Acceptance Sampling and Statistical Process Control

Probability Review Permutations (order matters) Number of permutations of n objects taken x at a time

Probability Review Permutations (cont.) Example: Number of permutations of 3 letters (A, B, C) taken 2 at a time (A,B,C)  AB, BA, AC, CA, BC, CB

Probability Review Combinations (order does not matter) Number of combinations of n objects taken x at a time

Probability Review (cont.) Combinations Example: Number of combinations of 3 letters (A, B, C) taken 2 at a time (A, B, C) = AB, AC, BC

Probability Review Binomial Distribution Sum of a series of independent, identically distributed Bernoulli random variables Probability of x defectives in n items p = probability of “success” (usually determined from long-term process average) 1-p = probability of “failure”

Probability Review Binomial Distribution (cont.) Example: What is the probability of 2 defectives in 4 items if p = 0.20?

Probability Review Binomial Distribution Exercises

Probability Review Hypergeometric Distribution Random sample of size n selected from N items, where D of the N items are defective Probability of finding r defectives in a sample of size n from a lot of size N

Probability Review Hypergeometric Distribution (cont.) Example: What is the probability of finding 0 defects in 10 items taken from a lot of size 100 containing 4 defects?

Probability Review Hypergeometric Distribution Exercises

Probability Review Binomial Approximation of Hypergeometric Use when n/N  0.10 P ~ D/N Example: What is the probability of finding 0 defects in 10 items selected from a lot of size 100 containing 40 defects? We got 0.652 from the straight hypergeometric.

Probability Review Poisson Distribution (to approximate Binomial) Use when n20 and p0.05 =np (average number of defects) Example: What is the probability of finding 2 defects in 4 items if p = 0.20? We got 0.1536 with the straight Binomial.

Acceptance Sampling Definition: process of accepting or rejecting a lot by inspecting a sample selected according to a predetermined sampling plan Notation: N = batch size n = sample size c = acceptance number p = proportion defective (known or long-term average) Pa = probability that a batch will be accepted

 and  Risks Type I Error: rejecting an acceptable lot (a.k.a. producer’s risk) P(Type I Error) =  Type II Error: accepting an unacceptable lot (a.k.a. consumer’s risk) P(Type II Error) = 

Operating Characteristic (OC) Curves OC Curves characterize acceptance sampling plans. OC Curves are complete plotting of Pa for a lot at all possible values of p.

OC Curves Steepness of OC curves indicates the power of the acceptance sampling plans to distinguish “good” lots from “bad” lots. Power = 1-P(accepting lot | p) = P(rejecting lot | p) For large values of p (i.e., large number of defects), we want the Power to be large (i.e., close to 1).

OC Curves Type A OC Curve Type B OC Curve Uses known value for p (i.e., lot composition is known) Can use hypergeometric distribution Type B OC Curve Uses process average for p Can use binomial distribution

Acceptance Sampling Plans Considerations: -risk -risk Acceptable Quality Level (AQL) Lot Tolerance Percent Defective (LTPD)

Acceptance Sampling Plans Acceptable Quality Level (AQL) Maximum percent defective that is acceptable  = P(rejecting lot | p = AQL) Corresponds to higher Pa (left-hand side of OC Curve) Lot Tolerance Percent Defective (LTPD) Worst quality that is acceptable (accepted with low probability)  = P(accepting lot | p = LTPD) Corresponds to lower Pa (right-hand side of OC Curve)

Acceptance Sampling Plans Single Sampling Take a single sample from the lot for inspection Quality of sampled work determines lot decision Double Sampling One small sample from the lot for inspection If quality of first sample is acceptable, that sample determines lot decision If quality of first sample is unacceptable or not clear, select a second small sample to make lot decision

Acceptance Sampling Plans Single-Sampling vs. Double-Sampling Plans Either plan can satisfy AQL and LTPD requirements Double sampling often results in smaller total sample sizes Double sampling can increase cost if second sample is required often Psychological advantage to double sampling (second chance)

Inspection Inspection involves verifying the quality of a work unit. Most inspection includes rectification of errors found. Rectification: 100% of defective work units are repaired or replaced Rejected lots are 100% verified and rectified Verifiers should have higher level of understanding to establish confidence in the inspection/rectification process.

Average Outgoing Quality Average Outgoing Quality (AOQ) = proportion defective after inspection and rectification AOQ curve relates outgoing quality to incoming quality Average Outgoing Quality Level (AOQL) = point where outgoing quality is worst Maximum AOQ over all possible values of p

Average Outgoing Quality Level (Table 1 provides values of y for c = 0,1,2,…,40) For small sampling fractions,

Average Outgoing Quality Level Determining sample size for desired outgoing quality (AOQL): Given desired AOQL Given desired acceptable number of defects, c Given y-value from Table 1 for desired c

Average Outgoing Quality Level Determining AOQL for known sample size and desired acceptable number of rejects, c: Given sample size, n Given desired c-value Given y-value from Table 1 for desired c-value Adjust c-value and n to manipulate the AOQL

Additional Acceptance Sampling Terminology Average Sample Number (ASN) Average number of sample units inspected to reach lot decision In single-sampling plan: ASN = n In double-sampling plan: n1  ASN  n1 + n2

Additional Acceptance Sampling Terminology Average Total Inspection (ATI): Average number of units inspected per lot In single-sampling plan: n  ATI  N ATI = n + (1-Pa)(N-n) Better incoming quality  less inspection

Additional Acceptance Sampling Terminology Average Fraction Inspected (AFI): Average fraction of units inspected per lot AFI = ATI/N In single-sampling plan: n/N  AFI  1

Acceptance Sampling Acceptance Sampling Plan Exercise

Other Sampling Plans Continuous Sampling Not possible to use lots/batches Level of inspection depends on perceived quality level Continuous Sampling Plan 1 (CSP1) Start at 100% inspection After i consecutive non-defectives, go to sample inspection Go back to 100% inspection when a defective is found

Other Sampling Plans Chain Sampling Skip-Lot Sampling Like standard sampling plans with c=0 However, allow 1 defect in a lot if previous i lots were defective-free Useful for small lots where c=0 is required Skip-Lot Sampling Lot-based continuous sampling Lots inspected 100% until i lots are defective-free, then go to sample inspection of lots

Acceptance Sampling Trade-Off Sampling Fraction vs. Acceptance Criteria Assuming a set quality level (AOQL), we can make choices regarding inspection rates and acceptance criteria To decrease inspection rate, we must tighten acceptance criteria To allow more defects, we must increase sample size

Acceptance Sampling Trade-Off Example: Desired AOQL = 5%, Batch Size 1-800 items 25% inspection  accept if 6 or fewer defects in a sample of size 60 10% inspection  accept if 5 or fewer defects in a sample of size 60 See Tables 1 and 2

Creating Work Units Homogeneity Batch Size Alike items within a batch Helps to keep samples representative Batch Size Rule-of-thumb: ½ to 1 day of work Very small batch  frequent QC, more paperwork Very large batch  delayed feedback, higher rework risk

Sample Selection Important for sample selection to be completely random Random Number Table Number batch items 1 through N Select random point in random number table Use next n numbers in the table as the batch items to select

Sample Selection Systematic Sampling Select a random start between 1 and 10 Select every xth item until n items are selected

U.S. Census Bureau Applications Address Canvassing Three random starts within listing pages One random start provides start for check of total listings Two random starts provide start for check of added housing units and deleted housing units No errors allowed QA form example from upcoming 2004 Census Test

U.S. Census Bureau Applications Review of Map Improvement Files Select map “features” for QC review Acceptance Sampling plan Global AQL requirement Sample size dependent on “batch” size Acceptance number dependent on sample size Uses stratified sampling

U.S. Census Bureau Applications Sample = 200 HIDs Acceptance Number = 14 0 to 10,000 HIDs AQL = 4% Matched Unmatched Added Road Water Other Sample = 315 HIDs Acceptance Number = 21 10,001 to 500,000 HIDs AQL = 4% Matched Unmatched Added Road Water Other

Process Control Allows us to plan quality into our processes Spend fewer resources on inspection and rework Observe processes, collect samples, measure quality, determine if the process produces acceptable results

Process Control Stresses prevention over inspection Traditional approach to QC: Most resources spent on inspection and rework Process Control approach to QC: Most resources spent on prevention with relatively little spent on inspection and rework Lower overall cost

Sources of Variation Random Sources Cause of variation is common or unassignable Process is still “in control” Difficult or impossible to eliminate Requires modification to the process itself

Sources of Variation Non-Random Sources Cause of variation is special and assignable Could be difficult to eliminate Causes process to be “out of control” Can address the specific cause of the variation

Sources of Variation Diagnosing non-random variation “2-out-of-3” – if two out of three consecutive points are out of control “4-out-of-5” – if four out of five consecutive points are out of control “7 successive” – if seven consecutive points are on one side of the process average

Control Charts Graphical device for assessing statistical control Plots data from a process in time order Three reference lines: Upper Limit Center Line (CL) Lower Limit

Control Charts CL represents the process average Upper and lower limits represent the region the process moves in under random variation

Control Charts Control chart limits can be Control Limits or Specification Limits Control Limits: based on quality capability of the process Control limits traditionally set at 3 standard deviations away from the CL Specification Limits: based on quality goal

Control Charts Example: Chemical process with assumed concentration of 3%, desired concentration  3.2% and  2.8% Specification Limits: LSL = 2.8, USL = 3.2 Control Limits: LCL = 2.5, UCL = 3.5 (based on true capability of process)

Control Charts A process is considered to be “in control” if the process data move randomly between the upper and lower limits

Control Charts

Control Charts Examples: Process: enumerator canvassing a block Measure: hours to canvass the block Is the process in statistical control?

Control Charts The process data in time order move randomly inside the control limits Therefore, the process IS in statistical control

Control Charts

Control Charts Some points above the UCL Therefore, process IS NOT in statistical control Might be hard-to-enumerate blocks (special cause) Field supervisor might want to send a more experienced lister to canvass those blocks

Control Charts

Control Charts Process has definite upward trend Therefore, process IS NOT in statistical control Perhaps lister has forgotten proper techniques (special cause) Field supervisor might want to consider re-training the lister

Control Charts

Control Charts Sometimes, patterns are hard to see This example seems to show a random distribution of points However, look what happens when we connect the points:

Control Charts

Control Charts The points are all inside the control limits So, the process IS in statistical control However, there is a definite cyclical pattern These types of patterns are not random and should be investigated

Control Charts Control Chart Exercises