Introduction to Statistical Quality Control_ 4th Edition_16_

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					        Chapter 7
Process and Measurement System
       Capability Analysis




       Introduction to Statistical Quality Control,
                       4th Edition
7-1. Introduction
• Process capability refers to the uniformity of the process.
• Variability in the process is a measure of the uniformity of
  output.
• Two types of variability:
   – Natural or inherent variability (instantaneous)
   – Variability over time
• Assume that a process involves a quality characteristic that
  follows a normal distribution with mean , and standard
  deviation, . The upper and lower natural tolerance limits
  of the process are
                         UNTL =  + 3
                          LNTL =  - 3
                    Introduction to Statistical Quality Control,
                                    4th Edition
7-1. Introduction

• Process capability analysis is an
  engineering study to estimate process
  capability.
• In a product characterization study, the
  distribution of the quality characteristic is
  estimated.


                Introduction to Statistical Quality Control,
                                4th Edition
     7-1. Introduction
Major uses of data from a process capability analysis

1.    Predicting how well the process will hold the tolerances.
2.    Assisting product developers/designers in selecting or
      modifying a process.
3.    Assisting in establishing an interval between sampling
      for process monitoring.
4.    Specifying performance requirements for new
      equipment.
5.    Selecting between competing vendors.
6.    Planning the sequence of production processes when
      there is an interactive effect of processes on tolerances
7.    Reducing the variability in a manufacturing process.
                       Introduction to Statistical Quality Control,
                                       4th Edition
 7-1. Introduction

Techniques used in process capability analysis

1. Histograms or probability plots
2. Control Charts
3. Designed Experiments




                  Introduction to Statistical Quality Control,
                                  4th Edition
7-2. Process Capability Analysis
     Using a Histogram or a
     Probability Plot
7-2.1 Using a Histogram
• The histogram along with the sample mean and
    sample standard deviation provides information
    about process capability.
   –   The process capability can be estimated as x  3s
   –   The shape of the histogram can be determined
   –   Histograms provide immediate, visual impression of
       process performance

                    Introduction to Statistical Quality Control,
                                    4th Edition
              Example 7-1
• Pgs. 353-354
• This procedure works if data are distributed
  normally




               Introduction to Statistical Quality Control,
                               4th Edition
 Reasons for poor process capability

• See Fig. 7-3
  – Poor process centering
     • Assume that this can be corrected
  – Excess process variability
     • Harder to correct




                  Introduction to Statistical Quality Control,
                                  4th Edition
7-2.2 Probability Plotting

•   Probability plotting is useful for
    –   Determining the shape of the distribution
    –   Determining the center of the distribution
    –   Determining the spread of the distribution.
•   Recall normal probability plots (Chapter 2)
    –   The mean of the distribution is given by the 50th
        percentile
    –   The standard deviation is estimated by
                   84th percentile – 50th percentile
                  ˆ


                      Introduction to Statistical Quality Control,
                                      4th Edition
7-2.2 Probability Plotting

Cautions in the use of normal probability plots
• If the data do not come from the assumed
   distribution, inferences about process capability
   drawn from the plot may be in error.
• Probability plotting is not an objective procedure
   (two analysts may arrive at different
   conclusions).


                  Introduction to Statistical Quality Control,
                                  4th Edition
                      Example
•   See Fig. 7-4
•   First, est = 260 psi
•   Then, est = 298 – 260 = 38 psi
•   Can also use normal probability plot to
    estimate fallout
    – If LSL = 200 psi
       • Then, from Fig 7-4, about 5% will be below that
         value

                   Introduction to Statistical Quality Control,
                                   4th Edition
7-3. Process Capability Ratios

7-3.1 Use and Interpretation of C p
• Recall
                 USL  LSL
            Cp 
                    6
    where LSL and USL are the lower and upper
    specification limits, respectively.



                 Introduction to Statistical Quality Control,
                                 4th Edition
7-3.1 Use and Interpretation of Cp

The estimate of Cp is given by


              ˆ  USL  LSL
              Cp
                     6ˆ
   Where the estimate  can be calculated using the sample
                      ˆ
      standard deviation, S, or R / d 2




                   Introduction to Statistical Quality Control,
                                   4th Edition
7-3.1 Use and Interpretation of Cp

Piston ring diameter in Example 5-1
• The estimate of Cp is

            ˆ  74.05  73.95
            Cp
                   6(0.0099)
                1.68


                Introduction to Statistical Quality Control,
                                4th Edition
7-3.1 Use and Interpretation of Cp

One-Sided Specifications

                          USL  
                   C pu 
                             3
                            LSL
                   C pl 
                             3
These indices are used for upper specification and
   lower specification limits, respectively

                  Introduction to Statistical Quality Control,
                                  4th Edition
            Example 7-2
• Pg. 359




            Introduction to Statistical Quality Control,
                            4th Edition
                 Table 7-3
• Process fallout for one- and two-sided
  specifications




               Introduction to Statistical Quality Control,
                               4th Edition
7-3.1 Use and Interpretation of Cp

Assumptions
The quantities presented here (Cp, Cpu, Clu) have some very
     critical assumptions:
1. The quality characteristic has a normal distribution.
2. The process is in statistical control
3. In the case of two-sided specifications, the process mean
     is centered between the lower and upper specification
     limits.
If any of these assumptions are violated, the resulting
     quantities may be in error.

                    Introduction to Statistical Quality Control,
                                    4th Edition
                  Table 7-4
• Recommended minimum values of the PCR
• For example, a new process with two-sided
  specifications has a recommended C p of
  1.50
  – This implies that process fallout would be 7
    ppm
• Six  would result in a Cp of 2.0

                Introduction to Statistical Quality Control,
                                4th Edition
7-3.2 Process Capability Ratio on
      Off-Center Process

•   Cp does not take into account where the
    process mean is located relative to the
    specifications.
•   A process capability ratio that does take
    into account centering is Cpk defined as
            Cpk = min(Cpu, Cpl)

                Introduction to Statistical Quality Control,
                                4th Edition
                  Figure 7-7
• All of the panels in the figure have Cp = 2.0
• But, when the process mean shifts, the
  capability of the process can change
  – Note that  does not shift




                 Introduction to Statistical Quality Control,
                                 4th Edition
             Figure 7-7, cont.
• For panel b, N(53, 22)
  – Cpk = min(Cpu, Cpl)
     • Cpu = (62-53)/[3(2)] = 1.5
     • Cpl = (53-38)/[3(2)] = 2.5
  – Cpk = 1.5




                  Introduction to Statistical Quality Control,
                                  4th Edition
7-3.3 Normality and the Process
      Capability Ratio
•   The normal distribution of the process
    output is an important assumption.
•   If the distribution is nonnormal, Luceno
    (1996) introduced the index, Cpc, defined
    as                 USL  LSL
               C pc 
                            
                          6   EXT
                            2

                Introduction to Statistical Quality Control,
                                4th Edition
                      Example
• USL = 90, LSL = 80
   – So, T = (90 + 80)/2 = 85
      • (T = target value)
   – Let X = 84
   – Then, Cpc = 1.33
• (Be careful with this result…I don’t trust it!)



                   Introduction to Statistical Quality Control,
                                   4th Edition
7-3.3 Normality and the Process
      Capability Ratio
•   A capability ratio involving quartiles of
    the process distribution is given by
                         USL  LSL
           C p (q ) 
                      x 0.99865  x 0.00135

•   In the case of the normal distribution
    Cp(q) reduces to Cp
                  Introduction to Statistical Quality Control,
                                  4th Edition
     Why does it reduce to Cp?
• In the case of the normal distribution
  – x.00135 =  – 3
  – x.99865 =  + 3




                 Introduction to Statistical Quality Control,
                                 4th Edition
7-4. Process Capability Analysis
     Using a Control Chart

•   If a process exhibits statistical control, then the
    process capability analysis can be conducted.
•   A process can exhibit statistical control, but may
    not be capable.
•   PCRs can be calculated using the process mean
    and process standard deviation estimates.



                   Introduction to Statistical Quality Control,
                                   4th Edition
                    Example
• Pgs. 373-375




                 Introduction to Statistical Quality Control,
                                 4th Edition
7-5. Process Capability Analysis
     Designed Experiments

•   Systematic approach to varying the
    variables believed to be influential on the
    process. (Factors that are necessary for
    the development of a product).
•   Designed experiments can determine the
    sources of variability in the process.

                 Introduction to Statistical Quality Control,
                                 4th Edition
                     Example
• Machine that fills bottles with a soft-drink
  beverage
  – Each machine has many filling heads that are
    independently adjusted
  – Quality characteristic measured is syrup content
    in degrees brix
  – Three possible causes of variabililty
     • Machines, heads, analytical tests

                  Introduction to Statistical Quality Control,
                                  4th Edition
               Example, cont.
•   Variability is B2 = M2 + H2 + A2
•   Conduct an experiment
•   Say the result is as shown in Fig. 7-12
•   Head-to-head variability is large
    – Improve the process by reducing this variance



                  Introduction to Statistical Quality Control,
                                  4th Edition
     7-7: Setting spec limits on
        discrete components
• Setting specifications to insure that the final
  product meets specifications
• As discussed previously, if normally
  distributed variables are linked, the result is
  normally distributed with mean the sum of
  the individual means, and variance the sum
  of the individual variances

                Introduction to Statistical Quality Control,
                                4th Edition
             Example 7-9
• Pgs. 388-389




                 Introduction to Statistical Quality Control,
                                 4th Edition
            Example 7-10
• Pgs. 389-390




                 Introduction to Statistical Quality Control,
                                 4th Edition
            Example 7-11
• Pgs. 391-392




                 Introduction to Statistical Quality Control,
                                 4th Edition
 7-8: Estimating tolerance limits
• Confidence limits
  – Provide an interval estimate of the parameters
    of a distribution
• Tolerance limits
  – Indicate the limits between which we can
    expect to find a specified proportion of a
    population


                Introduction to Statistical Quality Control,
                                4th Edition
 7-8.1: Tolerance limits based on
      the normal distribution
• Suppose x~N(, 2), both unknown
• Take a sample of size n and compute xbar and S2
• Natural tolerance limits might be estimated using:
   – Xbar + Za/2S
• Since xbar and S are only estimates, the interval
  may or may not always contain 100(1-a)% of the
  distribution


                    Introduction to Statistical Quality Control,
                                    4th Edition
 7-8.1: Tolerance limits based on
      the normal distribution
• However, we may use a constant K such
  that in a large number of samples a fraction
  g of the intervals xbar + KS will include at
  least 100(1-a)% of the distribution
• Values of K are tabulated in Appendix
  Table VII
  – 2<n<1000, g = .90, .95, .99, and a = .10, .05, .01


                   Introduction to Statistical Quality Control,
                                   4th Edition
            Example 7-13
• Pg. 396




             Introduction to Statistical Quality Control,
                             4th Edition
              Assignment
• Work odd-numbered exercises on the topics
  covered in class




              Introduction to Statistical Quality Control,
                              4th Edition
            End




Introduction to Statistical Quality Control,
                4th Edition

				
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