second order minimum conditions:
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A general continuous global approach to:- Optimal forest management with respect to theglobal warming problem and global economics
- One section of the lectures by Peter Lohmander at UPV, Polytechnical University of Valencia, Spain, February
2010
Peter LohmanderProfessor of forest management and economic optimization
SLU, Faculty of Forest SciencesDept. of forest economics
901 83 Umeå, Swedenhttp://www.Lohmander.com
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3
4
,min ( ) ( ) ( ) ( )
. .
u f a wu w
C C u C f C a v C w
s t
u f K f K u
w a K a K w
v u
5
,min ( ) ( ) ( ) ( )u f a w
u wC C u C K u C K u w C w
6
( ) ( ) ( ) 0
( ) ( ) 0
u f a
a w
CC u C K u C K u w
uC
C K u w C ww
u f a
a w
C C C
C C
7
2
2
2
2
2
2
( ) ( ) ( )
( )
( )
( ) ( )
u f a
a
a
a w
CC u C K u C K u w
u
CC K u w
u w
CC K u w
w u
CC K u w C w
w
8
Second order minimum conditions:
2
20
C
u
2 2
2
2 2
2
0
C C
u u w
C C
w u w
9
0u f aC C C
0u f a a
a a w
C C C C
C C C
10
0u f a u f aC C C C C C
2u f a a
u f a a w a
a a w
C C C CC C C C C C
C C C
11
2
2 2
0
u f a a w a
u a u w f a f w a a w a
u a u w f a f w a w
C C C C C C
C C C C C C C C C C C C
C C C C C C C C C C
12
Observations of the first and second order conditions:
( , )
0
0
( , ) 0
x
y
x y
f x y
f
f
df x y f dx f dy
13
2 2 2( , ) ( ) ( )xx xy yx yyd f x y f dx f dxdy f dydx f dy
xy yxf f
2 2 2
2 2 2
( , ) ( ) ( )
( , ) 2
xx xy yx yyd f x y f dx f dxdy f dydx f dy
d f x y au huv bv
14
2 2 22d f au huv bv
2 2 22h
d f a u uv bva
2 22 2 2 2 2
22
h h hd f a u uv v bv v
a a a
15
2 22 2h ab h
d f a u v va a
xxf a a
2xx xy
yx yy
f f a hab h
f f h b
16
2 20 0 0a ab h d f
2 20 0 0 0h
a ab h u v d fa
2 20 0 0 0a ab h v d f
2 20 0 0 0 0a ab h u v d f
17
So, if 0xxf and 0xx xy
yx yy
f f
f f
and 0 0dx dy
or 0 0dx dy
or 0 0dx dy
then 2 0d f
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Then, the solution to
represents a (locally) unique minimum.
0
0x
y
f
f
19
A numerically specified example:
1( ) 5
20uC u u
1( ) 10
300fC f f
1( ) 0
20aC K u w K u w
1( ) 14
100wC w w
20
21
Comparative statics analysis:
( ) ( ) ( ) 0
( ) ( ) 0
u f a
w a
CC u C K u C K u w
uC
C w C K u ww
22
u f a a f a
a w a a
C C C du C dw C C dK
C du C C dw C dK
23
1 1 1 1 1 1
20 300 20 20 300 20
1 1 1 1
20 100 20 20
du dw dK
du dw dK
24
31 1 16
300 20 300
1 6 1
20 100 20
du dw dK
du dw dK
25
16 1300 20
1 620 100 0.0007
0.1891891890.003731 1
300 20
1 620 100
du
dK
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31 16300 300
1 120 20 0.0025
0.6756756750.003731 1
300 20
1 620 100
dw
dK
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Explicit solution of the example for alternative values of K
( ) ( ) ( ) 0
( ) ( ) 0
u f a
w a
CC u C K u C K u w
uC
C w C K u ww
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1 1 15 10 ( ) 0 ( ) 0
20 300 20
1 114 0 ( ) 0
100 20
Cu K u K u w
u
Cw K u w
w
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31 15 16 15000
300 300 300 3005 6 5 1400
0100 100 100 100
Cu w K
uC
u w Kw
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0 31 15 16 1500 0
0 5 6 5 1400 0
Cu w K
u
Cu w K
w
31
0 31 15 1500 16
0 5 6 1400 5
Cu w K
u
Cu w K
w
32
33
1500 16 15
6 1500 16 15 1400 51400 5 6
31 15 31 6 5 15
5 6
30000 21
111
K
K KKu
Ku
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31 1500 16
5 1400 5 31( 1400 5 ) 5(1500 16 )31 15 31 6 5 15
5 6
50900 75
111
K
K K Kw
Kw
35
Dynamic approach analysis
du Cu
dt udw C
wdt w
1
36
eq
eq
x u u
y w w
37
u f a a
a w a
x C C C x C y
y C x C C y
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xx xy
yx yy
x m x m y
y m x m y
39
( ) ; ( )kt ktx t Ae y t Be
kt kt ktxx xy
kt kt ktyx yy
kAe m Ae m Be
kBe m Ae m Be
40
xx xy
yx yy
kA m A m B
kB m A m B
41
0
0xx xy
yx yy
k m m A
m k m B
42
• 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)).
43
One way to determine the value(s) of k is to use this equation:
0xx xy
xx yy xy yxyx yy
k m mk m k m m m
m k m
44
xy yxm m
20xx yy xyk m k m m
22 0xx yy xx yy xyk m m k m m m
45
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.
0
0
xxxx xy
xy
xy yx
xyxy yy
yy
k mBk m A m B
A m
m m
mBm A k m B
A k m
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2
22
0
0
xyxx
xy yy
xx yy xy
xx yy xx yy xy
mk mB
A m k m
k m k m m
k m m k m m m
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Lets us solve the equation!
2
2
2 2xx yy xx yy
xx yy xy
m m m mk m m m
2
2
2 2xx yy xx yy
xy
m m m mk m
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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.
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xx u f a
xy yx a
yy w a
m C C C
m m C
m C C
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2
2
2 2
u f a w au f a w a
a
C C C C CC C C C Ck C
2
22
2 2
u f w a u f w
a
C C C C C C Ck C
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• We may observe that
• As a consequence, both roots to to the equation are strictly negative.
• Therefore, divegence from the equilibrium solution is ruled out.
u f w u f wABS C C C ABS C C C
2
22
2 2
u f w a u f w
a
C C C C C C Ck C
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• 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.
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Derivation of the roots in the example:
2
21 1 1 1 1 1 1
120 300 100 10 20 300 1002 2 20
k
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249 13 1
600 600 400k
1
1
2
2
0.081667 0.054493
0.13616
0.081667 0.054493
0.027174
k
k
k
k
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1 2
1 2
1 2
1 2
( )
( )
k t k t
k t k t
x t A e A e
y t B e B e
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We may determine the path completely using the initial
conditions
0 0(0), (0) ,x y x y
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We also use the earlier derived results:
xyxx
xy yy
mk mB
A m k m
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Using the derived roots, we get:
11 1
xx
xy
k mB A
m
2 22
xy
xx
mA B
k m
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1 2
1 2
1 22
11 2
( )
( )
xyk t k t
xx
xx k t k t
xy
mx t A e B e
k m
k my t A e B e
m
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Let us use the initial conditions and determine the parameters!
0 1 22
10 1 2
xy
xx
xx
xy
mx A B
k m
k my A B
m
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2 01
021
1
1
xy
xx
xx
xy
m
k m xA
yBk m
m
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02
0 020
11
22
1
1
11
1
xy
xx xy
xx
xxxy
xxxx
xx
xy
mx
k m mx y
k myA
k mm
k mk m
k m
m
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0
1 10 0 0
21
22
1
1
11
1
xx xx
xy xy
xxxy
xxxx
xx
xy
x
k m k my y x
m mB
k mm
k mk m
k m
m
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Using the figures from the example, we get:
0 01
0.6565
1.431
x yA
0 02
0.6565
1.431
y xB
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The solutions to the numerically specified example
X(t) = -106.63·EXP(- 0.13612·t) + 6.61·EXP(- 0.02718·t)
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Y(t) = - 69.95·EXP(- 0.13612·t) - 10.07·EXP(- 0.02718·t)
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The cost function from different perspectives:
(based on the numerically specified example)
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Numerical solution of the example problem using direct
minimization:
Costmin Valencia Lohmander 2010-02-22
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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
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• Variable Value Reduced Cost• C 8975.806 0.000000• K 600.0000 0.000000• CU 4995.578 0.000000• CF 2516.909 0.000000• CA 1463.319 0.000000• CW 0.000000 0.000000• U 358.0645 0.000000• F 241.9355 0.000000• W 0.000000 1.903226• A 600.0000 0.000000• ANET 241.9355 0.000000• EQW -53.15315 0.000000• EQU 383.7838 0.000000
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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
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• Variable Value Reduced Cost• C 13952.25 0.000000• K 800.0000 0.000000• CU 6552.228 0.000000• CF 4022.401 0.000000• CA 2196.271 0.000000• CW 1181.353 0.000000• U 421.6216 0.000000• F 378.3784 0.000000• W 81.98198 0.000000• A 718.0180 0.000000• ANET 296.3964 0.000000• EQW 81.98198 0.000000• EQU 421.6216 0.000000
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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
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• Variable Value Reduced Cost• C 19357.66 0.000000• K 1000.000 0.000000• CU 7574.872 0.000000• CF 5892.379 0.000000• CA 2615.068 0.000000• CW 3275.339 0.000000• U 459.4595 0.000000• F 540.5405 0.000000• W 217.1171 0.000000• A 782.8829 0.000000• ANET 323.4234 0.000000• EQW 217.1171 0.000000• EQU 459.4595 0.000000
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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;
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• @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
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• Variable Value• CXX 0.1033333• CXY 0.5000000E-01• CYX 0.5000000E-01• CYY 0.6000000E-01
• X( 1) -100.0000• X( 2) -85.66667• X( 3) -73.30444• X( 4) -62.64442• X( 5) -53.45430• X( 6) -45.53346
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• X( 19) -3.638081• X( 20) -2.705807• X( 21) -1.912344• X( 22) -1.238468• X( 23) -0.6675825• X( 24) -0.1853596• X( 25) 0.2205703• X( 26) 0.5608842• X( 27) 0.8447973• X( 28) 1.080263
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• X( 40) 1.894182• X( 41) 1.881254• X( 42) 1.863429• X( 43) 1.841555• X( 44) 1.816360• X( 45) 1.788465
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• Y( 1) -80.00000• Y( 2) -70.20000• Y( 3) -61.70467• Y( 4) -54.33716• Y( 5) -47.94471• Y( 6) -42.39532• Y( 7) -37.57492• Y( 8) -33.38500
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• Y( 94) -0.7735111• Y( 95) -0.7524823• Y( 96) -0.7320263• Y( 97) -0.7121272• Y( 98) -0.6927697• Y( 99) -0.6739392• Y( 100) -0.6556210
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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.phphttp://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.pdfhttp://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
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• 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
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• 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.pdfPresentation as pdf: http://www.gip-ecofor.org/docs/nancy2008/ppt_des_presentations_orales/lohmander_session_3.1.pdfConference: 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
88
• 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 2009http://www.Lohmander.com/RuMa09/RuMa09.htm
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• 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.jpghttp://www.Lohmander.com/Information/ne2.jpghttp://www.Lohmander.com/Information/ne3.jpghttp://www.Lohmander.com/PL_SvSE_090205.pdfhttp://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
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• 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
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A general continuous global approach to:- Optimal forest management with respect to theglobal warming problem and global economics
- One section of the lectures by Peter Lohmander at UPV, Polytechnical University of Valencia, Spain, February
2010
Peter LohmanderProfessor of forest management and economic optimization
SLU, Faculty of Forest SciencesDept. of forest economics
901 83 Umeå
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