Chapter 2 of One- Theory of Control Chart

Chapter Three Statistical Process Control (SPC)

CHAPTER TWO - THEORY OF CONTROL CHARTS13Methods And Philosophy of Statistical Process ControlProcess And Measurement System Capability Analysis2Control Charts for VariablesControl Charts for AttributesThis is another option for an Overview slide.

IntroductionDr.Gere K2HistogramCheck sheetPareto ChartCause and effect diagram SPC can be applied to any process. Its seven major tools are:In a nutshell, the eventual goal of SPC is the elimination of variability in the process.While its key aspect is to obtain predictable processes that produce consistent results by quickly detecting the occurrence of assignable causes of process shifts, be it above or below control limits or unnatural patterns, so that investigation of the process and corrective action may be undertaken before many nonconforming units are manufactured. The application of statistical techniques to control a process.In market economy, the maintenance of quality has a profound importance in manufacturing and servicing environment. For doing so, one of the earliest tools is Statistical Process Control(SPC).5) Defect concentration diagram6) Scatter diagram7) Run chartWhile these tools , often called the magnificent seven, are an important part of SPC, they comprise only its technical aspects. SPC builds an environment in which it is the desire of all individuals in an organization for continuous improvement in quality and productivity.Dr.Gere K3Of the seven tools, the Shewhart control chart is probably the most technicallysophisticated. It was developed in the 1920s by Walter A. Shewhart of the Bell Telephone Laboratories. To understand the statistical concepts that form the basis of SPC, we must first describe Shewharts theory of variability.In any production process, regardless of how well designed or carefully maintained it is, a certain amount of inherent or natural variability will always exist. cont.VariationDr.Gere KCategories of variationWithin-piece variationOne portion of surface is rougher than another portion.Apiece-to-piece variationVariation among pieces produced at the same time.Time-to-time variationService given early would be different from that given later in the day.4There is no two natural items in any category are the same.Variation may be quite large or very small.If variation is very small, it may appear that items are identical, but precision instruments will show differences.Sources of Variation2. Assignable Causes or Special Causes

Dr.Gere K5Equipment:-Tool wear, machine vibration, Material:- Raw material qualityEnvironment:-Temperature, pressure, humidity, lightingOperator:-Operator performs- physical & emotional

Types of Variation1. Common or Chance CausesVariation due to chance (common) causes is inevitable in any process or product. They are difficult to trace and control even under best conditions of production. Since these variations may be due to some inherent characteristics of the process or machine which functions at random. W.E. Deming contended that only management can address common cause variation since it is inherent in the process as designed by management.Assignable (special) cause variation posses greater magnitude as compared to those due to chance causes and can be traced or detected. The power of Shewhart control chart lies in its ability to separate out the assignable causes of variations.Variable data Product characteristic that can be measuredLength, size, weight, height, time, velocity

Attribute dataProduct characteristic evaluated with a discrete choiceGood/bad, yes/no

Dr.Gere K6 Use of Control ChartsSeparate common and special causes of variationDetermine whether a process is in a state of statistical control or out-of-controlEstimate the process parameters (mean, variation) and assess the performance of a process or its capability.The control chart is an on-line process-monitoring technique widely used for this purpose. Control charts may also be used to estimate the parameters of a production process, and, through this information, to determine process capability. The control chart may also provide information useful in improving the process. Finally, remember that the eventual goal of statistical process control is the elimination of variability in the process.Types of Data6How to develop a control chart1.Select a quality characteristic to be measured Identify a characteristic to study - for example, part length or any other variable affecting performance.

2. Choose a subgroup size to be sampled Choose homogeneous subgroupsHomogeneous subgroups are produced under the same conditions, by the same machine, the same operator, the same mold, at approximately the same time.Try to maximize chance to detect differences between subgroups, while minimizing chance for difference with a group.3. Collect dataGenerally, collect 20-25 subgroups (100 total samples) before calculating the control limits.Each time a subgroup of sample size n is taken, an average is calculated for the subgroup and plotted on the control chart.

Dr.Gere K7Control ChartsControl charts for variablesControl charts for attributesDr.Gere K8The key instrument of SPC is the control chart invented by W. A. Shewhart in the 1920s. A control chart is a graphical comparison of performance data to computed control limits drawn as limit line on the chart.The primary function of control chart is to determine which type of variation is present and whether adjustments need to be made to the process. It can be as damaging to adjust a process which is operating in control (only common causes variation present) as it is to fail to adjust a process which is operating out of control (assignable causes of variation present). It is, therefore, important to be able to determine what type of variation is present in a process. Types of control charts:Dr.Gere K9The X chart plots sample means. It is a measure of between sample variations and is used to assess the centering and long-term variation of the process. The range chart and S chart measure the with in-sample variation and assess the short-term variation of the process.Control Charts for VariablesControl charts for variables enable the monitoring of the natural variability occurring in a process where the data is provided in measurable units rather than counted ones. The charts will then be used to reduce this variability around the nominal value. Charts are based on variability due to common causes and are used to determine the presence of special causes.Variable data are plotted on a combination of two charts. Using X bar chart and range (R) chart. However, S chart (standard deviation) chart should be used in place of a range chart for larger sample size (n>10).This is because; the range method loses efficiency relative to S2 as sample size increases. For a sample size of 2, the two methods are equivalent. Where as, for a sample size of 10, the range method efficiency is only 0.85% relative to S2. The constructing of an chart begins with the collection of a serious of samples from a process. The samples consist of two or more observations (sample size of 3 to 10 are best) each. The individual observations are averaged for each sample to determine the sample mean ( ). The averages of at least 25 to 30 sample means are calculated that is X-double bar ( ). The underlying distribution for the chart is the normal distribution. The centerline (CL) of the chart is . The upper control limit (UCL) is set at + 3sigma; the lower control limit (LCL) is set at - 3sigma. Appropriate placement of the upper and lower control limits is an economic issue. The intent would be to fix the limits in such a way as to balance the economic consequences of failing to detect a special cause when it does occur and wrongly identifying the presence of a special cause when it has really not occurred. Experience has led to the use of 3 limits as a good balance of these risks.

The 25 to 30 sample means are then plotted on the control chart. If none of the points fall outside the control limits and there are no discernible patterns in the plot, then process is said to be in control.

Dr.Gere K10 X Bar and R Control Charts

The centerline for the X-bar chart is X-double bar. The upper control limit and lower control limit are calculated using

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The constant A2 is tabulated for various sample sizes in Appendix Table VI.

The centerline for the range chart is . The UCL and LCL are calculated by:

The range chart is evaluated first, because the limits of the chart depends on the magnitude of the common causes variation of the process measured by , if it is in control then the chart is evaluated. An out of control signal on either chart is an indication that the process is out of control.

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12The constant d3 , and d3 is tabulated for various sample sizes in Appendix Table VI.

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17 If the mean and standard deviation of a process are known, or if they are specified by management, perhaps as goals to be achieved and R control charts can be constructed without analyzing past data. For 3-sigma limits, the control limits for are given by:

The control limits for the chart can then be written as

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Since = and , we can write

Defining the constants that depend only on the sample size, n, as

We obtain

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X and S control chartsThe 3-sigma limits for the S chart with the standards given are as follows

Where

The control limits can be rewritten using B6 and B5 as

While X-bar and R charts are widely used, it is occasionally desirable to estimate the process standard deviation directly instead of indirectly through the use of the range R. This leads to control charts for X-bar and S, where S is the sample standard deviation.

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Where

The control limits can then be written as

And for the X-bar chart

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Standard is given

Standard is given22 Cycles Cycles are short trends in the data that occur in repeated patterns. Causes of cycles on the X-bar chart include temperature and humidity changes, operator fatigue, rotation of operators and electrical fluctuations. While, operators fatigue, shift (day or night) and worn tools or dies are for the R chart. An example of a cycle is shown in Fig. A Mixture In a mixture pattern, the points tend to fall near the UCL and LCL with an absence of fluctuations near the middle. On the X-bar chart, mixtures can occur with over control. Where as, difference in materials and measuring equipment can cause a mixture on the R chart.Dr.Gere K23Detection of Patterns on X-bar and R chartsEven though, no points are above and below the control limits the process can be said out-of-control by looking on the unnatural patterns produced by special causes. Some of the patterns that are frequently seen are: Stratification Stratification is characterized by artificial constancy. Instead of fluctuating naturally inside limits, the points are very close to the CL. On the X-bar chart, this can be caused by incorrect calculation of the control limits. The pattern may occur on the R chart when the sampling process collects one unit from each of several underlying distributions. If the largest and smallest unit in each sample are similar, unnaturally small fluctuations will result.

Sudden Shift A sudden shift in level is shown by an instantaneous change in one direction or the other. On X-bar chart. On X-bar chart, a sudden shift could be caused by change to a new type of material, new operator, new inspector, new machines and so on. On the R chart, change in motivation of the operators, new operators and new equipment are few of the many causes of a sudden shift.Dr.Gere K24Dr.Gere K25TrendA trend is shown by a continuous movement in one direction. On the X-bar chart, a trend is caused due to gradual deterioration of equipment, worker fatigue and accumulation of waste products. On the R chart, improvement or deterioration of operators skill work fatigue and gradual change in homogeneity of incoming quality are come of the causes.Interpreting unnatural patterns is a challenge for quality control personnel. Some of the major problems associated with the analysis of control chart patterns can be summarized as follows. the random noise might contaminate the present pattern, the effect may change with the magnitude of the unnatural pattern.a pattern may sometimes resemble other patterns. For instance, a short trend may be a subset of other patterns.the problem is much more complicated if there is more than one pattern of interest, or if the signal-to-noise ratio is low.This shows pattern recognition is a crucial problem in statistical process control. Over the years, the zone test or run tests have been the major tool for interpreting control charts. Dr.Gere K26Although the zone tests or run tests have been proven to be effective in detecting out-of-control situations the interpretation of process data is still a very difficult task.The major difficulty lies in the fact that there is no one-to-one mapping between a supplementary rule and an unnatural pattern. In practice, the types of unnatural patterns that a process may experience are not known in advance. There might be several patterns associated with a particular rule. For instance, the possible patterns associated with the rule eight points in row on both sides of the centerline with none in zone C suggested by Nelson (1985), could be a mixture or a systematic variation. These tests might indicate that an unnatural pattern is present, but do not explicitly indicate which pattern really occurs. In addition, some of the supplementary rule results in more false alarms without significantly improving the performance of control charts. A typical example is the commonly used trend rule. Davis and Woodall (1988) evaluated the trend rule and concluded that the trend rule is not effective in detecting a linear trend. The limitations of supplementary tests have motivated interests in developing algorithms based neural networks for process data analysis. Dr.Gere K27

Dr.Gere K28Working Rules: Pattern Analysis Tests (PAT)PAT 1: One point plots beyond zone A on either side of the mean.

PAT 2: Nine points in a row plot on the same side of the mean.

PAT 3: Six consecutive points are strictly increasing or strictly decreasing.

PAT 4: Fourteen consecutive points which alternate up and down.

PAT 5: Two out of three consecutive points plot in zone A or beyond, and all three points plot on the same side of the mean.

PAT 6: Four out of five consecutive points plot in zone B or beyond, and all five points plot on the same side of the mean.

PAT 7: Fifteen consecutive points plot in zones C, spanning both sides of the mean.

PAT 8: Eight consecutive points plot at more than one standard deviation away from the mean with some smaller than the mean and Dr.Gere K29Dr.Gere K30

Process capability Process capability compares the output of an in-control process to the specification limits by using capability indices. The comparison is made by forming the ratio of the spread between the process specifications (the specification "width") to the spread of the process values, as measured by 6 process standard deviation units (the process "width").

Process Capability Indices We are often required to compare the output of a stable process with the process specifications and make a statement about how well the process meets specification. To do this we compare the natural variability of a stable process with the process specification limits.

A capable process is one where almost all the measurements fall inside the specification limits. This can be represented pictorially by the plot below:

Dr.Gere K31 There are several statistics that can be used to measure the capability of a process: Cp, Cpk, Cpm. Most capability indices estimates are valid only if the sample size used is 'large enough'. Large enough is generally thought to be about 50 independent data values. The Cp, Cpk, and Cpm statistics assume that the population of data values is normally distributed. Assuming a two-sided specification, if and are the mean and standard deviation, respectively, of the normal data and USL, LSL, and T are the upper and lower specification limits and the target value, respectively, then the population capability indices are defined as follows:

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To get an idea of the value of the Cp statistic for varying process widths, consider the following plot

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Definitions of various process capability indices:Dr.Gere K33Example The design specifications for component are 100 0.5. Whereas the process report shows that process average is 99.9mm and standard deviation is 0.18. Do these figures call for any action by any one?SolutionUSL = 100.5 mmLSL = 99.5mmX-bar = 99.9mm = 0.18Cp = 1/6 = 0.925

Defective products will always be there. It is therefore necessary to take action to reduce the number of defectives produced.Dr.Gere K

34Control Charts for Moving Averages and Ranges

Many organizations are involved in continuous processes, such as manufacturing steel, aluminum, paint, oil or chemicals. In such cases it is recognized that the ( , R) charts is inappropriate. So, in continuous processes extensive use is made of moving average charts.

Example The viscosity of an aircraft primer paint is an important quality characteristic. The product is produced in batches and as each batch takes several hours to produce, the production rate is too slow to allow sample sizes greater than one. The viscosity of the previous 15 batches is given belowDr.Gere K

UCL = 1.57CL= 0.48LCL = 0

U=34.8CL=33.52L=32.2435 Viscosity of Aircraft Primer PaintBatch #Viscosity (x)Moving Range (MR)133.75233.050.70334.000.95433.810.19533.460.35633.020.56733.680.34833.270.41933.490.221033.200.291133.620.421233.000.621333.540.541433.120.421533.840.7233.460.48Dr.Gere K36

Chart84.9854.974.964.995.015.025.055.085.03

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