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**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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**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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**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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**Types of Statistical Quality Control**

Process Process Acceptance Acceptance Control Control Sampling Sampling Variables Variables Attributes Attributes Variables Variables Attributes Attributes Charts Charts Charts Charts

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**Excellent review in exhibit TN8.5.**

Statistical Process Control (SPC) Charts UCL LCL Samples over time Normal Behavior Look for trends! UCL LCL Samples over time Possible problem, investigate Excellent review in exhibit TN8.5. UCL LCL Samples over time Possible problem, investigate

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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 LCL UCL 99.7%

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**Example of Constructing a p-Chart: Required Data**

Sample Sample size Number of defectives

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**Statistical Process Control Formulas: Attribute Measurements (p-Chart)**

Given: Compute control limits:

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**Example of Constructing a p-chart: Step 1**

1. Calculate the sample proportions, p (these are what can be plotted on the p-chart) for each sample.

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**Example of Constructing a p-chart: Steps 2&3**

2. Calculate the average of the sample proportions. 3. Calculate the standard deviation of the sample proportion

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**Example of Constructing a p-chart: Step 4**

4. Calculate the control limits. UCL = LCL = (0)

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**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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**R Chart Type of variables control chart Shows sample ranges over time**

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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**R Chart Control Limits From Table (function of sample size)**

Sample Range in sample i # Samples

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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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**R Chart Hotel Data Sample Day Delivery Time Mean Range**

Sample Mean =

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**R Chart Hotel Data Sample Day Delivery Time Mean Range**

Largest Smallest Sample Range =

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**R Chart Hotel Data Sample Day Delivery Time Mean Range**

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**R Chart Control Limits Solution**

From Table (n = 5)

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**R Chart Control Chart Solution**

UCL R-bar

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**X Chart Type of variables control chart Shows sample means over time**

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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**X Chart Control Limits**

From Table Mean of sample i Range of sample i # Samples

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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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**X Chart Hotel Data Sample Day Delivery Time Mean Range**

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**X Chart Control Limits Solution***

From Table (n = 5)

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**X Chart Control Chart Solution***

UCL X-bar LCL

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**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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**X AND R CHART EXAMPLE IN-CLASS EXERCISE**

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**Example of x-bar and R charts: Step 1**

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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**Example of x-bar and R charts: Step 2**

Example of x-bar and R charts: Step 2. Determine Control Limit Formulas and Necessary Tabled Values

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**Example of x-bar and R charts: Steps 3&4**

Example of x-bar and R charts: Steps 3&4. Calculate x-bar Chart and Plot Values

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**Example of x-bar and R charts: Steps 5&6: Calculate R-chart and Plot Values**

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**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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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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**Process Capability How do the limits relate to one another?**

You want: tolerance range > process range 1. Make bigger 2. Make smaller Two methods of accomplishing this: Implies having greater control over process Þ Good! Bad idea

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**Process Capability Measurement**

Cp index = Tolerance range / Process range What value(s) would you like for Cp? Þ Larger Cp indicates a more reliable and predictable process (less variability) The Cp index is based on the assumption that the process mean is centered at the midpoint of the tolerance range

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LTL UTL

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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: LTL UTL

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**Cpk Index = process mean (Unknown but can be estimated**

Refers to the LTL Refers to the UTL = process mean (Unknown but can be estimated with the grand mean) s = standard deviation (Unknown but can be estimated with the average range) Together, these process capability Indices show how well parts being produced conform to design specifications.

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Since Cp and Cpk are different we can conclude that the process is not centered, however the Cp index tells us that the process variability is very low LTL UTL

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**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 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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**Process capability Machined slot (inches) 0.497 inches 0.503 inches**

LTL 0.503 inches UTL = 0.001 inches Process mean 0.501 inches

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**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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**Computing Process Capability Indexes Using Control Chart Data**

Recall the following info from our in class exercise: Since A2 is calculated on the assumption of three sigma limits:

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**From the Central Limit Theorem:**

So, Therefore,

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**Suppose the Design Specs for the Gluing Process were 10. 7 **

Suppose the Design Specs for the Gluing Process were 10.7 .2, Calculate the Cp and Cpk Indexes: Answer:

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Note, multiplying each component of the Cpk 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 = 2.38 .597 * 3 = 1.79 .008 or .8% of the curve .036 or 3.6% of the curve

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**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 inches. What is the Cpk index for the supplier’s process?

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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 CP index is now 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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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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**Before: From Table From Table .004 .089 Total % outside**

(3x.88) =2.67s 3x.444 = 1.33s Total % outside Tolerance = = .093 or 9.3% -4

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After Since the process is centered then Cpk = Cp; Cp = UTL-LTL / 6s, so the tolerance limits are .75 x 6s = 4.5s apart each 2.25s from the mean From Table .012 2.25s 2.25s So % outside of Tolerance = .012(2) = .024 Or 2.4% -4

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**So the percentage decrease in defective parts is 1 – (2.4/9.3) = 74%**

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**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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**Acceptance Sampling Purposes Advantages 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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**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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**Risk Acceptable Quality Level (AQL) a (Producer’s risk)**

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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