Shewhart control charts. Algorithm for constructing Shewhart control charts Example of constructing a Shewhart chart

Plan:

10.1 Basics of Shewhart control charts

10.2 Types of Shewhart control charts

10.1 Basics of Shewhart control charts

The task of statistical process control is to ensure and maintain processes at an acceptable and stable level, ensuring that products and services meet established requirements. The main statistical tool used for this is the control chart. The control chart method helps determine whether a process has actually reached or remains in a statistically controlled state at a properly specified level, and then maintain control and a high degree of uniformity of the critical characteristics of a product or service by continuously recording product quality information during the production process. The use of control charts and their careful analysis lead to better understanding and improvement of processes.

Shewhart control charts (SCCH) are the main tool for statistical quality management. The CCS is used to compare the information obtained from samples about the current state of the process with control limits that represent the limits of the process’s own variability (scatter). CCS is used to assess whether a production process, service process or administrative control process is or is not in a statistically controlled state. Initially, KKSh were developed for use in industrial production. Currently, they are widely used in the service sector and other fields.

Control card is a graphical way of presenting and comparing information based on a sequence of samples reflecting the current state of the process, with boundaries established on the basis of the inherent variability of the process.

Control chart theory distinguishes two types of variability. The first type is variability due to “random (ordinary values), due to a countless variety of causes that are constantly present, which are not easy or impossible to identify. Each of these causes represents a very small proportion of the total variability, and none of them is significant in itself. However, the sum of all these causes is measurable and is assumed to be intrinsic to the process. Eliminating or reducing the influence of common causes requires management decisions and the allocation of resources to improve the process and system. The second type is real change in the process. They may be the result of some identifiable causes that are not inherent to the process internally, and can be eliminated. These identifiable causes are considered “non-random” or “special” causes of change. These may include tool failure, insufficient uniformity of material, production or control equipment, personnel qualifications, failure to follow procedures, etc.

The purpose of control charts is to detect unnatural variations in data from repeated processes and to provide criteria for detecting a lack of statistical control. The process is in a statistically controlled state if the variability is caused only by random reasons. In determining this acceptable level of variability, any deviation from it is considered to be the result of special causes that must be identified, eliminated or mitigated.

The Shewhart chart requires data obtained selectively from the process at approximately equal intervals. Intervals can be set either by time (eg hourly) or by quantity of product (each batch). Typically, each subgroup consists of the same type of units of products or services with the same controlled indicators, and all subgroups have equal volumes. For each subgroup, one or more characteristics are determined, such as the arithmetic mean of the subgroup and the range of the subgroup R or the sample standard deviation S. The Shewhart map is a graph of the values ​​of certain characteristics of subgroups depending on their numbers. It has a center line (CL) corresponding to the reference value of the characteristic. When assessing whether a process is in a statistically controlled state, the arithmetic mean of the data under consideration is usually used as a reference. In process control, the reference is the long-term value of the characteristic established in the technical specifications, or its nominal value based on previous information about the process, or the intended target value of the characteristic of the product or service. The Shewhart chart has two statistically definable control limits around the center line, called the upper control limit (UCL) and the lower control limit (LCL) (Figure 9).

Sample number

Figure 9 - View of the control card

The control boundaries on the Shewhart map are located at a distance of 3 from the center line, where - general standard deviation of the statistics used. Variation within subgroups is a measure of random variation. To get an estimate calculate the sample standard deviation or multiply the sample range by an appropriate factor. This measure does not include between-group variation and only assesses variability within subgroups.

Limits ±3 indicate that about 99.7% of the subgroup characteristic values ​​will fall within these limits, provided that the process is in a statistically controlled state. In other words, there is a risk of 0.3% (or an average of three in a thousand cases) that the plotted point will be outside the control limits when the process is stable. The word “approximately” is used because deviations from underlying assumptions, such as the distribution of the data, will affect the probability values.

Some consultants prefer a multiplier of 3.09 to provide a nominal probability of 0.2% (an average of two misleading observations per thousand), but Shewhart chose 3 to avoid having to consider exact probabilities. Similarly, some consultants use actual probability values ​​for maps based on non-normal distributions, such as range and discrepancy rate maps, in which case the Shewhart map also uses boundaries at distances of ±3 instead of probabilistic limits, simplifying empirical interpretation.

The probability that a boundary violation is truly a random event rather than a real signal is considered so small that when a point outside the boundary appears, certain actions should be taken. Since the action is taken precisely at this point, then control boundaries are sometimes called "action boundaries".

Often on the control map the boundaries are also drawn at a distance of 2 .Then any sample value falling outside the boundaries of 2a can serve as a warning about the impending situation of the process leaving the state of statistical control. Therefore, the limits are ±2 sometimes called "warning".

When using control charts, two types of errors are possible: type 1 and type 2.

An error of the first type occurs when the process is in a statistically controlled state, and the point jumps out of the control limits by chance. As a result, they incorrectly decide that the process has left the state of statistical control, and make an attempt to find and eliminate the cause of a non-existent problem.

An error of the second type occurs when the process under consideration is not controllable, and the points accidentally end up inside the control boundaries. In this case, they incorrectly conclude that the process is statistically controllable and miss the opportunity to prevent an increase in the yield of non-conforming products. The risk of a Type II error is a function of three factors: the width of the control limits, the degree of uncontrollability, and the sample size. Their nature is such that only a general statement about the magnitude of the error can be made.

The Shewhart chart system takes into account only type I errors, equal to 0.3% within limits 3 . Since in general it is impractical to make a full estimate of losses from a type II error in a specific situation, and it is convenient to arbitrarily take a small volume of a subgroup (4 or 5 units), it is advisable to use boundaries at a distance of ± 3 and focus primarily on managing and improving the quality of the process itself.

If the process is statistically controlled, control charts implement a method of continuously statistically testing the null hypothesis that the process has not changed and remains stable. But since the value of a particular deviation of a process characteristic from the target that might attract attention cannot usually be determined in advance, nor can the risk of a Type II error, and the sample size is not calculated to satisfy the appropriate level of risk, the Shewhart map should not be considered from the point of view of testing hypotheses . Shewhart emphasized the empirical usefulness of control charts for establishing deviations from the state of statistical control, and not their probabilistic interpretation. Some users use operating characteristic curves as a means to interpret hypothesis tests.

When the plotted value falls outside any of the control limits or the series of values ​​exhibits unusual patterns, the state of statistical control is questioned. In this case, it is necessary to investigate and detect non-random (special) causes, and the process can be stopped or corrected. Once special causes have been found and eliminated, the process is ready to continue again. When a Type I error occurs, no specific cause may be found. Then it is believed that the point going beyond the boundaries is a rather rare random phenomenon when the process is in a statistically controlled state.

When a process control chart is built for the first time, it often turns out that the process is statistically uncontrollable. Control limits calculated from data from such a process will sometimes lead to erroneous conclusions because they may be too broad. Therefore, before setting constant parameters of control charts, it is necessary to bring the process into a statistically controlled state.

Federal Agency for Education

State educational institution

higher professional education

"Kuzbass State Technical University"

Department of Plastics Processing Technology

Department of Chemical Technology of Inorganic Substances

Shewhart control cards

Guidelines for practical classes in the discipline

"Metrology, standardization, certification"

for students of specialties

250100 (240401) “Chemical technology of organic substances”

250200 (240301) “Chemical technology of inorganic substances”

250400 (240403) “Chemical technology of natural energy carriers

and carbon materials"

250600(240502) “Technology for processing plastics and elastomers”

Compiled by N. M. Igolinskaya

E. B. Silinina

M. A. Igolinskaya

Approved at a department meeting

educational and methodological commission

specialties 250200

Protocol No. 8 of March 30, 2006

An electronic copy is located

in the main building library

GU KuzGTU

Kemerovo 2006

OBJECTIVES OF PRACTICAL LESSONS

Familiarize yourself with the methods of constructing Shewhart control charts; according to the task option, calculate the boundaries and build a map for controlling the technological process.

Draw a conclusion about the smoothness of the process and its statistical control.

Perform procedures for bringing the map into the form of a statistically controlled process.

1. BASIC PROVISIONS OF THE THEORY

SHEWHART CONTROL CARDS

Control charts are graphical tools that use statistical approaches to control production processes. The purpose of such a control is to determine whether a statistically controlled state of the process has been achieved and whether it remains in this state while continuously obtaining information about the quality of the product.

Controlling the stability of the process allows you to reduce the cost of quality control of the finished product, choose the right raw material base and the price of the product as a product.

Control chart theory distinguishes two types of variability:

– variability due to random causes that are constantly present and cannot be identified and eliminated;

– variability, which represents real changes in the process due to certain reasons that can be identified and eliminated. Such variability is considered “non-random” (tool breakdown, heterogeneity of raw materials, violation of the technological regime, personnel qualifications, etc.).

Variability due to random causes is usually described by the parameters of the normal distribution and the Gaussian curve, which must be within the process tolerance. This situation is demonstrated in Fig. 1.

The ratio of the boundaries shown in the figure allows us to establish, based on the ratio of the areas of the ranges σ, the relationship between the hit frequency X 0 into and out of range. These frequencies are given in table. 1.

Rice. 1. The ratio of distribution boundaries (B) and technological tolerance (T) for an established statistically controlled process

Table 1

Relationship between the specified parameter deviation range X

and hit and miss rates X in this range

Consequently, if the requirements for the process are determined in such a way that the spread of control parameters does not exceed
, then the output of any given control parameter taken at random X i out of range
possible with probability 0.06, i.e. unlikely.

Let us introduce the characteristic I B – “process capability index”. This value determines the capabilities of the process and its statistical regulation. It is determined by the formula

, (1)

Where I B – process capability index;

T– process requirement;

IN– process capabilities.

If I B< 1, то процесс невозможен (не может быть обеспечено требуемое качество).

If I B = 1, then the process is on the edge of the possible. At the same time, despite the fact that the process under favorable conditions can provide a given quality, its statistical regulation is impossible.

If I B > 1, then the process is possible and statistical regulation of its quality can be realized.

A general view of one of the possible control charts is shown in Fig. 2.

Rice. 2. Control chart of distribution of current values ​​of the monitored parameter X for 18 measurement groups

Statistical process quality control is clearly shown in Fig. 3.

Rice. 3. Schematic representation of a statistically controlled process

Control charts are a way to track deviations from quality standards. Deviations that exceed established limits are called uncontrollable, and deviations that do not exceed established limits are called controllable. Looking ahead, we note that in Fig. Figure 2 shows measurements that fall outside both the lower control limit and the upper limit; this means that the corresponding process is out of control. Quality management theories state that only uncontrollable processes should be adjusted.

Control data is collected by taking regular measurements during a defined process. These measurements are recorded in a spreadsheet approximately as shown in Fig. 1.

In this example, we took the average of a sample of measurements and used standard deviation calculations to determine the upper and lower control limits for our process. The limited space of this article does not allow us to cover in detail the theory and formulas that are used in constructing a control chart. Let's focus better on building the diagram itself. Control chart based on the data shown in Fig. 1, is shown in Fig. 2.

To create a control chart, a simple line graph is used. First, highlight the data cells in columns A, E, F, I, and J (the data cells are in rows 2-15 of each column). When selecting columns, be sure to hold down the Ctrl key because the data being selected is not contiguous. Then click on the button Line(Graph) tabs Insert(Insert). In the menu that appears, click on any group icon 2D Line(Schedule). We clicked on the icon Line with Markers(Graph with markers). If you prefer a different display style, click on your chart and select the tab Design(Constructor). Then click on the small downward arrow button located in the lower right corner of the options group Chart Styles(Chart styles). A menu will appear on the screen with thumbnails of a variety of styles that can be applied to this type of chart (Figure 3).

Give this chart, as well as the horizontal and vertical axes, names as we did above. Change the chart legend as indicated in one of the earlier examples.

4. Examples of constructing Shewhart control charts using GOST R 50779.42–99

Shewhart control charts come in two main types: for quantitative and alternative data. For each control chart there are two situations:

a) standard values ​​are not specified;

b) standard values ​​are set.

Standard values ​​are values ​​established in accordance with some specific requirement or purpose.

The purpose of control charts for which no standard values ​​are specified is to detect deviations in the values ​​of characteristics (for example, or some other statistic) that are due to causes other than those that can be explained only by chance. These control charts are based entirely on data from the samples themselves and are used to detect variations that are due to non-random causes.

The purpose of control charts, given given standard values, is to determine whether the observed values ​​differ, etc. for several subgroups (each with a volume of observations) from the corresponding standard values ​​(or), etc. more than can be expected from the action of random causes alone. A special feature of maps with given standard values ​​is the additional requirement related to the position of the center and the variation of the process. Established values ​​may be based on experience gained from the use of control charts at specified standard values, as well as on economics determined after consideration of service needs and production costs, or specified in product specifications.

4.1 Control charts for quantitative data

Quantitative control charts are classic control charts used for process control where the characteristics or results of a process are measurable and the actual values ​​of the controlled parameter measured to the required accuracy are recorded.

Control charts for quantitative data allow you to control both the location of the center (level, mean, center of tuning) of the process and its spread (range, standard deviation). Therefore, control charts for quantitative data are almost always used and analyzed in pairs—one chart for location and the other for scatter.

The most commonly used pairs are and -cards, as well as -cards. Formulas for calculating the position of the control boundaries of these maps are given in Table. 1. The values ​​of the coefficients included in these formulas and depending on the sample size are given in Table. 2.

It should be emphasized that the coefficients given in this table were obtained under the assumption that the quantitative values ​​of the controlled parameter have a normal or close to normal distribution.

Table 1

Control limit formulas for Shewhart charts using quantitative data

table 2

Coefficients for calculating control chart lines

An alternative to maps are median control charts (– maps), the construction of which involves less computation than maps. This may make it easier to introduce them into production. The position of the central line on the map is determined by the average value of the medians () for all tested samples. The positions of the upper and lower control limits are determined by the relations

(4.1)

The values ​​of the coefficient, depending on the sample size, are given in table. 3.

Table 3

Coefficient values

Usually - map is used together with - map, sample size

In some cases, the cost or duration of measuring the controlled parameter is so great that it is necessary to control the process based on measuring individual values ​​of the controlled parameter. In this case, the sliding range serves as a measure of process variation, i.e. the absolute value of the difference in measurements of the monitored parameter in successive pairs: the difference between the first and second measurements, then the second and third, etc. Based on the moving ranges, the average moving range is calculated, which is used to construct control charts of individual values ​​and moving ranges (and -maps). Formulas for calculating the position of the control boundaries of these maps are given in Table. 4.

Table 4

Control limit formulas for individual value maps

The values ​​of the coefficients and can be indirectly obtained from Table 2 with n=2.

4.1.1 and -cards. No default values ​​specified

In table Figure 6 shows the results of measurements of the outer radius of the bushing. Four measurements were taken every half hour, for a total of 20 samples. The means and ranges of the subgroups are also shown in Table. 5. The maximum permissible values ​​for the outer radius are established: 0.219 and 0.125 dm. The goal is to determine the performance of the process and control it in terms of tuning and variation so that it meets the specified requirements.

Table 5

Manufacturing Data for Bushing Outer Radius

where is the number of subgroups,

The first step: constructing a map and determining the state of the process from it.

center line:

The values ​​of the factors and are taken from the table. 2 for n=4. Since the values ​​in the table. 5 are within the control limits, the map indicates a statistically controlled state. The value can now be used to calculate map control boundaries.

center line: g

The multiplier values ​​are taken from the table. 2 for n=4.

and -maps are shown in Fig. 5. Analysis of the map shows that the last three points are outside the boundaries. This indicates that some special causes of variation may be at work. If limits have been calculated based on previous data, then action must be taken at the point corresponding to the 18th subgroup.

Fig.5. Medium and large maps

At this point in the process, appropriate corrective action should be taken to eliminate the special causes and prevent their recurrence. Work with the maps continues after the revised control boundaries have been established without excluded points that went beyond the old boundaries, i.e. values ​​for samples No. 18, 19 and 20. The values ​​and lines of the control chart are recalculated as follows:

revised value

revised value

The revised map has the following parameters:

center line: g

revised –map:

center line:

(since the center line is: , then there is no LCL).

For a stable process with revised control limits, capabilities can be assessed. We calculate the opportunity index:

where is the upper maximum permissible value of the controlled parameter; – lower maximum permissible value of the controlled parameter; – estimated by the average variability within subgroups and expressed as . The value of the constant is taken from Table 2 for n=4.

Rice. 6. Revised and -maps

Since , the process capabilities can be considered acceptable. However, upon closer examination, it can be seen that the process is not set up correctly relative to the tolerance and therefore about 11.8% of units will fall outside the specified upper limit value. Therefore, before setting constant parameters of control charts, one must try to correctly configure the process, while maintaining it in a statistically controlled state.

The tool is used when processing is carried out with a tool whose design and dimensions are approved by GOST and OST or are available in industry standards. When developing technological processes for manufacturing parts, you should use a normalized tool as the cheapest and simplest. A special cutting tool is used in cases where processing with a normalized...

Such control is very expensive. Therefore, they move from continuous control to selective control using statistical methods for processing the results. However, such control is effective only when the technological processes, being in an established state, have accuracy and stability sufficient to “automatically” guarantee the production of defect-free products. Hence the need arises...

And organizing the control process. Inspection status In this course project, the technical assignment provides for the development of stages of the acceptance inspection process for a part of a cylindrical coaxial two-stage, two-flow gearbox - a gear wheel and active control during the hole grinding operation. Methods of active and acceptance control complement each other and are combined. Active...