# Taxonomy of Scheduling Problems with Learning and Deterioration Effects
## 3 Results
### Modeling Approaches for Learning and Deteriorating Effects
#### 3.3.1 Learning Effect
[58] also defined the actual processing time of a job in a single machine scheduling problem according to the start time of its processing, as well as the position of the job within the sequence, as computed in Equation (5). In their equation, \(\alpha\) and \(\theta\) are the learning and deterioration rate, respectively. These values can be estimated empirically, such that \(0<\alpha\leq 1\), \(0<\theta<1\). \[p_{jr}(t)=p_{j}(1-\theta t)\alpha^{r-1} \tag{5}\]  
Sum-of-Processing-Time-Based Learning Effect (ST-LE)  
The model presented by [PERSON] and [PERSON] [59] in a single machine scheduling problem is based on the proposition that the more a worker performs a task, the higher their performance will be. From this point of view, the learning effect is expressed in terms of the total processing time of the previously completed jobs. The actual job processing time is defined by Equation (6), where \(\alpha\leq 0\) is a constant learning rate. \[p_{jr}=\left(1+\sum_{k=1}^{r-1}p_{k}\right)^{\alpha}p_{j} \tag{6}\]  
An alternative approach to this model was developed by [PERSON] et al. [60], who, keeping in mind that learning is a declining phenomenon, proposed a sum-of-logarithm-processing-times-based learning effect. These authors studied a flowshop scheduling problem. The equation is given as:  
\[p_{ijr}=\left(1+\sum_{k=1}^{r-1}\text{ln}(p_{ik})\right)^{\alpha}p_{ij} \tag{7}\]  
where \(p_{ij}\) is the job processing time (without learning effect) and \(\alpha\leq 0\) is the learning rate. Truncated Sum-of-Processing-Time-Based Learning Effect (TST-LE)  
The use of the truncated function can indeed be justified since the learning is limited. For this model, the actual job processing time depends on the sum of processing times of jobs processed previously and a control parameter [61]. Therefore, for the case of a single machine as studied by these authors, the actual processing time of a job is defined as follows:  
\[p_{jr}=max\left\{\left(1+\sum_{k=1}^{r-1}p_{k}\right)^{\alpha},\beta\right\}p_{j} \tag{8}\]  
Another approach is proposed by [PERSON] et al. [62] as below:  
\[p_{jr}=max\left\{\left(1-\frac{\sum_{k=1}^{r-1}p_{k}}{\sum_{k=1}^{n}p_{k}} \right)^{\alpha},\beta\right\}p_{j} \tag{9}\]  
where \(\alpha\) is the learning index, and \(\beta\) the truncated parameter with \(\alpha\leq 0\) and \(0<\beta<1\). Exponential Learning Effect Based on Sum-of-Processing-Time (EST-LE)  
Some authors have included a learning effect based on an exponential sum of processing time. For instance, [PERSON] et al. [63] described this phenomenon using the following example: if human interactions have a significant impact during the processing of a job, workers will gain experience. The job processing time in a single machine scheduling problem is then as shown in Equation (10). Figure 7:0  
[PERSON] et al. [64] proposed that the actual processing time of a job be defined by an exponential function of the sum of the logarithm of the processing times of the jobs processed previously. Figure 7:1  
where \(\delta\geq 0\), \(\gamma\geq 0\) and \(0<\alpha\leq 1\) are parameters obtained empirically, and \(\delta+\gamma=1\). They use [PERSON]'s learning curve [65] as a starting point. Combining Position and Sum of Processing Time-Based Learning Effect (PST-LE)  
This type of model mixes both phenomena, combining the position-based learning effect to model machine learning effects with the sum-of-processing time [11]. In the case of [PERSON] et al. [66], the actual processing time of a job in the single machine problem is a function of the normal total processing time of previously processed jobs, as well as the position. The model is presented below. Figure 7:2  
where \(\alpha_{1}\) and \(\alpha_{2}\) indicate the learning rate, with \(\alpha_{1}\geq 1\) and \(\alpha_{2}<0\). On the other hand, [PERSON] et al. [67] propose a complex model for the job processing time in a single machine problem.