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1 Statistical Process Control (SPC) Stephen R. Lawrence Assoc. Prof. of Operations Mgmt Stephen.Lawrence@colorado.edu Leeds.colorado.edu/faculty/lawrence

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2 Process Control Tools

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3 Process tools assess conditions in existing processes to detect problems that require intervention in order to regain lost control. Check sheetsPareto analysis Scatter PlotsHistograms Scatter Plots Histograms Run Charts Control charts Cause & effect diagrams

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4 Check Sheets Check sheets explore what and where an event of interest is occurring. Attribute Check Sheet 27 15 19 20 28

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5 Run Charts time measurement Look for patterns and trends…

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6 SCATTERPLOTS Variable A Variable B x x x x x x xx x x x x xx x x x x x xx x x xx x x x x xx xxx x x x x xx xx x x x x xx x x x xxx xx x x x x xxx x x xx x x xx xx x x x x x xx x x xxx xx xx xxx x x xx xxx x x x x x x xx x x x x x x xx x x xx x x xx x x x Larger values of variable A appear to be associated with larger values of variable B.

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7 HISTOGRAMS A statistical tool used to show the extent and type of variance within the system. Frequency of Occurrences Outcome

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8 PARETO ANALYSIS A method for identifying and separating the vital few from the trivial many. Percentage of Occurrences Factor A B C D E F G I H J

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9 CAUSE & EFFECT DIAGRAMS Employees Procedures and Methods Training Speed Maintenance Equipment Condition Classification Error Inspection BAD CPU Pins not Assigned Defective Pins Received Defective Damaged in storage CPU Chip

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10 Example: Rogue River Adventures

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11 Process Variation

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12 Demings Theory of Variance Variation causes many problems for most processes Causes of variation are either common or special Variation can be either controlled or uncontrolled Management is responsible for most variation Management Employee Controlled Variation Uncontrolled Variation Common Cause Special Cause Categories of Variation

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13 Causes of Variation Natural CausesAssignable Causes What prevents perfection? Exogenous to process Not random Controllable Preventable Examples tool wear Monday effect poor maintenance Inherent to process Inherent to process Random Random Cannot be controlled Cannot be controlled Cannot be prevented Cannot be prevented Examples Examples –weather –accuracy of measurements –capability of machine Process variation...

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14 Specification vs. Variation Product specification desired range of product attribute part of product design length, weight, thickness, color,... nominal specification upper and lower specification limits Process variability inherent variation in processes limits what can actually be achieved defines and limits process capability Process may not be capable of meeting specification!

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15 Process Capability

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16 Process Capability LSLUSLSpec Process variation Capable process (Very) capable process Process not capable

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17 Process Capability Measure of capability of process to meet (fall within) specification limits Take width of process variation as 6 If 6 < (USL - LSL), then at least 99.7% of output of process will fall within specification limits LSL USL Spec 6 3 99.7%

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18 Variation -- RazorBlade

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19 Process Capability Ratio Define Process Capability Ratio Cp as If Cp > 1.0, process is... capable If Cp < 1.0, process is... not capable

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20 Process Capability -- Example A manufacturer of granola bars has a weight specification 2 ounces plus or minus 0.05 ounces. If the standard deviation of the bar-making machine is 0.02 ounces, is the process capable? USL = 2 + 0.05 = 2.05 ounces LSL = 2 - 0.05 = 1.95 ounces Cp = (USL - LSL) / 6 = (2.05 - 1.95) / 6(0.02) = 0.1 / 0.12 = 0.85 Therefore, the process is not capable!

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21 Process Centering LSLUSLSpec Capable and centered Capable, but not centered Not capable, and not centered not centered

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22 Process Centering -- Example For the granola bar manufacturer, if the process is incorrectly centered at 2.05 instead of 2.00 ounces, what fraction of bars will be out of specification? 2.0LSL=1.95USL=2.05 50% of production will be out of specification! Out of spec!

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23 Process Capability Index Cpk If C pk > 1.0, process is... Centered & capable If C pk < 1.0, process is... Not centered &/or not capable Mean Std dev

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24 Cpk Example 1 A manufacturer of granola bars has a weight specification 2 ounces plus or minus 0.05 ounces. If the standard deviation of the bar-making machine is = 0.02 ounces and the process mean is = 2.01, what is the process capability index? USL = 2.05 oz LSL = 1.95 ounces C pk = min [ ( -LSL) / 3 (USL- ) / 3 = min [ ( –1.95) / 0.06 (2.05 – 2.01) / 0.06 = min [ 1.0 0.67 = 0.67 Therefore, the process is not capable and/or not centered !

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25 Cpk Example 2 Venture Electronics manufactures a line of MP3 audio players. One of the components manufactured by Venture and used in its players has a nominal output voltage of 8.0 volts. Specifications allow for a variation of plus or minus 0.6 volts. An analysis of current production shows that mean output voltage for the component is 8.054 volts with a standard deviation of 0.192 volts. Is the process "capable: of producing components that meet specification? What fraction of components will fall outside of specification? What can management do to improve this fraction?

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26 Process Control Charts

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27 Process Control Charts Establish capability of process under normal conditions Use normal process as benchmark to statistically identify abnormal process behavior Correct process when signs of abnormal performance first begin to appear Control the process rather than inspect the product! Statistical technique for tracking a process and determining if it is going out to control

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28 Upper Control Limit Lower Control Limit 6 3 Target Spec Process Control Charts Upper Spec Limit Lower Spec Limit

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29 UCL Target LCL Samples Time In controlOut of control ! Natural variation Look for special cause ! Back in control! Process Control Charts

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30 When to Take Action A single point goes beyond control limits (above or below) Two consecutive points are near the same limit (above or below) A run of 5 points above or below the process mean Five or more points trending toward either limit A sharp change in level Other erratic behavior

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31 Samples vs. Population Population Distribution Sample Distribution Mean

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32 Types of Control Charts Attribute control charts Monitors frequency (proportion) of defectives p - charts Defects control charts Monitors number (count) of defects per unit c – charts Variable control charts Monitors continuous variables x-bar and R charts

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33 1. Attribute Control Charts p - charts Estimate and control the frequency of defects in a population Examples Invoices with error s (accounting) Incorrect account numbers (banking) Mal-shaped pretzels (food processing) Defective components (electronics) Any product with good/not good distinctions

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34 Using p-charts Find long-run proportion defective (p-bar) when the process is in control. Select a standard sample size n Determine control limits

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35 p-chart Example Chic Clothing is an upscale mail order clothing company selling merchandise to successful business women. The company sends out thousands of orders five days a week. In order to monitor the accuracy of its order fulfillment process, 200 orders are carefully checked every day for errors. Initial data were collected for 24 days when the order fulfillment process was thought to be "in control." The average percent defective was found to be 5.94%.

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36 2. Defect Control Charts c-charts Estimate & control the number of defects per unit Examples Defects per square yard of fabric Crimes in a neighborhood Potholes per mile of road Bad bytes per packet Most often used with continuous process (vs. batch)

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37 Using c-charts Find long-run proportion defective (c-bar) when the process is in control. Determine control limits

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38 2. c-chart Example Dave's is a restaurant chain that employs independent evaluators to visit its restaurants as secret shoppers to the asses the quality of service. The company evaluates restaurants in two categories, food quality, and service (promptness, order accuracy, courtesy, friendliness, etc.) The evaluator considers not only his/her order experiences, but also evaluations throughout the restaurant. Initial surveys find that the total number of service defects per survey is 7.3 when a restaurant is operating normally.

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39 3. Control Charts for Variables x-bar and R charts Monitor the condition or state of continuously variable processes Use to control continuous variables Length, weight, hardness, acidity, electrical resistance Examples Weight of a box of corn flakes (food processing) Departmental budget variances (accounting Length of wait for service (retailing) Thickness of paper leaving a paper-making machine

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40 x-bar and R charts Two things can go wrong process mean goes out of control process variability goes out of control Two control solutions X-bar charts for mean R charts for variability

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41 Problems with Continuous Variables Target Natural Process Distribution Mean not Centered Increased Variability

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42 Range (R) Chart Choose sample size n Determine average in-control sample ranges R-bar where R=max-min Construct R-chart with limits:

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43 Mean (x-bar) Chart Choose sample size n (same as for R-charts) Determine average of in-control sample means (x-double-bar) x-bar = sample mean k = number of observations of n samples Construct x-bar-chart with limits:

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44 x & R Chart Parameters

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45 R and x-bar Chart Example Resistors for electronic circuits are being manufactured on a high-speed automated machine. The machine is set up to produce resistors of 1,000 ohms each. Fifteen samples of 4 resistors each were taken over a period of time when the machine was operating normally. The average range of the samples was found to be R- bar=21.7 and the average mean of the samples was x-double-bar=999.1.

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46 When to Take Action A single point goes beyond control limits (above or below) Two consecutive points are near the same limit (above or below) A run of 5 points above or below the process mean Five or more points trending toward either limit A sharp change in level Other statistically erratic behavior

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47 Control Chart Error Trade-offs Setting control limits too tight (e.g., ± 2 ) means that normal variation will often be mistaken as an out-of-control condition (Type I error). Setting control limits too loose (e.g., ± 4 ) means that an out-of-control condition will be mistaken as normal variation (Type II error). Using control limits works well to balance Type I and Type II errors in many circumstances. 3 is not sacred -- use judgement.

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48 Video: SPC at Frito Lay

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49 Statistical Process Control

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