Lec 9 - Acceptance Sampling

Lecture 9

Acceptance Sampling Introduction

Michelle V. Mancenido 2007.09.25

Storytelling Time Once upon an afternoon, I was feeling lazy. So I asked my students to exchange papers and grade their seatmate’s problem set. I was very well aware of the risks associated with this method but because my laziness outweighed my usual passion for fair grading, I found myself relenting to my laziness. Later on, I decided to check if my students did not betray the trust that I had so generously awarded to them.

Storytelling Time I realized I had better things to do with my time. I couldn’t spend the whole day re-checking the papers and convincing myself that my 4th year Engineering students are truly trustworthy. Being an industrial engineer, I had to balance efficiency with effectiveness. I am also teaching Statistical Quality Control, and it would be a shame if I couldn’t practice what I preach. But before coming up with a solution, I had to develop a focused problem statement, so it would be clear to me what I would be solving.

Storytelling Time And so I wrote down my focused problem statement: Minimize the number of papers re-checked given that: (1) “My students” has a given level of “untrustworthiness” (2) Assuming this level of “untrustworthiness” is true, based on the number of papers I re-checked, I want a lower probability of concluding that the students are truly trustworthy.

I realize that this can be treated as an OR problem. But I decided not to treat it as such because my students are already having enough trouble with Statistics.

Storytelling Time My problem is to determine the number of papers to re-check per class. The decision factor is this: because I am not certain what the level of untrustworthiness is, I want a lower probability of concluding that my students are trustworthy, given different levels of untrustworthiness So here is what I did:   

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For each section, I counted the number of graded papers = N I defined several n = samples taken = rechecked papers I defined several x = number of papers in a sample where there was at least one error (intentional or unintentional) in checking I defined several p’s = level of untrustworthiness in a class, estimated to be the fraction of students who has the gall to cheat I computed the probability P for each combination. P is the probability that there will be at least x papers in the sample n, given that there is an untrustworthiness level p.

Storytelling Time To compute for the probability Pa is easy. Now is the time to ask my students: how do I compute the probability that there will be at least x papers with checking errors in them, in n number of papers that I recheck for each class, given that there is a certain untrustworthiness level p in that class?

Storytelling Time I set these criteria: (1) if there are no papers with errors in the samples that I rechecked, I will assume that there are no errors in the rest of the papers (2) if there is at most one paper with errors in the sample(s) that I rechecked, I will still assume that there are no errors in the rest of the papers. (3) If the two papers I rechecked have errors in checking, then I will make sure everyone suffers for this dishonesty. Problem is, I’m not also sure how discriminating these criteria are.

Storytelling Time 

So I came up with the table of probabilities in Microsoft Excel, one table for each criterion, assuming different number of papers re-checked.

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I am interested in these probabilities because they tell me the chance that I will conclude that my students are honest at a given level of dishonesty

When x = 0 p d 0 0 0.1 2 0.2 4 0.3 6 0.4 8 0.5 10 0.6 12 0.7 14 0.8 16 0.9 18 1 20

n=2 1.00 0.81 0.63 0.48 0.35 0.24 0.15 0.08 0.03 0.01 0.00

P (x=0) n=3 1.00 0.72 0.49 0.32 0.19 0.11 0.05 0.02 0.00 0.00 0.00

n=4 1.00 0.63 0.38 0.21 0.10 0.04 0.01 0.00 0.00 0.00 0.00

When x<=1 p d 0 0 0.1 2 0.2 4 0.3 6 0.4 8 0.5 10 0.6 12 0.7 14 0.8 16 0.9 18 1 20

P (x<=1) n=2 n=3 1.00 1.00 0.99 0.98 0.97 0.91 0.92 0.80 0.85 0.66 0.76 0.50 0.65 0.34 0.52 0.20 0.37 0.09 0.19 0.02 0.00 0.00

n=4 1.00 0.97 0.84 0.66 0.47 0.29 0.15 0.06 0.01 0.00 0.00

Storytelling Time My students love pictures, so I drew these curves for them.

n=2 n=3 n=4

Storytelling Time My students love pictures, so I drew these curves for them.

n=2 n=3 n=4

Storytelling Time Assuming that there are approximately 6 dishonest students in a class of 20, using the hypergeometric function: p = 0.30, d = 6

n=2

n=3

n=4

x=0

0.48

0.32

0.21

x≤1

0.92

0.80

0.66

If the level of untrustworthiness is truly this bad, I can stomach a 32% chance that I will conclude that the class is completely honest when there are about 6 bad seeds. And anyway, I have enough faith in my students, to believe that the rampancy of dishonesty is not even 30%, 20%, 0r 10%. It’s probably 5% or less. I will therefore re-check about 3 papers per class.

Acceptance Sampling 

Refers to the application of specific sampling plans to a designated lot or sequence of lots

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Concerned with inspection and decision-making regarding products - involved with randomly picking samples and “judging” whether the rest of the lot conform to specifications based on the samples

Acceptance Sampling 1. 2. 3.

Emphasis on lot sentencing, not estimation of lot quality Does not provide a direct form of quality control Most effective use - as audit tool

Types of Sampling According to Purpose Type A

Sampling to accept or reject immediate lot of product at hand

Type B

Sampling to determine if the process which produced the product at hand was within acceptable limits

Uses of Acceptance Sampling     

Testing is destructive Cost of 100% inspection high 100% inspection not feasible Supplier with excellent quality history Potentially serious product liability risk

Advantages of Acceptance Sampling      

Less expensive Less material/product handling Applicable to destructive testing Increased productivity (fewer personnel) Reduces inspection error Increased motivation to improve (rejection of lots)

Disadvantages of Acceptance Sampling 

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Risks of rejecting “good lots” and accepting “bad lots” Less information about the product or process Requires planning and documentation of the sampling schemes or systems

Progression from Assurance to Control Zero Inspection Audit Inspection

Supplier Certification

Process Control Reduced Inspection 100% inspection

Acceptance Sampling

Definition of Terms Terms commonly used in acceptance sampling:

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N

Number of units in a given lot or batch

n

Number of units in a sample

D

Number of defective or nonconforming units in a given lot size N

x

Number of defective or nonconforming units in a given sample size n

c

Acceptance number, the maximum allowable number of defective pieces in a sample of size n

p

Fraction defective (D/N) or (x/n)

µp

True process average fraction defective of a product submitted for inspection

p

Average fraction defective in observed samples

Pa

Probability of acceptance

β

Consumer’s risk

α

Producer’s risk

Definition of Terms 

Sampling Plan a specific plan which states the sample size(s) to be used and the associated criteria for accepting the lot

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Sampling Scheme a combination of acceptance sampling plans with rules for changing from one plan to another

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Sampling System unified collection of sampling schemes, together with criteria by which appropriate schemes may be chosen

Forms of Sampling Plans 

Attributes Plans 

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samples taken from the lot classified as conforming and nonconforming number nonconforming compared with acceptance number

Variables Plans  

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Samples taken measured on a specified characteristic Measurements summarized into a simple statistic and observed value compared with specifications percent defective based on conformance to measured specifications

Attributes vs. Variables Plans Feature

Attributes

Variables

Inspection

Defective or nondefective

Item measured. More sophisticated. Higher inspection cost.

Distribution of individual measurements

Need not be known

Must be known (normality assumption ideal)

Type of defect

Any number of defect types in one plan

Separate plan for each quality characteristic

Sample size

Depends on protection required

Smaller size for same protection provided by attributes (at least 30%)

Process Info

Percent defective

% defective + valuable information on average & variability

Severity

Weighs all defectives of a given kind equally

Weighs each unit against proximity to specifications

Evidence to supplier

Defectives available as evidence

Possible to reject lots without defectives

Types of Sampling Plans Type

Samples

Decision

Single

Number of samples = 1 Sample size = defined

Accept or reject lot based on single subgroup

Double

Number of samples = 2 Sample size = 1st smaller than 2nd

(1) (2)

Multiple

Sequential

Number of samples = more than 2 but defined and truncated Sample size = becomes smaller

(1)

Number of samples = theoretically unlimited until a decision is reached Sample size = depends on results previous samples

(1)

(2)

(2)

Accept or reject lot based on # defectives in 1st sample Take a second sample Accept or reject lot based on # defectives in (m-1) sample Take m samples Accept or reject lot based on # defectives in (m-1) sample Take m samples

Published Schemes

Inspection Procedures 

Lot formation   

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Lots should be homogenous Large lots over small ones Material handling efficiency

Random Samples  

Samples should be representative of the lot One technique: stratification