Acceptance sampling : Acceptance sampling Hassan Asif
Manager Quality Assurance
Packages, Ltd
What is acceptance sampling? : What is acceptance sampling? A statistical quality control technique, where a random sample is taken from a lot, and upon the results of appraising the sample, the lot will either be accepted or rejected. Accept or reject
Purpose of Acceptance sampling: : Purpose of Acceptance sampling: Determine the quality level of an incoming shipment or, at the end production.
Ensure that the quality level is within the level that has been predetermined
To avoid 100 percent inspection of incoming material that is humanly impossible.
How acceptance sampling works : How acceptance sampling works Attributes (“ok/not ok” inspection)
Defects – no. of defects per unit
Based on binomial distribution
Requires large sample size
Variables (continuous measurement)
Usually measured by mean and standard deviation
Based on normal distribution
Requires smaller sample size
Types of Acceptance sampling plans : Types of Acceptance sampling plans Single sampling plan
Double sampling plan
Multiple sampling plan
Rectifying sampling plan
Basic concept - Acceptance sampling : Basic concept - Acceptance sampling The decision to accept or reject the shipment is based on the following set standards:
Lot size = N
Sample size = n
Acceptance number = c
Defective items = d
If d <= c,
If d > c, Accept lot Reject lot
Lets go through a simple example How actually it works! : Lets go through a simple example How actually it works! You are an owner of a manufacturing plant and you receives a consignment of 5000 stainless steel plates. Your QA department inspect the sample of 100 plates. Predetermined acceptance no. “C” is 5 sheets. According to the report, total no. of defectives found are 11. What would be your decision? Accept? or Reject?
Two risks of acceptance sampling : Two risks of acceptance sampling 1- Producers risk:
It is the probability of rejecting a good lot that has an agreed acceptable quality level (AQL). Also known as Type 1 error and is indicated by the Greek letter “α” Why did it happen?
What is AQL?
Did my customer know it?
Acceptable quality level - AQL : Acceptable quality level - AQL The AQL is an acceptable proportion of defects in a lot to the customer.
Its an tangible factor that reflects the customer willingness to accept lots with a small proportion of defective items.
It might be given by producer or also can be dictated by customer i.e. AQL is negotiable.
AQL might be two defective items in lot of 500, or 0.004 or 0.4 %.
Two risks of acceptance sampling : 2- Consumers risk:
It is the probability of accepting a bad lot in which the fraction of defective items exceeds lot tolerance percent defective (LTPD). Also known as Type 2 error and is indicated by the Greek letter “β” Two risks of acceptance sampling Why did it happen?
What is LTPD?
Did my supplier know it?
Lot tolerance percent defective (LTPD) : Lot tolerance percent defective (LTPD) The LTPD is the maximum number of defective items a consumer will accept in a lot.
The customer would like the quality of a lot to be as good or better than the AQL but is willing to accept some lots with quality levels no worse than the LTPD.
It is also generally negotiated between the producer and consumer.
Slide 12: Very Important!
Be careful not to confuse “α” with the AQL or “β” with the LTPD
Operating characteristics curve: : Operating characteristics curve: OC curve describes how well an acceptance plan discriminates between good or bad lots. It actually measures the performance of an acceptance sampling plan.
It is a graph of the percentage defectives (P) in a lot Vs the probability that the sampling plan will accept the lot (Pa).
Aids in selection of plan that are effective in reducing risk.
Decreasing the acceptance no. “c” is preferred over increasing sample size “n”. Decreased acceptance no. makes life difficult for manufactures.
The steeper the OC curve, the better our sampling plan is for discriminating between good and bad lots.
Slide 14: An ideal OC curve: Figure (a) shows a perfect discrimination plan for a company that wants to reject all lots with more than 2.5 % defectives and accept all lots with less than or equal to 2.5 % defectives.
Unfortunately, the only way to assure 100% acceptance of good lots and 0% acceptance of bad lots is to conduct a full inspection, which is often very costly.
To avoid 100 % inspection we prefer to make sampling plan with optimum value of n and c to have best possible results with minimum risk (producer and consumer) involved.
Slide 15: A typical OC curve: An OC curve is typically used to represent the four parameters i.e.
“α” – Producers risk
“β” – Consumer risk
AQL – Acceptable quality level
LTPD – Lot tolerance percent defective.
Sample size “n” and acceptance no. “c” has a direct impact on the shape of OC curve
Constructing an OC curve : There are different ways of calculating and drawing OC curves.
Binomial distribution
Hyper geometric distribution
Poisson distribution
Larson nomogram Constructing an OC curve
Constructing an OC curve – Binomial distribution : Constructing an OC curve – Binomial distribution Binomial equation is as follow: Where,
Pa = probability of acceptance of lot
n = sample size
p = percent defective in a lot
c = acceptance no.
d = defects found in a lot
Constructing an OC curve – Poisson distribution : Mathematically, Poisson distribution approaches the Binomial distribution when “n” is large and “p” is small. Hence as a rule of thumb many text books suggest that poisson distribution can be used as an approximation of the binomial distribution when n > 20 & p < 0.05
Poisson formula can be used as follow:
Pa = [e-np (np)d]/d! Constructing an OC curve – Poisson distribution
Slide 19: Suppose we have a sampling plan defined by the following parameters: n = 89, c = 2.
What is the probability of accepting a lot with 0.5% defectives? Putting values in binomial equation we have,
Pa = {89!/0!(89-0)}*(0.005)0(1-0.005)89-0 +
{89!/1!(89-1)!}*(0.005)1 Putting values in binomial equation we have,
Pa = {89!/0!(89-0)}*(0.005)0(1-0.005)89-0 +
{89!/1!(89-1)!}*(0.005)1(1-0.005)89-1 +
{89!/2!(89-2)!}*(0.005)2(1-0.005)89-2
Pa = 0.6401 + 0.2862 + 0.0633
Pa = 0.989
Similarly we can find for other % def i.e. for 1% def Pa = 0.939, for 2% def Pa = 0.737
Slide 20: Probability of acceptance “Pa” of a lot can also be found by using Microsoft Excel as shown below:
OC curve for single sampling plan, n = 89 & c = 2 0.9897
Slide 21: Typical OC curve
n = 89
c = 2 Pa % age defective
Effect of “n” & “c” on OC curves : Effect of “n” & “c” on OC curves Figure indicates that for the same sample size (n = 100 in this case), a smaller value of c yields a steeper curve than does a larger value of c.
So one way to increase the probability of accepting only good lots and rejecting only bad lots with random sampling is to set very tight acceptance levels as c=1.
Probability of acceptance goes down with decrease in value of “c” OC Curves for Two Different Acceptable Levels of Defects (c = 1, c = 4) for the Same Sample Size (n = 100)
Effect of “n” & “c” on OC curves : Effect of “n” & “c” on OC curves OC Curves for Two Different Sample Sizes (n = 25, n = 100) but Same Acceptance Percentages (4%). Larger sample size shows better discrimination. A second way to develop a steeper, and thereby sounder, OC curve is to increase the sample size.
Figure illustrates that even when the acceptance number is the same proportion of the sample size, a larger value of n will increase the likelihood of accurately measuring the lot’s quality. In this figure, both curves use a maximum defect rate of 4% (equal to 4/100 = 1/25). If you see carefully, OC curve for n = 25, c = 1 rejects more good lots and accepts more bad lots than the second plan.
Slide 24: The probability of accepting a more than satisfactory lot (one with only 1% defects) is 99% for n = 100, but only 97% for n = 25. Likewise, the chance of accepting a “bad” lot (one with 5% defects) is only 44% for n = 100, whereas it is 64% using the smaller sample size.
Of course, were it not for the cost of extra inspection, every firm would opt for larger sample sizes. Also do try the OC curve example with different values of “n” and “c” and study its effect.
Slide 25: A shipment of 2,000 portable battery units for microcomputers is about to be inspected by a Malaysian importer. The Korean manufacturer and the importer have set up a sampling plan in which the “α” producer risk is limited to 5% at an acceptable quality level (AQL) of 2% defective, and the “β” consumer risk is set to 10% at Lot Tolerance Percent Defective (LTPD) = 7% defective.
We want to construct the OC curve for the plan of n = 120 sample size and an acceptance level of c ≤ 3 defectives. Both firms want to know if this plan will satisfy their quality and risk requirements.
By varying the percent defectives (p) from .01 (1%) to .08 (8%) and holding the sample size at n = 120, we can compute the probability of acceptance of the lot at each chosen level. Case study:
Slide 26: As this sampling plan has lager sample size “n” and smaller value of “p” so we will use poisson formula as an approximation of binomial distribution.
We will construct an OC curve to analyze the risks involved against the agreed AQL and LTPD.
Given data:
N=2000, n=120, c=3, AQL= 2%, LTPD=7%, e = 2.718
Putting values in Poisson formula to find Pa for 1% defective
Pa = {e-1.2 (1.2)3/3!} + {e-1.2 (1.2)2/2!} + {e-1.2 (1.2)1/1!} + {e-1.2 (1.2)0/0!}
Pa = 0.08669 + 0.2167 + 0.3612 + 0.301
Pa = 0.966
similarly for 2 % defective, Pa = 0.779 etc
Slide 27: 0.966
Slide 28: Pa % age defective AQL “α” – Producers risk = 22.1 % “β” – consumer risk = 3.2 % LTPD N = 2000
n = 120
c = 3
Slide 29: Conclusion:
For the AQL of p = 0.02 = 2% defects, the probability of acceptance of the lot = 0.779. This yields a producers risk “α” of 0.221, or 22.1%, which exceeds the 5% level desired by the producer. The consumer risk “β”of 0.032, or 3.2%, is well under the 10% sought by the consumer. It appears that new calculations are necessary with a larger sample size if the “α” level is to be lowered.
Slide 30: Hassan Asif
MS - Total Quality Management
BS - Chemical Engineering
Manager Quality Assurance
Packages Ltd.
Pakistan
00923334761836
hassanasif81@gmail.com