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8/22/2019 2002 An extension of Banerjee and Rahims model for economic
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Stochastics and Statistics
An extension of Banerjee and Rahims model for economic
design of moving average control chart for a
continuous flow process
Yun-Shiow Chen *, Yit-Ming Yang
Department of Industrial Engineering and Management, Yuan-Ze University, 135 Yuan-Tung Road, Nei-Li 32026, Taiwan, ROCReceived 14 June 2000; accepted 22 August 2001
Abstract
In this paper, we propose a model of a moving average control chart (MA control chart) with a Weibull failure
mechanism from an economic viewpoint. When the process-failure mechanism follows a Weibull model or other models
having increasing hazard rates, it is desirable to have the decreasing sampling interval with the age of the system. The
MA control chart is used to monitor quality characteristics of raw material or products in a continuous process. A cost
model utilizing a variable scheme instead of fixed sampling lengths in a continuous flow process is studied in this re-
search. The variable sampling scheme is used to maintain a constant integrated hazard rate over each sampling interval.
Optimal values for the design parameter, the moving subgroup size, the sampling interval, and the control limit co-efficient are determined by minimizing the loss-cost model. The performance of the loss cost with various Weibull
parameters is studied. A sensitivity analysis shows that the design parameters and loss cost depend on the model pa-
rameters and shift amounts.
2002 Elsevier Science B.V. All rights reserved.
Keywords: Economic design; Moving average control charts; Weibull shock model; Continuous flow process
1. Introduction
Duncan (1956) was the first researcher to designan xx-control chart from an economic viewpoint,
and the methodology he used was optimal for
solving the design parameters. The three design
parameters, the subgroup size (n), the sampling
interval h, and the control limit width (L), con-
sidered in the model were selected to optimally
minimize the average cost of a process. Duncan
(1956) assumed that the process had only a singleassignable cause and that the occurrence time of
an assignable cause was exponentially distributed.
This assumption has been widely used in subse-
quent work on the subject. Chiu and Wetherill
(1974) developed a simple approximate procedure
to optimize Duncans model, Taylor (1968) pre-
sented an economic design model for Cusum
charts, Saniga (1977) considered the joint eco-
nomic design of XX and R charts, Duncan (1971)
developed an economic design for the cases in
European Journal of Operational Research 143 (2002) 600610
www.elsevier.com/locate/dsw
* Corresponding author. Tel.: +886-3-463-8800; fax: +886-3-
463-8907.
E-mail address: ieyschen@saturn.yzu.edu.tw (Y.-S. Chen).
0377-2217/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved.
PII: S0 3 7 7 -2 2 1 7 (0 1 )0 0 3 4 1 -1
http://mail%20to:%20ieyschen@saturn.yzu.edu.tw/http://mail%20to:%20ieyschen@saturn.yzu.edu.tw/8/22/2019 2002 An extension of Banerjee and Rahims model for economic
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which multiple assignable causes, and Ho and
Case (1994) proposed the economically based
EWMA control charts. Banerjee and Rahim
(1987, 1988) studied a non-Markovian model, andthey applied a renewal theory to establish an
economic model when the process failure mecha-
nism follows a Weibull distribution. Banerjee and
Rahim (1988) incorporated Lorenzen and Vances
cost structure into a random sampling interval (cf.
Lorenzen and Vance, 1986). Rahim and Banerjee
(1993) extended Banerjee and Rahims model to
allow the possibility of age-dependent replacement
before failure when such a replacement yields
economic benefits (cf. Banerjee and Rahim (1988)).
Parkhideh and Case (1989) developed a six deci-
sion variables in their economic model for XX-
control chart. Ohta and Rahim (1997) proposed an
alternative and simplified design methodology that
reduces the number of six design variables to three.
Rahim and Costa (2000) presented the joint eco-
nomic design of XXR control charts when the oc-
currence time of assignable causes follow Weibull
distribution. Koo and Case (1990) first proposed
an economic design of XX-control charts for use in
monitoring a continuous flow process for the ex-
ponential process failure mechanism. Chen and
Yang (1999) extended Koo and Cases model tocases following a Weibull shock model (cf. Koo
and Case, 1990). All of the above work is either
focused on a piece part process for different kinds
of control charts or on a continuous flow process
for XX charts; little attention has been focused on
the moving average control chart (MA control
chart). The MA control chart is widely used in
continuous flow processes; for example, a chemical
plant may collect data, periodically, on the results
of analysis made to determine the percentages of
certain chemical constituents.In this paper, we adopt Banerjee and Rahims
cost structure to develop an economic design of an
MA control chart under a Weibull shock model in
a continuous flow process.
2. Definitions and assumptions
The moving average can be determined from a
time series of individual data values by finding the
mean of the first n consecutive values, then drop-
ping the oldest value and subsequently adding one
sample value from a new sample to form the suc-
cessive averages. The features of the model con-sidered in this article are as follows:
(1) The time that the process remains in an in-
control state follows a Weibull distribution. The
p.d.f. is given by
ft kttt1ektt ; t> 0; t=1; k > 0; 1where k is the scale parameter and t is the shape
parameter.
The Weibull hazard rate is derived as follows in
Eq. (2):
rt kttt
1
; t> 0; t=1: 2(2) The process is monitored by drawing a
random sample of size 1 at times, h1, h1 h2,h1 h2 h3; . . . ; let Wj
Pji1 hi for j 1; 2; . . .
W0 0, where j is the number of samples of size 1taken from the process. Fig. 1 depicts the sampling
scheme in a continuous flow process for the MA
chart. For non-uniform sampling, the optimal
sampling interval hj is chosen such that the prob-
ability of a process shift in any interval, given no
shift until the start of the interval, is a constant
for all intervals. According to this definition, theintegrated hazard rate over each sampling interval
is a constant, that is,ZWj1Wj
rtdtZh1
0
rtdt: 3
The hj and Wj can easily be obtained as follows:
hj bj1=t j 11=tch1; 4and
Wj
j1=t
h1:
5
The hj j 1; 2; . . . is a function of h1 andpossesses the following two properties: (I) h1P
h2 ; and (II) limm!1Pm
j1 hj 1.(3) The time to sample and chart one item is
negligible, and production ceases during the sear-
ches and repair.
(4) The process is normally distributed and
characterized by an in-control state. An assignable
cause occurring at random by a magnitude d re-
sults in a shift in the mean from u0 to either u0 dr
Y.-S. Chen, Y.-M. Yang / European Journal of Operational Research 143 (2002) 600610 601
8/22/2019 2002 An extension of Banerjee and Rahims model for economic
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or u0 dr, where u0, r, and d are, respectively, theprocess mean, the process standard deviation, and
the shift parameter.
Define Pj j 1; 2; . . . as the conditionalprobability that an assignable cause will occur
during the sampling interval hj, given that the
process is in an in-control state at time Wj
1, that is,
Pj R
Wj
Wj1ftdtR1
Wj1ftdt 1 e
kht1 : 6
The Pjs are the same for j 1; 2; . . . For conve-nience, let Pj P0 for j 1; 2; . . .
Define qj (j 1; 2; . . .) as the unconditionalprobability that an assignable cause occurs during
the sampling interval hj, and
qj Zwj
wj1ftdt 1 P0j1P0 for j 1; 2; . . .
73. Model components and cycle length
A complete description of the cost structure
model was presented by Lorenzen and Vance
(1986). The time factors in our model involve the
following terms:
1. Average search time per false alarm, denoted as
Z0.
2. Average time to discover the assignable cause
once an assignable cause detected, denoted as Z1.
3. Average time to repair the assignable cause
once the assignable cause discovered, denoted
as Z2.
Cost factors in the model are considered as fol-
lows:
1. Average quality cost per unit time when the
process is in-control, denoted as D0.
2. Average cost per false alarm when the process is
in-control, denoted as Y.
3. Average quality cost per unit time when the
process is out of control, denoted as D1.
4. Cost to locate and repair the assignable cause,
denoted as w.
5. The fixed sample cost and cost per unit sam-
pled, denoted as a and b, respectively.
A complete average cycle length is illustrated in
Fig. 2. The objective is to find the optimal design
parameters, including moving subgroup size (n),
the first sampling interval h1, and control limitcoefficient (L), which can minimize the loss cost for
the given cost and time parameters.
Since the time for occurrence of an assignable
cause follows a Weibull distribution, the average
process in-control time (denoted as AVGICT) is
Fig. 1. Sampling scheme and plotting of an MA control chart for a continuous flow process.
602 Y.-S. Chen, Y.-M. Yang / European Journal of Operational Research 143 (2002) 600610
8/22/2019 2002 An extension of Banerjee and Rahims model for economic
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AVGICT X1j0
ZWj1Wj
tftdt
1k
1=tC 1
1
t
; 8
where Cy is the gamma function and yP 1.Define sj (j 1; 2; . . .) as the expected duration
of the in-control period within the sampling in-terval hj, given that the shock occurred during
sampling interval, that is,
sj RWj
Wj1t Wj1ftdt
qj: 9
Thus, the expected in-control time during a sam-
pling interval in which the transition from an in-
control state to an out-of-control state occurred is
obtained using Eq. (10) as follows:
s X1j1
sjqj
1k
1=tC 1
1
t
h1P01 P0A1 P0;
10where Ax P1l0 l 11=txl, jxj < 1.
When the process is out of control, the process
mean will shift to u0 dr. Suppose that the shiftoccurs in the (j 1)th sampling interval, and let
the mean of this moving subgroup be u. Let Pjidenote the probability that the process mean is
shifted in the sampling interval hj1 and that theassignable cause is detected at the subsequent ith
moving subgroup. Then, j i g is the numberof samples before the assignable cause is detected.
Thus, Qji 1 Pji will be the probability that theassignable cause will not be detected in the fol-
lowing moving subgroups. Three different situa-tions for u and Pji, can be obtained as follows:
u u0 idg r if g< n;u0 idn r if gP n and i < n;u0 dr if gP n and iP n;
8>: 11
and
Pji
1 U L idffiffig
p
U L idffiffig
p
if g< n;
1
U L
idffiffinp U L
idffiffinp if gP n and i < n;
1 UL d ffiffiffinp UL d ffiffiffinp if gP n and iP n;
8>>>>>>>>>>>>>>>>>>>:
12
where UX is the CDF of the standard normaldistribution and L is the control limit. However,
when gP n and iP n in Eq. (12), Pji is independent
of j or i, let Pji P and Qji Q 1 P.From the three situations mentioned in Eq.
(12), the average time from a shift occurring in the
Fig. 2. A complete average cycle length.
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(j 1)th sampling interval to the assignable causedetected in a later subgroup is derived and denoted
as Ejt. Ejt consists of two parts, the averagetimes for the assignable cause to be detected in thesubsequent ith sample for i < n (denoted as Ptj1)and for iP n (denoted as Pt
j
2), i.e., Ejt Ptj1 Ptj2.Thus,
Ptj
1 Wj1 WjPj1
Xn1i2
Wij WjPjiYi1l1
Qjl
!
h1 jh
11=t j1=tiPj1
h1 Xn1i2
ih j1=t j1=tiPji Yi1l1
Qjl !
;
13
Ptj2
X1in
Wij Wj Yn1
l1Qjl
!QinP
h1Yn1l1
Qjl
!An
j;Q j1=t 11 Q
;
14
where Ak;x P1m0 m k1=txm for jxj < 1.Let Et be the expected time from an occur-
rence of a shift in the subgroup to the detection of
the assignable cause, and let it be formulated as
follows:
Et X1j0
Ejtqj1
X1
j0Ptj1 Ptj21 P0jP0: 15
Combining Eqs. (10), (13)(15), we obtain the
expected time to detect the assignable cause after
the process shift, denoted as AVGOOCT. Thus,
AVGOOCT Et s: 16The expected number of false alarms (ENF)
depends on Type I error a and the number ofsampled moving subgroups before an assignable
cause occurred. The ENF can be obtained as fol-
lows:
ENF aX1j1
jqj
1
! a 1 P
0
P0
; 17
where a 21 UL.The average time to search false alarms is
Z0ENF. If we let ATENF Z0ENF, then the ex-pected length of a cycle, ET, can be obtainedfrom Eqs. (9)(17). The result is as follows:
ET AVGICT AVGOOCT Z1 Z2 ATENF
Et h1P01 P0A1 P0
Z1
Z2
Z0a
1 P0
P0 :
18
The expected total cost incurred during a cycle
consists of the following cost components:
1. The expected cost to search false alarms is C1,
and C1 YENF.2. The expected quality cost for in-control process
is C2, and C2 D0AVGICT.3. The expected quality cost for out-of-control
process is C3, and C3 D1AVGOOCT.4. The cost to locate and repair the assignable
cause is C4, and C4 w.5. The expected sampling cost is C5, and C5 a br1 r2, where r1 is the number ofsampled units before an assignable cause oc-
curs, r2 is the number of sampled units before
the assignable cause is detected after a cause
occurs. Thus, r1 and r2 are derived as follows:
r1 X1j1
jqj 1P0
19
and
r2 X1j0
Xn1i2
i"
1PjiYi1l1
Qjl
!
Yn1l1
Qjl
!n
1 Q
P
#1 P0jP0:
20Combining the cost components mentioned
above, we obtain the expected total cost, EC, asfollows:
604 Y.-S. Chen, Y.-M. Yang / European Journal of Operational Research 143 (2002) 600610
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EC C1 C2 C3 C4 C5
aY
1 P0
P0
D1X1j0
Ptj1"
Ptj21 P0jP0#
D0 D1s D0h1P01 P0A1 P0
a b r2
1P0
w: 21
The optimum values for design parameters n, h1and L can be determined by minimizing the loss
cost function EA, that is, EA EC=ET.
4. Determination of optimal design parameters
Basing our classification on the characteristics
of the variables or parameters in the expected loss-
cost model, the parameters can be classified into
five categories. There are cost parameters Y;D0;a; b;D1;w; time parameters Z0;Z1;Z2; shift para-meter d; Weibull distribution parameters k; t and
Table 1The effect on the optimal design as t changes
Optimal design
t k n h1 L Loss cost
1 0.02 2 0.76 2.51 410.392
1.5 0.002425 2 2.96 2.65 378.730
1.8 0.0007078 2 4.99 2.65 375.228
2 0.000314 2 6.46 2.65 372.730
3 0.0000057 2 13.71 2.65 359.989
Note: The mean duration of time to failure for each pair k; t is same.
Table 2
The effect on the optimal design as k changes
Optimal design
k t n h1 L Loss cost
0.00001 2.859 2 12.74 2.65 351.750
0.0001 2.284 2 8.57 2.65 369.141
0.001 1.716 2 4.40 2.65 376.347
0.015 1.067 2 0.81 2.65 382.222
Note: The mean duration of time to failure for each pair k; t is same.
Table 3
The effect on the optimal design as either k (or t) changes while t (or k) is fixed
Optimal design
k t Mean n h1 L Loss cost
0.0001 2 88.62 2 9.8 2.66 336.257
0.001 2 28.03 2 4.24 2.63 416.556
0.015 2 7.24 2 1.60 2.59 524.906
0.001 1.5 90.33 2 4.34 2.66 339.338
0.001 1.8 41.29 2 4.36 2.65 389.346
0.001 3 8.93 2 3.41 2.60 482.747
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Table 4
The sensitivity results of changing one parameter from the base as t 3, k 0:001Quantity n h1 L Loss cost
d1 5 3.05 2.49 609.8571.5 3 3.23 2.57 532.597
1.8 3 3.25 2.70 502.427
2.2 2 3.42 2.67 466.810
2.5 2 3.45 2.77 448.753
3 2 3.51 2.87 428.870
Z0 0.625 2 3.39 2.66 487.813
0.9375 2 3.40 2.63 485.369
1.125 2 3.40 2.61 483.819
1.375 2 3.41 2.58 481.642
1.5625 2 3.42 2.56 479.915
1.875 2 3.43 2.52 476.835
Z1
Z2 1.625 2 3.43 2.58 552.136
2.4375 2 3.42 2.59 515.126
2.925 2 3.41 2.59 495.199
3.575 2 3.40 2.60 470.090
4.0625 2 3.40 2.60 454.185
4.875 2 3.39 2.61 428.806
Y 1000 2 3.53 2.17 454.623
1500 2 3.45 2.45 472.036
1800 2 3.42 2.55 478.909
2200 2 3.40 2.64 486.160
2500 2 3.38 2.69 490.679
3000 2 3.36 2.76 497.058
w 500 2 3.40 2.61 444.046
750 2 3.40 2.60 463.399950 2 3.41 2.60 478.878
1100 2 3.41 2.59 490.484
1250 2 3.41 2.59 502.089
1500 2 3.42 2.58 521.425
D0 105 2 3.39 2.62 410.154
157.5 2 3.40 2.61 446.460
189 2 3.40 2.60 468.234
231 2 3.41 2.59 497.256
262.5 2 3.42 2.59 519.014
315 2 3.43 2.58 555.260
D1 2000 2 3.96 2.60 399.862
3000 2 3.62 2.60 445.088
3600 2 3.48 2.60 468.363
4400 2 3.34 2.59 496.384
5000 2 3.26 2.59 515.659
6000 2 3.14 2.59 545.226
a b 20 3 2.81 3.02 428.95630 2 3.19 2.70 459.629
36 2 3.33 2.64 474.021
44 2 3.48 2.56 490.906
50 2 3.59 2.51 502.256
60 2 3.74 2.44 519.268
Note: Base case: d 2, Z0 1:25 h, Z1 Z2 3:25 h, Y $2000, w $1000, D1 $4000, a b $40, D0 $210.
606 Y.-S. Chen, Y.-M. Yang / European Journal of Operational Research 143 (2002) 600610
8/22/2019 2002 An extension of Banerjee and Rahims model for economic
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design parameters n; h1;L. Two numerical exam-ples are used to verify the capability of the model.
We modify Rahims BASIC computer program
to meet our research needs and solve the optimaldesign parameters to reach the minimum of the
EA (cf. Rahim, 1993).
Example 1. The cost, time, shift, Weibull distri-
bution parameters are set-up as follows: Y $2000, w $1000, D1 $4000, D0 $210, a $20,b $20, Z0 1:25 h, Z1 1:25 h, Z2 2 h, k 0:02, t 1, and d 2. (Note: h represents hours.)The optimal economic design parameters solved
from the program are n 2, h1 0:59 h andL
2:65, and the minimal loss cost is $383:011.
Example 2. Reset the parameters in Example 1 as
follows: Y $500, w $1100, D1 $950, D0 $50, a $20, b $20, Z0 0:25 h, Z1 Z2 1:0h, k 0:002, t 3 and d 2. The optimal designparameters are n 2, h1 4:13 h, L 2:08 andthe minimal loss cost is $297:972.
To study the effect on the design parameters
and loss cost of the various combinations of
Weibull parameters, we let the mean time to failure
be a constant 50 and the cost, time, and shift pa-rameters be the same as in Example 1. Numerical
results for the parameters n; h1;L and the losscost under different pairs of k; t are shown inTables 1 and 2. Table 1 shows that an increasing t
results in an increasing sampling interval h1, the
decreasing of the loss cost, and no significant
change in moving subgroup size n. The shortest
sampling interval and largest loss cost occur when
the failure time follows an exponential distribu-
tion, i.e., t 1. Table 2 reveals that as k increases,the h1 decreases, the loss cost increases and thereis little effect on L.
The numerical results of using the same set-up
of model parameters as in Example 1, except that
different pairs of k; t are used, are shown inTable 3. Table 3 indicates that the loss cost in-
creases when one of the k; t is fixed and the otherincreases. Also, the higher the value of mean time
to failure, the smaller the value of loss cost.
However, we cannot conclude that the loss cost is
a decreasing function of mean time to failure. An
increasing k (fixed t), results in a decreasing h1 and
L. There is a significant effect on n, and no extreme
change in L when t increases and k is fixed.
5. Sensitivity analysis
In this section, we discuss the robustness of the
model when the time, cost, and shift parameters
vary. If we use the parameter values given in Ex-
ample 1 and t 3, k 0:001 as our base case,each of the following parameters d; Z0; Z1 Z2; Y, w; D0; D1, and a b are perturbed by10%, 25%, and 50% from the base case. Theoptimal values of n, h1, L and the loss cost for all
48 cases are presented in Table 4. Several points
can be observed from an analysis of Table 4.
1. An increasing value ofd leads to a decreasing n
and loss cost, but an increasing h1 and L.
2. Z0;Z1 Z2 and w have little effect on n, h1 andL; however, the parameters Z1 Z2 and w havemore impact on the loss cost.
3. Y has a significant effect on h1 and L, but a less
significant effect on the loss cost.
4. When D1 is small, h1 is large and the loss cost is
small.5. When D0 varies, there is no significant effect on
n, h1 and L, but a significant effect on the loss
cost.
6. A large sample cost a b leads to a smallermoving subgroup size n, but has a significant ef-
fect on the h1, L, and loss cost.
6. Conclusion
This paper is a presentation of a detailed devel-opment of an economically based MA control chart
where the process-failure mechanism follows a
Weibull distribution. The loss-cost model is well
established and formulated, and the optimal design
parameters for the cost model are solved using a
BASIC program. The numerical and sensitivity
analyses are performed to reveal how the loss cost,
the design parameters, and the model parameters
relate. In this paper, we successfully extend the
economically based MA control chart design from a
Y.-S. Chen, Y.-M. Yang / European Journal of Operational Research 143 (2002) 600610 607
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uniform sampling scheme to a non-uniform sam-
pling scheme for use in a continuous flow process.
Acknowledgements
Part of this work was supported by the Na-
tional Science Council of Taiwan under grant No.
NSC89-2213-E-155-066.
Appendix A. Proof ofET and EC
The proposed procedure considers the system at
the end of the first sampling moving subgroup.
Systems of equations can be formulated by taking
into account all possible states of the system, the
residual times and cost in the cycle, and associated
probabilities. Let T and C, respectively, be the
residual time and cost in the first sampling moving
subgroup interval h1 whether an assignable causehas or has not occurred. Let T1 and C1 be the re-
sidual time and cost in the second moving sub-
group interval h2 whether an assignable causehas or has not occurred, given that no assignable
cause has occurred at the first moving subgroup
interval. In a similar fashion, we may define T2;T3; . . . and C2;C3; . . .
Let Tj and Cj (j 0; 1; 2; . . .) be the residualtimes and cost in the cycle beyond Wj, given that
the process is in the in-control state at time Wj.
Clearly, T0 T and C0 C.The proofs of ET and EC are illustrated in
the following lemmas.
Lemma A.1. The following statements are true:
ET P0Pt0
1 Pt0
2 Z1 Z2 1 P0h1 aZ0 ET1; A:1
ET1 P0Pt11 Pt12 Z1 Z2 1 P0h2 aZ0 ET2 A:2
for j 2; 3; . . . ; and
ETj1 P0Ptj11 Ptj12 Z1 Z2 1 P0hj aZ0 ETj: A:3
From (A.1)(A.3), we can obtain
ET P0Pt01 Pt02 Z1 Z2
1
P0
fh
1 aZ
0 P0
Pt1
1 Pt1
2 Z1 Z2 1 P0h2 aZ0 ET2g P0Pt01 Pt02 1 P0Pt11 Pt12
1 P02Pt21 Pt22 P0Z1 Z21 1 P0 1 P02 aZ01 P0 1 P02 1 P0h1 1 P02h2 1 P03h3
X1
j0 Ptj
1 Ptj
21 P0j
P0 Z1 Z2
aZ0 1 P0
P0 1 P0P0h1A1 P0;
A:4where Pt
j
1 andPtj
2 are the same as Eqs. (13) and (14).
Lemma A.2. The following statements are true:
EC a b
P0
a( b X
n1
i2 i" 1P0i Y
i1
l1Q0l !
Yi1l1
Q0l
!n
1 QP
#
D0 D1s1 D1Pt01 Pt02 w)
1 P0D0h1 aY EC1 A:5for j 2; 3; . . . ; and
E
Cj
1
a
b
P0
a( b X
n1
i2 i" 1
Pj1iYi1l1
Qj1l
!
Yi1l1
Qj1l
!
n
1 QP
# D0 D1sj
D1Ptj11 Ptj12 w)
1 P0D0hj aY ECj; A:6
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From (A.5) and (A.6), we can obtain
E
C
a
b
P0 a
( b
Xn1i2
i"
1P0iYi1l1
Q0l
!
Yn1l1
Q0l
!n
1 Q
P
#
D0 D1s1 D1Pt01 Pt02 w)
1
P0
D0h1( aY a b
P0 a(
bXn1i2
i"
1P1iYi1l1
Q1l
!
Yn1l1
Q1l
!n
1 Q
P
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References
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