09.2 integer programming.pdf

8/10/2019 09.2 Integer Programming.pdf

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Solving Combinatorial Optimization Problems: exact

techniques

Relaxation If we do not consider one or more constraints of the originalproblem P O, we can transform the problem in another problem that can be

simpler to solve P R. Considering that the problem is a minimization

problem, the optimal values of the objective function are related in the

following way:

f?PR f?

PO

(with less constraints the solution can only improve or stay the same).

Linear relaxation transform an integer problem in a problem with

continuous variables by dropping the constraint that the variables must beinteger (or Binary). This allows us to use the Simplex Algorithm instead of

the more complex, and far more time consuming, tree search.


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Solving the linear relaxation

Problem PL0:

max F = 3x+ 7y

subject to:

x 3.55x 4y 10

4

7x + 2y 9

x , y 0Optimal, non integer, solution:

x= 3.5 and y= 3.5; F = 35


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Branching in x: x 3

max F = 3x+ 7y

subject to:

x 3.5

5x 4y 10

4

7x + 2y 9

x , y 0

x 3

Solution (non integer):x= 3 and y= 3.6; F = 34.5


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Branching in x: x 4

max F = 3x+ 7y

subject to:

x 3.5

5x 4y 10

4

7x + 2y 9

x , y 0

x 4Without admissible solutions.


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Branching in y: y 4

max F = 3x+ 7y

subject to:

x 3.5

5x 4y 104

7x + 2y 9

x , y 0

x 3

y 4 Solution (non-integer):

x= 1.7 and y= 4; F = 33.2


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Branching in x: x 1

max F = 3x+ 7y

subject to:

x 3.5

5x 4y 10

47x + 2y 9

x , y 0

x 3

y 4

x 1

Solution (non-integer):

x= 1 and y= 4.2; F = 32.5


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Branching in x: x 2

max F = 3x+ 7y

subject to:

x 3.5

5x 4y 10

47x + 2y 9

x , y 0

x 3

y 4

x 2 Without feasible solutions.


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Branching in y: y 4

max F = 3x+ 7y

subject to:

x 3.5

5x 4y 10

47x + 2y 9

x , y 0

x 3

y 4

x 1y 4

Solution (integer):

x= 1 and y= 4; F = 31Best integer solution obtained so far!


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Branching in y: y 5

max F = 3x+ 7y

subject to:

x 3.5

5x 4y 10

47x + 2y 9

x , y 0

x 3

y 4

x 1y 5

,

,

-

I

h ; 4y I O

,

-,

/

/

-

,

/

-

- ;

4I

h y

9

,

/

/

,

,

, , ,

,

,

Without admissible solutions.


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Branch-and-Bound tree

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/ 0 1

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Upper and lower Bounds

the branch & bound is more efficient because it is possible to cut

nodes of the search tree without exploring them, because we are surenot to obtain better solutions than the ones we already have;

allow us to measure the distance, (considering the value of the

objective function) to the optimal solution.


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Bounds in a maximization problem:

alowerlimitLIis given by an

integer solution that has beenpreviously obtained the opti-

mal solution F? can never be

worse (lower) than the integer

solution that we already have;

an upperlimit LSis given bythe highest value of the objec-

tive function within all the no-

des that are not yet completely

explored (we hope to find there

an integer solution that is bet-ter than the one we got).

Maximization Problem

F*- LB UB - LB

F

Upper Bound UB

Value of the objective function for themost promising unexplored node

Optimal integer solution F*(unknown)

Lower bound LB

Best integer solution found(highest value of the objective

function)


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Bounds example

Consider a maximization problem where all the variables are integer. The

tree in figure1is obtained along the resolution process.

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09.2 integer programming.pdf

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