First-Stage Decisions and Recourse

In a two-stage stochastic program, decisions split by when they are made relative to the random outcome.

The first-stage decision $\mathbf{x}$ is fixed before the random vector $\boldsymbol{\xi}$ is observed -- it is the "here-and-now" commitment. Its cost $\mathbf{c}^T \mathbf{x}$ is paid for certain.

After $\boldsymbol{\xi}$ is observed, the decision maker reacts by choosing a recourse decision (or second-stage decision) $\mathbf{y}$, which is allowed to depend on the realized value. The cost of this reaction is captured by the recourse function

$$Q(\mathbf{x},\boldsymbol{\xi}) \;=\; \min_{\mathbf{y}\,\geq\,0}\,\Big\{\,\mathbf{q}(\boldsymbol{\xi})^T \mathbf{y}\;:\;T(\boldsymbol{\xi})\,\mathbf{x} + W\mathbf{y} = \mathbf{h}(\boldsymbol{\xi})\,\Big\}.$$

Notice that $Q$ depends on the first-stage choice $\mathbf{x}$, because the second-stage constraints involve $\mathbf{x}$.

The full two-stage stochastic program is

$$\min_{\mathbf{x}\,\geq\,0}\;\underbrace{\mathbf{c}^T \mathbf{x}}_{\text{first-stage cost}}\;+\;\underbrace{\mathbb{E}_{\boldsymbol{\xi}}\!\left[Q(\mathbf{x},\boldsymbol{\xi})\right]}_{\text{expected recourse cost}},\qquad A\mathbf{x}=\mathbf{b}.$$

Illustration. A bakery commits to baking $x$ loaves at \2$each in the morning. Demand$D$is random. If$x>D$, each unsold loaf is discarded for$$1$; if$x<D$, each unit of shortage triggers a$$5$emergency replenishment. Then$x$is the first-stage decision,$\mathbf{c}^T\mathbf{x}=2x$ is the first-stage cost, and the recourse function is

$$Q(x,D)\;=\;1\cdot\max(x-D,\,0)\;+\;5\cdot\max(D-x,\,0).$$