In this study, we develop a mathematical model for the master surgery scheduling problem, taking into account uncertainties in the length of stay in downstream units such as wards and ICU beds. Hospitals face constraints due to limited ward and ICU bed availability, with daily capacity determined by patient length of stay. By incorporating post-surgery length of stay as stochastic parameters in both the surgical unit and ICU, we reduce the possibility of surgery cancellations due to bed availability issues. Uncertain parameters are defined within a limited set of scenarios, and a two-stage stochastic programming model is proposed to address these uncertainties. Additionally, we develop a Two-phase matheuristic algorithm to efficiently solve the proposed model. In the first phase, an initial solution is generated, followed by a fix and optimize procedure in the second phase to improve the initial solution. The results obtained indicate the efficiency of the proposed solution method.

We consider a two-stage stochastic decision problem where the decision-maker has the opportunity to obtain information about the distribution of the random variables X through a set of discrete actions that we refer to as probing. Specifically, probing allows the decision-maker to observe components of a random vector Y that is jointly-distributed with X. We propose a three-stage optimization model for this problem, wherein the first-stage variables select components of Y to observe, and decisions in subsequent stages must be consistent with the obtained information. In the case that X and Y have finite support, Goel and Grossmann gave a mixed-integer programming formulation of this problem whose size is proportional to the square of cardinality of the sample space of the random variables. We propose to solve the model using bounds obtained from an information-based relaxation, combined with a branching scheme that enforces the consistency of decisions with observed information. The branch-and-bound approach can naturally be combined with sampling in order to estimate both lower and upper bounds on the optimal solution value. We demonstrate the scalability of our approach against the exact MIP formulation on instances of a stochastic facility location problem.

The diversion path problem is a network interdiction problem in which the leader modifies arc lengths, and the follower finds the shortest path from the source node to the sink node using the modified arc lengths. The leader's goal is to minimize the cost to have the follower choose a special path called the diversion path. When arc lengths are deterministic, this problem is solvable by linear programming. We study the situation where arc lengths are random, and the leader wishes to minimize the cost to assure that the follower follows the diversion path with probability at least a desired target. This chance-constrained stochastic diversion path problem has been studied by Baycik, Nguyen, and Smith in the special case when arc lengths are independent and normally distributed. We explore the use of sample average approximation(SAA) to approximately solve this problem with a general distribution. We first derive a compact mixed integer programming formulation of the SAA problem, but find that it is difficult to solve due to poor linear programming relaxation quality. We thus explore a method based on a branch-and-cut decomposition algorithm. Computational results will be presented.

Stochastic facility location problems with outsourcing costs (SFLPOC) optimize facility placement and customer assignment under demand uncertainty. Excess demand beyond a facility's capacity incurs outsourcing costs. This work addresses SFLPOC, aiming to minimize overall expected costs (installation, servicing, and outsourcing).
We model SFLPOC as a two-stage stochastic program. While prior work focused on specific assumptions or small scenario sets, we present methods suitable for general probability distributions. For discrete scenario sets, we improve upon classic Benders decomposition by exploiting the second-stage subproblem's structure.
To handle general distributions, we partition the probability space, enabling the computation of expected values with fewer scenarios. Coupled with Benders cuts, this provides an exact solution method for common distributions (e.g., Bernoulli, Gaussian).
Additionally, we introduce a compact formulation specifically for i.i.d. demand distributions, allowing us to solve even continuous distribution problems to optimality. Computational experiments on established benchmarks demonstrate that our compact formulation consistently finds optimal solutions, while the Benders approach provides strong solutions with proven optimality gaps for general distributions, outperforming sample average approximations.