Handbook of Healthcare Logistics: Bridging the Gap between Theory and Practice (International Series in Operations Research & Management Science, 302) 3030602117, 9783030602116

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International Series in Operations Research & Management Science

Maartje E. Zonderland Richard J. Boucherie Erwin W. Hans Nikky Kortbeek  Editors

Handbook of Healthcare Logistics Bridging the Gap between Theory and Practice

International Series in Operations Research & Management Science Volume 302

Series Editor Camille C. Price Department of Computer Science, Stephen F. Austin State University, Nacogdoches, TX, USA Associate Editor Joe Zhu Foisie Business School, Worcester Polytechnic Institute, Worcester, MA, USA Founding Editor Frederick S. Hillier Stanford University, Stanford, CA, USA

More information about this series at http://www.springer.com/series/6161

Maartje E. Zonderland • Richard J. Boucherie Erwin W. Hans • Nikky Kortbeek Editors

Handbook of Healthcare Logistics Bridging the Gap between Theory and Practice

Editors Maartje E. Zonderland Center for Healthcare Operations Improvement and Research, University of Twente Enschede, The Netherlands

Richard J. Boucherie Center for Healthcare Operations Improvement and Research, University of Twente Enschede, The Netherlands

Erwin W. Hans Center for Healthcare Operations Improvement and Research, University of Twente Enschede, The Netherlands

Nikky Kortbeek Center for Healthcare Operations Improvement and Research, University of Twente Enschede, The Netherlands Rhythm b.v. Amsterdam, The Netherlands

ISSN 0884-8289 ISSN 2214-7934 (electronic) International Series in Operations Research & Management Science ISBN 978-3-030-60211-6 ISBN 978-3-030-60212-3 (eBook) https://doi.org/10.1007/978-3-030-60212-3 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

Making an Impact on Healthcare Logistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maartje E. Zonderland, A. Gréanne Leeftink, Aleida Braaksma, Richard J. Boucherie, Erwin W. Hans, and Nikky Kortbeek

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Part I Overview of the State-of-the-Art A Survey of Literature Reviews on Patient Planning and Scheduling in Healthcare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maartje E. Zonderland and Richard J. Boucherie

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Theoretical and Practical Aspects of Outpatient Clinic Optimization . . . . . Maartje E. Zonderland

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Robust Surgery Scheduling: A Model-Based Overview . . . . . . . . . . . . . . . . . . . . . Maarten Otten, Jasper Bos, Aleida Braaksma, and Richard J. Boucherie

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Applications of Hospital Bed Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. J. (Thomas) Schneider and N. M. (Maartje) van de Vrugt

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Part II OR Applications in Healthcare Planning A Markov Modelling Approach for Surgical Process Analysis in Cataract Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maartje E. Zonderland, Siebe Brinkhof, Irene C. Notting, Richard J. Boucherie, Fredrik Boer, and Gré P.M. Luyten

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Emergency Operating Room or Not? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Nardo J. Borgman, Ingrid M. H. Vliegen, and Erwin W. Hans Implementing Algorithms to Reduce Ward Occupancy Fluctuation Through Advanced Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Peter T. Vanberkel, Richard J. Boucherie, Erwin W. Hans, Johann L. Hurink, Wineke A. M. van Lent, and Wim H. van Harten

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Bed Census Predictions and Nurse Staffing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Aleida Braaksma, Nikky Kortbeek, and Richard J. Boucherie Part III Case Studies Workload Forecasting and Demand-Driven Staffing: A Case Study for Post-operative Physiotherapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 R. F. M. Vromans, N. Kortbeek, L. Schoonhoven, B. van den Bosch, and M. Van Houdenhoven A Quantitative Analysis of Integrated Emergency Posts . . . . . . . . . . . . . . . . . . . . 201 M. R. K. Mes, I. M. H. Vliegen, and C. J. M. Doggen Minimizing Variation in Hospital Bed Utilization by Creating a Case Type Schedule for the Operating Room Planning . . . . . . . . . . . . . . . . . . . 231 Marc B. V. Rouppe van der Voort, Arvid J. Glerum, and Erwin W. Hans Case Study: Capacity Management in the General Hospital of ZGT, Almelo (NL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Michel M. Kats and Jasper H. Quik

Making an Impact on Healthcare Logistics Maartje E. Zonderland, A. Gréanne Leeftink, Aleida Braaksma, Richard J. Boucherie, Erwin W. Hans, and Nikky Kortbeek

Abstract This handbook provides our take on optimization of logistics processes in healthcare and on the gap that exists between theory and practice. We will bridge that gap as all theoretical results presented in this book have actually been implemented in the healthcare domain. We are driven by a desire to improve the healthcare system, by effectively making an impact with Operations Research (OR). We discuss specific projects that have addressed major challenges for healthcare Operations Research. We present our solution approaches, our approaches to implement the results in practice, and the impact on healthcare organizations. In addition, we discuss the problems we encountered when implementing the results in practice and how we addressed them. In this introductory chapter, we discuss the ecosystem of our research center CHOIR (Center for Healthcare Operations Improvement & Research) and demonstrate how we have an impact on healthcare logistics.

1 Introduction This handbook provides our take on optimization of logistics processes in healthcare and on the gap that exists between theory and practice. We will bridge that gap as all theoretical results presented in this book have actually been implemented in the healthcare domain. We are driven by a desire to improve the healthcare system, by effectively making an impact with Operations Research. We discuss specific

M. E. Zonderland () · A. G. Leeftink · A. Braaksma · R. J. Boucherie · E. W. Hans Center for Healthcare Operations Improvement and Research, University of Twente, Enschede, The Netherlands e-mail: [email protected] N. Kortbeek Center for Healthcare Operations Improvement and Research, University of Twente, Enschede, The Netherlands Rhythm b.v., Amsterdam, The Netherlands © Springer Nature Switzerland AG 2021 M. E. Zonderland et al. (eds.), Handbook of Healthcare Logistics, International Series in Operations Research & Management Science 302, https://doi.org/10.1007/978-3-030-60212-3_1

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projects that have addressed major challenges for healthcare Operations Research. We present our solution approaches, our approaches to implement the results in practice, and the impact on healthcare organizations. In addition, we discuss the problems we encountered when implementing the results in practice and how we addressed them. This book is targeted at (a.o.): • Operations Research teachers and students to have a standard reference text for state-of-the art Operations Research techniques that may be used in an advanced class on healthcare Operations Research; • Healthcare logistics researchers to have a standard reference text for state-ofthe art Operations Research techniques and valuable lessons on how to engage practice; • Healthcare logistics consultants to have an overview of available methods; • Healthcare managers to become aware of the potential, complexity, and (engineering) approach that come with applying healthcare Operations Research techniques; • Healthcare practitioners to become aware of the opportunities for logistical improvements that may be obtained invoking Operations Research techniques. In this introductory chapter, we will discuss the ecosystem of our research center CHOIR (Center for Healthcare Operations Improvement & Research) and demonstrate how we have an impact on healthcare logistics.

2 The Ecosystem of Education, Research, and Impact Since the early 2000s, CHOIR has invested in building sustainable relationships with healthcare institutions. Together with our stakeholders, we facilitate the CHOIR ecosystem of education, research, and impact. In this section we first explain the ecosystem in Sect. 2.1, followed by the network of healthcare organizations and other involved stakeholders in Sect. 2.2. We furthermore discuss the three main activities of CHOIR, education, research, and impact, in Sects. 2.3, 2.4, and 2.5, respectively.

2.1 The Ecosystem The CHOIR ecosystem is a unique collaboration between academia, healthcare organizations, and business partners. We leverage this collaboration from the onset of every project to ensure that (i) research starts from a question that is relevant to practice, (ii) research is executed as generically as possible, to advance its applicability to multiple healthcare organizations, and (iii) implementation is considered from the start of the project, to maximize the chances of actual implementation of the research results. Projects executed within the CHOIR ecosystem consist of five

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Fig. 1 The CHOIR ecosystem

phases, as shown in Fig. 1. Currently, CHOIR’s predominant involvement is with the first two phases of the ecosystem, starting with a question, followed by research and possibly a pilot study. However, after a successful research project, impact in practice is not guaranteed. Therefore, the ecosystem continues with a prototype and pilot, implementation, and evaluation phase, in which our spin-off company Rhythm is involved. Question: each project starts with a question, originating in healthcare practice. We stipulate involvement of clinical staff in every project, to ensure them taking ownership from the offset of the project. In many cases, an organization or department approaches us with a question, which is often formulated as a solution. This solution is regularly in the form of more capacity. For example, “The work pressure for our staff is way too high, can you calculate how much more capacity we need?” Most questions are raised on perceived issues, such as a high workload in our example. However, the human mind tends to remind those incidental situations over common situations in which no pressure was present. Furthermore, a rational interpretation of these issues is regularly not available, as performance is not quantitatively measured but subjectively interpreted. To address a question from practice in a systematic way and to find the root cause problem, we follow a systematic problem-solving method, such as the managerial problem-solving method [3]. We first determine the core problem(s). Furthermore, we analyze the initial performance of the system using quantitative and qualitative techniques. This is required to objectify the perceived problems, to establish whether any operational data is present, and in order to establish after the project whether performance has improved. For this purpose, we define KPIs and gather target performance levels from administrators. Using these systematic analyses, the perceived problem becomes a quantified and objectified problem. This more often

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than not causes a change in the research question. It is key to closely collaborate with the involved care providers, to make them aware of the (road to the) objectified problem and to maintain their ownership of the problem. Continuing the example, analysis might have shown that the work pressure for staff is indeed 20% above target in the mornings but below target in the afternoon. The research question then becomes: “How to allocate work over the day, to level the work pressure for staff?” or “How to optimally align the staff schedules to the workload over the day?” Research: after the question is well designed, the research goals are clear, the target performance is known, and stakeholders are involved and commit ownership, research starts. This starts with a (literature) search for approaches that can overcome the gap between the current and the desired situation. After this search, solutions are generated using a systematic solution design process, or tooling can be developed to design a(n optimal) solution. Tools that visualize (redesigned) processes can provide much insight to practitioners and lower the barrier for acceptance. Particularly discrete event simulation can greatly assist to demonstrate the expected performance of the solution in practice and to convince healthcare employees of the solution’s impact in practice. Prototype and pilot study: after a valid solution has been chosen, a prototype is developed, and/or a pilot study is started. A prototype is developed when software is required to assist in decision-making and includes the design of the new decision-making process, including the control mechanism, governance, information structures, and tooling. A pilot study is started to evaluate the use of the new decision-making process in practice. A pilot study is always conducted in a small setting, in order to evaluate the requirements of the implementation phase, to analyze the needed support during a full implementation, and to analyze and overcome the shortcomings of the solution in practice. Implementation: a successful prototype and pilot study is followed by a full implementation of the solution. This not only includes a well-designed implementation plan and support during the implementation but also aftercare and continuous development of the tooling to support the needs of practice. Evaluation: after the implementation of the solution in practice, the evaluation takes place to empirically assess whether the solution resulted in the expected performance improvement. From the evaluation, new questions come up, which makes these five phases act as a cycle.

2.2 Stakeholders Depending on the phase of the cycle, various stakeholders are involved to enable a successful impact in practice. Among these are the patients, frontline staff, healthcare administrators, business partners, bachelor, master, PDEng and PhD students, and faculty members.

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Gathering patient preferences is essential input for system redesign but is rarely reported in Operations Management (OM) publications. Patient inquiries, for example, using discrete choice experiments, may uncover operational constraints and performance indicators. Healthcare professionals are involved in all phases of the cycle, as the project is executed within the healthcare institute. Our network of healthcare professionals is involved after a first implementation and evaluation cycle in the healthcare organization under study, to show the results in practice and to enable further dissemination of the results among other healthcare institutes. Business partners play an important role in the implementation phase, by providing a complete business solution based on the research and prototype. A fruitful relation with such a partner is potentially mutually beneficial, as wellimplemented research leads to new research questions and more opportunities for implementation. Also, it may generate revenues, which preferably would (partly) flow back to fund research. For this reason, in 2014 CHOIR has started a spin-off company, called Rhythm. By collaborating with this spin-off, we may use part of the income of the partner for new research opportunities. The same can be obtained through a royalty or intellectual property contract with an external partner. Student-researchers are a main driver of our research projects. We find it essential for CHOIR researchers to be present in healthcare organizations, to lower the barrier for practitioners to approach us, to ensure relevant topics of study, and to promote involvement of frontline staff. Therefore, we are convinced that studentresearchers should be positioned in a healthcare organization for a major part of their research, ideally to be considered by practitioners as part of their own organization. Therefore, we follow the researcher-in-residence model in all stages of the research project. The researcher-in-residence model positions a student-researcher as a core member within a healthcare team of relevant care professionals [6]. In a context of process improvement, the researcher brings a new body of expertise, focused on data analysis, modeling, and structured decision-making, which is different from, but complementary to, the expertise of the existing team. Within CHOIR, we differentiate between three types of student-researchers, with their own skill set, related to the analytics framework of Gartner [2], as displayed in Fig. 2. Bachelor students excel in descriptive and diagnostic activities. Within a project team, they perform a thorough problem definition, together with a root cause analysis. Based on this analysis, they present improvement opportunities to assist decision-makers. Master students not only assist in descriptive and diagnostic activities but can also take on a predictive or prescriptive aspect to enhance decisionmaking support, depending on the organization’s need. PhD students collaborate in projects spanning all possible activities, including descriptive, predictive, and prescriptive activities. It is beneficial to include PhD students in projects from an early phase for a good understanding of the problem and its causes. Furthermore, the perception by clinicians and other healthcare staff that an outside person (the PhD student) is fully aware of all restrictions that apply leads to high acceptance and cooperation rates and therefore a better performance of the proposed solutions. As PhD students reside in a healthcare organization for a longer period of time,

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Fig. 2 Gartner’s analytics maturity model [2]

prototype, pilot study, and early implementation results can be evaluated to prove impact in practice not only analytically but also in an empirical way. Besides students that perform research activities, we employ PDEng students in the Professional Doctorate in Engineering (PDEng) program. This is a 2-year postgraduate program that focuses on applied research. Promising research results are translated into a prototype for a professional context, in close collaboration with a healthcare institute. A PDEng student focuses therefore on the prototype and pilot phase of the CHOIR ecosystem. PDEng students reside in a healthcare organization, in order to evaluate the requirements of the implementation phase, to set up the decision-making process, to analyze and overcome the shortcomings of the solution in practice, and to align this system with the institute’s information systems. Finally, faculty members are important contributors to the CHOIR ecosystem. They regularly visit healthcare institutes, and junior faculty members might even continue to participate in the researcher-in-residence model themselves. This prevents intellectual isolation of the junior staff and students, which is a common disadvantage of the researcher-in-residence model [6]. Being a researcher-inresidence requires great professional skills, which are best trained on site. Also, evidently it requires academic skills, which are best trained at the university, with faculty members and fellow researchers. The CHOIR researchers therefore reside at the university together for at least 2 days per week for this purpose and to share experiences and collaborate with other researchers-in-residence. Faculty members not only educate students and care professionals but also monitor the applied methodologies in the various projects, connect healthcare institutes with similar research questions, and link relevant people within the CHOIR network. Through their diverse activities, faculty members are involved in all three pillars of CHOIR (education, research, impact), which we elaborate upon in the upcoming sections.

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2.3 Education The first pillar of CHOIR is education. By educating both students and healthcare professionals, CHOIR aims to bridge the gap between theory and practice. As a research center within the University of Twente, CHOIR’s main educational focus is on BSc and MSc students of industrial engineering and management, health sciences, and applied mathematics. The teaching activities encompass Operations Research in healthcare through lectures and practical sessions but also professional and academic skills training and practical experience, through graduation projects and internships. We aim to educate students to become independent researchers within an organization that speaks a different language. After their studies, many CHOIR alumni continue their career in healthcare and spread the knowledge throughout the organizations they work in, which results in a growing healthcare logistics community. Aside from academic students, CHOIR also educates healthcare professionals, including managers, administrators, logistics staff, doctors, and (head) nurses. Most of them have an educational background in medicine or nursing and lack OM training. From experience, these professionals often know the practical constraints of process optimization but lack methodological knowledge and knowledge about theoretical (im)possibilities. Also, they typically find it hard to look at operational processes in an integrated way. Instead, they tend to only focus on their department or role in the system. The course encompasses not only theory. Parallel to the course, the participants have to perform a process improvement assignment within their own department, under supervision of CHOIR staff. In our experience, many former participants in the course become champions for our research approach. They often initiate new research projects, serve as in-company supervisors for our student projects, and guide the implementation of our research in practice.

2.4 Research The second pillar of CHOIR is research. We take on complex logistical challenges that are driven by practice to design or optimize the organization of healthcare processes. Herein, we aim to improve the quality of care, the quality of labor, and the efficiency of processes. We find it important to emphasize this, as process optimization is quickly solely associated with efficiency and working harder. As it is evidently undesirable to try out interventions in practice, in our research we make use of mathematical models and discrete event simulation to prospectively assess the performance of an intervention before actual implementation. We disseminate our research in the scientific community through two main channels. First, we present and publish our results in the OM/OR domain, where field experts can give us feedback on the methods used. Second, we present and publish our results in the medical domain, to show the potential of the use of

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OM/OR tools for optimizing healthcare processes. CHOIR receives its funding for research and PhD and PDEng projects both from practice, e.g., through funding from healthcare organizations, and from funding agencies, e.g., through national science programs.

2.5 Impact The focus of our education and research is to have an impact in practice, which is the third pillar of CHOIR. For a knowledge gathering and developing center such as CHOIR, impact is the dissemination and effective application of that knowledge in practice. We disseminate knowledge through the network of healthcare providers by the positioning of our students in healthcare organizations, through seminars and symposia at our university and at healthcare organizations, through teaching, through alumni, and through publications in professional and academic journals. By organizing meetings specifically targeted at professionals involved in capacity management and (patient) logistics, we collect new problems to work on but also share knowledge, tailored to the specific needs of this group. We apply knowledge in practice in various ways. Through the projects of bachelor, master, PDEng, and PhD students, we contribute to process optimization in practice. Although not every project results in implementation, the presence of someone with a different background that questions regular protocols raises an awareness for improvement potential in organizations. Also our spin-off Rhythm makes an impact in practice through various activities, such as consultancy, training, and knowledge and software development. Over the years, we have gathered much experience in effective and less effective application of knowledge in practice. In the next section, we therefore discuss some conditions for making an impact.

3 Deliberations for Impact In this section we reflect on the impact we have made in practice with CHOIR’s research projects and educational program and analyze the critical success factors. We discuss four deliberations that we encounter in our research projects, which we illustrate using real-life examples: 1. 2. 3. 4.

Theoretical projects vs. practical projects Theoretical solutions vs. solutions from practice Building relationships Bottom-up and top-down

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3.1 Theoretical Projects vs. Practical Projects The theoretical requirements for scientific research projects can be challenging. For example, modeling is one of the requirements for our BSc and MSc students in industrial engineering and management and applied mathematics to successfully finish their thesis projects. However, this theoretical requirement may contradict with the needs in practice, where a straightforward and easy-to-implement solution can have great impact already. Furthermore, there are projects with high potential for theoretical contributions to the scientific community, which are not relevant to practice. On the contrary, there are also projects with high practical relevance but with little significant contribution to science. For all our research projects, we always aim to combine a theoretical and practical perspective. We realized a fruitful synergy between theory and practice when we were involved in the rebuilding of the surgical inpatient units in the Academic Medical Center in Amsterdam. We used the Erlang loss model to quickly answer the hospital’s questions with respect to the number of beds required per unit given a prespecified blocking probability and simultaneously developed the much more detailed bed census prediction and flexible staffing models of chapter “Bed Census Predictions and Nurse Staffing”. By providing practical advice quickly, we convinced nurses, physicians, and management of the usefulness of Operations Research methods. In response, they committed to supporting our research for developing the more detailed models. Another example of finding the right balance between theory and practice is our work at the pre-anesthesia evaluation clinic at the University Medical Center in Leiden [7]. A mathematical model was developed to evaluate several alternative clinic designs, in order to overcome long patient waiting times, overcrowding, and high workload for staff. The model outcomes were discussed with clinic staff in several iterations, leading to improvement of the model and new scenarios to analyze. Already 4 months after the start of the project, an alternative clinic design, which was chosen unanimously by staff, was successfully implemented.

3.2 Theoretical Solutions vs. Solutions from Practice When problems increase in frequency or size, we find that problem owners rapidly advocate the necessity of more capacity as the solution. However, in our experience with improvement projects in healthcare settings, in hardly a handful of cases there was a proven capacity shortage. Increasingly, healthcare providers realize that the rising expenditures need to be countered and that more capacity (especially nursing staff) is unavailable. Instead, new process designs and new planning and control models are sought after to overcome their challenges. From a theoretical Operations Research perspective, exact methods to find or design an optimal solution are preferred over evaluation studies. Exact methods

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determine the best possible decision for the project team. As the decision is optimal, it shows the best possible performance that can be reached, which can serve as a benchmark for the organization’s performance. From a practical perspective, evaluation studies are preferred, for example, using computer simulation. Evaluation studies enable a researcher to test several interventions and scenarios. They give the involved healthcare staff more flexibility in testing those interventions which they consider promising to implement in practice. As evaluation studies often include a visual component, they are also relatively easy to follow and understand, which supports the acceptance of final recommendations even more. As mentioned in Sect. 2.1, the question posed by a healthcare organization may be introduced in the form of a solution. This proposed solution is typically evaluated as one of the possibilities to solve the problem in the research phase. This is a practical perspective, wherein the possibilities for implementation are considered more important than finding the optimal solution. Note that in practice, a straightforward, near-optimal, planning solution most likely leads to higher impact than an optimal solution, as employees adhere better to easier-to-understand planning solutions than to more complex solutions. In our experience, most successful studies from which one or more recommendations were implemented involved some kind of evaluation component. For example, in the histopathology laboratory of the University Medical Center in Utrecht, we were involved in a process improvement project [4]. During this project several interventions were evaluated. As decision-makers were involved in the design of the interventions themselves, they knew whether implementation of each intervention was realistic and possible. From a research perspective, it might be interesting to add several theoretically interesting solutions, or alternative solutions, for example, to serve as benchmarks for practice. Complex solutions, which are, for example, the outcome of an optimization study, have a higher chance of implementation when their implementation depends less on human input. An example is the schedule template in the outpatient clinics’ agendas of the same hospital [5]. Based on this research, an optimized blueprint schedule was programmed into the computer system used for appointment planning, which reduces the possibilities of misuse and nonadherence.

3.3 Building Relationships Research projects are not only successful due to the researcher but to a large extent also due to the project team surrounding the researcher. Many of the healthcare organizations that CHOIR collaborates with have a so-called champion. This champion supports the CHOIR ecosystem and is the connector between CHOIR researchers and the organization’s board. Besides champion individuals, also champion departments or organizations are present. For organizations to benefit most from process improvement projects, it is essential to grow from a champion individual to a champion department or organization to ensure the continuity of

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process improvement in the healthcare organization. This way, the implementation and continuation of the results of a project does not depend on a single individual. Collaborating with champion institutions allows for long-term relationships, which enables to take on larger research projects with long-term commitments. These projects can typically result in more impact, as not only quick wins are derived, and the project scope can be extended over multiple departments and hierarchical planning levels. If an organization has evolved into a champion organization, it is key to further invest in this organization to create a long-term relationship, to ensure the continuity of process improvement in healthcare organizations. This requires an understanding from both sides that research projects are typically time-consuming and thorough, whereas healthcare organizations are typically in need of solutions on the short-term that work just fine, when they face logistical challenges. Champion organizations should be supported with both: thorough research for the long-term (re)organization of their logistical processes and support for short-term challenges. The investment required is also related to visibility: the researchers should be present in the organization and ideally work from there several days a week. It is important to set up training programs for staff and the C-suite, in order to enable the organization to ultimately successfully complete improvement projects themselves. From the organization’s side, a financial investment is essential, since creating a financial relationship usually leads to enhanced commitment. One of our champion organizations is Sint Maartenskliniek (SMK), a Dutch hospital for movement and posture. In 2014, we started our formal collaboration with SMK trough a PhD project, initiated by a champion individual, and have expanded our collaboration ever since. So far, one PhD and one PDEng student have graduated, and numerous MSc and BSc thesis projects have been executed in various departments of SMK. Rhythm B.V. support SMK in handling their logistical challenges. In SMK, we often work in close collaboration with Rhythm B.V. This collaboration has resulted in valuable outcomes, for example, regarding translating the bed census predictions of chapter “Bed Census Predictions and Nurse Staffing” into tooling that is used on a day-to-day basis to predict the occupancy of the wards and assist tactical operating room management decisions and major advances in their multi-appointment scheduling [1]. Furthermore, through the close collaboration and prior experience with their processes and data, we were able to support SMK with their short-term challenges faced during the COVID-19 crisis, for example, related to restarting their in- and outpatient care.

3.4 Bottom-Up and Top-Down The bottom-up and top-down process improvement approaches are two complementary strategies for starting process improvement projects in practice. Bottom-up process improvement considers process improvement projects that are initiated by frontline staff and that focus on (iterative) process improvements on the

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operational level of control, for example, to promote a culture of continuous quality improvement. Given that the healthcare process improvement field is still growing, there are a lot of process improvement opportunities. Therefore, it is important that healthcare organizations create a safe atmosphere for bottom-up quality improvement, in which all staff is involved in the improvement efforts. Top-down process improvement considers process improvement projects initiated by the management of an organization or department and is often executed by a project group. Top-down approaches revolve around the redesign and optimization of processes, planning, and control. Top-down process improvement projects enable organizations to make structural changes to their organization and to invest in new solutions, especially on the strategic and tactical level of planning and control. These projects are often well organized and (financially) supported by top management as the expected impact is large. In a champion organization, the joint approach gives the highest probability of success. Bottom-up approaches aim to create an improvement culture, which creates a focus on operational processes and performance by frontline staff. Topdown approaches on the other hand can result in higher-level organizational changes. However, when operational level improvement opportunities are still abundant, frontline staff support for a top-down approach will be low. Therefore, the operational processes should operate sufficiently well before initiating a topdown approach. Furthermore, when top-down support is present, but problems are not perceived by frontline staff, implementation possibilities of project results are limited. Summarizing, it is essential to invest in champion individuals when starting research projects and to further extend this relationship to the development of a champion organization, to maximize impact. The research project at the pathology department of the academic hospital in Utrecht was initially top-down initiated [4]. Through the top-down commitment, the results therefore benefited strategic decision-making – ultimately leading to the acquisition of additional laboratory work of a regional hospital in their laboratory. However, to deliver a successful project, much effort was needed into getting the frontline employees involved. For this, being a researcher-in-residence was the key to success. By incorporating the frontline employees into the problem finding and solution design phases, they supported the final recommendations of the project and were eager to implement the new planning rules in practice.

4 The Handbook of Healthcare Logistics In this handbook of healthcare logistics, we highlight some of the results of the CHOIR ecosystem and demonstrate how our work has an impact on daily hospital operations. The book consists of three parts. In Part I, we provide an overview of the state of the art, starting with a survey of literature reviews on patient planning and scheduling in healthcare. Following the

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dominant patient routing in modern hospitals, we subsequently discuss outpatient clinic optimization, robust surgery scheduling, and hospital ward logistics. In Part II, four examples of Operations Research applications in healthcare planning are given, related to surgical process analysis, emergency operating rooms, ward occupancy leveling, and nurse staffing. In Part III, several case studies are presented: on clinical physiotherapy capacity planning, integrated emergency posts, master surgical scheduling, and the implementation of integral capacity management. We hope to inspire ambitious researchers, showing that it is possible to make an impact with healthcare Operations Research.

References 1. IA Bikker, MRK Mes, A Sauré, and RJ Boucherie. Online capacity planning for rehabilitation treatments: an approximate dynamic programming approach. Probability in the Engineering and Informational Sciences, pages 1–25, 2018. 2. Gartner. Gartner says advanced analytics is a top business priority. http://www.gartner.com/ newsroom/id/2881218, 2014. Online; Accessed 2017-04-06. 3. JMG Heerkens and A van Winden. Solving managerial problems systematically. Noordhoff Uitgevers, 2017. 4. AG Leeftink, RJ Boucherie, EW Hans, MAM Verdaasdonk, IMH Vliegen, and PJ Van Diest. Predicting turnaround time reductions of the diagnostic track in the histopathology laboratory using mathematical modelling. Journal of clinical pathology, 69(9):793–800, 2016. 5. AG Leeftink, IMH Vliegen, and Erwin W Hans. Stochastic integer programming for multidisciplinary outpatient clinic planning. Health care management science, 22(1):53–67, 2019. 6. M Marshall, C Pagel, C French, M Utley, D Allwood, N Fulop, C Pope, V Banks, and A Goldmann. Moving improvement research closer to practice: the researcher-in-residence model. BMJ Quality & Safety, 23(10):801–805, 2014. 7. ME Zonderland, F Boer, RJ Boucherie, A de Roode, and JW van Kleef. Redesign of a university hospital preanesthesia evaluation clinic using a queuing theory approach. Anesthesia & Analgesia, 109(5):1612–1621, 2009.

Part I

Overview of the State-of-the-Art

A Survey of Literature Reviews on Patient Planning and Scheduling in Healthcare Maartje E. Zonderland and Richard J. Boucherie

Abstract This chapter provides a classification of literature reviews on patient planning and scheduling decisions in healthcare. To this end, we provide a mathematical interpretation of hierarchical planning levels and then add the planning complexity dimension to an existing planning and scheduling framework, resulting in a 3D framework for patient planning and scheduling decisions in healthcare. Subsequently, we provide an overview on recent surveys and reviews on this topic and position these studies in the 3D framework.

1 Introduction This chapter provides a classification of literature reviews on patient planning and scheduling decisions in healthcare. To this end, first we provide a mathematical interpretation of the hierarchical planning levels as defined in [9]. Second, we add the planning complexity dimension to the planning and scheduling framework as introduced in [11], resulting in a 3D framework for patient planning and scheduling decisions in healthcare. Third, we provide an overview on recent surveys and reviews on this topic and position these studies in the 3D framework.

2 A 3D Classification of Planning Decisions in Healthcare The taxonomy of planning and control decisions in healthcare as presented in [11] has two axes. On the x-axis, services in healthcare are positioned, where we have selected ambulatory, emergency, surgical, inpatient and home care services, and

M. E. Zonderland () · R. J. Boucherie Center for Healthcare Operations Improvement and Research, University of Twente, Enschede, The Netherlands e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2021 M. E. Zonderland et al. (eds.), Handbook of Healthcare Logistics, International Series in Operations Research & Management Science 302, https://doi.org/10.1007/978-3-030-60212-3_2

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further services may be added, while the y-axis reflects the hierarchical nature of decision-making in resource capacity planning and control as introduced in [9]: strategic, tactical, operational offline and online levels. In this section, we provide a mathematical interpretation of these hierarchical planning levels via the level of uncertainty. Subsequently, we add a third dimension to the taxonomy: planning complexity, comprised of single activity planning, multidisciplinary planning and care pathway planning. Our aim is not to provide an in-depth description of these levels of complexity as these are covered in later chapters but to illustrate the levels that are extensively covered and those that are seldomly addressed in literature.

2.1 Hierarchical Planning Levels and Uncertainty Aligning supply of and demand for care aims for patients receiving their prescribed care at the right time, by the right professional, and at a suitable location, while staff workload is levelled, and other resources are efficiently utilised. This alignment is organised in three hierarchical planning levels: strategic, tactical and operational [9]. At the strategic level, the budget over a long time horizon (e.g. per year) is determined, which includes the total amount of available resources such as staff and rooms. At the operational level, the supply of care is known as well as the individual patients that have a demand for care; operational planning then amounts to booking a patient in available appointment time slots. Offline operational planning concerns the in-advance planning, whereas online operational planning involves control mechanisms that deal with last-minute unforeseen or unanticipated events. The tactical level is the intermediate level (e.g. a time horizon of several months) at which the available supply of care is known, but the individual patients and their demand for care are not yet (entirely) revealed. The available capacity is matched to specific activities in, e.g. a master schedule or blueprint. This planning level is often overlooked in healthcare and is tantamount to an adequate alignment of supply of and demand for care at the operational level where patients are booked in the time slots of the tactical master schedule. These planning levels may be more formally determined by the amount of randomness that is revealed in the planning process. At the operational level, the professional, the patient and the required treatment are matched. Randomness remains since, e.g. the treatment time might vary, the patient might not show up or arrive late, or the care-provisioner might be absent due to illness. At the tactical level, the schedule or capacity of the professional or type of professional is revealed, but the patient is not yet known. Thus, the number of patients that may be planned is known, but not which specific patients will be planned, which adds a second layer of randomness. At the strategic level, only the total capacity of the professionals is known, but not the number and type of the patients, adding a third layer of randomness to the planning problem.

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2.2 Planning Complexity A major part of the complexity of the decision-making process in healthcare planning lies in the type of appointments that must be planned. Usually, planning a single appointment, procedure or episode of inpatient care is less involved than planning several activities for one patient at once, especially when these are interrelated and of heterogeneous nature and need to be planned in a certain chronology. To this end, we introduce planning complexity as a third dimension in the framework and distinguish three types of complexity, single item planning, multidisciplinary planning and care pathway planning, which will be briefly described below based on the characteristics introduced in [16, 17].

2.2.1

Single Activity Planning

In single activity planning, appointments, procedures or episodes of inpatient care are planned subsequently, one by one, on a single resource. Characteristics which are important to consider are, a.o. [17] future demand from other patients for the same service period; the number and availability of resource types at the facility; and the horizon at which the scheduling decisions are made.

2.2.2

Multidisciplinary Planning

In multidisciplinary planning, coordinated packages of care for patients are planned. This can either be a single activity on multiple resources or multiple activities on a single or multiple resources [17]. Next to the characteristics important for single activity planning, additional characteristics to consider are, a.o. the number of activities to be scheduled in a certain time frame; requirements on the number of activities to be scheduled within a certain time entity of the time frame (e.g. a week); and the chronology and relationships among the activities to be scheduled.

2.2.3

Care Pathway Planning

Care pathway planning is in many ways similar to multidisciplinary planning. The main difference is that care pathways (sometimes also referred to as clinical pathways) are designed to exactly specify the care trajectory for an entire group of patients (e.g. patient types), while multidisciplinary trajectories are specified for a single patient. Since multiple patients receive the exact same care trajectory, care pathways reduce the process variability and thus also the planning complexity [16]. Note, however, that the adherence to these pathways is usually low [12], which may result in an increase in (operational) planning and scheduling complexity. Next to the characteristics important for single and multidisciplinary planning, additional

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characteristics to consider are, a.o. the magnitude of the patient group included in the care pathway and interference of care pathway planning with the planning process for other patients and/or patient groups.

2.3 A 3D Framework for Planning Decisions in Healthcare We introduced three dimensions for planning decisions in healthcare: services, the hierarchical level and planning complexity. We consider five services (ambulatory, emergency, surgical, inpatient and home care), four hierarchical levels (strategic, tactical, operational offline and operational online) and three complexity planning levels (single activity, multidisciplinary and care pathway), resulting in 5 × 4 × 3 = 60 subcategories that are graphically presented in Fig. 1.

3 Positioning of Overview Papers on Patient Planning and Scheduling Decisions in Healthcare in the 3D Framework This section provides a starting point for further reading,1 by identifying recent surveys and review papers on patient planning and scheduling and positioning these papers in our 3D framework in Fig. 1. Our aim is not to provide an in-depth description of these levels of complexity as these are covered in later chapters but to illustrate the levels that are extensively covered and those that are seldomly addressed in literature. Using Google Scholar we applied the following search strategy and selected 15 papers: • recent articles published since 2000; • focus on patient planning and scheduling decisions, not on staff scheduling or medical planning (i.e. topics such as advanced care planning or discharge planning are excluded); • should have either survey or review in the title; • should have either planning or scheduling in the title; and • one of the following words in title: healthcare, health care, patient, appointment, surgery, surgical, operating, inpatient, outpatient, ambulatory, emergency, home care. Our inclusion criteria did not result in surveys or reviews on patient planning and scheduling available for the emergency care services including facilities such as emergency departments, ambulances and trauma centres, but not including emer-

1 See

also the online reference database of OR/MS literature [10] provided by the Center for Healthcare Operations Improvement and Research.

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Fig. 1 The 3D framework with the 15 selected papers. The taxonomy presented in [11] falls into all categories

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gency patients within the OR environment. This is clear for the operational planning level, since emergency patient care cannot be planned on individual (patient) level. On the tactical and strategic level, however, patient planning and scheduling should support the decision-making process. Further observe that literature on ambulatory care, surgical care and inpatient care seems connected, but literature on home care services seem not to have a relationship with the other services covered in the framework. Regarding the hierarchical planning levels, the included papers focus either on all levels or on the operational level. When considering planning complexity, papers covering care pathways are scarce. This might be due to the lack of a uniform definition of care pathways from a mathematical perspective.

References 1. Ahmadi-Javid, A., Jalali, Z. and Klassen, K.J.: Outpatient appointment systems in healthcare: a review of optimization studies. Eur J Oper Res 258(1), 3–34 (2017) 2. Cardoen, B., Demeulemeester, E. and Belien, J.: Operating room planning and scheduling: A literature review. Eur J Oper Res 201(3), 921–932 (2010) 3. Cayirli, T. and Veral, E.: Outpatient scheduling in health care: a review of literature. Prod Oper Man 12(4), 519–549 (2003) 4. Cissé, M., Yalcindag, S., Kergosien, Y., Sahin, E., Lenté, C. and Matta, A.: OR problems related to home health care: a review of relevant routing and scheduling problems. Oper Res Health Care 13-14, 1–22 (2017) 5. Dantas, L.F., Fleck, J.L., Cyrino Oliveira, F.L. and Hamacher, S.: No-shows in appointment scheduling – a systematic literature review. Health Policy 122(4), 412–421 (2018) 6. Erdem, M. and Bulkan, S.: A literature review on home healthcare routing and scheduling problems. Eur J Health Tech Ass 2(1), 19–33 (2017) 7. Fikar, C. and Hirsch, P.: Home health care routing and scheduling: a review. Comp & Oper Res 77, 86–95 (2017) 8. Gupta, D. and Denton, B.: Appointment scheduling in health care: challenges and opportunities. IIE Trans 40, 800–819 (2008) 9. Hans, E.W., van Houdenhoven, M. and Hulshof, P.J.H.: A framework for health care planning and control. In: Hall, R.W. (ed.) International Series in Operations Research & Management, pp. 303-320. Springer, New York (2012) 10. Hulshof, P.J.H., Boucherie, R.J., et al: ORchestra: an online reference database of OR/MS literature in health care. Health Care Manag Sci 14(4), 383–384 (2011) 11. Hulshof, P.J.H., Kortbeek, N., Boucherie, R.J., Hans, E.W., Bakker, P.J.M.: Taxonomic classifcation of planning decisions in health care: a structured review of the state of the art in OR/MS. Health Syst 1, 129–175 (2012) 12. van de Klundert, J., Gorissen, P. and Zeemering, S.: Measuring clinical pathway adherence. J of Biomed Inf 43(6), 861-872 (2010) 13. Lamé, G., Jouini, O. and Stal-Le Cardinal, J.: Outpatient chemotherapy planning: a literature review with insights from a case study. IIE Trans Healthc Syst Eng 6(3) (2016) 14. Leeftink, A.G., Bikker, I.A., Vliegen, I.M.H, Boucherie, R.J.: Multi-disciplinary planning in health care: a review. Health Syst 1–24 (2018) 15. Marynissen, J. and Demeulemeester, E.: Literature review on multi-appointment scheduling problems in hospitals. Eur J Oper Res 272(2), 407–419 (2019) 16. Vanberkel, P.T., Boucherie, R.J., Hans, E.W., Hurink, J.L. and Litvak, N.: A survey of health care models that encompass multiple departments. Int J of Health Man and Information 1:1, 37–69 (2010)

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17. van de Vrugt, N.M., Braaksma, A. and Boucherie, R.J.: The state of the art in online appointment scheduling. In: van de Vrugt, N.M.: Efficient healthcare logistics with a human touch. Ph.D. thesis, University of Twente (2016) 18. Van Riet, C., Demeulemeester, E.: Trade-offs in operating room planning for electives and emergencies: a review. Oper Res Health Care 7, 52–69 19. Zhu, S., Fan, W., Yang, S., Pei, J. and Pardalos, P.M.: Operating room planning and surgical case scheduling: a review of literature. J Comb Opt 37, 757–805 (2019)

Theoretical and Practical Aspects of Outpatient Clinic Optimization Maartje E. Zonderland

Abstract The outpatient clinic is one of the most important departments of the hospital. Since most elective care trajectories start here, with a consultation between a care provider and a patient, the outpatient clinic functions as a gate to enter the hospital. Outpatient care is evolving rapidly around concentration of low- and highcomplex care and the digitization of care processes, thus introducing new challenges in the organization of care. In this chapter we introduce the concept of access and waiting time, briefly discuss outpatient clinic capacity management, study the planning and control framework in place, and discuss Operations Research models for outpatient clinic optimization, specifically those for patient flow analysis and appointment planning. We conclude this chapter with lessons learned during the implementation of our work and discuss current challenges in outpatient clinic management. We specifically aim to support the researcher who is starting in this field, by providing a comprehensive overview of the theoretical and practical aspects of outpatient clinic optimization.

1 Introduction The outpatient clinic is one of the most important departments of the hospital. Since most elective care trajectories start here, with a consultation between a care provider and a patient, the outpatient clinic functions as a gate to enter the hospital. Outpatient care is evolving rapidly around concentration of low- and high-complex care and the digitization of care processes, thus introducing new challenges in the organization of care. In this chapter we introduce the concept of access and waiting time, briefly discuss outpatient clinic capacity management, study the planning and control framework in place, and discuss Operations Research models for outpatient

M. E. Zonderland () Center for Healthcare Operations Improvement and Research, University of Twente, Enschede, The Netherlands e-mail: [email protected] © Springer Nature Switzerland AG 2021 M. E. Zonderland et al. (eds.), Handbook of Healthcare Logistics, International Series in Operations Research & Management Science 302, https://doi.org/10.1007/978-3-030-60212-3_3

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clinic optimization, specifically those for patient flow analysis and appointment planning. We conclude this chapter with lessons learned during the implementation of our work and discuss current challenges in outpatient clinic management. We specifically aim to support the researcher who is starting in this field, by providing a comprehensive overview of the theoretical and practical aspects of outpatient clinic optimization.

1.1 The Outpatient Clinic: The Gate to Elective Hospital Care Since the majority of acute patients are admitted through the emergency department and receive immediate treatment, this type of patient flow is usually not present at outpatient clinics. The elective outpatient clinic visits take place in the diagnostic phase and at a later stage for treatment and follow-up consultations. Optimization of outpatient clinic care requires state-of-the-art capacity management, focusing on outpatient clinic capacity and resource management and integration with other departments such as the operating rooms, cath lab, and inpatient wards. While the interaction with other departments is on tactical level, on an operational level there is a high interaction with other outpatient clinics and diagnostic facilities. Well known examples are that of surgical outpatient clinics with preanesthesia evaluation clinics and radiology departments, where the patient visits the other facilities directly after the outpatient clinic or leaves and then returns at a later point in time. Traditionally each elective care trajectory starts with a physical consultation and usually involves several outpatient clinic visits. Due to the transition of low-complex care to the GP setting, and e-health applications such as home monitoring, where the patient can be closely monitored from home and only visits the hospital if required, this typical organization of outpatient care is subject to change. Capacity will be redistributed among those patients who need it the most; at the same time the care provider activities will be focused increasingly on providing digital patient care. The COVID-19 pandemic has been a catalyst for these changes.

1.2 Crucial Concepts: Access and Waiting Time First we introduce two concepts which are essential for outpatient clinics: access and waiting time. Access time is defined as the time between the day of the appointment request and the appointment date, while waiting time is defined as the time between the physical arrival at the facility and the start of consultation and/or treatment [16]. Access time is usually measured in days or weeks; waiting time is measured in minutes or hours. Both can vary between patient groups and depend highly on the match between allocated capacity and patient demand and the ability to deal effectively with fluctuations in both entities. Outpatient clinics often have long

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access times and low utilization at the same time, albeit for different patient groups. Quotes such as “the waiting list is too long” or “the clinic is overcrowded” are therefore too general. Waiting lists are often quite intangible, and their reliability can be questionable. They exist in several forms: an order list in the hospital information system, one or several Excel spreadsheets, on paper, in an email program, etc. It is not always the case that patients on the waiting list are not planned for an appointment yet; it can also be that their access time is prolonged (i.e., it is longer than desirable from a medical – and sometimes economical – point of view), while they already have an appointment date set.

1.3 Outpatient Clinic Capacity Management Optimal alignment between patient demand and outpatient clinic capacity is essential, maybe even more than for other hospital departments, since often it is not yet clear what the patient’s diagnosis and thus the associated risk are. When treatment is delayed, the patient may choose another care provider. Also, the patient’s health condition may deteriorate which is a serious risk [24]. Patient noshows and healthcare provider availability influence this delicate equilibrium, and thus proper capacity management, starting at strategic level and working through the tactical to the operational level, is crucial. We will elaborate further on this in the following sections. Even though patient flow performance is better in hospitals that employ integrative practices, such as sharing waiting list and planning information, planning over several departments, and offering combination appointments, the overall level of integration in hospitals is usually low [7]. Currently, there is an increased focus on the link between the production levels on the outpatient clinic and the (elective) OR. However, integration of the planning processes from different departments is challenging for hospitals [8]. In literature, two-departmental interactions have been studied to a certain extent, but only a few studies relate to the interactions in the hospital as a whole [14, 29]. The ambiguity of patient care trajectories [29] and the segmented organizational structure of health care delivery [14] are identified as possible causes for this gap in literature. For the reader who wants to gain a deeper understanding of the organizational structures in hospitals and the logistic processes involved, we refer to [11].

2 Planning and Control of Outpatient Clinics The design and organization of processes is known as planning and control [14]. In this chapter we focus on the managerial function of outpatient clinic capacity planning as defined in [12]. Planning and control decision-making in

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healthcare comprises four hierarchical levels: the strategic, tactical, offline, and online operational level [12]. A short description of these planning levels and the planning decisions involved follows in the next paragraphs and is based on [14].

2.1 Strategic Planning Strategic planning involves structural decision-making on a long time horizon (typically 1 year or more) and is based on aggregated data and forecasts. For outpatient clinics, this involves: • Regional infrastructure: create an infrastructure in order to provide accessible outpatient care in an efficient manner by determining the amount, size, and location of outpatient facilities in a certain region. • Patient types (or case mix): determine the patient types and related volumes, for example, based on characteristics such as disease, diagnosis, severity, acuity, or age, which will be served by the facility. • Capacity dimensions: determine how many resources (local infrastructure, equipment, staff, and consultation time) are required to meet the demand for each patient type. This highly depends on the desired service levels for performance indicators such as costs, patient’s access and waiting time, and provider idle time. • Facility layout: decide on the layout for the outpatient clinic, given the resources required. A typical outpatient facility has at least a waiting room, reception desk, and consultation/examination rooms.

2.2 Tactical Planning Tactical planning translates the decisions made on the strategic level to the requirements needed to execute the healthcare delivery process, i.e., the what, where, how, when, and who. The time horizon is typically 3–12 months. Important challenges in the organization of outpatient care are: • Care pathways: for each patient type, determine the entire care pathway (sometimes referred to as clinical pathway) and relationships among the phases within the care pathways and in between the different care pathways. This also comprises visits and treatment at other hospital departments. Try to influence performance indicators such as length of the care trajectory, waiting and access time, and size of the waiting list by creating synergy among patient groups. This can, for example, be accomplished by combining multiple visits on a single day or mixing several patient groups in order to optimize resource utilization. • Fixed and flexible capacity: determine which amount of capacity (typically 20–40%) should be used flexibly, so that fluctuations in patient demand can be accounted for.

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• Waiting list management: determine which patient groups should be prioritized, determine acceptable access times for each patient group, and decide if walk-in or advanced access policies can be implemented at the facility. • Appointment schedules: develop blueprints providing specific times and dates for patient consultations. The following key decisions design an appointment schedule [14]: – – – – – – –

Number of patients per consultation session Patient overbooking Length of the appointment interval Number of patients per appointment slot Sequence of appointments Queue discipline in the waiting room Anticipation for unscheduled patients

• Staff scheduling: determine the staff shifts required and the number of employees to be scheduled on each shift.

2.3 Offline Operational Planning Offline operational planning relates to the short-term (1 day–3 months) decisionmaking related to the scheduling decisions made on an individual level, given the elective demand that is known at this point in time: • Patient assignment: schedule individual patients to appointment slots. This can be single appointments, combination appointments, or a series of appointments. • Staff assignment: schedule individual staff members to particular shifts.

2.4 Online Operational Planning Online operational planning comprises the control mechanisms required to deal with unplanned events, such as delay or urgent patients. The time horizon is 0–24 h. On this level, only decisions regarding unplanned events are made: • Patient reassignment: reschedule patients in order to handle unplanned events, such as delay, equipment failure, staff unavailability, or incoming acute patients. The new appointment can be on the same or on a different day. • Staff reassignment: reschedule staff in order to handle unplanned events, such as those defined above.

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3 Operations Research Models for Outpatient Clinic Optimization The outpatient clinic was one of the first hospital departments studied by Operations Research scholars [27, 31]. Within the literature considering outpatient clinic optimization, five main subjects can be distinguished: 1. 2. 3. 4. 5.

Patient flow analysis Appointment planning Panel sizing No-show modeling General organizational improvements

Patient flow analysis and appointment planning have been extensively studied and are therefore discussed in the next two paragraphs. The latter three subjects usually do not specifically use Operations Research models but are highly related to the development of Operations Research models for outpatient clinics: panel size and the no-show rate can be input parameters, while the general organization of the outpatient clinic defines patient flow and the appointment system used. Panel sizing is an important topic in mainly the USA and considers defining the population of patients who receive their care from a specific outpatient clinic [25]. Panel size, together with the number of appointments offered per day, determines patient access and waiting time to a high extent [32]. Patient no-shows influence clinic occupancy and financial results [13], and therefore it is important to be able to predict [4, 21] and influence [23] this behavior. General organizational improvements relate to the mapping of patient flows and other processes and the subsequent identification of possible measures to improve operations. Principles from the Lean methodology are often used in projects like these [5, 20]. Note that a thorough analysis of processes is usually required preceding the Operations Research modeling.

3.1 Patient Flow Analysis Simulation modeling is widely used to study patient flow problems at outpatient clinics. The famous Bailey-Welch rule [31] was derived using Monte Carlo simulation (see also Sect. 3.2). Since then, discrete event simulation has been the predominant approach (see, e.g., [9, 15, 33]). Setting up a simulation model requires detailed information on input distributions (e.g., of consultation length or patient arrivals), and building the model can be time-consuming. The possible level of detail, however, allows to model any system characteristic, such as opening hours, staff schedules, and appointment schedules. On the contrary, a queuing model requires fewer data and provides a robust insight in the underlying relationships of a system [34]. The choice to use either a simulation or queuing approach depends on

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the question at hand but also on the skills of the researcher. This probably explains why examples of queuing theory applications in outpatient clinics are scarce. A recent review on queuing theory applications in healthcare can be found in [17].

3.2 Appointment Planning Since the appointment system determines outpatient clinic organization to a high extent, appointment planning is the main topic studied in the Operations Research literature (see, e.g., [1, 3, 10, 14, 18, 22, 26] for reviews on this topic). Note that, despite all research done in the last two to three decades, the Bailey-Welch rule introduced in 1952 [31] is still used by many healthcare organizations to set up their appointment schedule. The Bailey-Welch rule implies that the first and second patient are both scheduled in the first slot, while subsequent patients are scheduled one at a time in the following slots. The Bailey-Welch rule is easy to implement and works well to prevent provider idle time during the first slots of the appointment schedule. However, it is very provider focused, since all but the first patient treated experience upon arrival already a waiting time of one slot. Examples demonstrating how to design advanced appointment schedules balancing several factors such as patient access time, waiting time, and provider idle time are given in this section. Following the planning complexity levels introduced in chapter “A Survey of Literature Reviews on Patient Planning and Scheduling in Healthcare” of this book, we discuss three types of patient planning in outpatient clinics: single appointment planning, multidisciplinary appointment planning, and care pathway planning.

3.2.1

Single Appointment Planning

In single appointment planning, appointments are planned for a single patient, one by one, on a single resource. Even though this seems relatively simple, the interference of an appointment with appointments for other patients on the same resource increases the complexity significantly. An example of this interdependence can be found in [16], which discusses appointment schedules for clinics where patients are seen both on walk-in and appointment basis. When designing appointment systems, there are two different allocation processes to consider: (1) the distribution of patient demand over available capacity in advance and (2) the appointment process during the day. OR models usually focus on the appointment process (see, e.g., [28] and [34]) or on an a combination of both the distribution of patient demand and related appointment schedules (see, e.g., [6], [16], and [32]). The allocation process determines the uncertainty modeled in patient arrivals and consultation length, plus the performance measures studied (Table 1). Note that the stochastic features of the appointment process influence clinic capacity as well.

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Table 1 Uncertainty in patient arrivals and consultation length, plus the performance measures studied for both allocation processes Allocation process Patient arrivals Consultation length Distribution of patient Stochastic Deterministic demand Appointment process Deterministic Stochastic (stochastic when considering walk-in or urgent patients and no shows)

3.2.2

Performance measures Patient access time; clinic utilization Patient waiting time; provider idle time; overtime; clinic utilization (on day and/or provider level)

Multidisciplinary Appointment Planning

In multidisciplinary planning, coordinated packages of care are planned for a single patient. This can either be a single activity on multiple resources or multiple activities on single or multiple resources [30]. Since the trajectories of care that need to be planned usually vary in both duration, and in the number, frequency, and type of appointments to be planned, focus is on the planning process and not so much on the performance of the appointment system during the day. See, e.g., [2].

3.2.3

Care Pathway Planning

Care pathway planning is in many ways similar to multidisciplinary planning. The main difference is that care pathways are designed to exactly specify the care trajectory for an entire group of patients, while multidisciplinary trajectories are specified for a single patient. Since multiple patients receive the exact same care trajectory, care pathways reduce the process variability and thus the planning complexity [29]. However, care pathways are notorious for complicating the planning for other patients, i.e., those who are not within the care pathway [35]. Instead of designing complicated planning systems, it is sometimes better to decrease access time for all patients. Related to this is the concept of advanced access [24], by “doing today’s work today.”

4 Lessons Learned from a Case Study In this section we discuss the practical aspects of a project, where we redesigned an anesthesia evaluation outpatient clinic. This project was carried out in Leiden University Medical Center (LUMC), a tertiary referral hospital in Leiden, the Netherlands. A detailed description of the project, the mathematical modeling involved, and the results can be found in [34]. Preoperative screening, where patients are reviewed from an anesthesiological perspective prior to elective surgery, is usually organized in an outpatient setting. In this case study, a project carried out at the pre-anesthesia evaluation clinic (PAC)

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of LUMC is described. When the study was conducted, 6,000 patients visited the PAC annually, of which about 70% were served on walk-in basis (i.e., without an appointment). These concerned mainly low-complex patients, while medium- to high-complex patients (10%) were usually given an appointment. The other 20% of patients that were seen on appointment basis initially visited the clinic as a walk-in patient, but at the moment of their arrival the clinic was overcrowded, and thus they were sent home with an appointment. The advantages of offering walk-in service are a higher level of accessibility and freedom for the patient to choose the moment of the hospital visit. Disadvantages are variability in demand and usually low utilization rates and long waiting times [16]. PAC management insisted on maintaining the walk-in option but was aware that from an organizational point of view a couple of improvements were necessary, namely, (1) reduce patient waiting time; (2) reduce the frequency and intensity of overcrowding (which has a direct relationship with variability in patient demand); and (3) reduce the time between the PAC consultation and the anesthesiologist’s approval for surgery. At the start of the project, a working group was set up with representatives from all employee groups working at the PAC (anesthesiologists, nurses, clinic assistants, and the secretary). The working group was supported by a mathematical analyst and chaired by the clinic director (an anesthesiologist). The initial design and its bottlenecks were discussed in several sessions by the working group. To eliminate bottlenecks, the working group commonly developed four alternative clinic designs which were evaluated using a multi-class open queuing network model. The results of the evaluation of the designs were extensively discussed in the working group. If necessary, it was possible to implement small changes in the model (like parameter values), such that changes proposed by the working group during the project meetings could be evaluated directly. The working group unanimously decided to implement a new clinic design, for which the queuing model predicted this would result to better overall performance of the clinic. After implementation, actual measured times of total patient length of stay before and after the intervention were compared. While the total length of stay did not significantly change, an unexpected increase in the number of patient visits of 16% was reported. It can be concluded that a considerable growth in patient inflow did not result in worsened performance. Looking back, there are a couple of lessons we learned from this case study which we would like to share. Setting up the working group and using a (mathematical) model, which could be used for instant evaluation of different alternative clinic designs and parameter/variable settings, were very useful. The use of the model decreased the subjectivity in the discussions and led the team away from pinpointing to specific individuals. Instead, employee roles and challenges associated with the setup of these roles were discussed, which was very valuable. Also, the involvement of the mathematical analyst, an unbiased individual without any prior relationship with the problem or the team, was very useful and improved the quality of the working group discussions. The implementation of the clinic redesign was quite smooth, probably because its possible (positive and negative!) effects were thoroughly analyzed and discussed. There were also a couple of points for improvement. The KPIs which were set by management were clear, but targets were missing on the desired reduction in waiting

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time, overcrowding, and time between the PAC consultation and anesthesiologist’s approval. This made it difficult to evaluate the project’s results afterward. When would it be good enough? Also, the implemented organizational change (i.e., the redesign) was not locked in: it was too easy to return to the initial design, which happened about 2 years later when a minor change in the workflow was effectuated, even though this resulted in prolonged waiting and access times for patients. Thus in the end, the intervention was not secured.

5 Current Challenges in Outpatient Clinic Management The world inside and outside of the hospital is changing rapidly. Overcrowding of waiting rooms is even more undesirable than it was before, making a well-designed appointment system an important asset in outpatient clinic management. One can think of several measures to decrease overcrowding, while patient access increases. Factors that influence the number of patients present in the waiting room are patient inflow, throughput, and outflow. For each category possible measures are given in Table 2. Table 2 Possible measures to influence patient access and clinic presence Category Inflow

Measure Distribution of consultations

Inflow

Digital consultations

Inflow

Home monitoring

In- & outflow Low-complex care by GP

Throughput

Additional waiting rooms

Throughput

Consultation preparation

Outflow

Shared decision-making

Description Distribute consultations evenly over the day and the days of the week Provide digital consultations for patients who require minimal physical examination or for whom the physical examination can be substituted by an (technological) innovation Provide home monitoring applications in order to provide immediate access when the patient’s condition deteriorates, while the patient does not need to visit the hospital when his/her condition remains stable Allocate low-complex treatment of patients to the GP in order to avoid hospital visits and medicalization of health status Use empty consultation rooms as waiting room to avoid overcrowding in the regular waiting room Prepare the consultation thoroughly by providing questionnaires, pre-diagnostics, and information about the consultation and treatment, in order to improve the consultation’s outcomes and avoid additional consultations Use shared decision-making to determine, together with the patient, what the best treatment options are. The overuse of treatment is likely to decrease [19], thus decreasing the demand for outpatient clinic capacity

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The increased usage of digital health applications requires different skill sets from hospital staff. While most of these initiatives are currently run from behind the counter in both in- and outpatient settings, a trend toward clustering in centers outside of hospitals is emerging. This shift in the organization of work, which was until very recently the domain of outpatient clinic staff, will have an enormous impact on the organization of outpatient clinics. Operations Research modeling can be of great value here.

References 1. Ahmadi-Javid, A., Jalali, Z., Klassen, K.J.: Outpatient appointment systems in healthcare: A review of optimization studies. Eur J of Oper Res 258, 3–34 (2017) 2. Braaksma, A., Kortbeek, N., Post, G.F. and Nollet, F.: Integral multidisciplinary rehabilitation treatment planning. Oper Res for Health Care 3(3), 145–159 (2014) 3. Cayirli, T. and Veral, E.: Outpatient scheduling in health care: a review of literature. Prod Oper Man 12(4), 519–549 (2003) 4. Chua, S.L. and Chow, W.L.: Development of predictive scoring model for risk stratification of no-show at a public hospital specialist outpatient clinic. Proc of Singapore Healthcare 28(2), 96–104 (2019) 5. Ciualla, T.A., Tatikonda, M.V., et al: Lean six sigma techniques to improve ophthalmology clinic efficiency. Retina 38(9), 1688–1698 (2018) 6. Creemers, S. and Lambrecht, M.: Queuing models for appointment-driven systems. Ann of Oper Res 178, 155–172 (2009) 7. Drupsteen, J., van der Vaart, T. and van Donk, D.P.: Integrative practices in hospitals and their impact on patient flow. Int J of Oper & Prod Man 33(7), 912–933 (2013) 8. Drupsteen, J., van der Vaart, T. and van Donk, D.P.: Operational antecedents of integrated patient planning in hospitals. Int J of Oper & Prod Man 36(8), 879–900 (2016) 9. Günal, M.M. and Pidd, M.: Discrete event simulation for performance modelling in health care: a review of the literature. J of Simulation 1, 42–51 (2010) 10. Gupta, D. and Denton, B.: Appointment scheduling in health care: challenges and opportunities. IIE Trans 40, 800–819 (2008) 11. van der Ham, A., van Merode, F., Ruwaard, D. and van Raak, A.: Identifying integration and differentiation in a hospital’s logistical system: a social network analysis of a case study. BMC Health Services Res preprint (2020) 12. Hans, E.W., van Houdenhoven, M. and Hulshof, P.J.H.: A framework for health care planning and control. In: Hall, R.W. (ed.) International Series in Operations Research & Management, pp. 303–320. Springer, New York (2012) 13. Huang, Z., Ashraf, M., Gordish-Dressman, H. and Mudd, P.: The financial impact of clinic noshow rates in an academic pediatric otolaryngology practice. Am J of Otolaryngology 28(2), 127–129 (2017) 14. Hulshof, P.J.H., Kortbeek, N., Boucherie, R.J., Hans, E.W., Bakker, P.J.M.: Taxonomic classification of planning decisions in health care: a structured review of the state of the art in OR/MS. Health Syst 1, 129–175 (2012) 15. Jun, J.B., Jacobson, S.H. and Swisher, J.R.: Application of discrete-event simulation in health care clinics: a survey. J of the Oper Res Soc 50, 109–123 (1999) 16. Kortbeek, N., Zonderland, M.E., Braaksma, A., Vliegen, I.M.H., Boucherie, R.J., Litvak, N., and Hans, E.W.: Designing cyclic appointment schedules for outpatient clinic with scheduled and unscheduled patient arrivals. Perf Eval 80, 5–26 (2014)

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17. Lakshmi, C. and Sivakumar, A.I.: Application of queuing theory in healthcare: a literature review. Oper Res for Health Care 2, 25–39 (2013) 18. Leeftink, A.G., Bikker, I.A., Vliegen, I.M.H, and Boucherie, R.J.: Mutli-disciplinary planning in health care: a review. Health Syst 1–24 (2018) 19. Légaré, F. and Thompson-Leduc, P.: Twelve myths about shared decision making. Pat Educ and Counseling 96(3), 281–286 (2014) 20. van Leijen-Zeelenberg, J.E., Brunings, J.W. et al: Using lean thinking at an otorhinolaryngology outpatient clinic to improve quality of care. The Laryngoscope 126(4), 839–846 (2016) 21. Liu, J., Xie, J., Yang, K.K. and Zheng, Z.: Effects of rescheduling on patient no-show behavior in outpatient clinics. Manufacturing & Service Operations Management 21(4), 780–797 (2019) 22. Marynissen, J. and Demeulemeester, E.: Literature review on multi-appointment scheduling problems in hospitals. Eur J Oper Res 272(2), 407–419 (2019) 23. Mohamed, K., Mustafa, A., Tahtamouni, S. et al: A quality improvement project to reduce the ‘no show’rate in a Paediatric Neurology clinic. BMJ Open Quality 5(1) (2016) 24. Murray, M., and Berwick, D.M.: Advanced access: reducing waiting and delays in primary care. J Am Med Assoc 289(8), 1035–1040 (2003) 25. Murray, M., Davies, M., and Boushon, B.: Panel size: How many patients can one doctor manage? Familiy Practice Manag 14(4), 44–51 (2007) 26. Palmer, R., Fulop, N.J., Utley, M.: A systematic literature review of operational research methods for modelling patient flow and outcomes within community healthcare and other settings. Health Syst 29–50 (2018) 27. Rising, E.J., Baron, R. and Averill, B.: A systems analysis of a university-health-service outpatient clinic. Oper Res 21(5), 1030–1047 (1973) 28. Sickinger, S. and Kolisch, R.: The performance of a generalized Bailey-Welch rule for outpatient appointment scheduling under inpatient and emergency demand. Health Care Manag Sci 12, 408–419 (2009) 29. Vanberkel, P.T., Boucherie, R.J., Hans, E.W., Hurink, J.L. and Litvak, N.: A survey of health care models that encompass multiple departments. Int J of Health Man and Information 1:1, 37–69 (2010) 30. van de Vrugt, N.M., Braaksma, A. and Boucherie, R.J.: The state of the art in online appointment scheduling. In: van de Vrugt, N.M.: Efficient healthcare logistics with a human touch. Ph.D. thesis, University of Twente (2016) 31. Welch, J.D., and Bailey, N.T.: Appointment systems in hospital outpatient departments. The Lancet 1(6718), 1105–1108 (1952) 32. Zacharias, C. and Armony, M.: Joint panel sizing and appointment scheduling in outpatient care. Manag Sci 63(11), 3978–3997 (2016) 33. Zhang, X.: Application of discrete event simulation in health care: a systematic review. BMC Health Services Research 18:687 (2018) 34. Zonderland, M.E., Boer, F., Boucherie, R.J., de Roode, A., and van Kleef, J.W.: Redesign of a university hospital preanesthesia evaluation clinic using a queuing theory approach. Anesth Analg 109(5), 1612–1621 (2009) 35. Zonderland, M.E., Boucherie. R.J. and Al Hanbali, A.: Appointments in care pathways: the Geox /D/1 queue with slot reservations. Queueuing Syst 79(1), 37–51 (2015)

Robust Surgery Scheduling: A Model-Based Overview Maarten Otten, Jasper Bos, Aleida Braaksma, and Richard J. Boucherie

Abstract In this chapter we give a model-based overview of robust surgery scheduling literature. A robust schedule maintains to perform well in case of disturbances affecting the schedule. We distinguish three types of disturbances that affect the surgery schedule. First, internal disturbances, such as variations in surgical time. Second, external disturbances, such as non-elective surgeries. Third, disturbances due to artificial variability, such as unavailable operating rooms. For each of these disturbances, we provide an overview of models, described in literature, which reduce the effect of the disturbance on the schedule by making it robust. Furthermore, we identify relevant open problems.

1 Introduction Performing surgeries is one of the key tasks of a hospital. A large share of the patients is treated by a surgeon during their hospital visit. Operating rooms (ORs) are among the most expensive resources of the hospital. Furthermore, hospitals are increasingly struggling to attract sufficient qualified OR personnel. Utilizing the ORs efficiently is therefore of key importance. In an ideal world, surgeries would be scheduled such that the ORs are perfectly utilized, but due to, among others, stochasticity, this is impossible. For various reasons surgeries can be advanced, delayed, or canceled, such that the daily practice at the ORs rarely matches the schedule made beforehand. Deviation from the original schedule can have many causes. In this chapter we divide these into three types. First, we consider internal causes due to natural variability of the processes at the OR, such as variability of OR setup times, of OR cleaning times, or of the surgery durations. These mostly affect the schedule

M. Otten () · J. Bos · A. Braaksma · R. J. Boucherie Center for Healthcare Operations Improvement and Research, University of Twente, Enschede, The Netherlands e-mail: [email protected] © Springer Nature Switzerland AG 2021 M. E. Zonderland et al. (eds.), Handbook of Healthcare Logistics, International Series in Operations Research & Management Science 302, https://doi.org/10.1007/978-3-030-60212-3_4

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with elective surgeries. Second, we consider external causes, such as no-shows, illness of patients, or non-elective surgeries. Since only elective surgeries can be scheduled beforehand, non-elective patients that arrive during the day will be scheduled within the existing schedule, possibly causing later surgeries to be delayed or even canceled. Similarly, elective surgeries that are canceled due to unavailability of the patient will cause either subsequent surgeries to be advanced or a gap in the OR schedule. Third, we consider causes due to artificial variability, i.e., inefficient and counteracting processes at the ORs, such as unavailability of clean ORs, well equipped ORs, or OR personnel. The distinction between natural and artificial variability is convenient in our model-based approach of the literature. Natural variability typically cannot be eliminated, and the approach for this type of disturbance will therefore be anticipatory, whereas for artificial variability the approach will be to optimize the processes. The literature on OR scheduling in general is abundant. Hulshof et al. [9] and Cardoen et al. [3] both provide extensive reviews of OR scheduling literature. Hans and Vanberkel [8] present a number of approaches to OR planning and scheduling on a strategic, tactical, and operational level. However, the amount of papers where variability is explicitly taken into account is limited. Van Riet and Demeulemeester [23] is a recent and up to date review of the literature where non-elective patients are taken into account. Ferrand et al. [5] also provide a review of literature where both elective and non-elective surgeries are considered. In this chapter we give an overview of several mathematical models described in literature to obtain a robust schedule, i.e., a (near) optimal schedule that under some deviation will still perform reasonably well. Our contribution with this chapter is threefold. First, we provide a model-based overview of the literature that takes the various sources of disturbances described above into account. For relevant combinations of one or more sources of disturbance, we state an objective and describe the models that are used in literature. Second, we not only describe the models but also state them in a concise way. Third, we give an overview of open problems. In Sect. 2 we discuss how internal disturbances caused by natural variation can be taken into account when scheduling the elective surgeries. In Sect. 3 we discuss the objectives and related models when external disturbances are taken into account. In Sect. 4 we discuss models described in literature that take artificial variation into account to optimize the processes at the OR. In Sect. 5 we discuss several models where combinations of the disturbances are considered. Finally in Sect. 6 we present an overview of the models we considered and draw conclusions based on our findings.

2 Internal Variability Most patients that are treated at the surgical department are scheduled beforehand. In addition to these elective patients, there are non-elective patients that arrive during

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the day and need treatment within a short time span. In this section we focus on scheduling elective surgeries. The duration of similar surgical procedures can fluctuate significantly due to various reasons. Some of these differences, such as variations between surgeons or between types of patients, can be accounted for. Other effects, like deviations from the original surgical procedure, cannot. In order to obtain a robust schedule, it is therefore important to take these variations into account when scheduling the elective surgeries. In this section we discuss several models for different objectives when the variability of surgery durations is taken into account.

2.1 Overtime Scheduling elective surgeries based on their expected duration only will generally lead to a high OR utilization. However, the probability of overtime (surgeries performed outside scheduled hours) and the number of surgeries canceled at the end of the day will increase with increasing variability of the surgery duration. Taking into account this variability is a trade-off between maximizing the OR utilization and minimizing the probability of OR overtime. This section describes several approaches and models that aim to assign surgeries to ORs such that the expected utilization is maximized while at the same time minimizing the probability of overtime. Scheduling surgeries can be divided into two related subproblems: the advance scheduling and the allocation scheduling problem. In the advance scheduling problem, surgeries are assigned to a date and an OR block. In the allocation problem, the sequence of the surgeries that are assigned to a certain date is determined. Hans and Vanberkel [8] use the term robust surgery loading for the variant of the advance scheduling problem they consider. Elective surgeries are scheduled in advance over a discrete planning horizon, t = 1, . . . , T . On each day t, each specialty z has a number of ORs at its disposal, denoted by the set Kzt . An element k of the set Kzt is termed an OR-day. The set of surgeries V must be allocated to the OR-days. Each element i ∈ V is a surgery of a certain specialty, and its duration has mean μi and variance σi2 . The set of surgeries of specialty z is denoted by Vz . To be determined are Vztj , the set of surgeries assigned by specialty z to OR j , at day t. Together with the surgeries, there is slack capacity assigned to each OR-day to accommodate for the effect of randomness in the surgery times. The amount of slack capacity assigned to OR-day Kzt is: δztj = β(p)



σi2 ,

(1)

i∈Vztj

which is the standard deviation of the sum of the surgery durations scaled with β(p) ≥ 0, a safety margin determined by the maximal tolerated probability of

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overtime p. For example, if we assume, by the central limit theorem, that the sum of the surgery durations is normally distributed, a probability p = 0.69 of having no overtime corresponds to β(p) = 0.50. Note that this approach assumes that the surgery durations are independent. Hans and Vanberkel [8] translate (1) into constraints on the number of surgeries that can be scheduled: 

μi + δztj ≤ ctj + Oztj

∀z, j ∈ Kzt , t,

(2)

i∈Vztj

where ctk is the capacity needed per OR-day and Oztk the maximum overtime for each specialty per OR-day. As objectives [8] use (in order of importance): minimizing the total overtime, maximizing the number of free OR-days, and maximizing the total free capacity. They use the base schedule, i.e., the schedule that the hospital scheduler makes, and propose several heuristics to improve the base schedule. They propose a list scheduling approach where surgeries are sorted in decreasing order of their expected duration. According to the order of the list, surgeries are added to the schedule. A surgery is added to the schedule when the constraints (2) are not violated; otherwise the next surgery on the list is added to the schedule. They furthermore propose local search methods to improve this base schedule. They find that in optimized schedules the portfolio effect (surgeries with similar variation characteristics are clustered on the same OR-day) plays an important role in reducing the amount of necessary slack capacity. Denton et al. [4], Landa et al. [13], and Molina-Pariente et al. [16] model the advance scheduling problem as a type of stochastic programming problem. This stochastic programming problem, a two-stage simple recourse model, divides the decision variables in two groups, those that are to be determined here and now (assigning surgeries to OR-days) and those that can be adjusted, at a cost, later on (amount of overtime at an OR). We state and discuss the problem as formulated by [4]. Suppose there are n surgeries to be scheduled at m ORs, where m itself is part of the decision. Let Di (ω) denote the duration of surgery i, a random variable where ω denotes a possible realization of the durations, and T the planned session length at each OR. g f denotes the fixed costs of opening an OR and g v the variable costs per time unit of keeping an OR open past time T . Let Xj be a binary decision variable whether OR j is opened, Yij whether surgery i is assigned to OR j and Oj the amount of overtime of OR j . Then the assignment of surgeries to ORs can be formulated as the following two-stage simple recourse problem: ⎧ ⎫ m  ⎨

⎬  Z ∗ = min g f Xj + E g v Oj (ω) , ⎩ ⎭

(3)

j =1

s.t. Yij ≤ Xj ,

∀i, j,

(4)

Robust Surgery Scheduling: A Model-Based Overview m 

Yij = 1,

41

∀i,

(5)

∀j, ω,

(6)

∀i, j,

(7)

∀j, ω.

(8)

j =1 n 

Di (ω)Yij − Oj (ω) ≤ T Xj ,

i=1

Yij , Xj ∈ {0, 1}, Oj (ω) ≥ 0,

The decisions whether an OR is opened and to which OR a surgery is assigned, i.e., the binary decision variables Xj and Yij , are first-stage decisions. The variable for overtime at the OR, Oj , is a recourse variable and is determined once the realization of the surgery durations, ω, is known. This is a two-stage simple recourse model which can be easily solved [1]. In addition to a stochastic programming approach, [4] add a robust programming approach for which they use a polyhedral uncertainty set. The stochastic programming approach is very suitable, provided there is enough statistical information available for the possible outcome scenarios of the realized surgery durations Di (ω). If there is only information about the possible outcomes of Di (ω), the robust programming approach is more suitable.

2.2 Deviation from the Schedule For the admission process to run smoothly, patients are asked to come to the hospital some time before their surgery is scheduled to start. In order to prevent that patients are not yet available for surgery or that their surgery is delayed for hours, it is important that the actual starting time does not deviate too much from the scheduled time. In this section we present the earliness/tardiness (E/T) model that aims to reduce the variability of the start times of surgeries. Suppose there are n surgeries to be scheduled in m ORs. Let nj denote the number of surgeries in OR j , and let Π = [π1 , π2 , . . . , πm ] denote the schedule for the m ORs for which πj = (πj,1 , . . . , πj,n ), ij 1, 2, . . . , m, is the sequence in j j which the surgeries are performed in OR j . Further, let s πj = (s1 . . . sn ) denote the scheduled start times of the surgeries in OR i and Di,j the duration of surgery i in OR j , a random variable. The realized start time of surgery i in OR j , also a random variable, is denoted by Sji =

i−1  k=1

Dπj k ,j .

(9)

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The earliness of surgery i at OR j is defined as Eji (sji ) = (sji −Sji )+ and its tardiness as Tji (sji ) = (Sji − sji )+ , where (x)+ = max{x, 0}. The unit earliness and tardiness costs parameters of surgery i in OR j are given by ji and τji , respectively. The E/T costs of surgery i in OR j are denoted by  Aij (sji ) = E ji Eji (sji ) + τji Tji (sji ) ,

(10)

for i = 1, . . . , n, j = 1, . . . m. The total expected E/T costs for a schedule Π with start times s Π is denoted by ET (Π, s Π ). The objective is to select a schedule Π and corresponding start times s Π in such a way that the total expected E/T costs are minimized. If the surgeries are preassigned to the ORs, it suffices to optimize the schedule for each OR separately. The single OR E/T problem is defined as min

(SET)

π,s

n   ET (π, s) = min E i Ei (si ) + τi Ti (si ) . π,s

(11)

i=1

When assigning the surgeries to an OR is also part of the decision, the joint schedule has to be optimized. The multiple OR E/T problem is defined as (MET)

min

Π,s Π

ET (Π, s Π ) = min

Π,s Π

n m    E ji Eji (sji ) + τji Tji (sji ) .

(12)

j =1 i=1

The problem is termed symmetric if the cost parameters are equal for each surgery, i.e., i =  and τi = τ , for i = 1, . . . , n. Weiss [25] describes an extension of this model in which the earliness costs are infinite, i.e., advancing a surgery is impossible. This variant of the problem is significantly harder because starting times of subsequent surgeries are not linked together in a straightforward way anymore due to the fact that idle time may occur between surgeries. Gupta [7] shows that for this variant of the E/T problem with one OR and two surgeries, it is optimal to schedule the surgeries in increasing order of the variances of the duration, the so-called Smallest Variance First (SVF) rule. For more than two surgeries and for multiple ORs, this remains an open problem. Whenever idle time is not allowed, the optimal scheduled start time of a surgery is fully determined by the probability distributions of the preceding surgeries. In this case, given a schedule Π , the optimal scheduled start time of surgery i in OR j with actual start time Sji is si∗

−1

=G



τi i + τi

 ,

(13)

where G is the distribution function of the realized start time, Sji . We now give a concise overview of the results described in literature and the open problems for the various versions of the E/T problem.

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For the case when the surgeries are preassigned to the ORs and when the earliness and tardiness costs are equal for each surgery, i.e., the symmetric single OR E/T problem, [6] obtain an optimal schedule under the assumption that the mean and variance of the surgery duration S are finite. In this case it is optimal to schedule the surgeries according to the SVF rule. Otten and Boucherie [17] consider an optimal schedule for the multiple OR version of the E/T problem under some technical assumptions on the distribution of the surgery duration, which they show to hold for at least the normal distribution. For the multiple OR E/T problem, they extend the SVF rule, i.e., they show that it is optimal to not only apply the SVF rule to the local sequence of surgeries at each OR but also to the global sequence of all surgeries at all ORs. In conclusion, we see that the SVF rule is optimal for several variants of the E/T problem when idle time is not allowed. For the nonsymmetric and multiple OR variants, it remains an open problem if this result holds for arbitrary distributions of the surgery duration. When idle time is allowed, very few results are known, but [7] and [6] show that the SVF rule still performs reasonably well. Furthermore, [17] show that an optimal schedule for the multiple OR variant is not unique. This leaves room for additional optimization of the schedule.

3 External Variability An important external cause of disturbance to the OR schedule is the arrival of non-elective patients during the day. In this section we discuss various relevant objectives when anticipating this type of disturbance, both from the perspective of OR utilization and from the perspective of the non-elective patients.

3.1 Overtime Ideally, the utilization of the ORs is high. However, as non-elective patients arrive during the day, a high OR utilization will inevitability lead to a high probability of overtime or cancellation, which is undesirable. When the objective is to maximize OR utilization but avoid overtime, the trade-off will be between using OR capacity for elective patients and reserving slack OR capacity for possible non-elective patients.

3.1.1

Non-elective Surgery Policy

The main decision is how non-elective surgeries are accommodated. We distinguish three possible policies [2, 23]: (1) a dedicated policy, where there are one or more dedicated ORs for non-elective patients; (2) a flexible policy, where slack capacity is

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fragmented and added to each of the regular ORs, and non-electives are then fitted into the elective schedule; and (3) a hybrid policy, where policies (1) and (2) are combined. Literature shows that it highly depends on the hospital’s characteristics which policy is optimal. For instance, [26] find a decrease in overtime using the flexible policy, while [23] report an increase in overtime using the same policy. Van Veen-Berkx et al. [24] find that the flexible policy is optimal by simulation while after implementation the opposite was the case. Borgman [2] carries out a simulation study in which he tests the different policies for many different hospital characteristics. His main conclusion is that the dedicated policy works best for small hospitals with at most eight ORs and that the flexible policy is best for larger hospitals. If the dedicated policy is adopted, the remaining question is how many ORs should be dedicated to non-elective patients. There is no research that addresses this question because this could easily be determined based on the expected capacity needed for non-elective surgeries. In the remainder of this section, we discuss some of the problems that hospitals face anticipating arriving non-electives during the day, when the flexible policy is adopted.

3.1.2

Required Capacity

Van Houdenhoven et al. [22] develop a model to determine the amount of required slack capacity for emergency patients. They assume that the elective surgeries are planned in blocks per specialty. Suppose that the expected number of emergency surgeries of specialty z is nel z at an arbitrary OR-day, each having a mean duration el μel elective surgeries in z and standard deviation σz . The expected duration of the  el el el 2 this block is therefore nz μz with a standard deviation of nel z (σz ) . The total required capacity to perform all the expected non-elective surgeries of specialty z is  el el 2 cz (pz ) = nel μ + β(p ) nel z z z z (σz ) ,

(14)

where pz is the allowed probability of overtime and β(pz ) is a function that gives a factor that yields a probability of overtime of pz , similar to the model by [8], described in Sect. 2.1. The function β(pz ) depends on the distribution of the sum of surgery durations. Van Houdenhoven et al. [22] assume that this sum is normally distributed, which implies that β(pz ) = Φ −1 (pz ), the standard-normal quantile function. For each block of surgeries of specialty z, an additional amount of capacity cz (pz ) is needed in order to have a probability of overtime of pz . For arbitrary distributions of the surgeries, this approach still works, albeit that the function β(pz ) will have to be approximated. Zonderland et al. [27] develop a queuing model to determine, on a strategic level, how much slack capacity needs to be reserved for semi-urgent patients, patients that

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do not need surgery immediately but within a few days. On a tactical level they develop a decision support tool, based on Markov decision theory, for scheduling these patients within the coming days.

3.1.3

Scheduling Elective Surgeries Anticipating Emergencies

Lamiri et al. [11] and Min and Yih [15] use a stochastic programming approach to model the scheduling of elective surgeries anticipating non-elective surgeries, under the flexible policy. We state and discuss the model as formulated by [11]. Suppose there are nel elective surgeries to be planned over a horizon of T days. To be determined is to which day each of the elective surgeries is assigned. Let ctel be the OR capacity in units of time available for elective surgeries on day t = 1, . . . , T and gto the cost of overtime per time unit. During the day emergency patients arrive and are to be treated at the same day. Ctem is a random variable having probability density function fCtem (x) and denotes the capacity needed for emergency patients on day t = 1, . . . , T . The elective surgeries have a duration di , a release date ri , and surgery costs git associated with them, which are assumed to be deterministic. Let Yit denote the binary decision variable whether elective surgery i is assigned to day t. The goal is to assign the elective patients such that the total surgery costs and the overtime costs are minimized. This is formulated in the following optimization problem: +1 n T  el

min

J (Y ) =

min

(git Yit ) +

i=1 t=ri

T 

 gto E

Ctem +



di Yit − ctel

+  ,

i

t=1

(15) T +1

s.t.

Yit = 1,

∀i,

(16)

∀i, t.

(17)

t=ri

Yit ∈ {0, 1},

Lamiri et al. [11] show that the solution can be analytically obtained by numerical integration: +1 n T  el

J (Y ) =

i=1 t=ri

(git Yit ) +

T  t=1

 gto

∞

qt

(z − qt )fCtem (z)dz,

(18)

 where qt = Ctel − i di Yit is the remaining regular capacity at day t. This approach is only tractable for small instances. For larger instances they propose a Monte Carlo approach:

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J (Y ) ≈ JL (Y ) =

(git Yit ) +

i=1 t=ri

T 

1 L L

gto

t=1

 +   em,l el Ct + di Yit − ct ,

l=1

i

(19) where Ctem,1 , . . . , Ctem,L are L independent samples generated for the random variable Ctem .

3.2 Waiting Time of Emergency Patients Emergency patients arriving at the OR need to be treated within a short time span. However, ongoing surgeries cannot be interrupted, so when the hospital does not have dedicated emergency ORs or these ORs are occupied, an emergency patient has to wait until the first of the ongoing surgeries is completed. It is therefore a reasonable idea to schedule the elective surgeries while taking into account the waiting time of emergency patients that will arrive during the day. In this section we discuss two models for the case when emergency patients are to be accommodated into the elective surgery schedule and the objective concerns the waiting time of these patients.

3.2.1

Maximum Waiting Time

Emergency patients of the most urgent category require surgery as soon as possible. In order to obtain a schedule that satisfies this requirement, the elective surgeries have to be sequenced in such a way that the maximum time until an emergency patient can be inserted into the schedule is minimized. Van Der Lans et al. [20] and Van Essen et al. [21] described this problem as a scheduling problem. In this section we state and discuss this model as described by [21]. Suppose there are n elective surgeries to be scheduled in m ORs. The surgeries are preassigned to an OR, Yi denotes the OR surgery i is assigned to and nj denotes the number of surgeries in OR j . Assume that at time t = 0 all elective patients are available for surgery and that preemption of surgeries and idle time for the ORs are not allowed. To be decided is the order in which surgeries are performed at each OR. Let Π = [π1 , . . . , πm ] denote the schedule, where πj = [πj,1 , . . . , πj,nj ] is the order of surgeries in OR j . The duration of surgery i is denoted by di and its completion time by πYi ,i

FiΠ

=

 k=1

dπYi ,k .

(20)

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47

A break-in moment (BIM) is any point in time when either a surgery starts or finishes. The set of BIMs is defined as   BΠ = t ∈ R+ |t = FiΠ , i = 1, . . . , n .

(21)

A break-in interval (BII) is the time between two consecutive BIMs. Since preemption and idling are not allowed, these are precisely the moments at which emergency patients can be fitted into the schedule, hence the name. See Fig. 1 for a visualization. The set of BIIs is defined as IΠ = {(a, b) ⊂ R+ |a, b ∈ BΠ , a ≤ b, c ∈ BΠ s.t.a ≤ c ≤ b} .

(22)

The objective is to minimize the maximum waiting time of an emergency patient. This is equivalent with minimizing the maximum BII over all possible schedules. The BIM problem is defined as  (BIM)

min Π

 max (|I |) .

I ∈IΠ

Note that BIIs are the intervals between BIMs; therefore, the set of BIIs forms a partition of the time interval that the ORs are in operation. The total time the ORs are in operation is not influenced by the sequencing of the surgeries because the surgeries are preassigned to the ORs and idle time is not allowed. This implies that if a schedule has a lot of relatively short BIIs, it will most likely have a long maximum BII. Therefore, intuitively we see that the goal of the BIM problem is to spread the BIMs as evenly as possible over the interval the ORs are in operation. Ideally, the BIMs are spread equidistantly over the interval, and all BIIs would have length   m · maxi FiΠ λ= . n

(23)

  The value λ is a lower bound for the optimal BIM value. Note that maxi FiΠ is the time that surgeries are performed at the OR(s), given a schedule Π . Van Essen et al. [21] show that the BIM problem is strongly NP-hard and therefore intractable for realistic size instances. In order to obtain reasonably good schedules for this problem fast enough for practical purposes, they describe several heuristics. First, they propose to sort the surgeries in increasing order of duration and add the first surgery to the schedule such that the new resulting BII is close enough to λ, where close enough depends on the scheduler’s preferences. Second, they propose to add the surgery for which the resulting BII is closest to λ to the schedule first. Third, they propose the previous heuristic with the addition to update the value of λ every time a surgery is added to the schedule. Based on simulation they conclude that the third heuristic performs best and results in a reduction of

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Fig. 1 Break-in moments and break-in intervals

about 10% of the maximum BII length compared to the base schedule provided by the hospital. The BIM model provides an elegant framework to reduce the maximum waiting time of emergency patients. However, it also has its shortcomings. First, the BIM model aims to minimize the maximum waiting time of an emergency patient, but it does not take into account that the schedule will change when an emergency patient is accommodated into the schedule. In a sense the BIM model only optimizes for the first arriving emergency patient. Second, the BIM model only takes the maximum waiting time into account. In Sect. 5 we discuss a way to address the first shortcoming, and in Sect. 3.2.2 we discuss a slightly modified problem that addresses the second shortcoming.

3.2.2

Average Waiting Time

The BIM problem described in the previous section provides an upper bound on the waiting time of arriving emergency patients, but it does not provide insight in the expected waiting time. The two schedules in Fig. 2 both have a maximum BII of length 2 and are therefore, according to the BIM problem, equivalent. However, the expected waiting time of a patient arriving at an arbitrary time during the day is 0.8 in the left schedule but only 0.6 in the right schedule. This example illustrates that only taking the largest BII into account may result in unnecessary waiting time for emergency patients. As we noted in Sect. 3.2.1, for the BIM problem it is optimal if all BIIs have length λ (see equation (23)). This also holds if we consider the average waiting time. The intuition for this is the following: suppose there is a large BII, then, because the sum of the lengths of the BIIs is constant, at least one of the other BIIs will be small. The probability that an emergency patient, who arrives at an arbitrary moment during the day, arrives during the large BII is higher than the probability of arriving during the small BII because this probability is proportional to the length of the interval. So we have that the expected waiting time for an emergency patient arriving at an arbitrary moment during the day increases if we deviate from a schedule with equidistant BIMs. The problem to find the schedule with the minimum expected average waiting time for emergency patients is therefore

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Fig. 2 Two schedules, both with a maximum BII of 2. The expected waiting time is 0.8 for the left schedule and 0.6 for the right schedule

equivalent to the problem of finding a schedule with a minimal sum of squared BII lengths. The minimal squared BII (MSB) problem is defined as

(MSB)

min Π

⎧ ⎨  ⎩

(a,b)∈IΠ

(b − a)2

⎫ ⎬ ⎭

.

Note that, although the BIM and MSB problems are quite similar and both try to approach the equidistant distribution of the BIMs, they are not equivalent. The reason for this is that the BIM problem focuses only on the maximum BII, whereas the MSB problem takes all BIIs into account. Unlike the BIM problem, the MSB problem cannot be formulated as an integer linear program (ILP), since the objective function is quadratic and not linear. This problem is not further described in literature. Therefore, further analysis of this problem and finding useful heuristics remains an open problem.

4 Artificial Variability A surgery can only be conducted as planned if all necessary resources are available at the right time. For example, if the surgical team or the anesthesiologist is unavailable, the surgery will be delayed or canceled. Also when the OR is not clean yet, the surgery will be delayed until the cleaning staff has finished. In this section we discuss several models that consider the variability in the OR schedule due to unavailability of necessary resources.

4.1 Blocking Time Usually, several resources necessary for surgery are shared between ORs. For example, an anesthesiologist that provides care at several ORs or an OR-cleaning

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team. If during the day the demand for such a resource is higher than the available capacity, e.g., if there are two cleaning teams but three ORs finish at the same time, some ORs will be blocked for some time, causing surgeries to be delayed. With this in mind, it is reasonable to schedule the surgeries in such a way that not too many surgeries either start or end at the same time. This problem is first described by [19]. In their paper they describe the Scheduling with Safety Distances (SSD) problem in which surgeries are scheduled such that the minimal time between start times is at least v. Below we state and discuss this problem. We use the notation and definitions stated in Sect. 3.2.1. Suppose there are n surgeries to be scheduled at m ORs. Before a surgery can start, a certain procedure has to be performed, e.g., anesthesiology. For this procedure there is only one available resource. The aim is to schedule the surgeries such that there is no blocking of the ORs due to unavailable resources. The Scheduling with Safety Distances (SSD) problem is to find an optimal schedule Π ∗ such that (SSD)

Π ∗ = arg min (|I |) ≥ v I ∈IΠ

for a certain threshold value v, where IΠ is the set of all BIIs in schedule Π (see equation (22)). Spieksma et al. [19] prove that the SSD problem is NP-complete. They identify some special cases for which the problem is solvable in polynomial time, but these are very stylized cases that are not useful in practice. The variant of the SSD problem described by [19] assumes that there is only one resource available that is necessary for each surgery (e.g., if there is only one cleaning team at the ORs). In practice, there are often multiple, albeit limited, resources. We can adjust the model for this situation by instead of requiring that all start times are at least v apart, requiring that at most n start times are less than v apart, when there are n available resources. If the threshold value v is not known or it is variable, we can adjust the problem such that it aims to schedule the surgery start times not at least v apart but as far apart as possible. This modified SSD problem has a lot of resemblance with the BIM problem. We use the name Scheduling with Maximum Safety Distances (SMSD) for this problem, and it is defined as  (SMSD)

max Π

 min (|I |) .

I ∈IΠ

Where the BIM problem minimizes the maximum BII and the SMSD problem maximizes the minimum BII. By the same reasoning as for the BIM and MSB problem, we see that all three problems in fact aim to spread the BIMs as evenly as possible over the day. However, the BIM, MSB, and SMSD problems are not equivalent. In Sect. 5 we will discuss the relation between these problems.

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5 Multiple Sources of Disturbance In the previous sections we discussed models that consider one type of disturbance to the OR schedule in isolation. However, in practice most, if not all, of the disturbances described will be of influence simultaneously. Furthermore, most of the objectives we considered are influenced by more than one source of disturbance. In this section we therefore discuss models that take multiple sources of disturbances into account.

5.1 Overtime In Sects. 2.1 and 3.1 we discussed models that minimize overtime anticipating emergency arrivals and variable surgery duration, respectively. All these models share the same objective, i.e., reducing the probability of overtime. Moreover, they use similar techniques, i.e., estimating the mean capacity needed and, based on the variance, additional slack capacity to reduce the probability of overtime. It is therefore reasonable to incorporate them in one model. Van Houdenhoven et al. [22] extend their model to determine the additional capacity necessary for emergency patients; see Sect. 3.1 to incorporate the variability of the elective surgeries. For this, suppose that there are nel z elective surgeries of specialty z, each having mean with standard deviation σzel . Then the formula for the slack capacity duration μel z needed (14) becomes  em el 2 em em 2 δz (pz ) = nem μ + β(p ) nel z z z z (σz ) + nz (σz ) ,

(24)

where nem z is the number of expected emergency surgeries of specialty z with em 2 expected duration μem z and variance (σz ) . So for each specialty z there are el nz μs time units of planned capacity, and additionally an amount of δz (pz ) of slack capacity is added to the schedule. Like in Sect. 3.1, β(pz ) is a factor depending on the probability distributions of the surgeries and the maximal tolerated probability of overtime pz . The model proposed by [11] (see equations (15)–(17)) can be extended to incorporate the variability in the durations of the surgeries by modeling the duration di as a random variable like the required capacity for emergency surgeries Ctem . This, however, reduces the tractability of the model. Lamiri et al. [10] propose a stochastic programming approach in which both the durations of the elective surgeries and the demand for emergency surgeries are modeled as a random variable. Lamiri et al. [12] and Razmi et al. [18] describe a column generation approach to solve this model efficiently.

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5.2 Waiting Time Emergency Patients In Sects. 2.1 and 3.1 we discussed several ways how a scheduler can determine the amount of slack capacity to anticipate emergency patients or variation in the surgery duration. Usually, the slack capacity is scheduled at the end of the day to act as a buffer to prevent overtime. However, if we take other objectives into account, it may be better to schedule the slack capacity at other moments. In Sect. 3.2.2 we discussed the BIM model. By dividing the slack capacity around these BIMs, the time until emergency patients can go into surgery can be reduced. Furthermore, by introducing slack intervals in the schedule, emergency patients arriving during these intervals will have zero waiting time.

5.3 Deviation from the Schedule Analysis of the models on minimizing deviation from the schedule, discussed in Sect. 2.2, shows that the Smallest Variance First (SVF) rule is in many cases optimal, and otherwise it still yields very good results. The intuition for this is that since surgeries are scheduled consecutively, the variability of earlier surgeries will accumulate for the later surgeries. Therefore, by scheduling highly variable surgeries near the end of the day, the effect on the other surgeries will be limited. Again, as discussed above, the accumulating effect of the variability can be further reduced by adding slack capacity at the end of each surgery, based on the variation of the surgery duration. The BIM problem discussed in Sect. 3.2.2 and the SMSD problem discussed in Sect. 4.1 both aim to spread the start and end times of the surgeries evenly over the day. However, since the BIM problem only considers the maximum BII, it accepts schedules where some of the BIIs are small as long as the maximum BII remains the same. Similarly the SMSD problem can produce schedules that have large BIIs. We can avoid this by extending the problem slightly. For this we define the following two BII vectors: + bΠ =[b1+ , b2+ , . . . , bn+ ], where b1+ ≥ · · · ≥ bn+ and bk+ =|Ik |, Ik ∈ IΠ , k=1 . . . n, (25) − bΠ =[b1− , b2− , . . . , bn− ], where b1− ≤ · · · ≤ bn− and bk− =|Ik |, Ik ∈ IΠ , k=1 . . . n, (26) + is a vector where IΠ is the set of all BIIs (see equation (22)). The vector bΠ containing the lengths of the BIIs corresponding to schedule Π in descending − order, and bΠ is the same vector only in the reverse order. With this notation we can define the BIM problem as finding a schedule Π for which the corresponding + vector bΠ has a minimum first element. The extended BIM and SMSD problems are

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considering not only the first element of the b vector but all entries. A schedule Π ∗ is optimal with respect to the lexicographical break-in moments (LBIM) problem if + its corresponding BII vector bΠ ∗ is lexicographically minimal, i.e., + + bΠ ∗ ≤lex bΠ

∀Π.

(27)

A schedule Π ∗ is optimal with respect to the lexicographical Scheduling with − Maximum Safety Distances (LSMSD) problem if its corresponding BII vector bΠ ∗ is lexicographically maximal, i.e., − − bΠ ∗ ≥lex bΠ

∀Π.

(28)

Although LBIM and LSMSD appear to be very similar, both aim to the schedule the BIMs as evenly spread over the day as possible, they are not equivalent. Suppose we have a set of surgeries with durations 2, 2, 2, 11, 11, 11, 11, 12, 20, and 20. Let Π1 denote the LBIM optimal schedule and Π2 the LSMSD optimal schedule for this set of surgeries; see Fig. 3. The corresponding BII vectors are + bΠ = [10, 10, 10, 10, 2, 1, 1, 1, 1], 1 − bΠ = [1, 1, 1, 1, 2, 10, 10, 10, 10], 1 + bΠ = [11, 11, 10, 7, 5, 4, 2, 2, 2, 2] and 2 − bΠ = [2, 2, 2, 2, 4, 5, 7, 10, 11, 11]. 2 + + − − ≤lex bΠ but also bΠ ≤lex bΠ . This shows that the LBIM We have that bΠ 1 2 2 1 problem and the LSMSD problem are not equivalent. As mentioned before there are clear similarities between the two problems; however, it remains an open problem what the implications of these similarities are for the optimal schedules for both problems.

Fig. 3 The LBIM optimal schedule, Π1 (left), and the LSMSD optimal schedule, Π2 (right)

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Table 1 Overview of the models of this chapter (between parenthesis are the subsections where the models are discussed) for each combination of objective and type of disturbance Overtime

Internal variability Slack capacity ILP and SP (Sect. 2.1)

Waiting time

E/T (Sect. 2.2)

Utilization

Slack capacity ILP and SP (Sect. 2.1) E/T (Sect. 2.2)

External variability Non-elective policy (Sect. 3.1.1) Non-elective capacity (Sect. 3.1.2) BIM (Sect. 3.2.1), MSB (Sect. 3.2.2) Non-elective policy (Sect. 3.1.1) Non-elective capacity (Sect. 3.1.2) Assigning surgeries (Sect. 3.1.3)

Artificial variability SSD (Sect. 4.1)

SSD (Sect. 4.1) SSD (Sect. 4.1)

6 Conclusion In this chapter we presented a model-based overview of literature on robust surgery scheduling. We showed that robustness is a key aspect to improve OR performance. A non-robust surgery schedule performs well when all processes at the OR are going as expected but can significantly reduce the OR performance when there are small disturbances in the schedule during the day. There are several types of disturbances which we divide into three groups. First, those that affect the scheduling of elective surgeries. Second, those that affect the effectuation of the schedule, e.g., non-elective surgeries. Third, those that affect the availability of resources needed. The best suited approach to anticipate one of the described disturbances is highly dependent on the objective the scheduler has. Table 1 gives an overview of models described in literature for combinations of a disturbance and an objective. In the large majority of papers on robust surgery scheduling, one specific source of disturbance is analyzed in isolation. In Sect. 5 we described some models that incorporate multiple sources of disturbances. Additional research is needed in order to develop models anticipating multiple sources of disturbances. Furthermore, models described in literature are mostly applied to a specific case which complicates the comparison of different approaches. It would therefore be beneficial to apply the various approaches to a standardized case, like [14], in order to develop a benchmark for adequate comparison of the models proposed in literature.

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References 1. Birge, J. R. and Louveaux, F. (2011). Introduction to Stochastic Programming. SpringerVerlag, New York. 2. Borgman, N. J. (2017). Managing urgent care in hospitals. PhD thesis, University of Twente, Enschede, The Netherlands. 3. Cardoen, B., Demeulemeester, E., and Beliën, J. (2010). Operating room planning and scheduling: A literature review. European Journal of Operational Research, 201(3):921–932. 4. Denton, B. T., Miller, A. J., Balasubramanian, H. J., and Huschka, T. R. (2010). Optimal allocation of surgery blocks to operating rooms under uncertainty. Operations Research, 58(4part-1):802–816. 5. Ferrand, Y. B., Magazine, M. J., and Rao, U. S. (2014). Managing operating room efficiency and responsiveness for emergency and elective surgeries, a literature survey. IIE Transactions on Healthcare Systems Engineering, 4(1):49–64. 6. Guda, H., Dawande, M., Janakiraman, G., and Jung, K. S. (2016). Optimal policy for a stochastic scheduling problem with applications to surgical scheduling. Production and Operations Management, 25(7):1194–1202. 7. Gupta, D. (2007). Surgical suites’ operations management. Production and Operations Management, 16(6):689–700. 8. Hans, E. W. and Vanberkel, P. T. (2012). Operating Theatre Planning and Scheduling, pages 105–130. Springer US, Boston, MA. 9. Hulshof, P. J. H., Kortbeek, N., Boucherie, R. J., Hans, E. W., and Bakker, P. J. M. (2012). Taxonomic classification of planning decisions in health care: a structured review of the state of the art in OR/MS. Health Systems, 1(2):129–175. 10. Lamiri, M., Dreo, J., and Xie, X. (2007). Operating room planning with random surgery times. In 2007 IEEE International Conference on Automation Science and Engineering, Scottsdale, AZ, USA, pages 521–526. 11. Lamiri, M., Grimaud, F., and Xie, X. (2009). Optimization methods for a stochastic surgery planning problem. International Journal of Production Economics, 120(2):400–410. 12. Lamiri, M., Xie, X., and Zhang, S. (2008). Column generation approach to operating theater planning with elective and emergency patients. IIE Transactions, 40(9):838–852. 13. Landa, P., Aringhieri, R., Soriano, P., Tánfani, E., and Testi, A. (2016). A hybrid optimization algorithm for surgeries scheduling. Operations Research for Health Care, 8:103–114. 14. Leeftink, G. and Hans, E. W. (2018). Case mix classification and a benchmark set for surgery scheduling. Journal of Scheduling, 21(1):17–33. 15. Min, D. and Yih, Y. (2010). Scheduling elective surgery under uncertainty and downstream capacity constraints. European Journal of Operational Research, 206(3):642–652. 16. Molina-Pariente, J. M., Hans, E. W., and Framinan, J. M. (2016). A stochastic approach for solving the operating room scheduling problem. Flexible Services and Manufacturing Journal, 30(1-2):224–251. 17. Otten, J. W. M. and Boucherie, R. J. (2018). Minimizing Earliness/Tardiness costs on multiple machines with an application to surgery scheduling. Submitted. 18. Razmi, J., Yousefi, M., and Barati, M. (2015). A stochastic model for operating room unique equipment planning under uncertainty. IFAC-PapersOnLine, 48(3):1796–1801. 19. Spieksma, F. C. R., Woeginger, G. J., and Yu, Z. (1995). Scheduling with safety distances. Annals of Operations Research, 57(1):251–264. 20. Van Der Lans, M., Hans, E. W., Hurink, J. L., Wullink, G., van Houdenhoven, M., and Kazemier, G. (2005). Anticipating urgent surgery in operating room departments. Number WP-158 in Beta working papers. BETA Research School for Operations Management and Logistics. 21. Van Essen, J. T., Hans, E. W., Hurink, J. L., and Oversberg, A. (2012). Minimizing the waiting time for emergency surgery. Operations Research for Health Care, 1(2-3):34–44.

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Applications of Hospital Bed Optimization A. J. (Thomas) Schneider and N. M. (Maartje) van de Vrugt

Abstract In this chapter we show typical bed capacity management decisions and how these can be supported using operations research (OR) models. During hospitalization, patients spend most of their time in a bed, situated at a ward. These wards, which include staff, beds, and equipment, are one of the most expensive resources of hospitals. Often patients who stay at a ward receive one or multiple treatments, which usually take place at different departments. Many wards still struggle to accommodate all incoming patients. Without aligned schedules, the flow of patients will fluctuate significantly, and therefore beds at wards will congest. As a result of this “disorganization,” staff will experience an unbalanced workload, and wards require more (buffer) capacity to accommodate all patients. With operations research techniques, planning and scheduling of both patient admissions and staff presence at wards can be optimized aiming to reduce variation in the bed occupancy. We also show three case studies using OR in bed management decision-making and discuss success and pitfalls.

A. J. (Thomas) Schneider () Center for Healthcare Operations Improvement and Research (CHOIR), University of Twente, Enschede, The Netherlands Department of Quality and Patient Safety, Leiden University Medical Center, Leiden, The Netherlands e-mail: [email protected] N. M. (Maartje) van de Vrugt Center for Healthcare Operations Improvement and Research (CHOIR), University of Twente, Enschede, The Netherlands Department of Strategy and Innovation, Amsterdam University Medical Center, Amsterdam, The Netherlands © Springer Nature Switzerland AG 2021 M. E. Zonderland et al. (eds.), Handbook of Healthcare Logistics, International Series in Operations Research & Management Science 302, https://doi.org/10.1007/978-3-030-60212-3_5

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1 Introduction In this chapter we show typical bed capacity management decisions and how these can be supported using operations research (OR) models. We illustrate practical problems and possible solutions. Furthermore, we show the potential impact of these type of models on capacity decisions in practice and highlight why implementation of the results was successful (see Sect. 4). In our previous work [45] we focused solely on bed occupancy modeling, while in this chapter we point out all capacity management-related decisions for hospital beds. During hospitalization, patients spend most of their time in a bed, situated at a ward. These wards, which include staff, beds and equipment, are one of the most expensive resources of hospitals and are defined as inpatient care facilities providing care by offering a room, a bed and board [24]. Often patients who stay at a ward receive one or multiple treatments, which usually take place at different departments. As an example, consider admissions at surgical wards, which are strongly determined by the operating room schedule. To optimize ward logistics, the planning and capacity availability should be aligned with other departments’ schedules. Many wards still struggle to accommodate all incoming patients. From our own experiences, we observe that these struggles are foremost a result of unnatural, selfinduced variation, caused by unbalanced and unaligned schedules between hospital departments. Without aligned schedules, the flow of patients will fluctuate significantly, and therefore beds at wards will congest. As a result of this “disorganization,” staff will experience an unbalanced workload, and wards require more (buffer) capacity to accommodate all patients. Not all variation is self-induced. Wards typically have unexpected daily fluctuations in the bed census, as a result of changes in patients’ health status, staff schedules and/or treatment plans. Admission planners often only take the average length of stay (LOS) into account. Good estimates of patient LOS, when based on multiple patient characteristics such as age and comorbidity, could potentially reduce the difference between the scheduled and realized bed census. With operations research techniques, planning and scheduling of both patient admissions and staff presence at wards can be optimized aiming to reduce variation in the bed occupancy. When variation can be further minimized, the staff schedules should be adapted accordingly to accommodate for these variations. For example, this may imply that more staff is scheduled to work every Monday or throughout the winter. Hospital ward logistics typically focuses on two key performance indicators: bed occupancy and blocking probability (e.g., when all beds are occupied). Bed occupancy is an important performance measure, but a universal definition of occupancy does not exist [45]. Next to occupancy and blocking probability, workload is an important performance measure for a ward. Workload also lacks a universal definition but is mainly based on patient (e.g., type of treatment or disease) and staff characteristics (e.g., junior vs. senior staff). Ward resources are scarce, and therefore

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hospital management often focuses on maximizing the bed occupancy. When bed occupancy rates are high, the blocking probability increases as well, which implies that patients more often become boarders (e.g., admissions at other wards then dedicated for their medical specialty) or have to be rescheduled. Additionally, it takes significantly more time to find a suitable bed to accommodate a patient. As a consequence, nurses have to treat patients of medical specialties they are not trained for, and doctors’ rounds take more time as they have to visit more wards to see their patients. Therefore, striving for high bed occupancy rates at wards competes with both quality of care and job satisfaction. The remainder of this chapter is organized as follows. We start by explaining the typical ward capacity management decisions in Sect. 2. To support the described capacity decision making, we present the related operations research models in Sect. 3. Next, we discuss case studies where operations research models have made practical impact for ward capacity decisions in Sect. 4. Finally, we discuss future developments and research opportunities in Sect. 5.

2 Ward Capacity Management The term “planning and control” is most often used for decisions on the acquisition and usage of capacity to efficiently satisfy customer demands [18]. Efficient realization of organizational goals (e.g., satisfied and healthy patients) requires hospitalwide coordination of capacity and flows, by continuously balancing demand and supply. Operations research can give insights to improve the efficiency of capacity and flows, which is increasingly important in these times of rising healthcare expenditures. To demarcate the scope of capacity management decisions and/or optimization interventions at wards, we use the four-by-four framework of [21] (Fig. 1). The framework hierarchically decomposes managerial levels on one axis, strategic, tactical, and operational (offline and online), and covers different managerial areas at the other axes. An important step in planning and control is setting the length of the scheduling horizon for the different hierarchical levels. At strategic level, decisions are made for at least 1 year but often for multiple years ahead. The operational level is maximally several weeks ahead. Therefore, the tactical level ranges from several weeks to 1 year ahead. On the other axis the framework integrates the managerial planning areas in healthcare: medical, resource capacity, materials, and financial planning. In this chapter we focus on resource capacity planning for both planning and control and operations research models for wards. Following this framework, we use a top-down approach explaining all planning and control decisions at the different hierarchical levels, as higher levels set boundaries for lower levels. Nevertheless, also bottom-up feedback should be in place in practice, so detected deviations and problems can be lifted one hierarchical level upwards for problem-solving.

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Fig. 1 The healthcare planning and control framework with applications

2.1 Strategic Ward Capacity Management At the strategic level, the hospital board decides on the hospital’s long-term “mission and strategy,” the areas on which the hospital aims to focus and excel. Important decisions at the strategic level are the desired case-mix, hospital layout, performance targets, bed capacity, and workforce planning.

2.1.1

The Desired Case-Mix of the Hospital

Based upon the hospital’s mission and strategy, the board and/or head of departments determines the case-mix of the hospital; a case-mix is the collection of patient groups a hospital treats. Some hospitals choose a very specific patient group to treat, for example, a breast cancer clinic, while general hospitals have a more diverse casemix. The preferred case-mix of a hospital determines to a large extent the required capacity. The case-mix of a hospital can be adjusted by attracting and/or deferring patient groups. As healthcare organizations and their professionals have the duty to care for their patients, a hospital is not allowed to defer a patient group until other regional or national providers agree to treat this patient group. Furthermore, the patient casemix could also change when a new doctor with a different specialization is hired. The same could happen when specialized doctors leave the hospital. Adjusting a hospital’s case-mix is a complex process as many factors should be taken into account. For example, case-mix decisions could not only affect the required capacity for care delivery but also the education and research possibilities. Another example is that patients not often are treated by a single specialty. So the decision to stop treatment for a specific patient group for a medical specialty could affect the case-mix of many other medical specialties.

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Hospital Layout Planning

Based on the mission and strategy, a hospital board decides at the strategic level on the type of wards and rooms that are available in the hospital. One example is the mix between single- and multi-person rooms. Single-person rooms ensure privacy for patients and their family but require more space and result in more walking distance for the staff, and patients monitoring could become more difficult. On the other hand, multi-person rooms are inefficient when patients have infectious diseases and when only same-sex rooms are put in place. Another aspect of strategic planning for wards is the decision to establish wards for special types of care; acute medical units, intensive care units, cardiac care units, and surgical admission lounges are examples of these dedicated wards. These wards serve a specific patient group based on severity, urgency, treatment or flow (e.g., elective versus acute admissions) and are mainly introduced to improve the quality of care and/or efficiency. As such decisions have high economic impact and take a long time to accomplish, these decisions will affect the hospital logistics for multiple years. After the global (idea of the) layout of the hospital is determined, all patient groups in the case-mix have to be assigned to wards. An important decision at this level is how many beds are considered and organized as one ward. From a logistical viewpoint, larger wards will result in economies of scale, leading to a higher occupancy with an acceptable blocking probability (see [45]). For this decision, the trade-off between medical and logistical perspectives should be taken into account. Purely from a logistical perspective, if all patient groups could be treated at each bed in the hospital, the bed census at this “single ward system” can be optimally balanced. As a result, nurses in this hospital have to be multiskilled (which is impossible for high-complex care), and doctors will spend more time to visit their patients at different areas of the hospital. From a medical perspective, a more differentiated distribution of patient groups over wards would be optimal, where patient types are clustered according to the skills required for their treatment. A balance between these two perspectives should therefore be found.

2.1.3

Setting Performance Targets

On a strategic level, the hospital board should set performance targets for the hospital. At hospital wards, logistical performance indicators are often the bed census or occupancy, where occupancy can be measured in many different ways [45]. Setting high occupancy targets for wards will, certainly for smaller wards, result in deferring more patients to other wards or hospitals. A different, better performance target would be an upper bound on the percentage of deferred patients and achieving the desired case-mix.

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The Number of Beds

When a hospital’s desired case-mix is determined, this information is used for strategic planning to forecast the demand for care for the hospital. This forecast is based on aggregated data, trends, and forecasts for the patient population. Using the forecasted demand for care and the set performance targets, a hospital can determine the required capacity to treat these patients. The required capacity is determined on an aggregated scale, such as the total number of required operating theater hours, outpatient clinic hours and ward hours for the upcoming year(s). Based on the aggregated data, the required ward capacity is (re)evaluated yearly. Typically, the number of physical beds at a ward is higher than the average number of used beds. Each ward should have buffer capacity, to accommodate unexpected peaks in patient arrivals. Often, not all physical beds at a ward are “staffed”; there is no nurse available to treat a patient in that bed. These extra beds ensure, for example, that patients with infection risks can be treated in isolation when the ward has multi-person rooms. Moreover, these beds form a buffer of clean beds if the time between one patient’s discharge and another patient’s admittance is short. A ward may also have overcapacity in the number of staffed beds, when the nurse-to-patient ratios do not perfectly match with the expected bed census. The ratio depends on the average “workload” of one patient and denotes the number of patients one nurse can take care of. For example, consider a ward with an expected bed census of 17 beds and the ratios per shift are as follows: day 1:3, afternoon 1:5, and night 1:8. As a result, the shifts requires at least 6, 4, and 3 nurses, respectively. On a strategic level this slack should be taken into account when the expected bed census is translated into the required number of full-time equivalent contracted nurses.

2.1.5

Workforce Planning

At hospital wards often not the physical beds but the number of nurses determines the ward capacity. Once the demand for beds is determined by the case-mix planning, the workforce should be aligned to it. An important capacity management decision that determines how many nurses are required is the nurse-to-bed ratio. For example, an ICU patient has a relatively high workload, and the nurse-topatient ratio is 1:1, while for a general ward during a night shift the ratio may be 1:16. Workload lacks a universally accepted definition but is generally considered as the relation between the demand (of patients) and the capacity available to fulfill this demand. Workload can be divided into objective (e.g., patient acuity metrics) and subjective (e.g., nurse workload perception) factors [43, 48]. Patient acuity metrics generally consist of activities of daily living, cognitive support,

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communication support, emotional support, safety management, patient assessment, injury or wound management, observational needs and medication preparation. Perceived workload is also not universally accepted as measurement for workload but mainly consists of staff characteristics such as age, experience, and educational level. The total workload at a ward is based on the patients’ acuity, shift (day, afternoon, or night) and the bed census. Nurses at a hospital are mostly assigned to one ward or to a few wards that accommodate patients with the same care requirements. At a strategic level, a hospital can decide to flexibly allocate a part of the capacity. For wards, this implies that not all beds are assigned to a certain medical specialty or specific patient group (e.g., organ transplantation patients), but part of the bed capacity will be assigned based on the actual demand for care for each patient group. Flexibility may also imply that a hospital creates a “flex-pool” of nurses; these nurses are often multiskilled and are allocated on a short term (e.g., each morning) to the most busy ward. The advantage of flexible capacity is that a hospital can better adapt to stochastic patient demand. Which part of the capacity is allocated in a flexible way, and on which KPIs the allocation decision is based, is a strategic capacity management decision. A hospital’s desired case-mix also determines the quantity and quality of the required staff to a large extent. However, the translation from case-mix to the number of staff-members is not the same for each hospital. The ever-continuous technological and medical innovations require more specialization from practitioners. In general, more specialization increases the number of involved specialists during diagnosis and treatment, as all specialists are specialized in a small part of a human body and/or specific diseases. Additionally, a hospital’s policy with respect to education may change the translation from case-mix to number of required staffmembers; junior students usually decrease the capacity due to supervision duties of the staff, while students who are about to graduate can often work without much supervision. Moreover, when the staff-members are involved in research projects, this typically decreases the available capacity for treating patients. Additionally, each hospital has differences in the workforce, which implies that strategic workforce planning should incorporate: influx (training, education, and immigrants) and outflux (retirement or retention) of staff, level of task differentiation (e.g., highly trained versus basic trained staff), and the in- or outsourcing of training and education programs. Strategic capacity management decisions are always long term and often require major monetary adjustments to accomplish. For economic value and employee satisfaction, yearly changing the ward layout of the hospital is not desirable. However, due to small changes in the case-mix, ward sizes may become inadequate over time. Here, both under- and overcapacity are a problem; see Sect. 4.1. Strategic decisions define the framework at the tactical and operational levels and are therefore to a large extent accountable for the performance of the hospital.

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2.2 Tactical Ward Capacity Management At the tactical level, capacity management decisions focus on organizing the desired case-mix, controlling patient access times and efficient capacity usage via generating master schedules, allocation of flexible capacity, and scheduling bounds which we will explain further in this section. As mentioned earlier, tactical capacity management decisions concern the organization of operations and processes on a “midterm.”

2.2.1

Master Schedules

The first step at the tactical level is to divide the total capacity among stakeholders and over the weeks of the year, resulting in a master schedule. Often a master schedule is set for an entire year, but hospitals can gain flexibility to adjust capacity to the patient demand when the scheduling horizon is shorter. For wards a master schedule may result in a weekly schedule where the capacity of each ward changes over time and beds are divided among different patient groups (e.g., elective and emergency admissions). Typically, a master schedule is different for each season. Temporary leaves of staff may require alterations to the master schedule too. Holidays, training, education or internships are examples of temporary leaves and should be planned on the tactical level. The master schedule of a ward should be aligned with the master schedules of other capacity, such as operating rooms and outpatient clinics, to create a stable flow of patients. Especially for wards that accommodate surgical patients, aligning the master schedules of the operating rooms and wards results in increased efficiency. This alignment is twofold: aligning holiday weeks and balancing bed census. Typically, a hospital has several holiday weeks per year in which both (elective) patients and staff are not available and capacity is reduced for elective (scheduled) care. To prevent a shortage or surplus of beds, the holiday weeks of the operating theater and wards should be aligned. Additionally, by optimizing the operating room master schedule, the postoperative bed census can be balanced better [14]. Balancing the bed census implies that a ward requires less buffer capacity and thus increases efficiency at a ward and reduces the risk of cancelling a surgery due to a lack of postoperative beds.

2.2.2

Flexible Allocation of Capacity

Patient demand is usually fluctuating; thus, it may be beneficial to adjust part of the capacity throughout the year. For the total flow in the hospital, admitting a similar number and type of patients each week is optimal. However, due to staff holidays and stochasticity in patient arrivals, this is a difficult aim. For each patient group (or aggregated for each medical specialty), it is therefore beneficial to periodically

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evaluate the available capacity and make minor adjustments where necessary and possible. For wards, this could imply asking nurses to work on a different ward for several weeks or doctors to, for example, help at the ward instead of working at the outpatient clinic. Hospital management can decide on the strategic level to partly reserve capacity and allocate this on a regular basis, e.g., monthly. At the tactical level, this capacity may be allocated for several weeks in advance. When applying flexible capacity allocation, it is important to have consensus among all stakeholders about the parameters and performance indicators upon which the allocation will be based and on the scheduling horizon on which the flexible capacity can be allocated. For wards, flexible capacity could imply that nurses from the flex-pool are assigned to a specific ward. Moreover, downstream resources should be taken into account when allocating staff from a flex-pool to align patient loads. For example, allocating flexible operating room time affects the bed census at the postoperative wards; thus, these decisions may require additional nurses at the postoperative wards. At most hospitals, staff rosters are generated several months in advance, and therefore staff planning is performed on tactical level too. These rosters only state which shifts and days an employee should work and do not specify the department, bed, and/or patients. The latter, detailed scheduling, is performed on the operational level. This provides the departments additional flexibility in staff allocation.

2.2.3

Regulating the Demand for Care

In order to balance patient flow and optimize efficiency, at the tactical level hospital management can decide to formulate rules for patient scheduling. Such rules may, for example, state the minimum and maximum number of elective patients that may be admitted to a ward per day. Another example is stating a maximum on the number of surgeries scheduled at the same day that require an ICU bed. These rules can be ward specific, medical specialty specific, or may hold for the entire hospital. Tactical capacity management decisions are crucial to efficiently organize patient care and flows, especially at the interface between different types of resources in a hospital. From our own experience, this level is still underdeveloped in many hospitals.

2.3 Operational Ward Capacity Management The operational level is divided in offline (service at a later point in time) and online (instant service) capacity management decisions. Compared to the other two levels, the operational level has very limited possibilities to adjust the capacity to the patient demand. The online level comprises the actual patient (room and bed) to staff scheduling and ad hoc decisions, such as replacing ill staff-members and admissions of emergency patients.

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Patient Scheduling

Patient scheduling on the operational level comprises deciding each elective patient’s admission date and ward. It should be taken into account that each admittance and discharge imply a workload peak for the nurses. Additionally, it is important to take the expected urgent patient admissions into account, as scheduling too many elective patients results in deference of emergency patients. Moreover, a patient schedule should minimize the number of in-hospital patient transfers, as each transfer could be a risk for the quality of care. Therefore, a patient schedule should, for example, take into account that some wards close beds during the weekends and therefore do not accept admissions that are expected to stay longer than Friday as the ward is closed during the weekends. Yet, when patients need to stay after Friday, they have to be transferred to other wards. Accurate predictions of the length of stay (LOS) are therefore crucial. Some hospitals/wards adjust the patient schedule one week in advance, based on the actual bed census and LOS predictions of the currently admitted patients. At the online level, a ward manager may decide to transfer a patient with relatively good health to a ward with a lower care level or to another hospital, to reserve capacity for high-care patients. In practice, patients are typically admitted to their medically preferred ward when there is available capacity, and ward managers start to transfer patients to “second-best” alternative (also called “overflow”) wards when capacity runs out. As a consequence, patients may wait for a long time at wards that are not medically preferred before a bed is available, as another patient needs to be transferred first. To decrease the time until a patient is assigned to a bed, ward managers may already transfer patients when there is still available capacity, to reserve enough capacity for new patients. By focusing on the expected discharge date at the moment of arrival, the LOS will decrease (this is also called discharge management). The decision on which patients should be transferred is often difficult, and optimizing this decision-making process may improve patient waiting time, quality of care, and even hospital revenues significantly. Hospitals may also apply admission control to make sure enough capacity is available for the patients that need it the most and/or benefit from it the most, especially when there are multiple hospitals in the proximity.

2.3.2

Staff Scheduling

At the operational offline level, staff is assigned to a specific ward several weeks in advance. When the hospital has a flex-pool of nurses, these nurses may be allocated to specific wards at the operational level. Some hospitals allocate these nurses several weeks in advance, based on long-term illness of staff, short-term staff leaves or forecasted patient demand. Hospitals may also decide to assign nurses from the flex-pool in an online way, which implies that a nurse is assigned to a ward at the beginning of a week or even each shift.

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At the online level, nurses are assigned to patients. This scheduling task is performed before each shift starts. Next to the number of patients present, also the “type” of patient is important to optimize the nurse-patient assignment. As patient acuities and staff characteristics vary over time, nurse-patient assignments should be optimized by distributing the workload among available nurses on the operational level. Although there is little room to adjust capacity to actual demand on the operational level, we have shown many capacity management decisions on this level that can further optimize patient care delivery. As improvement on this level requires relatively small adjustments in terms of work routine and/or investments, they are relatively easily to implement. Therefore, both management and staff can fulfill the potential of these improvements at any moment.

2.4 Feedback Between the Hierarchical Levels As common in literature, we have used a top-down approach for discussing all hierarchical control levels at wards. As mentioned earlier, healthcare processes and planning deal with stochasticity, and therefore unforeseen situations often occur. Monitoring systems should be in place to detect deviations from scheduled care processes. Using data from electronic health records, software can easily detect, present, and even predict these deviations. It is important to note that some data has to be put in manually (e.g., the expected discharge date) in order to accurately detect deviations. When a deviation is detected or predicted, planners and ward management can proactively arrange adjustments in capacity and/or demand. When detected deviations cannot be solved within the managerial boundaries of the level where the unforeseen situation occurred, the deviations should be escalated. Bottom-up feedback loops provide escalation channels to lift problemsolving to higher hierarchical levels. It must be clear for each level when detected deviations have to be escalated. An example for escalation could be regularly occurring peaks in postoperative elective patient arrivals; the master schedule of the operating theater should then be reconsidered in order to balance the postoperative arrivals at wards. In general, recurring problems may require structural redesign of processes and thus require decision-making on a higher hierarchical level. Therefore, escalation channels are an important component of the planning and control cycle for resource capacity planning.

3 Operations Research Models for Wards In this section we reflect on OR models that can be used for analyzing ward capacity management decisions. We follow the same hierarchical approach as the previous

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Queueing theory

Integer programming

Markov chains

Simulation

Dimensioning wards

3.1.1

3.1.2

3.1.3

3.1.4

Admission planning

3.3.2

3.3.1

Chain logistics or flow optimization

3.2.1

3.2.3

3.2.4

3.2.2

Patient scheduling and bed assignment

3.4.3

3.4.1

Nurse-to-patient assignment Length of stay and readmission forecast

Heuristics

3.4.5

3.4.2

3.6.3

3.6.1

Markov decision theory

3.4.4

3.5.1 3.6.2

Fig. 2 Overview of Sect. 3

section and show the capacity management decisions covered in this chapter and used OR techniques from literature to analyze these types of decisions in Fig. 2. One important optimization application at wards is nurse staffing. Although the physical capacity of a ward is determined by the number of beds that are present at the ward, in most hospitals the number of nurses present at the ward determines to a large extent the number of patients that can be accommodated. Many departments schedule the same number of nurses each shift or marginally adapt the nurse schedule to the bed demand. The topic is elaborated upon in chapter “Bed Census Predictions and Nurse Staffing” and is therefore not discussed in this chapter.

3.1 Dimensioning Wards Finding the optimal capacity of a ward by allocating patient groups among wards is a typical strategic decision. In the literature, dimensioning decisions are based upon queueing models, Markov chains, simulation, goal programming, and mixed integer programming models. Below we will evaluate these approaches and provide some examples from the literature.

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Queueing Theory

The Erlang loss and infinite server queueing models are by far the most-used models to determine the best dimension of hospital wards. With easy-to-use tools available, such as the Queuing Network Analyzer (see [56]), hospital practitioners are able to analyze decisions with queueing models. The examples provided by [45] and in the case study presented in this chapter in Sect. 4.2 demonstrate the value of these basic models for dimensioning hospital wards. Another advantage of the Erlang loss queue and infinite server queue is that these models are insensitive to the distribution of the length of stay; obtaining an average LOS from hospital data is enough for the analysis. Sophisticated data analysis to generate input data is therefore not required for these models. The basic queueing models do not encompass all hospital ward dynamics. For example, they do not encompass nonhomogeneous arrival and discharge rates, while in reality scheduled patients only arrive and discharge during the day. Another example of misrepresentation is that in practice patients are often not “blocked and lost” if all beds at their medically preferred ward are occupied upon their arrival, which implies that queueing models underestimate the bed occupancy. Gallivan and Utley [15] demonstrate that for an infinite server queue with piecewise stationary Poisson arrivals, the resulting model is easy to analyze. However, most queueing models become intractable with time-varying arrival and/or service rates. Additionally, feedback and overflow are typically difficult to analyze, as shown by, for example, [46], for a small network of an operating theater and an intensive care unit (ICU). To increase the predictive value of the model, Williams et al. [53] consider an Erlang loss queue in which the arrival rate depends on the number of occupied beds, to reflect that less patients are admitted to the ward when it is almost full. Bekker and de Bruin [3] analyze an infinite server queue with time-dependent arrival rate and use the square-root staffing rule to dimension an ICU.

3.1.2

Integer Programming

Queueing models alone require a trial-and-error approach to find optimal capacity. To overcome this problem, queueing models can be incorporated into a mixed integer programming approach. van Essen et al. [47] analyze three approaches assigning patient group clusters to wards. The exact approach uses the Erlang loss model to determine bed capacity given a blocking probability and an ILP is used to determine which patient groups should be clustered and assigned to a wards. The second approach uses an approximation of the Erlang loss model by a linear function for the required number of beds followed by the ILP for the clustering process. The last approach uses the exact formulation of the Erlang loss model for the number of required beds and a local search heuristic forming the clusters. Another example of combining queueing models with optimization models is given by [38], who use similar approaches as in [47] to determine the bed capacity for a network of maternity clinics. Pehlivan et al. [38] also linearize the blocking probability and bed

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census of the Erlang loss model and also analyze interactions between clinics with a mathematical model. The queueing model formulas can also be incorporated in a goal programming approach. Li et al. [32] use this approach to allocate a number of beds at each ward, to ultimately optimize multiple objectives set by the hospital management. Oddoye et al. [37] use simulation to relate the capacity (beds, nurses, and doctors) of a medical assessment unit to queue lengths for patients and incorporate this into a goal programming model.

3.1.3

Markov Chains

Predicting the bed census using Markov chains may result in higher accuracy compared to a queueing approach, as time-varying arrival and discharge rates may be incorporated in such models. Kortbeek et al. [30] invoke a Markov chain to predict the hourly bed census, which includes postoperative surgical patients, emergency admissions, and overflow patients to and from other wards. Using the steady-state distribution, the authors obtain an expression for the 95% percentile of the bed census. Markov chain models are also applicable in transient analyses; for example, Broyles et al. [8] predict the ICU bed census invoking a transient Markov chain analysis with maximum likelihood regression. Using Markov chains almost every desired detail can be modeled. However, including more details into the models quickly makes a Markov chain intractable.

3.1.4

Simulation

With simulation, all features of hospital wards imaginable can be incorporated, which makes this type of modeling sometimes the best or only option to model a ward. Holm et al. [23] use a simulation model to relate the capacity of a ward to the bed census for several wards for all possible numbers of beds and heuristically assign beds to the different wards using these relations. The model is evaluated using data of a university medical center. VanBerkel and Blake [49] analyze multiple scenarios to solve waiting list issues, of which redistributing beds among the wards is one. Transient analyses are also possible while using simulation models: Zhecheng [55] presents a simulation model to obtain short-term predictions based on the specific characteristics of the current patient population present at the ward. Simulation models require labor-intensive data analyses to generate input parameters, developing time, often require complete enumeration and output analyses. Furthermore, strategic analyses do not always require all details, and therefore simulation models are not an obvious first choice to determine the best capacity of a ward. Both academics and professionals should therefore keep in mind the trade-off between required level of detail and the required amount of building and running

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time and the value of the outcomes when using simulation techniques to analyze capacity management decisions.

3.2 Chain Logistics or Flow Optimization So far we have highlighted research that mainly focuses on a particular step in the patient care pathway: clinical treatment at inpatient wards. However, the inflow of inpatients is often determined by other hospital departments. Especially for wards that accommodate many surgical patients, the operating theater schedule determines to a large extent the bed census at the ward. For wards that accommodate many urgent patients, the emergency department and acute admission unit influence the bed census. Vanberkel et al. [50] provide a survey of healthcare models that encompass multiple departments. Interestingly, achieving optimal logistical flow through a hospital may result in suboptimal use of capacity of individual resources. In this section we will highlight literature on queueing, simulation, MIP, and Markov chain models that mimic the interaction between multiple departments.

3.2.1

Queueing Theory

Queueing networks are useful in relating different capacity levels to certain performance measures such as blocking probability. For example, Zonderland et al. [57] analyze multiple scenarios for a network of an emergency department, acute admission unit, and two wards, in which the acute admission unit may function as an overflow for the other three departments. They observe that with the used setting the arrivals of urgent patients can be increased at the expense of decreasing elective arrivals (the increased influx of emergency patients is higher than the decreased number of elective arrivals). Litvak et al. [33] tackle the problem of deferred acute intensive care patients as a result of capacity problems. They show that regional cooperation between multiple ICUs results in higher acceptance levels for these patients. The authors approximate the blocking probabilities in an overflow network with the equivalent random method and the Erlang loss queue. Setting a threshold to this blocking probability, they determine how many beds each ICU in a certain region should reserve (make available for regional patients) such that all acute intensive care patients in the region can be accommodated promptly.

3.2.2

Simulation

Simulation models are also used to analyze patient flows through multiple hospital departments and to determine the effect of changes in, for example, the capacity. Optimizing patient flows at individual departments of a hospital may lead to

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disturbed patient flows at other departments, as the bottleneck in the patient flow may shift to another department [29]. Schneider et al. [42] use simulation to analyze the flow of emergency patients among the three departments: ED, acute medical unit, and inpatient wards. The hospital of their case study has great difficulties accommodating all emergency admissions. Using heuristics they optimize the number of allocated beds per inpatient ward for emergency patients that need to stay longer than is intended at the acute medical unit. Another example of a simulation study in this area is Mustafee et al. [36], who investigate different strategies preventing patients to occupy high-care beds unnecessarily long due to the unavailability of beds with a lower level of care. Day et al. [12] investigate the effect of adding capacity and a different discharge policy on the patient flow at a pediatric surgical center.

3.2.3

Mixed Integer Programming

Mixed integer programming models are often developed to optimize the master surgery schedule (MSS) of the operating theater. Fügener et al. [14] optimize the MSS while minimizing the probability that overcapacity is necessary to accommodate all patients at the postoperative wards. Based on which surgical specialty is assigned to which time slot in the MSS, they analytically express the bed census distribution function for each ward. An operational offline approach is taken by Gartner and Kolisch [17], who invoke an MIP model to determine the admission dates for patients that require care at multiple departments.

3.2.4

Markov Chains

A Markov chain approach was invoked by Isken et al. [27] to model multiple patient pathways at an obstetrics department with multiple wards. The obtained expressions are incorporated into an MIP model to optimize the schedule of elective patients.

3.3 Admission Planning After the capacity dimensions are set for wards, demand and capacity can be optimized on a midterm horizon through admission planning and nurse rostering, respectively. As mentioned in the introduction, admission planning generates a blueprint schedule that schedules the different patient groups and not individual patients and/or treatments. For this type of optimization, mixed integer programming (MIP) models are mostly preferred in the literature.

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Mixed Integer Programming

In our previous work, we used a mixed integer programming approach to develop a tactical schedule for a weekday ward (see [45]). Weekday wards only admit elective patients during weekdays and close during the weekends. All patient care is delivered according to strict protocols, which results in highly accurate treatment time and LOS predictions. Typically, treatments with a longer LOS are scheduled at the beginning of a week and shorter treatments later during the week, to ensure that all patients are discharged before the weekend. Helm and Van Oyen [22] develop two infinite-server queueing models (one for emergency arrivals and one for elective arrivals) to determine the bed census that results from any admission plan for regular wards. Based on these bed census, an MIP model minimizes the blocking probability of emergency arrivals, the cancellation probability of elective arrivals, and the average number of boarders (patients who have to wait for their preferred ward) in this tactical admission plan. Bekker and Koeleman [4] use a quadratic program to obtain daily quota for the number of admissions to a ward to minimize the variability in the bed census (e.g., quota planning). In the quadratic program the bed census is modeled using a GI/G/∞ queue with a heavy traffic approximation, and the authors present an approximation for the bed census of a ward that experiences a non-Poisson arrival process. The aim is to generate rules of thumb for management and planners based on the results of their model. They conclude that quota planning has the most impact on the bed census variability (e.g., smooth bed census during the planning horizon). Using quota planning the arrival process of admissions at wards will be more stable. A stable arrival process results in a more stable bed census compared to high variable arrival rates. The next rule of thumb is to schedule arrivals given the number of available (or closed beds) during the planning horizon. Given the absence of admissions during weekends, beds can be closed, or patients with a longer LOS can be scheduled on Friday to improve the bed census during weekends. Typically, patients require other types of resources during hospitalization, such as the operating theater or diagnostic facilities. Therefore, the patient schedules at these resources affect each other. Incorporating many different capacity types and patients following uncertain treatment paths (see Hulshof et al. [25]) invoke an MIP approach to optimize the number of admitted patients per time period. The tractability of the MIP appears insufficient to optimize realistic instances, and therefore the authors turn to approximate dynamic programming [26].

3.3.2

Queueing Theory

Queueing approaches may be used to determine the best number of beds that should be reserved for a certain patient type. For example, Litvak et al. [33] investigate a network of ICUs that all reserve some capacity to admit emergency patients in the region using the equivalent random method. As ICU capacity is scarce and costly, it is typically utilized maximally, which results in blocked emergency patients and

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cancellations of scheduled patients. The analysis shows that when multiple regional ICUs cooperate as a network, they can increase the acceptance level of emergency patients with a smaller total number of beds compared to the setting of individual ICUs. Mandelbaum et al. [35] present another application of queueing models where they balance the bed census of wards with a similar level of care by considering routing algorithms for patients from the emergency department to wards.

3.4 Patient Scheduling and Bed Assignment The tactical admission planning results mainly in a blueprint schedule for patient admissions at wards. In the next planning phase (e.g., operational planning), actual patients are scheduled and assigned to available beds. The tactical blueprint serves as a guideline for scheduling patients. In some circumstances (e.g., patients that have been scheduled and/or availability of staff), management and planners can deviate from this blueprint. This would not be preferable as the blue print of downstream resources should also deviate. Using optimization, patient admission dates and bed assignments can be chosen such that the number of beds that is required to treat all patients is minimized, or the variation in bed usage is minimized. Additionally or alternatively, the number of patients that receive treatment within their preferred access time window can be maximized. Optimizing bed assignments will have the largest impact when the medically preferred ward has multi-person rooms or when there are several wards with adequate level of care. For multi-person rooms, for example, patients with infectious diseases and “same sex in one room,” rules may complicate the room assignments. Basically there are two decisions in this type of problem: (1) shifting admission dates and (2) transferring patients between wards according to medical preferences. Below, we highlight literature with respect to patient admission scheduling, bed assignment, and admission control. Here we exclude literature on (surgical) patient scheduling; for more insight on surgical scheduling, the reader is referred to [58]. Sets of benchmark instances for the offline optimization of bed assignments1 and the patient admission scheduling problem2 are available online.

3.4.1

Mixed Integer Programming

MIP models are incorporated in an online decision support system to optimize bed assignments. For example, Schmidt et al. [41] and Vancroonenburg et al. [52] determine the optimal ward and/or bed assignment for each patient with respect to the bed census among all wards, the adequate level of care for as many patients

1 https://people.cs.kuleuven.be/~wim.vancroonenburg/pas/ 2 http://satt.diegm.uniud.it/index.php?page=pasu

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as possible, and the number of transfers that are required during treatment. Ben Bachouch et al. [5] also apply an MIP approach to assign patients to beds, where elective patients request a time window in which they require treatment, whereas for emergency patients this window starts at the current time and is equal to the length of stay. Bed assignment decisions may also be optimized in an offline setting, where all patients scheduled for admission are assigned to beds in an optimal fashion. Braaksma et al. [6], for example, use an MIP to solve an operational offline patient scheduling and bed assignment problem at a weekday ward, in which they optimize over all medically preferred patient access times. Guido et al. [20] present a heuristic based on an MIP model to satisfy as many bed assignment constraints as possible in an offline optimization model while taking into account that some patients may require care from multiple medical specialties.

3.4.2

Heuristics

The patient admission scheduling problem including all constraints on bed assignments and patient access times is proven to be NP-hard by [51]. Therefore, recent literature is more frequently applying heuristics to improve the admission schedule. Kifah and Abdullah [28], for example, apply a “great deluge” algorithm to optimize admission dates and bed assignments in an offline setting. In this paper, the used neighborhood-search algorithm is compared to other heuristics known in the literature and concludes that this great deluge algorithm can compete with more familiar heuristics such as simulated annealing.

3.4.3

Queueing Theory

Griffiths et al. [19] investigate an operational admission control for an ICU using an Erlang loss queue with both elective and emergency arrivals. For analyzing the bed census, they show that both the arrival streams and service rates can be combined into a single queue with multiple servers (e.g., an M/G/c/c queue). The authors analyze the system by controlling the elective patient admission dates based on the bed census using Euler’s method to analyze the loss queue with time-dependent arrival rates. Using historical data, they show that it is possible to estimate the most probable level of bed occupancy for several days in advance, given the bed occupancy on the current day. In addition, the model is able to predict the expected split between emergency and elective patients over the forthcoming days. Based on the expected bed occupancy in the near future, staffing levels can be adjusted.

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Markov Decision Theory

Optimizing patient scheduling decisions using a Markov decision approach typically results in complicated scheduling policies that are difficult to implement in practice. For example, Barz and Rajaram [2] model patients with stochastic length of stay at multiple hospital resources (e.g., beds and operating rooms) such that emergency patients can always be admitted and elective patients are delayed or deferred. Even the approximate dynamic programming model was not solvable within the set time limits for realistically sized instances, and the authors evaluate some heuristics based upon the results for small instances. A Markov decision process approach is also used by Yang et al. [54] to decide which surgeries have to be rescheduled such that the ICU capacity is not exceeded. The authors base a heuristic solution approach upon the obtained optimal policy and apply it to cardiothoracic ICU data of surgery requests. It appears that the heuristic policy outperforms the current admission policy significantly. Markov decision models are also used to determine an optimal bed assignment policy in an online setting. For example, Thompson et al. [44] consider a hospital in which patients should be admitted to a bed at their medically preferred ward or one of the predetermined alternative wards. Dai and Shi [11] consider a similar problem and use approximate dynamic programming to optimize the decision whether to assign patients to their medically preferred ward or to the “second-best” ward. Transferring patients during their stay may optimize the bed assignments and shorten the time between admittance and bed assignment. These transfer decisions are often optimized together with the assignment of newly admitted patients. Transferring patients during their stay could be optimal from a bed census perspective. One could ask if other factors (e.g., quality of care, patient condition, and staff workload) should also be considered implementing such decision rules.

3.4.5

Simulation

Simulation models are used to investigate a number of bed assignment policies for specific hospital case studies. For example, Braaksma et al. [7] study different policies to reserve beds for patients that are about to be brought to the operating theater. Landa et al. [31] evaluate different policies of reserving beds for patients that are admitted to the hospital through the emergency department. In Sect. 4.1 we describe a simulation model to investigate how many beds should be reserved for high-care patients, which implies that patients with lower care requirements should be admitted to a ward that is not their medically preferred one.

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3.5 Nurse-to-Patient Ratio The physical beds at a ward are often not the limiting factor in the number of patients that can be accommodated. The number and type of nurses and the specific patients present at the ward determine whether there is capacity for new admissions. The nurse-to-patient ratio says how many “average” patients one nurse can take care of; if there are 5 patients with a high care demand, a ward can be full, while with 15 patients with a low care demand, there may still be available capacity. An acceptable workload is important for the well-being of nurses and the quality of care.

3.5.1

Integer Programming

To balance the workload fairly among the nurses, mostly linear programming approaches exist in the literature. For example, Acar and Butt [1] consider patient acuity scores and travel distances for the nurses in optimizing the nurse-patient assignments. Braaksma et al. [6] additionally consider the continuity of care, education, and patient or nurse preferences in the optimization. Pesant [39] applies a goal programming to optimize the nurse-patient assignments, extending an MIP approach from [43]. In this model, patients have a nurse-dependent acuity, motivated by differences in experience, training, or preferences of the nurses.

3.6 Length of Stay and Readmission Forecast The length of stay of patients in a hospital is typically not known exactly before the patient is admitted and sometimes even not known exactly the day before the patient may be discharged. Moreover, when a patient is discharged, there is always a possibility that the patient has to be readmitted for further treatment. The time a patient is medically ready to be discharged and the readmission probability are useful in the patient scheduling process, as these determine how many new patients may be admitted. In recent literature, we see queueing theory, simulation, machine learning, and regression approaches, of which we provide examples below.

3.6.1

Heuristics

An example of a machine learning approach (random forest model) is used to forecast the length of stay of obstetric patients, using information on a patient’s medical history from the electronic medical records [16]. Roumani et al. [40] forecast the readmission probability for a cardiac ICU, for which they compare a support vector machine, decision trees, and a logistic regression approach. The results of these studies may be implemented in a decision support tool and provide

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guidelines to practitioners which clinical measurements that indicate a relatively high risk of prolonged length of stay or an increased readmission probability.

3.6.2

Queueing Theory

Queueing theory is, for example, used to investigate the effects of different discharge policies at an ICU [34]. These authors investigate the practical implications of the best policies using simulation. When a patient needs to be admitted to the ICU at a moment all beds are occupied, typically the “most healthy” patient is discharged to a ward with a lower level of care; optimizing such decisions may improve the quality of care significantly. For a general ward, Chan et al. [9] develop an infinite server queue in which a server may only be released after an inspection, which mimics the final doctor visit before a patient may be discharged. The results of the queueing analysis indicate that inspections should be at equidistant times and additional inspections have a decreasing marginal reward.

3.6.3

Simulation

An example of a discrete event simulation model is given by Crawford et al. [10], who analyze the effect of different discharge strategies on the readmission rate and emergency department crowding for a complete hospital. The authors conclude that a more “aggressive” discharge policy that discharges patients as early as possible increases the readmission rate significantly.

3.7 Conclusion We have shown a broad overview of situations where patients and staff at hospital wards can benefit from operations research analyses. Obviously, each different research question may require a different modeling approach, but, as we have demonstrated above, many models are applicable to analyze similar capacity decisions. Queueing models are fast to obtain estimates and are therefore applied often as a first indicator of, for example, the required capacity. Typically, Markov chain models are less “general” compared to queueing models, as they are more difficult to reapply to other wards but are easier to model transient behavior and ward-specific patient admissions, discharges, and transfers. Similar to Markov chain analyses, mixed integer programming approaches are relatively case study specific. However, using MIP approaches processes and schedules may be optimized, although a complicating factor often is the stochasticity in healthcare processes. Machine learning and regression approaches are very useful to analyze large amounts of hospital data and are increasingly used to assist medical decision-making at wards.

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4 Impact in Practice of OR at Wards In this section we present three case studies that we conducted at our partnering hospitals. In all projects the hospital has implemented the results of the research. In these short case studies we focus on the practical approach that was taken and the insights from implementation. Each case study gives unique insights on success factors, pitfalls, and lessons learned.

4.1 Case Study I: Balancing Bed Census Two medical wards at the Jeroen Bosch Hospital (JBH) experienced unbalanced bed occupancies during 2012 and the first months of 2013.3 At the neurology department of the JBH, patients’ LOS was reduced significantly, resulting in more slack capacity. At the same time, the department of internal medicine experienced increasing numbers of patients, resulting in crowded wards and many patients being deferred to other wards. Both over- and under-capacity are a problem for wards. In case a ward has undercapacity, patients cannot always be accommodated at the medically preferred ward. As a consequence, patients of one medical specialty are placed at many different wards, and doctors spend much time visiting all their patients. Having overcapacity is a problem for hospital staff as many patients from other medical specialties are likely to be placed at the ward. As a consequence, nurses from the ward have to care for patients for which they were not fully trained and may experience a high workload if they feel incompetent to treat patients from other medical specialties. In both scenarios, patients do not always receive the best possible care, which increased the willingness of all stakeholders for solving this problem.

4.1.1

Project Organization

In accordance with the list of factors in [45], at the start of this project we commissioned a steering group consisting of all stakeholders in this problem: a neurologist, a specialist internal medicine, an administrator from the patient admission scheduling office, and all involved ward managers. The hospital management made this steering group responsible for finding a solution for the over- and under-capacity at the wards and made one organizational and a healthcare logistics advisor member of the steering group. One representative of the highest management level below the board of directors was made chair of the steering group. The neurologist and internist were selected based upon the trust and goodwill they had from their peers. 3 This

case study was conducted by the author, among others, N.M.(Maartje) van de Vrugt in the role of healthcare logistics advisor.

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These representatives were not necessarily the heads of department, since it was required that these doctors spent time on the wards and experienced the problems on a daily basis. The first meeting of the steering group started with getting to know all members of the group; although all stakeholders work on closely related topics, typically they do not often meet and/or talk to each other. The group discussed to what extent they experience a problem at the ward or during patient scheduling. Supporting this discussion, the logistical advisor presented the results of a data analysis with information on (1) the bed occupancy of all hospital wards, (2) the bed requirement per medical specialty, and (3) the number of patients per medical specialty that was not treated at their medically preferred ward. The (fully anonymized) data that was used for this analysis was routinely collected hospital data on admittance and discharge date, medical specialty, and ward. The data analyses objectified the discussion significantly. For example, at the neurology ward the nurses experienced a high workload, and the hospital data confirmed that the nurses had to take care of relatively many patients from other medical specialties, which increased the experienced workload.

4.1.2

Analysis of Possible Interventions

The result of the first session was that the steering group wanted to investigate two possible interventions: 1. Opening an acute medical unit (AMU), and 2. Reassigning medical specialties to wards, Invoking an M/G/s/s queue, the required bed capacity to achieve at most 5% blocking probability was determined for each specialty. This analysis confirmed the belief of the steering group that the distribution of beds among the specialties was not adequate but adding capacity to the total system was not necessary. For intervention 2, each of the possible scenarios required serious rebuilding of units or medical specialties being split up among multiple wards. Rebuilding several wards would be costly and would take several months. Therefore, the steering group decided to discard this intervention. The effects of intervention 1 were analyzed for several scenarios using an M(t)/M(t)/s/s queue [45]. The conclusion of this analysis was that the acute medical unit would not be beneficial for the hospital’s case-mix, and the steering group discarded this intervention. At this point in the project, the steering group was looking for new possible interventions and decided to investigate the possibility of creating an overflow ward for internal medicine at the neurology ward. In the analysis of intervention 1, each doctor had to determine a list of diagnoses for their specialty that were eligible to be treated at an acute medical unit. This list consisted of diagnoses that required a relatively low care level, and each acute patient with a diagnoses from the list would

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be admitted to the acute medical unit. With minor moderations to the list by the internist, the list was adequate for the eligible overflow patients. Since the admittance data was anonymized, an exact analysis of the overflow ward was not possible. Financial hospital data revealed which part of all internal medicine patients had diagnoses from the list, which was used to estimate the total overflow bed requirements. This number of beds was high enough to alleviate the pressure on the internal medicine ward and was low enough to be accommodated at the neurology ward.

4.1.3

Choosing an Intervention

Based on this promising result, the organizational advisor helped doctors and nurses from internal medicine and neurology to investigate what would be required, for example, in terms of skills, education, and doctors’ rounds at the wards. The most important decision at this level was how often the internists would visit the overflow patients, which medical decisions were allowed to be made by neurologists, and when an internist should be called for assistance. Based on these discussions, nurses and doctors were confident that the quality of the provided care would be good for the overflow patients. Additionally, the logistical advisor conducted a simulation study in which historical data was used to determine the best policy to start and stop overflowing patients. In this simulation, each patient was randomly eligible for the overflow ward. The steering group requested this additional research as the neurology ward manager feared that, due to the overflow patients, not enough beds would be available for neurology patients. Several overflow policies and their effects were presented to the steering group. Based on all gathered results, the steering group decided to implement the overflow ward, using the policy: only overflow if both (i) three or fewer beds are available at the internal medicine ward and (ii) two or more beds are available at the neurology ward. In September 2013 the intervention was implemented at the hospital.

4.1.4

After Intervention

In January 2014 data analysis and interviews with the staff showed that the intervention had the desired effect: the neurology ward accommodated more internal medicine patients (on average 2.5 beds) and less patients of other specialties. Both effects were statistically significant. Additionally, the internists reported a reduction of the time required for their rounds, and the neurology nurses experienced a reduction of the fluctuations in the workload and were confident to deliver high quality of care. A downside of the implementation appeared to be a higher workload at the internal medicine ward, as many of the “easier” patients were admitted to the neurology ward.

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Lessons Learned

The success of this case study relates to the fact that the interventions were proposed by clinical leaders and on objectifying the effects of these interventions prospectively using data. Based on these analyses, the steering group was able to choose the most promising intervention to implement. The higher management let the steering group choose what interventions to investigate but had set a clear target to find a solution for the problem. This autonomy was greatly appreciated by the steering group. A very important part of the project was checking all assumptions and data analyses with nurses and doctors working at the wards. The goal of many data validation discussions was to come to an agreement that the data indeed reflected what happened in reality at the wards. A lengthy discussion about data or assumptions during steering group meetings would be undesirable, as this would lead the group away from finding a solution. During the project there was an emphasis on finding a solution that all steering group members and involved staff would consider a clear improvement of the current situation. To this end, next to data analysis, a thorough risk analysis was done for every intervention the steering group suggested. It was important not to ignore any of the concerns of the steering group, as this would decrease the willingness to cooperate in implementing the intervention. For each of the concerns raised, if possible data analysis was performed, and the steering group took time to discuss all concerns thoroughly, until either the issue was alleviated or the corresponding intervention was discarded. One example is that the neurologists were concerned that the internal medicine patients would displace the neurology patients. This issue was alleviated by a simulation analysis with multiple scenarios, which eventually lead to a decision rule for the patient admission planners. Before the project started, higher management had emphasized with the steering group members that this project was initiated to improve both quality of care and employee satisfaction. When discussions within the steering group were boiling down to competing interests of the individuals in the steering group, the chair of the meeting reminded everyone to stay focused on the quality of care and employee satisfaction. In all discussions this reminder sufficed to find a common goal and, eventually, a solution to the problem. The autonomy of the steering group and the iterative process of testing possible interventions resulted in an intervention supported by all involved staff. This support was the key to the success of the intervention. Moreover, the intervention proved to be effective in reality, which was the ultimate goal of the project.

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4.2 Case Study II: Dimensioning Wards The Leiden University Medical Center (LUMC) dealt with multiple logistical problems at its wards.4 These problems related to the small wards in terms of number of beds and a medically illogical distribution of patient groups among wards. This resulted in rising numbers of refusals at the emergency department and increasing waiting lists. Small wards have more difficulties coping with variability such as arrivals and LOS as it has relatively more impact. Therefore, small wards will often have over- and under-capacity.

4.2.1

Project Organization

In 2014 the hospital board of directors decided to redistribute patient groups among wards and re-dimension wards. Therefore, a project was initiated with a steering group consisting of a management director (project lead), a project manager, all care managers, an organizational consultant, a change management consultant, a human resource consultant, and a healthcare logistics consultant. Furthermore, there were multiple topics for further analysis defined and one working group for each topic. To overcome the earlier mentioned logistical problems, the project had to implement the following interventions: • Redistribute patient groups among wards for long stay patients (e.g., a LOS of at least 5 days). • Introduce a ward for short stay patients (e.g., LOS of less than 5 days). • Introduce a ward for day treatments (e.g., LOS of at most 8 h). • Introduce a ward for acute admissions (e.g., an AMU). • Merge ward staff and management that have medical affinity so that beds are interchangeable at these wards (e.g., orthopedic surgery with traumatology or nephrology with endocrinology wards). • Introduce a new management consisting of a physician and nurse manager. The hospital management decided that the total capacity should not be increased, so all interventions should be achieved without increase in the number of beds and nurses.

4.2.2

Analysis of Possible Interventions

As boarders are a risk to the quality of care, the hospital wanted to minimize the probability of refusals at the medically preferred ward. As presented in Sect. 3, queueing models dominate strategic and tactical analyses. We therefore chose to 4 This

case study was conducted by the author, among others, A.J.(Thomas) Schneider in the role of healthcare logistics advisor.

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model multiple scenarios of assignments of patient groups among wards as an M/G/s/s queue. Based on [13], we analyzed each scenario (e.g., the patient load from selected medical specialties at a ward or merged wards) on two performance measures: (1) what is the blocking probability given an occupancy rate of 85%, and (2) what is the occupancy rate given with a blocking probability of 5%? Based on these insights we redistributed the patient groups. Furthermore, we developed a simulation model to analyze the flow of acute admissions via the acute medical unit (AMU). Patients stay at the AMU for at most 2 days. When further treatment is needed, they are transferred to the inpatient ward of their medical specialty. From this analysis it appeared that solely introducing an AMU would not solve the problem of emergency admission refusals. We therefore analyzed the number of beds at wards that should be allocated to minimize the number of refusals. We also showed the effect of bed shortage at wards (e.g., finally resulting in an overcrowded AMU and emergency department). To prevent flow congestion, we analyzed scenarios with different numbers of beds dedicated for these transfers at each inpatient ward. We used two heuristics to find the best number of beds for each ward or merged wards. This simulation study was executed by a healthcare logistics consultant and ward management (nurse manager and medical manager of the AMU).

4.2.3

Choosing an Intervention

The actual re-dimensioning of wards and redistribution of medical specialties over these wards were completely based on the queueing model results. This required significant effort to convince the stakeholders of the reliability of the model outcomes. We used pseudomized admission data of 2013 and 2014 as input for the model and invested several weeks in discussing the assumptions of the model, results, and data. This is an important step when using queueing models in practice. Although queueing models are based on straightforward formulas, it can be challenging for stakeholders to interpret. For acceptance, the model should be thoroughly discussed and not used as a “black box.” We planned special sessions with each medical specialty to discuss the data, showing first the current patient load at each ward. We showed individual patients records to medical staff in terms of admission and discharge dates. Next, we discussed the model assumptions and the used input and key performance indicators. Lastly, we showed the proposed redistribution of patient groups and the effect on the bed usage. Taking time to present the model and answering questions from all stakeholders in both plenary and individual settings, we finally convinced most stakeholders of the proposed redistribution and re-dimensioning.

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After Intervention

After the interventions have been completed, the hospital still struggles to create a schedule for patients for at the weekday ward such that this ward can be closed during the weekends. Furthermore, on a later point in time, some medical specialties are again redistributed among wards resulting from new insights to organize wards according to the new strategy of the hospital that implied more thematically care (e.g., oncology care and transplantation care). Again, a queueing model was used for this new distribution of medical specialties.

4.2.5

Lessons Learned

A pitfall in this type of analyses is the requests for more up-to-date data. Given the size of the project, we needed 6 months to discuss the analyses with all stakeholders. During this time, many things can change, and therefore some stakeholders requested a new analysis with up-to-date input data such that the model would be more reliable (e.g., real-time hype). These requests delayed the project by a year. In the end, we did not update the data any further and reasoned with stakeholders that we used highly aggregated data over a long time horizon, which means that mainly trends and/or strategic decisions are detectable in the results. We also showed the added value of merging wards given the performance measures. In practice merging wards in university hospitals has major implications for nursing staff. As nursing staff is highly trained for specific treatments and specialties, they now have to be trained in other and/or more fields of medical expertise as more patient types can be placed at merged wards. Using a simulation model for the introduction of the AMU, we were able to show stakeholders how the emergency admissions process evolved over time [42]. This visual representation and the implementation of tailored process characteristics significantly contributed in convincing stakeholders. Therefore achieving consensus was easier compared to the use of queueing models for re-dimensioning and redistribution of wards. This was also reflected in the time needed to convince all stakeholders (2 months). The simulation model was used as a tactical tool in the planning and control cycle; in the model we updated the distribution of dedicated beds in each quarter using data with a rolling horizon (e.g., adding the last quarter and deleting the first quarter of the data). The success of this project was a result of clear sense of urgency throughout the organization to become future-proof. Staff at the operational level dealt with the consequences of the suboptimal distribution of beds and small units on a daily basis. The determination of the board of directors and the persuasiveness of the management convinced all stakeholders to let go of strict bed allocation policies, resulting in larger units. Also the use of operations research models gave the management a safe environment to experiment with new bed distribution and their key performance indicators.

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4.3 Case Study III: Bed Assignment Optimization The Massachusetts General Hospital (MGH, USA) deals with operational bed occupancies between 95% and 100%.5 As a consequence, patients generally wait long before an inpatient bed is available upon admission or transfer. This results in flow congestion at the postanesthesia care unit (PACU) and the emergency department (ED). Especially for emergency patients long waiting times increase risks. The state of Massachusetts therefore has the policy “Code Help,” requiring hospitals to move all admitted inpatients out of the ED within a 30-min period after the ED’s maximum occupancy – influenced by the number of patients present and their acuity – is reached or exceeded. Activating Code Help causes the hospital to prioritize moving patients out of the ED, which results in delaying bed assignments for patients from other areas of the hospital, potentially requiring cancellation of elective surgeries and other activities. The consequences of Code Help require significant management attention and can affect hospital operations for several days. In 2015, notifications that the hospital was approaching or had reached Code Help frequently occurred multiple times per week.

4.3.1

Project Organization

The continuous lockdown gave rise to a hospital-wide redesign of admission scheduling. Under supervision of the CEO, a project was initiated that consisted of the head of the perioperative department, the head and a bed manager from the admitting department, the nurse managers and resource nurses of several clinical units, a professor, a postdoctoral fellow, a graduate student in healthcare operations research, and two advisors from the department of process improvement of the MGH.

4.3.2

Analysis of Possible Interventions

Before the intervention, elective surgical same-day admits were preassigned to beds that were occupied with patients who were to be discharged on that day. This was done to guarantee a continual patient flow from the PACU to inpatient units; by reserving a bed for a surgical patient, the bed could not be taken by a patient from the ED or another department. However, the exact timing of discharges was unknown and uncertain. As a consequence, patients were frequently waiting for their preassigned beds, while simultaneously other beds were waiting for their preassigned patients. Data analysis showed that the average time patients waited for a bed ranged between 2.5 and 26.9 h, while a subset of four surgical inpatient 5 This

case study was conducted by, among others, our CHOIR colleague Aleida Braaksma [7] in the role of postdocoral fellow.

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units (129 beds) experienced a total bed-wait-for-patient time of 11,181 h or 466 bed-days in 2015. To alleviate this problem, a simulation model was developed to investigate the effects of multiple interventions, among which is a just-in-time (JIT) bed assignment strategy. This strategy only assigns patients to empty beds just before the moment they are medically ready, and therefore beds could not be preassigned to patients. A second intervention that was investigated was virtually pooling the capacity of two surgical wards, as they were clinically similar. This intervention implies that admissions are not preassigned to one of the wards, but the ward is decided upon the moment the patient is assigned to a bed. The input for the simulation model is 1 year of hospital data including timestamps for admission and discharge. From the data, empirical distributions were determined for, for example, bed cleaning duration and patient transportation time. The model was made more realistic by implementing bed closures according to the hospital data (e.g., due to staffing shortages) and the hospital’s policy with respect to gender and infection precautions in semiprivate rooms. Additionally, the model improved patient cohorting by occasionally swapping a patient from one room to another, mimicking the policy that was used in practice.

4.3.3

After Intervention

Based on the simulation results and 2 earlier projects by graduate students in healthcare operations research, the hospital implemented the JIT and pooling policy on 12 surgical inpatient units. In the 5 months post-implementation, the average patient wait time for bed decreased by 18.1% for ED-to-floor transfers (P < 0.001), by 30.5% for PACU-to-floor transfers (P < 0.001), and by 10.0% for ICU-tofloor transfers (P < 0.05). As a consequence, patients receive their required care earlier, which improves the quality of care. Additionally, the intervention resulted in a smoothed workload for nurses and bed cleaners and less congestion in the ED and PACU. Another positive side effect is an increased focus on patient flow: due to the JIT, nurses wonder why a bed is empty for a long time, which may speed up for example handoffs and transportation.

4.3.4

Lessons Learned

For physicians and nurses, simulation is relatively easy to understand compared to mathematical modeling. Therefore, the project team was convinced the intervention would have a positive effect in practice. The determination of involved clinical leadership (e.g., the head of the perioperative department) was also key for success. In the first days of implementation, nurses were sometimes skeptical about the intervention. The project team leaders showed empathy for the struggles related to the new situation while simultaneously encouraging nurses to stay put. Daily short evaluation meetings gave the opportunity to quickly react to unforeseen side effects

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of or negative sentiments around the intervention, before these could evolve into larger problems. The new policies resulted in more stable admission and discharge rates throughout each day. These more stable rates were also noticed by the bed cleaning department as their peaks in workload were reduced. This was therefore also seen as a nice (unforeseen) result of the project.

4.4 Increasing Impact in Practice In this section we provide our viewpoint on how OR researchers can increase the likelihood of results being implemented in practice. In our previous work van de Vrugt et al. [45] we provided conditions that support a successful implementation in practice. One of the most important contributions to a successful implementation is the involvement of one or multiple so-called clinical leaders, who are important medical stakeholders in the process and have the respect of all their colleagues. This leader should be able to speak on behalf of his/her colleagues and should discuss the project often with peers. As a researcher, you should earn the trust of these clinical leaders such that they are convinced about the methodological approach and proposed interventions. Ultimately these clinical leaders can (and should) convince other colleagues. In any model, assumptions are necessary for tractability. Some assumptions may in reality be too unrealistic to be of practical relevance. Therefore, in all our projects we start with one or several observation rounds, in which we study the process in reality, get familiar with the practitioners and their decisions, get to know the assumptions that are important to strictly hold, and see what flexibility and stochasticity are present. Letting practitioners draw a typical patient process is often not accurate enough to obtain all modeling assumptions. Moreover, seeing the outcomes of the process in the data does not mean that the preferred medical process was followed. The time that is invested in making the assumptions and relations in the model more realistic will significantly reduce the time spent on data analysis. Additionally, making the model more realistic will increase the likelihood of adaptation in practice. Furthermore, to increase to likelihood of implementation, a researcher should be able to convey how the model works to healthcare practitioners and thereby earn the trust of practitioners. Expectation management is very important; practitioners should know what the model can and cannot do. Often when the mathematics behind the modeling approach becomes less complex, the results become easier to grasp and trust by practitioners. A graphical simulation model, in which practitioners should recognize their everyday working routines, is an easy way to earn trust and convince practitioners. After the project is completed for the practitioners, a researcher can decide to continue with thoroughly investigating the problem or extending the model, to make the modeling approach interesting enough to publish

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in a OR journal. Otherwise, or perhaps simultaneously, publishing together with the practitioners in medical journals may be considered. In academia we are used to thoroughly investigate every interesting feature of a model before we present our results. In practice, an iterative process will be more effective; first mimic the current process and let practitioners check it (and repeat this if necessary), and second iteratively investigate a few scenarios and discuss them with practitioners. The most promising for implementation is when the stakeholders actively participate in the iterative process by proposing the interventions to be investigated. Additionally, presenting the results in easyto-grasp graphics will increase their impact; checking the results with the involved clinical leader before they are presented to all practitioners improves adapting the presentation to the audience. One risk of this iterative process is that the project never ends as more and more scenarios are investigated. This risk can be avoided by setting clear performance targets first in the project and by keeping a strict project schedule. Discussing possible interventions can be challenging because desired outcomes are often based on extreme incidents. Exploring interventions mathematically and thus rationally often simplifies the discussions significantly. Conveying the chosen intervention to colleagues becomes easier for the clinical leaders, as the decision was based on rational arguments.

5 The Future State of OR for Wards As a result of continuing innovations, we expect the trend of decreasing LOS to continue. Decreasing LOS results in increasing turnover. With more admissions per bed and with the same turnover time for beds to become available again, the total downtime of beds (time between a discharge and the new admission on one bed) will increase. So minimizing the downtime will be a subject for future research, for example, by looking at the planning of bed cleaning or the number of spare beds at a ward. Another direction for future research will be the use of big data for trend predictions and incorporating these predictions in the strategic admission planning. This implies that the number of beds at a ward can vary throughout the year. An example is the prediction of the yearly influenza epidemic and allocating more beds for pneumonia diseases during the winter. Nowadays most hospitals have an electronic patient record, and this (anonymized) data can be used to improve planning and scheduling. Descriptive models (e.g., machine learning) can classify and/or cluster these data. The results of descriptive models can be used as input for operations research models. By combining these techniques, the reliability of results will increase. An example for wards is the prediction of a patient’s LOS, based on multiple (preferably routinely collected) patient characteristics.

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Finally, we see trends into more regional collaborations between hospitals for specific patient groups. For example, elderly patients may live in a nursing home but may still be treated by hospital doctors. Patients are treated and monitored outside the hospital for as much as possible, by nursing homes, home healthcare, or general practitioners. Using stochastic network techniques, these collaborations can be optimized, in order to, for example, determine adequate capacity levels at all network locations. Acknowledgments We are sincerely grateful to our colleagues, Aleida Braaksma and Maartje E. Zonderland, for their valuable input.

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Part II

OR Applications in Healthcare Planning

A Markov Modelling Approach for Surgical Process Analysis in Cataract Surgery Maartje E. Zonderland, Siebe Brinkhof, Irene C. Notting, Richard J. Boucherie, Fredrik Boer, and Gré P. M. Luyten

Abstract Variability of surgical times heavily affects efficiency and utilisation of the operating room. This chapter develops a data-driven mathematical model that characterises the actions of the surgical process and their contribution to the total surgical time, including variability. The model gives insight into the surgical process, without the need to analyse a large number of surgeries for application of statistical methods. A surgical flow chart of cataract surgery is constructed by observing 85 cataract surgeries performed at Leiden University Medical Center (LUMC), combined with expert opinion. Markov chain analysis, based on this flow chart, is used to analyse the surgical process. The model identifies the sources of delay and variability in the surgical process and provides a structured way of analysing surgeries. The obtained surgical time distribution approximates the empirical surgical time distribution of cataract surgeries performed in 2009 and 2010 at LUMC. The model developed in this chapter may be used to study the influence of modifications in the surgical process and to predict the resulting surgical times. It can easily be adapted to analyse future surgical processes or to represent different surgical procedures.

M. E. Zonderland () · S. Brinkhof · R. J. Boucherie Center for Healthcare Operations Improvement and Research, University of Twente, Enschede, The Netherlands e-mail: [email protected] I. C. Notting · G. P. M. Luyten Department of Ophthalmology, Leiden University Medical Center, Leiden, The Netherlands F. Boer Department of Anesthesiology, Leiden University Medical Center, Leiden, The Netherlands © Springer Nature Switzerland AG 2021 M. E. Zonderland et al. (eds.), Handbook of Healthcare Logistics, International Series in Operations Research & Management Science 302, https://doi.org/10.1007/978-3-030-60212-3_6

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1 Introduction Operating room (OR) scheduling is an important challenge for most hospitals and has received considerable attention in the healthcare logistics literature. OR scheduling based on mean values of surgery durations omitting variability results in high expected OR overtime and surgeon idle time [2]. Deterministic and stochastic mathematical programming models, queueing models, simulation models and heuristic approaches have been widely used to study and improve OR scheduling [1, 5]. An important aspect of OR scheduling is an adequate prediction of the surgical times, based on patient and surgeon characteristics that involve uncertainty [16]. Prediction of the total surgical time may be obtained from historical data recorded in the hospital information system [4]. A surgical process is defined as a succession of surgical actions performed between the first incision and closure of the wound(s). The surgical path is a realisation of this process, with total surgical time being the sum of the duration of the surgical actions. The surgeon decides which surgical actions are performed; the so-called decision points may have major influence on the surgical time [16]. Prior to surgery, it is uncertain which actions will be performed, in which order and how much time each action will consume. A surgical plan based on, among others, the patient’s physical status gives an indication of the surgical time. However, this estimation is subject to variation due to complications and other, usually uncontrollable, factors. These uncertainties may lead to multiple possible surgical paths, with different durations and likelihoods of occurrence [9, 19]. Adequate prediction of these surgical times is of utmost importance for OR scheduling. This chapter develops a data-driven Markov modelling approach to characterise the distribution of the surgical times based on observations of the surgical actions in the surgical process. In addition, our Markov modelling approach may be used to predict the distribution of surgical times for future, novel surgical processes. The model decomposes the surgical process into well-defined surgical actions and possible subsequent actions, where its parameters may be obtained from observation, expert opinion or a combination thereof. The model is validated using observed cataract surgeries performed at Leiden University Medical Center (LUMC). Cataract surgery is selected to validate the Markov modelling approach, since its surgical time is relatively short, its surgical process involves a well-defined set of surgical actions, and cataract surgeries are carried out in sufficient numbers to obtain a large set of observations, which can readily be obtained from the operating microscope. Numerical results indicate a good fit of the surgical times predicted by our Markov model with realised surgical times. This chapter is organised as follows. Section 2 provides a brief literature overview of surgical time variability and Markov models for surgical times. Section 3 provides a description of the cataract surgical process and our data collection approach resulting in the surgical flow chart that is the basis of our Markov modelling approach in Sect. 4. A detailed description of the surgical time distribution is included in Sect. 5, followed by our conclusions in Sect. 6.

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2 Literature The degree as well as the source of variability in surgical processes is a widely studied topic in literature since its first description in [14]. Since then, several statistical models have been proposed for modelling surgical procedure times, based on log-normal and normal distributions of the total surgical time or surgical actions [15]. Surgeons working at constant but different rates due to, for example, experience and the surgeon’s natural speed introduce another source of variability as shown in [16] via the analysis of 46,322 surgical processes. Variability may also be the result of surgical process disruptions, i.e. deviations from the natural progression of a surgical path [18]. Surgical process disruptions affect the mental readiness of the surgeon, which is regarded as an important factor affecting patient outcomes, probably even more than technical skills or physical readiness. The number of these (usually) uncontrollable factors depends on the type of procedure [9]. The effects of disruptions on surgical time are categorised into one of six groups in [19]: instrument change, surgeon position change, nurse duty shift, conversation, phone/pager answering and extraneous interruption. The frequency and duration of each type of disruptive event were recorded and analysed; focus is on disruptions of the surgical process and their effect on variability in surgical time, but the surgical process itself is not studied in detail. A few modelling approaches to analyse surgical processes were developed in literature. Hidden Markov models are developed in [6, 11, 12] to evaluate surgical skills in minimal invasive surgery. This is done by comparing the Markov model of experienced surgeons to that of residents at various levels of training [10]. The state space of the Markov model represents the possible different combinations of instruments used by each hand of the surgeon. The surgical actions are not specified; focus is on the combination of instruments used by residents, compared to staff surgeons. The resulting skill level is based on four equally weighted criteria: overall performance, economy of movement, tissue handling and the number of errors such as dropped needles. An approach based on sensor data for instrument use to predict the (remaining) surgical time is developed in [3], where the prediction uses data mining and process mining techniques to obtain the surgical flow chart. This chapter develops a data-driven Markov model that enables a statistical evaluation of surgical processes, based on the flow chart of surgeries taking into account all possible surgical actions and paths. This model can be used to estimate surgical times of existing surgical processes, as well as surgical times for novel surgeries.

3 Cataract Surgery and Data Collection Approach Age-related cataract is a very common cause of visual impairment in older adults. As the lens ages, it increases in weight and thickness. The centre of the lens (nucleus) undergoes compression and hardening, and the lens takes on a yellow

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or brownish hue with advancing age. This is a cataract, and over time, it may grow larger, resulting in poor vision. The lens may be replaced to restore vision. The exact surgical paths for cataract surgeries is of utmost importance for a complete description of the process. Therefore, for the cataract surgeries incorporated in the present paper, we provide a detailed description of these paths (see Fig. 1 for the flow chart). To remove the lens, a standard extracapsular cataract extraction is performed through a 3.0 mm beveled corneal or limbal incision with a disposable phaco knife. The anterior chamber is then filled with an Ophthalmic Viscosurgical Device (Healon OVD, Abbott Medical Optics, inc.), which can be done in multiple steps, and capsulorhexis is performed. The capsule of the lens is a layer of approximately 14 micrometres thick, which is very fragile. It is important that the remainder of the capsular bag is undamaged. If it tears, the lens will fall into the inner part of the eye. This is a major complication that delays the surgical process considerably. For the removal of the cataracteous lens, the Millennium Phacoemulsification equipment (Bausch&Lomb) is used. This technique uses an ultrasonically driven tip (phaco tip) to fragment the centre of the cataract. The lens fragments are removed by irrigation and aspiration. Here, it is even more important to leave the lens capsule undamaged since it is very sensitive in this stadium of the process, being unsupported due to the removed nucleus. Depending on the surgeon’s preferences and patient characteristics, such as lens hardness, a divide-and-conquer or stop-and-chop technique can be used. During phacoemulsification relatively high amounts of energy are delivered which can potentially damage the eye. It is therefore important to use as little phaco energy as possible. Nevertheless, smaller particles – in general resulting from more sculpting – are easier to remove, decreasing the risk of damaging the capsular bag. Finally, cortex remnants are removed with a bimanual irrigation and aspiration instrument. The OVD is used to implant the new intraocular lens and is removed thereafter with irrigation and aspiration of the canula. Suturing of the corneal wound is not necessary as a rule. A prospective case-series study was performed at the Department of Ophthalmology of the Leiden University Medical Center, Leiden, the Netherlands (LUMC). The study was performed in accordance with the principles of the Declaration of Helsinki. The medical ethical committee of LUMC (full name: Commissie Medische Ethiek Leids Universitair Medisch Centrum) approved this study (CME decision #CME09/028) and waived the requirement for written informed consent. All surgeries were performed by four experienced surgeons and three residents from the LUMC and were performed in 2009 and 2010. A total of 44 male and 41 female patients were included, with average age 71. Due to the last resort function of the LUMC, a high level of co-morbidity was present. Therefore, these patients might not represent the average cataract surgery population. Two cameras were set up in the OR, the first in the microscope of the surgeon and the second in a corner of the OR, giving a good overview of the surgical team and their interactions. Prior to surgery, patients were informed about the recordings, which were anonymised afterwards. Informed consent was given, and recordings started when the eye was draped and the eyelid speculum was placed. Solely the patient’s eye is visible on the video registration.

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#

Surgical acon

#

Surgical acon, decision point (>1 possible transions)

1

2

3 4 5 6 7 8

9

10

19 22

13

11

12

15

14

20

16

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17

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18

26

25

27 28

24

31

29 30

32

33

34

36 35 37

39

38

40

State 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Surgical acon Start of surgery Corneal incision, two side ports Healon in anterior chamber Scky iris, loosen with spatula Iris retracon Healon in anterior chamber, iris retracon/stricky iris Capsulorexis Hydrodissecon Enlarge main port in cornea for alternave lens Sculpt lens, 4 parts Sculpt lens, 2 parts Remove lens, without sculpt/crack Mixed sculpt/crack Remove lens surplusses, without sculpt/crack Crack lens, 2 parts Remove lens, 2 parts Remove lens surplusses, 2 parts Healon in capsular bag Crack lens, 4 parts Remove lens, 4 parts Remove lens surplusses, 4 parts Push out lens as a whole, without sculpt/crack Remove lens surplusses for whole lens Insert alternave lens Enlarge main port in cornea Remove surplusses aer Healon in capsular bag Fold lens prior to injecon Error lens injecon Prepare second aempt lens inseron Insert folded arficial lens Second aempt lens injecon Posion second lens Unfold and posioning of folded arficial lens Remove surplusses aer lens inseron Posion and flush with Irrigaon/Aspiraon Irrigaon and aspiraon Finish surgery, corneal incision WITHOUT stches Finish surgery, corneal incision WITH stches Finish surgery with enlarged main port WITH stches End of surgery

Fig. 1 Graphical representation of the cataract surgery flow chart, where arrows indicate possible transitions, and the table gives the corresponding surgical actions. The grey states indicate decision points, with more than one outgoing arrow

The process of cataract surgery in terms of surgical actions and their order of appearance is visualised by a surgical flow chart. Surgical actions are represented by circles and their order of appearance by arrows between actions. The time-action technique described in [7] is employed. Surgical actions that are performed by the

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surgeon are registered in chronological order, with the start and end times of each action determined by analysis of the video registrations. After each surgery, the surgical process is discussed with the surgeon. The flow chart was constructed from video observations and expert opinion. Discussions with surgeons declared welldefined surgical actions with start and end point indications such as usage of a different instrument. Each observation defines a surgical path, and a combination of these paths may result in unobserved, but possible surgical paths. The resulting surgical flow chart of cataract surgery is given in Fig. 1. The legend specifies the corresponding surgical actions as defined from discussions with the staff surgeons.

4 Markov Model Markov models are frequently used to analyse processes in which uncertainty plays an important role. For a comprehensive overview of Markov models, see [13, 17]. Our assumptions underlying the Markov modelling approach for surgical processes are the following: (i) Each surgical process consists of a number of well-defined actions with start and end points. (ii) Upon completion of a surgical action, the next action is selected from a welldefined set of possible actions, where selection of the next action is determined only by the current action. (iii) The sojourn times of surgical actions are independent random variables with a normally distributed duration. The surgical flow chart is used to develop a Markov model to analyse the surgical process. States represent surgical actions, and for each state the successive state is determined by transition probabilities. A possible sequence of states forms a surgical path. Each surgical action has a random duration with a corresponding probability distribution, for which we assume a normal distribution. The normal distribution is commonly used to characterise variables that tend to cluster around a mean with variability caused by small effects [13]. This assumption includes a positive probability of negative length. Nonetheless, the resulting errors may be negligible or within acceptable limits, allowing us to obtain a statistical estimation of surgical times with sufficient accuracy. Upon completion of a surgical action, there may be several possible subsequent actions from which the surgeon selects. Their fraction is modeled by transition probabilities. A more frequently selected action results in a higher value of the transition probability. Summarising, the Markov model has the following characteristics: S

State space of the Markov model containing all possible states of the process, representing the surgical actions of the surgical process.

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pi,j

τi

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Transition probabilities of the Markov process moving from state i to state j , representing the fraction of surgeries where surgical action j follows action i. This fraction may equal zero, which means that the two surgical actions cannot be in direct succession. Random variable representing the time spent in state i (sojourn time) before the process progresses to the next state. The duration of a surgical action is described by this parameter. The normal probability distribution is used to model the variability in this parameter.

Assumption (ii) might seem unrealistic as decisions in decision points may differ due to previous surgical actions. If that is the case, then decision states will be separated into multiple states with their separate surgical actions and transition probabilities. As an illustration, consider state 18 in Fig. 1. If selection of the next states 25, 26, 27, 28 and 30 depends on the surgical path leading to state 18, then state 18 would be separated into states 18a, b and so on, with their own set of transition probabilities. The surgical time equals the summation of the individual surgical action durations on a surgical path. For each (observed) surgical process, k the sequence of actions is registered in terms of the corresponding surgical flow chart. A sequence xk defines the surgical path sequence. The associated durations of each action are also registered as a sequence: the surgical time sequence tk . Let mk define the number of actions in surgical process k, and denote: Surgical path sequence of surgical process k : xk = {xk1 , . . . , xkj , . . . , xkmk }, (1) Surgical time sequence of surgical process k : tk = {tk1 , . . . , tkj , . . . , tkmk },

(2)

i.e. xkj and tkj are the j th action and time in surgical process k. Due to the assumption of normally distributed sojourn times for each surgical action, the surgical time for each surgical path is a summation of normally distributed sojourn times and therefore also has a normal distribution [13]. If the sojourn time of surgical action i has a normal distribution with mean μi and variance σi2 , then the surgical time Nxk of surgical path xk has a normal distribution with mean μxk and variance σx2k , given by μxk =



μi ,

(3)

σi2 .

(4)

i∈xk

σx2k =



i∈xk

The transition probabilities between surgical actions represent the fraction of times a surgeon selects a certain subsequent action. For example, if the transition probability from state i to state j equals 0.1, then surgeons select surgical action j following

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action i in 10% of the surgeries. The probability of occurrence oxk of a particular surgical path xk is obtained by multiplying the transition probabilities associated with the transitions in the surgical path. The probability of occurrence oxk of surgical path xk with mk surgical actions is given by 

mk−1

oxk =

j =1

pxkj ,xkj +1 .

(5)

The total surgical time distribution T is a weighted combination of the surgical times of all possible surgical paths. Note that these paths include but are not restricted to the paths observed to determine the surgical flow chart. The weight of path xk is its probability of occurrence oxk . Let X denote the set of all surgical paths (observed and unobserved). We thus obtain T =



ox · Nx .

(6)

x∈X

5 Data Analysis and Results From the observations of 85 cataract surgeries at the Department of Ophthalmology of the LUMC in 2009 and 2010, a total of 38 different surgical actions plus a start and end state are identified. Figure 1 gives all surgical actions and transitions between surgical actions that were observed. We identify 15 decision points indicated by the grey states, where multiple subsequent actions may be selected by the surgeon. To determine the characteristics of the Markov chain model for the surgical flow chart, from the observed data, we determine for each action i the mean μi and variance σi2 using the standard estimator for the sample mean and sample variance for the normal distribution. Table 1 gives for each action the number of times the action was observed and the estimated mean and variance of the sojourn times. For some states the standard deviation is large with respect to the mean due to a relatively small number of observations. States 26, 31 and 32 are observed only once, so for these states the standard deviation cannot be estimated and is set to 0. The transition probabilities pi,j between action i and action j are determined for the decision points using the standard estimator for transition probabilities as the fraction of times action i was followed by action j . Table 2 gives the transition probabilities. The number of possible paths in the flow chart of Fig. 1 is much larger than the observed number of 85 surgeries. Table 3 lists the number of possible paths from each decision state to the end of surgery in state 40. The total number of paths that may be identified from our observations is 7590, which clearly illustrates the complexity and the large number of possible surgical paths even for a surgical process with a small number of decision points such as cataract surgery.

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Table 1 Mean, standard deviation of sojourn times and number of observations for the states in the cataract surgery model State 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Description Start of surgery Corneal incision, two side ports Healon in anterior chamber Sticky iris, loosen with spatula Iris retraction Healon in anterior chamber, iris retraction/sticky iris Capsulorhexis Hydrodissection Enlarge main port in the cornea for alternative lens Sculpt lens, 4 parts Sculpt lens, 2 parts Remove lens, without sculpt/crack Mixed sculpt/crack Remove lens surpluses, without sculpt/crack Crack lens, 2 parts Remove lens, 2 parts Remove lens surpluses, 2 parts Healon in capsular bag Crack lens, 4 parts Remove lens, 4 parts Remove lens surpluses, 4 parts Push out lens as a whole, without sculpt/crack Remove lens surpluses for whole lens Insert alternative lens Enlarge main port in cornea Remove surpluses after Healon in capsular bag Fold lens prior to injection Error lens injection Prepare second attempt lens insertion Insert folded artificial lens Second attempt lens injection Position second lens Unfold and positioning of folded artificial lens Remove surpluses after lens insertion Position and flush with irrigation/aspiration Irrigation and aspiration Finish surgery, corneal incision WITHOUT stitches Finish surgery, corneal incision WITH stitches Finish surgery with enlarged main port WITH stitches End of surgery

x¯ 0.0 38.0 19.7 78.0 307.7 42.0 98.3 42.3 265.5 168.0 59.0 85.4 351.0 205.7 29.3 151.2 176.7 24.0 56.2 141.3 251.3 65.0 665.0 328.8 34.5 59.0 87.6 32.3 156.0 34.3 142.0 19.0 36.7 88.1 65.0 63.9 72.8 191.7 580.8 0.0

SD 0.0 29.0 9.2 23.3 179.1 44.0 82.2 41.6 71.5 103.3 17.7 32.6 260.8 66.8 14.3 73.4 75.8 12.5 56.9 97.1 144.8 39.0 102.0 97.1 4.5 0.0 16.0 17.4 55.5 30.6 0.0 0.0 70.4 53.2 29.3 32.1 60.8 107.9 318.5 0.0

N 85 85 84 3 3 5 85 83 2 59 13 7 4 7 13 13 13 81 57 63 62 2 2 4 2 1 7 3 3 80 1 1 45 7 46 28 60 14 4 85

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Table 2 Transition probabilities pi,j for the cataract surgery model State 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

pi,j p(1,2)=1.0000 p(2,3)=0.9882 p(3,4)=0.0238 p(4,6)=1.0000 p(5,6)=0.6667 p(6,7)=1.0000 p(7,8)=0.9765 p(8,9)=0.0120 p(9,22)=1.0000 p(10,19)=0.9661 p(11,15)=1.0000 p(12,14)=1.0000 p(13,20)=1.0000 p(14,18)=1.0000 p(15,16)=1.0000 p(16,17)=1.0000 p(17,18)=1.0000 p(18,25)=0.0123 p(19,20)=1.0000 p(20,18)=0.0159 p(21,18)=0.9677 p(22,23)=1.0000 p(23,24)=1.0000 p(24,39)=1.0000 p(25,24)=1.0000 p(26,27)=1.0000 p(27,28)=0.1429 p(28,29)=1.0000 p(29,30)=1.0000 p(30,33)=0.5625 p(31,32)=1.0000 p(32,36)=1.0000 p(33,34)=0.1111 p(34,35)=0.2857 p(35,37)=0.7609 p(36,37)=0.6429 p(37,40)=1.0000 p(38,40)=1.0000 p(39,40)=1.0000

pi,j

pi,j

p(2,4)=0.0118 p(3,5)=0.0357

p(3,7)=0.9405

pi,j

pi,j

p(8,12)=0.0843

p(8,13)=0.0482

pi,j

p(5,7)=0.3333 p(7,9)=0.0118 p(8,10)=0.6988

p(7,10)=0.0118 p(8,11)=0.1566

p(10,20)=0.0339

p(18,26)=0.0123 p(18,27)=0.0741

p(18,28)=0.0247 p(18,30)=0.8642 p(18,31)=0.0123

p(20,21)=0.9841 p(21,25)=0.0161 p(21,30)=0.0161

p(27,30)=0.8571

p(30,34)=0.0250 p(30,35)=0.4000

p(30,37)=0.0125

p(33,35)=0.2667 p(33,36)=0.6000 p(33,37)=0.0222 p(34,37)=0.7143 p(35,38)=0.1522 p(35,40)= 0.0870 p(36,38)=0.2500 p(36,40)=0.1071

In 2009 and 2010 a total of 2041 cataract surgeries was carried out in the LUMC. A histogram of the realised surgical times as well as the best-fit surgical time distribution is depicted in Fig. 2a, c, where in Fig. 2c the solid line represents surgeries carried out by all surgeons and the dashed line surgeries carried out by staff surgeons only. Figure 2b shows a histogram of the realised surgical times of the 85 observed cataract surgeries. To validate our Markov modelling approach, we compare our estimate of the surgical time distribution with that distribution for the realised surgeries in 2009

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Table 3 Number of paths from decision states to end state 40 State 1 2 3 4 5 6 7 8

# of paths 7590 7590 6072 1518 3036 1518 1518 1005

State 9 10 11 12 13 14 15 16

# of paths 1 512 118 118 256 118 118 118

State 17 18 19 20 21 22 23 24

# of paths 118 118 256 256 138 1 1 1

State 25 26 27 28 29 30 31 32

# of paths 1 38 38 19 19 19 3 3

State 33 34 35 36 37 38 39 40

# of paths 11 4 3 3 1 1 1

Table 4 Recorded length of cataract surgeries for different surgeons Surgeon 1 2 3 4 5 Total

Type Staff Staff Staff Staff Residents

Observations (N) 12 14 20 28 11 85

Surgeries in 2009–2010 (N) 251 541 231 382 144 1549

x¯ surgical time (s) 805 1384 939 900 1536 1058

SD surgical time (s) 496 643 299 428 432 515

Not all surgeons that carry out cataract surgeries of LUMC were included, so the total number of surgeries (1549) is smaller than the 2041 surgeries that were used for the histogram of Fig. 2a

and 2010. As surgeons perform a varying number of surgeries per year, the fraction of the number of observations per surgeon in our sample set of logged data do not fully match the fractions of surgeries carried out by each surgeon in 2009 and 2010. Comparison of the surgical time distribution from our 85 observations with that from the 2041 realised surgeries should take into account the differences between surgical processes over surgeons and residents as the decisions made during surgery and the speed of actions may vary over surgeons. Therefore, to compare with the logged data from 2009 and 2010, the registered surgeries are weighted, such that the weight of the fraction of surgeries carried out by each surgeon in the 85 observed surgeries represents the weight of the total number of surgeries carried out by each surgeon in the observed period. Table 4 gives the number of surgeries carried out by four staff surgeons and residents, where we do not discriminate between residents. Observe the large differences both in mean and standard deviation of the surgical times. Not all surgeons who perform cataract surgeries of LUMC were included in Table 4, so the total number of surgeries (1549) is smaller than the 2041 surgeries that were used for the histogram of Fig. 2a. Figure 2d shows the surgical time distribution for the 2041 cataract surgeries carried out in LUMC in 2009 and 2010 (solid line) and the surgical time distribution from our Markov modelling approach taking into account the differences in handling speed and preferences between surgeons, as well as the influence of number of surgeries carried out by different surgeons (dashed line). These graphs show a striking match which illustrates the validity of our Markov modelling approach.

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Fig. 2 Overview of realised and modeled surgical times. (a) Histogram of 2041 surgical time realisations in LUMC in 2009 and 2010. (b) Histogram of 85 surgical time realisations used to fit the model parameters. (c) Comparison of surgical time distributions of modeled surgical times for all surgeons (solid line) and staff surgeons only (dashed line). (d) Comparison of the observed surgical time distribution for the 2041 cataract surgeries in 2009 and 2010 (solid line) and the surgical time distributions from observations of 85 cataract surgeries taking into account the weighted fraction of surgeries carried out by different surgeons (dashed line)

The main characteristics of the surgical time distribution are captured by the Markov model. First, observe that each surgical path has a normally distributed surgical time. The surgical time distribution is a mixture of these surgical times. The mixture of random variables with a normal distribution is, in general, not normally distributed. Our model captures the skewness of the surgical time distribution (part (c) of Fig. 2). Second, the graphs of the predicted and observed surgical times (parts (a)–(c) of Fig. 2) do show a great level of similarity. The surgical time distribution obtained by the Markov model (part (c) of Fig. 2) and the histogram of realised surgical times (part (a) of Fig. 2), however, also show some discrepancy. There are two main explanations of this discrepancy. First, note that the number of observations used in this study is limited, so that estimates of times per surgical action and transition probabilities show considerable variance. Second, disruptions as categorised in [19] are only partly taken into account in the mathematical model. Most notably, interruptions of the surgical process are ignored; the recording was paused during interruptions. Such interruptions occur at random times with highly unpredictable duration. Incorporating these interruptions, which usually result in considerable delay, would likely have resulted in a more skewed surgical time distribution. Since the steps in the surgical process were used to obtain a model that allows for prediction of surgery times of (future) surgeries, it is legitimate to not take into account these interruptions.

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6 Conclusions The Markov model presented in this chapter reveals the structure of the surgical process for cataract treatment in terms of the possible sequences of surgical actions and their estimated durations. It provides a detailed description of the surgical process and predicts the total surgical time. The data- and knowledge-driven approach leads to a structured way of analysing surgeries in terms of process characteristics. The Markov model can be adapted to analyse future surgeries of different types. These models will give insight into the surgical process, without the need to analyse a large number of surgeries for application of statistical methods. Further analysis of the Markov model could enable the development of methods to decrease variability and skewness by standardising surgical paths, searching for optimal surgical paths and optimal solutions for complications during surgery. The Markov model appears to capture the total surgical time well. It requires detailed data about the surgical process, which is currently not often available. Healthcare data is exploding, and it is reasonable to assume that also data required for detailed Markov models for surgical processes will become available in the near future. If prior to a surgery it is known which paths are likely to be chosen by the surgeon, the expected surgical time can be estimated by analysing these paths. Differences in preferences and speed between surgeons can be analysed as well and could serve as a model for benchmarking and learning. Residents are able to compare their surgical skills and working pace with those of staff surgeons. Part (c) of Fig. 2 shows such an application of the model. Here the surgical times for all surgeons and only staff surgeons are compared. A more detailed comparison for different groups or individual surgeons may be undertaken using the model. This was beyond the scope of the present study. Categorising patients pre-surgically, for example, on gender or age, can be used to estimate their surgical time based on the patient characteristics and might lead to improved OR scheduling. In addition, during surgery the remaining duration of the surgical process might be estimated from the analysis of the Markov chain from the current action to end of surgery. By doing so, this structured approach suits the need of modern healthcare facilities by increasing both operating room efficiency and patient comfort, through increased insight in the surgical process. Measurement errors in the realised surgical time data were observed, due to unclear starting time indicators for the time registrations. Additional observations, to complete coverage of all possible surgical paths, would lead to a better validation of the model. Automation of the observation process would be interesting for further research, where recordings matched to the instruments that are used during the surgeries can be related to specific surgical actions. This can be used to automatically construct a state space and the definition of model parameters, as described in [8]. In particular, when analysing new surgical techniques, this approach may substantially reduce the efforts in obtaining logged data to construct a suitable Markov model.

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Emergency Operating Room or Not? Nardo J. Borgman, Ingrid M. H. Vliegen, and Erwin W. Hans

Abstract Hospital operating theaters often face the problem of unscheduled emergency arrivals that should be treated as soon as possible. In practice different policies are used to allocate these emergency patients to the operating rooms. These policies are (1) keeping operating rooms empty and available for emergency arrivals; (2) treating emergency patients in elective operating rooms, postponing elective patients; and (3) a mix of these two policies. The use of a specific policy affects performance (e.g., utilization, waiting times, overtime). Currently, these effects are not clear, and there is no agreement on what works ‘best’ for a specific hospital. Using discrete-event simulation, we evaluate the policies for many case characteristics such as hospital size, patient case mix, and fraction of (emergency) patients. We gathered the simulation results in a tool called OR analyzer. This tool is made available online and allows healthcare practitioners to gain insight into the effects of the scheduling policies in settings similar to their specific hospital setting. In addition, this tool allows others researching emergency scheduling policies to frame their hospital settings and compare results.

1 Introduction Hospitals that perform surgeries on both elective and emergency patients are faced with the problem of how to deal with arriving emergency patients [15]. In this chapter we study three emergency surgery scheduling policies, commonly

N. J. Borgman Gelre Ziekenhuizen, Apeldoorn, The Netherlands I. M. H. Vliegen Eindhoven University of Technology, Eindhoven, The Netherlands E. W. Hans () Center for Healthcare Operations Improvement and Research, University of Twente, Enschede, The Netherlands e-mail: [email protected] © Springer Nature Switzerland AG 2021 M. E. Zonderland et al. (eds.), Handbook of Healthcare Logistics, International Series in Operations Research & Management Science 302, https://doi.org/10.1007/978-3-030-60212-3_7

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encountered both in practice and in literature. The policies differ in the way emergency patients are allocated to operating rooms (ORs): 1. Reserve ORs for emergency patients (dedicated policy). 2. Treat emergency patients in elective ORs (flexible policy). 3. A mix of the above (hybrid policy). Policy 1 assigns all emergency patients to dedicated emergency operating rooms, thus preventing emergency patients from disrupting the elective program. If all emergency ORs are occupied, arriving emergency patients wait and are treated in the first dedicated emergency OR that becomes available. Policy 1 poses a trade-off between operating room utilization and emergency surgery waiting time: having more emergency ORs reduces waiting time but also reduces utilization. Policy 2 assigns all emergency patients to elective operating rooms. This is done by inserting emergency surgeries into the elective program without preempting elective surgeries. Consequently, an emergency surgery may start in any elective OR as soon as one becomes available, and elective surgeries must then be postponed or rescheduled. Furthermore, since preemption is not allowed, emergency patients may have to wait. Policy 3 combines advantages of both Policies 1 and 2; emergency patients may be treated in both elective and emergency ORs. If all emergency ORs are occupied and an elective OR is available, emergency patients are treated in this elective OR. In case all ORs are busy, emergency patients that arrive wait and are treated in the first OR that becomes available, regardless of OR type (i.e., elective or emergency OR). This policy aims to reduce the interruptions in the elective patient schedule caused by emergency arrivals while still allowing timely treatment of emergencies. Hospitals are thus faced with the question which policy to choose and, in case of Policies 1 and 3, how many ORs to dedicate to emergency patients (Fig. 1). The use of a specific policy affects a hospital’s performance (e.g., utilization, waiting times, overtime). However, these effects are not clear from literature, and there is no agreement on what works best and in which setting(s) to choose which

Fig. 1 Visualization of different emergency patient allocation policies

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policy. For example, using a flexible policy (2) is stated to both decrease [23] and increase [4] overtime of ORs. Differences in hospital characteristics may account for the difference in results among papers. For example, using a flexible policy may only be possible if there are sufficient ORs to ensure a timely start of treatment for emergency patients. Further contradictory results on policies are reported in [15]. Please check sentence starting "Using computer simulation we evaluate the different policies under many hospital settings, allowing us to evaluate the trends and effects of different characteristics on the performance of policies, which has previously not been done in literature. In addition, we take into account many often encountered performance indicators. Previous studies often use performance indicators relevant for their specific case study which may differ considerably from other studies, making comparison difficult. The simulation results are gathered in a tool called OR analyzer which is made available online. The tool allows healthcare practitioners to gain insights into the effects of the scheduling policies in settings similar to their specific hospital setting. In Sect. 2 we review the literature, Sect. 3 details our approach, and Sect. 4 describes the simulation model and OR analyzer tool. Sect. 5 discusses results, and Sect. 6 provides a discussion and conclusion.

2 Literature There is a considerable amount of literature on operating room (OR) planning and scheduling. As the ORs are often seen as the ‘core’ of hospitals, it is not surprising that this is a topic of much interest. Literature reviews on (among others) OR planning and scheduling are provided in [3, 7]and [10]. We consider the OR planning problem, where emergency surgery arrivals have to be taken into account. Work that specifically addresses non-elective surgeries and takes such surgeries into account on a tactical and/or operational level is less common. Two recent literature reviews that give an overview of OR planning and scheduling literature, taking into account non-electives and the trade-offs they pose, are [6, 15]. Van Riet et al. [15] note that it is still unclear what the effect is of different settings on the possible performance measures and that a key question is how to determine what level of demand and patient mix is required before a specific policy should be pursued. Starting from the aforementioned literature reviews, we selected studies that evaluated the use of one or more policies to deal with unexpected emergency patient arrivals from an operational research perspective. In addition, we also include more recent studies not covered in the latest literature reviews. Wullink et al. [23] compare the use of dedicated emergency ORs and general ORs, for a case study with 12 ORs in total. Looking at waiting time for emergencies, overtime, and utilization, they conclude that general ORs perform better on all performance measures when using a flexible policy. Based on their work, the hospital decided to close dedicated ORs. Van Veen-Berkx et al. [21] compare the operating theater performance before and after the dedicated OR was closed in the same hospital and find contradicting results in practice, concluding that overtime increased. It is unclear, however, whether

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other changes took place in between measurements. In addition, overtime is defined differently between studies. Wullink et al. [23] define overtime as the time used for surgeries after regular OR time ends. Van Veen-Berkx et al. [21] define overtime as the difference (in minutes) between scheduled and actual surgery end time of the last patient of the day. Under a flexible policy, however, less elective patients are scheduled in ORs (to account for the emergency arrivals), and the end time of the last scheduled patient occurs before the regular closing time of the OR. This difference between scheduled end time and regular OR closing time may account for the conflicting overtime results. Another work investigating a dedicated and flexible policy is by Ferrand et al. [4]. They consider a hospital with 20 operating rooms and, in case of the dedicated OR policy, vary the number of dedicated ORs. They also evaluate waiting time, utilization, and overtime. Contrary to the results of Wullink et al. [23], the average overtime per day increases under a flexible policy. Similarly, Persson and Persson [14] use simulation to model a Swedish orthopaedic department with two ORs and find a dedicated policy outperforms a flexible one. However, in their comparison, total available capacity also increased. More recent work by Ferrand et al. [5] investigates the effect of utilizing some ORs by both elective and emergency patients. Their case study again has 20 ORs, and they vary the number of dedicated and flexible ORs. Also, they perform a sensitivity analysis to evaluate the effect of increased emergency arrivals, nonstationary (emergency) arrivals, and different procedure times for elective patients. Lans et al. [11] use computer simulation to look at both tactical and operational planning levels when evaluating different policies to anticipate emergency surgeries. On the tactical level, they evaluate: (1) dedicated ORs, (2) planned slack in some ORs, and (3) planned slack in all ORs. Performance measures used are utilization, overtime, and waiting time of emergency surgeries. Accounting for all measures, they find that using planned slack in all ORs performs best (i.e., a flexible policy). A different policy is evaluated by Bowers and Mould [2] that investigate the effect of planning elective patients in an orthopaedic emergency OR session using simulation. These patients have a shorter access time; however there is a larger chance of cancellation. They find that throughput and utilization may increase, as long as elective patients are willing to accept a cancellation probability. Bhattacharyya et al. [1] report a retrospective analysis where two 1-year periods were compared before and after the use of a dedicated emergency OR for orthopaedic surgeries. They measured the utilization, cases done during the night (overtime), and the delay frequency of elective surgeries caused by emergencies. They conclude that overtime was considerably reduced by using a dedicated OR. Similarly, Wixted et al. [22] perform a retrospective review after a dedicated trauma OR was opened. They compare performance for a specific surgery type (isolated femur fractures) and find that less overtime and disruptions have taken place since a dedicated OR is used. However, during the study several changes occurred such as an increase in surgeons, as well the number of surgeries performed. Sandbaek et al. [16] carry out a before and after analysis of implementing a dedicated policy (from a flexible policy), where three ORs are reserved for emergency patients. They report

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a reduction in overtime for both elective and emergency ORs, as well as an increase in utilization. However, during this redesign of the ORs, a new patient classification and booking policy was also introduced, which may have affected the results. Under Policy 2 or 3, the way in which elective patients are scheduled may impact performance. Given that surgeries cannot be interrupted, emergency patients can only be treated if an OR is available and otherwise must ‘break in’ into the elective schedule when an elective surgery completes, before the next elective surgery starts. These completion times of surgeries are also called ‘break-in-moments’ (BIMs) [20]. As such, it may be beneficial for emergency patients to ensure that BIMs are spread evenly across the day. Van Essen et al. [20] take this into account and optimize surgery schedules such that the maximum time between BIMs is minimized. Most research into OR planning reported in literature started from a single case study, with specific hospital characteristics. This may lead to conflicting results and be of limited value to healthcare providers with their own specific case characteristics. Our contribution is that we fill this gap by investigating the performance of the different policies, under many different case characteristics, and evaluate the circumstances under which specific policies may perform best. In addition we develop, and make available, a tool that allows readers and healthcare practitioners to further evaluate the policies under different hospital characteristics.

3 Approach In this section we present the approach taken to investigate the different policies defined in Sect. 1. To compare policies not only among each other, but also under different case settings, we define broad hospital characteristics that make up a specific case setting. All possible case settings are then evaluated under all possible policies, including multiple settings per policy (e.g., Policy 1 having 1, 2, and so on emergency ORs) using computer simulation. Table 1 lists all the characteristics that make up a policy and scenario. Note that the first two settings define the policy. For example, having 0 emergency ORs and allowing to treat emergencies in elective ORs defines the flexible policy. The third setting indicates the use of break-in-moment (BIM) optimization. Using this elective surgeries are scheduled during the day such that the start and end times of surgeries and thus the possible break-in possibilities for emergencies are spread out as evenly as possible. As such, this may further improve the performance of the flexible and hybrid policy where break-ins are allowed. For details on the BIM approach, we refer to [20]. Note that the number of ORs are based on the overall system load (i.e., total number of patients). This is done to reduce the number of infeasible or irrelevant scenarios which are likely not seen in practice. For example, scheduling 20000 patients in only 5 ORs may be impossible regardless of policy, as the required capacity exceeds the total OR time available. We only evaluate scenarios where

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the load is between 40% and 100% (i.e., 0.4 ≤ systemload ≤ 1), where load is defined as

System load =

E[surgery duration] · Number of patients treated per year . total yearly capacity

(1)

To create the scenarios, for all combinations of settings 1, 3, 4, 5, and 7, all combinations of elective and emergency ORs are taken where the system load satisfies 0.4 ≤ systemload ≤ 1. This results in 320.000 different policy and case characteristics combinations to evaluate. Please check if all occurrences of period used as thousand separator should be changed to comma. Section 3.1 gives more details on the case mix characteristics, and Sect. 3.2 describes the used performance indicators to evaluate the different policies and scenarios.

3.1 Case Mix As observed in Sect. 2, most studies consider a specific hospital setting and case mix. The case mix describes the volume and properties of surgery types (i.e., sampled distributions and parameters) that a hospital performs. As the case mix may influence performance, we carry out an extensive scenario analysis, including different case mixes. To create distinct case mixes, we use the classification put forward by Leeftink and Hans [12]. They classify surgery types using two parameters: mean surgery duration (relative to total OR capacity) and coefficient of variation (CV) of the surgery duration, which both indicate the level of complexity involved in scheduling surgeries. Case mixes that include surgeries with a higher duration are more difficult to schedule, as schedules are more likely to have gaps which could not be filled.

Table 1 Case settings used to define policies and scenarios # Description 1 Treat emergency patients in elective ORs 2 Number of ORs reserved for emergency patients 3 Use of break-in-moment (BIM) optimization 4 Case mix of surgery types 5 Number of patients treated per year 6 Number of ORs that is available in total 7 Percentage of total patients that is an emergency patient

Values {True, false} {0, 1, 2,. . . , based on system load}, {True, false} {case mix A; case mix B; case mix C; case mix D} {1000, 2000,. . . , 20000} {5, 6, 7,. . . , based on system load} {0, 5, 10,. . . , 40}

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Fig. 2 Case mixes of three Dutch hospitals using classification by Leeftink and Hans [12]; sigma/mu denotes the CV, and mu/C denotes the mean surgery duration (relative to OR session size)

Conversely, low duration surgeries are easier to schedule, as gaps can be easily filled. The coefficient of variation indicates the variability of surgeries, which in turn leads to more uncertainty when carrying out planned schedules [19]. High variability may lead to overtime and cancellations. To account for the variability slack may be used when creating schedules [8]. Using these two parameters, it is possible to create different case mixes. To create multiple case mixes, we use a dataset that is based on historical data from both academic and nonacademic hospitals in the Netherlands over the past 10 years. In this dataset we categorize surgeries as high/low duration, as well as high/low coefficient of variation. Specifically, we denote a (relative) mean duration and coefficient of variation smaller than 0.5 as low and above as high. Figure 2 shows the case mix of three different hospitals that are included in the dataset. The figures show that case mixes can differ considerably between hospitals: hospital 1 (left) has surgery types with both high and low duration and CV, and hospital 2 (mid) almost exclusively has low duration and CV surgery types. Hospital 3 (right) has another case mix, where surgeries have both high and low CV, and only few high duration surgeries are carried out. The surgery types underlying the dataset used to construct scenarios and evaluate policies are the same as used by Leeftink and Hans [12]. This dataset contains surgery types, defined as three-parameter lognormal distributions, which are shown to fit well to surgery duration distributions [13, 17], and denotes the (relative) frequency with which these surgeries take place. To evaluate the influence surgery type case mixes have on the performance of the policies, we sample surgeries from different parts of the dataset. Figure 3 gives a visualization of the complete surgery type case mix and the different sample spaces to construct four theoretical case mixes. For example, by sampling surgeries from Fig. 3b, we have a case mix of surgeries with a low (average) duration and both high and low variability.

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Fig. 3 Case mixes based on historical data of Dutch hospitals categorized by CV and duration. (a) Low duration; low CV (b) Low duration; high and low CV (c) High and low duration; low CV (d) High and low duration and CV

3.2 Performance Indicators To evaluate the policies, we use performance measures most common from recent literature reviews [3, 7, 15], being utilization, overtime, and waiting time (for both elective and emergency patients). These represent the views of different stakeholders interacting with the operating theater, namely, patients, staff, and management. With respect to overtime, we evaluate both the average and the percentage of ORs that end in overtime, to evaluate the distribution of overtime over the OR days. In addition, we are interested in outliers with respect to emergency patients (e.g., patients that wait for longer than 30 min). While it may be acceptable for an emergency patient to wait for some time (e.g., some minutes), waiting a considerable amount of time may reduce health outcomes considerably. Finally, we also evaluate the number of cancelled surgeries (i.e., surgeries that could not start during regular hours). We use the performance indicators listed in Table 2.

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Table 2 Used performance indicators # 1 2 3 4 5 6 7

Description Percentage of time ORs are in use during regular time Percentage of ORs with work in overtime Average overtime per OR day (in minutes) Average waiting time of elective patients (in minutes) Average waiting time of emergency patients (in minutes) Percentage of emergency patients that wait longer than 30 minutes Percentage of surgeries that are cancelled due to time constraints

4 Simulation Model and OR Analyzer Tool In this section we first describe our simulation model, as well as settings used within the model in Sect. 4.1. Second, we present the OR analyzer tool that incorporates all results gained from the simulation model to quickly present and evaluate multiple policies and case settings in Sect. 4.2.

4.1 Simulation Model All simulations are done using discrete-event simulation (DES) with the ‘Operating Room Manager’ software [8, 9]. This software is developed using the Delphi programming language from CodeGear and allows for the strategic, tactical, and operational evaluation of an operating theater. Within the program, detailed operating theater reconstructions are possible, taking into account additional resources (e.g., diagnostic equipment) and constraints (e.g., scheduled lunch breaks). Figure 4 shows a screenshot of the program. In order to gain insights into the effects of the scheduling policies under different hospital characteristics, we do not model all complexities of a realistic operating theater. First, we assume the number of available (emergency) ORs is the only limiting resource. In practice, some resources besides the ORs may dictate when surgeries may start, such as available diagnostic equipment or surgical tools. Similarly, staff may not be available to immediately start a surgery. In the simulation model, these resources are not taken into account. Second, setup and changeover times are not considered. In practice, two surgical procedures may require changeover times between them, giving further importance to the elective patient schedule. We assume such changeover times are included in the surgery times. Also, we assume patient flow into and out of ORs may take place immediately (e.g., there are beds available at wards and pre-/post-anaesthesia care units) as we want to exclude effects other departments may have on the performance of the OR. To evaluate a scenario and policy, we simulate an operating theater for 50 weeks (each week containing 5 working days), with daily capacity of an OR being 8

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Fig. 4 Screenshot of the simulation model

hours. The elective surgeries that are sampled are spread evenly over the weeks (e.g., 5000 elective patients result in 100 patients to schedule per week). When creating the surgical schedules, surgeries are randomly selected and scheduled using a level fit policy, such that workload is spread over all available ORs. When using BIM optimization, surgeries are swapped (within their allocated ORs), such that the maximum break-in-interval is minimized over all ORs. ORs remain operational until 12 AM to complete an already started surgery (i.e., there is no cap on overtime, after which surgeries are postponed), but surgeries that have not started during regular hours are cancelled and not rescheduled. Emergency patients that arrive must be treated as soon as possible and are prioritized by their waiting time. Based on the policy, these patients are treated in the first available (elective or emergency) OR and are always prioritized over elective patients. When allocating patients to ORs, we do not specifically plan slack time. Given the evaluated scenario, all elective patients are scheduled, and if any remaining available OR time is left, this is then spread evenly across ORs, as patients are scheduled using a level fit policy. In addition, we assume all elective surgeries are available at the beginning of the day (i.e., can start as soon as possible) and that there are no no-shows of patients. Emergency patients are drawn from the same case mix underlying the elective surgeries (as noted in Sect. 3.1) and arrive at the ORs following a Poisson process. The (inter)arrival rates for each patient type are chosen such that the (expected) total number of arrivals corresponds to the number of patients arriving per year, as

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Fig. 5 Screenshot of the OR analyzer

detailed in Table 1. For example, with 10000 patients in total per year, of which 10% are emergency patients, there are 9000 elective patients to schedule, and (on average) 1000 emergency patients arrive during the year, with an overall rate parameter λ = 1000 2080 ≈ 0.48 per hour (assuming 52 weeks with 5 working days of 8 h).

4.2 OR Analyzer Tool The OR analyzer tool allows fast evaluation and comparison of two scenarios that were simulated using the simulation model and allows healthcare practitioners to gain insights into the effectiveness of the different policies for their specific hospital case. In addition, insights can be gained with respect to "what-if" analyses, such as the effectiveness of policies under expected future patient increases. The tool was developed using the Tableau visualization software package [18]. The tool contains all simulation results and can aggregate, display, and visualize the various performance indicators. Figure 5 shows a screenshot of the program. At the top a choice can be made between an introduction screen and screens for different stakeholders. In the stakeholder screens, their respective performance indicators are shown. The right columns can be used to set various scenarios and policy settings (e.g., number of patients, case mix, etc.). This may be done for at most two different scenarios (one column per scenario) simultaneously, which are both shown in the results graph.

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Each performance indicator graph displays its respective performance on the vertical axis, for a different number of dedicated emergency ORs displayed on the horizontal axis. This allows the user to compare different policies in multiple scenarios. For example, the figure shows all three policies for a hospital case with 18 ORs in total, 15.000 patients, of which 15% are emergencies; surgeries are drawn from case mix D, and there is no BIM optimization carried out. The OR analyzer is available online (available at: http://tabsoft.co/29jrDwM).

5 Results In this section we discuss some general findings on the effectiveness of policies under different settings across multiple performance indicators.

5.1 The Effect of Scale In smaller settings (i.e., few operating rooms) we find that Policy 1 performs best, as emergency patients that have to break in most likely have to wait for one of the few surgeries to finish. For larger settings we find the opposite, and Policies 2 and 3 perform better as breaking in becomes easier. Figure 6 shows the waiting times for a given policy and total number of ORs averaged over all other settings (e.g., percentage of patients that are emergency arrivals, BIM optimization, etc.). From Fig. 6 we see that there is always some waiting time for elective patients, regardless of emergencies, caused by the inherent stochasticity in surgery durations. Also, the elective waiting time under break-in and hybrid policies is considerably higher, as patients now may also wait for emergencies. In smaller settings (up to 8–9 ORs) there is no benefit in using a break-in over a dedicated policy, as waiting time for all patients is higher under the former policy. In addition, the average overtime and percentage of ORs with overtime is higher. The hybrid policy more quickly shows a trade-off, as waiting times for emergency patients are less in settings from 6 to 7 ORs at the cost of having longer elective waiting times. Nonetheless, in smaller settings there must be some dedicated OR capacity reserved for emergencies in order to ensure timely emergency care. While it seems that in larger settings there is little difference in emergency waiting time between a break-in and dedicated policy, Fig. 7 shows the average number of emergencies that wait longer than 30 min. Here we see that the waiting time is much less evenly distributed under a dedicated policy. This can be explained by the fact that the first few arrivals under a dedicated policy have zero waiting time as an emergency OR is ready for them, but subsequent arrivals wait much longer as they may have to wait for previous emergencies to finish. From this perspective it seems that both the break-in and hybrid policies outperform the dedicated setting for sufficiently large hospitals.

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Fig. 6 Waiting times per policy and number of ORs averaged over all other case settings. (a) Average waiting time of elective patients (b) Average waiting time of emergency patients

For all sizes the hybrid policy seems to perform the best. This can be explained as it allows fine-tuning between the extremes of the other two policies and it ensures not too many break-ins into the elective program; however, if there are many emergency arrivals at once, there is the possibility of breaking in which gives more flexibility. This flexibility allows a better use of the ORs, as effectively a dedicated policy is used, until there are too many emergency arrivals, at which point a break-in policy is used.

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Fig. 7 Number of times emergencies wait >30 min per policy and number of ORs averaged over all other case settings

Fig. 8 Percentage of cancelled elective surgeries per policy and system load averaged over all other case settings

5.2 The Effect of Load and Case Mix Variability Besides the effects of scale on policy effectiveness, we also evaluate the effect of overall system load (as denoted in Sect. 3) and case mix variability. Figures 8 and 9 show the average percentage of cancelled patients for different system loads. As the load increases, elective cancellations increase for all policies with larger percentages for the break-in and hybrid policies. This can be explained by the fact that under these policies, emergency patients are prioritized over electives, which decreases performance for elective patients. For the emergency cancellations, we find the opposite result. As the load increases, the number of emergency cancellations rapidly increase under a dedicated policy while remaining low under the break-in and hybrid policies. When evaluating performances individually for case mixes with high and low variability, we see a

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Fig. 9 Percentage of cancelled emergency surgeries per policy and system load averaged over all other case settings

similar pattern, only slightly more skewed towards lower loads. This makes sense as higher variability (and all else being equal) reduces system performance. The ability of the break-in and hybrid policies to accommodate higher loads may be explained by the fact that there is a (one-way) pooling effect between emergency and elective patients. In contrast, the dedicated policy has two distinct OR settings that do not interact (electives and emergencies).

5.3 The Effect of Break-in-Moment Optimization The use of break-in optimization may further improve the effectiveness of using a break-in or hybrid policy, as waiting times for emergency surgeries are minimized by reorganizing the elective program. More surprising is that the number of elective cancellations also decreases. Figure 10 shows the percentage of cancelled elective surgeries with and without BIM for the different policies and case mixes. The decrease in elective cancellations can be explained by the fact that, when using BIM, the break-in-moments are not only spread more evenly over the day, they are also spread more evenly across ORs. This in turn more evenly distributes work across the ORs and reduces cancellations. In addition, we see that the improvements are (relatively) larger when the case mix has higher variability or longer surgery durations. While the use of BIM improves Policies 2 and 3 and does not come at the expense of other performance measures, the increases in performance are not large enough to change the scale effect tipping points mentioned in Sect. 5.1. Even when using BIM, there needs to be a large enough number of total ORs for breaking in to be viable.

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Fig. 10 Percentage of cancelled elective surgeries with BIM (1) and without (0) per policy and case mix averaged over all other case settings

6 Conclusions In this chapter we have evaluated different policies that are used in practice to deal with emergency patients arriving at the operating theater. Three policies evaluated are the use of dedicated ORs for emergency patients (Policy 1), letting emergency patients break into the elective program (in regular ORs, Policy 2), and a combination of the two aforementioned policies (Policy 3). We evaluated the different policies using a discrete-event simulation model and incorporated the results in a tool to enable quick comparison of alternative policies and settings. This evaluation is carried out for an extensive set of scenario settings such as operating theater size and patient case mix, as well as multiple stakeholder performance indicators including utilization, elective and emergency waiting times, and overtime. In total, 320.000 scenario settings were simulated. Generally, we find that, given a sufficient operating theater size, there is a tradeoff in performance indicators between using a dedicated (1) or break-in (2) policy. Using dedicated emergency ORs leads to lower waiting times for elective patients, as there are no break-ins, but outliers in waiting times for emergency patients are more likely. The hybrid policy (3) outperforms the other policies as an intermediate solution that may be better tuned to the specific underlying hospital case. The main characteristic influencing the effectiveness of the policies is the operating theater size. We find that for smaller operating theaters, it is more beneficial to use dedicated emergency capacity, as there are not enough ORs to allow for a timely break-in into the elective program. The minimum required number of

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ORs lies around 8 to 9 ORs (in total) before a break-in policy becomes viable. Below this threshold, some dedicated emergency OR capacity is necessary regardless of all other characteristics. The use of BIM shows that the scheduling of patients during the day influences OR performance. As such, other more dynamic allocation policies of patients into the ORs may be interesting to investigate in future work. For example, inserting breaks into the elective program may further reduce emergency patient waiting times when using a break-in or hybrid policy. Alternatively, taking into account expected surgery completion times before allocating emergency patients to an elective OR may further complement break-in policies. By only allocating emergency patients to elective ORs when emergency ORs are not expected to be available soon may reduce emergency waiting time outliers while reducing elective program disruptions. To conclude, we have explored the relationship between policies to deal with arriving emergency patient arrivals at the OR and hospital characteristics such as case mix, hospital size, and fraction of emergency patients. To allow healthcare practitioners to evaluate policies and performance for their specific case setting, as well as perform what-if analyses, we made available the simulation results in the OR analyzer tool (available at: http://tabsoft.co/29jrDwM). We encourage readers to further explore and use the tool as they see fit.

References 1. T. Bhattacharyya, M. S. Vrahas, S. M. Morrison, E. Kim, R. A. Wiklund, R. M. Smith, and H. E. Rubash. The value of the dedicated orthopaedic trauma operating room. Journal of Trauma-injury Infection and Critical Care, 60(6):1336–1340, June 2006. 2. J. Bowers and G. Mould. Managing uncertainty in orthopaedic trauma theatres. European Journal of Operational Research, 154(3):599–608, May 2004. 3. B. Cardoen, E. Demeulemeester, and J. Belien. Operating room planning and scheduling: A literature review. European Journal of Operational Research, 201(3):921–932, March 2010. 4. Y. Ferrand, M. Magazine, and U. Rao. Comparing two operating-room-allocation policies for elective and emergency surgeries. In Proceedings of the 2010 Winter Simulation Conference, pages 2364–2374, Dec 2010. 5. Y. B. Ferrand, M. J. Magazine, and U. S. Rao. Partially flexible operating rooms for elective and emergency surgeries. Decision Sciences, 45(5):819–847, October 2014. 6. Yann B. Ferrand, Michael J. Magazine, and Uday S. Rao. Managing operating room efficiency and responsiveness for emergency and elective surgeries–a literature survey. IIE Transactions on Healthcare Systems Engineering, 4(1):49–64, 2014. 7. F. Guerriero and R. Guido. Operational research in the management of the operating theatre: a survey. Health Care Management Science, 14(1):89–114, March 2011. 8. E. Hans, G. Wullink, M. van Houdenhoven, and G. Kazemier. Robust surgery loading. European Journal of Operational Research, page EURO, 2008. 9. Erwin W. Hans and Tim Nieberg. Operating room manager game. INFORMS Transactions on Education, 8(1):25–36, 2007. 10. Peter J H Hulshof, Nikky Kortbeek, Richard J Boucherie, Erwin W Hans, and Piet J M Bakker. Taxonomic classification of planning decisions in health care: a structured review of the state of the art in or/ms. Health Systems, 1(2):129–175, 2012.

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11. van der MTC Lans, EW Hans, and JL Johann Hurink. Anticipating urgent surgery in operating room departments. Technical report, Enschede, 2006. 12. Granne Leeftink and Erwin Hans. Case mix classification and a benchmark set for surgery scheduling. Journal of Scheduling, September 2017. 13. J. H. May, D. P. Strum, and L. G. Vargas. Fitting the lognormal distribution to surgical procedure times. Decision Sciences, 31(1):129–148, 2000. 14. M. J. Persson and J. A. Persson. Analysing management policies for operating room planning using simulation. Health Care Management Science, 13(2):182–191, June 2010. 15. Carla Van Riet and Erik Demeulemeester. Trade-offs in operating room planning for electives and emergencies: A review. Operations Research for Health Care, 7:52–69, 2015. 16. B. E. Sandbaek, B. I. Helgheim, O. I. Larsen, and S. Fasting. Impact of changed management policies on operating room efficiency. Bmc Health Services Research, 14:224, May 2014. 17. P. S. Stepaniak, C. Heij, G. H. H. Mannaerts, M. de Quelerij, and G. de Vries. Modeling procedure and surgical times for current procedural terminology-anesthesia-surgeon combinations and evaluation in terms of case-duration prediction and operating room efficiency: A multicenter study. Anesthesia and Analgesia, 109(4):1232–1245, October 2009. 18. Tableau. Tableau software. tableau desktop (version 9.2) [computer software]. 19. D. C. Tyler, C. A. Pasquariello, and C. H. Chen. Determining optimum operating room utilization. Anesthesia and Analgesia, 96(4):1114–1121, April 2003. 20. J.T. van Essen, E.W. Hans, J.L. Hurink, and A. Oversberg. Minimizing the waiting time for emergency surgery. Operations Research for Health Care, 1(2–3):34–44, 2012. 21. E. van Veen-Berkx, S. G. Elkhuizen, B. Kuijper, and G. Kazemier. Dedicated operating room for emergency surgery generates more utilization, less overtime, and less cancellations. American Journal of Surgery, 211(1):122–128, January 2016. 22. J. J. Wixted, M. Reed, M. S. Eskander, B. Millar, R. C. Anderson, K. Bagchi, S. Kaur, P. Franklin, and W. Leclair. The effect of an orthopedic trauma room on after-hours surgery at a level one trauma center. Journal of Orthopaedic Trauma, 22(4):234–236, April 2008. 23. G. Wullink, M. Van Houdenhoven, E. W. Hans, J. M. van Oostrum, M. van der Lans, and G. Kazemier. Closing emergency operating rooms improves efficiency. Journal of Medical Systems, 31(6):543–546, December 2007.

Implementing Algorithms to Reduce Ward Occupancy Fluctuation Through Advanced Planning Peter T. Vanberkel, Richard J. Boucherie, Erwin W. Hans, Johann L. Hurink, Wineke A. M. van Lent, and Wim H. van Harten

Abstract As with many hospitals, NKI-AVL is eager to improve patient access through intelligent capacity investments. To this end, the hospital expanded its operating capacity from five to six operating rooms (ORs) and redesigned their master surgical schedule (MSS) in an effort to improve utilization and decrease hospital-wide congestion; an MSS is a cyclical schedule specifying when surgical specialties operate. Designing an efficient MSS is a complex task, requiring commitment and concessions on the part of competing stakeholders. There are many restrictions which need to be adhered to, including limited specialized equipment and physician availability. These restrictions are, for the most part, known in advance. The relationship between the MSS and the ward, however, is not known in advance and is plagued with uncertainties. For example, it may be known which patient type will be admitted to the ward after surgery; however, the number of patients changes from week to week, and it is not known with certainty how long each patient will stay in the hospital. Inpatient wards, furthermore, are one of the most expensive hospital resources and can be a major source of hospital congestion, as many departments rely on inpatient wards to receive and treat their patients prior to discharge from the hospital (e.g., the emergency department). This congestion leads to long waiting times for patients, patients receiving the wrong level of care, and extended lengths of stay for patients. Well-designed surgical schedules which take into account inpatient ward resources lead to reduced cancellations and higher and balanced utilization. We observed that peaks in the ward occupancy

P. T. Vanberkel () Department of Industrial Engineering, Dalhousie University, Halifax, NS, Canada e-mail: [email protected] R. J. Boucherie · E. W. Hans · J. L. Hurink Center for Healthcare Operations Improvement and Research, University of Twente, Enschede, The Netherlands W. A. M. van Lent · W. H. van Harten Netherlands Cancer Institute – Antoni van Leeuwenhoek (NKI-AVL) Hospital, Amsterdam, The Netherlands © Springer Nature Switzerland AG 2021 M. E. Zonderland et al. (eds.), Handbook of Healthcare Logistics, International Series in Operations Research & Management Science 302, https://doi.org/10.1007/978-3-030-60212-3_8

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are particularly dependent on the MSS, and, as a result, ward occupancies can be leveled through careful MSS design. Avoiding peaks and leveling ward occupancy across weekdays makes staff scheduling easier and limits the risk of exceeding capacity, which causes congestion and perpetuates inefficiencies throughout the hospital. Working with NKI-AVL we developed an operations research model to support the redesign of their MSS. The redesigned MSS improved the use of existing ward resources, thereby allowing an additional operating room to be built without additional investments in ward capacity. A post implementation review of beduse statistics validated our model’s projections. The success of the project served as proof-of-concept for our model, which has since been applied in several other hospitals.

1 Introduction Driven by an aging population, public opinion, increased health expenditures, and long waiting lists, a flood of changes in the health-care system have been set into motion. Health care constitutes the largest industry in many developed countries [9], and managing it is a complex task due to its importance to society and the often politically charged atmosphere within which it exists. Furthermore, the nature of health-care delivery does not allow the direct copying of success stories from the manufacturing industry, where logistical optimization has a long history. Healthcare processes and supply chains show considerable differences, such as a high degree of uncertainty, the medical autonomy of clinicians, and the fact that patients cannot be treated as products. As this research began in the Netherlands, we begin by discussing the state of the Dutch health-care system. Like most countries, the Dutch health-care system has struggled with poor quality and wasteful expenditures. This came to a head in 2006, when the country passed a new Health Insurance Act. The Act reformed the structure of health-care delivery with the intent of using competition to breed efficiencies and improve value for money. To ensure that all Dutch citizens have the same basic level of health insurance regardless of ability to pay, a number of regulatory elements were introduced. Significant competition was created at two different levels. Competition exists between health insurance companies, which vie for enrollees, and also health-care providers (new and existing), which vie for contracts with health insurance companies. Insurance companies compete mainly by offering extended coverage packages (e.g., additional dentistry, eyewear, cosmetic surgery, alternative medicine, etc.) at lower prices. Health-care providers compete mainly on the remuneration amounts (paid by insurers to providers) and quality of care (e.g., access times, treatment options) [35]. The crucial underpinning of this system is to use competitive markets and insurance companies to increase performance and create a more cost-effective health-care system. The extent to which this has worked can be debated; however, the concept has been generally lauded [26]. It has been our experience that this new

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competition has applied significant pressure on health-care providers, which has, as a result, significantly increased the use of (and demand for) operations research. Financial reforms are just one example of the many efforts by developed nations to eliminate poor quality and wasteful expenditures in health care. Perhaps not surprising, given that value for money was a guiding mandate in the reforms, they have acted as a catalyst for making operations research commonplace in many Dutch hospitals. This has led to an enormous increase in research questions motivated by health-care providers. Research results are influencing national health-care policies and changing the way health care is delivered across the country. Although the financial reforms are unique to the Netherlands, the operations research it has motivated has broader appeal and can support improvement efforts around the world. In this chapter we discuss a project motivated by a Dutch hospital which has broad applicability. The challenges and opportunities related to surgical scheduling are discussed below, but for now it is sufficient to say that it is a topic of significant academic study (see reviews [5] from 1997 and [8] from 2009) and is a challenge faced by health-care providers in many parts of the world. There has been limited reported success in terms of implementation and impact on health-care operations [8], and hence there is a need to develop solutions which can be readily applied and generalized for applicability in various hospital settings. The reported research project evolved from an operations research model to an application at the Netherlands Cancer Institute-Antoni van Leeuwenhoek (NKIAVL) to a standard practice. The structure of the chapter reflects this. The formal model is discussed in Sect. 2, immediately following the problem description (Sect. 1.1). The application is discussed in Sect. 3 and includes the post implementation analysis (Sect. 3.1.1) used to validate the model. Finally Sect. 3.2 provides concluding remarks and briefly discusses applications of the model at other hospitals. Much of this research has been reported before in [14, 30, 32–34]. Accordingly, the intent of this chapter is to summarize this research with a specific focus on how it has impacted NKI-AVL and generally how it has impacted other hospitals.

1.1 Problem Description No other single hospital department influences the workload of the hospital more than the Department of Surgery, in particular the activities in the operating room (OR) [20]. As such, its activities (or lack thereof) cause a ripple effect throughout the hospital. Upstream processes are less sensitive to changes, as there is often a waiting list for surgical operations which acts as a buffer, dampening the effect. For downstream processes, this is different as a buffer of post-surgery patients waiting to be admitted to a ward cannot exist. Since post-surgical activities are sensitive to the activities in the OR, it is important to derive one in terms of the other. As described in this section, the workload for downstream departments can be modelled as a function of the master surgical schedule (MSS).

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For a hospital, the OR accounts for more than 40% of its revenues and a similarly large portion of its costs [14]. An efficient OR thus significantly contributes to an efficient health-care delivery system as a whole. The planning and scheduling of OR time is discussed by many authors [2, 5, 6, 8, 17, 36] and is often described as a multiple stage process. The multiple stage process used by many hospitals starts with the long-term allocation of OR time to surgical specialties, e.g., the number of surgery hours per year. This allocation, referred to as Stage 1, is a strategic decision that reflects patient demand patterns and the priorities defined by hospital management. From this strategic decision, an MSS is developed for a shorter time horizon which divides OR time (aggregated into blocks) among the specialties, known as Stage 2. The specific assignment of patients to OR blocks within the MSS is commonly referred to as Stage 3. A fourth stage “addresses the monitoring and control of the OR activities” [23] on the day of surgery. In this chapter, we focus on the development of an MSS in Stage 2. The MSS is often specialty specific [2], meaning OR time is dedicated to a surgical specialty. In these MSSs, the decision of which patients (and consequently which surgeries) to schedule within each OR block is determined by the surgical specialty through consultation with the OR manager. Other MSSs are more specific with OR blocks being allotted to specific surgical procedures [18, 24]. Instead of using the term MSS, other authors refer to the distribution of OR time among surgical specialties as a surgical block schedule [25] and a timetable of operations [16]. The development of an MSS is often a complex balancing act. Since the OR is one of the hospital’s most expensive resources, hospitals wish to maximize its performance through high resource utilization, minimal overtime, minimal cancellations, and the elimination of conflicting equipment needs between rooms. Many authors describe methods for developing the MSS that take into account various resources within the OR such as staff, equipment, and instrument trays. For a review see [8]. Furthermore, the OR is often described as the engine that drives the hospital [20], implying many other departments are impacted by the MSS. The effect of the MSS on ward occupancy [1–4, 10, 11, 16, 18, 19, 24, 28, 29, 31], critical care resources [3, 10, 12, 18, 22, 24], and waiting lists [25, 31] has notably been studied. Three of these papers represent the relationship with deterministic models, while the remaining consider at least one variable as stochastic. The stochastic models are either simulation models, mathematical programming models, or queueing theory models. The analytical model presented in this chapter most closely resembles a queueing model. The model quantifies the effect of an MSS on admission/discharge rates, ward occupancy rates, and the workload of all departments treating inpatients. The robustness of this model and, as illustrated later, its ease of implementation are the main contributions of the model to surgical scheduling literature. Using our model, downstream workload distributions can be computed as a function of the MSS for all departments that provide care for recovering surgical patients. Specifically, the model computes the ward occupancy distributions, the

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patient admission/discharge distributions, and the distributions for the ongoing interventions/treatments required by recovering patients. Furthermore, the cumulative influence of multiple MSS cycles is considered. Since the MSS is identical from cycle to cycle, the overlapping of patients from one cycle to the next can be anticipated. A single MSS design is expected to remain in place for a long period of time leading to “steady-state” workload distributions for each day of the MSS cycle.

2 Methodology This section describes a model to determine the workload placed on hospital departments by recovering surgical patients. In the same way an MSS describes resource demands within the OR, we show how the resources of other departments can be seen as a function of the MSS. The method relies only on data which are easily extractable from typical patient management systems. The model is most easily described from a queueing theory perspective. The basic component of the model is a single OR block and its expected impact on the arrival rate to the hospital wards. The number of cases scheduled in such a block varies per specialty and is modelled as a specialty-specific random variable. This variable also represents the number of patients arriving to the ward (batch size). At the ward, each patient directly occupies a bed for a certain period of time. In the queueing model, the ward is seen as an infinite server system in which the patients occupy a server (ward bed) without delay. The time spent occupying a bed (length of stay, LOS) is the service time, which is modelled as a random variable. Again, this random variable is specific to the surgical specialty. Since patients occupying a server do not interfere with each other during their recovery, the aggregate number of patients for all OR blocks can be computed by adding the individual effects of all OR blocks. Finally, since the MSS is cyclical, the cumulative number of patients from recurring MSS cycles can be computed. The main output of the model is the distribution for the number of patients anticipated in the system on each day of the MSS. The model used for these calculations is explained in the following subsection. The three subsequent subsections explain how the model can be modified to obtain the distributions for (1) ward occupancies, (2) admissions/discharges, and (3) the number of patients in a specific day of their recovery. The time scale in the model is days; therefore, all metrics are considered on a daily basis.

2.1 Model Inputs An MSS represents a repetitive pattern over a certain number of days (say Q). For each day q ∈ {1, 2, . . . , Q} in the MSS, each of the I available ORs can be assigned to one of the available surgical specialties. More precisely, the MSS is

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described by the assignment of a surgical specialty j to each OR block bi,q where i ∈ {1, 2, . . . , I }. Using this notation, an empty MSS (i.e., before specialties have been assigned to OR blocks) is shown in Fig. 1 where each cell represents an OR block. It is possible for multiple blocks to be assigned to a single specialty on the same day. The way specialty j fills in an OR block is described by two parameters, cj and j dn . Parameter cj is a discrete distribution for the number of surgeries carried out in one block, i.e., P(cj = k) is the probability of k surgeries, k ∈ {0, 1, . . . , C j }, where C j is the maximum number of surgeries of specialty j that can be completed in one block. Specialties independently decide which patients to schedule during each block, meaning that the number of surgeries completed in one block does not j influence the number of surgeries completed in another. The second parameter dn is the probability that a patient, who is still in the ward on day n, is to be discharged that day (n ∈ {0, 1, . . . , Lj }, where Lj is the maximum LOS for specialty j ; a finite j LOS is used for numerical purposes). Note that d0 is the probability that the patient is discharged on the same day as surgery (i.e., an outpatient surgery or day-case j j surgery) and dLj = 1. The parameter dn is computed by dividing the probability that a patient’s total stay is exactly n days by the probability that the patient was not yet discharged before day n. Let P j (n) be the probability that the LOS of a patient j from specialty j is exactly n days long; then formally dn is computed as follows: P j (n)

j

dn =

(1)

.

j

L 

P j (k)

k=n

Days in MSS q=3

i=1

b1,1

b1,2

b1,3

i=2

b2,1

b2,2

b2,3

…

b2,Q

i=3

b3,1

b3,2

b3,3

…

b3,Q

…

q=Q …

…

q=2

… i=I

bI,1

bI,2

bI,3

…

bI,Q

Fig. 1 Empty MSS illustrating model notation

b1,Q

…

Operating rooms

q=1

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2.2 Recovering Patients in the Hospital j

Using cj and dn as model inputs, for a given MSS, the probability distribution for the number of recovering patients on each day q is computed in three steps. Step 1 computes the distribution of recovering patients from a single OR block of a specialty j , i.e., we essentially pre-calculate the distribution of recovering patients expected from an OR block of a specialty. In Step 2, we consider a given MSS and use the result from Step 1 to compute the distribution of recovering patients given a single cycle of the MSS. Finally, in Step 3, we incorporate recurring MSSs and compute the probability distribution of recovering patients on each day of the MSS. Step 1: Distribution of recovering patients from specialty j following from a single OR block In Step 1 we ignore the MSS and consider a single specialty j operating in a single OR block. The patient flow process is as follows: During the OR block patients receive surgery. The number of patients who undergo surgery in one OR block is given by the random variable cj . After surgery each patient still on the ward j on day n has the probability dn of being discharged that day. In the following, we j compute the probability P(hn = x) that n days after scheduling a block of specialty j , x patients of the block are still in recovery. Note that n ∈ {0, 1, ..., Lj } and j x ∈ {0, 1, ..., C j } and that, for example, P(h3 = 5) = 0.25 means that 3 days after surgery, there is a 25% chance that five patients are still recovering in the hospital. Day n = 0 is defined as the day of surgery, and it is assumed that patients occupy a bed all day on the day of surgery even though they may physically be in the OR. This is consistent with practice where patients have a recovery bed reserved for them before surgery. As such, the number of patients in recovery from specialty j on day n = 0 is by definition the number of surgeries performed that day by specialty j . It follows that the distribution for the number of recovering patients on day n = 0 is j h0 = c j . j Note that on day n, each patient still in the hospital has a probability dn of being j discharged that day and (1 − dn ) of staying. If there are k patients in recovery on day n, then the probability of s patients in recovery (where s ≤ k) on day n + 1 is  j j computed using the binomial distribution, ks (dn )k−s (1 − dn )s . Since we know the probability distribution for the number of patients at the end of day n = 0, we can iteratively use this formula to compute the probability of k patients at the end of all days n > 0. Summarizing, the distribution for the number of recovering patients on day n is recursively computed by ⎧ j ⎪ when n = 0 ⎪ P(c = x) ⎨ j   j C (2) P(hn = x) =  k j j j ⎪ (dn−1 )k−x (1 − dn−1 )x P(hn−1 = k) otherwise. ⎪ ⎩ x k=x

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Step 2: Aggregate distribution of recovering patients following from a single MSS j cycle In this step we consider the previously computed probability distribution hn and a given MSS as input. Although the MSS is cyclical and repeats after Q days, in this subsection, we consider only a single MSS cycle in isolation. The MSS defines when each specialty is assigned an OR block and thus the days on which patients of specialty j arrive to the ward. Based on these, we compute the total number of patients in recovery by means of discrete convolutions. To calculate the overall distribution of recovering patients, we first identify for each block bi,q the impact that this block has on the number of recovering patients in the hospital on days (q, q + 1, ...). If z denotes the specialty assigned to block bi,q i,q which follows from the MSS, then the distribution h¯ m for the number of recovering patients of block bi,q on day m (m ∈ {1, 2, ..., Q, Q + 1, Q + 2, . . .}) is given by i,q h¯ m =



hzm−q if q ≤ m < Lz + q, 0 otherwise

(3)

where 0 means P(h¯ m > 0) = 0. Note that specialties index j is no longer needed as specialties are accounted for by their designated OR block(s). Let Hm be a discrete distribution for the total number of recovering patients on day m resulting from a single MSS cycle. Since recovering patients do not interfere with each other, we can simply iteratively add the distributions of all the blocks impacting day m to get Hm . Adding two independent discrete distributions is done by discrete convolutions which we indicate by ∗. Let A and B be two independent discrete distributions. Then C = A ∗ B is computed by i,q

P(C = x) =

τ 

P(A = k)P(B = x − k)

k=0

where τ is equal to the largest x value with a positive probability that can result from A ∗ B. Using this notation, Hm is computed by ¯ 1,2 ¯ 1,Q ¯ 2,1 ¯ I,Q Hm = h¯ 1,1 m ∗ hm ∗ . . . ∗ hm ∗ hm ∗ . . . ∗ hm .

(4)

Step 3: Steady-state distribution of recovering patients In Step 3 we consider a series of MSSs to compute the steady-state probability distribution of recovering patients. The cyclic structure of the MSS implies that patients receiving surgery during one cycle may overlap with patients from the next cycle. In the case of a small Q, for example, patients from many different cycles can overlap. In Step 2 we have computed Hm for a single cycle of the MSS in isolation. Let M be the last day where there is still a positive probability that a recovering patient is present as computed by Hm . Thus M = maxj {Lj + x j } (where x j is the latest day q of a block assigned to specialty j ) indicates the range of the MSS. To calculate the overall distribution of recovering patients when the MSS is repeatedly

Implementing Algorithms to Reduce Ward Occupancy Fluctuation Through. . . q = 1, 2,…

Q, Q + 1,… q = 1, 2,…

M Q, Q + 1,… q = 1, 2,…

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Overlapping MSS Ranges M

Q, Q + 1,… q = 1, 2,…

M Q, Q + 1,…

M

Fig. 2 Consecutive MSS cycles illustrating overlapping recovering patients

executed, we must take into account M/Q consecutive cycles of the MSS (see Fig. 2). Let HqSS denote the probability distribution of recovering patients on day q of the MSS cycle, resulting from M/Q consecutive MSS cycles. Since the MSS does not change from cycle to cycle, HqSS is the same for all MSS cycles. Using discrete convolutions, HqSS is computed by HqSS = Hq ∗ Hq+Q ∗ Hq+2Q ∗ ... ∗ Hq+ M/QQ .

(5)

The relationship between the distribution HqSS and the workload associated with recovering patients is discussed in detail in the following three subsections.

2.3 Ward Occupancy Perhaps the most common measure of inpatient workload is ward occupancy. Ward occupancy, the distribution of the number of inpatients on a ward, follows easily from the basic model where we compute the distribution of all recovering patients. In practice patients tend to be segregated into different wards depending on the type of surgery they receive. To incorporate this segregation into the model and to consequently have recovering patient distributions for each ward, a minor modification needs to be made to the model. Let Wk be the set of specialties j whose patients are admitted to Ward k. Then in Step 2 we only have to consider those OR blocks assigned to a specialty in Wk .

2.4 Admission Rate/Discharge Rate Bed occupancy alone does not fully account for the workload associated with caring for recovering patients. During patient admissions and discharges, the nursing workload can increase. As such, in this subsection, we explain how to derive the probability distribution for daily admissions and discharges.

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The admission rate is the rate of arriving patients which we previously defined as the number of surgeries completed on day n = 0. For this metric we are only interested in a patient on the day of admission and wish to ignore them afterward. To modify the model to reflect this new purpose replace (3) with  j

hn =

cj when n = 0 0 otherwise.

(6)

With this modification, the resulting Hm represents the distribution for daily admission for each day q of the MSS. To have ward-specific results, we again can restrict this to blocks belonging to specialties of the specific ward. The discharge rate is the rate at which patients leave the ward and can be computed by adding an additional calculation in Step 1. The number of patients in j recovery on day n is distributed according to hn ; see (3). On day n, each patient has j j the probability dn of being discharged and the probability (1 − dn ) of staying. Let j Dn be a discrete distribution for the number of discharges from specialty j on day j j j n. Given hn and dn , Dn can be computed with a binomial distribution as follows: C    k j j j (dn )x (1 − dn )k−x P(hn = k). = x) = x j

j P(Dn

(7)

k=x

j

j

j

Finally, after computing Dn , one can set hn = Dn and continue with Step 2. By doing so, the resulting HqSS represents the distribution for daily discharges for each day q of the MSS. As with admissions, ward-specific results can also be obtained.

2.5 Patients in Day n of Their Recovery The final workload metric we consider is the distribution of patients in day n of their recovery. This is relevant for predicting workload for the many hospital departments who treat recovering patients. For example, a patient recovering from hip surgery may need to see a physiotherapist every other day during their recovery. This metric states the frequency of such visits. The analogy holds true for all types of services that take place on specific intervals during a patient’s recovery (e.g., chemotherapy, diagnostics, social work, discharge planning). In particular, this metric can help dimension capacity for clinical pathways patients whose recovery is intended to follow a strict regime. The metric requires substantial modifications to the original model, since we now must carry an index (n) for the “day of recovery” throughout the three steps. i,q Let h¯ m,n be a discrete distribution for the number of recovering patients from block i,q bi,q on day m in day n of their recovery. To compute h¯ m,n , replace (3) with the following:

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Table 1 Example results for the frequency of inpatient chemotherapy treatments Example results SS = 2) = 0.3 P(H1,3

n = 3, q = 1

SS = 3) = 0.5 P(H1,3

n = 3, q = 1

SS = 4) = 0.2 P(H1,3

n = 3, q = 1

SS = 2) = 0.4 P(H2,3

n = 3, q = 2

SS = 3) = 0.4 P(H2,3

n = 3, q = 2

SS = 4) = 0.2 P(H2,3

n = 3, q = 2

i,q h¯ m,n =

Interpretation 30% probability that exactly two treatments are required on the first day of the MSS cycle 50% probability that exactly three treatments are required on the first day of the MSS cycle 20% probability that exactly four treatments are required on the first day of the MSS cycle 40% probability that exactly two treatments are required on the second day of the MSS cycle 40% probability that exactly three treatments are required on the second day of the MSS cycle 20% probability that exactly four treatments are required on the second day of the MSS cycle



0 if m − q = n hzm−q otherwise,

(8)

and we replace (4) with the following: ¯ 1,2 ¯ 1,Q ¯ 2,1 ¯ I,Q Hm,n = h¯ 1,1 m,n ∗ hm,n ∗ ... ∗ hm,n ∗ hm,n ∗ ... ∗ hm,n

(9)

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To help to interpret this metric, consider the following fictitious example for patients who require chemotherapy treatment on day three of their recovery. The Chemotherapy Department would like to know how frequently they need to provide SS are illustrated in Table 1. this service. Example results for Hq,n

2.6 Assumptions Inherent to the model are a number of assumptions which are discussed in this subsection. One assumption resulting from the use of the infinite server system is that there is always a bed available for a patient after surgery. This implies that surgeries are never cancelled due to bed shortages. In practice this means that there is not a physical bed restriction and that additional staff can be called in when demand

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is higher than expected. The frequency of this occurring follows from the model. For example, if a hospital staffs 50 beds, then the probability of an additional staffed bed being needed on day q is P(HqSS = 51). In the current formulation, the model ignores seasonality. Of course, at certain times of the year, surgical blocks are cancelled to accommodate vacations and slowdowns, representing a change in supply. In this case, a modified MSS is temporarily used in breaking down the assumption that the same MSS repeats every Q days. However, given that the modifications to the MSS are typically cancellations of certain OR blocks, then the original result can act as an upper bound. Only elective surgeries are considered. To incorporate non-elective surgeries, it is possible to convolute a historic bed occupancy distribution for non-elective patients. Alternatively, it is possible to incorporate a virtual OR block into the model that represents emergency admissions. The inherent assumption of using the binomial distribution in this model is that all patients (experiments) have equal probability of each outcome and that the outcome is independent of other patients, i.e., it is assumed that the patients are independent and identically distributed. The independence assumption is natural as it implies that the amount of time one patient is in the hospital does not influence the amount of time another patient is in the hospital. The identically distributed requirement means that we must compute the number of beds needed tomorrow (and the number of case completed in one OR block), for all identically distributed cohorts of patients separately. In other words, the parameters of the binomial distribution must reflect all of the patients in a given cohort (for a discussion on defining statistically equivalent patient cohorts, see [15]). In our model we aggregate patient such that each surgical specialty is a patient cohort. It follows then that patients within each surgical specialty should be identically distributed. If a heterogeneous population is grouped together, this causes the ward census distribution to have longer tails (although the mean remains the same) and will overestimate the bed requirements when staffing for a certain percentile of demand. On the other hand, however, less aggregation (such as dividing a specialty by short- and long-stay patients) decreases the sample size from which to derive the parameters which in turn reduces the statistical confidence of the estimated parameters. In our case study that follows, we aggregate the data by specialty which allows for enough data to have a sufficient sample size and results in relatively homogeneous patient cohorts. In cases where patients of a surgical specialty are not identically distributed and cannot be aggregated into a single cohort, the model can still be used. First the heterogeneous specialty has to be divided into multiple homogeneous cohorts, and then these cohorts can be treated as if they were assigned their own OR block. Using this, the binomial distribution is applied as described above to determine the bed requirements of each cohort. Again, using the independence assumptions, these cohorts can be added (with discrete convolutions) to determine the total bed requirements for the complete surgical specialty.

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3 Application The Netherlands Cancer Institute-Antoni van Leeuwenhoek Hospital (NKI-AVL) is a comprehensive cancer center, which provides hospital care and research and is located in Amsterdam, The Netherlands. The hospital has 150 inpatient beds and sees about 24,000 new patients every year, making it approximately the size of a mid-sized general hospital. As with many Dutch hospitals, NKI-AVL is eager to improve access and increase capacity. To this end, the hospital has expanded its operating capacity from five to six operating rooms (ORs). The hospital welcomed this expansion as an opportunity to develop a new MSS. NKI-AVL distributes its surgical capacity to its six surgical specialties in the typical manner described previously. The yearly amount of operating time is first allotted to each specialty reflecting patient demand and hospital priorities. To implement this allotment, and to make the surgical department manageable, NKI-AVL divides OR time into OR blocks over a 1-week planning horizon. One OR block represents a full day of operating room time. The assignment of the surgical specialties to each OR block represents the MSS. An example MSS for five operating rooms is shown in Fig. 3. As previously discussed, the use of operations research in surgical scheduling is not new; however, the rate of implementation for this type of study is low [8]. To overcome this, Cardoen et al. [8] “encourage the provision of additional information on the behavioral factors that coincide with the actual implementation. Identifying the causes of failure or the reasons that lead to success, may be of great value to the research community.” In this section we describe the process of developing a new MSS for NKI-AVL and results observed after its implementation. The development process, which combined an operational research model and staff input, led to an MSS which was agreeable to staff from both the wards and the OR. Staff selected and implemented an MSS which the model predicted would result in a balanced ward occupancy. The development of the new MSS was completed over 3 months in an iterative manner. A team was formed consisting of a team leader from the surgical department, a team leader from the inpatient wards, the manager of both groups,

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and two of this chapter’s authors. The team members from the surgical department ensured MSS proposals did not cause conflicts within the OR, such as with physician schedules and available equipment. The projected impact that each MSS proposal would have on the wards was evaluated with the model described previously. Each new MSS proposal represented a new scenario to be evaluated by the model. From the model output, staff decided whether the MSS was acceptable or if further modifications to the MSS were necessary. The original MSS was roughly developed as follows. Based on production targets, the number of OR blocks to be assigned to each specialty during the 1week MSS cycle was determined. Next, the physicians’ commitments elsewhere in the hospital were determined, and their preferred operating days were considered. Potential equipment and resources conflicts were addressed, for example, it would be problematic to assign two specialties to the same operating day when both routinely require the same specialized OR. Considering these restrictions, OR staff proposed the original MSS. To determine how the original MSS impacted the wards, the model was used. As illustrated in the “Results” section, the original MSS results in an unbalanced ward occupancy (the motivation for this metric is also provided in the “Results” section). As such, the team decided the original MSS was not acceptable. Next, modifications to the original MSS were made, and a new MSS proposal was put forth. Given that in this project we were not asking surgical specialties to change how they operate (i.e., the number of surgeries they perform in an OR block and/or the invasiveness of their surgeries which can dictate length of stay), modifications to the MSS were limited to changes in the assignment of surgery specialties to OR blocks. Essentially, modifications consisted of swapping a specialty operating on one day with a specialty operating on a different day. Deciding which blocks to swap followed, first, from OR staff knowledge of what was possible within the constraints of the OR and, second, by intuition gained from seeing results from several MSS proposals. See Fig. 4 for an illustration of the type of modifications made. A number of MSSs were proposed, and the impact that each would have on the ward was evaluated by the model. This process of modifying and evaluating MSSs continued for several weeks until an MSS was found that satisfied staff from both the OR and the wards. A schematic overview for this process is displayed in Fig. 5. The MSS chosen by the team was implemented concurrently with the opening of the new OR in March of 2009. The new OR was phased in over several months, and once it became fully utilized, ward occupancy statistics were collected. The data, observed over a 33-week period when all six ORs were being regularly scheduled, was compared with what was projected from the model. The purpose, to ensure a more balanced ward occupancy, was indeed being achieved with the implemented MSS. In this way we could validate the model output and confirm the implemented MSS is resulting in the desired ward occupancy.

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3.1 Results This section is dived into two subsections. The first subsection discusses ward occupancy projected by the model during the MSS development process. We show ward occupancy projections from the original MSS proposal and from the MSS proposal that staff chose to implement (which we refer to as the implemented MSS). In the second subsection, we compare the ward occupancy projected by the

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model for the implemented MSS with the ward occupancy observed after it was implemented.

3.1.1

Projected Results

NKI-AVL has two wards for treating surgical patients, Ward A and Ward B, with a combined physical capacity of 100 beds. Management strives to staff enough beds such that for 90% of the days, there is sufficient coverage. In other words, they staff for the 90th percentile of demand; their accepted risk for needing to call in additional staff is thus 10%. Figure 6 illustrates the 90th percentile demand for staffed beds on each of the wards, resulting from the original MSS proposal. As is clear from the figure, the staffing requirements are relatively balanced across the weekdays (Monday to Friday) for Ward B. This is not the case for Ward A. On Ward A the occupancy is relatively low on Monday and Tuesday and relatively high on Thursday, Friday, and Saturday. This projected demand for staffed beds concerned the ward manager, as such an unbalanced demand profile makes staff scheduling, and ward operations, difficult. Early in the week, beds would be underutilized, whereas later in the week, beds would become highly utilized leading to significant problems, particularly as the wards approach peak capacity. For example, when inpatient wards reach their peak capacity and a patient admission is pending, staff often scramble to try and make a bed available. If one cannot be made available, additional staff are called in (or in rare cases, when additional staff cannot be found, the elective surgery is cancelled), which causes extra work for OR planners, wasted time for surgeons, and anxiety for patients. When a bed is made available, it often means a patient was transferred from one ward to another (often to a ward capable of caring for the patient but not the preferred one) or discharged. Either way, extra work is required by ward staff, and there is a disruption in patient care. Although completely eliminating such problems is not possible without an exorbitant amount of resources, sound planning ahead of time may help to minimize occurrences. After discussing the model output, all participating staff agreed that the original MSS, although appropriate for the OR, was not ideal for the wards. The discussion

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then moved to how to correct the imbalance across the weekdays by changing the assignment of OR blocks to specialties. Modifications to the original MSS were made by considering what changes were possible within the restrictions of the OR (e.g., physician schedules and equipment availability). Eventually, after considering several MSS proposals, the process led to an MSS (the implemented MSS) which was acceptable to all staff members. The implemented MSS fit within the restrictions of the OR and, as illustrated in Fig. 7, resulted in a more balanced ward occupancy. Comparing the implemented MSS with the original MSS, the implemented MSS dampened the fluctuation on Ward A by lowering occupancy on Thursday, Friday, and Saturday and increasing it on Monday and Tuesday. With the implemented MSS, the model predicted that no days would require more than 47 staffed beds, which reduced the maximum from 49 (predicted for the original MSS). Furthermore, the implemented MSS ensured the staffing requirements remained relatively balanced across the working days for both wards.

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Observed Results

The ward occupancy was observed over a 33-week period after the new OR was fully operational. From these data, probability distributions of beds used for each day of the MSS cycle were derived. Using chi-square goodness-of-fit tests [21], these observed distributions were compared to those projected by the model. For Ward B, six of the seven distributions (one for each day of the MSS cycle) were found to be statistically equivalent at a level α = 0.05, while the seventh day was statistically equivalent at a level α = 0.2. For Ward A, the tests revealed statistical equivalence at levels α = 0.15 (for three days), 0.25 (for two days), and 0.35 (for two days). At these alpha levels, we conclude that the observed ward occupancy is well predicted by the model. Explanations for the poorer fit of Ward A data are discussed in the following paragraphs where the 90th percentiles (desired staffing level) are compared for the observed and projected results.

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Figures 8 and 9 compare the projected ward occupancy with the observed ward occupancy during the 33-week period. Figure 8 displays results for Ward A and Fig. 9 for Ward B. As is observable in Figs. 8 and 9, the data indicates that both wards have balanced ward occupancies across the week days. However, it is also observable that our model overestimated the number of beds required in Ward A by approximately 16%. The overestimate is due to an unexpected increase in short-stay patients during the period of measurement. Had this change in patient mix been expected at the time the projections were made (and model input altered to reflect it), such an overestimate would likely not have been observed, and we would expect to have similarly accurate results as those for Ward B. As a final note on the model results, consider if hospital management decided to staff only for the average number of beds projected to be needed for six ORs. In this case, 32 beds would be assigned to Ward A and 29 beds to Ward B. This would have led to a bed shortage on 51% of the days, illustrating the importance of considering probability distributions in hospital planning.

3.2 Discussion With the approach discussed in this section, a new MSS was developed for NKIAVL which reduced the fluctuations in the daily ward census, creating a more balanced workload on the wards. The roll out of the new MSS corresponded with

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the opening of an additional OR which was expected to overwhelm the wards. By using the described process to develop an MSS that accounted for the inpatient wards, peaks in ward occupancy were reduced. As such capacity is used more efficiently, and the hospital has the means to support the additional OR without a major expansion in the wards. The main benefit of the model was the ability to quantify the concerns of ward staff, thereby providing a platform which they could begin to negotiate a solution. Staff was quick to embrace the model output, particularly after seeing several modifications to the original MSS, at which point they were able to roughly predict the model output intuitively. For example, on Thursdays and Fridays, the wards tended to be crowded with patients. To remedy this, specialties that completed many cases per OR block were removed from Thursday and Friday OR blocks and assigned to OR blocks earlier in the week. To accommodate these changes, specialties which complete a relatively small number of cases per OR block were moved to Thursday and Friday. Once staff could foresee the impact of swapping one surgical OR block assignment with another, the MSS which was eventually implemented came quickly. In the NKI-AVL application, we treated the equipment and physician schedule restrictions as unchangeable. It is possible that further improvements in the ward occupancy could have been achieved if these restrictions were relaxed. In this way the model can be used to illustrate the benefits of buying an extra piece of equipment or of changing physicians’ schedules. An additional restriction, which if relaxed may have allowed further improvements, is the assignment of wards to surgical specialties. In other words, in addition to changing when a specialty operates, it may prove advantageous to change which ward the patients are admitted to after surgery. At NKI-AVL, our model was used to solve the tactical surgical scheduling problem – a medium-term planning horizon with patients aggregated by surgical specialty. Alternatively, the same model can support decisions at an operational level – a shorter-term planning horizon without patient aggregation (for a discussion on levels of planning and control in health care, see [13]). Instead of computing the expected patients in recovery, the actual patients in recovery can be used as input. By combining this with the expected new arrivals from the OR, real-time workload projections can be used to identify upcoming staffing needs.

4 Conclusion Many good research projects conclude after the implementation of results with team members satisfied that anticipated improvements have been realized. This project, in some ways, merely began at this point. Variants of this model have been developed (and applied) at the request of three other Dutch hospitals, and the model forms the basis of a similar application at a German hospital. Below we discuss these other applications of our model.

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Leiden University Medical Center (LUMC), the Netherlands LUMC is one of eight University Medical Centers in the Netherlands and employs approximately 7000 professionals. Highly variable ward occupancy was proving problematic for staffing the inpatient wards at LUMC. Utilizing our model, staff proposed and evaluated a number of MSS proposals in order to reduce occupancy fluctuations. They found that with very little disruption to the current MSS (only four swaps of OR blocks), the maximum bed occupancy would reduce from 74 to 71. Additional reductions in the maximum bed occupancy were found only to be possible when additional OR time was made available [27]. Haga Hospital, The Hague, the Netherlands Haga Hospital is a top clinical teaching hospital in the Netherlands with 245 specialists, 729 beds, and 35,571 admissions in 2010. At Haga Hospital, management wanted to develop a new structured scheduling procedure in order to increase OR utilization and balance ward occupancy. To achieve this, the scheduling procedure was redesigned, and a formal process was put into practice supported by a decision support system. To model stochastic length of stays and to integrate bed leveling into this software, the model described in this chapter was used [7]. Technical University of Munich, Germany To make the model appropriate for a German hospital, modifications were made in collaboration with researchers at the Technical University of Munich. The first modification involved increasing the scope to include the Intensive Care Unit (ICU), a unit in which most acute surgical patients are admitted to receive one-on-one nursing care. This modification essentially amounted to changing the model from a single queue to a network of two queues, one for the ICU and one for the Ward. The second modification involves developing heuristics for determining good MSS proposals. A number of objectives are considered, including minimizing costs and determining the best improvement with the smallest change (disruption) of the existing MSS. The buy-in of other hospitals and the dissemination of our approach can be credited to the way in which the model was incorporated into the decision-making process and does not replace it. Supporting and not replacing the process allows for a less complex model and more staff engagement. This approach to problem-solving proved crucial for implementation and for making a meaningful impact.

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Bed Census Predictions and Nurse Staffing Aleida Braaksma, Nikky Kortbeek, and Richard J. Boucherie

Abstract Workloads in nursing wards depend highly on patient arrivals and lengths of stay, both of which are inherently variable. Predicting these workloads and staffing nurses accordingly are essential for guaranteeing quality of care in a costeffective manner. This chapter describes a stochastic method that uses hourly census predictions to derive efficient nurse staffing policies. The generic analytic approach minimizes staffing levels while satisfying so-called nurse-to-patient ratios. In particular, we explore the potential of flexible staffing policies that allow hospitals to dynamically respond to their fluctuating patient population by employing float nurses. The method is applied to a case study of the surgical inpatient clinic of the Academic Medical Center Amsterdam (AMC).

1 Introduction Societal developments and budget constraints demand hospitals to on the one hand increase quality of care and on the other hand efficiency [41]. This entails a strong incentive to reconsider the design and operations of inpatient care services that provide care to hospitalized patients by offering a room, a bed, and board [40]. Since the 1950s, the application of operational research methods yields significant contributions in accomplishing essential efficiency gains in healthcare delivery [30].

A. Braaksma () · R. J. Boucherie Center for Healthcare Operations Improvement and Research, University of Twente, Enschede, The Netherlands e-mail: [email protected]; [email protected] N. Kortbeek Center for Healthcare Operations Improvement and Research, University of Twente, Enschede, The Netherlands Rhythm b.v., Amsterdam, The Netherlands e-mail: [email protected] © Springer Nature Switzerland AG 2021 M. E. Zonderland et al. (eds.), Handbook of Healthcare Logistics, International Series in Operations Research & Management Science 302, https://doi.org/10.1007/978-3-030-60212-3_9

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This chapter, combining the results of [32, 33], presents an exact method to assist hospital management in adequately organizing their inpatient care services. The challenge in decision-making for inpatient care delivery is to guarantee care from appropriately skilled nurses and required equipment to patients with specific diagnoses while making efficient use of scarce resources [28, 50]. Deploying adequate nurse staffing levels is one of the prime responsibilities of inpatient care facility managers. Nursing staff typically accounts for the majority of hospital budgets [54], which means that every incidence of overstaffing is scrutinized during times when cost containment efforts are required [34]. Performance measures are required that reflect efficiency and quality of care to assess the quality of the logistical layout. Efficiency is often expressed in high bed occupancy. The drawback of high bed occupancy is that it may cause congestion and a threat to the provided quality of care [21, 22]: (i) patients may have to be rejected for admission due to lack of bed capacity, so-called admission refusals or rejections, and (ii) patients may be placed in less appropriate units, so-called misplacements [13, 27, 29]. Due to such misplacements, planning decisions regarding a specific care unit can affect the operations of other units [4, 12, 36]. At the same time, maintaining appropriate staffing levels is crucial to be able to provide high-quality care. Planning of the inpatient care facility should not only take into account the upstream departments, such as the emergency department and the operating rooms, but also the interrelationship between care units. In this chapter, following [32, 33], we present an exact method to assist healthcare administrators in ensuring safe patient care while also maintaining an efficient and cost-effective nursing service. We first present a generic exact analytical approach to achieve the required integral and coordinated resource capacity planning decision-making for inpatient care services, building upon the approach introduced in [49], which determines the workload placed on hospital departments by describing demand for elective inpatient care beds on a daily level as a function of the Master Surgical Schedule (MSS). The MSS is a (cyclic) block schedule that allocates operating time capacity among patient groups as typically used by hospitals to allocate operating room capacity [18, 24, 46]. Based on a cyclic arrival pattern of emergency patients and an MSS block schedule of surgical patients, we present demand predictions on an hourly level for several inpatient care units simultaneously for both acute and elective patients. Based on overflow rules, we translate the demand predictions to bed census predictions, since demand and census may differ due to rejections and misplacements. The combination of the hourly level perspective and the bed census conversion enables us to derive several performance measures, along which the effectiveness of different logistical configurations can be assessed. Subsequently, following [32], we incorporate the tactical decision that is referred to as ‘staff-shift scheduling’ in [30] into this integrated modeling framework. For each working shift, during a given planning horizon, we determine the number of employees that should be assigned to each inpatient care unit. The predictable fluctuation in inpatient population due to the operating room schedule and other predictable variabilities in patient arrivals (e.g., seasonal, day of week, and time of day effects) can be taken into account when determining the staffing levels for

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‘dedicated nurses’, which are nurses with a fixed assignment to a care unit. When only dedicated nurses are employed, the buffer capacity required to protect against random demand fluctuations can lead to regular overstaffing. When two or more care units cooperate by jointly appointing a flexible nurse pool, the variability of these random demand fluctuations balances out due to economies of scale, so that less buffer capacity is required. We explore the potential of flexible staffing policies that allow hospitals to dynamically respond to their fluctuating patient populations. This flexibility is achieved by employing a pool of cross-trained nurses, or ‘float nurses’ [19, 42], for whom assignments to specific care units are decided at the start of their shifts. To illustrate its potential, the method is applied to a case study that involves the care units in the surgical inpatient clinic of the Dutch university hospital, the Academic Medical Center Amsterdam (AMC), which serve the specialties of traumatology, orthopedics, plastic surgery, urology, vascular surgery, and general surgery. Inspired by the quantitative results, the AMC decided that the method will be fully implemented as part of the global redesign of its inpatient care services. This chapter is organized as follows: Sect. 2 provides a review of relevant literature; Sect. 3 presents the model to predict bed census; Sect. 4 presents the models for the fixed and the flexible staffing policies; Sect. 5 presents the numerical results for the case study; and Sect. 6 closes the chapter with a general discussion.

2 Literature Effectively designing inpatient care services requires simultaneous consideration of several interrelated strategic and tactical planning issues [30]. The inpatient care facility is a downstream department. The outflow of the operating theater and the emergency department are main drivers behind its workload. Therefore, it is highly desirable to apply coordinated planning: considering the inpatient care facility in isolation yields suboptimal decision-making [25, 48]. Smoothing patient inflow prevents large differences between peak and off-peak periods and so realizes a more efficient use of resources [1, 25, 51]. Although the control on the inflow of patients from the emergency department is inherently very limited due to its nature, anticipation for emergency admissions is possible, by statistically predicting the arrival process of emergency patients that often follows a cyclic pattern [22]. Hospitals typically allocate operating room capacity through a Master Surgical Schedule (MSS). Anticipation for elective surgical patients is possible as well, by taking the surgical schedule into account [1, 22, 25, 51]. In this chapter, we address these various patient flows and take the necessity of integral decision-making into account. Several analytical studies have addressed partial resource capacity planning within the inpatient care chain, for example, by dimensioning care units in isolation [5, 20, 22], balancing bed utilization across units [3, 12, 36], or improving the MSS to balance inpatient care demand [1, 6, 8, 46, 49]. More integral approaches can be

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found in simulation studies [25, 27, 44, 47]. The advantage of such approaches is their flexibility and therefore modeling power. However, the disadvantage is that the nature of such studies is typically context-specific, which limits the generalizability of application and findings. Personnel scheduling in general and capacity planning for nursing staff in specific have received considerable attention from the operations research community )see the extensive literature review [45] and the survey and classification articles [9, 14, 17]). The nurse staffing process involves a set of hierarchical decisions over different time horizons with different levels of precision. The first strategic level of decision-making is the workforce dimensioning decision which concerns both the number of employees that must be employed and is often expressed as the number of full-time equivalents and the mix in terms of skill categories [26, 35, 37]. The second tactical level concerns staff-shift scheduling, which deals with the problem of selecting which shifts are to be worked and how many employees should be assigned to each shift to meet the patient demand [17, 31]. The third operational offline decision level concerns the creation of individual nurse timetables, designed with the objective to meet the required shift staffing levels set on the tactical level while satisfying a complex set of restrictions involving work regulations and employee preferences. This planning step is often referred to as ‘nurse rostering’ [9– 11]. The fourth operational online decision level concerns the reconsideration of the staff schedule at the start of a shift. At this level, float nurses are assigned to specific care units [9, 42], and, based on the severity of need, on-call nurses, overtime, and voluntary absenteeism can be used to further align patient care supply and demand [23, 39]. The interdependence of the decision levels must be recognized to facilitate systematic improvements in nurse staffing. As expressed in the literature review by [39], each level is constrained by previous commitments made at higher levels, as well as by the degrees of flexibility conserved for later correction at lower levels. For a more elaborate exposition of the relevant decisions and considerations involved at each decision level and a detailed overview of relevant literature, see [30]. Tactical workforce decision-making in healthcare has received little attention. A spreadsheet approach is presented by [16], to retrospectively fit optimal shift staffing levels to historical census data. Simulation studies have shown to be successful in taking a more integral approach [23, 26]. Analytic yet deterministic approaches can be found in [7, 38, 52]. Stochastic approaches to determine shift staffing levels are available in [15, 54] and [55]. These references do not present an integral care chain approach, given that the demand distributions underlying the staffing decisions are not based on patient arrival patterns from the operating theaters and emergency departments. Concerning the operational online assignment strategy to place a given number of available float nurses in care units at the start of their shifts, [43] indicate that formulating such an assignment strategy requires the consideration of three issues: (1) a method for measuring of the urgency of need for an additional nurse; (2) a prediction per care unit of that urgency of need for an upcoming shift; and (3) development of a technique for the allocation of the available float nurses to care

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units in order to meet this need. Whereas [43] focus on the third issue by developing a branch-and-bound algorithm, our assignment strategy involves the consideration of all three steps. Nurse-to-patient ratios are commonly applied when determining staffing levels [2, 55]. These ratios indicate how many patients a registered nurse can care for during a shift, taking into account both direct and indirect patient care. Staffing according to nurse-to-patient ratios has received attention in the operations research literature [15, 54, 55]. In practice, setting the numerical values of the ratios seems to be more based on negotiation than on science [15, 54]. In this chapter, following [32, 33], we present an exact stochastic analytic approach to derive appropriate staffing levels, including the flexibility of float nurses, using nurse-to-patient ratios while taking an integrated care chain perspective.

3 Hourly Bed Census Predictions This section presents a general model to predict the workload at several care units due to patients arriving and departing according to a statistically predicted inflow and outflow pattern. Following [33], we will focus on the workload at an inpatient care facility on a timescale of hours, due to patients originating from the operating theater and emergency department. The basis for the operating room outflow prediction is the Master Surgical Schedule (MSS). The basis for the emergency department outflow prediction is a cyclic random arrival process, the Acute Admission Cycle (AAC). The cycles are combined into the Inpatient Facility Cycle (IFC), with length the least common multiple of the lengths of the MSS and the AAC. For the demand predictions, for both elective and acute patients, the impact of a single patient type in a single cycle (MSS or AAC) is determined, by which in the second step the impact of all patient types within a single cycle can be calculated. Since the IFC is cyclical, the predictions from the second step are combined to find the probability distributions of the number of recovering patients at the inpatient care facility on each time interval in the IFC. The resulting demand distributions are translated to bed census distributions, and performance measures are formulated based on the demand and census distributions. The operation of the inpatient care facility is as follows. Each day is divided in time intervals (hours). Patient admissions are assumed to take place independently at the start of a time interval. Elective patients are admitted to a care unit either on the day before or on the day of surgery. For acute patients we assume a cyclic (e.g., weekly) non-homogeneous Poisson arrival process corresponding to the unpredictable nature of emergency arrivals (see, e.g., [53]). Discharges take place independently at the end of a time interval. For elective patients we assume the length of stay to depend only on the type of patient and to be independent of the day of admission and the day of discharge. For acute patients the length of stay and time

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of discharge are dependent on the day and time of arrival, in particular to account for possible disruptions in diagnostics and treatment during nights and weekends. Patient admission requests may have to be rejected due to a shortage of beds, or patients may (temporarily) be placed in less appropriate units. As a consequence, demand predictions and bed census predictions do not coincide. Therefore, an additional step is required to translate the demand distributions into census distributions. This translation is performed by assuming that after a misplacement the patient is transferred to his preferred care unit when a bed becomes available, where we assume a fixed patient-to-ward allocation policy, which prescribes the prioritization of such transfers.

3.1 Demand Predictions for Elective Patients Model input Time. An MSS is a repeating blueprint for the surgical schedule of S days. Each day is divided in T time intervals. Therefore, we have time points t = 0, . . . , T , in which t = T corresponds to t = 0 of the next day. For each single patient, day n counts the number of days before or after surgery, i.e., n = 0 indicates the day of surgery. MSS utilization. For each day s ∈ {1, . . . , S}, a (sub)specialty j can be assigned to an available operating room i, i ∈ {1, . . . , I }. The OR block at operating room m i on day s is denoted by bi,s and is possibly divided in a morning block bi,s a , if an OR day is shared. The discrete distributions cj and an afternoon block bi,s represent how specialty j utilizes an OR block, i.e., cj (k) is the probability of k surgeries performed in one block, k ∈ {0, 1, . . . , C j }. If an OR block is divided j j in a morning OR block and an afternoon OR block, cM and cA represent the utilization probability distributions, respectively. For brevity, we do not include shared OR blocks in our formulation, since these can be modeled as two separate (fictitious) operating rooms. j Admissions. With probability en , n ∈ {−1, 0}, a patient of type j is admitted on day n. Given that a patient is admitted on day n, the time of admission is described j by the probability distribution wn,t . We assume that a patient who is admitted on the day of surgery is always admitted before or at time ϑj ; therefore, we have j w0,t = 0 for t = ϑj + 1, . . . , T − 1. Discharges. P j (n) is the probability that a type j patient stays n days after surgery, n ∈ {0, . . . , Lj }. Given that a patient is discharged on day n, the probability of j being discharged in time interval [t, t + 1) is given by mn,t . We assume that a patient who is discharged on the day of surgery is discharged after time ϑj , i.e., j m0,t = 0 for t = 0, . . . , ϑj .

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Single surgery block In this first step, we consider a single specialty j operating in a single OR block. Note that admissions can take place during day n = −1 and during day n = 0 until time t = ϑj . Discharges can take place during day n = 0 from time t = ϑj + 1 and during days n = 1, . . . , Lj . Therefore, the probability j hn,t (x) that n days after carrying out a block of specialty j , at time t, x patients of the block are still in recovery is  j hn,t (x)

=

j

an,t (x)

, n = −1 and n = 0, t ≤ ϑj ,

j dn,t (x)

, n = 0, t > ϑj and n = 1, . . . , Lj ,

j

where an,t (x) represents the probability that x patients are admitted until time t on j day n and dn,t (x) is the probability that x patients are still in recovery at time t on j j day n. The derivation of an,t is presented below and that of dn,t is by analogy and is presented in [33]. Observe that j

j an,t (x)

=

C 

j

an,t (x|y)cj (y),

y=x j

where an,t (x|y) is the probability that x patients are admitted until time t on day n, given that y admissions take place in total: ⎧  y j j ⎪ ⎪ (vn,t )x (1−vn,t )y−x ⎪ ⎪ x ⎪ ⎪  ⎪ x  ⎪ ⎪ y−g j j j ⎪ ⎪ (vn,t )x−g (1−vn,t )y−x an−1,T −1 (g|y) ⎪ ⎪ ⎪ x − g ⎨ g=0 j  x  an,t (x|y)=  y−g j j j ⎪ ⎪ (vn,t )xs−g (1−vn,t )y−x an,t−1 (g|y) ⎪ ⎪ x − g ⎪ ⎪ g=0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0

, n= − 1, t= −0, , n=0, t=0, , n= − 1, t=1, . . . , T − 1 and n=0, t=1, . . . , ϑj − 1, , n=0, t ≥ ϑj ,

j

where vn,t is the probability for a type j patient to be admitted at time t, given that he/she will be admitted at day n and is not yet admitted before t: j

j

vn,t =

j

en

T −1 k=t

j

wn,t en j

j

wn,k + e0 · 1(n=−1)

.

Single MSS cycle Now, we consider a single MSS in isolation. From the distribuj tions hn,t , we can determine the distributions Hm,t , the discrete distributions for the total number of recovering patients at time t on day m (m ∈ {0, 1, 2, . . . , S, S + 1, S + 2, . . .}) resulting from a single MSS cycle. We determine the overall

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probability distribution of the number of patients in recovery resulting from a single MSS, using discrete convolutions. If specialty j is assigned to OR block bi,s , then the distribution h¯ i,s m,t for the number of recovering patients of block bi,s present at time t on day m (m ∈ {0, 1, 2, . . . , S, S + 1, S + 2, . . .}) is given by  h¯ i,s m,t

=

0 j hm−s,t

, m < s − 1, , m ≥ s − 1,

¯ i,s where 0 means h¯ i,s m,t (0) = 1 and all other probabilities hm,t (x), x > 0 are 0. Then, Hm,t is computed by ¯ 1,2 ¯ 1,S ¯ 2,1 ¯ I,S Hm,t = h¯ 1,1 m,t ⊗ hm,t ⊗ . . . ⊗ hm,t ⊗ hm,t ⊗ . . . ⊗ hm,t ,

(1)

where ⊗ denotes the discrete convolution. Steady state In this step, the complete impact of the repeating MSS is considered. SS , the steady-state The distributions Hm,t are used to determine the distributions Hs,t probability distributions of the number of recovering patients at time t on day s of the cycle (s ∈ {1, . . . , S}). Since the cyclic structure of the MSS implies that the recovery of patients receiving surgery during one cycle may overlap with patients from the next cycle, SS can be the distributions Hm,t have to be overlapped in the correct manner. Hs,t computed as follows:  SS Hs,t

=

Hs,t ⊗ Hs+S,t ⊗ . . . ⊗ Hs+ M/SS,t , s = 1, . . . , S − 1, H0,t ⊗ HS,t ⊗ . . . ⊗ H M/SS,t

, s = S,

where M = max{m | ∃t, x with Hm,t (x) > 0}.

3.2 Demand Predictions for Acute Patients Model input Time. The AAC is the repeating cyclic arrival pattern of acute patients with a length of R days. For each single patient, day n counts the number of days after arrival. Admissions. An acute patient type is characterized by patient group p, p = 1, . . . , P , arrival day r, and arrival time θ , which is for notational convenience denoted by type j = (p, r, θ ). The Poisson arrival process of patient type j has arrival rate λj .

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Discharges. P j (n) is the probability that a type j patient stays n days, n ∈ {0, . . . , Lj }. Given that a patient is discharged at day n, the probability of being j j discharged in time interval [t, t + 1) is given by m ˜ n,t . By definition, m ˜ 0,t = 0 for t ≤ θ. Single patient type In this first step we consider a single patient type j . We compute j the probability gn,t (x) that on day n at time t, x patients are still in recovery. Admissions can take place during time interval [θ, θ + 1) on day n = 0 and discharges during day n = 0 after time θ and during days n = 1, . . . , Lj . Therefore, j we calculate gn,t (x) as follows:  j gn,t (x)

=

j

a˜ t (x) j d˜n,t (x)

, n = 0, t = θ, , n = 0, t > θ and n = 1, . . . , Lj ,

j

where a˜ t (x) represents the probability that x patients are admitted in time interval j [t, t + 1) on day n = 0 and d˜n,t (x) is the probability that x patients are still in j j recovery at time t on day n. The derivations of a˜ t and d˜n,t are by analogy with those probabilities for elective patients and may be found in [33]. Single cycle Now, we consider a single AAC in isolation. From the distributions j gn,t (x), we can determine the distributions Gw,t , the distributions for the total number of recovering patients at time t on day w (w ∈ {1, . . . , R, R +1, R +2, . . .}) resulting from a single AAC by analogy with those probabilities for elective patients (see [33]). Steady state In this step, the complete impact of the repeating AAC is considered. The distributions Gw,t are used to determine the distributions GSS r,t , the steady-state probability distributions of the number of recovering patients at time t on day r of the cycle (r ∈ {1, . . . , R}) by analogy with those probabilities for elective patients (see [33]).

3.3 Demand Predictions Per Care Unit To determine the complete demand distribution of both elective and acute patients, SS and GSS . In general, the we need to combine the steady-state distributions Hs,t r,t MSS cycle and AAC are not equal in length, i.e., S = R. This has to be taken into account when combining the two steady-state distributions. Therefore, we define the new IFC length Q = LCM(S, R), where the function LCM stands for least

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common multiple. Let Zq,t be the probability distribution of the total number of patients recovering at time t on day q during a time cycle of length Q: Zq,t = HqSSmod S+S·1(q mod S=0) ,t ⊗ GSS q mod R+R·1(q mod R=0) ,t . Let W k be the set of specialties j whose operated patients are (preferably) admitted to unit k (k ∈ {1, . . . , K}) and V k the set of acute patient types j that are (preferably) k , can be calculated admitted to unit k. Then, the demand distribution for unit k, Zq,t k k by only considering the patients in W in equation (1), and V may be obtained by analogy (see [33]).

3.4 Bed Census Predictions k , k = 1, . . . , K, into bed census We translate the demand distributions Zq,t distributions Zˆ q,t , the distributions of the number of patients present in each unit k at time t on day q. To this end, we require an allocation policy φ that uniquely specifies from a demand vector x = (x1 , . . . , xK ) a bed census vector xˆ = (xˆ1 , . . . , xˆK ), in which xk and xˆk denote the demand for unit k and the bed census at unit k, k respectively. Let φ(·) be the function that executes allocation policy φ. Let Zˆ q,t denote the marginal distribution of the census at unit k given by distribution Zˆ q,t . With Mk , the capacity of unit k in number of beds, we obtain:



1 K Zˆ q,t (ˆx) = Zˆ q,t (xˆ1 ), . . . , Zˆ q,t (xˆK ) =

 {x|ˆx=φ(x)}



K 

 k Zq,t (xk )

.

k=1

We do not impose restrictions on the allocation policy φ other than specifying a unique relation between demand x and census configuration xˆ . Recall that the underlying assumption is that a patient is transferred to his preferred unit when a bed becomes available. The policy φ also reflects the priority rules that are applied for such transfers. As an illustration, we present an example for an inpatient care facility with two care units of capacity M1 and M2 , respectively: ⎧ (x1 , x2 ) ⎪ ⎪ ⎪ ⎪ ⎨ (M1 , min{x2 + (x1 − M1 ), M2 }) φ(x) = ⎪ (min{x1 + (x2 − M2 ), M1 }, M2 ) ⎪ ⎪ ⎪ ⎩ (M1 , M2 )

, x1 ≤ M1 , x2 ≤ M2 , , x1 > M1 , x2 ≤ M2 , , x1 ≤ M1 , x2 > M2 ,

(2)

, x1 > M1 , x2 > M2 .

Under this policy patients are assigned to their bed of preference if available and are otherwise misplaced to the other unit if beds are available there.

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3.5 Performance Indicators k and the census distributions Z k we are ˆ q,t Based on the demand distributions Zq,t able to formulate a variety of performance indicators. We present a selection of such performance indicators, which will be used in Sect. 5 to evaluate the impact of different scenarios and interventions. k (α) and D k (α) be the α-th perˆ q,t Demand and bed census percentiles Let Dq,t centile of, respectively, demand and bed census at time t on day q:

 k Dq,t (α)

= arg min x



x 

k Zq,t (i)



≥α ,

k Dˆ q,t (α)

= arg min x

i=0

x 

 k (i) Zˆ q,t

≥α .

i=0

(Off-)Peak demand Reducing peaks and drops in demand will balance bed occupancy and therefore allows more efficient use of available staff and beds. Define k k P q (α) (P kq (α)) and P (α) (P k (α)) to be the maximum (minimum) α-th demand percentile per day and over the complete cycle, respectively:   k k P q (α) = max Dq,t (α) ,

 k  k P (α) = max P q (α) ,

  k P kq (α) = min Dq,t (α) ,

  P k (α) = min P kq (α) .

t

q

q

t

Admission rate Patient admissions may increase the nursing workload. Let Λkq,t be the distribution of the number of arriving patients during time interval [t, t + 1) on day q who are preferably admitted to care unit k. To obtain Λkq,t , we first determine j

a¯ n,t , the distribution of the number of elective type j arrivals during time interval [t, t + 1) on day n (n ∈ {−1, 0}): j

j a¯ n,t (x)=

C 

j

cj (y)a¯ n,t (x|y)

j

a¯ n,t (x|y)=

, with

y=0

  y j j j j (en wn,t )x (1−en wn,t )y−x . x

Λkq,t is then determined by taking the discrete convolution over all relevant arrival distributions of both elective and acute patient types:

Λkq,t =

⎧ ⎪ ⎨

I

⎧ ⎪ ⎨

⎪ ⎩ i=1 ⎪ ⎩j ∈W k :j ∈b

j

a¯ −1,t i,s 

⎫ ⎪ ⎬ ⎪ ⎭

⊗

⎧ ⎪ ⎨ ⎪ ⎩j ∈W k :j ∈b

j

a¯ 0,t i,s 

⎫⎫ ⎪ ⎬⎪ ⎬ ⎪ ⎭⎪ ⎭

⊗

⎧ ⎨ ⎩

j

a˜ t j ∈V k :r=r 

⎫ ⎬ ⎭

, (3)

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where s  = 1 + q ! mod S, s  = q mod S + S · 1(q mod S=0) , r  = q mod R + R · 1(q mod R=0) , and x∈X fx denotes the discrete convolution over the probability distributions fx , x ∈ X . The first term in the right-hand side of (3) represents the elective patients who claim a bed at unit k (j ∈ W k ), who are operated in any OR, and who are admitted on the day s  − 1 before surgery or on the day s  of surgery. The second term in the right-hand side of (3) represents the acute patients who claim a bed at unit k (j ∈ V k ) and who arrive on the corresponding day r  in the AAC. k , ρ k , and ρ k be the average bed utilization rate at Average bed occupancy Let ρq,t q care unit k respectively at time t on day q, on day q, and over the complete cycle: k

k ρq,t =

M 1  k x · Zˆ q,t (x), Mk

ρqk =

x=0

T −1 1  k ρq,t , T

1  k ρq . Q Q

ρk =

t=0

q=1

Rejection probability Let R φ,k denote the probability that under allocation policy φ an admission request of an arriving patient for unit k has to be rejected, because all beds at unit k are already occupied and none of the alternative beds (prescribed φ,k by φ) are available. To determine R φ,k , we first determine Rq,t : the probability of φ,k such an admission rejection at time t on day q. R is then calculated as follows: R φ,k = 

1



k q,t E[Λq,t ] q,t

φ,k

E[Λkq,t ]Rq,t .

Let n indicate the number of arriving patients who are preferably admitted to unit k and x = (x1 , . . . , xK ) the demand for each unit (in which these arrivals are already incorporated). Introduce R φ,k (x, n), the number of rejected patients under k (x |n) the probability allocation policy φ of the n arriving patients to unit k, and Zq,t k that at time t on day q in total xk patients demand a bed at unit k and n of them have φ,k just arrived. Then, Rq,t is calculated by φ,k

E[# rejections at unit k on time (q, t)] E[# arrivals to unit k on time (q, t)]   1  k = Zq,t (x ) R φ,k (x, n)Λkq,t (n)Zq,t (xk |n). k E[Λq,t ] x  =k n

Rq,t =

(4)

k (x |n), let us first introduce the concept cohort. A cohort For the derivation of Zq,t k is a group of patients originating from a single instance of an OR block (electives) or admission time interval (acute patients). Then

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k (x |n) = Zq,t k

P [Demand xk patients for unit k on time t on day q of which n are arriving in [t, t + 1)] P [n arrivals for unit k on day q in [t, t + 1)] ⎫ ⎧ Ω ⎬ ⎨   1 σ (i) fq,t (yσ (i) ) = k ⎭ ⎩ Λq,t (n) y ,...,y , σ (1)

σ (Ω)

nσ (1) ,...,nσ (ω) :   i yi =xk , j nj =n

"

i=ω+1



σ (j ) σ (j ) ω j =1 αq,t (yσ (j ) )aˇ q,t (nσ (j ) |yσ (j ) ) ,

where Ω is the total number of cohorts, ω the number of cohorts that do generate arrivals during time interval [t, t +1) on day q, and the permutation σ is such that the patient types σ (1), . . . , σ (ω) are the types that can generate those arrivals. Further, i as f i = hi for notational convenience we introduce the function fq,t q,t q,t for the j

i = g i for acute patient types. Also, we introduce α elective patients and fq,t q,t as q,t j

j

j

αq,t = aq,t for the elective patient types and αq,t =

(p,q mod R+R·1q mod R=0 ,t) a˜ t

j the acute patient types. It remains to define aˇ q,t (nj |yj ), arriving cohort, from the yj patients present in total, nj

for

the probability that for an arrivals occur during time

interval [t, t + 1):

 j

aˇ q,t (nj |yj ) =

 yj j j (νn,t )nj (1 − νn,t )yj −nj , nj j

j

where for elective patient types νn,t = j

j en

t

j

wn,t en

j j k=0 wn,k +e−1 ·1(n=0)

and for acute patient

types νn,t = 1. R φ,k (x, n) is uniquely determined by allocation policy φ. For example, for the case with K = 2 presented in (2), we have for unit k = 1: ⎧ min{n, x1 −M1 } ⎪ ⎪ ⎪ ⎪ ⎨ max{0, (x −M )−(M −x )} 1 1 2 2 R φ,1 (x, n)= ⎪ ⎪ n − max {0, min{n, (M2 −x2 −[x1 −M1 −n])}} ⎪ ⎪ ⎩ 0

, x1 ≥ M1 , x2 ≥ M2 , , x1 ≥ M1 , x2 < M2 , n ≥ (x1 −M1 ), , x1 ≥ M1 , x2 < M2 , n < (x1 − M1 ), , otherwise. (5)

In (5), the first case reflects the situation in which all beds at care unit 2 are occupied so that all arriving patients who do not fit in unit 1 have to be rejected. The second and third cases reflect the situation that (some of) the arriving patients can be misplaced to unit 2 so that only a part of the arriving patients have to be rejected.

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In the second case, the (x1 − M1 ) patients that do not fit at unit 1 are all arriving patients. In the third case, some of the (x1 − M1 ) patients were already present so that not all (M2 − x2 ) beds at unit 2 can be used to misplace arriving patients. Misplacement probability Let M φ,k denote the probability that under allocation policy φ a patient who is preferably admitted to care unit k is admitted to another unit. The derivation of M φ,k is equivalent to that of R φ,k . In (4), R φ,k (x, n) has to be replaced by M φ,k (x, n), which gives the number of misplaced patients under allocation policy φ of the n arriving patients to unit k and which is again uniquely determined by φ. Observe that for the two unit example presented in (2), we have: ⎧ , x1 > M1 , x2 < M2 , n ≥ (x1 − M1 ), ⎪ ⎨ min{x1 −M1 , M2 −x2 } φ,1 M (x, n)= max {0, min{n, (M2 −x2 −[x1 −M1 −n])}} , x1 > M1 , x2 < M2 , n < (x1 − M1 ), ⎪ ⎩ 0 , otherwise.

Productivity Let K be a set of cooperating care units, i.e., units that mutually allow misplacements. Let P K reflect the productivity of the available capacity at care units k ∈ K , defined as the number of patients that is treated per bed per year: PK =

 1 365 φ,k  (1 − Rq,t )E[Λkq,t ] k Q M k∈K q,t

(6)

k∈K

Remark 1 (Approximation) Observe that the misplacement and rejection probabilities are an abstract approximation of complex reality. In our model, we count each time interval how many of the arriving patients have to be misplaced or rejected. Since we do not remove rejected patients from the demand distribution, it is likely that we overestimate the rejection and misplacement probabilities. However, also in reality strict rejections are often avoided: by postponing elective admissions, predischarging another patient, or letting acute patients wait at the emergency department. These are all undesired degradations of provided quality of care. Therefore, our method provides a secure way of organizing inpatient care services. It is applicable to evaluate performance for care unit capacities that give low rejection probabilities, thus when high service levels are desired, which is typically the case in healthcare.

4 Flexible Nurse Staffing This section reviews two staffing models based on the bed census predictions described above as introduced in [32]. First, we discuss the requirements that need to be satisfied in setting appropriate staffing levels. Then we present the fixed staffing model and subsequently a model to find optimal staffing levels when float nurse pools are applied: the flexible staffing model.

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4.1 Staffing Requirements Recall that we consider a planning horizon of Q days (q = 1, . . . , Q), during which each day is divided in T time intervals (t = 0, 1, . . . , T − 1). The set of working shifts is denoted by T , where a shift τ is characterized by its start time bτ and its length τ . Within the time horizon, (q, t) is a unique time interval and (q, τ ) a unique shift. For notational convenience, t ≥ T indicates a time interval on a later day, e.g., (q, T + 5) = (q + 1, 5). For each of K inpatient care units, with the capacity of unit k being M k beds, staffing levels have to be determined for each shift (q, τ ). We consider two types of staffing policies: ‘fixed’ and ‘flexible’ staffing. Under fixed staffing, the number of nurses working in unit k during shift (q, τ ), denoted by k , is completely determined in advance. In the flexible case, ‘dedicated’ staffing sq,τ k per unit are determined, together with the number of nurses f levels dq,τ q,τ available in a flex pool. The decision regarding the particular units to which the float nurses are assigned is delayed until the start of the execution of a shift. We assign float nurses to one and the same care unit for a complete working shift, to avoid frequent handovers, which increase the risk of medical errors. Thus, we obtain staffing levels k = d k + f k , k = 1, . . . , K, where f k denotes the number of float nurses sq,τ q,τ q,τ q,τ assigned to unit k from the available fq,τ . Taking into account the current bed census and the predictions on patient admissions and discharges, the allocation of the float nurses to care units at the start of a shift is decided according to a predetermined assignment procedure. We denote such an assignment procedure by π . For both staffing policies, we assume shifts to be non-overlapping, and for the flexible policy, we assume shifts to be equivalent for each care unit. Our goal is to determine the most cost-efficient staffing levels such that certain quality-of-care constraints are satisfied. Because float nurses are required to be cross-trained, it is likely that these staff members are more expensive to employ. To be able to differentiate such costs, we therefore consider staffing costs ωd for each dedicated nurse who is staffed for one shift and ωf for each flexible nurse. Next, the k , indicating the nurse-to-patient ratio targets during shift (q, τ ) are reflected by rq,τ number of patients a nurse can be responsible for at any point in time. To keep track of the compliance to these targets, we define the concept ‘nurse-to-patient coverage’, or shortly ‘coverage’. With xtk the number of patients present at unit k at a certain k · s k /x k . time (q, t), bτ ≤ t < bτ + τ , the coverage at that time is given by rq,τ q,τ t Thus, a coverage of one or higher corresponds to a preferred situation. Starting from the following quality-of-care requirements as prerequisites, we will formulate the fixed and flexible staffing models by which the most cost-effective staffing levels can be found: (i) Staffing minimum. For safety reasons, at least S k nurses have to be present at care unit k at any time. (ii) Coverage minimum. The coverage at care unit k may never drop below β k . (iii) Coverage compliance. The long-run fraction of time that the coverage at care unit k is one or higher is at least α k . We denote the expectation of the coverage

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k (·); the arguments of this compliance at care unit k during shift (q, τ ) by cq,τ function depend on which staffing policy is considered. (Note that ‘coverage compliance’ is a measure defined for a shift, based on the measure ‘coverage’ that is defined for the time periods within that shift). (iv) Flexibility ratio. To ensure continuity of care, at any time, the fraction of nurses at care unit k that are dedicated nurses has to be at least γ k . (v) Fair float nurse assignment. The policy π , according to which the allocation of the available float nurses to care units at the start of a shift is done, has to be ‘fair’. Fairness is defined as assigning each next float nurse to the care unit where the expected coverage compliance during the upcoming shift is the lowest.

4.2 Fixed Staffing When only dedicated staffing is allowed, there is no interaction between care units. Therefore, the staffing problem decomposes in the following separate decision problems for each care unit k and each shift (q, τ ): k min zF = ωd sq,τ

(7)

k s.t. sq,τ ≥ Sk

(8)

$ # k k ≥ β k · M k /rq,τ sq,τ

(9)

 k  k k cq,τ sq,τ , rq,τ ≥ αk

(10)

The constraints (8), (9), and (10) reflect requirements (i), (ii), and (iii), respectively. k be the random variable with bed census distribution Z k counting the ˆ q,t Let Xq,t number of patients present on care unit k at time (q, t). Then, the coverage compliance in (10) can be calculated as follows: %  k  1 k k cq,τ sq,τ , rq,τ =E τ 1 = τ

bτ + τ −1 

&   k k k 1 Xq,t ≤ sq,τ · rq,τ

t=bτ

k s k ·rq,τ bτ + τ −1 q,τ  

t=bτ

k (x). Zˆ q,t

x=0

k ·r k sq,τ q,τ ˆ k k Zq,t (x) reflects the probability that with staffing level sq,τ Observe that x=0 k and under ratio rq,τ the nurse-to-patient ratio target is satisfied during time interval

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k satisfying [t, t + 1). The optimum of (7) is found by choosing the minimum sq,τ constraints (8) and (9) and increasing it until constraint (10) is satisfied.

4.3 Flexible Staffing The next step is to formulate the flexible staffing model. Note that for requirements (i) and (ii), the constraints are similar to those for fixed staffing. Under the k k assumption ωd ≤ ωf , we can replace sq,τ by dq,τ in (8) and (9). Due to the presence of a flex pool, the care units cannot be considered in isolation anymore. Hence, constraint (10) has to be replaced. An assignment procedure has to be formulated that fulfills requirement (v), and this assignment procedure influences the formulation of the constraint for requirement (iii). In addition, a constraint needs to be added for requirement (iv). For an assignment procedure π that allocates the float nurses to care units at π (d, f, y) = (g 1,π (d, f, y), . . . , g K,π (d, f, y)) be the start of a shift (q, τ ), let gq,τ q,τ q,τ the vector denoting the number of float nurses assigned to each care unit, when f flex nurses are available to allocate, the number of staffed dedicated nurses equals d = (d 1 , . . . , d K ), and the census at the different care units at time (q, bτ ) equals y = (y 1 , . . . , y K ). A vector of the type y reflects what we will call a census configuration. Let π ∗ denote the assignment procedure that ensures constraint (v). The assignk , k = 1, . . . , K and therefore ment procedure π ∗ depends on d q,τ , fq,τ , and rq,τ the coverage as well. Hence, requirement (v) gives a constraint of the form k (d k k ∗ cq,τ q,τ , fq,τ , rq,τ ) ≥ α . However, assignment procedure π depends on the census configuration y at time (q, bτ ), so calculation of the coverage compliance k (d k first requires the computation of cq,τ q,τ , fq,τ , rq,τ ; y), which describes the coverage compliance, given that at the start of shift (q, τ ) census configuration y is observed. Then, the coverage compliance is given by K     k   k k k w cq,τ d q,τ , fq,τ , rq,τ cq,τ d q,τ , fq,τ , rq,τ = ;y (y w ) . Zˆ q,b τ y

w=1

k (d k ∗ Using cq,τ q,τ , fq,τ , rq,τ ; y), the assignment policy π satisfying requirement (v) is the one that satisfies ∗

π (d q,τ , fq,τ , y) =  gq,τ

arg max 1 ,...,f K ) : (fq,τ q,τ



k

k =f fq,τ q,τ

  k k d , f , r ; y . min c q,τ q,τ q,τ q,τ  k

(11) k (y), the number of nurses staffed at care unit k Applying policy π ∗ provides sq,τ if census configuration y is observed at the start of shift (q, τ ). Hence, the flexible model for each shift (q, τ ) is the following, where constraints (13)–(17) reflect (i)– (v), respectively: :

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min

zE = ωf fq,τ + ωd



dk k q,τ

k s.t. dq,τ ≥ Sk ,

(12) for all k,

(13)

$ # k k , ≥ β k · M k /rq,τ dq,τ

for all k,

(14)

  k k cq,τ d q,t , fq,τ , rq,τ ≥ αk ,

for all k,

(15)

k k dq,τ ≥ γ k · sq,τ (y) ,

for all k, y,

(16)

  k k k,π ∗ sq,τ (y) = dq,τ + gq,τ d q,τ , fq,τ , y ,

for all k, y.

(17)

k (d, f k Finding the optimum for (12) requires the computation of cq,τ q,τ , rq,τ ; y) by considering every sample path of census configurations during a shift. For realistic instances, it is computationally expensive to find the optimal solution for 1 , . . . , dK , f dq,τ q,τ (see [32]). q,τ

5 Quantitative Results This section presents the experimental results. We present a selection of results taken from [32, 33].

5.1 Case Study Description The case study entails six surgical specialties of the university hospital AMC, which together have 104 beds in operation. The entire hospital has 20 operating rooms and 30 inpatient departments, with a total of 1000 beds. The following specialties are taken into account: traumatology (TRA), orthopedics (ORT), plastic surgery (PLA), urology (URO), vascular surgery (VAS), and general surgery (GEN). In the present setting, the patients of the abovementioned specialties are admitted to four different inpatient care departments. On Floor I, care unit A houses GEN and URO and unit B VAS and PLA. On Floor II, care unit C houses TRA and unit D ORT. The physical building is such that units A and B are physically adjacent (Floor I), as are units C and D (Floor II). For these specialties, we have historical data available over 2009–2010 on 3498 (5025) elective (acute) admissions, with an average length of stay (LOS) of 4.85 days (see Table 1). At the time of the original study, no cyclic MSS was applied. Each time, roughly 6 weeks in advance the MSS was determined for a period of 4 weeks. The capacities of units A, B, C, and D are 32, 24, 24, and 24 beds, respectively. The utilizations over 2009–2010 were 53.2%, 55.6%, 54.4%, and 60.6%, respectively (which includes some patients from other specialties that were placed in these care units). These utilizations reflect administrative bed census,

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Table 1 Overview of historical data 2009–2010 Specialty General surgery Urology Vascular surgery Plastic surgery Traumatology Orthopedics a Load = Expected

Acronym GEN URO VAS PLA TRA ORT

Care unit A A B B C D

Elective admissions 611 818 257 639 337 836

Acute admissions 901 1157 634 288 1200 845

Average LOS (in days) 3.31 3.68 8.30 2.29 5.88 6.23

Loada (# patients) 6.88 9.99 10.16 2.91 12.41 14.38

number of patient arrivals per day ∗ Average LOS

which means the percentage of time that a patient physically occupies a bed or keeps it reserved during the time the patient is at the operating theater or at the intensive care department. Unfortunately, no confident data was available on rejections and misplacements. Working days are divided in three shifts: the day shift (8:00–15:00), the evening shift (15:00–23:00), and the night shift (23:00–8:00). These time intervals indicate the times that nurses are responsible for direct patient care. Around these time intervals, the working shifts also incorporate time for patient handovers, indirect patient care, and professional development. At all times, there should be at least two nurses present at each care unit. According to agreements on working conditions for nurses in all university hospitals in the Netherlands, the contractual number of annual working hours per full-time equivalent (FTE) is 1872. The number of hours that one FTE can be employed for direct nursing care, after deduction of time reserved for professional development, holiday hours, and sick leave, is 1525.7 on average (also see [16]). The yearly cost per FTE, including all costs and bonuses, is roughly e53,000. The nurse-to-patient ratio targets prescribed by the board of the AMC for the care units of interest are 1:4 during the day shifts, 1:6 during the evening shifts, and 1:10 during the night shifts. At the time of the original study, the current staffing practice was based on the number of beds in service, independent of whether they were occupied, and no float nurse pools were employed. Thus, for example, for a care unit size of 24 beds and staffing ratio of 1:4, the number of dedicated nurses to staff was always 6. A scarcity of nursing capacity frequently leads to the expensive hiring of temporary nurses from external agencies, as well as to undesirable ad hoc bed closings. Also, the prescribed staffing levels cannot always be realized in practice. As a result, the inpatient care units experienced a lack of consistency in the delivered quality of nursing care. We have estimated the input parameters for our model based on historical data of 2009–2010 from the hospital’s electronic databases. The event logs of the operating room and inpatient care databases had to be matched. Since the data contained many errors, extensive cleaning was required. Patients of other specialties who stayed at departments A to D have been deleted. No cyclical MSS was applied in practice; therefore, in our model we set the MSS length at 2 years,

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following the surgery blocks as occurred in practice during 2009–2010. Elective surgery blocks are only executed on weekdays. For the elective patient types, the distributions for the number of surgeries and for the admission/discharge processes are estimated per specialty. We set the length of the AAC at 1 week. For the acute patients, the discharge distributions are estimated per specialty, and to have enough measurements, via the following clustering: admission time intervals 0–8, 8–18, and 18–24. Furthermore, for all patient types, the discharge distributions during a day are assumed to be equal for the days n ≥ 2.

5.2 Case Study Results: Bed Census To illustrate the potential of the presented staffing methodology for the case study, we present a selection of the interventions presented in [33]. For the interventions that are based on the current MSS, we run the model for the estimated 2-year MSS, and we calculate the performance measures only over the second year, to account for warm-up effects. To assess the effects of the interventions, we first evaluate the performance of the base case scenario. In all experiments, no ad hoc closings are allowed, as decided by the hospital board to be the near-future policy. Note that the calculated rejection and misplacement percentages are therefore most likely an underestimation of current practice (of which no reliable data is available). The productivity measure is calculated per floor, since the misplacement policy implies that capacity is ‘shared’ per floor. The following interventions are considered, of which the results are displayed in Tables 2 and 3: (0) Base case. To assess the effects of the interventions, we first evaluated the performance of a base case scenario, which is the situation that most closely resembles current practice. The base case involved the current bed capacities and misplacements between care units A and B (Floor I) and between units C and D (Floor II). (1) Rationalize bed requirements. The numbers of beds in the base case are a result of historical development. Given particular service requirements, we determine whether the number of beds can be reduced to achieve a higher bed utilization while a certain quality level is guaranteed. We consider rejection probabilities not exceeding 5%, 2.5%, and 1%. Often, there are different bed configurations with the same total number of beds per floor, satisfying a given maximum rejection probability. Per floor, from the available configurations, the one is chosen that gives the lowest maximum misplacement probability. It can be seen that a significant reduction in the number of beds is possible. However, the overall bed utilizations are still modest, because demand drops during weekend days when no elective surgeries take place. In addition, there is a correlation between moments of higher census and moments that patients arrive, which leads to higher rejection probabilities compared to, for instance, a stationary Poisson arrival process.

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Table 2 The numerical results for the base case, intervention 1, and intervention 2 (with the productivity-Δ% relative to the base case) Capacity Unit (# beds) A 32 B 24 C 24 D 24 1. Rationalize bed requirements Rejection