## 3 Case Study. The Applied Evaluation Processes
### Phase 3: Reliability Evaluation Models  
To evaluate the reliability of the human-machine system, some statistical models were formulated using a serial-parallel system approach, some concepts of reliability engineering, and statistics. According to these statistical models, it was necessary to have a database that recorded failures that occurred over time. Recording of product failures. When a failure is recorded, the sub-cause of failure and the place where the failure occurred are also analyzed. A list of the six components (sub-causes) and subsystems (where the failure occurred) is conveniently included to facilitate this analysis. Table 4 shows the database used to collect the product failure information and its respective analysis. The database from Table 4 was obtained from the scrap report by a machine, and it was helpful in failure control and the identification of component-subsystem relationships. Standardization and systematization of terminology. The process of standardization of the terminology used to feed the database consisted of the following:  
1. An awareness interview was conducted with operators, designers, and managers to establish a diagnosis to help design the survey. 2. The survey was designed and applied in each one of the three machining workshops to identify the terminology used by each person involved in the different functions, the failures found in the products in each area, the causes and sub-causes of the failures, and the places where failures occurred. 3. A matrix of relationships was constructed between the areas involved using the terminology identified in the application of the surveys in the different areas of the machine shops, reducing terms to obtain a standardization of the terminology used. It is important to ensure (through the commitment of the main actors in each machine shop) that the information-gathering process is carried out periodically and without interruption, and that the analysis of the collected data is constantly monitored (as required by the designed database). It is the responsibility of each machine shop that participates in this activity. Figure 4: Human–machine system of the machine tool. Determination of the periodicity of the analysis of the information. We previously selected the periodicity with which the analysis of the information collected would be carried out (monthly, weekly, daily, etc. ), depending on the response time we wanted to obtain and the amount of data available for the analysis and evaluation of reliability. Evaluation of the reliability using a statistical model. For this research, reliability is understood as the probability that the man-machine system works appropriately or does not produce defective parts after some time. The basis of the reliability evaluation is the defective parts that have been manufactured in the machine tool area. The subsystems considered are machines, tools, processes, measurement/instrumentation, programming/design, and materials, and the components considered are maintenance, admission profile, institutional conditions/attitude, information/communication, calibration, and supplier. Evaluation of the reliability of a component-subsystem combination. The first step in evaluating the reliability of a component-subsystem combination is to perform a goodness-of-fit test to determine the theoretical distribution that best describes the failure's behavior and its parameters. Here, the parameter estimation was performed by using the Weibull ++ Software. For data that follow a Weibull distribution, the reliability of each component-subsystem combination is evaluated by using Equation (1a). For data that follow an exponential distribution, their reliability is evaluated by using Equation (2). \[R(t)=e^{-(\frac{t}{\hbar})^{\beta}} \tag{1a}\]  
In the Weibull reliability function given in Equation (1a), \(R(t)\) is the reliability index, \(\beta\) is the shape parameter, and \(\eta\) is the scale parameter. In Equation (1b) is given the instantaneous Weibull time-dependent risk function.