QUALITY PLANNING II Study supports - vsb.czkatedry.fmmi.vsb.cz/Opory_FMMI_ENG/QM/Quality planning II.pdf · QUALITY PLANNING II Study supports ... (Advanced Product Quality Planning

PROJECT IRP

„Creation of English Study Supports for Selected Subjects of the Follow-up Master Study in the Quality Management Study Field“

IRP/2015/104

QUALITY PLANNING II

Study supports

Jiří Plura

Language review: Mark Landry

Title: Quality Planning II

Author: Prof. Ing. Jiří Plura, CSc.

Edition: first, 2015

Number of pages: 108

Study materials for the study support of Quality Planning II, the Faculty of Metallurgy and Material Engineering.

Intended for the IRP project of:

Creation of English Study Supports for Selected Subjects of the Follow-up Master Study in the Quality Management Study Field

Number: IRP/2015/104

Execution: VŠB – Technical University of Ostrava

The project is co-financed by the Ministry of Education, Youth and Sports of the Czech Republic

ISBN 978-80-248-3848-9

Jiří Plura Quality Planning II

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INSTRUCTIONS FOR STUDY

You have received a study support containing the main topics covered by the course of the

„Quality Planning II“ taught in the 2nd semester of the follow-up master's degree of the study

programme of Quality Management study field at the Faculty of Metallurgy and Material

Engineering.

Subject aims and outputs of learning

The aim of the course is to learn advanced techniques and the methods of quality planning.

The course builds on the knowledge gained in the course "Quality Planning I". Attention is

paid to modern methodological approaches to product quality planning, the requirements of

production part approval processes, to the advanced procedures of process capability

analysis, to the procedures of measurement system analysis and other specialized methods.

Knowledge outputs:

to characterize and classify advanced methods of quality planning

to identify appropriate methods for different situations Skill outputs:

to apply selected advanced procedures and the methods of quality planning

to interpret the results of the application of methods and propose appropriate actions.

Who is the course intended for The course is included in the follow-up master's study of the field of study of Quality Management, but it can be studied by an applicant from any other field of study, provided that he/she meets the required prerequisites. This study support is divided into chapters, which logically divide the studied matter, but are not equally comprehensive. The estimated study time of the chapters may vary considerably, which is why large chapters are further divided into numbered sub-chapters and they correspond to the structure described below. Method of communication with the educator This matter is presented to students within the frame of their lectures and practical exercises, where they practically learn the topic discussed during the theoretical lectures. But selected topics suppose self-learning and elaboration of the written seminar works discussed with the lecturer during the consultations and via internet.


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STRUCTURE OF THE CHAPTERS

Time to study

The time necessary to study the subject matter is given at the beginning of every chapter. The time is approximate and can serve as a rough guide for the study layout of the entire subject. The time may seem too long to some people or, on the contrary, too short to other ones. There are students who have never encountered this issue and, on the other hand, those who already have extensive experience in this field.

Goal

There are defined objectives given for you to achieve after reading the chapter and studying the topics - concrete skills, knowledge.

Lecture

This part is the actual presentation of the studied subject matter, the introduction of new terms, their definitions and explanation of the principles of the studied subject matter. All is accompanied by pictures, tables, solved examples.

Summary of terms

The main terms you should learn are repeated at the end of the chapter. If you still do not understand any of the terms, go back to them again.

Questions

There are several theoretical questions to verify that you have fully and well mastered the subject matter of the chapter.

References

There is a list of the used reference sources, from which you can draw additional information on the issue in question, at the very end of every chapter.

The author of this educational material wishes you a successful and pleasant study using this textbook.

Prof. Ing. Jiří Plura, CSc.


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CONTENTS

INSTRUCTIONS FOR STUDY .................................................... 3

STRUCTURE OF THE CHAPTERS ........................................... 4

1. ADVANCED APPROACHES TO PRODUCT QUALITY PLANNING ...... 7

1.1 Advanced Product Quality Planning (APQP) ....................................................................... 7

1.1.1 Planning ................................................................................................................................... 10

1.1.2 Product design and development ........................................................................................... 12

1.1.3 Process design and development ........................................................................................... 15

1.1.4 Product and process validation ............................................................................................... 17

1.1.5 Feedback, assessment and corrective action .......................................................................... 20

1.2. Production Part Approval Process – PPAP ....................................................................... 21

2. FAULT TREE ANALYSIS (FTA) .................................................... 29

2.1 Fault tree analysis procedure ................................................................................................ 32

3. ADVANCED APPROACHES TO PROCESS CAPABILITY ANALYSIS ... 36

3.1 Procedure of process capability analysis ............................................................................ 36

3.2 Process Capability Indices .................................................................................................... 39

3.3 The influence of process changes to capability indices .................................................... 42

3.3.1 Sensitivity of capability indices to the quality characteristic variability ................................. 42

3.3.2 Sensitivity of capability indices to the quality characteristic position .................................... 44

3.3.3 Ambiguity of process capability indices in relation to the distribution of the quality

characteristic .................................................................................................................................... 46

3.4 Factors affecting the results of process capability analysis ............................................. 49

3.5 Confidence intervals of process capability indices ............................................................ 50

4. THE PROCEDURES OF PROCESS CAPABILITY ANALYSIS IN NON-STANDARD SITUATIONS ............................................................... 56

4.1 Procedures of process capability analysis in the cases, when a process is „out of

control“ ............................................................................................................................................ 56

4.2 Procedures of process capability analysis when data normality is not met ................... 58

5. PROCESS CAPABILITY ANALYSIS IN THE CASE OF NON-MEASURABLE CHARACTERISTICS .................................................. 62

5.1 Process capability indicators in the case of monitoring nonconforming products ........ 65

5.2 Process capability indicators in the case of monitoring nonconformities ....................... 69

6. MEASUREMENT SYSTEM ANALYSIS. ANALYSIS OF DRIFT, BIAS AND LINEARITY ............................................................................ 72

6.1 Statistical properties of measurement systems.................................................................. 74

6.2 Analysis of measurement system drift ................................................................................. 76


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6.3 Analysis of measurement system bias ................................................................................ 78

6.4 Analysis of measurement system linearity .......................................................................... 82

7. MEASUREMENT SYSTEM ANALYSIS. REPEATABILITY AND REPRODUCIBILITY ANALYSIS (GRR ANALYSIS) ............................... 87

7.1 Range method ......................................................................................................................... 88

3.1 Average and Range Method ................................................................................................. 90

3.3 ANOVA ................................................................................................................................... 100

8. MEASUREMENT SYSTEM ANALYSIS. ATTRIBUTE MEASUREMENT SYSTEMS ANALYSIS .................................................................... 103

8.1 Evaluation of agreement between operators .................................................................... 104

8.2 Evaluation of the agreement between individual operators and reference values. .... 106

8.3 Effectiveness of attribute measurement system .............................................................. 106


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1. ADVANCED APPROACHES TO PRODUCT QUALITY PLANNING

Time for learning

3 hours

Goal

After studying this chapter you will be able to:

Explain the methodology of advanced product quality planning (APQP)

Describe the various inputs and outputs of APQP stages

Explain the course of the Production Part Approval Process and describe

individual requirements.

Lecture

1.1 Advanced Product Quality Planning (APQP)

Present development in the all fields of human activity is connected with increased

requirements for product quality. Quality planning has a total essential influence on a

product´s quality. According to the terminological standard of ISO 9000´s standards family

(ISO 9000, 2006) quality planning is defined as “part of quality management focused on

setting quality objectives and specifying necessary operational processes and related

resources to fulfill the quality objectives”. Quality planning represents many activities, which

decide about resulting quality. For example these partial activities are included in quality

planning [1]:

Quality objectives identification and their development in an organization

Product quality characteristics planning (the development of products, which meet customer (and other stakeholders) requirements

Quality plan processing


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Planning of methods, which will be used for the achievement of required product quality

Processes quality planning (the development of processes, which will be able to assure the required product quality and verification of their capability)

Planning preventive actions for possible problem risk minimization

Planning the methods of measuring and monitoring product and process quality

Planning measurement systems and verifying their suitability

Planning data collection and the necessary quality records, etc.

Quality planning is realized especially in the pre-production phase. Activities in these

phases decide about customer satisfaction, product competitiveness and an organization´s

profit. While in the past the production phase was regarded as a key phase for product

quality, at present it is generally recognized that the pre-production phase contributes to final

product quality approximately by eighty percent. This state is considerably influenced by the

increasing complexity of present products and the technologies used, competitive market

conditions and enhanced customer requirements.

The importance of quality planning is also connected with the fact that in pre-

production phases many more nonconformities arise than in production and other phases. In

addition, the removal of nonconformities during pre-production phases is much cheaper than

their removal after launching production. However, up to now many organizations pay

insufficient attention to these phases. Often there is a lack of time and money for sufficient

design processing and quality planning, but later there must be much more time and money

for the removal of problems occurring in the implementation phase.

Arguments for focusing on quality planning can be summarized by these points:

Quality planning principally influences customer satisfaction.

The way of product quality planning is an important attribute of an organization´s competitiveness.

Product quality planning is a way to prevent nonconformities during product implementation and use.

Most of nonconformities arise in the pre-production phases, where product quality planning activities are especially done.

The removal of nonconformities in the pre-production phases requires the lowest costs and the shortest time.

By using the procedures and methods of product quality planning an organization proves that it has utilized all means for achieving customer satisfaction and for preventing nonconformities.

Product quality planning results in increased customer reliance on the products of an organization.


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The success of product quality planning is significantly affected by the methodology

used. At present, methodologies used in the automotive industry are best processed. An

example is the APQP (Advanced Product Quality Planning and Control Plan) methodology.

This methodology was developed jointly by Chrysler, Ford and General Motors as part of the

Standard QS-9000.

In this methodology, product quality planning is defined as a „structured method of

defining and establishing the steps necessary to assure that a product satisfies the

customer“.

These main benefits of product quality planning are emphasized:

directing resources to satisfy a customer

promoting an early identification of the required changes

avoiding late changes.

Using APQP methodology simplifies the scheduling of product quality planning and

facilitates communication with suppliers.

Product quality planning, using APQP methodology is divided into five overlapping

stages (see Fig. 1.1).

1. Planning

2. Product design and development

3. Process design and development

4. Product and process validation

5. Feedback assessment and corrective action.

Before the APQP application preparatory phase is completed, it is necessary to

provide training for personnel who will be involved in product quality planning and establish

an interdisciplinary team of product quality planning. Members of the team should be, in

addition to representatives of the quality department, also representatives of design and

development, production, technical inspection, supply, sales, service, contractors,

customers, etc.

An important part of the preparatory phase is defining the action area of the team,

defining the methods of communication with other customer and supplier teams, establishing

a timetable for product quality planning and determining the necessary costs.


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Fig. 1.1 Product quality planning timing chart according to APQP methodology.

1.1.1 Planning

This initial phase of product quality planning should ensure a full understanding of the

requirements and expectations of customers. All activities must be carried out with respect to

the customer and to provide better products and services than the competitors.

Inputs:

Voice of a customer

The voice of the customer creates information obtained from internal and external

customers. When collecting this information there can be used, for example:

Market research

Historical warranty and quality information

Team experience.


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Business plan and marketing strategy

A business plan and marketing strategy set certain limitations that may affect product

quality planning (e.g. cost, schedule, resources for research and development, etc.).

Marketing strategy defines target customers, the main selling points and key competitors.

Product and process benchmark data

The use of benchmarking provides inputs in establishing product/process

performance targets.

Product and process assumptions

It is necessary to include among the inputs product and process assumptions

(such as the application of technical innovations, advanced materials, new technologies,

etc.).

Product reliability studies

This data considers the frequency of repairs or the replacement of components within

designated periods of time and the results of reliability tests.

Customer inputs

Valuable information about their needs and expectations can provide future users of

the product. This information should be used in developing criteria for evaluating customer

satisfaction.

Outputs:

Design goals

Design goals are a translation of the Voice of a Customer into measurable design

objectives. The proper selection of design goals assures that the Voice of a Customer is not

lost in subsequent design activities. The Voice of the Customer also includes regulatory

requirements.

Reliability and quality goals

Reliability goals are based on customer wants and expectations, program objectives

and reliability benchmarks. Quality goals should be based on metrics such as parts per

million or scrap reduction.


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Preliminary bill of material

The team should establish a preliminary bill of material. Part of it should be a list of

potential suppliers.

Preliminary process flow chart

The anticipated process of production should be described using a process flow

chart.

Preliminary listing of special product and process characteristics

On the basis of information from the customer and on the knowledge of the supplier

about the product and the process there should be established special characteristics of the

product and the process, which will require special attention.

Product assurance plan

This plan translates design goals into design requirements. It should include e. g.

outlining of program requirements, reliability goals and requirements, the assessment of new

technology, material, environment, packaging, service and manufacturing requirements,

etc., use of FMEA, the development of engineering requirements.

Management support

The final output of this phase is “Management Support” which can be understood as

"to inform management about the results and to obtain its support." It is one of the key

factors in the success of the product quality planning team. The team should officially inform

management about the results after completion of each phase of product quality planning (or

more often if necessary). The aim is to demonstrate that all quality planning requirements

have been met and all remaining problems are documented and solutions are planned. The

result should be the adoption of these results and obtaining support for the next phase

(release to the product design and development phase).

1.1.2 Product design and development

This phase is dealing with the elements of the planning process during which design

features and characteristics are developed into a near final form. A feasible design must

permit meeting production volumes and schedules, and be consistent with the ability to meet

engineering requirements, along with quality, reliability, investment cost, weight and timing

objectives. Inputs to this phase correspond to the outputs of the previous phase.


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a) Design Outputs:

Design Failure Modes and Effects Analysis (DFMEA)

Using the DFMEA application potential failures of a designed product are analysed

and their risks are evaluated. In the cases when the risk of potential failure is not acceptable

suitable actions for design quality improvement are proposed and implemented.

Design for manufacturability and assembly

Design for manufacturability and assembly is a simultaneous engineering process

designed to optimize the relationship between design function, manufacturability and

assembly. During this optimization, a product quality planning team should consider e. g. the

sensitivity to the variability of production conditions, dimensional tolerances, the number of

components, material handling, etc.

Design verification

Design verification confirms that product design meets a customer´s requirements

defined in the previous phase.

Design review

Design reviews are regularly scheduled meetings led by the organization´s design

engineering activity and must include other affected areas. Design reviews should include

e.g. an evaluation of design requirement considerations, reliability goals, component or

system duty cycles, computer simulations and bench test results, the review of design for

manufacturability and assembly, test failures, design verification progress, etc.

Prototype build – Control plan

The manufacture of prototype parts provides an opportunity for assessing how well

the product meets the voice of customer objectives. A prototype control plan is a document

which defines all measurements and material or functional tests applied during prototype

creation.

Engineering drawings (including mathematical data)

A product quality planning team should review whether drawings contain sufficient

information about the dimensions and other parameters of the individual parts. Drawings can

contain special (set by law or safety regulations) characteristics that must be put into the

control plan. Control and inspection points should be clearly defined, in order to propose

suitable measuring instruments and products. The dimensions on the drawings should be


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assessed in terms of manufacturability and compatibility with standards for production and

measurements.

Engineering specifications

A detailed review and understanding of the controlling specifications will help a

product quality planning team to identify the functional, durability and appearance

requirements of the subject component or assembly. Sample size, frequency, and

acceptance criteria of these parameters are generally defined as part of the engineering

specification. Otherwise, the sample size and frequency are to be determined by the

organization and listed in the control plan.

Material specifications

Material specifications should be reviewed for special characteristics relating to

physical properties, performance, environmental, handling, and storage requirements. These

characteristics should also be included in the control plan.

Drawing and specification changes

Where drawing and specification changes are required, the team must ensure that

the changes are promptly communicated and properly documented in all affected areas.

b) APQP outputs

New equipment, tooling and facilities requirements

The product quality planning team should include new equipment and facilities

requirements in the timing chart. The team should assure that there is a process to

determine that new equipment and tooling are capable and delivered on time. Facilities

progress should be monitored to assure completion before the start of the pilot production.

Special product and process characteristics

In the planning stage, the team identifies a preliminary list of special product and

process characteristics. The product quality planning team should build on this listing and

reach a consensus through the evaluation of the technical information. The organization

should refer to the appropriate customer-specific requirements for additional details on the

use of special product and process characteristics.


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Gages and testing equipment requirements

The product quality planning team should identify gages and testing equipment

requirements, add these requirements to the timing chart and monitor that the required timing

is met.

Team feasibility commitment and management support

The product quality planning team must assess the feasibility of the proposed design

(also in the cases when the design was submitted by a customer). The team must be

satisfied that the proposed design can be manufactured, assembled, tested, packaged, and

delivered in a sufficient quantity on schedule at an acceptable cost to the customer. The

team consensus that the proposed design is feasible should be documented as the Team

Feasibility Commitment.

1.1.3 Process design and development

This stage should ensure the comprehensive development of an effective

manufacturing system, which must assure that customer requirements, needs and

expectations will be met. The inputs to this stage are corresponding to the outputs of the

product design and development stage.

Outputs:

Packaging Standards and Specifications

Packaging Standards are usually defined by the customer, if not, the product quality

planning team should ensure that individual product packaging is designed and developed.

Packaging design should assure that the product characteristics remain unchanged during

packing, transit, and unpacking. The packaging should have compatibility with all identified

material handling equipment.

Product/Process Quality System Review

A product quality planning team should review the quality management system of the

manufacturing site. Any additional controls or procedural changes required to produce the

product should be updated, documented and included in the manufacturing control plan. This

is an opportunity for the improvement of an existing quality management system.


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Process Flow Chart

The process flow chart helps the team to analyze and optimize the proposed process.

It is especially useful in the implementation of the FMEA process and in developing a control

plan. It can be used to analyze the sources of variations of machines, materials, methods,

and manpower from the beginning to the end of a manufacturing or assembly process.

Floor Plan Layout

The floor plan should be developed and reviewed to determine the acceptability of

important control items, such as inspection points, control chart location, the applicability of

visual aids, interim repair stations, and storage areas to contain nonconforming material.

Characteristics Matrix

To analyze the relationship between the characteristics of the product and

manufacturing operations, it is recommended to apply the characteristics matrix. In this

matrix rows represent individual product quality characteristics and the columns individual

production operations. In the matrix cells there are used graphical symbols for identifying

whether a given operation decides about a given quality characteristic or whether a given

quality characteristic is important for a given operation performance. The more relationships

the given quality characteristics to the operations has, the more important is its control.

Process Failure Mode and Effects Analysis (PFMEA)

Within the process design and development the Process FMEA should be performed.

It is a team analysis of the proposed process, whose aim is to identify possible failures that

may arise during the production of the product and to analyze and minimize their risks.

Pre-Launch Control Plan

A pre-launch control plan specifies measurements and tests that will be done after a

prototype and before full production (especially during pilot production). The purpose of the

pre-launch control plan is to detect potential nonconformities during or prior to initiating the

production run. e.g. by the means of more in-process and final check points, more frequent

inspections, statistical evaluations, enhanced audits or identification of error-proofing

devices.

Process instructions

The product quality planning team should ensure that process instructions provide

sufficient understanding and detail for all personnel who have direct responsibility for the


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operation of the processes. These instructions should be developed on the basis of the

following sources: FMEA results, control plans, the process flow chart, engineering drawings

and material specifications, floor plan layout, the characteristics matrix, packaging

specifications, process parameters, experience and knowledge of the processes and

products, handling requirements, industry standards and operators´ qualification.

The process instructions for standard operating procedures should be published and

should include set-up parameters such as: machine speeds, feeds, cycle times and tooling,

and should be accessible to the operators and supervisors.

Measurement Systems Analysis Plan

The product quality planning team should ensure that a plan to accomplish the

required measurement systems analysis is developed. This plan should include e.g. the

responsibility to ensure bias, linearity, the repeatability and reproducibility of measurement

systems.

Preliminary Process Capability Study Plan

The product quality planning team should ensure the development of a preliminary

process capability analysis plan. More information can be found in PPAP (Production Part

Approval Process) [2] and SPC (Statistical Process Control) [3] methodologies.

Management Support

The product quality planning team should schedule a formal review designed to

reinforce management commitment at the conclusion of the process design and

development phase. This review is critical for keeping upper management informed as well

as gaining assistance in resolving any open issues. Management support includes the

confirmation of planning and providing the resources for product and process validation.

1.1.4 Product and process validation

Validation of the product and process is based on the evaluation of a significant

production run (pilot production). During a significant production run the product quality

planning team should validate that the control plan and process flow chart are being followed

and the products meet customer requirements. A significant production run may uncover

additional problems, which should be investigated and solved before starting serial

production. Inputs to this stage are corresponding to outputs from the stage "Process Design

and Development".


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Outputs:

Significant Production Run

The significant production run must be conducted using production tooling, production

equipment, the production environment (including production operators), facility, production

gages and production rate. The minimum quantity of products for a significant production run

is usually set by the customer. Outputs of the significant production run are used e. g. for:

Measurement systems analysis

Preliminary process capability study

Process review

Production validation testing

Production part approval

Packaging evaluation

Quality planning sign-off

Sample production parts.

Measurement Systems Analysis

Measurement of characteristics identified in the control plan should be performed

using specified measurement systems. The acceptability of these measurement systems

should be verified by the means of measurement system analyses during or before

significant production run.

Preliminary Process Capability Study

A preliminary process capability study should be performed on characteristics

identified in the control plan. The study provides an assessment of the readiness of the

process for production.

Production Part Approval

The Purpose of Production Part Approval Process (PPAP) is to provide evidence that

all customer engineering design records and specification requirements are properly

understood by the organization and that the manufacturing process has the potential to

produce a product consistently meeting these requirements during an actual production run

at the quoted production rate.


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Production Validation Testing

Production validation testing refers to engineering tests that validate that products

made using production tools and processes meet customer engineering standards including

appearance requirements.

Packaging Evaluation

All test shipments and test methods must assess the protection of the product from

normal transportation damage and adverse environmental factors. The product quality

planning team deals with evaluating the effectiveness of the packaging also in the cases

when packaging is defined by a customer.

Production Control Plan

The production control plan is a written description of the systems for controlling

production parts and processes. The production control plan is a living document and should

be updated to reflect the addition or deletion of controls based on experience gained by

producing parts. Approval of the authorized customer representative may be required.

Quality Planning Sign-Off and Management Support

The product quality planning team should perform a review at the manufacturing

locations and coordinate a formal sign-off. The product quality sign-off indicates to

management that the appropriate APQP activities have been completed. The sign-off occurs

prior to first product shipment and includes a review of the following:

Verification that process flow charts exist and are being followed

Verification that control plans exist, are available and are followed at all times for

all affected operations

Verification that process instructions contain all the special characteristics

specified in the control plan and all PFMEA recommendations have been taken

into consideration

Verification which special monitoring and measuring devices are required by the

control plan, verification gages, repeatability and reproducibility and their proper

usage

Demonstration of the required capacity.

Upon completion of the sign-off a review with management should be scheduled to

inform management of the program status and gain their support with any open issues. The


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Product Quality Planning Summary and Approval Report is an example of the documentation

required to support an effective quality planning sign-off.

1.1.5 Feedback, assessment and corrective action

Quality planning does not end with the process of validation and installation. It is a

component of the manufacturing stage where output can be evaluated when all special and

common causes of variation are present. This is also the time to evaluate the effectiveness

of the product quality planning efforts. Inputs to this stage are corresponding to outputs from

the stage "Product and Process Validation".

Outputs:

Reduced Variation

Control charts and other statistical techniques should be used as tools to identify

process variation. Analysis and corrective actions should be used to reduce variation.

Continual improvement requires attention, not only to the special cause of variation, but in

understanding common causes and seeking ways to reduce these sources of variation.

Proposals should be developed including costs, timing, and anticipated improvement for a

customer review.

Improved Customer Satisfaction

Quality planning activities and the demonstrated process capability are important for

customer satisfaction, however, the product or service still has to perform in the customer´s

environment. The effectiveness of product quality planning can be evaluated up to the stage

of the use of the product. The organization and customer should be partners in making the

changes necessary to correct any deficiencies and in improving customer satisfaction.

Improved Delivery and Service

There is continual improvement during the delivery and service stage of product

quality planning in which the organization and customer partnership solves problems. The

experience gained at this stage provides the customer and organization with the necessary

knowledge to reduce process variability, inventory, and quality costs and to provide the right

component or system for the next product.


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Effective Use of Lessons - Learned/Best Practices

A lessons learned or best practices portfolio are beneficial for capturing, retaining and

applying knowledge. Input to lessons learned and best practices can be obtained through a

variety of methods including:

• Review of Things Gone Right/Things Gone Wrong (TGR/TGW)

• Data from warranty and other performance metrics

• Corrective action plans

• Comparison with similar products and processes

• DFMEA and PFMEA results.

1.2. Production Part Approval Process – PPAP

The Production Part Approval Process (PPAP) [4] defines the generic requirements

for production part approval, including production and bulk materials. It is applied in the

automotive industry within Standard QS-9000. The purpose of PPAP is to determine if all

customer engineering design records and specification requirements are properly understood

by the organization and that the manufacturing process has the potential to produce a

product consistently meeting these requirements during an actual production run at the

quoted production rate. Within the framework of this process, the supplier presents to the

customer a range of evidence showing his readiness to begin serial production.

The production part approval process should be applied in the following situations:

• a new part or product

• correction of the discrepancy of a previously submitted part

• product modified by an engineering change (in design, specifications or materials)

• production using new or modified tools

• production after changing production equipment

• production in another manufacturing site

• change of subcontractor

• changing test methods

• tools for serial production were not used for more than a year.

Fulfillment of a number of requirements of the production part approval process into

serial production must be documented on the basis of the products produced during a

significant production run. A significant production run must be from a production, which lasts

from one to eight hours in which at least 300 consecutive parts must be made, unless


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otherwise specified by the authorized customer representative. A significant production run

must be manufactured under conditions corresponding to series production; at the production

site, at the production rate, using production tools, materials, gages and workers.

PPAP requirements

Fulfillment of the following requirements must be demonstrated within the framework

of the production part approval process:

1) Design Record

An organization must have the engineering documentation of a part, and must

provide proof of its material composition (IMDS = International Materials Data System can be

used).

2) Authorized Engineering Change Documents

In the case of any technical changes of a part that were not still reflected in the design

documentation relevant documents demonstrating the approval of these changes is required.

3) Customer Engineering Approval

If customer engineering approval is required, it must be documented as evidence.

4) Design Failure Mode and Effects Analysis (Design FMEA)

In the case when an organization is responsible for the design of the product, it must

apply the Design FMEA and document the results. Design FMEA can be processed for a

group of similar parts.

5) Process Flow Diagram(s)

The organization must show a process flow chart.

6) Process Failure Mode and Effects Analysis (Process FMEA)

The organization must apply the Process FMEA and document the results. Process

FMEA can be performed for a process producing a group of similar parts or materials.


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7) Control Plan

The organization must process a control plan that defines all methods used for

process control and is in accordance with customer requirements.

8) Measurement System Analysis Studies

The organization must perform measurement systems analysis (e.g. repeatability and

reproducibility, bias, linearity, stability) for all new or modified measurement systems.

9) Dimensional Results

The organization shall provide evidence that dimensional verifications required by the

design record and the control plan have been completed and the results indicate compliance

with specified requirements. Dimensional results must be for each unique manufacturing

process, e.g. for all production lines, cavities, molds, dies, etc. The organization shall identify

one of the measured parts as a master sample.

10) Records of Material / Performance Test Results

The organization must keep records of the tests prescribed in the design

documentation or control plan.

11) Initial Process Studies (preliminary (initial) process capability studies)

A preliminary process capability study is performed on the basis of data about the

quality of products produced during a significant production run. An acceptable level of

preliminary process capability must be achieved for all special characteristics. If no special

characteristics were identified, a customer reserves the right to request proof of a sufficient

level of preliminary process capability for other selected characteristics.

To evaluate the preliminary process capability at least 100 data in 25 subgroups

should be obtained from a significant production run. In some cases a process capability

study can be performed, with the consent of the customer, on the basis of long-term data

from the same or a similar process.

In the case of “out of control” processes the organization must identify, evaluate, and

if possible, eliminate the effect of special causes of variation prior to PPAP submission. The


24

organization shall notify the authorized customer representative of any unstable processes

and shall submit a corrective action plan.

Criteria of preliminary process capability:

Results Interpretation

Cpk >1,67 The process currently meets the acceptance criteria.

1,33 <= Cpk<=1,67 The process may be acceptable. Contact the authorized customer representative for a review of the study results.

Cpk < 1,33

The process does not currently meet the acceptance criteria. Contact the authorized customer representative for a review of the study results.

In the event that the acceptance criteria are not achieved, the organization must

submit a plan of corrective actions and modified control plan ensuring a hundred percent

inspection.

12) Qualified Laboratory Documentation

Inspection and testing for PPAP must be performed by a qualified laboratory as

defined by customer requirements (e.g. an accredited laboratory). This laboratory shall have

documentation that it is qualified for the type of measurements or tests conducted.

13) Appearance Approval Report (AAR)

In the case when design documentation includes requirements for appearance an

Appearance Approval Report (AAR) must be processed for each part (or a series of parts).

14) Sample Production Parts

The organization shall provide a sample products as specified by a customer.

15) Master Sample

The organization shall retain a master sample for the same period as production part

approval records. The master sample must be identified and must show the customer an

approval date on the sample.


25

16) Checking Aids

If the customer requests it, the organization must submit checking aids (fixtures,

gages, templates, etc.) for each part. The organization must certify that all aspects of these

checking aids agree with the requirements for dimensional requirements. The organization

shall provide preventive maintenance using checking aids for the life of the part.

17) Customer-Specific Requirements

The organization shall have records of compliance to all applicable specific

requirements of a customer.

18) Part Submission Warrant (PSW)

After fulfillment of all PPAP requirements the organization shall complete the Part

Submission Warrant (PSW). The organization shall verify that all of the measurement and

test results show conformance with customer requirements and that all required

documentation is available.

Organizations must fulfill all applicable requirements for part approval for serial

production. The way of submission can be different, because it depends on past experience

with a given supplier. They are used five levels of evidence of submitting within PPAP (see

Tab. 1.1). Level 3 is used as a starting level.

Based on the evaluation of the evidence submitted to meet the requirements of PPAP

there are three basic states of PPAP for the customer:

a) Approved

b) Interim approval (it permits the shipment of parts for a limited time or a limited

number of pieces)

c) Rejected.

In the case of interim approval or rejection the organization must remove the

shortcomings within the set deadline and submit new evidence of compliance. Records of

PPAP must be maintained for a period, when a given part is active plus one year.


26

Release of the production process and the product (PPF)

Similar to the Production Part Approval Process (PPAP), which is used within

Standard QS-9000, there is the Release of the Production Process and the Product (PPF)

applied within the VDA standard. PPF is a part of the VDA 2 "Quality Assurance of Supply"

(5th edition 2012). It defines a total of 22 requirements which approximately correspond to

the requirements of PPAP.

Tab. 1.1 PPAP submission levels.

Level 1 Warranty only (and for designated appearance items an Appearance Approval Report) submitted to the customer

Level 2 Warranty with product samples and limited supporting data submitted to the customer

Level 3 Warranty with product samples and complete supporting data submitted to the customer

Level 4 Warranty and other requirements as defined by the customer

Level 5 Warranty with product samples and complete supporting data reviewed by the organization manufacturing location

Summary of terms

Quality planning - part of quality management focused on setting quality objectives and

specifying necessary operational processes and related resources to fulfil quality objectives

APQP (Advanced Product Quality Planning and Control Plan) – product quality planning

methodology developed jointly by Chrysler, Ford and General Motors as part of Standard

QS-9000.

APQP stages - methodology APQP divides product quality planning into five overlapping

stages: 1) Planning, 2) Product design and development, 3) Process design and

development, 4) Product and process validation, 5) Feedback assessment and corrective

action.


27

Design for manufacturability and assembly - a simultaneous engineering process

designed to optimize the relationship between design function, manufacturability and

assembly.

Control plan - a document which defines all measurements and material or functional tests

applied during prototype creation or a significant production run or serial production

(Prototype control plan or Pre-launch control plan or Production control plan).

Special product and process characteristics – the designation of key product or process

characteristics used in the automotive industry.

Team Feasibility Commitment - a document which declares product quality planning team

consensus that the proposed design is feasible (can be manufactured, assembled, tested,

packaged, and delivered in a sufficient quantity on schedule at an acceptable cost to the

customer)

Significant Production Run - production run before series production under conditions

corresponding to the series production; at the production site, at the production rate, using

production tools, materials, gages and workers. A significant production run must be from a

production, which lasts from one to eight hours in which at least 300 consecutive parts must

be made, unless otherwise specified by the authorized customer representative.

PPAP (Production Part Approval Process) – a process which defines the generic

requirements for production part approval applied in the automotive industry. The purpose of

PPAP is to determine if all customer engineering design records and specification

requirements are properly understood by the organization and that the manufacturing

process has the potential to produce a product consistently meeting these requirements

during an actual production run at the quoted production rate.

Questions

1. Which activities are included in quality planning?

2. Why is quality planning a very important part of quality management?

3. What is the difference between APQP and PPAP?

4. What are main inputs of the APQP stage “Planning”?

5. What are DFMEA and PFMEA and what is the purpose of using them?

6. What is the content of the Pre-launch control plan?

7. In which situations should PPAP be applied?


28

8. How many products should be produced during a “significant production run”?

9. What are the PPAP acceptance criteria for preliminary (initial) process capability

analysis?

10. How many submission levels are used in PPAP?

References

[1] PLURA, J. Do you Plan Product Quality Effectively? Conradi Research Review, 2003,

No. 2, pp. 36-46, ISSN 1459-0980

[2] APQP. Advanced Product Quality Planning and Control Plan. 2nd edition. Chrysler

Corporation, Ford Motor Company, General Motors Corporation, 2008

[3] PPAP – Production Part Approval Process. 4th edition. AIAG, 2006

[4] GRYNA F. M., CHUA, R. C. H., De FEO, J. A. Juran's Quality Planning and Analysis:

for Enterprise Quality. 5th ed. New York: McGraw-Hill Higher Education, 2007, 774

pp., ISBN 978-0-07-296662-6


29

2. FAULT TREE ANALYSIS (FTA)

Time for learning

2 hours

Goal


Explain the principle of Fault Tree Analysis (FTA)

Explain the principles of Gate “AND” and Gate “OR”

Process a fault tree

Propose suitable actions for decreasing the probability of a fault event.

Lecture

Fault Tree Analysis (FTA) is a method of reliability analysis of complex systems,

which is based on the logical decomposition of a certain undesirable event (fault) on partial

(intermediate) or elementary (primary) events. Processing the fault tree analysis enables us

to analyse the mechanism of faults and to optimize the system with the objective to reduce

the probability of the undesirable event.

Fault Tree Analysis can be used as a tool of prevention or for the comprehensive

analysis of existing problems. The preventive use of Fault Tree Analysis is a recommended

part of a design review. Its application usually follows after FMEA, which represents basic

tool for the analysis of system behaviour.

FTA was firstly used in Bell Laboratories at the beginning of Sixties and later was

developed at the company Boeing. Initially it was mainly applied in the aerospace industry

and in nuclear energetics.


30

The main objectives of fault tree analysis are:

The identification of a cause or combination of causes leading to an undesirable (top)

event

The identification of causes, which can have the largest share on top event

occurrence

Estimating probability of events

An assessment whether the reliability of an analysed system is corresponding to the

requirements

Finding suitable actions for decreasing the probability of top event.

The basic tool of FTA is a fault tree which represents a graphical expression of the

relationships between partial events (partial failures) and specifically a final undesirable

event (top event). For fault tree processing there are used specific graphical symbols. At

present these graphical symbols are not unified, and also the standard for FTA permits the

use of various symbols depending on user preferences or the software used [1]. The

selected symbols for fault tree processing are given in Fig. 2.1.

Fault trees are displayed vertically or horizontally. In the case of vertical orientation

the top event is on the top and elementary events are lower, in the case of a horizontal

arrangement the top event can be on the left side or on the right side.

The basic elements of a fault tree are gates, which express the hierarchy of

undesirable events and define if a given event occurs only in the case of all input events

occuring or it occurs in the case of the occurence of any input event.

AND Gate

The AND gate is used in the cases when a given output event occurs only in the case

of the occurence of all input events together (logical product). The resulting probability of an

output event can be calculated (in the case when the input events are independent) on the

basis of the probability of the input events according to the formula:

nAND BPBPBPBPP 321"" (2.1)

where:

P“AND“ - the probability of an event exiting from AND gate


31

Bi - events entering the AND gate

P(Bi) - the probabilities of events entering the AND gate

With regard to the fact that the probabilities are expressed by values from 0 to 1, it is

possible to derive on the basis of this relationship that in the case of AND gate the probability

of an output event is lower or maximally equal to the probability of an input event which is

least probable:

ii

AND BPP min"" (2.2)

From this inequality it is evident that the presence of gates “AND” in the fault tree has

a positive effect on decreasing the probability of an undesirable event.

OR Gate

The OR gate is used in the cases, when an exiting event occurs in the case of the

occurence of any entry event (the logical sum). The resulting probability of an exiting event

can be calculated (in the case when the entry events are independent) on the basis of the

probability of entry events according to the formula:

nOR CPCPCPCPP 11111 321"" (2.3)

where:

P“OR“ - the probability of an event exiting from OR gate

Ci - events entering the OR gate

P(Ci) - the probabilities of events entering the OR gate.

From the given relationship it can be derived that in the case of the OR gate the

probability of an exiting event is greater or minimally equal to the probability of the most

probable entry event:

ii

OR CPP max"" (2.4)

From this inequality it is evident that the presence of OR gate in the fault tree has a

negative effect on the probability of an undesirable event.


32

The relationships between the probabilities of events exiting and entering the gates

„AND“ and “OR“ presented above are used both for estimating the probability of an

undesirable event and for the proposal of suitable actions for decreasing this probability.

Event (not elementary)

Elementary (primary) event

Gate „A“

Gate „OR“

Fig. 2.1 Basic graphical symbols used for failure tree processing.

2.1 Fault tree analysis procedure

The procedure of fault tree analysis can be divided into the following steps:

1. Defining the system which will be the object of analysis

2. A detailed understanding of a system, its functions and influencing factors

3. The identification of the possible failures of the system

1


33

4. The choice of the most severe potential failure of a system (the utilization of

FMEA results)

5. Identification of all partial events, which may participate in the occurrence of a

chosen failure event

6. Creating groups of partial events which must occur together for the occurrence of

a failure event

7. Fault tree processing using branching by the means of the AND and OR gates

8. Adding conditions for individual events occurrence if necessary

9. Next decomposition of partial events up to elementary events

10. Finishing the fault tree

11. Fault tree analysis and the proposal of changes of an assessed system if

necessary.

The initial step of fault tree analysis is a definition of the subject of analysis. It is

necessary to define the analysed system, to determine the purpose and extent of analysis

and to collect data about system functions and the conditions of its use.

The next step is focused on a detailed understanding of the system and the factors

influencing it. Fault tree processing supposes good knowledge of the functions of a system in

a normal state. All required functions of the system should be defined and the elements of

the system assuring these functions should be identified. At the same time all factors

influencing the system (including the human factor) should be analysed.

In the next step the possible failure events of the system are identified. The most

important event is chosen for a FTA application. In this stage the results of FMEA

applications are utilized.

The next step deals with the identification of all partial events which can directly

participate in chosen failure event occurrence. Branching of individual events in the fault tree

is performed using the gates “AND“ and „OR“. Partial events are gradually decomposed to

more concrete events up to elementary events.

The analysis of the processed fault tree can be qualitative or quantitative. In the case

of quantifying the probability of an undesirable top event the first step is the estimation of

elementary event probabilities. Experience with the occurrence of these events in other

systems and data declared by the producer of individual elements are used. On the basis of


34

the estimated probabilities of the elementary events it is possible, using the relationships for

individual gates, to calculate the probability of an analysed failure event. If this probability is

not corresponding to the expected reliability of the system, suitable actions for decreasing it

are sought.

An efficient way of decreasing probability of failure event is increasing the number of

AND gates in the system or decreasing the probabilities of elementary events. The most

considerable effect can be achieved by increasing the number of AND gates. It is possible to

assure this by a backup of the chosen elements of the system. Suitable actions for

decreasing the probability of a failure event can be proposed also in the cases when only

qualitative failure tree analysis is performed.

Summary of terms

Fault Tree Analysis (FTA) is a method of the reliability analysis of complex systems, which

is based on the logical decomposition of a certain undesirable event (fault) on partial

(intermediate) or elementary (primary) events. Processing the fault tree analysis enables to

analyse the mechanism of faults and to optimize the system with the objective to reduce the

probability of an undesirable event.

AND gate is used for the branching of a fault tree in the cases when a given exiting event

occurs only in the case of the occurring all entry events together (the logical product).

OR gate is used for the branching of a fault tree in the cases, when an exiting event occurs

in the case of the occurence of any entering event (the logical sum).

Questions

1. What are the differences between FTA and FMEA?

2. Explain the principle of the AND gate and the relationships between the probability of

an exiting event and the probabilities of entering events.


35

3. Explain the principle of the OR gate and the relationships between the probability of

an exiting event and the probabilities of entering events.

4. How is it possible to decrease the probability of a fault event?

References

[1] IEC 61025 Fault Tree analysis (FTA). Geneva: International Electrotechnical

Commission, 2006


36

3. ADVANCED APPROACHES TO PROCESS CAPABILITY ANALYSIS

Time for learning

5 hours

Objective


Analyze process capability

Interpret individual process capability indices

Analyze the relationship between process capability indices

Explain the effect of process variability changes on the process capability

indices

Explain the effect of process position changes on the process capability

indices

Explain which factors influence the information value of process capability

indices

Calculate confidence intervals of capability indices Cp and Cpk.

Lecture

3.1 Procedure of process capability analysis

Product quality is considerably influenced by process quality. Process quality is

evaluated by means of a process capability analysis. Process capability can be defined as

the process ability to permanently produce products meeting required quality criteria. As a

measure of a process capability various process capability indices are used.


37

Knowledge of process capability is very useful for both a customer and a producer.

For a customer it gives important information whether the conditions of production assure the

regular keeping of product specification limits. A producer can evaluate on the basis of this

knowledge the suitability of a given process for product completion, the risk of nonconforming

products occurrence, the efficiency of process improvement actions, etc. Process capability

analyses are required by quality management systems standards for the automotive industry

[1].

For correct process capability evaluation it is necessary to keep a correct procedure.

In the case of measurable quality characteristics it should include these steps [2]:

1. choice of the product quality characteristic

2. measurement system analysis

3. data collection from a running process

4. exploratory data analysis

5. process statistical stability verification

6. data normality verification

7. process capability indices calculation and its comparison to required values

8. the implementation of capability improvement actions if necessary.

1) Choice of a product quality characteristic

Process capability is evaluated on the basis of the chosen quality characteristic

values of produced products. An appropriate quality characteristic should be influenced by

an analysed process and requirements must be specified for it. The agreement of quality

criteria with customer requirements should be verified.

2) Measurement system analysis

Before data collection it is necessary to verify the properties of the measurement

system. Very important properties from process capability analysis point of view are bias,

which characterizes a systematic error of measurement and repeatability and reproducibility,

which characterize measurement variability. Unsatisfactory measurement system properties

can considerably distort results of the process capability analysis.

3) Data collection from a running process

Data about a chosen quality characteristic should be collected from a running process

during a satisfactory long period, which includes the influence of all common sources of

process variability. Product characteristic data are obtained by the measurement of product


38

subgroups taken from the process in regular intervals. There should be collected a hundred

pieces of data at minimum.

4) Exploratory data analysis

Exploratory analysis of collected data should be focused to outlier identification,

assessment of the character of measured characteristic distribution and data independence

analysis. It is very useful to apply graphical tools as histograms and box plots and

appropriate tests.

5) Process statistical stability verification

Process variability can be caused either by random causes only or by random and

assignable causes. Process capability can be evaluated only in cases, when a process is

influenced by random causes only. This state is classified as an „in control process“ and its

advantage is the predictability of a process behaviour. Verification of process statistical

stability is performed using a control chart, which can distinguish the influence of assignable

causes from the influence of random causes.

In the case when a process is „out of control“ it is possible to use two ways. The first

one is based on assignable causes identification (with the help of control chart analysis) and

their removal and the repetition of process capability analysis. The second one is based on

using original data for the assessment of „process performance“. Process performance

characterizes last process behaviour only and is not usable for process prediction.

6) Data normality verification

Standard capability indices are based on the assumption of data normality. Therefore

it is necessary to verify this assumption by using of suitable test of normality (e.g. Shapiro-

Wilk´s test, Chi square test, etc.). The situations when data are not normally distributed can

be dealt with by data transformation or by finding any other theoretical model suitable for the

description of experimental data distribution.

7) Process capability indices calculation and its comparison to required values

Process capability is evaluated by various process capability indices. It is necessary

to take into consideration the fact that each index characterizes a process capability in a

various manner. In practice, Cp and Cpk indices, which evaluate the potential and real ability

of a process to produce products meeting tolerance, are most frequently used. In the lower


39

extent Cpm index, which especially evaluates a process ability to achieve a target value of a

quality characteristic, and Cpmk index, which evaluates both aspects, are used.

8) Implementation of capability improvement actions if necessary

In the cases of finding that a process is not capable suitable improvement actions

should be put into effect. The type of appropriate action can be derived from the capability

indices values and from a graphical visualization of the situation.

3.2 Process Capability Indices

Cp index

The Cp index is defined as the ratio of maximum allowable range of a given quality

characteristic to the range over which the process is actually varying. It characterizes

potential process capability only, because it takes into account only characteristic variability

and not characteristic position with regard to tolerance limits [3]:

6

LSLUSL

pC

(3.1)

where: LSL - lower specification limit

USL - upper specification limit

- standard deviation

Actual variability is expressed by 6, which for normally distributed data represents

the range, in which a given quality characteristic will be with probability 0,9973.

Cpk index

The Cpk index takes into account not only characteristic variability, but also its position

with regard to tolerance limits. It characterizes actual process capability to meet tolerance

limits and due to this it is the most frequently used process capability index in practice. The

Cpk index is expressed as the ratio of the distance from characteristic mean to near tolerance

limit to half of the actual characteristic variability:

3;

3min;min

USLLSLCCC pUpLpk (3.2)


40

where:

μ - characteristic mean

The Cpk index is a crucial criterion of process capability assessment. As the only one

of the discussed capability indices the Cpk index is directly related to the expected occurrence

of non-conforming products. Processes are usually classified as capable in cases, when the

Cpk value is 1,33 at minimum.

The relation between Cp a Cpk can be expressed by this formula [4]:

3

2

LSLUSL

CC ppk (3.3)

Cpk and Cp indices have the same value only in the case, when characteristic mean

lies in the centre of tolerance limits, in the other cases the Cpk is lower. The higher is the

distance of the quality characteristic mean from the centre of tolerance limits, the higher is

the difference between Cpk and Cp. For example, when a characteristic mean lies one sigma

from the centre of tolerance, the difference between Cpk and Cp is 0,33.

Cpm index

The Cpm index compares the maximum allowable quality characteristic range with the

actual characteristic variability around the target value:

226 T

LSLUSLCpm

. (3.4)

where:

T – target value

This index takes into account both the given quality characteristic variability and the

rate of target value achievement. Its use is recommended only for cases when the target

value lies in the centre of tolerance limits.

It is possible to derive this relationship between Cpm and Cp:

2

1

T

CC

p

pm (3.5)


41

This relationship shows that the Cpm index is less than or equal to the Cp index.

Equality of these indices is achieved only when the mean value of quality characteristic

corresponds to the target value. For example in the case of moving the mean value by one

standard deviation from the target value, the value of Cpm index will be 1.41 times lower than

the value Cp index.

Cpm* index

The Cpm* index is a generalised Cpm index for cases when the target value doesn´t lie

in the centre of tolerance limits or only one-sided tolerance is specified. This index compares

the distance of a target value from the near tolerance limit with half of the actual variability of

the characteristic around the target value:

2222

*

3;

3min

T

TUSL

T

LSLTC pm

(3.6)

Cpmk index

The Cpmk index compares the distance from characteristic mean to the near tolerance

limit with the half of the actual variability of the characteristic around the target value:

2222 3;

3min

T

USL

T

LSLC pmk

(3.7)

The Cpmk index utilises the good property of Cpk, especially its ability to recognize,

whether values of a given characteristic actually lie inside tolerance limits, which combines

with the rate of target value achievement. It is possible to derive this relation between Cpmk

and Cpk indices:

2

pk

pmk

T1

CC

(3.8)

The ratio within brackets under the square root represents the distance from the

characteristic mean to the target value expressed by the number of standard deviations. For

example in the case of the quality characteristic mean shifting by one standard deviation


42

from the centre of tolerance limits the Cpm index will be 1.41 times lower than Cp index. Both

indices have the same value in the case, when the mean is equal to the target value.

For the cases when both tolerance limits and the target value of a given characteristic

are specified, the practical use of this procedure of capability indices determination can be

recommended:

1. Firstly the Cpk index should be determined for the evaluation of the real process

capability to meet tolerance limits

2. The Cp index should be determined (in the case of two-sided tolerance), because

its comparison to the Cpk index makes it possible to evaluate how potential

process capability is utilised and to find a suitable way for process improvement

3. The Cpmk index should be determined for obtaining information about the rate of

target value achievement. It makes sense especially in cases when a process is

capable to meet required tolerance limits.

Integrated information about process capability can be obtained using a suitable

combination of capability indices. For objective process capability assessment it is necessary

also to visualise the distribution of a quality characteristic (e.g. by using a histogram) with

regard to tolerance.

3.3 The influence of process changes to capability indices

Various capability indices assess process capability in various ways and their

interpretation should correspond to this fact. For the objective assessment of process

capability, suitable combinations of various capability indices must be used and always it is

necessary to append a graphical presentation of quality characteristic distribution, e. g. by

using of a histogram with drawn-out tolerance limits.

3.3.1 Sensitivity of capability indices to the quality characteristic variability

The values of all capability indices are a function of the standard deviation of the

monitored quality characteristic. If the standard deviation is in the denominator of appropriate

formulas, hyperbolic dependence can be expected.


43

The dependence of the process capability indices on the relative value of standard

deviation related to the half of tolerance width (d = (USL-LSL)/2) in the case where the mean

value of the quality characteristic is located in the centre of the tolerance (M=(USL+LSL)/2)

as well as when the target value corresponds to the centre of tolerance is shown in Fig. 3.1.

The figure shows that in this case the courses of capability indices decreasing with

increasing of relative standard deviation values are the same. The most considerable drop of

capability indices occurs for the relative standard deviations of less than about 0.15

(corresponding Cp index values are higher than about 2.2), further increasing of the standard

deviation has a milder effect on capability indices decrease.

In cases where the target value of the quality characteristic does not correspond to

the centre of tolerance, there can be expected lower values of capability indices which takes

into account the achievement of the target value (Cpm, Cpm* and Cpmk). Appropriate

dependences in the case when the target value lies at a distance of one-eighth of the

tolerance width from the centre tolerance (|T-M|=0,25d) are shown in Fig. 3.2 (instead of the

Cpm index which is not recommended in cases where the target value is not corresponding to

the centre of tolerance the course of Cpm* index is given).

Fig. 3.1 The dependence of process capability indices on the relative value of the standard

deviation of a quality characteristic (μ = M = T)

It can be seen from these dependences that in this case the indices Cpm* and Cpmk are

in the area of low variability especially much less sensitive to change of the process

variability than indices Cp and Cpk. The reason is the fact that the denominator in the formulas

for calculating these indices includes both the standard deviation and the distance of mean

value from the target value.


44

Fig. 3.2 The dependences of process capability indices on the relative value of the standard deviation of the quality characteristic (μ = M; |T-M| = 0,25d)

3.3.2 Sensitivity of capability indices to the quality characteristic position

If the presented capability indices assess the position of a monitored quality

characteristic towards the specifications in a different manner, there can be expected their

different sensitivity to changes in the position. Dependences of individual capability indices

on the relative distance of the mean value of the quality characteristic from the center of

tolerance ((μ-M)/d) are in the case of constant variability (σ/d = 0.25, which corresponds

Cp=1.33) and in the equality of a target value with a center of tolerance (T = M) shown in Fig.

3.3.

The dependence of the Cp index on the mean value of a quality characteristic is

corresponding to a parallel with the x axis. This confirms the fact that this index characterizes

only potential process capability (the ability to 'be in tolerance"), and does not reflect the

position of a quality characteristic to the tolerance limits.

The dependence of the Cpk index on the mean value of a quality characteristic is

characterized by the refracted shape of the curve. It is due to the fact that the Cpk index is the

smaller of the pair of indices CpL and CpU. By decreasing the absolute value of the mean

value distance from the centre of the tolerance, the Cpk index linearly increases and in a

middle of tolerance achieves the maximum value (the potential process capability is

maximally utilized in this case (Cpk = Cp)). When the mean value is reached some tolerance

limits the Cpk index reach zero and when the tolerance limits are exceeded it takes on

negative values.


45

Fig. 3.3 Dependences of process capability indices on the relative distance of the mean

value from the centre of tolerance (σ/d = 0,25; T= M).

The dependence of the Cpm index on the mean value has, in this case when the target

value is in the middle of tolerance, a similar course with a peak as the Cpk index, but this is

not a refracted curve, but a smooth curve with a peak corresponding to the target value. Fig.

3.3 shows that in the case of small deviations of the mean value from the target value the

decrease in the value of this index is greater than in the case of the Cpk index. On the

contrary, at higher distances of mean value from the target value, this decrease is much

milder and practically does not indicate exceeding the tolerance limits.

The dependence of the Cpmk index is similar as the dependence of the Cpm index but

its course is steeper. Unlike Cpm, the Cpmk index unambiguously indicates a situation where

the mean value of the quality characteristic overruns some tolerance limits, since the value of

this index similarly as in the case of Cpk index gets into the area of negative values.

In cases where the target value is not in the centre of tolerance there can be

expected a different behaviour of indices reflecting the degree of target value achievement.

The dependences of capability indices on the relative distance of the mean value of the

quality characteristic from the centre of tolerance in the case where the target value is at a

distance of one eighth of the width of the tolerance above the centre of the tolerance (T-M =

0.25d) are shown in Fig. 3.4. The features of dependences are essentially retained,

however in the case of Cpm* and Cpmk indices there is a shift of the curve maximum towards


46

the target value, which is accompanied by an asymmetry of indices changes when moving

the mean value to the upper and lower tolerance limits. There can be seen also a reduction

in the values of these indices compared to the previous case.

Fig. 3.4 Dependences of process capability indices on the relative distance of the mean

value from the centre of tolerance (σ/d = 0.25; T= M+0.25d)

The figure also shows that the maximum values of C*pm and Cpmk indices are not

achieved at the same position of the quality characteristic to the tolerance limits. While the

maximum value of the C*pm index is achieved when the mean value of the quality

characteristic reaches the target value, the maximum value of the Cpmk index is reached in

the region between the target value and the centre of the tolerance. It is connected with the

fact that the Cpmk index is basically a combination of Cpk, Cp and Cpm indices.

3.3.3 Ambiguity of process capability indices in relation to the distribution of the

quality characteristic

Closer analysis of the capability indices shows that none of them clearly identify the

actual distribution of the quality characteristic. This is most evident in the case of the Cp

index, which does not reflect the position of the quality characteristic. For other indices this

ambiguity is given by the fact that their values depend both on the position and on the

variability of the monitored quality characteristic. In the case of two sided tolerances

moreover other ambiguity is exhibited for these indices. It is given by the fact that their values


47

do not provide information to which the tolerance limit the process is closer, or whether the

mean value is above or below the target value. Examples of this ambiguity will be illustrated

in the case of the Cpk index.

The fact that the Cpk index value depends both on the mean value and standard

deviation may lead to a situation that the processes with a different distribution of the quality

characteristic may have the same value of Cpk index. If the changes in the mean value and

variability are proportional, so that the ratio of the distance of the mean value from closer

tolerance limits to the standard deviation is the same, it means that also the Cpk index is

same. An example of processes with the same Cpk index (Cpk = 1.33) is shown in Fig. 3.5.

Cases A, B and C or C, D and E represent the situations in which the distance of the

mean value from the closer tolerance limit and standard deviation are proportionally

changed. Cases A and E, or B and D represent situations which changes neither the

distance of mean value from the closer tolerance limit and standard deviation is changed, but

a closer tolerance limit becomes the opposite limit (see Tab. 3.1).

Fig. 3.5 Different situations of monitored quality characteristic distribution corresponding to the same Cpk index (Cpk = 1,33).

2 4 2 6 2 8 3 0 3 2 3 4 3 6

LSL USLA

B

C

D

E


48

Tab. 3.1 Process capability indices values and the expected occurrence of nonconforming products (ppm) in accordance with the situations in Fig. 3.5

A B C D E

μ 28 29 30 31 32

σ 0,5 0,75 1 0,75 0,5

Cpk 1,33 1,33 1,33 1,33 1,33

Cp 2,67 1,78 1,33 1,78 2,67

CpL 1,33 1,33 1,33 2,22 4,00

CpU 4,00 2,22 1,33 1,33 1,33

Cpm (T = 30) 0,65 1,07 1,33 1,07 0,65

Cpmk 0,32 0,80 1,33 0,80 0,32

ppm 33,05 33,05 66,10 33,05 33,05

For identification of changes in the situations A, B and C or C, D and E, there can be

used (in the case of two-sided tolerance) the Cp index which distinguishes the different

variation of the monitored quality characteristic. However, for distinguishing situations A and

E, or B and D this index is not usable.

Unambiguous information on the distribution of the quality characteristic in relation to

the tolerance limits in the case of two-sided tolerance provide values of both partial process

capability indices CpL and CpU. Although both of these indices depend on the distance of the

mean value from the tolerance limit and on the standard deviation, they unambiguously

determine to which tolerance limit the process is shifted.

Like in the case of the Cpk index the Cpm index value also does not clearly identify the

appropriate distribution of the monitored quality characteristics. If the standard deviation of

the quality characteristic and the distance of the mean value from the target value are

changed proportionally so that the value of the denominator in the calculation formula is the

same, the value of Cpm index is also the same.


49

3.4 Factors affecting the results of process capability analysis

During interpretation of the results of process capability analysis it is necessary to

take into consideration factors which affect process capability indices values. Some of the

most important factors are:

chosen quality characteristic

type of used capability indices

correspondence of used quality criteria with customer requirements

way of data collection

suitability of measurement system

the number of processed data

the fulfillment of limiting conditions (data normality, process stability).

Chosen quality characteristic

Calculated capability indices values should be always appended by information,

which product quality characteristic was used as a criterion of process capability. The

general term „capable process“ should be used only in cases when process capability was

evidenced with regard to all specified quality characteristics.

Type of used capability indices

Various capability indices assess process capability in various ways and their

interpretation should correspond to this fact. For an objective assessment of process

capability, suitable combinations of various capability indices must be used and always it is

necessary to append a graphical presentation of quality characteristic distribution, e. g. by

using a histogram with drawn-out tolerance limits.

Correspondence of used quality criteria with customer requirements

On the basis of practical experience it can be seen that in some cases incorrectly

defined tolerance limits are the cause of process incapability. Therefore, the verification of

tolerance limits correctness and their correspondence with customer requirements should be

an important part of process capability analysis.

Way of data collection

The method of data collection can considerably influence the determined process

capability indices values. It is especially important whether collected data describe all

common sources of variability influencing a process. In cases when data were collected


50

under limited conditions, determined process capability is related to these conditions and it is

necessary to distinguish it.

Suitability of measurement system

Process capability analysis is based on the data obtained by the measurement of a

selected product quality characteristic. The properties of the used measurement system

(bias, precision, repeatability, reproducibility, stability, linearity, etc.) can considerably

influence the quality of measured data and in this way also the values of determined process

capability indices.

Number of processed data

The number of processed data influences process capability indices values in two

ways. The first one is the fact that a small number of data insufficiently characterizes the real

variability of a product quality characteristic. The second one is connected with the fact that

process capability indices are calculated using a statistical estimation of mean value and

standard deviation, it means that they are also estimations. The smaller the number of

processed data is, the higher the width of the confidence interval of the process capability

indices estimation.

Fullfilment of limiting conditions

A very important aspect of process capability analysis is the fulfillment of limiting

conditions. A process must be in control (must be influenced only by the random causes of

variability) and data normality must be met. In cases when some of these conditions are not

fulfilled the calculated process capability indices can be completely valueless.

3.5 Confidence intervals of process capability indices

Process capability indices are calculated using a statistical estimation of the mean

value and standard deviation, it means that they are also estimations.

Two-sided confidence intervals

The width of the confidence interval of process capability indices considerably

depends on the number of processed data. In the case of the Cp index, the two-sided

confidence interval is expressed as follows:


51

p

2

,2

1

pp

2

,2 CCC

(3.9)

where:

2

,2

1

2

,2

;

- quantiles of 2 distribution for ν degrees of freedom

α - level of significance

ν - degrees of freedom.

For cases when the standard deviation is estimated on the basis of average variability

characteristics in the subgroups, degrees of freedom are calculated using the relation:

cf)1n(k (3.10)

where:

k - number of subgroups

n – subgroup size

fc - correction factor depending on the size of subgroups and the way of estimating

the standard deviation

In the case of standard deviation estimation using the average of the ranges in

subgroups and subgroup size n = 5 the correction factor fc takes the value fc = 0.906. In

cases where the standard deviation is estimated using the sample variances in the

subgroups, the correction factor is not considered.

If the standard deviation of the quality characteristic is estimated as the sample

standard deviation calculated from all collected data, the number of degrees of freedom is

calculated according to the relation:

1N (3.11)

where:

N – the number of data.

The dependence of the Cpk confidence interval width on the number of data in the

case when the Cp estimation is 1.33 and level of significance 0.05 can be seen in the Fig.


52

3.6. It is obvious that in the case of small number of processed data the confidence interval is

very wide and only with the higher number of processed data it narrows. For example, if we

calculate the Cp index on the basis of ten measured data and calculated value is 1.33, the

real Cp value lies with probability 0.95 in the interval from 0.73 to 1.93, if we calculate the Cp

index on the basis of a hundred measured data the real Cp value lies in the interval from 1.14

to 1.51.

Determination of the Cpk index interval estimation is a more complicated task since

the calculation of the index needs the estimation of both standard deviation and the mean

value of the quality characteristic. The two-sided confidence interval of the Cpk index can be

calculated according to the formula:

pk2

1

pkpk2

1

C)2

u

1(CC)2

u

1(

(3.12)

where:

21

u - quantile of standard normal distribution

ν - degrees of freedom

Fig. 3.6 Confidence limits of the two sided confidence interval of Cp for calculated Cp

value 1.33 depending on the number of processed data (α =0.05).

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

0 20 40 60 80 100 120 140

Cp c

onfidence inte

rval lim

its

number of data


53

Analysis of confidence interval of Cpk index estimation leads to similar conclusions as

in the case of the Cp index. For example, if we calculate the Cpk index on the basis of ten

measured data and the calculated value is 1.33, the real Cpk value lies with probability 0.95 in

the interval from 0.71 to 1.94. If we calculate the Cpk index on the basis of a hundred

measured data the real Cpk value lies in the interval from 1.14 to 1.52.

One-sided confidence intervals

For practical use it is more suitable to analyze the one-sided confidence intervals of

capability indices. With regard to the fact that the process can be considered as capable if

process capability indices are greater than the defined minimum value, it is sufficient to

calculate the one-sided confidence interval with a lower confidence limit. Information about

this confidence limit provides a "guarantee" (with the risk of error α) that the capability index

is not worse than the declared limit.

Appropriate one-sided confidence intervals for capability indices Cp and Cpk can be

expressed as:

pp CC ˆ2

,

(3.13)

ppk Cu

C ˆ2

1 1

(3.14)

The given relations can be used for calculating the minimum value of capability

indices estimation that needs to be achieved to confirm that a defined criterion was met. For

example, if the usual criterion for capable processes is Cpk ≥ 1.33 and we have processed

only ten data, we can classify a process as capable only in the cases when the estimated

value is greater than 2.17. If we have processed a hundred data we can classify a process

as capable in cases when the estimated value is greater than 1.51. The dependence of the

minimum value of Cpk index estimation on the number of processed data is shown in Fig. 3.7.


54

Fig. 3.7 Dependence of the minimum value of Cpk index estimation when a process

can be considered as capable on the number of processed data

Summary of terms

Process capability - process ability to permanently produce products meeting required

quality criteria

Process „in control“ (statistically stable process) - a process which is influenced by

random causes only

Confidence interval of process capability index - interval, in which the real value of a

given index is lying with a defined probability (usually with a probability 0.95)

Questions

1. How is it possible to verify whether the process is „in control“?

2. How is it possible to verify the normality of data?

3. What information does the Cpk index provide and what is its relation to the Cp index?

0

0,5

1

1,5

2

2,5

3

3,5

4

0 20 40 60 80 100 120 140

min

imal valu

e o

f C

pk e

stim

ation

number of data


55

4. What information does the Cpmk index provide and what is its relation to the Cpk index?

5. What type of function is corresponding to the dependence of Cp or Cpk on the

standard deviation of a monitored quality characteristic?

6. Explain the dependence of Cpk on the mean value of a given quality characteristic.

7. Under what conditions is the change of position of a given quality characteristic

towards tolerance limits not accompanied by a change of Cpk index?

8. Explain how the number of collected data affects the information value of the

capability indices.

9. Can the same process be classified in one case as capable and in the second case

as incapable?

10. We have a calculated two-sided confidence interval of Cpk. What condition should be

met for the capable process?

References

[1] KOTZ, S. – LOVELACE, C. R.: Process Capability Indices in Theory and Practice. New

York: Oxford University Press, 1998, ISBN 0-340-69177-8

[2] PLURA, J.: Procedures and Methods of Quality Planning and their Use for Forming

Process Optimization. In: Engineering the Future. Laszlo Dudas (editor). 1st edition.

Rijeka: Sciyo, 2010, Chapter 13, p. 257-279, ISBN 978-953-307-210-4

[3] KOTZ, S. – PEARN, V. L.: Encyklopedia and Hanbook of Process Capability Indices.

Singapore: World Scientific Publishing Co. Pte. Ltd. 2006, ISBN 981-256-759-3

[4] PLURA, J.: Practical Aspects of Process Capability Indices Use. In: Annals of DAAAM

& Proceedings of the 13th International DAAAM Symposium. Wienna: DAAAM

International Wienna, 2002, s. 433-434, ISBN 3-901-509-29-1


56

4. THE PROCEDURES OF PROCESS CAPABILITY ANALYSIS IN NON-STANDARD SITUATIONS

Time for learning

3 hours

Goal


Analyse process capability in the cases when a process is not “in control”

Analyse process capability in the cases when a monitored quality

characteristic is not normally distributed.

Lecture

As part of process capability analysis it is necessary to verify the compliance with the

assumptions. The process should be “in control” and the distribution of monitored quality

characteristic should correspond to normal distribution [1]. In the case that any of these

assumptions are not met, it is necessary to use special methods of analysis.

4.1 Procedures of process capability analysis in the cases, when a process is „out of control“

In the cases when an analysed process is „out of control“ it is possible to use these

procedure possibilities of process capability analysis:

a) Ensuring the statistical stability of a process and re-evaluation of its capability

b) Evaluating process performance.


57

Ad a)

If it is possible to identify the assignable causes influencing a process, it is suitable to

implement corrective actions for their removal. The whole procedure should have these

steps:

1. Identification of the assignable causes of process variability (using a control chart)

2. Permanent removal of the assignable causes of variability

3. The collection of new data from the running process

4. A new analysis of process capability (the first step should be an analysis of process

statistical stability).

Ad b)

In the cases when a process is out of control and the information for the analysis and

removal of assignable causes is not available, it is possible to evaluate „process

performance" using performance indices Pp, Ppk, etc. These indices characterise current

process behaviour only and it is not possible to use them for the prediction of future process

behaviour.

Process performance evaluation is used especially in the cases when:

• It is not possible to identify the assignable causes of process variability

• It is not possible to remove the assignable causes of process variability

• It is not possible to verify if the process is „in control“.

Process performance indices are calculated according to the same formulas as

process capability indices, the only difference is the way of estimating standard deviation.

While the standard deviation for process capability indices calculatation is estimated on the

basis of the average variability in the subgroups, for process performance indices calculation

it is estimated as a sample standard deviation for all data:

6

LSLUSLPp

(4.1)

3

USL;

3

LSLminP;PminP pUpLpk (4.2)


58

22pm

T6

LSLUSLP

(4.3)

2222

*

pm

T3

TUSL;

T3

LSLTminP

(4.4)

2222pmk

T3

USL;

T3

LSLminP

(4.5)

1n

xx

sˆ

n

1i

2

i

(4.6)

It is important to be aware that process performance indices as well as process

capability indices are based on the assumption of data normality. It means that before

calculating these indices it is necessary to verify this assumption. Regarding the fact that the

variability of these processes is influenced not only by random causes can be problematic for

the fulfilment of data normality.

4.2 Procedures of process capability analysis when data normality is not met

For process capability assessment there are standardly used process capability

indices based on the assumption of the normality of a monitored quality characteristic.

Situations when data normality is not met can be basically solved in these ways [2]:

a) Using another theoretical model of distribution

b) Data transformation to the variable corresponding to normal distribution

c) Using indicators which are not based on a concrete model of data distribution.

Before using some possible procedure it is necessary to pay attention to the causes

of data non-normality. The cause of data non-normality can be e.g. outliers caused by a

measurement error, however it is often data inhomogeneity caused by the step changes of

conditions of the process course during data collection. In these cases it is necessary to

eliminate appropriate causes and the affected data or to collect new data.


59

Using another theoretical model of distribution

One possibility of dealing with a situation when data are not normally distributed is

finding another suitable theoretical model of probability distribution for the description of the

distribution of a monitored quality characteristic. This procedure should be applied especially

in the cases when a monitored quality characteristic is naturally not normally distributed. The

choice and verification of a selected probability distribution model should be done using

some goodness of fit test. It is suitable to use some statistical software.

Similarly as in the case of normal distribution when values - 3 a + 3

correspond to the quantiles, for which the distribution function is 0.00135 and 0.99865, in the

case of another probability distribution model it is necessary to find quantiles corresponding

to these values of the distribution function. Appropriate process capability indices /

pC and

/

pkC can be calculated according to the formulas:

00135,099865,0

/

xx

LSLUSLC p

(4.7)

5,099865,0

5,0

00135,05,0

5,0/ ;minxx

xUSL

xx

LSLxC pk (4.8)

where: x 0,00135 - 0,135 % quantile of corresponding distribution

x 0,99865 - 99,865 % quantile of corresponding distribution

x 0,5 - median of corresponding distribution.

For marking the indices modified symbols are used, so calculated capability indices

do not validate a series of relationships that were derived on the basis of the assumption of a

normal distribution (for example, the estimate of the probability of non-conforming products).


60

Data transformation

In the case of data transformation the original variable data are recalculated using a

suitable transformation function into a new „transformed“ variable whose distribution can

correspond to normal distribution. Further calculations within the framework of process

capability analysis are done with this transformed variable.

For data transformation, various transformation functions are used. For example, the

often applied Box-Cox transformation uses this transformation function:

(4.9)

Another possibility is e. g. a power transformation which uses this function:

(4.10)

For finding a suitable transformation function it is suitable to use statistical software

which makes it possible to find the optimum values of parameters.

Process capability analysis using data transformation should be performed in these

steps:

1. Finding a suitable transformation function and the calculation of a transformed

variable

2. The verification of data normality of the transformed variable

3. Recalculating original tolerance limits using a given transformation function

4. Calculating process capability indices according to standard formulas for

normally distributed data.

It is necessary to append the obtained results with information that they were

obtained using data transformation.

Using indicators which are not based on a concrete model of data distribution

In the cases when both the finding of another theoretical model of distribution and

data transformation are not successful it is possible to assess process capability using some

other indicators, which are not based on a concrete model of data distribution. Often there


61

are used indicators for process capability evaluation in the cases of non-measurable

characteristics. Usually they are indicators derived from the proportion of nonconforming

products e.g. ppm, Sigma level, Cp equivalent, Cpk equivalent, etc. Besides, it is possible to

use some proposed special capability indicators, but they are not used in practice up to now.

Summary of terms

Data transformation is the recalculation of a given variable using a suitable transformation

function into a transformed variable, whose distribution is better corresponding to normal

distribution.

Process performance characterizes the ability of a process which is out of control to meet

tolerance limits.

Questions

1. What procedures of process capability analysis can be used in the case when a process

is “out of control”?

2. What procedures of process capability analysis can be used in the case when a

monitored quality characteristic is not normally distributed?

3. What is the purpose of data transformation?

References

[1] KOTZ, S. – LOVELACE, C.R.: Process Capability Indices in Theory and Practice.

New York: Oxford University Press, 1998, ISBN 0-340-69177-8

[2] PLURA, J.; ZEMEK, M.; KLAPUT, P. Approaches to the Process Capability Analysis

in the Case of Non-Normally Distributed Product Quality Characteristic. In: METAL

2013: 22 nd International Conference on Metallurgy and Materials. Ostrava:

TANGER, 2013, pp. 1700-1705. ISBN 978-80-87294-41-3.


62

5. PROCESS CAPABILITY ANALYSIS IN THE CASE OF NON-MEASURABLE CHARACTERISTICS

Time for learning

3 hours

Goal


Analyse process capability in the cases when a monitored quality

characteristic is not measurable.

Lecture

The procedures of process capability analysis mentioned above were based on the

fact that amonitored quality characteristic is a measurable variable. However there are many

situations when the quality of produced products can be evaluated only on the basis of

attributes and it is possible to distinguish conforming and nonconforming products or to

determine the occurrence of the nonconformities.

The average levels of nonconforming product occurrence or nonconformities

occurrence are most often used for process capability assessment in the case of attributes.

Sometimes they are recalculated to a Sigma level or Cp or Cpk equivalents.

Process capability analysis in the case of attributes should be performed using these

steps:

1. Choice of quality criterion of produced products

2. Measurement system analysis

3. Data collection from a running process


63

4. Exploratory data analysis

5. Verification of process statistical stability

6. The calculation of capability indicators and their comparison with required values

7. Proposal and implementation of actions for process capability improvement if

necessary.

1) Choice of the quality criterion of produced products

The quality of products is usually assessed on the basis of nonconforming products or

nonconformities. The proportion of nonconforming products or a number of nonconformities

in a unit are suitable criteria which make it possible to compare various processes. It is very

important to define exactly the nonconformities and nonconforming products in accordance

with customer requirements.

2) Measurement system analysis

Also in the cases of non-measurable quality characteristics it is necessary to verify

whether a measurement system (an inspection system) is providing sufficiently objective

data. For this purpose procedures for measurement system analyses for attributes are used

[1].

3) Data collection from a running process

The collection of data about chosen quality criteria of the products produced should

assure that data must describe the influence of all common factors influencing a given

process. The way of data collection should make it possible to process a suitable control

chart for attributes. Data about 25 subgroups of products should be collected as minimum.

4) Exploratory analysis of collected data

Exploratory data analysis, regarding non-measurable characteristics and the smaller

number of data (e.g. 25 proportions of non-conforming products), has limited possibilities.

Mostly it is sufficient to analyse collected data in a control chart.


64

5) Verification of process statistical stability

Verification of process statistical stability is performed using a suitable control chart.

In the case of monitoring nonconforming products a control chart for the proportion of

nonconforming units (p-diagram) is used. In the case of monitoring nonconformities control

chart for the number of nonconformities per unit (u-diagram) is used.

6) Calculation of the capability indicators and their comparison with required values

In the case of non-measurable characteristics various indicators are used for process

capability assessment. In the case of monitoring the occurrence of nonconforming products it

is most often the average proportion of nonconforming products, which is often presented by

the means of ppm (parts per million). For process capability assessment it is necessary to

compare this determined value with a maximal acceptable level. The average proportion of

nonconforming products can be also recalculated to a Sigma level or Cp or Cpk equivalents

for which limited values are usually defined.

In the case of monitoring the occurrence of nonconformities there can be used the

average number of nonconformities per unit - dpm (defects per million) as process capability

indicator. Often there is also used the indicator dpmo (defects per million opportunities),

which represents the number of defects per million opportunities to the defects. For the

decision if a process is capable it is also necessary to compare determined values with

defined criteria.

7) Proposal and implementation of actions for process capability improvement if

necessary

In the case of the conclusion that a process is not capable it is necessary to propose

a suitable improvement of the process. The choice of suitable actions is more complicated in

comparison with the process of capability analysis in the case of measurable characteristics

when it was possible to determine if appropriate actions should be focused on a better

centering of the process or on decreasing process variability. It is necessary to find suitable

actions for decreasing the occurrence of nonconforming products without closer information

how to do it.


65

5.1 Process capability indicators in the case of monitoring nonconforming products

Average proportion of nonconforming products

The average proportion of nonconforming products is the most often used process

capability indicator in this case. However if we want to use it as a process capability

indicator we must verify if the process is „in control“ (if the process is influenced only by the

random causes of variability). The statistical stability of the process is assessed by means of

a control chart for nonconforming products proportion (see Fig. 5.1). The levels of control

limits and the central line are calculated according to the relationships:

CL p

x

n

jj

k

jj

k

1

1 (5.1)

n

p1p3pLCL

(5.2)

n

p1p3pUCL

(5.3)

where: p - the average proportion of nonconforming products in the subgroup

xj - the number of nonconforming products in the j-group

nj - subgroup size

k - the number of subgroups

n - average subgroup size.

On the basis of the formula (5.1) it is obvious that the level of the central line in this

control chart is corresponding to the average proportion of nonconforming products. In the

case when control chart analysis results in the conclusion that a process is “in control” the

level of the central line characterizes the capability of a given process. For the final

conclusion about process capability it is necessary to compare this level with the value

required by the customer (maximally the acceptable proportion of non-conforming products

usually indicated as p0).


66

Fig. 5.1 Example of a control chart for the proportion of nonconforming products for a process which is „in control“ where the level of the central line characterizes the capability of a given process.

During decision-making about process capability it is necessary to take into

consideration that the calculated average proportion of nonconforming products is only a

point estimation of the real proportion. Appropriate process capability assessment should be

completed by information about the confidence interval of proportion or to apply a hypothesis

testing.

For the calculation of the upper limit of a one–sided confidence interval of non-

conforming products proportion there can be used the formula:

1n

)1n

d1()p1(p

u1n

d)p21(pp

**

1

*

*t,U

(5.4)

where:

1

1*

n

xp

pU,t - the upper limit of the one–sided confidence interval of

nonconforming products proportion

x - the number of nonconforming products in the sample

252321191715131197531

0,0014

0,0012

0,0010

0,0008

0,0006

0,0004

0,0002

0,0000

Sample

Pro

po

rtio

n

_P=0,000518

UCL=0,001199

LCL=0

p0


67

n - sample size

1u - the quantile of standardized normal distribution

- the level of significance

d - parameter depending on the level of significance (for α=0,05

d=0,677).

It is obvious from this formula that the width of the confidence interval depends on the

point estimation of the proportion of nonconforming products, on the chosen level of

significance and on the sample size. Especially dependence on the sample size is

significant. The process should be considered as capable only in the cases when the

determined upper limit of confidence interval does not exceed the maximal acceptable

proportion p0.

Using the formula for the calculation of the upper limit of a one-sided confidence

interval of non-conforming products proportion, it is possible to determine the maximal values

of the point estimation of nonconforming products proportion for which the process can be

considered as capable. Appropriate maximal values in the case of limited value p0 = 0.01 and

significance level α = 0.05 in the dependence on the sample size are displayed in Fig. 5.2.

The significant influence of the sample size on these maximal values can be seen.

On the basis of the course of a given dependence it is possible e.g. to determine that

a process can be considered as capable if for sample size 1200 units point estimation of

nonconforming products the proportion does not exceed the value 0.0058 or for sample size

6000 units point estimation of the nonconforming products proportion does not exceed value

0.0080.

Cpk a Cp equivalents

Equivalents of process capability indices represent the values of capability indices for

measurable quality characteristics which correspond to the same occurrence of non-

conforming products.

The Cpk equivalent characterizes a situation for which one-sided tolerance should be

defined in the case of a measurable characteristic. It is calculated according to the

relationship:


68

3

uCEkv

p1

pk

(5.5)

where:

pu

1 - the quantile of standardized normal distribution

p - the average proportion of nonconforming products for an “in control” process.

Fig. 5.2 Dependence of the maximal value of the point estimation of nonconforming products proportion for which in the case of the p0 = 0.01 process can be considered as capable for given sample size (α = 0.05).

The Cp equivalent characterizes a situation for which two-sided tolerance should be

defined in the case of a measurable characteristic and the mean value of a given quality

characteristic is centred to the middle of tolerance (the probability of overstepping both

tolerance limits is the same). It is calculated according to the relationship:

3

21p

p

u

CEkv

(5.6)

where:

21p

u

- the quantile of standardized normal distribution.

Determination of process capability indices equivalents is essentially the inverse task

of estimating the probability of nonconforming products occurrence on the basis of process


69

capability indices calculated for measurable quality characteristics. For the estimation of Cp

and Cpk equivalents there can be used the graphical dependence of the probability of

nonconforming products occurrence on the Cpk index (see Fig. 5.3). Using curve A which is

corresponding to the case of one-sided tolerance or to the case when the Cpk index is much

lower than the Cp index (a very badly centred process) it is possible to find a Cpk equivalent,

using curve B, which corresponds to the case when the Cpk index is equal to the Cp index (a

centred process), it is possible to find the Cp equivalent.

Fig. 5.3 Dependence of expected occurrence of the non-conforming products on the Cpk index value ( A: one-sided tolerance or Cpk<<Cp ; B: Cpk = Cp ).

5.2 Process capability indicators in the case of monitoring nonconformities

In the case of monitoring nonconformities the average number of nonconformities per

unit is most often used as a process capability indicator.

However, this indicator actually expresses process capability only in the case when

data are from a process which is “in control”, it means when the process is influenced by

random causes only. The statistical stability of the process is in this case assessed using a

control chart for the number of nonconformities per unit (u-diagram) - see Fig. 5.4. Control

limits and the central line in this chart are calculated according to formulas:


70

k

1j

j

k

1j

j

n

c

uCL (5.7)

n

u3uLCL (5.8)

n

u3uUCL (5.9)

kde:

u - the average proportion of nonconformities per unit

cj - the number of nonconformities in the i-subgroup

nj - subgroup size

k - the number of subgroups

n - average subgroup size

On the basis of formula (5.7) it can be seen that the level of the central line is

corresponding to the average proportion of nonconformities per unit. For the conclusion

about process capability it is necessary to compare this level with the maximal acceptable

value required by the customer (usually indicated as u0).

Fig 5.4 The example of a control chart for the number of nonconformities per unit in the case of “in control” process.

252321191715131197531

0,020

0,015

0,010

0,005

0,000

Sample

Sa

mp

le C

ou

nt

Pe

r U

nit

_U=0,00838

UCL=0,01932

LCL=0

U0

U Chart of C2

Tests performed with unequal sample sizes


71

Summary of terms

ppm (parts per million) – number of nonconforming products in millions of produced

products.

dpm (defects per million) – number of defects (nonconformities) in millions of produced

products.

dpmo (defects per million oportunities) – number of defects (nonconformities) in millions

of opportunities to the defects.

Cpk equivalent represents the values of the Cpk index for measurable quality

characteristics which corresponds to the same occurrence of nonconforming products. It

characterizes a situation for which one-sided tolerance should be defined in the case of a

measurable characteristic.

Cp equivalent represents the values of the Cpk index for measurable quality characteristics

which corresponds to the same occurrence of non-conforming products. It characterizes a

situation when the mean value of the monitored quality characteristic is lying in the center of

tolerance.

Questions

1. What assumptions must be verified for process capability analysis in the case of non-

measurable quality characteristics?

2. How is ppm calculated if we know the average proportion of nonconforming products?

3. What expresses the Cpk equivalent?

4. When in the case of monitoring the proportion of nonconformuing products is a process

considered capable?

References

[1] KOTZ, S. – LOVELACE, C.R.: Process Capability Indices in Theory and Practice.

New York: Oxford University Press, 1998, ISBN 0-340-69177-8


72

6. MEASUREMENT SYSTEM ANALYSIS. ANALYSIS OF DRIFT, BIAS AND LINEARITY

Time for learning

3 hours

Goal


Define measurement system properties

Explain and apply the procedures of drift, bias, and the linearity of

measurement system evaluation.

Lecture

Decision-making based on facts is one of the fundamental principles of contemporary

quality management. These facts are in most cases obtained from measurements. The

measured data then become an essential basis for important decisions, for example during

the quality inspection of products, in the control of processes, in the evaluation of the

effectiveness of corrective actions, during the implementation of improvement activities, etc.

The proposed or already used measurement system requires attention, because an

inadequate measurement system can provide distorted information that can lead to incorrect

decisions. Verification of the suitability of the measurement system to be used in collecting

the necessary data is an important part of quality planning and quality improvement and, for

example, in the automotive industry, it is strictly required [1].

Knowledge of measurement systems properties is very important in assessing the

conformity of products. The conformity or nonconformity of a product can be clearly

confirmed only if the entire interval of measurement uncertainty lies within or outside the


73

tolerance. If the uncertainty interval includes a tolerance limit, the compliance test result is

inconclusive.

Let us suppose that we would use four different products with values near the lower

limits of tolerance to make a set of repeated measurements and we would present the

distribution of the measured values using the Gaussian curve (see Fig. 6.1). The individual

situations should be interpreted as follows:

situation A: nonconformity with the requirements is guaranteed

situations B and C: the result of the conformity test is inconclusive

situation D: conformity with the requirements is guaranteed.

The figure clearly shows that the increased variability of the measurement system

significantly narrows the interval of values, in which the product conformity with the

requirements is guaranteed. In an extreme case, where the variability of measurements is

higher than the tolerance, the conformity with the requirements would not be guaranteed for

any product.

Fig. 6.1 Various situations of distribution of the repeated measurements of the monitored

characteristic.

8,5 9,5 10,5 11,5 12,5 13,5

x

A B C D

LSL USL


74

In the given example the normal distribution of a measured characteristic is

supposed. In the cases, when data are not normaly distributed, this problem is more

complex.

6.1 Statistical properties of measurement systems

Measurement system quality is evaluated by various statistical properties as bias,

precision, repeatability, reproducibility, drift, linearity and others.

Bias of the measurement system is expressed as the difference between the average

of repeating measurement of the same quality characteristic and accepted reference value. It

characterises the whole systematic measurement error (see Fig. 6.2).

Precision of the measurement system characterizes the variability of repeating

measurement results. It is expressed by means of imprecision using standard deviation or its

certain multiple (5,15σg or 6σg). Precision characterizes random measurement errors.

Fig. 6.2 Bias and precision of measurement system ( x average of repeated

measurement, xr – reference value)

BIAS xr x

PRECISION


75

Repeatability represents the precision of a measurement system in conditions when

a measurement is performed by one operating personnel, using the same gauge, in the

same place and during a short time.

Reproducibility represents the variability of the mean values of repeated

measurement sets performed in various conditions. Usually there are cases when

measurement is done by various operators, however there can be one operator and various

gauges or one operator, the same gauge and various places of measurement (see Fig. 6.3).

Drift of the measurement system represents the whole variability of measurement of

the same quality characteristic in a long term period (see Fig. 6.4).

Linearity is expressed as the difference between bias values in the supposed

working range of a measurement system.

Fig. 6.3 Reproducibility of measurement system.

Operator B

Operator B

Operator B

REPRODUCIBILITY


76

Fig. 6.4 Drift of measurement system.

6.2 Analysis of measurement system drift

The analysis of measurement system drift should be performed before the analyses

of other properties of a measurement system. Drift is evaluated on the basis of the

assessment of bias changes in the time of using a measurement system. Data needed for

drift analysis are obtained by repeating the measurement of the same sample at a suitable

chosen time interval.

Analysis of measurement system drift should be performed using these steps:

1. Sample choice

For the evaluation of measurement system drift there is chosen a sample of a product

or etalon, whose value is approximately corresponding to the centre of a production range. It

is suitable performing the same evaluation also for samples corresponding to the lower and

upper limit of a production range.

TIME 1 TIME 2

DRIFT


77

2. Repeated measurements of a sample at chosen time intervals

Repeated measurements of a chosen sample are performed at regular time intervals,

whose length depends on the type of measurement system and the frequency of its use. The

number of repeated measurements is chosen in the range from three to five.

3. Calculation of the sample characteristics of repeated measurements

After collection of a sufficient number of data (cca 25 subgroups of repeated

measurements) there are calculated sample characteristics in individual intervals (e.g.

averages and ranges or standard deviations).

4. Control limits calculation

On the basis of the average values of the sample characteristics in the subgroups

there are calculated control limits and central lines for the chosen pair of Shewhart´s control

charts (e.g. R,x or s,x ). For example in the case of control charts R,x the levels of

control limits and central lines are calculated according to formulas [2]:

R-diagram:

k

R

RCL

k

j

j

1

(6.1)

RDLCL 3 (6.2)

RDUCL .4 (6.3)

where:

k - number of subgroups

D3, D4 - coefficients depending on the subgroup size.

x - diagram:

k

x

xCL

k

j

j

1

(6.4)

RAxLCL 2 (6.5)

RAxUCL 2 (6.6)


78

where:

A2 - coefficient depending on the subgroup size.

5. Control charts processing and their analysis

A pair of control charts is processed and their analysis follows. Regarding the drift

definition it can be assessed on the basis of the control chart for averages. In the case, when

all displayed averages are inside control limits and a control chart doesn´t include any

unrandom patterns of points, the drift of measurement system is considered as acceptable.

The analysis of a control chart for the chosen measure of variability provides

information about measurement system consistency, which is defined as the whole variability

of the measurement of the same quality characteristic in a long term period (the changes of

repeatability in time). In the case when all values of the ranges (or standard deviations) in

subgroups are inside the control limits and a control chart doesn´t include any unrandom

patterns of points the consistency of a measurement system is considered as acceptable.

If the measurement system has an acceptable drift and consistency it means that the

variability of measurement results is influenced only by random (natural) causes of variability.

This measurement system (measurement process) can be classified as “in control”

(statistically stable).

Possible causes of unacceptable drift can be e.g.:

• a long interval between calibrations

• incorrect calibration

• the wearing out of measurement equipment

• the bad maintenance of measurement equipment

• the variability of environmental conditions.

6.3 Analysis of measurement system bias

Evaluation of measurement system bias should be performed using these steps:

1. Choice of a sample with a known reference value

Data needed for bias analysis are obtained by the measurement of a sample of a

product or etalon with a known reference value. The measured sample should approximately

correspond to the centre of the production range of monitoring quality characteristics.


79

Fig. 6.5 Control chart R,x .

2. Repeated measurements of the sample

Measurements of the same sample are done in repeatability conditions. It means that

the measurement is performed by one operator, by the same measurement equipment in the

same conditions and in the shortest time. It is necessary to obtain ten repeated

measurements as minimum.

3. Exploratory data analysis

Exploratory data analysis should be performed for collected data. The main objective

of this analysis is to assess whether measured data don´t include gross errors or the

influence of special causes of variability. It is possible to recommend outlier analysis using a

Subgroup

X-ba

r

CL = 9,99

UCL = 10,12

LCL = 9,86

0 5 10 15 20 25

9,8

9,9

10

10,1

10,2

10,3

Subgroup

Rang

e

CL = 0,22

UCL = 0,47

LCL = 0,00

0 5 10 15 20 25

0

0,1

0,2

0,3

0,4

0,5


80

box plot, the analysis of data in a run chart (dependence on the order) or in the case of a

sufficient number of data the processing of a histogram and its analysis.

4. Calculation of the average of repeated measurements

Average of repeated measurements is calculated according to the formula:

n

x

x

n

i

i 1 (6.7)

where:

xi - value of i - measurement of a given sample

n – the number of repeated measurements of a given sample.

5. Calculation of bias point estimation

Bias point estimation is calculated according to the formula:

rxxiB ˆ (6.8)

where:

x - the average of repeated measurements

xr - the reference value of a measured sample.

Bias point estimation doesn´t provide information whether a bias is statistically

significant. The statistical significance of a bias can be evaluated on the basis of the bias

confidence interval. Before confidence interval calculation it is necessary to verify if the

measurement system has an acceptable repeatability.

6. Calculation of the standard deviation of repeatability

The standard deviation of repeatability is calculated according to the formula:

(6.9)

where:

xi - i - measured value

n - number of measurements.

1n

)xx(sˆ

2

ie


81

7. Assessment of acceptability of measurement system repeatability

The acceptability of measurement system repeatability is assessed on the basis of

the percentage proportion of repeatability from the total variation, which is calculated

according to the formula:

100ˆ

100% TVTV

EVEV e

(6.10)

where:

EV - repeatability (Equipment Variation)

TV - total variation (Total Variation).

Total variation is expressed by the means of the standard deviation of given quality

characteristics of the produced products (preferably) or according to formula (7.22). The

percentage proportion of repeatability from the total variation should be lower than 10%.

8. Calculation of the bias confidence interval

The confidence interval of bias estimation is calculated according to the formula:

(6.11)

where:

- degrees of freedom ( 1n )

- quantile of Student´s distribution

α - level of significance (usually α=0.05).

In the cases when a calculated bias confidence interval includes a zero value the bias

of a measurement system is considered as statistical unsignificant, it means it is acceptable.

On the contrary, if a bias confidence interval doesn´t include a zero value the bias of a

measurement system is considered as statistical significant and it is necessary to do some

actions for its elimination (e.g. recalibration of measurement equipment).

Here are some examples of the causes of unacceptable bias:

• incorrect calibration

• the wearing out of measurement equipment

• unsuitable measurement equipment for a given measurement

• different measurement methods.

21,

t

21,

21,

ˆˆ

t

n

siBBit

n

siB


82

6.4 Analysis of measurement system linearity

The analysis of measurement system linearity is performed by using a similar

procedure as the analysis of measurement system bias. For assessment whether a bias is

changing with the size of a measured value it is necessary to perform repeated

measurements of several samples covering a production range.

The evaluation of measurement system linearity should be performed using these

steps:

1. Choice of samples

It is necessary to choose at minimum five samples with known reference values,

which represents and approximately evenly covers a production range.

2. Repeated measurements of samples

Similarly as in bias analysis it is necessary to perform ten repeated measurements for

each sample. All measurements should be performed in repeatability conditions, it means by

the same operator, with the same measurement equipment and at the same place. An

example of record of measured data is given in Fig. 6.6.

Part 1 2 3 4 5

xir x1r x2r x3r x4r x5r

1 x11 x21 x31 x41 x51

2 x12 x22 x32 x42 x52

3 x13 x23 x33 x43 x53

4 x14 x24 x34 x44 x54

5 x15 x25 x35 x45 x55

6 x16 x26 x36 x46 x56

7 x17 x27 x37 x47 x57

8 x18 x28 x38 x48 x58

9 x19 x29 x39 x49 x59

10 x110 x210 x310 x410 x510

Fig. 6.6 Example of record of measured data for measurement system linearity analysis.

3. Calculation of deviations

In this step the deviations of each measurement from the reference value are

calculated according to the formula:


83

irijij xxDev (6.12)

where:

xij - j-measured value of i-sample

xir - reference value of i-sample.

4. Calculation of bias for individual samples

The point estimations of biases for individual samples are the partial results of linearity

analysis. They are calculated according to the relation:

n

xx

iB

n

j

irij

i

1ˆ

(6.13)

where:

n - the number of repeated measurements.

There can be used also the definition relationship:

irii xxiB ˆ (6.14)

where:

ix - the average of repeated measurements of i-sample.

5. Displaying the dependence of deviations on the samples reference value

Measurement system linearity is firstly assessed graphically. It is a processed graph

of the dependence of deviations on the reference values of samples.

6. Regression analysis

In this step regression analysis is applied and the dependence of deviations on the

samples reference value is described by linear regression function:

r10 xbby (6.15)

where:

y – the deviation of a measured value from a reference value


84

xr – reference value

b0 – absolute term

b1 – regression coefficient.

Fig. 6.7 Example of the dependence of deviations on the reference values.

7. Testing of regression coefficient and absolute term

The final evaluation of measurement system linearity is based on the testing of a

regression coefficient and absolute term. The linearity of a measurement system is

considered as acceptable in the cases when regression coefficient b1 and also absolute term

b0, are not statistically significant (it is not possible to reject the hypothesis that they are equal

to zero) on the chosen level of significance (α=0.05). Evaluation can be performed by using

t-tests or it is possible to do it on the basis of the confidence intervals of b1 and b0.

For example the confidence interval of b1 estimation is calculated according to the

formula:

(6.16)

where:

21,

t

- quantile of Student´s distribution

y = -0,1317x + 0,7367 R2 = 0,7143

-1

-0,5

0

0,5

1

1,5

0 4 8 12

De

v

xr

11 b

21,

11b

21,

1 stbstb


85

- degrees of freedom ( 2n )

1bs - standard deviation of regression coefficient.

(6.17)

where s is standard deviation of residuals:

(6.18)

In the cases when the confidence interval of b1 includes the zero value it is possible to

consider that the regression coefficient is not statistically significant.

The confidence interval of absolute term estimation is calculated according to the

formula:

(6.19)

where:

0bs - standard deviation of the absolute term.

(6.20)

In the cases when the confidence interval of b0 includes the zero value it is possible to

consider that the regression coefficient is not statistically significant.

In the cases when b1 or b0 or both of them are statistically significant, the linearity of

the measurement system is not acceptable and it is necessary to solve it.

Unacceptable linearity can be caused e.g. by these causes:

• the calibration doesn´t cover the whole operational range of ameasurement system

• the bad maintenance of measurement equipment.

n

1i

2

i

b

xx

1ss

1

2n

yy

s

n

1i

2

ii

00 b

21,

00b

21,

0 stbstb

n

1i

2

i

n

1i

2

i

b

xxn

x

ss0


86

Summary of terms

Bias is expressed as the difference between the average of repeating measurement of the

same quality characteristic and the accepted reference value.

Precision characterizes the variability of repeating measurement results.

Repeatability represents the precision of a measurement system in the conditions when a

measurement is performed by one operating personnel, using the same gauge, at the same

place and during a short time.

Reproducibility represents the variability of mean values of repeated measurement sets

performed in various conditions.

Drift represents the whole variability of a measurement of the same quality characteristic in a

long term period.

Linearity is expressed as the difference between bias values in a supposed working range

of a measurement system.

Questions

1. What control chart is used for the evaluation of measurement system drift?

2. What can be the cause of the not acceptable drift of a measurement system?

3. What conditions should be assured for collecting data for bias evaluation?

4. What is the principle of measurement system linearity evaluation?

References

[1] Measurement System Analysis (MSA). 4th edition.Chrysler Group LLC, Ford Motor

Company, General Motors Corporation, 2010, 231 s., ISBN 978-1-60-534211-5

[2] MONTGOMERY, D. C.: Statistical Quality Control: A Modern Introduction. 6th Edition.

John Wiley & Sons, 2009, 734s. s., ISBN 978-0470-23397-9


87

7. MEASUREMENT SYSTEM ANALYSIS. REPEATABILITY AND REPRODUCIBILITY ANALYSIS (GRR ANALYSIS)

Time for learning

3 hours

Goal


Explain and apply the procedures of repeatability and the reproducibility of

measurement system evaluation

Propose suitable actions for measurement system improvement.

Lecture

The evaluation of the combined repeatability and reproducibility of the measurement

system (GRR) is implemented because it is mostly impossible to ensure constant conditions

(repeatability conditions) during the measurement itself in practice. The actual measurement

conditions usually vary whereas the operator taking the measurement is the one who is most

frequently changed. The study of the repeatability and reproducibility of a measurement

system can be done using several different procedures. The three most used methods are

the following [1,2]:

Range Method

Average and Range Method

ANOVA.


88

7.1 Range method

The Range Method is also called the “short method” and enables the quick

approximation of measurement variability. This method provides only the whole picture of the

measurement system because it does not decompose the variability into repeatability and

reproducibility. GRR analysis using the Range Method should be performed using these

steps:

1. Choice of samples

Data for this analysis are obtained by the measurement of product samples which

represents the production range. Standardly there are used five product samples.

2. Measurement of samples by two operators

The number of operators is usually minimized to two of them and each operator

measures each sample only once. The structure of measured data can be seen in the Table

7.1.

Tab. 7.1 Results of measurements for the Range Method application.

Sample Operator 1 Operator 2 Range

1 X11 X12 R1.

2 X21 X22 R2.

3 X31 X32 R3.

4 X41 X42 R4.

5 X51 X52 R5.

3. Calculation of the ranges of measured data for individual samples

The ranges of measured data for individual samples are calculated on the basis of

obtained data according to the formula:

ijj

ijj

i xxR minmax. (7.1)

4. Average range calculation

On the basis of the ranges for the individual samples average range is calculated:


89

r

R

R

r

i

i 1

.

(7.2)

where:

r – number of measured samples.

5. Calculation of combined repeatability and reproducibility

Combined repeatability and reproducibility (GRR) is calculated using this formula:

*

2d

RGRR (7.3)

where:

d2* - coefficient depending on the number of measurements of the individual

samples and on the number of measured samples [1].

6. Percentage of GRR evaluation

The final result of this analysis is the percentage proportion of GRR from the total

variation (TV). Total variation is estimated e.g. by the means of the standard deviation of a

given quality characteristic achieved in the production process. A calculated proportion of

%GRR is used as criterion of measurement system acceptability (see Tab.7.2).

Tab.7.2 Criteria of measurement system acceptability for the Range Method.

%GRR ≤ 10% the measurement system is acceptable

10% < %GRR ≤ 30%

the measurement system is conditionally acceptable owing to the

global variability of the process or the tolerance range, and it

depends on the proportion of the remedy cost and importance of

the quantity monitored.

%GRR > 30% the measurement system is unacceptable and it must be improved


90

3.1 Average and Range Method

The Average and Range Method is much better processed than the Range Method and

provides much more information. A very important is fact that this method makes it possible

to distinguish variability caused by repeatability and variability caused by the reproducibility

of measurement system. Application of this method should be performed using these steps:

1. Choice of measurement system

The first step of analysis should be verification if correct quality characteristic is

measured. A chosen measurement system should have a sufficient resolution. It is usualy

required that a measurement system should be able to resolute one tenth of the expected

variability of the monitored quality characteristics.

2. Definition of the basic parameters of analysis

During the preparatory phase it is necessary to define all basic parameters of the

measurement system analysis, i.e. the number of operators for performance of the

measurement system analysis, the number of measured samples and the number of

repeated measurements per each sample. There should be ten measured samples at least,

each sample shall be measured by the operator two times at least and the number of

operators shall be based on the real usage of the measurement system.

3. Choice of measured samples

The choice of measured samples is very important. The measured samples shall

evenly cover the whole production range of the monitored characteristic for the purpose of

standard analysis procedure. Chosen samples should be indicated for good identification.

4. Collecting data

The measurement of chosen samples should be performed in the place of using a

measurement system and all operators should apply the same procedure. The measurement

of samples is performed in random order and particular operators do not know the number of

particular samples, not even the results of previous measurement. Due to these reasons the

monitoring of a measurement and data recording are performed by the charged employee.

Measured data can be assigned in the form xijk, where:


91

h;1i - identification of operator (h –number of operators)

r;1j - identification of sample (r – number of samples)

n;1k - identification of trial (n – number of trials).

The structure of measured data in the case of three operators, ten samples and two

trials is shown in Table 7.3.

5. Verification of the statistical stability of the measurement process with regard to

the variability of repeated measurements

The first step of evaluation of collected data should be an assessment whether a

measurement process is statistical stable with regard to the variability of repeated

measurements. For this purpose it is necessary to process a control chart for the ranges of

repeated measurements.

Firstly, ranges of repeated measurements of individual samples by individual

operators are calculated according to the formula:

n;1k;xminxmaxR ijkk

ijkk

.ij (7.4)

where:

ijkkxmax - the maximal value of measurements of j-sample by i-operator

ijkkxmin - the minimal value of measurements of j-sample by i-operator.

Useful information about the ability of individual operators provides an average range

of the repeated measurements of all samples by an appropriate operator. It is calculated


r

R

R

r

1j

.ij

..i

(7.5)

where:

.ijR - the range of repeated measurements of j-sample by i-operator

r - the number of samples.


92

Tab. 7.3 Structure of measured data in the case of three operators, ten samples and two trials.

Op

era

tor

(i)

Trial (k

)

Measured sample (j)

1 2 3 4 5 6 7 8 9 10 k.ix

..ix

..iR

1

1 x111 x121 x131 x141 x151 x161 x171 x181 x191 x1101 1.1x

2 x112 x122 x132 x142 x152 x162 x172 x182 x192 x1102 2.1x

.ijx .11x .12x .13x .14x .15x .16x .17x .18x .19x .110x ..1x

Rij. R11. R12. R13. R14. R15. R16. R17. R18. R19. R110. ..1R

2

1 x211 x221 x231 x241 x251 x261 x271 x281 x291 x2101 1.2x

2 x212 x222 x232 x242 x252 x262 x272 x282 x292 x2102 2.2x

.ijx .21x .22x .23x .24x .25x .26x .27x .28x .29x .210x ..2x

Rij. R21. R22. R23. R24. R25. R26. R27. R28. R29. R210. ..2R

3

1 x311 x321 x331 x341 x351 x361 x371 x381 x391 x3101 1.3x

2 x312 x322 x332 x342 x352 x362 x372 x382 x392 x3102 2.3x

.ijx .31x .32x .33x .34x .35x .36x .37x .38x .39x .310x ..3x

Rij. R31. R32. R33. R34. R35. R36. R37. R38. R39. R310. ..3R

.j.x .1.x .2.x .3.x .4.x .5.x .6.x .7.x .8.x .9.x .10.x

The level of the central line in the control chart is corresponding to the average range of the

repeated measurement for all operators. It is calculated according to the formula:

h

R

R

h

1i

..i (7.6)

where:

..iR - the average range of repeated measurements of all samples by i-operator

h - the number of operators.


93

Control limits of the range control chart are calculated according to the formulas:

RDUCLR 4 (7.7)

RDLCLR 3 (7.8)

where:

D3, D4 - coefficients dependent on subgroup size.

The subgroup size for which it is necessary to find the values of the D3 and D4

coefficients is corresponding to the number of trials. Calculated control limits, central line and

the ranges of repeated measurements are displayed in the control chart and its analysis

follows. For next evaluation it is necessary to achieve a state when a measurement process

is “in control” from the point of view of repeated measurements variability, it means that all

ranges must be inside the control limits.

Fig. 7.6 Range control chart for the assessment of measurement process statistical stability from the point of view of repeated measurements variability.

In the cases when some ranges are outside the control limits it is necessary to

distinguish two situations. If ranges lying outside control limits were found for only one

operator, usually his/her measurement method is different from the others. It is necessary to

assure the training of a given operator in a measurement method and appropriate

measurements should be repeated. However, if ranges are outside control limits for all

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,09

0,1

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

ran

ge

sample number

Operator 1 Operator 3 Operator 2


94

operators, it usually means that a measurement system is too sensitive for operator activity

and it is necessary to improve it.

6. Suitability of a measurement system for the assessment of the variability of

measured samples

The suitability of ameasurement system for the assessment of variability of measured

samples can be evaluated by the means of a control chart for averages of repeated

measurements of individual samples by individual operators. It is a special use of a control

chart, because control limits are calculated on the basis of variability inside subgroups, which

is corresponding to the measurement system variability and displayed averages of repeated

measurements of individual samples are corresponding to production process variability. In

this way a given control chart compares the variability of a measurement system with the

varability of a production process.

Central line and control limits are calculated according to the formulas:

h

x

xCL

h

1i

..i (7.9)

RAxUCL 2 (7.10)

RAxLCL 2 (7.11)

where:

..ix - the averages of all measurements performed by individual operators

A2 – the coefficient depending on the number of trials (subgroup size)

R - average range of repeated measurement for all operators.

A measurement system is considered as suitable for the assessment of the variability

of measured samples in the cases when more than 50% of averages are lying outside

control limits. A control chart for averages also makes it possible to assess the conformity of

average values for individual samples determined by individual operators.

Information value of range and average control charts can be enhanced by ordering

of sample according to average values of given quality characteristic [3,4].


95

Fig. 7.7 Control chart for the averages assessing suitability of a measurement system for assessment of the variability of measured samples.

7. Evaluation of measurement system repeatability

If the measurement process is statistical stable from the point of view of repeated

measurements variability it is possible to calculate measurement system repeatability (EV –

Equipment Variation) according to the formula:

*

2

ed

REV (7.12)

where:

σe - the standard deviation of repeatability

*

2d - the coefficient depending on the subgroup size m (number of trials) and the

number of subgroups g (the number of samples multiplied by the number of

operators )

8. Evaluation of measurement system reproducibility

Firstly, it is necessary to calculate the averages of all measurements performed by

individual operators according to the relation:

r

x

x

r

1j

.ij

..i

(7.13)

where:

10,2

10,3

10,4

10,5

10,6

10,7

10,8

10,9

11

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

av

era

ge

sample number

Operator 1 Operator 2 Operator 3


96

r – the number of measured samples.

On the basis of the averages of individual operator measurements there is calculated

the range R0:

..ii

..ii

0 xminxmaxR ; h;1i (7.14)

where:

..iixmax - the maximal value of averages of all measurements by individual

operators

..iixmin - the minimal value of averages of all measurements by individual

operators

h - the number of operators.

Reproducibility of a measurement system (AV – Appraiser Variation) is calculated


rn

EV

d

R

rn

EVAV

22

*

2

0

22

0

(7.15)

where:

σ0 - standard deviation of reproducibility

r - the number of samples

n - the number of trials

*

2d - the coefficient depending on the number of operators (m – the number of

operators, g=1)

9. Evaluation of combined repeatability and reproducibility

The combined repeatability and reproducibility of a measurement system (GRR-

Gauge Repeatability & Reproducibility) can be calculated using this relationship:

22AVEVGRR (7.16)


97

The GRR value itself doesn´t provide information about the acceptability of a

measurement system, because it is not related to total variation. Total variation (TV) is

usually calculated on the basis of the variability of measured samples (PV – Part Variation)

which should represent the production range.

10. Part variation evaluation

For Part Variation determination it is necessary firstly to calculate the averages of all

the measurements of individual samples. Appropriate calculations can be performed on the

basis of the averages of repeated measurements of a given sample by individual operators:

h

x

x

h

1i

.ij

.j.

(7.17)

On the basis of the determined averages range of averages, (Rp) is calculated


r;1j;xminxmaxR .j.j

.j.j

p (7.18)

where:

.j.jxmax - the maximal average of all measurements of individual samples

.j.jxmin - the minimal average of all measurements of individual samples.

Part Variation can be calculated using the relationship:

*

2

p

pd

RPV (7.19)

where:

σp - the standard deviation of a given characteristic in measured samples

*

2d - the coefficient depending on the number of measured samples (m – the number of measured samples, g=1).


98

11. Total variation calculation

On the basis of Part Variation and combined repeatability and reproducibility Total

Variation (TV) can be calculated according to the formula:

22PVGRRTV (7.20)

In the cases when samples don´t represent a production range it is necessary to

estimate Total Variation in another way. If production process capability is known, Total

Variation is estimated as the standard deviation of monitored quality characteristics achieved

by a production process:

processTV (7.21)

If production process capability is unknown, it is possible to estimate Total Variation

as the maximal value of standard deviation, which still assures a potentially capable process

(usually Cp>=1,33). This maximal value of standard deviation can be calculated according to

the formula:

min,

max,6 P

processC

LSLUSLTV

(7.22)

Regarding the fact that the way of Total Variation determination considerably

influences the GRR study results it is necessary always specify it!

12. Calculation of the percentage proportion of repeatability and reproducibility from

the total variation

The next step of GRR study is the calculation of the percentage proportion of

repeatability, reproducibility and combined repeatability and reproducibility (and also part

variation) from the total variation. Appropriate proportions are calculated according to

formulas:

100TV

EVEV% (7.23)

100TV

AVAV% (7.24)

100TV

GRRGRR% (7.25)


99

100TV

PVPV% (7.26)

Percentage of combined repeatability and reproducibility from the total variation is

used as the first criterion of measurement system acceptability.

13. Number of distinct categories (ndc) calculation

On the basis of the determined Part Variation and combined repeatability and

reproducibility number of distinct categories, (ndc) is calculated according to the formula:

GRR

PV41,1ndc (7.27)

The calculated value is rounded down to the integers. The number of distinct

categories is used as a second criterion of measurement system acceptability.

14. Acceptability of measurement system assessment

Percentage proportion of GRR and ndc are used as criteria of measurement system

acceptability from the repeatability and reproducibility point of view. The limits for these

criteria are given in Table 7.4.

Important information is provided by the percentage proportions of separate

repeatability and separate reproducibility from the total variation. On the basis of these

values it is possible to find suitable actions for measurement system improvement. In the

case of a high percentage of repeatability, the appropriate causes of measurement system

variability should be found in the used measurement equipment, the used procedure of

measurement or in nonstable conditions of measurement. In the case of a high percentage

of reproducibility, appropriate causes of measurement system variability should be found in

the different approaches of individual operators or in their different skills.

A very important part of GRR studies are graphical tools which make it possible to

assure more a complex analysis of measured data and make it easier to find the causes of

measurement system variability.


100

Tab. 7.4 Criteria of measurement system acceptability.

%GRR ≤ 10% and ndc ≥ 5 the measurement system is acceptable

10% < %GRR ≤ 30% and ndc ≥ 5

the measurement system is conditionally

acceptable owing to the global variability of the

process or the tolerance range, and it depends

on the proportion of the remedy cost and the

importance of the quantity monitored.

%GRR > 30% or ndc < 5 the measurement system is unacceptable and it

must be improved

3.3 ANOVA

The last edition of MSA methodology places even more emphasis on the evaluation

of repeatability and reproducibility by the means of the analysis of variance (ANOVA). In the

case of using this method, the total variability can be divided into repeatability (EV),

reproducibility (AV), variability between parts (PV) and variability caused by the interaction

between operators and parts (INT). The evaluation of the GRR study using this method

makes it possible to obtain more information than in the case of the average and range

method, as it also provides additional information on how much of the total variability is

caused by the interaction between the individual operators and parts. If this interaction is

statistically significant, its value is recorded separately and the combined repeatability and

reproducibility is calculated as follows [1,5]:

(7.28)

If the interaction is not statistically significant, it is assigned to the value of

repeatability. The ANOVA method can therefore detect much more accurate estimates of the

variances, provided that the measurement errors are normally distributed. This assumption

can be verified using suitable graphical tools presented in work. The disadvantage of this

method includes more complicated calculations of the individual elements of variability, which

is why it is necessary to use suitable software for its application.

Interactions between the individual operators and parts can be detected with using

suitable graphical tools [6].

222INTAVEVGRR


101

Summary of terms

Repeatability represents the precision of a measurement system in conditions when

measurement is performed by one operating personnel, using the same gauge, in the same

place and during a short time.

Reproducibility represents the variability of the mean values of repeated measurement sets

performed in various conditions.

GRR (Gage Repeatability and Reproducibility) represents the combined repeatability and

reproducibility of a measurement system.

Range method represents simple GRR analysis which makes possible to evaluate GRR but

not separately repeatability and reproducibility.

Questions

1. What is the procedure for data collection for GRR analysis using the Average and Range

method?

2. What ways are used for estimation of Total Variation (TV)?

3. What tool is used for verification of statistical stability of the measurement process with

regard to the variability of repeated measurements?

4. What criteria are used for the assessment of acceptability of a measurement system from

a repeatability and reproducibility point of view?

5. What action for ithe mprovement of a measurement system should be proposed in the

case of high %AV?

6. What action for ithe mprovement of a measurement system should be proposed in the

case of high %EV?

7. What is the main advantage of using ANOVA method for GRR analysis?


102

References



[2] KLAPUT, P., PLURA, J.: Influence of Used Method on the Results of Gage

Repeatability and Reproducibility Study. In: Conference Proceedings of 21st

International Conference on Metalurgy and Materials METAL 2012, Brno, May 2012.

Ostrava: Tanger, 2012, p. 1756 – 1761, ISBN 978-80-87 294-31-4

[3] KLAPUT, P., PLURA, J.: The Importance of Graphical Tools for Measurement

Systems Analysis. In: QMOD Conference on Quality and Service Sciences 2011.

Pamplona (Španělsko): 2011, 19 s., ISBN 84-8081-211-7

[4] PLURA, J., KLAPUT, P.: Complex Analysis of Measurement Systems. In: 15th

QMOD - ICQSS Conference. Poznaň: Comprint, 2012, s. 1235-1250, ISBN 978-83-

89333-46-9

[5] MONTGOMERY, D. C.: Statistical Quality Control: A Modern Introduction. 6th Edition.

John Wiley & Sons, 2009, 734s. s., ISBN 978-0470-23397-9

[6] PLURA, J., KLAPUT, P.: Influence of the Interaction between Parts and Appraisers

on the Results of Repeatability and Reproducibility Analysis. Quality Innovation

Prosperity. 2012, Vol. 16, No. 1, p. 25 – 36, ISSN 1335-1745 (databáze SCOPUS)

8. MEASUREMENT SYSTEM ANALYSIS. ATTRIBUTE MEASUREMENT SYSTEMS ANALYSIS

Time for learning

2 hours

Goal


Explain and apply procedures for attribute measurement system analysis.

Lecture

In the cases when the results of a measurement system are attributes it is necessary

to apply the specific procedures of measurement system analysis. The simpliest case is a

situation when products are distinguished into conforming and nonconforming ones.

The preparation stages of attribute measurement system analysis and data collection

are similar as in the case of GRR analysis. Individual operators (appraisers) in random order

assess samples of products and every product should be assessed by one operator twice at

minimum [1].

An assessed set of products must contain not only conforming but also

nonconforming products and the number of assessed products should be greater than in the

case of GRR analysis (approximately 50 products). Reference assessment which represents

the correct assessment of a given product must be determined for each product.

The acheived results of product samples assessment are recorded in a table. An

example of part of this table in the case of three operators and three trials is given in Table


104

8.1. The decision about acceptance of a product is indicated by number one, the decision

about rejecting of product is indicated by zero.

8.1 Evaluation of agreement between operators

The evaluation of agreement between operators is performed using cross tables. An

example of a cross table, with analyses agreement between operator A and operator B is in

Table 8.2.

Tab. 8.1 Part of collected data table for three operators and three trials.

Díl A-1 A-2 A-3 B-1 B-2 B-3 C-1 C-2 C-3 Reference

1 1 1 1 1 1 1 1 1 1 1

2 1 1 1 1 1 1 1 1 1 1

3 0 0 0 0 0 0 0 0 0 0

4 0 0 0 0 0 0 0 0 0 0

5 0 0 0 0 0 0 0 0 0 0

6 1 1 0 1 1 0 1 0 0 1

7 1 1 1 1 1 1 1 0 1 1

8 1 1 1 1 1 1 1 1 1 1

9 0 0 0 0 0 0 0 0 0 0

10 1 1 1 1 1 1 1 1 1 1

11 1 1 1 1 1 1 1 1 1 1

12 0 0 0 0 0 0 0 1 0 0

13 1 1 1 1 1 1 1 1 1 1

14 1 1 0 1 1 1 0 0 1 1

15 1 1 1 1 1 1 1 1 1 1

16 1 1 1 1 1 1 1 1 1 1

17 1 1 1 1 1 1 1 1 1 1

18 1 1 1 1 1 1 1 1 1 1

19 1 1 1 1 1 1 1 1 1 1

20 1 1 1 1 1 1 1 1 1 1

The upper values in appropriate cells of the cross table indicated as X00, X01, X10, X11

represent the number of individual possible combinations of operators results (1-

acceptance, 0-rejection). The values named as “Total” and indicated as X0., X1., X.0, X.1

provide information how many times individual operators during assessing sets of products

made the decision “accepted” or “rejected”.

The lower values in appropriate cells of the cross table represent “Expected counts”

and they are calculated according to formulas:

0.

.0

00 XN

Xx (8.1)


105

1.

.0

01 XN

Xx (8.2)

0.

.1

10 XN

Xx (8.3)

1.

.1

11 XN

Xx (8.4)

Tab. 8.2 Cross table of agreement between operator A and operator B.

Operator B

Total 0 1

Op

era

tor

A

0

Count X00 X01 X0.

Expected count x00 x01 x0.

1

Count X10 X11 X1.


Total

Count X.0 X.1 N

Expected count x.0 x.1 n

To determine the level of agreement between operators, indicator κ (kappa) is used. It

is calculated according to the formula:

e

e0

p1

pp

(8.5)

Value p0 represents the proportion of cases, when both operators made the same

decision (both decided about the acceptance or about rejection of a product):

N

XXp 00110

(8.6)

Value pe represents the proportion of expected cases, when both operators make the

same decision (both decided about the acceptance or about the rejection of a product):

N

xxp 0011e

(8.7)

The value of the kappa indicator can be in the range from -1 to 1. A situation when

kappa is equal to 1 corresponds to the perfect agreement between operators. Values greater


106

than 0.75 are considered as good for excellent agreement, values less than 0.40 indicate

poor agreement between operators.

8.2 Evaluation of the agreement between individual operators and reference values.

The evaluation of agreement between operators provides information whether there

are not considerable differences between the assessments of operators but doesn´t provide

information on what is the agreement between the assessment of operators and reference

values. For this assessment there are used also cross tables. An example of a cross table,

which contains data for the evaluation of agreement between operator A and a reference is

given in Table 8.3. The evaluation of agreement between individual operators and reference

values is performed also on the basis of kappa indicator which is calculated using the same

procedure as in the case of agreement between operators, and also the used criteria are the

same.

Tab. 8.3 Cross table of agreement between operator A and a reference value.

Reference

Total 0 1

Op

era

tor

A

0 Count X00 X01 X0.


1 Count X10 X11 X1.


Total Count X.0 X.1 N

Expected count x.0 x.1 n

8.3 Effectiveness of attribute measurement system

Another possibility of evaluating an attribute measurement system is the evaluation of

measurement system effectiveness. The effectiveness of a measurement system is

expressed as a ratio of the number of correct decisions to the number of total opportunities

for a decision (the number of assessed products).


107

For evaluating measurement system effectiveness in a percentage expression, it is

possible to use the general relationship:

(8.8)

where:

B – total number of assessed products.

There are distinguished various types of evaluated effectivenesses, and the

appropriate type depends on the fact what is used as an A value. There are defined four

various variants:

a) Effectiveness of operator: A = the number of products for them given operator

agrees with him/herself on all trials

b) Effectiveness of operator towards reference value: A = number of products for

them given operator agrees with him/herself on all trials and results are same as the

reference value

c) Effectiveness of operators: A = number of products for them all operators agree

with themselves on all trials

d) Effectiveness of operators towards the reference value: A = the number of

products for them all operators agree with themselves on all trials and the results are

same as reference value.

An attribute measurement system is considered as fully acceptable in the case when

its effectiveness is minimally 90%.

Summary of terms

Cross table is a table for recording the results of the assessment of a set of products by

various operators. Data from a cross table are the basis for the evaluation of agreement

between operators and agreement between an operator and reference value.

100B

AU


108

Kappa indicator is a calculated indicator, which is used as criterion of agreement between

operators or agreement between an operator and reference value.

Questions

1. What is the procedure of data collection for attribute measurement system analysis?

2. What is indicated as X01?

3. Which value of the kappa indicator should be achieved for an acceptable agreement

between operator and reference value?

4. What is defined as effectiveness of attribute measurement system and how it is

evaluated?

References




109

Katedra, institut: Katedra managementu kvality, Fakulta metalurgie

a materiálového inženýrství, VŠB-TU Ostrava

Název: Quality Planning II

Autor: Jiří Plura

Místo, rok, vydání: Ostrava, 2015, 1. vydání

Počet stran: 108

Vydala: Vysoká škola báňská-Technická univerzita Ostrava

Neprodejné

ISBN 978-80-248-3848-9