STUDIES ON OPTIMAL DESIGNING OF CERTAIN

SPECIAL PURPOSE SAMPLING PLANS FOR

VARIABLES INSPECTION

A THESIS

Submitted by

M. USHA in partial fulfillment for the award of the degree

of

DOCTOR OF PHILOSOPHY

DEPARTMENT OF MATHEMATICS

KALASALINGAM UNIVERSITY

(Kalasalingam Academy of Research and Education)

ANAND NAGAR

KRISHNANKOIL 626 126

MAY 2015

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ABSTRACT

Acceptance sampling is one of the major areas of statistical

quality control (also familiarly known as Industrial Statistics). Acceptance

sampling is the methodology that deals with procedures by which decision to

accept or reject the lot based on the results of the inspection of samples.

Acceptance sampling prescribes a procedure that, if applied to a series of lots, will

give a specified risk of accepting lots of given quality. In other words, acceptance

sampling yields quality assurance. Implementation of acceptance sampling in

industries through the operation of sampling plan yields quality assurance. Use of

acceptance sampling is essential to secure ISO certification which gives a passport

for larger exports.

In general, the acceptance sampling plans are classified in to

attribute sampling plans and variables sampling plans. Many quality characteristics

cannot be conveniently represented numerically. In such cases, we usually classify

each item inspected as either conforming (non-defective) to the specification on

that quality characteristics or non-conforming (defective) to those specifications.

Quality characteristics of this type are called attributes. Sampling plans applied to

such quality characteristics are called attributes sampling plans. Several sampling

plans are available in the literature for the application of attributes quality

characteristics. For example, single sampling plan, double sampling plan, multiple

sampling plan etc.

Variables sampling plans specify the number of items to be

sampled and the criterion for sentencing lots when measurements data are

collected on the quality characteristic of interest. These plans are generally based

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on the sample average and sample standard deviations of the quality characteristic.

When the distribution of the quality characteristic in the lot or process is known,

variables sampling plans that have specified risks of accepting and rejecting lots of

given quality may be designed. The primary advantage of the variables sampling

plan is that the same operating characteristic (OC) curve can be obtained with a

smaller sample size than would be required by an attributes sampling plans. Thus a

variables acceptance sampling plan would require less sampling. The

measurements data required by a variables sampling plan would probably cost

more per observation than the collection of attributes data. However, the reduction

in sample size obtained may more than offset this increased cost. When

destructive testing is employed, variables sampling is particularly useful in

reducing the costs of inspection. Another advantage is that measurements data

usually provide more information about the manufacturing process or lot than do

attributes data. Generally, numerical measurements of the quality characteristics

are more useful than simple classification of the item as conforming or non-

conforming. A final point to be emphasized is that when acceptable quality levels

are very small, the sample size required by attributes sampling plans are very

large. Under these circumstances, there may be significant advantages in switching

to variables measurements. Thus as many manufacturers begin to emphasize

allowable numbers of non-conforming parts per million (ppm), variables sampling

plan becomes very attractive.

The special purpose sampling plans such as Chain Sampling Plan

(ChSP) of Dodge (1955), Quick Switching System (QSS) of Romboski (1969),

Tightened-Normal-Tightened (TNT) sampling scheme developed by Calvin (1977)

etc., so far developed only for application of attributes quality characteristics. But

for the inspection of measurable characteristics, no such special purpose plan is

available in the literature except chain sampling plan which was developed by

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Govindaraju and Balamurali (2000). So, this thesis is devoted to the optimal

designing of certain special purpose acceptance sampling plans for variables

inspection. These new plans will be very effective, efficient and attractive in terms

of reducing the cost of inspection. These plans will be particularly applied for

costly and destructive testing.

Chapter 1 of this thesis comprises of sections that consists of basic concepts

of quality control, acceptance sampling, reliability, lifetime distributions and the

review of sampling plans which are relevant to this thesis. In Chapter 2, the

optimal designing of variables quick switching system is proposed in which the

quality characteristic under study follows a normal distribution with known and

unknown standard deviations. The minimum sample size n is determined for the

predefined acceptance criteria Nk and Tk and are used to calculate the

probability of acceptance for different combinations of the consumer’s confidence

levels and the producer’s confidence levels. The results are presented in tables and

explained with figures and examples.

Chapter 3 of the thesis investigates the variables quick switching sampling

system when a measurable quality characteristic has double specification limits

beyond which an item is considered to be a non-conforming. The quality

characteristic of interest is assumed to follow the normal distribution. The optimal

parameters of the variables quick switching system are determined for both known

and unknown standard deviations which satisfy the producer’s and consumer’s

risks at the corresponding specified quality levels. Symmetric and asymmetric

cases based on the fraction non-conforming by the lower and the upper

specification limits are also considered. The problem is formulated as a nonlinear

programming where the objective function to be minimized is the average sample

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number and the constraints are related to lot acceptance probabilities at acceptable

quality level and limiting quality level under the operating characteristic curve.

The Chapter 4 of the thesis deals with the optimal designing of

Tightened-Normal-Tightened sampling scheme with sample sizes 1n , 2n and the

acceptance criteria k . The advantages of the proposed variables scheme over

variables single, double sampling plans and attributes sampling scheme are

discussed. Tables are also constructed for the selection and application of

parameters of known and unknown standard deviation variables sampling schemes

for specified two points on the operating characteristic curve. The problem is

formulated as a nonlinear programming with minimizing the average sample

number as the objective function and the constraints are related to lot acceptance

probabilities at acceptable quality level and limiting quality level based on the

operating characteristic curve.

In Chapter 5 of the thesis, we investigate the optimal designing of chain

sampling plan for the application of normally distributed quality characteristics.

The advantages of this proposed variables chain sampling plan over variables

single sampling plan and variables double sampling plan are discussed. Tables are

also constructed for the selection of optimal parameters of known and unknown

standard deviation variables chain sampling plan for specified two points on the

operating characteristic curve, namely, the acceptable quality level and the limiting

quality level, along with the producer’s and consumer’s risks. A non-linear

optimization problem is formulated in which the average sample number is

minimized subject to the constraints of satisfying the producer and consumer risks

at their respective quality levels.

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In Chapter 6, an optimal designing of variables sampling plan which can

be applied for sampling inspection of resubmitted lots when the quality

characteristic of interest follows the normal distribution is proposed. The

advantages of this proposed variables sampling plan over the existing single

sampling variables plan are discussed. Tables are also constructed for the selection

of optimal parameters of known and unknown standard deviation variables

resampling scheme for specified two points on the operating characteristic curve

namely the acceptable quality level and the limiting quality level along with the

producer and consumer’s risks. The optimization problem is formulated as a

nonlinear programming for finding the optimal parameters satisfying both

producer and consumer risks.

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ACKNOWLEDGEMENT

Thanks God, the merciful and the passionate, for providing me the

opportunity to step in the excellent world of science. To be able to step strong and

smooth in this way, I have also been supported by many people to whom I would

like to express my deepest gratitude.

I am deeply indebted to Dr. S. BALAMURALI, Ph.D., Professor,

Department of Mathematics, Kalasalingam University, Krishnankoil, TamilNadu,

India for his valuable Supervision, continuous guidance, expert consultancy,

unstinted support and enormous encouragement to bring out this thesis. In his I

experience a personification of excellence, dedication and commitment. I also

appreciate and thank, for his free availability and approach. I pray to God for his

well being.

I wish to record my gratitude to the Management and the authorities of

Kalasalingam University, for granting me permission and providing necessary

facilities to carry out the research. I am extremely grateful to the Chairman and

members of the Doctoral Committee for their support.

Many friends have helped me stay sane through these difficult years. I

greatly value their friendship and I deeply appreciate their belief in me. I would

like to thank Ms. M. Jeyadurga , who as a good friend, is always willing to help

and give her best suggestions.

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Most importantly, none of this would have been possible without the

love and patience of my family. I would like to express my heart-felt gratitude to

my family.

I would like to thank my spouse , Mr. T. Sermaraj, and my children

S. Divyamaki and S. Lakinyamaki for their constant love and support.

Last but not the least, I would like to thank my parents

Mr. G. Mahalingam and Mrs. M. Jeyasamvarthini , for giving birth to me at

the first place and supporting me spiritually throughout my life. I would also like

to thank my sister and brother. They were always supporting me and encouraging

me with their best wishes.

M. Usha

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TABLE OF CONTENTS

CHAPTER TITLE PAGE NO. NO. ABSRACT iv

LIST OF TABLES xvi

LIST OF FIGURES xix

LIST OF SYMBOLS AND ABBREVIATIONS xx

1. INTRODUCTION 1

Section 1 Basic Concepts of Quality Control 2

Section 2 Basic Concepts of Acceptance Sampling 6

Section 3 A Review of Variables Sampling Plans 20

Section 4 A Review on Certain Special Purpose Sampling 23

Plans by Attributes

Section 5 A Review on Special Purpose Sampling Plans 33

by Variables

2. OPTIMAL DESIGNING OF VARIABLES QUICK 34

SWITCHING SAMPLING SYSTEM (VQSS) BY MINIMIZING

THE AVERAGE SAMPLE NUMBER

2.1 Introduction 34

2.2 Conditions of Application 34

2.3 Operating Pocedure of Known Sigma Case 35

2.4 Operating Characteristic Function of VQSS 36

2.5 Designing of a Known Sigma VQSS 37

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CHAPTER TITLE PAGE

NO. NO.

2.6 Average Sample Number 37

2.7 Optimal Designing of Known Sigma VQSS 38

2.8 Optimal Designing of Unknown Sigma VQSS 38

2.9 Examples 42

2.9.1 Selection of Known Sigma VQSS Indexed by 42

AQL and LQL

2.9.2 Selection of Unknown Sigma VQSS Indexed 43

by AQL and LQL

2.10 Advantages of the VQSS 43

2.11 Comparison 44

3. OPTIMAL DESIGNING OF VARIABLES QUICK SWITCHING 53

SYSTEM WITH DOUBLE SPECIFICATION LIMITS

3.1 Introduction 53

3.2 Conditions for Application of VQSS 53

3.3 Operating Procedure of a Known Sigma VQSS 54

3.4 OC Function of a Known Sigma VQSS with Double 55

Specification Limits

3.4.1 Known Sigma VQSS with Symmetric Fraction 56

Non- conforming

3.4.2 Known Sigma VQSS with Asymmetric 58

Fraction Non-conforming

3.5 Designing of Unknown Sigma VQSS having Double 59

Specification Limits

3.6 Determination of the Optimal Parameters of VQSS 61

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CHAPTER TITLE PAGE NO. NO.

3.7 Illustrative Examples 63

3.7.1 Symmetric Fraction Non-conforming Case 63

3.7.2 Asymmetric Fraction Non-conforming Case 63

3.8 Industrial Application of the Proposed VQSS 65

3.9 Comparisons 68

3.10 Non-normality in VQSS 68

3.11 Conclusions 69

4. OPTIMAL DESIGNING OF VARIABLES TIGHTENED 81

NORMAL TIGHTENED (TNT) SAMPLING SCHEME BY

MINIMZING THE AVERAGE SAMPLE NUMBER

4.1 Introduction 81

4.2 Conditions of Application 81

4.3 Operating Procedure of Known Sigma Variables 82

TNT Scheme

4.4 OC Function of Known Sigma Variables TNT Scheme 83

4.5 Designing of a Known Sigma Variables TNT Scheme 85

4.6 Designing of Unknown Sigma Variables TNT Scheme 87

4.7 Examples 89

4.7.1 Selection of Known Sigma TNT Scheme Indexed 89 by AQL and LQL 4.7.2 Selection of Unknown Sigma Variables TNT 90 Scheme Indexed by AQL and LQL

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CHAPTER TITLE PAGE NO. NO. 4.7.3 Advantages of the Variables TNT Sampling 90

Scheme

4.8 Comparisons 91

4.8.1 Comparison Thorugh OC Curves 91

4.8.2 Comparison Through ASN 92

4.9 Conclusions 93

5. OPTIMAL DESIGNING OF VARIABLES CHAIN SAMPLING 100

PLAN BY MINIMZING THE AVERAGE SAMPLE NUMBER

5.1 Introduction 100

5.2 Conditions of Application 101

5.3 Operating Procedures of Variables ChSP 101

5.3.1 Known Sigma Case 101

5.3.2 Unknown Sigma Case 102

5.4 Designing Methodology of Variables ChSP 103

5.4.1 Known Sigma Case 103

5.4.2 Unknown Sigma Case 106

5.5 Designing Examples 109

5.5.1. Selection of Known Sigma Variables ChSP 109

for Specified AQL and LQL

5.5.2. Selection of Unknown Sigma Variables ChSP 110

for Specified AQL and LQL

5.6 Illustrative Example 110

5.7 Advantages of the Variables ChSP 112

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CHAPTER TITLE PAGE NO. NO.

5.8 Average Run Length of Variables ChSP 114

5.9 Conclusions 115

6. OPTIMAL DESIGNING OF VARIABLES SAMPLING 122

PLAN FOR RESUBMITTED LOTS

6.1 Introduction 122

6.2 Conditions of Application 122

6.3 Operating Procedure of the Known Sigma Variables 123

Resampling Scheme

6.4 Designing of Variables Resampling Scheme with 124

Known Standard Deviation

6.5 Operating Procedure of Variables Resampling 127

Scheme with Unknown Sigma

6.6 Designing of Variables Resampling Scheme with 128

Unknown Standard Deviation

6.7 Designing Examples 131

6.7.1. Selection of Known Sigma Variables 131

Resampling Scheme Indexed by AQL

and LQL

6.7.2. Selection of Unknown Sigma Variables 131

Resampling Scheme Indexed by AQL

and LQL

6.8 Merits of the Variables Resampling Scheme 132

6.9 Comparison with Attributes Resampling Scheme 134

6.10 Conclusions 134

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CHAPTER TITLE PAGE NO. NO.

7. CONCLUSIONS AND FUTURE WORK 141

REFERENCES 146

LIST OF PUBLICATIONS 159

CURRICULUM VITAE 161

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LIST OF TABLES

TABLE NO. TITLE PAGE NO.

2.1 Variables Quick Switching Sampling Systems 49

Indexed by AQL and LQL for =5% and =10%

Involving Minimum ASN

2.2 ASN Values of the Known Sigma Variables Single Sampling 54

Plan, Variables Double Sampling Plan and VQSS

2.3 ASN Values of the Unknown Sigma Variables Single 54

Sampling Plan, Variables Double Sampling Plan and VQSS

2.4 Parameters of Known Sigma Variables QSS for Some 55

Selected Combination of AQL and LQL Values

2.5 Parameters of Unknown Sigma Variables QSS for Some 55

Selected Combination of AQL and LQL Values

3.1 Optimal Parameters of Known Sigma VQSS with Double 70

Specification Limits (Symmetric Fraction Non-conforming)

3.2 Optimal Parameters of Known Sigma VQSS with Double 72

Specification Limits (Asymmetric Fraction Non-conforming)

3.3 Optimal Parameters of Unknown Sigma VQSS with Double 74

Specification Limits (Symmetric Fraction Non-conforming)

3.4 Optimal Parameters of Unknown Sigma VQSS with Double 76

Specification Limits (Symmetric Fraction Non-conforming)

3.5 Average Sample Number of Variables Single Sampling Plans 78

and VQSS with Double Specification Limits (Symmetric

Fraction Non- conforming)

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TABLE NO. TITLE PAGE NO. 4.1. Variables Tightened-Normal-Tightened Sampling Scheme 93

Indexed by AQL and LQL for =5% and =10% Involving

Minimum ASN

4.2 ASN Values of the Known Sigma Variables Single Sampling 97

Plan, Variables Double Sampling Plan and Variables TNT

Scheme

4.3 ASN Values of the Unknown Sigma Variables Single 97

Sampling Plan, Variables Double Sampling Plan and

Variables TNT Scheme

4.4 Parameters of Known Sigma Variables TNT Scheme 98

for different AQL and LQL Values

4.5 Parameters of Unknown Sigma Variables TNT Scheme 98

for different AQL and LQL Values

5.1 Variables Chain Sampling Plans Indexed by AQL 116

and LQL for =5% and =10%

5.2 Variables Chain Sampling Plans Indexed by AQL 119

and LQL for =1% and =1%

5.3 Variables Single Sampling Plans Indexed by AQL 119

and LQL for =1% and =1%

5.4 ASN Values of the Variables SSP, DSP and Variables 120

Chain Sampling Plans

6.1 Variables Resampling Scheme (with m=2) Indexed by 134

AQL and LQL for =5% and =10%

6.2 ASN Values of the Variables Single, Double Sampling 138

Plans and Resampling Schemes

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TABLE NO. TITLE PAGE NO.

6.3 Sample Size of the Attributes and Variables Resampling 138

Schemes

6.4 Average Sample Number of the Variables resampling 139

Schemes for different m values

6.5 Variables Resampling Schemes (with m=2) Involving 140

Minimum ASN Indexed by AQL and LQL

6.6 Variables Resampling Schemes (with m=2) Involving 141

Minimum Sum of ASN Indexed by AQL and LQL

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LIST OF FIGURES

FIGURE NO. FIGURE CAPTIONS PAGE NO.

1.1 Operating Characteristic Curve

2.1 OC Curves of Single Sampling Normal Plan (10, 1.754), 53

Quick Switching System (10; 1.754, 2.179) and

Single Sampling Tightened Plan (10, 2.179)

4.1 OC Curves of Single Sampling Normal Plan (12, 1.857), 96

TNT Scheme (63, 12; 1.857) and Single Sampling

\ Tightened Plan (63, 1.857)

5.1 OC Curves of a Variables Chain Sampling Plan for 121

Different iσ Values

6.1 ASN Curves of a Variables Single Sampling Plan and 142

Resampling Sampling Scheme

6.2 OC curves of a Variables Resampling Scheme for 143

different m values

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LIST OF SYMBOLS AND ABBREVIATIONS

The following is the glossary of symbols and abbreviations used in this thesis.

N Lot Size

n Sample Size

p Lot or process quality or fraction non-conforming

Pa (p) Probability of acceptance as function of lot quality.

p1 Acceptable Quality Level (AQL)

p2 Limiting Quality Level (LQL)

α Producer’s risk

β Consumer’s risk

n1 First stage sample size

n2 Second stage sample size

d Number of non-conforming items

c Acceptance number in attributes single sampling plan

Nk Acceptance criteria of normal inspection

Tk Acceptance criteria of tightened inspection

σ Population standard deviation

S2 Sample variance

U Upper specification limit

L Lower specification limit

nσ Sample size for known sigma plan

kσ Acceptance criteria for known sigma plan

k’σ Rejection criteria for known sigma plan

ns Sample size for unknown sigma plan

ks Acceptance criteria for unknown sigma plan

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k’s Rejection criteria for unknown sigma plan

푖휎 Number of preceding lots considered for accepting current lot for

known sigma plan

푖s Number of preceding lots considered for accepting current lot for

unknown sigma plan

v The value which is to be compared with acceptance criterion for

making decision

The population mean

X The sample mean

S The sample standard deviation

휎 The population standard deviation

Φ(⋅) The cumulative distribution function of standard normal

distribution

ASN Average Sample Number

SSP Single Sampling Plan

DSP Double Sampling Plan

QSS Quick Switching System

TNT Tightened – Normal - Tightened

ChSP Chain Sampling Plan

CHAPTER 1

INTRODUCTION

Inspection of raw materials, semi finished products, or finished products

are one aspect of quality assurance. Whenever a statistical technique is used to

control, maintain and improve the quality, it is termed as statistical quality control.

When inspection is for the purpose of acceptance or rejection of a

product, based on adherence to a standard, the type of procedure employed is

usually called acceptance sampling. Acceptance sampling is one of the major

components in the field of Statistical Quality control.

A company receives a shipment of product from a vendor. This product

is often a component or raw material used in the company’s manufacturing

process. A sample is taken from the lot, and some quality characteristic of the units

in the sample is inspected for a specified period of time. On the basis of the

information in this sample, a decision is made regarding lot disposition. Usually,

this decision is either to accept or to reject the lot. Accepted lots are put into

production; rejected lots may be returned to the vendor or may be subjected to

some other lot disposition action.

This chapter comprises of the following sections

SECTION 1 BASIC CONCEPTS OF QUALITY CONTROL

SECTION 2 BASIC CONCEPTS OF ACCEPTANCE SAMPLING

SECTION 3 A REVIEW ON ACCEPTANCE SAMPLING PLAN BY

VARIABLES

SECTION 4 A REVIEW ON CERTAIN SPECIAL PURPOSE SAMPLING

PLANS BY ATTRIBUTES

SECTION 5 A REVIEW ON SPECIAL PURPOSE SAMPLING PLANS

BY VARIABLES

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SECTION 1

BASIC CONCEPTS OF QUALITY CONTROL

1.1 QUALITY

In manufacturing sector, quality is a measure of excellence or a state

of being free from defects, deficiencies and significant variations. Quality of a

product is brought about by the strict and consistent adherence to measurable and

verifiable standards to achieve uniformity of output that satisfies specific consumer

or user requirements.

International Organization for Standardization (ISO) 8402-1986

standard defines quality as “the totality of features and characteristics of a product

or service that bears its ability to satisfy stated or implied needs”. ISO 9000

defines quality as "degree to which a set of inherent characteristics fulfills

requirements".

1.2 QUALITY CONTROL

American Society for Quality (ASQ) defines, Quality Control as the

operational techniques and activities used to fulfill requirements for quality.

1.3 STATISTICAL QUALITY CONTROL

Whenever a statistical technique is employed to control, improve and

maintain the quality or to solve quality problem it is termed as Statistical Quality

Control (SQC). The new era of quality control development began during the

World War II when SQC was much needed due to mass production. It is used

throughout the quality system at various stages of production such as

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Incoming inspection

Product moving from one stage to other

In – process

Machine start – up

Process monitoring

Process adjustment

Final product

Field surveillance

SQC is systematic as compared to guess – work of haphazard process

inspection. The mathematical and statistical approaches neutralize personal bias

and uncover poor judgment. The SQC consists of three general activities:

Systematic collection and graphic recording of accurate data.

Analyzing the data.

Practical engineering or management or management action, if the

information obtained indicates significant deviations from the specified

limits.

1.4 TOOLS OF SQC

The SQC is the term used to describe a set of statistical tools used by

quality professionals. The following are the statistical tools used generally for the

purpose of exercising control, improvement of quality, enhancement of

productivity, creation of consumer confidence and development of the industrial

economy of the country.

Frequency Distribution: It is a tabulation or tally of the number of times a

given quality characteristic occurs within the samples. Graphic

representation of frequency distribution will show the average quality,

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spread of quality, comparison with specific requirements and process

capability.

Control Chart: It is a graphical representation of quality characteristics,

which indicates whether the process is under control or not.

Acceptance Sampling: In order to maintain the quality of purchased lots,

two major alternatives are open to a buyer. One, complete inspection: every

single item in the lot is inspected and tested. Two, partial inspection: a

sample of items is taken, the sampled items are inspected and tested, and

the lot as a whole is accepted or rejected depending on whether few or

many non-conforming items are found in the sample. This type of sampling

called acceptance sampling which is the process of randomly inspecting a

sample of goods and deciding whether to accept the entire lot based on the

results. Acceptance sampling determines whether a batch of goods should

be accepted or rejected.

Analysis of the data: This includes techniques such as analysis of

correlation, analysis of variances, analysis for engineering design, problem

solving technique to eliminate cause of troubles.

1.5 BENEFITS OF STATISTICAL QUALITY CONTROL

SQC ensures rapid and efficient inspection at a minimum cost. It finds out

the cause excessive variability in manufactured products by forecasting

trouble before rejections occur and reducing the amount of spoiled work.

It exerts more effective pressure for quality improvement than that of a 100

percent inspection.

It easily detects faults. For example, using control charts one can easily

examine the deterioration in quality by verifying whether the points fall

above the upper control limits or below the lower control limits.

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So long as the statistical control continues, specifications can be accurately

predicted for future, by which it is possible to assess whether the production

processes are capable of producing the products with the given set of

specifications.

Increases output and reduces wasted machines and materials resulting in

higher productivity.

Better customer relations through general improvement in product and

higher share of the market.

It provides a common language that may be used by designers, production

personnel and inspectors.

It says when and where 100 percent inspection is required.

Creates quality awareness in employees.

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SECTION 2

BASIC CONCEPTS OF ACCEPTANCE SAMPLING

1.2.1 INTRODUCTION

Acceptance sampling is an important field of SQC that was

popularized by Dodge and Romig (1959) and originally applied by the U.S.

military to the testing of bullets during World War II. If 100 percent inspection

were executed in advance, no bullets would be left to shipment. If, on the other

hand, none were tested, malfunctions might occur in the field of battle, which may

result in potential disastrous result. Dodge proposed a “middle way” reasoning that

a sample should be selected randomly from a lot, and on the basis of sampling

information, a decision should be made regarding the disposition of the lot. In

general, the decision is either to accept or reject this lot. This process is called Lot

Acceptance Sampling or just acceptance sampling.

Single sampling plans and double sampling plans are the most basic

and widely applied testing plans when simple testing is needed. Multiple sampling

plans and sequential sampling plans provide marginally better disposition decision

at the expense of more complicated operating procedures. Other plans such as the

continuous sampling plan, bulk-sampling plan, and Tightened-Normal-Tightened

scheme etc., are well developed and frequently used in their respective working

condition.

1.2.2 NECESSITY OF ACCEPTANCE SAMPLING PLANS

Acceptance sampling plan is an essential tool in the SQC and is a

methodology which deals with quality contracting on product orders between the

producers and the consumers and thus allows the producers to take decision to

accept or reject the manufactured products based on the inspection of samples. It is

the process of evaluating a portion of the product/material in a lot for the purpose

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of accepting or rejecting the lot as either conforming or not conforming to a quality

specification

Acceptance sampling is necessary to limit the cost of inspection and is

the only available method to appraise the quality in destructive testing. Acceptance

sampling itself does not improve quality, but whenever the lot is rejected it

indicates the instability of the production process. Acceptance sampling is cost

efficient and the only admissible method of efficient tests with quick results.

1.2.3 MAJOR AREAS OF ACCEPTANCE SAMPLING

Acceptance sampling deals with procedures/algorithms by which

decision to accept or reject a lot is based on the results of the inspection of

samples.

According to Duncan (1986), an acceptance sampling plan is likely to

be implemented when the following holds:

When the cost of inspection is high and the loss arising from the passing of a

non-conforming unit is not great.

When a 100 percent inspection is fatiguing.

When inspection is destructive i.e., a situation where inspection is not

possible without destroying the article chemically or physically.

Where there are great quantities or areas to be inspected.

When it is desired to stimulate the maker and/or the buyer.

According to Dodge (1969), the major areas of acceptance sampling

may be classified under the following four broad categories,

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1. Lot-by-Lot acceptance sampling by the method of attributes, in which each

unit in a sample is inspected on a go-not-go basis for one or more

characteristics.

2. Lot- by-Lot sampling by the method of variables, in which each unit in a

sample is measured for a single characteristic, such as weight or strength, etc.

3. Continuous sampling of flow of units by the method of attributes and

4. Special purpose plans including chain sampling, skip-lot sampling and small

sample plans etc.

1.2.4 BASIC TERMINOLOGIES AND DEFINITIONS

SAMPLING PLAN, SCHEME AND SYSTEM

American National Standards Institute / American Society for Quality

Control (ANSI / ASQC) Standard A2 (1987) defines an acceptance sampling plan

as “a specific plan that states the sample size or sizes to be used and the associated

acceptance and non-acceptance criteria” It defines an acceptance-sampling scheme

as “a specific set of procedures which usually consists of acceptance sampling

plans in which lot sizes, sample sizes and acceptance criteria or the amount of

100% inspection and sampling are related”. The MIL-STD-105 D (1963) tables

and procedures are the examples for sampling scheme. Stephens and Larson

(1967) define a sampling system as “an assigned grouping of two or three

sampling plans and the rules for using (that is, switching between) these plans for

sentencing lots or batches of articles to achieve blending of the advantageous

features of the sampling plan”. Quick Switching System (QSS) of Romboski

(1969) is an example for a sampling system.

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CUMULATIVE AND NON – CUMULATIVE RESULTS SAMPLING PLANS

Dodge and Stephens (1966) defines a non – cumulative sampling plan as

one which uses the current sample information from the process or current product

entity in making decisions about process or product quality. Single and double

sampling plans are examples of non – cumulative sampling. Cumulative results

sampling inspection is one which uses the current and past information from the

process in making a decision about the process. Chain sampling plan of Dodge

(1955) is an example for cumulative results sampling plan.

INSPECTION ANSI / ASQC Standard A2 (1987) defines the term ‘inspection’ as

‘activities’, such as measuring, examining, testing, gauging one or more

characteristics of a product or service and comparing them with specified

requirements to determine conformity. A sampling scheme or a sampling system

may contain three types of inspections viz normal, tightened and reduced

inspection.

NORMAL INSPECTION

Inspection that is used in accordance with an acceptance sampling

scheme when a process is considered to be operating at or slightly better than its

acceptance quality level.

TIGHTENED INSPECTION

A feature of a sampling scheme using stricter acceptance criteria than

those used in normal inspection.

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REDUCED INSPECTION

A feature of a sampling scheme permitting smaller sample sizes than

those used in normal inspection.

OPERATING CHARACTERISTIC (OC) CURVE

The Operating Characteristic curve (OC) is a picture of a sampling

plan. Each sampling plan has a unique OC curve. The probability of acceptance

can be regarded as a function of the deviation of the specified value μ0 of the mean

from its true value μ. This function is called OC function of the sampling plan. An

OC curve is developed by determining the probability of acceptance for several

values of incoming quality.

Figure 1.1 Operating Characteristic (OC) Curve

11

The OC curves are generally classified as Type A and Type B OC

curves. ANSI/ASQC Standard A2 (1987) defines the terms as follows:

Type A OC curve is used for isolated or unique lots, or a lot from an

isolated sequence. “A curve showing, for a given sampling plan, the

probability of accepting a lot as a function of lot quality”.

Type B OC curve is used for a continuous stream of lots. “A curve

showing, for a given sampling plan, the probability of accepting a lot

as a function of the process average”.

In sampling systems or schemes, one will have a ‘composite OC curve”

which gives the steady state probability of acceptance under the switching rules of

the system or scheme as a function of process quality.

To evaluate the probability of acceptance, Pa (p), hypergeometirc model

is exact for type A situation (when sampling attribute characteristics from a finite

lot without replacement). Under type B situation, binomial model will be accurate

for the case of non-confirming units to calculate Pa (p). Binomial model is also

correct in case of sampling from a finite lot with replacement.

Poisson model is accurate in calculating Pa (p), which specifies a given

number of non-conformities per unit (or non-conformities per hundred units). In

case of variable sampling plans normal distribution (Gaussian) is widely used to

compute relevant measures of sampling plans. Hyper geometric, binomial, Poisson

and normal distributions are the distributions commonly used in the development

of acceptance sampling plans. Schilling (1982) has given the conditions under

which each of these is to be used.

12

HYPERGEOMETRIC MODEL

This is an exact model for the case of non-conforming units under

Type A situations and is useful for isolated lots. In this model the probability mass

function is given by

nN

xnmN

xm

xXP

where N is the population size

n is the size of the sample

k is the number of successes

BINOMIAL MODEL

This model is exact for the case of non-conforming units under type B

situations. This can also be used for type A situations for the case of non-

conforming units, whenever (n /N) ≤ 0.10. Under type B situation, for the case of

non conforming units, Poisson model can be used whenever n is large and p is

small such that np < 5.

The probability of getting exactly x defectives in a sample of size n is

given by the probability mass function:

nxppxn

xXP xnx .........3,2,1,0,1

where N is the lot size

n is the sample size taken from a lot

p is the proportion defective in the sample

13

POISSON MODEL

The Poisson distribution can be applied to systems with a large number

of possible events, each of which is rare to occur. The probability mass function is

given by

.......3,2,1,0,!

xx

exXPx

λ is a positive real number, equal to the expected number of occurrences during

the given interval or average.

NORMAL DISTRIBUTION

No area of statistics seems to have escaped the impact of the normal

distribution. This is certainly true of acceptance sampling where it forms the basis

of a large number of “variables” acceptance sampling plans. It has pervaded other

areas of acceptance sampling as well. The normal distribution is completely

specified by two parameters and .

The probability density of the normal distribution is:

0,

,,2

1,, 2

2

22

xexfx

Here, is the mean

is its standard deviation

AVERAGE SAMPLE NUMBER (ASN)

ANSI / ASQC Standard A2 (1987) defines ASN as “the average

sample units per lot used for making decisions either acceptance or non-

acceptance”. A plot of ASN against process quality is called ASN curve. ASN will

14

be affected according to the type of curtailment of inspection (on acceptance and

rejection decisions). Sampling inspection is to be called fully curtailed if sampling

is stopped whenever decision could be reached on acceptance (or rejection) before

reaching the prescribed sample size.

AVERAGE OUTGOING QUALITY (AOQ)

ANSI / ASQC Standard A2 (1987) defines AOQ as “the expected

quality” of outgoing product following the use of an acceptance sampling plan for

a given value of incoming product quality.

AVERAGE OUTGOING QUALITY LIMIT (AOQL)

“The maximum AOQ over all possible levels of incoming quality” is

termed as AOQL. The assumption underlying in this expression is that for all

accepted lots the average proportion non-conforming is assumed to be p and for all

rejected lots the entire units are being screened and non-conforming units are

replaced. A plot of AOQ against p is called AOQ curve.

AVERAGE TOTAL INSPECTION (ATI)

According to ANSI / ASQC Standard A2 (1987), ATI is “the average

number of units inspected per lot based on the sample size for accepted lots and all

inspected units in rejected lots”. ATI is not applicable whenever testing is

destructive. A plot of ATI against p is called ATI curve.

ACCEPTABLE QUALITY LEVEL (AQL)

ANSI/ASQC Standard A2 (1987) defines AQL as “the maximum

percentage or proportion of variant units in a lot or batch that, for the purpose of

acceptance sampling, can be considered satisfactory as a process average”.

15

LIMITING QUALITY LEVEL (LQL)

ANSI/ASQC Standard A2 (1987) defines LQL as “the percentage or

proportion of variant units in a batch or lot for which, for the purposes of

acceptance sampling, the consumer wishes the probability of acceptance to be

restricted to a specified low value”.

INDIFFERENCE QUALITY LEVEL (IQL)

The percentage of variant units in a batch or lot for which, for purposes

of acceptance sampling, the probability of acceptance to be restricted to a specific

value namely 0.50.

1.2.5 DESIGNING METHODOLOGY

In designing a sampling plan, one has to accomplish a number of

different purposes. According to Hamaker (1960), the most important are

1. To strike a balance between the consumer’s requirement, the producer’s

capabilities and the inspectors capacity.

2. To separate bad lots from good one.

3. Simplicity of procedures and administration.

4. Economy in number of observations.

5. To reduce the risk of wrong decisions with increasing lot size.

6. To use accumulated sample data as valuable source of information.

7. To exert pressure on the producer or supplier when the quality of the lot

received is unreliable up to standard.

8. To reduce sampling when the quality is reliable and satisfactory.

Hamaker (1960) also pointed out that these aims are partly conflicting

and all of them cannot be simultaneously realized.

16

The design methodologies of acceptance sampling may be categorized

as in the following.

Risk Based

Economical

Based

Non – Bayesian

1

2

Bayesian

3

4

Risk based sampling plans are traditional in nature, drawing upon

procedure and consumer type of risks as depicted by the OC curve. Economically

based sampling plans explicitly consider such factors as costs of inspections,

accepting a non conforming unit and rejecting a conforming unit in an attempt to

design a cost – effective plan. Bayesian plan design takes into account the past

history of similar lots submitted previously for inspection purposes.

Non – Bayesian plan design is not explicitly based upon the past lot history.

According to Peach (1947), the following are some of the major types

of designing the plans, based on the OC curves, which are classified according to

types of risk protection.

1. The plan is specified by requiring the OC curve to pass through two fixed

points. In some cases, it may be possible to impose certain additional

conditions also.

17

The two points generally selected are (p1, 1-α) and (p2, β) where,

p1= fraction non-conforming that is considered to be good so that

producer expects lot of quality p1 to be accepted most of the time.

p2= fraction non-conforming that is considered to be poor so that the

consumer expects lot of quality p2 to be rejected most of the time.

α = the producer’s risk of rejecting p1 quality and

β = the consumer’s risk of accepting p2 quality.

Sampling Plans of Cameron (1952) are the examples of this type of

designing. Schilling and Sommers (1981) term p1 as the Producer’s Quality Level

(PQL) and p2 as the Consumer’s Quality Level (CQL). Earlier literature calls p1 as

the Acceptable Quality Level (AQL) and p2 as the Limiting Quality Level (LQL)

or Rejectable Quality Level (RQL) or Lot Tolerance Percent Defective (LTPD).

Traditionally the values of α and β are assumed to be 95 % and 10 % respectively.

2. The plan is specified by fixing one point only through which the OC curve is

required to pass and one or more conditions, not explicitly in terms of the

OC curves. Dodge and Romig (1959) LTPD Sampling plans are the

examples for this type of designing.

3. The plan is specified by imposing upon the OC curve two or more

independent conditions none of which is explicitly involve the OC curves.

Dodge and Romig (1959) AOQL Sampling plans are the examples for this

type of designing.

18

1.2.5.1 DESIGNING METHODOLOGY FOLLOWED IN THIS THESIS In this thesis, ‘Search procedure’ has been followed. In this approach,

the parameters of a sampling plan are chosen, by trial and error by varying the

parameters in a uniform fashion depending upon the properties of OC function.

An example for this approach is the one followed by Guenther (1969,1970) while

determining the parameters of single and double sampling plans under the

conditions for application of binomial, Poisson and hyper geometric models of OC

curve. The advantage of search procedure is that the sample sizes need not be

rounded.

1.2.6 CLASSIFICATIONS OF ACCEPTANCE SAMPLING PLANS

The acceptance sampling plans are generally divided into two major

categories namely, attributes sampling plan and variables sampling plan.

1.2.6.1 ATTRIBUTES SAMPLING PLANS

Many quality characteristics cannot be conveniently represented

numerically. In such cases, we usually classify each item inspected as either

conforming (non-defective) to the specifications on that quality characteristics or

non conforming (defective) to those specifications. Quality characteristics of this

type are called attributes. Sampling plans applied to such quality characteristics are

called attributes sampling plans. Several sampling plans are available in the

literature for the application of attributes quality characteristics. For example,

single sampling plan, double sampling plan, multiple sampling plan, etc. (see Hald

(1981) and Schilling (1985)).

19

1.2.6.2 VARIABLES SAMPLING PLAN

Variables sampling plan specifies the number of items to be sampled

and the criterion for sentencing lots when measurements data are collected on

quality characteristic of interest. These plans are generally based on sample

average and sample standard deviations of the quality characteristic. When the

distributions of the quality characteristic in the lot or process is known, variables

sampling plans that have specified risks of accepting and rejecting lots of given

quality may be designed.

The main advantage of the variables sampling plan is that the same OC

curve can be obtained with a smaller sample size than would be required by an

attributes sampling plan. Thus, a variables acceptance sampling plan would require

less sampling. The measurements data required by a variables sampling plan

would probably cost more per observation than the collection of attributes data.

However, the reduction in sample size obtained may more than offset this

increased cost. When destructive testing is employed, variables sampling is

particularly useful in reducing the costs of inspection. Another advantage is that

measurements data usually provide more information about the manufacturing

process or lot than do attributes data. Generally, numerical measurements of

quality characteristics are more useful than simple classification of the item as

conforming or non-conforming. Another advantage of the variables sampling plan

is that when AQLs are very small, the sample size required by it is very less than

the attributes sampling plans. Under these circumstances, the variables sampling

plans have significant advantages. For compliance testing of a measurable

characteristic, a variable sampling plan may be preferred.

20

SECTION 3

A REVIEW OF VARIABLES SAMPLING PLANS

As many manufacturers begin to emphasize allowable numbers of non-

conforming parts per million (ppm), variables sampling becomes very attractive.

There are two cases in variables sampling plans.

(i) Sampling plans with known standard deviation

(ii) Sampling plans with unknown standard deviation

In these plans the decision on acceptance or rejection of the lot is

based on sample average alone. The decision specifications are associated with

each inspection characteristics. Many specification are one sided. (i.e.) The

specification merely states a lower specification limit L or an upper specification

limit U to apply to individual article.

When the standard deviation of the lot quality is known, the criteria for

acceptance and the associated mathematical computations get simplified. When

products are manufactured by automatic machinery whose inherent variation is

known and tested, we have an example where the lot standard deviation is known.

When we assume the lot standard deviation as known and give it a particular value

σ it is assumed as constant. We assume that the directly measurable quality

characteristic X follows the normal law of pattern of variation in the lot these

assumptions must be examined and reviewed from time to time when variables

plans with known sigma are in use. The n units in the sample are measured and the

values nxxx ,...., 21 are obtained. The mean is calculated. If the individual product

21

quality has an upper specification limit U then acceptance criteria for the lot based

on the single sampling results would be if Ukx , then accept the lot and if

Ukx , then reject the lot. In the case of unknown sigma variables sampling

plan, sample standard deviation S is used instead of .

1.3.1 LITERATURE REVIEW OF VARIABLES SAMPLING PLANS

Wallis (1947) suggested an approximation for finding the parameters

for unknown standard deviation plan from that of known plan. Military Standard

414, Department of Defense in 1957 issued “Sampling procedures and Tables for

Inspection by variables for Percent Defective” which was the culmination of the

developments in variables sampling plans. Lieberman and Resnikoff (1955)

developed tables and procedures for the selection of variable sampling plan

parameters for various AQL values given in the MIL-STD 414 scheme. They have

considered variables sampling plans for assuring the product quality when the

quality characteristic of the product follows normal distribution with unknown

standard deviation and provided a procedure for calculating the non-central t-

distribution. Owen (1967) developed variables sampling plans based on normal

distribution when the process standard deviation is unknown. Bender (1975)

considered variables sampling plans for assuring the product quality when the

quality characteristic of the product following normal distribution with unknown

standard deviation and provided a procedure for calculating the non-central t-

distribution. Hamaker (1979) has given a procedure of finding the parameters of

the unknown sigma variables sampling plans from the known sigma variables

sampling plans. Schneider and Wilrich (1981) investigated the robustness of

variables sampling plans. Govindaraju and Soundararajan (1986) developed tables

for selecting the parameters of variables single sampling plans that match with the

OC curves of MIL-STD 105D (1963) schemes. Kao (1971) provided the

comparison between the attribute acceptance sampling plans and the variable

22

acceptance sampling plans. Bravo and Wetherill (1980) developed a method for

designing variables double sampling plans with OC curves matching with the OC

curves of the equivalent single sampling plans. Sommers (1981) developed tables

for selecting variables double sampling plans and matched variables single

sampling plan having two fixed points on the OC curve. Schilling (1982) has

written an exclusive book on acceptance sampling. Bruhn Suhr and Krumbholz

(1990) studied the variables single sampling with double specification limits for

normally distributed quality characteristics. Collani (1990) criticized the variables

sampling plans and argued that the acceptance sampling by variables is

inappropriate if one is interested in the fraction non-conforming in incoming

batches. But, Seidel (1997) has proved that sampling by variables is always

optimal. Baillie (1992) developed tables for variables double sampling plans when

the process standard deviation is unknown. Hamilton and Lesperance (1995)

described the operating characteristics of the variables single sampling plans

having double specification limits. Govindaraju and Kuralmani (1998) have

studied the nature of the OC curve of known sigma single sampling variables plan.

Jun et al. (2006) developed variables acceptance sampling plans for Weibull

distributed items under sudden death testing.

Recently, there are developments in designing various variables

sampling plans. Pearn and Wu (2006) investigated the variables sampling plans

for very low fraction non-conforming. Pearn and Wu (2007) proposed an effective

decision making method for product acceptance based on measurement data. Sheu

et al. (2014) developed a variables sampling plan based on incapability index Cpp

proposed to deal with lot sentencing. Yen et al. (2014) developed variable

sampling plan using the exponentially weighted moving average (EWMA)

statistic based on the yield index for lot sentencing.

23

SECTION 4

A REVIEW ON CERTAIN SPECIAL PURPOSE SAMPLING PLANS BY

ATTRIBUTES

The special purpose sampling inspection plans often known as special

purpose plans is one of the major areas of acceptance sampling which is classified

under fourth category of Dodge’s classification (1969), are tailored for special

applications as against general or universal use.

Special purpose sampling plans are also known as conditional sampling

plans were developed to overcome the short comings of zero acceptance single

sampling plans whenever samples of small sizes only are practically possible for

disposition of lots. Zero acceptance single sampling plan results in rejection of a

lot even if there is only one nonconforming unit is observed in the sample thereby

resulting in a poor operating characteristic (OC) curve. This is applicable

whenever high quality product is desirable.

Some of the special purpose sampling plans are Chain sampling plan

(ChSP) of Dodge (1955), Repetitive group sampling (RGS) plan of Sherman

(1965), Multiple deferred/dependent state sampling plans of Wortham and Baker

(1976), Quick Switching System (QSS) of Romboski (1969), Tightened-Normal-

Tightened (TNT) sampling scheme developed by Calvin (1977), Skip-lot sampling

plan (SkSP) of Perry (1973) etc.

Balamurali and Kalyanasundaram (1997) determined a new sampling

scheme called an attribute single sampling scheme. Balamuali and

.Kalyanasundaram (1999) introduced conditional double sampling scheme and

they have made comparison with the single sampling scheme.

24

1.4.1 QUICK SWITCHING SYSTEM

A sampling system consists of two or more sampling plans and the

rules for switching between the sampling plans to achieve a blending of the

advantageous features of each of the sampling plans. In general, any sampling

system of inspection involving only normal and tightened inspection will be

referred to as a two-plan system.

Quick switching system (QSS) developed by Dodge (1967) is one of

the two-plan systems for the application of attributes quality characteristics. In

any two plan system, the tightened inspection can be used when the quality of a

product deteriorated and normal inspection is used when the quality is found to be

good. Dodge (1965), Hald and Thyregod (1965) and Stephen and Larson (1967)

have investigated the two-plan systems using different switching criteria to

achieve the desired discrimination on the operating characteristic (OC) curve.

Romboski (1969) has investigated the QSS of type QSS-1 by taking attributes

single sampling plan as the reference plan. Arumainayagam and Soundararajan

(1994, 1995) have constructed quick switching double sampling system by

tightening the acceptance number and tightening the sample sizes respectively.

Balamurali and Kalyanasundaram (1996) introduced procedures and constructed

tables for the selection of zero acceptance number quick switching systems.

Govindaraju (2011) designed zero acceptance number chained QSS. Balamurali

and Usha (2013) have investigated the QSS under the Weibull life time model.

The application of the system is as follows.

(1) Adopt a pair of sampling plans i.e., a normal plan (N) and a tightened plan (T).

(2) Use plan N for the first lot.

(3) For each lot inspected, if the lot is accepted, then use the plan N for the next

lot; if the current lot is rejected, then use plan T for the next lot.

25

OPERATING PROCEDURE OF QUICK SWITCHING SYSTEM

The operating procedure of attributes QSS-1 is as follows.

Step 1: Start with normal inspection. During normal inspection, take a random

sample of size n and inspect. Observe the number of non-conforming items

in the sample say d.

Step 2: Accept the lot if d ≤ cN and reject the lot if d > cN. If a lot is rejected on

normal inspection, then switch to tightened inspection as in Step 3.

Otherwise continue the normal inspection for the next lot.

Step 3: During tightened inspection, take a random sample of size n and inspect.

Observe the number of non-conforming items in the sample says d.

Step 4: Accept the lot if d ≤ cT and reject the lot if d > cT. If a lot is accepted on

tightened inspection, then switch to normal inspection as in Step 1.

Otherwise continue the same tightened inspection for the next lot.

( Note: TN cc )

MEASURES OF QSS

The important measures of QSS that describe the operation of an

acceptance sampling plan for various fraction nonconforming are,

1. The OC function (see Dodge (1967)) is

TN

Ta PP

PpP

1)(

where is the probability of accepting a lot based on a

single sample with parameters

and is the probability of accepting a lot based on a

single sampling plan with parameters

NN kvP Pr

TT kvP Pr

),( Nkn

),( Tkn

26

2. The Average Sample Number (ASN) is

3. The Average Outgoing Quality (AOQ) is

1.4.2 CHAIN SAMPLING PLANS (ChSP)

One of the cumulative results plans is the chain sampling plan (ChSP)

introduced by Dodge (1955), which made use of previous lots results, combining

with the current lot information, to achieve a reduction of sample size while

maintaining or even extending protection.

The ChSP was first conceived to overcome the problem of lack of

discrimination of the single sampling plan with acceptance number 0c , and had

been received wide applications in industries where the test is either costly or

destructive.

Soundararajan (1978a, 1978b) had carried out further evaluations of

ChSP-1 type sampling plans. Since the invention of ChSP-1, numerous works had

been done on the extensions to chain sampling plans. These included the plans

designated as ChSP-2 and ChSP-3, which were developed by Dodge (1958) but

kept unpublished, partly due to the complexities of its operating procedures.

Frishman (1960) developed extended chain sampling plans designated as ChSP-4

and ChSP-4A (perhaps contemplating publication of designations 2 and 3 by

Dodge). These plans were developed from an application in the sampling

inspection of torpedoes for Naval Ordnance as a check on the control of the

production process and test equipment (including 100% inspection). Features of

these plans included a basic acceptance number greater than zero, an option for

npASN

)( ppPpAOQ a

27

forward or backward accumulation of results for an acceptance-rejection decision

on the current lot, and provision for rejecting a lot on the basis of the results of a

single sample (ChSP-4A). Some variations of chain sampling for which

cumulative results were used in the sentencing of lots had also been developed by

Anscombe et al. (1947), Page (1954), Hill et al. (1959), Ewan and Kemp (1960),

Kemp (1962), Beattie (1962), Wortham and Mogg (1970), Soundarajan (1978a,

1978b) and Vaerst (1982).

Further extensions to a general family of chain sampling inspection

plans had been developed by Dodge and Stephens (1966) and published in

numerous technical reports, conference papers, and journal articles. Raju (1996a,

1996b, 1991,1995, 1997) did extensive research work on chain sampling plans

both cooperatively and independently. His contribution included extending idea of

ChSP-1 and devising tables based on the Poisson model for the construction of

two stage chain sampling plans ChSP-(0, 2) and ChSP-(1, 2) under difference sets

of criteria, outlining the structure of a generalized family of three- stage chain

sampling plans, which extended the concept of two-stage chain sampling plans of

Dodge and Stephens (1966). He also authored a series of 5 papers, which

presented procedures and tables for the construction, and selection of chain

sampling plans ChSP-4A (c1, c2). Balamurali et al. (2008) have explained the

concepts of skip-lot sampling and chain sampling. Balamurali and Palaniswamy

(2012) have determined the minimum variance outgoing quality limit (VOQL)

chain sampling plans for compliance testing.

OPERATING PROCEDURE

The operating procedure of the attributes ChSP-1 is as follows.

Step1: For each lot, select a sample of n units and test each unit for conformance

to the specified requirements

28

Step 2: Accept the lot if the observed number of non-conforming units d is zero,

reject the lot if d ≥ 2.

Step 3: Accept the lot if d is one and if no defective units are found in the

immediately preceding i samples of size n.

Thus a ChSP – 1 plan has two parameters namely n, the sample size for

each submitted lot and i, the number of previous samples on which the decision of

acceptance or rejection of the lot is based.

MEASURES OF ChSP

The important measures of ChSP that describe the operation of an

acceptance sampling plan are,

1. The OC function(see Dodge (1955)) is given by

innna PPPpP ,0,1,0

nP ,0 = Probability of getting exactly 0 defective in a sample of size n

nP ,1 = Probability of getting exactly 1 defective in a sample of size n

2. The Average Sample Number (ASN) is 3 The Average outgoing Quality (AOQ) is given by

npASN

)( ppPpAOQ a

29

1.4.3 TIGHTENED-NORMAL-TIGHTENED (TNT) SCHEME

The tightened-normal-tightened (TNT) sampling procedure developed

by Calvin (1977) is a particular case of the general two-plan system for the

inspection of attributes quality characteristics. This procedure is particularly

appropriate when the product is forthcoming in a stream of lots and a zero

acceptance number is to be maintained. This scheme utilizes two zero acceptance

number single sampling plans of different sample sizes namely n1 and n2 (< n1)

together with the switching rules and this scheme is designated as TNT-(n1, n2; 0).

Calvin (1977) has pointed out that, while increasing the protection to the

producer, the switching rules have no real effect on consumer’s quality level

namely LTPD or LQL which remains essentially that of the tightened plan. This

implies that the TNT scheme provides more protection to the producer while

safeguarding the consumer’s protection. Soundararajan and Vijayaraghavan

(1990) investigated the TNT scheme of type TNT-(n1, n2;c).

OPERATING PROCEDURE

The operating procedure of the attributes TNT scheme is as follows.

Step 1: Start with the tightened inspection level using the single sampling

attributes plan with sample size n1 and the acceptance number c. Accept

the lot if the number of non-conforming units, d ≤ c and reject the lot if

d > c. If ‘t’ lots in a row are accepted under tightened inspection, then

switch to normal inspection.

Step 2: During the normal inspection, inspect the lots using the single sampling

attributes plan with a sample size n2 and the acceptance number c .

Accept the lot if d ≤ c and reject the lot if d > c. Switch to tightened

inspection after a rejection of lot if an additional lot is rejected in the next

‘s’ lots.

30

MEASURES OF TNT SCHEME

The important measures of the TNT scheme that describe the operation

of an acceptance sampling plan for various fraction non-conforming.

1. The OC function (see Calvin (1977)) is given by

where

and

2. The Average Sample Number (ASN) is

1,)(

mmnnpASN

where 1s ,)1)(1(

2

22

2

s

s

PPP

and st ,)1(

1

11

1

PPP

t

t

3. The Average outgoing Quality (AOQ) and

1.4.4 RESAMPLING SCHEME

Govindaraju and Ganesalingam (1997) has proposed an attribute

sampling plan which can be applied in situations where resampling is permitted on

lots not accepted on original inspection. They have derived the performance

measures of the resampling scheme having single sampling attributes plan as the

reference plan. In this plan, it is assumed that during the course of resubmission,

12)( PPpPa

1s ,)1)(1(

2

22

2

s

s

PPP

s t , )1(

1

11

1

PPP

t

t

)( ppPpAOQ a

31

the quality of the lot is not improved by sorting etc. They have also discussed the

need for a provision for resampling of lots in case of zero acceptance sampling

plans. A resubmitted lot is defined in the ANSI/ASQC Standard A2-1987 (1987)

as the one which has been designated as not-acceptable and which is submitted

again for acceptance inspection after having been further tested, sorted,

reprocessed etc. If the lot is not accepted on original inspection, the producer may

test it and may also resubmit it without sorting or reprocessing it for resampling.

Recently, some of the authors have investigated the impact of

resampling scheme under various situations. For example, Aslam et al. (2011)

developed group acceptance sampling plan for resubmitted lots under Burr type

XII distribution. Aslam et. al .(2012) have developed Bayesian sampling

inspection for resubmitted lots under gamma-Poisson distribution.

OPERATING PROCEDURE

The operating procedure of the attributes resampling scheme is as follows.

Step 1: Perform original inspection. i.e., apply a reference (single) sampling plan

(with a sample size n and acceptance number c).

Step 2: On non acceptance on the original inspection, apply the reference plan

m times and reject the lot if it is not accepted on (m-1)st resubmission.

32

MEASURES OF RESAMPLING SCHEME

The important measures that describe the operation of an acceptance

sampling plan for various fraction non-conforming, p are

1. The OC function (see Govindaraju and Ganesalingam (1997)) is given by

)()(1....)()(1)()(1)()( 12 pPpPpPpPpPpPpPpL am

aaaaaa

ma pP )(11

where kvpPa Pr)(

2. The Average Sample Number (ASN) is npPnpPnpPnpASN m

aaa12 )(1....)(1)(1)(

)(

)(11pP

pPn

a

ma

3. The Average outgoing Quality (AOQ) ppPpAOQ a

33

SECTION 5

A REVIEW ON SPECIAL PURPOSE SAMPLING PLANS BY VARIABLES

The special purpose sampling plans were initially developed by Dodge

(1955) for attributes. Kuralmani and Govindaraju (1993) have investigated

conditional sampling plans for given AQL and LQL. Soundararajan and Palanivel

(1997)) have investigated on quick switching variables single sampling system

indexed by AQL and LQL by tightening acceptance criterion.

Govindaraju and Balamurali (1998) extended the idea of chain

sampling plans to variable inspection and examined the related properties and

listed the desired table. Balamurali and Jun (2006) have developed repetitive

group sampling procedure for variables inspection. Balamurali and Subramani

(2010) presented the procedures for designing of variables repetitive group

sampling plan indexed by indifference quality level and the relative slope on the

operating characteristic curve. Vijayaraghavan and Sakthivel (2011) have

developed chain sampling plans based on Bayesian methodology for variables

inspection.

Balamurali et al. (2005) have designed repetitive group sampling plan

for variables involving minimum average sample number. Balamurali and Jun

(2007) have developed multiple dependent state sampling plans for lot acceptance

based on the measurement data. Balamurali and Jun (2009) have designed a

variables two- plan system by minimizing the average sample number. Balamurali

and Subramani (2010) have designed of variables repetitive group sampling plans

indexed by point of control. Wu et al. (2012) and Aslam et al. (2013) investigated

the variables sampling plan for resubmitted lots based on the process capability

index Cpk. Balamurali et al. (2015) developed attribute-variable inspection lots

policy using resampling based on EWMA.

34

CHAPTER 2

OPTIMAL DESIGNING OF

VARIABLES QUICK SWITCHING SAMPLING SYSTEM

BY MINIMIZING THE AVERAGE SAMPLE NUMBER

2.1 INTRODUCTION

This chapter deals with optimal designing of variables quick switching

system (VQSS) where the quality characteristic under study follows normal

distribution and has upper specification limit or lower specification limit. The

known sigma as well as unknown sigma VQSS are designed by minimizing the

average sample number by formulating nonlinear programming problem where the

constraints are related to lot acceptance probabilities at AQL and LQL. Tables are

constructed for finding the optimal parameters of the known sigma as well as

unknown sigma VQSS. The results obtained are compared with that of the existing

plans and proved that the results obtained are optimal.

2.2 CONDITIONS OF APPLICATION

The following assumptions should be valid for the application of the

VQSS.

(i) Production is in a steady state, so that results of past, present and

future lots are broadly indicative of a continuing process.

(ii) Lots are submitted for inspection serially either in the order of

production or in the order of being submitted for inspection.

(iii) Inspection is by measurements, with quality is defined as the

fraction non-conforming, p.

In addition, the usual conditions for the application of variables single sampling

plans with known or unknown standard deviation should also be valid.

35

2.3 OPERATING PROCEDURE OF KNOWN SIGMA VQSS

The operating procedure of the VQSS is as follows.

Suppose that the quality characteristic of interest has the upper

specification limit U and follows a normal distribution with known standard

deviation σ. Then the following procedure of the VQSS is proposed.

Step 1: Start with normal inspection. During normal inspection, take a random

sample of size n, say nXXX ..., 21 and compute

XUv

, where

n

iiX

nX

1

1 .

Step 2: Accept the lot if Nkv and reject the lot if Nkv . If a lot is rejected on

normal inspection, then switch to tightened inspection as in Step 3.

Step 3: During tightened inspection, take a random sample of size n, say

nXXX ..., 21 and compute

XUv

, where

n

iiX

nX

1

1 .

Step 4: Accept the lot if Tkv and reject the lot if Tkv )( NT kk .

If a lot is rejected on tightened inspection, then immediately switch to

normal inspection as in Step 1.

Thus, the VQSS system is characterized by three parameters, namely

n, kN and kT. If kNσ=kTσ, then the system will reduce to the variables single

sampling plan.

36

2.4 OPERATING CHARACTERISTIC FUNCTION OF VQSS

The OC function of the VQSS, which gives the proportion of lots that

are expected to be accepted for given product quality, p under known sigma case

is given by

)Pr()Pr(1

)Pr(1

)(

TN

T

TN

Ta kvkv

kvPP

PpP

(2.1)

where NN kvP Pr is the probability of accepting a lot based on a single

sampling plan with parameters (n, kN) and TT kvP Pr is the probability of

accepting a lot based on a single sampling plan with parameters (n, kT). Under

Type B situation (i.e. a series of lots of the same quality), forming lots of N items

from a process and then drawing random sample of size n from these lots is

equivalent to drawing random samples of size n directly from the process. Hence

the derivation of the OC function is straightforward.

The fraction non-conforming in a lot will be determined as

)()(11 vvUp

(2.2)

where )(y is given by

y

dzzy2

exp21)(

2

, (2.3)

provided that the quality characteristic of interest is normally distributed with

mean µ and standard deviation σ, and the item is classified as non-conforming if it

exceeds the upper specification limit U.

37

Then its probability of acceptance is written as

)()(1

)()(TN

Ta ww

wpP

(2.4)

where nkvw TT and nkvw NN

2.5 DESIGNING OF A KNOWN SIGMA VQSS

The OC function of a known sigma VQSS is given in (2.4). If two

points on the OC curve namely, AQL(=p1), LQL(=p2), the producer’s risk α and

the consumer’s risk β are prescribed then the OC function can be expressed as

1)()(1

)(

11

1

TN

T

www (2.5)

and

)()(1

)(

22

2

TN

T

www (2.6)

Here wT1 is the value of wT at p=p1, wN1 is the value of wN at p=p1, wT2 is the value

of wT at p=p2 and wN2 is the value of wN at p=p2.

That is, nkvw TT )( 11 , nkvw NN )( 11

nkvw TT )( 22 and nkvw NN )( 22 (2.7)

where v1 is the value of v at AQL and v2 is the value of v at LQL. For given AQL

or LQL, the values of kN , kT and the sample size n are determined by using a

search procedure.

2.6 AVERAGE SAMPLE NUMBER

The average sample number (ASN), by definition, means the expected

number of sampled units required for making decisions about the lot. The concept

38

of ASN is meaningful under Type B sampling situations. It is also known that the

ASN of the known sigma VQSS is

nnnpASN

)( (2.8)

2.7 OPTIMAL DESIGNING OF KNOWN SIGMA VQSS

The ASN given above can be used as an objective function to solve for

the parameters (n, kNσ, kT). Since there are several choices to obtain the objective

function, it is considered here to minimize ASN at AQL. If the objective is to

minimize the ASN at AQL, then the problem will be reduced to the following

nonlinear optimization problem.

Minimize ASN(p1)= nσ

Subject to

1)( 1pPa

)( 2pPa

0 ,1 TNσ k kn (2.9)

where )( 1pPa and )( 2pPa are the lot acceptance probabilities at AQL and LQL

respectively and are given in (2.5) and (2.6) respectively.

2.8 OPTIMAL DESIGNING OF UNKNOWN SIGMA VQSS

Whenever the standard deviation is unknown, we should use the sample

standard deviation S instead of σ. In this case, the operation of the proposed

system is as follows.

39

Step 1: Start with the normal inspection level using the variables single sampling

plan with a sample size nS and the acceptance criterion kNS. Accept the

lot if NSkv and reject the lot if NSkv , where

SXU

v

,

Sn

ii

S

Xn

X1

1 and 1

)( 2

S

i

nXX

S . If a lot is rejected under normal

inspection, then switch to tightened inspection.

Step 2: During the tightened inspection, inspect the lots using the variables

single sampling plan with a sample size nS and the acceptance criterion

kTS(>kNS). Accept the lot if TSkv and reject the lot if TSkv , where

SXU

v

,

Sn

ii

S

Xn

X1

1 and 1

)( 2

S

i

nXX

S . If a lot is accepted

on tightened inspection, then immediately switch to normal inspection as

in Step 1.

Thus, the unknown sigma VQSS has the parameters namely the sample

size nS, and the acceptable criterion kNS and kTS. If kNS=kTS, then the VQSS will be

reduced to the variables single sampling plan with unknown standard deviation.

Hamaker (1979) has given an approximation for finding the parameters

of the unknown sigma single sampling plan from the parameters of the known

sigma single sampling plan. The relationship between known and unknown sigma

plan parameters is true only for single sampling plan.

40

Soundarajan and Palanivel (1997) have followed the same approximation

for selecting the parameters of unknown sigma VQSS. However it is to be

pointed out that Hamaker’s (1979) results must be extended to VQSS system

rather than wrongly assuming that the same approximation is valid for VQSS. So

the entire design of unknown sigma schemes provided in Soundararajan and

Palanivel (1997) seems faulty. So we will follow a different procedure for the

unknown sigma case.

The determination of parameters for the unknown sigma case namely

(nS, kNS, kTS) is slightly different from the known sigma case. It is known that

SkX NS is approximately normally distributed with mean )(SEkNS and

variance )(2

SVarkn NS

S

(see Duncan (1986)).

That is.,

SNS

SNSNS n

kn

kNSkX2

,~2

22

Therefore, the probability of accepting a lot under normal inspection is given by

pSkUXP NS pUSkXP NS

21)/(

2NS

S

NS

kn

kU

21

)( 2NS

SNS k

nkv

If we let

21

)( 2NS

SNSNS k

nkvw then the probability of acceptance under

tightened inspection is considered )( NSw .

41

Similarly if we let

21

)( 2TS

STSTS k

nkvw then the probability of acceptance

under tightened inspection is taken as )( TSw . Hence the lot acceptance

probability of the proposed system for sigma unknown case under two-points on

the OC curve is given by

)()(1

)()(

11

11

STSN

STa ww

wpP

(2.10)

and )()(1

)()(

22

22

STSN

STa ww

wpP

(2.11)

We obtain STSNSTSN wwww 2211 ,,, corresponding to 2211 ,,, TNTN wwww respectively by

21

)( 211NS

SNSSN k

nkvw ,

21

)( 211TS

STSST k

nkvw

21

)( 222NS

SNSSN k

nkvw and

21

)( 222TS

STSST k

nkvw

In this case, the optimization problem becomes,

Minimize ASN(p1) = nS

Subject to

1)( 1pPa

)( 2pPa

0 ,1 TSNSs k kn (2.12)

42

where )( 1pPa and )( 2pPa are the lot acceptance probabilities of the proposed

sampling system at AQL and LQL respectively and are described in (2.10) and

(2.11).

We may determine the parameters of the known sigma and

unknown sigma VQSS by solving the nonlinear equation given in (2.9) and

(2.12) respectively. There may exist multiple solutions since there are three

unknowns with only two equations. Generally a sampling would be desirable if

the required number of sampled is small. So, in this chapter, we consider the

ASN as the objective function to be minimized with the probability of

acceptance along with the corresponding producer’s and consumer’s risks as

constraints. To solve the above nonlinear optimization problems given in (2.9)

and (2.12), the sequential quadratic programming (SQP) proposed by Nocedal

and Wright (1999) can be used. The SQP is implemented in Matlab software

using the routine “fmincon”. By solving the nonlinear problem mentioned above,

the parameters (n, kN and kT) for known sigma plan and the parameters (nS, kNS

and kTS) for unknown sigma plan are determined and these values are tabulated in

Table 2.2.

2.9 EXAMPLES

2.9.1 SELECTION OF KNOWN SIGMA VQSS INDEXED BY AQL AND LQL Table 2.1 is used to determine the parameters of the known sigma

VQSS for specified values of AQL and LQL when = 5% and = 10%. For

example, if p1 = 2%, p2 = 8%, = 5% and = 10%, Table 2.1 gives the

parameters as n = 12, kN = 1.552 and kT = 1.817.

43

For the above example, the operation of the VQSS is explained as follows.

Step 1: Take a random sample of size 12 from the submitted lot for inspection

and compute

XUv , where

12

1121

iiXX . Accept the lot if 552.1v

and reject the lot if 552.1v . If a lot is rejected, then switch to tightened

inspection as in step 2.

Step 2: Select a random sample of size 12 and compute

XUv , where

12

1121

iiXX . Accept the lot if 817.1v and reject the lot if 817.1v .

Switch to normal inspection as in step 1, if a lot accepted in the tightened

inspection phase.

2.9.2 SELECTION OF UNKNOWN SIGMA VQSS INDEXED BY AQL AND LQL Table 2.1 can also be used for the selection of the parameters of the

unknown VQSS for given values of AQL and LQL. Suppose that AQL=1%,

LQL=5%, =5% and =10%. From Table 2.1, the parameters of the VQSS can be

determined as nS = 24, kNS = 1.729 and kTS = 2.214.

2.10 ADVANTAGES OF THE VQSS

In this section, we will discuss the advantages of the VQSS over attributes

QSS and variables single sampling plans. For the purpose of comparison, we will

consider the plans which have the same AQL and LQL.

44

Suppose that for given values of AQL=0.01, =5%, LQL=0.02 and =

10%, one can find the parameters of the attributes QSS under the application of

Poisson model as (i) n = 251, cT = 5 and cN = 9

For the same AQL and LQL, we can determine the parameters of the

variables single sampling plan (from Sommers (1981)) and VQSS (from Table

2.1) respectively as follows.

(ii) nσ = 116 and k = 2.17

(iii) n = 28, kNσ = 1.923 and kTσ = 2.418

By comparing the above, it is clear that the VQSS achieves a reduction

of over 89% in sample size than the attributes QSS and about 76% than the

variables single sampling plan with same AQL and LQL conditions. In order to

show the better efficiency of the VQSS, three OC curves are considered.

Figure 2.1 shows the OC curves of the variables single sampling plans

with parameters (10, 1.754) and (10, 2.179) and the VQSS with parameters (10;

1.754, 2.179). The VQSS (10; 1.754, 2.179) is selected in such a way that it

satisfies the two-points (p1 = 0.01, 1-α = 0.95) and (p2 = 0.045, β = 0.10) on the

OC curve.

2.11. COMPARISON

In this section, we compare the parameters of VQSS with those of

VQSS given in Soundararajan and Palanivel (1997). It is to be pointed out that the

ASNs of VQSS provided in Soundararajan and Palanivel (1997) are equal or

greater than the ASN of single sampling plans for some combinations of p1 and

p2. For example, for given p1=0.01 and p2=0.04, ASN of variables single sampling

plan is 506, but the ASN of VQSS given in Soundararajan and Palanivel (1997) is

1856. This is a contradiction, since in the attributes case, QSS will always have

45

minimum ASN than the attributes single sampling plan (see Romboski (1969) and

Soundararajan and Arumainayagam (1992)). This should be valid for variables

sampling also. Hence the entire design of variables QSS provided in

Soundararajan and Palanivel (1997) for both known and unknown sigma seems

faulty or doubtful. Hence the parameters given in this chapter are more reliable

and optimum.

Further, it is also to be pointed that the VQSS is economically superior

to the variables double sampling plan in terms of ASN. Obviously, a sampling

plan having smaller ASN would be more desirable. The variables double or

multiple sampling plans are not practically very useful. Variables sampling

Standards avoid presenting such plans due to increased complexity involved in

operating them.

Table 2.2 shows the ASN values of the variables single sampling plan

and the variables double sampling plan along with the VQSS for some arbitrarily

selected combinations of AQL and LQL under known sigma case. Table 2.3

gives the ASN values of the above said plans when sigma is unknown. These

ASN values are calculated at the producer’s quality level for both known and

unknown sigma plans. The sample size of the variables single sampling plan and

the ASN of the variables double sampling plan can be found in Sommers (1981).

Tables 2.4 and 2.5 apparently show that the VQSS will have minimum

ASN when compared to the variables single and double sampling plans for both

known and unknown sigma cases. Similar reduction in ASN can be achieved for

any combination of AQL and LQL values. This implies that VQSS will give

desired protection with minimum inspection so that the cost of inspection will

greatly be reduced. Thus the VQSS provides better protection than the variables

single sampling plans and variables double sampling.

46

Table 2.1. Variables Quick Switching Sampling Systems Indexed by AQL and LQL for =5% and =10% Involving Minimum ASN

p1

p2

MinASN(p1) Known Sigma

MinASN(p1) Unknown Sigma

n kTσ kNσ nS kTS kNS 0.001

0.0025

0.005

0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 0.012 0.015 0.025

0.004 0.005 0.006

0.0075 0.010 0.015 0.020 0.025 0.030 0.040

0.0075 0.010 0.012 0.015 0.020 0.030 0.035 0.040 0.060

134 45 31 14 11 10 14 8 7 9 6 4

96 35 26 23 14 9 8 6 8 5

100 31 23 19 9 9 7 7 6

2.998 2.958 2.899 2.994 2.985 2.939 2.769 2.897 2.899 2.710 2.751 2.672

2.841 2.917 2.878 2.773 2.762 2.693 2.572 2.569 2.353 2.374

2.626 2.683 2.641 2.549 2.657 2.377 2.385 2.298 2.109

2.943 2.833 2.784 2.599 2.535 2.519 2.639 2.457 2.409 2.525 2.381 2.222

2.596 2.432 2.398 2.413 2.302 2.193 2.182 2.079 2.213 2.039

2.361 2.183 2.146 2.139 1.897 1.982 1.895 1.913 1.884

234 108 104 71 60 45 38 35 32 28 24 16

326 156 113 82 56 37 33 28 19 15

330 123 89 63 45 28 23 20 15

3.193 3.166 2.999 2.998 2.955 2.985 2.982 2.947 2.926 2.880 2.819 2.715

2.917 2.945 2.910 2.861 2.809 2.710 2.563 2.490 2.564 2.485

2.679 2.705 2.664 2.617 2.533 2.443 2.424 2.394 2.224

2.773 2.666 2.704 2.628 2.600 2.525 2.482 2.467 2.446 2.415 2.379 2.260

2.542 2.445 2.410 2.371 2.309 2.235 2.233 2.200 2.074 2.010

2.329 2.205 2.169 2.122 2.073 1.978 1.929 1.894 1.834

47

Table 2.1. Contd….

p1

p2

MinASN(p1) Known Sigma

MinASN(p1) Unknown Sigma

n kTσ kNσ nS kTS kNS 0.0075

0.010

0.015

0.010 0.012 0.015 0.020 0.025 0.030 0.035 0.040 0.050 0.060 0.070

0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.060 0.070 0.080 0.090 0.100

0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.060 0.070 0.080

149 58 34 23 14 11 11 11 8 7 6

68 28 21 15 13 14 10 14 7 6 6 6 3

129 43 26 23 22 21 13 11 8 7

2.499 2.527 2.485 2.391 2.402 2.367 2.263 2.179 2.151 2.086 2.036

2.427 2.418 2.334 2.314 2.255 2.141 2.179 2.005 2.123 2.084 1.987 1.904 1.906

2.236 2.265 2.252 2.163 2.082 2.012 2.083 2.009 2.021 1.952

2.244 2.137 2.080 2.041 1.922 1.867 1.893 1.909 1.816 1.781 1.686

2.047 1.923 1.899 1.829 1.810 1.856 1.754 1.875 1.648 1.599 1.617 1.629 1.557

1.971 1.835 1.757 1.768 1.795 1.787 1.658 1.629 1.526 1.526

575 209 107 64 47 36 31 26 20 17 14

243 94 62 47 38 32 27 24 20 17 14 11 11

402 121 78 59 48 39 34 26 22 18

2.502 2.549 2.549 2.486 2.426 2.395 2.337 2.313 2.266 2.196 2.175

2.434 2.441 2.393 2.344 2.304 2.268 2.247 2.214 2.148 2.100 2.089 2.138 2.027

2.248 2.317 2.279 2.233 2.188 2.167 2.131 2.088 2.025 2.005

2.252 2.144 2.059 1.996 1.951 1.900 1.877 1.838 1.776 1.741 1.685

2.064 1.946 1.893 1.854 1.819 1.788 1.752 1.729 1.693 1.655 1.599 1.513 1.532

1.973 1.827 1.779 1.748 1.723 1.687 1.666 1.613 1.585 1.535

48

Table 2.1 Contd….

p1

p2

MinASN(p1) Known Sigma

MinASN(p1) Unknown Sigma

n kTσ kNσ nS kTS kNS 0.015

0.020

0.030

0.040

0.090 0.100

0.030 0.035 0.040 0.045 0.050 0.060 0.070 0.080 0.090 0.100 0.120

0.040 0.045 0.050 0.060 0.070 0.080 0.090 0.100 0.120 0.150 0.200

0.060 0.070 0.080 0.090 0.100 0.110 0.120 0.140 0.160

7 5

87 33 26 21 17 17 17 12 8 7 5

116 57 33 22 17 14 12 10 11 6 5

58 29 21 17 14 14 14 10 8

1.888 1.856

2.070 2.158 2.108 2.076 2.060 1.926 1.813 1.817 1.877 1.851 1.700

1.934 1.957 1.998 1.947 1.894 1.844 1.800 1.781 1.597 1.647 1.470

1.801 1.826 1.796 1.754 1.725 1.639 1.565 1.546 1.512

1.508 1.376

1.840 1.673 1.653 1.621 1.580 1.618 1.638 1.552 1.417 1.376 1.345

1.679 1.592 1.493 1.442 1.404 1.374 1.345 1.296 1.362 1.152 1.110

1.476 1.361 1.311 1.279 1.240 1.264 1.280 1.191 1.127

16 14

170 96 68 54 50 33 26 24 19 16 18

269 133 89 54 40 31 26 29 24 13 10

104 63 47 37 30 26 24 17 17

1.956 1.927

2.167 2.172 2.153 2.120 2.046 2.044 2.004 1.910 1.911 1.897 1.664

1.965 1.999 1.999 1.974 1.925 1.893 1.846 1.695 1.598 1.644 1.458

1.892 1.869 1.819 1.778 1.749 1.704 1.644 1.634 1.195

1.511 1.477

1.772 1.702 1.658 1.630 1.636 1.554 1.510 1.510 1.451 1.407 1.474

1.665 1.584 1.535 1.469 1.430 1.388 1.361 1.415 1.388 1.224 1.173

1.427 1.369 1.339 1.308 1.274 1.254 1.249 1.169 1.485

49

Table 2.1 Contd….

p1

p2

MinASN(p1) Known Sigma

MinASN(p1) Unknown Sigma

n kTσ kNσ nS kTS kNS

0.050

0.18 0.200

0.060 0.070 0.080 0.090 0.100 0.120 0.140 0.160 0.200 0.250

7 4

280 66 35 25 23 14 11 9 6 5

1.460 1.317

1.666 1.723 1.735 1.711 1.630 1.619 1.560 1.511 1.432 1.324

1.090 0.882

1.521 1.373 1.280 1.231 1.245 1.134 1.085 1.036 1.006 0.860

12 10

462 132 73 53 40 28 21 16 15 8

1.521 1.496

1.708 1.761 1.768 1.730 1.709 1.640 1.585 1.560 1.328 1.375

1.096 1.046

1.488 1.366 1.293 1.260 1.219 1.170 1.120 1.060 1.083 0.895

50

From Figure 2.1, it can be easily observed that, for good quality, i.e. for smaller

values of fraction nonconforming, the composite OC curve (OC curve of the

VQSS) coincides with the OC curve of the variables single sampling plan (10,

1.754). As quality deteriorates the OC curve of the composite OC curve moves

toward that for the single sampling plan (10, 2.179) and comes close to it beyond

the indifference quality level.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1Fraction Nonconforming, p

Prob

abili

ty o

f Acc

epta

nce,

Pa(

p)

Figure.2.1. OC Curves of Single Sampling Normal Plan (10, 1.754), Quick Switching System (10; 1.754, 2.179) and Single Sampling Tightened Plan (10, 2.179)

Variables Normal SSP

VQSS

Variables Tightened SSP

51

Table 2.2: ASN Values of the Known Sigma Variables Single Sampling Plan, Variables Double Sampling Plan and VQSS

p1

p2

ASN Variables

SSP Variables

DSP Variables*

QSS

Variables ** QSS

0.001 0.001 0.005 0.005 0.03 0.03 0.04 0.04 0.05 0.05

0.002 0.003 0.010 0.012 0.04 0.06 0.07 0.08 0.07 0.08

191 74 138 85 506 81 114 72 300 149

154.9 59.4

112.0 69.5

434.6 127.7 180.6 58.4

246.7 122.3

191 74

138 85

506 81

114 72

300 149

134 45 31 23

116 22 29 21 66 35

* ASN given in Soundararajan and Palanivel (1997) ** ASN given in Table 2.1

Table 2.3: ASN Values of the Unknown Sigma Variables Single Sampling Plan, Variables Double Sampling Plan and VQSS

p1

p2

ASN Variables

SSP Variables

DSP Variables*

QSS

Variables ** QSS

0.001 0.001 0.005 0.005 0.03 0.03 0.04 0.04 0.05 0.05

0.002 0.003 0.010 0.012 0.04 0.06 0.07 0.08 0.07 0.08

1032 381 547 327

1333 197 258 159 660 319

829.1 302.4 437.1 263.0 1138.7 316.5 417.6 125.8 535.4 258.0

1032 412 941 823 945 357 263 201 768 572

234 108 123 89

269 89

104 47

132 73

* ASN given in Soundararajan and Palanivel (1997) ** ASN given in Table 2.1

52

Table 2.4: Parameters of Known Sigma VQSS for Some Selected Combinations of AQL and LQL Values

p1

p2

*Parameters of VQSS

**Parameters of VQSS

n kNσ kTσ n kNσ kTσ 0.001 0.001 0.005 0.005 0.03 0.03 0.04 0.04 0.05 0.05

0.002 0.003 0.010 0.012 0.04 0.05 0.06 0.08 0.06 0.07

370 95 399 126

1856 515 965 121

1311 853

2.93 2.87 2.40 2.39 1.73 1.73 1.62 1.55 1.55 1.53

3.10 3.00 2.60 2.50 2.00 1.90 1.78 1.68 1.66 1.66

134 45 31 23

116 33 58 21

280 66

2.943 2.833 2.183 2.146 1.679 1.493 1.476 1.311 1.521 1.373

2.998 2.958 2.683 2.641 1.934 1.998 1.801 1.796 1.666 1.723

*Parameters from Soundararajan and Palanivel (1997) **Parameters from Table 2.1 Table 2.5: Parameters of Unknown Sigma VQSS for Some Selected Combinations of AQL and LQL Values

p1

p2

*Parameters of VQSS

**Parameters of VQSS

nS kNS kTS nS kNS kTS 0.001 0.001 0.005 0.005 0.03 0.03 0.04 0.04 0.05 0.05

0.002 0.003 0.010 0.012 0.04 0.05 0.06 0.08 0.06 0.07

2053 504

1647 502

5100 1365 2362 279

3001 1939

2.93 2.87 2.40 2.39 1.73 1.73 1.62 1.55 1.55 1.53

3.10 3.00 2.60 2.50 2.00 1.90 1.78 1.68 1.66 1.66

234 108 123 89

269 89

104 47

462 132

2.773 2.666 2.205 2.169 1.665 1.535 1.427 1.339 1.488 1.366

3.193 3.166 2.705 2.664 1.965 1.999 1.892 1.819 1.708 1.761

*Parameters from Soundararajan and Palanivel (1997) **Parameters from Table 2.1

53

CHAPTER 3

OPTIMAL DESIGNING OF VARIABLES QUICK SWITCHING SYSTEM

WITH DOUBLE SPECIFICATION LIMITS

3.1 INTRODUCTION

In this chapter, we have investigated VQSS when a measurable quality

characteristic has both upper and lower specification limits and follows normal

distribution. Most of the variables sampling plans are dealing with quality

characteristics having only one specification limit either lower or upper

specification limit. In the literature, there have been some studies available which

are dealing with sampling plans based on double specification limits. Military

Standard MIL-STD-414 (1957) described the procedure for the selection of a

variables single sampling plan involving the double specification limits. Schilling

(1982) suggested the use of two single sampling plans to handle the problem of

quality characteristics having double specification limits. Since the variables

sampling plans with double specifications are having wide applications, this

chapter deals with the designing methodology for determining the parameters of a

VQSS for normally distributed quality characteristics having double specification

limits.

3.2 CONDITIONS FOR APPLICATION OF VQSS

The following assumptions should be valid for the application of the VQSS.

(i) Production is in a steady state, so that results of past, present and future lots

are broadly indicative of a continuing process.

(ii) Lots are submitted for inspection serially either in the order of production or

in the order of being submitted for inspection.

54

(iii) Inspection is by measurements, with quality is defined as the fraction

nonconforming, p.

In addition, the usual conditions for the application of variables single

sampling plans with known or unknown standard deviation should also be valid.

3.3 OPERATING PROCEDURE OF A KNOWN SIGMA VQSS

Suppose that the standard deviation of the normal distribution is

known. Then, the operating procedure of VQSS with double specification limits is

shown below.

Step 1: Start with normal inspection. During normal inspection, take a random

Sample of size n, say nXXX ..., 21 and compute

1

X Lv

and

2

U Xv

, where

n

iiX

nX

1

1

Step 2: Accept the lot if 1 1Nv k and 2 2Nv k . Reject the lot if 1 1Nv k and/or

2 2Nv k . If a lot is rejected on normal inspection, then switch to tightened

inspection as in Step 3, otherwise continue the normal inspection.

Step 3: During tightened inspection, take a random sample of size n, say

nXXX ..., 21 and compute

1

X Lv

and

2

U Xv

, where

n

iiX

nX

1

1 .

55

Step 4: Accept the lot if 1 1Tv k and 2 2Tv k . Reject the lot if 1 1Tv k and/or

2 2Tv k 1 1 2 2( )T N T Nk k and k k . If a lot is accepted on tightened

inspection, then immediately switch to normal inspection as in Step 1.

Thus, the VQSS with double specification limits is characterized by

five parameters, namely 1 1 2 2( , , , )T N T Nn k k k and k . If 1 1 1 ( ) T Nk k k say and

2 2 2 ( )T Nk k k say , then the proposed system will reduce to the variables single

sampling plan with double specification limits. Also when the true mean is located

at the middle of double specification limits, that is, 2/)( UL , and if

1 2N Nk k and 1 2T Tk k , then it is called as the symmetric fraction

nonconforming case. If there is only one specification limit either L or U as in

Balamurali and Usha ( 2012a), then the proposed system can be used with L=-∞

( 1v ) or U=∞ ( 2v ). It is to be pointed out that the VQSS with single

specification limit presented in Chapter 2 is a special case of the VQSS with

double specification limits provided in this Chapter.

3.4 OC FUNCTION OF A KNOWN SIGMA VQSS WITH DOUBLE

SPECIFICATION LIMITS

The OC function of the VQSS, which gives the proportion of lots that

are expected to be accepted for given product quality, p under known sigma case is

given by

)()(1

)()(

pPpPpP

pPTN

Ta

(3.1)

Under double specification limits, the above can be written as

),(),(1

),()(22112211

2211

TTNN

TTa kvkvPkvkvP

kvkvPpP

(3.2)

56

where ),( 2211 NNN kvkvPP is the probability of accepting a lot under normal

inspection based on a sample with parameters 1 2, ,N Nn k k and

),( 2211 TTT kvkvPP is the probability of accepting a lot under tightened single

sampling plan with parameters 1 2, ,T Tn k k . Under type B situation (i.e. a series

of lots of the same quality), forming lots of N items from a process and then

drawing random sample of size n from these lots is equivalent to drawing random

samples of size n directly from the process. Hence the derivation of the OC

function is straightforward.

The distributions of 1v and 2v follows normal distributions with the same

variance and means /)( L and /)( U respectively. The fraction non-

conforming p for the quality characteristic with double specification limits will be

defined by

}{}{ UXPLXPp (3.3)

where X denotes the individual measurement of the quality characteristic under

study . When the fraction non-conforming beyond the lower limit and that beyond

the upper limit are specified separately, the means for 1v and 2v can be determined.

3.4.1 KNOWN SIGMA VQSS WITH SYMMETRIC FRACTION NON- CONFORMING In the case of symmetric fraction non-conforming, we first assume that

2/}{}{ pUXPLXP (3.4)

Then, for the specified fraction non-conforming p,

Uz p 2/ (3.5)

57

Lzz pp 2/2/1 (3.6)

where z is the standard normal variate corresponding to the tail probability of .

In the case of symmetric fraction non-conforming, we have

NNN kkk 21 and TTT kkk 21 . In this case, the design parameters will be

only n , Nk and Tk . Therefore, the probability of acceptance under tightened

inspection TP for the specified p reduces to

pkvkvPP TTT 21 , = pkUXkLP TT

1))((2 2/ nkzP TpT (3.7)

where )( is the cumulative standard normal distribution function and /2T pk z .

Similarly, the probability of acceptance under normal inspection NP for the specified p will be determined by

)])((1[2 2/ nkzP NpN (3.8)

where / 2N pk z .

Hence, the probability of acceptance of the proposed VQSS under double

specification limits is given by

1))((2))((2

1)])(([2)(

2/2/

2/

nkznkznkz

pPNpTp

Tpa

(3.9)

As mentioned earlier, the above OC function reduces to the OC

function of the variables single sampling plan with double specification limits

when kkk NT . Hence when kkk NT (3.9) becomes

58

1))((2)( 2/ nkzpP pa (3.10)

where n is the sample size of the variables single sampling plan and k is the acceptance criterion.

3.4.2 KNOWN SIGMA VQSS WITH ASYMMETRIC FRACTION NON-CONFORMING Generally, in some situations, the fraction non-conforming below the

lower specification limit and above the upper specification limit are different. In

such kind of situations, it will be assumed that

LpLXP }{ and UpUXP }{ such that ( ppp UL ) (3.11)

Then,

Uz

Up (3.12)

and

Lzz

LL pp1 (3.13)

Therefore, the probability of acceptance under tightened inspection for the

specified p reduces to

pkvkvPP TTT 2211 , = pkUXkLP TT 21

))(())(( 12 nzknkzPLU pTTpT (3.14)

Similarly, the probability of acceptance based on normal inspection for the

specified p becomes

))(())((1 12 nzknkzPLU pNNpN (3.15)

It should be noted that the condition ofUpTN zkk 22 and

LpTN zkk 11 are

needed for (3.14) and (3.15).

59

Therefore, the probability of acceptance of a lot under VQSS with double

specification limits is obtained by

)])(())(([])(())(([1

))(())(()(

1212

12

nzknkznzknkz

nzknkzpP

LULU

LU

pTTppNNp

pTTpa

(3.16)

The above function reduces to the OC function of the single sampling plan with

double specifications when 111 kkk NT (say) and 222 kkk NT (say). In this

case, (3.16) reduces to

))(())(()( 12 nzknkzpPLU ppa (3.17)

where n is the sample size of the variables single sampling plan and 1k and 2k are the acceptance criterion under double specification limits.

3.5 DESIGNING OF UNKNOWN SIGMA VQSS HAVING DOUBLE SPECIFICATION LIMITS

The standard deviation of a normal distribution is unknown in some of

the practical applications. In such cases, we should use the estimated standard

deviation from the sample. So, the unknown sigma VQSS with double

specification limits is operated as follows.

Step 1: Start with normal inspection. During normal inspection, take a random

sample of size n, say 1 2, ... nX X X and compute 1

X Lv

S

and

2

U Xv

S

, where 1

1 n

ii

X Xn

and 2

1

11

n

ii

S X Xn

.

60

Step 2: Accept the lot if NSkv 11 and NSkv 22 . Reject the lot if NSkv 11 and/or

NSkv 22 . If a lot is rejected on normal inspection, then switch to tightened

inspection as in Step 3, otherwise continue the normal inspection.

Step 3: During tightened inspection, take a random sample of size n, say

1 2, ... nX X X and compute 1

X Lv

S

and 2

U Xv

S

, where

1

1 n

ii

X Xn

and 2

1

11

n

ii

S X Xn

.

Step 4: Accept the lot if TSkv 11 and TSkv 22 . Reject the lot if TSkv 11 and/or

TSkv 22 ( NSTSNSTS kkandkk 2211 ). If a lot is rejected on tightened

inspection, then immediately switch to normal inspection as in Step 1.

Thus, the proposed sampling system under unknown standard deviation is also

characterized by five parameters n , NSk1 , NSk2 , TSk1 and TSk2 .

Here we consider only the symmetric case for the purpose of explaining

the designing methodology. That is, it will be assumed that NSNSNS kkk 21 and

TSTSTS kkk 21 .

Then, the probability of accepting a lot based on the tightened inspection becomes

pSkUXSkLPP TSTST

pLSkXPpUSkXP TSTS (3.18)

61

If we use the result that kSX (for a constant k) is approximately distributed as

normal as follows (see Duncan (1986)):

nk

nkNkSX

2,~

222 (3.19)

Then the equation (3.18) reduces to

12

2)(2 22/

TSTSpT k

nkzP (3.20)

Similarly, the probability of accepting a lot under normal inspection can be

obtained similarly as follows:

22/ 2

2)(22NS

NSpN knkzP (3.21)

The condition of 2/pTSNS zkk is needed in (3.20) and (3.21). Finally,

the probability of acceptance of a lot for the specified fraction non-conforming, p

will be obtained by using the equation (3.1), where PT and PN are given in (3.20)

and (3.21) respectively. As in the known sigma case, it reduces to the unknown

sigma single sampling plan with double specification limits when SNSTS kkk .

For asymmetric fraction non-conforming case, similar procedure can be adopted

for designing of unknown sigma VQSS.

3.6 DETERMINATION OF THE OPTIMAL PARAMETERS OF VQSS

The optimal parameters of the VQSS can be determined by using the

two-points on the OC curve approach that considers both the producer’s risk α

and the consumer’s risk β along with the corresponding quality levels. When the

product quality is at AQL, the probability of acceptance of a lot should be more

62

than 1-α, whereas the probability should be less than β when the quality is at the

specified LQL.

If the AQL and LQL are designated as 1p and 2p , respectively, then the

probability of acceptance should satisfy the following conditions.

1( ) 1aP p

2( )aP p (3.22)

There may exist multiple solutions to satisfy the above two inequalities,

so the objective function of minimizing the ASN will be considered. The ASN for

the VQSS at the quality level of p can be determined as

( )ASN p n (3.23)

We may evaluate the ASN at 1p or at 2p , since ASN for VQSS is a

constant nothing but the sample size only, irrespective of the quality levels. Hence,

the optimization problem for determining the optimal parameters for the known

sigma asymmetric case is given by

Minimize ( )ASN p

Subject to

1( ) 1aP p

2( )aP p

1n , 1 22 20 min( , )

U UN T p pk k z z

1 21 10 min( , )

L LN T p pk k z z (3.24)

To solve the above nonlinear optimization problem, the sequential

quadratic programming (SQP) proposed by Nocedal and Wright (1999) can be

63

used. The SQP has been implemented in Matlab Software using the routine

“fmincon”.

Four tables are developed and provided for the selection of optimal

parameters of the VQSS. Table 3.1 and Table 3.2 give the optimal parameters of

the known sigma VQSS for symmetric and asymmetric fraction non-conforming

respectively, whereas Table 3.3 and Table 3.4 provide the optimal parameters of

the VQSS for symmetric and asymmetric fraction non-conforming respectively

for the specified values of AQL and LQL when )1( =0.95 and =0.1.

3.7 ILLUSTRATIVE EXAMPLES

3.7.1 SYMMETRIC FRACTION NONCONFORMING CASE

Suppose a quality characteristic of interest follows a normal distribution

with =10 and double specification limits of L=75 and U=125. The inspector

wishes to adopt a VQSS, where AQL at =0.05 and LQL at =0.1 are specified

by 1p =0.01, 2p =0.05, respectively. Then, Table 3.1 gives the parameters as n =4,

559.1Nk and 932.1Tk .

Table 3.2 provides the design parameters of the known sigma VQSS

(asymmetric case) for the specified values of AQL and LQL when =0.05 and

=0.1. Here, Lp and Up were assumed to be 1/4 and 3/4, respectively, of AQL

or LQL.

3.7.2 ASYMMETRIC FRACTION NONCONFORMING CASE

Suppose one wants to determine the optimal parameters of VQSS when

the quality characteristic of interest follows a normal distribution with =15 and

double specification limits of L=75 and U=125 for the specified AQL and LQL

conditions, Table 3.2 can be used. For example, the AQL and LQL are specified as

64

1p =0.01 and 2p =0.05 at =0.05 and at =0.1 respectively. Then, Table 3.2 gives

the optimal parameters as n =4, 759.11 Nk , 443.12 Nk , 168.21 Tk

and 802.12 Tk .

For the above example, the system operates as follows.

Step 1: Start with normal inspection. During normal inspection, take a random

sample of size 4. Now, compute 1

X Lv

and

2

U Xv

, where

4

141

iiXX .

Step 2: Accept the lot if 759.11 v and 443.12 v . Reject the lot if 759.11 v and/or

443.12 v . If a lot is rejected on normal inspection, then switch to tightened

inspection as in Step 3.

Step 3: During tightened inspection, take a random sample of size 4, and compute

1

X Lv

and

2

U Xv

, where

4

141

iiXX .

Step 4: Accept the lot if 168.21 v and 802.12 v . Reject the lot if 168.21 v

and/or 802.12 v . If a lot is rejected on tightened inspection, then

immediately switch to normal inspection as in Step 1.

In a similar way, the unknown sigma VQSS can be developed and for

the easy selection of the optimal parameters Table 3.3 and Table 3.4 are also

constructed. Table 3.3 shows the design parameters of the unknown sigma VQSS

(symmetric case) and Table 3.4 gives the optimal parameters of unknown sigma

65

VQSS (asymmetric case) for the specified values of AQL and LQL for =0.05

and =0.1.

3.8 INDUSTRIAL APPLICATION OF THE PROPOSED VQSS

In order to apply the proposed VQSS with double specification limits in

real-life situations, we consider an industrial case study example as provided by

Wu and Pearn (2008).

Wu and Pearn (2008) stated the example as “Liquid crystals have been

used for display applications with various configurations. Most of the produced

displays recently involve the use of either twisted nematic (TN), or super twisted

nematic (STN) liquid crystals. The technology of the STN display was introduced

recently to improve the performance of LCD as an alternative to the TFT. A

larger twist angle can lead to a significantly larger electro-optical distortion. This

leads to a substantial improvement in the contract and viewing angles over TN

displays. An increasing number of personal computers are now network-ready

and multimedia-capable and are equipped with CD-ROM drives. Due to advances

in telecommunications’ technology, simple monochromatic displays are no longer

in popular demand. The next generation of telecommunication products will

require displays with rich, graphic quality images and personal interfaces.

Therefore, future display s must be clearer and sharper to meet these demands.

Until this point, STN-LCD has been used mainly to display still images, and

because of the slow response time needed to process still images, STN-LCD has

not been able to reproduce animated images at an adequate contrast level. Thus,

with the growing popularity of multimedia applications, there is a need for PCs

equipped with color STN-LCD capable of processing animated pictures instead of

still images. The space between the glass substrate is filled with liquid crystal

66

material and the thickness of the liquid crystal is kept uniform with glass fibers or

plastic balls as spacers. Thus, the STN-LCD is sensitive to the thickness of the

glass substrates”.

To illustrate how the proposed VQSS can be established and applied to the

actual data collected from the factories, we present a case study on STN-LCD

manufacturing process as proposed by Wu and Pearn (2008). The STN-LCD is

popularly used in making the PDA (personal digital assistant), Notebook personal

computer, Word Processor, and other Peripherals. The factory manufactures

various types of the LCD. For a particular model of the STN-LCD investigated,

the upper specification limit (U) of a glass substrate’s thickness is 0.77 mm, the

lower specification limit (L) of a glass substrate’s thickness is 0.63 mm. If the

product characteristic does not fall within the specification limits (L, U), the

lifetime or reliability of the STN-LCD will be discounted. In the contract, the

AQL and LQL are set to 0.05 and 0.1 with =5% and =10% respectively.

Therefore, the problem for quality practitioners is to determine the optimal

parameters of the proposed sampling system that provide the desired levels of

protection for both the producer and the consumer. Suppose that the quality

characteristic of interest has symmetric fraction non-conforming for specified

AQL and LQL conditions and the variance of the process is unknown. Based on

the proposed procedure, we can obtain the optimal parameters from Table 3.3 as

n =33, 632.1TSk and 329.1NSk . In this example, the proposed VQSS with

double specification limits can be implemented as follows.

67

Step 1: Start with normal inspection. From each submitted lot, take a random

sample of size 33. Suppose that the data may be as follows.

0.642 0.720 0.726 0.684 0.727 0.632 0.657

0.630 0.748 0.635 0.688 0.665 0.710 0.633

0.712 0.633 0.712 0.733 0.739 0.700 0.699

0.640 0.645 0.731 0.669 0.659 0.641

0.632 0.651 0.658 0.768 0.656 0.712

For this data, calculate 681424.011

Sn

iiX

nX and 041264.0

1)( 2

nXX

S i .

Also calculate 041264.0051424.0)(

1

S

LXv = 1.24622

and 041264.0088576.0)(

2

S

XUv = 2.14657

Step 2: Even though 329.114657.22 NSkv but, 329.124622.11 NSkv , the

lot is rejected. Since the lot is rejected on normal inspection, then immediately

switch to tightened inspection as in Step 3, for the next lot.

Step 3: During tightened inspection, take a random sample of size 33, from the

next consecutive lot. In this case, the data may be as follows.

0.695 0.764 0.786 0.699 0.757 0.732 0.657

0.730 0.764 0.695 0.744 0.765 0.671 0.653

0.718 0.693 0.742 0.735 0.679 0.753 0.739

0.655 0.623 0.751 0.699 0.639 0.691

0.642 0.675 0.739 0.748 0.666 0.742

68

For this data, calculate 68203.011

Sn

iiX

nX and 043865.0

1)( 2

nXX

S i .

Also calculate 043865.005203.0)(

1

S

LXv = 1.18614

and 043865.008797.0)(

2

S

XUv = 2.00547

Step 2: Since 632.100547.22 TSkv but, 632.118614.11 TSkv , the lot is

rejected. Because the lot is rejected on tightened inspection also, the same

tightened inspection will be done for the next lot.

3.9 COMPARISONS

For the purpose of comparing the proposed system with the other

sampling plans, we provide Table 3.5, which shows the parametric values of

known and unknown sigma variables single sampling plans with double

specification limits for symmetric fraction non-conforming. It can be observed that

the proposed VQSS results a smaller sample size than the sample size of single

sampling plan for both known sigma and unknown cases. The reduction ratios are

much higher for the sigma unknown case than for the sigma known case. It is also

to be pointed out that the sample size for the VQSS with double specification

limits is smaller than the VQSS with single specification limit (refer Balamurali

and Usha (2012 a)) for any specified combinations of AQL and LQL.

3.10 NON-NORMALITY IN VQSS

The variables sampling system developed in this chapter is based on the

assumption that the quality characteristic of interest follows a normal distribution.

Whenever the normality assumption is not true, using of any variables sampling

plans can be quite misleading (refer Sahli et al. (1997)). However it is to be

69

pointed out that the normal distribution can be justified due to the central limit

theorem as long as the statistics related to X is used. Obviously, there are some

situations in that the normal distribution is not suitable. If the distribution of

quality characteristic is known to follow any distribution other than the normal

distribution, sometimes we can utilize the analytical solutions to design the plans

but sometimes we cannot. It may depend on the statistics to be used. In such cases,

use of appropriate distribution is advisable. Hence, we can say that the use of

normal distribution is always an approximation only. Montgomery (1985)

investigated the effect of non-normality in the variables sampling plans. Some of

the authors have studied the effect of non-normality in variables sampling plans

and developed appropriate variables sampling plans depends upon the distribution

of the quality characteristic. For further details, readers are advised to refer

Srivastava (1961), Zimmer and Burr (1963), Das and Mitra (1964), Singh (1966),

Owen (1969), Takagi (1972), Kocherlakota and Balakrishnan (1986), Lam (1994),

Sahli et al. (1997), Suresh and Ramanathan (1997) and Chen and Lam (1999), Das

et al. (2002)).

3.11 CONCLUSION

In this chapter, we have developed a sampling system which can be

applied when the quality characteristic of interest has two specification limits

namely the lower and upper specification limits. Whenever the quality

characteristic involves double specification limits, separate sampling plan/system

should be developed in a different manner compared to the single specification

limit sampling plans. In this chapter, procedures and methodologies of determining

the optimal parameters of the VQSS have been developed for the inspection of

measurable characteristics having double specification limits. We have constructed

tables for both symmetric and the asymmetric fraction nonconforming cases for

both the known sigma VQSS and the unknown VQSS. We have also made a

70

comparison of the proposed system with variables single sampling plan having

double specification limits. It has been proved that the sample size required for the

proposed system is lesser than the sample size of the variables single sampling

plan with double specification limits.

71

Table 3.1. Optimal Parameters of Known Sigma VQSS with Double Specification Limits (Symmetric Fraction Non-conforming)

p1

p2

Optimal Parameters

n kTσ kNσ 0.001

0.0025

0.005

0.010

0.03

0.002 0.003 0.004 0.006 0.008 0.010 0.015 0.020

0.005 0.010 0.015 0.020 0.025 0.030 0.050

0.010 0.015 0.020 0.030 0.040 0.050 0.10 0.02 0.03 0.04 0.05 0.10 0.15 0.20

0.060 0.090 0.120

42 17 10 6 4 3 2 2

34 9 5 3 3 2 1

30 12 7 4 3 2 1 25 10 6 4 2 1 1

21 7 4

3.082 2.948 2.863 2.719 2.617 2.538 2.391 2.278

2.803 2.548 2.401 2.289 2.190 2.132 1.918

2.563 2.408 2.303 2.139 2.024 1.918 1.591 2.312 2.149 2.028 1.932 1.603 1.389 1.221

1.872 1.669 1.528

2.972 2.789 2.648 2.471 2.283 2.117 1.853 1.868

2.682 2.313 2.119 1.848 1.862 1.579 0.953

2.439 2.218 2.043 1.802 1.642 1.368 0.761 2.182 1.943 1.748 1.559 1.153 1.002 0.123

1.723 1.402 1.149

72

Table 3.1 Contd….

p1

p2

Optimal Parameters

n kTσ kNσ 0.03

0.05

0.150 0.300 0.100 0.150

0.200 0.250 0.500

3 1 16 6

4 1 1

1.401 0.991 1.632 1.408

1.249 1.088 0.591

1.001 0.121 1.452 1.128

0.963 0.072 0.032

73

Table 3.2 Optimal Parameters of Known Sigma VQSS with Double Specification Limits (Asymmetric Fraction Non-conforming)

p1

p2

Optimal Parameters n k1Tσ k2Tσ k1Nσ k2Nσ

0.001

0.0025

0.005

0.01

0.03

0.002 0.003 0.004 0.006 0.008 0.010 0.015 0.020

0.005 0.010 0.015 0.020 0.025 0.030 0.050

0.010 0.015 0.020 0.030 0.040 0.050 0.10

0.02 0.03 0.04 0.05 0.10 0.15 0.20

0.060 0.090 0.120

42 16 10 6 4 3 2 2

35 8 5 3 3 2 1

30 11 7 4 3 2 1

26 10 6 4 2 1 1

18 7 4

3.272 3.143 3.048 2.912 2.818 2.742 2.589 2.468

3.001 2.758 2.612 2.498 2.414 2.343 2.128

2.782 2.629 2.521 2.358 2.239 2.142 1.819

2.553 2.379 2.261 2.168 1.839 1.632 1.456

2.134 1.939 1.789

2.968 2.842 2.753 2.608 2.513 2.429 2.294 2.172

2.674 2.428 2.291 2.179 2.088 2.022 1.809

2.432 2.289 2.178 2.011 1.889 1.798 1.477

2.169 2.012 1.889 1.802 1.468 1.262 1.089

1.712 1.509 1.368

3.163 2.967 2.828 2.642 2.449 2.281 2.013 2.029

2.884 2.498 2.312 2.019 2.042 1.749 1.093

2.653 2.399 2.242 1.993 1.828 1.543 0.902

2.399 2.152 1.958 1.759 1.342 0.668 0.713

1.939 1.642 1.373

2.859 2.678 2.548 2.363 2.182 2.018 1.757 1.779

2.563 2.198 2.012 1.748 1.763 1.488 0.893

2.312 2.069 1.918 1.682 1.533 1.269 0.689

2.039 1.812 1.629 1.443 1.039 0.448 0.469

1.542 1.258 1.019

74

Table 3.2 Contd….

p1

p2

Optimal Parameters n k1Tσ k2Tσ k1Nσ k2Nσ

0.03

0.05

0.150 0.300

0.100 0.150 0.200 0.250 0.500

3 1

15 5 3 2 2

1.668 1.259

1.913 1.698 1.542 1.397 0.862

1.252 0.839

1.458 1.253 1.079 0.949 0.432

1.218 0.259

1.692 1.283 1.002 0.708 0.838

0.868 0.053

1.269 0.903 0.648 0.402 0.409

75

Table 3.3 Optimal Parameters of Unknown Sigma VQSS with Double Specification Limits (Symmetric Fraction Non-conforming)

p1

p2

Optimal Parameters

n kTS kNS 0.001

0.0025

0.005

0.010

0.002 0.003 0.004 0.006 0.008 0.010 0.015 0.020

0.005 0.010 0.015 0.020 0.025 0.030

0.050 0.010 0.015 0.020 0.030 0.040 0.050

0.10 0.02 0.03 0.04 0.05 0.10 0.15 0.20

200 72 42 23 15 12 8 6

147 30 16 11 8 6

4 112 39 22 12 8 6

3 83 28 16 11 4 3 2

3.084 2.952 2.848 2.712 2.608 2.528 2.383 2.264

2.788 2.548 2.401 2.278 2.193 2.122

1.892 2.562 2.407 2.301 2.129 2.012 1.911

1.569 2.312 2.149 2.022 1.918 1.576 1.358 1.188

2.859 2.627 2.473 2.289 2.052 2.029 1.903 1.772

2.568 2.147 1.949 1.823 1.649 1.338

1.419 2.319 2.059 1.863 1.714 1.548 1.432

1.188 2.062 1.767 1.612 1.489 1.088 1.067 0.849

76

Table 3.3 Contd….

p1

p2

Optimal Parameters

n kTS kNS 0.03

0.05

0.060 0.090 0.120 0.150 0.300

0.100 0.150 0.200 0.250 0.500

46 15 8 5 2

33 10 6 4 2

1.857 1.669 1.523 1.404 0.959

1.632 1.408 1.244 1.103 0.591

1.578 1.269 1.018 0.690 0.528

1.329 0.879 0.853 0.690 0.482

77

Table 3.4 Optimal Parameters of Unknown Sigma VQSS with Double Specification Limits (Asymmetric Fraction Non-conforming)

p1

p2

Optimal Parameters n k1TS k2TS k1NS k2NS

0.001

0.0025

0.005

0.01

0.03

0.002 0.003 0.004 0.006 0.008 0.010 0.015 0.020

0.005 0.010 0.015 0.020 0.025 0.030 0.050

0.010 0.015 0.020 0.030 0.040 0.050 0.10

0.02 0.03 0.04 0.05 0.10 0.15 0.20

0.060 0.090

228 80 44 24 15 12 8 6

162 30 16 10 8 7 4

126 43 22 12 9 6 3

90 31 16 12 5 3 2

48 16

3.262 3.129 3.028 2.892 2.793 2.712 2.549 2.438

2.989 2.738 2.584 2.469 2.384 2.303 2.084

2.768 2.609 2.521 2.334 2.219 2.102 1.769

2.533 2.362 2.229 2.128 1.789 1.572 1.388

2.113 1.913

2.969 2.848 2.758 2.628 2.528 2.449 2.229 2.187

2.678 2.452 2.291 2.193 2.108 2.029 1.809

2.442 2.229 2.188 2.034 1.913 1.819 1.477

2.192 2.028 1.912 1.808 1.488 1.268 1.113

1.722 1.532

3.139 2.937 2.793 2.572 2.442 2.318 2.073 1.909

2.864 2.469 2.242 2.098 1.958 1.828 1.469

2.623 2.393 2.214 1.979 1.822 1.687 1.109

2.369 2.108 1.923 1.779 1.359 0.978 1.018

1.919 1.603

2.878 2.698 2.564 2.369 2.242 2.129 1.903 1.753

2.583 2.229 2.028 1.943 1.768 1.648 1.313

2.332 2.116 1.968 1.762 1.613 1.489 0.958

2.069 1.839 1.669 1.553 1.149 0.812 0.818

1.594 1.308

78

Table 3.4 Contd….

p1

p2

Optimal Parameters n k1TS k2TS k1NS k2NS

0.03

0.05

0.120 0.150 0.300 0.100 0.150

0.200 0.250 0.500

8 5 2

35 12

6 4 2

1.749 1.618 1.179 1.849 1.659

1.478 1.332 0.786

1.388 1.268 0.859 1.473 1.268

1.112 0.969 0.468

1.378 1.182 0.752 1.679 1.348

1.132 0.957 0.658

1.119 0.948 0.532 1.329 1.049

0.848 0.702 0.369

79

Table 3.5. Average Sample Number of Variables Single Sampling Plans and VQSS with Double Specification Limits (Symmetric Fraction Non- conforming)

p1

p2

Average Sample Number Known Sigma Unknown Sigma

SSP VQSS SSP VQSS 0.001

0.0025

0.005

0.010

0.002 0.003 0.004 0.006 0.008 0.010 0.015 0.020

0.005 0.010 0.015 0.020 0.025 0.030 0.050

0.010 0.015 0.020 0.030 0.040 0.050 0.10

0.02 0.03 0.04 0.05 0.10 0.15 0.20

84 33 20 12 9 7 5 4

72 17 10 7 6 5 3

63 24 15 9 6 5 3

55 21 13 9 4 3 3

42 17 10 6 4 3 2 2

34 9 5 3 3 2 1

30 12 7 4 3 2 1

25 10 6 4 2 1 1

481 173 101 54 37 28 18 13

353 72 38 25 19 15 9

270 94 53 28 18 14 6

199 68 38 26 9 6 4

200 72 42 23 15 12 8 6

147 30 16 11 8 6 4

112 39 22 12 8 6 3

83 28 16 11 4 3 2

80

Table 3.5 Contd….

p1

p2

Average Sample Number Known Sigma Unknown Sigma

SSP VQSS SSP VQSS 0.03

0.05

0.060 0.090 0.120 0.150 0.300

0.100 0.150 0.200 0.250 0.500

41 15 9 7 3

34 13 8 6 3

21 7 4 3 1

16 6 4 1 1

110 36 20 13 4

79 25 13 9 3

46 15 8 5 2

33 10 6 4 2

81

CHAPTER 4

OPTIMAL DESIGNING OF VARIABLES

TIGHTENED-NORMAL-TIGHTENED SAMPLING SCHEME BY

MINIMZING THE AVERGE SAMPLE NUMBER

4.1 INTRODUCTION

The Tightened-Normal-Tightened (TNT) sampling scheme developed

by Calvin (1977), is a particular case of the general two-plan system for the

inspection of attributes characteristics. Balamurali and Jun (2009) developed a

designing methodology to determine the parameters of TNT sampling scheme

under variables sampling. Recently, Senthilkumar and Muthuraj (2010) provided

procedures for selecting parameters of the variables TNT scheme and constructed

tables for selecting parameters of the variables TNT scheme of type TNT (n1, n2;

k). However they didn’t follow any optimization techniques. So, this chapter

attempts to design a variables TNT (n1, n2; k) scheme by minimizing average

sample number (ASN) as done by Balamurali and Jun (2009). Obviously, any

sampling plan having smaller ASN would be more desirable. For the selection of

the parameters of the variables TNT scheme, the problem is formulated by a

nonlinear programming where the objective function to be minimized is the ASN

and the constraints are related to lot acceptance probabilities at acceptable quality

level (AQL) and LQL.

4.2 CONDITIONS OF APPLICATION

In order to apply the variables TNT sampling scheme, the following

assumptions should be valid.

(i) Production is in a steady state, so that results of past, present and

future lots are broadly indicative of a continuing process.

82

(ii) Lots are submitted for inspection serially either in the order of

production or in the order of being submitted for inspection.

(iii) Inspection is by measurements, with quality is defined as the

fraction nonconforming, p.

(iv) The distribution of the quality characteristic must be known and

follows normal distribution.

In addition, the usual conditions for the application of variables single sampling

plan with known or unknown standard deviation should also be valid (see for

further details Schilling (1982), Grant and Leavenworth (1996) and Montgomery

(2005)).

4.3 OPERATING PROCEDURE OF KNOWN SIGMA VARIABLES TNT SCHEME Suppose that the quality characteristic of interest has the upper specification limit

U and follows a normal distribution with known standard deviation σ.

Step 1: Start with the tightened inspection level using the single sampling

variables plan with a sample size n1 and the acceptance criterion k.

Accept the lot if kv and reject the lot if kv , where

1XUv and

1

111

1 n

iiX

nX .

Step 2: If t lots in a row are accepted under tightened inspection, then switch to

normal inspection.

Step 3: During the normal inspection, inspect the lots using the single sampling

variables plan with a sample size n2 (<n1) and the acceptance criterion k.

83

Accept the lot if kv and reject the lot if kv , where

2XUv and

2

122

1 n

iiX

nX .

Step 4: Switch to tightened inspection after a rejection of lot if an additional lot is

rejected in the next s lots.

Thus, the variable TNT scheme has five parameters namely the

tightened plan sample size n1, the normal plan sample size n2, the acceptance

criterion k and the switching parameters s and t. In the attributes case, the normal

plan sample size is taken as n (=n2) and the tightened plan sample size is

considered as mn (=n1) and m>1. We have followed the similar way in this

chapter. In this chapter, the value of m is considered in the interval 1.25(0.25)10.00

for constructing tables. Also the switching parameters are fixed as 4 and 5

respectively since, when s=4 and t=5, the sampling procedures correspond to the

procedures of MIL-STD 105D (1963) scheme involving only normal and tightened

inspections. Also Soundararajan and Vijayaraghavan (1992) observed that when

s=4 and t=5 gives more discriminating OC curve than any other combinations s

and t. It is to be pointed out that the VQSS presented in Chapter 2 is a special case

of the TNT scheme when s=0 and t=1 with reverse switching order. However, the

VQSS given in Chapter 2 has single sample size with two acceptance criteria

where as the TNT scheme proposed in this Chapter involves two sample sizes with

single acceptance criterion.

4.4 OC FUNCTION OF KNOWN SIGMA VARIABLES TNT SCHEME The OC function of the variables TNT scheme, which gives the

proportion of lots that are expected to be accepted for given product quality, p

under known sigma case is given by

12)(PP

pPa (4.1)

84

where 1s ,)1)(1(

2

22

2

s

s

PPP

s t , )1(

1

11

1

PPP

t

t

Here P1= Pr(v1 k) is the probability of acceptance under tightened inspection

and P2 = Pr(v2 k) is the probability of acceptance under normal inspection.

Under Type B situation (i.e. a series of lots of the same quality), forming lots of N

items from a process and then drawing random sample of size n from these lots is

equivalent to drawing random samples of size n directly from the process.

The fraction non-conforming in a lot will be determined as

)()(11 vvUp

(4.2)

where )( y is given by

y

dzzy2

exp21)(

2

, (4.3)

provided that the quality characteristic of interest is normally distributed with

mean µ and standard deviation σ, and the unit is classified as non-conforming if it

exceeds the upper specification limit U.

Then its probability of acceptance is written as

)()(

)( 12 wwpPa (4.4)

where

1s ,))(1))((1(

)(2

22

2

s

s

www

85

s t ,))(1()(

)(1

11

1

www

t

t

Here mnkvw 1 and nkvw 2 where mnn 1 and nn 2

4.5 DESIGNING OF A KNOWN SIGMA VARIABLES TNT SCHEME

The OC function of a known sigma variables TNT scheme is given in

(4.4). If two points on the OC curve namely, AQL(=p1), LQL(=p2), the producer’s

risk α and the consumer’s risk β are prescribed then the OC function can be

expressed as

1

)()( 1121 ww (4.5)

and

)()( 1222 ww (4.6)

Here w11 is the value of w1 at p=p1, w21 is the value of w2 at p=p1, w12 is the value

of w1 at p=p2 and w22 is the value of w2 at p=p2. That is,

mnkvw )( 111 , nkvw )( 121

mnkvw )( 212 and nkvw )( 222

where v1 is the value of v at AQL and v2 is the value of v at LQL.

The ASN, by definition, means the expected number of sampled units

required for making decisions about the lot. The concept of ASN is meaningful

under Type B sampling situations. The average sample number of the TNT

variables scheme is given by

86

mnnpASN )( , m>1 (4.7)

where 1s ,)1)(1(

2

22

2

s

s

PPP

s t ,)1(

1

11

1

PPP

t

t

where P1 and P2 are the probability of acceptance of tightened and normal plans

respectively. The ASN given above can be used as an objective function to solve

for the parameters (n1, n2, k). Since there are several choices to obtain the

objective function, it is considered here to minimize ASN at AQL. If the objective

is to minimize the ASN at AQL, then the problem will be reduced to the following

nonlinear optimization problem.

Minimize ASN(p1)

Subject to

1)( 1pPa

)( 2pPa

0,1 ,1 k mn (4.8)

where )( 1pPa and )( 2pPa are the lot acceptance probabilities at AQL and LQL

respectively and are given in (4.5) and (4.6) respectively and ASN(p1) is the ASN

at AQL.

87

4.6 DESIGNING OF UNKNOWN SIGMA VARIABLES TNT SCHEME

Whenever the standard deviation is unknown, we should use the sample

standard deviation S instead of σ. In this case, the operation of the scheme is as

follows.

Step 1: Start with the tightened inspection level using the single sampling

variables plan with a sample size n1 and the acceptance criterion kS.

Accept the lot if Skv and reject the lot if Skv , where S

XUv 1 ,

Sn

ii

S

Xn

X1

111

1 and 1

)(

1

21

S

i

nXX

S .

Step 2: If t lots in a row are accepted under tightened inspection, then switch to

normal inspection.

Step 3: During the normal inspection, inspect the lots using the single sampling

variables plan with a sample size n2σ (<n1σ) and the acceptance criterion kS.

Accept the lot if Skv and reject the lot if Skv , where S

XUv 2 ,

Sn

ii

S

Xn

X2

122

1 and 1

)(

2

22

S

i

nXX

S .

Step 4: Switch to tightened inspection after a rejection of lot if an additional lot is

rejected in the next s lots.

Thus, the unknown sigma variables TNT scheme has the parameters

namely the sample sizes n1S, n2S and the acceptable criterion kS. If n1S=n2S, then the

variables TNT scheme reduced to the variables single sampling plan with

88

unknown standard deviation. Hamaker (1979) has given an approximation for

finding the parameters of the unknown sigma single sampling plan from the

parameters of the known sigma single sampling plan. Senthilkumar and Muthuraj

(2010) have followed the similar approximation for selecting the parameters of

unknown sigma variables TNT scheme. In this chapter, we follow the same

approximation for finding the parameters of unknown sigma variables TNT

scheme. That is, one can determine the sample size for the unknown sigma TNT

scheme as

nkns 21

2

(4.9)

which is the normal plan sample size and the tightened plan sample size is

determined by multiplying the factor m with the normal plan sample size. The

acceptance criterion of unknown sigma TNT scheme is determined as

)54)44(

s

sS n

nkk (4.10)

Also the ASN of unknown sigma TNT scheme namely ASNs can be determined as

21

2s

SkASNASN (4.11)

One can determine the parameters of the known sigma TNT scheme by

solving the nonlinear equation given in (4.8). There may exist multiple solutions

since there are three unknowns namely nσ, kσ and m with only two equations.

Generally a sampling would be desirable if the required number of sampled is

small. So, in this chapter, we consider the ASN as the objective function to be

minimized with the probability of acceptance along with the corresponding

producer’s and consumer’s risks as constraints. To solve the above nonlinear

optimization problems given in (4.8), the sequential quadratic programming (SQP)

proposed by Nocedal and wright (1999) can be used. The SQP is implemented in

89

Matlab software using the routine “fmincon”. By solving the nonlinear problem

mentioned above, the parameters (n, m and k) for known sigma plan are

determined and tabulated in Table 4.1. The parameters (nS, m and kS) for unknown

sigma plan are also determined by using the approximation given in (4.9) and

(4.10) and also presented in Table 4.1.

4.7 EXAMPLES

4.7.1 SELECTION OF KNOWN SIGMA TNT SCHEME INDEXED BY AQL AND LQL Table 4.1 is used to determine the parameters of the known variables

TNT scheme for specified values of AQL and LQL when =5% and =10%. For

example, if p1=1%, p2=9%, =5% and =10%, Table 4.1 gives the parameters as

n = 5, m=5 and k = 1.598. The normal plan sample size is n2σ=n=5 and the

tightened plan sample size is obtained as n1σ=mxn= 5x5=25. The acceptance

criterion is same for both normal and tightened plans.

For the above example, the operation of the variables TNT scheme is as follows.

Step 1: Start with tightened inspection. Take a random sample of size 25 and

Compute

XUv , where

25

1251

iiXX . Accept the lot if 598.1v and

reject the lot if 598.1v . If t = 5 consecutive lots are accepted with the

same sample size and acceptance criterion, then switch to normal

inspection as in step 2.

90

Step 2: During normal inspection, select a random sample of size 5 and calculate

XUv , where

5

151

iiXX . Accept the lot if 598.1v and reject the

lot if 598.1v . If 2 out of (s+1)=5 lots are rejected on normal inspection

then immediately revert to tightened inspection as in step 1.

4.7.2 SELECTION OF UNKNOWN SIGMA VARIABLES TNT SCHEME INDEXED BY AQL AND LQL Table 4.1 can also be used for the selection of the parameters of the

unknown variables TNT scheme for given values of AQL and LQL. Suppose

that AQL=1%, LQL=7%, = 5% and =10%. From Table 4.1, the parameters of

the unknown sigma variables TNT scheme can be determined as nS = 14, kS =

1.694 and m = 8. The normal plan sample size of unknown sigma variables TNT

scheme is nS = 14 and the tightened plan sample size is obtained as m x nS =

8x14=112. The acceptance criterion for both normal and tightened plans is

kS=1.694.

4.7.3 ADVANTAGES OF THE VARIABLES TNT SAMPLING SCHEME

This section describes the advantages of the variables TNT scheme over

attributes TNT scheme, variables single and double sampling plans. For the

purpose of comparison, we will consider the plans which have the same AQL and

LQL. Suppose that for given values of AQL=0.02, =5%, LQL=0.07 and =

10%, one can find the parameters of the attributes TNT scheme from

Soundararajan and Vijayaraghavan (1992) under the application of Poisson model

as

(i) n1 = 76, n2=38, c = 2 and ASN=38.609

91

For the same AQL and LQL, we can determine the parameters of the variables

single, double sampling plans (from Sommers (1981)) and variables TNT scheme

(from Table 4.1) respectively as follows.

(ii) nσ = 26, kσ = 1.73 and ASN=26

(iii) n1σ = n2σ = 19, kaσ = 1.83, krσ = 1.67 and ASN=21.2

(iv) n = 13, kσ = 1.602, m = 8 and ASN=16.626

By comparing the above, it is clear that the variables TNT scheme

achieves a reduction of over 57% in sample size than the attributes TNT scheme

and about 36% than the variables single sampling plan with same AQL and LQL

conditions. Further, it is also to be pointed that the variables TNT scheme is

economically superior to the variables double sampling plan in terms of ASN. The

proposed scheme also achieves a reduction of 22% in ASN over the variables

double sampling plan. Obviously, a sampling plan having smaller ASN would be

more desirable. The variables double or multiple sampling plans are not practically

very useful. Variables sampling standards avoid presenting such plans due to

increased complexity involved in operating them.

4.8 COMPARISONS

4.8.1 COMPARISON THORUGH OC CURVES In order to show the better efficiency of the variables TNT scheme,

three OC curves are considered. Figure 4.1. shows the OC curves of the variables

TNT scheme with parameters nσ = 12, m = 5.25 and kσ = 1.857 along with two

variables single sampling plans (nσ =63, kσ =1.857) and (nσ = 12, kσ =1.857). The

variables TNT scheme is selected in such a way that it satisfies the two-points on

the OC curve condition (p1=0.01, 1-α = 0.95) and (p2 = 0.045, β = 0.10). From this

92

figure, it can be easily observed that, for good quality, i.e. for smaller values of

fraction nonconforming, the composite OC curve (OC curve of the variables TNT

scheme) coincides with the OC curve of the variables single sampling plan (12,

1.857). As quality deteriorates the OC curve of the composite OC curve moves

toward that for the single sampling plan (63, 1.857) and comes close to it beyond

the indifference quality level.

4.8.2 COMPARISON THROUGH ASN

Table 4.2 shows the ASN values of the variables single sampling plan

and the variables double sampling plan along with the two variables TNT schemes

(one given by Senthilkumar and Muthuraj (2010) and the other given in Table 4.1)

for some arbitrarily selected combinations of AQL and LQL under known sigma

case. Table 4.3 gives the ASN values of the above said plans when sigma is

unknown. These ASN values are calculated at the producer’s quality level for both

known and unknown sigma plans. The sample size of the variables single sampling

plan and the ASN of the variables double sampling plan can be found in Sommers

(1981). To strengthen this point, two more tables are provided in this chapter.

Table 4.4 gives the parameters of known sigma variables TNT scheme along with

parameters given by Senthilkumar and Muthuraj (2010) and Table 4.5 gives the

parameters of unknown sigma variables TNT scheme.

These tables apparently show that the variables TNT scheme provided

in this chapter will have minimum ASN when compared to the variables single,

double sampling plans and variables TNT scheme provided by Senthilkumar and

Muthuraj (2010) for both known and unknown sigma cases. Similar reduction in

ASN can be achieved for any combination of AQL and LQL values. This implies

that variables TNT scheme developed in this chapter will give desired protection

with minimum inspection so that the cost of inspection will greatly be reduced.

93

Thus the variables TNT scheme provides better protection than the variables single

sampling plans and variables double sampling plans. It is also to be noted that

Senthilkumar and Muthuraj (2010) have given several tables for finding the

parameters of variables TNT scheme for each value of m. But table developed in

this chapter will overcome this shortcoming and only one table given in this

chapter is enough for any combinations of AQL and LQL.

4.9 CONCLUSIONS

In this chapter, we have considered the designing of variables TNT sampling

scheme involving minimum average sample number for both known and unknown

standard deviation cases. In general, variables sampling plans require a smaller

sample size than do attributes sampling plans. This is also valid for the proposed

TNT sampling scheme. It has also been shown that the variables TNT scheme

provided in this chapter has smaller ASN than the ASN of the existing variables

single and double sampling plans. The variables TNT sampling scheme proposed

in this chapter also ensure the protection for the consumers in their point of view.

This variables TNT scheme will be effective and useful for compliance testing.

Further, tables provided in this chapter are compact and easy to apply for the

selection of parameters of variables TNT scheme for specified combinations of

AQL and LQL along with the producer’s and consumer’s risks.

94

Table 4.1. Variables Tightened-Normal-Tightened Sampling Scheme Indexed

by AQL and LQL for =5% and =10% Involving Minimum ASN

p1

p2

MinASN(p1) Known Sigma

MinASN(p1) Unknown Sigma

n m kσ ASNσ nS m kS ASNS 0.001

0.005

0.01

0.02

0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

0.007 0.008 0.009 0.01

0.012 0.015 0.020 0.03

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.03 0.04 0.05 0.06 0.07 0.08

97 36 22 16 13 12 9 8 8

294 153 94 67 42 27 16 9

56 22 13 9 7 6 5 5 4

140 46 26 18 13 10

7.75 9.0 9.25 9.0 8.0 5.25 9.5 9.25 6.0

10.0 8.75 9.75 10.0 8.5 7.25 7.75 8.0

10.0 7.75 8.5 9.5 9.5 8.0 8.75 5.0 7.75

9.5 9.25 8.0 7.0 8.0 9.5

2.925 2.819 2.742 2.683 2.638 2.619 2.548 2.515 2.512

2.481 2.444 2.408 2.376 2.325 2.262 2.169 2.032

2.108 1.979 1.873 1.784 1.712 1.661 1.599 1.598 1.512

1.916 1.813 1.734 1.669 1.602 1.537

123.244 47.538 29.121 21.100 16.605 14.008 12.071 10.647 9.558

400.85 200.36 127.01 90.486 54.707 33.683 20.327 11.487

75.757 27.914 16.809 12.067 9.399 7.668 6.557 5.798 5.076

187.378 60.920 33.213 22.251 16.626 13.324

512 179 105 74 58 53 38 33 33

1199 610 367 256 156 96 54 28

180 65 36 23 17 14 11 11 9

397 122 65 43 30 22

7.75 9.0

9.25 9.0 8.0

5.25 9.5

9.25 6.0

10.0 8.75 9.75 10.0 8.5

7.25 7.75 8.0

10.0 7.75 8.5 9.5 9.5 8.0

8.75 5.0

7.75

9.5 9.25 8.0 7.0 8.0 9.5

2.926 2.704 2.749 2.692 2.650 2.632 2.565 2.535 2.532

2.482 2.445 2.410 2.378 2.329 2.268 2.179 2.051

2.111 1.987 1.886 1.805 1.739 1.694 1.640 1.639 1.561

1.917 1.817 1.741 1.679 1.616 1.555

650.977 236.957 139.123 97.566 74.894 62.515 51.793 44.852 40.190

1635.09 797.959 495.754 346.408 203.051 120.313 68.599 35.649

244.553 83.005 46.720 31.713 21.613 18.663 15.374 13.584 11.258

531.761 161.455 83.536 53.613 38.332 29.444

95

Table 4.1. Contd….

p1

p2

MinASN(p1) Known Sigma

MinASN(p1) Unknown Sigma

n m kσ ASNσ nS m kS ASNS 0.02 0.03

0.04

0.05

0.09 0.1 0.11 0.12

0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.15

0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15

0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13

9 7 7 6

252 75 39 26 19 15 12 10 9 6

370 110 56 38 25 19 16 13 11 10 9

511 148 72 46 33 27 19 16

6.25 9.5 5.25 6.0 8.75 9.5 10.0 8.75 8.5 7.75 8.5 8.5 7.0 8.5

10.0 9.25 9.0 6.5 9.5 9.75 7.5 8.25 8.5 7.0 6.25

9.75 9.0 10.0 9.0 7.75 5.5 9.25 8.25

1.512 1.439 1.438 1.389

1.778 1.693 1.620 1.561 1.506 1.460 1.409 1.366 1.337 1.216

1.666 1.595 1.533 1.487 1.424 1.376 1.344 1.299 1.259 1.234 1.208

1.573 1.511 1.453 1.404 1.362 1.332 1.272 1.238

10.906 9.390 8.184 7.197

328.964 100.604 52.979 33.970 24.577 19.018 15.515 12.996 11.131 7.794

502.286 145.605 73.817 46.417 33.323 25.517 20.170 16.755 14.262 12.347 10.886

691.102 195.102 97.839 60.393 41.982 31.890 25.316 20.639

19 14 14 12

650 182 90 58 41 31 24 33 17 10

883 250 122 80 50 37 30 24 20 18 16

1143 317 148 91 64 51 34 28

6.25 9.5

5.25 6.0

8.75 9.5

10.0 8.75 8.5

7.75 8.5 8.5 7.0 8.5

10.0 9.25 9.0 6.5 9.5

9.75 7.5

8.25 8.5 7.0

6.25

9.75 9.0

10.0 9.0

7.75 5.5

9.25 8.25

1.533 1.467 1.466 1.421

1.779 1.695 1.625 1.568 1.515 1.472 1.424 1.385 1.358 1.251

1.666 1.597 1.536 1.492 1.431 1.386 1.356 1.313 1.276 1.252 1.228

1.573 1.512 1.455 1.408 1.367 1.339 1.282 1.250

23.726 19.496 16.981 14.467

849.342 245.182 122.889 75.723 52.798 39.630 31.255 25.464 21.398 13.891

1199.73 331.187 160.914

98.06 67.455 50.011 38.704 31.203 25.868 22.03

19.101

1546.47 418.172 201.468 120.248 81.231 60.464 46.109 36.751

96

Table 4.1. Contd….

p1

p2

MinASN(p1) Known Sigma

MinASN(p1) Unknown Sigma

n m kσ ASNσ nS m kS ASNS 0.05

0.06

0.07

0.14 0.15

0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15

0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.20

13 12

184 91 61 41 30 24 19 16

229 112 75 46 34 29 22 11

10.0 7.25

10.0 9.75 6.5

7.25 8.0 7.5

8.75 8.25

9.75 9.0 6.0

10.0 9.75 6.5

8.75 7.5

1.193 1.174

1.435 1.384 1.346 1.301 1.258 1.222 1.180 1.148

1.368 1.322 1.287 1.235 1.197 1.174 1.129 0.983

17.650 14.961

250.769 122.634 74.263 51.333 38.452 30.177 24.763 20.645

308.023 147.810 89.608 62.231 45.974 35.430 28.840 13.792

22 20

373 178 116 76 54 42 32 27

443 210 137 81 58 49 36 16

10.0 7.25

10.0 9.75 6.5 7.25 8.0 7.5 8.75 8.25

9.75 9.0 6.0 10.0 9.75 6.5 8.75 7.5

1.207 1.190

1.436 1.386 1.349 1.305 1.264 1.229 1.896 1.159

1.369 1.324 1.289 1.239 1.202 1.180 1.137 0.9996

30.514 25.548

509.305 240.413 141.827 95.065 69.167 52.985 42.284 34.514

596.562 277.280 164.092 109.986 79.200 60.101 47.484 20.682

97

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Fraction Nonconforming, p

Prob

abili

ty o

f Acc

epta

nce,

Pa(

p)

Figure 4.1: OC Curves of Single Sampling Normal Plan (12, 1.857), TNT

Scheme (63, 12; 1.857) and Single Sampling Tightened

Plan (63, 1.857)

Variables Normal SSP

Variables TNT Scheme

Variables Tightened SSP

98

Table 4.2: ASN Values of the Known Sigma Variables Single Sampling Plan, Variables Double Sampling Plan and Variables TNT Scheme

p1

p2

ASN Variables

SSP Variables

DSP Variables*

TNT Variables **

TNT 0.001 0.001 0.005 0.005 0.03 0.03 0.04 0.04 0.05 0.05

0.002 0.003 0.010 0.012 0.04 0.06 0.07 0.08 0.07 0.08

191 74 138 85 506 81 114 72 300 149

154.9 59.4

112.0 69.5

434.6 127.7 180.6 58.4

246.7 122.3

179.792 68.673 126.310 76.809 471.546 72.746 108.116 68.675 271.860 137.421

123.244 47.538 90.486 54.707 328.964 52.979 73.817 46.417 195.102 97.839

* ASN given in Senthilkumar and Muthuraj (2010) ** ASN given in Table 4.1 Table 4.3: ASN Values of the Unknown Sigma Variables Single Sampling Plan, Variables Double Sampling Plan and Variables TNT Scheme

p1

p2

ASN Variables

SSP Variables

DSP Variables*

TNT Variables **

TNT 0.001 0.001 0.005 0.005 0.03 0.03 0.04 0.04 0.05 0.05

0.002 0.003 0.010 0.012 0.04 0.06 0.07 0.08 0.07 0.08

1032 381 547 327

1333 197 258 159 660 319

829.1 302.4 437.1 263.0 1138.7 156.5 205.0 125.8 535.4 258.0

971.031 355.665 500.564 296.943

1238.397 176.451 245.252 151.353 598.751 294.171

650.977 236.957 346.408 203.051 849.342 122.889 160.914

98.06 418.172 201.468

* ASN given in Senthilkumar and Muthuraj (2010) ** ASN given in Table 4.1

99

Table 4.4: Parameters of Known Sigma Variables TNT Scheme for different AQL and LQL Values

p1

p2

*Parameters of Variables TNT

**Parameters of Variables TNT

n m kσ ASN n m kσ ASN 0.001 0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.002 0.007 0.020 0.03 0.04 0.06 0.07 0.08 0.10

146 451 83

224 387 163 224 328 163

2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0

2.955 2.503 2.150 1.941 1.796 1.621 1.541 1.460 1.351

151.596 469.248 86.288 233.170 401.699 169.272 233.170 341.510 169.301

97 294 56 140 252 110 148 184 112

7.75 10.0 10.0 9.5 8.75 9.25 9.0 10.0 9.0

2.925 2.481 2.108 1.916 1.778 1.595 1.511 1.435 1.322

123.244 400.85 75.757 187.378 328.964 145.605 195.102 250.769 147.810

*Parameters from Senthilkumar and Muthuraj (2010) **Parameters from Table 4.1 Table 4.5: Parameters of Unknown Sigma Variables TNT Scheme for different AQL and LQL Values

p1

p2

*Parameters of Variables TNT

**Parameters of Variables TNT

nS m kS ASN nS m kS ASN 0.001 0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.002 0.007 0.020 0.03 0.04 0.06 0.07 0.08 0.10

728 265 83

224 387 163 224 328 163

2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0

2.951 2.503 2.150 1.941 1.796 1.621 1.541 1.460 1.351

771.730 469.248 86.288 233.170 401.699 169.272 233.170 341.510 169.301

97 294 56 140 252 110 148 184 112

7.75 10.0 10.0 9.5 8.75 9.25 9.0 10.0 9.0

2.925 2.481 2.108 1.916 1.778 1.595 1.511 1.435 1.322

123.244 400.85 75.757 187.378 328.964 145.605 195.102 250.769 147.810

*Parameters from Senthilkumar and Muthuraj (2010) **Parameters from Table 4.1

100

CHAPTER 5

OPTIMAL DESIGNING OF

VARIABLES CHAIN SAMPLING PLAN BY MINIMIZING THE

AVERAGE SAMPLE NUMBER

5.1 INTRODUCTION

Some examples of the use of cumulative results to achieve a reduction

of the size while maintaining or even extending protection can be found in chain

sampling plans introduced by Dodge (1955). These plans were finally conceived to

overcome the problem of lack of discrimination in c = 0 sampling plans. The

procedure was developed to “chain” together the most recent inspections in a way

that would build up the shoulder of the OC curve of c = 0 plans. This is especially

desirable in situations in which small samples are demanded because of the

economic or physical difficulty of obtaining a sample. The chain sampling plan is

one of the conditional sampling procedures and this plan under variables

inspection will be useful when testing is costly and destructive. Govindaraju and

Balamurali (1998) extended the concept of chain sampling to variables inspection

but they have not provided any tables for the selection of parameters and also they

have dealt only with known standard deviation case. This chapter attempts to

provide tables for the easy industrial application of this plan for both known and

unknown standard deviation cases. It is also to be pointed out that Govindaraju and

Balamurali (1998) have given an approximate solution for finding the plans of

unknown standard deviation case. But in this chapter, we provide a different

procedure for finding the unknown sigma chain sampling plan. The major

advantage of this plan is to achieve better protection to the producer with

minimum ASN.

101

5.2 CONDITIONS OF APPLICATION

The following assumptions should be valid for the application of the

variables chain sampling plan.

(i) Lots are submitted for inspection serially, in the order of production from a

process that turns out a constant proportion non-conforming items.

(ii) The consumer has confidence in the supplier and there should be no reason to

believe that a particular lot is poorer than the preceding lots.

In addition, the usual conditions for the application of single sampling variables

plans with known or unknown standard deviation should also be valid.

5.3 OPERATING PROCEDURES OF VARIABLES ChSP 5.3.1 KNOWN SIGMA CASE

Suppose that the quality characteristic of interest has the upper

specification limit U and follows a normal distribution with unknown mean μ and

known standard deviation σ. Then the operating procedure of the variables chain

sampling plan is proposed as follows.

Step 1: From each submitted lot, take a random sample of size n , say

nXXX ..., 21 and compute

XUv

, where

n

iiX

nX

1

1 .

Step 2: Accept the lot if kv and reject the lot if 'kv . If kvk ' , then

accept the current lot provided that the preceding i lots were accepted on

the condition that kv but reject, otherwise. (Note: 'kk ).

102

In the case of lower specification limit L, the operating procedure is described as

follows.

Step 1: From each lot, take a random sample of size n , say nXXX ..., 21 and

compute

LXv , where

n

iiX

nX

1

1 .

Step 2: Accept the lot if kv and reject the lot if 'kv . If kvk ' , then

accept the current lot provided that the preceding i lots were accepted on

the condition that kv but reject, otherwise.

Thus, the proposed variables chain sampling plan is characterized by

four parameters, namely kin ,, and 'k . If 'kk , then the proposed plan will

reduce to the variables SSP. Also, when i tends to infinity, the proposed plan

becomes variables SSP with parameters n and k . It is to be pointed out that the

chain sampling plan can be applied for inspection of lots which are submitted

serially in the order of production or in the order of being submitted. The decision

of current lot depends on the results of preceding lots. So, when i ≥2, we need to

keep the records of results of previous lots. Of course, maintaining records of

preceding lots may be a drawback of the chain sampling plan over the single

sampling plan, however this can be compensated by minimizing the inspection

efforts in terms of minimum ASN with desired protection.

5.3.2 UNKNOWN SIGMA CASE

Whenever the standard deviation is unknown, we may use the sample

standard deviation S instead of σ. In this case, the plan operates as follows.

103

Step 1: From each submitted lot, take a random sample of size Sn , say

SnXXX ..., 21 and compute

SXUv

, where

Sn

ii

S

Xn

X1

1 and

1

)( 2

S

i

nXX

S .

Step 2: Accept the lot if Skv and reject the lot if SkV ' . If SS kvk ' , then

accept the current lot provided that the preceding Si lots were accepted

on the condition that Skv .

Thus, the proposed unknown sigma variables chain sampling plan is

characterized by four parameters namely SSS kin ,, and Sk ' . If SS kk ' , then the

proposed plan reduced to the variables single sampling plan with unknown

standard deviation.

5.4 DESIGNING METHODOLOGY OF THE VARIABLES ChSP 5.4.1 KNOWN SIGMA CASE

Generally, variables sampling plans are designed based on two points

on the OC curve namely )1,( 1 p and ),( 2 p , where 1p is called the AQL, 2p is

the LQL, is the producer’s risk and is the consumer’s risk. Any well

designed sampling plan which must provide at least )%1( probability of

acceptance of a lot when the process fraction nonconforming is at AQL level and

the sampling plan must also provide not more than % probability of acceptance

if the process fraction nonconforming is at the LQL level. Thus the acceptance

sampling plan must have its OC curve passing through two designated points

(AQL, 1 ) and (LQL, ). Some other strategies are also followed to design the

sampling plans besides the statistical based paradigm, which include Bayesian

104

approach and economic based approach. For further detail, readers may refer

Chen and Lam (1999), Ferrell and Chhoker (2002), Chen (2005), Chen et al.

(2007), Balamurali and Subramani (2010), Vijayaraghavan and Sakthivel (2011),

Balamurali et al. (2012) and Fallahnezhad and Aslam (2013). In this chapter, we

have followed the designing methodology based on two points on the OC curve

approach. The variables chain sampling plan is designed based on the two points

on the OC curve in the following manner.

The fraction non-conforming in a lot is given as

Up 1 (5.1)

where )(Y is the cumulative distribution function of standard normal

distribution and is given by

Y

dZZY2

exp21)(

2

, (5.2)

Here the quality characteristic of interest is normally distributed with

mean µ and standard deviation σ, and the unit is classified as non-conforming if it

exceeds the upper specification limit U. So, the unknown mean μ can be

determined if p is specified. Let us define the standardized quality characteristic

corresponding to the fraction conforming as

)1(1 pZ p (5.3)

Then the OC function of the variables chain sampling plan, which gives the

proportion of lots that are expected to be accepted for given product quality, p is

given by (see Govindaraju and Balamurali (1998)

ia kvkvkvkvpP )Pr( )Pr()'Pr()Pr()( (5.4)

105

where kv Pr is the probability of accepting a lot based on a single sample with

parameters ),( kn and 'Pr kv is the probability of rejecting a lot based on a

single sample with parameters )',( kn . Under type B situation (i.e. a series of lots

of the same quality), forming lots of N items from a process and then drawing

random sample of size n from these lots is equivalent to drawing random samples

of size n directly from the process. Hence the derivation of the OC function is

straightforward.

The probability of acceptance of the chain sampling plan can also be written as

ia wwwwpP )( )()()()( 2212 (5.5)

where nkZw p '1 and nkZw p 2

The OC function given in (5.5) under the specified AQL and LQL conditions can

be written as

1)( )()()( 21211121

iwwww

and iwwww )( )()()( 22221222

Here 11w is the value of 1w at 1pp , 21w is the value of 2w at 1pp , 12w is

the value of 1w at 2pp and 22w is the value of 2w at 2pp . That is,

nkZw p )'(111 , nkZw p )(

121

nkZw p )'(212 and nkZw p )(

222

where 1pZ is the value of pZ at AQL and

2pZ is the value of pZ at LQL.

106

The parameters of the known sigma variables chain sampling plan are

denoted by ( kkn ,', ).The following optimization problem is considered to

determine the optimal parameters of known sigma variables sampling plan such

as kin ,, and 'k .

Minimize npASN )( 1

Subject to 1)( 1pPa

)( 2pPa

,0,0' ,1i ,2 k kn n ЄN, i ЄN (5.6)

We may determine the parameters of the known sigma chain sampling

plan by solving the nonlinear equation given in (5.6).

5.4.2 UNKNOWN SIGMA CASE

The determination of parameters ( SSS kkn ,', ) of unknown sigma

variables chain sampling plan is explained as follows. It is known that for large

samples, SkX S is approximately normally distributed with mean )(SEkS

and variance )(2

SVarkn S (see Duncan (1986), Balamurali and Jun (2007)).

That is,

nk

nkNSkX SSS 2

,~2

22

Therefore, the probability of accepting a lot at each repetition is given by

pSkUXPkvP SS )(

pUSkXP S

107

21)/(

2S

S

S

kn

kU

21

)( 2S

SSp k

nkZ (5.7)

If we let,

21

)( 22S

SSpS k

nkZw then )()Pr( 2SS wkv . (5.8)

Similarly if we let,

2'

1)'( 21

S

SSpS k

nkZw then we have

)(1)'Pr( 1SS wkv (5.9)

Hence the lot acceptance probability for sigma unknown case is given by

SiSSSSa wwwwpP )( )()()()( 2212 (5.10)

where SSpS nkZw '1 and SSpS nkZw 2

If (AQL, 1 ) and (LQL, ) are prescribed then the OC function can be written

as

1)( )()()( 21211121Si

SSSS wwww

and SiSSSS wwww )( )()()( 22221222 (5.11)

108

Here Sw11 is the value of Sw1 at 1pp , Sw21 is the value of Sw2 at 1pp , Sw12 is

the value of Sw1 at 2pp and Sw22 is the value of Sw2 at 2pp .

We obtain SSSS wwww 22122111 ,,, respectively by

2'

1)'( 2111

S

SSpS k

nkZw ,

21

)( 2121S

SSpS k

nkZw

2'

1)'( 2212

S

SSpS k

nkZw and

21

)( 2222S

SSpS k

nkZw (5.12)

where 1pZ is the value of pZ at AQL and

2pZ is the value of pZ at LQL.

For given AQL and LQL, the parametric values of the unknown sigma

variables chain sampling plan namely SSS kin ,, and Sk ' are determined by satisfying

the required producer and consumer conditions. Alternatively, we can determine

the above parameters of the variables chain sampling plan to minimize the ASN at

AQL, which analogous to minimizing the average sample number in the variables

repetitive group sampling plans and multiple dependent state sampling plan (see

Balamurali et al. (2005), Balamurali and Jun (2007)). Some of the authors have

investigated the designing of sampling plans by using some other optimization

techniques which are available in the literature (see for example, Feldmann and

Krumbholz (2002), Krumbholz and Rohr (2006,2009), Krumbholz et al. (2012),

Duarte and Sariava (2010, 2013)). The ASN for the chain sampling plan is the

sample size only.

109

Therefore, the following optimization problem is considered to determine those

parameters.

Minimize SnpASN )( 1

Subject to 1)( 1pPa

)( 2pPa

,0,0' ,1 ,2 SSSS k kin Sn ЄN, Si ЄN (5.13)

We may determine the parameters of the unknown sigma chain

sampling plan by solving the nonlinear equation given in (5.13). There may exist

multiple solutions since there are four unknowns with only two equations.

Generally a sampling would be desirable if the required number of sampled is

small. So, in this chapter, we consider the ASN as the objective function to be

minimized with the probability of acceptance along with the corresponding

producer’s and consumer’s risks as constraints. To solve the above nonlinear

optimization problems given in (5.6) and (5.12), the sequential quadratic

programming (SQP) proposed by Nocedal and Wright (1999) can be used. The

SQP is implemented in Matlab software using the routine “fmincon”. By solving

the nonlinear problem mentioned above, the optimal parameters ( kin ,, and

'k ) for known sigma plan and the parameters ( SSS kin ,, and Sk ' ) for unknown

sigma plan are determined and these values are tabulated in Table 5.1.

5.5 DESIGNING EXAMPLES

5.5.1.SELECTION OF KNOWN SIGMA VARIABLES CHAIN SAMPLING PLAN FOR SPECIFIED AQL AND LQL Table 5.1 is used to determine the parameters of the known variables

chain sampling plan for specified values of AQL and LQL when = 5% and

110

= 10%. For example, if 1p = 2%, 2p = 7%, = 5% and = 10%, Table 5.1

gives the parameters as n = 18, i =3, 'k = 1.544 and k = 1.779.

For the above example, the plan is operated as follows.

From each submitted lot, take a random sample of size 18 and compute

XUv , where

18

1181

iiXX . Accept the lot if 779.1v and reject the lot if

544.1v . If 779.1544.1 v , then accept the current lot provided that the

preceding 3 lots were accepted on the condition that 779.1v with the sample

size of 18.

5.5.2. SELECTION OF UNKNOWN SIGMA VARIABLES ChSP FOR SPECIFIED AQL AND LQL

As mentioned earlier, the unknown sigma variables chain sampling plan

is operated as a known sigma variables chain sampling plan but the parameters

SSS kin ,, and Sk ' are used in the place of kin ,, and 'k respectively. Table 5.1

can also be used for the selection of the parameters of the unknown variables

chain sampling plan for given values of AQL and LQL. Suppose that

AQL=0.0075, LQL=0.035, =5% and =10%. From Table 5.1, the parameters

of the variables chain sampling plan can be determined as Sn = 47, Si = 1, Sk ' =

1.925 and Sk = 2.190.

5.6 ILLUSTRATIVE EXAMPLE

To illustrate the implementation of the proposed sampling plan for the

example given in section 5.5.2, we consider a case study data on STN-LCD

manufacturing process given in Wu and Pearn (2008). The upper specification

111

limit is given as 0.77 mm. Here we consider only 47 values randomly taken from

the original data given in Wu and Pearn (2008). The data are shown below.

0.717 0.698 0.726 0.684 0.727 0.688 0.708 0.703 0.694 0.713

0.730 0.699 0.710 0.688 0.665 0.704 0.725 0.729 0.716 0.685

0.712 0.716 0.712 0.733 0.709 0.703 0.730 0.716 0.688 0.688

0.712 0.702 0.726 0.669 0.718 0.714 0.726 0.683 0.713 0.737

0.740 0.706 0.726 0.688 0.715 0.704 0.724

The implementation of the plan is shown below.

Step 1: Take a random sample of size 47. The data are given above.

Step 2: For this data, we calculate 708915.011

Sn

iiX

nX and

017583.01

)( 2

nXX

S i .

Step3: Calculate 4741.3017583.0

)708915.077.0()(

SXUv

Step 4: Since 190.24741.3 Skv , we accept the current lot without

considering the result of past lots.

Just for sake of discussion, let us assume that the v value (for different

data set) is calculated as 2.15. In this case, we can accept the current lot provided

previous lot must have been accepted with the condition that 925.1' Skv .

Otherwise the current lot is rejected.

112

Further it is to be pointed out that the proposed variables chain

sampling plan is more efficient in terms of minimum ASN than the variables SSP

for low values of producer’s risk (α) and consumer’s risk (β). In order to prove

this, we provide two tables. Table 5.2 gives the optimal parameters of variables

chain sampling plan for some selected combinations of AQL and LQL and for

α=1% and β=1% and Table 5.3 gives the parameters of variables SSP under same

set of conditions. By comparing these two tables, one can easily observe that

variable chain sampling plan involves minimum ASN compared to the variables

SSP.

5.7 ADVANTAGES OF THE VARIABLES ChSP

This section describes the advantages of the variables chain sampling

plan over the conventional variables single sampling plan. Two acceptance

sampling plans will be called equivalent when they possess nearly identical OC

curves. A customary procedure for achieving such equivalency consists of

constructing the sampling plans so that their OC curves coincide in two suitably

chosen points namely )1,( 1 p and ),( 2 p . Suppose that for given values of

1p =0.5%, =5%, 2p =1.5% and =10%, one can find the parameters of the

known sigma variables chain sampling plan from Table 5.1 as

(i) n = 33, i =1, 'k = 2.211 and k = 2.421; ASN = 33

For the same values of the AQL and LQL, we can determine the parameters of the

single and double sampling variables plan (from Sommers (1981)) as

(ii) n = 53 and k = 2.35; ASN=53

(iii) n = 39, ak = 2.41 and rk = 2.31:ASN=43

113

It can be observed that variables chain sampling plan achieves a

reduction of over 38% in sample size than the variables SSSP and a reduction of

23% over DSP with same AQL and LQL conditions. It indicates that the variables

chain sampling plan achieves same OC curve with minimum sample size

compared to the variables single and double sampling plans.

Figure 5.1 shows the OC curves of the variables chain sampling plans

with parameters n = 18, 'k = 1.544 and k = 1.779 for different values of i . This

figure apparently shows that the variables chain sampling plan increases the

probability of acceptance in the region of principal interest, i.e. for good quality

levels and maintains the consumer’s risk at poor quality levels. This is also an

important feature of the variables chain sampling plan.

Further, it is also to be noted that the variables chain sampling plan is

economically superior to the double sampling plans in terms of average sample

number (ASN). Obviously, a sampling plan having smaller ASN would be more

desirable. The variables double or multiple sampling plans are not practically very

useful. Variables sampling Standards avoid presenting such plans due to increased

complexity involved in operating them, but the variables chain sampling plan has

no such complexity. Table 5.4 shows the ASNs for variables single sampling

plan, variables double sampling plan along with variables chain sampling plan for

some arbitrarily selected combinations of AQL and LQL. These ASN values are

calculated at the producer’s quality level for the known sigma plans. The ASN of

the variables single and double sampling plans can be found in Sommers (1981).

114

5.8 AVERAGE RUN LENGTH OF VARIABLES ChSP

Schilling (2005) has pointed out that average run length (ARL) is a

missing and meaningful measure for characterizing and evaluating the sampling

plans under Type B situations as in the process control procedures. The ARL

gives an indication of the expected number of samples until a decision is made.

The ARL can be easily calculated once the probability of acceptance (Pa(p)) of

the plan is known for any process fraction nonconforming, p. It is clear that the

distribution of the run length, L follows the geometric distribution with

probability mass function

)(1)()( 1 pPpPLf aL

aG (5.17)

Its mean and variance are respectively given by

)(11)(

pPLEARL

a (5.18)

2)(1

)()(

pPpPLVar

a

a

(5.19)

where Pa(p) is given in (5.5).

Table 5.5 gives the values of ARL of the chain sampling plan with

n = 16, 'k = 1.501 and k = 1.841 for different values of i . This table

apparently shows that when the process fraction nonconforming is small, the ARL

is high and for the increased values of fraction nonconforming the ARL is low.

By comparing the ARL values for different iσ values, when iσ increases, the ARL

values reduce even for the lower fraction nonconforming. Also it is clear from the

table that 95% of the lots will be accepted at fraction nonconforming 2% by the

variables chain sampling plan ( i =1) at an average rate of 20 inspections where as

115

with the single sampling plan ( i = ), at 2% nonconforming only 80% of lots

will be accepted at an average rate of 5 inspections. Also, 90% of the lots will be

rejected at the fraction nonconforming 7% by the variables chain sampling plan at

an average rate of 1.11 inspections and at the fraction nonconforming, 93% of the

lots will be rejected by the single sampling plan at the rate of 1.08 inspections.

5.9 CONCLUSIONS

The purpose of this chapter is to develop conditional sampling

procedures for the inspection of normally distributed quality characteristics.

Variables sampling plans generally require a smaller sample size than do

attributes plans. If the OC curve of the variables plan is unsatisfactory, then its

shape can be improved by chaining the past lot results. The proposed variables

chain sampling plan is one of the variables conditional sampling plans which also

ensure the protection against the consumer point of view. This plan is also simple

to apply rather than double and multiple sampling variables plans. Also this plan

provides better protection than the conventional single and double sampling

variables plans with minimum sample size. Such a variables chain sampling plan

will be effective and useful for compliance testing. However, it is also to be

pointed out that the variables chain sampling plan developed in this chapter is

based on the assumption that the quality characteristic of interest follows a normal

distribution. Whenever the normality assumption is not true or invalid, using of

this variables chain sampling plan can be quite misleading.

116

Table 5.1.Variables Chain Sampling Plans Indexed by AQL and LQL for =5% and =10%

1p

2p

Known Sigma Unknown Sigma

n i 'k k Sn Si Sk ' Sk 0.001

0.0025

0.005

0.0020 0.0025 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 0.012 0.015

0.004 0.005 0.006 0.0075 0.010 0.012 0.015 0.020 0.025 0.030 0.035

0.0075 0.010 0.012 0.015 0.020 0.025 0.030 0.035 0.040 0.050

119 68 46 31 21 19 15 13 12 12 9 7

230 105 64 39 26 19 16 12 10 8 7

259 86 53 33 24 16 14 12 10 8

1 2 1 2 2 4 3 3 3 6 4 3

2 2 2 2 3 2 4 4 6 5 6

1 1 1 1 4 3 6 6 4 5

2.883 2.810 2.776 2.740 2.610 2.597 2.534 2.495 2.521 2.517 2.331 2.237

2.658 2.598 2.530 2.441 2.379 2.335 2.296 2.260 2.206 2.114 2.307

2.434 2.326 2.277 2.211 2.161 2.046 1.989 2.002 1.976 1.893

3.013 2.965 2.961 2.885 2.860 2.807 2.789 2.765 2.736 2.697 2.686 2.657

2.738 2.703 2.675 2.641 2.579 2.555 2.491 2.425 2.366 2.334 2.297

2.524 2.486 2.457 2.421 2.316 2.281 2.224 2.182 2.156 2.098

650 361 242 147 103 84 72 62 51 45 40 34

1131 474 282 171 106 76 55 38 32 29 25

1113 347 211 128 75 55 43 36 31 25

1 1 1 1 1 2 3 3 2 2 3 4

2 2 1 1 1 1 1 1 2 5 5

2 1 1 1 2 2 3 3 3 4

2.888 2.834 2.774 2.716 2.637 2.599 2.532 2.518 2.470 2.425 2.414 2.387

2.676 2.585 2.533 2.464 2.430 2.321 2.237 2.139 2.163 2.107 2.092

2.442 2.348 2.299 2.233 2.100 2.083 1.974 1.979 1.964 1.925

3.013 2.984 2.964 2.921 2.897 2.829 2.792 2.768 2.765 2.750 2.694 2.637

2.736 2.705 2.698 2.669 2.610 2.601 2.572 2.529 2.413 2.342 2.302

2.512 2.483 2.454 2.418 2.345 2.293 2.249 2.209 2.174 2.105

117

Table 5.1. Contd….

1p

2p

Known Sigma Unknown Sigma

n i 'k k Sn Si Sk ' Sk 0.0075

0.010

0.02

0.010 0.012 0.015 0.020 0.025 0.030 0.035 0.040

0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.060 0.070

0.030 0.035 0.040 0.045 0.050 0.060 0.070 0.080 0.090 0.10 0.11 0.12 0.13

505 173 78 43 29 19 17 13

248 74 42 27 21 18 15 13 11 9

187 96 59 45 34 22 18 14 13 10 10 7 8

2 1 1 2 5 2 5 3 1 1 1 1 2 3 3 3 5 5 2 2 1 2 2 2 3 3 4 2 4 1 5

2.339 2.259 2.185 2.146 2.008 1.968 1.888 1.837

2.212 2.090 2.027 1.906 1.846 1.804 1.762 1.731 1.662 1.573

1.896 1.835 1.779 1.749 1.693 1.563 1.544 1.434 1.502 1.436 1.457 1.235 1.360

2.384 2.369 2.335 2.251 2.198 2.178 2.123 2.107

2.257 2.220 2.177 2.161 2.096 2.054 2.027 2.001 2.942 1.903

1.976 1.945 1.939 1.889 1.868 1.833 1.779 1.749 1.697 1.691 1.632 1.725 1.580

1817 690 286 133 83 61 47 41

786 247 135 93 66 53 46 39 30 26

528 272 169 117 88 63 44 36 27 22 23 17 17

1 2 1 1 1 1 1 3

1 1 1 2 1 2 3 2 3 4

1 2 2 1 1 2 2 3 2 1 4 1 4

2.329 2.278 2.182 2.083 2.003 1.984 1.925 1.851

2.187 2.071 2.011 1.954 1.900 1.846 1.811 1.824 1.730 1.728

1.886 1.831 1.770 1.734 1.683 1.670 1.577 1.525 1.423 1.386 1.476 1.379 1.356

2.394 2.353 2.337 2.293 2.258 2.214 2.190 2.111

2.267 2.226 2.186 2.124 2.125 2.066 2.026 2.004 1.955 1.898

1.991 1.946 1.920 1.919 1.903 1.820 1.792 1.745 1.737 1.766 1.636 1.699 1.596

118

Table 5.1. Contd….

1p

2p

Known Sigma Unknown Sigma

n i 'k k Sn Si Sk ' Sk 0.030

0.04

0.050

0.040 0.045 0.050 0.060 0.070 0.080 0.090 0.100 0.110 0.120 0.130 0.150

0.060 0.070 0.080 0.090 0.100 0.110 0.120 0.130 0.140 0.150 0.170

0.070 0.080 0.090 0.100 0.110 0.120 0.130 0.140 0.150 0.160 0.170

341 157 96 52 33 26 23 17 13 11 11 10

145 73 50 34 27 24 18 14 14 11 10

188 93 60 46 33 26 20 19 17 13 12

2 1 1 2 1 3 5 3 2 1 4 6 2 1 2 2 3 5 3 2 4 2 4 1 1 2 3 2 1 1 3 4 2 2

1.771 1.711 1.650 1.566 1.508 1.427 1.449 1.378 1.268 1.270 1.204 1.273

1.568 1.513 1.478 1.394 1.324 1.309 1.233 1.130 1.158 1.108 1.070

1.487 1.408 1.359 1.331 1.288 1.270 1.157 1.160 1.108 1.011 1.045

1.821 1.811 1.795 1.736 1.728 1.657 1.609 1.593 1.588 1.605 1.514 1.443

1.663 1.643 1.588 1.564 1.529 1.489 1.478 1.475 1.423 1.428 1.360

1.582 1.558 1.509 1.471 1.453 1.450 1.452 1.375 1.348 1.356 1.330

1027 416 253 126 81 62 44 36 29 26 23 18

339 166 102 71 53 42 34 31 27 23 17

427 203 127 86 63 49 45 34 30 25 24

5 2 1 1 1 2 2 2 2 3 3 3 2 1 1 1 1 1 1 2 3 3 2 2 1 2 1 1 1 3 2 2 1 3

1.771 1.704 1.679 1.585 1.539 1.514 1.401 1.384 1.305 1.274 1.289 1.230

1.564 1.507 1.425 1.374 1.315 1.294 1.234 1.271 1.193 1.130 1.061

1.485 1.422 1.365 1.331 1.261 1.213 1.235 1.148 1.165 1.134 1.112

1.816 1.799 1.789 1.760 1.724 1.659 1.641 1.609 1.590 1.549 1.519 1.475

1.664 1.647 1.625 1.600 1.580 1.554 1.539 1.461 1.433 1.415 1.396

1.570 1.557 1.510 1.506 1.491 1.473 1.395 1.393 1.365 1.384 1.312

119

Table 5.2:Variables Chain Sampling Plans Indexed by AQL and LQL for =1% and =1%

1p

2p

Variables Chain Sampling Plan (Known Sigma) n i 'k k )( 1pPa )( 2pPa ASN

0.001 0.0025 0.005 0.0075

0.01 0.02 0.03 0.04 0.05

0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.08 0.10

79 91 77 69 64

124 193 172 156

1 1 1 1 1 1 1 1 1

2.82801 2.56203 2.31004 2.15205 2.03504 1.84403 1.71300 1.57299 1.45798

2.83801 2.57203 2.32004 2.16205 2.04504 1.85403 1.73200 1.58299 1.46798

0.99007 0.99024 0.99012 0.99003 0.99007 0.99018 0.99007 0.9906 0.99017

0.00991 0.00956 0.00974 0.00976 0.00929 0.00997 0.00977 0.00986 0.00999

79 91 77 69 64

124 193 172 156

Table 5.3. Variables Single Sampling Plans Indexed by AQL and LQL for =1% and =1%

1p

2p

Variables Single Sampling Plan (Known Sigma)

n k )( 1pPa )( 2pPa ASN 0.001 0.0025 0.005 0.0075

0.01 0.02 0.03 0.04 0.05

0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.08 0.10

82 94 80 72 66 130 205 182 165

2.83305 2.56703 2.31505 2.15806 2.03907 1.84907 1.71806 1.57805 1.46305

0.99006 0.99001 0.99016 0.99004 0.99015 0.99018 0.99009 0.99007 0.99025

0.00992 0.00980 0.00970 0.00932 0.00957 0.00996 0.00971 0.00982 0.00984

82 94 80 72 66 130 205 182 165

120

Table 5.4: ASN Values of the Variables SSP, DSP and Variables Chain Sampling Plans

1p

2p

ASN Known Sigma Unknown Sigma

SSP DSP ChSP SSP DSP ChSP 0.001 0.0025 0.005 0.0075

0.01 0.02 0.03 0.04 0.05

0.003 0.0075 0.015 0.025 0.05 0.08 0.09 0.10 0.12

74 62 53 39 19 21 30 39 39

59.4 50.1 43.0 31.3 14.9 16.9 24.0 31.3 32.0

46 39 33 29 13 14 23 27 26

381 267 196 129 54 50 66 82 76

302.4 214.2 157.6 102.1 41.5 39.6 52.5 65.1 60.9

242 171 128 83 39 36 44 53 49

Table 5.5. ARL of Variables Chain Sampling Plan for Different i Values

p

ARL i = 1 i = 2 i = 3 i = 4 i =

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.10

88768.34

868.88 81.89 20.07 8.12 4.41 2.88 2.14 1.73 1.48 1.33 1.23 1.16 1.11 1.08 1.06 1.04 1.03 1.02

71697.5 554.66 50.25 12.66 5.42 3.14 2.19 1.73 1.47 1.32 1.22 1.15 1.11 1.08 1.06 1.04 1.03 1.02 1.01

60133.4 410.18 37.21 9.77 4.41 2.70 1.98 1.62 1.41 1.28 1.20 1.15 1.11 1.08 1.06 1.04 1.03 1.02 1.01

51781.5 327.19 30.13 8.25 3.92 2.50 1.89 1.58 1.39 1.28 1.20 1.14 1.11 1.08 1.06 1.04 1.03 1.02 1.01

607.80 38.29 10.64 5.06 3.15 2.29 1.83 1.56 1..39 1.28 1.20 1.14 1.11 1.08 1.06 1.04 1.03 1.02 1.01

121

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Fraction Non-conforming, p

Prob

abili

ty o

f Acc

epta

nce,

Pa(

p)

Figure 5.1: OC Curves of a Variables Chain Sampling Plan for Different iσ Values

iσ =1

iσ =2

iσ =3

iσ =4

iσ =∞

122

CHAPTER 6

OPTIMAL DESIGNING OF

VARIABLES SAMPLING PLAN FOR RESUBMITTED LOTS

6.1 INTRODUCTION

Lot resubmissions are permitted in situations where the original

inspection results are suspected or when the producer or supplier is allowed to opt

for resampling. Moreover, in testing certain products, the test procedures are not

always accurate and also obtaining a random sample is difficult etc. In such cases,

resampling is preferable. Based on this idea, this chapter attempted to develop a

variables sampling plan for the inspection of resubmitted lots of normally

distributed quality characteristics.

6.2 CONDITIONS OF APPLICATION

The following assumptions should be valid for the application of the

variables resampling scheme.

(i) It is required to discard the results of the original inspection that resulted in

non-acceptance of the lot. This may be by the provisions of a Contract or

Statute.

(ii) The consumer has confidence in the producer and producer will not

deliberately take advantage of the resampling option.

In addition, the usual conditions for the application of variables single sampling

plans with known or unknown standard deviation should also be valid.

123

6.3 OPERATING PROCEDURE OF THE KNOWN SIGMA VARIABLES RESAMPLING SCHEME

Suppose that the quality characteristic of interest has the upper

specification limit U and follows a normal distribution with unknown mean μ and

known standard deviation σ. Then the following procedure is proposed for the

variables resampling scheme .

Step 1: Perform original inspection by using variables single sampling plan as the

reference plan, i.e From each submitted lot, take a random sample of size

n, say nXXX ..., 21 and compute

XUv

, where

n

iiX

nX

1

1 .

Step 2: Accept the lot if kv and reject the lot if kv . On non-acceptance on

the original inspection, apply the variables single sampling plan for m

times and reject the lot if it is not accepted in the last stage of inspection.

In the case of lower specification limit L, the operating procedure is as follows.

Step 1: Perform original inspection by using variables single sampling plan as the

reference plan, i.e From each submitted lot, take a random sample of size

n, say nXXX ..., 21 and compute

LXv

, where

n

iiX

nX

1

1 .

Step 2: Accept the lot if kv and reject the lot if kv . On non-acceptance on

the original inspection, apply the variables single sampling plan for m

times and reject the lot if it is not accepted in the last stage of inspection.

124

Thus, the proposed variables resampling scheme is characterized by three

parameters, namely n, k and m. If m=1, then the proposed resampling scheme

will reduce to the variables single sampling plan.

6.4 DESIGNING OF VARIABLES RESAMPLING SCHEME WITH KNOWN STANDARD DEVIATION

The variables sampling plans are, in general, designed based on two

points on the OC curve namely (p1, 1-α) and (p2, β), where p1 is called the

acceptable quality level (AQL), p2 is the limiting quality level (LQL), α is the

producer’s risk and β is the consumer’s risk. A well designed sampling plan must

provide at least (1- α) probability of acceptance of a lot when the process fraction

nonconforming is at AQL and the sampling plan must also provide not more than

β probability of acceptance if the process fraction nonconforming is at the LQL.

Thus the acceptance sampling plan must have its OC curve passing through two

designated points (AQL, 1-α) and (LQL, β). The proposed variables resampling

scheme is designed based on the two points on the OC curve in the following

manner.

The fraction non-conforming in a lot will be determined as

Up 1 (6.1)

where )(y is given by

y

dzzy2

exp21)(

2

(6.2)

provided that the unit is classified as non-conforming if it exceeds the upper

specification limit U. So, the unknown mean μ can be determined if p is specified.

125

Let us define the standardized quality characteristic corresponding to

the fraction conforming as

)1(1 pz p (6.3)

Govindaraju and Ganesalingam (1997) derived the performance

measures such as OC function, average sample number (ASN), average outgoing

quality (AOQ) etc for the attributes resampling scheme. Based on this, the

performance measures of the variables resampling scheme are given as follows.

The OC function of the variables resampling scheme, which gives the

proportion of lots that are expected to be accepted for given product quality, p is

given by

)()(1....)()(1)()(1)()( 12 pPpPpPpPpPpPpPpL am

aaaaaa

ma pP )(11 (6.4)

where kvpPa Pr)( is the probability of accepting a lot based on a variables

single sampling plan with parameters (n, k). Under type B situation (i.e. a series

of lots of the same quality), forming lots of N items from a process and then

drawing random sample of size n from these lots is equivalent to drawing

random samples of size n directly from the process. Hence the derivation of the

OC function is straightforward.

The ASN function is given by

npPnpPnpPnpASN maaa

12 )(1....)(1)(1)(

)()(11 pPpPn am

a (6.5)

126

Govindaraju and Ganesalingam (1997) suggested that the preferred

value of m is two. That is, resampling is done on the resubmitted lot once if the lot

is not accepted on the original inspection. In this chapter, we also consider m=2

only for the proposed plan. So, when m=2, the probability of acceptance and ASN

of the resampling scheme become,

)(2)()( pPpPpL aa (6.6)

)(2)( pPnpASN a (6.7)

The above measures of the resampling scheme under variables inspection can also

be written as

)(2)()( wwpL (6.8)

)(2)( wnpASN (6.9)

where nkzw p

If (AQL, 1-α) and (LQL, β) are prescribed then we have

1)(2)( 11 ww

and )(2)( 22 ww (6.10)

Here w1 is the value of w at p=p1 and w2 is the value of w at p=p2. That is,

nkzw p )(11 and nkzw p )(

22 (6.11)

where 1pz is the value of pz at AQL and

2pz is the value of pz at LQL.

For given AQL and LQL, the parametric values of the variables

resampling scheme namely nσ, k, and m are determined by satisfying the required

127

producer’s and consumer’s conditions. Alternatively, we can determine the above

parameters of the variables resampling scheme to minimize the ASN at LQL,

which is analogous to minimizing the ASN in the variables repetitive group

sampling plans (see Balamurali et al. (2005). The ASN for the resampling scheme

is given in (6.5). Therefore, the following optimization problems are considered to

determine those parameters by minimizing the ASN at LQL and minimizing the

sum of ASN at AQL and LQL respectively.

Minimize )()(2 222 wwnpASN

Subject to 1)( 1pL

)( 2pL

0 ,1m ,1 kn (6.12)

and Minimize

)(

)(2)(

)(2)()(

2

2

1

121 w

ww

wnpASNpASN

Subject to 1)( 1pL

)( 2pL

0 ,1,1 kmn (6.13)

Here L(p1) and L(p2) are the probability of acceptance of the lot at AQL and LQL

respectively based on resampling scheme.

6.5 OPERATING PROCEDURE OF VARIABLES RESAMPLING SCHEME WITH UNKNOWN SIGMA Whenever the standard deviation is unknown, we may use the sample standard

deviation S instead of σ. In this case, the proposed scheme operates as follows.

Step 1: Perform original inspection by using variables single sampling plan as the

reference plan, i.e From each submitted lot, take a random sample of size

128

nS, say SnXXX ..., 21 and compute

SXUv

, where

Sn

ii

S

Xn

X1

1 and

1

)( 2

S

i

nXX

S .

Step 2: Accept the lot if Skv and reject the lot if Skv . On non-acceptance on

the original inspection, apply the variables single sampling plan for m

times and reject the lot if it is not accepted on (m-1)st inspection.

Thus, the proposed unknown sigma variables resampling scheme is characterized

by three parameters namely nS, kS and m.

6.6 DESIGNING OF VARIABLES RESAMPLING SCHEME WITH UNKNOWN STANDARD DEVIATION The determination of parameters (nS, kS, m) of unknown sigma plan is

slightly different from the known sigma case. It is known that SkX S is

approximately normally distributed with mean )(SEkS and variance

)(2

SVarkn S (see Duncan (1986)). That is,

nk

nkNSkX SSS 2

,~2

22

Therefore, the probability of accepting a lot at each repetition is given by

pSkUXPkvP SS )(

pUSkXP S

129

21)/(

2S

S

S

kn

kU

21

)( 2S

SSp k

nkz (6.13)

If we let,

21

)( 2S

SSpS k

nkzw then )()Pr( SS wkv . (6.14)

Hence the lot acceptance probability of the resampling scheme under sigma

unknown case is given by

)(2)()( SS wwpL

We obtain SS ww 21 and corresponding to AQL and LQL respectively by

21

)( 211S

SSpS k

nkzw and

21

)( 222S

SSpS k

nkzw (6.15)

In this case, the optimization problem becomes,

Minimize )()(2 222 wwnpASN S

Subject to 1)( 1pL

)( 2pL

0 ,1,1 SS kmn (6.16)

and

Minimize

)(

)(2)(

)(2)()(

2

2

1

121 w

ww

wnpASNpASN S

130

Subject to 1)( 1pL

)( 2pL

0 ,1,1 SS kmn (6.17)

Here L(p1) and L(p2) are the probability of acceptance of the lot at AQL and LQL

respectively.

We may determine the parameters of the known and unknown sigma

resampling schemes by solving the nonlinear equations given in [(6.12),(6.13)]

and [(6.16),(6.17)] respectively. Generally a sampling plan would be desirable if

the required number of sampled is small. So, this chapter considers the ASN as

the objective function to be minimized with the probability of acceptance along

with the corresponding producer’s and consumer’s risks as constraints. To solve

the above nonlinear optimization problems given in [(6.12),(6.13)] and

[(6.16),(6.17)], the sequential quadratic programming (SQP) proposed by Nocedal

and wright (1999) can be used. The SQP is implemented in Matlab software using

the routine “fmincon”. By solving the nonlinear problem mentioned above, the

parameters (n, k) for known sigma plan and the parameters (nS, kS) for unknown

sigma plan are determined and these values are tabulated in Table 6.1.

We have observed an interesting thing from Table 6.1 and Table 6.2 is

that we are getting almost same values of the parameters of the proposed variables

resampling scheme while minimizing ASN(p2) as well as minimizing the sum of

ASN at both AQL and LQL.

131

6.7 DESIGNING EXAMPLES

6.7.1. SELECTION OF KNOWN SIGMA VARIABLES RESAMPLING SCHEME INDEXED BY AQL AND LQL Table 6.1 is used to determine the parameters of the known and

unknown variables resampling schemes for specified values of AQL and LQL

when = 5% and = 10% with minimum ASN at LQL. Similarly Table 6.2 can

be used for obtaining the optimal parameters of the proposed scheme for both

known and unknown sigma with minimum sum of ASN at AQL and LQL. For

example, if p1 = 2%, p2 = 5%, = 5% and = 10%, Table 6.1 gives the

parameters as n = 35, k = 1.924 and m = 2. The ASN for this plan is 68.274. At

the same time, the unknown sigma resampling scheme under the same AQL and

LQL conditions, is determined as nS = 99, kS = 1.922 and m=2. The ASN at LQL

for this plan is 192.942.

For the above example, the plan is operated as follows.

Step 1: From each submitted lot, take a random sample of size 35 and compute

LXv , where

35

1351

iiXX .

Step 2: Accept the lot if 924.1v and reject the lot if 924.1v . On non-

acceptance on the original inspection, apply the variables single sampling plan for

2 times and reject the lot if it is not accepted on the 1st inspection.

6.7.2. SELECTION OF UNKNOWN SIGMA RESAMPLING SCHEME INDEXED BY AQL AND LQL

The unknown sigma resampling scheme is operated as a known sigma

resampling scheme but the parameters nS, kS and m are used in the place of nσ, kσ

and m respectively. Table 6.1 can also be used for the selection of the parameters

132

of the unknown variables resampling scheme for given values of AQL and

LQL. Suppose that AQL=1%, LQL=3%, =5% and =10%. From Table 6.1, the

parameters of the variables resampling scheme can be determined as nS = 99, kS =

2.183 and m = 2. The ASN for this plan is 192.950.

6.8 MERITS OF THE VARIABLES RESAMPLING SCHEME

In this section, we discuss the advantages of the variables resampling

scheme over the conventional variables SSP. Two acceptance sampling plans will

be called equivalent when they possess nearly identical OC curves. A customary

procedure for achieving such equivalency consists of constructing the sampling

plans so that their OC curves coincide in two suitably chosen points namely (p1,

1-α) and (p2, β). Suppose that for given values of p1=0.5%, =5%, p2=1% and

=10%, one can find the parameters of the known sigma variables resampling

scheme from Table 6.1 as

(i) n = 94, m=2, k = 2.497 and ASN (at LQL) = 183.401

For the same values of the AQL and LQL, we can determine the parameters of the

single sampling variables plan from (Sommers (1981)) as

(ii) nσ = 138 and kσ = 2.44, ASN=138

It can be observed that variables resampling scheme achieves a

reduction over 46% in sample size and about 25% reduction in ASN than the

variables SSP with same AQL and LQL conditions. It indicates that the variables

resampling scheme achieves same OC curve with minimum sample size or ASN

compared to the variables SSP. Further, Figure 6.1 gives the ASN curves of the

above mentioned variables resampling scheme and variables SSP. The resampling

scheme requires more sample size or ASN when the quality is poor that is, for

133

higher fraction non-conforming. This is logical since lots that are declared as not

acceptable are always resampled even though the original inspection showed the

evidence of poor quality. The main advantage and strength of the resampling

scheme lies in achieving smaller ASN at good quality that is, low fraction non-

conforming in which case the usual variables SSP requires a larger sample size or

ASN. This can easily be observed from Figure 6.1 and Figure 6.2 show the OC

curves of the variables resampling scheme with parameters n = 10 and kσ = 2.085

for different values of m such as m=2, m=3, m=4 and m=5. This figure shows that

when m increases the AQL values also increase.

Further, it is also to be pointed out that the variables resampling scheme

is economically superior to the variables single sampling plans in terms of ASN.

Obviously, a sampling plan having smaller ASN would be more desirable. The

variables double or multiple sampling plans are not practically very useful.

Variables sampling Standards avoid presenting such plans due to increased

complexity involved in operating them, but the variables resampling scheme has

no such complexity. Table 6.3 shows the ASNs for variable single and double

sampling variables plan along with variables resampling scheme for some

arbitrarily selected combinations of AQL and LQL. These ASN values are

calculated at the producer’s quality level for the known sigma plans. The ASN of

the variables single and double sampling plans can be found in Sommers (1981).

From this table, one can easily understand that the variables resampling scheme

will have minimum ASN when compared to the variables single sampling plans

and achieves the almost the same ASN as the variables double sampling plans.

Similar reduction in the ASN can be achieved for any combination of AQL and

LQL values. This implies that variables resampling scheme will give desired

protection with minimum inspection so that the cost of inspection will greatly be

134

reduced. Thus the variables resampling scheme provides better protection to the

producers than the variables single sampling plan.

6.9 COMPARISON WITH ATTRIBUTES RESAMPLING SCHEME

This section compares the proposed variables resampling schemes with

the attributes resampling scheme of Govindaraju and Ganesalingam (1997). For

this purpose, Table 6.4 is presented which gives the samples sizes of both

attributes and variables resampling schemes for different combinations of AQL

and LQL. The sample sizes of the attributes resampling schemes are taken from

Govindaraju and Ganesalingam (2007). This table apparently shows that variables

resampling scheme achieves a great reduction in sample size over the attributes

resampling scheme. Also when m increases for fixed values of AQL and LQL, the

sample size of the resampling scheme decreases but ASN(p2) will increase. This

can be easily observed from Table 6.5. Table 6.5 displays the optimal sample size

and ASN(p2) for different values of m and for fixed AQL and LQL values.

6.10 CONCLUSIONS

The proposed variables resampling scheme gives more protection to the

producer at the same time it also ensures the protection against the consumer

point of view. This plan is also simple to apply rather than double and multiple

sampling variables plans. Also this plan provides better protection than the

conventional single sampling variables plans with minimum ASN.

135

Table 6.1. Variables Resampling Schemes (with m=2) Involving Minimum ASN Indexed by AQL and LQL

p1

p2

MinASN(p2) Known Sigma Unknown Sigma

n kσ ASN(p2) nS kS ASN(p2) 0.001

0.005

0.01

0.02

0.03

0.04

0.05

0.0020 0.004 0.006 0.008

0.006 0.008 0.01

0.012

0.02 0.025 0.03

0.035

0.03 0.035 0.04 0.05

0.04

0.045 0.05 0.06

0.05 0.06 0.07 0.08

0.06 0.07 0.08 0.09 0.10

129 30 18 13

1445 208 94 57

78 43 29 22

194 99 63 35

342 168 104 54

512 151 76 49

711 201 101 63 44

3.023 2.951 2.902 2.862

2.555 2.523 2.497 2.475

2.240 2.210 2.184 2.163

1.999 1.977 1.957 1.924

1.839 1.822 1.805 1.777

1.717 1.688 1.663 1.639

1.616 1.591 1.568 1.547 1.528

251.552 58.477 35.118 25.335

2815.899 405.631 183.401 111.153

152.108 83.827 56.514 42.905

378.350 193.036 122.805 68.274

666.519 327.547 202.676 105.235

997.757 294.335 148.102 95.514

1385.524 391.712 196.871 122.797 85.753

713 160 90 64

6164 868 381 230

272 148 99 73

581 291 183 99

919 446 273 140

1266 364 182 113

1640 455 224 137 95

3.023 2.951 2.906 2.873

2.555 2.523 2.496 2.474

2.239 2.209 2.183 2.161

1.998 1.977 1.957 1.922

1.839 1.822 1.806 1.777

1.717 1.688 1.663 1.641

1.616 1.591 1.568 1.548 1.529

1389.883 311.820 175.410 124.766

12011.75 1692.537 742.489 448.209

530.048 288.427 192.950 142.264

1132.236 567.281 356.665 192.942

1790.912 869.483 532.261 272.895

2467.042 709.339 354.743 220.263

3195.917 886.678 436.525 267.022 185.177

136

Table 6.2.Variables Resampling Schemes (with m=2) Involving Minimum Sum of ASN Indexed by AQL and LQL

p1

p2

Min[ASN(p1)+ASN(p2)] Known Sigma Unknown Sigma

n

kσ

ASN(p1)+ ASN(p2)

nS

kS

ASN(p1)+ ASN(p2)

0.001

0.005

0.01

0.02

0.03

0.04

0.05

0.0020 0.004 0.006 0.008

0.006 0.008 0.01

0.012

0.02 0.025 0.03

0.035

0.03 0.035 0.04 0.05

0.04

0.045 0.05 0.06

0.05 0.06 0.07 0.08

0.06 0.07 0.08 0.09 0.10

129 30 18 13

1445 208 94 57

78 43 29 22

194 99 63 35

342 168 104 54

512 151 76 49

711 201 101 63 44

3.023 2.951 2.902 2.862

2.555 2.523 2.497 2.475

2.240 2.210 2.184 2.163

1.999 1.977 1.957 1.924

1.839 1.822 1.805 1.777

1.717 1.688 1.663 1.639

1.616 1.591 1.568 1.547 1.528

409.290 95.166 56.940 41.005

4571.700 660.100 298.320 180.887

247.511 136.414 91.948 69.788

615.683 314.102 199.763 111.029

1083.797 533.050 329.550 171.273

1624.070 478.664 241.006 155.159

2253.381 637.426 320.076 199.569 139.392

713 160 90 64

6164 868 381 230

272 148 99 73

581 291 183 99

919 443 273 140

1266 364 182 113

1640 455 224 137 95

3.023 2.951 2.906 2.873

2.555 2.523 2.496 2.474

2.239 2.209 2.183 2.161

1.998 1.977 1.957 1.922

1.839 1.821 1.806 1.777

1.717 1.688 1.663 1.641

1.616 1.591 1.568 1.548 1.529

2262.286 507.560 285.412 202.938

19500.28 2754.619 1207.84 729.240

862.196 469.126 313.654 231.305

1840.637 923.313 580.282 313.605

2912.349 1403.939 866.205 444.022

4015.786 1153.925 577.120 358.503

5197.656 1442.941 709.906 434.460 301.225

137

Table 6.3: ASN Values of the Variables Single, Double Sampling Plans and Resampling Schemes

p1

p2

ASN at AQL Known Sigma

Single Double Resampling Scheme 0.001

0.0025 0.005

0.0075 0.01 0.02 0.03 0.04 0.05

0.004 0.0075 0.015 0.030 0.03 0.05 0.09 0.13 0.11

45 62 53 29 44 52 30 22 49

36.8 50.1 43.0 23.2 35.0 42.3 24.0 18.3 40.0

36.69 50.15 42.81 23.22 35.48 42.82 24.45 18.35 40.35

Table 6.4: Sample Size of the Attributes and Variables Resampling Schemes

p1

p2

Sample size Attributes Resampling

Scheme Variables Resampling

Scheme 0.001 0.001 0.001

0.0025 0.0025 0.0025 0.004 0.004 0.004

0.0065 0.0065 0.0065

0.01 0.02 0.03 0.01 0.02 0.03 0.01 0.02 0.03 0.01 0.02 0.03

470 148 98 624 234 98

1177 311 156

* 249 207

10 6 4

25 11 7

231 65 38 927 117 58

* sample size greater than 5000

138

Table 6.5: ASN of the Variables Resampling Schemes for Different m values

p1

p2

m

Variables Resampling Scheme

Sample Size

ASN at LQL

0.001

0.002

0.01

0.004

0.005

0.02

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

226 160 136 121 112

439 306 255 228 209

388 272 227 201 186

226.0 311.822 394.123 465.480 537.145

439.0

596.375 739.072 877.211 1002.380

388.0

530.048 657.840 773.324 892.258

139

Figure. 6.1: ASN Curves of a Variables Single Sampling Plan and Resampling Sampling Scheme

140

FIGURE 6.2: OC CURVES OF A VARIABLES RESAMPLING SAMPLING SCHEME FOR DIFFERENT m VALUES

141

CHAPTER 7

CONCLUSIONS AND FUTURE WORK

This thesis is mainly focused on optimal designing of the special

purpose sampling plans for the application of variables inspection. In this thesis,

the optimal parameters of respective special purpose plans are determined based

on two-points on the OC curve approach namely (AQL, 1-α) and (LQL, β). In

chapter 1, we have presented certain basic concepts and review of literature which

are related to this thesis.

The quick switching system developed by Dodge (1967) is one of the

two-plan systems for the application of attributes quality characteristics. In any

two-plan system, the tightened inspection can be used when the quality of the

product deteriorated and normal inspection is used when the quality is found to be

good. In Chapter 2, we have investigated the optimal designing of variables quick

switching system by minimizing the average sample number, where the quality

characteristic of interest has the single specification limit and follows a normal

distribution. We have considered both known and unknown sigma cases for the

designing the variables quick switching system. The advantages of the variables

quick switching system over the variables single and double sampling plans and

attributes quick switching system have also been discussed. Tables have also been

constructed for the selection of parameters of known and unknown standard

deviation variables quick switching system for given AQL and LQL. Comparisons

have been made in terms of ASN with the existing plans and proved that ASN

given in the proposed system is optimum.

Chapter 3 of this thesis deals with the optimal designing of variables

quick switching system with double specification limits by minimizing the ASN.

142

We have investigated the variables quick switching system when a measurable

quality characteristic has double specification limits beyond which an item is

considered to be a non-conforming. The quality characteristic of interest is

assumed to follow a normal distribution. In this chapter, we have considered two

cases of fraction non-conforming namely symmetric fraction non-conforming and

asymmetric fraction non-conforming. The probability of acceptance of the

proposed variables quick switching system under double specification limits for

symmetric and asymmetric fraction non-conforming cases have also been derived.

The optimal parameters of the proposed system have been determined based on

two points on the OC curve approach by solving the optimization problem for the

known sigma asymmetric and asymmetric fraction non-conforming cases

separately. Necessary tables have also been constructed for the determination of

optimal parameters of the proposed sampling system with double specification

limits. Based on the comparisons, it has been proved that the proposed variables

quick switching system with double specification limits is optimum.

The Tightened-Normal-Tightened (TNT) sampling scheme procedure

developed by Calvin (1977) is a particular case of the general two plan system for

the inspection of attributes characteristics. In Chapter 4 of this thesis, we have

presented the optimal designing of variables TNT scheme by minimizing the ASN.

The proposed scheme can be applied for measurable characteristics, where the

quality characteristic follows normal distribution and has upper or lower

specification limit. We have considered known sigma and unknown sigma cases

separately for designing the proposed variables TNT scheme. Non-linear

optimization problems have been used to determine the optimal parameters of the

proposed scheme under known and unknown sigma cases. The advantages of the

proposed variables scheme over variables single, double sampling plans and

143

attributes sampling scheme have been discussed. Tables have also been

constructed for the application of the proposed scheme.

The concept of chain sampling was first introduced by Dodge (1955)

for the application of attribute quality characteristics. Govindaraj and Balamurali

(1998) extended the concept of chain sampling to variables inspection. Chapter 5

of the thesis has investigated the optimal designing of variables chain sampling

plan by minimizing the ASN. The chain sampling plan is one of the conditional

sampling plans and this plan under variables inspection will be useful for costly

and destructive testing. We have formulated an optimization problem for

determining the parameters of known and unknown sigma chain sampling plans.

The advantages of this proposed variables chain sampling plan over variables

single sampling plan and variables double sampling plan have also been discussed.

Tables have also been constructed for the selection of optimal parameters of

known and unknown standard deviation variables chain sampling plan for

specified two points on the operating characteristic curve namely the AQL and the

LQL along with the producer and consumer’s risks.

Govindaraju and Ganesalingam (1997) has proposed an attribute

sampling plan which can be applied in situations where resampling is permitted on

lots not accepted on original inspection. They have derived the performance

measures of the resampling scheme having single sampling attributes plan as the

reference plan. In this plan, it is assumed that during the course of resubmission,

the quality of the lot is not improved by sorting etc. They have also discussed the

need for a provision for resampling of lots in case of zero acceptance sampling

plans. Chapter 6 of the thesis has dealt with the optimal designing of variables

sampling plan for resubmitted Lots. This sampling plan can be applied for

inspection of resubmitted lots when the quality characteristic of interest follows

144

normal distribution and has single specification limit. Resubmission of lots for

inspection is allowed in some situations where the original inspection results are

suspected or when the supplier or producer is allowed to opt for resampling as per

the provisions of the contract etc. We have considered both known and unknown

sigma cases. Non-linear optimization problem has been considered for the

selection optimal parameters. The advantages of the proposed variables sampling

plan over the existing variables single sampling plan have also been discussed.

Useful tables have also been constructed for the selection of optimal parameters of

known and unknown standard deviation cases of the proposed variables sampling

plan.

In this thesis, we have developed five different sampling

systems/plans for variables inspection. The conditions of application of each

sampling procedure have been given in the respective chapter. All the sampling

plans provided in this thesis can be applied for the inspection of normally

distributed quality characteristics. However, the following may be an additional

guideline for applying appropriate sampling system.

(i) If the quality characteristic of interest has single specification limit and if we

want to use same sample size in both normal and tightened inspections but

with two different acceptance criteria, then the VQSS presented in Chapter 2

can be used.

(ii) If the quality characteristic of interest has single specification limit but we

want to use different sample sizes in normal and tightened inspections and

same acceptance criterion, then the TNT scheme proposed in Chapter 4 can be

applied.

(iii) If the quality characteristic of interest has double specification limits, then the

VQSS developed in Chapter 3 can be implemented.

145

(iv) Whenever we want to take a decision on the current lot submitted for

inspection based on the history of the previous lot quality, one can utilize the

chain sampling plan proposed in Chapter 5.

(v) Whenever the original inspection results are suspected or when the producer

or supplier has provision of opting resampling, the variables resampling

scheme proposed in Chapter 6 can be opted for quality inspection.

All the sampling systems/plans developed in this thesis are

applicable for the inspection of normally distributed quality characteristics. There

is no special purpose sampling plan available in the literature when the quality

characteristic under study follows other than a normal distribution. So, developing

special purpose sampling plans for other Gaussian family of distributions such as

Inverse Gaussian distribution, Half-nomal distribution and Folded Normal

distribution will be considered as future study.

146

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LIST OF PUBLICATIONS

1. Balamurali, S. and Usha, M. (2012),”Optimal Designing of a Variables

Quick Switching Sampling System by Minimizing the Average Sample

Number”, Journal of Applied Statistical Science, Vol.19, No.3, pp.51-66.

2. Balamurali, S. and Usha, M. (2012),”Variables Quick Switching System with

Double Specification Limits”, International Journal of Reliability,

Quality and Safety Engineering, Vol.19, No.2, pp.1250008-1-17.

3. Balamurali, S and Usha, M. (2013),”A New Failure Censored Variables

Sampling System for Weibull Distribution”, International Journal of

Performability Engineering, Vol.9, No.1, pp.3-12.

4. Balamurali, S. and Usha, M. (2013),”Optimal Designing of Variables Chain

Sampling Plan by Minimizing the Average Sample Number”,

International Journal of Manufacturing Engineering, Volume 2013,

Article ID 751807, 12 pages.

5. Balamurali, S. and Usha, M. (2014),”Optimal Designing of Variables Quick

Switching System Based on the Process Capability Index Cpk”, Journal

of Industrial and Production Engineering, Vol.31, No.2, pp.85-94.

6. Balamurali, S. and Usha, M.(2015),”Optimal Designing of Variables

Sampling Plan for Resubmitted Lots”, Communications in Statistics-

Simulation and Computation, Vol. 44, No.5, pp. 1210-1224.

160

7. Balamurali, S. and Usha, M. (2015),”Optimal Designing of Variables

Tightened-Normal-Tightened Sampling Scheme by Minimizing

the Average Sample Number”, International Journal of Industrial and

Systems Engineering, Vol.21, No.1, pp.99-118.

8. Balamurali, S. and Usha, M. (2014),”A New System of Skip-lot Resampling

Schemes”, Communications in Statistics-Simulation and Computation

(Accepted) .

9. Balamurali, S. and Usha, M. (2015),”Designing of Variables Quick

Switching Sampling System by Considering Process Loss Functions”,

Communications in Statistics-Theory and Methods (Accepted).

LIST OF PAPERS PRESENTED IN CONFERENCES

(1) Balamurali, S. and Usha, M (2011),”Evaluation of MLP-rx1Continuous

Sampling Plan Using a Markov Chain Model”, Paper Presented at the

International Conference on Mathematical Modelling and

Applications to Industrial Problems (MMIP 2011), NIT, Calicut,

March 28-31, 2011.

161

CURRICULUM VITAE

Mrs. M. Usha was an Assistant Professor at Kalasalingam University for 7

years. She has also more than 6 years of teaching experience at undergraduate and

postgraduate levels in various established Arts and Science Colleges in

Tamilnadu, India.

She has obtained her B.Sc (Mathematics) from Madurai Kamaraj

University with first class in the year 1997. She has obtained her Post Graduate

degree, M.Sc ( Mathematics) from Madurai Kamaraj University with first class in

the year 1999. She completed her M.Phil (Mathematics) with first class with

distinction in the year 2000.

She is a life member of Indian Society of Technical Education (ISTE).

Her areas of interest are Statistical Quality Control, Probability and Statistics and

Operations Research. She has published her research articles in refereed

International Journals and International conference.