Huang and Sellier introduced the concept of a robust subset of a matroid to propose approximate kernels for the matroid-constrained maximum vertex cover problem. We prove that the bound for the size of a robust subset of a transversal matroid given by Huang and Sellier can be improved.

Planning surgical activities is not an easy task due to many decision variables and uncertainty. A basic strategy to avoid surgery cancellation, over or under-utilization of operating rooms (OR) is to plan for the surgeries based on the OR resources in the first phase of OR Scheduling. This study proposes a concurrent model for case-mix planning and OR capacity assignment problems, in which the volume of elective surgeries and the number of OR blocks for surgical services are determined at the same time in such a way that the limited OR capacity is utilized based on the uncertain demands for surgeries. The model would be a stochastic Integer Goal Program and has been solved based on scenario-based programming using GAMS software for some real case studies. Based on the results of applying the model for the case study, the average of unsatisfed demands lies in [0.07%,1.49%] and the OR overtime is zero.