Google OR Tools CP-SAT Solver - 调度问题，目标是在没有硬约束的情况下平均轮班分布（每个周期的最大/最小值）

Question

I am using Google ORTools using the Python wrapper to solve a nurse scheduling problem but I am having trouble finding a way to implement a constraint that attempts to evenly distribute worked shifts without using hard constraints.

As an example, I working with a number of weeks, distinct shifts, and employees. For illustration I add a very simple constraint which is that no employee can work on average more than the others for the full time period.

In the example below, one of many solutions with 5 weeks, 5 shifts, and 5 employees is to just have employee 0 work all the shifts in week 0, employee 1 all the shifts in week 1, etc.

What I want to add, however, is a constraint that maximizes the number of distinct weeks each employee works WITHOUT using a weekly constraint such as each employee can only work up to 1 shift per week.

A few things that I have attempted but have failed to get working: Create a binary grid indexed by employee and week with a 1 if the employee has at least 1 shift that week and 0 otherwise and maximize the total sum of the grid.

Use a hard constraint such that each employee can only work so many shifts in a given week. This is what I want to avoid, I would rather the solver consider this as an objective than a hard constraint.

My sample code is below:

import os
import math
import pandas as pd
from ortools.sat.python import cp_model

num_weeks = 5
num_shifts = 5
num_employees = 5

all_weeks = range(num_weeks)
all_shifts = range(num_shifts)
all_employees = range(num_employees)

model = cp_model.CpModel()
assignments = {}

#Calculate a maximum number of shifts to balance everyone out for the full period 
max_total_shifts = math.ceil((num_weeks*num_shifts)/num_employees)

#Create a space of new boolean variables where the value is 1 if the employee is working that shift in that week, else 0
for w in all_weeks:
    for s in all_shifts:
        for e in all_employees:
            assignments[(w,s,e)] = model.NewBoolVar('w%i-s%i-e%i' % (w,s,e) )
        model.AddAtMostOne(assignments[(w,s,e)] for e in all_employees)

#Add the max constraint
for e in all_employees:
    model.Add(sum(assignments[w,s,e] for w in all_weeks for s in all_shifts) <= max_total_shifts)

#Assign as many shifts as possible
model.Maximize( sum(assignments[(w,s,e)] for w in all_weeks for s in all_shifts for e in all_employees) )


#Solve the model
solver = cp_model.CpSolver()
status = solver.Solve(model)        
print(status)

#Using pandas, view the solution
solution = pd.DataFrame()
data = []
for i,field in enumerate(model._CpModel__model.variables):
    model._CpModel__model.solution_hint.vars.extend([i])
    model._CpModel__model.solution_hint.values.extend([solver._CpSolver__solution.solution[i]])
    if solver._CpSolver__solution.solution[i]==1:
        data.append( [field.name,solver._CpSolver__solution.solution[i] ])
        #print("{} has value {}".format(field.name,solver._CpSolver__solution.solution[i]))  #         

solution = pd.DataFrame(data)

Answer 1

Fairness is the most complex question in OR. You need to try to capture what you want with equations as sample as possible.

Std deviation, variance are not simple. Minimizing max(worked per person) - min(worked per person);is simple.

Good luck, this is a tough question to come up with a business acceptable definition of a good schedule.