The output performance of the manufacturing system has a direct impact on the mechanical product quality. For guaranteeing product quality and production cost, many firms try to research the crucial issues on reliability of the manufacturing system with small sample data, to evaluate whether the manufacturing system is capable or not. The existing reliability methods depend on a known probability distribution or vast test data. However, the population performances of complex systems become uncertain as processing time; namely, their probability distributions are unknown, if the existing methods are still taken into account; it is ineffective. This paper proposes a novel evaluation method based on poor information to settle the problems of reliability of the running state of a manufacturing system under the condition of small sample sizes with a known or unknown probability distribution. Via grey bootstrap method, maximum entropy principle, and Poisson process, the experimental investigation on reliability evaluation for the running state of the manufacturing system shows that, under the best confidence level
Mechanical manufacturing process is an important link in the quality forming process of mechanical products. The good running state of the manufacturing system is a key aspect of the industrial production since it contributes to ensuring the manufacturing process to be reliable and further guarantees the quality of the products. Therefore, for a long time, the evaluation for the running state of the mechanical manufacturing system has been the important subject of much research that has been devoted to the theory and practice of the mechanical manufacturing, and many effort achievements have been performed on reliability investigations of the manufacturing system in recent years.
For example, considering the impact of various factors on the reliability of mechanical products, Schuh et al. [
Poor information means incomplete information, which indicates the characteristic information presented in the subject investigated is incomplete and insufficient and even the lack of a priori knowledge, such as, in the system analysis, a known probability distribution with only a small sample, an unknown probability distribution with few data only, and trends without any prior information. Poor information theory mainly includes the grey system theory, Bayesian theory, the fuzzy set theory, the bootstrap method, the maximum entropy method, and the chaos theory.
Viewing the existing research on poor information, the research and application of the problems involving poor information have drawn much attention and made remarkable progresses. For instance, based on poor information, Wang et al. [
Poisson process is a kind of the most basic independent increment processes with cumulative number of random events, belonging to a relatively simple stochastic process owning continuous time and discrete state. It plays an important role in the theory and application of stochastic processes. At present, Poisson process has been widely applied in the fields of physics, geology, biology, medicine, astronomy, automation system, service system, the reliability theory, and so forth [
Specific to the evaluation of the running state of the manufacturing system, poor information situations presented in the workpiece quality inspection process and in the manufacturing system adjustment process need to be concerned urgently.
There are many factors that interfere with the manufacturing process, which may lead to reducing the reliability of the manufacturing system. In order to ensure the high reliability of the running state of the manufacturing system, it is necessary to do the workpiece quality inspection regularly. Once the running state of the manufacturing system turns into or begins growing unreliable, it must terminate the manufacturing process and the manufacturing system should be conducted on adjustment or maintenance.
Workpiece quality detection is usually accomplished by sampling a few workpieces. It is performed to estimate the true value and confidence interval of the quality data by sampling workpieces, which can be used to timely assess that the running state of the manufacturing system is reliable or not.
Adjustment of the manufacturing system is usually done via trial cut several workpieces. It is to estimate the true value and confidence interval of the quality data by trial cut workpieces, so as to predict that the future running state of the manufacturing system is reliable or not.
In the manufacturing process that is either workpiece quality detection or adjustment of manufacturing system, the workpieces under investigation are very few and usually are only 4~10, and the quality data obtained belongs to the category of a small sample data. In addition, in the existing research, in order to realize to estimate the true value and confidence interval, the quality data are usually assumed as a set of data obeying the normal distribution. However, in practical production, the probability distribution function of mass quality data, such as roundness, parallelism, perpendicularity, concentricity, run-out, burns, and crack, conforms to the nonnormal distribution or unknown distribution. Therefore, it is difficult to solve this problem of reliability evaluation of the running state of manufacturing system using the existing achievements.
The reliability evaluation for the running state of the manufacturing system can be analyzed based on the output workpiece quality data in the manufacturing process. However, the workpiece quality data are a dynamic random process with unknown probability distribution, along with the trend of known or unknown disturbance, which belongs to a poor information system with uncertainty. At present, in the application of poor information theory, the bootstrap method [
The maximum entropy principle [
This paper recommends a new evaluation method based on poor information to solve the problem of reliability evaluation of the running state of manufacturing system under the condition of small sample size and unknown probability distribution. Using the grey bootstrap method, the maximum entropy principle, and Poisson process, it aims to realize reliability evaluation for the running state of manufacturing system with no variation and variation. Via the computer simulations and actual cases, the evaluation results of reliability of the running state of manufacturing system can be obtained. And the research method proposed in the paper provides scientific decisions and recommendations on how to decide properly on the running state of the manufacturing system, which can ensure product quality to be stable and reliable and realize the low manufacturing costs.
Suppose that the workpiece quality data is a random variable
The intrinsic data sequence characterizes the data sequence obtained in the optimal running state of the manufacturing system, which can satisfy the characteristics demands of population distribution of the workpiece quality parameter.
Based on small sample data in the raw intrinsic data sequence
The grey bootstrap method consists of the bootstrap method and the grey prediction model.
The bootstrap method can simulate a large number of bootstrap resampling samples via small sample data under investigation, and the grey prediction model can predict a large number of generated data via bootstrap resampling samples.
According to the bootstrap method, via an equiprobable sampling with replacement from the raw intrinsic data sequence
According to the grey system theory, suppose that the first-order accumulated generating operation (1-AGO) of
Via the grey prediction model,
Suppose that the generated mean vector is
Under the initial condition that
According to the inverse AGO in the grey system theory, the
A large number of generated data obtained by (
Based on the intrinsic generated data sequence
According to the maximum entropy principle in information theory, in all the feasible solutions, it is necessary to solve a problem; namely, the probability density function that maximizes the information entropy is the most unbiased estimation of the information source in the running state of the manufacturing system.
In the information theory, the information entropy
Let the information entropy maximum, namely,
According to the statistical theory, the estimated value of the
Hence, using the Lagrange multiplier method, the probability density function which stratifies (
Equation (
With the help of the principle of statistics, the true value estimate and the confidence interval estimate of the running state of the manufacturing system can be implemented using (
According to the principle of statistics, the estimated true value is given by
The estimated true value is a characteristic index to evaluate the running state of the manufacturing system, namely, an estimate of the size of the workpiece quality parameter.
Suppose that the significance level is
The confidence level
Under the confidence level
The confidence interval
According to (
The expanded uncertainty
In the end, the workpiece quality can be characterized as the estimated true value
Suppose that the output workpiece quality data in the manufacturing process falling out of the fluctuation range of the workpiece quality data in the reliable running state of the manufacturing system is defined as the event
In the actual manufacturing process, the number of the workpiece quality data out of the fluctuation range in the reliable running state of the manufacturing system is always greater than or equal to zero, and they are all the integers. When
For the counting process
According to Poisson process, the occurrence frequency of the event
For real time evaluation for the reliability of the manufacturing system in the normal run of the manufacturing system after being adjusted, it is essential to continuously inspect the workpiece quality. Suppose that the raw inspection data of the workpiece quality are obtained by inspecting a few workpieces and constitute the raw inspection data sequence, which is expressed as by
Generally, there are a few data in the raw inspection data sequence
If many raw inspection data in
Via using the maximum entropy principle mentioned in Section
Suppose there are
According to Poisson process, the variation intensity
When
In (
In the actual production, with the continuous accumulation of processing time, the running state of the manufacturing system becomes unknown; meanwhile the variation process of the manufacturing system becomes uncertain. However, on the whole, the manufacturing system itself follows the variation law from good to bad in the long-term manufacturing process. To ensure products meet the quality requirements, the dynamic analysis of the running state variation process of the manufacturing system is a demanding task.
According to Poisson process mentioned in Section
The experimental data under investigation are grouped to analyze the reliability of the running state variation process of the manufacturing system. The inspection data sequence
The inspection data subsequence
According to the grey bootstrap method and the maximum entropy principle and Poisson process, namely, (
On the basis of the raw intrinsic data sequence
Define the variation probability
The variation probability
There are a total of five cases studies involving two types of the evaluation issues on the running process of the manufacturing system. The former three cases are cases of evaluations for the running state of the manufacturing system, and the latter two cases are cases of evaluations for the running state variation process of the manufacturing system. It is worth mentioning that the problems of prior information with known or unknown probability distributions and trends are taken into account in the case studies.
This is a simulation case of evaluation for the running state of the manufacturing system which obeys a normal distribution. Based on the simulation data with respect to a processing quality parameter
Suppose that the mathematical expectation
Suppose that the mathematical expectation
Let
In Figure
In I region,
In III region,
In II region,
The intrinsic generated data sequence
The probability density function
The inspection data sequence
The relationship between
This is a simulation case of evaluation for the running state of the manufacturing system which obeys a Rayleigh distribution. Based on the simulation data with respect to a processing quality parameter
Suppose that the mathematical expectation
Suppose that the mathematical expectation
Let
In Figure
In I region,
In III region,
In II region,
The intrinsic generated data sequence
The probability density function
The inspection data sequence
The relationship between 2
This is an actual case of evaluation for the running state of the manufacturing system with unknown probability distribution. A rolling bearing inner raceway grinding machine is involved to grind the inner raceway of the tapered rolling bearing with 30204 in the case. By grinding and measurement after regulation of the machine, the 30 measured datasets of the inner raceway roundness of the bearing, in
1.08
0.90
1.06
1.28
0.88
1.87
1.16
1.06
0.97
1.01
0.70
1.15
0.72
1.08
0.67
1.10
0.98
1.15
1.14
1.64
0.73
0.87
1.91
1.95
1.19
0.78
1.51
1.39
1.39
3.28
Based on the above 30 measured datasets of the inner raceway roundness in regard to a processing quality parameter, namely, roundness, there is an evaluation for the running state of the grinding machine in the case.
The former 5 datasets of the above 30 measured datasets are selected as the elements of the raw intrinsic data sequence
Let
In Figure
In I region,
In III region,
In II region,
The intrinsic generated data sequence
The probability density function
The inspection data sequence
The relationship between 2
According to the evaluation results of the above 3 cases with respect to the running state of the manufacturing system, the comparative analysis of the above 3 cases is performed via the discussion and analysis of Figures
Cases
Via comparing Figures
On the whole, both can present the nonlinear linear relationship between 2
In I region, it is easy to see that both of the linear relationships between 2
In III region, both of the linear relationships between 2
In II region, both of the linear relationships between 2
That is, there are the roughly similar characteristic laws of 2
Case
The simulation cases and the actual case show that the research method of the reliability evaluation for the running state of the manufacturing system is feasible, and the research results are of theoretical value and practical significance.
With the increase of the processing time, the running state of the manufacturing system may emerge as random variation, which is regarded as the running state variation process of the manufacturing system. Because the variation trend is uncertain and the variation time is unknown, the reliability evaluation for the running state variation process of the manufacturing system is a demanding task, to ensure that the quality meets the requirements.
This is a practical case of evaluation for the running state variation process of the manufacturing system with unknown probability distribution. A rolling bearing roller diameter grinding machine is involved to grind the roller diameter of the tapered rolling bearing with 30204 in the case. The 30 raw datasets of the average diameter deviation of the roller are collected in turn according to the processing sequence, in
0.0118
0.0116
0.0102
0.0108
0.0106
0.0114
0.0057
0.0108
0.0100
0.0114
0.0114
0.0118
0.0115
0.0112
0.0112
0.0114
0.0130
0.0114
0.0114
0.0122
0.0121
0.0121
0.0115
0.0116
0.0109
0.0124
0.0117
0.0123
0.0114
0.0131
And it is easy to see the internal characteristic law of 30 raw datasets in Figure
On the basis of the 30 raw datasets in Figure
The former 6 raw datasets in Figure
Let the confidence level
In order to evaluate the running state variation process of the manufacturing system in real time, the latter 24 raw datasets in Figure
Based on the maximum entropy principle and Poisson process, by counting and calculating, the variation intensity
For further evaluating the running state variation process of the manufacturing system in real time, the inspection generated data sequences
In Figure
Let the probability density function of the intrinsic data sequence be equal to the probability density function of the inspection data sequence, and the abscissa values
In Figure
In Figure
The variation intensity
Number | Inspection data sequence | Variation intensity |
Reliability degree |
---|---|---|---|
1 |
|
0.36287 | 0.695679 |
2 |
|
0.26563 | 0.7667 |
3 |
|
0.2536 | 0.7760 |
4 |
|
0.4184 | 0.6581 |
The running state variation process of the manufacturing system (Case
Number | Inspection data sequence | Abscissa value |
Intersection area |
Variation probability |
---|---|---|---|---|
1 |
|
6.6334 | 0.3416 | 0.6584 |
2 |
|
8.1341 | 0.4142 | 0.5858 |
3 |
|
8.3794 | 0.4259 | 0.5741 |
4 |
|
6.0307 | 0.3117 | 0.6883 |
The 30 raw datasets of the roller average diameter deviation (Case
The intrinsic generated data sequence
The probability density function
The variation of the reliability of the manufacturing system (
The variation process of the intersection area
The variation process of the variation probability
This is a simulation case of evaluation for the running state variation process of the manufacturing system with unknown probability distribution. The case is further simulated as a manufacturing system with variation to assess the running state variation process based on Case
Based on 30 raw datasets in Figure
For 30 raw datasets in Figure
The former 6 raw datasets in Figure
In accordance with Case
In order to evaluate the running state variation process of the manufacturing system with variation in real time, referring to Case
Based on the maximum entropy principle and Poisson process, by counting and calculating, the variation intensity
In order to further evaluate the running state variation process of the manufacturing system in real time, the inspection generated data sequences
In Figure
Let the probability density function of the intrinsic data sequence be equal to the probability density function of the inspection data sequence, and the abscissa values
In Figure
Case
The simulated linear system can reflect the running state of the manufacturing system with variation by artificially adding the trace linear component
In Figure
Case
The simulated linear system can reflect the running state of the manufacturing system with variation by artificially adding the trace linear component
The variation intensity
Number | Inspection data sequence | Variation intensity |
Reliability degree |
---|---|---|---|
1 |
|
0.4914 | 0.6118 |
2 |
|
0.5352 | 0.5856 |
3 |
|
0.9860 | 0.3731 |
4 |
|
0.9341 | 0.3929 |
The running state variation process of the manufacturing system (Case
Number | Inspection data sequence | Abscissa value |
Intersection area |
Variation probability |
---|---|---|---|---|
1 |
|
5.4064 | 0.2805 | 0.7185 |
2 |
|
5.0985 | 0.2649 | 0.7351 |
3 |
|
3.3078 | 0.1728 | 0.8272 |
4 |
|
3.4398 | 0.1797 | 0.8203 |
The raw data sequence with a linear trend (Case
The simulation trace linear component (Case
The variation of the reliability of the manufacturing system (
The variation process of the intersection areas
The variation process of the variation probability
The sensitivity analysis is an approach to check the stability of the obtained results using the research method under a certain conditions. According to the research method presented in this paper, the variation intensity
Suppose that the mathematical expectation
The 2000 simulation datasets obeying the normal distribution.
The former 200 simulation datasets in Figure
The former 8 simulation datasets in Figure
In order to visually judge the sensitivity degree of the research method proposed with respect to the size of the samples, it is necessary to solve the intersection area
Let the probability density function of the large sample data be equal to the probability density function of the small sample data, by means of
Now on the basis of the research results of the large sample data, the comparative analysis of the research results of the large sample data and the small sample data can be put into effect to compute the relative error of the research results of them. In the light of the relative error of the research results of the large sample data and the small sample data, it is easy to judge that the sensitivity degree of the research method was proposed with respect to the large and small samples, which can realize the sensitivity analysis of the research method proposed regarding the size of the samples.
The related results of the comparative analysis are as follows: with the help of the analysis results of the large sample data, the variation intensity
The sensitivity analysis regarding the size of the samples shows that the size of the samples has no effect on the evaluation results of the research object. The sensitivity degree of the research method proposed in this paper with regard to the size of the samples is small, which is suffice to show that the research method proposed can solve the problem with small sample of the running state of the manufacturing system, and the evaluation results are trustworthy.
According to the results of the former 3 cases, the possibility of achieving high quality running state for the manufacturing system is small in I region. The possibility of achieving low quality running state for the manufacturing system is larger in III region. In II region, the running quality of achieving the running state is in line with the possibility of achieving the quality of the running state, and both are moderate in II region. It shows that II region is a region of the good running state for the manufacturing system.
For the former 3 cases, the relationship between the confidence level
The relationship between the confidence level
It can be found from the relationship of the reliability degree and the extended uncertainty that if the confidence level
The reliability evaluation for the running quality of the manufacturing system includes some elements, such as the confidence level, the confidence interval, 2 times the expanded uncertainty, and the reliability degree of achieving running quality with respect to the workpiece quality.
From the above discussion, the best running state of the manufacturing system in 3 cases can be evaluated, as follows: For Case For Case For Case
Based on the discussion results of the above 3 cases, Cases
In order to evaluate the running state variation process in real time, the reliability evaluation for the running quality of the manufacturing system includes 3 elements, such as the reliability degree of achieving running quality, the intersection area
In Case
In Case
By comparing the research results of Cases
In Case
The conclusions of this paper are as follows: A new evaluation method based on poor information is proposed to evaluate the reliability of the running state of manufacturing system under the condition of small sample size with known or unknown probability distributions in this paper. In the case of unknown and known probability distributions, small sample data obtained by detecting the workpiece quality are processed using the grey bootstrap theory and the maximum entropy principle to obtain the variation intensity of the running state of the manufacturing system by counting. With the help of Poisson process, the reliability model is established to realize reliability evaluation for the running state of the manufacturing system with no variation and variation. It is aimed to effectively determine the running quality of the manufacturing system, so as to ensure the product quality and reduce manufacturing costs. The evaluation results of the running state show that, based on the relationship between 2 times the expanded uncertainty and the reliability degree, II region is considered as the best choice of the good running state of the manufacturing system. Via hypothesis testing and contrastive analysis of the results, it is verified that the confidence level The evaluation results of the running state variation process show that, under the best confidence level The sensitivity analysis regarding the size of the samples indicates that the size of the research sample does not affect the evaluation results of the running state of the manufacturing system by the research method proposed in the paper. The research method proposed is feasible to assess the reliability of the running state of the manufacturing system, which can acquire favorable evaluation effect.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This project was funded by the National Natural Science Foundation of China (Grant nos. 51475144 and 51075123).