## 4 Numerical Example  
In this section, to verify the validity of the model, we investigate the production of six different sizes of cast iron pipes alternately in a centrifugal system in the literature [19], based on computer simulation principles. Consider this production system in the case of nonlinear degradation and perform production simulation for production lot sizing and PM thresholds. The production equipment is the key equipment in the cast iron process. After preparing the molten iron, the centrifuge is used for casting. The machine stops when different types of cast iron pipes are produced or there is an insufficient supply of molten iron. This is referred to as the make-ready time, which can be used for inspection and maintenance. Production systems may suffer from fatigue cracks, corrosion, and other degradation during the production process. Most of the current manufacturing equipment is complex composite equipment. To simplify the model and allow managers to make generalized decisions, our case studies are not limited to specific components or signals, but monitor the entire production system as a whole. Since no real data were collected from the production system, we made rough estimates of the parameters based on related work [30], respectively, \(\lambda=0.0458\) and \(b=1.463\). Different parameters have different effects on system reliability [41]. Although these parameters are not collected from real systems, numerical analysis provides useful insights for analyzing production systems with PM thresholds and production lot sizing. In this production process, the production quantity is in tons and the unit time is in days. As can be seen from Table 1, the system produces only 6 products, where the minimum demand is 2000. In practice, the number of lot sizing \(N\) is less than 100. This indicates that the search space in this case is not very large and the combinatorial optimization method is suitable for solving such optimization problems. In operations research, combinatorial optimization is the process of finding the optimal solution from the solution space to optimize the objective function [42]. Therefore, the combinatorial optimization method is used here to obtain the optimal solution. Assuming the failure threshold \(L_{f}=20.5\), the cost parameters are shown in Table 2.  
We first programmed the python algorithm to test the performance of the model based on the proposed model and parameter values. The maximum profit per unit of time was 20,997.21. Figure 2 shows the results of the profit per unit time when \(N\) goes from 5 to 40 and \(L_{P}\) goes from 5 to 20. To clearly illustrate the change in profit per unit time, the results are shown from four perspectives: (a) three-dimensional, (b) sectional view, (c) \(N=19\), and (d) \(L_{P}=16\). As can be seen from Figure 2, the profit per unit time rises gradually with different combinations of \(N\) and \(L_{P}\), obtaining a maximum value of 20,997.21 at \(N=19\) and \(L_{P}=16\). The results show that the production system can produce 19 production cycles (lots) and the optimal lot sizes for these six products are \(4500/19\approx 236\), \(2500/19\approx 131\), \(4000/19\approx 210\), \(3600/19\approx 189\), \(2000/19\approx 105\), \(3500/19\approx 184\). Excluding maintenance time and set-up time, the duration of a production cycle is approximately 18.9 days. As shown in Figure 3, it can be seen that when \(N=19\), \(L_{f}=20.5\), the overall trend of the amount of PM gradually decreases with an increase in \(L_{P}\) and the amount of CM gradually increases with an increase in \(L_{P}\). In particular, CM appears only at \(L_{P}=17\) and it appears more frequently when the value of \(L_{P}\) is closer to the value of \(L_{f}\). This can be explained in terms of management logic. For the model in this paper, if the value of \(L_{P}\) is set well below the failure threshold \(L_{f}\), then the probability that the magnitude of degradation exceeds \(L_{P}\) will increase significantly and PM measures will be implemented more frequently.