nurses.py¶
This example solves the problem of finding an optimal assignment of nurses to shifts.
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# Source file provided under Apache License, Version 2.0, January 2004,
# http://www.apache.org/licenses/
# (c) Copyright IBM Corp. 2015, 2018
# --------------------------------------------------------------------------
from collections import namedtuple
from docplex.mp.model import Model
from docplex.util.environment import get_environment
# ----------------------------------------------------------------------------
# Initialize the problem data
# ----------------------------------------------------------------------------
# utility to convert a weekday string to an index in 0..6
_all_days = ["monday",
"tuesday",
"wednesday",
"thursday",
"friday",
"saturday",
"sunday"]
def day_to_day_week(day):
day_map = {day: d for d, day in enumerate(_all_days)}
return day_map[day.lower()]
TWorkRules = namedtuple("TWorkRules", ["work_time_max"])
TVacation = namedtuple("TVacation", ["nurse", "day"])
TNursePair = namedtuple("TNursePair", ["firstNurse", "secondNurse"])
TSkillRequirement = namedtuple("TSkillRequirement", ["department", "skill", "required"])
NURSES = [("Anne", 11, 1, 25),
("Bethanie", 4, 5, 28),
("Betsy", 2, 2, 17),
("Cathy", 2, 2, 17),
("Cecilia", 9, 5, 38),
("Chris", 11, 4, 38),
("Cindy", 5, 2, 21),
("David", 1, 2, 15),
("Debbie", 7, 2, 24),
("Dee", 3, 3, 21),
("Gloria", 8, 2, 25),
("Isabelle", 3, 1, 16),
("Jane", 3, 4, 23),
("Janelle", 4, 3, 22),
("Janice", 2, 2, 17),
("Jemma", 2, 4, 22),
("Joan", 5, 3, 24),
("Joyce", 8, 3, 29),
("Jude", 4, 3, 22),
("Julie", 6, 2, 22),
("Juliet", 7, 4, 31),
("Kate", 5, 3, 24),
("Nancy", 8, 4, 32),
("Nathalie", 9, 5, 38),
("Nicole", 0, 2, 14),
("Patricia", 1, 1, 13),
("Patrick", 6, 1, 19),
("Roberta", 3, 5, 26),
("Suzanne", 5, 1, 18),
("Vickie", 7, 1, 20),
("Wendie", 5, 2, 21),
("Zoe", 8, 3, 29)
]
SHIFTS = [("Emergency", "monday", 2, 8, 3, 5),
("Emergency", "monday", 8, 12, 4, 7),
("Emergency", "monday", 12, 18, 2, 5),
("Emergency", "monday", 18, 2, 3, 7),
("Consultation", "monday", 8, 12, 10, 13),
("Consultation", "monday", 12, 18, 8, 12),
("Cardiac_Care", "monday", 8, 12, 10, 13),
("Cardiac_Care", "monday", 12, 18, 8, 12),
("Emergency", "tuesday", 8, 12, 4, 7),
("Emergency", "tuesday", 12, 18, 2, 5),
("Emergency", "tuesday", 18, 2, 3, 7),
("Consultation", "tuesday", 8, 12, 10, 13),
("Consultation", "tuesday", 12, 18, 8, 12),
("Cardiac_Care", "tuesday", 8, 12, 4, 7),
("Cardiac_Care", "tuesday", 12, 18, 2, 5),
("Cardiac_Care", "tuesday", 18, 2, 3, 7),
("Emergency", "wednesday", 2, 8, 3, 5),
("Emergency", "wednesday", 8, 12, 4, 7),
("Emergency", "wednesday", 12, 18, 2, 5),
("Emergency", "wednesday", 18, 2, 3, 7),
("Consultation", "wednesday", 8, 12, 10, 13),
("Consultation", "wednesday", 12, 18, 8, 12),
("Emergency", "thursday", 2, 8, 3, 5),
("Emergency", "thursday", 8, 12, 4, 7),
("Emergency", "thursday", 12, 18, 2, 5),
("Emergency", "thursday", 18, 2, 3, 7),
("Consultation", "thursday", 8, 12, 10, 13),
("Consultation", "thursday", 12, 18, 8, 12),
("Emergency", "friday", 2, 8, 3, 5),
("Emergency", "friday", 8, 12, 4, 7),
("Emergency", "friday", 12, 18, 2, 5),
("Emergency", "friday", 18, 2, 3, 7),
("Consultation", "friday", 8, 12, 10, 13),
("Consultation", "friday", 12, 18, 8, 12),
("Emergency", "saturday", 2, 12, 5, 7),
("Emergency", "saturday", 12, 20, 7, 9),
("Emergency", "saturday", 20, 2, 12, 12),
("Emergency", "sunday", 2, 12, 5, 7),
("Emergency", "sunday", 12, 20, 7, 9),
("Emergency", "sunday", 20, 2, 12, 12),
("Geriatrics", "sunday", 8, 10, 2, 5)]
NURSE_SKILLS = {"Anne": ["Anaesthesiology", "Oncology", "Pediatrics"],
"Betsy": ["Cardiac_Care"],
"Cathy": ["Anaesthesiology"],
"Cecilia": ["Anaesthesiology", "Oncology", "Pediatrics"],
"Chris": ["Cardiac_Care", "Oncology", "Geriatrics"],
"Gloria": ["Pediatrics"], "Jemma": ["Cardiac_Care"],
"Joyce": ["Anaesthesiology", "Pediatrics"],
"Julie": ["Geriatrics"], "Juliet": ["Pediatrics"],
"Kate": ["Pediatrics"], "Nancy": ["Cardiac_Care"],
"Nathalie": ["Anaesthesiology", "Geriatrics"],
"Patrick": ["Oncology"], "Suzanne": ["Pediatrics"],
"Wendie": ["Geriatrics"],
"Zoe": ["Cardiac_Care"]
}
VACATIONS = [("Anne", "friday"),
("Anne", "sunday"),
("Cathy", "thursday"),
("Cathy", "tuesday"),
("Joan", "thursday"),
("Joan", "saturday"),
("Juliet", "monday"),
("Juliet", "tuesday"),
("Juliet", "thursday"),
("Nathalie", "sunday"),
("Nathalie", "thursday"),
("Isabelle", "monday"),
("Isabelle", "thursday"),
("Patricia", "saturday"),
("Patricia", "wednesday"),
("Nicole", "friday"),
("Nicole", "wednesday"),
("Jude", "tuesday"),
("Jude", "friday"),
("Debbie", "saturday"),
("Debbie", "wednesday"),
("Joyce", "sunday"),
("Joyce", "thursday"),
("Chris", "thursday"),
("Chris", "tuesday"),
("Cecilia", "friday"),
("Cecilia", "wednesday"),
("Patrick", "saturday"),
("Patrick", "sunday"),
("Cindy", "sunday"),
("Dee", "tuesday"),
("Dee", "friday"),
("Jemma", "friday"),
("Jemma", "wednesday"),
("Bethanie", "wednesday"),
("Bethanie", "tuesday"),
("Betsy", "monday"),
("Betsy", "thursday"),
("David", "monday"),
("Gloria", "monday"),
("Jane", "saturday"),
("Jane", "sunday"),
("Janelle", "wednesday"),
("Janelle", "friday"),
("Julie", "sunday"),
("Kate", "tuesday"),
("Kate", "monday"),
("Nancy", "sunday"),
("Roberta", "friday"),
("Roberta", "saturday"),
("Janice", "tuesday"),
("Janice", "friday"),
("Suzanne", "monday"),
("Vickie", "wednesday"),
("Vickie", "friday"),
("Wendie", "thursday"),
("Wendie", "saturday"),
("Zoe", "saturday"),
("Zoe", "sunday")]
NURSE_ASSOCIATIONS = [("Isabelle", "Dee"),
("Anne", "Patrick")]
NURSE_INCOMPATIBILITIES = [("Patricia", "Patrick"),
("Janice", "Wendie"),
("Suzanne", "Betsy"),
("Janelle", "Jane"),
("Gloria", "David"),
("Dee", "Jemma"),
("Bethanie", "Dee"),
("Roberta", "Zoe"),
("Nicole", "Patricia"),
("Vickie", "Dee"),
("Joan", "Anne")
]
SKILL_REQUIREMENTS = [("Emergency", "Cardiac_Care", 1)]
DEFAULT_WORK_RULES = TWorkRules(40)
# ----------------------------------------------------------------------------
# Prepare the data for modeling
# ----------------------------------------------------------------------------
# subclass the namedtuple to refine the str() method as the nurse's name
class TNurse(namedtuple("TNurse1", ["name", "seniority", "qualification", "pay_rate"])):
def __str__(self):
return self.name
# specialized namedtuple to redefine its str() method
class TShift(namedtuple("TShift",
["department", "day", "start_time", "end_time", "min_requirement", "max_requirement"])):
def __str__(self):
# keep first two characters in department, uppercase
dept2 = self.department[0:4].upper()
# keep 3 days of weekday
dayname = self.day[0:3]
return '{}_{}_{:02d}'.format(dept2, dayname, self.start_time).replace(" ", "_")
class ShiftActivity(object):
@staticmethod
def to_abstime(day_index, time_of_day):
""" Convert a pair (day_index, time) into a number of hours since Monday 00:00
:param day_index: The index of the day from 1 to 7 (Monday is 1).
:param time_of_day: An integer number of hours.
:return:
"""
time = 24 * (day_index - 1)
time += time_of_day
return time
def __init__(self, weekday, start_time_of_day, end_time_of_day):
assert (start_time_of_day >= 0)
assert (start_time_of_day <= 24)
assert (end_time_of_day >= 0)
assert (end_time_of_day <= 24)
self._weekday = weekday
self._start_time_of_day = start_time_of_day
self._end_time_of_day = end_time_of_day
# conversion to absolute time.
start_day_index = day_to_day_week(self._weekday)
self.start_time = self.to_abstime(start_day_index, start_time_of_day)
end_day_index = start_day_index if end_time_of_day > start_time_of_day else start_day_index + 1
self.end_time = self.to_abstime(end_day_index, end_time_of_day)
assert self.end_time > self.start_time
@property
def duration(self):
return self.end_time - self.start_time
def overlaps(self, other_shift):
if not isinstance(other_shift, ShiftActivity):
return False
else:
return other_shift.end_time > self.start_time and other_shift.start_time < self.end_time
def solve(model, **kwargs):
# Here, we set the number of threads for CPLEX to 2 and set the time limit to 2mins.
model.parameters.threads = 2
model.parameters.timelimit = 120 # nurse should not take more than that !
sol = model.solve(log_output=True, **kwargs)
if sol is not None:
print("solution for a cost of {}".format(model.objective_value))
print_information(model)
print_solution(model)
return model.objective_value
else:
print("* model is infeasible")
return None
def load_data(model, shifts_, nurses_, nurse_skills, vacations_=None,
nurse_associations_=None, nurse_imcompatibilities_=None, verbose=True):
""" Usage: load_data(shifts, nurses, nurse_skills, vacations) """
model.number_of_overlaps = 0
model.work_rules = DEFAULT_WORK_RULES
model.shifts = [TShift(*shift_row) for shift_row in shifts_]
model.nurses = [TNurse(*nurse_row) for nurse_row in nurses_]
model.skill_requirements = SKILL_REQUIREMENTS
model.nurse_skills = nurse_skills
# transactional data
model.vacations = [TVacation(*vacation_row) for vacation_row in vacations_] if vacations_ else []
model.nurse_associations = [TNursePair(*npr) for npr in nurse_associations_]\
if nurse_associations_ else []
model.nurse_incompatibilities = [TNursePair(*npr) for npr in nurse_imcompatibilities_]\
if nurse_imcompatibilities_ else []
# computed
model.departments = set(sh.department for sh in model.shifts)
if verbose:
print('#nurses: {0}'.format(len(model.nurses)))
print('#shifts: {0}'.format(len(model.shifts)))
print('#vacations: {0}'.format(len(model.vacations)))
print("#associations=%d" % len(model.nurse_associations))
print("#incompatibilities=%d" % len(model.nurse_incompatibilities))
def setup_data(model):
""" compute internal data """
# compute shift activities (start, end duration) and stor ethem in a dict indexed by shifts
model.shift_activities = {s: ShiftActivity(s.day, s.start_time, s.end_time) for s in model.shifts}
# map from nurse names to nurse tuples.
model.nurses_by_id = {n.name: n for n in model.nurses}
def setup_variables(model):
all_nurses, all_shifts = model.nurses, model.shifts
# one binary variable for each pair (nurse, shift) equal to 1 iff nurse n is assigned to shift s
model.nurse_assignment_vars = model.binary_var_matrix(all_nurses, all_shifts, 'NurseAssigned')
# for each nurse, allocate one variable for work time
model.nurse_work_time_vars = model.continuous_var_dict(all_nurses, lb=0, name='NurseWorkTime')
# and two variables for over_average and under-average work time
model.nurse_over_average_time_vars = model.continuous_var_dict(all_nurses, lb=0,
name='NurseOverAverageWorkTime')
model.nurse_under_average_time_vars = model.continuous_var_dict(all_nurses, lb=0,
name='NurseUnderAverageWorkTime')
# finally the global average work time
model.average_nurse_work_time = model.continuous_var(lb=0, name='AverageWorkTime')
def setup_constraints(model):
all_nurses = model.nurses
all_shifts = model.shifts
nurse_assigned = model.nurse_assignment_vars
nurse_work_time = model.nurse_work_time_vars
shift_activities = model.shift_activities
nurses_by_id = model.nurses_by_id
max_work_time = model.work_rules.work_time_max
# define average
model.add_constraint(
len(all_nurses) * model.average_nurse_work_time == model.sum(nurse_work_time[n] for n in all_nurses), "average")
# compute nurse work time , average and under, over
for n in all_nurses:
work_time_var = nurse_work_time[n]
model.add_constraint(
work_time_var == model.sum(nurse_assigned[n, s] * shift_activities[s].duration for s in all_shifts),
"work_time_{0!s}".format(n))
# relate over/under average worktime variables to the worktime variables
# the trick here is that variables have zero lower bound
# however, thse variables are not completely defined by this constraint,
# only their difference is.
# if these variables are part of the objective, CPLEX wil naturally minimize their value,
# as expected
model.add_constraint(
work_time_var == model.average_nurse_work_time
+ model.nurse_over_average_time_vars[n]
- model.nurse_under_average_time_vars[n],
"average_work_time_{0!s}".format(n))
# state the maximum work time as a constraint, so that is can be relaxed,
# should the problem become infeasible.
model.add_constraint(work_time_var <= max_work_time, "max_time_{0!s}".format(n))
# vacations
v = 0
for vac_nurse_id, vac_day in model.vacations:
vac_n = nurses_by_id[vac_nurse_id]
for shift in (s for s in all_shifts if s.day == vac_day):
v += 1
model.add_constraint(nurse_assigned[vac_n, shift] == 0,
"medium_vacations_{0!s}_{1!s}_{2!s}".format(vac_n, vac_day, shift))
#print('#vacation cts: {0}'.format(v))
# a nurse cannot be assigned overlapping shifts
# post only one constraint per couple(s1, s2)
number_of_overlaps = 0
nb_shifts = len(all_shifts)
for i1 in range(nb_shifts):
for i2 in range(i1 + 1, nb_shifts):
s1 = all_shifts[i1]
s2 = all_shifts[i2]
if shift_activities[s1].overlaps(shift_activities[s2]):
number_of_overlaps += 1
for n in all_nurses:
model.add_constraint(nurse_assigned[n, s1] + nurse_assigned[n, s2] <= 1,
"high_overlapping_{0!s}_{1!s}_{2!s}".format(s1, s2, n))
#print('# overlapping cts: {0}'.format(number_of_overlaps))
for s in all_shifts:
demand_min = s.min_requirement
demand_max = s.max_requirement
total_assigned = model.sum(nurse_assigned[n, s] for n in model.nurses)
model.add_constraint(total_assigned >= demand_min,
"high_req_min_{0!s}_{1}".format(s, demand_min))
model.add_constraint(total_assigned <= demand_max,
"medium_req_max_{0!s}_{1}".format(s, demand_max))
for (dept, skill, required) in model.skill_requirements:
if required > 0:
for dsh in (s for s in all_shifts if dept == s.department):
model.add_constraint(model.sum(nurse_assigned[skilled_nurse, dsh] for skilled_nurse in
(n for n in all_nurses if
n.name in model.nurse_skills.keys() and skill in model.nurse_skills[
n.name])) >= required,
"high_required_{0!s}_{1!s}_{2!s}_{3!s}".format(dept, skill, required, dsh))
# nurse-nurse associations
# for each pair of associated nurses, their assignment variables are equal
# over all shifts.
c = 0
for (nurse_id1, nurse_id2) in model.nurse_associations:
if nurse_id1 in nurses_by_id and nurse_id2 in nurses_by_id:
nurse1 = nurses_by_id[nurse_id1]
nurse2 = nurses_by_id[nurse_id2]
for s in all_shifts:
c += 1
ctname = 'medium_ct_nurse_assoc_{0!s}_{1!s}_{2:d}'.format(nurse_id1, nurse_id2, c)
model.add_constraint(nurse_assigned[nurse1, s] == nurse_assigned[nurse2, s], ctname)
# nurse-nurse incompatibilities
# for each pair of incompatible nurses, the sum of assigned variables is less than one
# in other terms, both nurses can never be assigned to the same shift
c = 0
for (nurse_id1, nurse_id2) in model.nurse_incompatibilities:
if nurse_id1 in nurses_by_id and nurse_id2 in nurses_by_id:
nurse1 = nurses_by_id[nurse_id1]
nurse2 = nurses_by_id[nurse_id2]
for s in all_shifts:
c += 1
ctname = 'medium_ct_nurse_incompat_{0!s}_{1!s}_{2:d}'.format(nurse_id1, nurse_id2, c)
model.add_constraint(nurse_assigned[nurse1, s] + nurse_assigned[nurse2, s] <= 1, ctname)
model.total_number_of_assignments = model.sum(nurse_assigned[n, s] for n in all_nurses for s in all_shifts)
# model.nurse_costs = [model.nurse_assignment_vars[n, s] * n.pay_rate * model.shift_activities[s].duration
# for n in model.nurses for s in model.shifts]
def assignment_cost_f(ns):
n, s = ns
return n.pay_rate * model.shift_activities[s].duration
model.nurse_costs = model.scal_prod_f(nurse_assigned, assignment_cost_f)
model.total_salary_cost = model.sum(model.nurse_costs)
def setup_objective(model):
model.add_kpi(model.total_salary_cost, "Total salary cost")
model.add_kpi(model.total_number_of_assignments, "Total number of assignments")
model.add_kpi(model.average_nurse_work_time, "average work time")
total_over_average_worktime = model.sum(model.nurse_over_average_time_vars[n] for n in model.nurses)
total_under_average_worktime = model.sum(model.nurse_under_average_time_vars[n] for n in model.nurses)
model.add_kpi(total_over_average_worktime, "Total over-average worktime")
model.add_kpi(total_under_average_worktime, "Total under-average worktime")
total_fairness = total_over_average_worktime + total_under_average_worktime
model.add_kpi(total_fairness, "Total fairness")
model.minimize(model.total_salary_cost + total_fairness + model.total_number_of_assignments)
def print_information(model):
print("#shifts=%d" % len(model.shifts))
print("#nurses=%d" % len(model.nurses))
print("#vacations=%d" % len(model.vacations))
print("#nurse skills=%d" % len(model.nurse_skills))
print("#nurse associations=%d" % len(model.nurse_associations))
print("#incompatibilities=%d" % len(model.nurse_incompatibilities))
model.print_information()
model.report_kpis()
def print_solution(model):
print("*************************** Solution ***************************")
print("Allocation By Department:")
for d in model.departments:
print("\t{}: {}".format(d, sum(
model.nurse_assignment_vars[n, s].solution_value for n in model.nurses for s in model.shifts if
s.department == d)))
print("Cost By Department:")
for d in model.departments:
cost = sum(
model.nurse_assignment_vars[n, s].solution_value * n.pay_rate * model.shift_activities[s].duration for n in
model.nurses for s in model.shifts if s.department == d)
print("\t{}: {}".format(d, cost))
print("Nurses Assignments")
for n in sorted(model.nurses):
total_hours = sum(
model.nurse_assignment_vars[n, s].solution_value * model.shift_activities[s].duration for s in model.shifts)
print("\t{}: total hours:{}".format(n.name, total_hours))
for s in model.shifts:
if model.nurse_assignment_vars[n, s].solution_value == 1:
print("\t\t{}: {} {}-{}".format(s.day, s.department, s.start_time, s.end_time))
# ----------------------------------------------------------------------------
# Build the model
# ----------------------------------------------------------------------------
def build(context=None, verbose=False, **kwargs):
mdl = Model("Nurses", context=context, **kwargs)
load_data(mdl, SHIFTS, NURSES, NURSE_SKILLS, VACATIONS, NURSE_ASSOCIATIONS,
NURSE_INCOMPATIBILITIES, verbose=verbose)
setup_data(mdl)
setup_variables(mdl)
setup_constraints(mdl)
setup_objective(mdl)
return mdl
# ----------------------------------------------------------------------------
# Solve the model and display the result
# ----------------------------------------------------------------------------
if __name__ == '__main__':
# Build model
model = build()
# Solve the model and print solution
solve(model)
# Save the CPLEX solution as "solution.json" program output
with get_environment().get_output_stream("solution.json") as fp:
model.solution.export(fp, "json")
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