# Optimal Control Algorithms and Their Analysis for Short-Term Scheduling in Manufacturing Systems
## 5 Qualitative and Quantitative Analysis of the MS Scheduling Problem
Let us analyse some particular features of the models (1) (see Figure 1). During the conducted experiments, it was revealed that the following model parameters influenced the improvement of the general quality index:  
* the total number of operations on a planning horizon;
* a dispersion of volumes of operations;
* a ratio of the total volume of operations to the number of processes;
* a ratio of the amount of data of operation to the volume of the operation (relative operation density). On the basis of the obtained etalon solutions, we can methodically justify the usage and quality of certain heuristics for certain variants of initial data (see Figure 5). \begin{table}
\begin{tabular}{c c c} \hline \hline
**No** & **Results and Their Implementation** & **The Main Results of Qualitative Analysis of MS Control Processes** & **The Directions of Practical Implementation of the Results** \\ \hline \multirow{3}{*}{1} & Analysis of solution & Adequacy analysis of the control processes description in control models \\  & MS control & Analysis MS control \\ \hline \multirow{3}{*}{2} & Conditions of controllability and attainability in the problems of MS control & Analysis MS control \\  & & & technology realizability on the planning interval. Detection of main factors of MS goal and information-technology abilities. \\ \hline \multirow{3}{*}{3} & Uniqueness condition for optimal program controls in scheduling problems & Analysis of possibility of optimal schedules obtaining for MS functioning \\ \hline \multirow{3}{*}{4} & Necessary and sufficient conditions of optimality in MS control problems & Preliminary analysis of optimal control structure, obtaining of main expressions for MS scheduling algorithms \\ \hline \multirow{3}{*}{5} & \multirow{3}{*}{Conditions of reliability and sensitivity in MS control problems} & Evaluation of reliability and sensitivity of MS control processes with respect to perturbation impacts and to the alteration of input data contents and structure \\ \cline{1-1}  & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Application of optimal control approaches to scheduling. Having calculated optimal solutions for several points, it is possible to validate the decision to use either dynamic or heuristic planning algorithms. In Figure 4, the relative solution quality gained by the optimal control algorithm DYN is assumed to be 100%. The relative quality index of the heuristic solutions is calculated as a fraction of the optimal one, i.e., it can be observed that, in the case of a number of processes between 10 and 12, the quality of the heuristic and optimal solutions does not differ by more than 4%. In area 2, the DYN algorithm is preferable to the heuristics. If still using the heuristics, the FIFO algorithm is preferable to the Last-in-first-out one. The largest benefit from using the DYN algorithm is achieved in area 3. In this area, the LIFO algorithm is preferable to the FIFO algorithm. Finally, the proposed model and algorithm allows for the achievement of better results in many cases in comparison with heuristics algorithms. However, this point is not the most important. The most important point is that this approach allows the interlinking of planning and scheduling models within an adaptation framework. Therefore, the suggested results are important in the Industry 4.0 domain. Hence, the proposed modeling complex does not exist as a \"thing in itself\" but works in  
Figure 4: Representation of dynamic changes in schedule performance by attainable sets. Figure 5: Comparison of heuristic algorithms’ quality. the integrated decision-support system and guides the planning and scheduling decisions in dynamics on the principles of optimization and adaptation.