Presentation on theme: "Chapter 9A. Process Capability & Statistical Quality Control"— Presentation transcript:
1 Chapter 9A. Process Capability & Statistical Quality Control Outline:Basic StatisticsProcess VariationProcess CapabilityProcess Control ProceduresVariable dataX-bar chart and R-chartAttribute datap-chartAcceptance SamplingOperating Characteristic Curve
2 FocusThis technical note on statistical quality control (SQC) covers the quantitative aspects of quality managementSQC is a number of different techniques designed to evaluate quality from a conformance viewHow are we doing in meeting specifications?SQC can be applied to both manufacturing and service processesSQC techniques usually involve periodic sampling of the process and analysis of dataSample sizeNumber of samplesSQC techniques are looking for varianceMost processes produce variance in outputwe need to monitor the variance (and the mean also) and correct processes when they get out of range
3 Basic Statistics Mean Standard Deviation 99.7%μNormal Distributions have a mean (μ) and a standard deviation (σ)For a sample of N observations:Meanwhere:xi = Observed valueN = Total number of observed valuesStandard Deviation3
4 Statistics and Probability SQC relies on central limit theorem and normal dist.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.Acceptance sampling relies on Binomial and Hyper geometric probability concepts-3s-2s+2s+3s99.7%LCLUCLa/2a = Prob. of Type I error
5 Basic StatsUsing SQC,samples of a process output are taken, andsample statistics are calculatedThe purpose of sampling is to find when the process has changed in some nonrandom wayThe reason for the change can then be quickly determined and correctedThis allows us to detect changes in the actual distribution process3
6 VariationRandom (common) variation is inherent in the production process.Assignable variation is caused by factors that can be clearly identified and possibly managedUsing a saw to cut 2.1 meter long boards as a sample processDiscuss random vs. assignable variationGenerally, when variation is reduced, quality improves.It is impossible to have zero variability. T or F ?
7 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.IncrementalCost ofVariabilityHighZeroLowerSpecTargetUpperTraditional ViewIncrementalCost ofVariabilityHighZeroLowerSpecTargetUpperTaguchi’s ViewExhibits TN8.1 & TN8.230
8 Process Capability Tolerance (specification, design) Limits Bearing diameter inchesLTL = inches UTL = inchesProcess LimitsThe actual distribution from the processRun the process to make 100 bearings, compute the mean and std. dev. (and plot/graph the complete results)Suppose, mean = 1.250, std. dev = 0.002How do they relate to one another?28
9 Tolerance Limits vs. Process Capability Specification WidthActual Process WidthSpecification WidthActual Process Width
10 Process Capability Example Design Specs: Bearing diameter inchesLTL = inches UTL = inchesThe actual distribution from the process mean = 1.250, s = 0.002+- 3s limits (0.002) [1.244, 1.256]Anew process, std. dev. =
11 Process Capability Index, Cpk Capability Index shows how well parts being produced fit into design limit specificationsCompute the Cpk for the bearing example.Old process, mean = 1.250, s = 0.002What is the probability of producing defective bearings?New process, mean = 1.250, s= , re-compute the CpkWhen the computed (sample) mean = design (target) mean, what does that imply?
12 The Cereal Box ExampleRecall 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.Let’s assume that the government says that we must be within ± 5 percent of the weight advertised on the box.Upper Tolerance Limit = (16) = 16.8 ouncesLower Tolerance Limit = 16 – 0.05(16) = 15.2 ouncesWe go out and buy 1,000 boxes of cereal and find that they weight an average of ounces with a standard deviation of ounces.
13 Cereal Box Process Capability Specification or Tolerance LimitsUpper Spec = 16.8 oz,Lower Spec = 15.2 ozObserved WeightMean = oz, Std Dev = ozWhat does a Cpk of mean?Many companies look for a Cpk of 1.3 or better… 6-Sigma company wants 2.0!
14 Types of Statistical Sampling Sampling to accept or reject the immediate lot of product at hand (Acceptance Sampling).Attribute (Binary; Yes/No; Go/No-go information)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 applicationSampling to determine if the process is within acceptable limits (Statistical Process Control)Variable (Continuous)Usually measured by the mean and the standard deviation.X-bar and R chart applications
16 Control LimitsIf we establish control limits at +/- 3 standard deviations, then we would expect 99.7% of our observations to fall within these limitsxLCLUCL
17 Attribute Measurements (p-Chart) Item is “good” or “bad”Collect data, compute average fraction bad (defective) and std. dev. using:The, UCL, LCL using:Excel time!
18 Variable Measurements (x-Bar and R Charts) A variable of the item is measured (e.g., weight, length, salt content in a bag of chips)Note that the item (sample) is not declared good or badSince 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 & LCLFor 3-sigma limits, factors A2 , D3 , and D4 and are given in Exhibit 9A.6, p. 341Excel time!
19 Acceptance Sampling vs. SPC Sampling to accept or reject the immediate lot of product at hand (Acceptance Sampling).Determine quality levelEnsure quality is within predetermined (agreed) levelSampling to determine if the process is within acceptable limits - Statistical Process Control (SPC)33
20 Acceptance Sampling Advantages Economy Less handling damage Fewer inspectorsUpgrading of the inspection jobApplicability to destructive testingEntire lot rejection (motivation for improvement)DisadvantagesRisks of accepting “bad” lots and rejecting “good” lotsAdded planning and documentationSample provides less information than 100-percent inspection
21 n the sample size (how many units to sample from a lot) A Single Sampling PlanA Single Sampling Plan simply requires two parameters to be determined:n the sample size (how many units to sample from a lot)c the maximum number of defective items that can be found in the sample before the lot is rejected.
22 RISK RISKS for the producer and consumer in sampling plans: Acceptable Quality Level (AQL)Max. acceptable percentage of defectives defined by producer.a (Producer’s risk)The probability of rejecting a good lot.Lot Tolerance Percent Defective (LTPD)Percentage of defectives that defines consumer’s rejection point. (Consumer’s risk)The probability of accepting a bad lot.
23 Operating Characteristic Curve The OCC brings the concepts of producer’s risk, consumer’s risk, sample size, and maximum defects allowed togethern = 99c = 4AQLLTPD0.10.20.30.188.8.131.52.80.9123456789101112Percent defectiveProbability of acceptanceB =.10(consumer’s risk)a = .05 (producer’s risk)The shape or slope of the curve is dependent on a particular combination of the four parameters99
24 Example: Acceptance Sampling Zypercom, a manufacturer of video interfaces, purchases printed wiring boards from an outside vender, Procard. 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.1010
25 Developing A Single Sampling Plan Determine:AQL? a?LTPD? ?Divide LTPD by AQL 0.03/0.01 = 3Then find the value for “c” by selecting the value in the TN8.10 “n(AQL)”column that is equal to or just greater than the ratio above (3).Thus, c = 6From the row with c=6, get nAQL = and divide it by AQL3.286/0.01 = 328.6, round up to 329, n = 329Problems: 3, 4, 6, 7, 8, 9, 11, 13cLTPD/AQLn AQL44.8900.05253.5492.613110.9460.35563.2063.28626.5090.81872.9573.98134.8901.36682.7684.69544.0571.97092.6185.426Sampling Plan:Take a random sample of 329 units from a lot.Reject the lot if more than 6 units are defective.
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