chapter 8 – process capabilitysite.iugaza.edu.ps/aschokry/files/2014/09/sqc-chap-8-pc.pdf · 1...

1

Islamic University of Gaza - Palestine

Chapter 8 – Process Capability


1. Investigate and analyze Process Capability (Cp) usingcontrol charts,

2. Understand the differences between Cp and Processpotential,

3. Calculate and interpret Cp ratios,4. Understand the differences between Cp and Cpk,5. Estimate the natural limits of a process from a sample

of data from that process,

Learning Outcomes

2


Process Capability

• Process capability refers to the ability of a process toproduce products or provide services capable of meetingthe specifications set by the customer or designer.•(Cp is the ability of a process to make a feature within itstolerance.)• Specification limits are set by management in response tocustomers’ expectations• The upper specification limit (USL) is the largest value thatcan be obtained and still conform to customers’expectations• The lower specification limit (LSL) is the smallest valuethat is still conforming


Process Capability and objectives

• Enables successful manufacturing and sales• Prevents scrap, sorting, rework• Allows jobs to run well• Has major impact on cost and schedule

3


Process Capability

• Tolerances or specifications– Range of acceptable values established by engineeringdesign or customer requirements

• Process variability– Natural variability in a process

• Process capability– Process variability relative to specification


Process Capability

Cp represents the precision, but not the accuracy of theprocess in respect to the tolerance window.

High Accuracy but lowprecision

High Precision but lowAccuracy

4


© Wiley 2010

Process Capability

• Product Specifications– Preset product or service dimensions, tolerances– e.g. bottle fill might be 16 oz. ±.2 oz. (15.8oz.-16.2oz.)– Based on how product is to be used or what the customer expects

• Process Capability – Cp and Cpk– Assessing capability involves evaluating process variability relative

to preset product or service specifications– Cp assumes that the process is centered in the specification range

– Cpk helps to address a possible lack of centering of theprocess

6σLSLUSL

widthprocesswidthionspecificatCp

3σLSLμ,

3σμUSLminCpk


Process Capability & Tolerance

• When spec. established without knowing whetherprocess capable of meeting it or not serious situationscan result

• Process capable or not – actually looking at processspread, which is called process capability (6)

• Let’s define specification limit as tolerance (T) :T = USL -LSL

• 3 types of situation can result– the value of 6 < USL - LSL– the value of 6 = USL - LSL– the value of 6 > USL - LSL

5


Process Capability (6) And Tolerance

• Cp - Capability IndexT = U-LCp = 1 Case II 6 = TCp > 1 Case I 6 < TCp < 1 Case III 6 > TUsually Cp = 1.33 (de facto

std.)• Measure of process

performance• Shortfall of Cp - measure

not in terms of nominal ortarget value >>> must useCpk

Formulas

Cp = (T)/6

Cpk =3

Z(min)

Z (USL) = σxUSL

LSLx


How much of the curve is included in how many standard deviations?

From –1 to +1 is about 68% of the bell curve.From –2 to +2 is about 95%From –3 to +3 is about 99.73%From –4 to +4 is about 99.99%(NOTE: We usually show the bell from –3 to +3 to make it easier to draw, but in concept, the “tails” ofthe bell get very thin and go on forever.)

-6 -5 -4 -3 -2 -1 +1 +2 +3 +4 +5 +60

6



One Sigma Shift

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6Standard Deviations

LSL USL

Cp = 1.33

7


Relationship bet. Process Variability and Specification Width

• Possible ranges for Cp

– Cp < 1, as in Fig. (b),process not capable ofproducing withinspecifications

– Cp ≥ 1, as in Fig. (c), processexceeds minimal

specifications

• One shortcoming, Cp assumesthat the process is centered onthe specification range

• Cp = Cpk when process iscentered


Various Cp Values In Relationship to Specifications

LSL USL

Cp = 2.00

Cp = 1.33Cp = 1.00

Cp = 0.75

8


Computing the Cp Value at Cocoa Fizz: three bottling machinesare being evaluated for possible use at the Fizz plant. Themachines must be capable of meeting the design specification of15.8-16.2 oz. with at least a process capability index of 1.0 (Cp≥1)

• The table below shows the informationgathered from production runs oneach machine. Are they allacceptable?

• Solution:– Machine A

– Machine B

– Machine C

Machine σ USL-LSL 6σ

A .05 .4 .3

B .1 .4 .6

C .2 .4 1.2

1.336(.05).4

6σLSLUSLCp

67.06(.1).4

6σLSLUSLCp

0.336(.2).4

6σLSLUSLCp


Process Capability for Variable Control Charts

• Procedure (for S - Chart)1. Take subgroup size 4 for 20 subgroups

2. Calculate sample s.d., s, for each subgroup

3. Calculate avg. sample s.d. s = s/g

4. Calculate est. population s.d.

5. Calculate Process Capability =

4o csˆ

6

9


Process Capability for Variable Control Charts (cont.)

• R - Chart1. Take subgroup size 4 for 20 subgroups

1. Calculate R for each subgroup

2. Calculate avg. Range, = R/g

3. Calculate

4. Calculate 6

R

o2o dRσ


A

B

What is Cpk? Itis a measure ofhow wella process iswithin aspecification.

Cpk = A divided by BA = Distance from process mean to closest spec limitB = 3 Standard Deviations (also called “3 Sigma”)

A bigger Cpk is better because fewer units will be beyond spec.(A bigger “A” and a smaller “B” are better.)

SpecificationLimit

SpecificationLimit

Cpk =A divided byB

10


A

B

An Analogy

Analogy:The bell curve is your automobile.The spec limits are the edges of your garage door.If A = B, you are hitting the frame of your garage door with your car.

SpecificationLimit

SpecificationLimit

Cpk =A divided byB


A

B

How can we make Cpk (A divided by B) better?

1. Design the product so a wider tolerance is functional (“robust design”)2. Choose equipment and methods for a good safety margin (“process capability”)3. Correctly adjust, but only when needed (“control”)4. Discover ways to narrow the natural variation (“improvement”)

SpecificationLimit

SpecificationLimit

Cpk =A divided byB

11


A

B

What does a very good Cpk do for us?

This process is producing good units with a good safety margin.

Note that when Cpk = 2, our process mean is 6 standard deviationsfrom the nearest spec, so we say it has “6 Sigma Capability.”

SpecificationLimit

SpecificationLimit

This Cpk isabout 2.Very good!

Mean


A

B

What does a problem Cpk look like?

This process is in danger of producing some defects.It is too close to the specification limits.

(Remember: the bell curve tail goes further than B……we only show the bell to 3-sigma to make it easier to draw.)

SpecificationLimit

SpecificationLimit

This Cpk is justslightly greaterthan 1. Not good!

12


A

B

What does a very bad Cpk look like?

A significant part of the “tail” is hanging out beyond the spec limits.This process is producing scrap, rework, and customer rejects.

Notice that if distance “A” approaches zero……the Cpk would approach zero, and……the process would become 50% defective!

SpecificationLimit

SpecificationLimit

This Cpk is lessthan 1. We desirea minimum of 1.33and ultimately wewant 2 or more.


Example

Determine Cp and Cpk for aprocess with average 6.45, = 0.030, having USL =6.50 , LSL = 6.30 -- T = 0.2

SolutionCp= T/6= 0.2/6(0.03)=1.11Cpk = Z(min)/3Z(U) = (USL -x)/ =

(6.50-6.45)/0.03 = 1.67Z(L) = (x –LSL)/ =(6.45-6.30)/0.03 = 5.00

Cpk = 1.67/3 = 0.56Process NOT capable sincenot centered. Cp > 1 doesn’tmean capable. Have to checkCpk

UL T

6.30 6.506.45 =

x

13


Computing the Cpk Value at Coc Limon

• Design specifications call for atarget value of 16.0 ±0.2 OZ.(USL = 16.2 & LSL = 15.8)

• Observed process output hasnow shifted and has a µ of 15.9and a σ of 0.1 oz.

• Cpk is less than 1, revealing thatthe process is not capable

(1 L = 33.8 ounce (oz.)

.33.3.1Cpk

3(.1)15.815.9,

3(.1)15.916.2minCpk


Comments On Cp, Cpk

• Cp does not change when process center (avg.)changes

• Cp = Cpk when process is centred• Cpk Cp always this situation• Cpk = 1.00 de facto standard• Cpk < 1.00 process producing rejects• Cp < 1.00 process not capable• Cpk = 0 process center is at one of spec. limit (U or

L)• Cpk < 0 i.e. – ve value, avg outside of limits

14


Cpk(Defined as

distance fromprocess mean tothe nearest spec,

divided by 3Standard

Deviations)

"How ManySigma

Capability?"Distance of

Process Mean toSpec Limit in

Standard Deviations

Within Spec(Process Perfectly

Centered,Both Tails

Considered)Good Units Per

Million

PPM of the BellCurve

Out of Spec (Process Perfectly

Centered,Both Tails

Considered)


Out of Spec(Process Not

Centered,Only One TailConsidered)

The column ATLEFT equates to 1defective out of how

many total?


Out of Specwith Six-Sigma

Philosophyof 1.5 Standard

Deviation Penaltyfor AnticipatedUnfavorable

Process Mean Drift

0 0 0 1,000,000 500,000 2

0.17 0.5 382,925 617,075 308,538 3

0.33 1 682,689 317,311 158,655 6

0.5 1.5 866,386 133,614 66,807 15 500,000

0.67 2 954,500 45,500 22,750 44 308,538

0.83 2.5 987,581 12,419 6,210 161 158,655

1 3 997,300 2,700 1,350 741 66,807

1.17 3.5 999,535 465 233 4,298 22,750

1.33 4 999,937 63 32 31,560 6,210

1.5 4.5 999,993.2 6.8 3.4 294,048 1,350

1.67 5 999,999.4 0.6 0.3 3,483,046 233

1.83 5.5 999,999.96 0.04 0.02 52,530,944 32

2 6 999,999.998 0.002 0.001 1,009,976,693 3.42.17 6.5 999,999.99992 0.00008 0.00004 24,778,276,273 0.3

Cpk, PPM, and "Six Sigma"


Conclusion:Process Capability:Yes: No:

No:

No:

Yes:

Yes:

potentially capableif re-centered

potentially capableif re-centered

too wide

15


Examples:

Suppose that a feature tolerance is .750”/.760”, and theprocess average is .759”, and the process standarddeviation is .002”, is the process satisfactory and capable?

Suppose that a diameter tolerance is 8.010” to 8.060”, and theprocess average is 8.041”, and the process standarddeviation is .002”, is the process capable?


http://elsmar.com/Cp_vs_Cpk.html

16


End of Chapters (8)

chapter 8 – process capabilitysite.iugaza.edu.ps/aschokry/files/2014/09/sqc-chap-8-pc.pdf · 1...

Documents