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UNIVERSITEIT GENT
FACULTEIT ECONOMIE EN BEDRIJFSKUNDE
ACADEMIEJAAR 2015 – 2016
ANALYSIS, REDESIGN AND IMPLEMENTATION OF A DIALYSIS
PROCESS
Masterproef voorgedragen tot het bekomen van de graad van
Master of Science in de Toegepaste Economische Wetenschappen: Handelsingenieur
Gert De Baerdemaeker Nicolas Vanquickenborne
onder leiding van Prof. dr. Frederik Gailly
UNIVERSITEIT GENT
FACULTEIT ECONOMIE EN BEDRIJFSKUNDE
ACADEMIEJAAR 2015 – 2016
ANALYSIS, REDESIGN AND IMPLEMENTATION OF A DIALYSIS
PROCESS
Masterproef voorgedragen tot het bekomen van de graad van
Master of Science in de Toegepaste Economische Wetenschappen: Handelsingenieur
Gert De Baerdemaeker Nicolas Vanquickenborne
onder leiding van Prof. dr. Frederik Gailly
i
Permission
Ondergetekenden verklaren dat de inhoud van deze masterproef mag
geraadpleegd en/of gereproduceerd worden, mits bronvermelding.
Undersigned declares that the contents of this thesis may be consulted
and/or reproduced, provided acknowledgment.
Gert De Baerdemaeker
Nicolas Vanquickenborne
ii
iii
Nederlandse samenvatting
De kosten van de Belgische gezondheidszorg stijgen jaarlijks (Lapre,
Rutten, & Schut, 2001). Tegelijkertijd krimpt het budget (Federal department of
health, food chain safety and environment, 2015). Eenzelfde trend valt op te
merken bij de dialysecentra. Bovendien treden er ook nog andere problemen op in
dialysecentra: verpleegkundigen klagen over een ongebalanceerde werkdruk en
patiënten klagen over te lange wachttijden. Daarenboven, is er een te grote
tijdsoverlap tussen de vroege en de late werkshift van de verpleegkundigen. Dit
alles toont aan dat er binnen het dialyseproces een grote verbetering genoodzaakt
is.
Deze masterproef behelst een casestudie, uitgevoerd op de dialyseafdeling van AZ
Sint-Jan te Brugge en heeft als doel het dialysecentrum te analyseren en
verbeteringsvoorstellen aan te brengen.
Dit onderzoek start met het uitleggen van de toegepaste methodologieën.
Deze zullen ook verder in de thesis aan bod komen.
In een tweede hoofdstuk wordt er ingegaan op de werking van de nieren en
chronische nierinsufficiëntie. Evenals wordt er kort ingegaan op de mogelijke
behandelingen bij nierfalen – waarvan hemodialyse slechts één van de opties is.
Een derde hoofdstuk geeft een literatuuroverzicht. Vooreerst wordt gezocht
naar oorzaken van stress, absenteïsme en burnout bij (dialyse)verpleegkundigen.
Een te hoge patiënt-verpleegkundige ratio, werkdruk en aantal
verantwoordelijkheden worden aangeduid als mogelijke oorzaken. Bovendien
wordt aangetoond dat burnout bij verpleegkundigen leidt tot een lagere
patiënttevredenheid. Een tweede sectie behandelt de verwachte evolutie in het
aantal dialysepatiënten. In Brugge wordt geen stijging verwacht, een uitbreiding
van de dialysedienst wordt dan ook niet verder onderzocht. Een laatste onderdeel
geeft een overzicht aan literatuur wat betreft patiëntplanning. Gezien de literatuur
iv
over de planning van dialysepatiënten beperkt is, wordt er vaak beroep gedaan op
literatuur over de planning van een operatiekwartier.
Hoofdstuk vier behandelt het eigenlijke modelleren van het proces. Voor
het modelleren wordt een beroep gedaan op de Business Process Modeling cyclus
(Kim, 2015). In een eerste stap worden de problemen geïdentificeerd. Op basis
daarvan worden de processen geïdentificeerd. Voor deze thesis bleken het
patiëntenvervoer van en naar het ziekenhuis en het dialyseproces relevant. Het
dialyseproces wordt in dit hoofdstuk behandelt, het patiëntenvervoer is materie
voor het vijfde hoofdstuk en valt buiten deze cyclus. Verder wordt ook een goal
model opgesteld.
De tweede stap, de process discovery, verklaart de methodes waarmee
gegevens verzameld werden. Een documentanalyse, observatie en interviews
bleken de meest geschikte methodes. Op basis daarvan werd het dialyseproces
gemodelleerd. Daarbij werd gebruik gemaakt van de Business Process Modeling
Notation (Müller & Rogge-Solti, 2011), met Signavio als ondersteunende
modelleertool. De BPMN-techniek bleek uitermate geschikt omwille van zijn hoge
graad van verstaanbaarheid en eenvoud om te modelleren.
Een derde stuk beschrijft de procesanalyse. Het kwalitatieve aspect van
deze analyse voert een toegevoegde-waarde analyse uit waaruit blijkt dat de meeste
activiteiten waarde toevoegend zijn. De overige taken kunnen bovendien maar
moeilijk aangepast of geëlimineerd worden omwille van wetgeving en hygiëne.
Een tweede techniek, de 5-Why analyse, legt het pijnpunt van het dialyseproces
bloot: patiënten van eenzelfde blok worden allen op hetzelfde tijdstip verwacht op
de dialyseafdeling. De kwantitatieve analyse berekent enkele ratio’s. Zo blijkt er
voor acht patiënten gemiddeld 7 uur en 41 minuten werk te zijn, terwijl er twee
verpleegsters elk acht uur voor voorzien zijn. De doorlooptijdefficiëntie voor
patiënten (cycle time efficiency) blijkt dan weer hoog te zijn. Dit kan verklaard
worden door de lange dialyseduur in vergelijking met de wachttijden. De efficiëntie
is ongeveer 95%. Ook de benuttingsgraad (utilization) werd berekend, zoals
gesuggereerd door Cardoen, Demeulemeester en Belin (Cardoen,
Demeulemeester, & Belin, 2010). De benuttingsgraad ligt tussen 57% en 65%,
afhankelijk van de berekeningsmethode.
v
Een vierde sectie behandelt het eigenlijke herontwerp van het proces.
Hierbij wordt voornamelijk gefocust op de tijdsschema’s van de activiteiten om zo
de werkdruk beter te spreiden. Het eerste deel onderzoekt of er een relevante
onderverdeling van patiënten kan gemaakt worden. Er blijkt dat er een significant
verschil in tijdsduur is om een patiënt aan of af te sluiten naar gelang de patiënt
hulp nodig heeft om in of uit zijn bed te geraken. Dit tijdsverschil zal in verdere
modellen geïmplementeerd worden. De hypothese dat er een significant
tijdsverschil zou zijn tussen patiënten met een fistel en patiënten met een katheder,
kon niet weerhouden worden. Dit kan verklaard worden door een definiëring van
de aansluit- en afsluitactiviteiten. De aan- en afsluiting is immers inclusief de tijd
om een patiënt in of uit een bed te krijgen. Gezien er relatief gezien meer
kathederpatiënten hulp nodig hebben dan fistelpatiënten, wordt de tijdswinst van
een katheder – voor een katheder moet niet geprikt worden – tenietgedaan. Daarna
wordt een eerste optimalisatiemodel opgesteld. Hierbij worden drie activiteiten
voor elke patiënt gepland: de aansluiting, de afsluiting en het totaal aan activiteiten
tussen beide activiteiten. Twee technieken worden daarbij afgewogen ten opzichte
van elkaar. De eerste bestaat uit het minimaliseren van de som van de verschillen
tussen de werkdruk in een tijdsslot en de gemiddelde werkdruk over alle tijdssloten.
De tweede techniek focust op het minimaliseren van de hoogste werkdruk over alle
tijdssloten. Beide technieken resulteren in een gefaseerde aansluiting van patiënten.
Aangezien er gewerkt werd met tijdssloten van tien minuten, werd evenwel nog
geen onderscheid gemaakt tussen beide types van patiënten. De tweede techniek
had de meest toereikende resultaten en werd verder gebruikt voor het optimaliseren
van de aansluitingssessie. Hierbij werd wel een onderscheid gemaakt tussen
patiënten die zelfstandig in en uit bed geraken en zij die dat niet kunnen; er werden
tijdssloten van twee minuten gebruikt. Daarnaast werd er ook met buffers gewerkt.
Mocht een tijdsschema opgesteld worden op basis van gemiddelden, is de kans
groot dat patiënten alsnog moeten wachten, wat de werkdruk voor
verpleegkundigen verhoogt. Daarom werd de duurtijd van elke activiteit vergroot
met een tijdsbuffer gebaseerd op een servicelevel van 90%. Hieruit volgt dat
zelfstandige patiënten eerst gepland worden, met tijdsintervallen van 8 minuten.
Aansluitend volgen de patiënten die hulp nodig hebben, met tijdsintervallen van 14
minuten. Op basis van deze bevindingen werd een dagschema opgesteld waarbij
vi
bijgevolg de afsluitmomenten eveneens sequentieel gepland worden. Gezien de
werkdruk nu beter gespreid is, wordt de tijdsoverlap tussen de vroege werkshift en
de late werkshift als overbodig beschouwd. Het valt aan te raden om de tijdsoverlap
te beperken tot een minimum door het bestaande werkregime aan te passen.
In een vijfde hoofdstuk werd gekeken naar een optimalisatie binnen de
transportdiensten voor de patiënten. Momenteel is er geen duidelijke regelgeving
voor het organiseren van transport. Omwille van deze inefficiëntie is het verdelen
van patiënten per taxi dan ook moeilijk: ofwel rijdt een patiënt afzonderlijk in een
taxi, ofwel rijden patiënten samen zonder dat dit optimaal is. Concreet werd in dit
hoofdstuk eerst theoretisch gekeken naar het ‘Vehicle Routing Problem’, een
welgekend optimalisatieprobleem binnen de logistiek. Dit model laat toe de
goedkoopste route te vinden, waarbij iedere patiënt wordt getransporteerd naar het
ziekenhuis. Hier kunnen bepaalde restricties aan toegevoegd worden. De meest
markante restrictie voor dit transportprobleem, was het invoeren van
tijdsintervallen voor het ophalen van patiënten. Deze tijdsintervallen dienen een
menselijke component toe te voegen aan de optimalisatie. Indien ritten gedeeld
worden, dalen de kosten maar tegelijk stijgen voor sommige patiënten ook de
reistijden. Het invoeren van tijdsintervallen binnen het ‘Vehicle Routing Problem’
biedt hiervoor een oplossing door een limiet te zetten op deze extra reistijd. De
finale verbetering omtrent de verdeling van patiënten over verschillende
taxidiensten, werd getoetst op basis van vier criteria: het aantal voertuigen, de totale
reistijd, de totale extra reistijd voor patiënten en de totale gereden afstand. Wanneer
het nieuw model vergeleken werd met het oude schema, werd aangetoond dat drie
van deze vier criteria sterk verbeteren. Enkel de totale extra reistijd voor patiënten
stijgt, weliswaar binnen een gekozen limiet.
In een voorlaatste hoofdstuk werd gekeken naar mogelijk verder onderzoek
binnen deze unieke studie. Vooral een aanzet naar betere werkuren voor de
verpleegkundigen bleek interessant.
Hoofdstuk zeven is een algemene conclusie en sluit deze masterproef af.
vii
Preface
A master thesis is a piece of research which mostly requires lots of work, but often
it is only read by a minority of people. With the choice for an actual subject based
on a real case, we hope that several people will consider this thesis as fascinating
as we do. We are convinced this thesis could help several managers and doctors,
operative at dialysis centers, to critically analyze and eventually redesign their
dialysis center.
Before we report about this research, we would like to acknowledge some people
for their collaboration and advice. First, our promotor, Prof. dr. Frederik Gailly, has
to be acknowledged. He was always prepared to help and give advice, from the
choice of the subject until the writing of our thesis. After writing this dissertation,
we can look back on a satisfying collaboration.
We would also like to thank dr. An De Vriese. She gave us the opportunity to
analyze the hemodialysis department of AZ Sint-Jan in Bruges and offered useful
documents. Moreover, she also provided us with helpful advice. By extension, we
should also give credits to the personnel and patients of the dialysis center. They
were always open to our questions and were keen to help us during our many
observation sessions.
Moreover, we are thankful to everyone who provided us literature, advice and
answered our e-mails.
At last, we would also like to thank our family and friends. They always supported
us, even on the moments when it was difficult.
Gert De Baerdemaeker
Nicolas Vanquickenborne
Ghent, May, 2016
viii
ix
Table of Content
Permission........................................................................................................i
Nederlandsesamenvatting..............................................................................iii
Preface...........................................................................................................vii
Abbreviations.................................................................................................xv
Abstract........................................................................................................xvii
Chapter0:Introduction...................................................................................1
Chapter1:Methodology..................................................................................3
Chapter2:Functioningofkidneys,chronicrenalfailureandtreatment
possibilities.....................................................................................................7
2.1. Kidneyfunctioning.....................................................................................7
2.2. ChronicRenalFailure.................................................................................8
2.3. Treatmentpossibilities..............................................................................8
2.3.1. Peritonealdialysis................................................................................8
2.3.2. Hemodialysis.......................................................................................9
2.3.3. Transplantation...................................................................................9
Chapter3:Literaturereview...........................................................................11
3.1. Stress,absenteeismandburnoutatnurses...............................................11
3.2. Dialysispatients’evolutions.....................................................................12
3.3. Schedulingandoptimizationapproaches..................................................13
3.3.1. Schedulingwithoutuncertainty........................................................13
3.3.2. Schedulingwithuncertainty..............................................................14
3.4. VehicleRoutingProblem...........................................................................18
3.4.1. Differentclasses................................................................................18
3.4.2. Solutionmethods..............................................................................19
3.5. Conclusion................................................................................................21
Chapter4:ProcessModeling..........................................................................23
x
4.1. Processidentification................................................................................24
4.1.1. Valuechain........................................................................................25
4.1.2. Goalmodeling...................................................................................26
4.2. Processdiscovery......................................................................................27
4.2.1. Definitionofthesetting....................................................................27
4.2.2. Datacollection...................................................................................28
4.2.2.1. Evidence-baseddiscovery...........................................................284.2.2.2. Deductedinformation.................................................................29
4.2.3. Creationoftheprocessmodel..........................................................31
4.2.3.1. Mainprocess...............................................................................324.2.3.2. Subprocessconnection................................................................344.2.3.3. Subprocessdisinfectioncatheter................................................354.2.3.4. Subprocessdisconnection...........................................................354.2.3.5. EvaluationoftheBPMNtechnique.............................................36
4.2.4. Modelquality....................................................................................36
4.3. Processanalysis........................................................................................37
4.3.1. Qualitativeanalysis...........................................................................37
4.3.1.1. Valueaddedanalysis...................................................................374.3.1.2. Rootcauseanalysis......................................................................40
4.3.2. Quantitativeanalysis.........................................................................40
4.3.2.1. Performancedimensions.............................................................404.3.2.2. Utilization....................................................................................44
4.4. Processredesign.......................................................................................45
4.4.1. Durationofconnectionanddisconnectionactivity...........................46
4.4.1.1. Durationdifferencebetweencatheterandfistulapatients........474.4.1.2. Durationdifferencebetweendependentandindependent
patients …………………………………………………………………………………………………494.4.1.3. Conclusion...................................................................................514.4.1.4. Durationdifferencebetweenmorningandafternoonpatients..51
4.4.2. Optimizationmodel:optimizeworkloadlevelduringdialysis..........52
4.4.2.1. Assumptionsintheoptimizationmodel......................................524.4.2.2. Optimizationmodel.....................................................................554.4.2.3. Adaptedoptimizationmodel.......................................................614.4.2.4. Comparisonofbothtechniques..................................................63
4.4.3. Optimizationmodel:optimizeworkloadlevelduringconnection
activities.........................................................................................................63
xi
4.4.3.1. Experimentation..........................................................................654.4.3.2. Optimizationmodelincludingtimebuffers.................................67
4.4.4. Dayschedule.....................................................................................69
4.4.4.1. Composingdayschedule.............................................................694.4.4.2. Evaluationofdayschedule..........................................................71
4.4.5. Nursescheduling...............................................................................71
4.4.6. Redesignofindividualactivities........................................................72
4.5. Conclusion................................................................................................74
Chapter5:Transportation..............................................................................77
5.1. Motivation................................................................................................78
5.1.1. Transportationandlogistics..............................................................79
5.1.2. Passengertransportation..................................................................80
5.1.3. Externalcosts....................................................................................81
5.2. ClassesofVehicleRoutingProblems.........................................................81
5.2.1. TravelingSalesmanProblem.............................................................82
5.2.1.1. Problemformulation...................................................................825.2.1.2. Mathematicalformulation..........................................................83
5.2.2. MultipleTravelingSalesmanProblem...............................................865.2.2.1. Problemformulation...................................................................865.2.2.2. Mathematicalformulation..........................................................86
5.2.3. CapacitatedVehicleRoutingProblem...............................................88
5.2.3.1. Problemformulation...................................................................885.2.3.2. Mathematicalformulation..........................................................88
5.2.4. VariantsoftheVehicleRoutingProblem..........................................90
5.3. ComputationalComplexity.......................................................................93
5.4. Heuristicsolutionmethods.......................................................................94
5.4.1. Routeconstructionheuristics............................................................94
5.4.2. Routeimprovementheuristics..........................................................96
5.4.3. Metaheuristics...................................................................................97
5.5. PracticaltooltosolveVRPs.......................................................................98
5.5.1. VisualBasicforApplications..............................................................99
5.5.2. GeographicInformationSystem........................................................99
5.5.3. Heuristicalgorithm..........................................................................101
5.5.3.1. LargeNeighborhoodSearch......................................................1015.5.3.2. AdaptiveLargeNeighborhoodSearch.......................................102
xii
5.5.3.3. ImplementationoftheALNSheuristic......................................1035.6. VRPatthedialysiscenter........................................................................105
5.6.1. Currentstate...................................................................................105
5.6.2. Solutionapproach...........................................................................106
5.6.2.1. Timewindowsforpatients........................................................1075.6.2.2. VRPTool....................................................................................107
5.6.3. Results.............................................................................................108
5.7. Conclusion..............................................................................................110
Chapter6:FutureResearch..........................................................................111
Chapter7:Generalconclusion......................................................................113
Bibliography.....................................................................................................I
Appendix..........................................................................................................I
APPENDIXI:Goalmodel..............................................................................................I
APPENDIXII:BPMN(ASIS).........................................................................................II
APPENDIXIII:Valueaddedanalysis...........................................................................III
APPENDIXIV:Durationofactivitiesandefficiencyratios.........................................IV
APPENDIXV:Utilization.............................................................................................V
APPENDIXVI:CPLEX.................................................................................................VI
APPENDIXVII:BPMN(TOBE)..................................................................................VII
APPENDIXVIII:VRPusermanual............................................................................VIII
xiii
List of figures
Figure1:BusinessProcessManagementLifecycle..........................................................23
Figure2:Threemainprocessesforthedialysisprocess..................................................25
Figure3:Graphshowingsensitivityofc1........................................................................61
Figure4:Graphshowingsensitivityofc1(adapted).......................................................63
Figure5:Workloadduringconnectionactivity,withouttimebuffers.............................66
Figure6:Patientconnectionschedule.............................................................................68
Figure7:Workloadduringconnectionactivity,withtimebuffers..................................68
Figure8:Illustrationofthedayschedule........................................................................70
Figure9:GreenhousegasemissionsinEuropebysourcesector(2016).........................81
Figure10:IllustrationoftheTSP......................................................................................83
Figure11:TravelingSalesmanProblem,solutionwithtwosub-tours(n=9)...................85
Figure12:Illustrationofthem-TSP(n=12,m=4)............................................................88
Figure13:OverviewofseveralVRPclasses.....................................................................91
Figure14:Illustrationofthesweepalgorithm................................................................95
Figure15:Illustrationoftheλ-optoperator(λ=2)...........................................................96
Figure16:Illustrationofthevertexswapoperator,twoverticesareswapped..............97
Figure17:Illustrationofthevertexrelocationoperator,onevertexisrelocated..........97
Figure18:LocaloptimumvsGlobaloptimum.................................................................98
Figure19:Mapindicatinglocationofhospitaland168dialysispatients......................105
Figure20:Currenttransportationtimescheme............................................................106
Figure21:Illustrationoftimewindows.........................................................................107
Figure22:Illustrationofatimewindowsviolation.......................................................108
Figure23:Taxischeduleforshift1................................................................................109
xiv
List of tables
Table1:Numberofpatientsassignedtoeachshift........................................................29
Table2:Numberofpatientsperroomassignedtoeachshift........................................30
Table3:Sensitivityofc1..................................................................................................60
Table4:Sensitivityofc1(adapted).................................................................................62
Table5:OriginalLNSheuristic(Shaw,1998).................................................................102
Table6:AdaptiveLNSheuristic(Røpke&Pisinger,2006).............................................103
Table7:VariantoftheALNSheuristic,implementedatthetool..................................104
Table8:Numberofpatientspershift,usingtransportationservices...........................108
Table9:Performancecriteriaofoldandnewroutingschedule(shift1)......................109
xv
Abbreviations
We provide the reader with an overview of the abbreviations which will be
used throughout this thesis. Given in an alphabetical order:
ALNS Adaptive Large Neighborhood Search AV Arteriovenous Fistula BPM Business Process Management BPMN Business Process Management Notation BVA Business Value Adding CRF Chronic Renal Failure CVRP Capacitated Vehicle Routing Problem CVRPS Commercial Vehicle Routing Problem System DFJ Dantzig-Fulkerson-Johnson formulation DMAIC Define, Measure, Analyze, Improve and
Control FTL Full Truckload GPGP Generalized Partial Global Planning HD Hemodialysis HRQoL Health-Related Quality of Life LNS Large Neighborhood Search LTL Less than Truckload m-TSP multiple Traveling Salesman Problem MDVRP Multi-Depot Vehicle Routing Problem MILP Mixed Integer Linear Programming MTZ Miller-Tucker-Zemlin formulation NBVN Nederlandstalige Belgische Vereniging voor
Nefrologie NVA Non-Value Adding OVRP Open Vehicle Routing Problem VA Value Adding
xvi
VLSN Very Large Scale Neighborhood search VRP Vehicle Routing Problem VRPTW Vehicle Routing Problem with Time Windows PD Peritoneal Dialysis RTL Reformulation-Linearization Technique SEC Sub-tour Elimination Constraint SDVRP Site Dependent Vehicle Routing Problem TSP Traveling Salesman Problem VBA Visual Basic for Applications
xvii
Abstract
Background and problem: The health care faces big challenges in Belgium. Costs increase
while the budget is decreasing. This leads to the need to eliminate inefficiencies. The dialysis
center of AZ Sint-Jan in Bruges does not escape this trend. Nurses are complaining about an
unbalanced workload, patients about too long waiting times and inefficiencies are observed
concerning the personnel roster and transportation.
Design: A case study on the hemodialysis department of AZ Sint-Jan Bruges.
Methods: Two processes are studied: the dialysis process and the transportation of patients to
and from the hospital and back to their dwellings. The dialysis process will be analyzed based
on the principles of the BPM lifecycle. To redesign the process, a mixed integer linear
programming model will be built. The transportation services will be optimized using the
Vehicle Routing Problem with Time windows, a commonly used model in transportation and
logistics. The transportation issue is not included in the full BPM lifecycle.
Results and conclusion: The main problems of the dialysis process were observed at the
connection and the disconnection activity. Patients arrived at the same moment which led to a
summit in workload. Via the MILP model there was demonstrated that planning patients
sequentially balanced the workload. Therefore, waiting times could also be reduced. Because
nurses work according to a two-shift system and the dialysis center is only operative for 12
hours per day, there is a big time overlap in the shifts of nurses. There is recommended to
change the shift system in order to improve efficiency ratios. Additionally, for the
transportation services no clear methods existed to assign patients to share rides and reduce
costs. By finding a balance in the optimization approach between the cheapest route and the
extra driving duration imposed by sharing rides, clear business rules were set up. Implementing
these new business rules reduced the number of taxis, the total duration and the total distance.
The extra driving times imposed by the detour for picking up other patients only had a limited
increase, since a method was used that ensured a maximum on this extra time per patient.
1
Chapter 0
Introduction
Hemodialysis is an expensive treatment. Cleemput, Beguin, Gerkens,
Jadoul and Verpooten calculated the average yearly cost of hospital hemodialysis
and concluded that it amounts €48.000 (Cleemput, Beguin, Gerkens, Jadoul, &
Verpooten, 2010). This corresponds to a cost of €313 per dialysis session.
Moreover, the number of hemodialysis patients has increased over the last years.
In 1997, there were 1.987 patients. In 2013, this number grew up to 4.085. This
corresponds to a growth of 106%. It is needless to say that this comes with a large
cost (NBVN, 2012).
The Belgian government has to economize. In 2016, the federal department
of health, food chain safety and environment plans €408,3 millions of savings in
the health care domain (Federal department of health, food chain safety and
environment, 2015). In order to keep the Belgian health care system payable, it will
be necessary to allocate resources optimally and organize health care units
efficiently (Lapre, Rutten, & Schut, 2001).
Blindly increasing nurses’ workload will not be an option. As Karkar,
Dammang and Bouhaha state, an increased workload can lead to stress, burnout
and exhaustion (Karkar, Dammang, & Bouhaha, 2015). After observing the
dialysis center of AZ Sint-Jan in Bruges, there can be analyzed that there was a
level of absenteeism of 5.13% in 2013. Between 2011 and 2013, the absenteeism
increased by 25% (De Vriese, 2015). Hence, the research of Karkar, Dammang and
Bouhaha gets confirmed in the dialysis center at Bruges.
In addition, Horn, Buerhaus, Bergstrom and Smout have shown that
patients’ conditions improve when dialysis nurses spend more time on direct
2
patient care (Horn, Buerhaus, Bergstrom, & Smout, 2005). Atkins, Marshall and
Rajshekhar describe that employee dissatisfaction can have a negative effect on
patient loyalty and thus, hospitals’ earnings (Atkins, Marshall, & Rajshekhar,
1996). Thus, increasing the workload for nurses without taking other actions,
surely is not an option.
This dissertation focuses on the dialysis center of AZ Sint-Jan in Bruges-
Ostend. Several issues in this dialysis center were observed. First, nurses were
complaining about the workload when patients have to be connected and
disconnected. Moreover, patients had complaints about long waiting times between
the arrival at the hospital and the actual connection, but also between the
disconnection and leaving the hospital. Moreover, inefficiencies concerning nurse
staffing are present. There is big time overlap between the working hours of the
nurses of the early shift and the nurses of the late shift. Therefore, two processes
will be analyzed and redesigned: the hemodialysis process and the transportation
of patients. During the redesign of the hemodialysis process, there will be tried to
offer an answer to the present issues. There will be focused on efficiency, patient
satisfaction and nurse satisfaction. For the optimization of the transportation of
patients, financial criteria are compared with total traveling times.
This study will be organized as follows. In chapter 1, the methodology will
be elaborated. There is tried to give an overview of the techniques used to analyze
and redesign the dialysis center. In chapter 2, background information is given
about kidneys, chronic renal failure and treatment alternatives. Chapter 3 gives an
overview of the existing literature. Literature about stress, absenteeism, burnout,
patient evolutions, patient scheduling and transportation optimization will be
described. Chapter 4 constitutes the modeling of the dialysis process. In this
chapter, the process will be analyzed and optimized. Continuously, the
optimization of the current transportation services is described in chapter 5.
Chapter 6 describes recommendations for future research. In chapter 7, general
conclusions are elaborated.
3
Chapter 1
Methodology
This chapter provides an overview of the methodology used to try to
construct an answer for the research question. The research question is stated as
following: “How can patients’ arrival times, transportation services, personnel
staffing and the process itself best be scheduled in a dialysis process in order to
reduce nurses’ stress and be more cost-efficient?”. Our methodology is based on
a case study at AZ Sint-Jan Bruges-Ostend.
In order to solve this problem and to comprehend it fully, it is essential to
pore over the dialysis process as a whole. First, several interviews with dr. An De
Vriese – head of the dialysis department of AZ Sint-Jan in Bruges – were
conducted. The aim of these interviews was to understand the expectations of the
dialysis center about this master dissertation, the main characteristics and the most
important constraints. After this, observation sessions were initialized at the
dialysis center. During the first two observations, the intention was to understand
the process and the different activities which make up the entire process. In later
observation sessions, the duration of the activities was timed. These observations
were combined with ad hoc questions for the nurses and patients and the
information found on the website of the dialysis center.
To ensure that this dissertation is general and not only specific to the
dialysis center in Bruges as well as to ensure that all relevant constraints are
included, an additional interview was conducted with nephrologist dr. Chris
Luyckx of the dialysis center of AZ Alma in Eeklo.
On top of the observations and interviews, several documents were
collected to get a better understanding of the details of the process.
4
In order to understand the process and to discover its underlying problems,
a literature research was undertaken. To find appropriate articles, the widely-
known search engine ‘Google Scholar’ was used most of the time. Google Scholar
is an online and free-to-use search engine, which is recommended by Jascó as well
as Brophy and Bawden (Jacsó, 2005; Brophy & Bawden, 2005). To collect
information about the process, the search terms ‘dialysis process’, ‘dialysis process
issues’ and ‘dialysis process problems’ were used. The word ‘dialysis’ was
sometimes replaced by ‘renal’. In a succeeding stadium of the literature study, there
was also sought for ‘dialysis services’, ‘dialysis plan’ and ‘dialysis capacity’.
Again, ‘renal’ was used as an alternative for ‘dialysis’.
In a following stage, the expected patient evolution was investigated. A
change in the number of patients would have implications on the needed capacity
of the dialysis center. The evolution in the number of patients was estimated by
asking specifically for it during the interviews with the nephrologists and by
consulting the annual reports of the NBVN, which is a Dutch abbreviation and
stands for “Nederlandstalige Belgische Vereniging voor Nefrologie”.
Additionally, data was collected about patient scheduling. No specific
literature was found using search terms covering patient scheduling in dialysis
centers. For this reason, these search terms were slightly adapted. Alternatively,
there was searched for ‘scheduling in health care’, ‘patient scheduling’, ‘resource
allocation patient scheduling’ and ‘appointment scheduling health care’.
To get more information about uncertainty in patient scheduling, there was
also searched for ‘dynamic patient scheduling’ and ‘dynamic scheduling health
care’.
By reading more about these topics, there was found that intensive research
was performed about operation room planning. Although there are distinct
differences between operating room planning and dialysis unit planning, literature
about this topic was considered as value adding.
To learn more about the link between patient scheduling and the impact on
a patient’s quality of life, there was searched for academic literature about this topic
too. Moreover, there was searched for the impact of the duration of transportation.
The search terms ‘transportation quality of life hemodialysis patients’ and ‘patient
5
scheduling impact quality of life’ were used to find the appropriate academic
articles. There was also looked for research about the impact of nurse schedules on
their quality of life with the search terms ‘nurse staffing impact life’, ‘stress nurses’
and ‘burnout nurses’.
To analyze and redesign the process, the Business Process Management
Lifecycle was applied. To be able to confirm that this method was appropriate for
the specific problems of the hemodialysis center, academic articles were examined
to ensure that the choice for the BPM lifecycle method was justified.
During the redesign phase of the BPM Lifecycle, a model was built to
reschedule patients. To be able to build this model, there was made use of the theory
about Mixed Integer Linear Programming (MILP).
To achieve a better understanding of the transportation problem, present at
the dialysis center, observations were performed on patients using transportation
services arriving at and leaving from the hospital. Afterwards, the literature was
reviewed using the search terms ‘optimizing transportation’, ‘shared rides’ and
‘patient transportation’. Several of these papers referred to a mathematical
optimization problem concerning transportation optimization: the Vehicle Routing
Problem (VRP). Hence, search terms related to various models of this problem
were used: ‘classes of VRP’, ‘CVRP’, ‘VRPTW’ etc. Also, search terms related to
methods for solving a Vehicle Routing Problem were used. These resulted in search
terms such as ‘exact solution methods for VRP’, ‘heuristics for VRP’ and ‘tools for
solving VRP’.
During the VRPs forthcoming optimization exercise, a heuristic method
was found in the literature that was best capable of solving the case-specific
problem. The method used was called the Adaptive Large Neighborhood Search.
6
7
Chapter 2
Functioning of kidneys, chronic renal failure and treatment possibilities
This chapter gives a brief introduction on kidney functions and renal
diseases. It is present in this thesis to get a better understanding of the importance
of the discussed dialysis process in the next chapters. All of the information is taken
from the official site from AZ Sint-Jan concerning nephrology (AZ Sin-Jan, sd).
Hence, no further literature reference is put in this chapter.
While section 2.1 deals with the kidney’s functions, section 2.2 deals with
the kidney’s failures. Finally, treatment options are listed for these renal failures in
section 2.3.
2.1. Kidney functioning
Under normal circumstances, people are born with two kidneys. These are
located at the backside of the abdominal cavity. Well-functioning kidneys play an
important role in the metabolism of the human body. They are responsible for four
functions. First of all, they manage the amount of water in the body. Kidneys
balance the amount of water by dissipating excessive liquid via urine. Second,
kidneys remove wastes. They purify blood of by-products of the metabolism. These
wastes are removed together with the urine. The third function of the kidneys is
managing the composition of the blood. The amount of sodium and other
8
electrolytes (i.e. potassium, phosphorus) plus the acidity degree are managed by
the kidneys. At last, kidneys are also responsible for the production of certain
hormones. The three main hormones produced are renine, erythropoëtine and
vitamin D. Renine manages the blood pressure. Erythropoëtine stimulates the
production of red blood cells in the bone marrow. The insertion of calcium is
arranged by vitamin D.
2.2. Chronic Renal Failure
Chronic Renal Failure (CRF) is classified as a disease where the kidneys
are not functioning as they should for a certain period of time. Renal failure can
evolve over time where the condition deteriorates slightly, but it can also pop up
immediately. The most frequent causes of CRF are diabetes, high blood pressure,
inflammation of the urinary tract and hereditary kidney diseases. When the kidneys
are functioning less than 15% compared to healthy kidneys, the renal failure is life-
threatening. Then, it is necessary to start a treatment. There are two groups of
treatment possibilities: peritoneal dialysis and hemodialysis.
2.3. Treatment possibilities
2.3.1. Peritoneal dialysis Peritoneal dialysis (PD) makes use of a natural membrane, called the
peritoneum, and functions as a filter. The PD catheter, makes sure that the dialysis
liquid flows into the celiac. The dialysis fluid bags can be connected to the catheter.
The dialysis takes place when the liquid is flown in the celiac. Excessive and waste
liquids of the blood go through the peritoneum to the dialysis fluid. This liquid is
removed out of the celiac and new dialysate goes in. There are two different kinds
of PD: continual ambulant peritoneal dialysis and automatic peritoneal dialysis.
The first one is done manually during the day, while the second one is performed
by a machine during the night.
9
2.3.2. Hemodialysis Hemodialysis (HD) implicates that an artificial kidney removes waste and
excessive fluids. Blood is pumped through a filter where there is contact with the
dialysate. At this way the waste fluids are filtered and removed from the blood.
Then, the purified blood is transferred to the body again. The blood transportation
through the body can take place via an arteriovenous fistula or a catheter.
A fistula is placed on the forearm and links the vein with the artery. During
dialysis two needles are placed in the fistula. One is used to transfer blood to the
artificial kidney, the other brings the purified blood into the bloodstream again.
If an AV fistula is not possible or not yet ready to use, a catheter is used. A
catheter is a hollow tube which is connected to a main vein in the neck and chest
area. During the dialysis, the blood streams are linked to the catheter. A catheter is
a direct connection with the blood streams. This means that this area is vulnerable
to infections. Therefore, a fistula is preferred over a catheter.
2.3.3. Transplantation Some patients are eligible for a kidney transplantation. Because a
transplantation is not without risks, not everyone will be accepted as a possible
acceptor. Once accepted, the patient is placed on a long waiting list. A feasible
kidney can come from a recently died or from a living person. As long as there is
no kidney to transplant, a patient will be treated by peritoneal- or hemodialysis.
The analysis of this thesis is about the hemodialysis department located at AZ Sint-
Jan in Bruges. In the remaining text the terms hemodialysis and dialysis will be
cross-used.
10
11
Chapter 3 Literature review
Chapter three contains an overview of the literature used in formulating an
answer to our research question. The review was split into four main parts. First,
section 3.1 analyses the literature concerning nurses’ stress, absenteeism and
burnout. Following this, section 3.2 highlights the literature around the trend in the
number of dialysis patients. In section 3.3, the focus lays on literature around the
scheduling and optimization approaches. Finally, in section 3.4, distinct literature
about the Vehicle Routing Problem is searched. In addition, literature of solving
these problems is given.
3.1. Stress, absenteeism and burnout at nurses
As analyzed by Karkar, Dammang and Bouhaha a lot of nurses are subject
to moderate levels of stress (Karkar, Dammang, & Bouhaha, 2015). This stress is
mainly caused by work overload with extra responsibilities, excess number of
patients, sicker and older patients and the timing and duration of their working
hours. Nurses working at a hemodialysis department cope with additional types of
stress. The most important stressor is the intensity during initiation and termination,
but urgent interventions in case of life-threatening situations and verbally and
physically abusive patients can cause stress as well.
According to the research of Argentero, Dell’Olivo and Ferretti, burnouts
are characterized by three main feelings: emotional exhaustion, depersonalization
and reduced personal accomplishment (Argentero, Dell'Olivo, & Ferretti, 2008).
Because of the fact that patients have a strong emotional relationship with their
12
nurses, the emotional condition of nurses has an impact on patient satisfaction.
Argentero, Dell’Olivo and Ferretti also found that burnouts with nurses lead to less
satisfied patients.
Absenteeism and burnouts are serious problems at hemodialysis centers.
Numerous articles have researched the reason behind this. A main finding was that
a higher patient-nurse ratio, led to a higher possibility of burnout (Wolfe, 2011).
Moreover, the risk of infections with hepatitis C would raise (Saxena & Panhotra,
2003). Also, nurses were less attentive towards hand hygiene and made more
medication errors. On the other hand, Flynn found that other factors, like workload,
a non-supporting environment and the number of care tasks which cannot be
finished, are more important factors than the patient-to-RN ratio (RN is an
abbreviation for Registered Nurse) (Flynn, Thomas-Hawkins, & Clarke, 2009).
Nevertheless, it seems unwise to increase the patient-nurse ratio.
Analysis of the absenteeism at AZ Sint-Jan shows that the total absenteeism
is higher than the average of the hospital (5,13% vs. 4,78%). Especially, the short
and medium term absenteeism is higher than average (De Vriese, 2015). This
reinforces the necessity to spread the workload.
3.2. Dialysis patients’ evolutions
There is expected that the number of dialysis patients will remain stable the
next years. The fact that the number of patients in Bruges will stabilize can be
concluded from interviews with dr. De Vriese of AZ Sint-Jan. Moreover, also on
Flanders’ level the number of high-care hemodialysis patients remains the same.
As can be seen out of the data of NBVN, the number of dialysis patients has
remarkably increased starting from 2000 (NBVN, 2012). The reason behind this
can be found in the fact that before 2000, physicians did not start a renal treatment
for patients older than 80 years. After 2000, this opinion changed and it led to an
increase in patients. This growth is now again flattened out (Luyckx, 2015).
13
3.3. Scheduling and optimization approaches
3.3.1. Scheduling without uncertainty Holland describes the activities that must take place before the dialysis
session can start (Holland, 1994). He also stated that hemodialysis is unique
because the patients are repeaters who go to the hemodialysis center three times
per week and the length of the treatment is – in most cases – the same. These
characteristics have as a consequence that dialysis can be planned well. He
analyzed that in most centers, patients of the same batch arrive all at the same
moment. There are a couple of downsides of this system. It leads to an
underutilization of dialyzers, patients queuing and limited appointment choices for
new patients. Holland recommends to schedule patients to arrive at 15-minute
time-intervals. This leads to reduced waiting time for patients. If fewer patients
arrive at once, waiting automatically reduces. Moreover, dialysis machines have a
higher utilization degree and the operating hours have been reduced.
Vanholder, Veys, Van Biesen and Lameire link hemodialysis with
peritoneal dialysis (Vanholder, Veys, Van Biesen, & Lameire, 2002). They suggest
that hemodialysis sessions would perform better (i.e. better urea clearance, lower
mortality, lower blood pressure…) if hemodialysis would be scheduled daily or by
prolonging sessions to eight hours.
There is not much specific literature available about patient scheduling in
hemodialysis centers. Nevertheless, there is an extensive literature review about
patient scheduling in general. Cayirli and Veral state that the literature can be
classified into two categories (Cayirli & Veral, 2003). First, the appointment
system is considered static. Here, all decisions are made in advance because there
is no uncertainty. This model is most frequently used. On the other side, the system
can be seen dynamic. In this case, the schedule is revised continuously during the
day. The simplest case is when all patients arrive on time, only one doctor is present
and stochastic processing times are used. The model becomes more complicated
by increasing the number of services, doctors and appointments per clinic session.
It also becomes more complex by the uncertainty of the arrival processes, service
time distributions, queue disciplines and a lateness and interruption level of doctors
are taken into account. Cayirli and Veral also analyze several performance
14
measures. They classify four measures: cost-based, time-based, congestion and
fairness measures. Furthermore, they also elaborate on the appointment rules,
patient classifications and adjustments made to reduce the negative consequences
of walk-ins, no-shows and emergency patients. Patient classifications are used for
two purposes. First, they can sequence patients at the time of booking. Second, they
adapt the intervals of appointment because every patient class has different service
time characteristics.
A literature review was given by Cardoen, Demeulemeester and Beliën
regarding operation room planning and scheduling (Cardoen, Demeulemeester, &
Belin, 2010). They classify patients in two main classes: elective and non-elective
patients. Appointments for elective patients can be planned well in advance. Non-
elective patients are unexpected and need to be helped urgently. Elective patients
can be further classified as inpatients and outpatients. They also specify several
performance criteria, i.e. waiting time, throughput, utilization, leveling…
Gupta and Denton stress that the appointment scheduling problems can be
modeled as cost or penalty minimization problems or profit maximization
problems (Gupta & Denton, 2008). Often, fixed time slots are used to schedule
patients. Every patient is assigned to one or more of these slots, based on his
complaints. Further, Gupta and Denton recommend that there should be a patient-
specific resource allocation: patient specific information should be used to improve
decisions concerning resource allocation. At last, they also propose to take patient
preferences into account.
3.3.2. Scheduling with uncertainty Zhang et al. as well as Lamiri et al. both describe uncertainty in operation
room planning (Lamiri, Xie, Dolgui, & Grimaud, 2008; Zhang, Murali, Dessouky,
Belson, & Epstein, 2009). Zhang et al. use a mixed integer programming model to
allocate operating room capacity to specialties. The model is optimized for
patients’ length of stay. Also some of the randomness of the processes (i.e. surgery
time, arrival time, demand and no-shows) is taken into account. They also make
the distinction between emergency and non-emergency patients. Lamiri et al. made
a stochastic mixed integer programming model where they also classify patients
into an emergency and non-emergency group. They used Monte Carlo simulations
15
to optimize the model. The solutions generated by the Monte Carlo simulation were
compared with the results of a deterministic method. Their simulations performed
better.
Scheduling problems were also analyzed in other sectors, such as
manufacturing. Li and Ierapetritou made a literature review of process scheduling
under uncertainty (Li & Ierapetritou, 2008). They group uncertainty in four
categories: model-inherent uncertainty, process-inherent uncertainty, external
uncertainty and discrete uncertainty. Concerning uncertainty, there are three
methods described: bounded form, probability description and fuzzy description.
In scheduling, Li and Ierapetritou make distinction between reactive scheduling,
stochastic scheduling and robust optimization scheduling. A robust optimization
method focuses on building a preventive schedule to minimize consequences of
disruption. It tries to ensure that the predictive and realized model do not differ
much, while maintaining a high level of performance. An optimization is called
solution robust if it remains close to the optimum in all scenarios. It is model robust
if it remains feasible for most scenarios. The most important advantage of robust
scheduling over stochastic programming is that there is no probability distribution
of the underlying data assumed.
Balasubramanian and Grossman present a non-probabilistic approach to
optimize under uncertainty (Balasubramanian & Grossman, 2002). Their model is
based on the fuzzy set theory and interval arithmetic. They applied their approach
to a flow shop scheduling problem and a new product development scheduling
problem. The approach of Balasubramanian and Grossman consisted of three steps.
They first evaluated the schedule under uncertainty. This is followed by a
generalization of expressions for calculating the start-and end-times for tasks of a
certain schedule. At last, they optimize. Using a fuzzy set allows one to model
uncertainty without historical data. Moreover, it also reduces computation time to
solve the optimization problem.
Simulation-based methods have been widely used in patient scheduling and
evaluating appointments policies and uncertainties. Olugata, Cetik, Koyuncu and
Koyuncu developed an approach to schedule patients in a radiation oncology
department to minimize delays in treatments and maintain efficient use of capacity
(Ogulata, Cetik, Koyuncu, & Koyuncu, 2009). They advise to use the slack
16
capacity approach to schedule patients. According to this approach some patient
capacity is reserved as slack capacity. The dynamic discrete-event simulation was
analyzed as the most appropriate. Olugata, Cetik and Koyuncu found that treatment
delays are completely determined by the slack capacity, in case of high patient
frequency. In systems with low patient frequency, the main factor is the maximum
waiting time. Further, they analyzed that normal capacity usage ratio is highly
determined by the patient arriving frequency. At last, they stated that a slack
capacity higher than required causes inefficiency concerning total capacity.
An adaptive approach to automatic optimization of resource calendars was
made by Vermeulen et al. (Vermeulen, et al., 2009). They divide patients into
different groups based on departments, inpatients or outpatients, priority or
urgency and medical constraints. Their resource calendar consists of time slots of
varying sizes. Each patient group has its own reserved time slots. Nevertheless, if
a time slot is not used, it could be made available for other groups. Furthermore,
they stated that there are three different timeframes to be dynamic. First, there is
the long term view based on long-term expectations, which is a timeframe of
months. Second, medium term adjustments can be made weekly. These are made
for known future events. At last, day-to-day changes can be made. This is the short
term view. To evaluate the performance of their model, they calculated service
level as the percentage of patients who were scheduled on time. All patient groups
were given equal weights.
There exists extensive literature about agent-based scheduling. Mageshwari
and Kanaga elaborated on a review of this literature (Mageshwari & Kanaga,
2012). They briefly discuss wave scheduling. Three challenges are found between
inter-departmental coordination activities, which impacts patient workflow:
ineffectiveness of current information and communication technologies, lack of
common ground and breakdowns in information flow. They categorize patient
scheduling techniques as dynamic, distributed or coordinated scheduling. Dynamic
scheduling takes the dynamic changes of the hospital into account, i.e. patients may
come late, unavailable resources and equipment that needs repair. Patient
scheduling is called distributed when preferences on resources and patients are
inherently distributed and the model takes this into account. Third, a model is
coordinated to reduce the response time of the system in a distributed environment.
17
The dynamic patient scheduling is described by Paulussen et al. (Paulussen,
et al., 2004). They suggest a multi-agent system MedPage in which autonomous
agents represent patients and hospital resources. Via the market coordination
system of MedPaCo, patient agents negotiate on hospital resources which are
scarce. These negotiations are based on patients’ individual heath dependent cost
functions to minimize the stay time of the patients, which is equivalent to an overall
minimization of suffering for the patients. Delays are seen as risk by patient agents.
Paulussen, Jennings, Decker and Heinzl. also describe the distributed
patient scheduling (Paulussen, Jennings, Decker, & Heinzl, 2003). This model
focuses on the distributed organization structure of hospitals, with several
autonomous wards and ancillary units. Each unit has its own local view. They
propose a multi-agent based approach where patients and resources are
autonomous agents with their own goals, reflecting the decentralized structure of
the hospital. Again, a market mechanism is elaborated, in which each agent tries to
optimize their own situation via negotiations. The preferences should be based on
medical preferences. Not only the current health state of the patient is important
but also his health state development.
Decker and Jinjiang proposed a multi-agent solution (Decker & Jinjiang,
1998). To solve the patient scheduling problem, they made use of the Generalized
Partial Global Planning (GPGP) which is a task environment centered approach
to coordination. With this approach each agent constructs its own local view of the
structure and relationships of its tasks. Then, this view is extended with information
from other agents. The GPGP uses certain coordination mechanisms to construct
these partial views and to recognize and respond to certain task structure
relationships by committing to other agents. These commitments result in a more
coherent behavior. Their patient scheduling is based on finding priority functions
for agents, so the finish time of the whole task can be minimized. Therefore, the
coordinated relationships with the task and the start time of that task need both to
be considered. At this way, a task that facilitates many other tasks will have a higher
priority.
In manufacturing settings, the earliness-tardiness problem which a
particular job could experience, has been considered through job flow-shop models.
Ronconi and Birgin examine the flow-shop scheduling problem with no storage
18
constraints and with blocking in-process, which does not have any buffers between
consecutive machines (Ronconi & Birgin, 2012). They developed several mixed
integer programming models to minimize the sum of earliness and tardiness of the
jobs. CPLEX was used to solve the problem to optimality. Hooker combined mixed
integer linear programming and constraint programming in order to minimize
tardiness (Hooker, 2005). Tasks are assigned to facilities using MILP and
scheduled using constraint programming. These two are linked using Benders’
decomposition in a hybrid model. The hybrid model outperformed the individual
results of MILP and constraint programming, concerning speed. Hooker optimized
two objectives: minimize the total number of late tasks and minimize total
tardiness.
3.4. Vehicle Routing Problem
3.4.1. Different classes Dantzig and Ramser were the first who defined the Capacitated Vehicle
Routing Problem (CVRP) in an academic report and named it the “Truck
Dispatching Problem” (Dantzig & Ramser, 1959). The authors applied a primary
algorithmic approach to the petrol industry. In their research, they modeled how a
fleet of homogeneous trucks could serve the geographically spread demand for
petroleum from a central depot, while minimizing the driving distance. Stating that
the trucks were homogeneous is equal as saying that each vehicle on its own had
the same capacity as well as the same cost structure.
The original problem definition described by Dantzig and Ramser back in
1959 was as follows:
“A number of identical vehicles with a given capacity are located at a
central depot. They are available for servicing a set of customer orders. (…) Each
customer order has a specific location and size. Travel costs between all locations
are given. The goal is to design a least-cost set of routes for the vehicles in such a
way that all customers are visited once and vehicle capacities are adhered to.”
19
Five years later, Clarke and Wright improved on Dantzig and Ramser's
Truck Dispatching Problem by using a more generalized linear optimization model
to solve the problem (Clarke & Wright, 1964; Wen, Clausen, & Larsen, 2010).
They have expanded the model with trucks that had diverse capacities and offered
an efficient savings algorithm to solve the problem.
After these two influential papers, numerous solution approaches have been
proposed and benchmarked against each other (Eksioglu, Vural, & Reisman,
2009). The study by Eksioglu, Vural and Reisman showed that between 1959 and
2008, more than 1.000 articles with VRP as the main topic, were published. The
authors learnt that academic literature concerning VRP has known a yearly
exponential growing rate of roughly 6%. Several motives for this thriving interest
were given by Eksioglu, Vural and Reisman. One of the most meaningful reasons
was the advancement of resources with an improved computational power, which
opened up big prospects for more complex routing problems and hence, making
routing problems more capable to reflect real-world situations and with less
computational burdens. Another interesting and more recent explanation lies in the
forthcoming e-commerce business – where there is still an exponential growing
number of people who shop online and want their orders to ship as fast as possible
and at the lowest price. Transportation and logistics is one of the most vital aspects
for today’s e-commerce companies.
To make this problem more adapted to case-specific problems, various
authors have made specific variants of the Vehicle Routing Problem. Overviews of
some well-known variants were given by Toth and Vigo (Toth & Vigo, 2002).
Moreover, providing overviews of these Vehicle Routing Problems was the main
topic in several PhDs such as Nguyen, Wen as well as Clausen and Larsen
(Nguyen, 2014; Wen, Clausen, & Larsen, 2010).
3.4.2. Solution methods As Mariño described, vehicle Routing Problems belong to a class of
optimization problems that are extremely hard to solve to optimality, namely the
class of 𝒩𝒫-hard problems (Mariño, 2016). Hence, methods to solve these
problems exactly is a hot topic in today’s literature. Although significant progress
has been made concerning medium-sized instances of some general types of
20
routing problems, there is still an undisputable gap for larger instances or complex
VRPs, such as the VRPTW (Wen, Clausen, & Larsen, 2010).
Intensive research about the different solution algorithms and their
efficiencies has already been made by several writers, including Røpke and
Pisinger as well as Laporte and Semet (Røpke & Pisinger, 2006; Laporte & Semet,
2002). In their research papers, two groups of solution methods were distinct: exact
and approximate solution methods and heuristic solution methods. There is a vast
difference between the two.
An exact solution method tries to find the exact optimum for any instance
of the problem (Christofides, Mingozzi, & Toth, 1981). This solution method
requires a mathematical model that fits within the problem’s exact constraints. As
been studied by Fukasawa et al., widespread exact solution techniques for the
theoretical CVRP are the branch-and-bound, branch-and-cut and branch-and-price
techniques (Fukasawa, et al., 2004). A downside of these techniques are the
possibilities to only solve smaller-sized problems, the long processing times and
its inflexibility (Røpke, 2005). These methods also require sophisticated and costly
software, such as Gurobi or Matlab. While exact solution methods only terminate
when an exact solution is found, approximation methods do not. These solution
methods provide solutions of a certain quality. The gap between the current
solution and the exact optimal solution is known (Laporte, 1992).
A research by Røpke showed that state-of-the-art exact algorithms have
only solved limited-sized routing problems in a relative large computing time
(Røpke, 2005). Thus, for larger-sized routing problems or for problems that can
change in time and need to be computed several times, different solution strategies
need to be considered. Heuristic solution methods play an important role in solving
these complex problems. They are well-known in the area of operational research
and the literature provides various approaches in solving these problems. The aim
is at generating ‘reasonably good’ solutions within modest computing times by
performing only a reduced assessment of the entire search space. As stated by
Røpke, the term ‘relatively good’ is used because often there is no information
provided about the optimal solution (Røpke, 2005).
21
3.5. Conclusion
Section 3.1 learns that stress and absenteeism have several causes. There
seems to be an undeniable link between stress and burnout and the workload. Out
of section 3.2 follows that the number of dialysis patients will stagnate. Therefore,
there is chosen not to change the capacity of the dialysis center at AZ Sint-Jan.
Section 3.3 gives an extensive overview about patient scheduling literature. Several
of these techniques will be used to build a new patient schedule. Last of all, section
3.4 provides a first theoretical description of the Vehicle Routing Problem and
already suggests a group of suitable solution methods. The literature concerning
the routing problems will be used to build a better schedule concerning the
transportation of dialysis patients.
22
23
Chapter 4
Process Modeling
To model business processes, the Business Process Management lifecycle
can be applied. This lifecycle is described by Dumas, La Rosa, Mendling and
Reijers and is based on the research by Becker, Kugeler and Rosemann (Dumas,
La Rosa, Mendling, & Reijers, 2013; Becker, Kugeler, & Rosemann, 2011). There
are six stages in this lifecycle, as can be seen in Figure 1. This chapter is organized
according to this lifecycle. Section 4.1 handles process identification. The
following section, section 4.2, deals with the process discovery. The process
analysis is elaborated in section 4.3. In the last section, section 4.4, the process is
redesigned.
Figure 1: Business Process Management Lifecycle
24
4.1. Process identification
The first step identifies the process identification and the business
problems. The involved processes and their interdependence can be identified
based on the found problem. This stage results in a general sight at the
organizations’ processes and their relationships. There has to be focused on the key
processes. To evaluate which processes are most interesting to analyze and
redesign, an evaluation has to be made. This evaluation can be based on three
factors: importance, dysfunction and feasibility (Dumas, La Rosa, Mendling, &
Reijers, 2013).
The process as it is today leads to several problems:
• Nurses are dissatisfied because of the fact that they are confronted with peaks
in workload. The patients of the same shift arrive around the same moment.
They all have to be connected to a machine at the same time. There is a
patient/nurse ratio of 1/4 which means that each nurse has to connect four
patients at the same time. This leads to stress and eventually absenteeism and
burnout. On the other hand, at some other moments of the shift, nurses have
too much free time because the workload is too low.
• Patients are dissatisfied because of long waiting times. Waiting is created
because of several reasons. First, because patients arrive too early in the
hospital. This is mainly a problem in the afternoon shift. The taxi companies
first pick-up the patients who are scheduled in the afternoon, put them in the
waiting room of the hospital and then pick-up the disconnected patients of the
morning shift. Thus, the patients of the afternoon arrive before the patients of
the morning are disconnected. Second, patients also have to wait because all
patients have to be connected at the same moment. Moreover, they sometimes
have to wait for other patients who need further investigations and travel with
the same taxi.
• The nurse schedule is not optimal. As will be explained in the section about
process discovery, there is a two shift system. A time overlap is present
between the early and late shift between 9h45 and 15h30. There was observed
that there is not enough work for the nurses of both shifts. Nevertheless, the
25
overlap is advantageous for the disconnection of the patients of the morning
and the connection of the patients of the afternoon because those activities run
more smoothly.
4.1.1. Value chain In order to identify the involved processes Burns et al. as well as
Kawczynski and Taisch propose to make use of a value chain in healthcare (Burns,
et al., 2002; Kawczynski & Taisch, 2010). The value chain has the objective to
optimize the activities of cooperating firms to create a bundled service, manage the
chain end to end, develop highly competitive chains and create a portfolio approach
to work with suppliers and customer (Burns, et al., 2002).
This value chain is based on the chain as proposed by Michael Porter. Porter
makes the distinction between core processes and support processes. Core
processes are the processes that create value. Support processes facilitate the
processes’ execution.
The following model will focus on three core processes in order to avoid
complexity (Szelagowski, 2013). As a result of the evaluation follows that these
three core processes are the most important ones, since they directly have an impact
on the service perceived by the patients. Moreover, most complaints are about these
three processes. It is assumable that these processes will be feasible to change.
Figure 2 shows the value chain model of these three processes. First, the patients
have to be transported from their house to the hospital. Then, they get dialyzed for
four hours each. At last, they have to be transported back to their house.
As stated by Henkel, Johannesson and Perjons, a value chain model has to
be combined with a goal model and an action model (Henkel, Johannesson, &
Perjons, 2007). Value models are limited since they do not explicitly demonstrate
how a value relates to actions. Moreover, value chains cannot be used to discover
actions that can improve value creation.
Transportationtodialysiscenter Dialysis Transportationto
dwellings
Figure 2: Three main processes for the dialysis process
26
4.1.2. Goal modeling An essential part of process identification is goal modeling. An 𝑖∗ goal
model shows the different actors, their goals and softgoals and how these aims will
be achieved (Yu, 2001). The goals are attained by actors and their
interrelationships. There are two types of 𝑖∗ models. Strategic dependency
demonstrates how actors interrelate to achieve a goal. Strategic rationale shows
the reason why an actor uses the system and alternatives to attain goals (Yu, 2009).
The strategic rationale has five building blocks. First, actors are identified.
These can represent a role, an agent or a position. There are also goals, which are
situations a certain actor wants to achieve. Softgoals can be defined as well. These
highlight wishful conditions for the actor. However, the criteria for the condition
are not well defined. To model activities which can be completed by an actor, 𝑖∗
uses a task. At last, there are also resources. A resource, which can be physical as
well as informational, is required to perform a task. To connect these building
blocks, there are two types of links. First, there is a means-end link. This links a
mean, which is a task, with an end, which can be another task, resource, softgoal
or goal. Another link is the task decomposition link, which demonstrates the
softgoals, goals, other tasks and resources needed to perform a task (Yu, Social
Modelling and i*, 2009).
The strategic dependency is built around four types of dependencies. First,
the goal dependency shows how two actors interrelate to achieve a goal. The
depender depends on the dependee to perform the goal. The dependee can freely
decide between the alternatives to attain the goal. There is also a task dependency.
In this type of dependency, the depender relies on the dependee to perform an
activity. This relationship identifies the way to perform a task. A third link is the
resource dependency where the depender counts on the dependee for the
availability of a resource. Finally, the softgoal dependency shows the link between
the depender and the dependee to achieve a softgoal. There is no agreement
between the two actors beforehand on how this softgoal can be achieved (Yu,
2009).
To make up the goal model there is opt to make use of a combination of the
strategic rationale and the strategic dependency. This combination gives the best
27
possibilities to make an accurate goal model. The patient is located central in the
goal model, which is provided in Appendix I. Patients expect efficient and effective
care and it is the task of the nurses to give them the right care, consisting of
connecting, disconnecting, distributing meals… Furthermore, patients expect the
right medical treatment from the doctors, while doctors have the task to monitor
patients’ condition. The social workers have to help the patients with problems
about health insurance, transportation… The patients have to work towards a target
weight. The dietitians help patients to attain the target weight by making up a diet
and giving advice about the right food pattern. The patients depend on the
transportation firms to get efficient and effective transportation. The last actor
interacting with the patients, is the planner. The planner proposes a schedule with
the arrival times for all patients, based on their preferences and other remarks. The
planner also sets up the work scheme for the nurses. The goal of this timetable is
to increase or at least sustain the work satisfaction of the nurses. Moreover, there
is a balanced workload expected. The planning is the responsibility of the head
nurse.
4.2. Process discovery
During this phase, an as-is model is developed. This means that the existing
process is modeled. The model will be based on collected information about the
process. There are four subphases in the discovery step. First of all, the setting of
the process has to be defined. Second, the information has to be collected. Then,
the model has to be created. At last, the quality of the model has to be assured
(Dumas, La Rosa, Mendling, & Reijers, 2013).
4.2.1. Definition of the setting In the first phase, the team that will investigate the processes has to be
formed. It is clear that the team will consist of the authors of this master
dissertation. They can be called the process analysts since they have a thorough
knowledge of the business languages. As process analysts, they will have to
collaborate closely with the domain experts, which are the head of the dialysis
department, the head nurse and the other nurses.
28
4.2.2. Data collection The second step consists of gathering information. There are three general
methods to collect data: evidence-based, interview-based and workshop-based
discovery.
4.2.2.1. Evidence-based discovery
There is chosen for a document analysis and an observation as evidence-
based methods.
4.2.2.1.1. Document analysis
In the document analysis, the documents related to the process are
investigated. The documents concern the departments’ memorandum, the patient
schedules, the labor schedule, the distribution of patients over the dialysis rooms,
the mode of transportation of the patients and the division of tasks for the
personnel. Although, many of these documents summarize only specific parts of
the process, there is the belief that these documents are useful to better understand
the existing process. Another limitation on document analysis, is the fact that
documents are not always completely trustworthy. Nevertheless, the documents,
except for the memorandum, are not open to interpretation.
4.2.2.1.2. Observation
During the observation, the process from the viewpoint of the patients, is
observed to get a better insight at the process. The authors positioned themselves
as passive observers. A possible negative consequence of this method is the fact
that people behave differently when they are observed. To prevent this from
happening, there is chosen to observe at least four times. At this way, the
probability that the nurses act differently is smaller because there is more mutual
trust between the authors and the nurses.
The observation has shown that the duration of some activities was
dependent on the condition of the patients. It was mainly the connection and
disconnection activity which was subject to the situation of each patient. Based on
observations and interviews with nurses, there was concluded to statistically test
two factors on their impact on the duration of both the connection and
disconnection.
29
4.2.2.1.3. Interview-based discovery
The interview-based discovery is a method where domain experts are
interviewed in order to gather information. The head of the dialysis department in
AZ Sint-Jan was interviewed as well as the head of the dialysis department of AZ
Alma in Eeklo. At this way, a better and more general insight is gained on the
dialysis process and the constraints of the process. Moreover, during the
observation, many questions arose about some of the observed activities.
Interviews allow one to acquire an in-depth knowledge of the process, which is not
possible through observations.
Based on the documents, observations and interviews the following
information about patient scheduling, dialysis rooms, patient classification and
nurse scheduling was deducted.
4.2.2.2. Deducted information
4.2.2.2.1. Patient schedule
For the moment, the clinic has a patient pool of approximately 150 patients.
These patients are divided over a four shift system. Each patient gets dialysis three
times per week. The first patient group is scheduled on Monday, Wednesday and
Friday morning. The second group is scheduled in the afternoon on Monday,
Wednesday and Friday. The third and fourth group get dialysis on Tuesday,
Thursday and Saturday morning or afternoon. The distribution of patients over the
different shifts is included in Table 1. As can be seen, most patients are scheduled
on Monday, Wednesday and Friday. This is because nurses are not likely to work
on Saturday.
Shift 1
(M/W/F AM) Shift 2
(M/W/F PM) Shift 3
(T/T/S AM) Shift 4
(T/T/S PM) Number of patients 42 40 34 32
4.2.2.2.2. Dialysis rooms
In Bruges there are five dialysis rooms in total. Room 1 and 2 both have a
capacity of seven beds. Room 3 has 24 places. Four beds can be found at room 4.
Room 5 can receive nine patients. Room 2 houses the acute patients and the people
Table 1: Number of patients assigned to each shift
30
who need to be isolated due to the risk of infection. Room 4 is never used. The least
sick and most mobile patients are located in room 5. Table 2 shows the number of
patients per shift and per room.
Shift 1
(M/W/F AM) Shift 2
(M/W/F PM) Shift 3
(T/T/S AM) Shift 4
(T/T/S PM) Room 1 7 7 0 0 Room 2 2 1 1 1 Room 3 24 23 24 22 Room 4 - - - - Room 5 9 9 9 9
4.2.2.2.3. Condition of patients
In Belgium, dialysis patients are classified as low- or high-care. Low-care
patients are more independent and can connect and disconnect themselves up to a
certain point. Mostly, low-care patients are younger people. They only require a
doctor to visit them once every two weeks. On the other hand, high-care patients
require more help. They cannot install machines or take their own blood pressure.
They need a nephrologist to visit more frequently. Lots of high-care patients need
accessories (wheelchairs, walking sticks) to move. Some need help to get in or out
of their beds. Unfortunately, there are no official criteria in Belgium to classify
patients as low- or high-care. The classification of patients relies on the common
sense of the nephrologists. Moreover, the classification is also influenced by the
difference in nomenclature (NBVN, 2012; Luyckx, 2015). However, this falls
beyond the scope of this master thesis.
The hemodialysis unit of AZ Sint-Jan is officially a high-care unit, which
means that most of the patients are classified as high-care. A nephrologist visits all
the patients every session.
Around 55% of the patients in Bruges have an AV fistula. So, the other
45% have a catheter. Approximately three out of four catheters are permanent.
4.2.2.2.4. Shift system nurses
There are four different nurse shifts. The two most important shifts are the
early 8h shift and the late 8h shift. The early 8h shift starts at 6h45 and ends at
15h30. The late 8h shift starts at 9h45 and stops at 18h30. Moreover, there are also
Table 2: Number of patients per room assigned to each shift
31
employees who do not work full time. They work six hours per day. The early 6h
shift starts at 6h45 and ends at 12h45. The late shift starts at 12u30 and ends at
18h30.
4.2.3. Creation of the process model In the third phase, the collected information is processed into a model. The
model is created by making use of the Business Process Modeling Notation
(BPMN). This language is chosen because of several reasons. Rad, Benyoucef and
Kuziemsky state that it is important that a model can be understood by all model
users as well as by the people who do not have any process analysis skills (Rad,
Benyoucef, & Kuziemsky, 2009). Furthermore, the language has to be simple such
that it can bring clearness in complex processes. It also has to be easily adaptable
so it can be optimized for several purposes. BPMN has all these characteristics.
Jun, Ward, Morris and Clarkson listed the main modeling techniques and
their functionalities in the healthcare industry (Jun, Ward, Morris, & Clarkson,
2009). There was found that swim lane activity diagrams have the advantage of
giving a good understanding of roles in various tasks, while flowcharts are
favorable to get an initial understanding of the process. Since BPMN is a
combination of a swim lane activity diagram and a flowchart, it is a decent
language for the project at the hemodialysis department. Moreover, Jun, Ward,
Morris and Clarkson state that data flows are less important in modeling healthcare
processes.
Mendling, Recker and Reijer investigated that a business process modeling
language has to be auditory and visual (Mendling, Recker, & Reijers, 2010).
Auditory means that processes have to be documented with labels and text
annotations. Visual refers to the use of graphical constructs. Again, BPMN fulfils
these requirements.
Ruiz, Garcia, Calahorra and Lorente as well as Rojo et al. specifically
advocate the use of BPMN (Ruiz, Garcia, Calahorra, & Lorente, 2012; Rojo, et al.,
2008). Ruiz et al. refer to the fact that BPMN makes it possible to use subprocesses,
which make it possible to model different levels of detail. Ruiz et al. also see
advantages in the fact that BPMN is easy to re-use and that it is easy to understand.
This simplifies the communication between domain experts and process analysts.
32
Rojo et al. also refer to the fact that BPMN has a high understandability so
improvements can be implemented more easily by personnel.
Nevertheless, there are also doubts about BPMN being the most optimal
language to perform the process discovery phase. Müller and Rogge-Solti are more
critical towards BPMN (Müller & Rogge-Solti, 2011). The authors moot that
BPMN is not clear when there are many roles and thus many swim lanes or pools.
Müller and Rogge-Solti propose a new method. Nevertheless, this method is not
utilized in this work for two reasons. First, the hemodialysis process does not
include many roles. Second, the method is not mature enough and is not supported
in other academic work.
To make up the BPMN model, there can be relied on the method of Bruce
Silver (Silver, 2009). This method consists of six steps. First, the scope of the
process has to be set. This includes determining the way the process starts, the
different ways the process can end and when the process is complete. Second, the
main map has to be created. This embraces the identification of major activities
and the end state of each activity. Furthermore, these main activities become
subprocesses. Gateways are also included in the main map. In the fourth step, the
subprocesses are elaborated. Each subprocess has to start with a start event and end
with one or more end events for each end state. At last, message and data flows can
be included but these are not obligatory in BPMN. The BPMN-models of the
current process are included in Appendix II.
4.2.3.1. Main process
Around 6h45, the nurses arrive, install the machines and put the material
for the connection of the patients on the tables. At 7h15, the first patients arrive
and the nurses start connecting them to the machines. Because all patients are
scheduled at the same moment, this system can be classified as ‘single block’, as
described by Cayirli and Veral (Cayirli & Veral, 2003). Normally, each nurse is
assigned to four patients. In dialysis room 5, there are commonly nine patients
scheduled for two nurses because of the better condition of the patients. The
primary rule is that patients who need longer dialysis are connected first. Then,
patients are helped on a ‘first come, first served’ basis [implemented since February
2016]. Next, breakfast is distributed and the nurses help the patients who cannot
33
make their own sandwiches. If there is a blood collection in the afternoon, the
vignettes for the afternoon patients have to be printed. These vignettes state which
components of the blood that need to be investigated. These labels are placed on
the tubes. After breakfast, the material for the disconnection task has to be
prepared. Additionally, the material for the connection of patients of the following
shift is prepared. For each patient, a specific box is prepared. These boxes are
prepared per eight patients. For example, if nurse A is responsible for the patients
who lay on bed 1, 2, 3 and 4 and nurse B for patients on 5, 6, 7 and 8, those two
nurses will prepare the material for the patients who will lay on these eight beds in
the following shift. Before the end of the dialysis, the disconnection material is
distributed and put on the tables. These tasks occur sequentially. During the dialysis
of the patients the values of the patients, including their weight, have to be
recorded. This is repeated every 30 minutes, until the end of the dialysis for the
patient. Another task that has to be performed is the preparation of the medical files
of the next shift. A smaller task is the disinfection of the clips. Because of the higher
risk of infection, extra care is given to the disinfection and cleaning of catheters. In
normal circumstances, the catheter is cleaned every week. If non-water-resistant
plasters were used for the catheter, these plasters are changed every dialysis
session. When the doctor and the head nurse consult the patients, the nurse which
is responsible for a certain patient has to assist them. The observation also revealed
that several uncertain activities could occur. These activities happen regularly but
cannot be known in advance. There is opt to bundle these activities in the activity
‘Provide extra care for the patient’. It concerns patients becoming sick, heavy
bleedings, needles which are loosening and patients who do not show up. When
none of these activities occur, the remaining time can be spent on patients as extra
care (e.g. talk with patients).
Normally, the dialysis takes four hours. Sometimes the dialysis duration
will take longer, for example because the patient has skipped a previous session. A
dialysis duration can also be shorter than four hours, for example because the
patient is sick or needs additional investigations. After the dialysis, the patient will
be disconnected. This activity is further elaborated in a subprocess. After the
disconnection, the nurses still have to perform several tasks before the process is
finished. First the bed is prepared for the next patient: the bedding is removed, the
34
bed is disinfected and is made up. Each machine is also cleaned and new wires and
tubes are put in the machine for the following shift of patients. The medical files,
on which the values are recorded each 30 minutes, have to be inserted in the
computer system. Furthermore, the garbage bins have to be cleaned and there has
to be checked if the television screens are out.
The next shift is merely the same and will therefore not be discussed.
4.2.3.2. Subprocess connection
The connection is further modeled as a subprocess because it is considered
as an important activity where problems occur. The process starts with a check-up
of the condition of the patient. If the patient needs help from a nurse to get in or
out of his bed, a nurse will assist the patient. If the patient does not need any
assistance, a next task is immediately started. During this task, which also has to
be performed for patients who got assistance, a next check-up is executed. There
will be checked if a blood collection is needed, if the patient has a catheter or a
fistula, what the target weight of the session is and if there are special remarks from
the doctor. Then, patients are divided into catheter patients and fistula patients.
Although, some of the activities are similar, there is chosen to model both processes
separately in order to improve the readiness of the model.
Connection fistula. First, the cotton pads are disinfected and the blood pressure
monitor is put on. After this, the nurse places a bandage around the wrist. Then, the
area around the AV fistula is disinfected. Hereafter, the nurse pricks two times in
the fistula. In the next step, the nurse tightens a bandage around the upper arm. At
this way, the two wires are connected to the two needles more easily. An adhesive
bandage is placed on both wires. Then, the wires are attached to the machine and
tied up as comfortable as possible for the patient. At last, the dialysis machine is
set up, blood pressure and other values are recorded, and the table is cleaned.
Connection catheter. First, the patient puts a mouth mask on because of the
higher risk of infection. Then, the nurse puts medical gloves on and the blood
pressure monitor is placed around the arm of the patient. After that, a sterile cloth
is put on the patient. The sock is put off the catheter and the anti-clotting fluid is
removed out of the catheter. Hereafter, cotton pads are disinfected and are used to
35
disinfect the catheter and the area around it. Then some blood is removed to empty
out the catheter. Every two weeks or in case of additional investigations, there is a
blood collection needed. The rest of this subprocess looks more or less the same as
the subprocess of the fistula. The wires are connected to the catheter and the
machine. They are also tied up. The machine is installed, blood pressure and other
values are registered. At last, the table is cleaned and the sterile cloth is removed.
4.2.3.3. Subprocess disinfection catheter
The process starts with the patient who has to put on a mouth mask. Then,
the nurse puts medical gloves on and puts a sterile cloth on the patient. After this,
the adhesive plaster is removed and the cotton pads are disinfected. With these
pads, the catheter is cleaned. At last, a new adhesive plaster is placed on the catheter
and the mask, sterile cloth, medical gloves and cotton pads are removed.
4.2.3.4. Subprocess disconnection
The disconnection is considered as an important activity since there are
problems observed with this activity too. First there has to be figured out if the
patient has a fistula or a catheter. Again, both processes are merely modeled
separately to give a better overview and increase the understanding of the
processes.
Disconnection fistula. To start, there is a sterile cloth put under the
patients’ arm. Then, the bandage around the wrist is removed and the values, i.e.
weight, are recorded. Hereafter, the blood pressure monitor is removed. The wires
are removed one by one. In case of the fistulas, it can happen that there is an
uncontrolled bleeding. If there is such a bleeding a clip has to be put around the
arm of the patient. When the bleeding has stopped, an adhesive plaster can be put
on the fistula. When there was a heavy bleeding, a bandage will be used. Then, the
machine goes in setup mode for one hour and the table is cleaned. At last, the
condition of the patient is checked. The patient will need extra help if the patient
cannot get in and out of his bed independently or if the patient feels sick.
Disconnection catheter. First, the patient puts a mouth mask on and the
patient gets a sterile cloth around the catheter. The nurse puts medical gloves on.
36
Then, the bandage around the wrist is removed. Furthermore, the wires are
disconnected. Next, the blood pressure and other values are recorded. The blood
meter can be removed now. With the disinfected cotton pads, the catheter can be
made sterile. To prevent the catheter from being clogged, the anti-clotting fluid has
to be put in the catheter and the sock has to be moved over it. Then, the wires can
be removed out of the machine and the patients’ mask can be put off. After this,
the nurse takes her medical gloves off and sets the dialysis machine in a setup
mode. The table also has to be cleaned. The last steps are identical to the steps of
the fistula disconnection. There is checked if there is assistance needed. If there is
help needed, the nurse assists the patient.
4.2.3.5. Evaluation of the BPMN technique
As mentioned earlier, the BPMN technique is endorsed in several academic
papers to model health care processes. After modeling with this tool, there are some
clear advantages distinguished. BPMN is a simple language which implies that it
is rather easy to model but it also gives a clear overview of the process. At this
way, people without process modeling skills can easily understand the model.
Nevertheless, one disadvantage was experienced. It was found difficult to model
the uncertain activities.
4.2.4. Model quality The quality of the created model has to be evaluated based on three factors:
syntactic quality (verification), semantic quality (validation) and pragmatic quality
(certification). During the verification, where the syntactic quality is evaluated,
there is checked if the model is developed according to the syntactic BPMN rules
and guidelines. The validation is used to control if the statements in the model
match with the real situation and if all relevant statements are included in the
model. At last, the pragmatic quality of a model is assessed by evaluating the model
on three factors: understandability, maintainability and learning.
Understandability concerns the easiness of reading a model. Maintainability deals
with the changeability of a model. Learning relates to the degree of how good a
model demonstrates how a process works in reality.
37
Since the processes are modeled in Signavio, there is automatically checked
for syntactic quality. The modeled processes did not return any errors. The
semantic quality will be checked together with the head of the department as well
as with several nurses. In addition, the pragmatic quality will be controlled by
presenting the model to nurses. However, no problems are expected concerning the
certification since BPMN is a language with a high understandability and
adaptability (Ruiz, Garcia, Calahorra, & Lorente, 2012; Rojo, et al., 2008).
4.3. Process analysis
The process analysis phase consists of a qualitative analysis and a
quantitative analysis. During the qualitative analysis, a value added analysis and a
root cause analysis will be conducted. The quantitative analysis will consist of a
measurement of the process performance dimensions, efficiency ratios and
utilization (Dumas, La Rosa, Mendling, & Reijers, 2013).
4.3.1. Qualitative analysis
4.3.1.1. Value added analysis
The value added analysis consists of a value classification and a waste
elimination. During the value classification each step in the process is analyzed and
classified as value adding (VA), business value adding (BVA) or non-value adding
(NVA). A value adding activity delivers value to the customer. Business value
adding activities are necessary to support the business processes or is required due
to regulations. A non-value adding activity is not necessary for the business and
does not deliver any value. As can be seen in Appendix III, most of the activities
are classified as value adding. They can be distinguished in three categories of
tasks: contact moments with the patients, activities to prepare connection,
disconnection or other contact moments, and tasks to ensure hygiene. The tasks to
ensure hygiene are classified as value added because of the importance for the
patients. It is absolutely necessary to avoid transfers of infections.
38
The following activities are labeled as business value adding. Per activity,
the reason to classify it as business activity is explained. An alternative or
improvement for the activity will be handled in the section about process redesign.
Prepare dialysis: put on machine and distribute material for connection. Starting up a machine is not seen as value adding, however, it cannot be eliminated.
It is a necessary task to make the dialysis possible. Distributing material is also not
value adding. Nevertheless, it cannot be eliminated.
Determine if blood samples have to be taken. Determining if an event has
to take place is not value adding but it is also not possible to eliminate or automate
this step.
Distribute disconnection material (on tables). This task does not add
value. Nevertheless, it cannot be eliminated. It is also not advisable to integrate this
task with the previous activity (prepare material for disconnection in next shift).
There is expected that the time spent on the integrated activity would be higher
than the time spent on the activities separately. This can be explained by the fact
that if the activities are organized separately, the disconnection material for
multiple patients can be distributed together. This decreases the movement of the
nurses compared to the integrated activity where the material for a certain patient
is distributed as soon as its material is prepared.
Record patient values. The values are recorded every 30 minutes. This is
executed manually by the nurses. The nursing personnel looks at the dialysis screen
and writes down the patient values. This is non-value adding because this task can
be organized more efficiently. Nevertheless, this task cannot be seen as non-value
adding since it is necessary to monitor the patients’ values.
Prepare patient folder for next shift. The folders of the patients have to be
prepared. This implies that the folders have to be taken out of the cabinet, new files
have to be printed and patient’s data and remarks have to be filled in. In order to
be able to record values, the folders have to be prepared. This is not of real value
for the patients but it has to be performed in order to be able to record patient’s
values.
Evaluate if the catheter has to be cleaned. Normally, every week the catheter
has to be cleaned. If the catheter is not clean or if there is not made use of the
39
standard plaster, there is the possibility that the catheter has to be cleaned more
frequently. The nurse has to determine the necessity of cleaning. This does not
deliver direct value to the patient, but cannot be eliminated either.
Check if TV-screens are out. This task does not deliver value to the
patient but is necessary to perform. The biggest problem with this task lies in the
fact that the TV-screens are attached to the ceiling, which is too high for most
nurses.
Check if the patient can get in and out bed independently or needs help from
a nurse. This needs to be performed but does not deliver direct value to the patient.
Check-up patient data: blood collection needed, fistula/catheter, target
weight, doctor remarks. This check-up does not deliver direct value to the
patient but is necessary for the nurse to provide the most appropriate care.
Clean table. Removing waste is necessary, especially because of hygienic
reasons. Nevertheless, it does not deliver direct value to the patient.
Check-up patient data. At the end of the dialysis, the values of the patient
have to be evaluated. This is necessary, but does not deliver value to the patient.
Check if there is an uncontrolled bleeding. Again, this is a necessary step but
generates no direct value.
Put machine in setup mode. Putting a machine in setup mode is not value
adding, but needs to be performed in order to be able to dialyze the next shift of
patients.
Some activities were classified as non-value adding. Similar to the
business-value adding activities, there is elaborated on the reason why these tasks
were seen as non-value adding.
Assist doctor during consultation tour. This stage is seen as unnecessary. It
does not deliver value to the patient. All remarks and values observed by the nurse
are written down in the patient files. Communicating these remarks to the doctor is
not value adding.
40
Insert medical files in database. During dialysis, the values are filled in
manually by the nurse. After the dialysis, the nurse has to insert the values and
remarks in the medical file of the patient. This does not deliver value.
4.3.1.2. Root cause analysis
A second method which can be used in the qualitative process analysis, is
a root cause analysis. In this analysis the root causes of the problems are
investigated. There are several methods described in the literature. The 5-Why
analysis is preferred in making the root cause analysis. 5-Why is a technique which
is situated in the analysis phase of the Six Sigma DMAIC-cycle (DMAIC stands
for “Define, Measure, Analyze, Improve and Control”). The idea behind 5-Why is
that by repeatedly questioning why a problem happens, the root causes of the
problem can be found (Institute for Healthcare Improvement, 2015).
The main problem at the hemodialysis center can be found at the moments
the connection takes place in each patient block. Nurses are stressed and patients
have to wait to get connected.
Problem: Stressed nurses and patients who are confronted with waiting times
Why? Nurses have to connect several patients at the same time Why? Patients arrive at the same moment in time Why? Patients are scheduled at the same moment time
The 5-Why analysis results in the root cause of the problem. The root cause
lies in the fact that all patients of the same patient block are expected at the same
moment in time.
4.3.2. Quantitative analysis
4.3.2.1. Performance dimensions
During the quantitative analysis there are process performance measures
calculated. These measures can be determined for every process. Based on the data
of individual activities, process performance characteristics can be measured.
There are four process performance dimensions: time, cost, quality and flexibility.
41
4.3.2.1.1. Time
The dimension of time is expressed by the cycle time. The cycle time of a
process is the time needed for a case to go from the start of the process to the end.
The cycle time consists of two components: processing time and waiting time. The
processing time is the time that the resources of the process spend on effectively
handling an element. Applied to the dialysis case, the processing time can be called
service time and represents the time that a patient is being connected, dialyzed and
disconnected. The waiting time is the time that an element is in an idle mode.
Queuing time is the part of the waiting time where the element has to wait because
there are no resources available. In the dialysis case, the queuing time can mainly
be identified at the connection activities.
4.3.2.1.1.1. Cycle time
The cycle time was calculated based on the average duration of each
activity, which are represented in Appendix IV. The average duration of each
activity is expressed per patient. The average duration of the connection was based
on the fact that the average connection duration of patients who need help to get in
and out the bed is statistically different from the connection duration of patients
who can get in and out their bed independently. To get an average connection
duration, the two average durations were multiplied by the percentage of patients
they represent. 35% of the patients need help to get in and out their bed, the other
65% can do it independently. The printing and attaching of stickers for blood
samples only happens once every two weeks. This corresponds with one blood
collection out of six dialysis sessions. So, the probability of a blood collection
equals 17%. The duration of both tasks is multiplied by 17%. Recording patient
values happens every 30 minutes during the dialysis. A standard dialysis takes 4
hours. This means that the values will be recorded seven times. Disinfecting the
catheter is organized every week, which means once every three sessions. 45% of
the patients has a catheter, this implies that in 15% of the cases a catheter has to be
cleaned.
There is opt to foresee 10 minutes for each patient to provide extra care.
These 10 minutes are based on the uncertain activities which were observed during
the observation sessions, e.g. bleedings, patients who became nauseous, as
explained in the discovery section. There is a high probability that there does not
42
happen any uncertain events with a patient. Nevertheless, there is decided to still
provide 10 minutes of extra care for each patient. If there do not happen uncertain
events, these 10 minutes can be used to provide extra care to the patient. Horn,
Buerhaus, Bergstrom and Smout state that if nurses spend more direct care time
per patient, the condition of the patient improves. There are fewer pressure ulcers
and hospitalizations (Horn, Buerhaus, Bergstrom, & Smout, 2005).
The average time spent by a nurse on one patient equals the sum of all the
activities. On average, a nurse spends 58 minutes per patient. If the activity
‘provide extra care for patient’ is not considered, there are 48 minutes spent per
patient by one nurse. If a nurse is assigned to four patients, this implies that a nurse
has 3 hours and 50 minutes of work. Considering eight patients on the same four
chairs, then four of them will be scheduled in the morning and the other four will
be scheduled in the afternoon. Normally, two nurses would be assigned to these
eight patients. One nurse would have an early shift, the other a late shift. These
nurses work eight hours each, or 16 hours is total. For eight patients, there is only
7 hours and 41 minutes of work. This results in a ratio of 48%.
4.3.2.1.1.2. Cycle time efficiency
Cycle time efficiency is another metric that can be calculated. Cycle time
efficiency describes the ratio between the theoretical cycle time and the actual cycle
time. The theoretical cycle time is the sum of the processing times of all activities.
The actual cycle time is the sum of the cycle times of the activities, including
waiting time. Due to the fact that there is a dialysis time of four hours, there is
expected that the cycle time efficiency will be rather high. There are only three
relevant activities: connection, disconnection and the dialysis itself. The average
duration of the connection and disconnection activity can be calculated as
explained above. This leads to a theoretical cycle time of 4 hours and 16 minutes.
The actual cycle time of a patient, who is scheduled in the morning block, is based
on observations and is on average 4 hours and 32 minutes. There was an average
waiting time observed of 16 minutes between the arrival of the patient in the
dialysis room (at 7h15) and the connection of the patient. This leads to a cycle time
efficiency of 94.25%. The waiting time can be mostly found before the connection.
Normally, patients arrive at 7h15. A nurse is assigned to four patients but the nurse
can only connect one patient at the same time. In the afternoon, the cycle time lies
43
even closer to the theoretical cycle time. The cycle time is 270 minutes, which
corresponds to a cycle time efficiency of 94.95%. The waiting time before the
connection is now smaller. This can be explained by the fact that both nurses from
the early and the late shift are operative at the moment of connection. So, there are
twice as much nurses compared to the connection in the morning. Nevertheless,
there is waiting time observed during the disconnection. During the disconnection
in the afternoon, only the nurses of the late shift are operative. For four patients,
there were two nurses available during the connection. So, two patients could be
connected simultaneously. During the disconnection, there is only one nurse
available. This implies that one of both patients, who were connected
simultaneously, has to wait.
4.3.2.1.2. Financial measures
A second type of performance dimensions are the financial measures. Cost
and turnover are examples of financial measures. Because of the fact that a hospital
is a non-profit organization, yield and turnover are of less importance. Two types
of costs can be defined: fixed costs and variable costs. Fixed costs do not change
with the amount of goods or services produced, these are independent of any
business activity. Variable costs vary with the amount produced or serviced, these
increase or decrease with the production or service volume. Specifically, in dialysis
centers, fixed costs include the infrastructure costs such as the beds or chairs,
dialysis machines and the building. Variable costs are the direct labor of the nurses,
the wires for the dialysis machines and the food and drinks for the patients.
Nevertheless, the focus of this dissertation lies on leveling the workload for the
nurses. There will not be further elaborated on cost.
4.3.2.1.3. Quality
A third dimension is the dimension of quality. Two types of quality can be
distinguished: external and internal quality. External quality is defined as the
satisfaction of the client with the product or service and the process. Applied to the
case, the satisfaction with the service can be seen as the satisfaction of the patient
with the service delivered by the personnel of the center. The satisfaction of the
patient with the process is determined by the information and communication a
patient gets. The satisfaction will rise as the quantity, quality, relevancy and
44
timeliness improve. For example, a patient will appreciate when he gets the right
information about his condition or diet on time. The internal quality is about the
satisfaction of the other participants of the process. Internal quality is the level a
participant considers the process as in control.
4.3.2.1.4. Flexibility
A last performance dimension is the dimension of flexibility. Flexibility is
the level to which the process can react to changes. Applied to the case, the
flexibility will increase if the nurses are trained to do each activity of the process
at all types of patients. There was observed that the flexibility of the nurses is
already high since each nurse is responsible for four patients and has to perform
every activity for these patients. In other dialysis centers, there is worked based on
the principle of the separation of work. Each nurse is then responsible for certain
tasks, instead of certain activities. This has a negative effect on the flexibility.
4.3.2.2. Utilization
Additionally, the utilization of the chairs/beds was calculated, as suggested
by Cardoen, Demeulemeester and Belin (Cardoen, Demeulemeester, & Belin,
2010). In order to calculate the utilization, there has to be calculated how long the
chairs/beds are occupied. Based on the difference in the duration of connection and
disconnection between dependent and independent patients, the time each type of
patient occupies a chair/bed can be calculated. These durations are then multiplied
with the number of patients of each type in order to know the total time patients of
both types occupy chairs. These two numbers are then summed up. This results in
the total time chairs are taken in by patients on two days, namely 37.941 minutes.
Then, the time chairs/beds are available, is calculated. There are three ways
distinguished. The calculations are included in Appendix V.
4.3.2.2.1. Method 1: all chairs included
This first calculation method assumes that all chairs in the five rooms are
available as soon as the nurses of the early shift start until the nurses of the late
shift stop working. This corresponds with 11 hours and 45 minutes, from 6h45 until
18h30. There are 51 chairs. Based on the shift system of two days, there are 71.910
minutes available. This corresponds with a utilization of 53%. This number has to
45
be interpreted as the ratio between the time a chair is occupied on average and the
time nurses are present in the dialysis center.
4.3.2.2.2. Method 2: chairs of room 4 excluded
The calculations are similar to method 1. The difference lies in the number
of chairs taken into account. Since room 4 is never used for patients, the four chairs
in this room are excluded of the calculation. This results in a utilization of 57%.
4.3.2.2.3. Method 3: only chairs of open rooms included
To calculate the total available time only the rooms where there are nurses
assigned to, are included. This means that only room 3 and 5 are occupied during
the four blocks. Room 1 is occupied during the first two blocks, which implies that
there are nurses present during 705 minutes. Room 2 is only opened during block
1 and 3. Therefore, there are two early shifts needed to cover the dialysis of the
patients. This results in 525 minutes per day, from 6h45 until 15h30. Room 4 is
never operative. These calculations lead to a utilization of 65%.
4.4. Process redesign
Daily hemodialysis or prolonged hemodialysis, as proposed by Vanholder,
Veys, Van Biesen and Lameire is not optimal in our case (Vanholder, Veys, Van
Biesen, & Lameire, 2002). A first reason is that it would dramatically increase
costs, unless the hemodialysis is performed at home, as suggested by the authors.
Hemodialysis at home is not appropriate for the patients of AZ Sint-Jan Brugge
because of their high-care profile. A second reason is that daily hemodialysis would
also have a negative impact on the patients’ quality of life. The implications of
being away from home every day may not be underestimated. Besides that, a
prolonged dialysis can have a negative impact on patients’ mental state. It would
be hard to be away from home more than ten hours, three times a week. Therefore,
there will not be elaborated on prolonged or daily dialysis.
This chapter introduces an optimization method which aims to optimize the
patient scheduling. Actually, the optimization is focused around three concerns:
nurse satisfaction, patient satisfaction and cost control. The aim is to balance the
46
workload of the nurses. Therefore, the difference between the workload in a certain
time slot and the average workload has to be minimized. Furthermore, the time
between the first and the last connection in one-time block has to be minimized.
This is because all the connections have to succeed as soon as possible. The main
goal of the optimization method is to determine at which moment a patient has to
be connected. A penalty system, as proposed by Gupta and Denton is implemented
to optimize the following factors (Gupta & Denton, 2008):
• Balance in workload for the nurses;
• Connection period per block.
4.4.1. Duration of connection and disconnection activity There is assumed that the duration of the connection as well as the
disconnection are patient dependent. The duration of the connection and the
disconnection activity will vary with patients’ characteristics (Mageshwari &
Kanaga, 2012). Patient specific information will be used to assign patients to time
slots (Gupta & Denton, 2008). There will be tested if patient classes could be
created. As Cayirli and Veral describe, a classification can be made to sequence
patients and to adapt appointment intervals to the characteristics of the patient
classes (Cayirli & Veral, 2003).
There is hypothesized that these durations are dependent on two factors.
First, there is assumed that the type of connector, catheter or fistula, has an impact
on the duration. Based on observations and interviews with nursing personnel, we
expect that the connection of the patients with a fistula will be different compared
with the patients who have a catheter. At the disconnection activities, the time
differences were less obvious. A second factor that can have an influence on the
duration of both activities is the mobility of the patients. The distinction is made
between patients who are able to get in and out their beds independently and those
who need help from a nurse. There is expected that the duration of the connection
as well as the disconnection will be different for patients who need help from
nurses. There are only two factors taken into account because Cayirli and Veral
suggest to create a controllable number of patient groups.
47
4.4.1.1. Duration difference between catheter and fistula patients
4.4.1.1.1. Connection activity
First, the difference in the duration of the connections between catheter and
fistula patients will be tested. There is assumed that both populations are
independently and normally distributed. Moreover, there is assumed that both
variances are equal but not known. Vyncke suggests a statistical test (Vyncke,
2012). The null and alternative hypothesis can be stated as follows:
𝐻( ∶ 𝜇, = 𝜇. (4.1) 𝐻, ∶ 𝜇, ≠ 𝜇. (4.2)
Because the populations are normally distributed, the test statistic can be
expressed as the following:
𝑋1 −𝑋2
𝜎12𝑛1 +
𝜎22𝑛2
= 𝑋1 −𝑋2
𝜎 1𝑛1 +
1𝑛2
(4.3)
With probability variables:
𝑋1~𝑁 𝜇1, 𝜎12 , 𝑋2~𝑁 𝜇2, 𝜎2
2 (4.4)
Where 𝜎12 = 𝜎22 = 𝜎².
𝜎² is unknown but can be estimated by the pooled sample variance:
𝑆𝑝2 =
𝑋1𝑖 −𝑋1 2 + (𝑋2𝑖 − 𝑋2)²𝑛2𝑖=1
𝑛1𝑖=1
𝑛1 +𝑛2 − 2
(4.5)
The denominator has to be decreased by two. Two degrees of freedom have
to be given up because there are two averages calculated. At this way, the test
statistic can be expressed as the following:
𝑇 =
𝑋, −𝑋.
𝑆?1𝑛,+ 1𝑛.
~𝑡ABCADE. (4.6)
The test statistic follows the students’ 𝑡 distribution with 𝑛1 + 𝑛2 − 2
degrees of freedom. The test statistic is not normally distributed because the
variance has been estimated. Based on the degrees of freedom the critical values
can be calculated. If the calculated 𝑇-value lies out of the boundaries, set by the
48
critical values, the null hypothesis can be rejected. In that case, there can be
concluded that the averages are statistically different.
To test if the difference in duration of the connection activity between
catheter and fistula patients is statistically different, the null and alternative
hypothesis can be written as the following:
𝐻( ∶ 𝜇FF = 𝜇FG (4.7)
𝐻, ∶ 𝜇FF ≠ 𝜇FG (4.8)
Where 𝜇FF is the average duration of the connection activity for patients
with a catheter and 𝜇FG is the average duration of the connection activity for fistula
patients. There are 30 observations of durations. 14 were fistula observations, the
other 16 were durations of patients with a catheter. First, the averages of both
populations can be calculated:
𝑥𝑐𝑓 = 11.07, 𝑥𝑐𝑐 = 9.74
Then the difference between the two averages can be calculated:
𝑥𝑐𝑓 − 𝑥𝑐𝑐 = 1.33
To be able to calculate the test statistic, the pooled sample variance has to
be calculated:
𝑠𝑝,𝑐𝑐𝑓2 = 22.48
Based on the estimated variance, the test statistic can be calculated:
𝑡𝑐𝑐𝑓 = 0.7
As already mentioned, the degrees of freedom can be calculated as the
number of observations decreased by two. Since there are 30 observations, there
are 28 degrees of freedom. There is chosen for a level of significance of 5%, which
is a standard value. The critical value 𝑡28,0.95 can be read from the student’s 𝑡 table
and is 1.701. Because the calculated test statistic is smaller than the critical value,
the null hypothesis cannot be rejected. This implies that the duration of both groups
is not statistically different. At this way, the fact that a patient has a fistula or a
catheter will not be considered as a factor that has an impact on the duration of a
connection.
49
4.4.1.1.2. Disconnection activity
The same test and reasoning can be performed for the disconnection
activity. Again, there is assumed that both populations are normally and
independently distributed and that the variances are unknown but equal. There are
33 observations: 13 of patients with a fistula and 20 of patients with a catheter. The
null and alternative hypothesis can be written as the following:
𝐻( ∶ 𝜇TF = 𝜇TG (4.9)
𝐻, ∶ 𝜇TF ≠ 𝜇TG (4.10)
The calculations follow the same structure.
𝑥𝑑𝑓 = 7.81, 𝑥𝑑𝑐 = 8.54
𝑥𝑑𝑐 − 𝑥𝑑𝑓 = 0.73
𝑠𝑝,𝑑𝑐𝑓2 = 8.33
𝑡𝑑𝑐𝑓 = 0.71
Since there are 33 observations, the degrees of freedom are reduced to 31.
This implies a critical value of 1.796. Again, the test statistic is smaller than the
critical value. Therefore, the null hypothesis that the average duration of the
disconnection activity for patients with a catheter is statistically the same as the
average duration for fistula patients, cannot be rejected. Consequently, the fact that
a patient has a catheter or a fistula will not be a determining factor on the duration
of the disconnection.
4.4.1.2. Duration difference between dependent and independent patients
A second factor that will be statistically tested is the condition of the patient.
As already mentioned, the patients’ population will be separated in two groups:
patients who are able to get in and out their beds independently and patients who
need help from a nurse. About the connection and disconnection activity, there is
assumed that both populations are normally and independently distributed and that
the variances are unknown but equal.
50
4.4.1.2.1. Connection activity
Concerning the connection activity, there are 30 observations. 15 patients
were classified as independent, the other 15 were observed as needing help to get
in and out the bed. The following hypothesis will be tested:
𝐻( ∶ 𝜇FV = 𝜇FW (4.11) 𝐻, ∶ 𝜇FV ≠ 𝜇FW (4.12)
Where 𝜇FV is the average connection duration for patients who need help to
get in and out the bed and 𝜇FW is the average connection duration for patients who
do not need any help to get in or out the beds.
The calculations are the following:
𝑥𝑐ℎ = 11.53, 𝑥𝑐𝑠 = 6.42
𝑥𝑐ℎ − 𝑥𝑐𝑠 = 5.12
𝑠𝑝,𝑐ℎ𝑠2 = 28.12
𝑡𝑐ℎ𝑠 = 2.64
There are 28 degrees of freedom which results in a critical value of 1.701.
Since 2.64 is bigger than 1.701, the connection duration for patients who can get
in and out their beds independently is statistically different from the connection
duration for patients who need help.
4.4.1.2.2. Disconnection activity
Concerning the disconnection activity, there are 33 observations. 15
patients are classified as dependent of a nurse to get in or out their beds, the other
18 can get in or out their beds independently. The statistical test is the following:
𝐻( ∶ 𝜇TV = 𝜇TW (4.13) 𝐻, ∶ 𝜇TV ≠ 𝜇TW (4.14)
The averages, the pooled sample variances and the 𝑡 test statistic can be
calculated:
51
𝑥𝑑ℎ = 9.43, 𝑥𝑐𝑠 = 7.44
𝑥𝑑ℎ − 𝑥𝑑𝑠 = 1.99
𝑠𝑝,𝑑ℎ𝑠2 = 8.12
𝑡𝑑ℎ𝑠 = 2
The critical value 𝑡31,0.95, based on 31 degrees of freedom, is 1.796. The
test statistic is bigger than 𝑡31,0.95. This implies that the null hypothesis can be
rejected on a level of significance of 5%. The average disconnection duration of
both patient pools can be assumed as statistically different.
4.4.1.3. Conclusion
To conclude, the fact that the patient can get independently or dependently
in and out the bed is a determining factor for the duration of the connection and
disconnection. This confirms the hypothesis, based on observations.
Nevertheless, it is remarkable that the first hypothesis was not confirmed.
Lots of nurses thought the type of connector would be a determining factor for the
duration of the connection and disconnection. The fact that both durations are
statistically not different can possibly be explained by the fact that most of the
patients with a catheter are in a bad condition (as seen in the observations). A lot
of the patients with a catheter cannot independently get in and out of their beds.
The actual connection of a catheter patient will on average be shorter than the
connection with a fistula patient because to connect with a fistula there are two
pricks needed. These pricks are often technically difficult. The actual
disconnection for catheter patients will probably be shorter as well. The reason
behind this lies in the fact that fistula patients are often confronted with heavy
bleedings when disconnected. This asks for extra care from the nurse. Although,
there are time savings in the actual connection and disconnection of catheter
patients, these time savings are undone by the extra care the catheter patients need.
4.4.1.4. Duration difference between morning and afternoon patients
During the observation sessions, it seemed that the connections in the
afternoon would take longer than the connections in the morning. To test this
hypothesis, the same statistical test as before was used. Again, a normal and
52
independent distribution was assumed with equal and unknown variances. This led
to the following hypotheses:
𝐻( ∶ 𝜇FZ = 𝜇FZ (4.15) 𝐻, ∶ 𝜇FZ ≠ 𝜇F[ (4.16)
Where 𝜇FZ is the average connection duration in the morning and 𝜇F[ is
the average connection duration in the afternoon. The average, pooled sample
variance and test statistic can again be calculated:
𝑥𝑐𝑚 = 9.77, 𝑥𝑐𝑎 = 11.98
𝑥𝑐𝑎 − 𝑥𝑐𝑚 = 2.22
𝑠𝑝,𝑐𝑚𝑎2 = 21.92
𝑡𝑐𝑚𝑎 = 1.15
There are 30 observations, this results in a critical value of 1.701. The test
statistic is smaller than the critical value. Although, the average in the afternoon is
higher than the average in the morning, both values are not statistically different.
The null hypothesis, stating that both averages are equal, cannot be rejected.
4.4.2. Optimization model: optimize workload level during
dialysis
4.4.2.1. Assumptions in the optimization model
The hemodialysis scheduling problem can be categorized as an assignment
problem. In this assignment problem activities for patients will be assigned to time
slots, in order to balance the workload for the nurses.
In order to achieve an optimization, it is assumed that the following
information is known a priori:
• The set of dialysis patients.
• The set of activities and rules about a possible obligatory sequence between
certain activities.
53
• Duration of all activities. A distinction can be made between activities which
duration is patient specific and other activities which have an equal duration
for all patients. To estimate the duration of each patient’s activity, each patient
will be assigned to a specific patient class.
• The duration of the dialysis itself and the setup time of each machine after a
hemodialysis.
• Each patient’s preference of to be scheduled on a specific moment of the week.
• The number of possible blocks patients can be assigned to.
• The number of time slots available per day. The time slots all have an equal
length. They represent the time the dialysis center is available for high-care
hemodialysis and nurses are available to work.
• The pool of available nurses.
• The number of dialysis beds/chairs.
Based on the information gathered out of observation and interviews, the
optimization model is built considering the following assumptions:
• The number of hours that the dialysis center can be operational is fixed. The
possible operating hours are dependent on three factors. First, the night dialysis
takes place between 21h30 and 5h30. This results in the fact that the chairs
allocated to the night shift are unavailable between 21h00 and 6h00. Moreover,
the profile of the patients and their needs have to be taken into account. The
dialysis unit of AZ Sint-Jan is high-care which implies that the patients’
condition is delicate. For this reason, it is unwise to schedule patients in the
evening. The night’s rest of the patients is important and scheduling patients
too late in the evening would have an undeniable impact. Also scheduling
patients too early in the morning is undesirable because of the same reason.
Furthermore, also the nurses’ overall satisfaction has to be taken into account.
Their stress level is already high. Start working too early or stop working too
late would probably have a negative impact on their level of work satisfaction.
For this reason, the number of available operating hours is restricted to 14
hours per day, from 7h00 until 21h00.
54
• As already mentioned, the patient satisfaction is one of the main concerns of
this optimization approach. The dialysis patients have to eat according to a
strict diet. A warm meal every day is an essential element of their diet.
Nevertheless, a warm meal cannot be consumed during a dialysis session.
Moreover, most patients prefer to have a warm lunch at noon (Luyckx, 2015).
For this reason, a patient in the morning has to be connected between 7h00 and
9h00. Subsequently, these patients are disconnected at the latest around 13h00
and can have lunch afterwards. A patient in the afternoon can be connected as
from midday. At this way these patients can have lunch before midday,
possibly in the hospital’s restaurant.
• The assumption is made that are no differences between the chairs/beds. So,
every patient can be assigned to every chair.
• To reduce waiting time, the assumption is made that the dialysis has to start
immediately after the connection. The disconnection has to follow directly on
the dialysis as well as the setup, which has to be scheduled instantly after the
disconnection.
• The number of chairs is considered fixed. Based on the fact that there is no
increase expected in the number of patients, there is no increase in the number
of chairs planned.
• At the existing patient scheduling, patients are assigned to one of the four
blocks. Most patients are assigned to the block of their first preference.
Therefore, there is decided to let the patients in their block of preference. Thus,
each patient will remain assigned to the block in which he assigned. It is
important to include the patient preferences into the model and stick to them
as much as possible, as described by Gupta and Denton (Gupta & Denton,
2008).
The following restriction is present:
• There are four blocks which are very similar. Therefore, there is decided to
only consider one block. The results of this block will then be utilized to
schedule the other blocks.
55
4.4.2.2. Optimization model
The optimization method can be described in terms of the objective
function and the constraints as the following (Ferguson & Sargent, 1958):
OptimizationMinimize Weighted Penalties
𝑓 𝑏𝑎𝑙𝑎𝑛𝑐𝑒𝑖𝑛𝑤𝑜𝑟𝑘𝑙𝑜𝑎𝑑 , 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛
ConstraintsSubject to the following constraints for:
- Workload
- Sequence of activities
- Continuity of activities
- Duration of activities
- Latest start of activities
- End of activities
Sets𝐾 : set of activities, 𝑘𝜖𝐾
𝐼 : set of patients, 𝑖𝜖𝐼
𝑇 : set of time slots, 𝑡𝜖𝑇
Parameters𝑑jk : duration of activity 𝑘𝜖𝐾 for patient 𝑖𝜖𝐼
𝑐, / 𝑐. : cost factors in the objective function
𝑓 : latest moment a connection can start
𝑡j : time patient 𝑖𝜖𝐼 has to be dialyzed
𝑒 : latest time slot the connection activity can start
56
Decisionvariables𝑌jmn : 1 if activity 𝑘𝜖𝐾 for patient 𝑖𝜖𝐼 is executed in time slot 𝑡𝜖𝑇;
0 otherwise
𝐴jmn : 1 if activity 𝑘𝜖𝐾 for patient 𝑖𝜖𝐼 is executed in time slot 𝑡𝜖𝑇 as well as
in time slot 𝑡 − 1 𝜖𝑇;
0 otherwise
𝑤n : workload, over all patients and activities, per time slot 𝑡𝜖𝑇
𝑤[p : average workload over all time slots
𝑎n : difference between 𝑤[p and 𝑤n
𝑠jm : end of activity 𝑘𝜖𝐾for patient 𝑖𝜖𝐼
Model
𝑚𝑖𝑛𝑐, 𝑎n
q
nr,
+ 𝑐. 𝑠jm
s
mr,
t
jr,
(4.17)
subjected to
𝑤𝑡 = 𝑌𝑖𝑘𝑡𝐾
𝑘=1
𝐼
𝑖=1 , ∀𝑡 ∈ 𝑇 (4.18)
𝑤𝑎𝑣 =
( 𝑌𝑖𝑘𝑡)𝑇𝑡=1
𝐾𝑘=1
𝐼𝑖=1
𝑇 (4.19)
𝑎𝑡 ≥ 𝑤𝑡 −𝑤𝑎𝑣, ∀𝑡 ∈ 𝑇 (4.20)
𝑎𝑡 ≥ 𝑤𝑎𝑣 −𝑤𝑡, ∀𝑡 ∈ 𝑇 (4.21)
𝑒𝑖𝑘 ≥ 𝑌𝑖𝑘𝑡 ∙ 𝑡, ∀𝑖 ∈ 𝐼, ∀𝑘 ∈ 𝐾,∀𝑡 ∈ 𝑇 (4.22)
𝑌𝑖𝑘𝑡 = 𝑑𝑖𝑘, ∀𝑖 ∈ 𝐼, ∀𝑘 ∈ 𝐾
𝑇
𝑡=1
(4.23)
𝑌𝑖1𝑡 ∙ 𝑡 ≤ 𝑒, ∀𝑖 ∈ 𝐼 (4.24)
𝑌𝑖𝑘𝑡−1 +𝑌𝑖𝑘𝑡 ≥ 2 ∙ 𝐴𝑖𝑘𝑡, ∀𝑖 ∈ 𝐼, ∀𝑘 ∈ 𝐾,∀𝑡 ∈ 𝑇: 𝑡 ≥ 2 (4.25)
57
𝐴𝑖𝑘𝑡 = 𝑑𝑖𝑘 − 1, ∀𝑖 ∈ 𝐼, ∀𝑘 ∈ 𝐾
𝑇
𝑡=1
(4.26)
𝐴𝑖𝑘1 = 0, ∀𝑖 ∈ 𝐼, ∀𝑘 ∈ 𝐾 (4.27)
𝑠𝑖𝑘−1 ≤ 𝑠𝑖𝑘 − 𝑑𝑖𝑘, ∀𝑖 ∈ 𝐼, ∀𝑘 ∈ 𝐾 (4.28)
𝑠𝑖2 +𝑡𝑖 ≤ 𝑠𝑖4 −𝑑𝑖4, ∀𝑖 ∈ 𝐼 (4.29)
𝑌𝑖𝑘𝑡 ≤ 1
𝐾
𝑘=1 , ∀𝑖 ∈ 𝐼, ∀𝑡 ∈ 𝑇
(4.30)
𝑌jmn ∈ 0,1 , ∀𝑖 ∈ 𝐼, ∀𝑘 ∈ 𝐾,∀𝑡 ∈ 𝑇 (4.31) 𝐴jmn ∈ 0,1 , ∀𝑖 ∈ 𝐼, ∀𝑘 ∈ 𝐾,∀𝑡 ∈ 𝑇 (4.32) 𝑤n ≥ 0, ∀𝑡 ∈ 𝑇 (4.33)
The objective function minimizes the total penalty. This function consists
of two factors. The first factor levels the workload by minimizing the positive sum
of the differences between the workload in the different time slots and the average
workload over all time slots. The second factor makes sure that each task is
performed as early as possible. At this way, there is prevented that the total dialysis
would take longer than needed. Moreover, there is also ensured that the time
between different activities is minimized.
The objective function aims to give a schedule in which all activities for all
patients are assigned to time slots. The penalty system allows the decision maker
to control the tradeoff between the different factors in the objective function. The
decision maker can choose the penalties for all factors and thus, there can be chosen
on the importance of each factor.
In order to ensure an adequate scheduling and to meet the objectives, the
constraints have to be met. Constraint (4.18) defines the workload per time slot.
The workload is calculated as the sum of the binary variable 𝑌jmn over all patients
and activities. Constraint (4.19) defines the average workload over all time slots.
This average equals the sum of 𝑌jmn over all patients, activities and time slots,
divided by the number of time slots. The higher the deviation between the workload
per time slot and the average workload, the more the objective function has to
penalize. Thus, if the workload per time slot is bigger than the average workload
58
as well as in case the workload per time slot is smaller than the average workload,
there has to be penalized. This can be expressed through the absolute value:
𝑎𝑡 = 𝑤𝑡 −𝑤𝑎𝑣 (4.34)
Nevertheless, this expression is not linear. Based on Vanhoucke this can be
rewritten into a linear form through composing two constraints, (4.20) and (4.21)
(Vanhoucke, 2013). In constraint (4.22) the end of each activity for each patient is
calculated. The end 𝑠jm has to be bigger than all the time slots in which activity 𝑘
for patient 𝑖 is performed. The time slots in which activity 𝑘 for patient 𝑖 is
performed have to be equal to the duration of that activity for that patient. This is
expressed by constraint (4.23). Constraint (4.24) plans that the connection of each
patient has to be performed before 𝑒. In the morning block 𝑒 equals 9h00. At this
way, each dialysis can be finished before 13h00. This ensures that every patient
can consume a warm lunch. Constraint (4.25), (4.26) and (4.27) ensure the
continuity of the activities. Each activity has to be performed interruptedly. So, an
activity for a certain patient first has to be finished before another activity can start.
If 𝑌jm nE, and 𝑌jmn are both equal to one, 𝐴jmn also has to be one. This is defined by
(4.25) and (4.26). (4.27) is an extra constraint to ensure that 𝐴jm, is always zero.
Constraint (4.28) assures the sequence between the activities. Moreover, the
disconnection of a certain patient can only take place after the dialysis is finished,
as expressed by constrain (4.29). Thus, between the connection and the
disconnection of a certain patient there has to be a time difference which equals the
dialysis time. Normally, this will be four hours. The last constraint (4.30) excludes
that there could be two activities planned for the same patient during one time slot.
4.4.2.2.1. Experimentation
Restrictions• Only ten patients were involved.
• The activities were reduced to three activities: connection, disconnection and
the bundled activities between these two activities. Because the connection and
disconnection are the activities where the most problems are observed, the
activities before the connection and after the disconnection are excluded. The
59
duration of the activities between the connection and the disconnection are
summed up.
• A time slot represents 10 minutes. At this way, it was not possible to make a
distinction between the two types of patients. This distinction will be made in
further models.
Inputvalues• Cost 1: 2
• Cost 2: 0.5
• The duration of the connection and disconnection are both set equal to one
time slot.
• The duration of the bundled activities is rounded down to two time slots.
The model is solved by making use of the academic version of the CPLEX
solver, as proposed by Ronconi and Birgin as well as Hooker (Ronconi & Birgin,
2012; Hooker, 2005). The code used in CPLEX is provided in the Appendix VI.
4.4.2.2.2. Evaluation
The results will be evaluated based on the maximum of the differences
between the workload over all patients and the activities in a certain time slot 𝑡 and
the average workload over all time slots. The calculation of the average workload
is slightly adapted compared to the average workload calculated by the model.
Only the time slots which contain effectively executed activities are included in the
model.
𝐷Z[} = 𝑤n,Z[} − 𝑤′[p = 10 − 1.61 = 8.39
Additionally, the difference between the highest workload over all time
slots and the lowest workload over all time slots will be calculated. The lowest
workload is the workload during the dialysis session. Once the dialysis is
terminated the workload is zero. The workloads during these time slots were not
taken into account.
𝐷𝑚𝑎𝑥,𝑚𝑖𝑛 = 𝑤𝑡,𝑚𝑎𝑥 − 𝑤𝑡,𝑚𝑖𝑛 = 10 − 1 = 9
It is also essential that the activities are finished as soon as possible. It
remains important to schedule the connection of the patients not too diffused. If the
60
connection of the patients is scheduled close to each other, the nurses can focus on
the activities which have to be performed between the connection and the
disconnection. Moreover, disturbing patient transportation can be prevented. At
this way, it will be more calm and serene in the dialysis unit which is better for the
patients. An evaluation will be based on two factors: the end of the connection
activities and the end of the dialysis session. The last time slot a connection takes
place, will be indicated as 𝐸F. The last time slot a dialysis session takes place, will
be indicated by 𝐸. The last time slot a dialysis is terminated, is equal to the last
time slot a disconnection takes place, 𝐸T.
𝐸𝑐 = 1
𝐸 = 29 = 𝐸T
The values for the first two parameters are not satisfying. The workload is
high during the first time slot.
In order to search for better results, with a more balanced workload, the
first cost parameter 𝑐, will be increased stepwise. The original value will be raised
until 10.
𝑐, 2 4 5 6 7 8 10𝐷Z[} 8,39 7,75 5,82 1,95 0,97 0 0
𝐷Z[},ZjA 9 8 6 2 1 0 0𝐸F 1 2 4 8 9 10 10𝐸 29 32 34 38 39 40 40
Table 3 demonstrates the tradeoff between the factors of the objective
function. The weight of the first variable increases as the first cost factor goes up.
This results in a more balanced workload but also in a later end of the connection
and the whole dialysis session. Figure 3 provides an overview.
Table 3: Sensitivity of 𝑐,
61
4.4.2.3. Adapted optimization model
The model will be changed. To balance the workload, there was minimized
for the absolute difference between the workload per time slot and the average
workload over all time slots. In this adapted model, there will be optimized for the
maximum workload per time slot. Thus, the workload per time slot will be
calculated and the maximum of these will be minimized (Ferguson & Sargent,
1958).
A new variable is introduced:
𝑚 : maximum of workloads per time slot 𝑤n
The objective function is rewritten below.
𝑚𝑖𝑛𝑐1 ∙ 𝑚 +𝑐2 𝑠𝑖𝑘𝐾
𝑘=1
𝐼
𝑖=1 (4.35)
Constraints (4.20) and (4.21) are redundant. A new constraint is introduced:
𝑚 ≥ 𝑤n, ∀𝑡𝜖𝑇 (4.36)
0
2
4
6
8
10
12
2 4 5 6 7 8 10
Effectofcostratioonparameters
Dmax Dmax,min Ec
Figure 3: Graph showing sensitivity of 𝑐,
62
This constraint is introduced to determine the maximum workload. Since
𝑚 has to be bigger or equal to the workload in each time slot, 𝑚 will be the
maximum workload over all w�. Appendix VI displays the code used in CPLEX.
4.4.2.3.1. Experimentation
The restrictions and input values are almost all the same as in the original
model, except for the first cost factor 𝑐,. In the original model, the cost factor 𝑐,
was equal to 2 but it was multiplied by a bigger number because there was summed
over all 40 time slots. In this adapted model, there will be worked with a cost factor
𝑐, of 10. This cost factor gives the following results:
𝐷𝑚𝑎𝑥 = 4 − 1.38 = 2.62
𝐷𝑚𝑎𝑥,𝑚𝑖𝑛 = 4 − 0 = 9
𝐸𝑐 = 3
𝐸 = 29
In order to determine the optimal ratio between both cost factors, the first
cost factor will be changed. The cost factor will change between the range from 6
up to 18, stepwise by 2. This results in the following:
𝑐, 6 8 10 12 14 16 18𝐷Z[} 3,57 3,57 2,62 2,63 1,67 1,67 1,67
𝐷Z[},ZjA 5 5 4 4 3 3 3𝐸F 2 2 3 3 4 4 4𝐸 28 28 29 29 30 30 30
As in the first model, the same trend can be observed. If the first cost factor
increases, the workload will be more levelled but the dialysis will take longer.
Figure 4 gives a new overview.
Table 4: Sensitivity of 𝑐, (adapted)
63
4.4.2.4. Comparison of both techniques
When comparing both techniques, there can be concluded that the second
technique generates more satisfying results. With the first method a 𝐷Z[} of 2 can
only be achieved if the last dialysis ends at the 38th time slot. The last connection
ends at the 8th time slot. With the second method, a 𝐷Z[} of 2 – even 1.67 – can be
achieved with an 𝐸-value of 30. Therefore, the second method will be preferred
over the first method. The cost ratio of 14/0.5 will be used where cost factor 1 is
14 and the second cost factor is 0.5.
4.4.3. Optimization model: optimize workload level during
connection activities The previous dealt with optimizing the workload during the dialysis
session. Nevertheless, the time slots were restricted to 10 minutes. In order to know
the exact moment patients can be connected, there is made use of smaller time slots.
The time slots will consist of 2 minutes. The duration of the connection for patients
who need help to get in the bed, is 11.53 minutes which will be rounded to 12
minutes. The connection duration of the patients who get independently in their
beds, is 6.42 minutes. This will be rounded to 6 minutes. Since there will be worked
with time slots of 2 minutes, the model can be solved exactly.
0
1
2
3
4
5
6
6 8 10 12 14 16 18 20
Effectofcostratioonparameters
Dmax Dmax,min Ec
Figure 4: Graph showing sensitivity of 𝑐, (adapted)
64
The model is similar to the model used in the previous section. Since the
minimax optimization was preferred, this optimization method will again be
applied.
Sets𝐼 : set of patients, 𝑖𝜖𝐼
𝑇 : set of time slots, 𝑡𝜖𝑇
Parameters𝑑j : duration of the connection activity for patient 𝑖𝜖𝐼
𝑐, / 𝑐. : cost factors in the objective function
Decisionvariables𝑌jn : 1 if the connection activity for patient 𝑖𝜖𝐼 is executed in time slot 𝑡𝜖𝑇;
0 otherwise
𝐴jn : 1 if the connection activity for patient 𝑖𝜖𝐼 is executed in time slot 𝑡𝜖𝑇
as well as in time slot 𝑡 − 1 𝜖𝑇;
0 otherwise
𝑤n : workload, over all patients and activities, per time slot 𝑡𝜖𝑇
𝑤[p average workload over all time slots
𝑠j : end of the connection activity for patient 𝑡𝜖𝑇
𝑚 : maximum of workloads per time slot 𝑤n
Model
𝑚𝑖𝑛𝑐, ∙ 𝑚 +𝑐. 𝑠j
t
jr, (4.37)
65
subjected to
𝑤𝑡 = 𝑌𝑖𝑡𝐼
𝑖=1 , ∀𝑡 ∈ 𝑇 (4.38)
𝑤𝑎𝑣 =
𝑌𝑖𝑡𝑇𝑡=1
𝐼𝑖=1
𝑇 (4.39)
𝑒𝑖 ≥ 𝑌𝑖𝑡 ∙ 𝑡, ∀𝑖 ∈ 𝐼, ∀𝑡 ∈ 𝑇 (4.40)
𝑌𝑖𝑡 = 𝑑𝑖, ∀𝑖 ∈ 𝐼
𝑇
𝑡=1
(4.41)
𝑌𝑖 𝑡−1 +𝑌𝑖𝑡 ≥ 2 ∙ 𝐴𝑖𝑡, ∀𝑖 ∈ 𝐼, ∀𝑡 ∈ 𝑇: 𝑡 ≥ 2 (4.42)
𝐴𝑖𝑡 = 𝑑j − 1, ∀𝑖 ∈ 𝐼
𝑇
𝑡=1
(4.43)
𝐴𝑖1 = 0, ∀𝑖 ∈ 𝐼 (4.44)
𝑚 ≥ 𝑤n, ∀𝑡 ∈ 𝑇 (4.45)
𝑌𝑖𝑡 <
𝐼4, ∀𝑡 ∈ 𝑇
𝐼
𝑖=1
(4.46)
𝑌jn ∈ 0,1 , ∀𝑖 ∈ 𝐼, ∀𝑡 ∈ 𝑇 (4.47) 𝐴jn ∈ 0,1 , ∀𝑖 ∈ 𝐼, ∀𝑡 ∈ 𝑇 (4.48) 𝑤n ≥ 0, ∀𝑡 ∈ 𝑇 (4.49)
Constraint (4.46) is a new constraint. This constraint was added in order to
assign at most one patient to each nurse simultaneously. Therefore, the total
number of patients was divided by four, since there is a nurse-patient ratio of 1:4.
At this way, at most one patient can be assigned to each nurse in a certain time slot
t. Again, the codes are again added in Appendix VI.
4.4.3.1. Experimentation
Restrictions• Twelve patients were involved.
• A time slot represents 2 minutes.
66
Inputvalues• Cost 1: 2
• Cost 2: 0.5
• As already mentioned, 35% of the patients need help to get in their bed. The
remaining 65% can go independently in their bed. The connection duration of
four patients was set to three time slots, the remaining eight patients were
linked to a duration of 12 minutes. Four out of twelve patients corresponds to
a ratio of 33% which is considered close enough to the 35%.
Evaluationoftheresults
𝐷𝑚𝑎𝑥 = 3 − 2.67 = 0.33
𝐷𝑚𝑎𝑥,𝑚𝑖𝑛 = 3 − 1 = 2
𝐸𝑐 = 18
As can be observed in Figure 5, the workload is well balanced. During the
first 15 time slots, the workload is kept constant at three workloads. In time slots
16, 17 and 18 there is a workload of one because all patients, except for one, are
already connected. The other two nurses can already start with the other activities
which has to be performed between the connection and disconnection activity.
0
1
2
3
4
5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Workloadduringconnectionactivity(withouttimebuffers)
Figure 5: Workload during connection activity, without time buffers
67
4.4.3.2. Optimization model including time buffers
Although, these results are satisfying, there can be questioned if these
results are achievable. The assignment of patients to nurses was based on the
average duration of the connection activity. Since the standard deviation of the
connection duration for both patient groups is around 5, there would possibly be a
lot of waiting for patients. This would then again increase the stress and workload
for the nurses. There is imposed a service level of 90%. With a service level of
90%, the achieved duration of the connection will in 90% of the cases not exceed
the foreseen time for it (Vyncke, 2012). Moreover, if a certain nurse connects a
certain patient in less than the foreseen time, the nurse will have more time for the
following patient. At this way, it is highly probable in more than 90% of the cases
that the patients will be connected before the due moment1. The average duration
is multiplied by the product of a service level dependent factor and the standard
deviation of the activity duration. There was chosen for a service level of 90%,
hence, the duration is enlarged with a time buffer of 28%. The calculation of this
buffer was calculated based on the well-known invers normal probability
distribution. A service level higher than 90%, perhaps makes the time buffer too
large. Ogulata, Cetik, Koyuncu and Koyuncu discourage higher slack capacity than
needed (Ogulata, Cetik, Koyuncu, & Koyuncu, 2009). This leads to:
𝑥𝑐 1−𝛼 = 𝑥𝑐 + 𝑘 1−𝛼 ∙ 𝜎𝑐 (4.50)
With 𝑥𝑐 1−𝛼 is the new duration with the buffer, 𝑘 1−𝛼 is the service level
dependent factor and 𝜎𝑐is the standard deviation of the connection duration.
If there is chosen for a service level of 90% and the variance is estimated
by the pooled sample variance, this connection duration for both patient groups
becomes:
𝑥𝑐𝑠,90% = 𝑥𝑐𝑠 + 𝑘90% ∙ 𝜎𝑐𝑠 = 8.23
𝑥𝑐ℎ,90% = 𝑥𝑐ℎ + 𝑘90% ∙ 𝜎𝑐ℎ = 14.78
1 The due moment for a certain patient can be defined as the end of the last time
slot a patient is scheduled for the connection
68
These values are rounded to 8 and 14 which corresponds to 4 and 7 time
slots in the model. Again, twelve patients are involved: four of them need help, the
eight others can get in the bed independently.
Results
The workload during the first 19 time slots is constant at three (Figure 6
and Figure 7). During time slot 20, 21 and 22 the workload is 1. For these 12
patients, there are three nurses. These patients are randomly assigned to a nurse.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 301 1 1 1 1 1 1 12 1 1 1 1 1 1 13 1 1 1 1 1 1 14 1 1 1 1 1 1 15 1 1 1 16 1 1 1 17 1 1 1 18 1 1 1 19 1 1 1 110 1 1 1 111 1 1 1 112 1 1 1 1TOTAL3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 1 0 0 0 0 0 0 0 0
123
Patient5Patient6Patient12
Patient7Patient8Patient10
Patient9Patient11
Patient2Patient3Patient4
Patient1
0
1
2
3
4
5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Workloadduringconnectionactivity(withtimebuffers)
Figure 6: Patient connection schedule
Figure 7: Workload during connection activity, with time buffers
69
Evaluationoftheresults 𝐷𝑚𝑎𝑥 = 3 − 2.73 = 0.27
𝐷𝑚𝑎𝑥,𝑚𝑖𝑛 = 3 − 1 = 2
𝐸𝑐 = 22
These results are satisfying. During the connection the workload is
balanced.
The as-is process was classified as a ‘single block’. (Cayirli & Veral, 2003).
The newly elaborated system can be classified as a ‘multiple block, variable
interval’. There are two types of interval now: one of 8 minutes and one of 14
minutes. The biggest time intervals, those of 7 time slots, are more scheduled
towards the end of the connection session. This corresponds with the findings of
Cayirli and Veral to enlarge intervals towards the end of the session.
4.4.4. Day schedule
4.4.4.1. Composing day schedule
Based on the scheduling of the connection activities, the other activities can
be scheduled. This results in the schedule of one day. For the other activities there
will be buffered again, based on a service level of 90%. Since the activities between
the connection and disconnection and after the disconnection are executed in
parallel, these activities are again bundled. There are still some time slots in which
there is no work. These empty time slots can be found between the connection and
disconnection, because there has to be dialyzed for four fours. There is chosen to
let the connection of the afternoon begin at the same moment, even this creates six
time slots without workload for nurses 2 and 3. The patients are numbered.
Nevertheless, these numbers are not linked to real patients. The only distinction
taken into account is the difference in connection and disconnection duration.
Patient 1 to 4 are patients who need help to get in or out their beds, while patients
5-12 are not in need of any assistance.
As can be seen in Figure 8, the first connection is proposed to start at 7h05.
At this way, nurses can start working at 6h45 since there are 20 minutes of
preparation needed. The nurses can start working at the same moment as it is now.
70
So, patient 5, 7 and 12 are connected at 7h05. Four time slots later, at 7h13 patient
7, 8 and 10 are connected. Patient 2, 9 and 11 are scheduled at 7h21. Patient 3 and
4 can be connected at 7h29. The last patient, patient 1, can be connected at 7h35.
The patients who need help to get in their beds, patient 1, 2, 3 and 4, are thus
connected at the end of the connection session. At 7h49, the connection for all
patients is finished. Between the connection and the disconnection, the nurses can
perform the needed activities. The unused time slots can be utilized to provide extra
care to patients. A lot of the patients do not have many social contact, it is
recommended to give patients extra attention by talking with these patients (Horn,
Buerhaus, Bergstrom, & Smout, 2005). Since the connection was scheduled
sequentially, the disconnection activity will have the same pattern. The
disconnection is started at 11h13. All patients are disconnected at 11h57. After the
disconnection, the activities to finish the first block can start. At 12h49, the
following block of patients can be connected. To indicate these patients, there is
made use of the same numbers as the patients in the first block, but with an accent.
The same numbers were used to indicate that these patients have to be scheduled
on the corresponding chairs/beds of the patients of the morning block. The
connection in the afternoon starts 19 minutes later compared to the situation now.
This is a consequence of the better balanced workload. The last connection is
finished at 13h33. The rest of the process is similar to the morning block. The
disconnection starts four hours after the connection of the first patients of the
afternoon block, at 16h57. The last disconnection is finished at 17h41. To end the
day, the activities after the disconnection need to be performed. At this way, the
day ends at 18h33. This is 3 minutes later than the schedule now.
123
6:45 7:05 7:49 11:13 11:572 15 7
12 10
5 7 2 16 8 9 3 6 8
5,7,2,16,8,9,3
12,10,11,4
5,7,2,16,8,9,3
12,10,11,4 12,10,11,49 3
12 10 11 4 11 4
5,7,2,16,8,9,3
123
12:49 13:33 16:57 17:415',7',2',1'
6',8',9',3'12',10',11',4'
5' 7' 2' 1'6' 8'
5',7',2',1'6',8',9',3'
12',10',11',4'11' 4'
2' 1'6' 8' 9' 3'
12' 10' 11' 4'
5' 7'
12' 10'9' 3'
Figure 8: Illustration of the day schedule
71
4.4.4.2. Evaluation of day schedule
The schedule is advantageous compared to the existing schedule because
of several reasons. First, there is a better balance in workload. The maximum
workload for each nurse during a time slot is one. The connection, and as a
consequence the disconnection, was planned sequentially. Because of the time
buffers the waiting time for patients is minimized, even if the connection and
disconnection take longer than the average duration. This reduces the workload for
the nurses because patients do not have to queue anymore and do not have to wait
to be (dis)connected. The time buffers at the other activities have the same
advantage. If, for example, the activity before the connection in the morning takes
longer than planned, this can be neutralized by the time buffer.
This schedule also solves the problem of the relative absence of work
during the last time slots. In the new schedule there is work until the last time slots.
4.4.5. Nurse scheduling As already mentioned, the nurse schedule is not optimal because of the
overlap in time between the early and late shift of the nurses. In the proposed
schedule the work is balanced for one nurse per four patients. In the existing
schedule two nurses are assigned to four patients during the time overlap. The
overlap lays between 9h45 and 15h30. In the new schedule, the overlap has to be
minimized. It is recommended to elaborate on a new nurse schedule. There are
several possibilities which will be discussed:
Nurses work the whole day: from 6h45 until 18h30
• This results in a work day of 11 hours and 45 minutes. Literature states that
workdays longer than 8 hours have a negative impact on the quality of care,
because more errors are made (Scott, Rogers, Hwang, & Zhang, 2006;
Federale Overheidsdienst Werkgelegenheid, 2016). Moreover, this
proposition would also lead to legal issues because normal workdays consist
of eight hours at maximum. Work days longer than 8 hours are only possible
under specific circumstances (Federale Overheidsdienst Justitie, 2016).
72
Day is split in two parts
• Several combinations are possible. A work shift has to be at least three hours
(Federale Overheidsdienst Werkgelegenheid, 2016). Two options will be
discussed.
o Work shifts of 8 and 4 hours. In this system, half of the nurses could work
for eight hours, the other half could work for four hours. At this way, there
is almost no overlap. The few time that there is overlap between the shift,
is required to communicate about the situation of the patients who are
dialyzed between the shift switch. Nevertheless, there are some
disadvantages about this shift system. A certain degree of resistance can
be expected of the nurses to work only four hours on a day. Only 10% of
the nurses works half time (De Vriese, 2015). Moreover, nurses always
work at least 6 hours per day, also the nurses who work half time. Some
nurses would not be willing to shift to the new system because they would
have to work more days to accomplish the required number of hours per
month.
o Work shifts of 6 hours. In this system, every nurse works 6 hours per day.
Overlap can be reduced to a minimum, depending on the length of the
breaks2. The longer the breaks, the longer the overlap. This system also
leads to an additional advantage. The nurses are only responsible for the
patients of their shift. In the existing schedule, the nurse is responsible of
eight patients: four of the morning block and four of the afternoon block.
As the nurses of the early shift stop before the connection of the afternoon
patients, they are only responsible for the four patients of the morning
block but they still have to prepare the connection material and medical
files for the afternoon block. Nevertheless, there is expected resistance of
the nurses. 50% of the nurses works fulltime (De Vriese, 2015).
4.4.6. Redesign of individual activities The redesign of activities is difficult. Many tasks and sequences between
them are obligatory. The redesign of the tasks is based on the value added analysis
2InBelgiumitisobligatorytoplanabreakforeveryemployeewhoworkssixhoursormore.
73
performed in the process analysis. For each, there will be analyzed if there is a
possible improvement or if the activity can be eliminated. The adapted BPMN-
models are in Appendix VII.
Prepare dialysis: put on machine and distribute material for connection.
As mentioned in the analysis section, this task cannot be eliminated.
Determine if blood samples have to be taken. Elimination of this task is
not possible.
Distribute disconnection material (on tables). As explained in the analysis
section, elimination or improvement is not desirable.
Record patient values. On the long term, when the dialysis machines are
depreciated, there could be invested in dialysis machines which automatically link
the recorded values with the patients’ medical files.
Prepare patient folder for next shift. Again, there could be invested in
automation. If patient values do not have to be recorded manually anymore, the
need to print files and update folders disappears.
Evaluate if the catheter has to be cleaned. Moving this activity more upstream
or downstream into the process would not make any difference. Hence, this activity
is not changed.
Check if TV-screens are out. The problems with this task could be solved
by installing a switch which could be used to turn off all the screens in once.
Nevertheless, the cost of this investment would have to be weighed against the
advantages it would provide.
Check if the patient can go in and out bed independently or needs help form
nurse. An improvement is not necessary because it does not take much time and
is a habit for nurses.
Check-up patient data: blood collection needed, fistula/catheter, target weight, doctor remarks. This is a necessary task. A nurse has to be informed
about the condition of the patient.
Clean table. There will always be waste. It is difficult to prevent waste because
of hygienic reasons.
74
Check-up patient data. This task cannot be removed.
Check if there is an uncontrolled bleeding. This activity is necessary as well.
Put machine in setup mode. This task does not take much time; it is not
considered as a priority to change this task.
Assist doctor during consultation tour. There is advised to only assist the
doctor when the nurse has no other tasks to perform.
Insert medical files in database. Automation could help to eliminate this
step. As already mentioned, there could be invested in dialysis machines which link
the values automatically with the medical file of the patient. Alternatively, there
could be invested in laptops or tablets. These could be used to register the patient
values. One laptop/tablet could be used to register the values of several patients,
due to the mobility of the laptop/tablet. The values are automatically implemented
in the medical file of the patient.
4.5. Conclusion
Based on documents, interviews and observations, three main problems
were observed at the dialysis center. Patients were unsatisfied because of the long
waiting times. Nurses were unhappy because of the unbalance in workload.
Inefficiency raised because of the long time overlap between the early and late
shift. Based on the value added analysis, there could be concluded that not many
activities could be eliminated or reorganized. Out of the root cause analysis
followed that the cause of the issues lies in the fact that patients are scheduled
simultaneously.
To redesign the process, there was first looked at the duration of each
activity. For the duration of the connection and disconnection activity, there was
observed a significant difference between patients who can go in or out their
independently and patients who need help to get in or out their beds. The difference
between fistula and catheter patients was not significant. This was explained by the
fact that, relatively spoken, there are more catheter patients who need help than
fistula patients who need help. This increases the duration for the catheter patients.
75
Based on the durations, the patient schedule was reorganized. The schedule
was optimized in order to balance the workload. The technique where the
maximum workload per timeslot was minimized, was considered the most optimal.
An optimization of the connection session resulted in a patient schedule where
patients are planned sequentially. Patients who can go in and out their beds
independently, are planned with intervals of 8 minutes. Patients who need help, are
planned with intervals of 14 minutes. The independent patients are scheduled first,
the patients who need help at last. This corresponds with the findings of Cayitli and
Veral (Cayirli & Veral, 2003). Holland proposed to let dialysis patients arrive with
fixed 15-minute intervals (Holland, 1994). This dissertation also proposes to
schedule patients utilizing time intervals. However, contrary to Holland, there is
not made use of fixed time intervals. The time intervals are smaller and specific to
patient classes.
Relying on the adapted patient schedule, an overview was made of the
workload during the day. Between the connection and disconnection there were
time slots with less or no workload, caused by the obligatory dialysis duration of
four hours.
With a balanced workload, the time overlap between the early and late shift
becomes unnecessary. A new staff roster with less or no overlap is recommended,
but the success of a roster highly depends on the willingness of the nurses to
implement it.
At last the individual tasks were analyzed. Only minor improvements could
be booked since most tasks are obligatory due to hygienic reasons.
76
77
Chapter 5
Transportation
First, master the fundamentals. Larry Bird (1957–)
This chapter aims to propose a way to improve on the taxi services provided
to dialysis patients at AZ Sint-Jan Bruges-Ostend. Dialysis patients using private
transportation services need to go to and – after their treatment of on average four
hours – from the hospital back to their dwellings. Whilst some patients share rides,
no clear rules to determine the taxi routes are present and consequently sharing
rides is more of a standalone outcome instead of an integrated approach within
hospitals. In comparison with the current routes, finding a comprehensive approach
will lead to a more efficient use of these transportation services. To achieve this, a
variant of the Vehicle Routing Problem, which is the Open Vehicle Routing
Problem with Time Windows, will be applied.
A Vehicle Routing Problem, commonly abbreviated as the VRP, is a
generic name given to a whole class of problems that construct routes for a given
fleet of vehicles to service a set of customers, such that all customer’s requirements
and operational constraints are satisfied. The objective to evaluate the optimal
solution concerns minimizing cost, minimizing distance, minimizing total traveling
time and/or maximizing profit (Toth & Vigo, 2002). Slightly bluntly, we can say
that the Vehicle Routing Problem answers the question: “Given a set of customers
to service, what is the optimal set of routes for a fleet of vehicles to ride?”.
In the optimization literature, the Vehicle Routing Problem is one of the
most practically relevant and widely studied problems (Røpke, 2005). Since most
78
of the generic formulations of Vehicle Routing Problems are extensions of the
Multiple Traveling Salesman Problem, this chapter will gradually introduce some
basic models which forgo the class of Vehicle Routing Problems itself.
Section 5.1 opens by dealing with the relevancy of Vehicle Routing
Problems in today’s economy. Next, section 5.2 presents a comprehensive
overview of three preliminary variants of the Vehicle Routing Problem, namely the
Traveling Salesman Problem, the Multiple Traveling Salesman Problem and the
Capacitated Vehicle Routing Problem. After introducing these three variants, this
section continues by discussing more complex classes of Vehicle Routing
Problems. One of them is the routing problem used in the case of AZ Sint-Jan. The
next section, section 5.3, deals with the complexity of these problems. The aim is
to introduce the basics of the complexity theory and to highlight why routing
problems are so hard to solve. The group of heuristic solution methods for solving
these complex problems are discussed in section 5.4. Section 5.5 of this chapter
uncovers a tool created to solve smaller instances of the VRP as well as several
specific variants of the VRP. It is used in section 5.6. Here a description of and a
proposal for the transportation services at AZ Sint-Jan are given. For this proposal,
the tool from section 5.5 was used, as well as the theory from section 5.2.
5.1. Motivation
In today’s economy, transportation of goods as well as passenger
transportation forms a vital part in the global supply chain. Since enormous costs
are assigned to transportation in terms of vehicles, maintenance, wages and fuel
(but also the internal and external costs of emissions, such as CO2 and NOx), many
benefit from these type of optimizations (Eksioglu, Vural, & Reisman, 2009; Hall,
2016). The importance of an effective and efficient transport is only increasing due
to intense competition, several budget restrictions and a stronger focus on the
protection of the environment (Toth & Vigo, 2002). This is why Vehicle Routing
Problems are so widely studied and applied in academic literature.
Section 5.1.1 discusses the impact which VRP has on companies’
transportation and logistics operations. Next, in section 5.1.2, transportation of
79
passengers – more specifically transportation of dialysis patients – is discussed.
The impact an efficient transportation can have on external costs is discussed in the
last section, section 5.1.3.
5.1.1. Transportation and logistics Without any doubt, we can say that issues in transportation and logistics are
issues of all times and all places. They have a major economic and environmental
impact in most countries and regions over the world. Especially since governments
have put their focus on protecting the environment, these issues have gained
importance as well. Within the EU for example, the land transport policy is
promoting sustainable mobility that is efficient, safe and with a minimal of negative
effects on the environment (Steg & Gifford, 2005; Stantchev & Whiteing, 2006).
As transportation for most commodity products counts for a significant part
of the total costs (Notteboom & Rodrigue, 2013), there is a big incentive for
optimizing entire transportation processes and thus making the processes more
efficient. Moreover, Hasle and Kloster analyzed that using computer optimization
programs within routing problems, the use of it can save up to 5-20% of the
transportation cost (Toth & Vigo, 2002; Hasler & Kloster, 2007) and hence, a lot
of companies apply these programs to gain some competitive advantages in the
market. According to a survey conducted by Hall, several international
organizations have already developed a broad range of projects concerning
Commercial Vehicle Routing Problem Systems (CVRPSs) (Hall, 2016). As for
industries using this software, Drexl made a research about several software
providers and its customers. (Drexl, 2012). A quintessential finding was that the
following sectors made use of the software:
• Industry (raw materials and (semi-)finished goods transport);
• Wholesale and retail trade (consumer goods distribution);
• Full truckload (FTL) and less than truckload (LTL) forwarders;
• Parcel delivery and letter mail services;
• Reverse logistics and waste collection;
• Service technician, salesman, and other staff dispatching;
• Intra-plant logistics.
80
One can conclude that tours have to be planned in very diverse sectors and
in a very broad context. Therefore, routing problems have a central task in virtually
every enterprise concerned with physical goods or passenger transportation.
5.1.2. Passenger transportation Within a scheduling assignment for transporting people, human
components need to be taken into account. Especially when it involves patients
who suffer from a disease. Applied to dialysis patients, the literature has already
provided several answers concerning the general impact of transportation.
Diamant et al. stated that shorter travel times and distances are associated
with improved patient outcomes and a higher Health-Related Quality of Life
(HRQoL) (Diamant, et al., 2009). They found that patients who underwent dialysis
in satellite units3, compared to patients who are dialyzed in a hospital, reported
significantly lower rates on the stress part of the HRQoL. Moreover, satellite
patients conveyed significantly lower transportation costs and travel times. A
considerable proportion of them drives themselves to clinics. This can be explained
by the fact that satellite patients are rather low-care, compared to the patients going
to hospitals, which typically have more of a high-care profile. In 2012, Bello et al.
concluded that patients living on a remote location were less likely to receive
quality care and faced more risk to experience adverse health outcomes, compared
to those who lived closer (Bello, et al., 2012).
As stated by Moist et al., longer travel times are associated with greater
adjusted relative risk of death (Moist, et al., 2008). Patients traveling longer than
60 minutes had 20% more risk to death compared with those who traveled 15
minutes or less. Moreover, patients with longer travel times were more relying on
public and private transportation. Even 17% of the nurses agreed that patients who
arrived late did not get a full dialysis treatment.
All these studies write about the severe impact transportation has on a
dialysis patient’s life. Eventually, costs and health-related issues need to be
compared and a precise balance needs to be found in order to reduce costs but also
3Satelliteunitsarespecificdialysiscentersforlow-carepatients.Theseunitshaveasmallerpatient-to-RNratio.Anotheradvantageisthatthesecentersaremuchmorespreadaroundthecountry.
81
to reduce the negative impact on patients. Thus, VRP can help in finding the
optimal vehicle routes while keeping into account several time constraints. After
all, goods and passengers need to be approached in a different way.
5.1.3. External costs The impact on a macro-economic level should not be disregarded:
eliminating vehicles that could be prevented or vehicles with unnecessary long
routes lowers road utilization and congestions (Field & Field, 2012).
With regards to the environment, transportation counts for 22,2% of total
CO2 emission in Europe, as shown in Figure 9 (European Comission, 2016). Thus,
making transportation more efficient has a positive consequence on reducing
emissions and on the protection of the environment.
5.2. Classes of Vehicle Routing Problems
Coming back at the quote used at the beginning of this chapter “First,
master the fundamentals.” by Larry Bird, this section first enlightens the
fundamentals concerning the Vehicle Routing Problem. Afterwards, a better
understanding of more complex models and their corresponding solution
approaches is more achievable. A decent understanding of one of these variants,
Figure 9: Greenhouse gas emissions in Europe by source sector (2016)
82
the Vehicle Routing Problem with Time Windows, is necessary to solve the
transportation case study.
To categorize the specific routing problem for the dialysis patients, three
preliminary variants of the VRP are first presented. The basic variants give an idea
of the core problems present in more complex routing problems. The three basic
variants are the traveling salesman problem (section 5.2.1), the multiple traveling
salesman problem (section 5.2.2) and the capacitated vehicle routing problem
(section 5.2.3). Each of these sections first introduces the core problem in words,
afterwards their mathematical formulation is given. Succeeding this, a non-
exhaustive list of more complex routing problems is given in section 5.2.4.
5.2.1. Traveling Salesman Problem
5.2.1.1. Problem formulation
The most basic routing problem, but also one of the most studied
combinatorial optimization problems, is called the Traveling Salesman Problem
(TSP) (Cook, 2012). It can be seen as the easiest and most basic routing problem.
Dantzig, Fulkerson and Johnson published the first seminal paper regarding this
subject in 1954 (Dantzig, Fulkerson, & Johnson, 1954). Whilst no algorithmic
approaches for solving the TSP were given in this paper, it formed an indispensable
source of information to frame exact solution approaches afterwards (Lawler,
Lenstra, Rinnooy Kan, & Shmoys, 1985; Cook, 2012).
Finding an optimal solution for the Traveling Salesman Problem concerns
discovering the shortest distance tour, starting and ending at the salesman’s base
city, that visits all cities in which the salesman’s customers are located (Dantzig,
Fulkerson, & Johnson, 1954; Cook, 2012; Hoffman, Padberg, & Rinaldi, 2013). It
sounds simple enough, yet the traveling salesman problem is one of the most
intensely studied problems in applied mathematics and has defied large-scale
solutions to this day (Cook, 2012). In Figure 10 an instance of the TSP is shown.
On the left, the data is shown without any routes assigned. The optimal TSP
solution is shown at the right. Connections between each node are shown in
Euclidean distances.
83
5.2.1.2. Mathematical formulation
In the generic formulation for the TSP, we define nodes as points
representing the cities and arcs as the roads connecting these cities (Pataki, 2003).
If we let 𝒩 = 1, . . . , 𝑛 be the collection of cities (including the home city), then
we can identify a set 𝒜 which represents all possible arcs between all 𝑛 cities. To
each arc an arc-cost 𝑑jk gets assigned. These arc-costs represent the costs of moving
from node 𝑖 to node 𝑗. In the original TSP this cost showed the Euclidian distance
between the two cities 𝑖 and 𝑗 (Dantzig, Fulkerson, & Johnson, 1954). However,
other costs such as traveling distance or time can be used as well (Bektas, 2006).
The TSP is said to be symmetric when the arc-cost is the same in both directions,
meaning 𝑑jk = 𝑑kj (Hahsler & Hornik, 2007; Cruz, 2013; Hoffman, Padberg, &
Rinaldi, 2013). If and only if the arc 𝑖, 𝑗 ∈ 𝒜 is used in the optimal solution, the
vehicle flow binary decision variable 𝑥jk gets value one. Otherwise its value is zero.
The problem can be stated in an integer linear program instance. There are
a number of alternative formulations for the TSP (Bektas, 2006; University of
Wisconsin-Madison, 2016). The formulation used underneath is called the two-
index variable, assignment-based formulation and is an extension of the
formulation by Dantzig, Fulkerson and Johnson (Toth & Vigo, 2002).
Figure 10: Illustration of the TSP
84
𝑚𝑖𝑛 𝑑jk ∙ 𝑥jk�∈𝒩∖ �j∈𝒩
(5.51)
subjected to
𝑥jkj∈𝒩∖ k
= 1, ∀𝑗 ∈ 𝒩 (5.52)
𝑥jk�∈𝒩∖ �
= 1, ∀𝑖 ∈ 𝒩 (5.53)
𝑥jk ∈ 0,1 , ∀ 𝑖, 𝑗 ∈ 𝒜 (5.54)
The objective function in this integer program minimizes the distances
subjected to some constraints. Equations (5.52) and (5.53) are called degree
constraints and oblige that every city in 𝒩 will be visited exactly once by the
salesman (Bektas, 2006; Hasler & Kloster, 2007). Note that there is no need to
model a constraint that sets the salesman’s base city, which denotes his start and
end point. Since there is an implicit assumption that one salesman can visit all cities
without any capacity limits nor time window-constraints, only one directed cycle
that contains all 𝑛 cities will be determined. Hence, in this directed cycle the start
and end point can be any city – the tour will remain exactly the same.
In addition, sub-tours need to be eliminated by using so-called sub-tour
elimination constraints (SECs). It is a key part of a TSP to make sure no tours
between intermediate cities will be formed (Dantzig & Ramser, 1959; Miller,
Tucker, & Zemlin, 1960). This is done by making sure that every city visited
belongs to a route that is somehow connected with the home city of the salesman.
In order to truly comprehend the sub-tour elimination constraints, Figure 11 shows
a solution of a TSP (left) when these constraints are relaxed in the model. In the
solution network on the right side of the figure, the effect of implementing the
SECs is shown.
85
The literature proposes a number of alternative formulations to exclude
these degenerated tours. In the original generic formulation presented by Dantzig,
Fulkerson and Johnson, the formulation of the SEC was as following (Dantzig,
Fulkerson, & Johnson, 1954):
𝑥jkk∈𝒮j∈𝒮
≤ 𝒮 , ∀𝒮 ⊆ 𝒩 ∖ 1 (5.55)
However, adding this SEC implied adding an exponentially growing
number of constraints depending on the size of set 𝒩. Therefore, this formulation,
also denoted as the DFJ formulation, is impractical to use and alternative
formulations are at order (Miller, Tucker, & Zemlin, 1960; Pataki, 2003; Hahsler
& Hornik, 2007).
The most used formulation, as described in equations (5.56), (5.57) and
(5.58), is called the Miller-Tucker-Zemlin formulation, or in short the MTZ
formulation. Inserting these constraints implies only adding at maximum 𝑛 − 1 .
constraints (Hahsler & Hornik, 2007). However, a new set of variables 𝒰 =
𝑢j ∶ i ∈ 𝒩 ∖ 0 needs to be added. The variable 𝑢j is an integer variable and
denotes the position of node 𝑖 in a tour (Pataki, 2003).
𝑢( = 1 (5.56)
𝑢j − 𝑢k + 1 ≤ 𝑛 − 1 1 − 𝑥jk , ∀ 𝑖, 𝑗 ∈ 𝒜, ∀𝑖, 𝑗 ≠ 1 (5.57)
2 ≤ 𝑢j ≤ 𝑛, ∀𝑖 ∈ 𝒩 ∖ 0 (5.58)
As explained by Pataki, equation (5.56) sets 𝑢( equal to one – where index
zero indicates the base city. It is added to the MTZ formulation for limiting the
Figure 11: Traveling Salesman Problem, solution with two sub-tours (n=9)
86
number of constraints and thus, helping software packages in solving the TSP more
efficient (Hall, 2016).
In the conclusion of the research paper by Pataki primarily used to describe
this section about sub-tour elimination, the strengths and weaknesses of both
formulations are compared. He concluded that neither were efficient at solving
larger instances of the TSP using exact solution methods (Pataki, 2003). Other
formulations were at the order, such as combining these two formulations.
However, even at the day of writing this master thesis, these constraints have defied
large-scale solutions. Hence, SECs can be seen as a bottleneck to efficiently solve
the TSP with exact solution methods.
5.2.2. Multiple Traveling Salesman Problem
5.2.2.1. Problem formulation
Laporte and Norbert described the multiple Traveling Salesman Problem
(m-TSP) as an extension of the original TSP where more than one salesman is
allowed to start a directed cyclic route (Laporte & Nobert, 1980; University of
Wisconsin-Madison, 2016). This problem consists of finding the optimal tours for
all the salesmen such that all cities are visited exactly once, by only one salesman.
The standard m-TSP has 𝑚 salesmen which all start from and return to the same
single base city. In the original formulation, there were no costs involved other than
the costs related to the traveling distance (e.g. no fixed costs per salesman) (Bektas,
2006).
5.2.2.2. Mathematical formulation
Once again, the two-index variable, assignment-based formulation is used
to give a more formal approach to the problem. In the m-TSP, exactly 𝑚 salesmen
are present at one depot city, ready to visit 𝑛 cities in total. In the original problem,
each route assigned to salesman 𝑚 should not be empty, meaning that all the
salesmen present need to be on the road (Bektas & Kara, 2006). Compared to the
TSP, the only difference is that now there are numerous routes starting from the
base city instead of one route.
87
𝑚𝑖𝑛 𝑑jk ∙ 𝑥jkk∈𝒩∖ jj∈𝒩
(5.59)
subjected to
𝑥,kk∈𝒩∖ ,
= 𝑚, ∀(1, 𝑗) ∈ 𝒜 (5.60)
𝑥j,j∈𝒩∖ ,
= 𝑚, ∀(𝑖, 1) ∈ 𝒜 (5.61)
𝑥jkj∈𝒩∖ k
= 1, ∀𝑗 ∈ 𝒩 (5.62)
𝑥jk�∈𝒩∖ j
= 1, ∀𝑖 ∈ 𝒩 (5.63)
𝑥jk ∈ 0,1 , ∀ 𝑖, 𝑗 ∈ 𝒜 (5.64)
In this formulation, the constraints in equation (5.60) and (5.61) ensure that
𝑚 salesmen depart from and return to their base city. Just like in the TSP, the degree
constraints (5.62) and (5.63) enforce that only one route will enter and exit a city.
The MTZ formulations of the sub-tour elimination constraints for the m-
TSP are not precisely the same as it was for the TSP. However, the idea – and so
does their big disadvantage – remains exactly the same.
𝑢( = 1 (5.65)
𝑢j − 𝑢k + 𝑛 −m ∙ 𝑥jk ≤ 𝑛 −m − 1, ∀ 𝑖, 𝑗 ∈ 𝒜, ∀𝑖, 𝑗 ≠ 1 (5.66)
2 ≤ 𝑢j ≤ 𝑛, ∀𝑖 ∈ 𝒩 ∖ 0 (5.67) An illustration of this problem is given in Figure 12.
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5.2.3. Capacitated Vehicle Routing Problem
5.2.3.1. Problem formulation
The CVRP can be seen as a generalization of the m-Traveling Salesman
Problem by setting a limitation on the amount of customers a salesman can visit
(Laporte, 1992). Another difference merely lays in its more general vocabulary:
salesmen are changed into vehicles and the cities are more commonly referred to
as customers (Røpke, 2005). As stated by Laporte, the context of the CVRP is that
of a fleet of vehicles supplying customers using resources gathered from their
central depot. These vehicles can be homogeneous or heterogeneous. Stating that
the trucks are homogeneous is equal as saying that each vehicle on its own had the
same capacity as well as the same cost structure. On the other hand, stating that the
fleet is heterogeneous adds different vehicle parameters to the problem.
The CVRP already adds more detail to the problem, making it already
practical to use in some easygoing real-world problems. However, it does not take
into account the human component. As Diamant et al. as well as Bello et al. proved
within the transportation of dialysis patients, these human components are of major
importance (Diamant, et al., 2009; Bello, et al., 2012).
5.2.3.2. Mathematical formulation
In this section, the three-index directed, vehicle-flow formulation of a
heterogeneous CVRP with fixed costs is presented. This is a general used version
Figure 12: Illustration of the m-TSP (n=12, m= 4)
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of the CVRP and already highlights the complexity present in VRPs. The
formulation was based on the original formulation of Golden, Magnanti and
Nguyen and was later adjusted by Gheysens, Golden and Assad (Golden,
Magnanti, & Nguyen, 1977; Gheysens, Golden, & Assad, 1984). The CVRP is
either a pure commodity pick-up problem or a pure commodity delivery problem
(Wen, Clausen, & Larsen, 2010). The model is as follows:
𝑚𝑖𝑛 𝐹m ∙ 𝑥(kmk∈𝒩∖ j
s
mr,
+ 𝑐jkm ∙ 𝑥jkmk∈𝒩∖ jj∈𝒩
s
mr,
(5.68)
subjected to
𝑥jkmj∈𝒩
s
mr,
= 1, ∀𝑗 ∈ 𝒩 ∖ 0 (5.69)
𝑥j?mj∈𝒩
− 𝑥?kmk∈𝒩
= 0, ∀𝑘 ∈ 1, . . . , 𝐾 , ∀𝑝 ∈ 𝒩 ∖ 0 (5.70)
𝑥(kmk∈𝒩∖ (
≤ 𝑛m, ∀𝑘 ∈ 1, . . . , (5.71)
𝑑k ∙ 𝑥jkm ≤ 𝑦jk ≤ 𝐶m − 𝑑j ∙ 𝑥jkm , ∀ 𝑖, 𝑗 ∈ 𝒜, ∀𝑘 ∈ 1, . . . , 𝐾 (5.72)
𝑦jkj∈𝒩
− 𝑦kjj∈𝒩
= 𝑑k, ∀𝑗 ∈ 𝒩 ∖ 0 (5.73)
𝑥jkm ∈ 0,1 , ∀ 𝑖, 𝑗 ∈ 𝒜, ∀𝑘 ∈ 1, . . . , 𝐾 (5.74)
𝑦jk ≥ 0, ∀ 𝑖, 𝑗 ∈ 𝒜 (5.75)
The binary decision variable 𝑥jkm gets the value one only when an arc 𝑖, 𝑗 ∈
𝒜 is used in the tour of vehicle 𝑘. Otherwise it is zero. In addition, a second flow
variable 𝑦jk is introduced. The value specifies the quantity of goods a vehicle
carries when it leaves from customer 𝑖 and travels to serve customer 𝑗.
Again, a set 𝒩 of 𝑛 + 1 nodes represents the total set of nodes for the
current CVRP. Each node is associated with an index number from 0 to 𝑛, with 0
being the central depot 0 and the remaining 𝑛 nodes being the delivery points
1,⋯ , 𝑛 . Non-negative orders 𝑑j of some commodity are assigned to each
delivery node and hence, each delivery point 𝑖 represents a customer order.
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Servicing these customers’ orders is done by a fleet of 𝐾 heterogeneous vehicles
with a load capacity 𝐶m. Every vehicle of type 𝑘, has a number 𝑛m vehicles
available. There is also a fixed cost 𝐹m included for a vehicle to departure. The set
𝒜 is composed of precisely 𝑛 ∙ 𝑛 + 1 2 arcs and represents all the arcs possible
to connect each 𝑛 + 1 nodes from set 𝒩. Each arc is associated with a cost 𝑐jkm and
represents the cost of a vehicle 𝑘 connecting node 𝑖 to node 𝑗.
Equations (5.69) and (5.70) oblige that customers are visited exactly once
and that arriving at a node also means departing from the same node. The inequality
(5.71) ensures that a given amount of vehicles of type 𝑘 is not violated. The
inequality at (5.72) deals with the vehicle capacity limitations for an assigned tour.
In order to ensure that the quantity of goods a vehicle carries is adjusted according
to the customer’s demand after visiting him, constraint (5.73) is used.
This general model is already much more complex than its predecessors.
Moreover, sub-tour elimination still needs to happen in this model, making it even
more complex. Kulkarni and Bhave propose the following extension of the MTZ-
formulation (Kulkarni & Bhave, 1985):
𝑢j − 𝑢k + 𝐶m ∙ 𝑥jkms
mr,
≤ 𝐶m − 𝑞k, ∀ 𝑖, 𝑗 ∈ 𝒜, ∀𝑖, 𝑗 ≠ 1 (5.76)
qj ≤ 𝑢j ≤ 𝐶m ∙ 𝑥jkmj∈𝒩
s
mr,
, ∀𝑗 ∈ 𝒩 ∖ 0 (5.77)
In the context of the heterogeneous CVRP with fixed costs, still a lot of
effort is going on in the literature to improve these SECs. A good example is the
Reformulation-Linearization Technique (RLT) studied by Sherali and Adams
(Sherali & Adams, 1990). However, dealing with these approaches is no goal of
this dissertation topic. These problems are only reviewed to motivate the use of
different solution techniques.
5.2.4. Variants of the Vehicle Routing Problem Applied to the real-world, VRPs are much more complex than the CVRP
model described above. There exists a vast amount of extensions for the VRP in
order to be as close to the reality as possible. By adding complexity to the models,
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more constraints are added and hence understanding and solving these models is
even more complex. While it is not the subset of this master dissertation to discuss
each variant of the VRP in detail, it still seems important to give an overview of
these problems in words. Based on a research paper by Toth and Vigo, Figure 13
shows the variants discussed as well as their relationship with each other (Toth &
Vigo, 2002).
CVRP – Capacitated VRP. First, the CVRP is extended. Extensions
concerning limitations on the vehicle or driver can be driving distance limitations
and time limitations (Nguyen, 2014). In addition, the CVRP can be closed or open.
A CVRP is closed when the vehicles need to return to their base locations. The
open CVRP implies that vehicles do not need to return to their base city (Wen,
Clausen, & Larsen, 2010).
TDVRP – Time-Dependent VRP. In this VRP, vehicles are only allowed to
ride within a certain time frame. If for any reason, a vehicle gets a route assigned
that cannot be driven within its time limits, the solution is infeasible.
Figure 13: Overview of several VRP classes
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DDVRP – Distance-Dependent VRP. Here vehicles have distance
constraints. The idea is the same as the TDVRP but instead of a limited timespan
given to the drivers, there is a limitation on the distance traveled.
VRPTW – VRP with Time Windows. One important extension of the
classical CVRP is to include time windows for each customer. This introduces the
opportunity to model time intervals in which the vehicle must arrive at the
customer. Time windows are said to be hard when the route is infeasible when it
is set to visit customers not within their time window. Hence, hard time window
constraints force the vehicle to wait until the time slot opens (Nguyen, 2014). On
the other hand, time window constraints can be soft as well, meaning that a
violation only implies a certain penalty cost and no waiting of the vehicles. The
VRPTW can be seen as a crucial variant within the general routing problem, as this
includes many problems in real-life. Consequently, VRPTW has been subjected to
various intensive research efforts both for modeling the problem more efficiently
as well as for focusing on more efficient solution approaches.
SDVRP – Split Delivery VRP. In the split delivery vehicle routing
problem – introduced in the literature by Dror and Trudeau – each customer can be
visited more than once, allowing to split deliveries into several phases (Dror &
Trudeau, 1990). The authors motivated their study by showing that splitting
deliveries can generate significant logistic savings.
VRPB – VRP with Backhauls . The VRPB includes both a set of customers
to whom products are to be delivered, and a set of vendors whose goods need to be
transported back to the distribution center. In addition, on each route all deliveries
have to be made before any goods can be picked up to avoid rearranging the loads
on the vehicle.
VRPPD – VRP with Pick-up and Delivery. A pick-up and delivery problem is
initiated when a customer puts in a request. The request covers a pick-up location
together with a different delivery location and an order quantity (Røpke & Pisinger,
2006). The problem is ought to find the best route given a fleet of vehicles, knowing
that there needs to be a pick-up before there can be a delivery of that specific good.
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5.3. Computational Complexity
The computational complexity theory is a field in theoretical computer
science and mathematics, which deals with the resources required during
computation to solve a given problem. Most of the complexity theory deals with
the decision problems that can be answered by either ‘yes’ or ‘no’, and divides
these problems into different classes according to the difficulty of solving the
problems in terms of computational resources (Mariño, 2016). Since every single
optimization problem can be altered into a decision problem where the question is
‘Is there a feasible solution which returns an objective value better than the
previous best known value?’, these problems fall within the assumptions of the
complexity theory. Hence, optimization problems like the vehicle routing problems
can be assessed by its computational complexity (Wen, Clausen, & Larsen, 2010).
In mathematical programming, tractable problems that have at least one
algorithm to solve are called polynomial-time problems or just 𝒫-problems. As
stated by Mariño, these problems can be solved by algorithms with a number of
steps that is a polynomial function of the problem size 𝑛 (Mariño, 2016). Since the
number of steps is dependent on the computer’s computational power, the exact
computing time depends on which computer that is used.
Mariño recognizes that the complexity theory also differentiates hard
problems. These problems need solution methods with an exponential number of
steps dependent on the problem size 𝑛 and hence, these problems are considered to
be inefficient. A big incentive arises to model problems as efficient as possible.
Constraints such as the Sub-tour Elimination Constraints should be modeled as
efficient as possible considering the exponential growth in computing times (Hall,
2016). Since there is no clear rule to determine the time needed to solve these type
of problems, they are called non-deterministic polynomial hard problems,
abbreviated as 𝒩𝒫-hard problems. In relationship to this inefficiency, methods
exist that achieve near-optimal results in a relatively small timespan. These
solution methods are discussed in section 5.4.
Adding complexity to a certain model means adding constraints, making it
harder and harder to solve. Therefore, most of the real-world problems are 𝒩𝒫-
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hard problems. Lenstra and Ronnooy Kan have studied that even the TSP belongs
to the class of 𝒩𝒫-hard problems, mainly due to the efforts to eliminate sub-tours
(Lenstra & Rinnooy Kan, 1981). Their conclusion was that many variants of the
VRP are 𝒩𝒫-hard to solve. Since the VRPTW – used in the case study for AZ
Sint-Jan – is but an extension of the VRP, Desrochers, Desrosiers and Solomon
determined that it is 𝒩𝒫-hard as well (Desrochers, Desrosiers, & Solomon, 1992).
In relationship to these 𝒩𝒫-hard problems, alternative solution methods
are proposed. Making a complex mathematical model and applying exact solution
methods – which need relatively costly software packages – in order to solve, is
inefficient or perhaps infeasible. Rather, heuristic methods are recommended to
solve these type of problems (Røpke, 2005). These methods focus on generating
solutions within modest computing times by performing only a reduced assessment
of the entire search space. Besides the shorter running times, other advantages can
be listed as well (Wen, Clausen, & Larsen, 2010):
• Relatively easy to implement;
• Flexible enough to fit in specific problems;
• Sometimes create more robust solutions.
In the following section, an overview of these heuristic solution methods
applied to VRP is given. The goal is to have a solution method ready for solving
the transportation case study at AZ Sint-Jan, discussed in section 5.6.
5.4. Heuristic solution methods
Broadly seen, heuristic solution methods for the VRP can be divided into
three types: route construction heuristics (section 5.4.1), route improvement
heuristics (section 5.4.2) and metaheuristics (section 5.4.3).
5.4.1. Route construction heuristics As stated by Laporte and Semet, construction algorithms build initial
feasible solutions taking into account the problem’s objective function. However,
improving these newly formed solutions are in general not present in a construction
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algorithm (Laporte & Semet, 2002). Construction heuristics can be divided into
three groups: insertion heuristics, savings heuristics and clustering heuristics. In
what follows, we summarize the findings from Laporte and Semet about
construction heuristics.
Insertion heuristics. These algorithms construct solutions by adding unrouted
customers iteratively and greedily into the routes. The construction can be done
either sequential or parallel. Which customers to select is based on some chosen
priority rules, e.g. insert the closest customer or insert the customer that increases
the costs the least.
Savings heuristics. The first savings heuristic was introduced by Clarke and
Wright in 1964 (Clarke & Wright, 1964). They are also called greedy algorithms.
Within these types of heuristics, multiple routes are first formed back and from the
customer. Next, routes are merged one by one depending on a set of criteria.
Clustering heuristics. Clustering heuristics consists of two phases. In the first
phase the heuristic groups customers into so-called subsets. The second phase then
assigns vehicles to service within each subset and tries to calculate the best possible
route for each vehicle. An optional third phase can be applied when the subsets
formed are infeasible to be served by only one vehicle. A well-known example of
this clustering heuristic is the sweep algorithm and is illustrated in Figure 14.
Figure 14: Illustration of the sweep algorithm
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5.4.2. Route improvement heuristics Given a solution, for example generated by a route construction heuristic,
local search heuristics modify these solutions to produce new and enhanced routes
(Wen, Clausen, & Larsen, 2010). For this purpose, a lot of operators – or in the
literature also called neighborhoods – have been presented. Depending on the
number of routes simultaneously modified, the operators can be fitted into two
groups. A first group, called the intra-route operators, works on one route at a
time, while the second group, denoted as the inter-route operators, modifies more
than one route at a time (Laporte & Semet, 2002).
A widespread intra-route operator is the 𝜆-opt operator, proposed by Lin in
1965 (Lin, 1965). The 𝜆 indicates the amount of arcs that will be randomly removed
within the route. Since at least two arcs are needed to be exchanged, 𝜆 cannot be
smaller than two. Figure 15 gives an example of a 2-opt operator.
Popular examples of the inter-route operator are the vertex swap and the
vertex relocation procedures. In the vertex swap procedure, customers from
different routes are mutually exchanged. An illustration is given in Figure 16.
Vertex relocation just removes one customer from a route and inserts it into another
route. This procedure is illustrated in Figure 17.
Figure 15: Illustration of the λ-opt operator (λ=2)
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5.4.3. Metaheuristics A metaheuristic is a high-level, problem-independent technique that is in
general not greedy. It provides an algorithmic framework with a set of guidelines
to develop heuristic optimization algorithms (Laporte, Gendreau, Potvin, & Semet,
2002; Blum & Roli, 2003). Metaheuristics let the problem deviate away from a
ground solution which better allows to explore the solution space and avoid the
problem of achieving local optima, as illustrated in Figure 18. In fact, it may even
accept a temporarily infeasible solution. This exploration of different solution
spaces is sometimes termed as diversification. This is in contrast to the term
intensification, which implies improving local solutions found (Laporte, 2009).
Hence, a suitable tradeoff between diversification and intensification is crucial for
their success, both in terms of accuracy and in terms of running times.
Figure 16: Illustration of the vertex swap operator, two vertices are swapped
Figure 17: Illustration of the vertex relocation operator, one vertex is relocated
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Many different kinds of metaheuristics were proposed in the past decades
(e.g. simulated annealing, genetic algorithms, ant colony optimization) (Blum &
Roli, 2003). Some of them are inspired on principles in nature. To get a better
overview of the different kinds of metaheuristics and their advantages and
disadvantages, Blum and Roli give a comprehensive overview.
5.5. Practical tool to solve VRPs
Many different heuristic solution algorithms can be found in the academic
literature. The disadvantage of these theoretical solutions is that they are very
specific and difficult to adapt to real-world problems. Often the solution approach
is just written in a high-level programming language like C++ and are not
companioned with a graphical user interface, making these solutions not for the
faint of heart. Moreover, real-world situations are often much more complex than
the idealized problems described in the literature.
Of course commercial software applications capable of solving these
complex, real-world issues exist (e.g. Routing, eRoute Logistics, ArcGIS Network
Analyst, SpeedyRoute) (INFORMS, 2016). However, these commercial software
packages are not always that attractive as they induce a relatively large cost.
Several projects that offer a free or limited free VRP solution tool can be found on
the Internet (e.g. Open Door Logistics, OptaPlanner, OpenVRP). The drawback is
that these free offerings are rather hard to understand and hard to adapt to case-
Figure 18: Local optimum vs Global optimum
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specific problems. Because of this drawback, there is a need of a free and easy-to-
understand standard tool capable of solving several medium-sized variants of the
Vehicle Routing Problem (VeRoLog, sd). This lack was already acknowledged by
a working group on vehicle routing and logistics optimization, called VeRoLog.
They have programmed a flexible heuristic algorithm capable of solving some
classes of the VRP. It is an open-source project and the code can be found on
Github. This promising algorithm was used as a basic in order to develop a tool
capable of solving the problem at AZ Sint-Jan.
The tool’s code is uploaded on github.ugent.be/MIS/thesisdialyse/. For information
about Github, we refer to helpdesk.ugent.be/github/. A user guide, adapted to the
hospital case, is added as an appendix at the very end of this master dissertation.
This part opens with section 5.5.1 and gives a motivation for the chosen
programming language, Visual Basic for Applications. Section 5.5.2 discusses a
core component of any commercial software for VRPs, namely the geographic
information component. Lastly, the solution method embedded in the tool is
discussed in section 5.5.3.
5.5.1. Visual Basic for Applications Microsoft Excel has been around for quite some time. This spreadsheet tool
is accessible in practically every business, independent of the companies’ size or
activities. Visual Basic for Applications (VBA) is a programming language
embedded in Excel, but also in other products of Microsoft Office. It allows to
integrate the spreadsheet tool with a VBA-written program, called a macro
(Weterings, 2010). Using VBA, certain functions such as for and while loops which
Excel lacks are possible. Some benefits of using excel are its wide user-base, its
flexibility and the user-interface which is rather easy-to-understand. While high-
level programming languages still outperform VBA on many aspects, the primary
intention was to create a flexible and easy-to-understand tool for solving smaller
instances of the VRP in function of the case study.
5.5.2. Geographic Information System Solving a Vehicle Routing Problem starts by computing the arc-costs 𝑐jk
for every pair of locations possible. In most real-life cases the traveling costs
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considered are either distance or time, or both of them. Basically, it is the same as
what a GPS calculates: one can either set the GPS to choose the shortest route or
one can choose to drive on the fastest route to reach a destination. GPS-systems are
confronted with the so-called shortest-path problem, which also uses heuristics to
solve the problem (Singla & Chhilar, 2014).
To set up the time and distances between each node, the location data needs
to be accessed by a Geographic Information System (GIS). The capabilities of these
systems have significantly increased in the past few years (Berry, sd). The two
biggest Geographic Information Systems at the time of writing this dissertation,
are Google Maps and Microsoft’s Bing Maps. In the tool, Google Maps is
conducted, mainly because of Google Maps’ user-friendliness combined with its
limited free service. Bing Maps also provides a limited free service but it requires
a registered account with a key that can only take up to 25.000 requests a year. The
limitations of Google Maps on the other hand are much less strict. Google gives up
to 2.500 free requests per day.
The following functionalities from Google’s GIS are used: geocoding,
directions and My Maps.
Geocoding. This function translates each address into its equivalent latitude
and longitude pairs. The accuracy of these pairs depends on the amount of numbers
used. In the case of Google Maps, translating the latitude and longitude pairs back
to its original address is possible.
Directions. Requesting directions returns the real-world driving distance and
the corresponding driving time. Google’s Directions API facilitates – just like in
the web-application of Google Maps – various types of requests: shortest route or
fastest route, mode of transportation, avoiding tollways, …
My Maps. Here is referred to the functionalities of Google’s My Maps. This
service offers the possibility to create a custom map, import geographically-
specific data and draw lines or pinpoints on this map. It is mainly used to visualize
the proposed solution.
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5.5.3. Heuristic algorithm In the tool an extension of the Large Neighborhood Search (LNS) heuristic,
introduced by Shaw, is used (Shaw, 1998). The LNS belongs to the class of
metaheuristics, as seen in section 5.4.3. Røpke and Pisinger extended the large
neighborhood search heuristic of Shaw by allowing the use of multiple destroy and
repair operators within the same search process (Røpke & Pisinger, 2006). This
general framework is denoted as the Adaptive Large Neighborhood Search
(ALNS) heuristic and classifies in the class of Very Large-Scale Neighborhood
search (VLSN) algorithms (Røpke & Pisinger, 2010).
In a research paper by Røpke and Pisinger, the authors first presented their
extension of the LNS heuristic and applied it to the Pick-up and Delivery Vehicle
Routing Problem with Time Windows (Røpke & Pisinger, 2006). They tested the
heuristic on more than 350 benchmark instances with up to 500 requests and
concluded that it improved the best known solutions within the academic literature
in more than 50% of the problems. These benchmarks clearly showed that using
several competing subheuristics is very profitable. Hence, this explains the usage
of this heuristic in the tool.
In what follows, the base version of the heuristic is explained. Thereafter,
the adaptation of Røpke and Pisinger is introduced. Lastly, a pseudocode of the
implementation in the tool of the heuristic is presented.
5.5.3.1. Large Neighborhood Search
Local search heuristics are focused on making small changes to the current
solution in order to find better solutions. However, Shaw’s finding was that such
heuristics can have difficulties when it concerns moving away from one promising
solution area to another. Consequently, local search heuristics often get trapped in
local optima. Cordeau, Laporte and Mercier also recognized this problem
(Cordeau, Laporte, & Mercier, 2001). Their proposal consisted of relaxing several
constraints, allowing the algorithm to visit infeasible solutions. Shaw took another
approach in solving this problem. The initial heuristic described by Shaw is written
in pseudocode and is shown in Table 5.
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Framework: Large Neighborhood Search heuristic
1 Construct an incumbent solution s; s∗ ≔ s; 2 Repeat
3 Set s� = 𝑠; 4 Remove 𝑞 requests using NE from s�; 5 Reinsert 𝑞 request using NC into s�; 8 If 𝑓 𝑠� better than 𝑓 𝑠∗ then 9 Set s∗ ≔ s�; 6 If 𝑠� accepted then 7 Set s ≔ s�; 10 Until stop-criterion is met
11 Return s∗;
The pseudocode of the framework shows there is a main loop at the master
level between line 2 and line 10. The codes at line 4 and 5 are the most important
ones, as these allow to make bigger changes to the solution. Choosing good
removal and inserting neighborhoods is a vital part in achieving an efficient
algorithm. Parameter 𝑞 determines the size of these neighborhoods and establishes
the amount of intensification and diversification of the solution space to put into
the heuristic. The rest of the code considers whether the newly formed solution is
better than the current known solution and if it can be accepted. These steps are
repeated until a certain stop-criterion is met. These criteria can be several things,
such as the number of iterations performed or a time-limit (Shaw, 1998).
5.5.3.2. Adaptive Large Neighborhood Search
The ALNS-algorithm, as proposed by Røpke and Pisinger, represents a
unified heuristic, in which a certain number of simple heuristics compete to modify
the current solution, making it different than the original LNS-algorithm (Røpke &
Pisinger, 2006). Instead of just one remove and reinsert method, the ALNS
heuristic proposes several smaller methods to compete against each other. The
pseudocode is given in Table 6.
In each iteration, an initial solution gets gradually improved by alternately
destroying and then reconstructing a part of the solution using sets of appropriately
defined destruction and repair operators. Both neighborhoods are set in line 4,
Table 5: Original LNS heuristic (Shaw, 1998)
103
where the selection of the neighborhoods depend on their past performance score.
In a set with 𝑁 neighborhoods, where 𝜋j denotes the past performance of a
neighborhood 𝑖, the probability of a neighborhood 𝑗 getting selected is:
𝑝k =𝜋k𝜋j�
jr, (5.78)
Hence, the roulette wheel selection gives more probability to a
neighborhood being selected when their past performance scores are bigger. Note
that the destruction and repair operators are selected independently. This implies
the use of the roulette wheel selection to be used twice. The score 𝜋j itself can be
based on several criteria. In line 13 these scores get updated. The rest of the
algorithm remains the same when compared with the original LNS heuristic.
Framework: Adaptive Large Neighborhood Search heuristic
1 Construct an incumbent solution s; s∗ ≔ s; 2 Repeat
3 Set s� = 𝑠; 4
Choose a destroy neighborhood NE and a repair neighborhood NC using roulette wheel selection based on past performances (scores) π� ;
5 Remove 𝑞 requests using NE from s�; 6 Reinsert 𝑞 request using NC into s�; 7 If 𝑓 𝑠� better than 𝑓 𝑠∗ then 8 Set s∗ ≔ s�; 9 Update scores π� of NE and NC; 10 If 𝑠� accepted then 11 Set s ≔ s�; 12 Until stop-criterion is met 13 Return s∗;
5.5.3.3. Implementation of the ALNS heuristic
A variant of the ALNS heuristic is used in the macro. The pseudocode is
given in Table 7. Lines 7 until 17 indicate the main loop at the master level of the
heuristic, which gets repeated until the stop-criterion is met. The criterion used in
Table 6: Adaptive LNS heuristic (Røpke & Pisinger, 2006)
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the algorithm is a timer that starts when the algorithm is initialized. The code at
line 9 introduces a certain noise into the solution, dependent on the parameters 𝛼,
and 𝛼. set at line 3 and 4. After this mutation, locations are randomly inserted from
a list of possible candidates. The three local search heuristics, described at line 11,
compete against each other. These heuristics are described in section 5.4.2. The
rest of the code again considers whether the newly formed solution is better than
the current known solution and determines if it can be accepted.
Framework: Variant of the Adaptive Large Neighborhood Search heuristic
1 Construct an incumbent solution s; s∗ ≔ s; 2 Set iteration counter 𝑘 = 0; 3 Set minimum removal rate 𝛼,; 4 Set maximal removal rate 𝛼.; 5 Set candidate list size 𝛽; 6 Start timer;
7 Repeat 8 Set s� = 𝑠; 9 Randomly remove α, + U 0,1 ∙ α. − α, percent from s�; 10 Randomly reinsert locations from 𝛽 into s�; 11
Apply best among 2-opt, vertex swap and vertex relocation;
12 If 𝑓 𝑠� better than 𝑓 𝑠∗ then 13 Set s∗ ≔ s�; 14 If 𝑠� accepted then 15 Set s ≔ s�; 16 𝑘 + +; 17 Until timer hits zero 18 Return s∗;
Table 7: Variant of the ALNS heuristic, implemented at the tool
105
5.6. VRP at the dialysis center
This section opens with a map locating the patients’ addresses together with
the location of the hospital (Figure 19). The addresses were taken from a list of
168 dialysis patients, provided by dr. An De Vriese [this list is not embedded in
this thesis due to the privacy of the patients]. Besides patients’ addresses, the list
also offered the patients’ mode of transportation (including the taxi firms, if one is
used) and the patient’s assigned shift. Out of the 168 patients, more than 60% used
taxi services.
5.6.1. Current state The core problem concerning the transportation of patients to and from the
dialysis center is that no main method is known to determine how patients can share
rides in order to minimize costs. Hence, two inefficiencies occur:
• Too many patients using taxi services are transported alone;
• Even if patients share rides, it is not known if this is the cheapest solution.
Figure 20 illustrates a first conceptual transportation time scheme of the
current state. Two patients are picked up separately by a taxi and are driven to the
hospital. They arrive at the same time. This conceptual drawing will later be
adjusted to explain a new approach.
Figure 19: Map indicating location of hospital and 168 dialysis patients
106
5.6.2. Solution approach The aim for studying the theory around the Vehicle Routing Problem was
to propose an improvement on the taxi services provided to dialysis patients at AZ
Sint-Jan Bruges-Ostend. The solution approach can be split into two components.
• A first component consists of finding patients that can share taxi rides so that
global costs are minimized. This problem was identified in section 5.2.3 as the
Capacitated Vehicle Routing Problem.
• The second component deals with limiting the extra traveling times for
patients. These additional traveling times are imposed by the fact that sharing
rides will increase total traveling durations for certain patients. In section 5.1.2,
a recognition was made concerning the traveling duration and the impact on
the patient’s quality of life. To set limits on these extra traveling durations,
time windows for each patient are introduced. This extends the routing
problem to a Capacitated Vehicle Routing Problem with Time Windows
(section 5.2.4).
In addition, an assumption was made that there are no depots for the taxi
services to return to. This assumption was made because taxi riders continue their
working day after they released the patients at the dialysis center. Thus, the final
problem can be termed the ‘Open’ Capacitated Vehicle Routing Problem with Time
Windows.
Figure 20: Current transportation time scheme
107
5.6.2.1. Time windows for patients
Time windows indicate an interval in which transportation services can
pick-up or drop-off patients. In this context, it is used to set a limit on the extra
traveling duration imposed by sharing rides. This application of time windows is
new in the literature. It adds a human component to the otherwise, solely logistic
VRP. A conceptual figure illustrates the usage of these time windows (Figure 21).
When Figure 20 is compared with Figure 21, the extra driving time for
patient 1 becomes clear. Picking up patient 1 occurs within its time window. These
time windows for picking up patients and transport them to the hospital are
established using the following simple interval:
𝑡ZjA − 𝑡¡}n¢[, 𝑡ZjA (5.79)
The minimum time 𝑡ZjA indicates the shortest time possible for a patient to
go to the hospital. On the other hand, the extra traveling time 𝑡¡}n¢[ imposes a
maximum time added to the total traveling time.
5.6.2.2. VRP Tool
The tool discussed in section 5.5, is used in order to optimize the Open
Vehicle Routing Problem with Time Windows. A user manual on how to use this
tool is included in Appendix VIII. It is primarily based to help nurses use this tool.
Figure 22 shows a violation of these time windows. When these time
windows are violated, the tool indicates the solution as infeasible. However, if the
number of vehicles used in the schedule does not allow any better schedule, these
violations are minimized and the optimized, yet infeasible solution is given.
Figure 21: Illustration of time windows
108
5.6.3. Results From the dataset provided by dr. An De Vriese, four routing problems can
be identified. These routing problems correspond with the four shifts patients can
get assigned to (section 4.2.2.2.1). In Table 8 the number of patients per shift that
make use of transportation services are given.
Shift 1 (M/W/F
AM)
Shift 2 (M/W/F
PM)
Shift 3 (T/T/S AM)
Shift 4 (T/T/S PM)
Number of patients using transportation services
21 25 22 18
Figure 23 shows the outcome of the optimization of the first shift. Time
windows were set so that patients are assigned to a schedule with at most 15
minutes extra in driving time. In this context, each vehicle had the capacity of
transporting at most three patients.
The performance criteria to compare the new schedule with the old
schedule are the number of vehicles, total traveling duration, total extra duration
and total distance.
Figure 22: Illustration of a time windows violation
Table 8: Number of patients per shift, using transportation services
109
In Table 9 the different performance criteria of the old and new
transportation schedule of shift 1 are listed. Only the impact of the scenario where
patients are picked up from their homes to the hospital is given. However, returning
patients back to their dwellings is practically the same. Since one patient is in need
of the taxi service six times per week (patients need to go to the hospital and from
the hospital back to their homes again, this occurs three times a week), the impact
of this improved schedule should not go unnoticed.
Old NewNumber of vehicles 14 9Total traveling duration (hours) 6:00 5:06Total extra duration (hours) 0:37 1:15Total distance (km) 245.49 185.19
It is hard to quantify the impact in monetary terms. However, it can be
easily seen that the reduction in the number of vehicles, as well as the reduction in
total traveling time and the reduction in total driving distance will surely have a
lasting impact on the long term.
Based on feedback during the observations, it is acknowledged that several
patients opt not to use taxi services because the transportation services are too
expensive. Hence, this case study could be used as a financial persuasion for
patients. After all, sharing rides implies sharing costs.
Figure 23: Taxi schedule for shift 1
Table 9: Performance criteria of old and new routing schedule (shift 1)
110
5.7. Conclusion
The purpose of this chapter was to introduce a comprehensive approach for
the transportation problem at AZ Sint-Jan, together with a tool able to solve this
problem. Therefore, theoretical concepts concerning the Vehicle Routing Problem
were first introduced.
The transportation problem at AZ Sint-Jan was identified as the Open
Vehicle Routing Problem with Time Windows. This problem belonged to the class
of 𝒩𝒫-hard problems. In order to solve this problem efficiently, a heuristic method
was embedded in an Excel macro.
The outcome of the optimized transportation schedule was twofold. To
start, the number of vehicles, the total traveling duration and the total distance were
reduced. Yet, the traveling durations experienced by several patients increased. It
is up to the policy makers to assess these performance criteria and to highlight
which ones are the most important.
By making smart use of time windows – which method of usage was not
found anywhere else in academic literature – a limit on the extra traveling duration
was imposed for each patient. Meaning if a patient gets picked up from his home,
the detour in time to pick-up another patient before going to the hospital was
limited. Hence, the optimization problem attempts to find the best optimal routes
for the taxi services while limiting the extra times for the customers.
111
Chapter 6
Future Research
This dissertation aimed to analyze and redesign the hemodialysis process and the
according transportation process. Nevertheless, this thesis has its limitations.
Therefore, recommendations for future research will be elaborated.
In this dissertation topic, the workflow of nurses during the dialysis process
was analyzed. An optimal patient arrival schedule was conducted based on the
analysis of workflows and on the total waiting times for patients. Some new
staffing policies with a better fit with these new patient arrivals were already
formulated in this thesis. Nevertheless, this thesis lacks an operational nurse
scheduling roster. This problem is known in literature as the Nurse Scheduling
Problem (Maenhout, 2007). An in-depth research of this Nurse Scheduling
Problem, taking into account nurse preferences and working regulations is
recommended. The different staffing policies can be compared based on several
performance criteria. This problem was also one of the remarks that arose during
the feedback session with dr. De Vriese. This topic can be used to provide a best
known solution within the problem’s scheduling constraints. Additionally, this
future research can support decisions that lay outside the problem’s constraints,
such as different staffing policies.
A lot of surveys are conducted concerning the satisfaction of nurses
working at a dialysis unit. The implementation of a sequential connection of
dialysis patients is expected to have a positive outcome on nurses’ satisfaction. A
recommendation for future research is to study whether or not this is actual the
case. This study was not included in this thesis since such a research is only
112
meaningful on a longer term. If the outcome turns out to be positive, the sequential
connection planning can be exploited to other dialysis facilities.
Furthermore, there were also made assumptions. In this dissertation, there
was no distinction made between in- and outpatients. Though, there are some
differences between these two types of patients. Inpatients are hospitalized and are
transported through the internal transportation system. It was observed that the
internal transportation was often overwhelmed by patients who had to be carried to
another department. At this way, inpatients arrived late at the dialysis department.
Since these patients also need more investigations and their condition is less stable,
there is more uncertainty observed compared to outpatients. This introduces even
more stress for the nurses. In a more complex model, the distinction between these
types of patients could be incorporated.
113
Chapter 7
General conclusion
This final chapter will give a general conclusion regarding this master
dissertation. In this thesis, the hemodialysis process and transportation services
were analyzed and redesigned. Two perspectives were formed regarding these
topics.
- In the nurse’s perspective there was desired to find a more balanced workload.
The dialysis nurse’s workload is known to have several peaks and valleys. The
peaks are created by the arrival and connection of patients and their subsequent
disconnection and departure. To have a deeper understanding of the process
the BPM cycle was used. Here the processes were analyzed and the problems
inherent to these processes were described. To solve the observed issues, a
MILP model was composed. This led to a schedule in which patients were
planned sequentially. At this way, the workload is more balanced.
Furthermore, there is expected that this will have a positive effect on the
nurses’ satisfaction.
- The patient’s perspective was seen within a transportation context. It was
analyzed that a lot of patients arranged their own transport due to the fact that
taxi services are expensive and inefficient in terms of waiting. Nonetheless,
nobody really knew how the transportation firms could be optimized. A first
proposal was done using the theory about the Vehicle Routing Problem. The
new proposal outperformed the old transportation scheme in terms of total
number of vehicles, total traveling duration and total distance.
I
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Appendix I
Appendix
APPENDIX I: Goal model
Appendix II
APPENDIX II: BPMN (AS IS)
Appendix II
Subprocess connect to machine: AS IS
Appendix II
Subprocess disnnect from machine: AS IS
Appendix II
Subprocess disinfect catheter: AS IS
Appendix III
APPENDIX III: Value added analysis
MAINPROCESS TypePreparedialysis:putonmachine,distributematerialforconnection BVAConnectionofpatienttodialysismachine sub.Distributemealsand/ordrinkstopatients VADetermineifbloodsampleshavetobetaken BVAPrintstickersforbloodtubesfornextbatchofpatients VAPutstickersontubefornextbatchofpatients VAPreparematerialtodisconnectpatientsofmachine VAPreparematerialforconnectioninnextshift VADistributedisconnectionmaterial(ontables) BVARecordpatientvalues BVAPreparemedicalfilesfornextshift BVADisinfectclips VACheckifcatheterhastobecleaned BVADisinfectcatheter/fistula sub.Provideextracareforpatient VAAssistdoctorduringconsultationtour NVADisconnectpatientsofdialysismachine sub.Removebedding VADisinfectbed VAMakeupbed VACleandialysismachine VAPutwiresandtubesinmachinefornextshift VAInsertmedicalfilesindatabase NVACleangarbagebin VACheckifTV-screensareout BVA
SUBPROCESSCONNECTION TypeCheckifpatientcangoinandoutbedindependentlyorneedshelpfromnurse BVAHelppatientinbed VACheckpatientdata:bloodcollectionneeded,fistula/catheter,targetweight,doctorremarks… BVAPutbloodpressuremeteron VADisinfectcottonpads VAPutbandagearoundwrist VADisinfectareaaroundfistula VAPricktwotimesinfistula VAPutbandagearoundupperarm VAFastenwirestofistula VACollectblood VA
Appendix III
Connectwirestodialysismachineandtiewiresup VASetupmachine,registerbloodpressureandothervalues VACleantable BVAPutmaskonpatient VAPutgloveson VAPutbloodpressuremeteron VAPutsterileclothonpatient VARemovesockofcatheter VARemovefluidoutofcatheter VADisinfectcottonpads VARemovebloodtoemptyoutthecatheter VACollectblood VAConnectwirestodialysismachineandtiewiresup VASetupmachine,registerbloodpressureandothervalues VACleantable BVARemovesterilecloth,mouthmaskandputglovesoff VA
SUBPROCESSDISINFECTCATHETER TypePutmaskonpatient VAPutgloveson VAPutsterileclothonpatient VARemoveadhesiveplaster VADisinfectcottonpads VADisinfectcatheter VAPutnewadhesiveplasteron VARemovemask,sterilecloth,glovesandcottonpads VA
SUBPROCESSDISCONNECTION TypeCheck-uppatientdata BVAPutsterileclothunderpatients'arm VARemovebandageofwrist VARecordbloodpressureandothervalues VARemovebloodmeter VARemovewiresandbandagearoundwrist VACheckifthereisanuncontrolledbleeding BVAPutcliponarm VAPutplasterorbandageon/aroundfistula VAPutmachineinsetupmode BVACleantable BVAPutmaskonpatient VAPutsterileclotharoundcatheter VAPutgloveson VARemovebandagearoundwrist VA
Appendix III
Disconnectwires VARecordbloodpressureandothervalues VARemovebloodmeter VADisinfectcottonpads VADisinfectcatheterwithcottonpads VAPutfluidincatheterandclosecatheter VAPutsockovercatheter VARemovewiresofmachineandpatients'mask VAPutglovesoff VAPutmachineinsetupmode BVACleantable BVACheckconditionofpatient:mobilityandnausea VAAssistpatienttogetoutofthebed VA
Appendix IV
APPENDIX IV: Duration of activities and efficiency ratios
Activ
ity
Average
(min)
Ratio
Average
(min)
Occurrence
Prob
able
duratio
n
Bund
led
duratio
n
Duratio
nwith
buffer
Preparedialysis:putonmachine,distributematerialforconnection
4 4 4 5,12
Connectionofpatienttodialysismachine:help
11,53 35% 11,53 14,78
Connectionofpatienttodialysismachine:self
6,42 65% 8,22 8,22 6,42 8,23
Distributemealsand/ordrinkstopatients
1,46 1,45
25,36 32,50
Determineifbloodsampleshavetobetaken
0,4 17% 0,4
Printstickersforbloodtubesfornextbatchofpatients
0,61 0,10
Putstickersontubefornextbatchofpatients
1,33 0,22
Preparematerialtodisconnectpatientsofmachine
1,66 1,66
Preparematerialforconnectioninnextshift
4,09 4,09
Distributedisconnectionmaterial(ontables)
0,66 0,66
Recordpatientvalues 0,56 7% 3,92Preparemedicalfilesfornextshift
1 1
Disinfectclips 0,087 0,087Checkifcatheterhastobecleaned
0,4 15% 0,4
Disinfectcatheter 4,75 0,72Provideextracareforpatient
10
Assistdoctorduringconsultationtour
2,10 2,10
Disconnectpatientsofdialysismachine:help
9,43 35% 9,43 12,09
Disconnectpatientsofdialysismachine:self
7,44 65% 8,14 8,14 7,44 9,54
Removebedding 0,51 0,5110,49 13,44Disinfectbed 0,42 0,42
Makeupbed 2,55 2,55
Appendix IV
Cleandialysismachine 0,40 0,40Putwiresandtubesinmachinefornextshift
2,36 2,36
Insertmedicalfilesindatabase
3,72 3,72
Cleangarbagebin 0,13 0,13CheckifTV-screensareout
0,4 0,4
Timefor1patient 57,65 Timefor4patients 3,84 Timefor8patients 7,69 Operativehours2nurses
16
Efficiency 48,05% Dialysistime 240Theoreticalcycletime 256,3565Cycletimemorning 272Cycletimeefficiencymorning 94,25%Cycletimeafternoon 270Cycletimeefficiencyafternoon 94,95%
Appendix V
APPENDIX V: Utilization Type
ofp
atient
Duratio
nconn
ectio
n(m
in)
Duratio
ndiscon
nection
(min)
Duratio
ndialysis
(min)
Timepa
tienton
chair(min)
Ratio
Totalp
atients
Patie
ntsp
erclass
Timepe
rpatient
class(min)
Help 11,53 9,43 240 260,97 35% 148 52 13.559
Self 6,42 7,44 240 253,86 65% 96 24.381
Type
ofp
atient
Totaltim
echairs
areused
(min)
Totaln
umbe
rof
chairs
Timeavailable
(min)
Days
Timepe
r2days
(min)
Totaltim
echairs
areavailable
(min)
Availability
with
outroo
m4
(min)
Availabilitywith
used
room
s(m
in)
Help 37.941 51 705 2 1.410 71.910 66.270 58.815
Self 525 37.941 37.941 37.941
Method
1Method
2Method
3Utilization 53% 57% 65%
Appendix VI
APPENDIX VI: CPLEX
Optimization model
Model //sets int i=...; int k=...; int t=...; //ranges range pats=1..i; range acts=1..k; range slots=1..t; //parameters float duration[pats][acts]=...; float cost1=...; float cost2=...; float time[pats]=...; float latest=...; float time_total=...; //variables dvar boolean Y[pats][acts][slots]; dvar boolean A[pats][acts][slots]; dvar float+ workload_per_t[slots]; dvar float+ workload_average; dvar float dev_av[slots]; dvar float+ stop[pats][acts]; //objective function minimize cost1 * sum(t in slots)dev_av[t] + cost2 * sum(i in pats, k in acts)stop[i][k]; //constraints subject to { forall(t in slots) workload_per_timeslot: workload_per_t[t] == sum(i in pats, k in acts)Y[i][k][t]; define_average: workload_average == (sum(i in pats, k in acts, t in slots)Y[i][k][t]) / time_total; forall(t in slots) deviation_of_average_1: dev_av[t] >= workload_per_t[t] - workload_average; forall(t in slots) deviation_of_average_2: dev_av[t] >= workload_average - workload_per_t[t]; forall(i in pats, k in acts, t in slots) define_stop: stop[i][k] >= Y[i][k][t] * t;
Appendix VI
forall(i in pats, k in acts) activity_duration: sum(t in slots)Y[i][k][t] == duration[i][k]; forall(i in pats, t in slots) act_1: Y[i][1][t]*t <= latest; forall(i in pats, k in acts, t in slots: t>=2) continuity_1: Y[i][k][t-1] + Y[i][k][t] >= 2*A[i][k][t]; forall(i in pats, k in acts) continuity_2: sum(t in slots)A[i][k][t] == (duration[i][k]-1); forall(i in pats, k in acts) continuity_3: A[i][k][1] == 0; forall(i in pats) act_1_before_rest_1: stop[i][1] +1 <= stop[i][3] - duration[i][3]; forall(i in pats) act_2_after_rest_1: stop[i][2] - duration[i][2] >= stop[i][3] + 1; forall(i in pats) act_2_after_dialysis: stop[i][2] - duration[i][2] >= stop[i][1] + time[i] + 1; forall(i in pats, t in slots) max_workload_per_patient: sum(k in acts)Y[i][k][t] <= 1; } Data i=10; k=3; t=40 SheetConnection my_sheet("thesis 3 cplex.xlsx"); duration from SheetRead(my_sheet, "duration1"); cost1 from SheetRead(my_sheet, "cost1"); cost2 from SheetRead(my_sheet, "cost3"); time from SheetRead(my_sheet, "time1");
Appendix VI
Adapted optimization model
Model //sets int i=...; int k=...; int t=...; //ranges range pats=1..i; range acts=1..k; range slots=1..t; //parameters float duration[pats][acts]=...; float cost1=...; float cost2=...; float time[pats]=...; //variables dvar boolean Y[pats][acts][slots]; dvar boolean A[pats][acts][slots]; dvar float+ highest; dvar float+ workload_per_t[slots]; dvar float+ stop[pats][acts]; float latest=...; //objective function minimize cost1 * highest + cost2 * sum(i in pats, k in acts)stop[i][k]; //constraints subject to { forall(t in slots) workload_per_timeslot: workload_per_t[t] == sum(i in pats, k in acts)Y[i][k][t]; forall(t in slots) define_highest: highest >= workload_per_t[t]; forall(i in pats, k in acts, t in slots) define_stop: stop[i][k] >= Y[i][k][t] * t; forall(i in pats, k in acts) activity_duration: sum(t in slots)Y[i][k][t] == duration[i][k]; forall(i in pats, t in slots) act_1: Y[i][1][t]*t <= latest; forall(i in pats, k in acts, t in slots: t>=2) continuity_1: Y[i][k][t-1] + Y[i][k][t] >= 2*A[i][k][t]; forall(i in pats, k in acts)
Appendix VI
continuity_2: sum(t in slots)A[i][k][t] == (duration[i][k]-1); forall(i in pats, k in acts) continuity_3: A[i][k][1] == 0; forall(i in pats) act_1_before_rest_1: stop[i][1] +1 <= stop[i][3] - duration[i][3]; forall(i in pats) act_2_after_rest_1: stop[i][2] - duration[i][2] >= stop[i][3] + 1; forall(i in pats) act_2_after_dialysis: stop[i][2] - duration[i][2] >= stop[i][1] + time[i] + 1; forall(i in pats, t in slots) max_workload_per_patient: sum(k in acts)Y[i][k][t] <= 1; } Data i=10; k=3; t=40; SheetConnection my_sheet("thesis 3 cplex v2.xlsx"); duration from SheetRead(my_sheet, "duration1"); cost1 from SheetRead(my_sheet, "cost1"); cost2 from SheetRead(my_sheet, "cost3"); time from SheetRead(my_sheet, "time1");
Appendix VI
Optimization model during connection
Model //sets int i=...; int t=...; //ranges range pats=1..i; range slots=1..t; //parameters float duration[pats]=...; float cost1=...; float cost2=...; float time_total=...; float nurses=...; //variables dvar boolean Y[pats][slots]; dvar boolean A[pats][slots]; dvar float+ highest; dvar float+ workload_per_t[slots]; dvar float+ workload_average; dvar float+ stop[pats]; //objective function minimize cost1 * highest + cost2 * sum(i in pats)stop[i]; //constraints subject to { forall(t in slots) workload_per_timeslot: workload_per_t[t] == sum(i in pats)Y[i][t]; forall(t in slots) define_highest: highest >= workload_per_t[t]; define_average: workload_average == (sum(i in pats, t in slots)Y[i][t]) / time_total; forall(i in pats, t in slots) define_stop: stop[i] >= Y[i][t] * t; forall(i in pats) activity_duration: sum(t in slots)Y[i][t] == duration[i]; forall(i in pats, t in slots: t>=2) continuity_1: Y[i][t-1] + Y[i][t] >= 2*A[i][t]; forall(i in pats) continuity_2: sum(t in slots)A[i][t] == (duration[i]-1);
Appendix VI
forall(i in pats) continuity_3: A[i][1] == 0; forall(t in slots) max_workload: sum(i in pats)Y[i][t] <= nurses; } Data i=12; t=30; SheetConnection my_sheet("thesis 4 cplex.xlsx"); duration from SheetRead(my_sheet, "duration1"); cost1 from SheetRead(my_sheet, "cost1"); cost2 from SheetRead(my_sheet, "cost3");
Appendix VII
APPENDIX VII: BPMN (TO BE)
Appendix VIII
APPENDIX VIII: VRP user manual
This tool was mainly designed to solve the Vehicle Routing Problem with Time
Windows. It is seen in the context of transporting patients. The tool’s code can be
accesed by pressing ALT+F11 inside any Excel Window. The macro can be found
in the ribbon ‘ADD-INS’.
Please note that the macro will only work on Excel for Windows.
Overview of the file
Three main tabs are always present: Hospital list, Patient list and General
Information. These tabs are used as an input for the tool’s solver.
First of all, the tab ‘Hospital list’ should include the following information about
the hospital:
- Name; - Address; - Geocode.
The tab ‘Patient list’ lists all patients currently scheduled in the dialysis center.
Following information about the patients must be stored:
- Patient’s ID; - Name; - Address; - Geocode; - Distance and duration from the patient’s address to the hospital; - Service time; - Mode of transportation; - Dialysis shift.
In order to calculate the geocode as well as the distance and durations, two formulas
were added to the tool. The function for calculating the coordinates is
COORDINATES_google(). The two following functions, DISTANCE_google() and
DURATION_google() respectively calculate the distance and duration between two
locations. More information on how to use these funtions is provided in the tool.
Appendix VIII
The screen below shows a screenshot of the tab ‘General Information’. Here, the
user can enter some basic parameters for the VRP. The most notable ones are
‘closed route’ and ‘maximum extra time’. The former indicates if the vehicle needs
to return to his base depot. When it is an open route problem, the optimal solution
does not consider the vehicle’s ending location. The latter sets the maximum extra
time possible, endured by each customer. This extra time is compared with driving
directly to the destination.
Figure 1: General Information
Overview of the macro
A snapshot of the first function, called Worksheets, is given below. Each of these
four functions will create a new tab.
Figure 2: Main button - Worksheets
Locations. By clicking on the ‘Locations’-button a tab will be created asking
for the data of all the nodes that must be present for the routing problem. The user
is required to paste the names of the hospital and the patients as well as their
Appendix VIII
addresses. These data must be retrieved from the tabs ‘Hospital list’ and ‘Patient
list’. Here the time windows are also set for each location. This is done
automatically, based on the maximum extra time value from the ‘General
Information’-tab.
Taxis. This tab asks for specific information about the taxi services. Mainly, it’s
about the costs (fixed and variable), the vehicle’s capacity and the number of
vehicles.
Distance and duration. Once the ‘locations’-tab is filled in, it is used as
input for calculating the distance and duration matrix. The distance and duration
between each node needs to be known. The user has the option to choose a
distance/duration calculation method. Either it is based on Google Maps’ API or it
is based on Euclidean distances. The tool will ask which one you want to apply.
Please be patient when these values are calculated, as this can take some time.
Route. Finally, a tab called ‘route’ serves as a file that collects the solution. A
manual solution can be filled in or an automatically solution generated by the tool
itself can be filled in. In addition, this tab gives some feedback about several
performance criteria of the current solution.
The second button in the macro is called Solver. As its name might suggest,
clicking this button will engage a solution method that tries to find an optimal
solution for the given network. The CPU time needed is read from the ‘General
Information’-tab. It is important to note that every tab listed in the main button
‘Worksheets’ must be present in order for the solver to start.
Figure 3: Main button - Solver
Appendix VIII
When finished the macro will prompt a window giving feedback about the
feasibility of the solution. It also asks if the new solution must overwrite the current
solution present in the ‘Route’-tab.
Last but not least, a handy reset function is added under the main button Other.
This will clear all data present in the Excel file, except for the first two tabs:
Hospital List and Patient List.
Good luck!
