In this video, we'll design the optimization problem
that the operating room manager would need to solve.
The decision to be made is how many operating rooms
to assign each department on each day.
This means that we need to define integer decision
variables x_jk.
x_jk will represent the number of operating rooms
department j is allocated on day k.
So here j can take values such as ophthalmology, gynecology,
general surgery, and so on.
Whereas k would take values like Monday, Tuesday, Wednesday,
and so on.
The objective of this problem will
be to maximize the percent of target allocation hours
that each department is actually allocated.
So if target allocation hours are t_j for department j,
then we want to maximize the sum of 8*x_jk divided by t_j over
all departments and days of the week.
Let's break this down.
For example, if otolaryngology has
a target of 37.3 hours per week--
that means t_j equals to 37.3-- and we allocate them
4 operating rooms, then their percent of target allocation
hours is equal to 8 times 4, that
is, they got 32 hours, divided by the 37.3 hours
they asked for.
So they got 85.8% of what they wanted.
And we try to maximize this percentage value
for every department.
So that's why we sum over all departments
and all days of the week.
Let's continue on to the constraints.
Remember that Mount Sinai has 10 operating rooms.
Therefore, at most 10 operating rooms
can be assigned on any given day.
Also, the number of operating rooms allocated to a department
on a given day cannot exceed the number of surgery teams that
the department has available that day.
And as we said before, we must meet department daily minimums
and maximums as well as weekly minimums and maximums.
Let's turn these constraints into math.
So the first constraint is that at most 10 operating rooms
can be assigned on any given day.
Let's consider Monday for example.
That means that x for ophthalmology
on Monday plus x for gynecology on Monday
plus x for oral surgery on Monday
plus x for otolaryngology on Monday
plus x for general surgery on Monday
must add up to something less than or equal to 10.
We would have a similar constraint
for Tuesday, Wednesday, Thursday and Friday.
The next set of constraints is that the number of operating
rooms allocated to a department on a given day cannot exceed
the number of surgery teams that are available that day.
So for example, for gynecology on Friday
there are 3 teams available.
That means that x gynecology Friday must be between 0 and 3.
For oral surgery on Wednesday it must be between 0 and 0
because the oral surgeon is only in on Tuesdays and Thursdays.
The next set of constraints is that we
must meet department daily minimums and maximums.
For example, the general surgery department on Tuesday
requires between 0 and 6 operating rooms.
We would have a similar constraint for every department
and for every day of the week.
We also need to meet department weekly minimums and maximums.
So for example, the ophthalmology department
has a minimum of 3 and a maximum of 6 operating rooms
that it needs in any given week.