Acceptance Sampling May 2014 PDT155

Topics

Situations for Acceptance Sampling If Supplier have successfully implemented TQM philosophy…… SS

Application of Sampling Inspection. When product testing is : Destructive (e.g. tensile strength) Expensive ( e.g. welded metal) Time consuming (e.g. test parts to failure) When lot size is large (e.g 20,000units).

Definition Sampling inspection in which decisions are made to accept or reject product based on the results of the inspection of samples. Acceptance sampling used in manufacturing included making decisions about incoming raw material lots, in process sub lots and finished products lots. To ensure the quality characteristics(dimensional check, material check, functional check of a lot), the consumer will sample the items from the lots, test them and observe the number of rejection units.

Sampling Inspection A sample consisting of one or more units of product selected at random from the production process output and examines for one or more quality characteristics of lots or batches. Purpose is to obtain information in order to reach a decision regarding the acceptance of the lot. Generally, all industries are practicing with “zero defect “ concept for quite some time due to customers demand for better quality products. Goods which are expensive, no defective is allowed by original equipment customers because of high replacement costs.

Pros and Cons of Sampling Inspection Compare with 100% inspection, sampling inspection has the following: Advantages a. Economy due to inspecting certain part of the product.   b. Less damage due to handling during inspection. c. Fewer inspectors required. d. Upgrading the inspection job from piece to piece to lot-by- lot decision.

Pros and Cons of Sampling Inspection e. Applicable for testing.   Rejection on the entire lots rather than mere return of the defectives, thereby providing stronger motivation for improvement by vendor. Disadvantages Risk included in chance of bad lot “acceptance” and good lot “rejection” Sample taken provides lesser information than 100% inspection Requires planning and documentation No assurance that quality is perfect

Sampling Plan We considered acceptance sampling by attributes, to enable the inspector to decide the sample size to be used for inspection. A sampling plan is a statement of sample size to be drawn with its associated acceptance and rejection number. It is used with the associated acceptance and rejection criteria . These acceptance-sampling plans are either of attributes or variables types.

Inspection Processes Stage 1 : Raw Material received at incoming inspection Stage 2 : Work In Progress in manufacturing on parts produced Stage 3 : Out going inspection of finished goods before shipment

Types of Attribute Sampling Plans 1.Single sampling plan 2.Double-sampling plan 3.Multiple-sampling plan 4.Sequential-sampling 5.Dodge Romig Sampling plan

Single-Sampling Plan A single sampling plan is defined by sample size (n), acceptance number ( c ) & rejection number (c+1). N total items in a lot. Select n items at random. If c or less items are unacceptable, reject the lot. N = lot size n = sample size c = acceptance number The acceptance or rejection of the lot is based on the results from a single sample - thus a single-sampling plan. When a lot is rejected, it should be rescreened and rejected parts shall be replaced with good parts.

Single-Sampling Plan n(c/c+1) Lot of N Items Random Sample of n Items N - n Items Rescreen lot & replace bad parts with good ones d defectives found in sample Inspect n Items d > c d < c Accept Lot Reject Lot Accept Lot

Double-Sampling Plan explanation First sample (n1) is drawn from the lot (N). If the defectives in sample (n1)  c1, accept the lot. If the number of defectives  r1 , reject the lot. If the number of defectives in between c1 and r1, a second sample is drawn. If the combined samples (n1+n2) have defectives  c2 accept the lot otherwise reject the lot.

Double-Sampling Plan n1(c1/r1),n2,n1+n2(c2/c2+1) Lot of N Items Random Sample of n1 Items Rescreen and replace bad parts with good ones of the lot N – n1 Items d 1 defectives found in sample Inspect n1 Items d 1> r 1 d 1< c1 Accept Lot c1 <d1 < r1 Reject Lot (to next slide) Continue

n1(c1/r1),n2,n1+n2(c2/c2+1) cont. Continue (from previous slide) N – n1 Items Random Sample of n2 Items N – (n1 + n2) Items d 2 defectives Found in Sample Reject Lot Inspect n2 Items (d1+ d2) > c 2 + 1 (d1 + d2) < c2 Accept Lot

Operating Characteristic Curve (OC) The operating-characteristic (OC) curve measures the performance of an acceptance-sampling plan. It plots the percent of acceptance or probability of accepting the lot as a function of lot product quality. Two types of OC curves used for Acceptance sampling Type A (based on hypergeometric distribution from a finite lot size) Type B (based on Poisson & Binomial distribution from an infinite lot size) With low level of percent defective of the lots the probability of acceptance is high. (  0.95 -.98).

A Typical (OC) Curve for Proportions

Characterisitics of OC Curve AQL (Acceptable quality limit) is the worst tolerable process average when a series of lots is submitted for acceptance sampling. LTPD (Lot tolerance percent defective) is the upper limit of the percentage of defective items where the probability of the lot being accepted is 10%. Producer’s risk (α) is the probability of rejecting a lot that has a defect level equal to AQL. The producer suffers when this occurred because a lot with acceptable quality is rejected during inspection Type I error Consumer’s Risk (β) is the probability of acceptance of a lot that has a defect level equal to LTPD. The consumer suffers when a lot of unacceptable quality has been accepted during inspection. Type II error

Properties of OC curve Fig 14-3 shows an ideal OC curve will accept every lot with p  AQL and reject every lot with p > AQL. Fig. 14-4 shows the degree of discriminating power for any sampling plan with increase in sample size. Fig 14.5 shows steeper OC curves with reducing acceptance number, normally used in tightened inspection

Steps for construction of OC curve 1. Refer to a statistical table on cumulative Poisson Distribution for probability of “ r or more ”. 2. Look at the sampling plan rejection number (C+1) and across the row under the “m” columns select a value with a cum. probability (P)  0.90. The choice of =0.9 is selected to ensure a complete range of the OC curve can be plotted. Divide the “m” value as obtained from the table by the sample size (n) to get the approximate proportion defective to be plotted on the horizontal axis of the OC curve. Decide the number of points required for the OC curve. ( around 5 to 10)

Example OC Curve for 89(2/3) Sampling plan 89(2/3). Refer to poisson table ( Table C ) at r=3 across the row, select the m value with cum. probability = 0.7619. This correspond to m=4 at the column With m = n x p: i.e. 4 = 89 x p p= 4/89 = 0.045 Therefore the OC curve can be plotted from a range of p =0 to 0.05 with intervals of 0.01 ( 5 points) Tabulate a table with 4 columns for tabulation of the OC curve . (shown in next slide)

Exercise on OC Curve for 89(2/3) Rfhghghgh Prop def. (p) Col 1 M=n x p = 89 x p Col 2 P (rej) at r 3 Col3 P acc = 1-col 3 0.0000 1.000 0.01 .89 0.0629 0.9371 0.02 1.78 0.2694 0.7306 0.03 2.67 0.5064 ? 0.04 3.56 0.05 0.8264 0.1736 Exercise: Construction of OC curve (refer Table C) Refer to table above , fill up the value in the cells with the question mark. 2. Plot the OC curve with column (1) on x axis and column (4) on the y axis 3. M stand for mean of the poisson

Poisson Cumulative Probability Table ( Table C) Cumulative Poisson Probabilities Tables ( r or more success in n independent trial)   m= 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.0000 0.0952 0.1813 0.2592 0.3297 0.3935 0.4512 0.5034 0.5507 0.5934 0.6321 2 0.0047 0.0175 0.0369 0.0616 0.0902 0.1219 0.1558 0.1912 0.2275 0.2642 3 0.0002 0.0011 0.0036 0.0079 0.0144 0.0231 0.0341 0.0474 0.0629 0.0803 4 0.0001 0.0003 0.0008 0.0018 0.0034 0.0058 0.0091 0.0135 0.0190 5 0.0004 0.0014 0.0023 0.0037 6 0.0006 7 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 0.6671 0.6988 0.7275 0.7534 0.7769 0.7981 0.8173 0.8347 0.8504 0.8647 0.3010 0.3374 0.3732 0.4082 0.4422 0.4751 0.5068 0.5372 0.5663 0.5940 0.0996 0.1205 0.1429 0.1665 0.2166 0.2428 0.2694 0.2963 0.3233 0.0257 0.0338 0.0431 0.0537 0.0656 0.0788 0.0932 0.1087 0.1253 0.0054 0.0077 0.0107 0.0143 0.0186 0.0237 0.0296 0.0364 0.0441 0.0527 0.0010 0.0015 0.0022 0.0032 0.0045 0.0060 0.0080 0.0104 0.0132 0.0166 0.0009 0.0013 0.0019 0.0026 8 9

Poisson Cumulative Probability cont. (Table C) 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 1.0000 1 0.8775 0.8892 0.8997 0.9093 0.9179 0.9257 0.9328 0.9392 0.9450 0.9502 2 0.6204 0.6454 0.6691 0.6916 0.7127 0.7326 0.7513 0.7689 0.7854 0.8009 3 0.3504 0.3773 0.4040 0.4303 0.4562 0.4816 0.5064 0.5305 0.5540 0.5768 4 0.1614 0.1806 0.2007 0.2213 0.2424 0.2640 0.2859 0.3081 0.3304 0.3528 5 0.0621 0.0725 0.0838 0.0959 0.1088 0.1226 0.1371 0.1523 0.1682 0.1847 6 0.0204 0.0249 0.0300 0.0357 0.0420 0.0490 0.0567 0.0651 0.0742 0.0839 7 0.0059 0.0075 0.0094 0.0116 0.0142 0.0172 0.0206 0.0244 0.0287 0.0335 8 0.0015 0.0020 0.0026 0.0033 0.0042 0.0053 0.0066 0.0081 0.0099 0.0119 9 0.0003 0.0005 0.0006 0.0009 0.0011 0.0019 0.0024 0.0031 0.0038 10 0.0001 0.0002 0.0004 0.0007 11   12 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 0.9550 0.9592 0.9631 0.9666 0.9698 0.9727 0.9753 0.9776 0.9798 0.9817 0.8153 0.8288 0.8414 0.8532 0.8641 0.8743 0.8838 0.8926 0.9008 0.9084 0.5988 0.6201 0.6406 0.6603 0.6792 0.6973 0.7146 0.7311 0.7469 0.7619 0.3752 0.3975 0.4197 0.4416 0.4634 0.4848 0.5058 0.5265 0.5468 0.5665 0.2018 0.2194 0.2374 0.2558 0.2746 0.2936 0.3128 0.3322 0.3516 0.3712 0.0943 0.1054 0.1171 0.1295 0.1424 0.1559 0.1699 0.1844 0.1994 0.2149 0.0388 0.0446 0.0510 0.0579 0.0653 0.0733 0.0818 0.0909 0.1005 0.1107 0.0168 0.0198 0.0231 0.0267 0.0308 0.0352 0.0401 0.0454 0.0511 0.0047 0.0057 0.0069 0.0083 0.0117 0.0137 0.0160 0.0185 0.0214 0.0014 0.0018 0.0022 0.0027 0.0040 0.0048 0.0058 0.0008 0.0010 0.0013 0.0016 0.0023 0.0028 13 14

Solution to construction of OC curve N = 3000, n= 89, c= 2 Assume; p0=0.02 ; np0=89*0.02 = 1.8 Pa = 1-Prob of 3 or more = 1- 0.2694 (Table C) = 73.1 %

Case Study There are two sampling inspection plans. a) 10 (1/2) b) 25 (3/4) 1. Plot the operating characteristic curves for both sampling plans and make comments on them. 2. If the assembly plant using these plans can operate comfortably with 30% rejection rate of defective raw materials, taking into consideration the price, the operation can not operate efficiently if the rejection rate gets close to 40%. 3. Which sampling plan should be selected ?

Double Sampling Plan Determine using two sampling plans 1 st sampling plan ~ Prob. of acceptance on 1st sample with c or less failure. 2 nd Sampling plan ~ Prob. of acceptance on combined samples with failures in the range of between c1 and r1 in sample 1 and Prob. of acceptance equal to c2 failures in sample 2 minus the failure in sample 1.

Example 150(1/4) 200, 350 (5/6) Example 150(1/4) 200, 350 (5/6) Lot size N = 2400 1 st sample n1= 150, c1= 1, r1= 4 2 nd sample n2= 200 , c2= 5, r2= 6 (Pa)I = (P1 or less) in first sample I (Pa)II = (P2)I (P3 or less)II i.e. 2 rejects in sample I and 3 or less rejects in sample II Plus (P3)I (P2 or less)II i.e. 3 rejects in sample I and 2 or less rejects in sample II Adopt the 4 columns table format for computing of prob. of acceptance for samples I and II for the range of defect proportion to plot the OC curve.

Double sampling Plan 150(1/4) 200, 350(5/6)

Calculation of Pacc. on 150(1/4)200,350(5/6) (Pa) combined --- Possibility of accepting a lot would be = 86%

Calculation of Pacc. on 150(1/4)200,350(5/6)

Multiple Sampling Plans Continuation of double sampling plan samples until 7 samples to establish the acceptance and rejection decision . 1 st sample is 100% inspected until the inspection stopped when the rejection number is reached If at completion of any stage of sampling, the number of defectives  acceptance number , the lot is accepted . Lot shall be rejected at any stage of sampling, if the number of defectives  the rejection number. If the defectives are in between the acceptance and rejection numbers , select another sample for decision. Decision ends after the 7 th sample had been taken.

Sequential-Sampling Plan Units are randomly selected from the lot and tested unit by unit. Acceptance may not be allowed until a minimum number of units are sampled After each unit has been tested a decision has to be made on the following: i. Acceptance of the lot ii. Rejection iii. Continue-sampling. Sampling process continues until the lot is accepted or rejected. Uses smaller average sample size. Groups of units can be drawn , referred to as group sequential sampling

Background Information ANSI/ASQ Z1.4 1st devised in 1942 at Bell Telephone Labs Developed for government procurement Adopted by International Standards Organisation Has become the world standard for attribute inspection.

Applications The standard is applicable to the following: Inspection of finished products from a machine Raw components and raw materials checking Checking of units which are in process in production Inspection of supplies from the store Sampling of data or records Administrative procedures ( e.g. audit of document)

Sampling Standards for Attributes 1.ANSI/ASQ Z1.4 objective is to accept with high probability whose lots have a high quality equal to or better than the Acceptable Quality Limit. (AQL). 2. Acceptance of the lots is based on sampling by attributes 3. The AQL refers to the maximum percentage defective that for sampling purpose to be considered as satisfactory. 4. The AQL does not imply lots passing the sampling plan to have % def.  AQL neither it does not imply lots that have % def.  AQL will fail the sampling plan. 5.ANSI / ASQ Z1.4 contains sample size codes to be used with reference to lots size received and predetermined inspection levels: Level III , II, S4, S3, S2. S1 and I in descending order of discriminating power. Generally, Inspection level (II) is used.

Sampling Standards for Attributes 1. Level I gives the least samples and and less discrimination whereas, Level III gives largest samples and more discrimination. 2. S4 to S1 are for special inspection which required small sample sizes 3. Table 10 gives the code letters for various lot or batch sizes and inspection levels I, II, and III. This code letter is used on Table 14, 15 or 16 to find the appropriate sample size and acceptance number corresponding to the AQL desired. 4. Standard provides for three types of sampling plans: Single Sampling Plan, Double Sampling Plan and Multiple Sampling Plan

Table 10 Sample Size Code Inspection level II to be used unless otherwise specified.

Application of Switching Rules in Sampling Plans The switching rules are meant for consumers protection from bad lots using tightened inspection (mandatory). Provide an incentive for suppliers to use reduced inspection (optional) when good lots are received by consumers. Within each type of sampling plan provisions are made for switching of inspection based on Normal inspection Tightened inspection Reduced inspection

Conditions for Switching Rules Normal Inspection used when under agreement between supplier and consumer . Normally under Level II. Normal to Tightened inspection (mandatory) Used when producer’s quality has deteriorated. e.g. 2 out of 5 consecutive lots being rejected under normal inspection Acceptance requirements more stringent than under normal inspection. e.g. acceptance number of the sampling plan being reduced Normal to Reduced inspection (Optional) Used when producer’s quality has been consistently good with objective in saving inspection costs. Preceding 10 lots have been accepted, etc.

Conditions for Switching Rules cont. Tightened to Normal when 5 consecutive batches have been accepted Reduced to normal a lot has not been accepted, production irregular, etc

Consideration of Lot Quality Characterisitcs Start with normal inspection Lots should be homogeneous Larger lots are more preferable than smaller lots Inspected units should be selected at random Inspected units should represent all items in the lot Non-accepted lots are to be resubmitted

Acceptable Quality Limit (AQL) Acceptable Quality Level (AQL) has been replaced by the term Acceptable Quality Limit Maximum percent nonconforming that can be considered satisfactory as a process average. AQL is designated in the contract or designee AQL is a reference point on the OC curve, does not imply that nonconformities are tolerable AQL are determined from: Customers specified requirement. Quality inspection records of past 3 months. Records from longer period of time will not give an insight of the recent process status. Experimentation on the process Production capability

Average Outgoing Quality Limit (AOQL) The Average Outgoing Quality Limit (AOQL) is the maximum average outgoing quality over all possible values of incoming quality for a given sampling plan. It is the worst defective rate for the average outgoing quality. Regardless of the incoming quality level, the defective rate going to the customer should be less than the AOQL over an extended period of time. Individual lots might be worst than the AOQL but over the long run, the quality should not be worse than the AOQL.

Average Outgoing Quality Limit (AOQL) Computation of AOQ: AOQ = p x Pa  Pa = probability of acceptance p = % defectives in the submitted lot (s). AOQL is the maximum limit of the AOQ curve

Sample size and Inspection level determination Determined through the lot size and respective inspection level I, II and III The different levels provided is to provide protection to the consumers. reference Table 10-1 and Figures 10-2 Inspection levels Inspection level II is the normal Level I provides about one-half the amount of inspection level II Level III provide about twice the amount of inspection level II

Inspection Level continue Inspection levels Level II inexpensive items destructive testing Level III complex or expensive items production costs are high

Application of ANSI/ASQ Z1.4 Prerequisite Information : AQL lot size inspection level Sampling Plans: For single , double and multiple sampling plans. Select the appropriate table for use with each of the sampling plans.

Assignment 1 Refer to ANSI/ASQZ 1.4 , for a lot size of 2000, an AQL of 0.65%, and an inspection level of III, determine the single sampling plans for normal, tightened, and reduced inspection. Note: MIL STD 105E may be used in lieu of ANSI/ASQ 1.4

Dodge-Romig Sampling Tables Developed by H. F Dodge and H. G. Romig in the 1920’s. Inspection tables for lot-by-lot acceptance of product by sampling for attributes. Sampling tables based on Limiting Quality (LQ) and Average Outgoing Quality Limit (AOQL) Applicable to Single and Double sampling only Principle Advantage Minimum amount of inspection for a given inspection procedure

Dodge-Romig Tables Example: Based on LQ N=1500, process average = 0.25%, LQ = 1.0% From Dodge Romg Table: sampling plan, n= 490, c=2, AOQL = 0.21%

Assignment 2 Design the sampling plan (based on LQ) you want to use on the manufacturing shop floor for a lot size N=1500, process average = 0.25%, LQ = 1.0%

Assignment 3 Assignment 4 Draw the effect of changing acceptance number on the sampling plans below using the Operating Characteristics (OC) Curves with same lot size . Discuss the effect of changing the acceptance number, as reflected in the OC curves you have constructed. How will this affect the sampling plan you want to design to use on the shop floor? Sample (n) = 100 , c =0 Sample (n) = 100, c= 1 Sample (n) = 100, c = 2 Assignment 4 Draw the OC curve for different sample sizes. Sample (n) = 100, c = 1 Sample (n) = 200, c= 2 Sample (n) = 300 , c = 4

Solution to Assignment 1

Some Additional Notes on AQL and AOQL Stands for acceptable quality level. Customers are only willing to accept a small amount of defective items and this small percentage of defective items that the customers are willing to accept is called AQL.

Some Additional Notes on AQL and AOQL Stands for average outgoing quality level. It is the maximum or worst possible average quality that is expected to delivered to the customer. If the quality of lot or batches delivered to customer is worse than AQL and AOQL. This will cause customer dissatisfaction. Therefore, it is important in acceptance sampling that the sampling plan chosen is accurate enough in discriminating good and bad lots. So that the quality of lots not over those limits.

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