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Chapter 7 Process and Measurement System Capability Analysis Introduction to Statistical Quality Control, 4th Edition 7-1. Introduction • Process capability refers to the uniformity of the process. • Variability in the process is a measure of the uniformity of output. • Two types of variability: – Natural or inherent variability (instantaneous) – Variability over time • Assume that a process involves a quality characteristic that follows a normal distribution with mean , and standard deviation, . The upper and lower natural tolerance limits of the process are UNTL = + 3 LNTL = - 3 Introduction to Statistical Quality Control, 4th Edition 7-1. Introduction • Process capability analysis is an engineering study to estimate process capability. • In a product characterization study, the distribution of the quality characteristic is estimated. Introduction to Statistical Quality Control, 4th Edition 7-1. Introduction Major uses of data from a process capability analysis 1. Predicting how well the process will hold the tolerances. 2. Assisting product developers/designers in selecting or modifying a process. 3. Assisting in establishing an interval between sampling for process monitoring. 4. Specifying performance requirements for new equipment. 5. Selecting between competing vendors. 6. Planning the sequence of production processes when there is an interactive effect of processes on tolerances 7. Reducing the variability in a manufacturing process. Introduction to Statistical Quality Control, 4th Edition 7-1. Introduction Techniques used in process capability analysis 1. Histograms or probability plots 2. Control Charts 3. Designed Experiments Introduction to Statistical Quality Control, 4th Edition 7-2. Process Capability Analysis Using a Histogram or a Probability Plot 7-2.1 Using a Histogram • The histogram along with the sample mean and sample standard deviation provides information about process capability. – The process capability can be estimated as x 3s – The shape of the histogram can be determined – Histograms provide immediate, visual impression of process performance Introduction to Statistical Quality Control, 4th Edition Example 7-1 • Pgs. 353-354 • This procedure works if data are distributed normally Introduction to Statistical Quality Control, 4th Edition Reasons for poor process capability • See Fig. 7-3 – Poor process centering • Assume that this can be corrected – Excess process variability • Harder to correct Introduction to Statistical Quality Control, 4th Edition 7-2.2 Probability Plotting • Probability plotting is useful for – Determining the shape of the distribution – Determining the center of the distribution – Determining the spread of the distribution. • Recall normal probability plots (Chapter 2) – The mean of the distribution is given by the 50th percentile – The standard deviation is estimated by 84th percentile – 50th percentile ˆ Introduction to Statistical Quality Control, 4th Edition 7-2.2 Probability Plotting Cautions in the use of normal probability plots • If the data do not come from the assumed distribution, inferences about process capability drawn from the plot may be in error. • Probability plotting is not an objective procedure (two analysts may arrive at different conclusions). Introduction to Statistical Quality Control, 4th Edition Example • See Fig. 7-4 • First, est = 260 psi • Then, est = 298 – 260 = 38 psi • Can also use normal probability plot to estimate fallout – If LSL = 200 psi • Then, from Fig 7-4, about 5% will be below that value Introduction to Statistical Quality Control, 4th Edition 7-3. Process Capability Ratios 7-3.1 Use and Interpretation of C p • Recall USL LSL Cp 6 where LSL and USL are the lower and upper specification limits, respectively. Introduction to Statistical Quality Control, 4th Edition 7-3.1 Use and Interpretation of Cp The estimate of Cp is given by ˆ USL LSL Cp 6ˆ Where the estimate can be calculated using the sample ˆ standard deviation, S, or R / d 2 Introduction to Statistical Quality Control, 4th Edition 7-3.1 Use and Interpretation of Cp Piston ring diameter in Example 5-1 • The estimate of Cp is ˆ 74.05 73.95 Cp 6(0.0099) 1.68 Introduction to Statistical Quality Control, 4th Edition 7-3.1 Use and Interpretation of Cp One-Sided Specifications USL C pu 3 LSL C pl 3 These indices are used for upper specification and lower specification limits, respectively Introduction to Statistical Quality Control, 4th Edition Example 7-2 • Pg. 359 Introduction to Statistical Quality Control, 4th Edition Table 7-3 • Process fallout for one- and two-sided specifications Introduction to Statistical Quality Control, 4th Edition 7-3.1 Use and Interpretation of Cp Assumptions The quantities presented here (Cp, Cpu, Clu) have some very critical assumptions: 1. The quality characteristic has a normal distribution. 2. The process is in statistical control 3. In the case of two-sided specifications, the process mean is centered between the lower and upper specification limits. If any of these assumptions are violated, the resulting quantities may be in error. Introduction to Statistical Quality Control, 4th Edition Table 7-4 • Recommended minimum values of the PCR • For example, a new process with two-sided specifications has a recommended C p of 1.50 – This implies that process fallout would be 7 ppm • Six would result in a Cp of 2.0 Introduction to Statistical Quality Control, 4th Edition 7-3.2 Process Capability Ratio on Off-Center Process • Cp does not take into account where the process mean is located relative to the specifications. • A process capability ratio that does take into account centering is Cpk defined as Cpk = min(Cpu, Cpl) Introduction to Statistical Quality Control, 4th Edition Figure 7-7 • All of the panels in the figure have Cp = 2.0 • But, when the process mean shifts, the capability of the process can change – Note that does not shift Introduction to Statistical Quality Control, 4th Edition Figure 7-7, cont. • For panel b, N(53, 22) – Cpk = min(Cpu, Cpl) • Cpu = (62-53)/[3(2)] = 1.5 • Cpl = (53-38)/[3(2)] = 2.5 – Cpk = 1.5 Introduction to Statistical Quality Control, 4th Edition 7-3.3 Normality and the Process Capability Ratio • The normal distribution of the process output is an important assumption. • If the distribution is nonnormal, Luceno (1996) introduced the index, Cpc, defined as USL LSL C pc 6 EXT 2 Introduction to Statistical Quality Control, 4th Edition Example • USL = 90, LSL = 80 – So, T = (90 + 80)/2 = 85 • (T = target value) – Let X = 84 – Then, Cpc = 1.33 • (Be careful with this result…I don’t trust it!) Introduction to Statistical Quality Control, 4th Edition 7-3.3 Normality and the Process Capability Ratio • A capability ratio involving quartiles of the process distribution is given by USL LSL C p (q ) x 0.99865 x 0.00135 • In the case of the normal distribution Cp(q) reduces to Cp Introduction to Statistical Quality Control, 4th Edition Why does it reduce to Cp? • In the case of the normal distribution – x.00135 = – 3 – x.99865 = + 3 Introduction to Statistical Quality Control, 4th Edition 7-4. Process Capability Analysis Using a Control Chart • If a process exhibits statistical control, then the process capability analysis can be conducted. • A process can exhibit statistical control, but may not be capable. • PCRs can be calculated using the process mean and process standard deviation estimates. Introduction to Statistical Quality Control, 4th Edition Example • Pgs. 373-375 Introduction to Statistical Quality Control, 4th Edition 7-5. Process Capability Analysis Designed Experiments • Systematic approach to varying the variables believed to be influential on the process. (Factors that are necessary for the development of a product). • Designed experiments can determine the sources of variability in the process. Introduction to Statistical Quality Control, 4th Edition Example • Machine that fills bottles with a soft-drink beverage – Each machine has many filling heads that are independently adjusted – Quality characteristic measured is syrup content in degrees brix – Three possible causes of variabililty • Machines, heads, analytical tests Introduction to Statistical Quality Control, 4th Edition Example, cont. • Variability is B2 = M2 + H2 + A2 • Conduct an experiment • Say the result is as shown in Fig. 7-12 • Head-to-head variability is large – Improve the process by reducing this variance Introduction to Statistical Quality Control, 4th Edition 7-7: Setting spec limits on discrete components • Setting specifications to insure that the final product meets specifications • As discussed previously, if normally distributed variables are linked, the result is normally distributed with mean the sum of the individual means, and variance the sum of the individual variances Introduction to Statistical Quality Control, 4th Edition Example 7-9 • Pgs. 388-389 Introduction to Statistical Quality Control, 4th Edition Example 7-10 • Pgs. 389-390 Introduction to Statistical Quality Control, 4th Edition Example 7-11 • Pgs. 391-392 Introduction to Statistical Quality Control, 4th Edition 7-8: Estimating tolerance limits • Confidence limits – Provide an interval estimate of the parameters of a distribution • Tolerance limits – Indicate the limits between which we can expect to find a specified proportion of a population Introduction to Statistical Quality Control, 4th Edition 7-8.1: Tolerance limits based on the normal distribution • Suppose x~N(, 2), both unknown • Take a sample of size n and compute xbar and S2 • Natural tolerance limits might be estimated using: – Xbar + Za/2S • Since xbar and S are only estimates, the interval may or may not always contain 100(1-a)% of the distribution Introduction to Statistical Quality Control, 4th Edition 7-8.1: Tolerance limits based on the normal distribution • However, we may use a constant K such that in a large number of samples a fraction g of the intervals xbar + KS will include at least 100(1-a)% of the distribution • Values of K are tabulated in Appendix Table VII – 2<n<1000, g = .90, .95, .99, and a = .10, .05, .01 Introduction to Statistical Quality Control, 4th Edition Example 7-13 • Pg. 396 Introduction to Statistical Quality Control, 4th Edition Assignment • Work odd-numbered exercises on the topics covered in class Introduction to Statistical Quality Control, 4th Edition End Introduction to Statistical Quality Control, 4th Edition

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