## 3 Approach
### Human Skills Modeling Engine (Data-Driven)
#### 3.2.2 Complexity Function  
As was introduced in Section 3.2.1, the multi-attribute skill matrix is based on the skills that define the production process. For each skill, the complexity function is going to create a Gaussian distribution with the recorded data that relates the task that is associated with the skill measures, and with the skill level of the worker. The Gaussian distribution is chosen as the statistical model since it correctly adjusts to many natural phenomena (i.e., age, measurements, etc.) and applies to all data sets with finite variance. The recorded data is stored in the database and contains a batch of different measurements of different skill level workers. The competency measurements refer to the parameter that controls the skill. For example, if the skill is drilling, the skill measure can be in millimeters of material removed. Figure 8 presents the Gaussian distribution of the generic skill recorded data. Figure 7: Skill level identification [Source: inspired by the chapter “Skill management” (pages 113–119) in reference [12]]. Apart from the shape of the distribution model, Figure 8 has the following parameters that are important in the next steps of the human assessment:  
* Mean (u): Measure of the central tendency of a data set. Specifically, the sum of the values divided by the number of values. As the Gaussian distribution is a model representation of the real data, the shape of it is centered in the average of the values. * Nominal measured value (NMV): Theoretical measure in which the tasks related to the skill defined in the multi-attribute skill matrix should be carried out. Normally, this parameter is obtained from manufacturing optimization software and is specific to each skill on the production line. If the process is modeled precisely and is faithful to the reality of what happens on the production line, this value should be close or equal to the mean of the data. * Superior absolute limit (\(\mathsf{X_{sup,A}}\)): Maximum measured value in which an operator can make a product. This value is defined by an expert in the production line. * Inferior absolute limit (\(\mathsf{X_{inf,A}}\)): Minimum measured value that an operator must make a product. This value is defined by an expert in the production line. * Superior relative limit (\(\mathsf{X_{sup,S}}\)): Recommended maximum measured value for an operator to make a product according to his skill level, reflected in the multi-attribute skill matrix. This parameter is obtained by adding a certain percentage to the NMV, as shown in Equation (1). This percentage called threshold (Th) is unique for each skill level number and calculated based on the experience of an expert in the production line. \[X_{sup,\;S}=\text{NMV}\times\left(1+\frac{\text{Th}}{100}\times\left(X_{sup,\;A }-1\right)\right)\text{,}\] (1)
* Inferior relative limit (\(\mathsf{X_{inf,S}}\)): Recommended minimum measured value for an operator to make a product according to his skill level reflected in the skill matrix. This parameter is obtained by deducting a certain percentage from the NMV, as shown in Equation (2). This percentage called threshold (Th) is unique for each skill level number and calculated based on the experience of an expert on the production line. \[X_{inf,\;S}=\text{NMV}\times\left(1-\frac{\text{Th}}{100}\times\left(X_{inf,\;A }-1\right)\right)\text{,}\] (2) These parameters define the colored areas under the Gaussian distribution. * Acceptable area (blue color): Includes data between the limits \(\mathsf{X_{inf,S}}\) and \(\mathsf{X_{sup,S}}\) and its area defines the expected performance of the operator for its skill level. This acceptable area is also divided into three equal parts to differentiate the area closest to the NMV that is excellent performance, the closest to the marginal acceptable area that is admissible, and the one in the middle corresponds to a good performance. The limits of these intervals are shown in Figure 8. These parameters are defined by Equations (3)-(6).