Integer Programming 1

# Integer Programming 1 - Introduction to Integer Programming...

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MIT and James Orlin © 2003 1 Introduction to Integer Programming Integer programming models

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MIT and James Orlin © 2003 2 A 2-Variable Integer program maximize 3x + 4y subject to 5x + 8y 24 x, y 0 and integer What is the optimal solution?
The Feasible Region 0 1 2 3 4 5 0 1 2 3 4 5 Question: What is the optimal integer solution? What is the optimal linear solution? Can one use linear programming to solve the integer program?

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A rounding technique that sometimes is useful, and sometimes not. 0 1 2 3 4 5 0 1 2 3 4 5 Solve LP (ignore integrality) get x=24/5, y=0 and z =14 2/5. Round, get x=5, y=0, infeasible! Truncate, get x=4, y=0, and z =12 Same solution value at x=0, y=3. Optimal is x=3, y=1 , and z =13
MIT and James Orlin © 2003 5 Why integer programs? Advantages of restricting variables to take on integer values More realistic More flexibility Disadvantages More difficult to model Can be much more difficult to solve

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MIT and James Orlin © 2003 6 On 0-1 variables Integer programs: linear equalities and inequalities plus constraints that say a variable must be integer valued. We also permit “x j {0,1}.” This is equivalent to 0 x j 1 and x j integer.
MIT and James Orlin © 2003 7 The mystery of integer programming Some integer programs are easy (we can solve problems with millions of variables) Some integer programs are hard (even 100 variables can be challenging) It takes expertise and experience to know which is which It’s an active area of research

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MIT and James Orlin © 2003 8 The game of fiver. Click on a circle, and flip its color and that of adjacent colors. Can you make all of the circles red?
MIT and James Orlin © 2003 9 The game of fiver.

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MIT and James Orlin © 2003 10 The game of fiver.
MIT and James Orlin © 2003 11 The game of fiver. Let’s write an optimization problem whose solution solves the problem in the fewest moves.

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MIT and James Orlin © 2003 12 Optimizing the game of fiver. 1 2 3 4 5 1 2 3 4 5 Let x(i,j) = 1 if I click on the square in row i and column j. x(i,j) = 0 otherwise. Focus on the element in row 3, and column 2. To turn it red, we require that x(2,2) + x(3,1) + x(3,2) + x(3,3) + x(4,2) is odd
MIT and James Orlin © 2003 13 Optimizing the game of fiver (i, j) to be red for i = 1 to 5 and for j = 1 to 5 We want to minimize the number of moves. Minimize i,j=1 to 5 x(i,j) Subject to x(i, j) + x(i, j-1) + x(i, j+1) + x(i-1, j) + x(i+1, j) is odd for i = 1 to 5, j = 1 to 5 x(i, j) is 0 or 1 for i = 1 to 5 and j = 1 to 5 x(i, j) = 0 otherwise.

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