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FORS 8450 Advanced Forest Planning Lecture 21 Binary Search.

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1 FORS 8450 Advanced Forest Planning Lecture 21 Binary Search

2 Binary Search Allows one to find the maximum even-flow of volume or value that can be sustained over a planning horizon, given certain constraints. Used in conjunction with the appropriate forest inventory and growth and yield estimates. A process that attempts to find a solution to a problem by making progressively better guesses at the optimal value of the objective function. A simulation technique. FORS 4710 / 6710 Forest Planning

3 Fun Example FORS 4710 / 6710 Forest Planning

4 Binary Search Pick a number between 1 and 1000. I can tell you what it is in less than 11 guesses if you tell me whether my guess is higher or lower than your number. FORS 4710 / 6710 Forest Planning

5 Binary Search Process: 1) Set a target value 2) Determine the range of potential solution values in an ordered list 3) Select the mid-point of the ordered list 4) Compare the solution at the mid-point to the target 5) Make a decision a) increase the target b) decrease the target c) stop and report a solution 6) Return to Step 1 if necessary FORS 4710 / 6710 Forest Planning

6 Finance Example FORS 4710 / 6710 Forest Planning

7 Binary Search You have $1,000 in the bank, earning 3% interest. You want to withdraw, at the end of each of three years, the same amount of money. What do you decide to do.......? FORS 4710 / 6710 Forest Planning

8 Binary Search Start with 1/3 of what might be in the account at the end of the first year: What will be there at the end of the first year? $1,030.00 One third of that? $343.33 If this estimate leaves a surplus, increase it by $20. If this estimate leaves a deficit, decrease it by $10 (1/2 of the increase). FORS 4710 / 6710 Forest Planning

9 Binary Search You have $1,000 in the bank, earning 3% interest. You want to withdraw, at the end of each of three years, the same amount of money. First attempt: Beginning of year 1 balance End of year 1 balance End of year 1 withdrawal Beginning of year 2 balance End of year 2 balance End of year 2 withdrawal Beginning of year 3 balance End of year 3 balance End of year 3 withdrawal Surplus or deficit $1,000.00 1,030.00 343.33 686.67 707.27 343.33 363.94 374.86 343.33 + 31.53 (1/3 of 1,030.00) FORS 4710 / 6710 Forest Planning

10 Binary Search You have $1,000 in the bank, earning 3% interest. You want to withdraw, at the end of each of three years, the same amount of money. Second attempt: Beginning of year 1 balance End of year 1 balance End of year 1 withdrawal Beginning of year 2 balance End of year 2 balance End of year 2 withdrawal Beginning of year 3 balance End of year 3 balance End of year 3 withdrawal Surplus or deficit $1,000.00 1,030.00 363.33 666.67 686.67 363.33 323.34 333.04 363.33 - 30.29 (1/3 of 1,030.00 + $20) FORS 4710 / 6710 Forest Planning

11 Binary Search You have $1,000 in the bank, earning 3% interest. You want to withdraw, at the end of each of three years, the same amount of money. Third attempt: Beginning of year 1 balance End of year 1 balance End of year 1 withdrawal Beginning of year 2 balance End of year 2 balance End of year 2 withdrawal Beginning of year 3 balance End of year 3 balance End of year 3 withdrawal Surplus or deficit $1,000.00 1,030.00 353.33 676.67 696.97 353.33 343.64 353.95 353.33 + 0.62 (1/3 of 1,030.00 + $10) FORS 4710 / 6710 Forest Planning

12 Binary Search Why is this "even-flow" situation such a problem? FORS 4710 / 6710 Forest Planning

13 Binary Search in Forest Planning In forest planning, a binary search process that uses a simple harvest volume target and either stand-level volumes or strata-based volumes is relatively easy to perform. This type of planning model can be implemented within a spreadsheet environment if the attributes of the stands or strata that contribute to the objective function are available. Sorting the list and selecting the stands or strata for harvest is relatively straight-forward. This assumes that stands or strata can be assigned fractional values related to the harvesting decision, thus some are scheduled for harvest during more than one time period. When stands need to be modeled using integer decision variables, and when the adjacency of harvests must be recognized and accommodated, the use of binary search becomes more complex... FORS 4710 / 6710 Forest Planning

14 Binary Search in Forest Planning Binary search has traditionally been used to determine the highest timber volume that can be produced over the assumed time horizon. Caveats: 1. Since an individual stand will likely not produce as much volume as is necessary to achieve the highest even volume from a forest, stand-level harvest decisions are aggregated. It may take harvesting several stands in each year to produce the harvest level necessary. 2. The scheduling of individual stands begins with the first time period. Harvests are scheduled until the last one exceeds the harvest target for the year. Subsequently, harvests for the next time period are scheduled. This process continues until harvests have been scheduled for each time period, or we run out of harvesting options because no other stands are old enough, for example, to harvest. 3. The selection of harvest units is from a sorted list of harvesting options. What are the sorting rules? FORS 4710 / 6710 Forest Planning

15 Binary Search in Forest Planning Process: 1) Select a target harvest volume 2) Select an increment for the harvest volume 3) Schedule stands until the target has been reached in a time period 4) Move to the next time period and continue scheduling all time periods 5) Assess the solution. Was the target reached in all time periods? 6) Make a decision a) increase the target b) decrease the target c) stop and report a solution 7) Return to Step 1 if necessary FORS 4710 / 6710 Forest Planning

16 Binary Search in Forest Planning Definitions: Target harvest level: The level of harvest volume you expect to obtain each time period within the planning horizon. Step size: How much the target harvest volume will change with each iteration of a binary search model. Iteration: One complete schedule of activities for the entire planning horizon. Stopping rule: Stop the scheduling process when the step size is at least this "small." FORS 4710 / 6710 Forest Planning

17 Forestry Example FORS 4710 / 6710 Forest Planning

18 Timber yields AgeYield (ft 3 / acre) 201,750 303,440 404,290 504,740 605,070 705,330 805,560 Binary Search Example Loblolly pine forest 30-year rotation is desired Site index 90 30 acres Current age class structure: 20 acres of 55-year old loblolly pine 10 acres of 15-year old loblolly pine Planning assumptions: Plan in 10-year increments (10-year time periods) Develop a plan for 6 time periods Assume harvesting occurs in the middle of each time period Binary Search Assumptions: Target harvest level 40,000 ft 3 per time period Step size 5,000 ft 3 Harvest oldest aged timber first. FORS 4710 / 6710 Forest Planning

19 Binary Search Example - Iteration #1, Time Period 1 Middle of time period 1 (5 years from now) Age class structure before harvest 20 acres of 60-year old loblolly pine 10 acres of 20-year old loblolly pine AgeAcres Volume per acre (ft 3 ) Total volume (ft 3 ) 60 20 10 5,070 1,750 101,400 17,500 Harvest acres Harvest volume (ft 3 ) 7.89 0 40,000 0 Before harvest FORS 4710 / 6710 Forest Planning

20 Binary Search Example - Iteration #1, Time Period 2 Middle of time period 2 (15 years from now) Age class structure before harvest 12.11 acres of 70-year old loblolly pine 10 acres of 30-year old loblolly pine 7.89 acres of 10-year old loblolly pine AgeAcres Volume per acre (ft 3 ) Total volume (ft 3 ) 70 30 10 12.11 10.00 7.89 5,330 3,440 0 64,546 34,400 0 Harvest acres Harvest volume (ft 3 ) 7.50 0 40,000 0 Before harvest FORS 4710 / 6710 Forest Planning

21 Binary Search Example - Iteration #1, Time Period 3 Middle of time period 3 (25 years from now) Age class structure before harvest 4.61 acres of 80-year old loblolly pine 10 acres of 40-year old loblolly pine 7.89 acres of 20-year old loblolly pine 7.50 acres of 10-year old loblolly pine AgeAcres Volume per acre (ft 3 ) Total volume (ft 3 ) 80 40 20 10 4.61 10.00 7.89 7.50 5,560 4,290 1,750 0 25,632 42,900 13,808 0 Harvest acres Harvest volume (ft 3 ) 4.61 3.35 0 25,632 14,638 0 Before harvest FORS 4710 / 6710 Forest Planning

22 Binary Search Example - Iteration #1, Time Period 4 Middle of time period 4 (35 years from now) Age class structure before harvest 6.65 acres of 50-year old loblolly pine 7.89 acres of 30-year old loblolly pine 7.50 acres of 20-year old loblolly pine 7.96 acres of 10-year old loblolly pine AgeAcres Volume per acre (ft 3 ) Total volume (ft 3 ) 50 30 20 10 6.65 7.89 7.50 7.96 4,740 3,440 1,750 0 31,521 27,142 13,125 0 Harvest acres Harvest volume (ft 3 ) 6.65 2.46 0 31,521 8,479 0 Before harvest FORS 4710 / 6710 Forest Planning

23 Binary Search Example - Iteration #1, Time Period 5 Middle of time period 5 (45 years from now) Age class structure before harvest 5.43 acres of 40-year old loblolly pine 7.50 acres of 30-year old loblolly pine 7.96 acres of 20-year old loblolly pine 9.11 acres of 10-year old loblolly pine AgeAcres Volume per acre (ft 3 ) Total volume (ft 3 ) 40 30 20 10 5.43 7.50 7.96 9.11 4,290 3,440 1,750 0 23,295 25,800 13,930 0 Harvest acres Harvest volume (ft 3 ) 5.43 4.86 0 23,295 16,705 0 Before harvest FORS 4710 / 6710 Forest Planning

24 Binary Search Example - Iteration #1, Time Period 6 Middle of time period 6 (55 years from now) Age class structure before harvest 2.64 acres of 40-year old loblolly pine 7.96 acres of 30-year old loblolly pine 9.11 acres of 20-year old loblolly pine 10.29 acres of 10-year old loblolly pine AgeAcres Volume per acre (ft 3 ) Total volume (ft 3 ) 40 30 20 10 2.64 7.96 9.11 10.29 4,290 3,440 1,750 0 11,326 27,382 15,943 0 Harvest acres Harvest volume (ft 3 ) 2.64 7.96 0.74 0 11,326 27,382 1,292 0 Before harvest FORS 4710 / 6710 Forest Planning

25 Binary Search Example - Iteration #1, Summary Time period Acres cut 123456123456 7.89 7.50 7.96 9.11 10.29 11.34 40,000 Volume target was achieved each time period. Since we did not specify any ending inventory condition, we increase the volume target and try again. Volume target for iteration 2 is 45,000 ft 3 per time period. Decision and Actions: Volume target met Increase Target Volume Try again Planned harvest volume (ft 3 ) FORS 4710 / 6710 Forest Planning

26 Binary Search Example - Iteration #2, Time Period 1 Middle of time period 1 (5 years from now) Age class structure before harvest 20 acres of 60-year old loblolly pine 10 acres of 20-year old loblolly pine AgeAcres Volume per acre (ft 3 ) Total volume (ft 3 ) 60 20 10 5,070 1,750 101,400 17,500 Harvest acres Harvest volume (ft 3 ) 8.88 0 45,000 0 Before harvest FORS 4710 / 6710 Forest Planning

27 Binary Search Example - Iteration #2, Time Period 2 Middle of time period 2 (15 years from now) Age class structure before harvest 11.12 acres of 70-year old loblolly pine 10 acres of 30-year old loblolly pine 8.88 acres of 10-year old loblolly pine AgeAcres Volume per acre (ft 3 ) Total volume (ft 3 ) 70 30 10 11.12 10.00 8.88 5,330 3,440 0 59,270 42,900 0 Harvest acres Harvest volume (ft 3 ) 8.44 0 45,000 0 Before harvest FORS 4710 / 6710 Forest Planning

28 Binary Search Example - Iteration #2, Time Period 3 Middle of time period 3 (25 years from now) Age class structure before harvest 2.68 acres of 80-year old loblolly pine 10 acres of 40-year old loblolly pine 8.88 acres of 20-year old loblolly pine 8.44 acres of 10-year old loblolly pine AgeAcres Volume per acre (ft 3 ) Total volume (ft 3 ) 80 40 20 10 2.68 10.00 8.88 8.44 5,560 4,290 1,750 0 14,901 42,900 15,540 0 Harvest acres Harvest volume (ft 3 ) 2.68 7.02 0 14,901 30,099 0 Before harvest FORS 4710 / 6710 Forest Planning

29 Binary Search Example - Iteration #2, Time Period 4 Middle of time period 4 (35 years from now) Age class structure before harvest 2.98 acres of 50-year old loblolly pine 8.88 acres of 30-year old loblolly pine 8.44 acres of 20-year old loblolly pine 9.70 acres of 10-year old loblolly pine AgeAcres Volume per acre (ft 3 ) Total volume (ft 3 ) 50 30 20 10 2.98 8.88 8.44 9.70 4,740 3,440 1,750 0 14,125 30,547 14,770 0 Harvest acres Harvest volume (ft 3 ) 2.98 8.88 0.19 0 14,125 30,547 328 0 Before harvest FORS 4710 / 6710 Forest Planning

30 Binary Search Example - Iteration #2, Time Period 5 Middle of time period 5 (45 years from now) Age class structure before harvest 8.25 acres of 30-year old loblolly pine 9.70 acres of 20-year old loblolly pine 12.05 acres of 10-year old loblolly pine AgeAcres Volume per acre (ft 3 ) Total volume (ft 3 ) 30 20 10 8.25 9.70 12.05 3,440 1,750 0 28,380 16,975 0 Harvest acres Harvest volume (ft 3 ) 8.25 9.50 0 28,380 16,620 0 Before harvest FORS 4710 / 6710 Forest Planning

31 Binary Search Example - Iteration #2, Time Period 6 Middle of time period 6 (55 years from now) Age class structure before harvest 0.20 acres of 30-year old loblolly pine 12.05 acres of 20-year old loblolly pine 17.75 acres of 10-year old loblolly pine AgeAcres Volume per acre (ft 3 ) Total volume (ft 3 ) 30 20 10 0.20 12.05 17.75 3,440 1,750 0 668 21,088 0 Harvest acres Harvest volume (ft 3 ) 0.20 12.05 0 668 21,088 0 Before harvest FORS 4710 / 6710 Forest Planning

32 Binary Search Example - Iteration #2, Summary Time period Acres cut Planned harvest volume (ft 3 ) 123456123456 8.88 8.44 9.70 12.05 17.75 12.25 45,000 21,776 Volume target was NOT achieved in time period 6. We reduce the Step Size by (0.5 x Step Size), to 2,500 ft 3 then decrease the volume target by the Step Size, and try again. Volume target for iteration 3 is 42,500 ft 3 per time period. Decision and Actions: Volume target not met Reduce Step Size Reduce Target Volume Try again FORS 4710 / 6710 Forest Planning

33 Binary Search Example Iteration Target volume (ft 3 ) Step Size (ft 3 ) Planned harvest volume (ft 3 ) 123456 140,0005,00040,00040,00040,00040,00040,00040,000 245,0005,00045,00045,00045,00045,00045,00021,776 342,5002,50042,50042,50042,50042,50042,50038,707 441,2501,25041,25041,25041,25041,25041,25041,250 541,875 62541,87541,87541,87541,87541,87541,875 642,188 31242,18842,18842,18842,18842,18842,188 742,344 15642,34442,34442,34442,34442,34442,344 FORS 4710 / 6710 Forest Planning

34 Summary of binary search  Can provide a quick estimate of the planned harvest level for an unmanaged forest  Can allow you to assess whether the planned harvest level can be maintained through time  No control over how much area is planned to be harvested  Difficult to determine whether one is moving a forest toward a regulated state  There is generally no objective other than to achieve some level of volume harvested over time FORS 4710 / 6710 Forest Planning

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