Bootstrap Simulation & Response Surface
Optimization in Forest Management
Jingjing Liang
West Virginia University
Joseph Buongiorno
University of Wisconsin-Madison
Robert A. Monserud
USDA Forest Service
What is the BEST-
• target stand distribution
• cutting cycle
• target stand basal area
• financial return
for
• wood quality
?
• ecology
2
Models Involved
• forest growth
stand growth model
stumpage price model
• financial return
interest rate model
• wood quality
log grade model
• ecology
Shannon’s diversity
3
Stand growth model
100
100
Douglas-fir
Trees/ha
60
40
80
Trees/ha
Predicted, DF Model
Predicted, FVS
Observed
80
20
60
40
20
0
0
0
10
20
100
30
40 50 60 70
Diameter (cm)
80
90
100 110
0
10
20
30
100
Other shade-intolerant
40
50 60 70
Diameter (cm)
80
90
100 110
90
100 110
Other shade-tolerant
80
Trees/ha
80
Trees/ha
Western hemlock
60
40
60
40
20
20
0
0
0
10
20
30
40 50 60
Diameter (cm)
70
80
90
100 110
0
10
20
30
40
50
60
70
80
Diameter (cm)
Liang, Buongiorno, Monserud. 2005. Growth and Yield of All-aged Douglas-fir/western hemlock Stands: A
4
Matrix Model with Stand Diversity Effects. Can. J. For. Res. 35: 2369-2382.
Stand growth model

et 1  y t 1  y t 1
y t 1  G y t y t  R y t   u t 1
Bootstrapped from
60
60
Douglas-fir
40
30
20
10
0
40
30
20
10
0
0
30
60
90
120
60
150
0
30
60
A
Year
90
120
60
Other shade-intolerant species
150
B
Year
Other shade-tolerant species
50
Basal area (m2/ha)
50
Basal area (m2/ha)
Western hemlock
50
Basal area (m2/ha)
Basal area (m2/ha)
50
40
30
20
10
0
40
30
20
10
0
0
30
60
90
Year
120
150
C
0
30
60
90
Year
120
150
D
5
Stumpage price model
Stumpage price ($/m3)
800
600
400
200
0
1977 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003
Year
Oregon Quarterly stumpage prices of
Douglas-fir Grade I logs
Pt 1  0.6Pt  0.9et  et 1
6
Interest rate model
10
Interest rate
8
6
4
2
0
-2
1977 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001
Year
Interest rate in real term of AAA bonds
rt 1  0.72  0.81 rt   t 1
7
Log grade model
I: peeler
II: #1 sawmill
III: #2 sawmill
Dependent variable
 
ln  l    l 0   l1  D   l 2  S
m 
Coefficient
Douglas fir
π1/ π4
IV: #3 sawmill
V: other grades
Independent
variable
π3/ π4
Constant
2.91
*
D
0.04
**
S
-0.10
**
Constant
1.66
D
0.04
**
S
-0.10
*
Constant
-8.94
*
D
0.41
**
S
-0.40
*
Constant
-8.37
*
D
0.35
**
S
-0.30
*
Constant
-5.90
**
D
0.26
**
S
-0.13
**
Western hemlock
π1/ π4
π2/ π4
π3/ π4
8
Bootstrap simulation
Bootstrapped
random shock
stand growth model
stand growth model
stumpage price model
stumpage price model
interest rate model
interest rate model
a demonstration of Bootstrap procedure
9
Measure of performance
• land expectation value
LEV 
W
vh w
 (1  r ) w

w0
vy 50
(1  r )
50
 vy 0
• percentage of peeler logs in stock
• species/size diversity
Bi  Bi 
H s   ln
B 
i 1 B
 
m
10
Simulation parameters
• target stand distribution: BDq selection method
• Basal area:
60, 120, 180, 240, 300 (ft2/ac)
• Diameter:
40
• q-ratio:
1.2, 1.4, 1.6, 1.8
(in.)
• cutting cycle:
10, 15, 20, 25, 30 (yr)
• timespan:
50
(yr)
11
Initial stand distribution
500 plots bootstrapped from the 2,706 initial Douglas-fir/western hemlock
plots in the Pacific Northwest
9
Annual production (m3 /ha/y)
25
NPV ($1000/ha )
20
15
10
5
7
6
0
100
200
300
400
500
600
0
100
A
Number of plots
200
300
400
500
600
B
Number of plots
1.18
Average basal area (m 2/ha)
27
1.17
Species diversity
8
1.16
1.15
1.14
1.13
26
25
24
23
0
100
200
300
400
500
600
0
100
C
Number of plots
200
300
400
500
600
D
Number of plots
42
2.48
Percentage of peeler
Size diversity
41
2.46
2.44
2.42
40
39
38
37
0
100
200
300
400
Number of plots
500
600
E
0
100
200
300
400
Number of plots
500
600
F
F
12
Response surface
•
•
•
basal area:
q-ratio:
cutting cycle:
60, 120, 180, 240, 300 (ft2/ac)
1.2, 1.4, 1.6, 1.8
10, 15, 20, 25, 30
(yr)
5×4×5=100
management alternatives
response surface equations
R2
Equation
P=71.20** -1.61**C +0.21B -26.63**q +0.029C2 2.10B2 +5.62**q2 -0.32CB -2.08Bq +1.09**Cq
0.97
0
-2
-4
-6
-8
Hs=1.35** +0.012C -0.15**B -0.05q +0.0005C2
+0.11**B2 +0.02q2 -0.005CB -0.09**Bq -0.02**Cq
0.98
-10
-12
2
-2
1
-1
0
0
-1
1
-2
-0.005C2
Hd=3.24** -0.12**C -0.09B -0.23**q
2
2
0.03B -0.16**q -0.01CB +0.07Bq +0.11**Cq
2
0.99
13
Optimization
Management criterion
Maximum
Control variable
Cutting cycle
(year)
Basal area
(m2ha-1)
Target
q ratio
Land expectation value (1000$ha-1)
8.20
10
51
1.8
Annual Production (m3ha-1y-1)
9.00
10
51
1.8
Species diversity
1.27
10
14
1.2
Size diversity
2.74
10
14
1.2
Percentage of peeler logs
46.7
10
14
1.2
Stand basal area (m2ha-1)
28.3
30
51
1.7
14
LEV
Sensitivity analysis
10
10
10
5
5
5
0
0
0
-5
-5
P
10
15
20
C (yr)
25
30
-5
10
20
30
B (m 2/ha)
40
50
1.2
46
46
44
44
44
42
42
42
40
10
15
20
C (yr)
25
30
1.6
1.8
1.6
1.8
1.6
1.8
1.6
1.8
q
46
40
1.4
40
10
20
30
B (m 2/ha)
40
50
1.2
1.4
q
1.26
1.26
1.24
1.24
1.24
1.22
1.22
1.22
Hs
1.26
1.2
1.2
Hd
10
15
20
C (yr)
25
30
1.2
10
20
30
B (m 2/ha)
40
50
1.2
q
2.8
2.8
2.8
2.7
2.7
2.7
2.6
2.6
2.6
2.5
2.5
2.5
2.4
2.4
10
15
20
C (yr)
25
30
1.4
2.4
10
20
30
B (m 2/ha)
40
50
1.2
1.4
q
15
10
10
5
5
LEV($1000/ha)
LEV($1000/ha)
Inter-criterion correlation
0
-5
-10
0
-5
-10
2
4
6
8
10
39
Annual production (m 3/ha/y)
48
50
Percentage of peeler
Percentage of peeler
45
Percentage of peeler
50
47
44
41
38
1.14
42
1.16
1.18
1.20
1.22
1.24
Species diversity
1.26
1.28
47
44
41
38
2.37
2.47
2.57
2.67
2.77
Size diversity
16
Summary of study
•
Bootstrap simulation
•
Response Surface optimization
•
Adjusting B, q, and C could control for more than 97%
of the variability in species and size diversity, percentage
of peeler logs, and basal area, but could control for less
in LEV and annual production.
•
Strong positive correlation between LEV and annual
production, and between wood quality and size diversity
•
negative correlation between LEV and wood quality.
17
Acknowledgement
Jingjing Liang
jiliang@mail.wvu.edu
304-293-1577
The research leading to this paper was supported in part by the USDA Forest Service,
Pacific Northwest Forest Research Station, Human and Natural Resources Interaction
Program, NRICGP grant 2001-35108-10673, and by the School of Natural Resources,
University of Wisconsin-Madison.
We thank Dean Parry, Karen Waddell, and Susan Stevens Hummel for assistance with the
data.
18