document: The scheduling process we adopt matches a multiple stage stochastic programming approach. Standard two-stage stochastic programs with linear or convex functions are often solved using the L-shaped method or Bender's decomposition [44,6,7]. However, our recourse decision (scheduled cancellations) is still anticipative to further uncertainty, namely the second shift surgery durations, unavailability and cancellations. As such, the decision problem can be viewed as a three-stage recourse model [5,6]. Solving the scheduling problem is further complicated because the recourse function is integer. Laporte and Louveaux [26] propose modified L-shaped decomposition with adjusted optimal cuts for two stage stochastic program with integer recourse. Angulo et al. [1] alternately generate optimal cuts of the linear sub-problem and the integer sub-problem, which improves the practical convergence (see also [15,8]). We follow a sample average approximation approach (SAA) which uses this framework. Moreover, we prove and exploit a specific relationship between the first-stage realization and the optimal number of scheduled cancellations to speed up the computation of integer cuts. We use Jensen's inequality [17] to upper bound the minus second (and third) stage cost, a technique that was proposed by Batun et al. [3].
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