Transcript Document

A general continuous global approach to:
- Optimal forest management with respect to the
global warming problem and global economics
- One section of the lectures by Peter Lohmander at UPV,
Polytechnical University of Valencia, Spain, February
2010
Peter Lohmander
Professor of forest management and economic optimization
SLU, Faculty of Forest Sciences
Dept. of forest economics
901 83 Umeå, Sweden
http://www.Lohmander.com
1
2
3
min C  Cu (u )  C f ( f )  Ca (a  v)  Cw ( w)
u ,w
s.t.
u f K
 f  K u
wa  K
vu
a  K w
4
min C  Cu (u )  C f ( K  u )  Ca ( K  u  w)  Cw ( w)
u ,w
5
C
 Cu (u )  C f ( K  u )  Ca ( K  u  w)  0
u
C
 Ca ( K  u  w)  Cw ( w)
0
w
Cu  C f  Ca
Ca  Cw
6
C




C
(
u
)

C
(
K

u
)

C
(
K

u

w
)
u
f
a
2
u
2
C
 Ca ( K  u  w)
uw
2
C
 Ca ( K  u  w)
wu
2
C



C
(
K

u

w
)

C
a
w ( w)
2
w
2
7
Second order minimum
conditions:
C

0
2
u
2
C
2
u
2
C
wu
2
C
uw

0
2
C
2
w
2
8
Cu  C f   Ca  0

Cu  C f   Ca
 
Ca

 
Ca

Ca  Cw

0
9
Cu  C f   Ca  Cu  C f   Ca  0
C   C   C   C    C   C   C  C   C   C 





C  
C   C  
u
f
a
a
u
a
a
f
a
a
w
a
w
10
2
C   C   C  C   C    C  
u
f
a
a
w
a
2

   C C   C  
 CuCa  CuCw  C f Ca  C f Cw  Ca
2
a
w
a
 CuCa  CuCw  C f Ca  C f Cw  CaCw  0
11
2

Observations of the first and
second order conditions:
f ( x, y )
 fx  0

f

0
 y
df ( x, y )  f x dx  f y dy  0
12
d f ( x, y)  f xx (dx)  f xy dxdy  f yx dydx  f yy (dy)
2
2
2
f xy  f yx
d 2 f ( x, y )  f xx (dx) 2  f xy dxdy  f yx dydx  f yy (dy ) 2
d f ( x, y ) 
2
au
2

2huv

bv
13
2
d f  au  2huv  bv
2
2
2
h 
 2
2
d f  a  u  2 uv   bv
a 

2
 2
h
h 2
h 2
2
d f  a  u  2 uv  2 v   bv  v
a
a
a


2
2
2
14
2
h 

d f  au  v 
a 

2
 ab  h  2

v
 a 
2
f xx  a  a
f xx
f xy
f yx
f yy

a h
h b
 ab  h
2
15
 a  0    ab  h
2
 0  d f  0
2
h


2
 a  0   ab  h  0    u  v  0   d f  0
a


2
 a  0    ab  h
 a  0    ab  h
2
2
 0  v  0  d f  0
2
 0  u  0  v  0  d f  0
2
16
So, if
and
then
f xx  0
and
f xx
f yx
f xy
0
f yy
 dx  0  dy  0
or
 dx  0  dy  0
or
 dx  0  dy  0
d f 0
2
17
Then, the solution to
 fx  0

f

0
y

represents a (locally) unique minimum.
18
A numerically specified example:
1

Cu (u )  5  u
20
1

C f ( f )  10 
f
300
1

Ca ( K  u  w)  0   K  u  w 
20
1
Cw ( w)  14 
w
100
19
20
Comparative statics analysis:
C
 Cu (u )  C f ( K  u )  Ca ( K  u  w)  0
u
C
 Cw ( w)  Ca ( K  u  w)
0
w
21

  

  


 
 C   C   C  du  C  dw  C   C  dK
f
a
a
f
a
 u


Ca du  Cw  Ca dw  Ca dK

22
 1
1
1 
1 
 1 
 1
 20  300  20  du   20  dw   300  20  dK


 



1 
 1 
 1
 1 

du  
  dw    dK



 20 
 100 20 
 20 
23
 31 
 1 
 16 
du

dw

dK






 300


 20 
 300 

1
6
1







du  
dw    dK



  20 
 100 
 20 
24
 16 


 300 
 1 
 
du
 20 

dK  31 


 300 
 1 
 
 20 
 1 
 
 20 
 6 


 100  0.0007

 0.189189189
0.0037
 1 
 
 20 
 6 


 100 
25
 31 


 300 
 1 
 
dw
 20 

dK  31 


 300 
 1 
 
 20 
 16 


 300 
 1 
 
0.0025
 20 

 0.675675675
0.0037
 1 
 
 20 
 6 


 100 
26
Explicit solution of the example
for alternative values of K
C
 Cu (u )  C f ( K  u )  Ca ( K  u  w)  0
u
C
 Cw ( w)  Ca ( K  u  w)
0
w
27
C 
1  
1
1
 

  5  u   10 
( K  u )    0  ( K  u  w)   0
u 
20  
300
20
 

C 
1  
1

 14 
w    0  ( K  u  w) 
0
w 
100  
20

28
C 31
15
16
1500

u
w
K
0
u 300
300
300
300
C
5
6
5
1400

u
w
K
0
w 100
100
100
100
29
 C

 0   31u  15w  16 K  1500  0

 u

 C

 0   5u  6w  5K  1400
0

 w

30
 C

 0   31u  15w  1500  16 K

 u

 C

 0   5u  6w  1400  5K

 w

31
32
1500  16 K 15
6 1500  16 K   15  1400  5 K 
1400  5K 6
u

31 15
 31 6    515
5 6
30000  21K
u
111
33
31 1500  16 K
5 1400  5K 31(1400  5K )  5(1500  16 K )

w
31 15
 31 6    515
5 6
50900  75K
w
111
34
Dynamic approach analysis
C
du





u




u
dt

C

dw
w 




w
dt


  1
35
 x  u  ueq

y

w

w
eq

36

 
  







x


C

C

C
x

C
y
u
f
a
a


 y   Ca x
 Cw  Ca y


37

x


m
x

m
y

xx
xy

 y  myx x  myy y


38
x(t )  Ae ; y(t )  Be
kt
kt

 kAe   mxx Ae  mxy Be

kt
kt
kt
kBe


m
Ae

m
Be

yx
yy

kt
kt
kt
39
 kA  mxx A  mxy B

kB


m
A

m
B
yx
yy

40
 k  mxx
 m
 yx
mxy   A 0






k  myy   B  0
41
• k is selected in way such that the two
equations become identical. This way, the
equations only determine the ratio B/A, not
the values of A and B. This is necessary
since we must have some freedom to
determine A and B such that they fit the
initial conditions.
• With two roots (that usually are different),
we (usually) get two different ratios B/A.
This makes it possible to fit the parameters
to the (two dimensional) initial conditions
(x(0),y(0)).
42
One way to determine the value(s) of
k is to use this equation:
k  mxx
mxy
myx
k  myy
  k  mxx   k  myy   mxy myx  0
43
mxy  myx
 k  mxx   k  myy    mxy 
2
0
k   mxx  myy  k  mxx myy   mxy   0
2
2
44
Another way to get to the same equation, is to make
sure that the two equations give the same value to
the ratio B/A.
 B   k  mxx  
  k  mxx  A  mxy B  0   A  m 
xy


mxy  myx

B
mxy 

mxy A   k  myy  B  0   
 A  k  myy  



45
B 
 
A 
 mxy
  k  mxx 

mxy
 k  myy 
 k  mxx   k  m yy    mxy 
2
0
k   mxx  m yy  k  mxx m yy   mxy   0
2
2
46
Lets us solve the equation!
m

k 
xx
 myy     mxx  myy  
2

  mxx myy   mxy 

 
2
2


2
m

k 




m
m

m

2



xx
yy
xx
yy

   mxy 

 
2
2


2
47

No cyclical solutions!
• Observe that the expression within the
square root sign is positive.
• As a consequence, only real roots, k, exist.
• For this reason, cyclical solutions to the
differential equation system can be ruled
out.
48
mxx  Cu  C f   Ca
mxy  myx  Ca


myy  Cw  Ca
49




Cu  C f   Ca  Cw  Ca   Cu  C f   Ca  Cw  Ca
k 
 
2
2





Cu  C f   Cw  2Ca   Cu  C f   Cw

k 
 
2
2



2



  Ca


 
2

  C 
a



 
50
2
2
C   C   C   2C      C   C   C   


  C 
k 

 
 
2
2

2
u
f
w
a
u
f
w
a


• We may observe that



ABS Cu  C f   Cw  ABS Cu  C f   Cw

• As a consequence, both roots to to the equation are
strictly negative.
• Therefore, divegence from the equilibrium
solution is ruled out.
51
2
• With only strictly negative roots, we have a
guaranteed convergence to the equilibrium.
• However, this does not have to be monotone.
• With two different roots (k1 and k2) and with
parameters A1 and A2 with different signs (and/or
parameters B1 and B2 with different signs), the
sign(s) of the deviation(s) from the equilibrium
value(s) may change over time.
52
Derivation of the roots in the
example:
1
1
1
 1

 
 
20 300 100 10  

k 

2
2
 1
1
1 
  20  300  100    1  2
 

 
2

  20 




53
49   13 
1
k 



600   600  400
2
k1  0.081667  0.054493
k1  0.13616
k2  0.081667  0.054493
k2  0.027174
54
 x (t )  A1e  A2 e

k1t
k2t
y
(
t
)

B
e

B
e

1
2
k1t
k2t
55
We may determine the path
completely using the initial
conditions
 x(0), y(0)   x0 , y0 
56
We also use the earlier derived
results:
B

A
 mxy
  k  mxx 

mxy
 k  myy 
57
Using the derived roots, we get:
  k1  mxx 
B1 
A1
mxy
A2 
mxy
 k2  mxx 
B2
58
m

xy
k1t
k 2t
x
(
t
)

A
e

B
e
1
2

k

m

2
xx 



k

m

1
xx 
k1t
k2t
 y (t ) 
A1e  B2e

m
xy

59
Let us use the initial conditions and
determine the parameters!
mxy

x

A

B
1
2
 0
k

m

2
xx 



k

m


1
xx
y 
A1  B2
0

m
xy

60

1


 k  m 
1
xx

mxy




 k2  mxx    A1   x0 
 


  B2   y0 

1

mxy
61
x0
A1 

mxy
 k2  mxx 
y0
1
  k1  mxx 
mxy
1

mxy
 k2  mxx 
 mxy

x0  
 y0
 k2  mxx  


k1  mxx 

1
 k2  mxx 
1
62
1
B2 
  k1  mxx 
mxy
1
  k1  mxx 
mxy

x0
y0
mxy
 k2  mxx 
  k1  mxx  
y0  
x0

 m

xy



k1  mxx 

1
 k2  mxx 
1
63
Using the figures from the example,
we get:
x0  0.6565 y0
A1 
1.431
y0  0.6565 x0
B2 
1.431
64
The solutions to the numerically
specified example
X(t) = -106.63·EXP(- 0.13612·t) + 6.61·EXP(- 0.02718·t)
65
Y(t) = - 69.95·EXP(- 0.13612·t) - 10.07·EXP(- 0.02718·t)
66
67
The cost function
from different
perspectives:
(based on the numerically
specified example)
68
69
70
Numerical solution of the example
problem using direct
minimization:
Costmin Valencia Lohmander
2010-02-22
71
K = 600
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
model:
min = C;
k = 600;
C = cu + cf + ca + cw;
cu = 5*u+1/40*u^2;
cf = 10*f + 1/600*f^2;
ca = 1/40*(k-u-w)^2;
cw = 14*w+1/200*w^2;
f = k-u;
a = k-w;
@free(anet);
anet = a - u;
@free(eqw);
31*equ+15*eqw=1500 + 16*k;
5*equ+6*eqw = -1400+5*k;
end
72
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Variable
Value
C
8975.806
K
600.0000
CU
4995.578
CF
2516.909
CA
1463.319
CW
0.000000
U
358.0645
F
241.9355
W
0.000000
A
600.0000
ANET 241.9355
EQW -53.15315
EQU
383.7838
Reduced Cost
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1.903226
0.000000
0.000000
0.000000
0.000000
73
K = 800
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
model:
min = C;
k = 800;
C = cu + cf + ca + cw;
cu = 5*u+1/40*u^2;
cf = 10*f + 1/600*f^2;
ca = 1/40*(k-u-w)^2;
cw = 14*w+1/200*w^2;
f = k-u;
a = k-w;
@free(anet);
anet = a - u;
@free(eqw);
31*equ+15*eqw=1500 + 16*k;
5*equ+6*eqw = -1400+5*k;
end
74
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Variable
C
K
CU
CF
CA
CW
U
F
W
A
ANET
EQW
EQU
Value
13952.25
800.0000
6552.228
4022.401
2196.271
1181.353
421.6216
378.3784
81.98198
718.0180
296.3964
81.98198
421.6216
Reduced Cost
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
75
K = 1000
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
model:
min = C;
k = 1000;
C = cu + cf + ca + cw;
cu = 5*u+1/40*u^2;
cf = 10*f + 1/600*f^2;
ca = 1/40*(k-u-w)^2;
cw = 14*w+1/200*w^2;
f = k-u;
a = k-w;
@free(anet);
anet = a - u;
@free(eqw);
31*equ+15*eqw=1500 + 16*k;
5*equ+6*eqw = -1400+5*k;
end
76
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Variable
C
K
CU
CF
CA
CW
U
F
W
A
ANET
EQW
EQU
Value
19357.66
1000.000
7574.872
5892.379
2615.068
3275.339
459.4595
540.5405
217.1171
782.8829
323.4234
217.1171
459.4595
Reduced Cost
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
77
Numerical approximation of the
dynamics:
•
•
•
•
•
•
•
•
•
•
•
•
! dynsim;
! Peter Lohmander Valencia 20100222;
model:
sets:
time/1..100/:x,y,dx,dy;
endsets
cxx = 31/300;
cxy = 15/300;
cyx = 15/300;
cyy = 18/300;
x(1) = -100;
y(1) = -80;
78
• @FOR( time(t): dx(t)= -( cxx*x(t) + cxy*(y(t)) ));
• @FOR( time(t): dy(t)= -( cyx*x(t) + cyy*(y(t)) ));
• @FOR( time(t)| t#GT#1: x(t)= x(t-1) + dx(t-1) );
• @FOR( time(t)| t#GT#1: y(t)= y(t-1) + dy(t-1) );
•
•
•
•
@for(time(t): @free(x(t)));
@for(time(t): @free(y(t)));
@for(time(t): @free(dx(t)));
@for(time(t): @free(dy(t)));
• end
79
•
•
•
•
•
Variable
CXX
CXY
CYX
CYY
•
•
•
•
•
•
X( 1)
X( 2)
X( 3)
X( 4)
X( 5)
X( 6)
Value
0.1033333
0.5000000E-01
0.5000000E-01
0.6000000E-01
-100.0000
-85.66667
-73.30444
-62.64442
-53.45430
-45.53346
80
•
•
•
•
•
•
•
•
•
•
X( 19)
X( 20)
X( 21)
X( 22)
X( 23)
X( 24)
X( 25)
X( 26)
X( 27)
X( 28)
-3.638081
-2.705807
-1.912344
-1.238468
-0.6675825
-0.1853596
0.2205703
0.5608842
0.8447973
1.080263
81
•
•
•
•
•
•
X( 40)
X( 41)
X( 42)
X( 43)
X( 44)
X( 45)
1.894182
1.881254
1.863429
1.841555
1.816360
1.788465
82
•
•
•
•
•
•
•
•
Y( 1)
Y( 2)
Y( 3)
Y( 4)
Y( 5)
Y( 6)
Y( 7)
Y( 8)
-80.00000
-70.20000
-61.70467
-54.33716
-47.94471
-42.39532
-37.57492
-33.38500
83
•
•
•
•
•
•
•
Y( 94)
Y( 95)
Y( 96)
Y( 97)
Y( 98)
Y( 99)
Y( 100)
-0.7735111
-0.7524823
-0.7320263
-0.7121272
-0.6927697
-0.6739392
-0.6556210
84
References
•
•
•
•
•
Lohmander, P., Adaptive Optimization of Forest Management in a Stochastic World, in Weintraub A.
et al (Editors), Handbook of Operations Research in Natural Resources, Springer, Springer Science,
International Series in Operations Research and Management Science, New York, USA, pp 525-544,
2007 http://www.amazon.ca/gp/reader/0387718141/ref=sib_dp_pt/701-0734992-1741115#reader-link
Lohmander, P,. Energy Forum, Stockholm, 6-7 February 2008, Conference program with links to
report and software by Peter Lohmander:
http://www.energyforum.com/events/conferences/2008/c802/program.php
http://www.lohmander.com/EF2008/EF2008Lohmander.htm
Lohmander, P., Ekonomiskt rationell utveckling för skogs- och energisektorn i Sverige, Nordisk
Papper och Massa, Nr 3, 2008
Lohmander, P., Mohammadi, S., Optimal Continuous Cover Forest Management in an Uneven-Aged
Forest in the North of Iran, Journal of Applied Sciences 8(11), 2008
http://ansijournals.com/jas/2008/1995-2007.pdf
http://www.Lohmander.com/LoMoOCC.pdf
Lohmander, P., Guidelines for Economically Rational and Coordinated Dynamic Development of the
Forest and Bio Energy Sectors with CO2 constraints, Proceedings from the 16th European Biomass
Conference and Exhibition, Valencia, Spain, 02-06 June, 2008 (In the version in the link, below, an
earlier misprint has been corrected. ) http://www.Lohmander.com/Valencia2008.pdf
85
•
Lohmander, P., Economically Optimal Joint Strategy for Sustainable Bioenergy and Forest Sectors
with CO2 Constraints, European Biomass Forum, Exploring Future Markets, Financing and
Technology for Power Generation, CD, Marcus Evans Ltd, Amsterdam, 16th-17th June, 2008
http://www.Lohmander.com/Amsterdam2008.ppt
•
Lohmander, P., Ekonomiskt rationell utveckling för skogs- och energisektorn, Nordisk Energi, Nr. 4,
2008
•
Lohmander, P., Optimal resource control model & General continuous time optimal control model of
a forest resource, comparative dynamics and CO2 consideration effects, SLU Seminar in Forest
Economics, Umea, Sweden, 2008-09-18 http://www.lohmander.com/CM/CMLohmander.ppt
•
Lohmander, P., Tools for optimal coordination of CCS, power industry capacity expansion and bio
energy raw material production and harvesting, 2nd Annual EMISSIONS REDUCTION FORUM: Establishing Effective CO2, NOx, SOx Mitigation Strategies for the Power Industry, CD, Marcus
Evans Ltd, Madrid, Spain, 29th & 30th September 2008
http://www.lohmander.com/Madrid08/Madrid_2008_Lohmander.ppt
•
Lohmander, P., Optimal CCS, Carbon Capture and Storage, Under Risk, International Seminars in
Life Sciences, Universidad Politécnica de Valencia, Thursday 2008-10-16
http://www.lohmander.com/OptCCS/OptCCS.ppt
86
•
Lohmander, P., Economic forest production with consideration of the forest and energy industries, E.ON International
Bioenergy Conference, Malmo, Sweden, 2008-10-30 http://www.lohmander.com/eon081030/eon081030.ppt
•
Lohmander, P., Optimal dynamic control of the forest resource with changing energy demand functions and valuation
of CO2 storage, UE2008.fr, The European Forest-based Sector: Bio-Responses to Address New Climate and Energy
Challenges? Nancy, France, November 6-8, 2008 http://www.lohmander.com/Nancy08/Nancy08.ppt (See also later
versions 2009)
•
Lohmander, P., Optimal dynamic control of the forest resource with changing energy demand functions and valuation
of CO2 storage, The European Forest-based Sector: Bio-Responses to Address New Climate and Energy Challenges,
Nancy, France, November 6-8, 2008, Proceedings: (forthcoming) in French Forest Review (2009) Abstract: Page 65
of: http://www.gip-ecofor.org/docs/34/rsums_confnancy2008__20081105.pdf
Presentation as pdf: http://www.gipecofor.org/docs/nancy2008/ppt_des_presentations_orales/lohmander_session_3.1.pdf
Conference: http://www.gip-ecofor.org/docs/34/nancy2008englishprogramme20081106.pdf
•
ECOFOR, (in French) Summary of results by Peter Lohmander (on page 8) in “Evaluation du developpement de la
bioenergie”, in Bulletin d’information sur les forets europeennes, l’energie et climat, Volume 157, Numero 1, Lundi
10 novembre 2008 http://www.gip-ecofor.org/docs/34/nancy2008synthseiisd.pdf
•
IISD, Summary of results by Peter Lohmander (on page 6) in “Evaluation of Bioenergy Development”, in European
Forests, Energy and Climate Bulletin, Published by the International Institute for Sustainable Development (IISD)
http://www.iisd.org/ , Vol. 157, No. 1, Monday, 10 November, 2008
http://www.iisd.ca/download/pdf/sd/ymbvol157num1e.pdf
87
•
Lohmander, P., Integrated Regional Study Stage 1., Presentation at the E.ON - Holmen Sveaskog - SLU Research Meeting, Norrköping, Sweden, 2008-12-10 – 2008-12-11,
http://www.lohmander.com/NorrDec08/NorrDec08.ppt ,
http://www.lohmander.com/NorrDec08/NorrDec08.pdf ,
http://www.lohmander.com/NorrDec08/NorrDec08RawData.xls
•
Lohmander, P., Öka avverkningen och hjälp Sverige ur krisen, VI SKOGSÄGARE,
Debatt, Nr. 1, 2009 http://www.lohmander.com/PLdebattVIS2009nr1.pdf
•
Lohmander, P., Economic Forest Production with Consideration of the Forest and
Energy Industries (SLU 2009-01-29), http://www.lohmander.com/SLU09/SLU09.pdf
http://www.lohmander.com/SLU09/SLU09.ppt
•
Lohmander, P., Rational and sustainable international policy for the forest sector with
consideration of energy, global warming, risk, and regional development, SLU, Umea,
2009-02-18, http://www.lohmander.com/IntPres090218.ppt
•
Lohmander, P., Strategic options for the forest sector in Russia with focus on economic
optimization, energy and sustainability
(Full paper in English with short translation to Russian), ICFFI News, Vol. 1, Number
10, March 2009
http://www.Lohmander.com/RuMa09/RuMa09.htm
88
•
International seminar, ECONOMICS OF FORESTRY AND FOREST
SECTOR: ACTUAL PROBLEMS AND TRENDS, St Petersburg, Russia,
March 2009, http://www.lohmander.com/RuMa09/ProgramRuMa09.pdf
•
Lohmander, P., Satsa på biobränsle, Skogsvärden, Nr 1, 2009
http://www.Lohmander.com/PL_SV_1_09.jpg
•
Lohmander, P., Stor potential för svensk skogsenergi, Nordisk Energi, Nr. 2,
2009
http://www.Lohmander.com/Information/ne1.jpg
http://www.Lohmander.com/Information/ne2.jpg
http://www.Lohmander.com/Information/ne3.jpg
http://www.Lohmander.com/PL_SvSE_090205.pdf
http://www.Lohmander.com/PL_SvSE_090205.doc
•
Lohmander, P., Strategiska möjligheter för skogssektorn i Ryssland
Nordisk Papper och Massa, Nr 2, 2009
http://www.Lohmander.com/PL_NPM_2_2009.pdf
http://www.Lohmander.com/PL_RuSwe_09.pdf
http://www.Lohmander.com/PL_RuSwe_09.doc
89
• Lohmander, P., Economic forest production with consideration of the
forest- and energy industries, Project meeting presentation, Stockholm,
Sweden, 2009-05-11, http://www.lohmander.com/EON_090511.ppt
• Lohmander, P., Derivation of the Economically Optimal Joint Strategy
for Development of the Bioenergy and Forest Products Industries,
European Biomass and Bioenergy Forum, MarcusEvans, London, UK,
8-9 June, 2009,
http://www.lohmander.com/London09/London_Lohmander_09.ppt &
ttp://www.lohmander.com/London09.pdf
• Lohmander, P., Rational and sustainable international policy for the
forest sector - with consideration of energy, global warming, risk, and
regional development, Preliminary plan, 2009-08-05,
http://www.lohmander.com/ip090805.pdf
90
91
92
A general continuous global approach to:
- Optimal forest management with respect to the
global warming problem and global economics
- One section of the lectures by Peter Lohmander at UPV,
Polytechnical University of Valencia, Spain, February
2010
Peter Lohmander
Professor of forest management and economic optimization
SLU, Faculty of Forest Sciences
Dept. of forest economics
901 83 Umeå
93