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00:00:05,210 --> 00:00:06,990
Okay, so solving
successfully finished,

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00:00:06,990 --> 00:00:08,990
and we've got the results.

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00:00:08,990 --> 00:00:11,690
If you had any trouble
solving this problem,

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00:00:11,690 --> 00:00:15,280
then you should go into
Options in the Solver

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00:00:15,280 --> 00:00:19,240
and select that you want to
edit the length of solution time

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00:00:19,240 --> 00:00:22,480
and increase it
past 100 seconds.

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The LibreOffice Solver is
not incredibly powerful,

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00:00:25,680 --> 00:00:28,550
so it may need more time
to find the solution.

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00:00:28,550 --> 00:00:31,120
We do recommend that if
you have access to Excel,

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00:00:31,120 --> 00:00:34,590
that you use Excel instead for
integer optimization problems,

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00:00:34,590 --> 00:00:36,610
because Excel is a
little bit better

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00:00:36,610 --> 00:00:39,660
with integer optimization.

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00:00:39,660 --> 00:00:42,370
However, you probably got
the solution just fine.

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So you'll want to go ahead
and click Keep Result.

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00:00:46,500 --> 00:00:49,980
So now, we can take a
look at our solution.

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00:00:49,980 --> 00:00:53,470
So we've got the
objective of 4.46.

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00:00:53,470 --> 00:00:55,310
This is not particularly
interpretable,

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until you remember that
what our objective is,

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00:00:58,590 --> 00:01:03,560
is the total of all departments
percent of target allocation

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00:01:03,560 --> 00:01:05,570
hours reached.

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00:01:05,570 --> 00:01:07,730
So the best possible number
that we could get here

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would be 5, if
every department had

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100% of their target
allocation hours reached.

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So we've got 4.46.

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00:01:17,010 --> 00:01:18,970
So let's go up and take
a look at the solution.

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00:01:21,630 --> 00:01:24,690
So we're giving ophthalmology
a total of four operating

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00:01:24,690 --> 00:01:28,050
rooms per week, which
is 32 weekly hours,

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or 81% of their target.

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00:01:30,380 --> 00:01:33,420
We're giving
gynecology 14, which

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00:01:33,420 --> 00:01:37,410
is 112 weekly hours,
95% of their target.

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00:01:37,410 --> 00:01:40,050
We're giving oral surgery
two, and as we see,

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00:01:40,050 --> 00:01:42,240
the constraints restricted
us so that we only

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00:01:42,240 --> 00:01:45,190
gave oral surgery
operating rooms on Tuesdays

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00:01:45,190 --> 00:01:47,030
and Thursdays.

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This is 80% of the target.

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00:01:55,820 --> 00:01:59,180
We gave otolaryngology
three operating rooms

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and general surgery
23 operating rooms,

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00:02:02,010 --> 00:02:05,830
getting 91% and 97% of their
target allocation hours.

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00:02:08,570 --> 00:02:11,300
So suppose that you were the
operating room manager, who

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00:02:11,300 --> 00:02:13,760
knows integer optimization.

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00:02:13,760 --> 00:02:16,060
In practice, what would happen
is that you would go off

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00:02:16,060 --> 00:02:17,490
and you would
design what we just

43
00:02:17,490 --> 00:02:20,150
did here in the recitation.

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00:02:20,150 --> 00:02:21,760
You would have this
optimization model

45
00:02:21,760 --> 00:02:24,560
for operating room
scheduling in the hospital.

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00:02:24,560 --> 00:02:25,490
You'd solve it.

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00:02:25,490 --> 00:02:27,350
You'd find the optimal solution.

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00:02:27,350 --> 00:02:31,800
And you'd come back and present
it to the surgical departments.

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00:02:31,800 --> 00:02:33,560
And they'll look
at the schedule.

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00:02:33,560 --> 00:02:35,579
And all of a sudden,
they'll tell you

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00:02:35,579 --> 00:02:38,310
it doesn't work for them.

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And you'll say, why not?

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I incorporated everything
that you asked for.

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And it's true.

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You did.

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00:02:44,640 --> 00:02:47,030
But oftentimes, there are
actually other constraints

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that they forgot
to tell you about.

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So what would you
do in that case?

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00:02:51,590 --> 00:02:54,710
Well, you'd listen to their
additional constraints

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and go back and add
them to the model.

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00:02:57,300 --> 00:02:59,970
So that's exactly
what we'll do here.

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00:02:59,970 --> 00:03:01,750
Suppose the general
surgery department

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00:03:01,750 --> 00:03:04,210
took a look at the schedule
and said, "Oh, great, we

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have 23 operating
rooms per week.

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00:03:07,580 --> 00:03:12,120
That's exactly two less
than actually our maximum.

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00:03:12,120 --> 00:03:14,210
It's right in between our
minimum and our maximum.

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00:03:14,210 --> 00:03:16,280
And we're getting 97%
of our weekly target.

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00:03:16,280 --> 00:03:17,920
We think that's wonderful.

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00:03:17,920 --> 00:03:20,070
However, it's a little
bit uneven-- six

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on Wednesday, three on Thursday.

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00:03:23,150 --> 00:03:27,829
Can we just have at
least four per day?"

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00:03:27,829 --> 00:03:29,020
We can do that.

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00:03:29,020 --> 00:03:31,790
So let's go up to
the minimum number

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00:03:31,790 --> 00:03:33,540
of operating rooms per day.

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00:03:33,540 --> 00:03:37,960
And for general surgery, let's
change this from zero to four.

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00:03:37,960 --> 00:03:39,960
This will help them balance
throughout the week.

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00:03:43,200 --> 00:03:46,920
And potentially, ophthalmology
will look at their schedule,

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00:03:46,920 --> 00:03:48,300
and they'll think
the same thing.

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00:03:48,300 --> 00:03:49,670
They'll think,
"Well, we actually

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00:03:49,670 --> 00:03:52,329
would prefer to balance
it throughout the week.

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00:03:52,329 --> 00:03:55,280
We want at most one
operating room per day.

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00:03:55,280 --> 00:03:57,100
We don't want to
have two on Monday.

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Let's put one of those
on Thursday or Friday."

84
00:04:01,130 --> 00:04:01,770
So that's OK.

85
00:04:01,770 --> 00:04:03,720
We can do that too.

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00:04:03,720 --> 00:04:06,300
We just go to the maximum number
of operating rooms per day.

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00:04:06,300 --> 00:04:08,680
And for ophthalmology,
we change it to one.

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00:04:13,470 --> 00:04:17,709
Let's re-solve the model
and see what we get.

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00:04:17,709 --> 00:04:20,860
Again, we select the objective.

90
00:04:20,860 --> 00:04:24,040
Go to Tools, pull
down the Solver.

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00:04:24,040 --> 00:04:26,430
It should have everything
already loaded up,

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00:04:26,430 --> 00:04:27,640
and we just get to hit Solve.

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00:04:30,630 --> 00:04:31,470
So that was quick.

94
00:04:31,470 --> 00:04:33,600
Solving successfully finished.

95
00:04:33,600 --> 00:04:36,180
And it looks like we actually
have exactly the same results

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00:04:36,180 --> 00:04:38,180
as we had before.

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00:04:38,180 --> 00:04:42,290
Let's hit Keep Result and
take a closer look at this.

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00:04:42,290 --> 00:04:45,540
So our objective function
stayed exactly the same.

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00:04:45,540 --> 00:04:48,380
But we were able to incorporate
the new constraints,

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the new requirement that the
general surgery department have

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00:04:51,270 --> 00:04:53,930
at least four operating
rooms per day,

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00:04:53,930 --> 00:04:56,190
and that the ophthalmology
department have

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00:04:56,190 --> 00:04:59,670
at most one operating
room per day.

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00:04:59,670 --> 00:05:01,880
This means that
within the solution

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00:05:01,880 --> 00:05:05,770
we had before, there's actually
a little bit of wiggle room.

106
00:05:05,770 --> 00:05:08,760
We could move some things
around without changing

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00:05:08,760 --> 00:05:10,730
the optimality of the
solution, that is,

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00:05:10,730 --> 00:05:12,880
without changing the
percent of target

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00:05:12,880 --> 00:05:15,690
that each department received.

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00:05:15,690 --> 00:05:17,240
And as you see, this
is a little more

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balanced for the
ophthalmology department

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00:05:19,390 --> 00:05:20,850
and the general
surgery department.

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00:05:23,970 --> 00:05:29,020
So great-- so maybe seeing
that this was available,

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00:05:29,020 --> 00:05:31,140
then you'd go back,
you'd present it,

115
00:05:31,140 --> 00:05:35,580
and general surgery
would say, well, maybe

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00:05:35,580 --> 00:05:38,659
could we have at
least five per day?

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00:05:38,659 --> 00:05:40,840
So if you went up
and changed this,

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00:05:40,840 --> 00:05:45,490
made it five all the way
across instead of four

119
00:05:45,490 --> 00:05:52,270
and you solved it again,
what would happen?

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00:05:52,270 --> 00:05:53,730
You probably have
an idea already.

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00:05:57,540 --> 00:05:59,340
And this is exactly
what you were thinking.

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00:05:59,340 --> 00:06:01,100
No solution was found.

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00:06:01,100 --> 00:06:02,390
The model is infeasible.

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00:06:02,390 --> 00:06:04,560
Check limiting conditions.

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00:06:04,560 --> 00:06:06,750
So why is the model infeasible?

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00:06:06,750 --> 00:06:11,710
Well, let's take a look.

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00:06:11,710 --> 00:06:12,540
Scroll back over.

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00:06:15,210 --> 00:06:19,060
The model is infeasible
because, let's see--

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00:06:19,060 --> 00:06:21,130
so it's not because
of the minimum

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00:06:21,130 --> 00:06:24,850
and maximum weekly
department requirements,

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because general surgery said
that they were willing to have

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up to 25 operating
rooms per week.

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00:06:32,060 --> 00:06:37,710
However, if you look at the
weekly targets, 25 times 8

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00:06:37,710 --> 00:06:39,240
is 200.

135
00:06:39,240 --> 00:06:43,450
So if we were to actually assign
general surgery five operating

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00:06:43,450 --> 00:06:45,190
rooms per day, we
would be assigning

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00:06:45,190 --> 00:06:48,800
them 200 hours per week, which
exceeds their weekly target.

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00:06:48,800 --> 00:06:51,420
And our optimization model says
we cannot exceed the weekly

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00:06:51,420 --> 00:06:53,100
targets.

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00:06:53,100 --> 00:06:54,850
So just right there,
we can tell that it's

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00:06:54,850 --> 00:06:57,370
infeasible because
of this constraint.

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00:06:57,370 --> 00:06:58,880
So let's change
this back to four.

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00:07:04,630 --> 00:07:07,390
So suppose you've come
up with a solution,

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00:07:07,390 --> 00:07:09,860
you've gone back to the
surgical departments,

145
00:07:09,860 --> 00:07:12,670
you've presented your
solution, they've maybe

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00:07:12,670 --> 00:07:14,470
requested additional
changes, and we've

147
00:07:14,470 --> 00:07:16,800
made those changes here.

148
00:07:16,800 --> 00:07:19,810
So does this mean
that your job is done?

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00:07:19,810 --> 00:07:22,980
Well, no, not really.

150
00:07:22,980 --> 00:07:25,580
Of course, you could go back
to the surgical departments

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00:07:25,580 --> 00:07:27,540
and tell them that this
is the schedule they're

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00:07:27,540 --> 00:07:29,580
going to have every week.

153
00:07:29,580 --> 00:07:32,820
But I bet if did that,
you'd hear some complaints.

154
00:07:32,820 --> 00:07:37,330
Let's go into our solution and
take a closer look to see why.

155
00:07:37,330 --> 00:07:40,340
So for example, let's look at
the ophthalmology department.

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00:07:40,340 --> 00:07:42,990
In our solution,
we're assigning them

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00:07:42,990 --> 00:07:45,680
four operating rooms per week.

158
00:07:45,680 --> 00:07:48,380
That gives them 32
weekly hours, which

159
00:07:48,380 --> 00:07:53,250
is 81% of their target
allocation hours.

160
00:07:53,250 --> 00:07:58,310
If you assigned the schedule to
ophthalmology every week, then

161
00:07:58,310 --> 00:08:00,310
they would never
get more than 81%

162
00:08:00,310 --> 00:08:02,220
of their target
allocation hours.

163
00:08:02,220 --> 00:08:06,090
And over time, this would
lead to a real shortage.

164
00:08:06,090 --> 00:08:08,810
The reason why we never
assign them more than four

165
00:08:08,810 --> 00:08:12,130
in our optimization model is
because their weekly target

166
00:08:12,130 --> 00:08:15,740
number of hours was 39.4.

167
00:08:15,740 --> 00:08:18,340
If we had assigned them
five operating rooms,

168
00:08:18,340 --> 00:08:20,830
it would be giving
them 40 hours per week,

169
00:08:20,830 --> 00:08:23,440
which exceeds their
weekly target.

170
00:08:23,440 --> 00:08:25,490
So the way our optimization
model is set up,

171
00:08:25,490 --> 00:08:27,550
we will never give
them more than four

172
00:08:27,550 --> 00:08:30,610
operating rooms per week.

173
00:08:30,610 --> 00:08:33,409
But if you were
the OR manager, you

174
00:08:33,409 --> 00:08:35,970
would realize that this
is actually something

175
00:08:35,970 --> 00:08:38,730
that we should be a little
more flexible about.

176
00:08:38,730 --> 00:08:40,320
Perhaps you would
consider giving them

177
00:08:40,320 --> 00:08:42,760
five operating rooms
one week, or maybe

178
00:08:42,760 --> 00:08:48,110
five operating rooms every
two weeks out of the month.

179
00:08:48,110 --> 00:08:50,800
This is feasible because
we have extra operating

180
00:08:50,800 --> 00:08:56,510
rooms on Thursday and Friday,
as you can see down here.

181
00:08:56,510 --> 00:09:00,370
Plus, it seems like it's a more
efficient and fair solution.

182
00:09:00,370 --> 00:09:03,140
Yes, although they do
exceed their weekly target

183
00:09:03,140 --> 00:09:08,000
by 0.6 hours, it means that
on the weeks when they only

184
00:09:08,000 --> 00:09:13,230
get to have 81% of
their target allocation,

185
00:09:13,230 --> 00:09:14,950
they know that the
next week, they'll

186
00:09:14,950 --> 00:09:18,380
be able to make up for it.

187
00:09:18,380 --> 00:09:20,420
So it seems like a more
fair solution, as well as

188
00:09:20,420 --> 00:09:21,510
a more efficient solution.

189
00:09:24,670 --> 00:09:27,490
By combining the power
of integer optimization

190
00:09:27,490 --> 00:09:31,130
with the understanding and
flexibility of human judgment,

191
00:09:31,130 --> 00:09:33,640
you really get the
best of both worlds.