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FORS 8450 • Advanced Forest Planning
Lecture 21
BinarySearch
Binary Search
Allows one to find the maximum even-flow of volume or valuethat 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.
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FunExample
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Binary Search
Pick a number between 1 and 1000.
• I can tell you what it is in less than 11 guesses if you tellme whether my guess is higher or lower than your number.
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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 targetb) decrease the targetc) stop and report a solution
6) Return to Step 1 if necessary
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FinanceExample
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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.......?
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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).
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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 balanceEnd of year 1 balanceEnd of year 1 withdrawal
Beginning of year 2 balanceEnd of year 2 balanceEnd of year 2 withdrawal
Beginning of year 3 balanceEnd of year 3 balanceEnd of year 3 withdrawal
Surplus or deficit
$1,000.001,030.00
343.33
686.67707.27343.33
363.94374.86343.33
+ 31.53
(1/3 of 1,030.00)
FORS 4710 / 6710 Forest Planning
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 balanceEnd of year 1 balanceEnd of year 1 withdrawal
Beginning of year 2 balanceEnd of year 2 balanceEnd of year 2 withdrawal
Beginning of year 3 balanceEnd of year 3 balanceEnd of year 3 withdrawal
Surplus or deficit
$1,000.001,030.00
363.33
666.67686.67363.33
323.34333.04363.33
- 30.29
(1/3 of 1,030.00 + $20)
FORS 4710 / 6710 Forest Planning
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 balanceEnd of year 1 balanceEnd of year 1 withdrawal
Beginning of year 2 balanceEnd of year 2 balanceEnd of year 2 withdrawal
Beginning of year 3 balanceEnd of year 3 balanceEnd of year 3 withdrawal
Surplus or deficit
$1,000.001,030.00
353.33
676.67696.97353.33
343.64353.95353.33
+ 0.62
(1/3 of 1,030.00 + $10)
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Binary Search
Why is this "even-flow" situation such a problem?
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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...
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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?
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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 targetb) decrease the targetc) stop and report a solution
7) Return to Step 1 if necessary
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Binary Search in Forest Planning
Definitions:
Target harvest level: The level of harvest volume you expect to obtain eachtime period within the planning horizon.
Step size: How much the target harvest volume will change with each iterationof 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 atleast this "small."
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ForestryExample
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Timber yields
Age Yield (ft3 / acre)
20 1,75030 3,44040 4,29050 4,74060 5,07070 5,33080 5,560
Binary Search Example
Loblolly pine forest30-year rotation is desiredSite index 9030 acres
Current age class structure:20 acres of 55-year old loblolly pine10 acres of 15-year old loblolly pine
Planning assumptions:Plan in 10-year increments (10-year time periods)Develop a plan for 6 time periodsAssume harvesting occurs in the middle of each time period
Binary Search Assumptions:Target harvest level 40,000 ft3 per time periodStep size 5,000 ft3
Harvest oldest aged timber first.
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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 pine10 acres of 20-year old loblolly pine
Age Acres
Volumeper acre(ft3)
Totalvolume(ft3)
6020
2010
5,0701,750
101,40017,500
Harvestacres
Harvestvolume(ft3)
7.890
40,0000
Before harvest
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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 pine10 acres of 30-year old loblolly pine7.89 acres of 10-year old loblolly pine
Age Acres
Volumeper acre(ft3)
Totalvolume(ft3)
703010
12.1110.00
7.89
5,3303,440
0
64,54634,400
0
Harvestacres
Harvestvolume(ft3)
7.5000
40,00000
Before harvest
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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 pine10 acres of 40-year old loblolly pine7.89 acres of 20-year old loblolly pine7.50 acres of 10-year old loblolly pine
Age Acres
Volumeper acre(ft3)
Totalvolume(ft3)
80402010
4.6110.00
7.897.50
5,5604,2901,750
0
25,63242,90013,808
0
Harvestacres
Harvestvolume(ft3)
4.613.35
00
25,63214,638
00
Before harvest
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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 pine7.89 acres of 30-year old loblolly pine7.50 acres of 20-year old loblolly pine7.96 acres of 10-year old loblolly pine
Age Acres
Volumeper acre(ft3)
Totalvolume(ft3)
50302010
6.657.897.507.96
4,7403,4401,750
0
31,52127,14213,125
0
Harvestacres
Harvestvolume(ft3)
6.652.46
00
31,5218,479
00
Before harvest
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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 pine7.50 acres of 30-year old loblolly pine7.96 acres of 20-year old loblolly pine9.11 acres of 10-year old loblolly pine
Age Acres
Volumeper acre(ft3)
Totalvolume(ft3)
40302010
5.437.507.969.11
4,2903,4401,750
0
23,29525,80013,930
0
Harvestacres
Harvestvolume(ft3)
5.434.86
00
23,29516,705
00
Before harvest
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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 pine7.96 acres of 30-year old loblolly pine9.11 acres of 20-year old loblolly pine10.29 acres of 10-year old loblolly pine
Age Acres
Volumeper acre(ft3)
Totalvolume(ft3)
40302010
2.647.969.11
10.29
4,2903,4401,750
0
11,32627,38215,943
0
Harvestacres
Harvestvolume(ft3)
2.647.960.74
0
11,32627,382
1,2920
Before harvest
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Binary Search Example - Iteration #1, Summary
Timeperiod
Acrescut
123456
7.897.507.969.11
10.2911.34
40,00040,00040,00040,00040,00040,000
Volume target was achieved eachtime period.
Since we did not specify anyending inventory condition, weincrease the volume target andtry again.
Volume target for iteration 2 is45,000 ft3 per time period.
Decision and Actions:Volume target metIncrease Target VolumeTry again
Plannedharvestvolume
(ft3)
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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 pine10 acres of 20-year old loblolly pine
Age Acres
Volumeper acre(ft3)
Totalvolume(ft3)
6020
2010
5,0701,750
101,40017,500
Harvestacres
Harvestvolume(ft3)
8.880
45,0000
Before harvest
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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 pine10 acres of 30-year old loblolly pine8.88 acres of 10-year old loblolly pine
Age Acres
Volumeper acre(ft3)
Totalvolume(ft3)
703010
11.1210.00
8.88
5,3303,440
0
59,27042,900
0
Harvestacres
Harvestvolume(ft3)
8.4400
45,0000
Before harvest
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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 pine10 acres of 40-year old loblolly pine8.88 acres of 20-year old loblolly pine8.44 acres of 10-year old loblolly pine
Age Acres
Volumeper acre(ft3)
Totalvolume(ft3)
80402010
2.6810.00
8.888.44
5,5604,2901,750
0
14,90142,90015,540
0
Harvestacres
Harvestvolume(ft3)
2.687.02
00
14,90130,099
00
Before harvest
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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 pine8.88 acres of 30-year old loblolly pine8.44 acres of 20-year old loblolly pine9.70 acres of 10-year old loblolly pine
Age Acres
Volumeper acre(ft3)
Totalvolume(ft3)
50302010
2.988.888.449.70
4,7403,4401,750
0
14,12530,54714,770
0
Harvestacres
Harvestvolume(ft3)
2.988.880.19
0
14,12530,547
3280
Before harvest
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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 pine9.70 acres of 20-year old loblolly pine12.05 acres of 10-year old loblolly pine
Age Acres
Volumeper acre(ft3)
Totalvolume(ft3)
302010
8.259.70
12.05
3,4401,750
0
28,38016,975
0
Harvestacres
Harvestvolume(ft3)
8.259.50
0
28,38016,620
0
Before harvest
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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 pine12.05 acres of 20-year old loblolly pine17.75 acres of 10-year old loblolly pine
Age Acres
Volumeper acre(ft3)
Totalvolume(ft3)
302010
0.2012.0517.75
3,4401,750
0
66821,088
0
Harvestacres
Harvestvolume(ft3)
0.2012.05
0
66821,088
0
Before harvest
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Binary Search Example - Iteration #2, Summary
Timeperiod
Acrescut
Plannedharvestvolume
(ft3)
123456
8.888.449.70
12.0517.7512.25
45,00045,00045,00045,00045,00021,776
Volume target was NOT achieved in time period 6.
We reduce the Step Size by(0.5 x Step Size), to 2,500 ft3
then decrease the volume target by the Step Size, andtry again.
Volume target for iteration 3 is42,500 ft3 per time period.
Decision and Actions:Volume target not metReduce Step SizeReduce Target VolumeTry again
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Binary Search Example
Iteration
Targetvolume
(ft3)
Step Size(ft3)
Planned harvest volume (ft3)
1 2 3 4 5 6
1 40,000 5,000 40,000 40,000 40,000 40,000 40,000 40,000
2 45,000 5,000 45,000 45,000 45,000 45,000 45,000 21,776
3 42,500 2,500 42,500 42,500 42,500 42,500 42,500 38,707
4 41,250 1,250 41,250 41,250 41,250 41,250 41,250 41,250
5 41,875 625 41,875 41,875 41,875 41,875 41,875 41,875
6 42,188 312 42,188 42,188 42,188 42,188 42,188 42,188
7 42,344 156 42,344 42,344 42,344 42,344 42,344 42,344
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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
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