Stochastic_Programming

# Stochastic_Programming - EMSE 154 254 Applications of...

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EMSE 154 – 254 Applications of Linear and Nonlinear Optimization Instructor: Hernan Abeledo Source: LINGO documentation 1 EMSE 154-254 Applied Optimization Modeling Stochastic Programming

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EMSE 154 – 254 Applications of Linear and Nonlinear Optimization Instructor: Hernan Abeledo Source: LINGO documentation 2 Introduction The main reason multiperiod planning is difficult is because of uncertainty about the future. Typically, some action or decision must be taken today, which somehow strikes a compromise between the actions that would be, in retrospect, best for each of the possible “futures”. The Scenario Approach We will consider here planning problems with two periods. These situations consist of the following sequence of events: 1) We make a first-period decision. 2) Nature (or the marketplace) makes a random decision. 3) We make a second-period decision that attempts to repair the havoc wrought by nature in (2). The scenario approach assumes there are a finite number of decisions nature can make. We call each of these possible states of nature a “scenario”. For example, in practice, many people are satisfied with classifying demand for a product as being low, medium, or high; or classifying a winter as being severe, normal, or mild, rather than requiring a statement of the average daily temperature and total snowfall measured to six decimal places. General Motors has historically used low, medium, and high scenarios to represent demand uncertainty. Stochastic Programming: an LP based modeling approach for decision making under uncertainty.
EMSE 154 – 254 Applications of Linear and Nonlinear Optimization Instructor: Hernan Abeledo Source: LINGO documentation 3 Example 1 In this example only steps (1) and (2) are important. A farmer can plant corn, sorghum, or beans. He is willing to classify the growing season as being either wet or dry. For simplicity, assume the season can turn out to be only wet or dry, nothing in between. If it is wet, then corn is more profitable. Otherwise, beans are more profitable. The specifics are listed in the table below: In step (1), the farmer decides upon how much to plant of each crop. In step (2), nature decides if the season is wet or dry. Step (3) here is trivial (nothing left to decide). The farmer simply enjoys his profit or suffers his losses. There is no scenario in which beans is the best crop. You might be tempted to eliminate beans from further consideration, but we shall see this could be wrong.

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EMSE 154 – 254 Applications of Linear and Nonlinear Optimization Instructor: Hernan Abeledo Source: LINGO documentation 4 A situation with exactly two possible decisions by nature can be analyzed using a graph such as Figure 12.1: Let p denote the probability of a wet season. The three lines specify the expected profit for a given pure policy (all corn, all sorghum, or all soybeans) as a function of p . The expected profit for the various crops is then:
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Stochastic_Programming - EMSE 154 254 Applications of...

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