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Optimal Forest Management Under Risk
International Seminars in Life Sciences Universidad Politécnica de Valencia
Thursday 2007-02-22
Peter Lohmander
Professor of Forest Management and Economic OptimizationSLU, Swedish University of Agricultural Sciences
Umea, Sweden
http://www.Lohmander.com
Version 2007-02-18
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Schedule of Peter Lohmander from Claudio Benavent,International Officer ETSMRE - ETSIA
Thursday 2007-02-2210:30 Welcome at ETSMRE
10:45-13:45 Visits on campus and interviews with UPV Colleagues 14:00 Lunch on campus offered by ETSMRE 16:30 Conference in International Seminars 19:30 Free programme Friday 2007-02-23 10:30 Interview with Prof. Penny McDonald, Coordinator of the course "Preparation for International Study" 11:00-12:15 Institutional Presentation of your home Institution 12:30-13:30 International Office ETSMRE 14:00 Lunch on campus offered by ETSMRE 15:30 Free programme
Location: ETSMRE International Office (Avda. Blasco Ibáñez, 19-21).
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AMBITION
Research that leads to economically profitable and practical solutions and at the
same time leads to new discoveries of methods and practical general
approaches to problems in operations research in general and to forest
economics.
“Environmentally friendly” solutions are sometimes discovered also to be
economically optimal!
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Stochastic Dynamic Optimization
The future state of the world is hard to predict perfectly.
Some decisions must be made before the future is perfectly known.
Stochastic Dynamic Programming is the relevant approach.
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The unpredictable world and the
decision problems
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Figure 1. The real (inflation adjusted) stumpage price in Sweden. Source: Swedish Board of Forestry, Yearbook of Forest Statistics, 2000. We may regard the stumpage price as a stochastic processes. There is no method available which can predict future prices without error.
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Stochastic Price Export Price of Kraft Paper and Kraft Board
(Source: Statistics Sweden)
0,0
1000,0
2000,0
3000,0
4000,0
5000,0
6000,0
1996 1998 2000 2002 2004 2006
Year
SE
K/T
on
PP
PP minus 4500
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Stochastic Price PE Price of Electricity, Large Industrial
Consumers (li) 70 000 MWh (Source: Statistics Sweden)
0,0
10,0
20,0
30,0
40,0
50,0
60,0
70,0
-12,0 -10,0 -8,0 -6,0 -4,0 -2,0 0,0
Year - 2007
CS
EK
/kW
h
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Low Correlation between Energy Prices and Pulp Prices
(Source: Statistics Sweden)
0,0
500,0
1000,0
1500,0
2000,0
2500,0
3000,0
3500,0
1996 1998 2000 2002 2004 2006
Year
Pri
ces
in d
iffe
ren
t sc
ales
PP minus 4500
PE times 50
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Low Correlation between Energy Prices and Pulp Prices
Price of Electricity (li)
Export Price of Kraft Paper and Kraft Board
Price of Electricity (li) 1 0,2450
Export Price of Kraft Paper and Kraft Board
0,2450 1
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Low Correlation between Energy Prices and Pulp Prices
• It has been proved that the expected marginal capacity value of a production plant increases with price variation when different products are produced with the same type of raw material and the correlation between product prices is less than 1. (Lohmander 1989)
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Low Correlation between Energy Prices and Pulp Prices
• As a consequence, the most profitable investment level in production capacity, for instance a power plant, is higher with prices that are not perfectly predictable than according to what you find with traditional calculation.
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Joint probability density function with correlation 0.25 (which corresponds to the prices of electricity and kraft paper)
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Stochastic dynamic example with heating and pulp plants
P1
P2
P1
P2
P2 Time
The prices of electricity and kraft paper are not known many years in advance.
P1
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Stochastic dynamic example with heating and pulp plants
Time
The stock level can be changed over time. The most profitable extraction (harvest) in a particular period is affected by the prices of kraft paper and energy. This is one reason why it has to be sequentially optimized, based on the latest price information from the markets.
Stock level
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Stochastic dynamic example with heating and pulp plants
P1
P2
P1
P2
P2 Time
Time
Stock level
P1
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The stochastic dynamic optimization problem
1 2 1, 1 2, 1
1 2 1 2
1, 2, 1 2 1, 2, 1 2 1, 1 2, 1 1, 2, 1 1, 1 2, 1,
( , ) ( , , , )
( , , , ) ( , ; , , , ) ( , , , ) ( , , ) ( 1, , , )maxt t
t
r tt t t t t t t t t t t t t t
X X P P
X X S t i Cap Cap
f t i P P X X t i P P h t i X X e P P P P f t i P P
We maximize the expected present value of all future production.
The production of electricity and kraft paper in future periods is affected by the product prices and the stock of resources.
The stock of resources is dynamically optimized.
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The stochastic dynamic optimization problem
1, 2,( , , , )t t tf t i P P
The optimal expected present value, f, as a function of time, the stock level and the prices electricity and kraft paper.
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The stochastic dynamic optimization problem
1 2 1, 2,( , ; , , , )t t tX X t i P PThe profit in a particular period, t, as a function of the production levels of electricity and kraft paper, time, the stock level and the prices of electricity and kraft paper.
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The stochastic dynamic optimization problem
1 2( , , , )th t i X XThe cost of the stock in a period as a function of time, the stock level and the production levels of electricity and kraft paper.
(The production in period t affects the stock level in period t and in period t+1.)
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The stochastic dynamic optimization problem
1 2 1 2( , ) ( , , , )tX X S t i Cap Cap
The production of electricity and kraft paper in a period, t, is constrained by the production capacities in the kraft paper mill and the energy mill in that period and the entering resource stock level.
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The stochastic dynamic optimization problem
1, 1 2, 1
1, 1 2, 1 1, 2, 1 1, 1 2, 1( , , ) ( 1, , , )t t
r tt t t t t t t
P P
e P P P P f t i P P
The expected optimal objective function value of period t+1 is discounted to period t.
The probabilities of reaching different market state combinations at t+1 in the electricity market and in the kraft paper market are conditional on the prices in these markets at t.
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The stochastic dynamic optimization problem
1 2 1, 1 2, 1
1 2 1 2
1, 2, 1 2 1, 2, 1 2 1, 1 2, 1 1, 2, 1 1, 1 2, 1,
( , ) ( , , , )
( , , , ) ( , ; , , , ) ( , , , ) ( , , ) ( 1, , , )maxt t
t
r tt t t t t t t t t t t t t t
X X P P
X X S t i Cap Cap
f t i P P X X t i P P h t i X X e P P P P f t i P P
The total optimization problem is found above.
Now, we will illustrate this with a numerical program!
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X1
X2
Production capacity 2
Production capacity 1
Total wood supply
General illustration why the marginal value of production capacity increases with price risk (and connection to
heating plants)
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The economic optimization problem
1 1 2 2max PX P X
1 1 2 2
1 1
2 2
a X a X R
X Cap
X Cap
1 1 2 2 0d PdX P dX
2 1
1 2
dX P
dX P
Along the iso profit line we have:
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X1
X2
Production capacity 2
Production capacity 1
Total wood supply
Isoprofit line2 1
1 2
1dX P
dX P
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X1
X2
Production capacity 2
Production capacity 1
Total wood supply
Isoprofit line 2 1
1 2
1dX P
dX P
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X1
X2
Production capacity 2
Production capacity 1
Total wood supply Isoprofit line
2 1
1 2
1dX P
dX P
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Results:
• The expected economic value of one more unit of heating plant capacity is
17551 – 16461 = 1090.The economically optimal decision is this:If the investment cost of an extra unit of
capacity is less than 1090: Build this extra heating plant capacity!
No other investment calculation method would give the correct rule.
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Conclusions from the numerical model:
• It is possible to adaptively optimize all decisions over time including production of electricity, kraft paper and resource extraction.
• The approach makes it possible to determine the expected value of production capacity investments in heating plants and paper mills.
• The approach can be expanded to cover the complete energy and forest sector.
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Already in 1981
• World Bank Model” to study the Swedish forest sector. (Nilsson, S.)
• In the model, timber, pulp wood and fuel wood could be produced and harvested in all regions.
• The energy industry was considered as an option in all regions. It was possible too burn wood, not only fuel wood but also “pulp wood”.
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Capacity investments• The existing capacity in the saw mills, pulp
mills and paper mills was investigated and used in the model. It was possible to invest in more capacity of different kinds in the different regions.
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Structure in 1981• The forest sector of Sweden was modelled
as a linear programming problem. • The total economic result of all activities in
the forest sector of Sweden was maximized. • The wood based part of the energy sector
was considered as a part of this forest sector.
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Wood for energy in 1981• Among these results, we found that a large
proportion of the “pulpwood” should be used to produce energy.
• This was particularly the case in the north, at large distances from the coast.
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Surprise? Not really!• The cost of transporting pulpwood large
distances is very high. • If energy can be produced from pulpwood,
far away from the coast and the pulp industry, it is not surprising that this may be the most profitable alternative.
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Relevant model in 1981?• Of course, linear programming models are only
models of reality. This is true with all models.
• Of course, linear programming models do not capture all nonlinear and other “real” properties of the real world such as risk and integer constraints.
• Better options exist today to handle nonlinearities, risk, integer constraints and all kinds of other properties of the real world.
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Relevant result from 1981?
• The general finding that it may be optimal to use some of the wood for energy, still remains!
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Questions today (#1):• Can we combine the forest sector and
the energy sector in one modern optimization model for both sectors? The model should include relevant data for the heating and electricity plants and for all types of forest industry mills.
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President Lars Fritiof, E.ON (March 2006):
• ”During the next three years, the E.ON Group will invest about SEK 175 billion, of which SEK 155 billion will be invested in plants.”
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President Lars Fritiof, E.ON (March 2006):
• ”Secure energy supplies can no longer be taken for granted. Demand is increasing and European reserves of oil and gas are diminishing.The recently published EU Green Book reports that if nothing is done, Europe’s imports of energy will increase from 50 to 70% in the next 20 to 30 years.”
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Necessary Model Properties:
• The model should be dynamic and include the options to invest in new production capacity. Such new capacity could, when it comes to investments in energy plants, have different properties with respect to technological choices, possible fuels and degrees of flexibility.
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Why flexibility?• Prices and the availability of different
fuels are impossible to predict over horizons of the economic life time of a heating plant. That is why flexibility is valuable. In the old type of optimization models, such things could not be analyzed at all. Now, economic optimization of flexibility is possible.
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Dynamic options• In the model from 1981, one period was
analysed. In a new dynamic model, the use of the forest resources can also be optimized over time.
• In the model from 1981, the capacities of different mills was constant. In the dynamic model, the capacity investments can be optimized over time.
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The Option• A new generation of optimization models is
possible to construct. • We should not hesitate to develop this
generation!
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ConclusionsOptimal forest management under risk
contains a large number of topics.
The relevant general approach is stochastic dynamic programming.
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ConclusionsHere you may read more about these things:
http://www.lohmander.com/Information/Ref.htm
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ConclusionsOptimal forest management under risk contains a
large number of topics.
Here, you may instantly optimize some decisions!
http://www.lohmander.com/Program/Program.htm
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ConclusionsHere, you may study some courses:
http://www.lohmander.com/Kurser/Kurser.htm
In particular:Optimization in dynamic and stochastic decision problems (Graduate course) Forest Economics (Graduate course) Economic Forest Production (Advanced Undergraduate course) The Forest Sector from an International View (Advanced Undergraduate course)
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INFORMSThe Institute for Operations Research and the Management Sciences (INFORMS) is the largest professional society in the world for professionals in the field of operations research (O.R.).
It was established in 1995 with the merger of the Operations Research Society of America (ORSA) and The Institute of Management Sciences (TIMS).
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INFORMS 2007July 8-11, 2007INFORMS International Puerto RicoWestin Rio Mar Beach Resort & SpaRio Grande, Puerto Rico
O.R. in the Forest SectorO.R. in the Forest Sector is one of the clusters of INFORMS 2007. The sessions in this cluster will include the most interesting forest sector operations research applications from all countries.
Contact:Cluster Chair: Professor Peter Lohmander, SLU, Faculty of Forest Sciences, SE-901 83 Umea, Sweden. e-mail: [email protected]
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Thank you for listening!Here you may reach me in the future:
Peter LohmanderProfessor of Forest Management and Economic Optimization,SLU, Swedish University of Agricultural Sciences, Faculty of Forest Sciences, Dept. Of Forest Economics, SE-901 83 Umea, Sweden
http://www.Lohmander.com
Version 2007-02-18
