Building a schedule for employees can be an extremely complex optimization problem in which managers must balance employee preferences against schedule requirements. In this example, we show how a discrete quadratic model (DQM) and a constrained quadratic model (CQM) can be used to model this problem and how the hybrid solvers available in Leap can optimize over these competing scheduling and preference needs.
To run the CQM demo, type the command:
python demo.py
A prompt will appear asking for the number of employees:
Enter number of employees:
Type the number of employees to be considered and hit Enter. Note that the
CQM solver can take up to 5000 variables, or employees for this problem.
A second prompt for the number of shifts will appear:
Enter number of shifts:
Type the number of shifts and hit Enter. Note that we should have at least as
many employees as there are shifts.
Once these values have been entered, the program will randomly generate employee
preferences for the N shifts from most preferred (0) to least preferred (N-1). A
CQM is constructed (see below for details) and the problem is run using
LeapHybridCQMSampler.
Once the problem has run, two images are created. First, employee_schedule.png
illustrates the employee preference matrix alongside the schedule built.
Second, schedule_statistics.png shows how many employees are scheduled for
each shift, alongside a bar chart showing the employees' average preferences
for the shifts for which they have been scheduled.
At the end of the program's run, two statistics are printed to the command-line. Schedule score provides the energy value of the best solution found. Average happiness is the average of the employee preference values for the shifts that they are scheduled. A smaller average happiness score is better, and a score of 0.0 is a perfect score - everyone received their first choice shift.
To run the DQM demo, type the command:
python scheduler.py
Similar commands and output follow.
The employee scheduling problem consists of two components: a requirement that employees are evenly distributed across all shifts, and an objective to satisfy employees by scheduling them for their preferred shifts.
Since shift preferences are ranked from most preferred (smallest value) to least preferred (largest value), the rankings can be used for the biases in the quadratic model. Since the hybrid solvers look for minimum energy solutions, and minimum rank corresponds to most preferred, these naturally align.
An even distribution of employees across shifts would have approximately
num_employees/num_shifts scheduled employees per shift. To enforce this
requirement, both linear and quadratic biases must be adjusted in a specific
manner. We can do this either by building an equality constraint in our
constrained quadratic model, or using linear and quadratic biases to enforce
the constraint in the discrete quadratic model.
To determine the linear and quadratic bias adjustments, we must consider the underlying binary variables in our model. For a DQM with N shifts and M employees, each employee has a single variable constructed with N cases or classes. These are implemented as N binary variables per employee — one for each possible shift.
For a specific shift i, we require that exactly M/N employees are scheduled.
This is equivalent to saying that M/N employee variables are assigned case
i, or, returning to our binary variables, that M/N of the binary variables
corresponding to case i take a value of 1. In other words, the sum of all
employee case i binary variables should equal M/N. An equality over a
summation of binary variables can be converted to a minimization expression by
moving from the equality:
sum(shift i binary variables) = M/N
to a minimization expression:
min( ( sum(shift i binary variables) - M/N)**2 )
Expanding and simplifying this expression of binary variables, it becomes:
min( (-2*M/N+1)*sum(shift i linear biases) + 2*sum(shift i quadratic biases))
When this constraint is built into our DQM object, it is added in with a
coefficient gamma. This term gamma is known as a Lagrange parameter and can
be used to weight this constraint against the competing employee preferences.
You may wish to adjust this parameter depending on your problem requirements and
size. The value set here in this program was chosen to empirically work well as
a starting point for problems of a wide-variety of sizes. For more information
on setting this parameter, see D-Wave's Problem Formulation Guide.