Fuzzy inference to assess manufacturing process capability with imprecise data

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(1)ARTICLE IN PRESS. European Journal of Operational Research xxx (2007) xxx–xxx www.elsevier.com/locate/ejor. Production, Manufacturing and Logistics. Fuzzy inference to assess manufacturing process capability with imprecise data Bi-Min Hsu a, Ming-Hung Shu a. b,*. Department of Industrial Engineering & Management, Cheng Shiu University, 840 Cheng Cing Road, Niaosong, Kaohsiung 833, Taiwan, ROC b Department of Industrial Engineering & Management, National Kaohsiung University of Applied Sciences, 415 Chien Kung Road, Seng-Min District, Kaohsiung 807, Taiwan, ROC Received 22 November 2005; accepted 1 February 2007. Abstract Process capability indices provide numerical measures on whether a process conforms to the defined manufacturing capability prerequisite. These have been successfully applied by companies to compete with and to lead high-profit markets by evaluating the quality and productivity performance. The loss-based process capability index Cpm, sometimes called the Taguchi index, was proposed to measure process capability, wherein the output process measurements are precise. In the present study, we develop a realistic approach that allows the consideration of imprecise output data resulting from the measurements of the products quality. A general method combining the vector of fuzzy numbers to produce the membership function of fuzzy estimator of Taguchi index is introduced for further testing process capability. With the sampling distribution for the precise estimation of Cpm, two useful fuzzy inference criteria, the critical value and the fuzzy P-value, are proposed to assess the manufacturing process capability based on Cpm. The presented methodology takes into the consideration of a certain degree of imprecision on the sample data and leads to the three-decision rule with the four quadrants decision-making plot. The fuzzy inference procedure presented to assess process capability is a natural generalization of the traditional test, when the data are precise the proposed test is reduced to a classical test with a binary decision. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Process yield; Fuzzy sets; Fuzzy hypothesis testing; Critical value; Fuzzy P-value; Three-decision rule. 1. Introduction The loss-based process capability index Cpm, sometimes called the Taguchi index, proposed separately by Chan et al. (1988) and Hsiang and Taguchi (1985), was proposed to measure process capability. The index Cpm incorporates with the variation of production items with respect to the target value and the specification limits preset in the factory. It measures the ability of the process and reflects the density of the data about the target *. Corresponding author. Tel.: + 886 7 3814526x7105; fax: +886 7 3923375. E-mail addresses: workman@cc.kuas.edu.tw, mhshu0607@yahoo.com (M.-H. Shu).. 0377-2217/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.02.023. Please cite this article in press as: Hsu, B.-M., Shu, M.-H., Fuzzy inference to assess manufacturing process capability ..., European Journal of Operational Research (2007), doi:10.1016/j.ejor.2007.02.023.

(2) ARTICLE IN PRESS 2. B.-M. Hsu, M.-H. Shu / European Journal of Operational Research xxx (2007) xxx–xxx. Nomenclature C process capability requirement critical value c0 ^ pm calculated from the sample data value of C c* Taguchi index Cpm ^ 0 point estimators of Cpm ^ pm ; C C pm ~ ^ fuzzy estimator of Cpm C pm ^ pm ^cpm point estimate of C d half of the length of the specification interval ~ ~ 1; E ~ 2; . . . ; E ~q E q-dimensional fuzzy subsets vector E ~1; E ~2; . . . ; E ~q e q estimates e1 ; e2 ; . . . ; eq results in q fuzzy subsets E ^ pm fC^ pm ðxÞ probability density function (PDF) of C ^ pm F ^ ðxÞ cumulative distributed function (CDF) of C C pm. GðÞ CDF of the v2 distribution with degree of freedom n 1, v2n1 I A ðÞ indicator function of a classical set A LSL lower specification limit USL upper specification limit T target value l lower bound ~ lZ~ ðbÞ ¼ inf z2Z½b ~ ðzÞ lower bound of the b-cut of the fuzzy number Z ~^ lC~^ ðbÞ lower bound of the b-cut of the fuzzy number C pm pm lP~ ðbÞ lower bound of the b-cut of the fuzzy number P~ w upper bound ~ wZ~ ðbÞ ¼ supz2Z½b ~ ðzÞ upper bound of the b-cut of the fuzzy number Z ~^ wC^~ ðbÞ upper bound of the b-cut of the fuzzy number C pm pm wP~ ðbÞ upper bound of the b-cut of the fuzzy number P~ MLE maximum likelihood estimator m midpoint of the specification limits n sample size NC fraction of nonconformities P-value actual risk of misjudging an incapable process as a capable one PPM parts per million S 2 ; S 2n sample variances s2 ; s2n point estimates of sample variances 2 ~2 ~ S ; S n fuzzy estimators of sample variances UMVUE uniformly minimum variance unbiased estimator X random variable with the normal distribution x value of the random variable X X sample mean x point estimate of the sample mean X~ fuzzy estimator of the sample mean X Y random variable y value of the random variable Y ~ ¼ ½l ~ ðbÞ; w ~ ðbÞ b-cut of the fuzzy number Z~ Z½b Z Z ~^ ~ ^ pm ½b ¼ ½l ~ ðbÞ; w ~ ðbÞ b-cut of the fuzzy number C C pm ^ pm C. ^ pm C. P~ ½b ¼ ½lP~ ðbÞ; wP~ ðbÞ b-cut of the fuzzy number P~ z estimate of the fuzzy number Z~. Please cite this article in press as: Hsu, B.-M., Shu, M.-H., Fuzzy inference to assess manufacturing process capability ..., European Journal of Operational Research (2007), doi:10.1016/j.ejor.2007.02.023.

(3) ARTICLE IN PRESS B.-M. Hsu, M.-H. Shu / European Journal of Operational Research xxx (2007) xxx–xxx. 3. type-I error degree of a estimate belonging to a fuzzy number CDF of the standard normal distribution N(0, 1) PDF of the standard normal distribution N(0, 1) process mean process variance membership function of the fuzzy number Z~ ~ vector membership function of the vector fuzzy number E ~ by Zadeh’s extension principal membership function of the fuzzy number W ~^ membership function of the fuzzy number C pm by Zadeh’s extension principal pm membership function of the fuzzy number P~ by Zadeh’s extension principal g~p ðÞ lZ~ ðzÞ degree of a estimate z belonging to Z~ ~ lE~ ðeÞ degree of a Rq estimate e belonging to E ~ gW~ ðwÞ degree of a estimate w belonging to W ~^ ^ pm belonging to C gC~^ ð^cpm Þ degree of a estimate C pm pm ~ gP~ ðpÞ degree of a estimate p belonging to P. a b UðÞ /ðÞ l r2 lZ~ ðÞ lE~ ðÞ gW~ ðÞ gC~^ ðÞ. (see Hsiang and Taguchi, 1985; Chan et al., 1988; Kotz and Johnson, 1993; Kotz and Lovelace, 1998). The index Cpm is defined as follows: USL LSL C pm ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 6 r2 þ ðl T Þ2. ð1Þ. where USL is the upper specification limit, LSL is the lower specification limit, l is the process mean and r is the process standard deviation, and T is the target value usually set to the midpoint of the specification limits ðT ¼ m ¼ ðUSL þ LSLÞ=2Þ. For stable normal process, X N ðl; r2 Þ, the capability index Cpm is not primarily designed to provide an exact measure on the number of conforming items, i.e., the process yield. But, Cpm 2 considers the process departure ðl T Þ (rather than 6r alone) in the denominator of the definition to reflect 2 2 the degrees of process targeting. We note that r2 þ ðl T Þ ¼ E½ðX T Þ which is the denominator of Cpm. 2 Since E½ðX T Þ is the expected loss, where the loss of a characteristic X missing the target is often assumed 2 to be well approximated by the symmetric squared error loss function, lossðX Þ ¼ kðX T Þ , the capability index Cpm has been referred to as a loss-based index. Govaerts (1994) and Johnson and Kotz (1995) showed that pffiffiffi Þ 1 for C > 3 =3, or the fraction of nonconformities (NC) NC 6 2Uð3C pm Þ for Yield P 2Uð3C pm pm pffiffiffi C pm > 3=3, where UðÞ is the cumulative distributed function (CDF) of the standard normal distribution N(0, 1). Table 1 displays various values of C pm ¼ 1:00; . . . ; 2:00ð0:05Þ (the lower value 1.00 and upper value 2.00 with the values range 0.05), and the corresponding maximum possible nonconformities (in parts per million; PPM). For example, if a process has capability with C pm P 1:2, then the production yield would be at least 99.968% or say the number of the nonconformities is less than 318.2 PPM. Conceptually, in Eq. (1), the process mean l and the process standard deviation r are unknown and need to be estimated using the sample data, thus it is very important to obtain statistical properties of estimator of Cpm Table 1 Various values of Cpm and the maximum possible nonconformities Cpm. PPM. Cpm. PPM. Cpm. PPM. 1.00 1.05 1.10 1.15 1.20 1.25 1.30. 2699.796 1632.705 966.848 560.587 318.217 176.835 96.193. 1.35 1.40 1.45 1.50 1.55 1.60 1.65. 51.218 26.691 13.614 6.795 3.319 1.587 0.742. 1.70 1.75 1.80 1.85 1.90 1.95 2.00. 0.340 0.152 0.067 0.029 0.012 0.005 0.002. Please cite this article in press as: Hsu, B.-M., Shu, M.-H., Fuzzy inference to assess manufacturing process capability ..., European Journal of Operational Research (2007), doi:10.1016/j.ejor.2007.02.023.

(4) ARTICLE IN PRESS 4. B.-M. Hsu, M.-H. Shu / European Journal of Operational Research xxx (2007) xxx–xxx. and do hypothesis testing for assessing the manufacturing process capability. Ordinarily, the underlying manufacturing process data are assumed the output responses of continuous quantities to be precise numbers (Prasad and Calis, 1999; Shiau et al., 1999; Zimmer et al., 2001; Corbett and Pan, 2002; Pearn and Shu, 2003; Xekalaki and Perakis, 2004; Pearn and Wu, 2005; Hsu et al., 2007). In practical situations, however, it is much more realistic in general the output quality characteristics of continuous quantities are more or less non-precise (Filzmoser and Vertl, 2004; Viertl and Hareter, 2004; Sugano, 2006; Gulbay and Kahraman, 2007). The feasible description of such data is by so-called imprecise numbers. Such observations are also called fuzzy. The fuzziness is different from measurement errors and stochastic uncertainty (Filzmoser and Vertl, 2004). It is a feature of single observations from continuous quantities. Errors are described by statistical models and should not be confused with fuzziness. In general fuzziness and errors are superimposed. Some typical examples are data given by color intensity pictures (Sugano, 2006) or readings on the analogue measurement equipment. Other examples of imprecise data are data given by scarce sample data for estimating, observations with coarse scales, the linguistic data transformation into numeric data, and incomplete knowledge with vagueness of definitions (Gulbay and Kahraman, 2007). Also readings on digital measurement equipments are imprecise intervals since there are only a finite number of decimals available. Precise real numbers x0 2 R as well as intervals ½l; w R are uniquely characterized by their indicator functions I fx0 g ðÞ and I ½l;w ðÞ respectively, where the indicator function I A ðÞ of a classical set A is defined by 1 iff x 2 A; I A ðxÞ ¼ 0 iff x 62 A: Often the fuzziness of measurements implies that exact boundaries of interval data are not realistic. Therefore, it is necessary to generalize real numbers and intervals to describe fuzziness. This is done by the concept of imprecise numbers as generalization of real numbers and intervals. Imprecise numbers as well as imprecise subsets of R are described by generalizations of indicator functions, called membership functions (Zadeh, 1965). Hence, the traditional quality measurement proves to be difficult for evaluating the performance of the process directly (Zimmermann, 1991). Filzmoser and Vertl (2004) presented a preliminary approach for statistical testing at the basis of fuzzy values by introducing the fuzzy P-value. Taheri (2003) conducted a thorough review spanning the past 20 years of the development of fuzzy statistics which combines statistical methods and fuzzy set theory. For process capability analysis, Yongting (1996) proposed the concept of fuzzy quality and analysis on fuzzy probability and applied it to the computation of a process capability index, by which a fuzzy process can be controlled. Lee (2001) and Hong (2004) presented Cpk index estimation using fuzzy numbers when the measurement refers to the decision-maker’s subjective determination. As specification limits are fuzzy rather than precise, Parchami and Mashinchi (2005) proposed process capability indices as fuzzy numbers and obtained fuzzy confidence intervals for them. However, the fuzzy decision-making procedure for judging whether or not the process satisfies the preset requirement has been overlooked in the previous studies. The pioneering work of Buckley (2003, 2004, 2005a,b) introduced a new method in fuzzy statistics to estimate mean and variance parameters of normal distribution. The fuzzy situation of the sample mean X and the sample variance S 2n is established by a set of confidence intervals, one on top of the other, and triangular shaped fuzzy numbers of sample mean, X~ , and sample variance, S~2n , are formed. In this paper, we first review the traditional precise estimation of Cpm. Using the approach taken by Buckley (2005a,b) and Buckley and Eslami (2004) with some extensions, we introduce a general method combining the vector of fuzzy numbers X~ and S~2n to produce the membership function of fuzzy estimator of Cpm for further testing process capability. Two useful fuzzy inference criteria, the critical value and the fuzzy P-value, are proposed to assess manufacturing process capability based on Cpm. It turns out that the methodology provides the three-decision rule, accepting the manufacturing process, rejecting it, or recommending it for further study, with a four quadrant decision-making plot to assess process capability on a certain degree of imprecision on the sample data. From a four quadrants decision-making plot, practitioner can simultaneously observe several important features of manufacturing process capability based on Cpm and make a decision. The fuzzy inference for assessing process capability is a natural generalization of the traditional test, when the data are precise the proposed test is reduced to a classical test with a binary decision. A real-world example taken from microelectronics manufacturing process is presented to illustrate the applicability of the proposed approach. Please cite this article in press as: Hsu, B.-M., Shu, M.-H., Fuzzy inference to assess manufacturing process capability ..., European Journal of Operational Research (2007), doi:10.1016/j.ejor.2007.02.023.

(5) ARTICLE IN PRESS B.-M. Hsu, M.-H. Shu / European Journal of Operational Research xxx (2007) xxx–xxx. 5. 2. Estimation of Cpm Let X 1 ; X 2 ; . . . ; X n be a random sample of size n from stable normal process, N ðl; r2 Þ. The index Cpm can be rewritten as the following: d ð2Þ C pm ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 3 r2 þ ðl T Þ2 where d ¼ ðUSL LSLÞ=2 is half of the length of the specification interval. Chan et al. (1988) and Boyles (1991) proposed the following two estimators of Cpm, respectively, d ^ 0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3Þ C pm 2 n 3 S þ n1 ðX T Þ2 and d ^ pm ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi; ð4Þ C 2 2 3 S n þ ðX T Þ P P P ^0 where X ¼ ni¼1 X i =n, S 2 ¼ ni¼1 ðX i X Þ2 =ðn 1Þ and S 2n ¼ ni¼1 ðX i X Þ2 =n. In fact, the two estimators, C pm ^ and C pm , are asymptotically equivalent for sample data taken from a normally distributed population (Pearn et al., 1992). The statistics X and S 2n are the maximum likelihood estimators (MLEs) of l and r2, respectively, ^ pm is also the MLE of Cpm. Furthermore, the term S 2 þ ðX T Þ2 in the denominator therefore the estimator C n 2 ^ estimator (UMVUE) of the term r2 þ ðl T Þ in the of C pm is the uniformly minimum variance unbiased P 2 n 2 2 2 2 2 denominator of Cpm, where S n þ ðX T Þ ¼ i¼1 ðX i T Þ =n and r þ ðl T Þ ¼ E½ðX T Þ . Therefore, ^ pm , Eq. (4), to evaluate process capability with it is reasonable, for reliable purpose, that we use the estimator C precise data. 3. Fuzzy estimation 3.1. Fuzzy set theory In the traditional precise set, the degree of an element belongs to a set is either one or zero. In order to deal with the imprecise data, Zadeh (1965) proposed the fuzzy set theory. Definition. Let Z be a universe of discourse corresponding to an object whose current status is fuzzy, and the status value is characterized by a fuzzy set Z~ in Z. A membership function lZ~ ðzÞ : R ! ½0; 1 is called the membership function of Z~ with the following properties: (a) (b) (c) (d). 0 6 uZ~ ðzÞ 6 1, for 8z 2 R, Z~ is normal, iff sup uZ~ ðzÞ ¼ 1, for 9z 2 R, Z~ is convex subset of R, iff uZ~ ðpz1 þ ð1 pÞz2 Þ P min½uZ~ ðz1 Þ; uZ~ ðz2 Þ; 8z1 ; z2 2 R, for 8p 2 ½0; 1, ~ ¼ fzjuZ~ ðzÞ P bg ¼ ½lZ~ ðbÞ; wZ~ ðbÞ as an b-cut of Z~ which is a closed and bounded for 8b 2 ð0; 1, Z½b ~ and wZ~ ðbÞ ¼ supz2Z½b where lZ~ ðbÞ ¼ inf z2Z½b ~ ðzÞ as the lower bound of the b-cut of Z, ~ ðzÞ as the upper ~ bound of the b-cut of Z.. A triangular shaped fuzzy number has curves, not straight line segment, for the side of the triangle. 3.2. Buckley’s approach for fuzzy estimation We refer to Buckley’s approach (Buckley, 2005a,b) with some modifications to obtain fuzzy numbers for parameter estimation from a set of confidence intervals. Let Y be a random variable with probability density function (PDF) f ðy; hÞ for the single parameter h. In practical situations, h is unknown and it must be estimated from a random sample Y 1 ; Y 2 ; . . . ; Y n . Let pðY 1 ; Y 2 ; . . . ; Y n Þ be a statistic used to estimate h. Given Please cite this article in press as: Hsu, B.-M., Shu, M.-H., Fuzzy inference to assess manufacturing process capability ..., European Journal of Operational Research (2007), doi:10.1016/j.ejor.2007.02.023.

(6) ARTICLE IN PRESS 6. B.-M. Hsu, M.-H. Shu / European Journal of Operational Research xxx (2007) xxx–xxx. the values of these random variables Y i ¼ y i , 1 6 i 6 n, we obtain a point estimate ^h ¼ pðy 1 ; y 2 ; . . . ; y n Þ for h. We would never expect this point estimate be exactly equal h, so a (1 a)100% confidence interval for h is also required to compute. In this confidence interval, one usually sets a equal to 0.025, 0.05, or 0.1. Denote (1 a)100% confidence intervals for h as ½lðaÞ; wðaÞ, for 8a 2 ð0; 1Þ. Add to this the interval ½lð100%Þ; wð100%Þ ¼ ½^ h; ^ h as the 0% confidence intervals for h and the whole parameter space, H, ½lð0Þ; wð0Þ ¼ H as the 100% confidence interval. Now place these confidence intervals, one on top of the other, ~ to produce a triangular shaped fuzzy number ^h whose b-cuts are the confidence intervals. We have ~ ^ ¼ ½l~ ð0Þ; w~ ð0Þ ¼ H; ~h½100% ^ ^ h. ^ In this ^ ¼ ½l~ ðbÞ; w~ ðbÞ, for 8b 2 ð0; 1Þ; ~ h½0 ¼ ½l^h~ ð100%Þ; w^h~ ð100%Þ ¼ ½h; h½b ^ ^ ^ ^ h h h h ~ ^ ^ way we use more information in h than just a point estimate h, or just a single interval estimation. 3.3. Membership function of fuzzy estimations for X and S 2n By the inspiration of Buckley’s approach (Buckley, 2005a,b), we similarly establish the fuzzy situation of sample mean X and sample variance S 2n for sample data taken from normally distributed population. The triangular shaped fuzzy numbers with the b-cut of fuzzy estimator X~ and S~2n are shown as follows: Sn Sn ~ ð5Þ X ½b ¼ ½lX~ ðbÞ; wX~ ðbÞ ¼ X t1b=2;n1 pffiffiffiffiffiffiffiffiffiffiffi ; X þ t1b=2;n1 pffiffiffiffiffiffiffiffiffiffiffi ; for 8b 2 ð0; 1; n1 n1 " # 2 2 nS nS 2 n n ; S~n ½b ¼ ½lS~2n ðbÞ; wS~2n ðbÞ ¼ 2 ; for 8b 2 ð0; 1Þ; ð6Þ v1b=2;n1 v2b=2;n1 where t1b=2;n1 is the 1 b=2 percentile of the t distribution with n 1 degrees of freedom and v2b=2;n1 is the b=2 percentile of the ordinary central v2 with n 1 degrees of freedom. Now place these b-cut intervals, one on top up of the other, to produce triangular shaped fuzzy numbers X~ and S~2n , respectively. Fig. 1 shows the membership function of the fuzzy estimator X~ where x ¼ 5:187 with s2n ¼ 1:281 and n = 50 and Fig. 2 shows the membership function of the fuzzy estimator S~2n where s2n ¼ 1:281 with n = 50. ^ pm is a function of X and S 2 . In order to construct the cuts of fuzzy estimation From Eq. (4), we know that C n for Cpm, we present a general method combining the vector of fuzzy numbers ðX~ ; S~2n Þ to form the membership function of fuzzy estimator of Cpm for further testing process capability. 3.4. Statistic with fuzzy estimators ~ 2; . . . ; E ~ q . A sample of q esti~ 1; E Let us consider q fuzzy estimators. q fuzzy estimators are as fuzzy subsets E ~ 1; E ~ 2; . . . ; E ~ q . In a general setting, these q fuzzy subsets will have mates e1 ; e2 ; . . . ; eq results in q fuzzy subsets E β1 0.9 0.8. μ X∼ ( x ). 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 4.7. 4.8. 4.9. 5. 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. X. Fig. 1. The membership function plot lX~ ðxÞ with x ¼ 5:187 with s2n ¼ 1:281 and n = 50.. Please cite this article in press as: Hsu, B.-M., Shu, M.-H., Fuzzy inference to assess manufacturing process capability ..., European Journal of Operational Research (2007), doi:10.1016/j.ejor.2007.02.023.

(7) ARTICLE IN PRESS B.-M. Hsu, M.-H. Shu / European Journal of Operational Research xxx (2007) xxx–xxx. 7. β1 0.9 0.8. μ S∼n2 ( sn2 ). 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.8. 1. 1.2. 1.4. 1.6. 1.8. 2. 2.2. 2.4. 2.6. S n2. Fig. 2. The membership function plot lS~2n ðs2n Þ with s2n ¼ 1:281 with n = 50.. different membership functions denoted by lE~ 1 ðe1 Þ; lE~ 2 ðe2 Þ; . . . ; lE~ q ðeq Þ. We can combine these fuzzy subsets ~ which is determined by a so-called vector membership function into an q-dimensional fuzzy subsets vector E ~ evaluated a Rq estilE~ ðÞ. lE~ ðeÞ : Rq ! ½0; 1 where lE~ ðeÞ, for 8e 2 Rq , is the vector membership function for E ~ with the following properties: mate e indicating the degree to which e belongs to E (a) (b) (c) (d). 0 6 lE~ ðeÞ 6 1, for 8e 2 Rq , ~ is normal, iff supe2Rq l ~ ðeÞ ¼ 1, E E ~ is convex subset of Rq , iff l ~ ðpx þ ð1 pÞyÞ P min½l ~ ðxÞ; l ~ ðyÞ, 8x; y 2 Rq , for 8p 2 ½0; 1, E E E E ~ ¼ fejl ~ ðeÞ P bg ¼ ½l ~ ðbÞ; w ~ ðbÞ as an b-cut of E ~ which is a closed and bounded for 8b 2 ð0; 1, E½b E E E ~ where lE~ ðbÞ ¼ inf e2E½b ~ ðzÞ as the lower bound of the b-cut of E, and wE ~ ðzÞ as the upper ~ ðbÞ ¼ supe2E½b ~ bound of the b-cut of E.. Filzmoser and Vertl (2004) indicated that one way to combine the membership functions lE~ 1 ðe1 Þ; lE~ 2 ðe2 Þ; . . . ; lE~ q ðeq Þ into a vector membership function lE~ ðeÞ is the minimum combination rule, lE~ ðeÞ ¼ min½lE~ 1 ðe1 Þ; lE~ 2 ðe2 Þ; . . . ; lE~ q ðeq Þ;. for 8e 2 Rq :. ð7Þ. ~ ~ 2; . . . ; E ~ q, ~ 1; E Additionally, the b-cut E½b are Cartesian products of the b-cut of q fuzzy numbers E ~ ¼E ~ 2 ½b E ~ q ½b: ~ 1 ½b E E½b. ð8Þ. q. Given a function f : R ! R, it is possible to defined the membership induced by f on R by the extension principal developed by Zadeh (see Bandemer and Nather, 1992). Thus, supflE~ ðeÞ : f ðeÞ ¼ wg; if f 1 ðwÞ 6¼ 0 ; for 8w 2 R: ð9Þ gW~ ðwÞ ¼ 0; if f 1 ðwÞ ¼ 0 It is well known (Dubois and Prade, 1979, 1987, 2000; Otto et al., 1993) that if f is continuous and monotonic in each variable, and then one need only knows the value of the function at the endpoints of the b-cut for each lE~ i ðei Þ, for i ¼ 1; 2; . . . ; q in order to determine the endpoints for the b-cut interval for gW~ ðwÞ. Therefore, the function gW~ ðwÞ is indeed a membership function whose b-cuts are given as follows: ~ ½b ¼ ½lW~ ðbÞ; wW~ ðbÞ; W. ð10Þ. ~ , and wW~ ðbÞ ¼ supw2W~ ½b ðwÞ as the upper where lW~ ðbÞ ¼ inf w2W~ ½b ðwÞ as the lower bound of the b-cut of W ~ bound of the b-cut of W . ~^ Based on Eqs. (5)–(10), we can perform the calculation to obtain the b-cut of the fuzzy estimator C pm , ~ ~ ^ ^ C pm ½b, then place one on top of the other to construct the membership function of C pm ; gC~^ ð^cpm Þ. pm. Please cite this article in press as: Hsu, B.-M., Shu, M.-H., Fuzzy inference to assess manufacturing process capability ..., European Journal of Operational Research (2007), doi:10.1016/j.ejor.2007.02.023.

(8) ARTICLE IN PRESS 8. B.-M. Hsu, M.-H. Shu / European Journal of Operational Research xxx (2007) xxx–xxx. 3.5. Membership function of fuzzy estimator for Cpm From Eq. (4), choosing g 2 X~ ½b and h 2 S~2n ½b, we obtain d ~ ^ pm ðg; hÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi: C 3 h þ ðg T Þ2. ð11Þ. ~^ ~^ As g and h range through their interval, the b-cut of the fuzzy estimator C pm , C pm ½b, is defined as ~ ~ ^ pm ½b ¼ ½l ~ ðbÞ; w ~ ðbÞ. By Eq. (11), C ^ pm ðg; hÞ is increasing function of g and decreasing function of h, when C ^ pm ^ pm C C X 6 T . Therefore, we obtain lC~^ ðbÞðwC~^ ðbÞÞ by using the left (right) point of the interval X~ ½b in Eq. (5) and pm pm using the right (left) end point of the interval for S~2 ½b in Eq. (6). With similar method for case X > T , we can ~ ~ ^ pm , C ^ pm ½b for 8b 2 ð0; 1Þ as follows: formulate the b-cut of the fuzzy estimator C ~ ^ pm ½b ¼ ½l ~ ðbÞ; w ~ ðbÞ; C ^ ^ C C pm. where. ð12Þ. pm. 8 d lC~^ ðbÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 ; > pm > nS 2n > n ffiT > þ X t1b=2;n1 pSffiffiffiffi 3 2 < n1 v b=2;n1. d > wC~^ ðbÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 ; > pm 2 > nS Sffiffiffiffi > n n ffiT p þ X þt 3 : 1b=2;n1 n1 v2. if X 6 T ;. 1b=2;n1. 8 d lC~^ ðbÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 ; > pm > 2 nS > Sffiffiffiffi n n ffi p > þ X þt1b=2;n1 T 3 < n1 v2 b=2;n1. d > wC~^ ðbÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 ; > pm > 2 Sffiffiffiffi > nS n ffiT p þ X t 3 : 1b=2;n1 n1 v2. if X > T :. 1b=2;n1. ~ ^ pm ½b ¼ ½l ~ ðbÞ; w ~ ðbÞ as Eq. (12) are b-cut, one on top of the other, Proposition 1. The intervals C ^ pm ^ pm C C constructing a membership function gC~^ ð^cpm Þ. pm. β1 0.9. ηCˆ. 0.8. pm. ( cˆpm ). n = 30. 0.7. n = 75. 0.6. n = 150. 0.5 0.4 0.3 0.2 0.1 0. 0.8. 1. 1.2. 1.4. 1.6. 1.8. 2. Cˆ pm. Fig. 3. The membership function plots of ^cpm ¼ 1:143 with LSL ¼ 2:4, USL ¼ 3:4, T ¼ 2:9, x ¼ 2:825 ðx 6 T Þ, sn ¼ 0:125, and n = 30, 75, 150, respectively.. Please cite this article in press as: Hsu, B.-M., Shu, M.-H., Fuzzy inference to assess manufacturing process capability ..., European Journal of Operational Research (2007), doi:10.1016/j.ejor.2007.02.023.

(9) ARTICLE IN PRESS B.-M. Hsu, M.-H. Shu / European Journal of Operational Research xxx (2007) xxx–xxx. 9. β1 0.9 0.8. ηCˆ. pm. ( cˆpm ). n = 30. 0.7. n = 75. 0.6. n = 150. 0.5 0.4 0.3 0.2 0.1 0 0.7. 0.8. 0.9. 1. 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. Cˆ pm. Fig. 4. The membership function plots of ^cpm ¼ 1:221 with, LSL ¼ 3:3, USL ¼ 3:7, T ¼ 3:5, x ¼ 3:510 ðx > T Þ, sn ¼ 0:0537, and n = 30, 75, 150, respectively.. Proof. See Appendix for the proof.. h. ~^ ^ Figs. 3 and 4 show the membership function plots of fuzzy estimator C cpm ¼ 1:143 ~ pm ; gC ^ pm ðC pm Þ, where ^ with LSL ¼ 2:4, USL ¼ 3:4, T ¼ 2:9, x ¼ 2:825 ðx 6 T Þ; sn ¼ 0:125, and n = 30, 75, 150 and ^cpm ¼ 1:221 with, LSL ¼ 3:3, USL ¼ 3:7, T ¼ 3:5; x ¼ 3:510 ðx > T Þ; sn ¼ 0:0537, and n = 30, 75, 150, respectively. 4. Fuzzy inference for process performance 4.1. Hypothesis testing for process performance based on Cpm pffiffiffiffiffiffiffiffiffiffiffiffiffi ^ pm ¼ D=ð3 K þ LÞ, where D ¼ pffiffinffid=r, K ¼ nS 2 =r2 In fact, Eq. (4) can be alternatively expressed as C n 2 v2n1 , L ¼ nðX T Þ =r2 . Applying the integration technique used in Vannman (1995); Pearn and Shu (2003) ^ pm are described in ^ pm . The PDF and CDF of C obtained an exactly explicit form of the PDF and CDF of C 2 terms of a mixture of the v distribution and the normal distribution. 2 pffiffiffi pffiffiffi D2 D 2 t ½/ðt þ n nÞ þ /ðt n nÞ dt; ð13Þ 2 3 9x 9x 0. Z D=ð3xÞ 2 pffiffiffi pffiffiffi D 2 ð14Þ G t ½/ðt þ n nÞ þ /ðt n nÞ dt; F C^ pm ðxÞ ¼ 1 2 9x 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi for x > 0, where D ¼ nd=r ¼ 3C pm nð1 þ n2 Þ, GðÞ is the CDF of the v2 distribution with degree of freedom n 1, v2n1 , and /ðÞ is the PDF of the standard normal distribution N ð0; 1Þ. Boyles (1991) and ^ pm are quite accuShu et al. (2005) noted for the recommended sample sizes of n P 20, the PDF and CDF of C ^ rate approximations if we replace n with n ¼ ðX T Þ=S n . Figs. 5a and 5b show the PDF and CDF plots of ^ pm . C A hypothesis testing problem is regarded as a decision problem where decisions have to be made about the truth of two propositions, the null hypothesis H0 and the alternative H1. To assess process capability and make decisions in manufacturing process capability, the decision rules as [1] critical values, or [2] P-values are needed. fC^ pm ðxÞ ¼. Z. D=ð3xÞ. G. [1] Critical values are used for making decisions in manufacturing process capability testing with designated type-I error a, the risk of misjudging an incapable process (H0: C pm 6 CÞ as a capable one (H1: C pm > CÞ. Please cite this article in press as: Hsu, B.-M., Shu, M.-H., Fuzzy inference to assess manufacturing process capability ..., European Journal of Operational Research (2007), doi:10.1016/j.ejor.2007.02.023.

(10) ARTICLE IN PRESS 10. B.-M. Hsu, M.-H. Shu / European Journal of Operational Research xxx (2007) xxx–xxx f Cˆ. pm. (x ). 4 3.5. n = 80. 3. n = 50. 2.5. n = 20. 2 1.5 1 0.5 0. 0.8. 1. 1.2. 1.4. 1.6. 1.8. 2. Cˆ pm. ^ pm with LSL = 0, USL = 10, T = 5, x ¼ 5:187, s2 ¼ 1:281, ^n ¼ 0:1652, C pm ¼ 1:2, and n = 20, 50, 80, Fig. 5a. The PDF plots of C n respectively.. Cˆ pm 2. 1.8. 1.6. n = 20 n = 50 n = 80. 1.4. 1.2. 1. 0.8 0. 0.1. 0.2. 0.3. 0.4. 0.5. 0.6. FCˆ ( x ) pm. 0.7. 0.8. 0.9. 1. ^ pm with LSL = 0, USL = 10, T = 5, x ¼ 5:187, s2 ¼ 1:281, ^n ¼ 0:1652, C pm ¼ 1:2, and n = 20, 50, 80, Fig. 5b. The CDF plots of C n respectively.. Thus, if ^cpm > c0 then we reject the null hypothesis, and conclude that the process is capable with actual type-I error a. From Eq. (14), the critical value c0 can be obtained by solving the following Eq. (15) ^ pm P c0 jC pm ¼ CÞ ¼ PrðC. Z Cpffiffiffiffiffiffiffiffiffiffiffi nð1þ^ n2 Þ=c0 0. ! pffiffiffi pffiffiffi C 2 nð1 þ ^n2 Þ 2 G t ½/ðt þ ^n nÞ þ /ðt ^n nÞ dt ¼ a: 2 c0. ð15Þ. Using Eq. (15), we can compute the critical value c0 by an algorithm using the bisection search technique included the recursive adaptive Simpson quadrature integration method. (The program is available upon request.) [2] The P-values are used for making decisions in manufacturing process capability testing, which presents the actual risk of misjudging an incapable process (H0: C pm 6 CÞ as a capable one (H1: C pm > CÞ. Thus, if Please cite this article in press as: Hsu, B.-M., Shu, M.-H., Fuzzy inference to assess manufacturing process capability ..., European Journal of Operational Research (2007), doi:10.1016/j.ejor.2007.02.023.

(11) ARTICLE IN PRESS B.-M. Hsu, M.-H. Shu / European Journal of Operational Research xxx (2007) xxx–xxx. 11. p < a then we reject the null hypothesis, and conclude that the process is capable with actual type-I error ^ pm calculated from p (rather than a). From Eq. (14), the P-value corresponding to c , a specific value of C the sample data can be obtained by Eq. (16) that is: ! Z Cpffiffiffiffiffiffiffiffiffiffiffi nð1þ^ n2 Þ=c pffiffiffi pffiffiffi C 2 nð1 þ n^2 Þ 2 ^ P -value ¼ PrðC pm P c jC pm ¼ CÞ ¼ G t ½/ðt þ ^n nÞ þ /ðt ^n nÞ dt: 2 c 0 ð16Þ The P-value is easy to obtain by recursive adaptive Simpson quadrature integration method (The program is available upon request). Both programs require no more than 3 CPU seconds (1.66 GHz), to obtain the critical value c0 and P-value with accuracy up to 104 for all cases investigated. It can be noted that the description in [1,2] which the boundary between meeting and rejecting the quality requirement is too sharp through this way. Neyman and Pearson (1933) and Buckley and Eslami (2004) indicated the need for formulating a three-decision testing problem which (a) accept H0 and reject H1, (b) reject H0 and accept H1, (c) both H0 and H1 are neither accepted nor rejected. A typical example for assessing process capability is accepting the manufacturing process, rejecting it, or recommending it for further study. 4.2. Assessing manufacturing process performance by the critical value decision rule From proposition 1, since the gC~^ ð^cpm Þ is a membership function, all b-cuts are closed and finite interval pm ½lC~^ ðbÞ; wC~^ ðbÞ can be used in terms of estimates and compared with the critical value c0 of the test. The decipm pm sion is made according to a three-decision testing rule: (a) IF wC~^ ðbÞ < c0 THEN accept H0 and reject H1. pm (b) IF lC~^ ðbÞ > c0 THEN reject H0 and accept H1. pm (c) IF lC~^ ðbÞ 6 c0 6 wC~^ ðbÞ THEN both H0 and H1 are neither accepted nor rejected. pm. pm. It can be remarked that IF lC~^ ðbÞ ¼ wC~^ ðbÞ THEN the hypothesis testing is reduced to precise data with a pm pm two-decision testing rule. Testing procedure for critical value to assess process performance Step 1: Decide the definition of ‘‘capable’’ (set the value of C), and the a-risk (normally set to 0.025, 0.05, or 0.1), the chance of incorrectly concluding an incapable process as capable. ~ ^ pm ½b ¼ ½l ~ ðbÞ; w ~ ðbÞ as Eq. (12) from the sample data, one on top of the other, to conStep 2: Place C ^ pm ^ pm C C ~^ cpm Þ is produced. struct the triangular shaped fuzzy number C ~ pm . The membership function gC ^ pm ð^ Step 3: Find the critical value c0 from Eq. (15) given C, a-risk and n. Step 4: Determine b, a certain degree of imprecision on sample data. Step 5: Conclude that the process is capable C pm > C if lC~^ ðbÞ > c0 . Conclude that the process is incapable pm C pm 6 C if wC~^ ðbÞ < c0 . Conclude that the process is neither capable nor incapable, further study is pm needed, if lC~^ ðbÞ 6 c0 6 wC~^ ðbÞ. pm. pm. Figs. 6 and 7 show the PDF fC^ pm ðxÞ plot with x ¼ 68:27; s2n ¼ 16:56; ^n ¼ 0:8036; C ¼ 1:25, and n = 50 and the membership function gC~^ ð^cpm Þ. This is a one-sided test and critical values c0 with significance levels a, ðc0 ; aÞ, pm are (1.43, 0.1), (1.48, 0.05), (1.53, 0.025) obtained from Eq. (15). Based on the knowledge of production engi~^ neers, a particular grade of imprecise on sample data b is set to b ¼ 0:65. The b-cut, C pm ½b, for b ¼ 0:65 forms ~ ^ pm ½0:65 ¼ ½l ~ ð0:65Þ; w ~ ð0:65Þ ¼ ½1:4983; 1:6885 computed from Eq. (12) which is presented in Fig. 7 plot C ^ pm ^ pm C C at the intersection of the horizontal line with the membership function gC~^ ð^cpm Þ. If a-risk is set to 0.05, pm lC~^ ð0:65Þ ¼ 1:4983 > c0 ¼ 1:48, therefore we conclude that the process is capable C pm > 1:25 which means pm the production yield is at least 99.982% or says the number of the nonconformities is less than 176.8 PPM. Please cite this article in press as: Hsu, B.-M., Shu, M.-H., Fuzzy inference to assess manufacturing process capability ..., European Journal of Operational Research (2007), doi:10.1016/j.ejor.2007.02.023.

(12) ARTICLE IN PRESS 12. B.-M. Hsu, M.-H. Shu / European Journal of Operational Research xxx (2007) xxx–xxx f Cˆ. pm. (x). 3.5. 3. ( c0 = 1.43, α = 0.1 ). ( c0 = 1.48, α = 0.05 ). 2.5. ( c0 = 1.53, α = 0.025 ). 2. 1.5. 1. 0.5. 0. 1. 1.2. 1.4. 1.6. 1.8. 2. 2.2. Cˆ pm. ^ pm with x ¼ 68:27, s2 ¼ 16:56, ^n ¼ 0:8036, C ¼ 1:25, n = 50 and various ðc0 ; aÞ. Fig. 6. The PDF plot of C n β. 1. 0.9. Cˆ pm ( 0.65 ) = [ 1.4983,1.6885 ]. 0.8 0.7 0.6 0.5 0.4. ( c0 = 1.43, α = 0.1 ) ( c0 = 1.48, α = 0.05 ) ( c0 = 1.53, α = 0.025 ). 0.3 0.2 0.1 1. 1.2. 1.4. 1.6. 1.8. 2. 2.2. Cˆpm. ~^ ½0:65 ¼ ½1:4983; 1:6885. Fig. 7. The membership function gC~^ ð^cpm Þ with C pm pm. 4.3. Assessing manufacturing process capability by the fuzzy P-value decision rule Additionally, gC~^ ð^cpm Þ is a membership function, all b-cuts are closed and finite interval ½lC~^ ðbÞ; wC~^ ðbÞ. pm pm pm We can use these intervals for defining the corresponding intervals of fuzziness of P~ . Thus the intervals of fuzziness of P~ for a one-sided test are as follows: ~ ^ ^~ P~ ½b ¼ ½lP~ ðbÞ; wP~ ðbÞ ¼ ½PrðC for 8b 2 ð0; 1Þ: ~ ~ pm P wC ^ ðbÞjC pm ¼ CÞ; PrðC pm P lC ^ ðbÞjC pm ¼ CÞ; pm. pm. ð17Þ Proposition 2. The intervals P~ ½b ¼ ½lP~ ðbÞ; wP~ ðbÞ of Eq. (16) are b-cuts, one on top of the other, corresponding to a membership function gP~ ðpÞ. Proof. See Appendix for the proof.. h. In Fig. 8 we show in visual displays how P~ ½b ¼ ½lP~ ðbÞ; wP~ ðbÞ was obtained for b ¼ 0:65 indicated by two ~^ vertical lines. The exact P-value for lP~ ð0:65Þ ¼ PrðC ~ pm P wC ^ ð0:65ÞjC pm ¼ CÞ is shown by the dark area and pm. Please cite this article in press as: Hsu, B.-M., Shu, M.-H., Fuzzy inference to assess manufacturing process capability ..., European Journal of Operational Research (2007), doi:10.1016/j.ejor.2007.02.023.

(13) ARTICLE IN PRESS B.-M. Hsu, M.-H. Shu / European Journal of Operational Research xxx (2007) xxx–xxx fCˆ. pm. 13. ( x). ψP. ( 0.65 ). lP ( 0.65 ). Cˆ pm. β 1 0.9 0.9. lˆ. Cpm. 0.7. ψ Cˆ ( 0.65 ). ( 0.65 ). pm. 0.6 0.5 0.4 0.3 0.2 0.1. Cˆ pm. Fig. 8. P~ ½b ¼ ½lP~ ðbÞ; wP~ ðbÞ for b ¼ 0:65 by two black vertical lines and corresponding areas under the curve of fC^ pm ðxÞ.. ~ ^ pm P l ~ ð0:65ÞjC pm ¼ CÞ is shown by the dark and the light area under the exact P-value for wP~ ð0:65Þ ¼ PrðC ^ pm C curve of fC^ pm ðxÞ in the upper plot of Fig. 8. Both P-values form the b-cut P~ ½b for b ¼ 0:65, for example P~ ½0:65 ¼ ½0:0021; 0:0375, which is presented in the lower plot of Fig. 8 at the intersection of the horizontal line with the vertical lines through the exact P-values. Following this procedure for 8b 2 ð0; 1Þ, one on top of the other, the membership function gP~ ðpÞ can be constructed as Fig. 9. β. 1. 0.9 0.8 0.7. ηP ( p ). 0.6 0.5 0.4 0.3 0.2 0.1 0. 0.1. 0.2. 0.3. 0.4. 0.5. 0.6. 0.7. 0.8. 0.9. P. Fig. 9. The membership function gP~ ðpÞ.. Please cite this article in press as: Hsu, B.-M., Shu, M.-H., Fuzzy inference to assess manufacturing process capability ..., European Journal of Operational Research (2007), doi:10.1016/j.ejor.2007.02.023.

(14) ARTICLE IN PRESS 14. B.-M. Hsu, M.-H. Shu / European Journal of Operational Research xxx (2007) xxx–xxx β. 1. 0.9. P ( 0.65 ) = [ 0.0021,0.0375 ]. 0.8 0.7 0.6 0.5 0.4 0.3 0.2. α = 0.025 α = 0.05 α = 0.1. 0.1 0. 0.1. 0.2. 0.3. 0.4. 0.5. 0.6. 0.7. 0.8. 0.9. 1. P Fig. 10. The membership function gP~ ðpÞ with P~ ½0:65 ¼ ½0:0021; 0:0375 and a ¼ 0:1, 0.05, 0.025.. The decision is made according to a three-decision testing rule: (a) IF lP~ ðbÞ > a THEN accept H0 and reject H1. (b) IF wP~ ðbÞ > a THEN reject H0 and accept H1. (c) IF lP~ ðbÞ 6 a 6 wP~ ðbÞ THEN both H0 and H1 are neither accepted nor rejected. It can be remarked that IF lP~ ðbÞ ¼ wP~ ðbÞ THEN the hypothesis testing is similar to precise data with a twodecision testing rule. Testing procedure for fuzzy P-value to assess process performance Step 1: Decide the definition of ‘‘capable’’ (set the value of C), and the a-risk (normally set to 0.025, 0.05, or 0.1), the chance of wrongly concluding an incapable process as capable. ~ ^ pm ½b ¼ ½l ~ ðbÞ; w ~ ðbÞ as Eq. (12) from the sample data, one on top of the other, to Step 2: Place C ^ pm ^ pm C C ~^ construct the triangular shaped fuzzy number C cpm Þ is produced. ~ pm . The membership function gC ^ pm ð^ ^~ pm P w ~ ðbÞjC pm Step 3: Given C, a-risk and n, compute P-values by placing the P~ ½b ¼ ½PrðC ^ pm C ~ ^ pm P l ~ ðbÞjC pm ¼ CÞ from Eq. (16), one on top of the other, to construct the triangular ¼ CÞ; PrðC ^ C pm shaped fuzzy number P~ . The membership function gP~ ðpÞ is produced. Step 4: Determine b, a certain degree of imprecision on sample data. Step 5: Conclude that the process is capable C pm > C if wP~ ðbÞ < a. Conclude that the process is incapable C pm 6 C if lP~ ðbÞ > a. Conclude that the process is neither capable nor incapable, further study is needed, if lP~ ðbÞ 6 a 6 wP~ ðbÞ. Finally, in Fig. 10, we compare the resulting fuzzy P-value with a significance level a (e.g. a ¼ 0:05Þ which has to be fixed in advance, and conclude that H0 is rejected at the significance level a ¼ 0:05 form the threedecision rule ðwP~ ð0:65Þ ¼ 0:0375 < a ¼ 0:05Þ. 5. The four quadrants decision-making plot Similar to four quadrants plot popularized use in economic analysis, we integrate Figs. 5a, 5b, 6–10 and propose a four quadrant decision-making plot for assessing manufacturing process capability with imprecise data. From this plot, practitioners/engineers can simultaneously visualize several important features of the manufacturing process capability based on Cpm such as central tendency, spread or variability, skewness, Please cite this article in press as: Hsu, B.-M., Shu, M.-H., Fuzzy inference to assess manufacturing process capability ..., European Journal of Operational Research (2007), doi:10.1016/j.ejor.2007.02.023.

(15) ARTICLE IN PRESS B.-M. Hsu, M.-H. Shu / European Journal of Operational Research xxx (2007) xxx–xxx. fCˆ ( x ) pm. Cˆ pm. 3.5. 3. 15. a. b. 2.2. ( c0 = 1.43, α = 0.1 ) 2. ( c0 = 1.48, α = 0.05 ). 2.5. ( c0 = 1.53, α = 0.025 ). 1.8. 2. ( c0 = 1.53, α = 0.025 ) ( c0 = 1.48, α = 0.05 ). 1.6. 1.5. ( c0 = 1.43, α = 0.1 ). 1.4. 1 1.2. 0.5 1. 0 1. 1.2. 1.4. 1.6. 1.8. 2. 0. 2.2. 0.1. 0.2. 0.3. 0.4. Cˆ pm. β. 0.5. FCˆ ( x ) pm. 0.6. 0.7. 0.8. 0.6. 0.7. 0.8. 0.9. 1. β 1. 0.9. d. 1. c. 0.9. Cˆ pm ( 0.65 ) = [ 1.4983,1.6885 ]. 0.8. P ( 0.65 ) = [ 0.0021,0.0375 ]. 0.8. 0.7 0.7 0.6 0.5 0.4. 0.6. ( c0 = 1.43, α = 0.1 ) ( c0 = 1.48, α = 0.05 ) ( c0 = 1.53, α = 0.025 ). 0.5 0.4. 0.3. 0.3. 0.2. 0.2. 0.1. 0.1 1. 1.2. 1.4. 1.6. 1.8. 2. 2.2. α = 0.025 α = 0.05 α = 0.1 0. 0.1. 0.2. Cˆ pm. 0.3. 0.4. 0.5. 0.9. 1. P. ^ pm with x ¼ 68:27, s2 ¼ 16:56, ^n ¼ 0:8036, C ¼ 1:25, and n = 50, respectively. (c) The Fig. 11. (a) and (b) The PDF and CDF plots of C n membership function gC~^ ð^cpm Þ and critical values c0 ¼ 1:53, 1.48, 1.43. (d) The membership function gP~ ðpÞ and a ¼ 0:025, 0.05, 0.1. pm. kurtosis, P-values and critical values for making an immediate decision when the manufacturing process and measurements of products quality have different certain degrees of imprecise data and preset requirements. Fig. 11a in quadrant II (upper left) shows the PDF fC^ pm ðxÞ, Fig. 11b in quadrant I (upper right) shows the CDF F C^ pm ðxÞ, Fig. 11c in quadrant III (lower left) shows the membership function gC~^ ð^cpm Þ, the resulting pm membership function gP~ ðpÞ is presented in Fig. 11d in the quadrant IV (lower right). 6. An application example with the four quadrants decision-making plot We consider a case study for illustration purpose. Application of LEDs (light emitting diodes) is expanding rapidly since high intensity LEDs of wide range of colors have been recently developed and become available, which enabled application of LEDs in a wide variety of areas such as instrument cluster lighting, color displays, automotive backlighting in dashboards and switches, telecommunication indicator and backlighting in telephone and fax backlighting for audio and video equipment, etc. Please cite this article in press as: Hsu, B.-M., Shu, M.-H., Fuzzy inference to assess manufacturing process capability ..., European Journal of Operational Research (2007), doi:10.1016/j.ejor.2007.02.023.

(16) ARTICLE IN PRESS 16. B.-M. Hsu, M.-H. Shu / European Journal of Operational Research xxx (2007) xxx–xxx. With a focus on the critical characteristic, the luminous intensity of LED sources, we examine a particular LED product model, with the upper and the lower specification limits of luminous intensity are set to USL = 90 mcd, LSL = 40 mcd, and the target value is set to T = 65 mcd. The capability requirement in the company was set to C ¼ 1:10 (capable). To test if the process meets the capability (quality) requirement, we must determine whether the manufacturing process meets C pm > 1:10. The a-risk is set to be 0.05. We collected sample data of the luminous intensity from 50 LEDs. The histogram, and normal probability plot and the Shapiro–Wilk test of the 50 observations show the sample data appear to be normal. Thus, we conclude that the sample data can be regarded as taken from a normal process. In engineering applications, randomness is not the only aspect of uncertainty. All light measurements and rating systems till now depend on the perception of the human eye and are therefore subjective (Ryer, 1997). That leads to the fuzziness of the luminous intensity of LED sources data. Since data given by the luminous intensity of a particular LED product somehow degree of imprecision is inevitable, fuzzy inference to assess manufacturing process capability with imprecise data is suggested by pro-. fCˆ. pm. ( x). Cˆ pm 2. 3.5. a. b 1.8. 3. 1.6. 2.5. ( c0 = 1.31, α = 0.05 ). 2. 1.4. ( c0 = 1.31, α = 0.05 ). 1.5. 1.2 1. 1. 0.5. α = 0.05 0.8. 1. 1.2. 1.4. 1.6. 1.8. 2. 0.8. 0. 0.1. 0.2. 0.3. 0.4. Cˆ pm. β. β. 1. 0.9. c. 1. pm. ( x). 0.6. 0.7. 0.8. 0.9. 1. 0.6. 0.7. 0.8. 0.9. 1. d. 0.9. 0.8. 0.7. 0.6. 0.6. 0.5. 0.5. 0.4. 0.4. 0.3. 0.3. 0.2. 0.2. 0.1. 0.1. ( c0 = 1.31, α = 0.05 ) 1. 1.2. 1.4. Cˆ pm. P ( 0.70 ) = [ 0.0035, 0.0385 ]. 0.8. Cˆ pm ( 0.7 ) = [ 1.3257,1.4727 ]. 0.7. 0.8. 0.5. FCˆ. 1.6. 1.8. 2. ( c0 = 1.31, α = 0.05 ) 0. 0.1. 0.2. 0.3. 0.4. 0.5. P. ^ pm with x ¼ 68:27, s2 ¼ 24:56, ^n ¼ 0:6598, C ¼ 1:1, and n = 50, respectively. (c) The Fig. 12. (a) and (b) The PDF and CDF plots of C n membership function gC~^ ð^cpm Þ and critical values c0 ¼ 1:36, 1.31, 1.26. (d) The membership function gP~ ðpÞ and a ¼ 0:025, 0.05, 0.1. pm. Please cite this article in press as: Hsu, B.-M., Shu, M.-H., Fuzzy inference to assess manufacturing process capability ..., European Journal of Operational Research (2007), doi:10.1016/j.ejor.2007.02.023.

(17) ARTICLE IN PRESS B.-M. Hsu, M.-H. Shu / European Journal of Operational Research xxx (2007) xxx–xxx. 17. duction engineers. The sample mean and sample variance are calculated as x ¼ 68:27 and s2n ¼ 24:56. Placing above information to Eq. (4), we further obtain ^cpm ¼ 1:4035. The four quadrants decision-making plot is constructed as Fig. 12a–d. The critical value c0 ¼ 1:31 based on C ¼ 1:10, a-risk = 0.05, and n = 50, respectively. A certain degree of imprecise on sample data is set to b ¼ 0:70 based on the knowledge of production engineers. The ðlC~^ ðbÞ ¼ 1:3257Þ > ðc0 ¼ 1:31Þ from Fig. 12c and ðwP~ ðbÞ ¼ 0:0375Þ < ða ¼ 0:05Þ from Fig. 12d, pm the decision is made that we do have enough information to assure the production yield is at least 99.9133%, and the number of the nonconformities is less than 966.8 PPM. It should be noted that there are many factors such as color, device geometry, alignment of the LED into a text fixture, temperature, etc. that can induce imprecise data of the luminous intensity of LED measures. Due to the complexity of the light measurement, every company has their own limitations in measuring precise LEDs luminous intensity data. The production management department should develop its own measurement model, how well and precise data information could have been utilized or analyzed for each manufacturing process then build their own b-cut determination system that would result in an efficient and effective decision-making. The higher the value of b-cut is chosen, the better understood the manufacturing process and measurements of the products quality. 7. Conclusions In this paper, we contribute the fuzzy inference to assess manufacturing process capability with the situation of imprecise sample data. We introduce a general method combining the vector of fuzzy numbers X~ and S~2n to produce the membership function of fuzzy estimator of Taguchi index for further testing process capability. With the sampling distribution for the precise estimation of Cpm, two useful fuzzy inference criteria, the critical value and the fuzzy P-value, are presented to assess manufacturing process capability based on Cpm. The presented methodology can provide a particular degree of imprecise on the sample data and heads to the three-decision rule with the four quadrants decision-making plot. This proposed methodology is a natural generalization of the traditional test, when the data are precise the proposed test is reduced to the classical process capability test with a binary decision. A real-world application of the luminous intensity of LED sources data is investigated to illustrate the applicability of the proposed approach. Finally, we assume that measurements are taken from normally distributed populations in this research. However, non-normally distributed processes are common in real application. Using fuzzy inference to assess manufacturing process capability process with imprecise data under mild and severe departures from normality would be an interesting issue for further research. Acknowledgments The authors thank the referees, Professor Jatinder (Jeet) Gupta and Cynthia R. Lovelace for their constructive suggestions and comments that resulted in an improved present of our research. Work on this paper was partially funded by National Science Council of Taiwan, ROC under Grant NSC92-2213-E-251-005. Appendix ~ ^ pm ½b ¼ ½l ~ ðbÞ; w ~ ðbÞ as Eq. (12) are b-cut, one on top of the other, constructing Proposition 1. The intervals C ^ pm ^ pm C C a membership function gC~^ ð^cpm Þ. pm. Proof. Since lX~ ðxÞ and lS~2n ðs2n Þ are two membership functions, all b-cuts are closed and finite interval ½lX~ ðbÞ; wX~ ðbÞ and ½lS~2n ðbÞ; wS~2n ðbÞ for 8b 2 ð0; 1, respectively. lðX~ ;S~2 Þ ðx; s2n Þ, for 8ðx; s2n Þ 2 R2 , is the vector memn bership function for ðX~ ; S~2n Þ evaluated a R2 estimate ðx; s2n Þ indicating the degree to which ðx; s2n Þ belongs to ðX~ ; S~2n Þ. The properties are followed as: (a) 0 6 lðX~ ;S~2 Þ ðx; s2n Þ 6 1, for 8ðx; s2n Þ 2 R2 , n (b) supðx;s2n Þ2R2 lðX~ ;S~2 Þ ðx; s2n Þ ¼ 1, so ðX~ ; S~2n Þ is normal, n. Please cite this article in press as: Hsu, B.-M., Shu, M.-H., Fuzzy inference to assess manufacturing process capability ..., European Journal of Operational Research (2007), doi:10.1016/j.ejor.2007.02.023.

(18) ARTICLE IN PRESS 18. B.-M. Hsu, M.-H. Shu / European Journal of Operational Research xxx (2007) xxx–xxx. (c) lðX~ ;S~2 Þ ðpx þ ð1 pÞyÞ P min½lðX~ ;S~2 Þ ðxÞ; lðX~ ;S~2 Þ ðyÞ, 8x; y 2 R2 , for 8p 2 ½0; 1, thus ðX~ ; S~2n Þ is convex subset n. n. n. of R2 , (d) ðX~ ; S~2n Þ½b ¼ fðx; s2n ÞjlðX~ ;S~2 Þ ðx; s2n Þ P bg ¼ ½lðX~ ;S~2 Þ ðbÞ; wðX~ ;S~2 Þ ðbÞ as an b-cut of ðX~ ; S~2n Þ which is a closed and n n n bounded for 8b 2 ð0; 1. The membership functions lX~ ðxÞ and lS~2n ðs2n Þ are combined into a vector membership function lðX~ ;S~2 Þ ðx; s2n Þ n using the minimum combination rule, lðX~ ;S~2 Þ ðx; s2n Þ ¼ min½lX~ ðxÞ; lS~2n ðs2n Þ; n. for 8ðx; s2n Þ 2 R2 :. Additionally, the b-cuts ðX~ ; S~2n Þ½b are Cartesian products of b-cut of two fuzzy numbers X~ , S~2n ðX~ ; S~2n Þ½b ¼ X~ ½b S~2n ½b: Given a function f : R2 ! R, it is possible to defined the membership induced by f on R by the extension principal developed by Zadeh (see Bandemer and Nather, 1992). Thus, 2 2 1 ^ pm Þ ¼ supfmðX~ ;S~2n Þ ðx; sn Þ : f ðx; sn Þ ¼ wg if f ðwÞ 6¼ 0 ; for 8w 2 R: gC~^ ðC pm 0 if f 1 ðwÞ ¼ 0 Since f is continuous and monotonic in each variable, and then one need only knows the value of the function at the endpoints of b-cut for lX~ ðxÞ and lS~2n ðs2n Þ in order to determine the endpoints for b-cut intervals for gC~^ ð^cpm Þ. pm Therefore, the function gC~^ ð^cpm Þ is indeed a membership function whose b-cut are given as follows: pm. ~ ^ pm ½b ¼ ½l ~ ðbÞ; w ~ ðbÞ; C ^ ^ C C pm. pm. ~^ where lC~^ ðbÞ ¼ inf ðx;s2 Þ2ðX~ ;S~2 Þ½b ðwÞ as the lower bound of the b-cut of C ~ pm , and wC ^ pm ðbÞ ¼ supðx;s2 Þ2ðX~ ;S~2 Þ½b ðwÞ as pm n n n n ~ ^ the upper bound of the b-cut of C pm . h Proposition 2. The intervals P~ ½b ¼ ½lP~ ðbÞ; wP~ ðbÞ of Eq. (16) are b-cuts, one on top of the other, corresponding to a membership function g~p ðpÞ. Proof. Since gC~^ ð^cpm Þ is a membership function, all b-cuts are closed and finite interval ½lC~^ ðbÞ; wC~^ ðbÞ. The pm. pm. pm. properties are followed as: (a) (b) (c) (d). 0 6 g~p ðpÞ 6 1, for 8p 2 R, supp2R g~p ðpÞ ¼ 1, so P~ is normal, gP~ ðpp1 þ ð1 pÞp2 Þ P min½gP~ ðp1 Þ; gP~ ðp2 Þ, 8p1 ; p2 2 R, for 8p 2 ½0; 1, thus P~ is convex subset of R, ~ ~^ ^ pm P w ~ ðbÞjC pm ¼ CÞ; PrðC P~ ½b ¼ fpjgP~ ðpÞ P bg ¼ ½lP~ ðbÞ; wP~ ðbÞ ¼ ½PrðC ~ pm P lC ^ pm ^ pm ðbÞjC pm ¼ CÞ as an C ~ ~ b-cut of P which is a closed compact for 8b 2 ð0; 1. The intervals P ½b ¼ ½lP~ ðbÞ; wP~ ðbÞ are b-cuts, one on top of the other, corresponding to a membership function gP~ ðpÞ are proofed. h. Note that lP~ ðbÞ > 0 and wP~ ðbÞ 6 1. Therefore, the P~ ½b ¼ ½lP~ ðbÞ; wP~ ðbÞ can be interpreted in terms of probabilities and compared with the significance level of a of the test. References Boyles, R.A., 1991. The Taguchi capability index. Journal of Quality Technology 23, 17–26. Buckley, J.J., 2003. Fuzzy Probabilities: New Approach and Application. Physica-Verlag, Heidelberg. Buckley, J.J., Eslami, E., 2004. Uncertain probabilities II: The continuous case. Soft Computing 8, 193–199. Buckley, J.J., 2004. Uncertain probabilities III: The continuous case. Soft Computing 8, 200–206. Buckley, J.J., 2005a. Fuzzy statistics: Hypothesis testing. Soft Computing 9, 512–518.. Please cite this article in press as: Hsu, B.-M., Shu, M.-H., Fuzzy inference to assess manufacturing process capability ..., European Journal of Operational Research (2007), doi:10.1016/j.ejor.2007.02.023.

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Kluwer Academic Publishers, Reading, MA.. Please cite this article in press as: Hsu, B.-M., Shu, M.-H., Fuzzy inference to assess manufacturing process capability ..., European Journal of Operational Research (2007), doi:10.1016/j.ejor.2007.02.023.

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