我有 n 个任务要在给定的时间段内安排。每个任务具有最早开始时间、最佳开始时间、最晚结束时间、持续时间和优先权权重。任务不能重叠。要求是安排尽可能多的任务，尽可能接近它们的最佳开始时间，并优先考虑权重较高的任务，而不是所有任务都可以容纳。我已经阅读了间隔调度和加权间隔调度，但我没有遇到过包含最佳开始时间概念的算法。谁能给我指出一个可以做到这一点的 Python 库，或者我可以自己编码的合适算法的描述？ 【应用是天文成像的调度，起止时间是每个天体在天空中升起落下的时间，最佳时间是天体处于最大高度的时间；权重是天文学家分配给每个物体的优先级]。
【问题讨论】：
标签： python-3.x scheduling
使用 CP Optimizer (https://pypi.org/project/docplex/) 应该很容易建模和求解。使用此工具，您只需将问题表述为组合优化问题，自动搜索就会解决它（如果实例不是太大，则证明解决方案的最优性）。
这是我如何在 CP Optimizer 中制定此问题的示例（我给出了一个小实例）：
N = range(20)
EST = [166, 157, 118, 254, 213, 73, 100, 38, 113, 84, 43, 257, 74, 246, 73, 242, 207, 223, 242, 122]
OST = [205, 195, 134, 256, 252, 157, 106, 44, 167, 84, 85, 266, 121, 256, 110, 310, 229, 262, 286, 162]
LET = [259, 233, 172, 267, 269, 175, 111, 48, 197, 91, 97, 292, 147, 289, 127, 313, 238, 319, 346, 177]
D = [9, 7, 5, 1, 9, 9, 4, 2, 3, 2, 9, 8, 3, 1, 7, 2, 3, 4, 4, 2]
W = [0.254, 0.811, 0.479, 0.27, 0.968, 0.036, 0.373, 0.887, 0.855, 0.61, 0.855, 0.708, 0.376, 0.434, 0.834, 0.978, 0.354, 0.4, 0.128, 0.208]
P = [1.46, 1.47, 9.29, 0.32, 5.43, 5.02, 8.1, 8.35, 7.79, 0.79, 3.86, 4.33, 7.16, 0.86, 5.82, 1.88, 2.16, 0.04, 9.37, 9.36]
# PROBLEM FORMULATION
from docplex.cp.model import *
model = CpoModel()
# Decision variables: x[i] is the ith observation
x = [ interval_var(size=D[i], optional=True) for i in N]
# Cost expression
cost = sum([start_eval(x[i], CpoSegmentedFunction((-W[i],0),[(OST[i],0,W[i])]), P[i]) for i in N])
# Objective: minimize cost
model.add(minimize(cost))
# Constraints
model.add([EST[i] <= start_of(x[i], EST[i])] for i in N)
model.add([end_of(x[i]) <= LET[i]] for i in N)
model.add(no_overlap(x))
# PROBLEM RESOLUTION
sol = model.solve(trace_log=True, LogPeriod=1000000, TimeLimit=30)
# DISPLAY OF SOLUTION
for i in N:
s = sol.get_var_solution(x[i])
if s.is_absent():
print('Task ' + str(i) + ' not scheduled')
else:
print('Task ' + str(i) + ' scheduled on [' + str(s.get_start()) + ',' + str(s.get_end()) + ')')
在数据中：
执行如下：
! ----------------------------------------------------------------------------
! Minimization problem - 21 variables, 41 constraints
! LogPeriod = 1000000
! Initial process time : 0.00s (0.00s extraction + 0.00s propagation)
! . Log search space : 83.4 (before), 83.4 (after)
! . Memory usage : 476.1 kB (before), 476.1 kB (after)
! Using parallel search with 8 workers.
! ----------------------------------------------------------------------------
! Best Branches Non-fixed W Branch decision
0 21 -
+ New bound is 0
! Using iterative diving.
! Using temporal relaxation.
* 92.86000 21 0.04s 1 (gap is 100.0%)
* 7.209000 176 0.04s 1 (gap is 100.0%)
* 7.208000 231 0.04s 1 (gap is 100.0%)
* 5.831000 284 0.04s 1 (gap is 100.0%)
* 2.114000 470 0.04s 1 (gap is 100.0%)
2.114000 470 1 1 F -
+ New bound is 2.113788 (gap is 0.01%)
! ----------------------------------------------------------------------------
! Search completed, 5 solutions found.
! Best objective : 2.114000 (optimal - effective tol. is 0.0002114)
! Best bound : 2.113788
! ----------------------------------------------------------------------------
! Number of branches : 171602
! Number of fails : 9172
! Total memory usage : 4.3 MB (4.2 MB CP Optimizer + 0.0 MB Concert)
! Time spent in solve : 0.06s (0.06s engine + 0.00s extraction)
! Search speed (br. / s) : 2860024.7
! ----------------------------------------------------------------------------
Task 0 scheduled on [205,214)
Task 1 scheduled on [195,202)
Task 2 scheduled on [134,139)
Task 3 not scheduled
Task 4 scheduled on [252,261)
Task 5 scheduled on [153,162)
Task 6 scheduled on [106,110)
Task 7 scheduled on [44,46)
Task 8 scheduled on [167,170)
Task 9 not scheduled
Task 10 scheduled on [85,94)
Task 11 scheduled on [266,274)
Task 12 scheduled on [121,124)
Task 13 not scheduled
Task 14 scheduled on [110,117)
Task 15 scheduled on [310,312)
Task 16 scheduled on [229,232)
Task 17 scheduled on [262,266)
Task 18 scheduled on [286,290)
Task 19 scheduled on [162,164)
【讨论】：