Objective maximize s 7 example 5 integer programming

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Unformatted text preview: hat every pair of persons in S are friends. OBJECTIVE: maximize |S| 7 Example 5: Integer programming INPUT: a set of variables x1, …, xn and a set of linear inequalities and equalities, and a subset of variables that is required to be integer. FEASIBLE SOLUTION: a solution x’ that satisfies all of the inequalities and equalities as well as the integrality requirements. OBJECTIVE: maximize ∑i ci xi Example: maximize 3x + 4y subject to 5x + 8y ≤ 24 x, y ≥ 0 and integer 8 Which of the following is false? 1. The Traveling Salesman Problem is a combinatorial optimization problem. 2. Integer Programming is a combinatorial optimization problem. 3. Every instance of a combinatorial optimization problem has data, a method for determining which solutions are feasible, and an objective function value for each feasible solution. 4. Warren G. Harding was the greatest American ✓ President. 9 The advantages of integer programs Rule of thumb: integer programming can model any of the variables and constraints that you really want to put into an LP, but can’t. Binary variables – xi = 1 if we decide to do project i (else, it is 0) – xi = 1 if node i is selected in the graph (else 0) – xij = 1 if we carry out task j after task i (else, 0) – xit = 1 if we take subject i in semester t (else, 0) 10 Examples of constraints If project i is selected, then project j is not selected. If x1 > 0, then x1 ≥ 10. x3 ≥ 5 or x4 ≥ 8. x1, x2, x3, x4, x5, are all different integers in {1, 2, 3, 4, 5} x is divisible by 7 x is either 1 or 2 or 4 or 8 or 32 11 Nonlinear functions can be modeled using integer programming f(x) 7 5 3 y 0 2 4 x y = 2x y = 9–x 3 7 9 x if 3 ≤ x ≤ 7 y = -5 + x 0 if 0 ≤ x ≤ 3 if 7 ≤ x ≤ 9 12 You mean, that you can write all of those constraints in an integer program. That’s so easy. No. That’s not what we mean! We mean that we can take any of these constraints, and there is a way of creating integer programming constraints that are mathematically equivalent. It’s not so easy at first, but it gets easier after you see some examples. We’ll show you how to do this one step at a time...
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This note was uploaded on 03/18/2014 for the course MGMT 15.053 taught by Professor Jamesorli during the Spring '07 term at MIT.

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