9/17/2015IENG 486 Statistical Quality & Process Control 1 IENG Lecture 18 Introduction to Acceptance Sampling, Mil Std 105E

9/17/2015 IENG 486 Statistical Quality & Process Control2 Assignment  Reading: Chapter 9 Sections 9.1 – 9.1.5: pp Sections 9.2 – 9.2.4: pp Sections 9.3: pp  Homework:Due 03 DEC CH 9 Textbook Problems: 1a, 17, 26Hint: Use Excel!  Last Assignment: Download and complete Last Assign: Acceptance Sampling Requires MS Word for Nomograph Requires MS Excel for AOQ

9/17/2015 IENG 486 Statistical Quality & Process Control3 Acceptance Sampling

9/17/2015 IENG 486 Statistical Quality & Process Control4 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.

9/17/2015 IENG 486 Statistical Quality & Process Control5 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.

9/17/2015 IENG 486 Statistical Quality & Process Control6 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

9/17/2015 IENG 486 Statistical Quality & Process Control7 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

9/17/2015 IENG 486 Statistical Quality & Process Control8 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

9/17/2015 IENG 486 Statistical Quality & Process Control9 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

9/17/2015 IENG 486 Statistical Quality & Process Control10 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 lot 2. Generate / pick a random number for each unit / location in lot 3. Sort on the random number – reordering the lot / location pairs 4. Select first (or last) n items to make sample

9/17/2015 IENG 486 Statistical Quality & Process Control11 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

9/17/2015 IENG 486 Statistical Quality & Process Control12 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:

9/17/2015 IENG 486 Statistical Quality & Process Control13 Example  Lot fraction defective is p = 0.01, n = 89 and c = 2. Find probability of accepting lot.

9/17/2015 IENG 486 Statistical Quality & Process Control14 OC Curve  Performance measure of acceptance-sampling plan displays discriminatory power of sampling plan  Plot of: P a vs. p P a = P[Accepting Lot] p = lot fraction defective p = fraction defective in lotP a = P[Accepting Lot]

9/17/2015 IENG 486 Statistical Quality & Process Control15  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 OC Curve

9/17/2015 IENG 486 Statistical Quality & Process Control16 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:

9/17/2015 IENG 486 Statistical Quality & Process Control17 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.

9/17/2015 IENG 486 Statistical Quality & Process Control18 Effect of n on OC Curve  Precision with which a sampling plan differentiates between good and bad lots increases as the sample size increases

9/17/2015 IENG 486 Statistical Quality & Process Control19 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

9/17/2015 IENG 486 Statistical Quality & Process Control20 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 –  )

9/17/2015 IENG 486 Statistical Quality & Process Control21 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: p 1 = 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  p 1  That is,

9/17/2015 IENG 486 Statistical Quality & Process Control22 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 p 2 ” p 2 = LTPD = Lot Tolerance Percent Defective  is the probability a bad lot is accepted by the consumer when the lot really has a fraction defective  p 2  That is,

9/17/2015 IENG 486 Statistical Quality & Process Control23 Designing a Single-Sampling Plan with a Specified OC Curve  Use a chart called a Binomial Nomograph to design plan  Specify: p 1 = AQL (Acceptable Quality Level) p 2 = LTPD (Lot Tolerance Percent Defective) 1 –  = P[Lot is accepted | p = AQL] β = P[Lot is accepted | p = LTPD]

9/17/2015 IENG 486 Statistical Quality & Process Control24 Use a Binomial Nomograph to Find Sampling Plan (Figure 15-9, p. 643)  Draw two lines on nomograph 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  Example: Suppose p 1 = 0.01, α = 0.05, p 2 = 0.06, β = Find the acceptance sampling plan.

9/17/2015 IENG 486 Statistical Quality & Process Control25 Greek - Axis p - Axis p 1 = AQL =.01 1 –  = 1 –.05 =.95 p 2 = LTPD =.06  =.10 n = 120 c = 3 Take a sample of size 120. Accept lot if defectives ≤ 3. Otherwise, reject entire lot!

9/17/2015 IENG 486 Statistical Quality & Process Control26 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

9/17/2015 IENG 486 Statistical Quality & Process Control27 Rectifying Inspection Programs

9/17/2015 IENG 486 Statistical Quality & Process Control28 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

9/17/2015 IENG 486 Statistical Quality & Process Control29 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

9/17/2015 IENG 486 Statistical Quality & Process Control30 Development of AOQ  If lot accepted: Number defective units in lot:  Expected number of defective units:  Average fraction defective, Average Outgoing Quality, AOQ:

9/17/2015 IENG 486 Statistical Quality & Process Control31 Example for AOQ  Suppose N = 10,000, n = 89, c = 2, and incoming lot quality is p = Find the average outgoing lot quality.

9/17/2015 IENG 486 Statistical Quality & Process Control32 Questions & Issues