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**Chapter 9A. Process Capability & Statistical Quality Control**

Outline: Basic Statistics Process Variation Process Capability Process Control Procedures Variable data X-bar chart and R-chart Attribute data p-chart Acceptance Sampling Operating Characteristic Curve

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Focus This technical note on statistical quality control (SQC) covers the quantitative aspects of quality management SQC is a number of different techniques designed to evaluate quality from a conformance view How are we doing in meeting specifications? SQC can be applied to both manufacturing and service processes SQC techniques usually involve periodic sampling of the process and analysis of data Sample size Number of samples SQC techniques are looking for variance Most processes produce variance in output we need to monitor the variance (and the mean also) and correct processes when they get out of range

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**Basic Statistics Mean Standard Deviation**

99.7% μ Normal Distributions have a mean (μ) and a standard deviation (σ) For a sample of N observations: Mean where: xi = Observed value N = Total number of observed values Standard Deviation 3

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**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 +3s 99.7% LCL UCL a/2 a = Prob. of Type I error

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Basic Stats Using SQC, samples of a process output are taken, and sample statistics are calculated The purpose of sampling is to find when the process has changed in some nonrandom way The reason for the change can then be quickly determined and corrected This allows us to detect changes in the actual distribution process 3

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Variation Random (common) variation is inherent in the production process. Assignable variation is caused by factors that can be clearly identified and possibly managed Using a saw to cut 2.1 meter long boards as a sample process Discuss random vs. assignable variation Generally, when variation is reduced, quality improves. It is impossible to have zero variability. T or F ?

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**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. Incremental Cost of Variability High Zero Lower Spec Target Upper Traditional View Incremental Cost of Variability High Zero Lower Spec Target Upper Taguchi’s View Exhibits TN8.1 & TN8.2 30

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**Process Capability Tolerance (specification, design) Limits**

Bearing diameter inches LTL = inches UTL = inches Process Limits The actual distribution from the process Run the process to make 100 bearings, compute the mean and std. dev. (and plot/graph the complete results) Suppose, mean = 1.250, std. dev = 0.002 How do they relate to one another? 28

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**Tolerance Limits vs. Process Capability**

Specification Width Actual Process Width Specification Width Actual Process Width

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**Process Capability Example**

Design Specs: Bearing diameter inches LTL = inches UTL = inches The actual distribution from the process mean = 1.250, s = 0.002 +- 3s limits (0.002) [1.244, 1.256] Anew process, std. dev. =

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**Process Capability Index, Cpk**

Capability Index shows how well parts being produced fit into design limit specifications Compute the Cpk for the bearing example. Old process, mean = 1.250, s = 0.002 What is the probability of producing defective bearings? New process, mean = 1.250, s= , re-compute the Cpk When the computed (sample) mean = design (target) mean, what does that imply?

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The Cereal Box Example 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. 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 ounces Lower Tolerance Limit = 16 – 0.05(16) = 15.2 ounces We go out and buy 1,000 boxes of cereal and find that they weight an average of ounces with a standard deviation of ounces.

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**Cereal Box Process Capability**

Specification or Tolerance Limits Upper Spec = 16.8 oz, Lower Spec = 15.2 oz Observed Weight Mean = oz, Std Dev = oz What does a Cpk of mean? Many companies look for a Cpk of 1.3 or better… 6-Sigma company wants 2.0!

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**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 application Sampling 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

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**Evidence for Investigation…**

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Control Limits If we establish control limits at +/- 3 standard deviations, then we would expect 99.7% of our observations to fall within these limits x LCL UCL

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**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!

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**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 bad 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 A2 , D3 , and D4 and are given in Exhibit 9A.6, p. 341 Excel time!

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**Acceptance Sampling vs. SPC**

Sampling to accept or reject the immediate lot of product at hand (Acceptance Sampling). Determine quality level Ensure quality is within predetermined (agreed) level Sampling to determine if the process is within acceptable limits - Statistical Process Control (SPC) 3 3

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**Acceptance Sampling Advantages Economy Less handling damage**

Fewer inspectors Upgrading of the inspection job Applicability to destructive testing Entire lot rejection (motivation for improvement) Disadvantages Risks of accepting “bad” lots and rejecting “good” lots Added planning and documentation Sample provides less information than 100-percent inspection

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**n the sample size (how many units to sample from a lot)**

A Single Sampling Plan A 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.

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**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.

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**Operating Characteristic Curve**

The OCC brings the concepts of producer’s risk, consumer’s risk, sample size, and maximum defects allowed together n = 99 c = 4 AQL LTPD 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 10 11 12 Percent defective Probability of acceptance B =.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 parameters 9 9

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**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. 10 10

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**Developing A Single Sampling Plan**

Determine: AQL? a? LTPD? ? Divide LTPD by AQL 0.03/0.01 = 3 Then 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 = 6 From the row with c=6, get nAQL = and divide it by AQL 3.286/0.01 = 328.6, round up to 329, n = 329 Problems: 3, 4, 6, 7, 8, 9, 11, 13 c LTPD/AQL n AQL 44.890 0.052 5 3.549 2.613 1 10.946 0.355 6 3.206 3.286 2 6.509 0.818 7 2.957 3.981 3 4.890 1.366 8 2.768 4.695 4 4.057 1.970 9 2.618 5.426 Sampling 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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1 © The McGraw-Hill Companies, Inc., 2006 McGraw-Hill/Irwin Technical Note 8 Process Capability and Statistical Quality Control.

1 © The McGraw-Hill Companies, Inc., 2006 McGraw-Hill/Irwin Technical Note 8 Process Capability and Statistical Quality Control.

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