ACCEPTANCE SAMPLING FOR ATTRIBUTES Types Of Sampling Plans Mistakes To Avoid, And Their Statistical Equivalents AQL & LTPD Single-sampling Plans  Average Outgoing Quality ISO 2859 & Dodge-Romig Plans OPS 465 - Qual Mgmt

IS THIS SHIPMENT ANY GOOD? Can you trust your vendor's quality? If so, great! If not, inspect each shipment that arrives Sorting out good from bad shipments OPS 465 - Qual Mgmt

OPTIONS FOR VENDOR QUALITY Objective: ensure vendor delivers quality supplies Two ways to reach objective Inspect vendor's shipments “Acceptance" sampling: Have vendor performing QM Vendor supplies customer with relevant control charts Customer certifies vendor in QM The first approach is traditional, but the second is preferable in general and necessary for lean operations OPS 465 - Qual Mgmt

"ACCEPTANCE" SAMPLING Basic idea: Inspect a random sample of each lot Classify each item as ok/not ok Conclude entire lot is either Ok -- accept it Not ok -- reject it and/or sort it OPS 465 - Qual Mgmt

ATTRIBUTE VS. VARIABLE SAMPLING PLANS Simplest sampling plans are attribute Based on binomial (or hypergeometric) distribution May lose variable data on several QC'S Requires large sample size Plans based on variable data Based on normal distribution Provide more information on source of quality problems Require smaller samples for same a, b OPS 465 - Qual Mgmt

SINGLE, DOUBLE, & MULTIPLE SAMPLING PLANS Single sampling plans: Make accept/reject decision based on one sample Double sampling plans: Make accept/reject/take-another- sample decision based on first sample Make accept/reject decision based on second sample (if taken) Can have "triple", "quadruple", or any other multiple sampling plan Multiple sampling plans require more but smaller samples for same a, b OPS 465 - Qual Mgmt

SINGLE (ATTRIBUTE-BASED) SAMPLING PLANS Define: N -- number of parts in shipment n -- number of parts in a sample from shipment c -- acceptance number Acceptance sampling has 3 easy steps: For each shipment of N parts, a sample of size n is taken Inspect each of the n parts Reject the shipment if the number of defects exceeds c units. Otherwise, accept the shipment OPS 465 - Qual Mgmt

MISTAKES TO AVOID, & THEIR STATISTICAL EQUIVALENTS As with product quality control, there are two types of mistakes to avoid: Type I -- Conclude the shipment is bad when in fact it is good (false alarm) Type II -- Conclude the shipment is good when in fact it is bad (overlooked problem) Probability of each type of mistake: Type I -- a Type II -- b This is standard hypothesis testing with the following null hypothesis: H0: the shipment is good OPS 465 - Qual Mgmt

DOUBLE SAMPLING PLANS Define: Take sample of size n1 n1 -- sample size on first sample c1 -- acceptance number for first sample d1 -- defectives in first sample n2 -- sample size on second sample c2 -- acceptance number for both samples d2 -- defectives in second sample Take sample of size n1 Accept if d1 £ c1; reject if d1 > c2; Take second sample of size n2 if c1 < d1 £ c2 Accept if d1+d2 £ c2; reject if d1+d2 > c2 OPS 465 - Qual Mgmt

DEFINING GOOD AND BAD SHIPMENTS: AQL VERSUS LTPD Instead of simply "good" versus "bad", we will define "really good", "really bad", and "ok, but not great" shipments p -- True (unknown) percent defective in shipment AQL -- Acceptable quality level LTPD -- lot tolerance percent defective Then: A really good shipment has p <= AQL A really bad shipment has p >= LTPD Anything in between (AQL < p < LTPD) is ok, but not great OPS 465 - Qual Mgmt

THE OPERATING-CHARACTERISTIC (OC) CURVE For a given a sampling plan and a specified true fraction defective p, we can calculate Pa -- Probability of accepting lot If lot is truly good, 1 - Pa = a If lot is truly bad, Pa = b A plot of Pa as a function of p is called the OC curve for a given sampling plan OPS 465 - Qual Mgmt

THE OPERATING-CHARACTERISTIC (OC) CURVE The ideal sampling plan discriminates perfectly between good and bad shipments Both a and b are zero in this example! This requires a sample size equal to the population -- not feasible OPS 465 - Qual Mgmt

CONSTRUCTING AN (OC) CURVE For a specified single sampling plan, the OC curve may be constructed using a binomial distribution if n is small relative to the lot size p -- true fraction nonconforming n -- sample size c -- acceptance number We know that Excel OPS 465 - Qual Mgmt

CONSTRUCTING AN (OC) CURVE Suppose we have a sampling plan defined by the following parameters: n = 100 c = 2 What is the probability of accepting a lot with 0.5% defectives? OPS 465 - Qual Mgmt

CONSTRUCTING AN (OC) CURVE OPS 465 - Qual Mgmt

USING AN (OC) CURVE How do we find a and b using an OC curve? AQL = 0.01 LTPD = 0.05 Then a = 1 – Pa(p=0.01) = 1 - 0.9206 = 0.0794 And b = Pa(p=0.05) = 0.1183 OPS 465 - Qual Mgmt

AVERAGE OUTGOING QUALITY Consider a part with a long-term fraction nonconforming of p Samples of size n are taken from a lot of size N and inspected Any defectives in the sample of size n are replaced, accept or reject When a lot of is accepted, we expect p(N-n) defectives in the remainder of the lot When a lot is rejected, it will be sorted and defective units replaced, leaving N-n good units in the remainder This is referred to as "rectifying" inspection OPS 465 - Qual Mgmt

AVERAGE OUTGOING QUALITY If Pa is the probability of accepting a lot, then the average outgoing quality is: The worst possible AOQ is the AOQ Limit or AOQL Excel OPS 465 - Qual Mgmt

AVERAGE TOTAL INSPECTION Rectifying plans have greater inspection requirements The Average Total Inspections: OPS 465 - Qual Mgmt

ISO 2859 (ANSI/ASQC Z1.4) One of oldest sampling systems Covers single, double, & multiple sampling AQL-based: Type I error ranges 9%-1% as sample size increases Minimal control over Type II error Type II error decreases as general inspection level (I, II, III) increases “Special” inspection levels when small samples needed (and high Type II error probability tolerated) Mechanism for reduced or tightened inspection depending on recent vendor performance Tightened -- more inspection Reduced -- less inspection OPS 465 - Qual Mgmt

ISO 2859 (ANSI/ASQC Z1.4) A vendor begins at a "normal" inspection level Normal to tightened: 2/5 lots rejected Normal to reduced: Previous 10 lots accepted (NOT ISO 2859) Total defectives from 10 lots ok (NOT ISO 2859) If a vendor is at a tightened level: Tightened to normal: 5 previous lots accepted If a vendor is at a reduced level: Reduced to normal: a lot is rejected OPS 465 - Qual Mgmt

ISO 2859 A vendor begins at a "normal" inspection level Normal to reduced: “Switching score” set to zero If acceptance number is 0 or 1: Add 3 to the score if the lot would still have been accepted with an AQL one step tighter; else reset score to 0 If acceptance number is 2 or more: Add 3 to the score if the lot is accepted; else reset score to 0 If score hits 30, switch to reduced inspection OPS 465 - Qual Mgmt

USING ISO 2859 Choose the AQL Choose the general inspection level Determine lot size Find sample size code Choose type of sampling plan Select appropriate plan from table Switch to reduced/tightened inspection as required OPS 465 - Qual Mgmt

USING ISO 2859 OPS 465 - Qual Mgmt

USING ISO 2859 OPS 465 - Qual Mgmt

USING ISO 2859 OPS 465 - Qual Mgmt

USING ISO 2859 OPS 465 - Qual Mgmt

DODGE-ROMIG PLANS Developed in the 1920's Rectifying plans Requires knowledge of vendor's long-term process average (fraction non-conforming) Choice of LTPD or AOQL orientation Both minimize ATI for specified process average Type II error = 10%, OPS 465 - Qual Mgmt

DODGE-ROMIG PLANS AOQL plans: LTPD plans: 1) Determine N, p, and AOQL 2) Use table to find n and c Finds plan with specified AOQL which minimizes ATI Calculate resulting LTPD with Type II error = 10% LTPD plans: 1) Determine N, p, and LTPD Finds plan with specified LTPD which minimizes ATI Calculate resulting AOQL OPS 465 - Qual Mgmt

DODGE-ROMIG PLANS OPS 465 - Qual Mgmt

DODGE-ROMIG PLANS OPS 465 - Qual Mgmt