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We were able to solve our
sports scheduling problem

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with 4 teams, 24 decision
variables, and 22

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basic constraints,
pretty quickly.

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00:00:14,330 --> 00:00:18,140
However, the problem
size increases rapidly.

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The same problem
with 10 teams, would

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have 585 decision variables
and 175 basic constraints.

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For scheduling major
league baseball,

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the problem has 100,000
decision variables

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and 200,000 constraints.

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00:00:36,590 --> 00:00:40,000
For small problems, spreadsheet
softwares, like LibreOffice,

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are great.

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00:00:41,190 --> 00:00:43,170
But for large
problems like this,

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solving them in LibreOffice
would be impossible.

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So how are integer
optimization models like this

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solved in practice?

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Many different tricks are
used to solve large integer

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optimization problems.

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One is to reformulate
the problem.

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The sports scheduling
problem with more teams

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is often solved by
changing the formulation.

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Instead of the
decision variables

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we discussed in this
lecture, the variables

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are sequences of games.

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Additionally, the problem can
be split into three smaller

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problems that can each be self
separately and much faster

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than just solving
the whole problem.

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Another trick that's
often used, is

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what are called
Heuristic methods.

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These methods find good, but not
necessarily optimal decisions.

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A good decision is
sometimes accepted

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since the problem
is so much easier

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to solve using a
heuristic method.

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In addition to changing
the formulation

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in using heuristics, there are
general purpose optimization

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solvers that can
solve large problems.

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These include CPLEX,
Gurobi, GLPK and Cbc,

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a COIN-OR project.

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Most practitioners who solve
large optimization problems

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use one of these
software packages.

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And in the past 20 years, the
speed of integer optimization

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solvers has increased
by a factor of 250,000,

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which doesn't even include the
increasing speed of computers.

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Assuming a modest
machine speedup of 1,000,

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this means that a
problem that can

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be solved in one second
today, took seven years

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to solve 20 years ago.

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00:02:34,470 --> 00:02:36,470
Because of this
increase in speed

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00:02:36,470 --> 00:02:39,260
were able to solve much
larger and more complicated

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optimization problems today,
than just a few years ago.

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00:02:45,130 --> 00:02:48,020
So how about the sports
scheduling problem?

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00:02:48,020 --> 00:02:52,000
When the Sports Scheduling
Group was started in 1996,

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integer optimization software
was too slow to be useful.

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Now, they can use
powerful solvers

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to generate sports schedules.

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Even with these
solvers, it can take

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months to make the major
league baseball schedule.

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This is due to several reasons,
including the enormous list

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of ever changing
constraints that they

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have to account for, the
need to define priorities

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on the constraints to find a
feasible solution, and the fact

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that it takes several iterations
to get a good schedule.

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But even with these
challenges, analytics

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offers a significant edge
in sports scheduling.

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The use of optimization
allows for the addition

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of new constraints
or schedule changes.

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A new schedule can
easily be generated

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based on an updated
requirement or request.

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Now, all professional sports
and most college sports,

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construct their schedules
using optimization.

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In this lecture, we've seen
one powerful use of integer

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optimization, but this
method has a huge number

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of applications,
which you'll see

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more of in the second
lecture, The Recitation,

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and in the homework assignment.