Chapter 9 A. Process Capability & Statistical Quality Control Outline: u Basic Statistics u Process Variation u Process Capability u Process Control Procedures u u Variable data u X-bar chart and R-chart Attribute data u p-chart u Acceptance Sampling u Operating Characteristic Curve
Focus u This technical note on statistical quality control (SQC) covers the quantitative aspects of quality management u SQC is a number of different techniques designed to evaluate quality from a conformance view u How are we doing in meeting specifications? u SQC can be applied to both manufacturing and service processes u SQC techniques usually involve periodic sampling of the process and analysis of data Sample size u Number of samples u u SQC techniques are looking for variance u Most processes produce variance in output u we need to monitor the variance (and the mean also) and correct processes when they get out of range
Basic Statistics Normal Distributions have a mean (μ) and a standard deviation (σ) For a sample of N observations: Mean -3 s μ +3 s 99. 7% where: xi = Observed value N = Total number of observed values Standard Deviation 3
Statistics and Probability u SQC relies on central limit theorem and normal dist. u We establish the Upper Control Limits (UCL) and the Lower Control Limits (LCL) with plus or minus 3 standard deviations. Based on this we can expect 99. 7% of our sample observations to fall within these limits. u Acceptance sampling relies on Binomial and Hyper geometric probability concepts -3 s -2 s +3 s a/2 99. 7% a = Prob. of Type I error LCL UCL
Basic Stats u Using SQC, samples of a process output are taken, and u sample statistics are calculated u u The purpose of sampling is to find when the process has changed in some nonrandom way u The reason for the change can then be quickly determined and corrected u This allows us to detect changes in the actual distribution process 5
Variation u Random (common) variation is inherent in the production process. u Assignable variation is caused by factors that can be clearly identified and possibly managed u Using a saw to cut 2. 1 meter long boards as a sample process u Discuss random vs. assignable variation u Generally, when variation is reduced, quality improves. u It is impossible to have zero variability. T or F ?
Taguchi’s View of Variation Traditional view is that quality within the LS and US is good and that the cost of quality outside this range is constant, where Taguchi views costs as increasing as variability increases, so seek to achieve zero defects and that will truly minimize quality costs. High Incremental Cost of Variability Zero Lower Spec Target Spec Upper Spec Traditional View Lower Spec Exhibits TN 8. 1 & TN 8. 2 Target Spec Upper Spec Taguchi’s View 7
Process Capability u Tolerance (specification, design) Limits Bearing diameter 1. 250 +- 0. 005 inches u LTL = 1. 245 inches UTL = 1. 255 inches u u Process Limits The actual distribution from the process u Run the process to make 100 bearings, compute the mean and std. dev. (and plot/graph the complete results) u Suppose, mean = 1. 250, std. dev = 0. 002 u u How do they relate to one another?
Tolerance Limits vs. Process Capability Specification Width Actual Process Width 9
Process Capability Example u Design Specs: Bearing diameter 1. 250 +- 0. 005 inches u LTL = 1. 245 inches UTL = 1. 255 inches u The actual distribution from the process mean = 1. 250, s = 0. 002 u +- 3 s limits 1. 250 +- 3(0. 002) [1. 244, 1. 256] u Anew process, std. dev. = 0. 00083
Process Capability Index, Cpk u Capability Index shows how well parts being produced fit into design limit specifications u Compute the Cpk for the bearing example. u Old process, mean = 1. 250, s = 0. 002 u What is the probability of producing defective bearings? u New process, mean = 1. 250, s= 0. 00083, re-compute the Cpk u When the computed (sample) mean = design (target) mean, what does that imply?
The Cereal Box Example u Recall the cereal example. Consumer Reports has just published an article that shows that we frequently have less than 15 ounces of cereal in a box. u Let’s assume that the government says that we must be within ± 5 percent of the weight advertised on the box. u Upper Tolerance Limit = 16 + 0. 05(16) = 16. 8 ounces u Lower Tolerance Limit = 16 – 0. 05(16) = 15. 2 ounces u We go out and buy 1, 000 boxes of cereal and find that they weight an average of 15. 875 ounces with a standard deviation of 0. 529 ounces.
Cereal Box Process Capability u Specification or Tolerance Limits Upper Spec = 16. 8 oz, u Lower Spec = 15. 2 oz u u Observed Weight u Mean = 15. 875 oz, Std Dev = 0. 529 oz u What does a Cpk of 0. 4253 mean? u Many companies look for a Cpk of 1. 3 or better… 6 -Sigma company wants 2. 0!
Types of Statistical Sampling 1. Sampling to accept or reject the immediate lot of product at hand (Acceptance Sampling). u Attribute (Binary; Yes/No; Go/No-go information) u u u Defectives refers to the acceptability of product across a range of characteristics. Defects refers to the number of defects per unit which may be higher than the number of defectives. p-chart application 2. Sampling to determine if the process is within acceptable limits (Statistical Process Control) u Variable (Continuous) u u Usually measured by the mean and the standard deviation. X-bar and R chart applications
Evidence for Investigation… 15
Control Limits u If we establish control limits at +/3 standard deviations, then we would expect 99. 7% of our observations to fall within these limits LCL UCL x
Attribute Measurements (p-Chart) u Item is “good” or “bad” u Collect data, compute average fraction bad (defective) and std. dev. using: u The, UCL, LCL using: u Excel time!
Variable Measurements (x-Bar and R Charts) u A variable of the item is measured (e. g. , weight, length, salt content in a bag of chips) u Note that the item (sample) is not declared good or bad u Since the actual the standard deviation of the process is not known (and it may indeed fluctuate also) we use the sample data to compute the UCL & LCL For 3 -sigma limits, factors A 2 , D 3 , and D 4 and are given in Exhibit 9 A. 6, p. 341 u Excel time! u
Acceptance Sampling vs. SPC u Sampling to accept or reject the immediate lot of product at hand (Acceptance Sampling). u u Determine quality level Ensure quality is within predetermined (agreed) level u Sampling to determine if the process is within acceptable limits - Statistical Process Control (SPC) 3
Acceptance Sampling Advantages Economy Less handling damage Fewer inspectors Upgrading of the inspection job u Applicability to destructive testing u Entire lot rejection (motivation for improvement) u u u Disadvantages u Risks of accepting “bad” lots and rejecting “good” lots u Added planning and documentation u Sample provides less information than 100 -percent inspection 20
A Single Sampling Plan u A Single Sampling Plan simply requires two parameters to be determined: 1. n the sample size (how many units to sample from a lot) 2. c the maximum number of defective items that can be found in the sample before the lot is rejected.
RISK u RISKS for the producer and consumer in sampling plans: u Acceptable Quality Level (AQL) u Max. acceptable percentage of defectives defined by producer. u a (Producer’s risk) u The probability of rejecting a good lot. u Lot Tolerance Percent Defective (LTPD) u Percentage of defectives that defines consumer’s rejection point. u (Consumer’s risk) u The probability of accepting a bad lot.
Operating Characteristic Curve Probability of acceptance The OCC brings the concepts of producer’s risk, consumer’s risk, sample size, and maximum defects allowed together 1 0. 9 0. 8 0. 7 0. 6 0. 5 0. 4 0. 3 0. 2 0. 1 0 a =. 05 (producer’s risk) n = 99 c=4 B =. 10 (consumer’s risk) 1 2 3 AQL 4 5 6 7 8 The shape or slope of the curve is dependent on a particular combination of the four parameters 9 10 11 12 LTPD Percent defective 9
Example: Acceptance Sampling Zypercom, a manufacturer of video interfaces, purchases printed wiring boards from an outside vender, Procard has set an acceptable quality level of 1% and accepts a 5% risk of rejecting lots at or below this level. Zypercom considers lots with 3% defectives to be unacceptable and will assume a 10% risk of accepting a defective lot. Develop a sampling plan for Zypercom and determine a rule to be followed by the receiving inspection personnel. 10
Developing A Single Sampling Plan u Determine: a? ? AQL? u LTPD? u u Divide LTPD by AQL 0. 03/0. 01 = 3 u Then find the value for “c” by selecting the value in the TN 8. 10 “n(AQL)”column that is equal to or just greater than the ratio above (3). u Thus, c = 6 u From the row with c=6, get n. AQL = 3. 286 and divide it by AQL u 3. 286/0. 01 = 328. 6, round up to 329, n = 329 Sampling Plan: Take a random sample of 329 units from a lot. Reject the lot if more than 6 units are defective. c 0 1 2 3 4 LTPD/AQL n AQL 44. 890 10. 946 6. 509 4. 890 4. 057 0. 052 0. 355 0. 818 1. 366 1. 970 c 5 6 7 8 9 LTPD/AQL 3. 549 3. 206 2. 957 2. 768 2. 618 n AQL 2. 613 3. 286 3. 981 4. 695 5. 426
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