Capacity Planning for Effective Cohorting of Dialysis Patients during the Coronavirus Pandemic: A Case Study
Cem Bozkir, Cagri Ozmemis, Ali Kaan Kurbanzade, Burcu Balcik, Evrim Gunes, Serhan Tuglular
CCapacity Planning for Effective Cohorting of Dialysis Patientsduring the Coronavirus Pandemic: A Case Study
Cem D.C. Bozkir a • Cagri Ozmemis a • Ali Kaan Kurbanzade a Burcu Balcik a, ∗ • Evrim D. Gunes b • Serhan Tuglular c a Industrial Engineering Department, Ozyegin University, Istanbul, Turkey b Business Administration, College of Administrative Sciences and Economics,Koc University, Sariyer, Istanbul, Turkey c Medical Faculty, Department of Internal Medicine, Marmara University, Istanbul, Turkey ∗ Corresponding author: [email protected]
Abstract
Chronic dialysis patients have been among the most vulnerable groups of the society during thecoronavirus (COVID-19) pandemic as they need regular treatments in a hospital environment, facinginfection risk. Moreover, the demand for dialysis resources has significantly increased since manyCOVID-19 patients need acute dialysis due to kidney failure. In this study, we address capacityplanning decisions of a hemodialysis clinic located within a major hospital in Istanbul, designated toserve both infected and uninfected patients during the pandemic with limited resources (i.e., dialysismachines). The hemodialysis clinic applies a three-unit cohorting strategy to treat four types ofpatients in separate units and at different times to mitigate infection spread risk among patients.Accordingly, at the beginning of each week, the clinic needs to determine the number of availabledialysis machines to allocate to each unit that serves different patient cohorts. Given the uncertain-ties in the number of different types of patients that will need dialysis, it is a challenge to allocatethe scarce dialysis resources effectively by evaluating which capacity configuration would minimizethe overlapping treatment sessions of infected and uninfected patients over a week. We representthe uncertainties in the number of patients by a set of scenarios and present a two-stage stochasticprogramming approach to support capacity allocation decisions of the clinic. We present a case studybased on the real-world patient data obtained from the clinic to illustrate the effectiveness of theproposed modeling approach and compare the performance of different cohorting strategies.Keywords: Operations research in health services, COVID-19 pandemics, dialysis, patient co-horting, stochastic programming. a r X i v : . [ m a t h . O C ] F e b Introduction
The coronavirus disease (COVID-19) has caused a global pandemic, as declared by the World HealthOrganization in March 2020 (Sohrabi et al., 2020). It has created a significant burden on the capacity ofhealthcare systems all around the world due to a surge in demand for outpatient clinics, inpatient beds,and intensive care units. While many people could postpone their nonurgent hospital visits to protectfrom infection risk (Hafner, 2020), some patients with care needs due to chronic health conditions (suchas dialysis, chemotherapy, physical therapy) do not have the luxury to avoid or postpone visits to healthcare facilities since they are obliged to obtain medical care regularly, making them more vulnerable.It has also been a challenge for the hospitals to serve such chronic patients without exposing themto infection risk with ever-tighter resources. This paper addresses the capacity planning decisions of ahospital’s hemodialysis clinic that provides care to infected and uninfected patients during the COVID-19pandemic by implementing cohorting strategies to mitigate the risk of infection spread among patients.Dialysis is a treatment to remove waste products and excess fluid from the blood with the help ofa machine when the kidneys fail to work (The National Health Service, 2021). Most chronic hemodial-ysis patients must receive dialysis treatment three times a week. During the treatment, patients stayconnected to a dialysis machine for about four hours, which filters and purifies the blood. During theirfrequent and long hospital visits to get treatments, chronic hemodialysis patients can become highly sus-ceptible to COVID-19 transmission risk. Indeed, their infection risk is found as nearly two-fold greaterthan those receiving dialysis at home (Hsu and Weiner, 2020). Moreover, Chronic Kidney Disease (CKD)is the most prevalent risk factor for severe COVID-19 cases (ERACODA Working Group, 2021), andmortality rate of patients with CKD under COVID-19 virus infection is significantly higher compared tothose without CKD (Roper et al., 2020; Ozturk et al., 2020). Therefore, it is imperative for health careproviders to take precautions to ensure that the uninfected chronic dialysis patients are not in contactwith the COVID-19 infected patients during their clinic visits (Basile et al., 2020). Given that the globalprevalence rate of chronic kidney disease (all stages) is 9.1% (Bikbov et al., 2020), effective planning ofdialysis treatments during the pandemics by accounting for infection spread risk can protect the livesof millions of people in vulnerable conditions. However, even without a pandemic disturbance, planningand scheduling of treatments in a single dialysis unit is already a difficult problem (e.g., Liu et al. (2019);Holland (1994); Fleming et al. (2019)). When resources become tighter, such as the aftermath of anearthquake (Sever et al., 2009) or during a pandemic (Corbett et al., 2020), the increased demands anduncertainties make it ever more challenging to plan for dialysis services.In this study, motivated by a real case, we focus on dialysis services provided in a hospital during thepandemics where multiple patient types must be served. Specifically, besides the chronic hemodialysispatients, who need regular care, some patients need dialysis due to acute kidney failure for variousmedical conditions. These acute patients, who often have multiple health complications, are alreadyadmitted to the hospital, and transferring them to another hospital is usually impossible. During thepandemic, the hospital is additionally responsible for treating the suspected and infected COVID-19patients with dialysis needs that cannot be treated in unequipped small dialysis centers. We address thecapacity planning decisions of the hospital to treat different patient types by mitigating the transmissionrisk between infected and uninfected patients.Managing dialysis resources effectively during the pandemic is challenging for the hospital due toseveral reasons. First, the effective dialysis capacity can be reduced due to the additional time required2or cleaning the rooms in-between dialysis sessions as well as arrangements required to cohort patients.Specifically, the uninfected (clean), infected (confirmed), and suspected (shows symptoms or had contactwith a confirmed case) patients must be treated at different designated units; and if possible, duringdifferent (nonoverlapping) periods. Secondly, the demand for dialysis during a pandemic increases be-cause acute dialysis sessions may be needed for some infected patients due to the adverse effects ofCOVID-19 on kidneys (Sperati, 2020); indeed, 35.2% of COVID-19 patients needed dialysis (Klein et al.,2020; Smith, 2020). Thirdly, while the number of chronic patients on a treatment schedule is known,the number of infected and clean acute patients that need dialysis treatments each day can be highlyuncertain. In this resource-constrained environment, employing an operations research based approachcan be valuable to make effective cohorting, capacity planning and treatment scheduling decisions byusing limited resources efficiently.Treating the infected, suspected, and uninfected patients at different times and in separate areasare common approaches followed in healthcare facilities to mitigate the risk of infection transmissionamong patients (Kliger and Silberzweig, 2020). Different cohorting strategies, which involve definingpatient types and assigning them to separate areas, have been implemented by health care providersaround the world during the COVID-19 pandemic (e.g., Collison et al. (2020); Whiteside et al. (2020)),and there is no well-established standard yet. Park et al. (2020) and Meijers et al. (2020) summarizethe classification of different hemodialysis patients that are either infected by COVID-19 or under thesuspicion of infection and outline the operationalization of the dialysis clinics in Korea, Belgium, and Italy.Despite its importance, no study has yet proposed analytical tools to support the effective implementationof cohorting strategies in a dialysis clinic during the pandemic.In this paper, we focus on the operations of a dialysis clinic in a large public teaching and researchhospital located in Istanbul during the COVID-19 pandemic. This hospital is one of the designated“pandemic hospitals” in this large densely-populated city. The dialysis clinic is responsible for bothserving the uninfected chronic dialysis patients and acute inpatients, as well as the infected and sus-pected dialysis patients, often through referrals from other clinics/hospitals that cannot treat COVID-19patients. The clinic has been implementing a three-tier (unit) cohorting strategy since the beginning ofthe pandemic to mitigate infection spread risk among patients. To facilitate this, the clinic is dividedinto units separated by drywalls, where each unit has several machines. Accordingly, uninfected, sus-pected and infected patients are treated at different units. Depending on the estimated demands for theupcoming week, the hospital assigns dialysis machines to the units in the dialysis clinic thereby settingtheir capacity to serve different patient cohorts during the week. Moreover, it is desirable to schedulethe dialysis treatments of different patients at different times during the day to minimize physical inter-action and avoid infection spread. An alternative cohorting strategy could separate the clinic into twounits (Whiteside et al., 2020), one for the uninfected patients, and the other for suspected and infectedpatients, where these patients would be treated sequentially in the unit.Our collaboration with the dialysis clinic has two main objectives: (1) To develop mathematicalmodels for making capacity planning decisions to implement a given cohorting strategy effectively, (2)To evaluate the performance of alternative cohorting strategies with two and three tiers (units). Toachieve these objectives, we develop mathematical models to solve the capacity planning associatedwith each cohorting strategy. Specifically, for a given (i.e., two- or three-unit) cohorting strategy, werepresent the uncertainty in the number of dialysis patients in each cohort that will need dialysis forthe upcoming week by scenarios and develop two-stage stochastic programming models to decide on3he number of dialysis machines that must be allocated to each unit at the beginning of a week (firststage), based on the second stage decisions which determine a daily dialysis treatment schedule for eachunit by considering the number of patients in a scenario. The objective is to treat all patients in theclinic while minimizing the expected number of patients from different cohorts having dialysis at theoverlapping sessions. We test the proposed models on real-world data obtained from our collaboratinghospital regarding demands for different patient types and cohorting plans over a two-month periodduring the pandemic. We present results that illustrate the benefits of using the proposed models andcompare the performance of different cohorting strategies.The remainder of this paper is structured as follows. In §
2, we review the related literature. In § §
4, we present a case study anddiscuss the results. Finally, we present our conclusions in § We review the literature related to health care capacity planning problems in resource-constrained set-tings, and planning of hemodialysis services during pandemics.
There exists abundant literature that utilizes operations research methods to address capacity planningand resource allocation problems in health care settings. Existing studies address problems in differentsettings ranging from planning the operations of a single hospital or a unit (e.g., intensive care units,operating rooms) to a network of health providers (see reviews by Guerriero and Guido (2011); Rais andViana (2011); Hulshof et al. (2012); Ahmadi-Javid et al. (2017); Bai et al. (2018)).Catastrophic events (such as pandemics, natural and man-made disasters) can create unexpecteddemand surge and extreme resource scarcity in healthcare systems. Several studies address planningproblems faced in such settings and propose various strategies to manage the demand surge by usingavailable resources efficiently. For example, Repoussis et al. (2016) focus on regional emergency planningand present a mixed integer programming (MIP) model for the combined ambulance dispatching, patient-to-hospital assignment, and treatment ordering problem. The proposed model considers the triage levelsof casualties at each site and minimizes the maximum completion time of treatments. Caunhye andNie (2018) present a scenario-based stochastic programming model to determine the location of thealternative care facilities to be established following a disaster, and the number of patients to be directedto these centers by integrating triage and self movement of casualties. Ouyang et al. (2020) focus onbed allocation decisions in an intensive care unit (ICU) of a hospital during times of high demand andpresent a stochastic model to determine whether a new patient will be admitted to the ICU and whichpatients will be discharged from the ICU to minimize the long-run average mortality rate. Mills et al.(2020) evaluate mitigation and response strategies to create surge capacity for emergency patients in ahospital by taking disposition actions for different types of patients with varying severity. The authorspresent an optimization model and perform a simulation study to evaluate the performance of strategiesbased on the characteristics of individual hospitals.Several studies specifically focus on managing health care resources at the time of pandemics. Aroraet al. (2010) focus on satisfying the large amounts of need for antiviral medicines during an influenza pan-4emic. The authors formulate a two-stage model to first decide on the locations and amounts of suppliesto be prepositioned in the stockpile to prevent the disease spread, and then make redistribution decisionswith respect to transshipment and stockpiling costs in the treatment phase while considering the expectednumber of people affected. Sun et al. (2014) address allocation of inpatients among multiple hospitalsover several months during an influenza outbreak by considering different patient types with respect totheir needs for various medical equipment. The authors present a multi-objective model that minimizesthe total distance traveled by patients and the maximum distance of a patient to a hospital. Liu et al.(2015) develop an integer programming (IP) model to allocate medical resources to be transported fromsuppliers to hospitals through a set of distribution centers to control the spread of influenza pandemic.The authors base their demand estimations on a susceptible-exposed-infected-recovered (SEIR) modeland aim to minimize the total transportation cost while satisfying the dynamically changing demandfor the resources. Long et al. (2018) present a two-stage model that handles the allocation of treatmentunits used in Ebola response across different geographical regions according to the course of the epidemic,where the number of infected people is predicted by an SEIR model. The authors present alternativeapproaches to allocate health resources including a heuristic, a greedy policy, a linear program, and adynamic programming algorithm.Since the declaration of the COVID-19 pandemics in March 2020, there has been growing researchattention on the challenges faced in allocating and managing valuable health care resources during thepandemics. Klein et al. (2020) present a review of studies that present models and tools to estimate de-mand surge for hospital capacity and resources (such as hospital beds, ventilators) during the pandemics.Mehrotra et al. (2020) present a stochastic MIP model to allocate and share a critical resource duringthe COVID-19 pandemic, the ventilators, among risk-averse states by considering demand uncertainties.The authors apply their model to a case study by considering FEMA as a coordinator that controls theallocation of ventilators among the states in the USA. Parker et al. (2020) study a demand and resourceredistribution problem among multiple health care facilities. The authors develop deterministic androbust optimization models to determine the optimal demand and resource transfers among facilities tominimize the required surge capacity and resource shortage during a period of heightened demand.Although cohorting patients is a widely implemented method to prevent infection spread risk duringpandemics, a limited number of studies present analytical models to support the implementation ofcohorting strategies in health care settings. Pinker and Tezcan (2013) address capacity configurationdecisions in a hospital with the existence of patients needing isolation due to infectious diseases. Theauthors present a stochastic optimization model with a revenue maximization objective to decide onthe beds to be reserved for isolation and non-isolation patients, where the two groups have differentrequirements for spacing and admission criteria. Chia and Lin (2015) analyze the effectiveness of differentcohorting strategies in a pediatric hospital for allocating beds among different patient types. Wang et al.(2020) focus on allocating beds among different departments in a hospital in a setting with stochasticpatient arrival and service times and propose an approach that combines a MIP and a simulation modelto minimize the expected cost of rejection and waiting. Melman et al. (2020) develop a discrete eventsimulation model to evaluate alternative strategies for allocating limited resources effectively in a hospitalto prioritize the surgeries of COVID-19 and uninfected patients. The authors apply their model in ahospital and show that even though elective surgeries were canceled to use major hospital capacity forCOVID-19 treatment, it is better to prioritize such surgeries until a threshold to minimize total deaths.Different from the existing studies, we focus on weekly capacity planning decisions in a dialysis clinic5o treat multiple patient types in a pandemic setting. To the best of our knowledge, no study has yetfocused on modeling and evaluating alternative cohorting strategies where both rooms and treatmenttimes of different patient cohorts must be separated to mitigate infection spread, and the capacityconfiguration decisions are made to minimize expected overlapping sessions.
The literature that addresses hemodialysis planning during pandemics is limited to practical medicalguidelines to manage this process effectively (Naicker et al., 2020). For instance, Ikizler and Kliger(2020), Kliger and Silberzweig (2020) and Rubin (2020) highlight the importance of cohorting, screeningand disinfection in order to prevent infection in dialysis clinics. CMS (2020) suggests that uninfecteddialysis patients should wait for their sessions outside the hospital and there should be at least two metersamong patients in dialysis units. Roper et al. (2020) point out high infection risks in the waiting roomsof the clinics. Alberici et al. (2020) describe the layout changes made in a hospital to provide treatmentsto uninfected and infected patients with different kidney diseases. Kliger and Silberzweig (2020), Roperet al. (2020) and Corbett et al. (2020) recommend treating suspected or confirmed COVID-19 casesin isolated rooms and uninfected dialysis patients in regular rooms. Rubin (2020) cohort the dialysispatients into asymptomatic, suspected and confirmed COVID-19 cases.Although there exist a few operations research studies that address problems related to planninghemodialysis treatments, no study has yet addressed the particular challenges faced during a pandemicsetting. Liu et al. (2019) study a dialysis scheduling problem with different types of patients (conven-tional, hepatitis B, and hepatitis C) that must be treated in a dialysis center. The authors present an IPmodel to minimize night shifts and meet patient preferences. Fleming et al. (2019) address the problemof scheduling dialysis patients by considering each dialysis machine as a workstation. The proposed IPmodel minimizes the waiting time of patients arriving at the clinic for treatment and the scheduled finishtime of treatments each day. Yu et al. (2020) study appointment scheduling policies for patients thatneed a series of appointments such as chemotherapy or chronic dialysis patients. The authors present aMarkov decision process model that takes into account revenues per service per patient, and the costs ofstaffing, overtime, overbooking, and delay.Currie et al. (2020) express the need for scientific studies to prevent the infection rate of COVID-19 inhemodialysis units and highlight that this area is lacking in the literature. Corbett et al. (2020) discussthe challenges in planning the treatments of dialysis patients under a high level of uncertainty and stressthe need for an analytical method. Ikizler and Kliger (2020) highlight that the prolonged pandemicoverextends the healthcare capacities and creates shortages in equipment, which necessitates allocatingresources most efficiently by cohorting symptomatic and clean patients. Although several studies drawattention to the need for effective management of dialysis resources during a pandemic, no work in theliterature provides analytical methods for supporting the decision makers to make cohorting and thecapacity planning decisions in a dialysis unit to serve different types of patients. We contribute to theliterature by introducing a new problem and presenting mathematical models to support decision makingfor configuring the clinic’s capacity to effectively plan dialysis sessions during the pandemic. We focuson four types of patients and present models for alternative cohorting strategies that divide the clinicinto two and three units. The proposed models can be extended to other settings that consider differentcohorting strategies with different number of cohorts and patient types.6
Problem Description & Mathematical Modeling
In this section, we first describe the system based on the operations of our collaborating hemodialysisclinic located in a major hospital in Turkey ( § § § We focus on the operations of a hemodialysis clinic that treats different types of patients during theCOVID-19 pandemic. The clinic is located within the Marmara University’s Pendik Training and Re-search Hospital, which is a major public hospital located in Istanbul. In Turkey, all major public hospitalshave been declared as “pandemic hospitals” after COVID-19 was officially announced to be a pandemicin March 2020. A pandemic hospital is a designated hospital for treating COVID-19 infected or suspectedpatients. Nevertheless, the pandemic hospitals have continued providing their regular care services aswell, which. necessitated using resources efficiently more than ever. While all COVID-19 patients couldbe treated only at the pandemic hospitals during the initial months of the pandemic, other hospitalsstarted accepting COVID-19 patients in the later stages of the pandemic.In Turkey, chronic hemodialysis patients can get their treatment either in a public hospital or aprivate dialysis clinic. The dialysis treatments are covered by state-wide health insurance. However,some patients that are in special status (such as refugees) can receive treatments only at the dialysisclinics located within the public hospitals. All chronic dialysis patients receive treatments at theirregistered clinics three days a week, which can be either on the Monday-Wednesday-Friday (MWF) orthe Tuesday-Thursday-Friday (TTF) regime. Therefore, each clinic makes a schedule to treat its chronicpatients by following these regimes. All dialysis clinics in the country handle the transportation ofthe patients to and from the clinic to avoid delays in the schedule due to possible late arrivals of thepatients. Different from the small private dialysis centers, major public hospitals also have inpatients(i.e., admitted to various wards, intensive care units) who may need temporary dialysis treatment duringtheir stay at the hospital (e.g., after an operation). The number of such acute patients that need dialysistreatment varies and the hospital arranges the treatments of the acute patients along with the chronicpatients each day. However, the COVID-19 pandemic required making significant changes in this routinein our case hospital, which has additionally become responsible for serving the COVID-19 patients withthe same level of resources (dialysis machines, rooms).The private dialysis centers are only specialized in dialysis service and they are not fully equippedfor managing possible complications of dialysis patients with COVID-19. Therefore, chronic dialysispatients, who were normally treated in such dialysis centers, are referred to pandemic hospitals if theybecome infected with COVID-19. These patients are treated at the hospital until they are fully recoveredfrom COVID-19. Additionally, some COVID-19 patients who do not have a priori chronic kidney diseasemay need dialysis due to COVID-19 related complications. These patients are also treated at pandemichospitals. Our case hospital’s dialysis clinic has to admit all confirmed and suspected COVID-19 patientsfor dialysis treatment. The number of chronic or acute COVID-19 patients that will arrive at the dialysisclinic each day is not known beforehand. Hence, it is important for the hospital to make effective capacityplanning to accommodate the treatments of the COVID-19 patients along with its regular patients underuncertainty. In this study, we focus on the cohorting strategies applied in the hospital to treat infected,suspected and uninfected patients to minimize physical interaction among different groups.7he dialysis clinic in our case hospital has 14 dialysis machines. Starting from 8:00 am, four sessionscan be performed on each machine each day (Monday-Saturday). On Sundays, only emergency patientsare treated in the clinic. Since the beginning of the pandemic, the hospital cohorts the hemodialysis pa-tients into three cohorts (i.e., uninfected, infected, suspected), where each cohort is treated in a separateunit (standard, isolated and quarantine, respectively). Moreover, the units allocated to suspected andinfected patients are divided into single-patient rooms, where each room includes one dialysis machine.All machines and rooms are cleaned intensively between dialysis sessions. To divide a unit into rooms,the hospital can build drywalls easily (overnight). A nurse can handle the treatment of four patientssimultaneously in the same unit. Although the room arrangements within a unit can be made overnight,since all weekly plans (such as nurse shifts) must be fixed beforehand, the configuration of the units isarranged only at the end of the week using drywalls (i.e., Sunday). For instance, only two machineswere allocated to infected patients during the first couple of months of the pandemics, while there existfive rooms in the infected patient unit (isolated unit) during the period we gathered the data from thehospital (November-December 2020). A schematic of the dialysis clinic during this period is given inFigure 1. As shown in the figure, seven, five and two machines are currently allocated to standard,isolated and quarantine units, respectively.Figure 1: Dialysis Clinic Floor PlanIn addition to treating infected, suspected and uninfected patients in different units, the clinic sched-ules the treatments of different patient groups at different (nonoverlapping) sessions, as long as theircapacities permit. Since all units are along the same corridor, patients and their companions use thesame space before and after the dialysis sessions, and hence overlapping sessions may increase the risk ofinfection spread in the clinic. Therefore, the clinic arranges dialysis treatment schedules such that unin-fected (chronic and acute) patients are assigned to morning sessions, which are followed by the sessionsof suspected, and then the infected patients. Therefore, the number of rooms allocated to each unit atthe beginning of a week directly affects the number of overlapping sessions that will be incurred each day.8urrently, the hospital does not have any analytical tools that can support decisions for determining thesize of units by considering daily treatment schedules. Given the uncertainties in the number of patientsthat will need dialysis treatments in a day, it is challenging to make weekly capacity planning decisionsby considering their possible implications on daily treatment schedules and overlapping sessions. Ourfirst objective in this study is to develop a mathematical model that can assist the hospital to determinecohort capacities effectively under demand uncertainty.While our case hospital has separated the clinic into three units to treat three groups of patients,another cohorting strategy could be to divide the clinic into two units, where the noninfected patientsare treated in the first unit, and the suspected and infected patients can be sequentially treated in thesecond unit, as practiced in some applications (Whiteside et al., 2020). Depending on the number ofpatients over a week, which is not known beforehand (except the regular chronic patients), we investigatewhether this alternative cohorting strategy might work better for our clinic to avoid overlapping sessions.Hence, our second objective in this paper is to evaluate the performance of different cohorting strategiesand make recommendations to the hospital.We next define the problem and then present the mathematical models developed to achieve thestudy’s objectives.
We consider the capacity planning problem of a dialysis clinic in a hospital setting, which allocates thevaluable dialysis resources to serve different patient cohorts during a pandemic. Specifically, to mitigateinfection spread among infected and uninfected patients, the hospital cohorts the patients and treat themin different units, and if capacity allows, in nonoverlapping sessions. There exist four types (groups) ofpatients with different characteristics, summarized in Table 1.
Patient Types DescriptionUninfected Acute (Type 1) Uninfected patients that are admitted to the hospitaland need dialysis.
Uninfected Chronic (Type 2) Uninfected patients that receive dialysis treatmentregularly (every MWF or TTF).
Infected COVID-19 (Type 3) Infected patients with coronavirus that needdialysis.
Suspected COVID-19 (Type 4) Suspected patients with the possibility of havingcoronavirus that need dialysis.Table 1: Patient types treated in the dialysis clinic during the pandemicThe daily demand for chronic dialysis patients (Type 2) can be considered fixed and known as thesepatients have to receive regular treatments. The number of uninfected acute patients (Type 1) that needdialysis each day is uncertain and can depend on a variety of factors such as the number of admittedpatients in the hospital each day and their medical conditions which may cause temporary kidney failure.The daily demand by suspected and infected COVID-19 patients (Type 3 and 4) are also uncertain, which9epend on uncontrollable factors including the current infection rate in the population, referrals fromother clinics/hospitals, and the preferences of patients. Due to the high spread rate of COVID-19, apatient that was clean the day before can be suspected or infected the next day. Therefore, the dailypatient status is important to consider in assigning patients to different units and setting their treatmenttimes. In our model, we represent the uncertainties in the number of Type 1, 3 and 4 patients by a setof discrete scenarios. More specifically, each scenario specifies the estimated number of patients that willneed dialysis each day over a week. In our case study ( § i and Type j patients fromdifferent cohorts are treated in the same session, we call this an “overlap i x j ”. In particular, the clinicwants to avoid overlaps of uninfected patients with infected or suspected patients (overlaps 1x3, 1x4,2x3, 2x4). Avoiding each of these overlaps is of primary importance for the clinic. Moreover, in thethree-unit cohorting strategy where the infected and suspected patients are treated in different rooms,their overlap (overlap 3x4) should be avoided if possible; however, this overlap is considered to be lessserious compared to overlaps that involve clean patients. Note that in the two-unit cohorting strategy,Type 3 and 4 patients are sequentially treated in the same unit, and suspected patients are always treatedearlier than infected patients. Since there is enough time in between sessions to disinfect the rooms, weassume there is no risk for a Type 3 patient to infect a Type 4 patient if they are consecutively scheduledin the isolated unit, hence this is not considered an overlap.Figure 2 illustrates example daily treatment schedules for both cohorting strategies on a small in-stance with seven machines, and fifteen patients. The number of patients of Type 1-4 are three, five,four and three, respectively. As shown in Figure 2(a), when three, two, and two dialysis machines areallocated to standard, isolated and quarantine units, respectively, some patients from each type haveto be served in overlapping sessions. In Figure 2(b), three and four machines are allocated to standardand isolated units, respectively, and the sessions of some uninfected (Type 1 and 2) patients and sus-pected (Type 4) patients overlap. Therefore, both the cohorting policy and the capacity configurationdecisions can have a significant effect on the number of patients who are exposed to infection risk each day.10 nits Three-Unit Cohorting Two-Unit CohortingStandard Uninfected Acute (Type 1) Uninfected Acute (Type 1)Uninfected Chronic (Type 2) Uninfected Chronic (Type 2)
Isolated
Infected COVID-19 (Type 3) Infected COVID-19 (Type 3)Suspected COVID-19 (Type 4)
Quarantine
Suspected COVID-19 (Type 4)Table 2: Three-unit and two-unit cohorting strategy patient assignmentsFigure 2: Example daily treatment schedules under three-unit (a) and two-unit (b) cohorting.
IsolatedStandardQuarantine Session
End of the day
Type 4 PatientType 2 Patient Type 3 PatientType 1 Patient (a)
IsolatedStandard Session
End of the day (b)
Unit Unit
We consider a planning horizon of one week for the capacity planning decisions, which involve deter-mining the number of dialysis machines in each unit. Capacity planning must be done at the beginningof each week, facing uncertain demand from each patient type over the following week. We follow atwo-stage approach to formulate the clinic’s problem, where capacity planning is done in the first stage,accounting for the scheduling decisions under different demand scenarios, given in the second stage. Ac-cordingly, the capacity planning decisions are made at the beginning of each week, before uncertainties indemands are resolved by considering the implications of the second stage treatment scheduling decisions.Treatment scheduling decisions involve determining the timing of sessions allocated to each patient groupeach day given the predetermined first-stage decision variables and the realized patient demands.The objective is to minimize the total expected penalties that will occur over a week due to overlappingsessions. Specifically, a penalty is incurred for each patient that is scheduled in an overlapping sessionwith other patients treated in different units. We impose an overlapping penalty charge per patient toaccount for the fact that there might be a higher infection risk if a larger number of conflicting patientsare in the clinic at the same time. Additionally, if there are any patients that cannot be served dueto capacity insufficiency, this causes infeasibility, which is modeled by imposing a large penalty cost.11ifferent unit penalty charges might be set for not being able to treat different types of patients if itis more difficult to transfer certain types of patients to other facilities. Finally, to avoid alternativesolutions that spread patients unnecessarily over multiple sessions, which may arise when capacity isabundant, we impose a small penalty for each session performed over a day.The additional assumptions of our problem are as follows: • Each dialysis unit can have a capacity restriction imposed by physical constraints. • Only Type 1 and 2 patients can be served in the same unit during a session. Otherwise, a sessionmust be composed of patients of the same type. • The maximum number of sessions in each unit per day is equal. Moreover, each session has a fixedduration for all patient types. • The time required for disinfecting and preparing the machines between sessions is fixed and includedin a session’s duration. • One machine can serve one patient in each session. • The sessions run in parallel; that is, the starting and ending times of all sessions are the same. • In each unit, the dialysis sessions are scheduled consecutively to allow easier planning of otherresources (e.g., nurses). That is, there is a no-idling policy. • The sessions of uninfected patients are scheduled at earlier sessions starting from the beginningof the day, and if capacity allows, the sessions of infected and suspected patients start after allinfected patients are treated. • In the two-unit cohorting, the sessions of suspected patients precede those of infected patients inthe isolated unit.We next present stochastic programming models to solve the capacity planning problem of the hospitaldescribed here.
We present two integer programming models for the alternative cohorting strategies that separate thedialysis clinic into units. We first present the notation and then present the models for three- and two-unit models in § § Sets K : set of scenarios; k ∈ KI : set of patient types; i ∈ I ; i =1,..,4 J : set of dialysis units; j ∈ J ; j =1,..,3 (1: Standard unit, 2: Isolated unit, 3: Quarantine unit) T : set of days in the planning horizon; t ∈ T ; t =1,..,6 S : set of dialysis sessions on each day; s ∈ S ; s =1,..,4 Parameters p k : probability of scenario k H kit : number of type i patients who need to receive dialysis on day t of scenario kC j : maximum number of dialysis machines that can be allocated to unit j ˆ C : total number of available dialysis machines α : penalty (per patient) for uninfected patients and infected COVID-19 patients that are treated inoverlapping sessions α : penalty (per patient) for uninfected (Type 1, 2) patients and suspected COVID-19 (Type 4) patients12hat are treated in overlapping sessions α : penalty (per patient) for the conflict of suspected (Type 4) and infected (Type 3) patients that aretreated in overlapping sessionsΠ i : penalty (per patient) for infeasibility if a Type i patient cannot be treated due to capacity insuffi-ciency. (cid:15) : small penalty coefficient for starting a dialysis session (to avoid unnecessary sessions) First stage decision variables R j : number of dialysis machines allocated to dialysis unit j Second stage decision variables X kits : number of Type i patients scheduled to receive dialysis in session s on day t of scenario kF kit : number of Type i patients that cannot be served on day t of scenario kN kjts : , if dialysis unit type j is used for session s on day t of scenario k ,0 , otherwise .Q kts : number of uninfected (Type 1, 2) and infected (Type 3) patients treated in session s on day t ofscenario kG kts : number of uninfected (Type 1, 2) and suspected (Type 4) patients treated in session s on day t ofscenario kW kts : number of infected (Type 3) and suspected (Type 4) patients treated in session s on day t ofscenario kU kts : , if an overlap exists in session s on day t of scenario k for uninfected (Type 1, 2) and infected(Type 3) patients,0 , otherwise .D kts : , if an overlap exists in session s on day t of scenario k for uninfected (Type 1, 2) and suspected(Type 4) patients0 , otherwise .V kts : , if an overlap exists in session s on day t of scenario k for infected (Type 3) and suspected(Type 4) patients,0 , otherwise .Z u : auxiliary variable for the objective function The formulation for making capacity allocation decisions to implement the three-unit cohorting strategyis as follows.min Z u = (cid:88) k ∈ K p k (cid:20) α (cid:88) t ∈ T,s ∈ S Q kts + α (cid:88) t ∈ T,s ∈ S G kts + α (cid:88) t ∈ T,s ∈ S W kts + Π i (cid:88) i ∈ I,t ∈ T F kit + (cid:15) (cid:88) j ∈ J,t ∈ T,s ∈ S N kjts (cid:21) (1)subject to R j ≤ C j ∀ j ∈ J (2) (cid:88) j R j ≤ ˆ C (3) X k ts + X k ts ≤ R ∀ t ∈ T, s ∈ S, k ∈ K (4)13 k ts ≤ R ∀ t ∈ T, s ∈ S, k ∈ K (5) X k ts ≤ R ∀ t ∈ T, s ∈ S, k ∈ K (6) (cid:88) s X kits = H kit − F kit ∀ i ∈ I, t ∈ T, k ∈ K (7) X k ts + X k ts ≤ M × N k ts ∀ t ∈ T, s ∈ S, k ∈ K (8) N k ts ≥ N k t ( s +1) ∀ s ∈ S : s ≤ , t ∈ T, k ∈ K (9) X k ts ≤ M × N k ts ∀ t ∈ T, s ∈ S, k ∈ K (10) X k ts ≤ M × N k ts ∀ t ∈ T, s ∈ S, k ∈ K (11) S (cid:88) s (cid:48) = s +2 N kjts (cid:48) ≤ | S | (1 − N kjts + N kjt ( s +1) ) ∀ s ∈ S : s ≤ , t ∈ T, j ∈ J : j (cid:54) = 1 , k ∈ K (12)1 + U kts ≥ N k ts + N k ts ∀ t ∈ T, s ∈ S, k ∈ K (13) Q kts ≥ X k ts + X k ts + X k ts − M (1 − U kts ) ∀ t ∈ T, s ∈ S, k ∈ K (14)1 + V kts ≥ N k ts + N k ts ∀ t ∈ T, s ∈ S, k ∈ K (15) W kts ≥ X k ts + X k ts − M (1 − V kts ) ∀ t ∈ T, s ∈ S, k ∈ K (16)1 + D kts ≥ N k ts + N k ts ∀ t ∈ T, s ∈ S, k ∈ K (17) G kts ≥ X k ts + X k ts + X k ts − M (1 − D kts ) ∀ t ∈ T, s ∈ S, k ∈ K (18) R j ∈ Z + ∀ j ∈ J (19) X kits ∈ Z + ∀ i ∈ I, t ∈ T, s ∈ S, k ∈ K (20) F kit ∈ Z + ∀ i ∈ I, t ∈ T, k ∈ K (21) Q kts , W kts , G kts ∈ Z + ∀ t ∈ T, s ∈ S, k ∈ K (22) N kjts ∈ { , } ∀ j ∈ J, t ∈ T, s ∈ S, k ∈ K (23) U kts , V kts , D kts ∈ { , } ∀ t ∈ T, s ∈ S, k ∈ K (24)The first three terms of the objective function (1) minimizes the weighted sum of the penalties dueto overlapping sessions of patient groups that are not preferred to be in the clinic at the same times tomitigate the infections transmission risk. The fourth and fifth terms in (1) are for penalizing patientsthat cannot be treated in the clinic due to capacity insufficiency. The last term in the objective functionminimizes the number of dialysis sessions provided during a week to ensure that the sessions are scheduledin a compact way. Constraints (2) ensure that the number of machines allocated to each unit does notexceed its capacity. Constraint (3) ensures that the total number of dialysis machines assigned to unitsdoes not exceed the number of machines available in the clinic. Constraints (4)-(6) guarantee that forany unit, the number of patients assigned to a session cannot exceed the capacity reserved for that unit.Constraints (7) balance the daily number of dialysis sessions required and provided. Constraints (8)and (9) ensure that dialysis sessions of uninfected (Type 1 and 2) patients start from the beginningof the day, and multiple sessions are conducted consecutively in the standard unit. Constraints (10),(11) are for determining used sessions of suspected (Type 4) and infected (Type 3) patients anytimeduring a day; additionally, constraints (12) ensure that the sessions in their corresponding units must beconducted consecutively for the compactness of the schedule. In other words, by keeping track of the twoconsecutive sessions, (12) prevents leaving an idle session between any sessions in which suspected (Type4) and infected (Type 3) patients are treated. Constraints (13) determine the overlapping sessions among14tandard and isolated units. Accordingly, the number of uninfected (Type 1 and 2) and infected (Type3) patients that receive dialysis in the same session is determined by constraints (14). Similarly, thenumber of infected (Type 3) and suspected (Type 4) patients that receive dialysis in the same session isdetermined by constraints (15) and (16); further, the number of uninfected (Type 1 and 2) and suspected(Type 4) patients that receive dialysis in the same session is determined by constraints (17) and (18).Finally, the integer variables are defined by constraints (19)-(22), and binary variables are defined byconstraints (23) and (24).In our computations, we set the value of M in constraints (8), (10), and (11) equal to the number ofdialysis machines available, ˆ C . The M values in constraints (14), (16), and (18) are set as the number ofdialysis treatments that can be provided by the clinic, which equal to the multiplication of the numberof machines available and the number of sessions on a day ( | S | × ˆ C ). The model for the two-unit cohorting strategy has a few differences compared to the model presentedabove for the three-unit strategy. First, there exist two units, that is, j = 1 , ∈ J , where j = 1 denotesthe standard unit where the uninfected patients are treated, and j = 2 denotes the isolated unit wheresuspected and infected patients are treated sequentially. To sequence the treatments of infected patientsafter the suspected patients in the isolated unit, we introduce a new variable Y kts , which takes the valueof 1 when all suspected patients that can be treated in a day are assigned to a dialysis session, therebyspecifying the earliest session that the infected patients can be scheduled in the unit. Moreover, a dummysession is needed to schedule consecutive sessions of suspected and infected patients in the same cohortby using variables Y kts . We denote the dummy session by { } and let S = S ∪ { } . The W kts and V kts variables that are used before in the three-unit cohorting model are not relevant in the two-unit model,since the suspected and infected patients receive in the same unit in separate sessions. Finally, we define Z u as an auxiliary variable to keep the objective function value attained for the two-unit model.The model for the capacity planning problem of the clinic to implement a 2-unit cohorting strategyis presented below.min Z u = (cid:88) k ∈ K p k (cid:20) α (cid:88) t ∈ T,s ∈ S Q kts + α (cid:88) t ∈ T,s ∈ S G kts + Π i (cid:88) i ∈ I,t ∈ T F kit + (cid:15) (cid:88) j ∈ J,t ∈ T,s ∈ S N kjts (cid:21) (25)subject to(2) , (3) , (4) , (7) , (8) , (9) , (13) , (14)(17) , (18) , (19) , (20) , (21) , (23) X k ts + X k ts ≤ R ∀ t ∈ T, s ∈ S, k ∈ K (26) X k ts + X k ts ≤ M × N k ts ∀ t ∈ T, s ∈ S, k ∈ K (27) (cid:88) s (cid:48) = s +2 N k ts (cid:48) ≤ | S | (1 − N k ts + N k t ( s +1) ) ∀ s ∈ S : s ≤ , t ∈ T, k ∈ K (28) H k t − s (cid:88) s (cid:48) =0 X k ts (cid:48) − F k t ≤ M × Y kts ∀ t ∈ T, s ∈ S , k ∈ K (29) X k ts ≤ M × (1 − Y kt ( s − ) ∀ t ∈ T, s ∈ S, k ∈ K (30) X kit = 0 ∀ i ∈ I, t ∈ T, k ∈ K (31) Q kts , G kts ∈ Z + ∀ t ∈ T, s ∈ S, k ∈ K (32)15 kts , D kts , Y kts ∈ { , } ∀ t ∈ T, s ∈ S, k ∈ K (33)The objective function (25) is similar to (1), except that there is no penalty now associated withoverlapping Type 3 and 4 patients, which are sequentially treated in the same unit in the two-unitcohorting strategy. Moreover, constraints (5) and (6) in the three-unit model are consolidated in a singleconstraint (26) in the two-unit model since Type 3 and 4 patients are assigned to the same unit. Similar toconstraints (10) and (11), constraints (28) ensure that dialysis sessions for Type 3 and 4 infected patientscan start anytime during a day, but their sessions must be conducted consecutively. Constraints (29)and (30) specify the earliest session for Type 3 patients after Type 4 patients are scheduled. Constraints(33) define the domains of variables. The remaining constraints are identical to those of the three-unitmodel.In implementing the two-unit model, the value of M in constraints (27) is set equal to ˆ C . In constraints(29) and (30), M values are set to the largest number of Type 4 and Type 3 demand occurrences among | K | scenarios, respectively. In this section, we present a case study to test and illustrate the proposed models based on data obtainedfrom the dialysis clinic in our collaborating hospital, whose operations are described before ( § § § The hemodialysis clinic of our collaborating hospital has been applying a three-unit cohorting strategysince the beginning of the pandemic. The clinic does not have an advanced information system thatkeeps data regarding patient demands. Upon our request, a nurse working in the hemodialysis clinicof the Marmara University Pendik Hospital has recorded patient data for eight weeks in November andDecember 2020, which corresponds to a period around the second peak of the pandemic. The acquireddata from the clinic include the number of patients of each type treated in the clinic each day.We consider the eight weeks of data collected between November 2 to December 26 by excludingSundays, and obtain a data set for 48 days in total. The number of patients is very low on Sundayscompared to other days because only extremely urgent patients of Type 1 or Type 3 are treated onthose days, and unlike rest of the days with five nurses working in the clinic, only one nurse is workingon Sundays. The average number of patients being treated is four, on Sundays, which is extremely lowcompared to the weekdays and Saturdays. Therefore, we treat Sunday as an outlier and remove it fromthe whole data set. Figure 3 presents a plot of the daily number of patients for each type and weeklyintervals are indicated by the grey dashed lines. Besides, the raw data and descriptive statistics arepresented in Appendix A. We observe that the daily number of uninfected acute patients (Type 1) ishighly uncertain, as shown by fluctuations in the plot. On the other hand, the number of uninfectedchronic patients (Type 2) shows a periodic pattern and constant throughout the entire time horizon. Theinfected COVID-19 (Type 3) and suspected COVID-19 (Type 4) patients are a smaller percentage of the16verall demand, while the number of infected COVID-19 (Type 3) patients also varies considerably overthe observed period.We create demand scenarios based on the available data to illustrate our models. Since the unitconfiguration can be changed once per week, we consider a weekly (six days) planning horizon. There-fore, given the realized demands in the past, demand scenarios should be generated for the followingweek. One plausible approach to estimate demand coming from infected (Type 3) or suspected (Type4) COVID-19 patients is to consider the dynamics of the pandemic in the country. SIR-based diseaseprogression dynamic models have been used to estimate the demand for hospital services (cf. Weissmanet al. (2020)). An approach like this could be to use the aggregate prevalence estimates from such dy-namic models and assume a fraction of those patients will need dialysis. However, while these modelswork well at the aggregate level, they may not reliably estimate the patients of a single dialysis clinic,given that there exist other clinics serving the population. Therefore, for predicting the demand for eachtype of patient we use different methods according to the characteristics of the available data.Figure 3: Daily number of dialysis patients by typeFigure 4 shows the number of infected (Type 3) and suspected (Type 4) COVID-19 patients treatedin our case hospital and the daily reported COVID-19 infected patients in Turkey between November 2and December 26 (excluding Sundays). We observe that the number of Type 3 and 4 patients treated inthe pandemic hospital does not closely follow the same pattern with the number of cases in the country.When the COVID-19 cases peaked in the middle of November, the numbers of hemodialysis patientswith COVID-19 increased and peaked as well. However, after a curfew was announced at the beginningof December, which is indicated with the purple dashed line in the figure, the number of nationwidecases started to diminish, while the number of hemodialysis patients with COVID-19 seems stationary.There is also a higher variability in the demand for dialysis, which is at a smaller scale compared tothe aggregate number of cases at the national level. Time series methods are expected to work wellfor short-term forecasting during the pandemic (Doornik et al., 2020). Therefore, we use time seriesforecasting methods to estimate the demand for the upcoming week.17igure 4: Total number of Type 3 and 4 patients treated in the hemodialysis clinic and the reportednational COVID-19 cases (November-December 2020)Based on the clinic data, we generate scenarios that represent weekly patient demand estimates. Toobtain weekly demand estimates, we divide the data set with 48 observations (days) into two, namely,a training set and a testing set for validation purposes. The last week (six days) of the data points areused as a testing set whereas the first 42 data points are used as the training data set and the forecastingmethods are applied to first seven weeks of the data. The accuracy of the estimators is verified bycomparing the estimators with the testing set that contains six data points. Furthermore, this processis repeated by producing estimations for week six and seven based on the data that corresponds to thefirst five and six weeks, respectively. We next explain the procedure followed to predict demands foreach patient type. • Uninfected Chronic Patients (Type 2) . According to the data, the number of uninfected chronicpatients is constant through the entire horizon, 12 patients are treated according to the Monday-Wednesday-Friday (MWF) regime and eight patients are enrolled to Tuesday-Thursday-Saturday(TTS) regime. We set the future daily demands for Type 2 patients by considering these actualnumber of chronic patients regularly treated by the clinic. • Uninfected Acute (Type 1), Infected COVID-19 (Type 3) and Suspected COVID-19 (Type 4) Pa-tients . To predict the demand coming from these patient types, we use simple exponential smooth-ing method to minimize the Root Mean Square Errors (RMSE) and find the point estimate for thenext week’s daily demand for patient type i , ˆ Y i . The 80% and 90% prediction intervals are calcu-lated as ˆ Y i ± . σ i and ˆ Y i ± . σ i respectively, where the standard deviation σ i is approximatedby the RMSE. The RMSE values are multiplied by constant k , which are 1.28 and 1.64, in orderto create 80% and 90% prediction intervals (PI), respectively, to forecast Type i demands. Finally,assuming a uniform distribution with each prediction interval, we discretize the distributions byrounding negative values to 0 and the real numbers to the nearest integer. This process is appliedto first five, six and seven weeks to create discrete distributions for the following week (i.e., nextsix days). The estimated distributions are used to create instances to provide a basis for possi-ble scenarios that might occur in the dialysis clinic. The 80% and 90% prediction intervals canbe considered representing a less and more conservative approach in accounting for uncertainty,18espectively. The prediction intervals for each patient type for each week are presented in Table 3.Figure 5 depicts the data and the prediction intervals for Type 1, Type 3 and Type 4 patients re-spectively, using seven weeks of past data to predict week eight. The blue line represents the pointestimate, green dashed lines represent 80% PI approximation and red dashed lines represent 90%PI approximation. Discrete distributions derived for the demands of Type 3 and Type 4 patientsin week 8 are in Appendix B as an example. Week 6 Week 7 Week 8Patient Type 80% PI 90% PI 80% PI 90% PI 80% PI 90% PI
Type 1 (2.31, 11.94) (0.96, 13.29) (2.30, 11.40) (1.01, 12.68) (2.40, 11.12) (1.18, 12.65)Type 3 (1.87, 5.18) (1.41, 5.64) (2.20, 5.71) (1.70, 6.21) (1.87, 5.17) (1.40, 5.64)Type 4 (-0.73, 1.77) (-1.08, 2.21) (-0.76, 1.59) (-1.09, 1.92) (-1.01, 1.17) (-1.32, 1.48)
Table 3: Demand prediction intervals for weeks 6, 7 and 8Figure 5: Prediction intervals for Type 1 (a), Type 3 (b) and Type 4 (c) patients for week 8(a) (b)(c)We generate 30 equiprobable scenarios to test our models. Demands per day in each scenario are gen-erated by sampling from the corresponding probability distributions for each patient type (see AppendixB for the probability distributions). The other parameters used in our case instances are as follows. Thetotal number of machines, ˆ C is set at 14. The capacities of units are set as R =11, R =8 and R =519or the three-unit cohorting model, and as R =11 and R =8 for the two-unit cohorting model. In bothmodels, the objective function weights are set as α = α =1,000, α =100, Π i =100,000 and (cid:15) =2.The proposed models are solved on a computer with an Intel(R) Core (TM) i7-9750H CPU @ 2.60GHz processor and a 16 GB RAM. Gurobi 9.0.3 is used to solve the instances. We next present theresults. In this section, we present our results and analysis based on the case data. In § § § As described in § ixj , which is denoted by O ixj . For example, O (1 , x indicates the number of overlapsamong uninfected (Type 1 and 2) patients with infected (Type 3) patients.As shown in Table 4, except for week 3, there is a difference in resource allocation decisions, whichindicates that dialysis machines could be better allocated in the past, if the demand was better estimatedand the allocation was optimized. Results show that allocation decisions can significantly impact theoverlaps of uninfected and infected patients. We discuss the mechanisms in the following. For all weeksexcept week 3, the hospital reserves fewer dialysis machines for the uninfected patients than they should(i.e., optimal R is greater than 7). As a result, since capacity at earlier sessions is not enough, some ofthe uninfected patients receive dialysis at later sessions of the day, during which suspected and infectedpatients also receive their treatments. As a result, under the hospital’s allocation policy, the numberof overlaps in uninfected and suspected patients’ treatments increase, which results in greater objectivevalues. The hospital’s allocation is optimal only in week 3, in which demands from uninfected acutepatients and suspected patients are lower than average, whereas a larger number of sessions is neededfor infected patients. That is, high demands from infected patients in the third week appear to justifythe capacity allocation policy of the hospital. 20 erformance with Optimal Performance with optimalhospital’s allocation allocation allocation (known demand) Week O (1 , x O (1 , x O x Z u R R R O (1 , x O (1 , x O x Z (cid:48) u Z u − Z (cid:48) u Z u Table 4: Comparison of the overlaps under hospital’s current capacity allocation ( R = 7 , R = 5 , R = 2)with optimal allocation under perfect demand information.The average daily utilization of each unit provides more insights into the clinic’s current capacityallocation policy. Figure 6 presents the utilization of units in model’s and hospital’s allocations. Weobserve that for the hospital’s current allocation policy, other than the third week, the standard unit’sutilization is higher than that of the optimal allocation. In contrast, the average utilization of bothisolated and quarantine units is lower than the optimal. One can argue that this makes the systemmore resilient to surges in suspected and infected patient numbers. We can infer that the hospital re-served more buffer capacity for the infected and suspected patients to ensure a high service level to thesepatients. However, this cautious approach comes at the cost of increasing overlaps among patient cohorts.Figure 6: Utilization of units in hospital’s capacity allocation policy versus optimal allocation policy U t ili z a t i o n ( % ) R1 (hospital) R1 (model) R2 (hospital) R2 (model) R3 (hospital) R3 (model)
The results here, which hypothetically assume known demand, indicate that there might be anopportunity to make better capacity allocation decisions and reduce the number of overlapping patientsin the clinic. However, in reality, weekly capacity allocation decisions must be made under significantdemand uncertainty. In the next subsection, we evaluate the performance of the proposed stochasticoptimization model in making capacity allocation decisions by accounting for demand uncertainty.21 .2.2 Analysis of three-unit cohorting model’s solutions
We now analyze the solutions of the scenario-based stochastic model, which makes capacity allocation de-cisions for an upcoming week under demand uncertainty to implement the three-unit cohorting strategy.As described in § Three-unit cohorting Overlaps withmodel’s solutions (Expected overlaps) realized demand
Week PI (%) R R R E [ O (1 , x ] E [ O (1 , x ] E [ O x ] Z u O (1 , x O (1 , x O x Table 5: Performance of the three-unit stochastic modelTable 5 shows that the proposed stochastic optimization model tends to allocate a larger number ofdialysis machines to the standard unit ( R ) that treats uninfected (Type 1 and 2) patients compared tothe hospital’s current allocation policy. Moreover, the capacity configuration achieved by the stochasticmodel leads to a lower number of overlaps for these three weeks compared to those incurred underthe hospital’s allocation (Table 4). Specifically, in week 6, the hospital’s allocation causes an overlapof uninfected and infected patients, which could be avoided with the model’s solution. For week 8, themodel solution can decrease the overlaps of infected and suspected patients with uninfected patients. Thiscomes at a cost of some increase in the overlaps of suspected and infected patients, however since theseoverlaps are considered less critical, the model solution is an improvement over the hospital’s allocation.We also observe that while week 6 and 8 capacity allocation solutions are robust to the different PIsused to develop scenarios, week 7 solutions are affected by the different set of scenarios; specifically, inweek 7, the model allocates more machines to treat uninfected patients when more conservative demandrealizations are accounted for. Although the number of overlaps based on realized demands in week 7is not affected by the different capacity allocation solution, the results indicate that the prediction ofdemands and development of scenarios can be crucial factors affecting the performance of the capacityallocation decisions in the clinic.These results show that developing forecasts on the number of patients by using historical data andsolving the proposed scenario-based stochastic model can help the dialysis clinic reduce the overlappingsessions of uninfected and infected patients thereby making better cohorting during the pandemic.22 .2.3 Evaluation of different cohorting strategies We next investigate the effects of cohorting strategies on the performance of the system. We first comparethe solutions obtained by two- and three-unit cohorting models by considering the realized demands ofthe hospital for eight weeks, which are presented in Table 6. The last column of Table 6 presents thedifference in the objective functions of two- and three-unit models.
Optimal Performance with optimal Optimal Performance with optimalallocation allocation (known demand) allocation allocation (known demand)
Week R R R O (1 , x O (1 , x O x Z u R R O (1 , x O (1 , x Z u Z u − Z u Z u Table 6: Solutions of two-unit and three unit models under deterministic demandAs observed in Table 6, the solutions of the three-unit model dominate those of the two-unit modelin seven weeks out of eight. Since infected (Type 3) and suspected (Type 4) patients can be treatedin the same session in the three-unit model, the model assigns all of these patients to the last (i.e., thefourth) session of a day if possible. If the capacity of a single session is not sufficient, some patients areassigned to the third session, during which uninfected (Type 1 and 2) patients are usually treated as well.Therefore, by allowing overlaps in Type 3 and Type 4 patients’ dialysis treatments, the three-unit modelcan avoid the more serious overlaps with uninfected (Type 1 and 2) patients, thus incurring smallerpenalties.In the two-unit case, suspected (Type 4) and infected (Type 3) patients receive dialysis in the sameunit, but suspected patients’ sessions precede those of the infected ones. Consequently, since the wholeunit is allocated to a single patient group when the demand from one group of patients is low, the unitcapacity cannot be utilized fully. Moreover, assigning suspected patients to sessions earlier in the daymay cause increased overlaps with uninfected patients and result in worse objective values, as observedfrom Table 6.According to Table 6, the two-unit model slightly outperforms the three-unit model only in week 5,which is an exceptional case for the considered time horizon. Hence we examine the solutions achieved forthis week in further detail. We observe that the two-unit model allocates a greater amount of resourcesto the standard unit compared to the three-unit model. Specifically, the optimal allocations are R = 10, R = 4 for the two-unit model, and R = 8, R = 4, R = 2 for the three-unit model. In Figure 7,we present the resulting treatment schedules for two days from week 5, which include some overlaps. Inweek 5, the number of uninfected, infected and suspicious patient demands are 28, 4 and 2 on Monday(day 1), and 22, 1 and 3 on Wednesday (day 3). We show the number of patients assigned to each sessionby the alternate cohorting models in Figure 7. 23igure 7: Illustration of treatment schedules for days 1 and 3 of week 5 Day-1 , 3-unit
End of the day
SessionsUnits StandardIsolatedQuarantine (a) Day 1, three-unit model’s schedule(Daily penalty = 14,612) Day-1 , 2-unit
10 810
End of the day
SessionsUnits
StandardIsolated (b) Day 1, two-unit model’s schedule(Daily penalty = 10,010)
Day-3 , 3-unit
End of the day
SessionsUnits StandardIsolatedQuarantine (c) Day 3, three-unit model’s schedule(Daily penalty = 410) Day-3 , 2-unit
10 210
End of the day
SessionsUnits
StandardIsolated (d) Day 3, two-unit model’s schedule(Daily penalty = 3,010)
Type 1,2Patients Type 3Patient Type 4Patient
As shown in Figure 7(b), on day 1, since a larger number of machines are allocated to the standardunit in the two-unit model, uninfected patients’ dialysis treatments could be completed on the thirdsession, and they only overlap with two Type 3 patients on that session. On the other hand, as shownin Figure 7(a), due to the smaller number of dialysis machines allocated to the standard unit in thethree-unit case, Type 1 and 2 patients’ treatments continue until the last session of the day, and a largerpenalty is incurred due to several overlaps. In contrast, on day 3, the allocation made by the three-unitmodel results in a lower objective value (Figure 7 (c)). Since Type 3 and Type 4 patients are treated inparallel in the last session, they do not overlap with any uninfected patients in this case. However, asshown in Figure 7(d), overlapping treatments had to be scheduled in the two-unit solution.We also evaluate the performance of the two-unit cohorting strategy by solving the two-unit scenario-based stochastic model, in which the uncertainties inpatient demands are accounted for while deciding onthe number of machines for an upcoming week. Table 7 presents the results of the two-unit model, whichare compared with those obtained by the three-unit model for the same instances (Table 5). We observethat the three-unit cohorting strategy results in lower expected penalties than those of the two-unitcohorting strategy.These results show that determining the best cohorting policy and the number of dialysis machinesto allocate to different units can be challenging, even under the availability of perfect demand informa-tion. The demands for each patient type, which can be highly uncertain, can significantly affect the24 hree-unit cohorting Overlaps withmodel’s solutions (Expected overlaps) realized demand
Week PI (%) R R E [ O (1 , x ] E [ O (1 , x ] Z u O (1 , x O (1 , x Table 7: Performance of the two-unit stochastic modelperformance of cohorting strategies; and it is hard to foresee which cohorting strategy would performbetter without solving the stochastic optimization models proposed in this paper. The complexity ofthe cohorting, capacity allocation and treatment scheduling decisions, which increase under demand un-certainty, imply the need for analytical approaches to effectively manage dialysis treatments of variouspatient types during a pandemic. Using the methods proposed in this paper, dialysis clinics can evaluatethe performance of alternative cohorting policies and obtain the best allocation decision based on theforecasts for the upcoming time horizon.
The COVID-19 pandemic has shown that effective planning of health care resources is crucial. Moreover,treating vulnerable groups such as people with chronic diseases needs special attention to prevent infectionspread in health facilities. In this paper, we have focused on the operations of a hemodialysis clinic ina major pandemic hospital in Turkey, which is providing dialysis treatment to multiple patient groups.While the hospital has taken effective precautions to manage the treatment of different types of patientsthrough cohorting, they lack the analytical tools to make cohorting plans and to evaluate the effectivenessof different strategies. We present a two-stage stochastic programming modeling approach to help theclinic make more effective capacity planning and treatment scheduling decisions. We show with real datacollected during the COVID-19 pandemic that the clinic can make use of such an operations researchbased tool to mitigate the infection transmission risk at the hospital by decreasing overlapping sessionsamong infected and uninfected patient groups. Moreover, we present results that compare two alternativecohorting strategies based on the hospital’s data. We show that while three-unit cohorting is generallymore effective than two-unit cohorting to reduce overlaps among patient cohorts in our case study, theperformance of each strategy ican be highly dependent on the relative number of each patient type. Theproposed analytical methods can support the hospital to cohort patients during a pandemic effectivelyand manage scarce health resources efficiently.Although the hospital was able to treat all patients with the available dialysis resources since thebeginning of the pandemic, albeit incurring some overlapping sessions, the proposed models can also beused to assess when it might be impossible to treat all patients in the clinic with the existing capacityshould the demands from COVID-19 patients increase due to another pandemic wave. In such a case,25he model results can be presented to policymakers that can consider making arrangements to transferchronic dialysis patients to other private facilities in emergency cases, which is not currently allowed.Moreover, the non-medical operational solution proposed in this paper can be used to support cohorting,capacity planning and treatment scheduling decisions in different hospital units besides hemodialysissuch as to manage the treatments of chemotherapy, radiotherapy and physiotherapy patients.Given the scarce research attention on the planning and coordination of health care resources to treatchronic patients during extreme events such as natural disasters and pandemics, several future researchdirections exist. For example, the proposed models can be extended to develop tools for managingother types of chronic patients (such as chemotherapy patients) who may present different aspects andconstraints. While dialysis treatment session durations are fixed, chemotherapy or physical therapytreatment durations may be different for each patient. This makes the capacity per day dependenton the patients scheduled, and the scheduling problem becomes more challenging due to the additionalcomplexity of matching different treatment types. Another future research direction can be to incorporateuncertainty in patient arrivals during the day. We assumed that the demands from all patient types areknown at the beginning of the day while scheduling treatments to minimize overlaps. Arrivals of patientswith urgent treatment needs can be accommodated by reserving capacity for these patients, which is acommon practice in appointment systems. Our models can be extended in future research to account forsuch dynamics. Finally, different cohorting strategies can be analyzed with queueing models to obtaininsights on their effectiveness in different environments.
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Acknowledgements
This research has been supported by AXA Award Grant from AXA Research Fund.
A. Case Data
Monday Tuesday Wednesday Thursday Friday Saturday SundayPatient type: 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4Week 1
Week 2
13 12 4 0 8 8 4 1 11 12 4 0 7 8 6 0 9 12 2 1 8 8 4 1 1 0 2 0
Week 3
Week 4
12 12 2 3 6 8 5 1 3 12 4 3 4 8 5 0 4 12 4 2 4 8 4 1 3 0 1 0
Week 5
16 12 4 2 7 8 2 0 10 12 3 1 7 8 2 0 6 12 3 0 8 8 5 0 4 0 3 0
Week 6
Week 7
Week 8
13 12 7 0 10 8 3 0 11 12 3 1 5 8 3 0 16 12 4 1 9 8 2 0 6 0 0 0
Average
Minimum
Maximum
16 12 7 3 11 8 5 1 11 12 5 3 13 8 6 0 16 12 5 3 9 8 6 1 6 0 3 0
Std. Dev.
Table 8: Raw data and descriptive statistics on daily patients demands (November 2-December 28)
B. Discrete Distributions for Patient Types
P(X = x)Patient Type PI (%) 0 1 2 3 4 5 6