Forest and Agricultural Sector
Optimization Model (FASOM)
Basic Mathematical Structure
Linear Programming
Max
c X
j
a X
j

bi
for all i
Xj

0
for all j
j
j
s.t.
ij
j
FASOM can solve up to 6 Million Variables (j), 1 Million Equations (i)
Important Equations

Objective Function

Resource Restrictions

Commodity Restrictions

Intertemporal Transition Restrictions

Emission Restrictions
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
Variable
CROP
PAST
LIVE
FEED
TREE
HARV
BIOM
ECOL
LUCH
RESR
PROC
SUPP
DEMD
TRAD
EMIT
STCK
WELF
CMIX
Unit
1E3 ha
1E3 ha
mixed
mixed
1E3 ha
1E3 ha
1E3 ha
1E3 ha
1E3 ha
mixed
mixed
1E3 t
1E3 t
1E3 t
mixed
mixed
1E6 €
-
Type Description
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Free
0
Free
0
Crop production
Pasture
Livestock raising
Animal feeding
Standing forests
harvesting
Biomass crop plantations for bioenergy
Wetland ecosystem reserves
Land use changes
Factor and resource usage
Processing activities
Supply
Demand
Trade
Net emissions
Environmental and product stocks
Economic Surplus
Crop Mix
Index
Time Periods
Regions
Species
Symbol
t
r
s
Crops
c(s)
Trees
f(s)
Perennials
Livestock
Wildlife
Products
Resources/Inputs
Soil types
Nutrients
Technologies
b(s)
l(s)
w(s)
y
i
j(i)
n(i)
m
Site quality
Ecosystem state
Age cohorts
Soil state
Structures
Size classes
q
x(q)
a(q)
v
u
z(u)
Farm specialty
o(u)
Altitude levels
Environment
Policies
h(u)
e
p
Elements
2005-2010, 2010-2015, …, 2145-2150
25 EU member states, 11 Non-EU international regions
All individual and aggregate species categories
Soft wheat, hard wheat, barley, oats, rye, rice, corn, soybeans, sugar beet, potatoes, rapeseed,
sunflower, cotton, flax, hemp, pulse
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
Miscanthus, Switchgrass, Reed Canary Grass, Poplar, , Arundo, Cardoon, Eucalyptus
Dairy, beef cattle, hogs, goats, sheep, poultry
43 Birds, 9 mammals, 16 amphibians, 4 reptiles
17 crop, 8 forest industry, 5 bioenergy, 10 livestock
Soil types, hired and family labor, gasoline, diesel, electricity, natural gas, water, nutrients
Sand, loam, clay, bog, fen, 7 slope, 4 soil depth classes
Dry matter, protein, fat, fiber, metabolic energy, Lysine
alternative tillage, irrigation, fertilization, thinning, animal housing and manure management
choices
Age and suitability differences
Existing, suitable, marginal
0-5, 5-10, …, 295-300 [years]
Soil organic classes
FADN classifications (European Commission 2008)
< 4, 4 - < 8, 8 - < 16, 16 - < 40, 40- < 100, >= 100 all in ESU (European Commission 2008)
Field crops, horticulture, wine yards, permanent crops, dairy farms, grazing livestock, pigs
and or poultry, mixed farms
< 300, 300 – 600, 600 – 1100, > 1100 meters
16 Greenhouse gas accounts, wind and water erosion, 6 nutrient emissions, 5 wetland types
Alternative policies
Objective Function
Maximize
+ Area underneath demand curves
- Area underneath supply curves
- Costs
± Subsidies / Taxes from policies
The maximum equilibrates markets!
Market Equilibrium
Price
Supply
Consumer
Surplus
P*
Producer
Surplus
Demand
Q*
Quantity
Basic Objective Function
 

   CSt 
  t

TREE
Max WELF     t     RSt  


r, j,v,f ,u,a,m,p  TREE r,T, j,v,f ,u,a,m,p
 t
 r, j,v,f ,u,a,m,p
t 
 


C
   t 

  t

Terminal value of standing forests
Discount factor
Consumer surplus
Resource surplus
Costs of production and trade









Consumer and Resource Surplus
  DEMD

    r,t,y  DEMD r,t,y  d    
 r,y 



 SUPP
 

CSt  RSt     r,t,y  SUPPr,t,y  d   
 r,y 





 RESR
 
    r,t,i  RESR r,t,i  d    
 
 r,i 
Economic Principles
• Rationality ("wanting more rather than less of
a good or service")
• Law of diminishing marginal returns
• Law of increasing marginal cost
Demand function
price
Area
underneath
demand
function
• Decreasing marginal revenues
• uniquely defined by
• constant elasticity function
• observed price-quantity pair (p0,q0)
• estimated elasticity  (curvature)


q(p) p

p q
DEMD
r,t,y
,  p0 , q 0 
p0
Demand
function

sales
q00
q0
Economic Surplus Maximization
Land Supply
Forest Inventory
Processing Demand
Water Supply
CS
Labor Supply
PS
Implicit Supply and Demand
Animal Supply
National Inputs
Domestic Demand
Import Supply
Feed Demand
Export Demand


 
CROP
r,t, j,v,c,u,q,m,p  CROPr,t, j,v,c,u,q,m,p
 r, j,v,c,u,q,m,p

PAST
 
r,t, j,v,s,u,q,m,p  PASTr,t, j,v,s,u,q,m,p
 r, j,v,s,u,q,m,p

BIOM



r,t, j,v,b,u,q,m,p  BIOM r,t, j,v,b,u,q,m,p

 r, j,v,b,u,q,m,p
 
HARV
r,t, j,v,f ,u,a ,m,p  HARVr,t, j,v,f ,u,a,m,p
 r, j,v,f ,u,a,m,p

TREE
 
r,t, j,v,f ,u,a,m,p  TREE r,t, j,v,f ,u,a,m,p
 r, j,v,f ,u,a,m,p
Ct  
ECOL



r,t, j,v,s,u,x,m,p  ECOL r,t, j,v,s,u,x,m,p

 r, j,v,s,u,x,m,p
   LIVE
r,t,l,u,m,p  LIVE r,t,l,u,m, p
 r,l,u,m,p

FEED

PROC


   PROC

r,t,m
r,t,m
r,t,l,m  FEED r,t,l,m
r,l,m
 r,m

LUCH


  r,t, j,s,u,u  LUCH r,t, j,s,u,u
 r, j,u,u
TRADE



 r,r,y r,r,t,y  TRAD r,r,t,y
















































Production
and Trade
Cost

CROP


r,t, j,v,c,u,q,m,p,i  CROPr,t, j,v,c,u,q,m,p

 j,v,c,u,q,m,p
 
 PAST
r,t, j,v,s,u,q,m,p,i  PASTr,t, j,v,s,u,q,m,p
 j,v,s,u,q,m,p

 BIOM
 
r,t, j,v,b,u,q,m,p,i  BIOM r,t, j,v,b,u,q,m,p
 j,v,b,u,q,m,p

HARV



r,t, j,v,f ,u,a,m ,p,i  HARVr,t, j,v,f ,u,a,m,p

 j,v,f ,u,a,m,p
 
 TREE
r,t, j,v,f ,u,a,m,p,i  TREE r,t, j,v,f ,u,a,m,p
 j,v,f ,u,a,m,p

 ECOL
 
r,t, j,v,s,u,x,m,p,i  ECOL r,t, j,v,s,u,x,m,p
 j,v,s,u,x,m,p

LIVE


r,t,l,u,m,p,i  LIVE r,t,l,u,m, p
 
 l,u,m,p
PROC



r,t,m,i  PROC r,t,m
 m

FEED
    r,t,l,m,i  FEED r,t,l,m
 m

 
















Resource
Accounting
Equations
(r,t,i)









  RESR
r,t,i












Physical Resource Limits
(r,t,i)
RESR r,t,i  r,t,i

Commodity
Equations
(r,t,y)


Demand  Supply


    PROC
r,t,m,y  PROC r,t,m
 m

FEED


  r,t,l,m,y  FEED r,t,l,m
 m
   TRAD r,r,t,y
 r
  DEMD
r,t,y

 
 
 CROP
r,t, j,v,c,u,q,m,p,y  CROPr,t, j,v,c,u,q,m,p
 j,v,c,u,q,m,p

PAST




r,t, j,v,s,u,q,m,p,y  PASTr,t, j,v, s,u,q,m,p
 j,v,s,u,q,m,p

BIOM



r,t, j,v,b,u,q,m,p,y  BIOM r,t, j,v,b,u,q,m,p

j,v,b,u,q,m,p
 
  
 HARV
r,t, j,v,f ,u,a,m,p,y  HARVr,t, j,v,f ,u,a,m,p
  j,v,f ,u,a,m,p
 
 TREE
   
r,t, j,v,f ,u,a,m,p,y  TREE r,t, j,v,f ,u,a,m,p
  j,v,f ,u,a,m,p
ECOL
 


r,t, j,v,s,u,x,m,p,y  ECOL r,t, j,v,s,u,x,m,p
  j,v,s,u,x,m,p
 
    LIVE
r,t,l,u,m,p,y  LIVE r,t,l,u,m,p
 l,u,m,p

   TRAD r,r,t,y
 r
 SUPPr,t,y


































Industrial Processing (r,t,y)



PROC
r,t,m,y

 PROCr,t,m  0
m
• Processing activities can be bounded (capacity
limits) or enforced (e.g. when FASOM is linked
to other models)
Forest Transistion Equations
  TREE r,t 1, j,v,f ,u,a 1,m,p

t 1 a 1 
  TREE r,t, j,v,f ,u,a,m,p
 
a 1 


  TREE r,t 1, j,v,f ,u,a,m,p
t 1 a  A 
  HARV
 
r,t, j,v,f ,u,a,m,p a 1 

  INIT

r, j,v,f ,u,a,m,p t 1


• Standing forest area today + harvested area
today <= forest area from previous period
• Equation indexed by r,t,j,v,f,u,a,m,p

 
 
 CROP
r,t,j,v,c,u,q,m,p,e  CROPr,t, j,v,c,u,q,m,p
 j,v,c,u,q,m,p

 PAST
 
r,t,j,v,c,u,q,m,p,e  PASTr,t, j,v,c,u,q,m,p
 j,v,c,u,q,m,p

BIOM



r,t,j,v,b,u,q,m,p,e  BIOM r,t, j,v,b,u,q,m,p

 j,v,b,u,q,m,p
 
 TREE
r,t,j,v,f,u,a,m,p,e  TREE r,t, j,v,f ,u,a,m,p
 j,v,f ,u,a,m,p

 ECOL
 
r,t,j,v,s,u,x,m,p,e  ECOL r,t, j,v,s,u,x,m,p
j,v,s,u,x,m,p
 
LIVE



r,t,s,u,m,p,e  LIVE r,t,s,u,m,p

 s,u,m,p

 LUCH
 LUCH r,t,s,u,, 

 s,u, , r,t,s,u, , ,e

    PROC
r,t,m,e  PROC r,t,m
 m

FEED


  r,t,l,m,e  FEEDr,t,l,m
 m,l
  STCK  STCK  STCK
r,t,d
r,t 1,d
  r,t,d,e
 d

EMITr,t,e










































Emission
Accounting
Equation
(r,t,e)
Environmental Policy
EMITr,t,e  r,t,e
or

WELF  ()   r,t,e  EMITr,t,e
r,t,e

Duality restrictions (r,t,u)
Observed crop mixes
j,v,c,q,m,p
t
Crop Area Variable
•
•
•
•


CROPr,t, j,v,c,u,q,m,p   CMIX
 CMIXr,t,t ,u
r,t ,c,u
Past periods
Prevent extreme specialization
Incorporate difficult to observe data
Calibrate model based on duality theory
May include „flexibility contraints“

Crop Mix Variable
No crop (c) index!
Miscellaneous
•
•
•
•
GAMS
Systematic Model Check
Linearization
Alternative Objective Function
Linear Program Duality
Max
c X
a X
j
j

Z
ij
j

bi
for all i
Xj

0
for all j
U b
U a

Z
j
s.t.
j
Min
i
i
i
s.t.
i ij
i
Ui
 cj
for all j

for all i
0
Reduced Cost
Z
i a ij  Ui  c j   X
j
Shadow prices
Technical
Coefficients
Objective
Function
Coefficients
Variable Decomposition Example
(not from FASOM)
##
Landuse_Var(Bavaria,Sugarbeet)
SOLUTION VALUE
EQN
objfunc_Equ
Endowment_Equ(Bavaria,Land)
Endowment_Equ(Bavaria,Water)
Production_Equ(Bavaria,Sugarbeet)
TRUE REDUCED COST
1234.00
Aij
350.00
1.0000
250.00
-11.000
Ui
1.0000
90.000
0.0000
40.000
Z
i a ij  Ui  c j   X
j
Aij*Ui
350.00
90.000
0.0000
-440.00
0.0000
Variable Decomposition Example
(not from FASOM)
##
Landuse_Var(Bavaria,Wheat)
SOLUTION VALUE
EQN
objfunc_Equ
Endowment_Equ(Bavaria,Land)
Endowment_Equ(Bavaria,Water)
Production_Equ(Bavaria,Wheat)
TRUE REDUCED COST
0.00000
Aij
350.00
1.0000
250.00
-1.0000
Ui
1.0000
250.00
0.0000
89.000
Z
i a ij  Ui  c j   X
j
Aij*Ui
350.00
250.00
0.0000
-89.000
511.00
Complementary Slackness
Opt. Slack Variable Level
Shadow Price
*

 0
 U A  C X
*
 0
U
*'
 b  AX
*'
Reduced Cost
Opt. Variable Level
Solution Decomposition
Insights
• Why is an activity not used?
• How do individual equations contribute to the
variable‘s optimality?
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