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Quality Management Systems and Statistical Quality Control 2 Contents 2.1 Introduction to Quality Management Systems . . . 17 2.2 International Standards and Specifications ...... 19 2.3 ISO Standards for Quality Management and Assessment .............................. 19 2.3.1 Quality Audit, Conformity, and Certification 19 2.3.2 Environmental Standards ................. 21 2.4 Introduction to Statistical Methods for Quality Control ........................... 23 2.4.1 The Central Limit Theorem ............... 23 2.4.2 Terms and Definition in Statistical Quality Control ................................ 24 2.5 Histograms .................................. 25 2.6 Control Charts ............................... 25 2.7 Control Charts for Means ..................... 26 2.7.1 The R-Chart ............................ 26 2.7.2 Numerical Example, R-Chart ............. 29 2.7.3 The N x-Chart ........................... 29 2.7.4 Numerical Example, N x-Chart ............. 30 2.7.5 The s-Chart ............................ 30 2.7.6 Numerical Example, s-Chart and N x-Chart . . . 33 2.8 Control Charts for Attribute Data .............. 33 2.8.1 The p-Chart ............................ 35 2.8.2 Numerical Example, p-Chart .............. 36 2.8.3 The np-Chart ........................... 37 2.8.4 Numerical Example, np-Chart ............ 37 2.8.5 The c-Chart ............................ 37 2.8.6 Numerical Example, c-Chart .............. 39 2.8.7 The u-Chart ............................ 40 2.8.8 Numerical Example, u-Chart .............. 40 2.9 Capability Analysis ........................... 40 2.9.1 Numerical Example, Capability Analysis and Normal Probability .................. 42 2.9.2 Numerical Examples, Capability Analysis and Nonnormal Probability ............... 46 2.10 Six Sigma ................................... 48 2.10.1 Numerical Examples .................... 51 2.10.2 Six Sigma in the Service Sector. Thermal Water Treatments for Health and Fitness .... 51 Organizations depend on their customers and there- fore should understand current and future customer needs, should meet customer requirements and strive to exceed customer expectations... Identifying, un- derstanding and managing interrelated processes as a system contributes to the organization’s effective- ness and efficiency in achieving its objectives (EN ISO 9000:2006 Quality management systems – fun- damentals and vocabulary). Nowadays, user and consumer assume their own choices regarding very important competitive factors such as quality of product, production process, and production system. Users and consumers start making their choices when they feel they are able to value and compare firms with high quality standards by them- selves. This chapter introduces the reader to the main prob- lems concerning management and control of a qual- ity system and also the main supporting decision mea- sures and tools for so-called statistical quality control (SQC) and Six Sigma. 2.1 Introduction to Quality Management Systems The standard EN ISO 8402:1995, replaced by EN ISO 9000:2005, defines “quality” as “the totality of characteristics of an entity that bear on its ability to satisfy stated and implied needs,” and “product” as R. Manzini, A. Regattieri, H. Pham, E. Ferrari, Maintenance for Industrial Systems 17 © Springer 2010

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  • Quality Management Systemsand Statistical Quality Control 2

    Contents

    2.1 Introduction to Quality Management Systems . . . 17

    2.2 International Standards and Specifications . . . . . . 19

    2.3 ISO Standards for Quality Managementand Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.1 Quality Audit, Conformity, and Certification 192.3.2 Environmental Standards . . . . . . . . . . . . . . . . . 21

    2.4 Introduction to Statistical Methodsfor Quality Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4.1 The Central Limit Theorem . . . . . . . . . . . . . . . 232.4.2 Terms and Definition in Statistical Quality

    Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

    2.5 Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

    2.6 Control Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

    2.7 Control Charts for Means . . . . . . . . . . . . . . . . . . . . . 262.7.1 The R-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.7.2 Numerical Example, R-Chart . . . . . . . . . . . . . 292.7.3 The Nx-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.7.4 Numerical Example, Nx-Chart . . . . . . . . . . . . . 302.7.5 The s-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.7.6 Numerical Example, s-Chart and Nx-Chart . . . 33

    2.8 Control Charts for Attribute Data . . . . . . . . . . . . . . 332.8.1 The p-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.8.2 Numerical Example, p-Chart . . . . . . . . . . . . . . 362.8.3 The np-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.8.4 Numerical Example, np-Chart . . . . . . . . . . . . 372.8.5 The c-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.8.6 Numerical Example, c-Chart . . . . . . . . . . . . . . 392.8.7 The u-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.8.8 Numerical Example, u-Chart . . . . . . . . . . . . . . 40

    2.9 Capability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.9.1 Numerical Example, Capability Analysis

    and Normal Probability . . . . . . . . . . . . . . . . . . 422.9.2 Numerical Examples, Capability Analysis

    and Nonnormal Probability . . . . . . . . . . . . . . . 46

    2.10 Six Sigma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.10.1 Numerical Examples . . . . . . . . . . . . . . . . . . . . 512.10.2 Six Sigma in the Service Sector. Thermal

    Water Treatments for Health and Fitness . . . . 51

    Organizations depend on their customers and there-fore should understand current and future customerneeds, should meet customer requirements and striveto exceed customer expectations... Identifying, un-derstanding and managing interrelated processes asa system contributes to the organizations effective-ness and efficiency in achieving its objectives (ENISO 9000:2006 Quality management systems fun-damentals and vocabulary).

    Nowadays, user and consumer assume their ownchoices regarding very important competitive factorssuch as quality of product, production process, andproduction system. Users and consumers start makingtheir choices when they feel they are able to value andcompare firms with high quality standards by them-selves.

    This chapter introduces the reader to the main prob-lems concerning management and control of a qual-ity system and also the main supporting decision mea-sures and tools for so-called statistical quality control(SQC) and Six Sigma.

    2.1 Introduction to QualityManagement Systems

    The standard EN ISO 8402:1995, replaced by ENISO 9000:2005, defines quality as the totality ofcharacteristics of an entity that bear on its ability tosatisfy stated and implied needs, and product as

    R. Manzini, A. Regattieri, H. Pham, E. Ferrari, Maintenance for Industrial Systems 17 Springer 2010

  • 18 2 Quality Management Systems and Statistical Quality Control

    the result of activities or processes and can be tan-gible or intangible, or a combination thereof. Conse-quently, these definitions refer to production systemsboth in industrial sectors, such as insurance, banking,and transport, and service sectors, in accordance withthe conceptual framework introduced in Chap. 1. An-other synthetic definition of quality is the degree towhich a set of inherent characteristics fulfills require-ments (ISO 9000:2005).

    A requirement is an expectation; it is generally re-lated to the organization, customers, or other inter-ested, or involved, parties. We choose to name allthese entities, i. e., the stakeholders of the organiza-tion, as customers and, consequently, the basic key-word in quality management is customer satisfaction.Another basic term is capability as the ability of theorganization, system, or process to realize a productfulfilling the requirements.

    A quality management system is a particular man-agement system driving the organization with regardto quality. In other words, it assists companies and or-ganizations in enhancing customer satisfaction. Thisis the result of products capable of satisfying the ever-changing customer needs and expectations that conse-quently require the continuous improvement of prod-ucts, processes, and production systems.

    Quality management is a responsibility at all levelsof management and involves all members of an organi-

    Fig. 2.1 Process-based qual-ity management system(ISO 9000:2005)

    Continual improvement of the quality management system

    Requirements

    Interested parties

    Interested parties

    Managementresponsability

    Resourcemanagementrealization

    Mesaurementanalysis &

    improvement

    Product realization

    Product

    Satisfaction

    IN OUT

    Value adding activities

    Information flow

    zation. For this reason, in the 1980s total quality man-agement (TQM) as a business management strategyaimed at embedding awareness of quality in all orga-nizational processes found very great success. Accord-ing to the International Organization for Standardiza-tion (ISO) standards (ISO 9000:2006), the basic stepsfor developing and implementing a quality manage-ment system are:

    determination of needs and expectations of cus-tomers and other involved parties;

    definition of the organizations quality policy andquality objectives;

    determination of processes and responsibilities forquality assessment;

    identification and choice of production resourcesnecessary to attain the quality objectives;

    determination and application of methods to mea-sure the effectiveness and efficiency of each processwithin the production system;

    prevention of nonconformities and deletion of therelated causes;

    definition and application of a process for contin-uous improvement of the quality management sys-tem.

    Figure 2.1 presents the model of a process-based qual-ity management system, as proposed by the ISO stan-dards.

  • 2.3 ISO Standards for Quality Management and Assessment 19

    2.2 International Standardsand Specifications

    According to European Directive 98/34/EC of 22 June1998, a standard is a technical specification for re-peated or continuous application approved, withouta compulsory compliance, by one of the following rec-ognized standardization bodies:

    ISO; European standard (EN); national standard (e. g., in Italy UNI).

    Standards are therefore documents defining the char-acteristics (dimensional, performance, environmental,safety, organizational, etc.) of a product, process, orservice, in accordance with the state of the art, andthey are the result of input received from thousands ofexperts working in the European Union and elsewherein the world. Standards have the following distinctivecharacteristics:

    Consensuality: They must be approved with theconsensus of the participants in the works of prepa-ration and confirmed by the result of a public en-quiry.

    Democracy: All the interested economic/social par-ties can participate in the works and, above all,have the opportunity to make observations duringthe procedure prior to final and public approval.

    Transparency: UNI specifies the basic milestonesof the approval procedure for a draft standard, plac-ing the draft documents at the disposal of the inter-ested parties for consultation.

    Voluntary nature: Standards are a source of refer-ence that the interested parties agree to apply freelyon a noncompulsory basis.

    In particular CEN, the European Committee for Stan-dardization founded in 1961 by the national standardsbodies in the European Economic Community andEFTA countries, is contributing to the objectives of theEuropean Union and European Economic Area withvoluntary technical standards promoting free trade,safety of workers and consumers, interoperability ofnetworks, environmental protection, exploitation of re-search and development programs, and public procure-ment.

    CEN works closely with the European Commit-tee for Electrotechnical Standardization (CENELEC),the European Telecommunications Standards Institute

    (ETSI), and the ISO. CEN is a multisectorial organi-zation serving several sectors in different ways, as il-lustrated in the next sections and chapters dealing withsafety assessment.

    2.3 ISO Standards for QualityManagement and Assessment

    The main issues developed by the technical committeefor the area of quality are:

    1. CEN/CLC/TC 1 criteria for conformity assess-ment bodies;

    2. CEN/SS F20 quality assurance.

    Table 2.1 reports the list of standards belonging to thefirst technical committee since 2008.

    Similarly, Table 2.2 reports the list of standards be-longing to the technical committee CEN/SS F20 since2008, while Table 2.3 shows the list of standards cur-rently under development.

    Quality issues are discussed in several standardsthat belong to other technical groups. For example,there is a list of standards of the aerospace series deal-ing with quality, as reported in Table 2.4. Table 2.5presents a list of standards for quality managementsystems in health care services. Similarly, there areother sets of standards for specific sectors, businesses,or products.

    2.3.1 Quality Audit, Conformity,and Certification

    Quality audit is the systematic examination of a qual-ity system carried out by an internal or external qual-ity auditor, or an audit team. It is an independent anddocumented process to obtain audit evidence and to al-low its objective evaluation, in order to verify the ex-tent of the fulfillment of the audit criteria. In particular,third-party audits are conducted by external organiza-tions providing certification/registration of conformityto a standard or a set of standards, e. g., ISO 9001 orISO 14001. The audit process is the basis for the dec-laration of conformity.

    The audit process is conducted by an auditor, or anaudit team, i. e., a person or a team, with competence

  • 20 2 Quality Management Systems and Statistical Quality Control

    Table 2.1 CEN/CLC/TC 1 criteria for conformity assessment bodies, standards published since 2008

    Standard Title

    EN 45011:1998 General requirements for bodies operating product certification systems (ISO/IEC Guide65:1996)

    EN 45503:1996 Attestation Standard for the assessment of contract award procedures of entitiesoperating in the water, energy, transport and telecommunications sectors

    EN ISO/IEC 17000:2004 Conformity assessment Vocabulary and general principles (ISO/IEC 17000:2004)EN ISO/IEC 17011:2004 Conformity assessment General requirements for accreditation bodies accrediting

    conformity assessment bodies (ISO/IEC 17011:2004)EN ISO/IEC 17020:2004 General criteria for the operation of various types of bodies performing inspection

    (ISO/IEC 17020:1998)EN ISO/IEC 17021:2006 Conformity assessment Requirements for bodies providing audit and certification of

    management systems (ISO/IEC 17021:2006)EN ISO/IEC 17024:2003 Conformity assessment General requirements for bodies operating certification of

    persons (ISO/IEC 17024:2003)EN ISO/IEC 17025:2005 General requirements for the competence of testing and calibration laboratories

    (ISO/IEC 17025:2005)EN ISO/IEC 17025:2005/AC:2006 General requirements for the competence of testing and calibration laboratories

    (ISO/IEC 17025:2005/Cor.1:2006)EN ISO/IEC 17040:2005 Conformity assessment General requirements for peer assessment of conformity

    assessment bodies and accreditation bodies (ISO/IEC 17040:2005)EN ISO/IEC 17050-1:2004 Conformity assessment Suppliers declaration of conformity Part 1: General

    requirements (ISO/IEC 17050-1:2004)EN ISO/IEC 17050-2:2004 Conformity assessment Suppliers declaration of conformity Part 2: Supporting

    documentation (ISO/IEC 17050-2:2004)

    Table 2.2 CEN/SS F20 quality assurance, standards published since 2008

    Standard Title

    EN 45020:2006 Standardization and related activities General vocabulary (ISO/IEC Guide 2:2004)EN ISO 10012:2003 Measurement management systems Requirements for measurement processes and

    measuring equipment (ISO 10012:2003)EN ISO 15378:2007 Primary packaging materials for medicinal products Particular requirements for the

    application of ISO 9001:2000, with reference to good manufacturing practice (GMP)(ISO 15378:2006)

    EN ISO 19011:2002 Guidelines for quality and/or environmental management systems auditing(ISO 19011:2002)

    EN ISO 9000:2005 Quality management systems Fundamentals and vocabulary (ISO 9000:2005)EN ISO 9001:2000 Quality management systems Requirements (ISO 9001:2000)EN ISO 9004:2000 Quality management systems Guidelines for performance improvements

    (ISO 9004:2000)

    Table 2.3 CEN/SS F20 quality assurance, standards under development as of October 2008

    Standard Title

    ISO 15161:2001 Guidelines on the application of ISO 9001:2000 for the food and drink industry(ISO 15161:2001)

    prEN ISO 9001 Quality management systems Requirements (ISO/FDIS 9001:2008)prEN ISO 19011 rev Guidelines for auditing management systemsprEN ISO 9004 Managing for the sustained success of an organization A quality management

    approach (ISO/DIS 9004:2008)

  • 2.3 ISO Standards for Quality Management and Assessment 21

    Table 2.4 Aerospace series, quality standards

    Standard Title

    EN 9102:2006 Aerospace series Quality systems First article inspectionEN 9103:2005 Aerospace series Quality management systems Variation management of key

    characteristicsEN 9110:2005 Aerospace series Quality systems Model for quality assurance applicable to

    maintenance organizationsEN 9120:2005 Aerospace series Quality management systems Requirements for stockist distributors

    (based on ISO 9001:2000)EN 9104:2006 Aerospace series Quality management systems Requirements for Aerospace Quality

    Management System Certification/Registrations ProgramsEN 9111:2005 Aerospace series Quality management systems Assessment applicable to

    maintenance organizations (based on ISO 9001:2000)EN 9121:2005 Aerospace series Quality management systems Assessment applicable to stockist

    distributors (based on ISO 9001:2000)EN 9132:2006 Aerospace series Quality management systems Data Matrix Quality Requirements

    for Parts MarkingEN 4179:2005 Aerospace series Qualification and approval of personnel for nondestructive testingEN 4617:2006 Aerospace series Recommended practices for standardizing company standardsEN 9101:2008 Aerospace series Quality management systems Assessment (based on

    ISO 9001:2000)EN 9104-002:2008 Aerospace series Quality management systems Part 002: Requirements for Oversight

    of Aerospace Quality Management System Certification/Registrations Programs

    Table 2.5 CEN/TC 362, health care services, quality management systems

    Standard Title

    CEN/TR 15592:200 Health services Quality management systems Guide for the use ofEN ISO 9004:2000 in health services for performance improvement

    CEN/TS 15224:2005 Health services Quality management systems Guide for the use ofEN ISO 9001:2000

    to conduct an audit, in accordance with an audit pro-gram consisting of a set of one or more audits plannedfor a specific time frame. Audit findings are used to as-sess the effectiveness of the quality management sys-tem and to identify opportunities for improvement.Guidance on auditing is provided by ISO 19011:2002(Guidelines for quality and/or environmental manage-ment systems auditing).

    The main advantages arising from certification are:

    improvement of the company image; increase of productivity and company profit; rise of contractual power; quality guarantee of the product for the client.

    In the process of auditing and certification, the docu-mentation plays a very important role, enabling com-munication of intent and consistency of action. Severaltypes of documents are generated in quality manage-ment systems.

    2.3.2 Environmental Standards

    Every standard, even if related to product, service,or process, has an environmental impact. For a prod-uct this can vary according to the different stages ofthe product life cycle, such as production, distribu-tion, use, and end-of-life. To this purpose, CEN hasrecently been playing a major role in reducing envi-ronmental impacts by influencing the choices that aremade in connection with the design of products andprocesses. CEN has in place an organizational struc-ture to respond to the challenges posed by the devel-opments within the various sectors, as well as by theevolution of the legislation within the European Com-munity. The main bodies within CEN are:

    1. The Strategic Advisory Body on the Environment(SABE) an advisory body for the CEN TechnicalBoard on issues related to environment. Stakehold-ers identify environmental issues of importance

  • 22 2 Quality Management Systems and Statistical Quality Control

    to the standardization system and suggest corre-sponding solutions.

    2. The CEN Environmental Helpdesk provides sup-port and services to CEN Technical Bodies on howto address environmental aspects in standards.

    3. Sectors some sectors established a dedicatedbody to address environmental matters associatedwith their specific needs, such as the Construc-tion Sector Network Project for the Environment(CSNPE).

    4. Associates two CEN associate members providea particular focus on the environment within stan-dardization:

    European Environmental Citizens Organizationfor Standardization (ECOS);

    European Association for the Coordination ofConsumer Representation in Standardization(ANEC).

    Table 2.6 Technical committees on the environment

    Technical commitee Title

    CEN/TC 223 Soil improvers and growing mediaCEN/TC 230 Water analysisCEN/TC 264 Air qualityCEN/TC 292 Characterization of wasteCEN/TC 308 Characterization of sludgesCEN/TC 345 Characterization of soilsCEN/TC 351 Construction Products Assessment of release of dangerous substances

    Table 2.7 Committee CEN/SS S26 environmental management

    Standard Title

    EN ISO 14031:1999 Environmental management Environmental performance evaluation Guidelines(ISO 14031:1999)

    EN ISO 14001:2004 Environmental management systems Requirements with guidance for use(ISO 14001:2004)

    EN ISO 14024:2000 Environmental labels and declarations Type I environmental labeling Principles andprocedures (ISO 14024:1999)

    EN ISO 14021:2001 Environmental labels and declarations Self-declared environmental claims (Type IIenvironmental labelling) (ISO 14021:1999)

    EN ISO 14020:2001 Environmental labels and declarations General principles (ISO 14020:2000)EN ISO 14040:2006 Environmental management Life cycle assessment Principles and framework

    (ISO 14040:2006)EN ISO 14044:2006 Environmental management Life cycle assessment Requirements and guidelines

    (ISO 14044:2006)prEN ISO 14005 Environmental management systems Guidelines for a staged implementation of an

    environmental management system, including the use of environmental performanceevaluation

    Table 2.6 reports the list of technical committees onthe environment.

    There are several standards on environmental man-agement. To exemplify this, Table 2.7 reports the listof standards grouped in accordance with the commit-tee CEN/SS S26 environmental management.

    ISO 14000 is a family of standards supporting theorganizations on the containment of the polluting ef-fects on air, water, or land derived by their operations,in compliance with applicable laws and regulations. Inparticular, ISO 14001 is the international specificationfor an environmental management system (EMS). Itspecifies requirements for establishing an environmen-tal policy, determining environmental aspects and im-pacts of products/activities/services, planning environ-mental objectives and measurable targets, implemen-tation and operation of programs to meet objectivesand targets, checking and corrective action, and man-agement review.

  • 2.4 Introduction to Statistical Methods for Quality Control 23

    x

    x

    uniform n = 2 n = 10

    n = 10

    exponential

    n = 2

    x x

    n = 10n = 2

    x x

    Gauss

    x

    Fig. 2.2 Central limit theorem, examples

    2.4 Introduction to Statistical Methodsfor Quality Control

    The aim of the remainder of this chapter is the intro-duction and exemplification of effective models andmethods for statistical quality control. These tools arevery diffuse and can be used to guarantee also thereliability,1 productivity and safety of a generic pro-duction system in accordance with the purpose of thisbook, as illustrated in Chap. 1.

    2.4.1 The Central Limit Theorem

    This section briefly summarizes the basic result ob-tained by this famous theorem. Given a population orprocess, a random variable x, with mean and stan-dard deviation , let Nx be the mean of a random sam-ple made of n elements x1; x2; : : : ; xn extracted fromthis population: when the sample size n is sufficientlylarge, the sampling distribution of the random vari-

    1 Reliability, properly defined in Chap. 5, can be also defined asquality in use.

    able Nx can be approximated by a normal distribution.The larger the value of n, the better the approximation.

    This theorem holds irrespective of the shape of thepopulation, i. e., of the density function of the vari-able x.

    The analytic translation of the theorem is given bythe following equations:

    M. Nx/ D NNx D O; (2.1). Nx/ D Op

    n; (2.2)

    where O is the estimation of and O is the estimationof .

    Figure 2.2 graphically and qualitatively demon-strates these results representing the basis for the de-velopment and discussion of the methods illustratedand applied below. In the presence of a normal distri-bution of population, the variable Nx is normal too foreach value of size n.

    Figure 2.3 quantitatively demonstrates the centrallimit theorem starting from a set of random valuesdistributed in accordance with a uniform distribution0; 10: the variable Nx is a normally distributed vari-able when the number of items used for the calculusof mean Nxi is sufficiently large. In detail, in Fig. 2.3the size n is assumed be 2, 5, and 20.

  • 24 2 Quality Management Systems and Statistical Quality Control

    9.07.56.04.53.01.50.0

    0.16

    0.12

    0.08

    0.04

    0.009.07.56.04.53.01.5

    0.20

    0.15

    0.10

    0.05

    0.00

    87654321

    0.3

    0.2

    0.1

    0.06.66.05.44.84.23.6

    0.8

    0.6

    0.4

    0.2

    0.0

    DATA (n=1)

    Den

    sity

    Mean (n=2)

    Mean (n=5) Mean (n=20)

    Histogram of DATA (n=1) and means (n>1)

    Fig. 2.3 Central limit theorem, histogram of Nx for n D f1; 2; 5; 20g. Uniform distribution of variable x

    2.4.2 Terms andDefinition in StatisticalQuality Control

    Quality control is a part of quality management(ISO 9000:2005) focused on the fulfillment of qualityrequirements. It is a systematic process to monitor andimprove the quality of a product, e. g., a manufacturedarticle, or service by achieving the quality of theproduction process and the production plant. A list ofbasic terms and definitions in accordance with the ISOstandards follows:

    Process, set of interrelated activities turning inputinto output. It is a sequence of steps that results inan outcome.

    Product, result of a process. Defect, not fulfillment of a requirement related to

    an intended or specified use. Measurement process, set of operations to deter-

    mine the value of a quantity. Key characteristic, a quality characteristic the prod-

    uct or service should have to fulfill customer re-quirements and expectations.

    Value of a key characteristic. For several productsa single value is the desired quality level for a char-acteristic.

    Nominal or target value. It is the expected valuefor the key characteristic. It is almost impossible tomake each unit of product or service identical to the

    next; consequently it is nonsense to ask for identi-cal items having a key characteristic value exactlyequal to the target value. This need for flexibilityis supported by the introduction of limits and toler-ances.

    Specification limit, or tolerance, conformanceboundary, range, specified for a characteristic.The lower specification limit (LSL) is the lowerconformance boundary, the upper specificationlimit (USL) is the upper conformance bound-ary.The following equation summarizes the relation-ship among these terms:

    Specification limits D (nominal value) tolerance:(2.3)

    One-sided tolerance. It relates to characteristicswith only one specification limit.

    Two-sided tolerance. It refers to characteristics withboth USLs and LSLs.

    Nonconformity. It is a nonfulfillment of a require-ment. It is generally associated with a unit: a non-conformity unit, i. e., a unit that does not meet thespecifications.

    Nonconforming product or service. A product orservice with one or more nonconformities. A non-conforming product is not necessary defective, i. e.,no longer fit for use.

  • 2.6 Control Charts 25

    2.5 Histograms

    Histograms are effective and simple graphic tools forthe comprehension and analysis of a process behav-ior with regards to the target value and the specifica-tion limits. The histograms illustrate the frequency dis-tribution of variable data: the values assumed by thevariable are reported on the abscissa, while the verti-cal axis reports the absolute or relative frequency val-ues. The specification limits are generally included inthe graph and give warnings of possible process prob-lems. Figure 2.4 exemplifies a few histogram shapes.The control charts illustrated in the next section repre-sent a more effective tool for the analysis of a produc-tion process.

    2.6 Control Charts

    Control charts, introduced by W.A. Shewhart in 1924,are effective tools for the analysis of the variation ofrepetitive processes. They are able to identify possi-ble sources of process variation in order to control andeventually eliminate them. In a generic process, twodifferent kinds of variations can be distinguished:

    reasonsconfigurations

    process shifted to the "right" ormeasurements are out of calibration

    LSL USL

    mix of two different processes, e.g. datafrom two operators, two machines, orcollected at different points in time

    LSL USL

    LSL USL

    "Special (assignable) causes" ofvariation, i.e. errors of measurement or

    in the activity of data collection

    process shifted to the "left" ormeasurements are out of calibration

    LSL USL

    granularity, i.e. "granular process"

    LSL USL

    stable process within specifications

    LSL USL

    configurations reasons

    process variation too large for thespecification limits

    LSL USL

    "truncated" data

    LSL USL

    Fig. 2.4 Exemplifying histograms shapes. LSL lower specification limit, USL upper specification limit

    1. Common causes variations. They are the noise ofa production system and are uncontrollable varia-tions.

    2. Assignable (or special) causes variations. Theycan be properly identified and controlled. Someexamples are turnover in workman load, break-downs, machine or tool wear out, and tool change.

    Control charts are a family of tools for detecting theexistence of special causes variations in order to avoidthem, i. e., eliminate all anomalous controllable pat-terns, and bring the process into a state called of sta-tistical control, or simply in control, whose ran-dom behavior is justified by the existence of commoncauses variations. The in control state is necessary toobtain conforming products, as properly discussed inthe following sections on capability analysis and SixSigma.

    Control charts can be constructed by extracting suc-cessive samples from the variable output of the pro-cess. These samples, also called subgroups, all havesize n and have to be taken at regular intervals of time.For each group a summary statistic is calculated andplotted as illustrated in Fig. 2.5.

    Typical statistical measures calculated for each sub-group are reported in Table 2.8, where the related sta-tistical distribution is cited together with the values of

  • 26 2 Quality Management Systems and Statistical Quality Control

    Subgroup number

    Subg

    roup

    stat

    istic

    (e.g

    . R, p

    , u)

    1 2 3 4 5

    UCL

    LCL

    centerline

    Fig. 2.5 Control chart

    the centerline and control limits, as properly defined inthe next subsections.

    A control chart is made of three basic lines as illus-trated in Fig. 2.5:

    1. Centerline. It is the mean of the statistic quantifiedfor each subgroup, the so-called subgroup statistic(e. g., mean, range, standard deviation).

    2. Control limits. These limits on a control chart de-limit that region where a data point falls outside,thus alerting one to special causes of variation.This region is normally extended three standarddeviations on either side of the mean. The controllimits are:

    upper control limit (UCL), above the mean; lower control limit (LCL), below the mean.

    The generic point of the chart in Fig. 2.5 may representa subgroup, a sample, or a statistic. k different sam-ples are associated with k different points whose tem-poral sequence is reported on the chart. Control limitsare conventionally set at a distance of three standardserrors, i. e., three deviations of the subgroup statis-tic, from the centerline, because the distribution ofsamples closely approximates a normal statistical dis-tribution by the central limit theorem. Consequently,the analyst expects that about 99.73% of samples liewithin three standard deviations of the mean. This cor-responds to a probability of 0.27% that a control chartpoint falls outside one of the previously defined con-trol limits when no assignable causes are present.

    In some countries, such as in the UK, the adoptedconvention of three standard deviations is different.

    Figure 2.6 presents eight different anomalous pat-terns of statistic subgroups tested by Minitab Statis-

    tical Software to find reliable conditions for the in, orout, control state of the process.

    A process is said to be in control when all sub-groups on a control chart lie within the control lim-its and no anomalous patterns are in the sequence ofpoints representing the subgroups. Otherwise, the pro-cess is said to be out of control, i. e., it is not ran-dom because there are special causes variations affect-ing the output obtained.

    What happens in the presence of special causes?It is necessary to identify and eliminate them. Conse-quently, if a chart shows the possible existence of spe-cial causes by one of the anomalous behaviors illus-trated in Fig. 2.6, the analyst and the person responsi-ble for the process have to repeat the analysis by elim-inating the anomalous subgroups. Now, if all the testsare not verified, the process has been conducted to thestate of statistical control.

    2.7 Control Charts for Means

    These charts refer to continuous measurement data,also called variable data (see Table 2.8), becausethere are an infinite number of data between twogeneric ones.

    2.7.1 The R-Chart

    This is a chart for subgroup ranges. The range is thedifference between the maximum and the minimumvalues within a sample of size n:

    Ri D maxj D1;:::;nfxij g minj D1;:::;nfxij g; (2.4)

    where i is a generic sample and xij is the j th value inthe sample i .

    Consequently, the centerline is

    R D NR D 1k

    kX

    iD1Ri : (2.5)

    This value is a good estimation of the mean value ofvariable Ri , called R . We also define the statisticmeasure of variability of the variable Ri , the standarddeviation R. By the central limit theorem, the distri-bution of values Ri is normal. As a consequence, the

  • 2.7 Control Charts for Means 27

    Table2.8

    Stat

    isti

    calm

    easu

    res

    and

    cont

    rolc

    hart

    clas

    sific

    atio

    n

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    tica

    lSt

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    Stan

    dard

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    onti

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    aldi

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    ange

    RN R

    D1 k

    k X iD1

    Ri

    UC

    LD

    R

    C3

    R

    D4

    N RL

    CL

    D

    R

    3R

    D

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    sO S

    DNsD

    1 k

    k X iD1

    s iU

    CL

    DB

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    LC

    LD

    B3Ns

    Nx-ch

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    Var

    iabl

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    aldi

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    NxO

    DN N X

    D1 k

    k X iD1

    N X iU

    CL

    DOC

    3O p n

    N N X

    C3

    N R=d

    2p n

    DN N X

    CA

    2N R

    LC

    LD

    O

    3O p n

    N N X

    3

    N R=d

    2p n

    DN N X

    A

    2N R

    ODN R d 2

    UC

    LD

    OC3

    O p n

    N N XC

    3Ns=

    c4

    p nD

    N N XC

    A3Ns

    LC

    LD

    O

    3O p n

    N N X

    3

    Ns=c

    4p n

    DN N X

    A

    3Ns

    ODNs c 4

    p-c

    hart

    Att

    ribu

    teD

    iscr

    ete

    bi

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    ial

    dist

    ribu

    tion

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    conf

    orm

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    prop

    orti

    onp

    NpD

    1 k

    k X iD1

    pi

    UC

    LD

    NpC3s

    Np.1

    Np/

    n

    LC

    LD

    Np3s

    Np.1

    Np/

    n

    NpD

    x1

    Cx

    2C

    C

    xk

    1

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    k

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    Cn

    2C

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    k

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    Li

    DNpC

    3sNp.

    1

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    i

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    Np/n

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    -cha

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    np

    nNp

    UC

    LD

    nNpC

    3pn

    Np.1

    Np/

    LC

    LD

    nNp

    3pn

    Np.1

    Np/

    c-c

    hart

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    ribu

    teD

    iscr

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    NcD1 k

    k X iD1

    ci

    UC

    LD

    NcC3p

    NcL

    CL

    DNc

    3pNc

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    mit

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    per

    unit

    uNuD

    1 k

    k X iD1

    ui

    UC

    Li

    DNuC

    3sNu n i

    LC

    Li

    DNu

    3sNu n i

    UC

    Lup

    per

    cont

    roll

    imit

    ,LC

    Llo

    wer

    cont

    roll

    imit

    ,UC

    Li

    uppe

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    ei,

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    Li

    low

    erco

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    llim

    itfo

    rsa

    mpl

    ei

  • 28 2 Quality Management Systems and Statistical Quality Control

    Test 1 1 point beyond 3 std.dev. (zone A)

    Test 2 9 points in a row on same sideof the center line

    Test 3 6 points in a row all increasing(or decreasing)

    Test 4 14 points in a row alternangup and down

    Test 6 4 out of 5 points more than1 std.dev.

    Test 5 2/3 points in a row more than2 std.dev.

    Test 7 15 points in a row within1 std.dev. (either side)

    Test 8 8 points in a row more than1 std.dev. (either side)

    Fig. 2.6 Eight tests for special causes investigation, Minitab Statistical Software

    control limits are defined as

    UCLR D R C 3R D4 NR;LCLR D R 3R D3 NR;

    (2.6)

    where R is the standard deviation of the variable Rand D4 is a constant value depending on the size of

    the generic subgroup. The values are reported in Ap-pendix A.2.

    The following equation represents an estimation ofthe standard deviation of the variable and continuousdata xij :

    O DNRd2

    (2.7)

  • 2.7 Control Charts for Means 29

    2.7.2 Numerical Example, R-Chart

    This application refers to the assembly process ofan automotive engine. The process variable is a dis-tance, D, measured in tenths of millimeters, betweentwo characteristic axes in the drive shafts and heads.Table 2.9 presents the data collected over 25 days ofobservation and grouped in samples of size n D 5.

    By the application of Eqs. 2.5 and 2.6, we have

    NR D 1k

    kX

    iD1Ri D 1

    25.R1 C CR25/ 6:50;

    UCL D R C 3R D4 NR DD4.nD5/D2:114

    13:74;

    LCL D R 3R D3 NR DD4.nD5/D0

    0:

    The R-chart obtained is reported in Fig. 2.7. Previ-ously introduced tests for anomalous behaviors are notverified. As a consequence, the process seems to berandom and coherent with itself and its characteris-tic noise and variance. There are no special causes ofvariation.

    Table 2.9 Data 25 subgroups, numerical example

    Sample Month Day D (mm=10)

    1 7 25 0:387 5.192 1.839 0.088 1.7742 7 26 4.251 3.333 4.398 6.082 1.7063 7 27 2:727 2:806 4.655 0.494 2:8074 7 28 6.980 3.280 3.372 1:914 2.4785 7 29 3.978 3.479 7.034 4.388 1:7906 7 30 3.424 1.758 0.009 0:216 1.8327 7 31 4:285 2:369 2:666 2.639 3.0818 8 1 1:756 1:434 1.887 1:678 7.0609 8 2 4.184 1.005 0.825 6:427 4:598

    10 8 3 3:577 1:684 1.800 4.339 0.02711 8 4 2:467 2:752 4:029 2:798 2:15212 8 5 1.199 0.817 0:213 0:737 1:75713 8 6 4.312 1.127 2.534 1.618 0:66514 8 7 3.282 3.319 3:564 3.430 1.55615 8 8 2.000 3:364 1:996 1:830 0.01516 8 9 3.268 1.519 2.704 0.138 0:05017 8 10 3.356 3:335 3:358 4:302 2:85618 8 11 0:240 3:811 1:615 3:510 4:37719 8 12 4:524 0:091 1.945 4.515 1:66720 8 13 0.837 4:536 4.249 0.114 0:08721 8 14 1:016 2.023 4.539 0.075 2:72422 8 15 4.547 0.262 4:108 1:881 0:00423 8 16 0.159 3.786 1:951 6.315 5.16124 8 17 0.842 3:550 1:805 2:731 1:61025 8 18 4.435 1.730 0:185 0.242 4:689

    2.7.3 The Nx-Chart

    This is a chart for subgroup means. In the Nx-chart,also called x-chart, the problem is the estimationof the standard deviation of the population of val-ues. In Sect. 2.7.1, Eq. 2.7 is an effective estimation.Consequently, this chart is generally constructed afterthe creation of the R-chart and reveals the process tobe in the state of statistical control. The centerline ofthe statistic variable Nx is the average of the subgroupmeans:

    O D O. Nx/ D NNx D 1k

    kX

    iD1Nxi D

    kX

    iD1

    nX

    j D1xij : (2.8)

    The control limits are

    UCL Nx D OC 3 Opn

    NNx C 3NR=d2pn

    D NNx C A2 NR;

    LCL Nx D O 3 Opn

    NNx 3NR=d2pn

    D NNx A2 NR;(2.9)

    where d2 andA2 are constant values as reported in Ap-pendix A.2.

  • 30 2 Quality Management Systems and Statistical Quality Control

    DayMonth

    Sample

    18161412108642312927258888888887777252321191715131197531

    14

    12

    10

    8

    6

    4

    2

    0

    Sam

    ple

    Ran

    ge

    _R=6.50

    UCL=13.74

    LCL=0

    R Chart of D

    Fig. 2.7 R-chart, numerical example. Minitab Statistical Software. UCL upper control limit, LCL lower control limit

    2.7.4 Numerical Example, Nx-Chart

    Consider the application introduced in Sect. 2.7.2. ByEqs. 2.8 and 2.9,

    O D NNx D 1k

    kX

    iD1Nxi D 1

    25. Nx1 C C Nx25/ D 0:389;

    UCL D OC 3 Opn

    NNx C 3NR=d2pn

    D NNx C A2 NRD

    A2D0:5770:389 C 0:577 6:5 4:139;

    LCL D O 3 Opn

    NNx 3NR=d2pn

    D NNx A2 NRD

    A2D0:5770:389 0:577 6:5 3:361:

    The chart obtained is reported in Fig. 2.8. Test 6 foranomalous behaviors is verified in sample 5, month 7,and day 29, i. e., there are four of five points in zone Bor beyond. As a consequence, the process seems to beout of control. There is in fact a very scarce proba-bility of having a sample in those points when the pro-cess is in control. We assume we are able to properlyidentify this special cause of variation and to elimi-nate it. Figure 2.9 presents the charts obtained from thepool of samples without the anomalous subgroup 5.The chart shows another potential anomalous behav-

    ior regarding subgroup 4. In this way, assuming weidentify and eliminate new special causes, we obtainFigs. 2.10 and 2.11. In particular, Fig. 2.11 presentsa process in the state of statistical control: subgroups 2,4, and 5 have been eliminated.

    2.7.5 The s-Chart

    This chart for subgroup standard deviation can be usedto support the construction of the x-chart by the es-timation of the standard deviation of the continuousvariable xij . In particular, the control limits of the x-chart use the centerline of the s-chart.

    The average of standard deviation of subgroups, Os,is the centerline of the s-chart:

    OS D O.si / D Ns D 1k

    kX

    iD1si ; (2.10)

    where O.si / is the estimation of the mean of the vari-able si , the standard deviation of a subgroup.

    The control limits are

    UCLs D O.si /C 3 O.si /pn

    D B4 Ns;

    LCLs D O.si / 3 O.si /pn

    D B3 Ns;(2.11)

  • 2.7 Control Charts for Means 31

    DayMonth

    Sample

    18161412108642312927258888888887777252321191715131197531

    4

    2

    0

    -2

    -4

    Sam

    ple

    Mea

    n

    __X=0.389

    UCL=4.136

    LCL=-3.359

    DayMonth

    Sample

    18161412108642312927258888888887777252321191715131197531

    15

    10

    5

    0

    Sam

    ple

    Ran

    ge

    _R=6.50

    UCL=13.74

    LCL=0

    6

    Xbar-R Chart of D

    Fig. 2.8 x-chart from R. Numerical example (25 samples). Minitab Statistical Software

    Day [24]Month [24]

    Sample [24]

    1715131197531302725888888888777

    24222018161412108631

    4

    2

    0

    -2

    -4

    Sam

    ple

    Mea

    n

    __X=0.262

    UCL=3.984

    LCL=-3.459

    Day [24]Month [24]

    Sample [24]

    1715131197531302725888888888777

    24222018161412108631

    15

    10

    5

    0

    Sam

    ple

    Ran

    ge

    _R=6.45

    UCL=13.64

    LCL=0

    5

    Xbar-R Chart [rif.no sub.5]

    Fig. 2.9 x-chart from R. Numerical example (24 samples). Minitab Statistical Software

  • 32 2 Quality Management Systems and Statistical Quality Control

    Day [23]Month [23]

    Sample [23]

    18161412108642312725888888888777

    25232119171513119731

    4

    2

    0

    -2

    -4

    Sam

    ple

    Mea

    n

    __X=0.150

    UCL=3.847

    LCL=-3.546

    Day [23]Month [23]

    Sample [23]

    18161412108642312725888888888777

    25232119171513119731

    15

    10

    5

    0

    Sam

    ple

    Ran

    ge

    _R=6.41

    UCL=13.55

    LCL=0

    1Xbar-R Chart [no sub 4 and 5]

    Fig. 2.10 x-chart from R. Numerical example (23 samples). Minitab Statistical Software

    Day [22]Month [22]

    Sample [22]

    1715131197531302588888888877

    2422201816141210861

    4

    2

    0

    -2

    -4

    Sam

    ple

    Mea

    n

    __X=-0.022

    UCL=3.729

    LCL=-3.774

    Day [22]Month [22]

    Sample [22]

    1715131197531302588888888877

    2422201816141210861

    15

    10

    5

    0

    Sam

    ple

    Ran

    ge

    _R=6.50

    UCL=13.75

    LCL=0

    Xbar-R Chart [no sub 2, 4 and 5]

    Fig. 2.11 x-chart from R. Numerical example (22 samples). Minitab Statistical Software

  • 2.8 Control Charts for Attribute Data 33

    where O.si / is the estimation of the standard devia-tion of the variable si , the standard deviation of a sub-group, and B3 and B4 are constant values reported inAppendix A.2.

    The standard deviation of the process measurementis

    O D O. Nxi / D Nsc4: (2.12)

    As a consequence, the control limits of the x-chart are

    UCL Nx D OC 3 Opn

    NNx C 3 Ns=c4pn

    D NNx C A3 Ns;

    LCL Nx D O 3 Opn

    NNx 3 Ns=c4pn

    D NNx A3 Ns;(2.13)

    where A3 is a constant value reported in Ap-pendix A.2.

    2.7.6 Numerical Example, s-Chartand Nx-Chart

    Table 2.10 reports a set of measurement data madefor 20 samples of size n D 5. They are the outputof a manufacturing process in the automotive industry.The last three columns report some statistics useful forthe construction of the control charts and for verifica-tion of the status of the control of the process.

    With use of the values of the constant parameters inAppendix A.2, the following control limits and center-lines have been obtained.

    Firstly, we propose the results related to the R-chart. By Eq. 2.5 the centerline is

    R D NR D 1k

    kX

    iD1Ri 0:004155:

    By Eq. 2.6

    UCLR D4 NR D 2:114 0:004155 0:008784:LCLR D3 NR D 0 0:004155 D 0:

    These results are very close to those proposed by theR-chart, as constructed by the tool Minitab Statisti-cal Software (Fig. 2.12). From the R-chart the processseems to be in the state of statistical control.

    The x-chart is now created assuming the centerlineof theR-chart and in accordance with Eqs. 2.8 and 2.9:

    UCLx from R NNx CA2 NRD 0:009237 C 0:577 0:004155D 0:0116;

    LCLx from R NNx A2 NRD 0:009237 0:577 0:004155D 0:0068:

    The upper section of Fig. 2.12 presents the x-chartwhere some subgroups verify a few tests, as illustratedalso in Fig. 2.13. Consequently, the process is not ina state of control.

    Similarly, by the application of Eqs. 2.10, 2.11,and 2.13,

    OS D Ns D 1k

    kX

    iD1si 0:00170:

    UCLs B4 Ns D 2:089 0:00170 D 0:00355;LCLs B3 Ns D 0 0:00170 D 0;

    UCLx from s NNx C A3 NsD 0:009237 C 1:427 0:00170D 0:01166;

    LCLx from s NNx A3 NsD 0:009237 1:427 0:00170D 0:00681:

    All these values are also reported in Fig. 2.14, show-ing that the process is not in the state of statisticalcontrol. Consequently, a survey for the identificationand deletion of special causes of variations, and thesubsequent repetition of the control analysis, is re-quired.

    2.8 Control Charts for Attribute Data

    These charts refer to counted data, also called at-tribute data. They support the activities of monitoringand analysis of production processes whose productspossess, or do not possess, a specified characteristicor attribute. Attributes measurement is frequently ob-tained as the result of human judgements.

  • 34 2 Quality Management Systems and Statistical Quality Control

    Table 2.10 Measurement data and subgroup statistics. Numerical example

    ID sample i Measure X M.xi / Ri si

    1 0.0073 0.0101 0.0091 0.0091 0.0053 0.0082 0.0048 0.00192 0.0106 0.0083 0.0076 0.0074 0.0059 0.0080 0.0047 0.00173 0.0096 0.0080 0.0132 0.0105 0.0098 0.0102 0.0052 0.00194 0.0080 0.0076 0.0090 0.0099 0.0123 0.0094 0.0047 0.00195 0.0104 0.0084 0.0123 0.0132 0.0120 0.0113 0.0048 0.00196 0.0071 0.0052 0.0101 0.0123 0.0073 0.0084 0.0071 0.00287 0.0078 0.0089 0.0122 0.0091 0.0095 0.0095 0.0044 0.00168 0.0087 0.0094 0.0120 0.0102 0.0099 0.0101 0.0033 0.00129 0.0074 0.0081 0.0120 0.0116 0.0122 0.0103 0.0048 0.0023

    10 0.0081 0.0065 0.0105 0.0125 0.0136 0.0102 0.0071 0.002911 0.0078 0.0098 0.0113 0.0087 0.0118 0.0099 0.0040 0.001712 0.0089 0.0090 0.0111 0.0122 0.0126 0.0107 0.0037 0.001713 0.0087 0.0075 0.0125 0.0106 0.0113 0.0101 0.0050 0.002014 0.0084 0.0083 0.0101 0.0140 0.0097 0.0101 0.0057 0.002315 0.0074 0.0091 0.0116 0.0109 0.0108 0.0100 0.0042 0.001716 0.0069 0.0093 0.0090 0.0084 0.0090 0.0085 0.0024 0.001017 0.0077 0.0089 0.0091 0.0068 0.0094 0.0084 0.0026 0.001118 0.0076 0.0069 0.0062 0.0077 0.0067 0.0070 0.0015 0.000619 0.0069 0.0077 0.0073 0.0074 0.0074 0.0073 0.0008 0.000320 0.0063 0.0071 0.0078 0.0063 0.0088 0.0073 0.0025 0.0011

    Mean 0.009237 0.004155 0.0016832

    191715131197531

    0.011

    0.010

    0.009

    0.008

    0.007

    Sample

    Sam

    ple

    Mea

    n

    __X=0.009237

    UCL=0.011666

    LCL=0.006808

    191715131197531

    0.008

    0.006

    0.004

    0.002

    0.000

    Sample

    Sam

    ple

    Ran

    ge

    _R=0.004211

    UCL=0.008905

    LCL=0

    553

    266

    6

    Xbar-R Chart

    Fig. 2.12 R-chart and x-chart from R. Numerical example. Minitab Statistical Software

  • 2.8 Control Charts for Attribute Data 35

    Test Results for Xbar Chart

    TEST 2. 9 points in a row on same side of center line.Test Failed at points: 15

    TEST 3. 6 points in a row all increasing or all decreasing.Test Failed at points: 18

    TEST 5. 2 out of 3 points more than 2 standard deviations from center line (on one side of CL). Test Failed at points: 19; 20

    TEST 6. 4 out of 5 points more than 1 standard deviation from center line (on one side of CL). Test Failed at points: 12; 13; 14; 20

    Fig. 2.13 x-chart from R, test results. Numerical example. Minitab Statistical Software

    191715131197531

    0.011

    0.010

    0.009

    0.008

    0.007

    Sample

    Sam

    ple

    Mea

    n

    __X=0.009237

    UCL=0.011666

    LCL=0.006808

    191715131197531

    0.004

    0.003

    0.002

    0.001

    0.000

    Sample

    Sam

    ple

    StD

    ev

    _S=0.001702

    UCL=0.003555

    LCL=0

    553

    266

    6

    Xbar-S Chart

    Fig. 2.14 s-chart and x-chart from s. Numerical example. Minitab Statistical Software

    2.8.1 The p-Chart

    The p-chart is a control chart for monitoring the pro-portion of nonconforming items in successive sub-groups of size n. An item of a generic subgroup is saidto be nonconforming if it possesses a specified charac-teristic. Given p1; p2; : : : ; pk , the subgroups propor-tions of nonconforming items, the sampling random

    variable pi for the generic sample i has a mean anda standard deviation:

    p D ;

    p Dr.1 /

    n;

    (2.14)

    where is the true proportion of nonconforming itemsof the process, i. e., the population of items.

  • 36 2 Quality Management Systems and Statistical Quality Control

    The equations in Eq. 2.14 result from the binomialdiscrete distribution of the variable number of noncon-formities x. This distribution function is defined as

    p.x/ D n

    x

    !x.1 /nx ; (2.15)

    where x is the number of nonconformities and is theprobability the generic item has the attribute.

    The mean value of the standard deviation of thisdiscrete random variable is

    DX

    x

    xp.x/ D n;

    DsX

    x

    .x /2p.x/ D n.1 /:(2.16)

    By the central limit theorem, the centerline, as the esti-mated value of , and the control limits of the p-chartare

    Op D O.pi / D Np D 1k

    kX

    iD1pi ; (2.17)

    UCLp D Np C 3r Np.1 Np/

    n;

    LCLp D Np 3r Np.1 Np/

    n:

    (2.18)

    If the number of items for a subgroup is not constant,the centerline and the control limits are quantified bythe following equations:

    Np D x1 C x2 C C xk1 C xkn1 C n2 C C nk1 C nk ; (2.19)

    where xi is the number of nonconforming items insample i and ni is the number of items within the sub-group i , and

    UCLp;i D Np C 3s

    Np.1 Np/ni

    ;

    LCLp;i D Np 3s

    Np.1 Np/ni

    ;

    (2.20)

    where UCLi is the UCL for sample i and LCLi is theLCL for sample i .

    Table 2.11 Rejects versus tested items. Numerical example

    Day Rejects Tested Day Rejects Tested

    21=10 32 286 5=11 21 28122=10 25 304 6=11 14 31023=10 21 304 7=11 13 31324=10 23 324 8=11 21 29325=10 13 289 9=11 23 30526=10 14 299 10=11 13 31727=10 15 322 11=11 23 32328=10 17 316 12=11 15 30429=10 19 293 13=11 14 30430=10 21 287 14=11 15 32431=10 15 307 15=11 19 2891=11 16 328 16=11 22 2992=11 21 304 17=11 23 3183=11 9 296 18=11 24 3134=11 25 317 19=11 27 302

    2.8.2 Numerical Example, p-Chart

    Table 2.11 reports the data related to the number ofelectric parts rejected by a control process considering30 samples of different size.

    By the application of Eqs. 2.19 and 2.20,

    Np D x1 C x2 C C xk1 C xkn1 C n2 C C nk1 C nk D

    573

    9171

    0:0625;

    UCLp;i D Np C 3s

    Np.1 Np/ni

    0:0625 C 3s

    0:0625.1 0:0625/ni

    ;

    LCLp;i D Np 3s

    Np.1 Np/ni

    0:0625 3s

    0:0625.1 0:0625/ni

    :

    Figure 2.15 presents the p-chart generated byMinitab Statistical Software and shows that test 1(one point beyond three standard deviations) occursfor the first sample. This chart also presents the non-continuous trend of the control limits in accordancewith the equations in Eq. 2.20.

  • 2.8 Control Charts for Attribute Data 37

    17/1114/1111/118/115/112/1130/1027/1024/1021/10

    0.11

    0.10

    0.09

    0.08

    0.07

    0.06

    0.05

    0.04

    0.03

    0.02

    Day

    Pro

    port

    ion

    _P=0.0625

    UCL=0.1043

    LCL=0.0207

    1

    P Chart of Rejects

    Tests performed with unequal sample sizes

    Fig. 2.15 p-chart with unequal sample sizes. Numerical example. Minitab Statistical Software

    2.8.3 The np-Chart

    This is a control chart for monitoring the number ofnonconforming items in subgroups having the samesize. The centerline and control limits are

    Onp D n Np; (2.21)UCLnp D n Np C 3

    pn Np.1 Np/;

    LCLnp D n Np 3pn Np.1 Np/:

    (2.22)

    2.8.4 Numerical Example, np-Chart

    The data reported in Table 2.12 relate to a productionprocess similar to that illustrated in a previous applica-tion, see Sect. 2.8.2. The size of the subgroups is nowconstant and equal to 280 items. Figure 2.16 presentsthe np-chart generated by Minitab Statistical Soft-ware: test 1 is verified by two consecutive samples(collected on 12 and 13 November). The analyst hasto find the special causes, then he/she must eliminatethem and regenerate the chart, as in Fig. 2.17. Thissecond chart presents another anomalous subgroup:11=11. Similarly, it is necessary to eliminate this sam-ple and regenerate the chart.

    Table 2.12 Rejected items. Numerical example

    Day Rejects Day Rejects

    21=10 19 5=11 2122=10 24 6=11 1423=10 21 7=11 1324=10 23 8=11 2125=10 13 9=11 2326=10 32 10=11 1327=10 15 11=11 3428=10 17 12=11 3529=10 19 13=11 3630=10 21 14=11 1531=10 15 15=11 191=11 16 16=11 222=11 21 17=11 233=11 12 18=11 244=11 25 19=11 27

    2.8.5 The c-Chart

    The c-chart is a control chart used to track the numberof nonconformities in special subgroups, called in-spection units. In general, an item can have any num-ber of nonconformities. This is an inspection unit, asa unit of output sampled and monitored for determina-tion of nonconformities. The classic example is a sin-gle printed circuit board. An inspection unit can bea batch, a collection, of items. The monitoring activ-ity of the inspection unit is useful in a continuous pro-

  • 38 2 Quality Management Systems and Statistical Quality Control

    17/1114/1111/118/115/112/1130/1027/1024/1021/10

    35

    30

    25

    20

    15

    10

    Day

    Sam

    ple

    Cou

    nt

    __NP=21.1

    UCL=34.35

    LCL=7.85

    11

    NP Chart of Rejects

    Fig. 2.16 np-chart, equal sample sizes. Numerical example. Minitab Statistical Software

    19/1116/1111/118/115/112/1130/1027/1024/1021/10

    35

    30

    25

    20

    15

    10

    Day (no 12 & 13 /11)

    Sam

    ple

    Cou

    nt

    __NP=20.07

    UCL=33.02

    LCL=7.12

    1

    NP Chart of Rejects (no 12 & 13 /11)

    Fig. 2.17 np-chart, equal sample sizes. Numerical example. Minitab Statistical Software

    duction process. The number of nonconformities perinspection unit is called c.

    The centerline of the c-chart has the following av-erage value:

    Oc D O.ci / D Nc D 1k

    kX

    iD1ci : (2.23)

    The control limits are

    UCLc D Nc C 3pNc;

    LCLc D Nc 3pNc:

    (2.24)

    The mean and the variance of the Poisson distribution,defined for the random variable number of nonconfor-mities units counted in an inspection unit, are

    .ci / D .ci / D Nc: (2.25)The density function of this very important discreteprobability distribution is

    f .x/ D ex

    x; (2.26)

    where x is the random variable.

  • 2.8 Control Charts for Attribute Data 39

    2.8.6 Numerical Example, c-Chart

    Table 2.13 reports the number of coding errors madeby a typist in a page of 6,000 digits. Figure 2.18 showsthe c-chart obtained by the sequence of subgroups andthe following reference measures:

    Nc D 1k

    kX

    iD1ci D 6:8;

    UCLc D Nc C 3pNc D 6:8 C 3p6:8 14:62;

    LCLc D Nc 3pNc D maxf6:8 3p6:8; 0g 0;

    where ci is the number of nonconformities in an in-spection unit.

    From Fig. 2.18 there are no anomalous behaviorssuggesting the existence of special causes of variationsin the process, thus resulting in a state of statisticalcontrol.

    A significant remark can be made: why does thisnumerical example adopt the c-chart and not the p-chart? If a generic digit can be, or cannot be, an objectof an error, it is in fact possible to consider a binomialprocess where the probability of finding a digit with an

    28252219161310741

    16

    14

    12

    10

    8

    6

    4

    2

    0

    Day

    Sam

    ple

    Cou

    nt

    _C=6.8

    UCL=14.62

    LCL=0

    C Chart of errors

    Fig. 2.18 c-chart. Inspection unit equal to 6,000 digits. Numerical example. Minitab Statistical Software

    Table2.13 Errors in inspection unit of 6,000 digits. Numericalexample

    Day Errors Day Errors

    1 10 16 82 11 17 73 6 18 14 9 19 25 12 20 36 12 21 57 14 22 18 9 23 119 5 24 910 0 25 1411 1 26 112 2 27 913 1 28 114 11 29 815 9 30 12

    error is

    pi D cin

    D ci6;000

    ;

    where n is the number of digits identifying the inspec-tion unit.

    The corresponding p-chart, generated by Minitab

    Statistical Software and shown in Fig. 2.19, is verysimilar to the c-chart in Fig. 2.18.

  • 40 2 Quality Management Systems and Statistical Quality Control

    28252219161310741

    0.0025

    0.0020

    0.0015

    0.0010

    0.0005

    0.0000

    Day

    Pro

    port

    ion

    _P=0.001133

    UCL=0.002436

    LCL=0

    P Chart of errors

    Fig. 2.19 p-chart. Inspection unit equal to 6,000 digits. Numerical example. Minitab Statistical Software

    2.8.7 The u-Chart

    If the subgroup does not coincide with the inspectionunit and subgroups are made of different numbers ofinspection units, the number of nonconformities perunit, ui , is

    ui D cin: (2.27)

    The centerline and the control limits of the so-calledu-chart are

    Ou D O.ui / D Nu D 1k

    kX

    iD1ui ;

    UCLu;i D NuC 3s

    Nuni;

    LCLu;i D Nu 3s

    Nuni:

    (2.28)

    2.8.8 Numerical Example, u-Chart

    Table 2.14 reports the number of nonconformities asdefects on ceramic tiles of different sizes, expressed infeet squared.

    Figure 2.20 presents the u-chart obtained; five dif-ferent subgroups reveal themselves as anomalous. Fig-

    ure 2.21 shows the chart obtained by the elimination ofthose samples. A new sample, i D 30, is irregular.

    2.9 Capability Analysis

    A production process is said to be capable when it is instate of statistical control and products meet the spec-ification limits, i. e., the customers requirements. Inother words, the process is capable when it producesgood products. This is the first time the lower andupper specifications are explicitly considered in theanalysis of the process variations.

    Nonconformity rates are the proportions of pro-cess measurements above, or below, the USL, or LSL.This proportion can be quantified in parts per million(PPM), as

    PPM > USL D P.x > USL/ Pz >

    USL OO

    ;

    (2.29)

    PPM < LSL D P.x < LSL/ Pz USL 40000.00PPM Total 80000.00

    Observed PerformancePPM < LSL 53034.23PPM > USL 43586.49PPM Total 96620.72

    Exp. Within PerformancePPM < LSL 53675.43PPM > USL 44166.87PPM Total 97842.30

    Exp. Overall Performance

    WithinOverall

    Fig. 2.23 Capability analysis process 1, numerical example. Minitab Statistical Software

    Table 2.16 reports the process data as a result of theprocess improvement made for a new set of k D 20samples with n D 5 measurements each. Figure 2.25presents the report generated by the six-pack analysis.

    It demonstrates that the process is still in statisticalcontrol, centered on the target value, 600 mm, and witha Cpk value equal to 3.31. Consequently, the negligi-ble expected number of PPM outside the specification

  • 2.9 Capability Analysis 45

    191715131197531

    601

    600

    599

    Sam

    ple

    Mea

    n

    __X=599.972

    UCL=600.778

    LCL=599.165

    191715131197531

    3.0

    1.5

    0.0

    Sam

    ple

    Ran

    ge

    _R=1.398

    UCL=2.957

    LCL=0

    2015105

    601

    600

    599

    Sample

    Val

    ues

    601.5601.0600.5600.0599.5599.0598.5

    LSL Target USL

    LSL 599Target 600USL 601

    Specifications

    602600598

    Within

    Overall

    Specs

    StDev 0.60121Cp 0.55Cpk 0.54

    WithinStDev 0.603415Pp 0.55Ppk 0.54Cpm 0.55

    Overall

    Process Capability Sixpack of DATAXbar Chart

    R Chart

    Last 20 Subgroups

    Capability Histogram

    Normal Prob PlotAD: 0.481, P: 0.228

    Capability Plot

    Fig. 2.24 Six-pack analysis process 1, numerical example. Minitab Statistical Software

    Table 2.16 Measurement data process 2, numerical example

    Sample Data process 2 Mean value Range

    2.1 600.041 600.0938 600.1039 600.0911 600.1096 600.08788 0.06862.2 599.8219 599.9173 600.0308 600.07 600.0732 599.98264 0.25132.3 600.0089 600.075 600.0148 599.9714 600.0271 600.01944 0.10362.4 600.1896 600.1723 599.8368 600.0947 599.9781 600.0543 0.35282.5 600.1819 600.0538 599.9957 600.0995 599.9639 600.05896 0.2182.6 599.675 599.9778 599.9633 599.9895 599.8853 599.89818 0.31452.7 600.0521 600.1707 599.9446 599.8487 600.012 600.00562 0.3222.8 600.0002 600.0831 599.9298 599.9329 599.9142 599.97204 0.16892.9 600.02 599.9963 599.9278 599.9793 600.0456 599.9938 0.11782.10 600.1571 600.0212 599.9061 599.9786 600.0626 600.02512 0.2512.11 600.0934 599.9554 599.7975 600.0221 599.8821 599.9501 0.29592.12 599.8668 599.8757 600.0414 599.7939 600.1153 599.93862 0.32142.13 599.9859 599.9269 599.8124 600.0288 600.0261 599.95602 0.21642.14 599.9456 600.0405 600.0576 599.7819 600.0603 599.97718 0.27842.15 600.0487 600.0569 599.9321 599.9164 599.9984 599.9905 0.14052.16 599.8959 599.979 600.1418 600.1157 599.9525 600.01698 0.24592.17 600.1891 600.1168 600.1106 599.9148 600.0013 600.06652 0.27432.18 600.0002 600.1121 599.93 599.9924 600.0458 600.0161 0.18212.19 599.9228 600.092 599.9225 600.1062 600.1794 600.04458 0.25692.20 599.7843 599.9597 600.011 600.0409 600.0436 599.9679 0.2593

    Average 600.001124 0.23198

  • 46 2 Quality Management Systems and Statistical Quality Control

    191715131197531

    600.1

    600.0

    599.9

    Sam

    ple

    Mea

    n

    __X=600.0011

    UCL=600.1362

    LCL=599.8660

    191715131197531

    0.4

    0.2

    0.0

    Sam

    ple

    Ran

    ge

    _R=0.2342

    UCL=0.4952

    LCL=0

    2015105

    600.2

    600.0

    599.8

    Sample

    Val

    ues

    600.9600.6600.3600.0599.7599.4599.1

    LSL Target USL

    LSL 599Target 600USL 601

    Specifications

    600.4600.0599.6

    Within

    Overall

    Specs

    StDev 0.100682Cp 3.31Cpk 3.31

    WithinStDev 0.101659Pp 3.28Ppk 3.28Cpm 3.28

    Overall

    Process Capability Sixpack of DATA2Xbar Chart

    R Chart

    Last 20 Subgroups

    Capability Histogram

    Normal Prob PlotAD: 0.408, P: 0.340

    Capability Plot

    Fig. 2.25 Six-pack analysis process 2, numerical example. Minitab Statistical Software

    limits is quantified as

    Total PPM DPz >

    USL OO

    C Pz USL 6666.67PPM Total 6666.67

    Observed Performance

    PPM < LSL *PPM > USL 2344.33PPM Total 2344.33

    Exp. Overall Performance

    Process Capability of dataCalculations Based on Weibull Distribution Model

    Fig. 2.26 Capability analysis Weibull distribution, numerical example. Minitab Statistical Software

    151413121110987654321

    5

    3

    1

    Sam

    ple

    Mea

    n

    __X=3.098

    UCL=4.926

    LCL=1.270

    151413121110987654321

    3

    2

    1Sam

    ple

    StD

    ev

    _S=1.874

    UCL=3.217

    LCL=0.532

    15105

    10

    5

    0

    Sample

    Val

    ues

    9.07.56.04.53.01.50.0

    USL

    USL 10Specifications

    10.001.000.100.01

    Overall

    Specs

    Shape 1.70519Scale 3.47777Pp *Ppk 0.93

    Overall

    Process Capability Sixpack - Weibull distributionXbar Chart

    S Chart

    Last 15 Subgroups

    Capability Histogram

    Weibull Prob PlotAD: 0.248, P: > 0.250

    Capability Plot

    Fig. 2.27 Six-pack analysis Weibull distribution, numerical example. Minitab Statistical Software

    2.9.2.1 Weibull Distribution

    Table 2.17 reports data regarding the output of manu-facturing process of tile production in the ceramics in-dustry. This measurement refers to the planarity of thetile surface as the maximum vertical distance of cou-

    ples of two generic points on the surface, assuming asthe USL a maximum admissible value of 1 mm.

    Figures 2.26 and 2.27 present the report gener-ated by Minitab Statistical Software for the capabilityanalysis. The production process generates products,i. e., output, that are well fitted by a Weibull statisti-

  • 48 2 Quality Management Systems and Statistical Quality Control

    cal distribution, shape parameter D 1:71 and scaleparameter D 3:48. The process is therefore in sta-tistical control but it does not meet customer require-ments in terms of an admissible USL. In other words,the process is predictable but not capable. In par-ticular, the number of expected items over the USL isabout 6,667 PPM.

    2.9.2.2 Binomial Distribution

    This application deals with a call center. Table 2.18 re-ports the number of calls received in 1 h, between 3and 4 p.m., and the number of calls that were not an-swered by the operators. The measurement data can bemodeled by assuming a binomial distribution of val-ues. Figure 2.28 presents the results of the capabilityanalysis conducted on this set of values, called dataset 1. The process is not in statistical control becausesample 15 is over the UCL. As a consequence, it is notcorrect to quantify the production process capability.This figure nevertheless shows that the process is dif-ficultly capable, also in the absence of sample 15. Inorder to meet the demand of customers properly it isuseful to increase the number of operators in the callcenter.

    191715131197531

    0.26

    0.24

    0.22

    0.20

    Sample

    Pro

    port

    ion

    _P=0.22769

    UCL=0.25684

    LCL=0.19853

    2015105

    23.0

    22.5

    22.0

    21.5

    21.0

    Sample

    % D

    efec

    tive

    Upper CI: 0.7605

    %Defective: 22.77Lower CI: 22.35Upper CI: 23.19Target: 0.20PPM Def: 227686Lower CI: 223487

    Upper CI: 231926Process Z: 0.7465Lower CI: 0.7325

    (95,0% confidence)

    Summary Stats

    200019201840

    26

    24

    22

    20

    Sample Size

    % D

    efec

    tive

    24201612840

    8

    6

    4

    2

    0

    % Defective

    Freq

    uen

    cy

    Tar

    1

    Binomial Process Capability Analysis of no answerP Chart

    Tests performed with unequal sample sizes

    Cumulative % Defective

    Rate of Defectives

    Histogram

    Fig. 2.28 Binomial process capability, numerical example. Minitab Statistical Software

    Table 2.18 Number of calls and no answer, numerical ex-ample

    Sample No answer Calls Sample No answer Calls

    day 1 421 1935 day 11 410 1937day 2 392 1945 day 12 386 1838day 3 456 1934 day 13 436 2025day 4 436 1888 day 14 424 1888day 5 446 1894 day 15 497 1894day 6 429 1941 day 16 459 1941day 7 470 1868 day 17 433 1868day 8 455 1894 day 18 424 1894day 9 427 1938 day 19 425 1933day 10 424 1854 day 20 441 1862

    2.10 Six Sigma

    Six Sigma stands for six standard deviations andcan be defined as a business management strat-egy, originally developed by Motorola, that enjoyswidespread application in many sectors of industryand services. Six Sigma was originally developed asa set of practices designed to improve manufacturingprocesses and eliminate defects. This chapter presentsa synthetic recall of the basic purpose of Six Sigma,assuming that a large number of the models and meth-ods illustrated here and in the following can properly

  • 2.10 Six Sigma 49

    support it. Nevertheless, there are a lot of ad hoctools and models specifically designed by the theoristsand practitioners of this decisional and systematic ap-proach, as properly illustrated in the survey by Blackand Hunter (2003).

    Six Sigma is a standard and represents a measureof variability and repeatability in a production process.In particular, the 6 specifications, also known as SixSigma capabilities, ask a process variability to be ca-pable of producing a very high proportion of output

    Lower specification limit

    Upper specification limit

    capability 6

    x

    x

    x

    x

    capability 6

    a

    b

    c

    d

    Outside lower specification

    Out of specification

    Out of specification

    The process is very capable with 6

  • 50 2 Quality Management Systems and Statistical Quality Control

    i. e., 0.002 PPM:

    1 C6Z

    6f .x/dx D 21 .z D 6/

    0:00000000198024 2 109; (2.35)where is the standard deviation of the process, f .x/is the density function of the variable x, a measure

    Fig. 2.30 Four sigma (Cp DCpk D 1:33) versus SixSigma (Cp D Cpk D 2)

    5 10 155 10 155 10 15

    0 1 2 3 4 5 6-6 -5 -4 -3 -1-2

    six sigma design specification width

    mean

    LSULSL

    Cp=2 Cpk=1.5

    Cp= Cpk= 2

    Cp=2 Cpk=1.5 virtually zero defects

    5 10 150 1 2 3 4 5 6-6 -5 -4 -3 -1-2

    5 10 155 10 15

    LSU LSL

    four sigma design specification width

    defectdefect

    Cp=1.33 Cp=1.33

    1.5-1.5

    of the output of the process (process characteristic) x is assumed to be normally distributed and isa cumulative function of the standard normal distribu-tion.

    Figure 2.30, proposed by Black and Hunter (2003),compares the performance of a capable process withCp D Cpk D 1:33, known also as four sigma ca-pability, and a process with Cp D Cpk D 2, whichguarantees Six Sigma capability.

  • 2.10 Six Sigma 51

    2.10.1 Numerical Examples

    Among the previously illustrated numerical exam-ples only the one discussed in Sect. 2.9.1 (process 2)verifies the Six Sigma hypotheses, because Cp DCpk D 3:31.

    2.10.2 Six Sigma in the Service Sector.ThermalWater Treatmentsfor Health and Fitness

    In this subsection we present the results obtained bythe application of the Six Sigma philosophy to thehealth service sector of thermal water treatments. Thisinstance demonstrates how this methodological ap-proach is effective also for the optimization of ser-vice processes. In particular, in this case study severalhealth and fitness treatments are offered and they are

    Customer contact

    Boo

    king

    Boo

    king

    /Rec

    eptio

    nC

    usto

    mer E-mail

    deliveredRemote request

    Fax delivered

    Local request

    Call Missed request

    Recognize customer request

    Phonerequest?

    Unoccupiedcollegue?

    Forward to a collegue

    Luckycustomer?

    Call customer back

    Apply phone number for recall

    Acquaintcustomer

    yes

    no

    Consistentrequest?

    no

    yes

    no

    no

    no yes

    yes

    Being out tobook?

    Missed booking

    RetrieveCustomer

    Master Data

    no

    yes

    Customerin Master

    Data?

    Input Customer Data

    no

    noSingle customer?

    Request for wellness?

    Requestfor thermal treat-

    ment?

    Single hotel reservation

    Single thermal reservation

    Single wellness reservation

    yes yes

    noyes

    yes

    no no Is it a group?Request forwellness?

    no

    Group thermal reservation

    Group wellness reservation

    Group hotel reservation

    yes yes

    yes

    Tour operator reservation

    no

    Legend: Event Activity Note Decision Process Subprocess

    Take customer request

    Requestaccepted?

    yes

    Requestfor thermal treat-

    ment?

    Fig. 2.31 Booking procedure

    grouped in three divisions, each with a proper bookingoffice and dedicated employees: hotel, wellness, andthermal services.

    Employees are nominated to have contact with thecostumers, to identify their requirements, to accept therequests, and to finalize the booking process. Cus-tomers can have contact via telephone, e-mail, Website, or by presenting themselves at the reception. Ev-ery kind of service has its own booking procedure, de-pending on the customer request. Before the applica-tion of Six Sigma methodologies the process was di-vided into the following five subroutines, dependingon the service:

    single thermal booking; group thermal booking; single hotel booking; tour operator hotel booking; wellness booking.

  • 52 2 Quality Management Systems and Statistical Quality Control

    Once the booking procedure has been completed thestaff will wait for the customer. On his/her arrival, therelated booking data are recalled from the system andthe customer is sent to the so-called welcome pro-cess, which is common to hotel, wellness, and ther-mal services. By the next check-in stage the customeris accepted and can access the required service. Thereis a specific check-in stage with its own dedicated rulesand procedures for every kind of service. Once the cus-tomer has enjoyed the service, he/she will leave thesystem and go to the checkout stage, with its own pro-cedures too.

    The whole process, from the admittance to theexit of the customer, can be displayed as a flowchart;Fig. 2.31 exemplifies the detail of the booking proce-dure.

    The analysis of the whole process has emphasizedthe existence of significant improvement margins, re-lated to costs and time. For example, a particular ser-vice, e. g., thermal mud, may need a medical visit be-fore the customer is allowed to access the treatment.By the Six Sigma analysis it was possible to reduce thelead time of the customer during the visit, through theoptimization of the work tasks and processes. Some-times this can be performed by very simple tricks andexpedients.

    For example, the aural test can be invalidated bythe presence of a plug of ear wax in a patient. Teach-ing the technician how to recognize and remove thisobstruction reduces the probability of null tests, andconsequently there is a reduction of costs and leadtimes.

  • http://www.springer.com/978-1-84882-574-1