## 3 Materials and Methods
### Crossover Operator DPPX
In addition, the model is not confined to the production plant, since any machine may be viewed as a workplace, and the model and software offered can address, without change, the classic problem of labor scheduling in other vocations. In addition to optimizing execution time, the proposed technique also optimizes the machine layout. It displays a machine-oriented Gantt chart that serves as the foundation for production planning. In contrast to the models in the published literature, the DPPX operator minimizes the number of iterations and retains the variety of generations, hence limiting the convergence of GA to the local optimum while simultaneously reducing GA execution time. The program may also be used as a production planning tool completely independently of other commercial software, making it an ideal alternative for small- and medium-sized enterprises. However, the provided model has several drawbacks. The model excludes production disturbances such as cancellation of manufacturing orders, equipment malfunction, and urgent work orders. The inclusion of these limits might be one of the primary future research priorities. In addition, there are no intermediary warehouses between the machines in the provided model, thus waiting time and storage expenses are not accounted for. The most common disruptions in production, such as a machine breakdown or an urgent purchase order, do not greatly affect the practical applicability of the model. The proposed model is flexible and allows variations on a daily basis. A break point on the timeline in the operation schedule caused by, for example, a machine failure becomes the starting point of a new operation schedule in which the defective machine is removed from the schedule until it is repaired, so that the production process continues according to the newly formed operation schedule. The lack of inter-operational warehouses, however, limits the practical applicability of the proposed model in real conditions, since waiting time, transportation time, and storage costs are not taken into account. As one of the primary subjects of future study, the establishment of restricted intermediate warehouses would enable the implementation of the model and software solution given to large organizations. The second most important goal of future research is testing the proposed model in real conditions and testing its efficiency and effectiveness, as well as further improving the algorithm based on the results obtained from the real-world case study. Conceptualization, P.M. and M.E. ; methodology, P.M. and A.D.; software, P.M.; validation, S.P.S., E.S. and D.V. ; formal analysis, M.S. ; investigation, P.M.; resources, M.S. ; data curation, S.P.S. ; writing--original draft preparation, A.D.; writing--review and editing, P.M.; visualization, S.P.S. ; supervision, M.E. ; project administration, E.S. ; funding acquisition, P.M., E.S., D.V., A.D. and M.S. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The Delphi software solution data used to support the findings of this study are available from the corresponding author upon request. The authors declare no conflict of interest. ## References

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