Forest and Agricultural Sector Optimization Model (FASOM) Basic Mathematical Structure.
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Transcript of Forest and Agricultural Sector Optimization Model (FASOM) Basic Mathematical Structure.
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Forest and Agricultural Sector Optimization Model (FASOM)
Basic Mathematical Structure
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Linear Programming
j jj
ij j ij
j
c X
. . a X b for all i
X 0 for all j
Max
s t
FASOM can solve up to 6 Million Variables (j), 1 Million Equations (i)
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Important Equations
Objective Function
Resource Restrictions
Commodity Restrictions
Intertemporal Transition Restrictions
Emission Restrictions
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Parameter Description
Technical coefficients (yields, requirements, emissions)
Objective function coefficients
Supply and demand functions
Supply and demand function elasticities
Discount rate, product depreciation, dead wood decomposition
Resource endowments
Soil state transition probabilities
Land use change limits
Initial or previous land allocation
Alternative objective function parameters
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Variable Unit Type DescriptionCROP 1E3 ha 0 Crop productionPAST 1E3 ha 0 Pasture LIVE mixed 0 Livestock raisingFEED mixed 0 Animal feeding TREE 1E3 ha 0 Standing forestsHARV 1E3 ha 0 harvestingBIOM 1E3 ha 0 Biomass crop plantations for bioenergy ECOL 1E3 ha 0 Wetland ecosystem reservesLUCH 1E3 ha 0 Land use changesRESR mixed 0 Factor and resource usagePROC mixed 0 Processing activitiesSUPP 1E3 t 0 SupplyDEMD 1E3 t 0 DemandTRAD 1E3 t 0 TradeEMIT mixed Free Net emissionsSTCK mixed 0 Environmental and product stocksWELF 1E6 € Free Economic SurplusCMIX - 0 Crop Mix
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Index Symbol ElementsTime Periods t 2005-2010, 2010-2015, …, 2145-2150Regions r 25 EU member states, 11 Non-EU international regions Species s All individual and aggregate species categories
Crops c(s) Soft wheat, hard wheat, barley, oats, rye, rice, corn, soybeans, sugar beet, potatoes, rapeseed, sunflower, cotton, flax, hemp, pulse
Trees f(s) Spruce, larch, douglas fir, fir, scottish pine, pinus pinaster, poplar, oak, beech, birch, maple, hornbeam, alnus, ash, chestnut, cedar, eucalyptus, ilex locust, 4 mixed forest types
Perennials b(s) Miscanthus, Switchgrass, Reed Canary Grass, Poplar, , Arundo, Cardoon, Eucalyptus Livestock l(s) Dairy, beef cattle, hogs, goats, sheep, poultry Wildlife w(s) 43 Birds, 9 mammals, 16 amphibians, 4 reptilesProducts y 17 crop, 8 forest industry, 5 bioenergy, 10 livestockResources/Inputs i Soil types, hired and family labor, gasoline, diesel, electricity, natural gas, water, nutrients Soil types j(i) Sand, loam, clay, bog, fen, 7 slope, 4 soil depth classes Nutrients n(i) Dry matter, protein, fat, fiber, metabolic energy, Lysine
Technologies m alternative tillage, irrigation, fertilization, thinning, animal housing and manure management choices
Site quality q Age and suitability differences Ecosystem state x(q) Existing, suitable, marginal Age cohorts a(q) 0-5, 5-10, …, 295-300 [years]Soil state v Soil organic classesStructures u FADN classifications (European Commission 2008) Size classes z(u) < 4, 4 - < 8, 8 - < 16, 16 - < 40, 40- < 100, >= 100 all in ESU (European Commission 2008)
Farm specialty o(u) Field crops, horticulture, wine yards, permanent crops, dairy farms, grazing livestock, pigs and or poultry, mixed farms
Altitude levels h(u) < 300, 300 – 600, 600 – 1100, > 1100 meters
Environment e 16 Greenhouse gas accounts, wind and water erosion, 6 nutrient emissions, 5 wetland types
Policies p Alternative policies
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Objective Function
Maximize+ Area underneath demand curves- Area underneath supply curves- Costs± Subsidies / Taxes from policies
The maximum equilibrates markets!
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Market Equilibrium
Demand
Supply
Price
Quantity
P*
Q*
ProducerSurplus
ConsumerSurplus
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t
t
TREEt t r, j,v,f ,u,a ,m,p r,T, j,v,f ,u,a ,m,p
t t r, j,v,f ,u,a ,m,p
tt
CS
Max WELF RS TREE
C
Basic Objective Function
Terminal value of standing forests
Discount factor Consumer surplusResource surplusCosts of production and trade
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r,t ,yr,y
t t r,t ,yr,y
r,t ,ir,i
DEMD d
CS RS SUPP d
RESR d
DEMDr,t,y
SUPPr,t,y
RESRr,t,i
Consumer and Resource Surplus
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Economic Principles
• Rationality ("wanting more rather than less of a good or service")
• Law of diminishing marginal returns • Law of increasing marginal cost
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Demand function
Area underneath demand function
0 0, p ,qDEMDr,t,y
• Decreasing marginal revenues• uniquely defined by
• constant elasticity function• observed price-quantity pair (p0,q0) • estimated elasticity (curvature)
price
sales
Demand function
q00
p0
q0
q(p) p
p q
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Economic Surplus Maximization
Implicit Supply and Demand
Forest InventoryLand Supply
Water Supply
Labor Supply
Animal Supply
National Inputs Import Supply
Processing Demand
Feed Demand
Domestic Demand
Export Demand
CS
PS
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CROPr,t , j,v,c,u,q,m,p r,t , j,v,c,u,q,m,p
r, j,v,c,u,q,m,p
PASTr,t , j,v,s,u,q,m,p r,t , j,v,s,u,q,m,p
r, j,v,s,u,q,m,p
BIOMr,t , j,v,b,u,q,m,p r,t , j,v,b,u,q,m,p
r, j,v,b,u,q,m,p
r,t , j,v,f ,u,a
t
CROP
PAST
BIOM
C
HARV,m,p r,t , j,v,f ,u,a ,m,p
r, j,v,f ,u,a ,m,p
TREEr,t , j,v,f ,u,a ,m,p r,t , j,v,f ,u,a ,m,p
r, j,v,f ,u,a ,m,p
ECOLr,t , j,v,s,u,x,m,p r,t , j,v,s,u,x,m,p
r, j,v,s,u,x,m,p
LIVEr,t ,l,u,m,p r,t ,l,u,m,
HARV
TREE
ECOL
LIVE
pr,l,u,m,p
PROC FEEDr,t ,m r,t ,m r,t ,l,m r,t ,l,m
r,m r,l,m
LUCHr,t , j,s,u,u r,t , j,s,u,u
r, j,u,u
TRADEr,r,t ,y r,r ,t ,y
r,r ,y
PROC FEED
LUCH
TRAD
Production and Trade
Cost
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ResourceAccounting Equations
(r,t,i)
CROPr,t , j,v,c,u,q,m,p,i r,t , j,v,c,u,q,m,p
j,v,c,u,q,m,p
PASTr,t , j,v,s,u,q,m,p,i r,t , j,v,s,u,q,m,p
j,v,s,u,q,m,p
BIOMr,t , j,v,b,u,q,m,p,i r,t , j,v,b,u,q,m,p
j,v,b,u,q,m,p
r,t , j,v,f ,u,a ,m
CROP
PAST
BIOM
HARV,p,i r,t , j,v,f ,u,a ,m,p
j,v,f ,u,a ,m,p
TREEr,t , j,v,f ,u,a ,m,p,i r,t , j,v,f ,u,a ,m,p
j,v,f ,u,a ,m,p
ECOLr,t , j,v,s,u,x,m,p,i r,t , j,v,s,u,x,m,p
j,v,s,u,x,m,p
LIVEr,t ,l,u,m,p,i r,t ,l,u,m,
HARV
TREE
ECOL
LIVE
r,t ,i
pl,u,m,p
PROCr,t ,m,i r,t ,m
m
FEEDr,t ,l,m,i r,t ,l,m
m
RESR
PROC
FEED
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Physical Resource Limits(r,t,i)
r,t ,i r,t ,iRESR
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CROPr,t , j,v,c,u,q,m,p,y r,t , j,v,c,u,q,m,p
j,v,c,u,q,m,p
PASTr,t , j,v,s,u,q,m,p,y r,t , j,v,
PROCr,t ,m,y r,t ,m
m
FEEDr,t ,l,m,y r,t ,l,m
m
r,r ,t ,yr
r,t ,y
CROP
PAST
PROC
FEED
TRAD
DEMD
s,u,q,m,pj,v,s,u,q,m,p
BIOMr,t , j,v,b,u,q,m,p,y r,t , j,v,b,u,q,m,p
j,v,b,u,q,m,p
HARVr,t , j,v,f ,u,a ,m,p,y r,t , j,v,f ,u,a ,m,p
j,v,f ,u,a ,m,p
TREEr,t , j,v,f ,u,a ,m,p,y r,t , j,v,f ,u,a ,m,p
j,v,f ,u,
BIOM
HARV
TREE
a,m,p
ECOLr,t , j,v,s,u,x,m,p,y r,t , j,v,s,u,x,m,p
j,v,s,u,x,m,p
LIVEr,t ,l,u,m,p,y r,t ,l,u,m,p
l,u,m,p
r,r,t ,yr
r,t ,y
ECOL
LIVE
TRAD
SUPP
Commodity Equations (r,t,y)
Demand Supply
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PROCr,t ,m,y r,t ,m
m
PROC 0
Industrial Processing (r,t,y)
• Processing activities can be bounded (capacity limits) or enforced (e.g. when FASOM is linked to other models)
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Forest Transistion Equations
• Standing forest area today + harvested area today <= forest area from previous period
• Equation indexed by r,t,j,v,f,u,a,m,p
r,t 1, j,v,f ,u,a 1,m,p t 1 a 1r,t , j,v,f ,u,a ,m,p a 1
r,t 1, j,v,f ,u,a ,m,p t 1 a Ar,t , j,v,f ,u,a ,m,p a 1
r, j,v,f ,u,a ,m,p t 1
TREETREE
TREEHARV
INIT
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Emission Accounting
Equation(r,t,e)
CROPr,t,j,v,c,u,q,m,p,e r,t , j,v,c,u,q,m,p
j,v,c,u,q,m,p
PASTr,t,j,v,c,u,q,m,p,e r,t , j,v,c,u,q,m,p
j,v,c,u,q,m,p
BIOMr,t,j,v,b,u,q,m,p,e r,t , j,v,b,u,q,m,p
j,v,b,u,q,m,p
r,t,j
r,t ,e
CROP
PAST
BIOM
EMIT
TREE,v,f,u,a,m,p,e r,t , j,v,f ,u,a ,m,p
j,v,f ,u,a,m,p
ECOLr,t,j,v,s,u,x,m,p,e r,t , j,v,s,u,x,m,p
j,v,s,u,x,m,p
LIVEr,t,s,u,m,p,e r,t,s,u,m,p
s,u,m,p
LUCHr,t ,s,u, , ,e r,t ,s,u, ,
s,u, ,
TREE
ECOL
LIVE
LUCH
PROCr,t ,m,e r,t ,m
m
FEEDr,t ,l,m,e r,t ,l,m
m,l
STCKr,t ,d,e r,t ,d r,t 1,d
d
PROC
FEED
STCK STCK
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Environmental Policy
r,t ,e r,t ,eEMIT
r,t ,e r,t ,er,t ,e
WELF ( ) EMIT or
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Duality restrictions (r,t,u)
• Prevent extreme specialization• Incorporate difficult to observe data• Calibrate model based on duality theory• May include „flexibility contraints“
CMIXr,t , j,v,c,u,q,m,p r,t ,c,u r,t ,t ,u
j,v,c,q,m,p t
CROP CMIX
Past periods
Observed crop mixes
Crop Mix VariableNo crop (c) index!
Crop Area Variable
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Miscellaneous
• GAMS• Systematic Model Check• Linearization• Alternative Objective Function
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Linear Program Duality
i allfor 0 U
j allfor caUs.t.
ZbUMin
i
ji
iji
iii
jallfor0X
iallforbXa..
ZXc
j
ij
jij
jjj
ts
Max
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Reduced Cost
j
ji
iij X
ZcUa
Shadow prices
TechnicalCoefficients
ObjectiveFunctionCoefficients
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Variable Decomposition Example (not from FASOM)
## Landuse_Var(Bavaria,Sugarbeet) SOLUTION VALUE 1234.00
EQN Aij Ui Aij*Uiobjfunc_Equ 350.00 1.0000 350.00Endowment_Equ(Bavaria,Land) 1.0000 90.000 90.000Endowment_Equ(Bavaria,Water) 250.00 0.0000 0.0000Production_Equ(Bavaria,Sugarbeet) -11.000 40.000 -440.00TRUE REDUCED COST 0.0000
j
ji
iij X
ZcUa
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Variable Decomposition Example (not from FASOM)
## Landuse_Var(Bavaria,Wheat) SOLUTION VALUE 0.00000
EQN Aij Ui Aij*Ui objfunc_Equ 350.00 1.0000 350.00 Endowment_Equ(Bavaria,Land) 1.0000 250.00 250.00 Endowment_Equ(Bavaria,Water) 250.00 0.0000 0.0000 Production_Equ(Bavaria,Wheat) -1.0000 89.000 -89.000 TRUE REDUCED COST 511.00
j
ji
iij X
ZcUa
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Complementary Slackness
*' *
*' *
0U b AX
0U A C X
Reduced Cost Opt. Variable Level
Shadow Price Opt. Slack Variable Level
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Solution Decomposition
Insights
• Why is an activity not used?
• How do individual equations contribute to the variable‘s optimality?
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Current work
• Land management adaptation to policy & development
• Externality mitigation (Water, Greenhouse Gases, Biodiversity, Soil fertility)
• Stochastic formulation (extreme events)• Land use & management change costs• Learning and agricultural research policies• Investment restrictions