## 3 Case Study. The Applied Evaluation Processes
### Phase 4: Improvement
Monitoring of the parameters \(\beta\) and \(\eta\) and the parameter \(\lambda\) for activities with critical failures. The relationship between the reliability index and the Weibull \(\beta\) and \(\eta\) parameters shown in Equation (1a) is unique; for fixed \(t\) and \(R(t)\) values, unique \(\beta\) and \(\eta\) values exist. For activities that depend on time, addressed here as failures generated by a mechanical source, we propose to monitor the system's reliability by monitoring the corresponding \(\beta\) and \(\eta\) parameters. Therefore, since the parameter \(\beta\) represents the dispersion of the logarithm of the failure times, then when its value is lower compared with the previous period, the log-dispersion among the failure times will increase, implying that no improvement was made; that is to say, the higher the \(\beta\) value the lower the log-dispersion. Similarly, since \(\eta\) represents the product strength, then the higher the \(\eta\) value the better the product strength, implying that if its value is lower compared with the previous period, no improvement was made. On the other hand, for failures generated by human errors, by considering that human errors do not depend on time, their reliability was monitored by monitoring the \(\lambda\) parameter of Equation (2). Consequently, because \(\lambda\) is the inverse of the mean time to failure, if its  
Figure 6: Work instruction to avoid the recurrence of the failure. Source: SOMSAC. value is lower compared with the previous period, then this implies that no improvement was made. We started from the identification of the critical problem (FMEA analysis), as in the case of the cutting tool. Referring to the unidentified wear of the edge, we propose that an analysis of the life-strength relationship be carried out and a model be generated as a way of solving the problem. A specific material, a cutting tool, and a process/machine that were identified as critical were considered. From Table 14, the critical wear failure of the edge of the cutting tool was considered, whose NPR value is 30. Thus, to determine the parameters \(\beta\) and \(\eta\), the minimum and maximum forces generated by the machine were measured. Then, by using the method proposed in [32], the Weibull shape (\(\beta\)) and scale (\(\eta\)) parameters were determined based only on the observed maximal (\(\sigma_{1}\)) and minimal (\(\sigma_{2}\)) applied effort. The method's efficiency was based on the following facts:  
1. The square root of \(\sigma_{1}/\sigma_{2}\) represents the base life on which the Weibull lifetimes are estimated;
2. The mean of the logarithms of the expected lifetimes (g(\(x\))) is completely determined by the determinant of the analyzed stress matrix;
3. The Weibull distribution is a circle centered on the arithmetic mean (\(\mu\)), and it covers the entire span of the principal stresses;
4. \(\sigma_{1}/\sigma_{2}\) and \(g(x)\) completely determine the \(\sigma_{1i}\) and \(\sigma_{2i}\) values, which correspond to any lifetime in the Weibull analysis; and
5. \(\sigma_{1}/\sigma_{2}\) and \(\eta\) completely determine the minimal and maximal lifetime, which correspond to any \(\sigma_{1i}\) and \(\sigma_{2i}\) values. Additionally, the \(\beta\) and \(\eta\) parameters are used when the stress is either constant or variable. Life-effort relationship analysis as a method for improving the reliability of the man-machine system in the machining area. The data in Table 14 aim to generate a model of the life-effort relationship between the cutting tool used in the manufacturing process of different steel parts in the CNC machining center. The model was built as follows. The % strength represents the excess effort to which the machine is subjected in order to carry out the operation on the part for the \(x\) and \(y\) axes.