Nurse Scheduling

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:

• One, and only one, nurse has been assigned to each day (hard shift constraint)
• No nurse works two days in a row (hard nurse constraint)
• The nurses should work the same number of days

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:

• One, and only one, nurse has been assigned to each day
• No nurse works two days in a row
• Two nurses work 4 days, and one works three days. Because two nurses work one extra day each, the soft nurse constraint energy is unsatisfied.

An image of the schedule (shown below) is saved to the file schedule.png.

Usage

To run the demo, run the command

python nurse_scheduling.py

Code Overview

Here is a general overview of the Nurse Scheduling code:

• Assign the size of the problem (number of nurses and days) and parameters
• Compute the "penalty matrix" J
• Develop the QUBO matrix
• Run the problem (solve the QUBO)
• Calculate the hard nurse constraint sum, to check if the hard nurse constraints are satisfied
• Calculate the hard shift constraint sum, to check if the hard shift constraints are satisfied
• Calculate the soft nurse constraint sum, to check if the soft nurse constraints are satisfied
• Print the nurse schedule

Note that the total of the three constraint sums should equal the energy.

Code Specifics

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

References

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

License

Released under the Apache License 2.0. See LICENSE file.