Manufacturing Variability Measurement and Control

Control charts have been traditionally used as the method of determining the performance of manufacturing processes over time by the statistical characterization of a measured parameter that is dependent on the process. They have been used effectively to determine if the manufacturing process is in statistical control. Control exists when the occurrence of events (failures) follows the statistical laws of the distribution from which the sample was taken.

Control charts are run charts with a centerline drawn at the manufacturing process average and control limit lines drawn at the tail of the distribution at the 3 s_points. They are derived from the distribution of sample averages X, where s is the standard deviation of the production samples taken and is related to the population deviation through the central limit theorem. If the manufacturing process is under statistical control, 99.73% of all observations are within the control limits of the process. Control charts by themselves do not improve quality; they merely indicate that the quality is in statistical “synchronization” with the quality level at the time when the charts were created.

There are two major types of control charts: variable charts, which plot continuous data from the observed parameters, and attribute charts, which are discrete and plot accept/reject data. Variable charts are also known as X, R charts for high volume and moving range (MR) charts for low volume. Attribute charts tend to show proportion or percent defective. There are four types of attribute charts: P charts, C charts, nP charts, and U charts (see Figure 3.1).

The selection of the parameters to be control charted is an important part of the six sigma quality process. Too many parameters plotted tend to adversely affect the beneficial effect of the control charts, since they will all move in the same direction when the process is out of control. It is very important that the parameters selected for control charting be independent from each other and directly related to the overall performance of the product.

When introducing control charts to a manufacturing operation, it is beneficial to use elements that are universally recognized, such as temperature and relative humidity, or take readings from a process display monitor. In addition, the production operators have to be directly active in the charting process to increase their awareness and get them involved in the quality output of their jobs. Several shortcomings have been observed when initially introducing control charts. Some of these to avoid are:

• Improper training of production operators. Collecting a daily sample and calculating the average and range of the sample data set might seem to be a simple task. Unfortunately, because of the poor skill set of operators in many manufacturing plants, extensive training has to be provided to make sure the manufacturing operator can perform the required data collection and calculation.

• Using a software program for plotting data removes the focus from the data collection and interpretation of control charting. The issues of training and operating the software tools become the primary factors. Automatic means of plotting control charting should be introduced later in the quality improvement plan for production.

• Selecting variables that are outside of the production groups direct sphere of influence, or are difficult or impossible to control, could result in a negative perception of the quality effort. An example would be to plot the temperature and humidity of the production floor when there are no adequate environmental controls. The change in seasons will always bring an “out-of-control” condition.

In the latter stage of six sigma implementation, the low defect rate impacts the use of these charts. In many cases, successful implementation of six sigma may have rendered control charts obsolete, and the factory might switch over to TQM tools for keeping the quality level at the 3.4 PPM rate. The reason is that the defect rate is so low that only few defects occur in the production day, and the engineers can pay attention to individual defects rather than the sampling plan of the control charts.

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Control of Variable Processes and Its Relationship with Six Sigma

Variable processes are those in which direct measurements can be made of the quality characteristic in a periodic or daily sample. The daily samples are then compared with a historical record to see if the manufacturing process for the part is in control. In X, R charts, the sample measurements taken today are expected to fall within three standard deviations 3 s of the distribution of sample averages taken in the past. In moving range (MR) charts, the sample is compared with the 3 <r of the population standard deviation derived from an R estimator of cr. When the sample taken falls outside of the 3 s limits, the process is declared not in control, and a corrective action process is initiated.

Another type of charting for quality in production is the precontrol chart. These charts directly compare the daily measurements to the part specifications. They require operators to make periodic measurements, before the start of each shift, and then at selected time intervals afterward. They require the operator to adjust the production machines if the measurements fall outside a green zone halfway between the nominal and specification limits.

Precontrol charts ignore the natural distribution of process or machine variability. Instead, they require a higher level of operator training and intervention in manufacturing to ensure that production distribution is within halfway of the specification limits, on a daily basis. This is in direct opposition to six sigma concepts of analyzing and matching the process distribution to he specification limits only in the design phase, and thus removing the need to do so every time parts are produced.

Moving range charts (MR) are used in low-volume applications. They take advantage of statistical methodology to reduce the sample size. They will be discussed further in the Chapter 5. In high-volume manufacturing, where several measurements can be taken each day for production samples, X and R control charts are used to monitor the average and the standard deviation of production. It is important to note that 叉 control charts are derived from the sample average distribution, which is always normal, regardless of the parent distribution of the population σ, which is used for six sigma calculations of the defect rate, and is not always normal, as discussed in the previous chapter.

The X chart shows whether the manufacturing process is centered around or shifted from the historical average. If there is a trend in the plotted data, then the process value, as indicated by the sample average X, is moving up or down. The causes of X chart movements include faulty machine or process settings, improper operator training, and defective materials.

The R chart shows the uniformity or consistency of the manufacturing process. If the R chart is narrow, then the product is uniform. If the R chart is wide or out of control, then there is a nonuniform effect on the process, such as a poor repair or maintenance record, untrained operators, and nonuniform materials.

The variable control charts are generated by taking a historical record of the manufacturing process over a period of time. Shewhart, the father of control charts, recommends that “statistical control can not be reached until under the same conditions, not less than 25 samples of four each have been taken to satisfy the required criterion. ” These observations form the historical record of the process. All observations from now on are compared to this baseline.

From these observations, the sample average X and the sample range R, which is the absolute value of highest value minus the lowest value in the sample, are recorded. At the end of the observation period (25 samples), the average ofXs, designated as^ and the average of R% designated as R, are recorded.

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Variable Control Chart Limits

The control limits for the control charts are calculated using the following formulas and Table 3.1 for control chart factors. The control chart factors were designated with variables such as A2, Ds, and D4 to calculate the control limits of X and R control charts. The factor d2 is important in linking the average range and hence the standard deviation of the sample (s), to the population standard deviation a. The control chart factors shown in Table 3.1 stop at the number 20 of observations of the subgroup. Control charts are based on taking samples to approximate a large production output. If the sample be, comes large enough, there is no advantage to using samples and their associated normal distributions to generate variable control charts.

Instead, 100% of production could be tested to find out if the parts produced are within specifications.

Control and specification limits

Control chart limits indicate a different of conditions than the specification limits. Control limits are based on the distribution of sample averages，whereas specification limits are related to population distributions of parts. It is desirable to have the specification lim. It’s as large as possible compared to the process control limit.

The control limits represent the 3 5 points，based on a sample 〇f n observations. To determine the standard deviation of the product population, the central limit theorem can be used:

where

s = standard deviation the distribution of sample averages 

σ= population deviation 

n = sample size

Multiplying 173 the distance from the centerline of the X chart to one of the control limits by Vn will determine the total product population deviation. A simpler approximation is the use of the formula a = R/d2 from control chart factors in Table 3.1 to generate the total product standard deviation directly from the control chart data, d2 can be used as a good estimator for a when using small numbers of samples and their ranges.

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X, R variable control chart calculations example

Example 3.1

In this example, a critical dimension for a part is measured as it is being inspected in a machining operation. To set up the control chart, four measurements were taken every day for 25 successive days, to approximate the daily production variability. These measurements were then used to calculate the limits of the control charts. The measurements are shown in Table 3.2.

It should be noted that the value n used in Equation 3.5 is equal to 4, which is the number of observations in each sample. This is not to be confused with the 25 sets of subgroups or samples for the historical record of the process. If the 25 samples are taken daily, they represent approximately a one-month history of production.

During the first day, four samples were taken, measuring 9,12,11, and 14 thousands of an inch. These were recorded in the top of the four columns of sample #1. The average, or X was calculated and entered in column 5, and the R is entered in column 6.

X Sample 1 = (9 + 12 + 11 + 14)/4 = 11.50

The range, or R, is calculated by taking the highest reading (H in this case), minus the lowest reading (9 in this case).

R Sample 1 = 14 - 9 = 5

The averages of X and R are calculated by dividing the column totals of X and R by the number of subgroups.

X = (SUM OFXs)/number of subgroups

X=315.50/25 = 12.62

R = (SUM OF R's) number of subgroups

R=111/25 = 4.44

Using the control chart (Table 3.1), the control limits can be calculated using n = 4 as follows:

X Control limits

UCKX = X + A2 R = 12.62 + 0.73 • 4.44 = 15.86

UCLX =1-A2^= 12.62 - 0.73 • 4.44 = 9.38 

R Control limits

Upper control limit (UCL^) = D4R = 2.28 * 4.44 = 10.12 

Lower control limit (LCL/f) = = 0

Since the measurements were recorded in thousands of an inch, the centerline of the-X" control chart is 0.01262 and the control limits for X are 0.01586 and 0.00938. For the R chart, the centerline is set at 0.00444 and the limits are 0.01012 and 0.

These numbers form the control limits of the control chart. After the limits have been calculated, the control chart is ready for use in production. Each production day, four readings of the part dimension are to be taken by the responsible operators, with the average of the four readings plotted on the X chart，and the range or difference be- tween the highest and_ lowest reading to be plotted on the R chart. The daily numbers of X and R should plot within the control limits. If they plot outside the limits, the production process is not in control， and immediate corrective action should be initiated.

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Alternate methods for calculating control limits

The control limits are set to three times standard deviation of the sample distribution (s). s can be calculated from a the population standard deviation using the factor d2 according to the central limit theorem:

σ= R/d2 = 4.44/2.059 = 2.156

S = a/V^ = 2.156/2 = 1.078

± 3 s = 1.078.3 = 3.23, which is close to the A2 •R value of 3.24, which corresponds to the distance from the centerline to one of the control limits in the variable control charts.

It is interesting to note that of the total population of l〇〇 numbers (Table 3.2), then the standard deviation is a = 2.156, which is exactly the one predicted by the R estimator. If the specifications limits are given, then the Cp, Cpk, and reject rates can be calculated as in the example in the previous chapter.

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Examples of variable control chart calculations and their relationship to six sigma

These examples were developed to show the relationship of variable control charts and six sigma. They can be used as guidelines for communications between an enterprise and its suppliers.

Example 3.2a

A variable control chart for PCB surface resistance was created. There is only one minimum specification for resistance. The X bar was 20 megaohms (MH) and the UCL^ was 23 MH, with a sample size of 9. A new specification was adopted to keep resistance at a minimum of 16 MH. Assuming that the resistance measurement or process average =specification nominal (N), describe the Cp and Cpk reject rates and show the R chart limits.

Example 3.2a solution

Since the process is centered, Cp = Cpk. The distance from the X to UCLx = 35 = 3, therefore:

s=1

σ = s * Vn = 3 

LSL=16 MH 

Process average = 20 MH

Cp = Cpk = (LSL - process average)/3a = (20 -16)/3 * 3 = 4/9 = 0.444

z = (SL - average)/a = (16 - 20)/3 = 1.33 OTZ = 3 • Cpk = 1.33 Reject rate =f[-z) = 0.0976 = 91,760 PPM (one-sided rejects only, below LSL)

R= σ•d2 (n = 9) = 3 • 2.97 = 8.91 

UCLR = 1.82 • 8.91 = 16.22 MH 

LCLR = 0.X8-8.91 = 1.60 MH

Example 3.2b

A four sigma program was introduced at the company in Example 3.2a. For the surface resistance process, the lower specification limit (LSL) remained at 16 MH and the process a remained the same. Describe the Cp and Cpk reject rates and show the X and i? chart limits, using the same sample size of 9. Repeat for a six sigma program, with 1.5 σ shift, with the process average and sigma remaining the same.

Example 3.2b solution

The four sigma program implies a specification limit ofN±4(r = N±4•3 = N ± 12. The process average which is equal to the nominal N，is 4 tr away from the LSL，and is 16 + 12 = 28 MH，given LSL = 16 MH. Cp = Cpk = ± 4 CT/± 3 〇• = 2.33 and two-sided reject rate from the z table(Table 2.3) = 64 PPM.

The R chart remains the same as Example 3.4a, since the process variability σ did not change. The X chart is centered on X = 28 MH; ICL, = 28 - 3s = 25 MH; UCLX = 31 MH.

For six sigma, the same methodology applies, except that there is a *1.5 σ shift. The specification limits are N ± 6σ~ N ± 6 • 3 = N ± 18.

Given the LSL = 16 MH, the specification nominal N is 16 + 18 = 34 MH. Therefore, Cp = 2; Cpk « 1.5; reject rate from previous tables (±1.5 cr shift) = 3.4 PPM.

Assuming that the shift is toward the lower specification, then the process average could be +4.5 <T from the LSL or —1.5 (T from the nominal: 34 ~ 1.5 ■ 3 = 29.5 MH; or 16 + 4.5 3 = 29.5 MH. '

The R chart remains the same as_Example 3.4a, since the process variability <r did not change. If the X chart is centered on 5 = 29.5 then LCLV = 29.5 - 3 s = 26.5 MH and UCL^ = 32.5 MH. ’

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Attribute Charts and Their Relationship with Six Sigma

Attribute charts directly measure the rejects in the production operation，as opposed to measuring a particular value of the quality characteristic as in variable processes. They are more common in manufacturing because of the following:

1. Attribute or pass—fail test data are easier to measure than actual variable measurement. They can be obtained by devices or tools such as go/no-go gauges, calibrated for only the specification measurements, as opposed to measuring the full operating spectrum of parts.

2. Attribute data require much less operator training, since they only have to observe a reject indicator or light, as opposed to making several measurements on gauges or test equipment.

3. Attribute data can be directly collected from the manufacturing equipment, especially if there is a high degree of automation.

4. Storage and dissemination of attribute data is also much easier, since there is only the reject rate to store versus the actual measurements for variable data.

Attribute charts use different probability distributions than the normal distribution used in variable charts, depending on whether the sample size is constant or changing, as shown in Figure 3*1. For C and U charts, the Poisson distribution is used, whereas the P and nP charts use the binomial distribution.

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