FORS 8450 • Advanced Forest Planning Lecture 12 Tabu Search

1. FORS 8450 • Advanced Forest Planning Lecture 12 Tabu Search Change in the Value of a Medium-Sized Forest when Considering Spatial Harvest Scheduling Constraints Pete Bettinger Wise Batten, Jr.

2. Motivation for the Study A relatively rare spatial / temporal combination of resources 1. University of Georgia graduate course "Advanced Forest Planning" 2. HATT: Heuristic Algorithm Teaching Tool 3. Motivated student 4. Curiosity about impacts of potential green-up and adjacency restrictions

3. Methods Spatial Distribution of Pine Stands Conifer plantations Roads Streams

4. Methods Age Class Distribution

5. Methods Growth and Yield Projections 10 management regimes (prescriptions) were developed for each stand. SiMS 2003 (ForesTech International 2003) was used to model the prescriptions, providing NPV, harvest timing (thinning and clearcut), and potential revenues. Logical management prescriptions were devised for each of the stands, given the goals of the landowner (a preference for thinnings and medium-length rotations). The number of thinnings ranged from 0-2. Stumpage prices, taxes, and costs for silvicultural activities (e.g., site preparation and planting) were derived from current local knowledge.

6. Methods Forest Planning Problem Time horizon: 40 years Time periods: 1 year Objective: Maximize the net present value of future activities on the forest. Constraints: (1) A maximum clearcut area per period. (2) Adjacency restrictions for clearcuts (URM and ARM). Green-up periods assessed: 2-7 years

7. Methods Forest Planning Problem Types of spatial problems examined: Unit restriction model (URM): Green-up periods from 2-7 years Area restriction model (ARM) Green-up periods from 2-7 years Maximum clearcut sizes: 60, 120, 240 acres

8. Random feasible solution Methods Develop 1-opt neighborhood Develop 2-opt neighborhood Tabu Search 1-opt moves 2-opt moves Select candidate move Select candidate move No No Best solution ? Best solution ? Yes Yes Tabu ? Tabu ? No No Yes Yes Update solution Update solution 1-opt iterations complete? 2-opt iterations complete? No Yes No Yes Do another loop? No Report best solution Yes

9. Methods Tabu Search Parameters Aspiration criteria used. Tabu state for 1-opt moves applied to Stand / Prescription choices. Tabu state for 1-opt moves: 550 iterations (assessed 50-650). Tabu state for 2-opt moves applied to Stand / Stand swaps. Tabu state for 2-opt moves: 50 iterations (assessed 25-150). 50 independent runs of the heuristic were obtained for each type of spatial planning problem assessed.

10. Methods Linear Programming (LP) Solution A relaxed version of the problem was solved using linear programming. "Relaxed" = none of the spatial constraints are acknowledged. One could view these results as the "upper bound" on any solution that could be generated with the spatial constraints.

11. Methods Integer Programming (IP) Solutions Three of the spatial problems using the Unit Restriction Model of adjacency were solved using Integer programming. One could view these results as the "optimal" solutions to those scenarios that are assessed with the heuristic, since each will include the spatial constraints. A direct comparison of the IP and heuristic results helps us understand how well the heuristic performs for these types of problems.

12. Results Unit Restriction Model Change in NPV, as compared to "relaxed" LP solution: Green-up Change in NPV (years) 2 3 4 5 6 7 (%) - 0.6 - 1.0 - 3.3 - 7.6 - 13.3 - 15.2 • $12 / acre to $283 / acre

13. Results Area Restriction Model (240 acre maximum clearcut size) Change in NPV, as compared to "relaxed" LP solution: Green-up Change in NPV (years) 2 3 4 5 6 7 (%) - 0.1 - 0.1 - 0.1 - 0.1 - 0.1 - 0.2 • $1.50 / acre to $3.50 / acre

14. Results Area Restriction Model (120 acre maximum clearcut size) Change in NPV, as compared to "relaxed" LP solution: Green-up Change in NPV (years) 2 3 4 5 6 7 (%) - 0.1 - 0.1 - 0.2 - 0.4 - 0.4 - 1.3 • $1.50 / acre to $24 per acre

15. Results Area Restriction Model (60 acre maximum clearcut size) Change in NPV, as compared to "relaxed" LP solution: Green-up Change in NPV (years) 2 3 4 5 6 7 (%) - 0.1 - 0.4 - 1.0 - 1.4 - 2.5 - 4.8 • $2 / acre to $90 per acre

16. Results Performance Computer: Pentium IV, 2.4 GHz CPU URM solutions: 40 seconds each ARM solutions: 1.5 to 2.0 minutes each IP solution: 1 hour maximum

17. Results Validation of the Heuristic Model Solved the IP formulation of the URM model using LINDO 6.1. Pairwise adjacency constraints were used. Percent of solutions within 1% of IP solution 46 26 6 Average solution, compared to IP solution - 1.46 % - 2.68 % - 4.59 % Best solution, compared to IP solution - 0.25 % - 0.10 % - 0.24 % Green-up period 2 years 3 years 4 years Constraints 1,199 1,783 2,333

18. Discussion URM adjacency Technically, the URM model should only be used when all of the stands are about the same size as the maximum clearcut area allowed. Since they are not, the impacts are greater when using this model than when using the ARM model. Using this type of management model does not allow as much flexibility in harvest design as when using the ARM model.

19. Discussion Anticipatory Assessment of Impact The notion that net present value declines as green-up periods increase, or as maximum allowable clearcut sizes decrease, is not new. The level of impact is of interest, however, and should be assessed for a variety of landowner size classes and ownership distributions.

20. Discussion Drawbacks of the Heuristic A number of runs of the heuristic may be necessary for one to feel confident that they have developed a forest plan that could be close, in value, to the (perhaps unknown) global optimum solution. Speed of processing is a function of the computer programming language used and the potential speed of the computer's processor.

21. Conclusions Impact of green-up period length • A green-up period of 2-3 years did not seem to significantly affect the NPV of the resulting forest plans. • A longer green-up period (6-7 years) could reduce the NPV of the resulting forest plans 5-15%. Impact of maximum clearcut size restrictions • 240 acre maximum clearcut size does not affect NPV much at all. • 60 acre maximum clearcut size may affect NPV more dramatically (up to 5%), depending on the green-up period assumed.

22. Conclusions Value of using a heuristic over traditional integer programming • Solutions can be generated for problems difficult or impossible to solve with traditional integer programming. • The heuristic method for assessing the impacts of green-up and adjacency restrictions worked very well - the best heuristic solution was within 0.25% (of net present value) of the integer programming solution.

23. Further Information Study Results Batten, W.H., Jr., P. Bettinger, and J. Zhu. 2005. The effects of spatial harvest scheduling constraints on the value of a medium-sized forest holding in the southeastern United States. Southern Journal of Applied Forestry. 29: 185-193.