# A Variable Block Insertion Heuristic for the Blocking Flowshop Scheduling Problem with Total Flowtime Criterion

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## Abstract

**:**

## 1. Introduction

## 2. Blocking Flow Shop Scheduling Problem

## 3. Variable Block Insertion Heuristic

#### 3.1. Initial Solution

**else**$x=20$. Another distinction between $\mathrm{P}\mathrm{F}\_\mathrm{N}\mathrm{E}\mathrm{H}\left(\mathrm{x}\right)$ and $\mathrm{t}\mathrm{P}\mathrm{F}\_\mathrm{N}\mathrm{E}\mathrm{H}\left(\mathrm{x}\right)$ is about how to choose the best one amongst x number of sequences. Since even the partial NEH implementation may result in an inferior solution when compared to $\mathrm{t}\mathrm{P}\mathrm{F}$ heuristic, we check both solutions and keep the better one. Note that in all figures throughout the paper, $f\left(\pi \right)$ and $r$ corresponds to the total flowtime (TFT) value of a sequence $\pi $ and a uniform random number between 0 and 1, respectively. The $\mathrm{t}\mathrm{P}\mathrm{F}\_\mathrm{N}\mathrm{E}\mathrm{H}\left(\mathrm{x}\right)$ heuristic is outlined in Figure 4.

#### 3.2. Block Insertion Move Procedure

#### 3.3. Variable Local Search

## 4. Parameter Tuning

#### 4.1. Parameter Tuning of PFT_NEX(x) Heuristic

#### 4.2. Parameter Tuning of VBIH Algorithm

## 5. Computational Results

**DABC_RCT in [41].**The DABC_RCT algorithm is a very efficient algorithm and has three phases. In the employed bee phase, the TNO procedure is employed with the VLS local search. In the onlooker bee phase, the path-relinking approach is employed to generate the onlooker bees. In the scout bee phase, HPF2 is used to generate the scout bees. We refer to [41] for the details. We have coded the DABC_RCT algorithm in Visual C+13 to have a fair comparison. Note that the DABC_RCT algorithm uses HPF2 heuristic as an initial solution. Since our PFT_NEH(x) heuristic is substantially better than HPF2 heuristic, we employ the PFT_NEH(x) heuristic with $\mu =0.35$ and $\delta =15$ as one of the solution in the population. The rest of the population individuals are constructed randomly as suggested in the DABC_RCT algorithm and we denote it as the DABC*_RCT algorithm to have a fair comparison. The same parameters are also used which are suggested in the DABC_RCT algorithm [41]. Note that fast fitness calculation is employed to accelerate the insertion and swap neighborhood structures in the VLS local search they employed in the TNO procedure.**IG_RIS algorithm in [47,54,55,56,57,58,59].**To be fair again, we employ PFT_NEH(x) heuristic with $\mu =0.35$ and $\delta =15$ as an initial solution in the IG_RIS algorithm. IG_RIS algorithm relies on the destruction and construction procedure, where $dS$number of jobs is removed from a solution and they are reinserted to the partial solution sequentially. Then RIS local search is applied to the solution obtained after destruction and construction procedure. Note that fast fitness calculation is employed to accelerate the RIS insertion local search as in this paper. It is also employed in the destruction and construction procedure, too.**VBIH algorithms in this paper.**Since the PFT_NEH(x) heuristics provide very diversified initial solutions, we run the VBIH algorithm with $\mu $ values with 0.25, 0.30, 0.35, 0.40, 0.55 and 0.60 with $\delta =15$. Then we denote them as VBIH1, VBIH2, VBIH3, VBIH4, VBIH5 and VBIH6. In addition, when the maximum block size is equal to 1, the VBIH algorithm becomes an iterated local search. In other words, the current solution is perturbed with several insertion moves and then the VLS local search is applied to the solution after perturbation. Then, the acceptance criterion is imposed to the solution obtained. We denote this variant of the VBIH algorithm as IVLS algorithm.

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Conflicts of Interest

## References

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$\mathit{\mu}$ Values and CPU (s) | |||||||||||||||
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n × m | HPF2 [41] | NHPF1 [40] | NHPF2 [40] | $0.25$ | CPU (s) | $0.30$ | CPU (s) | $0.35$ | CPU (s) | $0.40$ | CPU (s) | $0.55$ | CPU (s) | $0.60$ | CPU (s) |

20 × 5 | 4.038 | 2.928 | 3.059 | 1.368 | 0.002 | 1.190 | 0.000 | 1.158 | 0.003 | 1.014 | 0.003 | 1.264 | 0.000 | 1.190 | 0.002 |

20 × 10 | 3.156 | 2.719 | 2.340 | 1.069 | 0.000 | 1.228 | 0.003 | 1.005 | 0.002 | 1.154 | 0.000 | 1.175 | 0.000 | 1.005 | 0.003 |

20 × 20 | 3.989 | 2.857 | 2.766 | 1.011 | 0.002 | 0.942 | 0.003 | 1.016 | 0.003 | 1.063 | 0.003 | 1.009 | 0.002 | 1.047 | 0.003 |

50 × 5 | 3.929 | 3.903 | 3.528 | 3.029 | 0.022 | 2.775 | 0.023 | 2.774 | 0.024 | 3.204 | 0.025 | 2.881 | 0.022 | 2.915 | 0.025 |

50 × 10 | 3.664 | 4.191 | 3.665 | 2.718 | 0.039 | 2.651 | 0.039 | 2.608 | 0.041 | 2.671 | 0.039 | 2.655 | 0.041 | 2.871 | 0.041 |

50 × 20 | 5.318 | 4.398 | 4.237 | 2.453 | 0.078 | 2.409 | 0.077 | 2.274 | 0.073 | 2.310 | 0.075 | 2.210 | 0.080 | 2.258 | 0.078 |

100 × 5 | 3.816 | 3.757 | 3.668 | 2.989 | 0.116 | 2.743 | 0.114 | 2.702 | 0.125 | 2.692 | 0.116 | 2.822 | 0.116 | 2.913 | 0.120 |

100 × 10 | 4.087 | 4.450 | 3.964 | 2.611 | 0.211 | 2.783 | 0.216 | 2.598 | 0.213 | 2.569 | 0.213 | 2.755 | 0.223 | 2.863 | 0.217 |

100 × 20 | 5.554 | 4.428 | 4.539 | 2.223 | 0.431 | 2.030 | 0.434 | 2.125 | 0.433 | 2.064 | 0.433 | 2.353 | 0.434 | 2.207 | 0.436 |

200 × 10 | 2.362 | 2.505 | 1.915 | 1.057 | 1.730 | 0.944 | 1.740 | 1.053 | 1.737 | 1.070 | 1.736 | 1.160 | 1.731 | 1.336 | 1.736 |

200 × 20 | 2.811 | 2.676 | 2.478 | 0.777 | 3.553 | 0.669 | 3.580 | 0.889 | 3.594 | 0.780 | 3.552 | 1.000 | 3.552 | 1.122 | 3.555 |

500 × 20 | 1.595 | 1.464 | 1.533 | −0.177 | 2.352 | −0.158 | 2.402 | −0.111 | 2.358 | −0.082 | 2.349 | 0.066 | 2.350 | 0.048 | 2.352 |

200 × 5 | 2.394 | 2.260 | 1.936 | 1.400 | 0.917 | 1.292 | 0.920 | 1.264 | 0.917 | 1.314 | 0.919 | 1.727 | 0.917 | 1.833 | 0.920 |

500 × 5 | 1.191 | 1.330 | 1.027 | 0.545 | 0.755 | 0.356 | 0.758 | 0.317 | 0.750 | 0.475 | 0.775 | 0.825 | 0.761 | 0.916 | 0.758 |

500 × 10 | 1.398 | 1.399 | 1.307 | 0.272 | 1.194 | 0.247 | 1.195 | 0.353 | 1.189 | 0.248 | 1.188 | 0.477 | 1.191 | 0.564 | 1.191 |

Average | 3.287 | 3.018 | 2.797 | 1.556 | 0.760 | 1.473 | 0.767 | 1.468 | 0.764 | 1.503 | 0.762 | 1.625 | 0.761 | 1.673 | 0.762 |

$\mathit{S}\mathit{o}\mathit{u}\mathit{r}\mathit{c}\mathit{e}$ | $\mathit{D}\mathit{F}$ | $\mathit{S}\mathit{e}\mathit{q}\mathit{S}\mathit{S}$ | $\mathit{A}\mathit{d}\mathit{j}\mathit{S}\mathit{S}$ | $\mathit{A}\mathit{d}\mathit{j}\mathit{M}\mathit{S}$ | $\mathit{F}$ | $\mathit{p}$ |
---|---|---|---|---|---|---|

$b{S}_{max}$ | 3 | 0.095370 | 0.095370 | 0.031790 | 98.55 | 0.00 |

$m{S}_{max}$ | 2 | 0.019873 | 0.019873 | 0.009937 | 30.80 | 0.00 |

$tP$ | 4 | 0.000676 | 0.000676 | 0.000169 | 0.52 | 0.72 |

$b{S}_{max}*m{S}_{max}$ | 6 | 0.039930 | 0.039930 | 0.006655 | 20.63 | 0.00 |

$b{S}_{max}*tP$ | 12 | 0.002395 | 0.002395 | 0.000200 | 0.62 | 0.81 |

$m{S}_{max}*tP$ | 8 | 0.001585 | 0.001585 | 0.000198 | 0.61 | 0.76 |

$Error$ | 24 | 0.007742 | 0.007742 | 0.000323 | - | - |

$Total$ | 59 | 0.167569 | - | - | - | - |

n × m | IG_RIS | DABC_RCT | IVLS | DABC*_RCT | VBIH1 | VBIH2 | VBIH3 | VBIH4 | VBIH5 | VBIH6 |
---|---|---|---|---|---|---|---|---|---|---|

20 × 5 | 0.049 | 0.004 | 0.068 | 0.006 | 0.001 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

20 × 10 | 0.016 | 0.021 | 0.151 | 0.031 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

20 × 20 | 0.010 | 0.015 | 0.032 | 0.002 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

50 × 5 | 0.901 | 0.788 | 0.713 | 0.490 | 0.418 | 0.354 | 0.346 | 0.376 | 0.296 | 0.325 |

50 × 10 | 0.819 | 0.752 | 0.916 | 0.752 | 0.562 | 0.455 | 0.457 | 0.424 | 0.463 | 0.502 |

50 × 20 | 0.521 | 0.561 | 0.652 | 0.490 | 0.318 | 0.378 | 0.316 | 0.355 | 0.330 | 0.321 |

100 × 5 | 1.555 | 1.135 | 1.133 | 0.937 | 0.650 | 0.764 | 0.709 | 0.650 | 0.696 | 0.714 |

100 × 10 | 1.639 | 1.301 | 1.358 | 1.285 | 1.070 | 1.179 | 1.126 | 1.098 | 1.112 | 1.118 |

100 × 20 | 1.147 | 1.279 | 0.927 | 0.983 | 0.759 | 0.833 | 0.799 | 0.903 | 0.796 | 0.750 |

200 × 10 | 0.633 | 0.559 | 0.236 | 0.312 | 0.427 | 0.243 | 0.377 | 0.293 | 0.297 | 0.308 |

200 × 20 | 0.337 | 0.581 | 0.193 | 0.201 | 0.146 | 0.227 | 0.211 | 0.255 | 0.430 | 0.421 |

500 × 20 | −0.229 | 0.426 | −0.381 | −0.274 | −0.353 | −0.374 | −0.395 | −0.303 | −0.192 | −0.185 |

200 × 5 | 0.854 | 0.470 | 0.373 | 0.358 | 0.392 | 0.325 | 0.341 | 0.372 | 0.388 | 0.401 |

500 × 5 | 0.259 | 0.397 | −0.014 | −0.022 | 0.150 | −0.012 | −0.053 | −0.042 | 0.299 | 0.369 |

500 × 10 | 0.234 | 0.607 | −0.033 | 0.092 | 0.000 | −0.031 | 0.035 | −0.046 | 0.174 | 0.282 |

Average | 0.583 | 0.593 | 0.422 | 0.376 | 0.303 | 0.289 | 0.285 | 0.289 | 0.339 | 0.355 |

**Table 4.**Paired t-test for variable block insertion heuristic (VBIH) variants versus discrete artificial bee colony (DABC)*_RCT algorithm.

Algorithm vs. algorithm | 95% CI for Mean Difference | p-Value |
---|---|---|

IVLS − DABC*_RCT | (0.0041, 0.0875) | 0.032 |

VBIH1 − DABC*_RCT | (−0.1162, −0.0298) | 0.001 |

VBIH2 − DABC*_RCT | (−0.1242, −0.0486) | 0.000 |

VBIH3 − DABC*_RCT | (−0.1193, −0.0639) | 0.000 |

VBIH4 − DABC*_RCT | (−0.1258, −0.0474) | 0.000 |

VBIH5 − DABC*_RCT | (−0.0843, 0.0117) | 0.137 |

VBIH6 − DABC*_RCT | (−0.0671, 0.0252) | 0.371 |

BKS [41] | IVLS | VBIH1 | VBIH2 | VBIH3 | VBIH4 | VBIH5 | VBIH6 | DABC_RCT | DABC*_RCT |
---|---|---|---|---|---|---|---|---|---|

Problem Set 20 × 5 | |||||||||

14953 | 14953 | 14953 | 14953 | 14953 | 14953 | 14953 | 14953 | 14953 | 14953 |

16343 | 16349 | 16343 | 16343 | 16343 | 16343 | 16343 | 16343 | 16343 | 16343 |

14297 | 14297 | 14297 | 14297 | 14297 | 14297 | 14297 | 14297 | 14297 | 14297 |

16483 | 16483 | 16483 | 16483 | 16483 | 16483 | 16483 | 16483 | 16483 | 16483 |

14212 | 14212 | 14212 | 14212 | 14212 | 14212 | 14212 | 14212 | 14212 | 14212 |

14624 | 14624 | 14624 | 14624 | 14624 | 14624 | 14624 | 14624 | 14624 | 14624 |

14936 | 14938 | 14936 | 14936 | 14936 | 14936 | 14936 | 14936 | 14938 | 14938 |

15193 | 15240 | 15193 | 15193 | 15193 | 15193 | 15193 | 15193 | 15193 | 15193 |

15544 | 15544 | 15544 | 15544 | 15544 | 15544 | 15544 | 15544 | 15544 | 15544 |

14392 | 14392 | 14392 | 14392 | 14392 | 14392 | 14392 | 14392 | 14392 | 14392 |

Problem Set 20 × 10 | |||||||||

22358 | 22537 | 22358 | 22358 | 22358 | 22358 | 22358 | 22358 | 22358 | 22358 |

23881 | 23881 | 23881 | 23881 | 23881 | 23881 | 23881 | 23881 | 23881 | 23881 |

20873 | 20873 | 20873 | 20873 | 20873 | 20873 | 20873 | 20873 | 20873 | 20873 |

19916 | 20020 | 19916 | 19916 | 19916 | 19916 | 19916 | 19916 | 19916 | 19916 |

20196 | 20196 | 20196 | 20196 | 20196 | 20196 | 20196 | 20196 | 20196 | 20196 |

20126 | 20126 | 20126 | 20126 | 20126 | 20126 | 20126 | 20126 | 20126 | 20126 |

19471 | 19471 | 19471 | 19471 | 19471 | 19471 | 19471 | 19471 | 19471 | 19471 |

21330 | 21369 | 21330 | 21330 | 21330 | 21330 | 21330 | 21330 | 21330 | 21330 |

21585 | 21585 | 21585 | 21585 | 21585 | 21585 | 21585 | 21585 | 21585 | 21585 |

22582 | 22582 | 22582 | 22582 | 22582 | 22582 | 22582 | 22582 | 22582 | 22582 |

Problem Set 20 × 20 | |||||||||

34683 | 34683 | 34683 | 34683 | 34683 | 34683 | 34683 | 34683 | 34683 | 34683 |

32855 | 32855 | 32855 | 32855 | 32855 | 32855 | 32855 | 32855 | 32855 | 32855 |

34825 | 34825 | 34825 | 34825 | 34825 | 34825 | 34825 | 34825 | 34825 | 34825 |

33006 | 33006 | 33006 | 33006 | 33006 | 33006 | 33006 | 33006 | 33006 | 33006 |

35328 | 35328 | 35328 | 35328 | 35328 | 35328 | 35328 | 35328 | 35328 | 35328 |

33720 | 33720 | 33720 | 33720 | 33720 | 33720 | 33720 | 33720 | 33720 | 33720 |

33992 | 33992 | 33992 | 33992 | 33992 | 33992 | 33992 | 33992 | 33992 | 33992 |

33388 | 33388 | 33388 | 33388 | 33388 | 33388 | 33388 | 33388 | 33388 | 33388 |

34798 | 34798 | 34798 | 34798 | 34798 | 34798 | 34798 | 34798 | 34798 | 34798 |

33174 | 33174 | 33174 | 33174 | 33174 | 33174 | 33174 | 33174 | 33174 | 33174 |

Problem Set 50 × 5 | |||||||||

72672 | 72758 | 72672 | 72696 | 72672 | 72696 | 72758 | 72827 | 73135 | 72768 |

78140 | 78707 | 78254 | 78332 | 78181 | 78181 | 78181 | 78284 | 78327 | 78295 |

72913 | 73211 | 73096 | 73224 | 73101 | 73224 | 72913 | 72994 | 72913 | 73224 |

77399 | 77711 | 77513 | 77571 | 77547 | 77586 | 77547 | 77547 | 77582 | 77607 |

78353 | 78705 | 78627 | 78579 | 78544 | 78363 | 78511 | 78511 | 78767 | 78620 |

75402 | 75402 | 75661 | 75606 | 75475 | 75593 | 75402 | 75514 | 76122 | 75615 |

73842 | 74322 | 73952 | 74202 | 73952 | 73890 | 73952 | 73891 | 73954 | 73890 |

73442 | 73964 | 73945 | 73442 | 73834 | 73442 | 73549 | 73549 | 73858 | 73442 |

70871 | 71360 | 70905 | 70871 | 70871 | 70883 | 70883 | 70883 | 71096 | 71105 |

78729 | 79271 | 78773 | 78729 | 78729 | 78807 | 78729 | 78729 | 78773 | 79093 |

Problem Set 50 × 10 | |||||||||

99674 | 100508 | 100373 | 99674 | 100299 | 100059 | 99721 | 100410 | 99900 | 100325 |

95608 | 95669 | 95907 | 96157 | 95669 | 96047 | 95876 | 95876 | 96565 | 96367 |

91791 | 92760 | 91956 | 92090 | 91791 | 92090 | 92090 | 92276 | 92588 | 92524 |

98454 | 98767 | 98475 | 98689 | 98454 | 98454 | 98507 | 98507 | 98692 | 98576 |

98164 | 98286 | 98243 | 98164 | 98230 | 98164 | 98228 | 98228 | 98610 | 98228 |

97246 | 97826 | 97637 | 97779 | 97431 | 97530 | 97558 | 97333 | 98029 | 97625 |

99953 | 100142 | 100030 | 99965 | 99965 | 99953 | 99971 | 100116 | 100440 | 100584 |

98027 | 98231 | 98149 | 98271 | 98476 | 98436 | 98543 | 98270 | 98723 | 98521 |

96708 | 97248 | 96708 | 96996 | 96996 | 96708 | 97142 | 96708 | 96978 | 97634 |

98019 | 99012 | 98316 | 98316 | 98316 | 98316 | 98053 | 98053 | 98316 | 98362 |

Problem Set 50 × 20 | |||||||||

136865 | 137240 | 136881 | 137075 | 137075 | 136968 | 137005 | 137005 | 136958 | 137161 |

129958 | 130333 | 130115 | 129975 | 130292 | 130244 | 130248 | 129975 | 130176 | 130303 |

127617 | 128784 | 127617 | 127957 | 127617 | 128073 | 127617 | 127617 | 128033 | 128393 |

131889 | 132452 | 132270 | 132169 | 132270 | 132103 | 131943 | 131943 | 132283 | 132169 |

130967 | 131159 | 130979 | 131217 | 131233 | 130967 | 131196 | 130979 | 131351 | 131064 |

131760 | 132121 | 131985 | 131760 | 131760 | 131929 | 131926 | 131921 | 131593 | 132007 |

134217 | 134857 | 134222 | 134534 | 134572 | 134534 | 134726 | 134451 | 134715 | 134636 |

132990 | 133502 | 132990 | 133210 | 133210 | 133210 | 133309 | 133309 | 133526 | 133210 |

132599 | 132819 | 132757 | 132715 | 132901 | 132757 | 132599 | 132599 | 132599 | 133120 |

135710 | 136166 | 135985 | 136162 | 136224 | 136363 | 136248 | 136146 | 136483 | 136473 |

Problem Set 100 × 5 | |||||||||

288332 | 289888 | 288446 | 289216 | 288765 | 288854 | 290346 | 288904 | 289463 | 288807 |

280491 | 282897 | 281066 | 280743 | 280853 | 280073 | 280873 | 280929 | 282857 | 282563 |

276228 | 277904 | 275863 | 276229 | 276322 | 277751 | 276589 | 276695 | 278117 | 277968 |

259596 | 262968 | 261462 | 261867 | 261601 | 261985 | 261715 | 261231 | 263990 | 261478 |

273086 | 275571 | 274651 | 274335 | 274451 | 274690 | 274005 | 274817 | 274804 | 275092 |

267381 | 271534 | 269418 | 267899 | 270406 | 269506 | 270408 | 269194 | 268899 | 269356 |

274744 | 277150 | 275884 | 277009 | 277342 | 276656 | 275491 | 276609 | 277163 | 277535 |

269689 | 272776 | 271187 | 270939 | 270945 | 270774 | 270668 | 271001 | 271916 | 271719 |

284816 | 286683 | 285308 | 285238 | 284901 | 284652 | 284952 | 284755 | 287494 | 284856 |

282005 | 282659 | 282969 | 283292 | 282814 | 282366 | 282367 | 282719 | 283596 | 283939 |

Problem Set 100 × 10 | |||||||||

354083 | 357361 | 354586 | 355794 | 356308 | 354892 | 353321 | 354624 | 354570 | 356911 |

333379 | 335775 | 335738 | 336636 | 335905 | 335268 | 336469 | 336601 | 337164 | 336403 |

343957 | 346543 | 344337 | 345157 | 345524 | 345069 | 344824 | 345654 | 344863 | 345889 |

359259 | 362441 | 361621 | 363410 | 360537 | 361230 | 360709 | 359680 | 361328 | 364095 |

338537 | 339455 | 339573 | 339976 | 338941 | 341468 | 341261 | 340741 | 340771 | 341207 |

327254 | 331594 | 329327 | 328482 | 330769 | 328075 | 328693 | 329377 | 330296 | 329454 |

335366 | 339300 | 338219 | 337094 | 339001 | 338948 | 338091 | 338091 | 337997 | 338643 |

343174 | 346305 | 345286 | 344609 | 344905 | 346013 | 345217 | 343843 | 344417 | 344886 |

344563 | 357006 | 354676 | 357664 | 356781 | 356050 | 355165 | 354659 | 356177 | 356628 |

347845 | 349966 | 349546 | 348910 | 350290 | 349351 | 350879 | 350580 | 350674 | 350693 |

Problem Set 100 × 20 | |||||||||

425224 | 427753 | 427810 | 426032 | 427688 | 426582 | 427549 | 426845 | 427899 | 426093 |

435289 | 438146 | 436000 | 436495 | 437478 | 436409 | 437908 | 436205 | 439020 | 437380 |

430634 | 431419 | 432953 | 433663 | 431713 | 434502 | 431905 | 432067 | 435222 | 434241 |

432314 | 436203 | 435708 | 435999 | 435001 | 437256 | 435401 | 435880 | 437725 | 438501 |

426405 | 429524 | 428737 | 429044 | 428845 | 430388 | 428200 | 429321 | 429886 | 429472 |

430308 | 434216 | 432099 | 431680 | 432345 | 433041 | 432139 | 434523 | 434105 | 431202 |

436642 | 440171 | 440174 | 441664 | 438725 | 441037 | 437488 | 439062 | 440081 | 442024 |

440930 | 445900 | 443867 | 445170 | 445219 | 443812 | 443648 | 444357 | 444299 | 445169 |

432876 | 435264 | 434452 | 435075 | 434655 | 433631 | 434701 | 434806 | 437318 | 435536 |

437286 | 440819 | 440918 | 438439 | 440992 | 438621 | 440484 | 438530 | 443779 | 439693 |

Problem Set 200 × 10 | |||||||||

1281633 | 1281292 | 1286077 | 1283314 | 1281947 | 1281401 | 1280745 | 1282445 | 1285819 | 1281919 |

1283164 | 1279913 | 1285268 | 1279947 | 1280799 | 1280548 | 1280432 | 1281886 | 1279240 | 1280107 |

1277933 | 1280412 | 1281905 | 1279882 | 1282228 | 1282447 | 1281843 | 1284700 | 1282812 | 1276690 |

1271502 | 1275865 | 1280417 | 1280027 | 1274711 | 1273979 | 1274071 | 1277414 | 1278833 | 1280871 |

1275901 | 1282570 | 1276110 | 1286907 | 1279126 | 1280511 | 1275710 | 1279823 | 1277954 | 1285519 |

1251213 | 1252110 | 1258463 | 1248655 | 1252536 | 1254724 | 1252145 | 1248058 | 1258054 | 1244667 |

1304158 | 1307602 | 1303545 | 1311626 | 1305541 | 1305201 | 1306954 | 1302585 | 1309672 | 1309810 |

1298900 | 1296591 | 1301184 | 1303103 | 1297501 | 1295844 | 1302459 | 1299302 | 1296469 | 1302829 |

1277801 | 1270883 | 1278023 | 1273946 | 1270118 | 1277145 | 1278259 | 1277646 | 1278808 | 1272519 |

1273794 | 1281472 | 1279452 | 1284374 | 1281680 | 1278887 | 1282278 | 1282763 | 1280799 | 1283774 |

Problem Set 200 × 20 | |||||||||

1499623 | 1505301 | 1509409 | 1506809 | 1503861 | 1508265 | 1516354 | 1511393 | 1512033 | 1508941 |

1541253 | 1538208 | 1540131 | 1531179 | 1538989 | 1538626 | 1538036 | 1536971 | 1538890 | 1534481 |

1546279 | 1546915 | 1553963 | 1557771 | 1550237 | 1553555 | 1558080 | 1558445 | 1559908 | 1558470 |

1540822 | 1542961 | 1537852 | 1545112 | 1544472 | 1544429 | 1548607 | 1547062 | 1541447 | 1544097 |

1514600 | 1515692 | 1514609 | 1520799 | 1515470 | 1517899 | 1520051 | 1521375 | 1518989 | 1520058 |

1528885 | 1535006 | 1533907 | 1532007 | 1535114 | 1534306 | 1533721 | 1533357 | 1543380 | 1528532 |

1532090 | 1535822 | 1534243 | 1532903 | 1536909 | 1534423 | 1540592 | 1540200 | 1539144 | 1529204 |

1543229 | 1537885 | 1540539 | 1542855 | 1540221 | 1538906 | 1541537 | 1542151 | 1543159 | 1543637 |

1524293 | 1522771 | 1517701 | 1516868 | 1522958 | 1525068 | 1524214 | 1519558 | 1524550 | 1514130 |

1535329 | 1533492 | 1528995 | 1534456 | 1534830 | 1535504 | 1538711 | 1536245 | 1540038 | 1534682 |

Problem Set 500 × 20 | |||||||||

8719682 | 8706102 | 8701976 | 8693435 | 8705502 | 8691397 | 8704588 | 8720565 | 8730777 | 8702888 |

8849228 | 8823698 | 8804700 | 8813687 | 8825577 | 8816790 | 8820512 | 8825084 | 8908220 | 8814543 |

8789777 | 8745145 | 8746129 | 8742781 | 8728918 | 8750421 | 8759822 | 8746826 | 8811943 | 8739388 |

8828454 | 8791325 | 8807711 | 8793936 | 8793088 | 8795242 | 8807649 | 8803254 | 8862426 | 8795162 |

8796337 | 8735014 | 8722394 | 8741122 | 8736988 | 8755137 | 8781374 | 8756332 | 8791516 | 8774397 |

8837577 | 8804938 | 8821880 | 8793126 | 8803399 | 8820331 | 8809748 | 8824192 | 8870293 | 8791898 |

8729909 | 8718042 | 8734168 | 8734722 | 8719324 | 8737033 | 8750115 | 8748411 | 8802463 | 8733839 |

8800506 | 8766044 | 8756165 | 8773136 | 8772308 | 8774052 | 8761218 | 8787988 | 8796375 | 8772084 |

8782791 | 8739864 | 8751027 | 8727721 | 8743741 | 8743940 | 8761953 | 8747533 | 8799104 | 8773353 |

8849551 | 8791671 | 8802420 | 8780952 | 8788594 | 8796622 | 8810047 | 8818606 | 8878995 | 8805877 |

Problem Set 200 × 5 | |||||||||

1071652 | 1073055 | 1072760 | 1076925 | 1071889 | 1073065 | 1073348 | 1071170 | 1071705 | 1070790 |

1026640 | 1026510 | 1021433 | 1025826 | 1025310 | 1023515 | 1028051 | 1024432 | 1019431 | 1025138 |

1059120 | 1064728 | 1062714 | 1064025 | 1064244 | 1060982 | 1065879 | 1064569 | 1062759 | 1061449 |

1044074 | 1048420 | 1051225 | 1042391 | 1048350 | 1049298 | 1042292 | 1047066 | 1048212 | 1044776 |

1064274 | 1064019 | 1064175 | 1060847 | 1064649 | 1061081 | 1060213 | 1062069 | 1060420 | 1062216 |

1021482 | 1024893 | 1027578 | 1029903 | 1025409 | 1027670 | 1029104 | 1030626 | 1026561 | 1029891 |

1082018 | 1081107 | 1079945 | 1082921 | 1081033 | 1081780 | 1083320 | 1084464 | 1083668 | 1081968 |

1043921 | 1047490 | 1048609 | 1050141 | 1045691 | 1048150 | 1051041 | 1048321 | 1050936 | 1049380 |

1057482 | 1057673 | 1058438 | 1056355 | 1058281 | 1058369 | 1056063 | 1055705 | 1058199 | 1059175 |

1037496 | 1043719 | 1043777 | 1039727 | 1042310 | 1045628 | 1039183 | 1043266 | 1036938 | 1039695 |

Problem Set 500 × 5 | |||||||||

6389122 | 6375325 | 6381517 | 6371100 | 6371117 | 6366762 | 6397170 | 6387506 | 6403589 | 6369864 |

6415066 | 6413469 | 6433713 | 6392620 | 6400361 | 6396807 | 6432279 | 6436552 | 6421048 | 6392856 |

6460745 | 6426771 | 6426591 | 6435399 | 6434882 | 6434628 | 6440953 | 6454047 | 6478507 | 6430821 |

6334201 | 6303859 | 6323236 | 6305175 | 6306089 | 6318146 | 6337465 | 6334555 | 6364065 | 6318682 |

6373873 | 6383164 | 6392640 | 6369007 | 6383774 | 6355801 | 6408507 | 6413732 | 6397351 | 6369219 |

6282522 | 6275452 | 6281428 | 6283594 | 6277932 | 6274362 | 6292695 | 6301826 | 6302507 | 6283635 |

6244926 | 6262136 | 6262957 | 6262423 | 6260089 | 6261444 | 6285376 | 6273743 | 6261620 | 6258574 |

6352627 | 6367281 | 6395544 | 6370417 | 6370566 | 6377367 | 6376438 | 6392365 | 6350755 | 6366156 |

6328390 | 6335967 | 6342425 | 6336154 | 6335220 | 6343154 | 6342796 | 6359365 | 6354617 | 6330549 |

6309180 | 6309639 | 6314591 | 6297997 | 6307224 | 6300714 | 6320912 | 6323148 | 6346828 | 6307074 |

Problem Set 500 × 10 | |||||||||

7552404 | 7514159 | 7534854 | 7522904 | 7519846 | 7523259 | 7533739 | 7549934 | 7577869 | 7541901 |

7665025 | 7632382 | 7633377 | 7649655 | 7635561 | 7622243 | 7642501 | 7652010 | 7658541 | 7643615 |

7626599 | 7590037 | 7599850 | 7580415 | 7588202 | 7590780 | 7622675 | 7622445 | 7652497 | 7603273 |

7626405 | 7618385 | 7600996 | 7633161 | 7619058 | 7615308 | 7645654 | 7623706 | 7635679 | 7635154 |

7479900 | 7484025 | 7468087 | 7472600 | 7468703 | 7478446 | 7464923 | 7496738 | 7504574 | 7476013 |

7537299 | 7548071 | 7546071 | 7563456 | 7551273 | 7551039 | 7572887 | 7566574 | 7586150 | 7562912 |

7510712 | 7505921 | 7502693 | 7490848 | 7504096 | 7482595 | 7478959 | 7514666 | 7534649 | 7491561 |

7562013 | 7577902 | 7599263 | 7598437 | 7588036 | 7598947 | 7599345 | 7635525 | 7607737 | 7599718 |

7550242 | 7538219 | 7537118 | 7533874 | 7539127 | 7547730 | 7577922 | 7574227 | 7581486 | 7536618 |

7549596 | 7577156 | 7596351 | 7588889 | 7580750 | 7562898 | 7611269 | 7589683 | 7662823 | 7594287 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tasgetiren, M.F.; Pan, Q.-K.; Kizilay, D.; Gao, K.
A Variable Block Insertion Heuristic for the Blocking Flowshop Scheduling Problem with Total Flowtime Criterion. *Algorithms* **2016**, *9*, 71.
https://doi.org/10.3390/a9040071

**AMA Style**

Tasgetiren MF, Pan Q-K, Kizilay D, Gao K.
A Variable Block Insertion Heuristic for the Blocking Flowshop Scheduling Problem with Total Flowtime Criterion. *Algorithms*. 2016; 9(4):71.
https://doi.org/10.3390/a9040071

**Chicago/Turabian Style**

Tasgetiren, Mehmet Fatih, Quan-Ke Pan, Damla Kizilay, and Kaizhou Gao.
2016. "A Variable Block Insertion Heuristic for the Blocking Flowshop Scheduling Problem with Total Flowtime Criterion" *Algorithms* 9, no. 4: 71.
https://doi.org/10.3390/a9040071