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Modern Acceptance Sampling

A tutorial on how to develop sampling plans that detect and prevent the use of unacceptable materials and services.
By Stan Hilliard
1999 H & H Servicco Corp.
Outline
PAGE TUTORIAL CONTENTS
2 Abstract
2 Using Process Knowledge and Market Requirements
3 Applying OC Curves
    1. To Evaluate Sampling Plans

    2. To Compare Sampling Plans
    3. To Design/Develop Sampling Plans
    4. Guidelines for Application
    5. For Matching various types of sampling plan
           a.  variables & attributes
           b.  fixed-n, & sequential
           c.  standard deviation known & unknown
13 Relationship of Acceptance Sampling to Control Charts
Relationship of Two-Point Plans to Mil-Standard Plans
15 Conclusion: Review of Application Considerations
16 Appendix
    1. A Nomograph for Determining Variables Plans
    2. Sequential Acceptance Sampling Plans

    3. References
20 Software Programs
  Links to examples of sampling plans

 

In-Person Presentations of this Tutorial
Date Location Event
03/05/96 Minneapolis, MN 43rd Annual Minnesota Quality Conference (ASQC)
10/30/96 Madison, WI 13th Annual Maintenance & Engineering Exposition (ASQC,IMI,SME)

 

ABSTRACT

Modern acceptance sampling involves a system of principles and methods. Their purpose is to develop decision rules to accept or reject product based on sample data. Factors are:

  1. The quality requirements of the product in the marketplace
  2. The capability of the process
  3. The cost and logistics of sample taking

This article uses the Two-Point method of sampling plan design, whereby the person designing the plan specifies two points on the operating characteristic curve. The following important visualization tools of this method will be explained:

  1. Operating characteristic curves (OC-Curves)
  2. Decision rules

It will show you how you can benefit by matching different types of sampling plan to the same oc-curve: attributes to variables, fixed-n to sequential, unknown to known .

USING MARKET REQUIREMENTS and PROCESS KNOWLEDGE

You can avoid the horrors of the product recall experience by applying the following principles. They involve methods of product acceptance and process control.

Rational Lots
Define homogeneous lots of product items. Use your knowledge of the patterns of the process to group the items into "natural lots". Lots based on purchase order alone must be subdivided into such natural lots for accurate acceptance decisions.
Known Standard Deviation
Apply known plans for variables where feasible. These plans use knowledge of the producer's process to reduce inspection cost. [ (sigma) = standard deviation].
Process Performance and Capability
Set an acceptable quality level (AQL) that "recognizes" the current process capability. This requires that you know the attainable levels of the process.
Market Requirement
Set a rejectable quality level (RQL) that enforces the customer requirement.
At rejection, you will know that:
  1. The lot is unacceptable, and
  2. The process has deviated from its known capability
    (Therefore start corrective action on the process.)

 

APPLYING OC-CURVE to EVALUATE SAMPLING PLANS


Figure 1.
OC-Curve. (Operating Characteristic)

The Operating Characteristic (OC) curve shows the probability of acceptance, Pa, for any level of lot quality. See Figure 1. On the horizontal axis is the quality characteristic.

This oc-curve enables you to evaluate the probability of acceptance for any true lot quality level-on a what-if basis. This way, you can design sampling plans that perform the way you want.

Interpret the curve according to this example:
  1. If the lot quality is 0.093 fraction defective, then the probability of acceptance, Pa, is 0.05.
  2. If the lot quality is 0.018 fraction defective, then the probability of acceptance, Pa, is 0.95.

APPLYING OC-CURVES to COMPARE SAMPLING PLANS


Figure 2.
Comparing Alternative Plans, A and B

You can use oc-curves to compare alternative plans. See Figure 2. Choose between the plans by their relative ability to detect rejectable lots. You should expect that the steeper the curve, the larger the sample size.

Complete this picture by comparing the costs of the sampling to the resulting performance.

 

WORKSHEET #1:
Develop an OC-Curve

Use program TP414 or the Variables Nomograph

Decision Rule of Current Known Sampling Plan

Sample Size: n = 10
Decision Limit: K =1.80

Your Task

Complete the oc-curve in the table below.
Use the program TP414 or the Variables Nomograph in the Appendix

Fraction Nonconforming p' Probability of Acceptance Pa
  0.95
  0.70
  0.50
  0.30
  0.05

Useful definitions: K = (Xbar-LISL) / sigma
                    
and: K = 3 * Cpk

Note: If you use the nomograph, use vertical n-lines for known sigma plans. (See scale at bottom of nomograph.)

A NOMOGRAPH FOR DETERMINING VARIABLES SAMPLING PLANS
See: "Nomograph for Determining Variables Sampling Plans", Lloyd S. Nelson,Journal of Quality Technology, Vol. 13, No. 4, October 1981. (Consult the references below)

APPLYING OC-CURVES to DESIGN -- THE TWO-POINT METHOD

The Two-Point method for developing acceptance sampling plans requires that you specify two points of the operating characteristic curve (oc-curve).

Producer's Point


Figure 3.
The Two-Point Method

The producer's point controls the acceptance of lots that are at an acceptable quality level. (See figure 3) The goal: prevent good lots from being rejected.

Consumer's Point

The consumer's point controls the rejection of lots that are at a rejectable quality level. (See figure 3) The goal: prevent bad lots from being accepted.

Decision Table Defines the Two Points


Figure 4. Decision Table

An OC Curve helps you to address the producer's and consumer's points in a visual, right brain way. On the other hand, the decision table (See figure 4) defines the situation in a logical, left brain way. The Type I and Type II errors correspond logically to the two points on the oc-curve.

Type I error -- Wrongful Rejection

A type I error is associated with the producer's point -- to reject when the true value of the quality characteristic is AQL. The risk of rejecting an AQL lot is the producer's risk (alpha risk)

Type II error -- Wrongful acceptance

A Type II error is to accept when the true value of the quality characteristic is RQL -- at the consumer's point. The risk of accepting a lot, if it is an RQL lot, is the consumer's risk ( = beta risk).

Summary of The Two-Point Method:

Choose alpha=0.05, beta=0.05, AQL, and RQL. These determine the sample size (n) and the decision limit(s). Then apply any process knowledge (within-lot standard deviation) and market requirements (ISLs).

WORKSHEET #2:
Develop a Fixed-n Sampling Plan

(Use of the program TP414 or the Variables Nomograph )

Determine the variables fixed-n sampling plan.

Product Requirement: Lower ISL=0
Process Capability: = 1 (known historical within-lot standard deviation)

Sampling Requirement

Producers Point: AQL=0.01, alpha=0.05
Consumers Point: RQL=0.10, beta=0.05

Your Task: use the program TP414 or the Variables Nomograph to determine:

Sample Size: n =_____
Decision Limit: K =_____

Useful definitions: K = (Xbar-LISL) / sigma
                    
and: K = 3 * Cpk

Note: If you use the nomograph, use vertical n-lines for known sigma plans. (See scale at bottom of nomograph.)

SOME GUIDELINES FOR APPLICATION

No principle is without exceptions, but the following guidelines are generally useful to get you started in a sampling application.

How to choose the Producer's Point in practice

  1. You should expect that lots at the producer's point quality level (AQL) will be accepted most of the time. Define AQL accordingly. Take into account historical quality levels.
  2. Choose the producers risk of rejecting a lot that is of AQL quality. Typical: = 0.05.

How to choose the Consumer's Point in practice

  1. You should expect that lots at the consumer's point quality level (RQL) will be rejected most of the time. Define RQL accordingly.
  2. Choose the consumers risk of accepting a lot that is of RQL quality. Typical: = 0.05.

Classify Quality Characteristics
You do not have to inspect all quality characteristics to the same sampling requirement. Some characteristics need less inspection than others. Some will not require testing at all, and others might require 100% inspection.

Resources for Two-Point Calculation
The Appendix contains references and nomographs for developing two-point variables and attribute sampling plans. The bibliography below refers to the Two-Point software programs TP105, TP414, and TP781 that facilitate these calculations.

The Advantages of Two-Point Decision Rules
The Two-Point method develops plans based on desired performance of the decision rule. Unlike Mil-Standard AQL plans, their method of selection does not involve esoteric 'inspection levels' and 'code letters'. The underlying statistics is the same.

A Philosophical Issues with Mil-Standard AQL Plans
The Mil-Standard AQL plans require you only to choose the producers point-not the consumer's. Actually, the consumer's point is usually more important. As a producer your main purpose is to prevent acceptance of rejectable or recallable quality lots.

A Technical Issue with Lot Size
Many sampling plan techniques do not use lot size to calculate the oc-curve. This includes Mil-Standard 105 and 414, and the software programs TP105 and TP414.. This practice has negligible effect when sampling large lots with small samples. A rule of thumb for accuracy of the oc-curve calculation is: N>10*n. (n=sample size, N=lot size)

MATCHING the OC CURVES of DIFFERENT TYPES OF PLAN

An important ability is to match sampling plans by their oc-curves. Two matched plans have the same operating characteristic curve, but different decision rules. You can safely choose between matched plans for economy, knowing they offer equal protection. The following table shows useful matches.

Plan A Plan B
Attribute Variables
Fixed-n Double, Multiple, Sequential
Variables - Known standard deviation Variables - Unknown standard deviation

The following series of sampling plan examples shows how various types of plan are matched to the same oc-curve. The oc-curve of this example is characterized by the two points: (AQL=0.01, =0.05) and (RQL=0.10, =0.05). For convenience we will refer to the curve as "OC-Curve X".

Attribute Fixed-n Plan
(Matched to OC-Curve X)

Decision rule produced by: TP105 for Attributes
(AQL=0.01, =0.05) and (RQL=0.10, =0.05)

n = 43 (sample size)
C = 1 (acceptance number)

Variables Fixed-n Plan, Unknown Sigma
(Matched to OC-Curve X)

Decision rule produced by: TP414 for Variables
(AQL=0.01, =0.05), (RQL=0.10, =0.05)

n = 27 (sample size)
K=1.80

Variables Fixed-n Plan, Known Sigma = 2.7
(Matched to OC-Curve X)

Decision rule produced by: TP414 for Variables
(AQL=0.01, =0.05) and (RQL=0.10, =0.05)

n = 10
A= 4.94

Attribute Sequential Sampling Plan
(Matched to OC-Curve X)

Sequential Probability Ratio method (SPR)

Decision rule produced by: TP105 for Attributes
(AQL=0.01, =0.05) and (RQL=0.10, =0.05)

Sequential Attributes
n Decision
From /To (Ac) (Re)

1

1

*

**

2

19

*

2

20

30

*

3

31

51

0

3

52

52

1

3

53

53

2

3

Variables Sequential Sampling Plan (SPR)
(Matched to OC-Curve X)

Known Sigma=2.7
SPR = Sequential Probability Ratio.

Decision rule produced by: TP414 for Variables
(AQL=0.01, =0.05) and (RQL=0.10, =0.05)

SPR Sequential
(n) (Ac) (Re)
1 ### 12.55
2 1.13 8.74
3 2.40 7.48
4 3.04 6.84
5 3.42 6.46
6 3.67 6.21
7 3.85 6.03
8 3.99 5.89
9 4.09 5.78
10 4.18 5.70
11 4.25 5.63
12 4.31 5.57
13 4.36 5.52

Variables Sequential Sampling Plan (TSS)
(Matched to OC-Curve X)

Known =2.7
TSS = Truncatable Single Sample

Decision rule produced by: TP414 for variables
(AQL=0.01, =0.05) and (RQL=0.10, =0.05)

TSS Sequential
(n) (Ac) (Re)
1 ### 11.87
2 0.38 9.50
3 1.52 8.36
4 2.25 7.63
5 2.79 7.09
6 3.24 6.64
7 3.62 6.26
8 3.98 5.89
9 4.35 5.53
10 4.94 4.94

Variables Sequential Sampling Plan, unknown Sigma
(Matched to OC-Curve X)

TSS = Truncatable Single Sample

Decision rule produced by: TP414 for Variables
(AQL=0.01, =0.05) and (RQL=0.10, =0.05)

 

Variables
TSS Sequential
Unknown Sigma
(n) K (Re) K (Ac)
2    
3 -3.83  
4 -1.03  
5 -0.20  
6 0.21  
7 0.46  
8 0.63 8.94
9 0.76 6.86
10 0.87 5.66
11 0.96 4.88
12 1.03 4.33
13 1.10 3.92
14 1.16 3.61
15 1.21 3.36
16 1.26 3.15
17 1.30 2.98
18 1.35 2.83
19 1.39 2.69
20 1.43 2.58
21 1.47 2.47
22 1.51 2.37
23 1.55 2.27
24 1.59 2.18
25 1.64 2.08
26 1.70 1.96
27 1.80 1.80

Translated Sampling Plans

Translated sampling plans have identical decision rules, but differently defined quality characteristics.

Plan A Plan B
Variables: ISL & Percent Nonconforming Variables: Mean
Exponential: MTBF Exponential: Reliable Life, & Reliability

RELATIONSHIP of SAMPLING to CONTROL CHARTS

Shewhart Control Charts are not Acceptance Plans

Shewhart Xbar and R charts analyze processes that involve a series of lots produced over time. They concern the relationship of the subgroups to each other, and not to any externally imposed specification. Use Xbar and R charts to discover the factors that contribute to process variability.

Shewhart charts cannot ensure against accepting poor or recallable lots. Even an in-control characteristic can have a substantial fraction of non-conformities. Xbar and R charts do not control the consumers risk () of accepting RQL or recallable lots.

Acceptance Control Charts.

Acceptance control charts are acceptance sampling plans that you convert into chart form for implementation. They control the producers point and the consumers point of the oc-curve. Acceptance charts provide a valid visible means for making acceptance sampling decisions.

Acceptance Sampling for Acceptance Decisions

The best way to make the accept/reject decision - whether a process is out-of-control or
in-control-is to use both the producer's risk and the consumer's risk. In other words, honor both the process capability and the product specifications.

RELATIONSHIP of TWO-POINT PLANS to MIL-STANDARD PLANS

You can match various types of sampling plan to Mil-Standard plans. For example, you can match a variables TSS sequential with known to a Mil-Standard 105E attribute plan. Match them at, say, the Pa=0.05 and Pa=0.95 points.

In instances like this, the two-point method provides a means to convert to more efficient and/or less costly sampling plans.

Regulatory, Consumer, and Litigation Issues

From the consumer, regulatory, and litigation standpoints, it may be better to know the two probabilities of a decision rule than simply to be able to say that you meet a published standard. Auditors may require that you know the probabilities of the oc-curve and that the calculations be valid. They do not specify what those probability levels must be.

The Equivalence of Various Types of Sampling plans

The two-point system for developing sampling plans differs from the "cook book" approach of the Military Standards. But the difference is in how you arrived at a plan, -- the statistical calculations are the same. The table below compares sampling plan development methods.

Equivalent Statistical Calculations for Acceptance Sampling Plans
Statistical Calculation Development Method Type of Decision Rule
Attribute, Binomial and Poisson Mil-Std-105E
ASQC/ANSI Z1.4
ISO-2859-1
ISO-8422
TP105 Software
Fixed-n, double, multiple, switching
Fixed-n, switching
Fixed-n, double, multiple, switching
Sequential
Fixed-n, double, multiple, sequential
Variables, Normal, Known , Unknown Mil-Std-414
ASQC/ANSI Z1.9
ISO-3951
ISO-8423
TP414 software
Fixed-n, switching
Fixed-n, switching
Fixed-n, switching
Sequential
Fixed-n, double, multiple, sequential
MTBF Exponential Mil-Std-781
TP781 software
Fixed-n, Sequential
Fixed-n, double, multiple, sequential

 

 

CONCLUSION: REVIEW of APPLICATION CONSIDERATIONS

Know the Performance of the Decision Rule

You should use the two-point method to focus on the performance of the accept/reject decision.

How does the decision rule effect the customer if the lot quality is poor?
(Know the consumer's risk .)

How does the decision rule effect the producer if the lot quality is good?
(Know the producer's risk.)

Consider Variables Sampling Plans

Consider variables sampling plans based on ISLs. They require smaller n than attribute plans.

Consider Sequential Decision Rules

Sequential sampling is the most statistically efficient type of acceptance sampling plan.

Use Known Plans

Consider the advantages of known versus unknown plans. If you choose known , use a test for variability to check that has not changed.

Choose Between Matched Plans to Minimize Sample Size and Inspection Cost

The following strategies will reduce sample size without increasing the and risks:

  1. Convert attribute plans to variables plans.
  2. Convert fixed-n plans to sequential plans.
  3. Convert unknown to known plans.
  4. Set the Target for the process mean to a region of lower average sample numbers.
bulletFor one-sided sequential sampling plans for either a lower or upper limit.
bulletFor plans with both lower and upper limits.

Nomograph
See the document "Modern Acceptance Sampling" in the references for a nomograph for designing variables sampling plans.

 

APPENDICES

Sequential Acceptance Sampling Plans

Sequential acceptance sampling plans are the most statistically efficient type of sampling plan. They are used when you do not want to take any more sample items than is necessary, but want to be confident that you have enough data to make a decision. For each decision, they allow the operator to increase the sample size one item at a time, or to form it into groups to match the logistics of the situation.

Mil-Std-105 for attributes contains double and multiple plans that are related to sequential plans. Mil-Std 414 for variables does not have a comparable scheme. Software programs TP105, TP414, and TP781 develop sequential and fixed-n plans for attributes, variables, and reliability.

Two Kinds of Sequential Plan for Variables: SPR and TSS

SPR Sequential Method
The SPR "sequential probability ratio" methodology was developed by the Statistical Research Group at Columbia University under the leadership of Abraham Wald. It was originally a Restricted war secret but was declassified in 1945 after world war II. See the bibliography: Abraham Wald, Thomas McWilliams.
TSS Sequential Method
The TSS "truncatable single sample" methodology was developed by Stan Hilliard (this author) during the 1970s and 1980s. TSS plans are a computerized numerical method that is implemented through software program TP414. See the bibliography for TP414.
The sample size of a TSS plan can not exceed the sample size of the fixed-n plan whose operating characteristic it matches.
The oc-curves of TSS plans match fixed-n oc-curves more closely than SPR.

Comparison of SPR and TSS Sequential Methods

We generally favor TSS plans over SPR plans, though both offer sound statistical risk protection. Simulations have demonstrated this. Our main reason for favoring the TSS method is its sample size limit of fixed-n. This property of TSS plans improves their acceptance among operators and inspectors, especially in manufacturing. The software program TP414 implements both TSS and SPR sequential plans.

 

References

Acceptance Sampling in Quality Control
Edward Schilling
Marcel Dekker, Inc.
Quality Control and Industrial Statistics
Acheson J. Duncan
Richard D. Irwin, Inc.
Volume 13, How to Use Sequential Statistical Methods
Thomas P. McWilliams
American Society for Quality Control
Sequential Analysis
Abraham Wald
New York: John Wiley & Sons
Nomograph for Determining Variables Sampling Plans
Lloyd S. Nelson
Journal of Quality Technology, Vol13, No. 4, October 1981
A Nomograph of the Cumulative Binomial Distribution
Harry R. Larson
Industrial Quality Control, Dec. 1966
Are Acceptance Sampling and SPC Complementary or Incompatible?
Sower, Motwani, and Savoie.
Quality Progress, September 1993
Product Acceptance Design Journey
Stan Hilliard, H&H Servicco Corp.
PO Box 9340
North St. Paul, MN 55109-0340
(612) 777-0152 Voice & Fax
Modern Acceptance Sampling
Stan Hilliard, H&H Servicco Corp.
PO Box 9340
North St. Paul, MN 55109-0340
(612) 777-0152 Voice & Fax
Software Programs TP105, TP414, TP781
H&H Servicco Corp.
PO Box 9340
North St. Paul, MN 55109-0340
(612) 777-0152 Voice & Fax

Software Programs

H & H Servicco Corp.
PO Box 9340
North St. Paul MN 55109-0340
Phone: (651) 777-0152 (Voice & Fax)
Email: service@samplingplans.com
http://www.samplingplans.com

H & H Servicco Corp. provides PC software that designs product acceptance sampling plans, process control plans, reliability test plans, and audit sample plans. These programs run on IBM PC or compatible computers.
The programs are available directly from H & H Servicco Corp.

TP105 - Two-Point Sampling plans for Attributes

Price: $245 (60 day satisfaction guarantee)

Software program TP105 designs and evaluates attribute sampling plans. Both fixed-n and sequential sampling plans meet user-specified consumer and producer risks. You input Alpha, Beta, AQL, and RQL, or alternatively, n and Ac. Users evaluate plan performance with OC, ASN, AOQ and ARL curves; evaluate sample data with confidence limits.

TP414 - Two-Point Sampling plans for Variables

Price: $245 (60 day satisfaction guarantee)

Software program TP414 designs and evaluates variables sampling plans. Users control either the lot/process mean or the percent non-conforming to specification(s). You input Alpha, Beta, AQL, and RQL, or alternatively, n and decision limit(s). Includes various types of plan: sequential, fixed-n, sigma-known, and sigma-unknown. Users evaluate performance with OC, ASN, AOQ, and ARL curves.

TP781 - Two-Point Sampling plans for Reliability

Price: $245 (60 day satisfaction guarantee)

Software program TP781 designs test plans for reliability qualification. Users specify acceptable and rejectable levels of either: (1) mean time between failures(MTBF) or, (2) reliability in reaching reliable life. TP781 calculates time/unit or number of units for fixed-n and sequential sampling plans. It uses the exponential distribution for OC, ASN, and ARL curves and confidence limits.

ASP - Audit Sample Planner

Price: $245 (60 day satisfaction guarantee)

Software program ASP helps you to plan audits and surveys for attribute (pass/fail) data:

  1. Planning: You develop a sample size depending on: 1) the margin of error, 2) the expected fraction of items "in error", and 3) the expected reply rate.
  2. Selection: ASP produces a random sample printed on a data sheet.
  3. Analysis: A report shows one-sided or two-sided confidence limits for the fraction and the total "in error" items in a population.

Links to Examples

These links will take you to examples of sampling plans:

bulletExample attribute sampling plans:
     defectives using the binomial distribution
     defects using the poisson distribution
bulletExample variables sampling plans for the mean:
     Sampling Bulk Liquids, Powders, Pellets
     Numerical Color Measurement
bulletExamples variables sampling plans for fraction nonconforming to specification
     Fixed-n and sequential variables examples
     Relationship of software TP414 to Mil-Std-414 and Z1.9
bulletExample reliability sampling plans for MTBF and for reliability at reliable life
     Strategy for reliability growth testing
     Wearout and sample size for MTBF test plans

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