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									MD 021 - Management and Operations

     Statistical Process Control
               Outline


    Definition of statistical process control (SPC)

    Process variation

    Characteristics of control charts

    Types of control charts

    Choosing an SPC method




                  1
                    Definition of Statistical Process Control



Statistical process control (SPC) is the application of statistical techniques to
determine whether the output of a process conforms to the product or service
design.

SPC is implemented via control charts that are used to monitor the output of the
process and indicate the presence of problems requiring further action.




                                          2
                                     Process Variation

All processes exhibit variation. For example, a machine that fills cereal boxes will not
put exactly the same amount in each box.

If we were to collect data over time on the amount of cereal in each box, a plot of the
data would be described as a distribution.

A distribution is characterized by its:
   Mean                                                                        Cereal Production

   Spread                                                      700


     Range
                                                                600




                                                 No. of boxes
                                                                500
     Standard deviation                                        400

                                                                300

   Shape                                                       200

     Symmetric                                                 100


     Skewed
                                                                 0
                                                                      9.7 9.75 9.8 9.85 9.9 9.95 10 10.1 10.1 10.2 10.2 10.3 10.3

                                                                                    Average weight (oz.)




                                             3
                           Quality Measurements


Control charts can be used to monitor processes where output is measured as
either variables or attributes.

Variables measures - Characteristics of a product or service that can be
measured on a continuous scale. Examples include length, width, and time.

Attributes measures - Characteristics of a product or service that can be
quickly counted for acceptable quality. Examples include the number of defects
in a product or service.




                                       4
                         Sources of Process Variation


Sources of process variation can be categorized as:

 Common causes - Random, or unavoidable, sources of variation within a
  process. A process with only common causes of variation is stable (i.e. the
  mean and spread do not change over time). Such a process is said to be “in a
  state of statistical control” or “in-control”.

 Assignable causes - Any cause of variation that can be identified and
  eliminated, originating from outside the normal process




                                        5
                      Characteristics of Control Charts


A control chart is a time-ordered diagram to monitor a quality characteristic,
consisting of:

   A nominal value, or center line – The average of several past samples

   Two control limits used to judge whether action is required, an upper
    control limit (UCL) and a lower control limit (LCL)

   Data points, each consisting of the average measurement calculated from a
    sample taken from the process, ordered over time. By the Central Limit
    Theorem, regardless of the distribution of the underlying individual
    measurements, the distribution of the sample means will follow a
    normal distribution. The control limits are set based on the sampling
    distribution of the quality measurement.



                                        6
                               Purpose of Control Charts


Control charts are intended to reflect only common causes of variation in order to detect
assignable causes of variation.


Question: If, at the time we are constructing a control chart, there are assignable causes
of variation in the process, how can we construct a meaningful control chart?
Answer: By carefully choosing our sample size so that only common causes are found
within a sample.


The control limits are set to:
   Usually detect when the process has gone out of control (narrow control limits
    work better)

   Usually not overreact to random variation (wider control limits work better)

  The control limits are set to strike a balance between these competing priorities.


                                             7
                                 Control Charts for Variables
                          Standard Deviation of the Process,  , Known


Control charts for variables (with the standard deviation of the process,  , known ) monitor the
mean, X , of the process distribution.

The control limits are:

     UCL  X  z X

      LCL  X  z X

     where: X = center line of the chart and the average of several past sample means, z is the
     standard normal deviate (number of standard deviations from the average),  X   / n and is
     the standard deviation of the distribution of sample means, and n is the sample size




                                                  8
2. An automatic filling machine is used to fill 1-liter bottles of cola. The machine’s output is
approximately normal with a mean of 1.0 liter and a standard deviation of 0.01 liter. Output is
monitored using means of samples of 25 observations.
a. Determine upper and lower control limits that will include roughly 97 percent of the sample means
when the process is in control.
x     = 1.0 liter,  = .01 liter, n = 25
                                      
      a. Control limits : x  2               ,         [z = 2.17 for 97%]
                                       n
                                      .01
         UCL is 1.0  2.17                 1.0043
                                       25
                                      .01
         LCL is 1.0  2.17                 .9957
                                       25

b. Given the sample means: 1.005, 1.001, 0.998, 1.002, 0.995 and 0.999, is the process in control?
                          1.006       out
         b.
                         1.0043   *                                  UCL
                          1.002

                          1.000                         
              (liters)     .998           

              Mean        .9957
                                                                 
                           .994
                                                   
                                                                     LCL

                                                            *
                                                            9
                                                  out
                                    Control Charts for Variables
                         Standard Deviation of the Process,  , Unknown

Control charts for variables monitor the mean ( X chart) and variability (R chart) of the process
distribution.

R chart: To calculate the range of the data, subtract the smallest from the largest measurement in
         the sample.

  The control limits are:
     UCLR  D4 R and LCLR  D3 R

     where: R = average of several past R values and is the central line of the control chart, and
     D3 , D4 = constants that provide three standard deviation (three-sigma) limits for a given sample
     size

X chart: The control limits are:

     UCL X  X  A2 R and LCL X  X  A2 R

     where: X = central line of the chart and the average of past sample means, and A2 = constant to
     provide three sigma limits for the process mean.



                                                 10
     Control Chart for Variables Example
                        X , R Charts

Webster Chemical Company produces mastics and caulking for the
construction industry. The product is blended in large mixers and
then pumped into tubes and capped. Webster is concerned whether
the filling process for tubes of caulking is in statistical control.
The process should be centered on 8 ounces per tube. Several
samples of eight tubes are taken and each tube is weighed in
ounces.

                              Tube number
Sample     1      2      3      4     5       6      7      8
  1      7.98   8.34   8.02   7.94 8.44     7.68   7.81   8.11
  2      8.23   8.12   7.98   8.41 8.31     8.18   7.99   8.06
  3      7.89   7.77   7.91   8.04 8.00     7.89   7.93   8.09
  4      8.24   8.18   7.83   8.05 7.90     8.16   7.97   8.07
  5      7.87   8.13   7.92   7.99 8.10     7.81   8.14   7.88
  6      8.13   8.14   8.11   8.13 8.14     8.12   8.13   8.14

a) Assuming that taking only 6 samples is sufficient, use the data
   in the table to construct three-sigma R-chart and X -chart
   control limits. Is the process in statistical control?

b) The process variability for the first and sixth samples appear to
   be out of control. Webster looks for assignable causes and
   quickly notes that the weighing scale was gummed up with
   caulking. Apparently, a tube was not properly capped. The
   sticky scale did not correctly read the variation in weights for
   the sixth sample.

  Delete that data and recalculate R , UCLR , and LCLR . Is the
  process in statistical control?



                                    11
            Solution to Control Chart for Variables Example

   a)

                              Tube number
Sample     1      2      3      4     5       6      7      8    Avg.    Range
  1      7.98   8.34   8.02   7.94 8.44     7.68   7.81   8.11   8.040   0.76
  2      8.23   8.12   7.98   8.41 8.31     8.18   7.99   8.06   8.160   0.43
  3      7.89   7.77   7.91   8.04 8.00     7.89   7.93   8.09   7.940   0.32
  4      8.24   8.18   7.83   8.05 7.90     8.16   7.97   8.07   8.050   0.41
  5      7.87   8.13   7.92   7.99 8.10     7.81   8.14   7.88   7.980   0.33
  6      8.13   8.14   8.11   8.13 8.14     8.12   8.13   8.14   8.130   0.03
                                                                 8.050   0.38
   X = 8.050, R = 0.38, n = 8
   From Table 7.1:
   UCLR  D4 R  1.864(0.38)  0.708
   LCLR  D3 R  0.136(0.38)  0.052
   UCLX  X  A2 R  8.050  0.373(0.38)  8.192
   LCLX  X  A2 R  8.050  0.373(0.38)  7.908

b) We delete the sixth observation and recalculate the control limits. The
   ranges, including the range for the first sample are now all within the
   revised control limits, and the process average for the second sample
   now falls just inside of the revised control limits.

      (0.76  0.43  0.32  0.41  0.33)
   R                                     0.45
                      5
      8.040  8.160  7.940  8.050  7.980
   X                                          8.034
                         5
   UCLR  D4 R  1.864(0.45)  0.839
   LCLR  D3 R  0.136(0.45)  0.061
   UCLX  X  A2 R  8.034  0.373(0.45)  8.202
   LCLX  X  A2 R  8.034  0.373(0.45)  7.866


                                     12
                         Control Charts for Attributes
                                   p-Chart

A p-chart is a commonly used control chart for attributes, whereby the quality
characteristic is counted, rather than measured, and the entire item or service can
be declared good or defective.

The standard deviation of the proportion defective, p, is:  p    p(1  p) / n ,
where n = sample size, and p = average of several past p values and central line
on the chart.

Using the normal approximation to the binomial distribution, which is the actual
distribution of p,
                    UCL p  p  z p and LCL p  p  z p

where z is the normal deviate (number of standard deviations from the average).


                                        13
               Control Chart for Attributes Example
                             p-Chart

A sticky scale brings Webster’s attention to whether caulking tubes
are being properly capped. If a significant proportion of the tubes
aren’t being sealed, Webster is placing their customers in a messy
situation. Tubes are packaged in large boxes of 144. Several
boxes are inspected and the following number of leaking tubes are
found:
     Sample     Tubes       Sample      Tubes    Sample      Tubes
        1          3            8          6        15          5
        2          5            9          4        16          0
        3          3           10          9        17          2
        4          4           11          2        18          6
        5          2           12          6        19          2
        6          4           13          5        20          1
        7          2           14          1
                                                  Total        72
Calculate p-chart three-sigma control limits to assess whether the
capping process is in statistical control.

                           72
Solution: n = 144, p             0.025
                         20(144)
       p(1  p)     0.025(1  0.025)
p                                  0.013
          n               144
UCL p  p  z p  0.025  3(0.013)  0.064
LCL p  p  z p  0.025  3(0.013)  0.014 (adjusted to zero)

The highest proportion of defectives occurs in sample #10, but is
still within control limits: p10  9 / 144  0.0625 . Therefore, the
process is in statistical control.



                                  14
                         Control Charts for Attributes
                                   c-Chart


A c-chart is another type of control chart for attributes, whereby the quality
characteristic is counted as the number of defects/ unit.

Using the normal approximation to the Poisson distribution, which is the actual
distribution of c,
                     UCLc  c  z c and LCLc  c  z c

where c is the average number of defects/unit and the center line of the c-chart.




                                         15
              Control Chart for Attributes Example
                            c-Chart

At Webster Chemical, lumps in the caulking compound could
cause difficulties in dispensing a smooth bead from the tube.
Testing for the presence of lumps destroys the product, so Webster
takes random samples. The following are the results of the study:

 Tube #       Lumps       Tube #        Lumps      Tube #   Lumps
   1            6           5             6           9       5
   2            5           6             4          10       0
   3            0           7             1          11       9
   4            4           8             6          12       2

Determine the c-chart two-sigma upper and lower control limits for
this process.


Solution:

     ( 6  5  0  4  6  4  1  6  5  0  9  2)
c                                                    4
                            12
c  c  4  2
UCLc  c  z c  4  (2)(2)  8
LCLc  c  z c  4  (2)(2)  0

The eleventh tube has too many lumps (9), so the process is
probably out of control.




                                   16
                   Process Capability Exercise

Webster Chemical’s nominal weight for filling tubes of caulk is
8.00 ounces  0.60 ounces. The target process capability ratio is
1.33. The current distribution of the filling process is centered on
8.054 ounces with a standard deviation of 0.192 ounces. Compute
the process capability index to assess whether the filling process is
capable and set properly.


Solution:

Process capability ratio:

       Upper specification - Lower specification 8.6  7.4
Cp                                                        10417
                                                              .
                          6                         .
                                                  6(0192)


Process capability index:

                     X - Lower specification Upper specification - X
C pk  miminum of                             ,
                                3                         3
                     8.054  7.400         8.600 - 8.054
                                    1135,
                                      .                   0.948
                           .
                        3(0192)                 .
                                             3(0192)


C pk  0.948

The process is not capable of consistently meeting specifications
according to the minimum capability level set by Webster.




                                 17
Supplementary Material:
1. Calculating P
     xi
pi 
     n
p i = proportion of defective products
x i = number of the defected products
n = sample size
m = number of samples
                                                                    m

                                           ( x1  x2  ...  xm )  i
                             x1 x2    x
                                 ... m                                x
   ( p1  p 2  ...  p m ) n n        n 
p                                                               i 1
             m                   m                 mn             mn


   total number of defected products
p
       total number of products




                                   18

								
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