## 4 Conclusions and Further Works  
Based on the results of the calculations for the four preceding cases, the mathematical model and proposed solution were successful in resolving the problem of delivery and production scheduling coordination. The four cases above illustrate a variety of possible real-world scenarios, including variations in the number of products, the number of containers, the number of planning periods, and various tentative delivery schedules. In the proposed mathematical model, there is a parameter called \(\alpha\), which is used to balance the production load across periods. The computational results over all cases considered concluded that the larger the \(\alpha\), the smaller the value of the optimum objective function. However, that causes a wider range of production loads. It is essential to communicate with the company's production manager in order to determine the value of \(\alpha\) used in the model so that the solution method can obtain the best objective function for the chosen \(\alpha\) value. This research can be expanded by considering some realistic aspects of production and delivery schedules that exist in the real world. These additional factors may impose constraints on the mathematical model. Simultaneous scheduling of delivery and production is another area of future research. Despite the success of mathematical programming in coordinating delivery and production schedules, the size of the resulting mathematical models will grow exponentially as more containers, products, and planning periods are introduced. Case 1 involves 428 integer variables and 11,081 constraints, whereas Case 2 involves 405 integer variables and \begin{table}
\begin{tabular}{c c c} \hline \(\alpha\) & **Objective Function Value** & **Weekly Production Load Range** \\ \hline
0.0025 & & 1257–1262 \\
0.0050 & 44 & 1238–1249 \\
0.0100 & 43 & 1232–1253 \\
0.0250 & 43 & 1216–1273 \\
0.0500 & 41 & 1190–1291 \\
0.1000 & 41 & 1190–1299 \\
0.4000 & 40 & 791–1685 \\
0.5000 & 39 & 625–1834 \\ \hline \end{tabular}
\end{table} 
Table 7: Results of Case 2 at various values of \(\alpha\). \begin{table}
\begin{tabular}{c c c} \hline \(\alpha\) & **Objective Function Value** & **Weekly Production Load Range** \\ \hline
0.0025 & 50 & 33,744–33,858 \\
0.0050 & 49 & 33,667–33,929 \\
0.0100 & 49 & 33,561–34,025 \\
0.0250 & 49 & 33,310–34,316 \\
0.0500 & 48 & 32,555–34,852 \\
0.1000 & 47 & 30,970–36,787 \\
0.4000 & 45 & 20,596–47,296 \\
0.5000 & 45 & 18,189–46,064 \\ \hline \end{tabular}
\end{table} 
Table 8: Results of Case 3 at various values of \(\alpha\). \begin{table}
\begin{tabular}{c c c} \hline \(\alpha\) & **Objective Function Value** & **Weekly Production Load Range** \\ \hline
0.0025 & & 4098–4117 \\
0.0050 & 44 & 4098–4117 \\
0.0100 & 44 & 4094–4131 \\
0.0250 & 43 & 4043–4207 \\
0.0500 & 42 & 4020–4297 \\
0.1000 & 42 & 3782–4461 \\
0.4000 & 39 & 2950–5464 \\
0.5000 & 39 & 2862–5473 \\ \hline \end{tabular}
\end{table} 
Table 9: Results of Case 4 at various values of \(\alpha\). 8048 constraints. The size of the model increases to 972 integer variables and 19,248 constraints if the planning period for a case like Case 1 is increased from 4 to 12 periods. This larger problem size will eventually cause the mathematical modeling solver's computation time to increase. For this reason, it is still necessary to develop the solution of mathematical models using other techniques, such as metaheuristics or other computational techniques, so that mathematical models can be solved more quickly. Conceptualization, T.J.A. and R.D.A. ; methodology, T.J.A. ; software, T.J.A. ; validation, T.J.A. and R.D.A. ; formal analysis, T.J.A. and R.D.A. ; investigation, T.J.A. and R.D.A. ; resources, R.D.A. ; data curation, R.D.A. ; writing--original draft preparation, T.J.A. ; writing--review and editing, T.J.A. and R.D.A. ; visualization, R.D.A. ; supervision, T.J.A. ; project administration, R.D.A. All authors have read and agreed to the published version of the manuscript.