A chance-constrained approach to stochastic line balancing problem
European Journal of Operational Research 180 (2007) 1098–1115 www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
A chance-constrained approach to stochastic line balancing problem Ku¨rsßad Ag˘pak a, Hadi Go¨kc¸en
b,*
a
b
Faculty of Engineering, Department of Industrial Engineering, Gaziantep University, Gaziantep, Turkey Faculty of Engineering and Architecture, Department of Industrial Engineering, Gazi University, Maltepe, 06570 Ankara, Turkey Received 18 November 2004; accepted 18 April 2006 Available online 11 July 2006
Abstract In this paper, chance-constrained 0–1 integer programming models for the stochastic traditional and U-type line balancing (ULB) problem are developed. These models are solved for several test problems that are well known in the literature and the computational results are given. In addition, a goal programming approach is presented in order to increase the system reliability, which is arising from the stochastic case. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Production; U-line balancing
1. Introduction An assembly line typically consists of several workstations, each of them being responsible for performing a specific set of tasks. Products stay at each workstation for the cycle time (C), which corresponds to the time interval between successively completed units. The assembly line balancing (ALB) problem is a prominent manufacturing problem, which originated in the early 1950s. The objective of this problem is to allocate the assembly tasks to a given number of workstations in equal portions so as to maximize the throughput. Furthermore, line balancing will reduce the differences between assigned workloads so that one operator is not very busy while other operators are idle. The studies related to the assembly line can be classified in two general groups. These are traditional straight assembly lines (with single and multi/mixed products) and U-type assembly lines (with single and multi/mixed products). The literature on assembly line balancing is relatively extensive. Studies on traditional assembly line balancing, the review papers of Baybars [1], Ghosh and Gagnon [2], Erel and Sarin [3] can be examined. In traditional line balancing problems the assembly line is designed straightly, and in U-type balancing problems the line is designed as U shape. In U-type design the entrance and the exit of the line are on the same position. *
Corresponding author. Tel.: +90 312 231 74 00/2839; fax: +90 312 230 84 34. E-mail address: [email protected] (H. Go¨kc¸en).
0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.04.042
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Fig. 1 illustrates a traditional straight assembly line and a U-type assembly line with a cycle time of 12 time units including 9 tasks. Each node in Fig. 1 represents the tasks and the numbers next to the nodes define the task times. There is a small and growing literature on ULB problem. SULB (Simple U Line Balancing) problem was first modelled by Miltenburg and Wijngaard [5]. Urban [6] proposed an integer programming formulation for determining the optimal balance. So, several techniques have been proposed for the solution of U-line balancing problems (see the paper of Scholl and Klein [7], Ohno et al. [8], Miltenburg [9], Sparling and Miltenburg [10], Miltenburg [11]). Another classification divides the assembly lines into two groups depending on the task times: deterministic and stochastic. In stochastic ALB problem, the task times are defined as probability distributions rather than being defined as determined and constant values. Among the reasons of this variability, the following factors can be mentioned: fatigue, loss of attention, workforce with insufficient qualifications, insufficient work satisfaction, erroneous entries, and breakdowns. In literature, normal distribution is widely used for task times [12–19]. For studies on stochastic traditional assembly line balancing, the review paper of Erel and Sarin [3] can be seen. The first study for stochastic case on U-type lines has been done by Ohno et al. [8]. In the mentioned study, there are analyses on random task times. Guerriero and Miltenburg [19] presented a dynamic programming approach to solve the stochastic U-type line balancing problem. Chance-constrained programming is one of the well-known techniques for modelling stochastic systems. Technique was developed by Charnes and Copper [20–22]. Urban and Chiang [23] formulated the stochastic U-line balancing problem as a chance-constrained programming model and used piecewise linear, integer program to find the optimal solution for the model. In this study, linear transformations are presented for finding the optimal and near optimal solution to chance-constrained programming model. Furthermore, a goal programming approach is presented in order to increase the assembly line system reliability arisen from the stochastic case. This paper is organized as follows. In Section 2, the Urban’s [6] integer programming formulation is given. The chance-constrained programming formulation for traditional and U-line balancing problem and its linear transformations are presented in Section 3. Also in Section 3, some definitions that are new for stochastic line balancing problem are given. Computational results and evaluation about the methods on test problems are given in Section 4. In Section 5, a goal programming approach and an illustrative example are presented for minimization of the mean station time and variance differences between stations. Finally, the paper is concluded with a summary of the approach.
4
2 1
2
5
3 3
4
Operator/Workstation #1
6
6 5
6
8
Operator #3
Operator #2
3
4 7
3 9
Operator # 4
a 2 1
2
4
Operator #1
3 9
3 3
Operator #2
3 8
5 4
5
Operator #3
4 7
6
6 6
b Fig. 1. Traditional (a) and U-type assembly line (b) with a cycle time of 12 [4].
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2. Urban’s integer programming formulation A simple ULB problem can be stated as follows: given set of tasks, n, the performance times of tasks, ti (i = 1, . . . , n), and the set of precedence relations, the problem is to assign the tasks to the workstations, (j = 1, . . . , m), such that the precedence relations are satisfied and some performance measures such as minimization of the cycle time or the number of workstation, etc., is achieved. The distinguishing feature of the ULB problem is that it must allow for the forward and backward assignment of tasks to workstations; for example, the first and the last task of an assembly can be placed in the same workstation on a U-line, but not on a traditional line [6]. Urban [6] accomplished this by establishing a ‘‘Phantom’’ network and appending it to the original precedence network. To illustrate this concept, he considered the classical test problem from Jackson [24]. Fig. 2, denotes the precedence network of this 11 task problem with the phantom network (dotted lines) attached to it. By starting in the middle of this extended network, assignments to the workstations can be made forward through the original network, backward through the phantom network, or simultaneously in both directions [6]. Basic difference between stochastic and deterministic cases is, in stochastic case task times are assumed to get values based on a probability distribution whereas in deterministic case task times are constant numbers. For this reason, while developing mathematical models for U-type and traditional assembly lines, integer-programming model of Urban [6] was used as a basis. Urban’s [6] integer programming model is formulated as follows: Objective function: m max X Min Sj
ð1Þ
j¼dmmin e
Constraints: m max X ðxij þ y ij Þ ¼ 1
for i ¼ 1; . . . ; n;
ð2Þ
j¼1 n X i¼1 n X
ti ðxij þ y ij Þ 6 C
for j ¼ 1; . . . ; dmmin e;
ti ðxij þ y ij Þ 6 CS j
ð3Þ
for j ¼ dmmin e þ 1; . . . ; dmmax e;
i¼1 m max X
ðmmax j þ 1Þðxrj xsj Þ P 0
ð4Þ
for all ðr; sÞ 2 P ;
ð5Þ
for all ðr; sÞ 2 P ;
ð6Þ
j¼1 m max X
ðmmax j þ 1Þðy sj y rj Þ P 0
j¼1
xij ; y ij ; S j 2 f0; 1g
2
1
6
for all i; j:
8
3
4
5
10
2
11
7
9
1
6
8
10
3
4
11
7
9
5
Fig. 2. Precedence network of the Jackson’s problem with phantom network appended [6].
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The notations used in the formulation are given below: n: total number of tasks, C: cycle time, ti: performance time of task i, i = 1, . . . , n, mmin: theoretical minimum number of workstation, mmax: maximum number of workstations which can be estimated from the operational setting or by utilizing U-line heuristics shown to perform well, xij: 1, if task i in the original precedence diagram is assigned to workstation j; 0 otherwise, i = 1, . . . , n, j = 1, . . . , mmax, yij: 1, if task i in the phantom precedence diagram is assigned to workstation j; 0 otherwise, i = 1, . . . , n, j = 1, . . . , mmax, Sj: 1, if there is any task assigned to workstation j; 0, otherwise, j = 1, . . . , mmax, P: the set of precedence relationships of tasks; for example, task r immediately precedes task s. Constraint (2) ensures that all tasks are assigned to a station and each task is assigned only once. Constraints (3) and (4) ensure that the work content of any station does not exceed the cycle time. Constraints (5) and (6) ensure that the precedence constraints are not violated on the original network and phantom network. As a result of objective function, the number of workstations will be minimized. If the yij variable and constraint 6 are omitted, the model can be used for traditional line balancing problems. Task times are used as coefficients only in constraints (3) and (4). Thus, only these two constraints will need to be transformed if the model is stochastic. 3. A chance-constrained 0–1 integer programming model for stochastic line balancing problem The objective of the model is to accomplish the balancing with minimum number of stations so that the probability of the total of task times assigned to stations being greater than cycle time (Pk) should stay under predetermined limits (a) (Pk 6 a). It is seen that (a) is greater than 0.8 in practice and literature. At the same time, (a) is assumed greater than 0.5 for valid and correct models in this study, too. In models, the task times are assumed to be normally distributed with known means and variances. Furthermore, each task’s time is independent of any other task’s time and the ordering of tasks into work assignments P [12–19,23]. P P 2 P Y l 2 p ffiffiffiffiffiffiffiffiffi If Y ¼ ti , while task time ti N ðli ; ri Þ, then Y N li ; ri . Using Z ¼ P 2i transformation, then ri
Z N(0, 1). Since the allowed maximum probability that station time exceeds cycle time is a, P(Y > C) 6 a can be written. After modifications, 1 P(Y 6 C) 6 a or P(Y 6 C) P 1 a. Using Z transformation we get P ! C li ffiffiffiffiffiffiffiffiffiffiffi P 1 a: P Z 6 pP r2i
ð7Þ
P(Z 6 z1a) = 1 a, z1a is 100(1 /) percentile of the standard normal distribution. Expression (7) is realized if and only if P C li z1a 6 pffiffiffiffiffiffiffiffiffiffi ð8Þ P 2ffi ri or X
li þ z1a
qffiffiffiffiffiffiffiffiffiffiffiffi X ffi r2i 6 C:
ð9Þ
When task variables (xij, yij) which are used in integer programming model for deterministic U-line, are added to inequality (9), then we can obtain the constraints (10) and (11). In the new model, these constraints will replace the cycle-time constraints (3) and (4) in the deterministic model.
K. Ag˘pak, H. Go¨kc¸en / European Journal of Operational Research 180 (2007) 1098–1115
1102 n X i¼1 n X
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X li ðxij þ y ij Þ þ z1a r2i ðxij þ y ij Þ 6 C
for j ¼ 1; . . . ; dmmin e;
ð10Þ
i¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X li ðxij þ y ij Þ þ z1a r2i ðxij þ y ij Þ 6 CS j
i¼1
for j ¼ dmmin e þ 1; . . . ; mmax :
ð11Þ
i¼1
Chance-constrained programming model for stochastic simple model U-type assembly line balancing (SSULB) problem is developed with the new constraints as follows: Objective function: mmax X Sj Min j¼dmmin e
Subject to: Constraints
ð2Þ; ð5Þ; ð6Þ; ð10Þ; ð11Þ; xij ; y ij ; S j 2 f0; 1g
for i; j:
The obtained new model is a non-linear 0–1 integer programming model. It is known that the non-linear nature is more complex than the linear nature. In the literature, solutions can be achieved for at most 45 tasks for deterministic case using linear 0–1 programming model [6]. Accordingly, the new model is expected to provide solutions for smaller size problems than the deterministic case. With linear-transformation, it is possible to solve the problems with greater size than the ones that can be solved using non-linear model. For linear-transformation, two approaches were used. First approach is linear approach and the second is pure linear transformation. 3.1. Linear approach for chance-constrained model of SSULB problem Linear approach (LA) gives an approximate solution rather than the optimal to the problems. sffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n n n n X X X X þ a2i 6 ai ; ai 2 R ) r2i ðxij þ y ij Þ 6 ri ðxij þ y ij Þ: i¼1
i¼1
i¼1
ð12Þ
i¼1
According to the inequality (12), the constraints (10) and (11) can be rewritten as follows: n X i¼1 n X i¼1
li ðxij þ y ij Þ þ z1a li ðxij þ y ij Þ þ z1a
n X i¼1 n X
ri ðxij þ y ij Þ 6 C ri ðxij þ y ij Þ 6 CS j
for j ¼ 1; . . . ; dmmin e; for j ¼ dmmin e þ 1; . . . ; mmax :
ð13Þ ð14Þ
i¼1
By doing so, the non-linear components of the model were linearized. With the last form of the model, upper limits for partially large-scale problems can be obtained. The approach used is based on a simple mathematical inequality and does not need a new variable or constraint added to the model. If (xij, yij) terms should be taken into account as multipliers in the left hand side of the constraints (13) and (14), then we can obtain the following inequalities, respectively. In inequalities (13) and (14), if the left hand side is bracketed using (xij, yij) then we can reach the equalities (15) and (16). n X ðli þ z1a ri Þðxij þ y ij Þ 6 C for j ¼ 1; . . . ; dmmin e; ð15Þ i¼1 n X ðli þ z1a ri Þðxij þ y ij Þ 6 CS j i¼1
for j ¼ dmmin e þ 1; . . . ; mmax :
ð16Þ
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If it is carefully examined, it can be seen that the constraints (15) and (16) are similar to the constraints (3) and (4) in the model given for deterministic case. With the new approach only the coefficients has been changed. Accordingly, we can say that the new model will have the same characteristics with the model developed for deterministic case. 3.2. Pure linear transformation for chance-constraint model of SSULB problem ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pP n 2 The reason why the developed chance-constraint model is not linear is the term i¼1 ri xij with task variable. For that reason by eliminating the square-root, linearizing will be possible. So, by squaring both side of the inequality (8), inequality (17) has been obtained. 2 Pn C i¼1 li xij ðz1a Þ 6 pP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi2 : n 2 r x ij i¼1 i 2
ð17Þ
pP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi n 2 While the non-linear situation caused by the term i¼1 ri xij has been eliminated by inequality (17), a new 2 Pn non-linear situation has been raised with term C i¼1 l2i xij . The open form of this term is given by Eq. (18). 0
C
X
li xij
2
¼ C 2 2C
X
l21 x21j þ l1 x1j l2 x2j þ þ l1 x1j ln xnj
1
B C B l1 x1j l2 x2j þ l22 x22j þ þ l2 x2j ln xnj C C: li xij þ B B C ... @ A l1 x1j ln xnj þ l2 x2j ln xnj þ þ l2n x2nj
ð18Þ
Since the variable xij is 0–1 integer, x2ij is equal to xij. The term (lixijlnxnj) is the non-linear part of the Eq. (18). So, a current transformation technique in the literature has been used to make this part linear. uinj ¼ xij xnj :
ð19Þ
After the variable transformation given in Eq. (19), the variable uij has been made dependent to xij and xnj by inequalities (20) and (21). xij þ xnj uinj 6 1;
ð20Þ
xij þ xnj 2uinj P 0:
ð21Þ
Another situation to be considered on inequality (17) is that the inequality is valid in two different cases. These can be expressed by the inequality below. C P l i ffiffiffiffiffiffiffiffiffiffiffi : ð22Þ jz1a j 6 pP r2i Assumed a value is greater than 0.5, therefore z1a is positive and right hand side of the inequality (22) must be positive, too. For that reason by using inequality (23) and (17) together, different cases for inequality (22) will be eliminated. This constraint is the same as the cycle time constraint at the basic deterministic model. X C li xij P 0:
ð23Þ
The model linearized with new constraints can be rewritten as follows. The model details (open form) for an example problem are given in Appendix.
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Objective function: mmax X
Min
Sj
j¼dmmin e
Subject to: Constraints
ð2Þ; ð5Þ and ð6Þ C 2 2C
n X
li pij þ
n X
i¼1
l2i pij þ 2
i¼1
for j ¼ 1; . . . ; mmax ; ! n X li pij P 0 CS j
n1 X n X
li lv uivj z21a
i¼1 v¼iþ1
n X
! r2i pij
P0
i¼1
ð24Þ for j ¼ 1; . . . ; mmax ;
ð25Þ
i¼1
pij þ pvj uivj 6 1
for j ¼ 1; . . . ; mmax ; i ¼ 1; . . . n 1; v ¼ i þ 1; . . . ; n;
pij þ pvj 2uivj P 0 xij þ y ij pij ¼ 0
for j ¼ 1; . . . ; mmax ; i ¼ 1; . . . n 1; v ¼ i þ 1; . . . ; n;
for j ¼ 1; . . . ; mmax ; i ¼ 1; . . . n;
xij ; y ij ; pij ; S j ; uivj 2 f0; 1g
ð26Þ ð27Þ ð28Þ
for i; j; v:
Cycle time, assignment and priority constraints take place in the new model like in basic deterministic model. The difference of the new model is in the constraints given by inequalities (24), (26), (27) and (28). All the mathematical models given at Section 3 are for U-type (shaped) assembly lines. If yij variables and constraint (6) are removed from the model, constructed models can be used for traditional assembly lines. mmin (Theoretical Minimum Number of Station), Ei (earliest station for task i) and Li (latest station for task i) parameters obtained for deterministic assembly line balancing problems are also a necessity either for U-type (shaped) or traditional assembly line balancing problems in stochastic cases. So, these parameters, which assure reducing the number of variables in the models, should be defined for the stochastic case. First, Urban and Chiang [23] defined lower bound (LB2) considering the stochastic aspect of the problem. In this study, the lower bound LB2 is called mmin and presented a different approach to proof of definition. Furthermore, inferences about Ei and Li parameters that will be new for the literature are given after mmin definition. Definition 1. When the task time ti (ti N ðli ; r2i ÞÞ, number of tasks n, the cycle time C and the upper limit of the allowed probability of exceeding the cycle time of the station time a are given, then the theoretical minimum number of stations mmin is & ’ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 Pn z1a i¼1 ri þ i¼1 li mmin ¼ ; ð29Þ C where dxe is the smallest integer greater than or equal to x. WePassumed that the m* is P the optimalP number of stations. If the station times for each station is Stk ¼ i2k ti , then Stk N lk ¼ i2k li ; r2k ¼ i2k r2i . Also, it is known that qffiffiffiffiffi lk þ z1a r2k 6 C ð30Þ for station k. The sum of inequality (30) for m* station is qffiffiffiffiffi n m n m X X X X li þ ðz1a r2k Þ 6 m C ) li þ z1a rk 6 m C: i¼1
k¼1
i¼1
k¼1
ð31Þ
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According to sffiffiffiffiffiffiffiffiffiffiffiffi n n X X a2i 6 ai ; i¼1
vffiffiffiffiffiffiffiffiffiffiffiffiffi u m m X uX ai 2 Rþ ) t r2k 6 rk k¼1
i¼1
ð32Þ
k¼1
we can write vffiffiffiffiffiffiffiffiffiffiffiffiffi u m n m X X uX t 2 li þ z1a rk 6 li þ z1a rk 6 m C:
n X
k¼1
i¼1
ð33Þ
k¼1
i¼1
It is known that X r2k ¼ r2i :
ð34Þ
i2k
If n X
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u m n m X X uX X 2 t li þ z1a ri 6 li þ z1a rk 6 m C k¼1
i¼1
i2k
k¼1
i¼1
ð35Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffi n n n m X X X X 2 ) li þ z1a ri 6 li þ z1a rk 6 m C i¼1
i¼1
k¼1
i¼1
then Pn mmin ¼
i¼1 li þ z1a C
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 i¼1 ri
6 m
&P or
mmin ¼
n i¼1 li
þ z1a C
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 ’ i¼1 ri
6 m :
ð36Þ
Definition 2. While cycle time is C and the upper limit of the probability of exceeding the cycle time of the allowed station time is a, the earliest station for processing the task i (Ei) is, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 P P li þ j2P j lj þ z1a r2i þ j2P i r2j 7: Ei ¼ 6 ð37Þ 6 7 C 6 7 The theoretical minimum number of station which can be obtained for an assembly line consisting of task i and its predecessors (Pi) is the earliest station which task i can be processed. So, mmin can be used as Ei. Definition 3. While cycle time is C and the upper limit of the allowed probability of exceeding the cycle time of the station time is a and the allowed maximum number of stations is K (K P m), then the latest station for processing the task i (Li) is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 P P lj þ j2Sci lj þ z1a r2i þ j2Sci r2j 7: Li ¼ K þ 1 6 ð38Þ 6 7 C 6 7 The theoretical minimum number of station which can be obtained for an assembly line consisting of task i and its successors (Sci) is the earliest station which task i can be processed. The difference between this value and K + 1 gives the latest station which task i can be assigned. Ei and Li definitions are developed for traditional lines. Therefore new models, which use the bounds, are given below. Firstly, MLAT (the model with linear approach for traditional lines) is presented.
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Objective function: mmax X Min Sj j¼dmmin e
Constraints: Li X
xij ¼ 1 for
i ¼ 1; . . . ; n;
ð39Þ
j¼EI_
X
ðli þ z1a ri Þxij 6 C
for j ¼ 1; . . . ; dmmin e;
ð40Þ
i2W j
X
ðli þ z1a ri Þxij 6 CS j
for j ¼ dmmin e þ 1; . . . ; mmax ;
ð41Þ
i2W j Ls Lr X X ðmmax j þ 1Þxrj ðmmax j þ 1Þxsj P 0 j¼Er
for ðr; sÞ 2 P ;
ð42Þ
j¼Es
xij ; y ij ; S j 2 f0; 1g
for all i; j:
Second model, MLTT (the model with pure linear transformation for traditional lines) is presented below. Objective function: mmax X Sj Min j¼dmmin e
Subject to: ð39Þ and ð42Þ;
Constraints 2
C 2C
X
li xij þ
i2W j
CS j
X
X
l2i xij þ 2
i2W j
X X
li lv uivj z21a
i2W j vj2W j ^>i
X
! r2i xij
P 0 for j ¼ 1;... ;mmax ;
i2W j
ð43Þ
! li xij
P 0 for j ¼ 1;.. .;mmax ;
ð44Þ
i2W j
xij þ xvj uivj 6 1 for j ¼ 1; ...; mmax ; i ¼ Ei ;. ..;Li ; v ¼ Ev ;... ;Lv ; i 6¼ v;
ð45Þ
xij þ xvj 2uivj P 0 for j ¼ 1;.. .;mmax ; i ¼ Ei ;... ;Li ;v ¼ Ev ;. ..;Lv ; i 6¼ v;
ð46Þ
xij ;y ij ;S j ; uivj 2 f0;1g for i;j;v: Wj represents subset of all tasks that can be assigned to workstation j. 4. Computational results and evaluation In this paper, four different new models are suggested for Chance Constraint Model of SSULB Problem. These are MLTT, MLTU (the model with pure linear transformation for U-type lines), MLAT and MLAU (the model with linear approach for U-type lines). Some comparisons on these models with respect to the number of variables and constraints are given in Table 1. While forming the comparison table, it is assumed that the models do not include any upper or lower varn2 þn 2
’’ more variables and ‘‘Kmax(n2 + 1)’’ constraints 2 than MLAU. In similar manner, MLTT requires ‘‘K max n 2n ’’ more variables and ‘‘Kmax(n2 1) + n’’
iable limits. For U-type line, MLTU contains ‘‘K max
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Table 1 Numbers of variables and constraints related to the models MLA
MLT
U-line
No. of variable No. of constraint
Kmax(2n + 1) n + Kmax + 2r
ðn2 þ5nþ2Þ K max 2 Kmax(n2 + 2) +
Traditional line
No. of variable No. of constraint
Kmax(n + 1) n + Kmax + r
ðn2 þnþ2Þ K max 2 Kmax(n2 n +
n + 2r
2) + n + r
constraints than MLAT. The reason of this is that pure linear transformation models give optimal and linear approach models also give approximate results. There are no determined test problem sets in stochastic line balancing literature for testing performance of presented models. However, there are test problem sets for deterministic case. In this study, a new test problem set which can be used in stochastic case, is formed by transforming the deterministic test problem set. To set up this test problem, Gueriero and Miltenburg [18] presented three different variabilities (low, high, random) in task completion time and determined the coefficient of variation (CV) intervals. Determined CV intervals are low (0 < CV < 0.3), high (0.3 < CV < 0.6) and random (0 < CV < 0.6). And then problem sets are established by producing random numbers in these intervals. The same interval descriptions are used in this study too. However in this study, test problem set for deterministic case in the literature is taken as a basis to gain a stochastic structure. Task times in deterministic case are accepted as mean task time(s). For determined variation type, random numbers are produced according to uniform distribution to obtain different CV values for each task. Variation of each task is obtained by multiplying these values with mean task times. In this way, deterministic test problem set is transformed into a usable structure in stochastic case and three different variance sets (low, high, random) according to CV intervals are obtained for one problem. Models are solved on test problems for three different a levels (low 0.15, medium 0.1, high 0.05) and three different cycle times by using GAMS-CPLEX solver on a Pentium IV 2.8 MHz personal computer. Four models are compared with each other with respect to the number of stations. The comparison results are given in Tables 2 and 3. Since the problem is in NP-Hard nature it takes too much time to reach a result for larger problems. So, the solver has been limited by 72,000 seconds (20 hours) in this study. In this study, 216 test problems for linear transformation models and 270 test problems for linear approach models were solved. If the results in Table 2 are observed, it can be seen that optimal solutions have been found in 137 of 216 problems by MLTT and in 95 problems by MLTU. With MLAT, in 57 of 137 problems known to be optimal exactly, the optimal solutions have been found. For 74 of 137, the solutions have been found with a deviation of 1 station and for 5 of 137, the solutions have been found with a deviation of 2 stations and lastly for 1 of 137, the solutions have been found with a deviation of 3 stations. With MLAU, for 49 of 95 problems known to be exactly optimal, the optimal solutions have been found. For 45 of 95, the solutions have been found with a deviation of 1 station and for 1 of 95, the solutions have been found with a deviation of 2 stations. In some problems with more than 25 tasks, optimal results generally cannot be reached in 72,000 seconds by MLTT or MLTU. Near optimal solutions have been generated up to 53 tasks by MLAT and MLAU. For large-scale problems, with MLTT in 5 of 54 problems and with MLTU in 6 of 54 problems, optimal solutions have been found (see Table 3). 5. Goal programming for balancing between stations With the developed models the result can be reached only for one goal. The results found do not guarantee the mean times or variances of the stations to be equal. In real world problems it is preferred the workloads of stations to be equal. At the same time, if we assume that incompletions of all the stations and station times are independent, equalizing workload between workstations will make a positive contribution to increase the total assembly line system reliability in stochastic case. In otherwise, probability that all workstations would complete their assigned tasks within the given cycle time can be increased. In this study, this probability is called
1108
Table 2 Results of comparison of the four models with respect to the number of stations No. of task
Cycle time
Type of variation
/ Level
Jackson
11
13
L
L M H L M H L M H
4 4 4 5 5 5 5 5 5
5 5 5 6 6 7 5 6 6
4 5 5 6 6 7 5 5 6
5 5 6 6 7 8 6 6 6
5 5 5 6 7 7 5 6 6
L M H L M H L M H
3 3 3 3 3 4 3 3 3
3 3 3 4 4 4 4 4 4
3 3 3 4 4 4 4 4 4
3 3 4 4 4 5 4 4 5
3 3 4 4 4 5 4 4 4
35
L M H L M H L M H
2 2 3 3 3 3 3 3 3
2 3 3 3 3 3 3 3 3
2 3 3 3 3 3 3 3 3
3 3 3 3 3 4 3 3 3
3 3 3 3 3 4 3 3 3
41
L M H L M H L M H
5 5 5 6 6 6 6 6 6
6 6 6 7 7 8 6 7 7
6a 6a 6a 7a 7a 8a 6 6 7a
6 6 6 8 8 10 7 7 8
6 6 6 7 8 9 7 7 7
L M H L M H L M H
4 4 5 5 5 5 5 5 5
5 5 5 5 6 6 5 5 6
5a 5a 5 5 6a 6a 5 5 5
5 5 5 6 7 8 6 6 6
5 5 5 6 7 7 5 6 6
H
R
19
L
H
R
25
L
H
R
Roszieng
25
27
L
H
R
33
L
H
R
mmin
Models MLTT
MLTU
MLAT
MLAU
Problem name
No. of task
Mitchell
21
Cycle time 27
Type of variation
/ Level
mmin
MLTU
MLAT
MLAU
L
L M H L M H L M H
5 5 5 5 5 5 5 5 5
5 5 5 6 6 7 5 5 5
5 5 5 6a 6a 7a 5 5 5
5 5 5 7 7 8 5 6 6
5 5 5 6 7 8 5 5 6
L M H L M H L M H
4 4 4 4 4 4 4 4 4
4 4 4 4 5 5 4 4 4
4 4 4 4 5a 5a 4 4 4
4 4 4 5 5 6 4 4 5
4 4 4 5 5 6 4 4 5
L M H L M H L M H
3 3 3 3 3 4 3 3 3
3 3 3 4 4 4 3 3 4
3 3 3 4 4 4 3 3 3
3 4 4 4 5 5 4 4 4
3 3 4 4 5 5 4 4 4
L M H L M H L M H
6 6 6 6 6 6 6 6 6
6 6 6 10a 8a 8a 7a 7a 8a
6 6 6 7a 8a 9a 7a 7a 8a
6 6 7 8 8 9 7 8 8
6 6 7 8 8 9 7 8 8
L M H L M H L M H
5 5 5 5 5 6 5 5 6
5 5 5 6a 6a 7a 6a 6a 7a
5 5 5 6a 6a 7a 6a 6a 7a
6 6 6 7 7 8 6 7 7
6 6 6 7 7 8 6 7 7
H
R
L
H
R
L
H
R
Heskia
28
205
L
H
R
235
L
H
R
Models MLTT
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Problem name
41
L
H
R
29
42
L
H
R
47
L
H
R
52
L
H
R
Lutz-1
32
2600
L
H
R
4 4 4 4 4 4 4 4 4
4 4 4 4 4 5 4 4 4
4 4 4 4 4 5a 4 4 4
4 4 4 5 6 6 5 5 5
4 4 4 5 5 6 4 5 5
265
L M H L M H L M H
8 9 9 9 9 9 9 9 9
9 9 10a 11a 0a 0a 10a 10a 11a
9a 10a 10a 0a 0a 0a 10a 11a 0a
10 10 11 12 13 14 10 11 12
10a 10 10 12 13 14 10 11 12a
L M H L M H L M H
8 8 8 8 8 8 8 8 8
8 8 9a 0a 0a 11a 9a 0a 0a
8 8 9a 10a 0a 0a 9a 0a 0a
9 9 9 11 12 12 9 10 10
9a 9 9 11a 11 11 9 10 10
60
L M H L M H L M H
7 7 7 7 7 8 7 7 7
7 8a 8a 0a 9a 0a 8a 10a 10a
7 9a 8a 0a 0a 0a 8a 8a 8a
8 8 8 10 10 12 9 9 10
8 8 8 10 10 11 8 9 9
75
L M H L M H L M H
6 6 6 6 7 7 6 6 7
6 7 7 8 8 8 7 7 8
0a 0a 7a 0a 8a 0a 7a 7a 0a
7 7 7 9 9 11 8 8 9
7 7 7 8 9 10 8 8 9
L
H
R
Sawyer
30
45
L
H
R
L
H
R
L
H
R
Gunther
35
81
L
H
R
L M H L M H L M H
5 5 5 5 5 5 5 5 5
5 5 5 5 6a 6a 5 5 6a
5 5 5 6a 6a 6a 5 5 6a
5 5 5 6 7 7 6 6 7
5 5 5 6 7 7 6 6 7
L M H L M H L M H
8 8 8 8 9 9 8 8 8
9a 9a 9a 0 0 0 10a 10a 11a
9a 9a 0 0 0 0 9a 0 0
9 9 10 12a 12 13 10 10 11
9 9 10 11 12 13 10 10 11
L M H L M H L M H
6 6 6 6 7 7 6 6 6
6 7a 7a 8a 8a 8a 7a 7a 7a
0 7a 7a 7a 8a 8a 7a 7a 7a
7 7 8 8 9 10 7 8 8
7 7 7 8 9 10 7 8 8
L M H L M H L M H
5 5 5 5 5 5 5 5 5
5 5 6a 6a 6a 7a 6a 6a 6a
5 5 5 6a 6a 0a 5 0 6a
6 6 6 7 7 8 6 6 7
6 6 6 7 7 8 6 6 7
L M H L M H L M H
7 7 7 7 7 7 7 7 7
7 7 8a 0a 9 0a 8a 9a 9a
7 0a 0a 0a 0a 0a 8a 0a 0a
8 7 8 8 8 8 9 9 10 10 11 11 9 8 9 9 10 10a (continued on next page)
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Buxey
L M H L M H L M H
1109
1110
Problem name
No. of task
Cycle time
Type of variation
/ Level
3100
L
L M H L M H L M H L M H L M H L M H
H
R
3600
L
H
R
a
Optimal results cannot be reached in 72,000 seconds.
mmin
Models
Problem name
No. of task
Cycle time
MLTT
MLTU
MLAT
MLAU
5 5 5 5 6 6 5 5 6
5 6 6 6 6 7 6 6 6
5 0a 6a 6a 6 7a 6a 6a 6
6 6 6 7 8 9 7 7 7
6 6 6 7 8 8 6 7 7
91
5 5 5 5 5 5 5 5 5
5 5 5 5 6 6 5 5 5
5 5 5 5 0a 6a 0a 5 5
5 5 5 6 7 7 6 6 6
5 5 5 6 7 7 6 6 6
101
Type of variation
/ Level
mmin
Models MLTT
MLTU
MLAT
MLAU
L
L M H L M H L M H
6 6 6 6 6 7 6 6 6
7a 7a 7a 7 8a 0a 7 7 8a
6 6 7a 0a 0a 0a 0a 7a 0a
7 7 7 8 9 10 8 8 9
7 7 7 8 9 10 8a 8 9a
L M H L M H L M H
5 5 6 6 6 6 6 6 6
6 6 6 7a 7a 8a 7a 7a 7a
6a 6a 6 0a 0a 0a 0a 0a 0a
6 6 6 8 8 9 7 7 8
6 6 6 7 8 9 7 7 8
H
R
L
H
R
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Table 2 (continued)
Table 3 Results of comparison of the two models with respect to the number of stations for large-scale problems No. of task
Cycle time
Type of variation
a Level
mmin
MLAT
MLAU
Kilbridge
45
103
L
L M H L M H L M H
6 6 6 6 6 7 6 6 7
7 7 7 8 9 10 8 8 9
7 7 7 8 9 10 8 8 9
L M H L M H L M H
5 5 5 6 6 6 5 6 6
6 6 6 7 8 8 7 7 8
6 6 6 7 8 8 7 7 8
3775
L M H L M H L M H
5 5 5 5 5 5 5 5 5
5 5 6 6 7 7 6 6 7
5 5 6 6 7 7 6 6 7
3925
H
R
120
L
H
R
137
L
H
R
Models
Problem name
No. of task
Cycle time
Type of variation
a Level
mmin
Models MLAT
MLAU
Hahn
53
3625
L
L M H L M H L M H
5 5 5 5 5 5 5 5 5
5 5 6 7 8 8 6 7 8
5 5 5 6 7 7 6 6 7
L M H L M H L M H
4 4 5 5 5 5 5 5 5
5 5 5 6 7 8 6 6 7
5 5 5 6 6 7 6 6 6
L M H L M H L M H
4 4 4 5 5 5 4 5 5
5 5 5 6 6 8 5 6 7
5 5 5 6 6 7 5 6 6
H
R
L
H
R
L
H
R
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Problem name
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the assembly line system reliability. For that reason, goal programming techniques has been used in order to equalize the workloads between workstations in respect of variances and/or mean times. Mean time goal: With this goal it is aimed to minimize the deviation of the mean times of the stations from a certain value. The goal constraints can be written as n X þ li pij þ dc for j ¼ 1; . . . ; mmax : ð47Þ j dcj ¼ lst i¼1
Variance goal: With this goal it is aimed to minimize the deviation of the variances of the stations from a certain value. The goal constraints can be written as n X þ r2i pij þ dv for j ¼ 1; . . . ; mmax : ð48Þ j dvj ¼ vst i¼1 þ The minimization of sum of the deviational variables dcþ j and dvj will assure the minimization of the deviation from lst and vst values for each workstation. In this way the differences of mean times and variances between stations will be minimized. But before applying the goal programming approach, the problem should be solved with linear transformation and/or linear approach models, and the number of stations (mg) should be found. Then according to the number of stations Pn 2 Pn r i¼1 li lst ¼ and vst ¼ i i values should be determined: mg mg
After this, the number of stations should be written as a constraint and the goal programming model should be solved. To illustrate the goal programming model discussed here, example with different goal sequences is solved. The precedence network of the problem was given in Fig. 2, Jackson problem. The mean task times and variations of the problem are also given in Table 4. A preemptive goal programming technique is used for solving example problem. In a preemptive goal programming model, the upper level goals are first optimized before lower level goals are considered. In a non-preemptive model, the goals are given some weights and considered simultaneously. We utilized the preemptive approach due to the difficulty associated with determining the weights for the various goals [25]. The cycle time and allowed incompletion level for the problem are taken as 19 and 0.1 respectively. The problem is solved using the GAMS-CPLEX mathematical programming package on a Pentium 4-2.0 GHz computer. Task assignments are shown in Table Qm5.g mg is found as four. The assembly line system reliability (ASR ¼ i¼1 Psti Þ is 0.863, lst is 12 and vst is 8.942 as dependent on mg. If the mean time goal takes the first priority (P1) and the variance goal the second (P2), then the balanced result on Table 6 is reached. If the priority sequence is changed the results on Table 7 can be reached. Especially for the goal sequence of P1 and P2, ASR value of 0.975 which is very high is obtained. Mean times and variances are distributed balanced between the stations. Table 4 Mean task times and variations Task
Mean time
Variance
Task
Mean time
Variance times
1 2 3 4 5 6
6 2 5 7 1 2
5.308 0.510 3.151 7.263 0.202 0.767
7 8 9 10 11
3 6 5 5 4
1.004 11.614 5.221 0.723 0.005
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Table 5 Task assignments of the sample problem Workstation
Tasks
Mean station time
Standard deviation
Probability of completion (Pst)
1 2 3 4
1, 2, 3, 5 11, 9 4, 7, 10 6, 8
14 9 15 8
3.028 2.286 2.998 3.518
0.950 0.999 0.909 0.999
Table 6 Balance results for the goal sequence P1 and P2 Workstation
Tasks
Mean station time
Standard deviation
Probability of completion (Pst)
1 2 3 4
1, 2, 8, 7,
12 11 11 12
2.943 2.922 3.512 2.495 OSR
0.991 0.997 0.989 0.998 0.975
3, 5 4, 6 10 9, 11
Table 7 Balance results for the goal sequence P2 and P1 Workstation
Tasks
Mean station time
Standard deviation
Probability of completion (Pst)
1 2 3 4
1, 3, 11 4, 10 5, 6, 7, 9 8
15 12 11 6
2.909 2.826 2.682 3.408 OSR
0.915 0.993 0.999 0.999 0.908
6. Conclusion In this paper, new mathematical models for stochastic traditional and U-type assembly lines have been developed by using chance-constrained programming technique. Since the mathematical model, which has been obtained, is non-linear, pure linear transformation and approximate-linear transformation (linear approach) have been utilized to construct 0–1 integer linear models. These models have been solved for several test problems well-known in the literature and the results have been compared with respect to the number of stations. The experimentation revealed that the MLAT and MLAU are capable of solving problems with more than 35 tasks (at most up to 53 tasks). MLTT and MLTU are generally capable of solving problems with up to 35 tasks. In most of the problems with more than 35 tasks, optimal results cannot be reached in 72,000 seconds. The models serve as a starting point for researchers in the field, and may be used as a validation tool for heuristic procedures. In addition, goal programming approach has been presented in order to increase the assembly line system reliability arised from the stochastic case. This suggested model provides increased flexibility to the decision maker in evaluating different alternatives. Appendix The sample MLTU constraints for illustrative example in Section 5 are given below. All constraints are not given due to greatness of the model. C = 19, a = 0.1, z1a = 1.28155 are given. mmin = 3 is found. mmax = 7 is assumed.
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Objective function: Min
7 X
Si ¼ S3 þ S4 þ S5 þ S6 þ S7
i¼3
Subject to: Constraint ð2Þ for task 1; ðx11 þ y 11 Þ þ ðx12 þ y 12 Þ þ ðx13 þ y 13 Þ þ ðx14 þ y 14 Þ þ ðx15 þ y 15 Þ þ ðx16 þ y 16 Þ þ ðx17 þ y 17 Þ ¼ 1: Constraint ð5Þ and ð6Þ for ð1; 2Þ relationships; 7x11 þ 6x12 þ 5x13 þ 4x14 þ 3x15 þ 2x16 þ x17 7x21 6x22 5x23 4x24 3x25 2x26 x27 P 0; 7y 11 6y 12 5y 13 4y 14 3y 15 2y 16 y 17 þ 7y 21 þ 6y 22 þ 5y 23 þ 4y 24 þ 3y 25 þ 2y 26 þ y 27 P 0: Constraint ð24Þ for workstation 1; 361 228p11 76p21 190p31 266p41 38p51 76p61 114p71 228p81 190p91 190p101 152p111 þ 36p11 þ 4p21 þ 25p31 þ 49p41 þ p51 þ 4p61 þ 9p71 þ 36p81 þ 25p91 þ 25p101 þ 16p111 þ 24u121 þ 60u131 þ 84u141 þ 12u151 þ 24u161 þ 36u171 þ 72u181 þ 60u191 þ 60u1101 þ 48u1111 þ 20u231 þ 28u241 þ 4u251 þ 8u261 þ 12u271 þ 24u281 þ 20u291 þ 20u2101 þ 16u2111 þ 70u341 þ 10u351 þ 20u361 þ 30u371 þ 60u381 þ 50u391 þ 50u3101 þ 40u3111 þ 14u451 þ 28u461 þ 42u471 þ 84u481 þ 70u491 þ 70u4101 þ 56u4111 þ 4u561 þ 6u571 þ 12u581 þ 10u591 þ 10u5101 þ 8u5111 þ 12u671 þ 24u681 þ 20u691 þ 20u6101 þ 16u611 þ 36u781 þ 30u791 þ 30u7101 þ 24u7111 þ 60u891 þ 60u8101 þ 48u8111 þ 50u9101 þ 40u9111 þ 40u10111 1:281552 ð5:308p11 þ 0:51p21 þ 3:151p31 þ 7:263p41 þ 0:202p51 þ 0:767p61 þ 1:004p71 þ 11:614p81 þ 5:221p91 þ 0:723p101 þ 0:005p111 Þ P 0: Constraint ð25Þ for workstation 1; 19S 1 6p11 2p21 5p31 7p41 p51 2p61 3p71 6p81 5p91 5p101 4p111 P 0: Constraint ð26Þ for workstation 1; task 1; j ¼ 2; 3; . . . ; 11; p11 þ p21 u121 6 1;
p11 þ p31 u131 6 1;
p11 þ p41 u141 6 1;
p11 þ p51 u151 6 1
p11 þ p61 u161 6 1;
p11 þ p71 u171 6 1;
p11 þ p81 u181 6 1;
p11 þ p91 u191 6 1
p11 þ p101 u1101 6 1;
p11 þ p111 u1111 6 1:
Constraint ð27Þ for workstation 1; task 1; j ¼ 2; 3; . . . ; 11; p11 þ p21 2u121 P 0;
p11 þ p31 2u131 P 0;
p11 þ p41 2u141 P 0;
p11 þ p51 2u151 P 0
p11 þ p61 2u161 P 0;
p11 þ p71 2u171 P 0;
p11 þ p81 2u181 P 0;
p11 þ p91 2u191 P 0
p11 þ p101 2u1101 P 0;
p11 þ p111 2u1111 P 0:
Constraint ð28Þ for task 1; x11 þ y 11 p11 ¼ 0;
x12 þ y 12 p12 ¼ 0;
x13 þ y 13 p13 ¼ 0;
x15 þ y 15 p15 ¼ 0;
x16 þ y 16 p16 ¼ 0;
x17 þ y 17 p17 ¼ 0:
x14 þ y 14 p14 ¼ 0;
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[7] A. Scholl, R. Klein, ULINO: Optimally balancing U-shaped JIT assembly lines, International Journal of Production Research 37 (1999) 721–736. [8] K. Ohno, K. Nakade, J.G. Shantikumar, Bounds and approximations for cycle times of a U-shaped production line, Operations Research Letters 21 (1997) 191–200. [9] G.J. Miltenburg, Balancing U-lines in a multiple U-line facility, European Journal of Operational Research 109 (1998) 1–23. [10] D. Sparling, G.J. Miltenburg, The mixed-model U-line balancing problem, International Journal of Production Research 36 (1998) 485–501. [11] G.J. Miltenburg, U-shaped production lines: A review of theory and practice, International Journal of Production Economics 70 (2001) 201–214. [12] C.L. Moodie, H.H. Young, A heuristic method of assembly line balancing for assumptions of constant or variable work element times, Journal of Industrial Engineering 16 (1965) 23–29. [13] J.F. Kottas, H.S. Lau, A cost oriented approach to stochastic line balancing, AIIE Transactions 5 (1973) 164–171. [14] J.F. Kottas, H.S. Lau, A total operating cost model for paced lines with stochastic task times, AIIE Transactions 8 (1976) 234–240. [15] J.F. Kottas, H.S. Lau, A stochastic line balancing procedure, International Journal of Production Research 9 (1981) 177–193. [16] G. Suresh, S. Sahu, Stochastic assembly line balancing using simulated annealing, International Journal of Production Research 32 (1994) 1801–1810. [17] D. Shin, An efficient heuristic for solving stochastic assembly line balancing problems, Computers and Industrial Engineering 18 (1990) 285–295. [18] H. Lau, A. Shtub, An exploratory study on stopping a paced line when incompletions occur, IIE Transactions 19 (1987) 463–467. [19] F. Guerriero, J. Miltenburg, The stochastic U-line balancing problem, Naval Research Logistics 50 (2003) 31–57. [20] A. Charnes, W.W. Cooper, Chance-constrained programming, Management Science 6 (1959) 73–79. [21] A. Charnes, W.W. Cooper, Chance constraints and normal deviates, The American Statistical Association Journal 57 (1962) 134–148. [22] A. Charnes, W.W. Cooper, Deterministic equivalents for optimizing and satisficing under chance constraints, Operations Research 11 (1963) 18–39. [23] T.L. Urban, W. Chiang, An optimal piece-wise linear program for the U-line balancing problem with stochastic task times, European Journal of Operational Research 168 (2006) 771–782. [24] J.R. Jackson, A computing procedure for a line balancing problem, Management Science 2 (1956) 261–272. [25] H. Gokcen, K. Agpak, A goal programming approach to simple U-line balancing problem, European Journal of Operational Research 171 (2006) 577–585.
Production, Manufacturing and Logistics
A chance-constrained approach to stochastic line balancing problem Ku¨rsßad Ag˘pak a, Hadi Go¨kc¸en
b,*
a
b
Faculty of Engineering, Department of Industrial Engineering, Gaziantep University, Gaziantep, Turkey Faculty of Engineering and Architecture, Department of Industrial Engineering, Gazi University, Maltepe, 06570 Ankara, Turkey Received 18 November 2004; accepted 18 April 2006 Available online 11 July 2006
Abstract In this paper, chance-constrained 0–1 integer programming models for the stochastic traditional and U-type line balancing (ULB) problem are developed. These models are solved for several test problems that are well known in the literature and the computational results are given. In addition, a goal programming approach is presented in order to increase the system reliability, which is arising from the stochastic case. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Production; U-line balancing
1. Introduction An assembly line typically consists of several workstations, each of them being responsible for performing a specific set of tasks. Products stay at each workstation for the cycle time (C), which corresponds to the time interval between successively completed units. The assembly line balancing (ALB) problem is a prominent manufacturing problem, which originated in the early 1950s. The objective of this problem is to allocate the assembly tasks to a given number of workstations in equal portions so as to maximize the throughput. Furthermore, line balancing will reduce the differences between assigned workloads so that one operator is not very busy while other operators are idle. The studies related to the assembly line can be classified in two general groups. These are traditional straight assembly lines (with single and multi/mixed products) and U-type assembly lines (with single and multi/mixed products). The literature on assembly line balancing is relatively extensive. Studies on traditional assembly line balancing, the review papers of Baybars [1], Ghosh and Gagnon [2], Erel and Sarin [3] can be examined. In traditional line balancing problems the assembly line is designed straightly, and in U-type balancing problems the line is designed as U shape. In U-type design the entrance and the exit of the line are on the same position. *
Corresponding author. Tel.: +90 312 231 74 00/2839; fax: +90 312 230 84 34. E-mail address: [email protected] (H. Go¨kc¸en).
0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.04.042
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Fig. 1 illustrates a traditional straight assembly line and a U-type assembly line with a cycle time of 12 time units including 9 tasks. Each node in Fig. 1 represents the tasks and the numbers next to the nodes define the task times. There is a small and growing literature on ULB problem. SULB (Simple U Line Balancing) problem was first modelled by Miltenburg and Wijngaard [5]. Urban [6] proposed an integer programming formulation for determining the optimal balance. So, several techniques have been proposed for the solution of U-line balancing problems (see the paper of Scholl and Klein [7], Ohno et al. [8], Miltenburg [9], Sparling and Miltenburg [10], Miltenburg [11]). Another classification divides the assembly lines into two groups depending on the task times: deterministic and stochastic. In stochastic ALB problem, the task times are defined as probability distributions rather than being defined as determined and constant values. Among the reasons of this variability, the following factors can be mentioned: fatigue, loss of attention, workforce with insufficient qualifications, insufficient work satisfaction, erroneous entries, and breakdowns. In literature, normal distribution is widely used for task times [12–19]. For studies on stochastic traditional assembly line balancing, the review paper of Erel and Sarin [3] can be seen. The first study for stochastic case on U-type lines has been done by Ohno et al. [8]. In the mentioned study, there are analyses on random task times. Guerriero and Miltenburg [19] presented a dynamic programming approach to solve the stochastic U-type line balancing problem. Chance-constrained programming is one of the well-known techniques for modelling stochastic systems. Technique was developed by Charnes and Copper [20–22]. Urban and Chiang [23] formulated the stochastic U-line balancing problem as a chance-constrained programming model and used piecewise linear, integer program to find the optimal solution for the model. In this study, linear transformations are presented for finding the optimal and near optimal solution to chance-constrained programming model. Furthermore, a goal programming approach is presented in order to increase the assembly line system reliability arisen from the stochastic case. This paper is organized as follows. In Section 2, the Urban’s [6] integer programming formulation is given. The chance-constrained programming formulation for traditional and U-line balancing problem and its linear transformations are presented in Section 3. Also in Section 3, some definitions that are new for stochastic line balancing problem are given. Computational results and evaluation about the methods on test problems are given in Section 4. In Section 5, a goal programming approach and an illustrative example are presented for minimization of the mean station time and variance differences between stations. Finally, the paper is concluded with a summary of the approach.
4
2 1
2
5
3 3
4
Operator/Workstation #1
6
6 5
6
8
Operator #3
Operator #2
3
4 7
3 9
Operator # 4
a 2 1
2
4
Operator #1
3 9
3 3
Operator #2
3 8
5 4
5
Operator #3
4 7
6
6 6
b Fig. 1. Traditional (a) and U-type assembly line (b) with a cycle time of 12 [4].
K. Ag˘pak, H. Go¨kc¸en / European Journal of Operational Research 180 (2007) 1098–1115
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2. Urban’s integer programming formulation A simple ULB problem can be stated as follows: given set of tasks, n, the performance times of tasks, ti (i = 1, . . . , n), and the set of precedence relations, the problem is to assign the tasks to the workstations, (j = 1, . . . , m), such that the precedence relations are satisfied and some performance measures such as minimization of the cycle time or the number of workstation, etc., is achieved. The distinguishing feature of the ULB problem is that it must allow for the forward and backward assignment of tasks to workstations; for example, the first and the last task of an assembly can be placed in the same workstation on a U-line, but not on a traditional line [6]. Urban [6] accomplished this by establishing a ‘‘Phantom’’ network and appending it to the original precedence network. To illustrate this concept, he considered the classical test problem from Jackson [24]. Fig. 2, denotes the precedence network of this 11 task problem with the phantom network (dotted lines) attached to it. By starting in the middle of this extended network, assignments to the workstations can be made forward through the original network, backward through the phantom network, or simultaneously in both directions [6]. Basic difference between stochastic and deterministic cases is, in stochastic case task times are assumed to get values based on a probability distribution whereas in deterministic case task times are constant numbers. For this reason, while developing mathematical models for U-type and traditional assembly lines, integer-programming model of Urban [6] was used as a basis. Urban’s [6] integer programming model is formulated as follows: Objective function: m max X Min Sj
ð1Þ
j¼dmmin e
Constraints: m max X ðxij þ y ij Þ ¼ 1
for i ¼ 1; . . . ; n;
ð2Þ
j¼1 n X i¼1 n X
ti ðxij þ y ij Þ 6 C
for j ¼ 1; . . . ; dmmin e;
ti ðxij þ y ij Þ 6 CS j
ð3Þ
for j ¼ dmmin e þ 1; . . . ; dmmax e;
i¼1 m max X
ðmmax j þ 1Þðxrj xsj Þ P 0
ð4Þ
for all ðr; sÞ 2 P ;
ð5Þ
for all ðr; sÞ 2 P ;
ð6Þ
j¼1 m max X
ðmmax j þ 1Þðy sj y rj Þ P 0
j¼1
xij ; y ij ; S j 2 f0; 1g
2
1
6
for all i; j:
8
3
4
5
10
2
11
7
9
1
6
8
10
3
4
11
7
9
5
Fig. 2. Precedence network of the Jackson’s problem with phantom network appended [6].
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The notations used in the formulation are given below: n: total number of tasks, C: cycle time, ti: performance time of task i, i = 1, . . . , n, mmin: theoretical minimum number of workstation, mmax: maximum number of workstations which can be estimated from the operational setting or by utilizing U-line heuristics shown to perform well, xij: 1, if task i in the original precedence diagram is assigned to workstation j; 0 otherwise, i = 1, . . . , n, j = 1, . . . , mmax, yij: 1, if task i in the phantom precedence diagram is assigned to workstation j; 0 otherwise, i = 1, . . . , n, j = 1, . . . , mmax, Sj: 1, if there is any task assigned to workstation j; 0, otherwise, j = 1, . . . , mmax, P: the set of precedence relationships of tasks; for example, task r immediately precedes task s. Constraint (2) ensures that all tasks are assigned to a station and each task is assigned only once. Constraints (3) and (4) ensure that the work content of any station does not exceed the cycle time. Constraints (5) and (6) ensure that the precedence constraints are not violated on the original network and phantom network. As a result of objective function, the number of workstations will be minimized. If the yij variable and constraint 6 are omitted, the model can be used for traditional line balancing problems. Task times are used as coefficients only in constraints (3) and (4). Thus, only these two constraints will need to be transformed if the model is stochastic. 3. A chance-constrained 0–1 integer programming model for stochastic line balancing problem The objective of the model is to accomplish the balancing with minimum number of stations so that the probability of the total of task times assigned to stations being greater than cycle time (Pk) should stay under predetermined limits (a) (Pk 6 a). It is seen that (a) is greater than 0.8 in practice and literature. At the same time, (a) is assumed greater than 0.5 for valid and correct models in this study, too. In models, the task times are assumed to be normally distributed with known means and variances. Furthermore, each task’s time is independent of any other task’s time and the ordering of tasks into work assignments P [12–19,23]. P P 2 P Y l 2 p ffiffiffiffiffiffiffiffiffi If Y ¼ ti , while task time ti N ðli ; ri Þ, then Y N li ; ri . Using Z ¼ P 2i transformation, then ri
Z N(0, 1). Since the allowed maximum probability that station time exceeds cycle time is a, P(Y > C) 6 a can be written. After modifications, 1 P(Y 6 C) 6 a or P(Y 6 C) P 1 a. Using Z transformation we get P ! C li ffiffiffiffiffiffiffiffiffiffiffi P 1 a: P Z 6 pP r2i
ð7Þ
P(Z 6 z1a) = 1 a, z1a is 100(1 /) percentile of the standard normal distribution. Expression (7) is realized if and only if P C li z1a 6 pffiffiffiffiffiffiffiffiffiffi ð8Þ P 2ffi ri or X
li þ z1a
qffiffiffiffiffiffiffiffiffiffiffiffi X ffi r2i 6 C:
ð9Þ
When task variables (xij, yij) which are used in integer programming model for deterministic U-line, are added to inequality (9), then we can obtain the constraints (10) and (11). In the new model, these constraints will replace the cycle-time constraints (3) and (4) in the deterministic model.
K. Ag˘pak, H. Go¨kc¸en / European Journal of Operational Research 180 (2007) 1098–1115
1102 n X i¼1 n X
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X li ðxij þ y ij Þ þ z1a r2i ðxij þ y ij Þ 6 C
for j ¼ 1; . . . ; dmmin e;
ð10Þ
i¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X li ðxij þ y ij Þ þ z1a r2i ðxij þ y ij Þ 6 CS j
i¼1
for j ¼ dmmin e þ 1; . . . ; mmax :
ð11Þ
i¼1
Chance-constrained programming model for stochastic simple model U-type assembly line balancing (SSULB) problem is developed with the new constraints as follows: Objective function: mmax X Sj Min j¼dmmin e
Subject to: Constraints
ð2Þ; ð5Þ; ð6Þ; ð10Þ; ð11Þ; xij ; y ij ; S j 2 f0; 1g
for i; j:
The obtained new model is a non-linear 0–1 integer programming model. It is known that the non-linear nature is more complex than the linear nature. In the literature, solutions can be achieved for at most 45 tasks for deterministic case using linear 0–1 programming model [6]. Accordingly, the new model is expected to provide solutions for smaller size problems than the deterministic case. With linear-transformation, it is possible to solve the problems with greater size than the ones that can be solved using non-linear model. For linear-transformation, two approaches were used. First approach is linear approach and the second is pure linear transformation. 3.1. Linear approach for chance-constrained model of SSULB problem Linear approach (LA) gives an approximate solution rather than the optimal to the problems. sffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n n n n X X X X þ a2i 6 ai ; ai 2 R ) r2i ðxij þ y ij Þ 6 ri ðxij þ y ij Þ: i¼1
i¼1
i¼1
ð12Þ
i¼1
According to the inequality (12), the constraints (10) and (11) can be rewritten as follows: n X i¼1 n X i¼1
li ðxij þ y ij Þ þ z1a li ðxij þ y ij Þ þ z1a
n X i¼1 n X
ri ðxij þ y ij Þ 6 C ri ðxij þ y ij Þ 6 CS j
for j ¼ 1; . . . ; dmmin e; for j ¼ dmmin e þ 1; . . . ; mmax :
ð13Þ ð14Þ
i¼1
By doing so, the non-linear components of the model were linearized. With the last form of the model, upper limits for partially large-scale problems can be obtained. The approach used is based on a simple mathematical inequality and does not need a new variable or constraint added to the model. If (xij, yij) terms should be taken into account as multipliers in the left hand side of the constraints (13) and (14), then we can obtain the following inequalities, respectively. In inequalities (13) and (14), if the left hand side is bracketed using (xij, yij) then we can reach the equalities (15) and (16). n X ðli þ z1a ri Þðxij þ y ij Þ 6 C for j ¼ 1; . . . ; dmmin e; ð15Þ i¼1 n X ðli þ z1a ri Þðxij þ y ij Þ 6 CS j i¼1
for j ¼ dmmin e þ 1; . . . ; mmax :
ð16Þ
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If it is carefully examined, it can be seen that the constraints (15) and (16) are similar to the constraints (3) and (4) in the model given for deterministic case. With the new approach only the coefficients has been changed. Accordingly, we can say that the new model will have the same characteristics with the model developed for deterministic case. 3.2. Pure linear transformation for chance-constraint model of SSULB problem ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pP n 2 The reason why the developed chance-constraint model is not linear is the term i¼1 ri xij with task variable. For that reason by eliminating the square-root, linearizing will be possible. So, by squaring both side of the inequality (8), inequality (17) has been obtained. 2 Pn C i¼1 li xij ðz1a Þ 6 pP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi2 : n 2 r x ij i¼1 i 2
ð17Þ
pP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi n 2 While the non-linear situation caused by the term i¼1 ri xij has been eliminated by inequality (17), a new 2 Pn non-linear situation has been raised with term C i¼1 l2i xij . The open form of this term is given by Eq. (18). 0
C
X
li xij
2
¼ C 2 2C
X
l21 x21j þ l1 x1j l2 x2j þ þ l1 x1j ln xnj
1
B C B l1 x1j l2 x2j þ l22 x22j þ þ l2 x2j ln xnj C C: li xij þ B B C ... @ A l1 x1j ln xnj þ l2 x2j ln xnj þ þ l2n x2nj
ð18Þ
Since the variable xij is 0–1 integer, x2ij is equal to xij. The term (lixijlnxnj) is the non-linear part of the Eq. (18). So, a current transformation technique in the literature has been used to make this part linear. uinj ¼ xij xnj :
ð19Þ
After the variable transformation given in Eq. (19), the variable uij has been made dependent to xij and xnj by inequalities (20) and (21). xij þ xnj uinj 6 1;
ð20Þ
xij þ xnj 2uinj P 0:
ð21Þ
Another situation to be considered on inequality (17) is that the inequality is valid in two different cases. These can be expressed by the inequality below. C P l i ffiffiffiffiffiffiffiffiffiffiffi : ð22Þ jz1a j 6 pP r2i Assumed a value is greater than 0.5, therefore z1a is positive and right hand side of the inequality (22) must be positive, too. For that reason by using inequality (23) and (17) together, different cases for inequality (22) will be eliminated. This constraint is the same as the cycle time constraint at the basic deterministic model. X C li xij P 0:
ð23Þ
The model linearized with new constraints can be rewritten as follows. The model details (open form) for an example problem are given in Appendix.
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Objective function: mmax X
Min
Sj
j¼dmmin e
Subject to: Constraints
ð2Þ; ð5Þ and ð6Þ C 2 2C
n X
li pij þ
n X
i¼1
l2i pij þ 2
i¼1
for j ¼ 1; . . . ; mmax ; ! n X li pij P 0 CS j
n1 X n X
li lv uivj z21a
i¼1 v¼iþ1
n X
! r2i pij
P0
i¼1
ð24Þ for j ¼ 1; . . . ; mmax ;
ð25Þ
i¼1
pij þ pvj uivj 6 1
for j ¼ 1; . . . ; mmax ; i ¼ 1; . . . n 1; v ¼ i þ 1; . . . ; n;
pij þ pvj 2uivj P 0 xij þ y ij pij ¼ 0
for j ¼ 1; . . . ; mmax ; i ¼ 1; . . . n 1; v ¼ i þ 1; . . . ; n;
for j ¼ 1; . . . ; mmax ; i ¼ 1; . . . n;
xij ; y ij ; pij ; S j ; uivj 2 f0; 1g
ð26Þ ð27Þ ð28Þ
for i; j; v:
Cycle time, assignment and priority constraints take place in the new model like in basic deterministic model. The difference of the new model is in the constraints given by inequalities (24), (26), (27) and (28). All the mathematical models given at Section 3 are for U-type (shaped) assembly lines. If yij variables and constraint (6) are removed from the model, constructed models can be used for traditional assembly lines. mmin (Theoretical Minimum Number of Station), Ei (earliest station for task i) and Li (latest station for task i) parameters obtained for deterministic assembly line balancing problems are also a necessity either for U-type (shaped) or traditional assembly line balancing problems in stochastic cases. So, these parameters, which assure reducing the number of variables in the models, should be defined for the stochastic case. First, Urban and Chiang [23] defined lower bound (LB2) considering the stochastic aspect of the problem. In this study, the lower bound LB2 is called mmin and presented a different approach to proof of definition. Furthermore, inferences about Ei and Li parameters that will be new for the literature are given after mmin definition. Definition 1. When the task time ti (ti N ðli ; r2i ÞÞ, number of tasks n, the cycle time C and the upper limit of the allowed probability of exceeding the cycle time of the station time a are given, then the theoretical minimum number of stations mmin is & ’ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 Pn z1a i¼1 ri þ i¼1 li mmin ¼ ; ð29Þ C where dxe is the smallest integer greater than or equal to x. WePassumed that the m* is P the optimalP number of stations. If the station times for each station is Stk ¼ i2k ti , then Stk N lk ¼ i2k li ; r2k ¼ i2k r2i . Also, it is known that qffiffiffiffiffi lk þ z1a r2k 6 C ð30Þ for station k. The sum of inequality (30) for m* station is qffiffiffiffiffi n m n m X X X X li þ ðz1a r2k Þ 6 m C ) li þ z1a rk 6 m C: i¼1
k¼1
i¼1
k¼1
ð31Þ
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According to sffiffiffiffiffiffiffiffiffiffiffiffi n n X X a2i 6 ai ; i¼1
vffiffiffiffiffiffiffiffiffiffiffiffiffi u m m X uX ai 2 Rþ ) t r2k 6 rk k¼1
i¼1
ð32Þ
k¼1
we can write vffiffiffiffiffiffiffiffiffiffiffiffiffi u m n m X X uX t 2 li þ z1a rk 6 li þ z1a rk 6 m C:
n X
k¼1
i¼1
ð33Þ
k¼1
i¼1
It is known that X r2k ¼ r2i :
ð34Þ
i2k
If n X
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u m n m X X uX X 2 t li þ z1a ri 6 li þ z1a rk 6 m C k¼1
i¼1
i2k
k¼1
i¼1
ð35Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffi n n n m X X X X 2 ) li þ z1a ri 6 li þ z1a rk 6 m C i¼1
i¼1
k¼1
i¼1
then Pn mmin ¼
i¼1 li þ z1a C
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 i¼1 ri
6 m
&P or
mmin ¼
n i¼1 li
þ z1a C
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 ’ i¼1 ri
6 m :
ð36Þ
Definition 2. While cycle time is C and the upper limit of the probability of exceeding the cycle time of the allowed station time is a, the earliest station for processing the task i (Ei) is, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 P P li þ j2P j lj þ z1a r2i þ j2P i r2j 7: Ei ¼ 6 ð37Þ 6 7 C 6 7 The theoretical minimum number of station which can be obtained for an assembly line consisting of task i and its predecessors (Pi) is the earliest station which task i can be processed. So, mmin can be used as Ei. Definition 3. While cycle time is C and the upper limit of the allowed probability of exceeding the cycle time of the station time is a and the allowed maximum number of stations is K (K P m), then the latest station for processing the task i (Li) is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 P P lj þ j2Sci lj þ z1a r2i þ j2Sci r2j 7: Li ¼ K þ 1 6 ð38Þ 6 7 C 6 7 The theoretical minimum number of station which can be obtained for an assembly line consisting of task i and its successors (Sci) is the earliest station which task i can be processed. The difference between this value and K + 1 gives the latest station which task i can be assigned. Ei and Li definitions are developed for traditional lines. Therefore new models, which use the bounds, are given below. Firstly, MLAT (the model with linear approach for traditional lines) is presented.
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Objective function: mmax X Min Sj j¼dmmin e
Constraints: Li X
xij ¼ 1 for
i ¼ 1; . . . ; n;
ð39Þ
j¼EI_
X
ðli þ z1a ri Þxij 6 C
for j ¼ 1; . . . ; dmmin e;
ð40Þ
i2W j
X
ðli þ z1a ri Þxij 6 CS j
for j ¼ dmmin e þ 1; . . . ; mmax ;
ð41Þ
i2W j Ls Lr X X ðmmax j þ 1Þxrj ðmmax j þ 1Þxsj P 0 j¼Er
for ðr; sÞ 2 P ;
ð42Þ
j¼Es
xij ; y ij ; S j 2 f0; 1g
for all i; j:
Second model, MLTT (the model with pure linear transformation for traditional lines) is presented below. Objective function: mmax X Sj Min j¼dmmin e
Subject to: ð39Þ and ð42Þ;
Constraints 2
C 2C
X
li xij þ
i2W j
CS j
X
X
l2i xij þ 2
i2W j
X X
li lv uivj z21a
i2W j vj2W j ^>i
X
! r2i xij
P 0 for j ¼ 1;... ;mmax ;
i2W j
ð43Þ
! li xij
P 0 for j ¼ 1;.. .;mmax ;
ð44Þ
i2W j
xij þ xvj uivj 6 1 for j ¼ 1; ...; mmax ; i ¼ Ei ;. ..;Li ; v ¼ Ev ;... ;Lv ; i 6¼ v;
ð45Þ
xij þ xvj 2uivj P 0 for j ¼ 1;.. .;mmax ; i ¼ Ei ;... ;Li ;v ¼ Ev ;. ..;Lv ; i 6¼ v;
ð46Þ
xij ;y ij ;S j ; uivj 2 f0;1g for i;j;v: Wj represents subset of all tasks that can be assigned to workstation j. 4. Computational results and evaluation In this paper, four different new models are suggested for Chance Constraint Model of SSULB Problem. These are MLTT, MLTU (the model with pure linear transformation for U-type lines), MLAT and MLAU (the model with linear approach for U-type lines). Some comparisons on these models with respect to the number of variables and constraints are given in Table 1. While forming the comparison table, it is assumed that the models do not include any upper or lower varn2 þn 2
’’ more variables and ‘‘Kmax(n2 + 1)’’ constraints 2 than MLAU. In similar manner, MLTT requires ‘‘K max n 2n ’’ more variables and ‘‘Kmax(n2 1) + n’’
iable limits. For U-type line, MLTU contains ‘‘K max
K. Ag˘pak, H. Go¨kc¸en / European Journal of Operational Research 180 (2007) 1098–1115
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Table 1 Numbers of variables and constraints related to the models MLA
MLT
U-line
No. of variable No. of constraint
Kmax(2n + 1) n + Kmax + 2r
ðn2 þ5nþ2Þ K max 2 Kmax(n2 + 2) +
Traditional line
No. of variable No. of constraint
Kmax(n + 1) n + Kmax + r
ðn2 þnþ2Þ K max 2 Kmax(n2 n +
n + 2r
2) + n + r
constraints than MLAT. The reason of this is that pure linear transformation models give optimal and linear approach models also give approximate results. There are no determined test problem sets in stochastic line balancing literature for testing performance of presented models. However, there are test problem sets for deterministic case. In this study, a new test problem set which can be used in stochastic case, is formed by transforming the deterministic test problem set. To set up this test problem, Gueriero and Miltenburg [18] presented three different variabilities (low, high, random) in task completion time and determined the coefficient of variation (CV) intervals. Determined CV intervals are low (0 < CV < 0.3), high (0.3 < CV < 0.6) and random (0 < CV < 0.6). And then problem sets are established by producing random numbers in these intervals. The same interval descriptions are used in this study too. However in this study, test problem set for deterministic case in the literature is taken as a basis to gain a stochastic structure. Task times in deterministic case are accepted as mean task time(s). For determined variation type, random numbers are produced according to uniform distribution to obtain different CV values for each task. Variation of each task is obtained by multiplying these values with mean task times. In this way, deterministic test problem set is transformed into a usable structure in stochastic case and three different variance sets (low, high, random) according to CV intervals are obtained for one problem. Models are solved on test problems for three different a levels (low 0.15, medium 0.1, high 0.05) and three different cycle times by using GAMS-CPLEX solver on a Pentium IV 2.8 MHz personal computer. Four models are compared with each other with respect to the number of stations. The comparison results are given in Tables 2 and 3. Since the problem is in NP-Hard nature it takes too much time to reach a result for larger problems. So, the solver has been limited by 72,000 seconds (20 hours) in this study. In this study, 216 test problems for linear transformation models and 270 test problems for linear approach models were solved. If the results in Table 2 are observed, it can be seen that optimal solutions have been found in 137 of 216 problems by MLTT and in 95 problems by MLTU. With MLAT, in 57 of 137 problems known to be optimal exactly, the optimal solutions have been found. For 74 of 137, the solutions have been found with a deviation of 1 station and for 5 of 137, the solutions have been found with a deviation of 2 stations and lastly for 1 of 137, the solutions have been found with a deviation of 3 stations. With MLAU, for 49 of 95 problems known to be exactly optimal, the optimal solutions have been found. For 45 of 95, the solutions have been found with a deviation of 1 station and for 1 of 95, the solutions have been found with a deviation of 2 stations. In some problems with more than 25 tasks, optimal results generally cannot be reached in 72,000 seconds by MLTT or MLTU. Near optimal solutions have been generated up to 53 tasks by MLAT and MLAU. For large-scale problems, with MLTT in 5 of 54 problems and with MLTU in 6 of 54 problems, optimal solutions have been found (see Table 3). 5. Goal programming for balancing between stations With the developed models the result can be reached only for one goal. The results found do not guarantee the mean times or variances of the stations to be equal. In real world problems it is preferred the workloads of stations to be equal. At the same time, if we assume that incompletions of all the stations and station times are independent, equalizing workload between workstations will make a positive contribution to increase the total assembly line system reliability in stochastic case. In otherwise, probability that all workstations would complete their assigned tasks within the given cycle time can be increased. In this study, this probability is called
1108
Table 2 Results of comparison of the four models with respect to the number of stations No. of task
Cycle time
Type of variation
/ Level
Jackson
11
13
L
L M H L M H L M H
4 4 4 5 5 5 5 5 5
5 5 5 6 6 7 5 6 6
4 5 5 6 6 7 5 5 6
5 5 6 6 7 8 6 6 6
5 5 5 6 7 7 5 6 6
L M H L M H L M H
3 3 3 3 3 4 3 3 3
3 3 3 4 4 4 4 4 4
3 3 3 4 4 4 4 4 4
3 3 4 4 4 5 4 4 5
3 3 4 4 4 5 4 4 4
35
L M H L M H L M H
2 2 3 3 3 3 3 3 3
2 3 3 3 3 3 3 3 3
2 3 3 3 3 3 3 3 3
3 3 3 3 3 4 3 3 3
3 3 3 3 3 4 3 3 3
41
L M H L M H L M H
5 5 5 6 6 6 6 6 6
6 6 6 7 7 8 6 7 7
6a 6a 6a 7a 7a 8a 6 6 7a
6 6 6 8 8 10 7 7 8
6 6 6 7 8 9 7 7 7
L M H L M H L M H
4 4 5 5 5 5 5 5 5
5 5 5 5 6 6 5 5 6
5a 5a 5 5 6a 6a 5 5 5
5 5 5 6 7 8 6 6 6
5 5 5 6 7 7 5 6 6
H
R
19
L
H
R
25
L
H
R
Roszieng
25
27
L
H
R
33
L
H
R
mmin
Models MLTT
MLTU
MLAT
MLAU
Problem name
No. of task
Mitchell
21
Cycle time 27
Type of variation
/ Level
mmin
MLTU
MLAT
MLAU
L
L M H L M H L M H
5 5 5 5 5 5 5 5 5
5 5 5 6 6 7 5 5 5
5 5 5 6a 6a 7a 5 5 5
5 5 5 7 7 8 5 6 6
5 5 5 6 7 8 5 5 6
L M H L M H L M H
4 4 4 4 4 4 4 4 4
4 4 4 4 5 5 4 4 4
4 4 4 4 5a 5a 4 4 4
4 4 4 5 5 6 4 4 5
4 4 4 5 5 6 4 4 5
L M H L M H L M H
3 3 3 3 3 4 3 3 3
3 3 3 4 4 4 3 3 4
3 3 3 4 4 4 3 3 3
3 4 4 4 5 5 4 4 4
3 3 4 4 5 5 4 4 4
L M H L M H L M H
6 6 6 6 6 6 6 6 6
6 6 6 10a 8a 8a 7a 7a 8a
6 6 6 7a 8a 9a 7a 7a 8a
6 6 7 8 8 9 7 8 8
6 6 7 8 8 9 7 8 8
L M H L M H L M H
5 5 5 5 5 6 5 5 6
5 5 5 6a 6a 7a 6a 6a 7a
5 5 5 6a 6a 7a 6a 6a 7a
6 6 6 7 7 8 6 7 7
6 6 6 7 7 8 6 7 7
H
R
L
H
R
L
H
R
Heskia
28
205
L
H
R
235
L
H
R
Models MLTT
K. Ag˘pak, H. Go¨kc¸en / European Journal of Operational Research 180 (2007) 1098–1115
Problem name
41
L
H
R
29
42
L
H
R
47
L
H
R
52
L
H
R
Lutz-1
32
2600
L
H
R
4 4 4 4 4 4 4 4 4
4 4 4 4 4 5 4 4 4
4 4 4 4 4 5a 4 4 4
4 4 4 5 6 6 5 5 5
4 4 4 5 5 6 4 5 5
265
L M H L M H L M H
8 9 9 9 9 9 9 9 9
9 9 10a 11a 0a 0a 10a 10a 11a
9a 10a 10a 0a 0a 0a 10a 11a 0a
10 10 11 12 13 14 10 11 12
10a 10 10 12 13 14 10 11 12a
L M H L M H L M H
8 8 8 8 8 8 8 8 8
8 8 9a 0a 0a 11a 9a 0a 0a
8 8 9a 10a 0a 0a 9a 0a 0a
9 9 9 11 12 12 9 10 10
9a 9 9 11a 11 11 9 10 10
60
L M H L M H L M H
7 7 7 7 7 8 7 7 7
7 8a 8a 0a 9a 0a 8a 10a 10a
7 9a 8a 0a 0a 0a 8a 8a 8a
8 8 8 10 10 12 9 9 10
8 8 8 10 10 11 8 9 9
75
L M H L M H L M H
6 6 6 6 7 7 6 6 7
6 7 7 8 8 8 7 7 8
0a 0a 7a 0a 8a 0a 7a 7a 0a
7 7 7 9 9 11 8 8 9
7 7 7 8 9 10 8 8 9
L
H
R
Sawyer
30
45
L
H
R
L
H
R
L
H
R
Gunther
35
81
L
H
R
L M H L M H L M H
5 5 5 5 5 5 5 5 5
5 5 5 5 6a 6a 5 5 6a
5 5 5 6a 6a 6a 5 5 6a
5 5 5 6 7 7 6 6 7
5 5 5 6 7 7 6 6 7
L M H L M H L M H
8 8 8 8 9 9 8 8 8
9a 9a 9a 0 0 0 10a 10a 11a
9a 9a 0 0 0 0 9a 0 0
9 9 10 12a 12 13 10 10 11
9 9 10 11 12 13 10 10 11
L M H L M H L M H
6 6 6 6 7 7 6 6 6
6 7a 7a 8a 8a 8a 7a 7a 7a
0 7a 7a 7a 8a 8a 7a 7a 7a
7 7 8 8 9 10 7 8 8
7 7 7 8 9 10 7 8 8
L M H L M H L M H
5 5 5 5 5 5 5 5 5
5 5 6a 6a 6a 7a 6a 6a 6a
5 5 5 6a 6a 0a 5 0 6a
6 6 6 7 7 8 6 6 7
6 6 6 7 7 8 6 6 7
L M H L M H L M H
7 7 7 7 7 7 7 7 7
7 7 8a 0a 9 0a 8a 9a 9a
7 0a 0a 0a 0a 0a 8a 0a 0a
8 7 8 8 8 8 9 9 10 10 11 11 9 8 9 9 10 10a (continued on next page)
K. Ag˘pak, H. Go¨kc¸en / European Journal of Operational Research 180 (2007) 1098–1115
Buxey
L M H L M H L M H
1109
1110
Problem name
No. of task
Cycle time
Type of variation
/ Level
3100
L
L M H L M H L M H L M H L M H L M H
H
R
3600
L
H
R
a
Optimal results cannot be reached in 72,000 seconds.
mmin
Models
Problem name
No. of task
Cycle time
MLTT
MLTU
MLAT
MLAU
5 5 5 5 6 6 5 5 6
5 6 6 6 6 7 6 6 6
5 0a 6a 6a 6 7a 6a 6a 6
6 6 6 7 8 9 7 7 7
6 6 6 7 8 8 6 7 7
91
5 5 5 5 5 5 5 5 5
5 5 5 5 6 6 5 5 5
5 5 5 5 0a 6a 0a 5 5
5 5 5 6 7 7 6 6 6
5 5 5 6 7 7 6 6 6
101
Type of variation
/ Level
mmin
Models MLTT
MLTU
MLAT
MLAU
L
L M H L M H L M H
6 6 6 6 6 7 6 6 6
7a 7a 7a 7 8a 0a 7 7 8a
6 6 7a 0a 0a 0a 0a 7a 0a
7 7 7 8 9 10 8 8 9
7 7 7 8 9 10 8a 8 9a
L M H L M H L M H
5 5 6 6 6 6 6 6 6
6 6 6 7a 7a 8a 7a 7a 7a
6a 6a 6 0a 0a 0a 0a 0a 0a
6 6 6 8 8 9 7 7 8
6 6 6 7 8 9 7 7 8
H
R
L
H
R
K. Ag˘pak, H. Go¨kc¸en / European Journal of Operational Research 180 (2007) 1098–1115
Table 2 (continued)
Table 3 Results of comparison of the two models with respect to the number of stations for large-scale problems No. of task
Cycle time
Type of variation
a Level
mmin
MLAT
MLAU
Kilbridge
45
103
L
L M H L M H L M H
6 6 6 6 6 7 6 6 7
7 7 7 8 9 10 8 8 9
7 7 7 8 9 10 8 8 9
L M H L M H L M H
5 5 5 6 6 6 5 6 6
6 6 6 7 8 8 7 7 8
6 6 6 7 8 8 7 7 8
3775
L M H L M H L M H
5 5 5 5 5 5 5 5 5
5 5 6 6 7 7 6 6 7
5 5 6 6 7 7 6 6 7
3925
H
R
120
L
H
R
137
L
H
R
Models
Problem name
No. of task
Cycle time
Type of variation
a Level
mmin
Models MLAT
MLAU
Hahn
53
3625
L
L M H L M H L M H
5 5 5 5 5 5 5 5 5
5 5 6 7 8 8 6 7 8
5 5 5 6 7 7 6 6 7
L M H L M H L M H
4 4 5 5 5 5 5 5 5
5 5 5 6 7 8 6 6 7
5 5 5 6 6 7 6 6 6
L M H L M H L M H
4 4 4 5 5 5 4 5 5
5 5 5 6 6 8 5 6 7
5 5 5 6 6 7 5 6 6
H
R
L
H
R
L
H
R
K. Ag˘pak, H. Go¨kc¸en / European Journal of Operational Research 180 (2007) 1098–1115
Problem name
1111
K. Ag˘pak, H. Go¨kc¸en / European Journal of Operational Research 180 (2007) 1098–1115
1112
the assembly line system reliability. For that reason, goal programming techniques has been used in order to equalize the workloads between workstations in respect of variances and/or mean times. Mean time goal: With this goal it is aimed to minimize the deviation of the mean times of the stations from a certain value. The goal constraints can be written as n X þ li pij þ dc for j ¼ 1; . . . ; mmax : ð47Þ j dcj ¼ lst i¼1
Variance goal: With this goal it is aimed to minimize the deviation of the variances of the stations from a certain value. The goal constraints can be written as n X þ r2i pij þ dv for j ¼ 1; . . . ; mmax : ð48Þ j dvj ¼ vst i¼1 þ The minimization of sum of the deviational variables dcþ j and dvj will assure the minimization of the deviation from lst and vst values for each workstation. In this way the differences of mean times and variances between stations will be minimized. But before applying the goal programming approach, the problem should be solved with linear transformation and/or linear approach models, and the number of stations (mg) should be found. Then according to the number of stations Pn 2 Pn r i¼1 li lst ¼ and vst ¼ i i values should be determined: mg mg
After this, the number of stations should be written as a constraint and the goal programming model should be solved. To illustrate the goal programming model discussed here, example with different goal sequences is solved. The precedence network of the problem was given in Fig. 2, Jackson problem. The mean task times and variations of the problem are also given in Table 4. A preemptive goal programming technique is used for solving example problem. In a preemptive goal programming model, the upper level goals are first optimized before lower level goals are considered. In a non-preemptive model, the goals are given some weights and considered simultaneously. We utilized the preemptive approach due to the difficulty associated with determining the weights for the various goals [25]. The cycle time and allowed incompletion level for the problem are taken as 19 and 0.1 respectively. The problem is solved using the GAMS-CPLEX mathematical programming package on a Pentium 4-2.0 GHz computer. Task assignments are shown in Table Qm5.g mg is found as four. The assembly line system reliability (ASR ¼ i¼1 Psti Þ is 0.863, lst is 12 and vst is 8.942 as dependent on mg. If the mean time goal takes the first priority (P1) and the variance goal the second (P2), then the balanced result on Table 6 is reached. If the priority sequence is changed the results on Table 7 can be reached. Especially for the goal sequence of P1 and P2, ASR value of 0.975 which is very high is obtained. Mean times and variances are distributed balanced between the stations. Table 4 Mean task times and variations Task
Mean time
Variance
Task
Mean time
Variance times
1 2 3 4 5 6
6 2 5 7 1 2
5.308 0.510 3.151 7.263 0.202 0.767
7 8 9 10 11
3 6 5 5 4
1.004 11.614 5.221 0.723 0.005
K. Ag˘pak, H. Go¨kc¸en / European Journal of Operational Research 180 (2007) 1098–1115
1113
Table 5 Task assignments of the sample problem Workstation
Tasks
Mean station time
Standard deviation
Probability of completion (Pst)
1 2 3 4
1, 2, 3, 5 11, 9 4, 7, 10 6, 8
14 9 15 8
3.028 2.286 2.998 3.518
0.950 0.999 0.909 0.999
Table 6 Balance results for the goal sequence P1 and P2 Workstation
Tasks
Mean station time
Standard deviation
Probability of completion (Pst)
1 2 3 4
1, 2, 8, 7,
12 11 11 12
2.943 2.922 3.512 2.495 OSR
0.991 0.997 0.989 0.998 0.975
3, 5 4, 6 10 9, 11
Table 7 Balance results for the goal sequence P2 and P1 Workstation
Tasks
Mean station time
Standard deviation
Probability of completion (Pst)
1 2 3 4
1, 3, 11 4, 10 5, 6, 7, 9 8
15 12 11 6
2.909 2.826 2.682 3.408 OSR
0.915 0.993 0.999 0.999 0.908
6. Conclusion In this paper, new mathematical models for stochastic traditional and U-type assembly lines have been developed by using chance-constrained programming technique. Since the mathematical model, which has been obtained, is non-linear, pure linear transformation and approximate-linear transformation (linear approach) have been utilized to construct 0–1 integer linear models. These models have been solved for several test problems well-known in the literature and the results have been compared with respect to the number of stations. The experimentation revealed that the MLAT and MLAU are capable of solving problems with more than 35 tasks (at most up to 53 tasks). MLTT and MLTU are generally capable of solving problems with up to 35 tasks. In most of the problems with more than 35 tasks, optimal results cannot be reached in 72,000 seconds. The models serve as a starting point for researchers in the field, and may be used as a validation tool for heuristic procedures. In addition, goal programming approach has been presented in order to increase the assembly line system reliability arised from the stochastic case. This suggested model provides increased flexibility to the decision maker in evaluating different alternatives. Appendix The sample MLTU constraints for illustrative example in Section 5 are given below. All constraints are not given due to greatness of the model. C = 19, a = 0.1, z1a = 1.28155 are given. mmin = 3 is found. mmax = 7 is assumed.
K. Ag˘pak, H. Go¨kc¸en / European Journal of Operational Research 180 (2007) 1098–1115
1114
Objective function: Min
7 X
Si ¼ S3 þ S4 þ S5 þ S6 þ S7
i¼3
Subject to: Constraint ð2Þ for task 1; ðx11 þ y 11 Þ þ ðx12 þ y 12 Þ þ ðx13 þ y 13 Þ þ ðx14 þ y 14 Þ þ ðx15 þ y 15 Þ þ ðx16 þ y 16 Þ þ ðx17 þ y 17 Þ ¼ 1: Constraint ð5Þ and ð6Þ for ð1; 2Þ relationships; 7x11 þ 6x12 þ 5x13 þ 4x14 þ 3x15 þ 2x16 þ x17 7x21 6x22 5x23 4x24 3x25 2x26 x27 P 0; 7y 11 6y 12 5y 13 4y 14 3y 15 2y 16 y 17 þ 7y 21 þ 6y 22 þ 5y 23 þ 4y 24 þ 3y 25 þ 2y 26 þ y 27 P 0: Constraint ð24Þ for workstation 1; 361 228p11 76p21 190p31 266p41 38p51 76p61 114p71 228p81 190p91 190p101 152p111 þ 36p11 þ 4p21 þ 25p31 þ 49p41 þ p51 þ 4p61 þ 9p71 þ 36p81 þ 25p91 þ 25p101 þ 16p111 þ 24u121 þ 60u131 þ 84u141 þ 12u151 þ 24u161 þ 36u171 þ 72u181 þ 60u191 þ 60u1101 þ 48u1111 þ 20u231 þ 28u241 þ 4u251 þ 8u261 þ 12u271 þ 24u281 þ 20u291 þ 20u2101 þ 16u2111 þ 70u341 þ 10u351 þ 20u361 þ 30u371 þ 60u381 þ 50u391 þ 50u3101 þ 40u3111 þ 14u451 þ 28u461 þ 42u471 þ 84u481 þ 70u491 þ 70u4101 þ 56u4111 þ 4u561 þ 6u571 þ 12u581 þ 10u591 þ 10u5101 þ 8u5111 þ 12u671 þ 24u681 þ 20u691 þ 20u6101 þ 16u611 þ 36u781 þ 30u791 þ 30u7101 þ 24u7111 þ 60u891 þ 60u8101 þ 48u8111 þ 50u9101 þ 40u9111 þ 40u10111 1:281552 ð5:308p11 þ 0:51p21 þ 3:151p31 þ 7:263p41 þ 0:202p51 þ 0:767p61 þ 1:004p71 þ 11:614p81 þ 5:221p91 þ 0:723p101 þ 0:005p111 Þ P 0: Constraint ð25Þ for workstation 1; 19S 1 6p11 2p21 5p31 7p41 p51 2p61 3p71 6p81 5p91 5p101 4p111 P 0: Constraint ð26Þ for workstation 1; task 1; j ¼ 2; 3; . . . ; 11; p11 þ p21 u121 6 1;
p11 þ p31 u131 6 1;
p11 þ p41 u141 6 1;
p11 þ p51 u151 6 1
p11 þ p61 u161 6 1;
p11 þ p71 u171 6 1;
p11 þ p81 u181 6 1;
p11 þ p91 u191 6 1
p11 þ p101 u1101 6 1;
p11 þ p111 u1111 6 1:
Constraint ð27Þ for workstation 1; task 1; j ¼ 2; 3; . . . ; 11; p11 þ p21 2u121 P 0;
p11 þ p31 2u131 P 0;
p11 þ p41 2u141 P 0;
p11 þ p51 2u151 P 0
p11 þ p61 2u161 P 0;
p11 þ p71 2u171 P 0;
p11 þ p81 2u181 P 0;
p11 þ p91 2u191 P 0
p11 þ p101 2u1101 P 0;
p11 þ p111 2u1111 P 0:
Constraint ð28Þ for task 1; x11 þ y 11 p11 ¼ 0;
x12 þ y 12 p12 ¼ 0;
x13 þ y 13 p13 ¼ 0;
x15 þ y 15 p15 ¼ 0;
x16 þ y 16 p16 ¼ 0;
x17 þ y 17 p17 ¼ 0:
x14 þ y 14 p14 ¼ 0;
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