  Motivated by kidney exchange, we study a stochastic cycle and chain packing
problem, where we aim to identify structures in a directed graph to maximize
the expectation of matched edge weights. All edges are subject to failure, and
the failures can have nonidentical probabilities. To the best of our knowledge,
the state-of-the-art approaches are only tractable when failure probabilities
are identical. We formulate a relevant non-convex optimization problem and
propose a tractable mixed-integer linear programming reformulation to solve it.
In addition, we propose a model that integrates both risks and the expected
utilities of the matching by incorporating conditional value at risk (CVaR)
into the objective function, providing a robust formulation for this problem.
Subsequently, we propose a sample-average-approximation (SAA) based approach to
solve this problem. We test our approaches on data from the United Network for
Organ Sharing (UNOS) and compare against state-of-the-art approaches. Our model
provides better performance with the same running time as a leading
deterministic approach (PICEF). Our CVaR extensions with an SAA-based method
improves the $\alpha \times 100\%$ ($0<\alpha\leqslant 1$) worst-case
performance substantially compared to existing models.
