1 Slides used in class may be different from slides in student pack Technical Note 8 Process Capability and Statistical Quality Control  Process Variation  Process Capability  Process Control Procedures – Variable data – Attribute data  Acceptance Sampling – Operating Characteristic Curve

2 Slides used in class may be different from slides in student pack Basic Causes of Variation  Assignable causes  Common causes  Key:

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

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

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

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%

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

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:

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

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

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

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

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 –  Monitors variability in process  Example: Weigh samples of coffee & compute ranges of samples; Plot

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)

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?

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

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

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

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

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

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

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

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?

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

25 Slides used in class may be different from slides in student pack  X Chart Control Limits Solution *

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

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.

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

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.

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

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

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

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?

34 Slides used in class may be different from slides in student pack Process Capability  Process limits -  Tolerance limits - –

35 Slides used in class may be different from slides in student pack Process Capability  How do the limits relate to one another?

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?

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

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

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

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. =  =  =

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

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?

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

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 :

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:

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

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:

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: non-conforming product from this process.002 or.2% of the curve.02 or 2% of the curve

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?

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

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)?

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

53 Slides used in class may be different from slides in student pack Before:

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

55 Slides used in class may be different from slides in student pack So the percentage decrease in defective parts is:

56 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  Sampling to determine if the process is within acceptable limits

57 Slides used in class may be different from slides in student pack Acceptance Sampling  Purposes – –  Advantages – – – – – –

58 Slides used in class may be different from slides in student pack Acceptance Sampling  Disadvantages – – – –

59 Slides used in class may be different from slides in student pack Risk  Acceptable Quality Level (AQL) –   (Producer’s risk) –  Lot Tolerance Percent Defective (LTPD) –   (Consumer’s risk) –