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For a single route
example, our problem

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is to find the optimal
number of discount seats

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and regular seats to
sell to maximize revenue.

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We'll assume that the price
of regular seats is $617,

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and the price of
discount seats is $238.

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Also, let's assume
that we forecasted

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the demand of regular
seats to be 100,

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and the demand of
discount seats to be 150.

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The capacity of our
airplane is 166 seats.

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Let's go ahead and formulate
this mathematically

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as a linear
optimization problem.

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The first step is to decide
what our decisions are,

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or the variables in our model.

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We need to decide how many
regular seats we went to sell.

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We'll call the number
of regular seats

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we sell R. We also need to
decide the number of discount

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seats we want to sell.

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We'll call the number of
discount seats we sell D.

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The second step
is to decide what

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our objective, or our goal, is.

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In this case, it's to
maximize the total revenue

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to the airline.

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The revenue from
each type of seat

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is equal to the number
of that type of seat

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sold times the seat price.

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In the case of
regular seats, this

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is $617 times R, the number
of regular seats we sell.

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And for discount seats,
this is $230 times D,

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the number of discount
seats we sell.

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We sum these together to
get the total revenue,

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and our objective is
to maximize this sum.

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The third step is to define
the constraints, or limits,

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of our decisions.

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One constraint is that American
Airlines can't sell more seats

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than the aircraft capacity,
which is 166 seats.

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So the total number of seats
sold, R + D has to be less than

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or equal to the capacity of 166.

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Additionally, American Airlines
shouldn't sell more seats

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than the demand for
each type of seat.

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So the regular seats,
R, shouldn't exceed 100.

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So R should be less
than or equal to 100.

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And the discount seats,
D, can't exceed 150.

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So D should be less
than or equal to 150.

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The final step is to make
sure our variables are

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taking reasonable values.

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In this case, it
wouldn't make sense

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to sell a negative
number of seats,

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so we need to make sure
that both R and D are

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greater than or equal to 0.

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So our entire problem is to
maximize total airline revenue,

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subject to the constraints
that seats sold can't exceed

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capacity, seats sold
can't exceed demand,

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and the seats sold
can't be negative.

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Mathematically, this can be
written as maximize 617*R +

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238*D, the total revenue,
subject to the constraints:

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R + D is less than or equal to
166, the capacity constraint;

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R less than or equal to 100,
and D less than or equal to 150,

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which are the
demand constraints;

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and R and D are both
greater than or equal to 0.

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This is called a linear
optimization problem.

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In the next video, we'll see
how to solve this problem using

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the software, LibreOffice.