a stochastic approach to optimize eucalypt stand...

Introduction Model Conclusions and Further work

A Stochastic Approach to Optimize EucalyptStand Management Scheduling, Under Fire Risk

Liliana FerreiraESTG*, Instituto Politecnico de Leiria

Joint work with:

M. ConstantinoCIO-DEIO, Faculdade de Ciencias da Universidade de Lisboa

J.G. Borges; J.Garcia-GonzalezCentro de Estudos Florestais, Instituto Superior de Agronomia

ALIOS-INFORMS Buenos Aires, Argentina 6-9 June*ESTG - Escola Superior de Tecnologia e Gestao

Liliana Ferreira IPL - ESTG

Eucalyptus Globulus

Outline

1 Introduction

2 ModelStochastic Dynamic ProgrammingResultsSensitivity Analysis

3 Conclusions and Further work

Eucalyptus Globulus

Outline

1 Introduction

Eucalyptus Globulus

Introduction

Forest Management:

Stand level;

Landscape level.

Eucalyptus Globulus

Introduction

Forest Management:

Stand level;

Landscape level.

Eucalyptus Globulus

Introduction

Forest Management:

Stand level;

Landscape level.

Eucalyptus Globulus

Concepts

Rotation length:

period of time since the stand is planted until it is clearcut;

Coppice cycle:

period of time since the stand regenerates until it is harvested;

Soil expectation value:

financial criterion used to make even-aged timbermanagement decisions;

present value of a perpetual series of even-aged rotations.

Eucalyptus Globulus

Concepts

Rotation length:

Coppice cycle:

Eucalyptus Globulus

Concepts

Rotation length:

Coppice cycle:

Eucalyptus Globulus

Research Aim

Development of an optimization forest management scheduling model:

at stand level;

with a stochastic element - wildfire risk;

Eucalyptus globulus Labill stand (even aged stand);

for short rotation coppice systems;

maximizes the soil expectation value;

finds the optimal harvest age in each cycle;

determines the optimal number of coppice cycles within a fullrotation.

Eucalyptus Globulus

Research Aim

at stand level;

Eucalyptus Globulus

Research Aim

at stand level;

Eucalyptus Globulus

Research Aim

at stand level;

Eucalyptus Globulus

Research Aim

at stand level;

Eucalyptus Globulus

Research Aim

at stand level;

Eucalyptus Globulus

Research Aim

at stand level;

Eucalyptus Globulus

Outline

1 Introduction

Eucalyptus Globulus

Stochastic Dynamic Programming

Dynamic Programming

Resolution of de subproblems, through a network, ina recursive form!

Subproblem 1

…Subproblem 2 Subproblem N‐1

Subproblem N

Eucalyptus Globulus

Stochastic dynamic programming

Stages

States

Decisions

Eucalyptus Globulus

Stages

States

Decisions

Eucalyptus Globulus

Stages

States

Decisions

Eucalyptus Globulus

Stages

States

Decisions

Eucalyptus Globulus

Stage:

characterized by the cumulative number of harvests during thelifetime of a plantation (n)

0 1 st …

Stage 1 Stage 2 Stage 3

2 nd 3 rd

harvest harvest harvest

Eucalyptus Globulus

State:

characterized by the number of years since the stand was planted(Tn) 0 1 st …

Stage 1 Stage 2 Stage 3

2 nd 3 rd

harvest harvest harvest

Eucalyptus Globulus

Decisions

Decisions:

definition of planned duration for coppice cycle (In);

number of selected sprouts by stool (Vn);

number of fuel treatments (Mn).

Início do estágio n

Início do estágio n+1

Crescimento / Incêndio

Estágio n

(Tn+1)

Corte talhadia ou finalLimpezas de mato (Mn)

Selecção de varasVn

3 anos

Beginningnth stage

Beginning(n+1)th stage

Growth / Fire

nth stage

(Tn+1)

Coppice or clear cutShrub cleaning (Mn)

Sprout selectionVn

3 years

Eucalyptus Globulus

Decisions

Decisions:

Estágio n

(Tn+1)

3 anos

Beginningnth stage

Growth / Fire

nth stage

(Tn+1)

Sprout selectionVn

3 years

Eucalyptus Globulus

Decisions

Decisions:

Estágio n

(Tn+1)

3 anos

Beginningnth stage

Growth / Fire

nth stage

(Tn+1)

Sprout selectionVn

3 years

Eucalyptus Globulus

Dynamic Programming Network

E1 E2 E31st harvest 2nd harvest

Number of harvests

3rd harvest

S32 S48

E4 4st harvest

Eucalyptus Globulus

DP network nodes translate the possible states for each stage;

DP network arcs take into consideration the range of decisionsthat have to be taken for each state;

Aim: determine the optimal policy - a function that associatesto each state one set of management decisions (e.g. fueltreatment, sprout selection, coppice cycles and rotationlength).

Eucalyptus Globulus

Fire Risk

Fire risk was incorporated through wildfire and damage scenarios

A scenario defines a particular typeof fire, characterized by:

the time of fire occurrence;

its intensity.

Eucalyptus Globulus

Fire Risk

Fire risk was incorporated through wildfire and damage scenarios

A scenario defines a particular typeof fire, characterized by:

the time of fire occurrence;

its intensity.

Eucalyptus Globulus

Fire Risk

Set of all possible scenarios J = J1∪ J2

J1 includes:

the scenario where a fire does not occur during a stage;

all the scenarios involving the occurrence of wildfires withoutmortality, that allow the stand’s growth;

J2 includes:

the set of scenarios involving wildfires with death of trees thatforce a clearcut of the stand.

Eucalyptus Globulus

Fire Risk

J1 includes:

J2 includes:

Eucalyptus Globulus

Fire Risk

J1 includes:

J2 includes:

Eucalyptus Globulus

Model Building

Model Functions

G FunctionG Function

F FunctionS Function

Eucalyptus Globulus

Model Building

The model was defined as follows:

Determine Z = F1(0)− CP (1)

Fn(Tn) = maxIn ∈ ΨnVn ∈ ΘnMn ∈ Πn

{Gn(Tn, In, Vn,Mn), Sn(Tn)} , n = 1, ...,N (2)

FN+1(TN+1) = SN+1(TN+1) (3)

Gn(Tn, In, Vn,Mn) =∑

j∈J1pj (Tn, In, Vn,Mn)

[B jn(Tn, In, Vn)− CV j

n(Tn)− Ljn(Tn, In,Mn) + Fn+1(Tn + In)]

j∈J2pj (Tn, In, Vn,Mn)

[B jn(Tn, In, Vn)− CV j

n(Tn)− Ljn(Tn,Hj ,Mn) + Sn(Tn + H j )

Tn+1 = Tn + In, with n = 1, ...,N and T1 = 0 (5)

Sn(Tn) =F1(0)− CR

(1 + i)Tn, with n = 2, ...,N + 1 and S1(T1) = 0 (6)

Eucalyptus Globulus

Model Building

Backward Method

Starts at the last stage;

Iterative process:

- to start the process it is necessary to use an estimate for the ”bareland nodes”(value of the rotations to perpetuity);

- Stopping condition: |Estimate value − solution| < small value;

- the convergence of the process can be proved using the fixed pointtheorem;

- the convergence is usually achieved after few iterations.

Eucalyptus Globulus

Model Building

Return Function

B jn(Tn, In,Vn) = R j

n(Tn, In,Vn) + Sal jn(Tn,Vn) (7)

R jn(Tn, In,Vn) =

(1 + i)Tn+In× Vol(n, In,Vn) , j ∈ J1

(1 + i)Tn+H j×(1− pmj

)× Vol(n,H j ,Vn), j ∈ J2

Sal jn(Tn,Vn) =

0 , j ∈ J1

(1 + i)Tn+H j× pmj × Vol(n,H j ,Vn), j ∈ J2

Eucalyptus Globulus

Model Building

Sprout selection cost:

CV jn(Tn) =

CV × NVn

(1 + i)Tn+3, if I jn > 3

0 , otherwise

CV = cost per sprout of doing sprout selection;

NVn = number of sprouts at nth stage.

Fuel treatment cost:

Ljn(Tn, Ijn,Mn) =

I jn∑l=1

(1 + i)Tn+l× PMn(l , I jn,Mn) (11)

PMn(l, I jn,Mn) =

1, if is carried out a fuel treatment, in lth year of jth scenario, with l = 1, ..., I jn

0, otherwise

Eucalyptus Globulus

Outline

1 Introduction

Eucalyptus Globulus

SDP model

Economic Parameters

Prices:

Timber stumpage price: 36 e/m3;Salvage price: 27 e/m3;Discount rate: 4%

Operational Costs:

( / )Operation Fixed cost (€/ha) Variable Cost

Shrub removal 167 ‐‐

l b f lPlantation cost 725 0.14€×number of plants

Sprout selection cost ‐‐‐ 0.15€×number of sprouts

i b f lConversion cost 1204 0.14€×number of plants

Table 1 ‐ Source: CAOF's Database of ANEFA

Eucalyptus Globulus

SDP model

States

Decision Possible values

Harvest age Ψn = {10, 11, …, 16}

Sprouts selected Θn = {1; 1.5; 2}

Shrub cleaning Πn = {1, 2, 3}

Table 2 ‐ Possible values for management decisions

Stage States

1 T = {0}

2 T = {1, 2, ..., 16}

3 T = {11, 12, ..., 32}

4 T = {21, 22, ..., 48}

5 T = {31, 32, ..., 64}

Table 3 ‐ Possible states for stochastic model

Eucalyptus Globulus

SDP model

Vegetation growth models

Globulus 3.0 - a growth and production model developed forPortuguese eucalypt stands (Tome et al.,2006);

Understory growth was estimated according to a modeldeveloped by (Botequim et al., 2009):

Biom = 17.745×(

1− exp−(0.085×understory age+0.004∗AB))

Eucalyptus Globulus

SDP model

Vegetation growth models

Globulus 3.0 - a growth and production model developed forPortuguese eucalypt stands (Tome et al.,2006);

Understory growth was estimated according to a modeldeveloped by (Botequim et al., 2009):

Biom = 17.745×(

1− exp−(0.085×understory age+0.004∗AB))

Eucalyptus Globulus

SDP model

Wildfire occurrence and damage models

1 Fire risk model: provides annual wildfire risk probability foreven aged eucalypt stands in Portugal

PfAge =1

1 + e−(−2.4368+0.0815×Biom−0.0671×Age−0.0671×AspSW+0.000613×N+0.0373×Dg)

2 Damage model: gives the proportion of dead trees in thestand, if mortality occurs

pam =1

1 + e−(0.8537+0.00244×Alt+0.0197×Slope−0.0851×AB+0.2246×Sd)

Eucalyptus Globulus

SDP model

Wildfire occurrence and damage models

1 Fire risk model: provides annual wildfire risk probability foreven aged eucalypt stands in Portugal

PfAge =1

1 + e−(−2.4368+0.0815×Biom−0.0671×Age−0.0671×AspSW+0.000613×N+0.0373×Dg)

2 Damage model: gives the proportion of dead trees in thestand, if mortality occurs

pam =1

1 + e−(0.8537+0.00244×Alt+0.0197×Slope−0.0851×AB+0.2246×Sd)

Eucalyptus Globulus

SDP model

Wildfire and damage scenarios probabilities

annual probabilities were assumed to be independent;

scenarios consider the probability of one wildfire occurrenceduring a cycle;

scenarios’ probabilities:

pj = Pij × (1− Pmort) , if j ∈ J1 − {0}pj = Pij × Pmort , if j ∈ J2

pj = 1−∑

j∈J1−{0}∪J2

pj , if j = 0

Eucalyptus Globulus

SDP model

pj = 1−∑

j∈J1−{0}∪J2

pj , if j = 0

Eucalyptus Globulus

SDP model

pj = 1−∑

j∈J1−{0}∪J2

pj , if j = 0

Eucalyptus Globulus

Results

Deterministic case

DP algorithm was programmed with C++;

test problem was solved with a desktop computer(CPU Duo P8400 with 3GB of RAM);

Deterministic case

1st cycle 2nd cycle 3rd cycle 4th cycleNPL

1 cycle 2 cycle 3 cycle 4 cycleSEV (€/ha)In Mn Vn In Mn Vn In Mn Vn In Mn Vn

1111 15 1 ‐ 16 1 2 16 1 2 16 1 2 4390.12

1250 15 1 ‐ 16 1 2 16 1 2 16 1 2 4584.98

1667 14 1 ‐ 14 1 2 16 1 2 14 1 2 5153.99

NPL = Number of planted trees

SEV = Soil expectation value

Table 4 ‐ Results for deterministic case

Eucalyptus Globulus

Results

Deterministic case

DP algorithm was programmed with C++;

test problem was solved with a desktop computer(CPU Duo P8400 with 3GB of RAM);

Deterministic case

1st cycle 2nd cycle 3rd cycle 4th cycleNPL

1 cycle 2 cycle 3 cycle 4 cycleSEV (€/ha)In Mn Vn In Mn Vn In Mn Vn In Mn Vn

1111 15 1 ‐ 16 1 2 16 1 2 16 1 2 4390.12

1250 15 1 ‐ 16 1 2 16 1 2 16 1 2 4584.98

1667 14 1 ‐ 14 1 2 16 1 2 14 1 2 5153.99

NPL = Number of planted trees

SEV = Soil expectation value

Table 4 ‐ Results for deterministic case

Eucalyptus Globulus

Results

Stochastic case

NPL 1st cycle 2nd cycle 3rd cycle 4th cycle SEV

(€/ha)In Mn Vn In Mn Vn In Mn Vn In Mn Vn

1111 16 1 16 1 2 16 1 2 16 1 2 2387 131111 16 1 ‐ 16 1 2 16 1 2 16 1 2 2387.13

1250 16 1 ‐ 16 1 2 16 1 2 16 1 2 2498.68

1667 16 1 ‐ 16 1 2 16 1 2 16 1 2 2813.84

Table 5 ‐ Results for stochastic dynamic programming model

Stage States

1st cycle 10.7

2nd cycle 10.46

3rd cycle 10.86

4th cycle 11.12

Table 6 ‐ Expected rotation length for SDP model with NPL=1111model, with NPL=1111

Eucalyptus Globulus

Sensitivity Analysis

Variations in discount rate

Stochastic case

Rate(%)

1st cycle 2nd cycle 3rd cycle 4th cycleSEV (€/ha)In Mn Vn In Mn Vn In Mn Vn In Mn Vn

2 16 1 ‐ 16 1 2 16 1 2 16 1 2 7307.97

4 16 1 ‐ 16 1 2 16 1 2 16 1 2 2387.13

6 15 1 ‐ 16 1 2 16 1 2 16 1 2 793.76

8 13 1 ‐ 14 1 2 14 1 2 16 1 2 73.08

Table 7 ‐ Sensitivity analysis for variations in discount rate, with NPL=1111

Eucalyptus Globulus

Sensitivity Analysis

Variations in prices

Stochastic caseStochastic case

P1€/m³

P2€/m³

1st cycle 2nd cycle 3rd cycle 4th cycleSEV

(€/ha)I M V I M V I M V I M VIn Mn Vn In Mn Vn In Mn Vn In Mn Vn

28.8 21.6 16 1 - 16 1 2 16 1 2 16 1 2 1345.76

36 27 16 1 - 16 1 2 16 1 2 16 1 2 2387.13

43.2 32.4 16 1 - 16 1 2 16 1 2 16 1 2 3428.49

P1 = timber stumpage price

P2 = salvage price

Table 8 ‐ Sensitivity analysis for variations in prices (changes of 20%), with NPL=1111

Eucalyptus Globulus

Outline

1 Introduction

Eucalyptus Globulus

Conclusions

In contrast to conventional deterministic DPapproaches, the solution by our formulationdoes not produce a predefined optimalprescription or an ”optimal path”;

It produces optimal stand managementpolicies, e.g. sprout selection, clearcut andfuel treatment options according to thestand state at any time;

Eucalyptus Globulus

Conclusions

In contrast to conventional deterministic DPapproaches, the solution by our formulationdoes not produce a predefined optimalprescription or an ”optimal path”;

It produces optimal stand managementpolicies, e.g. sprout selection, clearcut andfuel treatment options according to thestand state at any time;

Eucalyptus Globulus

Conclusions and further work

As a sequence of interrelated decisions ispresented, dynamic programming providesan efficient procedure to solve the problem;

The convergence of the stochastic model isquite good. Generally, not many iterationsare needed to get convergence;

Next step - examine the risk of fire and fueltreatment at landscape level.

Eucalyptus Globulus

a stochastic approach to optimize eucalypt stand...

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