We consider a dynamic network flow problem where the arc capacities are random variables. This gives a multistage stochastic linear program. We describe the randomness using a multi-scenario approach.

Richard M. Karp, Rajeev Motwani, Noam Nisan, Probabilistic Analysis of Network Flow Algorithms, Mathematics of Operations Research, Vol. 18, No. 1 (Feb., 1993), pp. 71-97 ...

Network matching would be an effective solution to the most common instances of surprise billing and compares favorably to many of the most commonly discussed alternative approaches.