Robust Hospital Management
Hospitals are under tremendous cost pressure and must achieve a balance between economic efficiency and a treatment that focuses on the patient. To improve clinical operations and patient safety, methods from economics, mathematical optimization and IT-driven management systems are imported into the operational management of hospitals. The goal is to maintain the high quality in medical care while lowering the costs. A major challenge in this optimization process is the changing demand arising from emergencies or patients without appointments, which are difficult to forecast, and thus are, in general, not integrated into the planning process. In this part of the project we will focus on the integration of such uncertainties into three main areas of hospital management:
1. the capacity planning and utilization of hospital beds,
2. the patient appointment scheduling, and
3. the transportation from patients to their appointments.
In the next subsection we will give a rough overview of existing scientific work to the mentioned subproblems. We will finally describe our approach to these problems in detail.
In 2012, Hulshof et al.  published a detailed bibliography and taxonomic classification on methods from operations research applied to problems in health care. Uncertainties are part of most decision problems in planning and controlling in health care. Mainly methods from queuing theory, Markov processes, and stochastic programming are used to include them into the optimization process, e.g., [1, 2, 3, 23, 40]. Next to dealing with uncertainties, Hulshof et al.  identify the challenge for researchers to develop integral models of different hierarchical planning levels and services in health care.
The location of beds and the assignment of patients to these beds in a hospital is studied in operations research at the strategical, tactical and operational level. To support the strategic planning queuing techniques, simulation and models from mathematical programming are already used. Traditionally these planning decisions are based on target occupancy levels. However, Green  points out that due to the high fluctuations different measurements as patient waiting time  or patient refusal rate  need to be integrated into the optimization process. In , Ma and Demeulemeester combine the allocation of beds with the appointment of elective patients. In order to integrate emergencies, they reserve a fixed capacity. The Patient-to-Bed Assignment Problem on an operational level has been formalized in 2010 by Demeester et al. . They use a combination of a patient-bed-suitability rating, the number of inpatient transfers and the number of mixed-gender-occupied rooms as the objective function and propose a hybrid tabu search algorithm for this problem. Later, the problem is reformulated to patient-to-room assignment, as it is generally assumed that all beds, located in the same room, are equal. Also more practical variants and other exact and heuristic approaches for patient-to-room assignment have been published, e.g., [7,8,50].
The scheduling of surgeries and the corresponding hospital’s operations theater are well studied . Only a limited number of papers take into account multiple resources . They mainly aggregate the scheduling decisions to half-day base or limit the scheduling horizon to one week. In order to incorporate uncertainties, Vissers et al.  build a stochastic discrete program and solve it with a sample average approximation method.
Vehicle routing problems are well-studied in discrete optimization . In the context of patient routing within the hospital, Hanne et al.  designed a computer-based planning system. Johnson et al.  introduced a simulation tool, and Bedaury et al.  a two-phase heuristic to solve the dynamic problem. Schmid and Doerner  solved the combination of operating room scheduling and transportation with a hybrid metaheuristic.
So far, we have concentrated on the operational patient-to-room assignment. Hospital beds are a special resource in a hospital: according to the number of beds the capacity of a hospital is measured; the size of wards and clinics are given by this number; and the corresponding budget on medical and nursing staff is determined by this number. Yet, the number of available beds fluctuates due to capacity changes in the nursing staff, patient demands and special needs of patients . These fluctuations primarily effect the scheduling of elective patients and the daily allocation of emergency patients to different wards and rooms. In case of a mismatch of available beds and admitted patients, a relocation of a bed or even of a patient to a different clinic or ward or the rejection of elective patients is possible. However, such means should only be used in extreme situations and not on a daily base. Contrary to all previously published work we do not regard a weighted combination of the patient-bed-suitability rating, the number of inpatient transfers and the number mixed-gender-occupied rooms as the objective function. Because choosing appropriate weights is very challenging and there also has no procedure yet been proposed to check afterwards if good weights have been chosen. Also, using a weighted combination prevents us from gaining better insights about how the different objectives influence each other. This is why we keep the three different aspects separated and treat them as independent objective functions. We compare and develop exact and heuristic approaches to solve the multi-objective patient-to-room assignment problem with a focus on robust solutions.