Value of the Stochastic Solution (VSS)

In a two-stage stochastic program we choose a first-stage decision $x$ before observing a random parameter $\xi,$ then incur cost $f(x,\xi).$ The recourse problem (RP) is the full stochastic optimization

$$RP = \min_{x}\; \mathbb{E}_\xi[f(x,\xi)],$$

whose optimizer $x^*$ is called the stochastic solution.

A much cheaper shortcut replaces $\xi$ with its mean $\bar\xi = \mathbb{E}[\xi]$ and solves the deterministic problem

$$EV = \min_{x}\; f(x,\bar\xi),$$

whose optimizer $\bar x$ is the expected-value solution. To see how that shortcut performs in the actual uncertain environment, plug $\bar x$ back into the stochastic objective:

$$EEV = \mathbb{E}_\xi[f(\bar x,\xi)].$$

The Value of the Stochastic Solution (VSS) measures the expected cost reduction obtained by solving the full stochastic program instead of using the mean-value shortcut:

$$\boxed{\;VSS = EEV - RP\;}$$

for a minimization problem. Since $x^*$ optimizes the stochastic objective and $\bar x$ is just one feasible choice, $RP \le EEV,$ so $VSS \ge 0$ always.

Quick illustration. If $EEV = 120$ and $RP = 95,$ then

$$VSS = 120 - 95 = 25.$$

Ignoring uncertainty would cost an extra $25$ in expectation.