## 3 Aggregated Fraction Model and Generalized Model: Comparison
### Reasons to Use the Generalized Model  
Suppose there is a resource leveling problem without any special requirements for synchronized resource occupation for one job, i.e., both models are acceptable. Is it reasonable to use generalized \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Data Set** & **Min** & **Q1** & **Median** & **Mean** & **Q3** & **Max** \\ \hline \(inst\_j10\_r5\) & 0.17 & 0.41 & 0.49 & 0.48 & 0.55 & 0.71 \\ \(inst\_j15\_r5\) & 0.25 & 0.42 & 0.51 & 0.50 & 0.56 & 0.72 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: \(X(I)\) values for instances of datasets \(inst\_j10\_r5\) and \(inst\_j15\_r5\). Figure 3: Time boxplots for datasets \(inst\_j10\_r5\) and \(inst\_j15\_r5\). model if it has worse performance? To answer this question, we make the following experiment: For each instance from dataset we run the generalized model with time limit which equals the execution time of aggregated fraction model. In other words, we compare the aggregated fraction model optimal solution with suboptimal solution of generalized model obtained in same time. Results are presented in Table 5.  
We can conclude that for this dataset the ratio of aggregated fraction model optimal solution objective value to generalized model suboptimal solution objective function obtained remains about two times more as without time limits. The value of \(X(I)\) depends on the instance parameters. For example, if the availability of resources is higher in periods, then the objective function value decreases for both models. In this case, we obtain lower values of \(X(I)\). We can demonstrate this on new datasets \(inst\_j10\_r5\_2\) and \(inst\_j15\_r5\_2\) with distribution \(L_{rt}\in[0,140]\) instead of \(L_{rt}\in[0,70]\), see Table 6.  
We also make new experiments. In addition to datasets presented above, we generate larger instances and vary the number of resource types: We generate two datasets with 5 and 10 resource types and 30 jobs. In Table 7 we present the ranges used to generate each instance parameter. Each dataset contains 30 instances. We note that there are several RLP benchmarks already introduced in the literature. [PERSON] et al. (2016) used two RLP benchmarks. The first benchmark was described by [PERSON] et al. (2017) and contained datasets with 10 and 20 jobs. The second benchmark contained datasets with 10, 20, and 30 jobs. It was presented by [PERSON] (2018). These datasets were prepared for the formulation with fixed job duration and intensity. For the variable duration case [PERSON] et al. enriched the data with the following assumption: Initial duration was considered as the maximal value, while the minimal duration was obtained by multiplying it by 0.75. Project deadline was calculated as the longest path from 0 to \(n+1\) job in the precedence graph. In some instances each job requires only one resource type. In both benchmarks the instances included 1, 3 or 5 resources. In our research, we generate instances with the same size, but various parameters such as minimal and maximal duration of jobs and resource allocation limits. It is also important to demonstrate the \begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline
**Time limit** & **Data set** & **Min** & **Q1** & **Median** & **Mean** & **Q3** & **Max** \\ \hline (a) Not fixed & \(inst\_j10\_r5\_2\) & 0.1 & 0.3 & 0.36 & 0.36 & 0.42 & 0.63 \\  & \(inst\_j15\_r5\_2\) & 0.04 & 0.26 & 0.32 & 0.32 & 0.39 & 0.52 \\ \hline (b) Aggregated fraction model & \(inst\_j10\_r5\_2\) & 0.1 & 0.33 & 0.4 & 0.41 & 0.47 & 0.97 \\ optimal solution construction time & \(inst\_j15\_r5\_2\) & 0.09 & 0.27 & 0.37 & 0.36 & 0.43 & 0.59 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: \(X(I)\) values for instances of datasets \(inst\_j10\_r5\_2\) and \(inst\_j15\_r5\_2\), in two cases: (a) without time limit and (b) with time limit equal to the optimal solution time for the aggregated fraction model.