sampling acceptance project

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Using Simulation to Compare and Evaluate Single & Double Sampling Plans

Asmar Farooq

Abstract. Using simulation as the main tool, operating characteristic curves are generated for

single and double sampling plan in order to analyze and select the optimum sampling plan.

Single Sampling plan turns out to be both simple and yields better quality lots, while double

sampling plan is found to be more useful when cost of sampling or timing is a constraint.

Keywords. Acceptance sampling, hyper geometric distribution, Operating Characteristic curve,

probability of acceptance, Type I and II errors, Average Outgoing Quality, Average Sampling

Number

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Introduction. Acceptance sampling is a statistical method that is applied on a batch of

production items to determine whether to accept or reject the lot. It is part of the quality control

process to ensure that the incoming items being received from the supplier satisfies

predetermined standards. Before the acceptance sampling procedure can begin, the criteria to

accept and reject the lot are clearly defined, and cannot be changed once the procedure begins. A

sample of size ‘n’ is randomly selected from the lot size ‘N’ and is tested for acceptance based

on the predefined criteria. If the lot is rejected, it may be subject to 100% inspection or it can be

returned back to the supplier for credit or replacement.

Acceptance sampling method is mainly useful under one or more of the following

conditions:

1. The time required to inspect all items in the lot is either not available or is too costly.

2. The process of testing negatively affects the unit’s lifetime or if testing is destructive to the

very item being tested.

3. The cost or time consequences of passing some percent of defective items is relatively lower

and/or manageable.

4. Testing large number of items can cause fatigue or boredom which can lead to unexpected

inspection errors.

It must be noted that the consequence of not testing the entire batch will lead to some likelihood

or risk in making wrong inference about the quality of the lot. These are termed as producer’s

and consumer’s risks.

Producer’s Risk. This is also known as Type I error and is denoted by α. It is the probability of

rejecting a lot given that the lot is good. In other words, it is risk that the supplier takes in that a

shipment with less than (1- α)% quality will be rejected. α is commonly set at 5.0%.

Consumer’s Risk. This is also known as Type II error and denoted by β. It is the probability of

accepting a lot given that the lot is bad. In other words, it is risk that the consumer takes in that a

shipment with more than (1- β)% quality will be accepted. β is commonly set at 10.0%.

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Types of sampling plans.

1. Single sampling plan. This is the most common and easiest plan to use. It follows the

flowchart below.

Figure 1

2. Double sampling plan. The procedure in this plan is very similar to the single sampling plan

except in case of an event d > b (Step 3), second random sample of size ‘n’ is selected and tested.

If d1 + d2 ≤ c, then the sample is accepted, otherwise rejected. ‘c’ is defined as the total number

of defective items allowed in a total of two random samples.

3. Multiple sampling plan. Multiple sampling plan is an extension of double sampling plan

where instead of two, more than two samples are allowed to be inspected from the same lot, if

the prior samples have failed the inspection test.

Simulation can be a great tool in determining the sampling plan that will give the optimal results

for any given event. One such arbitrary event is examined in this paper, and both the single and

double sampling plans are used and later analyzed for optimum plan selection.

No

Yes

STEP1: Randomly select

a sample of n from N

STEP 2: Inspect all items in

sample n for defects

STEP 3: d ≤ b ?

Reject Lot

N = number of items in a lot

n = number of items selected as a

sample from the lot.

d = total number of defective items

found in a sample ‘n’ after inspection

b =total number of defectives items

allowed in a sample for acceptance.

Accept lot

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Hypergeometric Distribution. Before we can begin the simulation of sampling plans, it is

necessary to introduce Hypergeometric distribution. It is a discrete probability distribution that

describes the probability of ‘d’ success in ‘n’ draw outs without replacement from a finite

population of size ‘N’ containing exactly ‘D’ successes. Hypergeometric distribution is closely

linked with sampling plans as it is demonstrate below. Success, in this case, translates to a

defective item.

𝑃(𝑋) = (𝐷

𝑑)(𝑁−𝐷𝑛−𝑑)

(𝑁𝑛)

where: n = sample size N = lot size

d = number of defective items in a sample D = total defective items present in a lot

then P(d) is the probability of finding ‘d’ defective items in the sample.

For example, from a lot of 100 items with 10 defective items, a sample of 20 is items is taken,

then the probability that there are no defective items in a sample is P(d=0)

Analytically, 𝑃(𝑑 = 0) = (10

0 )(100−1020−0 )

(10020 )

= 00.09511 or about 9.5%

Using simulation we get the same result. Please note that defective items are represented by 1.

set.seed(123) #Set seed at 123

m=10^6; n=20; d =numeric(m);

N =sample(c((numeric(90)),rep(1,10)),100) #Generates a random sample of

ninety 0’s and ten 1s. for(i in 1:m)

{

x = sample(N,n) # Using loop, a sample of n is taken

d[i]=sum(x) from N, and total # of defective

} items are stored in vector d

mean(d==0); # Probability of 0 defective items. 0.094814

cutp=(0:(max(d)+1))-.5;

hist(d,breaks=cutp,prob=T, xlab="# of defects", ylab="Probability",

main="Histogram");

lines(0:10,dhyper(0:10,10,90,20),col="blue"); #overlays the analytically

derived curve on top of the

simulated histogram

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Figure 2 demonstrates that the simulation can provide very close approximation to actual

probabilities. The blue curve is generated using the Hypergeometric distribution function.

Figure 2

Operating Characteristic Curve. The operating characteristic or OC curve is an important tool

which can help discriminate between the quality of the items in a lot being received with the

probability of its acceptance. The curve shows the probability that a given sampling plan will

result in lots with various fractions defective being accepted.

Analytically, we can compute the OC curve using Hypergeometric distribution function for any

given ‘N’, ‘n’ and ‘b’, however, it becomes difficult and complicated for double and multiple

sampling plans. Using simulation, it will be shown that simulation can be a quick and useful tool

to create OC curve chart.

An arbitrary situation is created where a QC Manager is seeking to determine a sampling plan

based on the OC curves for single and double sampling plan. Let the lot size ‘N’ be 500 units and

the sample size ‘n’ be 30. If a sample of 30 gets more than b=4 defective item, then the lot is

rejected, otherwise it is accepted. With these conditions, the event is simulated.

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Single Sampling plan simulation

set.seed(100) #Setting seed at 100

m = 10^5; N=500; n=30; b=4; #Input the constants

accept=numeric(m); #Stores information whether the lot is

accepted or rejected. (1=accept, 0= reject)

prop = seq(from=0, to=0.4, by=0.01); #Creates a sequence of proportion defective

item in a lot for which probability of

acceptance is generated. prob.accept = numeric(length(prop));

for(j in 1:length(prop)) #The function of this loop is to find

acceptance probability for each value of

{ defective proportion for(i in 1:m)

{ s=sample((rbinom(N,1,prop[j])),n) #Here a sample of n is generated out of a lot

N using binomial distribution function for

each defective proportion.

if((sum(s))<=b) { accept[i]=1} else {accept[i]=0} #Using the if statement, we

separate the accepted and rejected

} lot by assigning 0 and 1 values.

prob.accept[j] = mean(accept); } #This is probability of acceptance

plot(prop,prob.accept, type="l"); #Plot of OC Curve

abline(v=c(0,0.025,.05,.075,.1,.15,.2), col="grey");

abline(h=c(0.1,0.2,0.4,0.6,0.8,.85,.9,.95,.99), col="grey");

As mentioned earlier, the study of OC curve is useful tool for analysis of any sampling plan.

Using the code above, OC curve is generated and few points from the graph are given in the

table below.

Proportion defect 0.03 0.07 0.08 0.09 0.1 0.15 0.2 0.25 0.32

# of defect per lot 15 35 40 45 50 75 100 125 160

Acceptance probability 0.998 0.946 0.912 0.872 0.825 0.53 0.25 0.098 0.017

If the lot quality is 0.25 fractions defective, then the probability of acceptance is 10%

(Consumer’s Risk), on the other hand, if the lot quality is 0.07 fractions defective, then the

probability of acceptance is 95% (Producer’s risk).

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Figure 3 (left) & 4 (right)

Average Outgoing Quality (AOQ). In many sampling plans, if the lot is rejected, then the

entire lot is inspected and all defective items are replaced with the good ones. As the number of

defective items increases in the lot, it becomes more likely for it to be rejected and thus go

through the 100% inspection process. By using this replacement technique, the average outgoing

quality improves in terms of the percent defective. The equation for AOQ = (𝑃𝑎)(𝑃𝑑)(𝑁−𝑛)

𝑁

where Pa = probability of acceptance and Pd = Proportion defect or (D/N). Using this formula,

AOQ curve easily be generated. The maximum AOQ value is known as Average Outgoing

Quality Limit (AOQL) which is the highest defective percent expected to be present in a lot. The

AOQL in example above comes out to be 0.08.

Multiple Sampling plan simulation. Now, let’s run a different scenario where multiple

sampling plan is implemented. From a lot of N=500, instead of 30, 15 samples (S1) are tested,

half the amount from the single sampling plan. If the first sample has less than or equal to 2

defective items, then the lot is accepted, otherwise, second sample (S2) is taken from the

0.12

1-α

β

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remaining lot. If the total sum of defective items from samples S1 and S2 exceeds 4, then the lot

is reject, otherwise accepted.

set.seed(100)

m = 10^5; N=500; n=15; b=2; c=4; # c is set at 4 and will be used when the

accept=numeric(m); second sample is taken prop = seq(from=0, to=0.4, by=0.01);

prob.accept = numeric(length(prop));

count1.p=count2.p=numeric(length(prop)); # Counters are used to count the number of

time second sample is chosen as well as

number of time a lot is chosen given the

for(j in 1:length(prop)) second sample after the second sample is

taken. It is recorded for each defective prop. {

count1=count2=0 # Counter used for each loop of prop. Their

for(i in 1:m) sum after the end of each loop is recorded in

{ vectors count1.p and count2.p. lot=rbinom(N,1,prop[j]);

s1=sample(lot,n)

if((sum(s1))<=b) { accept[i]=1} else {

count1 = count1 + 1 #Counter for when the second sample is taken.

lot_count = c(N-sum(lot>0),sum(lot>0)) s1_count = c(n-sum(s1>0),sum(s1>0))

lot_after= lot_count - s1_count;

s2 = sample(c(rep(0,lot_after[1]), rep(1,lot_after[2])), n);

if((sum(s1+s2))<=c) { accept[i]=1; count2=count2+1} else {accept[i]=0}

}

}

prob.accept[j] = mean(accept);

count1.p[j] = sum(count1);


}

plot(prop,prob.accept, type="l");



plot(prop,count1.p/m, type="l") # Graphs of fraction of times second

abline(v=c(0,.1,.2,.3,.4), col="grey"); sample is accepted as well as the ratio

abline(h=c(0,.2,.4,.6,.8,1), col="grey"); between #of times samples is accepted

par(new=TRUE) to the number of times when the second

plot(prop,count2.p/count1.p, type="l") sample is chosen for inspection

#This code separates the first sample from the

lot, so the second sample can be taken. First, #

of defective and non-defective items are

computed for the lot and sample. The second

sample is taken from the difference of the two.

# The second counter is used when the lot is

accepted after the second sample is taken.

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Figure 5 (left) & 6 (right)

Proportion defect 0.03 0.07 0.08 0.09 0.1 0.15 0.2 0.25 0.32

# of defect per lot 15 35 40 45 50 75 100 125 160

Pa for single sampling 1.00 0.95 0.91 0.87 0.83 0.53 0.25 0.10 0.02

Pa for double sampling 1.00 0.97 0.95 0.92 0.89 0.68 0.45 0.26 0.10

Comparison. At 95% acceptance probability, the fraction defect is 0.08, which is slightly higher

than the Single sampling plan, but at 10% (consumer’s risk) the fraction defect is 0.32

(significantly higher than the single sampling plan which is 0.25). Furthermore, AOQL is also

higher by 2%.

If both plans are to be judged by their fraction defective rate at Type I and II errors as well as

AOQL values alone, then single sampling plan is certainly better. However, if inspection

requires items to be destroyed or sampling time is an issue, then sampling plan which requires

smaller sampling size is obviously better.

The sample size in single sampling plan always remains constant which in the given example is

30 units. However, in our example of double sampling plan, the first sample size is only 15, and

second sample is only required if the first sample fails to pass. Let’s find out how many times the

AOQL = 0.10 for Double sampling

plan

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second sample was taken, and what fraction of that sample resulted in acceptance of the lot.

Using the counter vector in the program code, following curves are plotted on the same graph.

Figure 7

sample size is a priority, then double sampling is certainly a better option.

The green curve represents

the probability of accepting

the lot during the first try.

The red curve represents the

probability of selecting the

second sample.

The blue curve represents the

conditional probability of

accepting the lot given that

the first try failed the

inspection test.

Figure 7 shows that at a

reasonable defect level of say

10% or 20%, the probability

of taking another sample is

20% and 60% respectively. It

turns out that if minimizing

Average Sample Number. It is the

average sample size required to make the

decision on acceptance at any given

defect level. It can be easily calculated by

the following formula.

ASN = n1+n2(1-Pa1) where

n1& n2 are 1st and 2nd sample size and Pa1

is the probability of accepting the lot after

the first trial.

In Figure 8, on average, only 18 samples

are required to accept the lot if the

defective rate is 10%.

Figure 8

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Summary and conclusion: The simulation approach to analyze a sampling plan is found to be

very successful with faster results than the analytical approach. Once the program code is

written, any number of events can easily be simulated by just changing the initial constants. In

the example used in this paper, it was found that single sampling plan is a better option when

quality is of utmost importance. The quality can be improved by increasing the sample size or by

choosing a more conservative value of ‘b’. On the other hand, if the inspection requires the

sample items to be destroyed or not enough time is available, then double sampling plan is a

better approach. The simulations method can also be easily extended to multiple sampling plan

with few adjustments.

In order to make the final decision between the two plans, one has to decide whether the average

sampling savings gained by the double sampling plan justifies the additional complexity and

increase in uncertainty in the quality of the lot. The judgment should be made on case by case

basis with factors such as available labor, time, cost and complexity in mind.

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Bibliography

Prins, Jack , Process or Product Monitoring and Control- Section#2 “Test Product for

Acceptability.” http://www.itl.nist.gov/div898/handbook/pmc/section2/pmc2.htm

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Appendix.

Complete Simulation 1 for Single Sampling plan.

set.seed(100)

m = 10^5; N=500; n=30; b=4;

accept=numeric(m);

prop = seq(from=0, to=0.4, by=0.01);


for(j in 1:length(prop))

{

for(i in 1:m)

{

s=sample((rbinom(N,1,prop[j])),n)

if((sum(s))<=b) { accept[i]=1} else {accept[i]=0}

}


}

plot(prop,prob.accept, type="l");



AOQ = prob.accept*prop*((N-n)/N)

plot(prop,AOQ, type="l", ylab="Average Outgoing Quality", xlab="Incoming

Percent Defective", main="AOQ Curve for Single Sampling Plan")

abline(v=c(.12))

abline(h=max(AOQ))

Complete Simulation 2 for Double Sampling plan.

set.seed(100)

m = 10^5; N=500; n=15; b=2; c=4;

accept=numeric(m);

prop = seq(from=0, to=0.4, by=0.01);


count1.p=count2.p=numeric(length(prop));

for(j in 1:length(prop))

{

count1=count2=0

for(i in 1:m)

{

lot=rbinom(N,1,prop[j]);

s1=sample(lot,n)

if((sum(s1))<=b) { accept[i]=1} else

{

count1 = count1 + 1

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lot_count = c(N-sum(lot>0),sum(lot>0))

s1_count = c(n-sum(s1>0),sum(s1>0))

lot_after= lot_count - s1_count;

s2 = sample(c(rep(0,lot_after[1]), rep(1,lot_after[2])), n);

if((sum(s1+s2))<=c) { accept[i]=1; count2=count2+1} else {accept[i]=0}

}

}




}

plot(prop,prob.accept, type="l", xlab="Proportion defect (D/N)",

ylab="Probability of acceptance (P)",main="OC curve of Double Sampling

Plan");



AOQ = prob.accept*prop*((N-n)/N) #For AOQ Graph plot(prop,AOQ, type="l", ylab="Average Outgoing Quality", xlab="Incoming

Percent Defective", main="AOQ Curve for Single Sampling Plan")

abline(v=c(.145))

abline(h=max(AOQ))

plot(prop,count1.p/m, type="l", xlab="Proportion defect (D/N)",

ylab="Fraction", main="2nd sample fraction & 2nd sample accepted vs percent

defective", col="red")

abline(h=seq(from=0, to=1, by=0.1), col="grey") #For Figure 7 abline(v=seq(from=0, to=0.4, by=0.05), col="grey")

par(new=TRUE)

lines(prop,count2.p/count1.p,col="blue")

lines(prop,1-count1.p/m,col="green")

plot(prop,(n+n*(count1.p/m)), type="l",main= "Average Sample Number",

ylab="Sample Size", xlab="Proportion defect (D/N))")

abline(v=seq(from=0, to=0.4, by=0.05), col="grey") #For ASN Graph abline(h=seq(from=15, to=30, by=2), col="grey")

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