1

Using Optimization in Public Forest

ManagementSociety for Industrial and Applied Mathematics

Pittsburgh, Pennsylvania

July 12, 2010

Michael Bevers

USDA Forest Service

Rocky Mountain Research Station

Fort Collins, Colorado

Email: mbevers@fs.fed.us

Mathematical programming

models

�Linear programs (LP)

�Integer programs (IP)

�Stochastic programs (SP)

�Chance-constrained programs

(CCP)

2

Linear Program (LP)

T

0

Minimize:

subject to:

n+∈ ℜ

=

c x

x

A x b

Mixed-Integer Linear Program

(MILP)

T

T1

T1 0

Minimize:

( , . . . , )

( , . . . , )

subject to:

pp

n pp n

x x

x x+ −

+

∈

∈ ℜ

=

c x

N

A x b

3

Two-Stage Stochastic Program

Deterministic Equivalent

1

T T

S

0

Min:

( , ,..., )

subject to:

s s s

s

nk

s s s

p

s

∈

+

+

∈ ℜ

=

+ = ∀

∑c x q y

x y y

A x b

T x Wy h

Chance-Constrained

Programming (CCP)

T

0

Minimize:

subject to:

Pr [ ( ) ( )]

n

j j jp j

+∈ ℜ

=

≥ ≥ ∀

c x

x

A x b

T ω x h ω

4

Linear Programming (LP)

Timber RAM

1970’s

Forplan

1980’s

Spectrum

1990’s

Timber RAM (LP)

1970’s

�Strata-based harvest scheduling

�Per-acre cost and production

�Timber management objectives

5

Forplan (LP)

1980’s

�Strata-based harvest scheduling

�Per-acre cost and production

�Multi-resource objectives

U.S. National Forest System Timber Harvest

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

1977

1980

1983

1986

1989

1992

1995

1998

2001

2004

2007

Bil

lio

n B

oa

rd F

ee

t

Sold

Cut

Reagan admin.

calls for 20 bbf in

forest plan

alternatives.

6

Spectrum (LP)

1990’s

�Strata-based harvest scheduling

�Per-acre cost and production

�Multi-resource objectives

�State and flow variables

�Linear difference equations

�No longer strictly per-acre

Photos by Luray Parker et al.

7

Potential Ferret Capacities

estimated 1994

3 2

1 1 2 2

2 1 2 1 1 2 3

1

2 1 1 6 5

1 2 3 3 3

1 1 1 4 3 1

4 4 1 2 1 2 2

3 10 1 3 2 1 4 3 4 3 1 2

1 1 2 1 1 5 7 4 5 10 4 13 1 5 1 2

1 3 7 8 8 11 5 2 10 2 11 11 3 4 4 3

1 1 7 8 4 11 7 9 8 9 8 1 12 9 4 2 3 6 4 2

1 2 4 3 5 4 5 8 7 7 7 2 4 6 10 2 4 4 3 2 2 1 1 4 1

3 6 6 6 3 3 1 5 8 11 7 5 8 3 5 6 3 2 1 4 1 3 1 1 2 1

1 6 1 2 7 4 7 9 11 12 9 8 2 1 3 2 9 3

1 2 1 2 4 13 13 11 12 6

Badlands

National Park

Buffalo Gap National Grassland

Maximize deterministic abundance

in a habitat-constrained problem:

Maximize: NT

s.t. Total Capacity ≤ c

Local population constraints

8

( ) ( )0

0

ƒ ; 1, . . . ,

0 ,

0

Ji

i i ij j i

j

ij ji

dnn n D n n i J

dt

D D i j

n

=

= + − =

= ≥ ∀

=

∑

Reaction-Diffusion Model

Heterogeneous Environment

Local Growth Dispersal

(Levin 1974, Allen 1987)

( ), 1 , 1

1

[1 ƒ ] ; 1, . . . ,J

it j t ji j t

j

n n g n i t T− −

=

= + ∀ =∑

Reaction-Diffusion Model

Discrete Space – Discrete Time

(Bevers and Flather 1999)

9

Reaction-Diffusion Model

Capacity-constrained Geometric Growth

(Bevers, Hof, Uresk & Schenbeck 1997)

, 1

1

(1 ) ; 1,...,J

it j ji j t

j

n r g n i t T−

=

≤ + ∀ =∑

; 1,...,it it itn c X i t T≤ ∀ =

0

100

200

300

400

500

600

700

Year

+100%

+80%

+60%

+40%

+20%

Current

Bla

ck-F

oo

ted

Fer

rets

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

10

+20% AlternativeExpected Population in Year 7

+20% AlternativeExpected Population in Year 7

1

11

1111

1

1

1

1

22

22

22

2

2

11

11

11

1

11

1

22

22

22

2

2

1

1

22

22

22

2

1

11

22

21

1

112

22

22

1

1

1

11

11

11

1

1

11

1111

11

1

1

1

1

11

1111

12

22

1

1

11

222

22

22

2

2

11

11

11

1

1

11

11

11

1

1

1

22

22

2

1

11

12

1

1

1

11

11111

1

11

1

1

11

11

1

1

11

1

1

11

11

1

11

1

11

11

11

11

1

1

11

1111

22

22

2

1

11

11

11

11

1

122

22

22

2

1

1

11

11

1

11

1

1

11

11

1

1

11

BadlandsNational Park(approximate

boundary)

+20% Long-Term Habitat AllocationWithout "Leave Areas"

+20% Long-Term Habitat AllocationWithout "Leave Areas"

1

12

12

42

1

2

1

1

23

43

----

21

12

22

2

21

1

22

3-----

--

--------

12

22

1-

-

--------

1

1

222-

11

2

1

12

22

21

12

2

1

-

1

11

11

11

13

32

2

-

11

2

23

44

3---

23

22

22

2

1

23

22

22

1

1

1

33

23

--

32

32

2

2

1

21

1

BadlandsNational Park(approximate

boundary)

2

211

1

12

1

1

11

22

2

1

22

1

1

21

12

1

12

2

11

12

12

22

1

1

32

21

11

44

33

--

11

3321

1--

1

2-------

2

22

22

1

1--

2

21

--

1

1

11

11

Mixed-Integer Linear

Programming (MILP)

�More common today than LP

�Helps with spatial relationships

�Fits frequent practice of

managing whole units

Fire Program Analysis

12

Stochastic

and

Chance-constrained

Programming

13

Consider the viability constraint in

this CCP problem:

Minimize: Cost

s.t. Pr (NT ≥ b) ≥ p

(plus management variables, land constraints and

population response constraints)

Or the “reliable abundance”

objective in this budget-constrained

problem:

Maximize: B

s.t. Pr (NT ≥ B) ≥ p

Cost ≤ c

14

B (%)

A Budget-constrained Reliable

Abundance Problem

B*

= Occupied existing territories

= Unoccupied existing territories

= Territories to be restored

= Unavailable land units

A Habitat Restoration Example

15

Management actions:

Restore habitat.

Management objective(s):

Maximize abundance.

Management constraints:

Restore 12 territories / year.

= Occupied existing territories

= Unoccupied existing territories

= Territories to be restored

= Unavailable land units

System Constraints ?

16

Probable Female Abundance under Optimal

Restoration Schedules

0

5

10

15

20

0 1 2 3 4 5

Year

Ad

ult

Fem

ale

s

p = 0.20

p = 0.34

p = 0.52

p = 0.69

p = 0.85

p = 0.92

p = 0.98

17

Closing observations�Systems being modeled often are

complex and poorly understood.

Am. J. Trop. Med. Hyg., 66(2), pp. 186–196Copyright 2002 by The American Society of Tropical Medicine and Hygiene

Russell E. Enscore et al.

“Increased precipitation during specific

periods resulted in increased numbers of

expected [human] cases [of plague in the

Southwest] . . . , as did the number of

days above certain lower thresholds

for maximum daily summer

temperatures.”

18

No. Days with Max Temp 87-90 Degrees F at

Fort Valley, Arizona

0

5

10

15

20

25

30

35

1900 1920 1940 1960 1980 2000 2020

Year

Da

ys

Flather, Bevers and Reynolds

Am. J. Trop. Med. Hyg., 66(2), pp. 186–196Copyright 2002 by The American Society of Tropical Medicine and Hygiene

Russell E. Enscore et al.

“The climatic variables found to be

important in our models are those that would

be expected to influence strongly the

population dynamics of the rodent hosts and

flea vectors of plague.”

19

Closing observations�Systems being modeled often are

complex and poorly understood.

�Qualitative interpretation of optimization

results often is most appropriate.

Closing observations�Systems being modeled often are

complex and poorly understood.

�Qualitative interpretation of optimization

results often is most appropriate.

�Managers often need expert help to

make good use of optimization models.

20

Closing observations�Systems being modeled often are

complex and poorly understood.

�Qualitative interpretation of optimization

results often is most appropriate.

�Managers often need expert help to

make good use of optimization models.

�Synthesis of results into simple

guidelines often is most useful.

Thanks!