## 2 Materials and Methods
### Equating All Pertinent Costs  
A critical component after \(H\) working hours presents a predominant failure mode described by a Weibull probability distribution of three parameters \(t_{0}\), \(\alpha\), and \(\beta\). The equipment unit to which this component belongs is scheduled to be overhauled at moment  
Figure 1: P-F curve (authors’ figure). \(T_{p}\) (\(T_{p}>H\)). To assess the condition of this component, expensive means of diagnosis are necessary. Each inspection \(n=1\), \(2\), \(3\),... costs \(C_{i}\) and will take place at moments \(t_{i1}\), \(t_{i2}\),... \(t_{in}\). The last inspection detected no sign of a failure in progress. Periods P-F and M-F are known. If a potential failure is not noticed or there is insufficient time to do anything if it is, functional failure will occur, and the equipment will come to a halt. The estimated opportunity cost is \(C_{op}\), which is obtained multiplying the standard production rate by the contribution margin. The contribution margin is the difference between the unit sale price of the product and its standard variable cost. If a potential failure is detected and there is enough time to prevent functional failure, the equipment unit is scheduled to stop as soon as possible, and the component will be repaired or replaced at a cost \(C_{rp}\); meanwhile, the production degradation cost \(C_{d}\) builds up slowly. \(R(t_{n}\!\upharpoonright\!t_{n-1})\) is the conditional reliability between inspections and the annual ITR is the company's current interest tax rate, which must be adjusted to conform to the base period of the analysis. If \(m\) represents the number of base periods in a year, the adjusted tax rate can then be derived from \(j=(1+\text{ITR})^{1/m}-1\). Once these four costs are known, they are multiplied twice: first, by the probability of each of them occurring at certain times up to moment \(T_{p}\) and second, by a conversion factor that allows calculating its value from moment \(H\) for each of the points mentioned above. After each of these four categories of costs is obtained, the corresponding expected costs are calculated. These are referred to in subsequent sections as:  
* \(\overline{C_{i}}\), the cost of each inspection;
* \(\overline{C_{rp}}\), the cost of repairing or replacing a potential failure at any time or at moment \(T_{p}\);
* \(\overline{C_{f}}\), the cost of a functional failure due to repair or replace and lost production; and
* \(\overline{C_{d}}\), the cost of degradation incurred over time. The sum of these four costs is \"the expected global cost (\(C_{g}\))\" Next, the expected life of component \(\overline{V}\) is calculated to allow the transformation of \(\overline{C_{g}}\) into a uniform series of payments \(\overline{C_{h}}\) spread over the period \(H\)-\(T_{p}\). This capital recovery, \(\overline{C_{h}}\), is the variable that can now be minimized by trial and error. Varying \(R(t_{n}\!\upharpoonright\!t_{n-1})\) progressively and maintaining \(T_{p}\) constant, a value of \(R(t_{n}\!\upharpoonright\!t_{n-1})\) exists which will turn \(\overline{C_{h}}\) into a minimum, the optimal value for \(R(t_{n}\!\upharpoonright\!t_{n-1})\). The same reasoning can be applied to keeping \(R(t_{n}\!\upharpoonright\!t_{n-1})\) constant and making \(T_{p}\) vary, a value of which exists that will turn \(\overline{C_{h}}\) into a minimum, the optimal value for \(T_{p}\). Each inspection \(n\) must then be performed at moments \(t_{n}\) from (2) until the period where the condition \((t_{n+1}-t_{n})<\) P-M (see Figure 1) is verified. After entering this period, inspections must be performed with P-M periodicity (see Figure 1). A sequence of safe time windows (STW) was given by \((M_{n}-P_{n})\) and unsafe time windows (UTW) by \((P_{n}-M_{n-1})\) (See Figure 2). The probability of a failure in progress not being detected is given by the sum of probabilities that point P will fall within any UTM.