使用OR-Tools连续改变“护士计划”问题？

问题描述

我正在使用OR-Tools尝试解决与Nurse Scheduling problem类似的问题，但是每个护士（或我的学生）的轮班时间不同。我目前的方法是将班次划分为最小公分母的块（在我的情况下为5分钟），并要求：

1. 如果当天安排了学生，则必须安排适当数量的区块
2. 这些块必须是连续的

我还在限制学生什么时候上课。我下面的当前代码似乎可以正常工作，但是我发现找不到解决方案的奇怪情况。例如，在下面的简单示例中，如果我将Alice约束为在块3和4（从零开始索引）期间不可用，则即使存在解决方案，求解器也会失败。任何其他约束集（采用可行的解决方案）都可以工作并符合上述要求。在定义约束时我做错什么了吗？有没有更好的方法来实现这一目标？我见过this example，但是我对此并不陌生，还不太了解发生了什么，因此希望实现一些更简单的方法。

from ortools.sat.python import cp_model

#assign student[0] to be a placeholder,stands for "no students during this time"
student_names = ['none','Alice'] 

#number of sessions per student; i.e. Alice needs 1 session
student_sessions = [0,1] 

#how many 5-minute blocks for each student's sessions; i.e. Alice's sessions are 3x5 minutes long
student_blocks = [0,3] 

#constraints,in the form (student,day,block); i.e. Alice can't be scheduled in block 3 on day 0
constraints = [(1,3)] 

#define the length of the day
num_blocks = 6

#define the number of days in the week
num_days = 1

num_students = len(student_sessions) 
print('Scheduling',num_students-1,'students') #recall that student[0] is just a placeholder
students = range(num_students)
blocks = range(num_blocks)
days = range(num_days)

#create the model
model = cp_model.CpModel()

#create the primary variable 'slots',a list of booleans with coordinates given by student,block
slots = {}
for s in students:
    for d in days:
        for b in blocks:
            slots[(s,d,b)] = model.NewBoolVar('block_s%id%ib%i' % (s,b))

#no more than 1 student per block
for d in days:
    for b in blocks:
        model.Add(sum(slots[(s,b)] for s in students)==1)

#require the total number of blocks per student = # sessions x blocks per session
for s in students[1:]: #we always skip over constraints for student[0]; they can fill in everything else
    total = 0
    for d in days:
        for b in blocks:
            total += slots[(s,b)]
    model.Add(total == int(student_sessions[s]*student_blocks[s]))

#each day,student should either have no blocks,or a number of blocks equaling their session length
for d in days:
    for s in students[1:]:
        blocked = model.NewBoolVar('blocking')
        noblock = model.NewBoolVar('noblock')
        model.Add(sum(slots[(s,b)] for b in blocks)==student_blocks[s]).OnlyEnforceIf(blocked)
        model.Add(sum(slots[(s,b)] for b in blocks)==0).OnlyEnforceIf(noblock)
        model.AddBoolOr([blocked,noblock])

# require session continuity,i.e. all a students blocks should be sequential,not spread out over a day
for d in days:
    for s in students[1:]:
        for b in blocks[:-(student_blocks[s])]:
            cont = model.NewBoolVar('cont')
            startBlock = model.NewBoolVar('startBlock')
            model.Add(sum([slots[(s,b)]])==0).OnlyEnforceIf(startBlock)
            model.Add(sum(slots[(s,b+i)] for i in range(student_blocks[s]))==student_blocks[s]).OnlyEnforceIf(cont)
            model.AddBoolXOr([startBlock,cont])

for ct in constraints:
    model.Add(slots[ct]==0)

solver = cp_model.CpSolver()
solver.Solve(model)
for d in days:
    print('Day',d)
    for b in blocks:
        for s in students:
            if solver.Value(slots[(s,b)])==1:
                print(student_names[s],'in block',b)

解决方法

下面的代码结合了Stradivari的建议，并且似乎有效：

from ortools.sat.python import cp_model

def negated_bounded_span(works,start,length):
    """Filters an isolated sub-sequence of variables assined to True.
  Extract the span of Boolean variables [start,start + length),negate them,and if there is variables to the left/right of this span,surround the span by
  them in non negated form.
  Args:
    works: a list of variables to extract the span from.
    start: the start to the span.
    length: the length of the span.
  Returns:
    a list of variables which conjunction will be false if the sub-list is
    assigned to True,and correctly bounded by variables assigned to False,or by the start or end of works.
  """
    sequence = []
    # Left border (start of works,or works[start - 1])
    if start > 0:
        sequence.append(works[start - 1])
    for i in range(length):
        sequence.append(works[start + i].Not())
    # Right border (end of works or works[start + length])
    if start + length < len(works):
        sequence.append(works[start + length])
    return sequence

#assign student[0] to be a placeholder,stands for "no students during this time"
student_names = ['none','Alice'] 

#number of sessions per student; i.e. Alice needs 1 session
student_sessions = [0,1] 

#how many 5-minute blocks for each student's sessions; i.e. Alice's sessions are 3x5 minutes long
student_blocks = [0,3] 

#constraints,in the form (student,day,block); i.e. Alice can't be scheduled in block 3 on day 0
constraints = [(1,5)] 

#define the length of the day
num_blocks = 6

#define the number of days in the week
num_days = 1

num_students = len(student_sessions) 
print('Scheduling',num_students-1,'students') #recall that student[0] is just a placeholder
students = range(num_students)
blocks = range(num_blocks)
days = range(num_days)

#create the model
model = cp_model.CpModel()

#create the primary variable 'slots',a list of booleans with coordinates given by student,block
slots = {}
for s in students:
    for d in days:
        for b in blocks:
            slots[(s,d,b)] = model.NewBoolVar('block_s%id%ib%i' % (s,b))

#no more than 1 student per block
for d in days:
    for b in blocks:
        model.Add(sum(slots[(s,b)] for s in students)==1)

#require the total number of blocks per student = # sessions x blocks per session
for s in students[1:]: #we always skip over constraints for student[0]; they can fill in everything else
    total = 0
    for d in days:
        for b in blocks:
            total += slots[(s,b)]
    model.Add(total == int(student_sessions[s]*student_blocks[s]))

#each day,student should either have no blocks,or a number of blocks equaling their session length
for d in days:
    for s in students[1:]:
        blocked = model.NewBoolVar('blocking')
        noblock = model.NewBoolVar('noblock')
        model.Add(sum(slots[(s,b)] for b in blocks)==student_blocks[s]).OnlyEnforceIf(blocked)
        model.Add(sum(slots[(s,b)] for b in blocks)==0).OnlyEnforceIf(noblock)
        model.AddBoolOr([blocked,noblock])

#require sessions be the correct length
for s in students[1:]:
    for d in days:
        sts = [slots[(s,b)] for b in blocks]
        for length in range(1,student_blocks[s]):
                for start in range(len(sts) - length + 1):
                    model.AddBoolOr(negated_bounded_span(sts,length))

for ct in constraints:
    model.Add(slots[ct]==0)

solver = cp_model.CpSolver()
solver.Solve(model)
for d in days:
    print('Day',d)
    for b in blocks:
        for s in students:
            if solver.Value(slots[(s,b)])==1:
                print(student_names[s],'in block',b)