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So we saw in the
previous video how

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we can solve linear optimization
problems in LibreOffice.

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Let's now try to get
some intuition for what's

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going on by visualizing
our problem.

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Since we only have two
decisions, R and D,

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we can visualize our
constraints in two dimensions.

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We'll plot D on the x-axis,
and R on the y-axis.

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We first have
non-negativity constraints,

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so R and D are both
greater than zero.

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We can plot the capacity
constraint, R + D less than

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or equal to 166, which is
shown as the red line here.

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Our solution has to be
to the left of this line

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according to this constraint.

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Now, let's add in our
demand constraints.

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The regular seats should be less
than the demand of 100, which

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requires the solution to
be below this blue line.

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And the discount seats should
be less than the demand of 150,

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which requires
the solution to be

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to the left of this green line.

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Taken together, our
constraints define

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what we call our "feasible
space" or the space

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of all possible values
that our decisions can

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take according to
our constraints.

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To find the optimal solution
now in our feasible space,

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we have to use the
objective, 617*R + 238*D.

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We can plot this objective
in our feasible space.

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So to know how many
seats we should

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sell to achieve a
certain revenue,

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we can see different
values of this line.

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So to achieve a
revenue of $20,000,

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our solution has to be somewhere
on this line in our feasible

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space.

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To achieve a revenue of
$40,000, our solution

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has to be somewhere on this
line in our feasible space.

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And to achieve a
revenue of $60,000,

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our solution has to be somewhere
on this line in our feasible

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space.

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Since the revenue is increasing
as we move this line up

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and our goal is to
maximize the revenue,

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our optimal solution
will be where

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this line can't go
any further and still

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be in our feasible space.

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So our optimal solution
is at this point

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with a revenue of $77,408.

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As we can see here,
the solution is

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dependent on how the
feasible space was defined.

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In the next video,
we'll see what

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happens if our capacity
and demands change.