## 2 Materials and Methods
### Biomechanic Fuzzy Logic Model
#### 2.3.1 Fuzzification  
Fuzzification was employed to convert task movements (universe of discourse) into degrees of biomechanical risk levels, expressed in linguistic terms. Each subset (MF, AF, \(\theta\), A, H, BMI, ET, R, and D) was represented by a fuzzy subset that determined the membership level for each category (Biomechanics, Anthropometrics, and Productivity) using a membership function, as illustrated in Figure 3. This process of fuzzification allows for the transformation of scalar values (crisp values) into linguistic variables, facilitating fuzzy reasoning [11]. The linguistic variables express quantities or values associated with specific levels of risk, as defined in Equations (1) and (2). The process of fuzzification required building one fuzzy equation for each subset. A total of nine fuzzy equations from Equation (A1) to Equation (A9) are provided in Appendix A, \"Linguistic Variables\", one for each subset. \[MV_{RL}(x)=\left\{ \begin{aligned} & 1.0\;IF\;x\;\in HRL\\ & 0.5\;IF\;x\;aceptable\;degree\;of\;\in MRL\\ & 0.0\;IF\;x\;\
otin Absence\;of\;RL\end{aligned} \right. \tag{1}\]  
Figure 2: The Universe of Discourse S contains the intersection of three sets: biomechanical, anthropometric, and productivity. Each set shares a grade of membership at its risk level. \[\mu MV_{HIGH}=0.x-2\qquad\text{OR} \tag{7}\]  
\[\mu MV_{HIGH}=1, \tag{8}\]  
Figure 3 is a simple graphical expression of straight lines that defines degrees of membership from 0.0 to 1.0 for each level of risk. The linguistic variables are represented by Equations (3) to (8). The crisp values represent the level of risk; therefore, 0 to 2 represents a low risk, 1 to 3 represents a medium risk, and 2 to 4 represents a high risk. Appendix B, \"Membership equations\", defines and provides fifty-four membership equations (six equations for each subset) and nine graphics to cover all linguistic variables. Four operations between fuzzy sets were required to combine sets in which we managed to minimise the level of risk and maximise productivity: containment, concentration, dilution, and intersection. Thus, these subsets have a shared risk level membership different from the individual risk level. * The subset \(\theta\) is part of two fuzzy subsets; it is contained in \(MF\) and \(AF\). The results of moment and force depend on \(\theta\). As a result, the subsets \(MF\) and \(AF\) have a conditional relationship with \(\theta\) defined as follows: \[CONDITIONATED\ SUBSETS=\theta\ \subset\ MF\ AND\ \theta\ \subset\ AF\] \[S=Arm\ Movement\ (Universe\ of\ Discourse)\] \[\theta=Fuzzy\ subset\ in\ MF\ and\ AF\] \[X,Y=are\ MF\ and\ AF\ in\ S\] If \(MF\cap AF\) is a binary relation \(X\times Y\) then \[F=\theta\ o\ MF\cap AF\] \[\mu_{F}(Y^{\prime})=MAX(MIN(X/Y),\ \mu_{\theta}(Y))\] (9)
* Considering the phrases maximising productivity and minimising risk level, from a fuzzy logic perspective, one needs to emphasise the members of the subset \(ET\), \(R\), and D and de-emphasise \(MF\) and \(AF\) to do these two definitions are presented:  
**Definition 1**. : _In the representation of linguistic modifiers (hedges), the operation of concentration is given by \(P^{2}\), normally called square law [11]._  
**Definition 2**. : _In the representation of linguistic hedges, the operation of dilution is given by \(P^{1\!\!