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1 Slides used in class may be different from slides in student pack Chapter 9A Process Capability and Statistical Quality Control Process Variation Process Capability Process Control Procedures – Variable data – Attribute data Acceptance Sampling – Operating Characteristic Curve

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2 Slides used in class may be different from slides in student pack Basic Causes of Variation Assignable causes are factors that can be clearly identified and possibly managed. Common causes are inherent to the production process. In order to reduce variation due to common causes, the process must be changed. Key: Determining which is which!

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3 Slides used in class may be different from slides in student pack Types of Control Charts Attribute (Go or no-go information) – Defectives refers to the acceptability of product across a range of characteristics. – p-chart application Variable (Continuous) – Usually measured by the mean and the standard deviation. – X-bar and R chart applications

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4 Slides used in class may be different from slides in student pack Types of Statistical Quality Control Statistical Quality Control Process Control Acceptance Sampling Variables Charts Attributes Charts VariablesAttributes Statistical Quality Control Process Control Acceptance Sampling Variables Charts Attributes Charts VariablesAttributes

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5 Slides used in class may be different from slides in student pack UCL LCL Samples over time Normal Behavior UCL LCL Samples over time Possible problem, investigate UCL LCL Samples over time Possible problem, investigate Statistical Process Control (SPC) Charts Excellent review in exhibit TN8.5. Look for trends!

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6 Slides used in class may be different from slides in student pack Control Limits 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. x LCLUCL 99.7%

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7 Slides used in class may be different from slides in student pack Example of Constructing a p-Chart: Required Data Sample Sample size Number of defectives

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8 Slides used in class may be different from slides in student pack Statistical Process Control Formulas: Attribute Measurements (p-Chart) Given: Compute control limits:

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9 Slides used in class may be different from slides in student pack 1. Calculate the sample proportions, p (these are what can be plotted on the p- chart) for each sample. Example of Constructing a p-chart: Step 1

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10 Slides used in class may be different from slides in student pack 2. Calculate the average of the sample proportions. 3. Calculate the standard deviation of the sample proportion Example of Constructing a p-chart: Steps 2&3

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11 Slides used in class may be different from slides in student pack 4. Calculate the control limits. UCL = LCL = (0) Example of Constructing a p-chart: Step 4

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12 Slides used in class may be different from slides in student pack Example of Constructing a p-Chart: Step 5 5. Plot the individual sample proportions, the average of the proportions, and the control limits

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13 Slides used in class may be different from slides in student pack 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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14 Slides used in class may be different from slides in student pack R Chart Control Limits Sample Range in sample i # Samples From Table (function of sample size)

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15 Slides used in class may be different from slides in student pack R Chart Example 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?

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16 Slides used in class may be different from slides in student pack R Chart Hotel Data Sample DayDelivery TimeMeanRange Sample Mean =

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17 Slides used in class may be different from slides in student pack R Chart Hotel Data Sample DayDelivery TimeMeanRange Sample Range = LargestSmallest

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18 Slides used in class may be different from slides in student pack R Chart Hotel Data Sample DayDelivery TimeMeanRange

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19 Slides used in class may be different from slides in student pack R Chart Control Limits Solution From Table (n = 5)

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20 Slides used in class may be different from slides in student pack R Chart Control Chart Solution UCL R-bar

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21 Slides used in class may be different from slides in student pack X Chart Type of variables control chart –Interval or ratio scaled numerical data Shows sample means over time Monitors process average Example: Weigh samples of coffee & compute means of samples; Plot

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22 Slides used in class may be different from slides in student pack X Chart Control Limits Range of sample i # Samples Mean of sample i From Table

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23 Slides used in class may be different from slides in student pack X Chart Example 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?

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24 Slides used in class may be different from slides in student pack X Chart Hotel Data Sample DayDelivery TimeMeanRange

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25 Slides used in class may be different from slides in student pack X Chart Control Limits Solution * From Table (n = 5)

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26 Slides used in class may be different from slides in student pack X Chart Control Chart Solution* UCL LCL X-bar

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27 Slides used in class may be different from slides in student pack X AND R CHART EXAMPLE IN-CLASS EXERCISE The following collection of data represents samples of the amount of force applied in a gluing process: Determine if the process is in control by calculating the appropriate upper and lower control limits of the X-bar and R charts.

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28 Slides used in class may be different from slides in student pack X AND R CHART EXAMPLE IN-CLASS EXERCISE

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29 Slides used in class may be different from slides in student pack Example of x-bar and R charts: Step 1. Calculate sample means, sample ranges, mean of means, and mean of ranges.

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30 Slides used in class may be different from slides in student pack Example of x-bar and R charts: Step 2. Determine Control Limit Formulas and Necessary Tabled Values

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31 Slides used in class may be different from slides in student pack Example of x-bar and R charts: Steps 3&4. Calculate x-bar Chart and Plot Values

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32 Slides used in class may be different from slides in student pack Example of x-bar and R charts: Steps 5&6: Calculate R-chart and Plot Values

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33 Slides used in class may be different from slides in student pack SOLUTION: Example of x-bar and R charts: 1. Is the process in Control? 2. If not, what could be the cause for the process being out of control?

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34 Slides used in class may be different from slides in student pack Process Capability Process limits - actual capabilities of process based on historical data Tolerance limits - what process design calls for - desired performance of process

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35 Slides used in class may be different from slides in student pack Process Capability How do the limits relate to one another? You want: tolerance range > process range Two methods of accomplishing this: 1. Make bigger2. Make smaller Bad ideaImplies having greater control over process Good!

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36 Slides used in class may be different from slides in student pack Process Capability Measurement C p index = Tolerance range / Process range What value(s) would you like for C p? Larger C p indicates a more reliable and predictable process (less variability) The C p index is based on the assumption that the process mean is centered at the midpoint of the tolerance range

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37 Slides used in class may be different from slides in student pack

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38 Slides used in class may be different from slides in student pack LTL UTL

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39 Slides used in class may be different from slides in student pack While the Cp index provides useful information on process variability, it does not give information on the process average relative to the tolerance limits. Note: UTL LTL

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40 Slides used in class may be different from slides in student pack C pk Index Together, these process capability Indices show how well parts being produced conform to design specifications. = process mean (Unknown but can be estimated with the grand mean) = standard deviation (Unknown but can be estimated with the average range) Refers to the LTLRefers to the UTL

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41 Slides used in class may be different from slides in student pack LTL UTL Since C p and C pk are different we can conclude that the process is not centered, however the C p index tells us that the process variability is very low

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42 Slides used in class may be different from slides in student pack An example of the use of process capability indices The design specifications for a machined slot is 0.5±.003 inches. Samples have been taken and the process mean is estimated to be.501. The process standard deviation is estimated to be.001. What can you say about the capability of this process to produce this dimension?

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43 Slides used in class may be different from slides in student pack Process capability inches LTL inches UTL Process mean inches Machined slot (inches) = inches

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44 Slides used in class may be different from slides in student pack Sampling Distributions (The Central Limit Theorem) Regardless of the underlying distribution, if the sample is large enough (>30), the distribution of sample means will be normally distributed around the population mean with a standard deviation of :

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45 Slides used in class may be different from slides in student pack Computing Process Capability Indexes Using Control Chart Data Recall the following info from our in class exercise: Since A 2 is calculated on the assumption of three sigma limits:

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46 Slides used in class may be different from slides in student pack From the Central Limit Theorem: So, Therefore,

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47 Slides used in class may be different from slides in student pack Suppose the Design Specs for the Gluing Process were 10.7 .2, Calculate the C p and C pk Indexes: Answer:

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48 Slides used in class may be different from slides in student pack Note, multiplying each component of the C pk calculation by 3 yields a Z value. You can use this to predict the % of items outside the tolerance limits: From Appendix E we would expect: =.044 or 4.4% non-conforming product from this process.792 * 3 = * 3 = or.8% of the curve.036 or 3.6% of the curve

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49 Slides used in class may be different from slides in student pack Capability Index – In Class Exercise You are a manufacturer of equipment. A drive shaft is purchased from a supplier close by. The blueprint for the shaft specs indicate a tolerance of 5.5 inches ±.003 inches. Your supplier is reporting a mean of inches. And a standard deviation of.0015 inches. What is the C pk index for the supplier’s process?

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50 Slides used in class may be different from slides in student pack

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51 Slides used in class may be different from slides in student pack Your engineering department is sent to the supplier’s site to help improve the capability on the shaft machining process. The result is that the process is now centered and the C P index is now.75. On a percentage basis, what is the improvement on the percentage of shafts which will be unusable (outside the tolerance limits)?

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52 Slides used in class may be different from slides in student pack To answer this question we must determine the percentage of defective shafts before and after the intervention from our engineering department

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53 Slides used in class may be different from slides in student pack Before: -4 x 3x.88) =2.67 From Table.089 From Table.004 Total % outside Tolerance = =.093 or 9.3%

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54 Slides used in class may be different from slides in student pack After Since the process is centered then C pk = C p ; C p = UTL-LTL / 6 so the tolerance limits are.75 x 6 = 4.5 apart each 2.25 from the mean -4 2.25 From Table.012 So % outside of Tolerance =.012(2) =.024 Or 2.4%

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55 Slides used in class may be different from slides in student pack So the percentage decrease in defective parts is 1 – (2.4/9.3) = 74%

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56 Slides used in class may be different from slides in student pack

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57 Slides used in class may be different from slides in student pack Basic Forms of Statistical Sampling for Quality Control Sampling to accept or reject the immediate lot of product at hand (Acceptance Sampling). Sampling to determine if the process is within acceptable limits (Statistical Process Control)

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58 Slides used in class may be different from slides in student pack Acceptance Sampling Purposes – Determine quality level – Ensure quality is within predetermined level Advantages – Economy – Less handling damage – Fewer inspectors – Upgrading of the inspection job – Applicability to destructive testing – Entire lot rejection (motivation for improvement)

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59 Slides used in class may be different from slides in student pack Acceptance Sampling Disadvantages – Risks of accepting “bad” lots and rejecting “good” lots – Added planning and documentation – Sample provides less information than 100- percent inspection – No information is obtained on the process. Just sorting “good” parts from “bad” parts

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60 Slides used in class may be different from slides in student pack Risk Acceptable Quality Level (AQL) – Max. acceptable percentage of defectives defined by producer. (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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