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Unformatted text preview: hat every pair of persons in S are friends. OBJECTIVE: maximize S 7 Example 5: Integer programming INPUT: a set of variables x1, …, xn and a set of
linear inequalities and equalities, and a subset of
variables that is required to be integer. FEASIBLE SOLUTION: a solution x’ that satisfies
all of the inequalities and equalities as well as the
integrality requirements. OBJECTIVE: maximize ∑i ci xi Example: maximize 3x + 4y subject to 5x + 8y ≤ 24
x, y ≥ 0 and integer
8 Which of the following is false?
1. The Traveling Salesman Problem is a
combinatorial optimization problem.
2. Integer Programming is a combinatorial
optimization problem.
3. Every instance of a combinatorial optimization
problem has data, a method for determining
which solutions are feasible, and an objective
function value for each feasible solution.
4. Warren G. Harding was the greatest American
✓
President.
9 The advantages of integer programs Rule of thumb: integer programming can model
any of the variables and constraints that you
really want to put into an LP, but can’t. Binary variables
– xi = 1 if we decide to do project i (else, it is 0)
– xi = 1 if node i is selected in the graph (else 0)
– xij = 1 if we carry out task j after task i (else, 0) – xit = 1 if we take subject i in semester t (else, 0)
10 Examples of constraints If project i is selected, then project j is not selected. If x1 > 0, then x1 ≥ 10. x3 ≥ 5 or x4 ≥ 8. x1, x2, x3, x4, x5, are all different integers in {1, 2, 3, 4, 5} x is divisible by 7 x is either 1 or 2 or 4 or 8 or 32 11 Nonlinear functions can be modeled using
integer programming
f(x)
7
5
3 y 0 2 4 x y = 2x
y = 9–x 3 7 9 x if 3 ≤ x ≤ 7 y = 5 + x
0 if 0 ≤ x ≤ 3 if 7 ≤ x ≤ 9
12 You mean, that
you can write
all of those
constraints in
an integer
program.
That’s so easy. No. That’s not what we mean! We
mean that we can take any of these
constraints, and there is a way of
creating integer programming
constraints that are mathematically
equivalent. It’s not so easy at first,
but it gets easier after you see some
examples. We’ll show you how to do
this one step at a time...
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This note was uploaded on 03/18/2014 for the course MGMT 15.053 taught by Professor Jamesorli during the Spring '07 term at MIT.
 Spring '07
 JamesOrli
 Management, Alice in Wonderland

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