Computational Results

To solve the PCBSP, two heuristic algorithms are coded in Virtual Basic 6.0 programming language, and run on a Pentium IV 3.2GHZPC. We first experiment with the two new algorithms on ten small size of the PCBSP, 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 minutes 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 3-6.

Table 3-6 indicates that the heuristic solutions receive 8 optimal solutions (out of 10) in term of total weight throughput. The average gap between the optimality and two heuristic algorithms, MGSA_TP and MPSA_TP, are 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.

Table 3-6A comparison between the optimal solutions and two heuristic algorithms (underlines indicate the optimal solutions).

1 10 6 1000 444000 501.64 444000 0.031 444000 0.016

2 11 6 950 444000 405.09 444000 0.031 444000 0.016

3 12 5 1100 514000 451.89 514000 0.031 514000 0.016

4 12 8 1075 557000 129.75 557000 0.031 557000 0.016

5 13 6 1000 488000 574.39 452000 0.047 471000 0.031

6 13 6 1100 537000 224.61 537000 0.031 537000 0.016

7 14 8 1300 660000 5920.83 660000 0.047 660000 0.031

8 14 7 1200 597000 1219.03 555000 0.047 584000 0.031

9 15 9 1440 670000 13430.4 670000 0.048 670000 0.032

10 15 10 1650 760000 61640.1 760000 0.063 760000 0.031

For large size of the PCBSP, solving the optimal solutions using integer programming model is computationally inefficiently. Therefore, in the following, we test the performance of the two new algorithms on the twenty-four problems described in Section 3.5. As the two parameters of the two algorithms, ₁ and ₁, have been applied by Pearn et al. [68], the preliminary comparison based on the two parameters analysis of MPSA_TP is summaries in Table 3-7. We first examine the three values of ₁ and five values of ₁, which are initially set to ₁ = 0, 0.5, 1 and ₁ =0, 0.5, 1, 1.5, 2,

respectively. Therefore, fifteen different combinations are obtained by various ₁ and ₁ values where ₁ is set to 0. Table 3-7 displays that the average values of the solutions obtained on 24 problems described previously. It appeared that the best parameter combination (₁, ₁) is (0.5, 1.5) in terms of average weighted throughput. Therefore, we limit the two parameters and further analysis the settings of parameter ₁ that is a new-added parameter for PCBSP. In this dissertation, parameter ₁ is examined to check whether parameter ₁ affects the solution for PCBSP (0 ₁ 1) or not ( ₁ 0).

We set  ₁ 0,0.05,0.1,...,0.95, and 1 and display the computational results in Table 3-8.

Table 3-8 indicates that the parameter combination, (₁,₁,₁) = (0.5, 0.05, 1.5), outperforms than the other settings. We therefore set the choice of parameter values of the MPSA_TP to (₁,₁,₁) = (0.5, 0.05, 1.5).

Table 3-7The preliminary comparison with various parameter settings.

1 ₁ _{1 throughput}^{weightedAverage} ₁ ₁ _{1 throughput}^{weightedAverage} ₁ ₁ _{1 throughput}^{weightedAverage}

0 0 0 5284083 0.5 0 0 5568292 1 0 0 5627813

0 0 0.5 5326438 0.5 0 0.5 5656500 1 0 0.5 5670188

0 0 1 5341271 0.5 0 1 5657396 1 0 1 5670521

0 0 1.5 5341271 0.5 0 1.5 5699958 1 0 1.5 5688750

0 0 2 5336375 0.5 0 2 5674292 1 0 2 5677563

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, and  ₁ 1.5; MPSA_TP2 denotes MPSA_TP with  ₁ 0.5,  ₁ 0, and

 ₁ 1.5). MPSA_TP2 represents that the weighted throughput ratio adding item of the savings calculation is not considered. Table 3-9 displays the detailed comparison within the two types of algorithms. It denotes the number of better solutions comparing to the two proposed heuristic procedures. In comparing the three algorithms, the test results showed that:

Table 3-8The comparisons with different parameter values of 1 when   .1 0 5 and

1 1 5

. (underlines indicate the best solutions for each problem instance).

1

Prob.

No. _{0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1} 1 493050 515050 505300 506050 506900 503900 502650 503050 499300 496500 497050 505300 505300 505300 505300 494300 503200 503200 503200 507300 507300 2 535100 547000 544400 543750 537400 544150 540350 538500 549200 548200 541500 540850 537350 533150 535500 535500 538500 538500 538450 527750 533950 3 603850 600950 584250 588600 582750 577300 573950 587000 576250 583750 572650 581050 577850 580750 585150 585150 581300 578050 578050 578050 578050 4 561450 591050 577500 579250 579100 576100 574850 580250 570500 566700 580450 578300 578300 578300 578300 565300 575400 575400 575400 581300 581300 5 577600 591400 589600 589700 597800 597900 587600 596600 585700 588700 588700 595600 592700 586050 586050 599150 580850 579900 579900 582050 582050 6 629850 625950 607250 609600 601750 600300 594950 610000 597250 604750 591650 603050 601850 601750 607150 607150 604300 600050 600050 600050 600050 7 501300 519300 523900 507800 507900 507900 511900 498750 499300 499300 507300 511200 499100 503200 503200 503200 495750 495750 495750 495750 495750 8 552700 558400 552700 545850 553000 557700 551100 545050 556400 557200 560400 557400 552150 545050 556050 553700 553700 553700 546300 545400 542200 9 574300 599700 595400 586150 589650 587800 583400 594100 581350 601950 578600 578600 580350 570450 575450 575450 584300 584300 584300 584300 575050 10 581500 590500 601100 581100 581100 581100 591100 570050 570500 570500 585500 585400 575400 575400 575400 575400 567750 567750 567750 567750 567750 11 599900 620900 580700 601150 599150 618000 613000 610000 604000 608000 593800 584600 587700 587150 587150 586550 589050 589050 589050 602900 577300 12 593300 622700 618400 608150 611650 609800 603400 616100 601350 625950 598600 598600 600350 589450 596450 596450 605300 605300 605300 605300 595050 13 497300 507350 513050 498350 506900 507900 511550 503050 512500 507300 505300 507100 507100 507050 507050 507100 511200 511200 511200 494300 494300 14 542800 556300 548400 548400 553000 549400 542600 551000 551000 547600 551200 551200 545700 544400 536050 536050 536950 536950 531450 536400 522750 15 592750 602850 587250 600050 582450 590700 588450 586750 585850 589850 587200 588650 584950 586100 586100 586100 581850 581650 581650 584550 583350 16 571700 585550 582250 574750 568800 581100 585750 580250 586700 580500 582500 580400 580400 585150 585150 580400 585400 585400 585400 565300 565300 17 586600 614700 610000 605900 596000 599800 595800 595800 601000 592950 595600 591400 605800 592700 592700 592050 587050 580850 580850 576850 576850 18 614750 627350 608250 625050 604450 614700 614450 606750 608850 615850 610200 613650 609250 607300 607300 607300 601850 604650 604650 608550 607350 19 506300 519200 510900 515900 511900 511900 506900 510950 507200 503200 507300 507000 507050 507050 511300 503200 503200 503200 503200 503200 503200 20 553700 564000 547050 563000 554100 557850 554200 553000 560400 554050 554050 552600 546950 550550 550550 549150 549150 549150 545200 546200 546200 21 601350 602700 594450 589450 595250 593650 594650 597950 601200 596850 596850 593050 587100 587100 582450 589050 583250 583250 583250 583250 583250 22 583500 595200 584100 591100 581100 581100 581100 580150 585500 581300 580500 585200 589050 589050 590800 579500 579500 579500 579500 579500 579500 23 599900 605200 597800 602700 605000 605800 606500 606900 598900 596050 602900 590050 586950 586950 591150 591150 597800 596150 597150 586050 586050 24 625350 626700 617450 613450 618250 615650 616650 620950 626200 620850 620850 615050 609100 609100 603450 611050 604250 604250 604250 604250 604250

* 2 12 3 0 0 0 0 1 2 2 1 0 0 0 0 1 0 0 0 0 0

The line labeled by symbol of * represents the number of best solutions for each problem instance selected from all of the parameter combinations.

Table 3-9Performance comparisons in the three algorithms.

Weighted throughput

Result with bold is the best solution among the three algorithms.

(1) The MPSA_TP received 19 better solutions (out of 24) than the MGSA_TP. The average improvement between the 19 improved problems of the MPSA_TP comparing with the MGSA_TP in terms of weighted throughput is 2.2%.

(2) In the MPSA_TP algorithm group, the MPSA_TP received 22 better solutions (out of 24) than the MPSA_TP2. The weighted throughput ratio term, in the savings calculation of the MPSA_TP, indeed improves the solutions.

By computing the mean of the solutions generated by each algorithm in total weighted throughput for the twenty-four problems, we could compare the performances among these algorithms. To further analyze 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 3-10. Since the factors such as tightness of due date, setup time variation, and variation of contract/spot weight 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 8 results each.

In table 3-10, the MPSA_TP outperforms the MGSA_TP and the MPSA_TP2 on all nine groups. Therefore, we can say the performance of the MPSA_TP is better than the MGSA_TP and the MPSA_TP2 stably in varied situations.

Table 3-10Results in means with different problem characteristic groups.

Algorithm MGSA_TP MPSA_TP MPSA_TP2

Tightness DD= Tight 12 5704250 5804583* 5672333

Tightness DD= Loose 12 5758250 5853750* 5727583

Contracted workload= Low 8 5423375 5529000* 5370125

Contracted workload= Middle 8 5697500 5822375* 5685375

Contracted workload= High 8 6072875 6136125* 6044375

Setup time variation = Small 12 5768583 5839250* 5730000

Setup time variation = Large 12 5693917 5819083* 5669917

Variation of weight ratio= Small 12 5502792 5577333* 5462083

Variation of weight ratio= Large 12 5959708 6081000* 5937833

Result with * is best solution among the three algorithms.

12

Finally, we compared performances generated by the three algorithms, which is presented in Table 3-11, 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 3.2GHZPC.

Table 3-11Performance 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

The results, displayed in Table 3-11, indicate that the run times of the three algorithms are quite fast for solving those problems containing five machines and 120 jobs with different problem characteristics. The results also reveal that the MPSA_TP outperformed the other algorithms in terms of average rank among the three algorithms and number of problems receiving the best solutions. It may be explained by that the MPSA_TP creates a multiple of K machine schedules simultaneously at the first phase which may cause that job pairs with the same product type are assigned to the same machines easily. Therefore, the setup times incurred from different product types can be reduced and then the values of weighted throughput will be increased. However, in Table 3-11, the run times of MPSA_TP are larger than the other two algorithms. The most likely explanation for this is that the number of job candidates which need to be checked their probable insertions and be determined the appropriate positions for insertions is more than the others.

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