## 4 Discrete Resource Case  
In the Objective Function (24), continuous overloading variables are used in order to calculate the cost of extra resources. It is compliant with such continuous resources as electricity or heat. However, in practice such resources as machines or human operators can be only available in discrete units. For this case, two possible models can be used. * Firstly, decision variable \(o_{rt}\) can be defined as integer with the minimal unit of each resource \(q_{r}\). New variables \(o_{rt}^{*}\) set the number of extra units used and they replace \(o_{rt}\) in Objective Function (24) and Constraint (23): \[Minimize\ \sum_{r\in R}\sum_{t\in T}e_{rt}o_{rt}^{*};\] (28) \[q_{r}o_{rt}^{*}\geq\sum_{j\in I}c_{jrt}-L_{rt},\ \forall t\in T,\ \forall r\in R.\] (29) We define this model as \(DO\) (discrete objective). This case can be used not only for discrete resources, but it also suits the usual practice when additional resources could be demanded in some packages, for example, the batteries. * Secondly, it is also possible to define other decision variables related to resource allocation as integers. This corresponds to the case when we have discrete resources and we allocate a discrete amount of workload to all periods. We define this model as \(DO\&R\) (discrete objective and resources). \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Data Set** & **Min** & **Q1** & **Median** & **Mean** & **Q3** & **Max** \\ \hline \(inst\_j30\_r5\) & 0.33 & 0.45 & 0.48 & 0.49 & 0.53 & 0.62 \\ \(inst\_j30\_r10\) & 0.42 & 0.49 & 0.51 & 0.51 & 0.54 & 0.58 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: \(X(I)\) value summary for instances of datasets \(inst\_j30\_r5\) and \(inst\_j30\_r5\), with a 5 min time limit. \begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Data Set** & **Model** & **Min** & **Q1** & **Median** & **Mean** & **Q3** & **Max** \\ \hline \(inst\_j30\_r5\) & Aggregated fraction & 0.004 & 0.011 & 0.016 & 0.018 & 0.023 & 0.053 \\ \(inst\_j30\_r10\) & Generalized & 0.05 & 0.11 & 0.16 & 0.16 & 0.19 & 0.3 \\ \(inst\_j30\_r10\) & Aggregated fraction & 0.007 & 0.01 & 0.012 & 0.014 & 0.019 & 0.032 \\ Generalized & 0.07 & 0.12 & 0.14 & 0.16 & 0.18 & 0.38 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Relative gap value summary for instances of datasets \(inst\_j30\_r5\) and \(inst\_j30\_r5\), with 5 min time limit. A computational experiment has been run to compare the behavior of continuous and discrete versions of the model. Seven models with different types and parameters of overload variables were considered: The original model with continuous overload variables, three versions of the discrete model with different resource unit size \(q_{r}=q\), \(\forall r\in R\), equal to 1, 3, and 5, defined as \(DO\), and the same values of \(q\) for the discrete resource allocation case defined as \(DO\&R\). Figure 4 presents the computational results for dataset \(inst\_j10\_r5\) with a 90 s time limit for all models. Each column is a boxplot that aggregates the data about the solution time, defining the median, lower and upper values, and quartiles. In addition, Table 10 provides the mean values of the objective function, solution time, and gap which was not presented in Figure 4. Here we also compare optimal solution objective function value provided by discrete model and continuous generalized model (\(V_{\text{C}}\)), and calculate relative delta. For example, for instance \(I\) and model \(DO\) it is  
\[\delta_{V_{DO}}(I)=\frac{V_{DO}(I)-V_{\text{C}}(I)}{V_{\text{C}}(I)}\]  
As could be expected, the computation time for the discrete model is higher than for the basic continuous model. For some part of the instances, no optimal solution was reached in 90 s for the discrete model, while all the instances were solved for the basic continuous model faster than in a second. However, the information about the gap provided by the solver shows that the main issue is in the proof of optimality: All instances had a very small gap value when the time limit was reached.