University of Groningen

Operations Research

Short term hospital bed capacity planning

Author:Roos HollanderS2590212

Professor:Prof. dr. R.H. Teunter

January 8, 2021

Master’s Thesis Operations ResearchSupervisor: Prof. dr. R.H. Teunter

Second assessor: Prof. dr. K.J. Roodbergen

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Short term hospital bed capacity planning

Author: Roos Hollander

Abstract

Hospital bed capacity planning is an important and challenging task. Bed occupancy highly dependson the length of stay (LoS) of patients in a hospital. LoS is known to be varying widely and theoptimal allocation of beds is complex due to several factors that play a role. However, bed capacityplanning is often kept very simple in practice. In this research, LoS of patients at the nursing wardof Internal Medicine in Martini Ziekenhuis is analyzed. A classification system for admissions inthis ward is introduced based on Diagnosis Related Groups (DRGs), such that the variance of LoSof patients within these categories is minimized. Using this classification system, a model is builtthat estimates the short term bed occupancy, approximated by a Normal distribution. The modelcalculates the number of beds needed in order to achieve a certain service guarantee. The modelcan steer hospital staff in making decisions about the optimal allocation of hospital beds.

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Contents

1 Introduction 41.1 Length of stay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Bed allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 The Dutch Healthcare system 6

3 Problem formulation 73.1 Martini Ziekenhuis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Research goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Literature review 94.1 Length of stay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Arrival process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.3 Bed allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Data 115.1 Data extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.2 Data cleaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

6 Analysis of LoS 14

7 Prediction of LoS 187.1 Factors that influence LoS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.2 Statistics as predictors for LoS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.3 Classification system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

8 Arrival process 26

9 Analysis of bed occupancy 28

10 Bed capacity model 2910.1 Simple model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.2 Model extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

11 Empirical results 34

12 Practical application 39

13 Conclusion 40

14 Restrictions and further research 41

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1 Introduction

Hospital capacity planning is an important issue in many countries due to the increasing costs ofinpatient care, constrained resources, and the growing demand for hospital care [19]. In order tooptimally allocate its resources, a hospital has to make policy decisions. Beds are a crucial butlimited hospital resource [21]. Bed allocation concerns the permanent number of beds assigned tothe different medical and surgical specialties in a hospital [16]. An underprovision of hospital bedsleads to patients in need of hospital care being refused, and the build-up of waiting lists or stresson another part of the hospital system [10]. Especially in the recent year, in which the COVID-19outbreak had a large impact on healthcare all over the world, the importance of hospital beds as aresource has been underpinned. The pressure on intensive care units increased a lot, and hospitalshad to reallocate beds in order to be able to give care to all COVID-19 patients. A fundamentalprinciple for appropriate bed allocation is that beds needed by a specialty should be determinedsuch that sufficient beds are available ‘most of the time’ [16].

1.1 Length of stay

Optimal allocation of hospital beds is hampered by the inherent uncertainty of a patient’s lengthof stay [21]. The length of stay (LoS) of patients is an important metric used within hospitals formaking their policy decisions. This metric is defined as the number of days in which a patient isadmitted onto a clinical ward of a hospital. However, there are several ways of defining the LoS.This research concerns the bed capacity planning on hospital nursing wards. According to the NZa,the Dutch Healthcare Authority, Dutch hospitals can register a care activity as an inpatient bedday at a nursing ward if that activity is part of a nursing period which minimally contains oneovernight stay. The period starts from the moment of admission until the moment of discharge,where the day of admission (provided this registration took place before 8:00 pm) and the day ofdischarge are both registered as bed days [26]. In this research, the definition of the NZa will beleading. Therefore, the definition of LoS that is considered during this research is as follows:

Definition 1: The length of stay (LoS) is the number of consecutive registrations of inpatient daysduring a single episode of hospitalization of a patient on a nursing ward.

Hence, this definition is based on financial registrations of care activities, also referred to as colddata. According to this cold data, LoS is the number of days that a patient is officially registered ata hospital bed, but this does not mean that this patient physically occupies that bed all those days.For example, suppose that a patient is admitted to a ward at 4:00 pm and that it is discharged thenext day at 11:00 am. The cold LoS of this patient is 2 days, but the physical, also referred to asthe warm, LoS for this patient is just 19 hours. The effect of using cold data is discussed further inthis research.

Furthermore, according to this definition, a hospital stay is defined as a minimum of two calendardays. Exceptions are possible for example if a patient is admitted to a ward after 8:00 pm. In thatcase, only one inpatient day is registered. Later in this research, other exceptions for hospitaliza-tions with a single registered inpatient day are discussed.

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It is generally known that LoS of patients in hospitals is varying widely, whether examined ongeographic area, on the level of hospitals, on the level of doctors or even at patient level [25]. Inorder to optimally allocate beds in a hospital, knowledge of LoS of patients is of great importance.

1.2 Bed allocation

Allocation of beds in a hospital is a challenging task since many factors play a role and there is alot of uncertainty. The admissions of patients depend on the arrival process of emergency cases andthe schedule of the surgical operations that take place in a hospital. Patients that have undergonesurgery are likely to need nursing care. Therefore, they will be admitted to a nursing ward forsome time, depending on the impact of these surgical operations and other patient related factorsthat might play a role. The LoS of patients that arrived with emergency also depends on similarpatient related factors and the reason of admission to a hospital. The allocation of patients tobeds on a ward depends on the current occupancy and the expected need of beds during a comingtime period. The latter mainly arises from intuition and preferences of surgeons. Since there areeven more factors that play a role and there is a lot of uncertainty concerning all these factors, theallocation of beds on nursing wards is a highly complex task.

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2 The Dutch Healthcare system

In February 2005, a new healthcare registration system was introduced in the Netherlands, a case-mix system based on a set of care related actions constituting a so-called DBC. A DBC is theabbreviation of the Dutch translation of a diagnosis treatment combination. This system was in-troduced for the registration and pricing of hospital and medical specialist care. Within a DBC,patients are expected to display a set of clinical responses which will, on statistical average, resultin similar use of hospital resources [17]. The reason for implementing this system was to makehealthcare costs more transparent. Healthcare providers and insurance companies can now betternegotiate about the quality, price and the amount of care. This makes comparison and renewal ofcare possible which contributes to better and affordable healthcare [27].

A DBC is, simply stated, a bundle of several care activities. When a patient enters a hospital, acare trajectory is opened by a medical specialist. Within this trajectory, all the care activities thata patient receives are registered. After a certain amount of time, for general care trajectories thisperiod is 120 days, the care trajectory is closed [27]. An application, a so-called grouper, then de-cides which specific DBC this trajectory belongs to based on decision trees specified by the NZa [27].

Hence, a DBC is a set of care activities with a specified combination of diagnose and treatment.It contains information about the compensation for the healthcare provider and the costs for thehealthcare facility [27]. A DBC is also often referred to as a care product. In total, there are 6489different DBCs recognized in the Netherlands. It is also possible that a specialist performs a careactivity not constituting a DBC, for example an X-ray. These activities are called OZPs, additionalcare products, and are independently reimbursed.

An internationally used system to classify healthcare products is based on Diagnosis Related Groups,DRGs. This system uses a classification of 467 illness categories, identified in the InternationalClassification of Diseases, Ninth Revision, Clinical Modification (ICD-9-CM) [17]. The system iscomparable to the DBC system used in the Netherlands. However, the Dutch system derives dif-ferent DBCs per speciality, while the international system classifies the products based on diagnoserather than specialty. Therefore, the Dutch system recognizes almost 14 times more different carecategories than the international system.

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3 Problem formulation

Although the optimal allocation of hospital beds is a complex task, bed capacity planning in hospi-tal nursing wards is often kept very simple in practice. This is possibly related to the fact that mosthospitals do not have a separate department for allocating patients to beds. Allocation is oftendone by physicians or nurses themselves. A certain amount of beds is often kept free for emergencycases and the occupancy of the other beds depends on the planning of surgical operations, wherevariance in LoS generally is not taken into account.

However, this may lead to an under or overprovision of hospital beds. An underprovision of bedsmay have serious impact on the functioning of a hospital. It is the primary cause of admissionsand surgical operation cancellations, delays in emergency admissions, early patient transfers fromintensive care units, delays in patient transfers between units and early patient discharge [19]. Fur-thermore, patients that are initially not able to receive healthcare might be needing even more andadvanced healthcare in the future, resulting in higher healthcare costs. The recent outbreak ofCOVID-19 showed large scale examples of underprovision of hospital beds in the Netherlands. Dueto the high pressure on hospitals and especially on intensive care units, regular care had to be scaleddown which affected lots of patients in need of healthcare. On the other hand, an overprovisionof beds is wasteful of scarce resources [10]. Excess bed capacity may lead to stagnant capital andadditional costs [19].

Hence, an improvement of the allocation of beds on nursing wards is important for both a costrelated point of view as well as clinical efficiency. It is of general interest to keep healthcare costslow and balanced, such that there is enough money to be able to provide all healthcare demanded.Since, in practice, bed allocation and bed capacity planning in hospitals is often kept simple, thereis still room for improvement. This research concerns the bed capacity planning for the ward ofInternal Medicine of Martini Ziekenhuis in Groningen, the Netherlands. Although some parts inthis research are specific for this ward of Martini Ziekenhuis, it is intended as a general guidelinefor bed capacity for other hospitals as well.

3.1 Martini Ziekenhuis

At the nursing ward of Internal Medicine of Martini Ziekenhuis in Groningen, there is a lot ofvariation in the amount of time that patients spend in a hospital bed. At this ward, there arepatients that have undergone a surgical operation belonging to the Internal Medicine department.However, patients from the Geriatric department in Martini Ziekenhuis that need nursing care arealso placed on this ward. The combination of these two specialties turns out to show high variancein the LoS of patients on this ward. Hence, in order to optimally allocate patients to the availablehospital beds, it is important to gain knowledge on the LoS of patients for this ward.

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3.2 Research goal

The goal of this research is to support a hospital in its short term bed capacity planning. A modelis built that helps hospital staff in making decisions about the optimal allocation of hospital beds.This model determines the number of beds needed on a nursing ward over a coming short timeperiod, given the scheduled surgical operations during a coming time period, in terms of likelihoodthat enough beds are available. It can serve as a warning for the staff to be able to anticipatein time. For example, if the model calculates that the current number of beds is not sufficient toserve all patient, the staff may decide to place an extra bed on the ward or to allocate a patientto another ward with more available beds. To be able to build this model, it is important to gainknowledge of the current bed capacity and the effects influencing bed occupancy. Therefore, thisresearch consists of two parts.

In the first part, the current bed capacity of the nursing ward of Internal Medicine of MartiniZiekenhuis is analyzed. Since the optimal allocation of beds is highly depending on the amount oftime patients spend in a hospital bed, this analysis is mainly focused on LoS. Factors that influencethe LoS of admitted patients on the ward are examined and prediction of LoS by several statisticalmetrics is compared. The purpose of this analysis is to introduce a new system of categorizinghospital admissions, such that the variance of the LoS of patients within these categories is mini-mized. These categories are based on bundling different care products showing similar distributionsof patient’s length of stay.

In the second part, a model is built that estimates the distribution of the number of hospital bedsneeded on a nursing ward when looking a number of days ahead. With the use of the categorizationsystem from the first part of this research, this model can steer bed allocation in order to achievea certain service guarantee.

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4 Literature review

4.1 Length of stay

The most common used statistic associated with LoS is the average length of stay, also sometimesreferred to as ALoS. However, multiple studies show that the distribution of LoS is generally non-symmetric. Under these circumstances, the average LoS, albeit easy to quantify and calculate, canbe misleading and not reflecting the nature of such underlying distribution [18].

Distributions that are found to be most suitable for fitting the heavy tailed LoS in hospitals are theLog-Normal, Gamma and Exponential distribution [11]. Also the Weibull distribution is commonlyused to fit LoS. Several researches suggest transformed distributions such as the Beta-Cauchy,Gamma-Pareto and Gamma-Exponential-Cauchy distribution proposed by Harini et al. (2018).Papi et al. (2014) introduce a generalization of the Phase-Type distribution and several otherstudies propose other mixtures or convolutions of distributions. However, the fit of most proposeddistributions depends on the data and the specific department or clinical characteristics of patients.Furthermore, the use of transformed distributions is not always convenient since they concern ge-ometric means instead of arithmetic means [9], and the need for retransformation makes them lesssuitable in practice.

There is no general consensus about the factors that influence the LoS of patients in certain hospitaldepartments. Studies find different factors that influence the LoS, such as geographic area, hospitaland different patient specific factors such as age and gender. However, all studies are limited indata availability and their outcomes are case and region dependent. Dada and Sule (2019) concludethat factors related to time such as season, month of the year and day of the week generally donot have any effect on the LoS [8]. Most studies agree that age and comorbidity have a significantinfluence on the LoS. Comorbidity is the state of a patient having multiple medical conditions atthe same time. A patient having a certain disease in combination with other conditions is generallyexpected to spend a longer time in hospital than a patient with a single condition.

4.2 Arrival process

Many arrival processes, especially unscheduled, have been shown to be well approximated by aPoisson process [7]. Therefore, the admissions of patients at a hospital are often modeled by thisprocess. For a nursing ward, this process is mostly used to describe the emergency admissionsof patients. However, patient flows within hospitals depend on the hierarchical structure betweendifferent departments a hospital. Operating room scheduling is observed to be ‘leading’ in thisstructure [24]. Hence, the arrival process of patients on a nursing ward also depends on the numberof surgical operations scheduled. Most researches on bed allocation on nursing wards thereforeseparate the flow of planned and emergency admissions.

4.3 Bed allocation

Bed allocation at a certain hospital ward concerns the permanent number of beds assigned to thatward. The relevant question to be answered is: ‘how many beds are needed to support a medicalor surgical unit under its current admission, treatment, and discharge practices?’ [16] In previous

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researches, several Operations Research (OR) techniques have been applied to answer this and otherrelated questions about capacity allocation in hospitals. One of the most frequent used OR modelsto determine the optimal bed allocation is queueing theory.

Gronescu et al. (2002) use classical queueing theory to describe the movement of patients througha hospital department. They assume that a hospital department can be described by a M/PH/cqueue in steady state, where the arrivals are assumed to be distributed by a Poisson process, theLoS follows a Phase-Type distribution and c is the number of beds. They also introduced theextended M/PH/c/N queueing model in which an extra waiting room is provided. De Bruin et.al. (2010) introduce a decision support system, based on the Erlang loss model, which can be usedto evaluate the current size of nursing units. As subsequent research by Belciug and Gorunescu(2014) proposes the use of a compartmental model, which offers a feasible structure of the hospitaldepartment in accordance to the queuing characteristics [5].

Other frequently occurring OR models in the literature to determine the optimal bed allocation onhospital wards are based on dynamic programming, simulation methods and Markov processes. Ak-cali et al. (2006) use mathematical programming to develop a network flow model that determinesthe optimal hospital bed capacity over a finite planning horizon. Bekker and Koeleman (2011)determine the optimal number of elective admissions per day, such that an average desired dailyoccupancy is achieved, using a quadratic programming model. The papers of Landa et al. (2014)and Keepers and Harrison (2009) provide simulation studies to balance emergency and planned ad-missions and determine that the occupancy of nursing wards is a good predictor for the frequencyof overflows. Broyles et al. (2010) present a Markov chain probability model that uses a maxi-mum likelihood regression to predict expectations and discrete distributions of transient inpatientinventories. Utley and Gallivan (2003) show that the bed occupancy distribution can easily beapproximated from their Markov model by a Normal distribution.

Different models prove to be modeling certain hospital processes, such as bed allocation, quite well.However, in practice, the distribution of hospital beds on nursing wards is to a great extent basedon historically obtained rights. A well-founded quantitative approach is often lacking when hospitalmanagement decides about the number of beds [7].

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5 Data

Data for this research is provided by LOGEX, a healthcare analytics company headquartered inAmsterdam, the Netherlands. The goal of LOGEX is to contribute to better and more affordablehealthcare. In 2017, LOGEX merged with MRDM and Value2Health and recently Prodacapo andIvbar also joined LOGEX to bundle their expertise to become the international leader in healthcareanalytics.

LOGEX analyses the production data of around 90% of all hospitals in the Netherlands, whichare about 60 hospitals. Every quarter, these hospitals provide their data consisting of four tables:registered activities, registered care products, patient data and operating room data of at least thethree most recent years. LOGEX stores and processes this data.

The tables of registered care products contain information on the diagnose and treatment of thepatient. In the DBC system, care products are derived per specialty. However, diagnoses fromdifferent specialties can lead to care products that are almost equivalent. Therefore, it can alsobe convenient to classify care products based on the diagnose of a patient rather than per specialism.

Next to the Dutch system of DBCs, there are some other classification systems of care products.The international DRG system to classify care products, as described in Section 2, classifies careproducts based on diagnose rather than specialty. The U.S. Department of Health and HumanServices has further developed this international system and invented a software tool named Clini-cal Classification Software. This tool assigns diagnoses in about 250 clinical relevant groups. Thisclassification is a process that will always continue being further developed. During recent years,LOGEX developed its own interpretation of this system in order to give their clients the most op-timal service. Based on the Clinical Classification Software, each product is therefore also assigneda CCS code, which are further divided over the so-called CCS clusters.

Table 1 shows the number of unique elements per classification system on the ward of InternalMedicine of Martini.

Category Number of unique elements in the data set

DBCs 260Diagnoses 199Diagnose groups 24CCS codes 115DRGs 96CCS clusters 9

Table 1: Unique elements per classification system in the data set

5.1 Data extraction

For this research, data from Martini Ziekenhuis from is extracted from the LOGEX database. Thenursing ward of the Internal Medicine Department of Martini Ziekenhuis is considered. The hospitalconsists of 2 units for this specialty, department 2C and 2D, containing 32 and 30 hospital beds

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respectively. Data from 2017 until 2019 is extracted from the LOGEX database. The first twoyears of data are used as training data. This data is used for the analysis of LoS. The last year ofdata is used as a test set.

The data is filtered on all registered bed days by the specialties Internal Medicine and Geriatrics.More than two thirds of all patients from the Geriatric department that need nursing care areplaced on either department 2C or 2D. The other patients are divided among other wards in thehospital. Since there are also patients of other specialties placed on one of the departments ofInternal Medicine, this exchange of patients is presumed to be nearly balanced. Therefore, it isassumed in this research that all admissions extracted from the database actually take place oneither department 2C or 2D in Martini Ziekenhuis.

Filtering the data per specialty, a hospitalization is measured as the number of subsequent careactivities of bed days within a specific care trajectory that started with a clinical admission. Inthis way, the LoS of a patient during its whole hospitalization is measured. Filtering stays onthe physical location of registration rather than specialty might cause a single hospitalization of apatient that is interrupted by a single day on an intensive care unit to be split into three separateadmissions. Of course, this exactly measures the occupancy of beds at the specific departments,but it makes analyzing the duration of hospitalization of that patient complicated.

The extraction of hospital stays from the financial registration records of care activities is referredto as cold data. It describes an integer number of days that a patient spends in a hospital. However,the warm LoS, the physical amount of time that a patient spends in a hospital, is described by anon-integer number. The number of warm bed days corresponding to an integer number of coldbed days is calculated by dividing the cold number of bed days by a factor. Martini Ziekenhuisuses a factor r that is specific for each department. The factors for nursing departments 2C and2D are given by r = 1.12 and r = 1.14 respectively.

For each admitted patient, age, gender, socioeconomic status are also subtracted from the LOGEXdatabase. The latter is a combined measure of both economic and sociological factors, indicatingthe position of a patient in relation to others. The socioeconomic status is often divided over threelevels: low, average and high. For each patient, the data set also contains a socioeconomic statusindicating either the highest 33%, the middle 33% or the lowest 33%.

5.2 Data cleaning

The data that hospitals provide to LOGEX is recorded by medical staff. In the process of admin-istration of the data, there is always the possibility that mistakes are made due to lack of time,lack of knowledge or human errors. When LOGEX receives a data set from a hospital, it validatesthis data to check on possible mistakes. However, it can be the case that some mistakes are notcaught during this validation. Therefore, it is important to keep these possible errors in mind whenanalyzing the data.

In the extracted data set, there are 3 registered stays of patients that do not contain the corre-sponding care traject, care product and diagnose. Furthermore, the data set contains 61 stays of

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which no patient related information is available such as socioeconomic status. These stays areremoved from the data set. For the same reasons, 1 admission is removed from the test data set.It is expected that these removed stays are not correlated with LoS and do not affect the purposeof this research.

Due to the fact that hospitalizations are measured per specialty rather than the physical locationof a registration in a hospital, it can occur that a registered stay contains some days at an intensivecare or that it contains registered care activities that afterwards turned out to be expired. Toaccount for this fact, the LoS used in this research is corrected by subtracting the intensive careregistrations and expired activities. This might cause some exceptions in the data set in which hos-pitalizations are measured as a single day, which is in contrast with the definition of LoS stated inSection 1. Such exceptions can also occur if, for example, a hospitalization of a patient is measuredas a single day at the ward of Internal Medicine, while within this hospitalization there are alsoinpatient days registered within another specialty at another ward. The LoS of this admission isthen also measured as a single bed day contrasting the definition of LoS. Other exceptions can becaused by patients being admitted at a ward after 8:00 pm. During this research, these exceptionsare kept in mind while analyzing LoS of patients.

It should be noted that, with this way of extracting hospital stays from the data, it might occurthat hospital stays with initially different durations are assigned the same LoS. For example, apatient that first spent 4 days on a nursing ward and subsequently 2 days on an intensive care unitis extracted from the data as having LoS equal to 4 days. However, since this research concernsthe bed capacity planning of a normal nursing ward, the number of interrupted stays is relativelylow so this difference is assumed to be not affecting the purpose of this research.

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6 Analysis of LoS

After cleaning the data, the data set that is left contains a total of 5366 stays of patients on thenursing ward of Internal Medicine at Martini Ziekenhuis. The classification system of care productsused by Martini Ziekenhuis is the DRG system described in Section 2. Therefore, the analysis inthis research is based on this system. Figure 1 shows the DRGs that have the highest average LoS.Note that only the frequently occurring DRGs are taken into consideration here, with more than50 admitted patients in the data set.

Figure 1: Frequently occuring DRGs with highest average LoS

For each registered stay, the data set contains information about whether the patient arrived withemergency or whether this patient was planned to be occupying a bed. The total of 5366 admissionscan therefore be split into planned and emergency admissions. The total number of admissions andsome statistics on the LoS are given in Table 2. It can be seen that the average LoS of patientsthat were planned on the ward is lower than the average LoS of patients arriving with emergency.The distribution of the emergency stays seems to have a longer right tail than the planned stays.Note that the standard deviation of LoS is almost equal to the mean of LoS for the total numberof admissions as well as for the planned and emergency admissions, which means that the data ofLoS is spread over a wide range of values. The longest hospitalization registered in the data set is64 days, which is a patient that arrived with an emergency case.

Nr. of admissions min 25% 50% 75% max mode µ σ

Total 5366 1 2 4 8 64 2 6.12 5.81Planned 1437 1 2 3 5 51 2 4.59 4.72Emergency 3929 1 2 5 9 64 2 6.69 6.06

Table 2: Statistics on LoS

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In Figure 2, the total number of registered bed days per month are shown. It can be seen thatthe largest part of the beds on the ward are occupied by patients arriving with emergency. Thepercentage of emergency admissions is around 70% of the total amount of admissions. The numberof registrations of bed days do vary between months, but a seasonal effect is not clearly visible.

Figure 2: Total number of registered bed days per month at Internal Medicine over 2017 and 2018

As described in Section 4, the distribution of LoS is generally non-symmetric. The most com-monly used distributions to fit LoS data are the Log-Normal distribution, Gamma, Exponentialand Weibull distribution. Figure 3 shows the fit of each of these distributions to the data of LoSfor all emergency admissions. The LoS of planned admissions shows a similar distribution. Notethat the histogram shows a skewed distribution for the LoS, as is expected from previous researches.

It can be observed that all distributions seem to fit the LoS data quite well. However, the fit of theExponential distribution has some analytical preference above the others because it has the specialproperty of being memoryless.

According to Ross [20], a random variable X is called memoryless if

P (X > s+ t | X > t) = P (X > s), ∀s, t ≥ 0.

This means that, if the random variable X is describing LoS, having an Exponential distribution,the future LoS of a patient does not depend on the time this patient already spent in a hospital.This property proves to be convenient later in this research, where remaining LoS of patients thatare already occupying a hospital bed is calculated.

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Figure 3: Fitting the Log-Normal, Gamma, Exponential and Weibull distribution to the data forLoS of emergency admissions

The Exponential distribution is often used for describing hospital LoS in previous literature, andalso for the data of the nursing ward of Internal Medicine of Martini Ziekenhuis it shows a good fit.A random variable X is Exponentially distributed if its probability density function is given by

f(x) = λe−λx for x ≥ 0.

This distribution is continuous and supported on the interval [0,∞). However, by the definitionstated in Section 1, LoS in this research is described by an integer number of days. Therefore, LoSin this research is a discrete random variable. This makes the Geometric distribution, the discreteanalogue of the Exponential distribution, a more appropriate choice for predicting LoS. A randomvariable Y is Geometrically distributed with parameter p if

P (Y = k) = p(1− p)k−1 for k = 1, 2, 3, ... (1)

The parameter p describes the probability that a patient occupying a bed today will not be oc-cupying a bed tomorrow. Then, the parameter q = 1 − p describes the probability that a patientcurrently occupying a bed will still be occupying that bed tomorrow. Rewriting the density functionof the Geometric distribution given in (1) in terms of parameter q yields

P (Y = k) = qk−1(1− q) for k = 1, 2, 3, ... (2)

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with the corresponding cumulative density function equal to

P (Y ≤ k) = 1− qk for k = 1, 2, 3, ...

The mean and variance of the Geometric distribution in terms of q are given by

E[Y ] =1

1− q,

Var[Y ] =q

(1− q)2.

Let the mean value of LoS be given by µ. An estimation of the parameter q can be obtained byequating the expected value of the Geometric distribution to µ. This yields

1

1− q= µ,

and rewriting this gives an estimation for q as

q = 1− 1

µ. (3)

Like its continuous analogue, the Geometric distribution is memoryless. Therefore, in the proceed-ings of this research, the Geometric distribution is used to describe and predict LoS of patients atthe nursing ward of Internal Medicine of Martini Ziekenhuis. From Table 2, it can be observed thatthe mean LoS for planned and emergency admissions equal µp = 4.59 and µe = 6.69 respectively.Using (3), the standard first moment estimators for the parameters of the Geometric distributiondescribing the LoS for planned and emergency admissions are given by qp = 0.78 and qe = 0.85.The fit of the Geometric distributions to the LoS data for planned and emergency admissions areshown in Figure 4 and 5 respectively.

Figure 4: Fit of the Geometric distribution withparameter qp = 0.78 to the histogram of LoS overplanned admissions

Figure 5: Fit of the Geometric distribution withparameter qe = 0.85 to the histogram of LoS overemergency admissions

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7 Prediction of LoS

7.1 Factors that influence LoS

As described in Section 4, there have been several studies on predicting LoS. These studies finddifferent outcomes on factors that have significant influence, depending for example on the surgicaloperation of which the length of a hospital stay is predicted. No specific literature on the predic-tion of LoS of patients on a nursing ward of Internal Medicine is found. Therefore, this researchfocuses on the factors that have been proven to generally influence hospital LoS, based on theiravailability in the data. For each admission, the diagnose and the urgency of the admission is takeninto account, where the urgency of an admission is the distinction between planned and emergencyadmissions. The patient related factors considered are age, gender and socioeconomic status. Thecorrelation between the LoS and these factors is described by the heatmap in Figure 6.

Figure 6: Correlation between LoS, urgency, diagnose, age, socioeconomic status (SES) and gender

Figure 6 shows that there is not much correlation between the gender and socioeconomic status ofa patient on the amount of time it spends on a hospital ward. However, urgency, diagnose and agedo show some positive correlation. From Table 2, it is concluded that the average LoS of patientarriving with emergency is larger than of patients arriving with a planned admission, which confirmsthe positive correlation between urgency and LoS. The positive correlation between diagnose andLoS is as expected from general intuition. The positive correlation between age and LoS suggeststhat, when one tries to predict the LoS of a patient, taking age into account might lead to a moreaccurate prediction.

18

Figure 7 shows how LoS is varying over different age categories. It can indeed be concluded thatover all admissions on the nursing ward of Internal Medicine in Martini Ziekenhuis, age has a pos-itive influence on the number of days patients spend in a hospital bed.

Figure 7: Boxplot of the LoS over different age categories

Although there have been several studies on fitting distributions on the data of LoS, the mostcommonly used statistic for prediction is ALoS, the average length of stay. This statistic is easy tocompute and therefore convenient to use as a predictor for LoS in practice. Since the distribution ofLoS is often positively skewed, one could also wonder whether it is worth examining the use of themedian and mode of LoS. It should be noted that, in practice, often no prediction on LoS is madeat all. Allocation of beds is regularly mostly based on the current occupancy of hospital beds andthe intuition of medical staff. In that respect, there is still scope of improvement in most hospitalsfor which the first part of this research could provide a guideline.

7.2 Statistics as predictors for LoS

To test the mean, median and mode as predictors of LoS, their performance is evaluated on the testdata set. The mean, median and mode of LoS are calculated over the training data set. Then, takingthese statistics as prediction for LoS over 2019, their errors are compared using the mean abso-lute error (MAE), the mean squared error (MSE) and the mean absolute percentage error (MAPE).

The MAE is a popular error measure since it is easy to compute and understandable. It is calcu-lated as the average of the absolute deviation from the mean. The MSE is also often used, although

19

it is more difficult to interpret and more sensitive to outliers. It is calculated as the average ofthe squared deviation from the mean. The root of the MSE, the RMSE, is also widely used andminimising this measure leads to a prediction of the mean. One of the most commonly used mea-sures for forecasting accuracy is the MAPE. It is calculated as the sum of the individual absoluteerrors divided by the demand for each prediction. Both MAE and MSE are scale dependent errormeasures. The MAPE is a unit free error measure, which measures the relative error. The inter-pretation of the MAPE is therefore very intuitive.

The performance of the mean, median and mode over all admissions as predictors of LoS is given inTable 3. Just looking at the mean absolute error (MAE), the median seems to be the best predictorof LoS. However, also taking the mean squared error (MSE) and mean absolute percentage error(MAPE) into account, one can conclude that the mean is the best predictor. It should be notedthat the error measures of this statistic are still quite large. This is explained by the large variancein LoS that is not taken into account when using a single parameter statistic as predictor.

MAE MSE (RMSE) MAPE

Mean of the LoS 3.96 30.08 (5.48) 0.65Median of the LoS 3.65 35.02 (5.92) 0.91Mode of the LoS 4.19 47.24 (6.87) 2.10

Table 3: Performance of mean, median and mode over all admissions as predictors of LoS

In Section 6, it is concluded that the difference of LoS between planned and emergency admissionsis considerably. The average LoS of patients arriving with emergency is higher than the LoS ofpatients that were planned to be staying on the ward and the heatmap in Figure 6 confirms arelatively strong positive correlation between urgency and LoS. Therefore, it is interesting to alsoconsider the performance of the mean, median and mode as predictors of LoS when calculatedover planned and emergency admissions separately. It is expected that these statistics yield moreaccurate predictions than statistics calculated over all admissions. The performance of the mean,median and mode over planned and emergency admissions is given in Table 4. It can be observedthat the prediction measures indeed lower than in Table 3. The prediction measures of the modeare similar, since the value of the mode over all admissions equals that of planned and emergencyadmissions.

MAE MSE (RMSE) MAPE

Mean of the LoS 3.83 29.68 (5.45) 0.63Median of the LoS 3.46 32.17 (5.67) 0.77Mode of the LoS 4.19 47.24 (6.87) 2.10

Table 4: Performance of mean, median and mode over planned and emergency admissions separatelyas predictors of LoS

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7.3 Classification system

The mean, median and mode of LoS are statistics that are easy to compute and therefore a practi-cally convenient choice for predicting the length of stay of a patient. However, using these statisticsdoes not take into account any variance in LoS. The actual amount of time a patient spends in ahospital bed may exceed the expected time a lot. Furthermore, previous researches have proventhat LoS is varying a lot whether examined on different levels.

Calculating the statistics for each DRG separately rather than over the total of all registrations,one would expect to obtain a more accurate prediction of LoS. In Table 5, it can be observed thatthe value of all prediction measures are lower when considering each DRG separately. This provesthat a prediction of LoS is more accurate when a DRG specific statistic is used.

MAE MSE (RMSE) MAPE

Mean of the LoS 3.67 28.46 (5.33) 0.61Median of the LoS 3.33 31.03 (5.57) 0.79Mode of the LoS 3.83 42.27 (6.50) 1.67

Table 5: Performance of mean, median and mode for each DRG separately as predictors of LoS

The data set extracted from the LOGEX database contains a total of 96 unique DRGs during theyears 2017 and 2018 on the Internal Medicine ward in Martini Ziekenhuis. Therefore, calculatingthe statistics for each DRG separately might be a too complicated procedure for medical staff touse in practice. Furthermore, it may not even be needed to calculate separate statistics for eachDRG, since some DRGs show similar distributions of LoS.

Therefore, a new classification system based on DRGs is introduced in this research. The goal isto divide admissions over a relatively small number of categories, such that the variance of LoSwithin these categories is reduced and a more accurate prediction of LoS is obtained. The relativelysmall number of categories makes the system easier to use in practice than considering each DRGseparately, and also assures that enough data is available for obtaining accurate estimates. It shouldbe noted that there might be other procedures of categorizing DRGs leading to the same or evena more accurate prediction of LoS. This research gives an example of a procedure applied to thespecific ward of Internal Medicine in Martini Ziekenhuis, which focuses on being effective and easilyto implement in practice.

7.3.1 Classification on occurrence of admissions

To introduce a new classification system, only the DRGs for which 10 or more registrations areavailable in the data set are taken into account. The remaining DRGs are placed in the firstcategory labeled S, sporadically occurring DRGs. This category contains all DRGs of which notenough data is available to draw relevant conclusions about their corresponding LoS. If, in practice,it occurs that a patient is admitted to the ward with a DRG that was not contained in the data

21

set, this DRG should also be placed in the category S.

7.3.2 Classification based on urgency

It is observed that LoS is more accurately predicted when considering planned and emergency ad-missions separately. Therefore, an obvious classification criterion is to separate admissions basedon their urgency. For emergency admissions, it is not possible to schedule the arrival of a patientand estimate its related factors in advance. However, recall from Section 4 that operating roomscheduling is often observed to be leading in the hierarchical structure of departments within ahospital. Therefore, it is to some extent possible to derive the planned admissions on a hospitalward for a coming (short) time period based on the scheduled surgical operations belonging tothe specialty of that ward. For an optimal bed allocation, it is of great advantage if LoS of thesepatients could be estimated beforehand.

7.3.3 Classification based on a combination of mean and standard deviation of LoS

After creating two data sets for planned and emergency admissions, the DRGs are further dividedover two categories based on a combination of their mean and standard deviation of LoS. For bothplanned and emergency admissions, the DRGs are divided over two groups of DRGs based on acombination of mean LoS and standard deviation of LoS per DRG. These categories are labeled Land H indicating relatively low mean and variance of LoS and relatively high mean and varianceof LoS. The division of DRGs over these two categories is graphically shown in the scatter plot inFigure 8.

Figure 8: Division of DRGs over categories L and H

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7.3.4 Classification based on influence of age

From Figure 6 and 7, it is concluded that age is positively correlated with the LoS. Taking age intoaccount might lead to a better prediction of LoS and a reduction in the variance. To determine theinfluence of age on the LoS of patients, a regression is used on each DRG.

Standard regression analysis assumes a Normal distribution of the error term. Furthermore, itpredicts continuous values of the dependent variable. However, in this research, the dependentvariable is LoS, which is a discrete, non-negative variable. The Poisson distribution is generally agood candidate to describe count values. The key assumption for this model is that the mean of thedata is equal to the variance. The data for LoS does not satisfy this assumption, since its mean ismuch smaller than its variance. This situation is called overdispersion. In the case of overdispersionand when the dependent variable is discrete and non-negative, the Negative Binomial regression isa good candidate.

For all DRGs labeled as L and H based on a combination of their mean and standard deviationof LoS, a negative binomial regression of the age on the LoS is performed. The DRGs are filteredon the significance of the p-value of the coefficient for age in these regressions. Both categoriesL and H are divided into two categories based on the influence of age per DRG. The DRGs thatwere categorized as L and for which age shows a significant influence, are now labeled as Lage. TheDRGs for which age does not have a significant coefficient in the regression remain in category L.The same procedure is performed for category H. Hence, all DRGs for which age has a significantinfluence on LoS are placed in either category Lage or Hage. For these categories, the influence ofage is graphically shown in Figure 9.

Figure 9: Influence of age for categories Lage and Hage

The positive correlation between age and LoS is recognized for the DRGs in category Lage but notfor Hage. The latter is not completely unexpected, since category Hage contains the relatively longhospital stays with large variation and therefore already contains the outliers for which influence ofage is less easy verifiable. It is decided to only consider different age categories for the admissionsof DRGs labeled as Lage.

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7.3.5 Performance of the new classification system

The DRGs are now labeled either as S, L, Lage or H based on their occurrence, a combination of themean and variance of LoS and on the influence of age on LoS. In Appendix A, an overview of whichDRG belongs to which category can be found. Each admission of a patient on the ward of InternalMedicine can be placed in either one of these categories, depending on its DRG. Admissions labeledas Lage are subsequently placed in category LA, LB , LC or LD depending on the age category ofthe patient. For the admissions not placed in category S, a division is made between planned andemergency admissions. An overview of all different categories with their description and statisticsis given in Table 6.

Category Description Mean Median Mode Standard Number ofdeviation admissions

S Admissions of DRGs with less 6.56 5 2 5.92 166than 10 registrations in the data set

PL Planned admissions of DRGs with 3.67 3 2 3.38 179relatively low mean and variance of LoS,

for which age has no influence

PLA Planned admissions of DRGs in Lage 3.42 2 2 3.21 43of patients younger than 40 years

PLB Planned admissions of DRGs in Lage 3.33 2 2 3.03 147of patients between 41 and 60 years

PLC Planned admissions of DRGs in Lage 3.95 3 2 3.53 308of patients between 61 and 80 years

PLD Planned admissions of DRGs in Lage 4.90 4 2 5.05 184of patients older than 80 years

PH Planned admissions of DRGs with 5.46 3 2 5.72 543relatively high mean and variance of LoS

EL Planned admissions of DRGs with 4.88 3 2 3.91 427relatively low mean and variance of LoS,

for which age has no significant influence

ELA Emergency admissions of DRGs in Lage 3.19 2 2 2.81 167of patients younger than 40 years

ELB Emergency admissions of DRGs in Lage 4.37 3 2 4.11 342of patients between 41 and 60 years

ELC Emergency admissions of DRGs in Lage 6.24 5 2 5.54 784of patients between 61 and 80 years

ELD Emergency admissions of DRGs in Lage 6.48 6 2 4.90 675of patients older than 80 years

EH Emergency admissions of DRGs with 8.57 7 2 7.33 1401relatively high mean and variance of LoS

Total/Mean 6.12 4 2 5.81 5366

Table 6: Overview of all different categories with their description and statistics

The performance of the statistics per category is evaluated in the same way as how the statistics onthe total data set and the statistics per DRG separately were evaluated. The performance of themean, median and mode per category is given in Table 7. The values of the prediction measures ofthe mean and median are lower when calculated per category rather than per DRG. The mode does

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not show a more accurate prediction, but this statistic does not prove to be an accurate predictorfor LoS at all. Hence, considering the mean as predictor of LoS, the introduced classification systemyields a more accurate prediction of LoS while being simpler and easy to use in practice. Therefore,the goal of the first part of this research is achieved.

MAE MSE (RMSE) MAPE

Mean of the LoS 3.60 28.01 (5.29) 0.60Median of the LoS 3.31 30.08 (5.48) 0.77Mode of the LoS 4.19 47.24 (6.87) 2.10

Table 7: Performance of mean, median and mode for each category as predictors of LoS

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8 Arrival process

For patients arriving with emergency, there is no knowledge beforehand on their moment of arrival.However, it is known from the literature that the arrival process of patients can often be welldescribed by a Poisson distribution [7]. To evaluate the arrival process of patients with emergencyadmissions, the number of emergency admissions on each day of the year is counted. Subsequently,the number of times a certain number of admissions occurs is registered. The count of each numberof admissions per day is given in Table 8.

Admissions per day 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Count 0 23 49 89 115 107 110 86 57 34 26 9 11 3 0 2 1

Table 8: Count of each number of admissions per day

In Figure 10, the frequency data of each number of admissions is graphically shown in a histogram.A Poisson distribution with parameter λ = 5.44 is fitted to this histogram. It can be concludedthat the arrival process can indeed be well described by a Poisson distribution.

Figure 10: Poisson distribution with parameter λ = 5.44 fitted to the data of arrivals of emergencyadmissions

In the first part of this research, a new classification system for admissions on the ward of InternalMedicine is introduced, in order to predict LoS and reduce the variance of LoS within each cate-gory. A logical thought would be to consider each of these categories that are labeled as emergencyadmissions, and analyze the arrival process for each category. However, unfortunately, these arrivalprocesses do not show clear Poisson arrivals probably because not enough data is available within

26

each category. Therefore, the arrivals of emergency cases is considered as one category. The cat-egories defined in the first part of this research will later be used to estimate the LoS of patientsthat are already occupying a bed on the ward.

Instead of analyzing the arrivals of emergency admissions as one category and estimating a singlePoisson parameter, it is also interesting to analyze the arrivals of patients over a week. It turns outthat the arrival rates vary remarkably over the days of the week. Especially in the weekends, thearrival rates turn out to be lower than the arrival rates on working days. The Poisson parameter forthe emergency arrivals during weekdays is given by λweek = 5.89. The parameter for the emergencyarrivals during the weekends is given by λweekend = 4.29. The Poisson parameters of the arrivalprocess for each day of the week are provided in Table 9.

Monday Tuesday Wednesday Thursday Friday Saturday Sunday

λ 5.33 6.03 5.79 5.93 6.39 4.25 4.32

Table 9: Parameters of the Poisson arrival process per day of the week

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9 Analysis of bed occupancy

As described in Section 5, the nursing ward of Internal Medicine in Martini Ziekenhuis has twodepartments 2C and 2D, containing 32 and 30 hospital beds respectively. Due to limitations ofthe data and the data extraction method, it is not known on which of these departments eachhospital stay takes place. Therefore, in this research, the two departments are considered as onelarge department containing 62 hospital beds.

The total number of beds needed according to the registrations of bed days on a nursing ward atsome day can be viewed as the demand for hospital beds. Then, the number of available beds on thatward can be seen as the inventory stock needed in order to supply the demand. Inventory stockingproblems with stochastic demand typically involve an estimate of some fractile of the demanddistribution, where the fractile is usually in the 0.8 - 0.99 range [14]. This fractile is termed the‘service level’. In this research, this service level is defined as the probability of not facing a shortageof beds on an arbitrary day. A histogram of the total bed occupancy per day is plotted in Figure11. The total bed occupancy can be roughly approximated by a Normal distribution with meanµ = 46 and standard deviation σ = 8.

Figure 11: Normal distribution fitted to the histogram of total bed occupancy

Note that this distribution is determined using cold data, based on the integer number of regis-trations of inpatient days. The average factor for the two departments 2C and 2D that MartiniZiekenhuis uses to transfer from cold to warm data, given by r = 1.13, is used in this research asthe factor for the total department. In that respect, the physical bed occupancy can be calculatedby dividing the cold number of occupied beds by this factor r. The number of beds needed in orderto supply the demand is calculated as a fractile of demand distribution divided by r. This will befurther discussed in Section 11.

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10 Bed capacity model

In this research, a model is built that estimates the distribution of the demand for beds for a numberof T days ahead. The model calculates the service level that is achieved for a certain number of bedsas a fractile of this estimated demand distribution. The other way around, the model calculates thenumber of beds needed in order to achieve a certain service level as a fractile of the inverse functionof the estimated demand distribution.

Let the start of the model, the current day, be given by t = 0. Then, the model assumes the numberof beds occupied at day T to be depending on the following occupancies:

– Beds that are already occupied at t = 0 from planned admissions and that are still occupiedat t = T

– Beds that are already occupied at t = 0 from emergency admissions and that are still occupiedat t = T

– Beds occupied by patients with planned admissions at t = 1, ..., T , that are still occupying abed at t = T

– Beds occupied by patients with emergency admissions at t = 1, ..., T , that are still occupyinga bed at t = T

Note that a patient arriving at T is certainly still occupying a bed at T , since LoS in this researchis described by an integer number of days. For each of the four occupancies defined above, themodel calculates the probability that a patient is still occupying a bed at T . The sum of theseprobabilities gives an estimation of the distribution of the number of beds occupied at day T . Theideology of the calculations is partly derived from the model of Utley and Gallivan (2003).

In this research, the LoS of each patient is assumed to be independently and identically distributed.Furthermore, complete knowledge on the schedule of surgical operations during a coming short timeperiod is assumed. The number of planned admissions during period T depends on the schedule ofsurgical operations during that period. In practice, this schedule is often only known for a maxi-mum of two weeks.

In Section 10.1, a simple version of the model is introduced. Then, in Section 10.2, three extensionsof this simple model are discussed, based on the classification system introduced in the first partof this research and the parameters estimated. The extended models are expected to yield a moreaccurate estimate of the bed occupancy distribution. The performances of the simple model andits extensions are discussed in Section 11.

10.1 Simple model

In the simple version of the model, the classification system of admissions introduced in the first partof this research is not used. Admissions are only separated on their urgency. A single parameterfor the arrival process of emergency admissions λ is used. The LoS of patients is described by therandom variable X having a Geometric distribution with parameter qp and qe for the planned andemergency admissions respectively.

29

Already occupied beds from planned admissions

Suppose that the number of beds occupied at day t = 0 from planned admissions is given by cp.Each patient will either be discharged today or it will stay for another day or more. Since theGeometric distribution is memoryless, the remaining LoS for the patients already occupying a bedat t = 0 from planned admissions is also Geometrically distributed with parameter qp. Therefore,the probability of staying another day is given by qp. The probability that a patient occupying abed at t = 0 from a planned admission that is still occupying a bed at day T is then described bya Bernoulli distribution with parameter bp, calculated by

bp = P (X > T ) = 1− P (X ≤ T ) = qTp .

The expected value and variance of the number of beds that is still occupied at day T from plannedadmissions are given by

E[No,p(T )] = cpbp = cpqTp ,

Var[No,p(T )] = cpbp(1− bp) = cpqTp (1− qTp ).

Already occupied beds from emergency admissions

Suppose that the number of beds occupied from emergency admissions at day t = 0 is given by ce.Then, similar to the calculations for beds already occupied from planned admissions, the expectedvalue and the variance of the number of beds that is still occupied at day T from emergencyadmissions are given by

E[No,e(T )] = ceqTe ,

Var[No,e(T )] = ceqTe (1− qTe ).

Planned admissions

During the coming time period of T days, it is assumed to be known what surgical operations areplanned. Hence, it is assumed to be known which patients are admitted on which day. Let thenumber of planned admissions at day t be given by ct, for t = 1, ..., T . In the same way as forpatients already occupying a bed at t = 0, the probability that a patient arriving at day t is stilloccupying a bed at day T is described by a Bernoulli distribution with parameter bt, calculated by

bt = P (X > (T − t)) = 1− P (X ≤ (T − t)) = q(T−t)p . (4)

The total number of beds occupied by planned admissions at day T is calculated by the sum ofpatients arriving during period T and staying until T . Therefore, the expected value and varianceof the number of beds still occupied at T by planned admissions are given by

E[Np(T )] =

T∑t=1

ctbt =

T∑t=1

ctq(T−t)p ,

Var[Np(T )] =

T∑t=1

ctbt(1− bt) =

T∑t=1

ctq(T−t)p (1− q(T−t)

p ).

30

Emergency admissions

The arrival process of emergency admissions is described by a Poisson distribution with parameterλ. As for the planned admissions, the probability of each patient arriving at t still occupying a bedat T is described by a Bernoulli distribution with parameter bt given by

bt = q(T−t)e .

Therefore, the number of beds occupied at day T by patients arriving with emergency during periodT is described by a Compound Poisson distribution of Bernoulli random variables. To calculatethe expected value and variance of this distribution, we condition on the number of emergencyadmissions at day t, for t = 1, ..., T , denoted by nt. The number of beds occupied at T is then givenby the sum over T of nt Bernoulli random variables, and it holds that

E[Ne(T )2 | nt] = Var[Ne(T ) | nt] + E[Ne(T ) | nt]2

=T∑t=1

ntq(T−t)e (1− q(T−t)

e ) +( T∑t=1

ntq(T−t)e

)2. (5)

The expected value and variance of the bed occupancy at day T of emergency patients arrivingwith rate λ is calculated by

E[Ne(T )] =

T∑t=1

λq(T−t)e ,

Var[Ne(T )] = E[Ne(T )2]− (E[Ne(T )])2

= Ent

[E[Ne(T )2 | nt]

]− Ent

[E[Ne(T ) | nt]2

].

Substituting (5) in the latter expression and using the fact that the expected value and variance ofthe number of arrivals at day t according to a Poisson distribution are given by E[nt] = Var[nt] = λ,the variance of the number of beds occupied at T yields

Var[Ne(T )] = Ent

[ T∑t=1

ntq(T−t)e (1− q(T−t)

e ) +( T∑t=1

ntq(T−t)e

)2]− Ent

[( T∑t=1

ntq(T−t)e

)2]= E[nt]

T∑t=1

q(T−t)e (1− q(T−t)

e ) + E[n2t ]( T∑t=1

q(T−t)e

)2− E[nt]

2( T∑t=1

q(T−t)e

)2= E[nt]

T∑t=1

q(T−t)e (1− q(T−t)

e ) +(

Var[nt] + E[nt]2)( T∑

t=1

q(T−t)e

)2− E[nt]

2( T∑t=1

q(T−t)e

)2= E[nt]

T∑t=1

q(T−t)e (1− q(T−t)

e ) + Var[nt]( T∑t=1

q(T−t)e

)2= λ

T∑t=1

q(T−t)e (1− q(T−t)

e ) + λ( T∑t=1

q(T−t)e

)2= λ

T∑t=1

q(T−t)e

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Hence, the expected value and variance of Ne(T ) are both given by the sum of the arrival rateλ times the probability of staying until day T . This is an obvious result, since each arrival of apatient at day t can be interpreted as rolling a dice for this patient either staying until T or beingdischarged before that day. Hence, the total number of beds occupied at T can be interpreted as a

sum of Poisson arrival processes with rate λq(T−t)e , which have expected value and variance equal

to λq(T−t)e , leading to the expectation and variance given by

E[Ne(T )] = Var[Ne(T )] =

T∑t=1

λq(T−t)e .

Distribution of total bed occupancy

The total expected value and variance of the number of beds occupied at day T is obtained byadding the expected values and variances of all already occupying patients and the planned andemergency admissions. Suppose there are cp and ce beds already occupied at day t = 0 fromplanned and emergency admissions respectively, and the number of planned admissions at time t isct, for t = 1, ..., T . Then, the expected value and variance of the number of occupied beds at dayT are given by

E[N(T )] = E[No,p(T )] + E[No,e(T )] + E[Np(T )] + E[Ne(T )]

= cpqTp + ceq

Te +

T∑t=1

ctq(T−t)p +

T∑t=1

λq(T−t)e (6)

Var[N(T )] = Var[No,p(T )] + Var[No,e(T )] + Var[Np(T )] + Var[Ne(T )]

= cpqTp (1− qTp ) + ceq

Te (1− qTe ) +

T∑t=1

ctq(T−t)p (1− q(T−t)

p ) +

T∑t=1

λq(T−t)e . (7)

10.2 Model extensions

In the simple model, the expected value and variance of the distribution of bed occupancy at timeT are calculated where only a distinction is made between planned and emergency admissions. Inthis section, three extensions of the simple model are discussed.

Model with LoS per category

In the first part of this research, a new classification system for admissions on the nursing ward ofInternal Medicine in Martini Ziekenhuis is introduced, based on urgency and DRG of the admissionand age of the patient. It is expected that, using this classification system rather than only distin-guishing between planned and emergency admissions, the model yields a more accurate predictionfor LoS. For each category i in the set of categories defined in Table 6, the parameter qi of stayinganother day is calculated as

qi = 1− 1

µi.

For the emergency admissions, the parameter qe is still used since it is not possible to assign anemergency admission to a category in advance. Assuming there are c0 beds already occupied at

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t = 0 and the number of planned admission at time t is given by ct for t = 1, ..., T , the expectedvalue and variance of the number of beds occupied at time T are given by

E[N(T )] = c0qTi +

T∑t=1

ctq(T−t)i +

T∑t=1

λq(T−t)e (8)

Var[N(T )] = c0qTi (1− qTi ) +

T∑t=1

ctq(T−t)i (1− q(T−t)

i ) +

T∑t=1

λq(T−t)e , (9)

where i is in the set of categories defined in Table 6.

Model with LoS per category and arrivals for week and weekend days

In Section 8, it is concluded that the difference between the arrival rates of emergency admissionsduring week days and weekend days is considerable. Therefore, the model can be extended by usingthe separate Poisson arrival parameters λweek and λweekend. The expected value and variance arecalculated according to (8) and (9), where the last terms are split up based on these two arrivalparameters. The arrivals of emergency admissions are expected to be more accurately predictedwhen using separate parameters for week and weekend days rather than a single arrival parameter.Therefore, this extension of the model is expected to outperform the simple model.

Model with LoS per category and arrivals per day of the week

The last discussed model can be further extended by estimating separate arrival parameters foremergency admissions for each day of the week, as given in Table 9. One could expect this model toyield an even more accurate prediction of the demand distribution. However, note that the modeloperates as if all parameters are perfectly describing the distributions of LoS and arrivals, but eachparameter estimation brings some estimation uncertainty. The performance of the simple modeland its three extensions are discussed in the next section.

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11 Empirical results

The accuracy of the model and its extensions, estimated using data from 2017 and 2018, is tested forpredicting the bed occupancy of 2019. The month January is used as a ‘warm up’ period. Duringthis month, the number of patients that are already occupying a bed is assumed to stabilize. Dataup to and including November is used in order to be able to look forward over the month December.For each day during this period, the distribution of bed occupancy when looking T days ahead isestimated. This estimated distribution is tested for both predicting the actual bed occupancy aspredicting the service level that a number of beds can guarantee.

The prediction performance of the simple models and its extensions is evaluated using the earlierdiscussed prediction measures MAE, MSE and MAPE. The prediction measures for these modelsare graphically shown in Figure 12.

Figure 12: Performance measures of the simple model and its extensions

It can be observed that the model, with separate LoS parameters per category based on the urgencyand DRG of the admission and age of the patient, as defined in the first part of this research, andseparate arrivals for week and weekend days, gives the highest accuracy. This model outperforms

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the simple model and the model with only separate LoS parameter per category, which is as ex-pected. However, the model with LoS parameters per category and arrival parameters per day ofthe week does not show better performance than the other models. This can be explained by thefact that estimating a separate parameter for each day of the week results in an uncertainty that istoo high to yield a more accurate prediction. Therefore, the model with categories and arrivals forweek and weekend days will be used in the remaining of this research, and is, from now on, referredto as the final model.

Figure 13 shows the expected values and the average bed occupancy for the already occupied beds,planned and emergency admissions for the final model. It also shows the total average predictionand actual occupancy.

Figure 13: Actual average bed occupancy and prediction of the final model

It can be seen that the final model still shows a small deviation in the prediction of total bed occu-pancy, mainly caused by the average underprediction of the occupancy by emergency admissions.This small deviation can be explained by uncertainty in parameter estimation. The parameters ofthe final model are estimated over the total data set of 2017 and 2018, while the model is evaluatedover the months February up to and including November from 2019, looking 30 days ahead. Themodel assumes the estimated parameters to be completely describing the distribution of arrivalsand LoS. Possible outliers might not be covered in these estimated parameters. Table 11 in Ap-pendix A indeed shows a slight difference in the parameters estimated over the different data sets.However, the total prediction differs from the actual occupancy with only 1 or 2 beds. Therefore,the model is considered to be predicting the short term bed occupancy well.

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Normal distribution of the estimated demand function

The model estimates the mean and variance of the distribution of bed occupancy when looking Tdays ahead by adding the expected values and variances of the bed occupancies by already occupiedbeds, planned and emergency admissions. The occupancy at day T by already occupied beds, fromboth planned and emergency admissions, is a sum of Bernoulli distributions. By the Central LimitTheorem, the sum of independent Bernoulli distributions is approximated by a Normal distribution.The occupancy by the planned admissions is calculated by a sum of a sum of independent Bernoullidistributions, which is therefore also Normal by approximation according to the Central Limit The-orem. The random variable of the number of beds occupied at T by the emergency admissions isdescribed by a Compound Poisson distribution, which is also Normal by approximation.

Thus, the total number of beds occupied at day T , given by N(T ), is a sum of random variables thatare all assumed to be Normally distributed. By the Central Limit Theorem as well as by summingrandom variables that are Normally distributed, N(T ) is assumed to have a Normal distribution.This assumption is in line with the paper of Utley and Gallivan (2003), which proves that the bedoccupancy distribution can easily be approximated by a Normal distribution [22].

To validate this assumption, the model is tested for predicting several service levels. Let the cumu-lative distribution of N(T ) be given by Φ. Then, given an intended service level of X, the numberof beds needed in order to achieve this service level is calculated as a fractile of the cumulativedemand distribution, given by

k = Φ−1(X).

Figure 14 shows the percentage of times that a service level is achieved when looking T days ahead,for several service levels. The distribution of N(T ) would be perfectly Normal if the percentage oftimes that a service level is achieved exactly coincides with the intended service level.

It can be observed that, after a small dip, the lines horizontally fluctuate at a level that is a bitlower than the intended service level. This is assumed to be caused by prediction errors of themodel, due to parameter estimation. To verify whether this assumption is correct, the model isperformed on the data of 2017 and 2018, over which the parameters are estimated. These resultsare shown in Figure 15. Now, the levels almost coincide with their intended service level. Hence,for 2017 and 2018, the distribution does approximately fit a Normal distribution. Therefore, thelower levels in Figure 14 can be explained by prediction errors of the model caused by parameterestimation.

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Figure 14: Probabilities of obtaining service levels using the final model over 2019

Figure 15: Probabilities of obtaining several service levels using the final model over 2017 and 2018

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The small dip of the lines in the beginning of both Figure 14 and 15 is probably caused by thepatients already occupying a bed at day t = 0. For these patients, a parameter for the Geometricdistribution of LoS is estimated over the whole data set. However, a patient that is already occu-pying a bed has a relatively higher probability of being a ‘difficult’ patient needing more hospitalcare than an average patient. This might cause the total occupancy by patients already occupyinga bed at t = 0 being larger than estimated by the parameters, causing a realized lower service levelthan intended. In Section 14, this possible drawback of the model is further discussed and relatedto a suggestion for further research.

In order to account for this dip and the fact that the service levels that are actually achieved arelower than intended, a hospital can decide to use adjusted fractiles for calculating service levels. Itshould be noted that prediction errors will always occur when using the model over another dataset. It is difficult to completely predict the effects of these prediction errors. Ideally, this effectshould be measured continuously and corrected for when using the model in practice.

Based on the prediction errors faced in this research, a suggestion for the adjusted fractiles for thenursing ward of Internal Medicine of Martini Ziekenhuis is made. The suggested adjusted servicelevels for some service levels mostly used in practice are given in Table 10.

Intended service Adjusted service level Qlevel X 1− 4 days ahead 5− 15 days ahead > 15 days ahead

0.98 0.997 0.995 0.9960.95 0.980 0.970 0.9700.90 0.960 0.940 0.9500.85 0.920 0.900 0.9200.80 0.880 0.840 0.870

Table 10: Suggested adjusted fractiles for some intended service levels, mostly used in practice

If the intended service level is X and the corresponding adjusted service level when looking T daysahead is given by Q, the number of beds needed in order to achieve the intended service level X iscalculated by

k =Φ−1(Q)

r,

where Φ is the cumulative distribution function of the number of beds estimated by the model andr is the factor to calculate from cold to warm data. The other way around, given a number ofavailable beds c, the adjusted service level when looking a certain number of days ahead can becalculated by

Q = Φ(rc).

Then, the actual service level can be approximated by interpolating the values in Table 10. In thenext section, the final model and the use of these adjusted service level are applied to a practicalexample.

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12 Practical application

In this section, the final model defined in the previous sections of this research is applied to aspecific practical example. The date March 10, 2019 is randomly drawn from the test data set andis used as the start of the model. It is assumed that at this date, so at day t = 0, the number ofavailable beds at the nursing ward of Internal Medicine equals c = 62.

At the start of the model, the total bed occupancy equals 48. According to the classification systemintroduced in the first part of this research, each admission can be placed in a category, based onthe urgency, DRG and age of the patient. For each of these categories, LoS is estimated by aGeometric distribution.

Suppose that the schedule for all planned admissions is known for the coming 2 weeks. Then, themodel can be used to estimate the distribution of the number of beds occupied when looking T daysahead. The expected value of this distribution and its 95% confidence range is shown in Figure 16.

Figure 16: Actual bed occupancy and the expected value with 95% confidence range

Figure 16 shows a peak in the expected occupancy where more than 62 beds are needed to achievea 0.95 service level. The staff of the ward could anticipate on this for example by, if possible,placing an extra bed on the ward or by deciding to transfer a patient that is already occupying abed to another nursing ward where enough beds are available. However, Figure 16 also shows thatthe actual bed occupancy never exceeded the currently available number of beds. Hence, settinga service level that is too high may be risky in the sense that it may lead to too unnecessary bedmovements.

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13 Conclusion

In the first part of this research, the current bed capacity of the nursing ward of Internal Medicineof Martini Ziekenhuis is analyzed. The largest part of the stays on this ward is caused by emer-gency admissions and LoS is spread over a wide range of values. A Geometric distribution isused to describe LoS, because of its discreteness and memoryless property. Factors that influenceLoS are analyzed and it turns out that urgency, diagnose and age are positively correlated with LoS.

A new classification system is introduced for admissions on the nursing ward of Internal Medicineof Martini Ziekenhuis. Each admission can be placed in a category based on its urgency, the DRGcorresponding to the admission, and the age of the patient. The performance of predicting LoSusing the mean, median and mode is evaluated. Compared to the median and mode, the meanyields most accurate predictions. Calculating the statistics per category yields a more accurateprediction than calculating the statistics over all admissions or over emergency and planned ad-missions separately. The variance of LoS within these categories is reduced, while the classificationsystem is simple and easy to use in practice.

In the second part of this research, a model is built that estimates the distribution of the numberof hospital beds when looking a number of T days ahead. For patients that are already occupyinga bed at the start of the model and for patients that arrive during period T , the probability ofstill occupying a bed at day T is calculated. A simple version of the model estimates a single LoSparameter for planned and emergency admissions and a single arrival parameter. Three extensionsof the model are discussed, based on the classification system introduced in the first part of thisresearch and the arrival parameters discussed in Section 8. The extension of the model that esti-mates LoS parameters per category and arrival parameters for week and weekend days proves topredict the bed occupancy most accurately.

The model assumes the estimated number of beds needed when looking T days ahead to be Normallydistributed. This assumption is verified over the data set of 2017 and 2018. Over the test data setof 2019, the distribution does not completely fit the Normal distribution. However, this is explainedby prediction errors caused by parameter estimation. Overall, the model is able to predict the shortterm bed occupancy for the nursing ward of Internal Medicine of Martini Ziekenhuis well.

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14 Restrictions and further research

The first part of this research focuses on categorizing admissions, mainly based on the internationalDRG system. Martini Ziekenhuis prefers the use of this system for their internal analyses. Hence,for this specific research about bed capacity at the nursing ward of Internal Medicine of MartiniZiekenhuis, the use of this system is most evident. However, a large part of the other Dutch hos-pitals does not use the DRG system. Therefore, LOGEX eventually wants to get rid of the use ofDRGs and prefers the use of the CSS system and its own interpretation of it.

The data used in this research is in some respects limited to the availability of information in theLOGEX database. Because the data base only consists of financially registered data, it is onlypossible to extract cold LoS data. Although this limitation is accounted for by using the factor rto calculate from cold to warm data, it still brings some uncertainty and immediately working withwarm data could give more clarifying results. However, not all hospitals keep track of warm data.Recording cold data is easier and obligated for hospitals, which makes the model introduced in thesecond part of this research broadly applicable.

The analysis of the factors that might influence LoS in this research is limited to availability ofdata. Comorbidity of patients, which is expected to be correlated with the duration of hospitalstays, is not available for each hospital stay in the data base. Another factor that might possiblyaffect LoS is the day of the week of the admission or discharge. For example, patients that areadmitted on Friday might not be discharged until Monday due to the lack of staff qualified for dis-charge decision during the weekends. The latter is not taken into account during this research dueto its complexity and the lack of available time. In further research, the analysis could be extendedwith these or even more factors, which could in place improve the introduced categorization system.

To determine the influence of age for each DRG in Section 7, a Negative Binomial regression is used.This regression also predicts zeros, which is not in line with the LoS data used in this research whichonly takes positive integers. Therefore, this regression could be transformed to a zero-truncatedmodel to make it more reliable. However, for the purpose of the regression in this research, the useof the Negative Binomial regression was sufficient.

The last and theoretically most interesting restriction and suggestion for further research followsfrom Section 11. Parameter estimation in this research is based on averages of historical data. Themodel assumes the estimated parameters to be fixed and correctly describing the distributions.However, especially for the LoS of patients that are already occupying a bed, it could be interestingto not only estimate a value for the parameters but also their distribution. This allows parametersto be varying, which could account for the fact that patients that are already occupying a bed havea relatively higher probability of staying longer than a patient just arriving on a ward. Further-more, a machine learning tool could be developed to calculate the probability that patients alreadyoccupying a bed at the start of the model will still be occupying a bed after T days. However, itshould be noted that such tool might need a lot of data data which makes it less achievable andnot relevant in practice.

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Appendix A

Description Parameter Data set2017-2018 2017-2018 (Feb-Nov) 2019 2019 (Feb-Nov)

Poisson arrival parameter λ 7.371 7.338 7.193 7.264λem 5.442 5.426 5.349 5.518λplan 2.490 2.473 2.688 2.648

Poisson arrival parameterper day of the week λmon 5.327 - 5.846 -

λtue 6.029 - 5.731 -λwed 5.789 - 5.846 -λthu 5.933 - 5.962 -λfri 6.394 - 6.077 -λsat 4.248 - 4.119 -λsun 4.325 - 3.780 -

Poisson arrival parameterper week or weekend day

λweek 5.894 5.842 5.892 6.056λweekend 4.286 4.350 3.950 4.132

Mean value of LoSµ 6.123 6.115 6.056 6.002µem 6.685 6.660 6.690 6.606µplan 4.585 4.595 4.254 4.118

Standard deviation of LoSσ 5.807 5.809 5.550 5.382σem 6.061 6.040 5.659 5.588σplan 4.720 4.793 4.794 4.157

Probability of stayingfor another day q 0.837 0.836 0.835 0.833

qem 0.850 0.850 0.851 0.849qplan 0.782 0.782 0.765 0.757

Table 11: Overview of all parameters

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Appendix B

DRG code Description

Category S B15-B19 Hepatitis viraalC22 Maligne neoplasma van leverC30-C39 Maligne neoplasma intrathoracaal overigC43-C44 Maligne neoplasma huidC45-C49 Maligne neoplasma weke delenC54-1 Maligne neoplasma van endometriumC57 Maligne neoplasma vrouwelijke geslachtsorganen excl. endometrium,

ovarium en tubaD34-D35 Benigne neoplasma schildklier/endocrienD86 SarcoıdoseE00-E04 HypothyreoıdieE05 HyperthyreoıdieE65-E68 ObesitasF03 DementieG00-G09 Meningitis/encefalitisG45 TIAH43-H44 Aandoening corpus vitreum/oogbolI20 Coronaire (ischemische) hartziekten-Angina pectorisI48 Hartritmestoornissen - AtriumfibrillerenI72 Aneurysmata excl. aortaI77 Aandoening arterien tot capillairen overigI87-I89 Aandoening venen/lymfevaten/-klieren overigJ94 Aandoening pleuraK21 Gastro-oesofageale refluxK29 GastritisK30 DyspepsieK50-K51 Niet-infectieuze enteritis en colitisK52 Overige niet-infectieuze gastro-enteritis en colitisK58 Prikkelbare darmsyndroomK75 Hepatitis niet viraalM01 Reumatoide artritis (RA)M07 Psoriatic arthritis (PsA)M25 Overige gewrichtsaandoeningen, niet elders geclassificeerdM32 Lupus erythematodes disseminatus [LED]M86-M94 Overige osteopathieen en chondropathieenN20-N23 UrolithiasisN30-N39 Overige aandoening urinewegen/prostaatR25-R29 Symptomen betreffende zenuwstelsel en botspierstelselZ13 Specifiek screeningsonderzoek

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DRG code Description

Category L C16 Maligne neoplasma van maagD12-D13 Benigne neoplasma spijsverteringsstelselD50-D61 Anemie en ziekte bloed/-vormende organenD65-D69 Stollingst/purpura/hemorr.aand/immunsystE70-E90 Stofwisselingsstoornissen-overigF05 DeliriumI26-I28 Pulmonale hartziekten en longcirculatieI77-6 Arteriitis temporalisJ45-J46 AstmaJ95-J99 Overige aandoeningen ademhalingstelselK22-K23 Aandoening slokdarm/maag/duodenum overigK56 DarmobstructieK80-K87 Aandoening galblaas/galwegen/pancreasR06-0 DyspnoeR07 Pijn in keel en borst

Category Lage ALLG-01 Allergie overigB99-B99 Overige infectieziekten, b.v. Lyme (excl. HIV en virale hepatitis)C18-C21 Maligne neoplasma van dikke darm (incl rectum en anus)C23-C25 Maligne neoplasma pancreas/galblaas/galwegenC50 Maligne neoplasma mammaE10-E13 Diabetes mellitusF10-F19 VerslavingF50 OndervoedingI10-I15 Hypertensieve aandoeningI30-I52 Hartaandoening overigI82 TromboseJ09-J18 LongontstekingK80-K81 Galsteenlijden en galblaasontstekingK90-K93 Aandoening spijsverteringsstelsel overigKGER-01 Klinische geriatrie overigL00-L08 Infectie huidN00-N08 Aandoening glomeruli en overig nier/uretherN39-0 Acute urineweginfectiesR10-R19 Algemene symptomen buikR50-R69 Algemene symptomenX40-X49 Intoxicatie

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DRG code Description

Category H A30-A49 SepsisC00-C75 Maligne neoplasma overigC15 Maligne neoplasma slokdarmC34 Maligne neoplasma longC56 Maligne neoplasma van ovarium en tubaC61 Maligne neoplasma prostaatC64-C66 Maligne neoplasma urinewegenC68 Maligne neoplasma nierC81-C96 Maligne neoplasma lymfoid en bloedvormend weefselE20-E35 Aandoening endocriene klieren overig (excl. schildklier en diabetes

mellitus)K25-K28 Zweren maag en twaalfvingerige darmK57 DivertikelziekteK65 PeritonitisK74 LevercirroseK76 Leveraandoeningen (excl. cirrose, hepatitis)KGER-02 Klinische geriatrie multipele orgaanstoornissenM05-M14 Artritis overigM30-M36 Bindweefsel - overige systeemziektenM46 Overige inflammatoire spondylopathieenM80-M85 OsteoporoseM95-M99 Overige aandoeningen botspierstelsel/bindweefselN17-N19 Nierinsufficientie

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