# Workforce Learning Curves for Human-Based Assembly Operations: A State-of-the-Art Review
## 2 Theoretical Background
### Learning Curves  
The first _learning curve theories_ were proposed back in 1885 by psychologist [PERSON] to describe the development of the skill level of a subject over time [8]. Starting in 1936, the first _learning curve models_ were formulated in a world recovering from World War I and looking to reduce the costs of military equipment manufacturing [10]. Since then, multiple diverse methods for estimating the learning curves and even different approaches have emerged for assessing different needs and areas. Multiple applications have been explored during the last decades for the use of learning curves to drive the performance development of individuals, teams, and technical systems in different domains. Moreover, in the manufacturing world, these _workforce learning curves_ refer to \"the mathematical description of the performance of a worker through time\" [10]. Due to the familiarity and expertise gained by a worker in a certain job, the number of units he/she can produce is usually expressed in either \"cost-per-unit-manufactured\" - understood as the monetary cost per unit manufactured, in the case of manual assembly operations the part of the salary of the operator allocated to the production of a single unit [11] or \"time-per-unit\" - expressed simply as the time required to complete an assembly operation for a single unit [11]. This last unit of measurement was reported by [PERSON] [8] to be the more effective one due to the change in monetary compensation in time for workers, while time remains a comparable unit across time. Based on their main mathematical model function, the existing _learning curve models_ in the scientific literature can be classified as:  
**Traditional Log-Linear Models**  
* _Cumulative Average Learning Model_--Developed in 1936 by [PERSON] [10] who observed that after doubling the produced parts, the production time was reduced by a constant rate. _[PERSON]'s learning curve model_ offers its calculations based on the following Equation (1): \[\overline{y}=aX^{-b}\] (1) which represents the _cumulative average cost or time_ of the first \(x\) unit, where \(A\) is the theoretical cost or time to produce the first unit, and \(b\) is the natural logarithm of the learning curve slope divided by the natural logarithm of 2. From [PERSON]'s observations, the learning curve slope parameter should be 80% to approximate a bomber manufacturing process behaviour (see Figure 1). Multiple variations of his model were created afterwards, accounting for process-specific variables observed by other authors and parameters not considered in his initial model. Some of the more popular examples are the Stanford-B model, [PERSON]'s unit learning model, and [PERSON]'s plateau model. * _Unit Learning Model_--Developed in 1947 by [PERSON] [12], which base formula is the same as the cumulative average learning model from [PERSON] [10], however, there is a  
Figure 1: Unit learning model graphic [12]. difference in interpretation of the values where \(\overline{y}\) represents the _individual cost or time_ of unit \(x\) where the rest of the variables remain the same (see Figure 2). These previous _log-linear models_ as presented by [PERSON] [10] and [PERSON] [12] had a main disadvantage in the long term; the _learning rate_ would keep on reducing the estimated cycle time indefinitely, which prompted several other authors to explore and develop better fitting models to account for the reduced learning rate. **Adapted Log-Linear Models**  
* _Plateau Model_--In 1957, [PERSON] & [PERSON] [13] first described the _plateau effect_ in their learning curve model to account for the reduction in the learning rate.