Managing the patient portfolio using mathematical programming: decision support guidelines using a real-world use case at a university hospital

Grieger, Milena; Heider, Steffen; McRae, Sebastian; Koperna, Thomas; Brunner, Jens O. Article — Published Version Managing the patient portfolio using mathematical programming: decision support guidelines using a realworld use case at a university hospital Journal of Business Economics Provided in Cooperation with: Springer Nature Suggested Citation: Grieger, Milena; Heider, Steffen; McRae, Sebastian; Koperna, Thomas; Brunner, Jens O. (2024) : Managing the patient portfolio using mathematical programming: decision support guidelines using a real-world use case at a university hospital, Journal of Business Economics, ISSN 1861-8928, Springer, Berlin, Heidelberg, Vol. 94, Iss. 9, pp. 1245-1260, https://doi.org/10.1007/s11573-024-01201-y This Version is available at: https://hdl.handle.net/10419/315428 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ ORIGINAL ARTICLE Journal of Business Economics (2024) 94:1245–1260 https://doi.org/10.1007/s11573-024-01201-y Abstract Many hospitals in Germany are facing escalating economic pressures. After several years of stagnation, the number of inpatient hospital treatments dropped by 13% in 2020 compared to the previous year. This negative tendency can also be seen in operating theaters (OTs). Strategic management of the case mix in hospital OTs now necessitates a solid data foundation. The case mix and the case mix index have become central economic indicators in contemporary hospital operations. In this work, we develop a mathematical model for case mix optimization at Augsburg University Hospital in Germany, which is based on an extensive data analysis with descriptive methods. The optimization model is subject to rigorous testing and evaluation through an extensive series of scenario analyses. The primary objective is to calculate a revenue-maximizing patient mix while respecting the available scarce personnel resources in the OT and intensive care unit. This research marks a pioneering effort in delineating the practical integration of case mix planning into a hospital’s routine operations using mathematical optimization. The analyses reveal a strong correlation between an upsurge in revenue and an increased number of cases. Furthermore, the results demonstrate that strategic planning of the patient mix has the potential to enhance revenue with existing resources. Even though the optimal patient mix may not be directly implementable in practice, the findings yield valuable insights for managerial decision-making. A critical examination of these results also fosters a nuanced discourse on the utilization of optimization models as decision support tools within hospital management. Keywords Case mix · Mathematical optimization · OT planning · Hospital · Decision support Accepted: 9 August 2024 / Published online: 28 August 2024 © The Author(s) 2024 Managing the patient portfolio using mathematical programming: decision support guidelines using a realworld use case at a university hospital MilenaGrieger1· SteffenHeider2· SebastianMcRae3· ThomasKoperna4· Jens O.Brunner1,5,6 Extended author information available on the last page of the article 1 3 M. Grieger et al. 1 Introduction Numerous hospitals in Germany are facing increased economic pressure. After years of stagnation, the number of inpatient hospital treatments declined by 13% in 2020 compared to the previous year, primarily due to the COVID-19 pandemic. Similarly, the total number of surgeries performed dropped by 10% to 6.4 million (Statistisches Bundesamt 2021). Furthermore, the growing shortage of medical professionals poses an increasingly daunting challenge for German hospital management (Osterloh 2018). Quantitative Operations Research/Management Science (OR/MS) methods offer a potential solution to support hospital management in dealing with these complex issues. Active strategic management of the service portfolio, commonly referred to as case mix, can help hospitals maintain a sustainable and economically viable service offering despite stagnant case numbers and limited (personnel) resources. The key challenge lies in identifying the optimal mix and volumes of patient categories, a complex planning task known as the case mix planning problem (CMPP) (Hof et al. 2017). The core of the CMPP is the optimal allocation of scarce resources (e.g., operating theater (OT) capacity) to strategically manage the case mix. The basic objective of strategic case mix planning is to fulfill the service mandate, increase the quality of care, and/or maximize (personnel) revenue (Hof et al. 2017; Waeschle et al. 2016). The specific planning objectives vary depending on the management perspective guiding the strategic control of the service portfolio. This paper examines the application of quantitative OR/MS methods for strategic service portfolio planning. Alongside in-depth analyses, the paper delves into the potential and limitations of applying these methods in real hospital practice. The insights gained are illustrated using the Augsburg University Hospital (UKA) as a practical case study. OR/MS is an interdisciplinary field concerned with developing and applying mathematical optimization models for decision support (Hulshof et al. 2012; van Wassenhove and Besiou 2013). In healthcare, OR/MS models and methods find diverse applications across various complex planning problems, including OT planning, shift scheduling, or service range planning (Erhard et al. 2018; Heider et al. 2022; Hof et al. 2017). Given the predominantly isolated consideration of surgical hospitals, making a universally applicable statement about resource allocation within a hospital is challenging. This study addresses this issue by systematically comparing personnel expenses and revenues for the entire hospital. Assuming that OT and intensive care unit (ICU) capacities remain relatively stable in the short to medium term, the optimized increase or reduction of specific surgery types and the distribution of available OT resources provide a redistribution of the surgical service portfolio. From a strategic management perspective, the results may signal the need to selectively reduce less profitable surgeries over the medium to long term. However, it should be noted that such reduction measures would require external cooperation, such as transferring patients to other hospitals as part of a cooperation model, while controlling patient inflow via referring parties. To implement CMPP through mathematical optimization models at the operational level, revenues generated from inpatient surgeries can be allocated to corresponding 1 3 1246 Managing the patient portfolio using mathematical programming:… cost categories, including personnel, material costs, and infrastructure, based on the Institut für das Entgeltsystem im Krankenhaus (InEK) matrix (Deutsche Krankenhausgesellschaft et al. 2016). Hospital information systems’ OT control tools, commonly used in practice, facilitate the comparison of directly assignable personnel expenses with these revenues. Through this data preparation and analysis, along with the help of optimization techniques, it becomes feasible to quantify the extent to which personnel expenses are covered by the revenues for various surgery types. These results can be calculated both for individual surgical departments and aggregated across the entire hospital. The purpose of this work is to introduce a mathematical optimization method that incorporates case mix planning seamlessly into the daily operation of hospitals and to develop decision support guidelines based on a use case at a university hospital. Our work is structured as follows. In Sect. 2, we introduce our case study and data, while Sect. 3 outlines our case mix optimization method. In Sect. 4, we provide the computational analysis results at the UKA. A discussion of the findings and decision support guidelines is provided in Sect. 5. Section 6 includes a summary and an outlook to future research. 2 Introducing the case study and data The model is built on a dataset comprising 22,657 cases encompassing 2,100 distinct surgery types. Various data cleaning steps were implemented to enhance patient flow control. The greater objective of this refinement process is to ensure greater efficiency. An overview of the data preparation process is presented in Fig. 1. Given the existence of numerous surgery types with minimal occurrences in the reference year, these types underwent categorization using an ABC analysis. For the further course of the study, solely surgery types falling within the A group are utilized. This group encompasses all surgery types that collectively represent 80% of case numbers, ordered in descending order of frequency. Consequently, 15,609 cases and 433 surgery types are included in the analysis. The determination of available scarce resources is then carried out based on the considered cases, as outlined in Sect. 2. A total of 12 different departments are considered. 3 Developing a simple mathematical model formulation for the CMPP The strategic management of the patient mix presents itself as a mathematical optimization challenge, with a detailed model provided below. The primary objective of case mix optimization is to ensure an economical service portfolio in the long term while increasing the quality of care. The framework for these decisions is shaped by the available resources, demand, and the hospital’s care mandate. Strategic case mix management operates on an aggregate annual basis, considering both resources and case numbers. Although Diagnosis-Related Groups (DRG) serve as a suitable classification system for retrospective analysis of the case mix, 1 3 1247 M. Grieger et al. they prove unsuitable for actively managing the service portfolio (Salge and Vera 2012). This is because assigning a surgery to a DRG group typically occurs retrospectively, and practical portfolio management based on DRG groups is challenging. Consequently, patient groups are characterized for optimization purposes using the surgery types commonly employed in internal surgical planning. These in-house surgery types, numbering more than 2,000 , effectively capture the treatment intent. Examples of the surgery types are reimplantation of a pacemaker, lung resection and pulmonary resection. The reimbursement, resource demands, and costs associated Fig. 1 Data preparation for the model 1 3 1248 Managing the patient portfolio using mathematical programming:… with patients in a surgery type are estimated based on the relative distribution of DRGs within that surgery type. In our study, we focus on maximizing the revenue generated from staffing. From the hospital’s perspective, material and infrastructure costs, often referred to as throughput costs, are considered temporary, and maximizing revenue or case mix points does not necessarily increase contribution margin. The rationale for this is that material and infrastructure costs can fluctuate in response to changes in patient demand and operational procedures, and are frequently adjusted to accommodate short-term requirements. Furthermore, certain medical procedures are associated with exceptionally high material costs (e.g. cochlear implants), which can distort the overall cost analysis if not carefully considered. In contrast, personnel costs have a significant impact on overall expenditure and are more stable over time (Statistisches Bundesamt 2023). Personnel revenue is determined based on the proportionate personnel revenue according to the InEK cost calculation as pursuant to Section 17b (5) KHG (Hospital Care Act) and the state prime rate. In the next step, the personnel revenue per surgery type is estimated based on the historical relative distribution of DRGs within each surgery type. The optimization of the case mix, as well as for the personnel revenue model, operates within the constraints defined by scarce resources. These resources encompass OT nursing, OT anesthesia nursing, OT physician anesthesia, OT time, and ICU staffing capacity. In this context, surgeon time is not considered a scarce resource because allocating personnel resources for the OT always takes precedence for surgeons, with other tasks outside the OT considered lower in priority. The capacity allocated per resource for a patient is approximated by the respective mean value of the observed resource commitments per surgery type. Furthermore, the available capacity per resource during the observation period, in this case, a calendar year, is determined. The sum of case numbers per surgery type and the previously determined committed capacity per surgery type is calculated for each scarce resource. This is based on the observation that scarce resources have not experienced significant idle times during the observation period. In addition to resource constraints, upper and lower limits on the number of patients per surgery type define the framework within which the case mix can be optimized. Upper limits are typically determined by demand, while lower limits are influenced by both the hospital’s service mandate and the need for quality assurance in services where treatment outcomes depend on service volume. For individual services, the lower limits are predefined based on the minimum quantity regulation pursuant to Section 136b (1) Sentence 1 Number 2 German Social Code V (SGB V). The described situation can be depicted using a mathematical CMPP model. Table 1 defines the sets, indices, parameter, and decision variables. The deterministic mathematical model contains an objective function with three constraints, and it can be formulated as follows. Ma x  s∈S ps·Xs (1) 1 3 1249 M. Grieger et al. s.t.  s∈S drs ·Xs≤cr∀r∈R (2) Xs≤us∀s∈S (3) Xs≥ls∀s∈S (4) xs≥0∀s∈S (5) The objective function (1) maximizes the total product of (proportionate) personnel revenue and the number of cases per surgery type. This optimization is aimed at maximizing personnel revenue across all departments. Constraints (2) guarantee compliance with capacity limits per resource type, preventing any deviation in the case mix from causing the defined scarce resources to exceed their available capacities. Constraints (3) and (4) set upper and lower limits on the resulting number of cases for each surgery type. It is also possible to define individual limits for each surgery type if the necessary data is accessible. Constraints (5) are the non-negativity constraints. 4 Solving the CMPP at Augsburg University Hospital This section discusses the results of the case study described in Sect. 2. It considers three scenarios involving potential deviation of 10%,15% , and 20% from the case numbers of the reference year. In our model, these percentages of 10%,15% , and 20% are represented by the upper and lower limits us and ls . For example, in the 10% scenario, the number of cases per surgery type can vary between 90% and 110% of the number of cases in the reference year. In other words, any number of cases between these two limits can be realized if the hospital plans for it. For example, if a surgery type originally has a case number of 505 , the lower limit in the 10% scenario is 454 , and the upper limit is 556 . This means that any case number between 454 and 556 can theoretically be realized for this surgery type. The model selects the number of cases that is most profitable according to the objective function. A distinction is made between personnel revenue, proportionate revenue, and case mix. Sets and indices s∈S Set of surgery types r∈R Set of resource types Parameter ps Average personnel revenue per surgery with surgery type s drs Average resource requirement per surgery per resource type r and surgery type s us Case upper limit per surgery type s ls Case lower limit per surgery type s cr Capacity limit per resource r Decision variable Xs Case number per surgery type s Table 1 Sets, indices, parameter, and decision variables 1 3 1250 Managing the patient portfolio using mathematical programming:… 4.1 Personnel revenue Figure 2 illustrates that the application of the optimization model yields higher values for both the overall number of cases to be treated and personnel revenue in all three scenarios when compared to the current status. Depending on the specific scenario, the increase in total personnel revenue ranges from +2.3% and +4.5% , while the total number of cases increases by a margin of +3.4% and +6.7% , across the scenarios, all while maintaining the same or lower capacity utilization when compared to the reference year’s values. An increase in the total number of cases and personnel revenue does not necessarily translate to an increase for each department. Figure 3 shows the relative changes in the number of cases and personnel revenue per department in the 10% scenario. While most departments experience growth in both case numbers and personnel revenue, there are also departments where, from an overall optimization perspective, it would be more economically sound to reduce case numbers and personnel revenue. One of the primary factors contributing to these reductions is the elevated material costs associated with surgeries at these departments. Although material costs positively impact case mix points per department, they represent solely temporary expenses not factored in this model. Another contributing factor is an unfavorable ratio of personnel revenue to committed capacity per scarce resource when compared to other types of surgeries. In the developed optimization model, this ratio per surgery type is assessed across departments and optimally employed with the objective of maximizing personnel revenue for the entire hospital. Nonetheless, changes in both key metrics are not solely confined to increases or decreases. Considering department 10, for example, it would be possible to decrease the case numbers by adjusting the case mix without affecting the department’s personnel revenue. A detailed overview of the evolution of personnel revenues per department can be found in the left part of Table 2. To enable the monitoring of the proposed changes at the department level, the results have been organized for each department and scenario (10%,15%,20%) . Fig. 2 Total relative change in number of cases and personnel revenue per scenario 1 3 1251 M. Grieger et al. Figure 4 presents an excerpt of the outcomes provided to the respective department, using one department as an illustrative example. In this representation, both the various scenarios and the different surgery types are displayed. For each type of surgery, it is evident whether there should be an increase or decrease in the number of cases within the department. For example, personnel revenue is more advantageous to have a higher number of pacemakers implanted in the department than in the reference year. Thus, it is apparent that for each scenario and each department, a customized decision is made regarding the ideal case mix. The model optimally adjusts the number of cases per surgery type to either the upper or lower limit (i.e. either exactly Table 2 Development of personnel revenue per department and scenario (personnel revenue model) Personnel revenue Proportionate personnel revenue ±10% ±15% ±20% ±10% ±15% ±20% Dep. 1 +8.78% +13.17% +17.56% +8.45% +12.67% +16.90% Dep. 2 +0.14% +0.22% +0.29% −0.68% −1.01% −1.35% Dep. 3 +7.85% +11.77% +15.69% +7.78% +11.68% +15.57% Dep. 4 −4.85% −7.27% −9.70% −5.47% −8.20% −10.94% Dep. 5 +7.67% +11.50% +15.34% +6.57% +9.85% +13.13% Dep. 6 +2.95% +4.42% +5.90% +0.61% +0.92% +1.22% Dep. 7 −2.18% −3.27% −4.36% −3.73% −5.60% −7.47% Dep. 8 +4.91% +7.37% +9.83% +3.51% +5.27% +7.03% Dep. 9 +10.00% +15.00% +20.00% +10.00% +15.00% +20.00% Dep. 10 −0.07% −0.10% −0.14% −0.70% −1.04% −1.39% Dep. 11 −0.71% −1.07% −1.42% −1.52% −2.27% −3.03% Dep. 12 +7.71% +11.56% +15.41% +5.09% +7.63% +10.17% Total +2.26% +3.39% +4.52% +0.32% +0.47% +0.63% Fig. 3 Relative change in case numbers and personnel revenue per department in 10% scenario (Dep.: Department) 1 3 1252 Managing the patient portfolio using mathematical programming:… Funding The authors declare that no funds, grants, or other support were received during the preparation of this manuscript. Open Access funding enabled and organized by Projekt DEAL. Data availability The data that support the findings of this study are not openly available due to reasons of sensitivity and are available from the corresponding author upon reasonable request. Declarations Competing interests The authors have no relevant financial or non-financial interests to disclose. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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Authors and Affiliations MilenaGrieger1· SteffenHeider2· SebastianMcRae3· ThomasKoperna4· Jens O.Brunner1,5,6 Milena Grieger [email protected] 1 Health Care Operations/Health Information Management, Faculty of Business and Economics, University of Augsburg, Universitätsstraße 16, 86159 Augsburg, Germany 2 Digitalization and Business Analytics Department, Augsburg University Hospital, Stenglinstraße 2, 86156 Augsburg, Germany 3 Klinikum rechts der Isar, Ismaninger Straße 22, 81675 Munich, Germany 4 Operating Theater Management, Ludwig-Maximilian University Hospital, Marchioninistraße 15, 81377 Munich, Germany 5 Department of Technology, Management, and Economics, Technical University of Denmark, Kongens Lyngby, Denmark 6 Next Generation Technology, Region Zealand, Denmark 1 3 1260