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Solving Integer Programming with BranchandBound Technique
This is the divide and conquer method. We divide a large problem into a few smaller ones. (This is the
“branch” part.) The conquering part is done by estimate how good a solution we can get for each smaller
problems (to do this, we may have to divide the problem further, until we get a problem that we can handle),
that is the “bound” part.
We will use the
linear programming relaxation
to estimate the optimal solution of an integer programming.
* For an integer programming model
P
, the linear programming model we get by dropping the require
ment that all variables must be integers is called the linear programming relaxation of
P
.
The steps are:
•
Divide a problem into subproblems
•
Calculate the LP relaxation of a subproblem
—
The LP problem has no feasible solution, done;
—
The LP problem has an integer optimal solution; done. Compare the optimal solution with the
best solution we know (the incumbent).
—
The LP problem has an optimal solution that is worse than the incumbent, done.
In all the cases above, we know all we need to know about that subproblem. We say that subproblem
is fathomed.
—
The LP problem has an optimal solution that are not all integer, better than the incumbent. In
this case we would have to divide this subproblem further and repeat.
Example 1
Max
Z
=
−
x
1
+4
x
2
Subject to
−
10
x
1
+20
x
2
≤
22
5
x
1
+10
x
2
≤
49
x
1
≤
5
x
i
≥
0
,x
i
’s are integers
1
0
1
2
3
4

1
123456
(3.8,3)
1
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 Spring '11
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