Title: Decision%20Analysis
1Slides Prepared by JOHN S. LOUCKS St. Edwards
University
2Statistical Methods for Quality Control
- Statistical Process Control
- Acceptance Sampling
3Quality Terminology
- Quality is the totality of features and
characteristics of a product or service that
bears on its ability to satisfy given needs.
4Quality Terminology
- Quality assurance refers to the entire system of
policies, procedures, and guidelines established
by an organization to achieve and maintain
quality. - The objective of quality engineering is to
include quality in the design of products and
processes and to identify potential quality
problems prior to production. - Quality control consists of making a series of
inspections and measurements to determine whether
quality standards are being met.
5Statistical Process Control (SPC)
- The goal of SPC is to determine whether the
process can be continued or whether it should be
adjusted to achieve a desired quality level. - If the variation in the quality of the production
output is due to assignable causes (operator
error, worn-out tooling, bad raw material, . . .
) the process should be adjusted or corrected as
soon as possible. - If the variation in output is due to common
causes (variation in materials, humidity,
temperature, . . . ) which the manager cannot
control, the process does not need to be adjusted.
6SPC Hypotheses
- SPC procedures are based on hypothesis-testing
methodology. - The null hypothesis H0 is formulated in terms of
the production process being in control. - The alternative hypothesis Ha is formulated in
terms of the process being out of control. - As with other hypothesis-testing procedures, both
a Type I error (adjusting an in-control process)
and a Type II error (allowing an out-of-control
process to continue) are possible.
7Decisions and State of the Process
- Type I and Type II Errors
- State of Production Process
- Decision
H0 True In Control
Ha True Out of Control
Correct Decision
Type II Error Allow out-of-control process to
continue
Continue Process
Correct Decision
Type I Error Adjust in-control process
Adjust Process
8Control Charts
- SPC uses graphical displays known as control
charts to monitor a production process. - Control charts provide a basis for deciding
whether the variation in the output is due to
common causes (in control) or assignable causes
(out of control).
9Control Charts
- Two important lines on a control chart are the
upper control limit (UCL) and lower control limit
(LCL). - These lines are chosen so that when the process
is in control, there will be a high probability
that the sample finding will be between the two
lines. - Values outside of the control limits provide
strong evidence that the process is out of
control.
10Types of Control Charts
- An x chart is used if the quality of the output
is measured in terms of a variable such as
length, weight, temperature, and so on. - x represents the mean value found in a sample of
the output. - An R chart is used to monitor the range of the
measurements in the sample. - A p chart is used to monitor the proportion
defective in the sample. - An np chart is used to monitor the number of
defective items in the sample.
11x Chart Structure
x
UCL
Center Line
Process Mean When in Control
LCL
Time
12Control Limits for an x Chart
- Process Mean and Standard Deviation Known
-
13Example Granite Rock Co.
- Control Limits for an x Chart Process Mean
- and Standard Deviation Known
- The weight of bags of cement filled by
Granites packaging process is normally
distributed with a mean of 50 pounds and a
standard deviation of 1.5 pounds. - What should be the control limits for samples
of 9 bags?
14Example Granite Rock Co.
- Control Limits for an x Chart Process Mean
- and Standard Deviation Known
- ????? 50, ? 1.5, n 9
- UCL 50 3(.5) 51.5
- LCL 50 - 3(.5) 48.5
15Control Limits for an x Chart
- Process Mean and Standard Deviation Unknown
-
- where
- x overall sample mean
- R average range
- A2 a constant that depends on n taken
from - Factors for Control Charts table
_
16Factors for x and R Control Charts
17Control Limits for an R Chart
- UCL RD4
- LCL RD3
- where
- R average range
- D3, D4 constants that depend on n
found in Factors for Control
Charts table
_
_
_
18Factors for x and R Control Charts
19Example Granite Rock Co.
- Control Limits for x and R Charts Process Mean
- and Standard Deviation Unknown
- Suppose Granite does not know the true mean and
standard deviation for its bag filling process.
It wants to develop x and R charts based on
twenty samples of 5 bags each. - The twenty samples resulted in an overall
sample mean of 50.01 pounds and an average range
of .322 pounds.
20Example Granite Rock Co.
- Control Limits for R Chart Process Mean
- and Standard Deviation Unknown
- x 50.01, R .322, n 5
- UCL RD4 .322(2.114) .681
- LCL RD3 .322(0) 0
_
_
_
21Example Granite Rock Co.
22Example Granite Rock Co.
- Control Limits for x Chart Process Mean
- and Standard Deviation Unknown
- x 50.01, R .322, n 5
- UCL x A2R 50.01 .577(.322) 50.196
- LCL x - A2R 50.01 - .577(.322) 49.824
23Example Granite Rock Co.
24Control Limits for a p Chart
- where
- assuming
- np gt 5
- n(1-p) gt 5
- Note If computed LCL is negative, set LCL
0
25Example Norwest Bank
- Every check cashed or deposited at Norwest Bank
must be encoded with the amount of the check
before it can begin the Federal Reserve clearing
process. The accuracy of the check encoding
process is of utmost importance. If there is any
discrepancy between the amount a check is made
out for and the encoded amount, the check is
defective.
26Example Norwest Bank
- Twenty samples, each consisting of 250 checks,
were selected and examined when the encoding
process was known to be operating correctly. The
number of defective checks found in the samples
follow.
27Example Norwest Bank
- Control Limits for a p Chart
- Suppose Norwest does not know the proportion of
defective checks, p, for the encoding process
when it is in control. - We will treat the data (20 samples) collected
as one large sample and compute the average
number of defective checks for all the data.
That value can then be used to estimate p.
28Example Norwest Bank
- Control Limits for a p Chart
- Estimated p 80/((20)(250)) 80/5000 .016
29Example Norwest Bank
30Control Limits for an np Chart
-
- assuming
- np gt 5
- n(1-p) gt 5
- Note If computed LCL is negative, set LCL 0
31Interpretation of Control Charts
- The location and pattern of points in a control
chart enable us to determine, with a small
probability of error, whether a process is in
statistical control. - A primary indication that a process may be out of
control is a data point outside the control
limits. - Certain patterns of points within the control
limits can be warning signals of quality
problems - Large number of points on one side of center
line. - Six or seven points in a row that indicate either
an increasing or decreasing trend. - . . . and other patterns.
32Acceptance Sampling
- Acceptance sampling is a statistical method that
enables us to base the accept-reject decision on
the inspection of a sample of items from the lot. - Acceptance sampling has advantages over 100
inspection including less expensive, less
product damage, fewer people involved, . . . and
more.
33Acceptance Sampling Procedure
Lot received
Sample selected
Sampled items inspected for quality
Results compared with specified quality
characteristics
Quality is not satisfactory
Quality is satisfactory
Accept the lot
Reject the lot
Send to production or customer
Decide on disposition of the lot
34Acceptance Sampling
- Acceptance sampling is based on
hypothesis-testing methodology. - The hypothesis are
- H0 Good-quality lot
- Ha Poor-quality lot
35The Outcomes of Acceptance Sampling
- Type I and Type II Errors
- State of the Lot
- Decision
H0 True Good-Quality Lot
Ha True Poor-Quality Lot
Correct Decision
Type II Error Consumers Risk
Accept H0 Accept the Lot
Correct Decision
Type I Error Producers Risk
Reject H0 Reject the Lot
36Probability of Accepting a Lot
- Binomial Probability Function for Acceptance
Sampling -
- where
- n sample size
- p proportion of defective items in lot
- x number of defective items in sample
- f(x) probability of x defective items in
sample
37Example Acceptance Sampling
- An inspector takes a sample of 20 items from a
lot. - Her policy is to accept a lot if no more than 2
defective - items are found in the sample.
- Assuming that 5 percent of a lot is defective,
what is - the probability that she will accept a lot?
Reject a lot? - n 20, c 2, and p .05
- P(Accept Lot) f(0) f(1) f(2)
- .3585 .3774 .1887
- .9246
- P(Reject Lot) 1 - .9246
- .0754
38Example Acceptance Sampling
- Using the Tables of Binomial Probabilities
39Selecting an Acceptance Sampling Plan
- In formulating a plan, managers must specify two
values for the fraction defective in the lot. - a the probability that a lot with p0 defectives
will be rejected. - b the probability that a lot with p1 defectives
will be accepted. - Then, the values of n and c are selected that
result in an acceptance sampling plan that comes
closest to meeting both the a and b requirements
specified.
40Operating Characteristic Curve
41Multiple Sampling Plans
- A multiple sampling plan uses two or more stages
of sampling. - At each stage the decision possibilities are
- stop sampling and accept the lot,
- stop sampling and reject the lot, or
- continue sampling.
- Multiple sampling plans often result in a smaller
total sample size than single-sample plans with
the same Type I error and Type II error
probabilities.
42A Two-Stage Acceptance Sampling Plan
Inspect n1 items
Find x1 defective items in this sample
Yes
Accept the lot
x1 lt c1 ?
No
Yes
Reject the lot
x1 gt c2 ?
No
Inspect n2 additional items
Find x2 defective items in this sample
Yes
No
x1 x2 lt c3 ?
43End of Chapter