quality control and reliability inspection and sampling · •from the oc curve in above figure,...

Quality Control and Reliability

Inspection and Sampling

Prepared by

Dr. M. S. Memon

Dept. of Industrial Engineering & Management

Mehran UET, Jamshoro, Sindh, Pakistan

1

• Introduction to Inspection concepts

• Introduction of sampling

– Advantages and disadvantages of sampling

– Types of sampling plans

• Operating characteristic curve

• Evaluating Sampling Plans

• Lot-by-lot attribute sampling plans

Chapter Objectives

2

• Sampling is a process used in statistical analysis in which a

predetermined number of observations are taken from a larger

population.

• Acceptance sampling plans where inspection is by attributes are

discussed.

• In these plans, a product item is classified as conforming or not, but

the degree of conformance is not specified.

• In certain sampling plans, the terms defect and defective are used

interchangeably with nonconformity and nonconforming items.

Introduction to Acceptance Sampling

3

• Acceptance sampling can be performed during inspection of

incoming raw materials, components, and assemblies, in

various phases of in-process operations, or during final product

inspection.

• Acceptance sampling does not control or improve the quality

level of the process.

• Quality cannot be inspected into a product or service; quality

must be designed and built into it.

Introduction to Acceptance Sampling

4

• Sampling is advantageous in that:

1. If inspection is destructive, 100% inspection is not feasible.

2. Sampling is more economical and causes less damage due tohandling. If inspection cost is high or if inspection time is long,limited resources may make sampling preferable.

3. Sampling reduces inspection error. In high-quantity, repetitiveinspection, such as 100% inspection, inspector fatigue can preventthe identification of all nonconformities or nonconforming units.

4. Sampling provides a strong motivation to improve quality becausean entire batch or lot may be rejected.

Advantages and Disadvantages of Sampling

5

• Sampling plans are disadvantageous in that:

1. There is a risk of rejecting "good" lots or accepting "poor"

lots, identified as the producer's risk and consumer's risk,

respectively.

2. There is less information about the product compared to that

obtained from 100% inspection.

3. The selection and adoption of a sampling plan require more

time and effort in planning and documentation.

Advantages and Disadvantages of Sampling

6

• In acceptance sampling, units are randomly chosen from a

batch, lot, or process. There are two types of risk inherent in

any sampling plan:

– Producer's Risk

– Consumer's Risk

Producer and Consumer Risks

7

• Producer's Risk: The risk associated with rejecting a "good" lot,

due to the inherent nature of random sampling, is defined as a

producer's risk.

• The notion of the quality level of lots that defines acceptable

level or "good" product will be influenced by the needs of the

customer.

• Acceptable quality level (AQL) is the terminology used to

define this level of quality.


8

• Consumer's Risk: The risk associated with accepting a "poor" lot,due to the inherent nature of random sampling, is defined as aconsumer's risk.

• Further, norms of customer requirements will govern the definitionof a "poor" lot. Limiting quality level (LQL) or rejectable qualitylevel (RQL) is the terminology used to defined this level ofunacceptable quality.

• An alternative terminology, when the quality level is expressed inpercentage nonconformance, is lot tolerance percent defective(LTPD).


9

• Thus, when we state a producer's risk in a sampling plan, we mustcorrespondingly state a desirable level of quality that we prefer toaccept.

• For example, if we state that the producer's risk is 5% for an AQL of0.02, it means that we consider batches that are 2% nonconformingto be good and prefer to reject such batches no more than 5% of thetime.

• If the consumer's risk is 10% for an LQL of 0.08, this means thatbatches that are 8% nonconforming are poor and we prefer to acceptthese batches no more than 10% of the time.


10

• The operating characteristic (OC) curve measures theperformance of a sampling plan.

• It plots the probability of accepting the lot versus the proportionnonconforming of the lot.

• It shows the discriminatory power of the sampling plan.

• For all sampling plans, we want to accept lots with a low proportionnonconforming most of the time and we do not want to acceptbatches with a high proportion nonconforming very often.

• The OC curve indicates the degree to which we achieve thisobjective.

Operating Characteristic Curve (OC Curve)

11

• Suppose that we have chosen aproportion nonconforming levelp0 such that if a lot has aproportion nonconforming lessthan or equal to p0, we consider itto be a good lot and we accept it.

• On the other hand, if theproportion nonconforming of thelot exceeds P0, we consider thelot to be poor and we reject it.

• The ideal OC curve for thesecircumstances is shown in Figure.


Fig. Ideal operating characteristic curve.

12

• In practice, however, the shape of the OC curve is not ideal.

• To construct the OC curve for a single sampling plan, let N

denote the lot size, n the sample size, and c the acceptance

number.

• A random sample of size n is chosen from the lot of size N.

• If the observed number of nonconforming items or

nonconformities is less than or equal to c, the lot is accepted.

Otherwise, the lot is rejected.


13

• To construct a type A OC curve, we assume that the sample is chosen from an isolated

lot of finite size.

• The probability of accepting the lot is calculated based on a hypergeometric distribution.

The probability of finding x nonconforming items in the sample is given by

• where D represents the number of nonconforming items in the lot. Since the lot will be

accepted if c or fewer nonconforming items are found, the probability of lot acceptance

is:


14

• To construct a type B OC curve, we assume that a stream of lots is produced by the

process and that the lot size is large (at least 10 times) compared to the sample size. A

binomial distribution can be used to find the probability of observing x

nonconforming items in a sample of size n.

• Assuming the lot proportion nonconforming is p, this probability is given by


15

• If the lot size is large and the probability of a nonconforming item is

small, a Poisson distribution can be used as an approximation to the

binomial distribution.

• The probability of x nonconforming items in the sample is found from

• where λ = np represents the average number of nonconforming items in

the sample.

• The probability of lot acceptance, Pa, can then be found from same

previous equation.


16

• Construct an OC curve for a single samplingplan where the lot size is 2000, the samplesize is 50, and the acceptance number is 2.

Solution

• We are given N = 2000, n = 50, and c = 2.

• The probability of lot acceptance isequivalent to the probability of obtaining 2or fewer nonconforming items in the sample.

• Let's suppose that p is 0.02 (i.e., the batch is2% nonconforming).

• Since np = (50) (0.02) = 1.0, the probabilityPa of accepting the lot (using Appendix A-2)is 0.920.

• A plot of these values, the OC curve, isshown in Figure.

Example on OC Curve

Fig. OC curve for the sampling plan in Example.

17

Example on OC CurveTable. Lot Acceptance Probabilities for Different Values of Proportion Nonconforming for the Sampling Plan N —

2000, n = 50, c = 2.

18

• Producer and consumer risk can also be demonstrated through the OCcurve.

• Suppose that our numerical definition of good quality (indicated by theAQL) is 0.01 and that of poor quality (indicated by the LQL) is 0.11.

• From the OC curve in above Figure, the producer's risk α is 1 - 0.986 =0.014. We consider batches that are 1 % nonconforming to be good.

• If our sampling plan is used, such batches will be rejected about 1.4% ofthe time.

• Batches that are 11% nonconforming, on the other hand, will be accepted8.8% of the time. The consumer's risk is therefore 8.8%.

Example on OC Curve

19

• The parameters n and c of the samplingplan affect the shape of the OC curve.

• As long as the lot size N is significantlylarge compared to the sample size n, thelot size does not have an appreciableimpact on the shape of OC curve.

• For fixed values of N and c, as thesample size becomes larger, the slope ofthe OC curve becomes steeper, implyinga greater discriminatory power.

• Figure shows the OC curves for thosesampling plans.

Effect of the sample size and the acceptance number

Fig. Effect of the sample size on the shape of the OC

curve

20

• Figure 10-4 shows the OCcurves for four sampling plans.

• Note that the probability ofacceptance decreases for a givenlot quality as the acceptancenumber c decreases.

• The chosen values of n and cshould be such that they matchthe goals of the user.

Effect of the sample size and the acceptance number

Fig. Effect of the acceptance number on the shape of the

OC curve

21

• There are, generally speaking, three types of attribute sampling plans:single, double and multiple.

• In a single sampling plan, the information obtained from one sample isused to make a decision to accept or reject a lot.– There are two parameters in this sampling plan: the sample size n and the

acceptance number c.

• A double sampling plan involves making a decision to accept the lot,reject the lot, or take a second sample.– If the inference from the first sample is that the lot quality is quite good, the lot is

accepted. If the inference is poor lot quality, the lot is rejected.

– If the first sample gives an inference of neither good nor poor quality, a secondsample is taken.

Types of sampling plans

22

• The parameters of a double sampling plan are as follows:


23

• Let's consider the following double sampling plan where attribute inspection is conducted tofind the number of nonconforming items:

• The working procedure for this plan is initially, to select a random sample of 40 items fromthe lot of size 5000.– If 1 or fewer nonconforming items are found, the lot is accepted, but if 4 or more nonconforming

items are found, the lot is rejected.

– If the observed number of nonconforming items is 2 or 3, a second sample of size 60 is selected.

– If the combined number of nonconforming items from both samples is less than or equal to 5, thelot is accepted; if it is 6 or more, the lot is rejected.


24

• Multiple sampling plans are an extension of double samplingplans.

– Three, four, five, or as many samples as desired may be needed to make adecision regarding the lot.

– The sampling plan can be terminated at any stage once the acceptance orrejection criteria have been met.

• The ultimate extension of the multiple sampling plan is thesequential sampling plan, which is an item-by-item inspection plan.

– After each item is inspected, a decision is made to accept the lot, reject thelot, or choose another item for inspection


25

• As far as simplicity is concerned, the single sampling plan is the best, followedby double and then multiple sampling plans.

• Administrative costs for record keeping, training, and inspection are the least forsingle and the highest for multiple sampling plans.

• On average, for equivalent plans, the number of items inspected to make adecision regarding the lot is usually more for a single sampling plan.– This is because double and multiple sampling plans use fewer items in their samples, so

if the lots are of very good or poor quality, a decision to accept or reject them is madequickly.

• Inspection costs will therefore be the most for single, and the least for multiplesampling plans.

• Single sampling plans provide the most information, and multiple sampling plansthe least.

Advantages and Disadvantages of sampling plans

26

• The OC curve is one measure of the performance of a samplingplan.

• We also use other measures to evaluate the goodness of a samplingplan.

• These involve

– the average quality level (AQL) of batches leaving the inspection station,

– the average number of items inspected before making a decision on the lot,and

– the average amount of inspection per lot if a rejected lot goes through 100%inspection.

Evaluating sampling plans

27

• First consider the concept of rectifying inspection as it appliesto lots that are rejected through sampling plans.

• Usually, such lots go through 100% inspection, known asscreening, where nonconforming items are replaced withconforming ones.

• Such a procedure is known as rectification inspection because itaffects the quality of the product that leaves the inspectionstation.

• Nonconforming items found in the sample are also replaced.

Rectifying Inspection

28

• The average outgoing quality (AOQ) is the average quality level of aseries of batches that leave the inspection station, assuming rectifyinginspection, after coming in for inspection at a certain quality level p.

• The AOQ measures the average quality level of a large number of batchesof incoming quality p, the proportion nonconforming in the lots, assumingrectification.

• Taking N as the lot size, n as the sample size, p as the incoming lot quality,and Pa as the probability of accepting the lot using the given sampling plan,the average, outgoing quality is given by

Average Outgoing Quality

29

• The value of AOQ depends on the incoming quality level p of the batches.

• Thus, an AOQ curve that evaluates the effectiveness of the sampling planfor various levels of incoming quality is usually constructed.

• Let's, consider the single sampling plan N = 2000, n = 50, c = 2. Supposethat the incoming quality of batches is 2% nonconforming.

• From the Poisson cumulative distribution tables in Appendix A-2, theprobability Pa of accepting the lot using the sampling plan is 0.920. Theaverage outgoing quality is

• Thus, if batches come in as 2% nonconforming, the average outgoingquality is 1.79%.

Average Outgoing Quality

30

Example: Construct the AOQ curve forthe sampling plan N = 2000, n = 50, c =2.

Solution

• The probability of lot acceptance forvarious values of the incoming lotquality p is already computed andlisted in Table (Slide# 18).

• Using these values of Pa and p, thevalues of AOQ are calculated fordifferent values of p.

Example on AOQ

31

Fig. AOQ curve for the sampling plan

• Note that when the incoming qualityis very good, the average outgoingquality is also very good.

• When the incoming quality is verypoor, the average outgoing quality isgood because most of the lots arerejected by the sampling plan and gothrough screening.

• In between these extremes, the AOQcurve reaches a maximum, AOQL.

Example on AOQ

32

Fig. AOQ curve for the sampling plan

• The average outgoing quality limit (AOQL) is the maximum value,

or peak, of the AOQ curve.

• It represents the worst average quality that would leave the

inspection station, assuming rectification, regardless of the

incoming lot quality.

• The AOQL value is also a measure of goodness of a sampling plan.

• Note that the protection offered by the sampling plan, in terms of

the AOQL value, does not apply to individual lots. It holds for the

average quality of a series of batches.

Average Outgoing Quality Limit

33

• Consider previous Example and the AOQ curve.

• The AOQL value is approximately 0.0265, or 2.65%.

• This means that for the sampling plan above, N = 2000, n = 50, c = 2, wehave, some protection against the worst quality for a series of batches thatleave the inspection program.

• The average quality level will not be poorer than 2.65% nonconforming.

• The AOQL value and the shape of the AOQ curve depend on the particularsampling plan.

• Sampling plans are designed such that their AOQL does not exceed acertain specified value.

Average Outgoing Quality Limit

34

• If rectifying inspection is conducted for lots rejected by thesampling plan, another evaluation measure is the average totalinspection (ATI).

• The ATI represents the average number of items inspected per lot.

• If a lot has no nonconforming items, it will obviously be acceptedby the chosen sampling plan, and only n items (the sample size) willbe inspected for a lot.

• At the other extreme, if the lot has 100% nonconforming items, thenumber inspected per lot will be N (the lot size) assuming thatrejected lots are screened.

Average Total Inspection (ATI)

35

• For single sampling plans, the average total inspection per lot forlots with an incoming quality level p is given by

• For a double sampling plan, the ATI is given by

• where Pa1 represents the probability of accepting the lot on the firstsample, and Pa2 represents the probability of lot acceptance on thesecond sample.


36

• Example: Construct the ATI curve for the sampling plan where

N = 2000, n = 50, c = 2.

Solution

• Consider the calculations for a given value of the lot quality p

of 0.02. As shown in Table (slide# 18), the probability of

accepting such a lot using the sampling plan is Pa = 0.920. The

ATI for this value of p is


37

• For other values of p, the ATI is

found in the same manner.

• The ATI curve is plotted in

Figure.

• Given the unit cost of inspection,

the ATI curve can be used to

estimate the average inspection

cost if the quality level of

incoming batches is known.


38

Fig. ATI curve for the sampling plan

• Attribute sampling plans are designed to make a decision

regarding items that are submitted for inspection in lots.

• The objective is to find suitable sample sizes and acceptance

numbers of sampling plans that meet certain levels of stipulated

risks (such as the producer's risk, consumer's risk, or both).

Lot-by-Lot Attribute Sampling Plans

39

Single Sampling Plans

• Single sampling plans deal with making a decision regarding a

lot of size N based on information contained in one sample of

size n.

• The acceptance number c of the sampling plan represents the

number of nonconforming items or nonconformities, depending

on the circumstances, that cannot be exceeded in the sample in

order for the lot to be accepted.


40

The OC Curve

• The OC curve of a single sampling plan has beendescribed in detail.

• It represents the probability of accepting the lot,Pa, as a function of the lot quality, which issimply the proportion nonconforming p if itemsare classified only as conforming or not.

• The effects of the parameters n and c on theshape of the OC curve have also been discussed.

• A study of these effects enables us to chooseappropriate values of n and c, given desirablelevels of protection against the producer's andconsumer's risks.


41

Fig. OC curve showing (AQL, 1-α) and (LQL, β)

for a sampling plan

• The OC curve in Figure shows the relationshipbetween AQL and LQL parameters.

• For a sampling plan specified by n and c, lotswith a proportion nonconforming level of AQLthat come in for inspection should be accepted100(1-α)% of the time.

• Similarly, if the proportion nonconforming ofbatches coming in for inspection is LQL, theyshould be accepted 100β% of the time.

• A suitable choice of n and c ensures that goodlots will be accepted a large percentage of thetime and that bad lots will be acceptedinfrequently.


42

Fig. OC curve showing (AQL, 1-α) and (LQL, β)

for a sampling plan

• Now we will discuss several approaches for designing single

sampling plans.

• Basically, these approaches involve determining the sample size

n and acceptance number c of the plan. The criteria selected

influences the parameters of the plan.

– Stipulated Producer's Risk

– Stipulated Consumer's Risk

– Stipulated Producer and Consumer Risk

Design of Single Sampling Plans

43

Stipulated Producer's Risk

• Let's suppose the producer's risk α and its

associated quality level p1, which is the

acceptable quality level (AQL), are specified.

• We desire single sampling plans that will

accept lots of quality level p1, 100(l-α)% of

the time.

• Figure shows the OC curves of sampling

plans that meet this stipulated criteria. Note

that several plans may satisfy this criteria. We

want to find a sampling plan whose OC curve

passes through the single point (AQL, 1- α).


44

Stipulated Producer's Risk

• To find the appropriate sampling plan, first select an acceptance number

c.

• The mean number of nonconforming items in the sample is given by λ =

np. Hence, for a probability of lot acceptance Pa equal to 1 - α at p=p1,

the value of λ is found in Appendix A-2.

• Because λ = np1 =n (AQL), the sample size n is found by dividing the

value of n(AQL) by AQL.

• Fractional computed values of the sample size are always rounded up to

be conservative.


45


46

Table. Values of np for a Producer's Risk of 0.05 and a Consumer's Risk of 0.10

• Example: Find a single sampling plan thatsatisfies a producer's risk of 5% for lots that are1.5% nonconforming.

• Solution We are given a = 0.05 and AQL =0.015. If we choose an acceptance number c =1,for which previous Table gives np1 = 0.355, thesample size is


47

• Note that all three plans satisfy the producer's risk of5% at the AQL value of 1.5%.

• However, they have varying degrees of protectionagainst acceptance of poor quality lots, which would beof interest to the consumer.

• Of the three plans shown, n = 220, c = 6 provides thebest protection to the consumer because it has thelowest probability of accepting poor quality lots.

• However, we must also consider the increasedinspection costs associated with this plan, because thesample size for c = 6 is the largest of the three.

• Note: Other values of c could be selected as well.


48

Stipulated Consumer's Risk

• Let's suppose that the consumer's risk β andits associated quality level p2, which is thelimiting quality level (LQL), are given.

• We want to find sampling plans that willaccept lots of quality level p2, 100β % of thetime.

• Here again, a number of sampling plans willsatisfy this criterion. Figure shows the OCcurves for three sampling plans that meet thecriterion.


49

Fig. OC curves of single sampling plans for stipulated

consumer's risk and LQL

Stipulated Consumer's Risk

• The procedure is similar to that used with producer's risk.

• A value of the acceptance number c is chosen. Based on theprobability of acceptance of β, for lots of quality p2 = LQL.

• The value of λ = np2 is found in Appendix A-2.

• If the value of β is 0.10, we can use same previous Table toobtain the value of np2.

• The sample size is calculated by dividing the value of np2 by p2.


50

Example: Find a single sampling plan that will satisfy a consumer's risk of 10%for lots that are 8% nonconforming.

Solution We are given β = 0.10 and p2 = LQL = 0.08.

• If we select an acceptance number of 1, Previous Table (slide#46) gives np2 =3.890. The sample size is


51

• Figure shows the OC curves for these samplingplans.

• All three pass through the point (p2, β), thussatisfying the consumer's stipulation. The degreeof protection for extremely good batches, as far asthe producer is concerned, is different.

• The plan n = 132, c = 6 will reject good batches(say, 1% nonconforming) the least frequently ofthe three plans. Of course, it has the largestsample size, which may cause the inspection costto be high.

• Other values of the acceptance number could beselected as well.


52

Fig. OC curves of single sampling plans for stipulated

consumer's risk and LQL

Stipulated Producer and Consumer Risk

• We desire sampling plans that satisfy a producer's risk a (given anassociated quality level p1 = AQL) and a consumer's risk â (given anassociated quality level p2 = LQL).

• Good lots, with quality level given by AQL, are to be rejected nomore than 100α% of the time.

• Poor lots, with quality level specified by LQL, are to be accepted nomore than 100β% of the time.

• It can be difficult to find a sampling plan that exactly satisfies boththe producer's and consumer's stipulation.


53

Stipulated Producer and Consumer Risk

• Let's consider the plans shown in Figure.

• Two plans meet the producer's stipulation

exactly and come close to meeting the

consumer's stipulation.

• Two other plans meet the consumer's

stipulation exactly and come close to meeting

the producer's stipulation. Of these four plans,

one must be selected based on additional

criteria of concern to the user.

• It may be of interest, for example, to choose the

plan with the smallest sample size to minimize

inspection costs, or the one with the largest

sample size to provide the most protection.


54

Fig. OC curves of sampling plans for stipulated

producers' and consumer's risks.

Example: Find a single sampling plan that satisfies a producer's

risk of 5% for lots that are 1.8% nonconforming, and a

consumer's risk of 10% for lots that are 9% nonconforming.

Solution

• We have α = 0.05, p1 = AQL = 0.018, β = 0.10, and p2 = LQL =

0.09. First, we compute the ratio np2/np1, which is the ratio

p2/p1 because n cancels out:


55

• For values of α = 0.05 and β = 0.10, we use the last column in Table (slide#46) to

determine the possible acceptance numbers.

• We find that the ratio 5.00 falls between 6.51 and 4.89, corresponding to acceptance

numbers of 2 and 3, respectively.

• Two plans (one for c = 2 and one for c = 3) satisfy the producer's stipulation exactly:


56

For c = 2, np 1= 0.818, and the sample

size isFor c = 3, np 1= 1.366, and the sample

size is

So, the plans n = 45, c = 2 and n = 76, c = 3 both satisfy the producer's stipulation

exactly.

• Next, we find that two plans (c = 2 and c = 3) satisfy the consumer'sstipulation exactly: For c = 2, np2 = 5.322, and the sample size is

• For c = 3, np2 = 6.681, and the sample size is

• The plans n = 60, c = 2 and n = 75, c = 3 both satisfy theconsumer's stipulation exactly.


57

• The four candidates are as follows:

Plan 1: n = 45, c = 2 Plan 3: n = 60, c = 2

Plan 2: n = 76, c = 3 Plan 4: n = 75, c = 3

• Now let's see how close plans 1 and 2 (which satisfy theproducer's stipulation) come to satisfying the consumer'sstipulation.

• For a target value of the consumer's risk β of 0.10, we find theproportion nonconforming p2 of batches that would be accepted100 β % of the time.


58

• For n = 45 and c = 2 (plan 1), if β =0.10, then np2 = 5.322. Thus,

• For n = 76 and c = 3 (plan 2), if β =0.10, then np2 = 6.681. So

• Plan 1 accepts batches that are 11.83%nonconforming 10% of the time.


59

• On the other hand, plan 2 acceptsbatches that are only 8.79%nonconforming 10% of the time.

• Our goal is to find a plan thataccepts batches that are 9%nonconforming 10% of the time.

• Given that the target value of p2(the specified LQL) is 0.09, wefind plan 2's value of p2 = 0.0879is closer to the target value thanplan l's value 0.1183.

• Now let's find a plan that satisfies the consumer's stipulation

exactly and comes as close as possible to satisfying the

producer's stipulation.

• For plans 3 and 4, we need to determine the proportion

nonconforming p1 of batches that would be accepted 95% of the

time.

• This satisfies the producer's risk a = 0.05.


60

• For n = 60 and c = 2 (plan 3), if α

= 0.05, then np1 = 0.818. So

• For n = 75 and c = 3 (plan 4), if α

= 0.05, then np1 = 1.366. So


61

• Plan 3 rejects batches that are 1.36%nonconforming 5% of the time.

• On the other hand, plan 4 rejectsbatches that are 1.82%nonconforming 5% of the time.

• Since plan 4's value of p1= 0.0182 iscloser to the target value p = 0.0180,plan 4 is selected.

• Note that plan 4 is more stringentthan our goal.

• Another criterion we could use to select a sampling plan is tochoose the one with the smallest sample size in order to minimizeinspection costs. Of the four candidates plan 1 would be selectedwith n = 45, c = 2. This plan satisfies the producer's stipulationexactly.

• Alternatively, we could select a plan with the largest sample size,which provides the most information. Here we would choose plan 2with n = 76, c = 3.

• As discussed previously, this plan satisfies the producer's stipulationexactly and comes as close as possible to the consumer's stipulation.


62

• In the preceding sections we discussed several methods for determiningsampling plans.

• Many organizations prefer to use existing plans, known as standardizedsampling plans, rather than compute sampling plans of their own.

• They simply select a set of criteria and determine the standardized plansthat best match this criteria.

• Although standardized plans use predefined criteria, companies cangenerally adjust their criteria to match the standardized plan.

• The advantage here is that plans can be selected with very little effort.

• Moreover, characteristics and performance measures of the plans arealready calculated and tabulated.

Standard Sampling Plans

63

• There are two common lot-by-lot attribute sampling plans.– Sampling Procedures and Tables for Inspection by Attributes

(ANSI/ISO/ASQ Z 1.4-2003)

– The Dodge-Romig system

• ANSI/ISO/ASQ Z1.4 is used as an acceptable quality level system.– This means that the quality level of good lots should be rejected

infrequently.

– If the process average proportion nonconforming is less than the AQL, thesampling plans in ANSI/ISO/ASQ Z1.4 are designed to accept the majorityof the lots.

– However, protecting the consumer by not accepting poor lots (that is, thelimiting quality level) was not a key criterion in ANSI/ISO/ASQ Zl.4 plans.


64

• In this course, we will only discuss the Dodge-Romig plans indetails

Dodge-Romig Plans

• Dodge and Romig (1959) designed a set of plans based onachieving a certain overall level of quality for products sent to theconsumer.

• Although ANSI/ISO/ASQ Zl .4 is a system based on AQL, it haslittle impact on the overall quality level because the sample sizesare quite small compared to the lot sizes and only thenonconforming items in the sample are detected.

• Dodge-Romig plans, however, are based on rectifying inspection.


65

Dodge-Romig Plans

• There are two sets of plans.

– One is based on satisfying a given limiting quality level (LQL) based

on a consumer's risk β, the target value of which is 0.10.

• The other is based on meeting a certain value of the average

outgoing quality limit.

• For both sets of plans, the objective is to minimize the average

total inspection.


66

Plans Based on LQL

• These plans are used when protection is desired for the

acceptance of individual lots of a certain quality level.

• The Dodge-Romig LQL-based plans accept lots with a quality

level given by LQL 100β% of the time (a β of 0.10 was used to

develop these plans).

• Plans exist for LQL values of 0.5,1.0,2.0,3.0,4.0,5.0,7.0, and

10.0% nonconforming.

Dodge-Romig plans

67

Plans Based on LQL

• To use the Dodge-Romig tables, an estimate of the process

average nonconforming ҧ𝑝 is necessary.

• Recent data from the process can be used to develop this

estimate.

• If no data is available for the process, the largest value of the

process average nonconforming found in the table can be used

as a conservative estimate.

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Plans Based on LQL

• The Dodge-Romig table for a single sampling plan with an LQL of5% is shown in Table (next slide).

• Note that the process average in Table lists values to 2.5%.

• For values over 2.5% (which is half the LQL), sampling plans maynot be preferable because 100% inspection becomes moreeconomical.

• The table also provides a value of the AOQL (in percentage) for agiven sampling plan.

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Dodge-Romig plans

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Example: Find a Dodge-Romig plan when the lot size is 700, the LQL is 5%,and the process average is 1.30% nonconforming. A single sampling plan isdesired.

Solution

• Using above Table to index the lot size and process average, the singlesampling plan is found to be n = 130, c = 3.

• For this sampling plan, the AOQL is 1.2%.

• This means that the worst average outgoing quality, regardless of incomingquality, will not exceed 1.2%.

• If the process average were not known, the maximum listed value of theprocess average would be used (in this case, the range 2.01 to 2.5%), andthe plan would be n = 200, c = 6.

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Plans Based on AOQL

• When we need to provide a level of protection for the average

quality level of a stream of batches, a plan based on the average

outgoing quality limit is often appropriate.

• A specified value of AOQL is selected.

• The objective is to choose plans such that the worst average

outgoing quality for a stream of lots, regardless of incoming

quality, will not exceed this AOQL value.

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• Dodge-Romig AOQL-based plans are designed to meet thiscriterion and also to minimize the average total inspection.

• The plans are tabulated for AOQL values of 0.10, 0.25, 0.50, 0.75,1.00, 1.50, 2.00, 2.50, 3.00, 4.00, 5.00, 7.00, and 10.00%.

• Both single and double sampling plans are available for theseAOQL values.

• As in the previous set of plans, the lot size and process averagemust be known in order to use the tables.

• The tables also provide the LQL values for a consumer's risk β of0.10.

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Dodge-Romig AOQL-based plans

• The Dodge-Romig table for a single sampling plan with an

AOQL of 3.0% is shown in Table (next slide).

• Note that the process average is listed to a value of 3.0% (equal

to the AOQL value of 3.0%).

• For process averages exceeding this value, 100% inspection

becomes economical.

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Example: Find a Dodge-Romig single sampling plan when the lot sizeis 1200, the average outgoing quality limit is 3%, and the processaverage is 1.4% nonconforming.

Solution

• From above Table 10-9, indexing the lot size of 1200 and processaverage of 1.4% nonconforming, the single sampling plan is foundto be n = 65, c = 3.

• For this plan, the LQL is 10.2%.

• This means that for individual lots with a nonconformance rate of10.2%, the probability of accepting such lots would be 10%, theconsumer's risk.

quality control and reliability inspection and sampling · •from the oc curve in above figure,...

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