IENG 486 Lecture 18 Introduction to Acceptance Sampling

IENG 486 - Lecture 18 Introduction to Acceptance Sampling, Mil Std 105 E 10/30/2020 IENG 486 Statistical Quality & Process Control 1

Assignment w Reading: n Chapter 9 l l l Sections 9. 1 – 9. 1. 5: pp. 399 - 410 Sections 9. 2 – 9. 2. 4: pp. 419 - 425 Sections 9. 3: pp. 428 - 430 w Homework: n Due 03 DEC CH 9 Textbook Problems: l 1 a, 17, 26 Hint: Use Excel! w Last Assignment: n Download and complete Last Assign: Acceptance Sampling l l 10/30/2020 Requires MS Word for Nomograph Requires MS Excel for AOQ IENG 486 Statistical Quality & Process Control 2

Acceptance Sampling 10/30/2020 IENG 486 Statistical Quality & Process Control 3

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. 10/30/2020 IENG 486 Statistical Quality & Process Control 4

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. 10/30/2020 IENG 486 Statistical Quality & Process Control 5

Acceptance Sampling Used When: w Testing is destructive w 100% inspection is not technologically feasible w 100% inspection error rate results in higher percentage of defectives being passed than is inherent to product w Cost of 100% inspection extremely high w Vender has excellent quality history so reduction from 100% is desired but not high enough to eliminate inspection altogether w Potential for serious product liability risks; program for continuously monitoring product required 10/30/2020 IENG 486 Statistical Quality & Process Control 6

Advantages of Acceptance Sampling over 100% Inspection w w w 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 10/30/2020 IENG 486 Statistical Quality & Process Control 7

Disadvantages of Acceptance Sampling (vs 100% Inspection) w Always a risk of accepting “bad” lots and rejecting “good” lots n Producer’s Risk: chance of rejecting a “good” lot – n Consumer’s Risk: chance of accepting a “bad” lot – w Less information is generated about the product or the process that manufactured the product w Requires planning and documentation of the procedure – 100% inspection does not 10/30/2020 IENG 486 Statistical Quality & Process Control 8

Lot Formation w Lots should be homogeneous n Units in a lot should be produced by the same: l l n 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. w Larger lots preferred to smaller ones – more economically efficient w Lots should conform to the materials-handling systems in both the vendor and consumer facilities n Lots should be packaged to minimize shipping risks and make selection of sample units easy 10/30/2020 IENG 486 Statistical Quality & Process Control 9

Random Sampling w IMPORTANT: n n w Watch for Salting: n w Units selected for inspection from lot must be chosen at random Should be representative of all units in a lot Vendor may put “good” units on top layer of lot knowing a lax inspector might only sample from the top layer Suggested technique: 1. 2. 3. 4. Assign a number to each unit, or use location of unit in lot Generate / pick a random number for each unit / location in lot Sort on the random number – reordering the lot / location pairs Select first (or last) n items to make sample 10/30/2020 IENG 486 Statistical Quality & Process Control 10

Single Sampling Plans for Attributes w Quality characteristic is an attribute, i. e. , conforming or nonconforming n n n N - Lot size n - sample size c - acceptance number w Ex. Consider N = 10, 000 with sampling plan n = 89 and c = 2 n n n From lot of size N = 10, 000 Draw sample of size n = 89 If # of defectives c = 2 l n Accept lot If # of defectives > c = 2 l 10/30/2020 Reject lot IENG 486 Statistical Quality & Process Control 11

How to Compute the OC Curve Probabilities w Assume that the lot size N is large (infinite) w d - # defectives ~ Binomial(p, n) where n n p - fraction defective items in lot n - sample size w Probability of acceptance: 10/30/2020 IENG 486 Statistical Quality & Process Control 12

Example w Lot fraction defective is p = 0. 01, n = 89 and c = 2. Find probability of accepting lot. 10/30/2020 IENG 486 Statistical Quality & Process Control 13

OC Curve w Performance measure of acceptance-sampling plan n displays discriminatory power of sampling plan w Plot of: Pa vs. p n n 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 10/30/2020 IENG 486 Statistical Quality & Process Control 14

OC Curve w 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 10/30/2020 IENG 486 Statistical Quality & Process Control 15

Ideal OC Curve w Suppose the lot quality is considered bad if p = 0. 01 or more w A sampling plan that discriminated perfectly between good and bad lots would have an OC curve like: 10/30/2020 IENG 486 Statistical Quality & Process Control 16

Ideal OC Curve w In theory it is obtainable by 100% inspection IF inspection were error free. w Obviously, ideal OC curve is unobtainable in practice w But, ideal OC curve can be approached by increasing sample size, n. 10/30/2020 IENG 486 Statistical Quality & Process Control 17

Effect of n on OC Curve w Precision with which a sampling plan differentiates between good and bad lots increases as the sample size increases 10/30/2020 IENG 486 Statistical Quality & Process Control 18

Effect of c on OC Curve w Changing acceptance number, c, does not dramatically change slope of OC curve. w Plans with smaller values of c provide discrimination at lower levels of lot fraction defective 10/30/2020 IENG 486 Statistical Quality & Process Control 19

Producer and Consumer Risks in Acceptance Sampling w Because we take only a sub-sample from a lot, there is a risk that: n a good lot will be rejected (Producer’s Risk – a ) and n a bad lot will be accepted (Consumer’s Risk – b ) 10/30/2020 IENG 486 Statistical Quality & Process Control 20

Producer’s Risk - a w Producer wants as many lots accepted by consumer as possible so n Producer “makes sure” the process produces a level of fraction defective equal to or less than: p 1 = AQL = Acceptable Quality Level a is the probability that a good lot will be rejected by the consumer even though the lot really has a fraction defective p 1 w That is, 10/30/2020 IENG 486 Statistical Quality & Process Control 21

Consumer’s Risk - b w Consumer wants to make sure that no bad lots are accepted n Consumer says, “I will not accept a lot if percent defective is greater than or equal to p 2” p 2 = LTPD = Lot Tolerance Percent Defective b is the probability a bad lot is accepted by the consumer when the lot really has a fraction defective p 2 w That is, 10/30/2020 IENG 486 Statistical Quality & Process Control 22

Designing a Single-Sampling Plan with a Specified OC Curve w Use a chart called a Binomial Nomograph to design plan w Specify: n p 1 = AQL (Acceptable Quality Level) n p 2 = LTPD (Lot Tolerance Percent Defective) n 1 – = P[Lot is accepted | p = AQL] n β = P[Lot is accepted | p = LTPD] 10/30/2020 IENG 486 Statistical Quality & Process Control 23

Use a Binomial Nomograph to Find Sampling Plan (Figure 15 -9, p. 643) w Draw two lines on nomograph n n n Line 1 connects p 1 = AQL to (1 - ) Line 2 connects p 2 = LTPD to Pick n and c from the intersection of the lines w Example: Suppose n n p 1 = 0. 01, α = 0. 05, p 2 = 0. 06, β = 0. 10. Find the acceptance sampling plan. 10/30/2020 IENG 486 Statistical Quality & Process Control 24

p 1 = AQL =. 01 p - Axis Greek - Axis p 2 = LTPD =. 06 n = 120 =. 10 1 – = 1 –. 05 =. 95 c = 3 Take a sample of size 120. Accept lot if defectives ≤ 3. Otherwise, reject entire lot! 10/30/2020 IENG 486 Statistical Quality & Process Control 25

Rectifying Inspection Programs w Acceptance sampling programs usually require corrective action when lots are rejected, that is, n Screening rejected lots l Screening means doing 100% inspection on lot w In screening, defective items are n n Removed or Reworked or Returned to vendor or Replaced with known good items 10/30/2020 IENG 486 Statistical Quality & Process Control 26

Rectifying Inspection Programs 10/30/2020 IENG 486 Statistical Quality & Process Control 27

Where to Use Rectifying Inspection w Used when manufacturer wishes to know average level of quality that is likely to result at given stage of manufacturing w Example stages: n n n Receiving inspection In-process inspection of semi-finished goods Final inspection of finished goods w Objective: give assurance regarding average quality of material used in next stage of manufacturing operations 10/30/2020 IENG 486 Statistical Quality & Process Control 28

Average Outgoing Quality: AOQ w Quality that results from application of rectifying inspection n Average value obtained over long sequence of lots from process with fraction defective p w N - Lot size, n = # units in sample w Assumes all known defective units replaced with good ones, that is, n n If lot rejected, replace all bad units in lot If lot accepted, just replace the bad units in sample 10/30/2020 IENG 486 Statistical Quality & Process Control 29

Development of AOQ w If lot accepted: Number defective units in lot: w Expected number of defective units: w Average fraction defective, Average Outgoing Quality, AOQ: 10/30/2020 IENG 486 Statistical Quality & Process Control 30

Example for AOQ w Suppose N = 10, 000, n = 89, c = 2, and incoming lot quality is p = 0. 01. Find the average outgoing lot quality. 10/30/2020 IENG 486 Statistical Quality & Process Control 31

Questions & Issues 10/30/2020 IENG 486 Statistical Quality & Process Control 32