# Integration of Production Planning and Scheduling Based on RTN Representation under Uncertainties
## 5 Conclusions  
This paper introduces and analyzes uncertain factors involved in production planning and scheduling. According to the property of time scale, these uncertain factors are dealt with distinctively in the planning and scheduling layers. The planning layer and scheduling layer models are formulated by discrete-time and continuous-time modeling methods, respectively. Chance constrained programming and fuzzy theory are introduced into planning layer model and scheduling layer model, respectively, to describe the uncertainties of demand and utility. Rolling horizon optimization strategy is utilized to solve the bi-level integrated model iteratively. Through comparison, simulation results show that the proposed model and algorithm are feasible and effective, can calculate the consumption of utility in every period, decrease the effects of uncertain factors on optimization results, and more accurately reflect the actual production process.  
**Author Contributions:** [PERSON] conceived, formulated and verified the scheduling model. [PERSON], [PERSON] and [PERSON] analyzed the results, discussed the manuscript and corrected the manuscript. All authors have read and approved the final manuscript.  
**Funding:** This work was supported by Scientific Research Fund of Liaoning Provincial Education Department (No. L2017 LQN030, No. L2017 LQN032), supported by Talent Scientific Research Fund of LSHU (No. 2016 XJJ-101 and No. 2016 XJJ-102), and supported by Youth Innovation Talent Program by Department of Education of Guangdong Province, China (601821K42050).  
**Conflicts of Interest:** The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Modeling Method** & **Objective Function (\$)** & \(S_{8}^{plan}(kg)\) & \(S_{9}^{plan}(kg)\) \\ \hline Deterministic method & 99,688.52 & 1330 & 1550 \\ Uncertain method & 97,285.11 & 1296.998 & 1514.416 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparative analysis of the planning layer.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Modeling Method** & **Objective Function (\$)** & \(S_{8}^{plan}(kg)\) & \(S_{9}^{plan}(kg)\) \\ \hline Deterministic method & 128.9 & 1201.1 & 1550 \\ Uncertain method & 87.277 & 1209.721 & 1514.416 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Comparative analysis of the scheduling layer.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline _utility_ & \(use_{1}(kg)\) & \(use_{2}(kg)\) & \(use_{3}(kg)\) & \(use_{4}(kg)\) & \(use_{5}(kg)\) & \(use_{all}(kg)\) \\ \hline HS & 352.31419 & 345.83433 & 383.03704 & 370.31418 & 353.44445 & 1804.94419 \\ CW & 337.26051 & 339.90991 & 357.33333 & 322.22298 & 328.33334 & 1685.06007 \\ \hline \hline \end{tabular}
\end{table}
Table 8: The consumption amount of each period of utility under the deterministic method.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline _utility_ & \(use_{1}(kg)\) & \(use_{2}(kg)\) & \(use_{3}(kg)\) & \(use_{4}(kg)\) & \(use_{5}(kg)\) & \(use_{all}(kg)\) \\ \hline HS & 339.07849 & 371.35303 & 340.01582 & 343.52171 & 341.69782 & 1735.66687 \\ CW & 340.84803 & 359.11669 & 372.16621 & 330.04896 & 330.19241 & 1732.3723 \\ \hline \hline \end{tabular}
\end{table}
Table 9: The consumption amount of each period of utility under the uncertain method.