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1. Introduction
Consider a system that can be characterized as being in any of a previously specified set of states. Suppose that the system evolves or changes from one state to another over time according to a certain law of motion, and let X be the state of the system at time t [1]. If how the system evolved is not deterministic, but caused by some random mechanism, then Xt can be a random variable for each value of the index t. This collection of random variables is the definition of a stochastic process and serves as a model to represent the random evolution of a system over time [2]. In general, the random variables that make up a process are not independent of each other but are related to each other in some particular way [3]. A continuous random variable can take any value in an interval and it is represented by a set of real numbers. It is usually associated with measurements that generate physical quantities. The density function, the cumulative function, the complement function, the expected value of the average, the expected value of the variance, and the moments [4]. The random variable in the Weibull Distribution always takes values greater than or equal to 0 since it always represents time. The density function, the cumulative function, the expected value of the mean, and the variance are
To evaluate the reliability of a random variable with a Weibull distribution, the following equation is used:
For the case where no failure times are available, it is proposed to use the method given in Piña [5]. The method allows estimating the parameters β and η of the Weibull distribution, from the observed or predicted principal stresses σ1 (maximum) and σ2 (minimum), thus allowing the calculation of the reliability R(t) for any desired stress value [5].
In the machining work, when performing the analysis of the causes of non-conforming parts, some variables are generating failures due to the applied mechanical stress or strength, which due to tool wear varies over time. These variables are represented using the Weibull distribution. On the other hand, the variables that are generating the failures and that are not mechanical (human, procedural, administrative factors, etc.) or time-dependent, are represented by the Exponential distribution.
The exponential distribution is applied in reliability when the failure function is independent of time. This property of the exponential distribution is known as forgetfulness. The density function, the cumulative function, the expected value of the mean and variance are:
when t ≥ 0
In the machining work, when performing the analysis of the causes of non-conforming parts, some variables are generating failures due to the applied mechanical stress or strain, which due to tool wear varies over time. These variables are represented using the Weibull distribution. On the other hand, the variables that are generating the failures and that are not of a mechanical nature (human, procedural, administrative factors, etc.) or time-dependent, are represented by the Exponential distribution [6].
Because the production process of parts in this type of machining shop is by high-mix, low-volume batches, the reliability of the man-machine system is affected by various products or part numbers, different types and characteristics of the material used, the various machines involved, as well as the different faults or defects that occur in the parts and produce scrap [7]. For this reason, reliability monitoring should be based on the causal factors of low reliability and the parameters of the distributions that describe the behavior of failures within the system components [8].
Some machining shops in the city of Chihuahua in the metal-mechanic sector have problems that do not allow them to meet delivery times by not completing production orders due to the presence of non-conforming parts in the scrap area. This generates high costs for them, in addition to the loss of their customers' confidence because they are not able to fulfill orders in due time and form. It was proposed to develop a method to monitor the reliability of the man-machine system, monitoring the parameters β (shape parameter) and η (scale parameter) of the component-subsystem combinations that follow a Weibull distribution [9].
This is to incorporate the monitoring information into the Series-Parallel statistical model, for the control and improvement of the reliability of the man-machine system, reducing the number of failures that generate scrap and can thus fulfill orders promptly. The monitoring of the changes in the Weibull parameters was done by monitoring the magnitudes of the stresses applied when manufacturing the parts [10].
2. Methodology
2.1 Man-machine system
The worker operates powered equipment, such as a tool or machine for production. This configuration is the most widely used in manufacturing systems. It involves combinations of one or more workers on one or more pieces of equipment; workers and machines combine to take advantage of their respective strengths and attributes [11].
2.2 Reliability
Suppose that T > 0 is a continuous random variable that records the failure time of a certain component that is in operation at time zero. We will denote by F(t) and f(t) the distribution and density function of this random variable. In reliability theory, it is of interest to know the probability that the component works correctly at least until any time t or the probability that it stops working at the next instant of time, given that the component has worked well until a certain time [12]. For this reason, the following two functions are defined: The reliability function is defined as R(t) = P (T > t), while the failure rate function is r(t) = f(t)/ (1 - F(t)). The reliability function is also called the survival function and the failure rate function is also known as the hazard function and is so called because a component that has survived to time t will fail in the interval (t, t + ∆t] with conditional probability r(t) ∆t + o(∆t), [13] in effect:
Implying that the reliability function in terms of the hazard function is given by: R(t) = (- ∫t r(s) ds) exp. In addition, the mean time to failure E(T) can be written in terms of the function R(t) as:
(
2.3 Reliability for serial systems
Consider a system of n components placed in series as shown in Figure 1. We will assume that each of these components works independently of the other. It is intuitively clear that such a system as a whole work if every component is in good condition [14]. We are interested in finding the reliability and failure rate functions of such a system. Note that the failure time T of such a system is the smallest failure time of each of the components, i.e., T =min {T1, Tn}. Thus, physical schemes of this type help to justify asking the question of finding the probability distribution of the minimum of n independent random variables, which, for time-dependent homogeneous systems, is the Weibull distribution [15].
The reliability and failure rate functions for series systems are presented below. We will denote by R1 (t), Rn (t) the reliability functions of each component in the system, and the failure rate functions will be r1(t), rn(t).
The reliability and failure rate functions of the series system are:
2.4 Reliability for Parallel Systems
In a system of n components placed in parallel, each of these components operates independently of the other, as shown in Figure 2. Such types of systems work as a whole if at least one of the components is in good condition. Systems of this type are used when it is desired to maintain a high level of operational reliability, such as aircraft flight systems, or nuclear reactor management systems. The failure time T of such a system is the largest failure time of the components, i.e., T = max {T1, . . . . , Tn}, where the event (T ≤ t) is equivalent to all components failing before time t. The reliability function of the parallel system is:
2.5 Weibull distribution
The Weibull distribution is used in Reliability Engineering for failure time data and is useful to determine the probability that a part will fail after a certain time t. It is described by 2 or 3 parameters: shape (β), scale (η), and location (γ), which are present in the probability density function (pdf), where the failure time is given by:
(3 parameters) o
(2 parameters)
2.6 Properties of the Weibull Distribution
The Weibull reliability function starts at β =1 since it is assumed that at the beginning of the mission, all the equipment is in good condition, and as time goes by the reliability decreases. For values of β less than 1 the reliability function decreases asymptotically. Like the density function, for values of β =1, the reliability function assumes the exponential form. The properties of the Weibull distribution are:
For β = 1 the curve decreases monotonically faster than for 0 < β <1.
2.7 Parameters of the Weibull distribution
It is useful first to know the effect of the different parameters of the Weibull distribution to be able to act:
Effect of the shape parameter. The shape parameter describes the way the data are distributed. A shape of β =3 approximates a normal curve. A low shape value, e.g., β =1, gives a curve with skewness to the right. A high shape value, e.g., β =10, gives a curve with skewness to the left.
Effect of the scale parameter. The scale, or characteristic life, is the 63.2 failure percentile of the data. The scale defines the position of the Weibull curve concerning the threshold value, which is similar to the way the mean defines the position of a normal curve.
Effect of the threshold value parameter. The threshold value parameter describes a shift of the distribution away from 0. A negative threshold value shifts the distribution to the left, while a positive threshold value shifts the distribution to the right [16].
3. Study case introduction
Case Study in a Machining Workshop, which starts with the process of collecting and analyzing the scrap generated, the evaluation of its reliability, and the identification of the worst reliability, as well as the monitoring of parameters. Research hypotheses were proposed and, to test them, different probabilistic and statistical tools were implemented. The direction of this research was guided by the testing of four issues that are shown below:
In the man-machine system there are activities whose risk or probability of failure depends on time and/or the failure is generated by mechanical stresses (friction, wear, etc.); that is, the damage accumulates over time.
In the man-machine system, there are activities whose risk or probability of failure is constant and does not depend on time, of which it is possible to determine their mean and variance.
It is possible to combine activities and determine the reliability of each combination, in a series-parallel system, to determine the partial and total reliability of the system.
It is possible to monitor the parameters of each component-subsystem combination of the man-machine system, incorporating them into the series-parallel statistical model, due to its configuration, and detect the worst reliability.
3.1 Study Case
A machining shop in the metal-mechanical sector, located in the city of Chihuahua, whose production system is batch, high-mix-low volume, has semi-automated CNC machines for the manufacture of parts. The company has some problems that do not allow them to meet delivery times, because they cannot complete production orders due to the presence of nonconforming parts in the scrap area, which generates high costs, in addition to the loss of confidence of its customers. The workshop wants to improve its reliability, reducing the number of failures that generate scrap and thus be able to fulfill orders promptly.
3.2 Data Analysis
To test the hypotheses formulated and taking as reference the data from the machining shop, the following was carried out: Classification of faults or defects that occur in the manufacture of parts. Information was collected on the types of failures or defects that occurred in the manufacture of parts in the machining shop, from April 27 to June 2, 2021 [17]. These failures were classified by their origin (failure mechanism), using the Cluster Analysis Technique, developed through brainstorming generated by a work team made up of 2 operators, 1 maintenance technician, and a production supervisor. Table 1 shows the resulting classification of the type of failure by its origin.
Table 1 Classification of failure type by origin. Own design.
| ORIGIN OF THE FAULTS | |||||
|---|---|---|---|---|---|
| Mechanic | Mechanic Error | Human Error | Thermal Treatments | Material | Improvements |
| 3 | 22 | 1 | 43 | 26 | 42 |
| 4 | 23 | 2 | 49 | 30 | 45 |
| 5 | 24 | 10 | 50 | 33 | |
| 6 | 25 | 16 | 51 | 34 | |
| 7 | 28 | 17 | 52 | 58 | |
| 8 | 29 | 27 | 53 | ||
| 9 | 31 | 32 | 55 | ||
| 11 | 35 | 36 | 56 | ||
| 12 | 37 | 39 | 57 | ||
| 13 | 38 | 44 | |||
| 14 | 40 | 54 | |||
| 15 | 41 | ||||
| 18 | 46 | ||||
| 19 | 47 | ||||
| 20 | 48 | ||||
| 21 | |||||
In Table 1, the failure origins resulting from the analysis were identified as follows: mechanical, human error, heat treatments, material, and unforeseen events. Table 2 shows the frequency of failures by origin.
Table 2 Frequency table of failures by origin.
| ORIGIN | NUMBER OF FAILURES | % RELATIVE | % ACCUMULATED |
|---|---|---|---|
| MECHANIC | 31 | 53.45% | 53.45% |
| HUMAN ERROR | 11 | 18.97% | 72.42% |
| THERMAL TREATMENTS | 9 | 15.51% | 87.93% |
| MATERIAL | 5 | 8.62% | 96.55% |
| IMPROVEMENTS | 2 | 3.45% | 100% |
| Total | 58 | 100% |
Table 2 shows that one of the origins with the highest number of failures is the mechanical type, with 31 out of a total of 58 (53.54%). Figure 3 shows the Pareto Diagram of the origin of the type of failure, using the data from Table 2.
3.3 Man-machine system model, for the identification of the combinations present in the failures of non-conforming parts for scrap.
The man-machine system is made up of the component-subsystem combinations and the subsystems that participated in the failures during the period analyzed. As a reference to identify the component-subsystem combinations that participated in the man-machine system, Figure 4 is presented. There are 24 possible component-subsystem combinations possible to occur in a certain period. Only those combinations that have contributed to the generation of failures are considered for the reliability assessment, using the corresponding Weibull or exponential distribution, depending on the origin of the failure.
3.4 Component-subsystem combinations are present in failures
The times to failure generated by a certain component-subsystem combination together form a data distribution, either Weibull or Exponential, with their corresponding parameters.
Table 3 Combinations are present in the scrap.
| WEIBULL/EXPONENTIAL PARAMETERS | ||||
|---|---|---|---|---|
| COMBINATION | DESCRIPTION | β | η | 1/λ |
| 1,1 | MACHINE MAINTENANCE/SETUP | 1 | 3.1667 | |
| 1,6 | MACHINE-SUPPLIER | 1 | 9.5 | |
| 2,1 | TOOL MAINTENANCE/SETUP | 1 | 9.5 | |
| 3,1 | PROCESS//OPERATION-MAINTENANCE/SET UP | 1.0688 | 3.1268 | |
Considering the parameters in Table 3, these are used to evaluate the reliability of each component-subsystem combination of the subsystems and the system. As can be seen, all participating combinations were analyzed with the Weibull distribution, since only failures of mechanical origin were involved.
3.5 Evaluation of the Reliability of the Subsystems and the Man-Machine System
Considering Equation 11 (series configuration) the reliability of each subsystem participating in the failures was evaluated. Considering Equation 13 (parallel configuration) the reliability of the man-machine system was evaluated. Table 5 shows the values of the reliability of the subsystems that were present in the scrap and the reliability of the man-machine system during the reviewed period, for 5 periods of time considered within the 9.5 hr. shift.
Table 4 Reliability of the combinations presents in the scrap from January 25 to 30, 2021, for 5 time periods.
| RELIABILITY OF THE SUBSYSTEMS (Rij) AND THE SYSTEM RS | ||||
|---|---|---|---|---|
| PERIODS (hr.) | R1 | R2 | R3 | RS |
| 1 | 0.6563 | 0.9 | 0.645 | 0.9878 |
| 3 | 0.2827 | 0.7292 | 0.2683 | 0.8579 |
| 5 | 0.1218 | 0.5907 | 0.1116 | 0.6807 |
| 7 | 0.0524 | 0.4786 | 0.0464 | 0.5289 |
| 9 | 0.0226 | 0.3877 | 0.0193 | 0.4131 |
Table 4 shows that 7 hours after the start of the shift, in subsystem 1 (machine), the probability of not having 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%, being the worst reliability of subsystem 3.86% and in subsystem 3 (process/operation) it was 4.64%, the worst reliability being that of subsystem 3. Concerning the reliability of the man-machine system, it showed a probability of 52.89% that no failures for scrap would occur in 7 hours after the start of the shift.
3.6 Component-subsystem combinations are present in failures
The times to failure, generated by a certain component-subsystem combination, together form a data distribution, either Weibull or Exponential, with their corresponding parameters. Table 5 shows the component-subsystem combinations that were present in the failure analysis.
Table 5 Reliability of the combinations present in the scrap.
| WEIBULL/EXPONENTIAL PARAMETERS | ||||
|---|---|---|---|---|
| COMBINATION | DESCRIPTION | β | η | 1/λ |
| 1,1 | Machine-Manto/set up | 3.4615 | 7.9737 | |
| 1,6 | Machine-Supplier | 2.1839 | 7.1972 | |
| 2,1 | Tool-Manto/set up | 1.5263 | 5.3284 | |
| 3,1 | Process/Operation-Manto/set up | 1.9091 | 2.3472 | |
| 3,3 | Process/Operation-Institutional Condition/Attitude | 9.5 | ||
As can be seen in Table 5, four of the five combinations were analyzed with the Weibull distribution, because they involve aspects of mechanical origin. The last combination was analyzed with the Exponential distribution because it involves aspects of human error origin. The Weibull ++ software was used to calculate the parameters of both the Weibull and Exponential distributions.
3.7 Evaluation of the reliability of the component-subsystem combinations
Considering the statistical model to evaluate the reliability with each of the Weibull and Exponential distributions of Equations 5 and 10 respectively, the reliabilities of each combination were evaluated. Table 7 shows the values of the reliabilities of each component-subsystem combination that were present in the scrap, during the analyzed period, for 5 time periods considered within the 9.5 hr shift.
Table 6 Reliability of the combinations present in the scrap.
| RELIABILITY OF COMPONENT-SUBSYSTEM COMBINATIONS rij | |||||
|---|---|---|---|---|---|
| PERIODS (hr.) | r1,1 | r1,6 | r2,1 | r3,1 | r3,3 |
| 1 | 0.6478 | 0.7382 | 0.7509 | 0.4433 | 0.9 |
| 3 | 0.2718 | 0.4023 | 0.4232 | 0.0871 | 0.7292 |
| 5 | 0.1141 | 0.2193 | 0.2387 | 0.0171 | 0.5907 |
| 7 | 0.0478 | 0.1195 | 0.1346 | 0.0037 | 0.4786 |
| 9 | 0.02 | 0.0651 | 0.0759 | 0.0006 | 0.3877 |
Table 6 shows that the component-subsystem combinations with the lowest reliability value were 3.1 and 1.1, referring to Process/Operation-Maintenance/Set up and Machine-Maintenance/Set up, respectively, with Maintenance/Set up being the component in common.
3.8 Evaluation of the reliability of the subsystems and the man-machine system
Considering Equation 11, which considers a series configuration, the reliability of each subsystem participating in the failures was evaluated. Considering Equation 13 (parallel configuration) the reliability of the man-machine system was evaluated. Table 8 shows the values of the reliability of the subsystems that were present in the scrap and the reliability of the man-machine system, during the period reviewed, for 5 periods of time considered within the 9.5 hrs. shift.
Table 7 Reliability of subsystems and system, present in the scrap.
| SUBSYSTEM RELIABILITY (Rij) AND SYSTEM | ||||
|---|---|---|---|---|
| RELIABILITY RS | ||||
| PERIODS | R1 | R2 | R3 | RS |
| (hr.) | ||||
| 1 | 0.4782 | 0.7509 | 0.399 | 0.9219 |
| 3 | 0.1094 | 0.4234 | 0.0635 | 0.5191 |
| 5 | 0.025 | 0.2387 | 0.0101 | 0.2653 |
| 7 | 0.0057 | 0.1346 | 0.0016 | 0.1409 |
| 9 | 0.0013 | 0.0759 | 0.0002 | 0.0773 |
Table 7 shows that 7 hours after the start of the shift, in subsystem 1 (machine), the probability of not having 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 worst reliability was that of subsystem 3. Concerning the reliability of the man-machine system, it showed a probability of 14.09% that no scrap failures would occur in 7 hours after the start of the shift.
4. Results
For the verification of the research, these were based in a theoretical-practical way, using mathematical and statistical models, as well as the Case Study. The procedure performed is described below. In the man-machine system, there are activities whose risk or probability of failure depends on time, and/or the failure is generated by mechanical efforts (by friction, wear, etc.), that is, the damage accumulates over time. time. For the evaluation of the reliability of the man-machine system, the Weibull distribution is used, since this distribution is appropriate when the variables that generate the failure are based on mechanical efforts, whose damage accumulates over time. To prove that in the man-machine system, there are activities whose probability of failure depends on time, and are due to mechanical stress, Table 8 is presented.
Table 8 Activities of the man-machine system are dependent on time and mechanical effort.
| Activities | Time-dependent | Mechanical stresses |
|---|---|---|
| Cut | The cutting tool removes material in the form of chips from the workpiece and the sharpness of the tool suffers wear over time. | The pressure of the cutting tool against the metal of the workpiece. |
| Drilling | The material is cut by rotating a drill bit, which removes chips from the material, making a hole. | |
| Turning | The cutting tool approaches the workpiece radially, which, when rotating, is provided with the cutting motion. | |
| Milling | The rotary motion of a tool for cutting, feeding, or depth. | |
| Grinding | The grinding wheel wears the workpiece as it advances towards the grinding wheel, by abrasive means. |
Table 8 describes 5 activities carried out in the Machining Workshop that depend on the edge of the tool. When the cutting tool is installed new or just sharpened, after some time of machining parts and it loses its edge, it makes the efficiency of the cut depend on time. In addition, the failure is generated by efforts of the mechanical type, since the activity is carried out by the pressure of the cutting tool against the metal of the piece; therefore, the analysis used is the Weibull.
In the man-machine system, there are activities whose risk or probability of failure is constant and does not depend on time, from which it is possible to determine their mean and variance. With the use of mathematical and statistical models, it is shown that the probability of failure (risk) of the system may not be a function of time (Exponential). For the evaluation of the reliability of the man-machine system, the Exponential distribution is used, since this distribution is appropriate when the variables that generate the failure are not based on time. To prove that in the man-machine system, there are activities whose probability of failure does not depend on time, Table 9 is presented.
Table 9 Failures in the man-machine system are not dependent on time.
| Activities | Not time-dependent |
|---|---|
| Process errors due to unfamiliarity with new processes and equipment | They occur per event and can be quantified. |
| Non-compliance with requirements by subcontracted suppliers | |
| Insufficient power capacity to keep machinery running | |
| Use of incorrect material due to carelessness | |
| Different finishes due to operator carelessness |
Table 9 describes 5 failures that occur in the man-machine system, caused by human error and unforeseen events, which do not depend on time and can be quantified to obtain the probability of failure; then the analysis to use is Exponential. Table 2 shows that the origin of human error type, has a frequency of 11, out of a total of 58, contributing 18.97%. Figure 3 shows the origin of the type of failure, where failures of human-error origin are identified, together with failures of mechanical origin, as vital causes, contributing 72.5%. For reliability monitoring, the parameters of the Weibull and/or Exponential distributions were used, based on the occurrence of failures, according to the data recorded during 2 different periods of the year 2021. Table 3 shows the parameters of the component-subsystem combinations that were present in the failure analysis for the period from January 25 to 30, 2021, while Table 6 shows the parameters of the component-subsystem combinations that were present in the failure analysis for the period from May 17 to 28, 2021. Combinations 1,1;1,6; 2,1 and 3,1 contributed to the failures during the 2 periods analyzed, which involved machine-maintenance/set up; machine-supplier; tool-maintenance/set up, as well as process/operation-maintenance/set up. These combinations were considered to be mechanical in origin and time-dependent. Combination 3.3 occurred only during the last period analyzed and involved process/operation-institutional conditions/attitude.
This combination was considered to be of human error origin and not time-dependent. To evaluate the reliability of the combinations of both periods, the statistical models corresponding to the Weibull (Equation 5) and Exponential (Equation 10) distribution were used, depending on the origin of the failures. Table 4 presents the values of the reliabilities of each component-subsystem combination that was present in the scrap in the period from January 25 to 30, 2021, for 5 time periods considered within the 9.5 hrs. shift, while Table 6 presents the values of the reliabilities of each component-subsystem combination that was present in the scrap in the period from May 17 to 28, 2021, for 5 time periods considered within the 9.5 hours shift. For the 2 periods analyzed, the worst reliabilities were those of the combinations 3,1 and 1,1, referring to Process/Operation-Maintenance/Set up and Machine-Maintenance/Set up.
To evaluate the reliability of the subsystems and the man-machine system for both periods, the statistical model for a serial configuration was used (Equation 11). Table 5 shows the values of the reliability of the subsystems and the system for the period from January 25 to 30, 2021, for 5 time periods considered within the 9.5 hours shift, while Table 7 shows the values of the reliability of the subsystems and the system for the period from May 17 to 28, 2021. Table 5 shows that 7 hours after starting the shift, in subsystem 1 (machine), the probability of not having 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%, with the worst reliability in subsystem 3. Table 8 shows that 7 hours after starting the shift, in subsystem 1 (machine), the probability of not having 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 worst reliabilities were those of subsystem 3 for the 2 periods analyzed, referring to the Process/Operation.
5. Conclusions
With the verification of hypothesis 1, it can be concluded that, with the use of mathematical and statistical models, it is demonstrated that the probability of failure (risk) of the man-machine system is time-dependent and is generated by mechanical type stresses, which occur in the manufacture of parts; the failures of mechanical origin presented in Table 2 and Figure 3, are an example of this. The man-machine system, as it participates in the failures that occur in the machining area, presents a risk rate or probability of failure that grows steadily. The Weibull distribution used to evaluate its reliability, represents the damage in an additive way, with a growth rate at a constant rate, through its shape parameter β. This parameter remains constant and is also representative of the process variation and reliability that the data represents. The man-machine system is a dynamic process, so the location parameter γ of the Weibull distribution is not considered, only the parameters β and η. Failures of human error origin actively participate in the failures of the machining area due to the nature of the study, since it is a workshop that produces high-mix, low-volume batches, where operators, technicians, programmers, etc., make voluntary and involuntary errors when making changes to the process/operation, machines, tools and materials, produced by the frequent changes in the number of parts to be manufactured. For the above, on the man-machine system, there is a risk or probability of failure, which is not dependent on time, being of human error origin, reason enough for which the Exponential distribution is used for the evaluation of the reliability of the man-machine system. Mixing production batches, varying the type of material and/or the type of tool to be used, is when wear on the cutting tool can occur and this is what happens in reality in machining workshops, since large batches of the same part are generally not produced. It is also advisable to use a measuring tool that allows one to appreciate the wear on the cutting edge of the cutting tool, such as a microscopic optical comparator since with the vernier or the micrometer it is not always possible to appreciate the variation in the cutting edge of the tool.
