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Unformatted text preview: . But
first, we’ll review what we
mean by integer programs. 13 Integer Programs
Integer programs: a linear program plus the
additional constraints that some or all of the
variables must be integer valued.
We also permit “xj ∈{0,1},” or equivalently,
“xj is binary”
This is a shortcut for writing the constraints:
0 ≤ xj ≤ 1 and xj integer. 14 Types of Integer Programs
Mixed integer linear
programs
(MILPs or MIPs) Pure Integer Programs 01 Integer
Programs xj ≥ 0 and integer for some or all j. xj ≥ 0 and integer for every j. xj ∈ {0,1} for every j. Note, pure integer programming instances that are unbounded
can have an infinite number of solutions. But they have a
finite number of solutions if the variables are bounded. 15 Goals of lectures on Integer Programming. Lectures 1 and 2 – Introduce integer programming
– Techniques (or tricks) for formulating
combinatorial optimization problems as IPs Lectures 3 and 4.
– How integer programs are solved (and why
they are hard to solve).
• Rely on solving LPs fast
• Branch and bound and cutting planes Lecture 5. Review and modeling practice
16 A 2Variable Integer program maximize 3x + 4y subject to 5x + 8y ≤ 24
x, y ≥ 0 and integer What is the optimal solution? 17 The Feasible Region 5 Question: What is the
optimal integer solution? 3 4 What is the optimal linear
solution? 0 1 2 Can one use linear
programming to solve the
integer program? 0 1 2 3 4 5 A rounding technique that sometimes
is useful, and sometimes not. 0 1 2 3 4 5 Solve LP (ignore
integrality) get x=24/5,
y=0 and z =14 2/5.
Round, get x=5, y=0,
infeasible!
Truncate, get x=4, y=0,
and z =12
Same solution value at
x=0, y=3.
Optimal is x=3, y=1, and
z =13
0 1 2 3 4 5 0 0 1 1 2 2 3 3 4 4 5 5 Consider the feasible regions for the
two integer programs on this slide. 0 1 2 3 4 max 3x + 4y s.t. 5 5x + 8y ≤ 24 x, y ≥ 0 and integer 0 1 2 3 max x+ 5 3x + 4y s.t. 4 y≤ 4 2x + 3y ≤ 9
x, y ≥ 0 and integer 20 Which of the following is false for the two
integer programs on the previous slide?
1. The two models are the same in that they
have the same feasible regions a...
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 Spring '07
 JamesOrli
 Management, Alice in Wonderland

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