STEPS FOR STUDYING PROCESS CAPABILITY - CONTROL LIMIT METHOD


STEPS FOR STUDYING PROCESS CAPABILITY ”CONTROL LIMIT METHOD

Control chart data may be used to answer process capability questions. Use the following procedure:

  1. Assess process stability. Make a control chart to assess process stability. Unless the process is in control, the sample data cannot be generalized to the population. When a process is out of control, it is changing and has more than one population. Figure 12.6 shows a completed control chart for some data.

    click to expand
    Figure 12.6: An Xbar and R chart with some hypothetical data.

  2. Gather the required information. Three types of information are needed. The location (Xdouble bar) and spread (Rbar) of the process distribution must be described, as well as the customers' wants (upper specification limit and lower specification limit; USL and LSL).

    For the hypothetical data,

    Xdouble bar = 17.77,

    Rbar = 5.56,

    LSL = 10, and

    USL = 20.

  3. Estimate the population standard deviation. A capability analysis describes the population in terms of the specifications. Xbar and R charts are inappropriate for capability analysis because the range underestimates the spread of the population. To better estimate the process spread, the range must be converted to an estimated standard deviation. Use this formula:

    where

    =

    estimated population standard deviation for individuals

    =

    centerline of the range chart

    d 2

    =

    a standard conversion based on the sample size (see Table 8.1)

    The estimated standard deviation for the hypothetical data is:

    = 5.56/2.236

    = 2.390

    This procedure is appropriate only for data that follow the normal distribution. If these calculations are performed for data that do not resemble the normal distribution (are not symmetrical), any conclusions about capability are invalid.

  4. Assess the normality of the distribution. Build a frequency distribution and compare its shape with that of the normal distribution. (We will discuss more precise methods later including testing for normality.) Figure 12.7 shows a frequency distribution for the hypothetical data.

    click to expand
    Figure 12.7: Frequency distribution based on the hypothetical data.

  5. Estimate the spread of the manufacturing process. To define the boundaries of the population, use the formula

    where

    Xdouble bar

    =

    centerline of the process

    =

    estimated population standard deviation

    Three estimated population standard deviation represents 49.87% of the population. This amount above and below the centerline defines 99.73% of the population. To fulfill the capability requirements, 99.73% of the population must be within specifications.

    The estimated spread of the process in our example is as follows :

  6. Graphically summarize the capability analysis.

    1. Draw a normal distribution curve to represent the population. Use Xdouble bar as the center of the curve. Use ±3 standard deviations as the ends of the curve.

    2. Scale the curve with standard deviations. Mark the locations of ±1 and ±2 standard deviations from the mean (Xdouble bar) by drawing a vertical line from the horizontal axis to the curve. Label each line. The capability analysis of the hypothetical data is summarized in Figure 12.8.

      click to expand
      Figure 12.8: A graphical representation of capability analysis.

  7. Apply the specifications to the population curve. Locate the upper (USL) and lower (LSL) specification limits on the horizontal axis and draw vertical lines up to the curve for each. Label each line and shade the area beyond the specification limits.

    The upper and lower specifications are applied to the hypothetical data (see Figure 12.9).

    click to expand
    Figure 12.9: Specifications to the population curve.

  8. Assess process capability. Most companies require that at least 99.73% of all products be within specifications before a process is considered capable. If ±3 standard deviations of the curve are within the specifications, this requirement is satisfied, and the process is labeled "capable."

    The process is not capable if the population curve extends beyond the specifications. The reasons that the process is not capable need to be studied. The location or the spread of the distribution can cause the process to be outside of specifications. If the mean is too close to one of the specification limits, the process needs to be adjusted so that the mean is closer to the midpoint between the specification limits. If the spread of the distribution is too large, there is excessive variation in the process. The sources of this excessive variation must be identified and removed. If the spread is too large and the mean is too close to either specification limit, the spread should be reduced before the location of the population is adjusted.

    The spread of the hypothetical process in our example distribution is too large and must be reduced before the population is adjusted to target dimension.

  9. Quantify the percentages in and out of specifications.

    1. Calculate standardized Z. Use the formulas below to calculate the distance (in terms of standard deviations) from the mean to each of the specifications.

      where

      Z LSL

      =

      the distance from the mean to the lower specifica tion limit

      Z USL

      =

      the distance from the mean to the upper specification limit

      USL

      =

      upper specification limit

      LSL

      =

      lower specification limit

      Xdouble bar

      =

      centerline of the Xbar chart

      =

      estimated population standard deviation

      For our example, the Z values are calculated as follows:

      Z LSL = (10 - 17.77)/2.39 = -3.2

      Z USL = (20 - 17.77)/2.39 = +.93

      These distances will be used to find the percentages of the population in and out of specifications.

    2. Find the percentage out of specifications for each z value.

      Use the table for the Area under the Normal Curve (see Table 5.1) to find the percentage out of specifications.

      The percentages out of specifications for each Z value for the example's data are:

      Proportion beyond LSL = .00058

      Percentage beyond LSL = .058%

      Proportion beyond USL = .1762

      Percentage beyond USL = 17.62%

    3. Summarize the percentage of the population out of specifications.

      Add the percentage out of specifications for each Z value. The total percentage out of specifications for the example's data = .058 + 17.62 = 17.678%

    4. Calculate the percentage in specifications.

      Subtract the total percentage out of specifications from 100%. For the data in our example, the total percentage of products in specifications = 100 - 17.678 = 82.322%.

  10. Interpret process capability.

    To meet capability requirements, the absolute value of the standardized z values must be 3 or greater (i.e., ±3 standard deviations must be within specifications).

Another way to determine capability is to study the percentage in and out of specifications. At least 99.73% of the population must be within specifications (i.e., no more than 0.27% can be out of specifications). If the percentage out of specifications for the two sides of the curve are unequal , the mean of the process must be shifted to the midpoint between the specification limits.

The spread of the data in our example population is too large. No adjustment of the process average will cause this process to be capable. The average range (Rbar) must be reduced, then the mean (Xdouble bar) must be centered between the specification limits.

Remember that the process is already in control; therefore, any movement in the level of process capability requires systematic changes. Noncapability is due to common causes of variation, and a solution must involve the design of the product, the setup of the equipment, or the characteristics of the materials.

In the last example, we assumed that the use of a normal distribution model was appropriate for the analysis of the process in question. This is not always the case, however. Suppose, for example, that a process was brought under control at

  • Xdouble bar = .952955, Rbar = .0045

  • USL = .9559, LSL = .9516, n = 5, k = 25, and X min = .9515

Suppose further, however, that a hypothesis of normality is rejected in an analysis of the process data. Further, let us assume that a hypothesis of exponentiality is accepted. In this case,

  • The use of Rbar/ d 2 = is irrelevant to the process capability study.

  • The table of normal values is inappropriate for use in the capability study. Rather, a table of exponential values must be used (see Selby, 1969).

It is important to note that to use the exponential distribution values table, one may be required to transform the data. Recall that a z score is used in conjunction with the normal table, because the table is based on the assumption that ¼ = 0 and ƒ = 1. As this is rarely the case with actual data, the z value transforms the distribution of observed data into one that fits these requirements, allowing us to use the normal table.

The table of exponential values also requires that certain assumptions be met. Unlike the normal table, the value of ¼ is irrelevant to the exponential table values. In this case, it is the value of the origin parameter on which the table is based. Specifically, this parameter reflects the smallest value, or origin, of the distribution and is assumed to be equal to 0.

In some cases, such as that of measuring roundness, the inherent origin parameter or value of the distribution is 0. In essence, the minimum possible value is 0, and all other values are greater than 0.

In other cases, such as in measuring a point-to-point distance between electrical leads, the minimum value ( X min ) of the distribution will be positive, but not equal to 0. In still other cases, the distribution might have a minimum value that is negative, and range to positive upper values.

In actual production situations in which exponential distributions do not originate with a value of 0 (for example, measuring initial voltage from batteries after repeated discharges), these data must be transformed (analogous to the use of the z score for the normal table) so that the origin parameter is modeled on the assumption that it ( not ¼ ) is equal to 0. To accomplish this, we use the value X min , the minimum observed value for the process. In all cases, the result of the transformation of the data set would result in a minimum value (i.e., transformed value) of 0. This would be true of data sets in which the raw data were all positive; and it would also be true where all or some of the observed data were negative, even if 0 were included in the spread of the scores. In cases such as these, the minimum negative score would be transformed to 0, and the original value of 0 would take on a positive value.

In our example, we also know that the area above any value on the curve for an exponential distribution corresponds to X/ ¼ , where X = score/value/observation. We are also aware, based upon the central limit theorem, that Xdouble bar = ¼ , regardless of the fact that the underlying process is not normally distributed. Given, therefore, the values ¼ = = .952955, USL = .9559, LSL = .9516, and X min = .9515, we follow the following steps:

  • Step 1. Convert the ¼ and specifications to exponential distribution values:

    ¼ E

    =

    ¼ - X min

    ¼ E

    =

    .952955 - 9515

     

    =

    .001455

    USL E

    =

    USL - X min

     

    =

    .9559 - .9515

     

    =

    .0044

    LSL E

    =

    LSL - X min

     

    =

    .9516 - .9515

     

    =

    .0001

  • Step 2. Obtain the X/ ¼ values for the USL and LSL:

    USL E / ¼ E = .0044/.001455 = 3.02

    and

    LSL E / ¼ E = .0001/.001455 = .069

  • Step 3. Using a table of exponential distribution values, find the area of the curve (% of parts ) outside of the specifications:

    USL = 3.02 ‰ˆ .0498

    LSL = .069 ‰ˆ .0676

    Total% out of specification = 4.98 + 6.67 = 11.74




Six Sigma and Beyond. Statistical Process Control (Vol. 4)
Six Sigma and Beyond: Statistical Process Control, Volume IV
ISBN: 1574443135
EAN: 2147483647
Year: 2003
Pages: 181
Authors: D.H. Stamatis

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