1. 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

2. 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

3. Relationship between population and sample values • Assume Normal Dist. • ‘Estimate’ popu. std. dev. • c4  ; n = 84 • (c4 = 0.99699) = 4.17 = 2.09

4. 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 =

5. 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

6. 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

7. Control limits, Process spread, Dist of averages, & distribution of individual values are interdependent. – determined by the process • C. Charts CANNOT determine process meets spec.

8. 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 result the value of 6 < USL-LSL the value of 6 = USL - LSL the value of 6 > USL - LSL

9. Case I and Case II situations

10. Case 3 situation

11. Process Capability • Procedure (s – method) • Take subgroup size 4 for 20 subgroups • Calculate sample s.d., s, for each subgroup • Calculate avg. sample s.d. s= s/g • Calculate est. population s.d. • Calculate Process Capability = • R - method • Same as 1. above • Calculate R for each subgroup • Calculate avg. Range, = R/g • Calculate • Calculate Calculate 6

12. Cp - Capability Index T = U-L Cp = 1  Case II 6 = T Cp > 1  Case I 6 < T Cp < 1  Case III 6 > T Usually 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 = Process Capability (6) And Tolerance Z (USL) = Z (LSL) =

14. Determine Cp and Cpk for a process with average 6.45, = 0.030, having USL = 6.50 , LSL = 6.30 -- T = 0.2 Solution Cp= T/6= 0.2/6(0.03)=1.11 Cpk = Z(min)/3 Z(U) = (USL -x)/  = 6.50-6.45)/0.03 = 1.67 Z(L) = (x –LSL)/  = 6.45-6.30)/0.03 = 5.00 Cpk = 1.67/3 = 0.56 Process NOT capable since not centered. Cp > 1 doesn’t mean capable. Have to check Cpk L U T 6.50 6.45 = 6.30 Example

15. 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

17. Find Cp, Cpk x = 129.7 (Length of radiator hose)  = 2.35 Spec. 130.0  3.0 What is the % defective? Find Cp, Cpk Spec.U = 58 mm L = 42 mm  = 2 mm When = 50 When = 54 Find Cp, Cpk U = 56 L = 44  = 2 When = 50 When = 56 Exercise