4. Solutions for the Aging Test Scheduling Problem
4.5. Computational Results and Comparisons
4.5.1. Analysis of Results from Small and Moderate Size Problems
The experimental design involves two essential characteristics, ready time variation and processing time variation. These two variations are characterized by two magnitudes, large (L) and small (S). Accordingly, the ready times in a problem are generated from uniform distributions in [0,300], [0,100] for large and small variations, respectively. The processing times are generated from uniform distributions in [90,300], [100,200] for large and small variations, respectively. The structure and data of the test problems are generated covering a wide variety of scheduling problems encountered in industrial practice. A number of machines and a number of jobs are alternated in order to get twenty-four problem configurations. For each problem configuration, five problem
instances are randomly generated. Thus, 120 different problem instances are generated which are either small or moderate in size. The four different experimental factors are listed in Table 4-3. The aging oven can process a batch in which the total number of pieces from all the jobs in that batch does not exceed 450 pieces. Without loss of generality, we assume that the size of each job is less than the machine capacity (i.e. 450 pieces of panel). Once the batch processing begins, it is non-preemptive until the batch is completely processed. Processing and ready times are measured in minutes. All jobs should be formed as batches and be processed completely by the minimum makespan.
Table 4-3Experimental factors for small and moderate sized problems.
Factor Value considered Number of values
Number of jobs (N) 7, 15, 20 3
Ready time variation L, S 2
Processing time variation L, S 2
Number of machines (K) 2, 3 2
Total problem configurations 24
Instances per configuration 5
Total problem instances 120
Table 4-4 presents the solutions generated by the all the proposed algorithms on the eight small problem configurations with seven jobs in each. The values of the optimal solutions are obtained by solving the MILP model (Model P), which is formulated in Section 4.2. In Table 4-4, the problem configuration “7LL2”represents the 7 jobs with large ready time and large processing time variations, which are processed on two batch machines. In this testing, the run times of Model P and the CMA may vary for problem instances with different configurations. However, the run times of the CMA are significantly faster than for the original MILP model (Model P). All the solutions of the CMA are equal to the values obtained from Model P; hence the solutions are optimal. In the three heuristic algorithms, their performances are sensitive to the values of the
parameters and (as concluded also by Lee and Uzsoy [51]). This section provides the experiments involving the three heuristic algorithms to the testing problem instances using several values of , which are initially set to 0, 0.2, 0.4, 0.6, 0.8, and 1 (0 1) and , which are initially set to 0, 0.2, 0.4, …, 2.6, 2.8, and 3 (0 3). The best solution, obtained using one of the parameter combinations, is selected as the final solution to the heuristic algorithm. It is worthwhile to note that the MixedH obtains 34 (out of 40) optimal solutions within 0.8 CPU seconds for each problem instance.
Table 4-5 displays the results for the problem instances with fifteen jobs and different configurations and their performance comparisons in terms of the makespan obtained using mathematical and heuristic algorithmic solutions. In the MILP model (Model P) and Model N of the CMA, the depth-first search strategy (Wolsey [93]) is implemented by choosing the most recently created node. To avoid the CPLEX routine which requires a tremendous amount of computation time, the maximum run time is set at 28800 CPU seconds. Furthermore, the nodes created cannot be greater than 1E06 in Phase II of the CMA in order to check the subsequent batch numbers repeatedly. CPLEX could stop at the pre-determined time without guaranteeing optimality for problems with high computational complexity. However, the depth-first search strategy can incorporate the strong branching rule (Wolsey [93]) causing the variable selection based on partially solving a number of sub-problems with tentative branches in order to find the most promising branch. Table 4-5 shows that the performances of the CMA are reasonably good; that is, with all the problem instances it achieved better solutions than the original MILP model (Model P). Furthermore, the three heuristic algorithms perform well and
efficiently. As seen in Table 4-5, due to its solution strategy, the MixedH provides the best solutions of all three heuristic algorithms for all problem configurations.
Furthermore, table 4-6 displays the results for the problem instances with twenty jobs and different configurations and their performance comparisons in terms of the makespan obtained using mathematical and heuristic algorithmic solutions. In table 4-6, the run times of MILP model (Model P) and Model N of the CMA almost approach to 28800 seconds, which is the maximum run times allowed. Therefore, these mathematical methods, including MILP and CMA models, are both computationally inefficient.
Notably, the MixedH receives 15 better solutions (out of 40) than those of the two mathematical based algorithms. Thus, as the number of job increases to exceed 20 jobs, the MixedH not only runs fast but also obtains satisfactory solutions.
Table 4-7 displays the performance comparisons among the five algorithms in terms of (1) average rank, (2) average run times, (3) number of problem instances receiving the best solutions, and (4) number of optimal solutions. The results indicate that the CMA can obtain 40 (out of 40) optimal solutions for the 7-job test problem instances and it can perform remarkably well for the 15-job and 20-job problem instances. The CMA significantly speeds up the original MILP model in test problem instances. However, when the number of jobs is increased to 20 jobs, the computation times required by CMA are also increased and inefficient. It should also be noted that all three heuristic algorithms run very fast. With the 120 problem instances tested, it was found that none of them required more than 2 CPU seconds on a Pentium IV 3.2GHZ PC. To access the accuracy of the heuristic solutions, the best solution selected from the CMA and the
Table 4-4Run times and makespan results for 7-job problem instances.
1 7LL2 459 2496 459 11.2 459 0.672 486 0.641 459 0.688
7LL3 379 8742 379 71.9 379 0.625 379 0.625 379 0.656
7LS2 395 501.2 395 13.7 395 0.609 401 0.609 395 0.672
7LS3 345 1032 345 21.4 365 0.625 345 0.625 345 0.656
7SL2 430 1041.3 430 11.2 430 0.656 480 0.625 430 0.676
7SL3 370 648 370 1.7 370 0.625 370 0.688 370 0.719
7SS2 346 1268 346 12.1 346 0.625 393 0.625 346 0.656
7SS3 289 15475 289 66.3 289 0.609 306 0.625 289 0.672
2 7LL2 607 463 607 5.5 607 0.613 631 0.625 607 0.656
7LL3 540 15.8 540 6.2 568 0.609 540 0.609 540 0.656
7LS2 488 482 488 6.4 488 0.625 502 0.672 488 0.712
7LS3 418 2899.8 418 32.6 418 0.625 422 0.625 418 0.672
7SL2 561 450 561 18.3 570 0.641 575 0.609 570 0.719
7SL3 435 28052 435 150.3 435 0.609 435 0.609 435 0.656
7SS2 348 1376 348 25.2 348 0.641 367 0.703 348 0.756
7SS3 267 20714 267 115.9 267 0.625 267 0.609 267 0.688
3 7LL2 522 1.6 522 1.1 522 0.625 599 0.609 522 0.734
7LL3 522 7.5 522 5.3 522 0.641 522 0.594 522 0.672
7LS2 455 1.3 455 1.1 455 0.609 471 0.594 455 0.672
7LS3 455 10 455 3 455 0.641 455 0.609 455 0.672
7SL2 424 693.3 424 8.7 424 0.641 468 0.609 424 0.734
7SL3 332 7389.2 332 4.4 358 0.625 332 0.672 332 0.692
7SS2 330 461.8 330 11 345 0.625 343 0.594 343 0.656
7SS3 303 23341 303 85 318 0.625 303 0.609 303 0.813
4 7LL2 630 1606 630 13.1 656 0.609 656 0.609 656 0.672
7LL3 543 41721 543 64.8 619 0.625 543 0.625 543 0.672
7LS2 411 7.7 411 2.9 411 0.625 530 0.594 411 0.656
7LS3 411 12.4 411 6.8 411 0.672 411 0.609 411 0.734
7SL2 512 4209.9 512 7.7 512 0.625 534 0.672 512 0.756
7SL3 425 31346.8 425 117.8 465 0.625 425 0.703 425 0.741
7SS2 313 462.8 313 9.6 317 0.625 336 0.609 317 0.641
7SS3 258 22398 258 61.6 261 0.625 258 0.609 258 0.641
5 7LL2 574 1306.8 574 6.4 592 0.641 590 0.641 590 0.656
7LL3 555 7.6 555 5.3 555 0.625 555 0.609 555 0.655
7LS2 453 1.6 453 0.7 454 0.625 453 0.703 453 0.741
7LS3 453 10.9 453 4.8 453 0.609 453 0.719 453 0.741
7SL2 498 3310 498 15.6 498 0.625 534 0.609 498 0.766
7SL3 398 43339 398 134.2 398 0.719 398 0.609 398 0.752
7SS2 371 3748 371 23.1 393 0.609 373 0.672 373 0.741
7SS3 294 22360 294 96.1 298 0.625 294 0.609 294 0.719
The underlined values represent the best solutions for each problem instance from among all of the algorithms.
Table 4-5Run times and makespan results for 15-job problem instances.
1 15LL2 592 28800 579 786.8 610 0.844 612 0.797 610 0.947
15LL3 552 28800 552 15.3 552 0.781 552 0.781 552 0.931
15LS2 467 250.6 467 5.1 467 0.781 467 0.766 467 0.916
15LS3 467 7471.3 467 19.8 467 0.797 467 1.031 467 1.041 15SL2 532 28800 532 1096.2 532 0.813 575 0.766 532 0.994 15SL3 382 28800 382 2789.5 382 0.875 382 0.875 382 0.947 15SS2 372 28800 372 1252.5 409 0.828 392 0.766 392 0.931 15SS3 276 28800 275 2225 303 0.938 275 0.766 275 0.994 2 15LL2 614 28800 605 3756.1 629 0.813 713 0.766 629 0.978
15LL3 565 642 565 17.4 565 0.797 565 0.750 565 0.947
15LS2 498 28800 498 3063 516 0.797 551 0.797 516 0.931 15LS3 478 28800 478 31.6 478 0.813 478 0.750 478 0.931 15SL2 500 28800 475 1085.9 475 0.813 537 0.750 475 0.963 15SL3 380 28800 380 150.7 380 0.797 380 0.750 380 0.916 15SS2 380 28800 374 8478 380 0.781 403 0.766 380 0.947 15SS3 293 28800 293 183.9 293 0.797 293 0.750 293 0.931
3 15LL2 505 28800 505 902.3 505 0.797 596 0.750 505 0.963
15LL3 442 28800 442 48.2 442 0.781 442 0.750 442 0.916 15LS2 455 28800 455 872.4 467 0.781 498 0.750 467 0.931 15LS3 424 28800 424 18.8 426 0.875 426 0.813 426 0.963
15SL2 425 28800 425 999 428 0.797 466 0.750 428 0.931
15SL3 323 28800 314 1089 323 0.797 314 0.766 314 0.931 15SS2 375 28800 375 1290 393 0.781 420 0.750 393 0.916 15SS3 318 28800 306 3012.9 315 0.797 306 0.750 306 0.947
4 15LL2 495 28800 495 1185 516 0.797 546 0.750 516 0.931
15LL3 445 28800 445 16.9 445 0.859 469 0.813 445 0.978 15LS2 435 28800 422 997.7 435 0.781 456 0.828 435 0.947 15LS3 370 28800 370 1275.5 375 0.781 390 0.734 375 0.892 15SL2 459 28800 459 1323 473 0.797 473 0.750 473 0.910
15SL3 369 28800 369 219 369 0.781 369 0.766 369 0.947
15SS2 384 28800 383 1437 403 0.766 410 0.734 403 0.916 15SS3 327 28800 327 3684 331 0.781 331 0.750 331 0.916 5 15LL2 553 28800 553 1014.7 586 0.828 590 0.766 586 0.963
15LL3 538 813 538 13.1 538 0.797 538 0.781 538 0.916
15LS2 454 28800 454 472.2 464 0.875 488 0.828 464 0.994 15LS3 451 28800 451 22.7 451 0.859 451 0.828 451 0.993 15SL2 517 28800 477 1483.9 505 0.797 517 0.750 505 0.916 15SL3 363 28800 350 697.5 363 0.828 363 0.750 363 0.916 15SS2 374 28800 368 3138 390 0.797 390 0.750 390 0.900 15SS3 324 28800 321 4298 368 0.813 326 0.750 326 0.947 The underlined values represent the best solutions for each problem instance from among all of the algorithms.
Table 4-6Run times and makespan results for 20-job problem instances.
1 20LL2 821 28800 757 28800 790 1.094 875 1.094 790 1.109
20LL3 595 28800 573 774.2 601 1.125 625 1.078 601 1.250 20LS2 656 28800 592 28800 575 1.063 635 1.031 575 1.141 20LS3 498 28800 459 147.48 459 1.063 514 1.047 459 1.453 20SL2 790 28800 682 28800 677 1.109 745 1.094 677 1.766 20SL3 527 28800 489 28800 559 1.063 508 1.063 508 1.094 20SS2 573 28800 529 28800 527 1.063 548 1.063 527 1.094 20SS3 483 28800 380 28800 388 1.078 395 1.047 388 1.094
2 20LL2 939 28800 759 28800 792 1.078 918 1.109 792 1.188
20LL3 663 28800 588 28800 609 1.078 613 1.078 609 1.094 20LS2 613 28800 551 28800 566 1.078 618 1.063 566 1.109 20LS3 455 28800 455 76.16 455 1.094 455 1.094 455 1.125 20SL2 805 28800 744 28800 741 1.094 809 1.078 741 1.141 20SL3 631 28800 544 28800 575 1.094 558 1.094 558 1.125 20SS2 601 28800 519 28800 518 1.063 526 1.047 518 1.094 20SS3 461 28800 371 28800 382 1.063 393 1.063 382 1.203 3 20LL2 1000 28800 876 28800 903 1.078 894 1.094 894 1.188 20LL3 703 28800 622 28800 645 1.078 687 1.078 645 1.203 20LS2 641 28800 568 28800 555 1.078 690 1.078 555 1.109 20LS3 484 28800 471 28800 473 1.078 480 1.078 473 1.094 20SL2 887 28800 793 28800 770 1.094 886 1.078 770 1.172 20SL3 603 28800 579 28800 585 1.125 594 1.078 585 1.219 20SS2 664 28800 521 28800 545 1.094 569 1.125 545 1.172 20SS3 458 28800 383 28800 382 1.078 397 1.078 382 1.109
4 20LL2 925 28800 805 28800 776 1.109 948 1.109 776 1.141
20LL3 671 28800 639 28800 656 1.141 680 1.125 656 1.188 20LS2 604 28800 578 28800 598 1.094 670 1.078 598 1.141 20LS3 478 28800 455 28800 456 1.078 507 1.063 456 1.219 20SL2 826 28800 748 28800 765 1.094 822 1.094 765 1.109 20SL3 611 28800 531 28800 513 1.094 570 1.063 513 1.109 20SS2 632 28800 522 28800 514 1.078 580 1.047 514 1.109 20SS3 450 28800 395 28800 410 1.078 414 1.094 410 1.109
5 20LL2 950 28800 775 28800 769 1.094 830 1.094 769 1.109
20LL3 673 28800 578 28800 567 1.078 655 1.063 567 1.094 20LS2 638 28800 549 28800 552 1.078 714 1.078 552 1.094 20LS3 475 28800 443 967.66 443 1.078 475 1.109 443 1.156 20SL2 880 28800 748 28800 693 1.109 816 1.078 693 1.397 20SL3 643 28800 530 28800 490 1.094 559 1.063 490 1.344 20SS2 577 28800 478 28800 495 1.078 524 1.063 495 1.297 20SS3 432 28800 347 28800 354 1.078 374 1.109 354 1.266 The underlined values represent the best solutions for each problem instance from among all of the algorithms.
original MILP model (Model P) is used for each problem instance as a convenient reference point. The average percentage deviations between the MixedH and the selected best solutions of the two mathematical based algorithms are 0.36%, 1.8%, and 0.64% for 7-job, 15-job, and 20-job problem instances, respectively. The percentage deviation is defined as
VMixedH V V
, where VMixedH and V are the values for each problem instance which is obtained using the MixedH and the two mathematical based algorithms, respectively. The CMA may perform inefficiently as the number of job increases to the large scale usual in real-world factories. Thus, if the computational time is a primary concern, MixedH can solve real-world problems well.Table 4-7Comparisons of the five algorithms.
Problem MILP CMA H1 H2 MixedH
7-job Average rank among the five algorithmsa 1 1 2.23 2.9 1.3
Average run times (in CPU seconds) 7335.3 31.5 0.629 0.630 0.696 Number of problem instances receiving the best solutions 40 40 25 19 34
Number of optimal solutions 40 40 25 19 34
15-job Average rank among the five algorithmsa 1.6 1 2.3 3.1 1.95
Average run times (in CPU seconds) 26149 1362 0.810 0.776 0.943 Number of problem instances receiving the best solutions 26 40 15 15 19
Number of optimal solutions 4 4 4 4 4
20-job Average rank among the five algorithmsa 4.55 1.7 1.75 4.025 1.575 Average run times (in CPU seconds) 28800 25969 1.086 1.079 1.181 Number of problem instances receiving the best solutions 1 25 18 1 18
Number of optimal solutions 1 4 3 1 3
aThe smallest rank value indicates the best solutions among all algorithms.