Two-Stage Stochastic Programs: Here-and-Now vs. Wait-and-See

Introduces the two-stage stochastic programming framework. Defines first-stage (here-and-now) decisions made before uncertainty is revealed and second-stage (recourse) costs. Distinguishes the recourse problem RP from the wait-and-see value WS, and computes the expected value of perfect information EVPI = RP - WS.

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The Two-Stage Structure

A two-stage stochastic program splits a decision problem into two stages separated by the revelation of uncertainty.

First stage (here-and-now). The decision $x$ must be chosen before the random scenario is observed. We are committed to $x$ no matter what happens next.
Second stage (recourse). A scenario $s\in S=\{s_1,s_2,\ldots,s_K\}$ is realized with probability $p_s$ . Given $x$ and $s$ , the total cost incurred is $q(x,s)$ .

The expected total cost of committing to first-stage decision $x$ is the probability-weighted sum of scenario costs:

\mathbb{E}_s[q(x,s)]=\sum_{s\in S} p_s\cdot q(x,s).

For example, suppose demand is Low ( $p=0.3$ ) or High ( $p=0.7$ ), and producing $x=150$ units yields $q(150,\text{Low})=80$ and $q(150,\text{High})=50.$ Then

\mathbb{E}_s[q(150,s)]=0.3\cdot 80+0.7\cdot 50=24+35=59.