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Statistical Process Control Statistical

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					Statistical Process
Control
Evolution of Quality Control


      Acceptance            Process        Continuous
       Sampling             Control       Improvement

Inspection of materials    Corrective        Quality
  before and product      action during     built into
   after production        production      the process



       reactive                    proactive
Cereal Manufacturer Example

   Your company produces boxes of cereal
   Each box of cereal is supposed to be 16.0 oz
   How cereal boxes are filled:
      an empty box is moved forward on a conveyer until it is positioned
       under the filling funnel
      a door at the base of the funnel is opened
      after a certain period of time the door is closed
      the conveyer moves the boxes forward again and the process repeats


   Will each box be exactly 16.0 oz, or will there be at least a little
    variation in the weight?
   How can we tell if the box filling machine is working properly or not?
Getting Started

 Mark out this section of the notes:
 “How does variation relate to quality?
     “unusual”variation
     “too much” variation
What we will cover …

   Statistical process control
       Used during processing by comparing measurements to
        standards
       Used to detect problems with the process (by looking at output of
        process) so process can be corrected
   Process capability
       Used before processing
       Used to determine if a particular process is able to meet
        customer specifications


   Difference between them?
       Purpose & when used
       Basis (what is examined)
Getting started …
    Penny placement – round 1
         Place penny on paper and trace outline of it
         Now pick it up, hold it about 6 inches above the paper and try to
          place it in exactly the same place 5 times
         Was it in exactly the same place every time?

    Penny placement – round 2
         Close your eyes and pick up the penny
         Now try to place the penny in exactly the same place 5 times in a
          row (open your eyes to check your placement each time but close
          them before picking it up again)
         Was it in exactly the same place every time?

    Is there a difference between the 1st and 2nd rounds?
    What accounts for that difference?
Why use samples in SPC?

Individual observations tend to be too erratic
        to make trends quickly visible
    Only natural variation present

                                                Output from process forms
                                                 distributions that are stable and
                                                 predictable over time
                                                   Same mean and standard deviation
                                                   Day 1 – Day 4 samples from the
                                                    same population


       Day 1         Day 2   Day 3   Day 4
                                                Process is considered “in control”
                                                 (operating as expected)




  Distribution of
sample (n=5) means
    from day 1
Assignable cause of variation present

                                   Output from process forms
                                    distributions that are not stable or
                                    predictable over time
                                      Vary in central tendency (mean),
                                       standard deviation (variation) and
                                       shape
                                      Day 4 samples not from the same
                                       population as Day 1
Day 1   Day 2   Day 3   Day 4
                                   Process considered “out of control”
                                    (not performing as expected)
Statistical Process Control Chart
      A graphic representation of process data over time
      Time-ordered sequence of sample statistics


                                            Upper limit
                                              Target

                                            Lower limit


      SPC – monitor a process by comparing sample
       statistics to standards (limits) that are based on
       natural variation being present
What sample statistic?

   X -chart (mean)            X
   R-chart (range)            R
   P-chart (% defective)      p
   C-chart (# defective)      c
Statistical Process Control Chart

                                       3σ X           UCL
                                       2σ X
                                                 Natural or inherent
                                       1σ X         variation

                                                      X
                                       1σ X     Natural or inherent
                                                     variation
                                       2σ X
                                       - 3σ X           LCL

     U,LCL = target value + “acceptable” variation

           = overall mean + (# of std deviations)(std deviation)
Statistical Process Control Chart

  Day 1   Day 2   Day 3   Day 4

                                  If only natural variation present
                                     where should most sample
                                             values fall?




 Day 1    Day 2   Day 3   Day 4

                                  If assignable causes of variation
                                    present where should at least
                                      some sample values fall?
Statistical Process Control Chart

1. Establish control limits based on prior samples

2. Use control chart to monitor process by
   comparing sample values to established control
   limits
Let’s play “Is it a variable or an attribute?”


   Diameter of a steel bar
   # of lawn mowers per 20 that start on 1st pull
   # of complaints at a fast-food joint per week
   Length of a piece of metal
   # of bolts of fabric with too little fabric per
    week
Which chart?
                       How is quality
                        assessed?
      Measured                              Defects are
    (variable data)                           counted
   (weight, length,                       (attribute data)
     Density, etc.)
 X and R-charts                         Is the count out of some
                                         known or set number?


                      Yes                                    No
           (# of defects out of a                (# of defects each day or
           sample of n units – so              # of defects in a windshield -
           you can determine the                  so you do not know the
                % defective)                       sample size and can’t
                                                calculate the % defective)
               P-chart                                  C-chart
Control Charts for Variables

           shift in mean



                           shift in
                           variation
Variable Charts

   X-chart – detect shift in central tendency
     If process (population) standard deviation known
     If process (population) standard deviation not known
      (not enough data)


   R-chart – detect shift in the variation
What is a standard deviation?
      95.45% of values will lie within + 2 standard deviations
      99.73% of values will lie within + 3 standard deviations




                         95.45% of area
                         under the curve


                          99.73% of area
                          under the curve


            -3     -2    -1     0     +1    2     +3
In-class example

   Based on 100 samples of 5 cereal boxes each collected
    over time, you have estimated the population mean as
    16.0 oz and the population standard deviation as 1 oz.
   Determine 3s control limits for an X-chart

     X-chart:
                                            1 
                 U,LCL= μ  zσ X  16.0  3   
                                            5
                                  16.0  1.3
                                              Hint:
                                     If you have 5 samples,
In-class example                    what is your sample size?

   You have 5 samples that have been collected for the
    cereal-box filling line
         Sample                 Weights
          Time      Box 1   Box 2       Box 3   Box 4
          9 am      15.8     16.2       15.9    16.2
         10 am      16.1     16.2       15.9    15.8

         11 am      15.9     16.0       16.3    16.1

         Noon       15.7     15.8       16.1    16.2

          1 pm      15.7     16.1       15.9    16.1

   Determine control limits for all necessary charts
In-class example
 Sample                      Weight
  Time    Box 1   Box 2   Box 3   Box 4   Mean    Range
  9 am    15.8    16.2    15.9    16.2    16.02    0.4
  10 am   16.1    16.2    15.9    15.8    16.00    0.4
  11 am   15.9    16.0    16.3    16.1    16.08    0.4    n4
  Noon    15.7    15.8    16.1    16.2    15.95    0.5    X  16.0
  1 pm    15.7    16.1    15.9    16.1    15.95    0.4
                                                          R  0.42
Factors for 3s Control Limits
 Sample Size   Mean Factor   Upper Range   Lower Range
     n             A2            D4            D3
      2          1.880          3.268             0
      3          1.023          2.574             0
      4           .729          2.282             0
      5           .577          2.115             0
      6           .483          2.004             0
      7           .419          1.924         0.076
      8           .373          1.864         0.136
      9           .337          1.816         0.184
     10           .308          1.777         0.223
     12           .266          1.716         0.284
In-class example
 Sample                      Weight
  Time                                                    n4
          Box 1   Box 2   Box 3   Box 4   Mean    Range
  9 am    15.8    16.2    15.9    16.2    16.02    0.4    X  16.0
  10 am   16.1    16.2    15.9    15.8    16.00    0.4    R  0.42
  11 am   15.9    16.0    16.3    16.1    16.08    0.4    A 2  0.729
  Noon    15.7    15.8    16.1    16.2    15.95    0.5    D 4  2.282
  1 pm    15.7    16.1    15.9    16.1    15.95    0.4
                                                          D3  0

            U, LCLX  X  A 2 R  16.0  .729.42
            UCL R  D 4 R  2.282(.42) 0.96
            LCLR  D3 R  0(.42)  0
In-class Example – Munchkins

   What reflects quality?

   Set up control chart
       Take samples when you think process is working properly
       Calculate appropriate sample statistics
       Determine control limits for all appropriate charts
   Using the control chart
       Draw samples of pre-defined size at pre-determined time
        intervals
       Plot appropriate sample statistics on control chart
       Draw conclusion – what type of variation is present?
            If “out of control”, what could be assignable cause of variation?
             What can be done to fix it?
            Did corrective action remove assignable cause?
“In control” Process
      both central tendency and variation must be “in
       control” for process to be “in control”

                                     Mean = 1.0


                                     Mean = 1.0



                                     Variation =   0.2


                                     Variation = 0.2
Using the Control Chart

√   Set control limits based on data collected in prior
    periods
   Use control charts to monitor process
     Take  sample of size n at pre-specified time intervals
     Calculate the appropriate sample statistic for that
      sample
     Plot sample values on appropriate charts
     Draw conclusion and take appropriate action
     Repeat process

             What are you trying to accomplish
                     with SPC charts?
Which chart?
                       How is quality
                        assessed?
      Measured                              Defects are
    (variable data)                           counted
   (weight, length,                       (attribute data)
     Density, etc.)
 X and R-charts                         Is the count out of some
                                         known or set number?


                      Yes                                    No
           (# of defects out of a                (# of defects each day or
           sample of n units – so              # of defects in a windshield -
           you can determine the                  so you do not know the
                % defective)                       Sample size and can’t
                                                Calculate the % defective)
               P-chart                                  C-chart
Examples
   Telephone inquiries of IRS customers are monitored daily (5 days in total).
    Incidents of incorrect information or other nonconformities (such as
    impoliteness to customers) are recorded. The data for 5 sample days are
    below. Construct the 3s control chart for nonconformities.

       Day                           1       2       3    4      5

       # of nonconformities          5       10      23   20     15


   Detroit Central Hospital is trying to improve its image by providing a positive
    experience for its patients and their relatives. A 100-patient sample of
    questionnaires that accompanied meals had the following results.
    Construct 3s control limits.

        Day                      1       2       3    4   5     6      7

        # of unsatisfied
                                 8       5       4    7   8     4      6
        customers
Examples

   Telephone inquiries of IRS customers are monitored daily (5 days in total).
    Incidents of incorrect information or other nonconformities (such as
    impoliteness to customers) are recorded. The data for 5 sample days are
    below. Construct the 3s control chart for nonconformities.

           Day                         1      2      3      4      5

           # of nonconformities        5     10     23     20     15



                       U , LCL  c  z c
                               14.6  3 14.6
                       UCL  26.06
                       LCL  3.14
Examples
   Detroit Central Hospital is trying to improve its image by providing a positive
    experience for its patients and their relatives. A 100-patient sample of
    questionnaires that accompanied meals have the following results.
    Construct 3s control limits.

    Day                            1     2      3          4   5     6     7
    # of unsatisfied customers     8     5      4          7   8     4     6
    p                             .08   .05    .04     .07     .08   .04   .06


                           U , LCL  p  z
                                               
                                              p 1 p   
                                                 n
                                             .06.94 
                                  0.06  3
                                               100
                                  .06  3(.0237 )
                           UCL  .13
                           LCL  .01  0
Control Charts for M&M’s

Two situations – what chart for each?
   monitor M&M processing based on # of defects per bag
   monitor mix of colors per sample of 55 M&M’s
Managerial Considerations

   where to use control charts
       where might process go “out of control”?
       where in the process is critical?
   type of chart
       variable or attribute data
       trade-off between them – ease vs richness of data
   when to stop process
       a single sample value outside limits
       trends or patterns
   sample size
       time and cost
       effectiveness
Sample Size
In-Class Demonstration

   No food this time
   Need 8-10 volunteers to come up front and bring
    something with which to write

   Customer specifications: all rolls must be
    between 4 and 10, inclusive
   Is the process capable of meeting customer
    specifications essentially 100% of the time?
   What is needed to make it capable?
Number Production

   Process with only natural variation

         2   3   4   5   6   7   8   9    10   11   12




   Cannot meet customer’s specifications of 4-10 – process
    is not capable
 Side note: this is what Deming meant by 85% of quality
problems have to do with materials and/or processes, not
                  employee performance
Number Production

   Change process to reduce natural variation


               4   5   6   7   8    9   10




   Every roll will meet specifications – process is capable
Process Control vs Capability

   In control
     only   natural variation present


   Capable
     Process with natural variation is able to meet specs –
      process variation is “small” enough such that output
      meets specifications essentially 100% of the time
Process Variation

   Natural process variation – range for all units produced




process distribution


            -3s                                                       +3s
                                           6s
                       99.73% of area under curve falls within + 3s


   Measure as process width = + 3s (= 6s)
Specifications

   range of acceptable values established by design
    engineer or customers

   Measure as specification width
       + x (= 2x)
       upper specification – lower specification


                        x                    x
 Process Capability

    Natural variability compared to specifications– can we meet specs
     given the inherent variability?


       Lower spec                    2x                  Upper spec




                                      99.74%
process distribution


            -3s                                               +3s
                         Specification width           spec width
                  Cp =                            =
                            process width                   6s
                                        Customer specs
Process is “not capable”

                                                                         That means 4.6%
                                                                         of the units produced
                                    Only 95.4% of the area under         will not meet specs
 process distribution               the curve is between specs



               -3s         -2s       -1s                   +1s     +2s       +3s

                                           C = 0.67
          You make improvements to the process to
               reduce the natural variation p
       Process as it currently operates
            (ex: manual welding)
                                        Customer specs

Process is “capable”
                                                                         That means 0.3%
                                       Cp = 1.00


                                       Customer specs

Process is “capable”
                                                                        That means 0.27%
                                                                        of the units produced
                                                                        will not meet specs
                                   Now 99.73% of the area under
         process distribution       the curve is between specs



                           -3s   -2s   -1s          +1s    +2s    +3s




         Make improvements to process
         to reduce the natural variation
             (ex: automated welding)
                                                                               of the units produced
                                          Now 99.7% of the area under          will not meet specs
          process distribution            the curve is between specs



                            -3s   -2s    -1s    +1s            +2s       +3s
                                        Cp = 2.00
     You make more improvements to the process
       to further reduce the natural variation


                                         Customer specs
Process is “capable”
                                                                               That means 0.0003%
 and almost perfect
                                                                               of the units produced
  (Motorola’s goal)                     Now 99.9997% of the area under         will not meet specs
          process distribution          the curve is between specs


                            -6s   -4s     -2s          +2s    +4s       +6s




                               Make further
                         Improvements to process
                       (ex: improved equipment for
                            automated process)
Process Control and Capability

Process Control                       Process Capability
   Based on sample values and           Based on individual observations
    sampling distribution                 and process distribution
   Done on an on-going basis to         Done to evaluate a process before
    monitor process                       production (e.g., new specification,
   Natural vs assignable variation       new technology available)
                                         Natural variation in process
                                          compared to specs – is process
                                          capable of meeting specs given
                                          the natural variation of the process
          Process Control and Capability
            Control Chart
                                   UCL

Sample
statistic

                                   LCL




Each Xi                                     Design
                                         specifications
Process Control and Capability


 Process is …         Capable            Not capable


                                        Try to reduce
   In control          Good!
                                       natural variation


                   Find assignable
 Out of control                             Yuck!
                  cause of variation