Introduction
Article | Multi-objective | Proactive scheduling | Reactive scheduling | Varying task processing time | Transfer time |
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[12] | ✕ | ✕ | ✕ | ✕ | ✕ |
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This article | ✓ | ✕ | ✓ | ✓ | ✓ |
Symbols | Description |
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i, j | Index of tasks, \(i, j = 0, 1, 2,\ldots , N, N + 1\) |
l | Index of skill type, \(l = 1, 2,\ldots , LN\) |
k | Index of information collection agent, \(k = 1, 2,\ldots , K\) |
t | Index of the time step |
N | The number of non-dummy tasks |
m | The number of objectives |
\(V=\{0, \ldots ,i, \ldots ,j, \ldots , N+1 \}\) | Task set, task 0 and N+1 is start and end task, respectively |
\(V^*\) | Incomplete tasks set after the disruption occurs |
\(R=\{1, \ldots , k, \ldots K\}\) | Agent or resource set |
K | The number of agents which can be used |
\(R^*\) | Agent or resource set after the disruption occurs |
\(L=\{1, \ldots , l, \ldots LN\}\) | Set of information collection skills |
\(P_j, P_j^{I} \) | The indirect and direct predecessor of task j |
\(S_j, S_j^{I}\) | The indirect and direct predecessor of task j |
\(F_j, F_j^*\) | The finish time of task j in the initial and repaired schedule |
\(ST_j, ST^*_j\) | The start time of task j in the initial and repaired schedule |
\(L_j \) | The skills required by task j |
\(L^k\) | Agent k’s skill capacity |
\(V_k\) | The set of tasks that can use resource k |
\(A_j^l\) | The area size in task j that needs skill l to perform |
\(A_j^{*l}\) | The area size in task j which needs skill l to perform |
\(R_j\) | The set of agents that can be used in task j |
\(RA_j^l\) | The set of agents that are allocated to task j to perform skill l |
\(r_l^{\rho }(t)\) | The total skill consumption of skill l in a given time t within the initial schedule |
\(r_l^{*\rho }(t)\) | The total skill consumption of skill l in a given time t within the repaired schedule |
\(MR_j\) | The maximum number of agents that can be used in task j |
Dt | The time point when the disruption happens |
\(\Delta _{ij}\) | The time cost for transfer agents from task i to task j |
UB | The maximum makespan for the information collection mission |
\(T=\{0, \ldots , t, \ldots , UB\}\) | Set of time steps |
\(ES_j, LS_j\) | Task j’s earliest and latest time to start |
\(p_j\) | Task j’s processing time |
\(p_j^{\max }\) | Task j’s maximum processing time |
\(\Gamma _j\) | The preparation time for the agents which are allocated to task j |
\(u_{kl}\) | The number of skill l that the agent k can provide |
\(c_{kl}\) | The cost for agent k to use skill l per time step |
\(q_{kl}\) | The quality contribution for agent k to use skill l per time step |
\(s^\prime _j\) | The actual start time of task j |
\(s_{jt}\) (decision variable) | Equals 1 if task j is started at time t, 0 otherwise |
\(x_{jklt}\) (decision variable) | Equals 1 if agent k is allocated to task j to perform skill l at time t, 0 otherwise |
\(z_{ijk}\) (decision variable) | Equals 1 if agent k is transferred from task i to task j, 0 otherwise |
Problem formulation
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Task 0 denotes the dummy start task, and task \(N + 1\) denotes the dummy end task.
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Preemption is allowed when disruption occurs.
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Each agent can only contribute one type of the skills it masters in a task.
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The cost and quality are pre-defined for each agent corresponding to each task.
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Only after all the allocated resources are transferred to the starting point of the task, the task can be started.
Notations
Name | nAct | K | L | NC | SF | RSS | MD |
---|---|---|---|---|---|---|---|
1 | 30 | 25 | 3 | 2.0 | 0.75 | [0.4, 0.5, 0.5] | 3 |
2 | 35 | 30 | 3 | 2.0 | 0.75 | [0.5, 0.4, 0.4] | 3 |
3 | 40 | 30 | 3 | 2.0 | 0.75 | [0.4, 0.4, 0.4] | 3 |
4 | 45 | 30 | 3 | 2.5 | 0.75 | [0.3, 0.4, 0.4] | 4 |
5 | 50 | 30 | 3 | 2.5 | 0.75 | [0.3, 0.3, 0.4] | 4 |
6 | 55 | 35 | 3 | 2.5 | 0.75 | [0.3, 0.4, 0.3] | 4 |
7 | 60 | 40 | 3 | 3.0 | 1.0 | [0.3, 0.3, 0.3] | 5 |
8 | 65 | 40 | 3 | 3.0 | 1.0 | [0.4, 0.4, 0.4] | 5 |
9 | 70 | 45 | 3 | 3.5 | 1.0 | [0.3, 0.4, 0.4] | 5 |
10 | 75 | 45 | 3 | 3.5 | 1.0 | [0.3, 0.3, 0.4] | 6 |
Algorithm | Parameter | Parameter level | ||
---|---|---|---|---|
Level 1 | Level 2 | Level 3 | ||
PPLS | W | 14 | 16 | 18 |
\(\beta \) | 0.95 | 0.97 | 0.99 | |
\(\epsilon \) | 0.30 | 0.35 | 0.40 | |
Mn | 15 | 16 | 17 | |
\(\sigma _{i}\) | 0.04 | 0.06 | 0.08 | |
MOTLA | SPo | 150 | 200 | 250 |
PCr | 0.15 | 0.25 | 0.35 | |
PMu | 0.02 | 0.04 | 0.06 | |
EMOIS | SPo | 150 | 200 | 250 |
PCr | 0.1 | 0.2 | 0.3 | |
PMu | 0.01 | 0.03 | 0.05 | |
MOIWO | PS | 150 | 200 | 250 |
Initial sigma | 0.1 | 0.2 | 0.3 | |
Final sigma | 0.01 | 0.03 | 0.05 | |
S-min | 2 | 3 | 4 | |
S-max | 6 | 8 | 10 |
Basic formulation for normal schedules
PPLS | EMOIS | MOIWO | MOTLA | |
---|---|---|---|---|
Instance-1 | 337.289 (19.225) | 254.545 (17.818)\(\wr \) | 392.454 (45.132)\(\dagger \) | 290.945 (33.75)\(\wr \) |
Instance-2 | 335.945 (31.579) | 347.225 (19.445)\(\approx \) | 388.568 (41.577)\(\dagger \) | 305.936 (26.31)\(\wr \) |
Instance-3 | 236.876 (17.055) | 253.771 (14.972)\(\dagger \) | 313.026 (36.624)\(\dagger \) | 244.106 (22.458)\(\approx \) |
Instance-4 | 614.829 (70.705) | 602.858 (39.789)\(\approx \) | 640.366 (49.949)\(\approx \) | 624.096 (52.424)\(\approx \) |
Instance-5 | 676.131 (66.261) | 704.918 (64.852)\(\approx \) | 818.136 (90.813)\(\dagger \) | 705.26 (54.305)\(\approx \) |
Instance-6 | 728.668 (82.34) | 841.264 (72.349)\(\dagger \) | 841.156 (58.04)\(\dagger \) | 855.684 (48.774)\(\dagger \) |
Instance-7 | 790.837 (61.685) | 999.644 (109.961)\(\dagger \) | 945.279 (72.786)\(\dagger \) | 1010.093 (54.545)\(\dagger \) |
Instance-8 | 1442.284 (128.363) | 1737.017 (166.754)\(\dagger \) | 1797.055 (122.2)\(\dagger \) | 1838.744 (187.552)\(\dagger \) |
Instance-9 | 1605.757 (176.633) | 1913.732 (130.134)\(\dagger \) | 2011.649 (120.699)\(\dagger \) | 1970.588 (100.5)\(\dagger \) |
Instance-10 | 1615.725 (93.712) | 1878.023 (161.51)\(\dagger \) | 1850.601 (142.496)\(\dagger \) | 2084.211 (106.295)\(\dagger \) |
\(\dagger \)/\(\wr \)/\(\approx \) | – | 6/1/3 | 9/0/1 | 5/2/3 |
PPLS | EMOIS | MOIWO | MOTLA | |
---|---|---|---|---|
Instance-1 | 296.889 (29.689) | 230.612 (20.294)\(\wr \) | 355.844 (40.922)\(\dagger \) | 271.226 (27.394)\(\wr \) |
Instance-2 | 294.533 (21.206) | 297.767 (21.141)\(\approx \) | 354.779 (21.287)\(\dagger \) | 281.94 (26.502)\(\approx \) |
Instance-3 | 368.881 (35.413) | 374.959 (34.496)\(\approx \) | 472.723 (31.2)\(\dagger \) | 347.678 (26.771)\(\approx \) |
Instance-4 | 656.057 (76.759) | 666.31 (44.643)\(\approx \) | 736.49 (58.919)\(\dagger \) | 688.686 (42.699)\(\approx \) |
Instance-5 | 1356.051 (115.264) | 1291.477 (78.78)\(\approx \) | 1398.77 (116.098)\(\approx \) | 1258.062 (124.548)\(\wr \) |
Instance-6 | 1216.13 (125.261) | 1376.378 (94.97)\(\dagger \) | 1507.744 (117.604)\(\dagger \) | 1462.239 (99.432)\(\dagger \) |
Instance-7 | 1626.66 (190.319) | 1779.951 (177.995)\(\dagger \) | 1832.006 (122.744)\(\dagger \) | 1865.787 (106.35)\(\dagger \) |
Instance-8 | 1432.015 (104.537) | 1683.063 (102.667)\(\dagger \) | 1853.527 (174.232)\(\dagger \) | 1760.33 (191.876)\(\dagger \) |
Instance-9 | 1764.954 (125.312) | 2174.526 (213.104)\(\dagger \) | 2177.77 (132.844)\(\dagger \) | 2048.478 (219.187)\(\dagger \) |
Instance-10 | 1665.306 (156.539) | 2004.089 (140.286)\(\dagger \) | 2067.712 (215.042)\(\dagger \) | 2146.815 (251.177)\(\dagger \) |
\(\dagger \)/\(\wr \)/\(\approx \) | – | 5/1/4 | 9/0/1 | 5/2/3 |
PPLS | EMOIS | MOIWO | MOTLA | |
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Instance-1 | 7.270 (0.785) | 7.377 (0.73)\(\approx \) | 8.019 (0.473)\(\wr \) | 8.498 (0.833)\(\wr \) |
Instance-2 | 7.366 (0.759) | 7.811 (0.43)\(\wr \) | 8.141 (0.464)\(\wr \) | 8.843 (0.964)\(\wr \) |
Instance-3 | 7.981 (0.495) | 7.273 (0.662)\(\dagger \) | 8.12 (0.438)\(\approx \) | 6.629 (0.703)\(\dagger \) |
Instance-4 | 9.850 (0.561) | 9.107 (0.61)\(\dagger \) | 11.56 (0.728)\(\wr \) | 10.767 (0.775)\(\wr \) |
Instance-5 | 9.593 (0.835) | 10.269 (0.657)\(\wr \) | 9.005 (1.009)\(\dagger \) | 10.916 (0.72)\(\wr \) |
Instance-6 | 15.97 (1.182) | 18.588 (1.487)\(\wr \) | 15.869 (1.063)\(\approx \) | 13.202 (0.858)\(\dagger \) |
Instance-7 | 13.163 (1.369) | 13.708 (0.836)\(\approx \) | 11.846 (0.746)\(\dagger \) | 14.147 (1.075)\(\wr \) |
Instance-8 | 14.001 (1.442) | 13.809 (1.16)\(\approx \) | 13.026 (0.808)\(\dagger \) | 13.875 (0.971)\(\approx \) |
Instance-9 | 12.776 (0.779) | 12.953 (1.282)\(\approx \) | 13.422 (1.248)\(\approx \) | 12.626 (1.503)\(\approx \) |
Instance-10 | 19.016 (1.864) | 20.507 (1.128)\(\wr \) | 21.545 (1.163)\(\wr \) | 15.781 (1.294)\(\dagger \) |
\(\dagger \)/\(\wr \)/\(\approx \) | – | 2/4/4 | 3/4/3 | 3/5/2 |
PPLS | EMOIS | MOIWO | MOTLA | |
---|---|---|---|---|
Instance-1 | 6.893 (0.503) | 9.479 (0.891)\(\wr \) | 5.735 (0.539)\(\dagger \) | 7.187 (0.446)\(\approx \) |
Instance-2 | 6.977 (0.733) | 8.059 (0.467)\(\wr \) | 5.695 (0.513)\(\dagger \) | 7.478 (0.733)\(\wr \) |
Instance-3 | 10.261 (0.759) | 11.148 (0.557)\(\wr \) | 9.851 (0.847)\(\approx \) | 11.91 (1.167)\(\wr \) |
Instance-4 | 10.347 (1.159) | 11.478 (0.872)\(\wr \) | 11.248 (0.686)\(\wr \) | 11.834 (1.325)\(\wr \) |
Instance-5 | 8.143 (0.708) | 7.998 (0.904)\(\approx \) | 8.713 (0.889)\(\wr \) | 7.432 (0.81)\(\dagger \) |
Instance-6 | 12.797 (1.472) | 13.581 (1.589)\(\wr \) | 11.711 (0.843)\(\dagger \) | 12.353 (1.136)\(\approx \) |
Instance-7 | 13.346 (0.867) | 11.29 (0.948)\(\dagger \) | 15.218 (0.791)\(\wr \) | 13.778 (1.571)\(\approx \) |
Instance-8 | 15.856 (1.823) | 12.449 (1.108)\(\dagger \) | 20.719 (2.445)\(\wr \) | 13.278 (1.407)\(\dagger \) |
Instance-9 | 13.683 (0.93) | 13.523 (1.339)\(\approx \) | 14.317 (0.802)\(\approx \) | 15.479 (1.099)\(\wr \) |
Instance-10 | 16.81 (1.328) | 17.773 (1.102)\(\approx \) | 13.782 (1.502)\(\dagger \) | 17.922 (0.914)\(\wr \) |
\(\dagger \)/\(\wr \)/\(\approx \) | – | 2/5/3 | 4/4/2 | 2/5/3 |
PPLS | EMOIS | MOIWO | MOTLA | |
---|---|---|---|---|
Instance-1 | 8.36e+12 (9.78e+11) | 8.20e+12 (7.46e+11)\(\approx \) | 7.46e+12 (5.37e+11)\(\dagger \) | 5.90e+12 (5.55e+11)\(\dagger \) |
Instance-2 | 7.66e+12 (8.81e+11) | 5.99e+12 (3.47e+11)\(\dagger \) | 7.38e+12 (4.14e+11)\(\approx \) | 5.92e+12 (4.14e+11)\(\dagger \) |
Instance-3 | 8.52e+12 (7.84e+11) | 6.26e+12 (7.51e+11)\(\dagger \) | 8.16e+12 (8.16e+11)\(\approx \) | 4.96e+12 (3.87e+11)\(\dagger \) |
Instance-4 | 1.65e+14 (1.72e+13) | 1.03e+14 (1.05e+13)\(\dagger \) | 1.46e+14 (1.27e+13)\(\dagger \) | 1.13e+14 (1.26e+13)\(\dagger \) |
Instance-5 | 4.41e+13 (3.00e+12) | 1.93e+13 (9.83e+11)\(\dagger \) | 4.71e+13 (2.59e+12)\(\wr \) | 4.36e+13 (3.05e+12)\(\approx \) |
Instance-6 | 1.69e+14 (2.01e+13) | 1.90e+14 (1.65e+13)\(\wr \) | 1.48e+14 (1.69e+13)\(\dagger \) | 1.34e+14 (1.50e+13)\(\dagger \) |
Instance-7 | 9.14e+14 (1.02e+14) | 7.96e+14 (4.93e+13)\(\dagger \) | 4.39e+14 (2.37e+13)\(\dagger \) | 9.23e+14 (6.83e+13)\(\approx \) |
Instance-8 | 2.43e+14 (2.21e+13) | 2.08e+14 (1.08e+13)\(\dagger \) | 1.93e+14 (9.64e+12)\(\dagger \) | 2.98e+14 (2.92e+13)\(\wr \) |
Instance-9 | 5.60e+14 (3.14e+13) | 5.23e+14 (4.71e+13)\(\dagger \) | 3.65e+14 (3.03e+13)\(\dagger \) | 4.49e+14 (5.38e+13)\(\dagger \) |
Instance-10 | 8.33e+14 (6.41e+13) | 8.25e+14 (8.00e+13)\(\approx \) | 6.76e+14 (5.68e+13)\(\dagger \) | 8.82e+14 (4.59e+13)\(\approx \) |
\(\dagger \)/\(\wr \)/\(\approx \) | – | 7/1/2 | 7/1/2 | 6/1/3 |
PPLS | EMOIS | MOIWO | MOTLA | |
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Instance-1 | 9.04e+12 (7.50e+11) | 7.59e+12 (7.74e+11)\(\dagger \) | 8.86e+12 (1.01e+12)\(\approx \) | 7.37e+12 (8.40e+11)\(\dagger \) |
Instance-2 | 9.12e+12 (5.84e+11) | 7.87e+12 (6.14e+11)\(\dagger \) | 8.84e+12 (9.28e+11)\(\approx \) | 7.51e+12 (5.86e+11)\(\dagger \) |
Instance-3 | 1.51e+13 (1.65e+12) | 1.39e+13 (1.36e+12)\(\dagger \) | 1.41e+13 (7.45e+11)\(\dagger \) | 1.20e+13 (1.20e+12)\(\dagger \) |
Instance-4 | 1.85e+14 (1.83e+13) | 1.87e+14 (1.19e+13)\(\approx \) | 1.38e+14 (1.31e+13)\(\dagger \) | 1.54e+14 (1.65e+13)\(\dagger \) |
Instance-5 | 3.46e+13 (3.15e+12) | 3.00e+13 (3.42e+12)\(\dagger \) | 3.33e+13 (2.33e+12)\(\approx \) | 2.43e+13 (2.58e+12)\(\dagger \) |
Instance-6 | 1.65e+14 (1.15e+13) | 1.80e+14 (1.91e+13)\(\wr \) | 1.59e+14 (1.37e+13)\(\approx \) | 1.67e+14 (1.59e+13)\(\approx \) |
Instance-7 | 9.58e+14 (9.58e+13) | 8.49e+14 (5.60e+13)\(\dagger \) | 7.75e+14 (5.97e+13)\(\dagger \) | 8.26e+14 (8.67e+13)\(\dagger \) |
Instance-8 | 1.78e+14 (1.80e+13) | 1.09e+14 (9.58e+12)\(\dagger \) | 1.11e+14 (1.19e+13)\(\dagger \) | 1.46e+14 (1.74e+13)\(\dagger \) |
Instance-9 | 3.29e+14 (3.25e+13) | 3.44e+14 (3.68e+13)\(\approx \) | 2.08e+14 (1.97e+13)\(\dagger \) | 2.88e+14 (2.94e+13)\(\dagger \) |
Instance-10 | 1.36e+15 (1.00e+14) | 1.28e+15 (6.65e+13)\(\approx \) | 1.12e+15 (5.60e+13)\(\dagger \) | 1.12e+15 (1.06e+14)\(\dagger \) |
\(\dagger \)/\(\wr \)/\(\approx \) | – | 6/1/3 | 6/0/4 | 9/0/1 |
Preempt-repeat condition | Preempt-resume condition | |||
---|---|---|---|---|
PPLS | PPLS-WN | PPLS | PPLS-WN | |
Instance-1 | 7.270 (0.785) | 7.119 (0.736) | 6.893 (0.503) | 7.366 (0.496) |
Instance-2 | 7.366 (0.759) | 7.541 (0.736) | 6.977 (0.733) | 6.549 (0.780) |
Instance-3 | 7.981 (0.495) | 7.645 (0.477) | 10.261 (0.759) | 9.649 (0.806) |
Instance-4 | 9.850 (0.561) | 10.296 (0.525) | 10.347 (1.159) | 9.669 (1.239) |
Instance-5 | 9.593 (0.835) | 9.219 (0.869) | 8.143 (0.708) | 8.395 (0.753) |
Instance-6 | 15.970 (1.182) | 14.922 (1.222) | 12.797 (1.472) | 11.937 (1.455) |
Instance-7 | 13.163 (1.369) | 12.479 (1.452) | 13.346 (0.867) | 14.085 (0.915) |
Instance-8 | 14.001 (1.442) | 14.472 (1.471) | 15.856 (1.823) | 14.851 (1.932) |
Instance-9 | 12.776 (0.779) | 12.496 (0.796) | 13.683 (0.930) | 12.750 (0.929) |
Instance-10 | 19.016 (1.864) | 18.116 (1.741) | 16.810 (1.328) | 15.794 (1.318) |
Preempt-repeat condition | Preempt-resume condition | |||
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PPLS | PPLS-WN | PPLS | PPLS-WN | |
Instance-1 | 8.36e+12 (9.78e+11) | 8.86e+12 (9.76e+11) | 9.04e+12 (7.50e+11) | 9.54e+12 (7.75e+11) |
Instance-2 | 7.66e+12 (8.81e+11) | 7.41e+12 (8.48e+11) | 9.12e+12 (5.84e+11) | 8.48e+12 (6.05e+11) |
Instance-3 | 8.52e+12 (7.84e+11) | 8.05e+12 (7.32e+11) | 1.51e+13 (1.65e+12) | 1.47e+13 (1.72e+12) |
Instance-4 | 1.65e+14 (1.72e+13) | 1.61e+14 (1.67e+13) | 1.85e+14 (1.83e+13) | 1.94e+14 (1.79e+13) |
Instance-5 | 4.41e+13 (3.00e+12) | 4.55e+13 (3.01e+12) | 3.46e+13 (3.15e+12) | 3.65e+13 (3.23e+12) |
Instance-6 | 1.69e+14 (2.01e+13) | 1.62e+14 (1.90e+13) | 1.65e+14 (1.15e+13) | 1.63e+14 (1.23e+13) |
Instance-7 | 9.14e+14 (1.02e+14) | 8.85e+14 (1.09e+14) | 9.58e+14 (9.58e+13) | 1.02e+15 (9.51e+13) |
Instance-8 | 2.43e+14 (2.21e+13) | 2.27e+14 (2.08e+13) | 1.78e+14 (1.80e+13) | 1.73e+14 (1.89e+13) |
Instance-9 | 5.60e+14 (3.14e+13) | 5.81e+14 (3.24e+13) | 3.29e+14 (3.25e+13) | 3.38e+14 (3.40e+13) |
Instance-10 | 8.33e+14 (6.41e+13) | 8.91e+14 (6.47e+13) | 1.36e+15 (1.00e+14) | 1.27e+15 (1.03e+14) |
Disruption recovery model
Preempt-repeat condition
Preempt-resume condition
Parallel pareto local search algorithm
Solution representation
Parallel pareto local search
Decomposition method
Framework of the proposed method
Individual local search
Acceptance criterion based on constraint handling
Acceptance criterion based on negative correlation
Experiment
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EMOIS [41], which uses an improved NSGA-II to solve multi-objective multi-skill RCPSP.
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MOIWO [42], in which a modified multi-objective invasive weeds optimization algorithm is proposed to solve multi-mode multi-skill RCPSP.
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MOTLA [43], which uses teaching–learning-based optimization algorithm to solve multi-objective multi-skill RCPSP.
Performance metric
Test instance
Parameters’ tuning
Performance evaluation
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The framework of PPLS is designed specifically to fit the information collection mission reactive scheduling problem’s characteristics.
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The proper design of the solution representation and decoding scheme contributes to reducing the search space.
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For the information collection mission reactive scheduling problem within a short time limit, the algorithm with stronger local search ability tends to have better performance, and the importance of global search ability is relatively weak.