Modern problems of science and education. Quality management Quality assessment based on distribution density
4. Indicators of accuracy and stability of technological processes. Methods for assessing technological processes. Basic conditions for intensification technological process.
Statistical methods of product quality management have, in comparison with continuous product control, such an important advantage as the ability to detect deviations from the technological process not when the entire batch of parts is manufactured, but during the process (when it is possible to timely intervene in the process and correct it).
The main areas of application of statistical methods for product quality management are presented in Fig. 1.
Rice. 1. Statistical methods for product quality management
Let us briefly explain the concepts used in the figure.
Statistical analysis of process accuracy and stability- this is the establishment by statistical methods of the values of indicators of accuracy and stability of the technological process and the determination of the patterns of its occurrence over time.
Statistical process control- this is an adjustment of the values of technological process parameters based on the results of selective monitoring of controlled parameters, carried out to technologically ensure the required level of product quality.
Statistical acceptance control of product quality- this is control based on the use of mathematical statistics methods to verify compliance of product quality with established requirements and acceptance of products.
Statistical method for assessing product quality - This is a method in which the quality values of product quality indicators are determined using the rules of mathematical statistics.
The term "statistical acceptance inspection" should not necessarily be associated with inspection finished products. Statistical acceptance control can be used in incoming control operations, in procurement control operations, in operational control, in control of finished products, etc., i.e. in cases where it is necessary to decide whether to accept or reject a batch of products.
The scope of application of statistical methods in problems of product quality management is extremely wide and covers the entire life cycle of products (development, production, operation, consumption, etc.).
Statistical methods of analysis and assessment of product quality, statistical methods regulation of technological processes and statistical methods of acceptance control of product quality are components of product quality management.
Methods for assessing technological processes.
Quality assessment by distribution density
One of the graphical representation methods is a histogram (bar histogram), which reflects the state of quality of the tested batch of products and helps to understand the state of quality of products in the general population, to identify the position of the average value and the nature of dispersion in it.
Rice. 2. Pareto histogram
Although the histogram allows you to recognize the quality status of a batch of products by appearance distribution, it does not provide all the information about the magnitude of the latitude, the symmetry between the right and left sides of the distribution, the presence or absence of the center of the distribution in quantitative terms.
Assessing the accuracy of technological processes
After the shape and width of the distribution have been determined based on comparison with the tolerance, it is examined whether it is possible to produce high-quality products using this technological process. In other words, it becomes possible to quantify the accuracy of technological processes based on the survey results.
For this purpose, you can use the following formula:
where is the accuracy coefficient of the technological process;
- product approval;
Standard deviation.
The accuracy of the technological process is assessed based on the following criteria:
The technological process is accurate and satisfactory;
- requires careful observation;
Unsatisfactory. In this case, it is necessary to immediately find out the reason for the appearance of defective products and take control measures.
Fig.3. Process accuracy factor
Rice. 3.a - accuracy is stable because it has a margin of accuracy;
Rice. 3.b - the tolerance field is completely filled, there is a fear that defective products will appear;
Rice. 3.c - defective products appear on both sides of the tolerance.
In order to construct a normal distribution curve together with a histogram, it must be converted to the scale in which the histogram and the empirical curve are made.
STATISTIKA can do all this, and with only the initial data for the histogram.
Rice. 4. Histogram in STATISTICA
The red line on the graph shows the fitted normal distribution curve.
Exist different kinds distributions of random variables: normal, binomial, range distribution, Poisson distribution, etc. Very often the normal distribution is used as a model, since many sets of measurements have a distribution approaching normal. Conventionally, the area under the normal distribution curve is relatively equal to unity (Fig. 5.).
Fig.5. Bell curve
An abbreviated table of areas under the normal curve can be presented in Table 1.
This table presents the area values at standard deviations from to Z. In order to determine the area value between two Z values, you need to subtract the corresponding values given in the table. For example, the area between
Z=-1 and Z=2 is equal to 0.9773 - 0.1587 = 0.8186.
Using tables of the normal distribution function, you can determine the value or percentage of defective products.
Let's assume that the technological process is established; it is known that = 0.501, = 0.022, in addition, in accordance with the requirements of regulatory and technical documentation, the upper and lower values are equal to 0.500 0.005.
Let us determine the deviations of the upper and lower permissible values from the average, multiples of the value:
The probabilities of a normally distributed random variable falling into the intervals 0-1.82 and 0-2.52, respectively, are 0.9656 - 0.5 = 0.4656 and 0.5 - 0.0059 = 0.4941.
Therefore, we expect to receive approximately the following data:
0.4656 + 0.4941 = 0.9597 = 95.97% of products meet the established requirements;
0.500 - 0.4656 = 0.0344 = 3.44% of products have a size exceeding the upper tolerance;
0.500 - 0.4941 = 0.0059 = 0.59% of products have a size below the minimum tolerance.
Histograms in STATISTIKA allow you to fit a number of distributions to the data. When constructing a histogram, you simply select the desired distribution from the list.
The presented methodology allows us to evaluate any technological process, allows us to quantify the accuracy of the process, and determine the values of parameters that go beyond acceptable limits.
Basic conditions for intensifying the technological process. The following synchronization methods are used: connection, separation of operations, change of operating mode, intensification of processes (catalysis, use of high pressures, high temperatures, selection of technological modes, etc.), reconstruction of equipment (change of operating speed), reduction of interruptions in equipment operation, change of techniques the work of workers, changes in the degree and nature of the division of labor, etc.
Critical role in reduction production cycles belongs to technical progress. Application of new types of high-performance equipment, mechanization of labor-intensive work, comprehensive development of comprehensive mechanization and automation production processes, improvement of technology, modes and methods of work, intensification of production processes, introduction of scientific organization of labor and production make it possible to reduce the duration of production (technological) operations, as well as breaks between them and, on this basis, reduce the time spent on manufacturing products and, accordingly, improve the use working capital.
Accelerating the turnover of working capital depends primarily on increasing production output per unit of time. The most important role here belongs to technological progress. The use of new types of high-performance equipment, the comprehensive development of comprehensive mechanization and automation of production processes, the improvement of technology, the mechanization of auxiliary work, the intensification of production processes, the introduction of scientific organization of labor and production make it possible to reduce the duration of production (technological) operations, as well as the breaks between them. As a result, the time spent on production is reduced and the use of working capital is improved.
The most important role in increasing production output per unit of time belongs to technical progress. The use of new types of high-performance equipment, the comprehensive development of comprehensive mechanization and automation of production processes, the improvement of technology, the mechanization of auxiliary work, the intensification of production processes, the introduction of scientific organization of labor and production make it possible to reduce the duration of production (technological) operations, as well as the breaks between them. On this basis, the time spent on production is reduced and the use of working capital is accordingly improved; usage
highly qualified specialists; the labor market is emerging (the market for unskilled and skilled labor; there is a reduction in demand for a number of categories of specialists); organizational and economic relations between the state and higher education (the system of planned distribution of university graduates was eliminated); the development of entrepreneurship leads to an outflow of the best specialists from higher education. As a result, the number of applicants is decreasing, and the need for new specialties appears; there is a redistribution of the number of students by specialty and region; there is an intensification of learning processes and a reduction in their terms, instability of state funding for higher education; There is a need to reorganize the structure and activities of universities. A way out of this situation is possible by increasing the degree of independence of universities, however, the state is not eliminated from participating in higher education.
The intensification of development processes depends on the degree of readiness of the enterprise to produce a large range of new products, which, in turn, is significantly influenced by the quality of technological preparation of production, which is the decisive phase of the production process.
Naturally, the insufficient level of concentration and integration of many production facilities and the difficult financial condition prevent them from implementing structural modernization. In order to get out of such a situation, first of all, rational financial policy. The industrial financing policy should become an integral part of the mechanism for the structural modernization of the PC economy. It logically complements the organizational measures associated with the course of reforms aimed at developing market relations. In this regard, one of the most important features of the mechanism of structural modernization of the PC is the intensification of the processes of intersectoral integration of finance and the organization of “logistics” of investment flows. This process is designed to ensure significant changes in the nature of the turnover of all social capital in the republic's PC.
At the same time, under the conditions of scientific and technological revolution, there was an intensification of the processes of division of labor between industrial developed countries. In the context of the growth of mass automated production, its further complication and increase in the variety of products, a gigantic increase in R&D costs, the domestic market of even large industrialized countries turns out to be relatively narrow. The specialization of countries in the production of certain types of products and the acquisition of other products in foreign countries, mutual trade increases significantly. An important role in this process is played by the growth of capital migration between industrialized countries, the formation of giant transnational corporations that develop specialization and cooperation between their enterprises located in different countries, integration of the economies of industrialized countries.
The development of technology and the associated intensification of processes are not limited. Therefore, the possibilities for intensively increasing the use of fixed assets and production capacities are not limited.
The main goal of expansion, reconstruction and technical re-equipment of an existing enterprise is to further intensify production, increase production capacity, product output and improve its quality while ensuring an increase in labor productivity, reducing material intensity of production, increasing capital productivity and introducing low-waste (or waste-free) technology.
The intensification of production and the growth of its efficiency are objectively reflected in the growth clean products. To the greatest extent, the growth of net products exceeds the growth of marketable products in oil refining. The smallest deviations in the growth of marketable and net products are observed in sub-sectors where extensive development factors predominate - tire repair, asbestos, and mechanical engineering.
As the main factor for increasing economic efficiency intensification of production.
Intensification of production - systematic introduction of achievements of scientific and technological progress, leading to better use production resources and increased production efficiency.
Production intensification is achieved in the following areas:
The intensification of production is expressed in indicators of its efficiency: growth in labor productivity; increasing capital productivity; reducing material consumption; improving the quality of products and work.
Intensification of production based on accelerating scientific and technological progress is the main link in solving the problems facing the countries of the socialist community. Based on this, they agreed to concentrate their efforts and organize close, comprehensive cooperation in the development of five priority areas that underlie modern revolutionary changes in science, technology and production: electronization of the national economy; complex automation; new materials and technologies for their production and processing; nuclear energy; biotechnology.
The most important requirement for intensification construction production is to ensure that the growth rate of the volume of construction and installation work outstrips the rate of quantitative growth of the machine fleet by equipping construction organizations with high-performance machines. Consequently, the intensification of production is accompanied by an increase in the replacement of machines of outdated designs with expired service life with new, high-performance ones.
When the level of productive forces of society was not yet very high, and the exhaustibility of natural resources and environmental problems had not yet become noticeable, quantitative intensification of production prevailed, i.e. without regard to the state of nature.
Intensification of production was also carried out through the creation and widespread introduction of high-performance integrated technological installations and units; rational combination and combination of several processes in one technological unit; improving catalytic systems and using new highly efficient catalysts; organization of highly specialized large-tonnage production.
Laboratory work No. 4
Statistical process control involves preliminary analysis of accuracy and stability.
Stability can be assessed by constructing and analyzing histograms and control charts. To assess the accuracy of the technological process (with a normal distribution of the quality indicator), the probable proportion of defective products is found q and accuracy coefficient K t, and also evaluate the distribution parameters - mathematical expectation m and standard deviation s. To do this, they usually take a sample of at least 100 in volume. It is advisable to select product units not in a row, but, for example, every fifth, tenth, etc., which will allow a more correct assessment of the state of the technological process.
With the correct setup of the technological process, the mathematical expectation must correspond to the middle of the tolerance field, specified (usually in the regulatory and technical documentation for the product) by the upper and lower limits of Tv and Tn. In this case m = m 0. When m deviates from m 0, the proportion of defective products increases.
An increase in the standard deviation leads to a greater dispersion of the quality indicator, as a result of which the proportion of defective products also increases.
Probable proportion of defective products q(or the probable proportion of suitable products p = 1-q) can be calculated based on the properties of the integral distribution function (Fig. 3.1.), according to which
P(x<Т н) = F(Т н) And
P(T n
If only a lower tolerance is specified for a product, then the product whose quality indicator is X
Rice. 3.1. Determination of the share of defective products
by the cumulative distribution function.
If only an upper tolerance is specified for a product, then a product with a quality indicator x>T in and, therefore, will be defective.
p = F(T in)
q =1 - F(T in)
If upper and lower tolerances are specified for a product, then the product whose quality indicator is Tn<х
p = F(T in) - F(T n)
q =1 + F(T n) - F(T c)
The technological process accuracy coefficient K t allows you to quantify the accuracy of the technological process.
Where is the clearance? T= T in - T n, S– sample standard deviation.
At Kt £ 0.75, the technological process is quite accurate.
At K t = 0.76...0.98, the technological process requires careful monitoring.
At Kt > 0.98 the accuracy is unsatisfactory.
Example 3.1. A preliminary analysis of the technological process for producing paper by breaking length showed that m = 2500 m and s = 100 m. It was established that the distribution of breaking length approximately corresponds to normal. The technical specifications indicate that the breaking length of the paper must be at least 2300 m. Determine the likely proportion of defective products.
Open a new file. Enter the title of the work “Lab. work 3. Analysis of the accuracy of the technological process.” In accordance with the properties of the cumulative distribution function
q = F(T n)
Calculation using the statistical function NORMIDIST gives the value q = 0.02275 (Figure 3.2).
Figure 3.2. Calculation of the probable proportion of defective products in example 3.1.
Thus, the probable proportion of defective products is about 2.3%.
Exercise
1. Perform calculations according to the example.
2. The technical specifications specify a shaft diameter of 80±0.4 mm. It has been established that in the production of shafts the mathematical expectation of the diameter is 79.8 mm, the standard deviation is 0.18 mm. Find the probable proportion of defective products and the accuracy rate of the technological process. Is the process accurate enough?
1A method for assessing the accuracy and stability of technological processes is proposed, based on checking the homogeneity of two independent samples (extracted from the same population), namely, comparing their distribution functions. When implementing this method, one sample is accepted as the base one, when the quality of the manufactured product meets the requirements of regulatory and technical documentation, and the second sample is a study sample and is necessary for subsequent analysis of the quality of the process according to any indicator. It is proposed to use the Wilcoxon test as a criterion to check the homogeneity of two independent samples. In the example under consideration, an analysis of the stability of the concrete production process was carried out by comparing two different samples obtained as a result of collecting and analyzing statistical information about product quality. The proposed method allows you to obtain reliable information about product quality and process stability without the use of control charts and histograms.
quality control
construction products
methods of mathematical statistics
1. Gmurman V.E. Probability theory and mathematical statistics: textbook. manual for universities / V. E. Gmurman. – Ed. 11th – M.: Higher. school, 2005. – 479 p.
2. Kozitsyna A.V. Quality tools as an effective way to improve product quality [Text] / A.V. Kozitsyna, L.V. Makarova, R.V. Tarasov // Modern scientific research and innovation. – April 2014. - No. 4 [Electronic resource]. URL: http://web.snauka.ru/issues/2014/04/33360 (date of access: 04/09/2014).
3. Loganina V.I. Development of a quality management system at enterprises [Text]: textbook / V.I. Loganina, O.V. Karpova, R.V. Tarasov. - M: KDU, 2008. -148 p.
4. Makarova L.V. Methodological approach to ensuring stability and quality of technological processes [Text] / L.V. Makarova, R.V. Tarasov, D.V. Tarasov, O.F. Petrina // Scientific and theoretical journal Vestnik BSTU im. V.G. Shukhova. - No. 1. - 2015. - P. 120–124.
5. Orlov A.I. Mathematics of Case: Probability and Statistics - Basic Facts: Study Guide. – M.: MZ-Press, 2004. – 110 p. – URL: http://www.aup.ru/books/m155/
In modern competitive conditions, the manufacturer must provide high quality products at an affordable price. Achieving these goals is impossible without creating optimal production conditions aimed at improving technological processes and the control system at the enterprise. The control system at enterprises in the construction industry, as a rule, includes three components: incoming control, operational control and acceptance control. Improving these control methods can significantly reduce production costs while constantly improving product quality. Of particular interest in these conditions is the analysis of the accuracy and stability of technological processes, which today is unthinkable without the use of statistical methods.
Statistical methods have proven themselves as quality tools and are used in cases where, based on the results of a limited number of observations, it is necessary to establish the reasons for the improvement or deterioration in the accuracy and stability of technological processes or the operation of technological equipment. The accuracy of a technological process is understood as a property of a technological process that determines the proximity of the actual and nominal values of the parameters of the manufactured product. The stability of a technological process is understood as a property of a technological process that determines the constancy of probability distributions for its parameters over a certain period of time without outside intervention. In turn, ensuring the stability and accuracy of the production process affects the quality of the finished product.
Enterprise or process management systems in the field of quality require the use of statistical methods:
Methods for analyzing product quality assessment;
Methods for regulating technological processes;
Acceptance quality control methods, etc.
The use of these methods allows:
Identify random and systematic indicators that can lead to defects;
Check compliance with the requirements of GOSTs, SNIPs and regulatory documents;
Identify potential production reserves;
Determine technical standards and tolerances of manufactured products;
Correctly select process equipment and test plan.
There are several “classic” problems.
1. Identify the correspondence between the quality indicators of manufactured products and the reference product. This task comes down to the analysis of mathematical expectations, and consists of testing the null hypothesis: , where
X is a random variable, the values of which determine the result of tests (observations);
a is the value of the reference product.
2. Identify the difference between the dispersion of the quality indicator of manufactured products and the reference product. This task comes down to comparing variances and consists of testing the null hypothesis: 
.
In this work, to analyze the stability of the technological process, it is proposed to check the homogeneity of two independent samples, namely, to compare their distribution functions and test the null hypothesis: 
.
Formulation of the problem
In factory laboratories and quality departments of enterprises, as a rule, to assess the stability of the technological process, they resort to constructing histograms for the random variable under study, drawing up control charts for the reporting period (for example, a week or month) and their subsequent analysis.
The proposed technique can be reduced to checking the homogeneity of two independent samples (extracted from the same population), namely, to comparing their distribution functions.
In this case, one sample can be considered basic, when the quality of the products met all technical and regulatory requirements (and the numerical characteristics of this sample can be determined), and the second sample is a study sample and is intended to identify improvement (deterioration, stability) of the technological process according to some indicator.
Let's consider an example of the implementation of the proposed methodology for assessing the stability of the technological process for the production of M150 concrete. The analysis was carried out on the basis of data on the compressive strength () of control samples at a hardening age of 28 days (table).
Independent samples from the general population
|
Serial number |
Base sample |
Study sample |
|
Sample size |
In the Statistica 10 environment, to make the test results clearer, histograms of the distributions of the base and study samples were constructed with normal distribution densities superimposed on them and the values of the statistics were obtained (the smaller the values of the Kolmogorov-Smirnov statistics, the closer the distribution of the random variable is to normal).
Histograms of distributions of the base and study samples
As a criterion for checking the homogeneity of two independent samples, we will use the Wilcoxon test, the undoubted advantage of which is the possibility of application to random variables with an unknown distribution law (only the requirement of continuity of random variables is required).
This criterion at a given significance level consists of testing the null hypothesis 
on the homogeneity of two independent samples of volumes and () under a competing hypothesis 
. The process of testing the null hypothesis varies slightly depending on the sample size and is conditionally divided into two cases:
1) the size of both samples does not exceed 25;
2) the size of at least one of the samples exceeds 25.
In the example under consideration, the size of both samples does not exceed 25.
Wilcoxon test
At the first stage of testing the criterion, it is necessary to arrange the options of both samples (table) in ascending order, i.e. in the form of one variation series:
151, 151, 151, 151, 151, 151, 151, 152, 152, 152, 152, 152, 153,
154, 154, 154, 154, 157, 158, 158, 158, 158, 158, 158, 158, 159, 159,
159, 160, 160, 160, 160, 160, 160, 160, 161, 161, 161, 161
(here the first selection options are highlighted in bold),
and find in this series the observed value of the criterion - the sum of the ordinal numbers of the variant of the first sample:
The second stage is to determine the upper and lower critical points at a given significance level (for example, ):
1) lower critical point 
is found according to the tables of critical points of the Wilcoxon test:
2) the upper critical point is determined by the formula:
If 
or 
- the null hypothesis is rejected. If - there is no reason to reject the null hypothesis.
From the above calculations it is clear that
and there is no reason to reject the null hypothesis.
Consequently, the reference and study samples have the same distribution functions, and the technological process for the production of M150 concrete is stable.
conclusions
The proposed method does not require the construction of histograms and control charts and makes it possible to quickly analyze the accuracy and stability of technological processes while ensuring high reliability of the results. However, one should take into account the fact that if, according to the results of the analysis, the process turns out to be unstable, then the sample under study needs to be studied in more detail in order to identify the causes of process instability and deterioration in product quality.
Reviewers:
Loganina V.I., Doctor of Technical Sciences, Professor, Head. Department of Quality Management and Trade and Trade, Federal State Budgetary Educational Institution of Higher Professional Education "Penza State University of Architecture and Construction", Penza;
Danilov A.M., Doctor of Technical Sciences, Professor of the Department of Mathematics and Mathematical Modeling, Penza State University of Architecture and Construction, Penza.
Bibliographic link
Tarasov D.V., Tarasov R.V., Makarova L.V., Slepova I.E. METHODOLOGY FOR ASSESSING THE STABILITY OF A TECHNOLOGICAL PROCESS IN THE PRODUCTION OF CONSTRUCTION PRODUCTS // Modern problems of science and education. – 2015. – No. 1-1.;URL: http://science-education.ru/ru/article/view?id=17674 (access date: 02/01/2020). We bring to your attention magazines published by the publishing house "Academy of Natural Sciences"
Statistical methods of product quality management have, in comparison with continuous product control, such an important advantage as the ability to detect deviations from the technological process not when the entire batch of parts is manufactured, but during the process (when it is possible to timely intervene in the process and correct it).
Main areas of application of statistical methods for product quality management
Rice. 1. Statistical methods for product quality management
Let us briefly explain the concepts used in the figure.
Statistical analysis of process accuracy and stability- this is the establishment by statistical methods of the values of indicators of accuracy and stability of the technological process and the determination of the patterns of its occurrence over time.
Statistical process control- this is an adjustment of the values of technological process parameters based on the results of selective monitoring of controlled parameters, carried out to technologically ensure the required level of product quality.
Statistical acceptance control of product quality- this is control based on the use of mathematical statistics methods to verify compliance of product quality with established requirements and acceptance of products.
Statistical method for assessing product quality - This is a method in which the quality values of product quality indicators are determined using the rules of mathematical statistics.
The term "statistical acceptance control" should not necessarily be associated with the control of finished products. Statistical acceptance control can be used in incoming control operations, in procurement control operations, in operational control, in control of finished products, etc., i.e. in cases where it is necessary to decide whether to accept or reject a batch of products.
The scope of application of statistical methods in problems of product quality management is extremely wide and covers the entire life cycle of products (development, production, operation, consumption, etc.).
Statistical methods for analyzing and assessing product quality, statistical methods for regulating technological processes and statistical methods for acceptance control of product quality are components of product quality management.
Quality assessment by distribution density
One of the graphical representation methods is a histogram (bar histogram), which reflects the state of quality of the tested batch of products and helps to understand the state of quality of products in the general population, to identify the position of the average value and the nature of dispersion in it.
Rice. 2. Pareto histogram
Although a histogram allows you to recognize the quality status of a batch of products by the appearance of the distribution, it does not provide all the information about the magnitude of the latitude, the symmetry between the right and left sides of the distribution, the presence or absence of a distribution center in quantitative terms.
Assessing the accuracy of technological processes
After the shape and width of the distribution have been determined based on comparison with the tolerance, it is examined whether it is possible to produce high-quality products using this technological process. In other words, it becomes possible to quantify the accuracy of technological processes based on the survey results.
For this purpose, you can use the following formula:
where is the accuracy coefficient of the technological process;
Product approval;
Standard deviation.
The accuracy of the technological process is assessed based on the following criteria:
The technological process is accurate and satisfactory;
- requires careful observation;
Unsatisfactory. In this case, it is necessary to immediately find out the reason for the appearance of defective products and take control measures.
Fig.3. Process accuracy factor
Rice. 3.a - accuracy is stable because it has a margin of accuracy;
Rice. 3.b - the tolerance field is completely filled, there is a fear that defective products will appear;
Rice. 3.c - defective products appear on both sides of the tolerance.
In order to construct a normal distribution curve together with a histogram, it must be converted to the scale in which the histogram and the empirical curve are made.
STATISTICA can do all this, and with only the initial data for the histogram.
Rice. 4. Histogram in STATISTICA
The red line on the graph shows the fitted normal distribution curve.
There are different types of distribution of random variables: normal, binomial, Poisson distribution, etc.
Very often the normal distribution is used as a model, since many sets of measurements have a distribution that approaches normal. Conventionally, the area under the normal distribution curve is relatively equal to unity (Fig. 5.).
Fig.5. Bell curve
An abbreviated table of areas under the normal curve can be presented in Table 1.
This table presents the area values at standard deviations from to Z. In order to determine the area value between two Z values, you need to subtract the corresponding values given in the table. For example, the area between Z=-1 and Z=2 is 0.9773 - 0.1587 = 0.8186.
Using tables of the normal distribution function, you can determine the value or percentage of defective products.
Let's assume that the technological process is established; it is known that = 0.501, = 0.022, in addition, in accordance with the requirements of regulatory and technical documentation, the upper and lower values are equal to 0.500 0.005.
Let us determine the deviations of the upper and lower permissible values from the average, multiples of the value:
The probabilities of a normally distributed random variable falling into the intervals 0-1.82 and 0-2.52, respectively, are 0.9656 - 0.5 = 0.4656 and 0.5 - 0.0059 = 0.4941.
Therefore, we expect to receive approximately the following data:
0.4656 + 0.4941 = 0.9597 = 95.97% of products meet the established requirements;
0.500 - 0.4656 = 0.0344 = 3.44% of products have a size exceeding the upper tolerance;
0.500 - 0.4941 = 0.0059 = 0.59% of products have a size below the minimum tolerance.
Histograms in STATISTICA allow you to fit a number of distributions to the data. When constructing a histogram, you simply select the desired distribution from the list.
Fig.6. Window for constructing histograms in STATISTICA
The presented methodology allows us to evaluate any technological process, allows us to quantify the accuracy of the process, and determine the values of parameters that go beyond acceptable limits.
where Ki are private quality indicators,
P – product sign.
In turn, private indicators are defined as
where Kf is the actual quality level,
Ke is the level of the best sample (standard).
During a comprehensive assessment of the quality of p
production, a weighted arithmetic average can also be used when the averaged initial relative indicators Ki differ relatively little from each other:
, (2.7)
where Ki is a private relative quality indicator;
Wi – coefficients of weight of indicators (determined by experts).
If the value of the summary quality indicator is greater than one, then we can conclude that the product sample in question is better in quality than the base sample.
Much more often, the method of relative linear assessments is used to assess the quality level. In this case, the integral assessment of the quality level is found by the formula:
, (2.8)
where Kfi is the actual quality level,
Kei – reference (normative) level.
Formula (2.6) can also be used to assess the instability of the technological process, and the formula for calculating the summary indicator of instability (Kn) takes the following form:
, (2.9) AAAAAAAAAAAAAAAAAAAAAAAAAAA
where Кнi are the actual process parameters,
Рнi – standard (specified by technological regulations) parameters;
i – number of parameters;
n – number of measurements.
The considered approaches can also be used in tasks where it is necessary to give a summary assessment of the quality of an enterprise’s work, taking into account many indicators. For their use, a necessary condition is the availability of normative (reference) values with which actual levels of indicators can be compared.
Example 1. Using the method of generalized quality assessment of the State Standard of Russia, check the compliance of the quality of electric lamps with the standard. The average burning time of electric lamps of a certain power manufactured by the company is 420 hours. The reference lifespan is 450 hours. The efficiency has a reference value of 20 lm/W, and the actual efficiency is 19 lm/W.
The actual quality level of manufactured electric lamps is 11.3% lower than the standard.
Example 2. There is data on the quality levels of the same type of automatic washing machines manufactured by Vesta (Vyatka-Alenka) and Ariston companies according to passport data. Give a comparative assessment of the quality levels of machine tools if the weight coefficients of each factor determined by experts are 0.31, 0.29, 0.03, 0.07, 0.3, respectively.
|
Level of quality washing machine |
Units |
"Alenka" |
"Ariston" |
|
Water consumption per main wash cycle | |||
|
Time of the longest washing cycle at 90 0C when filling only with cold water | |||
|
Power consumption | |||
|
Guaranteed shelf life |
In order to determine the relative level of quality of washing machines, a summary quality coefficient is calculated using the method proposed by Professor V.A. Trapeznikov. When calculating coefficients, the nature of the indicators is also taken into account. For “positive” indicators, with increasing values of which the quality increases, formula (2.4) is chosen, and for “negative” indicators, with increasing values of which the quality of the product decreases, the inverse formula is used.
The relative quality level of an automatic washing machine of the Ariston brand is 11% higher than the quality level of an automatic washing machine of the Vyatka-Alenka brand.
Example 3. There are data on the results of measurements of concentrated parameters of the technological process during a work shift.
According to the technological regulations, the standard values are: pressure – 100 kPa, acidity – 6.0.
Determine the summary relative indicator of instability of the technological process using the method of relative linear estimates.
|
Measurement number |
Pressure |
Acidity |
Sum of relative deviations |