Development of Fuzzy Extreme Value Theory. Populations

Transcription

1 Applied Mathematial Sienes, Vol. 6, 0, no. 7, Development of Fuzzy Extreme Value Theory Control Charts Using α -uts for Sewed Populations Rungsarit Intaramo Department of Mathematis, Faulty of Siene King Mongut s University of Tehnology Thonburi Bango, Thailand loju@hotmail.om Adisa Pongpullponsa* Department of Mathematis, Faulty of Siene King Mongut s University of Tehnology Thonburi Bango, Thailand adisa.pon@mutt.a.th Abstrat Reent studies have demonstrated that the adaptive (i.e., variable sample sizes, sampling intervals, and/or ation limit oeffiients) x harts are quier than standard Shewhart (SS) x ontrol harts in deteting proess mean shifts. The usual assumption for designing a ontrol hart is that the data or measurements are normally distributed. However, this assumption may not be true for some proesses. In this paper fuzzy extreme value (FEV) theory ontrol harts have been developed from extreme value (EV) ontrol harts using α -uts with the unertain data whih is evaluated under non-normality. For many problems, ontrol harts ome from unertain data suh as human, measurement devies or environmental onditions. In this ontext, fuzzy set theory is useful to help in solving the data problems aused by this unertainty. The data for the experiment will be transformed to fuzzy ontrol harts by using membership funtions. The effiieny of ontrol harts are determined by average run length (ARL). * Corresponding author.

2 58 R. Intaramo and A. Pongpullponsa Keywords: non - normality distribution data, α-uts, α-level fuzzy midrange Introdution The ontrol hart originated in the early 90s, it has beome a powerful tool in statistial proess ontrol (SPC) that is the most widely used in industrial proesses. Control harts are designed to monitor the proess of hange in mean and variane, they also reflet the ability of the proess. Control harts have two types: variable and attribute. Tehniques of statistial proess ontrol are widely used by the manufaturing industry to detet and eliminate defets during prodution. Control hart tehnique is well-nown as a ey step in prodution proess monitoring. The ontrol hart has a major funtion in deteting the ourene of assignable auses, so that the neessary orretion an be made before non-onforming produts are manufatured in a large amount. The ontrol hart tehnique may be onsidered as both the graphial expression and operation of statistial hypothesis testing. It is reommended that if a ontrol hart is employed to monitor proess, some test parameters should be determined suh as the sample size, the sampling interval between suessive samples, and the ontrol limits or ritial regions of the hart. SPC is an effiient tehnique for improvement of a firm s quality and produtivity. The main objetive of SPC is similar to that of the ontrol hart tehnique, that is, to rapidly examine the ourrene of assignable auses or proess shifts.

3 Development of fuzzy extreme value theory ontrol harts 583 Many studies were done to ombine statistial methods and fuzzy set theory. Fuzzy sets theory was first introdued by Zadeh (965). In 005 Zadeh outlined generalized theory of unertainty (GTU) whih presented a hange of perspetive and diretion in thining about the system and unertainties. Buley and Eslami (004) introdued the theory of estimation of the mean and variane of the onfidene intervals using triangular numbers as the estimator. M. B. Vermaat ET AL(003) studied the omparison of ontrol harts based on normal, non-parametri ontrol harts and extreme value (EV) ontrol harts. A.Pongpullponsa, W. Suraheriati and R. Intaramo, (006) used the onept of EV theory of M. B. Vermaat ET AL(003) to develop EV theory ontrol harts whih data are Weibull, lognormal and Burr s distributions by omparing with its effiieny of weighted variane method (WV), saled weighted variane method (SWV) ontrol harts of A.Pongpullponsa, W. Suraheriati and P.Kriweradehahai (004). There is limited information available on fuzzy attribute ontrol harts and their appliations: Wang and Raz (990) proposed some approahes by assigning a fuzzy set to eah linguisti term and then ombining these for eah sample using the rules of fuzzy arithmeti. Kanagawa et al. (993) introdued a ontrol hart based on the probability density funtion for linguisti data. Gullbay et al. (004) suggested the α -ut fuzzy ontrol harts for linguisti data. Gullbay and Kahraman (006) developed fuzzy ontrol harts for determining the unnatural

4 584 R. Intaramo and A. Pongpullponsa patterns. Information on fuzzy variable ontrol harts and their appliations are also limited: Roland and Wang (000) introdued fuzzy SPC theory based on the appliation of fuzzy logi to the SPC-zone rules. El-Shal and Morris (000) modified SPC-zone rules to redue false alarm and detet the real error. Zarandi et al. (008) presented a new hybrid method based on a ombination of fuzzified sensitivity riteria and fuzzy adaptive sampling rules to determine the sample size and sample interval of the ontrol harts in order to determine the sample size and sample interval of the ontrol harts. In fat, the problem with ontrol harts is aused by unertain data i.e. human, measurement devies or environmental onditions. The studies of A. Pongpullponsa, W. Suraheriati and and R. Intaramo, (006) are important as they indiate the ambiguity data of the hart. Thus, fuzzy set theory is useful in helping to solve the problems aused by unertain data by applying fuzzy to EV theory to develop a new hart (FEV), in order to ontrol and improve proess effiieny at its best. It was disovered by Sentur and Erginel (009) that ontrol harts ould be used to solve the problem of unertain data by using fuzzy theory. The topi of the researh studied was fuzzy ~ X R % and ~ ~ X S ontrol harts using α -ut. The methods used in the transformation of fuzzy sets into salars are fuzzy mode, fuzzy median and α -level fuzzy midrange. Whih one you hoose to use depends on the diffiulty of the omputation or preferene as in Wang (990). The aim of this study is to introdue the framewor of FEV theory ontrol harts whih are Weibull, lognormal and Burr s distributions, using α -ut with

5 Development of fuzzy extreme value theory ontrol harts 585 the methods of α -level fuzzy midrange. First of all, we transform EV theory ontrol harts to FEV theory ontrol harts. To obtain FEV theory ontrol hart, triangular fuzzy numbers (a,b,) are used. Seondly α -ut FEV ontrol harts are developed by using α -ut approah. Thirdly α -level fuzzy midrange for FEV ontrol harts are alulated by using α -level fuzzy midrange transformation tehniques. Finally, we an use the ARL to determine the effiieny of the hart. This paper is organized as follows: non-normal distributions as Weibull, lognormal and Burr s, EV ontrol harts andα -level fuzzy midrange are introdued in the seond setion. FEV ontrol harts are developed in setion 3. The effiieny of FEV ontrol harts are examined in setion 4. The onlusions are presented in the final setion. Model Consideration In this study, we will onsider FEV theory ontrol harts whih are developed from EV theory ontrol harts studied by Pongpullponsa, A., Suraheriati, W. and Intaramo, R. (006). These harts have non-normal distribution data whih are Weibull, lognormal and Burr s.. Weibull distribution Weibull is ontinuous distribution that is used widely. Let X be ontinuous random variables that are Weibull distribution with θ > 0 and > 0. Density funtion

6 586 R. Intaramo and A. Pongpullponsa ( x/ θ) f( x; θ, ) = x e x > 0 θ Cumulative distribution funtion ( x/ θ ) F( x; θ, ) = e x > 0 Where θ is sale parameter is shape parameter In this study θ = and are relevant, with a oeffiient of sewness at α3 { 0.,0.5,,,3,4,5,6,7,8,9} shown in table. Table represents a oeffiient of sewness and shape parameter of Weibull distribution Coeffiient of sewness ( α 3) Shape parameter ( )

7 Development of fuzzy extreme value theory ontrol harts 587. Lognormal distribution Lognormal is orrelated with normal distribution but random variables have positive values. Let X equal ontinuous random variables that are lognormal distribution. Density funtion ln x μ f( x; μσ, ) = exp xσ π σ x > 0 where μ is sale parameter σ is shape parameter In this study μ = 0.,0.5,,,3, 4,5,6,7,8,9 and σ are relevant with a oeffiient of sewness at α3 { 0.,0.5,,,3,4,5,6,7,8,9} shown in table. Table represents a oeffiient of sewness and shape parameter of lognormal distribution Coeffiient of sewness 3 ( ) α Shape parameters ( ) σ

8 588 R. Intaramo and A. Pongpullponsa Table (ontinued) Coeffiient of sewness ( α 3) Shape parameters ( σ ) Burr s distribution Burr s is a type of ontinuous distribution. Let X equal ontinuous random variables that are Burr s distribution with parameter and m. Density funtion mx f( x) = ( + x ) 0 x < 0 Cumulative distribution funtion where m, x 0 m+ m F( X) = ( + x ) x > 0 Burr s distribution an be represented as Weibull distribution when m inreases. α 3 and α 4 an be obtained by m > 3 and m > 4 respetively, where α 3 is the oeffiient of sewness and α 4 is the oeffiient of urtosis. This study is onfigured as shown in table 3.

9 Development of fuzzy extreme value theory ontrol harts 589 Table 3 represents onstants α 3, m,, μσ, for a oeffiient of sewness of Burr s distribution Coeffiient of sewness ( α 3) Coeffiient of urtosis ( α 4) m μ σ * 5 * 6 * 7 ** 8 ** Note μ denotes mean of the population. σ denotes standard deviation of the population. * ** denotes no oeffiient of urtosis beause m < 4. denotes without any onstant.

10 580 R. Intaramo and A. Pongpullponsa.4 Extreme value ontrol hart : EV ontrol hart EV theory that deals with the tail behaviour of distribution, an be modelled using EV distribution by Deer (989), whih an be monitored as an index of extreme values. Beause we an t mae assumptions regarding the value of γ, we an use the moment estimator to alulate an approximate value as follows : () () ( M ) γ = M + () M () and γ = M + _ () _ () ( M ) _ () M () The study of Deer (989), F is the q-quantile of the distribution funtion, so γ ( m/ q)) () F ( q; γ ) X( m) ( ( γ 0)) X( m) M γ = + (3) with 0 < q <. x y and x y denotes the minimum and maximum respetively. Define r M (log X log X ) (4) m ( r) = m n= ( n+ ) ( m) and _ ( r ) m r M = (log X log X ) (5) ( n) ( m+ ) m n=

11 Development of fuzzy extreme value theory ontrol harts 58 where an integer taes the values r = or, and m is the number of upper and lower order statistis respetively used in the estimation of the ontrol limits. From EV theory ontrol harts are γ ( m/ q)) () ( m) ( ( γ 0)) ( m) γ UCL = X + X M (6) γ _ ( m/ α /)) LCL = X + ( ( γ 0)) X M (7) ( m+ ) ( m+ ) γ () where _ ( r ) M = n = M n ( r ) n is the number of lass, m is number of sample, is the number of sample size and ( r) M is the moment estimator. Hene, we must approximate the value of M by using binomial theorem of ( r ) sewed populations whih are Weibull, lognormal and Burr s distributions, see equation (9), (0) and () respetively. Estimator of M of Weibull distribution ( r ) By binomial theorem so ( r) M = E( xμ) r M ( ) ( ) (8) ( r) i i = μ E x i= 0 i r

12 58 R. Intaramo and A. Pongpullponsa Find i i ( x/ θ) EX ( ) = x x e dx θ 0 + i ( x/ θ) x e dx 0 = θ Let x x y = = θ θ x = yθ x= / y θ dx = y θdy + i y i E( X ) = y θ e y θdy θ 0 + i y = y θ e y dy θ 0 + i + i y θ 0 θ = y e y dy i = θ 0 + i y y y e dy i = θ 0 + i + y y e dy i = θ 0 + i y y e dy

13 Development of fuzzy extreme value theory ontrol harts 583 i i θ τ + = From equation (8) so M + i τ (9) ( r) i i = ( μ) θ i= 0 i r Estimator of M of lognormal distribution ( r ) By binomial theorem so ( r) M = E( xμ) r M ( ) ( ) ( r) i i = μ E x i= 0 i r Find Let ln xμ σ i i = 0 EX ( ) x e dx xσ π y = ln x x = e y dx = y e dy yμ σ i y i = y 0 EX ( ) ( e) e dx e σ π = M ( i) y

14 584 R. Intaramo and A. Pongpullponsa = e σ ( i) μ ( i) + From equation (8) so M (0) σ ( i) ( i) ( r) μ + i = ( μ) e i= 0 i r Estimator of M of Burr s distribution ( r ) By binomial theorem so ( r) M = E( xμ) r M ( ) ( ) ( r) i i = μ E x i= 0 i r Find i i mx EX ( ) = x dx 0 ( + x ) m+ Let i = v y = + x,0< y < y x = y y J = dx = dy y y ( v+ ) + m y y = my dy y y y 0

15 Development of fuzzy extreme value theory ontrol harts 585 v m+ y = my 0 dy y y 0 v m+ = my ( y) dy v 0 v v m + = my ( y) dy from v= i From equation (8) so v v = m m,+ i i = m m,+ i i mτ m τ + = τ ( m + ) M ( r) i i mτ m τ + i = ( μ) () i= 0 i τ ( m + ) r.5 α -level fuzzy midrange Fuzzy transformation tehniques have four types : fuzzy mode, fuzzy median, fuzzy average and α -level fuzzy midrange. In this study, the α -level fuzzy midrange transformation tehnique is used for FEV theory ontrol harts. The α -level fuzzy midrange f α mr is defined as the midpoint of the α -level

16 586 R. Intaramo and A. Pongpullponsa uts. Let A α isα -level uts, nonfuzzy sets that onsist of any elements whose membership is greater than or equal toα. If a α and b α are end points of A α then α ( α α fmr = a + ) () In fat the fuzzy mode is a speial ase of α -level fuzzy midrange whenα =. α -level fuzzy midrange of sample j S α, is determined by, mr j ( aj + j) + α ( bj aj) ( j bj) S α mr, j = (3) The definition of α -level fuzzy midrange of sample j for fuzzy ~ x ontrol hart is S α mrx, j ( xa + x ) ( ) ( ) j + α x j b x j a x j x j b j = (4) Then, the ondition of proess ontrol for eah sample an be defined as mr x mr x, j mr x Pr in ontrol for LCL α oess ontrol S α UCL α = out of ontrol for otherwise (5) 3 Fuzzy extreme value theory ontrol hart 3. Fuzzy extreme value theory ontrol hart By studying EV theory ontrol harts, it was disovered that unertain data was a problem, so we use fuzzy theory to solve these problems. The studies of Sentur, S., Erginel N. (009) used fuzzy theory in ontrol harts. Then we modified EV theory ontrol harts to FEV theory ontrol harts, whih use

17 Development of fuzzy extreme value theory ontrol harts 587 membership represented by a triangular fuzzy number (a,b,) as shown in Fig. Therefore, the FEV theory ontrol limits are γ ( m/ q)) () ( m) ( ( γ 0)) ( m) γ UCL= X + X M (6) γ a, γ b, ( m/ q)) () ( m/ q)) () ( m), a ( ( γ a, 0)) ( m), a, a, ( m), b ( ( γ b, 0)) ( m), b, b γ a, γ b, = X + X M X + X M γ, ( m/ q)) () ( m), + γ, ( m),, γ,, X ( ( 0)) X M ( m+ ) ( m+ ) γ _ () γ ( m/ α /)) LCL = X + ( ( γ 0)) X M (7) γ a, _ () γ b, _ () ( m/ α/) ( m/ α/) = X + ( ( γ 0)) X M, X + ( ( γ 0)) X M ( m+ ), a a, ( m+ ), a a, ( m+ ), b b, ( m+ ), b b, γ a, γ b, γ, ( m/ α /) ( m+ ), + γ, ( m+ ), γ,, X ( ( 0)) X M _ (), μ α 0 a a α b α Fig represents of a sample of triangular fuzzy number

18 588 R. Intaramo and A. Pongpullponsa 3. α -ut fuzzy extreme value theory ontrol hart An α - ut onsists of any elements whose membership is greater than or equal to α. Applying α - ut of fuzzy sets, the values of X ( m), a, X ( m),, M, () a, M are determined as follows: (), X = X + α( X X ) (8) α ( m), a ( m), a ( m), b ( m), a X = X α( X X ) (9) α ( m), ( m), ( m), ( m), b ( M ) = ( M ) + α(( M ) ( M )) (0) () α () () () a, a, b, a, ( M ) = ( M ) α(( M ) ( M )) () () α () () (),,, b, Therefore, the α -ut FEV theory ontrol limits are γ α α ( m/ q)) () ( m) ( ( 0)) α ( m) ( ) α γ γ UCL = X + X M () γ _ () α α ( m/ α /)) ( m ) ( ( 0)) α ( m ) ( ) α + γ + γ LCL = X + X M (3) α -ut ontrol limits are shown in Fig. 3.3 α -level fuzzy midrange for α -ut FEV theory ontrol hart An α -level fuzzy midrange is one of four transformation tehniques used to determine the FEV ontrol harts. In this study α -level fuzzy midrange is used as the fuzzy transformation method while alulating α -level fuzzy midrange forα -ut FEV theory ontrol limits

19 Development of fuzzy extreme value theory ontrol harts 589 UCL % α mr α α γ () () ( m), a ( m), mq (, ) α a (, ) α + α + γ X ( m) γ X X ( / )) M M = + ( ( 0)) ( ) (4) α α γ () () α α ( m ), a ( m ), ( / α /) α ( a, ) + (, ) γ X ( m+ ) γ LCL % X X m mr ( ( 0)) ( M M α = + ) (5) For an approximation of M of Weibull, lognormal and Burr s distributions ( r ) see equations (9),(0) and () respetively. μ UCL CL LCL UCL UCL α CL α 3 3 CL 3 UCL α 3 LCL α UCL CL α CL LCL3 3 UCL α LCL Fig α-ut ontrol hart ( CL %, LCL % and UCL % ) 0 4 Simulation studies The purpose of this study is to ompare the effiieny of FEV theory ontrol harts for sewed populations i,e., Weibull, lognormal and Burr s distributions whih have various values of the oeffiient of sewness whih are 0., 0.5,.0,.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, number of lass n = 300, number of sample size = 0 are randomly generated from Weibull, lognormal and Burr s distributions with θ =, relevant with a oeffiient of sewness shown in table

20 5830 R. Intaramo and A. Pongpullponsa, μ = 0, σ relevant with a oeffiient of sewness shown in table and m, is shown in table 3 respetively. The proedure is repeated 0,000 times for shift sizes of 0.5 σ,.0 σ,.0 σ,.5σ and3.0σ. From this study, the results are as : Table 4 represents the ARL orresponding to a different oeffiient of sewness. Coeffient of Weibull Lognormal Burr s sewness ( α 3) distribution distribution distribution After determining the UCL and LCL, using equations (4) and (5), the ARL results are given in Table 4, it shows that right sew inreases and the ARL

21 Development of fuzzy extreme value theory ontrol harts 583 dereases. Lognormal distribution is most effiient at a oeffiient of sewness 0. and the ARL is maximum. Burr s distribution is most effiient at a oeffiient of sewness 0.5,.0 and.0. Weibull distribution is most effiient at a oeffiient of sewness 3.0,4.0,5.0,6.0,7.0,8.0 and 9.0, see Figure 3. Fig 3 represents omparision of ARL of Weibull, Lognormal and burr s distribution. 4. If data is shifted, right sew inreases and the ARL dereases. In this study, Weibull distribution is most effiient at a oeffiient of sewness.0. Burr s distribution is most effiient at a oeffiient of sewness 0., 0.5,.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0 and Conlusions This study is to alulate the ARL of FEV theory ontrol harts, using α -ut with the methods of α -level fuzzy midrange for sewed populations whih are Weibull, lognormal and Burr s distributions. The result of the study is, the ARL of

22 583 R. Intaramo and A. Pongpullponsa FEV theory ontrol harts whih have lognormal distribution is most effiient at a oeffiient of sewness 0.. Burr s distribution is most effiient at a oeffiient of sewness 0.5,.0 and.0. Weibull distribution is most effiient at a oeffiient of sewness 3.0,4.0,5.0,6.0,7.0,8.0 and 9.0. The results of the ARL alulation of FEV theory ontrol harts at a oeffiient of sewness 0. of Weibull, lognormal and Burr s distributions are, ARL =.0, 5.3 and respetively. In this study, the ARL using FEV theory is greater than when using EV theory studied by A.Pongpullponsa, W. Suraheriati and R. Intaramo, (006). It shows that when fuzzy theory is applied to ontrol harts, the performane is better. For further researh, we may be able to develop ontrol harts by using other methods suh as weighted variane method, saled weighted variane method and empirial quantile method. These ould then be ompared with the results in this study, or, we may study data under other distributions suh as student s t distribution et. Referenes [] Buley J.J., Eslami E, Unertain Probabilities II, The Continuous Case,Soft Computing 8(004), [] Deers A.L.M., Einmahl J.H.J., De Haan L, A Moment Estimator for the Index of an Extreme-value Distribution, Annals of Statistis 7(989), [3] El-shal S.M., Morris A.S, A Fuzzy Rule-based Algorithm to Improve the Performane of Statistial Proess Control In Quality Systems, Journal of Intelligene Fuzzy Systems 9(000), 07-3.

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