenarios where binary values ( or \(X\)) might not fully capture the degree of student performance. Keeping up with Equations (72)-(74), the linguistic terms \(B_{k}^{L}\), \(B_{k}^{P}\) and \(B_{k}^{A}\) yield fuzzy sets defined by their membership functions, which map the crisp inputs \(z(j)_{L}\), \(z(j)_{P}\) and \(z(j)_{A}\) to a degree of membership in the interval \([0,1]\). Moreover, for \(L\), we can consider membership functions  
\[\mu_{B_{1}^{L}}(z(j)_{L}),\mu_{B_{2}^{L}}(z(j)_{L}),\ldots,\mu_{B_{m(L)-1}^{L }}(z(j)_{L}),\mu_{B_{m(L)}^{L}}(z(j)_{L}). \tag{77}\]  
Similarly, for \(P\), we take membership functions  
\[\mu_{B_{1}^{P}}(z(j)_{P}),\mu_{B_{2}^{P}}(z(j)_{P}),\ldots,\mu_{B_{m(P)-1}^{P} }(z(j)_{P}),\mu_{B_{m(P)}^{P}}(z(j)_{P}). \tag{78}\]  
And for \(A\),  
\[\mu_{B_{1}^{A}}(z(j)_{A}),\mu_{B_{2}^{A}}(z(j)_{A}),\ldots,\mu_{B_{m(A)-1}^{A }}(z(j)_{A}),\mu_{B_{m(A)}^{A}}(z(j)_{A}). \tag{79}\]  
The output of the fuzzification step is a fuzzy representation of each input variable \(z(j)_{\star}\) across its relevant fuzzy sets. This representation serves as the basis for subsequent processing, including the application of fuzzy rules and aggregation procedures. Rule Evaluation Procedure  
The FIS of the ST scheme operates the rules to compute a degree of activation \(\alpha_{r}\) for each output evaluation \(y_{r}\). The inference engine evaluates the rules in the knowledge base using fuzzy logic operators to combine membership degrees. For a given input \(Z=\{z(j)_{L},z(j)_{P},z(j)_{A}\}\) set (cf. Equation (70)), the FIS calculates the activation level \(\alpha_{r}\) of each rule \(R_{r}\) using the 'and' (\(min\)) or 'or' (\(max\)) operators, let us say  
\[\alpha_{r}=min\Big{(}(\mu_{B_{k}^{L}}(z(j)_{L}),\mu_{B_{k}^{P}}(z(j)_{P}),\mu_ {B_{k}^{A}}(z(j)_{A})\Big{)} \tag{80}\]  
or  
\[\alpha_{r}=max\Big{(}(\mu_{B_{k}^{L}}(z(j)_{L}),\mu_{B_{k}^{P}}(z(j)_{P}),\mu_ {B_{k}^{A}}(z(j)_{A})\Big{)} \tag{81}\]  
The aggregation process combines the outputs \(y_{r}\) of all activated rules to produce an overall fuzzy set \(Y\). This procedure relies on the THEN clauses composing the inference rules \(R_{r}\) that jointly specify the output fuzzy set \(Y\):  
\[Y=\Big{\{}B_{1}^{Y},B_{2}^{Y},\ldots,B_{j}^{Y},\ldots,B_{m(Y)}^{Y}\Big{\}}. \tag{82}\]Each output fuzzy set \(B_{j}^{Y}\) in \(Y\) is defined by its membership function \(\mu_{B_{j}^{Y}}(y)\). For each rule \(R_{r}\), the degree of activation \(\alpha_{r}\) (calculated during the rule evaluation step) is used to scale the membership function of the corresponding output fuzzy set \(B_{r}^{Y}\). This scaling modifies the membership function values to reflect the strength of the contribution of rule \(R_{r}\), to the global output \(Y\). Then, the aggregation process combines these scaled membership functions \(\alpha_{r}\mu_{B_{r}^{Y}}(y)\) from all rules using the \(max\) operator. This operator ensures that the highest degree of membership across all rules is retained for each value of \(y\). The resulting aggregated membership function is expressed as  
\[\mu_{Y}(y)=\underset{r}{max}\Big{\{}\alpha_{r}\mu_{B_{r}^{Y}}(y)\Big{\}}. \tag{83}\]  
This aggregated fuzzy set represents the combined output of the FIS, capturing all relevant contributions from the activated rules.