fors 8450 advanced forest planning lecture 21 binary search

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.

FORS 4710 / 6710 Forest Planning

FunExample


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.


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


FinanceExample


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.......?


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).


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



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)


Binary Search


Second attempt:




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)


Binary Search


Third attempt:




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)


Binary Search

Why is this "even-flow" situation such a problem?


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...



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?



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



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."


ForestryExample


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.


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





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





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





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






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






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


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)





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





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





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






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





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





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


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


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


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 8450 advanced forest planning lecture 21 binary search

Documents

balanceend of year

withdrawalbeginning

forest planningfunexamplefors

appropriate forest inventory

ordered list3

ordered list4

better guesses

optimal value