Reducing access and waitingtime for orthopedics

Deventer ziekenhuis

Master Thesis inIndustrial Engineering and Management

ARJAN PANNEKOEK

Supervisors UniversityDr.ir. A.G. LeeftinkProf.dr.ir. E.W. Hans

By order ofS. Koemans

M. BrillemanDeventer Ziekenhuis

Management summaryThis research focuses on the reduction of the access and waiting time for the orthopedicdepartment of Deventer Ziekenhuis (DZ).

Problem descriptionThe department experiences seasonality, which leads to a stressful period in the second halfof the year. In this period, secretaries indicate that it is difficult to find empty patient slotswithin 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 andinefficient sessions. The department also feels that there is sometimes a mismatch betweentheir doctor capacity and the capacity of the OR department.

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

ApproachWe want to obtain practical plannings rules to allocate blocks over the available weeklydoctor 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 usea 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 approachto 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 indicatehow blocks should be allocated over the weekly available doctor capacity and to indicatea capacity mismatch between the orthopedic capacity and the OR capacity. In the thirdstep, we evaluate the performance of our planning rules by performing a Discrete EventSimulation (DES) model.

ResultsOur MIP model provides a quantitative substantiation for the feeling that there is sometimesa mismatch between the doctor and OR capacities. The cause is that some doctors live inanother region of the Netherlands where holidays are timed differently and therefore werecommend the department to align capacities on forehand since this positively influencesthe access and/or waiting time.

We formulate practical planning rules how the department should handle in case of (1) newpatients arrivals, (2) holiday season and (3) non-holiday season. We conclude that the modeldoes not heavily react to new patient arrivals and therefore we advice the department tointroduce our new OC blocks and to limit changing the type of patients slots of OC blocksto a minimum. We recommend to update the OC blocks yearly with the use of the createdExcel tool. We formulate the planning rules in days (1 day contains 2 blocks) since this isappropriate for practice and they are included below.

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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 us,t = 4.(b) Plan a maximum of 3 OR days if the weekly doctor capacity us,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 us,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 us,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 everydetail of the department is incorporated. The outcomes of the DES model show a morestable 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 theyensure that blocks can be divided without the use of experience and/or feelings. Becausethe differences between the performance indicators for the current and suggested situationare positive, we advise the department to use them.

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

Contribution to practiceThis 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 spreadsheetregarding the flow of patients towards the ward is experienced as helpful and the side-projectas potential for further research. We contribute also to practice since we extend the IntegralCapacity Management (ICM) support within DZ.

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Contribution to theoryThis research tackles a resource scheduling problem for elective care where patients needmultiple appointments at multiple resources. By combining and expanding the models ofHulshof 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 lifecase by using a simulation model. The results are promising, and the research is interestingfor surgical specialties that need to allocate capacity to minimize the weighted number ofwaiting patients.

The side-project contributes to theory since we created a self made Excel spreadsheet modelthat 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.

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Management samenvattingDit onderzoek focust zich op het verlagen van de toegangs- en wachttijd voor de afdelingorthopedie in het Deventer ziekenhuis (DZ).

ProbleemomschrijvingDe afdeling orthopedie ervaart seizoenspatronen en dit resulteert in een stressvolle periodein de tweede helft van het jaar. In deze periode vinden secretaresses het moeilijk om vrijepatiëntplekken te vinden binnen een aanzienlijk tijdsbestek. De planners weten niet of zeblokken moeten inplannen voor de polikliniek (poli) of de operatiekamer (OK) en de artsenhebben het gevoel dat ze overvolle en inefficiënte sessies werken. Tevens heeft de afdelinghet 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 dehuidige poliblokken reflecteren niet de huidige patiëntvraag. Wanneer de planningshorizonverkleint 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 poliblokken om te voldoen aan de patiëntvraag. De directe effecten van deze twee procedureszijn positief voor de toegangs- en wachttijd, echter zijn de indirecte effecten onbekend enintroduceren de procedures variabiliteit in de patiëntenstroom.

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

ResultatenDoor 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 oorzaakhiervoor is dat sommige artsen in een andere regio van Nederland wonen waar vakantiesanders getimed zijn. We bevelen de afdeling aan de arts en OK capaciteit op voorhand afte 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) nieuwepatiënt aankomsten, (2) in vakantieseizoen en in (3) niet-vakantieseizoen. We concluderendat het model niet abrupt reageert op nieuwe patiënt aankomsten en daarom adviseren wijde nieuwe poli blokken in gebruik te nemen en het omboeken van patiëntsloten van poliblokken te beperken tot een minimum. We adviseren de poli blokken jaarlijks te updatenmet het gebruik van de hiervoor gemaakte Excel tool. We formuleren de planningsregels indagen (1 dag bevat 2 blokken) omdat dit bruikbaar is voor de praktijk.

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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 us,t = 4.(b) Plan maximaal 3 OK dagen als de wekelijkse arts capaciteit us,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 us,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 us,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 planningsregelwaarbij niet elk detail van de afdeling is meegenomen. De uitkomsten van het DES modellaten 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, makkelijkte implementeren en garanderen dat blokken verdeelt kunnen worden zonder het gebruikvan ervaring en/of gevoelens. Omdat de verschillen tussen de performance indicatoren vande huidige situatie en de voorgestelde situatie positief zijn, adviseren wij de afdeling deplanningsregels te hanteren.

Side projectIn de praktijk wordt de flow van patiënten richting de kliniek niet in acht genomen. Dedoelstelling 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 blokkenalloceert voor elke dag van de week. Naast de Excel spreadsheet formuleren we arts en letselafhankelijke planningsregels die meer sturing geven aan de flow van patiënten.

Bijdragen aan de praktijkDit onderzoek draagt bij aan de praktijk gezien het feit dat de nieuwe poli blokken en deplanningsregels al in gebruik zijn in de praktijk. De Excel spreadsheet voor het plannen vanOK blokken wordt ervaren als een handig hulpmiddel dat in gebruik is en het side projectals potentieel voor vervolgonderzoek. We dragen ook bij aan de praktijk doordat we meerdraagvlak creëren voor het integraal capaciteitsmanagement binnen DZ.

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Bijdragen aan de theorieDit onderzoek richt zich op een plannings probleem van resources voor electieve zorg waar-bij patiënten meerdere afspraken bij meerdere resources hebben. Door het combineren enuitbreiden van de modellen van Hulshof et al. [2013] en Nguyen et al. [2015] hebben weeen 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 eenDES model. De resultaten zijn veelbelovend en het onderzoek is interessant voor snijdendespecialismen die capaciteit moeten alloceren om op deze wijze het gewogen aantal wachtendepatiënten te minimaliseren.

Het side-project draagt bij aan de theorie doordat we een eigengemaakt Excel spreadsheetmodel hebben gecreëerd die OK blokken alloceert waarmee de variabiliteit in het aantalbezette 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 waarbijpatiënten een bed in de kliniek vereisen.

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

I would like to thank Gréanne Leeftink not only for guiding the research but also for herenthusiasm and critical feedback which have lifted me to a higher level. I also want tothank Erwin Hans as a discussion partner who gave the research a good direction. WithinDeventer 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 wantto thank her for the pleasant cooperation and the good coordination. Saskia Koemans, Iwant to thank you for your support, pleasant cooperation and positive feedback regardingthe research. Vanessa Souilljee, thanks for your enthusiasm, studious attitude and pleasantcooperation. Besides Deventer ziekenhuis, I want to thank Benjamin Lubach as colleagueand friend during the research. Benjamin, thanks for the good and pleasant collaboration.

Arjan PannekoekEpe, July 2018

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ContentsManagement summary i

Management samenvatting iv

Preface vii

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Research goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Research questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Current Situation 42.1 Process description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Resource planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Patient planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 New patients arrivals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 Capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7 Access and waiting time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.8 Doctor analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Literature review 153.1 Queuing theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Research positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Solution design 204.1 Conceptual model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Data gathering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 Technical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4 Modeling approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.5 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Experiment results 285.1 Experiment approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2 Experiment setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.3 Experiment outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.4 Simulation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.5 Weekly planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6 Conclusion and recommendations 426.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.3 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.5 Further research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.6 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 IntroductionThe orthopedic department within Deventer Hospital (DZ) experiences problems with theiraccess time to the outpatient clinic and the waiting time for surgeries. The norms for theaccess and waiting times are frequently exceeded. This report covers the research that isperformed 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 problemdescription, followed by the research goal and questions in respectively Section 1.3 and Sec-tion 1.4.

1.1 BackgroundThe hospital in Deventer, named Deventer Ziekenhuis, has been active with Lean Six Sigmasince 2009. The goal is to organize the hospital in such a way that costs are saved, quality ofcare 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 hospitalswith the focus on training and development. The staff of DZ consists of more than 2200employees, of which 158 medical specialists treating more than 300.000 patients each yeardivided over the 82 departments.

1.2 Problem descriptionThe orthopedic department provides mainly elective care for a wide variety of patients. Thedoctors 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 belowa norm set by the Dutch government, the so-called ’treeknorm’. The treeknorm for theaccess time to an OC is 4 weeks [Ministerie van Volksgezondheid, 2014]. The departmentwants 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 ORday, there is no such ’treeknorm’. The department aims for a maximum waiting time of 6weeks.

The department experiences seasonality which leads to a stressful period in the second halfof the year. Especially in this season, the secretaries indicate that it is difficult to findempty patient slots within reasonable time. The planners do not know if they should planOC blocks or OR blocks and the doctors feel that they work overcrowded and inefficientsessions. For several years, they try to avoid the peak pressures but without results.

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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 theOC, 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 patientsis the ratio between new and recurrent patients in an OC block. New patients might havea higher probability than recurrent patients to undergo surgery, consequently the waitingtime increases. Sometimes, working more OR blocks seems the solution but the number ofORs is bounded based on the capacity of the OR department. The phenomena that occurscan be described as the bullwhip effect. The bullwhip effect is known from manufacturingsituations, 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 recurrentpatients in an OC block. Both proceedings positively influence the direct access and waitingtime but the (indirect) effects are unknown.

According to the manager, it is difficult to know how to anticipate in situations where boththe access and the waiting time are high and there are still patients to schedule. In thisresearch we aim to gain insight into the effects of the current way of working and to advisethe 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 variabilitywhich we have to deal with. Therefore, the new patient arrival and the OR departmentcapacity are not further explained within this report and are assumed as input.

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The ratio of patient types in an OC block and the allocation of OC and OR blocks containartificial variability. In this research, we focus on the allocation of blocks over the availabledoctor capacity while taking into account the four main factors for inducing the bullwhipeffect.

1.3 Research goalThis research focuses on the reduction of the access and waiting time. We define the goalof the research as follows:

Reduce the access and waiting time for the orthopedic department.

1.4 Research questionsThe 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 theaccess and waiting time. The remainder of the chapter contains a review of the accessand 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 aliterature review on methods to solve capacity allocation problems.

3. Which method is suitable to model the orthopedic department and are the results ofthe method a true reflection of reality?In Chapter 4, based on the outcomes of Chapter 3, we propose a model to improvethe identified problem at the orthopedic department. We validate the model by usinga blackbox validation.

4. What is the performance of the proposed situation? And how good is this compared tothe 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 inpractice, we provide recommendations regarding implementation in Section 6.4.

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2 Current SituationWe 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 respectivelySections 2.2 and 2.3. The arrival intensity of new patients is covered in Section 2.4. Weprovide information regarding the production and capacities in respectively Sections 2.5 and2.6. The access and waiting time are covered in Section 2.7 followed by the doctor analysisin Section 2.8. Section 2.9 gives an overview of the current situation at the department.

2.1 Process descriptionThis section provides some general information, information about doctor activities andcovers the various patient types.

2.1.1 General information

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

An orthopedic workweek consists of 5 working days, from Monday to Friday. Surgeries areonly 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 between1:30pm - 4:10pm. There is no walk-in because patients can only get an appointment if theyhave a referral from a General Practitioner (GP). Emergency patients are treated at traumaOCs 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 andOR. An OC block can be of different types because the doctors have different specialties asincluded in Table 1.

Table 1: Outpatient clinic block typeDoctor Standard Shoulder Knee Back Sport Rijssen Raalte #

A x x 2B x x 2C x x 2D x x x 3E x x x x 4F x x 2

The standard OC block is performed by every doctor. During a standard OC session, thedoctor treats patients with different types of injuries. The shoulder, knee, back and sportare OC sessions for specific injuries. External sessions in Rijssen and Raalte are equal to astandard block but performed outside DZ.

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An OR block is not divided into different types. Each doctor performs the surgeries whichhe 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 andrecurrent patients as explained in Appendix B. These new and recurrent patients follow thecare 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 Radiologyand Nuclear Medicine (CRN). A diagnosis is discussed at the OC and from there on, thereare two possible care path ways: surgical or conservative (non-surgical).

1. SurgicalA 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. ConservativeA conservative patient gets a conservative treatment such as an injection, a referralto another hospital or specialist, a checkup or a resignation.

2.2 Resource planningThe resource planning concerns the planning of the doctors. The resource planning isdivided into the strategic, tactical and operational level as we successively discuss in thenext subsections.

2.2.1 Strategic resource planning

The outcome of the strategic resource planning is a yearly blueprint roster. Every doctorhas a four-week repeating master schedule which contains the days at which the doctorworks at the OC (including block type) and the OR. The planner copies this four-weekmaster schedule 13 times to generate the yearly roster as schematically visualized in Figure3. The master schedule is the basis for the yearly roster and made in the past, static andnot updated based on current and future demand.

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Figure 3: Structure yearly roster

2.2.2 Tactical resource planning

The yearly blueprint roster is the basis for the tactical resource planning. The tacticalresource planning has a time span of 20 weeks. The planner requests 15 weeks in advancefor the number of ORs needed in a certain month. The number of obtained ORs can varybecause 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 forproduction days. The planner needs to know these non-productive days as soon as possiblebecause a roster is finalized 6 weeks in advance after which no more changes are allowed fordoctors.

2.2.3 Operational resource planning

The operational resource planning has a horizon of 6 weeks. In the finalized 6 weeks, onlythe planner is able to make changes in the roster. Changes are only made if doctors becomeill, or if OC/OR blocks are not completely filled. If changes need to be made, the goal isto positively influence the access and/or waiting time. If, for example, an OC block is notcompletely filled, the access time is low and the waiting time high, an OC block can changein 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 wayof how the department tries to control the access and waiting time. We cover the secondproceeding in Section 2.3.

2.3 Patient planningThe 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 blockconsists of patient slots of 10 minutes, except for second opinions (20 minutes). A predefinedblock contains slots for various patient types as schematically shown in Table 2. In thecurrent 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 currentand future demand. We explain the actual planning of patients in patient slots in the nextsubsection.

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Table 2: Outpatient clinic gridPredefined OC block Changed OC block

8:00 New patient New patient8:10 Recurrent patient New patient8:20 New patient New patient8:30 Recurrent patient Recurrent patient8:40 Second opinion Second opinion9:00 ... ...

2.3.2 Operational patient planning

At the operational patient planning, patient are assigned to patient slots. The rollingplanning horizon for patients is 13 weeks. This planning horizon opens weekly and patientsare 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 anavailable slot within the planning horizon that matches the patient type and the patientspreference. If there are no available slots for the patient, secretaries do have two options:

1. Register for waiting listThis is done for patients with less urgency, for example in case of a yearly checkup.

2. Change slot typeThe 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 andwaiting time. The choice for one of the two options varies per secretary and is based onintuition and experience. Based on the data, we cannot find out how many slots have beenchanged. However, the manager and planner indicates that this happens in almost everyOC block.

2.4 New patients arrivalsA 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 aggregatedfor all doctors. These requests are independent of the available capacity.

Figure 4: New patient arrivals(Source: Hix, n=5860, Jan - Dec 2016)

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

2.5 ProductionIn 2016, the doctors treated 15404 patients and they performed 1887 surgeries as Table3 shows. The majority of the treated patients are recurrent patients (65.5%). The 15404patients are 82% of the total OC visits. The remaining patients are treated by doctors intraining and assistants.

Table 3: Production statistics(Source: Hix, n=17291, Jan - Dec 2016)

Non-shared capacityDoctor OR NP CPown Non-CPown

A 253 767 820 654B 401 1016 933 634C 273 1047 1195 564D 362 779 769 835E 330 878 1079 820F 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 patientsare CPown patients. We define CPown patients as a recurrent patient that visits the samedoctor as during their first visit. A percentage of 59.6% means that 59.6% of the recurrentpatients were initially seen as a new patient by this doctor. This indicates that despite thefact that doctors have different specialties, OC capacity is shared. On average, 73.5% of theOC capacity per doctor is devoted to new and CPown patients described as the non-sharedOC capacity.

There are many causes for shared capacity. A medical reason, a capacity issue, a patientthat is seen by an assistant, a patients preference, or a combination of all. Based on theavailable data, we cannot determine the exact reason for patients to change doctors. Figure5 shows that sharing capacity depends on the time of the year. As we expect, capacity isshared more during summer holidays.

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

2.6 CapacitiesThe 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 actualreceived OR capacity. The total doctor capacity is the doctor capacity that is available tofill 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 weeksthat the doctors want to perform surgeries but there is no OR capacity and vice versa. Acause for this problem is that some doctors live in another region of the Netherlands whereholidays 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 week9 as the capacities are reduced in different weeks. Also during summer holidays, the ORdepartment capacity is longer reduced than the doctor capacity. In the rest of the year, thecapacities do follow the same pattern.

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2.7 Access and waiting timeThe 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 toprevent coincidence that there is a spot available in short-term.

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

The actual waiting time is calculated based on the average time between the planning of thesurgery 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 arescheduled. Therefore, we include the monitored waiting time instead of the actual waitingtime.

Table 4 shows the actual access time, the monitored waiting time and their correspondingparameters. 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 weeksDoctor 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 timeThe average access time, as shown in Table 4, is below the treeknorm of 4 weeks for everydoctor, 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 accesstime and the probability of exceeding are high. These extremes were caused by a 6-weekholiday which is an incident.

Waiting timeThe 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 probabilitiesfor 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 timesas included in Table 4. The minimum access and waiting time should be above one week toavoid idle time.

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2.8 Doctor analysisThe remainder of this chapter is dedicated to an analysis of one doctor, which representsthe other doctors. The analysis of doctor B consists of two parts: Subsection 2.8.1 analysesthe 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 overthe capacity of doctor B. After that, we look at the arrival request of new patients and thecorresponding access time and we zoom in at the utilization of OC blocks and the alignmentof supply and demand.

The yearly roster for every doctor is generated by copying the master schedule (4 weeks) 13times as explained in Section 2.2. The blueprint schedule for doctor B consists of four equalweeks. One week is shown in Table 5. The abbreviation ’AD’ stands for ’AdministrationDay’ 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 SunMorning 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 andthe procedure of switching between OC and OR blocks, doctor B performed 160 OC blocksand 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 withthe number of blocks results in the number of patient slots that are reserved for new andrecurrent patients. These reserved slots (theoretical) are compared with the situation inpractice as Table 6 shows.

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

Doctor B Theoretical PracticeNew 1143 1016Recurrent 1543 1567

2686 2579

In theory, 2686 patients can be treated in the 160 OC blocks. 42.6% of these slots werereserved for new patients. The situation in practice shows that 96.0% of the reserved slotsare used but the ratio between new and recurrent patients differs, which indicates that theOC blocks are not aligned with patient demand.

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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 andthe maximum access time is 5.3 weeks. In the beginning of 2017, the access time fluctuatesaround 3.5 weeks. A remark is that the access time increases during the year. An increasingaccess time could indicate insufficient capacity. Table 6 shows that there are more reservedslots than treated patients, which indicates sufficient capacity. This means that the timingof patient demand and offered patient type slots are not aligned. We assume that thishappens due to the flexibility of changing slots types were future capacity and demand arenot incorporated, as we discuss in Section 2.3. To indicate how the capacity is used, weinclude 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 isthat buffer slots are not included in the calculations. These slots are used frequently, whichresults 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 performedagainst 78 (-7%) in the second half of the year. Note that these utilizations are lower inpractice since we cannot subtract the buffer slots.

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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 9.4weeks. In the beginning of 2016, the waiting time fluctuates around 4 weeks, whereas afterthe 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 OCblocks. However, the waiting time is less reliable than the access time because the methodof ’the third free spot’ uses only one measurement per week. On top of that, the planninghorizon 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 thewaiting time. Even if we could use the actual waiting time, we are unable to relate activitychanges.

The waiting time for doctor B increases between week 25 and 26. The waiting time in week25 is 2.5 weeks because the third free spot for a surgery is in week 28. In week 26, thewaiting time increases because there is no capacity in week 30 - 32 and the third free spot isin 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 noextra capacity to lower the waiting time.

We conclude that due to a 3-week holiday the waiting time increases and stays fluctuatingaround 8 weeks because knee prostheses were not available and there is no extra capacityto lower the waiting time. In the beginning of 2017, there is more capacity and the waitingtime decreases to 4 weeks again.

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2.9 ConclusionIn 2016, the orthopedic department treated 15404 patients and they performed 1887 surg-eries. The arrival of new patient demand contains much variability since the standarddeviation is 27.9 patients per week. In Section 2.5, we conclude that patients do changedoctor 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 mismatchin capacities between the orthopedic department and the OR department. The cause forthe mismatch is that some doctors live in another region of the Netherlands where holidaysare timed differently. Section 2.7 indicates that there is much variability within the accessand waiting times and that for some doctors the probabilities of exceeding the norms arehigh. The average exceedance probability of the access time norm is equal to 19.5% and theaverage 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, thedepartment tries to control the access and waiting time by (1) switching between OC andOR blocks (Section 2.2) and (2) by changing the type of patient slots in OC blocks (Section2.3) to fulfill demand. This directly affects access and waiting times, however the indirecteffects are unknown and the procedures introduce variability in the flow of patients withinthe 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 variabilityon the waiting time is high [Silvester et al., 2004].

The problem that can be identified is a tactical resource scheduling problem with variabilityin patient arrivals. To tackle the problem, we strive for an approach that allocates blocksover 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 literaturestudy is performed to find an approach applicable for this research.

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3 Literature reviewAs the current situation shows, the orthopedic department uses a static approach that resultsin a cyclic plan. These cyclic plans are manually adjusted to align capacity with variabilityin 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 Section3.2. Methods to solve capacity allocation problems are covered in Section 3.3 and differentstudies are discussed in Section 3.4. We provide the conclusion of the literature review inSection 3.5.

3.1 Queuing theoryWaiting 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 initiativesincluding 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 waitingtimes, 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 mismatchbetween 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. Whencapacity is dimensioned to cover average demand, a queue will develop due to the natureof fluctuation of demand and capacity [Silvester et al., 2004]. Especially when the systemis running near its maximum capacity, the impact of variability on the access and waitingtime is high [Hopp and Spearman, 2011].

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

3.2 Research positioningPlanning and control within health care is unique due to many stakeholders, rising ex-penditures and the various involved departments. To ensure completeness and coherenceof responsibilities for every managerial area, Hans et al. [2011] developed a framework forhealth care planning and control. Figure 10 shows the four-by-four matrix that is useful toposition the research and to demarcate the scope.

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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 ofrenewable resources. Within this managerial area, there are multiple time horizons definedas 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 blockplanning, staffing and admission planning. According to Hulshof et al. [2012], these threeproblems are covered under the term ’capacity allocation’ problems. The goal of capacityallocation problems is to align the capacity of the department with the expected demandto control performance measures such as access and waiting times.

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

Figure 11: Ways to study a system[Law et al., 2007]

A common way to study systems in theOR/MS field is by the use of a mathemat-ical models. Mathematical models are dif-ferentiated by Law in models that providean analytical solution and models that usesimulation. If an analytical (exact) solutionis available and computationally efficient, itis the most obvious choice. However, manysystems 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].

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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 onquantitative 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 allocationproblems are Mathematical Programming (MP) and computer simulation. Ernhard et al.confirm this and state that 80% of the capacity allocation problems applied modeling derivedfrom MP approaches like Linear Programming (LP), Integer Programming (IP) and MixedInteger Programming (MIP). They showed that queuing theory and computer simulationare less frequently used methods.

3.4 StudiesIn the remainder of this chapter, we want to discuss studies that use the above-mentionedmethods to solve capacity allocation problems in a health care environment. The discussedstudies 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 planson the intermediate term, for multiple resources and multiple care processes. The goal is toachieve 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 bedeterministic and the outcome is a plan that allocates resource capacity over care processesand 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 tacticallevel 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 modelleads to the optimal allocated capacity between new and recurrent patients for each timeunit.

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 overdifferent care processes and over different patient types.

A specific characteristic of the method of Hulshof et al. is that it keeps track of the numberof time units a patient waits at a specific queue. By tracking the number of time units apatient 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 revisitappointments.

A limitation of the model of Hulshof et al. is that it uses a predefined set of care processeswith each a specific number of stages and resources defined as a care path. This leads to afinite number of stages in the care process. The model of Nguyen et al. uses an infinite set of

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stages in the care process because patients can recur with a certain probability independentof how often they visited the department before. An advantage of the model of Hulshof etal. 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 ofsmall instances. For real life sized problems as in hospitals, DP is generally difficult andpossibly intractable [Hulshof et al., 2016]. Therefore, Hulshof et al. [2016] propose a methodto develop a tactical resource allocation and patient admission plan by using ApproximateDynamic Programming (ADP). Their objective is to achieve equitable access and treatmentduration 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 fromeach queue that have been waiting a certain amount of time. The model incorporatesstochasticity in patient arrivals and patient transitions between different stages. Compu-tational results show that the approach provides an accurate approximation of the valuefunctions and that it is suitable for large problem instances.

Tsai [2017] performed a master project continuing on the work of Hulshof et al. [2016]. Shemodeled 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 ofthe model is to minimize the waiting time of the patients in the OR queue and to keep theamount of unused OR time below a certain level.

In the model of Tsai, each doctor has a budgeted amount of yearly sessions to divide. Byusing 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 iscomputational expensive. After approximations and simplifications, computation time isreduced but Tsai was unable to create a yearly roster.

A queuing network analysis is very effective to balance a system fast and easily [Vanberkelet al., 2010]. Calculations are exact and fewer data is needed. Queuing models can help toestimate the number of required staff in each time unit to achieve a performance as indicatedin 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 wherepatients have to (re-)visit specific care providers in a network of care queues still involvemodelling complications.

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3.4.2 Simulation studies

A model must be studied by the means of simulation when the system is too complex toevaluate it analytically [Law et al., 2007]. A simulation model is not used to obtain anoptimal solution but to evaluate inputs of a model numerically in question to see how theyaffect 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 totest 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 verycomplex, time-consuming to write and require detailed information [Law et al., 2007].

3.5 ConclusionThe primary cause for waiting times is the mismatch between supply and demand. Whencapacity 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 Hulshofet 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 resultsare 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 ofHulshof et al. provides an accurate approximation of the value functions and is suitable forlarge problem instances whereas Tsai was unable to create a yearly roster due to a modelthat 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 specificcare providers in a network of care queues still involve modelling complications [Lakshmiand Iyer, 2013].

Computer simulation is not used to obtain the optimal solution but creates the possibility toexperiment 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 isfrequently used, we aim for an optimal solution, and the studies of Hulshof et al. [2013] andNguyen et al. [2015] show promising results. These studies also have the most similaritieswith our problem. By combining the advantages of both studies, we obtain a model thatoptimally allocates blocks over the available doctor capacity while incorporating future ca-pacity and demand.

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4 Solution designIn this chapter we propose a model to tackle the problem as identified in Chapter 2. Tostructure 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 reviewperformed in Chapter 3. Section 4.2 explains the way of data gathering followed by theexplanation 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 thetechnical explanation of our MIP model. We explain our modeling approach in Section 4.4and the validation of the model is done in Section 4.5. The last section, Section 4.6, providesa conclusion.

4.1 Conceptual modelIn order to improve the situation at the orthopedic department, we propose a MIP modelbased 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 doctorcapacity. The goal is to allocate blocks in such a way that the weighted number of waitingpatients is minimized.

The model of Hulshof et al. [2013] is the basis for our MIP. Nevertheless, we should makeadjustments to fit it to our situation. First, we adjust the number of patients that areserved. Hulshof et al. determine the number of patients to serve from a particular queue ata particular stage. However, due to practical reasons, it is in our case not possible that theratio between new and recurrent patients differs per OC block. Therefore, we create oneOC block per doctor that reflects current care demand. Consequently, we know how manypatients 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 patientmoves to the next stage. This probability is independent of how often the patient visitedthe 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 inthe care process. This deterministic delay is described as the advised return time betweenstages based on historical data.

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When constructing the capacity allocation plan, we incorporate the natural variability ofdemand and the natural variability in OR department capacity. We assume service ratesto be deterministic. The arrival rate of new patients is based on historical data. We fitprobability distributions to the patient arrival data and use random numbers to generatepatients for each doctor. We decide that doctors can only work a complete day of two equalblocks (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 manyyears as possible with a fixed doctor capacity. Each year has different new patients arrivalsand therefore we can analyze the systems behaviour. The analyses are provided in Chapter 5.

4.2 Data gatheringWe gather our data from the hospital information system Hix. From Hix, we use the OCand OR data from several periods. The exact periods, the way of calculating and the valuesfor the different parameters are included in Appendix C. An important remark regardingthe data is that we divided the different patient types into new and recurrent patients asexplained in Appendix B.

4.3 Technical modelIn this section, we explain the technical model as introduced in Section 4.1. This section isdivided into two parts. First, we explain in Subsection 4.3.1 how we obtain the transitionrates. This part is described as ’Phase 1’ and input for the MIP model which we explain inSubsection 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 andmust therefore reflect practice as close as possible. We define the transition rate as thefraction 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 carepath, which results in shared OC capacity. There are many reasons for patients to changedoctor, e.g., because of the absence of a doctor. Based on the current data, we are notable to find out the exact reason that causes patients to change doctor. Therefore, weassume that patients do not change doctors in our model. This gives us two possible waysto calculate the transition rates:

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1. Based on the shared and non-shared OC capacityIf 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 theyear as Figure 5 shows in Section 2.5.

2. Based on the non-shared OC capacityIf we calculate the transition rates based on the non-shared OC capacity, we assumedoctors 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 accountas explained in Section 2.5.

We choose to calculate the transition rates based on the non-shared OC capacity. Althoughwe can say only something about 73.5% of the OC capacity, we approximate the currentsituation as close as possible by extrapolating the non-shared OC capacity with the use ofa Monte Carlo simulation. The input for the Monte Carlo simulation are the new patientsarrivals based on the period of 2012 - 2016 (see Appendix C.1). By changing the transitionrates, 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 workindependently. Due to this assumption, we are able to run our model per doctor. Theoutcomes of an individual run can be merged and used as initial solution for the completesystem. 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 simulationresults of 1500 experiments. The confidence interval indicates that the new transition ratesresult in production numbers that lay within the interval with a probability of 95%. Thestandard 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 errorNP 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 theconfidence interval with a probability of 95%. Therefore, we can say that the new transitionrates 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 blockcontains 18 slots. The filling of the slots is based on the ratio between NP and CP thatresults from the Monte Carlo simulation. Consequently, an OC block of doctor B contains7.5 NP slots and 11.5 CP slots. We use fractional values to decrease our computation time.

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As a result of Phase 1, we obtain the transition rates for every doctor such that we can rep-resent the production of 2016. We also obtain the probability distributions for new patientarrivals based on the period of 2012 - 2016 and the service rates per queue per doctor. Theabove-mentioned parameters are used as input for the MIP model which we explain in thenext subsection.

Besides the OC blocks that we use as input for the MIP model (which only contain new andrecurrent patient slots), we also updated the OC blocks that are used in current practice.We updated these blocks such that they reflect patient demand based on our calculations.An updated OC block is included in Appendix E. These updated OC blocks are approvedby the doctors and planners and are already in use in practice.

4.3.2 Phase 2: MIP model

The MIP model schedules OC and OR blocks over the available doctor capacity while in-corporating future capacity and demand. We define S as set of doctors. Set T is the setof weeks in the planning horizon. Furthermore, we consider set I as the set of queues foreach doctor s, A as the set of stations and N as set of time periods a patient is waiting. Wedefine both the routing matrix and the delay matrix, as the Cartesian product of |I| · |I| foreach doctor s. This implies that internal transitions between doctors are not allowed. Therouting matrix is filled with the transition rates that result from Phase 1. We fill the delaymatrix with the deterministic delays as calculated based on historical data as we explain inAppendix C.2. The weight-factor matrix consist of the Cartesian product of |N | · |I| for alldoctors s. As a starting point, we choose to let the weight-factors increase linearly with theform Ax+ b where A = 2 and b = −1. Initially, we do not distinguish between queues.

The weekly doctor capacity, denoted by integer parameter as,t, should be divided over sta-tions A. The number of times a doctor works at a certain station is denoted by the integervariable Xs,t,i. If a doctor works at a station, he serves patients from one or multiple queueswith a service rate φs,i,a. Let variable Pn

s,t,i denote the number of patients that are waitingn time periods at the beginning of week t and variable Cn

s,t,i the number of patients servedthat are waiting n time periods at the beginning of week t. Table 8 shows the used sets,indices, variables and parameters.

In line with both, Hulshof et al. [2013] and Nguyen et al. [2015], patients arrive at thesystem with arrival rate λs,t,i. Once served, patients can move from queue i to queue j orleave the system. Parameter qs,i,j , represent the transition rate and denotes the fraction ofpatients that move from queue i to queue j and the value 1 -

∑j qs,i,j denotes the fraction

of patients that leave the system. We introduce integer parameter ds,i,j as a deterministicdelay between queues i and j for each doctor s.

In accordance with our research goals, we allocate OC and OR blocks over available doctorcapacity to minimize the weighted number of waiting patients. As stated in Section 1.2, weincorporate the natural variability of the patient arrival and the OR department capacity.The objective of the proposed MIP is included in equation 1.

minimize

S∑s

T∑t

I∑i

N∑n

A∑a

βns,iP

ns,t,i

1φs,i,a

(1)

23

We introduce parameter βns,i as a weight to prioritize queues and patients. The service

rate φs,i,a is incorporated to ensure that patients are treated equal between doctors. Theconstraints to model the orthopedic department are given below.

Constraint 2 calculates the number of patients that enter queue i from outside the systemor after being served at queue j. Parameter ls,t,i, indicates the number of initial waitingpatients at the start of the model and parameter λs,t,i the arriving patients from outside thesystem. The double summation covers the interval flow of patients that move from queuesj to queue i.

P 0s,t,i = ls,t,i + λs,t,i +

∑j

∑n

qs,j,i ∗ cns,t−ds,j,i,j ∀s, t, i, n (2)

Constraint 3 updates the number of waiting patients. Doctors cannot serve more patientsthan there are waiting, therefore we introduce Constraint 4. The number of time periods apatients waits must be at least 1 week, therefore n must be larger than zero.

Pns,t,i = Pn−1

s,t−1,i − Cn−1s,t−1,i ∀s, t, i n > 0 (3)

Cns,t,i ≤ Pn

s,t,i ∀s, t, i n > 0 (4)

Constraint 5 ensures that if a doctor is assigned to station a, he serves patients from one ormultiple queues i. ∑

n

Cns,t,i ≤

∑a

Xs,t,a ∗ φs,i,a ∀s, t, i, a n > 0 (5)

By introducing the summation over a we ensure that if a session is completely filled, itcontains the ratio between new and recurrent patients as resulted from the Monte Carlosimulation.

Due to capacity restrictions, we introduce Constraint 6 and Constraint 7. Let parameteras,t be the weekly availability of doctor s that can be filled with OC and OR blocks. Thecapacity ut is the number of blocks that can be performed at the OR department and thesemust be shared among all doctors.∑

a

Xs,t,a ≤ as,t ∀s, t (6)

∑s

Xs,t,a ≤ ut ∀t a = 2 (7)

It is not desirable that a doctor works a complete week full of OC or OR blocks. Thereforewe introduce Constraint 8 and Constraint 9. These constraints ensure that the number ofOC and OR blocks per week t differ at most 3.

Xs,t,1 ≤ Xs,t,2 + 3 ∀s, t (8)

Xs,t,1 ≥ Xs,t,2 − 3 ∀s, t (9)

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The number of yearly blocks per doctor is limited due to agreements. Each year consistsof 52 weeks t. Constraint 10 ensures that the number of blocks per year does not exceedthe agreed number of blocks. We introduce Constraint 10 for every year in the planninghorizon.

52∑t=0

∑a

Xs,t,a ≤ bs ∀s (10)

Table 8: Sets, indices, variables and parametersSets and indices

y ∈ Y Yeart ∈ T Weeks ∈ S Doctora ∈ A Stationi, j ∈ I Queuen Time period (to indicate waiting time)

Decision variablesCn

s,t,i The number of patients served from queue i in week t,who have been waiting n weeks for doctor s

Xs,t,a The number of sessions doctor s works at station a in week t

Auxiliary variablePn

s,t,i The number of patients in queue i at the start of week t,who have been waiting n weeks for doctor s

Parametersβn

s,i Objective function weight of patients in queue i,who have been waiting n weeks for doctor s

λs,t,i New demand at queue i in week t for doctor sφs,i,a The number of patients served from queue i,

if doctor s works a session at station aqs,i,j Probability that a patient moves from queue i to queue jds,i,j Number of time periods to move from queue i to queue jus,t Capacity of doctor s in week tvt Capacity of the OR department in week tbs The maximum number of blocks per year per doctor s

4.4 Modeling approachBecause the model size is too large to solve a real life instance within reasonable time, wefirst determine an initial solution. Due to the assumption that doctors work independently,we can solve the problem for each doctor separately. We use these outcomes as an initialsolution for the complete department.

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To solve the problem per doctor, the OR capacity should be divided over doctors on fore-hand. The most obvious choice is to divide the available OR capacity in six equal portions.However, OR capacity cannot be fractional and should therefore be rounded. We divide theOR capacity over the doctors, which we denote by os,t; the obtained OR capacity per doctors in week t. Note that os,t should always be smaller than or equal to 3 due to Constraint 8and Constraint 9. The steps of the self-made heuristic are included below.

1. Create two basis OR rosters for 52 weeks. Divide the OR capacity vt by 3. Roundthe resulting capacity once up and once down. The result is that we have two basisOR rosters. One with 136 OR blocks a year and one with 140 OR blocks per year.

2. Divide the basis OR rosters over the doctors. The three doctors that need the mostOR blocks per year, denoted as Criteria1 below, obtain the OR roster with 140 blocks.

Criteria1 =∑52

t=0∑N

n Pns,t,i

φs,i,a∀s i = 2, a = 2

3. For each doctor, we check if us,t = 0. If so, and vt > 0, we set os,t to 0.

4. Divide the weekly remaining OR capacity based on Criteria2:

Criteria2 = 1− os,tφs,i,a∑52

t=0∑N

n Pns,t,i

∀s i = 2, a = 2

Start at t = 0 and assign a remaining OR block to the doctor with the highest criteria.Assign the next OR block to the doctor with the second highest criteria (excludingthe doctor that already received a block) until there are no remaining OR blocks left.Increase t with 1 and divide the remaining OR blocks up to t = 52.

5. Solve the MIP model for each doctor separately.

6. Solve the MIP model for the complete department with the use of the initial solution.

4.5 ValidationWe use a blackbox validation to check if the model is an accurate representation of theactual system. Blackbox validation is a useful method because the validation can be per-formed even though there is no knowledge of the internal structure of the object to be tested.

Together with several stakeholders, we checked the input distributions and we comparedthe outcomes of the model with our expectations. We create different scenarios to performa sensitivity analysis and despite the fact that the outcomes of our model are an improvedsituation of reality, we conclude that the values of the variables and the behaviour of thesystem do match our expectations.

As described in Phase 1 in Section 4.3, we not only created OC blocks that are useablefor our MIP model (which only contains new and recurrent patients slots), we also createdOC blocks useable for practice based on our calculations. These OC blocks are approved bythe doctors and planners and are already in use. This is another step that validates our work.

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4.6 ConclusionIn this chapter, we proposed a MIP model which allocates capacity in such a way that theweighted number of waiting patients is minimized. The model of Hulshof et al. [2013] isthe basis for our MIP. We adjust the model to fit it to our situation and we introduced aninfinite set of stages in the care process as proposed in Nguyen et al. [2015].

As the current situation shows, there are patients that change doctors during their carepath which results in shared OC capacity. We obtain the transition rates by extrapolatingthe non-shared OC capacity with the use of a Monte Carlo simulation. We assume doctorsto work independently and therefore patients do not change doctors in our model whichresults in transition rates that reflect reality.

The model size is too large to solve a real life instance within reasonable time. Due tothe assumption that doctors work independently we are able to solve the problem for eachdoctor separately. We create a heuristic that divides OR capacity beforehand. We use theoutcomes of the heuristic as initial solution for the MIP model which reduces the computa-tion time.

A blackbox validation shows that the model does behave like our expectations. We createddifferent scenarios and the values of the variables do match practice. Therefore we can con-clude, together with several stakeholders, that the created MIP model is a true reflection ofreality.

We also updated the OC blocks that are used in practice based on our calculations. Theseupdated OC blocks are approved by the doctors and planners and are already in use inpractice. This step also validates our work.

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5 Experiment resultsIn Chapter 4 we propose a MIP model to minimize the weighted number of waiting patients.In this chapter, we determine how the system behaves in different circumstances to formu-late practical planning rules and to identify a capacity mismatch.

In Section 5.1, we explain the experiment approach. The experiment setup, which we cover inSection 5.2, indicates how we setup the experiments followed by the experiment outcomes inSection 5.3. Section 5.3 is divided into two subsections. We start with a capacity mismatchin Subsection 5.3.1 followed by different planning rules in Subsection 5.3.2. In Section 5.4,we briefly explain our simulation model and we analyze the outcomes. Since the allocation ofsessions within a week is also important, we introduce and explain a side-project in Section5.5. We provide a conclusion of Chapter 5 in Section 5.6.

5.1 Experiment approachThe goal of the research is to reduce the access and waiting time at the orthopedic de-partment. We want to obtain practical planning rules that are applicable in general andwe want to indicate where the doctor capacity us,t and OR capacity vt can be better aligned.

We perform experiments for two scenarios. The two scenarios we use are (1) a realistic sce-nario and (2) an idealistic scenario. For a better substantiation of our analysis, we choose toperform two experiments for each scenario. The two experiments we use are (1) the doctorcapacity us,t and OR capacity vt of 2016 and (2) the doctor capacity us,t and OR capacityvt of 2017. Table 9 shows an overview of the scenarios and experiments.

Table 9: Scenarios and experimentsExperiment Scenario1 Realistic scenario1.1 Capacity of 2016 with OR restrictions1.2 Capacity of 2017 with OR restrictions2 Idealistic scenario2.1 Capacity of 2016 without OR restrictions2.2 Capacity of 2017 without OR restrictions

In the realistic scenario we use the OR capacity vt as it was in practice. In the idealisticscenario, we remove the OR capacity vt limitation. By performing experiments for bothscenarios, we can analyze how the system has to work and how it wants to work.

5.2 Experiment setupWe start each experiment with an empty system. Because our MIP model contains thecharacteristics of a non-terminating simulation, we are interested in the behaviour understeady-state. Before the system reaches it steady-state, it contains a warm up period l. Weuse the graphical procedure of Welch to determine the warmup period.

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The computation time of our model is mainly influenced by the size of two sets: set t andset n. The number of elements in set t should be at least 4 times our warmup period accord-ing to the graphical procedure of Welch. Set n contains a sufficient number of elements ifPN

s,t,i = 0 for the last element of n. We should make t and n as small as possible to decreaseour computation time.

The idea is to generate as much data as possible. However, due to the large variance inour system we need a long warm up period. Consequently, we should increase set t and ourcomputation time increases. After an iterative process of determining the sizes of set t andset n we found out that we should make a trade off between reliability of the output dataand the computation time as our model is not solvable within reasonable time.

As a trade off, we choose to use the batch means method and to perform two replications foreach experiment where we set the stopping criteria equal to 48 hours. We set the runlengthof the model equal to 20 years (t = 1040), the size of set n equal to 10 and we use an initialsolution as suggested in Section 4.4.

One replication results in 1040 observations Y1, ...Y1040. We want to maximize the numberof observations to use and we want to minimize the correlation between observations. AsLaw et al. [2007] suggest, if batch size b is large enough it is reasonable to treat observationsas if they are independent and identically distributed random variables. We choose to set bequal to 1040−l

5 and assume that these batches are approximately uncorrelated. From eachbatch, we take the first year as observations for our analysis.

The parameters we use for the modelling are included in Appendix C. The warmup lengthfor the MIP model is elaborated in Appendix D and the outcomes are also included inSection 5.3. Because we cannot solve the replications to optimality within 48 hours, we alsoinclude the integrality gap for each experiment in the next section.

5.3 Experiment outcomesTable 10 shows the general information of each replication. The warmup length is elabo-rated in Appendix D. Each experiment results in 1560 usable observations.

Table 10: The specifications of the replicationsExperiment Run Integrality gap Warmup length

1.1 1 9.07% 520 weeks1.1 2 13.22% 520 weeks1.2 1 11.58% 520 weeks1.2 2 10.65% 520 weeks2.1 1 6.68% 520 weeks2.1 2 6.01% 520 weeks2.2 1 7.41% 520 weeks2.2 2 6.76% 520 weeks

The remainder of this section is divided into two subsections. First we cover the strategicallevel in Subsection 5.3.1, where we indicate a capacity mismatch. Subsection 5.3.2 coversthe tactical level in which we explain the planning rules.

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5.3.1 Capacity mismatch

In the idealistic scenario, the orthopedic department can organize the allocation of blocksin such a way that they are not limited by other departments. We want to analyze how thesystem behaves in an idealistic scenario by removing limitations.

There are two capacity limitations, the doctor capacity us,t and the OR capacity vt. Anincrease in doctor capacity us,t is practically not feasible therefore we choose to increase theOR capacity vt. Due to the existence of Constraint 8 and Constraint 9, the maximum num-ber of OR blocks per doctor s per week t is 6 (3 days of 2 blocks). We set the OR capacityvt for each week t equal to 36 blocks (6 blocks a week times 6 doctors). The consequence isthat the OR capacity vt is not a limiting parameter anymore and we can analyze how themodel wants to allocate blocks.

Figure 13 and Figure 14 show how the model allocates OR capacity in case the OR capacityis not a limiting parameter for respectively 2016 and 2017. The allocation of blocks is theaverage of the different replications.

Figure 13: Average OR allocation without capacity limitations of 2016(Source: MIP model, n=420)

Figure 14: Average OR allocation without capacity limitations of 2017(Source: MIP model, n=400)

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A first remark is that the total yearly OR capacity is 1% more than the allocated numberof OR days (outside and within the OR capacity). This indicates that there is sufficient ORcapacity available. However, as Figure 13 and Figure 14 show, we can identify mismatches.The dark grey bars indicate that, based on the available doctor capacity us,t, more ORcapacity vt in the corresponding week positively influences the access and/or waiting time.The white bars indicate that the department could work with less OR capacity in the cor-responding week if they receive the capacity of the dark grey bars.

We are interested in the weeks where there is always capacity allocated outside the obtainedOR capacity. These weeks are remarked as the crucial weeks and included in Figure 15 andFigure 16 for respectively 2016 and 2017.

Figure 15: Crucial weeks in 2016(Source: MIP model, n=420)

Figure 16: Crucial weeks in 2017(Source: MIP model, n=400)

There are four weeks (9, 11, 33 and 47) in which always extra OR capacity is desired basedon the available doctor capacity us,t and regardless of the number of new patients arrivals.Figure 17 shows the capacities of the OR department and the doctors as we saw in Chapter2, but we also included the crucial weeks for the capacities of 2016 in Figure 17.

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Figure 17: Crucial weeks in 2016 facing the capacities(Source: DES model and MedSpace)

We can see in Figure 17 that the crucial weeks are timed around holidays and occur if thecapacities do not follow the same pattern which we remark as a mismatch in capacities.Week 11 is a special case since the capacities do follow the same pattern, however week 11 isthe anesthesia week. In this week, there is less anesthesia capacity and therefore remarkedas an crucial week.

Initially, the department feels that there is sometimes a mismatch in capacities aroundholidays due to different holiday regions as indicated in Chapter 2. Due to the use of theMIP model, we have a quantitative substantiation for their feelings and we recommend thedepartment to align the capacities on forehand since this positively influences the accessand/or waiting time.

5.3.2 Planning rules

The manager and planner indicate that they want to know how to handle in case of (1)different new patient arrivals, in (2) holiday season and (3) non-holiday season. We treatthese cases in chronological order. We use both, the data resulting from the idealistic andrealistic scenario, to formulate practical planning rules applicable for the tactical level.

1. Different new patient arrivalsTo indicate how the department should handle in case of different new patient arrivals, weanalyze how the model allocates OC blocks if we only incorporate new patient arrivals.Table 11 shows the outcomes of the MIP model and we explain the table below. In theleft upper corner we show the outcomes of case 1 (Arrivals

∑t−1t−1 λs,t,1). In this case, we

look at the number of allocated OC blocks in week t if we incorporate a new patient arrivalperiod of 1 week (from t − 1 up to t − 1). The y-axis contains the number of allocatedOC blocks in week t and the x-axis the arrival intensity divided into a low, medium andhigh intensity including their corresponding patient arrival boundaries. We can see in Table11 that, in case of a low new patient arrival intensity, the model chooses in 38% of thesituations to allocate 2 OC blocks per doctor in week t, in 47% of the situations for 4 OCblocks per doctor in week t and in 15% of the situations for 6 OC blocks per doctor in week t.

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Table 11: Allocation of OC blocks depending on new patient arrivals(Source: Outcome of MIP model, n=5067)

1. Arrivals∑t−1

t−1 λs,t,1 2. Arrivals∑t−1

t−2 λs,t,1Low Medium High Low Medium High

OC blocks 0 10 20 OC blocks 0 20 40in week t 10 20 999 in week t 20 40 999

2 38% 33% 29% 2 39% 34% 28%4 47% 49% 51% 4 47% 49% 51%6 15% 18% 20% 6 14% 17% 21%

3. Arrivals∑t−1

t−3 λs,t,1 4. Arrivals∑t−1

t−4 λs,t,1Low Medium High Low Medium High

OC blocks 0 30 60 OC blocks 0 40 80in week t 30 60 999 in week t 40 80 999

2 38% 37% 30% 2 43% 34% 26%4 55% 46% 51% 4 46% 48% 53%6 7% 17% 19% 6 11% 18% 20%

If we compare the different cases from Table 11 we conclude that, regardless of the length ofthe incorporated time horizon and the intensity level of new patient arrivals, the model doesnot allocate OC blocks differently. Therefore we can say that the model does not heavilyreact to the arrival intensity of new patient arrivals.

As we conclude in Section 2.3, secretaries have the ability to change the type of patient slotsin OC blocks to fulfill demand. The manager and planner indicate that in almost everyOC block slot types are changed as the secretaries react heavily to new patient arrivals.Therefore we advice the department to introduce our new OC blocks and to limit changingthe type of patient slots of OC blocks to a minimum. We recommend to update the OCblocks yearly with the use of the created Excel tool.

2. Holiday seasonTo indicate how the department should handle in case of holidays, we analyze how the MIPmodel allocates blocks before and after holidays and in case of different holiday lengths.Figure 18 summarizes the outcomes of the realistic scenario of the MIP model aggregatedfor all doctors and based on the capacities of 2016 and 2017. We include the bars thatrepresent the percentage of available capacity that is allocated to OC blocks. One minusthat percentage is the capacity allocated to OR blocks.

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Figure 18: Allocation of blocks in holiday season - Realistic scenario(Source: MIP model, n=5067)

Figure 18 shows that, regardless of the length of the holiday, the model does not anticipatedifferent before holidays. However, the model allocates more capacity to the OC after theholiday if the length of the holiday increases as Figure 18 shows. One could argue that morecapacity is allocated to OC blocks because of OR capacity limitations that are present inthe realistic scenario. Therefore we also analyze how the model allocates capacity aroundholidays in the idealistic scenario (without OR capacity limitations) as Figure 19 shows.

Figure 19: Allocation of blocks in holiday season - Idealistic scenario(Source: MIP model, n=5067)

We still can conclude that the model does not anticipate before holidays. However, themodel does allocate more capacity to OC blocks after a 2-week holiday. Two weeks after a3-week or longer holiday, the model does allocate less capacity to OC blocks. We can for-mulate the following planning rule: Allocate more than 50% of the available doctor capacityus,t to OC blocks after a holiday of minimal 2 weeks.

3. Non-holiday seasonNext to the planning rules for new patient arrivals and holidays, we formulate rules for non-holiday season which are applicable in general and divided into planning rules for OC, ORand holidays. We formulate the rules in days (2 blocks) since this is the most appropriatefor practice. These rules are based on the different experiments and included below.

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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 us,t = 4.(b) Plan a maximum of 3 OR days if the weekly doctor capacity us,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 us,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 us,t to OC days

the week after a holiday of minimal 2 weeks.

5.4 Simulation modelTo evaluate the performance of our planning rules we use a Discrete Event Simulation (DES)model. First, we want to simulate the current situation in Plant Simulation by using the OCand OR blocks as they were planned in 2016. Second, we apply our planning rules for OC,OR and holidays and evaluate the situation again. Based on the performance indicators,which we clarify in Subsection 5.4.1, we can compare the current and suggested situation.

We use the improved OC blocks that result from Phase 1 in Section 4.3 as input for theDES model. The reason we use these blocks is that (1) we cannot represent practice as thefilling of the blocks in practice varies per planner due to the use of intuition and experienceas concluded in Chapter 2 and (2) we used these OC blocks in our MIP model to formulatethe planning rules.

Idealistically, we want to model the department at once in Plant Simulation. However, dueto size limitations we are unable to create such a big model at once. Therefore we per-form the DES model for each doctor separately. We switch OC and OR blocks based onour rules. As, for example, the rules indicate that we should perform more OC blocks, wechange an OR block in an OC block. Consequently, the access time decreases, the waitingtime increases and we have one OR block left to divide to other doctors. In practice, thisOR block can be divided to other doctors, however, since we model the situation per doctor,we do not know which doctor needs the OR block the most. Therefore, we are unable todivide this remaining OR block in our DES model. The consequence is that the outcomesare pessimistic since in practice remaining OR blocks can be divided.

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We start each run with an empty system and determine the warmup length with Welch’graphical procedure. For each doctor, we perform replications to construct a 95% confidenceinterval for the mean access and waiting time by using the replication/deletion approach.In Appendix G, we further elaborate and validate the DES model and explain the processflow, assumptions, simplifications, warmup length and replications.

5.4.1 Performance indicators

We use performance indicators to compare the current situation with the suggested situ-ation. The performance indicators are based on our objective of reducing the access andwaiting time for the orthopedic department. Besides the access and waiting indicators, weintroduce the utilization of the OC and OR as performance indicators. Table 12 summarizesthe performance indicators.

Table 12: Performance indicatorsSymbol Performance indicatorX Average access and waiting timeσ Standard deviation in access and waiting timeMin Minimum access and waiting timeMax Maximum access and waiting timeUOC Average outpatient clinic utilizationUOR Average operating room capacity utilization

We calculate the average utilizations by dividing the number of treated patients by thenumber of available slots. The standard deviations are the deviation within the year.

5.4.2 Current situation

We simulate the current situation by using the OC and OR blocks as they were plannedin 2016. The outcomes of the current situation of the DES model are included in Table 13and validated in Appendix G. Note that the minimums cannot be lower than one becausewe assume that patients have to wait at least one week.

Table 13: Access and waiting time statistics of the current situation(Source: DES model, n=4680)

Access time in weeks Waiting time in weeksDoctor X Min Max σA X Min Max σW

A 2.8 1.1 5.4 1.5 6.2 4.4 9.1 1.4B 2.7 1.7 4.7 0.7 5.7 3.8 8.5 1.4C 2.5 1.6 4.3 0.5 3.8 2.6 6.7 1.0D 3.4 1.6 7.4 1.3 5.7 3.3 9.0 1.9E 3.4 2.2 5.5 0.8 5.8 4.1 9.1 1.6F 2.6 1.4 5.2 1.0 3.9 1.8 7.4 1.7Averages 2.9 1.6 5.4 1.0 5.2 3.3 8.3 1.5

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Figure 20 and Figure 21 show the corresponding OC and OR utilization of the departmentfor the current situation following from the DES model. Note that the OR utilization ishigh because the OR utilization is defined as the time between the start of the first surgeryand the end of the last surgery divided by 8 hours. This means that we incorporate thetotal turn overtime and therefore the utilization can be this high.

Figure 20: Average OC utilization of the current situation(Source: DES model, n=520)

Figure 21: Average OR utilization of the current situation(Source: DES model, n=520)

Based on Figure 20 and Figure 21, we conclude that the utilizations are lower in the firsthalf of the year. We include the exact specifications of the utilizations in Table 14. The col-umn ’1st’ indicates the average utilization in the first half of the year and the column ’2nd’the average utilization in the second half of the year. The column ’σ’ shows the standarddeviation of the utilization.

Table 14: Utilization specifications of the current situation(Source: DES model, n=520)

1st 2nd Average σOC 87.8% 91.0% 89.4% 6.2%OR 95.2% 98.2% 96.7% 3.2%

The values in Table 14 do correspond with practice since the department experiences astressful period in the second half of the year as we concluded in Subsection 2.8.1. In theperiods with a lower utilization, the department is able to lower their access and waitingtimes again.

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5.4.3 Suggested situation

In the suggested situation, we change blocks based on the planning rules we formulate inSubsection 5.3.2. The outcomes of the suggested situation are included in Table 15.

Table 15: Access and waiting time statistics of the suggested situation(Source: DES model, n=4680)

Access time in weeks Waiting time in weeksDoctor X Min Max σA X Min Max σW

A 2.6 1.1 5.3 1.4 8.5 7.1 10.3 1.0B 2.2 1.4 4.4 0.6 8.9 7.3 11.2 1.2C 2.6 1.6 4.3 0.5 3.3 2.2 6.3 1.0D 3.6 1.7 7.5 1.4 5.0 2.8 8.2 1.7E 3.0 2.0 4.8 0.6 7.5 6.1 10.1 1.3F 2.2 1.3 4.6 0.7 5.9 3.7 9.3 1.7Averages 2.7 1.5 5.2 0.9 6.5 4.9 9.2 1.3

From Table 15 we can conclude that the average access time is decreased and the waitingtime increased. The shift in averages is in line with our expectations since remaining ORblocks, in total 10 blocks, can not be divided. The minimum and maximum access timesare equal or decreased while the minimum and maximum waiting times are increased. Mostimportant, we remark that the deviations in access and waiting time are decreased. Theaverage standard deviation of the access time is decreased from 1.0 weeks to 0.9 weeks (-10%) and the average standard deviation of the waiting time is decreased from 1.5 to 1.3weeks (-13%). This means that due to the usage of our planning rules, both the access andthe waiting time become more stable since they contain less variability. Note that thereare 10 OR blocks remaining that can be divided in practice, which decreases the averagewaiting time. However, if these sessions are divided it can influence the other performanceindicators but we are unable to test this.

Figure 22 and Figure 23 show the utilization of the suggested situation. Table 16 containsthe exact specifications of the utilization of the suggested situation.

Figure 22: Average OC utilization of the suggested situation(Source: DES model, n=520)

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Figure 23: Average OR utilization of the suggested situation(Source: DES model, n=520)

Table 16: Utilization specifications of the suggested situation(Source: DES model, n=520)

1st 2nd Average σOC 87.0% 90.0% 88.5% 6.0%OR 96.3% 98.8% 97.5% 2.4%

If we compare Table 14 and Table 16, we conclude that the average utilization of the OC isdecreased with 0.9% and the average utilization of the OR increased by 0.8%. This shift isin line with our expectations. The standard deviations of the utilization of the OC and ORare decreased with respectively 3% and 25%. The combination of a lower average utiliza-tion and less variance in the utilization for the OC means that we introduce a more stableworkload for the OC if we use our planning rules. For the OR, there are 10 OR blocksremaining to divide and therefore the average utilization increases and since the utilizationis bounded (100%) we can lower the standard deviation in OR utilization by 25%. Notethat this reduction of 25% will be different in practice since remaining OR blocks can bedivided and the utilizations are not bounded.

We use the DES model to obtain the performance of the planning rules while not everydetail of the department is incorporated. However, the planning rules are substantiated,easy to implement and they ensure that blocks can be divided without the use of experienceand/or feelings. Because the differences between the performance indicators for the currentand suggested situation are positive, we advise the department to use them.

5.5 Weekly planningSince we provide the planning rules and the mismatch in capacities on a weekly basis, theplanner knows how many blocks are required during a week. However, it is also importantwhen these blocks are planned during the week when we consider the flow of patients to-wards the ward. Currently, the department does not take into account the flow of patientstowards the ward. The ward works reactive by adjusting their workforce. Therefore we in-troduce a side-project to obtain guidelines and helpful practical tools for planning OR blocks.

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The goal of the side-project is to minimize the variation in the number of used beds usedby orthopedic patients in the ward between Monday to Friday. We choose for Monday toFriday since elective surgeries are only performed during weekdays.

We introduce W as week and Ud as the number of used beds by orthopedic patients on dayd. Our objective can be translated into the following equation:

minimize

W∑w

∑5d=1(Ud − Uw)2

5 (11)

Where Uw is equal to the average beds in use by orthopedic patients within week w (fromMonday to Friday).

There are three important parameters per doctor to take into account:

1. The average number of clinical patients that are operated by doctor s during an ORblock.

2. The average length of stay of patients that undergo a surgery by doctor s.

3. The number of available blocks per day d for each doctor s.

Another parameter to include is the number of OR blocks and their corresponding outputof previous week as the length of stay can be longer than 4 days (patients operated in pre-vious week still require a bed this week). The last parameter to include is the available ORcapacity for each day d.

We covered the above-mentioned scheduling problem in an Excel spreadsheet were we alsoinclude restrictions regarding operating tools as the number of tools are limited. The spread-sheet can be used to obtain the optimal allocation of OR blocks for the corresponding week.However, the planner indicates that she also uses the spreadsheet to see what the conse-quences are regarding the flow of patients if doctors want to change OR blocks.

Since several managers experience the side-project as important and easy to implement, weextended the spreadsheet. In this extended spreadsheet, we also include other specialismsthat provide a flow of patients towards ward ’D2’. As the orthopedic department providesthe largest supply of patients into ward ’D2’, we use their allocation of OR blocks as basis.The extended spreadsheet optimally allocates OR blocks for the other specialisms such thatwe can obtain a more stable number of used beds at the ward during the weekdays.

Note that we fill the spreadsheet with history based averages. The actual filling of the ORblocks can still introduce peaks and since there are no elective surgeries performed duringweekends, the number of used beds in the beginning of the week remains low. Therefore weintroduce plannings rules for filling orthopedic OR blocks to give more direction to the flowof patients. Since these planning rules are doctor and injury type specific, we include thesein Appendix F.

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5.6 ConclusionFirst, our MIP model identifies a mismatch between the doctor and OR capacity. We pro-vide a quantitative substantiation to confirm the feelings of the department that they feela mismatch in capacities during holidays as identified in Section 2.6. We recommend thedepartment to align the capacities during holidays on forehand since a good alignment pos-itively influences the access and/or waiting time.

Second, we use the outcomes of the MIP model to indicate how the department shouldhandle in case of (1) new patient arrivals, (2) holiday season and (3) non-holiday season.The outcomes show that, regardless of the length of the incorporated time horizon and theintensity level of new patients arrivals that the model does not allocate OC blocks differ-ently. Therefore we can say that the model does not heavily react to the arrival intensityof new patient arrivals. We advice the department to introduce our new OC blocks andto limit changing the type of patient slots of OC blocks to a minimum. We recommend toupdate the OC blocks yearly with the use of the created Excel tool.

After analyzing the outcomes of the MIP model, we conclude that the model does not antic-ipate before holidays. However, after holidays the following planning rule applies: Allocatemore than 50% of the available doctor capacity us,t to OC days after a holiday of minimal 2weeks. We formulate general applicable planning rules divided into planning rules for OC,OR and holiday as we include in Section 5.3.2.

We use the DES model to evaluate the performance of the planning rules while not everydetail of the department is incorporated. We are unable to simulate the exact performancesince we are not able to divide 10 remaining OR blocks. Therefore the mean waiting timeis pessimistic. 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 workload for the OC (σ -3%). Theplanning rules are substantiated, easy to implement and they ensure that blocks can bedivided without the use of experience and/or feelings. Because the differences between theperformance indicators for the current and suggested situation are positive, we advise thedepartment to use them.

Since the timing of blocks within a week is important for the flow of patients towards theward, we introduce a side-project to obtain guidelines and helpful practical tools for plan-ning OR blocks. Currently, the flow of patients towards the ward is not incorporated andthe ward works reactive by adjusting their workforce. The goal of the side-project is tominimize the variation in the number of used beds used by orthopedic patients in the wardbetween Monday to Friday. We covered the OR scheduling problem in an Excel spreadsheetfor the orthopedic department and we extended this to also incorporate the flow of patientsof other specialisms towards ward ’D2’. Since the actual OR filling can still introduce peaks,we formulate doctor and injury type specific planning rules to give more direction to theflow of patients as included in Appendix F.

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6 Conclusion and recommendationsThis chapter covers the conclusion and recommendations regarding our research. In Section6.1, we cover the conclusions followed by the contributions to both, practice and theoryin Section 6.2. Besides the recommendations that we cover in the conclusion, we havegeneral recommendations for the department as we include in Section 6.3. To implementour suggestions in practice, we delight our thoughts on how to implement them in Section6.4. Since there is always room for more improvement, we offer several options for furtherresearch in Section 6.5 followed by a discussion of the research in Section 6.6.

6.1 ConclusionThe orthopedic department experiences a stressful period in the second half of the year. Inthis period, secretaries indicate that it is difficult to find empty patient slots within reason-able time. The planners do not know if they should plan OC blocks or OR blocks and thedoctors feel that they work overcrowded and inefficient sessions. The department also feelsthat there is sometimes a mismatch between their doctor capacity and the capacity of theOR department.

We conclude in Chapter 2 that the average exceedance probability of the access time normis equal to 19.5% and the average exceedance probability of the waiting time norm is equalto 30.5%. The department uses a static allocation of blocks and the filling of OC blockscontains predefined patients slots that are not based on current demand. As the planninghorizon decreases, the department tries to control the access and waiting time by (1) switch-ing between OC and OR blocks (Section 2.2) and (2) by changing the type of patient slotsin 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 theflow of patients within the department.

We develop a MIP model (Chapter 4) that allocates blocks in such a way that the weightednumber of waiting patients is minimized. We use the MIP model as simulation-optimizationapproach to obtain practical planning rules and to indicate where doctor and OR capacitycan be better aligned. Since the MIP model is too large to solve a real life instance withinreasonable time, we first determine an initial solution by using a self-made heuristic (Section4.4) that divides OR capacity on forehand to reduce computation time.

The outcomes of the MIP model provide a quantitative substantiation to confirm the feel-ings of the department that they feel a mismatch in capacity during holidays. The cause isthat some doctors live in another region of the Netherlands where holidays are timed differ-ently. We recommend the department to align capacities on forehand since this positivelyinfluences the access and/or waiting time.

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We formulate practical planning rules how the department should handle in case of (1) newpatients arrivals, (2) holiday season and (3) non-holiday season. We conclude that the modeldoes not heavily react to new patient arrivals and therefore we advice the department tointroduce our new OC blocks and to limit changing the type of patients slots of OC blocksto a minimum. We recommend to update the OC blocks yearly with the use of the createdExcel tool. We formulate the planning rules in days (1 day contains 2 blocks) since this isappropriate 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 us,t = 4.(b) Plan a maximum of 3 OR days if the weekly doctor capacity us,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 us,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 us,t to OC days

the week after a holiday of minimal 2 weeks.

We use the DES model (Appendix G) to obtain the performance of the planning rules whilenot every detail of the department is incorporated. The outcomes of the DES model showa more stable access time (σ -10%), a more stable waiting time (σ -13%) and a more stableworkload for the OC (σ -3%). The planning rules are substantiated, easy to implementand 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 suggestedsituation are positive, we advise the department to use them.

The planning rules ensure guidelines for every week. However, since the daily allocation ofOR blocks is important for the flow of patients towards the ward, we introduce a side-projectto obtain practical guidelines and helpful practical tools for planning OR blocks within aweek. The goal of the side-project is to minimize the variation in the number of beds usedin the ward within the corresponding week. The side-project has resulted in an easy to useExcel spreadsheet for the orthopedic and OR planners. For the actual filling of OR blocks,we formulate doctor and injury specific planning rules to give more direction to the flow ofpatients towards the ward (Appendix F).

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6.2 ContributionsIn this section, we discuss the contributions to theory and practice.

Contributions to theoryThis research tackles a resource scheduling problem for elective care where patients needmultiple appointments at multiple resources. By combining and expanding the models ofHulshof et al. [2013] and Nguyen et al. [2015] we generate a model that allocates blocksover the available doctor capacity. The planning rules that we formulate have been testedin a real life case by using a DES model. The results are promising, and the research isinteresting for surgical specialties that need to allocate capacity to minimize the weightednumber of waiting patients.

The side-project contributes to theory since we created a self made Excel spreadsheet modelthat minimizes the variability in used beds in the ward. The model is general applicable,incorporates real life constraints and is interesting for surgical specialties where patientsrequire a bed at the ward.

Contribution to practiceOur research contributes to practice in several ways. First, we created new OC blocks forevery doctor that do reflect patient demand. These blocks are not only created, but alsoalready in use and are experienced as pleasant to work with.

Second, we contribute to practice as we translated the outcomes of the mathematical modelsinto easy applicable practical rules. These rules are used in practice and the planner indi-cates that she also uses the planning planning rules to negotiate with the OR department.Before, it was difficult to obtain extra OR capacity if it is required. However, with theplanning rules she has a substantiation to negotiate.

Third, we created two Excel tools. The first Excel tool is explained in Section 5.5. Anothertool we created is an Excel tool that is helpful in case changes have to be made to OCblocks. The tool helps to validate whether patient types are in proportion and the planneris able to update OC blocks regarding patient demand.

The fourth contribution to practice is that we extended the Integral Capacity Management(ICM) support. ICM is in the initial stage within DZ. We performed several presentationsfor different organizational layers to inform, but also try to activate the audience to thinkhow they can contribute to ICM.

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6.3 RecommendationsBesides the recommendations that follow from our research, we define some general recom-mendations that we think are interesting and or useful for the department. We describethese general recommendations below.

Preference patientsA situation that occurs in practice is that a fraction of the patients does have a prefer-ence for their surgery date. Currently, the department experience that patients prefer tobe operated after the summer. However, there is no data available. During the research,we introduced a check mark in the planning process such that preference patients can befiltered. We recommend the department to analyze the data and to gain insight into theexact behaviour and wishes of preference patients.

Consultation timesCurrently, each OC consult takes 10 minutes independent of the patient type (except sec-ond opinions (20 minutes)). The length of 10 minutes is experienced as sufficient. Still,it is good to gain insight into the realization. By monitoring the consultation times, thedepartment can benchmark between doctors and assistants which can result in an improvedperformance. Also, one may expect a relation between the length of the consult and thenumber of visits which can be analyzed once consultation times are monitored.

Access timesEvery week, the access times for each doctor and injury type are monitored. By hand, asecretary searches for the third free spot. The third free spot is considered as the accesstime. The way of monitoring is error prone and time consuming, therefore we recommendthe department to automate the access time monitoring because decisions are based on thisinformation.

Shared capacityIn Section 2.5, we conclude that doctors share OC capacity. Sharing capacity is efficient toremain flexible. However, we have noticed in practice that this flexibility can be abused.Since there is a doctor specialized in feet, idealistically patients with feet injuries are plannedat his sessions. However, if his access time is high secretaries can plan this patient at anotherdoctor to guarantee a low access time. Since the non-specialized doctor is not specializedenough in feet, the probability is high that the patient still should visit the specialized doc-tor. This way of sharing capacities is not effective and should be avoided.

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6.4 ImplementationTo implement our suggestions in practice, we indicate how we think that the suggestionsshould be implemented. We divide this section into (1) the capacity mismatch, (2) theplanning rules and (3) the weekly planning.

Capacity mismatchIn Subsection 5.3.1 we provide a quantitative substantiation for the mismatch in doctor ca-pacity and OR department capacity. A cause for the mismatch is that some doctors live inanother region of the Netherlands where holidays are timed differently than in the region ofthe hospital. We recommend the department to align the capacities on forehand since thispositively influences the access and/or waiting time. The yearly OR capacity is dimensionedin the last quarter of the previous year. In order to ensure proper coordination and align-ment of capacity, it is important that the capacities for both, the orthopedic departmentand the OR department are dimensioned at the same time.

Planning rulesThe planning rules for OC, OR and holidays are covered in Subsection 5.3.2. The plan-ning rules are quantitatively substantiated and it is important that the planner, doctorsand manager are aware of the substantiation since this can convince people. Therefore wegive a presentation for these people so that it becomes clear where the rules are based on.The planner is the person that should actually use the planning rules, therefore we have tomake sure that she understands and knows how to apply them. That is why we organize aseparate session in which all rules are discussed once again and in which questions can beasked. We are also available for questions at all times.

Weekly planningThe side-project, where we focus on the daily planning of OR blocks, is important on a dailybasis. Since there are many stakeholders involved, we must ensure that all these stakehold-ers are properly informed and be aware of the consequences of the project. Therefore, wealready formed a project group at the beginning of the side-project. The project groupconsist of the involved stakeholders and frequently and together we discussed the Excelspreadsheet. We have also looked at how we can create extra support and how the variousdepartments can help each other in order to ensure that the variability in the number of usedbeds in the ward is minimized. The hospital must ensure that periodic meetings will continue

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6.5 Further researchThe care path of an orthopedic patient (Section 2.1) is very versatile and therefore the or-thopedic activities do influence various departments. Within this research, we focused onthe access and waiting time. However, there is room for improvement on several aspects,therefore we offer a number of options for further research below.

Patients in wardIn Section 5.5, we introduce a side-project where we focus on the timing and filling of ORblocks where the goal is to minimize the variability in used beds in the ward. In currentpractice, the wards work reactive by adjusting their workforce. The idea behind the side-project is to work proactive, however, due to the use of history based averages the approachis quite simplistic. We recommend to start a research which focuses entirely on the fillingand timing of OR blocks such that we can reduce the variability of used beds in the ward.An example of a research that focuses on an improved patient flow towards the ward byadjusting the Master Surgical Schedules (MSS) is the research of Vanberkel et al. [2011]

Patients to CRNAt the Center for Radiology and Nuclear Medicine (CRN), imaging and or diagnostics scansare made. The number of orthopedic patients at CRN largely depends on the activities per-formed at the orthopedic department. In current practice, the CRN works reactive andthe alignment between both departments can be improved. Currently, the number of slotsreserved for orthopedic patients at CRN is independent on the number of performed OCblocks at the orthopedic department. The consequence is a mismatch in supply and demandand long waiting times. Also the timing of orthopedic patient slots at CRN can be improvedsince orthopedic patient slots at CRN are offered in one timezone of the day but the in-flow of patients is divided over the day. We believe that a research in this area has potential.

Patients during summerDuring the research we introduced a check mark in the planning process such that preferencepatients can be filtered. When data is available, one can start a study on how to influenceand how to plan preference patients. This means that both, qualitative and quantitativeresearch should be carried out. Qualitative research should ensure that we get insight intohow we can steer and influence patients such that they want a surgery when we have capac-ity. Quantitative research should indicate how we should plan preference patients once weknow the number of preference patients and their behaviour.

Booking limitsThe goal of the orthopedic and OR department is to get the highest possible OR utilization.However, the planners indicate that they find it difficult to make a decision when they shouldor should not plan a patient with regard to the OR utilization. The described problemcontains similarities to the planning of aircraft passengers known as revenue management.Revenue management uses queuing theory in order to substantiate choices at any momentin time. By the use of revenue management, booking limits and useful guidelines can beintroduced to help the planners. An example of a research that uses revenue managementin order to manage OR capacity is the study of Stanciu et al. [2010].

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6.6 DiscussionWe discuss the research in this section. In Subsection 6.6.1 we cover the expectations of ourresearch followed by the limitations of our research in Subsection 6.6.2.

6.6.1 Expectations of the research

VarianceThe research focuses on the allocation of orthopedic doctors. In some periods, there areorthopedic specialists (in Dutch: chef de clinique) hired to help the orthopedic doctors.These specialists treat patients and because of the presence of the specialist, an orthopedicdoctor treats less patients. The consequence is that more variability is introduced in thenew patient arrival pattern of an orthopedic doctor since we neglect the specialists and theirtreated patients in our research. An opportunity to reduce the variability and acceleratemodeling was to make a correction for these specialists. However, we did not chose forthis because each specialist is different and has a certain specialty which should increasecomplexity. Beforehand, we did not expect this variability.

Patient transitionThe orthopedic doctors do have different specialties but a certain overlap. In the beginningof the research, we expected and assumed that only a small fraction of patients changebetween doctors during holidays. Therefore, we calculated the transition rates while ne-glecting capacity sharing. We found out that we were unable to reflect current practice.After searching for different causes, we found out that doctors do share more capacity aswe concluded in Section 2.5. Section 2.5 confirms our expectations that capacity sharingis higher during holidays, however, we did not expect that on average 26.5% of the OCcapacity was shared.

MIP modelOn forehand, we wanted to extend our knowledge of linear programming. Since linear pro-gramming was also suitable for modeling the problem, it became a challenge. However, wedid not expect that complexity was such an issue regarding the computation time as we ex-plain in Section 5.2. The two sets that influence our computation the most where set t andset n. However, we found out that values of our weight factor βn

s,i also influence the com-putation time a lot. It took a lot of time to get a feel for the model which we underestimated.

FlexibilityAs stated in Section 2.3, the patient planning is flexible since secretaries can change the typeof patient slots to fulfill demand. Since we cannot retract data about slots changes from Hix,we are unable to quantify the number of changes. We were expecting secretaries to changethe type of slots in exceptional cases, however, according to the manager and planner thishappens in almost every OC block. The planned OC blocks do confirm the statement andone can directly see that most of the OC blocks do not match with the predefined grids.We were not expecting this flexibility.

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6.6.2 Limitations of the research

Despite the fact that we want to include as much detail as possible in the research, thereare some limitations which we cover in this section.

Patients’ preferenceIn practice, a fraction of the patients does have a preference for their surgery date. Sincemost of the surgeries are elective, patients are dependant on informal caregivers and as therehabilitation process takes some months, patients prefer to be operated after the summer.This phenomena is experienced in practice, however, the department has no insight in thenumber of patients and their exact wishes and preferences. As there is no data present, wewere unable to include patients’ preferences in our model.

Patient typesIn the database of the hospital, there are many patient types. However, in practice wemostly talk about new and recurrent patients. Consequently, we decided to divide the dif-ferent patient types into new and recurrent patients as we approach the research from ahighly aggregated level. One could argue that we introduce a certain inaccuracy by dividingpatients types as for example, a yearly checkup and a diagnosis discussion are both remarkedas recurrent patients.

MIP modelIn Section 4.5, we validate our MIP model by using a blackbox validation. However, thereare some limitations as we explain below.

• Warmup lengthWe use the graphical procedure of Welch to determine the warmup length of the MIPmodel. As stated by Welch, the runlength of each replication must be at least fourtimes the warmup length. Further, Welch state that the number of replications fordetermining the warmup length must be at least five.

Due to the complexity of our MIP model, we were unable to solve a runlength longerthan 1040 weeks, even with the use of an initial solution. As our MIP model was notcompletely stable after 260 weeks, we assumed the warmup length to be 520 weeks.This assumption is based on 2 replications instead of 5 or more due to time limitationsas each replication takes 50 hours to solve.

• Integrality gapIdealistically, we want to solve the MIP model many times to optimality when we usethe simulation optimization approach. Nonetheless, we were unable to obtain optimalsolutions as our remaining gap was on average 8% after 50 hours of solving.

DES modelEven though we cannot represent the filling of OC blocks as practice in our DES model, weare still able to represent practice (Appendix G). However, there are still some limitationsregarding the DES model which we cover below.

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• First-Come-First-Serve (FCFS) policy. In practice, patients arrive and in consultationwith the secretary they receive a patient slot that fits their preference. However, inour DES model each patient is planned at the first free patient slot which results in aFCFS queuing principle. Other queuing principles may result in different outcomes.

• Per doctor instead of aggregated. As we explain in Section 5.4, idealistically we wantto model the department at once in Plant Simulation. However, due to student licenselimitations we are unable to create such a big model at once. The consequence is thatoutcomes are pessimistic since we cannot divide remaining OR blocks which we canin practice. Therefore results can deviate.

• Blocking probability OC slots. Since patient slots can be blocked in practice due tomany reasons, we introduce a probability that one patient slot per OC block per pa-tient type per doctor is blocked as we explain in Appendix G. In practice however, thenumber of blocked OC patient slots is more variable. Consequently, practice containsmore variability and therefore results can deviate.

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Appendices

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A Secondary activitiesSecondary activities are all the activities where the doctor does not work at the OC or OR.Table 17 shows the secondary activities.

Table 17: Secondary activitiesNumber Secondary activity1 Holiday2 Not present3 Compensation4 Special leave5 Management6 Examination7 Research8 Hip/knee presentation9 Illness10 First aid11 Education12 Exam13 Department consultation14 Regional day15 Conference

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B Patient typesThis research focuses on a highly aggregated level. Therefore, the different appointmentcodes are translated into new and recurrent patients. Most choices are obvious, however, inconsultation with the department we choose to divide emergency patients (appointment code’Spoed’) over new and recurrent patients. Emergency patients are no real emergency pa-tients but patients with a slightly higher priority. Table 18 shows the regarding information.

Table 18: Appointment codes (patient types)Appointment code Translated intoAFSP10 RecurrentAFSP20 RecurrentAFSP30 RecurrentCP RecurrentCPFYS RecurrentCPO RecurrentCPSCHOUD RecurrentCPSPA RecurrentCPSPF RecurrentCPVERP RecurrentCPT RecurrentHECHT RecurrentNP NewNPFYS NewNPSCHOUD NewNPSCO1 NewNPSPA NewNPSPF NewSCHOENCP RecurrentSCHOENNP NewSCOLIOSE RecurrentSECOPINI NewSPOED New/Recurrent (50/50)SUPERV RecurrentTEL RecurrentUITSC1 RecurrentUITSC2 RecurrentUITSL RecurrentUITSLSCH Recurrent

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C Data gatheringThe input for the MIP model are parameters. For each parameter we obtained raw datafrom the hospital database named ’Hix’. For each parameter, the raw data is transformedinto useable data for our model. Each subsequent section contains a brief explanation ofhow we transform the data for each parameter.

C.1 New patient arrivalsWe use new patient arrivals for the input of our MIP model. These new patient arrivals arebased on historical data from the period of 2012 - 2016. We use this period because datacan be retrieved from 2012. In consultation with the department, each year is divided into4 periods that differ in arrival intensity. For each doctor and period, with a significancelevel of 95%, we fail to reject the hypothesized probability distribution. Table 19 shows thecorresponding parameters.

Table 19: Arrival parameters per doctor(Source: Hix, n=1352, Jan 2012 - Dec 2016)

Doctor Period Distribution n Avg Min Max σ σ2 Alpha BetaA 1 13 Gamma 65 15.84 0 34 6.6 43.7 5.7 2.8A 14 26 Gamma 65 12.96 2 35 6.2 38.2 4.4 2.9A 27 42 Normal 80 17.32 2 39 8.1 65.2 - -A 43 52 Normal 50 11.07 0 30 7.4 55.5 - -B 1 13 Gamma 65 20.36 0 51 8.8 77.8 5.3 3.8B 14 26 Gamma 65 19.95 5 40 8.1 65.2 6.1 3.3B 27 42 Gamma 80 19.30 0 37 8.9 78.5 4.7 4.1B 43 52 Gamma 50 17.08 0 35 8.3 69.7 4.2 4.1C 1 13 Gamma 65 20.26 0 40 7.7 59.3 6.9 2.9C 14 26 Gamma 65 20.59 4 38 8.9 79.8 5.3 3.9C 27 42 Gamma 80 18.93 1 34 8.9 78.9 4.5 4.2C 43 52 Gamma 50 21.14 0 41 9.7 94.3 4.7 4.5D 1 13 Gamma 65 15.97 0 37 7.8 61.0 4.2 3.8D 14 26 Gamma 65 15.35 2 49 9.3 86.7 2.7 5.6D 27 42 Gamma 80 14.71 1 47 10.1 101.5 2.1 6.9D 43 52 Gamma 50 13.07 0 28 7.3 53.7 3.2 4.1E 1 13 Gamma 65 17.53 0 41 10.6 111.3 2.7 6.5E 14 26 Gamma 65 16.18 2 40 9.4 87.7 3.0 5.4E 27 42 Gamma 80 16.68 1 45 10.6 113.2 2.5 6.8E 43 52 Gamma 50 15.53 0 33 7.8 60.8 4.0 3.9F 1 13 Gamma 13 17.58 0 35 9.0 80.5 3.8 4.6F 14 26 Gamma 13 18.62 4 39 10.2 104.6 3.3 5.6F 27 42 Gamma 16 13.13 5 29 7.3 52.9 3.3 4.0F 43 52 Gamma 10 14.79 0 31 9.1 82.0 2.7 5.5

An important remark regarding the probability distribution of doctor F is that these distri-butions are based on insufficient data points to be statistically significant. This is the casebecause doctor F works since halfway 2015 at the orthopedic department. However, we usethe available data points to obtain probability distributions and their parameters.

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C.2 Delays between queuesWe introduce ds,i,j as delay between queues. The delay is specified as the time betweenan appointment and the next subscription. A delay between queues may for example bespecified by a doctor for e.g. medical reasons. We calculate these delays based on historicaldata of 2016. Figure 24 shows the visual representation of the delay.

Figure 24: Patient delay

At t1, a new patient arrives and makes an appointment for t2. The time between t1 and t2is the access time (w1). After the appointment at t2, the patient makes the subscription forthe next appointment at t3. The time between the appointment at t2 and the subscriptionat t3 is defined as the delay ds,i,j .

The delay ds,i,j differs between queues and doctors. Table 20 shows the corresponding de-terministic delays as we use for our MIP model.

Table 20: Delay between queues per doctor in weeks(Source: Hix, n=17818, Jan - Dec 2016)Doctor NP CP OR

A NP - 5 8A CP - 13 8A OR - 1 17B NP - 6 8B CP - 12 7B OR - 2 3C NP - 3 6C CP - 9 6C OR - 6 1D NP - 6 7D CP - 13 7D OR - 1 1E NP - 4 7E CP - 11 8E OR - 1 4F NP - 2 7F CP - 7 6F OR - 6 1

NP = New patientCP = Recurrent patient

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For the use of our simulation model, we use stochastic delays; with a certain probability thedelay is equal to ds,i,j . By generating a random number between 0 and 1 from the uniformdistribution we determine the delay. Table 21 shows the corresponding delays for doctor B.We choose to bound the maximum delay to 20 weeks.

Table 21: Stochastic delay between queues for doctor B(Source: Hix, n=1633, Jan - Dec 2016)

Delay in weeks db,1,3 db,1,2 db,3,3 db,3,2 db,2,3 db,2,21 0.02 0.00 0.03 0.09 0.01 0.882 0.05 0.06 0.04 0.03 0.00 0.043 0.15 0.03 0.08 0.11 0.03 0.004 0.14 0.09 0.05 0.08 0.05 0.005 0.13 0.12 0.09 0.07 0.05 0.046 0.06 0.06 0.04 0.09 0.04 0.007 0.08 0.07 0.07 0.04 0.22 0.008 0.05 0.08 0.03 0.06 0.14 0.009 0.02 0.12 0.03 0.11 0.19 0.0010 0.03 0.10 0.03 0.06 0.05 0.0411 0.02 0.08 0.03 0.08 0.03 0.0012 0.01 0.06 0.02 0.07 0.03 0.0013 0.02 0.04 0.03 0.03 0.05 0.0014 0.03 0.00 0.07 0.01 0.02 0.0015 0.02 0.02 0.02 0.01 0.01 0.0016 0.01 0.02 0.02 0.00 0.01 0.0017 0.01 0.01 0.01 0.02 0.01 0.0018 0.01 0.00 0.02 0.01 0.01 0.0019 0.01 0.00 0.02 0.00 0.02 0.0020 0.13 0.04 0.27 0.03 0.03 0.00

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C.3 Monte Carlo simulationA Monte Carlo simulation is characterized by simulating the same instance many times, butwith different start solutions. We create different start solutions by generating new patientarrivals from the obtained probability distributions by using random numbers as explainedin Appendix C.1.

For the Monte Carlo simulation, we make the assumption that every doctor has sufficientcapacity to treat his patients. We treat patients directly and we incorporate the time be-tween two consecutive appointments, based on historical data, as delay.

The goal of the Monte Carlo simulation is to obtain transition rates such that the productionnumbers of the simulation do reflect the production numbers of 2016. We follow the nextsteps:

1. Calculate the transition rates based on the non-shared OC capacity per doctor. Thesetransition rates are used as start for the iterative process. Go to step 2.

(a) If the production number lay not within the confidence interval, adjust the tran-sition probabilities.

2. Generate new patient arrivals for every week of the next 20 years based on the obtainedprobability distributions by using random numbers.

3. Run the Monte Carlo simulation 100 times.

4. Check if the production numbers of 2016 lay within the 95% confidence interval re-sulting from the simulation run. We calculate the confidence interval based on thesteady-state situation (years 6 - 20) resulting in 1500 data points.

(a) If the resulting production numbers lay within the confidence interval, these ratesare the new transition rates.

(b) If the resulting productions numbers lay not within the confidence interval, goback to step 1.a.

Table 22 shows the resulting transition rates which we use as input for the MIP model.

Table 22: Transition rates between queue i and queue j for every doctor sQueue j Queue j

Doctor Queue i 1 2 3 Doctor Queue i 1 2 3A 1 0 0.08 0.89 D 1 0 0.12 0.80A 2 0 0.13 0.60 D 2 0 0.10 0.59A 3 0 0.11 0.45 D 3 0 0.17 0.48B 1 0 0.12 0.58 E 1 0 0.06 0.94B 2 0 0.10 0.59 E 2 0 0.09 0.62B 3 0 0.15 0.48 E 3 0 0.13 0.47C 1 0 0.04 0.79 F 1 0 0.05 0.84C 2 0 0.14 0.68 F 2 0 0.07 0.65C 3 0 0.12 0.43 F 3 0 0.11 0.53

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Table 23 shows the resulting production numbers of the Monte Carlo simulation.

Table 23: Outcomes Monte Carlo simulation (n=1500)Simulation results

Doctor Patienttype 2016 Average Confidence interval Standard errorA NP 767 765.2 762.7 767.8 0.05%A CP 1474 1477.9 1473.3 1482.4 0.03%A OR 253 252.5 251.3 253.6 0.11%B NP 1016 1014.3 1011.1 1017.4 0.04%B CP 1567 1563.2 1558.5 1567.9 0.03%B OR 401 400.6 399.0 402.2 0.08%C NP 1047 1043.1 1039.9 1046.3 0.04%C CP 1759 1759.7 1754.6 1764.7 0.03%C OR 273 273.7 272.4 274.9 0.11%D NP 779 778.0 774.6 781.3 0.04%D CP 1604 1599.3 1593.6 1605.1 0.02%D OR 362 361.7 360.0 363.4 0.08%E NP 878 875.3 871.8 878.8 0.04%E CP 1899 1895.2 1888.4 1902.0 0.02%E OR 330 331.1 329.5 332.6 0.08%F NP 825 829.0 825.7 832.2 0.04%F CP 1789 1787.1 1780.6 1793.5 0.02%F OR 268 268.7 267.4 270.0 0.10%

NP = New patient, CP = Recurrent patient

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C.4 Service ratesWe denote φs,i,a as the number of patients served from queue i if doctor s works a sessionat station a. We describe φs,i,a as the service rate which we use as input for the MIP model.A service rate is based on the number of slots per block and the filling of these slots.

Each OC block contains 19 slots. Because doctors can only work complete days of twoblocks in our MIP model, each doctor can serve 38 patients if he works a day at the OC.

The slots of an OC block can be filled with new and recurrent patients. It is not desirablethat the ratio between new and recurrent patients differ per OC block, therefore we reserveslots for patient types. The number of slots reserved for a certain patient type depends onthe outcomes of the Monte Carlo simulation (Appendix C.3). Table 24 shows the number ofreserved slots per patient type per OC block per doctor based on the Monte Carlo simulation.

An OR block does not have slots. Therefore, we use the historic average number of surg-eries per OR block per doctor to calculate the service rate. According to the manager, theaverage historic utilization of the OR is 95%. This OR utilization is defined as the timebetween the start of the first surgery and the end of the last surgery divided by 8 hours. Wedivide the average number of surgeries per session by the utilization of the OR to obtainthe effective service rate per doctor as Table 24 shows.

Table 24: Service rate per block per queue per doctorQueue

Doctor NP CP ORA 6.5 12.5 2.23B 7.5 11.5 2.83C 7.1 11.9 2.52D 6.2 12.8 2.83E 6.0 13.0 2.82F 6.0 13.0 2.25

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D Warm up length MIPWe use the graphical procedure of Welch to determine the warmup length for each exper-iment. Due to time limitations, we perform 2 replications per experiment as the runningtime for 1 replication is 48 hours and 2 hours for generating an initial solution. We use thenumber of patients in our system (

∑s,i,n P

ns,t,i for every week t) as performance indicator

since this performance indicator is in line with our objective of minimizing the weightednumber of waiting patients.

Each replication has a runlength of 20 years (t = 1040) as this is our maximum runlengthdue to the complexity of our model as explained in Section 5.2. As Welch indicates, therunlength must be at least 4 times as big as the warmup length l. Since we cannot run longerthan 1040 weeks, our maximum warmup length is 260 weeks according to the procedure ofWelch. We can see in Figures 25, 26, 27 and 28 that our performance indicators are notcompletely stable after 260 weeks. As a trade off between reliability of the output data andcomputation time of our model, we set our warmup length equal to 520 weeks.

Figure 25: Warmup length realistic scenario - Capacity 2016 - W=260

Figure 26: Warmup length realistic scenario - Capacity 2017 - W=260

Figure 27: Warmup length idealistic scenario - Capacity 2016 - W=260

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Figure 28: Warmup length idealistic scenario - Capacity 2017 - W=260

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E OC blockFor each doctor and each OC type there are different OC blocks generated during the re-search. As a new doctor starts working, we use a standard OC block that reflects theaverages of all doctors. For imaging, Figure 29 shows the most standard OC block.

Figure 29: Standard OC block

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F Daily planning rulesWe introduce planning rules to introduce more direction to the flow of patients towards theward. These planning rules are formulated together with several stakeholders and we takeinto account the average length of stay and the number of surgeries per day.

Knee surgeriesThe total number of knee surgeries to plan in multiple OR blocks on one specific day. Onlyin the following combinations:

1x oxford knee1x patella femoralis prothese3x total knee prosthese

2x oxford knee (per OR block)2x total knee prosthese

1x oxford knee1x patella femoralis prosthese3x total knee prosthese

1x oxford knee3x total knee prosthese2x patella femoralis prosthese (only for doctor D)

3x total knee prosthese (only for doctor D)2x patella femoralis prosthese (only for doctor D)

Hip surgeriesThe total number of hip surgeries to plan in multiple OR blocks on one specific day.

• Plan not more than one revision per day.

• Plan no hips for doctor D.

• Plan oxford hips and anterior cruciate ligaments for doctor D and doctor E.

• Plan side position hips for doctor B and doctor F.

• Plan front position hips for doctor B.

ProsthesesThe preference of the number of prostheses per OR block in the following combination:1. Two total knee prosthese and one hip.2. One total knee prosthese and two hips.

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G Simulation modelWe use a DES model to evaluate the current and suggested situation. We aim to create acurrent situation in Plant Simulation where the performance indicators do match practiceas close as possible. For the suggested situation, we use the current situation as basis andwe switch blocks based on our planning rules. A brief explanation and some assumptionsare explained below.

For the MIP model, we use transition rates. These transition rates are included in AppendixC.3. Instead of transition rates, we use transition probabilities (Appendix C.3) in our DESmodel to introduce stochasticity. By generating a random number between 0 and 1 fromthe uniform distribution we determine the next stage in the care process of a patient.

Between each stage in the care process of a patient, we use a stochastic delay between queuesby also generating a random number between 0 and 1 from a uniform distribution. Thecorresponding delays ds,i,j for doctor B are included in Appendix C.2. New patients arriveweekly according to the probability distributions as explained in Appendix C.1.

Equal to our MIP model, doctors do not share capacity and therefore patients cannot changedoctors. We use the OC and OR blocks for each doctor as they were planned in 2016. Wemake some assumptions for the filling of the blocks as we explain below.

In current practice, planners can change the type of a patient slot to fulfill demand. Theresult is that each OC block can differ in patient type ratios. As these changes are based onintuition, experience and without rules, we are unable to create equal fillings of OC blocks.Therefore, we choose to use the improved OC blocks that result from Phase 1 (Subsection4.3.1).

In practice, OC patient slots can be blocked due to secondary activities. Based on the avail-able data, we are unable to obtain specifications. Therefore, we introduce a probability thatone patient slot per OC block per patient type per doctor is blocked. After experimenting,we found out that we can create matching access times by introducing this probability perdoctor.

For the OR blocks, we use the service rates as we used for the MIP model. As these servicerates are fractional, we choose to introduce a probability that this service rate is roundeddown and one minus that probability that the service rate is rounded up. After experiment-ing with these probabilities per doctor, we found out that we can reflect the waiting timeclose to practice.

Since we want to reflect current practice as close as possible, we calculate the access andwaiting time exactly the same as we did in Section 2.7. The access time is calculated basedon the average time between the request data and the appointment date of all patientsscheduled in the corresponding week. We calculate the waiting time based on the third freespot since we could not include the realized waiting time in Section 2.7 as there are weeks inwhich no patients are scheduled. Note that the waiting time is integer since patients arriveweekly in our DES model. We also introduce that patients should wait for at least one week.

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Process flowFigure 30 shows the process flow of the DES model.

Figure 30: Process flow DES model

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Input and outputTable 25 shows the input and output of the DES model.

Table 25: Input and output of the DES modelInput OutputPatient arrival distributions Access timeTransition probabilities Waiting timeDelays UtilizationOriginal blocks of 2016Adjusted blocks of 2016

Warmup lengthWe use the graphical procedure of Welch to determine the warmup length for each doctor forboth, the access and waiting time. For each case we use 50 replications and each replicationhas a runlength equal to 22.000 weeks. We take the longest warmup period of each case andset this as standard for all doctors. The longest warmup period is 5000 weeks. We set thewarmup length for all doctors equal to 5200 weeks since this corresponds with 100 years.Because we perform the same procedure for each doctor, we again only include the warmuplength graphs for doctor B, as Figure 31 and Figure 32 show.

Figure 31: Warmup length access time doctor B

Figure 32: Warmup length waiting time doctor B

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ReplicationsWe perform replications to create a 95% confidence interval for the mean access and waitingtime by using the replication/deletion approach. Table 26 shows the number of replications(n) we perform per doctor to obtain the confidence interval. Because simulations are alsocarried out during night, we sometimes performed much more replications than needed fora 95% confidence interval, therefore we also include y′ (corrected target value) in Table 26.

Table 26: Number of replications DES modelCurrent Suggested

Doctor n y’ n y’A 7379 0.0220 3617 0.0230B 7042 0.0239 5358 0.2316C 8481 0.0239 9396 0.0231D 10057 0.0235 11597 0.0234E 7528 0.0216 5569 0.0209F 9566 0.0237 7785 0.0237

ValidationTo validate the DES model, we compare the access and waiting times outcomes of the DESmodel with the access and waiting time in practice for doctor B. Figure 33 shows the lower-bound (LB) and upperbound (UB) of the 95% confidence interval of the mean access timeand we include the access time from practice.

Figure 33: Access time validation doctor B(Source: Outcomes DES model)(Source: Hix, n=5860, Jan - Dec 2016)

We can remark the same pattern during the year and the access time peak in week 27 isequally timed. The access time from practice contains more variability since this is just oneobservations and the outcomes of the DES model are a 95% confidence interval.

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Figure 34 shows the lowerbound (LB) and upperbound (UB) of the 95% confidence intervalof the mean waiting time and the waiting time from practice.

Figure 34: Waiting time validation doctor B(Source: Outcomes DES model)(Source: MedSpace, n=325, Jan - Dec 2016)

We can remark the same pattern during the year. The waiting time peaks in week 27 areequally timed. The waiting time from practice contains more variability since this is justone observation. Table 27 shows the different performance indicators for the situation inpractice and the outcomes of the DES model.

Table 27: Comparison of practice and DES model outcomes for doctor BAccess time Waiting time

X Min Max Sigma X Min Max SigmaPractice 2.8 0.8 5.3 1.2 5.6 2.0 9.4 2.5DES model 2.7 1.7 4.7 0.7 5.7 3.8 8.5 1.4

Table 27 shows that the averages are almost equal and that the maximums have the sameorder size. The minimums are different because in our DES model patients must wait atleast for one week. Since the access and waiting time outcomes of the DES model do followthe same pattern as the access and waiting time in practice and the holidays are timedequally, we can conclude that our DES model is a good representation of practice.

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