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Let us demonstrate
the fundamentals

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of the approach that is called
integer optimization that

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is used in the eHarmony
matching algorithm.

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Suppose we have three
men and three women.

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Their compatibility scores range
between 1 and 5 for all pairs,

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and they are shown
in the figure.

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For example, the first
man and the first woman

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have a compatibility score of 1.

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The first man and
the second woman

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have a compatibility
score of 3, and so forth.

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So how should we
match pairs together

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to maximize compatibility?

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The red line indicates a match.

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In this case man one is
matched to the third woman

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with a compatibility score of 5.

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The second man to
the first woman

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with a compatibility score of 4.

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And the third man to the second
woman with a compatibility

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score of 5.

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And a total compatibility
score of 14.

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Let us now formulate
this matching problem

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in the language of
integer optimization.

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We denote the data for the
matching problem as w_ij.

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These numbers represent
the compatibility score

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between user i and user j.

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In the example shown in
the figure, w_13 is 5.

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We first define
decision variables x_ij.

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These are binary
variables taking

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value 1 if we match user i and
user j together, and value 0

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otherwise.

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We next define the
objective function,

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which is to maximize
compatibility between matches.

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In this case, we maximize
the sum of (w_11*x_11) +

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(w_12*x_12) and so
forth, up to (w_33*x_33).

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We next define the constraints
for the decision variables

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x_ij.

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Each man-- say, man one --should
match to exactly one woman.

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And we represent this by
the mathematical constraint

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as follows-- x_11
+ x_12 + x_13 = 1.

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Similarly, each
woman-- say, woman one

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--should match to
exactly one man.

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And we represent this
constraint mathematically

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as x_11 + x_21 + x_31 = 1.

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The full optimization
problem is indicated next.

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First we have the objective
function we indicated earlier.

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Next we have the
set of constraints

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that indicate that every man
should match with exactly one

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woman.

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Next we have the
constraints that every woman

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should match with one man.

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And finally that the
variables are binary.

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Let us now extend the
methodology to other areas.

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Suppose, for example, we want
to show each woman her top two

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matches.

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This is represented
by the constraint x_11

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+ x_21 + x_31 = 2.