This article was originally published in the Fall 2012 edition of OnAnalytics, published by the Institute for Business Analytics at Indiana University’s Kelley School of Business.
This article focuses on insights from Kurt Bretthauer, professor of Operations & Decision Technologies. Find more information about this research in the article, “Blocking In Healthcare Operations: A New Heuristic and an Application“
Hospitals face the dual imperative of delivering the most effective medical care while attempting to minimize costs in an environment of limited resources and increasing expenditures. One crucial determination is the mix of inpatient beds provided for different levels of care. Viewed in isolation, post-acute or skilled nursing units may not appear as vital or as cost-effective as more intensive treatment units. If these post-acute care units fill, however, patient flow will be blocked out of – and into – more acute care. With this study, the researchers developed a model for minimizing this blocking while also addressing a variety of management considerations such as overall revenue, patient acuity, and the objective of not turning patients away from the hospital.
Statement of the problem
Within a hospital setting, patients typically move through several levels or units of care. Hospitals must accurately predict patient flow in order to determine the appropriate number of beds in each service unit. Too many beds in a given unit waste valuable resources, while too few can result in patients being turned away or, conversely, retained at a higher level of care than they require. Modeling patient flow is difficult, however, not only because patients vary in their length of stay but also because their movements among the different treatment levels are not always sequential. While many will begin in critical care and move toward post-acute treatment one step at a time, others will enter or leave the system from its midpoints and/or return to more acute care after spending time in less-intensive treatment. Hospital administrators must also take into consideration the relative importance their organization places on keeping patients flowing throughout the system, treating the sickest patients, producing as much revenue as possible, and not turning patients away.
Data Sources Used
The researchers based their model on a large US hospital system on the West Coast, which could be aggregated into four units of inpatient treatment: Intensive Care (ICU), Step Down (SD), Acute Care (AC), and Post-Acute Care (PAC). Throughout the period of data collection, there were 28, 76, 125, and 56 beds in the respective units. The data contained detailed information on patient arrivals, lengths of stay at each unit, and patient routing.
In order to build the patient flow model, the researchers began with an n-stage tandem capacitated queuing system in which patients sequentially visited each stage of treatment, capturing the blocking in patient flow that would occur at each stage within such a system. The next phase of analysis involved developing a heuristic to estimate the amount of blocking in the system, and testing the researchers’ heuristic against the exact solution and against other heuristics from the literature. For these analyses, the researchers used the MATLAB computing environment.
Using the tandem heuristic as a building block, the researchers went on to model the more complex hospital setting in which patients may move non-sequentially among the units (general patient routing). This algorithm became the Capacity Allocation Model, which could be used to solve for the minimum probability of blockages at each stage of service, subject to a budget constraint. The model also allowed for different weights to be placed on the various hospital units in order to capture the relative importance to administrators of admitting external patient arrivals, freeing up capacity in the most-intensive care units, and affecting revenue.
Applying the model to the data from the US hospital, the researchers found that the blocking probabilities obtained from their heuristic method compared well with those obtained from the MedModel simulation package. Because the main concern of the hospital was the number of beds in the PAC unit, the researchers tested blocking probabilities with 105, 90, 70, and 56 PAC beds, which were the numbers of beds the unit had at different times during the previous decade. The researchers used the heuristic to solve the optimization problem under different weighted scenarios: minimizing blocking equally at all stages (Scenario I); minimizing blocking with priority weighted toward the most intensive units (Scenario II); minimizing the number of patients turned away from the hospital (Scenario III); minimizing per-server per-day costs, an inverse proxy for revenues (Scenario IV); and minimizing the revenue lost from turning patients away from the hospital (Scenario V).
Lastly, the researchers conducted a sensitivity analysis to determine the robustness of the model against varying parameters, specifically those of patient arrival rate and budget.
The performance of the researchers’ tandem system heuristic for estimating blocking was found to be robust against the exact solution, and to represent a significant improvement in accuracy over existing heuristics from the literature. By taking into account settings with severely constrained capacity and those in which there is no buffer or waiting room in front of the various stages, the new heuristic outperformed prior approximation methods.
When applied to the hospital data, the model indicated that the initial reduction in PAC beds from 105 to 90 had hardly any effect on service, increasing blocking from 0.0% to just 0.3%. The second round of reductions to 70 beds, however, not only led to a significant increase in PAC blocking but also to a slight increase in blocking probabilities upstream in the AC and SD units. The final reduction to 56 beds, moreover, led to significant blocking problems in the PAC along with large increases in the blocking probabilities of the AC and SD units. This final cut appeared to decrease blocking probability to the ICU, but further analysis revealed that this reduction was due to external patient arrivals at the SD and AC being turned away, reducing overall patient flow throughout the hospital.
Solving for the different management objectives under consideration, the researchers observed that the optimal capacity for all five scenarios suggested an increase in PAC capacity. Additionally, all but Scenario III (minimizing number of patients turned away) also called for an increase in ICU beds. The resources necessary for these transitions came from decreases in SD and/or AC capacity.
Interestingly, the number of PAC beds was close to optimal for four out of the five scenarios before its last reduction from 70 to 56 beds, indicating that the final cut had significantly increased blocking at the hospital.
The sensitivity analysis revealed that the model was robust unless patient arrival rate increased drastically. In that scenario, as in that of a significant increase to the hospital’s budget, it became advantageous to shift more resources upstream toward the more-acute care units.
Within a hospital setting, this research provides important insight into the value post-acute care provides by reducing blocking in other units. Hospitals can use their own patient flow data to apply the Capacity Allocation Model and determine their optimal mix of beds while taking into account their particular budget constraints and specific management objectives.
The model can be applied to other settings as well, enabling an analysis of flow between different units or stations within such sectors as manufacturing, customer service, or case management. In these and other multi-stage treatment or production situations, managers and administrators can use the Capacity Allocation Model to determine the optimal number of servers to minimize blocking, maximize revenue, or avoid turning people or orders away.
This research demonstrates how analytics can be used to solve complex business problems using highly accurate simplified models. Additionally, these models can be used to determine solutions that meet a wide range of management objectives.