Batch production prediction for the mechanical cutting industry based on process capability

Abstract

In the mechanical cutting industry, trial production is used for predicting and evaluating the quality of product processes before batch production, and it can be expressed through the qualification rate. However, it cannot objectively and comprehensively evaluate the quality of product processes. This study optimizes the analysis of outliers and stability in mathematical statistics to better apply it in the mechanical cutting industry; then, it combines them with process capability analysis. Simultaneously, considering the non-normal distribution of process parameters, a batch production-prediction model is proposed. The reliability of batch production-prediction model is verified by the diameter, roundness and roughness of structural common samples. Meanwhile, for other mechanical parts in the mechanical cutting industry, the model proposed in this paper can be used to quickly and accurately predict and evaluate batch production.

Introduction

In industrial production, mechanical cutting plays an important role as one of the most common machining methods for mechanical parts. The batch production of important mechanical parts also relies on the mechanical cutting industry. Therefore, understanding the process parameters of important mechanical parts that require batch production are important. Before the batch production of important mechanical parts, it is necessary to predict and evaluate batch production through a trial production. Chen et al. proposed a comprehensive evaluation model that considered statistical process control to assist in evaluating and improving the batch production of mechanical parts¹. Li et al. used grey models to establish reliability models and used the latest data to achieve accurate predictions². Guo et al. proposed a dynamic quality prediction method for manufacturing processes based on machine learning³. In addition, to predict product quality in multi-variety and small-batch production, Xiao et al. developed a prediction framework consisting of data-driven and mechanism-based methods⁴. Through the above literatures research, it is found that scholars have taken into account more factors such as statistical process control, data-driven, etc. in batch production prediction, and used different models and methods such as grey models, reliability models, and dynamic quality prediction methods to accurately predict and evaluate batch production. Compared with the qualification rate, the models and methods in the literatures are more comprehensive and accurate, and can better predict and evaluate batch production.

A process capability analysis can effectively analyze process parameters related to machining accuracy and surface quality, thereby providing a prediction and evaluation for batch production. The process capability index is typically used to express process capability⁵. The process capability index expresses the process capability at its best state, whereas the process performance index expresses the process capability at its current state^6,7 . Wang et al. investigated the difference between process capability and performance indices for single-value data⁸. In addition, for studying the process capability indices, Abdolshah et al. proposed a method for calculating process capability indices using fuzzy data⁹. Erfanian et al. constructed a new process capability index based on loss and non-normal distribution¹⁰. In addition, there are two main research methods for analyzing the process capability of non-normal distributions. One method involves the conversion of non-normal distribution data into normal distribution data. The current mainstream conversion method is the Johnson transformation method^11,12, which has a high success rate and significantly higher significance level after conversion than other methods. The other method involves finding a formula for calculating the process capability or performance index under a non-normal distribution. Maravelakis derived process capability index-calculation formulas for Poisson and binomial distributions¹³. Vännman proposed a method for calculating the process capability index when the tolerance is one-sided¹⁴. In the above literatures, scholars have studied the process capability index for single-value data and fuzzy data; Meanwhile, scholars have proposed the Johnson transformation method to calculate the process capability index for non-normal distributions. In addition, for data with different distribution characteristics or situations such as poisson distribution, binomial distribution, and one-sided tolerance, scholars have proposed corresponding process capability index calculation formulas. Compared with traditional calculation formulas⁵, scholars' research is more comprehensive and specific.

The presence of outliers can affect the individual distribution test of process parameters and the results of the process capability analysis. Therefore, before conducting the process capability analysis¹⁵, it is necessary to detect and remove outliers from all measured values. Mandhare et al. discussed existing outlier detection methods, which are mainly divided into two categories: statistical-based and bias-based¹⁶. Nooghabi analyzed the process capability under three conditions with outliers: outliers in the mean, outliers in the standard deviation, and outliers in the mean and standard deviation¹⁷. In addition, process stability is one of the prerequisites for process capability analysis and batch production¹⁸, and the control chart of the analysis and judgment process can be used to determine whether the process is stable¹⁹. Ho et al. proposed an attribute control chart for monitoring process variability²⁰. Balamurali et al. designed a new variable batch-attribute control chart²¹. In the above literatures, scholars have summarized existing outlier detection methods and analyzed process capability under three outlier conditions. In addition, scholars have designed an attribute control chart for detecting process variability and a new variable batch-attribute control chart to face actual industrial processing situations. Compared to traditional outlier detection methods and control charts, the research in the literatures has more practical application value and can provide new methods and ideas for industrial production.

This study aimed to optimize the analysis of outliers and stability in mathematical statistics to make it more suitable for industrial production. Simultaneously, the case of the non-normal distribution of process parameters was considered. Taking truncated normal distribution, absolute value distribution, and one-sided tolerance as examples, a batch production-prediction model based on a process capability analysis is proposed.

The remainder of this paper is organized as follows. Section "Batch production-prediction model" presents an overview of the proposed model. Section "Measurement methods after mechanical cutting" presents the measurement methods used in this study. Sections "Results and batch production-prediction model validation" and "Discussion" present the study findings and their discussion, respectively. Section "Conclusions" summarizes the results and presents the conclusions drawn from them.

Batch production-prediction model

Before batch production, the process must be stable and able to satisfy the specified requirements of the product²². Process capability analysis is a good tool²³. Before conducting the process capability analysis, the following three conditions must be satisfied: process stability, individual independence, and sample data following normal distribution. Therefore, before conducting the process capability analysis, the process must be stable. Second, it is inevitable that there will be products with outliers in trial production, which can lead to errors or even stability analysis failures. Moreover, individual distribution testing can also be affected, causing process parameters to no longer obey the theoretical distribution characteristics¹⁵. Finally, many process parameters do not strictly obey a normal distribution; thus, for this type of process parameter, the corresponding formula for calculating the process capability index under the distribution should be found. In this section, the analysis of outliers and stability in mathematical statistics is optimized to better apply them to process parameters. Additionally, the outlier, stability, and process capability analyses considering the specific distribution of process parameters are combined to propose a batch production-prediction model, which can more objectively and comprehensively predict and evaluate batch production through trial production.

Outliers and stability analysis

Outliers and stability analysis are the foundation of process capability analysis. According to literature²⁴, in mathematical statistics, when the deviation between the measured value x_i and average value µ exceeds two times the standard deviation of all measured values S, it is called an outlier, the mathematical expression for outliers is:

$$\left| {x_{i} - \mu } \right| > 2S.$$

(1)

When the deviation between the measured value x_i and average value µ exceeds three times the standard deviation of all measured values S, it is called a high outlier, and the mathematical expression is:

$$\left| {x_{i} - \mu } \right| > 3S,$$

(2)

where $S = \sqrt {\frac{1}{N - 1}\sum\limits_{i = 1}^{N} {\left( {x_{i} - \mu } \right)^{2} } }$, and N is the number of measured values.

Equations (1) and (2) correspond to two different detection standards. For process parameters, the selection of outlier detection standards should be based on the tolerance size and importance of the process parameters. For important process parameters or process parameters with small tolerance sizes, the testing standard of Eq. (2) should be used. Important process parameters or process parameters with small tolerance sizes mean that the allowable fluctuation range during processing is small, and the standard deviation will be small. In this case, if the testing standards of Eq. (1) are used, many normal measured values will be classified as outliers. Similarly, for process parameters with low importance or larger tolerance sizes, Eq. (1) should be used as the detection standard.

When conducting an outlier analysis on process parameters, as outliers are eliminated, the standard deviation and mean of all measured values will also change. At this time, retesting may result in new outliers, and even new outliers may exist after several outlier analyses. To prevent this phenomenon from occurring, conditions should be set for detecting outliers of process parameters. When this condition is reached, the outlier analysis should be stopped. For process parameters, there are mainly two conditions: 1, No outliers exist; 2. The process parameters obey their corresponding theoretical distribution (for example, for radial dimensions of shaft parts, an outlier analysis should be stopped when following a truncated normal distribution).

After conducting the outlier analysis, mean and standard deviation control charts can be used to analyze the stability of the process¹⁹. The upper and lower limits (UTL and LTL) and centerline (CL) of the mean and standard deviation control charts are presented in Table 1. In Table 1, A3, B3, and B4 are control chart parameters, as presented in the Table 2: “Factors for controlling limits in metrological control charts” in literature²⁵ “Control charts-Part 2: Shewhart control charts”. $\overline{S}$ is the control chart standard deviation, $\overline{S} = {{\sum\limits_{I = 1}^{A} {\left( {\sqrt {\frac{1}{n - 1}\sum\limits_{i = 1}^{n} {\left( {x_{in} - \overline{{x_{n} }} } \right)^{2} } } } \right)}_{I} } \mathord{\left/ {\vphantom {{\sum\limits_{I = 1}^{A} {\left( {\sqrt {\frac{1}{n - 1}\sum\limits_{i = 1}^{n} {\left( {x_{in} - \overline{{x_{n} }} } \right)^{2} } } } \right)}_{I} } A}} \right. \kern-0pt} A}$, where n is the subgroup capacity, x_in is the i-th measurement value in the I-th subgroup, $\overline{{x_{n} }}$ is the mean of the I-th subgroup, and A is the number of subgroups. The mean of the measured values within each subgroup is displayed in the mean control chart, and the trend of the mean changes between subgroups can be obtained through the mean control chart to determine the stability of the process.

Table 1 Upper and lower control limits and center line²⁵.

Table 2 A₃, B₃, and B₄ when the subgroup capacity is "3 ≤ n ≤ 6"²⁵.

When there are subgroups higher than UCL or lower than LCL in the mean control chart, the process is unstable. The standard deviation control chart is used to assist the mean control chart. Only when the standard deviation control chart reaches stability, that is, there are no subgroups higher than UCL or lower than LCL in the standard deviation control chart, can the mean control chart be used to judge the stability of the process. In addition, as the capacity of subgroups changes, A₃, B₃, and B₄ also change accordingly. The values of A₃, B₃, and B₄ are listed in Table 2 when the subgroup capacity is 3 ≤ n ≤ 6.

According to Table 2, when the subgroup capacity is "3 ≤ n", then A₃ < 2. At this point, in the stability analysis of process parameters with larger tolerances, subgroups within the $\left( {\mu + {\text{A}}_{3} \overline{S} ,\mu + 2S} \right)$ and $\left( {\mu - 2S,\mu - {\text{A}}_{3} \overline{S} } \right)$ intervals exceed the upper control limit UCL or be lower than the lower control limit LCL. In the stability analysis of process parameters with small tolerances, subgroups within the $\left( {\mu + {\text{A}}_{3} \overline{S} ,\mu + 3S} \right)$ and $\left( {\mu - 3S,\mu - {\text{A}}_{3} \overline{S} } \right)$ intervals exceed the UCL or be lower than the LCL. At this time, the control chart cannot fully express whether the process is stable. The standard deviation of all measurement values S and tolerance T must be combined to conduct a comprehensive stability analysis. According to the quality 10-times rule, when

$$S < \frac{UTL - LTL}{{10}} = \frac{T}{10},$$

(3)

it can be considered that the process is stable; thus, a process capability analysis can be conducted.

Process capability analysis and batch production-prediction model

The normal distribution of data is an important condition. However, in practice, owing to factors such as processing ideas and data types, the probability density function of process parameters is not strictly normally distributed, resulting in the inability to achieve subsequent process capability analyses. This section considers the truncated normal distribution²⁶, absolute value distribution, and one-sided tolerance to derive the process capability and performance index formulas for the three situations and provides solutions when the process parameters obey other distributions¹⁴. Finally, a batch production-prediction model based on process capability and performance analyses is proposed.

Process capability analysis under normal distribution

When the process parameters of mechanical parts obey normal distribution, the process capability and performance index formulas are shown in Eqs. (4–7)⁵. In Eq. (4) and Eq. (5), $\sigma$ is the intra-group standard deviation, $\sigma = {{\overline{S} } \mathord{\left/ {\vphantom {{\overline{S} } {C_{4} }}} \right. \kern-0pt} {C_{4} }}$, C₄ is a control chart parameter, as presented in the Table 2: “Factors for controlling limits in metrological control charts” in literature²⁵ “Control charts-Part 2: Shewhart control charts”.

$$C_{p} = \frac{UTL - LTL}{{6\sigma }},$$

(4)

$$C_{pk} = \min \left\{ {\frac{UTL - \mu }{{3\sigma }},\frac{\mu - LTL}{{3\sigma }}} \right\},$$

(5)

$$P_{p} = \frac{UTL - LTL}{{6S}},$$

(6)

$$P_{pk} = \min \left\{ {\frac{UTL - \mu }{{3S}},\frac{\mu - LTL}{{3S}}} \right\},$$

(7)

Equations (4–7) can be used to calculate process capability and performance indices to predict and evaluate batch production.

Process capability analysis under non-normal distribution

Process parameters of many products do not strictly obey the normal distribution. Processors are susceptible to certain ideas when processing, which makes process parameters obey the truncated normal distribution. For example, the radial dimensions of shaft parts are often subject to the idea of " better repair than exceed the tolerance." during processing, and finally obey the truncated normal distribution. When process parameters obey the truncated normal distribution, the process capability and performance indices change. John demonstrated that when the measured values of process parameters obey a truncated normal distribution²⁶, the probability density function is shown in Eq. (8). In Eq. (8), $\mu_{t}$ is the mean value of the original normal distribution of truncated normal distribution and S_t is the overall standard deviation of the original normal distribution of truncated normal distribution.

$$f\left( x \right) = \frac{{{\text{e}}^{{ - \frac{{\left( {x - \mu_{t} } \right)^{2} }}{{2S_{t}^{2} }}}} }}{{S_{t} \left( {\int_{ - \infty }^{{\frac{{UTL - \mu_{t} }}{{S_{t} }}}} {{\text{e}}^{{ - \frac{{t^{2} }}{2}}} } dt - \int_{ - \infty }^{{\frac{{LTL - \mu_{t} }}{{S_{t} }}}} {{\text{e}}^{{ - \frac{{t^{2} }}{2}}} } dt} \right)}}\left( {LTL \le x \le UTL} \right),$$

(8)

The mean $\mu_{t}$ and standard deviation S_t of the original normal distribution of the truncated normal distribution are determined from the mean $\mu$ and standard deviation S of all measured values.

$$\mu = E\left( x \right) = \mu_{t} + \frac{{{\text{e}}^{{ - \frac{{\left( {LTL - \mu_{t} } \right)^{2} }}{{2S_{t}^{2} }}}} - {\text{e}}^{{ - \frac{{\left( {UTL - \mu_{t} } \right)^{2} }}{{2S_{t}^{2} }}}} }}{{\int_{ - \infty }^{{\frac{{UTL - \mu_{t} }}{{S_{t} }}}} {{\text{e}}^{{ - \frac{{t^{2} }}{2}}} } dt - \int_{ - \infty }^{{\frac{{LTL - \mu_{t} }}{{S_{t} }}}} {{\text{e}}^{{ - \frac{{t^{2} }}{2}}} } dt}}S_{t} ,$$

(9)

$$\begin{gathered} S^{2} = D\left( x \right) = \mu_{t}^{2} + S_{t}^{2} + \frac{{S_{t} }}{{\int_{ - \infty }^{{\frac{{UTL - \mu_{t} }}{{S_{t} }}}} {{\text{e}}^{{ - \frac{{t^{2} }}{2}}} } dt - \int_{ - \infty }^{{\frac{{LTL - \mu_{t} }}{{S_{t} }}}} {{\text{e}}^{{ - \frac{{t^{2} }}{2}}} } dt}} \times \hfill \\ \left[ {\left( {\mu_{t} + LTL} \right){\text{e}}^{{ - \frac{{\left( {LTL - \mu_{t} } \right)^{2} }}{{2S_{t}^{2} }}}} - \left( {\mu_{t} + UTL} \right){\text{e}}^{{ - \frac{{\left( {UTL - \mu_{t} } \right)^{2} }}{{2S_{t}^{2} }}}} } \right] - \mu^{2} . \hfill \\ \end{gathered}$$

(10)

In addition to truncated normal distribution, many process parameters of products in industrial production obey the absolute value distribution, such as geometric tolerances of shaft parts. The absolute value distribution is explained using roundness as an example.

Roundness (Fig. 1) is the degree to which the cross section of a workpiece is close to the theoretical circle. Assuming R_tc is the theoretical radius of a circle, $R_{\max }$ is the radius of the maximum radius circle including the contour of the measured circle and $R_{\min }$ is the radius of the minimum radius circle including the contour of the measured circle. If $R_{\max } - R_{tc} > R_{tc} - R_{\min }$, the mathematical expression of roundness can be expressed as follows:

$$f = + \left( {\left( {R_{\max } - R_{tc} } \right) - \left( {R_{tc} - R_{\min } } \right)} \right),$$

(11)

if $R_{\max } - R_{tc} < R_{tc} - R_{\min }$, the mathematical expression of roundness is

$$f = - \left( {\left( {R_{\max } - R_{tc} } \right) - \left( {R_{tc} - R_{\min } } \right)} \right),$$

(12)

Figure 1

Measurement of roundness.

The positive sign represents the outward direction of the circular section deviating from the theoretical circle, and the negative sign represents the inward direction of the circular section deviating from the theoretical circle. However, in actual application, the roundness only represents the deviation degree. Therefore, the absolute values are considered for the Eqs. (11) and (12). At this time, the probability density function of the process parameters will also change and obey the absolute value distribution. The mathematical expression is shown in Eq. (13). In Eq. (13), μ_a is the mean value of the original normal distribution of absolute value distribution and S_a is the overall standard deviation of the original normal distribution of absolute value distribution.

$$f\left( x \right) = \frac{1}{{\sqrt {2\pi } S_{a} }}\left[ {{\text{e}}^{{ - \frac{{\left( {x - \mu_{a} } \right)^{2} }}{{2S_{a}^{2} }}}} + {\text{e}}^{{ - \frac{{\left( {x + \mu_{a} } \right)^{2} }}{{2S_{a}^{2} }}}} } \right]\left( {x \ge 0} \right),$$

(13)

Similar to the truncated normal distribution, the mean value μ_a and standard deviation S_a of the original normal distribution of the absolute value distribution are calculated by the mean $\mu$ and standard deviation S of all measured values. Let $\mu_{a} = bS_{a}$, the following can be obtained through calculations:

$$\mu = E\left( x \right) = S_{a} \left( {\sqrt {\frac{2}{\pi }} {\text{e}}^{{ - \frac{{b^{2} }}{2}}} + b - \sqrt {\frac{2}{\pi }} b\int_{b}^{ + \infty } {{\text{e}}^{{ - \frac{{t^{2} }}{2}}} dt} } \right),$$

(14)

$$S = \sqrt {D\left( x \right)} = S_{a} \sqrt {\left( {b^{2} + 1} \right) - \left( {\sqrt {\frac{2}{\pi }} {\text{e}}^{{ - \frac{{b^{2} }}{2}}} + b - \sqrt {\frac{2}{\pi }} b\int_{b}^{ + \infty } {{\text{e}}^{{ - \frac{{t^{2} }}{2}}} dt} } \right)^{2} } .$$

(15)

In modern industrial products, the requirements for many process parameters are not within a certain range, but not lower than or higher than a certain value. Because the one-sided tolerance process parameters have only a one-sided tolerance, process capability and performance indices will also change. According to the literature¹⁴, there are four cases, as shown in Fig. 2, and the process capability and performance index calculation formulas are presented in Table 3. In Table 3, M is the nominal value of the process parameter and $\varepsilon$ is the offset, $\varepsilon = \left| {\mu - M} \right|$.

Figure 2

Four cases of one-sided tolerance.

Table 3 Process capability and performance index formulas of one-sided tolerance.

In addition to the three special distributions, process parameters may also obey other non-normal distributions. The calculation formulas for process capability and performance indices under the corresponding distribution must be found. If they cannot be found, Johnson transformation method can also be applied to convert non-normal distribution data into normal distribution data¹². Although this method is simple and efficient, it has two drawbacks: the Johnson transformation method may not be successful because conversion failures may exist, and the reliability of the data will decrease after Johnson conversion²⁷. At this point, the obtained process capability and performance indices may not be completely convincing.

Batch production-prediction model based on the process capability analysis

According to Sections "Outliers and stability analysis", and "Process capability analysis under normal distribution", we can obtain a batch production-prediction model based on the process capability analysis, as shown in Fig. 3.

Figure 3

Batch production-prediction model.

This model optimizes outliers and the stability analysis in mathematical statistics, considering the non-normal distribution of process parameters and predicts and evaluates batch production through process capability and performance indices.

The process capability index is typically greater than the process performance index, and the process capability index is typically used to reflect the best state of the process, while the process performance index is used to reflect the current state of the process. For batch production, more attention should be paid to the process performance index, but the process capability index cannot be completely ignored. When the difference between the process capability and performance indices is large, it indicates that there are some problems that must be optimized. At this time, the cost and benefits brought by problem solving should be comprehensively considered, and the most cost-effective optimization solution should be selected to improve process performance. In addition, for batch production, the standards for the process performance index are presented in Table 4.

Table 4 Evaluation criteria for mass production⁵.

Process performance index P_p and process performance index P_pk, considering the center position, can both be used to predict and evaluate batch production. When the process parameters do not obey a normal distribution, the peak may not be at the center of the distribution function. In this case, P_p should be used to predict and evaluate batch production. When the process parameters obey a normal distribution, the peak must be at the center of the distribution function, and P_pk should be used to predict and evaluate batch production.

Measurement methods after mechanical cutting

Structural common samples, as the most basic components in industrial production, are widely used in mechanical, electrical, aerospace, and other fields requiring mass production. 18CrNiMo7-6 is a low-alloy steel with high strength and toughness and is suitable for manufacturing parts such as gears, bearings, and transmission shafts that withstand high impact loads. In addition, it has good wear and corrosion resistances and can be used under harsh conditions. Its chemical composition is presented in Table 5²⁸. This study used 18CrNiMo7-6 alloy steel as the raw material and processed it through wire cutting and turning to produce structural common samples with machining allowances, as shown in Fig. 4. The cutting process was divided into rough and fine machining, and rough machining was performed on a CNC lathe (CAK4085; Shenyang Machine Tool Co. Ltd, China) at a spindle speed of 1000 r/min, feed speed of 60 mm/min, and back draft of 0.63 mm. A universal tool-grinding machine WS11 was used for finishing at a spindle speed of 6200 r/min, feed speed of 0.125 mm/min, and workpiece speed of 600 r/min. Subsequently, the important process parameters (dimensions, geometric tolerances, and roughness) of the structural common samples were measured. A sample in which the number of subgroups A ≥ 30 was called a large sample, at which the point data analysis has a certain degree of reliability; therefore, 30 trial production structural common samples were processed.

Table 5 Chemical composition of 18CrNiMo7-6 gear steel²⁸.

Figure 4

Processing of structural common samples. (a) CNC lathe, (b) lathe tools, (c) grinder, (d) photograph of the sample (semi-finished product), (e) photograph of the sample (finished product), and (f) process parameter diagram showing the sample dimensions (scale: mm).

Measurement methods for dimensional and geometric tolerance

Dimensions and geometric tolerances, as the most fundamental process parameters of mechanical parts, have a significant effect on the mechanical properties of the parts and the fit between the parts²⁹. A three-coordinate measuring machine can accurately measure the dimensions and geometric tolerances of parts³⁰. This study used a three-coordinate measuring machine (ZEISS CONTURA) to measure the diameter $\phi {13} \pm {0}{\text{.02}}$ mm and roundness O0.01 mm of the structural common specimens. Owing to the symmetry of the center of the structural common specimen, the A and B end faces were randomly selected. A schematic of the diameter and roundness measurement process of a sample is shown in Fig. 5.

Figure 5

The diameter and roundness measurement process of a sample.

To prevent measurement randomness, measurements of the same process parameter were typically taken at different positions. The measurement positions for the diameter $\phi {13} \pm {0}{\text{.02}}$ mm and roundness O0.01 mm of structural common specimens are listed in Table 6, and the measurement results are provided in Appendix 1.

Table 6 Positions of different elements (Diameter $\phi {13} \pm {0}{\text{.02}}$ mm and Roundness O0.01 mm).

Measurement method for roughness

Surface roughness is commonly used to evaluate the surface machining quality of parts and has a significant effect on the stress concentration coefficient of parts. Research has shown that the lower the surface roughness, the lower the stress concentration coefficient of parts³¹. The new BRUKER 3D optical topographer (NPFLEX white-light interferometer) was used to measure the surface roughness R_a0.4 $\mu {\text{m}}$ of structurally common specimens. A schematic of the surface-roughness measurement process of a sample is shown in Fig. 6.

Figure 6

Setup for roughness measurement. (a) the surface-roughness measurement process; (b) enlarged view of the measurement process.

Similarly, to prevent measurement randomness, measurements were taken at different positions of surface roughness R_a0.4 $\mu {\text{m}}$. The different positions of surface roughness R_a0.4 $\mu {\text{m}}$ are presented in Table 7, and the measurement results are shown in Appendix 2.

Table 7 Position of roughness R_a0.4 $\mu {\text{m}}$.

Results and batch production-prediction model validation

The measurement processes of the three important process parameters, diameter $\phi {13} \pm {0}{\text{.02}}$ mm, roundness O0.01 mm, and surface roughness R_a0.4 $\mu {\text{m}}$, of the structural common samples are presented in Section "Measurement methods after mechanical cutting". In this section, the application of the batch production-prediction model proposed in Section "Batch production-prediction model" is presented. The proposed model is used to analyze the outliers and stability of diameter $\phi {13} \pm {0}{\text{.02}}\;{\text{mm}}$, roundness O0.01 mm, and roughness R_a0.4 $\mu {\text{m}}$ of the structural common samples, as well as to calculate the process capability and performance indices to predict batch production.

Batch production prediction of diameter $\phi {13} \pm {0}{\text{.02}}$ mm and roundness O0.01 mm

The batch production-prediction model as shown in Fig. 3 is used to predict the diameter $\phi {13} \pm {0}{\text{.02}}$ mm and roundness O0.01 mm of the structural common samples in Section "Measurement methods after mechanical cutting", which is divided into 5 steps:

Step 1: measurement and sequencing of process parameters

The measurement of diameter $\phi {13} \pm {0}{\text{.02}}$ mm and roundness O0.01 mm have been completed in Section "Measurement methods after mechanical cutting". The results are shown in Appendix 1.

Diameters A1, A2, B1, and B2 are quality characteristics of the same process parameters at different measurement positions on a sample. Therefore, the measured values of diameters A1, A2, B1, and B2 were grouped. Similarly, roundness 30, 34, 38, 42, 46, and 50 are quality characteristics of the same process parameters at different positions on a sample. Therefore, the measured values of roundness 30, 34, 38, 42, 46, and 50 were grouped. Corresponding to the 30 samples, there are 30 sets of sample data.

Step 2: outlier analysis and stability analysis

The tolerance size of the diameter $\phi {13} \pm {0}{\text{.02}}$ mm was 0.04, which is the process parameter with the maximum tolerance size in the dimension diagram (Fig. 4). Therefore, Eq. (1) should be used for outlier analysis. However, the tolerance size of roundness O0.01 mm was 0.01, which is the process parameter with the minimum tolerance in the dimension diagram (Fig. 4). Therefore, Eq. (2) should be used for outlier analysis. Thus, for diameter $\phi {13} \pm {0}{\text{.02}}$ mm, two outlier analyses were performed, and for roundness O0.01 mm, one outlier analysis was performed. The results are presented in Tables 8 and 9.

Table 8 Outlier analysis results of diameter $\phi {13} \pm {0}{\text{.02}}$ mm.

Table 9 Outlier analysis results of roundness O0.01 mm.

The results showed that outliers existed in the intra group data of Nos. 3, 8, 10, and 17 of the process parameter diameter $\phi {13} \pm {0}{\text{.02}}$ mm. Similarly, outliers existed in the intra group data of Nos. 24, 28, and 30 of the process parameter roundness O0.01 mm. The entire group of measured values containing outlier samples were eliminated, and a stability analysis was conducted on the remaining 26 samples with process parameter diameter $\phi {13} \pm {0}{\text{.02}}$ mm and 27 samples with process parameter roundness O0.01 mm. After calculation, the upper and lower control limits and centerlines of the control chart for process parameter diameter $\phi {13} \pm {0}{\text{.02}}$ mm and roundness O0.01 mm are presented in Table 10. The mean and standard deviation control charts for diameter $\phi {13} \pm {0}{\text{.02}}$ mm and roundness O0.01 mm based on the calculated results are shown in Figs. 7 and 8.

Table 10 UCL, CL, and LCL of the control chart (diameter $\phi {13} \pm {0}{\text{.02}}$ mm and roundness O0.01 mm).

Figure 7

(a) Mean control chart and (b)standard deviation control chart for diameter $\phi {13} \pm {0}{\text{.02}}$ mm.

Figure 8

(a) Mean control chart and (b) standard deviation control chart for roundness O0.01 mm.

The results from Figs. 7 and 8 indicate that the variation trend of diameter $\phi {13} \pm {0}{\text{.02}}$ mm and roundness O0.01 mm is random, but the process has not reached stability. However, upon careful observations of Figs. 7a and 8a, it is found that in the mean control chart, the upper control limit of diameter $\phi {13} \pm {0}{\text{.02}}$ mm UCL < UTL, and the lower control limit LCL ≫ LTL; the upper control limit of roundness O0.01 mm UCL ≪ UTL, and the lower control limit LCL ≫ LTL, which will inevitably result in many measured values not being within the [LCL, UCL]. The intra group standard deviation reflects the fluctuation changes of the same sample at different measurement positions. Through Figs. 7b and 8b, it is found that although the maximum intra group standard deviation of diameter $\phi {13} \pm {0}{\text{.02}}$ mm is 0.0055 mm and the maximum intra group standard deviation of roundness O0.01 mm is 0.00067 mm, which exceeds the control upper limit of the standard deviation, there is no situation of exceeding the tolerance within the group, and the standard deviation of all diameter $\phi {13} \pm {0}{\text{.02}}$ mm measured values $S = 0.0032{\text{mm}} < \frac{T}{10} = 0.004{\text{mm}}$, and the standard deviation of all roundness O0.01 mm measured values $S = 0.0009{\text{mm}} < \frac{T}{10} = 0.001{\text{mm}}$. Therefore, by comprehensively analyzing the mean control chart, standard deviation control chart, maximum group standard deviation, standard deviation of all measured values, and tolerance, it can be demonstrated that the process of diameter $\phi {13} \pm {0}{\text{.02}}$ mm and roundness O0.01 mm has reached stability, and a process capability analysis can be conducted.

Step 3: normal distribution test

Kolmogorov–Smirnov test method is used to test the normal distribution of diameter $\phi {13} \pm {0}{\text{.02}}$ mm and roundness O0.01 mm, the results are as shown in Table 11.

Table 11 Normal distribution test results of diameter $\phi {13} \pm {0}{\text{.02}}$ mm and roundness O0.01 mm.

The results in Table 11 show that the significances of diameter $\phi {13} \pm {0}{\text{.02}}$ mm and roundness O0.01 mm are less than the significance level 0.05, so do not follow the normal distribution, and step 4 is required.

Step 4: find corresponding distribution and perform individual distribution test

In Section "Process capability analysis under non-normal distribution", it is known that the diameter should follow the truncated normal distribution and the roundness should follow absolute value distribution. Before conducting the process capability analysis, the individual distribution test should be carried out because the measured values of diameter $\phi {13} \pm {0}{\text{.02}}$ mm and roundness O0.01 mm do not follow normal distribution, as shown in Fig. 9.

Figure 9

Individual distribution tests of (a) the diameter and (b) roundness.

The results show that diameter $\phi {13} \pm {0}{\text{.02}}$ mm obeys truncated normal distribution, and roundness O0.01 mm obeys absolute value distribution, which can be used for the process capability analysis.

Step 5: process capability analysis and batch production prediction

The calculation formulas for truncated normal and absolute value distributions presented in Section "Process capability analysis under non-normal distribution" are applied to calculate process capability and performance indices of diameter $\phi {13} \pm {0}{\text{.02}}$ mm and roundness O0.01 mm, and the results are presented in Table 12.

Table 12 C_p and P_p for diameter $\phi {13} \pm {0}{\text{.02}}$ mm and roundness O0.01 mm.

The results indicate that process performance indices of diameter $\phi {13} \pm {0}{\text{.02}}$ mm and roundness both satisfy the condition for batch production. However, there is still a certain gap between the process performance and capability indices of the diameter $\phi {13} \pm {0}{\text{.02}}$ mm, indicating that there are some problems that must be optimized. At this time, the cost of problem optimization and the benefits of optimization should be comprehensively considered, and the most cost-effective optimization plan should be selected to further improve the process performance index and make batch production more secure. The process performance index of roundness O0.01 mm is 3.8883, indicating that the process capability in the current state is excellent and should continue to be maintained.

Batch production prediction of roughness R _a0.4 μm

Similar to verification in Section "Batch production prediction of diameter$\phi {13} \pm {0}{\text{.02}}$ mm and roundness O0.01 mm", the batch production-prediction model as shown in Fig. 3 is used to predict the roughness R_a0.4 µm of the structural common samples in Section "Measurement methods after mechanical cutting", which is divided into 5 steps:

Step 1: measurement and sequencing of process parameters

The measurement of roughness R_a0.4 µm have been completed in Section "Measurement methods after mechanical cutting". The results are shown in Appendix 2.

In the roughness R_a0.4 $\mu {\text{m}}$ of the process parameters, square areas 1, 2, and 3 represent the quality characteristics of different measurement positions with the same process parameters on a sample. Therefore, the measured values of square areas 1, 2, and 3 are grouped. There 30 samples; therefore, there are 30 sets of sample data.

Step 2: outlier analysis and stability analysis

The tolerance size of surface roughness is 0.4, which is the process parameter with the maximum tolerance size in the dimension diagram shown in Fig. 4. Therefore, Eq. (1) should be selected for outlier analysis. Two outlier analyses were conducted, and the results are presented in Table 13.

Table 13 Outlier analysis results of roughness R_a0.4 µm.

The results showed that outliers existed in the intra group data of Nos. 1, 8, 11, and 22. All measured values of the samples with outliers were eliminated, and a stability analysis was performed on the remaining 26 samples. After calculation, the upper and lower control limits and centerlines of the surface roughness R_a0.4 $\mu {\text{m}}$ control chart were obtained, as presented in Table 14. Based on the calculated results, the mean and standard deviation control charts of the surface roughness R_a0.4 $\mu {\text{m}}$ were plotted, as shown in Fig. 10.

Table 14 UCL, CL, and LCL of control charts (roughness R_a0.4 $\mu {\text{m}}$).

Figure 10

(a) Mean control chart and (b) standard deviation control chart for roughness R_a0.4 $\mu {\text{m}}$.

The results in Fig. 10 indicate that the entire process of surface roughness R_a0.4 $\mu {\text{m}}$ is random but has not reached stability. Figure 10a shows that in the mean control chart, the upper control limit of surface roughness R_a0.4 $\mu {\text{m}}$ UCL ≪ UTL, and the lower control limit LCL ≫ LTL. Although the maximum standard deviation within the group is 0.0329 $\mu {\text{m}}$, as shown in Fig. 10b, it has exceeded the upper control limit of the standard deviation control chart; however, there are no measured values that exceed the tolerance within the group. Moreover, the standard deviation of all measured values for surface roughness R_a0.4 $\mu {\text{m}}$ was $S = 0.0256\mu {\text{m}} < \frac{T}{10} = 0.04\mu {\text{m}}$. Therefore, a comprehensive analysis of the mean control chart, standard deviation control chart, maximum group standard deviation, standard deviation of all measured values, and tolerance demonstrate that the process of surface roughness R_a0.4 $\mu {\text{m}}$ has reached stability; thus, the process capability analysis can be conducted.

Step 3: normal distribution test

Kolmogorov–Smirnov test method is used to test the normal distribution of roughness R_a0.4 µm, the results are as shown in Table 15:

Table 15 Normal distribution test results of roughness R_a0.4 µm.

The result in Table 15 shows that the significance of roughness R_a0.4 µm is less than the significance level 0.05, so does not follow the normal distribution, and step 4 is required.

Step4: find corresponding distribution and perform individual distribution test

Because roughness R_a0.4 $\mu {\text{m}}$ obeys one-sided tolerance, there is no specific distribution feature and individual distribution testing is not required.

Step 5: process capability analysis and batch production prediction

The surface roughness R_a0.4 $\mu {\text{m}}$ belongs to the one-sided tolerance with an average of 0.2496 $\mu {\text{m}}$, and the nominal value M = 0 $\mu {\text{m}}$, which belongs to Case 2 in Fig. 2. Process capability and performance indices can be calculated using Eqs. (18) and (19), and the results are presented in Table 16.

Table 16 C_p and P_p for roughness R_a0.4 $\mu {\text{m}}$.

The results indicate that the process performance index of surface roughness R_a0.4 $\mu {\text{m}}$ is 3.5878, indicating that the process capability in the current state is excellent and should continue to be maintained. Although there is a significant difference between the process performance and capability indices, the cost of optimizing existing problems is much greater than the benefits of optimization. Therefore, the maintenance of the current machining accuracy must continue.

Discussion

The effect of outliers

As mentioned earlier, outliers have a significant impact on the process capability analysis of process parameters and are reflected during stability analysis and in the process capability index. This section explains the effect of outliers using process parameter roundness O0.01 mm, mainly from the perspective of the effect of outliers on the mean and standard deviation control charts, individual distribution test, and process capability and performance indices.

Figure 11 shows the standard deviation control chart of the process parameter roundness O0.01 mm before and after eliminating outliers. The results show that there was a serious deviation for the intra-group standard deviation before eliminating outliers, and there were measured values within the group exceeding the roundness O0.01 mm tolerance. The standard deviation control chart can be directly used to determine that the process is not in a stable state. However, after eliminating outliers, the maximum intra-group standard deviation was only 0.00067 mm, and there were no measured values within the group exceeding the roundness O0.01 mm tolerance. At this point, the stability of the process can be further determined through the mean control chart, standard deviation of all measured values, and tolerance.

Figure 11

(a) Standard deviation control chart of roundness O0.01 mm before eliminating the outliers, and (b) standard deviation control chart of roundness O0.01 mm after eliminating the outliers.

Table 17 presents the mean and standard deviation of the process parameter roundness O0.01 mm before and after eliminating outliers. The results show that the mean change before and after eliminating outliers is insignificant, while the standard deviation change is more significant. The standard deviation of all measured values before removing outliers $S = 0.0014\;{\text{mm}} > \frac{T}{10} = 0.001\;{\text{mm}}$, and the conclusion that the process is unstable can be obtained using Eq. (3). However, after eliminating the outliers, the standard deviation of all measured values $S = 0.0009\;{\text{mm}} < \frac{T}{10} = 0.001\;{\text{mm}}$, the process stabilized.

Table 17 Mean and standard deviation of roundness O0.01 mm before and after eliminating the outliers.

In summary, outliers affect the stability analysis results by affecting the mean, standard deviation, and control chart. Before eliminating outliers, analyzing the control chart, standard deviation of all measured values, and tolerances led to the conclusion that the process is unstable. However, after eliminating outliers, analyzing the control chart, standard deviation of all measured values, and tolerances revealed that the process had stabilized.

Moreover, outliers have a certain effect on the individual distribution test of process parameter roundness O0.01 mm. Figure 12 shows the individual distribution test before and after eliminating outliers for roundness O0.01 mm.

Figure 12

Distribution test (a) before and (b) after eliminating outliers for roundness O0.01 mm.

The results indicate that the individual distribution test failed before eliminating outliers, and it was impossible to verify whether the process parameter roundness O0.01 mm obeys the absolute value distribution. However, after eliminating the outliers, the individual distribution test can be used to verify that roundness O0.01 mm obeys the absolute value distribution.

Finally, outliers also affect process capability and performance indices. Table 18 presents the process capability and performance indices of the process parameter roundness O0.01 mm before and after eliminating the outliers.

Table 18 C_p and P_p of roundness O0.01 mm before and after eliminating outliers.

The results presented in Table 18 indicate that there is a certain difference between the process capability and performance indices before and after eliminating the outliers. Although this difference does not affect the result, it leads to the underestimation of the process performance, and severe underestimation of the process capability. If outliers are not eliminated during the stability analysis, it will be concluded that the mechanical parts are unsuitable for batch production. In summary, the outliers must be detected and eliminated.

Method without outlier analysis, optimized stability analysis and distribution characteristics

In the previous analysis, outliers, specific distribution characteristics of process parameters were not considered and only control charts were used to judge if the process was stable. In this section, the previous methods are discussed and compared with the analytical methods presented in this paper. For the convenience of explanation, the method without outlier analysis, optimized stability analysis and distribution characteristics is referred to as the traditional method, and the method with outlier analysis, optimized stability analysis and distribution characteristics is referred to as the new method. The specific comparison is as follows.

As mentioned in Section "Batch production-prediction model", before conducting the process capability analysis, the following three conditions must be satisfied: process stability, individual independence, and sample data following normal distribution. Traditional method can also apply process capability/performance indices to predict and evaluate batch production, as shown in Fig. 13.

Figure 13

Batch production-prediction model based on traditional method.

Using traditional method to predict batch production for the three process parameters in Section "Results and batch production-prediction model validation" of the paper, the steps are as follows:

Step 1: measurement and sequencing of process parameters

The measurement of process parameters and the ranking of measured values have been completed in Section "Measurement methods after mechanical cutting". The results are shown in Appendix 1 and Appendix 2.

Step 2: stability analysis

According to Table 1, the mean control charts of diameter $\phi {13} \pm {0}{\text{.02}}$ mm, roundness O0.01 mm and roughness R_a0.4 µm are drawn, and the results are shown in Figs. 14, 15 and 16.

Figure 14

Mean control chart by traditional method (Diameter $\phi {13} \pm {0}{\text{.02}}$ mm).

Figure 15

Mean control chart by traditional method (Roundness O0.01 mm).

Figure 16

Mean control chart by traditional method (Roughness R_a0.4 µm).

The results of Figs. 14, 15 and 16 show that although the mean control charts of diameter $\phi {13} \pm {0}{\text{.02}}$ mm, roundness O0.01 mm and roughness R_a0.4 µm are in a random process, there are subgroups that exceed the control upper and control lower limits of the mean control charts, so the processes of diameter $\phi {13} \pm {0}{\text{.02}}$ mm, roundness O0.01 mm and roughness R_a0.4 µm are unstable.

Step 3: normal distribution test

When the process of process parameters is unstable, only the process performance index can be calculated⁶. The process performance index is calculated using the standard deviation of all measured values S, so there is no requirement on whether the process is stable and whether the data obey the normal distribution. According to the characteristics of the process performance index, when the process is unstable, skip step 3 and go directly to step 4.

Step 4: process capability analysis and batch production prediction

Process performance indices of diameter $\phi {13} \pm {0}{\text{.02}}$ mm, roundness O0.01 mm and roughness R_a0.4 µm are calculated by Eq. (6), and the results are shown in the Table 19.

Table 19 Process performance index results using traditional method.

The calculation results show that the process performance of Diameter $\phi {13} \pm {0}{\text{.02}}$ mm and roughness R_a0.4 µm is excellent, which can conduct batch production, but the roundness O0.01 mm process is insufficient, which is not suitable for batch production. Table 20 summarizes the batch production evaluation results of traditional and new methods.

Table 20 The batch production evaluation results of traditional and new methods.

The results in Table 20 indicate that there are differences in conclusions between the traditional method and the new method. For the stability analysis results, traditional methods may have some shortcomings as they do not consider the existence of outliers and only rely on control charts to determine the stability of the process. The impact of outliers on the stability analysis and process capability analysis has been detailed in Section "The effect of outliers".

But the new method takes into account outliers and comprehensively judges the stability of the process through outlier analysis, control charts, tolerance sizes of process parameters, and standard deviation of all measured values S, which can solve the shortcomings of traditional methods that can’t comprehensively and objectively analyze the stability of process parameters.

In addition, traditional method can’t give process capacity indices, which leads to the failure to understand the best capacity of the processing process of the process parameters, and cannot provide suggestions for the follow-up optimization and improvement scheme; Meanwhile, traditional method does not consider the specific distribution of process parameters. As shown in Table 20, without considering the specific distribution of process parameters, the lower process performance index will be obtained, resulting in the underestimation of process performance of process parameters and even the wrong batch production-prediction evaluation results, which will have a negative impact on industrial production.

Conclusions

This study optimized the outliers and stability analysis in mathematical statistics to better apply it to the process parameters of industrial production machinery parts. To this end, the non-normal distribution of process parameters of machinery parts was considered, a batch production-prediction model based on a process capability analysis was proposed and verified by three specific examples. The experimental verification revealed that the proposed model is simple, fast, and has good reliability and potential applications in batch production in the mechanical cutting industry. The main conclusions drawn from the study findings are as follows.

• 1. Outliers and stability analysis in mathematical statistics were optimized. By conducting outlier and stability analyses on important process parameters of trial production mechanical parts, batch production can be predicted. According to the quality 10-times rule, for diameter $\phi {13} \pm {0}{\text{.02}}$ mm, the standard deviation of all measured values was $S = 0.0032\;{\text{mm}} < \frac{T}{10} = 0.004\;{\text{mm}}$. Similarly, for roundness O0.01 mm, the standard deviation of all measured values was $S = 0.0009\;{\text{mm}} < \frac{T}{10} = 0.001\;{\text{mm}}$, and for roughness R_a0.4 $\mu {\text{m}}$, the standard deviation of all measured values was $S = 0.0256\;\mu {\text{m}} < \frac{T}{10} = 0.04\;\mu {\text{m}}$. These results indicate that all processes reached stability; therefore, batch production can be performed.

• 2. The process capability and performance index formulas were derived under truncated normal distribution, absolute value distribution, and one-sided tolerance, and a batch production-prediction and evaluation model based on process capability analysis was proposed. The results revealed that the process performance indices for diameter $\phi {13} \pm {0}{\text{.02}}$ mm, roundness O0.01 mm, and surface roughness R_a0.4 $\mu {\text{m}}$ were 2.5157, 3.8883, and 3.5878, respectively, all of which satisfied the condition for batch production.

• 3. Specific optimization suggestions are given through the calculation of process capability index and process performance index of diameter $\phi {13} \pm {0}{\text{.02}}$ mm, roundness O0.01 mm, and roughness R_a0.4 µm: for diameter $\phi {13} \pm {0}{\text{.02}}$ mm, the cost of problem optimization and the benefits of optimization should be comprehensively considered, and the most cost-effective optimization plan should be selected; For roundness O0.01 mm, as the process performance is enough excellent, continue to maintain excellent process performance; For roughness R_a0.4 µm, due to the cost of optimizing existing problems is much greater than the benefits of optimization, therefore, the maintenance of the current machining accuracy should continue.

• 4. The effects of outliers on process parameter roundness O0.01 mm, specifically on the mean, standard deviation, control chart, individual distribution tests, and process capability and performance indices, were discussed. The results indicate that the presence of outliers can cause changes in the mean and standard deviation of the process parameters, leading to errors in the stability analysis results. Second, the presence of outliers can also lead to the failure of individual distribution tests for process parameters. Finally, the presence of outliers can significantly underestimate process capability and performance of process parameters.

• 5. By comparing the differences between the traditional method and the new method, it is found that the traditional method has some defects: the traditional method does not take into account outliers and the stability analysis is only judged by the control charts, which may lead to the inability to comprehensively and objectively evaluate the process stability of the process parameters. In addition, the traditional method does not consider the distribution characteristics of the process parameters, which may lead to the inability to obtain the process capability index, and the lower process performance index may be obtained, and then the gap between the current processing capacity and the best processing capacity cannot be understood, and no more suggestions can be provided; At the same time, underestimation of process performance may lead to wrong batch production prediction conclusions, with serious negative impact on industrial production.

This study provides a fast and efficient batch production prediction model for the mechanical cutting industry through trial production, which has certain application value. However, there are still certain limitations: this study only predicted batch production of process parameters that obey truncated normal distribution, absolute value distribution, and one-sided tolerance. In addition, there are many process parameters that do not obey normal distribution or the three non-normal distributions studied in this study. The calculation formulas for the process capability and performance index of these process parameters need to be discovered.