Class_Notes_Part_1B

Class_Notes_Part_1B - (2) Piecewise-Linear Functions In...

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(2) Piecewise-Linear Functions In general, we can minimize piecewise-linear convex functions, and we can maximize piecewise-linear concave functions. What's a piecewise-linear convex function!? 1. Piecewise-linear: the function is linear for a while, then there's a kink, then it's linear again, then another kink, then it's linear again, and so on. But it's continuous, and there are never curvy parts. 2. Convex: In this case, the slope of each linear segment keeps getting larger and larger as we look at the function from left to right. 3. Concave: In this case, the slope of each linear segment keeps getting smaller and smaller as we look left-to-right. But what if we're minimizing something that's not convex? 23
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(2) Piecewise-Linear Functions Example problem. We have a fire station that must be added to a town with a grid road network. Neighborhoods exist at coordinates (0,0), (2,8), (4,2), and (10,10). Our objective is to minimize the sum of travel times from the ¯ re station to each neighborhood. Fire trucks have to use the grid network; they cannot go directly from their station to a neighborhood in a straight line. Each city block coordinate takes a minute to drive. So if the fire station is located at (4,5), it will take 9 minutes to get to (0,0), 5 minutes to get to (2,8), 3 minutes to get to (4,2), and 11 minutes to get to (10,10). Where's the piecewise-linear function here?
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Model: Model 1: Model 2: 24
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(2) Piecewise-Linear Functions Bonus: what if we have a police station, with the same problem as above? Where would it go? Now, suppose that we want the fire station and police station to be at least 10 minutes away from one another. Can we use the piecewise-linear modeling trick? 25
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(2) Other piecewise-linear functions “Your call is free for the first 5 minutes, then $3.95 for every minute afterwards!" (Same idea as slacks; special case of piecewise-linear functions.) Travel time between two points is 10 minutes as long as there are not more than 100 cars on that segment. Then, each car adds another minute of travel time until 150 cars are reached. Then, each additional car adds another three minutes of travel time. Our objective function minimizes (
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Class_Notes_Part_1B - (2) Piecewise-Linear Functions In...

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