Introduction to Statistical Quality Control, 4th Edition

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Transcript Introduction to Statistical Quality Control, 4th Edition

Chapter 7

Process and Measurement System Capability Analysis

Introduction to Statistical Quality Control, 4th Edition

7-1. Introduction

• • Process capability

Variability

refers to the uniformity of the process.

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 =  LNTL =  + 3  - 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

Predicting how well the process will hold the tolerances.

Assisting product developers/designers in selecting or modifying a process.

Assisting in Establishing an interval between sampling for process monitoring.

Specifying performance requirements for new equipment.

Selecting between competing vendors.

Planning the sequence of production processes when there is an interactive effect of processes on tolerances Reducing the variability in a manufacturing process.

Introduction to Statistical Quality Control, 4th Edition

7-1. Introduction Techniques used in process capability analysis

Histograms or probability plots Control Charts 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  3 s The shape of the histogram can be determined (such as if it follows a normal distribution) Histograms provide immediate, visual impression of process performance.

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 50 th percentile – The standard deviation is estimated by  84 th percentile – 50 th percentile 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

7-3. Process Capability Ratios

•

7-3.1 Use and Interpretation of C p

Recall C p  USL  6  LSL 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 C p The estimate of C p is given by

p  USL  6  LSL Where the estimate can be calculated using the sample standard deviation, S, or R / d 2 Introduction to Statistical Quality Control, 4th Edition

7-3.1 Use and Interpretation of C p

• •

Piston ring diameter in Example 5-1

USL=74.05, LSL=73.95,

   0 .

023 ,

2  2 .

326

2  0 .

023 2 .

326  0 .

0099 p The estimate of C p is  74 .

 1 .

68 05  73 .

6 ( 0 .

0099 ) 95 Introduction to Statistical Quality Control, 4th Edition

7-3.1 Use and Interpretation of C p One-Sided Specifications

C pu C pl   USL     3  LSL 3  These indices are used for upper specification and lower specification limits, respectively Introduction to Statistical Quality Control, 4th Edition

7-3.1 Use and Interpretation of C p Assumptions

The quantities presented here (C p , C pu , C lu ) have some very

critical

assumptions: 1.

The quality characteristic has a normal distribution.

The process is in statistical control 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

7-3.2 Process Capability Ratio an Off-Center Process

• • C p 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 C pk defined as C pk = min(C pu , C pl ) 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, C pc , defined as C pc  6 USL  E  LSL X  T 2 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 C p ( q )  USL  LSL x 0 .

99865  x 0 .

00135 • In the case of the normal distribution C p (q) reduces to C p Introduction to Statistical Quality Control, 4th Edition

7-3.4 More About Process Centering

• • C pk should not be used alone as an measure of process centering.

C pk depends inversely on  large as  approaches zero. and becomes (That is, a large value of C pk does not necessarily reveal anything about the location of the mean in the interval (LSL, USL) Introduction to Statistical Quality Control, 4th Edition

7-3.4 More About Process Centering

• An improved capability ratio to measure process centering is C pm .

C pm  USL  6  LSL where  is the squre root of expected squared deviation from target: T =½(USL+LSL),  2  E   x  T  2    2  (   T ) 2 Introduction to Statistical Quality Control, 4th Edition

•

7-3.4 More About Process Centering

C pm can be rewritten another way: C pm  6 USL  2   (  LSL  T ) 2  C p 1   2 where   T    Introduction to Statistical Quality Control, 4th Edition

•

7-3.4 More About Process Centering

A logical estimate of C pm is: ˆ pm  p 1  V 2 where V  T  x S Introduction to Statistical Quality Control, 4th Edition

7-3.4 More About Process Centering

•

Example 7-3.

Consider two processes A and B.

For process A: C pm  C p 1   2  1 .

0 1  0  1 .

0 since process A is centered.

• For process B: C pm  C p 1   2  2 .

0  0 .

63 1  (  3 ) 2 Introduction to Statistical Quality Control, 4th Edition

• •

7-3.4 More About Process Centering

third generation

process capability ratio, proposed by Pearn et. al. (1992) is C pkm   1  C pk    T C pk 1   2 2 C pkm has increased sensitivity to departures of the process mean from the desired target.

Introduction to Statistical Quality Control, 4th Edition

7-3.5 Confidence Intervals and Tests on Process Capability Ratios

•

C p

Ĉ p is a point estimate for the true C p , and subject to variability. A

100(1-



) percent confidence interval

on C p is p  1 2   / 2 , n  1 n  1  C p  p  2  / 2 , n  1 n  1 Introduction to Statistical Quality Control, 4th Edition

7-3.5 Confidence Intervals and Tests on Process Capability Ratios

Example 7-4.

USL = 62, LSL = 38, n = 20, S = 1.75, The process mean is centered. The point estimate of C p  is 62  38 p 6 ( 1 .

75 )  95% confidence interval on C p is 2 .

29  2 1   / n 2 , n  1  1  C p  p  2  n / 2 , n  1  1 1 2 .

29 .

57  8 .

91 19 C p  Introduction to Statistical Quality Control,  C p 3 .

01  4th Edition 2 .

29 32 .

85 19

7-3.5 Confidence Intervals and Tests on Process Capability Ratios

•

C pk

Ĉ pk is a point estimate for the true C pk , and subject to variability. An approximate

100(1-



) percent confidence interval

on C pk is pk    1  Z  / 2 1 9 n C pk  2 ( n 1  1 )     C pk  pk    1  Z  / 2 1 9 n C pk  2 ( n 1  1 )    Introduction to Statistical Quality Control, 4th Edition

7-3.5 Confidence Intervals and Tests on Process Capability

•

Ratios Example 7-5.

n = 20, Ĉ pk = 1.33. An approximate 95% confidence interval on C pk Cˆ pk     1  Z  / 2 9 1 n Cˆ pk  2 ( n 1  is 1 )      C pk  Cˆ pk     1  Z  / 2 1 9 n Cˆ pk  2 ( n 1  1 )     1 .

33    1  1 .

96 1 9 ( 20 ) 1 .

33  1 2 ( 19 )     C pk  1 .

33    1  1 .

96 1 9 ( 20 ) 1 .

33  1 2 ( 19 )    0 .

99 

C pk

 1 .

67 The result is a very wide confidence interval ranging from below unity (bad) up to 1.67 (good). Very little has really been learned about actual process capability (small sample, n = 20.) Introduction to Statistical Quality Control, 4th Edition

7-3.5 Confidence Intervals and Tests on Process Capability Ratios

•

C pc

Ĉ pc is a point estimate for the true C pc , and subject to variability. An approximate

100(1-



) percent confidence interval

on C pc is 1  pc t  2 , n  1   c s c n    C pc  1  pc t  2 , n  1   c s c n   where c  1 n i n   1 x i  T Introduction to Statistical Quality Control, 4th Edition

7-3.5 Confidence Intervals and Tests on Process Capability Ratios Testing Hypotheses about PCRs

• • • May be common practice in industry to require a supplier to demonstrate process capability.

Demonstrate C p meets or exceeds some particular target value, C p0 .

This problem can be formulated using hypothesis testing procedures Introduction to Statistical Quality Control, 4th Edition

7-3.5 Confidence Intervals and Tests on Process Capability Ratios

• • •

Testing Hypotheses about PCRs

The hypotheses may be stated as H H 0 0 : C : C p p   C C p0 p0 (process is not capable) (process is capable) We would like to reject H o Table 7-5 provides sample sizes and critical values for testing H 0 : C p = C p0 Introduction to Statistical Quality Control, 4th Edition

7-3.5 Confidence Intervals and Tests on Process Capability Ratios

• Table 7-5 Introduction to Statistical Quality Control, 4th Edition

7-3.5 Confidence Intervals and Tests on Process Capability Ratios

• • • • •

Example 7-6

H 0 : C p = 1.33

H 1 : C p > 1.33

High probability of detecting if process capability is below 1.33, say 0.90. Giving C p (Low) = 1.33

High probability of detecting if process capability exceeds 1.66, say 0.90. Giving C p (High) = 1.66

 =  = 0.10.

Determine the sample size and critical value, C, from Table 7-5.

Introduction to Statistical Quality Control, 4th Edition

7-3.5 Confidence Intervals and Tests on Process Capability Ratios

• • •

Example 7-6

Compute the ratio C p (High)/C p (Low): C p ( High ) C p ( Low )  1 .

66 1 .

33  1 .

25 Enter Table 7-5, panel (a) (since  =  = 0.10). The sample size is found to be n = 70 and C/C p (Low) = 1.10

Calculate C: C  Cp ( Low )( 1 .

10 )  1 .

33 ( 1 .

10 )  1 .

46 Introduction to Statistical Quality Control, 4th Edition

7-3.5 Confidence Intervals and Tests on Process Capability Ratios

•

Example 7-6

– Interpretation: To demonstrate capability, the supplier must take a sample of n = 70 parts, and the sample process capability ratio must exceed 1.46.

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

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

7-6. Gage and Measurement System Capability Studies

•

7-6.1 Control Charts and Tabular Methods

Two portions of total variability : – product variability which is that variability that is inherent to the product itself – gage variability or measurement variability which is the variability due to measurement error  2 Total   2 product   2 gage Introduction to Statistical Quality Control, 4th Edition

7-6.1 Control Charts and Tabular Methods

• X

and R Charts

interpreted as that due to the ability of the gage to distinguish between units of the product • The variability seen on the R chart can be interpreted as the variability due to operator.

Introduction to Statistical Quality Control, 4th Edition

7-6.1 Control Charts and Tabular Methods

• Example 7-7 Introduction to Statistical Quality Control, 4th Edition

Introduction to Statistical Quality Control, 4th Edition

7-6.1 Control Charts and Tabular Methods

• •

Precision to Tolerance (P/T) Ratio

An estimate of the standard deviation for measurement error is  ˆ

gage



R d

2  1 .

0 1 .

128  0 .

887 The P/T ratio is

 6  ˆ

gage USL



LSL

 6  0 .

887  60  5  0 .

097 Introduction to Statistical Quality Control, 4th Edition

7-6.1 Control Charts and Tabular Methods

• From the actual sample measurements in Table 7-6, we can calculate s = 3.17. This is an estimate of the standard deviation of total variability, including both product  ˆ variability and gage variability. Therefore, 2

Total



2  3 .

17 2  10 .

05  2

Total

  2

product

  2

gage

 ˆ 2

gage

 0 .

887 2  ˆ 2

product

  ˆ 2

Total

 0 .

79   ˆ 2

gage

 10 .

05  0 .

79  9 .

26  ˆ

product

 9 .

26  3 .

04 Introduction to Statistical Quality Control, 4th Edition

7-6.1 Control Charts and Tabular Methods

•

Percentage of Product Characteristic Variability

A statistic for process variability that does not depend on the specifications limits is the percentage of product characteristic variability:  ˆ  ˆ

gage product

 100  0 .

887  100 3 .

04  29 .

2 % Introduction to Statistical Quality Control, 4th Edition

7-6.1 Control Charts and Tabular Methods

•

Gage R&R Studies

Gage repeatability and reproducibility studies involve breaking the total gage variability into two portions: (R&R) – repeatability which is the basic inherent precision of the gage – reproducibility is the variability due to different operators using the gage.

Introduction to Statistical Quality Control, 4th Edition

7-6.1 Control Charts and Tabular Methods

•

Gage R&R Studies

Gage variability can be broken down as  2 measuremen t error   2 gage   2 reproducib ility   2 repeatabil ity • More than one operator (or different conditions) would be needed to conduct the gage R&R study.

Introduction to Statistical Quality Control, 4th Edition

7-6.1 Control Charts and Tabular Methods

• •

Statistics for Gage R&R Studies (The Tabular Method)

Say there are

operators in the study The standard deviation due to repeatability can be found as  R repeatabil ity d 2 where R  R 1  R 2    R p p and d 2 is based on the # of observations per part per operator.

Introduction to Statistical Quality Control, 4th Edition

7-6.1 Control Charts and Tabular Methods

•

Statistics for Gage R&R Studies (the Tabular Method)

The standard deviation for reproducibility reproducib ility  R d x 2 where R x  x max  x min is given as x max  max( x 1 , x 2 ,  x p ) d 2 x  min( x , x ,  x ) min 1 2 p is based on the number of operators, p Introduction to Statistical Quality Control, 4th Edition

7-6.1 Control Charts and Tabular Methods

• Example 7-8 Introduction to Statistical Quality Control, 4th Edition

Introduction to Statistical Quality Control, 4th Edition

• •

7-6.2 Methods Based on Analysis of Variance

The analysis of variance (Chapter 3) can be extended to analyze the data from an experiment and to estimate the appropriate components of gage variability.

For illustration, assume there are

parts and

operators, each operator measures every part

times. Introduction to Statistical Quality Control, 4th Edition

Introduction to Statistical Quality Control, 4th Edition

7-6.2 Methods Based on Analysis of Variance

• The measurements, y ijk , could be represented by the model

y ijk

  

P i



O j

 (

)

 

ijk

  

k i j

 1 , 2 ,...

 1 , 2 ,...,

where i = part, j = operator, k = measurement.

Introduction to Statistical Quality Control, 4th Edition

•

7-6.2 Methods Based on Analysis of Variance

The variance of any observation can be given by

(

y ijk

)   2

  2

  2  2

,  2

,  2 are the

variance components.

Introduction to Statistical Quality Control, 4th Edition

•

7-6.2 Methods Based on Analysis of Variance

Estimating the variance components can be accomplished using the following formulas  ˆ 2 

MS E

 ˆ 2

  ˆ ˆ 2

 

MS PO



MS E MS O



n MS PO



MS P pn



MS PO on

Introduction to Statistical Quality Control, 4th Edition

7-6.2 Methods Based on Analysis of Variance

 ˆ 2 

MS E

 0 .

51  ˆ

2 

MS O



MS PO pn

 19 .

63  2 .

70 10  3  0 .

56  ˆ 2

 ˆ 2

 

MS P



MS PO MS on



PO MS E n

  437 .

33  2 .

3 70   3 3 0 .

2 .

07 51   48 .

29 0 .

73 Introduction to Statistical Quality Control, 4th Edition

Introduction to Statistical Quality Control, 4th Edition

Transcript Introduction to Statistical Quality Control, 4th Edition

Chapter 7

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