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# lect23 - ISE 536Fall03 Linear Programming and Extensions...

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ISE 536–Fall03: Linear Programming and Extensions November 26, 2003 Lecture 23: Integer Programming, Modeling Lecturer: Fernando Ord´ o˜nez 1 Mixed Integer Programming Problem Here we study linear programming problems with some variables constrained to be integer. For example: min c t x + d t y s . t . Ax + By = b x, y 0 x integer Mixed integer programming problem: Integer programming problem: Binary programming problem: Example (zero-one knapsack problem). We are given n items, each has weight w j and value c j . Given a bound K on the weight that we can carry on the knapsack, we would like to select the items that maximize the total value. Formulate this as an integer programming problem. 1

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Things that can be represented with integer variables: x y and x, y binary: x 1 + x 2 + x 3 1, x i binary: x 1 + x 2 + x 3 = 1, x i binary: The vectors a i 0, i = 1 , . . . , m and a t i x b i y i i = 1 , . . . , m m i =1 y i k y i { 0 , 1 } i = 1 , . . . , m x = m j =1 a j y j , m j =1 y j = 1, and y j binary: How can we use integer variables to represent a piecewise linear function? (consider
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lect23 - ISE 536Fall03 Linear Programming and Extensions...

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