This is a demo of a nurse scheduling model developed by Ikeda, Nakamura and Humble (INH).
The nurse scheduling problem seeks to find an optimal assignment for a group of nurses, under constraints of scheduling and personnel. INH developed a model which is a simplified representation of a real-world nursing facility.
In the general nurse scheduling problem, there are three types of constraints, which are mentioned here to provide background for INH's constraints. These types of constraints, in the general problem, are:
1) Both upper and lower limits on the number of breaks. 2) The number of nurses on duty for each shift slot. 3) For each individual nurse, upper and lower limits on the time interval between days of duty.
These three types of constraints combine to ensure sufficient nurses on duty at all times, without overworking any particular nurse.
INH formulated a QUBO from a simplification of these constraints, discussed below, that tries to achieve reasonable results for nurse scheduling.
INH's three types of constraints are:
1) "hard shift" constraint: requires that at least one nurse is assigned for each working day.
2) "hard nurse" constraint: requires that no nurse works two or more consecutive days.
3) "soft nurse" constraint: promotes that all nurses should have roughly even work schedules.
This demo seeks to obtain reasonable results for a nurse schedule, based on
INH's model. Our implementation attempts to find a schedule for a number
n_nurses
of nurses and a number n_days
of days that satisfies the following
conditions:
Running the demo results in the following output, at the command-line:
Building binary quadratic model...
Sending problem to hybrid sampler...
Building schedule and checking constraints...
Hard shift constraint: Satisfied
Hard nurse constraint: Satisfied
Soft nurse constraint: Unsatisfied
Schedule:
Nurse 2 X X X X
Nurse 1 X X X X
Nurse 0 X X X
0 1 2 3 4 5 6 7 8 9 10
Schedule saved as schedule.png.
The results show the following:
An image of the schedule (shown below) is saved to the file schedule.png
.
To run the demo, run the command
python nurse_scheduling.py
Here is a general overview of the Nurse Scheduling code:
Note that the total of the three constraint sums should equal the energy.
Some notes on the code:
We use a two-dimensional QUBO matrix, Q[i
, j
], in which both indices i
and j
are composite indices. Each composite index is used to represent the
combinations of the variables nurse
and day
. The (nurse
, day
) tuples
are placed into the one-dimensional index in the following order, where
nurse
is first index, and day
is second index, in the tuples:
(0, 0) (0, 1) (0, 2)... (0, D) (1, 0) (1, 1)... (1, D)
The methods get_index
and get_nurse_and_day
are used to convert back and
forth between (nurse
, day
) tuples and the composite indices.
The three constraint sums are separated out in order to be able to confirm the individual effects manually. For example, if a nurse was assigned to two successive days, the hard nurse constraint sum would be nonzero.
We have not yet confirmed Ikeda's results with reverse annealing
Ikeda, K., Nakamura, Y. & Humble, T.S. Application of Quantum Annealing to Nurse Scheduling Problem. Sci Rep 9, 12837 (2019). https://doi.org/10.1038/s41598-019-49172-3
Released under the Apache License 2.0. See LICENSE file.