2. www.helsinki.fi/yliopisto
• Aim is to:
• Integrate uncertainty into the development of forest
plans
‒ Inventory, growth models, climate change...
• Produce a robust solution which meets the demands of
the decision maker(s), and can accommodate
preferences towards risks
• One method is through stochastic programming
‒ issues of tractability can become an issue
27.1.2015 2Kyle Eyvindson
Introduction
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• Mathematical optimization where some parameters are
uncertain.
• Depending on the structure of the problem, different
problem formulation alternatives are available
‒ simple recourse
‒ two-stage (multi stage) recourse
1/27/2015 3Kyle Eyvindson
Stochastic programming:
Briefly
Determine
optimal time
to conduct
inventory to
maximize ...
Maximize First
period harvest
volume, s.t.
non-declining
harvest.
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• From LP to SP – a 2 stand example:
H – harvest, T – Thin, N – Do nothing.
1/27/2015 4Kyle Eyvindson
SP formulated through a deterministic
approximation of the uncertainties.
(Birge and Louveaux 2011)
t=0
t=1
t=2
H NT
N N H T N
NT
N H T N
H NT
N N H T N
NT
N H T N
H NT
N N H T N
NT
N H T N
H NT
N N H T N
NT
N H T N
H NT
N N H T N
NT
N H T N
H NT
N N H T N
NT
N H T N
Each scenario is a representation of the current and future forest resources
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• This requires the known (or estimated) distribution of
the error.
• A number of scenarios are developed to approximate
the distribution. (King and Wallace 2012)
‒ A need for balance:
too many scenarios – tractability issues
too few scenarios – problem representation issues
Kyle Eyvindson
Incorporating uncertainty into
the planning problem
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• It depends on:
• the formulation used,
• the risk preferences involved,
• the amount of uncertainty under consideration
• the accuracy required
• One way to determine an appropriate number of scenarios
is through the sample average approximation (SAA,
Kleywegt et al. 2001.)
Kyle Eyvindson
How many scenarios is enough?
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• A method for evaluating the quality of a stochastic
solution.
• The algorithm simply:
‒ Select the size of the samples (N and N’), and number of
replications (M)
‒ For each m in M:
‒ Solve the problem
‒ This provides an estimate of the objective function (using N), and with
this solution, evaluate the problem using N’
‒ Evaluate the optimality gap and variance of the estimator – if
gap is too high, increase N and/or N’
Kyle Eyvindson
Sample Average Approximation
(Kleywegt et al. 2001)
N’>>N
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• A forest where the DM wishes
to
• maximize first period income
‒ subject to:
‒ even flow constraints;
‒ and an end inventory constraint.
• Small forest holding
‒ 47.3 hectares, 41 stands
Forest planning problem
27.1.2015 8Kyle Eyvindson
22%
17%
20%
9%
32%
Age Class Distribution (years)
0-20
20-40
40-60
60-80
80+
30%
8%
9%
6%
31%
16%
Diameter Distribution (m)
0-5
5-10
10-15
15-20
20-25
25+
0
10
20
30
40
50
60
Pine Spruce Birch
Wood Volume (m3/ha)
9. www.helsinki.fi/yliopistoKyle Eyvindson
• Two cases are studied:
• The case where only the inventory uncertainty is
included
• and where both inventory uncertainty and growth model
errors are included.
• A few assumptions were made:
1. A recent inventory was conducted
2. The inventory method was assumed to have an error
which was normally distributed, mean zero and a standard
deviation of 20% of the mean height and basal area.
Scenario generation approach:
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• For each inventory error, a set of 50 growth model error
scenarios were simulated.
• The growth model errors were generated using a one
period autoregressive process [AR(1)], using the same
models as Pietilä et al. 2010.
• Forest simulation was done using SIMO (Rasinmäki et al.
2009)
• Created a set of 528 schedules for the 41 stands (~13 schedules per
stand) for each scenario.
27.1.2015 10Kyle Eyvindson
Scenario generation approach:
(2)
11. www.helsinki.fi/yliopistoKyle Eyvindson
• A standard even flow problem.
• Maximize: 1st period incomes
‒ subject to even flow and end inventory constraints
Using both hard and soft constraints
• For application in a stochastic setting this problem needs slight
modification:
• Maximize: Expected 1st period incomes – sum of scenario
based negative deviations
‒ subject to soft even flow an end inventory constraints
Having strict constraints is not the real intention behind the even-flow
problem.
The soft constraints allow for a ‘more or less’ even flow in all scenarios.
Sample problem:
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• The size of the stochastic problem need not be
enormous.
• The size of the problem depends upon:
‒ the amount of uncertainty under consideration,
‒ the importance the uncertainty has in the problem
formulation, and
‒ the acceptability of selecting a ‘sub-optimal’ solution.
• A stochastic program with a sizable optimality gap still
outperform the deterministic equivalent.
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Conclusions:
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• Birge, J.R., and Louveaux, F. 2011. Introduction to stochastic programming. Second
edition. Springer, New York. 499 p.
• Kangas, A., Hartikainen, M., and Miettinen, K. 2013. Simultaneous optimization of
harvest schedule and measurement strategy. Scand. J. Forest Res.(ahead-of-print),
1-10. doi: 10.1080/02827581.2013.823237.
• Kleywegt, Shapiro, Homem-de-Mello. 2001. The sample average approximation for
stochastic discrete optimization. SIAM. J. OPTIM. (12:2) 479-502.
• King, A.J., and Wallace, S.W. 2012 Modeling with Stochastic Programming,
Springer, New York
• Krzemienowski, A. & Ogryczak W. 2005. On extending the LP computable risk
measures to account downside risk. Computational Optimization and Applications
32:133-160.
• Rasinmäki, J., Mäkinen, A., and Kalliovirta, J. 2009. SIMO: an adaptable simulation
framework for multiscale forest resource data. Comput. Electron. Agric. 66(1): 76–
84. doi: 10.1016/j.compag.2008.12.007.
• Pietilä, Kangas, Mäkinen, Mehtätalo. 2010. Influence of Growth Prediction Errors on
the Expeced Loses from Forest Decisions. Silva Fennica 44(5). 829:843.
27.1.2015 17Kyle Eyvindson
References:
Editor's Notes
One scenario represents one deterministic version of the possible future. Inventory, growth errors and climate change errors can be incorporated into the scenario development.
Here I’ll describe the difference between the result and the scenarios.
Here I’ll describe what is going on with the solutions. A much steadier flow of income over the periods – some negative deviations, yes, but a much flatter profile.
