Developing a Method for Increasing Accuracy and Precision in Measurement System Analysis: A Fuzzy Approach

Transcription

1 Journal of Inustrial Engineering 6( Developing a Metho for Increasing Accuracy an Precision in Measurement System Analysis: A Fuzzy Approach Abolfazl Kazemi a, Hassan Haleh a, Vahi Hajipour b, Seye Habib A. ahmati b a Islamic Aza University, Qazvin Branch, Department of Inustrial an Mechanical Engineering, Qazvin, Iran b Islamic Aza University, Qazvin Branch, Member of Young esearchers Club, Qazvin, Iran eceive 3 Jul., 00; evise 4 Jul., 00; Accepte Aug., 00 Abstract Measurement systems analysis (MSA has been applie in ifferent aspect of inustrial assessments to evaluate various types of quantitative an qualitative measures. Qualification of a measurement system epens on two important features: accuracy an precision. Since the capability of each quality system is severely relate to the capability of its measurement system, the weakness of the two mentione features can reuce the reliance on the qualitative ecisions. Consequently, since in the literature fuzzy MSA is not consiere as an inepenent stuy, in this paper, a fuzzy metho is evelope for increasing metho accuracy an precision by encountering the impreciseness of some measures of MSA. To o so, bias, capability, an gauge repeatability an reproucibility (G& inices are consiere as triangular fuzzy numbers. The application of the propose metho is illustrate through a case stuy taken from an automotive parts inustry. All rights reserve. Keywors: MSA; Fuzzy numbers; G&; Quality techniques.. Introuction Currently, there are many quality control techniques for recognizing causes of errors an preventing their occurrence. Measurement system analysis (MSA is consiere as an important one. Measurement is efine as the assignment of numbers or values to material things to represent the relationship among them with respect to particular properties (C.Eisenhart [5]. It is the process of evaluating an unknown quantity an expressing it into numbers that is usually consiere as preceence of any statistical process control. Moreover, MSA has been recognize as one of the key requirements in the ol QS9000 Quality Stanar, Six Sigma technique, an even new stanars such as ISO TS6949:009. These extensive applications are ue to avantages of MSA incluing promoting the compatibility of the measurement system for the given process an reucing the contamination of measurement variation in the total process variation. MSA is base on an important philosophy which argues that measurement errors creep into any process easurement Corresponing author abkaazemi@qiau.ac.ir metho. Therefore, it shoul be consiere as the preceent process of any quality measurement system (Harry & Lawson [0]. MSA quantifies a measurement error via the examination of multiple sources of variation in a process, incluing the variation resulting from the measurement system, from the operators, an from the parts []. Since statistical measures are estimate by ata which are obtaine through sampling, they are usually unreliable [7]. In this case, it is helpful to think of a measure value as the sum of two variables; the quantity of measure value an its error as Eq.(. Y ( Measure _ value ( True _ value e ( i i i The measurement system increases the total observe variability ( obs of the measure parts. In any measuring, some of the observe variability is ue to variability in the 5

2 Abolfazl Kazemi et al./ Dveloping a Metho for Increasing Acurracy an Precision in process (, whereas the rest variability is ue to the p measurement error or gauge variability (. The variance of the total observe measurements can be expresse as Eq. ( [7]. It means that total variability equals to the sum of process variability an measurement variability. obs p ( In this equation inclues two major types of error by itself which are calle repeatability an reproucibility. epeatability ( epeatability which can be etermine by measuring a part for several times, quantifies the variability in a measurement system, resulte from its gauge, [7], [9], [4]. eproucibility ( eproucibility which is etermine from the variability create by several operators measuring a part for several times, quantifies the variation in a measurement system resulte from the operators of the gauge an environmental factors, [], [9], [3], [5]. Square root of is calle gauge repeatability an reproucibility (G& that as it mentione moels the all error relate to the gauge. It can be shown as Eq. (3. repeatability reproucibility (3 Foster [6] propose some proceures for calculating ifferent inices of MSA in orer to calculate G& as major output of MSA. In orer to istinguish prouct variance from evice variance, Grubbs [7], Juran an Gyrna [9] carrie out MSA stuies on two or more measurement evices an propose a proceure for estimating the sensitivity of the measurement evices. Senol [] statistically evaluate MSA metho by the means of esigne experiments to minimize α-β risks an n (sample size. A G& stuy which was introuce in [9] estimates the repeatability an reproucibility components of measurement system variation with the primary objective of assessing whether or not the gauge is appropriate for the intene applications. Evaluating measurement an process capabilities by G& with four quality measures presente by Al-efaie an Bata []. As it mentione, one reason that causes inices calculate by sample ata are unreliable is the uncertainty of the ata. Therefore, statistical calculations such as stanar eviation, point an interval estimation, hypothesis testing an other similar one are use. Besies, unreliability of inices has another reason which is resulte from the impreciseness of ata. In the literature to eal with impreciseness usually fuzzy concept (introuce by Zaeh [6] is use. However, in the literature of the quality issues any work with the legen of fuzzy MSA (FMSA was not foun an we have just reviewe the most relevant work such as the application of fuzzy moelling in ifferent quality inices an quality control charts. Lee [4] an Hong [] propose C pk inex estimation using fuzzy numbers. Parchami et al. [0] using fuzzy specification limits rather than precise ones propose a new fuzzy number for all types process capability inices. Parchami et al. [] introuce a new metho that uses confience interval of capability inices to prouce fuzzy number for them. Faraz an Bameni Moghaam [4] an Gulbay an Kahraman [8] propose efficient methos for creating fuzzy quality control charts. Accoring to what was mentione, MSA help to juge about compatibility of the measurement system with the given measurement process an provies conitions for more reliable ecision. This stuy makes important contribution to the MSA literature an evelops fuzzy concept on MSA metho to create inices of MSA much more accurate. The rest of the paper is organize as follows. The next section illustrates classical MSA an its ifferent inices. Section 3 evelops fuzzy concept to create FMSA metho. Section 4 escribes a case stuy in automotive parts inustry in Kachiran Company. Finally, in Section 5 some notes about concluing remarks an future research are presente.. Measurement System Analysis (MSA Measurement system is the collection of instruments or gauges, stanars, operations, methos, fixtures, software, personnel, environment an assumption use to quantify a unit of measure or fix assessment [5]. Corresponingly, MSA is a collection of statistical methos for the analysis of measurement system capability (Automotive Inustry Action Group (AIAG []; Smith et al. [4]. It seeks to escribe, categorize, evaluate the quality of measurements; improve the usefulness, accuracy, precision, meaningfulness of measurements; an propose methos for eveloping better measurement instruments by Montgomery, unger [6]. Some state goals of MSA are estimate components of measurement error, estimate the contribution of measurement error to the total variability of a process or equipment parameter, etermine stability of a metrology tool over time, an to compare an correlate multiple metrology tools. Measurement process is a kin of prouction process that its output is number. Fig. presents measurement process with its inputs an outputs, [5]. Accoring to the type of ata, MSA has two categories of measurements; quantitative measurement an qualitative one. In this paper, quantitative measurements are iscusse. In the rest of the section, we illustrate the steps require to execute MSA. 6

3 Journal of Inustrial Engineering 6( Bias There are the ifference between the observe average of measurements an the master average of the same parts using precision instruments [5]. Actually, bias is a measure that represents the ifference between the averages value of the measurement an certifie value of a specific part. Fig.. Measurement system analysis process [5] Fig. represents bias concept schematically, [5]. In orer to compute this inex, we measure a part with an instrument for at least ten times. Then we shoul acquire the average of these observations an compare it with the true value of the part. We can obtain the value of bias by Eq. (3. B x g x m (3 xg Where B is bias value, is the average of measure ata, x is a traceable stanar. If a traceable stanar C g m is not available, you can measure the part ten times in a controlle environment an then the average of values is use to etermine the reference value. This sample will be consiere as the Master Sample. Decision making base on only bias value is unusual. Actually, after obtaining, we can recognize appropriate level of bias. the observe ata. However these measures o not consier customer requirements an it is not suitable for general comparisons among processes Leung PK, Spiring F, [5]. Capability inices C p, C pk, C nm, an C nmk have been propose in the manufacturing an service inustries, proviing numerical measures on whether or not a process is capable of reproucing items within the specification limits, Shishebori, Hamaani []. Similarly, C g an C gk are use to show the capability of the measurement gauge. The inices an C g an C gk are efine by Eq.(4, (5. C g 0.T (4 6S g 0.T xg xm Cgk (5 3Sg Where T is part tolerance an S g shows stanar eviation of observe values using measurement instrument. The minimum acceptance criteria for C g an C gk are equal to.33 []..3. Gauge epeatability an eproucibility (G& As it was mentione earlier, the total measurement variation is the sum of variation ue to repeatability an reproucibility Eq. (3. epeatability an reproucibility can influence the precision an accuracy respectively..4. epeatability The same characteristic of the prouct shoul be measure repeately in orer to etermine the sensitivity of the measurement process [6]. When an inspector uses the same gauge to measure a prouct several times uner the same conitions, several ifferent values of measurement may occur. This error, calle repeatability, comes from the gauge itself [8]. epeatability is compute as Eq.(6. EV 5.5 (6 Fig.. Scheme of bias [5].. Capability Capability is a measure of process ability to consistently prouce a result that meets the specification requirements. The term process capability was synonymous with process variation measures such as stanar eviation or range of 7 Fig.3. Precision of epeatability [5] Where is the average of variation range, is obtaine from a specific table, an EV is tool variation.

4 Abolfazl Kazemi et al./ Dveloping a Metho for Increasing Acurracy an Precision in Also 5.5 interval involves 99 percents of ata in normal istribution. Fig.3 represents the variation among successive measurements of the same characteristic, by the same person using the same instrument [5]..5. eproucibility This error occurs when ifferent inspectors measure a prouct uner the same conition. Practically, it is ue to eficient traine inspectors or out of stanar measuring methos [8]. It is compute as Eq. (7. AV ( EV DIF (5.5 nr. (7 Where DIF max( i min( i ; i=,,,numbers of operator, obtaine from same table within previous x DIF subsection with g=, m is number of operators, stanar eviation of reproucibility, EV is repeatability value, AV is appraiser variation, n is number of use parts an r is number of trials that each piece is measure. If DIF ( EV then AV DIF an nr. otherwise reproucibility value is equal to zero. Fig.4. Precision of eucibility [5] Fig.4 represents the stanar eviation of the averages of the measurements mae by ifferent persons, machines, an tools [5] when measure the ientical characteristic on the same part..5.. Gauge & A G& stuy is a metho of etermining the suitability of a gauge system for measuring a particular process. Every measurement has some associate error, an if this error is large compare to the allowable range of values (the tolerance ban, the measuring evice will frequently accept ba parts an reject goo ones [5]. is 8 Total G& is the estimation of the combine estimate variation from repeatability an reproucibility []. In a G& stuy, we try to quantify measurement variation as a percent of process variation. G& inex is compute as (8. & EV AV (8 An ieal measurement system shoul not have any variation. However, this is impossible an we have to be satisfie with a measurement system that has variation less than 0% of the process variation. As the portion of variation ue to measurement system increases, the value of measurement system reuces. If this proportion is more than 30%, the measurement system is unacceptable. Table summarizes system status for obtaine %G& []. Table System status after computing %G& %G& Decision Guieline <%0 Acceptable measurement system %0 to %30 This nees to be agree with the customer >%30 Unacceptable measurement system 3. Fuzzy Measurement System Analysis (FMSA In this section, some basic concept of fuzzy sets an fuzzy numbers are reviewe. Fuzzy set theory which was introuce by Professor Lofti Zaeh in 965 [6] is a typical metho for encountering with ambiguity an imprecision. Since most practical an inustrial methos an problems are encountere with imprecise ata or lack of ata consiering fuzzy techniques help us to make our methos much more accurate. Nowaays we eal with ifferent quality problems that all of them are precee by MSA. In this situation, if the quality of measurement system is low, it can be expecte that process analysis will not be vali. Therefore, we consier MSA with fuzzy numbers to have a more precise an accurate measurement system an ata analysis. This precise ata analysis will lea to a more accurate ecision making an quality system. Experimental results of a case stuy will represent performance of this new type of MSA in an inustrial real worl instance an show how MSA execute with fuzzy calculations in etail. Meanwhile, first some mathematical operations in fuzzy concept is reviewe, then they are expane on ifferent inices in MSA in the rest of this section. Fuzzy calculations are hanle with fuzzy numbers. A fuzzy number is a convex fuzzy subset of the real line an is completely efine by its membership function. Different type of membership functions are consiere in the literature of fuzzy numbers. This paper uses triangular membership function [3] for ifferent inices of MSA an

5 Journal of Inustrial Engineering 6( shows etaile calculations of MSA in the environment of fuzzy concept. The rest of the section illustrates all of these calculations. Denoting the triangular fuzzy number M by a triplet (a, b, c an N by a triplet (, e, f [6], the aition, subtraction, multiple, an ivision of the two triangular fuzzy numbers can be shown as Eq.(9 to Eq.( respectively. M N ( abc,, ( e,, f ( ab, ec, f M N ( abc,, ( e,, f ( a f, bec, M N ( abc,, ( e,, f ( ab, ec, f M N ( abc,, ( e,, f (,, (9 (9 f e EV 5.5 (,, ( a,, b c ( EV, EV, EV (3 Then, accoring to the corresponing ranking approach the results are as follows. We suppose that i= is the largest member of a fuzzy number an i= is the smallest member of a fuzzy number. Our metho for making a fuzzy triangular number is efine in Eq.(0 [3]. b ( b b b ; 0.999,.00 (0 The corresponing shape of the fuzzy number is plotte as Fig.5. max( i min( i max(,, i i i min(,, i i i ( ( a c b b c a ( ( (4 Fig. 5. Membership function of b fuzzy number It shoul be mentione that in this stuy for ranking two fuzzy numbers like ( a, b, c an (, e, f the thir member of each set is consiere as the criteria of comparison. For instance among fuzzy M an N (efine at Eq.(4, since max(,=, fuzzy M is selecte. N (6,8, & M (5,9, ( The inices of fuzzy measurement system are presente below. Equations (-(6 are fuzzy inices of (3, (6, (7, an (8, respectively. For the sake of completeness, we have provie the complete set of equations. B x g xm ( x a xm, x b xm, x c xm ( B, B, B ( ( EV AV (5.5 DIF nr. ( ( EV, EV a b, EV (5.5 c nr (, a nr b b nr (,, c a nr.. AV a AV b AV c (5 Finally, fuzzy G& is efine as (8. All of the membership functions of these inices are presente in next section in a case stuy. & EV AV ( &, &, & ( EVa, EVb, EV ( AVa, AVb, AVc, (6 9

6 Abolfazl Kazemi et al./ Dveloping a Metho for Increasing Acurracy an Precision in Table Case stuy ata Part Number Operator Measurements (mm Operator Measurements (mm Operator 3 Measurements (mm M M M M M M (6.4,6.,6.6 (6.,6.6,6. (6.09,6.5,6. (6.08,6.4,6. (6.09,6.5,6. (6.,6.6,6. (6.3,6.9,6.5 (6.3,6.9,6.5 (6.3,6.9,6.5 (6.3,6.9,6.5 (6.3,6.9,6.5 (6.4,6.,6.6 3 (6.05,6.,6.7 (6.06,6.,6.8 (6.05,6.,6.7 (6.04,6.0,6.6 (6.04,6.,6.6 (6.05,6.,6.7 4 (6.,6.7,6.3 (6.,6.7,6.3 (6.,6.7,6.3 (6.,6.7,6.3 (6.,6.7,6.3 (6.,6.7,6.3 5 (6.9,6.5,6.3 (6.9,6.5,6.3 (6.9,6.5,6.3 (6.0,6.6,6.3 (6.9,6.5,6.3 (6.9,6.5,6.3 6 (6.06,6.,6.8 (6.06,6.,6.8 (6.06,6.,6.8 (6.06,6.,6.8 (6.06,6.,6.8 (6.06,6.,6.8 7 (6.07,6.3,6.9 (6.08,6.4,6. (6.08,6.4,6.0 (6.07,6.3,6.9 (6.07,6.3,6.9 (6.07,6.3,6.9 8 (6.4,6.,6.6 (6.4,6.,6.6 (6.4,6.,6.6 (6.4,6.,6.6 (6.4,6.,6.6 (6.4,6.,6.6 9 (6.4,6.3,6.36 (6.4,6.3,6.36 (6.4,6.3,6.36 (6.3,6.9,6.35 (6.3,6.9,6.35 (6.4,6.3, (6.,6.8,6.34 (6.,6.8,6.34 (6.,6.8,6.34 (6.,6.8,6.34 (6.,6.8,6.34 (6.,6.7, Numerical Example In this section a case stuy which investigates housing clutch in automotive parts inustry in Kachiran Company in Asia within crisp environment is consiere. Then, to extract more accurate inices an make a more reliable ecision making we bring the crisp case stuy into the fuzzy environment an propose FMSA. In our case stuy, we have 0 parts, 3 operators, trials an the tolerance of corresponing part is Therefore m is 6. an T is 0.. Also, the coefficient for right han sie an left han sie corresponing fuzzy number is Table ( shows case stuy ata an Table (3, (4 represent results of MSA inices with fuzzy number. It shoul be mentione that stability an linearity of the ata ha teste in an exact environment before the ata become fuzzy. It means that our non fuzzy ata ha the both basic features which are stability an linearity. Finally, Fig. 6 to Fig. 0 plots the membership function of MSA inices. Table 3 esults within fuzzy environment Inex Operator Operator Operator 3 (6.3,6.9,6.6 (6.3,6.9,6.5 (6.3,6.9,6.5 B (-0.07,-0.0,0.06 (-0.07,-0.0,0.05 (-0.07,-0.0,0.05 (-0.,0.0,0.3 (-0.,0,0.3 (-0.,0.0,0.3 operators o not have any ifferences, that shows reprouctively of the measurement system is in a high level. Therefore, having appropriate reprouctively an repeatability causes a goo G&. Table 4 proves our mentione expectations are right. Our graphical outputs in Table 4 MSA inices with fuzzy numbers Inices Fuzzy Number (-0.,0.0,0.3 E V (-0.53,0.03,0.6 (-0.3,0,0. A V (0,0.0,0.9 & (0,0.04,0.67 Fig.6 to Fig.0 illustrates inices schematically. Through this figures or fuzzy numbers, vagueness of the inices is moele. Figure 0 as an example moels an removes vagueness of the G& criterion. This figure not only inclues crisp G& in the point 0.04, but also is consiste of any other number aroun this value that is in the oubtful area. Therefore, consiering MSA inices as fuzzy numbers causes to improve quality of measurement system an corresponing ecision will make with more accurate information. As it can be seen in table, for each operator, integer parts of all observations are the same an ecimal part of the ata has little variation. This achievement shows that the repeatability of the gauges is appropriate. On the other han, accoring to the table 3, results of the ifferent 30

7 Journal of Inustrial Engineering 6(005-3 Fig.6. Membership function of Fig.0. Membership function of & 5. Conclusion an Future Works Fig.7. Membership function of Fig.8. Membership function of Fig.9. Membership function of E V A V MSA is a collection of statistical methos for analyzing measurement systems. Since most practical systems are encountere with imprecise ata, consiering fuzzy concept help us to make MSA much more accurate. It means that, the unerlying ata are usually assume to be precise numbers, but it is much more realistic to consier them through fuzzy values. However, although FMSA can be a consierable issue, in the literature any corresponing paper has not been foun. Thus, in this paper MSA was moele in fuzzy environment an triangular fuzzy numbers were propose for MSA inices. The output of fuzzy inices were illustrate through ifferent tables an figures an shown that all the crisp concepts an efinitions of MSA can be evelope in fuzzy environment. Finally, a real-worl example taken from a housing clutch manufacturing process was investigate to explain efficient performance of FMSA more explicitly. Interpretations of the FMSA were also one accoring to tables just like interpretation of the crisp MSA to explain relationship among these two areas. For future works, following options can be suggeste: Using other types of membership functions that can leas to better results. Expaning this metho for qualitative measurement ata Consiering some variable together in the problem an eveloping a fuzzy control system with various rules Developing a new ranking approach other than the ranking approach of this paper Consiering other aspects of MSA such as fuzzy stability or fuzzy linearity 3

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