On the sampling distributions of the estimated process loss indices with asymmetric tolerances

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On the Sampling Distributions of the

Estimated Process Loss Indices with

Asymmetric Tolerances

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

Department of Industrial Engineering and Management , Ching Yun University , Taiwan

b

Department of Industrial Engineering and Management , National Chiao Tung University , Taiwan

c

Department of Industrial Engineering and Systems Management , Feng Chia University , Taiwan

Published online: 21 Nov 2007.

To cite this article: Y. C. Chang , W. L. Pearn & Chien-Wei Wu (2007) On the Sampling Distributions of the Estimated Process Loss Indices with Asymmetric Tolerances, Communications in Statistics -Simulation and Computation, 36:6, 1153-1170, DOI: 10.1080/03610910701569168

To link to this article: http://dx.doi.org/10.1080/03610910701569168

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ISSN: 0361-0918 print/1532-4141 online DOI: 10.1080/03610910701569168

Distributions and Applications

On the Sampling Distributions of the Estimated

Process Loss Indices with Asymmetric Tolerances

Y. C. CHANG

_{, W. L. PEARN}

_{, AND CHIEN-WEI WU}

1_{Department of Industrial Engineering and Management,}

Ching Yun University, Taiwan

2_{Department of Industrial Engineering and Management,}

National Chiao Tung University, Taiwan

3_{Department of Industrial Engineering and Systems Management,}

Feng Chia University, Taiwan

Pearn et al. (2006a) proposed a new generalization of expected loss index L

_e to handle processes with both symmetric and asymmetric tolerances. Putting the loss in relative terms, a user needs only to specify the target and the distance from the target at which the product would have zero worth to quantify the performance of a process. The expected loss index L

e may be expressed as L

e= L

ot+ L

pe, which

provides an uncontaminated separation between information concerning the process accuracy and the process precision. In order to apply the theory of testing statistical hypothesis to test whether a process is capable or not under normality assumption, in this paper we first derive explicit form for the cumulative distribution function and the probability density function of the natural estimator of the three indices

L

_ot
L

_pe, and L

_e We have proved that the sampling distributions of
L

_pe and
L

_otmay be expressed as the chi-square distribution and the normal distribution, respectively. And the distribution of
L

_e can be described in terms of a mixture of the chi-square distribution and the normal distribution. Then, we develop a decision-making rule based on the estimated index
L

e. Finally, an example of testing L

e is also presented

for illustrative purpose.

Keywords Asymmetric tolerances; Decision-making rule; Process capability

indices; Process loss indices; Sampling distributions.

Mathematics Subject Classification Primary 62E15; Secondary 62P30.

1. Introduction

Process capability indices (PCIs), including C_p
C_a
C_pk
C_pm, and C_pmk (see Chan et al., 1988; Kane, 1986; Pearn et al., 1992, 1998), are convenient and powerful tools

Received May 28, 2005; Accepted March 23, 2007

Address correspondence to Y. C. Chang, Department of Industrial Engineering and Management, Ching Yun University, 229 Chien-Hsin Road, Jung-Li 320, Taiwan; E-mail: ycchang@cyu.edu.tw

1153

for measuring performance from many different perspectives. Those indices convey critical information regarding whether a process is capable of reproducing items satisfying customers’ requirement. In recent years, PCIs have received substantial research attention in quality assurance and statistical literatures as well. The use of PCIs in industry did not begin in the United States until the early 1980’s. Soon after, this explosion of use expanded into other industries such as automated, semiconductor, and IC assembly manufacturing industries, to measure product qualities that meet specification. Based on analyzing the PCIs, a production department can trace and improve a poor process so that the quality level can be enhanced and the requirements of the customers can be satisfied. These well-known PCIs have been defined respectively as:

C_p= USL− LSL 6
Ca= 1 −  − m d
(1) C_pk= min  USL−  3
− LSL 3 
C_pm= USL− LSL 62+  − T2
(2) C_pmk= min  USL−  32+  − T2
− LSL 32+  − T2 
(3)

where  is the process mean,  is the process standard deviation, USL is the upper specification limit, LSL is the lower specification limit, m= USL + LSL/2 is the mid-point of the specification interval, T is the target value, and d= USL − −LSL/2 is half length of the specification interval.

1.1. Loss Measure with Symmetric Tolerances

Johnson (1992) developed the so-called relative expected loss Lefor symmetric case,

which is defined as the ratio of the expected quadratic loss and the square of the half specification width:

L_e=   −  x− T2 d2  dFx=  − T d 2 +   d 2
(4)

where Fx is the cumulative distribution function (cdf) of the measured characteristic. If we denote the first term − T/d2 _{by L}

ot and the second

term /d2 _{by L}

pe, then Le can be rewritten as Le= Lot+ Lpe. Unfortunately,

the index L_e inconsistently measures process capability in many cases, particularly for processes with asymmetric tolerances, and thus reflects process potential and performance inaccurately.

1.2. Loss Measure with Asymmetric Tolerances

To remedy for this, Pearn et al. (2006a) proposed a modification of expected loss index, which referred to as L

_e, to handle processes with both symmetric and asymmetric tolerances. Regardless of whether the tolerances are symmetric or asymmetric, the new index obtains the minimal value at = T. Additionally, the

half specification width d is substituted by d∗. This generalization of expected loss index may be expressed as follows:

L

_e =  A d∗ 2 +   d∗ 2
(5)

where A= max  − T · d/Du
T−  · d/Dl, Du = USL − T
Dl= T − LSL, and

d∗= min D_u
D_l. Note that L

_e is sensitive to target value T and it obtains larger value when T is away from the mid-point between the upper and the lower specification limits. We denoted A/d∗2 _{by L}

ot
/d∗ 2 _{by L}

pe, and hence L

e =

L

_ot+ L

_pe. Obviously, if the tolerances are symmetric T = m, then L

_pe reduces to the original index Le.

A process is said to have a symmetric tolerance if the target value T is set to be the mid-point of the specification interval (LSL, USL). In general, asymmetric tolerances (T= m) simply reflect that deviations from target are less tolerable in one direction than the other (see Wu and Tang, 1998). Recent research and advances made in this subject are Boyles (1994), Vännman (1997), Jessenberger and Weihs (2000), and the more recent Pearn et al. (2006a,b). Asymmetric tolerances can also arise from a situation where the tolerances are symmetric to begin with, but the process follows a non normal distribution and the data is transformed to achieve approximate normality as shown by Chou et al. (1998).

For statistical inferences problems, in order to develop a successfully decision-making rule based on the estimated index
L

_e to test whether a normally distributed process is capable or not, the cdf of
L

_e is needed. In this article, we first derive explicit forms for the cdf and probability density function (pdf) of the natural estimators of L

_ot, L

_pe, and L

_e when sampling is drawn from a normal distributed data. Those sampling distribution results greatly simplify the complexity on analyzing the statistical properties of the estimated indices. Then, we develop a reliable decision-making rule based on the estimated index
L

_e, which can be used to test whether the process is capable or not.

2. Contour Plots of L

We investigate some effects of the process mean  and the process variance 2_{on L}

e

when the specification tolerances are symmetric or asymmetric. From the inequality L

_e ≥ A/d∗2_{, it is not difficult to show a necessary condition for L}

e ≤ C is T−Dd ∗√_C d ≤  ≤ T + D_ud∗√C d
(6)

where C is a constant. If the tolerances are symmetric, the necessary condition for L

_e ≤ C reduces to

m− d√C≤  ≤ m + d√C (7)

Figures 1 and 2 display the contours of L

_e in the 
 plane for the symmetric case (LSL, T , USL= 20
30
40 and the asymmetric case (LSL, T, USL = 20
35
40, respectively. Contours are shown for L

_e = 011
006
005
004, and 0.03 from top to bottom in each plot.

Figure 1. Contours of L

_e in 
 plane with L

_e = 011, 0.06, 0.05, 0.04, 0.03 (top to bottom) for symmetric case D_u= D_l.

To obtain the estimated value of L

_e, sample data must be collected, and a great degree of uncertainty may be introduced into capability assessments owing to the sampling errors. The approach by simply looking at the calculated values of the estimated indices and then make a conclusion on whether the given process is capable, is highly unreliable since the sampling errors have been ignored. As the use of the capability indices grows more widespread, users are becoming educated and sensitive to the impact of the estimators and their sampling distributions on constructing confidence intervals and performing hypothesis testing.

Figure 2. Contours of L

_e in 
 plane with L

_e = 011, 0.06, 0.05, 0.04, 0.03 (top to bottom) for asymmetric case 3Du= Dl.

3. Sampling Distributions of the Estimated Process Loss Indices

Let X1
X2
  
Xnbe a random sample of size n from a normally distributed process

N
2_{with mean  and standard deviation . To estimate the generalization L}

e,

Pearn et al. (2006a) proposed the natural estimator
L

_e, which is defined as:

L

_e =
A d∗ 2 +  S_n d∗ 2
(8)

where
A= maxX− T · d/D_u
T− X· d/D_l , the mean  is estimated by the sample mean, X=n_i=1X_i/n, and the variance 2_{by S}2

n =

n i=1

X_i− X2/n, the maximum likelihood estimator. By letting
L

_ot =
A/d∗2 and
L

_pe= S_n/d∗2_,

the relationship
L

_e =
L

_ot+
L

_pe may be established.

For the case where the production tolerance is symmetric,
Amay be simplified asX − T. Therefore, the estimator
L

_e reduces to
L_e = n−1d−2·n_i=1X_i− T2_{, the}

natural estimator of Le discussed in Johnson (1992). Consequently, we may view

the estimator
L

_e as a direct extension of
L_e. In the following, we focus on sampling distributions of the estimated process loss indices
L

_pe

L

_ot, and
L

_e.

In attempt to derive the cdf and pdf of
L

_pe,
L

_ot, and
L

_e, we first introduce the following notation: (1) B= nd∗2/2_; (2) K= nS2 n/ 2_{
which is distributed as} 2 n−1;

(3) Z= n1/2_{X− T/
which is distributed as N
1, where = n}1/2_{− T/;}

(4) Y = max2 _d

uZ
−dlZ
where du= d/Du
dl= d/Dl.

After some algebraic manipulations, the following expressions
L

_pe= K/B,
L

_ot = Y/B, and
L

_e = Y + K/B can be established.

3.1. Sampling Distribution of
L

_pe

Theorem 3.1. Let X1
X2
  
Xn be a random sample of size n from a normally

distributed process N
2_{. Then
L}

pe is distributed as 2/nd∗2 times a chi-square

distribution with n− 1 degrees of freedom. And the pdf and cdf of
L

_pe can be expressed respectively as: f
L

pex= nd∗2 2 fK  nd∗2 2 x = BfKBx
(9) F
<sub>L

pe</sub>x= FK  nd∗2 2 x = FKBx
for x >0
(10)

where F_K· and f_K·, respectively, denote the cdf and the pdf of K, which is distributed as 2

n−1.

Proof. Since
L

_pe= K/B, the cdf of
L

_pecan be obtained directly as: F
_L

pex= P
L

pe≤ x  = PK ≤ Bx
for x > 0 (11)

Differentiate (11) with respect to x, the pdf of
L

_pemay be derived.

If the specification tolerance is symmetric (i.e., T = m), then d∗= d and the inconsistency loss index estimator
L

_pereduces to
L_pe. Therefore, the pdf and the cdf of
L

_pe can be simplified, respectively, as:

f
_L pex= nd2 2 fK  nd2 2 x
(12) F
_L pex= FK  nd2 2 x
for x > 0 (13)

Let · and · denote the cdf and the pdf of the standardized normal distribution N0
1, respectively. Then we can express the cdf of Z as F_Zz= z − and the pdf of Z as f_Zz= z − , respectively. The cdf of Y can be written as:

F_Yy= PY ≤ y = PY ≤ y
Z < 0 + PY ≤ y
Z ≥ 0 = P−d−1  √ y≤ Z < 0 + P0 ≤ Z ≤ d−1_u √y = d−1  √ y+  + d−1_u √y−  − 1
for y > 0 (14) Taking the derivative of F_Yywith respect to y to obtain the pdf of Y as:

f_Yy= 1 2√y

d−1 d−1 √y+  + d_u−1d−1_u √y− 
for y > 0 (15) If the tolerance is symmetric, then d_u= d_l= 1, and the corresponding pdf of Y can be simplified as:

f_Yy= 1 2√y

√y+  + √y−  (16)

For symmetric case, since Y = Z2_{reduces to the non central chi-square distribution}

with one degree of freedom and non centrality parameter = n1/2_{− T/. The pdf}

of Y , an alternative form of (16), can be expressed as: f_Yy=   j=0 P_j/2fYjy
y >0
(17) where Y_j is distributed as 2 1+2j
Pj/2= e−/2/2j/j! = PW = j, and W

follows a Poisson distribution with expected value /2, where = 2_.

3.2. Sampling Distribution of
L

_ot

Theorem 3.2. Let X₁
X₂
  
X_n be a random sample of size n from a normally distributed process N
2_{. Then the pdf and the cdf of
L}

ot can be expressed, respectively, as: f
_L

otx= √ B/x 2  d−1 d−1 √Bx+ + d−1_u d−1_u √Bx− 
(18) F
<sub>L

ot</sub>x=   d−1√Bx+ + d_u−1√Bx− − 1
for x > 0
(19)

where · and · are the cdf and the pdf of the standardized normal distribution N0
1, respectively.

Proof. Since
L

_ot = Y/B, the cdf of
L

_ot can be derived easily by (14): F
_L

otx= P
L

ot ≤ x  = PY ≤ Bx = d−1√Bx+ + d_u−1√Bx− − 1
for x > 0 (20) The pdf of
L

_ot follows by differentiate (20) with respect to x.

If the specification tolerance is symmetric, then D_u = D_l= d
d_u = d_l= 1, and the off-target loss index estimator
L

_ot reduces to
L_ot. The pdf and the cdf of
L

_ot therefore can be simplified, respectively, as:

f
_L otx= √ B/x 2  √Bx+ + √Bx− 
(21) F
_L otx=  √ Bx+ + √Bx− − 1
for x > 0 (22) 3.3. Sampling Distribution of
L

e

Theorem 3.3. Let X₁
X₂
  
X_n be a random sample of size n from a normally distributed process N
2_{. Then the pdf and the cdf of
L}

e can be expressed, respectively, as: f
<sub>L

e</sub>x=  1 0  B3_x/t 2 fKBx1− t ×d−1 d−1 √Bxt+ + d_u−1d_u−1√Bxt− dt
(23) F
_L

ex=  Bx 0 F_KBx− y 1 2√y ×d−1 d−1 √y+ + d_u−1d_u−1√y− dy
for x > 0
(24) where f_K· and F_K·, respectively, denote the pdf and the cdf of K, which is distributed as 2

n−1.

Proof. Since
L

_e = Y + K/B, the cdf of
L

_e can be expressed as: F
<sub>L

e</sub>x= P
L

e ≤ 0x  = PY + K ≤ Bx =  0 PK≤ Bx − Y Y = yf_Yydy = Bx 0

F_KBx− yf_Yydy
for x≥ 0 (25)

The last equality is valid since Bx− y ≥ 0 for 0 ≤ y ≤ Bx, and Bx − y < 0 for y > Bx. Thus, F_KBx− y = 0 for y > Bx. Using (15) the distribution function F
<sub>L

e</sub>(x) can be expressed as (24).

On the other hand, since d dx  ux 0 fx
tdt = ux 0  xfx
tdt+ fx
uxu
_x

(see Varberg and Purcell, 1992) and f
<sub>L

e</sub>x=

d

dxF
L

ex

we may obtain the pdf of
L

_e as:

f
<sub>L

e</sub>x=

 Bx 0

Bf_KBx− yf_Yydy+ BF_K0f_YBx Note that K is distributed as 2

n−1, so FK0= 0. By changing variable t = y/Bx in

the above integral, we have y= Bxt and dy = Bx dt. Hence,

f
_L

ex=

 1 0

B2xf_KBx1− tfYBxtdt

Using (15) the pdf f
L

excan be expressed as (23).

For the case when the specification tolerance is symmetric, then Du = Dl= d,

d_u= d_l= 1, and estimator of the expected loss index
L

_e reduces to
L_e. The pdf (23) and the cdf (24) of
L

_e reduce to those of
L_e. Hence, the cdf and the pdf of
L_e can be expressed, respectively, as:

f
_L ex=  1 0  B3_x/t 2 fKBx1− t  √Bxt+ + √Bxt− dt
(26) F
_L ex=  Bx 0 F_KBx− y 1 2√y  √y+ + √y− dy
for x > 0 (27)

Furthermore, for symmetric tolerances since Y follows a non central chi-square distribution with one degree of freedom and non centrality parameter , we can substitute the pdf of Y as expressed in (17) into (25). Hence, the cdf of
L_e can be expressed in an alternative form as:

F
_L ex=   j=0 P_j/2  Bx 0 F_KBx− yf_Y jydy
for x > 0
(28) where Y_j is distributed as 2 1+2j, = 2, and Pj/2= e−/2/2j/j!.

Taking the derivative of F
_L

ex in (28) with respect to x to obtain the pdf of

L_e as: f
_L ex=   j=0 P_j/2  Bx 0 Bf_KBx− yf_Y jydy
for x > 0 (29)

Now by changing variable t= y/Bx in the above integral, we have y = Bxt and dy= Bxdt. Hence, the result follows:

f
_L ex=   j=0 P_j/2  1 0 B2xf_KBx1− tf_Y jBxtdt
for x > 0 (30) Since f_KBx1− t = 2 −n−1/2 n− 1/2Bx1− t n−3/2_e−Bx1−t/2
f_YjBxt= 2 −2j+1/2 2j+ 1/2Bxt 2j−1/2_e−Bxt/2
we have f
_L ex= 2−n/2Bn/2_xn/2−1 n− 1/2   j=0 P_j   2 e−Bx/2Bx/2j 2j+ 1/2  1 0 t2j+1/2−11− tn−1/2−1_dt = B/2n/2xn/2−1 n− 1/2   j=0 P_j   2 P_j  Bx 2 j+ 1 2j+ 1/2 2j+ 1/2n − 1/2 2j+ n/2 = B/2n/2_xn/2−1 j=0 P_j   2 P_j  Bx 2 j+ 1 2j+ n/2
for x > 0
(31) where P_j/2= e−/2/2j_/j!

Figure 3. The pdf of
L

_e with a= −1, b = 3, and n = 10
30
50
100
300 (bottom to top in plot) for 3Du= Dl.

Figure 4. The pdf of
L

_e with a= −05
b = 3, and n = 10
30
50
100
300 (bottom to top in plot) for 3D_u= D_l.

Figure 5. The pdf of
L

e with a= 05
b = 3, and n = 10
30
50
100
300 (bottom to top in

plot) for 3Du= Dl.

Figure 6. The pdf of
L

e with a= 1
b = 3, and n = 10
30
50
100
300 (bottom to top in

plot) for 3Du= Dl.

4. Distribution Plots of

L

We plot the pdf of
L

_e when the underlying process is normal for several selected cases. Figures 3–6 depict the plots of the pdf of
L

_e for four levels of L

_e index value with parameter a set to a=  − T/ = −10 L

_e = 016
a= −05L

_e = 012
a= 05L

_e = 022, and a= 10L

_e = 056, respectively. The asymmetric case is considered by setting 3USL− T = T − LSL
b = d∗/= 3, and sample size n = 10, 30, 50, 100, and 300 from bottom to top in each figure.

From Figures 3–6, we discover that as the value of L

_e increases, the spread of the distribution also increases. For small sample size n= 10 as example, the distributions are skew to the right (have positive skewness) and have large spread. As sample size n increases, the spread decreases and so does the skewness. We also observe that
L

_e is approximately unbiased and bell-shaped for sample size n greater than 50.

5. A Decision-Making Rule for Testing L

Under normality assumption, we proved that the cdf and the pdf of
L

_e can be represented in terms of a mixture of the central chi-square distribution and the normal distribution. Using the index L

_e, the engineers can assess the process performance and monitor the manufacturing processes on routine basis. To obtain an effective decision-making rule, we consider a testing hypothesis with the null hypothesis and the alternative hypothesis, respectively, as

H₀ L

_e ≥ C (incapable) versus H₁ L

_e < C (capable)

The null hypothesis H0 will be rejected if
L

e < c, where the constant c, called

the critical value, is determined so that the significance level of the test is , i.e., P
L

_e< c_ L

_e = C= . The decision-making rule to be used is then stated as follows: for given  (the probability of wrongly reject null hypothesis when it is true) and sample size n, the process will be considered capable if
L

_e < c_ and incapable if
L

_e ≥ c_.

We note that, by setting a=  − T/ and b = d∗/, the indices L

_ot and L

_pe can be rewritten as L

_ot= d_la/b2 _{for a < 0, L}

pe= dua/b2 for a > 0, and
L

pe=

1/b2_{, where d}

u = d/Du
dl= d/Dl . Hence, the value of L

e = L

ot+ L

pe can be

calculated when values of a, b, d_u, and d_lare given. For example, if a
b
d_u
d_l= 1
3
2/3
2 then L

_e = 2 × 1/32_{+ 1/3}2_{= 5/9 If L}

e = C, from L

e = L

ot+ L

pe, we have C= da 2 b2 + 1 b2 for a≤ 0 and C = d_ua2 b2 + 1 b2 for a > 0
then b2₌_d a2+ 1  /C for a≤ 0 and b2₌_d ua2+ 1  /C for a > 0. In addition, we have B= nd∗2/2_{= nb}2_{. Therefore, if L}

e = C, then

B= nda2+ 1/C for a≤ 0 and B = nd_ua2+ 1/C for a > 0 (32) Furthermore, we have the equality = n1/2_{− T/ = n}1/2_{a. From the result}

of Theorem 3.3, we can use the central chi-square distribution and the normal distribution to find the critical value c_ satisfying P
L

_e < c_ L

_e = C= , i.e., F
L

ec=  given L

e = C, or equivalent to  Bc 0 F_KBc_− y 1 2√y  d−1d−1 √y+√na+ d_u−1d_u−1√y−√nady= 
(33) where B is given in (32).

We note that the distribution characteristic parameter a=  − T/ in (33) is usually unknown, which has to be estimated in real applications, naturally by substituting  and 2_{by the sample mean X and the maximum likelihood estimator}

S2 n=

n i=1

X_i− X2/n. To realize the relationship between c and a, we examine

the behavior of c_against a= −3(0.05)3, which covers a wide range of applications with process capability analysis. The results indicate that the critical value reaches its minimum at a= 05 in all cases with accuracy up to 10−3. Figures 7–10 plot the curves of c_ vs. a for some selected cases.

5.1. Making Decision by Critical Value

Tables 1–2 display the critical values c_for 3D_u= D_l, 3D_u= 2D_l, respectively, with C= 005, sample size n = 100, a = −3(0.2)3, and  = 001, 0.05, 0.10. To test if the process meets the capability requirement, we first determine the value of C and , then estimate the index L

_e and parameter a from the collected sample. If the calculated value of
L

_e is smaller than the critical value c 
L

e < c

, then we conclude that the process meets the capability requirementL

_e < C. Otherwise, we do not have sufficient information to conclude whether the process meets the preset capability requirement. In this case, we would believe that L

_e ≥ C (the process is incapable).

Figure 7. Plots of cvs. a for L

e = 011, 3Du= Dl, n= 25
50
75
100
300 (bottom to top

in plot).

Figure 8. Plots of c_vs. a for L

_e = 005, 3D_u= D_l, n= 25
50
75
100
300 (bottom to top in plot).

Figure 9. Plots of c vs. a for L

e = 011, 3Du= 2Dl, n= 25
50
75
100
300 (bottom to

top in plot).

Figure 10. Plots of c_ vs. a for L

_e = 005, 3D_u= 2D_l, n= 25
50
75
100
300 (bottom to top in plot).

Table 1

Critical values c_for C= 005 with n = 100, a= −3(0.2)3, and  = 001
005
010 under 3D_u= D_l a = 001 = 005 = 010 −3.0 0.0431 0.0450 0.0461 −2.8 0.0427 0.0447 0.0458 −2.6 0.0423 0.0444 0.0456 −2.4 0.0418 0.0441 0.0453 −2.2 0.0413 0.0437 0.0450 −2.0 0.0408 0.0433 0.0447 −1.8 0.0402 0.0428 0.0443 −1.6 0.0395 0.0423 0.0439 −1.4 0.0388 0.0418 0.0435 −1.2 0.0380 0.0412 0.0430 −1.0 0.0373 0.0406 0.0425 −0.8 0.0365 0.0401 0.0420 −0.6 0.0358 0.0395 0.0416 −0.4 0.0353 0.0391 0.0412 −0.2 0.0349 0.0388 0.0410 0.0 0.0352 0.0392 0.0414 0.2 0.0334 0.0376 0.0400 0.4 0.0304 0.0351 0.0379 0.6 0.0300 0.0350 0.0379 0.8 0.0313 0.0362 0.0389 1.0 0.0331 0.0376 0.0401 1.2 0.0348 0.0389 0.0412 1.4 0.0363 0.0400 0.0421 1.6 0.0375 0.0410 0.0429 1.8 0.0386 0.0418 0.0435 2.0 0.0396 0.0425 0.0441 2.2 0.0404 0.0431 0.0445 2.4 0.0411 0.0436 0.0449 2.6 0.0417 0.0440 0.0453 2.8 0.0422 0.0444 0.0456 3.0 0.0427 0.0447 0.0459

5.2. Making Decision by p-value

We also can calculate the p-value, i.e., the probability that
L

_e does not exceed the observed index given the values of C, du, dl, a, and sample size n, and then compare

this probability with the significance level . If the estimated index value is l0, then

the p-value can be calculated as:

p-value =  Bl0 0 F_KBl₀− y 1 2√y  d−1 d−1 √y+√na+ d−1_u d−1_u √y−√nady
(34)

Table 2

Critical values c_ for C= 005 with n = 100
a = −3023, and = 001
005
010 under 3Du = 2Dl a = 001 = 005 = 010 −3.0 0.0431 0.0450 0.0461 −2.8 0.0427 0.0447 0.0458 −2.6 0.0423 0.0444 0.0456 −2.4 0.0418 0.0441 0.0453 −2.2 0.0413 0.0437 0.0450 −2.0 0.0407 0.0433 0.0447 −1.8 0.0401 0.0428 0.0443 −1.6 0.0394 0.0423 0.0439 −1.4 0.0387 0.0418 0.0435 −1.2 0.0379 0.0412 0.0430 −1.0 0.0372 0.0406 0.0425 −0.8 0.0364 0.0400 0.0420 −0.6 0.0358 0.0395 0.0416 −0.4 0.0353 0.0391 0.0413 −0.2 0.0350 0.0389 0.0411 0.0 0.0350 0.0390 0.0412 0.2 0.0349 0.0389 0.0411 0.4 0.0344 0.0385 0.0408 0.6 0.0344 0.0385 0.0408 0.8 0.0349 0.0389 0.0411 1.0 0.0357 0.0395 0.0417 1.2 0.0367 0.0403 0.0423 1.4 0.0377 0.0410 0.0429 1.6 0.0386 0.0417 0.0435 1.8 0.0394 0.0423 0.0439 2.0 0.0402 0.0429 0.0444 2.2 0.0408 0.0434 0.0448 2.4 0.0415 0.0438 0.0451 2.6 0.0420 0.0442 0.0454 2.8 0.0424 0.0446 0.0457 3.0 0.0429 0.0449 0.0460

where B is given in (32). The numerical calculations can be easily carried out using the computer software, to integrate the function based on the chi-square distribution and the normal distribution. If the p-value is smaller than , then we conclude that the process meets the capability requirement.

5.3. An Example of Testing L

_e

Due to low-power consumption, high reliability, and high brightness, Light Emitting Diode (LED) lamps have many applications in traffic signals, full-color displays, etc. As an illustrative example, we consider an LED manufacturing process. Suppose a customer has told his LED supplier that, in order to quality for business with his

company, the supplier must demonstrate that his process capability L

_e is less than 0.05. This problem may be formulated as a hypothesis-testing problem:

H0 L

e ≥ 005 (incapable) versus H1 L

e <005 (capable)

In statistical hypothesis testing, rejection of H₀ is always a strong conclusion. The supplier would like to reject H₀, thereby demonstrating that his process is capable. Moreover, he wants to be sure that if the process capability is below 0.05 there will be a high probability of judging the process capable (say, 0.95).

With a focus on the critical characteristic, the luminous intensity of LED sources, we examine a particular LED product model to test whether the production process of LED is capable or not. Historical data based on routine process monitoring shows that the process is under statistical control and the process distribution is shown to be fairly close to the normal distribution. The upper and the lower specification limits of luminous intensity are set to USL= 40 mcd, LSL = 20 mcd, and the target value is set to T = 35 mcd. Some calculations are made to obtain d= 10, Du= 5
Dl= 15, d∗= 5
du= 2
dl = 2/3. A random sample of

size n= 100 is taken, and calculated statistics are X = 3525
S_n= 03125,
A= 05, ˆa = 08, and
L

_e = 00325 Using Table 1 based on n = 100, we obtain c_= 00362 Since the calculated
L

_e ≤ 00362, we may claim that the process is capable at the significant level = 005.

Alternatively, we obtain the p-value = 0.015 via (34). We would conclude that the process meets the capability requirement if  is set to be larger than 0.015. Otherwise, we do not have sufficient information to make a conclusion. We note that in the illustrative example, the estimated off-target loss index
L

_ot = 002 which occupies 61.5% of
L

_e value, and the estimated inconsistency loss index
L

_pe= 00125 which occupies 38.5% of
L

_e value. Obviously, it can be seen that the variability is contributed mainly by the process departure in this case.

6. Conclusions

Pearn et al. (2006a) introduced a new generalization of expected loss index L

_e to handle processes with both symmetric and asymmetric tolerances. The relative expected loss L

_e = L

_ot+ L

_pe, which provides an uncontaminated separation between information concerning the relative inconsistency loss L

_pe and the relative off-target loss L

_ot. In this article, we considered the three indices, and derive the sampling distributions of their natural estimators under normality assumption. In addition, the theory of testing statistical hypothesis was used to determine whether a process is capable or not. For illustrative purpose, we demonstrated the use of derived results by presenting a case study on LED manufacturing process, to evaluate the process performance.

Acknowledgments

The authors are grateful to the anonymous referees, the Associate Editor, and the Editor for their insightful comments and suggestions, which substantially improve the presentation of this article. The research of Dr. Y. C. Chang was supported by the National Science Council of Republic of China (Grant No. NSC 95-2221-E-231-024).

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