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WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 16 Integer Programming

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Example: land Allocation An area of land, divided into three types Three land uses: Timber, Forage, and Recreation Maximum budget Costs and revenues for each land use option Some external requirements Objective: maximize net profits Oct 15, 2012Wood 492 - Saba Vahid2 Example 8

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Oct 15, 20123 Why Integer Programming? Discrete inputs and outputs –e.g. selecting the number of shifts for a production facility (1,2, etc.) –Assigning equipment or personnel to production tasks (can’t assign 1.5 machines or half a person to do a task!) Wood 492 - Saba Vahid

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Oct 15, 20124 Binary (yes/no) variables –Variables are either 1(yes) or 0 (no) –Facility Location problem (a location is either selected or not) –Road building –Harvesting a block –Network problems (selecting a minimum distance/cost path from A to B in a network) Why Integer Programming? Wood 492 - Saba Vahid

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Oct 15, 20125 Logical conditions: if {x}, then {y} –If product A is made, then product B should be made too –If an activity is selected, it should be performed completely (all of a harvest block must be harvested) –Select one of a few possible options (selecting a cutting pattern) Why Integer Programming? Wood 492 - Saba Vahid

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Oct 15, 20126 Solve the LP relaxation and round the answers –effective when solution values are sufficiently large (errors may be ignored) –Normally the rounded answers are not feasible, or are far from optimal (example)example Exhaustive search of all feasible points –Computationally infeasible due to exponential growth of the number of answers Solution Approach Wood 492 - Saba Vahid

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Oct 15, 20127 Rounding LP Solutions Rounded solutions are not feasible (1,2) or (2,2) 01234 1 2 x2x2 x1x1 LP relaxation feasible region Optimal solution for LP relaxation (1.5,2) 0123 x1x1 1 2 x2x2 Z (objective function, Max) Optimal solution for LP relaxation (2,1.8) Rounded solution is not optimal (2,1) or infeasible (2,2) Optimal Integer solution (0,2) Back Z (objective function, Max) Wood 492 - Saba Vahid

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Oct 15, 20128 Branch and Bound –Divide and conquer! –Divide problem into smaller problems by portioning the feasible solution region Cutting Planes –Solve the LP relaxation of the problem –If answers are integer : Done! –Otherwise, add constraints until you reach an integer answer Solution Approach Wood 492 - Saba Vahid

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Next Class Integer formulation examples Branch and bound Oct 15, 20129Wood 492 - Saba Vahid

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Branch and Bound See Beale paper. Example: Maximize z=x1+x2 x2 x1.

Branch and Bound See Beale paper. Example: Maximize z=x1+x2 x2 x1.

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