Computational test

In this section, three computational tests are presented. The purpose of the first test is to show the results for the problems of small or moderate size. The second test focuses the computational efficiency for the heuristic algorithm for the problems of larger size. The third test focuses on solving the scheduling problem based on real-world applications.

4.6.1. Computational test 1

In this test, computational results were presented by a set of randomly generated test problems, with similar characteristics to industrial data. 12 jobs are to be completed within two days. Thus, the machine capacity is set to 2880 minutes. Table 4-4 shows the data set used to generate the test problems. We consider two values of number of job clusters (I=3, 6), two sets of level of priority (H=3, 5), and three values of number of machines (K=3, 4, 5). The unit processing time for the product types are 40, 45, and 50.

The lot size of each job was generated using uniform[10, 15] for 3 machines, uniform[14, 19] for 4 machines, and uniform[18, 23] for 5 machines. Thus, we have a total of 12 combinations of problem parameters. For each combination, we generate 10 instances, yielding a total of 120 problems.

Table 4-4. Data Set

Factor Values considered Total values

Number of job clusters ( I ) 3, 6 2

Levels of job priority ( H ) 3, 5 2

Number of machines ( K ) 3, 4, 5 3

Number of jobs (∑Ji ) 12 1

The IP model was tested using a computer program coded in ILOG OPL language and solved with ILOG CPLEX on a Pentium IV 3.0 GHz PC. The heuristic algorithm

was coded in Compaq Visual Fortran 6.6. For evaluating the solution quality, percentage error e=[(S_h −S_opt)/S_opt]×100 is employed, where S_h is the average setup

time of the heuristic solution and S_opt is the optimal average setup time obtained from the IP model. Table 4-5 lists the results. The proposed heuristic is effective and each percentage error is less than 3%. The efficiency of the models is also reported based on the average CPU time (in seconds). For the IP model, the computation time increases with increasing the number of machines, while the heuristic algorithm is able to obtain the solutions within almost instant time for every problems in this test.

Table 4-5. Summary Results

IP model Heuristic

I H K

Sopt Avg. run

time (sec) S^h Avg. run time (sec)

e* (%)

3 3 3 93 839.41 93 0.0015 0.00

6 3 3 240 38.23 240 0.0015 0.00

3 5 3 81 133.51 81 0.0015 0.00

6 5 3 267 39.31 273 0.0015 2.25

3 3 4 93 1974.55 93 0.0015 0.00

6 3 4 186 65.61 186 0.0015 0.00

3 5 4 120 1225.81 120 0.0015 0.00

6 5 4 201 41.85 204 0.0015 1.49

3 3 5 96 1722.96 96 0.0015 0.00

6 3 5 168 247.97 171 0.0015 1.79

3 5 5 117 2110.11 120 0.0015 2.56

6 5 5 177 65.48 177 0.0015 0.00

*: e=[(S_h −S_opt)/S_opt]×100

Using the combination of depth-first search strategy and strong branching rule showed to be powerful. For every test problems of 3 machines and 4 machines in the data set, the optimal solution was reached within the 15,000 nodes created (within 5 minutes of execution time). For every test problem of 5 machines in the data set, the

optimal solution was reached within the 17,000 nodes created (within 10 minutes of execution time).

4.6.2. Computational test 2

In this test, computational results were presented by six larger-size problems, with similar characteristics to industrial data. The jobs are to be completed within two days.

Thus, the machine capacity is set to 2880 minutes. We consider two values of number of job clusters (I=8, 10), two sets of level of priority (H=3, 5), and three values of number of machines (K=8, 9, 10).

The IP model with the combination of depth-first search strategy and strong branching rule (IP_DFS) was tested using a computer program coded in ILOG OPL language and solved with ILOG CPLEX on a Pentium IV 3.0 GHz PC. The heuristic algorithm was coded in Compaq Visual Fortran 6.6. For evaluating the solution quality, percentage error e=[(S_h −S_dfs)/S_dfs]×100 is employed, where S_h is the average setup

time of the heuristic solution and S_dfs is the average setup time obtained from the IP_DFS model.

For the six problems, the solution values obtained by the heuristic algorithm were compared with those obtained by IP_DFS. According to the computational results, the heuristic algorithm outperformed IP_DFS both in solution quality and runtime consumed.

Table 4- 6 Summary Results for larger-size problem

IP_DFS Heuristic I H K

Sdfs Run time (sec)

Sh Run time (sec)

e* (%)

10 3 8 660 43953.80 360 0.0015 -45.45

8 3 9 1200 45894.81 540 0.0015 -55.00

8 5 9 1560 47395.84 660 0.0015 -57.69

10 3 9 1740 51290.69 630 0.0015 -63.79

10 5 9 1320 52408.55 750 0.0015 -43.18

8 5 10 2340 61162.38 420 0.0015 -82.05

*: e=[(S_h −S_dfs)/S_dfs]×100

4.6.3. Computational test 3

In this section, we consider the following example taken from an IC assembly shop-floor in an IC manufacturing factory located in the Science-based Industrial Park at Tainan, Taiwan. For the case we investigated, there are 20 product types of TSOP2 packaging being processed on 33 parallel LOC die bonders.

This real example contains 105 wafer lots with job priority, lot size, and unit processing time, which would be die bonding under certain size of chop table, mount stage and mount head, as shown in Table 4-6. These jobs are to be completed on the 33 parallel die bonders within two days. Therefore, the machine capacity is set to 2880 minutes.

The setup time required for switching one product type to another depends on the chop table changes, mount stage and mount head changes, and parameter settings is shown in Table 4-5. The time to change chop table is 240 minutes, the time to change mount stage and mount head is 120 minutes, and parameter settings is 30 minutes in this case.

Solving the real-world ICASP by our proposed algorithm (the program codes of the algorithm are written in Compaq Visual Fortran 6.6), the sets of machine schedule are generated. The proposed algorithm takes only 0.07 CPU seconds to obtain the solution with total machine workload 87602 with setup time 6480 and processing time 81122 on 33 die bonders, as shown in Figure 4-6.

Table 4-7. The product types, processing time, and job priority code in the real-world

Table 4-6. The product types, processing time, and job priority code in the real-world

Table 4-8. Setup times required for switching one product type to another in the real-world example

To

From U 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20

U 0 270 270 150 150 150 270 150 150 150 150 150 150 150 150 150 270 150 150 150 270 01 0 0 30 270 270 270 30 270 270 270 270 270 270 270 270 270 150 390 270 270 150 02 0 30 0 270 270 270 30 270 270 270 270 270 270 270 270 270 150 390 270 270 150 03 0 390 390 0 30 30 390 30 30 30 30 150 150 150 150 150 390 150 150 150 390 04 0 390 390 30 0 30 390 30 30 30 30 150 150 150 150 150 390 150 150 150 390 05 0 390 390 30 30 0 390 30 30 30 30 150 150 150 150 150 390 150 150 150 390 06 0 150 150 270 270 270 0 270 270 270 270 270 270 270 390 390 150 390 390 390 150 07 0 390 390 30 30 30 390 0 30 30 30 150 150 150 150 150 390 150 150 150 390 08 0 390 390 30 30 30 390 30 0 30 30 150 150 150 150 150 390 150 150 150 390 09 0 390 390 30 30 30 390 30 30 0 30 150 150 150 150 150 390 150 150 150 390 10 0 390 390 30 30 30 390 30 30 30 0 150 150 150 150 150 390 150 150 150 390 11 0 390 390 30 30 30 270 30 30 30 30 0 30 30 150 150 390 150 150 150 390 12 0 390 390 30 30 30 270 30 30 30 30 30 0 30 150 150 390 150 150 150 390 13 0 390 390 30 30 30 270 30 30 30 30 30 30 0 150 150 390 150 150 150 390 14 0 270 270 30 30 30 270 30 30 30 30 30 30 30 0 30 390 150 30 30 390 15 0 270 270 30 30 30 270 30 30 30 30 30 30 30 30 0 390 150 30 30 390 16 0 30 30 270 270 270 30 270 270 270 270 270 270 270 270 270 0 270 270 270 30 17 0 270 270 30 30 30 270 30 30 30 30 30 30 30 30 30 270 0 30 30 270 18 0 270 270 30 30 30 270 30 30 30 30 30 30 30 30 30 390 150 0 30 390 19 0 270 270 30 30 30 270 30 30 30 30 30 30 30 30 30 390 150 30 0 390 20 0 30 30 270 270 270 30 270 270 270 270 270 270 270 270 270 30 270 270 270 0

Figure 4-6. The schedule for the real-world ICASP application example.

Figure 4-5. The schedule for the real-world ICASP application example (continued).