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Dynamic Programming

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

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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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