WOOD 492 MODELLING FOR DECISION SUPPORT

1. WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 16 Integer Programming

2. 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 Example 8 Wood 492 - Saba Vahid

3. 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

4. Why Integer Programming? • 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) Wood 492 - Saba Vahid

5. Why Integer Programming? • 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) Wood 492 - Saba Vahid

6. Solution Approach • 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) • Exhaustive search of all feasible points • Computationally infeasible due to exponential growth of the number of answers Wood 492 - Saba Vahid

7. Rounding LP Solutions Rounded solutions are not feasible (1,2) or (2,2) x2 2 Optimal solution for LP relaxation (1.5,2) Z (objective function, Max) 1 LP relaxation feasible region 0 1 2 3 4 x1 Optimal Integer solution (0,2) x2 Optimal solution for LP relaxation (2,1.8) 2 Z (objective function, Max) 1 Rounded solution is not optimal (2,1) or infeasible (2,2) 0 1 2 3 x1 Back Wood 492 - Saba Vahid

8. Solution Approach • 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 Wood 492 - Saba Vahid

9. Next Class • Integer formulation examples • Branch and bound Wood 492 - Saba Vahid