# Taxonomy of Scheduling Problems with Learning and Deterioration Effects
## 3 Results
### Modeling Approaches for Learning and Deteriorating Effects
#### 3.3.1 Learning Effect  
The learning effect in manufacturing systems was introduced in the work of [PERSON] [5], who stated the increase in productivity as a function of the number of units completed. Thus, monotonous activities have the potential to be improved to achieve a high level of productivity thanks to practical experience [52]. As a result of the invention of the learning equation, certain models were proposed, enhanced, and brought closer to reality. The first study of scheduling with learning effects did not appear until 1999, with the seminal work of [PERSON] [6] and then [PERSON] and [PERSON] [53], whose articles have become the baseline on the subject. Several models have been proposed in the literature and have been grouped together for their analysis. [PERSON] [8] classified them in two approaches: (i) position-based learning effects, and (ii) sum-of-the-processing-time learning. [PERSON] et al. [11] subsequently updated this classification and analyzed some works on the topic up to 2017. The current paper presents an analysis of articles that parts of the one proposed by [PERSON] et al. [11], with some modifications. The definition and certain reference models for each category are described below. Position-Based Learning Effect (P-LE)  
In this approach, the learning effect is affected by the number of jobs processed previously [8]. Job processing time therefore decreases with the number of repetitions rather than accumulated time [11]. In 1999, [PERSON] [6] demonstrated that the single machine scheduling problem can be solved in polynomial time and introduced the equation to calculate the actual job processing time:  
\[p_{jr}=p_{j}r^{\alpha} \tag{1}\]  
where \(p_{jr}\) is the processing time of the job \(j\) located in position \(r\) of the schedule. The constant learning index is \(\alpha\) (\(\alpha<0\)). Even though some authors have modified [PERSON]'s equation or proposed their own equations, they are all based on the principle of the number of repetitions. This approach is the one most commonly used in the short-listed articles. Truncated Position-Based Learning Effect (TP-LE)  
This phenomenon is intended to bring the learning effect closer to reality. For a truncated position learning effect, the processing time of the job depends on its position  
Figure 7: Taxonomy visualization. and a control parameter, which indicates the learning limitation. In the work of [PERSON] and [PERSON] [53] the existence of a learning threshold value is already mentioned. An approach in a single machine scheduling problem, where the actual processing time of a job is defined according to Equation (2). \(\alpha\) is a parameter of learning effect (\(\alpha<0\)) and \(\beta\) is a control parameter for learning truncation with \(0<\beta<1\). The latter parameter introduces a lower bound on the processing time. \[p_{jr}=p_{j}\text{max}\{r^{\alpha},\beta\} \tag{2}\]  
Some variations have been proposed, such as the one reported by [54], incorporating the consumption of convex resources. In their equation, \(\alpha_{j}\leq 0\) is the job-dependent learning factor, \(\eta\) is a positive constant [55; 56], and \(u_{j}\) is the amount of resource that can be allocated to job \(j\). \[p_{j}\big{(}r,u_{j}\big{)}=\left(\frac{p_{j}\text{max}\{r^{\alpha_{j}},\beta\} }{u_{j}}\right)^{\eta} \tag{3}\]  
Exponential Learning Effect Based on Position (EP-LE)  
Unlike position-based approaches, the exponential learning effect based on position explains a learning process where the job processing time decreases progressively, thus ensuring the quality of the products. Equation (4) shows the model proposed in a flowshop scheduling problem by [PERSON] [57], where \(p_{jir}\) denotes the actual processing time of job \(j\) on the \(r\)-th position of the machine or worker \(i\). \[p_{jir}=p_{ji}\alpha^{r-1} \tag{4}\]  
[PERSON] et al.