## 2 Production, Maintenance, and Quality Problem
### Description of the Problem  
In this study, we present a joint optimization of production, maintenance, and quality control problem. The considered supply chain comprises of a single machine producing only one type of item, which will be stored in a principal store and delivered to multi-purchases warehouses (\((w_{0},w_{1},\ \ldots\ w_{L})\) with a different delivery time, \(\tau_{i}\) (\(i=0,\ \ldots,\ L\))), from which the customers receive their demand products. Under a given service level and quality specification limit, each purchase warehouse wants to satisfy numerous random demands over a finite horizon, \(H\) (Figure 1). In the considered supply chain, the manufactured parts are transported from top stock to the warehouses (\(w_{0},w_{1},\ \ldots\ w_{L}\)) by a conveyance with capacity \(Q_{v}\). The machine of process is subject to random failures and repairs. The deterioration of the process machine is impacted both by time and production cadences. Figure 1: Supply chain management. Consequently, the production process' capability and reliability are affected by the production rate, \(u(k)\), which causes the production of non-conforming units. Failure rate of process machine, \(\lambda(t)\), increases with both time and use, and concerning the quality control, we supposed that the process machine is inspected and controlled every \(h\) time unit during the production interval, \(\Delta t\), by a quantifiable quality statistic, \(X_{t}\). with \(n\) sizes for the sample at each inspection, \(t\). Assume that the specific statistical measures (average and standard deviation) for each sampling must all be between the \"upper specification limit\" (_USL_) and the \"lower specification limit\" (_LSL_). The results of each measurement are recorded in a measurement card with a first species risk. This type of quality control requires that the samples follow the Gaussian distribution, and assumes that the production process is stable if the Gaussian law, \(X_{t}\), is constant with a mean, \(\mu_{0}\), and standard deviation, \(\sigma_{0}\), which are known or well estimated. When all statistics are within control limits, the process is considered under control, since the average sample of the quality indicator measurements is at its target value between the \"lower control limit\" (_LCL_) and the \"upper control limit\" (_UCL_). On the other hand, several factors can influence the stability of processes that come from several causes, such as measurements, materials, personnel, environment, methods, and machines, which can be studied using Ishikawa diagrams. In this study, we are interested in the influence of the machine's degradation on the stability of the process. To reduce the chances of being out of control and the stability/capability of the production process resulting from degradation and the failure rate, a maintenance policy characterized by a preventive maintenance activity with minimal repair (PMMR) is performed through the finite production horizon. Perfect preventive maintenance activities are periodically scheduled at \(T=a\). \(\Delta t=a\cdot b\cdot h\) (\(a\) and \(b\) are integer numbers) with \(\Delta t\) is the production period length, and at each failure between two successive preventive maintenance activities, a corrective maintenance with minimal repair is performed. Therefore, we consider two scenarios as follows (Figure 2). Scenario 1: This scenario is adopted when the process is \"under control\" or under a \"stable\" state. We apply the PMMR planning with negligible duration for each performed preventive and corrective maintenance action in this situation. Scenario 2: This scenario is adopted when the process is \"out of control\" between \(j\)th and \((j+1)\)th sampling. The out of control occurs following an assignable cause, affecting the product quality due to machine degradation. Consequently, the indicator measurements of the sample are situated outside the control limits or the mean value changes.