GE330_lect16 - Lecture 16 Integer and Mixed Integer...

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Lecture 16 Integer and Mixed Integer Problems March 30, 2009
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Application: Set Covering To promote on-campus safety, the U of A Security Department is in the process of installing emergency telephones at selected locations. The department wants to install the minimum number of telephones, provided that each of the campus main streets is served by at least one telephone. It is reasonable to place the telephones at street intersections so that each telephone will serve at least two streets. In the following picture, the eight candidate locations are denoted by numbers 1 to 8. 7
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Application: Set Covering In this example, we have 8 candidates corresponding to which we define 8 binary variables. x j = ( 1 if a telephone is installed in location j 0 otherwise It’s easy to see that the objective function is just: min 8 X j =1 x j . To make sure every street is served by at least one telephone, we need to make sure one of the two ends of a street is chosen to install a telephone. For example, for street A , we need to make sure at least one of location 1 and location 2 is chosen. This yields the constraint x 1 + x 2 1 . 8
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Application: Set Covering The overall ILP formulation is: max z = x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 s.t. x 1 + x 2 1 x 2 + x 3 1 x 4 + x 5 1 x 7 + x 8 1 x 6 + x 7 1 x 2 + x 6 1 x 1 + x 6 1 x 4 + x 7 1 x 2 + x 4 1 x 5 + x 8 1 x 3 + x 5 1 x j ∈ { 0 , 1 } The optimal solution is x 1 = x 2 = x 5 = x 7 = 1 all others are 0.
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This note was uploaded on 08/31/2010 for the course IESE GE 330 taught by Professor Nedich during the Spring '09 term at University of Illinois at Urbana–Champaign.

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GE330_lect16 - Lecture 16 Integer and Mixed Integer...

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