Editing Stochastic programming (section)

==== Deterministic equivalent of a stochastic problem====
With a finite number of scenarios, two-stage stochastic linear programs can be modelled as large linear programming problems. This formulation is often called the deterministic equivalent linear program, or abbreviated to deterministic equivalent. (Strictly speaking a deterministic equivalent is any mathematical program that can be used to compute the optimal first-stage decision, so these will exist for continuous probability distributions as well, when one can represent the second-stage cost in some closed form.)
For example, to form the deterministic equivalent to the above stochastic linear program, we assign a probability <math>p_k</math> to each scenario <math>k=1,\dots,K</math>. Then we can minimize the expected value of the objective, subject to the constraints from all scenarios:

<math>
\begin{array}{lccccccccccccc}
\text{Minimize} & f^\top x & + & g^\top y & + & p_1h_1^\top z_1 & + & p_2h_2^Tz_2 & + & \cdots & + & p_Kh_K^\top z_K &  &  \\ 
\text{subject to} & Tx & + & Uy &  &  &  &  &  &  &  &  & = & r \\ 
 &  &  & V_1 y & + & W_1z_1 &  &  &  &  &  &  & = & s_1 \\ 
 &  &  & V_2 y &  &  & + & W_2z_2 &  &  &  &  & = & s_2 \\ 
 &  &  & \vdots &  &  &  &  &  & \ddots &  &  &  & \vdots \\ 
 &  &  & V_Ky &  &  &  &  &  &  & + & W_Kz_K & = & s_K \\ 
 & x & , & y & , & z_1 & , & z_2 & , & \ldots & , & z_K & \geq & 0 \\ 
\end{array}
</math>

We have a different vector <math>z_k</math> of later-period variables for each scenario <math>k</math>. The first-period variables <math>x</math> and <math>y</math> are the same in every scenario, however, because we must make a decision for the first period before we know which scenario will be realized. As a result, the constraints involving just <math>x</math> and <math>y</math> need only be specified once, while the remaining constraints must be given separately for each scenario.