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Forest Thinning Optimization with Dynamic Programming

Learn about solving forest management problems using dynamic programming with examples and principles of optimality. Explore stages, decisions, and recursive relations for efficient resource allocation.

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Forest Thinning Optimization with Dynamic Programming

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  1. Dynamic Programming Lecture 13 (5/21/2014)

  2. - A Forest Thinning Example - Age 10 Age 20 Age 30 Age 10 Stage 1 Stage 2 Stage 3 5 850 850 6 750 0 750 150 2 650 200 7 650 650 0 0 100 1 260 3 535 50 11 260 8 600 600 175 100 0 4 410 500 75 9 500 150 400 10 400 Source: Dykstra’s Mathematical Programming for Natural Resource Management (1984)

  3. Dynamic Programming Cont. • Stages, states and actions (decisions) • Backward and forward recursion

  4. Solving Dynamic Programs • Recursive relation at stage i (Bellman Equation):

  5. Dynamic Programming • Structural requirements of DP • The problem can be divided into stages • Each stage has a finite set of associated states (discrete state DP) • The impact of a policy decision to transform a state in a given stage to another state in the subsequent stage is deterministic • Principle of optimality: given the current state of the system, the optimal policy for the remaining stages is independent of any prior policy adopted

  6. Examples of DP • The Floyd-Warshall Algorithm (used in the Bucket formulation of ARM):

  7. Examples of DP cont. • Minimizing the risk of losing an endangered species (non-linear DP): $2M $2M $0 $2M $1M $2M $0 $2M $1M $1M $2M $1M $0 $1M $2M $1M $0 $0 $0 $0 Stage 1 Stage 2 Stage 3 Source: Buongiorno and Gilles’ (2003) Decision Methods for Forest Resource Management

  8. Minimizing the risk of losing an endangered species (DP example) • Stages (t=1,2,3) represent $ allocation decisions for • Project 1, 2 and 3; • States (i=0,1,2) represent the budgets available at • each stage t; • Decisions (j=0,1,2) represent the budgets available • at stage t+1; • denotes the probability of failure of Project t if • decision j is made (i.e., i-j is spent on Project t); and • is the smallest probability of total failure from stage t • onward, starting in state i and making the best • decision j*. Then, the recursive relation can be stated as:

  9. Markov Chains(based on Buongiorno and Gilles 2003) Transition probability matrix P • Vector pt converges to a vector of steady-state probabilities p*. • p* is independent of p0!

  10. Markov Chains cont. • Berger-Parker Landscape Index • Mean Residence Times • Mean Recurrence Times

  11. Markov Chains cont. • Forest dynamics (expected revenues or biodiversity with vs. w/o management:

  12. Markov Chains cont. • Present value of expected returns

  13. Markov Decision Processes

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