Among statistical quality control methods, the most common are the so-called seven quality control tools:

1) Pareto chart;

2) Ishikawa cause-and-effect diagram;

3) control card;

4) histogram;

5) scatter diagram;

6) layering method;

7) check sheets.

Taken together, these methods form an effective system of quality control and analysis methods. The seven simple methods can be used in any sequence, in any combination, in various analytical situations; they can be considered both as an integral system and as individual analysis tools. In each specific case, it is proposed to determine the composition and structure of the working set of methods. The “Seven Tools of Quality Control” are actively used by Japanese firms.

1. Pareto chart allows you to visualize the amount of losses depending on various objects; is a type of bar chart used to visually display the factors under consideration in order of decreasing importance.

In 1897, the Italian economist V. Pareto proposed a formula describing the uneven distribution of benefits. The same idea was graphically illustrated in a diagram by the American economist M. Lorenz in 1907. Both scientists showed that most often the largest share of income or wealth belongs to a small number of people. The famous American quality management specialist J. Juran applied this approach in the field of quality control. This made it possible to divide the factors influencing quality into a few essentially important and numerous non-essential. It turned out that, as a rule, the overwhelming number of defects and associated losses arise due to a relatively small number of reasons. J. Juran called this approach Pareto analysis.

To construct a Pareto chart, the initial data is presented in the form of a table, in the first column of which the analyzed factors are indicated, in the second - absolute data characterizing the number of cases of detection of the analyzed factors in the period under review, in the third - the total number of factors by type, in the fourth - their percentage , in the fifth - the cumulative (accumulated) percentage of cases of detection of factors

The construction of a Pareto chart begins by plotting the data from column 1 on the x-axis, and the data from column 2 on the ordinate axis, arranged in descending order of frequency of occurrence. “Other factors” are always placed last on the y-axis; if the share of these factors is relatively large, then it is necessary to decipher them, highlighting the most significant ones. Based on these initial data, a bar chart is constructed, and then, using the data in column 5 and an additional ordinate indicating the cumulative percentage, a Lorenz curve is drawn. It is possible to construct a Pareto diagram when the data from column 4 is plotted on the main ordinate; in this case, to draw the Lorenz curve, there is no need to include an additional ordinate in the diagram (this is the version of the diagram that is most common in practice).

The defining advantage of the Pareto diagram is that it makes it possible to divide factors into significant (occurring most often) and minor (occurring relatively rarely). For example, analysis of the diagram presented in Fig. (as well as the Lorenz curve), shows that shrinkage cavities, gas porosity and other cracks in cast parts account for 89.5% of all non-conformities. Therefore, work to ensure the quality of parts should begin with the elimination of precisely these inconsistencies.

Drawing a Pareto chart often reveals a pattern called the 80/20 Rule, which is based on the Pareto principle, which states that most effects are caused by relatively few causes. In relation to the analysis of inconsistencies, this pattern can be formulated as follows: usually 80% of detected inconsistencies are associated with only 20% of all possible causes.

In addition to identifying and ranking factors and their significance, the Pareto diagram is successfully used to clearly demonstrate the effectiveness of certain measures in the field of quality assurance: it is enough to construct and compare two Pareto diagrams - before and after the implementation of any measures.

2. Cause and effect diagram proposed in 1953 by K. Ishikawa (“Ishikawa diagram”). A diagram is a graphical arrangement of factors influencing the object of analysis.

The main advantage of the Ishikawa diagram is that it gives a visual representation not only of those factors that influence the object being studied, but also of the cause-and-effect relationships of these factors.

When constructing an Ishikawa diagram, large primary arrows are drawn to the central horizontal arrow depicting the object of analysis, indicating the main factors (groups of factors) influencing the object of analysis. Next, second-order arrows are added to each primary arrow, which, in turn, are followed by third-order arrows, etc., until all the arrows are plotted on the diagram, indicating factors that have a noticeable impact on the object of analysis in a particular situations. Each of the arrows on the diagram, depending on its position, represents either a cause or an effect: the previous arrow in relation to the next one always acts as a cause, and the next one as a consequence.

The main task when constructing a diagram is to ensure correct subordination in the interdependence of factors, as well as its clear design.

When structuring a diagram at the level of primary arrows of factors in many real situations, you can use the rule of the “five M” proposed by Ishikawa himself (materials, machines, methods, measurements, people). This rule is that in general there are five possible causes of certain results associated with causal factors.

A detailed Ishikawa diagram can serve as the basis for drawing up a plan of interrelated measures that provide a comprehensive solution to the problem posed in the analysis.

3. Control card was proposed in 1924 by W. Shewhart. It is built on a form (form), on which a grid of thin vertical and horizontal lines is applied. The selected statistical characteristic of the observed parameter is marked vertically on the map (for example, individual or arithmetic mean value, median, range, etc.), and horizontally - the time or number of the control sample. Thus, on the map of arithmetic average values ​​the following is first drawn: a horizontal central line corresponding to the value of the tolerance center (TC) (at this value the technological operation is considered optimally adjusted); two horizontal lines of the limits of the technological tolerance established by the regulatory documentation (upper - Tv and lower - Tn); two horizontal lines, which are the boundaries for regulating the values ​​of the controlled parameter (upper - Рв and lower - Рн). The control limits limit the range of values ​​of the regulated sample characteristic corresponding to the satisfactory adjustment of the technological operation (if the controlled parameter is specified by a unilateral norm, then only one control limit is plotted on the control chart). For a better perception of the control chart, it is advisable to designate its central line and boundaries in different colors, for example, the central line - green, tolerance limits - red, control boundaries - black.

The control limits are calculated taking into account the accepted distribution of values ​​of the controlled parameter and the additional probability of receiving a false warning signal about an operation disorder. A confidence interval indicates within which limits the true value of a statistical characteristic is expected.

Working with a control chart comes down to the fact that, based on observation of the values ​​of the controlled parameter, it is established whether this parameter is within the control limits, and on the basis of this a decision is made on whether the technological operation is adjusted or not.

The decision to disrupt the operation is made when at least one observation, recorded on the map as a point, goes beyond the regulation boundaries. However, even before the points go beyond the regulation boundaries, the control chart makes it possible to judge emerging violations of the technological operation based on the following signs:

♦ several successive values ​​of the controlled parameter appear near the control boundaries;

♦ the values ​​are distributed on one side of the center line, i.e. the average value shifts relative to the center of the setting (the presence of a systematic deviation is indicated, for example, by the location of seven values ​​in a row above or below the center line, as well as the location of 10 out of 11, 12 out of 14, 14 out of 17 and 16 out of 20 values ​​on one side of the midline);

♦ the values ​​of the controlled parameter are widely scattered;

♦ there is a tendency for the values ​​of the controlled parameter to approach one of the control limits.

Plotting the acceptable values ​​of a parameter determines how often the parameter falls within or outside the acceptable range.

The histogram is constructed in the following sequence:

a) a table of initial data is compiled;

b) the range of the analyzed parameter is estimated;

c) the span width is determined;

d) the starting point of the first interval is established;

e) the final number of intervals is selected.

The appearance of the histogram depends on the sample size, the number of intervals, and the starting point of the first interval. The larger the sample size and the smaller the width of the interval, the closer the histogram is to a continuous curve.

5. Scatter diagram(scatter diagram) is used to identify the dependence of one variable (product quality indicator, technological process parameter, quality costs, etc.) on another. The diagram does not answer the question of whether one variable causes another, but it can clarify whether a cause-and-effect relationship exists in a given case at all and what its strength is.

The most common statistical method for identifying such a relationship is correlation analysis, based on estimating the correlation coefficient. The relationship between the studied quantities can be complete, that is, functional, when the correlation coefficient is equal to one (+1), if the variables simultaneously increase or decrease, and (-1), if when one variable increases, the other decreases. An example of a functional relationship is the hardness of the workpiece material: the higher the hardness, the greater the wear.

In the case where there is no relationship at all, the correlation coefficient is zero. An intermediate case is also possible when the dependence of related quantities is incomplete, since it is distorted by the influence of extraneous additional factors. An illustration of this kind of correlation can be seen in the dependence of workers’ labor productivity on their length of service under the influence of additional factors such as education, health, etc. The greater the influence of these additional factors, the less close the connection between experience and labor productivity

Correlation connections are described by the corresponding equations. In cases where it is necessary to find out the dependence of one parameter on several others, regression analysis is used. To identify the influence of individual factors on the parameter under study, analysis of variance is used, which assumes that the significance of each factor in individual conditions is characterized by its contribution to the variance of the experimental result.

6. Layering (stratification) method used to identify the causes of variations in product characteristics. The method consists in dividing (stratifying) the obtained characteristics depending on various factors: the quality of the source materials, work methods, etc. In this case, the influence of one or another factor on the characteristics of the product is determined, which makes it possible to take the necessary measures to eliminate their unacceptable scatter.

In Fig. 8.7.6 shows an example of stratification of the Pareto diagram by factors A and B with the simplest detailed analysis (“unraveling of connections”) of the diagram. In this case, delamination allows you to get an idea of ​​the hidden causes of defects.

7. Checklists used for control based on qualitative and quantitative characteristics. A control sheet is a paper form on which the names of the controlled indicators are given and their values ​​​​obtained during the control process are recorded.

The following types of checklists are used:

♦ check sheet for recording the distribution of the measured parameter during the production process;

♦ checklist for recording types of nonconformities;

♦ checklist for assessing the reproducibility and performance of the technological process.

Statistical methods play a major role in ensuring product quality.

The purpose of statistical control methods is to exclude random changes in product quality. Such changes are caused by specific reasons that need to be identified and eliminated. Statistical quality control methods are divided into:

statistical acceptance control based on an alternative criterion;

selective acceptance control based on varying quality characteristics;

statistical acceptance control standards;

system of economic plans;

continuous sampling plans;

methods of statistical regulation of technological processes.

It should be noted that statistical control and regulation of product quality are well known in our country. In this area, our scientists have an undoubted priority. Suffice it to recall the works of A.N. Kolmogorov on unbiased assessments of the quality of accepted products based on the results of sampling inspection, development of an acceptance inspection standard using economic criteria.

There are various methods of product quality control, among which statistical methods occupy a special place. Statistical methods of quality control are currently used not only in production, but also in planning, design, marketing, logistics, etc. The sequence of application of the seven methods may be different depending on the goal set for the system. Likewise, the quality control system used does not necessarily have to include all seven methods. There may be fewer, or there may be more, since there are other statistical methods. However, we can say with complete confidence that the seven quality control tools are necessary and sufficient statistical methods, the use of which helps solve 95% of all problems arising in production. Many of the modern methods of mathematical statistics are quite difficult to understand, and even more so for widespread use by everyone participants in the quality management process. Therefore, Japanese scientists selected from the entire set seven methods that are most applicable in quality control processes. The merit of the Japanese is that they provided simplicity, clarity, visualization of these methods, turning them into quality control tools that can be understood and effectively used without special mathematical training. At the same time, despite their simplicity, these methods allow you to maintain a connection with statistics and give professionals the opportunity to improve them if necessary. So, the seven main methods or tools of quality control include the following statistical methods:

check sheet

· histogram

· scatter diagram

Pareto chart

· stratification (stratification)

· Ishikawa diagram (cause-and-effect diagram)

· control card

Rice. 2.

The listed quality control tools can be considered both as individual methods and as a system of methods that provide comprehensive control of quality indicators. They are the most important component of a comprehensive Total Quality Management control system. The implementation of the seven quality control tools should begin with training in these methods for all participants in the process. For example, the successful implementation of quality control tools in Japan was facilitated by training company management and employees in quality control techniques. Speaking about seven simple statistical methods of quality control, it should be emphasized that their main purpose is to control the ongoing process and provide the process participant with facts for adjusting and improving the process. Knowledge and practical application of the seven quality control tools underlie one of the most important requirements of TQM - constant self-monitoring. In industries, statistical methods are used to analyze product and process quality. Quality analysis is an analysis by which, using data and statistical methods, the relationship between the exact and the replaced quality characteristics is determined. Process analysis is an analysis that allows us to understand the relationship between causal factors and results such as quality, cost, productivity, etc. Process control involves identifying causal factors that affect the smooth functioning of the production process. Quality, cost and productivity are the results of the control process. Statistical methods for product quality control are currently becoming increasingly recognized and widespread in industry. Scientific methods of statistical control of product quality are used in the following industries: mechanical engineering, light industry, and public services. The main objective of statistical control methods is to ensure the production of usable products and the provision of useful services at the lowest cost. Statistical methods of product quality control provide significant results in the following indicators: · improving the quality of purchased raw materials; · saving of raw materials and labor; · improving the quality of manufactured products; · reduction of control costs; · reduction in the number of defects; · improving the relationship between production and consumer; · facilitating the transition of production from one type of product to another. The main task is not just to increase the quality of products, but to increase the quantity of products that would be suitable for consumption. Two basic concepts in quality control are the measurement of controlled parameters and their distribution. In order to judge the quality of a product, it is not necessary to measure parameters such as the strength of the material, paper, weight of the item, quality of coloring, etc. The second concept of distribution of values ​​of a controlled parameter is based on the fact that there are no two parameters of the same products that are absolutely identical in value; As measurements become more precise, small discrepancies are found in the parameter measurements. The variability of the “behavior” of the controlled parameter is of two types. The first case is when its values ​​constitute a set of random variables formed under normal conditions; the second when the set of its random variables is formed under conditions different from normal under the influence of certain reasons. Personnel managing the process in which the controlled parameter is formed must determine from its values: firstly, under what conditions they were obtained (normal or different from them); and if they are obtained under conditions other than normal, then what are the reasons for the violation of normal process conditions. Then a control action is taken to eliminate these causes. When applying statistical control methods, it is important to establish what pattern the distribution of controlled product parameters obeys (the Gaussian normal distribution curve, the distribution characteristic of the Maxwell distribution curve, etc.). A change in the value of a specific controlled parameter of a product or technological mode is manifested in a change in the distribution function. Comparing the actual distribution function with the normal one allows you to control the technological process or product quality. The general scheme of statistical quality control consists of the following stages: 1) small samples of products are selected periodically or according to a special algorithm; 2) sample products are checked to determine the value of a specific feature X for each product; 3) selected values ​​of X (X 1, X 2, ..., X n) are entered into a control chart, which indicates the permissible specific limits of change in attribute X; 4) based on the distribution of points X on the control chart relative to neutral boundaries, a decision is made on the suitability of products or defects during acceptance statistical control or on the need for intervention in technological process with statistical process control. The statistical quality control map is shown in Fig. 3.

Rice. 3.

The horizontal axis indicates the sample numbers (per shift, day, week, month); the vertical axis displays the size of the selected characteristic X, the controlled parameter, the lower and upper tolerance limits (NGD, IOP); lower and upper warning limits (LPKG, VPKG).

Test No. 2

The second test aims to practically solve problems related to various issues of quality management.

Task No. 1

The manufactured product is subject to selective quality control. Calculate the number 1 in the sample<А<9 (7) дефектных изделий, если вероятность появления годного изделия равна В= 0,93, а выборка равна N=21. Построить графики плотности вероятности и кумулятивной вероятности. Дано:

B = 0.93 - probability of a suitable product appearing.

N = 21 - sample number.

1. A = ? - number of defective products, if

Build:

1. Probability density plot.

2. Cumulative probability plot.

To solve this problem I will use Bernoulli's formula:

1. According to our data, we calculate the probability:

N = 21 - sample number;

B = p = 0.93 - probability of appearance of a suitable product;

q = 1 - 0.93 = 0.07 - probability of defects.

1,47*0,234=0,344;

210 * 0,0049 * 0,252 = 0,2593;

1330 * 0,000343 * 0,2708 = 0,1235;

5985 * 0,00002401 * 0,2912 = 0,04185

20349 * 0,00000011764 * 0,3131 = 0,00075;

2. We calculate the cumulative probability, i.e. accumulation of probability according to the formula:

A is the number of defective products for which the calculation is performed, then knowing the values, we can find

3. Let’s enter all the received data into the table:

4. Let's build a probability density graph and a cumulative probability graph:

Task No. 2

During the metrological certification of a voltmeter with the declared accuracy class A=1, 10 measurements of the reference value U=1.5 were performed, with a final measurement limit of N=2. Determine compliance with the accuracy class declared during production, using the largest values ​​of the relative and reduced error. Assess the quality of multiple measurements by processing the measurement result. The changes are considered direct, equally accurate, free from correction.

Given: A = 1 - declared accuracy class.

N = 2 is the final measurement limit.

Define:

1. Compliance with the accuracy class declared during production, using the largest values ​​of relative and reduced errors.

2. Assess the quality of multiple measurements by processing the measurement result.

1. Determine the relative error and select the max value:

2. Determine the reduced error and determine max

meaning:

max 5.7 = 0.015.

3. max 5.7 = 0.02 and max 5.7 = 0.015< A (A = 1)

4. Let's determine the standard unit deviation.

Average value of x.

5. Define multiple deviation:

Introduction

The most important source of growth in production efficiency is the constant improvement of the technical level and quality of products. Technical systems are characterized by strict functional integration of all elements, so they do not contain secondary elements that can be poorly designed and manufactured. Thus, the current level of development of scientific and technological progress has significantly tightened the requirements for the technical level and quality of products in general and their individual elements. A systematic approach allows you to objectively select the scale and direction of quality management, types of products, forms and methods of production that provide the greatest effect of the efforts and funds spent on improving product quality. A systematic approach to improving the quality of products makes it possible to lay the scientific foundations of industrial enterprises, associations, and planning bodies.

In industries, statistical methods are used to analyze product and process quality. Quality analysis is an analysis by which, using data and statistical methods, the relationship between the exact and the replaced quality characteristics is determined. Process analysis is an analysis that allows us to understand the relationship between causal factors and results such as quality, cost, productivity, etc. Process control involves identifying causal factors that affect the smooth functioning of the production process. Quality, cost and productivity are the results of the control process.

Statistical methods for product quality control are currently becoming increasingly recognized and widespread in industry. Scientific methods of statistical control of product quality are used in the following industries: mechanical engineering, light industry, and public services.

The main objective of statistical control methods is to ensure the production of usable products and the provision of useful services at the lowest cost.

Statistical methods for product quality control provide significant results in the following indicators:

· improving the quality of purchased raw materials;

· saving of raw materials and labor;

· improving the quality of manufactured products;

· reduction of control costs;

· reduction in the number of defects;

· improving the relationship between production and consumer;

· facilitating the transition of production from one type of product to another.

The main task is not just to increase the quality of products, but to increase the quantity of products that would be suitable for consumption.

Two basic concepts in quality control are the measurement of controlled parameters and their distribution. In order to judge the quality of a product, it is not necessary to measure parameters such as the strength of the material, paper, weight of the item, quality of coloring, etc.

The second concept - distribution of values ​​of a controlled parameter - is based on the fact that there are no two parameters of the same products that are absolutely identical in value; As measurements become more precise, small discrepancies are found in the parameter measurements.

The variability of the “behavior” of the controlled parameter is of 2 types. The first case is when its values ​​constitute a set of random variables formed under normal conditions; the second is when the set of its random variables is formed under conditions different from normal under the influence of certain reasons.

1. Statistical acceptance control based on an alternative criterion

The consumer, as a rule, does not have the opportunity to control the quality of products during the manufacturing process. However, he must be sure that the products he receives from the manufacturer meet the established requirements, and if this is not confirmed, he has the right to demand that the manufacturer replace the defective product or eliminate the defects.

The main method of monitoring raw materials, supplies and finished products supplied to consumers is statistical acceptance control of product quality.

Statistical acceptance control of product quality– selective control of product quality, based on the use of mathematical statistics methods to check product quality to established requirements.

If the sample size becomes equal to the volume of the entire controlled population, then such control is called continuous. Complete control is possible only in cases where the quality of the product does not deteriorate during the control process, otherwise selective control, i.e. control of a certain small part of the total production becomes forced.

Complete control is carried out if there are no special obstacles to this, in the event of the possibility of a critical defect, i.e. a defect, the presence of which completely precludes the use of the product for its intended purpose.

All products can also be tested under the following conditions:

· the batch of products or material is small;

· the quality of the input material is poor or nothing is known about it.

You can limit yourself to checking part of the material or products if:

· the defect will not cause serious equipment malfunction and does not pose a threat to life;

Products are used in groups;

· Defective products can be detected at a later stage of assembly.

In the practice of statistical control, the general share q is unknown and should be estimated based on the results of control of a random sample of n products, of which m are defective.

A statistical control plan is understood as a system of rules indicating methods for selecting products for testing, and the conditions under which a batch should be accepted, rejected, or continued control.

There are the following types of plans for statistical control of a batch of products based on an alternative criterion:

one-stage plans, according to which, if among n randomly selected products the number of defective m is no more than the acceptance number C (mC), then the batch is accepted; otherwise the batch is rejected;

two-stage plans, according to which, if among n1 randomly selected products the number of defective m1 is no more than the acceptance number C1 (m1C1), then the batch is accepted; if m11, where d1 is the rejection number, then the batch is rejected. If C1 m1 d1, then a decision is made to take a second sample of size n2. Then, if the total number of products in two samples (m1 + m2) is C2, then the batch is accepted, otherwise the batch is rejected according to the data of the two samples;

multi-stage plans are a logical continuation of two-stage plans. Initially, a batch of volume n1 is taken and the number of defective products m1 is determined. If m1≤C1, then the batch is accepted. If C1p m1 d1 (D1C1+1), then the batch is rejected. If C1m1d1, then a decision is made to take a second sample of size n2. Let there be m2 defectives among n1 + n2. Then, if m2c2, where c2 is the second acceptance number, the batch is accepted; if m2d2 (d2 c2 + 1), then the batch is rejected. When c2 m2 d2 the decision is made to take the third sample. Further control is carried out according to a similar scheme, with the exception of the last k-th step. At the k-th step, if among the inspected products of the sample there were mk defective and mkck, then the batch is accepted; if m k ck, then the batch is rejected. In multi-stage plans, the number of steps k is assumed to be n1 =n2=…= nk;

sequential control, in which the decision on the controlled batch is made after assessing the quality of samples, the total number of which is not predetermined and is determined in a process based on the results of previous samples.

Single-stage plans are simpler in terms of organizing production control. Two-stage, multi-stage and sequential control plans provide greater accuracy of decisions with the same sample size, but they are more complex in organizational terms.

The task of selective acceptance control actually comes down to statistical testing of the hypothesis that the proportion of defective products q in a batch is equal to the permissible value qo, i.e. H0:q = q0.

The goal of choosing the right statistical control plan is to make errors of the first and second types unlikely. Let us recall that errors of the first type are associated with the possibility of mistakenly rejecting a batch of products; errors of the second type are associated with the possibility of mistakenly missing a defective batch.

2. Statistical acceptance control standards

For the successful application of statistical methods for product quality control, the availability of appropriate guidelines and standards, which should be available to a wide range of engineering and technical workers, is of great importance. Standards for statistical acceptance control provide the ability to objectively compare quality levels of batches of the same type of product both over time and across different enterprises.

Let us dwell on the basic requirements for standards for statistical acceptance control.

First of all, the standard must contain a sufficiently large number of plans with different operational characteristics. This is important, as it will allow you to choose control plans taking into account the characteristics of production and consumer requirements for product quality. It is desirable that the standard specify different types of plans: single-stage, two-stage, multi-stage, sequential control plans, etc.

The main elements of acceptance control standards are:

1. Tables of sampling plans used in normal production conditions, as well as plans for enhanced control in conditions of disturbances and to facilitate control when achieving high quality.

2. Rules for selecting plans taking into account control features.

3. Rules for the transition from normal control to enhanced or lightweight control and the reverse transition during the normal course of production.

4. Methods for calculating subsequent assessments of quality indicators of the controlled process.

Depending on the guarantees provided by acceptance control plans, the following methods for constructing plans are distinguished:

set the values ​​of the supplier's risk and the consumer's risk and put forward the requirement that the operational characteristic P(q) pass through approximately two points: q0, α and qm, where q0 and qm are, respectively, acceptable and rejection quality levels. This plan is called a compromise plan, since it ensures protection of the interests of both the consumer and the supplier. For small values ​​of α and β, the sample size should be large;

select one point on the operating characteristic curve and accept one or more additional independent conditions.

The first system of statistical acceptance inspection plans to be widely used in industry was developed by Dodge and Rolig. Plans for this system provide for continuous control of products from rejected batches and the replacement of defective products with suitable ones.

The American standard MIL-STD-LO5D has become widespread in many countries. The domestic standard GOST-18242–72 is close in structure to the American one and contains plans for one-stage and two-stage acceptance inspection. The standard is based on the concept of acceptable quality level (AQL) q0, which is considered as the maximum percentage of defective products permissible by the consumer in a batch manufactured during normal production. The probability of rejecting a batch with a share of defective products equal to q0 is small for standard plans and decreases as the sample size increases. For most plans it does not exceed 0.05.

When inspecting products based on several criteria, the standard recommends classifying defects into three classes: critical, significant and minor.

3. Control cards

One of the main tools in the vast arsenal of statistical quality control methods is control charts. It is generally accepted that the idea of ​​the control chart belongs to the famous American statistician Walter L. Shewhart. It was proposed in 1924 and described in detail in 1931. Initially, they were used to record the results of measurements of the required properties of products. If the parameter went beyond the tolerance range, it indicated the need to stop production and adjust the process in accordance with the knowledge of the specialist managing the production.

This provided information about when someone, on what equipment, received defects in the past.

However, in this case, the decision to adjust was made when the defect had already been received. Therefore, it was important to find a procedure that would accumulate information not only for retrospective research, but also for use in decision making. This proposal was published by the American statistician I. Page in 1954. Maps that are used in decision making are called cumulative.

A control chart consists of a center line, two control limits (above and below the center line), and characteristic (performance indicator) values ​​plotted on the map to represent the condition of the process.

At certain periods of time, n manufactured products are selected (all in a row; selectively; periodically from a continuous flow, etc.) and the controlled parameter is measured.

The measurement results are plotted on a control chart, and depending on these values, a decision is made to adjust the process or to continue the process without adjustments.

Signals of a possible problem with the technological process can be:

the point goes beyond the control limits (point 6); (the process got out of control);

the location of a group of consecutive points near one control boundary, but not going beyond it (11, 12, 13, 14), which indicates a violation of the level of equipment settings;

strong scattering of points (15, 16, 17, 18, 19, 20) on the control map relative to the center line, which indicates a decrease in the accuracy of the technological process.

Upper limit

Central line

Lower limit

6 11 12 13 14 15 16 17 18 19 20 Sample number

Conclusion

Increasing development of a new economic environment for our country of reproduction, i.e. market relations dictates the need for constant improvement of quality using all possibilities, all achievements of progress in the field of technology and organization of production.

The most complete and comprehensive quality assessment is ensured when all the properties of the analyzed object are taken into account, manifested at all stages of its life cycle: during manufacturing, transportation, storage, use, repair, etc. service.

Thus, the manufacturer must control the quality of the product and, based on the results of sampling, judge the state of the corresponding technological process. Thanks to this, he promptly detects problems in the process and corrects them.

List of used literature

1. GembrisS. Herrmann J., “Quality Management”, Omega-L SmartBook, 2008.

2. Shevchuk D.A., “Quality Control”, Gross-Media., M., 2009.

3. Electronic textbook “Quality Control”

Statistical quality control

Statistical quality control means control in which not all products of a manufactured batch are checked, but only a sample from it. At the same time, the quality of the entire batch is judged based on the results of the control.

There are two types of statistical control: control on a qualitative basis, the most common special case of which is control on an alternative basis, and control on a quantitative basis.

When monitoring by alternative criteria, all products in a batch are divided into two groups: suitable and defective. The batch is assessed based on the percentage of defective products in the sample.

The main characteristic of the quality of a batch when monitoring by an alternative criterion is the proportion of defective products in the batch:

where M is the number of defective products in the batch;

N - batch size.

When checking a sample of volume N, M defective products are identified. Based on the value of q, a decision is made on acceptance or rejection of the batch.

Basic terms of statistical control

Unit of production is a separate copy of piece products or a quantity of non-piece or piece products determined in the established order.

Note. Products may be completed or unfinished, in the process of being manufactured, mined or repaired.

Product is a unit of industrial product, the quantity of which can be calculated in pieces or copies.

A controlled batch of products is a batch intended to control a set of units of production of the same name, standard rating or standard size and design, produced over a certain period of time under the same conditions.

Note. Manufactured products may be in the process of manufacturing, mining or repair.

Batch volume - the number of units of product that make up the batch.

Product flow - products of the same name, standard or standard size and design, in motion on the production line.

Sampling - a product or a certain set of products selected for control from a batch or product flow.

Note. Depending on the degree of product completion, completed and unfinished production items, including blanks, may be classified as products.

Sample size - the number of products that make up the sample.

Instantaneous sampling is a sampling from the product flow, which consists of the products that were last produced at the time of selection within a fairly short time interval.

A pooled sample is a sample consisting of a series of instantaneous samples.

Random sample - a sample, when compiled for any product in the controlled population, they provide the same probability of selection.

Purposeful sampling is a sampling in which items are selected with a specific tendency to change the probability of selecting defective items.

Systematic sampling is a sample in which the inclusion of products is determined by its number or position in a pre-ordered controlled population.

A representative sample (RDP representative sample) is a sample in which such a number of products are selected from each part of the controlled population to sufficiently reflect the properties of this population as a whole.

Sample - a certain amount of non-piece products selected for control.

Sample volume - the number of units of non-piece products that make up the sample.

A spot sample (NDS - one-time sample) is a sample taken simultaneously from a certain part of a non-piece product.

A combined sample (NPS - total sample) is a sample consisting of a series of point samples.

Sampling period - the time interval between the moments of taking adjacent samples or samples from the product flow.

Sampling control is control in which a decision on the quality of the controlled product is made based on the results of checking one or more samples or samples from a batch or stream of products.

Statistical acceptance control of product quality (statistical acceptance control) - selective control of product quality, based on the use of mathematical statistics methods to verify compliance of product quality with established requirements.

The share of defective units of production is the ratio of the number of defective units of production to the total number of units of production in the batch.

Defectivity level is the proportion of defective units of production or the number of defective units per hundred units of production.

Acceptance number is a control standard, which is a criterion for acceptance of a batch of products and is equal to the maximum number of defective units (defects) in a sample or sample in the case of statistical acceptance control.

Rejection number is a control standard, which is a criterion for rejecting a batch of products and is equal to the minimum number of defective units (defect) in a sample or sample in the case of statistical acceptance control.

The decisive rule is an instruction intended for making a decision regarding the acceptance of a batch of products based on the results of its control.

Note. To make a decision, a certain set of decision rules may be provided.

A control plan is a set of data on the type of control, the volume of a controlled batch of products, samples or samples, control standards and decisive rules.

Statistical acceptance control scheme (acceptance control scheme) - a complete set of statistical acceptance control plans in combination with a set of rules for applying these plans,

Operational characteristic of a statistical acceptance control plan (operational characteristic) - expressed by an equation, graph or table and conditioned by a specific control plan, the dependence of the probability of acceptance on a value characterizing the quality of this product.

Supplier risk is the probability of rejecting a batch of products that has an acceptable level of defectiveness.

Consumer risk is the probability of accepting a batch of products that has a defective level of rejection.

Single-stage control (NDP - single-sample control; single-sample control; single-sample control) - statistical acceptance control, characterized by the fact that the decision regarding the acceptance of a batch of products is made based on the results of control of only one sample or sample.

Reduced control (RDP reduced control) - statistical acceptance control used in the case when the result of control of a given number of previous batches of products provides sufficient grounds for the conclusion that the actual level of defects is lower than the acceptance level, and is characterized by a smaller sample size than with normal control .

Enhanced control is a statistical acceptance control used in the case when the results of control of a given number of previous batches of products provide sufficient grounds for the conclusion that the actual level of defects is higher than the acceptance level, and is characterized by more stringent control standards than with normal control.

Selection of samples for testing is carried out using various methods. In the first method of submitting products for control, units of products subject to control are ordered and numbered with continuous numbering; they are submitted for control in the form of a certain limited set, formed independently of the production process. From this population, a sample is selected using a uniform random number generator or a table of uniformly distributed random numbers. A random number generator can be a rotating circle with numbers printed on the division points. The number of division points is determined by the required number of random numbers, i.e. the number of units of product in the controlled batch. Another version of the generator is a lottery machine with a number of renumbered balls, the number of which is equal to the number of units of the controlled lot.

There are computational procedures for obtaining uniformly distributed random numbers, including those based on the use of tables of uniformly distributed random numbers.

A table of uniformly distributed random numbers is the result of a statistical experiment recorded in the form of a table, carried out using a sensor (generator) of uniformly distributed random numbers.

Suppose we have a table of random numbers uniformly distributed between 0 and 10,000.

To get random numbers X 4 , evenly distributed in the range from 0 to 1, you need to divide all these numbers by 10,000.

Random numbers uniformly distributed on the interval (0, b) are determined by the formula

As numbers of products included in the sample, you need to take the whole part of the obtained random numbers [yy]. With each new selection of samples, you need to randomly select the first of these numbers, and then the next n - 1 number after it, n the sample size. If some numbers are repeated, then you need to increase the number of selected random numbers by the number of repetitions.

The procedure for randomly selecting products into a sample using tables of uniformly distributed random numbers consists of renumbering all products of the batch subject to control, compiling a relatively short series of random numbers in the range from 1 to N, where N is the volume of the batch, and selecting the first n different numbers from this series . These numbers determine the products included in the sample of volume N.

Examples of products submitted for control using the “row” method: engines, refrigerators, washing machines.

The second way to submit products for inspection is “scatter”.

In this case, when selecting units for the sample, the “method of greatest objectivity” is used. When applying this method, the sample includes units of production from different parts of the controlled lot.

The third way of presenting products for control is called “flow”. In this case, units of products enter control in a continuous flow simultaneously with the release of products. Product units are ordered, you can find a unit of any given number. This method is typical for the case when products are controlled immediately after they come off the assembly line.

In this case, the method of systematically selecting units of production in the sample is used. The next task after selecting samples for testing is choosing a control plan, i.e. establishing the volume of the controlled batch, sample size, acceptance number, and decisive rule. This problem is solved by the methods considered, taking into account the established values ​​of errors of the first and second types, as well as economic factors.

Basic standardized concepts used in quality control, including certification.

Permissible deviation - deviation of the value of a product quality indicator or its parameter from the nominal value, which is within the limits established by regulatory documentation.

A defect is each individual non-compliance of a product with the requirements established by regulatory documentation.

An obvious defect is a defect for the detection of which the relevant rules, methods and controls are provided in the regulatory documentation.

A hidden defect is a defect for which the regulatory documentation does not provide the necessary rules, methods and controls to identify it.

A critical defect is a defect in the presence of which the use of the product for its intended purpose is practically impossible or is excluded in accordance with safety requirements.

A significant defect is a defect that significantly affects the intended use of the product or its durability, but is not critical.

A minor defect is a defect that does not significantly affect the intended use of the product or its durability.

The division of defects into critical, significant and minor is used when analyzing the level of product quality and its manufacturing technology.

A correctable defect is a defect, the elimination of which is technically possible and economically feasible.

An irreparable defect is a defect the elimination of which is technically impossible or economically impractical.

A defective unit of production is a unit of production that has at least one defect.

A defective product is a product that has at least one defect.

A defect is a defective unit of production or a set of such units.

A correctable marriage is a marriage in which all defects are correctable.

An irreparable defect is a defect consisting of such units of production, each of which has at least one irreparable defect.

Product grade - gradation of a certain type of product according to one or more quality indicators, established by regulatory documentation.

Statistical control plans. The product manufacturer is obliged to ensure that quality indicators comply with the values ​​​​established in the specifications. In the future, during quality control, those products whose parameter is lower (or higher, or goes beyond the upper or lower limits) of the established value are considered defective.

As already noted, a parameter is usually understood as a target indicator. The use of this term is traditional for products in many industries: electrical and radio elements, engines, mechanical parts. In addition to the parameter going beyond the established limits, the cause of product defects can be design and manufacturing defects, for example, dents on the body, car doors that won’t close, non-functioning indicators, etc.

In a comprehensive product quality management system, statistical control methods are among the most progressive. They are based on the application of mathematical statistics methods to systematic control of the quality of products and the state of the technological process in order to maintain its stability and ensure a given level of quality of manufactured products.

Statistical methods for monitoring production and quality of products and services have the following advantages over other methods:

1) are preventive in nature;

2) in many cases they allow a reasonable transition to selective control and thereby reduce the labor intensity of control operations;

3) provide a visual representation of the dynamics of changes in product quality and the mood of the production process, which allows timely measures to prevent defects not only by inspectors, but also by workshop workers - workers, foremen, technologists, adjusters, foremen at the production stage.

Statistical methods for managing the quality of products and services assume:

1) statistical analysis of the accuracy of the technological process in order to bring it to the required tune, accuracy and statistically stable state;

2) ongoing monitoring in order to regulate and maintain the process in a state that ensures the specified quality parameters;

3) selective statistical acceptance control of the quality of finished products.

Statistical analysis of the accuracy of technological processes is a one-time examination of process reliability by studying the quality characteristics of a large number of products processed under certain conditions in a given operation. This type of analysis makes it possible to determine the actual accuracy of the process and compare it with the specified one, evaluate the quality and stability of the process setup, identify the probable percentage of defects, and determine economically feasible tolerances.

The most common methods for statistical analysis of the accuracy of technological processes are:

· comparison of average parameter values ​​with nominal values;

· comparison of variances;

· assessment of correlation coefficients;

· regression analysis, etc.

Method for comparing average parameter values ​​with nominal values used in cases where it is necessary to establish the compliance of a manufactured product with a standard and in other cases when comparing the values ​​of the same quality indicators for several groups of products.

Variance Comparison Method used in cases where it is necessary to characterize the variability of quality indicators, their dispersion depending on the processing method or other factors.

Correlation coefficient used when assessing the degree of dependence of quality indicators on other indicators.

TO regression analysis are used in cases of assessing a quality indicator based on the results of observations of other indicators.

Statistical control of a technological process is an adjustment of the values ​​of technological process parameters based on the results of selective monitoring of the parameters of manufactured products in order to ensure the required level of quality. In the process of statistical regulation of a technological process, a small quantity (5–10 units) of manufactured products is periodically checked at a specific operation, the statistical quality parameter corresponding to the distribution is calculated and compared with its nominal value. This control ensures continuous monitoring of the stability of the operation and uniformity of quality, which makes it possible to promptly signal an upcoming deviation and thereby prevent the occurrence of defects and defects, ensuring a given level of product quality.

The distribution of a qualitative parameter can be represented in the form of a normal distribution curve (Figure 1), subject to the law of normal distribution of random variables:

Where y– probability density or frequency of occurrence of a random variable;

X– value of a random variable;

– center of distribution (grouping) of deviations, at which the value at greatest;

– standard deviation of a random variable X.

Figure 1 – Normal distribution curve of random variables

Here are the most important statistical characteristics of the law of normal distribution:

1) arithmetic mean value of a qualitative characteristic, characterizing the accuracy of the process,

Where n− number of units of products in the sample (number of measurements);

x i− measurement of the controlled parameter i-th product in the sample;

2) standard deviation of a random variable (the value of a qualitative parameter characterizing the magnitude of the field of actual dispersion of the dimensions of the controlled parameter),

; (3)

3) range of dispersion of qualitative characteristics R, which is the difference between the largest and smallest actual dimensions,

The control results (calculation of the given characteristics) are displayed graphically on the statistical control map (Figure 2). Based on the obtained parameters, the process is controlled and decisions are made on the quality of products produced during the period between two samples.

Number of samples

Control parameters

Rejection zone

R

2,75

3,25

2,25

3,25

2,75

2,75

2,25

2,25

C = 4.2

TBR

C = 3,864

PBR 4

δ’=4.2

C = 0.479

PHR 1

C=0

THR

Figure 2– Map of statistical quality control of capacitors

The control chart is intended for statistical control based on one quality indicator. In its upper part the values ​​are marked with dots arithmetic average quality indicators X . Four boundaries are drawn here: two external ones, limiting the tolerance field, − T in (upper technical tolerance) and T n (lower technical tolerance), outside of which there is a defective zone, and two internal ones - R in (upper warning tolerance) and R n (lower warning tolerance), between which is the nominal size of the controlled parameter R nom.

External boundaries T in and T n are determined based on the permissible relative value of deviation of the controlled parameter from the nominal value:

T in = X nom + ∆ X f; (5)

T n = X nom − ∆ X f, (6)

Where ± ∆X f - permissible absolute value of deviation from the nominal size,

where is the permissible deviation from the nominal value, %.

Internal boundaries are determined by the formulas:

; , (8)

where is the tolerance field for the value of the parameter being studied (based on the lower and

upper limits of the nominal value);

n– number of product units in the sample.

The arithmetic mean value of the parameter being studied in j th sample

Where x i– value of the controlled parameter i-th study in j th sample.

Position of span control lines R V R And R n R determined by the formulas:

R V R = V 1 d; (10)

R n R = V 2 d, (11)

Where V i and V 2 are taken from tables compiled on the basis of correlation analysis of the measured parameter.

Below are the results of sample measurements (5−10 products) and the arithmetic average for each sample X.At the bottom of the map, for each sample number, the values ​​of the range of variation are plotted and a lower solid boundary is plotted (usually T n R is taken equal to zero, and T in R is equal to the tolerance field), the upper limit of regulation of the ranges P in R (limiting the zone of permissible values ​​of the ranges R in samples), as well as the solid line T in R (upper tolerance limit).

The technological process proceeds satisfactorily if the arithmetic average values ​​of the samples do not exceed the regulation limits R in and R n , and the scope R don't go beyond their borders T V R. In this case, the entire batch prepared between the current and previous samples is considered valid and is removed from the workplace. If a defect is detected in the sample or statistical analysis indicates the possibility of its occurrence in a given state of the technological process, all products accumulated at the machine over the last period of time are subject to sorting, and the machine is stopped for readjustment.

Precautionary boundaries R in and R n are established in such a way that the departure of certain values ​​beyond these limits under the influence of errors that disrupt the normal course of the process does not yet mean the appearance of a defect, but only a preliminary signal of the possibility of its occurrence if these errors are not immediately eliminated. In such cases, the inspector, marking the obtained values ​​on the map and comparing them with the position of the regulation boundaries, must warn the administration of the site or workshop about the possibility of defects and the need to make adjustments to the equipment.

From the above example it is clear that in the period between the first and third samples there was a systematic detuning of the equipment. As a result, in the third sample it was found that the value X exceeded the permissible value R V . The process has been stopped, which is indicated on the card with a sign (↓) , and the equipment was reconfigured. The parts manufactured between the second and third samples were subjected to continuous inspection.

After the resumption, the process proceeded within the established limits, however, in the eighth sample it was discovered that the range R exceeded the permissible value T V R. The equipment was stopped again (↓). Parts manufactured between the seventh and eighth samples were subjected to continuous inspection. After identifying and eliminating random factors that worsened product quality, the process was resumed and continued within the precautionary limits until the eleventh sample.

Based on the results of calculations (15) – (17), the conclusion is drawn: if l f < l d, then the process setup is good if l f > l d - unsatisfactory.

Statistical acceptance control of products is used as a selective method when accepting large batches of products, raw materials, materials, and semi-finished products. It is based on the use of mathematical statistics methods to verify that product quality meets the established standard. Based on the quality of the sample taken for control, the quality of the entire batch can be assessed with sufficient reliability.

The advantages of acceptance statistical control are a reduction in the labor intensity of control compared to 100% product inspection, guaranteed provision of the specified product quality, and reliability of assessment of the specified quality level.

Two methods can be used for statistical acceptance control:

1) control based on an alternative criterion, when the share of defects in the sample is taken as a quality indicator;

2) control based on quantitative characteristics, when the statistical characteristics of the distribution of the measured parameter in the sample are determined (average value and variance σ), and the quality of the entire batch of products is assessed based on the obtained values.

During acceptance control, the actual values ​​of the measured parameter for all products in the sample and the arithmetic average values ​​of these parameters are determined by quantitative criteria. X and variance d, after which inequalities (15) – (17) are solved.

If all inequalities are true, the game is accepted. Otherwise, the batch goes to sorting. The advantage of this method is a significantly smaller sample volume with the same reliability of the batch assessment (sample volume is reduced by 3-10 times), which is especially important for control, which is associated with the destruction of products.