Acceptance Sampling Terminology Acceptable Quality Level LTPD or RQL Producer’s risk Consumer’s risk Factors affecting these risks Simple single sampling Rectification based sampling plans Average out going quality (AOQ) Average Sample number (ASN) Average total inspection (ATI)

How will you find the consumer’s risk?

The general approach N (Lot) n Count Number Conforming Accept or Reject Lot Specify the sampling plan For a lot size N, determine the sample size (s) n, and Select acceptance criteria for good lots Select rejection criteria bad lots Accept the lot if acceptance criteria are satisfied Specify course of action if lot is rejected

Sampling plans are based on sample statistics and the theory says that since we inspect only a sample and not the whole lot, there is a chance of making an error. NOT MEASURING QUALITY, REJECTING OR ACCEPTING A LOT DEPENDING ON CERTAIN CIRCUMSTANCES.

We need to consider two types of errors that result in wrong decisions Reject Accept Good lot Bad lot T R U H DECISION

The Ideal OC Curve This OC curve represents a sampling plan which has maximum discriminating power. How can we achieve this ? Pa p AQL 1.00 0.00

This is possible only when we have 100 % inspection The Ideal OC Curve Pa p AQL 1.00 0.00 Eventhough this is the most desirable sampling plan, it is not practical to achieve this. This is possible only when we have 100 % inspection

Operating Characteristic Curve AQL LTPD  = 0.10  = 0.05 Probability of acceptance, Pa { 0.60 0.40 0.20 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.80 Proportion defective p 1.00 OC curve for n and Ac

A Typical OC Curve n1> n2 > n3 Ac = constant AQL LTPD 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 10 11 12 Percent defective Probability of acceptance  = consumer’s risk = producer’s risk n1 n2 n3  

A Typical OC Curve Percent defective Ac1< Ac2 < Ac3 n = constant AQL LTPD 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 10 11 12 Probability of acceptance  = consumer’s risk = producer’s risk Ac1 Ac2 Ac3  

= average number of defectives Poisson Distribution The probability of obtaining x nonconforming units (defectives) on a single inspection unit, when the average number of defectives is some constant, λ is: where, = average number of defectives

A process is operating at a nonconformance level of 1% A process is operating at a nonconformance level of 1%. What is the probability that a sample of size 100 will have 2 defective units ? p = 0.01, n = 100, x = 2  = np = 1 = 0.367

A process is operating at a nonconformance level of 1% A process is operating at a nonconformance level of 1%. What is the probability that a sample of size 100 will have 2 or less defective units ? p = 0.01, n = 100, x = 2  = np = 1

From the Poisson distribution table above, P (x  2 | np = 1) = 0.92 Using standard tables np = 100 (0.01) = 1 Cumulative Poisson distribution table From the Poisson distribution table above, P (x  2 | np = 1) = 0.92

From the Poisson distribution table , find Calculating producer’s risk (a) for a given plan and p N = 2000, n = 50, Ac = 2, p = AQL= 0.02 np = 50 (0.02) = 1 From the Poisson distribution table , find Prob (no. of defective units  2, given that np = 1) = Probability of acceptance = 0.92 = 92 % Producer’s risk = Probability of rejection = 1 – 0.92 = 0.08 = 8 %

From the above Poisson distribution table , Calculating consumer’s risk (b) for a given plan and p N = 2000, n = 50, Ac = 2, p = LTPD = 0.04 np = 50 (0.04) = 2 From the above Poisson distribution table , P (x  2 | np = 2) = probability of acceptance = 0.677 Consumer’s risk = 67.7 %

We calculated producer’s and consumer’s risk for a given plan. Example We calculated producer’s and consumer’s risk for a given plan. How do we select a sampling plan if we were to have a producer’s risk of 2 % for an AQL of 0.01 ? Use the following table.

We calculated producer’s and consumer’s risk for a given plan. Example We calculated producer’s and consumer’s risk for a given plan. How do we select a sampling plan if we were to have a producer’s risk of 2 % for an AQL of 0.01 ? Use the following table.

Average Outgoing Quality for rectification based sampling plans AOQ is the average quality level of a series of lots that leave the inspection station, assuming ‘rectification’, after coming in for inspection at a certain quality level p.

What is the AOQ for p = AQL ? Example What is the AOQ for p = AQL ? N = 2000, n = 50, Ac = 2, AQL = 2 % = 0.02 np = 50 x 0.02 = 1 AOQ = [ Pa. p (N-n)] / N AOQ = [ 0.92 x 0.02 x (2000 - 50)] / 2000 AOQ = 0.0179 = 1.79 %

Average outgoing quality curve

Average outgoing quality Limit (AOQL) It represents the worst average quality that would leave the inspection station, assuming rectification, regardless of incoming quality

It is the peak of the AOQ curve. Average outgoing quality limit It is the peak of the AOQ curve. AOQL

How much do I need to inspect ? ATI

ATI can be used to calculate the average cost of inspection. Average Total Inspection (ATI) It is the average number of items inspected per lot if rectifying inspection is conducted. ATI can be used to calculate the average cost of inspection.

Average Total Inspection Curve ATI = n + (1- Pa) (N-n) N=1000 N=25 D=2

A manufacturer has selected a supplier to supply the raw material in the form of forged pieces in lots of 2000 units at a price of Rs 200/unit. Due to the nature of the forging process, a variation in the surface hardness is expected in the supplied pieces. The production manager knows that higher values of hardness lead to a poor tool life and hence sees the need to use an acceptance sampling plan to check the hardness of the incoming units. She decides to use a rectification based sampling plan with an acceptable quality level of 1%, a sample size of 20 and an acceptance number of 0. The variable cost of inspection is Rs 10 per unit and the fixed cost is Rs 100 per sample. What is the contribution to cost per piece.?

QA