# Workforce Learning Curves for Human-Based Assembly Operations: A State-of-the-Art Review
## 2 Theoretical Background
### Learning Curves
To account for this effect, the authors based their model on [PERSON]'s learning curve [10], modifying it by adding a second process parameter \(c\), which represents the standard time estimated from empirical data (the long-term value of the cycle time), thus generating a _plateau_ on the learning curve on the value of \(c\), via the Equation (2): \[\overline{y}=(A-c)x^{-b}+c\] (2) where the values of \(A\), \(x\), and \(b\) are the same as the ones found in [PERSON]'s model [10] (see Figure 3). Another notable example of the _plateau geometry model_ is presented by [PERSON] in 1979 [14], accounting for the limits a machine presents on the learning curve of an operator, meaning he/she cannot lower the cycle time below the standard operating time of a machine. * _Stanford-B Model_--After World War II, the Stanford Research Institute was commissioned by the US Air Force to develop a new cost estimating learning curve for a process with previous similar production, meaning a previous workforce experience. To assess the presented requirement, the Institute developed the _Stanford-B model_, named after the additional \(B\) parameter, corresponding to the previous \(B\) number of units of experience, shifting the original log-linear along the time axis [15] (see Figure 3). To achieve this, the new learning curve model was expressed as Equation (3): \[\overline{y}=A(x+B)^{-b}\] (3)
* _[PERSON]'s Model_--[PERSON] realized in 1957 [16] that certain operations where machines were involved had a reduced learning effect on the operator, and a fraction of the cycle time came from the standard operation of the machine. He developed a model to assess this effect on burden-heavy operations with the mathematical modelling of Equation (4): \[\overline{y}=A\Big{[}M+(1-M)x^{-b}\Big{]}\] (4) where \(M\) represents the fraction of the work performed by a machine. When \(M=1\), meaning a total automated process, the learning effect becomes 0, leaving a constant  
Figure 2: Cumulative average learning model graphic [10]. work rate, and when \(M=0\) the model simply becomes [PERSON]'s log-linear model [10] (see Figure 3). * _S-Curve Model_--Named after the particular shape of the graph of the model [17], it is a combined form of both _[PERSON]'s assessment_ of the machine involvement in the learning process [16] and _[PERSON]'s_ previous experience effect [15]. Thus, the resulting model is Equation (5): \[\overline{y}=A\Big{[}M+(1-M)(x+B)^{-b}\Big{]}\] (5) where the parameters keep the same meaning as in the origin model. The result is an _S-shaped curve_ with a more precise but harder-to-estimate model of four parameters (see Figure 3). Other _adapted function models_ have been created following the base log-linear model of [PERSON] [10] such as _Levy's Adapted Function_[18] or _[PERSON]'s_[19] modification of [PERSON]'s model [18]. Although these models are not usually discussed in the broader literature, it is to note that _Levy's Adapted Function_[18] was presented as the first step towards an exponential model, the later _Levy's Adapted Function Model_ in the same article [18]. Derived from all the variations of the log-linear model, different models started adding new learning parameters, which lead to more complex mathematical models, mainly _hyperbolic and exponential models_ to assess different particularities found in certain groups of work processes. **Exponential Models**  
To assess the diverse learning effects on workforce skills development, different _exponential models_ were developed using the existing _log-linear models_ as their foundation but adding a \"damping\" effect on their learning curves, thus an exponential behaviour. The main exponential models discussed in the scientific literature are:  
* _[PERSON]'s Adapted Function Model_--From the adapted function model developed by [PERSON] [18], the author added a parameter \(\lambda\) representing the speed at which the model approaches the maximum estimated throughput (_P max_).