SPC by pengxiang

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									      BUSA 3110
Statistics for Business
  Statistical Process Control
     Kim I. Melton, Ph.D.
Transition
 Population                  Process
   We have a fixed             We have an on-going
    population (a group of       process that produces
    interest)                    output
   We want to use              We want to assess if
    information from a           the observed output
    sample from the              from the process can
    existing group to draw       be considered as a
    a conclusion about the       sample from a
    entire group.                population and
 Goal                          We want to be able to
   Describe— “What”             “infer” to future output
                                 from the process
                              Goal
                                Explain— “Why?”
      Population and Sample
Population:
All of the items/measurements of         Inference is back
interest                                 to the population
                                         that produced the
                                         sample




                                   Sample:
                                   The items/measurements
                                   actually obtained
Process and “Sample”
          Two issues arise:
            Changes can occur in an on-
             going process while you are
             collecting data—i.e., you don’t
             know if all of your data is
             coming from the same
             population
            Although describing past output
             may be useful, this is
             descriptive (history). You really
             want to be able to know what
             to expect in the future—i.e.,
             you aren’t trying to make an
             inference about the process as
             it existed while you were
             collecting data.
Classical Statistical Inference?
Suppose you could obtain data on the
process of delivering shipments to a
customer (at much cost in terms of dollars
and time). You would like to say something
about the expected delivery time for the near
future. How can you use the data?
Looking at the SAME Data as
Output from a Process…

                              ?
                              ?
                              ?
                              ?
                              ?
                              ?
Process Based Studies
           We don’t know if there is a
            population. Conditions may not
            be the same for repeated
            observations from the process.
           The goal is to make decisions
            that will relate to the future
            output from the process; to do
            this we must understand the
            underlying cause system.
           If the process is stable, the
            history of the process (combined
            with subject matter knowledge)
            can be used to predict the future
            output. No numerical measure
            of confidence can be calculated!
Aims:
Monitoring and Improvement
 Monitoring
   Determine if conditions were similar enough over the entire
    period to make some kind of descriptive statement about
    output from the process
   Determine if the variation in the process comes from the
    design of the process or from a combination of the design of
    the process and some other sources of variation
 Improvement
   Determine what type of action is called for to influence
    future outcomes
   Improvement comes from
     Changes to the process (not just to the numbers)
       To change the average
       To reduce variation
       To make the process more robust to variation
       Innovation (change the problem!)
SIPOC Model
 Suppliers provide
 Inputs to the
 Process which produces
 Output for
 Customers
Voices of the Process
 Voice   of   the   Customer
 Voice   of   the   Product/Service
 Voice   of   the   Process
 Voice   of   the   System
 Example: Bottle Closures
 Voice of the Customer –What the customer wants
  (may be the end customer or the bottler)
  Fits
    e.g., The bottler may provide specs to the closure
     manufacturer:
     E dimension - target: 1.003; LSL = .995; USL 1.011
  Clean
  Safe
    e.g., The FDA says that no more than 3 in 1000 TE bands can
     fail
  Reusable (can be taken off and put back on)
  Easy to open
  Doesn’t leak
  Stays on
    Bottle Closure
 Voice of the Product – Data from output from the
  process about the product that relates to what
  the customer wants:
  Fits  actual results on various diameters and
   other dimensions
  Clean color and lack of debris
  Safe actual number of TE bands that fail
  Reusable (can be taken off and put back on) 
   Threads
  Easy to open  actual torque required to open
  Doesn’t leak  data on liners or clearances
  Stays on  results of heat tests
  Bottle Closure
 Voice of the Process
   The factors that can be controlled during the
    production process that might impact the
    product/service characteristic
   Fit/Reusable/Doesn’t leak/Stays on dimensions 
    Machine settings: temperatures, pressure, time
    Resin: type, color
   Clean free from contamination
    Screening of resin prior to injection into the mold
   Safe TE Band
    Amount of resin forced into the mold (to control the size of
     the “bridges”)
   Doesn’t leak Liner
    Temperature of extruder head
Voice of the System
 Research and Design (design of closure and
  molds)
 Hiring practices
 Maintenance practices
 Training
   Stability
   (aka: Statistical Control)

 Stability depends on the consistency of output
  (in terms of the range of variation seen); a
  stable process could consistently produce
  product that is too small, too large, or within
  the acceptable range.
 Variation is considered stable when there are
  only common causes of variation present; if
  there are both common and special causes of
  variation present the process is not considered
  stable (i.e., it is not in a state of statistical
  control)
Stability, Acceptability, and
Desirability
 Stability: What is consistently happening
   A process is called stable if the variation in the
    outputs is predictable (may be predictably small
    or predictably large—the issue is that the range
    can be predicted)
 Acceptability: What is considered “OK” from
  the process
   Output that meets the requirements set by the
    customer is considered “acceptable.”
 Desirability: The ideal output from the process
   A process that is producing acceptable output can
    still be improved by reducing variation around a
    target.
Stability, Accuracy, Precision
Stability, Accuracy,   Acceptable vs.
Precision              Desirable
 Bottle Filling Process
 All three process
  could be stable
 Only the third
  process might be
                      1
  considered
  acceptable
 Even the third
  process could be
  made more           2
  desirable (by
  filling every
  bottle to the
  reported fill
  volume)             3
     Zero Defects vs.
     Continual Improvement
 E Dimension on a closure
   Operational Definition identifies how “deep” in the closure to
    measure, how to orient the closure on the measurement
    machine, how to obtain the “diameter,” and where to start
    and stop measuring (the threads spiral down the closure)
 Acceptable (Specifications):
   Closure: LSL .995     USL 1.011
   Bottle:  LSL .972     USL .997
 Desirable (Nominal):
   Closure: 1.003
   Bottle: .984
 If all product were made to nominal, clearance would be
  .019
 If “meeting specs” is good enough
   Clearances could range from -.002 to .039    [Some closures
    won’t fit on the bottles!!!]
____ Limits
 Spec Limits (specifications)
   Measurements that are “acceptable” to the
    customer
 Control limits
   Statistically calculated limits used to assess
    stability and estimate the range of values for
    the characteristic plotted on a control chart
 Natural Process Limits
   Range of measurements for individual items
    expected to be seen from a process (depends
    on stability)
Run Charts
              Measurements for a
               process characteristic
               are plotted in time
               order
              Patterns in the data
               indicate special causes
               of variability
  time ==>
                  Trends
                  Clusters
                  A repeating pattern
                  No variability in plotted
                   points
Run Charts
        If the process is stable all of the
        following will be true:
        • Most points will plot near some
          central value
        • Some variation will exist
        • Individual points will not be
          predictable, but the overall
          clustering of points will be
          predictable
        • No patterns will show up
Control Charts
              A Run Chart with statistically
               calculated limits
              Limits are based on data
               collected from the process
              If points plot within the limits
               and show no patterns, the
               process is said to be "in
               control" or stable
              If the process is considered
               stable, limits represent the
  time ==>     expected range of variation
               for the value plotted
              A process that is in control is
               predictable (it may or may
               not be producing desirable
               output)
  Setting Up Control Charts
Step    Determine question(s) to be answered
       1:
Step    Design data collection plan and collect data
       2:
Step    Plot run chart and look for obvious patterns
       3:
Step    If no patterns, calculate control limits (using
       4:
        formulas for the appropriate type of chart)
Step 5: Conduct runs tests
Step 6: Interpret the chart
Step 7: Determine appropriate type of action and
        take steps to accomplish this
   Runs Tests-Melton
A control chart fails to show
stability
if any of the following occur:
   • at least one point plots       A
     outside the control limits     B
   • two of three consecutive       C
     points in the same A zone      C
   • fifteen consecutive points     B
     plot in the C zones            A
   • more than seven
     consecutive points on the
     same side of the center line
   • seven or more consecutive
     increases (or decreases)
                                    Note: Other runs tests
   • fifteen consecutive points
     alternating up and down        are available in other
                                    books.
  LCL = 6.387
  CL = 15.467
  UCL = 24.547
Width = 3.02666
Insurance Quotes

   An insurance company staffs quote lines
so that independent agents can call in for
quotes on insurance. The following data
represent the time (in seconds) for one
operator to respond to five consecutive calls
from the same state. One subgroup (of five
observations) is collected each day.
The Data
  Day Obs. 1   Obs. 2 Obs. 3   Obs. 4   Obs. 5
   1   197      190     162     159      194
   2   200      192     177     227      180
   3   186      178     209     197      190
   4   206      168     209     208      182
   5   182      175     158     207      226
   6   195      179     216     213      193
   7   197      195     213     198      217
   8   208      248     193     158      177
   9   184      166     224     186      180
  10   203      185     212     214      161
  11   189      183     207     176      207
  12   223      175     196     213      200
  13   200      168     193     233      164
  14   186      161     179     155      203
  15   199      218     211     217      230
X-bar and R charts
X-Bar and R Charts
Location and Spread
Control Limits




If the process
appears to be stable,
then:
Using Excel to Calculate X-Bar and R

Assume that the data for the first subgroup are in
cells B2, C2, D2, E2, and F2

If you want to put X-Bar for this subgroup in cell I2,
then click on cell I2 and type:
            =average(B2:F2)

If you want to put R for this subgroup in cell J2,
then click on cell J2 and type:
            =max(B2:F2)-min(B2:F2)

Copy the formulas down the column to find the
corresponding values for each subgroup
X-Bars and Rs
   X-Bar    R
   180.4   38
   195.2   50
     192   31
   194.6   41
   189.6   68
   199.2   37
     204   22
   196.8   90
     188   58
     195   53
   192.4   31
   201.4   48
   191.6   69
   176.8   48
     215   31
Control Chart Constants


      n   D3     D4      A2      d2
      2   None 3.267     1.880   1.128
      3   None 2.574     1.023   1.693
      4   None 2.282     .729    2.059
      5   None 2.114     .577    2.326
      6   None 2.004     .483    2.534
      7   .076   1.924   .419    2.704
R Chart
 Calculating Control Limits
  CL: 47.667
  UCL: 2.114*47.667 = 100.767
  LCL: none (NOTE: this is different from 0)
 Conducting Runs Tests
  Width of zones: (100.767 – 47.667)/3 = 17.7
  Heights:
      100.767
      83.067
      65.367
      47.667
      29.967
      12.266
R Chart
                                    R Chart                                            Runs:

          100




          80




          60
R (n=5)




          40




          20




           0
                1   2   3   4   5   6     7   8      9   10   11   12   13   14   15
                                        Subgroup #
R Chart
(What does this tell us?)
 Remember that R looks at within subgroup
  variation.
 R does not address the magnitude of the
  observations—just the difference between the
  highest and lowest in a subgroup.
   For example, the following two subgroups would
    have the same range:
    5, 10, 15, 20
    105, 110, 115, 120
 R addresses stability with respect to spread (or
  within subgroup variation)
 Lack of stability relates to changes within a
  subgroup.
  X-Bar Chart
 Calculating Control Limits
   CL: 194.133
   LCL: 194.133 - .577(47.667) = 166.63
   UCL: 194.133 + .577(47.667) = 221.637
 Conducting Runs Tests
   Width of zones: (221.637 – 194.133)/3 = 9.168
   Heights:
      221.637
      212.469
      203.301
      194.133
      184.965
      175.798
      166.63
X-Bar Chart
                                     X-Bar Chart                                            Runs:
               230



               220



               210



               200
 X-Bar (n=5)




               190



               180



               170



               160
                     1   2   3   4   5   6     7   8      9   10   11   12   13   14   15
                                             Subgroup #
X-Bar Chart
(What does this tell us?)
 Remember that the X-Bar Chart plots
  subgroup averages.
 For the limits on the X-Bar Chart to have
  meaning, you need stability with respect to
  spread—the average R needs to be
  meaningful.
 The X-Bar chart assesses if there are
  additional sources of variation between
  subgroups (above and beyond the variation
  captured within the subgroups).
Interpreting Results
When the Process is Judged to be Stable

 Recall that control limits deal with the expected
  range of variation for the characteristic plotted on
  the control chart.
 Therefore, the limits on the X-bar chart deal with
  the expected range of variation for subgroup
  averages.
 Natural process limits, expected ranges for the
  individual values, can be calculated as:



  *based on Empirical Rule (and assumes mound
  shaped distribution for individual measurements)
            Can we use the Empirical Rule?
     Stability allows                                           Calculating Natural
      us to consider                                              Process Limits appears
      the observations                                            reasonable.
      as coming from a
      single
      distribution.
                      Histogram of
                     Individual Obs.
            16
            14
            12
                                                                 LNPL=194.133-3(20.493)
            10                                                        = 132.654
Frequency




             8
             6
             4
                                                                 UNPL=194.133+3(20.493)
             2
             0
                                                                      = 255.612
                 9

                     9

                         9

                              9

                                   9

                                        9

                                             9

                                                  9

                                                       9

                                                            9
            15

                   16

                        17

                             18

                                  19

                                       20

                                            21

                                                 22

                                                      23

                                                           24




                 Time in Seconds (max in bin given)
     Limits
 Control Limits (LCL and UCL)
   calculated from data collected from the process
   used to assess stability of the process
   related to the characteristic plotted (e.g., X-Bar or R)
 Natural Process Limits (LNPL and UNPL)
   only calculated if the process appears to be stable
   related to measurements for individual items
   provides information about range of measurements for individual
    items that can be expected
   Based on the Empirical Rule (so assumes mound shaped
    distribution of individual measurements
 Specification Limits - “Specs” (LSL and USL)
   determined by the user based on desired or needed measurements
    for an item
   related to desired (acceptable) range of measurements for
    individual items
Call Center Goals
 Suppose that management of the insurance
  company wants quotes times to be
  between two minutes and four minutes.
  (They believe any times below two minutes
  will be rushed and unfriendly, and times
  above four minutes would discourage
  future business.)
 With the current process, what proportion
  of the calls would you expect to take more
  than four minutes?
How are we doing?

 The manager’s expectation that calls will be
  between 2 and 4 minutes provides “specs” that
  we will write in terms of seconds:
 LSL = 120; USL = 240
 We just found Natural Process Limits of:
 LNPL = 132.654; UNPL = 255.612

                        120                240


                           132.654               255.612



       PROBLEM: The process is expected to
       produce some calls that are too long.
 How many calls will be too long?
 Based on a comparison of our Natural
  Process Limits and the Specs, we would
  expect to find a few calls that are too long.
 Calculating, we see
   P(X > 240) =
   =1-NORMDIST(240,194.133,20.493,1) =
    .0126
   About one and a quarter percent (1.26%) of
    the calls will be too long.
 How many calls will be too short?
 We can compare our Natural Process Limits and
  the specs to estimate that very few calls will be
  too short.
 To find the proportion of calls that would be
  expected to be too short, we find:
  P(X < 120) =
  =NORMDIST(120,194.133,20.493,1) = .00015
  Less than .02% would be expected to be too short.
Call Center Ads

Suppose the manager wants to create an
ad telling people how quickly they can
provide a quote. The manager wants to
include a statement along the lines of:

“If you call us, our agent will provide you
with a quote in less than ________
minutes.”

What number should be placed in the
blank?
  Graphically…
 Logically, the
  advertised length will
  be high enough that
  “most” of the calls will
  be handled in the
  advertised time.
 The shaded in area
  represents the calls
  that are completed
  within the advertised
  time.
 The advertised time
  will depend on the
  proportion of calls that   Where z is the value of z that
  we want to complete        has an area  to the left and is
  within the claim           found with
  number of seconds.         =normsinv()
 Process Capability
 To talk about capability of a process, we must have
  stability
 Capability refers to the “voice of the process”
 The capability of the process tells the range of
  values that can be expected for the measurements
  of some process characteristic
 Specifications (specs) provide a “voice of the
  customer.”
 Capability indexes are a fairly common way of
  communicating the relationship between
  specifications and process performance.
 Capability indexes attempt to compare the “voice
  of the process” with the “voice of the customer.”
Two Approaches
to Talking about Capability

 Approach 1:
  Comparison of Engineering Tolerances to
   Natural Tolerances
    Engineering Tolerances refer to the specifications
     for the characteristic
     ET = USL - LSL
    Natural Tolerances refer to the natural process
     limits for the characteristic
     NT = UNPL - LNPL (where natural process limits
     are calculated as m ± 3s and s is estimated by
     Rbar/d2 from a stable process)
  If NT < ET, we say the process is capable
  If NT < ET and the natural process limits are
   within the specification limits, we say the
   process is capable and meeting spec.
Capable but Not Meeting Specs???

 When capability is described in terms of the amount
 of variation (without looking at location), a process
 with very little variation could consistently produce
 unacceptable product.

 Example:
    Nails are sold by weight, but builders need to know how
    many nails are contained in boxes of a given weight.
    Suppose a builder specifies that each box of nails should
    contain 990 to 1010 nails
    (i.e., 1000 ± 10).

    The producer has reduced the variation in the weight of nails
    to the point where there is only a difference of 1 to 4 nails
    from one box to another—but, boxes actually contain 983 to
    987 nails.
  Capability (cont.)
 Approach 2:
  Capability Indexes
    Cp tells us if the natural variation is smaller than the
     allowed variation. Cp does not look at process location;
     therefore it is possible to have a ‘good’ Cp and be
     making large amounts of unacceptable product.



    Cpk tells us if the natural variation is ‘small enough’ and
     ‘far enough’ from the specifications for most product to
     meet specs for the characteristic. Cpk cannot be larger
     than Cp.
Capability Formulas
          USL - LSL             ET
  Cp =                      =
                 6s             NT




  Cpk = min   {       USL - X
                          3s    ,   X - LSL
                                      3s      }
 If the process is centered between the specs:

         Cpk =
   Selecting the Appropriate Type
   of Control Chart

 Since control limits are
  calculated from data
  collected from the process,
  we need to know which
  formulas to use!
Control Charts–Variables Data

 X-bar and R charts (used together)
  X-bar (the average of n observations)
   attempts to
   assess location
  R (the range of n observations) attempts to
   assess spread

 X and Moving Range charts (used together)
  When there is no logical grouping, individual
   values are plotted on the X chart
  A moving range is used to assess spread
Control Charts–Attributes Data
 p or np charts
   "n" items are studied
   each item is classified in one of two categories
   we are counting the number in one of the
    categories
     a p chart plots the proportion in one category
     an np chart plots the number in one category
 c or u charts
   an inspection unit (IU) is defined
   the number of occurrences are counted and
    plotted
  Determine
 characteristic
    to study
                                         Counting or
                        counting          Measuring      measuring



             Attributes Data                                   Variables Data


              Classifying                                       How many
                                    no
               into two                                         items per
       yes    categories?                                       subgroup?

                                                         one                    2 to 8

      Constant                      Constant
      subgroup                       area of
        size?                      opportunity?          Consider       Consider
                                                          X/mR            X/R
yes                no
                                   yes          no


Consider     Consider       Consider          Consider
 np or p        p              c                 u
Examples
 Potholes per mile
 Time of delivery of the meal cart
  at lunch at a nursing home
 Complaints per day (lunch meal) at a
  nursing home
 1:1s in a mental health facility
 Confiscated items per day at Atlanta
  Hartsfield Jackson Airport
 Late arriving flights per day by Delta at ATL
 Weight of contents of cans of tomato soup
 Food Service

   A nursing home serves meals to 100
patients each day. Meals are prepared in a
central kitchen facility, served onto trays,
placed in specially heat carts, and delivered
to the staff on the units for distribution to
patients. Lunch is scheduled to be served at
noon. In order to start to address
complaints about cold food being delivered
to patients, some data was collected.
     The Data
Day of   Number of Time of Cart    Day of   Number of Time of
Cart
Week     Complaints     Delivery   Week     Complaints   Delivery
Mon.     39 11:30      Tues.                30 11:58
Tues.    31 11:35     Wed.                  38 11:50
Wed.     30 12:00     Thurs.                24 12:20
Thurs.   27 12:15     Fri.                  25 12:03
Fri.     38 11:45     Sat.                  28 12:04
Sat.     33 11:32     Sun.                  39 12:56
Sun.     35 12:00     Mon.                  31 11:35
Mon.     26 12:15


                                     Total number of complaints:
                                     474
Complaints Data
(actually data on the # of “complainers”)

    Day of   Number of                 Day of    Number of
    Week     Complaints                Week      Complaints
    Mon.     39 Tues.     30
    Tues.    31 Wed.      38
    Wed.     30 Thurs.         24
    Thurs.   27 Fri.      25
    Fri.     38 Sat.      28
    Sat.     33 Sun.      39
    Sun.     35 Mon.      31
    Mon.     26

                               Total number of complaints: 474
Formulas
             For this example:
            Formulas:
                                             Complaints




                              15
                                   20
                                        25
                                                30
                                                          35
                                                               40
                                                                    45
                                                                         50
               M
                   on
                          .
              Tu
                 e      s.
               W
                ed
                          .
              Th
                   ur
                        s.

                   Fr
                      i.

               Sa
                 t.

               Su
                  n       .
               M
                   on
                          .
              Tu
                 e




Day of Week
                        s.
               W
                ed
                          .
                                                                              # of Complaints




              Th
                   ur
                        s.

                   Fr
                      i   .
               S
                    at
                       .
               Su
                  n       .
               M
                   on
                          .
                                                                                                np Chart for # of Complaints
Using Data to Generate Theories
Revised Complaint Data
Suppose the data had been:
 Day # Complaints# Meals Served   pi
  1        39           95      .411
  2        31          102      .304
  3        30           87      .345
  …         …            …      …
 15        31          110      .282

 Totals   474      1500
p chart for Complaints
                 Point 1:
                Formulas




                Point 2:
p Chart for Complaints
  1:1's in a Mental Hospital

      When patients 'act-out' in ways that can be
hazardous to themselves or to others, they may
be placed on one-to-one (1:1). When a patient
is on 1:1 a staff member must remain within
arm's distance of the patient at all times. This
requires one staff member to have no other
duties other than the supervision of this one
patient. Obviously, this is very expensive.
      In an attempt to improve service and
reduce costs, the number of 1:1's is being studied.
To start, data from the past quarter were
studied.
The Data (Day Shift)
    Week   1:1’s          Week      1:1’s
      1     23             10        18
      2     18             11        19
      3     27             12        17
      4     35             13        16
      5     10             14        14
      6     19             15        21
      7     24             16        11
      8     20             17        14
      9     21             18        15




       Total number of 1:1's: 342
Formulas for c Chart
                 For this example:
                Formulas:
c Chart for 1:1s
                                   1:1s for Day Shift
          40


          35


          30


          25
  1:1's




          20


          15


          10



          5


          0
               1   2   3   4   5    6   7   8    9   10   11   12   13   14   15   16   17   18

                                                Week
Improvement?
                                     1:1s for Day Shift
        40



        35



        30


        25
1:1's




        20



        15



        10



        5



        0
             1   2   3   4   5   6   7   8   9   10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
                                                     Week

								
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