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Statistical Process Control Operations Management Dr. Ron Lembke

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Designed Size 10 11 12 13 14 15 16 17 18 19 20

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Natural Variation 14.5 14.6 14.7 14.8 14.9 15.0 15.1 15.2 15.3 15.4

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Theoretical Basis of Control Charts 95.5% of all X fall within ± 2 Properties of normal distribution

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Theoretical Basis of Control Charts Properties of normal distribution 99.7% of all X fall within ± 3

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Skewness Lack of symmetry Pearson’s coefficient of skewness: Skewness = 0 Negative Skew < 0 Positive Skew > 0

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Kurtosis Amount of peakedness or flatness Kurtosis < 0 Kurtosis > 0 Kurtosis = 0

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Heteroskedasticity Sub-groups with different variances

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Design Tolerances Design tolerance: Determined by users’ needs USL -- Upper Specification Limit LSL -- Lower Specification Limit Eg: specified size +/- 0.005 inches No connection between tolerance and completely unrelated to natural variation.

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Process Capability and 6 A “capable” process has USL and LSL 3 or more standard deviations away from the mean, or 3σ. 99.7% (or more) of product is acceptable to customers LSLUSL 33 66 LSLUSL

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Process Capability LSLUSL LSL USL CapableNot Capable LSLUSL LSLUSL

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Process Capability Specs: 1.5 +/- 0.01 Mean: 1.505 Std. Dev. = 0.002 Are we in trouble?

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Process Capability Specs: 1.5 +/- 0.01 LSL = 1.5 – 0.01 = 1.49 USL = 1.5 + 0.01 = 1.51 Mean: 1.505 Std. Dev. = 0.002 LCL = 1.505 - 3*0.002 = 1.499 UCL = 1.505 + 0.006 = 1.511 1.499 1.511.491.511 Process Specs

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Capability Index Capability Index (C pk ) will tell the position of the control limits relative to the design specifications. C pk >= 1.33, process is capable C pk < 1.33, process is not capable

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Process Capability, C pk Tells how well parts produced fit into specs Process Specs 33 33 LSLUSL

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Process Capability Tells how well parts produced fit into specs For our example: C pk = min[ 0.015/.006, 0.005/0.006] C pk = min[2.5,0.833] = 0.833 < 1.33 Process not capable

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Process Capability: Re-centered If process were properly centered Specs: 1.5 +/- 0.01 LTL = 1.5 – 0.01 = 1.49 UTL = 1.5 + 0.01 = 1.51 Mean: 1.5 Std. Dev. = 0.002 LCL = 1.5 - 3*0.002 = 1.494 UCL = 1.5 + 0.006 = 1.506 1.4941.511.491.506 Process Specs

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If re-centered, it would be Capable 1.4941.511.491.506 Process Specs

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Packaged Goods What are the Tolerance Levels? What we have to do to measure capability? What are the sources of variability?

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Production Process Make Candy PackagePut in big bags Make Candy Mix Mix % Candy irregularity Wrong wt.

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Processes Involved Candy Manufacturing: Are M&Ms uniform size & weight? Should be easier with plain than peanut Percentage of broken items (probably from printing) Mixing: Is proper color mix in each bag? Individual packages: Are same # put in each package? Is same weight put in each package? Large bags: Are same number of packages put in each bag? Is same weight put in each bag?

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Weighing Package and all candies Before placing candy on scale, press “ON/TARE” button Wait for 0.00 to appear If it doesn’t say “g”, press Cal/Mode button a few times Write weight down on form

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Candy colors 1. Write Name on form 2. Write weight on form 3. Write Package # on form 4. Count # of each color and write on form 5. Count total # of candies and write on form 6. (Advanced only): Eat candies 7. Turn in forms and complete wrappers

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Peanut Candy Weights Avg. 2.18, stdv 0.242, c.v. = 0.111

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Plain Candy Weights Avg 0.858, StDev 0.035, C.V. 0.0413

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Peanut Color Mix website Brown 17.7%20% Yellow 8.2%20% Red 9.5%20% Blue15.4%20% Orange26.4%10% Green22.7%10%

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Classwebsite Brown12.1%30% Yellow14.7%20% Red11.4%20% Blue19.5%10% Orange21.2%10% Green21.2%10% Plain Color Mix

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So who cares? Dept. of Commerce National Institutes of Standards & Technology NIST Handbook 133 Fair Packaging and Labeling Act

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Acceptable?

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Package Weight “Not Labeled for Individual Retail Sale” If individual is 18g MAV is 10% = 1.8g Nothing can be below 18g – 1.8g = 16.2g

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Goal of Control Charts See if process is “in control” Process should show random values No trends or unlikely patterns Visual representation much easier to interpret Tables of data – any patterns? Spot trends, unlikely patterns easily

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NFL Control Chart?

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Control Charts UCL LCL avg Values Sample Number

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Definitions of Out of Control 1. No points outside control limits 2. Same number above & below center line 3. Points seem to fall randomly above and below center line 4. Most are near the center line, only a few are close to control limits 1. 8 Consecutive pts on one side of centerline 2. 2 of 3 points in outer third 3. 4 of 5 in outer two-thirds region

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Control Charts NormalToo LowToo high 5 above, or belowRun of 5 Extreme variability

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Control Charts UCL LCL avg 1σ1σ 2σ2σ 2σ2σ 1σ1σ

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Control Charts 2 out of 3 in the outer third

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Out of Control Point? Is there an “assignable cause?” Or day-to-day variability? If not usual variability, GET IT OUT Remove data point from data set, and recalculate control limits If it is regular, day-to-day variability, LEAVE IT IN Include it when calculating control limits

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Attribute Control Charts Tell us whether points in tolerance or not p chart: percentage with given characteristic (usually whether defective or not) np chart: number of units with characteristic c chart: count # of occurrences in a fixed area of opportunity (defects per car) u chart: # of events in a changeable area of opportunity (sq. yards of paper drawn from a machine)

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Attributes vs. Variables Attributes: Good / bad, works / doesn’t count % bad (P chart) count # defects / item (C chart) Variables: measure length, weight, temperature (x-bar chart) measure variability in length (R chart)

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p Chart Control Limits # Defective Items in Sample i Sample i Size

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p Chart Control Limits # Defective Items in Sample i Sample i Size z = 2 for 95.5% limits; z = 3 for 99.7% limits # Samples

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p Chart Control Limits # Defective Items in Sample i # Samples Sample i Size z = 2 for 95.5% limits; z = 3 for 99.7% limits

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p Chart Example You’re manager of a 1,700 room hotel. For 7 days, you collect data on the readiness of all of the rooms that someone checked out of. Is the process in control (use z = 3)? © 1995 Corel Corp.

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p Chart Hotel Data # RoomsNo. NotProportion DaynReady p 11,300130130/1,300 =.100 2800 90.113 340021.053 435025.071 530018.06 640012.03 760030.05

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p Chart Control Limits

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p Chart Solution

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Hotel Room Readiness P-Bar

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R Chart Type of variables control chart Interval or ratio scaled numerical data Shows sample ranges over time Difference between smallest & largest values in inspection sample Monitors variability in process Example: Weigh samples of coffee & compute ranges of samples; Plot

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You’re manager of a 500- room hotel. You want to analyze the time it takes to deliver luggage to the room. For 7 days, you collect data on 5 deliveries per day. Is the process in control? Hotel Example

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Hotel Data DayDelivery Time 17.304.206.103.455.55 24.608.707.604.437.62 35.982.926.204.205.10 47.205.105.196.804.21 54.004.505.501.894.46 610.108.106.505.066.94 76.775.085.906.909.30

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R & X Chart Hotel Data Sample DayDelivery TimeMeanRange 17.304.206.103.455.555.32 7.30 + 4.20 + 6.10 + 3.45 + 5.55 5 Sample Mean =

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R & X Chart Hotel Data Sample DayDelivery TimeMeanRange 17.304.206.103.455.555.323.85 7.30 - 3.45Sample Range = LargestSmallest

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R & X Chart Hotel Data Sample DayDelivery TimeMeanRange 17.304.206.103.455.555.323.85 24.608.707.604.437.626.594.27 35.982.926.204.205.104.883.28 47.205.105.196.804.215.702.99 54.004.505.501.894.464.073.61 610.108.106.505.066.947.345.04 76.775.085.906.909.306.794.22

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R Chart Control Limits Sample Range at Time i # Samples Table 10.3, p.433

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Control Chart Limits

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R Chart Control Limits

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R Chart Solution UCL

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X Chart Control Limits Sample Range at Time i # Samples Sample Mean at Time i

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X Chart Control Limits A 2 from Table 10-3

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Table 10.3 Limits

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R & X Chart Hotel Data Sample DayDelivery TimeMeanRange 17.304.206.103.455.555.323.85 24.608.707.604.437.626.594.27 35.982.926.204.205.104.883.28 47.205.105.196.804.215.702.99 54.004.505.501.894.464.073.61 610.108.106.505.066.947.345.04 76.775.085.906.909.306.794.22

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X Chart Control Limits

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X Chart Solution* 0 2 4 6 8 1234567 X, Minutes Day UCL LCL

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