# Heuristic Method for Decision-Making in Common Scheduling Problems
## 8 Conclusions  
The paper includes a generalized method to optimize discrete processes that involve complex dependencies or constraints. It is particularly useful for a common problem of decision-making: \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Problem Example** & **GR-1a** & **GR-1b** & **GR-2** & **GR-3** & **Gtr-1** & **Gtr-2** & **Gtr-3** & **Gtr-4** & **Gtr-5** \\ \hline _Alg._\(q1-\widehat{s1}\) & 1 & - & 1 & 2 & 32 & 23 & 2 & 6 & 9 \\ _Alg._\(q1-\widehat{s2}\) & 1 & - & 1 & 2 & 15 & 5 & 1 & 2 & 7 \\ _Alg._\(q1-\widehat{s3}\) & 12 & 7 & 18 & 20 & 38 & 40 & 40 & 18 & 31 \\ _Alg._\(q1-\widehat{s4}\) & 9 & - & 2 & 1 & 38 & 37 & 5 & 3 & 28 \\ _Alg._\(q1-\widehat{s5}\) & 5 & - & 2 & 1 & 40 & 40 & 40 & 36 & 24 \\ \hline _Alg._\(q2-\widehat{s1}\) & 13 & 22 & 15 & 11 & 31 & 3 & 27 & 2 & - \\ _Alg._\(q2-\widehat{s2}\) & 5 & 6 & 6 & 8 & 22 & 3 & 27 & 2 & - \\ _Alg._\(q2-\widehat{s3}\) & 14 & 40 & 40 & 19 & 35 & 4 & 33 & 8 & - \\ _Alg._\(q2-\widehat{s4}\) & 6 & 6 & 6 & 4 & 20 & 3 & 30 & 2 & - \\ _Alg._\(q2-\widehat{s5}\) & 16 & 17 & 12 & 18 & 34 & 3 & 31 & 2 & - \\ \hline _Alg._\(q3-\widehat{s1}\) & 19 & 6 & 15 & 14 & 27 & 9 & 6 & 1 & - \\ _Alg._\(q3-\widehat{s2}\) & 8 & 4 & 3 & 9 & 20 & 3 & 6 & 1 & - \\ _Alg._\(q3-\widehat{s3}\) & 31 & 13 & 20 & 30 & 31 & 19 & 8 & 1 & - \\ _Alg._\(q3-\widehat{s4}\) & 8 & 4 & 3 & 10 & 20 & 6 & 9 & 1 & - \\ _Alg._\(q3-\widehat{s5}\) & 22 & 10 & 7 & 24 & 29 & 2 & 7 & 1 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Number of admissible solutions (trajectories). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Problem Example** & **Obtained Total Cost** & **Percentage Error** & **Simulation Time (ms)** \\ \hline _Alg._\(q1\) & 77,288.44 & - & 9546 \\ _Alg._\(q1-\widehat{s1}\) & 77,288.44 & - & 20,171 \\ _Alg._\(q1-\widehat{s2}\) & 77,288.4 & - & 15,921 \\ _Alg._\(q1-\widehat{s3}\) & 77,146.08 & - & 23,906 \\ _Alg._\(q1-\widehat{s4}\) & 77,146.08 & - & 23,687 \\ _Alg._\(q1-\widehat{s5}\) & 77,288.44 & - & 22,234 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Obtained results for irregular network _Gtr-5_ (suboptimal cost). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Problem Example** & **GR-1a** & **GR-1b** & **GR-2** & **GR-3** & **Gtr-1** & **Gtr-2** & **Gtr-3** & **Gtr-4** & **Gtr-5** \\ \hline Average obtained total cost & 16,819.90 & 16,833.03 & 16,914.23 & 15,719.80 & 22,281.54 & 30,663.77 & 30,780.96 & 30,785.37 & 77,240.98 \\ \hline Average percentage error & 2.04\% & 1.78\% & 2.36\% & -9.91\% & -3.68\% & - & - & -1.15\% & - \\ \hline Average simulation time (ms) & 234 & 225 & 243 & 235 & 382 & 413 & 484 & 1304 & 19,244 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Summary of obtained results. an NP-hard scheduling problem with limitations related to tasks and resources dependent on the current state of the process. The paper presents and discusses its key elements, i.e., the preliminary analysis of the problem, the technique of decision-making at a given state, the selection technique for the initial state of the trajectory's final part, the parameters modification, and the pruning of non-perspective trajectories. The algorithm scheme shows the relationship between these elements in detail. An assumption of the approach under consideration is constructing the state graph by the final part of the trajectory generation, but not the whole graph and only its most perspective part. This is important because of the complexity of the real scheduling problem. The application of the presented approach to scheduling problems with deadlines and machine setups dependent on the process state shows the usefulness of the method. It should be emphasized that developing and applying the auxiliary coefficient selection algorithms in the local decision choice criterion or machine learning method can bring even better results. It is planned in further work to develop automated procedures for changing the criterion of the state selection and hybrid approach.