Stochastic modeling of patient flow in hospitals

Patient flow is the movement of patients through a hospital system during their time of care. The uncertainty in the healthcare system makes it challenging to utilize resources optimally so as to improve the quality of care. Patients could face long waiting time to access services, and as the result, long queues may form, which then puts pressure on the system and requires decisions from doctors, nurses, managers, and administrators. These decisions need to be made carefully and on time, otherwise, bottlenecks and delays will be propagated back through the system, eventually leading to impairment of patient flow, congestion, and imbalance of demand and available resources in the hospital. Therefore, patient flow optimization is a complex problem that requires advanced modelling and solution techniques to assist with the management of patient flow, provide insights, and suggest improvements.
Stochastic modeling is an effective technique to capture the random nature of healthcare systems. In this thesis, we focus on the patient admission scheduling problem (PAS) and surgery scheduling problem, which are some of the planning processes related to patient flow, and construct several stochastic models to capture the random dynamics of the hospital system. We model the Length of Stay (LoS) using probability distributions and use stochastic objective functions. In our models of patient flow, we include the effect of transfers and being assigned to a less-suitable ward on the LoS distribution. We also develop an integrated model for the PAS and surgery scheduling problem combined.
In Chapter 1, we review the relevant literature on patient flow, discuss current gaps and summarise our contributions to this research. We describe the key mathematical techniques we have used in this study and give the outline of the thesis.
In Chapter 2, we construct an integer programming model for the PAS problem, in which a stochastic objective function aims to minimize the expected costs of assignment, delay, overcrowding, and transfer, and also considers gender policy. We model the LoS using a discrete phase-type (PH) distribution. Previously, the PAS problem was studied in a deterministic setting. Here, we argue that a stochastic component is essential for better modelling of real-world problems which are typically stochastic. We support our claim with numerical examples and show that the optimal solutions obtained from deterministic models are inadequate when compared with the solutions of our stochastic model.
In Chapter 3, we build on the ideas from our initial study on the PAS problem in Chapter 2. We construct a more advanced model for the PAS problem, in which we capture the effect of transfers and ward suitability on the LoS distribution for the individual patients. We apply an integer linear programming as the key component of the model and define a stochastic objective function, which is a linear combination of relevant costs. We apply a discrete phase-type distribution whose parameters depend on transfers and ward suitability, to model the LoS, to see the effect of decision-making on the system. Next, we construct numerical examples to illustrate the performance of the model and compare it with the initial model developed in Chapter 2 which does not have these features. We find that the effect of transfers and ward suitability on the LoS and on the system should not be ignored.
In Chapter 4, we develop a model for surgery scheduling problems, and an integrated stochastic model for surgery scheduling and the PAS problems combined. First, we develop a stochastic model for the surgery scheduling problem (Model I), in which the stochastic objective function aims to minimize costs related to surgery, over-time, under-time, waiting time, admission delay, and surgery cancellation. We use a continuous phase-type distribution to model distributions related to the duration of surgery, over-time, under-time, and surgery waiting time. We use simulated data to illustrate the performance of this model. Second, we develop an integrated model for the surgery scheduling and the PAS problems combined (Model II), in which the stochastic objective function aims to minimize the above mentioned costs related to surgery as well as the costs related to the assignment to the intensive care unit (ICU), the allocation to wards, transfers, and admission delay. We use a continuous phase-type distribution to model the LoS in the ICU and wards. Next, we apply a five-year data from one of the Australian tertiary referral hospitals to the phase-type and Poisson distributions to estimate parameters related to the LoS and random arrivals, respectively. We construct numerical examples to illustrate the application of the two models.
Finally, in Chapter 5, we summarize the main findings and contributions of the thesis. We comment on how stochastic modelling can help in better management of patient flow, and outline our ideas for future research.