Quality Management: Total Quality Management and Statistical by 420qmn

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									UCL                    UCL
                                          UCL



                                          LCL
                       LCL                Additional improvements
LCL                    Process centered   made to the process
Process not centered   and stable
and not stable


           Quality Management:
    Total Quality Management and
      Statistical Process Control
                       S. Cholette
                        BUS786
                                                                    1
              Outline of Quality Module
•       Quality Management is very important for Operations
        Management. It is the entire focus of one DS class.
•       However, can only introduce many of the ideas and
        methods in our limited time. The emphasis in this class
        will be on Total Quality Management (TQM) and
        Statistical Process Control (SPC)
•       We skip around parts of 2 chapters of Stevenson:
    1.    Chapter 9: Management of Quality
          –   Read all for background, but focus on Quality Tools
    –      Chapter 10: Quality Control- read about Statistical Process
           Control
          – We will not study acceptance sampling or run tests
          • Focus is on identifying, creating and interpreting
             x, R, p and c charts
                                                                         2
    Introduction to Quality- Chapter 9

•   What does the term quality mean?




     Quality is the ability of a product or service to
     consistently meet or exceed customer expectations
                                                         3
         Dimensions of Quality
1. Performance - main characteristics of the
    product/service
   • Conformance - how well product/service conforms
       to specifications and to customer’s expectations
2. Aesthetics - appearance, feel, smell, taste
3. Special features - extra characteristics
4. Safety - Risk of injury
5. Reliability - consistency of performance
6. Durability - useful life of the product/service
7. Perceived Quality - indirect evaluation of quality (e.g.
   reputation)
8. Service after sale - handling of customer complaints
   or checking on customer satisfaction
                                                          4
               Costs of Quality
• Failure Costs - costs incurred by defective
  parts/products or faulty services
   – Internal Failure Costs
       • Costs incurred to fix problems that are detected before the
         product/service is delivered to the customer.
   – External Failure Costs
       • All costs incurred to fix problems that are detected after the
         product/service is delivered to the customer.
• Appraisal Costs
   – All product and/or service inspection costs
• Prevention Costs
   – All quality training, quality planning, customer
     assessment, process control, and quality
     improvement costs to prevent defects from
     occurring
• In general, increasing spending (appropriately) on
  appraisal and prevention will lower failure costs                 5
The Consequences of Poor Quality
1.       Loss of business
2.       Liability
3.       Productivity
     –     Rework consumes




                             Cost of detection (dollars)
           capacity
4.       Costs

•        Furthermore, the
         cost of poor
         quality increases
         with later
         detection
                             Process                          Final testing        Customer
                                                           When defect is detected       6
     Real Life Examples of Quality Failures:
     Thousands Given Wrong STD Results
                          10/30/03 Yahoo News


• CRANBROOK, British Columbia - About 3,000 people got the wrong
  results when they were tested for gonorrhea and chlamydia over an
  18-month period, health officials say. Because of a faulty diagnostic
  machine in this southeastern British Columbia town, test results for
  the two sexually transmitted diseases were reversed …. Officials
  were notified of the defective BD ProbeTec in July by the
  manufacturer, Becton, Dickenson and Co. of Paramus, N.J…. said.
  "The machine was flipping the tests results,… In other words, if you
  were a positive, you would have received a negative reading. If you
  were a negative, you would have received a positive reading."

• About 3,000 people were tested between Nov. 1, 2000, and May 24,
  2002, and about 83 were told they were clean when they actually
  had one of the diseases. Most of the 83 have been contacted but
  not all, Paine said. The rest of the 3,000 were told they were
  infected and were given treatment although they did not have the
  diseases, Paine said.
                                                                      7
             Ethics and Quality
• Substandard work
  –   Defective products
  –   Substandard service
  –   Poor designs
  –   Shoddy workmanship
  –   Substandard parts and materials




      Having knowledge of this and failing to correct
      and report it in a timely manner is unethical.

                                                        8
           Some “Quality” Humor
• The emphasis on quality has changed over the past few
  decades, with Japan leading the West in terms of overall
  defect levels

• Back in the 1970’s IBM Canada Ltd. of Markham, Ont.,
  ordered some parts from a new supplier in Japan. The
  company noted in its order that acceptable quality
  allowed for 1.5 percent defects (a fairly high standard in
  North America at the time). The Japanese sent the
  order, with a few parts packaged separately in plastic.
  The accompanying letter said: "We don't know why you
  want 1.5 per cent defective parts, but for your
  convenience, we've packed them separately."
                                                               9
                             A Case for Quality
  •    Many modern manufactured goods, especially computers and electronic
       goods, have hundreds or thousands of component parts
  •    Assume that for the device to work: a) All of the components have to perform
       to specification, and b) All components have the same probability of failure,
       with the chance of failure for one component independent of all others
  •    The table shows the effect of improving quality (lowering probability of
       defective parts) on overall product yield
Probability a                         Chance of device              Chance of
part is       Chance one   # parts in with 100 parts     # parts in device with 1000
defective     part works   a device working              a device parts working      Comments

                                                                                   1% defect rate generally acceptable
 1.0000%       99.000%          100            36.60%      1000          0.004%    in old-style manufacturing
                                                                                   The minimum for a "capable"
                                                                                   process- Cp = 1, where 99.74% of
 0.2600%       99.740%          100            77.08%      1000          7.402%    output in spec
                                                                                   +/- 4-sigma, process is capable,
 0.0032%       99.997%          100            99.68%      1000        96.881%     but has not achieved 6-sigma
                                                                                   Motorola's famous 6-sigma goal (Cp
                                                                                   = 2), fewer than 3.4 defective parts
 0.0003% 100.000%               100            99.97%      1000        99.661%     per million
                                                                                                                10
    Total Quality Management
TQM: A philosophy that involves everyone in an
organization in a continual effort to improve quality and
achieve customer satisfaction

TQM Approach
  •Find out what the customer wants
  •Design a product or service that meets or exceeds
  customer wants
  •Design processes that facilitates doing the job
  right the first time
  •Keep track of results
  •Extend these concepts to suppliers
                                                            11
      Continuous Improvement
• Philosophy that seeks to make never-ending
  improvements to the process of converting inputs
  into outputs.
• Commonly practiced in Japan before US
   – Kaizen: Japanese word for continuous
     improvement
   – However, the quality guru attributed with helping
     Japanese Manufacturers engage in these practices was
     American
   – And his name is…




                                                            12
                   Deming’s 14 Points
1.     Create constancy of purpose
2.     Adopt a new philosophy- we no longer have to accept defects as a
       way of life
3.     Cease dependence on mass inspection
4.     End the practice of buying on price alone (and buying from suppliers
       who can’t meet standards)
5.     Use statistical methods to find problems and continually improve
      •     SPC will be a tool in quest for continual improvement
6.     Institute modern methods of on-the-job training
7.     Improve supervision
8.     Drive out fear
9.     Break down departmental barriers
10.    Eliminate numerical goals and slogans (at least arbitrary ones…)
11.    Eliminate work standards that simply set quotas
12.    Remove barriers between employees and their pride of workmanship
13.    Institute a vigorous program of education and retraining
14.    Structure management to continually support these points          13
 Process Improvement and Tools
• Process improvement - a systematic
  approach to improving a process
  – Process mapping
  – Analyze the process
  – Redesign the process
• Tools
  – There are a number of tools that can be used for
    problem solving and process improvement
  – Tools aid in data collection and interpretation, and
    provide the basis for decision making




                                                       14
               Basic Quality Tools
• Flowcharts – useful for representing the process and
  identifying potential weak links
• Check sheets – tallies by defect/time/attribute
• Histograms
• Pareto Charts – like histograms, except ordered by
  decreasing # of defects. Usually include a cumulative
  problem line.
• Scatter diagrams – shows the correlation between two
  variables (remember Forecasting?)
• Control charts (talk about later in Ch 10)
• Cause-and-effect diagrams
• Run charts- More general than Control Charts.
   – E.g. plot # of defects over time

                                                          15
                Check Sheet
Can make it multi-dimensionally by adding element of time or
another variable

               Product Defects
     Defect type               Tally                 Total
A. Tears in fabric             ////                   4
B. Discolored fabric           ///                    3
C. Broken fiber board          //// //// //// ////
                               //// //// //// /       36
D. Ragged edges                //// //                 7
                                         Total        50

                                                               16
                        Pareto Chart
  • 80% of the problems may be from 20% of the causes!
  • What percent of complaints are caused by the coffee
    machine under filling the cups?
                                      Coffee Vending Machine Complaints, Week1

                        140
Underfilled cup    50
Too Hot            36   120

Burnt Taste        21   100
Grainy Residue     15
                        80
Other               8
total             130   60

                        40

                        20

                         0
                              Underfilled cup   Too Hot   Burnt Taste   Grainy Residue   Other



                                                                                                 17
                       Control Chart
We will study these in greater detail in SPC – Chapter 10


1020
                                                                 UCL
1010
1000
 990
 980
                                                                 LCL

 970
       0   1   2   3   4   5   6   7   8   9 10 11 12 13 14 15



                                                                  18
                Tracking Improvements
                   via Control Chart
                Week 1                Week 2                 Week 7
        UCL                    UCL
                                                      UCL

                                                                             1000ml
990ml

                                                      LCL
                               LCL
                                                      Additional improvements
        LCL                  Process centered         made to the process
        Process not centered and stable
        and not stable



        In a bottling plant that fills 1-liter bottles of soda, we can see
        that have successfully engaged in continuous improvement!
                                                                                19
                             Tracking Improvements
                                via Pareto Chart
              Coffee Vending Machine Complaints, Week1                         Coffee Vending Machine Complaints, Week 2

140                                                                      140

120                                                                      120

100                                                                      100

80                                                                       80

60                                                                       60

40                                                                       40

20                                                                       20


 0                                                                        0
      Underfilled cup   Too Hot   Burnt Taste   Grainy Residue   Other         Too Hot   Burnt Taste    Grainy   cup overflows   Other
                                                                                                       Residue




         • Improved complaint levels from 130/week to 80/week.
         • Predominant problem has changed
         • What should we focus on next?                                                                                         20
    Examples: Using TQM Tools
•   (# 2, Stevenson, p. 422) An air-conditioning repair shop is trying to determine
    the primary reasons for service calls in the previous week. Calls are
    recorded by whether they originate from residential or commercial customers
    and are categorized by problem type (Noisy, Warm, Odor, Failed)
•   Create a checklist by problem type and customer type, as well as a Pareto
    chart for each customer type. What problems should we look into?
Call#   problem   r/c   Call#   problem   r/c   Call#   problem   r/c   Call#   problem        r/c
301       f       r     311       n       r     321       f       r     331       n            r
302       o       r     312       f       c     322       o       r     332       w            r
303       n       c     313       n       r     323       f       r     333       o            r
304       n       r     314       w       c     324       n       c     334       o            c
305       w       c     315       f       c     325       f       r     335       n            r
306       n       r     316       o       c     326       o       r     336       w            r
307       f       r     317       w       c     327       w       c     337       o            c
308       n       c     318       n       r     328       o       c     338       o            r
309       w       r     319       o       c     329       o       c     339       f            r
310       n       r     320       f       r     330       n       r     340       n            r
                                                                        341       o            c
                                                                                          21
 Examples: Using TQM Tools
• Our checklist is organized by problem type and customer type.
  From here we can see that Residential customer complaints
  outnumber commercial complaints and that residential customers
  are more likely to complain about noise problems, where
  commercial customers are more likely to complain about odor.


                                 resident        commerc
                                 r               c        total
              noisy        n                10          3         13
              failed       f                 7          2          9
              odor         o                 5          7         12
              warm         w                 3          4          7
              total defect                  25         16         41



                                                                       22
 Examples: Using TQM Tools
• We create two pareto charts, one for each type of customer. The
  pareto charts show that these two different types of customers
  experience a different frequency of problems. (Note that the order
  of the attributes is different for each pareto chart)



                 Complaints: Residential

            30
            25                                                 Complaints: Commercial

            20                                            20
  # calls




            15                                            15


                                                # calls
            10                                            10


            5                                             5


            0                                             0
                                                               o        w          n      f
                 n        f          o      w                         type of complaint
                                                                                              23
                        type of complaint
      Chapter 10: Quality Control
•   As you might expect, this entails using statistical
    methods to assist with Quality Control
•   Two topics are traditionally covered

    1. Inspection
    • Acceptance Sampling is in the
      supplement to Chapter 10
      which we do not cover in this
      class

    2. Statistical Process
       Control (SPC)

                                                          24
                  Inspection
  • How Much/How Often
  • Where/When
  • Centralized vs. On-site



 Inputs            Transformation   Outputs




Acceptance             Process      Acceptance
sampling               control      sampling

                                                 25
                     Inspection Costs
Too little inspection and we incur costs from not catching defective items
Too much inspection and incur large inspection costs
 - especially costly if need to use destructive sampling (fizz testing -beer cans)
Cost




                                                                       Total Cost
                                                                        Cost of
                                                                        inspection

                                                                     Cost of
                                                                     passing
                                                                     defectives
                         Optimal
                                                                               26
                      Amount of Inspection
        Statistical Process Control
• Definition: statistical evaluation of the output of a
  process during production
   – Take periodic samples and measure to a standard. If results
     are unacceptable, stop the process and take corrective
     action

• The Control Process
   –   Define
   –   Measure
   –   Compare to a standard
   –   Evaluate
   –   Take corrective action (if necessary)
   –   Evaluate the corrective action (Did we fix it?)


                                                                   27
    Statistical Process Control
• Variations and Control
  – Random variation: Natural variations in the output
    of process, created by countless minor factors and
    unable to be eliminated
  – Assignable variation: (Causal, non-random) A
    variation whose source can be identified and
    eliminated




                                                     28
                Sampling Distribution
• Because our sample has multiple observations, it will have a
tighter distribution (less variation) than that of the process we
are sampling from
    • Many factors determine how large the sample size should be, but for
    the purposes of this class, it will be provided
• Also thanks to the Central Limit Theorem, the sampling
distribution will be normal, even if the underlying process isn’t
normally distributed                     Sampling
                                                distribution


                                                               Process
                                                               distribution




                                                                              29
                                   Mean
         Review: Normal Distribution
Unlike in inventory management, we are now concerned about both being
too high and too low. (Goldilocks Phenomenon.)
    • There is a .0228 probability a value falls 2 below the mean and a 1-.9772 =
    .0228 probability a value falls 2 above the mean. This results in a .0456
    probability that a value falls outside the interval of [m-2,m+2]
    • If we expand the interval to [m-3,m+3], the probability of being outside
    drops to .0026 (an order of magnitude less probable!)




      Standard deviation




                           -3     -2                          +2      +3
                                              Mean
                                              95.44%

                                              99.74%                              30
        Example: Using Normal
        Distributions for Quality
Lard-Os potato chips guarantees that all snack-sized bags of chips are
between 16 and 17 ounces. The machine that fills the bags has an output
that can be approximated by a normal distribution, with a mean of 16.5
ounces and a standard deviation of .25 oz. What is the percentage of
bags that violate Lard-O’s promise?
    • Given m16.5,.25 We are above/below specified weight if we go
   over/under it by .5 oz.
   • We can compute the Z-values, where z = (spec – m)/
        (16-16.5)/.25 = -.5/.25 -> z = -2
        (17-16.5)/.25 = .5/.25 -> z = 2
   • Using a table lookup on p.888 for z = -2, yields .0228 = P(x<16oz) and a
   lookup for z = 2 gives .9772
        Remember: (P(x>17) = 1 - P(x<17) = 1 - .9772 = .0228.
        Hint: If mean is centered between upper and lower bounds, you only
        need to determin z’s once and multiply by 2!
   • Therefore Lard=O’s violate their guarantee just under 5% of the time
   (2*.0228 = .0456)                                                          31
Control Limits and Errors
Type I error:
Probability of searching for
a cause when none exists

                                                    UCL



                                                   Process
                                                   average



                                                    LCL

                               (a) Three-sigma limits
                                                          33
Control Limits and Errors
Type I error:
Probability of searching for
a cause when none exists

                                                      UCL



                                                      Process
                                                      average



                                                      LCL

                               (b) Two-sigma limits
                                                            34
Control Limits and Errors
               Type II error: Probability of
               concluding that nothing has
               changed when process has!


                                     UCL

                       Shift in process
                       average        Process
                                      average



                                     LCL

                (a) Three-sigma limits
                                               35
Control Limits and Errors
               Type II error: Probability of
               concluding that nothing has
               changed when process has!



                                        UCL

                        Shift in process
                        average        Process
                                       average



                                        LCL

                (b) Two-sigma limits
                                               36
   Selecting Control Limits: Summary

• Using wider control limits, a 3-sigma (3) instead of 2-
  sigma (2) range, will decrease chance of Type I errors.
  But it will increase chance of incurring Type II errors

• In this class you will told which control limits to use.
  Typically 3-sigma limits will be used, but we may also
  look at 2-sigma




                                                             37
                     Control Chart

     Abnormal variation                         Out of
     due to assignable sources                  control
                                                                UCL

                                                                Mean
        Normal variation
        due to chance
                                                                LCL
    Abnormal variation
    due to assignable sources

0   1   2   3   4    5     6    7   8   9   10 11 12 13 14 15
                           Sample number


                                                                 38
           Control Charts for Variables
•    We use two different charts to monitor measurable
     variables
       •    Examples: diameter of a tube, length of a piston rod, weight of a
            package, temperature of a cup of coffee
       •    Ideally, we would like the process to be invariant. Deviations
            from specification in either direction are usually undesirable.
            (Too hot or too cold, Too small or too big, Too hard or too soft)

    1. Mean control chart (X-bar, x ) measures the central
       tendency of a process
    2. Range control chart (R) measures process dispersion
•    Later we will also discuss attribute charts (p-Charts and c-
     Charts) when we can’t measure

•    If any of the sample means fall outside the control limits,
     the process is considered Out of Control!
                                                                        39
                 X-Bar / Mean Charts
•     We have two different choices for computing control limits
1.    If process std. deviation,  is known (or can be estimated
      from historical data)

                       UCL  x + z x

                       LCL  x - z x
     Where:
     x  /√n   (standard deviation of distribution of sample means)
     n = sample size
     z = standard normal deviate

     x = mean of sample means or target process value
          -sometimes called x-bar-bar.
         - if we know the true process mean we would use that instead     40
        Example: X-bar Charts
          When  is Known
• Stevenson, #12: A hotel is investigating the speed of the check- in
  process. Traditionally, the process standard deviation () has
  been.146 minutes. Below are 39 samples (each sample has 14
  observations) Is the process in control (using 3-sigma control limits)?

     sample        avg time sample    avg time sample    avg time
               1         3.86      14       3.81      27       3.81
               2          3.9      15       3.83      28       3.86
               3         3.83      16       3.86      29       3.98
               4         3.81      17       3.82      30       3.96
               5         3.84      18       3.86      31       3.88
               6         3.83      19       3.84      32       3.76
               7         3.87      20       3.87      33       3.83
               8         3.88      21       3.84      34       3.77
               9         3.84      22       3.82      35       3.86
              10          3.8      23       3.89      36        3.8
              11         3.88      24       3.86      37       3.84
              12         3.86      25       3.88      38       3.79
              13         3.88      26        3.9      39       3.85    41
         Example: X-bar Charts
           When  is Known
•   Given we know , we can use method 1
•   We have been given the control limits of 3-sigma, so z = +/- 3
•   Determine the mean of means (x-bar-bar) = 3.85
•   Now since we know x-bar-bar, , and N, we use the formulae:

                UCL  x + z x      = 3.85 + 3*.146/(sqrt(14)) = 3.97

                 LCL  x - z x     = 3.85 - 3*.146/(sqrt(14)) = 3.73

• Remember to factor in that sqrt(N), for N = sample size.
     – We are actually taking into account 14*39 = 546 data points!



                                                                        42
      Example: X-bar Charts
        When  is Known
• Plotting the observations against the control limits shows
  that sample 29 is above the UCL. Therefore we would
  conclude the process is out of control.
                  3- Sigma Control Chart: Check-in Timing

       4.00
       3.95
       3.90
       3.85                                                             LCL
       3.80                                                             data
       3.75                                                             UCL
       3.70
       3.65
       3.60
          1

              4

                  7
                      10

                           13

                                16

                                     19

                                          22

                                               25

                                                    28

                                                         31

                                                              34

                                                                   37
                                                                               43
       Example extension:
      2 vs. 3 Control Limits
• 1 observation out of 39 might not suggest process was
  greatly out of control, but if we had used stricter limits (2-
  sigma) we would be even more alert.
       for 2 sigma,         UCL = 3.85 + 2*.146/(sqrt(14)) = 3.93
                            LCL = 3.85 - 2*.146/(sqrt(14)) = 3.77
                    2-sigma Control Limits for Check-in Process

            4.000
            3.950
            3.900
                                                                              lcl
            3.850
                                                                              data
            3.800
                                                                              ucl
            3.750
            3.700
            3.650
                                                                                     44
                1

                    4
                        7
                            10
                                 13

                                      16
                                           19

                                                22
                                                     25

                                                          28
                                                               31
                                                                    34

                                                                         37
      X-Bar / Mean Charts, Revisited
2. If process standard deviation is unknown, (perhaps the process is
     new or recently revised) or if we are satisfied with using the
     Average Range (works fine for sample sizes up to 10), we can
     use these, instead:
              UCL  x + A2 R
              LCL  x - A2 R
              Where
              x  average of sample means or target value
              R  average of sample Ranges (max - min)
              A2  control factor   (see chart, p. 429)

  Note: this approach assumes that the process dispersion is in control
  • if not, our control limits would be too loose
                                                                          45
  • to make sure, we would graph the forthcoming R-chart first
           Mean and Range Charts

                             (process mean is
                             shifting upward)
Sampling
Distribution


                     UCL




     x-Chart                         Detects shift

                     LCL
               UCL



                                     Does not
     R-chart
                                     detect shift
               LCL
                                                46
          Mean and Range Charts


Sampling
Distribution              (process variability is increasing)



               UCL




   x-Chart                       Does not
                                 reveal increase
               LCL
               UCL




   R-chart                       Reveals increase

               LCL
                                                     47
                     Range Chart
• Range control limits are more straightforward to compute:

          UCL  D4 R
          LCL  D3 R
          Where
          R  average of sample Ranges (max - min)
          D3 , D4  control factors (see chart)
• For small sample sizes, (n < 7), note that LCL will be 0
• Generally we are more concerned about increased
  dispersion. (Exceeding UCL)
   – However if process variability decreases below LCL, it is worth
     investigating – perhaps to use as a process improvement           48
       Bookmark: Factors for
       3-Sigma Control Limits
Also found in Table 10-3 (I will provide a copy for the final)
   Note that the values result in tighter CLs as sample size increases
         n            A2               D3              D4
         2          1.880              0              3.267
         3          1.023              0              2.575
         4          0.729              0              2.282
         5          0.577              0              2.115
         6          0.483              0              2.004
         7          0.419            0.076            1.924
         8          0.373            0.136            1.864
         9          0.337            0.184            1.816
         10         0.308            0.223            1.777
        Etc….                                                        49
    Example: Mean and Range
             Charts
• We wish to determine if screw production is in statistical
  control. We have no prior information, other than the 5
  samples (where each sample has 4 observations)


                        1        2        3        4
              s1     0.5014   0.5022   0.5009   0.5027
              s2     0.5015   0.5031   0.5029   0.5012
              s3     0.5018   0.5026   0.5035   0.5023
              s4     0.5010   0.5034   0.5011   0.5012
              s5     0.5041   0.5056   0.5034   0.5035




                                                           50
        Example: Mean and Range
                 Charts
1. First compute the sample range and mean for each of the 5
   samples
       -e.g. s1 varies between .5009 to .5027, so the range for s1 = .0018 and
       the average of the 4 observations in S1 is .5018, so that is the sample mean
2. Next we can compute
          x or x-bar-bar, the average of sample means and
          R or R-bar, the average of sample ranges
             1           2          3             4        Range        Mean
  s1         0.5014      0.5022     0.5009        0.5027   0.0018       0.5018
  s2         0.5015      0.5031     0.5029        0.5012   0.0019       0.5022
  s3         0.5018      0.5026     0.5035        0.5023   0.0017       0.5026
  s4         0.501       0.5034     0.5011        0.5012   0.0024       0.5017
  s5         0.5041      0.5056     0.5034        0.5035   0.0022       0.5042
                                             R-bar=        0.002
                                             X-bar=                     0.5025
                                                                                 51
 Example: Mean and Range
          Charts
• Since we have no prior information on process metrics,
we cannot use the first method to establish control limits
for the mean.
• Before we can use the second method to set control
limits for the mean, we first need to determine if process
variation is in control with the R – Chart
- Factor table lookup shows that sample size 4, D3 = 0, D4 = 2.28

   R = 0.0020

  UCLR = D4R = 2.282 (0.0020) = 0.00456
  LCLR = D3R = 0 (0.0020) = 0
                                                               52
              Example: Range Chart

              0.005
                                                  UCLR = 0.00456
              0.004
Range (in.)




              0.003

              0.002                               R = 0.0020

              0.001

                 0                                LCLR = 0
                      1    2    3   4     5   6
                          Sample number                        53
Example: Control Limits for X-Chart
Given the Range is in control, we can use the second
approach to find the control limits for the mean chart


     Control limits for
       x - Chart           R = 0.0020      A2 = 0.729
                           x = 0.5025

       UCLx = x + A2R
       LCLx = x - A2R
     UCLx = 0.5025 + 0.729 (0.0020) = 0.5040
     LCLx = 0.5025 - 0.729 (0.0020) = 0.5010
                                                         54
       Example: Process Out of Control
          (with Respect to Mean)

                0.5050

                0.5040                               UCLx = 0.5040
Average (in.)




                0.5030
                                                     x = 0.5025
                0.5020

                0.5010                               LCLx = 0.5010


                         1    2    3   4     5   6
                             Sample number                           55
     Control Charts for Attributes
•        For some processes, we may need to count, rather
         than measure the samples.
     –     Use the term Attributes instead of Variables
1.       p-Chart - Control chart used to monitor the
         proportion of defectives in a process
     •     Requires that all observations can be placed in 1 of 2
           categories: i.e. fresh –vs.- spoiled
2.       c-Chart - Control chart used to monitor the number
         of defects per unit
     •     Use only when number of occurrences per unit of measure
           can be counted; non-occurrences cannot be counted.
           i.e. Number of accidents at an intersection in 1 month.



                                                                    56
               Use p-Charts When:
1.       When observations can be placed into one of two
         categories
     –     Good or bad
     –     Pass or fail
     –     Operate or don’t operate

2.       When the data consists of multiple samples of
         several observations each

•        Most common example: Number of defects/errors in
         a fixed sample size.
     –     8 out of 200 apples rotten-> 4% of sample is bad


                                                              57
      Control Limits for the p-Chart
• Error rates technically should be modeled with Binomial
  distribution, but instead are approximated by Normal
• p-bar = average fraction defective in observations

          = defects observed / total observations
   – if we know the true population tendency, p, we can use it in place of p-bar
   – We estimate the standard deviation of the sample, p


  UCLp = p + zp
   LCLp = p - zp
   p =          p(1 - p)/n
                                                                            58
                      Example: p-Chart
• Modified from Stevenson, #5: In checking several samples (where a
  sample is 200 statements) of credit card statements for errors, an
  auditor found the following. (Statements were either classified as
  erroneous or correct):
                 Sample            1        2        3        4
        # erroneous statements | 4          2        5        9


• What is the fraction defective in each sample?
        4/200 = .02        2/200 = .01      5/200 = .025      9/200 = .045
• If the true fraction is unknown, what is your estimate for it?
    – Mean of sample means = (.02+ .01 + .025 + .045)/4 = .025
• What is estimate of mean and standard deviation of sampling
  distribution of fractions defective for samples of this size?
      m= .025 and p = sqrt[(1-p)(p)/N] = sqrt(.975)(.025)/200) = .011
                                                                             59
 Example: p-Chart continued
• What are the 2-sigma control limits
  Re-compute UCL/LCL for new m,p
    UCL = m+ z*p = .025 + 2*.011= .047
     LCL = m- z*p = .025 - 2*.011 = .003
• Is the process in control given the 2-sigma control limits?
    Yes, because all 4 sample means fall within the interval of [.003, .047]

• Now assume that the long-term fraction erroneous of the process is
  known to be 2%. What are the values of the mean and standard
  deviation of the sampling distribution?
   We now use p = .02 instead of p = .025 to calculate mand p
   m.02and p = sqrt(p(1-p)/N) = sqrt(.02*.98/200) = .0099

• What would the 2-sigma control limits now be, given the above info?
  Re-compute UCL/LCL for new m,p
    UCL = m+ z*p = .02 + 2*.0099 = .0398
     LCL = m- z*p = .02 - 2*.0099 = .0002

• Is process in control given the new 2-sigma control limits?
    No. Last sample mean, (#4 at .045) is above UCL                            60
                  Use of c-Charts
• Use only when the number of occurrences per
  unit of measure can be counted
• Non-occurrences cannot be counted
  – Scratches, chips, dents, or errors per item
       • Example: You lend you car out to different people every
         week. At the end of 5 weeks, the number of new dings per
         week was 3,4,2,9,2. Was the person in week 4 “out of
         control” or is this the sort of variation you might expect? (try
         for both 2 and 3 sigma CLs)
  –   Cracks or faults per unit of distance
  –   Breaks or Tears per unit of area
  –   Bacteria or pollutants per unit of volume
  –   Calls, complaints, accidents, failures per unit of time
  –   Number of typos on a page of text
                                                                       61
                        c-Chart
• In theory, should use a Poisson distribution,
  but, in practice, this can be approximated
  by a normal distribution
•          UCL  c + z c
          LCL  c - z c
          Where
          c  process   mean number of defects

          (use sample mean c if   c unknown)
                                                 62
                Example: c-Chart
• A history professor is grading a midterm essay question. To
  take her mind off the tedium, she’s decided to count the
  number of egregious grammar mistakes. (Note that as this
  is an essay, rather than a multiple choice test, she can’t
  count proportion right/wrong and then use a p-Chart)
• Is the process in control, given 3-sigma control limits?

              test    #errors   test    #errors
               1        2        8        0
               2        3        9        2
               3        1        10       1
               4        0        11       3
               5        1        12       1
               6        3        13       2
               7        2        14       0               63
                  Example: c-Chart
• Since we need to use a c-chart, and we have not been
  given any historical information for the process mean, c,
  we need to the sample mean instead.
      c-bar = average of 14 observations = 1.5

• Now compute UCL and LCL
   UCL = c-bar + z*sqrt(c-bar ) = 1.5 + 3*sqrt(1.5) = 5.17
   LCL = c-bar - z*sqrt(c-bar ) = 1.5 - 3*sqrt(1.5) = -2.17 -> 0
   Remember to truncate negative LCL’s to 0

• Looking back at all 14 observations, none exceed the 5.17
  UCL, so process is in control.


                                                                   64
                   More Practice
                 with Control Charts
• ( Part of a question from the 412 Midterm from another professor’s
  class) A clothing manufacturer is monitoring sock output. Although
  many things can go wrong in the process of making socks, they are
  only recording socks as either defective or non-defective. A sample is
  composed of 100 randomly selected socks. Here are the results:

sample             1          2         3          4          5            6
#defective         4          7         2          3          2            6




                                                                      65
         More Practice with Control Charts
•   What is the sample size? How many samples are we taking?
•    100 socks per sample, 6 samples in total
•   What type of control chart would you use and why?
•    p-Chart, (Are counting attributes and can sort each observation into 1 of 2
    categories)
•   Determine the fraction defective for each sample
•   S1- .04 S2- .07 S3- .02 S4- .03 S5-.02 S6- .06 (= #defects / #observations!)
•   If the true fraction defective for the process is unknown, what is your estimate
    for it?
•   .04 (mean of means- either average the 6 numbers above, or tally
    total_defects/600)
•   If the true fraction defective is known to be 3%, what is the mean and standard
    deviation of the sampling distribution?
•   Mean = .03. Standard Dev. = sqrt(p(1-p)/N) = sqrt(.03*.97/100) = .017
•   What would the three-sigma control limits for this process be?
     =   m+/- 3* =   .03 +/- 3*(.017) = 0 to .081 ( truncate -.02 to 0 for LCL)
•   Is this process under control, given the three-sigma control limits?
•   Yes (all of the “fraction defective” for each of the 6 samples is less than .081)
                                                                                    66
     Examples: HW Problems
• Modified from Ch10, #2: An automatic filling
  machine is used to fill 1-liter bottles of cola.
  The machine’s output is known to be
  approximately normal a process standard
  deviation of = 10ml. The output is monitored
  using means of samples of 25 observations.
• Determine the upper and lower control limits that will
  include about 95.5% of the sample means when the
  process is in control. (z =2)

• Given these sample means: 1005, 1001, 998, 1002,
  995, 999, is the process in control? Why/why not?


                                                           67
               More HW Problems
• Ch10, #3, p. 462: Checkout time at a supermarket is
  monitored using a mean and a range chart. (i.e. we don’t
  know the standard deviation of the process) Six
  samples of n=20 observations have been obtained and
  the sample means and ranges computed (so we don’t
  have calculate these ourselves, like we will in #4!)
       sample     mean               range
           1          3.06            .42
           2          3.15            .50
           3          3.13            .41
           4          3.11            .46
           5          3.06            .46
           6          3.09            .45
•   Using the factors in Table 10-2, determine upper and lower limits for mean
    and range charts
•   Is the process in control?                                                   68
       Practice Old Exam Problems
  • 1# Quiz : A program at SF State is concerned about their
    graduation rates. Every year they enroll 400 junior-level
    students, and they’ve been tracking the percent who
    manage to finish the program in a timely manner. Below are
    the data for the past 8 years. Express all calculations to 3
    decimal places: ie. 12.3% or 0.123.
        – Calculate the proportion of the class that graduates ontime each year
        – What kind of control chart should be used in this situation? Justify
          Why.
        – Compute the 3 sigma control limits
        – Are we in control, give the control limits? Why/Why Not?

year           2000   2001    2002    2003    2004    2005    2006    2007
ontime grads   230    215      250     225     245     220     235     205
                                                                         69
          More Test Problem Practice
For her quality control project, a student considers the height
  of the women players in the basketball teams in her school’s
  league. Each of the 6 teams has exactly 8 players. She’s
  recorded both the average height and the range of heights
  in inches- and she has no other information available.
   – What type chart should she use to see if the differences in height
     ranges between these teams is within what would be expected with
     random variation? Compute the control limits.
   – What type chart should she use to see if the differences in average
     heights between these teams is within what would be expected with
     random variation? Compute the control limits.
   – Are there any school teams that appear to be outliers?
                                       Range
             Team      Mean (inches)   (inches)
             school1              70              15
             school2              76              10
             school3              69              13
             school4              75              14
             school5              71              11                   70
             school6              77              15
 Yet More Old Test Questions
• #3: Odwalla’s OJ is packaged in 250 ml bottles and has a
  process standard deviation of 10 ml. In monitoring the fill
  process, 6 samples (of 25 bottles each) were collected
  and averaged.
                                                 Sample   Sample mean
  • What is the total number of bottles           s1          249
  sampled?                                        s2          252
  • What is the standard deviation of the         s3          253
  distribution of sample means
                                                  s4          248
  • Compute the 3-sigma control limits. Is
  the process in control, given these control     s5          245
  limits? Why/Why not?                            s6          253
  • Compute the 2-sigma control limits. Is the
  process in control, given these control
  limits? Why/Why not?                                           71
      Last of the Test Problems
• Exam question: An avant-garde clothing manufacturer runs a
  series of high profile, risqué ads on a billboard on Hwy101 and
  regularly collects protest calls from people who are offended by
  them. They have no idea how many people in total see the ad,
  but they have been collecting statistics on the number of phone
  calls from irate viewers.
       type        Description                   #complaints
       R         Offensive racially/ethnically         10
       M         Demeaning to men                        4
       W         Demeaning to women                    14
       I         Ad is Incomprehensible                  6
       O         other                                   2

• Depict this data with a pareto chart. Also depict the cumulative
  complaint line.
• What percent of the total complaints can be attributed to the
  most prevalent complaint?
                                                                     72
       HW Problem Continued
• The ad agency also tracks the complaints by week received.
       week       #complaints
              1            4
              2            5
              3            4
              4           11
              5            3
              6            9
• What type of control chart would we use to monitor this process?
  (a) X-bar chart           (b) R-chart   (c) p-chart   (d) c-chart
• And why?
• What are the 3-sigma control limits for this process? Assume the
  historical complaint rate is unknown.
• Is the process mean in control, according to the control limits?
  Y/N Why/Why not?
• BONUS Assume now that the historical complaint rate has been
  4 calls a week. What would the 3-sigma control limits for this
  process be now? Is the process in control, according to the         73
  control limits? Y/N (Circle one)

								
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