www.palgrave-journals.com/jors/
Fast and effective algorithms for the liquid crystal
display module (LCM) scheduling problem with
sequence-dependent setup time
SH Chung∗, WL Pearn and YT Tai
National Chiao Tung University, Hsinchu, Taiwan, ROC
The liquid crystal display module scheduling problem (LCMSP) is a variation of the classical parallel machines scheduling problem, which has many real-world applications, particular, in the thin film transistor liquid crystal display (TFT-LCD) manufacturing industry. In this paper, we present a case study on the LCMSP, which is taken from a final liquid crystal display module (LCM) shop floor in a TFT-LCD industry. For the case we investigated, the jobs are clustered by their product types and the machine setup time is sequentially dependent on the product types of the jobs processed on the machine. In LCMSP, the objective is to maximize the total profit subject to fulfilling contracted quantities without violating the due date and machine capacity restrictions. The LCMSP can be modelled as a multi-level optimization problem. The sub-problem of LCMSP can be transformed into the vehicle routing problem with time window (VRPTW). One can therefore solve the LCMSP efficiently using existing VRPTW algorithms. We present two new algorithms based on the savings algorithms with some modifications to accommodate the LCMSP. Based on the characteristics of the LCM process, a set of test problems is generated covering most of the real-world applications for test purposes. Computational results and performance comparisons show that the proposed algorithms solved the LCMSP efficiently and near-optimally.
Journal of the Operational Research Society (2009) 60, 921 – 933. doi:10.1057/palgrave.jors.2602604 Published online 4 June 2008
Keywords: parallel machines scheduling; profit; sequence dependence setup time
Introduction
The liquid crystal display module scheduling problem (LCMSP) is a variation of the classical parallel-machine scheduling problem considered by So (1990), Shin and Leon (2004), and Jeong et al (2001), which has many real-world applications, particularly, in the thin film transistor liquid crystal displays (TFT-LCD) manufacturing industry. The major three stages of processes for the TFT-LCD manufac-turing include TFT array fabrication, LCD assembly, and liquid crystal display module (LCM). The LCM is the final stage of the whole process. It assembles customer-specified components into the cells, which are received from the LCD assembly stage. In the LCM stage, the varied contracted quan-tities, due date, product profits, and long setup time of critical resources (printed circuit board bonding) result in the deter-mination of jobs and setups sequence becomes more difficult. Therefore, developing fast and efficient scheduling algorithms to maximize the profit without violating customer due date and enhance the utilization of the critical resources is essential.
∗Correspondence: SH Chung, Department of Industrial Engineering and Management, National Chiao Tung University, Hsinchu, Taiwan, ROC. E-mail: shchung@mail.nctu.edu.tw
In this paper, we consider a case study on the LCMSP in the LCM factory with sequence dependence setup time and multiple profits. The case we investigated is taken from a module assembly process of a TFT-LCD factory in Hsinchu science-based industrial park, Taiwan. For the case we inves-tigated, the jobs are clustered by their product types, which must be processed on any parallel machines and be completed before the due date. Setup times between two consecutive jobs of different product types on the same machine are sequentially dependent. The job processing time may vary, depending on the product type of the jobs processed on. Furthermore, the hybrid market consists of contract and spot markets in TFT-LCD factories. The contract jobs, associated with contracted prices and due dates, have been placed. The remaining capacity is to be sold into the spot market. The further spot prices are uncertain and high price volatility due to today’s fierce competitive environment. Since the LCMSP involved different product profits, with constraints on sequence dependence setup time, product-type-dependent processing time, due date and machine capacity, it is more difficult to solve than the classical parallel machine scheduling problem.
So (1990) considered a simple version of the LCMSP with sequence-independent setup time. He presented three
COG process
FPC attachment TAB process
PCB bonding AssemblyBacklight Aging test Inspection
Front end Back end
Figure 1 The six steps in the LCM process.
Figure 2 PCB bonding illustration.
heuristics to solve the problem approximately. However, the setup time in his model was only characterized by two magni-tudes, minor and major. He only tackled a problem involving minor setup times between jobs of the same family and major setup times between jobs of different groups. Further, he did not consider the due date restriction, which is essential in practical factories. Tovia et al (2004) considered a simple version of the LCMSP, which is a parallel machine scheduling problem with the objective of maximizing throughput. Tovia
et al (2004) also presented a mathematical programming
model and a rule-based heuristic approach. Unfortunately, their model only includes processing time without consid-ering setup time and due date restrictions, which may not reflect the real situations accurately.
Few works have considered the scheduling problem in TFT-LCD plants. Jeong et al (2001) addressed the LCD assembly process, presented mathematical models, and proposed two heuristic algorithms considering mean flow time and demands fulfillment. Unfortunately, their model does not consider the due date restriction and setup time that are essential and critical in real-world situations. Shin and Leon (2004) discussed an analogous version of the LCMSP, which considers family setup time and due date. They proposed two heuristic approaches based on the MULTIFIT method and tabu search to solve the LCMSP approximately. We note that Shin and Leon (2004) have simplified the model by considering family setup time ignoring the fact that the setup times are sequentially dependent on hundreds of product types of job processed. However, none of these investigations considered sequence-dependent setup time and job profits under contract and spot market simultaneously.
The LCMSP we investigated can be modelled as a multi-level optimization problem. At the first multi-level, we schedule
the contract jobs to minimize the total setup time without violating machine capacity and customer due date restric-tions, which can be solved using algorithms for vehicle routing problem with time windows (VRPTW) (see Pearn et
al, 2002a,b, 2004a,b). At the second level, we apply a greedy
concept to choose a subset of spot jobs then insert into the schedules constructed in the first level. In this paper, we use basic technologies for VRPTW algorithms including parallel, generalized savings algorithms and provide two modifica-tions to solve the LCMSP efficiently. To further analyse the impact of the problem characteristics on the performance of those savings algorithms, we provide a set of test problems, considering the workload level of contract jobs, tightness of due dates, setup time variation, and variation of profit ratio. Exact solutions are used here as convenient reference points for evaluating the accuracy and effectiveness of our heuristic algorithms. The computational experiments and comparisons demonstrate the performance of the three phases of the modi-fied parallel savings algorithm (PSA) outperform the other algorithms.
LCM process and problem description
LCM process
The TFT-LCD applications include monitors, notebook PCs, mobile phones, portable DVDs, LCD TVs, and many others. Such applications are hundreds of product types of job processed with sizes ranging from 1.6 to 46 inches. The manufacturing process of the LCM assembly gener-ally consists of front end and back end two main segments. It contains the following six processes: (1) COG (chip on glass)/TAB (tape automated bonding) process, (2) flexible printed circuit board (FPC) attachment, (3) printed circuit
Figure 3 COG structure.
Figure 4 TAB structure.
board (PCB) bonding, (4) back light assembly, (5) aging test, (6) inspection, as presented in Figure 1. For the case investigated, the critical resource is PCB bonding, which has the longest setup time and using anisotropic conductive film (ACF) to connect the PCB and FPC, as illustrated in Figure 2. The processes of PCB bonding are determined by the types of mount technology of LCD drive IC and controller IC, namely, the COG (chip on glass) and TAB (tape automated bonding). The COG is a technology that mounts the LCD driver to the contact edge of the LCD glass, which is depicted in Figure 3. The TAB is the LCD driver or controller elec-tronics are encapsulated in a thin film, like package, with metal leads extension from the IC chips, which is depicted as Figure 4. The process is also called OLB (Outer Lead Bonding). In general, large size LCD applications (ranging from 15 to 46 inches) adopt the TAB structure and small size LCD applications (ranging from 1.6 to 6 inches) adopt the COG structure.
Setup time between different technologies in PCB bonding is complicated and time consuming. The longest setup time may consume six hours. The setup activities of PCB bonding include the following: cool down temperature, replacement of the appropriate mold, and the rising to a suitable temper-ature and voltage. Furthermore, setup times between two consecutive jobs of different product types on the same PCB bonding machine are sequentially dependent. In a similar product family, the setup activity may adjust the temperature and voltage. Different LCD panel sizes must replace the mold.
LCMSP
The scheduling problem in this paper can be stated as follows. Let machine group M = {mk|k = 1, 2, . . . , K }, containing K machines, and product types R= {Ri|i = 1, 2, . . . , I },
containing I clusters of jobs. Because the hybrid contract and spot sales environment is considered, one product type could be divided into two subsets RP
i and R q
i. The contract job set
can be shown as RP i = {r P i j| j = 1, 2, . . . , J contract i }, and the
spot sales job set can be shown as Rqi = {ri jq| j = Jcontract
i + 1,
Jcontract
i + 2, . . . , Ji}. Term Jicontract and Ji is the number
Table 1 The job information for the seven-job example Job ID Product Processing Profit Due Contract/
type time date spot
rA1 RA 21 50 80 C* rA2 RA 21 40 100 S† rA3 RA 21 40 100 S rB1 RB 25 60 93 C rB2 RB 25 50 100 S rC1 RC 28 63 100 C rC2 RC 28 63 60 C
∗Term C indicates the contract job.
†Term S indicates the spot job.
Table 2 Setup times required for switching one product type to another for RA, RB, and RC
To From U RA RB RC U – 15 15 15 RA 0 0 10 7 RB 0 8 0 5 RC 0 3 16 0
of contract jobs and total jobs of product type Ri,
respec-tively. Each job in their associated product type is a candi-date to be processed without preemption on a set of parallel machines K .
Each job ri jcarries with it processing time denoted by pi j,
and profit denoted by profiti j, where i = {1, 2, . . . , I } and
j = {1, 2, . . . , Ji}. For each job ri j, bi j represent the ready
time of the job to be processed on a machine and ei j represent
the latest starting time to process job ri j, which relates to
the due dates di j and can be computed as ei j = di j − pi j. A
setup time is incurred in the different product types. When job ri j immediately succeeds job rij on machine mk, a setup
time si i happens. The setup time is sequentially dependent
on the product types of the jobs processed on the machine. The objective is to maximize the total profit without violating contracted due date and machine capacity, W .
An illustrative example
Consider the following example with two machines and seven jobs clustered into three product types. Table 1 displays the product type, processing time, job profit, due date, and contracted status for each job. Furthermore, the setup times are incurred for switching one product type to another for the three product types RA, RB, and RC. Table 2 shows the required setup times and the term U denotes that the machine is in idle status. The capacity is set to 95 for each machine in the illustrative example.
The objective in the illustrative example is to find a schedule for the subset of jobs, which satisfies the due date restrictions without violating the constraints of machine capacity, while maximizing the total profits.
An integer programming formulation
We formulate the MILP model for the LCMSP. The appli-cability of the MILP model associated with a sequence-dependent setup time of parallel machine scheduling has been demonstrated by Pearn et al (2002b). We modify the objec-tive and some constraints of MILP model to accommodate the LCMSP.
Let xi j kbe the variable indicating whether job ri j is
sched-uled on machine mk, with xi j k = 1 if job ri j is scheduled
to be processed on machine mk, and xi j k= 0 otherwise. Let yi j ijk be the precedence variable, where yi j ijk should be
set to 1 if job rij is scheduled following job ri j (not
neces-sarily directly), and where yi j ijk= 0 otherwise. Let zi j ijk
be the direct-precedence variable, with zi j ijk= 1 if job rij
is scheduled immediately following job ri j on machine mk,
and zi j ijk=0 otherwise. Further, the starting processing time ti j k should not be less than the ready time and not be greater
than the latest starting time ei j. The exact formulation for the
LCMSP is as follows. Maximize Z= K k=1 I i=1 Ji j=1 xi j kprofiti j (1) subject to K k=1 xi j k= 1 for i = 1, 2, . . . , I, j= 1, 2, . . . , Jicontract (2) K k=1 xi j k1 for i = 1, 2, . . . , I, j = Jicontract+ 1, Jicontract+ 2, . . . , Ji (3) Capacity constraints: I i=0 Ji j=1 xi j kpi j + I i=0 Ji j=1 ⎛ ⎝I i=0 Ji j=1 zi j ijksi i ⎞ ⎠ W for all k (4)
Due date constraints:
ti j kbi jxi j k for all i, j, k (5) ti j kei jxi j k for all i, j, k (6) Precedence constraints:
ti j k+ pi j + si i− tijk+ Q(yi j ijk− 1)0 for all i, j, k
(7) ti j k+ pi j + si i− tijk+ Q(yi j ijk+ zi j ijk− 2)0 for all i, j, k (8) (yi j ijk+ yiji j k) − Q(xi j k+ xijk− 2)1 for all i, j, k (9) (yi j ijk+ yiji j k) + Q(xi j k+ xijk− 2)1 for all i, j, k (10) (yi j ijk+ yiji j k) − Q(xi j k+ xijk)0 for all i, j, k (11) (yi j ijk+ yiji j k) − Q(xijk+ xi j k+ 1)0 for all i, j, k (12) (yi j ijk+ yiji j k) − Q(xi j k+ xijk+ 1)0 for all i, j, k (13) yi j ijkzi j ijk for all i, j, k (14) I i=0 Ji j=1 xi j k− ri j=ri j zi j ijk= 1 for all k (15) yi j i∗j∗k+ zi j i∗j∗k− Q(yi j i∗j∗k+ zi j i∗j∗k− 2) − Q(yi j ijk − zi j ijk− 1)2 for all i, j, k (16) Binary variables: xi j k∈ {0, 1} for all i, j, k (17) yi j ijk∈ {0, 1} for all i, j, k (18) zi j ijk∈ {0, 1} for all i, j, k (19)
The objective function (1) states that the total profit is to be maximized over all machines. Constraint (2) ensures that each contract job is scheduled on one machine exactly. Constraint (3) ensures that each spot job is scheduled on at most one machine. Constraint (4) is the capacity constraint, which forces the sum of processing time and setup time for each machine within available capacity. Constraints (5) and (6) state that the starting time ti j kfor each job ri jscheduled on
machine mkshould not be less than the earlier starting time bi j
and not be greater than the latest starting time ei j. Constraints
(7) and (8) are the starting time constraints. The time of the following job starts after the proceeding job and related setup is complete. As usual, Q is a ‘sufficiently large’ positive number, so that constraints (9)–(13) are satisfied for yi j ijk=
0 or 1. Constraints (9)–(13) are the precedence constraints and constraints (14)–(16) are the direct constraints. These constraints state their sequence relation. In constraint (16), variable zi j i∗j∗kstates that there is a job ri∗j∗existing after job ri j when yi j ijk=1 and zi j ijk=0. Constraints (17)–(19)
indi-cate that xi j k, yi j ijk, and zi j ijk are binary integer variables.
The total number of variables is 2NI2K , and the total number
of equations is NI3K− (5/2)NI2K− (3/2)NIK+ NI+ 2K ,
where NI =iI=1Ji.
Algorithms for the LCMSP
In this section, we present two heuristic procedures to solve the LCMSP efficiently. Some basic technologies have been used for developing algorithms for machine scheduling. We modify the conventional savings algorithms to minimize the total setup time for the contract jobs and apply the greedy
concept for the spot jobs to enhance the total profits. We first review two savings algorithms, the PSA proposed by Golden (1977) and the (GSA) provided by Christofides et al (1979). We then present two modified algorithms, which are referred to as the three-phase modified parallel savings algo-rithm (MPSA TP) and the three-phase modified generalized savings algorithm (MGSA TP).
Parallel savings algorithm
Golden (1977) proposed the PSA to solve the TSP approxi-mately. The PSA, initially calculates the savings of all pairs of jobs and sorts those savings in descending order. The PSA creates the multiple of K machines simultaneously at the initial stage, where K is the number of machines. Note that a selected pair of jobs is feasible if it does not violate the machine capacity constraints. The PSA then searches down-ward from the savings list, for a job which could be merged into one of current K schedules (at the first endpoint or last endpoint) with the most savings. The algorithm terminates when no more jobs can be inserted. Algorithm details are presented as follows.
Step 1: (Initialization) Calculate the savings S Ai i, defined
as the following, for all pairs of two jobs associated with product type Riand Ri, where U denotes the
idle status.
S Ai i= sU i+ siU− si i (20)
Step 2: Sort the savings list in descending order of
magnitude.
Step 3: (Schedule initial construction) Choose the first K pairs of jobs on the list satisfying the machine capacity constraints, to start K new schedules simultaneously.
Step 4: Starting from the top of the savings list. Find the first feasible pair on the list to add to one of the two ends of a currently constructed schedule with the most savings.
Step 5: The chosen jobs form a feasible machine schedule. Repeat Step 4 until all schedules are full and cannot be expanded.
Generalized savings algorithm
In contrast to the PSA, which constructs a multiple of K machine schedules simultaneously at the initial stage, the GSA proposed by Christofides et al (1979) creates one schedule at a time and considers not only the end points but also posi-tions between two consecutive jobs when inserting a new job into the current partial schedule. Besides, the insertion costs are calculated for every unscheduled job at every possible position. The chosen job, which maximizes savings while minimizing insertion costs, is used to avoid the algorithm to create a new schedule on another machine with a high setup time.
New algorithms
To effectively apply these technologies, we modify them to develop fast and effective algorithms. The two new algorithms essentially consist of three phases. Phase I applies network algorithms to schedule sub-problem of contract jobs with minimum total setup time. Phase II inserts spot jobs near the same constructed product type so as no extra setup time is needed. Phase III sorts the remaining spot jobs with profit ratio (ie job profit is divided by job processing time) and chooses a subset of high profit ratio jobs which are then inserted into the constructed schedules sequentially until all machine capaci-ties are full. It is noted, however, that the contract jobs in the current constructed schedule should be pushed backward when a spot job is inserted into this schedule in Phase II or Phase III. Thus, the contract jobs, following the inserted spot job, should be re-checked according to their due dates to meet customer deadlines. The new solution procedures can be described as follows.
Three-phase modified parallel savings algorithm (MPSA TP)
The MPSA TP calculates savings by adding two terms, profit ratio and time window restrictions. The profit ratio term is added in the savings calculation in order to choose the pairs of jobs with higher profit ratios as the first job pairs for higher savings values than the pairs of jobs with lower ones. By doing this, the jobs with higher profit ratios are forced to be processed earlier than other jobs with lower ones. Further-more, the jobs with the same profit ratios are forced to be processed closer to each other than the other jobs. The other added term, time window restrictions, takes job ri j whose
latest starting time(ei j) is earlier than a later job (eij) and
tends to place it before another job (rij). Time window
restrictions have been used by Pearn et al (2004a); however, they only considered the value 011 and did not system-atically examine the parameter with the value11. In this paper, three parameters,1,1, and1, and the three ranges 011, 011, and 012, are added to the savings function to weight the ‘savings term’, ‘profit ratio term’, and ‘time window restrictions term’, respectively. Parameter 1 represents weight of savings, which results from the setup time between two jobs ri j and rijin a job pair. It can help the
savings term to avoid a long setup time being incurred. Para-meter1 is used to weight the summation of the profit ratios calculated involving two jobs ri j and rijin a pair in order to achieve higher profits. Finally, parameter1can assist the time windows restrictions term to prevent the jobs being processed after their due date in order to enhance customer satisfaction. Term W represents the machine predetermined capacity.
Phase I (Modified savings algorithm)
Step 1: Calculate the savings, PSAr i jr ij, for all pair of jobs ri j and rij, where U denotes the idle status. si i, is
the required setup time for switching product type
job profit, job processing time, and latest starting time, respec-tively, to process job ri j.
PSAri jri j = ⎧ ⎨ ⎩ 0 1(sU i+siU−si i)+1 profit i j pi j + profitij pij + 10W 1 ei j − (2 −1) eij if PSAri jri j < 0 or i = i , j = j otherwise
Step 2: Sort the savings and create a list in descending order of magnitude.
Step 3: Choose the first feasible K pairs of jobs on
the list satisfying the machine capacity and due date constraints and remove these pairs from the savings list. Then, start the K new schedules simultaneously.
Step 4: Starting from the top of the remaining savings
list, find the first feasible pair on the list and check which job of the pair is able to be added to one of the two ends of a currently constructed schedule.
Step 5: The chosen jobs form a feasible machine schedule. Repeat Step 4 of Phase I until all the contract jobs are scheduled.
Phase II (spot jobs assigned without extra setup) Step 1: Calculate the profit ratio for the spot jobs.
Step 2: Sort the profit ratio and create a profit ratio list in descending order of magnitude.
Step 3: Choose the job with highest profit ratio as the job to be inserted. Find the machine, which has scheduled the jobs with the same product type as the job to be inserted. Whether the job is inserted or not, it should be removed from the profit ratio list.
Step 4: If the machine for insertion is found, then insert the chosen job by the side of the job with same product type without violating the machine capacity and due date constraints.
Step 5: Repeat Steps 3 and 4 of Phase II until no more jobs can be found in the profit ratio list.
Table 3 The value of each pair of contract jobs
Pair ID Job pair Values of savings Pair ID Job pair Values of savings
1 (rA1, rB1) 14.75 7 (rC1, rA1) 19.14 2 (rA1, rC1) 18.29 8 (rC1, rB1) 10.89 3 (rA1, rC2) 13.57 9 (rC1, rC2) 16.21 4 (rB1, rA1) 22.76 10 (rC2, rA1) 33.28 5 (rB1, rC1) 24.54 11 (rC2, rB1) 25.02 6 (rB1, rC2) 19.83 12 (rC2, rC1) 35.06
Phase III (Remaining spot jobs assigned with extra setup) Step 1: Calculate the profit ratio for the remaining spot
jobs.
Step 2: Sort the profit ratio in descending order of magni-tude.
Step 3: Choose the top job on the list as the job to be
inserted.
Step 4: Schedule the job by applying the cheapest inser-tion algorithm (Rosenkrantz et al, 1977) sequen-tially to construct a feasible schedule without violating machine capacity and due date restric-tions. Whether the job is inserted or not, it should be removed from the profit ratio list.
Step 5: The algorithm will terminate with no job in the
profit ratio list. Otherwise, return to Steps 3 and 4 of Phase III.
To illustrate how the MPSA TP algorithm may be applied, we consider the LCMSP example with two machines and three product types described in the previous section enti-tled ‘LCM process and problem description’. In Phase I, the savings value for each pair between two contract jobs is calculated by Step 1 and shown in Table 3 while the values of1,1, and1, are set to 0.5, 0.05, and 1.5, respec-tively. The savings are sorted in descending order of magni-tude using Step 2, and the first two feasible pairs (12th pair and 4th pair) are chosen and scheduled in the two machines initially in Step 3. After processing using the algorithm and by repeating Steps 4 and 5 until no more contract jobs can be found, two schedules are constructed: job rC2precedes job rC1 directly on Machine 1 and job rB1precedes job rA1directly on Machine 2.
In Phase II, the three spot jobs (rA2, rA3, and rB2) are inserted into the two constructed schedules without violating machine capacity and due date constraints. In Step 1 of this phase, the profit ratios of the three jobs are computed: 1.9, 1.9, and 2 for job rA2, job rA3, and job rB2, respectively. The profit ratios are sorted using Step 2. It was found that rB2 is the highest ratio and is, therefore, chosen and removed from the profit ratio list in Step 3. Machine 2 has the same product type as job rB2; therefore, job rB2can be inserted into Machine 2 in Step 4. However, the insertion of job rB2 into the two possible positions (preceding or following job rB1
0 setup time process time 15 95 m1 m2 rA1 rA2 rA3 70 40 rB1 rC2 rC1
Figure 5 Gantt chart for the example problem.
immediately) will cause the job rA1 to violate its due date. Hence, job rB2 is not inserted into this machine. Then, Step 3 of the algorithm is repeated and the other spot job rA2 is considered. In Step 4, Machine 2 is the inserted machine. Job
rA2 is scheduled following job rA1 without any extra setup time and without violating any restrictions. Furthermore, by repeating Step 3, the third spot job rA3is chosen but is not suit-able for insertion due to its violation of due date and machine capacity.
Finally, in Phase III, the remaining spot jobs (rB2and rA3) are considered and scheduled although extra setup times are required. The profit ratios of the two spot jobs are computed and sorted in descending order in Step 1 and Step 2 of Phase III, respectively. In Step 3, job rB2 is chosen and removed from the list, but it cannot be inserted into the two machines. However, after processing is repeated in Step 3, the other job,
rA3, is inserted into Machine 1 using the cheapest insertion algorithm in Step 4. Owing to the absence of more jobs on the profit ratio list and fullness of the two machines, the final solution from the PSA TP algorithm is therefore 316 and is depicted in Figure 5.
Three-phase modified generalized savings algorithm
(MGSA TP)
The MGSA TP constructs the schedules sequentially in contrast to the MPSA TP which creates a multiple of K machine schedules simultaneously. In addition, the differ-ence in Phase I with MPSA TP is that MPSA TP considers only two end points when merging a new job into the current partial schedule, while MGSA TP considers not only the end points but also the positions between two consecutive jobs when merging a new job into the current partial schedule. The savings function is expressed as follows:
GSAri jri j = ⎧ ⎪ ⎨ ⎪ ⎩ 0 if GSAri jri j < 0 or i = i , j = j 2(sU i+siU = si i)+2 profit i j pi j + profitij pij + 10W 2 ei j − (2 −2) eij otherwise
Based on the new savings function, the initial partial schedule can be selected from the top of the savings
list. In addition, schedules can be expanded based on parameters 1, 2, and formulas (21)–(24) (designed by Christofides et al, 1979). Let PS be the current schedule,
PS= (uU, . . . , ug−1, ug, . . . , uG, uU), where uU and uU are
virtual jobs (machine idle status). The insertion costs are computed using formulas (21)–(22) for every unscheduled job ri j at every possible position of PS. Let (ug−1, ri j, ug)
be the additional setup cost when job ri j is inserted between
position(g − 1) and g in schedule PS. Let∗ (ug− 1, ri j, ug)
be the minimal insertion cost value.
(ug−1, ri j, ug) = sug−1i+ si ug−1× sug−1ug, 12
(21)
∗(u
g−1, ri j, ug) = min
g=1,...,G[(ug−1, ri j, ug)] (22) Job ri j is chosen, which maximizes the savings∗(ug−1, ri j, ug) while minimizing the insertion cost∗(ug−1, ri j, ug),
and which avoids the algorithm to create a new schedule on another machine with a high setup time1×sU i. Furthermore,
in addition to taking into account machine capacity, the due date constraints of all jobs must also be examined for violation before a job is inserted. The procedure is repeated until all schedules are full and cannot be expanded. Phase II and Phase III are same as the corresponding phases of the MPSA TP algorithm.
(ug−1, ri j, ug) =2× sU i−∗(ug−1, ri j, ug), 122 (23)
∗(u
g−1, ri j, ug) = max[(ug−1, ri j, ug)] (24)
The same illustrative example described in the previous section entitled ‘LCM process and problem description’ with two machines and three product types is also used to illus-trate Phase I of MGSA TP. First, the savings are computed when the values of1,1, and 1, are set to 0.5, 0.05, and 1.5, respectively. The savings values for MGSA TP are the same as those obtained by MPSA TP, as shown in Table 3. In contrast to the MPSA TP algorithm, one job pair (rC2, rC1) with the largest savings is chosen and scheduled on Machine 1 initially. Second, the other two contract jobs (rA1 and
rB1) should be considered with their insertion and savings cost using formulas (21)–(24) in order to decide which one should be inserted into the current partial schedule. For the insertion costs, there are three possible positions for each candidate contract job on the current partial schedule,
PS= (uU, rC2, rC1, uU). Therefore, six insertion costs are
computed while1= 2 and2= 1 and are presented in Table 4. In Table 4, minimal insertion costs obtained for rA1and rB1 are−8 and −10, respectively. The following savings costs are then obtained, (ug−1, rA1, ug) = 1 × 15 − (−8) = 23
and(ug−1, rB1, ug) = 1 × 15 − (−10) = 25 to rA1 and rB1, respectively. Therefore,∗(ug−1, ri j, ug) = max[23, 24] = 25,
Table 4 The insertion cost of each job at every possible position Job ID Possible insertion Insertion cost Values
positions rA1 (uU, rA1, rC2) SUA+ SAC− 2 × SUC −8 rA1 (uC2, rA1, rC1) SCA+ SAC− 2 × SCC 10 rA1 (uC1, rA1, rU) SCA+ SAU− 2 × SCU 3 rB1 (uU, rB1, rC2) SUB+ SBC− 2 × SUC −10 rB1 (uC2, rB1, rC1) SCB+ SBC− 2 × SCC 21 rB1 (uC1, rB1, rU) SCB+ SBU− 2 × SCU 16
Third, job rB1is the candidate job and will be inserted on Machine 1 preceding rC2. However, this insertion would cause
rC2and rC1to be out of their due dates. Therefore, job rB1is not inserted in the machine. As in this example there is only one other possible contract job(rA1), the process needs to be repeated to determine if it is suitable for insertion. However, it was found that job rA1also cannot be inserted in Machine 1 because of due date and machine capacity constraints. There-fore, the algorithm steps are repeated on Machine 2, jobs rB1 and rA1are scheduled on Machine 2 as they were found not to violate any restrictions. Phase I of MGSA TP is terminated when all contract jobs are scheduled.
Test problems design
For the purpose of testing and comparing the performance of the proposed two new algorithms on various LCMSP with different data characteristics, we generate a set of 24 prob-lems based on a case taken from an LCM factory located on the science-based industrial park in Taiwan. For the case investigated, jobs of 26 product types contain contract and spot jobs. The jobs are scheduled to five identical parallel machines. The contract jobs must be completed on the parallel machines before the given due dates. The machine capacity is set to 4320 min, which is set to equal to the planning period (three days). ‘Minute’ here is used as the time unit for the job process time, setup time, due date, and machine capacity.
The structure and data of the generated test problems are generated covering most real-world applications. These characteristics include: (1) workload level of contract jobs, (2) tightness of due dates, (3) setup time variation, and (4) variation of (contract/spot) profit ratio. These problem sizes range from low workload level of contract jobs, loose due date tightness, small setup time variation, low variation of (contract/spot) profit ratio, to high workload level of contract jobs, tight due date tightness, high setup time variation, and high variation of (contract/spot) profit ratio.
Workload level of contract jobs
In the real production environment, different workload levels of contract jobs result from different market demand or sale seasons, such as hot seasons in electronic industries. There-fore, we need to evaluate the impact of different workload levels of contract jobs on the performance of the solution
algorithms. Owing to varied workload levels of contract jobs, the number of contract jobs is different. Let ES be the estimated setup time required in the problem. AVGi[Si i] is
the average setup time from product type Ri to other types.
And finally, AVG[SiU] is the average setup time to switch
to idle status, AVG[SU i] be the average setup time from idle
status to process. The estimated setup time can be expressed as follows. ES= K (AVG[SU i] + AVG[SiU]) +(I − 1) I all i AVGi[Si i] (25)
The workload calculation formula in our investigation can be expressed in Equation (26). In the 24 testing prob-lems, each problem contains 120 jobs carrying specific contract jobs and spot jobs. Taking problem 4, 5, and 6 (see Table 5) for example, when the number of contract jobs is 25, 50, and 75, the workload level of contract jobs will be 42, 64, and 86%, respectively. Problem 6 is used to illus-trate how the calculation of workload level be applied, a setup time matrix is presented in Table A1 and detailed job information is shown in Table A2 (see the Appendix). The average setup times required for switching from a job with idle status to process(AVG[SU i]) is 120 min, while the
reverse (AVG[SiU]) only requires 0 min. The average setup
time requires for switching from all contract jobs of product type Rito type Ri (AVG[Si i]) is equal to 3496.9 min.
There-fore, the estimated setup time is 3962.4 min. Furthermore, the total processing time of the contract jobs in Problem 6 is 14 451 min. The workload level of Problem 6 is then obtained using Equation (26) when the number of machines(K ) is 5 and each machine capacity(W) is 4320 min.
Workload level= ES+ I i J j pi j K× W × 100%
for i= 1, 2, . . . , I and j = 1, 2, . . . , Jicontract (26)
Tightness of due dates
To analyse the impact of the tightness of due dates on the performance of scheduling algorithms, we include two levels of the tightness of due dates. Here, we apply the tightness index formula proposed by Pearn et al (2004a) and Equation (25) to estimate the setup time. In the tight situation, the number of jobs during the due dates of day 1 and day 2 is greater than day 3. In the loose situation, the number of jobs during the due date of day 1 would be less than the number of jobs during the due dates of day 2 and day 3.
Setup time variation
In LCMSP, we reduce setup time by scheduling contract and spot jobs without violating the contracted due dates. Thus, the setup time is one of the critical factors for increasing the impact of results. However, the setup time could be varied because of different considerations of setup operations. For
Table 5 Summarized information of 24 problems
Problem number Tightness of Workload level Setup time Variation
due date of contract jobs variation (contract/spot)
profit ratio
Parameter Tight Loose Low Middle High Small Large Small Large
1 * * * * 2 * * * * * 3 * * 4 * * * 5 * * * 6 * * 7 * * * 8 * * * 9 * * 10 * * * 11 * * * 12 * * 13 * * * * 14 * * * * 15 * * * 16 * * * * 17 * * * * 18 * * * 19 * * * * 20 * * * * 21 * * * 22 * * * * 23 * * * * 24 * * *
instance, the cool down and the rapid rising of temperature and voltage are good examples of differing conditions. In our test, we include two levels of setup time variation. The high setup time variation is 10519.4 and the low setup time variation is 3990.3.
Variation of (contract/spot) profit ratio
The contract/spot profit ratio is the division of the contract job unit profit by the corresponding spot job unit profit for each of the product types. The variation of the contract/spot profit ratio is to analyse the variance among different product types. In the real-world application, the profit ratio among different product types should be varied owing to the market competition. The high variation of contract/spot profit ratio is set to 0.01 and the low variation of contract/spot profit ratio is set to 0.001.
The problem sets of the four considered factors have 24 different problems. The problem lists with the four different factors are listed as Table 5. Take problem 6 for example, the setup time matrix is presented in Table A1 and the detailed job information is shown in Table A2 (see the Appendix).
Computational results
To solve the LCMSP case, two heuristic algorithms are coded in Virtual Basic 6.0 programming language, and run on a Pentium IV 3.2 GHz PC. We first experiment with the two
new algorithms on 10 small size of LCMSP, where the optimal solutions are available. The size of the problems range from 10 to 15 total jobs, 6 to 10 contract jobs, and 950 to 1650 min machine capacity with various workload levels of contract jobs. For these problems, we write a C+ + programming code to generate the constraints and variables of the models. In addition, we solve them using the IP software CPLEX 7.5 to obtain optimal solutions. The computational results are displayed in Table 6.
Table 6 indicates that the heuristic solutions receive eight optimal solutions (out of 10) in term of total profits. The average gap between the optimality and two heuristic algorithms, MGSA TP and MPSA TP, is 1.4 and 0.5%, respectively. The average run times (in CPU seconds) for the two heuristic algorithms are indeed significantly faster than the run time of optimality.
For large size of the LCMSP, solving the optimal solutions using integer programming model is computationally ineffi-cient. Therefore, in the following, we test the performance of the two new algorithms on the 24 problems described in the previous section entitled ‘Test problems design’. We obtained 72 computational results, which include 48 results for MPSA TP with two types of parameter combination (MPSA TP denotes MPSA TP with 1 = 0.5, 1 = 0.05, and1= 1.5; MPSA TP2 denotes MPSA TP with 1= 0.5,
1= 0, and 1= 1.5). MPSA TP2 represents that the profit ratio adding item of the savings calculation is not considered.
Table 6 A comparison between the optimal solutions and two heuristic algorithms
Prob. no. J Jcontract W Optimal value CPU (s) MGSA TP MPSA TP
Profit CPU Profit CPU
1 10 6 1000 444 000 501.64 444 000 0.031 444 000 0.016 2 11 6 950 444 000 405.09 444 000 0.031 444 000 0.016 3 12 5 1100 514 000 451.89 514 000 0.031 514 000 0.016 4 12 8 1075 557 000 129.75 557 000 0.031 557 000 0.016 5 13 6 1000 488 000 574.39 452 000 0.047 471 000 0.031 6 13 6 1100 537 000 224.61 537 000 0.031 537 000 0.016 7 14 8 1300 660 000 5920.83 660 000 0.047 660 000 0.031 8 14 7 1200 597 000 1219.03 555 000 0.047 584 000 0.031 9 15 9 1440 670 000 13430.4 670 000 0.048 670 000 0.032 10 15 10 1650 760 000 61640.1 760 000 0.063 760 000 0.031
Underlines indicate the optimal solutions.
Table 7 Performance comparison in the three algorithms Profit
Parameter MGSA TP MPSA TP MPSA TP2 Prob. no. MGSA TP MPSA TP2 MPSA TP2
2= 0.3 1= 0.5 1= 0.5 subtracted from subtracted from subtracted from
2= 0.2 1= 0.05 1= 0 MPSA TP MPSA TP MGSA TP
2= 0.4 1= 1.5 1= 1.5 Prob. 1= 2 no. 2= 1 1 5 056 000 5 150 500 4 930 500 1 94 500* 220 000* 125 500* 2 5 385 000 5 470 000 5 351 000 2 85 000* 119 000* 34 000* 3 5 796 500 6 009 500 6 038 500 3 213 000* −29 000 −242 000 4 5 776 000 5 910 500 5 614 500 4 134 500* 296 000* 161 500* 5 5 791 000 5 914 000 5 776 000 5 123 000* 138 000* 15 000* 6 5 986 500 6 259 500 6 298 500 6 273 000* −39 000 −312 000 7 5 099 000 5 193 000 5 013 000 7 94 000* 180 000* 86 000* 8 5 440 000 5 584 000 5 527 000 8 144 000* 57 000* −87 000 9 6 020 000 5 997 000 5 743 000 9 −23 000 254 000* 277 000* 10 5 826 000 5 905 000 5 815 000 10 79 000* 90 000* 11 000* 11 5 915 000 6 209 000 5 999 000 11 294 000* 210 000* −84 000 12 6 236 000 6 227 000 5 933 000 12 −9000 294 000* 303 000* 13 5 102 000 5 073 500 4 973 000 13 −28 500 100 500* 129 000* 14 5 499 000 5 563 000 5 428 000 14 64000* 135 000* 71 000* 15 6 037 000 6 028 500 5 927 500 15 −8500 101 000* 109 500* 16 5 786 000 5 855 500 5 717 000 16 69 500* 138 500* 69 000* 17 5 959 000 6 147 000 5 866 000 17 188 000* 281 000* 93 000* 18 6 277 000 6 273 500 6 147 500 18 −3500 126 000* 129 500* 19 5 016 000 5 192 000 5 063 000 19 176 000* 129 000* −47 000 20 5 577 000 5 640 000 5 537 000 20 63 000* 103 000* 40 000* 21 6 006 000 6 027 000 6 013 500 21 21 000* 13 500* −7500 22 5 726 000 5 952 000 5 835 000 22 226 000* 117 000* −109 000 23 6 014 000 6 052 000 5 999 000 23 38 000* 53 000* 15 000* 24 6 224 000 6 267 000 6 253 500 24 43 000* 13 500* −29 500
Result in bold is the best solution among the three algorithms. Result with * is better than the former.
Table 7 displays the detailed comparison within the two types of algorithms. It denotes the number of better solutions comparing to the two proposed heuristic procedures and the three pairs of subtraction among three different results. Each cell represents the value by subtracting the former from the latter and the cell with the symbol * indicates that the former is better. In comparing the three algorithms, the test results
showed that:
(1) The MPSA TP received 19 better solutions (out of 24) than the MGSA TP. The average improvement between the 19 improved problems of MPSA TP comparing with MGSA TP in terms of profit is 2.2%.
Table 8 Results in means with different problem characteristic groups
Algorithm MGSA TP MPSA TP MPSA TP2
2= 0.3,2= 0.02,2= 0.4 1= 0.5,1= 0.05, 1= 0.5,1= 0,
Parameter 1= 2,2= 1 1= 1.5 1= 1.5
n Mean Mean Mean
Total 24 5 731 250 5 829 167* 5 699 958
Tightness DD= Tight 12 5 704 250 5 804 583* 5 672 333
Tightness DD= Loose 12 5 758 250 5 853 750* 5 727 583
Contracted workload= Low 8 5 423 375 5 529 000* 5 370 125
Contracted workload= Middle 8 5 697 500 5 822 375* 5 685 375
Contracted workload= High 8 6 072 875 6 136 125* 6 044 375
Setup timevariation= Small 12 5 768 583 5 839 250* 5 730 000
Setup time variation= Large 12 5 693 917 5 819 083* 5 669 917
Variation of profit ratio= Small 12 5 502 792 5 577 333* 5 462 083
Variation of profit ratio= Large 12 5 959 708 6 081 000* 5 937 833
Result with * is best solution among the three algorithms.
Table 9 Performance comparisons of the three algorithms (24 problems)
MGSA TP MPSA TP MPSA TP2
Average rank among the three algorithms 2.125 1.25 2.625
Number of problems receiving the best solutions 5 17 2
Average run times CPU seconds 5.11 11.9 8.08
(2) In the MPSA TP algorithm group, the MPSA TP received 22 better solutions (out of 24) than MPSA TP2. The profit ratio term, in the savings calculation of MPSA TP, indeed improves the solutions.
By computing the mean of the solutions generated by each algorithm in total profit for the 24 problems, we could compare the performances among these algorithms. To further analyse the performance of those algorithms on problems with different characteristics, we grouped the results with the four problem factors and factor levels, which is shown in Table 8. Since the factors such as tightness of due date, setup time variation, and variation of profit ratio, contain two levels of values, these groups each include 12 results. Because the factor of workload level of contract jobs contains three levels of values, these groups include eight results each.
In Table 8, MPSA TP outperforms MGSA TP and MPSA TP2 on all nine groups. Therefore, we can say the performance of the MPSA TP is better than the MGSA TP and MPSA TP2 stably in varied situations.
Finally, we compared performances generated by the three algorithms, which is presented in Table 9, with respect to (1) average rank among the three algorithms, (2) number of problems receiving the best solutions, and (3) average run times in CPU seconds on a Pentium IV 3.2 GHz PC.
The results, displayed in Table 9, indicate that the run times of the three algorithms are quite short for solving those prob-lems containing five machines and 120 jobs with different
problem characteristics. In particular, the MPSA TP signifi-cantly outperformed the other algorithms.
Conclusions
In this paper, we considered the LCMSP with sequence dependence setup time and multiple product profits to sequence jobs on identical parallel machines, which has many real-world applications, particular, in the TFT-LCD manu-facturing industry. For the LCMSP case investigated, the jobs are clustered by 26 product types, which are processed on five identical parallel machines and must be completed before the due dates. The LCMSP involves constraints on multiple product profit, sequence dependence setup times, due date, product-type-dependent processing time, and machine capacity, it is more difficult to solve than the classical parallel machines scheduling problem. In this paper, we investigated two network algorithms and developed two modifications to solve the LCMSP efficiently. To test the performances of those algorithms, a set of test problems was designed. The design of test problems involves four critical factors including the workload level of contract jobs, tightness of due date, setup time variation, and variation of profit ratio. The compu-tational test results reveal that the MPSA TP perform better than the MPSA TP2 and MGSA TP stably in varied situa-tions. All proposed algorithms solve the large-scale LCMSP effectively.
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Table A1 Setup times matrix for 26 product types in problem 6 (unit: minutes) To/ From U 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 U 0 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 1 0 0 15 20 20 360 360 180 30 30 180 360 250 250 180 30 30 150 120 50 50 50 130 20 50 30 20 2 0 15 0 20 20 360 320 180 30 30 180 360 30 30 170 170 250 30 50 50 150 100 100 150 15 250 30 3 0 20 20 0 120 30 30 170 170 250 250 250 130 150 50 260 80 30 50 50 150 100 100 150 15 80 30 4 0 280 280 360 0 130 20 50 260 80 80 280 130 150 50 260 80 30 50 50 150 100 100 150 15 80 30 5 0 280 280 360 15 0 20 50 260 80 80 280 130 150 50 260 80 30 50 50 150 100 100 150 15 80 30 6 0 280 280 360 15 20 0 50 260 80 80 280 15 80 150 150 120 80 250 250 250 100 100 270 30 120 80 7 0 20 20 120 50 15 80 0 150 120 120 30 15 80 150 150 120 100 150 150 150 80 80 30 30 120 100 8 0 20 20 15 320 30 120 120 0 250 250 250 50 150 150 260 80 250 170 170 250 250 250 250 30 80 20 9 0 280 280 200 50 50 150 150 260 0 80 280 250 100 250 150 30 250 50 260 80 80 80 280 30 30 20 10 0 150 150 80 80 250 250 250 270 30 0 20 150 80 280 30 250 15 170 130 250 250 250 250 150 30 50 11 0 270 270 100 100 150 150 280 30 30 100 0 50 180 120 120 250 150 170 130 250 250 250 250 270 50 150 12 0 15 15 30 30 180 180 120 120 50 150 150 0 180 120 120 250 280 250 250 100 30 30 280 250 250 280 13 0 20 20 360 360 260 80 80 280 250 250 100 50 0 120 150 150 15 360 360 320 320 120 15 280 150 15 14 0 280 280 15 15 260 80 80 280 150 150 80 20 15 0 150 208 15 320 320 320 320 120 15 280 30 15 15 0 20 20 50 50 130 250 250 250 150 80 80 150 80 80 0 80 250 50 260 80 80 80 280 30 250 250 16 0 20 20 208 208 280 250 30 30 280 280 280 280 100 30 30 0 130 150 150 120 120 120 30 280 50 150 17 0 280 280 50 50 150 150 80 80 120 250 280 150 80 80 80 80 0 20 20 20 50 50 150 280 250 250 18 0 150 150 80 80 270 270 100 100 50 150 280 270 100 100 100 100 15 0 130 250 50 250 250 150 150 150 19 0 30 50 30 30 15 280 120 120 30 30 15 15 280 280 120 120 30 280 0 250 250 100 30 15 180 180 20 0 80 250 30 280 15 15 120 120 30 30 15 15 360 360 120 120 30 280 280 0 250 100 30 150 30 30 21 0 100 150 30 280 15 50 250 250 80 80 250 250 360 360 120 120 250 250 100 30 0 30 30 30 30 30 22 0 150 150 80 150 50 50 150 150 100 100 150 15 20 20 250 250 250 250 100 30 30 0 250 30 170 250 23 0 270 270 100 150 50 50 50 180 120 120 180 15 20 20 250 250 250 250 100 30 30 30 0 260 260 80 24 0 30 50 30 30 360 280 120 120 30 30 15 250 280 280 120 120 30 280 280 250 250 100 30 0 180 180 25 0 80 250 30 280 360 360 120 120 30 30 15 250 280 280 120 120 30 280 280 250 250 100 30 150 0 180 26 0 80 250 30 280 360 360 120 120 30 30 15 250 280 280 120 120 30 280 280 250 250 100 30 150 0 0
Table A2 The job information of the 120 jobs in LCMSP
Job ID Product Processing Profit Due Contract/ Job ID Product Processing Profit Due Contract/
type time (min) date Spot type time (min) date Spot
1 1 213 50 000 1440 C 61 12 213 75 000 1440 C 2 1 213 50 000 1440 C 62 12 213 75 000 1440 C 3 1 213 50 000 1440 C 63 12 213 65 000 4320 S 4 1 213 40 000 4320 S 64 12 213 65 000 4320 S 5 1 213 40 000 4320 S 65 13 183 72 500 2880 C 6 2 192 60 000 2880 C 66 13 183 72 500 2880 C
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Table A2 (Continued)
Job ID Product Processing Profit Due Contract/ Job ID Product Processing Profit Due Contract/
type time (min) date Spot type time (min) date Spot
7 2 192 60 000 2880 C 67 13 183 72 500 4320 C 8 3 213 63 000 1440 C 68 13 183 62 500 4320 S 9 3 213 63 000 1440 C 69 13 183 62 500 4320 S 10 3 213 63 000 1440 C 70 14 213 65 000 2880 C 11 3 213 53 000 4320 S 71 15 183 60 000 2880 C 12 3 213 53 000 4320 S 72 15 183 60 000 2880 C 13 4 175 50 000 4320 C 73 15 183 60 000 2880 C 14 5 183 59 000 1440 C 74 15 183 60 000 4320 C 15 5 183 59 000 1440 C 75 15 183 50 000 4320 S 16 5 183 59 000 1440 C 76 15 183 50 000 4320 S 17 5 183 49 000 4320 S 77 15 183 50 000 4320 S 18 5 183 49 000 4320 S 78 16 175 78 500 4320 C 19 5 183 49 000 4320 S 79 16 175 78 500 4320 C 20 5 183 49 000 4320 S 80 16 175 78 500 4320 C 21 6 213 76 000 2880 C 81 16 175 78 500 4320 C 22 6 213 76 000 2880 C 82 16 175 68 500 4320 S 23 6 213 76 000 2880 C 83 17 213 60 000 2880 C 24 6 213 76 000 2880 C 84 17 213 60 000 2880 C 25 6 213 76 000 2880 C 85 18 183 54 000 4320 C 26 6 213 66 000 4320 S 86 18 183 54 000 4320 C 27 6 213 66 000 4320 S 87 18 183 44 000 4320 S 28 6 213 66 000 4320 S 88 18 183 44 000 4320 S 29 6 213 66 000 4320 S 89 19 200 70 000 2880 C 30 6 213 66 000 4320 S 90 19 200 70 000 2880 C 31 7 183 70 000 1440 C 91 20 200 60 000 2880 C 32 7 183 70 000 1440 C 92 20 200 60 000 4320 C 33 7 183 70 000 1440 C 93 20 200 60 000 4320 C 34 7 183 60 000 4320 S 94 20 200 60 000 4320 C 35 7 183 60 000 4320 S 95 20 200 50 000 4320 S 36 7 183 60 000 4320 S 96 21 192 50 000 4320 C 37 7 183 60 000 4320 S 97 21 192 50 000 4320 C 38 8 167 66 000 2880 C 98 21 192 50 000 4320 C 39 8 167 66 000 2880 C 99 21 192 50 000 4320 C 40 8 167 66 000 4320 C 100 21 192 50 000 4320 C 41 8 167 66 000 4320 C 101 21 192 50 000 4320 C 42 8 167 66 000 4320 C 102 22 213 62 500 2880 C 43 9 192 75 000 1440 C 103 22 213 62 500 2880 C 44 9 192 75 000 1440 C 104 23 213 58 000 4320 S 45 9 192 75 000 1440 C 105 23 213 58 000 4320 S 46 9 192 65 000 4320 S 106 23 213 58 000 4320 S 47 9 192 65 000 4320 S 107 24 200 71 000 2880 C 48 9 192 65 000 4320 S 108 24 200 71 000 2880 C 49 9 192 65 000 4320 S 109 24 200 71 000 2880 C 50 10 183 60 000 4320 C 110 24 200 71 000 2880 C 51 10 183 60 000 4320 C 111 25 183 61 000 2880 C 52 10 183 50 000 4320 S 112 25 183 61 000 2880 C 53 10 183 50 000 4320 S 113 25 183 61 000 4320 C 54 11 167 50 000 2880 C 114 25 183 61 000 4320 C 55 11 167 50 000 2880 C 115 25 183 51 000 4320 S 56 11 167 40 000 4320 S 116 25 183 51 000 4320 S 57 11 167 40 000 4320 S 117 26 192 65 500 1440 C 58 11 167 40 000 4320 S 118 26 192 55 500 4320 S 59 12 213 75 000 1440 C 119 26 192 55 500 4320 S 60 12 213 75 000 1440 C 120 26 192 55 500 4320 S Received July 2006; accepted January 2008 after two revisions
