Quality Control and Reliability Inspection and Sampling Prepared by Dr. M. S. Memon Dept. of Industrial Engineering & Management Mehran UET, Jamshoro, Sindh, Pakistan 1
Chapter Objectives 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 2
Introduction to Acceptance Sampling 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. 3
Introduction to Acceptance Sampling 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. 4
Advantages and Disadvantages of Sampling 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 to handling. 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, repetitive inspection, such as 100% inspection, inspector fatigue can prevent the identification of all nonconformities or nonconforming units. 4. Sampling provides a strong motivation to improve quality because an entire batch or lot may be rejected. 5
Advantages and Disadvantages of Sampling 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. 6
Producer and Consumer Risks 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 7
Producer and Consumer Risks 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
Producer and Consumer Risks Consumer's Risk: The risk associated with accepting a "poor" lot, due to the inherent nature of random sampling, is defined as a consumer's risk. Further, norms of customer requirements will govern the definition of a "poor" lot. Limiting quality level (LQL) or rejectable quality level (RQL) is the terminology used to defined this level of unacceptable quality. An alternative terminology, when the quality level is expressed in percentage nonconformance, is lot tolerance percent defective (LTPD). 9
Producer and Consumer Risks Thus, when we state a producer's risk in a sampling plan, we must correspondingly state a desirable level of quality that we prefer to accept. For example, if we state that the producer's risk is 5% for an AQL of 0.02, it means that we consider batches that are 2% nonconforming to be good and prefer to reject such batches no more than 5% of the time. If the consumer's risk is 10% for an LQL of 0.08, this means that batches that are 8% nonconforming are poor and we prefer to accept these batches no more than 10% of the time. 10
Operating Characteristic Curve (OC Curve) The operating characteristic (OC) curve measures the performance of a sampling plan. It plots the probability of accepting the lot versus the proportion nonconforming of the lot. It shows the discriminatory power of the sampling plan. For all sampling plans, we want to accept lots with a low proportion nonconforming most of the time and we do not want to accept batches with a high proportion nonconforming very often. The OC curve indicates the degree to which we achieve this objective. 11
Operating Characteristic Curve (OC Curve) Suppose that we have chosen a proportion nonconforming level p 0 such that if a lot has a proportion nonconforming less than or equal to p 0, we consider it to be a good lot and we accept it. On the other hand, if the proportion nonconforming of the lot exceeds P 0, we consider the lot to be poor and we reject it. The ideal OC curve for these circumstances is shown in Figure. Fig. Ideal operating characteristic curve. 12
Operating Characteristic Curve (OC Curve) 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
Operating Characteristic Curve (OC Curve) 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
Operating Characteristic Curve (OC Curve) 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
Operating Characteristic Curve (OC Curve) 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
Example on OC Curve Construct an OC curve for a single sampling plan where the lot size is 2000, the sample size is 50, and the acceptance number is 2. Solution We are given N = 2000, n = 50, and c = 2. The probability of lot acceptance is equivalent to the probability of obtaining 2 or fewer nonconforming items in the sample. Let's suppose that p is 0.02 (i.e., the batch is 2% nonconforming). Since np = (50) (0.02) = 1.0, the probability P a of accepting the lot (using Appendix A-2) is 0.920. A plot of these values, the OC curve, is shown in Figure. Fig. OC curve for the sampling plan in Example. 17
Example on OC Curve Table. Lot Acceptance Probabilities for Different Values of Proportion Nonconforming for the Sampling Plan N 2000, n = 50, c = 2. 18
Example on OC Curve Producer and consumer risk can also be demonstrated through the OC curve. Suppose that our numerical definition of good quality (indicated by the AQL) 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% of the time. Batches that are 11% nonconforming, on the other hand, will be accepted 8.8% of the time. The consumer's risk is therefore 8.8%. 19
Effect of the sample size and the acceptance number The parameters n and c of the sampling plan affect the shape of the OC curve. As long as the lot size N is significantly large compared to the sample size n, the lot size does not have an appreciable impact on the shape of OC curve. For fixed values of N and c, as the sample size becomes larger, the slope of the OC curve becomes steeper, implying a greater discriminatory power. Figure shows the OC curves for those sampling plans. Fig. Effect of the sample size on the shape of the OC curve 20
Effect of the sample size and the acceptance number Figure 10-4 shows the OC curves for four sampling plans. Note that the probability of acceptance decreases for a given lot quality as the acceptance number c decreases. The chosen values of n and c should be such that they match the goals of the user. Fig. Effect of the acceptance number on the shape of the OC curve 21
Types of sampling plans 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 is used 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 second sample is taken. 22
Types of sampling plans The parameters of a double sampling plan are as follows: 23
Types of sampling plans Let's consider the following double sampling plan where attribute inspection is conducted to find the number of nonconforming items: The working procedure for this plan is initially, to select a random sample of 40 items from the 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, the lot is accepted; if it is 6 or more, the lot is rejected. 24
Types of sampling plans Multiple sampling plans are an extension of double sampling plans. Three, four, five, or as many samples as desired may be needed to make a decision regarding the lot. The sampling plan can be terminated at any stage once the acceptance or rejection criteria have been met. The ultimate extension of the multiple sampling plan is the sequential sampling plan, which is an item-by-item inspection plan. After each item is inspected, a decision is made to accept the lot, reject the lot, or choose another item for inspection 25
Advantages and Disadvantages of sampling plans As far as simplicity is concerned, the single sampling plan is the best, followed by double and then multiple sampling plans. Administrative costs for record keeping, training, and inspection are the least for single and the highest for multiple sampling plans. On average, for equivalent plans, the number of items inspected to make a decision 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 made quickly. Inspection costs will therefore be the most for single, and the least for multiple sampling plans. Single sampling plans provide the most information, and multiple sampling plans the least. 26
Evaluating sampling plans The OC curve is one measure of the performance of a sampling plan. We also use other measures to evaluate the goodness of a sampling plan. 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. 27
Rectifying Inspection First consider the concept of rectifying inspection as it applies to lots that are rejected through sampling plans. Usually, such lots go through 100% inspection, known as screening, where nonconforming items are replaced with conforming ones. Such a procedure is known as rectification inspection because it affects the quality of the product that leaves the inspection station. Nonconforming items found in the sample are also replaced. 28
Average Outgoing Quality The average outgoing quality (AOQ) is the average quality level of a series of batches that leave the inspection station, assuming rectifying inspection, after coming in for inspection at a certain quality level p. The AOQ measures the average quality level of a large number of batches of incoming quality p, the proportion nonconforming in the lots, assuming rectification. Taking N as the lot size, n as the sample size, p as the incoming lot quality, and P a as the probability of accepting the lot using the given sampling plan, the average, outgoing quality is given by 29
Average Outgoing Quality 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 plan for various levels of incoming quality is usually constructed. Let's, consider the single sampling plan N = 2000, n = 50, c = 2. Suppose that the incoming quality of batches is 2% nonconforming. From the Poisson cumulative distribution tables in Appendix A-2, the probability P a of accepting the lot using the sampling plan is 0.920. The average outgoing quality is Thus, if batches come in as 2% nonconforming, the average outgoing quality is 1.79%. 30
Example on AOQ Example: Construct the AOQ curve for the sampling plan N = 2000, n = 50, c = 2. Solution The probability of lot acceptance for various values of the incoming lot quality p is already computed and listed in Table (Slide# 18). Using these values of P a and p, the values of AOQ are calculated for different values of p. Fig. AOQ curve for the sampling plan 31
Example on AOQ Note that when the incoming quality is very good, the average outgoing quality is also very good. When the incoming quality is very poor, the average outgoing quality is good because most of the lots are rejected by the sampling plan and go through screening. In between these extremes, the AOQ curve reaches a maximum, AOQL. Fig. AOQ curve for the sampling plan 32
Average Outgoing Quality Limit 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. 33
Average Outgoing Quality Limit 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, we have, some protection against the worst quality for a series of batches that leave 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 particular sampling plan. Sampling plans are designed such that their AOQL does not exceed a certain specified value. 34
Average Total Inspection (ATI) If rectifying inspection is conducted for lots rejected by the sampling plan, another evaluation measure is the average total inspection (ATI). The ATI represents the average number of items inspected per lot. If a lot has no nonconforming items, it will obviously be accepted by the chosen sampling plan, and only n items (the sample size) will be inspected for a lot. At the other extreme, if the lot has 100% nonconforming items, the number inspected per lot will be N (the lot size) assuming that rejected lots are screened. 35
Average Total Inspection (ATI) For single sampling plans, the average total inspection per lot for lots with an incoming quality level p is given by For a double sampling plan, the ATI is given by where P a1 represents the probability of accepting the lot on the first sample, and P a2 represents the probability of lot acceptance on the second sample. 36
Average Total Inspection (ATI) 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 P a = 0.920. The ATI for this value of p is 37
Average Total Inspection (ATI) 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. Fig. ATI curve for the sampling plan 38
Lot-by-Lot Attribute Sampling Plans 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). 39
Lot-by-Lot Attribute Sampling Plans 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
Lot-by-Lot Attribute Sampling Plans The OC Curve The OC curve of a single sampling plan has been described in detail. It represents the probability of accepting the lot, P a, as a function of the lot quality, which is simply the proportion nonconforming p if items are classified only as conforming or not. The effects of the parameters n and c on the shape of the OC curve have also been discussed. A study of these effects enables us to choose appropriate values of n and c, given desirable levels of protection against the producer's and consumer's risks. Fig. OC curve showing (AQL, 1-α) and (LQL, β) for a sampling plan 41
Lot-by-Lot Attribute Sampling Plans The OC curve in Figure shows the relationship between AQL and LQL parameters. For a sampling plan specified by n and c, lots with a proportion nonconforming level of AQL that come in for inspection should be accepted 100(1-α)% of the time. Similarly, if the proportion nonconforming of batches coming in for inspection is LQL, they should be accepted 100β% of the time. A suitable choice of n and c ensures that good lots will be accepted a large percentage of the time and that bad lots will be accepted infrequently. Fig. OC curve showing (AQL, 1-α) and (LQL, β) for a sampling plan 42
Design of Single Sampling Plans 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 43
Design of Single Sampling Plans Stipulated Producer's Risk Let's suppose the producer's risk α and its associated quality level p 1, which is the acceptable quality level (AQL), are specified. We desire single sampling plans that will accept lots of quality level p 1, 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
Design of Single Sampling Plans 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 P a equal to 1 - α at p=p 1, the value of λ is found in Appendix A-2. Because λ = np 1 =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
Design of Single Sampling Plans Table. Values of np for a Producer's Risk of 0.05 and a Consumer's Risk of 0.10 46
Design of Single Sampling Plans Example: Find a single sampling plan that satisfies a producer's risk of 5% for lots that are 1.5% 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 np 1 = 0.355, the sample size is 47
Design of Single Sampling Plans Note that all three plans satisfy the producer's risk of 5% at the AQL value of 1.5%. However, they have varying degrees of protection against acceptance of poor quality lots, which would be of interest to the consumer. Of the three plans shown, n = 220, c = 6 provides the best protection to the consumer because it has the lowest probability of accepting poor quality lots. However, we must also consider the increased inspection costs associated with this plan, because the sample size for c = 6 is the largest of the three. Note: Other values of c could be selected as well. 48
Design of Single Sampling Plans Stipulated Consumer's Risk Let's suppose that the consumer's risk β and its associated quality level p 2, which is the limiting quality level (LQL), are given. We want to find sampling plans that will accept lots of quality level p 2, 100β % of the time. Here again, a number of sampling plans will satisfy this criterion. Figure shows the OC curves for three sampling plans that meet the criterion. Fig. OC curves of single sampling plans for stipulated consumer's risk and LQL 49
Design of Single Sampling Plans 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 the probability of acceptance of β, for lots of quality p 2 = LQL. The value of λ = np 2 is found in Appendix A-2. If the value of β is 0.10, we can use same previous Table to obtain the value of np2. The sample size is calculated by dividing the value of np 2 by p 2. 50
Design of Single Sampling Plans 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 p 2 = LQL = 0.08. If we select an acceptance number of 1, Previous Table (slide#46) gives np 2 = 3.890. The sample size is 51
Design of Single Sampling Plans Figure shows the OC curves for these sampling plans. All three pass through the point (p 2, β), thus satisfying the consumer's stipulation. The degree of protection for extremely good batches, as far as the producer is concerned, is different. The plan n = 132, c = 6 will reject good batches (say, 1% nonconforming) the least frequently of the three plans. Of course, it has the largest sample size, which may cause the inspection cost to be high. Other values of the acceptance number could be selected as well. Fig. OC curves of single sampling plans for stipulated consumer's risk and LQL 52
Design of Single Sampling Plans Stipulated Producer and Consumer Risk We desire sampling plans that satisfy a producer's risk a (given an associated quality level p 1 = AQL) and a consumer's risk â (given an associated quality level p 2 = LQL). Good lots, with quality level given by AQL, are to be rejected no more than 100α% of the time. Poor lots, with quality level specified by LQL, are to be accepted no more than 100β% of the time. It can be difficult to find a sampling plan that exactly satisfies both the producer's and consumer's stipulation. 53
Design of Single Sampling Plans 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. Fig. OC curves of sampling plans for stipulated producers' and consumer's risks. 54
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 Design of Single Sampling Plans We have α = 0.05, p 1 = AQL = 0.018, β = 0.10, and p 2 = LQL = 0.09. First, we compute the ratio np 2 /np 1, which is the ratio p 2 /p 1 because n cancels out: 55
Design of Single Sampling Plans 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: For c = 2, np 1 = 0.818, and the sample size is For 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. 56
Design of Single Sampling Plans Next, we find that two plans (c = 2 and c = 3) satisfy the consumer's stipulation exactly: For c = 2, np 2 = 5.322, and the sample size is For c = 3, np 2 = 6.681, and the sample size is The plans n = 60, c = 2 and n = 75, c = 3 both satisfy the consumer's stipulation exactly. 57
Design of Single Sampling Plans 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 the producer's stipulation) come to satisfying the consumer's stipulation. For a target value of the consumer's risk β of 0.10, we find the proportion nonconforming p 2 of batches that would be accepted 100 β % of the time. 58
Design of Single Sampling Plans For n = 45 and c = 2 (plan 1), if β = 0.10, then np 2 = 5.322. Thus, For n = 76 and c = 3 (plan 2), if β = 0.10, then np 2 = 6.681. So Plan 1 accepts batches that are 11.83% nonconforming 10% of the time. On the other hand, plan 2 accepts batches that are only 8.79% nonconforming 10% of the time. Our goal is to find a plan that accepts batches that are 9% nonconforming 10% of the time. Given that the target value of p2 (the specified LQL) is 0.09, we find plan 2's value of p 2 = 0.0879 is closer to the target value than plan l's value 0.1183. 59
Design of Single Sampling Plans 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 p 1 of batches that would be accepted 95% of the time. This satisfies the producer's risk a = 0.05. 60
Design of Single Sampling Plans For n = 60 and c = 2 (plan 3), if α = 0.05, then np 1 = 0.818. So For n = 75 and c = 3 (plan 4), if α = 0.05, then np 1 = 1.366. So Plan 3 rejects batches that are 1.36% nonconforming 5% of the time. On the other hand, plan 4 rejects batches that are 1.82% nonconforming 5% of the time. Since plan 4's value of p 1 = 0.0182 is closer to the target value p = 0.0180, plan 4 is selected. Note that plan 4 is more stringent than our goal. 61
Design of Single Sampling Plans Another criterion we could use to select a sampling plan is to choose the one with the smallest sample size in order to minimize inspection costs. Of the four candidates plan 1 would be selected with n = 45, c = 2. This plan satisfies the producer's stipulation exactly. Alternatively, we could select a plan with the largest sample size, which provides the most information. Here we would choose plan 2 with n = 76, c = 3. As discussed previously, this plan satisfies the producer's stipulation exactly and comes as close as possible to the consumer's stipulation. 62
Standard Sampling Plans In the preceding sections we discussed several methods for determining sampling plans. Many organizations prefer to use existing plans, known as standardized sampling plans, rather than compute sampling plans of their own. They simply select a set of criteria and determine the standardized plans that best match this criteria. Although standardized plans use predefined criteria, companies can generally 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 are already calculated and tabulated. 63
Standard Sampling Plans 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, the sampling plans in ANSI/ISO/ASQ Z1.4 are designed to accept the majority of the lots. However, protecting the consumer by not accepting poor lots (that is, the limiting quality level) was not a key criterion in ANSI/ISO/ASQ Zl.4 plans. 64
Standard Sampling Plans In this course, we will only discuss the Dodge-Romig plans in details Dodge-Romig Plans Dodge and Romig (1959) designed a set of plans based on achieving a certain overall level of quality for products sent to the consumer. Although ANSI/ISO/ASQ Zl.4 is a system based on AQL, it has little impact on the overall quality level because the sample sizes are quite small compared to the lot sizes and only the nonconforming items in the sample are detected. Dodge-Romig plans, however, are based on rectifying inspection. 65
Standard Sampling Plans 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
Dodge-Romig plans 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. 67
ҧ Dodge-Romig plans Plans Based on LQL To use the Dodge-Romig tables, an estimate of the process average nonconforming p 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. 68
Dodge-Romig plans Plans Based on LQL The Dodge-Romig table for a single sampling plan with an LQL of 5% 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 may not be preferable because 100% inspection becomes more economical. The table also provides a value of the AOQL (in percentage) for a given sampling plan. 69
Dodge-Romig plans 70
Dodge-Romig plans 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 is desired. Solution Using above Table to index the lot size and process average, the single sampling 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 incoming quality, will not exceed 1.2%. If the process average were not known, the maximum listed value of the process average would be used (in this case, the range 2.01 to 2.5%), and the plan would be n = 200, c = 6. 71
Dodge-Romig plans 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. 72
Dodge-Romig plans Dodge-Romig AOQL-based plans are designed to meet this criterion 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 these AOQL values. As in the previous set of plans, the lot size and process average must be known in order to use the tables. The tables also provide the LQL values for a consumer's risk β of 0.10. 73
Dodge-Romig plans 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. 74
Dodge-Romig plans 75
Dodge-Romig plans Example: Find a Dodge-Romig single sampling plan when the lot size is 1200, the average outgoing quality limit is 3%, and the process average is 1.4% nonconforming. Solution From above Table 10-9, indexing the lot size of 1200 and process average of 1.4% nonconforming, the single sampling plan is found to be n = 65, c = 3. For this plan, the LQL is 10.2%. This means that for individual lots with a nonconformance rate of 10.2%, the probability of accepting such lots would be 10%, the consumer's risk. 76