2002 an extension of banerjee and rahim’s model for economic

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: [email protected] (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:%[email protected]/http://mail%20to:%[email protected]/


2/11

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


3/11

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


4/11

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.



5/11

(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:



6/11

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



7/11

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.



8/11

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



9/11

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



10/11

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

#


1

P0

D0h1( aY a b

P0 a(

bXn1i2

i"

1P1iYi1l1

Q1l

!

Yn1l1

Q1l

!n

1 Q

P

#


1 P0D0h2 aY EC2) a b1 1 P0 1 P02

a bP0Xn1i2

i(

1P0iYi1l1

Q0l

!

Yn1l1

Q0l

!n

1 Q

P

1

P0

Xn1

i2 i"

1

P1i Y

i1

l1Q1l !

Yn1l1

Q1l

!n

1 Q

P

#

)

D0 D1P0s1 1 P0s2

1 P02s3

wP01 1 P0 1 P02

aY1 P01 1 P0 1 P02 D1P0Pt01 Pt02 1 P0Pt11 Pt12 D0h11 P0 h21 P02

a b 1P0

a bX1j0

Xn1i2

i"

1PjiYi1l1

Qjl

!

Yn1l1

Qjl

!n

1 Q

P

#1 P0jP0

D0

D1

s

w

aY

1 P0P

0 D1

X1j0

Ptj1 Ptj21 P0jP0

D01 P0P0h1A1 P0: A:7

References

Banerjee, P.K., Rahim, M.A., 1987. The economic design of

control charts: A renewal theory approach. Engineering

Optimization 12, 6673.

Banerjee, P.K., Rahim, M.A., 1988. Economic design of xx

control charts under Weibull shock models. Technometrics

30, 407414.

Chen, Y.S., Yang, Y.M., 1999. Economic design of xx control

charts for continuous flow process with Weibull in-control

time. In: The 13th Asia Quality Symposium 99 Osaka, pp.

1720.

Chiu, W.K., Wetherill, G.B., 1974. A simplified scheme for the

economic design ofxx-charts. Journal of Quality Technology

6, 6369.

Duncan, A.J., 1956. The economic design of xx charts used to

maintain current control of a process. Journal of the

American Statistical Association 51, 228242.

Duncan, A.J., 1971. The economic design of xx-chartswhen there is a multiplicity of assignable causes.

Journal of the American Statistical Association 66,

107121.

Ho, C., Case, K.E., 1994. The economically-based EWMA

control chart. International Journal of Production Research

32, 21792186.

Koo, T.Y., Case, K.E., 1990. Economic design of x-bar control

charts for use in monitoring continuous flow processes.

International Journal of Production Research 28, 2001

2011.

Lorenzen, T.J., Vance, L.C., 1986. The economic design of

control charts: A unified approach. Technometrics 28, 310.



11/11

Ohta, H., Rahim, M.A., 1997. A dynamic economic model for

an xx-control chart design. IIE Transactions 29 (6), 481486.

Parkhideh, B., Case, K.E., 1989. The economic design of a

dynamic xx-control chart. IIE Transactions 21, 313323.

Rahim, M.A., 1993. Economic design of xx control chartsassuming Weibull in-control times. Journal of Quality

Technology 25, 296305.

Rahim, M.A., Banerjee, P.K., 1993. A generalized model for

the economic design of xx-control charts for production

system with increasing failure rate and early replacement.

Naval Research Logistics 40 (6), 787809.

Rahim, M.A., Costa, A.F.B., 2000. Joint economic design of xx

and R charts under Weibull shock models. International

Journal of Production Research 29 (3), 845873.Saniga, E.M., 1977. Joint economically optimal design of xx and

R control charts. Management Science 24, 420431.

Taylor, H.M., 1968. The economic design of cumulative sum

control charts. Technometrics 10, 479488.


2002 an extension of banerjee and rahim’s model for economic

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