326. Stochastic Programming
Optimization with uncertain parameters whose probability distribution is known. Distinguishes:
- Here-and-now decisions made before the random data is revealed
- Wait-and-see / recourse decisions made after observing the random outcome
326.1. Two-stage stochastic LP
where:
- : first-stage (here-and-now) decision — fixed before is observed
- : random parameter
- : second-stage (recourse) decision — chosen after observing
- : optimal recourse cost given and
The expectation aggregates over the distribution of .
326.2. Subtopics
- Two-stage recourse — the core model
- Scenario trees — multi-stage extension
- Chance constraints — probabilistic feasibility
- Sample average approximation — solve via Monte Carlo
- EVPI vs VSS — value of information / value of stochastic modeling
326.3. When to use
- Decisions must be made before uncertainty resolves — capacity planning, network design, hiring
- Some adjustments are possible later — production levels, shipping, hiring temps
- You want explicit risk management not just nominal expected-value optimization
326.4. Alternatives
- Dynamic programming: same as stochastic programming for sequential decisions, but explodes with state-space dimension. SP scales better in continuous variables.
- Robust optimization: hedges against worst-case uncertainty instead of expected case. Less data-hungry; more conservative.
- Scenario analysis: pick a few representative scenarios; solve each deterministically. Crude but easy.