8 Nature of Stoch Sols

# 8 Nature of Stoch Sols - 1 Expected Returns for Stochastic...

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1 Expected Returns for Stochastic Versus Deterministic Policies An interesting aspect of stochastic problems as they are handled by dynamic programming has to do with the expected return for the optimal policy. The solution policies that result from a stochastic dynamic programming analysis are not predetermined. By this we mean that we can not say a priori the observation of the random variable at a given decision point what specific decision is to be taken. The observation of the random variable generally specifies the state into which the system transitions in the next decision period. Once the state is observed, then the optional decision based on the state is taken. This is the same for deterministic problems, except that we know exactly where the system will reside at each decision period and, therefore, the future decisions can be pre-specified at the start of the actual process evolution. We can not pre-specify the decision to be made in advance for stochastic problems; we can only specify the decision to be made in a given period conditioned on the state of the system upon entering that period. While this might seem a limitation of the stochastic solution, it is actually beneficial in that the expected return from such wait-and-see policies is higher than that which can be obtained from any fixed policy a priori the system evolution. To illustrate this property, we analyze a simple three period stochastic model and compare the result with the best solution obtained by fixing the decisions before the process evolves. Consider a summer growing season in which we plan to harvest an alfalfa crop three times during the season. The summer months have highly variable amounts of rainfall and as a result this field has the capability to be irrigated. Due to the nature of this crop, if irrigation is applied and the following growth period is dry then an increased growth is obtained and the profit is higher than if no irrigation was applied. However, if the crop is irrigated at the start of the period and it turns out that the month is also a wet month, then most likely the crop will have too much water and the yield decreased. In addition, the profit will be even lower due to the cost of irrigation.

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## This note was uploaded on 10/16/2010 for the course ISEN 689 taught by Professor Klutke during the Spring '07 term at Texas A&M.

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8 Nature of Stoch Sols - 1 Expected Returns for Stochastic...

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