329. Chance Constraints

Constraints required to hold with high probability rather than always. Useful when:

• Always-feasibility is too expensive
• The data is uncertain and 100% reliability isn’t needed (or even possible)
• Service-level requirements are stated probabilistically (“95% of orders shipped on time”)

329.1. Formulation

For uncertain parameters :

The constraint must hold with probability at least . is the risk level — typically or .

Equivalent: violate the constraint with probability at most .

329.2. Joint vs individual chance constraints

Individual: each constraint holds with high probability, separately:

Joint: all constraints hold simultaneously with high probability:

Joint is stronger — and harder to handle. Often individual is what’s actually intended.

329.3. Reformulation under normality

If is normally distributed (with mean and variance ):

The chance constraint becomes a deterministic constraint involving — possibly convex (e.g., if is linear in and ). This is a second-order cone constraint, solvable by interior-point methods.

329.4. Reformulation via sampling

Without normality, use scenarios :

Enforcing the chance constraint on the sample becomes an MILP with binary indicators per scenario — exponential growth, manageable up to .

329.5. Equivalence to VaR / CVaR

In risk management, chance constraints relate to:

• Value at Risk (VaR): the -quantile of the loss distribution. The chance constraint is essentially .
• Conditional VaR (CVaR): expected loss beyond VaR. Tighter, convex, easier to optimize.

In practice, CVaR-based reformulations of chance constraints are more numerically tractable.

329.6. Where chance constraints matter

• Service-level inventory: 95% in-stock probability per cycle
• Robust facility location: feasible under high-probability demand realizations
• Portfolio optimization:
• Power systems: probabilistic generation-load balance
• Network design: links carry expected traffic with high probability

329.7. See also

• Stochastic Programming
• Two-stage Recourse
• SAA
• Cycle Service Level — inventory application