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Let's look at how
sports scheduling can

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be done with optimization by
starting with a small example.

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Suppose we're trying to schedule
a tournament between four

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teams-- Atlanta, Boston,
Chicago, and Detroit.

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We'll call these
teams A, B, C, and D.

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These teams are divided
into two divisions.

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Atlanta and Boston
are in one division,

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and Chicago and Detroit
are in the second division.

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Each team plays one game a
week for a total of four weeks.

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During the four weeks, each
team plays the other team

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in the same division
twice, and each team

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plays the teams in the
other divisions once.

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The team with the most
wins from each division

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will play in the
championship game.

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For this reason, teams prefer
to play divisional games later.

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If team A plays team C and
D in the first two weeks

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and wins both games while
team B also plays team C and D

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and loses both games,
A knows that they only

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need to win one of
the games against B

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to beat B in terms of wins
and to go to the championship.

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We can schedule this tournament
by using an optimization model.

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Our objective is to maximize
team preferences, which

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are that teams would like to
play divisional games later.

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Our decisions are
which teams should

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play each other each week.

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And we have three
types of constraints.

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Each team needs to play the
other team in their division

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twice, each team needs to play
the teams in the other division

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once, and each team should play
exactly one team each week.

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Let's start by discussing
our decision variables.

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We need to decide which teams
will play each other each week.

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To do this, let's define
decision variables

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which we'll call x_ijk.

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If team i plays
team j in week k,

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then x_ijk will be equal to 1.

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Otherwise, x_ijk equals 0.

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As an example, suppose team
A plays team C in week 2.

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Then x_AC2 would equal 1.

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Since A only plays C
once, we should have then

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that x_AC1, or A playing C in
week 1, should be equal to 0.

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Similarly, x_AC3 should equal
0, and x_AC4 should equal 0.

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This is called a binary
decision variable

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since it's a decision
variable that

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can only take two
values, 0 and 1.

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This is a new type
of decision variable,

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and it's what makes
integer optimization

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different from
linear optimization.

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The decision variables
in integer optimization

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can only take integer values.

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This includes binary
decision variables,

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like the ones we have here,
that can only be either 0 or 1.

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These variables
can model decisions

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like where to build
a new warehouse,

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whether or not to
invest in a stock,

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or assigning nurses to shifts.

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Integer optimization
problems can also

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have integer decision variables
that take values 1, 2, 3, 4, 5,

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etc.

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These variables
can model decisions

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like the number of new
machines to purchase,

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the number of workers
to assign for a shift,

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and the number of items
to stock in a store.

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Other than the new types of
variables, integer optimization

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is exactly like
linear optimization.

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But we'll see this week how
integer optimization variables,

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and especially binary variables,
can increase our modeling

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capabilities.

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Now let's go back
to our formulation.

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As we said before,
our decisions are

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which teams should play
each other each week.

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We'll model this with the
binary decision variables

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we just discussed--
x_ijk which equal 1

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if team i plays
team j in week k.

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Now let's use these
decision variables

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to form our constraints.

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The first constraint
is that each team

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should play the other team
in their division twice.

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So teams A and B should
play each other twice

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in the four weeks.

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This can be modeled with the
constraint x_AB1 + x_AB2 +

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x_AB3 + x_AB4 = 2.

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This will force two of
these decision variables

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to be equal to 1, and the
other two decision variables

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to be equal to 0.

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The ones that are equal to 1
will correspond to the weeks

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that A and B will
play each other.

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We'll have a similar
constraint for teams C and D.

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Our next constraint
is that each team

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should play the teams in
the other division once.

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So teams A and C should
play each other once

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in the four weeks.

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This can be modeled with a
constraint x_AC1 + x_AC2 +

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x_AC3 + x_AC4 = 1.

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This is very similar to
the previous constraint,

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except this time only one
of the decision variables

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will have value 1.

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We'll have similar
constraints for teams A and D,

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teams B and C, and teams B and
D. Our last type of constraint

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is that each team
should play exactly one

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other team each week.

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This means that A should
play B, C or D in week 1.

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This can be modeled
with a constraint x_AB1

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+ x_AC1 + x_AD1 = 1.

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Exactly one of these decision
variables will be equal to 1,

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meaning that A will play
that team in week 1.

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We'll have a similar constraint
for every team and week pair.

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Now, let's model our objective.

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Let's assume that teams
have a preference of 1

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for playing divisional games
in week 1, a preference of 2

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for playing divisional games
in week 2, a preference of 4

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for playing divisional games in
week 3, and a preference of 8

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for playing divisional
games in week 4.

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So the preference doubles
with each later week.

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Then we can model our objective
as x_AB1 + 2*x_AB2 + 4*x_AB3 +

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8*x_AB4 + x_CD1 + 2*x_CD2
+ 4*x_CD3 + 8*x_CD4.

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Then if team AB plays in week
3, we'll add 4 to our objective.

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If they play in
week 1, then we'll

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only add 1 to our objective.

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If they don't play in a
week, that term will be 0

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and will not contribute
to the objective.

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Now that we've set
up our problem,

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we're ready to solve it.

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In the next video, we'll
set up and solve our problem

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in LibreOffice.