Quality yield measure for processes with asymmetric tolerances

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Quality yield measure for processes with asymmetric

tolerances

W. L. Pearn a , P. C. Lin b , Y. C. Chang c & Chien-Wei Wu d a

Department of Industrial Engineering & Management , National Chiao Tung University , Taiwan ROC E-mail:

b

Department of Distribution Management , National Chin-Yi Institute of Technology , Taiwan ROC E-mail:

c

Department of Industrial Engineering & Management , Ching Yun University , Jung-Li, ROC Taiwan E-mail:

d

Department of Industrial Engineering and Systems Management , Feng Chia University , Taiwan ROC E-mail:

Published online: 23 Feb 2007.

To cite this article: W. L. Pearn , P. C. Lin , Y. C. Chang & Chien-Wei Wu (2006) Quality yield measure for processes with asymmetric tolerances, IIE Transactions, 38:8, 619-633, DOI: 10.1080/07408170600692150

To link to this article: http://dx.doi.org/10.1080/07408170600692150

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ISSN: 0740-817X print / 1545-8830 online DOI: 10.1080/07408170600692150

Quality yield measure for processes with asymmetric

tolerances

W. L. PEARN1,∗_{, P. C. LIN}2_{, Y. C. CHANG}3 _{and CHIEN-WEI WU}4

1_{Department of Industrial Engineering & Management, National Chiao Tung University, Taiwan, ROC} E-mail: [email protected]

2_{Department of Distribution Management, National Chin-Yi Institute of Technology, Taiwan, ROC} E-mail: [email protected]

3_{Department of Industrial Engineering & Management, Ching Yun University, Jung-Li, Taiwan, ROC} E-mail: [email protected]

4_{Department of Industrial Engineering and Systems Management, Feng Chia University, Taiwan, ROC} E-mail: [email protected]

Received June 2003 and accepted May 2005

Process capability indices provide numerical measures on whether or not a process is able to produce products that meet prespecified quality targets and are often used by manufacturers to evaluate manufacturing performance. Although process yield is the primary focus of the performance criteria, a formula that combines the yield and the average process loss, called the quality yield index, has been developed. This index, the quality yield, can be viewed as the conventional process yield minus the truncated expected relative process loss within the specifications. Although cases with symmetric tolerances dominate in practical situations, cases with asymmetric tolerances can also occur. In this paper, we generalize the quality yield index for asymmetric tolerances. The generalization technique is justified, and some statistical properties of the estimated generalization are investigated. An application example on high-density light emitting diodes is also presented to illustrate the applicability of the generalization.

1. Introduction

Process capability indices (PCIs) are widely used in man-ufacturing industries, to provide a numerical measure on whether or not a process is capable of producing items that meet a preset quality requirement. Kane (1986) considered the two basic indices Cp and Cpk, and investigated some properties of their estimators. The two basic indices Cpand

Cpkare deﬁned as:

Cp = USL− LSL 6σ , (1) Cpk = min{USL − µ, µ − LSL} 3σ , (2)

where USL and LSL are the upper and lower speciﬁcation limits,µ and σ are the process mean and the standard de-viation of the characteristic, respectively. Since the designs of Cp and Cpkare independent of the target value T, they can fail to account for process loss incurred by a departure from the target value. For this reason, two more advanced indices Cpmand Cpmkwere developed by Chan et al. (1988),

∗_{Corresponding author}

and Pearn et al. (1992), which are deﬁned as:

Cpm = USL− LSL 6σ2+ (µ − T)2, (3) Cpmk = min{USL − µ, µ − LSL} 3σ2_{+ (µ − T)}2 . (4) For more details about PCIs, see the books by Kotz and Johnson (1993) and Kotz and Lovelace (1998) for a good overview of the literature. The recent review paper by Kotz and Johnson (2002) provides a compact survey with in-terpretations and comments on some 170 publications on PCIs, that were published during 1992–2000.

There are three measures that are of considerable interest to us and we will now highlight their properties.

Yield measure: Boyles (1991) noted that Cpk is a yield-based index. Yield, the proportion of conforming items, is a commonly accepted measurement criterion for the pro-cess capability. Suppose that the proportion of conforming items is the primary concern, then the natural measure is the proportion itself called the yield, which we refer to as

Y and is deﬁned as: Y =

USL LSL

dFX(x), (5)

0740-817XC2006 “IIE”

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where FX(x) is the cumulative distribution function of the measured characteristic X . The disadvantage of the yield measure is that it does not distinguish between the prod-ucts that fall inside the speciﬁcation limits. To remedy this disadvantage, the quadratic loss function can be used to dis-tinguish between the products by increasing the penalty as the departure from the target value increases. However, the quadratic loss function does not provide any comparison with the speciﬁcation limits.

Loss measure: To remedy this problem, Johnson (1992)

developed the so-called relative expected loss Le for the symmetric case, which is deﬁned as the ratio of the ex-pected quadratic loss to the square of the half-speciﬁcation width:

Le=

σ2_{+ (µ − T)}2

d2 , (6)

where d= (USL − LSL)/2 is the half-speciﬁcation width. This measure has a direct relationship with Cpm because

Le= (3Cpm)−2. The advantage of Leover Cpmis that the es-timator of the former has better statistical properties than that of the latter, since the former does not involve a re-ciprocal transformation of the process mean and variance. The disadvantage of the Leindex is the difﬁculty in setting a standard for the measure since its value ranges from zero to inﬁnity.

Quality yield measure: Tsui (1997) proposed the quality

yield index to incorporate the average process loss obtained with the conventional yield measure. The quality yield in-dex, which has been referred to as the Q-yield and denoted by Yqis defined as: Yq= USL LSL 1− (x− T) 2 d2 dFX(x). (7) A process is said to have a symmetric tolerance if the target value T is set to be the midpoint of the specification interval [LSL,USL], i.e., T= M = (USL + LSL)/2. Most research in the quality assurance literature is focused on cases in which the manufacturing tolerance is symmetric. Examples include Kane (1986), Chan et al. (1988), Choi and Owen (1990), Boyles (1991), Pearn et al. (1992), Vännman (1995), Vännman and Kotz (1995), and Spiring (1997). Al-though cases with symmetric tolerances are common in practical situations, cases with asymmetric tolerances of-ten occur in manufacturing industries.

In general, asymmetric tolerances simply reﬂect that deviations from the target value are less tolerable in one direction than in the other direction (Boyles, 1994; V¨annman,1997; Wu and Tang, 1998). Asymmetric ances can also arise from a situation in which the toler-ances are symmetric to begin with, but the process follows a non-normal distribution and the data are transformed to achieve approximate normality, as shown by Chou et al. (1998) who have used Johnson’s curves to transform non-normal process data. Unfortunately, there has been

com-paratively little research published on cases with asymmet-ric tolerances. Exceptions include Boyles (1994), V¨annman (1997), Chen (1998), Pearn and Chen (1998), Chen et al. (1999), and Pearn et al. (1999).

In this paper, we consider the quality yield index for pro-cesses with asymmetric tolerances. We consider an asym-metric loss function, and the corresponding truncated worth function to generalize the quality yield index. Com-parisons among the yield, the quality yield, and some pop-ular process capability indices are examined. Distributional properties of the estimated Yqare also investigated. A con-ﬁdence interval for Yqis constructed to estimate the manu-facturing capability. Finally, an application example using the index Yqto assess the manufacturing capability of light emitting diodes is presented to illustrate the applicability of the proposed approach.

2. Quality yield with asymmetric tolerances

Yield is currently deﬁned as the percentage of processed units that pass inspection. Therefore, the yield index Y can be deﬁned mathematically as the expected value of the worth W (X ) where W (x)= 1 for LSL < x < USL and

W (x)= 0 for x ≤ LSL or x ≥ USL, that is, Y = E[W(X)].

The disadvantage of the yield measure is that it does not distinguish the worth of the products that fall inside the speciﬁcation limits, i.e., they are equally good.

Taguchi championed the concept of the process loss (product’s worth) when the quality characteristic departs from the customers’ ideal value T. The cost of a character-istic X missing the target is often assumed to be well ap-proximated by the symmetric squared error loss function (Hsiang and Taguchi, 1985):

L(x)= k(x − T)2, (8)

where k is a positive constant. A product has the maxi-mal worth WT when the corresponding characteristic X has the target value T (Johnson, 1992). Using the loss func-tion given by Equafunc-tion (8), the worth of the product with characteristic X is:

W (x)= WT − k(x − T)2. (9) Therefore, as the deviation of X from T increases, the worth becomes less, eventually becoming zero and then negative. 2.1. Asymmetric loss function

Now, for a process with the manufacturing speciﬁcation (LSL, T, USL), we can redeﬁne W (x)= 0 for x ≤ LSL or x≥USL, and W(x) = WT− k(x − T)2 for LSL <x<

USL. Using W (LSL)= 0, we obtain k = WT/(dl)2, where

dl= T − LSL. On the other hand, using W(USL) = 0, we obtain k= WT/(du)2, where du= USL−T. For the sym-metric case, both of the values of k reduce to WT/(d)2. Without loss of generality, we can set WT= 1. Therefore,

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for a process with a manufacturing speciﬁcation of (LSL,

T, USL), we can deﬁne a general truncated loss function of x as: L(x)=      [(T− x)/dl]2 LSL< x ≤ T, [(x− T)/du]2 _T _{≤ x < USL,} 1 otherwise. (10)

Hence, the corresponding general truncated worth function of x becomes: W (x)=      1− [(T − x)/dl]2 _LSL_{< x ≤ T,} 1− [(x − T)/du]2 T ≤ x < USL, 0 otherwise. (11)

Then, the expected loss Le, deﬁned as E[L(X )], can be expressed as:

Le= ∞

−∞L(x)dFX(x)= 1 + FX(LSL)− FX(USL) + (dl)−2_E[(T_{− X)}2_{|LSL < X ≤ T]P[LSL < X ≤ T]} + (du)−2_E[(X_{− T)}2_{|T ≤ X < USL]P[T ≤ X < USL].}

(12) Figure 1 is a plot of L(x) for a process with an asymmet-ric manufacturing speciﬁcation of (LSL, T, USL)= (10, 40, 50). Figure 2 is a plot of W (x) for a process with an asym-metric tolerance of (LSL, T, USL)= (10, 40, 50). Now, using the worth function, we can distinguish between the product worths of products that fall inside of the speciﬁca-tion limits.

Consider two items x1 and x2 with x1 > T and x2<

T, satisfying the relationship (x1− T)/du= (T − x2)/dl (equal departure ratios). In this case, the worth values

Fig. 1. The plot of L(x), the loss function for an asymmetric spec-iﬁcation (LSL, T, USL)= (10, 40, 50).

Fig. 2. The plot of W (x), the worth function for an asymmetric speciﬁcation (LSL, T, USL)= (10, 40, 50).

given to items x1 and x2 are the same. For example, we note that for the midpoint of the left-hand side tolerance,

x1 = (T + LSL)/2, and the midpoint of the right-hand side tolerance, x2 = (T + USL)/2, the corresponding worth can be calculated as:

W (x1)= 1 − [(T − x1)/dl]2

= 1 − {[T − (T + LSL)/2]/(T − LSL)}2_{= 3/4,}

W (x2)= 1 − [(x2− T)/du]2

= 1 − {[((T + USL)/2) − T]/(USL − T)}2 _{= 3/4.} Obviously, the two points x1and x2have the same departure ratio (relative departure) k= (T − x1)/dl = (x2− T)/du= 1/2. Checking the process loss at x1 and x2, we have that

L(x1)= L(x2)= 1/4 and the equal worth value is 3/4. In fact, 0< W(x) < 1 for LSL < x < USL and W(T) = 1. On the other hand, W (x)= 0 if x falls outside the speciﬁcation limits.

2.2. Quality yield with asymmetric tolerances

Suppose that a process characteristic X follows a distribu-tion with the cumulative distribudistribu-tion funcdistribu-tion FX(x) and the probability density function fX(x). FW(w), the cumula-tive distribution function of W (X ), can be expressed as (see Appendix): FW(w) = 1 + FX(T− dl √ 1− w) − FX(T+ du √ 1− w), 0≤ w ≤ 1. (13)

Particularly, the fraction of nonconforming items, the prob-ability of an item falling outside the speciﬁed tolerance lim-its, can be calculated as:

FW(0)= 1 + FX(LSL)− FX(USL). (14)

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Table 1. Normal distribution with µ = T compared to Yq= 0.5(0.1)0.9 Case 1 Yq(%) µ σ 50 T 3.558 213 60 T 2.782 604 70 T 2.176 123 80 T 1.651 2655 90 T 1.121 61

Hence, fW(w), the probability density function for W(X), can be expressed as:

fW(w) = 1 2√1− w{dlfX(T− dl √ 1− w) + dufX(T+ du √ 1− w)}, 0 < w < 1. (15) The mean value and variance of W (X ) can be calculated as: E[W (X )]= 1 0 wdFW(w), = 1 0 w 2√1− w{dlfX(T− dl √ 1− w) + dufX(T+ du √ 1− w)}dw, (16) E[W (X )]2 = 1 0 w2_dF W(w), = 1 0 w2 2√1− w{dlfX(T− dl √ 1− w) + dufX(T+ du √ 1− w)}dw, (17) Var[W (X )]= E[W(X)]2− E2[W (X )]. (18) Now we can deﬁne the Q-yield as E[W (X )], the expected value of the worth W (X ). The Q-yield will be between zero and one, and can be used as an index of the ability of a process when considering process yield and process loss. The Q-yield index Yqcan be interpreted as the proportion of “perfect” items whereas the yield index Y is the proportion of conforming items. As with the existing process capability indices, the Q-yield index Yqalso has the larger-the-better property.

Table 2. Normal distribution withµ shifted from T to USL by

du/6, du/4, and du/3, respectively, compared to Yq= 0.5(0.1)0.9

Case 2 Case 3 Case 4

Yq (%) µ σ µ σ µ σ 50 T+ du/6 3.593 474 T + du/4 3.551 352 T + du/3 3.465 2255 60 T+ du/6 2.824 0045 T + du/4 2.767 893 T + du/3 2.651 555 70 T+ du/6 2.221 167 T + du/4 2.144 3699 T + du/3 1.981 3995 80 T+ du/6 1.690 9245 T + du/4 1.575 1335 T + du/3 1.316 363 90 T+ du/6 1.111 1475 T + du/4 0.852 496 T + du/3 —

Table 3. Normal distribution withµ shifted from T to LSL by

dl/6, dl/4, and dl/3, respectively, compared to Yq= 0.5(0.1)0.9

Yq (%) µ σ µ σ µ σ 50 T− dl/6 3.440 189 T − dl/4 3.345 944 T − dl/3 3.221 025 60 T− dl/6 2.630 8625 T − dl/4 2.503 9585 T − dl/3 2.326 2755 70 T− dl/6 1.985 113 T − dl/4 1.818 3015 T − dl/3 1.576 054 80 T− dl/6 1.419 7015 T − dl/4 1.216 756 T − dl/3 0.930 123 90 T− dl/6 0.850 78 T− dl/4 0.587 4915 T − dl/3 — This quality yield index differs from the expected relative worth index deﬁned in Johnson (1992) in that it truncates the deviation outside the speciﬁcations. With this trunca-tion, the quality yield index will be between zero and one and, thus it provides a standardized measure. Also, by re-lating it to the yield measure, which is widely accepted in manufacturing industries, it will be better understood and accepted as a capability measure. The advantage of the Yq index over the Leindex is the value of the former goes from zero to one. Similar to the yield index Y , an ideal value of Yq is one, which provides the user a clear guide about the standard. Similarly to the yield Y , the yield index Yq requires no normality assumption.

To illustrate some basic properties of the quality yield Yq compared to a normal distribution for various application cases, we consider the parameter settings listed in Tables 1– 4. For a process with an asymmetric tolerance (LSL, T,

USL)= (3, 0, 4.5) (so that dl= 3, du= 4.5), ﬁve levels of Yq,

0.5(0.1)0.9, are selected in each case. The studied cases are arranged in the following manner. In case 1, we setµ = T and calculated the corresponding σ for each Yq level. In cases 2–4,µ is shifted from T toward USL by du/6, du/4, and du/3, respectively. We then solve for σ in each setting. In cases 5–7,µ is shifted from T toward LSL by dls/6, dl/4, and dl/3, respectively. We then solve for σ in each case. Finally, in cases 8–10, σ is ﬁxed at three levels, 1/3, 1/2, and 1. The corresponding values ofµ in each setting have again been computed.

Figures 3–6 display four selected normally distributed processes, which are N(µ = T, σ ), N(µ = T + du/4, σ),

N(µ = T − dl/4, σ ) and N(µ, σ = 1/2) respectively, with the quadratic loss function and ﬁve levels of quality yield (see cases 1, 3, 6, and 9). The quality yield could be treated

Table 4. Normal distribution withσ ﬁxed in three levels, 1/3, 1/2, and 1, respectively, compared to Yq= 0.5(0.1)0.9

Yq (%) µ σ µ σ µ σ 50 3.164 4764 1/3 3.143 2 0.5 3.076668 1 60 2.826 46245 1/3 2.801 8575 0.5 2.689652 1 70 2.442 1075 1/3 2.413 5035 0.5 2.2613755 1 80 1.984 6635 1/3 1.949 3365 0.5 1.744542 1 90 1.383 308 1/3 1.332 17 0.5 0.960625 1

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Fig. 3. Distribution plots of normal distribution N(µ = T, σ) with the loss function for variousσ.

as the traditional yield minus the truncated expected rel-ative loss within the speciﬁcations to quantify how well a process can reproduce product items to meet customer re-quirements. Whereas yield is the proportion of conforming products, Q-yield can be interpreted as the average degree of products reaching “perfect” or “on target” states. 3. Comparison of yield, Q-yield, and PCIs

To illustrate the basic differences between the yield Y , the quality yield Yq, and the four well-known process capability

Fig. 4. Distribution plots of normal distribution N(µ = T +

du/4, σ ) with the loss function for various σ .

Fig. 5. Distribution plots of normal distribution N(µ = T −

dl/4, σ) with the loss function for various σ.

indices Cp, Cpk, Cpmand Cpmk, we compare values measured on some processes based on the yield Y , quality yield Yq, and the four indices.

3.1. Comparison of Q-yield and yield

Both the Q-yield index and the conventional yield index can be applied to processes with any distribution. The con-ventional yield, however, does not distinguish between the products that fall inside the speciﬁcation tolerance. For ex-ample, if X follows the uniform distribution U(LSL, USL)

Fig. 6. Distribution plots of normal distribution N(µ, σ = 1/2) with the loss function for variousµ.

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with target T where LSL< T < USL, then yield Y = 1.00 and Q-yield Yq = 0.667, respectively. From a manufac-turing perspective all the produced units are good prod-ucts, however, the consumer would consider the process to be of a low quality even though the yield Y= 1.00. To further demonstrate the difference between yield and Q-yield, we consider a set of triangular-distributed pro-cesses with a< x < b and mode c. Table 5 lists the quality yield measure of those triangular-distributed processes with modes c= 11(1)49, (a, b) = (LSL, USL) = (10, 50), and tar-get value T= 30(5)45. For these processes, the yield value given to all processes is Y= 1.00. On the other hand, the Q-yield obtains its maximum of 0.833 (in bold type) not at

Table 5. Comparisons of the Q-yield measure for triangular pro-cesses with mode c= 11(1)49, (LSL, USL) = (10, 50), and T = 30(5)45 c µ T= 30 T= 35 T= 40 T= 45 11 23.667 0.6828 0.6404 0.5979 0.5551 12 24.000 0.6981 0.6557 0.6118 0.5679 13 24.333 0.7129 0.6687 0.6250 0.5821 14 24.667 0.7259 0.6832 0.6388 0.5936 15 25.000 0.7396 0.6952 0.6515 0.6059 16 25.333 0.7517 0.7088 0.6627 0.6179 17 25.667 0.7611 0.7208 0.6760 0.6282 18 26.000 0.7733 0.7322 0.6877 0.6411 19 26.333 0.7834 0.7430 0.6990 0.6524 20 26.667 0.7899 0.7532 0.7100 0.6633 21 27.000 0.7992 0.7630 0.7203 0.6739 22 27.333 0.8073 0.7715 0.7304 0.6809 23 27.667 0.8128 0.7779 0.7399 0.6943 24 28.000 0.8180 0.7880 0.7489 0.7040 25 28.333 0.8229 0.7963 0.7591 0.7133 26 28.667 0.8257 0.8022 0.7665 0.7232 27 29.000 0.8297 0.8067 0.7740 0.7307 28 29.333 0.8314 0.8142 0.7829 0.7355 29 29.667 0.8328 0.8188 0.7887 0.7490 30 30.000 0.8333 0.8219 0.7930 0.7560 31 30.333 0.8328 0.8267 0.8018 0.7645 32 30.667 0.8314 0.8290 0.8080 0.7709 33 31.000 0.8297 0.8325 0.8130 0.7790 34 31.333 0.8257 0.8329 0.8163 0.7853 35 31.667 0.8229 0.8333 0.8222 0.7918 36 32.000 0.8180 0.8329 0.8260 0.7980 37 32.333 0.8128 0.8314 0.8284 0.8043 38 32.667 0.8073 0.8284 0.8314 0.8066 39 33.000 0.7992 0.8259 0.8331 0.8147 40 33.333 0.7899 0.8199 0.8333 0.8203 41 33.667 0.7834 0.8157 0.8326 0.8242 42 34.000 0.7733 0.8071 0.8305 0.8275 43 34.333 0.7611 0.8001 0.8272 0.8308 44 34.667 0.7517 0.7906 0.8226 0.8327 45 35.000 0.7396 0.7796 0.8147 0.8333 46 35.333 0.7259 0.7686 0.8065 0.8321 47 35.667 0.7129 0.7549 0.7958 0.8289 48 36.000 0.6981 0.7409 0.7824 0.8206 49 36.333 0.6828 0.7254 0.7674 0.8085

Fig. 7. Plots of Yqfor processes with c= 11(1)49, and T = 30(5)45

(left to right).

µ = T but at the mode c = T for the triangular-distributed

processes. The plots of Yqversus mode c= 11 (1)49 with

T = 30(5)45 are displayed in Fig. 7. The ﬁgure shows that Yqalways attains its maximum value of 0.833 at the mode, as the target value moves from 30 to 45 in steps of ﬁve.

For a normally distributed process with mean µ and standard deviation σ, we can write X ∼ N(µ, σ2). Using Equation (13), the corresponding cumulative distribution function of W (X ) is:

FW(w) = 1 + ((T − µ − dl √

1− w)/σ)

− ((T − µ + du√1− w)/σ), 0 ≤ w ≤ 1, (19) where is the cumulative distribution function of the stan-dard normal distribution. The corresponding probability density function of W (X ) is:

fW(w) = 1 2σ√1− w{dlφ((T − µ − dl √ 1− w)/σ ) + duφ((T − µ + du √ 1− w)/σ)}, (20) where 0< w < 1, and φ is the probability density function of the standard normal distribution. The corresponding Q-yield, deﬁned as the expected value function of W (X ), therefore, can be expressed as:

Yq= 1 0 w 2σ√1− w{dlφ((T − µ − dl √ 1− w)/σ ) + duφ((T − µ + du √ 1− w)/σ )}dw. (21) Table 6 is a comparison of the Q-yield for normally dis-tributed processes withµ = 10(1)50, σ = 10/3 and 20/3 re-spectively, where (LSL, USL)= (10, 50) and T = 30(5)45. For symmetric cases (T= 30), the maximal Yq occurs at

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Table 6. Comparisons of the Q-yield measure for normal processes withµ = 10(1)50, σ = 10/3, 20/3, (LSL, USL) = (10, 50), and T = 30(5)45 σ = 10/3 σ = 20/3 µ T = 30 T = 35 T = 40 T = 45 T = 30 T = 35 T = 40 T = 45 10 0.119 0.098 0.082 0.071 0.210 0.177 0.153 0.134 11 0.167 0.137 0.116 0.101 0.249 0.210 0.182 0.159 12 0.223 0.184 0.156 0.136 0.290 0.246 0.213 0.187 13 0.286 0.236 0.201 0.175 0.334 0.285 0.247 0.218 14 0.352 0.292 0.250 0.218 0.381 0.326 0.283 0.250 15 0.420 0.350 0.300 0.262 0.428 0.368 0.321 0.284 16 0.487 0.409 0.351 0.307 0.476 0.412 0.360 0.319 17 0.552 0.466 0.401 0.352 0.524 0.456 0.400 0.355 18 0.613 0.521 0.451 0.396 0.572 0.500 0.440 0.392 19 0.670 0.573 0.498 0.439 0.617 0.543 0.480 0.429 20 0.722 0.622 0.543 0.481 0.661 0.586 0.520 0.466 21 0.770 0.669 0.587 0.521 0.702 0.627 0.559 0.502 22 0.812 0.712 0.628 0.559 0.740 0.666 0.597 0.538 23 0.850 0.752 0.667 0.596 0.774 0.703 0.634 0.573 24 0.882 0.789 0.703 0.631 0.804 0.737 0.669 0.606 25 0.910 0.822 0.738 0.664 0.830 0.769 0.702 0.639 26 0.932 0.853 0.770 0.696 0.851 0.797 0.733 0.670 27 0.950 0.880 0.800 0.726 0.868 0.822 0.761 0.699 28 0.962 0.904 0.828 0.755 0.880 0.843 0.787 0.727 29 0.970 0.924 0.853 0.782 0.887 0.860 0.811 0.753 30 0.972 0.942 0.877 0.807 0.890 0.873 0.831 0.777 31 0.970 0.955 0.898 0.831 0.887 0.882 0.849 0.799 32 0.962 0.965 0.916 0.853 0.880 0.887 0.862 0.818 33 0.950 0.970 0.933 0.873 0.868 0.886 0.872 0.835 34 0.932 0.971 0.947 0.892 0.851 0.881 0.878 0.848 35 0.910 0.966 0.958 0.909 0.830 0.870 0.879 0.858 36 0.882 0.956 0.965 0.925 0.804 0.854 0.875 0.865 37 0.850 0.938 0.968 0.938 0.774 0.833 0.866 0.866 38 0.812 0.914 0.966 0.949 0.740 0.806 0.850 0.863 39 0.770 0.881 0.957 0.957 0.702 0.775 0.829 0.854 40 0.722 0.840 0.939 0.961 0.661 0.738 0.802 0.839 41 0.670 0.791 0.910 0.959 0.617 0.697 0.769 0.818 42 0.613 0.734 0.868 0.947 0.572 0.653 0.731 0.790 43 0.552 0.669 0.813 0.923 0.524 0.605 0.687 0.756 44 0.487 0.598 0.744 0.882 0.476 0.555 0.640 0.716 45 0.420 0.520 0.663 0.823 0.428 0.503 0.588 0.671 46 0.352 0.440 0.573 0.743 0.381 0.451 0.534 0.621 47 0.286 0.360 0.477 0.646 0.334 0.399 0.479 0.567 48 0.223 0.283 0.381 0.538 0.290 0.349 0.424 0.512 49 0.167 0.213 0.290 0.426 0.249 0.301 0.370 0.455 50 0.119 0.153 0.210 0.320 0.210 0.256 0.319 0.398

µ = T. However for asymmetric cases (T = 30), the

maxi-mal Yqoccurs not atµ = T, but at a value between the tar-get value T and 30 (the center of the speciﬁcation interval). This is reasonable, because the on-target process (µ=T) has a larger proportion of low-quality products than the process with maximal Yqvalue. For example, let’s compare two pro-cesses A and B withµA= 40, µB = 45, σA = σB = 10/3, and (LSL, T, USL)= (10, 45, 50). In Table 6, we have

Yq= 0.961 for process A and Yq= 0.823 for process B; the result corresponds to the fact that on-target process B has a larger proportion of low-quality products than process A.

As we mentioned earlier, two items with equal departure ratios have equal worth. However, for two processes, A and B, with equal departure ratios (µA− T)/du= (T − µB)/dl and σA = σB, there are not equal average worths for the two samples produced in processes A and B. For example, normally distributed processes A and B withµA = USL,

µB = LSL and σA= σB have an equal yield Y, with the proportions of conforming items being 50%, but the Yq values given to processes A and B are different for asym-metric cases. In fact, Table 6 also displays that the Yqvalue given to process B is less than that given to process A, since the average quality of products coming from process A is better than that coming from process B for cases in which

T > M.

3.2. Comparison of Q-yield and PCIs

Most of the investigations performed on the existing PCIs,

Cp, Cpk, Cpm, and Cpmk, depend heavily on the assumption of a normal variability. If the underlying distributions are non-normal, then the capability calculations are highly un-reliable since the conventional estimator S2_of_σ2_{is sensitive} to departures from normality, and estimators of those in-dices are calculated using S2_{( Somerville and Montgomery,} 1997). Table 7 displays comparisons of the six indices: the yield Y , Q-yield Yq, Cp, Cpk, Cpm, and Cpmk using normal processes for various values ofµ with ﬁxed σ = 20/3, and (LSL, T, USL)= (10, 30, 50). For the symmetric case, all the six indices obtain their maximum atµ = T.

Figures 8 and 9 display plots of Yq, Y againstµ and the four PCIs Cp, Cpk, Cpm, and Cpmk against µ respectively. With a ﬁxedσ = 20/3, and (LSL, T, USL) = (10, 30, 50), µ is varied from 10 to 50 in unit steps to examine the sensitivity of these indices with respect toµ. For the symmetric case, all the six indices attain their maximum atµ = T = 30 as can be easily seen in the plots. However, asµ departs from

T, all (except Cp) decrease, as one may expect.

Tables 8 and 9 are comparisons of the six indices in nor-mal processes for various values ofµ with a fixed σ = 10/3 and (LSL, T, USL)= (10, 40, 50) and a fixed σ = 20/3 and (LSL, T, USL)= (10, 40, 50) respectively. For the asymmet-ric case, with a fixedσ = 10/3 and (LSL, T, USL) = (10, 40, 50), Figs. 10 and 11 display plots of Yq, Y againstµ and four PCIs Cp, Cpk, Cpm, and Cpmk, againstµ respectively. Sim-ilarly, Figs. 12 and 13 display plots of Yq, Y againstµ and the four PCIs Cp, Cpk, Cpmand Cpmk, againstµ respectively. The specification limits are set to (LSL, T, USL)= (10, 40, 50) andσ = 20/3 is fixed. In this setting, µ is varied from 10 to 50 in unit steps to examine the sensitivity of these indices with respect toµ, for processes with asymmetric tolerances. For the asymmetric cases, none (except Yq) among the six indices accurately reflects the process performance. In fact, the Y index only reflects the quantity and not the qual-ity of the conforming items, Cp cannot reflect the shift of the process mean, Cpk being a yield-based index cannot reflect the departure of the process meanµ from the tar-get value T. The index Cpmattains its maximum atµ = T,

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Table 7. Comparisons among the six indices for normal processes with variousµ, ﬁxed σ = 20/3, and (LSL, T, USL) = (10, 30, 50)

µ Y Yq Cp Cpk Cpm Cpmk 10 0.500 0.210 1.000 0.000 0.316 0.000 11 0.560 0.249 1.000 0.050 0.331 0.017 12 0.618 0.290 1.000 0.100 0.347 0.035 13 0.674 0.334 1.000 0.150 0.365 0.055 14 0.726 0.381 1.000 0.200 0.385 0.077 15 0.773 0.428 1.000 0.250 0.406 0.102 16 0.816 0.476 1.000 0.300 0.430 0.129 17 0.853 0.524 1.000 0.350 0.456 0.160 18 0.885 0.572 1.000 0.400 0.486 0.194 19 0.912 0.617 1.000 0.450 0.518 0.233 20 0.933 0.661 1.000 0.500 0.555 0.277 21 0.951 0.702 1.000 0.550 0.595 0.327 22 0.964 0.740 1.000 0.600 0.640 0.384 23 0.974 0.774 1.000 0.650 0.690 0.449 24 0.982 0.804 1.000 0.700 0.743 0.520 25 0.988 0.830 1.000 0.750 0.800 0.600 26 0.992 0.851 1.000 0.800 0.857 0.686 27 0.994 0.868 1.000 0.850 0.912 0.775 28 0.996 0.880 1.000 0.900 0.958 0.862 29 0.997 0.887 1.000 0.950 0.989 0.939 30 0.997 0.890 1.000 1.000 1.000 1.000 31 0.997 0.887 1.000 0.950 0.989 0.939 32 0.996 0.880 1.000 0.900 0.958 0.862 33 0.994 0.868 1.000 0.850 0.912 0.775 34 0.992 0.851 1.000 0.800 0.857 0.686 35 0.988 0.830 1.000 0.750 0.800 0.600 36 0.982 0.804 1.000 0.700 0.743 0.520 37 0.974 0.774 1.000 0.650 0.690 0.449 38 0.964 0.740 1.000 0.600 0.640 0.384 39 0.951 0.702 1.000 0.550 0.595 0.327 40 0.933 0.661 1.000 0.500 0.555 0.277 41 0.912 0.617 1.000 0.450 0.518 0.233 42 0.885 0.572 1.000 0.400 0.486 0.194 43 0.853 0.524 1.000 0.350 0.456 0.160 44 0.816 0.476 1.000 0.300 0.430 0.129 45 0.773 0.428 1.000 0.250 0.406 0.102 46 0.726 0.381 1.000 0.200 0.385 0.077 47 0.674 0.334 1.000 0.150 0.365 0.055 48 0.618 0.290 1.000 0.100 0.347 0.035 49 0.560 0.249 1.000 0.050 0.331 0.017 50 0.500 0.210 1.000 0.000 0.316 0.000

but the corresponding on-target process is not the process with the best average quality (proportion of “perfect” items, when considering both process yield and loss) for asym-metric cases, as we pointed out earlier. The index Cpmk can-not accurately distinguish the average quality of items pro-duced using different processes. For example, although the value of Cpmk is zero for both the A and B processes with

µA = USL, µB= LSL and σA = σB, the average quality (measured by Yq) of items produced by process A is better than that produced by process B for cases where T> M (as mentioned earlier).

Fig. 8. Plots of Y and Yq(top to bottom) versusµ = 10(1)50 for

processes with ﬁxedσ = 20/3, and (LSL, T, USL) = (10, 30, 50).

4. Distributional properties of the estimated Yq

We now investigate some of the distributional properties of an estimator of Yq. A conﬁdence interval for Yqis con-structed. An approximate process performance testing pro-cedure is also investigated.

Fig. 9. Plots of Cp, Cpk, Cpm and Cpmk (top to bottom) ver-sus µ = 10(1)50 for processes with ﬁxed σ = 20/3, (LSL, T,

USL)= (10, 30, 50).

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Table 8. Comparisons among the six indices for normal processes with variousµ, ﬁxed σ = 10/3, and (LSL, T, USL) = (10, 40, 50) µ Y Yq Cp Cpk Cpm Cpmk 10 0.500 0.082 2.000 0.000 0.221 0.000 11 0.618 0.116 2.000 0.100 0.228 0.011 12 0.726 0.156 2.000 0.200 0.236 0.024 13 0.816 0.201 2.000 0.300 0.245 0.037 14 0.885 0.250 2.000 0.400 0.254 0.051 15 0.933 0.300 2.000 0.500 0.264 0.066 16 0.964 0.351 2.000 0.600 0.275 0.083 17 0.982 0.401 2.000 0.700 0.287 0.100 18 0.992 0.451 2.000 0.800 0.300 0.120 19 0.997 0.498 2.000 0.900 0.314 0.141 20 0.999 0.543 2.000 1.000 0.329 0.164 21 1.000 0.587 2.000 1.100 0.346 0.190 22 1.000 0.628 2.000 1.200 0.364 0.219 23 1.000 0.667 2.000 1.300 0.385 0.250 24 1.000 0.703 2.000 1.400 0.408 0.286 25 1.000 0.738 2.000 1.500 0.434 0.325 26 1.000 0.770 2.000 1.600 0.463 0.371 27 1.000 0.800 2.000 1.700 0.497 0.422 28 1.000 0.828 2.000 1.800 0.535 0.482 29 1.000 0.853 2.000 1.900 0.580 0.551 30 1.000 0.877 2.000 2.000 0.633 0.632 31 1.000 0.898 2.000 1.900 0.695 0.660 32 1.000 0.916 2.000 1.800 0.769 0.692 33 1.000 0.933 2.000 1.700 0.860 0.731 34 1.000 0.947 2.000 1.600 0.971 0.777 35 1.000 0.958 2.000 1.500 1.109 0.832 36 1.000 0.965 2.000 1.400 1.281 0.896 37 1.000 0.968 2.000 1.300 1.487 0.966 38 1.000 0.966 2.000 1.200 1.715 1.029 39 1.000 0.957 2.000 1.100 1.916 1.054 40 0.999 0.939 2.000 1.000 2.000 1.000 41 0.997 0.910 2.000 0.900 1.916 0.862 42 0.992 0.868 2.000 0.800 1.715 0.686 43 0.982 0.813 2.000 0.700 1.487 0.520 44 0.964 0.744 2.000 0.600 1.281 0.384 45 0.933 0.663 2.000 0.500 1.109 0.277 46 0.885 0.573 2.000 0.400 0.971 0.194 47 0.816 0.477 2.000 0.300 0.860 0.129 48 0.726 0.381 2.000 0.200 0.769 0.077 49 0.618 0.290 2.000 0.100 0.695 0.035 50 0.500 0.210 2.000 0.000 0.633 0.000

4.1. Estimation of the Q-yield

If the process parametersµ and σ are unknown, then Yq must be estimated from a sample. Let X1, X2, . . . , Xn be a random sample taken from the process, and W1, W2, . . . ,Wn be the corresponding worth. To estimate the Q-yield, we can consider the following estimator:

ˆ Yq= n i=1 Wi n = ¯W. (22)

It is easy to verify that E( ˆYq)= Yq. Therefore, ˆYqis an un-biased estimator of Yqwith Var( ˆYq)= n−1Var(W1). Use of

Table 9. Comparisons among the six indices for normal processes with variousµ, ﬁxed σ = 20/3, and (LSL, T, USL) = (10, 40, 50) µ Y Yq Cp Cpk Cpm Cpmk 10 0.500 0.153 1.000 0.000 0.217 0.000 11 0.560 0.182 1.000 0.050 0.224 0.011 12 0.618 0.213 1.000 0.100 0.232 0.023 13 0.674 0.247 1.000 0.150 0.240 0.036 14 0.726 0.283 1.000 0.200 0.248 0.050 15 0.773 0.321 1.000 0.250 0.258 0.064 16 0.816 0.360 1.000 0.300 0.268 0.080 17 0.853 0.400 1.000 0.350 0.278 0.097 18 0.885 0.440 1.000 0.400 0.290 0.116 19 0.912 0.480 1.000 0.450 0.303 0.136 20 0.933 0.520 1.000 0.500 0.316 0.158 21 0.951 0.559 1.000 0.550 0.331 0.182 22 0.964 0.597 1.000 0.600 0.347 0.208 23 0.974 0.634 1.000 0.650 0.365 0.237 24 0.982 0.669 1.000 0.700 0.385 0.269 25 0.988 0.702 1.000 0.750 0.406 0.305 26 0.992 0.733 1.000 0.800 0.430 0.344 27 0.994 0.761 1.000 0.850 0.456 0.388 28 0.996 0.787 1.000 0.900 0.486 0.437 29 0.997 0.811 1.000 0.950 0.518 0.492 30 0.997 0.831 1.000 1.000 0.555 0.555 31 0.997 0.849 1.000 0.950 0.595 0.566 32 0.996 0.862 1.000 0.900 0.640 0.576 33 0.994 0.872 1.000 0.850 0.690 0.587 34 0.992 0.878 1.000 0.800 0.743 0.595 35 0.988 0.879 1.000 0.750 0.800 0.600 36 0.982 0.875 1.000 0.700 0.857 0.600 37 0.974 0.866 1.000 0.650 0.912 0.593 38 0.964 0.850 1.000 0.600 0.958 0.575 39 0.951 0.829 1.000 0.550 0.989 0.544 40 0.933 0.802 1.000 0.500 1.000 0.500 41 0.912 0.769 1.000 0.450 0.989 0.445 42 0.885 0.731 1.000 0.400 0.958 0.383 43 0.853 0.687 1.000 0.350 0.912 0.319 44 0.816 0.640 1.000 0.300 0.857 0.257 45 0.773 0.588 1.000 0.250 0.800 0.200 46 0.726 0.534 1.000 0.200 0.743 0.149 47 0.674 0.479 1.000 0.150 0.690 0.104 48 0.618 0.424 1.000 0.100 0.640 0.064 49 0.560 0.370 1.000 0.050 0.595 0.030 50 0.500 0.319 1.000 0.000 0.555 0.000

the unbiased estimator ˆYqdoes not require any knowledge of the process distribution. However, if the distribution of the characteristic X is given with cumulative distribution function FX, then the cumulative distribution function of the corresponding worth FWcan be calculated, and the cu-mulative distribution function of ˆYqcan be expressed as the

n-fold convolution of FW: FYˆq(y)= P( ˆYq≤ y) = P Wi≤ ny = G(ny), (23) where G is the n-fold convolution of FW. The complexity of the cumulative distribution function of ˆYq comes from

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Fig. 10. Plots of Y (top) and Yq(bottom) versusµ = 10(1)50, for

processes withσ = 10/3, (LSL, T, USL) = (10, 40, 50).

the truncation property of the worth function. There is no analytic closed-form solution for FYˆq(y). However, for a

large sample size n, we can show that: √

n( ˆYq− Yq)

S → N(0,1), (24)

Fig. 11. Plots of Cp(top), Cpk(left), Cpm (right) and Cpmk (bot-tom) versusµ = 10(1)50, for processes with σ = 10/3, (LSL, T,

USL)= (10, 40, 50).

Fig. 12. Plots of Y (top) and Yq(bottom) versusµ = 10(1)50, for processes withσ = 20/3, and (LSL, T, USL) = (10, 40, 50).

where the sample variance S2₌_(W

i− ¯W )2/(n − 1). Consequently, an approximate (1− α)100% conﬁdence in-terval of Yqcan be established as:

( ˆYq− z1−α/2S/ √

n, ˆYq+ z1−α/2S/ √

n), (25) where z1−α/2 is the (1− α/2) quantile value of the stan-dard normal distribution N(0, 1). We note that a lower

Fig. 13. Plots of Cp(top), Cpk(left), Cpm(right) and Cpmk(bottom) versusµ = 10(1)50, σ = 20/3, (LSL, T, USL) = (10, 40, 50).

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(1− α)100% confidence limit can be obtained from the lower (one-sided) confidence limit. If the calculated lower confidence limit is greater than the predetermined index value, then we would judge that the process is capable. Oth-erwise, the process is considered to be incapable, and some quality improvement activities must be initiated.

4.2. Distribution plot of the Q-yield estimator

Monte Carlo simulations were performed to investigate the behavior of the sampling distribution of the estimated Yq, for several selected cases, where the underlying process dis-tributions are normal, skewed, or heavy tailed. A true value of the quality yield Yq= 0.6 is picked, with the underlying process distributions being set to:

1. A normal distribution N(µ, σ2) with probability density function:

f (x)= (√2πσ)−1exp[−(x − µ)2/2σ2], (26) with meanµ and variance σ2_{, for}_{−∞ < x < ∞.} 2. A lognormal distribution LN(µ, σ2_{) with probability}

density function:

f (x)= (x√2πσ )−1exp[−(ln x − µ)2/2σ2], (27) with mean exp(µ + σ2_{/2) and variance exp(2µ + 2σ}2₎₋ exp(2µ + σ2_{), for x}_{> 0.}

3. A Student’s t distribution tk with degree of freedom k, where the probability density function is:

f (x)= [((k + 1)/2)/ (k/2)](√kπ)−1

× (1 + x2_/k)−(k+1)/2_{, −∞ < x < ∞, (28)} with mean µ = 0, for k > 1 and variance σ2 = k/ (k− 2), for k > 2.

4. A chi-square distributionχ2

k with degree of freedom k, where the probability density function is:

f (x)= [1/ (k/2)](1/2)k/2xk/2−1e−x/2, x > 0, (29)

with meanµ = k and variance σ2 _{= 2k, k = 1, 2, . . . .} 5. A Weibull distribution W (α, β) with probability density

function:

f (x)= αβxβ−1exp(−αxβ), (30) with mean µ = α−1/β(1 + β−1) and variance σ2₌

α−2/β_[_{(1 + 2β}−1₎₋2₍₁_{+ β}−1_)]_{, for x > 0.}

We randomly generated N= 10 000 samples of sizes

n= 25, 50, 75, 100, 150, 200, 250, and 300 for each

dis-tribution and then calculated the estimate value of Yqfor each sample. Figures 14–21 plot the distribution of ˆYqfor the eight levels of sample size with Yq= 0.6, respectively. In each figure, five underlying process distributions includ-ing normal, lognormal, Student’s t, chi-square, and Weibull are drawn with fixed sample size in order to investigate how the sample size affects the distribution of ˆYq. From those

Fig. 14. Distribution plots of ˆYqfor N(µ, σ2),χ_k2, tk, LN(µ, σ2),

W (α, β) (bottom to top) with n = 25.

plots, one may observe that for a moderate sample size n (about 100) the distributions of the estimated Q-yield index all appear quite close to normal. Therefore, for practical purposes, normal approximations may be used for capabil-ity testing of Yq.

5. An application example

We consider a case study for illustration purpose. The use of Light Emitting Diodes (LEDs) has rapidly expanded

Fig. 15. Distribution plots of ˆYqfor N(µ, σ2),χk2, tk, LN(µ, σ2),

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since the development of high-intensity LEDs with a wide range of colors that has led to their application in a wide variety of areas. LEDs are considerably different from lamps in terms of their physical size, flux level, spec-trum, and spatial intensity distribution. LED technology provides a number of benefits over incandescent bulbs. Some benefits of LEDs for instrument cluster lighting are:

1. LEDs have a lower power consumption: a LED instru-ment cluster uses approximately 1/5 of the electrical cur-rent of an incandescent instrument cluster.

2. LEDs generate less heat: interior thermal measure-ments within the instrument cluster case indicate that the LED design operates 10–15◦C cooler than an incandescent light design. Interior thermal measure-ments within the cavity airspace indicate that the LED

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design operates 25–50◦C cooler than an incandescent design.

3. LEDs provide an equivalent or better lighting: some comparative performances are that red LEDs are 3x brighter and amber LEDs are 2x brighter.

4. LEDs have a better reliability: LEDs are capable of with-standing high degrees of mechanical shock and vibration without failure. LEDs are capable of withstanding over 1000 temperature cycles between 40/100◦C, without failing.

5. LEDs allow for smaller telltales: since LEDs are avail-able in sizes less than 1/8in diameter, LED telltales can be placed on spacing of 0.25–0.30, if desired.

6. LEDs are dimmable using a potentiometer: LEDs are normally wired in series with a current limiting resistor. In general, LEDs can be dimmed with a single poten-tiometer, as long as all series strings use the same num-ber of LEDs. LEDs can also be dimmed through pulse width modulation. In this case, the number of lamps in each series string is not critical.

7. LEDs provide direct cost savings: potentially, LEDs al-low for less expensive drive circuits. LEDs operate at lower currents (20 mA instead of 255 mA). Also, LEDs do not have a high inrush current when ﬁrst turned on. In general, LEDs outperformed the incandescent bulbs for all gauge colors.

With a focus on a critical characteristic of the luminous intensity of LED sources, we examine a particular LED product model, with the upper and lower speciﬁcation lim-its of luminous intensity being set to USL= 90 mcd and

LSL= 40 mcd with the target value being set to T = 60

mcd. We note that this is an asymmetric tolerances case. The LED is said to be defective if the characteristic data does not fall within the specification limits (LSL, USL). To make the use of the methodology more convenient and ac-celerate the computation, an integrated S-PLUS computer program was developed (available from the authors) to cal-culate the lower confidence bounds. We only need to input the manufacturing specification limits, USL, LSL, target value T, and the collected sample data of size n. Then the estimated values ˆY , ˆYqand the lower confidence bounds of

ˆ

Yqcan be obtained easily. Thus, whether or not the process is capable may be determined.

A total of 150 observations were collected from a sta-ble process in the factory, which are displayed in Fig. 22. Figure 23 is a histogram of the sample data. From Fig. 23, it is evident from the density line that the underlying pro-cess distribution is far from normal. Refering to the dis-tribution plots of the Q-yield estimator, a random sample of size n= 150 seems to be large enough to apply the nor-mal approximation approach to the capability testing of

Yq. Proceeding with the calculations by running the in-tegrated S-PLUS program with a 95% conﬁdence level,

Fig. 22. A sample of observations of size n= 150.

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Fig. 23. Histogram of the sample data.

we obtain ˆYq= 0.8082 and the corresponding lower conﬁ-dence bound as 0.7768. We note that the estimated ˆYqindex value is about 0.81. In fact, all 150 observations fall within the speciﬁcation interval (LSL, USL) so that the sample es-timator of yield is ˆY= 1. From the producer’s point of view,

the proportion of conforming products is 100%. However, to quantify how well a process can meet customer require-ments a lower confidence bound of ˆYq of approximately 0.78 can be interpreted as a degree of satisfaction with the products, of at least 78%, with a 95% confidence level. From the corresponding lower confidence bound on Yq, 0.7768, an example of capability testing is that if the Q-yield re-quirement preprint on the contract Yq is set to 0.78, we may only conclude that the process is marginally capable, with a 95% confidence level.

6. Conclusions

In this paper, we ﬁrst reviewed the Q-yield Yq, proposed by Tsui (1997) for processes with symmetric tolerances. We then used the worth function to generalize the concept of the Q-yield to processes with asymmetric tolerances. The anal-ysis and comparisons showed that the new generalization incorporates the asymmetry of the manufacturing tolerance (with an asymmetric loss function), which reﬂects process performance more accurately. We also proposed an unbi-ased estimator of Yqto access the ability of the considered process, which does not require the assumption of normal variability. Some Monte Carlo simulations were conducted to investigate the behavior of the sampling distribution of the estimated Yq. The result showed that for moderate sam-ple size n of no greater than 300 the distributions of the es-timated Yqall appear to be normal. Therefore, normal ap-proximation may be used to perform the capability testing.

Acknowledgements

We would like to express our sincere appreciation to the referees for their constructive comments and carefully read-ings on earlier versions that improved this paper. The third author thanks National Science Council of the Republic of China for funding the project (NSC 94-2213-E-231-017).

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Appendix

Suppose a process characteristic X follows a distribution with the cumulative distribution function FX(x) and the probability density function fX(x). The fraction of noncon-forming items, i.e., the probability of an item falling outside speciﬁed tolerance limits, can be derived as:

FW(0)= P[W(X) = 0],

= P[X ≤ LSL] + P[X ≥ USL],

= FX(LSL)+ 1 − FX(USL). (A1) For the case where w > 0, the cumulative distribution function of W (X ), can be obtained as:

FW(w) = P[W(X) ≤ w], = P[W(X) = 0] + P[(0 < W(X) ≤ w) ∩ (LSL < X ≤ T)] + P[(0 < W(X) ≤ w) ∩ (T ≤ X < USL)], = P[W(X) = 0] + P[(0 < 1 − [(T − X)/dl]2 _{≤ w)} ∩(LSL < X ≤ T)] + P[(0 < 1 − [(X − T)/du]2 _{≤ w)} ∩(T ≤ X < USL)], = P[W(X) = 0] + P 0< dl2− (T − X)2≤ dl2w ∩(LSL < X ≤ T)+ P 0< du2− (X − T)2 ≤ du2w ∩(T ≤ X < USL), = P[W(X) = 0] + P[(d2 l(1− w) ≤ (T − X) 2_{< d}2 l) ∩(LSL < X ≤T)+ P d_u2(1− w) ≤ (X − T)2< d_u2) ∩(T ≤ X < USL), = P[W(X) = 0] + P[(dl√1− w ≤ (T − X) < dl) ∩(LSL < X ≤ T)] + P[(du√1− w≤ (X − T)<du) ∩(T ≤ X < USL)], = P[W(X) = 0] + P[LSL < X ≤ T − dl√1− w] + P[T + du√1− w ≤ X < USL], = [FX(LSL)+ 1 − FX(USL)]+ [FX(T− dl √ 1− w) −FX(LSL)]+ [FX(USL)− FX(T+ du √ 1− w)], = 1 + FX(T− dl √ 1− w) − FX(T+ du √ 1− w), 0≤ w ≤ 1. (A2) Biographies

W. L. Pearn is a Professor in the operations research and quality as-surance group of the Department of Industrial Engineering and Man-agement, National Chiao Tung University, Taiwan. His research areas include process capability analysis, network optimization, queuing ser-vice management, applied statistics, and semiconductor manufacturing scheduling.

P. C. Lin is a Professor in the Department of Distribution Management, National Chin-Yi Institute of Technology, Taichung, Taiwan. He received his M.S. degree in Statistics from the National Chung Hsing University, and his Ph.D. degree in Quality Management from the National Chiao Tung University, Taiwan.

Y. C. Chang received his Ph.D. degree in Industrial Engineering and Management from National Chiao Tung University, Taiwan. Currently, he is an Assistant Professor in the quality management and operations research group of the Department of Industrial Engineering and Man-agement, Ching Yun University,Taiwan. His research interests include process capability analysis and queuing systems management.

Chien-Wei Wu is an Assistant Professor in the Department of Industrial Engineering and Systems Management, Feng Chia University, Taiwan. He received his Ph.D. degree in Industrial Engineering and Management from the National Chiao Tung University and the M.S. degree in Statistics from the National Tsing Hua University, Taiwan. His research interests include statistical quality control, process capability analysis and data analysis.