9/17/2015ieng 486 statistical quality & process control 1 ieng 486 - lecture 18 introduction to...

04/19/23 IENG 486 Statistical Quality & Process Control 1

IENG 486 - Lecture 18

Introduction to Acceptance Sampling,

Mil Std 105E


Assignment

Reading: Chapter 9

Sections 9.1 – 9.1.5: pp. 399 - 410 Sections 9.2 – 9.2.4: pp. 419 - 425 Sections 9.3: pp. 428 - 430

Homework: Due 03 DEC CH 9 Textbook Problems:

1a, 17, 26 Hint: Use Excel!

Last Assignment: Download and complete Last Assign: Acceptance Sampling

Requires MS Word for Nomograph Requires MS Excel for AOQ


Acceptance Sampling

Company receives shipment fromvendor

Sample taken from lot,Quality characteristic inspected

Lot Sentencing:Accept lot?

YES

Return lotto vendor

NO

Use lot inproduction


Three Important Aspects of Acceptance Sampling

1. Purpose is to sentence lots, not to estimate lot quality

2. Acceptance sampling does not provide any direct form of quality control. It simply rejects or accepts lots. Process controls are used to control and systematically improve quality, but acceptance sampling is not.

3. Most effective use of acceptance sampling is not to “inspect quality into the product,” but rather as audit tool to insure that output of process conforms to requirements.


Three Approaches to Lot Sentencing

1. Accept with no inspection

2. 100% inspection – inspect every item in the lot, remove all defectives

Defectives – returned to vendor, reworked, replaced or discarded

3. Acceptance sampling – sample is taken from lot, a quality characteristic is inspected; then on the basis of information in sample, a decision is made regarding lot disposition.


Acceptance Sampling Used When:

Testing is destructive 100% inspection is not technologically feasible 100% inspection error rate results in higher percentage of

defectives being passed than is inherent to product Cost of 100% inspection extremely high Vender has excellent quality history so reduction from 100% is

desired but not high enough to eliminate inspection altogether Potential for serious product liability risks; program for

continuously monitoring product required


Advantages of Acceptance Sampling over 100% Inspection

Less expensive because there is less sampling Less handling of product hence reduced damage Applicable to destructive testing Fewer personnel are involved in inspection activities Greatly reduces amount of inspection error Rejection of entire lots as opposed to return of defectives

provides stronger motivation to vendor for quality improvements


Disadvantages of Acceptance Sampling (vs 100% Inspection)

Always a risk of accepting “bad” lots and rejecting “good” lots Producer’s Risk: chance of rejecting a “good” lot – Consumer’s Risk: chance of accepting a “bad” lot –

Less information is generated about the product or the process that manufactured the product

Requires planning and documentation of the procedure – 100% inspection does not


Lot Formation

Lots should be homogeneous Units in a lot should be produced by the same:

machines, operators, from common raw materials, approximately same time

If lots are not homogeneous – acceptance-sampling scheme may not function effectively and make it difficult to eliminate the source of defective products.

Larger lots preferred to smaller ones – more economically efficient

Lots should conform to the materials-handling systems in both the vendor and consumer facilities

Lots should be packaged to minimize shipping risks and make selection of sample units easy


Random Sampling

IMPORTANT: Units selected for inspection from lot must be chosen at random Should be representative of all units in a lot

Watch for Salting: Vendor may put “good” units on top layer of lot knowing a lax inspector might

only sample from the top layer

Suggested technique:1. Assign a number to each unit, or use location of unit in lot2. Generate / pick a random number for each unit / location in lot3. Sort on the random number – reordering the lot / location pairs4. Select first (or last) n items to make sample


Single Sampling Plans for Attributes

Quality characteristic is an attribute, i.e., conforming or nonconforming

N - Lot size n - sample size c - acceptance number

Ex. Consider N = 10,000 with sampling plan n = 89 and c = 2 From lot of size N = 10,000 Draw sample of size n = 89 If # of defectives c = 2

Accept lot If # of defectives > c = 2

Reject lot


How to Compute the OC Curve Probabilities

Assume that the lot size N is large (infinite)

d - # defectives ~ Binomial(p,n)where

p - fraction defective items in lot n - sample size

Probability of acceptance:

0

P 1c

n iia

i

nP d c p p

i


Example

Lot fraction defective is p = 0.01, n = 89 and c = 2. Find probability of accepting lot.


OC Curve

Performance measure of acceptance-sampling plan displays discriminatory power of sampling plan

Plot of: Pa vs. p Pa = P[Accepting Lot] p = lot fraction defective

p = fraction defective in lot Pa = P[Accepting Lot]

0.005 0.9897

0.010 0.9397

0.015 0.8502

0.020 0.7366

0.025 0.6153

0.030 0.4985

0.035 0.3936


OC curve displays the probability that a lot submitted with a certain fraction defective will be either accepted or rejected given the current sampling plan

Probability of Acceptance, Pa

0.00.20.40.60.81.0

0.00 0.02 0.04 0.06 0.08 0.10

Lot fraction defective, p

Pa

n=89c=2

OC Curve


Ideal OC Curve

Suppose the lot quality is considered bad if p = 0.01 or more A sampling plan that discriminated perfectly between good

and bad lots would have an OC curve like:

1.00

0.040.01 0.02 0.03




Ideal OC Curve

In theory it is obtainable by 100% inspection IF inspection were error free.

Obviously, ideal OC curve is unobtainable in practice

But, ideal OC curve can be approached by increasing sample size, n.


Effect of n on OC Curve

Precision with which a sampling plan differentiates between good and bad lots increases as the sample size increases


0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.02 0.04 0.06 0.08 0.10


Pan=50, c=1

n=100, c=2

n=200, c=4

n=1000, c=20


Effect of c on OC Curve

Changing acceptance number, c, does not dramatically change slope of OC curve.

Plans with smaller values of c provide discrimination at lower levels of lot fraction defective


0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.02 0.04 0.06 0.08 0.10


Pa

n=89, c=2

n=89, c=1

n=89, c=0


Producer and Consumer Risks in Acceptance Sampling

Because we take only a sub-sample from a lot, there is a risk that:

a good lot will be rejected (Producer’s Risk – )

and a bad lot will be accepted

(Consumer’s Risk – )


Producer’s Risk -

Producer wants as many lots accepted by consumer as possible so Producer “makes sure” the process produces a level of fraction defective

equal to or less than:

p1 = AQL = Acceptable Quality Level

is the probability that a good lot will be rejected by the consumer even though the lot really has a fraction defective p1

That is,

Lot is rejected given that process

has an acceptable quality levelP

Lot is rejectedP p AQL


Consumer’s Risk -

Consumer wants to make sure that no bad lots are accepted Consumer says, “I will not accept a lot if percent defective is greater

than or equal to p2”

p2 = LTPD = Lot Tolerance Percent Defective

is the probability a bad lot is accepted by the consumer when the lot really has a fraction defective p2

That is,

Lot accepted given that lot

has unacceptable quality levelP

Lot acceptedP p LTPD


Designing a Single-Sampling Plan with a Specified OC Curve

Use a chart called a Binomial Nomograph to design plan

Specify: p1 = AQL (Acceptable Quality Level)

p2 = LTPD (Lot Tolerance Percent Defective)

1 – = P[Lot is accepted | p = AQL]

β = P[Lot is accepted | p = LTPD]


Use a Binomial Nomograph to Find Sampling Plan

(Figure 15-9, p. 643)

Draw two lines on nomograph Line 1 connects p1 = AQL to (1- ) Line 2 connects p2 = LTPD to Pick n and c from the intersection of the lines

Example: Suppose p1 = 0.01, α = 0.05, p2 = 0.06, β = 0.10.

Find the acceptance sampling plan.


Greek - Axis

p - Axisp1 = AQL = .01

1 – = 1 – .05 = .95

p2 = LTPD = .06

= .10

n = 120

c = 3

Take a sample of size 120.

Accept lot if defectives ≤ 3.

Otherwise, reject entire lot!


Rectifying Inspection Programs

Acceptance sampling programs usually require corrective action when lots are rejected, that is,

Screening rejected lots Screening means doing 100% inspection on lot

In screening, defective items are Removed or Reworked or Returned to vendor or Replaced with known good items


Rectifying Inspection Programs

InspectionActivity

Rejected Lots: 100%

Inspected

AcceptedLots

FractionDefective

Incoming Lots:Fraction Defective

FractionDefective = 0

Outgoing Lots:Fraction Defective

0p

0p

1 0p p


Where to Use Rectifying Inspection

Used when manufacturer wishes to know average level of quality that is likely to result at given stage of manufacturing

Example stages: Receiving inspection In-process inspection of semi-finished goods Final inspection of finished goods

Objective: give assurance regarding average quality of material used in next stage of manufacturing operations


Average Outgoing Quality: AOQ

Quality that results from application of rectifying inspection Average value obtained over long sequence of lots from process with

fraction defective p

N - Lot size, n = # units in sample Assumes all known defective units replaced with good ones,

that is, If lot rejected, replace all bad units in lot If lot accepted, just replace the bad units in sample

aP p N nAOQ

N


Development of AOQ

If lot accepted:Number defective units in lot:

Expected number of defective units:

Average fraction defective,Average Outgoing Quality, AOQ:

# units

fraction remaining

defectivein lot

p N n

Lot # defectiveProb

accepted units in lotaP p N n

aP p N nAOQ

N


Example for AOQ

Suppose N = 10,000, n = 89, c = 2, and incoming lot quality is p = 0.01. Find the average outgoing lot quality.


Questions & Issues

9/17/2015ieng 486 statistical quality & process control 1 ieng 486 - lecture 18 introduction to...

Documents

quality improvements

quality characteristic

direct form of quality

output of process

process controls

excellent quality history

product required ieng

lot qualityacceptance