QualityControl Chapter 9- Lot-by-Lot Acceptance Sampling By Attributes PowerPoint presentation to accompany Besterfield Quality Control, 8e PowerPoints created by Rosida Coowar
Outline Fundamental Aspects Statistical Aspects Sampling Plan Design
Learning Objectives When you have completed this chapter you should be able to: Know the advantages and disadvantages of sampling; the types of sampling plans and selection factors; criteria for formation of lots; criteria for sampling selection; and decisions regarding rejected lots. Determine the OC Curve for a single sampling plan.
Learning Objectives-cont’d. When you have completed this chapter you should be able to: Determine the equations needed to graph the OC Curve for a double sampling plan. Know the properties of OC Curves. Know the consumer-producer relationships of risk, AQL, and LQ.
Learning Objectives-cont’d. When you have completed this chapter you should be able to: Determine the AOQ curve and the AOQL for a single sampling plan. Determine single sampling plans for stipulated producers risk and for stipulated consumers risk.
Fundamental Aspects Acceptance Sampling is a form of inspection applied to lots or batches of items before or after a process to judge conformance to predetermined standards.
Acceptance Sampling Acceptance Sampling is very useful when: Large numbers of items must be processed in a short amount of time. The cost of “passing defectives” is low. Fatigue/boredom is caused by inspecting large numbers of items.
Acceptance Sampling Acceptance Sampling is very useful when: Destructive testing is required
Acceptance Sampling • Three important aspects of sampling: • Involves random sampling of the entire lot • Accept and Reject Lots (does not improve the quality) “Lot Sentencing” • Audit Tool • Three approaches to “lot sentencing”: • Accept with no inspection • 100% inspection • Acceptance Sampling
Acceptance Sampling • Advantages • Less expensive • Reduced damage • Reduces the amount of inspection error • Disadvantages • Risk of accepting “bad” lots and rejecting “good” lots • Less information generated • Requires planning and documentation
Sampling Plans • Sampling Plans specify the lot size, sample size, number of samples and acceptance/rejection criteria. • Sampling plans involve: • Single sampling • Double sampling • Multiple sampling Random sample Lot
Sampling Plans • Single Sampling Plan • N = lot size • n = sample size • C=acceptance number • If c or less non-conforming units are found in the sample, the lot is accepted, else it is rejected.
Single Sampling Plan • A single sampling plan is one where: • A representative sample of n items is drawn from a lot size of N items • Each item in the sample is examined and classified as good/defective • If the number of defective exceeds a specified rejection number (c) the whole lot is rejected; otherwise the whole lot is accepted
Double Sampling Plan • A Double Sampling Plan allows to take a second sample if the results of the original sample are inconclusive. • Specifies the lot size, size of the initial sample, the accept/reject/inconclusive criteria for the initial sample (N, n1, c1 (Ac), r1(Re)) • Specifies the size of the second sample and the acceptance rejection criteria based on the total number of defective observed in both the first and second sample (n2,c2,r2)
Double Sampling Plan Lot First Random sample First sample inconclusive, take second sample Accept Lot Reject Lot C1 r1 Compare number of defective found in the first random sample to C1 and r1 and make appropriate decision.
Double Sampling Plan Lot First Random sample Second Random sample Accept Lot Reject Lot C2 Compare the total number of defective in both lots to C2 and make the appropriate decision
Double Sampling Plan • A Multiple Sampling Plan is similar to the double sampling plan in that successive trials are made, each of which has acceptance, rejection and inconclusive options. • Which Plan you choose depends on: • Cost and time • Number of samples needed and number of items in each sample
Lot Formation Considerations before inspection: • Lots should be homogeneous • Larger lots are more preferable than smaller lots • Lots should be conformable to the materials-handling systems used in both the vendor and consumer facilities
Random Sampling • Units selected for inspection should be chosen at random • If random samples are not used, bias can be introduced • If judgment methods are used to select the sample, the statistical basis of the acceptance-sampling procedure is lost
Statistical Aspects The Operating Characteristic Curve: • Measures the performance of an acceptance sampling plan • Plots the probability of accepting the lot versus the lot fraction defective • Shows the probability that a lot submitted with a certain fraction defective will be either accepted or rejected
Acceptable Quality Level (AQL) • The AQL is a percent defective that is the base line requirement for the quality of the producer's product. The producer would like to design a sampling plan such that there is a high probability of accepting a lot that has a defect level less than or equal to the AQL.
Lot Tolerance Percent Defective • The Lot Tolerance Percent Defective LTPD or LQ is a designated high defect level that would be unacceptable to the consumer. The consumer would like the sampling plan to have a low probability of accepting a lot with a defect level as high as the LTPD.
Type I Error (Producer’s Risk) • This is the probability, for a given (n,c) sampling plan, of rejecting a lot that has a defect level equal to the AQL. The producer suffers when this occurs, because a lot with acceptable quality was rejected. The symbol α is commonly used for the Type I error and typical values for range from 0.2 to 0.01.
Type II Error (Consumer’s Risk) • This is the probability, for a given (n,c) sampling plan, of accepting a lot with a defect level equal to the LTPD. The consumer suffers when this occurs, because a lot with unacceptable quality was accepted. The symbol β is commonly used for the Type II error and typical values range from 0.2 to 0.01.
Operating Characteristic Curve • This curve plots the probability of accepting the lot (Y-axis) versus the lot fraction or percent defectives (X-axis). The OC curve is the primary tool for displaying and investigating the properties of a Lot Acceptance Sampling Plan.
OC Curves There are two types of OC curves: • Type A • Gives the probability of acceptance of an individual lot coming from finite production • Type B • Gives the probability of acceptance for lots coming from a continuous production
OCCs for Single Sampling Plans Under this sampling plan, if the lot has 3% defective . the probability of accepting the lot is 90% . the probability of rejecting the lot is 10% If the lot has 20% defective . it has a small probability (5%) of being accepted . the probability of rejecting the lot is 95% An Operating Characteristic Curve (OCC) is a probability curve for a sampling plan that shows the probabilities of accepting lots with various lot quality levels (% defectives). 1 0.9 0.8 0.8 0.7 0.7 0.6 0.5 Probability of acceptinglot 0.4 0.3 0.2 0.1 0 Lot quality (% defective) 0 .05 .10 .15 .20
OCC, AQL and Producer’s Risk 1 Producer’s Risk = probability acceptable lot is rejected 0.9 0.8 0.7 0.6 0.5 AQL - percentage level of defects at which a customer is willing to accept Probabilityof accepting lot 0.4 0.3 0.2 0.1 0 Lot quality (% defective) 0 .10 .15 .20 .05 “Acceptable Lot”
OCC, LTPD and Consumer’s Risk 1 0.9 0.8 0.7 LTPD - upper limit on the percentage of defectives that a customer is willing to accept. 0.6 0.5 Probability of accepting lot 0.4 0.3 0.2 0.1 Consumer’s Risk = probability unacceptable is accepted 0 Lot quality (% defective) 0 .10 .15 .20 .05 “Unacceptable Lot”
Double Sampling Plan Inspect a sample of 150 from lot of 2400 If 1 or less Nonconforming units accept lots and stop If 4 or more Nonconforming units the lot is not accepted and stop If 2 or 3 nonconforming units, inspect a second sample of 200 If 5 or less Nonconforming units On both samples, Accept the lot If 6 or more Nonconforming units On both samples The lot is not accepted Figure 9-5 Graphical description of the double sampling plan: N=2400,n1=150,c1=1 r1=4, n2=200, c2=5, and r2=6
Average Outgoing Quality (AOQ) A common procedure, when sampling and testing is non-destructive, is to 100% inspect rejected lots and replace all defectives with good units. In this case, all rejected lots are made perfect and the only defects left are those in lots that were accepted.
Average Outgoing Quality The Average Outgoing Quality (AOQ) is the average of rejected lots (100% inspection) and accepted lots ( a sample of items inspected)
AOQ and Acceptance Sampling 11 lots 2% nonconforming 15 lots 2% nonconforming Producer N=3000 n=89 c=2 Consumer 4 lots 2% nonconforming 4 lots 0% nonconforming Figure 9-15 How acceptance Sampling works
AOQ and Acceptance Sampling Figure 9-15 cont’d.
Average Quality of Inspected Lots Typically the term (N-n)/N is very close to 1; therefore, the equation most often used is:
Average Outgoing Quality Level • A plot of the AOQ (Y-axis) versus the incoming lot p (X-axis) will start at 0 for p = 0, and return to 0 for p = 1 (where every lot is 100% inspected and rectified). In between, it will rise to a maximum. This maximum, which is the worst possible long term AOQ, is called the Average Outgoing Quality Level AOQL.
Average Total Inspection (ATI) When rejected lots are 100% inspected, it is easy to calculate the ATI if lots come consistently with a defect level of p. For a LASP (n,c) with a probability pa of accepting a lot with defect level p, we have: ATI = n + (1 - pa) (N - n) where N is the lot size.
Average Sample Number (ASN) For a single sampling (n,c) we know each and every lot has a sample of size n taken and inspected or tested. For double, multiple and sequential plans, the amount of sampling varies depending on the number of defects observed.
Average Sample Number (ASN) For any given double, multiple or sequential plan, a long term ASN can be calculated assuming all lots come in with a defect level of p. A plot of the ASN, versus the incoming defect level p, describes the sampling efficiency of a given plan scheme. ASN = n1 + n2 (1 – P1) for a double sampling plan.
Sampling Plan Design Suppose α is known and the AQL is also known then : • Sampling plan with stipulated producer’s risk • Sampling plan with stipulated consumer’s risk • Sampling plan with stipulated producer’s and consumer’s risk can be designed.
Sampling Plan Design • Stipulated Producer’s Risk • α = 0.05 AQL = 1.2% • Pa=0.95 P0.95= 0.012 • Assume values for C, find np0.95 for this c value, calculate n
Sampling Plan Design • Stipulated Consumer’s Risk • β = 0.10 LQ = 6.0% • Pa=0.10 P0.10= 0.060 • Assume values for C, find np0.95 for this c value, calculate n
Sampling Plan Design • Stipulated Producer’s and Consumer’s risk • α = 0.10 β = 0.10 • AQL=0.9 LQ= 7.8 • Find the ratio of P0.10/P0.95. From table 9-4 C is between 1 and 2. Find n for c =1 and n for c =2 .
Sampling Plan Design • Have 4 plans. • Select plan based on: • Lowest sampling size • Greatest sampling size • Plan exactly meets consumer’s stipulation and is as close as possible to producer’s stipulation • Plan exactly meets producer’s stipulation and is as close as possible to consumer’s stipulation