Stochastic Programming - Stochastic Linear Program

Stochastic Linear Program

A stochastic linear program is a specific instance of the classical two-stage stochastic program. A stochastic LP is built from a collection of multi-period linear programs (LPs), each having the same structure but somewhat different data. The two-period LP, representing the scenario, may be regarded as having the following form:

$\begin{array}{lccccccc} \text{Maximize} & f^T x & + & g^T y & + & h_k^Tz_k & & \\ \text{subject to} & Tx & + & Uy & & & = & r \\ & & & V_k y & + & W_kz_k & = & s_k \\ & x &, & y &, & z_k & \geq & 0 \\ \end{array}$

The vectors and contain the first-period variables, whose values must be chosen immediately. The vector contains all of the variables for subsequent periods. The constraints involve only first-period variables and are the same in every scenario. The other constraints involve variables of later periods and differ in some respects from scenario to scenario, reflecting uncertainty about the future.

Note that solving the two-period LP is equivalent to assuming the scenario in the second period with no uncertainty. In order to incorporate uncertainties in the second stage, one should assign probabilities to different scenarios and solve the corresponding deterministic equivalent.

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