# Integration of Production Planning and Scheduling Based on RTN Representation under Uncertainties
## 4 Case Study
### Integrated Model Simulation Analysis  
The order data of two products in nine planning periods and the corresponding average supply of two utilities are given. Based on the RTN representation, integrated optimization solving method and rolling horizon optimization strategy were used to iteratively solve the integrated model to complete the planning and scheduling tasks of the first five planning periods. The software Lingo 11.0 was used for simulation calculation on a 2.20 GHz computer, where the planning horizon is 5 days and the planning period and scheduling horizon are both 1 day. The number of event points is 12. Figures 8 and 9 respectively show the gantt chart and utility consumption level of the deterministic method and the uncertainty method. From Figures 8 and 9, we can intuitively get the respective task number and the corresponding specific utility consumption under the deterministic method and the uncertain method. Under the deterministic method, the solution time of the rolling horizon algorithm is 28 s. Under the uncertain method, the solution time of the rolling horizon algorithm is 34 s.  
In the comparative analysis between the deterministic method and the uncertain method shown in Tables 6 and 7, \(S_{8}^{plan}\) and \(S_{9}^{plan}\) respectively represent the planned production of \(S_{8}\) and \(S_{9}\); \(S_{8}^{schedule}\) and \(S_{9}^{schedule}\) represent the scheduled production of \(S_{8}\) and \(S_{9}\) respectively. The objective function of the planning layer of the deterministic method is $99,688.52, and the output of products \(S_{8}\) and \(S_{9}\) in the planning layer are, respectively, 1330 and 1550 kg. In the uncertain method with the confidence factor of 0.9, the objective function of the planning layer is $97,285.11, and the output of products \(S_{8}\) and \(S_{9}\) are respectively 1296.998 and 1514.416 kg. The objective function in the scheduling layer of the deterministic method is 128.9 kg, and the output of products \(S_{8}\) and \(S_{9}\) are, respectively, 1201.1 and 1550 kg. The objective function in the scheduling layer of the uncertain method is 87.277 kg, and the output of products \(S_{8}\) and \(S_{9}\) are, respectively, 1209.721 and 1514.416 kg. The final profit of the enterprise under the deterministic method is $86,792.96, and the final profit of the enterprise under the uncertain method is $88,557.392. The comparison of the final profit shows that the uncertain method reflects the actual production process more realistically due to considering the uncertainty of the demand and the utility, which brings greater profits to the enterprise. Tables 8 and 9 respectively represent the consumption amount of utility in each period under the deterministic method and uncertain method, where \(use_{i}\) (\(i=1\cdots 5\)) represents the consumption amount of utility in the planning period \(i\) and \(use_{all}\) represents the total usage amount of utility. The total amounts of utility HS used under the deterministic method and the uncertain method are 1804.94419 and 1735.6687 kg, respectively. The total amounts of utility CW used under the deterministic method and the uncertain method are 1685.06007 and 1732.3723 kg, respectively. Backlog order quantity of 87.277 kg under the uncertain method is significantly less than that of 128.9 kg under the deterministic method. These comparative results show that introducing the uncertain programming method into the planning and scheduling layers, respectively, can greatly reduce the generation of backlog order quantity, and the scheduling layer can more effectively complete the tasks assigned by the planning layer. Based on the RTN representation, the uncertain method adopted in the scheduling layer can effectively reduce the impact of uncertainty factors on the optimization  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{w}_{1}\) & \(\mathbf{w}_{2}\) & \(\mathbf{w}_{3}\) & \begin{tabular}{c} **Objective** \\ **Function (kg)** \\ \end{tabular}