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- 1. www.helsinki.fi/yliopisto Assessment of Scenario Generation Approaches for Forest Management Planning through Stochastic Programming Kyle Eyvindson and Annika Kangas 27.1.2015Kyle Eyvindson 1
- 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
- 3. www.helsinki.fi/yliopisto • 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.
- 4. www.helsinki.fi/yliopisto • 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
- 5. www.helsinki.fi/yliopisto • 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
- 6. www.helsinki.fi/yliopisto • 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?
- 7. www.helsinki.fi/yliopisto • 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
- 8. www.helsinki.fi/yliopisto • 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:
- 10. www.helsinki.fi/yliopisto • 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:
- 12. www.helsinki.fi/yliopisto • Deterministic solution Kyle Eyvindson A visualization: Soft constraints Hard constraints
- 13. www.helsinki.fi/yliopisto • Stochastic solution Kyle Eyvindson A visualization: Light weight on negative deviations Strong weight on negative deviations
- 14. www.helsinki.fi/yliopistoKyle Eyvindson Results of the SAA: Light weight on negative deviations: Only inventory errors Inventory and Growth model errors
- 15. www.helsinki.fi/yliopistoKyle Eyvindson Results of the SAA: Strong weight on negative deviations: Only inventory errors Inventory and Growth model errors
- 16. www.helsinki.fi/yliopisto • 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. 27.1.2015 16Kyle Eyvindson Conclusions:
- 17. www.helsinki.fi/yliopisto • 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: