FORS 8450 • Advanced Forest Planning Lecture 6 Threshold Accepting

1. FORS 8450 • Advanced Forest Planning Lecture 6 Threshold Accepting

2. Threshold Accepting

3. Threshold Accepting Background Threshold accepting was introduced by Dueck and Scheuer (1990), and has been used in a number of fields, including forestry, to plan activities. For forest planning, the general implementation goes something like this: During the scheduling of activities, a management unit and time period are selected at random (proposed). Constraints are assessed. If the proposed solution is feasible, the threshold accepting algorithm accepts it if it is not much worse than the previous current (or best) solution. The approach is a form of Monte Carlo method that uses a local search, in which a subset of solutions is explored by moving from one solution to a neighboring solution. To avoid converging and becoming stuck in a local maximum (or minimum) the procedure provides for an occasional acceptance of an inferior solution, which allows it to move away from a local maximum.

4. Threshold Accepting Necessary parameters 1) A beginning threshold level (in objective function units). 2) An ending threshold level. 3) An assumption of the number of iterations per threshold. 4) A rate of change for the threshold. Other assumptions 1) Do infeasible solutions or unsuccessful suggested changes count toward the number of iterations per threshold? 2) Can the initial threshold be taken out of the hands of the user and estimated based on the quality of the initial solution? 3) Is the threshold change linear or non-linear? 4) Does the threshold apply to the previous current or the best solution?

5. Threshold Accepting A general note for this and other algorithms Threshold accepting is in a class of algorithms that generally operate in a 1-opt or 2-opt random decision mode. 1-opt: A change is made to a single decision unit (e.g., the timing of a clearcut for a stand, or the assignment of a management regime to a stand). • One stand is chosen at random • A decision is chosen at random for the stand 2-opt: A change is made simultaneously to two decision units, and they swap assignments. • Two stands are chosen at random • The decisions assigned to each are swapped

6. Threshold Accepting Characteristics of the algorithm 1) A solution is improved upon as the algorithm operates. 2) Changes that lead to higher quality solutions are always welcome. 3) Changes that lead to lower quality solutions are acceptable as well, as long as they are not too far away from the previously-saved best solution. Example: A change is suggested that leads to a lower quality solution - $1,500 in NPV lower that the previously saved "best" solution. As long as the decrease in quality is smaller than the current threshold, the change is acceptable. 4) The threshold begins as a large number (user-defined), but decreases in size as the algorithm operates. 5) The algorithm stops and reports the best solution when the threshold is very small (user-defined).

7. Select a choice at random Threshold Accepting Evaluate goals Basic Process Feasible, better solution? Accept the choice Yes No Feasible, Within threshold? Accept the choice Yes No Have we reached the stopping criteria? No Yes Stop and report the best solution found during search

8. Read data Threshold Accepting A Specific Forest Planning Process Four broad steps. Step 3 is described in more detail next. Clear arrays Step 1 Develop initial random solution Step 2 Schedule activities Step 3 Δ Th? Change threshold Step 4 Done? Report best solution

9. Threshold Accepting Select unit and period A Specific Forest Planning Process Step 3 Yes Violated? Check adjacency constraint Develop a solution No Best? Save as best solution Calculate solution value Yes No Pass? Save as current solution Check TA criteria Yes (Return) No Exhausted? Revert to previous solution Yes (Return) No

10. Threshold Accepting Change in solution value over about 8,000 iterations for a specific forest planning problem with a maximization objective.

11. Threshold Accepting Change in solution value over about 8,000 iterations for a specific forest planning problem with a maximization objective.