Chapter 9A

Process Capability and SPC

McGraw-Hill/Irwin © 2011 The McGraw-Hill Companies, All Rights Reserved

Learning Objectives

 Explain what statistical quality control is.

 Calculate the capability of a process.

 Understand how processes are monitored with control charts.

 Recognize acceptance sampling concepts.

9A-2

Types of Situations where SPC can be Applied

LO 1  How many paint defects are there in the finish of a car?

 How long does it take to execute market orders?

 How well are we able to maintain the dimensional tolerance on our ball bearing assembly?

 How long do customers wait to be served from our drive-through window?

9A-3

What Is Quality?

    How do you define quality?

Durability, reliability, long warrantee Fitness for use, degree of conformance Maintainability     Measures of quality Grade —measurable characteristics, finish Consistency —good or bad, predictability Conformance —degree product meets specifications  Consistency versus conformance

9A-4

LO 1

Basic Forms of Variation



Assignable variation

: caused by  factors that can be clearly identified and possibly managed Example: a poorly trained employee that creates variation in finished product output  

Common variation

: variation that is inherent in the production process Example: a molding process that always leaves “burrs” or flaws on a molded item

9A-5

Variations Around Us

LO 1  When variation is reduced, quality is improved  However, it is impossible to have zero   variation Engineers assign acceptable limits for variation The limits are know as the

upper and lower specification limits



A

lso known as

upper and lower tolerance limits 9A-6

LO 1

Taguchi’s View of Variation

  Traditional view is that quality within the range is good and that the cost of quality outside this range is constant     Taguchi views costs as increasing as variability increases, so seek to achieve zero defects and that will truly minimize quality costs Society loses (pays) for poor quality Design products/processes impervious to variations Use experimental/robust design Shoot for target not conformance to specifications

9A-7

LO 2

Process Capability

 Taguchi argues that tolerance is not a yes/no decision, but a continuous function  Other experts argue that the process should be so good the probability of generating a defect should be very low

9A-8

Process Capability

     Process (control) limits Calculated from data gathered from the process It is natural tolerance limits Defined by ±3σ (standard deviation) Used to determine if process is in statistical control    Tolerance (specification) limits Often determined externally, e.g., by customer Process may be in control but not within specification  How do the limits relate to one another?

9A-9

LO 2

Process Capability

9A-10

Process Capability

C p



USL

 6 

LSL

     Case 1: Cp > 1 USL-LSL > 6 sigma Process quality higher than customer’s Situation desired Defacto standard is 1.33+

LSL LNTL UNTL USL

6  

9A-11

Process Capability

C p



USL

 6 

LSL

    Case 2: Cp = 1 USL-LSL = 6 sigma Approximately 0.27% defectives will be made Process is unstable

LSL LNTL USL UNTL

6  

9A-12

Process Capability

C p



USL

 6 

LSL

     Case 3: Cp < 1 USL-LSL < 6 sigma Situation undesirable Process is yield sensitive Could produce large number of defectives

LNTL LSL USL UNTL

6  

9A-13

Process Capability Index,

C pk

  Most widely used capability measure Measures design versus specification relative to the nominal value    Based on worst case situation Defacto value is 1 and processes with this score is capable  Scores > 1 indicates 6-sigma subsumed by the inspection limits Scores less than 1 will result in an incapable process

9A-14

LO 2

Capability Index (C

pk

)

 Capability index (C pk ) shows how well parts being produced fit into design limit specifications C pk = min  X 3 LTL  or UTL 3  X    Also useful to calculate probabilities

Z LTL



LTL

 

X Z UTL



UTL

 

X 9A-15

LO 2

Example: Capability

   Data Designed for an average of 60 psi  Lower limit of 55 psi, upper limit of 65 psi Sample mean of 61 psi, standard deviation of 2 psi  Calculate C pk

C pk

 min   

x



LSL

3  ,

USL

3  

x

     min min    1 , 61  55 3   0 .

6667 ,  65  61 3      0 .

6667

9A-16

What does a C

pk

0.6667 mean?

of

 An index that shows how well the units being produced fit within the specification limits.  This is a process that will produce a relatively high number of defects.

 Many companies look for a

C pk

of 1.3 or better… 6-Sigma companies want 2.0!

9A-17

Example: Probabilities

Less than 55 psi

Z



P

(

Z X

 

X

  3 )   55  2 61 0 .

00135   3 More than 65 psi

Z



P

(

Z X

   2 )

X

  65  61 2 0 .

02275  2 LO 2

P

(

Z

  3 or

Z

 2 )  0 .

00135  0 .

02275  0 .

02410

9A-18

LO 3

Process Control Procedures

    Attribute (Go or no-go information) Defectives refers to the acceptability of product across a range of characteristics.

Defects refers to the number of defects per unit which may be higher than the number of defectives.

p-chart application  Variable (Continuous)  Usually measured by the mean and the standard deviation.

 X-bar and R chart applications

9A-19

LO 3

Control Chart Evidence for Investigation

9A-20

LO 3

Process Control with Attribute Measurement: Using ρ Charts

  Created for good/bad attributes Use simple statistics to create the control limits

p



s p UCL

  Total number

p

Number   of of defects from all samples samples  Sample size

n p



zs p LCL



p



zs p 9A-21

LO 3

Example: Control Chart Design

9A-22

Example: Calculations

LO 3

p

 Total number of defects from all samples Number of samples x Sample size  91 3 , 000  0 .

03033

s p



p n p

 0 .

03033  1  0 .

03033   0 .

00990 300

UCL



LCL



p

 3

s p p

 3

s p

 0 .

03033  3  0 .

00990   0 .

06003  0 .

03033  3  0 .

00990   0 .

00063

9A-23

LO 3

Process Control with Attribute Measurements: Using c Charts

 With ρ charts, each item was either good or bad  With a

c

chart, each item can have multiple defects

c

 Average number of defects per unit

s p



c UCL



c



z LCL



c



z c c 9A-24

LO 3

Example: Lumber Yard

 Lumber yard expects four knotholes per eight foot board

c

 4

s p



c



UCL



c



z



s p LCL



c



z



s p

4  2   4  3    10 4  3     2  0

9A-25

Process Control with Variable Measurements: Using

x

and R Charts

LO 3  In variable sampling, we measure actual values rather than sampling attributes    Generally want small sample size Quicker Cheaper   Samples of 4-5 are typical Want 25 or so samples to set up chart

9A-26

LO 3

How to Construct

x

Charts if Standard Deviation Known

UCL X 

X



zs X LCL

X 

X



zs X

where s X s 

s

 Standard deviation

n

 Standard deviation of the of sample means process distributi on n  Sample size X  Average of sample means or a target va lue set for the process z  Number of standard deviations for a specific confidence level

9A-27

LO 3

How to Construct

x

Charts and R

X

Chart UCL X 

X



A

2

R

LCL X 

X



A

2

R R

Chart UCL R LCL R 

D

4

R



D

3

R 9A-28

LO 3

Example: The Data

9A-29

LO 3

Example: Calculations and Chart

9A-30

Acceptance Sampling

LO 4  Acceptance sampling is sampling to accept or reject the immediate lot of   product at hand Does not always “Determine quality level” Results subject to sampling error    Purposes Make decision about (sentence) a product Otherwise, ensures quality is within predetermined level

9A-31

Acceptance Sampling

       Advantages Economy Less handling damage Fewer inspectors Upgrading of the inspection job Applicability to destructive testing Entire lot rejection (motivation for improvement)     Disadvantages R isks of accepting “bad” lots (

consumer’s risk

) and rejecting “good” lots (

producer’s risk

) Added planning and documentation Sample provides less information than 100-percent inspection

9A-32

Single Sampling Plan

   Defined by

n

and

c n

is sample size —how many to sample at a time

c

is the acceptance number —the maximum number of defective items that can be found in the sample before the lot is rejected LO 4  Values for

n

and

c

are determined by the     interaction of four factors

AQL α LTPD β

or acceptable quality level or lot tolerance percent defective

9A-33

Risk

LO 4   Acceptable quality level (

AQL

)   Maximum acceptable percentage of defectives defined by producer The  (producer’s risk) The probability of rejecting a good lot   Lot tolerance percent defective (

LTPD

)   Percentage of defectives that defines consumer’s rejection point The  (consumer’s risk) The probability of accepting a bad lot

9A-34

Standard Table of Sampling Plans

   MIL-STD-105D For attribute sampling plans Needs to know:   The lot size N The inspection level (I, II, III)    The AQL Type of sampling (single, double, multiple) Type of inspection (normal, tightened, reduced)  Find a code letter then read plan from Table

9A-35

Standard Table of Sampling Plans:

Single Sampling Plan

Example: If N=2000 and

AQL

=0.65% find the

normal

,

tightened

, and

reduced

single sampling plan using inspection level II.

Example: If N=20,000 and AQL=1.5% find the

tightened

, and

reduced normal

double sampling plan using , inspection level I.

9A-36