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European Journal of Operational Research 159 (2004) 636–650 www.elsevier.com/locate/dsw
Production, Manufacturing and Logistics
A Markovian approach to determining optimum process target levels for a multi-stage serial production system Shannon R. Bowling a, Mohammad T. Khasawneh b, Sittichai Kaewkuekool c, Byung Rae Cho a,* a
b
Department of Industrial Engineering, Clemson University, Clemson, SC 29634-0920, USA Department of Systems Science and Industrial Engineering, State University of New York at Binghamton, Binghamton, NY 13902, USA c Department of Production Technology Education, King MongkutÕs 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 decision 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. Ó 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 processes is the determination of optimum process target (mean). Selecting the optimum target for a process is critically 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 produced 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 Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(03)00429-6
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focuses primarily on the cost of non-conformance. Consider a certain quality characteristic, where a product 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 specification 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 determining 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 limits are 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 under the 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) determined 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 can be 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
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machine setting when two machines are connected in series. Das (1995) used maximization of expected profits 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 model considering 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 reduction 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, Teeravaraprug 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 consideration 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 EðBFÞ EðPCÞ EðSCÞ EðRCÞ SP PCi SCi RCi n Xi li r2i Li Ui UðxÞ
expected profit per item expected benefit per item expected processing cost per item expected scrap cost per item expected rework cost per item selling price per item processing cost associated with stage i scrap cost associated with stage i rework cost associated with stage i number of stages quality characteristic associated with stage i process mean setting for machine i process variance setting for machine i lower specification limit associated with stage i upper specification limit associated with stage i cumulative normal function
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P Q R A O M F pij mij fij
639
transition probability matrix square matrix containing transition probabilities of going from any non-absorbing state to any other non-absorbing state matrix containing all probabilities of going from any non-absorbing state to an absorbing state (i.e., finished or scrapped product) an identity matrix representing the probability of staying in an state matrix representing the probabilities of escaping an absorbing state (always zero) fundamental matrix containing the expected number of transitions from any non-absorbing state to any other non-absorbing state before absorption occurs absorption probability matrix containing the long run probabilities of the transition from any nonabsorbing state to any absorbing state the probability of going from state i to state j in a single step expected number of transitions from any non-absorbing state (i) to any other non-absorbing state (j) before absorption occurs 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 stage i þ 1. 3.3. Rationale for screening inspection Recent advances in technology have motivated the automation of many of todayÕs complex manufacturing 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 systems 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
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item is then reworked, accepted or scrapped. Therefore, the expected profit per item can be expressed as follows: 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 production system. 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: 2 Z 1 x l 12 1r 1 1 1 pffiffiffiffiffiffi e p11 ¼ dx1 ¼ 1 UðU1 Þ; ð2aÞ 2pr1 U1
Fig. 1. A single-stage production system.
Fig. 2. Illustration of accepted, reworked, and scrapped probabilities.
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p12 ¼
Z
U1 L1
p13 ¼
Z
L1 1
1 1 pffiffiffiffiffiffi e 2 2pr1
1 1 pffiffiffiffiffiffi e 2 2pr1
x1 l1 r1
x1 l1 r1
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2 dx1 ¼ UðU1 Þ UðL1 Þ;
ð2bÞ
dx1 ¼ UðL1 Þ:
ð2cÞ
2
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: 1
M ¼ ðI QÞ
¼ m11 ¼
1 ; ð1 p11 Þ
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 initial state 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 per item (SP) multiplied by the absorption probability of an item being accepted (i.e., f12 ). The expected processing 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 of these 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 state 1 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 that transient state 1 is occupied for the reworking purposes until absorption is m11 1. Consequently the
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expected rework cost is given by RC1 ðm11 1Þ. Therefore, the expected profit per item for a single-stage production system can be expressed as a function of f12 , f13 and m11 as follows: EðPRÞ ¼ SP f12 PC1 SC1 f13 RC1 ðm11 1Þ: ð3Þ Substituting for f13 and m11 , the expected profit equation can be rewritten as follows: p13 p13 p11 RC1 EðPRÞ ¼ SP 1 PC1 SC1 : 1 p11 1 p11 1 p11 The equation can then be rewritten in terms of the cumulative normal distribution as follows: UðL1 Þ UðL1 Þ 1 UðU1 Þ RC1 EðPRÞ ¼ SP 1 PC1 SC1 : UðU1 Þ UðU1 Þ UðU1 Þ
ð4Þ
ð5Þ
The terms UðL1 Þ and UðU1 Þ are functions of the decision variable l1 , which is the process mean. Obviously, one would like to find the value of l1 that maximizes the expected profit. This can be performed numerically 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 a product at stage i, and pinþ2 is the probability of scrapping a product at stage i, where n is the number of stages. 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
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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 by the long-term percentage of scrapped products at stage 2 (i.e., ð1 f14 Þ 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 expressed as follows: EðPRÞ ¼ ½SPð1 f14 Þð1 f24 Þ ½PC1 þ PC2 ð1 f14 Þ ½SC1 f14 þ SC2 ð1 f14 Þf24 ½RC1 ðm11 1Þ þ RC2 ðm22 1Þð1 f14 Þ ;
ð6aÞ
p14 p12 p24 p24 þ 1 EðPRÞ ¼ SP 1 ð1 p11 Þ ð1 p11 Þð1 p22 Þ ð1 p22 Þ
p14 p12 p24 þ PC1 þ PC2 1 ð1 p11 Þ ð1 p11 Þð1 p22 Þ
p14 p12 p24 p14 p12 p24 p24 SC1 þ þ SC2 1 þ ð1 p11 Þ ð1 p11 Þð1 p22 Þ ð1 p11 Þ ð1 p11 Þð1 p22 Þ ð1 p22 Þ
p11 p22 p14 p12 p24 þ ; þ RC2 1 ð6bÞ RC1 1 p11 1 p22 ð1 p11 Þ ð1 p11 Þð1 p22 Þ
UðL1 Þ ½UðU1 Þ UðL1 Þ UðL2 Þ UðL2 Þ EðPRÞ ¼ SP 1 þ 1 UðU1 Þ UðU1 ÞUðU2 Þ UðU2 Þ
UðL1 Þ ½UðU1 Þ UðL1 Þ UðL2 Þ þ PC1 þ PC2 1 UðU1 Þ UðU1 ÞUðU2 Þ
UðL1 Þ ½UðU1 Þ UðL1 Þ UðL2 Þ UðL1 Þ ½UðU1 Þ UðL1 Þ UðL2 Þ UðL2 Þ þ þ SC2 1 þ SC1 UðU1 Þ UðU1 ÞUðU2 Þ UðU1 Þ UðU1 ÞUðU2 Þ UðU2 Þ
1 UðU1 Þ 1 UðU 2Þ UðL1 Þ ½UðU1 Þ UðL1 Þ UðL2 Þ þ RC2 1 þ : RC1 ð6cÞ UðU1 Þ UðU2 Þ UðU1 Þ UðU1 ÞUðU2 Þ
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The terms UðL1 Þ, UðU1 Þ, UðL2 Þ, and UðU2 Þ are functions of the decision variables l1 and l2 , which are the process 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.
:
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645
Therefore, the expected profit per item for an n-stage serial production system can be expressed as follows: ! " !)# ( n n i Y Y X EðPRÞ ¼ SP ð1 finþ2 Þ PC1 þ PCi 1 fj1nþ2 "
i¼1
SC1 f1nþ2 þ
n X
i¼2
( SCi
i¼2
"
RC1 ðm11 1Þ þ
n X i¼2
(
i Y
j¼2
!
)#
ð1 fj1nþ2 Þ finþ2
j¼2
RCi ðmii 1Þ
i Y
!)# ð1 fj1nþ2 Þ
:
ð7Þ
j¼2
Combining the terms, the above model can be further simplified to the following: ! n Y EðPRÞ ¼ SP ð1 finþ2 Þ PC1 SC1 f1nþ2 RC1 ðm11 1Þ i¼1
n X
"
i¼2
i Y
# ð1 fj1nþ2 ÞfPCi þ SCi finþ2 þ RCi ðmii 1Þg :
ð8Þ
j¼2
As it can be seen from Eq. (8), mii and finþ2 are the only terms required in order to obtain the expected profit per item. These terms can be represented in the following general forms: mii ¼
fknþ2
1 ; 1 pii nk X pknþ2 ¼ þ 1 pkk i¼1
ð9aÞ (
) nk Y pniþ1nþ2 pnjnjþ1 : 1 pniþ1niþ1 j¼i 1 pnjnj
ð9bÞ
5. Numerical example 5.1. Single-stage system 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 l 1 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 the specified range of ½L1 ¼ 8; U1 ¼ 12 . 5.2. Two-stage system 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. Using the GRG method, the expected profit is maximized at l 1 ¼ 10:5708 and l 2 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 .
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Expected Profit
90 80 70 60 50 40 30 20
µ1* = 10.6144 8
8.5
9
9.5
10
10.5
11
11.5
12
Process Mean
Fig. 5. Expected profit versus process mean.
8 9 10
µ2 11 12 50
µ2* = 10.5708 25 Expected 0
µ1* = 15.6301
Profit
-25 17
16 15 14
µ1
13
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 singlestage 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 RC SC r L U
25 15 10 1.0 8 12
20 12 17 1.0 13 17
12 8 5 1.0 10 14
15 10 12 1.0 7 11
4 2 3 1.0 18 22
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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
l 1 l 2 l 3 l 4 l 5 Expected profit
10.6144
10.5708 15.6301
10.4937 10.6821 10.7982
10.4944 15.5795 12.9347 9.8413
93.4377
70.8264
49.8332
40.0821
10.4792 15.5485 12.9392 9.8299 21.2054 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 in the following sections. 6.1. Single-stage system
Optimum Process Mean.
10.66
93.55
10.65
Mean Expected Profit
10.64
93.5 93.45
10.63
93.4
10.62 93.35
10.61 10.6
93.3
10.59
93.25
10.58 0
5
10
15
20
25
30
35
Optimum Expected Profit
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.
93.2 40
Scrap Cost
Fig. 7. Effect of scrap cost on optimal value of l1 and expected profit.
Optimum Process Mean
11.4 11.2 11.0 10.8 10.6 10.4 10.2 0
5
10
15
20
25
30
35
Rework Cost
Fig. 8. Effect of rework cost on optimal value of l1 .
40
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Optimum Expected Profit
648
95.0 94.5 94.0 93.5 93.0 92.5 92.0 91.5 91.0 0
5
10
15
20
25
30
35
40
Rework Cost
Fig. 9. Effect of rework cost on optimal value of expected profit.
6.2. Two-stage system 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 for any 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 2 3 4 5
7 11 15 19 23
10.5534 10.5623 10.5708 10.5790 10.5870
15.6223 15.6262 15.6301 15.6339 15.6377
70.9089 70.8673 70.8264 70.7862 70.7466
SC2
6 7 8 9 10
4 8 12 16 20
10.5709 10.5708 10.5708 10.5708 10.5707
15.6223 15.6263 15.6301 15.6339 15.6376
70.8637 70.8449 70.8264 70.8080 70.7898
RC1
11 12 13 14 15
2 6 10 14 18
10.9458 10.6911 10.5708 10.4910 10.4311
15.6301 15.6301 15.6301 15.6301 15.6301
71.7178 71.1974 70.8264 70.5222 70.2587
RC2
16 17 18 19 20
9 13 17 21 25
10.5726 10.5716 10.5708 10.5701 10.5694
15.7794 15.6933 15.6301 15.5800 15.5386
71.6735 71.2192 70.8264 70.4756 70.1559
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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 deviation, the sensitivity analysis showed the behavior of the optimum process target under different conditions.
References Al-Fawzan, M.A., Rahim, M.A., 2001. Optimal control of deteriorating process with a quadratic loss function. Quality and Reliability Engineering-International 17 (6), 459–466. Al-Sultan, K.S., 1994. An algorithm for determination of the optimum target values for two machines in series with quality sampling plan. International Journal of Production Research 32, 37–45. Al-Sultan, K.S., Pulak, M.F.S., 2000. Optimum target values for two machines in series with 100% inspection. European Journal of Operational Research 120, 181–189. Arcelus, F.J., Banerjee, P.K., 1985. Selection of the most economical production plan in a tool-wear process. Technometrics 27 (4), 433–437. Arcelus, F.J., Rahim, M.A., 1990. Optimal process levels for the joint control of variables and attributes. European Journal of Operations Research (45), 224–230. Bettes, D.C., 1962. Finding an optimal target value in relation to a fixed lower limit and an arbitrary upper limit. Applied Statistics 11, 202–210. Bisgaard, S., Hunter, W.G., Pallesen, L., 1984. Economic selection of quality of manufactured product. Technometrics 26, 9–18. Boucher, T.O., Jafari, M.A., 1991. The optimum target value for single filling operations with quality plans. Journal of Quality Technology 23 (1), 44–47. Carlsson, O., 1984. Determining the most profitable process level for a production process under different sales conditions. Journal of Quality Technology 16, 44–49. Chen, S.L., Chung, K.J., 1996. Selection of the optimal precision level and target value for a production process: The lowerspecification-limit case. IIE Transactions 28, 979–985. Cho, B.R., 2002. Optimum process target for two quality characteristics using regression analysis. Quality Engineering 15 (1), 37–47. Das, C., 1995. Selection and evaluation of most profitable process targets for control of canning quality. Computers and Industrial Engineering 28 (2), 259–266. Golhar, D.Y., 1987. Determination of the best mean contents for a canning problem. Journal of Quality Technology 19, 82–84. Golhar, D.Y., Pollock, S.M., 1988. Determination of the optimal process mean and the upper limit for a canning problem. Journal of Quality Technology 20, 188–192. Hong, S.H., Elsayed, E.A., 1999. The optimal mean for processes with normally distributed measurement error. Journal of Quality Technology 31 (3), 338–344. Hunter, W.G., Kartha, C.P., 1977. Determining the most profitable target value for a production process. Journal of Quality Technology 9, 176–181. Kim, Y.J., Cho, B.R., Philips, M.D., 2000. Determination of the optimum process mean with the consideration of variance reduction and process capability. Quality Engineering 13 (2), 251–260. Liu, W., Taghavachari, M., 1997. The target mean problem for an arbitrary quality characteristic distribution. International Journal of Production Research 35 (6), 1713–1727. Nelson, L.S., 1978. Best target value for a production process. Journal of Quality Technology 10, 88–89. Nelson, L.S., 1979. Nomograph for setting process to minimize scrap cost. Journal of Quality Technology 11, 48–49. Pakkala, T.P.M., Rahim, M.A., 1999. Determination of an optimal setting and production run using Taguchi loss function. International Journal of Reliability, Quality and Safety Engineering 6 (4), 335–346. Phillips, D.M., Cho, R.B., 2000. A nonlinear model for determining the most economical process mean under a Beta distribution. International Journal of Reliability, Quality and Safety Engineering 7 (1), 61–74. Pollock, S.M., Golhar, D., 1998. The canning problem revisited: The case of capacitated production and fixed demand. European Journal of Operations Research 105, 475–482.
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Rahim, M.A., Al-Sultan, K.S., 2000. Joint determination of the target mean and variance of a process. Journal of Quality Maintenance Engineering 6 (3), 192–199. Rahim, M.A., Bhadury, J., Al-Sultan, K.S., 2002. Joint economic selection of target mean and variance. Engineering Optimization 34 (1), 1–14. Rahim, M.A., Shaibu, A.B., 2000. Economic selection of optimal target values. Process Control and Quality 11 (5), 369–381. Schmidt, R.L., Pfeifer, P.E., 1991. Economic selection of the mean and upper limit for a canning problem with limited capacity. Journal of Quality Technology 23 (4), 312–317. Shao, Y.E., Fowler, J.W., Runger, G.C., 2000. Determining the optimal target for a process with multiple markets and variable holding cost. International Journal of Production Economics 65 (3), 229–242. Springer, C.H., 1951. A method for determining the most economic position of a process mean. Industrial Quality Control 8, 36–39. Teeravaraprug, J., Cho, B.R., 2002. Designing the optimal target levels for multiple quality characteristics. International Journal of Production Research 40 (1), 37–54. Teeravaraprug, J., Cho, B.R., Kennedy, W.J., 2000. Designing the most cost-effective process target using regression analysis: A case study. Process Control and Quality 11 (6), 469–477. Usher, J.S., Alexander, S.M., Duggines, D.C., 1996. The filling problem revisited. Quality Engineering 9 (1), 35–44.
Production, Manufacturing and Logistics
A Markovian approach to determining optimum process target levels for a multi-stage serial production system Shannon R. Bowling a, Mohammad T. Khasawneh b, Sittichai Kaewkuekool c, Byung Rae Cho a,* a
b
Department of Industrial Engineering, Clemson University, Clemson, SC 29634-0920, USA Department of Systems Science and Industrial Engineering, State University of New York at Binghamton, Binghamton, NY 13902, USA c Department of Production Technology Education, King MongkutÕs 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 decision 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. Ó 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 processes is the determination of optimum process target (mean). Selecting the optimum target for a process is critically 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 produced 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 Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(03)00429-6
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focuses primarily on the cost of non-conformance. Consider a certain quality characteristic, where a product 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 specification 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 determining 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 limits are 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 under the 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) determined 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 can be 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
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machine setting when two machines are connected in series. Das (1995) used maximization of expected profits 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 model considering 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 reduction 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, Teeravaraprug 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 consideration 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 EðBFÞ EðPCÞ EðSCÞ EðRCÞ SP PCi SCi RCi n Xi li r2i Li Ui UðxÞ
expected profit per item expected benefit per item expected processing cost per item expected scrap cost per item expected rework cost per item selling price per item processing cost associated with stage i scrap cost associated with stage i rework cost associated with stage i number of stages quality characteristic associated with stage i process mean setting for machine i process variance setting for machine i lower specification limit associated with stage i upper specification limit associated with stage i cumulative normal function
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P Q R A O M F pij mij fij
639
transition probability matrix square matrix containing transition probabilities of going from any non-absorbing state to any other non-absorbing state matrix containing all probabilities of going from any non-absorbing state to an absorbing state (i.e., finished or scrapped product) an identity matrix representing the probability of staying in an state matrix representing the probabilities of escaping an absorbing state (always zero) fundamental matrix containing the expected number of transitions from any non-absorbing state to any other non-absorbing state before absorption occurs absorption probability matrix containing the long run probabilities of the transition from any nonabsorbing state to any absorbing state the probability of going from state i to state j in a single step expected number of transitions from any non-absorbing state (i) to any other non-absorbing state (j) before absorption occurs 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 stage i þ 1. 3.3. Rationale for screening inspection Recent advances in technology have motivated the automation of many of todayÕs complex manufacturing 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 systems 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
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item is then reworked, accepted or scrapped. Therefore, the expected profit per item can be expressed as follows: 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 production system. 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: 2 Z 1 x l 12 1r 1 1 1 pffiffiffiffiffiffi e p11 ¼ dx1 ¼ 1 UðU1 Þ; ð2aÞ 2pr1 U1
Fig. 1. A single-stage production system.
Fig. 2. Illustration of accepted, reworked, and scrapped probabilities.
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p12 ¼
Z
U1 L1
p13 ¼
Z
L1 1
1 1 pffiffiffiffiffiffi e 2 2pr1
1 1 pffiffiffiffiffiffi e 2 2pr1
x1 l1 r1
x1 l1 r1
641
2 dx1 ¼ UðU1 Þ UðL1 Þ;
ð2bÞ
dx1 ¼ UðL1 Þ:
ð2cÞ
2
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: 1
M ¼ ðI QÞ
¼ m11 ¼
1 ; ð1 p11 Þ
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 initial state 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 per item (SP) multiplied by the absorption probability of an item being accepted (i.e., f12 ). The expected processing 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 of these 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 state 1 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 that transient state 1 is occupied for the reworking purposes until absorption is m11 1. Consequently the
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expected rework cost is given by RC1 ðm11 1Þ. Therefore, the expected profit per item for a single-stage production system can be expressed as a function of f12 , f13 and m11 as follows: EðPRÞ ¼ SP f12 PC1 SC1 f13 RC1 ðm11 1Þ: ð3Þ Substituting for f13 and m11 , the expected profit equation can be rewritten as follows: p13 p13 p11 RC1 EðPRÞ ¼ SP 1 PC1 SC1 : 1 p11 1 p11 1 p11 The equation can then be rewritten in terms of the cumulative normal distribution as follows: UðL1 Þ UðL1 Þ 1 UðU1 Þ RC1 EðPRÞ ¼ SP 1 PC1 SC1 : UðU1 Þ UðU1 Þ UðU1 Þ
ð4Þ
ð5Þ
The terms UðL1 Þ and UðU1 Þ are functions of the decision variable l1 , which is the process mean. Obviously, one would like to find the value of l1 that maximizes the expected profit. This can be performed numerically 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 a product at stage i, and pinþ2 is the probability of scrapping a product at stage i, where n is the number of stages. 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
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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 by the long-term percentage of scrapped products at stage 2 (i.e., ð1 f14 Þ 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 expressed as follows: EðPRÞ ¼ ½SPð1 f14 Þð1 f24 Þ ½PC1 þ PC2 ð1 f14 Þ ½SC1 f14 þ SC2 ð1 f14 Þf24 ½RC1 ðm11 1Þ þ RC2 ðm22 1Þð1 f14 Þ ;
ð6aÞ
p14 p12 p24 p24 þ 1 EðPRÞ ¼ SP 1 ð1 p11 Þ ð1 p11 Þð1 p22 Þ ð1 p22 Þ
p14 p12 p24 þ PC1 þ PC2 1 ð1 p11 Þ ð1 p11 Þð1 p22 Þ
p14 p12 p24 p14 p12 p24 p24 SC1 þ þ SC2 1 þ ð1 p11 Þ ð1 p11 Þð1 p22 Þ ð1 p11 Þ ð1 p11 Þð1 p22 Þ ð1 p22 Þ
p11 p22 p14 p12 p24 þ ; þ RC2 1 ð6bÞ RC1 1 p11 1 p22 ð1 p11 Þ ð1 p11 Þð1 p22 Þ
UðL1 Þ ½UðU1 Þ UðL1 Þ UðL2 Þ UðL2 Þ EðPRÞ ¼ SP 1 þ 1 UðU1 Þ UðU1 ÞUðU2 Þ UðU2 Þ
UðL1 Þ ½UðU1 Þ UðL1 Þ UðL2 Þ þ PC1 þ PC2 1 UðU1 Þ UðU1 ÞUðU2 Þ
UðL1 Þ ½UðU1 Þ UðL1 Þ UðL2 Þ UðL1 Þ ½UðU1 Þ UðL1 Þ UðL2 Þ UðL2 Þ þ þ SC2 1 þ SC1 UðU1 Þ UðU1 ÞUðU2 Þ UðU1 Þ UðU1 ÞUðU2 Þ UðU2 Þ
1 UðU1 Þ 1 UðU 2Þ UðL1 Þ ½UðU1 Þ UðL1 Þ UðL2 Þ þ RC2 1 þ : RC1 ð6cÞ UðU1 Þ UðU2 Þ UðU1 Þ UðU1 ÞUðU2 Þ
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The terms UðL1 Þ, UðU1 Þ, UðL2 Þ, and UðU2 Þ are functions of the decision variables l1 and l2 , which are the process 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.
:
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645
Therefore, the expected profit per item for an n-stage serial production system can be expressed as follows: ! " !)# ( n n i Y Y X EðPRÞ ¼ SP ð1 finþ2 Þ PC1 þ PCi 1 fj1nþ2 "
i¼1
SC1 f1nþ2 þ
n X
i¼2
( SCi
i¼2
"
RC1 ðm11 1Þ þ
n X i¼2
(
i Y
j¼2
!
)#
ð1 fj1nþ2 Þ finþ2
j¼2
RCi ðmii 1Þ
i Y
!)# ð1 fj1nþ2 Þ
:
ð7Þ
j¼2
Combining the terms, the above model can be further simplified to the following: ! n Y EðPRÞ ¼ SP ð1 finþ2 Þ PC1 SC1 f1nþ2 RC1 ðm11 1Þ i¼1
n X
"
i¼2
i Y
# ð1 fj1nþ2 ÞfPCi þ SCi finþ2 þ RCi ðmii 1Þg :
ð8Þ
j¼2
As it can be seen from Eq. (8), mii and finþ2 are the only terms required in order to obtain the expected profit per item. These terms can be represented in the following general forms: mii ¼
fknþ2
1 ; 1 pii nk X pknþ2 ¼ þ 1 pkk i¼1
ð9aÞ (
) nk Y pniþ1nþ2 pnjnjþ1 : 1 pniþ1niþ1 j¼i 1 pnjnj
ð9bÞ
5. Numerical example 5.1. Single-stage system 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 l 1 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 the specified range of ½L1 ¼ 8; U1 ¼ 12 . 5.2. Two-stage system 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. Using the GRG method, the expected profit is maximized at l 1 ¼ 10:5708 and l 2 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 .
646
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Expected Profit
90 80 70 60 50 40 30 20
µ1* = 10.6144 8
8.5
9
9.5
10
10.5
11
11.5
12
Process Mean
Fig. 5. Expected profit versus process mean.
8 9 10
µ2 11 12 50
µ2* = 10.5708 25 Expected 0
µ1* = 15.6301
Profit
-25 17
16 15 14
µ1
13
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 singlestage 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 RC SC r L U
25 15 10 1.0 8 12
20 12 17 1.0 13 17
12 8 5 1.0 10 14
15 10 12 1.0 7 11
4 2 3 1.0 18 22
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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
l 1 l 2 l 3 l 4 l 5 Expected profit
10.6144
10.5708 15.6301
10.4937 10.6821 10.7982
10.4944 15.5795 12.9347 9.8413
93.4377
70.8264
49.8332
40.0821
10.4792 15.5485 12.9392 9.8299 21.2054 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 in the following sections. 6.1. Single-stage system
Optimum Process Mean.
10.66
93.55
10.65
Mean Expected Profit
10.64
93.5 93.45
10.63
93.4
10.62 93.35
10.61 10.6
93.3
10.59
93.25
10.58 0
5
10
15
20
25
30
35
Optimum Expected Profit
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.
93.2 40
Scrap Cost
Fig. 7. Effect of scrap cost on optimal value of l1 and expected profit.
Optimum Process Mean
11.4 11.2 11.0 10.8 10.6 10.4 10.2 0
5
10
15
20
25
30
35
Rework Cost
Fig. 8. Effect of rework cost on optimal value of l1 .
40
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Optimum Expected Profit
648
95.0 94.5 94.0 93.5 93.0 92.5 92.0 91.5 91.0 0
5
10
15
20
25
30
35
40
Rework Cost
Fig. 9. Effect of rework cost on optimal value of expected profit.
6.2. Two-stage system 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 for any 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 2 3 4 5
7 11 15 19 23
10.5534 10.5623 10.5708 10.5790 10.5870
15.6223 15.6262 15.6301 15.6339 15.6377
70.9089 70.8673 70.8264 70.7862 70.7466
SC2
6 7 8 9 10
4 8 12 16 20
10.5709 10.5708 10.5708 10.5708 10.5707
15.6223 15.6263 15.6301 15.6339 15.6376
70.8637 70.8449 70.8264 70.8080 70.7898
RC1
11 12 13 14 15
2 6 10 14 18
10.9458 10.6911 10.5708 10.4910 10.4311
15.6301 15.6301 15.6301 15.6301 15.6301
71.7178 71.1974 70.8264 70.5222 70.2587
RC2
16 17 18 19 20
9 13 17 21 25
10.5726 10.5716 10.5708 10.5701 10.5694
15.7794 15.6933 15.6301 15.5800 15.5386
71.6735 71.2192 70.8264 70.4756 70.1559
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649
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 deviation, the sensitivity analysis showed the behavior of the optimum process target under different conditions.
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