ARCUS Clinics Pforzheim Using mathematical optimisation to develop automated hospital bed assignments

ARCUS hospitals and medical practices are working with inovex to continue the digitalisation of their administration and patient services. Once digital solutions for patients and hospital administration had been developed, the teams turned their attention to using mathematical modelling to optimise bed occupancy planning. The new system assigns patients to hospital beds by taking into account a series of important parameters – and it does so in a fully automated, intelligent manner.

The collaboration between ARCUS and inovex has already produced the myARCUS digital patient portal.

 This tool provides value-adding digital services for patients, including enabling them to schedule appointments and operations from the comfort of their homes. The ARCUSflow internal scheduling and resource management tool was developed at the same time in order to make operating theatre planning more efficient, flexible, and reliable. ARCUSflow (partially) automatically incorporates all the parameters relevant for surgical scheduling, including medical requirements, staff availability, operating theatre capacities – and now optimised, fully automated bed assignments.

Excel planning is a thing of the past

In the past, assigning hospital beds was a complicated process, one which evolved over time and was initially digitalised using Excel. The challenge lies in the complexity of the parameters to be taken into account.

Planning bed occupancy involves answering a number of questions. These include: How long will each patient stay? Is monitoring required? Does the patient have an infection which requires them to be separated from other patients? Is the patient entitled to a private room, a semi-private room, or a larger shared room? Does the patient have private or state health insurance? Do they need to have someone accompanying them? How many staff are available at this time?

Staff previously addressed this complexity using Excel documents coded in a variety of different ways with colours, comments, and symbols. This system, however, was confusing, unintuitive, and time-consuming, and a lack of centralised processing led to errors and multiple document versions.

Partial automation of bed assignments

In the first step, the newly developed bed occupancy plan automatically took into account the most important parameters from operating theatre planning: the prescribed length of stay, the medical requirements (such as monitoring, for example), and the desired room (size, type of insurance). If there was a room available, patients were automatically booked into it. If not, they were added to a worklist, from where department staff could manually assign them to an available room using a drag-and-drop function.

In the second step, “downgrades” were added in order to accommodate patients whose initial preferences could not be honoured. This meant, for example, that the system proposed rooms in alternative categories (such as semi-private rather than private rooms) or stays involving room transfers.

This (partially) automated bed occupancy assignment solution resulted in enormous time savings, more efficient bed utilisation, and fewer errors. But even with this system, there were limits: once a patient was assigned a bed, they remained there. This prevented the “last percent” of available hospital capacity from being utilised because – to give an example – eight patients would have had to be transferred from different wards in order to free up an additional bed for two days. Such problems are too complex to be resolved manually in everyday clinical practice. In the third step, therefore, a mathematical optimisation model was used to automatically generate an optimal bed occupancy plan for the coming week.

Assigning beds using mathematical modelling

In order to apply mathematical optimisation to bed occupancy planning, the “admissible set” of bed assignments  must first be defined. These require, for example, that there can never be more than two patients in a semi-private room, or that only patients of the same gender can be assigned to a shared room.

The second part of the mathematical modelling process involves describing the optimisation goals. This process is intended to identify and minimise unfavourable occurrences. While, for example, medically unnecessary transfers of patients between rooms should be avoided, they should not be completely ruled out if they result in greater gains elsewhere.

An essential aspect of problem modelling involves considering which aspects are hard constraints describing the admissible set of possible assignments and which are only desirable and thereby the optimization’s target. The latter must be weighted to reflect reality as closely as possible.

One key benefit of precise mathematical modelling is that it enables statements to be made about the existence and uniqueness of optimal solutions. A mathematically optimal solutions an admissible one which can not be further improved under any circumstances.

Continuous reduction of complexity

In order to measure the efficiency of the software combined with the mathematical model, a proof of concept (PoC) was used to study an entire past month’s manual room assignments and determine what could have been improved. The mathematical problem formulation (number of operations multiplied by possible rooms per day per patient multiplied by length of stay), however, turned out to be too complex to solve within an acceptable timeframe. Even using state-of-the-art software and hardware, it was generally impossible in general for solutions to be calculated in less than 24 hours.

Through working with hospital staff, it became clear that finding admissible solutions should take precedence over reaching optimality. While the gender separation requirement, for example, must always be met, it was sometimes possible to assign a patient who requested a private room to a shared one. Prioritising the assignment criteria in this way made it possible to develop a methodology which focuses primarily on achieving admissible solutions and then makes improvements as time allows.

In order to further reduce complexity, only randomly selected patient groups were considered variable. The remaining patients were assumed to be already occupying acceptable rooms. This made the resulting “sub-problems” significantly less complex and enabled them to be solved iteratively.

The result is an admissible bed occupancy plan which is created sufficiently rapidly to be useful and which is improved iteratively in accordance with selected goals. The unpredictable runtime of the algorithm can thus be controlled by adjusting how close to the theoretical optimum solutions should be. For the first real-world deployment of the optimisation tool, the next five days were chosen as the time period, and the software was run once per night at midnight.

Close collaboration creates improvements

Since inpatient scheduling is subject to short-term changes throughout the day, it was important that the next iteration of the tool be able to react to these changes in real time. In order to facilitate this, the mathematical model was given a previously optimised solution as initial guess and new information was taken into account in the subsequent iterations. This enables significant time to be saved, allowing, for example, another operation to be added to the schedule in a matter of minutes – or even seconds.

During regular consultations with hospital staff, a number of lessons were learned and improvements to the model and its methods were made possible. It quickly became clear, for example, that there were numerous reasons for staff to reject certain aspects of the mathematically optimised solution. In specific cases, for example, manual room assignments may contradict the modelled constraints, thus abandoning the models admissible set. To resolve this, staff were given the option of “pinning” assignments in the software’s user  interface. This causes the corresponding variables to be a constant in the mathematical model and the conflicting constraints to be removed.

Greater available capacity, more efficient workflows

Now in daily use, the automatically optimised bed assignment solution saves a great deal of operational capacity – which can now be appropriately maximised in other areas. The optimised plan also highlights opportunities for increasing hospital occupancy by creating efficient room assignments which would be too complex for humans to calculate. This has led to an increase in both patient satisfaction and hospital efficiency.

Project note