# Taxonomy of Scheduling Problems with Learning and Deterioration Effects
## 3 Results
### Modeling Approaches for Learning and Deteriorating Effects
#### 3.3.2 Deterioration Effect  
In the real world, processing times could increase if producers and resources become less efficient over time. The literature refers to this phenomenon as the deterioration of the job or resources. The categories in which deterioration effects are grouped in the present review correspond to those mentioned by [PERSON] et al. [81] and [PERSON] [51], such as (i) starting-time deterioration effect, (ii) positional deterioration effect, and (iii) cumulative deterioration effect. The first relates mainly to job deterioration, while the last two refer to resource deterioration. With respect to workers, the deterioration effect may include aging and forgetting, for example. Starting-Time Deterioration Effect (ST-DE)  
In this case, jobs deteriorate. [PERSON] and [PERSON] [82] introduced the case where jobs deteriorate while waiting for a service, and their processing times increase while waiting. An example of job deterioration is when an ingot is waiting to be inserted into the rolling machine. The ingot cools down and has to be heated before rolling. Similarly, the time and effort required to control a fire increases when the wait to put out the fire is longer [83]. [PERSON] et al. [84] proposed the following equation for a linear deterioration. They applied the model to a flowshop problem. \[p_{ijr}=p_{ij}+b_{j}t \tag{15}\]  
where \(p_{ij}\) is the job processing time, \(b_{j}>0\) is the deterioration ratio of job \(j\), and \(t\) is the starting time. Position-Based Deterioration Effect (P-DE)  
This model attempts to address the real context in which machines and equipment deteriorate with the number of jobs handled [85]. For a flowshop scheduling problem, [PERSON] et al. [86] used a position-dependent deterioration model defined in (16) where \(\alpha_{ij}\geq 0\) is the deterioration factor. \[p_{ijr}=p_{ij}r^{\alpha_{ij}} \tag{16}\]  
[PERSON] and [PERSON] [87] consider two different models in a parallel machine scheduling problem. One of them is a declining convex model by which the actual processing time  
Figure 8: (**a**) System configuration vs. learning effect approach. (**b**) Learning effect approach vs. objective function types. is given by 17, where \(u_{ij}>0\) is the amount of resource allocated to the job \(j\), and \(\eta\) is a positive constant (see [PERSON] et al. [55]). \[p_{ijr}=\left(\frac{\overline{p}_{ij}r^{\alpha}}{u_{ij}}\right)^{\eta} \tag{17}\]  
This category also includes the aging effect, as this is a type of deterioration. [PERSON] and [PERSON] [88] defined the actual job processing time in a single machine as:  
\[p_{j}(r)=p_{j}+b_{j}\big{(}max\big{\{}(r-1)-g_{j},0\big{\}}\big{)}^{\sigma_{j}} \tag{18}\]  
where \(\sigma_{j}\) denotes various aging ratios provided by individual jobs, \(r\) is the number of performed jobs (or position), and \(g_{j}\) and \(b_{j}\) are the aging threshold and aging ratio of job \(j\). Cumulative Deterioration Effect (C-DE)  
The job processing time is an increasing function of the total processing time, load or difficulty related to the completion of previously scheduled works [81]. For instance, [PERSON] et al. [45] focus on a job-shop scheduling problem and define the work processing time, taking into account the cumulative deterioration effect of the speed-scaling machines. \[p_{jo(r)}=\frac{p_{io}}{V_{il}}\left(a_{j}+b_{j}\sum_{k=1}^{r-1}p_{k}^{\alpha MA }\right) \tag{19}\]  
\(p_{jo}\) is the actual processing time of the operation \(o\) of job \(j\), and \(V_{il}\) is the selected speed level \(l\) of the machine \(i\) for the operation. \(a_{i}\) and \(b_{i}\) are constants related to the deterioration of the machine, and \(\sum_{k=1}^{r-1}p_{k}^{\xi MA}\) refers to the volume of work that the machine has carried out just before the task and since the last maintenance activity. This approach incorporates speed as a key point, as one of their objectives is to minimize energy consumption (and it is true that the higher the speed, the more energy is needed).