topic 4. statistical process control (control charts) and acceptance sampling

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

Sampling

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

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

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

In control Signals

• In control: plots Normally distributed, unbiased, no patterns– indicating no assignable variations exist

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.

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

X

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

nz

*

nz

*


– 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%)

Construct and Use Control Charts (Example 1 for X-bar Chart

by Method 1)

• Example 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.


• 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

inii xxx ,,, 21

n

xxxx iniii

21

iniiiniii xxxxxxR ,,,min,,,max 2121


– 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

k

xxxx k

21

k

RRRR k

21

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

zero) (otherwise ,02

2

RAXLCL

RAXUCL

XCL

2A

Construct and Use Control Charts

Sample Size (n) Mean Factor (A2) Upper Range (D4)

Lower Range (D3)

2 1.880 3.268 0

3 1.023 2.574 0

4 0.729 2.282 0

5 0.577 2.115 0

6 0.483 2.004 0

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

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).

Construct and Use Control Charts (R Chart)

• Construct R chart (based on past samples)

are functions of sample size n

RDLCL

RDUCL

RCL

3

4

34 ,DD

Construct and Use Control Charts (Example 2 for X-bar and R

Charts by Method 2)

• Example 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.

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

9.9 9.8 9.9 10.3 10.2

9.8 9.9 9.9 10.1 10.3

10.1 10.4 10.1 10.5 9.7

Construct and Use Control Charts (Use X- bar chart and R

chart) • Use X-bar chart and R chart

– Calculate averages and ranges of new samples

– Plot on the X-bar chart and R chart, respectively

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 9.9

9.8 9.9 9.9 10.4 10.4

9.9 9.9 9.9 10.6 10.5

10.3 10.5 10.3 10.5 9.9

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.id


– 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

n

dp ii

k

dddd k

21

n

d

k

pppp k

21


– P Chart is

• Again, z is normal score (two tails)

zero otherwise ,0

1

1

n

ppzpLCL

n

ppzpUCL

pCL


• Use p-chart– Calculate defective rates of new samples– Plot on the p-chart

Construct and Use Control Charts (Example 3 for p Chart)

• Example 3 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?

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

ic

k

cccc k

21


• c Chart is

– Again, z is normal score (two tails)

zero otherwise ,0

czcLCL

czcUCL

cCL


• Use c-chart– Count number of defectives in new samples– Plot on the c-chart

Construct and Use Control Charts (Example 4

for c Chart)• Example 4

– 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 0

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

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

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.

Acceptance Sampling

• Operating Characteristic (OC) Curves

– to evaluate how well a single acceptance sampling plan discriminates between good and bad lots

Acceptance Sampling

• Draw an OC curve approximately for a given n and c

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.

Acceptance Sampling

c

i

i

i

e

pn

cP

cPP

0

!

mean th Poisson wion based

, defectives of #

defectives of #acceptance

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.

Acceptance Sampling

• Example 5 for OC curve: A single sampling plan with n=100 and c=3 is used to inspect a shipment of 10000 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)

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

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.

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?

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

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

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

aa Pp

N

nNPpAOQ

aP

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.

Acceptance Sampling

• Other Sampling Plans

– Double sampling plan• Given n: sample size• : acceptable level of the first sample• : acceptable level of both samples

1c

2c

Acceptance Sampling

• Example: • n = 100, = 4, = 7, Number of defective in the

first sample = 5.1c 2c

Acceptance Sampling

• Sequential sampling plan– Given:

• n: sample size and upper and • lower limits of number of defectives allowed

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.

Acceptance Sampling

• Advantages of double and sequential samplings:– Psychologically:– Cost: less inspection for the same accuracy

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.


Sample

1 2 3 4 5 6

79.1 80.5 79.6 78.9 80.5 79.7

78.9 78.8 79.7 79.4 79.5 80.6

80.0 81.0 80.4 79.8 80.4 80.5

78.4 80.4 80.3 80.3 80.7 80.0

81.0 80.1 80.8 80.6 78.8 80.1


• 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?


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 81.0

80.0 81.5 83.1 78.5

82.5 82.2 80.0 79.6

79.0 82.3 79.5 82.0


• 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)


Sample

1 2 3 4 5 6 7 8 9 10 11 12 13

# of defectiv

es

1 2 2 4 2 3 2 0 2 10 3 2 1


• Problem 3The 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.


Day

1 2 3 4 5 6 7 8 9 10 11 12 13 14

# ofcomplaints

4 10 14 8 5 6 3 12 9 7 5 4 2 10


• Problem 4Answer 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 classIf 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?

topic 4. statistical process control (control charts) and acceptance sampling

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