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PROFESSOR GOODCHILD CEE 587 APRIL 15, 2009 Basic Optimization

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f(x) x objective function constraints

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Optimization Problem Elements Objective function The thing you want to minimize or maximize Typically total cost Constraints The complexities of the problem E.g. capacities, demands, supply An unconstrained optimization leads to 0 or infinity

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f(x) x objective function (nonlinear) minimum Constraints can’t improve the solution

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Route choice Objective function Minimize k c k y k Subject to: k y k =1 y k =0 or 1 where, c k = cost of travel on link k y k =1 if link k is used, 0 otherwise

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c1c1 c2c2 c3c3 y1y1 y2y2 y3y3

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c 1 =4 y1y1 y2y2 y3y3 c 2 =2 c 3 =3 y1y1 y2y2 y3y3 Objective function 1004 1106 1119 0102 0013

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Assigning Customers to Vehicles Minimize i k c ik y ik Subject to: k y ik =1 i a i y ik ≤b k y ik =0 or 1 where, c ik = cost of serving customer i with vehicle k a i =order size from customer i b k =capacity of vehicle k y ik =1 if customer i is assigned to vehicle k, 0 otherwise

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