Chapter 5a Process Capability

This chapter introduces the topic of process capability studies. The theory behind

process capability and the calculation of Cp and Cpk is presented

Specification Limit &Process Limit Look at indv. values and avg. values of x’s Indv x’s values n = 84 - considered as population Avg’s n = 21 - sample taken

= same (in this case)

Normally distributed individual x’s and avg. values having same mean, only the spread is different >

Relationship = popu. Std. dev. of avg’s

If n = 5 = 0.45 = popn std dev of indiv. x’s SPREAD OF AVGS IS HALF OF SPREAD FOR INDV.

VALUES

xXandX

x

n

σσx

x

Relationship between population and sample values Assume Normal Dist. ‘Estimate’ popu. std. dev.

c4 ; n = 84

(c4 = 0.99699)

= 4.17

= 2.09

417.4

nx

4cs

ˆ

3n4)1n(4

0.99699

4.16σ

Central Limit Theorem

‘If the population from which samples are taken is NOT normal, the distribution of SAMPLE AVERAGES will tend toward normality provided that sample size, n, is at least 4.’

Tendency gets better as n Standardized normal for distribution of

averages Z = nσ

μx

Central Limit Theorem is one reason why control chart works

No need to worry about distribution of x’s is not normal, i.e. indv. values.

Averages distribution will tend to ND

Control Limits & Specifications

Control limits - limits for avg’s, and established as a func. of avg’s

Specification limits - allowable variation in size as per design documents e.g. drawing

for individual values estimated by design

engineers

Control limits, Process spread, Dist of averages, & distribution of individual values are interdependent. – determined by the process

C. Charts CANNOT determine process meets spec.

Process Capability & Tolerance When spec. established without knowing

whether process capable of meeting it or not serious situations can result

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

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

3 types of situation can resultthe value of 6 < USL-LSLthe value of 6 = USL - LSLthe value of 6 > USL - LSL

Case I and Case II situations

Case 3 situation

Process Capability Procedure (s – method)1. Take subgroup size 4 for 20 subgroups2. Calculate sample s.d., s, for each subgroup3. Calculate avg. sample s.d. s = s/g4. Calculate est. population s.d.5. Calculate Process Capability = R - method1. Same as 1. above2. Calculate R for each subgroup3. Calculate avg. Range, = R/g4. Calculate 5. Calculate Calculate 6

4o csˆ 6

R

o2o dRσ

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 or target value >>> must use Cpk

Formulas

Cp = (T)/6

Cpk = 3

Z(min)

Z (USL) = σ

xUSL

LSLxZ (LSL) =

ExampleDetermine Cp and Cpk for a

process 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 = 5.00Cpk = 1.67/3 = 0.56

Process NOT capable since not centered. Cp > 1 doesn’t mean capable. Have to check Cpk

UL T

6.30 6.506.45 = x

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

Exercise

1. Find Cp, Cpkx = 129.7 (Length of radiator hose) = 2.35Spec. 130.0 3.0What is the % defective?

2. Find Cp, CpkSpec.U = 58 mmL = 42 mm = 2 mmWhen = 50When = 54

3. Find Cp, Cpk

U = 56

L = 44

= 2

When = 50

When = 56