~TATISTICALLY AND ECONOMICALLY BASED ATTRIBUTE ACCEPTANCE SAMPLING MODELS WITH INSPECTION ERRORS

by

Rufus D. Collins Jr.

Dissertation Submitted to the Graduate Faculty of the

Virginia Polytechnic Institute and State University

in Partial Fulfillment of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY

in

INDUSTRIAL ENGINEERING AND OPERATIONS RESEARCH

APPROVED:

K. E. Case, Chairman

G. K. Bennett M. R. Reynolds

J. W. Schm1dt' H. L. Snylt'er

May, 1974

Blacksburg, Virginia

ACKNOWLEDGEMENTS

The author wishes to express sincere appreciation to the individuals

who made the completion of this research possible. I am deeply indebted to my

major advisor, Dr. K. E. Case, for bis able counsel and supervision. Thanks

are also due the other members of my committee, Dr. G. K. Bennett, Dr.

M. R. Reynolds, Dr. J. W. Schmidt, and Dr. H. L. Snyder for their valuable

criticisms and suggestions. For the encouragement of Mr. C. O. Brooks, I am

indeed grateful. Also, I owe much to the skill of Mrs. Kathy Mercier who typed

this final manuscript.

ii

TABLE OF CONTENTS

Page

CHAPTER I. INTRODUCTION ..•.•••••...•.••...•.•••• 1

Attributes Acceptance Sampling . • . • • • • • • • • • • • • • • . • . • . 2

Survey of Existing Research. • • • • • • • • • • . • • • • • • • • • . • . . 6

Research Purpose and Scope . . • • • • • • • . • • . • • • • • • • • • • • 14

CHAPTER II. ERROR EFFECTS ON SINGLE SAMPLING ATTRIBUTE ACCEPTANCE PLANS . • • • • • • • • • • 18

Performance Measures . • . • • • • • • • • • . • • • • • . • • • • • • • • • 18

Sensitivity Analysis • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 30

CHAPTER III. ECONOMICALLY BASED ATTRIBUTE ACCEPT-ANCE SAMPLING MODEL •••••••••••••••••• 38

Distributional Considerations • • • • • • • • • • • • • • • • • • • • • • • • 39

Formulation of the Model. • • • • • • • • • • • • • • • • • • • • • • • • • • 49

CHAPTER IV. COST MODEL EVALUATION: MIXED BINOMIAL PRIOR DISTRIBUTION • • • • • • • • • • • • • • • • • • • • 61

Summary of Equations. • • • • • • • • • • • • • • • • • • • • • • • • • • • • 62

Input Data for the Cost Model Evaluation . • • • • • • • • • • • • • • • 63

Selection of An Optimal Sampling Plan Without Errors. • • • • • • 65

Expected Cost of an Optimal Plan When Error is Present. • • • • 72

Optimal Sampling Plan Designed for an Error Prone Process • • 79

iii

TABLE OF CONTENTS ( Concluded)

Page

CHAPTER V. COST MODEL EVALUATION: POLYA PRIOR DISTRIBUTION • • • • • • • • • • • • • • • • • • • • • • • • • 102

Primary Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Alternate Method . • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 127

CHAPTER VI.

Summary

SUMMARY AND RECOMMENDATIONS •••••••••

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

146

Areas for Future Research • • • • • • • • • • • • • • • • • • • • • • • • • 159

APPENDIX A. THE INCREMENTAL SEARCH PROCEDURE • • • • • 162

APPENDIX B. DEVELOPMENT OF RELATED FORMULAE...... 166

APPENDIX C. DOCUMENTATION OF COMPUTER PROGRAM. • • • 176

REFERENCES. • . . . . . . . • • • . • . . . . . • . • • . . . . • . . . . . • • . . • 214

BIBLIOGRAPIIY. • • . • • • . • . . . • . • • . . . • . • • • • . • • . . • • . • • • . 216

VITA . . . . • • • . . . . . . . . . . • • . . • • • • • • . • . . . • . . . • • • • • • . • 222

iv

LIST OF TABLES

Table Title Page

IV-1 Input Data for Cost Example. • • • • • • • • . • • • • • • • • • • • • 64

IV-2 Total Expected Cost for Varied Mixed Binomial Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

IV-3 Effect of a Change in the Process Variance ••••••••••• 70

IV-4 Performance Measure~ 1 When the Optimal Sampling Plan

(n = 122, c = 9) is Subject to Error • • • • • • • • • • • • • • • • 73

IV-5 Performance Measure~ 2 of Optimal Plans Designed for

Error Prone Process • • • • • • • • • • • • • • • • • • • • • • • • • • 81

IV-6

IV-7

IV-8

Performance Measure 3 for Selected Error Pairs ••••••

Mixed Binomial Parameter Sets •••••••••••••••••••

Cost Data for Mixed Binomial Parameter Sets ••••••••••

V-1 Performance Measure L\ 1 When the Optimal Sampling Plan

87

96

97

(n = 122, c = 6) is Subject to Error • • • • • • • • • • • • • • • • 106

V-2 Performance Measure ~ 2 of Optimal Plans Designed for

Error Prone Process • • • • • • • • • • • • • • • • • • • • • • • • • • 116

V-3

V-4

Performance Measure ~ 3 for Selected Error Pairs ••••••

Cost Data Obtained with Polya Prior and Sampling Plans Derived from Mixed Binomial Parameter Sets ••••••••••

V-5 Performance Measure 1 When the Optimal Sampling

118

125

Plan ( n = 122, c = 6) is Subject to Error • • • • • • • • • • • • 132

V-6 Performance Measure 6. 2 of Optimal Plans Designed

for Error Prone Process . . . . . . . . . . . . . . . . . . . . . . . 138

V

LIST OF TABLES ( Concluded)

Table Title Page

V-7 Performance Measure .6. 3 for Selected Error Pairs. • • • • • 143

VI-1 Summary of Selected Cost Data from Example Problems • • 153

VI-2 Summary of Performance Measures for Example Problems in Chapters IV and V. • • • • • • • • • • • • • • • • • • • • • • • • • • 155

vi

Figure

2-1.

2-2.

LIST OF ILLUSTRATIONS

Title

Effect of Errors on the Probability of Accepting a Lot

Effect of Errors on the Probability of Accepting a Lot

Page

31

33

2-3. Effect of Errors on the Average Outgoing Quality. • • • • • • • • 34

2-4. Effect of Errors on the Average Total Inspection With Replacement . • • • . • • • • • • • • • • • • • • • • . • • • • • • • • • • • 35

2-5. Effect of Errors on the Average Total Inspection Without Replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3-1. Typical Process Curve for Percent Defective • • • • • • • • • • • 40

4-1. Total Expected Cost for Varied Mixed Binomial Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4-2. Effect of a Change in the Process Variance • • • • • • • • • • • • 71

4-3. Effect of Type I Error on the Total Expected Cost • • • • • • • • 75

4-4. Effect of Type I Error on the Performance Measure~ 1 • • • • 76

4-5. Effect of Type II Error on the Total Expected Cost. • • • • • • • 77

4-6. Effect of Type II Error on the Performance Measure 1 • • • 78

4-7. Effect of Type I Error on the Total Expected Cost of an Optimal Plan Designed for Error Prone Process •••••••• 82

4-8. Effect of Type I Error on the Performance Measure~ 2 • • • • 83

4-9. Effect of Type II Error on the Total Expected Cost of an Optimal Plan Designed for Error Prone Process. • • • • • • • • 84

4-10. Effect of Type II Error on the Performance Measure~ 2 • • • 85

4-11. Effect of Type I Error on the Performance Measure~ 3• • • • 88

vii

LIST OF ILLUSTRATIONS ( Continued)

Figure Title Page

4-12. Effect of Type II Error on the Performance Measure .6.3 89

4-13. Relationships of Sample Size and Acceptance Number to the Observed Fraction Defective for Specified Type I Error Rates . . • • • • . . . • • • • • • • • • • • • • • • . • • • • • • . • • . . • • 93

4-14. Relationships of Sample Size and Acceptance Number to the Observed Fraction Defective for Specified Type II Error Rates . . • • . • • • . • • • • . • • • • • . • . • • • • • • . • • • • • • • • • 94

4-15. Total Expected Cost of :Mixed Binomial Parameter Sets for Specified Error Pairs • • • • • • • • • • • • • • • • • • • • • • • . • • • 98

4-16. Total Expected Cost of :Mixed Binomial Parameter Sets for Specified Error Pairs when MBl is Treated as Actual Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5-1. Effect of Type I Error on the Total Expected Cost • • • • • • • • 107

5-2. Effect of Type I Error on the Performance Measure .6. 1 • • • • 108

5-3. Effect of Type II Error on the Total Expected Cost. • • • • • • • 109

5-4. Effect of Type II Error on the F erformance Measure 1 • • • 110

5-5. Effect of Type I Error on the Total Expected Cost of an Optimal Plan Designed for Error Prone Process. • • • • • • • • 112

5-6. Effect of Type I Error on the Performance Measure .6. 2 • • • • 113

5-7. Effect of Type II Error on the Total Expected Cost of an Optimal Plan Designed for Error Prone Process. • • • • • • • • 114

5-8. Effect of Type II Error on the Performance Measure .6. 2 • • • 115

5-9. Effect of Type I Error on the Performance Measure .6. 3 . • • • 119

5-10. Effect of Type II Error on the Performance Measure .6.3. • • • 120

viii

Figure

LIST OF ILLUSTRATIONS ( Concluded)

Title

5-11. Relationship of Sample Size and Acceptance Number to the Observed Fraction Defective for Specified Type I Error

Page

Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5-12. Relationship of Sample Size and Acceptance Number to the Observed Fraction Defective for Specified Type II Error Rates • • • • . • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 122

5-13. Comparison of Total Expected Cost for Assumed Prior at Selected Error Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5-14. Effect of Type I Error on the Total Expected Cost •••••••• 133

5-15. Effect of Type I Error on the Performance Measure 1 •••• 134

5-16. Effect of Type II Error on the Total Expected Cost •••••••• 135

5-17. Effect of Type II Error on the Performance Measure ~l •••• 136

5-18. Effect of Type I Error on the Performance Measure 2 •••• 139

5-19. Effect of Type I Error on the Total Expected Costs of an Optimal Plan Designed for Error Prone Process ••••••••• 140

5-20. Effect of Type II Error on the Total Expected Cost of an Optimal Plan Designed for Error Pr~me Process • • • • • • • • • 141

5-21. Effect of Type II Error on the Performance Measure ~ 2 • . • • 142

5-22. Effect of Type I Error on the Performance Measure ~ 3 . • • . 144

5-23. Effect of Type II Error on the Performance Measure ~ 3 • • • 145

ix

CHAPTER I

INTRODUCTION

Quality control has existed throughout history whenever there has

been an interest in the quality of manufactured goods. The most significant

advancements have occurred, however, since 1927 with the development of

statistical quality control methods. Statistical quality control is that aspect

of total quality control that couples statistical theory with quality control

objectives to enhance the decision processes.

One of the major areas of statistical quality control is acceptance

sampling. Acceptance sampling permits the determination of a course of

action by establishing the risk of accepting lots of given quality. A common

procedure is to consider each submitted lot separately and to base the

decision action on the evidence provided by inspection of one or more ran-

dom samples chosen from the lot. When the decision action is based on

evidence provided by only one sample the procedure is referred to as a

single sampling plan.

Acceptance sampling schemes are often established for quality

characteristics which are measurable on a continuous scale; such schemes

are described by variable sampling plans. When the acquired evidence is

based on one or more quality characteristics from which the product is

graded as defective or non-defective, it is said that acceptance sampling

is by attributes.

1

2

Decisions formulated from such evidence can be affected by many dif-

ferent factors. One factor of particular interest is the accuracy ( or inaccuracy)

of evidence concerning the conformance of inspected items to prescribed

standards. The fraction defective derived from statistical data pertaining to

observed defectives, rather than known defectives, is often used to represent

the actual process fraction defective. Errors occurring in the inspection

process, regardless of their cause, are therefore not considered in the deci-

sion to accept or reject a batch of manufactured goods. Consequently, without

explicit consideration of such inspection errors, the statistical or economic

objectives desired from an acceptance sampling plan may be vastly different

from the results actually achieved.

This dissertation evaluates the consequences which result from quality

decisions based upon inaccurate inspection results. Consideration is given

to both the statistical and economic factors involved with lot-by-lot accep-

tance using single sampling by attributes. Compensation methods are sought

whereby sampling plans may be adjusted to offset the adverse effects of

inspection inaccuracies.

Attributes Acceptance Sampling

The simplicity of attributes acceptance sampling plans was a signifi-

cant factor in the selection of such plans as a representative model for the

evaluation of error effects in this research. Further, it was anticipated

3

that the results would be of general interest to a broader spectrum of potential

users because of the extensive use of sampling by attributes throughout industry.

Among the reasons attributed to the popularity of attributes acceptance sam-

pling plans are those identified by Grant [ 1] as follows:

( a) many quality control characteristics are recognizable only as

attributes.

(b) the cost of inspection per item is generally less than that

incurred when inspection is by variables.

( c) the data recording and clerical costs are generally less than

those incurred when inspection is by variables.

( d) acceptance criteria do not necessarily have to be applied to

each quality characteristic for sampling by attributes as does sampling by

variables.

( e) attributes sampling usually permits the use of less skilled

inspectors than variable sampling schemes.

(f) the distribution of quality characteristics need not be known

for sampling by attributes.

The tendency to use sampling by attributes is likely to continue for

most products unless there exists a decisive argument which favors sampling

by variables.

4

Attributes acceptance plans are but a subset of the available proce-

dures comprising sampling theory. The procedure of interest in this research

is where a decision is made for each lot from evidence obtained by inspection

of a randomly selected sample from the lot. Such a procedure employs a

systematic plan which is described by three numbers. One is the number of

items, N, in the lot from which the inspection sample is drawn. The second

is the number of items, n, in the sample which was randomly formed from

the lot awaiting sentencing. The third is the acceptance number, c, which

defines the maximum allowable number of defective items that can exist in a

lot and the lot be sentenced as acceptable. More than c defectives in a lot

results in the "lot being rejected or sentenced as non-acceptable. When a

lot is sentenced as non-acceptable the entire lot may be scrapped or a rectifi-

cation scheme enforced. Rectification schemes are inspection programs

which attempt to eliminate a sufficient number of defectives in the remainder

of the lot to attain the desired quality. The aim of statistical quail ty control,

prior to implementing an inspection plan, is to establish the specific values

of N, n and c that afford the best protection of the desired quality objectives.

Associated with this category of plans are theoretical measures of per-

formance. Typical of these measures are the probability of lot acceptance,

average outgoing quality and average total inspection. These measures are

described in detail by Duncan [ 2] for the error free inspection process.

5

Development of these measures and others are illustrated in this text for the

error prone process.

Cost Based Acceptance Sampling

When the significant costs associated with each possible lot decision

are identifiable, a sampling plan can be determined on the basis of economic

criterion alone. The sample size and acceptance number are sought that

will minimize the total expected cost for a specified lot size. Since the

quality decisions are based on expected costs, the distribution of all possible

values which the number of defectives in the lot may assume must be known.

This distribution is known as the prior distribution and is chosen before

sampling is performed. The prior is of a discrete form since the number

of defective items in a lot can only assume an integer value within the range

from zero to the lot size.

The expected cost for a specified sampling plan is obtained by weight-

ing the cost associated with the implementation of each lot disposition by the

probability that a specific number of defectives is observed in the sample.

The total expected cost is obtained by summing the expected costs thus

obtained for each possible value which the number of defectives may assume.

This approach to cost based acceptance sampling will be termed the Bayesian

decision theory approach.

6

The optimal sampling plan can be obtained by a two step procedure.

The total expected cost is first established for a given sample size by deter-

mining the acceptance number which minimizes the total expected cost. The

second step completes the optimization by examining all possible sample

sizes and selecting the one which minimizes the total expected cost.

Survey of Existing Research

Information gathered from many different sources clearly establishes

that inspection is seldom 100% accurate [3, 4, 5, 6, 7). This observation

focuses attention on the need for better understanding of error causes and

effects.

Errors are caused by many varied factors but their effect can be con-

veniently catagorized into classes for the purpose of analysis. Class I errors

are those attributable to the erroneous classification of good items as noncon-

forming to a specified standard. Conversely, Class II errors are attributable

to the erroneous classification of defective items as conforming to the speci-

fied standard. The observed fraction defective may differ significantly from

the actual fraction defective as a result of such errors. When this situation

exists, the probability increases that the decision maker will arrive at a

decision other than the one he would have made had he known the actual frac-

tion defective. Decisions such as these have serious implications which

should be considered in the interest of optimizing economic and quality

objectives.

7

The factors attributing to inspection accuracy can be classified in

many different ways. Harris and Chaney [8] stated that three basic factors

are involved which comprise the individual ability of the inspectors, the

physical nature of the inspection task and its environment, and those related

to the organization and its methods for accomplishing the inspection task.

McKenzie [6] suggests that additional factors result because of personal

relations and interactions with others in the quality environment.

The inspection process cannot at this time be reliabily adjusted to

account for these factors; their effects and interactions are too complex

and difficult to isolate. As a result, studies have been conducted under

controlled conditions which provide useful information regarding the effects

of specific factors.

The Effect of Defect Rate

A study reported by Harris [ 4] relates that as the incoming quality

of the lot improves, the inspection accuracy decreases. An opposing argu-

ment was presented by Lambert and Burford [9] in which it was reported

that inspection accuracy was highest for high incoming quality and decreased

as the incoming quality worsened. Despite contradictory results, both studies

indicated that the incoming quality significantly affected the inspection

accuracy.

8

The Effect of Complexity

Harris [ 3] investigated the effect of equipment complexity on inspection

performance. Complexity was considered to be primarily a function of the

number of parts which make up the equipment item and the way in which the

parts are arranged. On the basis of these considerations, an equipment com-

plexity number was assigned to each of the test items. A measure of inspec-

tor performance was obtained for each equipment item by dividing the number

of defects detected by the total number of known defects present. The results

indicated that the inspection performance varied inversely with the equipment

complexity. The less complex an equipment item, the larger the percentage

of defects that were detected.

The Effect of Vigilance

Vigilance is necessary for successful performance in almost any job

situation. This is particularly true of the inspection task where the arrival

of a defective is assumed to occur randomly and infrequently. Most of the

studies performed in vigilance research have dealt with monitoring tasks of

a simplified form [8]. Typically, the defect to be discovered was easily

detected and required little, if any, judgement by the inspector regarding

conformance to a standard. Inspection tasks in industrial situations are

often more complex and require decisions of greater difficulty by the inspec-

tor. As a general observation, one might conclude that inspection accuracies

9

may be limited by the degree to whfoh the state of attention of the inspector

can be controlled.

The Effects of Individual Abilities

McKenzie [6] suggests that physiological and psychological factors

may limit the accuracy of the individual inspector. Much useful research has

been done on the selection of inspectors based upon their psychological and

physiological abilities. The study of visual acuity, perceptual discrimination

and general intelligence, among other factors, provides additional information

to aid in the selection of inspectors. Unfortunately, a common factor between

inspection tasks and psychological functions has not been acquired. McKenzie

[6] attributes this to the fact that different inspector capabilities may be

required for different inspection tasks.

Other Inspection Factors Involved

The preceding factors contributing to inaccuracies in the inspection

process are by no means complete. Additional factors, such as the inspec-

tion environment, methods of inspection, norms and biases of the inspec-

tors also promote inspection errors. Continued studies will be necessary

if the factors which contribute to inspection inaccuracies are to be better

understood and accounted for by improved inspection techniques and policies.

10

Minimization of Error Effects by Analytical Methods

It has been stated that total elimination of inspection inaccuracies is,

practically speaking, an unrealizable goal. Even if the knowledge were avail-

able, it would likely prove to be economically infeasible to remove all sources

of error. This conclusion suggests a course of action that eliminates errors

at their source, when practical, and compensates for the error prone situation

when removal of errors is impracticable.

Statistical Based Acceptance Sampling with Error

A review of the literature disclosed that the research reported by

Ayoub, Walvekar and Lambert [ 10] is representative of prior investigations

with the forementioned objective. In summary, the effects of inspection

error were investigated for four parameters. These parameters were

producer risk, consumer risk, average outgoing quality and average total

inspection. The average outgoing quality was investigated for the inspec-

tion scheme where the observed defective items were replaced by items

assumed to be free of defects. The investigation of the average total inspec-

tion dealt only with an inspection policy that did not provide for replacement

of defective items. Subsequently, as a part of the research reported herein,

the above investigations were extended to each of the above measures of per-

formance for either a nonreplacement policy, a replacement policy or repair

policy [11, 12]. It was assumed that the repaired or replacement items were

11

also subject to an error prone inspection. More recently, the AOQ with

error has been summarized by Case, Bennett and Schmitt [ 13] for nine

different rectifying inspection policies.

In most investigations reported to date, it has been assumed that the

distribution of observed defectives in the sample is binomially distributed

with an observed fraction defective. While this appears to be intuitively

correct, it remains to be shown in this dissertation that the observed defec-

tives are in fact distributed in this manner.

Economic Based Acceptance Sampling

The ec_onomic considerations relative to acceptance sampling plans

has been the subject of extensive research. The majority of the published

research in this area has been confined to inspection processes that are

free of error. Among the more significant research efforts, of interest

in this dissertation, are those dealing with the cost model proposed by

Guthrie and Johns [ 14). This model assumes that each pertinent cost

factor can be defined as a linear function of the number of defectivE items

in the lot. The asymptotic properties of the model allows the elimination

of lengthy search procedures designed to locate near optimal values of the

sample size and acceptance number. The authors have shown that the

optimal sample size and acceptance number can be approximated by analy-

tical expressions whose accuracy improves as the lot size increases.

12

Lane [ 15] examined the modeling of single attribute quality control

systems, operating in an error free environment, from the perspective of

formal Bayesian decision theory. The reported research had two primary

goals. The first goal being to establish a method of analysis which could

be applied to the general problem of single attribute cost modeling. The

second goal was to develop a model for the situation where three possible

lot dispositions exist. Specifically, the circumstance where a lot can be

accepted, rejected and screened, or rejected and subsequently scrapped.

Supporting examples were provided to demonstrate the application of the

formal Bayesian approach to the developed models.

The research reported by Bennett, Case and Schmitt [16] is a major

contribution to the field of cost based acceptance sampling where the inspec-

tion process is error prone. Both type I and type II inspection errors, as

previously defined, were considered. The Bayesian decision theory approach

was employed to model an error prone quality system consisting of a single

sampling plan design involving several cost components. Each of the cost

terms in the resulting models were linear functions of the number of

defectives remaining in the unsampled portion of the lot. Inherent in the

formulation of the models were the terms describing the expected number

of observed defectives in the sample and in the lot. Several numerical

examples were provided to illustrate the effects of inspection errors;

optimal plans were obtained for both error free and error prone situations.

13

Performance measures were provided as means of estabrishing the sensitivity

of the optimal plans to inspection errors. The results clearly revealed that

any error is costly. The cost incurred by the introduction of errors into the

system could, however, be partially compensated for by developing an optimal

sampling plan that accounted for the particular error situation.

A different approach was reported by Ayoub, Lambert and Walvekar [ 17)

which required the comparison of two plans before arriving at a decision. One

plan resulted from the assumption of an error free process and a second plan

obtained by statistically adjusting the error free plan to compensate for antici-

pated errors. A cost comparison was made of these two plans and a decision

derived on the basis of which plan yielded the lessor cost. The decision maker

would be indifferent as to which alternative is selected when the two costs are

equal. It does not appear that a near optimal economic plan would result since

the original plan, without error compensation, was based upon statistical con-

siderations alone. Other authors have developed cost models of linear, two

action, quality control systems operating in an error free process. This

brief review is by no means exhaustive but does represent the ideas that are

germane to the research reported herein. A selected bibliography is provided

for the reader that wishes to acquire additional material related to the subject.

14

Research Purpose and Scope

It is apparent that many factors in the inspection task contribute to

inaccurate sampling results, thus establishing an obvious need to understand

the effects of errors in the inspection process. Ideally, one would like to

remove all causes of errors in the inspection process and thus remove the

effects in their entirety. This is not presently feasible nor practical for

most inspection situations, thus means are desired that will compensate

fully or partially for error effects.

Statistical Considerations

Many sampling plans are based solely on statistical considerations.

Economic factors are sometimes excluded for various reasons. The require-

ment of high quality items for a specific application may reduce economic

factors to secondary importance. In certain situations reliable cost factors

may be extremely difficult to acquire. Product lines may change frequently

and little, if any, historical data accumulated that is applicable to the current

product. The management responsible for establishing quality schemes may

conclude that minimizing economic losses at the present time is not' in keeping

with iong term objectives. Whatever the reasons, many quality inspection

schemes are seemingly based on statistical criteria alone.

This dissertation investigates the situation where the number of defec-

tives in a lot is a random variable and random samples from the lot are formed

in a manner described by the hypergeometric distribution. The conditional

15

distribution of observed defectives in a sample given the actual number of

defectives in the sample will be developed. In addition, the marginal distri-

bution of observed defectives will be formulated. The marginal distribution

will then be used to evaluate the probability of lot acceptance for various

combinations of Type I and Type II errors.

The effects of inspection error on the average outgoing quality and

the average total inspection are also evaluated for several replacement

schemes. The replacement schemes considered either elect to replace or

not replace items which are observed to be defective. When items are to

be replaced it is assumed that the replacements are also subject to inspec-

tion error. Further the effects of inspection error on the design of single

sampling plans, based on the measures of lot tolerance percent defective and

acceptable quality level, are considered. Since single sampling plans are

often determined by specifying these consumer and producer risk points,

a method is provided whereby desired quality risks can be maintained even

though sampling is subject to inspection error. This procedure permits

management to achieve quality goals even when the product lots are sub-

mitted to error prone inspection tasks.

Economic Considerations

The application of inspection schemes affect the economic situation

of a performing organization whether the consequences are realized or ignored.

16

The managen1ent that wishes to minin;.ize the costs associated with a quality

program should consider the long term implications of its inspection policies.

This is a meaningful objective when the various costs associated with the

disposition of each possible lot decision can be assessed with reasonable

accuracy. This allows a sampling plan to be determined that minimizes the

expected costs ( losses) resulting from the chosen quality scheme.

To determine the expected costs a Bayesian decision approach was

adopted to describe the costs incurred in the inspection process. This approach

assumes a prior distribution of defectives in a lot that represents the distri-

bution of defectives for all previously inspected lots. It is desired to establish

a method for selecting a sampling plan that will minimize the expected cost

over its long term application. Since the plan is to be selected before the

inspection scheme is implemented, the expected cost of a particular lot dispo-

sition will be obtained by weighting the cost associated with each possible

sample outcome. The mixed binomial distribution with two components is

representative of the typical process curve for fraction defectives [ 18] and

will be used extensively in the research reported herein. The Polya prior

distribution is also utilized to permit comparison with other reported results.

The necessary distributional relationships will be fully developed for each of

the assumed prior distributions to permit evaluation of a cost model which

incorporates type I and type II errors. The cost model wi 11 be a variation

of the Guthrie and Johns [14] cost model discussed previously. A new

17

distribution will be developed that describes the distribution of observed defec-

tives in a sample given the number of actual defectives in the lot being inspec-

ted. This will permit the model to be formulated in terms of the sample results

which the inspector may observe. Each possible number of observed defectives

will be considered in determining the total expected cost for each lot disposition.

The cost model that results will be evaluated to establish the optimal sample

size and acceptance number for specified statistical and cost parameters. The

sensitivity of the expected cost of the optimal plan derived without error will be

obtained for selected combinations of Type I and Type II errors. Performance

measures will be defined that depict the percent change in total expected costs

relative to the optimal plan, both with and without error considerations.

Documentation of Results

The findings from this research are documented herein and inferences

drawn regarding the effects of error prone inspection processes for both

statistical and economic based sampling plans. The computer programs devel-

oped in the course of this research are included in the Appendix for the readers

use of perusal.

CHAPTER II

ERROR EFFECTS ON SINGLE SAMPLING ATTRIBUTE ACCEPTANCE PLANS

In this Chapter the effects of inspection errors on single sampling

attribute acceptance plans are investigated. Performance measures are

formulated for typical inspection schemes. Example problems are presented

that permit inferences to be drawn regarding the sensitivity of the performance

measures for varied error rates. A method is discussed whereby quality

risks can be maintained even though sampling is subject to inspection error.

Performance Measures

Single sampling plans involving attribute inspection are characterized

by two decision variables, the sample size, n, and the acceptance number,

c. In these plans a sample of n items is drawn from a lot size, N. Each

item is inspected and classified as either good or defective. If the number of

items classified as defective exceeds c, the lot is rejected. Otherwise,

it is accepted.

Two types of errors are possible in attribute sampling. An item which

is good may be classified as defective (type I error), or an item which is

defective :r;nay be classified as good (type II error).

Let

E1 = the event that a good item is classified as a defective,

18

and

Then,

19

E 2 = the event that a defective item is classified good,

A == the event that an item is defective,

B == the event that an item is classified as a defective.

By defining the quantities

p == P(A), true fraction defective,

p e == P ( B), apparent fraction defective,

ei == P(E1), the probability that E1 occurs,

and

e 2 == P(E 2), the probability that E 2 occurs,

the expression for the apparent fraction defective may be more meaningfully

expressed as

( 2. 1)

Distributional Considerations

When lots of size N are formed from a process which is operating in

a state of statistical control with a process fraction defective, p, the distri-

bution of defectives, X, in a lot is described by the binomial distribution as

( 2. 2)

20

When samples are randomly formed from a lot without replacement

the distribution of defectives, x, in a sample of size, n, given X defectives

in the lot is hypergeometric and is defined as I

1 (xi X) n

( 2. 3)

It follows that the marginal distribution of actual defectives in the

sample, gn (x), is given by

g (x) = n

N-n+x 1 ln (xi X) fN (X) X=x

( 2. 4)

Hald [ 19] has shown that when the prior distribution, fN (X), is of

the hypergeometric, binomial, Polya, rectangular or mixed binomial families

the marginal distribution (x) will be of the same family as the prior

distribution. The distribution gn (x) assumes the same form as fN (X)

except n and x replaces N and X respectively. This property is referred

to as the property of reproducibility. Accordingly

(n) x n-x gn (x) = x p (1-p) ( 2. 5)

21

In Appendix Bit is shown that the conditional distribution of observed defec-

. tives, Ye,

l(y Ix) = e

given the actual defectives, x, can be described by

min [x,ye]

i=max [y -(n-x), OJ e ( 2. 6)

The marginal distribution of observed defectives, gn (ye>, is given by

n gn(ye) = 1 (y Ix) g (x)

x=O n e n ( 2. 7)

Substituting the expressions from Equation 2. 5 and Equation 2. 6 into Equation

2. 7 results in

min [x, ye] . n-x-y + i

~' ( n-x) (x) Ye -l e x-i i = l.J y -i i el (1-el) e2 ( l-e2)

x=O i=max e [y -(n-x), OJ

e

( 2. 8)

Factoring and regrouping terms in Equation 2. 8 yields

Ye Y [p(l-e2)] i

[ ] y -1 gn (ye> =c) (.·) (1-p)el e . ye i=O 1

n-y [ ] n-x-y + I e

(n-~e) 1 x-i (1-p) ) 1-e1) e • (pe2) x-i=O X-1

22

Utilizing the Binomial Theorem which states that

m . . L (~) aJ bm-J = j=O J

it follows that

m (a+ b) ( 2. 9)

n-y • [pe2 + ( 1-p) ( 1-e1)] e

( 2. 10)

It was previously shown in Equation 2.1 that the observed fraction defective,

can be described as

thus the complement of p e can be described as

( 2. 11)

Substituting p and ( 1-p ) into the preceeding expression for e e

g (y ) yields n e

(1-p ) e

n-y e ( 2. 12)

This expression is the binomial distribution and is identical to the

marginal distribution g (x) except y has been substituted for x and p n e e

has been substituted for p •

Probability of Acceptance

The lot will be accepted when the number of observed defectives in

the sample is less than or equal to the acceptance number. Assuming perfect

23

inspection, the probability of lot acceptance is given by

Pa - L (:) Px ( 1-p) n-x x-=O

( 2. 13)

The probability of lot acceptance for the general case which also encompasses

the situation where errors are present is described as

Pa e

:::

n

L y =O e

y n-y ( n) e ( l-p ) e

Ye Pe e

Average Outgoing Quality with Replacement

( 2. 14)

The performance measure described by the average outgoing quality,

AOQ, is defined by the ratio

AOQ = expected number of defective iteJlls remaining after inspection total number of items in the lot ·

If perfect inspection is assumed with replacement of the defective items, the

ratio becomes

( 2.15)

However, it was previously observed that perfect inspection is seldom achieved.

Therefore, an expression is sought for the AOQ with replacement which involves

inspection error.

Knowing that a lot will be replenished to its original size, N, the

expression can be obtained for the expected number of defectives in the lot

following inspection. The expected number of defectives remaining in the lot

24

after final acceptance may be segmented into categories ::i.s follows:

p ( N-n) Pa = ( the number of defectives remaining in the un-e inspected portion of an accepted lot) ( the

DITR

probability of lot acceptance)

= (the number of defective items classed as good in the screened portion of a rejected lot) (the probability of lot rejection)

= the number of defective items classed as good in the sample,

= the number of defectives introduced through replacement.

To determine DITR, first consider the expected number of defective

items replaced in the lot given that the lot is accepted. In this situation,

replacement items are introduced only for those items in the sample which

are classified as defective. The expected number of such items is simply

y = np e ( 2.16)

Thus, Y replacement items must be classified as good in order to replenish

the lot.

Let p denote the probability that an item is classified as good. Then g

from Equation 2. 11 it is observed that

( 2. 17)

It follows that the number of items inspected, in order to obtain Y apparently

good replacement items, has a negative binomial distribution with mean Y[·P . ' g

25

Thus, the expected number of items examined in order to obtain Y apparently

good items is Y/p • Of these, only those items which are incorrectly classi-g

fied as good are actually defective. Hence, the expected number of defectives

introduced through replacement when the lot is accepted is given by

DITR a ( 2. 18)

If the lot is rejected and subsequently screened, DITR is given by the

above expression plus the expected number of defectives introduced through

screening inspection. Following a development analogous to the one above,

the defectives introduced in a lot that undergoes screening inspection, DITR , s

is described by

where

DITR s

Z :::- (N-n)p e

( 2. 19)

( 2. 20)

is the expected numb9r of items to be replaced in the screened portion of the

lot. In summary,

DITR == DITR + DITR ( 1-Pa ) a s e

( 2. 21)

Having identified the components of the expression for the expected

number of defective items which may remain in a lot after completion of sample

26

inspection, possible screening and replacerr,ent, the AGQ can now be described

as

p( N-n) Pa + p( N-n) ( 1-Pae) e 2 + npe2 + DITR AOQ = e

N ( 2. 22)

Substituting the expression for DITR from Equation 2. 19 into Equation 2. 20

and simplifying results in a computationally efficient expression for the AOQ:

AOQ = [npe 2 + p(N-n)(l-p )Pa + p(N-n)(l-Pa )e2 ] -.:.. N(l-p) e e e e ( 2. 23)

Average Total Inspection with Replacement

The average total inspection per lot ( ATI) represents the long run

average of total items inspected per lot. It includes the original sample, the

screened portion of rejected lots, and all items from the process inspected

for replenishment of the lot. The ATI may be segmented into the following

categories:

Thus,

n = the sample size

( N-n) ( 1-Pa ) = the expected number of items screened, e

ATI

y - the expected number of items inspected to obtain

pg Y replacements in the sample,

the expected number of items inspected to obtain Z replacements in the screened portion of the lot.

y = n + (N-n) ( 1-Pa ) + - + e Pg

z - (1-Pa ) P e g

( 2. 24)

27

Substituting for Y, Z, and p from Equations 2. 16, 2. 2 0 and 2.17 , respec-g

tively, permits the A TI to be stated as

which reduces to the convenient expression

ATI = n + (1-Pa ) (N-n) + [np e e 2 + (N-n)(l-Pa )p] [1+ p + p c ••• ] e e e e

= n+ (1-Pa)(N-n)

e 1-p

e

Average Outgoing Quality without Replacement

( 2. 25)

The average outgoing quality without replacement is again the ratio of

the number of defective items in the lot to the lot size; however, no defectives

are introduced through replacement and there is no replenishment of lot size.

The expected number of defectives remaining in a lot after inspection

is the sum of those defectives described for the AOQ with replacement, less

DITR. The lot size, now diminished through the exclusion of observed defec-

tives, is equal to N - Y - Z (1-Pa ) where Y and Z are as previously e

determined. Thus,

AOQ = npe2 + p(N-n) Pae + p(N-n) (1-Pa ) e

N- np(l-e) - (1-p)ne - p(N-n) (1-Pa) (1-e) - (1-p) (N-n) (1-Pa )e 2 1 e 2 e 1

which reduces to

AOQ =

28

npe2 + p(N-n) Pae + p(N-n) ( 1-Pae) e 2 N-np - (1-Pa ) (N-n)p e e e

Average Total Inspection without Replacement

( 2. 26)

The average total inspection without replacement is just the average

inspection per lot:

ATI = n + ( 1-Pa ) (N-n) e

( 2, 27)

Adjusting a Single Sampling Plan

Each single sampling plan has associated with it an operating character-

istic curve. T~e design of single sampling plans is often based on the choice of

two points on this theoretical curve. These points may be the AQL and LTPD,

often associated with the producer's risk and consumer's risk, respectively.

For example, define AQL and LTPD as follows:

AQL = p = the actual fraction defective considered to be an 1-a

LTPD = p /3

acceptable quality level and at which it is desired to accept ( 1-a) • 100 percent of such lots,

= the actual fraction defective considered to be a lot tolerance limit and at which it is desired to have a (3 • 100 percent probability of accepting such a lot.

Consider the selection of pl-a and p /3 and the development of a

sampling plan assuming perfect inspection. Under conditions of inspection

error, the desired AQL and LTPD fractions defective no longer have probabili-

ties of acceptance of 1-a and (3, respectively. The actual OC curve may be

29

forced however, to fit the desired points. In order to actually attain levels

of p1_"' and p/3 it is only necessary to design the sampling plan for p uc e, 1-'.l!

and p /3 where e,

and

AQL e = Pe, 1-a ( 2. 28)

( 2. 29)

If the observed OC curve then fits p and p /3 , the actual OC curve e, 1-a e,

will fit p1 and p • -a /3

Actual Fraction Defective Equals Observed Fraction Defective

As a matter of interest, consider the point at which the actual fraction

defective is equal to the apparent fraction defective, e.g., p = p • Then, . e

(2.30)

and upon solving for p , the resulting expression is

p = ( 2. 31)

When the actual fraction defective is zero the quality is observed to be perfect

only if e1 is equal to zero. Similiarly, when the actual fraction defective is

one the items are all observed to be defective only if e 2 equals zero.

30

Sensitivity Analysis

In order to illustrate the effects of inspection error, a typical sampling

plan was selected and evaluated. Specifically, the lot size, sample size, and

acceptance number were selected as N = 4000, n = 150, and c = 5, respectively.

For this plan, four error pairs were used: ( e1, c 2) = (. 0, . 0), (. 01, • 0) ,

(. 0, . 15) and (. 01, • 15). The probability of acceptance, AOQ with and without

replacement, and ATI with and without replacement were determined as a func-

tion of incoming fraction defective for each error pair. In addition, the effect

of error pairs (e1,e2) = (.01, .15), (.01, .20), (.01, .25) and (.01, .30)

were evaluated with respect to the probability of acceptance in order to illustrate

the equality of true and observed fractions defective at

Figure 2-1 illustrates that the effect of a type I error is to reduce the probability

of acceptance for all possible values of p. Similarly, the effect of a type II

error is to increase the probability of acceptance. These results are to be

expected since a type I error occurs when a good item is incorrectly classified

as defective, and a type II error occurs when a defective item is incorrectly

classified as good.

When e1 > 0 and e 2 > 0 , the observed probability of acceptance is

less than that when e1 = e 2 = 0 for values of p less than

p' =

31

O.I

Ill u z -c I-L Ill u u u -c II. 0

I-:::; iii 0.4 -c 1111 0 a.

., 0.2 11=-., + '2

0 .02 ,Q6 ,08 .10 ,12

IHCOMiHC QUALITY (p)

Ftpre 2-1. Effect of. Errors on the Probability of Accepttng a Lot

32

For values of p > p' the observed probability of acceptance is greater than that

without error. That is,

Pa < Pa if p < p' and Pa > Pa if p > p' • e e

These observations may be witnessed in Figure 2-2.

Figure 2-3 examines the average outgoing quality as a function of frac-

tion defective and error. Incorrect classification of a good item reduces the

average outgoing quality due to the fact that more screening inspection takes

place while incorrect classification of a defective item has the effect of causing

higher AOQ values for all values of p. Type II error also causes a significant

change in the shape of the AOQ curve. Near the point at which Pa - 0 , e

the AOQ curve rises due to the increased number of defective items classified

good as p increases.

Therefore, for any given sampling plan encompassing type II errors,

the conventional concept of the AOQL is not meaningful.

Figures 2-4 and 2-5 illustrate the average total inspection as a function

of fraction defective and error for the replacement and non-replacement policies

respectively. As intuitively expected, the general effects of type I and type II

errors are to increase or decrease the ATI, respectively, for any specified

incoming fraction defective.

For the example shown, the policy of replacement or nonrcplacement

did not significantly affect the AOQ. The distinguishing difference between the

policies of replacement and nonreplacement are in the ATI values as the fraction

defective increases. Under a nonreplacement policy, it can be seen that as

Ill u :z: I-

0.8

0.6 u

LL 0 >-1-::::; iii o., 113 0 It: Cl.

0.2

o.

33

.02 .o, .06 .08 .10 .12 INCOMING QUALITY (p)

Figure 2-2. Effect of Errors on the Frobability of Accepting a Lot

g ... :; :::, 0 Cl :z 0 Cl ... :::, 0

.02

.01 Ill: w >

0 .02

34

::-,... , ,15)

(.0, .0)

(,01, .0) --WITH REPLACEMENT

- - WITHOUT REPLACEMENT

.CM ,06 .08 ,10 INCOMING QUALITY (p)

Figure 2-3.

Effect of. Errors on the ATerage Out,ot111 Quality

.12

35

4000 .:: :z: w w u ,c ..I Q. 3000 w Ill: ... :::, 0 :z:: ... :z: 2000 e ... u w Q.

"' ..I ,c ... 1000 0 ... w C, ,c Ill: w > ,c

0-+-----.....-----------------------...J 0 ,02 .06 ,10 ,12

IHCOMIHC QUALITY (p)

Figure 2-4. Effect of Errors on the Average Total Inapection With Replacement

36

5000

;:: % w :::E II.I 4000 u

•1 ..J A. w (.01, .0) •1 + e2 °' :z: ... z 3000 (.0, .0) 2 ... u w A. .,, (.0, .15) (.01, ,15) ..J

2000 ... 0 ... w C)

a,: II.I >

1000

0+------.------T-----T-------,,-------,-----1 0 .02 .04 .06 .08 .10 ,12

INCOMING QUALITY (p)

Figure 2-5. Effect of Errors on the Average Total Iruspectlon Without Replacement

37

Pa - 0 , ATI - N. Thus, for large fractions defective and normal ranges of e

error the A TI essentially equals the lot size. Under a replacement policy, as

the observed fraction defective p - 1 , Pa - 0 and the ATI increases with-e e

out bound. Since the policy chosen affects the A TI and thus the incremental

quality costs, it will be desirable to consider both policies in the selection of

an acceptable plan.

CHAPTER III

ECONOMICALLY BASED ATTRIBUTES ACCEPTANCE SAMPLING MODEL

In the previous chapter, the effects of inspection errors were evaluated

for a number of commonly used performance measures. An adjusted sampling

plan was obtained that compensated for the error effects and possessed the

desired quality characteristics. The selection of the desired quality character-

istics is left to the decision maker who draws upon the information available to

him. Often times the information available is limited and the selection of

quality characteristics is confined to an intuitive evaluation of the factors at

hand. While intuition may be satisfactory in many situations, it may introduce

unwarranted costs in other situations.

In profit motivated enterprises, the real basis of selecting among

alternatives is the financial worth of each alternative in terms of costs and

revenues. Where the appropriate cost factors can be identified and expressed

in monetary terms, then economic based quality modeling can be used to enhance

the decision making process. This situation is not likely to exist unless a con-

centrated effort is promoted within the business that enlists the support of all

related organizations. It simply cannot be left to the inspection department

alone, but must include the organizational elements responsible for repair,

handling, packaging, customer relations, warranties and other significant

activities related to quality costs. In many situations, such an action can

38

39

not be economically justified because of the additional cost which is incurred in

the collection of data. Still other situations exist that suggest substantial bene-

fits could be acquired.

This chapter will establish a representative economic based attributes

acceptance sampling model which will be used in subsequent chapters to evaluate

the effects of error prone inspection processes. Particular attention will be

devoted to specific distributional considerations of interest which govern the

behavior of the model.

Distributional Considerations

The fraction of defective items produced over time is governed by a

probabilistic process. The long run distribution of process fraction defective

is described by the process curve. One typical process curve is illustrated in

Figure 3-1. The curve pictured indicates that the process described normally

operates at a low fraction defective. On occasion, however, a relatively high

fraction defective is realized. Although the curve drawn is continuous, process

distributions are often represented as discrete mass functions.

Lots of size N are formed from the process in a sequence of Bernoulli

trials. That is, each item used in forming the lot is either good or bad, inde-

pendent of the quality of other items in the lot. The distribution of the number

of defectives in the lot, assumed before conducting the sampling inspection, is

called the prior distribution and will be of prime interest in the Bayesian formu-

lation of the model that follows.

40

p

Figure 3-1.

Typical Curre for Percent Def ectiTe

41

Mixed Binomial Prior Distribution of Lot Defectives

When the process distribution consists of m different states, in which

the defectives are formed as a sequence of Bernoulli trials, the resulting prior

distribution is a mixed binomial. The mixed binomial is a rich distribution,

capable of assuming many different shapes. It is also a distribution for which

there is much support as a prior distribution [18, 20).

The situation to be represented by the mixed binomial in this research

can be visualized in the following manner. A manufacturing process is operating

in statistical control with a given process average, p1 • Periodically, a shift

occurs in the production cycle. The process remains in statistical control but

with a different process average, p2, than previously observed. Past experi-

ence indicates that p1 occurs with a frequency of a and p2 with a frequency

( 1-a). This situation is depicted by the mixed binomial distribution with two

components as

( 3. 1)

When samples are randomly formed from the lots without replacement,

the conditional distribution 1 (xi X) is hypergeometric. Thus, n

or

1 (xi X) n

1 (xi X) = n

42

( 3. 2)

It follows that the marginal distribution of actual defectives in the sample is

given by

g (x) = n

N-n+x ln (xi X) fN(X)

X=x ( 3. 3)

Hald [ 19] has shown that when the prior distribution fN ( X) is from

the mixed binomial family, the marginal distribution will also be of the same

family. Thus, by the property of reproducibility to hypergeometric sampling,

the distribution gn (x) assumes the same form as fN(X) as described in

Equation 3.1 except n and x replaces N and X respectively. Accordingly,

(3. 4)

This mass function describes the unconditional distribution of actual defectives

in a sample of size n •

In practice, when inaccuracies exist, the inspector observes what

appears to be defective items. The relationship between the observed defectives,

ye , and the actual defectives, x , is given by the conditional expression

1 (y Ix) = n e

min [ x, Ye]

i=max [ y -(n-x),O]

e

(n-~x~) y -1 l e

( 3. 5)

43

It follows that the marginal distribution of observed defectives, ye' is

given by

n gr/Ye) = I: ln(Yelx) gn(x)

x=O ( 3. 6)

Substituting the expressions for 1 (y Ix) and g (x) from Equations 3. 4 and n e n

3. 5, respectively, into the Equation 3. 6 yields

min n [ x,ye]

= \' \' (n-x)(x) ye-i(l- )n-x-ye+i x-i(l- )i Li Li -i i e el e2 e2

x=O i=max ye

or

= a

[y -(n-x),O] e

n

I: x=O

min l x, y el

I: i=max

[ y -(n-x), OJ e

• ( n) x ( 1_ ) n-x x P2 P2

[min ] x,y e

+ ( 1-a) I i=max

[y -(n-x),O] e

. n-x-y +i y -1 e e ( 1-e1)

el

• x-i (l- )i(n) x (l-p )n-x e2 e2 x P2 2 ( 3. 7)

44

Proceeding with the development in the same manner as that in the previous

chapter for the binomial prior results in

a a ( (;J~ (: e )[pl (l-e1)] i

•

n-y e

( :: e) [(ple)x-i 1(1-pl) (l-e1)] n-x-y e -i-i) x-i=O j

(:e) [pp-e2)r

From the Binomial Theorem it follows that

[(l-p2) ( 1-el)] n-x-y e +i) ( 3. 8)

n-y (1-p2) (l-e1)] e •

( 3. 9)

The apparent fraction defective was described in Equation 2. 1 as

p = p( 1 - e ) + ( 1 - p) e e 2 1

The apparent fraction of good items is the complement of the apparent fraction

defective and is described in Equation 2. 11 as

(1-p) = pe + (1-p) (1-e) e 2 1

45

Substituting p and (1-p ) as appropriate into Equation 3. 9 for g (y ) yields e e n e

gn (ye) = a ( ;e) P:~l (1-pe, 1/-y e + (1-a) ( :.) P :.•2 (1-pe, /-ye

(3.10)

Thus the marginal distribution of observed defectives is from the same family

as the distribution for actual defectives in the sample. Substitution of y , e

Pe,l, and Pe, 2 , for x, p1 and p2 respectively in gn(x) results in

Polya Prior Distribution of Defectives

The effect of inaccuracies in an inspection process was investigated by

Bennett, Case and Schmidt [16]. A Polya Prior distribution was used as the

underlying distribution governing the arrival of actual defectives in the lots at

the inspection station. This distribution will be investigated herein to permit a

comparison with the data derived from the assumption of a mixed binomial prior

distribution of lot defectives. The conditional and marginal distributions for

observed defectives will be developed in an analogous manner to that used with

the mixed binomial prior.

The distribution of actual defectives in the incoming lots is

r (t+N-X) r (t)

r (s+ t) r (s+t+ N)

( 3.11)

46

Hald [ 19) showed that if the prior distribution of fN(X) is Polya, then the

marginal distribution gn (x) will be of exactly the same form as fN(X) but

with N and X replaced by n and x •. Therefore,

r (t+ n-x) r (t)

r (s+ t) r (s+t+ n)

The marginal distribution of observed sampled defectives is

n = I:

x=O 1 (y Ix) g (x) n e n

where 1 (y ! x) is as defined in the previous section. Substituting into n e

(3.12)

( 3. 13)

Equation 3. 13 the applicable distribution expressions from Equations 3. 5 and

3.12 for 1 (y Ix) and g (x) , respectively, results in n e n

n = I:

x=O

min [x,ye] .

. I: ( n-: )(~) e/ e -1 1=max ye

[ y -(n-x), O] e

r (s+x) r (s)

r (t+ n-x) r ( t)

r ( s+t) r (s+t+n)

( 3. 14)

It is desirable to simplify the preceeding equation. Factoring and

redefining the Polya distribution in the form of a continuous integral function

yields

47

min

r (s+t) r (s)r (t)

n [x,ye] y -i n-x-y +i . ( n-:)( )e1 e (1-el) e

x=O 1=max Ye [y -(n-x),O]

e

• e x-i(l-e /(n) J ps+x-1 (l-p)t+n-x-1 dp 2 2 X O

Regrouping terms in t~e preceding equation

r (s+t) r(s)r(t)

n

x=O i=max [y -(n-x),O]

e

x-i s-1 t-1 • (pe2) p ( 1-p) dp

which reduces to

= Ue) r (s+t) 1 1

e C H r-1 gn (ye) / i~O ie (1-p) el e r (s)r (t)

n-y ( n-ye) [ ]n-x-y +I e • (1-p)(l-e1) e

x-i=O x-i

( x-i s-1 t-1 • pe2) P (1-p) dp

[p(l-e2)r

( 3.15)

48

Employing the Binomial Theorem it follows that

[ ]n-y e s-1 t-1 • (1-p) (l-e1) + pe2 p (1-p) dp ( 3.16)

Using the relationships previously developed for p and ( 1-p ) from e e

Equations 2.1 and 2.11, respectively, permits Equation 3.16 to be written

simply as

g (y ) = ( n) n e y e

r ( s+t) r(s)r(t)

1

J 0

Y n-y e e s-1 t-1 Pe (1-pe) P (1-p) dp

( 3. 17)

Note that when both type I and type II errors are equal to zero, then

ye equals x and

1 J x n-x s-1 t-1 p (1-p) p (1-p) dp 0

Collecting terms under the integral yields

Since

1

1

J 0

J s+ x-1 t+ n-x-1 p (1-p) dp = 0

Equation 3. 19 can be written as

s+x-1 t+n-x-1 p (1-p) dp

r ( s+x) r (t+n-x) r (s+t+n)

(3.18)

( 3. 19)

( ) = ( n) r ( s+ x) gn Ye x r (s)

49

r (t+ n-x) r (t)

r ( s+t) r ( s+t+n) (3. 20)

which is the same expression as g (x) in Equation 3.12. n

Formulation of the Model

The error free cost model developed by Guthrie and Johns [ 14] will be

used as a basis for developing a more general model that includes the error

prone inspection process. The notation that follows will be used to describe the

Guthrie and Johns model. Let

n = sample size

C ;;:: acceptance number

X ;;:: actual defectives in a sample

X :::: actual defectives in a lot

N = lot size

s1 = cost per item of sampling and testing

s2 = repair cost for a defective item found in sampling

Al ::;:: cost per item associated with handling the N-n items

not inspected in an accepted lot

A2 ::: cost associated with a defective item which is accepted

Rl :=: cost per item of inspecting the remaining N-n items in a

rejected lot

and

R2 = repair cost associated with a defective item in the

remaining N-n items of a rejected lot.

50

Cost of Lot Acceptance with Inspection Errors

The effect of inspection errors is to increase the likelihood that the

wrong decision will be made regarding the sentencing of a lot. Even when the

right decision is made as to acceptance or rejection of the lot, inferences

regarding the cost consequences may still be in error. Whereas type I errors

suggest that the product quality is worse than it actually is, type II errors tend

to promote a better image of product quality that is warranted.

The economic objectives of the decision makers may be enhanced

through better understanding of the error effects and their economic

consequences. With this as the objective, the error free model depicted in

Equations 3. 21 and 3. 22 will be modified to account for the introduction or

errors.

The use of several terms introduced in Chapter II will again be required

to supplement those used in the Guthrie and Johns model. These terms are

=

and

::::

probability that a good item will be erroneously classified

as non-conforming to the prescribed standard,

probability that a defective item will be erroneously classified

as conforming to the prescribed standard,

the number of items observed to be defective as a result of

both proper and improper classification of the sample items.

51

A fixed size sampling inspection scheme is employed which randomly

selects a sample of n items from a lot of size N. The sample is inspected

and each item either classified as good or defective. If the number of defectives,

x , is less than or equal to the acceptance number, c , the lot is accepted;

otherwise, the lot is rejected. The cost consequences which may result from

sentencing a lot to either of the two possible outcomes is delineated as the sum

of specific costs which may be incurred. Mathematically, the Guthrie and Johns

cost model describes the total cost when the inspection results are known as

TC(N,n,X,x,c) == nS1 + xS 2 + (N-n)A1 + (X-x)A2

for

X S C (3. 21)

and

TC(N,n,X,x,c) = nS1 + xS2 + (N-n)R1 + (X-x)R 2

for

X > C ( 3. 22)

Equation 3. 21 is the cost of accepting a lot when the actual number of

defectives in the sample is observed to be less than or equal to the acceptance

number for a specified plan. Equation 3. 22 depicts in a similar manner the

sum of the elemental costs which are incurred when the number of defects in the

sample exceed the acceptance number and the lot is rejected on the basis of

sample results.

52

The introduction of error conditions is easily accomplished by considering each

element cost depicted in Equation 3. 21. The specific cost elements that result

are as follows:

nS1 =

yeS2 =

xe2A2 =

cost of sampling and testing n items

repair costs for they items observed defective in the sample e

cost associated with the expected number of defectives in the

sample that are classified as good and accepted

cost associated '\\ith handling the N-n items not inspected in

an accepted lot

cost associated with the defective items accepted in the

uninspected portion of the lot.

The total expected cost given that the lot is accepted is obtained by summation

of the above elemental costs. That is,

y :S C • e ( 3. 23)

Taking the expected value with respect to X yields

y :S C • e ( 3. 24)

It has been shown by Hald that for prior distributions which are repro-

ducible to hypergeometric sampling that

E(X!X) = (N-n) (x+ 1)

(n+l)

g (x+ 1) n+l

g (x) n + X ( 3. 25)

53

This may be rewritten for convenience as

where

E(X! x) = ( N-n) f(x) + x

f (x) --(x+l) (n-+ 1)

g 1 (x+ 1) n+ g (x) n

( 3. 26)

( 3. 27)

Substituting the expression for E(XI x) from Equation 3. 26 into Equation 3. 24

results in a total expected acceptance cost of

Taking the expected value with respect to x yields

y C e

y C • e

( 3. 28)

( 3. 29)

The preceeding expression for the cost of acceptance may be further

simplified by allowing

( 3. 30)

( 3. 31)

and

C = A (N-n) 3 2 ( 3. 32)

The total expected cost can now be expressed as

(3. 33)

54

Thus the total expected cost may be obtained for the situation where no more

than c defectives are observed in the sample and the lot is accepted.

Specific terms in Equation 3. 33 may be obtained from the relationships

that follow. Let

and

n

L E(x;y) = e x=O

E[f(x);y] = e f(x) h ( xJ y ) n e

where, from Bayes Theorem,

g (x) 1 (y Ix) h (xi ) - n n e n ye - g (y )

n e

( 3. 34)

( 3. 35)

( 3. 36)

The distribution 1 (y Ix) is defined previously in Equation 3. 5 • The distri-n e

butions g (x) and g (y ) are dependent upon the choice of prior distributions n n e

as discussed earlier in this chapter.

When the mixed binomial as defined in Equation 3. 1 is chosen as the

prior distribution, the general Equation 3. 27 for f(x) may be specified as

f(x) = (x+ 1)

(n+ 1)

( 3. 37)

55

Cancelling terms permits Equation 3. 37 to be written as

f(x) =

x+ 1 n-x x+ 1 n-x a P1 ( 1-pl) + ( 1-a) P2 ( 1-p2)

x n-x x n-x ap1 (1-p) + (l-a)p2 (l-p2) ( 3. 38)

Further simplification of f(x) may be obtained by defining

and

Then

( 3. 39)

and

( 1-p ) = k ( 1-p ) 2 2 1 ( 3. 40)

Substituting the values for p2 and ( 1-p2) from Equations 3. 39 and 3. 40,

respectively, into Equation 3. 38 results in

f(x) =

( 3. 41)

Factoring, f(x) reduces to

f(x) (3.42)

56

When the prior distribution fN(X) is of the Polya form, the marginal distribu-

tion, g (x) , is also of the Polya form as previously shown in Equation 3. 12 • n

Substituting this expression for g (x) into the general Equation 3. 27 for n

f (x) yields

f(x) = ( x+ 1) [-(_:_:_~_)_r_(_;_+(_:+-'-) _1> __ r_i_t_; t..:..~_-x_> __ r...;(_~_+_~:_:_!_\.....;_)] (n+l) ( xn) r (s+x) r (t+n-x) r (s+t)

r (s) I (t) r (s+t+n)

( 3. 43)

which conveniently reduces to

f( S + X x) := s + t + n ( 3. 44)

Cost of Lot Rejection with Error

When the inspection process is free of errors and the number of defec-

tives observed in the sample is greater than the acceptance number c , the lot

is rejected and the cost incurred described by Equation 3. 22 • Different cost

results may be anticipated, however, when the process is prone to error. One

additional term is required to supplement previous definitions before the element

costs can be fully developed. That term represents the observed number of

defectives when the entire lot is subject to inspection, and is denoted as Y • e

The elemental cost may then be identified as follows:

nSl =

Ye82 =

(N-n)R1 ==

cost of sampling and testing n items

repair cost of the y items observed defective in the sample e

cost of inspecting the remaining N-n items in a rejected lot.

57

cost associated with the expected number of defectives in

the lot that are classified as good and accepted

(Y -y )R 2 = e e repair cost for items observed defective in the remaining

N-n items of a rejected lot.

The total expected cost of rejection is the sum of these element costs

and is given by

where

and

y 2:: C e

Taking the expected value with respect to Y yields e

y > C e

E=l-e-e 1 2

(3.45)

( 3. 46)

The expression for E (YI X) is obtained in a manner analogous to that shown in

Appendix B for E (y! x) • Substituting the relationships for E (Y IX) and E e

into Equation 3. 46 yields

y > C e

58

Regrouping terms,

y > C e

Taking the expected value with respect to X yields

where

E(XI x) = ( N-n) f (x) + x

y > C e

as defined in Equation 3. 25 during development of the acceptance cost

expr.ession. Thus,

Taking the expected value with respect to x yields

y > C e

( 3. 47)

( 3. 48)

y > C • e ( 3. 49)

(3.50)

The expression for the cost of rejection may be further simplified by

allowing

( 3. 51)

59

(3.52)

and

( 3. 53)

This permits the total expected cost of rejection to be simply stated as

( 3. 54)

The terms E(x!y ) and E[f(x)I y ] are as defined previously in Equations 3.34 e e and 3. 35 respectively for the acceptance cost expression.

Decision Criteria

When a lot is subjected to inspection and ye defectives arc observed,

the lot will be accepted when the expected cost of acceptance is less than or

equal to the expected cost of rejection or

( a. 55)

It is apparent that the expected cost, after ye bas been observed, of either

acceptance or rejection is a function of E(xl y ) and E [f (x) I y ] . It is assume<l e e

that the expected cost of rejection and the expected cost of acceptance

intersect at only one specific value of y • At this point of intersection neither e

the decision to accept nor reject the lot infers a cost advantage. This point of

intersection may be appropriately termed the breakeven point. The minimal

total expected cost will be obtained in the long term when the course of action

60

is adopted that results in the lesser cost for each value of y obtained. Thus e

the lot should be accepted for all values of y less than or equal to that at e

the breakeven point and rejected for all values of y greater than that at the e

breakeven point. The value of y that corresponds to the breakeven point is e

defined as the acceptance number c • Therefore, after inspection of the sample

the lot will be accepted if y c and will be rejected if y > c • e e

In determining the long run strategy, the occurrence of each value

of y is probabilistic and is described by the appropriate probability e

mass function g ( y ) • Thus, the long term expected cost per lot is an n e

average of the weighted cost for each value of y • Mathematically this cost e

relationship is described as

TC(n,c) = C

y =0 e

TC(n,c,y ) g (y ) + e n e

n

y =c+l e

TC(n,c,y ) g (y ) e n e

(3.56)

where TC(n, c, y ) is defined in Equations 3. 33 and 3. 54 for the accept and e

reject situations, respectively.

This cost relationship and other formulae developed°in this chapter

are the basis for the research results reported in Chapters IV and V. The

results are those derived from extensive analyses to establish the cost con-

sequences of an error prone inspection process.

CHAPTER IV

COST MODEL EVAI.UATION: MIXED BINOMIAL PRIOR DISTRIBUTION

The cost model described in Chapter III will be evaluated, in this chapter,

for the case where the underlying distribution of defectives in a lot is described

by the mixed binomial distribution. A similar evaluation will be reported in

Chapter V for the case where the underlying distribution of defectives in a lot

is described by the Polya distribution. The mixed binomial and the Polya are

rich distributions, each capable of assuming many different shapes. The most

significant difference is that the mixed binomial distribution used in this research

is bimodal, whereas the Polya distribution is unimodal. A comparison of the

results obtained from evaluation of these distributions will permit inferences to

be drawn regarding the importance of accurately describing the prior distributional

form.

In this chapter an optimal sampling plan will be obtained for an error free

inspection process. The optimal plan obtained will be evaluated to determine the

expected cost where inspection is known to be error prone. Selected combina-

tions of type I and type II errors will be evaluated to establish the sensitivity of

the optimal plan to error conditions. Further, optimal plans will be sought, for

error prone situations, that will partially compensate for the costs introduced by

error conditions. Performance measures will be established to aid in evaluation

of the results.

61

62

Summary of Equations

The major equations, developed in Chapter Ill, that pertain specifically

to the objectives of this chapter are reiterated for the reader's convenience.

The total expected cost per lot for a specified sampling plan, TC(n, c) , was

given in Equation 3. 56 as

C

TC(n, c) = Li y =O

e

TC(n,c,y) g (y) + e n e

n '\' TC(n, c, y ) g (y ) l.J e n e y cc:c+ 1

e ( 4. 1)

where TC(n, c,y ) was described in Equations 3. 29 and 3. 50, respectively, as e

or

y > C e

( 4. 2)

( 4. 3)

The distribution of observed defectives in a sample, g0 (ye) , was completely

developed in Chapter III for the mixed binomial distribution. A simplified

expression for g (y ) was given in Equation 3.10 as n e

63

Equations4.2and4.3requiretheuseof E(xly) and E[f(x)I y] which are e · e

defined in Equations 3. 34 and 3. 35, respectively, for the general case. The

specific expression of f (x) for the mixed binomial distribution, with two com-

ponents, was described in Equation 3. 38 as

f(x) = x n-x a P1 (1-p) +

( 4. 5)

Other equations will be introduced throughout the chapter, as necessary, to aid

in evaluating the suitability of the mixed binomial distribution as the underlying

prior distribution of defectives.

Input Data for the Cost Model Evaluation

Table IV-1 lists the numerical value assigned to each cost parameter

appearing in Equations 4. 2 and 4. 3 • It was assumed that the lots were formed

from a homogeneous source and contain 1,000 items each. Other parameters

which must be specified, before an evaluation of the model can be initiated, are

the mixed binomial parameters a, p1 , and p2 . It was noted earlier that a

comparison will be made, in a subsequent chapter, of the results obtained when

the prior distribution is either a mixed binomial or Polya distribution. A mean,

µ , and variance, c/ , of 50 and 825. 4098, respectively, were chosen as

common parameters for the two distributions of interest. These values were

substituted into the following expressions for µ

distribution:

2 and u of the mixed binomial

( 4. 6)

64

Table IV-1

Input Data for Cost Examples

Cost Numerical Parameter Value

s1 $ 2.00

s2 $ 1. 90

Al Cl> .p o. 00

A2 $ 40. DO

Rl $ 2. 00

R2 $ 1.90

65

and

( 4. 7)

A search procedure was conducted to select values of the mixed binomial para-

meters QI, p1 , and p 2 that satisfy the relationships in Equations 4. 6 and 4. 7 •

The procedure utilized assigned a value to QI and then solved for values of p1

and p 2 • The computer program employing this procedure is included in

Appendix C. A feasible solution was obtained when the values of QI , p1, and

p 2 were all positive and in the range between zero and one. Eleven typical

parameter sets for the mixed binomial distribution which satisfied the pre-

scribed conditions are listed in Table IV-2.

Selection of An Optimal Sampling Plan Without Errors

An incremental search was performed on the cost model in Equations 4.1,

4. 2, and 4. 3 • The decision variables involved in the search procedure were the

sample size, n, and the acceptance number, c . The values of n and c were

sought that would minimize the total expected cost per lot for the specified input

parameters. Briefly stated, the procedure searched over different values of n

and c until an expected cost per lot was obtained that represented the global

minimum. Additional details of the search procedure are provided in

Appendix A.

a

0.02

0.04

0.06

o. 08

0.10

o. 20

o. 30

0.40

0.50

o. 60

0.70

66

Table IV-2

Total Expected Cost for Varied Mixed Binomial Parameters

pl P2 (n, c)

o. 2453340 0.0460136 50,6

0.1867078 o.0443038 72,7

0.1604486 o. 0429501 90, 8

0.1446281 0.0417714 99,8

0.1337159 0.0406982 117, 9

0.1058078 o. 0360480 143,9

o. 0926283 0.0317307 146,8

0.0841787 o. 0272142 146,7

0.0779089 0.0220911 143,6

0.0727891 0.0158163 141, 5

o. 0682693 o. 0073717 91,2

TC(n, c)

1869.89

1826. 35

1796. 83

1774.38

1756.44

1699.62

1669.61

1652. 96

1645.61

1647.37

1658.32

67

The optimal plan represented by the sample size, n , and acceptance

number, c, and the corresponding total expected cost, TC(n, c) , were

obtained for each of the specified mixed binomial parameter sets. These data

are tabulated in Table IV-2 with the corresponding input parameters for the

mixed binomial distribution. The TC(n, c) is displayed graphically in Figure

4-1 as a function of the mixed binomial parameter a • It is apparent that the

TC(n, c) is significantly affected by the parameters chosen for the mixed

binomial distribution. It is insufficient to know only that the form of the dis-

tribution is a mixed binomial with a mean and variance of 50 and 825. 4098,

respectively. This observation suggests that considerable information may be

required regarding the precise shape of the mixed binomial prior distribution.

This inference will be investigated further in a subsequent section of this chapter

that pertains to the shape of the prior distribution.

Since the TC(n,c) varied for different mixed binomial parameter sets

having the same mean and variance, it was necessary to find a specific set of

parameters such that the optimal cost would be the same as that acquired with a

Polya prior distribution. The procedure developed for selecting mixed binomial

parameter sets was continued in search of a set of parameters that would yield

an optimal cost close to that reported by Bennett, Case and Schmitt [ 16] for a

Polya prior distribution having the same data inputs. The optimal sample size,

n, acceptance number, c, and total expected cost, TC(n, c) , reported in the

aforementioned reference were 122, 6, and $ 1745. 61 respectively. The

-" .. C u t-

68

1900

NOTE: µ = 50 <1 2 = 825.4098 el = 0.0

1800 •2 = 0.0

1700

1600-+----,.-----,---,---------,,-----.-----~ 0 .2 .6

ALPHA (a) WEIGHING FACTOR

Figure 4-1.

.8

Total Expected Coat for Varied Mtxec'! Binomial I arameters

69

numerical values of a, p1, and p2 obtained for the mixed binomial prior

were, in the same order, 0.115, 0.1274089 and o. 0399412. These para-

meters resulted in comparative values of the sample size, n, acceptance

number, c , and total expected cost, TC(n, c) , for the mixed binomial

prior distribution of 122, 9, and $ 1744. 89, respectively.

Shift in the Process Variance

The effect of a shift in the process variance was evaluated. The mean,

1-1 = 50, and alpha, a= 0.115, were held constant and new values of p1 and p2

computed as the variance, ,,2 = 825. 4098, was incremented over a range of

approximately ± ninety units. The results are tabulated in Table IV-3. The

first column lists the incremental change in the process variance. The second

and third columns list the value of the fraction defective, for each of the two

mixed binomial components, which satisfy the relationships of Equations 4. 6

and 4. 7 • Columns four and five list, respectively, the optimal sampling plan,

(n, c) , and the total expected cost, TC(n, c) , that correspond to the new

values of p1 and p2 • The total expected cost, TC(n, c) , is depicted graphi-

cally in Figure 4-2 as a function of the incremental change in variance.

The total expected cost was observed to decrease linearly as a variance

change of approximately five percent resulted in a total expected cost change of

approximately one percent. It was also apparent that the sample size, n, and

the acceptance number, c, decreased as the variance was increased. The

70

Table IV-3

Effect of a Change in the Process Variance

Increment pl P2 (n, c) TC(n, c)

90.0 0.1317683 o. 0393747 109,8 1724.59

80.1 o. 1312984 0.0394358 109,8 1726. 78

70. 2 0.1308285 0.0394969 l·G9, 8 1728. 97

60. 2 0.1303485 0.0395592 109,8 1731. 21

50.2 o. 1298086 o. 0396216 109, 8 1733. 46

40. 2 o. 1293786 o. 0396853 1J9,8 1735. 76

30. 2 0.1288987 o. 0397476 110, 8 1738. JO

20. 2 0.1284088 o. 0398113 110, 8 1740. 29

10. 0 0.1279088 0.0398763 122, 9 1742. 59

o. 0 o. 1274089 o. 0399412 122, 9 1744. 89

-10.0 o. 1269090 o. 0400062 122, 9 1747. 20

-20. 0 o.1264090 o. 0400711 122, 9 1749. 51

-30.1 0.1258991 o. 0401374 123, 9 1751.87

-40.1 0.1253892 o. 0402037 123, 9 1754. 23

-50.1 0.1248792 o. 0402699 123, 9 1756.59

-60.1 0.1243593 o. 0403375 123, 9 1759.00

-70.1 o. 1238393 o. 0404051 124, 9 1761. 40

-80.1 0.1233193 o. 0404726 124, 9 1763. 81

-90.1 0.1227893 o. 0405415 124, 9 1766. 20

71

1800

1790. NOTE: a 2 : 825"4 µ = 50

1780 el = 0.0 e2 = 0.0

1770

1760

--.. 1750 -u I-

1740

1730

1720

1710

1700 -100 -80 -60 -40 -20 0 +20 +40 +60 +80 +100

IMCREMEMT OF VARIANCE

Figure 4-2. Effect of a Change in the Process Variance

72

fraction defective for the binomial component which occurred most frequently, p 2 ,

decreased in value as the variance increased. The fraction defective for the

binomial component which occurred less frequently, p1 , increased in value as

the variance increased.

Expected Cost of an Optimal P Ian When Error is Present

The optimal sampling plan, n = 122 and c = 9, obtained in the previous

section was designed for an error free inspection process. This section will inves-

tigate the economic consequences of utilizing this sampling plan in an inspection

environment that is error prone. The selected error pairs which will be used

in this evaluation are listed in the first column of Table IV-4. The error pairs

are comprised of a type I error rate, e1 , and a type II error rate, e 2 • The

total expected cost of the optimal plan, established for an error free environment,

but operating in an error prone situation will be described as TC ( n, c) el,e2

This is the same nomenclature used previously; except subscripts have been

added to denote the prevailing error situation.

The total expected cost of the optimal sampling plan, TC ( 122, 9) , el,e2

was evaluated for each of the selected pairs of error rates. The results are tabu-

lated in column two of Table IV-4.

It is apparent that the total expected cost was less for the error free situ-

ation than for any error prone situation. The performance measure 1 was

73

Table IV-4

Performance Measure 1 When the Optimal Sampling P Ian

(n=l22, c=9) is Subject to Error

(el, e2) TC(n, c) el,e2 ~1

• 00, • 00 1744.89 o. 00

• 00, • 05 1766.33 1. 23

• 00, .15 1810.98 3.79

• 03, Q 00 1872.17 7. 29

• 03, • 05 1902. 23 9.02

• 03, .15 1950.06 11.76

• 10, • 00 2261.76 29.62

• 10, • 05 2352. 74 34.84

• 10, .15 2531.81 45.10

74

formulated to depict the percentage variation in TC(l22, 9) as compared el,e2

with TC( 122, 9) O, 0 • The performance measure A 1 is defined as

[TC(n, c) - TC(n, c) 0 oJ

el,e2 ' 6. = ------------ • 100 · 1 TC(n,c) 0 0

' ( 4. 8)

For the optimal sampling plan of interest,

6. 1 == $ 1 744. 89 • 100 (4. 9)

The value of A 1 is tabulated in column three of Table IV-4 with the

corresponding error pair, (e1, e 2), and total expected cost, TC(l22,9) • el,e2

Figures 4-3 and 4-4 depict the total expected cost, TC(l22, 9) , and the el,e2

performance measure o. 1 when e 2 is held constant and e1 is varied over the

range from O. 0 to 10. 0 percent. Similarly, Figures 4-5 and 4-6 depict the total (>

expected cost, TC( 122, 9) , and the performance measure A 1 when e1 el,e2

is held constant and e2 is varied over the range from O. Oto 15. 0 percent.

It was observed that the total expected cost was greater for any error

situation than the cost incurred when errors were absent. An increase in per-

cent error, either type I or type II, added to the total expected cost for the

specified optimal plan. This increase was apparent for type I and type II errors

singularily or in combination. A type I error resulted in a more significant

change than type II errors for a given percentage. This is intuitively obvious,

N • .. ..... • .. N N ..... u t-

75

2600

2500

2400

2300

2200

2100

2000

1900

1800

1700 o.o 0.05

TYPE I ERROR (e 1)

Figure 4-3. Effect of Type I Error on the Total

Expected Cost

0.10

76

40

30

<l 20

10

o.o 0.05 0.10

TYPE I ERROR (e,.1)

Figure 4-4.

Effect of Type I Error on the Ferformance Meuure 1

2600

2500

2-400

2300 N • .. 2200 • 0:-..

N N 2100 --u t-

2000

1900

1800

1700 o.o

77

<•1=0.10)

<•1= 0.03)

( •1 = 0.0)

0.05 0.10

TYPE II ERROR (e 2)

Figure 4-5. Effect of Type n Error on the Total

Expected

0.15

40

30

<J 20

10

0.0

78

(e 1 :0.10)

0,05 0.10

TYPE II ERROR (e 2 )

Figure 4-6. Effect of Type n Error on the Performance

Meuure 6. 1

0.15

79

since the fraction of good items is generally large when compared with the

fraction of bad items. This is certainly the situation in the example case being •

considered. Type I errors increase the inspection cost by promoting the rejec-

tion of good items and causing the unwarranted inspection of others. Type II

errors add to the inspection cost by causing the acceptance of items ( or lots)

that should rightly be rejected, thus promoting bad will among consumers.

Optimal Sampling Plan Designed for an Error Prone Process

There exists the typical situation where all practical actions have been

taken to reduce errors in the inspection process, yet known errors continue to

exist. This section evaluates the total expected cost of an optimal sampling

plan, ( n, c) , which is designed specifically for the inspection process that is

known to be error prone. The total expected cost of such a plan will be desig-

nated as TC(n, c)' • The nomenclature is the same as that used previously el,e2

except the accent ( ') was added to denote that the optimal plan has been designed

for the error conditions specified in the subscript. The absence of the accent

( ') indicates that the optimal plan was developed for an error free inspection

process.

A comparison of the optimal plans, for each of the design conditions,

should be helpful in understanding the economic consequences involved. The

performance measure 2 was used to compare the percentage change of an

optimal plan designed for inspection error with an optimal plan designed for

80

an error free process. The performance measure 2 is defined as

[TC(n,c)' - TC(n,c) 0 o]

el,e2 ' == ------------- • 100 2 TC(n, c) 0 0

' ( 4. 10)

It has been shown earlier in this chapter (see Table IV-4) that the TC(l22, 9) 0, 0

is $ 1744. 89 • Substituting this value of TC( 122, 9) 0 0 into Equation 4. 10 yields '

el,e2 [TC(n,c)' - $1744.89]

2 == $ 1744. 89 • lOO ( 4. 11)

The incremental search procedure, discussed in Appendix A, was used to deter-

mine the optimal plan (n, c)' , and the associated TC(n, c)' , for selected el,e2

error pairs ( e1, e2) • The value of 2 was calculated for each value of the

TC(n, c)' obtained. The values of (n, c)' , TC(n, c)' , and 2 el,e2 el,e2

are tabulated in Table IV-5 for each error pair, ( e1, e 2). The total expected

cost, TC ( n, c) ' , is presented graphically in Figures 4-7 and 4-9 as a el,e2

function of type I and type II error rates, respectively. The performance mea-

sure 2 is depicted in Figure 4-8 as a function of type I error rates. and in

Figure 4-10 as a function of type II error rates.

Several observations can be readily drawn from the acquired data. The

performance measure 2 was always less than 1 for corresponding error

pairs. It may be inferred that the total expected cost can be reduced by selecting

an optimal plan that compensates for the error prone environment. Not only

81

Table IV-5

Performance Measure 2 of Optimd I lane Designed for

Error I rone r rocesses

(el, e2) (n, c)' TC(n,c}' el,e2 ~2

• 00, • 00 122, 9 1744. 89 o.oo • oo, • 05 136,10 1755.90 1. 20

• 00, .15 178,12 1801.32 3. 23

• 03, • 00 132, 14 1776. 20 1.79

• 03, • 05 162, 17 1796. 05 2.93

• 03, .15 208, 21 1830.72 4.92

• lJ, • 00 153, 27 1843. 67 5.66

• 10, • 05 176,31 1865. 23 6.90

• 10, .15 200,35 1898.16 8.78

N • .. .... - II -u .. C -u ....

82

2000--r-------------,

1900

1800

17004---------.---------,

0.0 0.05

TYPE I ERROR <•1)

Figure 4-7.

0.10

Effect of Type I Errar on the Total Expected Cost of an Optimal PlanDe1igned for Error Frone Frocess

83

10

N

<J

o.o 0.05

TYPE I ERROR (e 1)

Figure 4-8.

0.10

Effect of Type I Error on the F erformance Measure~ 2

84

N 1900 • .. ... . -" .. C u t-

1800

1700-t--------r-------,.---------1 o.o 0,10

TYPE II ERROR <•2)

Figure 4-9.

Effect of Type U Error on the Total Expected Cost of an Optimal Plan for Error Prone Frocess

0.15

85

10

8 ( •1 =0.10)

4

2

0-l""';;...._--------.--------,....----------1 o.o 0.05 0.10

TYPE II ERROR (e2)

Figure 4-10.

Effect m Type n Error on the Performance Measure~ 2

0.15

86

would the cost situation be improved by selecting an optimal plan that considers

the inspection error, but the expected costs encountered would be significantly

less as the error rates increase. A type I error continued to impact the expec-

ted cost to a greater extent than a type II error for a similar percentage value.

The performance measures 6. 1 and 6. 2 have compared the total

expected cost, TC(n, c) and TC(n, c)' , respectively, with the el,e2 el,e2

total expected cost TC(n, c) O, 0 • A third performance measure, 6. 3 , com-

pares the total expected cost of the optimal sampling plans, (n, c) and (n, c)' ,

for the same error prone situation. This performance measure may be the

most informative of the three performance measures since it can be used to

evaluate the relative cost differences between the two optimal plans. The per-

formance measure 6. 3 is described as

• 100 (4.12)

The data necessary to determine 6. 3 have been reported previously in Tables

IV-4 and IV-5. The data have been reiterated in Table IV-6 with the related

values of 6. 3 obtained from Equation 4.12 .

Observation of the data reveals that the value of .6. 3 was positive for

each of the error conditions evaluated. This situation can exist only if

TC(n, c)' is less than TC(n, c) • Thus, it may be inferred that el,e2 el,e2

87

Table IV-6

Performance Measure 3 for Selected Error Pairs

(el, e2) TC(n, c) el,e2

TC(n,c)' el,e2 ~3

• 00, • 00 1744.89 1744. 89 0.000

• 00, • ~5 l 7C6. 33 1765.90 0.002

• OJ, .15 1810. 98 1801. 32 o. J54

• 03, • 00 1872.17 1776. 20 5.403

• 03, • 05 1902. 23 1796.05 5. 912

• 03, .15 1950. '.)6 1830. 72 6.519

.10, • 00 2261. 76 1843.67 22. 677

• 10, • 05 2352. 74 1865. 23 26.137

• 10, .15 2531.81 1898.16 33.382

M

<J

88

,0----------------

30

20

10

0.0 0.05 TYPE I ERROR (e 1)

Figure 4-11.

0.10

Effect of Type I Error on the Ferformance Measure~ 3

89

40--------------------

30

20

10

0.0 0.05

( e 1 = 0.03)

( e = 0.00) 1

0.10

TYPE II ERROR (e 2)

Figure 4-12.

Effect d. Type II Error on the Performance Measure~ 3

0.15

90

when the inspection process is error prone, the lesser cost will be incurred by

using the optimal plan, (n, c)' , which is designed specifically for the error

prone environment. Examination of Figures 4-11 and 4-12 indicates that a

type I error had a greater effect than a type II error on the relative expected

cost of the optimal sampling plans, (n, c) and (n, c)' for similar error prone

situations.

Relationship of Error Rate, Sample Size, and Acceptance Number

In designing a sampling plan, a search procedure has been used to iden-

tify the sample size and acceptance number that will minimize the expected quality

costs. An understanding of the relationships between the error rates, sample

size and acceptance number is desirable if one is to visualize how these variables

interact one with the other. Since many parameters are involved, it is desirable

to have a common parameter to aid in the assimilation of data. The apparent

fraction defective will be used for this purpose. The data recorded in Table IV-5

will be used as a basis for the evaluation. The acceptance number, c, sample

size, n, and type I error rate, e1 , will be plotted as a function of the apparent

fraction defective, p . The type II error rate will be held constant for this e

analysis. Similarily, the acceptance number, sample size and type II error

rate will be graphed as a function of the apparent fraction defective as the type I

error rate is maintained constant.

The apparent fraction defective for the error free situation, p, is

defined for the mixed binomial distribution with two components as

91

= ap + p - ap 1 2 2 ( 4.13)

The apparent fraction defective for the error prone situation, pe, has not yet

been defined and must be developed. The relationship between the apparent

fraction defective and the actual fraction defective was provided in Equation 2.1 •

In a similar manner, the apparent fraction defective for each component of the

mixed binomial is described as

p = p ( 1-e ) + ( 1-p1) e1 e1 1 2 (4.14)

and

( 4.15)

The apparent fraction defective in an error prone situation is dependent upon the

frequency with which p e 1

and p , as described by Equations 4. 14 and 4. 15 , e2

respectively, occur. Thus, the apparent fraction defective, p , for the general e

case may be written as

Expanding Equation 4.16 results in the expression

pe = ap1(1-e2) + a(l-p1)e1 + p2(1-e2)

+ (l-p2)e1 - ap2(1-e2) - a(l-p2)e1

( 4. 16)

( 4.17)

92

Regrouping terms in Equation 4. 17 permits p e to be described as

( 4.18)

It follows from the definition of p ( see Equation 4.13) that Equation 4. 18 can

be expressed in a simpler form as

( 4.19)

The values of p were calculated that correspond to the data in Table IV-5. e

The results are shown in Figures 4-13 and 4-14.

It is observed that n and c were made larger by increasing the type I

and type II error rates singularly or in combination. Type II errors had a lesser

effect on the acceptance number than a corresponding type I error. This trend

was less noticeable however, in establishing the sample size. Both type I and

type II errors contributed significantly to variations in the optimal sample size.

In general, as the apparent fraction defective increased for a specified type I

error rate, the type II error rate, sample size and acceptance number decreased.

Similarly, as the apparent fraction defective increased for a specified type II

error rate, the type I error rate, sample size and acceptance number increased.

Shape of the Prior Distribution

It was noted earlier in this chapter that considerable information may be

required regarding the precise shape of the mixed binomial prior disbrituion, if

the total expected costs are to be minimized. It was observed, for the data in

-JL 20 0,: 25 35 0,: w 0,:

w w <XI <XI <XI ::E ::E ::E ::, ::, ::, 15 % 20- % 30 % w w w u u u % % z ,c( ,c( ,c( 10 15- 25 0.. 0.. 0.. w w w u u u u u u 5 ,c( 10 ,c( 20 ,c( I I

250 250 250 -C -=- -=-- w 200 w 200 !:::! 200 N in in

.,, w w w ..J ..J

..J 150 0.. 150 0.. 150 0.. ::E ::E ::E ,c( ,c( I I co ,c( .,, .,, .,, c.., 100 100 100

0,15 _ 0.15 0.15 N N -N .!!- ..!!. ..!!.

0,: 0,10 0.10 9 1 =0.03 0.10 0 0,: °" °" I.,._ I>! Di:: w w w

0.05 = 0.05 : 0.05 w w w 0.. 0.. a. ...

0.0 0.0 0.0 .o.c

Pe .05 .07 Pe

.08 .135 .us Pe

Figure 4-13.

Relationships of Sample Size and Acceptance Number to the ObserTed Fraction Defective for Specified Type I Error Rates

- ..co - .co - 40 u u u -- - a,: • a,: w w 30 w 30 Ill 30 cG Ill :::E :::E :::E ::, ::, ::, z z z 20 w 20 w 20 w u u u z z z .,c .,c .,c I-I- I- 10 a. 10 a. 10 a. w w w u u u u u u .,c .,c 0 .,c 0 0

'; 250 - 250 - 250 C C - - -w w w 200 200 9l 2007----.,.

w w --------- w ..J ..J ..J 150 - 150 150 I .,c .,c .,. .,. 100 100 100 - -£ .10 11:'.:1~ 1F:1~ !-"j~ :: .o ,_.o~

.o.c .10 ,16 .o.c ,10 .16 .o.c ;;. Pe

Figure 4-14.

Relationships of Sample Size and Acceptance Number to the Observed Fr~ction DefectiTe for Specified Type ll Error Rates

,10

;;.

I U) •

I .16

95

Table IV-2, that many mixed binomial parameter sets have the same mean and

variance. A different optimal plan, (n, c) , and total expected cost, TC(n, c)

were acquired for each of three representative parameter sets.

A single mixed binomial parameter set has been used throughout this

chapter. This parameter set will be designated as MBl and two additional

parameter sets selected, for additional analyses, that will be designated as

MB2 and MB3. The added parameter sets, MB2 and MB3, were chosen from

Table IV-2. Table IV-7 reiterates the chosen values of a, p1 , and p2 for

MBl, MB2, and MB3, respectively. Note that the value of a for MB2 is less

than the value of a for MBl, whereas, the value of a for MB3 is greater than

the value of a for MBl.

The optimal sampling plan, ( n, c)' , and total expected cost,

TC(n, c)' , were determined, in the manner previously discussed, for el,e2

each of the subject parameter sets and selected error pairs. This information

is tabulated in Table IV-8. The analysis was continued by re-evaluating each

of the optimal plans as though MBl represented the true prior distribution.

The new total expected cost was obtained by using an optimal sampling plan,

{n, c)' derived for either MB2 or MB3, but using the values of a, p1 , and

p 2 which characterize MBl. This total expected cost was designated as

TC(n, c)" and is tabulated in the right hand column of Table IV-8. The el,e2

information in Table IV-8 has been displayed graphically in Figures4-15 and

4-16.

96

Table "f'V-7

Mixed Binomial Parameter Sets

a pl p2

MB! 0.115 0.1274089 0.0399412

MB2 o. 060 0.1604486 o. 0429501

MB3 0.300 o. 0926283 o. 0317307

97

Table IV-8

Cost Data for Mixed Binomial Parameter Sets

Parameter er el, e2 (n, c) 1 TC(n, c) 1 TC( n, c)" Set el,e2 el,e2

• 00, • 00 122,9 1744. 89 1744.89

MBl 0.115 • 00, • 05 136,10 1765. 90 1765. 90

• 03, , • 00 132, 14 1776. 20 1776.30

.10, .15 200,35 1898.16 Hl9G.16

• 00, • 00 90,8 1796.83 1735.75

MB2 o. 060 • 00, • 05 103, 9 1807. 80 l 78S. 77

• 03, • 00 107, 13 1816.18 1797.02

• 10, .15 190,35 1886.77 1904.68

• 00, • 00 146,8 1669.61 1781.78

MB3 0.300 • 00, • 05 150,8 1717. 08 1805.66

• 03, • 00 173, 15 1728.34 1813.89

• 10, .15 192, 32 1944.67 1906. 71

98

2000----------------------

N1900 • .. -• • "y 1800 .. I: u ...

1700

(.10,.15)

(.03,0) (0,.05) (0,0)

1600 _..,_ ____________________ ---4

MB2 Mil MB3

ACTUAL PRIOR DISTRIBUTIONS

Figure 4-15. Total Expected Coet of Mixed Binomial

F arameter Seta for Speclfted Error Fairs

99

2000 ---------------------

N19()0 • .. .... = •

1800 C u

1-

1700

1600

_--.;. _______ (.10,.15)

- (0,.15) (0, .05) (0,0)

NOTE: ACTUAL PRIOR Ml 1; (el' e 2 ) AS SPECIFIED

MB2 MBl MB3

ASSUMED PRIOR DISTRIBUTIONS

Figure 4-16. Total Expected Cost of Mixed Binomial

Parameter Sets for Specified Error Fairs when MBl is Treated aa Actual Prior

100

Figure 4-15 is a plot of the TC(n, c)' for each of the parameter el,e2

sets MBl, MB2 and MB3. Each parameter set represents the actual prior

distribution assumed in establishing TC(n, c)' • It is apparent that the el,e2

total expected cost differs for each of the mixed binomial priors. The cost dif-

ference between the plans, for a given error pair ( e1, e2) , increases as the

difference in a increases. One may also observe a greater dispersion in the

TC(n, c)' , for the specified error pairs, as the bimodal characteristics el,e2

of the mixed binomial parameter sets become more pronounced.

Figure 4-16 presents the cost consequences of assuming an incorrect

shape for the prior distribution though its mean and variance be known. Each

optimal plan, ( n, c) ' , was obtained for a mixed binomial distribution whose

shape was described either by MB2 or MB3. The total expected cost,

TC( n, c)" , · was computed for each optimal plan thus obtained, but the el,e2

true prior distribution was assumed to be described by MBl. The total expected

cost of either a MB2 or MB3 seemingly optimal plan was always greater than that

incurred by the true MBl optimal plan. It may be inferred, on the basis of the

example data, that an increase in the total expected cost can be anticipated if

the shape of the mixed binomial prior distribution is inaccurately described.

This increase in the total expected cost did not exceed three percent for the

example data though the values assigned to the three parameter sets varied

widely.

101

The significance of the increased total expected cost, which may result

from an erroneous selection of parameters, must be established by the decision

maker for a given quality situation. When the inspection costs are high and

many similar lots are to be processed using the assumed optimal plan, unneces-

sary costs will be experienced. If however, the inspection costs are low and

only a few lots are to be processed using the assumed plan, the added cost may

be insignificant. The judgement of the decision maker will establish what future

action is required, if any, to alter the parameters of the assumed mixed

binomial distribution.

CHAPTER V

COST MODEL EVALUATION: POLY A PRIOR DISTRIBUTION

This chapter will evaluate the use of a Polya distribution to represent

the underlying distribution of defectives in the incoming lots. The Polya distri-

bution is of particular interest in this research; the chief reasons being that the

Polya distribution is unimodal and is the representative distribution for similar

research applications to that reported herein. This will permit a comparison of

the results obtained for the distributional forms represented by the unimodal

Polya prior in this chapter and the bimodal mixed binomial prior in Chapter IV.

Of equal interest is a comparison of the methods and results presented by

Bennett, Case and Schmitt [ 16] with those reported in this research.

Primary Method

The evaluations performed in this part of the chapter are analogous to

those in Chapter IV where the mixed binomial prior distribution was utilized.

Summary of Equations

Many of the equations discussed in Chapter IV are of a general form

and are equally applicable for the evaluations of this section. The reader is

referred to Equations 4.1, 4. 2, and 4. 3 for a description of the cost model.

The referenced cost model requires an expression for g (y ) and f(x) which n e

are dependent upon the prior distribution of defectives in the lot. Equations 3. 17

102

103

and 3. 44 describe g (y ) and f(x) when the prior distribution of defectives in n e

the lot is Polya distributed. These equations for g (y ) and f(x) were n e

described as

and

f(x) =

respectively.

r ( s+t) r(s)r(t)

s +x s + t + n

Y n-y e e s-1 t-1 ( 1 - p ) p ( 1 - p) dp e ( 5.1)

( 5. 2)

The performance measures .6. 1 , .6. 2 and .6. 3 , as described in Equa-

tions 4.8, 4.11, and 4.12, respectively, will again be used to evaluate the effec-

tiveness of the derived optimal sampling plans.

Input Data for the Cost Model Evaluation

The numerical values assigned to each of the cost parameters in Equa-

tions 4.1, 4. 2, and 4. 3 are the same as those used in the mixed binomial distri-

bution. These values are listed in Table IV-1. It was again assumed that the

lots are formed from a homogeneous source and contain 1,000 items each.

Other inputs which are required, before proceding with an evaluation of the

cost model, are the values of the Polya parameters, s and t, in E.quation

5. 2 • The parameter values which correspond to the mixed binomial example

2 ( µ = 50 and u = 825. 4098) are s = 3 and t = 57.

104

Selection of an uptimal I-lan Without Error

A search was performed on the cost model described in Equations 4. 1,

4. 2, and 4. 3. The decision variables involved in the search procedure were

the sample size, n, and the acceptance number, c • The procedure used to

determine the optimal values of the decision variables, which would minimize

the total expected cost, are described in Appendix A. The optimal sampling plan,

(n, c) , for the input data in the preceding section was computed to be ( 122, 6).

The total expected cost, TC(n, c) , corresponding to this plan was $1745. 61 •

Expected Cost of an Optimal Plan when Error is Present

This section will investigate the economic consequences of utilizing

the optimal sampling plan ( 122, 6) in an error prone inspection process. The

reader is reminded that the optimal sampling plan ( 122, 6) was obtained for an

error free inspection process.

The total expected cost, TC(n, c) , was obtained for selected el,e2

sets of error rates ( e1, e 2) • The performance measure 1 , described in

Equation 4. 8, was used to depict the percentage variation in TC( 122, 6) . el,e2

as compared with TC(122, 6) 0 0 • The performance measure 1 can be '

expressed, for the optimal sampling plan of interest, as

100 (5.3)

105

The values of TC(l22,6) and A 1 were computed for each of el,e2

the selected error pairs (e1, e2) • These values are recorded in Table V-1.

The data contained in Table V-1 is presented graphically in Figures 5-1, 5-2,

5-3, and 5-4. Figures 5-1 and 5-2 depict TC(l22,6) and A 1, respec-el,e2

tively, when e 2 is held constant and e1 is varied over the range from o. O to

10. O percent. Figures 5-3 and 5-4 depict TC(l22,6) and 1 , respec-el, e2

tively, when e1 is held constant and e 2 is varied over the range o. 0 to

15. 0 percent.

Similar trends to those obtained for the mixed binomial distribution

were observed. The total expected cost of the optimal plan ( 122, 6) increased

as the type I and type Il error rates were increased, either singularly or in

combination. A type I error resulted in a more significant change than type Il

errors for the same percentage.

Optimal Sampling Plan Designed for an Error Prone Process

It was apparent that the total expected cost of an optimal plan, derived

for an error free process, is significantly increased by the introduction of

errors. This section investigates the total expected cost of an optimal plan,

(n, c)' , designed for a specific error prone process. The total expected costs,

TC(n, c)' and TC(l22, 6) 0 0 , are compared for selected error pairs el,e2 •

( e1, e 2) • The performance measure A 2 , as described in Equation 4.11,

106

Table V-1

Ferformance Measure 1 When the Optimal Sampling Flan

(n = 122, c = 6) is Subject to Error

(el' e2) TC( 122, 6) el,e2 1

• 00, • 00 1745. 61 o.oo • 00, • 05 1795.11 2. 84

• 00, .15 1878.33 7. 60

• 03, • 00 1927.05 10.39

• 03, • 05 1996.83 14.39

• 03, .15 2126. 08 21. 80

• 10, • 00 2271.10 30.10

• 10, • 05 2365.19 35.49

• 10, .15 2553.28 46.27

N II .. -II

N"' N -u

2600

2500

2-400

2300

2200

2100

2000

1900

1800

1700

107

OoO 0.05 .0.10

TYPE I ERROR (e1)

Figvre 5-1.

Effect of Type I Error on the Total Expected Coat

<]

108

40

30

20

10

0-¥------~------...J 0.0 0.05 0.10

TYPE I ERROR (e1)

Figure 5-2. Effect of Type I Error on the Performance

Meaaure 1

N • .. . ....

41 -"O .. N N .... -u I-

109

2600

2500

2400

2300

2200

2100

2000

1900 ( •1 =0.0)

1800

1700-'--------.-------,---------1 0,0 0.05 0.10

TYPE II ERROR (e2)

Figure 5-3.

Effect ol Type Il Error on the Total Expected Cost

0.15

<1

0.0

110

0.05 0,10

TYPE II ERROR (e2)

Figure 5-4.

Effect of Type n Error on the Performance Meuure l

0.15

111

useci to make this comparison. The performance measure A 2 \vas described

as

[TC(n, c)' - $ 1745. 61] el,e2

$ 1745. 61 • 100 ( 5. 4)

The values of TC(n, c)' and A 2 were computed for each of the selected el,e2

error paris (e1, e 2). These values are recorded in Table V-2. The data con-

tained in Table V-2 is presented graphically in Figures 5-5, 5-6, 5-7, and 5-8.

Figures 5-5 and 5-6 depict TC( n, c) ' and A 2 respectively, when e2 is el,e2

held constant and e1 is varied over the range from 0. 00 to 10. 0 percent.

Figures 5-7 and 5-8 depict TC(n, c)' and A 2 respectively, when e1 el,e2

is held constant and e2 varied over the range o. 0 to 15. 0 percent.

Similar trends to those obtained with the mixed binomial prior were

observed. The performance measure A 2 was always less than A 1 for the

same error pair ( e1, e 2) • It can be inferred from this data that the minimum

cost of inspection will be obtained when the optimal sampling plan, ( n, c) ' , is

derived for the error prone situation that exists.

The performance measures A 1 and A 2 have been used to compare

the total expected costs TC(n, c) and TC(n, c)' respectively, el,e2 el,e2

with the total expected cost TC(n, c) O, 0 • The performance measure A 3 was

introduced in Equation 4.12 to compare the total expected cost of the optimal

N ., ., ... -___ ., u ..

-=-u I-

112

2000---------------

1900

1800

o.o

(e 2 = 0.0)

0.05

TYPE I ERROR (el)

Figure 5-5.

0.10

Effect of Type I Error on the Total E;x:pected Cost of an Optimal Plan Designed for Error r rone I rocess

N

<]

0.0

113

0.05

TYPE I ERROR (el)

Figure 5--6.

0.10

Effect of TytMt I Error on the I erformance Meaaure .:1 2

N .. .. . .. --u .. C --u ....

114

2000

1900

1800

1700-L---------------..---------1 0,0 0.05 0,10 0.15

TYPE II ERROR (e2)

Figure 5-7.

Effect of Type n Error on the Total Expecte~ Cost of en 0ptimal F Ian De11igned for Error Prone I rocess

115

12

10

8 N

<] 6

2

0-¥;__------..-------------f 0.0 0.05 0.10

TYPE II ERROR (e2)

Figure 5-8. Effect of Type II Error on the Perf o:rmance

Meuare 2

0.15

116

Table V-2

Performance Measure 2 of Optimal Plans Designed for

Error :r rone Process

(e1, e2) (n, c)' TC(n, c)' el,e2 ~2

• 00, • 00 122, 6 1745. 61 o. 00

• 00, • 05 132, 7 1793.92 2. 77

• 00, .15 180, 11 1865.40 6.86

• 03, • 00 143,12 1794. 68 2. 81

• 03, • 05 159,15 1837.50 5. 25

• 03, .15 200,19 1898. 29 8. 74

• 10, • 00 147, 24 1883.14 7. 88

.10, • J5 160, 27 1915.57 9. 74

• 10, .15 200,35 1953. 20 11.89

117

sampling plans, (n, c) and (n, c)' , for the same error prone situation. The

performance measure .6. 3 is described as

• 100 ( 5. 5)

The data necessary to determine .6. 3 have been reported previously in Tables

V-1 and V-2. The data have been reiterated in Table V-3 with the related values

of .6. 3 obtained from Equation 5. 5 • Figure 5-9 and 5-10 display the values of

.6. 3 , from Table V-3, as a function of e1 and e 2 , respectively. The value of

.6. 3 is observed to be positive for each of the error conditions evaluated. The

same observation was noted in Chapter IV where the prior distribution was des-

cribed by the mixed binomial distribution. This situation can only exist when

TC(n, c)' is less than TC(n, c) • Thus, when the inspection process el,e2 el,e2

is error prone, the lesser cost will be incurred by using the optimal plan, ( n, c)

which is designed specifically for the error prone environment. Increasing the

value of e1 or e 2 , singularly or in combination, served to increase the value

of .6. 3 • This increase in .6. 3 is a measure of the additional cost which would

be incurred if the sampling plan (122, 6) was used in lieu of (n, c)' for a

specific error prone process.

Relationship of Error Rate, Sample Size and Acceptance Number

The data recorded in Table V-2 were used in Figures 5-11 and 5-12 to

evaluate the relationship of error rates, sample size and acceptance number

118

Table V-3

Performance Measure 3 for Selected Error Prirs

(el, e2) TC(l22,6) el,e2

TC(n, c)' el,e2 ~3

• oo. • 00 1745. 61 1745. 61 o.oo • 00, • 05 1795.11 1793. 92 0.07

• 00, .15 1878.33 1865. 40 o. 69

• 03, • 00 1927.05 1794. 70 7. 37

• 03, • 05 1996.83 1837.31 8. 68

• 03, .15 2126. 08 1898.29 11.99

• 10, • 00 2271.10 1883.14 20.60

• 10, • 05 2365,19 1915.57 23.47

.10, .15 2553. 28 1953.20 30.72

M

<l

119

,0--------------,

30

( e 2 = 0,05)

20

10

o~-----.--------1 0.0 0.05 0.10

TYPE I ERROR (e 1)

Figure 5-9. Effect of Type I Error on the Performance

Measure~ 3

120

40---------------------

30

M

<J 20

10

(e 1 = 0.0)

0 "===================~'.=======::i o.o 0.05 0.10

TYPE II ERROR (e 2)

Figure 5-10.

Effect of Type n Error on the Ferformance 3

0.15

:; 20 - 25 - 35 Ill: a,: a,: w w w 10 10 110

::E 15 ::E 20 ::l:i 30 ::, ::, ::, z :z: % w w w u V u % 10 :z: 15 25 < < -~ ... ... I-CL a. a. w w i w u 5 u 10

u u u u 20 < < <

250 250 250

5 5 5 w 200 w 200- ------ w 200 N N N .;; .;; .;; w w w ..J 150 ..J 150- i 150 CL CL ::E ::E < < ... "' "' 100 100 100 I I ....

' I I I I "" .... 0,15 0.15 0.15 -]' N -N

Ill: 0.10 0.10 •1: 0.03 0.10 0 Ill: a,: Ill: a,: 0,:

0,: w w w = 0.05 = 0.05 = 0.05 w w w CL a. >- >- a.

>-... I-1- 0.0 o.o 0.0

.04 .cs .07 .08 .135 .145

Pe Pe Pe

Figure 5-11.

Relationahlp of Sample Size and Acceptance Number to the Obsen-ed Fraction Defective for Specified Type I E::-ror Rates

- 40 - 40 - 40

« « a: w w 30 w 30 11D 30 11D 11D ::e :!: :::e ::, ::, ::, z :z: z

w 20 w 20 w 20 u u z

< I- I- I- 10 a. 10 II. 10 II. w w "' u u u V u u

0 < 0 < 0

250 250 250 - - c , w 200 w 200

2001---- I N N ;;; ;;; w w ..I 150 ..I 150 -- i 150 a. II. ::e --- :::e ::e

< .,, .,, .,, 100 100 100

- -r--------------r ::1 1r:1~ 0 0 .0 I

I- ·°' .10 .16 1- .04 .10

f:::1~ l .O I

,16 1- .04 .10 .16

Pe Pe Pe

Figure 5-12.

Relationship of Sample Size and Acceptance Number to the Observ~action Defective for Specified_ Type II Er:sor Rates

I-' NI NI

123

to the apparent fraction defective. The apparent fraction defective of the Polya

prior distribution is expressed, using the relationship in Equation 2.1, as

( 5. 6)

The apparent fraction defective, p , for the Polya distribution is analogous to e

the apparent fraction defective, p e, for the mixed binomial distribution ( see

Equation 4. 19).

In Figure 5-11 the parameters being investigated were plotted as a

function of the apparent fraction defective; type I error rates were held constant

while type TI error rates were varied. Figure 5-12 is a graphical presentation

of the same parameters with the type TI error rates being held constant as the

type I error rates were varied. Comparison of Tables IV-5 and V-2 for the

mixed binomial and Polya distributions, respectively, suggest that the selection

of an optimal plan is influenced by the prior distribution. The significance of

the differences will be the subject of a subsequent section. It will suffice, at

this time, to simply note that the trend data for the unimodal Polya model

behaves in a like manner to that observed for the bimodal binomial model. The

sample size, acceptance number and type II error rates decrease in value for

a constant type I error rate and increasing apparent fraction defective. Con-

versely, the sample size, acceptance number and type I error rates increase

in value for a constant type II error rate and increasing apparent fraction

defective.

124

Form of the Prior Distribution

The optimal sampling plans derived from the mixed binomial parameter

sets MBl, MB2, and MB3 were reevaluated under the assumption the the Polya

prior evaluated in this section was the true prior. Reference is made to Table

IV-8 for the specific values of the three mixed binomial parameter sets. The

term TC(il, c) 11 is interpreted in the same manner as was used previously el,e2

for the mixed binomial distribution. It represents the total expected cost of a

seemingly optimal plan that is governed by the true prior distribution. The

seemingly optimal plan can result from an incorrect assumption of the true prior

distributional shape. For the case of interest here, the seemingly optimal plan

is that represented either by MBl, MB2, or MB3. The true prior is represented

by a Polya distribution. The data for ( e1, e2) , ( n, c)' and TC( n, c)' el,e2

which pertains to the mixed binomial distribution, is reiterated in Table V-4

for easy reference. The last column of Table V-4 contains the total expected

cost, TC(n, c) 11 , of the seemingly optimal plan, (n, c)' , when operating el,e2

in a process described by the Polya prior distribution.

The TC(n, c)' and TC(n, c) 11 are shown in Figure 5-13 . el,e2 el,e2

for each of the optimal plans derived from the mixed binomial parameter sets.

It is observed that TC(n, c)" is greater than TC(n, c)' for any el,e2 el,e2

mixed binomial optimal plan, ( n, c) ' , or error pair, ( e1, e 2) • The

125

Table V-4

Cost Data Obtained with Polya Prior and Sampling Plans Derived from Mixed Binomial F arameter Sets

MIXED BINOMIAL PRIORS POLYA PRIOR (ASSUMED) (TRUE)

Parameter a (el• e2) ( n, c)' TC(n, c)' TC(n, c)" Set el,e2 el,e2

• 00, • 00 122, 9 1744. 89 1795.39

MBl 0.115 • 00, • 05 136,10 1765. 90 1828. 32

• 03, • 00 132, 14 1776. 20 1831. 76

.10, .• 15 200,35 1898.16 1953.20

• 00, • 00 90,8 1796.83 1845.19

MB2 o. 060 • 00, • 05 103,9 1807.80 1866.99

• 03, • 00 107, 13 1816.18 1873.12

• 10, .15 190,35 1886.77 1955.2C

• 00, • 00 146,8 1669.61 1749.90

MB3 0.300 • 00, • 05 150,8 1717. 08 1794.81

• 03, • 00 173, 15 1728.34 1798.19

• 10, .15 192, 32 1944.67 1958.92

126

2000

1900-

1800 ~R

---1700- MB PRIOR .........__ .........__

1600 I I

M

" 20:. -.. ,... :: " u 1900-..

C -u I- 1800 0:: ----0 ----0:: 1700-0..

MB PRIOR ->-..J 1600 0 0..

"- 2000 M • .. HOT E: (el = 0.03, e 2 = O. 0 ) ,... 1900

' • -u .. C u 1800-I-0::

-- ---MB PRIOR --0 1700-0:: 0.. co 1600

2000 __________ POLY.A PRIOR

1900 ------MB PRIOR

1800

1700 NOTE : (e 1 = 0.10, e 2 ::0.15)

1600

Figure 5-13 .• Comparison of_ Total Expected Coat for Assumed Prior at

Selected Error Pairs

127

difference between TC(n, c)' and TC(n, c)" never exceeded five el,e2 el,e2

percent of the true expected cost, TC(n, c)" , though the true prior was el,e2

Polya distributed. This cost difference appears to be insignificant for most

quality control situations. One should recall, however, that TC(n, c)" e1,c2

does not reflect the total expected cost of an optimal plan derived for the true

prior distribution. The TC(n, c)' obtained for the true Polya prior el,e2

(see Table V-2) was always less than TC(n,c)'" • A comparison of the el,e2

performance measures a 1 , a 2 and a 3 will be made in Chapter VI to aid

in drawing inferences pertaining to the distributional form.

Alternate Method

The method reported by Bennett, Case and Schmidt [16] represents a

different approach than that discussed throughout this research. This approach

will be referred to as the alternate method to distinguish between the primary

method reported in this document. The same general cost functions are utilized

for both methods. The basic differences in the two methods rests in the treat-

ment of observed defectives in the lot and in the sample. The primary method

employs a probability mass function which describes all the possible values

which the number of observed defectives can assume in a sample subjected to

inspection. The alternate method uses the expected number of defectives in the

lot ancl in the s~.mple.

128

A brief development of the equations used in the alternate method is

provided in terms of the nomenclature used in this document. The reader is

referred to the original publication by Bennett, Case and Schmidt [16 J for the

detailed developments.

Summary of Equations

The notation used in formulation of the primary method is applicable to

the following expression for the alternate method ( see Formulation of the Model,

Chapter IIl) • The total expected cost given that the lot is accepted is described

as

where x is defined [16) as e

X = E(y ) e e

and

e:=1-e -e 1 2

X C

Taking the expectation of Equation 5. 7 with respect to X yields

where E ( X: x) is described in Equation 3. 26 as

E(XI x) = (N-n) (s + x)

(s + t + n) + X

( 5. 7)

( 5. 8)

( 5. 9)

( 5. 11)

Similarly, the total expected cost given that the lot is rejected is described as

129

X > C

X > C ( 5.12)

where

X = E(Y ) e e

(5.13)

Taldng the expectation of Equation 5.12 with respect to X yields

X > C (5.14)

where E(XI x) is defined in Equation 5.11.

The decision to accept or reject a lot is based on the number of observed

defectives relative to the acceptance number, c • Since the expressions for the

total expected cost are written in terms of actual defectives rather than observed

defectives, the acceptance number, c, will be. redefined as c'. The relation-

ship of c' and c is given by

C = nel + €C 1

Solving for c' yields

( 5.15)

c - ne1 c' = ---- ( 5. 16)

€

The total expected cost for a specified sampling plan, ( n, c) ' , may now be

expressed· as

c' TC(n, c', x) = L,

x=O TC(n,c',x)g (x) + n

n L, TC(n, c', x) gn (x)

x=c'+l ( 5.17)

where TC(n, c' ,x) is defined in Equations 5. 10 and 5.14 for the accept and

reject situations, respectively.

130

The alternate method is computationally simpler than the primary

method developed in this research. Results can be obtained from the alternate

method without the extensive summations and integral approximations required

by the primary method. If similar results can be obtained by using the alternate

method in lieu of the primary method, the problem will be greatly simplified

and considerable savings in computational time may be realized.

Input Data for the Cost Model Evaluation

The input data for evaluation of the alternate method is identical to that

used to evaluate the primary method. A Polya prior distribution is assumed with

the parameters s = 3 and t = 57 • The numerical values assigned to each of the

cost parameters in Equations 5.10 and 5.14 are listed in Table IV-1. Lots are

assumed to contain 1, 000 items each.

The results computed with this input data will be presented primarily

as tables and figures. A summary of the numerical performance measures will

be compiled in Chapter VI for each method of interest. A comparison will be

made of the summarized performance measures and inferences drawn concern-

ing the applicability of the two subject methods.

Selection of an Optimal Plan Without Error

An incremental search was performed of the cost model described in

Equations 5.10, 5.14, and 5.17 • The decision variables involved in the search

were the sample size, n, and the acceptance number, c. The search

131

procedure is defined in Appendix A. The optimal sampling plan, (n, c) , for an

error free inspection process was computed to be ( 122, 6) • The total expected

cost, TC(n, c), corresponding to this plan was $1745. 61 . These values are

identical to those obtained by the primary method for an error free inspection

process.

Expected Cost of an Optimal Plan When Error is Present

This section will investigate the cost consequences of utilizing the

optimal sampling plan, ( 122, 6), in an error prone inspection process. The

total expected cost, TC(l22,6) , of the plan optimized for an error el,e2

free process, but operating in an error prone environment was established.

The performance measure A 1 , defined in Equation 5. 3, was calculated for

each computed value of TC(l22, 6) • The selected error pairs, (e1, e 2) , el,e2

are the same as those used previously for all reported analyses.

The values of TC( 122, 6) , A 1 , and ( e1, e 2) are recorded in el,e2

Table V-5. The information contained in Table V-5 has been graphically illus-

trated in Figures 5-14, 5-15, 5-16 and 5-17. The results are presented in the

same manner as the results for all previous analyses. The trends are similar

to those obtained from the cost evaluations employing the primary method and a

Polya prior distribution.

132

Table V-5

Performance Measure 1 W11en the Optimal Sampling Flan

(n = 122, c = 6) is Subject to Error

(el, e2) TC( 122, 6) el'e2

~l

• 00, • 00 1745. 61 o.oo • 00, • 05 1804.84 3.39

• 00, .15 1907.61 9.28

• 03, • 00 1950.17 11. 72

• 03, • 05 2037. 62 16. 73

• 03, .15 2212. 52 26. 75

.10, • 00 2229.51 27.72

.10, • 05 2323. 81 33.12

.10, .15 2512. 41 43.93

133

2600

2500

2400

2300 (e 2 = 0.15)

N II 2200 .. .... • ..

N 2100 N .... u I-

2000

1900

1800

1700 0.0 0.05 0.10

TYPE I ERROR (el)

Figure 5-14.

Effect of Type I Error on the Total Expected Cost

134

40

(e 2 = 0.15)

30

<] 20 (e 2 = 0.0)

10

0-+--------r-------' 0.0 0.05 0.10

TYPE I ERROR (el)

Figure 5-15.

Effect d Type I Error on the F erf ormanee Measure 1

N QI .. .... • ..

N N .... u ...

2600

2500

2400

2300

2200

2100

2000

1900

1800

1700

(el= 0.03)

0.0

135

0.05 0.10

TYPE II ERRORw2)

Figure 5-16.

0.15

Effect of Type Il Error on the Tot2.l Expectec Cost

40

30

<] 20

0.0

136

0.05 0.10

TYPE II ERROR (e2)

Figure 5-17.

0.15

Effect d. Type II Error on the F erf ormanee Meaaure l

137

Optimal Sampling Plan Designed for an Error Prone l'rocess

Optimal sampling plans, ( n, c) ', will be established for specific error

conditions, (e1, e 2). The total expected cost, TC(n, c)' h t e1 , e 2 , t a corres-

ponds to the plan (n, c)' will be computed. The performance measure A 2 ,

described in Equation 5. 4 will be used to compare TC(n, c)' and el,e2

TC( 122, 6) 0 0 • The performance measure 3 , described in Equation 5. 5 '

will be used to compare TC(n, c)' and TC( 122, 6) . el,e2 el,e2

listed in Table V-6. The information in Table V-6 is shown graphically in

Figures 5-18, 5-19, 5-20 and 5-21. The values of ( e1 , e2) , TC( 122, 6) , el,e2

TC ( n, c) ' , and 3 are recorded in Table V-7. The information con-el, e2

tained in Table V-7 is illustrated in Figures 5-22 and 5-23. The trend results

are comparable with those obtained from the evaluation of A 3 which employed

the primary method and a Polya prior distribution. As was noted earlier, a

comparison of the numerical values of A 3 for the primary and alternate methods

will be accomplished in Chapter VI.

/

138

Table V-6

F erformance Measure 6. 2 of Optimal I' lans Designed for

Error Prone r rocess

(el• e2) ( n, c) i:- TC(n,c)' el,e2 ~2

• 00, • 00 122, 6 1745. 61 O.JO • 3J, • 05 111, 6 1802.13 3. 24

• ~o. .15 78,4 1899.47 8. Cl

• 03, • 00 117, 9 1769.65 1. 38

• 03, • 05 1J6,9 1823. 75 4.48

• 03, .15 75,6 1915. 28 9. 72

.10, • 00 106, 16 1820.88 4.31

.10, • 05 83,13 1869.10 7. 07

.10, .15 65,10 1947. 62 11. 57

139

12

10

8

N 6 <J

4

2

0 0.0 0.05 0.10

TYPE I ERROR (e 1)

Figure 5-18. Effect ol Type I Error on the Performance Measure 2

140

N 1900 II .. .---u ..

C -u .... 1800

1700-+-------------o.o 0.05 0.10

TYPE I ERROR (e 1)

Figure 5-19. Effect of Type I Error on the Total Expected Costa d an

Optimal I lan Deaigned for Error F rone I roceH

N .• "' -... -u ..

C: -u t-

1900

1800

0.0

141

0.05 0.10

TYPE II ERROR (e2)

Figure 5-20.

0.15

Effect of Type II Error on the Total Expected Cost of an Optimal Flan Designed for Error I rone I rocess

N

<]

12

10

8

6

4

2

0 0.0

142

0.05 0.10

TYPE II ERROR (e2)

Figure 5-21.

0.15

Effect <:i Type II Error on the PerforIIcance Measure 2

143

Table V-7

Performance Measure 3 for Seldeted Erro~ Fairs

( 01' 82) TC(l22,6) TC(n, c)' ~. el,e2 el,e2 3

• 00, • 00 1745. 61 1745. 61 o.oo • 00, • 05 1804. 84 1802.13 0.15

• 00, .15 1907.61 1899.47 0.43

• 03, • 00 1950.17 1769.65 10. 20

• 03, • 05 2037. 62 1823. 75 11.73

• a3, • 15 2212. 52 1915.28 15.52

• 10, • 00 2229.51 1820.88 22.44

.10, • 05 2323. 81 1869.10 24.32

• 10, .15 2512. 41 1947.62 29.00

144

40,-------------

30

M

<J 20

10

0-;--------,,-------1

0.0 0.05 0.10

TYPE I ERROR (e1)

Figure 5-22. Effect of Type I Error on th• Performance Measure 1. :-l

145

40-.------------------~

30

M <J 20

10

0 4=======;:============::;::::========~ 0.0 0.05 0.10

TY PE II ERROR (e2)

Figure 5-23.

0.15

Effect of Type Il Error on the Performance Measure 3

CHAPTER VI

SUMMARY AND RECOMMENDATIONS

The research reported in this dissertation has sought to identify the

effects of inspection errors on quality decisions. The strategy being that if

the effects were understood, perhaps, means could be developed that would com-

pensate for inaccuracies in the inspection process. Two basic areas of research

were identified and investigated. These areas were concerned with statistically

and economically based quality control plans. This chapter summarizes the

major findings of the research reported in earlier chapters. Potential areas

for further research, which promise to be rewarding, are also discussed.

Summary

A review was first conducted of the reported investigations of other

researchers. It was apparent that many factors in an inspection process con-

tribute to inaccurate sampling results. This research has assumed that sources

in the inspection process which generate errors have been reduced to only a

few. The few sources of errors that remain are either unknown or impractical

to remove. When this situation prevails, the decision maker can either ignore

the consequences of the remaining errors or compensate for their effects.

Each of these situations was investigated for single sampling attribute accept-

ance plans.

146

147

Distributional Considerations

The primary approach, used throughout this investigation, assumed

that the quality planner had knowledge of the prior distribution of defectives,

fX (N) , which described the lot fraction defective arriving at an inspection

station. Samples were assumed drawn without replacement from the arriving

lots. The distribution of defectives in the sample given the defectives in the

lot, ln (x! X), was defined by the hypergeometric distribution. From these

distributional considerations the marginal distribution of defectives in the

sample, g (x) , was obtained. At this point in the research, error had not n

been considered and the distributional expressions used were those commonly

described in existing literature.

To extend this research it was necessary to develop a new probability

mass function, 1 (y !x) , that related the number of observed defectives in an n e

error prone process to the number of actual defectives. Such an expression

was developed and became the basis for establishing the marginal distribution

of observed defectives, g (y ) , in a sample. Further, it was shown that if n e

the prior distribution was from the binomial or mixed binomial families that

the expression for g (y ) was of the same form. The expression for g (y ) n e n e

differed from fX( N) only in that the number of items in the sample, n, and

the apparent fraction defective, p , were substituted for the number of items e

in the lot, N, and actual fraction defective, p , respectively. While this

seemed intuitively obvious, the proof had not been documented in the existing

literature.

148

Statistical Consideratio11s

The preceding approach was used with a binomial prior distribution to

investigate the effect of errors on statistically based quality plans. Performance

measures were identified, or developed, for typical inspection schemes under

the assumptions of error free environments and error prone environments.

These performance measures included the probability of lot acceptance, F , a

average outgoing quality, AOQ, with and without replacement, and the average

total inspection, ATI, with and without replacement. The performance measures

were then c-valuated for a representative example and selected combinations of

type I and type II error rates.

It was shown that the apparent fraction defective, p , was equal to e

the actual fraction defective, p, for one specific value defined by the expression

p' =

where e1 and e 2 are type I and type II error rates, respectively. It is obvious

from this relationship that both the actual and apparent fraction defectives can

equal zero. Further, both the actual and apparent fraction defectives can equal

one only when e 2 equal zero.

Consider the situation where e1 > 0 and e2 > 0. The observed prob-

ability of acceptance, Pa was less than that for the error free situation when e'

p < p'. Conversely, the observed probability of acceptance, Pa , was greater e

than that for the error free situation when p > p'. That is,

Pa < Pa if p < p' e and Pa > Pa if p > p' e

149

The average outgoing quality, AOQ, was examined as a function of

type I and type II errors. It was found that type I errors reduced the AOQ

since a greater number of lots were rejected and subsequently screened. Type

II errors resulted in higher AOQ values because of the incorrect classification

of defective items. Examination of the results obtained from the example prob-

len: s revealed a very small difference in the values of AOQ for either a replace-

ment or nonreplacement policy. However, it was observed that the conventional

concept of the AOQ is not meaningful when type II errors are present in the

inspection process. Near the point where Pa approached zero, the slope of e

the AOQ curve increased positively as p increased. This observation was

attributed to the increasing number of defective items which were incorrectly

classified as good items.

As intuitively expected, the general effects of type I and II errors were

to increase or decrease, respectively, the average total items inspected per

lot, ATI, for any given fraction defective, p. The values of ATI were signifi-

cantly affected by the choice of either a replacement policy or nonreplacement

policy. Under a nonreplacement policy, it was observed that the ATI approached

the lot size, N , as the incoming fraction defective, p , increased. When a

replacement policy was applicable, the ATI increased without bound as the

incoming fraction defective, p, increased.

A method for designing a sampling plan which accounts for an error

prone environment was presented. Adjustments for errors were made to the

150

desired fractions defective described as the acceptable quality level, AQL, and

the lot tolerance percent defective, LTPD. A sampling plan was then designed,

for the adjusted values of AQL and LTPD, that would yield similar results to an

optimal plan designed for and used in an error free environment.

Economic Considerations

The cost functions developed by Guthrie and Johns (14) were modified

to account for an error prone inspection process. Several prior distributions

were assumed which descri~d the frequency with which defective items could

occur in the incoming inspection lots. The mixed binomial prior distribution

with two components was chosen as the probability mass function of primary

interest. Its bimodal characteristics and recognition among researchers made

it an excellent choice. It was desired to compare the results obtained with this

bimodal prior distribution and a representative unimodal prior distribution.

The Polya prior was selected for comparison with the mixed binomial prior.

A common mean, µ , and variance, u2 , were identified for use with either

the Polya or mixed binomial distribution and appropriate parameter values

were calculated using the method of moments. Typical cost data were identi-

fied for example purposes.

Techniques were developed to enable a search of the cost functions for

a plan that would minimize the total expected cost. The optimal plan, ( n, c) ,

151

and total expected cost for an error free process, TC(n, c) 0 0 , were sought , for each prior distribution of interest. It was shown that there are many mixed

binomial parameter sets which result in the same mean, µ , and variance, a 2 •

A parameter set was chosen for the mixed binomial prior distribution that would

result in approximately the same TC(n, c) O, 0 as that determined using the Polya

prior distribution. The total expected cost in an error prone environment,

TC(n, c) , was then computed for the sampling plan (n, c) and selected el,e2

combinations of type I and type II error rates, ( e1, e 2) • A search was then

performed on the cost model to establish an optimal plan, ( n, c) ' , for each

prevailing error situation depicted by ( e1, e 2). The total expected cost for an

error prone environment, TC(n, c)' , was computed for each optimal plan el,e2

(n,c)'. Theprocedureforcalculating TC(n,c)O,O' TC(n,c) , and el,e2

TC ( n, c) ' was repeated for an alternate method of analysis which also el,e2

assumed a Polya prior distribution. The alternate method was proposed by

Bennett, Case and Schini tt [ 16] and utilized the expected number of observed

defectives in a sample. The alternate method differed from that employed in

this research where a distribution was developed to describe the number of

items observed defective by an inspector. A comparison of the two methods

was desired since the primary approach is more descriptive of an actual situa-

tion and the alternate method is computationally easier to employ.

152

A summary of the data from Chapters IV and V is provided as an aid in

comparing the results. The data were grouped according to the assumed prior

distribution and methods of analyses as follows:

Group A

Group B

Group C

Data obtained with a mixed binomial prior and the

method of analysis developed in this research.

Data obtained with a Polya prior and the method of

analysis developed in this research.

Data obtained with a Polya prior and the alternate

method of analysis.

The optimal plan, (n, c)', and total expected cost, TC(n, c)' , are shown el,e2

in Table VI-1 for each of the preceding data groups. Nine different error con-

ditions are represented. The lesser cost is observed to occur for the error

free situation, ( e1 = 0, e 2 = 0), regardless of the prior distribution or method

of analysis. In each data grouping the TC(n, c)' increases as the type I el,e2

and type II error rates increase. The number of items in the sample, n, and

the acceptance number, c, tend to increase in value for the primary method

of analysis as the error rates increase. An opposite trend is apparent for the

alternate method; the values of n and c become smaller as the error rates

increase. A general observation is that the total expected cost of the optimal

plans, for any specified error situation, are relatively close in value for any of

the data groups. This is a subjective assessment of the different data groups;

153

Table VI-1

' Summary of Selected Cost Data from Example Problema

PRIMARY METHOD AL TERNA TE METHOD

A B C

(el• e2) ( n, c)' TC(n,o)' ( n, c)' TC(n,c)' (n.c)'' TC(n, c)' el,e2 el,e2 . el• e2

• 00, • 00 122,9 $1744. 89 122,6 $1745.61 122,6 $1745.61

• 00, • 05 136,10 $1765. 90 133,7 $1793. 92 111,6 $ 1802.13

• 00, .15 178, 12 $ 1801. 32 180,11 $1865.40 78,4 $1899.47

• 03, • 00 132,14 $1776. 20 143,12 $1794. 68 117,9 $1769.65

• 03, •. 05 162, l 7 $1796. 05 159,15 $ 1837. 50 106,9 $1823. 75

• 03, .15 208,21 $ 1830. 72 200,19 $1898. 29 75,6 $1915. 28

.10; • 00 153,27 $1843. 67 147,24 $ iJd3.14 106,16 $1820. 88

• 10, • 05 176, 31 $ 1865. 23 160,27 $ 1915. 57 83,13 $ 1869.10

• 10, .15 200,35 $ 1898.16 200,35 $1953. 20 65,10 $1947.62

154

quantitative procedures were established that will enaule a better comparison to

he performed.

Three performance measures were established to provide for an assess-

ment of the total expected cost depicted by either TC(n, c) 0 0 , TC(n, c) e , ' el, 2

or TC(n, c)' . These performance measures were described as . el, e2

[TC(n,c) - TC(n,c) 0 0]

el,e2 ' = ----------- • 100 TC(n, c) 0 0

'

[TC(n, c)' - TC(n, c) 0 0]

el,e2 ' = ------------- • TC(n,c) 0 0 ,

100

and

The values of 6. 1, .6. 2 , and 6. 3 from Chapters IV and V are presented in

Table VI-2 for each of the data groups A, B, and C. A number of inferences

may be drawn from the data as shown. It is apparent from 6. 1 that neglecting

error in the design of a cost model and sampling plan for use in an error prone

environment results in a very significant cost increase over that expected. It

is equally apparent from 6. 2 that sampling plans can be designed which partial-

ly compensate for inspection error and result in costs much closer than those

expected in an error free environment. Performance measure 6. 3 , perhaps

el,e2

• OD, • 00

• oo, • 05

• 00, .15

• 03, .oo • 03, • 05

• 03, • .15

• 10, .. oo • 10, • 05

,.10, .15

155

Table VI-2

Summary of F erformance Measures for Example i roblems in Chapters IV and V

~1 ~2 ~3

A B C A B C A B

o.oo o.oo o.oo o.oo o.oo o. 00 o.oo o. 00

1. 23 2.84 3.39 1.20 2. 77 3. 24 o. 00 o. 07

3.79 7.60 9. 28 3. 23 6.86 8.81 0.05 0.69 ' 7. 29 10.39 11.72 1.79 2. 81 1.38 5.40 7.37

9.02 14.39 16. 73 2. 93 s. 25 4.48 5.91 8.68

11.76 21.80 26. 75 4.92 8. 74 9. 72 6.52 11.99

29.62 30.10 27. 72 5.66 7. 88 4.31 22.68 20.60

34.84 35.49 33.12 6.90 9. 74 7. 07 26.14 23.47

45.10 46.27 43.93 8.78 11.89 11.57 33.38 30.72

' C

o. 00

0.15

0.43

10. 20

11.73

15. 52

22.44

24.32

29.00

156

the most important of the three sensitivity measures, indicates the savings

which may be realized by properly designing a plan for an imperfect inspection

process.

Groups A and B utilized the same method of analysis but different

prior distributions, with resulting differences in all the performance measures.

The maximum difference in the numerical values of the data in groups A and B

were 10. 04%, 3. 82%, and 5. 41% for 1 , 2 , and 3, respectively.

I

This implies that the choice of a prior distribution affects the computation of an

optimal plan and its resultant predicted total expected cost. It was shown in

Chapter IV, for an error free inspection process, that a precise knowledge of

the mixed binomial parameters is required if the total expected cost is to be

accurately projected. The effect which the prior distribution has on the total

expected cost, TC(n, c)' , is illustrated for the example problems by . el,e2

the comparative values of 2 in Table VI-2. Recall that the mean, µ , and

variance, cr2 , were the same for both distributions; further, a parameter set

was computed for the mixed binomial prior that would result in approximately

the same TC(n, c) 0 0 as obtained with the Polya prior distribution. The value '

of 2 , for example problems, differed between the mixed binomial and Polya

prior distribution less than 4% for all error prone conditions. This difference

may seem inconsequential when one considers the uncertainty involved in the

accuracy of the cost factors and prior distribution parameters. The difficulty

of such an assumption lies in the fact that not only were the mean and variance

157

matched for the distributions of interest, but the parameters of the distributions

were chosen that would yield the same total expected cost for an error free

environment. Thus, it may be inferred that the difference in TC(n, c)' cl,e2

for assumed distributions will be inconsequential only when these stated con-

ditions are satisfied.

Groups B and C utilized different methods of analysis but the same

Polya prior distributions. The maximum differences in the numerical values

of the data in groups B and C were 4. 95% , 3. 57% , and :3. 53% for D,,. 1 , D.. 2 ,

and D.. 3 , respectively. The differences obtained by the two methods of analysis

were, however, substantially less for most error conditions. One could infer

from the results obtained, that the alternate method will yield an optimal plan,

(n,c)', whose total expected cost, TC(n,c)' , is sufficiently accurate el,e2

for most inspection situations. The method of analysis proposed in this research

is believed to be more representative of the inspection situation than the alternate

method, but is computationally more difficult and time consuming to evaluate.

Additional investigations were performed to better determine the

significance of the prior distributional shape. Three mixed binomial parameter

sets were chosen that represented distributions having the same mean and vari-

ance. The optimal plan, (n, c) ', and the total expected cost, TC(n, c)' , el,e2

were computed for selected error pairs, ( e1, e 2) • One of the mixed binomial

parameter sets was selected to represent the actual prior distribution. The

158

total expected cost for the error prone environment, TC(n c)" , was ' C C l' 2

obtained by using the seemingly optimal plans for the two remaining parameter

sets with the values of a , p1 , and p 2 for the actual prior. The resultant

TC(n, c)" was compared with TC(n, c)' . A similar evaluation was el,e2 el,e2

made with a Polya prior distribution. The Polya prior distribution was evaluated

with the optimal plan, ( n, c) ', obtained from each of the three mixed binomial

priors. Again, the resultant TC(n, c)" was compared with TC(n, c)' • el,e2 el,e2

It was observed that the assumption of the prior distribution does affect the

quality costs that will be incurred. In the case of a mixed binomial prior distri-

bution, the choice of parameters which do not precisely define the distribution's

shape will result in an added expense. A greater cost was experienced, in each

example, where the assumed plan from a mixed binomial prior was used with the

real Poly a prior.

Other considerations which affect the selection of an optimal plan and

its total expected cost are included in the text. Such considerations are too

numerous to enumerate in this section. It will suffice to note that a large

number of figures are provided throughout the text as aids to understanding the

subtleties inherent in the error prone inspection process.

In conclusion, this research has shown that errors in the inspection

process invariably result in additional quality costs. Analytical methods were

developed that are capable of partially compensating for the error effects.

159

However, it was not possible to obtain the optimal plans (n,c) and (n,c)' for

error free and error prone conditions, respectively, which would result in the

same total expected cost. This observation emphasizes the necessity of cou-

pling analytical techniques with quality actions that remove the source of

errors from an inspection process.

Areas for Future Research

The investigation of inspection errors, on both statistically and economi-

cally based quality plans, suggests that unnecessary quality costs are often

incurred. Even though the causes of errors may not be apparent, there is no

need to ignore such errors and suffer the penalties. Compensation can be

obtained that will partially offset the losses introduced by errors. An area

for future research, that would extend the investigations reported herein, invol-

ves the identification of valid input parameters. Very little information is avail-

able that identifies the type and extent of errors for different classes of items.

This information would be of immense value to the future researcher, as well

as the quality department that is desirous of investigating error compensated

plans but has only superfical knowledge of the expected error rates. A

further extension of the models developed in this research would be to depict

the error rates as random variables. This would be more realistic of an actual

inspection situation.

Throughout this research a single sample attribute acceptance plan was

employed. Only the sample size and acceptance numbers were treated as

160

decision variables. In many situations, other sampling schemes are more appro-

priate and different variables may comprise the decision problem. The analytical

methods, discussed herein, could logically be extended to include multiattribute

acceptance plans. The general approach is not necessarily confined to attribute

acceptance plans, but could be extended to encompass variables inspection

schemes as well. Further, this research has differentiated between statistically

based plans and economically based plans. A natural outgrowth of research in

these areas would be the coupling of techniques in the design of semi-economic

sampling plans. Both statistical and economic criteria would be the basis for

selecting the optimal plan that minimizes the total expected cost per lot formed

for inspection.

This investigation revealed that the choice of a prior distribution is

important. Often times, the mean and variance of the prior distribution is

known but the specific form of the mass function is unknown. Additional research

would be helpful that examines this situation to determine when a reliable answer

can be obtained for an assumed prior distribution.

One of the biggest handicaps in the implementation of an economic based

model is the acquisition of reliable cost parameters. Research efforts that would

identify a simple methodology or practices whereby such costs could be identified

accurately would be most beneficial. This would undoubtably involve the consider-

ation of elements other than quality department.

161

The research reported in this investigation has established a precedent

for dealing with the observed inspection results. This research and subsequent

extensions to other applications, could conceivably alter the entire approach to

quality control decision theory. Future researchers are encouraged to build

upon this research to explore new avenues of coping with error prone inspection

processes.

APPENDIX A

THE INCREMENTAL SEARCH PROCEDURE

162

163

The incremental search procedure used in Chapters IV and V to locate

an optimal sampling plan, for each of the selected example problems, is dis-

cussed in the paragraphs that follow. The method is not guaranteed to yield

an optimal plan due to the nature of the cost response surfaces; however, it

is believed that the solutions are nearly always optimal.

A unidimensional search technique is employed to evaluate a cost

model describing the total expected cost per lot formed for inspection. The

cost components comprising the total expected cost are defined in Equations 3. 33

and 3. 54 • A change in notation, from that used in the text, will be used to

simplify this discussion of the search procedure. The cost of acceptance,

Equation 3. 33, will be noted as ECA. The cost of rejection, Equation 3. 54 ,

will be noted as ECR. The decision variables involved in the search are the

number of items in a sample, n , and the acceptance number, c •

The total expected cost, TC, is first determined for a non-inspection

policy. Next, the sample size, n , is incremented by some predetermined

value INCR and TC is again calculated. A specific procedure is followed for

each value of n employed in the search technique. The number of observed

defectives, y , is assigned the value of zero; the values of ECA and ECR e

are computed. If ECA is less than ECR, the value of y is increased by 1. e

Again, the values of ECA and ECR are determined and ECA compared with ECR.

This procedure is repeated until a value of ECA is found that is greater than

ECR. The current value of y is then decreased by 1 and assigned as the e

164

acceptance number, c • Note that y can not exceed n • Should y be e e

incremented to the value of n and E CR remain less than E CA, the acceptance

number, c , is assigned the value of n • When the value of c has been

established, the sum of ECA for all y less than or equal to c is obtained. e

The procedure is then continued and the sum obtained for all ECR correspond-

ing to values of y from c+ 1 to n • Application of this procedure establishes e

the optimal c and the minimum TC for the specified n .

The value of n is again increased an increment, INCR and a new c

and TC determined. The procedure is continued until a TC is obtained at a

new n , which is greater than the TC at N-INCR. The search procedure is

continued, beginning at n-( 2) (INCR). The value of INCR is now redefined as

INCR - DELl . The stated procedure is continued until a new value of TC is

obtained at n , which is greater than the TC at n - INCR • The search pro-

cedure is begun anew at n - ( 2) (INCR) • The value of INCR is again redefined

as INCR - DEL2 • This terminal value of INCR is assigned the value of unity

since that is the smallest discriminating value which can be assigned to n • The

search procedure is again initiated and the search continued until the values of

n and c are located which yields the minimum TC. Ideally the search would

be terminated at this point.

Further checks are made to assure that TC for the seemingly optimal

plan, ( n, c) , actually represents the global minimum rather than a local

minimum. The n is increased positively and negatively in increments of 1

165

until the values of c obtained are greater than or less than the optimal c by

a specified value. When the local minimum obtained to each side of the sus-

pected global minimum is greater than the suspected global minimum, one can

safely assume the optimal sampling plan has been identified. If it is found that

the suspected global minimum is in fact a local minimum, the search is continued.

The value of n is incremented until a global minimum is obtained which has a

local minimum to each side that is greater than the global minimum. The search

is terminated at this point.

APPENDIX B

DEVELOPMENT OF RELATED FORMULAE

166

167

This appendix contains the development of formulae which are pertinent

to the research presented in the main text. The formulae describe

(i) the distribution of observed defectives in a sample, given the

actual defectives in the sample.

(ii) the expected value of observed defectives, in a sample, given the

actual defectives in a sample.

( iii) the expected value of observed defectives, in a sample, that are

binomially distributed.

(iv) the expected value of observed defectives, in a sample, that are

distributed per a mixed binomial distribution with two components.

Formulae (i) and (ii) were used in the reported research, but documented

in this appendix since their inclusion in the main text would not have added to

the clarity of the presented material. Formulae (iii) and (iv) were not

used in the main text but are included for future reference since they are

directly related to the subject material.

Distribution of Observed Defectives, in a Sample, Given the Actual Defectives in the Sample

The situation which will be considered consists of a sample of size

n in which there are exactly x defectives. The inspection is known to be

prone to type I and type II errors. The distribution of observed defectives in a

sample given the actual number of defectives in a sample is desired. A specific

situation where n==5 and x=3 will be used to identify a general equation for the

desired distribution.

168

Let the number of observed defectives in a sample, y , assume all e

possible values from O to n and write the corresponding probability of lot

acceptance. If

where

and

If

If

If

ye = 0 , then

P ( y I x) = P ( o I 3) e

e1 = type II error rate = probability of classifying a good item as being defective,

e 2 = type II error rate = probability of classifying a defective item as being good.

Ye = 1 ' then

2 2 3 (l-e1) e1 P(y ! x) = P (113) = 3 ( 1-e1) ( 1-e2) e 2 + 2e2 e

y = 2 ' then e

2 3 2 P(y Ix) = P ( 2 ! 3) = e e + 6e1 (1-e1) (1-e2)e2 e 1 2

2 2 + 3 (1-e1) (1-e2) e 2

Ye =3 ' then

P(y Ix) = P(3!3) e

If

Y = 4 then e '

P(y Ix) = P(4!3) e

If

Y = 5 then e ,

P(y Ix) e

169

The information available in the preceeding expressions is tabulated in

the following table to aid in identifying the distribution form. Note the high

degree of symmetry within the table.

Multification Exponent

Ye Factor el ( l-e1) e2 ( 1-e2)

0 1 0 2 3 0 1 2 1 1 3 0

3 0 2 2 1 2 1 2 0 3 0

3 0 2 1 2 6 1 1 2 1

3 1 0 2 0 3 3 2 0 2 1 6 1 1 1 2

4 2 1 1 0 3 3 2 0 1 2

5 1 2 0 0 3

170

From the data thus arranged, it is observed that the distributio,1 is of

the form

By trial and observation

1 (y Ix) = n e

min

It is desired to formulate a general expression from that derived

using a particular example. Substituting x and n-x for 3 and 2 , respectively;

in the preceeding equation yields

1 (y Ix) = n e

min [ x,y e]

i=max [y -(n-x), 0) e

The limits thus obtained are intuitively obvious since all exponents

are integers and must be positive to have meaning. Since the exponents

Ye - i 0 and x - i O, it follows that min (i) = 0 and max (i)

= min(x, Ye). Similarly for the remaining exponents,

171

n - x - ye + i 0 and i O. It follows that min(i) = max[ye-(n-x), O]

and max(i) = "". To satisfy the conditions simultaneously for

all four exponents, min(i) = max[ye-(n-x), O] and max(i) =

min(x, Ye).

The conditional distribution of Ye given x may also be determined

by considering the joint probability that i of the x truly defective

items are classified as defective and Ye - i of the n - x good items

are classified as defective. Since these are independent events, their

joint probability is the product of the two respective binomial mass

functions. Summing over all possible values of i, recognizing that i

must not exceed either x or Ye, that Ye - i must not exceed n - x, and

that i must be nonnegative, results in

min [x, Ye]

L i=max

[ye-(n-x), O]

( X) . i X-i i 0-ez) ez

which is identical to the expression for ln(Ye \ x) developed above.

E(y Ix) = e

=

=

172

Expected Value of Observed Defectives, in a Sample, Given the Actual Defectives in a Sample

n-x x e e ( ) ( ) y -i n-x-y + i

. . e1 ( 1-e1) Y -1 1 i=max e

[y -(n-x),0] e

x-i i • e ( 1-e ) 2 2

min n [x,y e1 I: I: (y -i)

0 . e y = 1=max e [y -(n-x),0]

e

)( ) y -i n-x-y +i

( n-x x e ( 1_ ) e

. . el el y -1 1 e

= '\"' X X-1 (l )1 X ( ) • • l.J . e2 -e2 . 0 1 1=

y -i n-x-y +i e e • e ( 1-e ) 1 1

Since

and

173

n-x L;

y -i=O e

(n-~) y -1 e

= 1

it follows that

Let

then

€ = 1 - e - e 1 2

Expected Value of Observed Defectives, in a Sample, That are Binomially Distributed

Given the distribution

n-y e (1-p) e

174

it follows that the

Let

E=l-e -e 2 1

then

E(y ) = ne1 + npE . e

Since

E(x) = np

it follows that

E(y ) = ne1 + E E(x) e

Expected Value of Observed Defectives, in a Sample, That are Distributed by a Mixed Binomial Distribution

With Two Components

Given the distribution

n-y (1-p ) e

e,1

175

it follows that the

E(y) = anp 1 + (1-a)np 2 e e, e,

Expanding

and collecting terms yields

APPENDIX C

DOCUMENTATION OF COMPUTER PROGRAMS

176

177

The computer programs that follow were used in the investigations

reported in the text of this dissertation. Each main program is preceded by

a brief description of its purpose and definition of terms. All programs were

processed with an IBM 7044 or IBM 7094 computer at the Marshall Space

Flight Center.

L *******~************** L MAIN PROGRAM NUMB~R l l C C L l. C l.

********************** ltfIS P~OGHAM AN INCREMENTAL SEARCH TU LOCATE THE orTIMAL S~MPLING ~LA~ ~~IICH WILL MINI~IZ~ THE EXPECTEU QUALITY COSTS PER l 1~SP~CTION LUT.lHE UI~TRiriUTir~ 0~ DtFiCTIVES IN THE SAMPLE A~D THE

JL5tRVED UfFECTIVL~ l~ THE SAMPLE ARE DESCRIBEU BY MlXEO Bl~OMIAL ~ITH TWLl COMPONENlS. COMPEN~~TION FOR

L, JNSPECTIO~ ERRORS IS PROVIDED. C l. C GtFINITION OF TERMS C Al=CUST PER ITEM ASSOCIATED WITH HA~DLING THE: L-"11 ITEMS NOT INSPECTED C IN-AN ACCfPTFO LOT L A2=COST PcR ITEM AS~OCIATED WITH A DEFECTIVE ITEM WHICH IS ACCEPTEU l. ALPHA=~EIGHTI~G fACTUR (PROBABILITY OF OCCURANCE) OF FIRST BINOMIAL L COMPU~f~T

AMAX=LO~~R LIMIT I~ THE DISTRIBUTION FUNCTION FOR OBSERVED DEFECTIV~S GIV~N THE ACTUAL UEFECTIVE5

i... A M I 1 ~-= u PP.: R LI M I T I I\J THE O I TR 1 BUT I Ll N FU i~ CTI ON F OR O 8 S ERV E D DEF E CT I V ES C GIVEN THE 4CTUAL DEFECTIVES l. C=ACCEPTANC~ ~UMbtR C ll=TYPE l l ·iSPcCTION tRKOR. \, E2=TYPi:: 2 11·-.SPtCT I ON t:RRuR C EC=~XPECTED CO~T UF SAMPLING C :- l. .'\ = t- X :> C T:: u C n S T ,-, t= AC C E P T AN CE c t: L 1~ = "= x P 1.: c T:: 1, cos r t1 F- Kt J t c r 1 ni,i C. t:FX=t:XPi:CTl::U VALUc: Uf- A SPEClfltU F-UNCTION GF THE ACTUAL DEFl::CTIVE~ l. Glvi_,-.i THf: iU~br:R OF uL~E=RVtll OEFECfIVf-S C t:.X=~APi:CTfiJ VALUt: CH- ALTUAL l>EF~CTIVt:S GIVEN THE r,iuM~ER OF \, UbS~RV~u U~F~CTIVfS

.... 00

C C (.,

C C C C. C C C (..

C C C C C (.,

C C C C

L=LOT SILE N=SAMPLc: SIZE Pl=FkACTIO~ DEFECTIVE Of FIRST BINOMIAL COMPONENT P2=FRACTIUN DEFECTIVt UF SECOND BINOMIAL COMPONENT PEl=FRACTI~,~ DEFECTIVE Of FIRST blNOMIAL COMPONENT WITH COMPENSATION FDR ERROR PE2=FRACTION UEFECTlVE UF S::COND BI1'40MIAL COMPONENT wlTH COMPENSATION FOR ERRUR PYEb=PRObAdILITY OF EXACTLY YE DEFECTIVES OCCURRING IN A SAMPLE AS CALCULATED FROM A MODIFIED ~IXEU BINOMIAL PYEC=PROOUCT OF THE PROBABILITY OF YE GIVEN KAND THE PROBABILITY OF X Rl=COST PER ITEM OF INSPECTING THE REMAJNl~G L-N ITEMS IN A REJECTED LDT R2=REPAIR COST ASSOCIATED WITH A DEFECTIVE ITEM IN THE REMAINING L-N OF A REJECTED LOT Sl=COST PER ITEM OF SAMPLING AND TESTING S2=K~PAIR COST FOR A DEFECTIVE ITEM FOUND IN SAMPLING X=ACTUAL DEFECTIVES IN THE SAMPLE YE=OBSERV~D OEFtCTIVtS IN THE SAMPLE

COMMO:--J N,YE COMMO~/BLOK1/X,AL,ALM,E2,XP1,XP2 COMMUN/bLOK2/PYEB,EX,ErX,L,Sl,S2 REAL L,N,INLR fJRMAT(4FlO.C,Fl0.2)

J FORMAT(4Fl0.7) 6 FURMAT(6Fl0.2) 9 FORMAT(lHi,4Fl:.O,Fl0.3//)

12 FDKMAT(jX,4Fl0.7//) 15 rURMAT(6F10.2)

123 FORMAT(3X,F7.0,10X,F7.0,14X,F9.2,l3X,F9.2, l3X,F9.2) 1~3 FURMAT(lHl,llHSAMPLE SILE,5X,l6HOPTIMAL ACC. N0.,5X,18HEXPECTED CO

1ST ACC.,5X,l8HlXPECTEU COST REJ.,5X,19HTOTAL EXPECTED COST//)

...

319 FORMAT(3X,F7.0,31X,F9.2,35X,F9.2) C READ INPUT DATA

Rc~D(5,o)Sl,S2,Al,A2,Rl,R2 2j COi"4TlNUE

REA0(5,l)L,INCR,DEL1,DEL2,ALPHA RcA0(5,3)Pl,P2,El,c2 IF(L)8o,8d,34

34 CONT I NUE WRITf.(6,9)L,INCR,DEL1,DEL2,ALPHA WRITE(o,12)Pl,P2,~l,E2 W~ITE(6,l5)Sl,S2,Al,AZ,Rl,R2 WRITE(o,193) J=O JJ =l "!=O.O PEl=FRCP(Pl,El,E2) PE2=FRCP(P2,El,E2) AL=ALOG(ALPHA) ALM=ALOG(l.O-ALPHA) EC~l=ALPHA*A2*L*PEl+(l.O-ALPHA)*A2*L*PE2 EC=fCNI WRITE(6,319)N,ECN1,EC N=!~+l.u

313 CJNTINUE K=G YE=O.O ECR=O.O ECA=O.u SYEB=O.O

13 CON.TI NUf: 1F(SYEb.GT.0.9999)GO TO 113 PYtB=PYEbL(PE1,PE2) SYt:b=SYE.8+PYEE1

.... I

IF(El.EQ.o.o.AND.E2.EQ.O.O)GO TO 24 IF(El.EQ.O.J)GO TO 53 X=O.O GO TO 63

53 X=YE oJ CONTINUE

EX=O.O E:FX=O.O SYEC=O.O

33 CONTINUE FX=FNCX(ALPHA,Pl,P2) XPl=XP(Pl) XP.:!=XP(P2) IF(El.EQ.O.O)GO TO 25 IF(E2.E~.O.O)GO TO 26 PYEC=PYf:C~(El) GO TO 43

25 PYEC=PYEC2(DUM) GO TO 43

2o PYEC=PYECl(El) 43 CONTINUE

CALL EXPV(FX,PYEC) SYEC=SYi::C+PYcC DELTA=PYtB-SYEC lF(DELTA.LT.O.OOOl)GO TO 93 IF(X.EW.N)GO TO 93

153 X=X+l.O GO TO 33

24 CU -.ITINUE X=YE FX=FNCX(ALPHA,Pl,P2) EF.X=FX 1:X=YE

9 j CONT I NUE:

... CII ...

IF(K.E~.l)GO TO 243 ACOST=COSTA(Al,A2,E2) RC05T=COSTR(A2,Rl,R2,El,E2) lf(ACOST.GT.RCOST)GO TO 233 ECA=ECA+ACOST

203 YE=YE+l.O GO TO 13

233 C=YE:-1.u K=l GO TO 253

243 RCOST=COSTR(A2,Rl,R2,El,E2) 253 ECR=ECR+RCOST

YE=Yl:+l.O GO TO 13

113 CONTINUE IF(K.E'1.0)C=N EC=ECA+ECR WRITE(6,l23)N,C,ECA,ECR,EC IF(J.EQ.O)GO TO 334 IF(J.EW.l)GO TO 293 IF(J.EQ.2)GO TU 333 IF(J.EQ.3)GO TO 333 IF(J.E~.4)GO TO 423

334 CONTINUE ,\J=N-1.C, J =J+ 1 GO TO 263

333 lf(EC.LE.REC)GO TO 263 lf(~.GT.RN.ANO.EC.~T.REC)GO TO 273 \l=N+INCR GO T0-313

29J r~rEC.LE.~ECJGO TO 263 RN=N-1 NCR

273 J=J+l \J=RN-1:~CR

.... GQ ..

IF(J.EQ.2)1NCR=INCR-DEL1· IF(J.E~.3)1NCR=INCR-DEL2 IFCJ.EQ.3)RRN=N JF(J.E~.4)GO TU 423 GO TO 313

263 R1:C=EC RN=N RC=C N=N+INCR JF(N.GT.L)GO TO 323 GO TO 313

423 CONTINUE IF(C.EQ.CRC+2.0))JJ=2 1F(JJ.EU.2)GO TO 433 N=I-J+l.O GO TO 313

433 CONTINUE IF(C.EQ.CRC-2.0))GO TO 323 RRN=RRN-1.0 N=RRN GO TO 313

323 CONTINUE GO TO 23

88 CONTINUE STOP END

... r:

C *********************** C MAIN PROGRAM NUMBER 2 C C C C C C C C C C L C C C C C C C C C C C C C C C C C C C C C

*********************** THIS IS THE MAIN PROGR~M FOR CALCULATING THE EXPECTED QU~LITY COSTS PER LOT FOR A GIVEN SAMPLIN~ PLAN. THE OISTRIUTION OF OEFECTI~ES IN THE SAMPLE ANO THE DISTRlbUTION OF OBSFRVED DEFECTIVES IN T~t SAMPLE ARE OESCRlbED BY MlXEO olNOMIAL DISTRIBUTIONS WITH TWO COMPO~~~Ts.

DEFINITION OF TERMS Al=COST PER ITEM ASSOCIATED WITH HA~DLING THE L-~ ITEMS NOT INSPECTED IN AN ACCEPTED LOT A2=COST PER ITEM AS~OCIATEO WITH A DEFECTIVE ITEM ~HICH IS ACCEPTED ALPHA=WtlGHTING FACTOR (PROBABILITY OF OCCJRANCE) OF FIRST BINOMIAL

COMPONENT AMAX=LOWtR LIMIT IN THE DISTRIBUTION FUNCTION FOR OBSERVED DEFECTIVES GIVEN THE ACTUAL DEFECTIVES AMIN=UPPER LIMIT IN THE DISTRIBUTION FUNCTION FOR OBSERVED DEFECTIVES GIVEN THE ACTUAL UEFECTIVcS C=ACCEPTANCE NUMBtR El=TYPE l INSPECTION ERROR E2=TYPE 2 INSPECTION ERROR EC=EXPECTEO COST OF SAMPLING ECA=EXPECTcU COST OF ACCEPTANCE tCR=EXPECTED COST OF REJECTION EFX=EXPcCTED VALUE OF A SPECIFIED FUNCTION OF THE ACTUAL DEFECTIVES GIVEN THE NUMBER OF OdSERVEu DEFECTIVES EX=EXPECTEO VALUE OF ACTUAL DEF~CTl¥ES GIV:N THE NUM6ER OF OBSERVED DEFECTIVES L=LOT SIZE N=SAMPLE SIZE Pl=FRlCTION DEFECTIVE OF FIRST BINOMIAL COMPONENT P2=FRACTION DEFECTIVE Of SECOND tlINOMIAL CJMPONENT

UEFECTIVE OF FIRST tllNOMIAL CJMPONENT WITH COMPENSATION

.... :

C FDR ERROR C PE2=FRACTION DEFECTIVE OF SECO~D BINOMIAL WITH COMPE~SATION C FOR ERROR C PYEB=PROBABILITY Of EXACTLY YE DEFECTIVES JCCURRING IN A SAMPLE AS C CALCULATEU FROM A MODIFIED MIXED BINOMIAL C PYEC=PROBA~ILlTY OF EXACTLY YE DEF~CTIVES OCCURRING IN A SAMPLE C GIVEN THAT EXACTLY X DEFECTIVES EXISTS IN THE SAMPLE C Rl=COST PER ITEM OF INSPECTING THE REMAINI~G L-N ITEMS IN A REJECTED C LOT C R2=REPAIR COST ASSOCIATED WITH A DEFECTIVE ITEM IN THE REMAINING L-N C OF A REJECTED LOT C Sl=COST PER ITEM OF SAMPLING ANU TtSTING C S2=REPAIR CO~T FOR A DEFECTIVE ITEM FOU~O IN SAMPLING C X=ACTUAL DEFECTIVES IN THE SAMPLE C YE=DBSERVED DEFECTIVES IN THE SAMPLE C

COMMON N,YF. COMMON/BLOKl/X,AL,ALM,E2,XPl,XP2

CDMMON/BLOK2/PYEB,EX,EFX,L,Sl,S2 REAL L,N,1

l FORMAT(3FlO.O,Fl0.2) 3 FORMAT(2Fl0.7) o FORMAT(6Fl0.2) 9 FJKMAT(lHl,l8X,lOHINPUT OATA/l9X,l~H----------//4X,15HCOST PARAMET

lE~S,lOX,22HSTATISTICAL PARAMETEKS/4X,15H---------------,lOX,22H---2-------------------//4X,4HSl=S,Fl0.2,14X,2HL=,4X,Fl0.0/4X,4HS2=$,f 3l0.2,14X,2H~=,4X,FlC.0/4X,4HAl=j,rl0.2,14X,2HC=,4~,Fl0.0/4X,4HA2=S 4,Fl0.2,l4X,6HALPHA=,FlD.3/4X,4HRl=S,Fl0.2,14X,3HPl=,3X,Fl0.7/4X,4H ~R2=S,Fl0.2,14X,3HP2=,3X,Fl0.7)

12 FJRMAT(lHl,2X.l7H( El , E2 ),4X,18HEXPECTED COST ACC.,4X,18H lEXPECTED COST ~EJ.,4X,l9HTOTAL EXPECTEO COST,5X,14HPERCENT CHANGE/ 2/)

15 FQRMAT(3X,lH(,F5.2,2X,lH,,F5.2,2X,lH),6X,F9.2,l3X,F9.2,l4X,F9.2, 113X,F7.2)

I

223 f-ORMAT(l4) C READ INPUT DATA

READ(5,l)L,N,C,ALPHA READ(5,6)Sl,S2,Al,A2,Rl,R2 REA0(5,3)Pl,P2 WRITE(6,9)Sl,L,S2,N,Al,C,A2,ALPHA,Rl,Pl,R2,P2 WR.ITE(6,12)

C READ THE NUMBER OF ERROR PAIRS TO BE EVALUATED READ(5,223)K J=l

313 CONTINUE READ(5,3)El,E2 YE=O.O ECR=O.O t:CA=O.O SYEB=O.O PEl=FRCP(Pl,El,E2) PE2=FRCP(P2,El,E2) AL=ALOG(ALPHA) ALM=ALOG(l.O-ALPHA)

13 CONTINUE IF(SYEB.GT.0.9999)~0 TO 113 PYEB=PYE8l(PE1,PE2) SYtB=SYEB+PYEB IF(SYEB.LT.C.OOOl)GO TO 203 IF(El.EQ.O.O.AND.E~.EQ.O.O)GO TO 24 IF(El.EQ.O.O)GO TO 53 X=O.O GO TO 63

53 X=YE 63 CONTINUE

EX=O.O EFX=C.O SYEC=O.O

... =

33 CO~TINUE FX=F~CX(ALPHA,Pl,P2) XPl=XP(Pl) XP2=XP(P2) IF(El.EQ.O.O)GO TO 25 IF(E2.E~.O.O)GO TO 26 PYEC=PYEC.3(El) GO TO 43

25 PYEC=PYEC2(DUM) GO TO 43

26 PYEC=PYECl(Elt 4J CONTINUE

CALL EXPV(FX,PYEC) SYEC=SYEC+PYEC DEL TA =PYEB-SYEC IF(OELTA.LT.O.OOOl)GO TO g3 IF(X.E~.N)GO TO 93

153 X=X+l.O GO TO 33

24 CONTINUE X=YE FX=FNCX(ALPHA,Pl,P2) EFX=FX EX=YE

'-13 CONTINUE IF(Y~.GT.C)GO TO 103 COST=COSTA(Al,A2,E2) ECA=ECA+COST YE=YE+l.O GO TO 13

103 COST=COSTk(A2,Rl,R2,El,E2) ECR=ECR+COST

203 YE=YE+!.O GO TO 13

11.3 CONT 1 N lJE

.... •

C CALCULATE THE TOTAL EXPECTED COST OF SAMPLIN; INSPECTION EC=ECA+ECR IF(El.EQ.O.O.AND.E2.EQ.O.O)REC=EC DELl=((cC-REC)/KEC)*luO.O WRITE(o,15)El,E2,ECA,EC~,EC,DEL1 IF(J.EQ.K)GO TO 323 J=J+l GO TO 313

323 CONTINUE RETURN END

C ********************** SUBROUTINE EXPV(FX,PYEC)

C ********************** COMMON/BLOKl/X COMMON/BLOK2/PYEB,tX,EFX IF(X.GT.O.O)GO TO 133 EXX=O.O GO TO 143

C CALCULATE THE EXPECTED VALUE OF ACTUAL DEFECTIVES GIVtN THE NUMBER C OF OBSERVED DEFECTIVES

133 EXX=(X*PYfC)/PYEB E:X=EX+EXX

C CALCULATE TH= EXPECTEO VALUE OF THE SPECIFIED FuNCTION OF ACTUAL DEFECTIVES C GIVEN THE NUMBER OF OtlSERVEO DEFECTIVES

143 EFXX=(FX*PYEC)/PYEb EFX=EFX+EFXX RETURN ENU

01> CD

C ********************** FUNCTION PYEBl(PEl,PE2)

C ********************** C CALCULATE THE PROBABILITY OF VE DEFECTIVES I~ A SAMPLE OF SIZE N

COMMON N,YE COMMON/BLOKl/DUM,AL,ALM REAL N YEB=ALGAMA(N+l.O)-ALGAMA(N-YE+l.0)-ALGAMA(YE+l.O) YEPl=YE*ALOG(PEl)+(N-YE)*ALOG(l.O-PEl) YEP2=YE*ALOG(PE2)+(N-YE)*ALOG(l.O-PE2) YEBl=AL+YEB+YEPl YcB2=ALM+YEB+YEP2 YE 1.>l = E XP ( YE B 1 ) YEb2=EXP(YEtl2) PYEtH=YEBl+YEB2 RETURN END

C ********************** FUNCTION PYECl(El>

C ********************** C CALCULATES PYEC WHEN TYPE l ERRORS ONLY ARE PRESENT.

COMMON N,YE COMMON/tiLOKl/X,AL,ALM,DUM,XPl,XP2 REAL N YECw=ALGAMA(N+l.O)-ALGAMA(N-YE+l.O)-ALGAMA(YE-X+l.O)-ALGAMA(X+l.O)

l+(YE-X)*ALUG(El)+(N-YE)*ALOG(l.O-El) YECl=AL+YcCW+XPl YE~2=ALM+YECW+XP2 YE.Cl=EXP(Y!::Cl) YEC2=EXP(YEC2) YEC=YECl+YEC2 PYl:::Cl=YEC RETURN E~o

.... oe -

C ********************** FUNCTION PYEC2(DUM)

C ********************** C CALCULATES PYEC WHEN TYPE 2 ERRORS ONLY ARE PRESENT.

COMMON N,YE CuMMON/BLOKl/X,AL,ALH,E2,XPl,XP2 REAL N YECW=ALGAMA(N+l.0)-ALGAMA(N-X+l.O)-ALGAMA(X-YE+l.O)-ALGAMACYE+l.O)

l+(X-YE)*ALOG(E2)+VE*ALOG(l.O-E2) YECl=AL+YECW+XPl YEC2=ALM+YECW+XP2 YECl=EXP(YEC.l) YEC2=EXP(YEC2) YEC=YECl+YEC2 PYEC2=YEC RETURN END

C ********************** FUNCTION PYEC3(El)

C ********************** t CALCULATES PYEC WHEN TYPE 1 ANU TYPE 2 ERRORS ARE PRESENT.

COMMON 'i,YE CU~MON/BLOKl/X,AL,ALH,E2,XPl,XP2 REAL N, I PYi::C=O.O CONST=l.OE+36 AMAX=AMAXl(YE-(N-X),Q.O) AMIN=AMINl(X,YE) I=AMAX

23 CONTINuE A=N-X-Yl:+l+l.O 6=YE-I+l.O F=X-1+1.0 O=N+l.Q i:=l+l.C

.... '°· 0

YECW =ALGAMA(D)-ALGAMA(A)-ALGAMA(B)-ALGAMA(F)-ALGAMA(E)+(B-1.0)* 1ALOG(El)+(A-l.ul*ALOG(l.O-El)+(F-l.O)*ALOG(E2)+1*ALOG(l.O-E2)

YECl=AL+YECW+XPl+ALOG(CONST) YtC2=ALM+YECW+XP2+ALOG(CONST) IF(YEC1.LT.62.89)GO TO 13 YECl=EXP(YECl)/CO~ST GO TO 15

13 YEC.:l=O.O l~ IF(YEC2.LT.62.89)GO TO 14

YEC2=EXP(YEC2)/CONST GO TO 16

14 Yi:C2=0.0 lb YEC=YECl+YEC2

PYEC=PYEC+YEC IF(I.EQ.AMIN)GO TO 43 I=I+l.O GO TO 23

43 CONTINUE PYEC3=PYEC 'tETURN END

C ********************** FUNCTION FNCX(ALPHA,Pl,P2)

C *•******************** C CALCULATE THt VALUE OF THE SPECIFIED FUNCTIO~ OF ACTUAL DEFECTIVES C FOR A GIVEN VALUE OF X

COMMON N CO~MON/SLOKl/X REAL N DOUBLE PRECISION FX1,FX2,FX3,FX4 CUNST=l.Ot:+36 FXl=ALOG(ALPHA)+(X+l.O)*ALOG(Pl)+(N-X)*ALOG(l.O-Pl)+ALOG(CONST) FX2=ALOG(l.O-ALPHA)+(X+l.O)*ALOG(P2)+(N-X)*ALOG(l.C-P2)+ALOG(CONST

1 )

.... ., ...

FX3=ALOG(ALPHA)+X*ALO~(Pll+(N-X)*ALOG(!.O-Pl)+ALOG(CON~T) FX4=ALOG(l.,:-ALPhA)+X*ALOG(P2)+(N-X)*ALOG(i.J-P2)+ALOG(CO~ST) FXl=EXP(FXl) FJ<2=EXP(FA2) FX3=EXP(F/43) FX4==~XP(FX4) FNCX=(FXl+FX2)/(FXJ+~X4) RETURN E i'JO

C ********************** FUNCTIO~ FRCP(P,tl,E2)

C **********~*********** FRCP=P'-'(l.J-E2)+(l.O-P)*El Rt:TUR~ E 1--J D

C *~******************** F U i\j C T I CJ N X P ( P )

C ********************** COMMON N COMMON/BLJKl/X REAL N XI-' = X * ALO G ( P ) + ( I~ - X t *ALO G ( 1 • -:, - P ) RETUR~ E~~u

ts:,

C ********************** FUNCTION COSTA(Al,A2,E2)

C ********************** C CALCULATES THc COST OF ACCEPTANCE WHEN THE OBSERVED DEFECTIVES ARE LESS C THA~ OR EQUAL TO THt ACCEPTANCE NUMBtR.

COMMON N,Yt CUMMUN/~LUK2/PYE8,tX,tFX,L,Sl,S2 REAL L,~ COSTA=PYE8*(N*Sl+Yt*S2-A2*E2*EX+(L-N)*Al+(L-N)*A2*EFX) RETURN E~D

C ********************** FUNCTION CUSTR(A2,Rl,R2,El,E2)

C ********************** C CALCULATES THE COST OF REJECTION WHEN THE OBSERVED DEFECTIVES ARE GREATER C THAN THE ACCEPTANCE NUMBER. ·

COMMON N,YE LOMMON/BLOK2/PYE~,EX,EFX,L,Sl,S2 REAL L,N COSTR=PYEB*(N*~l+Yt*(S2-R2)+(L-N)*Rl+(E2*A2+R2*(1.0-El-E2))*EX+

ll*El*R2+(E2*A2+R2*(1.0-El-E2))*(L-N)*EFX) RETURN END

... a

L ****************** C MAI~ PROGRAM NUMBER 3 C ****************** C THIS PROGRAM CALCULATES THE PARA~ETERS FOR THE Ml(ED til~OMIAL C OIST~IHUTIO~ WHEN THE FIRST ANO SECOND MOME~T~ ARE SPECIFIED. L C C C C C C C C C C C C

****************** UEFINITION OF TERMS

ALPHA=~EI;HTI~G FACfOR A~SW=THE CALCULATEu SECOND MOMENT UELTA=OIFFERENCE BETWEEN THE CALCULATED SECONU MOMENT AND THE DESIRED SECOND MOMENT EAN=THE DESIRED FIRST MOMENT Pl=THE FRACTION DEFECTIVE OF THE FIRST BINOMIAL COMPONENET P2=THE FRACTION DErECTIVE OF THE StCONO BINOMIAL COMPONENT RKl=THE RATIO OF P2 TO Pl RK2=TH~ RATIO OF (l.O-P2) TO (l.O-Pl) S=SIZE OF LOT VAR=THE DESIRED SECOND MOMENT

C ****************** l

1A,9X,2HK1,12X,2HK2) 2 FORMAT(4X,F6.0,3X,Fb.3,2Fll.8,3X,Fl2.7,3X,Fl2.7,3X,Fl0.5,3X,Fl0.5) 5

jQ FORMAT(l3,Fl0.3,Fl0.6,Fl0.6,FlO.b,Fl0.3) bl FORMAT(lHl)

REA0(5,5)S,EAN,VAR 3 CONTI~uE

READ (5,50)IL,OELA,Pl,PF,OELP,ALPHA IF(ll)88,88,4

4 CONTINUE WRITt:(6,6!) WRITE(6,5)S,EAN,VAR WRITE(6,50)1L,DELA,PI,PF,OELP,ALPHA WRITE(6,l)

... :

JL=(PF-Pl)/DELP DO 10 l=l,IL Pl=PI DO 2C J=l,Jl P2=((EAN/Sl-(ALPHA*Pl))/(l.O-ALPHA) A=S*(ALPHA$Pl*(l.O-Pl)+(l.C-ALPHA)~P2*(1.)-P2)) d=S**2.C*(ALPHA*(AciS(Pl-ALPHA*Pl-(l.O-ALPHA)*P2))**2•C) C=S**2.0*((1.0-ALPHA)*(ABS(P2-ALPHA*Pl-(l.O-ALPHA)*P2))**2•0) Ar'iSW =A+B+C OELTA=ANSw-\/AR RK1=P2/Pl RK2=(1.0-P2)/(l.O-Pll IF(UELTA.GT.200.0)GO TO 20 IF(OELTA.LT.(-200.0))GO TO 20 WRITE(6,2)S,ALPHA,Pl,P2,ANSW,DELTA,RK1,RK2

20 Pl=Pl+DELP 30 CONTINUE 10 ALPHA:ALPHA+DELA

GO TO 3 88 CONTINUE

STOP ENO

... u, OI

C *********************** C MAIN PROGRAM NU~BER 4 C *********************** C C THIS PROGRAM CALCULATES THE PROBABILITY OF LOT ACCEPTANCE, AVERAGE C OUTGOING QUALITY ANG AVERAGE TOTAL INSPECTION FCR INSPECTION SCHE~ES C SUBJECT TO INSPECTION ERRORS. RECTIFICATION W(TH REPLACE~ENT AND C RECTIFICATION wlTHOUT REPLACEM=NT IS CONSIDERED. C

8 FORMAT(12,I3,12) 19 FORMAT(I2) 65 FORMAT(2F3.2) 90 FORMAT(9X,F6.3,5X,F7.4,7X,F7.4,7X,F8.2,6X,F7.4,7X,F8.2)

120 FORMAT(lhl,9X,3HSL=,F6.0,3X,2HN=,I3,3X,2HC=,12,3X,3HE1=,F5.2,3X, l3HE2=,F5.2//10X,8HINCOMING,3X,6HP(ACC),8X,SHAO~WR,9X,5HAT[WR,9X.6H 2AOQWOR,8X,6HATIWOR/1CX,7HQUALITY//)

INTEGER C,D,l,Y C INPUT THE NUMBER OF SAMPLING PLANS TO BE EVALUATED (Z).

READ(5,l':1) l SL=4000. Y=l

29 CONTINUE C INPUT THE NU~BER OF ERROK DATA CARDS (M), SAMPLE SIZE (N), ANC THE C ACCEPTANCE NUMHcR (C).

READ(5,8) M,N,C L=l

15 CONTINUt: C INPUT TYPE l ERROR (El) AND TYPE 2 ERROR (E2).

REA0(5,65) El,E2 WRITE(6,l20)SL,N,C,El,t2

C INCREMENT INCOMING QUALITY FROM C TO .10 IN STEPS OF .005. P=O. DO 50 K=l,20C D=C+l

... i

C CORRECT THE INCOMING QUALITY FOR ~RROR. PE=((l.-t2)~P)+(tl*(l.-P)) SN=N IF(PE.EQ.0.0)GO TO 33 BINOM=O. BINOX=O.

C CALCULATE THE PROBABILITY OF LOT ACCEPTANCE DO 2C 1==1,0 X=l-l iF(X.GT.0.01 GO TO 3C BINOM=(l.O-PE)**SN GO TO 40

30 CONTINUE BINOX=ALGAMA(SN+l.0)-ALGAMA(SN-X+l.O)-ALGA~A(X+l.O)+X*ALOG(PE)+

l(SN-X)*ALOG(l.O-PE) BINOX=EXP(dINOX)

40 BINOM=BINOM+BINOX 20 CONTINUE

GO TO 35 33 BINOM=l.O 35 CONTINUE

C AVERAGE OUTGOING QUALITY WIT~ REPLACEME~T AO~WR=(SN*P*E2+P*(SL-SN)*(l.-PE)*B1NOM+P*(SL-SN)*(l.-BINOM)*E2)/(S ll*(l.-PE))

C AVERAGE OUTGOING QUALITY WIThOUT REPLACE~ENT/REPAIR AOQWOR=(SN*P*E2+P*(SL-SN)*BINOM+P*(SL-SN)*(l.-BI~CM)*E2)/(SL-SN*PE

1-(l.-BINOM)*(SL-SN)*PE) C AVERAGE TOTAL l~SPECTIO~ WIT~OUT REPLACE~ENT/REPAIR

ATIWOR=SN+(l.-BINOM)*(SL-SN) C AVERAGE TOTAL l~SPECTION WIT~ REPLACEMENT

ATIWR=ATlWOR/(l.-PE) · WRITE(6,SC)P,8INOM,AOCwR,ATI~K,A0QwOR,ATlwOR P=P+.005 IF(BINOM.LT •• OCl) GO TO 47

50 CONTINUE

... • --a

47 CONTINUE L=L+l IF(L.LE.M) GO TO 15 Y=Y+l IF(Y.LE.l) GO TO 29 STUP

_ END C *************~******** C MA PROGRAM NUMt3~R 5 C ********************** C (,

C C C l (.

THIS PROGRAM CONDUCTS AN I~CREMENTAL SEARCH TO LOCATE THE OPTIMAL ~AMPLING PLA~ WHICH WILL MINIMIZE THE EXPECTED QUALITY COSTS PER INSPECTION LOT.THE O[STRl8UTION OF DEFECTIVES IN THE ~AMPLE A~D THE DISTRIBUTION Of OBSERVED UEFECTIVE~ IN THE SAMPLE ARE DESCRIB~D BY THE PDLYA UISTRIBUTION. COMPENSATION FOR INSPECTION ERRORS IS PROVIDED.

C DEFINITION OF TERMS L Al=COST PtR ITEM AS50CIATED WITH HA~DLING THE L-N ITEMS NOT INSPECTED C IN AN ACCEPTFD LOT C A2=COST PER ITEM ASSOCIATED WITH A DEFECTIVE ITEM WHICH IS ACCEPTED L AMAX=LOWER LIMIT IN THE D15TRIBUTION FUNCTION FOR OBSERVED DEFECTIVES L G I Vt: N T E A C TU A L U E F E C T I VE S C AMIN=UPPt:R LIMIT I," THE DISTRIBlJTIO~ FUNCTION FOR Jt3SERVED DEFECTIVES C GIVEN THc A:TUAL DEFtCTIVES C C=ACCEPTA~CE NUMBER C tl=TYPE 1 l!\JSPECTION ,::RRO~ C E2=TYPE 2 INSPECTION ERRO~ C E:=tXPECTED COST OF 5AMPLING C ~CA=EXPrCT~u COST Of ACCEPTANCE C tCK=EXPECTED COST Of REJECTION C EFX=EXPECTEL1 VALUE OF A SPECIFIED FlJNCTION OF THE ACTUAL DEFECTIVES C GIVEN THc NUMtiER OF OoSERVED DEFECTIVES C EX=LXPECTEu VALUE fJF ACTUAL DEFECTIVES GIV=-'4 THE NJMBER OF L OBSERVED UEFECTIVES

... ., OD

C L=L(lT SIZE ~=SAMPLE SIZE

C PEl=MEAN 08SERVEO F~ACTION OEFECTIVt C PYEo=PROBA81LITY OF EXACTLY VE DEFECTIVES JCCURRING IN A SAMPLE C PYEC=PRODUCT OF THE PROBABILITY OF Ye GIVE~ X AND THE PROBABILITY OF X L Rl=COST PER ITEM OF I~SPECTING THE REMAINl~G L-N ITEMS I~ A REJECTED

' C LOT L R2=~EPAIR CCJST ASSOCIATED WITH A DEFECTIVE ITcM IN THE ~EMAINl~G L-N C OF A REJ~CTED LOT C S=POLYA UISTRIBlJTION PARAMETER C Sl=COST PER ITEM OF SAMPLING AND TESTING C S2=~EPAIR COST FOR A DEFECTIVE ITEM FOU~D I~ SAMPLING

T=POLYA DISTRIBUTION PARAMETER C X=ACTUAL DEFECTIVES IN THE SAMPLE C YE:OBSf:R\lED lJEFECTIVf:S I~ THE SAMPLE

COMMON ~,YE COMMON/dLOKl/X,c2,~,T COMMON/BLOKZ/PYEB,EX,tFX,L,Sl,S2

E: A L L , i\J , I NC R i FORMAT(4FlO.O,FlO.~) 3 FukMAT(2Fln.2,2Fl0.7) c.; FORMAT(6Fl0.2) 7 FO~MAT(8F~.2,Fl0.2,IlJ) 8 FORMAT(5X,8F5.2,FlO.i,IlO) 9 FJRMAT(lHl,18X,lOHINPUT DATA/l9X,~3H----------//4X,15HCOST PARAMET

1EKS,10X,22HSTATIST1CAL PARAMETERS/4X,15H---------------,lOX,22H---2-------------------//4X,4HSl=$,Fl0.2,14X,2HL=,4X,Fl0.0/4X,4HS2=S,F 3l0.2,14X,3HE1=,3X,Fl0.2/4X,4HAl=$,Fl0.2,14X,3HE2=,iX,Fl0.2/4X,4HA2 4=$,Fl0.2,14X,2HS=,~X,Fl0.0/4X,4HRl=$,Fl0.2,i4X,2HT=,4A,Fl0.0/4X,4H 5R2=$,Fl0.2.////)

123 FJRMAT(3X,F7.0,luX,F7.0,14X,Fg.z,13x,F9.2,13X,F9.2) 193 FORMAT(lHl,llHSAMPLE SIZE,5X,l6HOPTIMAL ACC. N0.,5X,18HEXP~CTEO CO

1ST ACC.,5X,18HEXPECTED COST REJ.,5X,l9HTOTAL EXPECTED COST//)

... I

319 FJRMAT(3X,F7.0,31X,F9.2,35X,Fg.z) C READ INPUT DATA

RE~D(5,6)S1,S2,Al,A2,Rl,R2 RcAD(5,7)AL,UL,CL,UL,EL,FL,GL,HL,TOL,ITFI~

23 C,01'JTI/\JUt: c AD ( 5 , l ) L , Ii~ CR, u EL 1 , DEL 2

REA0(5,3)S,T,~1,E2 IF(S)8b,8d,34

34 Co~'! TI "4UE W~ITE(6,9)Sl,L,S2,El,Al,E2,A2,S,Rl,T,R2 ~RITE(b,8)AL,BL,CL,DL,EL,FL,GL,HL,TOL,ITFI~ WRITE(o,1Y3) J=O JJ=l ~=o.o Pl=(S/(S+T)) PEl=FRCP(Pl,El,E2> ECNI=A2*L*PE1 c:C=EC"4I WRITt(o,3i9)N,ECNI,EC \J="J+l .c

313 C01~TINUE K=O YE=O.O Ec~=o.o ECi.\=O.C SYEb=O.O

13 CONTI;~UE IF(SYEB.GT.0.9Q99)GO TO 113 IF(El.EQ.~.O.AND.E2.EQ.O.O)GO TO 55 RI~T=GAUS~(AL,BL,S,T,El,E2,YE,TOL,ITFIN)+~AUSS(CL,DL,S,T,El,E2,YE,

lTOL,ITFIN)+GAUSS(EL,FL,S,T,El,f2,YE,TOL,ITFIN)+GAUSS(GL,HL,S,T,El, 2E~,YE,TUL,ITFIN)

I

IF(RINT.E~.0.0)~0 TO 113 TIME=ALGAMA(N+l.O)-ALGAMA(N-YE+l.O)-ALGAMA(YE+l.O)+ALGAMA(S+T)-

lALGAMA(S)-ALGAMA(T) PYEB=(EXP(ALOG(Rl~T)+TIME)J/(l.CE+30) GO TO b~

5~ PYEB=ALGAMA(N+l.O)-ALGAMA(N-YE+l.0)-ALGAMA{YE+l.O)+ALGAMA(S+YE) l+ALGAMA(T+N-YE)+ALGAMA(S+T)-ALGAMA(S)-ALGA~A(T)-ALGAMA(S+T+~)

PYEB=EXP(PYEU) o.:> cor\JT I "JUt:

SYE8=SYE8+-PYEB IF(El.Eu.o.O.ANO.E2.EJ.O.O)GO TO 24 IF(El.EQ.8.0)GO TO 53 x=o.o GO TO 63

~3 X=YE 63 CONTINUE

EX=O.O EFX=O.O SYf.C=O.O

33 CONTINUE F X = F -~ C ;< ( S , T ) IF(El.tQ.O.O)GO TO 25 IF(E2.FQ.J.u)GO TO 26 PYEC=PYcC3(El) GD TO 43

i~ PYEC=PYEC2(DUM) GU TO 43

2b PYcC=PYECl(El) 43 CO\JTINUE:

CALL EXPV(FX,PYeC) SYEC=SYEC+PYt:C OELTA=t->Yt:o-SYEC IF(DELTA.LT.O.OOul)GO TO 93 IF(X.E:.N)Gu TO 93

....

153 X=X+l.C GU TO 33

241- CONTINUE X=YE FX=FNCX(S,T) EFX=FX EX=YE

9 3 C U i\J T I NU t: lF(K.EQ.l)GO TU 243 ACOST=LOSTA(Al,A2,E2) RCOST=COSTR(A2,Rl,R2,El,E2) IF(ACUST.GT.RCOST)GO TO 233 ECA=ECA+ACOST

20J YE=YE+l.O GO TO 13

23J C=YE-1.0 K=l GO TO 253

243 RCO~T=COSTR(A2,Rl,R2,El,E2) 253 ECR=ECR+RCOST

YE=YE+l.O GO TO 1"3

113 CO~Tl~lJt: IF(K.Et.c.O)C=N EC=ECA+ECR WK1TE{6,l23JN,C,ECA,ECK,EC IF(J.EQ.O)GO TO 334 IFCJ.ElJ.l)·GO TO 21~ IF(J.E~.2)GO TO 33~ IF(J.EW.3)GU TO 333 IF(J .• EU.4)GO TO 423

3; 4 CO i~ TI N lJE ~~:N-1.0 J=J+l

0 N

~OTO 263 333 lF(EC.LE.REC)GO Tu 2oJ

IF(N.GT.R~.A~O.EC.GT.REC)GO TO 273 N::N+INLR GU TO 313

293 IF(EC.LE.RfC)GO TO 263 R~=~-l~CR

27J J=J+l \J=RN-INCR IF(N.LT.C.u)N=O.O IF(~.LT.O.O)RN=l.O IF(J.EQ.2)INCR=INCR-DEL1 IF(J.EQ.3)1NCR=INCR-DEL2 IF(J.EQ.3)RRN=N IF(J.c~.4.AND.RN.E~.1.Q)GO TO 3l3 If(J.EW.4)GO TO 423 \J=N+INCR GO TO 313

263 REC=EC RI~ =i\J RC=C N=r~+INCR IF(N.GT.L)GO TO 323 GlJ TO 313

4i3 CO\JTINUE IF(C.EW.(RC+2.0))JJ=2 IF(JJ.EQ.2)GO TU 433 N = r,J + 1 • Ci GO TO 313

43J CfJ1·~T I NU[ IF(C.E~.(RC-2.0))GU TO 323 RRN=RRN-1.0 ~=RKN

I

GO TO jl3 323 CO\JTl'\JUE

GO TO 23 88 CUi'"4T I NUE

STOP ENO

C *************¥********* C MAIN PROGRAM NUMBER 6 C C C C C C C

*********************** THIS IS THE MAIN PROGRAM FOR CALCULATING THE EXPECTED QUALITY COSTS PER LOT FOR A GIVEN ~AMPLING PLAN. THE OISTRlUTION OF DEFECTIVES IN THt SAMPlt AND THE DISTRIBUTION OF OBSERVED DEFECTIVES IN THE SAMPLE ARE DESCRIBED BY A POLYA PRIOR DISTRIUUTION.

C u t: F I ,~ I T I O ,'J OF TE R MS C Al=CDST PER ITEM ASSOCIATED WITH HANDLING THE L-~ ITEMS NOT INSPECTED C IN AN ACCEPTED LOT C A2=COST PER ITEM ASSOCIATED WITH A DEFECTIVE ITEM WHICH IS ACCEPTEU C AMAX=LOWER LIMIT 1:~ THE DISTRIBUTION FUNCTION FOR OBSERVED DEFECTIVES C GIV:N THE ACTUAL OEFECTIVE5 C AMIN=UPPER LIMIT IN THt DISTRIBUTION FUNCTION FOR OBSERVED DEFECTIVES C GIV~N THE ACTUAL OEFFCTIVES l C=ACCEPTANCE NUMBER C El=TYP~ l INSPECTION ERROR C E2=TVPE 2 INSPECTION ERROR C EC=cXPECTEi) COST OF SAMPLIN(; C E C A = E X P EC T t: 0 C OS T OF AC C E P T A :-.JC E: C ECR=EXPECTED CUST Of REJECTION C EFX=E-XP~CT~O VALUE OF A SPECIFIED FUNCTION OF THE ACTUAL DEFECTIVE~ C GIVEN THE NUMBER OF OBSERVED DEFECTIVES C EX=~XPECTE) VALUE 0~ ACTUAL DEFECTIVES GIVEi~ THE NUMBER OF C OBS~RVEO DEFtCTIVES C l=LOT Silt

I

C N=SAMPLE Sile C PYEb=PRUuA8ILITY OF EXACTLY YE DEFECTIVES OCCURRING IN A SAMPLF C PYtC=PRODUCT OF THc PR08ABILITY OF YE GIVEN X AND THE PROBABILITY OF X C Rl=COST PER ITtM OF INSPECTING THE REMAINING L-N· ITEMS IN A REJECTED C LOT C R2=REPAIR COST AS50CIATEO WITH A DEFECTIVE ITEM IN THE REMAINING L-N C OF A REJECTED LOT C S=POLYA DISTRIBUTION PARAMETER C Sl=COST PER ITEM OF SAMPLING AND TESTING C S2=REPAIR COST FOR A DEFECTIVE ITEM FOUND I~ SAMPLING C T=POLYA DISTRIBUTION PARAMETER C X=ACTUAL DEFECTIVES IN THE SAMPLE C YE=fl8SERVED DEFECTIVES IN THE SAMPLE

COMMON N,YE COMMON/8LOK1/X,E2,S,T COMMON/bLOK2/PYEB,EX,cFX,L,Sl,S2 REAL L,N

l FORMAT(3FlO.O,Fl0.2) _j FORMAT(2F1.0.7) o FURMAT(oFl0.2) 7 FORMAT(8F5.2,Fl0.2,Il0)

FORMAT(1Hl,18X,10HINPUT OATA/l9x,lOH----------//4X,15HCOST PARAMET 1ERS,10X,22HSTATISTICAL PARAMETERS/4X,lSH---------------,lOX,22H---2-------------------//4X,4HSl=S,Fl0.2,14X,2HL=,4X,Fl0.0/4X,4HS2=$,F 31C.2,14X,2HN=,4X,FlO.0/4X,4HAl=$,FlO.2,14X,2HC=,4X,FlO.O/4X,4HA2=S 4,Fl0.2,14X,2HS=,4X,Fl0.0/4X,4HRl=i,Fl0.2,14X,2HT=,4X,Fl0.0/4X,4HR2 5=1-,Fl0.2)

12 FORMAT(lHl,2X,17H( El , EZ ),4X,18HEXPECTEO COST ACC.,4X,18H ltXPfCTEO COST REJ.,4X,l9HTOTAL EXPECTEU COST,5X,l4HPERCENT CHANGE/ 2./) .

1~ FURMAT(3X,1H(,F5.?.,2X,1H7,F5.2,2X,1H),8X,F9.2,13X,f9.2,l4X,F9.2, ll3X,F7.2)

223 FOt{MAT(l4) C REAU l~PUT DATA

! •

REA0(5,b)Sl,S2,Al,A2,Rl,R2 READ(j,3)S,T RcA0(5,7)AL,BL,CL,UL,EL,FL,GL,HL,TOL,ITFIN

C READ THE NUMBER OF ERROR PAIRS TO aE EVALUATED REAU(5,223)K J=l

313 CONTINU;: Rt:AD(S,l)L,N,C ~RITE(o,9)S1,L,S2,N,Al,C,A2,S,Rl,T,R2 WRITt:(6,12) READ(5,3)El,E2 YE=C.O ECR=C.C ECA=O.O SYEB=0.0

lj CONTINUE IF(SYEti.GT.0.9999)GO TO 113 IF(El.tu.o.O.AND.E2.Ew.o.O)GO TO 55 RINT=GAUSS(AL,BL,S,T,El,E2,YE,TOL,ITFIN)+GAUSS(CL,DL,S,T,El,E2,YE,

lTOL,ITFIN)+GAUSS(El,FL,S,T,El,~2,YE,TOL,ITFIN)+GAUSS(GL,HL,S,T,El, 2E2,YE,TUL,ITFIN)

IF(~INT.EQ.O.J)GO TO 113 TIME=ALGAMA(r~+l.0)-ALGAMA(N-YE+l.O)-ALGAMA(YE+l.O)+ALGAMA(S+T)-

lALGAMA(S)-ALGAMA(T) PYEH=(EXP(ALUG(RI~T)+TIME))/(l.OE+30) GU TO 65

~5 PYEb=ALGAMA(N+l.0)-ALGAMA(~-YE+l.J)-ALGAMA(VE+l.O)+ALGAMA(S+YE) l+ALGAMA(T+N-YE)+AL~AMA(S+T)-ALGAMA(S)-ALGAMA(T)-ALGAMA(S+T+N)

PYt:B=F.XP (PYE ti) 6S co:~T I \JUE

SYEB=SYEB+PYEti IF(El.EQ.0.0.ANO.E2.t~.O.O)GO TO 24 l~(El.EQ.O.O)GO TO 53 X=G.O GO TO o3

I

5J X=YE 63 C01'JT I NUc

EX=O.O EFX=C.O SYEC=O.O

33 CONTINUE FX=FNCX(S,T) IF(tl.EQ.O.O)GO TO 25 IF(E2.EQ.O.O)GU TO 26 PYEC=PYtC3(El) GO TO 43

2S PYEC=PYEC2(0UM) GU TO 43

26 PYEC=PYECl(El) 43 COi~Tl"JUE

CALL EXPV(FX,PYEC) SYEC=SYEC+PYEC DELTA=PYEB-SYEC IF(DELTA.LT.O.OOOl)GO TO 93 IF(X.EQ.N)GO TO 93

l~_j X=X+l.O GO TO 33

24 CONTINUE X=YE FX=FNCX(S,T) EFx=FX EA=YE

93 COf'iTINUE IF(YE.GT.C)GU TO 103 CO~T=CO~TA(Al,A2,e2) c:C A=E CA+·co ST YE=YE+l.O GO TO 13

103 COST=CUSTR(A2,Rl,Ri,El,E2)

• -ii

ECR=ECR+COST 203 YE:=YE+l.O

GO TU 13 113 CONTI~UE

CALCULATE THE TOTAL EXPECTED COST OF SAMPLING INSPECTION EC=ECA+ECR IF(El.EQ.O.O.AND.E2.EQ.O.O)REC=EC DELl=((EC-REC)/REC)*lOC.O wRITt(6,l5)El,E2,ECA,ECR,EC,OELl IF(J.EQ.K)Gu TO 323 J=J+l GO TO 313

323 CONTINUE RETURN ENLJ

C **¥******************* SUBROUTINE EXPV(FX,PYtC)

C ********************** COMMON/bLOKl/X C8MMON/BLJK2/PYEB,EX,EFX IF(X.GT.O.O)GU TO 133 EXX=O.C GO TO 143

C CALCULATE THE EXPECTEu VALUE OF ACTUAL DEFECTIVES GIVEN THE NUMBER C OF OBSERVED DEFECTIVE~

133 EXX=(X*~YEC)/PYE8 EX=Ex+c=xx

C C~L~ULATE TH= EXPECTEU VALUE OF THE SPECIFIED FUNCTION OF ACTUAL DEFECTIVES C G I \/ E: i~ T H E l"J U M B E R OF O o 5 E R V E D D E F E CT I v E 5

l4J EFXX=(FX*PYEC)/PYEb EFX=·E:FX+EFXX RETURN t:~U

I CD

C ********************•* FUNCTIO~ PYECl(El)

C ********************** ~. CALCULATES PYEC WHEN TYPE 1 ERROR ONLY IS PRESENT

COMMON N,YE :o~MON/BLUK1/X,E2,S,T RcAL N A=ALGAMA(N-X+l.O)-ALGAM~(N-YE+l.O)-ALGAMA(YE-X+l.O)

l+(YE-X)*ALOG(ElJ+(N-YE)*ALOG(l.O-El) B=ALGAMA(r~+l.O)-ALGAMA(N-X+l.0)-ALGAMA(X+l.O)+ALGAMA(S+X)

l+ALGAMA(T+N-X)+ALGAMA(S+T)-ALGAMA(S)-ALGAMA(T)-ALGA~A(S+T+~) PYECl=EXP(A+B) RETURN E~D

C ********************** FUNCTION PYEC2(0UM)

C ********************** C CAL:ULATES PYEC WHEN TYPE 2 ERROR ONLY lS PRESENT

COMMON N,VE COMMON/BLOKl/X,E2,S,T REAL N A=ALGAMA(X+l.O)-ALGAMA(X-YE+l.C)-ALGAMA(YE+l.O)+(X-YE)*ALOG(E2)+

1Yt*ALOG(l.G-E2) B=ALGAMA(~+l.O)-ALGAMA(N-X+l.0)-ALGAMA(X+l.O)+ALGAMA(S+X)

l+ALGAMA(T+N-X)+ALGAMA(S+T)-ALGAMA(S)-ALGA~A(T)-ALGAMA(S+T+~) PYEC2=EXP(A+B) RETURN ENU

I

C ••••••**************** FUr~CTIO;~ PYEC3(El)

C **~******************* C CALCULATES PYEC ~HEN TYPE 1 AND TYPE 2 ERRORS ARE PRESENT

COMMON N,YF CO~MON/8LUKl/X,E2,~,T Re Al N, I PYi:C=O.O AMAX=A~AXl(YE-(N-XJ,0.0) AMIN=AMINl(X,YE) I=AMAX

23 CONTINUE A=ALGAMA(~-X+l.O)-ALGA~A(N-X-YE+l+l.0)-AL;AMA(YE-I+l.O)

l+ALGAMA(X+l.O)-ALGAMA(X-1+1.0)-ALGAMA(l+l.O) 2+(YE-I)*ALOG(El)+(N-X-YE+I)•ALOG(l.O-El)+(X-l)*ALOG(E2)+1*ALOG(l.O 3-t:2)

u=~LGAMA(~+l.O)-ALGAM4(N-X+l.O)-ALGAMA(X+l.O)+ALGA~A(S+X) l+ALGAMA(T+N-X)+ALGAMA(S+T)-ALGAMA(S)-ALGA~A(T)-ALGAMAIS+T+~)

YtC=E:XP(A+b) PYEC=PYEC+VEC IF(I.E~.AMIN)GO TO 43 l=I+l.0 GO TO 23

43 cor~TINUE PYEC3=PYEC ~l:TURN E.~o

N ... 0

C ********************** FUNCTION Fi~CX(S,T)

C ********************** C ~ALCULATE THc VALUE OF T~E SPECIFIEU FUNCTIO~ Of ACTUAL DEFECTIVES C FU~ A GIVEN VALUE OF X

COMMON N COMMflN/BLOKl/X RtAL N FNCX=(S+X)/(S+T+N) RETURN END

C ********************** FUNCTIO~ FRCP(P,El,E2)

C ********************** FRCP=P*(l.O-E2)+(1.Q-P)*El R~TUR~ E ,'JI)

C ********************** FUNCTION COSTA(Al,A2,E2)

C ********************** L CALCULATES THE COST OF ACCEPTANCE wHEN THE OBSERVED DEFECTIVES ARE LESS L THA\J UR EQUAL TO THE ACCEPTANCE NUMBC:R.

COMMON N,YE CJMMON/8LOK2/PYEB,~X,tFX,L,Sl,S2 REAL L,N . :J5TA=PYEJ*(N*Sl+YE*S2-A2*E2*EX+(L-N)*Al+(L-N)*A2*fFX) RETURN E1~LJ

N ...

C ********************** FUNCTION COSTR(A2,Rl,R2,El,E2)

C ********************** C CALCuLATES TH~ COST 0~ REJECTION WHEN THE OBSERVED DEFECTIVE~ ARE GREATER C THA'l THE ~CCEPTA:~CE NUMB~R.

COMMON N,YE COMMON/BLJK2/PYEB,EX,EFX,L,Sl,S2 R~AL L,N COSTR=PYtb*(N*Sl+YE*(S2-R2)+(L-N)*Rl+(E2*A~+R2*(1.0-El-E2))*EX+

ll*fl*R2+(f2*A2+R2*(l.O-El-E2))*(L-N)*EFX) RETURN ENU

C ****************~******* FUNCTION GAUSS(AL,BL,S,T,tl,E2,YE,TOL,ITFIN)

C **********************•* C INTEGRATES THE 5PECIFIED FU~CTION F OVER THE LIMITS AL TD Bl

COMMON N t(f:AL N DIMENSION V(ll),R(ll) DOUBLE ~RECISION V,R DATA V,R/-.978228658146O57DO,-.887Ob25997b8O95UO,

1-.730152 O!·_, 55 74049U0, -. 31909612 92 06812 DO,-. 26954315 5952 34500, 2O.OO0,.2l]543155952345uO,.519O9ol292O6812DC,.73O1520C5574O49OO, 3.887O6Z599768O9~DO,.g7a228058l4bO57OO,.O55bo8567ll6l74UO, 4.l2558O3614649O5UO,.l8629O21O~27734OO,.23319376459l99OOO, 5.2b28O454451O247DC,.272925G867779JlO0,.262HO45445lC247UO, 6.2331937b~59199ODO,.lB62YOilO'J27734DO,.125~BO3694649O~OO, 7.O5566b567ll6174uO /

~II·~ T = l IT=O

l IT=IT+l DcL=(BL-AL)/(FLuAT(NI~T)) 55=O.O D J 11 I = l , 1·J I N T B=AL+D~L*(FLOAT(I))

N ... N

A=6-0EL Z=C.0 00 12 J=l,11 X=((B+A)+(B-A)*V(J))*~.5 F=(l.O~+3C)*((X*(l.O-E2)+(1.O-X)*El)**YE)*((l.O-(X*(l.O-E2)+(1.O-X

1 ) * E l ) ) ** ( r'-J - YE ) ) * ( X * * ( S-1 • C ) ) * ( ( l • :J - X ) * * ( T - l • 0 ) ) 12 Z=Z+R(J)*F

l = ( B - A ) ,.,c Z :::: U • 5 SS=SS+l

l~ CONTINUE IF(ITFIN.EQ.l)GO TO 14 IF(IT.GT.l)GO TO 2

3 SAVE=SS NI~T=2*NI ·JT GO TU 1

2 IF(ABS(5AVE)-TOL)5,5,o X=A8S( 5AVc-SS) GO TO 7

o X=ABS((SAVt-SS)/SAVE) I IF(X.GT.TOL.ANO.IT.LE.ITFIN)GO TO 3

lFtX.GT.TOL)GO TO o~ 14 GAUSS=S~ 99 Rc:TURN o _, WK I TE ( 6, 4) I T, I T F I '", X , S S

CALL E..<IT 4 FJRMAT(lH0,27HGAUSS IT,ITFIN,UIF,FINAL ,214,2E20.8)

ENu

= w

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

REFERENCES

Grant, E. L., Statistical Quality Contol, Mc-Graw-Hill Book Company New York, 1964.

Duncan, A. J. , Quality Control and Industrial Statistics, Richard D. Irwin, Inc., Homewood, Illinois, 1965.

Harris, D. H., "Effect of Equipment Complexity on inspection Performance", Journal of Applied Psychology, Vol. 50, No. 3, 1966, pp. 236-237.

Harris, D. H., "Effect of Defect Rate on Inspection Accuracy", Journal of Applied Psychology, Vol. 52, No. 5, 1968, pp. 377-379.

Mccornack, R. L. , "Inspector Accuracy: A Study of the Literature", Sandia Laboratories, SCTM 53-61 ( 14) Case No. 418. 04, Albuquerque, New Mexico, February, 1961.

McKenzie, R. M., "On the Accuracy of Inspectors", Ergonomics, Vol. 1, 1957-58, pp. 258-270.

McKnight, K. A. , "An Investigation of the Effects of Two Types of Inspector Error on Sampling Inspection Plans", A Thesis in Industrial Engineering for Texas Technological College, June, 1967.

Harris, D. H. , and F. B. Chancy, Human Factors in Quality Assur-ance, John Wiley and Sons, Inc., New York, 1969.

Lambert, D. K., and C. L. Burford, "Effects of Inspection Variables on Five Measures of Inspection Accuracy", Society of Manufacturing EngineersL IQ 70-811 Technical Paper, 1970.

Ayoub, M. M. , A. G. Walvekar, and B. K. Lambert, "Statistical Quality Control Under Inspection Errors'', Technical Paper, Society of Manufacturing Engineers, 1970, pp. 70-284.

Q

Collins, R. D., Jr., K. E. Case, and G. K. Bennett, "The Effect of Inspection Accuracy in Statistical Quality Control", Proceedings, 23rd Annual AIIE Conference and Convention, Anaheim, California, May 31 - June 3, 1972, pp. 423-430. ·

214

12.

13.

14.

15.

16.

17.

18.

19.

20.

215

REFERENCES ( Concluded)

Collins, R. D., Jr., K. E. Case, and G. K. Bennett, "The Effects of Inspection Error on Single Sampling Inspection Plans", International Journal of Production Research, Vol. II, No. 3, July, 1973, pp. 289-298.

Case, K. E., G. K. Bennett, and J. W. Schmidt, "The Effect of Inspection Error on Average Outgoing Quality", Working paper, Vlrginia Polytechnic Institute and State University, Blacksburg, Virginia, February, 1974.

Guthrie D., Jr., and M. V. Johns, Jr., "Bayes Acceptance Sampling Procedures for Large Lots", Annuals of Mathematical Statistics, Vol. 30m 1959, pp. 896-925.

Lane, Robert W., "Single Attribute Cost Modeling & A Formal Bayesian Approach", A Thesis in Industrial Engineering and Operations Research, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, December, 1972.

Bennett, G. Kemble, K. E. Case, and J. W. Schmidt, "The Economic Effects of Inspector Error on Attribute Sampling Plans", Accepted for publication: Naval Research Logistics Quarterly, 1974.

Ayoub, M. M~ , B. K. Lambert, and A. G. Walvekar, "Effects of two types of Inspector Error on Single Sampling Inspection Plans", pre-sented at Human Factors Society, San Francisco, October, 1970.

Wetherill, G. B., Sampling Inspection and Qt.!alicy Control, Methrien and Co., LTD, London, 1969.

Hald, A., "The Compound Hypergeometric Distribution and a·system of Single Sampling Inspection Plans Based on Prior Distributions anci Costs", Technometrics, Vol. 2, No. 3, August 1960, pp. 275-340.

Barnard, G. A. , "Sampling Inspection and Statistical Decisions'', Journal of the Royal Statistical Sociecy, Series B (methodological), Vol. XVI, No. 2, 1954, pp. 151-174.

BIBLIOGRAPHY

Ayoub, M. M., A. G. Walvekar, and B. K. Lambert, "Statistical Quality Con-trol Under Inspection Errors", Technical Paper, Society of Manufactu'ring Engineers, 1970, pp. 70-284.

Ayoub, M. M., B. K. Lambert, and A. G. Walvekar, "Effects of two types of Inspector Error on Single Sampling Inspection Plans", presented at Human Factors Society, San Francisco, October, 1970.

Barnard, G. A. , "Sampling Inspection and Statistical Decisions", Journal of the Rolal Statistical Society, Series B (methodological), Vol. XVI, No. 2, 1954, pp. 151-174.

Bennett, C. A., "Effects of Measurement Error on Chemical Process Control'', !p.dustrial Quality Control, Vol. X, No. 4, January 1954, pp. 17-20.

Bennett, G. K., K. E. Case, and J. W. Schmidt, ''The Economic Effects of Inspector Error on Attribute Sampling Plans", Accepted for Publication: Naval Research Logistics Quarterly, 1974.

Budne, T. A., "The Error in Measurement", Industrial Quality Control, Vol. VIIl, No. 6, May 1961, pp. 97-100.

Caplen, R. H., ''A Contribution to the Problem of Choosing a Sampling Inspec-tion Plan", The ~ality E!YP:neer, July-August, 1962, pp. 103-107.

Carroll, J. M. , "Estimating Errors in the Inspection of Complex Products", AIIE Transactions, Vol. I, No. 3, September 1969, pp. 229-235.

Case, K. E. , G. K. Bennett, and J. W. Schmidt, "The Effect of Inspection Error on Average Outgoing Quality", Working paper, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, February, 1974.

Case, K. E., J. W. Schmidt and G. K. Bennett, "Cost-Based Acceptance Sampling", Industrial Engineering, Vol. 4, No. 11, November 1972, pp. 26-31.

Champerowne, D. G. , ''The Economics of Sequential Sampling Procedures for Defectives", Appliep Statistics, Vol. 2, 1953, pp. 118-130.

216

217

BIBLIOGRAPHY ( Continued)

Collins, R. D., Jr. "A Cost Based Quality Control Model With Compensation .. for Inspection Errors", EG 799 Status Report, July 1972.

Collins, R. D., Jr., K. E. Case, and G. K. Bennett, " The Effects of Inspec-tion Error on Single Sampling Inspection Plans", International Journal of Production Research, forthcoming.

Collins, R. D., Jr., K. E. Case, and G. K. Bennett, "The Effect of Inspec-tion Accuracy in Statistical Quality Controls'', Proceedings, 23rd Annual AIIE Conference and Convention, Anaheim, California, May 31-June 3, 1972, pp. 423-430.

Cox, D. R., "Serial Sampling Acceptance Schemes Derived from Bayes 's Theorem", Technometrics 2 Vol. 2, No. 3, August 1960, pp. 353-360.

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Diviney, T. E., and N. A. David, "Simplified Sampling Technique for Large Lots", The Journal of Industrial Engine~'.!:_1.ng, November-December, 1962, pp. 511-513.

Duncan, A. J. , "Quap.j:y _Control and Industrial Statistics, Richard D. Irwin, Inc., Hotnewood, Illinois, 1965.

Eagle, A. R., "A fviethod for Handling Errors in Testing and Measuring'', Industrial Quality Control, Vol, X, No. 5, March 1954, pp. 10-15.

Fruewirth, M. A., "An Economic Criterion for Minimizing Overall Inspection and Repair Cost", ( Source Unknown) • pp. 27-31.

Grant, E. L., Statistical Quality Control, Mc-Graw-Hill Book Company, ----------New York, 1964.

Guenther, W. C. , "On the Determine.ti.on of Single Sampling Attribute Plans Based upon a linear Cost Model and a Prior Distribution", Technometrics, --Vol. 13, No. 3, August, 1971, pp. 483-498.

218

BIBLIOGRAPHY ( Continued)

Guthrie, D., Jr., and M. V. Johns, Jr., "Bayes ~.\cceptance Sampling Proced-ures for Large Lots'', Annuals of Mathematical Statitics, Vol. 30, 1959, ---- ----~--pp. 896-925.

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219

BIBIJIOGRAPHY ( Continued)

Lambert, D. K., and C. L. Burford, "Effects of Inspection Variables on Five .. Measures of Inspection Accuracy", Society of Manufacturing Engineers,

IQ 70-811 Technical Paper, 1970.

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220

BIBLIOGRAPHY ( Continued)

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221

BIBLIOGRAPHY ( Concluded)

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The vita has been removed from the scanned document

S'rJ\··rISTIC'.!\l IJy· ~t\ND EC()NOl\,IICAI.1 Y BASED ATTP.IBlTTE 1\(-\_~l·~l>'T':\~C'E S .. AlvlIJLli\(} l\l()DEI~S \\c-ITH INSPECTION ERRORS

By

Ilufus D. Collins, Jr.

(ABSTRACT)

T'his dissertation exan1ines the quality control situation where lots are

either accepted or rejected on the basis of a single sample which is randomly

selected from the lot. Judgen1ent regarding the quality of an item is based on

defined attributes \Vhich either co11forn1 to or deviate from prescribed standards.

Traditionally, it has been assun1ed that no errors are introduced into the inspec-

tion process. While such as assumption eases the computational procedures

involved in the evaluation of sampling schemes, it ignores considerable evidence

that n1ost inspection operations are error prone.

This research investigates the statistical and economical considerations

which may be involved in an inspection process subject to such inaccuracies.

In the context of this research errors are introduced into the process when an

iten1 is erroneously classified as either good or bad. Good items classified as

bad items are referred to as type I errors; conversely, bad items classified

as good iterr1s are referred to as type II errors. Perforn1ance measures have

been identified that are meaningful in establishing trend data from which to

draw inferences concerning the effects of inspection errors.

The purely statistical quality n1odel is first evaluated for both the error

free and error prone situations. The formulas for the average outgoing quality,

AC)Q, average total inspection, A 11I, and the probability of accepting a lot, P , a

are established for both situations of interest. the error prone situation, a

ne\v probability 1nass function is defined which describes the conditional distri-

bution governing the occurrence of observed defectives in a sample given the

actual nun1ber of defectives in a sample. Other pertinent distributional con-

siderations are developed and discussed. Typical numerical examples are used

to illustrate the effects of both type I and type II errors which may occur either

alone or in co1nbination.

The second part of this dissertation assun1es that many quality schemes

originate within the structure of an economic framework; in such cases the

inspection criteria should appropriately be based on economic criteria. An

economic based quality control model is forn1ulated that is applicable to

either the error free environment or the error prone environment. For the

purpose of this research it is assumed that the prior distribution of defectives

in a lot before the sample is formed is either described by a mixed binomial

distribution or a Polya distribution. The distributional considerations per-

tinent to the model are fully developed within the text.

Several numerical examples are evaluated to illustrate the selection

of optimal sampling plans for inspection schemes subject to inaccuracies.

The expected cost penalty incurred when using an optimal plan designed for

an error free environment when errors actually exist was also investigated.

Considerable data are presented which permits one to draw inferences pertain-

ing to the importance of the assumed distribution. The data were presented in

numerous tables and figures to aid in establishing the significant trends from

which the consequences of inspection errors could be inferred.