Organizations whose employees work multiple shifts need to schedule sufficient workers for each daily shift. Typically, the schedules will have constraints, such as "no employee should work two shifts in a row". Finding a schedule that satisfies all constraints can be computationally difficult.

The following sections present two examples of employee scheduling problems, and show how to solve them using the CP-SAT solver.

For a more sophisticated example, see this shift scheduling program on GitHub.

## A nurse scheduling problem

In the next example, a hospital supervisor needs to create a schedule for four nurses over a three-day period, subject to the following conditions:

- Each day is divided into three 8-hour shifts.
- Every day, each shift is assigned to a single nurse, and no nurse works more than one shift.
- Each nurse is assigned to at least two shifts during the three-day period.

## A Python solution.

The following sections present a Python solution to the nurse scheduling problem.

### Data for the example

The following code creates the data for the example.

num_nurses = 4 num_shifts = 3 num_days = 3 all_nurses = range(num_nurses) all_shifts = range(num_shifts) all_days = range(num_days)

### Create the variables

The following code an array of variables for the problem.

shifts = {} for n in all_nurses: for d in all_days: for s in all_shifts: shifts[(n, d, s)] = model.NewBoolVar('shift_n%id%is%i' % (n, d, s))

The array defines assignments for shifts to nurses as follows:

`shifts[(n, d, s)]`

equals 1 if shift s is assigned to nurse n on day d, and
0 otherwise.

### Assign nurses to shifts

Next, we show how to assign nurses to shifts subject to the following constraints:

- Each shift is assigned to a single nurse per day.
- Each nurse works at most one shift per day.

Here's the code that creates the first condition.

for d in all_days: for s in all_shifts: model.Add(sum(shifts[(n, d, s)] for n in all_nurses) == 1)

The last line says that for each shift, the sum of the nurses assigned to that shift is 1.

Next, here's the code that requires that each nurse works at most one shift per day.

for n in all_nurses: for d in all_days: model.Add(sum(shifts[(n, d, s)] for s in all_shifts) <= 1)

For each nurse, the sum of shifts assigned to that nurse is at most 1 ("at most" because a nurse might have the day off).

### Assign shifts evenly

Next, we show how to assign shifts to nurses as evenly as possible. Since there are nine shifts over the three-day period, we can assign two shifts to each of the four nurses. After that there will be one shift left over, which can be assigned to any nurse.

The following code ensures that each nurse works at least two shifts in the three-day period.

# min_shifts_per_nurse is the largest integer such that every nurse # can be assigned at least that many shifts. If the number of nurses doesn't # divide the total number of shifts over the schedule period, # some nurses have to work one more shift, for a total of # min_shifts_per_nurse + 1. min_shifts_per_nurse = (num_shifts * num_days) // num_nurses max_shifts_per_nurse = min_shifts_per_nurse + 1 for n in all_nurses: num_shifts_worked = sum( shifts[(n, d, s)] for d in all_days for s in all_shifts) model.Add(min_shifts_per_nurse <= num_shifts_worked) model.Add(num_shifts_worked <= max_shifts_per_nurse)

Since there are `num_shifts * num_days`

total shifts in the schedule period,
you can assign at least

(num_shifts * num_days) // num_nurses

shifts to each nurse, but some shifts may be left over. (Here
`//`

is the Python integer division operator, which returns the floor of the usual
quotient.)

For the given values of `num_nurses = 4`

, `num_shifts = 3`

,
and `num_days = 3`

, the expression
`min_shifts_per_nurse`

has the value
`(3 * 3 // 4) = 2`

, so you can assign at least two
shifts to each nurse. This is guaranteed by the constraint

model.Add(min_shifts_per_nurse <= num_shifts_worked)

Since there are nine total shifts over the three-day period, there is one remaining shift after assigning two shifts to each nurse. The extra shift can be assigned to any nurse.

The final line

model.Add(num_shifts_worked <= max_shifts_per_nurse)

ensures that no nurse is assigned more than one extra shift. The constraint isn't necessary in this case, because there's only one extra shift. But for different parameter values, there could be several extra shifts, in which case the constraint is necessary.

### Call the solver and display the results

The following code calls the solver and displays the first five solutions.

solver = cp_model.CpSolver() solver.parameters.linearization_level = 0 # Display the first five solutions. a_few_solutions = range(5) solution_printer = NursesPartialSolutionPrinter( shifts, num_nurses, num_days, num_shifts, a_few_solutions) solver.SearchForAllSolutions(model, solution_printer)

Here are the first five solutions.

Solution 0 Day 0 Nurse 0 does not work Nurse 1 works shift 0 Nurse 2 works shift 1 Nurse 3 works shift 2 Day 1 Nurse 0 works shift 2 Nurse 1 does not work Nurse 2 works shift 1 Nurse 3 works shift 0 Day 2 Nurse 0 works shift 2 Nurse 1 works shift 1 Nurse 2 works shift 0 Nurse 3 does not work Solution 1 Day 0 Nurse 0 works shift 0 Nurse 1 does not work Nurse 2 works shift 1 Nurse 3 works shift 2 Day 1 Nurse 0 does not work Nurse 1 works shift 2 Nurse 2 works shift 1 Nurse 3 works shift 0 Day 2 Nurse 0 works shift 2 Nurse 1 works shift 1 Nurse 2 works shift 0 Nurse 3 does not work Solution 2 Day 0 Nurse 0 works shift 0 Nurse 1 does not work Nurse 2 works shift 1 Nurse 3 works shift 2 Day 1 Nurse 0 works shift 1 Nurse 1 works shift 2 Nurse 2 does not work Nurse 3 works shift 0 Day 2 Nurse 0 works shift 2 Nurse 1 works shift 1 Nurse 2 works shift 0 Nurse 3 does not work Solution 3 Day 0 Nurse 0 does not work Nurse 1 works shift 0 Nurse 2 works shift 1 Nurse 3 works shift 2 Day 1 Nurse 0 works shift 1 Nurse 1 works shift 2 Nurse 2 does not work Nurse 3 works shift 0 Day 2 Nurse 0 works shift 2 Nurse 1 works shift 1 Nurse 2 works shift 0 Nurse 3 does not work Solution 4 Day 0 Nurse 0 does not work Nurse 1 works shift 0 Nurse 2 works shift 1 Nurse 3 works shift 2 Day 1 Nurse 0 works shift 2 Nurse 1 works shift 1 Nurse 2 does not work Nurse 3 works shift 0 Day 2 Nurse 0 works shift 2 Nurse 1 works shift 1 Nurse 2 works shift 0 Nurse 3 does not work Statistics - conflicts : 215 - branches : 46931 - wall time : 0.320325 s - solutions found : 5184

The total number of solutions is 5184. The following counting argument explains why.

First,
there are 4 choices for the one nurse who works an extra shift.
Having chosen that nurse, there are 3 shifts the nurse can be assigned to on each of the
3 days, so the number of possible ways to assign the nurse with the extra shift is
4 · 3^{3} = 108. After assigning this nurse, there are two remaining
unassigned shifts on each day.

Of the remaining three nurses, one works days 0 and 1, one works days 0 and 2, and one works days 1 and 2. There are 3! = 6 ways to assign the nurses to these days, as shown in the diagram below. (The three nurses are labeled A, B, and C, and we have not yet assigned them to shifts.)

Day 0 Day 1 Day 2 A B A C B C A B B C A C A C A B B C A C B C A B B C A B A C B C A C A B

For each row in the above diagram, there are 2^{3} = 8 possible
ways to assign the remaining shifts to the nurses (two choices on each day).
So the total number of possible assignments is 108·6·8 = 5184.

### Entire program

Here is the entire program for the nurse scheduling problem.

from __future__ import print_function from ortools.sat.python import cp_model class NursesPartialSolutionPrinter(cp_model.CpSolverSolutionCallback): """Print intermediate solutions.""" def __init__(self, shifts, num_nurses, num_days, num_shifts, sols): cp_model.CpSolverSolutionCallback.__init__(self) self._shifts = shifts self._num_nurses = num_nurses self._num_days = num_days self._num_shifts = num_shifts self._solutions = set(sols) self._solution_count = 0 def on_solution_callback(self): if self._solution_count in self._solutions: print('Solution %i' % self._solution_count) for d in range(self._num_days): print('Day %i' % d) for n in range(self._num_nurses): is_working = False for s in range(self._num_shifts): if self.Value(self._shifts[(n, d, s)]): is_working = True print(' Nurse %i works shift %i' % (n, s)) if not is_working: print(' Nurse {} does not work'.format(n)) print() self._solution_count += 1 def solution_count(self): return self._solution_count def main(): # Data. num_nurses = 4 num_shifts = 3 num_days = 3 all_nurses = range(num_nurses) all_shifts = range(num_shifts) all_days = range(num_days) # Creates the model. model = cp_model.CpModel() # Creates shift variables. # shifts[(n, d, s)]: nurse 'n' works shift 's' on day 'd'. shifts = {} for n in all_nurses: for d in all_days: for s in all_shifts: shifts[(n, d, s)] = model.NewBoolVar('shift_n%id%is%i' % (n, d, s)) # Each shift is assigned to exactly one nurse in the schedule period. for d in all_days: for s in all_shifts: model.Add(sum(shifts[(n, d, s)] for n in all_nurses) == 1) # Each nurse works at most one shift per day. for n in all_nurses: for d in all_days: model.Add(sum(shifts[(n, d, s)] for s in all_shifts) <= 1) # min_shifts_per_nurse is the largest integer such that every nurse # can be assigned at least that many shifts. If the number of nurses doesn't # divide the total number of shifts over the schedule period, # some nurses have to work one more shift, for a total of # min_shifts_per_nurse + 1. min_shifts_per_nurse = (num_shifts * num_days) // num_nurses max_shifts_per_nurse = min_shifts_per_nurse + 1 for n in all_nurses: num_shifts_worked = sum( shifts[(n, d, s)] for d in all_days for s in all_shifts) model.Add(min_shifts_per_nurse <= num_shifts_worked) model.Add(num_shifts_worked <= max_shifts_per_nurse) # Creates the solver and solve. solver = cp_model.CpSolver() solver.parameters.linearization_level = 0 # Display the first five solutions. a_few_solutions = range(5) solution_printer = NursesPartialSolutionPrinter( shifts, num_nurses, num_days, num_shifts, a_few_solutions) solver.SearchForAllSolutions(model, solution_printer) # Statistics. print() print('Statistics') print(' - conflicts : %i' % solver.NumConflicts()) print(' - branches : %i' % solver.NumBranches()) print(' - wall time : %f s' % solver.WallTime()) print(' - solutions found : %i' % solution_printer.solution_count()) if __name__ == '__main__': main()

## Scheduling with shift requests

In this section, we take the previous example and add nurse requests for specific shifts. We then look for a schedule that maximizes the number of requests that are met. For most scheduling problems, it's best to optimize an objective function, as it is usually not practical to print all possible schedules.

This example has the same constraints as the previous example.

### Data for the example

The data for this example is shown below.

num_nurses = 5 num_shifts = 3 num_days = 7 all_nurses = range(num_nurses) all_shifts = range(num_shifts) all_days = range(num_days) shift_requests = [[[0, 0, 1], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 1]], [[0, 0, 0], [0, 0, 0], [0, 1, 0], [0, 1, 0], [1, 0, 0], [0, 0, 0], [0, 0, 1]], [[0, 1, 0], [0, 1, 0], [0, 0, 0], [1, 0, 0], [0, 0, 0], [0, 1, 0], [0, 0, 0]], [[0, 0, 1], [0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 0], [1, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 0]]]

In addition to the variables from the previous example, the data also contains a set of triples, corresponding to the three shifts per day. Each element of the triple is 0 or 1, indicating whether a shift was requested. For example, the triple [0, 0, 1] in the fifth position of row 1 indicates that nurse 1 requests shift 3 on day 5.

### Objective for the example

We want to optimize the following objective function.

model.Maximize( sum(shift_requests[n][d][s] * shifts[(n, d, s)] for n in all_nurses for d in all_days for s in all_shifts))

Since `shift_requests[n][d][s] * shifts[(n, d, s)`

is 1 if shift `s`

is assigned to nurse `n`

on day `d`

*and* that nurse requested that
shift (and 0 otherwise), the objective is the number shift of assignments that meet a request.

### Call the solver and display the results

The following code calls the solver and displays the following output, which contains an optimal schedule (although perhaps not the only one). The output shows which shift assignments were requested and the number of request that were met.

solver = cp_model.CpSolver() solver.Solve(model) for d in all_days: print('Day', d) for n in all_nurses: for s in all_shifts: if solver.Value(shifts[(n, d, s)]) == 1: if shift_requests[n][d][s] == 1: print('Nurse', n, 'works shift', s, '(requested).') else: print('Nurse', n, 'works shift', s, '(not requested).') print() print() print('Statistics') print(' - Number of shift requests met = %i' % solver.ObjectiveValue(), '(out of', num_nurses * min_shifts_per_nurse, ')') print(' - wall time : %f s' % solver.WallTime())

When you run the program, it displays the following output:

Day 0 Nurse 1 works shift 0 (not requested). Nurse 2 works shift 1 (requested). Nurse 3 works shift 2 (requested). Day 1 Nurse 0 works shift 0 (not requested). Nurse 2 works shift 1 (requested). Nurse 4 works shift 2 (requested). Day 2 Nurse 1 works shift 2 (not requested). Nurse 3 works shift 0 (requested). Nurse 4 works shift 1 (requested). Day 3 Nurse 2 works shift 0 (requested). Nurse 3 works shift 1 (requested). Nurse 4 works shift 2 (not requested). Day 4 Nurse 0 works shift 2 (requested). Nurse 1 works shift 0 (requested). Nurse 4 works shift 1 (not requested). Day 5 Nurse 0 works shift 2 (not requested). Nurse 2 works shift 1 (requested). Nurse 3 works shift 0 (requested). Day 6 Nurse 0 works shift 1 (not requested). Nurse 1 works shift 2 (requested). Nurse 4 works shift 0 (not requested). Statistics - Number of shift requests met = 13 (out of 20 ) - wall time : 0.003571 s

### Entire program

Here is the entire program for scheduling with shift requests.

from __future__ import print_function from ortools.sat.python import cp_model def main(): # This program tries to find an optimal assignment of nurses to shifts # (3 shifts per day, for 7 days), subject to some constraints (see below). # Each nurse can request to be assigned to specific shifts. # The optimal assignment maximizes the number of fulfilled shift requests. num_nurses = 5 num_shifts = 3 num_days = 7 all_nurses = range(num_nurses) all_shifts = range(num_shifts) all_days = range(num_days) shift_requests = [[[0, 0, 1], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 1]], [[0, 0, 0], [0, 0, 0], [0, 1, 0], [0, 1, 0], [1, 0, 0], [0, 0, 0], [0, 0, 1]], [[0, 1, 0], [0, 1, 0], [0, 0, 0], [1, 0, 0], [0, 0, 0], [0, 1, 0], [0, 0, 0]], [[0, 0, 1], [0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 0], [1, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 0]]] # Creates the model. model = cp_model.CpModel() # Creates shift variables. # shifts[(n, d, s)]: nurse 'n' works shift 's' on day 'd'. shifts = {} for n in all_nurses: for d in all_days: for s in all_shifts: shifts[(n, d, s)] = model.NewBoolVar('shift_n%id%is%i' % (n, d, s)) # Each shift is assigned to exactly one nurse in . for d in all_days: for s in all_shifts: model.Add(sum(shifts[(n, d, s)] for n in all_nurses) == 1) # Each nurse works at most one shift per day. for n in all_nurses: for d in all_days: model.Add(sum(shifts[(n, d, s)] for s in all_shifts) <= 1) # min_shifts_assigned is the largest integer such that every nurse can be # assigned at least that number of shifts. min_shifts_per_nurse = (num_shifts * num_days) // num_nurses max_shifts_per_nurse = min_shifts_per_nurse + 1 for n in all_nurses: num_shifts_worked = sum( shifts[(n, d, s)] for d in all_days for s in all_shifts) model.Add(min_shifts_per_nurse <= num_shifts_worked) model.Add(num_shifts_worked <= max_shifts_per_nurse) model.Maximize( sum(shift_requests[n][d][s] * shifts[(n, d, s)] for n in all_nurses for d in all_days for s in all_shifts)) # Creates the solver and solve. solver = cp_model.CpSolver() solver.Solve(model) for d in all_days: print('Day', d) for n in all_nurses: for s in all_shifts: if solver.Value(shifts[(n, d, s)]) == 1: if shift_requests[n][d][s] == 1: print('Nurse', n, 'works shift', s, '(requested).') else: print('Nurse', n, 'works shift', s, '(not requested).') print() # Statistics. print() print('Statistics') print(' - Number of shift requests met = %i' % solver.ObjectiveValue(), '(out of', num_nurses * min_shifts_per_nurse, ')') print(' - wall time : %f s' % solver.WallTime()) if __name__ == '__main__': main()