# nurses.py¶

This example solves the problem of finding an optimal assignment of nurses to shifts.

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527 528 529 530 # -------------------------------------------------------------------------- # 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")