INT 506/706: Total Quality Management

Lec #6, Process Capability

Outline

• Individual values vs. averages• Control limits vs. Specs• 6σ spread versus specs• Calculating Process Capability Indices

2

Process Capability

Individual vs. averages

Individual vs. averages

nX

The ability of X-bar to approximate μ is measured by the following equation:

Individual vs. averages

24

ˆor ˆd

R

c

s

IF the process can be assumed to be normal, the population standard deviation can be estimated from either

the sample standard deviation (s) or the range (R).

Control limits vs Specifications

A process under control exhibits:

1. 2/3 of points near center

2. Few points close to the center

3. Points around centerline

4. Points balanced on both sides

5. No points beyond control limits

6. No patterns or trends

Control limits vs Specifications

6σ Spread versus Specs

24

ˆor ˆd

R

c

s

3 Possible CASES1) 6σ < USL – LSL2) 6σ = USL – LSL3) 6σ > USL - LSL

CASE #1 6σ < USL – LSL

CASE #2 6σ = USL – LSL

CASE #3 6σ > USL – LSL

Calculating 6σ

Assuming process is STABLE and IN CONTROL

1. 20 subgroups of n=4

2. Calculate sample std dev, si, for each sample

3. Calculate average sample std dev,

4. Calculate pop std dev

5. Multiply pop std dev *6

m

ss

m

ii

1

4

ˆc

s

Example

Example

126.0)021.0(6ˆ6

021.09400.0

02.0ˆ

02.021

414.021

015.0...029.0031.0

4

1

c

s

m

ss

m

ii

Capability Index Cp

6Cp

LSLUSL

Capability Index Cp

794.0126.0

95.1105.12ˆ6

Cp

LSLUSL

EXAMPLE

INTERPRETATION:Process NOT capable (case 3)

Capability Ratio, Cr

LSLUSL

6Cr

Inverse of Cp so larger the ratio, the LESS capable the process is of meeting specifications

Cpk

3

(min)Cpk

Z

Cp and Cr do not take into account the centering of the process

ˆ)(

ˆ)(Z

LSLXLSLZ

or

XUSLUSL

Where Z(min)=

Cpk

Cpk

Review

• Review pdf file in blackboard – “CapabilityCaseStudy”– Put into Minitab and run capability modules to see

what you come up with• Is it the same as the case study?• Do you understand the interpretation of the case study?