In this phase, two methods that can help to improve the reliability of the human–machine system in the machine tool area are applied. These methods depend on the type of origin of the failures that occurred and their effect on the system’s reliability. Failures were analyzed to identify their origin. A case study was used for this purpose, so the data that appear were obtained from a machine shop.
Origin and contribution of failures. The analysis consists of a classification of kinds of failures by origin using a cluster analysis to detect the kinds of origins of failures in the machine tool area according to similar characteristics of each identified failure that occurred in the machine tool.
The frequency of each kind of failure by origin is presented in
Table 6. Through the Pareto Diagram shown in
Figure 5, we identified the sources of failure that most affected their appearance and made the greatest contribution to or had the highest impact on the reliability of the man–machine system.
Table 6 shows that the origin with the highest number of failures is the mechanical type, with 31 out of 58 (53.54%). Using data from
Table 6, a Pareto Diagram of the origins of failures and their contribution from
Figure 5 was drawn. In
Figure 5, failures of mechanical and human-error origin are vital causes, with 72.4% of the contribution. In contrast, failures of heat treatment, material, and other origin were identified as trivial causes, with a 27.6% contribution to the failures in the products of the machine tool area. After identifying that the failures of both mechanical and human-error origin contribute most to the appearance of failures in the products, two improvement methods, namely monitoring the parameters
β and
η of the Weibull distribution for time-dependent failures, such as those of a mechanical origin, and monitoring the parameter
λ of the exponential distribution for non-time-dependent failures, were developed.
Monitoring of Parameters between Two Sequential Periods
The objective of monitoring the parameters is to identify the lowest reliabilities in the human–machine system during two sequential periods. A case study from a machine shop was carried out to develop this method. The steps were as follows:
Component–subsystem relationships present in failures. This refers to the time to failure generated by a specific component–subsystem combination from a data distribution, either Weibull or exponential, with their corresponding parameters.
Table 7 and
Table 8 show the parameters for each component–subsystem relationship presented in the failure analysis for the period from 25 January to 30 January 2021 and for the period from 17 May to 28 May 2021, respectively. These parameters were helpful in evaluating the reliability of each component–subsystem relationship and they were obtained using the software Weibull
++.
Evaluation of the reliability of component–subsystem relationships. To evaluate the reliability from the period of 25 January to 30 January 2021, the statistical model represented by Equation (1) (Weibull distribution) was used because only failures of mechanical origin were present. To evaluate the reliability from the period of 17 May to 28 May 2021, both the Weibull distribution (Equation (1a)) and the exponential distribution (Equation (2)) were used because there were failures of mechanical and human-error origin.
Table 9 shows the reliability indices for the component–subsystem relationships present in the machine tool during the period of 25 January to 30 January 2021 for five time periods considered within the 9.5-h shift.
Table 10 shows the reliability indices for the component–subsystem relationships present in the machine tool during the period of 17 to 28 May 2021 for five time periods considered within the 9.5-h shift.
Evaluation of the reliability of the subsystems and the man–machine system. Using Equation (3), which considers a series configuration, the reliabilities of the subsystems that contributed to the failures were evaluated. Considering Equation (4), which considers a parallel configuration, the reliability of the man–machine system was evaluated.
Table 11 shows the reliability indices of the subsystems that contributed to the failures and the reliability indices of the man–machine system for five time periods considered within the 9.5-h shift during the period from 25 to 30 January 2021.
Table 11 shows that, in the 7-h period after the start of the shift, in subsystem 1 (machine), the probability that there were no failures for scrap was 5.24%, while in subsystem 2 (tool) it was 47.86% and in subsystem 3 (process/operation) it was 4.64%. The lowest reliability was obtained for subsystem 3. Regarding the reliability of the man–machine system, the analysis showed a probability of 52.89% that no scrap failures would occur in the 7-h period after the shift started.
Table 12 shows the reliability indices of the subsystems present in the failure analysis and the reliability indices of the man–machine system for five time periods considered within the 9.5-h shift during the period from 17 May to 28 May 2021.
Table 12 shows that in the 7-h period after the start of the shift, in subsystem 1 (machine), the probability that there were no failures for scrap was 0.57%, while in subsystem 2 (tool) it was 13.46% and in subsystem 3 (process/operation) it was 0.16%.
The lowest reliability was obtained for subsystem 3. Regarding the reliability of the man–machine system, the analysis showed a probability of 14.09% that no scrap failures would occur in the 7-h period after the shift started. Once the lowest reliabilities of the subsystems (process/operation) and the components (maintenance/setup) were identified, to determine which component or components had an influence on the low reliability, the causes that generated it, the origin of these causes, and the actions to be taken, we performed a FMEA analysis. The recommended actions were documented as a work instruction. This action is described below.
Failure Mode and Effect Analysis (FMEA). The analysis developed in the previous section for the case study yielded the causes that generated the low reliability of the subsystem (process/operation) and the components (maintenance/setup and institutional conditions/attitude). With this information as an input, we generated a FMEA analysis with a work team of two operators, one maintenance technician, and the production supervisor.
Table 13 shows the FMEA analysis used to control and monitor the activities that affect the reliability of the man–machine system.
Table 13 presents some activities identified as being of a mechanical and human-error origin. The corresponding activity for assembling, welding, and polishing, whose defect was the presentation of porous pieces due to contaminated tungsten, was the one that presented the highest NPR value. This particular action was monitored and controlled, as a work instruction, to prevent recurrence as shown in
Figure 6. The work instruction presented in
Figure 6 was used to reduce or eliminate failures due to this failure mode and evaluate the system’s improvement.
Monitoring of the parameters β and η and the parameter λ for activities with critical failures. The relationship between the reliability index and the Weibull β and η parameters shown in Equation (1a) is unique; for fixed t and R(t) values, unique β and η 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 β and η parameters. Therefore, since the parameter β 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 β value the lower the log-dispersion. Similarly, since η represents the product strength, then the higher the η 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 λ parameter of Equation (2). Consequently, because λ is the inverse of the mean time to failure, if its 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
β and
η, the minimum and maximum forces generated by the machine were measured. Then, by using the method proposed in [
32], the Weibull shape (
β) and scale (
η) parameters were determined based only on the observed maximal (
σ1) and minimal (
σ2) applied effort. The method’s efficiency was based on the following facts:
- (1)
The square root of σ1/σ2 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 (μ), and it covers the entire span of the principal stresses;
- (4)
σ1/σ2 and g(x) completely determine the σ1i and σ2i values, which correspond to any lifetime in the Weibull analysis; and
- (5)
σ1/σ2 and η completely determine the minimal and maximal lifetime, which correspond to any σ1i and σ2i values. Additionally, the β and η 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.
Table 15 shows the values captured from the screen of the machining center board referring to the effort generated in the
z axis for drilling the pieces with a 3/8″ drill according to the number of accumulated cycles.
Table 16 shows the values captured from the screen of the machining center board referring to the efforts generated in the
z axis for drilling the pieces with a 1/4″ center drill according to the number of accumulated cycles.
A comparison among
Table 14,
Table 15 and
Table 16 shows that efforts presenting the most variation occur when roughing the part on the
x axis, which range from 4% to 21%. Second, there are efforts in roughing the part on the
y axis, which range from 2% to 13%. The efforts exerted on the
z axis to drill the pieces, with two types of drill bits, range from 22% to 29%.
Considering the most significant variation in the effort values, corresponding to that exerted on the
x axis (minimum 4% and maximum 21%), these values were used to calculate the parameters
β and
η given in
Table 17 according to [
28]. Additionally, the 95% confidence interval was constructed in order to monitor the maximal machine resistance.
Table 5,
Table 6,
Table 7,
Table 8,
Table 9,
Table 10,
Table 11,
Table 12 and
Table 13 are given In
Table 17.
The above-mentioned analysis was performed by using the Weibull parameters fitted from the data as
where
and
are the maximum and minimum efforts at which the machine is performing, and
is the mean of the median rank approach estimated from the elements of the Y vector determined by using the sample size from Piña-Monarrez et al., [
32] given as
The
n elements of the
Y vector used to estimate
were determined based on the median rank approach as
From these
Y elements, the corresponding standardized base lifetime
values that allowed us to determine both the applied effort and its corresponding minimal resistance (see
Table 17) are given as
Therefore, from Equations (5), (6), and (9) the applied effort and the minimal resistance are given as
Finally, the mean and standard deviation for the maximum values of resistance and the corresponding 95% confidence interval are given as
As shown
Table 17, the corresponding reliability indicator was established for each machine resistance. For a machine that has a minimum resistance of 21% over its nominal value, it has a reliability of 0.704 (the minimum resistance that the machine must have to have this reliability). When the effort is 9.1651 (
η) and the process has a resistance of 21, the reliability is 0.7040. When the resistance is higher, the reliability is higher too. This method consists of monitoring the critical value of 21% through the confidence interval, hoping that the maximum value of the machine effort does not exceed the upper limit of 22.4353. It is also possible to evaluate whether the maximum efforts produced by the machine exceed the standard efforts considered in the confidence interval.