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Published on October 4, 2007

Author: Gulkund

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Stochastic Optimization ESI 6912:  Stochastic Optimization ESI 6912 Instructor: Prof. S. Uryasev NOTES 2: FARMING EXAMPLE Farming Example :  Farming Example 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 Outline Initial Data:  Initial Data Variables:  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 Deterministic Setup of Optimization Problem:  Deterministic Setup of Optimization Problem Optimal Solution for Deterministic Optimization problem:  Optimal Solution for Deterministic Optimization problem Scenario Analysis:  Scenario Analysis Optimal solution based on expected yields Scenario 2 Scenario Analysis (cont’d):  Scenario Analysis (cont’d) Optimal solution based on above average yields (+20%) Scenario 1 Scenario 3 Optimal solution based on below average yields (-20%) Extensive Form of Stochastic Problem:  Extensive Form of Stochastic Problem Scenario approach: +20% 0% -20% 1st scenario 3d scenario 2nd scenario - the first stage Decision Yield (W, C, SB) (2.5, 3.0, 20) (3.0, 3.6, 24) (2.0, 2.4, 16) the second stage - probability 1/3 1/3 1/3 Extensive Form of Stochastic Problem (cont’d):  Extensive Form of Stochastic Problem (cont’d) 1st scenario 2nd scenario 3d scenario 1st scenario 2nd scenario 3d scenario Extensive Form of Stochastic Problem (cont’d):  Extensive Form of Stochastic Problem (cont’d) Optimal Solution: Recourse Reformulation of Stochastic Program:  Recourse Reformulation of Stochastic Program - s-th scenario Random matrix Recourse Reformulation of Stochastic Program (cont’d):  Recourse Reformulation of Stochastic Program (cont’d) stochastic constraints (2nd stage) deterministic constraints (1st stage) stochastic part (2nd stage) deterministic part (1st stage) Recourse Reformulation of Stochastic Program (cont’d):  Recourse Reformulation of Stochastic Program (cont’d) Recourse function: Expected value of the recourse function: Recourse Reformulation of Stochastic Program (cont’d):  Recourse Reformulation of Stochastic Program (cont’d) General model formulation: Slide16:  1. - distributed independently 2. - uniformly distributed Uncertain variables with continuous distributions density: Extensive Formulation of the Stochastic Program:  Extensive Formulation of the Stochastic Program stochastic constraints (2nd stage) deterministic constraints (1st stage) stochastic part (2nd stage) deterministic part (1st stage) Decomposition of Stochastic Program:  Decomposition of Stochastic Program 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) corn sugar beets wheat Recourse Functions:  Recourse Functions Corn Sugar beets Wheat Explicit Form for Recourse Functions:  Explicit Form for Recourse Functions Sugar beets: Wheat: Corn: Recourse Formulation of Stochastic Program:  Recourse Formulation of Stochastic Program Calculation of Expected Values for Recourse Functions:  Calculation of Expected Values for Recourse Functions Wheat: yield is uniformly distributed 1. In case when : where is the expected value of Calculation of Expected Values for Recourse Functions (cont’d):  Calculation of Expected Values for Recourse Functions (cont’d) 2. In the case when : 3. In the case when : Calculation of Expected Values for Recourse Functions (cont’d):  Calculation of Expected Values for Recourse Functions (cont’d) Wheat: Corn: Sugar beets: Expected Recourse Value for Wheat as a Function of Acres Planted:  Expected Recourse Value for Wheat as a Function of Acres Planted Global Formulation of Stochastic Program:  Global Formulation of Stochastic Program are continuous convex functions depending only upon decision vector Convex optimization problem Derivation of Optimal Solution:  Derivation of Optimal Solution Karush-Kuhn-Tucker conditions: Notations: - the dual variable Calculation of Derivatives:  Calculation of Derivatives Wheat: Corn: Sugar beets: Optimal solution:  Optimal solution Assume: Using enumerative technique, it can be established that optimal solution should satisfy System: Optimal values: Deterministic Optimization:  Deterministic Optimization = random variable = expected value of random variable Scenario Analysis:  Scenario Analysis Optimization problem corresponding to scenario = realization of random variable in the scenario = vector in the scenario Extensive Form of Stochastic Program:  Extensive Form of Stochastic Program = probability of the scenario = expected loss Variables:  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 Deterministic Setup of Optimization Problem:  Deterministic Setup of Optimization Problem f(x,y) Optimal Solution for Deterministic Optimization problem:  Optimal Solution for Deterministic Optimization problem Scenario Analysis:  Scenario Analysis Optimal solution based on expected yields Scenario 2 Scenario Analysis (cont’d):  Scenario Analysis (cont’d) Optimal solution based on above average yields (+20%) Scenario 1 Scenario 3 Optimal solution based on below average yields (-20%) Extensive Form of Stochastic Problem:  Extensive Form of Stochastic Problem Scenario approach: +20% 0% -20% 1st scenario 3d scenario 2nd scenario - the first stage Decision Yield (W, C, SB) (2.5, 3.0, 20) (3.0, 3.6, 24) (2.0, 2.4, 16) the second stage - probability 1/3 1/3 1/3 Extensive Form of Stochastic Problem (cont’d):  Extensive Form of Stochastic Problem (cont’d) 1st scenario 2nd scenario 3d scenario 1st scenario 2nd scenario 3d scenario f(x,y1) f(x,y2) f(x,y3) Optimal Solution for Stochastic Problem Formulated in Extensive Form:  Optimal Solution for Stochastic Problem Formulated in Extensive Form Problem with Recourse:  Problem with Recourse Recourse function: Problem with Recourse (cont’d):  Problem with Recourse (cont’d) where Recourse Reformulation of Stochastic Program:  Recourse Reformulation of Stochastic Program - s-th scenario Random matrix Expected Value of Perfect Information (EVPI):  Expected Value of Perfect Information (EVPI) is an optimal solution of Solution with perfect information = expected performance with perfect information Stochastic programming solution is an optimal solution of EVPI Value of Stochastic Solution (VSS):  Value of Stochastic Solution (VSS) is an optimal solution of Expected value solution Stochastic programming solution is an optimal solution of VSS Stochastic Programming Solution:  Stochastic Programming Solution Scenario Analysis:  Scenario Analysis Optimal solution based on expected yields Scenario 2 Scenario Analysis (cont’d):  Scenario Analysis (cont’d) Optimal solution based on above average yields (+20%) Scenario 1 Scenario 3 Optimal solution based on below average yields (-20%) Farming Example, EVPI:  Farming Example, EVPI Solution with perfect information Stochastic programming solution Farming Example, VSS:  Farming Example, VSS is an optimal solution of Expected value solution Stochastic programming solution VSS

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