Reducing access and waiting time for orthopedics

Reducing access and waiting time for orthopedics

Deventer ziekenhuis

Master Thesis in

Industrial Engineering and Management

ARJAN PANNEKOEK

Supervisors University

Dr.ir. A.G. Leeftink

Prof.dr.ir. E.W. Hans

By order of

S. Koemans

M. Brilleman

Deventer Ziekenhuis

Management summary

This research focuses on the reduction of the access and waiting time for the orthopedic department of Deventer Ziekenhuis (DZ).

Problem description

The department experiences seasonality, which leads to a stressful period in the second half of the year. In this period, secretaries indicate that it is difficult to find empty patient slots within reasonable time. The planners do not know if they should plan outpatient clinic (OC) blocks or Operating Room (OR) blocks and the doctors feel that they work overcrowded and inefficient sessions. The department also feels that there is sometimes a mismatch between their doctor capacity and the capacity of the OR department.

The department uses a static allocation of blocks over the year and the predefined OC blocks do not reflect current demand. As the planning horizon decreases, the department tries to control the access and waiting time by (1) switching between OC and OR blocks and (2) by changing the type of patient slots in OC blocks to fulfill demand. This directly affects access and waiting times, however the indirect effects are unknown and the procedures introduce variability in the flow of patients within the department.

Approach

We want to obtain practical plannings rules to allocate blocks over the available weekly doctor capacity. To obtain the end result we divide our approach into three steps. First, we generate new OC blocks for each doctor that reflect patient demand. Second, we use a Mixed Integer Linear Programming (MIP) model based on the studies of Hulshof et al.

[2013] and Nguyen et al. [2015]. The MIP model is used as simulation-optimization approach to allocate blocks such that the weighted number of waiting patients is minimized. We an- alyze the outcomes of the MIP model to formulate practical planning rules that indicate how blocks should be allocated over the weekly available doctor capacity and to indicate a capacity mismatch between the orthopedic capacity and the OR capacity. In the third step, we evaluate the performance of our planning rules by performing a Discrete Event Simulation (DES) model.

Results

Our MIP model provides a quantitative substantiation for the feeling that there is sometimes a mismatch between the doctor and OR capacities. The cause is that some doctors live in another region of the Netherlands where holidays are timed differently and therefore we recommend the department to align capacities on forehand since this positively influences the access and/or waiting time.

We formulate practical planning rules how the department should handle in case of (1) new

patients arrivals, (2) holiday season and (3) non-holiday season. We conclude that the model

does not heavily react to new patient arrivals and therefore we advice the department to

introduce our new OC blocks and to limit changing the type of patients slots of OC blocks

to a minimum. We recommend to update the OC blocks yearly with the use of the created

Excel tool. We formulate the planning rules in days (1 day contains 2 blocks) since this is

appropriate for practice and they are included below.

1. Planning rules for OC per doctor:

(a) Plan a minimum of 1 OC day every week.

(b) Plan a maximum of 3 OC days up to 1 week in a row.

(c) Plan a maximum of 3 OC days up to 2 weeks in a row if these weeks are just after a holiday of minimal 3 weeks.

(d) Never plan 3 OC days in 3 consecutive weeks.

2. Planning rules for OR per doctor:

(a) Plan a minimum of 1 OR day if the weekly doctor capacity u s,t = 4.

(b) Plan a maximum of 3 OR days if the weekly doctor capacity u s,t = 4, but this week may not be just before or after a holiday of minimal 3 weeks.

(c) Never plan 3 OR days if the weekly doctor capacity u s,t = 3.

3. Planning rules for holidays per doctor:

(a) Plan a minimum of 1 OR day and 1 OC day in the week before a holiday.

(b) Plan a minimum of 1 OR day and 1 OC day two weeks and one week before a holiday of minimal 2 weeks.

(c) Allocate more than 50% of the available weekly doctor capacity u s,t to OC days the week after a holiday of minimal 2 weeks.

We use the DES model to obtain the performance of the planning rules while not every detail of the department is incorporated. The outcomes of the DES model show a more stable access time (σ -10%), a more stable waiting time (σ -13%) and a more stable work- load for the OC (σ -3%). The planning rules are substantiated, easy to implement and they ensure that blocks can be divided without the use of experience and/or feelings. Because the differences between the performance indicators for the current and suggested situation are positive, we advise the department to use them.

Side project

Currently, the flow of patients towards the ward is not incorporated. The objective of the side-project is to minimize the variation in the number of used beds in the ward. We have developed and introduced an Excel spreadsheet that allocates OR blocks for every day of the week while incorporating the objective. Besides the spreadsheet, we formulate doctor and injury type specific planning rules to give more direction to the flow of patients towards the ward.

Contribution to practice

This research contributes to practice since we generate new OC blocks and practical plan-

ning rules that are already in use practice. The developed and introduced Excel spreadsheet

regarding the flow of patients towards the ward is experienced as helpful and the side-project

as potential for further research. We contribute also to practice since we extend the Integral

Capacity Management (ICM) support within DZ.

Contribution to theory

This research tackles a resource scheduling problem for elective care where patients need multiple appointments at multiple resources. By combining and expanding the models of Hulshof et al. and Nguyen et al. we generate a model that allocates blocks over the avail- able doctor capacity. The planning rules that we formulate have been tested in a real life case by using a simulation model. The results are promising, and the research is interesting for surgical specialties that need to allocate capacity to minimize the weighted number of waiting patients.

The side-project contributes to theory since we created a self made Excel spreadsheet model that allocates OR blocks such that it minimizes the variability in the used beds in the ward.

The model is general applicable, incorporates real life constraints and is interesting for sur-

gical specialties where patients require a bed at the ward.

Management samenvatting

Dit onderzoek focust zich op het verlagen van de toegangs- en wachttijd voor de afdeling orthopedie in het Deventer ziekenhuis (DZ).

Probleemomschrijving

De afdeling orthopedie ervaart seizoenspatronen en dit resulteert in een stressvolle periode in de tweede helft van het jaar. In deze periode vinden secretaresses het moeilijk om vrije patiëntplekken te vinden binnen een aanzienlijk tijdsbestek. De planners weten niet of ze blokken moeten inplannen voor de polikliniek (poli) of de operatiekamer (OK) en de artsen hebben het gevoel dat ze overvolle en inefficiënte sessies werken. Tevens heeft de afdeling het gevoel dat er een mismatch aanwezig is tussen de arts capaciteit en de OK capaciteit.

De afdeling gebruikt een statische allocatie van blokken (poli en OK) over het jaar en de huidige poliblokken reflecteren niet de huidige patiëntvraag. Wanneer de planningshorizon verkleint probeert de afdeling de toegangs- en wachttijd te beïnvloeden met (1) het wis- selen tussen poli en OK blokken en (2) het omboeken van de types van patiëntsloten in poli blokken om te voldoen aan de patiëntvraag. De directe effecten van deze twee procedures zijn positief voor de toegangs- en wachttijd, echter zijn de indirecte effecten onbekend en introduceren de procedures variabiliteit in de patiëntenstroom.

Probleemaanpak

We willen praktische planningsregels verkrijgen voor het alloceren van blokken over de we- kelijks beschikbare arts capaciteit. De aanpak hiervoor is verdeeld in drie stappen. Allereerst herzien we de poli blokken voor elke arts zodat deze voldoen aan de patiëntvraag. Ten tweede gebruiken we een Mixed Integer Linear Programming (MIP) model gebaseerd op studies van Hulshof et al. [2013] en Nguyen et al. [2015]. We gebruiken het MIP model als simulatie optimalisatie aanpak waarbij het model blokken alloceert zodat het gewogen aantal wach- tende patiënten wordt geminimaliseerd. We analyseren de uitkomsten van het MIP model zodat we praktische planningsregels kunnen formuleren die aangeven hoe blokken gealloceert moeten worden over de wekelijks beschikbare capaciteit en we willen een mismatch tussen de arts en OK capaciteit kwantificeren. In de derde stap maken we een Discrete Event Simulation (DES) model waarmee we de performance van onze planningsregels evalueren.

Resultaten

Door het gebruik van het MIP model hebben we een kwantitatieve onderbouwing verkre- gen voor het gevoel van de aanwezigheid van een mismatch in capaciteiten. De oorzaak hiervoor is dat sommige artsen in een andere regio van Nederland wonen waar vakanties anders getimed zijn. We bevelen de afdeling aan de arts en OK capaciteit op voorhand af te stemmen aangezien dit een positief effect heeft op de toegangs- en/of wachttijd.

We formuleren planningsregels hoe gehandeld moet worden in het geval van (1) nieuwe

patiënt aankomsten, (2) in vakantieseizoen en in (3) niet-vakantieseizoen. We concluderen

dat het model niet abrupt reageert op nieuwe patiënt aankomsten en daarom adviseren wij

de nieuwe poli blokken in gebruik te nemen en het omboeken van patiëntsloten van poli

blokken te beperken tot een minimum. We adviseren de poli blokken jaarlijks te updaten

met het gebruik van de hiervoor gemaakte Excel tool. We formuleren de planningsregels in

dagen (1 dag bevat 2 blokken) omdat dit bruikbaar is voor de praktijk.

1. Planningsregels voor poli per arts:

(a) Plan minimaal 1 poli dag per week.

(b) Plan maximaal één week achter elkaar 3 OK dagen in.

(c) Plan maximaal twee weken achter elkaar 3 OK dagen in wanneer deze weken direct na een vakantie van minimaal 3 weken vallen.

(d) Plan nooit 3 weken achter elkaar 3 OK dagen in.

2. Planningsregel voor OK per arts:

(a) Plan minimaal 1 OK dag als de wekelijkse arts capaciteit u s,t = 4.

(b) Plan maximaal 3 OK dagen als de wekelijkse arts capaciteit u s,t = 4, maar deze week mag niet voor of na een vakantie van minimaal 3 weken zijn.

(c) Plan nooit 3 OK dagen wanneer de wekelijkse arts capaciteit u s,t = 3.

3. Planningregels voor vakanties per arts:

(a) Plan minimaal 1 OK dag en 1 poli dag in de week voor een vakantie.

(b) Plan minimaal 1 OK dag en 1 poli dag twee weken en één week voor de vakantie wanneer de vakantie minimaal 2 weken duurt.

(c) Alloceer meer dan 50% van de beschikbare wekelijkse arts capaciteit u s,t aan polidagen wanneer deze week valt na een vakantie van minimaal 2 weken.

We gebruiken het DES model voor het verkrijgen van de performance van de planningsregel waarbij niet elk detail van de afdeling is meegenomen. De uitkomsten van het DES model laten een stabielere toegangstijd (σ -10%), een stabielere wachttijd (σ -13%) en een sta- bielere werkdruk voor de poli (σ -3%) zien. De planningsregels zijn onderbouwt, makkelijk te implementeren en garanderen dat blokken verdeelt kunnen worden zonder het gebruik van ervaring en/of gevoelens. Omdat de verschillen tussen de performance indicatoren van de huidige situatie en de voorgestelde situatie positief zijn, adviseren wij de afdeling de planningsregels te hanteren.

Side project

In de praktijk wordt de flow van patiënten richting de kliniek niet in acht genomen. De doelstelling van het side project is het minimaliseren van de variabilteit in het aantal ge- bruikte bedden gedurende de week. We introduceren een Excel spreadsheet die OK blokken alloceert voor elke dag van de week. Naast de Excel spreadsheet formuleren we arts en letsel afhankelijke planningsregels die meer sturing geven aan de flow van patiënten.

Bijdragen aan de praktijk

Dit onderzoek draagt bij aan de praktijk gezien het feit dat de nieuwe poli blokken en de

planningsregels al in gebruik zijn in de praktijk. De Excel spreadsheet voor het plannen van

OK blokken wordt ervaren als een handig hulpmiddel dat in gebruik is en het side project

als potentieel voor vervolgonderzoek. We dragen ook bij aan de praktijk doordat we meer

draagvlak creëren voor het integraal capaciteitsmanagement binnen DZ.

Bijdragen aan de theorie

Dit onderzoek richt zich op een plannings probleem van resources voor electieve zorg waar- bij patiënten meerdere afspraken bij meerdere resources hebben. Door het combineren en uitbreiden van de modellen van Hulshof et al. [2013] en Nguyen et al. [2015] hebben we een model gecreëerd dat blokken alloceert over de beschikbare arts capaciteit. De plan- ningsregels die hieruit voort komen zijn getest in een real life case door het gebruik van een DES model. De resultaten zijn veelbelovend en het onderzoek is interessant voor snijdende specialismen die capaciteit moeten alloceren om op deze wijze het gewogen aantal wachtende patiënten te minimaliseren.

Het side-project draagt bij aan de theorie doordat we een eigengemaakt Excel spreadsheet

model hebben gecreëerd die OK blokken alloceert waarmee de variabiliteit in het aantal

bezette bedden in de kliniek geminimaliseerd kan worden. Het model is algemeen toepas-

baar, neemt real life restricties in acht en is interessant voor snijdende specialismen waarbij

patiënten een bed in de kliniek vereisen.

Preface

Now that the complete report is in front of me, this brings me a feeling of satisfaction and I look back on a challenging and complicated research. I have learned that the best idea is not always the right one to present and I have gained experience in translating practical problems into mathematical models.

I would like to thank Gréanne Leeftink not only for guiding the research but also for her enthusiasm and critical feedback which have lifted me to a higher level. I also want to thank Erwin Hans as a discussion partner who gave the research a good direction. Within Deventer hospital there are multiple persons that I want to thank. At first, Machteld Brille- man. Machteld gave me the opportunity to perform the research within DZ and I want to thank her for the pleasant cooperation and the good coordination. Saskia Koemans, I want to thank you for your support, pleasant cooperation and positive feedback regarding the research. Vanessa Souilljee, thanks for your enthusiasm, studious attitude and pleasant cooperation. Besides Deventer ziekenhuis, I want to thank Benjamin Lubach as colleague and friend during the research. Benjamin, thanks for the good and pleasant collaboration.

Arjan Pannekoek

Epe, July 2018

Contents

Management summary i

Management samenvatting iv

Preface vii

1 Introduction 1

1.1 Background . . . . 1

1.2 Problem description . . . . 1

1.3 Research goal . . . . 3

1.4 Research questions . . . . 3

2 Current Situation 4 2.1 Process description . . . . 4

2.2 Resource planning . . . . 5

2.3 Patient planning . . . . 6

2.4 New patients arrivals . . . . 7

2.5 Production . . . . 8

2.6 Capacities . . . . 9

2.7 Access and waiting time . . . 10

2.8 Doctor analysis . . . 11

2.9 Conclusion . . . 14

3 Literature review 15 3.1 Queuing theory . . . 15

3.2 Research positioning . . . 15

3.3 Methods . . . 16

3.4 Studies . . . 17

3.5 Conclusion . . . 19

4 Solution design 20 4.1 Conceptual model . . . 20

4.2 Data gathering . . . 21

4.3 Technical model . . . 21

4.4 Modeling approach . . . 25

4.5 Validation . . . 26

4.6 Conclusion . . . 27

5 Experiment results 28 5.1 Experiment approach . . . 28

5.2 Experiment setup . . . 28

5.3 Experiment outcomes . . . 29

5.4 Simulation model . . . 35

5.5 Weekly planning . . . 39

5.6 Conclusion . . . 41

6 Conclusion and recommendations 42

6.1 Conclusion . . . 42

6.2 Contributions . . . 44

6.3 Recommendations . . . 45

6.4 Implementation . . . 46

6.5 Further research . . . 47

6.6 Discussion . . . 48

A Secondary activities 54

B Patient types 55

C Data gathering 56

D Warm up length MIP 62

E OC block 64

F Daily planning rules 65

G Simulation model 66

1 Introduction

The orthopedic department within Deventer Hospital (DZ) experiences problems with their access time to the outpatient clinic and the waiting time for surgeries. The norms for the access and waiting times are frequently exceeded. This report covers the research that is performed to reduce the access and waiting time within the orthopedic department.

This chapter introduces the problem starting with some background information of the hos- pital and the orthopedic department in Section 1.1. In Section 1.2, we provide the problem description, followed by the research goal and questions in respectively Section 1.3 and Sec- tion 1.4.

1.1 Background

The hospital in Deventer, named Deventer Ziekenhuis, has been active with Lean Six Sigma since 2009. The goal is to organize the hospital in such a way that costs are saved, quality of care is improved and the efficiency increases. On top of that, Integral Capacity Management (ICM) is included in the strategy of the hospital. This research is part of ICM.

1.1.1 Deventer ziekenhuis

DZ is a member of the ’Samenwerkende Topklinische opleidingsziekenhuizen’ (STZ), in En- glish the ’cooperative top clinical teaching hospitals’. STZ is a cooperation between hospitals with the focus on training and development. The staff of DZ consists of more than 2200 employees, of which 158 medical specialists treating more than 300.000 patients each year divided over the 82 departments.

1.2 Problem description

The orthopedic department provides mainly elective care for a wide variety of patients. The doctors can work in the Outpatient Clinic (OC) or the Operating Room (OR). The depart- ment faces problems with their access time for OC and waiting time between OC and OR.

The access time, defined as the time of request until the actual OC consult, has to be below a norm set by the Dutch government, the so-called ’treeknorm’. The treeknorm for the access time to an OC is 4 weeks [Ministerie van Volksgezondheid, 2014]. The department wants to distinguish themselves and therefore aims for a maximum access time of 2 weeks.

For the waiting time, the time between the last OC consult before OR and the actual OR day, there is no such ’treeknorm’. The department aims for a maximum waiting time of 6 weeks.

The department experiences seasonality which leads to a stressful period in the second half

of the year. Especially in this season, the secretaries indicate that it is difficult to find

empty patient slots within reasonable time. The planners do not know if they should plan

OC blocks or OR blocks and the doctors feel that they work overcrowded and inefficient

sessions. For several years, they try to avoid the peak pressures but without results.

The orthopedic department is a complex system as Figure 1 shows.

Figure 1: Schematic overview orthopedic department We explain the complexity by the following example:

When a peak of new patients arrives, the access time increases. If a doctor works at the OC, his access time decreases but he generates recurrent patients and patients for the OR, resulting in an increasing waiting time. An important factor regarding the flow of patients is the ratio between new and recurrent patients in an OC block. New patients might have a higher probability than recurrent patients to undergo surgery, consequently the waiting time increases. Sometimes, working more OR blocks seems the solution but the number of ORs is bounded based on the capacity of the OR department. The phenomena that occurs can be described as the bullwhip effect. The bullwhip effect is known from manufacturing situations, but is also present in the health care sector [Sethuraman and Tirupati, 2005].

There are four main factors that induce the bullwhip effect:

1. New patient arrivals.

2. The ratio between OC and OR blocks per doctor.

3. The ratio of new and recurrent patients in an OC block.

4. The capacity of the OR department.

Currently, the orthopedic department tries to control the access and waiting time by (1) switching between OC and OR blocks and (2) by changing the ratio of new and recurrent patients in an OC block. Both proceedings positively influence the direct access and waiting time but the (indirect) effects are unknown.

According to the manager, it is difficult to know how to anticipate in situations where both the access and the waiting time are high and there are still patients to schedule. In this research we aim to gain insight into the effects of the current way of working and to advise the department on how the access and waiting time can be reduced.

To demarcate the problem, we make a distinction between natural and artificial variability.

Both the new patient arrival and the OR department capacity contain natural variability

which we have to deal with. Therefore, the new patient arrival and the OR department

capacity are not further explained within this report and are assumed as input.

The ratio of patient types in an OC block and the allocation of OC and OR blocks contain artificial variability. In this research, we focus on the allocation of blocks over the available doctor capacity while taking into account the four main factors for inducing the bullwhip effect.

1.3 Research goal

This research focuses on the reduction of the access and waiting time. We define the goal of the research as follows:

Reduce the access and waiting time for the orthopedic department.

1.4 Research questions

The research goal is translated into the following main research question:

How can the orthopedic department reduce their access and waiting time?

There are sub-questions formulated to answer the main research question:

1. What is the current situation at the orthopedic department?

In Chapter 2, we explain the current situation. With the use of a visual overview, we identify and explain dependencies and important causes for the variability in the access and waiting time. The remainder of the chapter contains a review of the access and waiting time and an analysis of the current situation.

2. Which methods are applicable to model the orthopedic department?

We use methods for an objective substantiation of the research. Chapter 3 contains a literature review on methods to solve capacity allocation problems.

3. Which method is suitable to model the orthopedic department and are the results of the method a true reflection of reality?

In Chapter 4, based on the outcomes of Chapter 3, we propose a model to improve the identified problem at the orthopedic department. We validate the model by using a blackbox validation.

4. What is the performance of the proposed situation? And how good is this compared to the current situation?

We perform experiments to gain insight into the performances. We analyze the out- comes in Chapter 5 to make recommendations for the department.

5. How can the proposed situation be implemented on a tactical and operational level?

We formulate a suggested situation in Chapter 5. To implement our suggestions in

practice, we provide recommendations regarding implementation in Section 6.4.

2 Current Situation

We explain the current situation at the orthopedic department in this chapter. In Section 2.1, a process description is included, followed by the resource and patient planning in respectively Sections 2.2 and 2.3. The arrival intensity of new patients is covered in Section 2.4. We provide information regarding the production and capacities in respectively Sections 2.5 and 2.6. The access and waiting time are covered in Section 2.7 followed by the doctor analysis in Section 2.8. Section 2.9 gives an overview of the current situation at the department.

2.1 Process description

This section provides some general information, information about doctor activities and covers the various patient types.

2.1.1 General information

The orthopedic department is a department that treated 18776 patients at their OC and performed 2504 surgeries in 2016. There are six orthopedic doctors, two doctor assistants and several doctors in training. An operational manager is responsible for the assisting functions such as the planner and secretaries. The department has 5 OC rooms which can be extended to 10 rooms in an exceptional case.

An orthopedic workweek consists of 5 working days, from Monday to Friday. Surgeries are only performed during the workweek. Patients can be treated at the OC during two blocks:

(1) in the morning block between 8:10am - 12:20pm and (2) in the afternoon block between 1:30pm - 4:10pm. There is no walk-in because patients can only get an appointment if they have a referral from a General Practitioner (GP). Emergency patients are treated at trauma OCs and trauma ORs which are performed separately from the orthopedic department.

2.1.2 Doctor activities

Each orthopedic doctor must work 174 production days per year divided over the OC and OR. An OC block can be of different types because the doctors have different specialties as included in Table 1.

Table 1: Outpatient clinic block type

Doctor Standard Shoulder Knee Back Sport Rijssen Raalte #

A x x 2

B x x 2

C x x 2

D x x x 3

E x x x x 4

F x x 2

The standard OC block is performed by every doctor. During a standard OC session, the

doctor treats patients with different types of injuries. The shoulder, knee, back and sport

are OC sessions for specific injuries. External sessions in Rijssen and Raalte are equal to a

standard block but performed outside DZ.

An OR block is not divided into different types. Each doctor performs the surgeries which he is authorized for. Beside the 174 production days, the doctors have secondary activities (e.g. holidays, administration days, conferences) as included in Appendix A.

2.1.3 Patient types

According to the system of DZ, there are 33 patient types which we divide into new and recurrent patients as explained in Appendix B. These new and recurrent patients follow the care path as Figure 2 shows.

Figure 2: Care path orthopedic patient

Before an OC consult, imaging and or diagnostics scans are made at Center for Radiology and Nuclear Medicine (CRN). A diagnosis is discussed at the OC and from there on, there are two possible care path ways: surgical or conservative (non-surgical).

1. Surgical

A surgical patient requires a surgery. Within this care path, the patient visits consecu- tively the admission office, the pre-operative screening, the physiotherapist (if needed), the OR and the ward.

2. Conservative

A conservative patient gets a conservative treatment such as an injection, a referral to another hospital or specialist, a checkup or a resignation.

2.2 Resource planning

The resource planning concerns the planning of the doctors. The resource planning is divided into the strategic, tactical and operational level as we successively discuss in the next subsections.

2.2.1 Strategic resource planning

The outcome of the strategic resource planning is a yearly blueprint roster. Every doctor

has a four-week repeating master schedule which contains the days at which the doctor

works at the OC (including block type) and the OR. The planner copies this four-week

master schedule 13 times to generate the yearly roster as schematically visualized in Figure

3. The master schedule is the basis for the yearly roster and made in the past, static and

not updated based on current and future demand.

Figure 3: Structure yearly roster

2.2.2 Tactical resource planning

The yearly blueprint roster is the basis for the tactical resource planning. The tactical resource planning has a time span of 20 weeks. The planner requests 15 weeks in advance for the number of ORs needed in a certain month. The number of obtained ORs can vary because of available OR capacity. The results are changes in the yearly roster.

In consultation with the planner, the doctors indicate which days they are not available for production days. The planner needs to know these non-productive days as soon as possible because a roster is finalized 6 weeks in advance after which no more changes are allowed for doctors.

2.2.3 Operational resource planning

The operational resource planning has a horizon of 6 weeks. In the finalized 6 weeks, only the planner is able to make changes in the roster. Changes are only made if doctors become ill, or if OC/OR blocks are not completely filled. If changes need to be made, the goal is to positively influence the access and/or waiting time. If, for example, an OC block is not completely filled, the access time is low and the waiting time high, an OC block can change in an OR block. These changes are made by the planner based on intuition and experience, without rules and in cooperating with the corresponding doctor.

The above mentioned proceeding of switching between OC and OR blocks is a first way of how the department tries to control the access and waiting time. We cover the second proceeding in Section 2.3.

2.3 Patient planning

The patient planning can be divided into tactical and operational level as we explain below.

2.3.1 Tactical patient planning

With regard to tactical patient planning, blocks are distributed over the available capacity.

For every OC block, doctor and shift combination, predefined blocks are available. A block

consists of patient slots of 10 minutes, except for second opinions (20 minutes). A predefined

block contains slots for various patient types as schematically shown in Table 2. In the

current situation, the slots within a block do differ between doctors for the same block type

(see Table 1 in Section 2.1). As the blocks are not updated, they are not based on current

and future demand. We explain the actual planning of patients in patient slots in the next

subsection.

Table 2: Outpatient clinic grid

Predefined OC block Changed OC block

8:00 New patient New patient

8:10 Recurrent patient New patient

8:20 New patient New patient

8:30 Recurrent patient Recurrent patient 8:40 Second opinion Second opinion

9:00 ... ...

2.3.2 Operational patient planning

At the operational patient planning, patient are assigned to patient slots. The rolling planning horizon for patients is 13 weeks. This planning horizon opens weekly and patients are scheduled in patients slots according to a First-Come-First-Serve (FCFS) policy. First, waiting list patients are scheduled. If the secretary schedules a patient, she searches for an available slot within the planning horizon that matches the patient type and the patients preference. If there are no available slots for the patient, secretaries do have two options:

1. Register for waiting list

This is done for patients with less urgency, for example in case of a yearly checkup.

2. Change slot type

The type of a slot can be changed such that a patient can be scheduled (see column

’Changed OC block’ in Table 2).

The procedure of changing slot types is a second procedure that influences the access and waiting time. The choice for one of the two options varies per secretary and is based on intuition and experience. Based on the data, we cannot find out how many slots have been changed. However, the manager and planner indicates that this happens in almost every OC block.

2.4 New patients arrivals

A new patient can request an appointment after he/she has a referral from a general practi- tioner or another specialist. Figure 4 shows the number of new patient requests aggregated for all doctors. These requests are independent of the available capacity.

Figure 4: New patient arrivals

(Source: Hix, n=5860, Jan - Dec 2016)

On average, every week 112 new patients request an appointment. The standard deviation of the number of requests is 27.9 patients per week. The number of new patient appointment requests varies during the year and we can remark holidays.

2.5 Production

In 2016, the doctors treated 15404 patients and they performed 1887 surgeries as Table 3 shows. The majority of the treated patients are recurrent patients (65.5%). The 15404 patients are 82% of the total OC visits. The remaining patients are treated by doctors in training and assistants.

Table 3: Production statistics

(Source: Hix, n=17291, Jan - Dec 2016) Non-shared capacity

Doctor OR NP CP own Non-CP own

A 253 767 820 654

B 401 1016 933 634

C 273 1047 1195 564

D 362 779 769 835

E 330 878 1079 820

F 268 825 1221 568

1887 5312 6017 4075

NP = New patient, CP = Recurrent patient

A remark regarding the production statistics is that only 59.6% of the recurrent patients are CP own patients. We define CP own patients as a recurrent patient that visits the same doctor as during their first visit. A percentage of 59.6% means that 59.6% of the recurrent patients were initially seen as a new patient by this doctor. This indicates that despite the fact that doctors have different specialties, OC capacity is shared. On average, 73.5% of the OC capacity per doctor is devoted to new and CP own patients described as the non-shared OC capacity.

There are many causes for shared capacity. A medical reason, a capacity issue, a patient

that is seen by an assistant, a patients preference, or a combination of all. Based on the

available data, we cannot determine the exact reason for patients to change doctors. Figure

5 shows that sharing capacity depends on the time of the year. As we expect, capacity is

shared more during summer holidays.

Figure 5: Shared OC capacity of the department (Source: Hix, n=15404, Jan - Dec 2016)

2.6 Capacities

The capacities we consider within the research are the doctor capacity and the OR depart- ment capacity. Figure 6 shows both capacities. The obtained OR capacity is the actual received OR capacity. The total doctor capacity is the doctor capacity that is available to fill with OC and OR blocks.

Figure 6: Capacities of 2016

(Source: MedSpace, Jan - Dec 2016)

The capacities need to be aligned in order to function properly. The department feels that, especially during holidays, there is sometimes a mismatch in capacities. There are weeks that the doctors want to perform surgeries but there is no OR capacity and vice versa. A cause for this problem is that some doctors live in another region of the Netherlands where holidays are timed differently than in the region of the hospital.

We can easily remark the holidays. We can identify a mismatch between week 8 and week

9 as the capacities are reduced in different weeks. Also during summer holidays, the OR

department capacity is longer reduced than the doctor capacity. In the rest of the year, the

capacities do follow the same pattern.

2.7 Access and waiting time

The access and waiting time are monitored weekly by the planner. At the moment of mon- itoring, the third available slot is used to measure the access and waiting time. This to prevent coincidence that there is a spot available in short-term.

The actual access time is calculated based on the average time between the request date and the appointment date of all patients scheduled in the corresponding week. We only include patients with an access time less than 9 weeks because we assume that patients with a longer access time are patients with a preference date. We choose for 9 weeks because the maximum monitored access time is 9 weeks.

The actual waiting time is calculated based on the average time between the planning of the surgery and the surgery itself of all patients scheduled in the corresponding week. However, it is not always possible to calculate a waiting time due to weeks in which no patients are scheduled. Therefore, we include the monitored waiting time instead of the actual waiting time.

Table 4 shows the actual access time, the monitored waiting time and their corresponding parameters. We draw our conclusions below the table.

Table 4: Access and waiting time statistics in weeks

(Access Time: Source: Hix, n=4879, Jan - Dec 2016)

Actual access time in weeks Monitored waiting time in weeks Doctor X Min Max σ P(X>4) X Min Max σ P(X>6)

A 2.7 0.7 4.9 1.3 15.5% 6.1 1.6 11.6 2.8 46.0%

B 2.8 0.8 5.3 1.2 14.9% 5.6 2.0 9.4 2.5 38.6%

C 2.6 0.9 4.7 1.2 11.7% 3.8 1.0 8.0 1.9 12.7%

D 3.4 1.0 7.1 1.5 29.8% 5.7 1.4 9.4 2.1 40.5%

E 3.4 0.9 6.4 1.4 30.2% 5.7 1.6 10.1 2.3 39.0%

F 2.6 0.7 5.7 1.4 15.0% 3.9 2.2 6.4 1.2 6.0%

Average 2.9 0.8 5.7 1.3 19.5% 5.1 1.6 9.2 2.1 30.5%

Access time

The average access time, as shown in Table 4, is below the treeknorm of 4 weeks for every doctor, but higher than the target access time of 2 weeks set by the department. The prob- ability of exceeding the treeknorm is included in Table 4. For doctor D, the average access time and the probability of exceeding are high. These extremes were caused by a 6-week holiday which is an incident.

Waiting time

The target waiting time, as set by the department, is a waiting time of 6 weeks. The prob- ability of exceeding the norm of 6 weeks is included in Table 4. The exceeding probabilities for doctor C and doctor F are lower since their patient mix is treated more conservatively.

The goal of the research is to reduce the average and maximum access and waiting times

as included in Table 4. The minimum access and waiting time should be above one week to

avoid idle time.

2.8 Doctor analysis

The remainder of this chapter is dedicated to an analysis of one doctor, which represents the other doctors. The analysis of doctor B consists of two parts: Subsection 2.8.1 analyses the access time and Subsection 2.8.2 the waiting time.

2.8.1 Access time

For the analysis of the access time, we first look at the way how blocks are distributed over the capacity of doctor B. After that, we look at the arrival request of new patients and the corresponding access time and we zoom in at the utilization of OC blocks and the alignment of supply and demand.

The yearly roster for every doctor is generated by copying the master schedule (4 weeks) 13 times as explained in Section 2.2. The blueprint schedule for doctor B consists of four equal weeks. One week is shown in Table 5. The abbreviation ’AD’ stands for ’Administration Day’ and ’OCRa’ stands for ’OC Raalte’ (a standard OC block performed outside DZ).

Table 5: Blueprint week doctor B (Source: Medspace)

Blocks Mon Tue Wed Thu Fri Sat Sun

Morning OR OCRa AD OR OC - -

Afternoon OR OCRa AD OR OC - -

The blueprint week for doctor B contains 4 OC blocks (2 OCRa and 2 OC) and 4 OR blocks, a ratio of 1:1. The resulting yearly roster ratio is also 1:1. Due to secondary activities and the procedure of switching between OC and OR blocks, doctor B performed 160 OC blocks and 165 OR blocks in 2016; a ratio of 0.97:1.00.

Every OC block has a predefined grid of patient slots. Multiplying the slots per block with the number of blocks results in the number of patient slots that are reserved for new and recurrent patients. These reserved slots (theoretical) are compared with the situation in practice as Table 6 shows.

Table 6: Patients slots and treated patients (Source: Hix, n=2579, Jan - Dec 2016) Doctor B Theoretical Practice

New 1143 1016

Recurrent 1543 1567

2686 2579

In theory, 2686 patients can be treated in the 160 OC blocks. 42.6% of these slots were

reserved for new patients. The situation in practice shows that 96.0% of the reserved slots

are used but the ratio between new and recurrent patients differs, which indicates that the

OC blocks are not aligned with patient demand.

Figure 7 shows the access time that corresponds with the above mentioned situation.

Figure 7: Access time for doctor B

(Source: Hix, n=5860, Jan - Dec 2016)

The average access time for doctor B is 2.8 weeks, the minimum access time is 0.8 weeks and the maximum access time is 5.3 weeks. In the beginning of 2017, the access time fluctuates around 3.5 weeks. A remark is that the access time increases during the year. An increasing access time could indicate insufficient capacity. Table 6 shows that there are more reserved slots than treated patients, which indicates sufficient capacity. This means that the timing of patient demand and offered patient type slots are not aligned. We assume that this happens due to the flexibility of changing slots types were future capacity and demand are not incorporated, as we discuss in Section 2.3. To indicate how the capacity is used, we include the utilization of OC blocks in Figure 8.

Figure 8: Utilization of the OC blocks

(Source: Medspace, n=160, Jan - Dec 2016)

We calculate the weekly utilization by dividing the number of treated patients by the num- ber of reserved slots. The average utilization is 99%. A remark regarding the utilization is that buffer slots are not included in the calculations. These slots are used frequently, which results in more capacity and therefore a lower utilization in practice.

It stands out that the utilization in the first half of the year is lower than the second half,

respectively 95% and 102%. In the first half of the year there are 84 OC blocks performed

against 78 (-7%) in the second half of the year. Note that these utilizations are lower in

practice since we cannot subtract the buffer slots.

2.8.2 Waiting time

Figure 9 shows the waiting time and the number of OC and OR blocks for doctor B in 2016.

The average waiting time is 5.6 weeks, with a minimum of 2.0 weeks and a maximum of 9.4 weeks. In the beginning of 2016, the waiting time fluctuates around 4 weeks, whereas after the summer period the average waiting time fluctuates around 8 weeks.

Figure 9: Monitored waiting time including OC and OR blocks (Source: Medspace, n=325, Jan - Dec 2016)

The number of patients waiting for an OR depends on the number of OC blocks performed.

As a consequence, we expect a longer waiting time if the period before contains more OC blocks. However, the waiting time is less reliable than the access time because the method of ’the third free spot’ uses only one measurement per week. On top of that, the planning horizon for surgeries is not extended every week but every month. Due to these uncertain- ties, we are unable to relate activity changes, as for example more OC blocks, directly to the waiting time. Even if we could use the actual waiting time, we are unable to relate activity changes.

The waiting time for doctor B increases between week 25 and 26. The waiting time in week 25 is 2.5 weeks because the third free spot for a surgery is in week 28. In week 26, the waiting time increases because there is no capacity in week 30 - 32 and the third free spot is in week 34. From week 26 till week 52, the waiting time fluctuates around 8 weeks because (1) in the last quarter of the year, knee prostheses were not available and (2) there is no extra capacity to lower the waiting time.

We conclude that due to a 3-week holiday the waiting time increases and stays fluctuating

around 8 weeks because knee prostheses were not available and there is no extra capacity

to lower the waiting time. In the beginning of 2017, there is more capacity and the waiting

time decreases to 4 weeks again.

2.9 Conclusion

In 2016, the orthopedic department treated 15404 patients and they performed 1887 surg- eries. The arrival of new patient demand contains much variability since the standard deviation is 27.9 patients per week. In Section 2.5, we conclude that patients do change doctor between visits. The result is that doctors share on average 26.5% of their OC capacity.

In Section 2.6, we found out that the department feels that there is sometimes a mismatch in capacities between the orthopedic department and the OR department. The cause for the mismatch is that some doctors live in another region of the Netherlands where holidays are timed differently. Section 2.7 indicates that there is much variability within the access and waiting times and that for some doctors the probabilities of exceeding the norms are high. The average exceedance probability of the access time norm is equal to 19.5% and the average exceedance probability of the waiting time norm is equal to 30.5%.

We conclude that the allocation of blocks is static. As the planning horizon decreases, the department tries to control the access and waiting time by (1) switching between OC and OR blocks (Section 2.2) and (2) by changing the type of patient slots in OC blocks (Section 2.3) to fulfill demand. This directly affects access and waiting times, however the indirect effects are unknown and the procedures introduce variability in the flow of patients within the department. As known from basic queuing theory, variability is fateful for waiting times.

Especially when a system is running near its maximum capacity, the impact of variability on the waiting time is high [Silvester et al., 2004].

The problem that can be identified is a tactical resource scheduling problem with variability

in patient arrivals. To tackle the problem, we strive for an approach that allocates blocks

over the available doctor capacity. This approach must incorporate future demand and ca-

pacity such that the access and waiting time are minimized. In the next chapter, a literature

study is performed to find an approach applicable for this research.

3 Literature review

As the current situation shows, the orthopedic department uses a static approach that results in a cyclic plan. These cyclic plans are manually adjusted to align capacity with variability in demand as good as possible. The goal of this literature review is to find a dynamic ap- proach that results in planning rules that respond to the variability in demand and supply.

In Section 3.1, we provide some queuing theory followed by the research positioning in Section 3.2. Methods to solve capacity allocation problems are covered in Section 3.3 and different studies are discussed in Section 3.4. We provide the conclusion of the literature review in Section 3.5.

3.1 Queuing theory

Waiting times for elective treatments largely reduced in the last decade in the Netherlands [Schut and Varkevisser, 2013]. This is caused by introducing a range of policy initiatives including higher spending, waiting-times target schemes and incentive mechanisms [Sicil- iani et al., 2014]. The increase of capacity is a common used approach to reduce waiting times, which is associated with high costs. However, studies show that the lack of capac- ity is typically not the major issue for waiting times. The primary cause is the mismatch between supply and demand [Van Rooij, 2001; Siciliani and Hurst, 2003; Martin et al., 2003].

A common mistake is that most of the capacity plans are based on average demand. When capacity is dimensioned to cover average demand, a queue will develop due to the nature of fluctuation of demand and capacity [Silvester et al., 2004]. Especially when the system is running near its maximum capacity, the impact of variability on the access and waiting time is high [Hopp and Spearman, 2011].

As concluded by Probst et al. [1997], the waiting time has an emphatical effect on patients satisfaction. Chung et al. [1999] confirms this as he states that the clinic waiting time is the most important predictor of patient satisfaction related to efficient clinic operation. To control access and waiting times, resource planning and control receives more attention.

3.2 Research positioning

Planning and control within health care is unique due to many stakeholders, rising ex-

penditures and the various involved departments. To ensure completeness and coherence

of responsibilities for every managerial area, Hans et al. [2011] developed a framework for

health care planning and control. Figure 10 shows the four-by-four matrix that is useful to

position the research and to demarcate the scope.

Figure 10: Framework for health care planning and control [Hans et al., 2011]

This research focuses on the managerial area resource capacity planning. The resource ca- pacity planning addresses the dimensioning, planning, scheduling, monitoring and control of renewable resources. Within this managerial area, there are multiple time horizons defined as included on the vertical axis of Figure 10. For this research, the tactical level is of inter- est. Examples that correspond with the tactical planning and control problems are block planning, staffing and admission planning. According to Hulshof et al. [2012], these three problems are covered under the term ’capacity allocation’ problems. The goal of capacity allocation problems is to align the capacity of the department with the expected demand to control performance measures such as access and waiting times.

3.3 Methods

The department can be seen as a system and there are different ways to study these. Law et al. [2007] make a clear distinction as Figure 11 shows. If it is possible to experiment with the actual system, it is probably desirable. However, it is rarely possible because such an experiment would often be too costly or to disruptive to the system [Law et al., 2007].

Figure 11: Ways to study a system A common way to study systems in the

OR/MS field is by the use of a mathemat-

ical models. Mathematical models are dif-

ferentiated by Law in models that provide

an analytical solution and models that use

simulation. If an analytical (exact) solution

is available and computationally efficient, it

is the most obvious choice. However, many

systems are highly complex such that math-

ematical models of them are complex them-

selves, precluding any possibility of an ana-

lytical solution [Law et al., 2007].

To create an overview of the used mathematical methods within healthcare, Hulshof et al.

[2012] provide a review of planning and control problems and their corresponding meth- ods to solve these problems. Erhard et al. [2017], provides the first review that focuses on quantitative methods for physician scheduling in hospitals. They reviewed 68 relevant publi- cations in the OR/MS field and described the different problem types and modeling features.

According to the taxonomy of Hulshof et al. [2012], methods to solve capacity allocation problems are Mathematical Programming (MP) and computer simulation. Ernhard et al.

confirm this and state that 80% of the capacity allocation problems applied modeling derived from MP approaches like Linear Programming (LP), Integer Programming (IP) and Mixed Integer Programming (MIP). They showed that queuing theory and computer simulation are less frequently used methods.

3.4 Studies

In the remainder of this chapter, we want to discuss studies that use the above-mentioned methods to solve capacity allocation problems in a health care environment. The discussed studies are divided into analytical and simulation studies.

3.4.1 Analytical studies

Hulshof et al. [2013] provide a MIP model to develop tactical resource and admission plans on the intermediate term, for multiple resources and multiple care processes. The goal is to achieve equitable access and treatments duration for patients groups and to serve a strate- gically agreed number of patients. Demand and treatment duration are considered to be deterministic and the outcome is a plan that allocates resource capacity over care processes and determines the number of patients to serve at a particular stage.

Nguyen et al. [2015] introduce a MIP model to plan future required capacity at the tactical level for a re-entry system. Their objective is to minimize the maximum required capacity.

Nguyen et al. assume demand and treatment duration to be deterministic and the model leads to the optimal allocated capacity between new and recurrent patients for each time unit.

Both models optimally allocate resource capacity. In case of Nguyen et al., capacity is al- located over patient types in one care process. In Hulshof et al., capacity is allocated over different care processes and over different patient types.

A specific characteristic of the method of Hulshof et al. is that it keeps track of the number of time units a patient waits at a specific queue. By tracking the number of time units a patient waits, Hulshof et al. are able to calculate the average waiting time. Nguyen et al.

introduce an allowable range for the appointment lead-time, the time between two revisit appointments.

A limitation of the model of Hulshof et al. is that it uses a predefined set of care processes

with each a specific number of stages and resources defined as a care path. This leads to a

finite number of stages in the care process. The model of Nguyen et al. uses an infinite set of

stages in the care process because patients can recur with a certain probability independent of how often they visited the department before. An advantage of the model of Hulshof et al. is that it incorporates a deterministic delay between stages (e.g., for medical reasons).

Both studies do not incorporate no-shows.

Computational results show that the proposed MIP model by Hulshof et al. improves com- pliance with access time targets, care process duration and the number of patients served.

The study of Nguyen et al. found the optimal planned capacity.

Dynamic Programming (DP) is a method for solving complex problems but only in case of small instances. For real life sized problems as in hospitals, DP is generally difficult and possibly intractable [Hulshof et al., 2016]. Therefore, Hulshof et al. [2016] propose a method to develop a tactical resource allocation and patient admission plan by using Approximate Dynamic Programming (ADP). Their objective is to achieve equitable access and treatment duration for patient groups and to serve the strategically agreed number of patients.

At each stage, the model of Hulshof et al. [2016] decides how many patients to treat from each queue that have been waiting a certain amount of time. The model incorporates stochasticity in patient arrivals and patient transitions between different stages. Compu- tational results show that the approach provides an accurate approximation of the value functions and that it is suitable for large problem instances.

Tsai [2017] performed a master project continuing on the work of Hulshof et al. [2016]. She modeled an orthopedic department as a Markov Chain and uses Stochastic Dynamic Pro- gramming (SDP) to optimally allocate capacity over OC and OR blocks. The objective of the model is to minimize the waiting time of the patients in the OR queue and to keep the amount of unused OR time below a certain level.

In the model of Tsai, each doctor has a budgeted amount of yearly sessions to divide. By using a recursion formula, she tries to find the optimal action for each state of the system.

Tsai assumes that there is always sufficient demand and she does not incorporate the lim- ited capacity of the OR department. Another limitation is that the initial SDP model is computational expensive. After approximations and simplifications, computation time is reduced but Tsai was unable to create a yearly roster.

A queuing network analysis is very effective to balance a system fast and easily [Vanberkel et al., 2010]. Calculations are exact and fewer data is needed. Queuing models can help to estimate the number of required staff in each time unit to achieve a performance as indicated in for example Yankovic and Green [2008], Jennings and de Vericourt [2007] and Green et al.

[2006]. However, as concluded from the review of Lakshmi and Iyer [2013], clinics where

patients have to (re-)visit specific care providers in a network of care queues still involve

modelling complications.

3.4.2 Simulation studies

A model must be studied by the means of simulation when the system is too complex to evaluate it analytically [Law et al., 2007]. A simulation model is not used to obtain an optimal solution but to evaluate inputs of a model numerically in question to see how they affect the output measures of performance. Several studies with capacity allocation prob- lems use simulation as method, for example Nguyen et al. [2005]; VanBerkel and Blake [2007]; Vermeulen et al. [2009]. These studies have in common that they use simulation to test operational interventions to see how they affect the performance indicators. A simula- tion model is also often generated to validate or evaluate performances [van de Vrugt, 2016;

Ma and Demeulemeester, 2013]. Drawbacks of simulation are that they tend to be very complex, time-consuming to write and require detailed information [Law et al., 2007].

3.5 Conclusion

The primary cause for waiting times is the mismatch between supply and demand. When capacity is dimensioned to cover average demand, the influence of variation develops a queue.

The research problem is summarized as a capacity allocation problem. According to Hulshof et al. [2012] and Erhard et al. [2017], modeling derived from MP is most frequently used.

Queuing theory and computer simulation are less frequently used methods.

We reviewed the MIP models of Hulshof et al. [2013] and Nguyen et al. [2015]. Both mod- els are used to solve a capacity allocation problem in a similar setting and their results are promising. We also highlighted two studies that used programming derived from DP, namely the study of Hulshof et al. [2016] (ADP) and Tsai [2017] (SDP). The ADP model of Hulshof et al. provides an accurate approximation of the value functions and is suitable for large problem instances whereas Tsai was unable to create a yearly roster due to a model that is computational expensive.

Queuing models can help to estimate the number of required staff in each time period.

Nevertheless, current literature shows that clinics where patients have to (re-)visit specific care providers in a network of care queues still involve modelling complications [Lakshmi and Iyer, 2013].

Computer simulation is not used to obtain the optimal solution but creates the possibility to experiment with model inputs and observe how they affect output measures of performance.

Simulation models are also often generated to validate or evaluate performances.

We want to model our problem as a MIP model because modelling derived from MP is

frequently used, we aim for an optimal solution, and the studies of Hulshof et al. [2013] and

Nguyen et al. [2015] show promising results. These studies also have the most similarities

with our problem. By combining the advantages of both studies, we obtain a model that

optimally allocates blocks over the available doctor capacity while incorporating future ca-

pacity and demand.

4 Solution design

In this chapter we propose a model to tackle the problem as identified in Chapter 2. To structure this chapter, we follow the sound modeling steps as proposed by Law et al. [2007].

Figure 12 shows the corresponding sections.

Figure 12: Steps in a modeling study

In Section 4.1 we explain the conceptual model that we create based on the literature review performed in Chapter 3. Section 4.2 explains the way of data gathering followed by the explanation of the technical model in Section 4.3. Section 4.3 is divided into two subsections:

Subsection 4.3.1 explains how we obtain the transition rates and Subsection 4.3.2 covers the technical explanation of our MIP model. We explain our modeling approach in Section 4.4 and the validation of the model is done in Section 4.5. The last section, Section 4.6, provides a conclusion.

4.1 Conceptual model

In order to improve the situation at the orthopedic department, we propose a MIP model based on specific characteristics from the study of Hulshof et al. [2013] and Nguyen et al.

[2015]. The outcome of the model is a plan that allocates blocks over the available doctor capacity. The goal is to allocate blocks in such a way that the weighted number of waiting patients is minimized.

The model of Hulshof et al. [2013] is the basis for our MIP. Nevertheless, we should make adjustments to fit it to our situation. First, we adjust the number of patients that are served. Hulshof et al. determine the number of patients to serve from a particular queue at a particular stage. However, due to practical reasons, it is in our case not possible that the ratio between new and recurrent patients differs per OC block. Therefore, we create one OC block per doctor that reflects current care demand. Consequently, we know how many patients are served from a queue if a doctor works at the OC.

Second, we adjust the predefined care path that Hulshof et al. use by introducing an in- finite set of stages in the care process as proposed in Nguyen et al. [2015]. Nguyen et al.

use an infinite set of stages in the care process by introducing a probability that the patient moves to the next stage. This probability is independent of how often the patient visited the department before. The result is a transition of patients that reflects reality.

Another addition to the model is the use of a deterministic delay between the queues in

the care process. This deterministic delay is described as the advised return time between

stages based on historical data.

When constructing the capacity allocation plan, we incorporate the natural variability of demand and the natural variability in OR department capacity. We assume service rates to be deterministic. The arrival rate of new patients is based on historical data. We fit probability distributions to the patient arrival data and use random numbers to generate patients for each doctor. We decide that doctors can only work a complete day of two equal blocks (only OC blocks or only OR blocks) because single blocks a day are not desirable.

We do not take into account no-shows.

We use the MIP model as simulation-optimization method. The idea is to simulate as many years as possible with a fixed doctor capacity. Each year has different new patients arrivals and therefore we can analyze the systems behaviour. The analyses are provided in Chapter 5.

4.2 Data gathering

We gather our data from the hospital information system Hix. From Hix, we use the OC and OR data from several periods. The exact periods, the way of calculating and the values for the different parameters are included in Appendix C. An important remark regarding the data is that we divided the different patient types into new and recurrent patients as explained in Appendix B.

4.3 Technical model

In this section, we explain the technical model as introduced in Section 4.1. This section is divided into two parts. First, we explain in Subsection 4.3.1 how we obtain the transition rates. This part is described as ’Phase 1’ and input for the MIP model which we explain in Subsection 4.3.2. The MIP model is remarked as ’Phase 2’.

4.3.1 Phase 1: Transition rates

The transition rates have a big influence on the flow of patients through the system and must therefore reflect practice as close as possible. We define the transition rate as the fraction of patients that move to the next stage in their care process or leave the system.

These rates are used as input for the MIP model.

We conclude in Section 2.5 that there are patients that change doctors during their care

path, which results in shared OC capacity. There are many reasons for patients to change

doctor, e.g., because of the absence of a doctor. Based on the current data, we are not

able to find out the exact reason that causes patients to change doctor. Therefore, we

assume that patients do not change doctors in our model. This gives us two possible ways

to calculate the transition rates:

1. Based on the shared and non-shared OC capacity

If we calculate the transition rates based on the shared and non-shared OC capacity, we must assume that each doctor shares the same amount of capacity and that capac- ity is shared equal during the year. However, capacity is not shared equal during the year as Figure 5 shows in Section 2.5.

2. Based on the non-shared OC capacity

If we calculate the transition rates based on the non-shared OC capacity, we assume doctors to work independently and therefore patients do not change doctors. However, with this method we only take 73.5% of the total available OC capacity into account as explained in Section 2.5.

We choose to calculate the transition rates based on the non-shared OC capacity. Although we can say only something about 73.5% of the OC capacity, we approximate the current situation as close as possible by extrapolating the non-shared OC capacity with the use of a Monte Carlo simulation. The input for the Monte Carlo simulation are the new patients arrivals based on the period of 2012 - 2016 (see Appendix C.1). By changing the transition rates, we create a steady-state system that reflects the production of 2016.

Another advantage of the second way of calculating is that we can assume doctors to work independently. Due to this assumption, we are able to run our model per doctor. The outcomes of an individual run can be merged and used as initial solution for the complete system. This method decreases the run time of our MIP model.

For the explanation of the Monte Carlo simulation, we take doctor B as example again.

Table 7 shows the production numbers of doctor B next to the Monte Carlo simulation results of 1500 experiments. The confidence interval indicates that the new transition rates result in production numbers that lay within the interval with a probability of 95%. The standard error indicates the precision of the average that results from the simulation.

Table 7: Monte Carlo simulation results (n=1500)

2016 Simulation

Production Average Confidence interval Standard error

NP 1016 1016.3 1013.1 1019.5 0.04%

CP 1567 1563.4 1558.6 1568.2 0.03%

OR 397 397.2 395.7 398.8 0.08%

We conclude that the new transition rates result in production numbers that lay within the confidence interval with a probability of 95%. Therefore, we can say that the new transition rates result in production numbers that reflect the production of 2016.

For the model, we use one OC block per doctor that reflects patient demand. Each block

contains 18 slots. The filling of the slots is based on the ratio between NP and CP that

results from the Monte Carlo simulation. Consequently, an OC block of doctor B contains

7.5 NP slots and 11.5 CP slots. We use fractional values to decrease our computation time.