tut7 - Transformations in Integer Programming In this...

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1 Transformations in Integer Programming Cleaver, an MIT Beaver In this tutorial, we explain a variety of transformations in integer programming.
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2 But first, a hard question. Tim, the turkey Why haven’t there been any tutorials in such a long time? I missed them too. But we have created lots of other educational materials.
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3 Transforming Logical Conditions Max 16x 1 + 22x 2 + 12x 3 + 8x 4 + 11x 5 + 19x 6 5x 1 + 7x 2 + 4x 3 + 3x 4 + 4x 5 + 6x 6 14 x j {0,1} for each j = 1 to 6 Nooz, the most trusted name in fox. Here is the integer program used in the first integer programming lecture. As you recall, it’s based on a game show that I was on. So that we can learn modeling of logical constraints, I’m going to pretend to add conditions on what I will choose.
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4 Max 16x 1 + 22x 2 + 12x 3 + 8x 4 + 11x 5 + 19x 6 5x 1 + 7x 2 + 4x 3 + 3x 4 + 4x 5 + 6x 6 14 x j {0,1} for each j = 1 to 6 Nooz, the most trusted name in fox. Suppose that I refuse to select item 5 if I have also selected item 1. In this case, the feasible solutions for Nooz will permit x 1 = 1 or x 5 = 1, but not both. This can be modeled via the linear constraint, x 1 + x 2 1
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5 Tim I don’t get it. The linear constraint doesn’t look anything like the logical constraint. Ollie Well, Tim. The important thing isn’t whether it “makes sense” as a logical constraint. The important thing is that an optimal solution for the integer program will produce an optimal solution for the original problem. But how will we know that? In this case, we need only focus on variables x 1 and x 5 to see that the constraint works. We don’t need to think about the other variables. You see, Nooz just wants to eliminate the possibility that items 1 and 5 are both selected. In other words, we cannot have x 1 = 1 and x 5 = 1. The linear constraint accomplishes the same thing, taking into account that x 1 and x 5 are both binary.
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6 Another logical constraint Max 16x 1 + 22x 2 + 12x 3 + 8x 4 + 11x 5 + 19x 6 5x 1 + 7x 2 + 4x 3 + 3x 4 + 4x 5 + 6x 6 14 x j {0,1} for each j = 1 to 6 Nooz Suppose in this illustration that I will take item 3 if and only if I take item 4. How do we model that. In this case, the feasible solutions for Nooz will permit x 3 = 1 and x 4 = 1, or else x 3 = 0 and x 4 = 0, We can model it as the linear constraint x 3 = x 4 .
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7 Ollie, can you explain it to me.
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This note was uploaded on 01/18/2012 for the course MGMT 15.053 taught by Professor Jamesorli during the Spring '07 term at MIT.

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tut7 - Transformations in Integer Programming In this...

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