# Production and Maintenance Planning for a Deteriorating System with Operation-Dependent Defectives
## 9 Conclusions  
The number of scientific publications in the new field that analyses the strong interaction between production logistics, quality, and maintenance design is increasing steadily, reflecting the major relevance of this subject. However, most of the literature is based on systems that optimize either the production-maintenance or the production-quality relationships, leading to sub-optimal solutions. By contrast, this paper investigates the problem of determining the optimal production and repair/major maintenance switching maintenance strategies in the context of reliability and quality deterioration. In the manufacturing system under study, the rate of defectives depends on the production pace of the machine, defining operation-dependent defectives. We developed a stochastic optimization model where the joint production and maintenance control policies were determined by the resolution of the Hamilton-Jacobi-Bellman equations. From the obtained results it has been found when the rate of defectives depends on the production pace where the production policy  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Point** & \((x,a)\) & \(Z\) & \(u^{*}(\cdot)\) & \(w^{*}(\cdot)\) \\ \hline \(P_{1}\) & \((-2,20)\) & \(Z_{3}\) & \(u_{max}=10\) & \(0\) \\ \(P_{2}\) & \((5,40)\) & \(Z_{1}\) & \(u_{1}=2.5\) & \(0\) \\ \(P_{3}\) & \((3,60)\) & \(Z_{2}\) & \(u_{2}=5\) & \(1\) \\ \(P_{4}\) & \((8,70)\) & \(Z_{1}\) & \(0\) & \(0\) \\ \(P_{5}\) & \((0,90)\) & \(Z_{3}\) & \(u_{max}=10\) & \(1\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Implementation of the control policy.  
Figure 17: Implementation chart. (**a**) Trace of the production policy; (**b**) Trace of the repair/major maintenance policy.  
defines a multi-hedging point policy with several production thresholds. Additionally, the results shows that in order to reduce the total incurred cost, it is beneficial to progressively decrease the production pace of the unit from its maximal value to inferior rates when there are no backlog in the system and there is a positive level of inventory. The results obtained promote major maintenance activities when the production unit reaches a level of deterioration that justifies the cost of an expensive maintenance. We illustrated and validated our proposed approach through a numerical example and an extensive sensitivity analysis. We discussed managerial implications for the decision maker at implementing our approach in real production systems. To further validate the model, a comparative study has been performed, where we noted that our approach represents a valuable alternative for controlling modern production units. Since we obtained cost economies of around 36%, at comparing our joint control policy with other strategies based on common assumptions of the literature that disregard the major interactions among production, quality, and maintenance. As a subject of future research, imperfect maintenance strategies could be integrated to the developed model in order to reduce the total cost and improve the profitability of the company.  
The authors would like to acknowledge the financial support of PRODEP of Mexico.  
[PERSON] and [PERSON] conceived and designed the mathematical model, [PERSON] and [PERSON] developed the numerical approach; [PERSON] performed the sensitivity analysis and [PERSON] wrote the paper.  
The authors declare no conflict of interest.