capability analysis of the variable measurement system with fuzzy data

Applied Mathematical Modelling xxx (2014) xxx–xxx

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Capability analysis of the variable measurement systemwith fuzzy data

http://dx.doi.org/10.1016/j.apm.2014.03.0170307-904X/� 2014 Elsevier Inc. All rights reserved.

⇑ Tel.: +1 (919) 274 6937.E-mail address: [email protected]

Please cite this article in press as: H. Moheb-Alizadeh, Capability analysis of the variable measurement system with fuzzy data, AppModell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.017

Hadi Moheb-Alizadeh ⇑Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27695, USA

a r t i c l e i n f o

Article history:Received 10 November 2012Received in revised form 21 December 2013Accepted 11 March 2014Available online xxxx

Keywords:Measurement system analysis (MSA)Extension principleFuzzy numberRanking methodNon-linear programming

a b s t r a c t

The aim of this paper is to propose an approach to analyze capability of the variable mea-surement system in fuzzy environment, where the data acquired from the measurementprocess under study are assumed fuzzy numbers. To accomplish this goal, a pair of non-linear programming problems is formulated based on Zadeh’s extension principle to com-pute a-level cuts of assessment criteria, which are frequently used to analyze capability ofthe variable measurement system in practice. The membership functions of these criteriaare then constructed analytically by numerating different values of a. The capabilityassessment criteria discussed in this paper include repeatability, reproducibly, GRR% andCgk. In the next step, a method for ranking fuzzy numbers is exploited to evaluate whethercapability of the variable measurement system is satisfactory in fuzzy environment or not.Since fuzzy measures are gathered from the measurement system in a more realistic situ-ation in which all variations and unexpected conditions are taken into account, it is shownusing an empirical example that incorporating fuzziness into measurement data results ina more accurate capability analysis.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

All decisions made about the status of a manufacturing process, including accepting or rejecting production parts, initialsetting up or re-setting up production equipments, inspecting samples and so forth, are based on data drawn from the pro-cess. This reflects the importance of quality of obtained data. Nowadays, almost all manufacturing organizations calibrate thecontrol instruments used to extract the required data, whereas the measurement instrument is just one component of ameasurement system. The measurement system includes appraisers, production parts, measurement instruments, methodsand physical environment (such as light, heat, etc.). Hence, the adequacy of just measurement instrument does not solitarilyassure the correctness of a measurement system. Each of the abovementioned components may significantly influence themeasurement system performance and consequently, the quality of collected data. Measurement system analysis (MSA) is astatistical approach to investigate effects of the components and their interactions on the measurement system performance.

Based on the type of its outcome, a measurement system is generally categorized into two classes including variable andattribute measurement systems. A variable measurement system which is the subject of the present study takes into accounta continuum value as measurement outcome. On the other hand, one of a finite number of categories is regarded asmeasurement outcome in an attribute measurement system. Capability of the variable measurement systems is examinedby several criteria such as repeatability, reproducibility, GRR% and Cgk.

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2 H. Moheb-Alizadeh / Applied Mathematical Modelling xxx (2014) xxx–xxx

Measurement systems are traditionally analyzed using precise data. However, there may be many situations in which weare not able to collect precise and certain measurement data. As JCGM 100:2008 [1] mentions, even if all of the known orsuspected components of measurement errors have been evaluated and the appropriate corrections have been applied, therestill remains an uncertainty about the correctness of the stated result, that is, a doubt about how well the measurementresult represents the precise value of the quantity being measured. In practice, such an uncertainty may be resulted fromseveral sources including: incomplete definition of the part being measured (measurand), imperfect realization of themeasurand definition, inadequate knowledge of the effects of environmental conditions on the measurement, inexact valuesof measurement standards and reference materials, approximations and assumptions incorporated in the measurementmethod and procedure [1], special structure or shape of the measurand, inherent variability in the measurand, humanjudgment and subjectivity and so forth. One way to take into account such an uncertainty in analyzing capability of a mea-surement system is to deal with imprecise or vague data. In this case, the outcome of a measurement system can be repre-sented as a linguistic term, such as either ‘‘a range between 270.15 and 270.3’’ in a variable-measured production process, or‘‘good’’, ‘‘medium’’ and ‘‘bad’’ in an attribute-measured production process. This uncertain situation can be properly modeledby fuzzy set theory initially proposed by Zadeh [2]. In this case, the data drawn from the variable measurement system arerepresented as fuzzy numbers instead of common precise quantities.

To the best of our knowledge, no research studying the variable measurement system in fuzzy environment is observed inthe literature, while other quality engineering techniques such as failure mode and effect analysis (FMEA), quality functiondeployment (QFD), control charts and statistical quality control (SPC) have been comprehensively addressed and investi-gated in fuzzy environment. For example, Xu et al. [3] presented a fuzzy logic-based method for FMEA to address interde-pendencies among various failure modes with uncertain and imprecise information. Liu et al. [4] proposed an FMEA usingthe fuzzy evidential reasoning (FER) approach and grey theory to improve attaining various opinions of FMEA team membersand determining risk priorities of the failure modes. Chen and Ko [5] proposed fuzzy nonlinear programming models basedon Kano’s concept to determine the accomplishment levels of parts characteristics in quality function deployment (QFD)technique. Liu [6] integrated fuzzy QFD and the prototype product selection model to develop a product design and selection(PDS) approach. He adopted the a-cut operation to calculate the fuzzy set of each component and considered engineeringcharacteristics and the factors involved in prototype product selection. In statistical quality control area, Moheb-Alizadehand Fatemi-Ghomi [7] explored the impact of various transformation methods on power of control charts and concluded thatfuzzy mode and fuzzy average transformation methods lead to the most and least powerful control charts, respectively. Shuand Wu [8] proposed the fuzzy �X and R control charts, whose fuzzy control limits are obtained on the basis of the results ofresolution identity. The interested reader is referred to [9–13] to investigate more quality engineering techniques in fuzzyenvironment.

As the sole study, this paper endeavors to analyze capability of the variable measurement system in fuzzy environment Inthis regard, it is supposed that the data obtained from such a measurement system are fuzzy numbers. Applying Zadeh’sextension principle [14,15], a pair of non-linear mathematical programming problems is formulated to compute the lowerand upper bounds of a-level cuts of assessment criteria, which are prevalently used for capability analysis of the variablemeasurement system in practice. These criteria include repeatability, reproducibility, GRR% and Cgk. Afterward, the member-ship functions of these criteria in fuzzy environment are derived numerically by counting different values of a. In order toevaluate whether capability of the variable measurement system is acceptable with fuzzy data or not, a method for rankingfuzzy numbers is exploited. Since use of fuzzy data in capability analysis is more realistic and practical, it is shown using anempirical example that the proposed approach provides a more accurate tool for examining capability of the variable mea-surement system. The developed approach can be employed in any measurement laboratory on any type of measurementdevice, where taking into account the measurement uncertainty is concerned. Hence, the application of the proposedapproach may widely rang from pharmaceutical and biomedical industries, auto-industry and aerospace industry tonano-manufacturing. In all these industries, it is required to derive the most precise and correct measures from a measure-ment system to have a fair judgment on whether the product under measurement is hitting the acceptability criteria or not.

The rest of this paper is organized as follows: Section 2 gives preliminary concepts about variable measurement system.Section 3 describes how to use extension principle to derive a-level cuts of assessment criteria of capability in fuzzy envi-ronment. A ranking method for fuzzy numbers used to evaluate capability criteria with fuzzy data is given in Section 4. Then,Section 5 represents a numerical example to show how to perform the proposed approach in a practical context. Finally,some conclusions and recommendations for future works are given in Section 6.

2. Measurement systems analysis

This section presents the theory of variable measurement system analysis when crisp measurement data are applied.Measurement systems analysis (MSA) is viewed as an important component for many quality initiatives. As a part of ISO/TS 16949 and AIAG [16] standards, it is defined as an experimental and mathematical approach for determining how muchthe variation resulted from measurement process contribute to total process variability. Measurement systems are generallyclassified into two distinct groups including attribute measurement system and variable measurement system, on which thepresent paper concentrates. Similar to all other processes, the measurement systems are influenced by both random and sys-tematic sources of variation. The acronym S.W.I.P.E is used to represent the six potential sources of variation in measurement

Please cite this article in press as: H. Moheb-Alizadeh, Capability analysis of the variable measurement system with fuzzy data, Appl. Math.Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.017


H. Moheb-Alizadeh / Applied Mathematical Modelling xxx (2014) xxx–xxx 3

processes. It stands for Standard, Work piece (part), Instrument, Person and procedure, and Environment. These sources ofvariation may result in data with low quality. Capability of the variable measurement system to provide precise data ischiefly analyzed by assessing several criteria such as repeatability, reproducibility, GRR% and Cgk [16]. Repeatability is viewedas inherent variability in the measurement system. It quantifies the variation in outcome of the variable measurement sys-tem resulted from repeated measures taken in the same laboratory and in the same environmental condition. On the otherhand, reproducibility measures the variation in the outcome obtained from the same measurement instrument on identicalparts with different operators. An estimate for the combined variation of repeatability and reproducibility is called gaugerepeatability and reproducibility (GRR). GRR is usually compared with total variation resulted from both measurementand production processes in order to assess its magnitude. In this case, the resulted statistic is named GRR%. Moreover,Cgk is another capability estimate comparing variation of the variable measurement system with specification limits or tol-erance of production part.

The variable measurement system analysis has attracted considerable attentions in precise (non-fuzzy) environment. Forinstance, Senol [17] first re-identified the variability sources leading to errors in the measurements, then re-established amodel with designed experiments including laboratory factor as a measurement variability factor, and finally identifiednew producer (a)-consumer (b) risks with the required minimum sample size (n). Al-Refaie and Bata [18] proposed a pro-cedure for evaluating a measurement system and manufacturing process capabilities using gauge repeatability and repro-ducibility designed experiments with four quality measures including precision-to-tolerance ratio (PTR), signal-to-noiseratio (SNR), discrimination ratio (DR), and process capability index (Cp or Cpk). Shishebori and Zeinal Hamadani [19] studiedthe behavior of multivariate process capability index (MCp) in the presence of gauge measurement errors and examined theeffect of correlation coefficient and measurement capability on the statistical properties of the estimated MCp. Larsen [20]extended the univariate single measurement variable to a common manufacturing test scenario where multiple parametersare tested on each device with a sequence of tests. Burdick et al. [21] reviewed methods for conducting and analyzing mea-surement systems capability studies with emphasis on ANOVA approach and studied designed experiments with bothcrossed and nested factors. Woodall and Borror [22] discussed several commonly used gauge repeatability and reproducibil-ity (GRR) acceptance criteria and derived the relationships existed among them.

In the following, it is briefly explicated how to quantify the assessment criteria of the variable measurement system inprecise (non-fuzzy) environment.

2.1. Repeatability, reproducibility and GRR%

Capability of the variable measurement system quantified by repeatability and reproducibility is principally assessed bythree different techniques including range method, average and range method, and ANOVA method [16]. The latter tech-nique is elaborated in the present paper. In ANOVA method, the variance of a measurement system can be divided into fourcategories: part, appraiser, interaction between parts and appraisers, and replication error to the instrument [16]. Therewarding points of ANOVA method rather than average and range method are: it can handle any experimental setup, itcan assess the variance more accurately and derive more information from experimental data [16].

To run an ANOVA study, the selected parts covering production tolerance should be given to appraisers randomly. In thiscase, an ANOVA table with two factors under study including part and appraiser is represented as Table 1, where xijk is themeasure of ith part; i ¼ 1; . . . ; p, measured by jth appraiser; j ¼ 1; . . . ; a, in the kth replication; k ¼ 1; . . . ;n. Therefore, basedon the theory of ANOVA, the variances of repeatability and reproducibility are estimated as follows [23]:

Table 1An ANO

Facto

PleaseModel

r2repeatability ¼ MSR; ð1Þ

r2reproducibility ¼ ½MSA þ ðp� 1ÞMSAP � pMSR�=pn; ð2Þ

where, MSA, MSAP and MSR are mean squares of the factors appraiser, appraiser and part interaction, and replication errorrespectively. These mean squares are computed using Eqs. (62), (64), and (65) given in Appendix A. In this case, repeatability(EV) and reproducibility (AV) are calculated as [23]:

EV ¼ 5:15rrepeatability; ð3Þ

AV ¼ 5:15rreproducibility: ð4Þ

VA table with two factors of part and appraiser.

Factor B (appraiser)

1 2 . . . a

r A (part) 1 x111; x112; . . . ; x11n x121; x122; . . . ; x12n . . . x1a1; x1a2; . . . ; x1an

2 x211; x212; . . . ; x21n x221; x222; . . . ; x22n . . . x2a1; x2a2; . . . ; x2an

..

. ... ..

. ... ..

.

p xp11; xp12; . . . ; xp1n xp21; xp22; . . . ; xp2n . . . xpa1; xpa2; . . . ; xpan

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Now according to definition, GRR, an overall estimate for capability of the variable measurement system, is defined asfollows [16]:

PleaseMode

GRR ¼ ðEV2 þ AV2Þ0:5¼ 5:15 ½MSA þ ðp� 1ÞMSAP þ pðn� 1ÞMSR�=pn½ �0:5: ð5Þ

In order to have a meaningful understanding about the magnitude of variation contributed by measurement system, GRRis compared with total variation (TV) resulted from both measurement and production processes. The variance of productionprocess is conveniently estimated using ANOVA table as [23]:

r2p ¼ ðMSP �MSAPÞ=an; ð6Þ

where, MSP is mean square of production process computed by Eq. (63). In this case, production process (part) variation (PV)is assessed by:

PV ¼ 5:15rp: ð7Þ

Now, total variation (TV) is computed as follows [16]:

TV ¼ ðGRR2 þ PV2Þ0:5: ð8Þ

Therefore, GRR is compared with total variation as GRR% ¼ 100ðGRR=TVÞ to examine its magnitude [16]. The currentequation can be equivalently written as follows:

GRR% ¼ 100aMSA þ aðp� 1ÞMSAP þ apðn� 1ÞMSR

aMSA þ ðap� p� aÞMSAP þ apðn� 1ÞMSR þ pMSp

" #0:5

: ð9Þ

If GRR% is less than or equal to 30, it is said that capability of the variable measurement system under study is reasonable[16].

2.2. Capability index of gauge (Cgk)

Cgk is an overall assessment criterion for capability analysis of the variable measurement system. To compute it, apart with known reference value is firstly measured several times. The index is then calculated using the followingequation:

Cgk ¼ 0:1T � j�X � RV j� �

=3r; ð10Þ

where, T, �X and r are production tolerance, the average and standard deviation of collected measures respectively. RV standsfor reference value [24]. This criterion should be greater than or equal to 1.33 for the variable measurement system to besupposed capable.

3. Developing capability assessment criteria with fuzzy data

In this section, we elucidate how to develop assessment criteria described in preceding section in fuzzy environment,where the measurement data are supposed to be fuzzy numbers. As mentioned before, the main idea is to apply Zadeh’sextension principle [14,15] defined as follows:

Extension principle: Assume X1; . . . ;Xn and Y are ordinary (crisp) sets and f is a point mapping from ½X1; . . . ;Xn� to Y. The

extension principle implies that f can be extended to act on fuzzy subsets of X1; . . . ;Xn such that if ~Ai; i ¼ 1; . . . ;n and ~B arefuzzy subsets of X1; . . . ;Xn and Y, respectively, then:

l~BðyÞ ¼ supx

minfl~AiðxiÞ; i ¼ 1; . . . ;nj y ¼ f ðx1; . . . ; xnÞg: ð11Þ

In Eq. (11), n membership functions are involved. Therefore, it is hardly possible to extract l~BðyÞ in an explicit form.According to this equation, l~BðyÞ is the minimum of l~Ai

ðxiÞ; i ¼ 1; . . . ;n. Hence, we need l~AiðxiÞP a; i ¼ 1; . . . ;n and at least

one l~AiðxiÞ; i ¼ 1; . . . ;n equal to a such that y ¼ f ðx1; . . . ; xnÞ to satisfy l~BðyÞ ¼ a. To extract the membership function of ~B, it is

sufficient to derive the left and right shape functions of l~BðyÞ, which is equivalent to finding the lower bound ð~BÞLa and upper

bound ð~BÞUa of the a-level cuts of ~B. Since ð~BÞLa and ð~BÞUa are the minimum and maximum of f ðx1; . . . ; xnÞ, respectively, they canbe expressed as:

ð~BÞLa ¼min f ðx1; . . . ; xnÞ ð~xiÞLa 6 xi 6

�� ð~xiÞUa ; i ¼ 1; . . . ;nn o

; ð12Þ

ð~BÞUa ¼ max f ðx1; . . . ; xnÞ ð~xiÞLa 6 xi 6

�� ð~xiÞUa ; i ¼ 1; . . . ;nn o

: ð13Þ

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Since all a-level cuts construct a nested structure with respect to a [15], i.e. if 0 < a2 < a1 6 1, then

ð~xiÞLa1; ð~xiÞUa1

h i# ð~xiÞLa2

; ð~xiÞUa2

h i, the feasible region defined by a1 in optimization problems (12) and (13) is smaller than that

defined by a2. As a result, the a-level cuts of ~B at a1 is contained in that defined by a2, i.e. ð~BÞLa1; ð~BÞUa1

h i# ð~BÞLa2

; ð~BÞUa2

h i. Con-

sequently, the left shape and right shape functions are non-decreasing and non-increasing respectively, which assure the

convexity of l~BðyÞ. To approximate the shape of l~BðyÞ, the numerical solutions for ð~BÞLa and ð~BÞUa are collected at different val-ues of a.

The optimization problems (12) and (13) are customized to derive a-level cuts of assessment criteria for capability of thevariable measurement system in fuzzy environment.

3.1. Fuzzy repeatability

Now, repeatability criterion is developed in fuzzy environment using Zadeh’s extension principle. According to Eq. (1), thevariance of repeatability is a function of MSR, i.e. r2

Repeatability ¼ f ðMSRÞ. Hence, based on the optimization problems (12) and(13), the following programming problems should be optimized to derive its a-level cuts:

PleaseModel

~r2Repeatability

� �L

a¼min f ðMSRÞ ðM~SRÞ

L

a 6 MSR 6 ðM~SRÞU

a

��n o; ð14Þ

~r2Repeatability

� �U

a¼max f ðMSRÞ ðM~SRÞ

L

a 6 MSR 6 ðM~SRÞU

a

��n o; ð15Þ

where, f ðMSRÞ ¼ MSR. It is simply concluded that:

~r2Repeatability

� �L

a¼ ðM~SRÞ

L

a; ð16Þ

~r2Repeatability

� �U

a¼ ðM~SRÞ

U

a : ð17Þ

Clearly, ~r2Repeatability

� �L

aand ~r2

Repeatability

� �U

aconstruct the lower and upper bounds of a-level cuts of ~r2

Repeatability, because M~SR

has hit its bounds in each level, i.e. lM~SR¼ a.

Now, it should be examined how to derive ðM~SRÞL

a and ðM~SRÞU

a in Eqs. (16) and (17). According to Eq. (65) in Appendix A,the variable MSR is itself a function of SSR. Therefore, referring to problems (12) and (13), we have

ðM~SRÞL

a ¼ ðS~SRÞL

a=apðn� 1Þ; ð18Þ

ðM~SRÞU

a ¼ ðS~SRÞU

a=apðn� 1Þ: ð19Þ

Moreover, based on Eq. (69) in Appendix A, it is known that SSR is a function of SST , SSAP , SSA and SSP , i.e.SSR ¼ f ðSST ; SSAP; SSA; SSPÞ. Hence, the lower and upper bounds of its a-level cuts can be extracted as follows based on prob-lems (12) and (13):

ðS~SRÞL

a ¼ ðS~STÞL

a � ðS~SAPÞU

a � ðS~SAÞU

a � ðS~SPÞU

a ; ð20Þ

ðS~SRÞU

a ¼ ðS~STÞU

a � ðS~SAPÞL

a � ðS~SAÞL

a � ðS~SPÞL

a: ð21Þ

Now, ðS~SAÞL

a; ðS~SAÞU

a

h i, ðS~SAPÞ

L

a; ðS~SAPÞU

a

h i, ðS~SPÞ

L

a; ðS~SPÞU

a

h iand ðS~STÞ

L

a; ðS~STÞU

a

h iare computed in a similar way. According to Eq.

(66) in Appendix A, SSA is a function of x:j: and x:::. Therefore, two optimization problems

ðS~SAÞL

a ¼min f ðx:j:; x:::Þ x:j: ; x::: 2 ASSA

�� and ðS~SAÞ

U

a ¼ max f ðx:j:; x:::Þ x:j:; x::: 2 ASSA

�� are used to extract a-level cuts of S~SA,

where, f ðx:j:; x:::Þ is defined by Eq. (66) and ASSA denotes ð~x:j:ÞLa 6 x:j: 6 ð~x:j:ÞUa , ð~x:::ÞLa 6 x::: 6 ð~x:::ÞUa . In this case, we have:

ðS~SAÞL

a ¼Xa

j¼1

ð~x:j:ÞLah i2

pn� ð~x:::ÞUa

h i2

apn; ð22Þ

ðS~SAÞU

a ¼Xa

j¼1

ð~x:j:ÞUah i2

pn� ð~x:::ÞLa

h i2

apn: ð23Þ

The current approach can be followed to derive a-level cuts of S~Sp, S~SAP and S~ST . Excluding computational details andbased on Eqs. (67), (68), and (70) in Appendix A, we have the following equations to calculate the lower and upper boundsof a-level cuts of S~Sp, S~SAP and S~ST , respectively:




PleaseMode

ðS~SPÞL

a ¼Xp

i¼1

ð~xi::ÞLah i2

an� ð~x:::ÞUa

h i2

apn; ð24Þ

ðS~SPÞU

a ¼Xp

i¼1

ð~xi::ÞUah i2

an� ð~x:::ÞLa

h i2

apn; ð25Þ

and

ðS~SAPÞL

a ¼Xp

i¼1

Xa

j¼1

ð~xij:ÞLah i2

n� ð~x:::ÞUa

h i2

apn� ðS~SAÞU

a � ðS~SPÞU

a ð26Þ

ðS~SAPÞU

a ¼Xp

i¼1

Xa

j¼1

ð~xij:ÞUah i2

n� ð~x:::ÞLa

h i2

apn� ðS~SAÞL

a � ðS~SPÞL

a; ð27Þ

and finally,

ðS~STÞL

a ¼Xp

i¼1

Xa

j¼1

Xn

k¼1

ð~xijkÞLah i2

� ð~x:::ÞUah i2

apn; ð28Þ

ðS~STÞU

a ¼Xp

i¼1

Xa

j¼1

Xn

k¼1

ð~xijkÞUah i2

� ð~x:::ÞLah i2

apn: ð29Þ

In Eqs. (22)–(29), it is required to have ð~xi::ÞLa; ð~xi::ÞUah i

, ð~x:j:ÞLa; ð~x:j:ÞUa

h i, ð~xij:ÞLa; ð~xij:ÞUah i

and ð~x:::ÞLa; ð~x:::ÞUa

h i. Since based on Eqs.

(71)–(74), xi::, x:j:, xij: and x::: all are functions of just xijk, it is derived using extension principle that ð~xi::ÞLa ¼Pa

j¼1

Pnk¼1ð~xijkÞLa,

ð~xi::ÞUa ¼Pa

j¼1

Pnk¼1ð~xijkÞUa , ð~x:j:ÞLa ¼

Ppi¼1

Pnk¼1ð~xijkÞLa, ð~x:j:ÞUa ¼

Ppi¼1

Pnk¼1ð~xijkÞUa , ð~xij:ÞLa ¼

Pnk¼1ð~xijkÞLa, ð~xij:ÞUa ¼

Pnk¼1ð~xijkÞUa , ð~x:::ÞLa ¼Pp

i¼1

Paj¼1

Pnk¼1ð~xijkÞLa and ð~x:::ÞUa ¼

Ppi¼1

Paj¼1

Pnk¼1ð~xijkÞUa .

Finally, a-level cuts of repeatability are extracted using Eqs. (3), (16), and (17) as follows:

ðE~VÞLa ¼ 5:15 ðM~SRÞL

a

h i0:5; ð30Þ

ðE~VÞUa ¼ 5:15 ðM~SRÞU

a

h i0:5: ð31Þ

The membership function of E~V can be derived using above equations for various values of a ¼ 0; . . . ;1.

3.2. Fuzzy reproducibility

Referring to Eq. (6), the variance of reproducibility can be regarded as a function of three variables including MSA, MSAP

and MSR. Hence, the following optimization problems should be considered in order to derive its a-level cuts in fuzzy envi-ronment based on problems (12) and (13):

~r2Reproducibility

� �L

a¼min f ðMSA;MSAP;MSRÞ jMSA;MSAP;MSR 2 AReproducibility

� �; ð32Þ

~r2Reproducibility

� �U

a¼max f ðMSA;MSAP ;MSRÞ jMSA;MSAP;MSR 2 AReproducibility

� �; ð33Þ

where, f ðMSA;MSAP;MSRÞ is defined as Eq. (6) and AReproducibility indicates bounds for the variables as ðM~SAÞL

a 6 MSA 6 ðM~SAÞU

a ,

ðM~SAPÞL

a 6 MSAP 6 ðM~SAPÞU

a , ðM~SRÞL

a 6 MSR 6 ðM~SRÞU

a . Because of linear and simple structure of the objective function in aboveoptimization problems, they can be solved as follows:

~r2Reproducibility

� �L

a¼ ðM~SAÞ

L

a þ ðp� 1ÞðM~SAPÞL

a � pðM~SRÞU

a

h i.pn; ð34Þ

~r2Reproducibility

� �U

a¼ ðM~SAÞ

U

a þ ðp� 1ÞðM~SAPÞU

a � pðM~SRÞL

a

h i.pn: ð35Þ

Now, it is required to calculate ðM~SAÞL

a; ðM~SAÞU

a

h iand ðM~SAPÞ

L

a; ðM~SAPÞU

a

h i. The values of ðM~SRÞ

L

a; ðM~SRÞU

a

h iare computed

using Eqs. (18) and (19). According to Eqs. (62) and (64), MSA and MSAP are functions of SSA and SSAP , respectively. Therefore,following the same procedure as what was elaborated earlier, the lower and upper bounds of their a-level cuts are calculatedas:




PleaseModel

ðM~SAÞL

a ¼ ðS~SAÞL

a=ða� 1Þ; ð36Þ

ðM~SAÞU

a ¼ ðS~SAÞU

a=ða� 1Þ; ð37Þ

ðM~SAPÞL

a ¼ ðS~SAPÞL

a=ða� 1Þðp� 1Þ; ð38Þ

ðM~SAPÞU

a ¼ ðS~SAPÞU

a=ða� 1Þðp� 1Þ; ð39Þ

where, ðS~SAÞL

a, ðS~SAÞU

a , ðS~SAPÞL

a and ðS~SAPÞU

a are computed using Eqs. (22), (23), (26), and (27), respectively.

Finally, according to Eqs. (4), (34), and (35), the lower and upper bounds of a-level cuts of reproducibility are extracted asfollows:

ðA~VÞLa ¼ 5:15 ½ðM~SAÞL


a � pðM~SRÞU

a �=pnh i0:5

; ð40Þ

ðA~VÞUa ¼ 5:15 ½ðM~SAÞU


a � pðM~SRÞL

a�=pnh i0:5

: ð41Þ

In this case, the membership function of A~V is constructed using above equations for a ¼ 0; . . . ;1.

3.3. Fuzzy GRR and GRR%

In this section, GRR and GRR% assessment criteria are developed when using fuzzy data. Based on Zadeh’s extension prin-ciple, the lower and upper bounds of a-level cuts of G~RR can be derived using the following optimization problems:

ðG~RRÞLa ¼min f ðMSA;MSAP;MSRÞ jMSA;MSAP;MSR 2 AGRRf g; ð42Þ

ðG~RRÞUa ¼max f ðMSA;MSAP;MSRÞ MSA;MSAP;MSR 2 AGRRjf g; ð43Þ

where, f ðMSA;MSAP;MSRÞ is defined by Eq. (6) and AGRR indicates bounds on the variables as ðM~SAÞL

a 6 MSA 6 ðM~SAÞU

a ,

ðM~SAPÞL


a , ðM~SRÞL

a 6 MSR 6 ðM~SRÞU

a . As it is clear, AGRR is the same as AReproducibility. The above optimization prob-lems can be conveniently solved as follows:

ðG~RRÞLa ¼ 5:15 ðM~SAÞL


a þ pðn� 1ÞðM~SRÞL

a

h i.pn

h i0:5; ð44Þ

ðG~RRÞUa ¼ 5:15 ðM~SAÞU


a þ pðn� 1ÞðM~SRÞU

a

h i.pn

h i0:5; ð45Þ

where, ðM~SAÞL

a; ðM~SAÞU

a

h i, ðM~SAPÞ

L

a; ðM~SAPÞU

a

h iand ðM~SRÞ

L

a; ðM~SRÞU

a

h iare defined by Eqs. (36), (37), (38), (39), (18), and (19),

respectively.Before going through computing the bounds of a-level cuts of G~RR%, the lower and upper bounds of a-level cuts of M~Sp

are calculated as:

ðM~SPÞL

a ¼ ðS~SPÞL

a

.ðp� 1Þ; ð46Þ

ðM~SPÞU

a ¼ ðS~SPÞU

a

.ðp� 1Þ: ð47Þ

Now, Zadeh’s extension principle requires us to optimize the following programming problems toward obtaining a-levelcuts of G~RR%:

ðG~RR%ÞLa ¼min f ðMSA;MSAP;MSR;MSPÞ MSA;MSAP ;MSR;MSP 2 AGRR%jf g; ð48Þ

ðG~RR%ÞUa ¼max f ðMSA;MSAP;MSR;MSPÞ MSA;MSAP;MSR;MSP 2 AGRR%jf g; ð49Þ

where, f ðMSA;MSAP;MSR;MSPÞ is defined by Eq. (9) and AGRR% sets bounds on the variables as ðM~SAÞL

a 6 MSA 6 ðM~SAÞU

a ,

ðM~SAPÞL


a , ðM~SRÞL

a 6 MSR 6 ðM~SRÞU

a and ðM~SPÞL

a 6 MSR 6 ðM~SPÞU

a .

In optimization problems (48) and (49), f is a fractional function. Hence, contrary to previous cases, the optimal solutionscannot be derived in a closed form. Instead, the following two fractional non-linear programming problems should be ana-lytically optimized to derive a-level cuts of G~RR%:




PleaseMode

ðG~RR%ÞLa ¼min 100aMSA þ aðp� 1ÞMSAP þ apðn� 1ÞMSR

aMSA þ ðap� p� aÞMSAP þ apðn� 1ÞMSR þ pðM~SpÞU

a

24

35

0:5

;

s:t:

ðM~SAÞL

a 6 MSA 6 ðM~SAÞU

a ;

ðM~SRÞL

a 6 MSR 6 ðM~SRÞU

a ;

ðM~SAPÞL


a ;

ð50Þ

ðG~RR%ÞUa ¼max100aMSA þ aðp� 1ÞMSAP þ apðn� 1ÞMSR

aMSA þ ðap� p� aÞMSAP þ apðn� 1ÞMSR þ pðM~SpÞL

a

24

35

0:5

;

s:t:

ðM~SAÞL

a 6 MSA 6 ðM~SAÞU

a ;

ðM~SRÞL

a 6 MSR 6 ðM~SRÞU

a ;

ðM~SAPÞL


a ;

ð51Þ

In above problems, the variable MSP is directly set to the upper and lower bounds of its a-level cut respectively, toassure lG~RR% ¼ a as required by extension principle. The abovementioned models are a pair of non-linear programmingproblems that can be effectively solved via the constrained variable metric method and generalized reduced gradient meth-od [25]. The membership function of G~RR% is then derived solving above two optimization problems for various values ofa ¼ 0; . . . ;1.

3.4. Fuzzy Cgk

According to Eq. (10), Cgk can be viewed as a function of S and �X. Therefore, the following two optimization problemsshould be considered in order to compute the lower and upper bounds of a-level cuts of ~Cgk:

ð~CgkÞL

a ¼min f ð�X; SÞ ð~�XÞL

a

�� 6 �X 6 ð~�XÞU

a ; ð~SÞL

a 6 S 6 ð~SÞU

a

�; ð52Þ

ð~CgkÞU

a ¼max f ð�X; SÞ ð~�XÞL

a

�� 6 �X 6 ð~�XÞU

a ; ð~SÞL

a 6 S 6 ð~SÞU

a

�; ð53Þ

where, f ð�X; SÞ is defined as Eq. (10). Setting the variable S directly to the upper and lower bounds of its a-level cut respec-tively in above optimization problems to assure l~Cgk

¼ a, we have:

ð~CgkÞL

a ¼ min 0:1T � j�X � RV j� ��

3ð~SÞU

a ;

s:t:

ð~�XÞL

a 6�X 6 ð~�XÞ

U

a ;

ð54Þ

ð~CgkÞU

a ¼max 0:1T � j�X � RV j� ��

3ð~SÞL

a;

s:t:

ð~�XÞL

a 6�X 6 ð~�XÞ

U

a :

ð55Þ

In order to tackle the absolute term j�X � RV j in objective functions of the above problems, the transformation methodproposed by Lu and Wang [26] is applied. In this case, the following programming problems are derived:

ð~CgkÞL

a ¼ min 0:1T � ðy1 þ y2Þ½ �=3ð~SÞU

a ;

s:t:

y1 � y2 6 ð~�XÞU

a � RV ;

y1 � y2 P ð~�XÞL

a � RV ;

y1; y2 P 0;

ð56Þ




PleaseModel

ð~CgkÞU

a ¼max 0:1T � ðy1 þ y2Þ½ �=3ð~SÞL

a;

s:t:

y1 � y2 6 ð~�XÞU

a � RV ;

y1 � y2 P ð~�XÞL

a � RV ;

y1; y2 P 0:

ð57Þ

The new problem is to calculate the values of ð~SÞL

a and ð~SÞU

a in the above programming problems. The standard deviation

defined as S ¼Pn

i¼1ðXiÞ2 � n�X2� �.

n� 1h i0:5

is a function of Xi; i ¼ 1; . . . ;n and �X. Hence, the following non-linear program-

ming problems should be optimized to derive the lower and upper bounds of its a-level cuts:

ð~SÞL

a ¼minXn

i¼1

ðXiÞ2 � n ð~�XÞU

a

�2 !,

n� 1

" #0:5

;

s:t:

ð~XiÞL

a 6 Xi 6 ð~XiÞU

a ; i ¼ 1; . . . ;n;

ð58Þ

ð~SÞU

a ¼maxXn

i¼1

ðXiÞ2 � n½ð~�XÞL

a�2

!,n� 1

" #0:5

;

s:t:

ð~XiÞL

a 6 Xi 6 ð~XiÞU

a i ¼ 1; . . . ;n;

ð59Þ

where the variable �X is directly set to the upper and lower bounds of its a-level cuts to assure l~S ¼ a.Up to now, we have been able to compute a-level cuts of assessment criteria for capability of the variable measurement

system. Membership functions of the respective criteria are then extracted by accumulating the results from differenta values. In the next section, a procedure is introduced based on a fuzzy number ranking method to judge about acceptabilityof the variable measurement system under study.

4. Evaluating capability of the variable measurement system

In this section, a procedure is described to see whether capability of the variable measurement system under study with

fuzzy data is acceptable. In fuzzy environment, the assessment criteria G~RR% and ~Cgk should be compared with their corre-sponding critical values. It necessitates applying a method for ranking fuzzy numbers. There are a lot of methods in the lit-erature in this area [27–31]. However, most of them need membership functions of the fuzzy numbers to be ranked. Themethod proposed by Chen and Klein [28], on the other hand, is very appropriate for the present study because it is basedon a-level cuts.

To describe the method proposed by Chen and Klein [28], suppose the fuzzy numbers ~Aj; j ¼ 1; . . . ;m are going to beranked. Let h be the maximum height of l~Aj

; j ¼ 1; . . . ;m. Assume h is split into n equal intervals such that

ai ¼ ih=n; i ¼ 0; . . . ;n. Chen and Klein [28] proposed the following index for ranking fuzzy numbers:

Ij ¼Xn

i¼1

ð~AjÞU

ai� c

h i Xn

i¼1

ð~AjÞU

ai� c

h i�Xn

i¼1

ð~AjÞL

ai� d

h i" #,; n!1 ð60Þ

where, c ¼mini;j

ð~AjÞL

ai

n oand d ¼max

i;jð~AjÞ

U

ai

n o. The larger (smaller) value of index Ij, the greater (smaller) the fuzzy number ~Aj

is. While this method is authentic when n advances infinity, Chen and Klein [28] proposed that n = 3 or 4 suffices to discrim-inate the differences.

In studying the variable measurement system with fuzzy data, the fuzzy assessment criteria G~RR% and ~Cgk are comparedwith their respective critical values. The critical values may be regarded as either fuzzy or crisp numbers. First, suppose deci-sion maker determines a crisp critical value (as it is used in non-fuzzy sense). In this case, we know that the real number pcan be viewed as a fuzzy number with membership function of lPðpÞ : X ! f0;1g with P ¼ fpg and

lPðpÞ ¼1 if and only if x ð2 XÞ ¼ p;

0 if and only if x ð2 XÞ – p;

ð61Þ

where, X is a universal set. Therefore, the aforementioned ranking method can be suitably used for evaluating acceptability of

G~RR% and ~Cgk. In this regard, if IG~RR% 6 I30 and I~CgkP I1:33, it is concluded that G~RR% � 30 and ~Cgk � 1:33, respectively. Con-

sequently, capability of the variable measurement system is acceptable in fuzzy environment.




On the other hand, as a more general case, if the critical values are defined by decision maker as fuzzy numbers, then

G~RR% and ~Cgk are compared with the fuzzy critical values 3~0 and 1:~33 respectively. In this case, if IG~RR% 6 I~30 and

I~CgkP I1:~33, then G~RR% � ~30 and ~Cgk � 1:~33. As a result, capability of the variable measurement system is acceptable.

Moreover, it is known that a primary application of MSA is to make various kinds of comparisons. For instance, the mea-surement system A is said to be more capable than the measurement system B if ðGRR%ÞA 6 ðGRR%ÞB and ðCgkÞA P ðCgkÞB. Infuzzy environment, the abovementioned method for ranking fuzzy numbers can be conveniently applied to make such com-parisons. In this case, the measurement system A is more capable than the measurement system B if IðG~RR%ÞA 6 IðG~RR%ÞB andIð~CgkÞA

P Ið~CgkÞB.

In summary, in order to implement the proposed approach and analyze capability of the variable measurement systemwith fuzzy data, the lower and upper bounds of a-level cuts of the assessment criteria are first obtained using given pairs ofmathematical programming problems. The computed bounds are then accumulated to build up membership function ofeach assessment criterion in fuzzy environment. As the final step, the obtained membership functions are compared withtheir respective critical values using Chen and Klein [28] ranking method in order to decide whether the variable measure-ment system is acceptable or not.

5. Empirical example

In this section, an example is given to demonstrate how to perform capability analysis of the variable measurement sys-tem with fuzzy data. In this regard, the primary shaft of a gearbox represented in Fig. 1 is selected to be studied. This shaft isundergone a heat treatment process to reach a predetermined surface hardness. The length of its specific section as shown inFig. 1 is measured by an appraiser in the shop environment, where is normally warmer than its operational position. Theinfluence of this warmer surrounding on the true measure of shaft’s length cannot be easily examined, i.e. the length ofits specific section may unforeseeably change after reaching the regular temperature. Hence, as mentioned in Section 1, thisinadequate knowledge of the effects of environmental conditions on the measurement results in an uncertainty whichshould be taken into account in analyzing the measurement system. Consequently, it is supposed that the measures acquiredfrom the measurement system are fuzzy numbers with triangular membership functions, called triangular fuzzy numbers

shown as ðxa; xb; xcÞ. An a-level cut of the triangular fuzzy number ~X ¼ ðxa; xb; xcÞ is represented as ~X ¼ ð~XÞLa; ð~XÞU

a

h i, where

ð~XÞLa ¼ xb � ð1� aÞðxb � xaÞ and ð~XÞUa ¼ xb þ ð1� aÞðxc � xbÞ. The measurement system under consideration includes twotrained appraisers, a micrometer to measure the length of the shaft’s specific section and a measurement guideline usedin the heat treatment shop.

To quantify the assessment criteria, 10 production parts covering production tolerance are randomly given to twoappraisers, each to be approximately measured two times. The results are presented as triangular fuzzy numbers in Table 2.The center value of each fuzzy measure given in this table denotes the most possible measure of each production part in eachreplication. On the other hand, the end points indicate the least possible upper and lower measures. In other words, thelength of each production part lies certainly within upper and lower bounds of its respective fuzzy measure and is most pos-sibly equal to the center value. The end points represent the uncertainty involved in measurement system. The more uncer-tainty involved, the wider fuzzy measures are supposed to be gathered. To assess repeatability, reproducibility, GRR and

GRR%, it is necessary to have a-level cuts of M~SA, M~SP , M~SAP and M~SR. Table 3 gives a-level cuts of these fuzzy mean squares

for a ¼ 0; . . . ;1, where ðM~SRÞL

a; ðM~SRÞU

a

h i, ðM~SAÞ

L

a; ðM~SAÞU

a

h i, ðM~SApÞ

L

a; ðM~SApÞU

a

h iand ðM~SpÞ

L

a; ðM~SpÞU

a

h iare computed using Eqs.

(16), (17), (36), (37), (38), (39), (46), and (47), respectively. As it is observed, the values of ðM~SAÞL

a, ðM~SAPÞL

a and ðM~SRÞL

a fora ¼ 0; . . . ;0:9 represented in parentheses are negative in the present example. It is very often in crisp environment to obtaina negative value for variance components when using ANOVA method in measurement systems analysis. This issue has beenaddressed before [16,32] to find out a solution. When negative estimates of variance components emerge, they are usually

Measurement section

Fig. 1. The measurement section of the primary shaft before undergoing heat treatment.



Table 2Fuzzy measures gathered for evaluating fuzzy repeatability, reproducibility, GRR and GRR%.

Part no. Operator 1 Operator 2Replication Replication

1 2 1 2

1 (470.34,471,471.50) (483.82,484,484.08) (484.06,485,485.32) (479.99,480,480.42)2 (764.83,765,765.54) (741.51,742,742.84) (777.96,778,778.52) (806.51,807,807.47)3 (327.68,328,328.50) (325.71,326,326.94) (327.67,328,328.71) (313.71,314,314.78)4 (445.69,446,446.79) (432.89,433,433.12) (454.38,455,455.80) (449.25,450,450.09)5 (455.73,456,456.13) (453.24,454,454.62) (469.24,470,470.24) (460.32,461,461.24)6 (442.91,443,443.94) (449.05,450,450.25) (459.65,460,460.11) (454.25,455,455.67)7 (551.75,552,552.38) (556.91,557,557.30) (550.79,551,551.30) (546.19,547,547.95)8 (476.62,477,477.04) (478.04,479,479.49) (498.83,499,499.13) (491.19,492,492.77)9 (508.17,509,509.82) (507.22,508,508.72) (512.49,513,513.63) (539.87,540,540.76)

10 (383.44,384,384.96) (370.40,371,371.24) (389.18,390,390.20) (384.76,385,385.70)

Table 3a-level cuts of fuzzy mean squares.

a M~SA M~SP M~SAP M~SR

ðM~SAÞLa ðM~SAÞ

Ua ðM~SPÞ

La ðM~SPÞ

Ua ðM~SAPÞ

La ðM~SAPÞ

Ua ðM~SRÞ

La ðM~SRÞ

Ua

0 0 (�36495.2) 39004.73 52766.62 61118.35 0 (�12375.0) 12707.34 0 (�11221.4) 11347.660.1 0 (�32719.0) 35230.70 53182.29 60698.82 0 (�11121.0) 11452.98 0 (�10091.4) 10220.690.2 0 (�28942.9) 31456.70 53597.96 60279.29 0 (�9867.09) 10198.62 0 (�8961.39) 9093.730.3 0 (�25166.7) 27682.73 54013.62 59859.77 0 (�8613.16) 8944.28 0 (�7831.40) 7966.780.4 0 (�21390.9) 23908.78 54429.29 59440.25 0 (�7359.24) 7689.94 0 (�6701.42) 6839.830.5 0 (�17614.8) 20134.86 54844.95 59020.73 0 (�6105.32) 6435.61 0 (�5571.44) 5712.900.6 0 (�13838.4) 16360.96 55260.61 58601.23 0 (�4851.42) 5181.30 0 (�4441.46) 4585.970.7 0 (�10062.4) 12587.09 55676.27 58181.72 0 (�3597.52) 3926.99 0 (�3311.50) 3459.060.8 0 (�6286.35) 8813.24 56091.93 57762.22 0 (�2343.63) 2672.69 0 (�2181.53) 2332.160.9 0 (�2510.35) 5039.42 56507.58 57342.73 0 (�1089.75) 1418.40 0 (�1051.58) 1205.261 1265.63 1265.63 56923.24 56923.24 164.13 164.13 78.38 78.38

Table 4a-level cuts of fuzzy assessment criteria.

a E~V A~V G~RR G~RR%

ðE~VÞLa ðE~VÞUa ðA~VÞLa ðA~VÞUa ðG~RRÞLa ðG~RRÞUa ðG~RR%ÞLa ðG~RR%ÞUa0 0 548.61 0 593.46 0 594.87 0 75.580.1 0 520.65 0 563.24 0 564.76 0 73.180.2 0 491.11 0 531.30 0 532.95 0 70.480.3 0 459.67 0 497.31 0 499.11 0 67.430.4 0 425.92 0 460.82 0 462.81 0 63.930.5 0 389.26 0 421.18 0 423.4 0 59.860.6 0 348.76 0 377.41 0 379.94 0 55.050.7 0 302.89 0 327.83 0 330.81 0 49.170.8 0 248.71 0 269.29 0 272.97 0 41.690.9 0 178.79 0 193.80 0 198.98 0 31.271 45.59 45.59 50.97 50.97 68.39 68.39 11.08 11.08


accompanied by non-significant model sources of variability and this is an evidence that the variance component under

study is really zero [32]. Following this notion, we have substituted the values of ðM~SAÞL

a, ðM~SAPÞL

a and ðM~SRÞL

a fora ¼ 0; . . . ;0:9 by zero in Table 3. In this regard, it should be noted that when the lower bound of an a-level cut becomes neg-ative in fuzzy environment, this is in fact the respective variance component that has become negative in that specific a-le-vel. Using these quantities, a-level cuts of repeatability as well as reproducibility, GRR and GRR% are calculated as presented

in Table 4. The Eqs. (30), (31), (40), (41), (44), and (45) are used to compute ðE~VÞLa; ðE~VÞUah i

, ðA~VÞLa; ðA~VÞUah i

, ðG~RRÞLa; ðG~RRÞUah i

,

respectively. Furthermore, the optimization problems (50) and (51) are utilized to obtain ðG~RR%ÞLa; ðG~RR%ÞUah i

. In the present

paper, the non-linear programming solver Lingo 14 [33] is used to solve these non-linear programming problems. In these

optimization problems, ðM~SpÞL

a and ðM~SpÞU

a are extracted from Table 3. Using the obtained a-level cuts, the membership



(a)

0 100 200 300 400 500 6000.0

0.2

0.4

0.6

0.8

1.0

Mem

bers

hip

Deg

ree

GRR(c)

(b)

0 10 20 30 40 50 60 70 800.0

0.2

0.4

0.6

0.8

1.0

Mem

bers

hip

Deg

ree

GRR%(d)

0 100 200 300 400 500 6000.0

0.2

0.4

0.6

0.8

1.0M

embe

rshi

p D

egre

e

Repeatability0 100 200 300 400 500 600

0.0

0.2

0.4

0.6

0.8

1.0

Mem

bers

hip

Deg

ree

Reproducibility

Fig. 2. Membership functions of (2-a) Repeatability, (2-b) Reproducibility, (2-c) GRR and (2-d) GRR%.


functions of these assessment criteria are conveniently constructed. Fig. 2(a)–(d) represent the membership functions of E~V ,A~V , G~RR, G~RR%, respectively.

Now, the capability criterion G~RR% should be compared with its critical value to see whether it is satisfactory or not. Asmentioned before, the critical value can be regarded as either the crisp number 30 or the fuzzy number 3~0. In this example,the fuzzy critical value ð28;30;33Þ is defined by decision maker to be compared with G~RR%. This fuzzy critical value is de-picted by a triangle in Fig. 2-d. Based on the presented ranking method, we have IG~RR% ¼ 0:4219 and I3~0 ¼ 0:4034. Since

IG~RR% > I~30, it is concluded that G~RR% � 3~0. Subsequently, this assessment criterion for capability of the variable measure-ment system is not acceptable. If the measurement system under consideration is analyzed in crisp environment, i.e.GRR% is computed using just the center values of fuzzy measures in Table 2 and Eq. (10), then we have GRR% ¼ 11:08. Inthis case, since GRR% 6 30, capability of the measurement system is unrealistically satisfactory in crisp environment. It

should be noted that the crisp GRR% computed is in fact equal to ðG~RR%ÞLa¼1 and ðG~RR%ÞUa¼1. However, fuzzy measures arederived from the measurement system in a more real condition in which all likely variations are taken into account. There-fore, it can be certainly said that GRR% analysis in fuzzy environment is more accurate and realistic rather than that in crispenvironment. As a result, GRR% is realistically unacceptable with fuzzy data.

As a complementary consideration on the fuzzy numbers presented in Table 2, it may be of interest to compare the aver-age of their widths with some meaningful measures of variability in crisp environment. This helps understand how wide thegiven fuzzy numbers are in comparison with reasonable measures of variability. Such measures may be Standard Deviation(SD) of repeatability and part-to-part components. If we regard the central values of fuzzy numbers in Table 2 and run GRRstudy on them using a suitable statistical software, the standard deviations of repeatability and part-to-part are then derivedas 8.853 and 119.121, respectively. On the other hand, the average of fuzzy numbers widths is computed as 0.968. As it isobserved, the average width of fuzzy numbers is only 10.9 and 0.81 percent of standard deviations of repeatability and



Table 5Fuzzy measures and a-level cuts of ~Cgk (reference value = 410).

Replication no. Fuzzy measures a ð~CgkÞL

a ð~CgkÞU

a

1 (409.35,409.54,410.28) 0 0.0115 39.5572 (409.32,409.72,410.47) 0.1 0.0116 37.5293 (410.25,410.51,411.04) 0.2 0.0118 35.3844 (410.63,410.94,411.79) 0.3 0.0119 33.1025 (409.27,409.68,410.27) 0.4 0.0120 30.6506 (409.27,410.10,410.88) 0.5 0.0121 27.9847 (409.78,409.93,410.32) 0.6 0.0122 25.0368 (408.58,408.58,409.35) 0.7 0.0122 21.6919 (409.75,410.04,410.50) 0.8 0.0123 17.727

10 (408.34,408.75,408.84) 0.9 0.0124 12.53211 (410.93,411.44,411.50) 1 1.2879 1.287912 (408.77,408.85,409.09)13 (409.64,410.08,410.26)14 (411.38,411.39,411.53)15 (409.07,409.58,410.07)

0 10 20 30 400.0

0.2

0.4

0.6

0.8

1.0

Mem

bers

hip

Deg

ree

Cgk

Fig. 3. Membership function of Cgk .

Table 6Comparisons between resulted of capability analysis in fuzzy and crisp environments.

Environment Assessment criteria Critical value Result

GRR% Cgk GRR% Cgk GRR% Cgk

Fuzzy Fig. 2-d Fig. 3 (28,30,33) (1.2,1.33,2) Reject AcceptCrisp 11.08 1.2879 30 1.33 Accept Reject


part-to-part. It means that the width of fuzzy numbers in this empirical example is not really large in comparison with nat-ural variability behind measurement system (repeatability) and production process (part-to-part).

The last capability criterion that should be examined with fuzzy data is the overall capability index. To do this, a test blockwith known reference value of 410 is approximately measured by an appraiser 15 times. The collected measures are delin-

eated in Table 5 as triangular fuzzy numbers. Then a-level cuts of ~Cgk are computed using two optimization problems (56)

and (57) as given in the right hand side of Table 5. These values can be also used to establish the membership function of ~Cgk,

which is depicted in Fig. 3. As mentioned formerly, ~Cgk can be compared with either the fuzzy critical value 1:~33 or the crispcritical value 1.33. In this example, the fuzzy critical value ð1:2;1:33;2Þ is determined by decision maker to be comparedwith ~Cgk. This fuzzy critical value is represented by a triangle on Fig. 3. In this case, since I~Cgk

¼ 0:394 and I1:~33 ¼ 0:0414, it

is concluded ~Cgk � 1:~33 and consequently, the overall capability index in fuzzy environment is acceptable. Here, it is alsoworth comparing this conclusion with that in crisp environment. If the center values of fuzzy measures in Table 5 are re-

garded, the value of Cgk in crisp environment is computed as 1.288, which is equal to ð~CgkÞL

a¼1 and ð~CgkÞU

a¼1. As it is observed,the capability index in crisp environment is slightly less than 1.33. Therefore, a practitioner rejects overall capability of thegauge. As mentioned earlier, since fuzzy measures are gathered from the measurement system more realistically, i.e. all




variations and unexpected conditions are taken into account, it can be said that the overall capability index of the gauge isactually acceptable based on the results obtained in fuzzy environment. Table 6 summarizes the results attained from capa-bility analysis of the variable measurement system in the current example for both fuzzy and crisp environments.

6. Conclusions

This paper attempted to analyze capability of the variable measurement system in fuzzy environment, where the mea-surement data were assumed to be fuzzy numbers. The main idea was to utilize Zadeh’s extension principle. Application ofthis principle to capability criteria studied including repeatability, reproducibility, GRR% and Cgk results in a pair of non-linearmathematical programming problems. Optimal solutions of these non-linear programming problems gave the lower andupper bounds of a-level cuts of each capability criterion in fuzzy environment. The membership function of each capabilitycriterion was then constructed by counting diverse values of a. In the next step, a ranking method for fuzzy numbers wasexploited in order to evaluate acceptability of G~RR% and ~Cgk. The advantage of incorporating fuzzy measures in analyzingcapability of the variable measurement system was presented using a numerical example. This advantage is to make a moreaccurate decision about capability of the variable measurement system, because fuzzy measures pose to be gathered in amore realistic and practical situation in which all variations and unexpected conditions are included.

As mentioned in Section 1, the other type of measurement system is called attribute measurement system, which has anoutcome of either zero or one. In such a measurement system, zero quantity means the production part is rejected whereasone quantity denotes it is accepted. As a future study, the attribute measurement system can be developed in fuzzy environ-ment where status of a production part can be categorized as ‘‘good’’, ‘‘medium’’, ‘‘bad’’ and so on. Moreover, it is possible tofind confidence intervals for variance component estimates, which can be viewed as a solution to deal with their negativevalues. Such confidence intervals have been comprehensively discussed by Montgomery and Runger [34] and Burdicket al. [35]. Developing confidence intervals for variance component estimates in fuzzy environment is the other attractiveextension on classical capability analysis of the variable measurement system.

Appendix A

According to Table 1, mean squares of a two-way analysis of variance with two factors including appraiser (A) and part (P)are calculated as follows [36]:

PleaseMode

MSA ¼ SSA=ða� 1Þ; ð62Þ

MSP ¼ SSP=ðp� 1Þ; ð63Þ

MSAP ¼ SSAP=ða� 1Þðp� 1Þ; ð64Þ

MSR ¼ SSR=apðn� 1Þ; ð65Þ

where, sum of squares SSA, SSP , SSAP and SSR are computed as:

SSA ¼Xa

j¼1

x2:j:

,pn� x2

:::=apn; ð66Þ

SSP ¼Xp

i¼1

x2i::

,an� x2

:::=apn; ð67Þ

SSAP ¼Xp

i¼1

Xa

j¼1

x2ij:

,n� x2

:::=apn� SSA � SSP; ð68Þ

SSR ¼ SST �Xp

i¼1

Xa

j¼1

x2ij:

,n� x2

:::=apn; ð69Þ

and

SST ¼Xp

i¼1

Xa

j¼1

Xn

k¼1

x2ijk � x2

:::=apn: ð70Þ

Moreover, in the above equations, we have:

xi:: ¼Xa

j¼1

Xn

k¼1

xijk; ð71Þ




PleaseModel

x:j: ¼Xp

i¼1

Xn

k¼1

xijk; ð72Þ

xij: ¼Xn

k¼1

xijk; ð73Þ

and

x::: ¼Xp

i¼1

Xa

j¼1

Xn

k¼1

xijk: ð74Þ

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.apm.2014.03.017.

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capability analysis of the variable measurement system with fuzzy data

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