The second major class of models in stochastic programming is chance-constrained programs (CCP).  A CCP model is a) similar to general stochastic programs in that the model contains random quantities with known distributions, but b) simpler in that the model has just a single decision stage and a single random outcome stage.

The goal in CCP is to make an optimal decision prior to realization of random data, while controlling the chance that constraints are violated.  Consider an LP with random matrix Ξ and random right-hand-side ω:

Min c x

Ξxi ≥ ωi   i=1...m

If we required all m realizations of Ξ x ≥ ω to be satisfied, then we would get a very conservative/expensive solution x, or no feasible solution at all.  The distinctive feature of CCP is that we require that Ξ x ≥ ω be satisfied with some prespecified probability, 0 < p < 1, as opposed to it being satisfied for all possible realizations of (Ξ,ω).

An example of a CCP model would be a blending model where the quality level of the raw materials is not known with certainty, but have known probability distributions.  Given this variability in raw material quality, it may not always be practical to satisfy quality requirements in the final blend 100 percent of the time.  Instead, we seek to find a blend of the raw materials that will satisfy quality constraints to a specified precentage, say 90 percent, of the time.  We present this example later in the chapter in section A CCP Fuel Blending Model.