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In this video, we'll design
the optimization problem

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that the operating room
manager would need to solve.

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The decision to be made is
how many operating rooms

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to assign each
department on each day.

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This means that we need
to define integer decision

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variables x_jk.

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x_jk will represent the
number of operating rooms

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department j is
allocated on day k.

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So here j can take values such
as ophthalmology, gynecology,

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general surgery, and so on.

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Whereas k would take values
like Monday, Tuesday, Wednesday,

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and so on.

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The objective of
this problem will

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be to maximize the percent
of target allocation hours

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that each department
is actually allocated.

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So if target allocation hours
are t_j for department j,

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then we want to maximize the sum
of 8*x_jk divided by t_j over

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all departments and
days of the week.

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Let's break this down.

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For example, if
otolaryngology has

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a target of 37.3
hours per week--

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that means t_j equals to
37.3-- and we allocate them

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4 operating rooms, then their
percent of target allocation

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hours is equal to
8 times 4, that

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is, they got 32 hours,
divided by the 37.3 hours

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they asked for.

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So they got 85.8%
of what they wanted.

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And we try to maximize
this percentage value

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for every department.

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So that's why we sum
over all departments

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and all days of the week.

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Let's continue on
to the constraints.

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Remember that Mount Sinai
has 10 operating rooms.

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Therefore, at most
10 operating rooms

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can be assigned
on any given day.

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Also, the number of operating
rooms allocated to a department

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on a given day cannot exceed
the number of surgery teams that

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the department has
available that day.

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And as we said before, we must
meet department daily minimums

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and maximums as well as
weekly minimums and maximums.

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Let's turn these
constraints into math.

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So the first constraint is
that at most 10 operating rooms

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can be assigned
on any given day.

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Let's consider
Monday for example.

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That means that x
for ophthalmology

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on Monday plus x for
gynecology on Monday

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plus x for oral
surgery on Monday

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plus x for
otolaryngology on Monday

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plus x for general
surgery on Monday

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must add up to something
less than or equal to 10.

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We would have a
similar constraint

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for Tuesday, Wednesday,
Thursday and Friday.

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The next set of constraints is
that the number of operating

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rooms allocated to a department
on a given day cannot exceed

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the number of surgery teams
that are available that day.

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So for example, for
gynecology on Friday

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there are 3 teams available.

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That means that x gynecology
Friday must be between 0 and 3.

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For oral surgery on Wednesday
it must be between 0 and 0

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because the oral surgeon is only
in on Tuesdays and Thursdays.

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The next set of
constraints is that we

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must meet department daily
minimums and maximums.

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For example, the general
surgery department on Tuesday

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requires between 0
and 6 operating rooms.

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We would have a similar
constraint for every department

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and for every day of the week.

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We also need to meet department
weekly minimums and maximums.

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So for example, the
ophthalmology department

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has a minimum of 3 and a
maximum of 6 operating rooms

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that it needs in any given week.