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An assembly scheduler for TFT LCD manufacturing
Computers & Industrial Engineering 41 (2001) 37±58
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An assembly scheduler for TFT LCD manufacturing Bongju Jeong*, Si-Won Kim, Yun-Jun Lee Department of Industrial Systems Engineering, Yonsei University, 134 Shinchon-dong, Seodaemun-ku, Seoul 120-749, South Korea
Abstract This paper presents development of a scheduling system for Thin Film Transistor Liquid Crystal Display (TFT LCD) assembly process. With given daily production plan that is rough-cut capacitated, the scheduler generates shift-based capacity feasible schedules at each assembly step. The scheduling problem is a parallel-machine scheduling problem with sequence dependent setups and some manufacturing environments, which make the problem considerably dif®cult. The scheduling objective is to minimize the mean ¯ow time and maximize the production progressiveness. Two scheduling heuristics are proposed for each production step of assembly process. The ®rst scheduling heuristic is applicable for the ®rst three steps of assembly process, i.e. coating, rubbing, and attaching step, and for the later steps another similar heuristic is employed. We also propose a simple method for computing production requirements for each step where daily production demand is unknown. The proposed scheduling procedures are implemented in a real TFT LCD manufacturing line and their performances are investigated. The experimental results show that the proposed scheduler provides quite better schedules than the current practice employed in the line in terms of both mean ¯ow time and ful®llment of production demands. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: TFT LCD manufacturing; Scheduling; Sequence-dependent setup
1. Introduction Thin Film Transistor Liquid Crystal Display (TFT LCD) manufacturing process consists of two basic processes, TFT fabrication and LCD assembly process. The TFT fabrication process is very similar to semiconductor wafer fabrication, but it is much simpler. The assembly process of TFT LCD consists of about 15 assembly steps and usually takes about three days. In this paper, we are concerned with development of scheduling system for the assembly process. Most TFT LCD * Corresponding author. E-mail address: [email protected] (B. Jeong). 0360-8352/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S0 3 6 0 - 8 3 5 2 ( 0 1 ) 0 0 04 1 - 9
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Fig. 1. TFT LCD assembly process.
manufacturing ®rms, unlike fabrication process, have paid little attention to assembly process because its production ¯ow time is shorter, the process looks much simpler and less variable, and the equipment are much less expensive than those of TFT fabrication. From the scheduling viewpoint, however, a lot of opportunities for improvement still exist in the assembly process so that even simple improved scheduling procedure can result in the signi®cant improvement of line productivity. A TFT LCD assembly line consists of many stations (or production steps) and each station usually has parallel identical machines. As shown in Fig. 1, throughout the assembly process, each TFT and color ®lter (CF) glasses are separately coated and rubbed with chemical called poly-imid, then attached together, hot pressed, ®lled with liquid crystal, cut into cell units, assembled with module components, and ®nally tested before outgoing. Usually CF glasses are purchased from outside vendors while most TFT LCD ®rms manufacture TFT glasses in their own fabrication facilities. Each TFT and CF glass should be exactly matched in size and type so that each pair of two glasses can be attached in attaching step and ®nally become a TFT LCD product unit after processed in the subsequent steps. Each glass contains four or six cells according to the size of glass and each cell represents a LCD product unit. Glasses are typically grouped into lot sizes of 20 and released into coating step that is the ®rst step of assembly process. After attaching step, the attached glasses of TFT and CF are processed until they are cut into cells in cutting step. Different types or sizes of products are simultaneously manufactured in a line and change of product types requires sequence-dependent setups. Since the setup time
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occurs only when the product types are changed and each product type is entered in a lot (i.e. batch), these setups actually require the batch setup times. Throughout this paper, we use `setup time' with the meaning of batch setup time. In this paper, we assume that there is a daily production planner for TFT LCD manufacturing line. The daily planner provides rough-cut capacitated daily production plans for input and output of TFT fabrication and assembly process. The assembly scheduler is designed to generate shift-based schedules for each step of assembly process. The scheduling goals are to minimize the mean ¯ow time of cell and to maximize production progressiveness that is a measure for ful®llment of production demands. No previous research works are found on the problem of scheduling TFT LCD assembly process. However, the literature on scheduling problems with sequence-dependent setup is extensive and too broad to be fully covered here. There is a comprehensive review paper on scheduling problem with setup time (Allahverdi, Gupta, & Aldowaisan, 1999). Many research works are found on problems with sequence-dependent setup time on parallel machines. Guinet (1990) proposed a heuristic algorithm using the mixed integer programming to minimize the mean ¯ow time. Elmaghraby, Guuinet, and Schellenberger (1993) extended Guinet's results to develop an improved heuristic to minimize makespan. Sumichrast and Baker (1987) proposed a heuristic method using integer programming to minimize total setup time. Ovacik and Uzsoy (1993) studied the problem of minimizing maximum completion time and maximum lateness in semiconductor testing facilities. Also they proposed the rolling horizon heuristic to minimize maximum lateness (Ovacik & Uzsoy, 1995). Guinet and Dussauchoy (1993) developed a heuristic algorithm by extending Hungarian method. Lee and Pinedo (1997) proposed a three-phase heuristic that combines a dispatching rule and a simulated annealing procedure to solve minimizing weighted total tardiness. In the special cases of parallel machines, Marsh and Montgomery (1973) proposed heuristic algorithms to minimize change-over time. Deane and White (1975) also extended Marsh and Montgomery results to develop a branch-and-bound algorithm which performs workload balancing and minimizes total setup cost. This paper is organized as follows: Section 2 describes the scheduling problems and mathematical models. Section 3 proposes two scheduling heuristics for the early steps and later steps of assembly process, respectively. Section 4 presents the experimental results for evaluating performances of the proposed scheduling procedure. Finally, concluding remarks and direction for future research are addressed in Section 5.
2. Mathematical models As indicated in Section 1, the early two steps (i.e. coating and rubbing step) process TFT and CF glasses separately and two matched glasses of TFT and CF are attached in attaching step. After then the attached glasses are processed in the subsequent steps of assembly process. Considering this situation, the appropriate matching of TFT and CF glasses is very important in scheduling the ®rst three steps because the unmatched glasses can be no longer processed and may cause starvation in the subsequent steps. Thus, we decompose the scheduling problem into two different scheduling problems for the early and later steps, respectively.
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2.1. Model for scheduling early steps In order to formulate the scheduling problem for the early three steps, let us use the following notations: Indices: i, j
product index. Products indicate both TFT and CFs. If i is TFT product, then 2i indicates the matched CF product lot index machine index index for the assigned position of lot on a machine production step index
a, b m k s
8 0 > > > > > <1 > > 2 > > > : 3
for a bank for coatingstep for rubbingstep for attachingstep
Parameters:
a ,b N Li L
relative weight factor the number of products the number of lots of product i N X i1
Li the total number of lots
Ms the number of machines of step s T scheduling horizon processing time of product i on step s. This parameter equals zero for s 0 pis K bT= min {pi;s }c the maximum number of lots that can be processed on a machine during T where i;s
dis cij
bwc is the largest integer that does not exceed w production demand for product i at step s setup time from product i to product j Decision variables: (
xi;a;m;k;s
1
if lot a of product i is assigned to machine m in kth position at step s
0 otherwise This variable also equals zero for k 0 or s 0.
B. Jeong et al. / Computers & Industrial Engineering 41 (2001) 37±58
Ci,a,s SLis
41
completion time of lot a of product i at step s.At the bank, Ci;a;0 0. slack variable for product i at step s
The scheduling problem P for the early three steps of assembly process is stated as follows: Model P: Minimize
Li N X N X 3 X aX Ci;a;3 1 b uSLis u L i1 a1 i1 s1
1
Subject to: Ms X K X
xi;a;m;k;s # 1
;i; a; s
2
xi;a;m;k;s # 1
for s 1; 2 and ;m; k
3
xi;a;m;k;s # 2
for s 3 and ;m; k
4
m1 k1 Li N X X i1 a1 Li N X X i1 a1
Ms X K X m1 k1
xi;a;m;k;s #
Ms X K X m1 k1
xi;a;m;k;s21
for s 1; 2; 3 and ;i; a
5
xi;a;m;k;3 x2i;a;m;k;3 Ms X Li X K X a1 m1 k1
Ci;a;s
6
xi;a;m;k;s 1 SLis dis
;i; s
9 8 Lj N X =
Ci;a;3 C2i;a;3 xi;a;m;k;s 0 or 1
;i; a ;i; a; m; k; s
7
for s 1; 2; 3 and ;i; a
8 9 10
The objectives of model P given in Eq. (1) are to minimize the mean ¯ow time of lot and deviation from the production demand. These two objectives can be prioritized using the weight factors a; b: Constraint set (2) assures that each lot should be assigned to only one machine. Constraint sets (3) and (4) ensure that each position in the sequence on a machine cannot have more than one lot. Note that in attaching step s 3; a lot actually consists of two lots of TFT and CF, which is an attached unit. Constraint set (5) ensures that each lot should be processed in
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the order of production steps. Constraint set (6) enforces the exact match of TFT and CF lots at attaching step. Constraint set (7) is the production demand balance for each product. Constraint set (8) computes the completion time of each lot at each production step. Constraint set (9) also enforces that the attached TFT and CF lots should have the same completion time at attaching step. Eq. (10) indicates xi;a;m;k;s are binary integer variables. In order to refer to the scheduling problems under consideration, we adopt the terminology of Pinedo (1995). The terminology consists of three ®elds, aubug; where a indicates the machine (processor) environments, b the job characteristics, g the optimality criteria. Then the problem P can be classi®ed as Rurj ; sduMFT which represents a parallel machine scheduling problem with sequence dependent setups, job release times, and the objective of minimizing the mean ¯ow time. This problem is known to be a notorious NP-hard problem (Allahverdi et al., 1999). Moreover, problem P has the additional complicated manufacturing environments such as multi-objectives, multi-production stages (steps), and the matching constraints, which make the problem extremely dif®cult. No research is known about scheduling algorithms for this type of problem.
2.2. Model for scheduling later steps Throughout the later steps of assembly process, an attached product of TFT and CF is the processed unit. Since we do not need to consider the match of TFT and CF lots in the later steps, the scheduling problem Q for later steps becomes simpler than problem P. Unlike the problem P, problem Q is formulated for each of later steps, not for whole steps. In order to formulate the scheduling problem for each later step, we can use the same notations used in problem P except the followings (no station index is used): Parameters: processing time of product i production demand for product i setup time from product i to product j
pi di cij
Decision variables: ( xi;a;m;k
1
if lot a of product i is assigned to machine m in kth position
0
otherwise
This variable also equals zero for k 0 Cia SLi
completion time of lot a of product i slack variable for product i
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Fig. 2. The early steps of assembly process.
The scheduling problem Q for each of the later steps of assembly process is stated as follows: Model Q: Minimize
Li N X N X aX Cia 1 b uSLi u L i1 a1 i1
11
Subject to: M X K X
xi;a;m;k # 1
;i; a
12
xi;a;m;k # 1
;m; k
13
m1 k1 Li N X X i1 a1
Li X M X K X a1 m1 k1
xi;a;m;k 1 SLi di
;i
8 9 Lj N X
;i; a; m; k
14
;i; a
15 16
The objective of model Q is the same as that of problem P. Constraint set (12) assures that each lot should be assigned to only one machine. Constraint set (13) ensures that each position in the sequence on
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a machine cannot have more than one lot. Constraint set (14) is the production demand balance for each product. Constraint set (15) computes the completion time of each lot. Eq. (16) indicates xi;a;m;k are binary integer variables. 3. Scheduling algorithms 3.1. Scheduling heuristics for the early steps We consider a scheduling procedure for coating, rubbing, and attaching steps, which are the ®rst three steps of TFT LCD assembly process, as shown in Fig. 2. TFT glasses made in the fabrication process and CF glasses are entered into coating step and they are rubbed with rolling machines in rubbing station, and they are matched and attached in attaching step. TFT and CF inventory are stocked in the `bank' before they are processed into coating step. There are also two buffers for WIP, called `stocker'. One is located between coating and rubbing step, and the other between rubbing and attaching step. The primary objective in the early steps is to match the TFTs and the corresponding CF glasses as many as possible before the attaching step where only the matched pair of TFT and CF is allowed. Since one of the objectives of problem P is to minimize the mean ¯ow time, we need to seek the minimization of WIP in the system according to Little's law (Little, 1961). Therefore, in order to reduce WIP, we have to reduce the unmatched TFT and CF stocks waiting at each stocker so that the WIP in the stockers can be reduced. Also, the reduction of the unmatched stocks can reduce the chance of starvation in the attaching step, which consequently reduces mean ¯ow time. Another way of reducing mean ¯ow time is to minimize setup times for product changes by optimally sequencing the input products. In order to describe the scheduling procedure, we de®ne the following notations: p i; s processing time of product i at step s c i; j setup time needed for changing product i to j d i daily production demand for product i TFT(i) cumulative scheduled amount of TFT for product i S_TFT(i) current TFT stocks of product i in stocker CF(i) cumulative scheduled amount of CF for product i S_CF(i) current TFT stocks of product i in stocker UM(i) unmatchness of product i {S_TFT i 1 TFT i}-{S_CF i 1 CF i} MAXU maximum allowed unmatchness ConL available amount of continuous input at current scheduling time LAP(m) the last assigned product on machine m EST(i) the earliest starting time of product i MEST minimum earliest starting time mini {EST i} EAT(m) the earliest available time of machine m MEAT minimum earliest available time minm {EAT m} AP(i) amount of the current actual production for product i The heuristic algorithm for scheduling problem at each step is described as follows.
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Fig. 3. Flowchart of scheduling algorithm E_SCHED.
3.1.1. Algorithm E-SCHED Step 1. [Initial assignment] For each machine m, if the current loaded product i is included in the set of devices in daily production demand, then assign product i with the amount of min:ConL; d i on the same machine m. Step 2. [Scheduling for unmatchness reduction] 1. Select a machine q with the most earliest available time, i.e. EAT q min{EAT m} for all m. 2. Select a product j with the maximum uUM iu for all product i. 3. If uUM ju , MAXU; then go to Step 3. Otherwise, schedule product j as follows: If UM j . 0 and d j . CF j; then schedule the CF with amount of min:{ConL; d j 2 CF j; uUM ju} on machine q If UM j , 0 and (d j . TFT j; then schedule the TFT with amount of min:{ConL; d j 2 TFT j; uUM ju} on machine q Update UM(j), CF(j), TFT(j), and EAT(q). 4. Go to Step 2-(1). Step 3. [Regular scheduling]
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Fig. 4. Illustration of reallocation.
(1) Select a machine q with the most earliest available time, i.e. EAT(q) min{EAT(m)} for all m. Let I set of products i such that EST i # AT q: (Case 1)I ± f Select a product i with mini[I {c LAP q; i}: Schedule a product i on machine q with the amount of min{ConL; d i 2 AP i}: Update EAT, MEAT, and LAP: EAT q à EAT q 1 c LAP q; i 1 p ip min{ConL; d i 2 AP i}; MEAT à min{EAT m} for all m LAP q à i: (Case 2) I f If there exists a product i assigned on machine m such that EAT m 2 MEAT . 2p MEST 2 MEAT and stock of i in buffer 1 MEST 2 MEAT=p i; s 2 1 $ 2p MEST 2 MEAT=p i; s; then move i from machine m to machine q with amount of b MEST 2 MEAT=p i; sc where bwc is the largest integer that does not exceed w. Update EAT(q), MEAT, EAT(m), and LAP(q) as follows: EAT q à EAT q 1 c LAP q; i 1 p i; sp b MEST 2 MEAT=p i; sc; MEAT à EAT q; EAT m à EAT m 2 p i; sp b MEST 2 MEAT=p i; sc; LAP q à i: (2) If all planned products have been scheduled or no further scheduling for them is possible, go to Step 4. Otherwise, repeat Step 2 and 3. Step 4. [Scheduling unplanned products]
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T1
s-1
T2
s
47
s+1
D1a t A D1b
B
t
D2a
C D2b
(a) t1 + t2 ≤ 1
s-1 D1a t2 D1b
T1
T2
s
s+1
t1 A B C
D2a
D2b (b) t1 + t2 > 1 Fig. 5. Computing production requirements.
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B. Jeong et al. / Computers & Industrial Engineering 41 (2001) 37±58
If there still exist available machines during the scheduling horizon (a day), schedule the unplanned products with d i 0 using Step 2 and 3. Step 5. [Conversion to shift schedule] Decompose the resultant daily schedule into 3 shift schedules. In Step 1, the same products as the current loaded products are considered in initial assignment so that unnecessary setups can be avoided. Step 2 tries to reduce the current unmatchness for the products that exceed MAXU (maximum allowed unmatchness). If TFT's are over-loaded, the matched CF's are scheduled with the amount of min{available CF quantity, unful®lled amount of demand, unmatched amount}. The algorithm repeats Step 2 until all products have the allowable unmatchness, i.e. uUM iu , MAXU for all i. Step 3 schedules all planned products according to MST rule (minimization of setup time). Case 1 indicates the case with no idle times where the product can be scheduled on any available times of machines. On the contrary, in Case 2, the machine idle times can occur because there exist a product that is not available on the earliest available time of machine. In this case, the algorithm tries to schedule the available products on the inserted idle times. To do this, we check if there exists a prescheduled product that can be reallocated to the idle time period. If such a product is found, reallocate it with the available amounts of the product. Consequently, the reallocated product is assigned on two machines simultaneously and the idle times are eliminated. This case is illustrated in Fig. 3. The algorithm repeats Steps 2 and 3 until all planned products are scheduled or the scheduling horizon is over. And then, if any available machine still exists, Step 4 schedules the unplanned products by applying Steps 2 and 3 again. Step 5 converts the daily schedule to three shift schedules by simply decomposing one day period. Fig. 4 shows the summarized ¯owchart of algorithm E_SCHED. For P complexity analysis of algorithm, let M Ms and D Ni1 di (Note that the algorithm is applied to each production step, not whole three steps at once.) The computational complexity of Step 1 is given as O(MDL). Steps 2, 3, and 4 have the same complexities of O(MDKL) and Step 5 has a constant running time. Thus, algorithm E_SCHED has the polynomial time complexity of O(MDKL). 3.2. Scheduling heuristic for the later steps As in the early steps, the scheduling objectives are to minimize the mean ¯ow time and unful®lled production requirements of each step. Suppose the ¯ow time consists of only the processing time and setup time. Given ®xed WIP in the system, in order to achieve the ®rst objective, we have no choice but minimize the total setup time because the processing time is ®xed and cannot be reduced. Unlike the early steps, since we do not have to consider the matching problem of TFT and CFT glasses, the proposed heuristic L_SCHED for the later steps does not include Step 2 of E_SCHED and the other steps are all applicable, too. 3.2.1. Algorithm L-SCHED Step 1. [Initial assignment]: same as Step 1 of E_SCHED. Step 2. [Scheduling planned products]: same as Step 3 of E_SCHED. Step 3. [Scheduling unplanned products]: same as Step 4 of E_SCHED. Step 4. [Conversion to shift schedule]: same as Step 5 of E_SCHED.
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3.3. Computing production requirements One of the major input data for assembly scheduler is the production requirement for each step. Unfortunately, the daily production planner only provides input and out plans for a whole assembly process, not for each step. Thus, somehow we need to compute the production requirements for every step. In this section, we propose a method for computing the production requirements of each step. This problem is generalized as a problem to compute the production requirement for a speci®c step s which lies between any two steps s 2 1 and s 1 1 with known production requirements. As shown in Fig. 5, the production requirement (input) of step s 2 1 may be different from the production requirement (output) of step s 1 1: Then the problem is to determine the intermediate requirement at step s. The proposed method uses the linear interpolation technique to proportionally distribute the production demand according to the ratio of output to input. Let us de®ne the following notations: T1 lead time from step s 2 1 to step s T2 lead time from step s to step s 1 1 t1 T1 2 bT1 c where bwc is the largest integer that does not exceed w t2 T2 2 bT2 c D1a production requirement for step s at bT1 c 1 1 days before the current scheduling day D1b production requirement for step s at bT1 c days before the current scheduling day D2a production requirement for step s 1 1 at bT2 c days after the current scheduling day D2b production requirement for step s 1 1 at bT2 c 1 1 days after the current scheduling day D production requirement for step s at the current scheduling day Then we can derive the following equations for computing D: DA1B1C where (Case 1) t1 1 t2 # 1 in Fig. 5(a)
A
T2 D1a 1 T1 D2a t1 T1 1 T2
B
T2 D1b 1 T1 D2a 1 2 t1 2 t2 T1 1 T2
C
T2 D1b 1 T1 D2b t2 T1 1 T2
17
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Table 1 Processing time data Production step
Product type
Interarrival time between glasses (min)
Processing time per glass (min)
Lot size (glasses)
Processing time per lot (min)
Coating
1,2,3,4 5 1,2,3,4 5 All
0.8 1.1 1.0 1.3 1.5
17.5 21.9 38.8 50.6 40.0
20 20 20 20 20
32.7 42.8 57.8 75.3 68.5
Rubbing Attaching
(Case 2) t1 1 t2 . 1 in Fig. 5(b) A
T2 D1a 1 T1 D2a 1 2 t2 T1 1 T2
B
T2 D1a 1 T1 D2b t1 1 t2 2 1 T1 1 T2
C
T2 D1b 1 T1 D2b 1 2 t1 T1 1 T2
Table 2 Setup time data Production step
From-product
To-product
Setup time (min)
Coating
1, 2, 3 4 4 (TFT) 4 (CF) 1, 2, 3, 4 5
4 1, 2, 3 4 (CF) 4 (TFT) 5 1, 2, 3, 4
55 55 55 55 60 75
Rubbing
1, 2, 3 4 4 (TFT) 4 (CF) 1, 2, 3, 4 5
4 1, 2, 3 4 (CF) 4 (TFT) 5 1, 2, 3, 4
37 37 37 37 60 60
Attaching
1 2 3 1, 2, 3 4 1, 2, 3, 4 5
2, 3 1, 3 1, 2 4 1, 2, 3 5 1, 2, 3, 4
30 30 30 30 30 300 300
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Table 3 Matching table TFT type
CF type
TFT1
CF1 CF2 CF3 CF4 CF5 CF6 CF7 CF8
TFT2 TFT3 TFT4 TFT5
CF1 Group CF2 Group CF3 Group CF4 Group CF5 Group
4. Experimental evaluation 4.1. Data preparation In order to investigate the ef®ciency of the proposed scheduling system, we compared it to the current practice in a real TFT LCD assembly line of A-company which is known to be one of the largest TFT LCD manufacturers in the world (The company name is anonymous due to con®dentiality.) Using sets of actual data collected during entire three weeks (i.e. 21 days, 63 shifts), the scheduling algorithms were applied for the assembly process. Although the scheduling system was designed to generate shift Table 4 TFT fabrication out plan (unit: cell) Day
TFT1
TFT2
TFT3
TFT4
TFT5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
3028 3122 3122 3122 3177 3177 3177 96 2400 1440 0 2208 0 1440 0 0 0 1440 0 0 1152
4953 5473 6089 6535 7101 7433 7606 3840 0 0 2016 0 2976 0 2592 1440 1920 0 768 0 960
7140 7315 7344 7750 7924 7924 7924 0 0 0 0 960 672 0 0 1056 1056 1056 960 1152 960
21375 22431 23580 24408 24875 25899 26309 4896 7008 7968 8544 5760 6240 9504 5856 6336 6240 4608 5760 7296 7584
940 940 940 940 940 940 940 0 960 1440 0 960 480 0 0 0 0 0 0 0 0
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Table 5 Daily assembly production plan (unit: cell) Day
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Assembly In
Assembly Out
TFT1
TFT2
TFT3
TFT4
TFT5
TFT1
TFT2
TFT3
TFT4
TFT5
640 640 640 640 640 800 1600 2400 0 0 2400 0 2560 0 1440 0 1424 0 1408 0 1408
1600 1440 1600 1600 2400 1600 1600 640 2000 2000 0 2480 0 2480 0 1920 0 1440 0 1440 0
800 960 800 0 0 0 0 960 0 0 0 0 0 0 1120 1120 1120 1120 1120 1120 1120
4960 4960 4960 4960 4960 4960 6400 5600 7200 7200 7200 7200 7200 7200 6080 6080 6080 6080 6080 6080 6080
0 0 0 800 0 800 0 0 1200 1200 0 0 0 0 0 0 0 0 0 0 800
1200 1040 1080 584 584 584 584 720 1448 2176 0 0 2176 0 2320 0 1280 0 1280 0 1200
560 608 1448 1288 1448 1448 2176 1448 1448 576 1816 1816 0 2248 0 2248 0 1736 0 1304 0
160 0 872 1016 1016 0 0 0 0 872 0 0 0 0 0 0 1016 1016 1016 1016 1016
5600 5800 5600 5256 4496 4496 4496 4496 5800 5056 6520 6520 6520 6520 6520 6520 5512 5512 5512 5512 5512
0 0 0 0 0 0 624 0 624 0 0 880 880 0 0 0 0 0 0 0 0
schedules for every step of assembly process, we applied our system to only the ®rst three steps because of some practical reasons. In reality it has been generally recognized throughout the company that the performance of the early steps dominates those of entire assembly process. This is because the later steps are all non-bottleneck steps and have enough capacities compared to the early steps. Also, like a transfer line, there are no buffers between the later steps so that changing the product sequences determined in the early steps can hardly happen. We prepared four kinds of data as follows: time data such as processing and setup times, matching information for TFT and CF glasses, daily TFT fabrication out plan, and daily assembly production plan (input and output plan). Since all actual data are con®dential according to company policy, all ®gures shown in this paper were appropriately scaled from the original ®gures without affecting the performance results. 4.1.1. Time data There are two kinds of time data: processing and setup time. Table 1 shows the processing time data for three production steps. This line currently produces ®ve types of TFT LCD product, i 1±5: The processing times are different for products. In coating and rubbing step, the product type shown in Table 1 indicates the corresponding TFT and CF types. The interarrival time ta and raw processing time t0 are applicable to glass unit. Each glass consists of six cells. The lot size (l) is the number of glasses in a lot. Then the processing time (p) for a lot is computed as follows: p t0 1 l 2 1ta :
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Table 6 Comparisons of mean ¯ow time (MFT) Step
Statistic
Scheduler
Actual
Throughput (cell/day)
WIP (cell)
Throughput (cell/day)
WIP (cell)
Throughput (cell/day)
WIP (cell)
Coating
Average Stdev.
17,367 2002
3657 522
0.21 0.011
18,315 4622
4032 1017
0.22 0.00
Rubbing
Average Stdev.
17,399 2239
2941 859
0.17 0.040
20,288 3677
5481 993
0.27 0.00
Attaching
Average Stdev.
9705 2151
1833 489
0.2 0.074
10,065 2017
2315 464
0.23 0.00
Table 2 shows setup time data for product change-over. In most of cases, changes between TFT and CF with a same product type do not require setups. For product 4, however, the change-over time occurs even between TFT and CF in coating and rubbing step. 4.1.2. Matching information Matching of TFT to CF glass is one-to-many relationship, as shown in Table 3. For example, TFT1 can be matched to either CF1 or CF2 while TFT2 is only matched to CF3. The CF types matched to a same TFT are grouped together and assigned to a same group. Then we have ®ve CF groups corresponding to ®ve products, as shown in Table 3. 4.1.3. TFT fabrication out plan Without loss of practicality, we assume that there are enough CF stocks that are purchased from outside vendors. Actually, A-company considers CF to be a long-term purchasing item so that shortage of CF inventory hardly happens. On the contrary, TFT glasses are manufactured in the company own facilities and a lot of efforts are made to meet the production demands. Therefore, the performance of assembly process is strongly affected by that of TFT fabrication process. Table 4 shows daily TFT fabrication out plans during three weeks and we assume that there exists the exact amount of the matched CF stock in the bank. Table 7 Comparisons of average production progressiveness for ®ve product types Step
Statistic
Actual
Scheduler
Coating
Average Stdev.
0.633 0.105
0.952 0.073
Rubbing
Average Stdev.
0.825 0.095
0.951 0.060
Attaching
Average Stdev.
0.824 0.111
0.942 0.133
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Table 8 Comparisons of variability of over-production (unit: cells) Step Coating
Product TFT1 TFT2 TFT3 TFT4 TFT5 CF1 Group CF2 Group CF3 Group CF4 Group CF5 Group
Actual 326.79 557.70 249.33 676.63 206.19 268.29 303.29 157.87 760.80 161.04
Scheduler 114.54 132.10 7.92 39.06 38.47 74.67 112.39 5.90 19.58 38.38
Rubbing
TFT1 TFT2 TFT3 TFT4 TFT5 CF1 Group CF2 Group CF3 Group CF4 Group
196.81 363.11 239.11 335.95 159.49 206.26 370.50 232.10 273.19
100.48 125.93 20.18 75.30 18.73 42.12 164.13 15.66 31.02
Attaching
CF5 Group Product 1 Product 2 Product 3 Product 4 Product 5
147.82 202.50 310.09 206.75 383.38 142.81 297.51
31.93 104.34 158.50 98.12 266.42 18.66 74.18
Average
4.1.4. Daily assembly production plan Table 5 shows a daily assembly In/Out plan that is generated by the daily production planner of Acompany. This plan is a rough-cut capacitated production plan that is constructed according to the sales plan, monthly production plan, and overall line capacities. 4.2. Experimental results In this section, we compared the performances of our proposed scheduling system to those of current practice employed in A-company. The scheduling logic used in the line of A-company is somewhat adhoc based rules. The rules are frequently changed by line workers according to manufacturing environments such as order cancellation, demand changes, line disciplines, and so on. However, the most frequently rules are First-In-First-Out (FIFO) and minimum setup time (MST) rules. Here, MST rule is executed in the manner of continuous production, which does not allow setup as possible and keep producing until inventory shortage occurs. Also, the production managers and line workers in A-company are much more concerned about meeting the monthly production plan rather than daily plans because of ease of control, and that they are used to this practice.
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For three-week experiments, we set the maximum allowed unmatchness (MAXU) to be 10 lots which is empirically determined. Table 6 shows the comparison results of the proposed scheduling system versus the current practice in terms of mean ¯ow time of cells. Since throughput rate (cells per day) and average WIP are the direct results of scheduling system, mean ¯ow time can be computed indirectly using Litte's law (Little, 1961) as follows: Flow time WIP=Throughput rate:
18
However, A-company has not maintained daily WIP data ever. Instead, the company maintains average ¯ow time data for each step, which are updated every two months, and we could get daily throughput. In applying our scheduler, we also considered real manufacturing environments such as the probable machine breakdowns, preventive maintenance time, and other losses. The experimental results in Table 6 show that the mean ¯ow time by the proposed scheduler is apparently smaller than the actual results at all three steps during most of 21 days. The total mean ¯ow time for the early steps by the scheduler is 0.58 days and the actual one is 0.72 days. Thus, we have achieved 19.4% reduction of mean ¯ow time by the proposed scheduler. As another performance criterion, production progressiveness (or demand ful®llment) is computed against given daily production plans. Since the daily production planner of A-company gives only the input and output plan of the whole assembly line for each product, we computed the out plans for each step using Eq. (17). Production progressiveness (PR) for product i is computed as follows: 8 > < Pi if di 2 Pi . 0 under production demand ; 19 PRi di > : 1 otherwise over production demand where di is the daily production demand for product i and Pi is the daily actual production for product i. As shown in Table 7, the scheduling system gives much better production progressiveness of average 94.8% during 21 days than the current practice of average 76.1% at all three steps. Thus, the average 18.7% increase of production progressiveness has been achieved. Note that the biggest improvement by the scheduler has been made in coating step. This indicates that in coating step there is little concern about meeting production demands and FIFO rules are mostly used according to TFT fabrication out. On the other hands, in rubbing and attaching steps, the operators begin to be more conscious on production demand and try to control the production using WIP in stocker. This means that the rubbing and attaching steps have the higher WIP level than the coating step as shown in Table 6. From production management viewpoint, production smoothing is one of the most important issues in the company because it makes it easier to respond to demand ¯uctuations quickly. Production smoothness can be measured by standard deviation of over-production as shown in Table 8. For all products and steps, the proposed scheduler provides the smaller variability of over-production than the actual practice. On the average, 75.1% reduction of variability has been achieved by the proposed scheduler. For TFT1 product at coating step, Fig. 6 shows the comparison results in variability between the proposed scheduler and actual practice. The proposed scheduling system has been developed using Delphi 5 and the pilot version is currently running on Pentium II-400 MHz computer in a TFT LCD assembly line of A-company. The signi®cant improvements in mean ¯ow time and production progressiveness are expected after full implementation of the proposed scheduler as shown in the three-weeks experimental results. The computing time for
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Fig. 6. Comparisons of over-production for TFT1 at coating step.
scheduling the early steps is less than a second so that rescheduling can be possible at any time without computational burden, which is one of the strong requirements of the production planning group of Acompany. Fig. 7 shows a sample screen for parameter inputs and illustrates Gantt charts according to the schedules generated by the proposed system. 5. Conclusions In this paper, we present an assembly scheduler for each production step of TFT LCD assembly process. The scheduler includes two heuristic procedures, which are employed separately for the early and later production steps. We assume that the daily production plans are provided for TFT fabrication and assembly process. The scheduling problem under consideration is a parallel-machine scheduling problem with sequence dependent setups, multi-stage production, multi-objectives, and the matching constraints. The scheduling objectives are to minimize the mean ¯ow time and maximize the ful®llment of production demands. Since the problem is a NP-hard, we propose practical heuristic algorithms. In TFT LCD assembly process, the early steps require processing both TFT and CF glasses independently with identical setup while the later subsequent steps deal with the attached unit of two matched glasses. Thus the major concern for scheduling the early steps is to synchronize TFT and CF glasses at the subsequent steps so as to minimize the number of unmatched TFT or CF glasses, which in turn reduce the unmatched WIP in stocker and consequently reduce the mean ¯ow time. The other way of
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(a) A Sample Screen for Inputs
(b) Gantt Charts
Fig. 7. TFT LCD assembly scheduler: sample screens. (a) A sample screen for inputs. (b) Gantt charts.
minimizing mean ¯ow time is to minimize the total setup time by optimally sequencing products. These concepts were included in the scheduling algorithms. Also, we propose a simple method for computing the production requirements of each step because the daily production planner provides only input and output plans for the whole assembly process, not for every step of assembly process. The concept behind the method is to proportionally distribute the production demands according to the ratio of output to input production demands. The proposed scheduler was implemented in a real TFT LCD assembly line.
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Implementation results show that the scheduler generates quite better schedules than the current practice employed in the line in terms of the scheduling objectives. This is the ®rst research on the scheduling problem for TFT LCD assembly process. The scheduling heuristics were designed to focus on the practical requirements of production planning people in the TFT LCD assembly line. The requirements were as follows: (1) to provide a capacitated shift schedule for every step of assembly process, (2) to reduce WIP in the line, (3) to maximize the ful®llment of production demands, and (4) quick run time for rescheduling. The second requirement is equivalent to minimizing the mean ¯ow time, which is the primary objective of our scheduler. We believe that all these requirements have been considerably satis®ed in the proposed scheduler and the company people consider the full implementation of the system. Possible directions for future research may include the improvement of algorithms using ef®cient search techniques and optimal determination of MAXU which is determined empirically in the current scheduler. References Allahverdi, A., Gupta, J. N. D., & Aldowaisan, T. (1999). A review of scheduling research involving setup considerations. Omega, International Journal of Management Science, 27, 219±239. Deane, R. H., & White, E. R. (1975). Balancing workloads and minimizing setup costs in the parallel processing shop. Operation Research Quarterly, 26, 45±53. Elmaghraby, S. E., Guinet, A., & Schellenberger, K. W. (1993). Sequencing on parallel processors: an alternate approach. OR Technical Report, No. 273. Raleigh, NC: North Carolina State University. Guinet, A. (1990). Textile production systems: a succession of non-identical parallel processor shops. Journal of Operations Research Society, 42, 655±671. Guinet, A., & Dussauchoy, A. (1993). Scheduling sequence dependent jobs on identical parallel machines to minimize completion time criteria. International Journal of Production Research, 31, 1579±1594. Lee, Y. H., & Pinedo, M. (1997). Scheduling jobs on parallel machines with sequence dependent setup times. European Journal of Operation Research, 100, 464±474. Little, J. D. C. (1961). A proof for the queuing formula: L lW. Operations Research, 9, 383±387. Marsh, J. D., & Montgomery, D. C. (1973). Optimal procedures for scheduling jobs with sequence-dependent changeover times on parallel processors. AIIE Technical Papers., 279±286. Ovacik, I. M., & Uzsoy, R. (1993). Worst-case error bounds for parallel machine scheduling problems with bounded sequencedependent setup times. Operational Research Letters, 14, 251±256. Ovacik, I. M., & Uzsoy, R. (1995). Rolling horizon procedures for dynamic parallel machine scheduling with sequencedependent setup times. International Journal of Production Research, 33, 3173±3192. Pinedo, M. (1995). Scheduling: Theory, Algorithms, and Systems, Englewood Cliffs, NJ: Prentice Hall. Sumichrast, R., & Baker, J. R. (1987). Scheduling parallel processors: an integer linear programming based heuristic for minimizing setup time. International Journal of Production Research, 25, 761±771.
www.elsevier.com/locate/dsw
An assembly scheduler for TFT LCD manufacturing Bongju Jeong*, Si-Won Kim, Yun-Jun Lee Department of Industrial Systems Engineering, Yonsei University, 134 Shinchon-dong, Seodaemun-ku, Seoul 120-749, South Korea
Abstract This paper presents development of a scheduling system for Thin Film Transistor Liquid Crystal Display (TFT LCD) assembly process. With given daily production plan that is rough-cut capacitated, the scheduler generates shift-based capacity feasible schedules at each assembly step. The scheduling problem is a parallel-machine scheduling problem with sequence dependent setups and some manufacturing environments, which make the problem considerably dif®cult. The scheduling objective is to minimize the mean ¯ow time and maximize the production progressiveness. Two scheduling heuristics are proposed for each production step of assembly process. The ®rst scheduling heuristic is applicable for the ®rst three steps of assembly process, i.e. coating, rubbing, and attaching step, and for the later steps another similar heuristic is employed. We also propose a simple method for computing production requirements for each step where daily production demand is unknown. The proposed scheduling procedures are implemented in a real TFT LCD manufacturing line and their performances are investigated. The experimental results show that the proposed scheduler provides quite better schedules than the current practice employed in the line in terms of both mean ¯ow time and ful®llment of production demands. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: TFT LCD manufacturing; Scheduling; Sequence-dependent setup
1. Introduction Thin Film Transistor Liquid Crystal Display (TFT LCD) manufacturing process consists of two basic processes, TFT fabrication and LCD assembly process. The TFT fabrication process is very similar to semiconductor wafer fabrication, but it is much simpler. The assembly process of TFT LCD consists of about 15 assembly steps and usually takes about three days. In this paper, we are concerned with development of scheduling system for the assembly process. Most TFT LCD * Corresponding author. E-mail address: [email protected] (B. Jeong). 0360-8352/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S0 3 6 0 - 8 3 5 2 ( 0 1 ) 0 0 04 1 - 9
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Fig. 1. TFT LCD assembly process.
manufacturing ®rms, unlike fabrication process, have paid little attention to assembly process because its production ¯ow time is shorter, the process looks much simpler and less variable, and the equipment are much less expensive than those of TFT fabrication. From the scheduling viewpoint, however, a lot of opportunities for improvement still exist in the assembly process so that even simple improved scheduling procedure can result in the signi®cant improvement of line productivity. A TFT LCD assembly line consists of many stations (or production steps) and each station usually has parallel identical machines. As shown in Fig. 1, throughout the assembly process, each TFT and color ®lter (CF) glasses are separately coated and rubbed with chemical called poly-imid, then attached together, hot pressed, ®lled with liquid crystal, cut into cell units, assembled with module components, and ®nally tested before outgoing. Usually CF glasses are purchased from outside vendors while most TFT LCD ®rms manufacture TFT glasses in their own fabrication facilities. Each TFT and CF glass should be exactly matched in size and type so that each pair of two glasses can be attached in attaching step and ®nally become a TFT LCD product unit after processed in the subsequent steps. Each glass contains four or six cells according to the size of glass and each cell represents a LCD product unit. Glasses are typically grouped into lot sizes of 20 and released into coating step that is the ®rst step of assembly process. After attaching step, the attached glasses of TFT and CF are processed until they are cut into cells in cutting step. Different types or sizes of products are simultaneously manufactured in a line and change of product types requires sequence-dependent setups. Since the setup time
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39
occurs only when the product types are changed and each product type is entered in a lot (i.e. batch), these setups actually require the batch setup times. Throughout this paper, we use `setup time' with the meaning of batch setup time. In this paper, we assume that there is a daily production planner for TFT LCD manufacturing line. The daily planner provides rough-cut capacitated daily production plans for input and output of TFT fabrication and assembly process. The assembly scheduler is designed to generate shift-based schedules for each step of assembly process. The scheduling goals are to minimize the mean ¯ow time of cell and to maximize production progressiveness that is a measure for ful®llment of production demands. No previous research works are found on the problem of scheduling TFT LCD assembly process. However, the literature on scheduling problems with sequence-dependent setup is extensive and too broad to be fully covered here. There is a comprehensive review paper on scheduling problem with setup time (Allahverdi, Gupta, & Aldowaisan, 1999). Many research works are found on problems with sequence-dependent setup time on parallel machines. Guinet (1990) proposed a heuristic algorithm using the mixed integer programming to minimize the mean ¯ow time. Elmaghraby, Guuinet, and Schellenberger (1993) extended Guinet's results to develop an improved heuristic to minimize makespan. Sumichrast and Baker (1987) proposed a heuristic method using integer programming to minimize total setup time. Ovacik and Uzsoy (1993) studied the problem of minimizing maximum completion time and maximum lateness in semiconductor testing facilities. Also they proposed the rolling horizon heuristic to minimize maximum lateness (Ovacik & Uzsoy, 1995). Guinet and Dussauchoy (1993) developed a heuristic algorithm by extending Hungarian method. Lee and Pinedo (1997) proposed a three-phase heuristic that combines a dispatching rule and a simulated annealing procedure to solve minimizing weighted total tardiness. In the special cases of parallel machines, Marsh and Montgomery (1973) proposed heuristic algorithms to minimize change-over time. Deane and White (1975) also extended Marsh and Montgomery results to develop a branch-and-bound algorithm which performs workload balancing and minimizes total setup cost. This paper is organized as follows: Section 2 describes the scheduling problems and mathematical models. Section 3 proposes two scheduling heuristics for the early steps and later steps of assembly process, respectively. Section 4 presents the experimental results for evaluating performances of the proposed scheduling procedure. Finally, concluding remarks and direction for future research are addressed in Section 5.
2. Mathematical models As indicated in Section 1, the early two steps (i.e. coating and rubbing step) process TFT and CF glasses separately and two matched glasses of TFT and CF are attached in attaching step. After then the attached glasses are processed in the subsequent steps of assembly process. Considering this situation, the appropriate matching of TFT and CF glasses is very important in scheduling the ®rst three steps because the unmatched glasses can be no longer processed and may cause starvation in the subsequent steps. Thus, we decompose the scheduling problem into two different scheduling problems for the early and later steps, respectively.
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B. Jeong et al. / Computers & Industrial Engineering 41 (2001) 37±58
2.1. Model for scheduling early steps In order to formulate the scheduling problem for the early three steps, let us use the following notations: Indices: i, j
product index. Products indicate both TFT and CFs. If i is TFT product, then 2i indicates the matched CF product lot index machine index index for the assigned position of lot on a machine production step index
a, b m k s
8 0 > > > > > <1 > > 2 > > > : 3
for a bank for coatingstep for rubbingstep for attachingstep
Parameters:
a ,b N Li L
relative weight factor the number of products the number of lots of product i N X i1
Li the total number of lots
Ms the number of machines of step s T scheduling horizon processing time of product i on step s. This parameter equals zero for s 0 pis K bT= min {pi;s }c the maximum number of lots that can be processed on a machine during T where i;s
dis cij
bwc is the largest integer that does not exceed w production demand for product i at step s setup time from product i to product j Decision variables: (
xi;a;m;k;s
1
if lot a of product i is assigned to machine m in kth position at step s
0 otherwise This variable also equals zero for k 0 or s 0.
B. Jeong et al. / Computers & Industrial Engineering 41 (2001) 37±58
Ci,a,s SLis
41
completion time of lot a of product i at step s.At the bank, Ci;a;0 0. slack variable for product i at step s
The scheduling problem P for the early three steps of assembly process is stated as follows: Model P: Minimize
Li N X N X 3 X aX Ci;a;3 1 b uSLis u L i1 a1 i1 s1
1
Subject to: Ms X K X
xi;a;m;k;s # 1
;i; a; s
2
xi;a;m;k;s # 1
for s 1; 2 and ;m; k
3
xi;a;m;k;s # 2
for s 3 and ;m; k
4
m1 k1 Li N X X i1 a1 Li N X X i1 a1
Ms X K X m1 k1
xi;a;m;k;s #
Ms X K X m1 k1
xi;a;m;k;s21
for s 1; 2; 3 and ;i; a
5
xi;a;m;k;3 x2i;a;m;k;3 Ms X Li X K X a1 m1 k1
Ci;a;s
6
xi;a;m;k;s 1 SLis dis
;i; s
9 8 Lj N X =
Ci;a;3 C2i;a;3 xi;a;m;k;s 0 or 1
;i; a ;i; a; m; k; s
7
for s 1; 2; 3 and ;i; a
8 9 10
The objectives of model P given in Eq. (1) are to minimize the mean ¯ow time of lot and deviation from the production demand. These two objectives can be prioritized using the weight factors a; b: Constraint set (2) assures that each lot should be assigned to only one machine. Constraint sets (3) and (4) ensure that each position in the sequence on a machine cannot have more than one lot. Note that in attaching step s 3; a lot actually consists of two lots of TFT and CF, which is an attached unit. Constraint set (5) ensures that each lot should be processed in
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B. Jeong et al. / Computers & Industrial Engineering 41 (2001) 37±58
the order of production steps. Constraint set (6) enforces the exact match of TFT and CF lots at attaching step. Constraint set (7) is the production demand balance for each product. Constraint set (8) computes the completion time of each lot at each production step. Constraint set (9) also enforces that the attached TFT and CF lots should have the same completion time at attaching step. Eq. (10) indicates xi;a;m;k;s are binary integer variables. In order to refer to the scheduling problems under consideration, we adopt the terminology of Pinedo (1995). The terminology consists of three ®elds, aubug; where a indicates the machine (processor) environments, b the job characteristics, g the optimality criteria. Then the problem P can be classi®ed as Rurj ; sduMFT which represents a parallel machine scheduling problem with sequence dependent setups, job release times, and the objective of minimizing the mean ¯ow time. This problem is known to be a notorious NP-hard problem (Allahverdi et al., 1999). Moreover, problem P has the additional complicated manufacturing environments such as multi-objectives, multi-production stages (steps), and the matching constraints, which make the problem extremely dif®cult. No research is known about scheduling algorithms for this type of problem.
2.2. Model for scheduling later steps Throughout the later steps of assembly process, an attached product of TFT and CF is the processed unit. Since we do not need to consider the match of TFT and CF lots in the later steps, the scheduling problem Q for later steps becomes simpler than problem P. Unlike the problem P, problem Q is formulated for each of later steps, not for whole steps. In order to formulate the scheduling problem for each later step, we can use the same notations used in problem P except the followings (no station index is used): Parameters: processing time of product i production demand for product i setup time from product i to product j
pi di cij
Decision variables: ( xi;a;m;k
1
if lot a of product i is assigned to machine m in kth position
0
otherwise
This variable also equals zero for k 0 Cia SLi
completion time of lot a of product i slack variable for product i
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43
Fig. 2. The early steps of assembly process.
The scheduling problem Q for each of the later steps of assembly process is stated as follows: Model Q: Minimize
Li N X N X aX Cia 1 b uSLi u L i1 a1 i1
11
Subject to: M X K X
xi;a;m;k # 1
;i; a
12
xi;a;m;k # 1
;m; k
13
m1 k1 Li N X X i1 a1
Li X M X K X a1 m1 k1
xi;a;m;k 1 SLi di
;i
8 9 Lj N X
;i; a; m; k
14
;i; a
15 16
The objective of model Q is the same as that of problem P. Constraint set (12) assures that each lot should be assigned to only one machine. Constraint set (13) ensures that each position in the sequence on
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a machine cannot have more than one lot. Constraint set (14) is the production demand balance for each product. Constraint set (15) computes the completion time of each lot. Eq. (16) indicates xi;a;m;k are binary integer variables. 3. Scheduling algorithms 3.1. Scheduling heuristics for the early steps We consider a scheduling procedure for coating, rubbing, and attaching steps, which are the ®rst three steps of TFT LCD assembly process, as shown in Fig. 2. TFT glasses made in the fabrication process and CF glasses are entered into coating step and they are rubbed with rolling machines in rubbing station, and they are matched and attached in attaching step. TFT and CF inventory are stocked in the `bank' before they are processed into coating step. There are also two buffers for WIP, called `stocker'. One is located between coating and rubbing step, and the other between rubbing and attaching step. The primary objective in the early steps is to match the TFTs and the corresponding CF glasses as many as possible before the attaching step where only the matched pair of TFT and CF is allowed. Since one of the objectives of problem P is to minimize the mean ¯ow time, we need to seek the minimization of WIP in the system according to Little's law (Little, 1961). Therefore, in order to reduce WIP, we have to reduce the unmatched TFT and CF stocks waiting at each stocker so that the WIP in the stockers can be reduced. Also, the reduction of the unmatched stocks can reduce the chance of starvation in the attaching step, which consequently reduces mean ¯ow time. Another way of reducing mean ¯ow time is to minimize setup times for product changes by optimally sequencing the input products. In order to describe the scheduling procedure, we de®ne the following notations: p i; s processing time of product i at step s c i; j setup time needed for changing product i to j d i daily production demand for product i TFT(i) cumulative scheduled amount of TFT for product i S_TFT(i) current TFT stocks of product i in stocker CF(i) cumulative scheduled amount of CF for product i S_CF(i) current TFT stocks of product i in stocker UM(i) unmatchness of product i {S_TFT i 1 TFT i}-{S_CF i 1 CF i} MAXU maximum allowed unmatchness ConL available amount of continuous input at current scheduling time LAP(m) the last assigned product on machine m EST(i) the earliest starting time of product i MEST minimum earliest starting time mini {EST i} EAT(m) the earliest available time of machine m MEAT minimum earliest available time minm {EAT m} AP(i) amount of the current actual production for product i The heuristic algorithm for scheduling problem at each step is described as follows.
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45
Fig. 3. Flowchart of scheduling algorithm E_SCHED.
3.1.1. Algorithm E-SCHED Step 1. [Initial assignment] For each machine m, if the current loaded product i is included in the set of devices in daily production demand, then assign product i with the amount of min:ConL; d i on the same machine m. Step 2. [Scheduling for unmatchness reduction] 1. Select a machine q with the most earliest available time, i.e. EAT q min{EAT m} for all m. 2. Select a product j with the maximum uUM iu for all product i. 3. If uUM ju , MAXU; then go to Step 3. Otherwise, schedule product j as follows: If UM j . 0 and d j . CF j; then schedule the CF with amount of min:{ConL; d j 2 CF j; uUM ju} on machine q If UM j , 0 and (d j . TFT j; then schedule the TFT with amount of min:{ConL; d j 2 TFT j; uUM ju} on machine q Update UM(j), CF(j), TFT(j), and EAT(q). 4. Go to Step 2-(1). Step 3. [Regular scheduling]
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B. Jeong et al. / Computers & Industrial Engineering 41 (2001) 37±58
Fig. 4. Illustration of reallocation.
(1) Select a machine q with the most earliest available time, i.e. EAT(q) min{EAT(m)} for all m. Let I set of products i such that EST i # AT q: (Case 1)I ± f Select a product i with mini[I {c LAP q; i}: Schedule a product i on machine q with the amount of min{ConL; d i 2 AP i}: Update EAT, MEAT, and LAP: EAT q à EAT q 1 c LAP q; i 1 p ip min{ConL; d i 2 AP i}; MEAT à min{EAT m} for all m LAP q à i: (Case 2) I f If there exists a product i assigned on machine m such that EAT m 2 MEAT . 2p MEST 2 MEAT and stock of i in buffer 1 MEST 2 MEAT=p i; s 2 1 $ 2p MEST 2 MEAT=p i; s; then move i from machine m to machine q with amount of b MEST 2 MEAT=p i; sc where bwc is the largest integer that does not exceed w. Update EAT(q), MEAT, EAT(m), and LAP(q) as follows: EAT q à EAT q 1 c LAP q; i 1 p i; sp b MEST 2 MEAT=p i; sc; MEAT à EAT q; EAT m à EAT m 2 p i; sp b MEST 2 MEAT=p i; sc; LAP q à i: (2) If all planned products have been scheduled or no further scheduling for them is possible, go to Step 4. Otherwise, repeat Step 2 and 3. Step 4. [Scheduling unplanned products]
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T1
s-1
T2
s
47
s+1
D1a t A D1b
B
t
D2a
C D2b
(a) t1 + t2 ≤ 1
s-1 D1a t2 D1b
T1
T2
s
s+1
t1 A B C
D2a
D2b (b) t1 + t2 > 1 Fig. 5. Computing production requirements.
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If there still exist available machines during the scheduling horizon (a day), schedule the unplanned products with d i 0 using Step 2 and 3. Step 5. [Conversion to shift schedule] Decompose the resultant daily schedule into 3 shift schedules. In Step 1, the same products as the current loaded products are considered in initial assignment so that unnecessary setups can be avoided. Step 2 tries to reduce the current unmatchness for the products that exceed MAXU (maximum allowed unmatchness). If TFT's are over-loaded, the matched CF's are scheduled with the amount of min{available CF quantity, unful®lled amount of demand, unmatched amount}. The algorithm repeats Step 2 until all products have the allowable unmatchness, i.e. uUM iu , MAXU for all i. Step 3 schedules all planned products according to MST rule (minimization of setup time). Case 1 indicates the case with no idle times where the product can be scheduled on any available times of machines. On the contrary, in Case 2, the machine idle times can occur because there exist a product that is not available on the earliest available time of machine. In this case, the algorithm tries to schedule the available products on the inserted idle times. To do this, we check if there exists a prescheduled product that can be reallocated to the idle time period. If such a product is found, reallocate it with the available amounts of the product. Consequently, the reallocated product is assigned on two machines simultaneously and the idle times are eliminated. This case is illustrated in Fig. 3. The algorithm repeats Steps 2 and 3 until all planned products are scheduled or the scheduling horizon is over. And then, if any available machine still exists, Step 4 schedules the unplanned products by applying Steps 2 and 3 again. Step 5 converts the daily schedule to three shift schedules by simply decomposing one day period. Fig. 4 shows the summarized ¯owchart of algorithm E_SCHED. For P complexity analysis of algorithm, let M Ms and D Ni1 di (Note that the algorithm is applied to each production step, not whole three steps at once.) The computational complexity of Step 1 is given as O(MDL). Steps 2, 3, and 4 have the same complexities of O(MDKL) and Step 5 has a constant running time. Thus, algorithm E_SCHED has the polynomial time complexity of O(MDKL). 3.2. Scheduling heuristic for the later steps As in the early steps, the scheduling objectives are to minimize the mean ¯ow time and unful®lled production requirements of each step. Suppose the ¯ow time consists of only the processing time and setup time. Given ®xed WIP in the system, in order to achieve the ®rst objective, we have no choice but minimize the total setup time because the processing time is ®xed and cannot be reduced. Unlike the early steps, since we do not have to consider the matching problem of TFT and CFT glasses, the proposed heuristic L_SCHED for the later steps does not include Step 2 of E_SCHED and the other steps are all applicable, too. 3.2.1. Algorithm L-SCHED Step 1. [Initial assignment]: same as Step 1 of E_SCHED. Step 2. [Scheduling planned products]: same as Step 3 of E_SCHED. Step 3. [Scheduling unplanned products]: same as Step 4 of E_SCHED. Step 4. [Conversion to shift schedule]: same as Step 5 of E_SCHED.
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3.3. Computing production requirements One of the major input data for assembly scheduler is the production requirement for each step. Unfortunately, the daily production planner only provides input and out plans for a whole assembly process, not for each step. Thus, somehow we need to compute the production requirements for every step. In this section, we propose a method for computing the production requirements of each step. This problem is generalized as a problem to compute the production requirement for a speci®c step s which lies between any two steps s 2 1 and s 1 1 with known production requirements. As shown in Fig. 5, the production requirement (input) of step s 2 1 may be different from the production requirement (output) of step s 1 1: Then the problem is to determine the intermediate requirement at step s. The proposed method uses the linear interpolation technique to proportionally distribute the production demand according to the ratio of output to input. Let us de®ne the following notations: T1 lead time from step s 2 1 to step s T2 lead time from step s to step s 1 1 t1 T1 2 bT1 c where bwc is the largest integer that does not exceed w t2 T2 2 bT2 c D1a production requirement for step s at bT1 c 1 1 days before the current scheduling day D1b production requirement for step s at bT1 c days before the current scheduling day D2a production requirement for step s 1 1 at bT2 c days after the current scheduling day D2b production requirement for step s 1 1 at bT2 c 1 1 days after the current scheduling day D production requirement for step s at the current scheduling day Then we can derive the following equations for computing D: DA1B1C where (Case 1) t1 1 t2 # 1 in Fig. 5(a)
A
T2 D1a 1 T1 D2a t1 T1 1 T2
B
T2 D1b 1 T1 D2a 1 2 t1 2 t2 T1 1 T2
C
T2 D1b 1 T1 D2b t2 T1 1 T2
17
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Table 1 Processing time data Production step
Product type
Interarrival time between glasses (min)
Processing time per glass (min)
Lot size (glasses)
Processing time per lot (min)
Coating
1,2,3,4 5 1,2,3,4 5 All
0.8 1.1 1.0 1.3 1.5
17.5 21.9 38.8 50.6 40.0
20 20 20 20 20
32.7 42.8 57.8 75.3 68.5
Rubbing Attaching
(Case 2) t1 1 t2 . 1 in Fig. 5(b) A
T2 D1a 1 T1 D2a 1 2 t2 T1 1 T2
B
T2 D1a 1 T1 D2b t1 1 t2 2 1 T1 1 T2
C
T2 D1b 1 T1 D2b 1 2 t1 T1 1 T2
Table 2 Setup time data Production step
From-product
To-product
Setup time (min)
Coating
1, 2, 3 4 4 (TFT) 4 (CF) 1, 2, 3, 4 5
4 1, 2, 3 4 (CF) 4 (TFT) 5 1, 2, 3, 4
55 55 55 55 60 75
Rubbing
1, 2, 3 4 4 (TFT) 4 (CF) 1, 2, 3, 4 5
4 1, 2, 3 4 (CF) 4 (TFT) 5 1, 2, 3, 4
37 37 37 37 60 60
Attaching
1 2 3 1, 2, 3 4 1, 2, 3, 4 5
2, 3 1, 3 1, 2 4 1, 2, 3 5 1, 2, 3, 4
30 30 30 30 30 300 300
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Table 3 Matching table TFT type
CF type
TFT1
CF1 CF2 CF3 CF4 CF5 CF6 CF7 CF8
TFT2 TFT3 TFT4 TFT5
CF1 Group CF2 Group CF3 Group CF4 Group CF5 Group
4. Experimental evaluation 4.1. Data preparation In order to investigate the ef®ciency of the proposed scheduling system, we compared it to the current practice in a real TFT LCD assembly line of A-company which is known to be one of the largest TFT LCD manufacturers in the world (The company name is anonymous due to con®dentiality.) Using sets of actual data collected during entire three weeks (i.e. 21 days, 63 shifts), the scheduling algorithms were applied for the assembly process. Although the scheduling system was designed to generate shift Table 4 TFT fabrication out plan (unit: cell) Day
TFT1
TFT2
TFT3
TFT4
TFT5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
3028 3122 3122 3122 3177 3177 3177 96 2400 1440 0 2208 0 1440 0 0 0 1440 0 0 1152
4953 5473 6089 6535 7101 7433 7606 3840 0 0 2016 0 2976 0 2592 1440 1920 0 768 0 960
7140 7315 7344 7750 7924 7924 7924 0 0 0 0 960 672 0 0 1056 1056 1056 960 1152 960
21375 22431 23580 24408 24875 25899 26309 4896 7008 7968 8544 5760 6240 9504 5856 6336 6240 4608 5760 7296 7584
940 940 940 940 940 940 940 0 960 1440 0 960 480 0 0 0 0 0 0 0 0
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Table 5 Daily assembly production plan (unit: cell) Day
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Assembly In
Assembly Out
TFT1
TFT2
TFT3
TFT4
TFT5
TFT1
TFT2
TFT3
TFT4
TFT5
640 640 640 640 640 800 1600 2400 0 0 2400 0 2560 0 1440 0 1424 0 1408 0 1408
1600 1440 1600 1600 2400 1600 1600 640 2000 2000 0 2480 0 2480 0 1920 0 1440 0 1440 0
800 960 800 0 0 0 0 960 0 0 0 0 0 0 1120 1120 1120 1120 1120 1120 1120
4960 4960 4960 4960 4960 4960 6400 5600 7200 7200 7200 7200 7200 7200 6080 6080 6080 6080 6080 6080 6080
0 0 0 800 0 800 0 0 1200 1200 0 0 0 0 0 0 0 0 0 0 800
1200 1040 1080 584 584 584 584 720 1448 2176 0 0 2176 0 2320 0 1280 0 1280 0 1200
560 608 1448 1288 1448 1448 2176 1448 1448 576 1816 1816 0 2248 0 2248 0 1736 0 1304 0
160 0 872 1016 1016 0 0 0 0 872 0 0 0 0 0 0 1016 1016 1016 1016 1016
5600 5800 5600 5256 4496 4496 4496 4496 5800 5056 6520 6520 6520 6520 6520 6520 5512 5512 5512 5512 5512
0 0 0 0 0 0 624 0 624 0 0 880 880 0 0 0 0 0 0 0 0
schedules for every step of assembly process, we applied our system to only the ®rst three steps because of some practical reasons. In reality it has been generally recognized throughout the company that the performance of the early steps dominates those of entire assembly process. This is because the later steps are all non-bottleneck steps and have enough capacities compared to the early steps. Also, like a transfer line, there are no buffers between the later steps so that changing the product sequences determined in the early steps can hardly happen. We prepared four kinds of data as follows: time data such as processing and setup times, matching information for TFT and CF glasses, daily TFT fabrication out plan, and daily assembly production plan (input and output plan). Since all actual data are con®dential according to company policy, all ®gures shown in this paper were appropriately scaled from the original ®gures without affecting the performance results. 4.1.1. Time data There are two kinds of time data: processing and setup time. Table 1 shows the processing time data for three production steps. This line currently produces ®ve types of TFT LCD product, i 1±5: The processing times are different for products. In coating and rubbing step, the product type shown in Table 1 indicates the corresponding TFT and CF types. The interarrival time ta and raw processing time t0 are applicable to glass unit. Each glass consists of six cells. The lot size (l) is the number of glasses in a lot. Then the processing time (p) for a lot is computed as follows: p t0 1 l 2 1ta :
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Table 6 Comparisons of mean ¯ow time (MFT) Step
Statistic
Scheduler
Actual
Throughput (cell/day)
WIP (cell)
Throughput (cell/day)
WIP (cell)
Throughput (cell/day)
WIP (cell)
Coating
Average Stdev.
17,367 2002
3657 522
0.21 0.011
18,315 4622
4032 1017
0.22 0.00
Rubbing
Average Stdev.
17,399 2239
2941 859
0.17 0.040
20,288 3677
5481 993
0.27 0.00
Attaching
Average Stdev.
9705 2151
1833 489
0.2 0.074
10,065 2017
2315 464
0.23 0.00
Table 2 shows setup time data for product change-over. In most of cases, changes between TFT and CF with a same product type do not require setups. For product 4, however, the change-over time occurs even between TFT and CF in coating and rubbing step. 4.1.2. Matching information Matching of TFT to CF glass is one-to-many relationship, as shown in Table 3. For example, TFT1 can be matched to either CF1 or CF2 while TFT2 is only matched to CF3. The CF types matched to a same TFT are grouped together and assigned to a same group. Then we have ®ve CF groups corresponding to ®ve products, as shown in Table 3. 4.1.3. TFT fabrication out plan Without loss of practicality, we assume that there are enough CF stocks that are purchased from outside vendors. Actually, A-company considers CF to be a long-term purchasing item so that shortage of CF inventory hardly happens. On the contrary, TFT glasses are manufactured in the company own facilities and a lot of efforts are made to meet the production demands. Therefore, the performance of assembly process is strongly affected by that of TFT fabrication process. Table 4 shows daily TFT fabrication out plans during three weeks and we assume that there exists the exact amount of the matched CF stock in the bank. Table 7 Comparisons of average production progressiveness for ®ve product types Step
Statistic
Actual
Scheduler
Coating
Average Stdev.
0.633 0.105
0.952 0.073
Rubbing
Average Stdev.
0.825 0.095
0.951 0.060
Attaching
Average Stdev.
0.824 0.111
0.942 0.133
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Table 8 Comparisons of variability of over-production (unit: cells) Step Coating
Product TFT1 TFT2 TFT3 TFT4 TFT5 CF1 Group CF2 Group CF3 Group CF4 Group CF5 Group
Actual 326.79 557.70 249.33 676.63 206.19 268.29 303.29 157.87 760.80 161.04
Scheduler 114.54 132.10 7.92 39.06 38.47 74.67 112.39 5.90 19.58 38.38
Rubbing
TFT1 TFT2 TFT3 TFT4 TFT5 CF1 Group CF2 Group CF3 Group CF4 Group
196.81 363.11 239.11 335.95 159.49 206.26 370.50 232.10 273.19
100.48 125.93 20.18 75.30 18.73 42.12 164.13 15.66 31.02
Attaching
CF5 Group Product 1 Product 2 Product 3 Product 4 Product 5
147.82 202.50 310.09 206.75 383.38 142.81 297.51
31.93 104.34 158.50 98.12 266.42 18.66 74.18
Average
4.1.4. Daily assembly production plan Table 5 shows a daily assembly In/Out plan that is generated by the daily production planner of Acompany. This plan is a rough-cut capacitated production plan that is constructed according to the sales plan, monthly production plan, and overall line capacities. 4.2. Experimental results In this section, we compared the performances of our proposed scheduling system to those of current practice employed in A-company. The scheduling logic used in the line of A-company is somewhat adhoc based rules. The rules are frequently changed by line workers according to manufacturing environments such as order cancellation, demand changes, line disciplines, and so on. However, the most frequently rules are First-In-First-Out (FIFO) and minimum setup time (MST) rules. Here, MST rule is executed in the manner of continuous production, which does not allow setup as possible and keep producing until inventory shortage occurs. Also, the production managers and line workers in A-company are much more concerned about meeting the monthly production plan rather than daily plans because of ease of control, and that they are used to this practice.
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For three-week experiments, we set the maximum allowed unmatchness (MAXU) to be 10 lots which is empirically determined. Table 6 shows the comparison results of the proposed scheduling system versus the current practice in terms of mean ¯ow time of cells. Since throughput rate (cells per day) and average WIP are the direct results of scheduling system, mean ¯ow time can be computed indirectly using Litte's law (Little, 1961) as follows: Flow time WIP=Throughput rate:
18
However, A-company has not maintained daily WIP data ever. Instead, the company maintains average ¯ow time data for each step, which are updated every two months, and we could get daily throughput. In applying our scheduler, we also considered real manufacturing environments such as the probable machine breakdowns, preventive maintenance time, and other losses. The experimental results in Table 6 show that the mean ¯ow time by the proposed scheduler is apparently smaller than the actual results at all three steps during most of 21 days. The total mean ¯ow time for the early steps by the scheduler is 0.58 days and the actual one is 0.72 days. Thus, we have achieved 19.4% reduction of mean ¯ow time by the proposed scheduler. As another performance criterion, production progressiveness (or demand ful®llment) is computed against given daily production plans. Since the daily production planner of A-company gives only the input and output plan of the whole assembly line for each product, we computed the out plans for each step using Eq. (17). Production progressiveness (PR) for product i is computed as follows: 8 > < Pi if di 2 Pi . 0 under production demand ; 19 PRi di > : 1 otherwise over production demand where di is the daily production demand for product i and Pi is the daily actual production for product i. As shown in Table 7, the scheduling system gives much better production progressiveness of average 94.8% during 21 days than the current practice of average 76.1% at all three steps. Thus, the average 18.7% increase of production progressiveness has been achieved. Note that the biggest improvement by the scheduler has been made in coating step. This indicates that in coating step there is little concern about meeting production demands and FIFO rules are mostly used according to TFT fabrication out. On the other hands, in rubbing and attaching steps, the operators begin to be more conscious on production demand and try to control the production using WIP in stocker. This means that the rubbing and attaching steps have the higher WIP level than the coating step as shown in Table 6. From production management viewpoint, production smoothing is one of the most important issues in the company because it makes it easier to respond to demand ¯uctuations quickly. Production smoothness can be measured by standard deviation of over-production as shown in Table 8. For all products and steps, the proposed scheduler provides the smaller variability of over-production than the actual practice. On the average, 75.1% reduction of variability has been achieved by the proposed scheduler. For TFT1 product at coating step, Fig. 6 shows the comparison results in variability between the proposed scheduler and actual practice. The proposed scheduling system has been developed using Delphi 5 and the pilot version is currently running on Pentium II-400 MHz computer in a TFT LCD assembly line of A-company. The signi®cant improvements in mean ¯ow time and production progressiveness are expected after full implementation of the proposed scheduler as shown in the three-weeks experimental results. The computing time for
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Fig. 6. Comparisons of over-production for TFT1 at coating step.
scheduling the early steps is less than a second so that rescheduling can be possible at any time without computational burden, which is one of the strong requirements of the production planning group of Acompany. Fig. 7 shows a sample screen for parameter inputs and illustrates Gantt charts according to the schedules generated by the proposed system. 5. Conclusions In this paper, we present an assembly scheduler for each production step of TFT LCD assembly process. The scheduler includes two heuristic procedures, which are employed separately for the early and later production steps. We assume that the daily production plans are provided for TFT fabrication and assembly process. The scheduling problem under consideration is a parallel-machine scheduling problem with sequence dependent setups, multi-stage production, multi-objectives, and the matching constraints. The scheduling objectives are to minimize the mean ¯ow time and maximize the ful®llment of production demands. Since the problem is a NP-hard, we propose practical heuristic algorithms. In TFT LCD assembly process, the early steps require processing both TFT and CF glasses independently with identical setup while the later subsequent steps deal with the attached unit of two matched glasses. Thus the major concern for scheduling the early steps is to synchronize TFT and CF glasses at the subsequent steps so as to minimize the number of unmatched TFT or CF glasses, which in turn reduce the unmatched WIP in stocker and consequently reduce the mean ¯ow time. The other way of
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(a) A Sample Screen for Inputs
(b) Gantt Charts
Fig. 7. TFT LCD assembly scheduler: sample screens. (a) A sample screen for inputs. (b) Gantt charts.
minimizing mean ¯ow time is to minimize the total setup time by optimally sequencing products. These concepts were included in the scheduling algorithms. Also, we propose a simple method for computing the production requirements of each step because the daily production planner provides only input and output plans for the whole assembly process, not for every step of assembly process. The concept behind the method is to proportionally distribute the production demands according to the ratio of output to input production demands. The proposed scheduler was implemented in a real TFT LCD assembly line.
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Implementation results show that the scheduler generates quite better schedules than the current practice employed in the line in terms of the scheduling objectives. This is the ®rst research on the scheduling problem for TFT LCD assembly process. The scheduling heuristics were designed to focus on the practical requirements of production planning people in the TFT LCD assembly line. The requirements were as follows: (1) to provide a capacitated shift schedule for every step of assembly process, (2) to reduce WIP in the line, (3) to maximize the ful®llment of production demands, and (4) quick run time for rescheduling. The second requirement is equivalent to minimizing the mean ¯ow time, which is the primary objective of our scheduler. We believe that all these requirements have been considerably satis®ed in the proposed scheduler and the company people consider the full implementation of the system. Possible directions for future research may include the improvement of algorithms using ef®cient search techniques and optimal determination of MAXU which is determined empirically in the current scheduler. References Allahverdi, A., Gupta, J. N. D., & Aldowaisan, T. (1999). A review of scheduling research involving setup considerations. Omega, International Journal of Management Science, 27, 219±239. Deane, R. H., & White, E. R. (1975). Balancing workloads and minimizing setup costs in the parallel processing shop. Operation Research Quarterly, 26, 45±53. Elmaghraby, S. E., Guinet, A., & Schellenberger, K. W. (1993). Sequencing on parallel processors: an alternate approach. OR Technical Report, No. 273. Raleigh, NC: North Carolina State University. Guinet, A. (1990). Textile production systems: a succession of non-identical parallel processor shops. Journal of Operations Research Society, 42, 655±671. Guinet, A., & Dussauchoy, A. (1993). Scheduling sequence dependent jobs on identical parallel machines to minimize completion time criteria. International Journal of Production Research, 31, 1579±1594. Lee, Y. H., & Pinedo, M. (1997). Scheduling jobs on parallel machines with sequence dependent setup times. European Journal of Operation Research, 100, 464±474. Little, J. D. C. (1961). A proof for the queuing formula: L lW. Operations Research, 9, 383±387. Marsh, J. D., & Montgomery, D. C. (1973). Optimal procedures for scheduling jobs with sequence-dependent changeover times on parallel processors. AIIE Technical Papers., 279±286. Ovacik, I. M., & Uzsoy, R. (1993). Worst-case error bounds for parallel machine scheduling problems with bounded sequencedependent setup times. Operational Research Letters, 14, 251±256. Ovacik, I. M., & Uzsoy, R. (1995). Rolling horizon procedures for dynamic parallel machine scheduling with sequencedependent setup times. International Journal of Production Research, 33, 3173±3192. Pinedo, M. (1995). Scheduling: Theory, Algorithms, and Systems, Englewood Cliffs, NJ: Prentice Hall. Sumichrast, R., & Baker, J. R. (1987). Scheduling parallel processors: an integer linear programming based heuristic for minimizing setup time. International Journal of Production Research, 25, 761±771.