1. Topic 4. Statistical Process Control (Control Charts) and Acceptance Sampling

2. Statistical Process Control
I. Statistical Process Control:
graphical presentation of samples of process output over time
used to monitoring (production) process and detect quality problems

3. Two Types of Variations
Nature vs. Assignable Variations
Nature
Assignable
Characteristics
Stable
Unstable
Caused by
Technology Limits
Malfunction of machine or people
Actions
Technology Innovation
Find the cause, correct it

4. Idea Behind Control Charts
If (production) process is normal
only natural variations exist
samples of output is Normally distributed
within 3 std. 99.7% of time
Therefore,
If not within 3 std. ==> assignable variations exist!
UCL (Upper Control Limit) and LCL (Lower Control Limit) are set to correspond to the 3 std. lines if no specification

5. In control Signals
In control: plots Normally distributed, unbiased, no patterns
indicating no assignable variations exist

6. Out of control Signals one plot outside UCL or LCL (for all charts)
2 of 3 consecutive plots out of 2 std. Line (for X-bar chart)
7 consecutive plots on one side (for X-bar chart)
 indicating assignable variations exist, sign of quality problems.

7. Types of control chart Variable Charts: for continuous quality measure
X-bar ( ) chart: process average
R chart: process dispersion and variation
Attribute Charts: for attribute quality measure
p chart: defective rate
c chart: number of defectives

8. Construct and Use Control Charts (X-bar Charts)
Construct X-bar chart
1. based on some process information: If process (population) mean and standard deviation are known.
CL =
UCL =
LCL =
n: sample size
z: normal score (two tails), equals 3 without specification

9. Construct and Use Control Charts (X-bar Charts)
Some important normal scores
Z= 3 (99.7%)
Z= 2.5 (98.75%)
Z= 2.33 (98%)
Z= 2.17 (97%)
Z= 2 (95.5%)
Z= 1.96 (95%)
Z=1.645 (90%)

10. Construct and Use Control Charts (Example 1 for X-bar Chart by Method 1)
Samples taken from a process for making aluminum rods have an average of 2cm. The sample size is 16. The process variability is approximately normal and has a std. of 0.1cm. Design an X-bar chart for this process control.

11. Construct and Use Control Charts (X-bar Charts)
Construct X-bar Chart
2. If process mean and standard deviation are unknown, X-bar Chart can be constructed based only on past samples
Assume k past samples with sample size n.
Sample i (i = 1, 2, …, k) has n observations
Sample mean for sample i is
Sample range for sample is

12. Construct and Use Control Charts (X-bar Charts)
2. If process mean and standard deviation are unknown, X-bar Chart can be constructed based only on past samples (continuous)
The average of past k samples is
The range average of past k samples is

13. Construct and Use Control Charts (X-bar Chart)
2. If process mean and standard deviation are unknown, X-bar Chart can be constructed based only on past samples (continuous)
X-bar chart is:
Mean factor is a function of sample size n

14. Construct and Use Control Charts
Sample Size (n)
Mean Factor (A2)
Upper Range (D4)
Lower Range (D3) 2
1.880
3.268 3
1.023
2.574 4
0.729
2.282 5
0.577
2.115 6
0.483
2.004 7
0.419
1.924
0.076 8
0.373
1.864
0.136 9
0.337
1.816
0.184 10
0.308
1.777
0.223

15. Construct and Use Control Charts (X-bar Chart)
3. Differences between method 1. and method 2.
Method 1 is based known process (target or standard) information, while method 2 is based on past data information (target or standard unknown)
Therefore, if there is a out of control signal by method 1, we can say it is different from the target (standard). However, if there is a out of control signal by method 2, we cannot say it is different from the target (standard).

16. Construct and Use Control Charts (R Chart)
Construct R chart (based on past samples)
are functions of sample size n

17. Construct and Use Control Charts (Example 2 for X-bar and R Charts by Method 2)
Five samples of drop-forged steel handles, with four observations in each sample, have been taken. The weight of each handle in the samples is given below (in ounces). Use the sample data to construct an X-bar chart and an R-chart to monitor the future process.

18. Construct and Use Control Charts (Example 2 for X-bar and R Charts by Method 2)
Continuous
Sample 1
Sample 2
Sample 3
Sample 4
Sample 5
10.2
10.3
9.7
9.9
9.8
10.1
10.4
10.5

19. Construct and Use Control Charts (Use X- bar chart and R chart)
Calculate averages and ranges of new samples
Plot on the X-bar chart and R chart, respectively

20. Construct and Use Control Charts (Example 2 Continuous)
Five more samples of the handles are taken. Is the process in control (changed)?
Sample 6
Sample 7
Sample 8
Sample 9
Sample 10
10.4
10.5
9.9
10.3
9.8
10.6

21. Construct and Use Control Charts (p Chart)
Construct p chart (defective rate chart) based past samples
Assume there are k past samples with sample size n.
Each item in a sample may only have two possible outcomes: good or defective
In sample i (i = 1,2, …, k), there are number of defectives out of n items.

22. Construct and Use Control Charts (p Chart)
The defective rate for sample i is
The average number of defectives in past k samples is
And the average defective rate in the past samples is

23. Construct and Use Control Charts (p Chart)
P Chart is
Again, z is normal score (two tails)

24. Construct and Use Control Charts (p Chart)
Use p-chart
Calculate defective rates of new samples
Plot on the p-chart

25. Construct and Use Control Charts (Example 3 for p Chart)
A good quality lawnmower is supposed to start at the first try. In the third quarter, 50 craftsman lawnmowers are started every day and an average of 4 did not start. In the fourth quarter, the number of lawnmower did not start (out of 50) in the first 6 days are 4, 5, 4, 6, 7, 6, respectively. Was the quality of lawnmower changed in the fourth quarter?

26. Construct and Use Control Charts (c Chart)
Only used when sample size is unknown
Number of complaints received in a postmaster office
Assume there are k past samples with unknown sample size
Sample i has number defectives
The average defective number in past samples is

27. Construct and Use Control Charts (c Chart)
c Chart is
Again, z is normal score (two tails)

28. Construct and Use Control Charts (c Chart)
Use c-chart
Count number of defectives in new samples
Plot on the c-chart

29. Construct and Use Control Charts (Example 4 for c Chart)
There have been complaints that the sports page of the Dubuque Register has lots of typos. The last 6 days have been examined carefully, and the number of typos/page is recorded below. Is the process in control?
Day
Mon.
Tues.
Wed.
Thurs.
Fri.
Sat.
Typos 2 1 5 3 4

30. Acceptance Sampling II. Acceptance Sampling
Acceptance Sampling: Accept or reject a lot (input components or finished products) based on inspection of a sample of products in the lot
Tool for Quality Assurance

31. Acceptance Sampling Role of Inspection
Involved in all stages of production process
Inspection itself does not improve quality
Destructive and nondestructive inspection
Why sampling instead of 100% inspection?
Destructive test
Worker's morale
Cost consideration

32. Acceptance Sampling Single Acceptance Sampling Plan:
Take a sample of size n from a lot with size N
Inspect the sample 100%
If number of defective > c, reject the whole lot; otherwise, accept it.
need to determine n and c.

33. Acceptance Sampling Operating Characteristic (OC) Curves
to evaluate how well a single acceptance sampling plan discriminates between good and bad lots

34. Acceptance Sampling
Draw an OC curve approximately for a given n and c

35. Acceptance Sampling Idea:
The number of defectives in a sample of size n with defective rate p follows a Poisson distribution approximately with parameter  = n*p, when p is small, n is large, and N is even larger.

36. Acceptance Sampling

37. Acceptance Sampling Procedure: 1. Create a series of p = 1% to 10%.
2. Calculate  = n*p for each p.
3. Use the Poisson table of Appendix B to find P(acceptance) for each  and c.
4. Link P(acceptance) to form a curve.

38. Acceptance Sampling
Example 5 for OC curve: A single sampling plan with n=100 and c=3 is used to inspect a shipment of computer memory chips. Draw the OC curve for the sampling plan.
P(%) 1 2 3 4 5 6 7 8 9 10
 = n*p
P(accpt)

39. Acceptance Sampling Concepts related to the OC Curve
AQL: Acceptable quality level, the defective rate that a consumer is happy to accept (considers as a good lot)
LTPD: Lot tolerance percent defective, the maximum defective rate that a consumer is willing to accept

40. Acceptance Sampling Concepts related to the OC Curve (continued)
Consumer's risk: the probability that a lot containing defective rate exceeding the LTPD will be accepted.
Producer's risk: the probability that a lot containing the AQL will be rejected.

41. Acceptance Sampling
Example 5 continued: The buyer of the memory chip requires that the consumer’s risk is limited to 5% at LTPD = 8%. The producer requires that the producer’s risk is no more than 5% at AQL = 2%. Does the single sampling plan meet both consumer and producer’s requirements?

42. Acceptance Sampling
Sensitivity of OC curve, consumer's risk, and producer's risk to N, n, c.
Changing n, keeping c constant:
n increase, and c constant, tougher
Changing c, keeping n constant:
c increase, and n constant, easier

43. Acceptance Sampling
Sensitivity of OC curve, consumer's risk, and producer's risk to N, n, c.
Changing both n and c, keeping c/n constant:
Both n and c increase, more accurate
Changing N:
N increase, less accurate

44. Acceptance Sampling Average Outgoing Quality (AOQ)
the quality after inspection (by a single sampling plan), measured in defective rate, assuming all defectives in the rejected lot are replaced
= P (acceptance for a lot with defective rate p), can be found from the OC curve

45. Acceptance Sampling
Example 5 continued: The average defective rate of the memory chip is about 5% (based on the past data). Calculate the AOQ of the memory chip after it is inspected by the sampling plan in Example 5.

46. Acceptance Sampling Other Sampling Plans Double sampling plan
Given n: sample size
: acceptable level of the first sample
: acceptable level of both samples

47. Acceptance Sampling Example:
n = 100, = 4, = 7, Number of defective in the first sample = 5.

48. Acceptance Sampling Sequential sampling plan Given:
n: sample size and upper and
lower limits of number of defectives allowed

49. Acceptance Sampling Procedure:
Count # of total defectives found in all previous samples
If # of defectives > upper boundary, reject the lot
If # of defectives <= lower boundary, accept the lot
Otherwise, take a new sample and repeat.

50. Acceptance Sampling Advantages of double and sequential samplings:
Psychologically:
Cost: less inspection for the same accuracy

51. Homework for SPC and Acceptance Sampling
Problem 1
A manufacturing company wants to use control charts to monitor a continuous process to cut plastic tubes into standard lengths. Samples of five observations each were taken yesterday, and the results are in the table below.
1. Using these sample data to construct appropriate control charts to monitor the future cutting process.

52. Homework for SPC and Acceptance Sampling
Sample 1 2 3 4 5 6
79.1
80.5
79.6
78.9
79.7
78.8
79.4
79.5
80.6
80.0
81.0
80.4
79.8
78.4
80.3
80.7
80.1
80.8

53. Homework for SPC and Acceptance Sampling
2. Four more samples of the same size have been taken today, and the results are given below. Based on the control charts you constructed, did you notice any major changes in today’s cutting process?

54. Homework for SPC and Acceptance Sampling
Sample 1 (today)
Sample 2 (today)
Sample 3 (today)
Sample 4 (today)
78.0
81.0
79.0
79.1
82.5
82.0
81.2
80.0
81.5
83.1
78.5
82.2
79.6
82.3
79.5

55. Homework for SPC and Acceptance Sampling
Problem 2
An automatic screw machine produces hex nuts. If a hex nut does not meet the quality standard, it is considered as defective. Samples of 200 hex nuts each were taken to monitor the production. The number of defective hex nuts from the past 13 samples is listed below. Construct an appropriate control chart and determine whether or not the process is in control. (Hint: The quality here is measured by defective rate)

56. Homework for SPC and Acceptance Sampling
Sample 1 2 3 4 5 6 7 8 9 10 11 12 13
# of defectives

57. Homework for SPC and Acceptance Sampling
Problem 3
The postmaster of a small western city receives a certain number of complaints about mail delivery each day. The number of complaints in the past 14 days is given below. Construct a control chart to see if the quality of mail delivery is in control (stable)?
Hint: For Problem 2 and 3, use the same data to construct control charts and plot on the charts constructed.

58. Homework for SPC and Acceptance Sampling
Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14
# of
complaints

59. Homework for SPC and Acceptance Sampling
Problem 4
Answer the following questions for a single sampling plan with sample size n = 80 and c = 4.
Draw the OC curve for the sampling plan, using the Poisson table distributed in class
If AQL = 2% and LTPD = 8%, what would be the producer's and consumer's risks associated with the sampling plan?
If the sampling plan is used to inspect a lot of 10,000 products with an average defective rate of 5%, what would be the average quality after inspection, assuming all the defectives will be replaced if the lot is rejected?