amarkovianapproachtodeterminingoptimumprocess...

European Journal of Operational Research 159 (2004) 636–650

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Production, Manufacturing and Logistics

A Markovian approach to determining optimum processtarget levels for a multi-stage serial production system

Shannon R. Bowling a, Mohammad T. Khasawneh b, Sittichai Kaewkuekool c,Byung Rae Cho a,*

a Department of Industrial Engineering, Clemson University, Clemson, SC 29634-0920, USAb Department of Systems Science and Industrial Engineering, State University of New York at Binghamton, Binghamton, NY 13902, USAc Department of Production Technology Education, King Mongkut�s University of Technology Thonburi, Bangkok 10140, Thailand

Received 11 December 2002; accepted 2 June 2003

Available online 26 September 2003

Abstract

Consider a production system where products are produced continuously and screened for conformance with their

specification limits. When product performance falls below a lower specification limit or above an upper limit, a de-

cision is made to rework or scrap the product. The majority of the process target models in the literature deal with a

single-stage production system. In the real-world industrial settings, however, products are often processed through

multi-stage production systems. If the probabilities associated with its recurrent, transient and absorbing states are

known, we can better understand the nature of a production system and thus better capture the optimum target for a

process. This paper first discusses the roles of a Markovian approach and then develops the general form of a

Markovian model for optimum process target levels within the framework of a multi-stage serial production system.

Numerical examples and sensitivity analysis are performed.

� 2003 Elsevier B.V. All rights reserved.

Keywords: Quality control; Process target levels; Markov chain; Multi-stage serial production

1. Introduction

One of the most important decision-making problems encountered in a wide variety of industrial pro-

cesses is the determination of optimum process target (mean). Selecting the optimum target for a process iscritically important since it can affect the process defective rate, processing cost, and scrap and rework

costs. Furthermore, the process target may need to be reset frequently and promptly due to unpredictable

random variation in many manufacturing processes. For a production process where products are pro-

duced continuously, specification limits are usually implemented based on a quality evaluation system that

* Corresponding author. Tel.: +1-864-656-1874; fax: +1-864-656-0795.

E-mail address: [email protected] (B.R. Cho).

0377-2217/$ - see front matter � 2003 Elsevier B.V. All rights reserved.

doi:10.1016/S0377-2217(03)00429-6

mail to: [email protected]

S.R. Bowling et al. / European Journal of Operational Research 159 (2004) 636–650 637

focuses primarily on the cost of non-conformance. Consider a certain quality characteristic, where aproduct is rejected if its product performance associated with the quality characteristic of interest either

falls above an upper specification limit or falls below a lower specification limit. If product performance is

higher than the upper limit, the product can be reworked, whereas the product is scrapped if it falls below

the lower limit. The proportion of rejected products largely depends on the levels and tolerance of speci-

fication limits. If the process target is set too low, then the proportion of non-conforming products becomes

high, but the manufacturer may experience high rejection costs associated with non-conforming products.

The question then becomes how to determine the optimum process.

As discussed in the next section, a number of models have been proposed in the literature for deter-mining an optimum process target. Our investigation indicates that the majority of the methodologies

reported in the research community deal with determining optimum process target within a single-stage

production system. In the real-world industrial settings, however, products are often processed through

multi-stage production systems, where raw material is transformed into the final product in a series of

distinct processing stages. Product items deemed to be non-conforming may be scrapped or reworked, while

conforming items are allowed to continue through the system. The primary objective of this paper is to

determine optimal process target levels by employing Markovian properties in order to maximize the total

profit associated with a multi-stage serial production system, in which lower and upper specification limitsare given at each stage. In addition, it is assumed that each quality characteristic is governed by a normal

distribution, and screening (100%) inspection is performed.

The remainder of this paper is organized as follows. Section 2 discusses the literature review. After

introducing the notation and assumptions, the rationale for screening inspection is discussed in Section 3.

In Section 4, Markovian models for single-stage and two-stage production systems are first developed and

then a general model for an n-stage production system is derived. Numerical examples and sensitivity

analyses are given for illustrative purposes in Sections 5 and 6. The conclusion follows in the last section.

2. Related literature

The initial work probably began with Springer (1951) who considered the problem of determining an

optimal process target with specified upper and lower specification limits. Bettes (1962) considered a similar

problem with a fixed lower specification limit and arbitrary upper specification limit. In some situations,

however, the products that do not meet the minimum requirement for product performance may be sold at

a reduced price. Hunter and Kartha (1977) presented a model to determine the optimal process target underthe assumption that the products meeting the requirement are sold in a regular market at a fixed price, while

the underachieved products are sold at a reduced price in a secondary market. Nelson (1978, 1979) de-

termined approximate solutions to the Hunter and Kartha model (1977) and developed a nomograph for

the Springer model (1951). The Hunter and Kartha model (1977) was later modified by Bisgaard et al.

(1984) who assumed that underachieved products are sold at a price proportional to their performance, and

by Carlsson (1984) who included a more general income function. In addition, Arcelus and Banerjee (1985)

extended the work of Bisgaard et al. (1984), assuming a linear shift in the process mean. Golhar (1987)

developed a model for the optimal process target under the assumptions that underachieved products canbe reprocessed. Golhar and Pollock (1988) modified this model by treating both the upper specification

limit and the process mean as control variables.

Arcelus and Rahim (1990) presented a model for the most profitable process target where both variable

and attribute quality characteristics of a product are considered simultaneously, while Boucher and Jafari

(1991) addressed the same problem by extending the line of research under the context of a sampling plan.

Schmidt and Pfeifer (1991) extended the models of Golhar (1987) and Golhar and Pollock (1988) by

considering a limited process capacity. Al-Sultan (1994) developed an algorithm to find the optimal

638 S.R. Bowling et al. / European Journal of Operational Research 159 (2004) 636–650

machine setting when two machines are connected in series. Das (1995) used maximization of expectedprofits as a criterion for selecting an optimal process target when lower specification limit is given. Chen

and Chung (1996) and Hong and Elsayed (1999) studied the effects of inspection errors. Usher et al. (1996)

considered the process target problem in a situation where demand for a product does not exactly meet the

capacity of a filling operation. Liu and Taghavachari (1997) studied the general problem of determining

both optimal process target and upper specification limit when a quality characteristic follows an arbitrary

continuous distribution. Pollock and Golhar (1998) reconsidered the process target problem under the

environment of capacitated production and fixed demand. Pakkala and Rahim (1999) presented a model

for the most economical process target and production run. Al-Sultan and Pulak (2000) proposed a modelconsidering a manufacturing system with two stages in series to find the optimum target values with a lower

specification limit and application of a 100% inspection policy.

Most researchers assume that process variance is given. The problem of jointly determining a process

target and a variance was studied by Rahim and Shaibu (2000), Rahim and Al-Sultan (2000), and Rahim

et al. (2002). Along the same line, Al-Fawzan and Rahim (2001) applied the Taguchi loss function to

determine the optimal process target and variance. Shao et al. (2000) examined several methods for process

target optimization when several grades of customer specifications are sold within the same market. Kim

et al. (2000) proposed a model for determining the optimal process target while considering variance re-duction and process capability. Phillips and Cho (2000) proposed a model for the optimal process target

under the situation in which a process distribution is skewed. There are situations in which empirical data

concerning the costs associated with product performance are available. Under this situation, Teerava-

raprug et al. (2000) developed a model for the most cost-effective process target using regression analysis.

Finally, Cho (2002) and Teeravaraprug and Cho (2002) studied the process target problem with the con-

sideration of multiple quality characteristics.

3. Preliminaries

Notation and assumptions are summarized below, and then the rationale for screening inspection is

discussed.

3.1. Notation

EðPRÞ expected profit per item

EðBFÞ expected benefit per item

EðPCÞ expected processing cost per item

EðSCÞ expected scrap cost per item

EðRCÞ expected rework cost per item

SP selling price per item

PCi processing cost associated with stage iSCi scrap cost associated with stage iRCi rework cost associated with stage in number of stages

Xi quality characteristic associated with stage ili process mean setting for machine ir2i process variance setting for machine iLi lower specification limit associated with stage iUi upper specification limit associated with stage iUðxÞ cumulative normal function


P transition probability matrix

Q square matrix containing transition probabilities of going from any non-absorbing state to any

other non-absorbing state

R matrix containing all probabilities of going from any non-absorbing state to an absorbing state

(i.e., finished or scrapped product)

A an identity matrix representing the probability of staying in an state

O matrix representing the probabilities of escaping an absorbing state (always zero)

M fundamental matrix containing the expected number of transitions from any non-absorbing stateto any other non-absorbing state before absorption occurs

F absorption probability matrix containing the long run probabilities of the transition from any non-

absorbing state to any absorbing state

pij the probability of going from state i to state j in a single stepmij expected number of transitions from any non-absorbing state (i) to any other non-absorbing state

(j) before absorption occursfij long run probability of going from any non-absorbing state (i) to any absorbing state (j)

3.2. Assumptions

1. Products are produced continuously.

2. All product items are subject to inspection.

3. When product performance falls below a lower specification limit or above an upper specification limit, a

product is reworked or scrapped, respectively. If product performance falls within the limits, the product

goes on to the next stage.4. Each product requires the same inspection cost, which is included in the processing cost.

5. The quality characteristic, Xi, is a random variable and is normally distributed with mean li and variance

r2i .6. The process is under control.

7. The machine sequence is fixed. That is, products have to be processed at stage i first and then at stageiþ 1.

3.3. Rationale for screening inspection

Recent advances in technology have motivated the automation of many of today�s complex manufac-turing systems. In general, automated systems, computerized machines, and specialized robots can perform

rigorous procedures while providing consistent results and superior performance. In these circumstances,

product inspection is one of the major functions that ensure quality of products and customer satisfaction.

To achieve best performance and consistent quality of outgoing products, screening (100%) inspection in

modern manufacturing systems is becoming more attractive than traditional sampling techniques. Highly

automated inspection systems have found increasing applications in quality control processes. These sys-tems are very useful in reducing error rates, inspection times, and inspection costs.

4. Model development

Consider a multi-stage serial production system in which products are being produced continuously.

Each stage is defined as having a single machine and a single inspection station. At each stage, the item is

processed and the quality characteristic associated with the stage is examined at an inspection station. The


item is then reworked, accepted or scrapped. Therefore, the expected profit per item can be expressed asfollows:

EðPRÞ ¼ EðBFÞ � EðPCÞ � EðSCÞ � EðRCÞ: ð1Þ
The purpose of this paper is to develop a Markovian model for determining the optimum process target
value for each production stage. The paper starts by developing the models for single-stage, two-stage, and

three-stage serial production systems. The paper then generalizes the model for an n-stage serial productionsystem.

4.1. Single-stage system

Consider a single-stage production system as shown in Fig. 1.

The single-step transition probability matrix can be expressed as follows:

;

where p11 is the probability of an item being reworked, p12 is the probability of an item being accepted, and

p13 is the probability of an item being scrapped. Assuming a normally distributed quality characteristics as

shown in Fig. 2, these probabilities can be expressed as follows:

p11 ¼Z 1

U1

1ffiffiffiffiffiffi2p

pr1e�12

x1�l1r1

� �2dx1 ¼ 1� UðU1Þ; ð2aÞ

Fig. 1. A single-stage production system.

Fig. 2. Illustration of accepted, reworked, and scrapped probabilities.


p12 ¼Z U1

L1


pr1e�12

x1�l1r1

� �2dx1 ¼ UðU1Þ � UðL1Þ; ð2bÞ

p13 ¼Z L1

�1


pr1e�12

x1�l1r1

� �2dx1 ¼ UðL1Þ: ð2cÞ

As it can be observed, the matrix P is an absorbing Markov chain with states 2 and 3 being absorbing

and state 1 being transient. Analyzing this absorbing Markov chain requires the rearrangement of the

single-step probability matrix in the following form:

:

Rearranging the P matrix in the latter form yields the following matrix:

:

The fundamental matrix M, which is a one-by-one matrix in this case, can be obtained as follows:

M ¼ ðI �QÞ�1 ¼ m11 ¼1

ð1� p11Þ;

where I is the identity matrix. The value m11 represents the expected number of times in the long run that

the transient state 1 is occupied before absorption occurs (i.e., accepted or scrapped), given that the initialstate is 1. The long-run absorption probability matrix, F, can be calculated as follows:

:

The elements of the F matrix, f12 and f13 represent the probabilities of an item being accepted and

scrapped, respectively. The expected profit per item can be obtained by using Eq. (1), in which it consists of

the benefit, processing costs, scrap cost, and rework cost per item. The expected benefit is a selling price peritem (SP) multiplied by the absorption probability of an item being accepted (i.e., f12). The expectedprocessing cost per item is PC1. The expected scrap cost per item is the scrap cost (SC1) multiplied by the

absorption probability of a product being scrapped (i.e., f13). Note that once a product goes into one ofthese two absorbing states (i.e., states 2 and 3), the product cannot go back to state 1. Hence, the number of

visits to the absorbing states is 1. When a product is reworked, the expected rework cost for a single visit to

the rework state (i.e., state 1) is RC1 � ðm11 � 1Þ. Since the expected number of times that the transient state1 is occupied before absorption occurs (i.e., accepted or scrapped) is m11 and since the reworking process

occurs after the item is initially processed only one time at state 1, the expected number of time thattransient state 1 is occupied for the reworking purposes until absorption is m11 � 1. Consequently the


expected rework cost is given by RC1 � ðm11 � 1Þ. Therefore, the expected profit per item for a single-stageproduction system can be expressed as a function of f12, f13 and m11 as follows:

EðPRÞ ¼ SP � f12 � PC1 � SC1 � f13 �RC1ðm11 � 1Þ: ð3Þ
Substituting for f13 and m11, the expected profit equation can be rewritten as follows:
EðPRÞ ¼ SP 1

�� p131� p11

�� PC1 � SC1

p131� p11

�RC1

p111� p11

� �: ð4Þ

The equation can then be rewritten in terms of the cumulative normal distribution as follows:

EðPRÞ ¼ SP 1

�� UðL1Þ

UðU1Þ

�� PC1 � SC1

UðL1ÞUðU1Þ

�RC1

1� UðU1ÞUðU1Þ

� �: ð5Þ

The terms UðL1Þ and UðU1Þ are functions of the decision variable l1, which is the process mean. Ob-viously, one would like to find the value of l1 that maximizes the expected profit. This can be performednumerically using a number of nonlinear optimization software packages.

4.2. Two-stage system

Consider a two-stage serial production system as shown in Fig. 3.

The single-step transition probability matrix can be expressed as follows:

;

where pii is the rework probability associated with stage i, piiþ1 is the probability associated with accepting aproduct at stage i, and pinþ2 is the probability of scrapping a product at stage i, where n is the number ofstages. Rearranging the P matrix and applying the procedure used for the single-stage system yields the

following fundamental and absorption matrices:

;

P12 P23

;

where mii � 1 is the long-term percentage of reworked products, and finþ2 is the long-term percentage of

scrapped products. The expected profit can be obtained by using Eq. (1). As can be seen, Eq. (1) consists of

the benefit, processing costs, scrap cost, and rework cost per item. The expected benefit is simply the selling

Fig. 3. A two-stage serial production system.


price per item (SP) multiplied by the long-term percentage of accepted products at stage 1 (i.e., 1� f14)multiplied by the percentage of accepted products at stage 2 (i.e., 1� f24).The expected processing cost for a two-stage system is the expected processing cost per item at stage 1

(i.e., PC1) plus the expected processing cost at stage 2, which is PC2 multiplied by the long-term percentage

of products accepted at stage 1 (i.e., 1� f14). Similarly, the expected scrap cost per item is the scrap cost

(SC1) multiplied by the long-term percentage of scrapped products at stage 1 (i.e., f14) plus S2 multiplied bythe long-term percentage of scrapped products at stage 2 (i.e., ð1� f14Þ f24).The expected rework cost per item is the rework cost (RC1) multiplied by the long-term percentage of

reworked products at stage 1 (i.e., m11 � 1) plus RC2 multiplied by the long-term percentage of reworked

products at stage 2 (i.e., m22 � 1) multiplied by the long-term percentage of accepted products at stage 1

(i.e., 1� f14). Therefore, the expected profit per item for a two-stage serial production system can be ex-pressed as follows:

EðPRÞ ¼ ½SPð1� f14Þð1� f24Þ� � ½PC1 þ PC2ð1� f14Þ� � ½SC1f14 þ SC2ð1� f14Þf24�� ½RC1ðm11 � 1Þ þRC2ðm22 � 1Þð1� f14Þ�; ð6aÞ

EðPRÞ ¼ SP 1

�� p14

ð1� p11Þ

�þ p12p24ð1� p11Þð1� p22Þ

�1

�� p24ð1� p22Þ

�

� PC1

�þ PC2 1

�� p14

ð1� p11Þ

�þ p12p24ð1� p11Þð1� p22Þ

�

� SC1

p14ð1� p11Þ

��þ p12p24ð1� p11Þð1� p22Þ

þ SC2 1

�� p14

ð1� p11Þ

�þ p12p24ð1� p11Þð1� p22Þ

�p24

ð1� p22Þ

� RC1

p111� p11

� ��þRC2

p221� p22

� �1

�� p14

ð1� p11Þ

�þ p12p24ð1� p11Þð1� p22Þ

�; ð6bÞ

EðPRÞ ¼ SP 1

�� UðL1Þ

UðU1Þ

�þ ½UðU1Þ � UðL1Þ�UðL2Þ

UðU1ÞUðU2Þ

�1

�� UðL2Þ

UðU2Þ

�

� PC1

�þ PC2 1

�� UðL1Þ

UðU1Þ


UðU1ÞUðU2Þ

�

� SC1

UðL1ÞUðU1Þ

��þ ½UðU1Þ � UðL1Þ�UðL2Þ

UðU1ÞUðU2Þ

þ SC2 1

�� UðL1Þ

UðU1Þ


UðU1ÞUðU2Þ

�UðL2ÞUðU2Þ

� RC1

1� UðU1ÞUðU1Þ

� ��þRC2

1� UðU2ÞUðU2Þ

� �1

�� UðL1Þ

UðU1Þ


UðU1ÞUðU2Þ

�: ð6cÞ


The terms UðL1Þ, UðU1Þ, UðL2Þ, and UðU2Þ are functions of the decision variables l1 and l2, which are theprocess mean for machines 1 and 2, respectively.

4.3. A general model for an n-stage system

Consider an n-stage serial production system as shown in Fig. 4.

The single-step transition probability matrix, fundamental matrix, and long-term absorption probability

matrix can be expressed as follows:

;

; :

Fig. 4. An n-stage serial production system.


Therefore, the expected profit per item for an n-stage serial production system can be expressed as follows:

EðPRÞ ¼ SPYni¼1

ð1

� finþ2Þ!

� PC1

"þXni¼2

PCi

Yij¼2

1� (

� fj�1nþ2�!)#

� SC1f1nþ2

"þXni¼2

SCi

Yij¼2

ð1 (

� fj�1nþ2Þ!finþ2

)#

� RC1ðm11

"� 1Þ þ

Xni¼2

RCiðmii

(� 1Þ

Yij¼2

ð1

� fj�1nþ2Þ!)#

: ð7Þ

Combining the terms, the above model can be further simplified to the following:

EðPRÞ ¼ SPYni¼1

ð1

� finþ2Þ!

� PC1 � SC1f1nþ2 �RC1ðm11 � 1Þ

�Xni¼2

Yij¼2

ð1"

� fj�1nþ2ÞfPCi þ SCifinþ2 þRCiðmii � 1Þg#: ð8Þ

As it can be seen from Eq. (8), mii and finþ2 are the only terms required in order to obtain the expected profitper item. These terms can be represented in the following general forms:

mii ¼1

1� pii; ð9aÞ

fknþ2 ¼pknþ21� pkk

þXn�k

i¼1

pn�iþ1nþ2

1� pn�iþ1n�iþ1

Yn�k

j¼i

pn�jn�jþ1

1� pn�jn�j

� �( ): ð9bÞ

5. Numerical example


The above model can be illustrated by solving a numerical example for a single-stage production system.

Consider a single-stage production system and the following parameters: SP ¼ 120, PC1 ¼ 25, RC1 ¼ 10,

SC1 ¼ 15, r1 ¼ 1:0, L1 ¼ 8:0 and U1 ¼ 12. Using the generalized reduced gradient (GRG) method, the

expected profit is maximized at l1 ¼ 10:6144 and the profit per item is 93.4377. Fig. 5 shows the expected

profit as a function of the process mean. As it can be seen, the expected profit is a concave function over thespecified range of ½L1 ¼ 8;U1 ¼ 12�.


Consider a two-stage production system and the following parameters: SP ¼ 120, PC1 ¼ 25, PC2 ¼ 20,

RC1 ¼ 10, RC2 ¼ 17, SC1 ¼ 15, SC2 ¼ 12, r1 ¼ 1:0, L1 ¼ 8:0, L2 ¼ 13:0, U1 ¼ 12:0 and U2 ¼ 17:0. Usingthe GRG method, the expected profit is maximized at l

1 ¼ 10:5708 and l2 ¼ 15:6301 with an expected

profit of 70.8264.Fig. 6 shows the expected profit as a function of the process means (l1 and l2). The expected profit is a

concave function over the specified range of ½L1 ¼ 8; L2 ¼ 13; U1 ¼ 12;U2 ¼ 17�.

20

30

40

50

60

70

80

90

100

8 8.5 9 9.5 10 10.5 11 11.5 12

Process Mean

Exp

ecte

d P

rofi

t

µ1* = 10.6144

Fig. 5. Expected profit versus process mean.

89

10

11

12

13

14

15

16

17-25

0

25

50

Expected Profit

µ1

µ2

µ2* = 10.5708

µ1* = 15.6301

Fig. 6. Effect of changing process means on the expected process.


5.3. Multi-stage system

In order to illustrate the use of the general model, analyses will be performed to optimum process means

for a three-stage, four-stage, and five-stage serial production system, based on the parameters shown in

Table 1. The optimum process means and expected profit for these cases, including the results of a single-

stage and two-stage systems, is summarized in Table 2.

Table 1

Data for a multi-stage serial production system

Parameter Stage 1 Stage 2 Stage 3 Stage 4 Stage 5

PC 25 20 12 15 4

RC 15 12 8 10 2

SC 10 17 5 12 3

r 1.0 1.0 1.0 1.0 1.0

L 8 13 10 7 18

U 12 17 14 11 22

Table 2

Optimum process means and expected profit for a multi-stage serial production system

Parameter Stage 1 Stage 2 Stage 3 Stage 4 Stage 5

l1 10.6144 10.5708 10.4937 10.4944 10.4792

l2 15.6301 10.6821 15.5795 15.5485

l3 10.7982 12.9347 12.9392

l4 9.8413 9.8299

l5 21.2054

Expected profit 93.4377 70.8264 49.8332 40.0821 35.0708


6. Sensitivity analysis

It is very beneficial to perform sensitivity analysis of the proposed model parameters to illustrate the

possible impact of estimated parameters on the optimal process mean and the optimal expected profit. The

rework and scrap cost were varied in the single-stage and two-stage systems and their effects are shown inthe following sections.


Figs. 7–9 show the behavior of the optimum process mean and the optimum expected profit with the

variation of the scrap and rework costs. Notice that in all cases the optimum process mean and expected

profit are sensitive to changes in the rework and scrap cost values.

10.58

10.59

10.6

10.61

10.62

10.63

10.64

10.65

10.66

0 5 10 15 20 25 30 35 40Scrap Cost

Opt

imum

Pro

cess

Mea

n.

93.2

93.25

93.3

93.35

93.4

93.45

93.5

93.55O

ptim

um E

xpec

ted

Pro

fit

MeanExpected Profit

Fig. 7. Effect of scrap cost on optimal value of l1 and expected profit.

10.2

10.4

10.6

10.8

11.0

11.2

11.4

Rework Cost

Opt

imum

Pro

cess

Mea

n

0 5 10 15 20 25 30 35 40

Fig. 8. Effect of rework cost on optimal value of l1.

91.0

91.5

92.0

92.5

93.0

93.5

94.0

94.5

95.0

Rework Cost

Opt

imum

Exp

ecte

d P

rofi

t

0 5 10 15 20 25 30 35 40

Fig. 9. Effect of rework cost on optimal value of expected profit.



Table 3 shows the behaviors of the optimum process mean and the optimum expected profit with the

variation of the scrap and rework costs for a two-stage production system. For cases 1–5 as scrap cost for

stage 1 increases the optimum means for both stages increase slightly. For cases 6–10 as scrap cost for stage

2 increases the optimum mean for stage 1 remains relatively constant and that of stage 2 increases. For

cases 11–15 as rework cost for stage 1 increases the optimum means for stage 1 decreases and that of stage 2

remains constant. For cases 16–20 as rework cost for stage 2 increases the optimum mean for both stages

decrease. It is observed that the optimum expected profit decreases as scrap and rework costs increase forany of the stages.

Table 3

Sensitivity analysis for a two-stage production system

Cost parameter Case # Parameter

value

Optimum

process mean1

Optimum

process mean2

Optimum

expected profit

SC1 1 7 10.5534 15.6223 70.9089

2 11 10.5623 15.6262 70.8673

3 15 10.5708 15.6301 70.8264

4 19 10.5790 15.6339 70.7862

5 23 10.5870 15.6377 70.7466

SC2 6 4 10.5709 15.6223 70.8637

7 8 10.5708 15.6263 70.8449

8 12 10.5708 15.6301 70.8264

9 16 10.5708 15.6339 70.8080

10 20 10.5707 15.6376 70.7898

RC1 11 2 10.9458 15.6301 71.7178

12 6 10.6911 15.6301 71.1974

13 10 10.5708 15.6301 70.8264

14 14 10.4910 15.6301 70.5222

15 18 10.4311 15.6301 70.2587

RC2 16 9 10.5726 15.7794 71.6735

17 13 10.5716 15.6933 71.2192

18 17 10.5708 15.6301 70.8264

19 21 10.5701 15.5800 70.4756

20 25 10.5694 15.5386 70.1559


7. Discussion and conclusions

In this paper, the optimum process target (mean) levels for a multi-stage serial production system have

been determined numerically using a Markovian approach. The paper starts by developing a general model

for the expected profit per item by taking into account processing, scrap, and rework costs. A general model

for the expected profit for an n-stage serial production system was presented. The model was then used to

determine the optimum process target levels for three-stage, four-stage, and five-stage production systems.

By varying the cost parameters, such as scrap cost, rework cost, process mean, and process standard de-viation, the sensitivity analysis showed the behavior of the optimum process target under different con-

ditions.

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