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## Chapter 5a Process Capability

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**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**Relationship between population and sample values**• Assume Normal Dist. • ‘Estimate’ popu. std. dev. • c4 ; n = 84 • (c4 = 0.99699) = 4.17 = 2.09**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 =**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 result the value of 6 < USL-LSL the value of 6 = USL - LSL the value of 6 > USL - LSL**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**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) =**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**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**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