Stochastic Optimization ESI 6912

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Stochastic OptimizationESI 6912
NOTES 2 FARMING EXAMPLE

• Instructor Prof. S. Uryasev

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Farming Example
Outline

• 1. Deterministic Setup of The Optimization
  Problem
• 2. Extensive Form of The Stochastic Program
• 3. Recourse Reformulation of The Stochastic
  Problem.
• 4. Expected Value of Perfect Information (EVPI)
• 5. Value of Stochastic Solution

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Initial Data
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Variables
- acres of land devoted to wheat - acres of land
devoted to corn - acres of land devoted to sugar
beets - tons of wheat sold - tons of corn sold -
tons of sugar beets sold at favorable price -
tons of sugar beets sold at lower price - tons
of wheat purchased - tons of corn purchased
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Deterministic Setup of Optimization Problem
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Optimal Solution for Deterministic Optimization
problem
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Scenario Analysis
Optimal solution based on expected yields
Scenario 2
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Scenario Analysis (contd)
Optimal solution based on above average yields
(20)
Scenario 1
Optimal solution based on below average yields
(-20)
Scenario 3
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Extensive Form of Stochastic Problem
Scenario approach
Yield (W, C, SB)
the second stage -
1st scenario
20
(3.0, 3.6, 24)
probability
1/3
2nd scenario
1/3
0
(2.5, 3.0, 20)
1/3
3d scenario
-20
(2.0, 2.4, 16)
Decision
- the first stage
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Extensive Form of Stochastic Problem(contd)
1st scenario
2nd scenario
3d scenario
1st scenario
2nd scenario
3d scenario
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Extensive Form of Stochastic Problem(contd)
Optimal Solution
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Recourse Reformulation of Stochastic Program
- s-th scenario
Random matrix
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Recourse Reformulation of Stochastic Program
(contd)
deterministic part (1st stage)
stochastic part (2nd stage)
deterministic constraints (1st stage)
stochastic constraints (2nd stage)
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Recourse Reformulation of Stochastic Program
(contd)
Recourse function
Expected value of the recourse function
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Recourse Reformulation of Stochastic Program
(contd)
General model formulation
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Uncertain variables with continuous distributions
1. - distributed independently
2. -
uniformly distributed
density
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Extensive Formulation of the Stochastic Program
deterministic part (1st stage)
stochastic part (2nd stage)
deterministic constraints (1st stage)
stochastic constraints (2nd stage)
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Decomposition of Stochastic Program
sugar beets
corn
wheat
depend only upon decision and random yield
(wheat)
depend only upon decision and random yield
(corn)
depend only upon decision and random yield
(sugar beets)
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Recourse Functions
Wheat
Corn
Sugar beets
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Explicit Form for Recourse Functions
Wheat
Corn
Sugar beets
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Recourse Formulation of Stochastic Program
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Calculation of Expected Values for Recourse
Functions
Wheat
yield is uniformly distributed
1. In case when
where is the expected value
of
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Calculation of Expected Values for Recourse
Functions (contd)
2. In the case when

3. In the case when
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Calculation of Expected Values for Recourse
Functions (contd)
Wheat
Corn
Sugar beets
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Expected Recourse Value for Wheat as a Function
of Acres Planted
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Global Formulation of Stochastic Program
Convex optimization problem
are continuous convex functions depending only
upon decision vector
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Derivation of Optimal Solution
Notations
- the dual variable
Karush-Kuhn-Tucker conditions
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Calculation of Derivatives
Wheat
Corn
Sugar beets
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Optimal solution
Assume
Using enumerative technique, it can be
established that optimal solution should satisfy
System
Optimal values
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Deterministic Optimization
random variable
expected value of random variable
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Scenario Analysis
Optimization problem corresponding to scenario
realization of random variable in the
scenario
vector in the scenario
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Extensive Form of Stochastic Program
probability of the scenario
expected loss
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Variables
- acres of land devoted to wheat - acres of land
devoted to corn - acres of land devoted to sugar
beets - tons of wheat sold - tons of corn sold -
tons of sugar beets sold at favorable price -
tons of sugar beets sold at lower price - tons
of wheat purchased - tons of corn purchased
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Deterministic Setup of Optimization Problem
f(x,y)
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Optimal Solution for Deterministic Optimization
problem
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Scenario Analysis
Optimal solution based on expected yields
Scenario 2
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Scenario Analysis (contd)
Optimal solution based on above average yields
(20)
Scenario 1
Optimal solution based on below average yields
(-20)
Scenario 3
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Extensive Form of Stochastic Problem
Scenario approach
Yield (W, C, SB)
the second stage -
1st scenario
20
(3.0, 3.6, 24)
probability
1/3
2nd scenario
1/3
0
(2.5, 3.0, 20)
1/3
3d scenario
-20
(2.0, 2.4, 16)
Decision
- the first stage
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Extensive Form of Stochastic Problem(contd)
f(x,y1)
1st scenario
f(x,y2)
2nd scenario
f(x,y3)
3d scenario
1st scenario
2nd scenario
3d scenario
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Optimal Solution for Stochastic
ProblemFormulated in Extensive Form
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Problem with Recourse
Recourse function
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Problem with Recourse (contd)
where
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Recourse Reformulation of Stochastic Program
- s-th scenario
Random matrix
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Expected Value of Perfect Information (EVPI)
Solution with perfect information
is an optimal solution of
expected performance with perfect
information
Stochastic programming solution
is an optimal solution of
EVPI
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Value of Stochastic Solution (VSS)
Expected value solution
is an optimal solution of
Stochastic programming solution
is an optimal solution of
VSS
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Stochastic Programming Solution
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Scenario Analysis
Optimal solution based on expected yields
Scenario 2
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Scenario Analysis (contd)
Optimal solution based on above average yields
(20)
Scenario 1
Optimal solution based on below average yields
(-20)
Scenario 3
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Farming Example, EVPI
Solution with perfect information
Stochastic programming solution
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Farming Example, VSS
Expected value solution
is an optimal solution of
Stochastic programming solution
VSS