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S6 - 1© 2011 Pearson Education, Inc. publishing as Prentice Hall S6 Statistical Process Control PowerPoint presentation to accompany Heizer and Render Operations Management, 10e Principles of Operations Management, 8e PowerPoint slides by Jeff Heyl

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S6 - 2© 2011 Pearson Education, Inc. publishing as Prentice Hall Statistical Process Control The objective of a process control system is to provide a statistical signal when assignable causes of variation are present

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S6 - 3© 2011 Pearson Education, Inc. publishing as Prentice Hall Variability is inherent in every process Natural or common causes Special or assignable causes Provides a statistical signal when assignable causes are present Detect and eliminate assignable causes of variation Statistical Process Control (SPC)

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S6 - 4© 2011 Pearson Education, Inc. publishing as Prentice Hall Natural Variations Also called common causes Affect virtually all production processes Expected amount of variation Output measures follow a probability distribution For any distribution there is a measure of central tendency and dispersion If the distribution of outputs falls within acceptable limits, the process is said to be “in control”

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S6 - 5© 2011 Pearson Education, Inc. publishing as Prentice Hall Assignable Variations Also called special causes of variation Generally this is some change in the process Variations that can be traced to a specific reason The objective is to discover when assignable causes are present Eliminate the bad causes Incorporate the good causes

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S6 - 6© 2011 Pearson Education, Inc. publishing as Prentice Hall Types of Data Characteristics that can take any real value May be in whole or in fractional numbers Continuous random variables VariablesAttributes Defect-related characteristics Classify products as either good or bad or count defects Categorical or discrete random variables

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S6 - 7© 2011 Pearson Education, Inc. publishing as Prentice Hall Control Charts for Variables For variables that have continuous dimensions Weight, speed, length, strength, etc. x-charts are to control the central tendency of the process R-charts are to control the dispersion of the process These two charts must be used together

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S6 - 8© 2011 Pearson Education, Inc. publishing as Prentice Hall Steps In Creating Control Charts 1.Take samples from the population and compute the appropriate sample statistic 2.Use the sample statistic to calculate control limits and draw the control chart 3.Plot sample results on the control chart and determine the state of the process (in or out of control) 4.Investigate possible assignable causes and take any indicated actions 5.Continue sampling from the process and reset the control limits when necessary

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S6 - 9© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Chart Limits For x-Charts when we don’t know Lower control limit (LCL) = x - A 2 R Upper control limit (UCL) = x + A 2 R whereR=average range of the samples A 2 =control chart factor found in Table S6.1 x=mean of the sample means

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S6 - 10© 2011 Pearson Education, Inc. publishing as Prentice Hall Control Chart Factors Table S6.1 Sample Size Mean Factor Upper Range Lower Range n A 2 D 4 D 3 21.8803.2680 31.0232.5740 4.7292.2820 5.5772.1150 6.4832.0040 7.4191.9240.076 8.3731.8640.136 9.3371.8160.184 10.3081.7770.223 12.2661.7160.284

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S6 - 11 Setting Control Limits Process average x = 12 ounces, Average range =.25 Sample size n = 5 Data for the - and R -Charts: Ounces of Super Cola in each bottle Sample12345 R 111.9011.8311.9911.7112.0011.8860.29 212.1012.1912.1012.1912.1012.1360.09 312.20 12.0112.2912.2012.1800.28 411.8011.9911.90 11.8980.19 512.0011.8012.1011.7011.9011.9000.40 Average =12.0000.25 1 = (11.90 + 11.83 + 11.99 + 11.71 + 12.00)/5 = 11.886 R 1 = 12.00 – 11.71 = 0.290

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S6 - 12© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Control Limits Process average x = 12 ounces Average range R =.25 Sample size n = 5

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S6 - 13© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Control Limits UCL x = x + A 2 R = 12 + (.577)(.25) = 12 +.144 = 12.144 ounces Process average x = 12 ounces Average range R =.25 Sample size n = 5 From Table S6.1

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S6 - 14© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Control Limits UCL x = x + A 2 R = 12 + (.577)(.25) = 12 +.144 = 12.144 ounces LCL x = x - A 2 R = 12 -.144 = 11.857 ounces Process average x = 12 ounces Average range R =.25 Sample size n = 5 UCL = 12.144 Mean = 12 LCL = 11.857

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S6 - 15© 2011 Pearson Education, Inc. publishing as Prentice Hall Restaurant Control Limits For salmon filets at Darden Restaurants Sample Mean x Bar Chart UCL = 11.524 x – 10.959 LCL – 10.394 ||||||||| 1357911131517 11.5 – 11.0 – 10.5 – Sample Range Range Chart UCL = 0.6943 R = 0.2125 LCL = 0 ||||||||| 1357911131517 0.8 – 0.4 – 0.0 –

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S6 - 16© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Chart Limits For x-Charts when we know Upper control limit (UCL) = x + z x Lower control limit (LCL) = x - z x wherex=mean of the sample means or a target value set for the process z=number of normal standard deviations x =standard deviation of the sample means = / n =population standard deviation n=sample size

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S6 - 17© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Control Limits Hour 1 SampleWeight of NumberOat Flakes 117 213 316 418 517 616 715 817 916 Mean16.1 =1 HourMeanHourMean 116.1715.2 216.8816.4 315.5916.3 416.51014.8 516.51114.2 616.41217.3 n = 9 LCL x = x - z x = 16 - 3(1/3) = 15 ozs For 99.73% control limits, z = 3 UCL x = x + z x = 16 + 3(1/3) = 17 ozs

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S6 - 18© 2011 Pearson Education, Inc. publishing as Prentice Hall 17 = UCL 15 = LCL 16 = Mean Setting Control Limits Control Chart for sample of 9 boxes Sample number |||||||||||| 123456789101112 Variation due to assignable causes Variation due to natural causes Out of control

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S6 - 19© 2011 Pearson Education, Inc. publishing as Prentice Hall R – Chart Type of variables control chart Shows sample ranges over time Difference between smallest and largest values in sample Monitors process variability Independent from process mean

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S6 - 20© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Chart Limits For R-Charts Lower control limit (LCL R ) = D 3 R Upper control limit (UCL R ) = D 4 R where R=average range of the samples D 3 and D 4 =control chart factors from Table S6.1

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S6 - 21© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Control Limits UCL R = D 4 R = (2.115)(5.3) = 11.2 pounds LCL R = D 3 R = (0)(5.3) = 0 pounds Average range R = 5.3 pounds Sample size n = 5 From Table S6.1 D 4 = 2.115, D 3 = 0 UCL = 11.2 Mean = 5.3 LCL = 0

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S6 - 22© 2011 Pearson Education, Inc. publishing as Prentice Hall Patterns in Control Charts Normal behavior. Process is “in control.” Upper control limit Target Lower control limit Figure S6.7

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S6 - 23© 2011 Pearson Education, Inc. publishing as Prentice Hall Upper control limit Target Lower control limit Patterns in Control Charts One plot out above (or below). Investigate for cause. Process is “out of control.” Figure S6.7

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S6 - 24© 2011 Pearson Education, Inc. publishing as Prentice Hall Upper control limit Target Lower control limit Patterns in Control Charts Two plots very near lower (or upper) control. Investigate for cause. Figure S6.7

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S6 - 25© 2011 Pearson Education, Inc. publishing as Prentice Hall Upper control limit Target Lower control limit Patterns in Control Charts Trends in either direction, 5 plots. Investigate for cause of progressive change. Figure S6.7

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S6 - 26© 2011 Pearson Education, Inc. publishing as Prentice Hall Upper control limit Target Lower control limit Patterns in Control Charts Run of 5 above (or below) central line. Investigate for cause. Figure S6.7

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S6 - 27© 2011 Pearson Education, Inc. publishing as Prentice Hall Upper control limit Target Lower control limit Patterns in Control Charts Erratic behavior. Investigate. Figure S6.7

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S6 - 28 Type I and Type II error Actual process state In control Out of control Decision based on SPC chart In control (Don't stop production) No error Type II error Out of control (Stop production) Type I error No error

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S6 - 29© 2011 Pearson Education, Inc. publishing as Prentice Hall Control Charts for Attributes For variables that are categorical Good/bad, yes/no, acceptable/unacceptable Measurement is typically counting defectives Charts may measure Percent defective (p-chart) Number of defects (c-chart)

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S6 - 30© 2011 Pearson Education, Inc. publishing as Prentice Hall Control Limits for p-Charts Population will be a binomial distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics UCL p = p + z p ^ LCL p = p - z p ^ wherep=mean fraction defective in the sample z=number of standard deviations p =standard deviation of the sampling distribution n=sample size ^ p(1 - p) n p = ^

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S6 - 31© 2011 Pearson Education, Inc. publishing as Prentice Hall p-Chart for Data Entry SampleNumberFractionSampleNumberFraction Numberof ErrorsDefectiveNumberof ErrorsDefective 16.06116.06 25.05121.01 30.00138.08 41.01147.07 54.04155.05 62.02164.04 75.051711.11 83.03183.03 93.03190.00 102.02204.04 Total = 80 (.04)(1 -.04) 100 p = =.02 ^ p = =.04 80 (100)(20)

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S6 - 32© 2011 Pearson Education, Inc. publishing as Prentice Hall.11 –.10 –.09 –.08 –.07 –.06 –.05 –.04 –.03 –.02 –.01 –.00 – Sample number Fraction defective |||||||||| 2468101214161820 p-Chart for Data Entry UCL p = p + z p =.04 + 3(.02) =.10 ^ LCL p = p - z p =.04 - 3(.02) = 0 ^ UCL p = 0.10 LCL p = 0.00 p = 0.04

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S6 - 33© 2011 Pearson Education, Inc. publishing as Prentice Hall.11 –.10 –.09 –.08 –.07 –.06 –.05 –.04 –.03 –.02 –.01 –.00 – Sample number Fraction defective |||||||||| 2468101214161820 p-Chart for Data Entry UCL p = p + z p =.04 + 3(.02) =.10 ^ LCL p = p - z p =.04 - 3(.02) = 0 ^ UCL p = 0.10 LCL p = 0.00 p = 0.04 Possible assignable causes present

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S6 - 34© 2011 Pearson Education, Inc. publishing as Prentice Hall Control Limits for c-Charts Population will be a Poisson distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics wherec=mean number defective in the sample UCL c = c + Z cLCL c = c - Z c

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S6 - 35© 2011 Pearson Education, Inc. publishing as Prentice Hall c-Chart for Cab Company c = 54 complaints/9 days = 6 complaints/day |1|1 |2|2 |3|3 |4|4 |5|5 |6|6 |7|7 |8|8 |9|9 Day Number defective 14 – 12 – 10 – 8 – 6 – 4 – 2 – 0 – UCL c = c + 3 c = 6 + 3 6 = 13.35 LCL c = c - 3 c = 6 - 3 6 = 0 UCL c = 13.35 LCL c = 0 c = 6

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S6 - 36© 2011 Pearson Education, Inc. publishing as Prentice Hall Which Control Chart to Use Table S6.3 Variables Data Using an x-Chart and R-Chart 1.Observations are variables 2.Collect 20 - 25 samples of n = 4, or n = 5, or more, each from a stable process and compute the mean for the x-chart and range for the R- chart 3.Track samples of n observations each.

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S6 - 37© 2011 Pearson Education, Inc. publishing as Prentice Hall Which Control Chart to Use Table S6.3 Attribute Data Using the p-Chart 1.Observations are attributes that can be categorized as good or bad (or pass–fail, or functional–broken), that is, in two states. 2.We deal with fraction, proportion, or percent defectives. 3.There are several samples, with many observations in each. For example, 20 samples of n = 100 observations in each.

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S6 - 38© 2011 Pearson Education, Inc. publishing as Prentice Hall Which Control Chart to Use Table S6.3 Attribute Data Using a c-Chart 1.Observations are attributes whose defects per unit of output can be counted. 2.We deal with the number counted, which is a small part of the possible occurrences. 3.Defects may be: number of blemishes on a desk; complaints in a day; crimes in a year; broken seats in a stadium; typos in a chapter of this text; or flaws in a bolt of cloth.

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S6 - 39© 2011 Pearson Education, Inc. publishing as Prentice Hall Process Capability The natural variation of a process should be small enough to produce products that meet the standards required A process in statistical control does not necessarily meet the design specifications Process capability is a measure of the relationship between the natural variation of the process and the design specifications

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S6 - 40© 2011 Pearson Education, Inc. publishing as Prentice Hall Process Capability Index A capable process must have a C pk of at least 1.33 A capable process is not necessarily in the center of the specification, but it falls within the specification limit at both extremes C pk = minimum of, Upper Specification - x Limit Lower x -Specification Limit

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S6 - 41© 2011 Pearson Education, Inc. publishing as Prentice Hall Process Capability Index New Cutting Machine New process mean x =.250 inches Process standard deviation =.0005 inches Upper Specification Limit =.251 inches Lower Specification Limit =.249 inches

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S6 - 42© 2011 Pearson Education, Inc. publishing as Prentice Hall Process Capability Index New Cutting Machine New process mean x =.250 inches Process standard deviation =.0005 inches Upper Specification Limit =.251 inches Lower Specification Limit =.249 inches C pk = minimum of, (.251) -.250 (3).0005

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S6 - 43© 2011 Pearson Education, Inc. publishing as Prentice Hall Process Capability Index New Cutting Machine New process mean x =.250 inches Process standard deviation =.0005 inches Upper Specification Limit =.251 inches Lower Specification Limit =.249 inches C pk = = 0.67.001.0015 New machine is NOT capable C pk = minimum of, (.251) -.250 (3).0005.250 - (.249) (3).0005 Both calculations result in

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S6 - 44© 2011 Pearson Education, Inc. publishing as Prentice Hall Interpreting C pk C pk = negative number C pk = zero C pk = between 0 and 1 C pk = 1 C pk > 1 Figure S6.8

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S6 - 45© 2011 Pearson Education, Inc. publishing as Prentice Hall Process Capability Ratio C p = Upper Specification - Lower Specification 6 not capable A capable process must have a C p of 1.0 is considered not capable Does not look at how well the process is centered in the specification range A target value of C p = 1.33 is used to allow for off-center processes Six Sigma quality requires a C p = 2.0

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S6 - 46© 2011 Pearson Education, Inc. publishing as Prentice Hall Process Capability Ratio C p = Upper Specification - Lower Specification 6 Insurance claims process Process mean x = 210.0 minutes Process standard deviation =.516 minutes Design specification = 210 ± 3 minutes

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S6 - 47© 2011 Pearson Education, Inc. publishing as Prentice Hall Process Capability Ratio C p = Upper Specification - Lower Specification 6 Insurance claims process Process mean x = 210.0 minutes Process standard deviation =.516 minutes Design specification = 210 ± 3 minutes = = 1.938 213 - 207 6(.516)

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S6 - 48© 2011 Pearson Education, Inc. publishing as Prentice Hall Process Capability Ratio C p = Upper Specification - Lower Specification 6 Insurance claims process Process mean x = 210.0 minutes Process standard deviation =.516 minutes Design specification = 210 ± 3 minutes = = 1.938 213 - 207 6(.516) Process variance is small enough

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S6 - 49© 2011 Pearson Education, Inc. publishing as Prentice Hall Acceptance Sampling Form of quality testing used for incoming materials or finished goods Take samples at random from a lot (shipment) of items Inspect each of the items in the sample Decide whether to reject the whole lot based on the inspection results Only screens lots; does not drive quality improvement efforts

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S6 - 50© 2011 Pearson Education, Inc. publishing as Prentice Hall Acceptance Sampling An Accepting Sampling plan is defined by n,c, where, n = Sample size c = Critical number of defectives in the sample N = Lot size

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S6 - 51© 2011 Pearson Education, Inc. publishing as Prentice Hall Acceptance Sampling Form of quality testing used for incoming materials or finished goods Take samples at random from a lot (shipment) of items Inspect each of the items in the sample Decide whether to reject the whole lot based on the inspection results Only screens lots; does not drive quality improvement efforts Rejected lots can be: Returned to the supplier Culled for defectives (100% inspection)

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S6 - 52© 2011 Pearson Education, Inc. publishing as Prentice Hall Operating Characteristic Curve For a given (n,c), OC Curve is a plot of P d on the x-axis and P a on the y-axis P d = Probability of defectives in the lot P a = Probability of accepting the lot, i.e. P(X<=c), x = number of defectives in the sample Poisson probability tables can be used to develop OC-Curve

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S6 - 53 Determining P a np012.40.670.938.992.45.638.925.989.50.607.910.986.55.577.894.982.60.549.878.977.65.522.861.972 P a =.977 Let the sampling plan be n = 60, and c = 2 For P d = 0, P a = 1.0 For P d = 1%, compute nPd = (60 x.01) = 0.6

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S6 - 54 OC Curve for n=60, c=2 PdPd nP d PaPa 0%0.01.000 1%0.60.977 2%1.20.879 3%1.80.731 4%2.40.570 6%3.60.303 8%4.80.143 10%6.00.062 12%7.20.025 14%8.40.010

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S6 - 55© 2011 Pearson Education, Inc. publishing as Prentice Hall AQL and LTPD Acceptable Quality Level (AQL) Poorest level of quality we are willing to accept Lot Tolerance Percent Defective (LTPD) Quality level we consider bad Consumer (buyer) does not want to accept lots with more defects than LTPD

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S6 - 56© 2011 Pearson Education, Inc. publishing as Prentice Hall Producer’s and Consumer’s Risks Producer's risk ( ) The probability of rejecting a good lot (i.e. P d <= AQL) based on the acceptance sampling plan (Type I error = a) Consumer's risk ( ) The probability of accepting a bad lot (i.e. P d >= LTPD) based on the acceptance sampling plan (Type II error = )

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S6 - 57 Producer’s and Consumer’s risks The lot state P d <= AQL (The lot is good) P d >= LTPD (The lot is bad) Decision based on n,c Sample defectives <= c Accept the lot No error Accept the lot P a = Consumer’s risk (Type II error) Sample defectives > c Reject the lot 1- P a = Producer’s risk (Type I error) Reject the lot No error

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S6 - 58© 2011 Pearson Education, Inc. publishing as Prentice Hall An OC Curve Probability of Acceptance Percent defective |||||||||012345678|||||||||012345678 100 – 95 – 75 – 50 – 25 – 10 – 0 – = 0.05 producer’s risk for AQL = 0.10 Consumer’s risk for LTPD LTPDAQL Bad lots Indifference zone Good lots Figure S6.9

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S6 - 59© 2011 Pearson Education, Inc. publishing as Prentice Hall OC Curves for Different Sampling Plans n = 50, c = 1 n = 100, c = 2

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S6 - 60© 2011 Pearson Education, Inc. publishing as Prentice Hall Average Outgoing Quality 1.If a sampling plan replaces all defectives 2.If we know the incoming percent defective for the lot We can compute the average outgoing quality (AOQ) in percent defective The maximum AOQ is the highest percent defective or the lowest average quality and is called the average outgoing quality level (AOQL)

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S6 - 61© 2011 Pearson Education, Inc. publishing as Prentice Hall Average Outgoing Quality where P d = true percent defective of the lot P a = probability of accepting the lot N= number of items in the lot n= number of items in the sample AOQ = (P d )(P a )(N - n) N

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S6 - 62 Average Outgoing Quality P d = 2%, nP d =.02(100) = 2, from Table P a = 0.857 AOQ = (P d )(P a )(N - n) N Suppose that the sampling plan is n = 100, c = 3, and the lot size N = 1000. Determine the AOQ value for an incoming lot defective of 2%.

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S6 - 63 AOQ for n=100, c=3 PdPd nP d PaPa AOQ 0%01.0 0.000 1%10.86 0.009 2%20.65 0.015 3%30.43 0.017 4%40.15 0.016 6%60.04 0.008 8%80.01 0.003 AOQL = Average Outgoing Quality Limit = 1.7%

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