# Term Paper: Stochastic Modeling Is a Mathematical

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The optimal policy from such a model is a single first-stage policy and a collection of recourse decisions (a decision rule) defining which second-stage action should be taken in response to each random outcome [5].

Solution approaches to stochastic programming models are driven by the type of probability distributions governing the random parameters. A common approach to handling uncertainty is to define a small number of scenarios to represent the future. In such cases, it is possible to compute a solution to the stochastic programming problem by solving a deterministic equivalent linear program. These problems are typically very large-scale problems, and so, much research effort in the stochastic programming community has been devoted to developing algorithms that exploit the problem structure, in particular in the hope of decomposing large problems into smaller more tractable components. When the probability distributions of random parameters are continuous, or there are many random parameters, one is faced with the problem of constructing appropriate scenarios to approximate the uncertainty. One approach to this problem constructs two different deterministic equivalent problems, the optimal solutions of which provide upper and lower bounds on the optimal value z* of the original problem.

Stochastic programming has been applied to a wide variety of areas in engineering. Some of the engineering problems such as electrical generation capacity planning, machine Scheduling, timber management, traffic management, automobile inventory management, and lake level management are complex indeterminate problems that require stochastic solutions.

References

[1] A. Prekopa. Stochastic Programming. Kluwer Academic Publishers, Netherlands, 1995.

[2] S.R. Tayur, R.R. Thomas, and N.R. Natraj. An algebraic geometry algorithm for scheduling in the presence of setups and correlated demands. Mathematical Programming, 69(3):369-401, 1995.

[3] C.C. Caroe and R.Schultz. Dual decomposition in stochastic integer programming. Operations Research Letters, 24:37-45, 1999.

[4] R. Hemmecke and R. Schultz. Decomposition of test sets in stochastic integer programming. Mathematical Programming, 94:323-341, 2003.

[5] H.D. Sherali and B.M.P. Fraticelli. A modification of Benders' decomposition algorithm for discrete subproblems: An approach for stochastic programs with integer…
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