# Workforce Learning Curves for Human-Based Assembly Operations: A State-of-the-Art Review
## 2 Theoretical Background
### Learning Curves
The resulting mathematical Equation (6): \[iP(x)=\big{(}1-e^{\wedge}(-\lambda x)\big{)}\big{(}+\big{(}x^{\wedge}b/y_{-}1 \big{)}e^{\wedge}(-\lambda x)\] (6) where the variables \(b\), \(x\) and \(y\) keep the same interpretation as in [PERSON]'s model [10], being \(b\) the natural logarithm of the learning curve slope divided by the natural logarithm of two, \(x\) the current production unit in the lifecycle and y the initial cycle time of the process (see Figure 4). * _Two and Three Parameter Models_--Both models were proposed by [PERSON] in 1978 [20], the _Two Parameter model_ offers an approximation of the skill development behaviour in the workforce using two new parameters; \(k\) which represents the maxi  
Figure 3: Adapted log-linear learning curve models graphics [10; 13; 14; 15; 16; 17]. mum throughput of a worker after the learning curve slope approaching zero, and the parameter \(r\) representing the learning rate in time units. The resulting Equation (7) is:  
\[y=k\Big{(}1-e^{-\frac{x}{r}}\Big{)} \tag{7}\]  
After t developing the _Two Parameter model_, [PERSON] [20] improved the fitting accuracy by adding a third parameter, \(p\) which corresponds to the worker's prior experience, expressed in time units (see Figure 5). The mathematical Equation (8) for this variation is:  
Figure 1:0  
The authors establish that their _Three Parameter model_ has a good fit with operations where the operator has previous experience but performs poorly with operations that frequently require new and complex tasks (see Figure 6). * _Constant Time Model_--In 1990, [PERSON] [21] validated a similarly structured model to the _Two Parameter_ model. He found that an exponential model with a starting constant value is better suited to estimate the learning curve of processes after the steepest learning curve period, meaning after they have adapted to a new task or job. The model's mathematical Equation (9) is represented as: Figure 1:1 (9) where \(y_{c}\) represents the worker's initial performance, \(y_{f}\) is the highest achievable performance after learning, \(x\) remains the same as previous models (meaning how many units of experience the worker has), and \(\tau\) is the time constant for a particular curve (see Figure 7). Figure 1:2 [PERSON]’s adapted function model profile graph [18]. Figure 1:3 Two-parameter model graphic [20]. **Hyperbolic Models**  
Another group of models that came after the log-linear models were the _hyperbolic models_. These novel models have become relevant for the study and further advancement in recent years of learning curves due to their capacity of showing both the increase and decrease of cycle time due to the parameters they use to measure work performance. There is one main group of hyperbolic models developed by [PERSON] in 1978 [20]. These are the _Two_ and _Three Parameter hyperbolic learning curve models_, both of which have a similar mathematical formulation as the Two and Three Parameter exponential learning curve models, but these differentiate themselves by including a parameter \(p\) to assess the prior experience (see Figure 8). The resulting _Two Parameter hyperbolic model_ Equation (10) is:  
Figure 1:4  
where initially the parameter \(r\) refers to the total nonconforming units, \(x\) refers to the conforming units, and \(k\) refers to a constant production level. The _Three Parameter hyperbolic model_ follows a similar Equation (11):  
with the inclusion of the parameter \(p\) representing the previous units of experience (see Figure 9). Figure 1:5 Two-parameter hyperbolic learning curve model [20]. Figure 1:6 Three-parameter model graphic [20]. Figure 1:7 Constant time model graphic [21]. Figure 1:8  
One of the main topics of discussion in the scientific literature is the estimation of the parameters of the learning curves. In most of the case studies used to validate the previous models presented, the data was collected using work study techniques, or in some cases just approximations of the cycle times.